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Reggeon field theory: Formulation and use
REGGEON FIELD THEORY: FORMULATION AND USE
Henry DI. ABARBANEL Fermi NationalAccelerator Laboratory, Batavia, Ill. 60510, U.S.A.
John D. BRONZAN Department of Physics, Rutgers, The State University, New Brunswick, N.J. 08903, U.S.A.
Robert L. SUGAR Department of Physics, University of California, Santa Barbara, California 93106, U.S.A.
and Alan R. WHITE Department of Physics, University of California, Berkeley, California 94720, U.S.A.
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 21, no.3 (1975) 119182. NORTH-HOLLAND PUBLISHING COMPANY
REGGEON FIELD THEORY: FORMULATION AND USE* Henry D.I. ABARBANEL Fermi National A ccelerator Laboratory, Batavia, 111. 60510, (J.,S’.A
John D. BRONZAN Department of Physics, Rutgers, The State University, New Brunswick, N.J. 08903, ILS.A.
Robert L. SUGAR Department of Physics, University of California, Santa Barbara, California 93/06, U.S.A.
Alan R. WHITE Department of Physics, University of (‘alifórnia, Berkeley, California 94720, U.S.A Received 5 May 1975 Contents: 1. Introduction 2. Hybrid Feynman graphs as motivation for Reggeon Field Theory 3. Reggeon unitarity relations 4. Theoretical developments in Reggeon Field Theory 4.1. Opening remarks 4.2. Formulation of the RFT 4.3. Early developments in RFT fur P 4.4. Recent developments in the RFT for the Pomeron
121 126 135 149 149 150 153 158
4.5. Secondary trajectories and ,siulti-Pomeron corrections 162 4.6. Higher point P couplings 164 4.7. The formal status of the RFT 165 4.8. Ideas about a~> 1 167 4.9. Other “weak coupling” ideas l6lt 5. Inelastic processes in RFT 169 6. Conclusions, outlook and critical problems 176 Referencv~ 180
Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 21, No. 3 (1975) 119182. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher, Orders must be accompanied by check. Single issue price DII. 2O.~-,postage included. ~
*Work supported by the Atomic Energy Commission and the National Science Foundation.
H.D.I. Abarbanel et al., Reggeon field theory: formulation and use
121
Abstract: We formulate and discuss Reggeon field theory, which enables one to systematically analyze the exchange of Regge poles and associated branch points in high energy hadron scattering. The field theory is first motivated by a consideration of hybrid Feynman graphs, and then a more general derivation from crossed-channel multiparticle unitarity relations is given. Rules for Reggeon interaction and propagation are formulated. We treat in some detail the problem of the Pomeron or vacuum pole which has s(O) = 1 and is responsible for diffractive processes. In particular the renormalization group analysis of Reggeon field theory is presented and the structure of Pomeron partial wave amplitudes is elucidated. Also the question of Pomeron or absorptive corrections to secondary trajectories (both fermion and boson) is considered. We make some comments on important problems yet remaining in Reggeon field theory; in particular, we stress the study of its s-channel content.
1. Introduction The development of the theory of complex angular momentum (J) and the notion of moving singularities in the J-plane has provided for many years an important framework for the theoretical and phenomenological analysis of high energy (s) and small momentum transfer (t) hadron scattering processes. An elastic amplitude TIB(z, t) has the Sommerfeld—Watson representation —l T~B(z,t)=.~-_f ~‘°°
dJ(~+ 1) sinirj [PJ(—z)+rPJ(z)]F~B(J, t)
(1.1)
where z is the cosine of the center of mass scattering angle, and so is linearly related to s, while r = ±1 and denotes signature. The initial hope, based on non-relativistic potential theory, was that the only singularities of the partial-wave amplitude FXB(J, t) would be simple poles [1—3] Regge poles, whose position depends on t (fig. 1.1). It is well-known that in a relativistic theory a Regge pole gives both resonance poles in the cross-channel (positive t) and high-energy power behavior, for fixed t, in the direct or s-channel. Both of these properties of a Regge pole can easily be obtained from (1. 1) and their use to relate experimental results in channels related by crossing was one of the early triumphs of complex angular momentum theory. Unfortunately it soon became clear, following the theoretical work of Amati, Fubini and A
~utl
Fig. 1.1. Exchange of a Regge pole in an elastic process giving rise to s~(t) behavior of the elastic amplitude TAB(S, f). The residue of the pole factorizes.
,,,.....___...~
A
S~
Fig. 1.2. Exchange of n Regge poles with trajectory2)—1]. a(t) giving rise to a branch point at cs(fl)(t)—1 n[cs(tIn
—
122
/1.1)1. A barbans’l et al., Reggeon field theon: .lormulation and use
Stanghellini 141 Mandelstam [5 and Polkinghorne 161 that in a relativistic theory Regge Poles t) must he accompanied by further branch-point singularities Regge cuts. These in 1~’~B(~. branch-points can be viewed as resulting directly l’rom the exchange of two or more Regge poles at high energy. Alternatively they can he thought of’ as resulting from unitarity in the cross-channel, since this requires branch-points at the production thresholds for two or more Regge poles. (The branch—points do not produce singularities in the t-channel physical partial—waves and so, unlike Regge poles, Regge cuts are not directly observable in this channel.) The exchange of’,’i Regge poles (fig. 1 .2). all of which have the same trajectory a( t) gives a branch-point at 0(t): ,
~°kt)
I
=
,
n[~(t/n2)
--
II.
(1.2)
If we assume the existence of’ a l~omeroiipole (P), which carries vacuum quatituni numbers and has intercept a~(O)= I then the branch—points involving a Regge pole a 1~( t) ( which may or may not be the P) and many accompanying P’s become particularly significant. This is because these branch-points lie to the right of’ the Regge pole in the .1—plane for negative t ( this is clear from (1.2) if we take aR(t) = ~~(t)), and so, in the absence of special arguments to the contrary, ought to provide the dominant contribution to high—energy scattering. From a phenomenological point of view, the branch—points are very unattractive. While a pole has a residue, which, depending on t as it must, factorizes, a branch-point is characterized by a function off and t representing the jump or discontinuity across the cut attached to the branch—point. In general there is no ‘actorization and considerable f’reedom of parametrization. Not surprisingly theret’ore, phenomenologists have avoided Regge cuts as far as possible, and in t’act in some areas the experimental success of Regge poles is more striking today than at any time in the past decade. In particular the parametrization of uN charge exchange by a simple p-Regge pole exchange
(1.3)
2
(up
~n)
=
~(t)s2oP(t)_
has recently been shown to hold over an enormous range of’ laboratory momentum [71 Nevertheless it has also become clear recently that both experimentally and theoretically Regge cuts are unavoidable and must be accounted for. Tile rise of total cross-sections through Fermi-Lab and ISR energy ranges [8 ~l0I may well he paranietrized by a single Porneron pole (P) but this requires a~(0)> I and we know this is inconsistent with the Froissart hound. Therefore, any consistent parametrization of asyn~ptoticallyrising cross-sections must make essential use of Regge cuts. The advent of the active study of inclusive reactions, in particular the combination of’ the Mueller theorem [II -141 with tile sum rules relating dif’ferent inclusive cross—sections [1 5 1 6 I has also shown that an asymptotically constant total cross-section associated with an isolated P pole, is inconsistent. It is first argued that if’ tile total cross-section goes to a constant, then the triple P coupling observed in tile triple Regge region of the one-particle inclusive cross-section must vanish at zero momentum transfer [1 5, 171 A further sequence of arguments then heads to the conclusion that the P couplings (to particles) which appear in total cross-sections should also vanish [18, 19] Heiice the total cross-section cannot go to a constant. Thus as a matter of principle, Regge cuts must play an essential role if the total cross-section is to rise indefinitely or approach a constant at asymptotic energies. We have already pointed out that (1 .2) implies the importance of multi-P cuts if ~~(O) = I We emphasize that from a purely theoretical viewpoint multiparticle t-channel unitarity 120. 211 (if .
.
.
H. V.1. A barbanel et al., Reggeon field theory: formulation and use
123
nothing else) requires, that if we wish to describe the Pomeron as a Regge pole then nP cuts with c$’)(O) = 1 must also be present. The issue has been their relative experimental and theoretical importance. What we are now arguing is that for a completely self-consistent picture of the P the cuts are absolutely essential. This is true whether we are discussing the theoretical constraints of s and t-channel unitarity on the P or whether we are discussing experimental properties. In this article we shall review the derivation and current status of the one theory whose aim is to correctly assess the effect of all multi-P cuts. This theory was developed by V.N. Gribov [221 and his collaborators and has been variously called the Reggeon diagram technique or the Reggeon calculus. We have decided to call it Reggeon Field Theory (RFT) instead because this name accurately states that we are dealing with a field theory for quasi-particles (Reggeons) with a Lagrangian, field operators, Green’s functions etc. In fact most of the recent progress in the subject has come from the use of currently popular field-theoretic techniques based on the renormalization group. We should perhaps add that with our present understanding we believe the fieldtheoretical picture of Reggeons is only valuable in the scattering region. We do not expect it to be
adequate to describe the t-channel creation of particles and resonances. Our aim is to give a coherent account of RFT rather than a historical survey or a detailed account of technical points. We have attempted to discuss those subjects which seem central so that a general reader can learn the scope and technical rudiments of the theory. We hope that references will lead both the general reader and the expert to the current literature. In RFT the t-channel or cross channel is stressed. In our view the best starting point for a model independent derivation of the RFT is the multi-particle unitarity relation for the t-channel partial wave amplitude. That Regge cuts can be studied more directly in the t-channel unitarity relation than in the s-channel relation was first pointed out by Mandelstam [51.In a fundamental paper Gribov, Pomeranchuk and Ter-Martirosyan [201 (GPT) extrapolated and generalized Mandelstam’s work to show that a complete structure for Regge cuts could be obtained from the multiparticle unitarity relation for the t-channel partial-wave amplitude. (The ambiguities in the GPT work associated with problems of signature and complex helicity continuations have since been resolved by White [21, 23, 241.) The result is that, in the angular momentum plane Reggeons look very much like quasi-particles. The multi-Reggeon branch-points can be regarded as Reggeon production thresholds. The discontinuities across the attached cuts are given by formulae very similar to conventional unitarity relations and we therefore call them Reggeon unitarity relations. It is our belief that a proper treatment of Reggeon unitarity is crucial for the study of scattering at large s, fixed t. Our conviction arises because all of the multi-P channels are coupled by the discontinuity formulae, and for t 0, the thresholds of these channels approach each other. (In eq. (1.2) when c~(0)= 1, a’~(U) F.) The resulting strong coupling Of many P channels is a vital effect whose treatment is unique to the RFT. Some time after the GPT work Gribov [22] showed, by studying classes of hybrid Feynman graphs, how an underlying field theory of strong interactions could be expected to satisfy Reggeon unitarity. He found a J-plane perturbation expansion which is analogous to the Feynman—Dyson perturbation solution of conventional unitarity. This was Gribov’s original justification for the RFT. However the RFT is probably best thought of as a device for ensuring that the Reggeon unitarity relations are satisfied. From this point of view it is clear that the RFT may also be applicable to theories of strong interactions that are not simple local field theories (dual models, for example). -+
124
/1.1).]. A barhanel et of.. Reggeon field theore formulation and use
Reggeons are treated directly as quasi—particles by associating eadi of them with a field. No attempt is made to understand the spectrum of Regge trajectories~instead one attempts to study their interactions given that they exist. The philosophy here is tile same as in tile study of interactions among excitations such as phonons and plasmons in solid—state physics. While care is taken to enforce t—channel un itarity, the constraints of s—channel unitarity are not explicitly built into the RFT. As a result one mnst verify that tile theory does not violate these constraints. (Because of the limited phase—space region described by RFT a complete check of s—channel unitarity in tile form “SS~= i is probably not possible.) When the renormalized P singularity has intercept one. tile P interactions apparently remove the violation of the inclusive sum rules found for poles alone F 25 I Furthermore when one tries to increase the P intercept above one, model calculations suggest that after all cuts are su nl Illed, tIle Iroissart hon nd is at most saturated, in t’airly general circumstances 126. 271 As a result we believe tilat tile tlleory is complete enough to satisfy the constraints of s—cllannel unitarity. Tile above philosophy should be contrasted with that of s—channel models, In these models the constraints of s-channel unitarity are htnit into the elastic scattering amplitude by repeated iteration of t-channei exchanges. For example. in elkonal models. wilen tue eikonal phase is given by Regge pole exchange [28, 29] one is summing the graphs of fig. 1.2, which do not include any interactions among the Reggeons. Such interactions can he put in by lland. one at a time 1301 hut not in such a way as to satist’y Reggeon unitarity. Note that the graphs of the s-channel models form a subset of the RFT graphs. which may help to explain why the RFT does appear to satisfy’ tile constraints of s-channel unitarity. Models which satisfy full multiparticle s-channel unitarity have also been constructed [31—33] but again they do not satisfy Reggeon unitarity. The RFT has one apparent limitation which would be fatal if it were unmitigated. There are interactions among any number of P’s or among P’s and Regge poles carrying quantum numbers. Each of these interactions is specified by an arbitrary function of the momentLim and angular momentum of the P’s and Regge poles coming together at a point. In the same way. arbitrary functions are involved in the production and absorption of P’s by particles. The theory therefore has infinitely many parameters. It is possible that these parameters can be calculated in terms of a smaller set of parameters of the (assumed) underlying strong interaction theory. We favor the view that all these parameters are unimportant if the energy is high enough, while at current energies a phenomenology involving only a few parameters can probably be constructed. Tile former statement is defended by renormalization group arguments to be discussed in detail later. These arguments tell us that total and elastic cross-sections should have the asymptotic torms ‘‘
.
.
.
OAB(s)
g~g~(log s)~
‘~
(1 .4)
=
(~)
and
dt
g~g~(1ogs)2~(p/p0),
AB
where
p
t/(logs)°.
(1.6)
H.D.1. A barbanel et al., Reggeon field theory: formulation and use
125
Fig. 1.3. The dominant term in the solution to Rl:T which sums all the multi’P cuts.
This asymptotic behavior arises from diagrams in which the external particles couple through poles (fig. 1.3) called enhanced graphs in the Soviet literature. Tile nummbersg~,gB and the scale parameter Po in the “scaling function” ~pwill depend on the parameters of the RFT, but u~and r’ and the functional form of ~pwill not. This was found to he the case in the original papers of Abarbanel and Bronzan [34]who considered the theory with just a triple P interaction, and of Migdal, Polyakov and Ter-Martirosyan [25],who considered both the pure triple P theory and the general theory obtained by adding higher couplings. Actually r~and v are two of several exponents in the theory, that are analogous to the critical exponents which govern second-order phase transition theory [35]. The universality principle that the critical exponents of the theory are independent of the (unknown) underlying parameters is also directly analogous to the universality of critical exponents in phase transitions [35]. The universality of p offers the possibility that we can calculate the shape (if not the scale) of the diffraction peak at very high energies. We are fortunate that in asking for the form of the high energy behavior of hadron scattering processes we are asking a question which can have an answer which is both simple and universal. Our plan of presentation will be this: We begin in section 2 with a brief review of Gribov’s derivation of Reggeon calculus rules from hybrid Feynman graphs. These rules underlie the RFT one abstracts from the graphs. We believe the hybrid graph approach provides the best physical motivation for the RFT and this is why we begin with it. The hybrid graph approach has, however, suffered from criticisms of double counting (which are difficult, although, we believe, possible to resolve), and one can certainly question the generality of the rules extracted from a particular model field theory. As a matter of principle therefore we prefer to introduce the RFT as a solution of Regge cut discontinuity formulae, that is Reggeon unitarity. The derivation of Regge cut discontinuity formulae using only the hallowed S-matrix principles of unitarity and analyticity is discussed in section 3. The discussion concentrates on the twoReggeon cut, with the extension to the n-Reggeon cut covered briefly. We also discuss some points that are independent of RFT, for example the sign of the 2P cut and the use of a sum rule to relate its magnitude to inclusive reaction data. Section 4 is devoted to the problem of the Pomeron and its self-consistency within the RFT. We first review early work by Soviet workers and others which is based on the Schwinger—Dyson equations of the theory and is directed towards a “weak coupling” P (that is asymptotically constant total cross-sections). We then discuss the application of the renormalization group to the problem. We show that an explicit solution for the P exists which is actually a “strong-coupling” or “scaling” solution and gives(l.4) and(l.5). We then discuss the interaction of the “scaling” P —
1 26
/1.1)1. A barbanel et al., Reggeon field theory: formulation and use
with boson and fermion Regge poles. This ~ectiomicloses on a speculative note about the existence of further solutions to the RFT. in section 5 we review tile U5C of the RF’U to describe inelastic processes. We discuss tile “ctltting rules” of Abramovskii, Gribov and Kanchelli [361 whicil can be used to derive an RFT l’or inclusive production processes from that for tIle elastic amplitude bypassing both the hybrid Feynman graph and tile Reggeon unitarity approaches. We discuss 2 -N production processes using tile hybrid grapil approach and briefly discuss tile triple Regge region from both this approach and Reggeon unitarity. Tile rest of this section is devoted to tile general issue of the s-channel content of Reggeon field theories which we address incompletely while stressing its imnportance. Section 6 is a discussion of conclusions and views towa’-d a phenomenology using tile RFT. We discuss some orders of magnitude to indicate our conception of where “asymptopia” may lie, Our prejudice is that what we primarily see at present accelerator and colliding beam energies is a “bare” P. together with its “bare” couplings. We discuss briefly how one might attempt to make a phenomenology out of this observation. Tile final part of tilis section is a resumé of our article
emphasizillg the important conclusions and giving our outlook f’or the future of Reggeon Field Fheory.
2. Hybrid Feynman graphs as motivation for Reggeon Field Theory RFT was first abstracted from a study of Ilybrid Feynman diagrams by Gribov [221in I 967. Examples of such diagrams are shown in figs. 2.1 and 2.3. Their importance is that they represent explicit examples of the general form of amplitudes tllat we expect to represent multiple Regge exchanges ill a relativistic field theory. Tile circles represent off mass-shell two-body amplitudes tilat will eventually be represented by their Regge asymptotic form. Tile first step. however, is to find the relation between the asymptotic behavior of tile circles and that of the entire graph. One can then study tile interaction between f-plane Poles and cuts by making appropriate choices for tile asymptotic behavior of the circles. In order to illustrate Gribov’s procedure let us outline his discussion of tile diagram of f’ig. 2. 1 We wish to calculate the asymptotic hehavi@r of this diagram for large s = (p, + P2)2 alld fixed t = (p~ ~ For simplicity we take all tile particles to be spinless and to have equal mass. in. It is convenient to introduce Sudakov variables [371 by writing a general t’our-vector, k. in the !‘ormlm —
(2.1) where in2 P~Pi~ ~P2’ S
_.
in2
P2P2~Pl
S
(2.2)
and ~ aild ~ can range from plus to Illinuls infinity. Tile vectors Pi and P2 have length ~
2m~/s+ m6/s2
so that at large s we make an error of O( l/s) by setting
(2.3)
H.D.1. A barbanel et al., Reggeon field theory: formulation and use
127
~
Fig. 2.1. The hybrid Feynman graph which produces the two Reggeon cut contribution to the elastic process P1 + P2 -~ p1 2’ fixed t = (p~—~ p’~)2.The blobs represent generalized ladder graphs and have power behavior in their subat larger = (pi + p2) energies; for example,fi [(k 2). All momentum transfers and particle masses are presumed to remain
p~p~0.
(2.4)
The vector k~is a two-dimensional space-like vector orthogonal to j.~and to ñ
2• In particular
2q2
q
(pi—p’i)
=
—
(P2—Pt) +
q 1
(2.5)
2 q~.We shall nearly always neglect terms of order (m2 or t)/s. (This means that in the and so qany results we derive are accurate only within a neighborhood of at most one unit from J-plane the leading singularity.) In terms of the Sudakov variables we shall, however, write d4k in the form d4k
=
-.~/~p dct
d~3d2k
1
(2.6)
2)”2. Keeping the factor p ~ s”2 + O(1/s) will enable us to make our treatment where p = (~sparts rn straightforward at a later stage. of absorptive The two-body amplitudes in fig. 2.1 will be denoted byf 1(k1, k, k2) and 2 f2(p,—k,, 2k,~k q—k, p2—k2). It will be assumed thatf, andf2 are large when their energies s, = (k,+k2) 2 and 2 2(p,—k,) (p 2 and (q—k)2that andisthe masses (p,—k 2, k~, that become 52 = (p,k1+p2k2) 2—k2) are large, 0(s). It willk~, also be assumed they fall offlarger whenthan the momentum transfers k are certainly reasonable when, the asymptotic 1) much rn2. These assumptions behavior of the f 1 is dominated by Regge pole exchange. In this case the f, can be written in factorized form; for example 2 k2)g(k~,(k+k 2 k2) G(k2, 2k f1(k1, k, k2) = g(k~,(k—k1) 2) 1 k2). (2.7) —
-
...
The Reggeon propagator G, has the Sommerfeld—Watson representation 2, 2k,~k G(k
-_J’
C+i°°
2)
=
dl
~
2)(2(k~k ~,
G,(k
2)Y
(2.8)
1 28
Jl.D.I. A barbanel et al., Reggeon field theory: formulation and use
wilere as usual tile comltour of integration rums to tile right of all singularities of G1 and crosses tile real axis between I = I and 1 = 0. exp~ ~iu(1+ ~(i =
—-—-
-
sin~ 7r(/
e
--
—~-~
+ ~
(1
=
--~
~
+ y
(2.9)
~~---
sinurl
T)) ~
with T = ±I for even or odd signature. In general. of course, cult contributions to tile f~cannot be writtell ill factorized form. However. as we shall see cults which arise from the interplay of two or more poles can he expressed as illtegrals over f’actorized forms. As a result it is sufficient to study’ llyhrid diagrams using only’ cxpressions of tile form of’ eq. (2.7) for tile J,. We are now in a position to read otf tile higll—energy hellavior of the diagram of fig. 2. 1 First consider tile denominators arising from tile left-hand cross. For s ~ .
2+ie k~--m (p,—k,)2—rn2+i
a
2+ie. 1(~, l)s+k~---rn (a, a )(~, -~)s+(k k ,n2+ie,
(k--k 2—rn2+ie 1) (~j -k q+k)2-in2+ic
(a,—a)(~, I ~)s+2q2(a a,
,--
(2, I 0) ~-
1
2
~+~,)+(q
ifl2+~.
1~ k1+k1~) Notice that the four-momenta squared which enter 2. 1 0) also appear as mass variables in the ,t’,. The requirement that they and tile momentum transt’ers be less than or order in2 gives
k~
2,
k,~~ m2,
1~ m a,
rn2/s.
~
I.
rn2/s.
a
I.
(2.11)
A similar analysis of tile right-hand cross gives
~ 132
~ m2.
k~~ in2. /3
rn2/s
I.
(2.i2)
Finally the requirement that 2k 1’ k2 S
/3ia2s and 2(p,—k,) (p~—k~) (1 ‘
—-/3,)(
I
a2)s he of order
gives
~
I.
a2~ 1.
(2.13)
Since /3 ~ it can be neglected ill tile denominators of’ (2. 1 0). Similarly a can be neglected in the corresponding terms arising from tile right-hand cross. Putting together all of the above results we obtain a complete t’actorization of the integral represented by fig. 2. 1 which can now he written in the t’orm /3,
2k
2)
T(s, q
=
iu
dl,
d12
~ d
1
~,
~ 12iV~2tq, ~
(2.14)
H.D.J. A barbanel et al., Reggeon field theorv:formulation and use
129
where X2(~f~2p)2da, d~,da d2k1~, 2 k~)g((p N,,X,2(q, 4(4~)l/2 /3’,’( k5) 1 /3,)l2[k~_rn2+ie]
2, (p
2 (q—k 2)
(2~)~ 2—rn2+iel g(k~, 1—k,) 1—k,+k—q) 1) 1 [(p -,(k—k1) [(k,—k)2—rn2+ieY’[(p,—k,+k—q)2—rn2+ieI 1
1—’k,)
(2.15)
A is the coupling constant for the three-particle vertices. N, 112(q, k1) can, of course, also be expressed in terms of the quantities associated with the right-hand crosses. Several features of eq. (2.15) are worth noticing. the g’s depend onthe a,, ie prescription and a only for 2, (k—k,)2 andFirst (p,—k,+k—q)2. Since through the mass variables k~, ( p,—k,) the singularities in these variables is the same as for the propagators we see from (2.10) that the a, and a integrals vanish unless 0 ( /3, ~ 1. For /3, outside of this range, all singularities of the a, integrand lie on one side of the contour. We also see that if the coupling between the two-body amplitudes, which determines N, werelie planar, allsame singularities in the a-plane As (in athe region 2/s)112, would on the side of the a contour. result, tile where a, and a are of order rn graph would be negligible. asymptotic contribution of the This last result is the connection between Regge cut asymptotic behavior and the presence of a third double spectral function in the associated “two-particle/two Reggeon amplitude” which we shall find from a different point of view in the next section. Note that sinceM2 = (p,—k)2 —aS+rn2, we can pick out the a-integration in (2.15) and write /3k,
I
N,,, 2(q, k1)
~°°~j~~j2
f
~
—‘—
2, q, k T,1J2(M
1)
(2.16)
2, q, k 2-contour can wherebeT,,,2(M 1) is defined by the remainder of the integrand in (2.15). The M now rotated to give 1
N, 112(q, k1)
—~
~2
°°dM2
f —A,112(M2, q, k1~
(2.17)
2iT
4m
2, q, k 2, q, k 2-variable. We shall comwhere A,,(M on (2i7) 1) isshortly. the absorptive part of T,,,2(M 1) in the M ment further Since the important singularities in a and a,, in (2.15), occur when these variables are of order rn2/s, each of these integrals give rise to a factor of i/s. Consequently, N,,, 2 becomes independent 2 and k~negative, which is the region of interest, N,,, of s at high energies. In addition, for q 2 is real. If the simple crosses of fig. 2.1 are replaced by more general couplings we still expect (2.14) to hold. Of course, N,112 will no longer be given by (2.1 5) but we still expect it to be independent of s asymptotically and real in the s-channel physical region (this is provided that the couplings do not themselves contain Regge singularities a circumstance we consider shortly). Next note since ‘~J~p does not change sign under —s, thefactors signature 2) we given by that (2.14) is determined by the product of the ssignature E,,~,of the amplitude T(s, q 2.This is a generalized form of the result that the signature of the two-Reggeon cut is given by the product of the signatures of the contributing Regge poles. —
-~
130
11.1)1. A harbanel et al., Reggeon field theory: formulation and use
The Reggeon calculus has as its objective a set of rules for tile calculation of’ the signatured partial-wave amplitude I’i.J, q~) which is related to T(s, q~) by the Sommerfeld Watson integral (‘+1=
T(s, q~)=
tU ~~~‘F(f, q~).
(2.18)
~ being the signature factor corresponding to tile signature of 7’(s, q~ (2.18) give ).
Hf. q~)=
be jllverted to
call
q~) (s’) t~~~A(s’,
th’
(2.19)
where
A(s, q~)= Abs T(s, q~)= ~ IT(s+ie, q~) T(s-- ic, q~)I
(2.20)
(2.18) is equivalent to (1 I ) to leading order in s. The Mellin transform (2.19) is more convenient for our purposes here than tile Froissart Grihov projection WhicIl appears in (1 I In the next section we shall use the Froissart Gribov projection because it diagonalizes I-channel unitarity equations. Using the facts that .
.
).
Abs(~ 1s’)= s’,
Abs(~-~) =
(2.21
-
we see from (2.14) tilat 2k
=—f—-~
A(s, q~)
ir
dl,
d12
d
~
1
y~N~,(q1, k~)G11(k~) (/2((q~ k1)2)s~~~’2
(2.22)
witil
(I -r,) 71112 = ~
cosL_.(i,+12 +
---~-)1 (I -r2)
+
(2.23)
and =
51~[_-Q~ L~_~)J +
(2.24)
.
Finally. (2.19) gives 2k
I~’(f, q2)f~ d11 d12 d 1 y
N2 (q
--~
‘
2iri
27Ti
(27r)2
1112
1~2
,
~
k
2) (.1, (k~JG,((q1---k~)
)--..~-.-—~—--~~
f+ I ----1,
12
If the amplitudesf, andf 2 are dominated asymptotically by Regge pole exchange then
(2.25)
H.D.I. Abarbanel et al., Reggeon field theory: formulation and use
131
Fig. 2.2. The Reggeon graph contribution to the t-channel partial wave amplitude coming from the hybrid graph of fig. 2.1. N is a two particle-two Reggeon transition amplitude; G is a Reggeon propagator. See eq. (2.14).
2) = 1
G,,(k~)= l1—a,(k~)’
G,2((q1—k1)
2_a2((q±_ks)2)
(2.26)
In this case 2)
.121
~12
= Pu “A~
(2.27)
~ (2ir)2 J+l—a,(k~)—a
F(J q
2) 2((q1—k1)
d2k
—J 2iri
1 (1\2
2
~‘~1~2
,,(
01a2
‘~‘-~“
(
2
~)
~
q1
2
~
)
.
)
where the 1,-contour ruins to the right of the pole in G1, and to the left of the pole in G~+1 (2.28) now corresponds directly to the Reggeon (Feynman) diagram of fig. 2.2. The vertices where two particles produce or absorb two Reggeons represent the factor N0102~,/y~,,22 (we absorb the signature factor into the vertices). The propagators represent the Reggeon propagators G11, ~ and the loop represents the 1, and k~,integrations. There is conservation of both one minus angular momentum and k1 at the vertices if one pair of external particles is regarded as a source for angular momentum; and transverse momentum q1, and the other pair is regarded as a sink for the same quantities. Both angular momentum and transverse momentum then flow through the diagram. One is ordinarily interested in the behavior of F(J, q~)near the f-plane branch-point generated by (2.28). For the P—P cut, where r, = = 1, ~ —l in this domain. One sees then that the effective two particle 2P coupling \/ya,o2Na,02 is pure imaginary. Similarly we see from (2.23) that the two particle P + meson trajectory coupling is also pure imaginary independent of the signature of the meson trajectory. Note that if we put 1, = a,, 12 = a2 in (2.17) we have expressed the two-particle two Reggeon coupling Na,a2 as 2, an q,integral over the absorptive part of the two particle/two Reggeon scattering k amplitude Ta,02(M 1). Comparing this with (2.19) evaluated atJ —l we see that (2.16) is analogous to evaluating the two particle/two Reggeon partial-wave amplitude at a nonsense point — in this casef= a,+a2—l. This is the connection of the couplingN0,,~2with a “nonsense fixed-pole residue” which we shall find in the next section. It is straightforward to generalize the above results to the three-Reggeon cut diagram shown in fig. 2.3. One merely uses forf,(k,, k1k2) the two-Reggeon cut amplitude of (2.14). Proceeding as before one finds that the contribution of fig. 2.3 to the elastic amplitude is given by ,,.
-~
-÷
-~
T(s
2k 2k~ r. . . . . . dl1 di3 d 1 d 2~2iri 2iri (2ir)2 (2ir)2
n2) = 7T — ‘~
X
—
1
_____
N~,,
S1~’~~2
~‘1l”12”13
2)G, 213(q,k1, k~)G,,(k~,
2)G, 2((k1—k~)
2) 3((q—k1)
(2.29)
1 32
1 barbanel
/1.1).].
et al., Reggeon held theory. forniulatioi, aiid u 0
A ~ l:iy. 2.3. A ivhrid tevrinisu erapli svhjyh
eIvy~S tliiee Reeccon
cut c~ntrjhuti’~iit~’ IlL
where A
1,1213 is obtained f’roni (2. I 5) by replacing g( k~,(k--k, )2 A ~) by A’1 by I,. Calculating tile absorptive part of 1’) s, q/) as before tllell gives 2/~ d2k d/, dl~d 2 1(1, q~) (~)2(~)2 Y~,0,0 ) G~(Is~:~‘J~2 /~
I, by l,+12 I and h
~.
=f~
~
whicil corresponds to tile Reggeon of n-Reggeons is immediate
/‘(.J, q~)= 2Tiffl~ I/ ~T
2k ~
1
~
/5(~i
--
k1
(1’ ey’nman ) tllagrani of fig. 2.4. ‘l’he generalitation to the case
..~,[p,G,. A~)]6(11
~
(/
I
+~
(I,
I
))(
2~)2 62
(q
~ A 1
where in both (2.30) and (2.31)
7/,
,~ = (
- I
)“
(i,
1sin[~-(
(2.31
+
A±))1:/n
(2.32)
~.
Note that we have written (2.3 1) in a symmetric form by introducing 6—lu netions conserving angular nlomentum and transverse Illonlentuin. Tile interpretation of ( 2.3 1) is as follows: two particles act as a source of two nlolnentum £/ and ‘‘energy’’ I — J~from this source31,.iH n—Reggeons with nloniellttllll k,1 with and energ~’1 1,. / =G(1,. I n. Each Reggeon propagates the a,llplitude A~ emerge with to amplitude A’,,12are!11\~ appropriate it umtil they all absorbed by’ two particles acting as a sink. At each stage energy and momentum have been conserved tIle overall energy and momentuim consel-vation 6—functions .
.
...
,
have been factored of’f’. Ibis rather attractive description of’ the n—Reggeon diagram is supported by the study ol’ mole
~E2k21a2Nt3t
E 3, k31,a.
1g. 2.4. The Reggeun graph which comes
Irom
the hybrid lcynman graph ol
fig.
2.3.
H.D.I. Abarbanel et al., Reggeon field theory: formulation and use
133
Fig. 2.5. A hybrid Feynman graph which introduces Reggeon interactions.
complicated hybrid diagrams. Consider fig. 2.5 next. An extension of the foregoing Sudakov analysis shows that this corresponds to the Reggeon diagram of fig. 2.6, and yields for F(J, q~) d2k,1 d2k212 ~ dl, dl2 (2ir)2~2(q F(J, q~)= (27r) 1—k,1—k21) (2.33)
f2~)2
X 2~i~( 1 —J—( I —l~)—(1—12)) N,,,2(k1±,k2j)y,112r1,,2~(k,1,k21, q1) G,(l,, k,1) G2(l2, k2±)) G3(J, q1). The additional ingredient here is the triple Reggeon vertex r shown in the present calculation as fig. 2.7. More complicated contributions will change the detailed form of r but leave F(J, ?j~)as in (2.33). Since the coupling of two particles to one P is real and that of two particles to two P’s is pure imaginary, the triple P coupling of fig. 2.6 must be pure imaginary for k,1, k21, q1 0, since F(J, q~)must be real below the t-channel threshold. This means that ~ must be pure imaginary and so r,,,2~must be real. The coupling of a meson trajectory to a P plus a meson must similarly be pure imaginary. To derive Reggeon rules from a general hybrid Feynman graph one identifies familiar elements, for example, G, r the N’s, and writes the desired contribution to F(J, q1) as products of the basic building blocks put together with energy and momentum integrations. The final outcome is that to obtain the full partial-wave amplitude we must write down the complete set of “Feynman” Reggeon graphs including general interactions in which arbitrary numbers of Reggeons are destroyed and created. Of course, hybrid diagrams corresponding to all possible graphs have not been considered. The general rules are obviously an extrapolation of the results obtained for simple diagrams. An infinite set of diagrams involving four-Reggeon interactions has been considered by Landshoff and Polkinghorne [38]. The corresponding contributions to the partial-wave amplitude are then constructed by writing down for all couplings, propagators 2k vertex2)functions for each internal Reggeon loop. Energy, for each internal Reggeon line, and f(dl/2lri)(d 1/(27r) E = (1—i) and momentum being conserved at each vertex. Notice that because the Reggeon propagator is linear in E we must specify the direction of propagation of the Reggeons. Diagrams which are topologically identical, but which have one or more Reggeons propagating in opposite directions are distinct [22].
Fig. 2.6. The Reggeon graph contribution to F(J,q2 1) from the hybrid Feynman diagram of fig. 2.5. The triple Reggeon coupling T~,a2~3 appears here.
134
11.1)1.
.1 barhanel s’t al.
,
Reg,~’eonfield theore - formulation and use
F- ~, k~
1,a~ ‘.,/
~2
Fig.
—
(5,
\E~-J,q~,a3
k,1
2.7. Ihe simplest contribution to the triple Rc’ggeon coipling ~o u2u~-1 his will appear as a building block in nor’ cniiplicated Rcgc’ean graphs.
Clearly we have arrived at a set ot’”Feynman’ rules for writing down our partial—wave a~nplitude and we shall formalize this turtiler in section 4. We f’inish our description of’ tile derivation of tilese rules from hybrid graphs by noting tile rules for the phases of the P couplings that we have found. From (2.32) and similar signature f’actors for Reggeon interaction (liagranls we find that if Nm is tile coupling (treated as a constant) of two particles to in P’s and ~ is tile cotiplmg of in P’s to n P’s then
1. A,nn (2.34) ~ (i)’°— We have now given a bare outline of the arguments of’ Grihov to motivate a field theoretic description of Reggeons. We do not intend to attempt the complete decomposition of’ a relativistic ~
(i)i?i+i
2
,
field theory into hybrid diagrams. Tllis would be required to turll our present heuristic justification of the RFT into a proper derivation. (‘learly we would need as a starting point some picture u)f~the initial process wilich generates an /solaled P pole with a~(0) I The amplitude f’or this process would then provide the input two-body amplitude folr our ilybrid (liagrallls. (Conceptually it is sometimes useful to picture this orocess as tile familiar multiperipheral or ladder mecilanisni 1361 but this is not essential.) However, there are still problems associated with tile breaking up of hybrid diagrams wilich correspond to Reggeon interactions, such as that of fig. 2.5. into a part corresponding to tile full Reggeon diagram (that of fig. 2.6 for fig. 2.5) and parts corresponding to renormalization of tile original, or bare. P0k and or vertices. To handle this problem correctly it is necessary to introduce a cut-off ill tIle / and k integrations in Reggeon uliagranls and to relate this cut-off directly to the break-up of the phase-space ill hybrid diagranls. A (liscussioll of’ how this can he done and how the accusations of double-counting at this stage can be avoided, has ‘~
.
,
been given by DeTar [391 Fortunately we can avoid tilis problem by ulsing Reggeon nnitarity as a basis for tile RFT. as we disculss in the next section. Tile Reggeon unitarity equations are applicable once we know. or rather assume, that there is an isolated P pole in tile neighbourhood of .J I.1 0. These equlations reassure us that we have not been double-counting in discussing hybrid diagranls. Bef’ore leaving tile counting problem altogether however, we note that we shall discuss the use of a cutoft’ RFT in relation to tile renormalization group in section 4. It is worth noting from tile above discussion that tile cuit-off may well he an essential feature ill ensuring that tile RFT is welI-def’ined botil mathematically and physically. Tile generality of the Reggeon unitarity approach to RFT gives a powerful logical basis to the apparently more model dependent, albeit rather more pllysically motivated j~rocessdescribed above. To compare tile results of this section directly with those of the next section we briefly discuss taking the discontinuity across the cuts attaciled to the Reggeon branch points occurring in tile partial-wave amplitudes we have obtained.
H.D.J. A barbanel eta!., Reggeon field theory: formulation and use
135
Consider first the n-Reggeon branch point generated in (2.31). This can be viewed as a “threshold” singularity generated by the poles of the propagators G,.(k,21). The discontinuity is obtained by the usual “Cutkosky rules” of putting each Reggeon on its “spin” (mass) shell: 2k d2k disc~F(J,q~)= 1)’~’2iri sin[-~ (.~_~ 1 d ~ ö2 (qj — ~ k 2 1~)X (2ir) ...
(—
x ~
(2.35)
this expression being simply obtained from (2.31) by writing G,.(k~ 1) 2iri ~(l1—a,(k~)). For more complicated Reggeon diagrams there will be a contribution to the n-Reggeon discontinuity from every n-Reggeon state that can be found by “cutting” the diagram vertically. (There will, as in a conventional field theory, be a renormalization of the pole position in each propagator by “self-Energy” insertions. This renormalization will only be unambiguously defined from hybrid diagrams once the cut-off is introduced as discussed above.) To take all these contributions into account we follow the usual unitarity prescription of complex-conjugating that part of the diagram to the right of the cutting line. More technically, the part of the diagram to the left of the cutting line is evaluated above its f-plane branch cut, while that to the right is evaluated below the branch cut. After summing all contributions the final result is that the complete discontinuity has the same form as (2.35), except that —~
2 =Nai... ~
1’~o1...an(~)’
0n(J+~)N0i...0i,(J~i)
(2.36)
where Nai ~~(J)is the complete two particle n Reggeon amplitude. The complete imaginary part of F(J, q~)arising from its Reggeon branch-points is given by the Reggeon unitarity relation -*
disc~F(J,q~)=
~
n2
disc~F(J,q~)
(2.37)
‘~
where disc~F(J,q~)is given by (2.35) with the substitution (2.36). The purpose of the next section is”to derive (2.37) directly from multiparticle t-channel unitarity. We can then argue that
the procedure of the latter part of this section can be reversed and the RFT viewed simply as a solution of (2.37).
3. Reggeon unitarity relations In this section we discuss the derivation and implication of Reggeon cut discontinuity formulae from the point of view of I-channel unitarity. We use standard S-matrix methods and the results depend on conventional S-matrix assumptions of analyticity put together with the existence of moving poles in the J-plane. Briefly speaking we argue that branch points in J arise from the presence of two or more moving poles in J. If there are additional singularities in the f-plane, we will not find them. Indeed, in the spirit of “maximal analyticity in the f-plane” we assume them
1 36
H.D.J.
A barbanel ci at., Reggeon
field theory: formulation and use
11111111 >IIIIIIII(
11111 >/lIIIIIIi\<
1’ (a)
(h(
Fig. 3.1.
(a) A planar F’eynnian ~g’aphwith two Reggeons, represented by ladders, in the t’channel, l’hts planar graph does not give rise to a branch point in the f-plane at a(2)(t), eq. (1.2). (h) The simplest non-planar 1’cynn,an graph which does s’,eld the two Reggeon branch point.
to be absent. The full treatment o!~multiparticle i—channel unitarity ill tile i—plane is rather conlplicated but we shall try to minimize the technical details without depriving tile reader of all insight into the procedure. To motivate the use of the i-channel as tile correct place for an S-matrix analysis off-plane branch points let uis look at the Feynman graphs winch were the original indication of the presence of cuts [4, 5], (fig. 3.la and fig. 3.Ib). Both graphs have many intermediate states in the s-channel gotten by cutting the ladders which give rise to the Regge behavior. A priori all possible nunibers of particles in tile s-channel must be considered at the same time to discuss the large s. fixed t behavior of these diagrams [40] Since we are, Ilowever, interested ill a fixed I limit, by continuing t from the scattering region, t ~ 0, to some t’inite time-like point we can restrict our attention to particular i-channel intermediate states. In the diagrams in fig. 3.1 the lowest intermediate state in t is the four particle state. This suggests we may be able to study tile two Reggeon cut in the full partial wave amplitude by looking at the four particle unitarity integral in the i-channel. Indeed, this is so [20. 21] provided, of course, that the Reggeons involved couple to some two-particle states. Mandelstam [5] actually analyzed in detail the diagrams of fig. 3.2 where one of the ladders (Regge poles) of fig. 3.1 is replaced by a single particle. Neitiler t’igs. 3. Ia nor 3.2a has a Regge cut while 3.lb and 3.2b do. Mandelstam showed this for figs. 3. 2a and 3.2h by considering the three particle unitarity relation in the i-channel. He argued that only in tile case tilat the five point amplitude shown in fig. 3.3 has botil a right and left-hand cut in the sub-energy s, could a f-plane branch point be present in tile~eIasticamplitudes of fig. 3.2. Fig. 3.2a has only a righthand cut and does not yield aJ-plane cut; fig. 3.2b has both right and left and gives rise to a f-plane cut. In other words, a f-plane cut will only occur when the “three particle-Reggeon aniplitude” of fig. 3.3 has a “third double spectral function”. This part of Mandelstam’s argument is .
,
independent of Feynman graphs as is the final form for the contribution of the cut to tile partial
(a)
(b)
Fig. 3.2. (a) A planar Feynman graph with one particle and one Reggeon in the i-channel, This planar graph does not give rise to a branch point in J. (b) The simplest non-planar graph which does give a Reggeon-partiele branch point in J. It is the presence of a left and right hand cut in the sub-energy s~which guarantees the existence of the cut. This Reggeon-particle cut is shielded by the two Reggeon cut (fig. 3.lb) in the scattering region, t ~ 0.
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
137
titi
Fig. 3.3. The two particle
-+
one particle
+
Reggeon graph which is contained in the two particle Reggeon is represented by a ladder.
—*
three particle amplitude. The
wave amplitude F(J, t)
—‘f
(.~./~_m)2
di,
X”2(t,t, m2)
C(t, t~) f+l’—a(t,)
(3.1)
where C(t, I,) is a smooth function of its arguments, ct(t) is the pole trajectory coming from the ladder graphs, and X(x, y, z) = (x+y—z)2—4xy is the familiar kinematic factor. Subsequently, Gribov, Pomeranchuk, and Ter-Martirosyan (GPT) [20] compared the form of (3. 1) with the contribution to the two particle unitarity relation for F(f, t) coming from a spinless particle of mass m and a particle of mass m,, spinf,, helicity n,. This is +Ji
X”~(t,m~,m2) F(J+1—n,)
Im 5F(J, t)
~
n,—J1
i
\ I’,
I
flj)
f~,~,(t)ff’,~,(t),
(3.2)
where f~,~~(t) is a helicity partial wave amplitude for the transition: two spinless particles -÷spinless particle + particle of spinf,, helicity n1. This formulae contains a fixed pole atff,—1 arising from the helicity state with n1 f1, and near this point strongly resembles (3.1). Further, it is known that the residue of this fixed pole will vanish unless the amplitude giving f1~(t)has a third double spectral function. This is the usual connection between nonsense wrong-signature fixed poles and third double spectral functions. {There is a rather thick layer ofjargon in this, and we are required to ask the reader’s patience and refer him to standard references [41] for an explanation of terms. } This comparison suggested to GPT that f-plane branch points could be viewed as resulting from the presence of a fixed-pole in Reggeon particle intermediate states all along the Regge trajectory. This observation was the key to subsequent analysis off-plane cuts using t-channel unitarity. A complete analysis of three particle unitarity together with a discussion of the Reggeon-particle cut in (3.1) has been given by White [21].This last cut moves off the physical sheet of the f-plane around the two Reggeon branch point when t becomes small. Since it, therefore, does not contribute to the large s, fixed t behavior we turn our attention to the two Reggeon cut. To study the two-Reggeon we consider across not thethe complete t-channel unitarity rela2, but insteadbranch-point isolate the discontinuity four-particle threshold. This tion for I 2 1 6m eliminates the need to discuss the two and three particle contributions to the unitarity relation. The Feynman graphs of fig. 3.1 do not have two or three particle intermediate states in the t-channel and so for them the four-particle discontinuity would be the complete unitarity relation for 1 6m2 ~ I ~ 25m2. In general the four-particle threshold discontinuity can be written in a fairly conventional S-matrix form [42, 43]
1 38
lJ.D.I. ~1harbanel ci at.,
Reggeon field theors’: fdrmulation and use
P1 ONE
4rn?
m
Pm?
1-ig. 3.4. ‘the path ut coiitini,ation in the i—plane to reach, the .iniphitudv’ ‘T~ needed in the unitarits Irniulic eq
=
-~
~
~.:
i3.3
J1II.~and ~ are physical amplitudes, bitt the other amplitudes are (let illeul bE analytic con I tutution of these physical amplitudes.~~ is i~+ contillued around tile four-partIcle tilm’esilold as illustrated in fig. 3.4. is~ analytically con tintteul around both the four-particle threshold in the total energy t (this is denoted by tile 4) and around all phase—space thresholds in tile stlb— channels (this is denoted by ~ and involves going around two—particle tilresllolds ill all two— particle channels and three-particle thresholds in all tilree-particle cilanllelS). The ltegration implied by tile right-hand side of’ (3.3) is over normal f’our-particle phase-space. Tile next step is to project tllis unitarity relation onto partial waves. In fig.3.5 a complete set of partial wave variables is given. Since the external particles are spinless we need only the total angular momentum f and (mass)2 = I = Q2 of the (‘our—particle state. i1. t~. Q~lot’ the two pairs 4~
ii~tile hehcity of each pair. The total helicity ii = n~+ n2. One catl now use these variables to construct the unitarity relation for partial waves coming from (3.3): 4(J, I) = (3.4) F(J, i) F fdp ~ A(f, f,, f 4(J, ,Ji, J 2, ni, in, I. in2i I EU, f,, J2, n~,n2, I, t,. t2)F 2, il’, /12. 1, t~,~
of particles and
*
—
J2 2~In~ I
with A(f,
J,, J2, t, t,, t2)
(~,+
=
----
I)
(2J2+ 1) [U, -n ,+l) [(J2-n2+ I I [(f- /1+1)
F(J,+n ,+ I) F(f2+n2+ I) F(f+n+ I)
-
-
~
c)
an(I
Fig. 3.5.
A set
of variables for the partial wave projection of the 2 -~4 amplitude needed in the study of the four particle contribution to the partial wave unitarity relatiun. f is angular momentum and n is helicity.
*(3~4)can be obtained directly from (3.3) by diagonalizing the momentum-space integrations using partial-wave projections of the six-point function. The details of this procedure are given in the appendices of ref. 1211.
H.D.J. A barbanel eta!., Reggeon field theori’.’ formulation and use
fdp
=
ffdi 4,ii2
(2ir)5264!
2 X”2(i i i,,_ta)[(ti_4m2)(i L 1112 2~~~4m2)1O
~ 4m2
139
(3.6) J
Eq. (3.4) gives the discontinuity of the usual partial wave amplitude F(f, t) across the four-par-
ticle threshold. The amplitude F(f, f,, J 2, n,, n2, 1, 1,, 12) is the helicity partial wave amplitude for two spinless particles to go to a particle mass I,, spinamplitude J,, hehicity a particlecontinued of (12, J2, n2). 4” of the same arguments is the helicityofpartial wave of n, theplus analytically six F point function. We now focus on the “kinematic factor” A in (3.4). it contains the fixed pole atf= n—i. This is the singularity GPT expected to be present in order for the two Reggeon cut to appear in F(J, t) when continued to complex J. GPT did not try to find the exact form of this continuation. They essentially tailored their assumptions on how to continue the sums in (3.4) to achieve the expected results. Since their work there has been considerable insight into the technical problems associated with analytic continuations of multiparticle amplitudes in angular momentum as well as helicity [44]. For our purposes it is enough to know that the GPT program can be carried through although we shall need the exact form of the continuation of (3.4) to complex f. This has been given by White [211 and we follow his development. To continue F(f, J,, J 2, n,, n2, I, I,, t2) away from integer f, f1, and n, we must introduce signature as well as labels> or < telling us whether we have continued n, into the right or left half helicity planes. We will collect these various labels into an index q and refer to them only when essential. The signature labels naturally refer to continuation in f—n, call the signature ~J~—n~, ~ n~,r. The signaturesr, r,, r2 forf, f ,, f2 are r~r’,r’2,
r,
~
r2
(3.7)
‘i2r~.
The contribution to the discontinuity in (3.4) from n,, n2> 0 is now written —sin{~ir(J—fl}Cd 16
X
~
~
f___________
~ ~r.±i
Cj
sin {~ir(n,—~) } sin{~ir(n2—~) }sin{~ir(J—n—~) }
T(J, n)FT4(J, n) A(J, n)F
7)
dn,dn2
(3.8)
where J = (J, f,, J 2), n = (n,, n2), c = (r, r,, r2, r’,, r~),and ~ = ~-(l—r), ~ = ~(l—r). Clearly the sums over n, and n2 in (3.4) have been converted to a contour integral over the contour C~.Fig. 3.6 shows the projection of C~in then, plane at fixed n2 and for fixed n, it would have the same form in the n2 plane. That the contour is asymptotically parallel to both the n, and n2 imaginary axes ensures that the integral converges and that the problem of divergent helicity sums in this context is avoided [45—47]. At integerf the two series of poles shown by ciosses in fig. 3.6 come together and pinch the contour C,. The double 1ntegral over n, and n2 develops a pole because of the pinching of the contour by the three sets of poles in the integrand coming from the factors of sin{~7r(n,~i!,)}, sin{~ir(n2—f~)}, and sin~ir(J—n—fl}. The pole of the integral is cancelled by the “signature factor” sin{~ir(f—fl},which multiplies the integral and thus singles out the pole
140
lID.!. A barbanel ci at,, Reggeomi field theory: k,rniulation and use
n
a It I F
I
~
L
HELICITY PLANE --
‘i~
-
—
-n
1~r; --n~r1’+2 Fig. 3.6. The contour in the helicity plane is required in eq. (3.8) for the four particle contribution
I
partial wave unitaritv.
residue as the value of’ (3.8) at integer f. ‘l’his residue contains just the requisite finite sttm over helicity amplitudes. We can now study the two Reggeon cut which arises from the presence of Regge poles at f1 = a~(i~), with signatures r~,in the .J~channels. Poles occur at a1 a~(i~)in the term with f1—n1 0 in (3.8) provided i~= + I As sllown in t’ig. 3.6 these poles lie on opposite sides of tile ( contour to the pole of A at J = n- I. 4(J,n), there there is aintegral compensating “nonsense zero” eitllerofJ’~J. ii) or L’~of the Regge willTherefore, be a pole unless in the helicity at .J = a,(ti)+02(t2)--I as ainresult the pinching .
poles and the nonsense pole. For there to be no nonsense
zero
f
=
a ,+n 2 I must he a wrong-sig.
naiure nonsense point. This requires
T=+l
~TTT~
(~))
and since =
T1T1
=
T2 = T2T2
=
+1
(3.10)
we must have (3.11)
TT,T2.
This is the S-matrix origin of tile rule that the signature of a cut is the product of the signatures of the Regge poles building tile cuit. Isolating the contribution of the pole at f = ct~(t,)+ 02(12)-— Ito (3.8). and taking = = r = +i for simplicity, we obtain A’(f, ~)R~(J, t)I’T~4”(f,
2
~—sin(~ir~fdp—-~ ir 4
~
~,
-.
--
.
-
-
sin { ~ira ,(t,) } sin { ~ira
I) ----
(3.12)
2(t2) ~(f--a ~(t~)---a2(t2 )+ I)
where A’(J, ~) is the residue of A(J, n) at tile pole at f = n ,+n2--- 1 and R~(f, I) ~5 the double Regge pole residue of F~(J,f, = n,, J2 = n2, a1, a2). Note that we are using the fact that the Regge 4’ trajectories acquire is not singular at f, imaginary a,(ti), f parts at the lowest thresholds in the I, and 12 channels so that F~ 2 = 02(12), but instead has poles at f, = a~(i,),.1’2 = a~’(t2). (3.12) already resembles (3.1) and we can improve the resemblance and simplify (3.12) by using two-particle unitarity m tile I, and 12 channels. It can be shown that we may write 2 4 2 1/2 2 1/2 4.\~N~ (3.13)
(i-~)
(~
(i-_~_) R~a~
H,D,I. Abarbanel et a!., Reggeon field theory.’ formulation and use
t
1
141
PLANE
—
C TWO PARTICLE
-~:
THRESHOLD
Fig. 3.7. The t~or t2 plane in the four particle phase space showing how the t~ integration is deformed to enclose the two particle threshold.
in (3.12), where N~(f, t) is defined from R~(f,t) by factorizing off the two-particle/Reggeon vertex functions, if we replace the i, and 12 integrations in fdp by contour integrals enclosing the two-particle thresholds in those channels see fig. 3.7. N~ is simply the analytic continuation of N~around the four-particle thresholds in the t-channel. If we make tile further simplification of absorbing ~JA”~ into we obtain for ImF(f, t) —
‘~
1
44! sin(~irJ)f Mt
,‘~,
t 2)>o
ir2
i/2j’.~
1~.
di, dt2
\~,~
r..ir Ir t,,av~ \ p..jr4 71
1”.ar,’.”~
‘~
‘2J
~—~-~-
.-———-_____
(3.14)
.
t sin{~ira,(t,)}sin{~ira2(t2)}(f—a,(i,)—a2(t2)+l)
(3.12) is now clearly analogous to (3.1) except that the Reggeon-particle “fixed-pole” at J a(t,)—-i has been replaced by the two-Reggeon fixed-pole atf= cm,(t,) + a2(t2)—l. The Reggeon-particle branch-point occurs in (3.1) when the fixed-pole hits the end-point of the t,-integration this gives a branch point at 2)---i. (3.15) f a((’fi—m) There are various end-point singularities generated in (3.14) but, apart from the two-Reggeon branch-point, they are not singular on the physical sheet of the full amplitude near t = 0 [48, 491. The two-Reggeon branch-point is generated when the pole atf= a,(t,)+a 2(t2)—l is tangential to the integration boundary at X(t, t,, t2) = 0. The position of the branch-point can therefore be formed by solving the Landau equations —
a
0
(3.16)
±(~,[a,(t,)+a2(t2)]+~2~(t,t,, t2)) = 0
(3.17)
(ti,
[a ,(t,)+ct2(t2)1 +i2X(t, I,, ta))
=
at2 together with the conditions f,—a,(t,)—a2(t2)+l
=
0,
X(t,
I,,
t2)
=
0.
If the two Regge trajectories are identical, the solution of these equations is t,
(3.18) =
~ t, with the
branch-point occurring at fa(2)2a(lt) 1.
(3.19)
For non-identical trajectories the branch-point trajectory will be more complicated. Since the2,integration in (3. 14) is two-dimensional and the trajectory functionsinare(3.14). complex for it is not straightforward to take the two-Reggeon cut discontinuity HowI,, 12> 4m becomes much easier to work with if we analytically continue it to I < 0 [49] The ever, (3.14) details of this continuation are complicated but the net result is that if we extract the “threshold” .
III).!. A harbanel ci at. - Reu’i,’eon field theory: formulation and us
142
heilavior of ~ \/7 -
Ar)
if
fi=(’T ‘
1) at X( 1. 1,. 1/
~
til
rX( 1,
12) = 0
by writine
1,, t~)1c’ o~ I
02)2
J
‘
i~
-~
then We can simplY rotate the integration contottr in (3.14) to (lie region X(t, 1~.t~)K 0 and Write [A(t, t~,t2)J
‘2
U A(t,
1,, 12)1
(3.21)
‘2,
(For the purpose of studying the two-Reggeon bratlcil—poillt we can igncre tile fact Illat the original colltour encircled the positive I~and ‘2 thresholds.) i~ort K 0. then we obtain f’roni (3. I 4)
f
(,~
dt,d121 ir2~ 4!~<~
X(i. 1~,
~
12fl
-
‘
(‘~
I
a(f)
sIn/a-I(tIs1n~sira2(t2)U,J
~2(/2)+l)
.
--
(3.22)
If tile trajectory f’unctions are real tor negative I, and 12 (lien the collision of the pole at f = a,(t, )+a2(t2)-- 1 with A( t, 1,, i~)= 0 is 110W straightf’orwarul see fig. 3.8 and if and are both non—singular at .J = we obtani Ironi (3.22)
disc,I’~(.J,I) =
sili(~ir.I) 2~4! A
—~
J
5(.I
ult~(115
-
cu1(I,)-
I
a2(t2)+l)(~(’~ -,
-
.
‘
>3,23>
,
Alt, t~,t2) I’ sin( ~ ira )sin( ira2)
The discontinuity we have taken is disc~F(f.t)
=
F(J—~ic,t)—-F(J+ie, t)
(3.24)
since tile sign of this discontinuity is the sign of the contribution of the cut to the total cross-seetioll. To obtain tile complete discontinuity across the two-Reggeon cut we should actually multiply (3.23) by a f’actor of six. Firstly I’Ll, .J,. f2. a. a2) is syillilietric under iii. ~ji. n~alld so we obtain an exactly similar contribution trotil that part of in which Fl,. ~12 K 0. Also the phase-space Jdp we have used is that for f’our identical scalar particles. This was for simplicity. but it means that tile same Reggeons call appear in channels defined by regrouping the four-pat’tides to define new angular momentttm states. Since there are three possible Ways of pairing four particles, by repeating tile f’oregoing analysis using different angular niollielituIll states we obtain two more distinct contributions to tile two—Reggeon cut. ~2
-
---
(3.4)
fl
II ~
“~\~aIt1+a2o2
- I
Fig. 3,8. The position of the f’aetors in eq. (3.22) in the I~. t~Plane. the intersection of rise to the bramic It point at J ~(t ) -
=
Alt,
t , , t2 I and
J --a,)!, (
~2 (i2
(+1
givc~
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
Fig. 3.9. The two Reggeon
143
two Reggeon a,nplitude MA,A’(J, t).
—~
By considering the exact form for the f-plane continuations of the parts of (3.4) for which n,> 0, n2 < 0 or n, < 0, n2> 0 we can show that no two-Reggeon cut is generated [21]. Essentially this is because the helicity-Regge poles and fixed-pole lie on the same side of the helicity contour and cannot generate a pole atf
a,(t,)+a2(i2)—l.
The next problem is to take account of the presence of the two-Reggeon branch-point in ex2, and we shall G~(f, t). c~(J, I) will singular at fin=C~(J, a~,rather = a~, i> l6mcondition for ploit this. To study the be branch-point i) wethan havef to use when the unitarity F(f, f,, f 2, ni, n2). The four-particle threshold discontinuity formula that has to be partial-wave projected is [42,
_______
431 —
=
(3.25)
.
Proceeding through the analogous steps to those above we are finally led to study the twoReggeon cut in a four-Reggeon amplitude Maai(f, I) shown pictorially in fig. 3.9. This amplitude is defined as a double fixed-pole residue of a four-Reggeon amplitude defined by factorization at Regge-helicity poles. (The fixed-poles occur atf= a’,(t’,)+a’2(t~)—1andf= a,(t,)+a2(t2)—l, and the Regge-helicity poles occur atf1 = n1 = a~(t1),I = 1,2, 1’, 2’.) To study the two-Reggeon cut in M~a~ we have to go to the discontinuity relation for the eight-point function [42, ~ —
-
=
.
(3.26)
The procedure of projecting (3.23) and (3.24) onto partial-wave and continuing to complex helicity and angular momentum involves even more complicated expressions than those we have already considered [211,mainly because of the proliferation of variables and indices. Therefore we shall not attempt to give the full details of the manipulations of these equations, but instead illustrate only the formal structure that is used. We can write (3.14) in the form 4(J) = N(f) F(J)N4(J) (3.27) F(J) F where N(f) N~(f, t) and 17(f) is a formal expression for the integration in (3.14) which generates the two-Reggeon cut discontinuity given by (3.23). From (3.25) and (3.26) we obtain analogous equations to (3.27) —
N(J) —N4(J’)=N4(J’)F(f)N(J)
(3.28)
M(J)
(3.29)
—
M4(J)
=
M4(J) F(f)M(J).
The appearances of the same 17(f) in all three of (3.27)—(3.29) is vital. (3.27)—(3.29) are initially obtained for t ~1 6m2 and ReJ> Rea~.If we continue them to Ref < Rea~and use ± to denote a ± ie prescription in the f-plane with respect to the cut we obtain from (3.29) M(f~)—M4(J)=M4(f)17(J~)M(P).
(3.30)
144
lID.]. Abarbanel eta!., Rs’ggeon field theor”; formulation and use
From whidil we obtain
6M(J)
=
M(.J~)---M(J)= M4(f) [~(J~)6M(J)+M4(f) SF6J) M(f)
>3.3 I
and so I ---M~(J)F(J~))~M(J)
=
M4(J) 6[(J) M(J~
>3.32>
=(l---M4(J)F(f~))MfJ~)6l’6J)M(J) which has the
8MCJ)
=
(333)
sOlt,ltiotl
M(J~)~F(J) MCJ).
>3.34)
Similarly we can obtain*
~N(J)= N(J~)6F(J)M6J).
>3.35)
a tid
8F(f)
=
NCJ~)~F(f) .‘VCJ).
3.36)
For t < 0. 517(J) is given by (3.23) (apart from the factor of six) and so writing out (3.36) in full 6(J--a,(t,) -a
disc F(J,
1) =
-~
sin(~if)
f di, di2 [ -
A
<0
2(i2)+l)C~(Jt1) C’~r)(J~ 1) 2sin~ira ,(t X(t, i,, t2)] “ 1)} sin {~ lra2( } -
-
--
--
,
(3.37)
2)
It is clearly straightforward to write out detailed versions of (3.34) and (3.35) giving the discontinuities across the two-Reggeon cut in N~r)(f~ i) and M~A~,l(J, i). There are several important properties of (3.37) which we should note. The most immediate point is that since C~(f, i) is real-analytic ~ I) and so the sign of the two-Reggeon cut discontinuity is determined. If both a, and a2 are Pomerons so that in (3.37)
J
2a(t)
--
1
a(i)
1
fort
0.
(3.38)
the integrand is positive definite and the overall negative sign persists. A detailed analysis [491 shows that this negative sign can be traced to the fact that the “signature factor” sin ~ ir(f—n ,—n2—- [f +~‘~] ) } in (3.8) is evaluated at a nonsense wrong-signature point where it gives Tile next point we note is that the signature factor sin(~if) in t’ront of the integral in (3.37) ensures that the branch-point is not present (as a fixed singularity) in tile physical i-channel partialwaves that is F(f, i) evaluated at even integer f. Another point is that the factors sin( ira,), sin(~7ra2)lead to the getieration tilet~, Reggeon2),ofX(t, i particle branch-point in (3.37). It is generated when a, or a2 = 0 (i, or = ,n 2) = 0, andf= a, + a2 ---1. Because of this the two-Reggeon cut is able to shield the Reggeon-particle cut from the physical sheet of the f-plane when t 0, so that the Reggeon-particle cut does not contribute to the high-energy scattering. -—-(~
--
1.
--
12
‘~
*‘fl~,~ procedure for isolating a discontinuity is the conventional one used in S-Matrix theory. In particular it is analogous to that used in non-relativistic scattering theory for finding the discontinuity across the elastic cut in the T-matrix, The analogy is drawn in ref. 1501 where the two Reggeon cut discontinuity is discussed this way.
H.D.I. A barbane! eta!., Reggeon field theory: formulation and use
145
2
M
~
Fig. 3.10. The two particle-, two Reggeon amplitude. The net helicity in the t-channel is A 1(t1)
+
A further point with important physical significance is that (3.37) shows that the cut discontinuity is entirely controlled by the fixed-pole residue of N~(f,i). This residue has been defined in terms of the f-plane continuation of F~U,n) by first continuing to Regge poles at = n, = a,(t1),f2 = n2 = a2(i2), and then to the fixed-pole atf= a,+a2—l. However, we can also regard N~(f,I) as the Froissart—Gribov projection of a momentum-space heiicity amplitude which the scattering particles into two Reggeonsamplitude. see fig. This 3.10.isThis amplitude 2,describes t) is commonly referredoftotwo as the maximum helicity-flip because in the Aa(M t 1 and 12 channels n, = J, and n2 = f2 would be the maximum values n, and n2 could take if f, andf2 were physical spins. (In the symmetric notation we have used, if n1 is defined as the helicity of the f,-state then, —n2 must be defined as the helicity of Froissart—Gribov the f2-state.) 2, I) satisfies the usual formula The Froissart—Gribov of Aa(M (omitting i~labels and theprojection signature labels associated with ~ for simplicity) —
Na(f,
t)
=
~ZO
~
Q~A2(z)
ImA~(z,I) + lj
dz
Q~t~2(_z)
z being the center of mass scattering angle. At J = a 1+a2—i, Q~,~2(z) since it is Fa(M2, t)
ImA~(z,t), 2 =
~
—
(3.39)
i)(a ,+c12), and
(z
~(z2_ l)(a,+ct
=
2)Aa(z,
(3.40)
i)
which has no kinematic singularities at z = ±2,1, the second in (3.39) be first written t) and then integral transformed to ancan integral over the as an integral cut around left-hand cut of Fa(M right-hand usingthe Cauchy’s theorem. Thus we obtain the usual expressions for fixed-pole residues of hehicity amplitudes NaUa,+a
2, 2—
i,t)~f
(3.41)
1).
dzlmFa(M
Now the Reggeon particle imaginary part is experimentally accessible in the single particle inclusive process shown in fig. 3. Ii. In the regime where s with t and M2 fixed, the inclusive cross section measures ImF~(M2,0) given in (3.41). Inclusive cross section data provides us with a phenomenological handle on the strength of the discontinuity across Regge cuts [5 11. Our final point about these formulae for the two Reggeon cut concerns the phase space. In the neighborhood of cs -~ f — 1 which is the regime of the P, all factors besides the phase space are smooth. The phase space integration is two dimensional so we take advantage of this by defining -*
146
11.1)1. A barbanel vial., Reggeon field theory: formulation and use
~M2~
_t~ ~=~-
ZERO MOMENTUM TRANSFER
Jig. 3.11. ‘i’he single particle inclusive process in the limit
=,
t..112 Used. [he Reggeon—particle absorplis~’part measured here -i
two vectors q1 for each momentum transfer t~= .1 can be written
561 a,(t,) -a2(t2)+l) -
= 5(1
.
Further we note that
-J (1 -a,(q1fl (I a2(q2))),
tile
liii..I~l.l12
UI is
delta function on (3.42~
which encourages us to write Reggeon energies L’, = I a,(q ) and I: = I .1. Ihe discontinuity across the two Reggeon cut in LIE, q ) is then (absorbing signature aild such (‘actors) (see fig, 3. I2 discEflE, q) X
=
---
2iJ’d2q,
N(L’+ic, E,. q,,
1:2.
2q 2(q- -q,- q (IL’, d 2dE2S(L’ -E1 -L’2) 5 2> 5(EI (I a1(q10) 5(E2 fa q2)N(’~ie.F,. q,, L2.
~2).
(3.43)
where A- is the Reggeon particle amplitude as shown in fig .3. I 2. This discontinuity torillula is identical to that given by (2.32) witll a = 2. It suggests that we treat the Reggeon as a quasi—particleS lit-’ing in [ito s~)acea/Ic! one ti/lie thniension carrying energ~ E(q) = I a(q) when it is on the ‘‘mass shell’’, In this sense (3.43) is (apart from the negative sigh an ordinary unitarity relation giving the two quasi-particle intertilediate state contribution to tile imaginary part of PIE, q). Note -tilat we do not associate an energy Witil individual particles, instead pairs of particles are treated as sources of F and q (‘or Reggeons wit idil tilen propagate. Similar relations can he written for N(E) and M( F) appearing in (3.35) and ( 3.36). Perhaps the best viewpoint ~n tile Reggeon field theories is that they are a conlpact device I’or providing automatically the correct pilase space structure (‘or Re-ggeons and satisfying these Reggeon ullitarity relations [34, 52 It is also possible to discuss tile solution of (3.34) ( 3.37) using “S-Matrix” nlethods rather than tile field—theoretic methods which are tile subject ot tiliS review.Such methods are likely to be limited in use to the situation where (lilly a small number of Regge cuts need he considered. This may well be the situation for tile weak-coupling P (if it exists) but is clearly not the case for the strong-coupling P. (The reader is referred to the next section for precise definitions of weak and strong coupling.) We shall not discuss these methods here. hut instead ref’er thie reader to refs. (50 and 54—56 1 for a detailed discussion of this subject. 541
147
H.D.J. A barbanel et al., Reggeon field theory: formulation and use
E~
= E 2)
discs F(E,q
=
=
Fig. 3.12. The discontinuity across the two Reggeon cut in the particle partial wave amplitude F(E 1—f, t -—1q12). The vertical dotted line across the blob in the left-hand part of the figure indicates a discontinuity in F for fixed q has been taken. In the righthalf of the figure the Reggeons carrying two momentum qj are “on-shell” and thus have “energy” equal to 1 —~j(qj).The vertical dotted line indicates this.
Tile treatment we have given here of the two-Reggeon cut can be extended to all multi-Reggeon cuts [20,21]. The N-Reggeon cut can be studied through the 2N unitarity integral. As we have seen the four-particle unitarity integral is sufficiently complicated to handle technically and the higher unitarity integrals are worse. However, we can illustrate the general structure by looking briefly at the three-Reggeon cut (more details can be found in the GPT paper). We analyze the six-particle unitarity integral using partial-wave amplitudes corresponding to the coupling scheme shown in fig. 3.13. The three-Reggeon cut is generated by Regge poles at f. = n 1 = a1(t~), = 1, 3,4, together with nonsense wrong-signature poles atf2 = + n4 1,f n1 + n2 1 in the amplitude with J2 = n2( n3 + n4). A fixed-pole is then generated at —‘
J = a,(i,)
+
a3(t3)
+
cs4(t4)
-—
2.
(3.44)
When this fixed-pole is tangential to the phase-space boundary at 2 i~’2 t~’~ i~2= 0, i” the three-Reggeon cut is generated. For identical trajectories this occurs at i~2= —
—
—
(3.45)
—
giving a branch-point at = 3a(t/9) —
2.
t~/2=
t~’2= t/3,
(3.46)
In general the N-Reggeon cut is generated by N Regge-pole N—i nonsense wrong-signature poles giving a fixed-pole at
Fig. 3.13. The angular momentum (.T~)and helicity
(n
1) and mass (t1) configuration used in studying the six particle contribution to the partial wave amplitude. This is the way to study the three Reggeon cut generated at J a1 (t1) + a2(t2) + a3(t3)—2.
=
148
.1
lID.!. A barbanel etal., Reggeon field theory: formulation and use
a,(t,)+
,,.
a~r(t.~)(N
I)
(3.47)
which collides with the pilase—space boundary at 112
t~t2
-
..
-t~2=
13.48)
0
and, (‘or identical trajectories gives a branch—point at -
1
=
N[a(t/A°)
II.
(3.4~))
l’o obtain tile total discontinuity across the N-Reggeon cut ~t is, ot’ course, necessary to add tile colltrihutions from all possible partial-wave coupling schetnes. ‘File lorIll of the discontinuity is analogous to (3.37) being expressed as an integral of a sum of products of two multiple fixed-pole residues one evaluated above the N—Reggeon cut and one below (boundary values of cuts in sub—cilannels must also be specified as for momentum space (hilitarity). The ntost important point is that the discontinuity call be expressed a~tile N— ( quasi—)particle contribution to the quasi— unitarity relation f’or the scattering of quasi—particle Reggeons, The sign of the .-V-Pomeron cut is 1 )~‘ ‘ because tile cut is generated by (N—- I ) nonsense poles and there will be (A-’---- 1) factors of I coming from the corresponding signature factot’s. ‘Ihe extension of Reggeon unitarity to production processes is coiltpitcated by tIle necessity to perform ilelicity as well as angular tlionientthm coiltihlthations. Reggeon unitarlty will initially (Ic— termine Reggeon field tileory rules for production procc~eswhen all azimuthal angles ( the cottju— gate variables to ilelicities) are large. i~or2--*N processes tile azimuthal angles can not be taken large in the physical region and analytic continuation must he used to obtain the rules . I-low— ever, (‘or inclusive processes this is not necessary and it can be siloWll tilat Reggeon unitarity dcternlines tile Reggeon rules t’or the triple Regge limit of the one-particle inclusive cross-section in a straightforward way [58. FilIally WC nlention briefly some technical points that we ilave glossed over in our derivation of tile two-Reggeon cut discontinuity formula. Firstly, because of physical region singularities of the six-point function in channels overlapping the I, and [2 channels, the amplitude --
157
I
5~1
F(J, J,, J 2, n,, a2, t, t~.t2) will not be a single analytic function of t, and
throughout the physical region. This problenl can be ehminated by using tile Steint.lann relations together with a multiple dispersion relation to separate out that hart of the amplitude having singularities in the 1, and ~2 cilannels. Secondly it can be shown that even when we separate out that part of the amplitude, tile Froissart----Grihov amplitudes 1”~(J,n) only satisfy tile (‘arison colldition as a tunetion off for part of the physical region ranges of t, and i2. Fortunately we caii cover the whole physical region by using all possible partial-wave coupling schemes. Further tile regioli in which F~(J,n) does satisfy the Carlson condition includes tile region where the two-Reggeon cut ,s generated. As a final point we note that if’ the Pomeron has a~(0) = 1 - tilen a~(1) will ilave a cohllplex part for t < 0 and so we cannot write tile two-Porneron discontinuity as a real iltegral for I < 0. as in (3.37). Therefore, there will be n~range oft where tile discontinuity can be expressed as a simple real integral. However, in a perturbatiotl solution of tile discontinuity formulae (such as the Reggeon field theory) which begins with bare analytic trajectory functions, perturbative unitarity will involve real discontinuity formulae. 12
H.D,I. .4barbane! eta!,, Reggeon field theory: formulation and use
149
4. Theoretical developments in Reggeon Field Theory 4.1. Opening remarks This section is devoted to a discussion first of the P pole and all its cuts taken by themselves, and then the consequences of those P interactions for the structure of secondary trajectories. It is useful to begin by explaining why the P and its interactions must be studied first. The essential reason is contained back in eqs. (3.47)—(3.49), for the position of branch points in J. When one has multiple exchange of P with cs(0) = 1, the cuts and pole collide at t = 0. This is certainly true for P’s alone. It is equally true for one secondary trajectory with LSR(O) * I when it is exchanged in the t-channel with any number of P’s. In that case the cuts and pole collide at t = 0 at J = aR(O). The notation developed in section 2 is useful in this regard. There we learned that in the Reggeon unitarity relations a two-momentum q and an “energy” E = 1—f were conserved for each Reggeon. For Reggeons ~n shell the E, q relation is E(q)= I —a(q)
(4.1)
so that E(0)
= 0 for the P. This is just the E, q relation of a quasi-particle with no energy gap. Every other Reggeon has an energy gap of z~= 1 —ct(0)> 0. The addition of any number of P’s to some exchange process in the i-channel, then, does not alter the threshold for that process in the
E plane since at q = 0 a zero energy quantum may be added. The addition of any other trajectory to an already existing i-channel exchange alters the threshold (position of branch point) because finite energy, is required at zero momentum. All this is analogous to the positions of thresholds in conventional quantum field theory when massless particles are present. The P is analogous to a massless particle, and the solution to the problem of the J plane singularity arising from multiple P cuts is the solution to an infrared problem. Now we understand how the thresholds of multiple P exchange stand out and further we see why in the case of P alone the thresholds all occur at J = 1 (E = 0) at t = 0 (q = 0). From a physics point of view it is useful to recall at this juncture the significance of J 1, t = 0 before we go on to the P itself. The Froissart—Grihov formula for signatured partial wave amplitudes gives us a connection between the absorptive part of two body amplitudes AB -÷ A’B’ and the i-channel partial wave amplitude F(J, t). Schematically, ~,
F(J,
—f dss~~’Abs TAB,A’B~(s,t). 2~
t) =
(4.2)
Furthermore it is known for a variety of processes, called diffractive that ‘~AB—~A’B’(~) s large ~2a(O)~~
2(log ~
(4.3)
with cs(0) 1, ~3small, and gAA’ or hBB~numbers which depend only on the properties of A and A’ or only on B and B’ respectively. In short, the behavior of diffractive amplitudes 1. is almost energy independent; cs(0) 1; 2. shows factorization of vertices as expected from an elementary t-channel exchange. On further inspection of the amplitudes which have diffractive behavior one learns
150
lID.!. A barha,ii’l et a!,. Reggeon field theory: formulation and us,-
3. 110 ciuantum numbers are eXcllanged in the I—channel. From an operational this is the Pomeron. Equation (4.3) translates into tIle .1 plane as
F6. 0)
=
~
[J---a(
0)1
point
of view
(4.4)
‘~
that is. a brancil point at .J = a( 0) = I whose detailed nature governs tile behavior ill log s of’ elastic or quasi-elastic cross sections. The optical theorem relates tile beilavior of elastic amplitudes to total cross scctiohl5~lot (4.3) we Ilave AI3~ ~ ,.~ /1BB5 o(O) T ~S, ,s iarge ~AA
- ‘‘I ‘,
,,
.1/3
-
—
and we know fronl tile Froissart hound thai ct(0)~ I.
(4±11
and
~
2, wilell~(0)
1.
(4.71
Tile experimental beilavior of amplitudes tilen chooses a(0) = 1 as tile point where we ought to concentrate our attention. The result of’ sunlming all colliding cuts at i = I will be to determine the fine structure in .1 at .1 = I and thus tile logs behavior of cross sections. Eurtherlllore any singularities ill .1 away from J = I will give contributions to cross sections whicll are negligible by powers ofs wilen P can be present~that is. vacuum quantum nunlbers are allowed in the I—channel. So we turn our attention to tile problem of’ a P singularity ill tile J plane which lies at.! = 1 at I = 0. 4.2.
Formulation of the RET
We have learned in sections 2 and 3 that Reggeons propagate in two space, x, and one tillle, 1’. dimensions. These are conjugate to the two momentum, q. and energy, 1 --f. and are actually the impact parameter and rapidity of the Reggeon. The RFT describes this by a field amplitude ~p(x,r) for finding the Reggeon at x and ‘7’. In the absence of interactions tile Reggeon field satisfies the SchrOdinger equation
/
i— 1p(x,r)
a?-
~l
/1 --
\\
cs0{--v)) ~p(x.r).
(4.8)
\h
where ~0(q) is given by tile choice of bare ellergy momentum relation; i.e., tile non-interacting Reggeon. This function ~0(q) is a priori at our disposal it specifies the nature of tile P or other Reggeon before interactions are accotinted for. We lllthSt turn to physics to help us here. The appropriate physics is tile connection between bound states in the t-cilannel atid lai’ge s behavior in tile S channel. Fifteen years ago it became clear that the Regge pole at .1 = a(i) represents a bound state of hadrons whose mass squared has been continued fronl ln~> 0 to the value i occurring in the scattering process, and, at tile sanie time, whose spin J~ilas been continued to ~(t), [60] . This connection betweetl bound states and crossed channel large s behavior is one of the deep attractive features of Regge theory. Now, what is important l’or us is that it
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
151
follows from i-channel partial wave dispersion relations that a(i) is analytic at i = 0 unless at i = 0 there is a collision of this pole trajectory with some other singularity in the f-plane [601. For a Reggeon in isolation, then, one expects cs(t) to be regular at i = 0. For the purposes of our present concern, which is the behavior near q = 0, J = 1, we may expect that the non-interacting Reggeon is adequately represented by (4.9) cxo(q) ct0—ct~q2, where ct~and ct’ 0 are parameters which it may be possible to extract from a detailed knowledge of the way in which a Reggeon is constructed in the “correct” underlying field theoretic or S-matrix model. The free energy momentum relation is then 2 + (I—ct E(q) = cs’0q 0), (4.10) which is like a non-relativistic quasi-particle with mass gap (1—ct0) = z~and mass = (2ct~”. The mobility of a Reggeon in x, r space depends on its effective mass (2ct~’, and the smaller ct~,the less mobile is the Reggeon. Since ct’ is known to be smaller for P by about ~ compared to, say, the p trajectory, we may expect the P to be very ponderous. If we now associate a field with this Reggeon, the free action will be A0 fd2x dr £0(x, r),
(4.11)
with E0(x,r)—p~’(x,r)
(4.12)
~
2 the Lagrangian density. The next issue is tile interaction.
To determine what interactions allowed we studied turn to to models which generate inter3 field theoryaremodels were extract the RFT rules.Reggeon Other natural actions. inInwhich sectionto2examine p models this question are multiperipheral and dual models or perhaps more realistic field theories. In each of these it is possible to have four and five and higher point P interactions as well as the simplest triple P coupling discussed before. Four P couplings arise, for example, in a ~p3field theory, where the P is generated via ladder graphs, as in fig. 4. 1. Here two P’s can scatter, or one P can make a transition into three P’s. In any given model each N P coupling, N = 3, 4, ..., is in general a function of the energies (I —f,) and momenta of the P’s involved. This means that if we are to incorporate directly the content of an underlying theory into a RFT, we will have to construct a field theory which is both non-local and involves an infinite number of coupling functions, one for each independent n P -~ m P transition. For 4 P couplings, for example, we will have two independent functions E 14(x,
r)
fd2x,
2x dr, ...d 4dr4 ~
r,)~p’~(x2,r2) p~(x3,r3) ~p(x4, r4)+h.c.}
X F,(x—x,, r—r1,
...
x—x4, T—T4) + p”(x,, r,) p~(x2,T2) ~p(x3,?-~)~p(x4,
X F2(x—x,, r—r,,
...
x—x4, ‘r—r4).
T4)
(4.13)
Unless there is some simplification, we are faced with the prescription of an infinite number of arbitrary functions.
lID.!. --I harbanc’l et a!. - Ri’~geonfield theory. formulation and use
152
_~i~ _j I~~±ii
j-~
~
-~
4.1. }-~uUrP couplings allowed in a Rl’l’ of P’s. Thes- are abstracted troii~dOriA le~Oman graphs, or the niultiperiphieral node!, my theory hiavine
Reege pole’s.
ultimately interested in the in frared ( L’, 0. q, 0) behavior of ally Rfl’ we write down, as a f’irst step we expand each of tile coupling functions about this point. We begin the discussion by retainillg only the first non—vanishing terms. This reduces the non-local theory to a local theory with coupling constants and maybe a f’ew derivative interactions. ihis procedure applied to 3 P and 4 P ititeractions gives Since we are
--
~ 3~x, r) + he.
--—{~(x.r)
-~-~
~(x,
r)2~(x. TY
(4.14)
f’or the interaction Lagrangian. This has illlproved tile situation in tile sense that we 110W ilave a local theory with an infinite number of’ coupling constants. However, there is Ohle hll0~C level of heuristic argument we may use to eliminate from our prelilllinary consideration any but tile 3 P coupling. Essentially we argue that to emit and absorb P’s inside any diagranl of’ our RFT costs a phase space factor dE d2q each time. We are ultimately interested in the E 1 0. q. 0 hmit of’ the RFT so each additional phase space t’actor decreases tile importance of any given grapil. N P couplings with N > 3 require more phase space than N = 3; therefore. one may ignore tilehll. Later in this section we will return to reconsider tile wilole matter of iligiler point P couplings aild derivative couplings, and justit’y this heuristic argument. We must incorporate a final point troill sections 2 and 3 before we begin tile disctisSlOhl of tile various attacks on RFT. In those sections we learned that tile two P cut carries a mi~ussign relative to the contribution of a single P exchange. A grapll like fig. 4.2 IlILIst cohltaibl a Illiiltis sign. This is naturally incorporated as part of tile Fevilman rules of the RFT if tue coupling A0 in (4.14) is pure imaginary: A0 = ir0. Tile RFT now to be discussed has tile Lagrange fuilctiohl -~
-~
H.D.l. A barbane! eta!,, Reggeon field theory: formu!ation and use
153
Fig. 4.2. A Reggeon graph contribution to the elastic particle amplitude. This contains the two P cut and must contribute negatively to the total cross section. We arrange this by writing the triple P coupling to be purely imaginary ir.
=
i
ir0
‘
2
3r
(4.15)
2
The solution of this field theory with an anti-Hermitean interaction in the infrared limit is the central problem of RFT. The spectrum of the theory determines the Pomeron. This was first written down and studied by Gribov and Migdai [61—631.it is with their work that we now concern ourselves. 4.3. Early developments in RET for
P
To review in detail the work of the Leningrad school would be a monumental task. Leaving this pleasure to the historians we will sketch the essential results of their calculations. To begin we established some notation that we will employ frequently. The objects of primary interest in RFT, as in any field theory, are the Green’s functions for the m P -+ n P transition T~1, ...y,~,T~) =
(0lT~(p~(y1, Tyt)...~(Ym, r~)p(x1,r~1)...ip(x~, r~))iO>
(4.16)
where 0) is the no P state. There is no crossing in RFT so both n and m must be given. The Fourier transform of G(~~.m) is of interest also (fig. 4.3): n
n+m
E1
—
ii
~
q~
)62(~
—
jn+1
i=1
2ym ~
exp{—iq
fd2xi dr~1..d
n+rn
fl
E1
~
q1) G(nm)(E1, q1)
(4.17)
=
1n+i
1x 1+iE l?-x1
—
...
~
Tym }G(n.m)(xi,
r~1,.. .y,.~,~
The spectrum of the theory is specified by the zeroes of the inverse P propagator
En+m,~’n+m
Fig. 4.3. The notation for a n —~m Reggeon Green’s function.
lID.!. .1 barbanel et a!., R,-~m,’eo,,field theory: formulation and us,’
154
1-ig.
4.4.
t 1,1> is the html! P propagator md I~‘~ ~ the Imiple
fhe Schwinger Dyson equation for the Poimmeron proper wit energy. G I’ proper s’ertes unction,
F~’”~(E, q2)= IG(11)(E, q2)L.
(4.i~)
It is coilvenient to disctiss the one P irreducible proper vertex f’unctions [0’ tim)~L,, q,) gotten by amputatilig tile external legs of’ tile one P irreducible part of’ (~°‘ ~. l~I .1) or I - ~ and 1~ .2) the 3 P vertex tunction, are promilleilt ill tile analysis of Grihov and Migdal. I
In the non-illteracting theory
G~U(E,q2)
=
--~--
~(i
takes tile valtIe
,
~---
L--a 2--0q
.i)
(4,1’))
~O+lC
where the ie prescription means only retarded propagation. Its origin is tile requirenlent that. ill tile Sommerfeld—Watson integrals def’illing tile tileory. tile iultegration contour in .J lies to tile right of all singularities in F(J, t). When all interactions of P’s are summed, tile inverse propagator takes the form
iFt”~(E,q2)
=
E—ct~q2-—~
2), 0---~(E, q
where the proper self energy
~(E, q2)
=
~
must
satisf’y
(4.20) the Sch winger—Dysoll equation (t’ig. 4.4)
(q--q’)2) 1
wofd~ dE’ [F( “1~(E’,q2) [t”°(E--E’, 2 (_~)
X FD~(E’,F—F’, q’. q—q’) The Soviet workers discussed
+
ill
(terms due to i’~ ~
-i
with ii ~ 3).
(4.21)
detail two possible solutions to tile RFT we have Posed
(a) The weak coupling solution. wherein tor snlall L’ and q2 ~w.cfL’,
q2) ~ F, q2.
so that the resulting propagator is very much
4.22)
tile
the P spectrum is a simple pole still, constrained iF~’~(E, q2) E---ct~q2. Eq2
same as tile unperturbed propagator (4.1 ~) and 110W to have ~ = I -
ct(
0).
50
(4.23)
=0
This cannot come about if the higher order vertex functions are arbitrary since they are linked to 1) via the representation. In particular it’ l’~(~2) the three P vertex fuiiction took 011 tile -
-
constant value of lowest order perturbation theory, tile right-hand side of(4.2 I ) would be infrared divergent. N1’2~must therefore vanish at the inf’ra-red point where all of’ its arguments are zero.
H.D.J. Abarbanel eta!., Reggeon field theory: formulation and use
~(
~
155
~
Fig. 4.5. The graphical representation of the triple P vertex function ~
2)
Gribov and Migdal suggested that the zero of pO~2)should be sufficiently strong that the infra-red regions in (4.21) and the analogous equation for F~,2) shown gr’aphically in fig. 4.5, be completely suppressed. This requires the leading behavior of p~,2) to be analytic at the infra-red point, since the only possible source of non-analyticity the infra-red regions are suppressed in the integral equations it satisfies. Therefore —
q
—
2+c(q~+q~) + higher order terms,
(4.24)
1, E2, q2) = a(E1+E2)+b(q1+q2) and the detailed form of F0’t~is then q2)
iF~’0(E,
=
E—ct’ 2 0q
+
(aE+bq24CE/ot’ 2 , 0) log(~cs~q2_E)
(4.25)
—
l6irct
0
where a and b appear in F~~2)and contributions further terms when in (4.21) negligible[20] in the 2’2~are also from required to vanish theirare arguments infrared limit. Note that [~(i,3) and F’ are zero if (4.24) and (4.25) hold. (b) The strong coupling solution [63] in which for small E and q2 ~
q2) ~ E, q2
(4.26)
and both the propagator and vertex function have a scaling form for small values of their arguments
iF~’~
—
(_E)Y~~(q2/(~~E)Z),~
(_EY~F((~)Z~ (E)Z’
(_E)Z’
)
(4.27)
where 3’y+z—2~z= 2. In this case the infra-red regions of the Schwinger—Dyson equations are not suppressed, but instead the scaling forms of F~’,’)and ~ combine so that these regions scale consistently throughout the equations. The weak coupling solution leaves the P pole essentially unchanged. The gap ~o (or intercept of the P pole a 0) is shifted to zero (or ct(0) = 1), but basically a pole at t = 0 is left a pole. The three P coupling function, however, changes quite drastically. In lowest order perturbation theory one has 312, =
(4.28)
r0/(2ir)
while the fully interacting solution to the RFT vanishes at E 1, q, -÷ 0, as in (4.24). Such a phenomenon is possible because of the imaginary triple P coupling. Formally, the full [~i1,2) is defined by an infinite series in r0 with terms of alternating signs, so a cancellation or extinction of the lowest order term is possible at isolated points in momentum space. Physically one argues that P exchange is absorptive, a fact which is embodied in the imaginary nature oi’ the coupling
lID.!. A harbanel et a!., Ri s’gm’on field do ore - formulation audits
156
~
I ig.
4.6. The integral equation br lime triple P vertex unction. i~ .21 ~ time Iwo P irreducible Itelhe S.mht,’Ier kernel.
three P’s. and that tile Shill of all absorptive corrections ( f’ig. 4.5) cancels 1111 the leading colislan I ‘[‘his solution is also called a quasi—stable I’ontcroii because the rate f’or the tieca of’ a P 11110 t ~ P’s vanislles abnorillaily rapidly as energy tends to zero. [‘his argu nlent of (ribov and M igdal em I was tile earliest argument for the vanislling of’ tile triple vcrtcs. The dynamics underlying a quasi—stable P was later studied ill a ladder graph Illodel iii RF I b~ Bronzan 1641 . He wrote an integral equation (‘or 1A .2) of tile s~itihol c 101111 (see fig. 4.() of’
I
P
F2)=ro+
l2)G,G.1)I~(22) (421)) tile two P irreducible part of [(22) It is the RF’I’ counterpart of the l3ethe S~tInetc where V~ kernel encouiltered ill conventional field theory. File model coilsisted of appro\illlatlne irreducible V~2’2~ by the one P excilange graphs of f’ig. 4.7. Tilis yields [~2) as an ilfinite series of RIT ladder graphs. Tile potential, y(2.2) ill this model is very singular ill tile infrared limit. Ihen (4.20) can hold only if LII 2) vanishes as L’~,q 1 -÷ 0 from almost all directions, ill tileaectllllulate weak eouphng t .1) all ot as wilicil at solution. There is also shakes an infinite of new P poles G F = 0. This last result one’snumber confidence ill the weakillcoupling solution. It is true tllat bare vertices have been used ill fig. 4.7. so V(22) is probably overly singular. On tile other ilalld, if (4.24) holds, and complete vertices are used ill t’ig. 4.7. Vt22~~5 11011 singular. and there is 110 reason for Ft’ i) to vanish. Either way, tile weak coupling solution does Ilot occur iui detail in this model. Tile weak coupling solution is subject to other difficulties. One line of argtilllellt notes that il~ the P is a pole passing through j = 1 = 0. tilen tile vanishing of [(1.2), which may be demonstrated in marvelous generality using tools wilieh blossomed ill tile study of’ inclusive reactiolls [1 7 requires tile vanishing of large numbers of other P couplings at t = 0 1 I S , including tile coupling to particles [19] which is supposed to set tile scale for asymptotic hadron total cross sections. This apparent disaster has been circumvented by arguments by Grihov [65 I and by (‘ardy aild White [52, 661 . The former examines tile vertex function for two iladrolls ailCi a P when tile hadrons are considered as composite systems. ile finds tilat tilere is a very neat cancellation title to P interaction with the constituents SO that the particle particle P coupling Ileed ilOt vanisil at t = 0. One of the conclusions he draws, however, is that all hadron cross seetioiis illtlst he asyntptotically equal as well as constant. If tilis is indeed the case, then the asymptotic regime where
22>
15
.
t
.
this takes place is a long way away. Cardy and White pointed out that if’, as in Bronian’s model, an uifra—red singular
[(2.2)
was
responsible for the zero of [~,2) then the decoupling arguments based on the relation of’ pole contributions in inclusive cross-sections [71 via the inclusive sum rules, woLlid not go tilrough. This is because the singular potential will also contribute to the vertices appearing iii cut contributions to the two-particle inclusive cross-section. Consequelltly tile cut coiltribLltioilS are enhanced auld not
lI.D.I. Abarbanel et al., Reggeon field theory: ,fbrmulation and use
Fig. 4.7. The lowest order contributions to
157
vA2,2)
suppressed relative to the pole contributions. The sum rules can no longer be used to relate pole contributions. Therefore tile vanishing of the triple-P vertex no longer requires the couplings of the P to other Reggeons or to hadrons to vanish. A further point noted by Cardy and White was that if the complete two particle/two P amplitude is separated into its one P irreducible and reducible parts then the one P irreducible part satisfies the same integral equation as ~(1,2), that is (4.29), except that the inhomogeneous term is different. Therefore an infra-red singular V(2,2) will also produce an infra-red zero of the one P irreducible amplitude. The leading contribution of the two P cut to the total cross-section then comes from the one P reducible amplitude that is the cut couples through the pole and factorises, hence 2). (4.30) u~(s) ~ + O(1/(logs) —
However, as noted above v~2’2~ is only singular in Bronzan’s model when vertex renormahisation is ignored. In fact since we have no complete weak coupling model we cannot say whether a singular [7(22) is the answer to the decoupling problems. If it is, it seems that the higher-order p couplings must play an essential role in producing the singularity. The weak coupling or quasi-stable solution to the P problem in RFT is attractive for a number of physical reasons: (1) It is rather simple; (2) It has an immediate implication for the detailed processes in the s-channel which produce it; namely, some sort of generalized ladder or multiperipheral graph. The single particle inclusive spectrum will have a rapidity plateau and the cross sections for n particles, an(s), will be more or less that of the multiperipheral model with small corrections due to the two P exchange cut [361. One may hope to parametrize the cut in terms of a small number of undetermined constants. Since the weak coupling solution satisfies t-channel unitarity and apart from the decoupling problem seems to be otherwise self-consistent, it is certainly attractive. We will argue, however, that the weak coupling “solution” is not the solution to the problem set by eq. (4.1 5). The strong coupling scaling solution was not favored by Gribov and Migdal [61, 631 . Their argument began with the observation that near F = q2 = 0, one has iF~~1)(E,q2) =
—~(E,q2).
(4.31)
Let us accept eq. (4.31) and then examine eq. (4.21) together with the Schwinger—Dyson equation for 17(1,2). We now substitute 1”~’~-* 1”~’~’) = This transforms the Schwinger—Dyson equations into those for a strong coupled field theory with Hermitean coupling. (The role of eq. (4.31) in this is that when it holds, changing the sign of J”0~1) is equivalent to changing the sign of ~.) For the Hermitean theory we can invoke the Kallen—Lehmann representation to learn that ImF(1,1) has a definite sign. The sign is such that the contribution of fig. 4.9a to cr~is negative. Clearly this is unacceptable and were this the way a strong coupling, scaling form for F~’ q2) is in fact achieved by the RFT we consider, one would be forced to reject it. _p(1~~•
.i)(E,
11,1),!. -1 barbanel y’t a!.. Reggeon field t/leors’: formulation avid us,’
158
Development beyond the work of Gribov and M igdah ilas been possible because there is an alternative infrared behavior to tilat of eqs. (4.22) and (4.26)
~(E. q2)
=
E--a~q2---~>-o’(E, q2).
u(Ii’. q2) ~ F, q2
(4.32)
.
The hare P pole is extingtliSiled, amid a stronger singularity is present in G~I ,~at I: = q2 = 0. 11115 singularity represents the confluence of the P poie and cuts at E = q2 = 0. (For q2 ~ 0. there is still a P pole.) Note that the weak coupling solution has a pole as tile leading singularity at = q2 = 0, wilile the Gribov Migdal strong couphng soltit ion, eq. (4.3 1) has a weaker singularity tilan a pole. We now turn to what may be termed “recent developments’’ of RF’[’ to see how eq. (4.32) enlerges. 4.4. Recent dei’elopments in the RF’T~.irthe /‘onieron In tile work of Gribov and Migdai 16 1 63 I it was assumed that tile interacting Pomeron has = I = 0. Because there is no energy gap. one must s’itller SUill the full perturbation series to learn tile infrared behavior of ~ or Ilave compelling argunlents for any omissions. Faced with this. Grihov and Migdal studied the Sellwinger Dyson equations for the complete proper vertex functions. Tile solutions they found were tllerelore not calculated dii’eetlv from tilt’ Lagrangian, eq. (4.1 5). but were based on self—consistency conditions. Tilere is another way to calculate the infrared behavior of tile which avoids perturbation theory in the three—P coupling. Tilis approach uses tile constraints of tile renormalization group to yield the allowed forms of tile ~0. m) ill certain regions of’ E~.q 1 phase space. This approach was used by Migdal, Polyakov, and Ter-Martirosyan [251, alld by Abarbanel and Bronzan [341. In describing it we will use tile notation of tile latter atltilors. although tile Soviet results are identical where they overlap. The pililosophy of the renormahization group approacil to tile study of quantulll t’ield tileories is described at some length in summer school lectures by Coleman [671 amitl by Abarbanel [65] Here we confine ourselves to the outlines and concentrate on tile results for tile P. Our RFT has in tile Lagrangian of (4. 1 5) four parameters: tile scale of tile term involving ~ which is taken to he I the slope parameter a~ the bare mass gap ~ and the bare coupling coi’stant r0. When the full proper vertex functions are evaluated they again may he parametrized in terms ot’ four numbers: I. a’, & alld r which replace the bare parameters. These nuni herscon2 derivatives and the value of f’otir 2) at some are specified by the value of of [~1) F, qand its FWe andfollow q venient. but arbitrary. point space. ref. [34] by taking L’~ ---E~.< 0. alld - - a(0)
RFT
~(I
q
1 = 0 as this point. Pilysies dictates that a change in /i’,~ E,v + &L’N must have no consequences (‘or tile physical content of’ tile RFT. We can compensate for and the rcilange in L~.by tile substitutions 6FN/EN. r + ~E, a a + ~ ~LN/Erg, ~ -÷~ + ~ I + Y 5i/Ey.. which are ellosen in such a way tilat tile vertex functions computed from our Lagrangian are unchanged. How is this done in practice? The fields ~ which enter the Lagrangian have their scale (nomnlallzation) altered by the interaction so that the unrenormalized ~ becomes --~
-~
—~
--~
I
(433) while a~, and r0 become a’, andr respectively. Tile ~~mm0 expressed in terms of~u are ftmctions of a~, r0 and a possible cutoff to define tile integrals in perturbation theory. Tile [(nm) ~
~,
~
H.D.J. A barbanel et al., Reggeon field theory: formulation and use
159
expressed in terms of p are functions of a’, ~, rand EN. The relation between them is 2F(n.m)(E. q., a~,~ r F(nm)(E. q,, a’, ~ r, EN) = z(n+m~ 0, A)
(4.34)
where A represents some form of cutoff. Since 1”~~,m) has no knowledge of EN, it must not change when EN -÷ EN + 5Ev. 17(n,m) however, must satisfy a a a a (n+m) ,~ [EN —+~(r,a )—+~(r,a’)~ ~+~(r, a ) --~-~y(r,a )jF(n~m)(L. q~,a’, ~,r, EN) = 0. ‘~
~—,
(4.35) This clearly puts a constraint on the way the parameters can enter 17(n,m), The P problem requires z~= 0 = 1 —a(0) for the fully interacting P. This, as in tile earlier Soviet work, is taken as given and is not derived. Clearly this requirement demands a special relation between a~,r0, and z~.We will return to that relation. For the moment, however, we imagine that this relation can be and has been arranged, and continue to seek the consequences of a theory with ~ = 0. The solution to the constraint (4.35) known as the renormalization group equation is —
q~,g, a’, EN)
=
—
F~a~m)(E1, q1, ~(—log~),~‘(—log~)) exp
f
~‘
[1
-
~
(4.36)
—loge
where we have introduced the dimensionless coupling 2E~j’2 g = (a’)~ and the auxiliary functions ~ d~/d17= —f3(g~(rj)),
(4.37) and ~‘(r~)satisfy
,~(0)= g
(4.38)
and 1
d&’(r~) ____
=
a(~) d~
1—
~(r~))
~ ~,
a(~)
z(g(~)),
a(0)a.
(4.39)
In these we find the crucial function p3(g) which measures the response of the dimensionless coupling constant to a change in the normalization point. The utility of the renormalization group approach is now explicit. We want to study [‘(E 1,q,) as F. -* 0 or in (4.36) as -~ 0. This means we need to know ~(i~) as ri = —loge + For this we need to know j3(g~(r~)). The key observation is that if j3(g) = 0 at g = g1, say, then g~(r~) approaches g1 since d~/d~ vanishes there. Whether ~ goes to g1 as ~ ± depends on the slope !3~= d13/dg~g1 For the infrared, r~-÷+ oo, limit, we require /3k> 0. In this case ‘-~
°°.
‘÷
.~(—log~) = g1 as
-~
The
+
(g_g1)~t
0. ~~m)
are constrained by eq. (4.36) to then have the form
(4.40)
11.1). 1. I ba,’hanel
160
~(n.om)~~’•
q~.g. a-’.
L~)=
J+z(L~l)(2-
‘m om)/2
-
/
it
(~/:v
a!. - R-ggc’omm field theorm’ - fornmm,latiomm amid mis,
,)
(2
oi)/
2
(4.41)
(Omf’mm),’2)y)(’i I
/
)
Lv
0
-Porn
L
.
/
( F~ F .
-
E\ -
—)
)
-
,
Si
(,,csq1’q1 -
.
-
1/
~,‘m
-
~vlicrc
F and
= ~
~‘l
f
(4,42)
and
(‘,,
are some constants while
2,g. o’,L’~) i[°’°(E, q
~
is
— c’~F~(;)
~)~i
an tmknown ftinef ion. l”or
[(II
this means
2~L / (‘,5a’q ~ ~
~
5j’)-\z(s’
i)
Ri -
.
(4.43)
wilicil is precisely the sealing form suggested in the earlier strong coupling soltitioii. Now we have to inquire into tile possible values of’ g . Suppose g = 0: then the tileor\’ is a tree theory in tile in framed limit and y(g 1 ) = 0 while z(g1) = I . This is tile weak coupling or quasi-stable P. II’ g1 0. only the strong coupling solution is chosen. From a eOnlptltafional point of’ view. tllell, the nature of the P reduces to tile study of tile ieroes of’ ~(g) and tile evaluation of y(g1 ) and zIg,). Tilis is a task not mucil less complicated tilan solving the full RFT. There are two modes of attack wilich we know. One is likely to he unreliable: tile second is complicated in procedure, hut likely to be reliable. Tile first technique is to use perturbation tileory to learn ~(g). In lowest order we need the graphs shown in f’ig. 4.5 and fig. 4.8 which yields (4----D) g+Kg~.
K> 0,
(444)
when the phase space ilas D space dimensions. This ~(g) Ilas a zero with positive slope at 4---D --
.
(4,45)
4K
It’D 4, tilen g, would be very small and having evaluated only the lowest graphs f’or ~(g) would be extremely reliable. Physics requ.nres D = 2. however and the accuracy of keeping oillv these graphs is at best problematic. Indeed. only the graphs silowil lead to 0.17.
(4.46)
and =
(‘or D
= 2.
1.08
To next order in an expansion in 4- 1)
(4.47> it
is found that 169. 701
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
161
+ Fig. 4.8. The lowest order contributions to
4—D
—y(g 1) =
4—D\2 257
[~
4
i~(1 ,i)
371
(~-~)+(~~ log~+~j+
...
(4.48)
and 2 I’155
z(g 1)
1
+
‘4—D\ 1 ‘4—D\ ~—)—+~——“) \l2
-
2
12
L” 24
4
791
log+”j+ 3
...
(4.49)
48
or —~y(g1)
(4.50)
0.38,
and z(g1)~
1.18.
(4.51)
It would seem, therefore, that this series in 4--D or equivalently in g~is unreliable at 4—D = 2. Such series expansions are employed in statistical3)physics for thebecritical second orderto terms might small, indices but we in know no way phase transitions. It is possible that the O((4—.D) be sure except by direct (laborious) calculation. From this procedure, then, one has definite, albeit unreliable, values for ‘y and z. If one reformulates RFT on a lattice, then one can envision using techniques developed in statistical mechanics to evaluate ‘y and z directly in two-dimensions [71—73]. It has been possible to show that if there is an infra-red stable fixed point of the type discussed above, —y ~ z ~ 2; so the Froissart bound is satisfied and a 0tot [731. Numerical calculations of y and z are 51 ~ presently in progress [71--731 Here let us comment on the physical implications of the existence of a zero of p3(g) at g 1 ‘‘ 0. First of all, of course, we have the scaling form which is similar to the strong coupling solution of Gribov and Migdal [63]. Indeed, the weak coupling or quasi-stable P would appear to be out. There is a very significant difference, however, since in the present solution 170 is not equal to but has the value —a, where a is small, as in (4.32). By computation, the problem of negative total cross sections does not arise. Second, even though the bare Pomeron pole is extinguished there is a moving pole, on the trajectory ~
—~,
a(t)
(4.52)
= 1 + ctl/z(gi)
This can be shown either by a perturbative evaluation of Ipii, [25, 341 or by a more accurate evaluation using the full power of the renormalization group [59]. In either evaluation of z(g1) given above, z(g1)> 1, so any trajectory has a cusp at t = 0. Third, when one couples particles back into the theory, one finds a hierarchy of contributions to ar(s) for A + B -~ anything (see fig. 4.9): ar(s)
‘~
g~g~(l0gr)’
—f~8(logrY’+ k~~(logs)_2 (ii)
+
(4.53)
111)1.
162
A harbanel eta!., Rm:geo;m field 1!meori’: formulation amid misc
(a)
s(s)
g
5 g~(icq
-
~-:~S
h~-~Ieg 0
I ii.’. 4.9. ‘The imierareim~of contributions to the total cross ser’tjuIm fur 5 4 u —--‘aim timing v-ommliime ruin lime strulme eoupli me, Sealimme Soiutlofl to time Pummmeromm RI-I, ‘T IS tim,’ imnoimiaiomms dimension ma’ time P field. in an expansiomi of’ the RI-I about seale inv:mrialmt theory
-~is
found (po~sihivmmnreliahiy) tim be LmhOmmt
‘~-
It is iniportant note that the dominance, of’ logs, of the generalited graph of 9a. yieldsto a f~ictori:ed as)’m/)to tic totalb~’powel’s (‘ros,s’ set-lion. Sines’ y(g, 1 > 0. this ‘‘pole’’ cross section rises. fig. 4. 4.5. Secondary trajectories and multi—J’onueron corrections Having in iland a theory of tile P by itself it becomes quite natural o ask wllat will he tile structure of’ partial wave amplitudes with quantum number exchange or, like the P’with a( 0) > I This has direct application to experimental fact in tile ease of the p trajectory with a~(0) since experiments oii irp 71°Il for a Lirge range of’ incident beam niomentum show an s and I dependence consistent with the exchange of a sinlple p Regge pole with trajectory I 7 I —~
a,~(t)=
+
t/(GeV/c)2.
(4.54)
Also in the case of fermion trajectories there is the long staiidi ig problem of parity doublets we will C0il1~ to this. Since tile excilange of any Reggeon witil a-R (0) < I requires an energy gap E( 0) 1 a,~(0) -> 0. the important processes wilichl determine the sillgularity near.! = a 1(( 0) are one Reggeon-multiple P excilanges. All such processes have an infinite number of tilresholds at F = I a,~(0). If’ we restrict ourselves to triple cotipliigs by phase space or simplicity argtulleilts, then only graphs such as fig. 4. 1 0 will occur and we have two couplings: r0 from our earlier work and a Reggeon Reggeon P (RRP) coupling which must be itire imaginary for tile sanie sigilature reason. Because Reggeons with aR(O) < I are to occur only once, tlleir number operator is conserved and we illay solve (‘or the coupled P-Reggeon Green’s functions as all P energies = I --J~--~0 and all shifted Reggeon energies (~= First we consider tile case of boson trajectories (P. A2. p. w, or wilat have you.) lilis was studied ‘
H.D.I. A barbanel et al., Reggeon field theory: formulation and use
163
Fig. 4.10. Reggeon graphs for the propagator of a secondary trajectory, dotted line, coming from its interaction with the P, wiggly line. The P interacts with itself. For the usual reasons only triple couplings are considered,
in the early Soviet work by Gribov, Lenin and Migdal [74].. They rather casually treated the ratio of slopes for the P and Reggeon and neglected the triple P coupling altogether. This latter point is certainly in the spirit of the quasi-stable P. They found that by using a “Ward identity” they were able to determine almost on dimensional grounds that an originally linear Reggeon trajectory aRO(t) = aRO + a~
(4.55)
0t
was strongly modified by the RRP interaction to (4.56)
T11’~,
aR(t)
= aR(O) ±~K’y
or if there were no
RRP
aR(t) = aR(O)
a~t
+
—
coupling but only a RRPP quartic coupling, the resulting trajectory was Ct/(logt)3.
(4.57)
Both these solutions are at best true in the weak coupling case. In the case of strong coupling no solution was presented. The renormalization group approach [75] does not presuppose the absence of a triple P coupling; indeed, it is a crucial element. From a field theory of P’s and Reggeon one finds scaling laws for the Green’s functions as before. There turn out to be three zeros of the beta function for the P-Reggeon vertex which could yield the infra-red behavior of the coupled P-Reggeon vertex functions. Two of these yield Reggeon trajectories which have cusps at t = 0. One gives a linear trajectory, with renormalized intercept and slope, near t = 0 aR(t)
=
aR(O)
+
a’Rt + 0(t2)
(4.58)
which in light of the experimental facts on irN charge exchange is an attractive result. The amplitude, for charge exchange for example, would read in this last case TAB~A~B1(s,t)
gAA’R
(t) gBB’R (t) s’~R(t) (exp {—ilTaR(t) }
+
r~)(logS)’TR,
(4.59)
t small
where TR is the Reggeon signature and aR(t) is given in (4.58). The index “y~is in the expansion about D = 4 described above = 1/12.
(4.60)
/I,D.I. ‘1 barbanel eta!, - R,’,i,’~eommfield the’orm- - /oromulatiomm amid
164
mOo
In any case it is not unreasoilable to expect y14 to he small in niagnitude, i’ltis leaves the result of the boson trajectory witil all P cori’eetions 111 fine shape as far as experitilcu i goes, It is quite sum’-prising that tile collision of’ all tile multiple P en ts has so niild an effect Tile problenl ot’ term ion trajectories is cnorniouslv more contplicated in detail ~tltiiotigli I cutl— ceptually is tile same as the boson secondary trajectory 76 I - The key C\ pert ttletltal fact is that f’ernl ion trajectories are essentially linear in tile Matitlelstani variable u appropriate for backward scattering =
a1,-( 0)
±
u.
t -Lu
It
and 5111CC botll positive and negative parity trajectories in ust be present to tnauit aiil the Mandelstam allalyticity of’ the scattering amplitude. 110111 trajectories would scent to be pic~~’i1 1. Only trajectories carrying eitiler positive or negative parity are ktiown: never, built. A successi uI theory ot’ femniion trajectories ilitist, therefore. avoid these parity doublets. Starting witil the standard P theory described before and positive a/ic! negative parit~-’trajectories of the f’orm ±~~/7+
a01,-il.
(4,02)
finds. tilrougll tile usual renormalitation group procedure. a i’enoriiiali.’ed lrajec-tu/’t’ a/must linear in a. l’urtllerlllore f’or a < 0 in the scattering region both parity poies am’,’ 1)11 tile physietl sheet of’ tile ~ plane. For a > 0 in tile l’egime where pai-ticles lie on tile Regge 1ra~ectories.Oils’ I)! tile parity partners slips onto an unphysical sheet tilrougll the P F cuts, Built ititportant observed properties ot’ ternhioll trajectories are thus achieved. An “unnatural’’ asl-leet of’ ills’ treatment (it fermion trajectories is that the t’ixed points in the space of coupling constants ale onl coilsi itional is’ stable, ‘[his means that the results we have stated univ if tliers’ is a certain i’eiation aillong tilt’ couplings and slopes. ‘file results of the calcuilations of the multiple P corrections to secondary trajectories, by their accord with observed facts give support to tile solution to the P problem by itself. To test any of these in detail is difficult because of the present uileertailties ut (lie actual values of tile indices sucil as y, not even to meiition the experimental problems in diff’erentiating among various variations iii logs. Clearly the whole package ilasan attractiveness and coileretlee \Vilieil is quite pleasilIg. OilC
4.u. Higher p0/lit
P
couplings’
In setting up the RFT as sumnlarized in tile Lagrange 1 nlletioll (4.151 we briefly’ diseusset-l and theil ignored four and more point P couplings as well as derivative couplings An heuristic argunient based on phase space was given for tilis. It is possible to make a stronger case. Within the context of tile weak coupling or quasi—stable P. Gribov and Migdal were able to use ordinar pci’turhation theory to evaluate corrections to their leading P structure [61 I . Here tile pilase space arguments are both correct aild persuasive. Typical of the results obtaiiled are the correeti( us to 2 logt correctile inverse propagator as given in (4.25) where a linear trajectory is modified by a t tion a harnlless addition, In tile case of tile latter day renornialization group strong coupling solution, as discussed above. the problem is again non-perturbative. Migdal. Poiyakov and Ter-Martirosyan gave a iluillber of -
H.D.I. .4 barbanel et al., Reggeon field theory: formulation and use
165
arguments why four point and higher and derivative couplings would become negligible in the
infrared limit [251. Abarbanel and Bronzan showed that when the triple P coupling vanishes, the fixed value of the renormalized four P coupling is zero, so that the P is weak coupling [77] . This is consistent with the notion of a dominant triple P coupling. Finally Brower and Ellis demonstrated that the simplest derivative couplings were harmless in the infrared limit [78]. Basically the task of each of these calculations was to show that the effective coupling constants, like the ~ of the three P case, for higher point or derivative couplings were driven to zero in the infrared limit while the triple P coupling approached the same zero g, of ~3(g).in short, both the stability of the original calculation and the ignorability of higher order couplings
-—
all in tile infrared limit
—
were suggested by these exercises.
A much more complete treatment of this matter was given by Jengo and Calucci [79, 80]. They considered, at once, an infinite set of n P couplings, n = 3, 4, ..., and all derivative couplings. If tile E~,q. phase space integrals were always extended to infinity, then such a theory would certainly be non-renormalizable. However, since we are here involved with an infrared phenomenon it is natural to introduce a cut-off. As we discussed in section 2 such a cut-off is almost certainly required to consistently extract the RFT from an underlying theory. The action of this theory is A =fdrd2x{[~ ~
.
rr0
V~_~o~t~_ (V2~t)(V2~)+...]
r’0 +
~
[~~t2~2+~
...]+ ...
(~t2v2~+~2v2~t)+
(4.63)
with the prescription that behavior in momentum integrals to be cutoffofatthe Iq, cutoff I = A, A. Now 2. The infrared of the space theoryallought to beareindependent lE~ I = ct~,A one scales the cutoffs to lq,l = A/A, IE~I = ct~,A2/Band expresses the scaled theory in terms of new constants ~ ~o, ~‘o,etc. which are functions of A, B, ct~,~ (Scaling the cutoffs, as iildicated above, is equivalent to holding the cutoffs fixed and scaling momenta and energies; thus A or B play the role of ~ in (4.36).) Carrying out this procedure it is found that if one only requires ~o to be stable at zero, that is, = I, then the renormahization group strong coupling solution is reproduced with all couplings except 7~,the three P coupling, going to zero in the infrared limit. If one puts further constraints on the couplings, then it is possible to find other solutions to the RFT including, strikingly enough, the quasi-stable P. To achieve that, however one must essentially require that r 0, be specially chosen so that it maps into zero under the scalings of the cutoff. From the point of view of physics there is no special motivation to further constrain the theory in this way. Indeed, if we do so and launch upon the pathway of the quasi-stable P, then we are plagued with the various decouphings of P. This rather general treatment of all couplings at once provides a posteriori support for the model Lagrangian of (4. 1 5). ~
4.7. The formal status of the RFT
There are two points that are essential to our understanding of the solution of the RFT with the
166
11,1)1, -l barhanc’l it a!,, Re
comm fim’ld thi’orm’, Joromimlatiomm amid mice
Lagrangian of (4. I 5) provided by tile reilornlalisation group. Firstly we tieeul to know that the (Ileory is renorillalisable in tile eotlveiltional sense. so that the forillalisnl of the renornialisation group is applicable. In particular we wouilsl like to he sure that we can safely set ~ = 0. since this is esseiltial to tile dmlensioilal analysis uised. Secondly we need to ktlow how (lie theory can be constructed perturhatively since thus is tIle oilly’ way we can explicitly check Reggeon unitarity, Tilese two points are not uilrelatesi, In the original stuidy of the RFT it was t’m,uisl tisefuil to generalise (lie two space shilliensions to D dimensions. There are two reasons for this. Firstly in a general. non—i ntegs’r number of di— mensions all Feynnlan graphs can he defined by’ analytic continuation from I) < 2 the ultra—violet thvergenees are relegated to poles itl the variable D and a simple regutlarisatioti proceslure is provided in principle by suihtraetillg such poles, with their (real) i’esidues. froni tile Greetl’s f’uuietionx il wilicil tlley occur. However, the RFT is actuall super—renormalisable iti terms of simple power counting (‘or D < 4. This means that (lie subti’aetion oI the poles has to be equuvaleilt to an intercept (mass) renormalisation anti it is not eleai- then that this suihiraction is coinpat ible with the = 0 condition. Tile second reason for varying D is that at D = 4. a clearly uilphysieal situation, the theory possesses a (broken) scale invariance itl separate space. .v. anti “tuue r scalings. Illis ntanil’ests itself in tile fact that tile dii’nensionless coupling constant. g = (r/a’°”4)L’~?~j4 - has no explicit dependence on tile nornlalisation point lay, As a ecnsequence tilere is a iero 0! [3(g) at g, = 0 which governs the infra—red behavior of the theory. It is this latter t’aet which motivates the expansion of tIle theory around D = 4 in powers of 4- -1), The poles fronl tile uilti-a-vioiet divergences occur at .
4
D = 2/n.
ii
and a series expansion
--~
ill
...
(4.64)
powers of (4 D) actually hides (lie poles since
n24-- D)
/1 =
= 1, 2.
---
~----...
(4.65)
‘fileref’ore the expansion of’ tile theory around I) = 4 sidesteps tile problem of the intercept renornlalisation. In f’act forD < 4 tile problem of the ultra-violet divergences of (lie tlleory cannot he separated from the infrared behaviour if we set ~ = 0. The theory cannot he reilornlalised order by order in perturbation tileory. fllis problem has bceil stuidied in detail by Sugar ansi White, both for the present RFT [81, 82] and for conventional Xcp4 tileOries [81]. where tile problems are very similar. Tile essence of the dit’ficulty is that t’or ~ = 0, the iilterccpt renormalizaijomi eotinter-(erill, ö~ ilas tile forni =
~
“Thj~ 4--D).
(4,66)
where f is dimensionless. Tilis follows froill dimensional analysis only. Clearly ö~canilot have a power series expansion in r0, and any perturhative construction of tile propagator must ilecessarily involve a re-ordering of standard pertuirhation theory. Sugar and White provided such a scheme rising renormahzation group apparatuis. Tile derivative renormalization condition on .1> is
H.D.I.
Abarbanel et al., Reggeon field theory: formulation and use 11 Ztx “ N’
—‘—iF~”~~E 01 E’EN /
167
p467
where Z is tile wave function renormahization constant and
4] 4/(4_D)E~,1 . 0/a~,’~ Eq. (4.67) can be integrated to give x~,=
[r
(4.68)
ir~i)(E,0) = _L~xNfdxNx~Z(xNyi with x
=
xNEN-/(’--L).
(4.69)
When this integral representation is combined with the formulae
Z(g) = exp[fdg’~g’)/P(g’)],
(4.70)
(r 0/a’~~)E~ 4)/4
=
g exp[_jdg’(g’ ~ +~(g’)~)]
which are fairly easily derived from the renormalization group equations [821, we obtain a complete set of equations for calêulating F”(E, 0) given y(g) and ~(g). Provided that ~(g) has a zero with positive slope this set of equations solves all of the above problems. ‘Tile infra-red behavior of pO,i) follows immediately and when the representation of p~~’) is compared with perturbation theory we find =
(r0/a~)~4-D)
f
~NXN
[I Z(xN)~]
(4.71)
-
which does indeed contain all of the ultra-violet poles (as divergences at XN = theory 0). When equa2 they can be combined with perturbation forthe p(’2) to tions are generalised to non-zero q provide a complete iterative scheme for constructing p~’)in such a way that it always satisfies = 0 and has the correct infra-red scaling behavior. Finally the representation of F(11)(E, q2) allows us to show that perturbation theory can be used for large (—E) and (—q2) showing that F~”0(E,q2) has a leading P pole and Reggeon unitarity is satisfied. Note also that when the representation of 5z~is modified to allow for the presence of a cut-off it can be shown that S~is positive and so (to the extent that higher P couplings can be ignored) we must have a~,>1 to obtain a renormalised intercept at one. In this case each term of the perturbation expansion will appear to violate the Froissart bound, even though the sum respects it. 4.8.
Ideas about a 0> 1
We have seen that the requirement ~ = 0 is met only if r0 and a’~,are related in a special manner (eq. (4.71)). Since this relationship is not automatically satisfied, the requirement ~ = ~,
0
lID,!, -I harbanel ci a!, -
165
!ie~,’,5’eo/mlie/cl
theorm’ - formmmmm/atiomm
ammo’ mice
be ‘‘unnatural’’ even thougil it is indicated experinlentailv. One alternative to thus situa(ioii is to ask wilat llappens when ~ hecoilles arbitrarily large and negative. In model ealeulatiorls in which all ifltilite set of cuits is sunlnled, tIle F’roissart bound is not violated, even thouigh every iiidividual cuit violates the bound [26. 27] . I nstead. the new “P’’ is a pair of branch points at illigilt
a(t)
= I ± 21a~
ni] m 2
1(a0
tmonieron of (lie Regge-e ikonal mould
)~_
2’). 33
I
lii
sotlie respects.
It
:2t
comits’s
hue “P” reseni bles from a pole with a tile h 11 > I . and in mi pact n~u’anls’terspas’s’ it is a shsk whose t’aslnis nil )ws like Imi s. I-iowever, tile disk is gray. not black. and total doss sections faetoriie in tlis’ It igli cilergy lint ii rather than approaching a common value, ci1 (,s’) (lu,s ~2 has the miiaxuii 11111 rate (if increase ul loss esl by’ the broissart bout 1151, TIus seems to be !‘ortui tons because s--channel uut itarit\ is miot in posesi iii RiTh On the oilier hansl, it nay not he accidental that it is i mu possibls’ to have a rettornialited ~ -- (I (a > I I within (lie Ri”~l’.even thouigh s-channel unitarits’ is 2not ~
(4,73)
where a
11 and ~7oare pi’opoi’tional to ~ ansi sleps’tid in sls’tail ott this’ Lagi’angiait before sluftitig. ‘[his new “f’ree” Lagratigiati has E( q) sped ru itt 2 = (I’ cc(q))2 = (77~ 2)24a1 lf(q)
1+a~1q
where when a(q)
T~=
= I ±2\/a~)a
)47~)
2 0q
for small q2. This, in general f’ea(uires, resemu hhes the results of refs. I 26. The more general problem of’ why ~ = 0 or how one can incorporate the RF’I’ remains open ansi inviting,
27
it
I
in a “nattiral” wa
ni 15)
Other ‘weak coupling’’ ic/eat
4,5),
The idea that tile soiutioil to ouir RF’l’ is stroilg eouiphing in (lie iltrared lint it. namely
[3(g 1)=
g,
0, has certain uinattractive features t’ront an aesthetic Ibm I of’ view even though (lie physics is certainly sounsl. The reasoning is more or less that simlee we had to begm ouir RF’i’ by a choice of’ bare trajectory correct ns’ar .1 = I and I = 0 it is perhaps discomfortmg that we did ilot reproduice that singuilaritv after stimmi ig all graplls. To do so wouiltl requnre that tile f3(g) (‘or tile RFT’ muist hlave a zero at g = 0 wluchi governs the inf’rared behavior. for then all Green’s f’uulctions hecoille tile original Green’s ftuictions plus sillall corrections. Sumch a RFT could he said to bootstrap itself near E1, q1 0. Some examples of such tileories ilave been giveil hy Aharbanel 185 1 . One 0.
H.D.I. A barbanel et al., Reggeon field theory: formulation and use
169
which has the scaling form similar to the Moscow—Batavia strong coupling theory is G~”~(E, q2)” = E~(l+a0q2/E3P~), where the power p is undetermined. This theory bootstraps itself in the sense stated above when a constant triple P coupling is present. A serious fault of these and any other theories which begin with non-linear bare trajectories is that demonstrating that multiparticle t-channeh unitarity is satisfied is a difficult task. it should be recalled from section 3 that t-channel unitarity is nothing less than the foundation of RFT. (4.76)
5. Inelastic processes in
RFT
In this section we consider particle production processes in the RFT. This is a very important problem in its own right, and it is crucial for studying the internal consistency of the RFT. In constructing tile RFT for the elastic amplitude t-channel unitarity was built in from the start, but the constraints of s-channel unitarity were not. One must verify a posteriori that these constraints are satisfied, and in most cases this requires some information about production processes. One approach to the problem is to derive RFT rules for the production amplitudes and then use the techniques of section 4 to study their asymptotic behavior. As indicated in section 2, it is
difficult to apply Reggeon unitarity to 2—N production amplitudes; however, it has been possible to study the asymptotic behavior of these amplitudes in the multi-Regge region of phase space using the hybrid Feynman diagram procedure. Drummond [86] and Campbell [87] have studied hybrid diagrams which head to Reggeon graphs with one closed loop, and recently Bartels [881 has obtained the rules for a general Reggeon diagram. Here we will briefly discuss the results for the two-to-three amplitude. First, one must identify the counterparts of the signatured partial wave amplitudes encountered in the elastic process. We refer to the kinematics of fig. 5.1. In the double Regge limit we consider the five point amplitude A 5i2~ Sf3, ~23 -÷oo and t 5 with 1 = Q~,t2 = Q~,and 77i2 = Si2/Si~,S23fixed. When we have Regge poles ct1(t,) in the t1 channel, study of a hybrid graph as in fig. 5.2 leads to A5(s12, Si3, =
523,
7112, t1, t2)
!31(t~)(32(t2)
{SS~”~i ~cii
~02_01~(t1, t2, h1i2)+s~s~”~2 ~2L1~~52 v2(t1, t2, llu) }
(5.1)
~‘~~~I)sm3
Fig.
5.1.
Kinematics associated
with the
double Regge limit of the five point amplitude. In this limit s12, si 3 Q~ and ~i12 “S12/Si 3S23 held fixed.
~2 3
~
with Q~,
1
/1,1),!, .4 barbaric! ci a!,, Re,m,’g-omm (ie!d theorm’: Jornmulatio,m ammo’
70
N
m/sc’
/
/ fig. 5.2. A ii’ hod l-’eynmnan graph and
its Reggemmn gra pim contribution to time slimmmbie Reese iinm ii ot’ time t’ic’e point t’simmc tim 5i 2 moist ‘5i ~ tue othem. fl “t 2 and ~2 3 distinct partial cease terms enter; one has simultaneous stiseontmnuities ~
‘icc
-
with —
cxp{ i~(ai---a’ 2)}+riT2
~0i02
—
a2)
Si117T(a1
The deconlhbosition ol’ A illdicated ill dtl. 1 5. 1) is consistent with tile Steinmaii relations, which tell us that A5 cannot hlave simultalleouis diseontinuities in ‘~i3 aild s23. The two particle-Reggeon coupling [3is real. The two Reggeon-particle coumplings ~“1 are real and, in models. are analytic at 17i2 = 0. This suggests that the appropriate partial wave amplitudes are tile generalizations of V
1 and V2. it also suggests than one will requure two kimlds of two Reggeon-particle couplings to build a calculus for evaluating Reggeon cuit contributions to the 2 3 process. fhese expectations are borne out by the study of more eonlphicated llyhrid graphs~for example, thioss’ wilich correspond to tile Reggeon graph of fig.5.3. Thus aild fancier graphs yield (lie form for A, —~
A,
=f~ ~‘ ~
1t’ t2.77i2)+~~
~
~
~m ~‘1~.’2,
‘J 2~~
J2 V26’m..J2.
ti
t2,77i2)~.
(5.2) the double Regge limit. The Reggeon graph rules yiehuls V1 ~uid “2’ Bartels has generalized the above results to (lie muitiparticle production amphituides. A4~0,I 881 Tile only new difficulty is that tile decomposition of’ tile amplitudes requured by (lie Steinman relations becomes cuimbersonie. However, for processes in wilicil onlypartial the P singuilarity is impor2 beilavior of tile multiparticie wave amplitudes. In tailt, one only needs the small L’ and k ill
E 1z!-
J~
Fig. 5.3. A Rcggeon graph with branch points in J1 and J2 which contributes to the double Regge himmii of time 2
3 process,
H.D.I. A barbanel et al,, Reggeon field theory: formulation and use
171
~I=I Fig. 5.4. The picture of the basic Pomeron (P) as generalized ladder graph.
this case it sufficies to approximate the bare vertices and coupling functions by constants, and
only one independent amplitude enters for each n. In some cases the analysis of inclusive cross sections is even cleaner than that of exclusive cross
sections. For example, the RFT rules for the single particle inclusive cross section in the triple Regge limit have been obtained both from the hybrid graph approach [59] and from the Reggeon unitarity relations [58]. Further work on inclusive cross sections is presently in progress. Another approach to the s-channel content of a Reggeon calculus has been given by Abramovskii, Gribov and Kanchelhi (AGK) [36]. They bypass RFT by studying the s-channel absorptive part of two-body amplitudes directly. Since such absorptive parts are immediately related to production processes in the s-channel, we gain directly the information desired. However, one must pay a certain price for this splendid efficiency since it is necessary to specify how one cuts through Reggeons in extracting absorptive parts. The formulation of AGK assumes that the
Reggeon, in particular the P is given by a set of generalized ladder graphs as in fig. 5.4. The cut through this P is, up to considerations of signature to be treated in a moment, just as in the fat1iihiar multiperipheral model shown in fig. 5.5. It gives rise to a uniform distribution in produced particle rapidity in the central region for single particle inclusive production; it yields multiplicity of produced particles proportional to log s, etc. With cutting rules in hand, one may cut up any given Reggeon graph to find the contribution to the desired s-channel process. Clearly this method is most attractive when only a small number of Reggeon graphs need be treated. As such it lends itself well to the so-called weak coupling P discussed in the last section. That P is without terribly high regard at the time of this writing and summation of infinite sets of P graphs appears rather necessary. These caveats exposed we turn to a description of the AGK cutting rules and some consequences thereof. Given the rules for cutting a P, AKG argue that the details of the basic production mechanism
Fig. 5.5. The cut across the P reveals a multiperipheral production process in the s-channel.
II,D.J.
172
.4 barbanel eta!,, Rem~y011 Jie!d thc’oru’,’ /ormmmmilation amid mice
UT JP
TO ~
jY
(bl CUT UP
9
9
+
TO STUDY
I ig. 5.6. (a) Absorptive corrections to the basic production process. production t roimm ts5’cu cim~mins.and production with a iarpe rapidity cap these amphtudes are revealed by the cmmt of the tsvuu Reggeon graph contrihimtion to time elastis’’~imnphtmmde.One cuts one, tcvo, or zero Reggeons respectively. (b) ‘the prodmmctiomm iirocessc~revealed by cut tiny ~mpthe Reizgeiun miraph of’ tip. 2.6,
need not be specified. Withl the cuittimlg rumles they study the effects omi ~ie generahlied ladsier P arising from multiple P exchange ails! from P iteractions. Two low order graphs are silown in fig. 5.6 along with tile contributions to produictioil anlphi(uides generated by cumttillg thelll. Tile features revealed in this n~annerare absorptive corrections to the laduler-hike produietiori aillplituiu!e of t’ig. 5.5, produiction l’rom more (Flail one ( lllultiperiphleral like) chaill, aild the possibility of’ large rapidity gaps between produced particles. One cai~already make some quiahita(ive statenlents from fig. 5.6. First, if the average iluniher of particies arising fromli the exchange of’ tile basic P is n1 a log s, (hell ii front the exehailge of’
k non—interacting P’s will he of order ku
~. In the absence of’ P iilteraetiolls then, tile iliuihtiphicitV 5 ~. P interaetiomls will t’ill ill (lie uhistribution will be as shown in f’ig. 5.7a with a,l/uk,l (log S) gaps and smooth ouit to look more like f’ig. 5.7h. The actual possibility of seeing tile peaks in a, 1 at kn rests on model dependent couiphings not estimated by AGK or anyone else. Furthlernlore
a
Ii
I
it~ C
/
~~
3~
~
lip. 5.7. (a) The distribution in number (n) of particles produced by time graphs ~itli nmuitipie non’interaeting exchanged P. Fig. 1.2 is an example of this, (b) The expected smoothing of the o,~distribution of (a) which will coimie froimi P interactions.
1-ID.I. A barbanel eta!., Reggeon field theory: formulation and use
173
/~IIIHIHK
Fig. 5.8. A cut part way through a Reggeon which leaves a hanging chain with rapidity spread v. The particle line carrying momen2 ~ cY, which must be large since a Reggeon is cut. Since large q2 is presumed to be absent, so is the partial tum q will have mass q cut of this Reggcon.
production from k independent P’s is expected to give rise to the order of ka particles per unit rapidity interval. In individual events one expects to encounter long range fluctuations in the
rapidity distribution. Calculation of fluctuation probabilities have been made by AGK for the weak coupling P [361. The detailed cutting rules depend crucially on the observation that for a particular discontinuity to be non-negligible, the cut must go completely through any Reggeon in the graph or hot cut it at all. As an example of this consider the cut of the two Reggeon graph in figs. 2.1 and 2.2 as shown in fig. 5.8. In order for it to be interpreted as a cut through a Reggeon the rapidity spread y across the cut lines must be large. In that case the particle line carrying momentum q will have an enormous mass q2 = e~.However, it is a basic assumption of the hybrid graph approach that any diagram is negligible when a line carries a large mass. For small values of y the cut is interpreted as passing through the Regge vertex functions Mn). This argument applies to any Reggeon diagram, for a partially cut Reggeon leaves a hanging chain and yields an unacceptably large q2 somewhere.
For diagrams involving the exchange of k non-interacting P’s the analysis is straightforward. The contribution to the amplitude T(s, q~)from k P’s is
T~(s, q~)= _i~sffl~~
~2(qj~
qJj)N~~k(qll,
...q~ 1)2fl
~G1(s,q~1),
(5.3)
where G1(s, q~)=
_50/(q~)
~
sin{~lrct1(q1)
}
.
(5.4)
Cutting through N~ should have no effect on them. Recall that N(2) can be written as in eq. (3.41) as an integral over the Reggeon-particle absorptive part. Since such an absorptive part involves only on-shell intermediate states, cutting does nothing. For N~,k> 2, the argument is more formal, requiring an elaborate excursion into Sudakov land. The conclusion is that cut N’s are the same as uncut N’s. The s-channel absorptive part of T~(s,q~)is now reduced to a combinatorial problem in enumerating the ways of and weights associated with cutting 0, 1, ... k P’s in all possible ways. Take T(2)(s, q~)for example:
174
liD.!. .1 harhanc’I et al.. Reggeon tie!d ihcori’: formulation ammo’ use
2~(s, q~)= ~ir Abs T~
s
d2k11 dk (ir)
k
-
k 21 ~
(~ir)~
x {1 G1G~+G2G’~’ I + [AbsG1
( iGs)*+iG’2 .AbsG’1+AhsG2 ( iG )“+iGAbsG-21 +1 2AhsGm AbsG2i
where AbsG’~= 1mG1 =s°i.
(5,6)
‘File ternis in square brackets restilt respectively Ironi cuitting f’irs( 0, then I . ansi finally’ 2 P’s. Note that for identical Reggeons (lie signs and weights of these eontribtitions are +2, 8. ansi +4 yielding an overall negative two P cuit term ui (lie elastic process. Diagranis involving only triple P vertices can also be easily dealt with since s’uitting across thhis vertex leaves it unchanged. However. the diseontinuuty across (lie gemleral nP niP vertex cannot be expressed in terms of tile vertex itself when iitn > I . As a resumit in order to calculate indivisitial contributions to s-chaiinel discontinutities ff0111 Reggeon diagrams cotitaining such vertices, it is necessary to have greater knowledge of tile vertex than is required to evaltiate tile Reggeon graph itself. (A tileory witil only triple P vertices as building blocks for n i-n transitions is thus qtiite attractive.) Tile cutting rules were employed by AGK in stuidying single particle inclusive cross sections in tile central legion. This is the regime of’ tile process a + b e + 1’a X where iii the cc titer of mass = ~ logs or of h, t’b = ~log .c frame of’ a + b, the rapidity i’ of e is finite as the rapidity of a. grows large. Taking tile uiiagrani of fig. 5.9a ails! cuitting it gives the leading eontrihuition to the (histribution in y and P-i , the transverse illolllelltunll of c: -—~
—~
~
da(a+b - c+X) dvd2p = gagh!~(pT)lmG(3’ --
5~--p,0) lmG(j-’
~b’
0).
(5,7)
which is precisely the flat rapidity distrihuition5~h expected 0) =uip1 for (lieproportional P. (‘orreetions and 5.Qe.when Tileya(add to he to to (5.7) result from Reggeon graphs as in figs. 5.
_______________
go
3
Pa
.
‘
P 0
2
/y~,:.))
-
~
(P1) 2b
(a)
~b’0~
>
(b) —-—/*—-
)
(C) - - -~
lip, 5.9. Reggeon graph contributions to the 3 —~ 3 amplitude whmise discontinuity in = (P inclusive distribcmtimun a -f b —~ c + X in time central region,
+ ~h
~)2
gives time single particle
H.DI. A barbanel eta!., Reggeon field theory: formulation and use
1
f~(i4)[
(Ya”Y)
±
1
175
(5.8)
(YYb)
after a variety of marvelous cancellations. It is worth emphasizing once again that results like (5.8) are useful only when a small number of Reggeon diagrams make the major contribution to some process. The signal of the weak coupling P, which allows a small number of graphs, is the appearance of corrections as in (5.8) which are only of order (log s)~or equivalently (rapidity)’ The corresponding calculations with the strong coupling P are far more difficult. In this case one must sum an infinite set of cut Reggeon diagrams, and as yet no convincing method has been presented for identifying the ones which make the leading contributions to the inclusive cross section. A start in this direction has been made by the Moscow group [251, and by Caneschi and Jengo [89], who consider a theory with both cut and uncut P’s. The Russians argue that the leading contribution to the single particle inclusive cross section is again given by eq. (5.7) except that is now to be interpreted as the full P propagator. For large values of y ImG(y, 0) grows like y”t, so in this case the rapidity distribution is not flat. One of the most important open questions concerning the strong coupling solution of the RFT is whether the constraints of s-channel unitarity are in fact satisfied. Although a definitive answer cannot be given at the present time, the preliminary indications are very encouraging. First, the Froissart bound in D dimensions requires that G
u~11~ ( c(logs)’~,
(5.9)
where c is a constant. The bound is satisfied for small values of c (see eqs. (4.46) and (4.53)), and it will be satisfied at D = 2 provided an infra-red stable fixed point exists [73]. The crucial test must await the completion of the direct calculations in two dimensions which are now in progress [71—731. A closely related test comes from the calculation of the exclusive cross sections in the multiRegge region of phase space. Long ago Finkelstein and Kajantie [90] showed that if the P were a simple pole with intercept one and if the P—P-particle coupling did not vanish at zero momentum transfer, then one was led directly to a violation of the Froissart bound. The Moscow group has repeated this calculation for the strong coupling P and found [251 “~
c,~(lns)°~,
(5.10)
where u0~2is the cross section to produce n+2 particles, c~is a constant, and to first order in cs=2—7/12 ~=
1
—/4.
(5.11) (5.12)
So, at least for small values of , there is no Finkeistein—Kajantie disease. The calculation leading to eq. (5.10) starts with a constant bare P—P-particle vertex function, but because of the absorptive nature of the P the renormahized vertex function vanishes when all transverse momenta go to zero. Similarly when one calculates the single particle inclusive cross section in the triple Regge limit [25, 58, 591, a constant bare triple P vertex leads to a renormalized one which vanishes when
176
11,1)1, -i barbanel it a!,, Reggm’omm field tht’orm-: formulation ammo’ mice
the nionientuim transfer does. For small values of there is no violation oh’ the etiergy conservation sumn~rutle. Finally (‘anesehi ansi Jeilgo [89 I have ealetilated tue moments of’ the tiluiltiphici tv u!istrihuitioti in models with cuit ansi uineui( P’s. They limit! no eotltrauhetion with (lie constraints arising from tile positivity of the partial cross sections.
6.
Conclusions,
outlook and critical problems
Up to now our diseussioll has been prilllarilv concerned with (lie theoretical shevelOpillent of Reggeon field theories rather than with their eoi~sequiences [‘oi’ pllenolllenologv. illere are several reasons f’or this. First. unltil tile development of the Moscow Batavia strong coupling solution it was not at all clear tilat a self—consistent theory of (lie Ponteron, its self’ interactions. ansi its illter— actions with other Reggeons was available, even in tile RFT. It is an exceedingly slouihtftil busitiess to do phenomenology witil SO uncertain a f’otmndation. Second, a real feeling t’or the size of P interactions has only been available since the stuidy of higll energy incluisive processes began at (lie CERN—ISR and at the Fermi National Accelerator Laboratory. We are not going to attelllpt here eitbler a review of or a eonstructiot-i of a phenoiliemlology of hadron i’eae(ions at very high energies on the basis of RFT, huit we will indicate wilat iii our opinion are tile lessotis we have learnesi that wihl play a basic role in any sumell uheserrptioii of phiellonlena. Tile first issue concerns the size of tile triple P cotipling and tile energy doniaui in which it is necessary to sum tile lull series of P graphs. We will concentrate on the grapils in fig. 6. 1 which we know will coiitribute to tile dominant term in the total cross section (see fig. 4.8). The order of magnituide of the corrections to tue single P exchange is set by tile dimensioilleSs parameter g0 = r~/a~E’. In terms of s this means that -
(r~/~)logs
(6.1)
sets tile scale for the convergence of the P series. ‘Flie best estinlate of’ the size of the hare triple coupling, r0, comes from the2,single B A + is X small, (see f’ig. s/rn2 particle -÷ withinclusive t fixed).experiment If the tripleA P+ coumpling one6.2) tllavin tile triple Regge region (s, m use the lowest order in the P series to estimate its size. Under this assuimption one finds frolll °°
JIll
-~
r~/
+ X
and pd
-~
d
+ X
P
—~
data tilat
1/50,
(ft2)
-+
)~y~ -I_i,-
6.1. Reggeon gTaphs which contribute to time leading behavior of the total cross section. The expansion parameter imere is (ro/oo) lug a. When it is of order OflC, the whole series must he summed to give time strong coupling scaling solution of RFT. Indications arc that at present energies this parameter is only of order 1/8 or so, Fig.
H.D.I. A barbanel et al., Reggeon field theory: formulation and use
177
btAAtA
~M2~
Fig. 6.2. The triple Rcgge region of the inclusive process A + B
—~
A + anything, where one may estinmate the triple Regge coupling.
and guessing that when (r~/ct~)lns ~ we will need a large number of terms learn that logs
ill
25
the P series, we
(6.3)
is where the whole sum given by the Moscow—Batavia scaling solution, will certainly be necessary. Now the logs values available at the CERN-ISR are only 8 or 8.5 at the most, and at FNAL they range up to about 7. At the energies of these devices the full sum of P graphs would not appear to be necessary. This conclusion is certainly in line with the fashionable phenomenology based on s-channel approaches such as the multiperipheral model, which is motivated by the apparent “short-range rapidity correlation” nature of production processes observed experimentally. We cannot completely rule out the possibility that the scaling solution is applicable at ISR—FNAL energies. If it were, then in the calculation of the inclusive cross section the quantity r~/a~ would be replaced by an effective coupling which is independent of r0, [25]. This alternative seems unlikely because we expect this effective coupling to be of order of magnitude unity; how-
ever, this question cannot be fully resolved until we have reliable calculations of the critical exponents and scaling functions in two-dimensions. Work in this direction is presently in progress. When only a few terms in the P series are required for a good numerical estimate of the process under study then in the small log s regime one must face squarely the matter of secondary contributions from N P coupling with N> 3 and from possible derivative couplings. We know of no way to choose among the alternatives which present themselves except by a trial and error approach. However, we are encouraged by the fact that it has been possible to fit the pp, pp, ~ and K~pelastic scattering data over a wide range of energies with only a few terms in the P series [91, 921. There is a nice feature to having only a finite number of terms to address. In such a case the AGK construction, even with its need for a specific assumption on the result of cutting P’s [40], will be a useful tool in relating specific s-channel production processes to the f-plane physics. Another lesson that we learned in section 4 was that when we required Z~=
l—a(0) = 0, as we desire for the full sum of P graphs, the bare gap, z~= 1 —a0(0) < 0, so the bare P has ao(0)> 1. (Happily the best first in refs. [91, 921 also require ~~(0)> 1, with i~ having the right order of magnitude [93].) In making the estimate in eq. (6.3) we followed the approach where the perturbation series was developed using propagators with the renormalized intercept gap, L~= 0. Since we expect the bare P to be observable at present accelerator energies, it may be more appropriate to rearrange the perturbation series so that the bare gap enters the propagators. In this case each term in the series violates the Froissart bound, but the sum is constructed to satisfy
11,1).!, ,.1 barbanel eta!., Rr’ggco,i field theorm’: formulation and
178
t. Actually tile value of ~ is on tlìe order of’ 5
misc
1 0-2 wInch means a very slow growth in s from my P graph. Taking thus eft’ect into aecoulilt we niay boldly ililagille that tile estimate in eq. ( ~g.3 5 too large by a f’actor of as much as two. ‘filen the iced f’or the (‘till P series might set in at ogs 1 2 5. It’ this were to be so, the use of’ (lie scaling forni f’or tile P propagator might well e an attractive, compact expression to use at t’illite energies. To really be of use, however, we would again need reliable valuies of the sealing inslices and reliable knowledge of’ (lie sealing (tine— Lions, Despite our eatitiouis ahllost pessinl istis’ view towam’sl the ui tihitv of a Reggeon ealculuts phienoni— Inology at tile present stage of theoretical sleveloplllent. thlere have been several at tempts at fitting real data. Serious evaluation of’ these. often attractive. phenolllenological essays is difficuilt. We pass tilis task on to tile reader by providing a f’ew references l’ronl wilicil he or she can begin [91. 92, ~4I Here we (tim oumr attentioll to a resumllC~of (lie ideas covered in this article and to a elleerluil view of tile progress achieved in the slevelopments we have reported. We began by recahhimlg time necessity of braneil points in the .J—plane arising froth (lie combination of moving poles ill .1 with unitarity. We assummed that tile Ollly branch points were those coming from tile presence of mtiltipie moving poles in the 1-channel, fills assumptiomi is actumahly extremely conservative. It resembles almost in detail (he experience of many years in hoeatimlg the position of brailehi p0111(5 in the energy planes arisimlg in conventional f’ield theories. There, of eoumrse. one has a variety of’ dispersion relatiomis and suim rumles wInch allow a direct positive assessnien( of’ the validity of’ this assumption. We are not so fortumnate here. We are forced to f’all back on (lie ( lever entirely convincing, however persuasive and attractive) observation that satisf’yiiig t-eilanllel muiltipartiele unitarity is straightforward and natural wilen branch points arise via moving poles and dil’ficult. if possible at all, otilerwise. Clearly if’ tllere are hralleh points in .1 arising from otiier sources, we have missed them. We next saw how to obtain expressions (‘or (lie discontinumities across the f-plane cuts froni tile inultiparticle 1—channel relations. These Reggeon unitarity relations are cruicial for an understanding of the P near J = 1 and I = U since they lead to strong couphings among tile multi-P channels whose thresholds all collide at tins point. Ill order to insure that the full Reggeon unitarity relations are satisfied, we introduced a field theory to describe (he emission, absorption aild propagation of Reggeons. Tile Reggeon field mp(x. T) operates in a world with two space and one time dimension. The branch points in the .1-plane arise naturally as singularities in the Feynman integrals of tile perturbation expansion of tile field tlleory. These singularities represent, as iii conventional field theories, thresholds f’or tile prosluiction of tile quanta (Reggeons) described by X
-—- 1
ip.
When tile renormahized intercept gap valhisiled, we (‘ouncl that (lie RFT exhibited infra-red behavior analogous to that of conventional fielsi theories with massless particles. En this case oi~e could utse the renormalization group to study the behavior of the theory near J = I and I = 0. Wilen an infra-red stable fixed point exists, the Reggeon Green’s functions amid tile total anti elastic cross sections satisfy scaling laws which are given in eqs. (1.4), (1.5) and (4.41). We saw tilat such a fixed point does exist in D = e’ space dimensions, and attempted to extrapolate to the physically interesting ease, D = 2. How are we to view the f’ield p(x, i-)? Since the Reggeons it describes are composite states of’ the observed hadrons, it represents some averaged or mean behavior of (lie umnderiying iladroils. This mean behavior is likely to be ratiler independent of the constituent liadrons if the distamiees --
H.D.I. A barbanel eta!,, Reggeon field theory: formulation and use
179
IxI and “times” r represented in the field ~o(x, r) are large compared to the scales of the hadrons. The natural scales for hadrons are lxi (rnprotonY1 and r = logs I when s is measured in units of(rnproton)2. So far lxi (mprotoni’ and r~.’I, the average field maybe expected to be a good representative of the collective behavior of the underlying hadronic matter. Now this translates into small momentum, IqI, and energy E for the quanta described by the field. This is just the limit where we employed our RFT to learn in detail about P interactions and amplitudes. in a -“
~.‘
sense this is very attractive, and in another sense this is terribly disappointing. The latter comes because we are saying that in large s, small t processes we will not be learning about the basic
structure of hadrons; indeed, we are averaging over the hadron coordinates in a grand fashion. There is a very persuasive analogy for this point of view. In many body problems near critical points one describes the free energy of the system in terms of mean fields which average over large blocks of local, more fundamental coordinates. These mean fields interact; their quanta are emitted, are absorbed, and propagate. They are the clear analogue of our Reggeon field. One never produces the quanta of the mean field as free states outside the medium (electrons and ion cores) which gives them life. One never produces Reggeons as free states outside the medium (particles acting as sources) which gives them life. At the critical temperature the correlation length for static correlation functions goes to infinity. This as usual, is the signal for a long range interaction mediated by a massless particle. We, too, have a massiess excitation called the Pomeron. It provides for infinite range correlations in rapidity, the time dimension. The universality of scaling functions and critical exponents in the theory of second order phase transitions carries over directly into our Reggeon language. We, when a(O) = 1, sit precisely at the analogue of T= Tcriticai. This marvelous universality means that phase transition phenomena near critical points will not teach one about the detailed dynamics underlying the observed phenomena. So, too, are we not learning aboumt the detailed dynamics underlying the Pomeron by studying very large s, small t elastic amplitudes. This brings us to the first of our critical problems now open for discussion in Reggeon physics; namely, the s-channel content of the theory we have built with quite explicit 1-channel unitarity. Since the Reggeons reflect an infinite number of s-channel production processes, tilere is clearly a rich well of information on the structure of the Reggeons to be plumbed by the detailed study of s-channel phenomena. We can say this in a language rather well adapted to the view of the generation of long range correlations in rapidity by Reggeon interactions. In the elastic amplitudes which have been the primary concern of our report, one probes the amplitude for a source (two particles) to emit Reggeons and then at a later time (rapidity) for another set of two particles acting as a sink to absorb them (fig. 6.3). In an s-channel production process such as single particle inclusive reactions in the triple Regge region there are three times involved log(mpro 2= 1on) logM2, and logs (fig. 6.4). [The energy scale is always mproton I GeV/c~.]We, by studying this inclusive process, are probing intermediate times, which requires the Pomeron to reveal some of its short range (in rapidity) structure. Other examples of intermediate time probes will come directly to mind. Each has its counterpart in an s-channel process which is part of the building up of the Reggeons. Study of these many time correlation functions should provide a systematic method to learn how to put together Reggeons. It has the incidental, highly non-trivial attractiveness of discussing experimentally accessible s-channel processes. The elucidation of these s-channel properties is then an issue of the first importance in Reggeon theories. Another issue, slightly more elusive perhaps, is that of the nature of bare Reggeons. The bare Pomeron is an important example. This is the quantity that is probed at present machine and col-
I,
11.1)1. .4 barhammel ct
180
a!., Rc’ggc’omm field timeort’: Jorrmmulation and use
x e~oqM~
Y
~~:Iog
S
i’ig. 6.3. I’iastic amplitudes study the two time correlation mmnetnmn for sources (called particles) tsm emit Repgcsmns ammd reabsorb tlmenm, Time timm~esare 0 aimd log a for time elastis’ process, I- or long timmmes time scaling solo lion of RI”’F is appl ir’:mble, It averages over enormmmous numnbers of s-channel inelastic prmcesses and over the lmadromm coordinates which are at time imearI nt Reggeon bmmil ding.
~ lip. 6.4, Single particle inclusive anmplitudes (as in fig. 6.21 in’ 2 , and log .r. ‘flme study of many voive t liree times: 0, log M timmmr’ correiatiomm tuns’tions reveals Imow Poimmeromms beha~e at internmediatc t immmes. Sucim correl:m tion fcmnutions’,mrc eqmmiv:mtemm
to learning about a’—cimanrmcl physics in
RI
1’.
hiding beam emiergies. Its strtmeture is stire to reflect the detailesi hadroil slynamiues of wliatevei’ thieoi’y, omle cilooses as an attractive candidate. We havemi’t really mumch of an imif’orniative natuire to say about tile building of’ Reggeons. Despite decades of clever work on the miiatter, (lie issume of’ how hadromls bind to f’orm hadrons amid Reggeons remains open amid enticing. The importance of’ the m’natter clearly transcends the somewllat cireumserihesi set of problems tins report has been able to brimig oum( ails!, insleesi. seems a ehallengillg note oii winch to f’imiish.
Ackno wiedgenients We woumid like to tilank our Reggeon Fielsi Tileory colleagues (‘or countless shscussiomis which clarified oumr thinking on the ideas presented here. Unforttmmla(ely they are too miumnieroums to cite imidividually. We are particularly indebted to Estiler Singer f’or 11cr extraordinary ef’l’orts in P~’~paring this mam’iuscript. Finally, three of ums (J.B.B., R.L.S. amid A.R.W.) woumhd like ts) express our appreciation for the hospitality extended to us at Fermilab, where tins work was begumn.
References Ill ‘F. Regge, Nuovo iinento 14 (1959) 951; 18 12] (.F. Chew and S. i”rautschi, Phys. Rev. 1.ett,
(1960> 947, 7 (1961) 394: R. ulankenbecler and ML. Goldberger, Plmys. Rev, 126 (1962) 766. 131 V.N. Grihov, Soviet Phys. IETP 14 (1962) 478. 1395. 141 1). Amati, A. Stanghelhni and S. Fuhini, Nuovo Ciinento 26 (1962) 896; S. Mandeistam, Nuovo Ciniento 30 (1963) 1113, 1127. 1148, 61 J.C. Polkinghorne, J. Math, Phys, 4 (1963)1396. 171 Di. Melienaa Ct ai,, Proc. XVII Intern. Conf, on Higim Energy Physics, 1-37, ed, JR. Smith, Loimdon (1974), Ss’e alsim V. harper-Plenary, Report on Reaction Mechanismmms at high Energy. ibid. 181 U. Amm’ialdi ci ai., Plmys. Leit, 44B (1973)112. 191 5.R. Aimiendolia ci ai.. Plays. Lett. 4413 (1973) 119. 110] ‘1K. Kycia et al,, Proc. XVII Intern. Conf, on High Energy Pimysics. 1-32. ed, J.R. Snmith, London (1974) Ill] AU. Mueller, Phys. Rev. D2 (1970) 2963.
H.D.I. A barbanel eta!., Reggeon field theory: formulation and use
181
[12] CI. Tan, Phys. Rev. 04 (1971) 2412. 1131 I-l.P. Stapp, Phys. Rev, D3 (1971) 3177. iC. Polkinghorne, Nuovo Cimento 7A (1972) 555. Freedman and G. Veneziano, Plays. Rev. D4 (1971) 906. 1161 G. Veneziano, Phys. Lett. 36B (1971) 397. [171l-I.D.I. Abarbanel, G.E. Chew, M.L. Goldberger and L.M. Saunders, Phys. Rev. Lett. 26 (1971) 937. 1181 C.E. Jones, F.E. Low, S.H. Tye, 0. Veneziano and I.E. Young, Pl’tys. Rev. D6 (1972) 1033. 1191 R.C. Brower and J.H. Weiss, Phys. Lett. 4lB (1972) 631. 1201 V.N. Gribov,I.Ya. Pomeranchuk and K.A. Ter-Martirosyan, Phys. Rev. 139 (1965) B184. 1211 A.R. White, NucI. Phys. B50 (1972) 93, 130. 1221 V.N. Gribov, Soviet Phys. JETP 26 (1968) 414. 1231 P. Goddard and AR. White, Nuovo Cinmento 1A (1971) 645. [241AR. White, NucI. Phys. B39 (1972) 432, 461. [251 A.A. Migdal, A.M. Polyakov and K.A. Ter-Martirosyan, Phys. Lett. 48B (1974) 239; Zh. Eksp. Teor. Fiz. 67 (1974) 84. [261 J.B. Bronzan, Phys. Rev. D9 (1974) 2397. [271J.L. Cardy, NucI. Phys. B75 (1974) 413. [281S. Frautschi and B. Margolis, Nuovo Cimento 56A (1968) 1155. [291 H. Cheng and ‘r-T. Wu, Phys. Rev. Lett. 24 (1970) 1456. [301 R. Blankenbecler, J.R. Fulco and R.L. Sugar, Phys. Rev. D9 (1974) 736. [311 G. Calcucci, R. Jengo and C. Rebbi, NuovoCimento4A(197l) 330;6A (1971) 601. [321 R. Aviv, R. Blankenbecler and R.L. Sugar, Phys. Rev. 05 (1972) 3252; S. Auerbach, R. Aviv, R. Blankenbecler and R.L. Sugar, Phys. Rev. D8 (1972) 2216. 1331 SI. Chang and TM. Yan, Phys. Rev. D4 (1971) 537. 1341 H.D.I. Abarbanel and J.B. Bronzan, Phys. Lett. 48B (1974) 345; Phys. Rev. D9 (1974) 2397. [351For a review of this subject and further references see J. Kogut and KG. Wilson, Physics Report 1 2C (1974) 77. [361 V.A. Abramovskii, V.N. Gribov and O.V. Kanchelli, Yad. Fiz. 18 (1973) 595 ISov. J. NucI. Phys. 18 (1974) 3081. [371V.V. Sudakov, Soviet Phys. JETP 3 (1956) 65. [381 P.V. Landshoff and J.C. Polkinghorne, Phys. Rev. 181 (1968) 1989. [391 C.E. DeTar, M.I.T. Preprint, MIT-CTP 421 (1974). [401 A detailed study of these graphs in the weak coupling limit by I.T. Halliday and Ci. Sachrajda, Phys. Rev. D8 (1973) 3598, has shown that there is some simplification and the cutting rules of ref. [36] hold. There is, however, still some dispute about what s-channel states contribute to the two-Reggeon cut in general. (See T. DeGrand MIT preprint, 1975.) 1411 P.D.B. Collins, Physics Reports 1C (1971) 104, section 2.8. [421R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The Analytic S’Matrix (Cambridge University Press, 1966). [431 J. Coster and H.P. Stapp, J. Math. Phys. 11(1970)1441. [441 The results directly relevant for our purpose here can be found in refs, [23, 241. Other related papers on complex helicity are: C.E. Jones~,F.E. Low and J.E. Young, Phys. Rev. D4 (1971) 2358; J.H. Weis, Phys. Rev. D6 (1972) 2823; H.D.I. Abarbanel and A. Schwimmer, Phys. Rev. D6 (1972) 3018; A.R. White, Nuch. Phys. B67 (1973) 189. [451R.L. Omnes and V.A. Alessandrini, Phys. Rev. 136B (1964) 1137. [461 Ya.I. Azimov, A.A. Anselm, V.N. Gribov, G.S. Daniov and 1.1. Dyatlov, Soviet Phys. JETP 21(1965)1189; 22 (1966) 383. [471IT. Drummond and GA. Winbow, Phys. Rev. 161 (1967) 1401. [48] Ya.A. Simonov, JETP 21(1965)160. [491A.R. White, Phys. Rev. 010 (1974) 1236. [501 H.D.l. Abarbanel, Phys. Rev. D10 (1972) 2788. [51] This observation was first made by V.N. Gribov and A.A. Migdal, Soviet 1. Nucl. Phys. 8 (1969) 583 on the basis of eq. (2.17) which is sometimes referred to as the Gribov-Migdal sum rule. It was first used by A.B. Kaidalov, Soviet J. NucI. Phys. 13 (1971) 226 and later by others including l.J. Muzinich, FE. Paige, T.L. Trueman and L.L. Wang, Phys. Rev. D4 (1972) 1048. [52] J.L. Cardy and AR. White, Noel. Phys. B80 (1974) 12. [53] H.D.I. Abarbanel, Proc. XVII Intern. Conf. on High Energy Physics, 1-136, ed, J.R. Smith, London (1974). 1541 J.B. Bronzan, Phys. Rev. D4 (1971) 1097, 2569. [551 J.B. Bronzan and C.S. Hui, Phys. Rev. D5 (1972) 964. [561 K. Kang and A.R. White, Phys. Rev. D10 (1974) 983. [571 A complete description of the cross-channel unitarity constraints on multiparticle partial-wave amplitudes together with the related Reggeon unitarity relations will be given in a forthcoming series of papers: A.R. White, in preparation. [141
[151CE. DeTar, D.Z.
182
11.1).!. ,‘i barbane! eta!,, Rr’ggeomm field theorm’: Jormimu!ation and
I 58~iL, Cardy, R.L. Sugar aimd AR. W’imite, U.C. Bcrkeie~ Sammta Barbara preprint 159] 11.1)1. Abarbancl, J , Bands, lB. llronzaim aimd I). Sidlmmm, to l,e published. ~-
use
(1974>. Pimys, Letters, to be puhlislmed.
1601 See’, ‘or example. R. Oehimmc, Complex Angular Mommientummi in Elementary P:mrticle Scattering, in: Strong Interactions and High Energy Physics, ed. R.G. Moorhousc (Oliver and Boyd, Edinburgh and London, 1964) pp. 129~222. [611 V.N. Gribov and A.A. Mipdal. Yad, liz, 8 (1968) 1002 lSmviet J, Nuci, i’hys. 8 (1969) 583]. 162] V.N. Gribov and A.A. Migdal, Yad, Fiz. 8 (1968)1 21 3 [Soviet J. Nuel, Plmys. 8 (1969) 71(3] [631 V.N. Gribov and A.A. Migdal. Zh. 1’,ksp, Tcor. lii. 55>1968) 1498 Soviet Plmys JET 1’ 28 (1969) 7841. [641 iii. ilrmmmzan, Phys. Rev, 07 (1973> 480. [651 \‘.N. Grihov. in: Proc. XVI lntermm. Con!’. 0mm ihigli I.nerg~Ptmvsics, Vol. 3, eric, ii). iav’kson. A. Rcmtmcrts mmmd R. l)mmmm:mldsomm (N:mtmonal Acceieratmmr l,:mbor:mtory, Batavia, Ill.. 1973) e. 491. 1661 J.l,. Cardy and AR. White, Phmas. Letters 4711 (1973) 445, 1671 S. (‘oleimman, Lectures’ at time 197(3 Intern. Scmimmimmer Schommi cmi Physics I’.ttore Majorana. 168] ll.l).I . Abarhanel, Proc. Sumnmrmer Institute on Particle i’imysies. Vol. I, SI,AC PUB-I 79. t 974. p. 399, 69] M. Baker, PImyc. Letters 51 B (1974)158: :mnd Moscmmsv (l’]’EP) prcprint. 1701 ill. Brmmnzan and i.W. Dash, Pimys. Letters SIB (1974) 496; Pimys. Rev. 1)11)) 1974) 4208. 1711 RI’. hlremwer, i. Ellis. R. Small amid Wi. Zakrzewski, Reggeon l’ield ‘I’tmeory on a lattice: A t’ormimrmlation, CERN preprimm! ‘Fli. 1973 (February 1975). 172] J. Ellis and R. Savit. Pimase ‘l’rammsitionc and time hligIm Temmmperature Expansion ‘or the Reggcon (‘:mteulims, (‘I’ RN preprini I’ll. 1974 (Marclm 1975). [731 J.I~.(“ardy amid R.L. Sugar, Reggeon Held ‘i’imeory on a Lattice. I. U(’Sll prcpnimmt (Marcim 1975). 74] V.N. Gribov, F.M. levin and A.A. N’ligdal, Yad, liz, I 2 (197(3)173 lSmmviet J, Nmicl. Pimys. 12 (1971) m)3 I 1751 Il.D.I. Abarbanel and R.L. Sugar, Phys. Rev. 010 (1974) 721. 1761 i. Bartels and R. Savit, Fermilab preprints 1”NAL-Pub-74/60-TUY and I NAL-Pub-74/6 i-Ti 1Y. 1771 11.0.1. Abarbanel and 1.11. Bronzan, Plmys. Rev. D9 (1974) 3304. [78] R.C, llrower and J. l’:mlis, Phvs. Letters 5111(1974) 242. 1791 R. Jengo, Plmys. Letters SIB (1974) 143. 1801 U. Calucei and R. iengo, Nuci. Plmys, 1184 (1975) 413. 1811 R.L. Sugar and AR. Wimite, Plays. Rev. 1)1(1(1974) 4063. 1821 R.L. Sugar and AR. White, Pimys, Rev. D10 (1974)41)74. [831 ll.D.1. Abarbanel, Plmys. Letters 4911 (1974) 61. [84] S.S. Pinsky and V. RabI, Ohio State University preprint (‘00-1545-14(1 (1974). 185] H.D.I. Aharbancl, Fermnila b preprint NAL’Pub’74/42’TI IV (1974). 1861 Er. Drunmnmond, Plays. Rev. 176 (1968) 20(3. 1871 O.K. Campbell, Plays. Rev, 188 (1969) 2471. (88] i. Bartels, I”erimiilab-i’ub. 74/94-THY and 74/95-’l’HY. 1891 L. Caneschi and R. Jengo, CERN preprint, [h. 1939-CERN (1974). [90] J. }“inkelstein and K. Kajantie, Phys. Letters 26B (1968) 305: Nuovo Cimmiento 56A (1968) 659. [911P.0.13. (‘ollins, I’D. Gault and A. Martin, Nuci, Pimys. B83 (1974) 241. 92] A. Capella and J. Kaplan. Phys. Letters 5211 (1974) 448, 1931 A.R. Wimite, Proc. SLAC Summmmmmer Institute on Strong Interactions (Admgust 1974). 1941 N.S. Craige and G. Preparata, Plmys, I.etters 4513 (1973) 487: C. Pajarcs and I). Schit’f, Nuovo (‘im. Edt. 8 (1973) 237; J.N. Ng and V.P. Suklmatmnc, Nuci. Plmys. 1355 (1973) 253: I’,l).li. Collins, 1.1). Gault and A. Martin, Plmys. Letters 4713 (1973) 171; Nuel. Plmys. B80 (1974) 135.
Henry DI. ABARBANEL Fermi NationalAccelerator Laboratory, Batavia, Ill. 60510, U.S.A.
John D. BRONZAN Department of Physics, Rutgers, The State University, New Brunswick, N.J. 08903, U.S.A.
Robert L. SUGAR Department of Physics, University of California, Santa Barbara, California 93106, U.S.A.
and Alan R. WHITE Department of Physics, University of California, Berkeley, California 94720, U.S.A.
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Section C of Physics Letters) 21, no.3 (1975) 119182. NORTH-HOLLAND PUBLISHING COMPANY
REGGEON FIELD THEORY: FORMULATION AND USE* Henry D.I. ABARBANEL Fermi National A ccelerator Laboratory, Batavia, 111. 60510, (J.,S’.A
John D. BRONZAN Department of Physics, Rutgers, The State University, New Brunswick, N.J. 08903, ILS.A.
Robert L. SUGAR Department of Physics, University of California, Santa Barbara, California 93/06, U.S.A.
Alan R. WHITE Department of Physics, University of (‘alifórnia, Berkeley, California 94720, U.S.A Received 5 May 1975 Contents: 1. Introduction 2. Hybrid Feynman graphs as motivation for Reggeon Field Theory 3. Reggeon unitarity relations 4. Theoretical developments in Reggeon Field Theory 4.1. Opening remarks 4.2. Formulation of the RFT 4.3. Early developments in RFT fur P 4.4. Recent developments in the RFT for the Pomeron
121 126 135 149 149 150 153 158
4.5. Secondary trajectories and ,siulti-Pomeron corrections 162 4.6. Higher point P couplings 164 4.7. The formal status of the RFT 165 4.8. Ideas about a~> 1 167 4.9. Other “weak coupling” ideas l6lt 5. Inelastic processes in RFT 169 6. Conclusions, outlook and critical problems 176 Referencv~ 180
Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 21, No. 3 (1975) 119182. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher, Orders must be accompanied by check. Single issue price DII. 2O.~-,postage included. ~
*Work supported by the Atomic Energy Commission and the National Science Foundation.
H.D.I. Abarbanel et al., Reggeon field theory: formulation and use
121
Abstract: We formulate and discuss Reggeon field theory, which enables one to systematically analyze the exchange of Regge poles and associated branch points in high energy hadron scattering. The field theory is first motivated by a consideration of hybrid Feynman graphs, and then a more general derivation from crossed-channel multiparticle unitarity relations is given. Rules for Reggeon interaction and propagation are formulated. We treat in some detail the problem of the Pomeron or vacuum pole which has s(O) = 1 and is responsible for diffractive processes. In particular the renormalization group analysis of Reggeon field theory is presented and the structure of Pomeron partial wave amplitudes is elucidated. Also the question of Pomeron or absorptive corrections to secondary trajectories (both fermion and boson) is considered. We make some comments on important problems yet remaining in Reggeon field theory; in particular, we stress the study of its s-channel content.
1. Introduction The development of the theory of complex angular momentum (J) and the notion of moving singularities in the J-plane has provided for many years an important framework for the theoretical and phenomenological analysis of high energy (s) and small momentum transfer (t) hadron scattering processes. An elastic amplitude TIB(z, t) has the Sommerfeld—Watson representation —l T~B(z,t)=.~-_f ~‘°°
dJ(~+ 1) sinirj [PJ(—z)+rPJ(z)]F~B(J, t)
(1.1)
where z is the cosine of the center of mass scattering angle, and so is linearly related to s, while r = ±1 and denotes signature. The initial hope, based on non-relativistic potential theory, was that the only singularities of the partial-wave amplitude FXB(J, t) would be simple poles [1—3] Regge poles, whose position depends on t (fig. 1.1). It is well-known that in a relativistic theory a Regge pole gives both resonance poles in the cross-channel (positive t) and high-energy power behavior, for fixed t, in the direct or s-channel. Both of these properties of a Regge pole can easily be obtained from (1. 1) and their use to relate experimental results in channels related by crossing was one of the early triumphs of complex angular momentum theory. Unfortunately it soon became clear, following the theoretical work of Amati, Fubini and A
~utl
Fig. 1.1. Exchange of a Regge pole in an elastic process giving rise to s~(t) behavior of the elastic amplitude TAB(S, f). The residue of the pole factorizes.
,,,.....___...~
A
S~
Fig. 1.2. Exchange of n Regge poles with trajectory2)—1]. a(t) giving rise to a branch point at cs(fl)(t)—1 n[cs(tIn
—
122
/1.1)1. A barbans’l et al., Reggeon field theon: .lormulation and use
Stanghellini 141 Mandelstam [5 and Polkinghorne 161 that in a relativistic theory Regge Poles t) must he accompanied by further branch-point singularities Regge cuts. These in 1~’~B(~. branch-points can be viewed as resulting directly l’rom the exchange of two or more Regge poles at high energy. Alternatively they can he thought of’ as resulting from unitarity in the cross-channel, since this requires branch-points at the production thresholds for two or more Regge poles. (The branch—points do not produce singularities in the t-channel physical partial—waves and so, unlike Regge poles, Regge cuts are not directly observable in this channel.) The exchange of’,’i Regge poles (fig. 1 .2). all of which have the same trajectory a( t) gives a branch-point at 0(t): ,
~°kt)
I
=
,
n[~(t/n2)
--
II.
(1.2)
If we assume the existence of’ a l~omeroiipole (P), which carries vacuum quatituni numbers and has intercept a~(O)= I then the branch—points involving a Regge pole a 1~( t) ( which may or may not be the P) and many accompanying P’s become particularly significant. This is because these branch-points lie to the right of’ the Regge pole in the .1—plane for negative t ( this is clear from (1.2) if we take aR(t) = ~~(t)), and so, in the absence of special arguments to the contrary, ought to provide the dominant contribution to high—energy scattering. From a phenomenological point of view, the branch—points are very unattractive. While a pole has a residue, which, depending on t as it must, factorizes, a branch-point is characterized by a function off and t representing the jump or discontinuity across the cut attached to the branch—point. In general there is no ‘actorization and considerable f’reedom of parametrization. Not surprisingly theret’ore, phenomenologists have avoided Regge cuts as far as possible, and in t’act in some areas the experimental success of Regge poles is more striking today than at any time in the past decade. In particular the parametrization of uN charge exchange by a simple p-Regge pole exchange
(1.3)
2
(up
~n)
=
~(t)s2oP(t)_
has recently been shown to hold over an enormous range of’ laboratory momentum [71 Nevertheless it has also become clear recently that both experimentally and theoretically Regge cuts are unavoidable and must be accounted for. Tile rise of total cross-sections through Fermi-Lab and ISR energy ranges [8 ~l0I may well he paranietrized by a single Porneron pole (P) but this requires a~(0)> I and we know this is inconsistent with the Froissart hound. Therefore, any consistent parametrization of asyn~ptoticallyrising cross-sections must make essential use of Regge cuts. The advent of the active study of inclusive reactions, in particular the combination of’ the Mueller theorem [II -141 with tile sum rules relating dif’ferent inclusive cross—sections [1 5 1 6 I has also shown that an asymptotically constant total cross-section associated with an isolated P pole, is inconsistent. It is first argued that if’ tile total cross-section goes to a constant, then the triple P coupling observed in tile triple Regge region of the one-particle inclusive cross-section must vanish at zero momentum transfer [1 5, 171 A further sequence of arguments then heads to the conclusion that the P couplings (to particles) which appear in total cross-sections should also vanish [18, 19] Heiice the total cross-section cannot go to a constant. Thus as a matter of principle, Regge cuts must play an essential role if the total cross-section is to rise indefinitely or approach a constant at asymptotic energies. We have already pointed out that (1 .2) implies the importance of multi-P cuts if ~~(O) = I We emphasize that from a purely theoretical viewpoint multiparticle t-channel unitarity 120. 211 (if .
.
.
H. V.1. A barbanel et al., Reggeon field theory: formulation and use
123
nothing else) requires, that if we wish to describe the Pomeron as a Regge pole then nP cuts with c$’)(O) = 1 must also be present. The issue has been their relative experimental and theoretical importance. What we are now arguing is that for a completely self-consistent picture of the P the cuts are absolutely essential. This is true whether we are discussing the theoretical constraints of s and t-channel unitarity on the P or whether we are discussing experimental properties. In this article we shall review the derivation and current status of the one theory whose aim is to correctly assess the effect of all multi-P cuts. This theory was developed by V.N. Gribov [221 and his collaborators and has been variously called the Reggeon diagram technique or the Reggeon calculus. We have decided to call it Reggeon Field Theory (RFT) instead because this name accurately states that we are dealing with a field theory for quasi-particles (Reggeons) with a Lagrangian, field operators, Green’s functions etc. In fact most of the recent progress in the subject has come from the use of currently popular field-theoretic techniques based on the renormalization group. We should perhaps add that with our present understanding we believe the fieldtheoretical picture of Reggeons is only valuable in the scattering region. We do not expect it to be
adequate to describe the t-channel creation of particles and resonances. Our aim is to give a coherent account of RFT rather than a historical survey or a detailed account of technical points. We have attempted to discuss those subjects which seem central so that a general reader can learn the scope and technical rudiments of the theory. We hope that references will lead both the general reader and the expert to the current literature. In RFT the t-channel or cross channel is stressed. In our view the best starting point for a model independent derivation of the RFT is the multi-particle unitarity relation for the t-channel partial wave amplitude. That Regge cuts can be studied more directly in the t-channel unitarity relation than in the s-channel relation was first pointed out by Mandelstam [51.In a fundamental paper Gribov, Pomeranchuk and Ter-Martirosyan [201 (GPT) extrapolated and generalized Mandelstam’s work to show that a complete structure for Regge cuts could be obtained from the multiparticle unitarity relation for the t-channel partial-wave amplitude. (The ambiguities in the GPT work associated with problems of signature and complex helicity continuations have since been resolved by White [21, 23, 241.) The result is that, in the angular momentum plane Reggeons look very much like quasi-particles. The multi-Reggeon branch-points can be regarded as Reggeon production thresholds. The discontinuities across the attached cuts are given by formulae very similar to conventional unitarity relations and we therefore call them Reggeon unitarity relations. It is our belief that a proper treatment of Reggeon unitarity is crucial for the study of scattering at large s, fixed t. Our conviction arises because all of the multi-P channels are coupled by the discontinuity formulae, and for t 0, the thresholds of these channels approach each other. (In eq. (1.2) when c~(0)= 1, a’~(U) F.) The resulting strong coupling Of many P channels is a vital effect whose treatment is unique to the RFT. Some time after the GPT work Gribov [22] showed, by studying classes of hybrid Feynman graphs, how an underlying field theory of strong interactions could be expected to satisfy Reggeon unitarity. He found a J-plane perturbation expansion which is analogous to the Feynman—Dyson perturbation solution of conventional unitarity. This was Gribov’s original justification for the RFT. However the RFT is probably best thought of as a device for ensuring that the Reggeon unitarity relations are satisfied. From this point of view it is clear that the RFT may also be applicable to theories of strong interactions that are not simple local field theories (dual models, for example). -+
124
/1.1).]. A barhanel et of.. Reggeon field theore formulation and use
Reggeons are treated directly as quasi—particles by associating eadi of them with a field. No attempt is made to understand the spectrum of Regge trajectories~instead one attempts to study their interactions given that they exist. The philosophy here is tile same as in tile study of interactions among excitations such as phonons and plasmons in solid—state physics. While care is taken to enforce t—channel un itarity, the constraints of s—channel unitarity are not explicitly built into the RFT. As a result one mnst verify that tile theory does not violate these constraints. (Because of the limited phase—space region described by RFT a complete check of s—channel unitarity in tile form “SS~= i is probably not possible.) When the renormalized P singularity has intercept one. tile P interactions apparently remove the violation of the inclusive sum rules found for poles alone F 25 I Furthermore when one tries to increase the P intercept above one, model calculations suggest that after all cuts are su nl Illed, tIle Iroissart hon nd is at most saturated, in t’airly general circumstances 126. 271 As a result we believe tilat tile tlleory is complete enough to satisfy the constraints of s—cllannel unitarity. Tile above philosophy should be contrasted with that of s—channel models, In these models the constraints of s-channel unitarity are htnit into the elastic scattering amplitude by repeated iteration of t-channei exchanges. For example. in elkonal models. wilen tue eikonal phase is given by Regge pole exchange [28, 29] one is summing the graphs of fig. 1.2, which do not include any interactions among the Reggeons. Such interactions can he put in by lland. one at a time 1301 hut not in such a way as to satist’y Reggeon unitarity. Note that the graphs of the s-channel models form a subset of the RFT graphs. which may help to explain why the RFT does appear to satisfy’ tile constraints of s-channel unitarity. Models which satisfy full multiparticle s-channel unitarity have also been constructed [31—33] but again they do not satisfy Reggeon unitarity. The RFT has one apparent limitation which would be fatal if it were unmitigated. There are interactions among any number of P’s or among P’s and Regge poles carrying quantum numbers. Each of these interactions is specified by an arbitrary function of the momentLim and angular momentum of the P’s and Regge poles coming together at a point. In the same way. arbitrary functions are involved in the production and absorption of P’s by particles. The theory therefore has infinitely many parameters. It is possible that these parameters can be calculated in terms of a smaller set of parameters of the (assumed) underlying strong interaction theory. We favor the view that all these parameters are unimportant if the energy is high enough, while at current energies a phenomenology involving only a few parameters can probably be constructed. Tile former statement is defended by renormalization group arguments to be discussed in detail later. These arguments tell us that total and elastic cross-sections should have the asymptotic torms ‘‘
.
.
.
OAB(s)
g~g~(log s)~
‘~
(1 .4)
=
(~)
and
dt
g~g~(1ogs)2~(p/p0),
AB
where
p
t/(logs)°.
(1.6)
H.D.1. A barbanel et al., Reggeon field theory: formulation and use
125
Fig. 1.3. The dominant term in the solution to Rl:T which sums all the multi’P cuts.
This asymptotic behavior arises from diagrams in which the external particles couple through poles (fig. 1.3) called enhanced graphs in the Soviet literature. Tile nummbersg~,gB and the scale parameter Po in the “scaling function” ~pwill depend on the parameters of the RFT, but u~and r’ and the functional form of ~pwill not. This was found to he the case in the original papers of Abarbanel and Bronzan [34]who considered the theory with just a triple P interaction, and of Migdal, Polyakov and Ter-Martirosyan [25],who considered both the pure triple P theory and the general theory obtained by adding higher couplings. Actually r~and v are two of several exponents in the theory, that are analogous to the critical exponents which govern second-order phase transition theory [35]. The universality principle that the critical exponents of the theory are independent of the (unknown) underlying parameters is also directly analogous to the universality of critical exponents in phase transitions [35]. The universality of p offers the possibility that we can calculate the shape (if not the scale) of the diffraction peak at very high energies. We are fortunate that in asking for the form of the high energy behavior of hadron scattering processes we are asking a question which can have an answer which is both simple and universal. Our plan of presentation will be this: We begin in section 2 with a brief review of Gribov’s derivation of Reggeon calculus rules from hybrid Feynman graphs. These rules underlie the RFT one abstracts from the graphs. We believe the hybrid graph approach provides the best physical motivation for the RFT and this is why we begin with it. The hybrid graph approach has, however, suffered from criticisms of double counting (which are difficult, although, we believe, possible to resolve), and one can certainly question the generality of the rules extracted from a particular model field theory. As a matter of principle therefore we prefer to introduce the RFT as a solution of Regge cut discontinuity formulae, that is Reggeon unitarity. The derivation of Regge cut discontinuity formulae using only the hallowed S-matrix principles of unitarity and analyticity is discussed in section 3. The discussion concentrates on the twoReggeon cut, with the extension to the n-Reggeon cut covered briefly. We also discuss some points that are independent of RFT, for example the sign of the 2P cut and the use of a sum rule to relate its magnitude to inclusive reaction data. Section 4 is devoted to the problem of the Pomeron and its self-consistency within the RFT. We first review early work by Soviet workers and others which is based on the Schwinger—Dyson equations of the theory and is directed towards a “weak coupling” P (that is asymptotically constant total cross-sections). We then discuss the application of the renormalization group to the problem. We show that an explicit solution for the P exists which is actually a “strong-coupling” or “scaling” solution and gives(l.4) and(l.5). We then discuss the interaction of the “scaling” P —
1 26
/1.1)1. A barbanel et al., Reggeon field theory: formulation and use
with boson and fermion Regge poles. This ~ectiomicloses on a speculative note about the existence of further solutions to the RFT. in section 5 we review tile U5C of the RF’U to describe inelastic processes. We discuss tile “ctltting rules” of Abramovskii, Gribov and Kanchelli [361 whicil can be used to derive an RFT l’or inclusive production processes from that for tIle elastic amplitude bypassing both the hybrid Feynman graph and tile Reggeon unitarity approaches. We discuss 2 -N production processes using tile hybrid grapil approach and briefly discuss tile triple Regge region from both this approach and Reggeon unitarity. Tile rest of this section is devoted to tile general issue of the s-channel content of Reggeon field theories which we address incompletely while stressing its imnportance. Section 6 is a discussion of conclusions and views towa’-d a phenomenology using tile RFT. We discuss some orders of magnitude to indicate our conception of where “asymptopia” may lie, Our prejudice is that what we primarily see at present accelerator and colliding beam energies is a “bare” P. together with its “bare” couplings. We discuss briefly how one might attempt to make a phenomenology out of this observation. Tile final part of tilis section is a resumé of our article
emphasizillg the important conclusions and giving our outlook f’or the future of Reggeon Field Fheory.
2. Hybrid Feynman graphs as motivation for Reggeon Field Theory RFT was first abstracted from a study of Ilybrid Feynman diagrams by Gribov [221in I 967. Examples of such diagrams are shown in figs. 2.1 and 2.3. Their importance is that they represent explicit examples of the general form of amplitudes tllat we expect to represent multiple Regge exchanges ill a relativistic field theory. Tile circles represent off mass-shell two-body amplitudes tilat will eventually be represented by their Regge asymptotic form. Tile first step. however, is to find the relation between the asymptotic behavior of tile circles and that of the entire graph. One can then study tile interaction between f-plane Poles and cuts by making appropriate choices for tile asymptotic behavior of the circles. In order to illustrate Gribov’s procedure let us outline his discussion of tile diagram of f’ig. 2. 1 We wish to calculate the asymptotic hehavi@r of this diagram for large s = (p, + P2)2 alld fixed t = (p~ ~ For simplicity we take all tile particles to be spinless and to have equal mass. in. It is convenient to introduce Sudakov variables [371 by writing a general t’our-vector, k. in the !‘ormlm —
(2.1) where in2 P~Pi~ ~P2’ S
_.
in2
P2P2~Pl
S
(2.2)
and ~ aild ~ can range from plus to Illinuls infinity. Tile vectors Pi and P2 have length ~
2m~/s+ m6/s2
so that at large s we make an error of O( l/s) by setting
(2.3)
H.D.1. A barbanel et al., Reggeon field theory: formulation and use
127
~
Fig. 2.1. The hybrid Feynman graph which produces the two Reggeon cut contribution to the elastic process P1 + P2 -~ p1 2’ fixed t = (p~—~ p’~)2.The blobs represent generalized ladder graphs and have power behavior in their subat larger = (pi + p2) energies; for example,fi [(k 2). All momentum transfers and particle masses are presumed to remain
p~p~0.
(2.4)
The vector k~is a two-dimensional space-like vector orthogonal to j.~and to ñ
2• In particular
2q2
q
(pi—p’i)
=
—
(P2—Pt) +
q 1
(2.5)
2 q~.We shall nearly always neglect terms of order (m2 or t)/s. (This means that in the and so qany results we derive are accurate only within a neighborhood of at most one unit from J-plane the leading singularity.) In terms of the Sudakov variables we shall, however, write d4k in the form d4k
=
-.~/~p dct
d~3d2k
1
(2.6)
2)”2. Keeping the factor p ~ s”2 + O(1/s) will enable us to make our treatment where p = (~sparts rn straightforward at a later stage. of absorptive The two-body amplitudes in fig. 2.1 will be denoted byf 1(k1, k, k2) and 2 f2(p,—k,, 2k,~k q—k, p2—k2). It will be assumed thatf, andf2 are large when their energies s, = (k,+k2) 2 and 2 2(p,—k,) (p 2 and (q—k)2that andisthe masses (p,—k 2, k~, that become 52 = (p,k1+p2k2) 2—k2) are large, 0(s). It willk~, also be assumed they fall offlarger whenthan the momentum transfers k are certainly reasonable when, the asymptotic 1) much rn2. These assumptions behavior of the f 1 is dominated by Regge pole exchange. In this case the f, can be written in factorized form; for example 2 k2)g(k~,(k+k 2 k2) G(k2, 2k f1(k1, k, k2) = g(k~,(k—k1) 2) 1 k2). (2.7) —
-
...
The Reggeon propagator G, has the Sommerfeld—Watson representation 2, 2k,~k G(k
-_J’
C+i°°
2)
=
dl
~
2)(2(k~k ~,
G,(k
2)Y
(2.8)
1 28
Jl.D.I. A barbanel et al., Reggeon field theory: formulation and use
wilere as usual tile comltour of integration rums to tile right of all singularities of G1 and crosses tile real axis between I = I and 1 = 0. exp~ ~iu(1+ ~(i =
—-—-
-
sin~ 7r(/
e
--
—~-~
+ ~
(1
=
--~
~
+ y
(2.9)
~~---
sinurl
T)) ~
with T = ±I for even or odd signature. In general. of course, cult contributions to tile f~cannot be writtell ill factorized form. However. as we shall see cults which arise from the interplay of two or more poles can he expressed as illtegrals over f’actorized forms. As a result it is sufficient to study’ llyhrid diagrams using only’ cxpressions of tile form of’ eq. (2.7) for tile J,. We are now in a position to read otf tile higll—energy hellavior of the diagram of fig. 2. 1 First consider tile denominators arising from tile left-hand cross. For s ~ .
2+ie k~--m (p,—k,)2—rn2+i
a
2+ie. 1(~, l)s+k~---rn (a, a )(~, -~)s+(k k ,n2+ie,
(k--k 2—rn2+ie 1) (~j -k q+k)2-in2+ic
(a,—a)(~, I ~)s+2q2(a a,
,--
(2, I 0) ~-
1
2
~+~,)+(q
ifl2+~.
1~ k1+k1~) Notice that the four-momenta squared which enter 2. 1 0) also appear as mass variables in the ,t’,. The requirement that they and tile momentum transt’ers be less than or order in2 gives
k~
2,
k,~~ m2,
1~ m a,
rn2/s.
~
I.
rn2/s.
a
I.
(2.11)
A similar analysis of tile right-hand cross gives
~ 132
~ m2.
k~~ in2. /3
rn2/s
I.
(2.i2)
Finally the requirement that 2k 1’ k2 S
/3ia2s and 2(p,—k,) (p~—k~) (1 ‘
—-/3,)(
I
a2)s he of order
gives
~
I.
a2~ 1.
(2.13)
Since /3 ~ it can be neglected ill tile denominators of’ (2. 1 0). Similarly a can be neglected in the corresponding terms arising from tile right-hand cross. Putting together all of the above results we obtain a complete t’actorization of the integral represented by fig. 2. 1 which can now he written in the t’orm /3,
2k
2)
T(s, q
=
iu
dl,
d12
~ d
1
~,
~ 12iV~2tq, ~
(2.14)
H.D.J. A barbanel et al., Reggeon field theorv:formulation and use
129
where X2(~f~2p)2da, d~,da d2k1~, 2 k~)g((p N,,X,2(q, 4(4~)l/2 /3’,’( k5) 1 /3,)l2[k~_rn2+ie]
2, (p
2 (q—k 2)
(2~)~ 2—rn2+iel g(k~, 1—k,) 1—k,+k—q) 1) 1 [(p -,(k—k1) [(k,—k)2—rn2+ieY’[(p,—k,+k—q)2—rn2+ieI 1
1—’k,)
(2.15)
A is the coupling constant for the three-particle vertices. N, 112(q, k1) can, of course, also be expressed in terms of the quantities associated with the right-hand crosses. Several features of eq. (2.15) are worth noticing. the g’s depend onthe a,, ie prescription and a only for 2, (k—k,)2 andFirst (p,—k,+k—q)2. Since through the mass variables k~, ( p,—k,) the singularities in these variables is the same as for the propagators we see from (2.10) that the a, and a integrals vanish unless 0 ( /3, ~ 1. For /3, outside of this range, all singularities of the a, integrand lie on one side of the contour. We also see that if the coupling between the two-body amplitudes, which determines N, werelie planar, allsame singularities in the a-plane As (in athe region 2/s)112, would on the side of the a contour. result, tile where a, and a are of order rn graph would be negligible. asymptotic contribution of the This last result is the connection between Regge cut asymptotic behavior and the presence of a third double spectral function in the associated “two-particle/two Reggeon amplitude” which we shall find from a different point of view in the next section. Note that sinceM2 = (p,—k)2 —aS+rn2, we can pick out the a-integration in (2.15) and write /3k,
I
N,,, 2(q, k1)
~°°~j~~j2
f
~
—‘—
2, q, k T,1J2(M
1)
(2.16)
2, q, k 2-contour can wherebeT,,,2(M 1) is defined by the remainder of the integrand in (2.15). The M now rotated to give 1
N, 112(q, k1)
—~
~2
°°dM2
f —A,112(M2, q, k1~
(2.17)
2iT
4m
2, q, k 2, q, k 2-variable. We shall comwhere A,,(M on (2i7) 1) isshortly. the absorptive part of T,,,2(M 1) in the M ment further Since the important singularities in a and a,, in (2.15), occur when these variables are of order rn2/s, each of these integrals give rise to a factor of i/s. Consequently, N,,, 2 becomes independent 2 and k~negative, which is the region of interest, N,,, of s at high energies. In addition, for q 2 is real. If the simple crosses of fig. 2.1 are replaced by more general couplings we still expect (2.14) to hold. Of course, N,112 will no longer be given by (2.1 5) but we still expect it to be independent of s asymptotically and real in the s-channel physical region (this is provided that the couplings do not themselves contain Regge singularities a circumstance we consider shortly). Next note since ‘~J~p does not change sign under —s, thefactors signature 2) we given by that (2.14) is determined by the product of the ssignature E,,~,of the amplitude T(s, q 2.This is a generalized form of the result that the signature of the two-Reggeon cut is given by the product of the signatures of the contributing Regge poles. —
-~
130
11.1)1. A harbanel et al., Reggeon field theory: formulation and use
The Reggeon calculus has as its objective a set of rules for tile calculation of’ the signatured partial-wave amplitude I’i.J, q~) which is related to T(s, q~) by the Sommerfeld Watson integral (‘+1=
T(s, q~)=
tU ~~~‘F(f, q~).
(2.18)
~ being the signature factor corresponding to tile signature of 7’(s, q~ (2.18) give ).
Hf. q~)=
be jllverted to
call
q~) (s’) t~~~A(s’,
th’
(2.19)
where
A(s, q~)= Abs T(s, q~)= ~ IT(s+ie, q~) T(s-- ic, q~)I
(2.20)
(2.18) is equivalent to (1 I ) to leading order in s. The Mellin transform (2.19) is more convenient for our purposes here than tile Froissart Grihov projection WhicIl appears in (1 I In the next section we shall use the Froissart Gribov projection because it diagonalizes I-channel unitarity equations. Using the facts that .
.
).
Abs(~ 1s’)= s’,
Abs(~-~) =
(2.21
-
we see from (2.14) tilat 2k
=—f—-~
A(s, q~)
ir
dl,
d12
d
~
1
y~N~,(q1, k~)G11(k~) (/2((q~ k1)2)s~~~’2
(2.22)
witil
(I -r,) 71112 = ~
cosL_.(i,+12 +
---~-)1 (I -r2)
+
(2.23)
and =
51~[_-Q~ L~_~)J +
(2.24)
.
Finally. (2.19) gives 2k
I~’(f, q2)f~ d11 d12 d 1 y
N2 (q
--~
‘
2iri
27Ti
(27r)2
1112
1~2
,
~
k
2) (.1, (k~JG,((q1---k~)
)--..~-.-—~—--~~
f+ I ----1,
12
If the amplitudesf, andf 2 are dominated asymptotically by Regge pole exchange then
(2.25)
H.D.I. Abarbanel et al., Reggeon field theory: formulation and use
131
Fig. 2.2. The Reggeon graph contribution to the t-channel partial wave amplitude coming from the hybrid graph of fig. 2.1. N is a two particle-two Reggeon transition amplitude; G is a Reggeon propagator. See eq. (2.14).
2) = 1
G,,(k~)= l1—a,(k~)’
G,2((q1—k1)
2_a2((q±_ks)2)
(2.26)
In this case 2)
.121
~12
= Pu “A~
(2.27)
~ (2ir)2 J+l—a,(k~)—a
F(J q
2) 2((q1—k1)
d2k
—J 2iri
1 (1\2
2
~‘~1~2
,,(
01a2
‘~‘-~“
(
2
~)
~
q1
2
~
)
.
)
where the 1,-contour ruins to the right of the pole in G1, and to the left of the pole in G~+1 (2.28) now corresponds directly to the Reggeon (Feynman) diagram of fig. 2.2. The vertices where two particles produce or absorb two Reggeons represent the factor N0102~,/y~,,22 (we absorb the signature factor into the vertices). The propagators represent the Reggeon propagators G11, ~ and the loop represents the 1, and k~,integrations. There is conservation of both one minus angular momentum and k1 at the vertices if one pair of external particles is regarded as a source for angular momentum; and transverse momentum q1, and the other pair is regarded as a sink for the same quantities. Both angular momentum and transverse momentum then flow through the diagram. One is ordinarily interested in the behavior of F(J, q~)near the f-plane branch-point generated by (2.28). For the P—P cut, where r, = = 1, ~ —l in this domain. One sees then that the effective two particle 2P coupling \/ya,o2Na,02 is pure imaginary. Similarly we see from (2.23) that the two particle P + meson trajectory coupling is also pure imaginary independent of the signature of the meson trajectory. Note that if we put 1, = a,, 12 = a2 in (2.17) we have expressed the two-particle two Reggeon coupling Na,a2 as 2, an q,integral over the absorptive part of the two particle/two Reggeon scattering k amplitude Ta,02(M 1). Comparing this with (2.19) evaluated atJ —l we see that (2.16) is analogous to evaluating the two particle/two Reggeon partial-wave amplitude at a nonsense point — in this casef= a,+a2—l. This is the connection of the couplingN0,,~2with a “nonsense fixed-pole residue” which we shall find in the next section. It is straightforward to generalize the above results to the three-Reggeon cut diagram shown in fig. 2.3. One merely uses forf,(k,, k1k2) the two-Reggeon cut amplitude of (2.14). Proceeding as before one finds that the contribution of fig. 2.3 to the elastic amplitude is given by ,,.
-~
-÷
-~
T(s
2k 2k~ r. . . . . . dl1 di3 d 1 d 2~2iri 2iri (2ir)2 (2ir)2
n2) = 7T — ‘~
X
—
1
_____
N~,,
S1~’~~2
~‘1l”12”13
2)G, 213(q,k1, k~)G,,(k~,
2)G, 2((k1—k~)
2) 3((q—k1)
(2.29)
1 32
1 barbanel
/1.1).].
et al., Reggeon held theory. forniulatioi, aiid u 0
A ~ l:iy. 2.3. A ivhrid tevrinisu erapli svhjyh
eIvy~S tliiee Reeccon
cut c~ntrjhuti’~iit~’ IlL
where A
1,1213 is obtained f’roni (2. I 5) by replacing g( k~,(k--k, )2 A ~) by A’1 by I,. Calculating tile absorptive part of 1’) s, q/) as before tllell gives 2/~ d2k d/, dl~d 2 1(1, q~) (~)2(~)2 Y~,0,0 ) G~(Is~:~‘J~2 /~
I, by l,+12 I and h
~.
=f~
~
whicil corresponds to tile Reggeon of n-Reggeons is immediate
/‘(.J, q~)= 2Tiffl~ I/ ~T
2k ~
1
~
/5(~i
--
k1
(1’ ey’nman ) tllagrani of fig. 2.4. ‘l’he generalitation to the case
..~,[p,G,. A~)]6(11
~
(/
I
+~
(I,
I
))(
2~)2 62
(q
~ A 1
where in both (2.30) and (2.31)
7/,
,~ = (
- I
)“
(i,
1sin[~-(
(2.31
+
A±))1:/n
(2.32)
~.
Note that we have written (2.3 1) in a symmetric form by introducing 6—lu netions conserving angular nlomentum and transverse Illonlentuin. Tile interpretation of ( 2.3 1) is as follows: two particles act as a source of two nlolnentum £/ and ‘‘energy’’ I — J~from this source31,.iH n—Reggeons with nloniellttllll k,1 with and energ~’1 1,. / =G(1,. I n. Each Reggeon propagates the a,llplitude A~ emerge with to amplitude A’,,12are!11\~ appropriate it umtil they all absorbed by’ two particles acting as a sink. At each stage energy and momentum have been conserved tIle overall energy and momentuim consel-vation 6—functions .
.
...
,
have been factored of’f’. Ibis rather attractive description of’ the n—Reggeon diagram is supported by the study ol’ mole
~E2k21a2Nt3t
E 3, k31,a.
1g. 2.4. The Reggeun graph which comes
Irom
the hybrid lcynman graph ol
fig.
2.3.
H.D.I. Abarbanel et al., Reggeon field theory: formulation and use
133
Fig. 2.5. A hybrid Feynman graph which introduces Reggeon interactions.
complicated hybrid diagrams. Consider fig. 2.5 next. An extension of the foregoing Sudakov analysis shows that this corresponds to the Reggeon diagram of fig. 2.6, and yields for F(J, q~) d2k,1 d2k212 ~ dl, dl2 (2ir)2~2(q F(J, q~)= (27r) 1—k,1—k21) (2.33)
f2~)2
X 2~i~( 1 —J—( I —l~)—(1—12)) N,,,2(k1±,k2j)y,112r1,,2~(k,1,k21, q1) G,(l,, k,1) G2(l2, k2±)) G3(J, q1). The additional ingredient here is the triple Reggeon vertex r shown in the present calculation as fig. 2.7. More complicated contributions will change the detailed form of r but leave F(J, ?j~)as in (2.33). Since the coupling of two particles to one P is real and that of two particles to two P’s is pure imaginary, the triple P coupling of fig. 2.6 must be pure imaginary for k,1, k21, q1 0, since F(J, q~)must be real below the t-channel threshold. This means that ~ must be pure imaginary and so r,,,2~must be real. The coupling of a meson trajectory to a P plus a meson must similarly be pure imaginary. To derive Reggeon rules from a general hybrid Feynman graph one identifies familiar elements, for example, G, r the N’s, and writes the desired contribution to F(J, q1) as products of the basic building blocks put together with energy and momentum integrations. The final outcome is that to obtain the full partial-wave amplitude we must write down the complete set of “Feynman” Reggeon graphs including general interactions in which arbitrary numbers of Reggeons are destroyed and created. Of course, hybrid diagrams corresponding to all possible graphs have not been considered. The general rules are obviously an extrapolation of the results obtained for simple diagrams. An infinite set of diagrams involving four-Reggeon interactions has been considered by Landshoff and Polkinghorne [38]. The corresponding contributions to the partial-wave amplitude are then constructed by writing down for all couplings, propagators 2k vertex2)functions for each internal Reggeon loop. Energy, for each internal Reggeon line, and f(dl/2lri)(d 1/(27r) E = (1—i) and momentum being conserved at each vertex. Notice that because the Reggeon propagator is linear in E we must specify the direction of propagation of the Reggeons. Diagrams which are topologically identical, but which have one or more Reggeons propagating in opposite directions are distinct [22].
Fig. 2.6. The Reggeon graph contribution to F(J,q2 1) from the hybrid Feynman diagram of fig. 2.5. The triple Reggeon coupling T~,a2~3 appears here.
134
11.1)1.
.1 barhanel s’t al.
,
Reg,~’eonfield theore - formulation and use
F- ~, k~
1,a~ ‘.,/
~2
Fig.
—
(5,
\E~-J,q~,a3
k,1
2.7. Ihe simplest contribution to the triple Rc’ggeon coipling ~o u2u~-1 his will appear as a building block in nor’ cniiplicated Rcgc’ean graphs.
Clearly we have arrived at a set ot’”Feynman’ rules for writing down our partial—wave a~nplitude and we shall formalize this turtiler in section 4. We f’inish our description of’ tile derivation of tilese rules from hybrid graphs by noting tile rules for the phases of the P couplings that we have found. From (2.32) and similar signature f’actors for Reggeon interaction (liagranls we find that if Nm is tile coupling (treated as a constant) of two particles to in P’s and ~ is tile cotiplmg of in P’s to n P’s then
1. A,nn (2.34) ~ (i)’°— We have now given a bare outline of the arguments of’ Grihov to motivate a field theoretic description of Reggeons. We do not intend to attempt the complete decomposition of’ a relativistic ~
(i)i?i+i
2
,
field theory into hybrid diagrams. Tllis would be required to turll our present heuristic justification of the RFT into a proper derivation. (‘learly we would need as a starting point some picture u)f~the initial process wilich generates an /solaled P pole with a~(0) I The amplitude f’or this process would then provide the input two-body amplitude folr our ilybrid (liagrallls. (Conceptually it is sometimes useful to picture this orocess as tile familiar multiperipheral or ladder mecilanisni 1361 but this is not essential.) However, there are still problems associated with tile breaking up of hybrid diagrams wilich correspond to Reggeon interactions, such as that of fig. 2.5. into a part corresponding to tile full Reggeon diagram (that of fig. 2.6 for fig. 2.5) and parts corresponding to renormalization of tile original, or bare. P0k and or vertices. To handle this problem correctly it is necessary to introduce a cut-off ill tIle / and k integrations in Reggeon uliagranls and to relate this cut-off directly to the break-up of the phase-space ill hybrid diagranls. A (liscussioll of’ how this can he done and how the accusations of double-counting at this stage can be avoided, has ‘~
.
,
been given by DeTar [391 Fortunately we can avoid tilis problem by ulsing Reggeon nnitarity as a basis for tile RFT. as we disculss in the next section. Tile Reggeon unitarity equations are applicable once we know. or rather assume, that there is an isolated P pole in tile neighbourhood of .J I.1 0. These equlations reassure us that we have not been double-counting in discussing hybrid diagranls. Bef’ore leaving tile counting problem altogether however, we note that we shall discuss the use of a cutoft’ RFT in relation to tile renormalization group in section 4. It is worth noting from tile above discussion that tile cuit-off may well he an essential feature ill ensuring that tile RFT is welI-def’ined botil mathematically and physically. Tile generality of the Reggeon unitarity approach to RFT gives a powerful logical basis to the apparently more model dependent, albeit rather more pllysically motivated j~rocessdescribed above. To compare tile results of this section directly with those of the next section we briefly discuss taking the discontinuity across the cuts attaciled to the Reggeon branch points occurring in tile partial-wave amplitudes we have obtained.
H.D.J. A barbanel eta!., Reggeon field theory: formulation and use
135
Consider first the n-Reggeon branch point generated in (2.31). This can be viewed as a “threshold” singularity generated by the poles of the propagators G,.(k,21). The discontinuity is obtained by the usual “Cutkosky rules” of putting each Reggeon on its “spin” (mass) shell: 2k d2k disc~F(J,q~)= 1)’~’2iri sin[-~ (.~_~ 1 d ~ ö2 (qj — ~ k 2 1~)X (2ir) ...
(—
x ~
(2.35)
this expression being simply obtained from (2.31) by writing G,.(k~ 1) 2iri ~(l1—a,(k~)). For more complicated Reggeon diagrams there will be a contribution to the n-Reggeon discontinuity from every n-Reggeon state that can be found by “cutting” the diagram vertically. (There will, as in a conventional field theory, be a renormalization of the pole position in each propagator by “self-Energy” insertions. This renormalization will only be unambiguously defined from hybrid diagrams once the cut-off is introduced as discussed above.) To take all these contributions into account we follow the usual unitarity prescription of complex-conjugating that part of the diagram to the right of the cutting line. More technically, the part of the diagram to the left of the cutting line is evaluated above its f-plane branch cut, while that to the right is evaluated below the branch cut. After summing all contributions the final result is that the complete discontinuity has the same form as (2.35), except that —~
2 =Nai... ~
1’~o1...an(~)’
0n(J+~)N0i...0i,(J~i)
(2.36)
where Nai ~~(J)is the complete two particle n Reggeon amplitude. The complete imaginary part of F(J, q~)arising from its Reggeon branch-points is given by the Reggeon unitarity relation -*
disc~F(J,q~)=
~
n2
disc~F(J,q~)
(2.37)
‘~
where disc~F(J,q~)is given by (2.35) with the substitution (2.36). The purpose of the next section is”to derive (2.37) directly from multiparticle t-channel unitarity. We can then argue that
the procedure of the latter part of this section can be reversed and the RFT viewed simply as a solution of (2.37).
3. Reggeon unitarity relations In this section we discuss the derivation and implication of Reggeon cut discontinuity formulae from the point of view of I-channel unitarity. We use standard S-matrix methods and the results depend on conventional S-matrix assumptions of analyticity put together with the existence of moving poles in the J-plane. Briefly speaking we argue that branch points in J arise from the presence of two or more moving poles in J. If there are additional singularities in the f-plane, we will not find them. Indeed, in the spirit of “maximal analyticity in the f-plane” we assume them
1 36
H.D.J.
A barbanel ci at., Reggeon
field theory: formulation and use
11111111 >IIIIIIII(
11111 >/lIIIIIIi\<
1’ (a)
(h(
Fig. 3.1.
(a) A planar F’eynnian ~g’aphwith two Reggeons, represented by ladders, in the t’channel, l’hts planar graph does not give rise to a branch point in the f-plane at a(2)(t), eq. (1.2). (h) The simplest non-planar 1’cynn,an graph which does s’,eld the two Reggeon branch point.
to be absent. The full treatment o!~multiparticle i—channel unitarity ill tile i—plane is rather conlplicated but we shall try to minimize the technical details without depriving tile reader of all insight into the procedure. To motivate the use of the i-channel as tile correct place for an S-matrix analysis off-plane branch points let uis look at the Feynman graphs winch were the original indication of the presence of cuts [4, 5], (fig. 3.la and fig. 3.Ib). Both graphs have many intermediate states in the s-channel gotten by cutting the ladders which give rise to the Regge behavior. A priori all possible nunibers of particles in tile s-channel must be considered at the same time to discuss the large s. fixed t behavior of these diagrams [40] Since we are, Ilowever, interested ill a fixed I limit, by continuing t from the scattering region, t ~ 0, to some t’inite time-like point we can restrict our attention to particular i-channel intermediate states. In the diagrams in fig. 3.1 the lowest intermediate state in t is the four particle state. This suggests we may be able to study tile two Reggeon cut in the full partial wave amplitude by looking at the four particle unitarity integral in the i-channel. Indeed, this is so [20. 21] provided, of course, that the Reggeons involved couple to some two-particle states. Mandelstam [5] actually analyzed in detail the diagrams of fig. 3.2 where one of the ladders (Regge poles) of fig. 3.1 is replaced by a single particle. Neitiler t’igs. 3. Ia nor 3.2a has a Regge cut while 3.lb and 3.2b do. Mandelstam showed this for figs. 3. 2a and 3.2h by considering the three particle unitarity relation in the i-channel. He argued that only in tile case tilat the five point amplitude shown in fig. 3.3 has botil a right and left-hand cut in the sub-energy s, could a f-plane branch point be present in tile~eIasticamplitudes of fig. 3.2. Fig. 3.2a has only a righthand cut and does not yield aJ-plane cut; fig. 3.2b has both right and left and gives rise to a f-plane cut. In other words, a f-plane cut will only occur when the “three particle-Reggeon aniplitude” of fig. 3.3 has a “third double spectral function”. This part of Mandelstam’s argument is .
,
independent of Feynman graphs as is the final form for the contribution of the cut to tile partial
(a)
(b)
Fig. 3.2. (a) A planar Feynman graph with one particle and one Reggeon in the i-channel, This planar graph does not give rise to a branch point in J. (b) The simplest non-planar graph which does give a Reggeon-partiele branch point in J. It is the presence of a left and right hand cut in the sub-energy s~which guarantees the existence of the cut. This Reggeon-particle cut is shielded by the two Reggeon cut (fig. 3.lb) in the scattering region, t ~ 0.
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
137
titi
Fig. 3.3. The two particle
-+
one particle
+
Reggeon graph which is contained in the two particle Reggeon is represented by a ladder.
—*
three particle amplitude. The
wave amplitude F(J, t)
—‘f
(.~./~_m)2
di,
X”2(t,t, m2)
C(t, t~) f+l’—a(t,)
(3.1)
where C(t, I,) is a smooth function of its arguments, ct(t) is the pole trajectory coming from the ladder graphs, and X(x, y, z) = (x+y—z)2—4xy is the familiar kinematic factor. Subsequently, Gribov, Pomeranchuk, and Ter-Martirosyan (GPT) [20] compared the form of (3. 1) with the contribution to the two particle unitarity relation for F(f, t) coming from a spinless particle of mass m and a particle of mass m,, spinf,, helicity n,. This is +Ji
X”~(t,m~,m2) F(J+1—n,)
Im 5F(J, t)
~
n,—J1
i
\ I’,
I
flj)
f~,~,(t)ff’,~,(t),
(3.2)
where f~,~~(t) is a helicity partial wave amplitude for the transition: two spinless particles -÷spinless particle + particle of spinf,, helicity n1. This formulae contains a fixed pole atff,—1 arising from the helicity state with n1 f1, and near this point strongly resembles (3.1). Further, it is known that the residue of this fixed pole will vanish unless the amplitude giving f1~(t)has a third double spectral function. This is the usual connection between nonsense wrong-signature fixed poles and third double spectral functions. {There is a rather thick layer ofjargon in this, and we are required to ask the reader’s patience and refer him to standard references [41] for an explanation of terms. } This comparison suggested to GPT that f-plane branch points could be viewed as resulting from the presence of a fixed-pole in Reggeon particle intermediate states all along the Regge trajectory. This observation was the key to subsequent analysis off-plane cuts using t-channel unitarity. A complete analysis of three particle unitarity together with a discussion of the Reggeon-particle cut in (3.1) has been given by White [21].This last cut moves off the physical sheet of the f-plane around the two Reggeon branch point when t becomes small. Since it, therefore, does not contribute to the large s, fixed t behavior we turn our attention to the two Reggeon cut. To study the two-Reggeon we consider across not thethe complete t-channel unitarity rela2, but insteadbranch-point isolate the discontinuity four-particle threshold. This tion for I 2 1 6m eliminates the need to discuss the two and three particle contributions to the unitarity relation. The Feynman graphs of fig. 3.1 do not have two or three particle intermediate states in the t-channel and so for them the four-particle discontinuity would be the complete unitarity relation for 1 6m2 ~ I ~ 25m2. In general the four-particle threshold discontinuity can be written in a fairly conventional S-matrix form [42, 43]
1 38
lJ.D.I. ~1harbanel ci at.,
Reggeon field theors’: fdrmulation and use
P1 ONE
4rn?
m
Pm?
1-ig. 3.4. ‘the path ut coiitini,ation in the i—plane to reach, the .iniphitudv’ ‘T~ needed in the unitarits Irniulic eq
=
-~
~
~.:
i3.3
J1II.~and ~ are physical amplitudes, bitt the other amplitudes are (let illeul bE analytic con I tutution of these physical amplitudes.~~ is i~+ contillued around tile four-partIcle tilm’esilold as illustrated in fig. 3.4. is~ analytically con tintteul around both the four-particle threshold in the total energy t (this is denoted by tile 4) and around all phase—space thresholds in tile stlb— channels (this is denoted by ~ and involves going around two—particle tilresllolds ill all two— particle channels and three-particle thresholds in all tilree-particle cilanllelS). The ltegration implied by tile right-hand side of’ (3.3) is over normal f’our-particle phase-space. Tile next step is to project tllis unitarity relation onto partial waves. In fig.3.5 a complete set of partial wave variables is given. Since the external particles are spinless we need only the total angular momentum f and (mass)2 = I = Q2 of the (‘our—particle state. i1. t~. Q~lot’ the two pairs 4~
ii~tile hehcity of each pair. The total helicity ii = n~+ n2. One catl now use these variables to construct the unitarity relation for partial waves coming from (3.3): 4(J, I) = (3.4) F(J, i) F fdp ~ A(f, f,, f 4(J, ,Ji, J 2, ni, in, I. in2i I EU, f,, J2, n~,n2, I, t,. t2)F 2, il’, /12. 1, t~,~
of particles and
*
—
J2 2~In~ I
with A(f,
J,, J2, t, t,, t2)
(~,+
=
----
I)
(2J2+ 1) [U, -n ,+l) [(J2-n2+ I I [(f- /1+1)
F(J,+n ,+ I) F(f2+n2+ I) F(f+n+ I)
-
-
~
c)
an(I
Fig. 3.5.
A set
of variables for the partial wave projection of the 2 -~4 amplitude needed in the study of the four particle contribution to the partial wave unitarity relatiun. f is angular momentum and n is helicity.
*(3~4)can be obtained directly from (3.3) by diagonalizing the momentum-space integrations using partial-wave projections of the six-point function. The details of this procedure are given in the appendices of ref. 1211.
H.D.J. A barbanel eta!., Reggeon field theori’.’ formulation and use
fdp
=
ffdi 4,ii2
(2ir)5264!
2 X”2(i i i,,_ta)[(ti_4m2)(i L 1112 2~~~4m2)1O
~ 4m2
139
(3.6) J
Eq. (3.4) gives the discontinuity of the usual partial wave amplitude F(f, t) across the four-par-
ticle threshold. The amplitude F(f, f,, J 2, n,, n2, 1, 1,, 12) is the helicity partial wave amplitude for two spinless particles to go to a particle mass I,, spinamplitude J,, hehicity a particlecontinued of (12, J2, n2). 4” of the same arguments is the helicityofpartial wave of n, theplus analytically six F point function. We now focus on the “kinematic factor” A in (3.4). it contains the fixed pole atf= n—i. This is the singularity GPT expected to be present in order for the two Reggeon cut to appear in F(J, t) when continued to complex J. GPT did not try to find the exact form of this continuation. They essentially tailored their assumptions on how to continue the sums in (3.4) to achieve the expected results. Since their work there has been considerable insight into the technical problems associated with analytic continuations of multiparticle amplitudes in angular momentum as well as helicity [44]. For our purposes it is enough to know that the GPT program can be carried through although we shall need the exact form of the continuation of (3.4) to complex f. This has been given by White [211 and we follow his development. To continue F(f, J,, J 2, n,, n2, I, I,, t2) away from integer f, f1, and n, we must introduce signature as well as labels> or < telling us whether we have continued n, into the right or left half helicity planes. We will collect these various labels into an index q and refer to them only when essential. The signature labels naturally refer to continuation in f—n, call the signature ~J~—n~, ~ n~,r. The signaturesr, r,, r2 forf, f ,, f2 are r~r’,r’2,
r,
~
r2
(3.7)
‘i2r~.
The contribution to the discontinuity in (3.4) from n,, n2> 0 is now written —sin{~ir(J—fl}Cd 16
X
~
~
f___________
~ ~r.±i
Cj
sin {~ir(n,—~) } sin{~ir(n2—~) }sin{~ir(J—n—~) }
T(J, n)FT4(J, n) A(J, n)F
7)
dn,dn2
(3.8)
where J = (J, f,, J 2), n = (n,, n2), c = (r, r,, r2, r’,, r~),and ~ = ~-(l—r), ~ = ~(l—r). Clearly the sums over n, and n2 in (3.4) have been converted to a contour integral over the contour C~.Fig. 3.6 shows the projection of C~in then, plane at fixed n2 and for fixed n, it would have the same form in the n2 plane. That the contour is asymptotically parallel to both the n, and n2 imaginary axes ensures that the integral converges and that the problem of divergent helicity sums in this context is avoided [45—47]. At integerf the two series of poles shown by ciosses in fig. 3.6 come together and pinch the contour C,. The double 1ntegral over n, and n2 develops a pole because of the pinching of the contour by the three sets of poles in the integrand coming from the factors of sin{~7r(n,~i!,)}, sin{~ir(n2—f~)}, and sin~ir(J—n—fl}. The pole of the integral is cancelled by the “signature factor” sin{~ir(f—fl},which multiplies the integral and thus singles out the pole
140
lID.!. A barbanel ci at,, Reggeomi field theory: k,rniulation and use
n
a It I F
I
~
L
HELICITY PLANE --
‘i~
-
—
-n
1~r; --n~r1’+2 Fig. 3.6. The contour in the helicity plane is required in eq. (3.8) for the four particle contribution
I
partial wave unitaritv.
residue as the value of’ (3.8) at integer f. ‘l’his residue contains just the requisite finite sttm over helicity amplitudes. We can now study the two Reggeon cut which arises from the presence of Regge poles at f1 = a~(i~), with signatures r~,in the .J~channels. Poles occur at a1 a~(i~)in the term with f1—n1 0 in (3.8) provided i~= + I As sllown in t’ig. 3.6 these poles lie on opposite sides of tile ( contour to the pole of A at J = n- I. 4(J,n), there there is aintegral compensating “nonsense zero” eitllerofJ’~J. ii) or L’~of the Regge willTherefore, be a pole unless in the helicity at .J = a,(ti)+02(t2)--I as ainresult the pinching .
poles and the nonsense pole. For there to be no nonsense
zero
f
=
a ,+n 2 I must he a wrong-sig.
naiure nonsense point. This requires
T=+l
~TTT~
(~))
and since =
T1T1
=
T2 = T2T2
=
+1
(3.10)
we must have (3.11)
TT,T2.
This is the S-matrix origin of tile rule that the signature of a cut is the product of the signatures of the Regge poles building tile cuit. Isolating the contribution of the pole at f = ct~(t,)+ 02(12)-— Ito (3.8). and taking = = r = +i for simplicity, we obtain A’(f, ~)R~(J, t)I’T~4”(f,
2
~—sin(~ir~fdp—-~ ir 4
~
~,
-.
--
.
-
-
sin { ~ira ,(t,) } sin { ~ira
I) ----
(3.12)
2(t2) ~(f--a ~(t~)---a2(t2 )+ I)
where A’(J, ~) is the residue of A(J, n) at tile pole at f = n ,+n2--- 1 and R~(f, I) ~5 the double Regge pole residue of F~(J,f, = n,, J2 = n2, a1, a2). Note that we are using the fact that the Regge 4’ trajectories acquire is not singular at f, imaginary a,(ti), f parts at the lowest thresholds in the I, and 12 channels so that F~ 2 = 02(12), but instead has poles at f, = a~(i,),.1’2 = a~’(t2). (3.12) already resembles (3.1) and we can improve the resemblance and simplify (3.12) by using two-particle unitarity m tile I, and 12 channels. It can be shown that we may write 2 4 2 1/2 2 1/2 4.\~N~ (3.13)
(i-~)
(~
(i-_~_) R~a~
H,D,I. Abarbanel et a!., Reggeon field theory.’ formulation and use
t
1
141
PLANE
—
C TWO PARTICLE
-~:
THRESHOLD
Fig. 3.7. The t~or t2 plane in the four particle phase space showing how the t~ integration is deformed to enclose the two particle threshold.
in (3.12), where N~(f, t) is defined from R~(f,t) by factorizing off the two-particle/Reggeon vertex functions, if we replace the i, and 12 integrations in fdp by contour integrals enclosing the two-particle thresholds in those channels see fig. 3.7. N~ is simply the analytic continuation of N~around the four-particle thresholds in the t-channel. If we make tile further simplification of absorbing ~JA”~ into we obtain for ImF(f, t) —
‘~
1
44! sin(~irJ)f Mt
,‘~,
t 2)>o
ir2
i/2j’.~
1~.
di, dt2
\~,~
r..ir Ir t,,av~ \ p..jr4 71
1”.ar,’.”~
‘~
‘2J
~—~-~-
.-———-_____
(3.14)
.
t sin{~ira,(t,)}sin{~ira2(t2)}(f—a,(i,)—a2(t2)+l)
(3.12) is now clearly analogous to (3.1) except that the Reggeon-particle “fixed-pole” at J a(t,)—-i has been replaced by the two-Reggeon fixed-pole atf= cm,(t,) + a2(t2)—l. The Reggeon-particle branch-point occurs in (3.1) when the fixed-pole hits the end-point of the t,-integration this gives a branch point at 2)---i. (3.15) f a((’fi—m) There are various end-point singularities generated in (3.14) but, apart from the two-Reggeon branch-point, they are not singular on the physical sheet of the full amplitude near t = 0 [48, 491. The two-Reggeon branch-point is generated when the pole atf= a,(t,)+a 2(t2)—l is tangential to the integration boundary at X(t, t,, t2) = 0. The position of the branch-point can therefore be formed by solving the Landau equations —
a
0
(3.16)
±(~,[a,(t,)+a2(t2)]+~2~(t,t,, t2)) = 0
(3.17)
(ti,
[a ,(t,)+ct2(t2)1 +i2X(t, I,, ta))
=
at2 together with the conditions f,—a,(t,)—a2(t2)+l
=
0,
X(t,
I,,
t2)
=
0.
If the two Regge trajectories are identical, the solution of these equations is t,
(3.18) =
~ t, with the
branch-point occurring at fa(2)2a(lt) 1.
(3.19)
For non-identical trajectories the branch-point trajectory will be more complicated. Since the2,integration in (3. 14) is two-dimensional and the trajectory functionsinare(3.14). complex for it is not straightforward to take the two-Reggeon cut discontinuity HowI,, 12> 4m becomes much easier to work with if we analytically continue it to I < 0 [49] The ever, (3.14) details of this continuation are complicated but the net result is that if we extract the “threshold” .
III).!. A harbanel ci at. - Reu’i,’eon field theory: formulation and us
142
heilavior of ~ \/7 -
Ar)
if
fi=(’T ‘
1) at X( 1. 1,. 1/
~
til
rX( 1,
12) = 0
by writine
1,, t~)1c’ o~ I
02)2
J
‘
i~
-~
then We can simplY rotate the integration contottr in (3.14) to (lie region X(t, 1~.t~)K 0 and Write [A(t, t~,t2)J
‘2
U A(t,
1,, 12)1
(3.21)
‘2,
(For the purpose of studying the two-Reggeon bratlcil—poillt we can igncre tile fact Illat the original colltour encircled the positive I~and ‘2 thresholds.) i~ort K 0. then we obtain f’roni (3. I 4)
f
(,~
dt,d121 ir2~ 4!~<~
X(i. 1~,
~
12fl
-
‘
(‘~
I
a(f)
sIn/a-I(tIs1n~sira2(t2)U,J
~2(/2)+l)
.
--
(3.22)
If tile trajectory f’unctions are real tor negative I, and 12 (lien the collision of the pole at f = a,(t, )+a2(t2)-- 1 with A( t, 1,, i~)= 0 is 110W straightf’orwarul see fig. 3.8 and if and are both non—singular at .J = we obtani Ironi (3.22)
disc,I’~(.J,I) =
sili(~ir.I) 2~4! A
—~
J
5(.I
ult~(115
-
cu1(I,)-
I
a2(t2)+l)(~(’~ -,
-
.
‘
>3,23>
,
Alt, t~,t2) I’ sin( ~ ira )sin( ira2)
The discontinuity we have taken is disc~F(f.t)
=
F(J—~ic,t)—-F(J+ie, t)
(3.24)
since tile sign of this discontinuity is the sign of the contribution of the cut to the total cross-seetioll. To obtain tile complete discontinuity across the two-Reggeon cut we should actually multiply (3.23) by a f’actor of six. Firstly I’Ll, .J,. f2. a. a2) is syillilietric under iii. ~ji. n~alld so we obtain an exactly similar contribution trotil that part of in which Fl,. ~12 K 0. Also the phase-space Jdp we have used is that for f’our identical scalar particles. This was for simplicity. but it means that tile same Reggeons call appear in channels defined by regrouping the four-pat’tides to define new angular momentttm states. Since there are three possible Ways of pairing four particles, by repeating tile f’oregoing analysis using different angular niollielituIll states we obtain two more distinct contributions to tile two—Reggeon cut. ~2
-
---
(3.4)
fl
II ~
“~\~aIt1+a2o2
- I
Fig. 3,8. The position of the f’aetors in eq. (3.22) in the I~. t~Plane. the intersection of rise to the bramic It point at J ~(t ) -
=
Alt,
t , , t2 I and
J --a,)!, (
~2 (i2
(+1
givc~
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
Fig. 3.9. The two Reggeon
143
two Reggeon a,nplitude MA,A’(J, t).
—~
By considering the exact form for the f-plane continuations of the parts of (3.4) for which n,> 0, n2 < 0 or n, < 0, n2> 0 we can show that no two-Reggeon cut is generated [21]. Essentially this is because the helicity-Regge poles and fixed-pole lie on the same side of the helicity contour and cannot generate a pole atf
a,(t,)+a2(i2)—l.
The next problem is to take account of the presence of the two-Reggeon branch-point in ex2, and we shall G~(f, t). c~(J, I) will singular at fin=C~(J, a~,rather = a~, i> l6mcondition for ploit this. To study the be branch-point i) wethan havef to use when the unitarity F(f, f,, f 2, ni, n2). The four-particle threshold discontinuity formula that has to be partial-wave projected is [42,
_______
431 —
=
(3.25)
.
Proceeding through the analogous steps to those above we are finally led to study the twoReggeon cut in a four-Reggeon amplitude Maai(f, I) shown pictorially in fig. 3.9. This amplitude is defined as a double fixed-pole residue of a four-Reggeon amplitude defined by factorization at Regge-helicity poles. (The fixed-poles occur atf= a’,(t’,)+a’2(t~)—1andf= a,(t,)+a2(t2)—l, and the Regge-helicity poles occur atf1 = n1 = a~(t1),I = 1,2, 1’, 2’.) To study the two-Reggeon cut in M~a~ we have to go to the discontinuity relation for the eight-point function [42, ~ —
-
=
.
(3.26)
The procedure of projecting (3.23) and (3.24) onto partial-wave and continuing to complex helicity and angular momentum involves even more complicated expressions than those we have already considered [211,mainly because of the proliferation of variables and indices. Therefore we shall not attempt to give the full details of the manipulations of these equations, but instead illustrate only the formal structure that is used. We can write (3.14) in the form 4(J) = N(f) F(J)N4(J) (3.27) F(J) F where N(f) N~(f, t) and 17(f) is a formal expression for the integration in (3.14) which generates the two-Reggeon cut discontinuity given by (3.23). From (3.25) and (3.26) we obtain analogous equations to (3.27) —
N(J) —N4(J’)=N4(J’)F(f)N(J)
(3.28)
M(J)
(3.29)
—
M4(J)
=
M4(J) F(f)M(J).
The appearances of the same 17(f) in all three of (3.27)—(3.29) is vital. (3.27)—(3.29) are initially obtained for t ~1 6m2 and ReJ> Rea~.If we continue them to Ref < Rea~and use ± to denote a ± ie prescription in the f-plane with respect to the cut we obtain from (3.29) M(f~)—M4(J)=M4(f)17(J~)M(P).
(3.30)
144
lID.]. Abarbanel eta!., Rs’ggeon field theor”; formulation and use
From whidil we obtain
6M(J)
=
M(.J~)---M(J)= M4(f) [~(J~)6M(J)+M4(f) SF6J) M(f)
>3.3 I
and so I ---M~(J)F(J~))~M(J)
=
M4(J) 6[(J) M(J~
>3.32>
=(l---M4(J)F(f~))MfJ~)6l’6J)M(J) which has the
8MCJ)
=
(333)
sOlt,ltiotl
M(J~)~F(J) MCJ).
>3.34)
Similarly we can obtain*
~N(J)= N(J~)6F(J)M6J).
>3.35)
a tid
8F(f)
=
NCJ~)~F(f) .‘VCJ).
3.36)
For t < 0. 517(J) is given by (3.23) (apart from the factor of six) and so writing out (3.36) in full 6(J--a,(t,) -a
disc F(J,
1) =
-~
sin(~if)
f di, di2 [ -
A
<0
2(i2)+l)C~(Jt1) C’~r)(J~ 1) 2sin~ira ,(t X(t, i,, t2)] “ 1)} sin {~ lra2( } -
-
--
--
,
(3.37)
2)
It is clearly straightforward to write out detailed versions of (3.34) and (3.35) giving the discontinuities across the two-Reggeon cut in N~r)(f~ i) and M~A~,l(J, i). There are several important properties of (3.37) which we should note. The most immediate point is that since C~(f, i) is real-analytic ~ I) and so the sign of the two-Reggeon cut discontinuity is determined. If both a, and a2 are Pomerons so that in (3.37)
J
2a(t)
--
1
a(i)
1
fort
0.
(3.38)
the integrand is positive definite and the overall negative sign persists. A detailed analysis [491 shows that this negative sign can be traced to the fact that the “signature factor” sin ~ ir(f—n ,—n2—- [f +~‘~] ) } in (3.8) is evaluated at a nonsense wrong-signature point where it gives Tile next point we note is that the signature factor sin(~if) in t’ront of the integral in (3.37) ensures that the branch-point is not present (as a fixed singularity) in tile physical i-channel partialwaves that is F(f, i) evaluated at even integer f. Another point is that the factors sin( ira,), sin(~7ra2)lead to the getieration tilet~, Reggeon2),ofX(t, i particle branch-point in (3.37). It is generated when a, or a2 = 0 (i, or = ,n 2) = 0, andf= a, + a2 ---1. Because of this the two-Reggeon cut is able to shield the Reggeon-particle cut from the physical sheet of the f-plane when t 0, so that the Reggeon-particle cut does not contribute to the high-energy scattering. -—-(~
--
1.
--
12
‘~
*‘fl~,~ procedure for isolating a discontinuity is the conventional one used in S-Matrix theory. In particular it is analogous to that used in non-relativistic scattering theory for finding the discontinuity across the elastic cut in the T-matrix, The analogy is drawn in ref. 1501 where the two Reggeon cut discontinuity is discussed this way.
H.D.I. A barbane! eta!., Reggeon field theory: formulation and use
145
2
M
~
Fig. 3.10. The two particle-, two Reggeon amplitude. The net helicity in the t-channel is A 1(t1)
+
A further point with important physical significance is that (3.37) shows that the cut discontinuity is entirely controlled by the fixed-pole residue of N~(f,i). This residue has been defined in terms of the f-plane continuation of F~U,n) by first continuing to Regge poles at = n, = a,(t1),f2 = n2 = a2(i2), and then to the fixed-pole atf= a,+a2—l. However, we can also regard N~(f,I) as the Froissart—Gribov projection of a momentum-space heiicity amplitude which the scattering particles into two Reggeonsamplitude. see fig. This 3.10.isThis amplitude 2,describes t) is commonly referredoftotwo as the maximum helicity-flip because in the Aa(M t 1 and 12 channels n, = J, and n2 = f2 would be the maximum values n, and n2 could take if f, andf2 were physical spins. (In the symmetric notation we have used, if n1 is defined as the helicity of the f,-state then, —n2 must be defined as the helicity of Froissart—Gribov the f2-state.) 2, I) satisfies the usual formula The Froissart—Gribov of Aa(M (omitting i~labels and theprojection signature labels associated with ~ for simplicity) —
Na(f,
t)
=
~ZO
~
Q~A2(z)
ImA~(z,I) + lj
dz
Q~t~2(_z)
z being the center of mass scattering angle. At J = a 1+a2—i, Q~,~2(z) since it is Fa(M2, t)
ImA~(z,t), 2 =
~
—
(3.39)
i)(a ,+c12), and
(z
~(z2_ l)(a,+ct
=
2)Aa(z,
(3.40)
i)
which has no kinematic singularities at z = ±2,1, the second in (3.39) be first written t) and then integral transformed to ancan integral over the as an integral cut around left-hand cut of Fa(M right-hand usingthe Cauchy’s theorem. Thus we obtain the usual expressions for fixed-pole residues of hehicity amplitudes NaUa,+a
2, 2—
i,t)~f
(3.41)
1).
dzlmFa(M
Now the Reggeon particle imaginary part is experimentally accessible in the single particle inclusive process shown in fig. 3. Ii. In the regime where s with t and M2 fixed, the inclusive cross section measures ImF~(M2,0) given in (3.41). Inclusive cross section data provides us with a phenomenological handle on the strength of the discontinuity across Regge cuts [5 11. Our final point about these formulae for the two Reggeon cut concerns the phase space. In the neighborhood of cs -~ f — 1 which is the regime of the P, all factors besides the phase space are smooth. The phase space integration is two dimensional so we take advantage of this by defining -*
146
11.1)1. A barbanel vial., Reggeon field theory: formulation and use
~M2~
_t~ ~=~-
ZERO MOMENTUM TRANSFER
Jig. 3.11. ‘i’he single particle inclusive process in the limit
=,
t..112 Used. [he Reggeon—particle absorplis~’part measured here -i
two vectors q1 for each momentum transfer t~= .1 can be written
561 a,(t,) -a2(t2)+l) -
= 5(1
.
Further we note that
-J (1 -a,(q1fl (I a2(q2))),
tile
liii..I~l.l12
UI is
delta function on (3.42~
which encourages us to write Reggeon energies L’, = I a,(q ) and I: = I .1. Ihe discontinuity across the two Reggeon cut in LIE, q ) is then (absorbing signature aild such (‘actors) (see fig, 3. I2 discEflE, q) X
=
---
2iJ’d2q,
N(L’+ic, E,. q,,
1:2.
2q 2(q- -q,- q (IL’, d 2dE2S(L’ -E1 -L’2) 5 2> 5(EI (I a1(q10) 5(E2 fa q2)N(’~ie.F,. q,, L2.
~2).
(3.43)
where A- is the Reggeon particle amplitude as shown in fig .3. I 2. This discontinuity torillula is identical to that given by (2.32) witll a = 2. It suggests that we treat the Reggeon as a quasi—particleS lit-’ing in [ito s~)acea/Ic! one ti/lie thniension carrying energ~ E(q) = I a(q) when it is on the ‘‘mass shell’’, In this sense (3.43) is (apart from the negative sigh an ordinary unitarity relation giving the two quasi-particle intertilediate state contribution to tile imaginary part of PIE, q). Note -tilat we do not associate an energy Witil individual particles, instead pairs of particles are treated as sources of F and q (‘or Reggeons wit idil tilen propagate. Similar relations can he written for N(E) and M( F) appearing in (3.35) and ( 3.36). Perhaps the best viewpoint ~n tile Reggeon field theories is that they are a conlpact device I’or providing automatically the correct pilase space structure (‘or Re-ggeons and satisfying these Reggeon ullitarity relations [34, 52 It is also possible to discuss tile solution of (3.34) ( 3.37) using “S-Matrix” nlethods rather than tile field—theoretic methods which are tile subject ot tiliS review.Such methods are likely to be limited in use to the situation where (lilly a small number of Regge cuts need he considered. This may well be the situation for tile weak-coupling P (if it exists) but is clearly not the case for the strong-coupling P. (The reader is referred to the next section for precise definitions of weak and strong coupling.) We shall not discuss these methods here. hut instead ref’er thie reader to refs. (50 and 54—56 1 for a detailed discussion of this subject. 541
147
H.D.J. A barbanel et al., Reggeon field theory: formulation and use
E~
= E 2)
discs F(E,q
=
=
Fig. 3.12. The discontinuity across the two Reggeon cut in the particle partial wave amplitude F(E 1—f, t -—1q12). The vertical dotted line across the blob in the left-hand part of the figure indicates a discontinuity in F for fixed q has been taken. In the righthalf of the figure the Reggeons carrying two momentum qj are “on-shell” and thus have “energy” equal to 1 —~j(qj).The vertical dotted line indicates this.
Tile treatment we have given here of the two-Reggeon cut can be extended to all multi-Reggeon cuts [20,21]. The N-Reggeon cut can be studied through the 2N unitarity integral. As we have seen the four-particle unitarity integral is sufficiently complicated to handle technically and the higher unitarity integrals are worse. However, we can illustrate the general structure by looking briefly at the three-Reggeon cut (more details can be found in the GPT paper). We analyze the six-particle unitarity integral using partial-wave amplitudes corresponding to the coupling scheme shown in fig. 3.13. The three-Reggeon cut is generated by Regge poles at f. = n 1 = a1(t~), = 1, 3,4, together with nonsense wrong-signature poles atf2 = + n4 1,f n1 + n2 1 in the amplitude with J2 = n2( n3 + n4). A fixed-pole is then generated at —‘
J = a,(i,)
+
a3(t3)
+
cs4(t4)
-—
2.
(3.44)
When this fixed-pole is tangential to the phase-space boundary at 2 i~’2 t~’~ i~2= 0, i” the three-Reggeon cut is generated. For identical trajectories this occurs at i~2= —
—
—
(3.45)
—
giving a branch-point at = 3a(t/9) —
2.
t~/2=
t~’2= t/3,
(3.46)
In general the N-Reggeon cut is generated by N Regge-pole N—i nonsense wrong-signature poles giving a fixed-pole at
Fig. 3.13. The angular momentum (.T~)and helicity
(n
1) and mass (t1) configuration used in studying the six particle contribution to the partial wave amplitude. This is the way to study the three Reggeon cut generated at J a1 (t1) + a2(t2) + a3(t3)—2.
=
148
.1
lID.!. A barbanel etal., Reggeon field theory: formulation and use
a,(t,)+
,,.
a~r(t.~)(N
I)
(3.47)
which collides with the pilase—space boundary at 112
t~t2
-
..
-t~2=
13.48)
0
and, (‘or identical trajectories gives a branch—point at -
1
=
N[a(t/A°)
II.
(3.4~))
l’o obtain tile total discontinuity across the N-Reggeon cut ~t is, ot’ course, necessary to add tile colltrihutions from all possible partial-wave coupling schetnes. ‘File lorIll of the discontinuity is analogous to (3.37) being expressed as an integral of a sum of products of two multiple fixed-pole residues one evaluated above the N—Reggeon cut and one below (boundary values of cuts in sub—cilannels must also be specified as for momentum space (hilitarity). The ntost important point is that the discontinuity call be expressed a~tile N— ( quasi—)particle contribution to the quasi— unitarity relation f’or the scattering of quasi—particle Reggeons, The sign of the .-V-Pomeron cut is 1 )~‘ ‘ because tile cut is generated by (N—- I ) nonsense poles and there will be (A-’---- 1) factors of I coming from the corresponding signature factot’s. ‘Ihe extension of Reggeon unitarity to production processes is coiltpitcated by tIle necessity to perform ilelicity as well as angular tlionientthm coiltihlthations. Reggeon unitarlty will initially (Ic— termine Reggeon field tileory rules for production procc~eswhen all azimuthal angles ( the cottju— gate variables to ilelicities) are large. i~or2--*N processes tile azimuthal angles can not be taken large in the physical region and analytic continuation must he used to obtain the rules . I-low— ever, (‘or inclusive processes this is not necessary and it can be siloWll tilat Reggeon unitarity dcternlines tile Reggeon rules t’or the triple Regge limit of the one-particle inclusive cross-section in a straightforward way [58. FilIally WC nlention briefly some technical points that we ilave glossed over in our derivation of tile two-Reggeon cut discontinuity formula. Firstly, because of physical region singularities of the six-point function in channels overlapping the I, and [2 channels, the amplitude --
157
I
5~1
F(J, J,, J 2, n,, a2, t, t~.t2) will not be a single analytic function of t, and
throughout the physical region. This problenl can be ehminated by using tile Steint.lann relations together with a multiple dispersion relation to separate out that hart of the amplitude having singularities in the 1, and ~2 cilannels. Secondly it can be shown that even when we separate out that part of the amplitude, tile Froissart----Grihov amplitudes 1”~(J,n) only satisfy tile (‘arison colldition as a tunetion off for part of the physical region ranges of t, and i2. Fortunately we caii cover the whole physical region by using all possible partial-wave coupling schemes. Further tile regioli in which F~(J,n) does satisfy the Carlson condition includes tile region where the two-Reggeon cut ,s generated. As a final point we note that if’ the Pomeron has a~(0) = 1 - tilen a~(1) will ilave a cohllplex part for t < 0 and so we cannot write tile two-Porneron discontinuity as a real iltegral for I < 0. as in (3.37). Therefore, there will be n~range oft where tile discontinuity can be expressed as a simple real integral. However, in a perturbatiotl solution of tile discontinuity formulae (such as the Reggeon field theory) which begins with bare analytic trajectory functions, perturbative unitarity will involve real discontinuity formulae. 12
H.D,I. .4barbane! eta!,, Reggeon field theory: formulation and use
149
4. Theoretical developments in Reggeon Field Theory 4.1. Opening remarks This section is devoted to a discussion first of the P pole and all its cuts taken by themselves, and then the consequences of those P interactions for the structure of secondary trajectories. It is useful to begin by explaining why the P and its interactions must be studied first. The essential reason is contained back in eqs. (3.47)—(3.49), for the position of branch points in J. When one has multiple exchange of P with cs(0) = 1, the cuts and pole collide at t = 0. This is certainly true for P’s alone. It is equally true for one secondary trajectory with LSR(O) * I when it is exchanged in the t-channel with any number of P’s. In that case the cuts and pole collide at t = 0 at J = aR(O). The notation developed in section 2 is useful in this regard. There we learned that in the Reggeon unitarity relations a two-momentum q and an “energy” E = 1—f were conserved for each Reggeon. For Reggeons ~n shell the E, q relation is E(q)= I —a(q)
(4.1)
so that E(0)
= 0 for the P. This is just the E, q relation of a quasi-particle with no energy gap. Every other Reggeon has an energy gap of z~= 1 —ct(0)> 0. The addition of any number of P’s to some exchange process in the i-channel, then, does not alter the threshold for that process in the
E plane since at q = 0 a zero energy quantum may be added. The addition of any other trajectory to an already existing i-channel exchange alters the threshold (position of branch point) because finite energy, is required at zero momentum. All this is analogous to the positions of thresholds in conventional quantum field theory when massless particles are present. The P is analogous to a massless particle, and the solution to the problem of the J plane singularity arising from multiple P cuts is the solution to an infrared problem. Now we understand how the thresholds of multiple P exchange stand out and further we see why in the case of P alone the thresholds all occur at J = 1 (E = 0) at t = 0 (q = 0). From a physics point of view it is useful to recall at this juncture the significance of J 1, t = 0 before we go on to the P itself. The Froissart—Grihov formula for signatured partial wave amplitudes gives us a connection between the absorptive part of two body amplitudes AB -÷ A’B’ and the i-channel partial wave amplitude F(J, t). Schematically, ~,
F(J,
—f dss~~’Abs TAB,A’B~(s,t). 2~
t) =
(4.2)
Furthermore it is known for a variety of processes, called diffractive that ‘~AB—~A’B’(~) s large ~2a(O)~~
2(log ~
(4.3)
with cs(0) 1, ~3small, and gAA’ or hBB~numbers which depend only on the properties of A and A’ or only on B and B’ respectively. In short, the behavior of diffractive amplitudes 1. is almost energy independent; cs(0) 1; 2. shows factorization of vertices as expected from an elementary t-channel exchange. On further inspection of the amplitudes which have diffractive behavior one learns
150
lID.!. A barha,ii’l et a!,. Reggeon field theory: formulation and us,-
3. 110 ciuantum numbers are eXcllanged in the I—channel. From an operational this is the Pomeron. Equation (4.3) translates into tIle .1 plane as
F6. 0)
=
~
[J---a(
0)1
point
of view
(4.4)
‘~
that is. a brancil point at .J = a( 0) = I whose detailed nature governs tile behavior ill log s of’ elastic or quasi-elastic cross sections. The optical theorem relates tile beilavior of elastic amplitudes to total cross scctiohl5~lot (4.3) we Ilave AI3~ ~ ,.~ /1BB5 o(O) T ~S, ,s iarge ~AA
- ‘‘I ‘,
,,
.1/3
-
—
and we know fronl tile Froissart hound thai ct(0)~ I.
(4±11
and
~
2, wilell~(0)
1.
(4.71
Tile experimental beilavior of amplitudes tilen chooses a(0) = 1 as tile point where we ought to concentrate our attention. The result of’ sunlming all colliding cuts at i = I will be to determine the fine structure in .1 at .1 = I and thus tile logs behavior of cross sections. Eurtherlllore any singularities ill .1 away from J = I will give contributions to cross sections whicll are negligible by powers ofs wilen P can be present~that is. vacuum quantum nunlbers are allowed in the I—channel. So we turn our attention to tile problem of’ a P singularity ill tile J plane which lies at.! = 1 at I = 0. 4.2.
Formulation of the RET
We have learned in sections 2 and 3 that Reggeons propagate in two space, x, and one tillle, 1’. dimensions. These are conjugate to the two momentum, q. and energy, 1 --f. and are actually the impact parameter and rapidity of the Reggeon. The RFT describes this by a field amplitude ~p(x,r) for finding the Reggeon at x and ‘7’. In the absence of interactions tile Reggeon field satisfies the SchrOdinger equation
/
i— 1p(x,r)
a?-
~l
/1 --
\\
cs0{--v)) ~p(x.r).
(4.8)
\h
where ~0(q) is given by tile choice of bare ellergy momentum relation; i.e., tile non-interacting Reggeon. This function ~0(q) is a priori at our disposal it specifies the nature of tile P or other Reggeon before interactions are accotinted for. We lllthSt turn to physics to help us here. The appropriate physics is tile connection between bound states in the t-cilannel atid lai’ge s behavior in tile S channel. Fifteen years ago it became clear that the Regge pole at .1 = a(i) represents a bound state of hadrons whose mass squared has been continued fronl ln~> 0 to the value i occurring in the scattering process, and, at tile sanie time, whose spin J~ilas been continued to ~(t), [60] . This connection betweetl bound states and crossed channel large s behavior is one of the deep attractive features of Regge theory. Now, what is important l’or us is that it
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
151
follows from i-channel partial wave dispersion relations that a(i) is analytic at i = 0 unless at i = 0 there is a collision of this pole trajectory with some other singularity in the f-plane [601. For a Reggeon in isolation, then, one expects cs(t) to be regular at i = 0. For the purposes of our present concern, which is the behavior near q = 0, J = 1, we may expect that the non-interacting Reggeon is adequately represented by (4.9) cxo(q) ct0—ct~q2, where ct~and ct’ 0 are parameters which it may be possible to extract from a detailed knowledge of the way in which a Reggeon is constructed in the “correct” underlying field theoretic or S-matrix model. The free energy momentum relation is then 2 + (I—ct E(q) = cs’0q 0), (4.10) which is like a non-relativistic quasi-particle with mass gap (1—ct0) = z~and mass = (2ct~”. The mobility of a Reggeon in x, r space depends on its effective mass (2ct~’, and the smaller ct~,the less mobile is the Reggeon. Since ct’ is known to be smaller for P by about ~ compared to, say, the p trajectory, we may expect the P to be very ponderous. If we now associate a field with this Reggeon, the free action will be A0 fd2x dr £0(x, r),
(4.11)
with E0(x,r)—p~’(x,r)
(4.12)
~
2 the Lagrangian density. The next issue is tile interaction.
To determine what interactions allowed we studied turn to to models which generate inter3 field theoryaremodels were extract the RFT rules.Reggeon Other natural actions. inInwhich sectionto2examine p models this question are multiperipheral and dual models or perhaps more realistic field theories. In each of these it is possible to have four and five and higher point P interactions as well as the simplest triple P coupling discussed before. Four P couplings arise, for example, in a ~p3field theory, where the P is generated via ladder graphs, as in fig. 4. 1. Here two P’s can scatter, or one P can make a transition into three P’s. In any given model each N P coupling, N = 3, 4, ..., is in general a function of the energies (I —f,) and momenta of the P’s involved. This means that if we are to incorporate directly the content of an underlying theory into a RFT, we will have to construct a field theory which is both non-local and involves an infinite number of coupling functions, one for each independent n P -~ m P transition. For 4 P couplings, for example, we will have two independent functions E 14(x,
r)
fd2x,
2x dr, ...d 4dr4 ~
r,)~p’~(x2,r2) p~(x3,r3) ~p(x4, r4)+h.c.}
X F,(x—x,, r—r1,
...
x—x4, T—T4) + p”(x,, r,) p~(x2,T2) ~p(x3,?-~)~p(x4,
X F2(x—x,, r—r,,
...
x—x4, ‘r—r4).
T4)
(4.13)
Unless there is some simplification, we are faced with the prescription of an infinite number of arbitrary functions.
lID.!. --I harbanc’l et a!. - Ri’~geonfield theory. formulation and use
152
_~i~ _j I~~±ii
j-~
~
-~
4.1. }-~uUrP couplings allowed in a Rl’l’ of P’s. Thes- are abstracted troii~dOriA le~Oman graphs, or the niultiperiphieral node!, my theory hiavine
Reege pole’s.
ultimately interested in the in frared ( L’, 0. q, 0) behavior of ally Rfl’ we write down, as a f’irst step we expand each of tile coupling functions about this point. We begin the discussion by retainillg only the first non—vanishing terms. This reduces the non-local theory to a local theory with coupling constants and maybe a f’ew derivative interactions. ihis procedure applied to 3 P and 4 P ititeractions gives Since we are
--
~ 3~x, r) + he.
--—{~(x.r)
-~-~
~(x,
r)2~(x. TY
(4.14)
f’or the interaction Lagrangian. This has illlproved tile situation in tile sense that we 110W ilave a local theory with an infinite number of’ coupling constants. However, there is Ohle hll0~C level of heuristic argument we may use to eliminate from our prelilllinary consideration any but tile 3 P coupling. Essentially we argue that to emit and absorb P’s inside any diagranl of’ our RFT costs a phase space factor dE d2q each time. We are ultimately interested in the E 1 0. q. 0 hmit of’ the RFT so each additional phase space t’actor decreases tile importance of any given grapil. N P couplings with N > 3 require more phase space than N = 3; therefore. one may ignore tilehll. Later in this section we will return to reconsider tile wilole matter of iligiler point P couplings aild derivative couplings, and justit’y this heuristic argument. We must incorporate a final point troill sections 2 and 3 before we begin tile disctisSlOhl of tile various attacks on RFT. In those sections we learned that tile two P cut carries a mi~ussign relative to the contribution of a single P exchange. A grapll like fig. 4.2 IlILIst cohltaibl a Illiiltis sign. This is naturally incorporated as part of tile Fevilman rules of the RFT if tue coupling A0 in (4.14) is pure imaginary: A0 = ir0. Tile RFT now to be discussed has tile Lagrange fuilctiohl -~
-~
H.D.l. A barbane! eta!,, Reggeon field theory: formu!ation and use
153
Fig. 4.2. A Reggeon graph contribution to the elastic particle amplitude. This contains the two P cut and must contribute negatively to the total cross section. We arrange this by writing the triple P coupling to be purely imaginary ir.
=
i
ir0
‘
2
3r
(4.15)
2
The solution of this field theory with an anti-Hermitean interaction in the infrared limit is the central problem of RFT. The spectrum of the theory determines the Pomeron. This was first written down and studied by Gribov and Migdai [61—631.it is with their work that we now concern ourselves. 4.3. Early developments in RET for
P
To review in detail the work of the Leningrad school would be a monumental task. Leaving this pleasure to the historians we will sketch the essential results of their calculations. To begin we established some notation that we will employ frequently. The objects of primary interest in RFT, as in any field theory, are the Green’s functions for the m P -+ n P transition T~1, ...y,~,T~) =
(0lT~(p~(y1, Tyt)...~(Ym, r~)p(x1,r~1)...ip(x~, r~))iO>
(4.16)
where 0) is the no P state. There is no crossing in RFT so both n and m must be given. The Fourier transform of G(~~.m) is of interest also (fig. 4.3): n
n+m
E1
—
ii
~
q~
)62(~
—
jn+1
i=1
2ym ~
exp{—iq
fd2xi dr~1..d
n+rn
fl
E1
~
q1) G(nm)(E1, q1)
(4.17)
=
1n+i
1x 1+iE l?-x1
—
...
~
Tym }G(n.m)(xi,
r~1,.. .y,.~,~
The spectrum of the theory is specified by the zeroes of the inverse P propagator
En+m,~’n+m
Fig. 4.3. The notation for a n —~m Reggeon Green’s function.
lID.!. .1 barbanel et a!., R,-~m,’eo,,field theory: formulation and us,’
154
1-ig.
4.4.
t 1,1> is the html! P propagator md I~‘~ ~ the Imiple
fhe Schwinger Dyson equation for the Poimmeron proper wit energy. G I’ proper s’ertes unction,
F~’”~(E, q2)= IG(11)(E, q2)L.
(4.i~)
It is coilvenient to disctiss the one P irreducible proper vertex f’unctions [0’ tim)~L,, q,) gotten by amputatilig tile external legs of’ tile one P irreducible part of’ (~°‘ ~. l~I .1) or I - ~ and 1~ .2) the 3 P vertex tunction, are promilleilt ill tile analysis of Grihov and Migdal. I
In the non-illteracting theory
G~U(E,q2)
=
--~--
~(i
takes tile valtIe
,
~---
L--a 2--0q
.i)
(4,1’))
~O+lC
where the ie prescription means only retarded propagation. Its origin is tile requirenlent that. ill tile Sommerfeld—Watson integrals def’illing tile tileory. tile iultegration contour in .J lies to tile right of all singularities in F(J, t). When all interactions of P’s are summed, tile inverse propagator takes the form
iFt”~(E,q2)
=
E—ct~q2-—~
2), 0---~(E, q
where the proper self energy
~(E, q2)
=
~
must
satisf’y
(4.20) the Sch winger—Dysoll equation (t’ig. 4.4)
(q--q’)2) 1
wofd~ dE’ [F( “1~(E’,q2) [t”°(E--E’, 2 (_~)
X FD~(E’,F—F’, q’. q—q’) The Soviet workers discussed
+
ill
(terms due to i’~ ~
-i
with ii ~ 3).
(4.21)
detail two possible solutions to tile RFT we have Posed
(a) The weak coupling solution. wherein tor snlall L’ and q2 ~w.cfL’,
q2) ~ F, q2.
so that the resulting propagator is very much
4.22)
tile
the P spectrum is a simple pole still, constrained iF~’~(E, q2) E---ct~q2. Eq2
same as tile unperturbed propagator (4.1 ~) and 110W to have ~ = I -
ct(
0).
50
(4.23)
=0
This cannot come about if the higher order vertex functions are arbitrary since they are linked to 1) via the representation. In particular it’ l’~(~2) the three P vertex fuiiction took 011 tile -
-
constant value of lowest order perturbation theory, tile right-hand side of(4.2 I ) would be infrared divergent. N1’2~must therefore vanish at the inf’ra-red point where all of’ its arguments are zero.
H.D.J. Abarbanel eta!., Reggeon field theory: formulation and use
~(
~
155
~
Fig. 4.5. The graphical representation of the triple P vertex function ~
2)
Gribov and Migdal suggested that the zero of pO~2)should be sufficiently strong that the infra-red regions in (4.21) and the analogous equation for F~,2) shown gr’aphically in fig. 4.5, be completely suppressed. This requires the leading behavior of p~,2) to be analytic at the infra-red point, since the only possible source of non-analyticity the infra-red regions are suppressed in the integral equations it satisfies. Therefore —
q
—
2+c(q~+q~) + higher order terms,
(4.24)
1, E2, q2) = a(E1+E2)+b(q1+q2) and the detailed form of F0’t~is then q2)
iF~’0(E,
=
E—ct’ 2 0q
+
(aE+bq24CE/ot’ 2 , 0) log(~cs~q2_E)
(4.25)
—
l6irct
0
where a and b appear in F~~2)and contributions further terms when in (4.21) negligible[20] in the 2’2~are also from required to vanish theirare arguments infrared limit. Note that [~(i,3) and F’ are zero if (4.24) and (4.25) hold. (b) The strong coupling solution [63] in which for small E and q2 ~
q2) ~ E, q2
(4.26)
and both the propagator and vertex function have a scaling form for small values of their arguments
iF~’~
—
(_E)Y~~(q2/(~~E)Z),~
(_EY~F((~)Z~ (E)Z’
(_E)Z’
)
(4.27)
where 3’y+z—2~z= 2. In this case the infra-red regions of the Schwinger—Dyson equations are not suppressed, but instead the scaling forms of F~’,’)and ~ combine so that these regions scale consistently throughout the equations. The weak coupling solution leaves the P pole essentially unchanged. The gap ~o (or intercept of the P pole a 0) is shifted to zero (or ct(0) = 1), but basically a pole at t = 0 is left a pole. The three P coupling function, however, changes quite drastically. In lowest order perturbation theory one has 312, =
(4.28)
r0/(2ir)
while the fully interacting solution to the RFT vanishes at E 1, q, -÷ 0, as in (4.24). Such a phenomenon is possible because of the imaginary triple P coupling. Formally, the full [~i1,2) is defined by an infinite series in r0 with terms of alternating signs, so a cancellation or extinction of the lowest order term is possible at isolated points in momentum space. Physically one argues that P exchange is absorptive, a fact which is embodied in the imaginary nature oi’ the coupling
lID.!. A harbanel et a!., Ri s’gm’on field do ore - formulation audits
156
~
I ig.
4.6. The integral equation br lime triple P vertex unction. i~ .21 ~ time Iwo P irreducible Itelhe S.mht,’Ier kernel.
three P’s. and that tile Shill of all absorptive corrections ( f’ig. 4.5) cancels 1111 the leading colislan I ‘[‘his solution is also called a quasi—stable I’ontcroii because the rate f’or the tieca of’ a P 11110 t ~ P’s vanislles abnorillaily rapidly as energy tends to zero. [‘his argu nlent of (ribov and M igdal em I was tile earliest argument for the vanislling of’ tile triple vcrtcs. The dynamics underlying a quasi—stable P was later studied ill a ladder graph Illodel iii RF I b~ Bronzan 1641 . He wrote an integral equation (‘or 1A .2) of tile s~itihol c 101111 (see fig. 4.() of’
I
P
F2)=ro+
l2)G,G.1)I~(22) (421)) tile two P irreducible part of [(22) It is the RF’I’ counterpart of the l3ethe S~tInetc where V~ kernel encouiltered ill conventional field theory. File model coilsisted of appro\illlatlne irreducible V~2’2~ by the one P excilange graphs of f’ig. 4.7. Tilis yields [~2) as an ilfinite series of RIT ladder graphs. Tile potential, y(2.2) ill this model is very singular ill tile infrared limit. Ihen (4.20) can hold only if LII 2) vanishes as L’~,q 1 -÷ 0 from almost all directions, ill tileaectllllulate weak eouphng t .1) all ot as wilicil at solution. There is also shakes an infinite of new P poles G F = 0. This last result one’snumber confidence ill the weakillcoupling solution. It is true tllat bare vertices have been used ill fig. 4.7. so V(22) is probably overly singular. On tile other ilalld, if (4.24) holds, and complete vertices are used ill t’ig. 4.7. Vt22~~5 11011 singular. and there is 110 reason for Ft’ i) to vanish. Either way, tile weak coupling solution does Ilot occur iui detail in this model. Tile weak coupling solution is subject to other difficulties. One line of argtilllellt notes that il~ the P is a pole passing through j = 1 = 0. tilen tile vanishing of [(1.2), which may be demonstrated in marvelous generality using tools wilieh blossomed ill tile study of’ inclusive reactiolls [1 7 requires tile vanishing of large numbers of other P couplings at t = 0 1 I S , including tile coupling to particles [19] which is supposed to set tile scale for asymptotic hadron total cross sections. This apparent disaster has been circumvented by arguments by Grihov [65 I and by (‘ardy aild White [52, 661 . The former examines tile vertex function for two iladrolls ailCi a P when tile hadrons are considered as composite systems. ile finds tilat tilere is a very neat cancellation title to P interaction with the constituents SO that the particle particle P coupling Ileed ilOt vanisil at t = 0. One of the conclusions he draws, however, is that all hadron cross seetioiis illtlst he asyntptotically equal as well as constant. If tilis is indeed the case, then the asymptotic regime where
22>
15
.
t
.
this takes place is a long way away. Cardy and White pointed out that if’, as in Bronian’s model, an uifra—red singular
[(2.2)
was
responsible for the zero of [~,2) then the decoupling arguments based on the relation of’ pole contributions in inclusive cross-sections [71 via the inclusive sum rules, woLlid not go tilrough. This is because the singular potential will also contribute to the vertices appearing iii cut contributions to the two-particle inclusive cross-section. Consequelltly tile cut coiltribLltioilS are enhanced auld not
lI.D.I. Abarbanel et al., Reggeon field theory: ,fbrmulation and use
Fig. 4.7. The lowest order contributions to
157
vA2,2)
suppressed relative to the pole contributions. The sum rules can no longer be used to relate pole contributions. Therefore tile vanishing of the triple-P vertex no longer requires the couplings of the P to other Reggeons or to hadrons to vanish. A further point noted by Cardy and White was that if the complete two particle/two P amplitude is separated into its one P irreducible and reducible parts then the one P irreducible part satisfies the same integral equation as ~(1,2), that is (4.29), except that the inhomogeneous term is different. Therefore an infra-red singular V(2,2) will also produce an infra-red zero of the one P irreducible amplitude. The leading contribution of the two P cut to the total cross-section then comes from the one P reducible amplitude that is the cut couples through the pole and factorises, hence 2). (4.30) u~(s) ~ + O(1/(logs) —
However, as noted above v~2’2~ is only singular in Bronzan’s model when vertex renormahisation is ignored. In fact since we have no complete weak coupling model we cannot say whether a singular [7(22) is the answer to the decoupling problems. If it is, it seems that the higher-order p couplings must play an essential role in producing the singularity. The weak coupling or quasi-stable solution to the P problem in RFT is attractive for a number of physical reasons: (1) It is rather simple; (2) It has an immediate implication for the detailed processes in the s-channel which produce it; namely, some sort of generalized ladder or multiperipheral graph. The single particle inclusive spectrum will have a rapidity plateau and the cross sections for n particles, an(s), will be more or less that of the multiperipheral model with small corrections due to the two P exchange cut [361. One may hope to parametrize the cut in terms of a small number of undetermined constants. Since the weak coupling solution satisfies t-channel unitarity and apart from the decoupling problem seems to be otherwise self-consistent, it is certainly attractive. We will argue, however, that the weak coupling “solution” is not the solution to the problem set by eq. (4.1 5). The strong coupling scaling solution was not favored by Gribov and Migdal [61, 631 . Their argument began with the observation that near F = q2 = 0, one has iF~~1)(E,q2) =
—~(E,q2).
(4.31)
Let us accept eq. (4.31) and then examine eq. (4.21) together with the Schwinger—Dyson equation for 17(1,2). We now substitute 1”~’~-* 1”~’~’) = This transforms the Schwinger—Dyson equations into those for a strong coupled field theory with Hermitean coupling. (The role of eq. (4.31) in this is that when it holds, changing the sign of J”0~1) is equivalent to changing the sign of ~.) For the Hermitean theory we can invoke the Kallen—Lehmann representation to learn that ImF(1,1) has a definite sign. The sign is such that the contribution of fig. 4.9a to cr~is negative. Clearly this is unacceptable and were this the way a strong coupling, scaling form for F~’ q2) is in fact achieved by the RFT we consider, one would be forced to reject it. _p(1~~•
.i)(E,
11,1),!. -1 barbanel y’t a!.. Reggeon field t/leors’: formulation avid us,’
158
Development beyond the work of Gribov and M igdah ilas been possible because there is an alternative infrared behavior to tilat of eqs. (4.22) and (4.26)
~(E. q2)
=
E--a~q2---~>-o’(E, q2).
u(Ii’. q2) ~ F, q2
(4.32)
.
The hare P pole is extingtliSiled, amid a stronger singularity is present in G~I ,~at I: = q2 = 0. 11115 singularity represents the confluence of the P poie and cuts at E = q2 = 0. (For q2 ~ 0. there is still a P pole.) Note that the weak coupling solution has a pole as tile leading singularity at = q2 = 0, wilile the Gribov Migdal strong couphng soltit ion, eq. (4.3 1) has a weaker singularity tilan a pole. We now turn to what may be termed “recent developments’’ of RF’[’ to see how eq. (4.32) enlerges. 4.4. Recent dei’elopments in the RF’T~.irthe /‘onieron In tile work of Gribov and Migdai 16 1 63 I it was assumed that tile interacting Pomeron has = I = 0. Because there is no energy gap. one must s’itller SUill the full perturbation series to learn tile infrared behavior of ~ or Ilave compelling argunlents for any omissions. Faced with this. Grihov and Migdal studied the Sellwinger Dyson equations for the complete proper vertex functions. Tile solutions they found were tllerelore not calculated dii’eetlv from tilt’ Lagrangian, eq. (4.1 5). but were based on self—consistency conditions. Tilere is another way to calculate the infrared behavior of tile which avoids perturbation theory in the three—P coupling. Tilis approach uses tile constraints of tile renormalization group to yield the allowed forms of tile ~0. m) ill certain regions of’ E~.q 1 phase space. This approach was used by Migdal, Polyakov, and Ter-Martirosyan [251, alld by Abarbanel and Bronzan [341. In describing it we will use tile notation of tile latter atltilors. although tile Soviet results are identical where they overlap. The pililosophy of the renormahization group approacil to tile study of quantulll t’ield tileories is described at some length in summer school lectures by Coleman [671 amitl by Abarbanel [65] Here we confine ourselves to the outlines and concentrate on tile results for tile P. Our RFT has in tile Lagrangian of (4. 1 5) four parameters: tile scale of tile term involving ~ which is taken to he I the slope parameter a~ the bare mass gap ~ and the bare coupling coi’stant r0. When the full proper vertex functions are evaluated they again may he parametrized in terms ot’ four numbers: I. a’, & alld r which replace the bare parameters. These nuni herscon2 derivatives and the value of f’otir 2) at some are specified by the value of of [~1) F, qand its FWe andfollow q venient. but arbitrary. point space. ref. [34] by taking L’~ ---E~.< 0. alld - - a(0)
RFT
~(I
q
1 = 0 as this point. Pilysies dictates that a change in /i’,~ E,v + &L’N must have no consequences (‘or tile physical content of’ tile RFT. We can compensate for and the rcilange in L~.by tile substitutions 6FN/EN. r + ~E, a a + ~ ~LN/Erg, ~ -÷~ + ~ I + Y 5i/Ey.. which are ellosen in such a way tilat tile vertex functions computed from our Lagrangian are unchanged. How is this done in practice? The fields ~ which enter the Lagrangian have their scale (nomnlallzation) altered by the interaction so that the unrenormalized ~ becomes --~
-~
—~
--~
I
(433) while a~, and r0 become a’, andr respectively. Tile ~~mm0 expressed in terms of~u are ftmctions of a~, r0 and a possible cutoff to define tile integrals in perturbation theory. Tile [(nm) ~
~,
~
H.D.J. A barbanel et al., Reggeon field theory: formulation and use
159
expressed in terms of p are functions of a’, ~, rand EN. The relation between them is 2F(n.m)(E. q., a~,~ r F(nm)(E. q,, a’, ~ r, EN) = z(n+m~ 0, A)
(4.34)
where A represents some form of cutoff. Since 1”~~,m) has no knowledge of EN, it must not change when EN -÷ EN + 5Ev. 17(n,m) however, must satisfy a a a a (n+m) ,~ [EN —+~(r,a )—+~(r,a’)~ ~+~(r, a ) --~-~y(r,a )jF(n~m)(L. q~,a’, ~,r, EN) = 0. ‘~
~—,
(4.35) This clearly puts a constraint on the way the parameters can enter 17(n,m), The P problem requires z~= 0 = 1 —a(0) for the fully interacting P. This, as in tile earlier Soviet work, is taken as given and is not derived. Clearly this requirement demands a special relation between a~,r0, and z~.We will return to that relation. For the moment, however, we imagine that this relation can be and has been arranged, and continue to seek the consequences of a theory with ~ = 0. The solution to the constraint (4.35) known as the renormalization group equation is —
q~,g, a’, EN)
=
—
F~a~m)(E1, q1, ~(—log~),~‘(—log~)) exp
f
~‘
[1
-
~
(4.36)
—loge
where we have introduced the dimensionless coupling 2E~j’2 g = (a’)~ and the auxiliary functions ~ d~/d17= —f3(g~(rj)),
(4.37) and ~‘(r~)satisfy
,~(0)= g
(4.38)
and 1
d&’(r~) ____
=
a(~) d~
1—
~(r~))
~ ~,
a(~)
z(g(~)),
a(0)a.
(4.39)
In these we find the crucial function p3(g) which measures the response of the dimensionless coupling constant to a change in the normalization point. The utility of the renormalization group approach is now explicit. We want to study [‘(E 1,q,) as F. -* 0 or in (4.36) as -~ 0. This means we need to know ~(i~) as ri = —loge + For this we need to know j3(g~(r~)). The key observation is that if j3(g) = 0 at g = g1, say, then g~(r~) approaches g1 since d~/d~ vanishes there. Whether ~ goes to g1 as ~ ± depends on the slope !3~= d13/dg~g1 For the infrared, r~-÷+ oo, limit, we require /3k> 0. In this case ‘-~
°°.
‘÷
.~(—log~) = g1 as
-~
The
+
(g_g1)~t
0. ~~m)
are constrained by eq. (4.36) to then have the form
(4.40)
11.1). 1. I ba,’hanel
160
~(n.om)~~’•
q~.g. a-’.
L~)=
J+z(L~l)(2-
‘m om)/2
-
/
it
(~/:v
a!. - R-ggc’omm field theorm’ - fornmm,latiomm amid mis,
,)
(2
oi)/
2
(4.41)
(Omf’mm),’2)y)(’i I
/
)
Lv
0
-Porn
L
.
/
( F~ F .
-
E\ -
—)
)
-
,
Si
(,,csq1’q1 -
.
-
1/
~,‘m
-
~vlicrc
F and
= ~
~‘l
f
(4,42)
and
(‘,,
are some constants while
2,g. o’,L’~) i[°’°(E, q
~
is
— c’~F~(;)
~)~i
an tmknown ftinef ion. l”or
[(II
this means
2~L / (‘,5a’q ~ ~
~
5j’)-\z(s’
i)
Ri -
.
(4.43)
wilicil is precisely the sealing form suggested in the earlier strong coupling soltitioii. Now we have to inquire into tile possible values of’ g . Suppose g = 0: then the tileor\’ is a tree theory in tile in framed limit and y(g 1 ) = 0 while z(g1) = I . This is tile weak coupling or quasi-stable P. II’ g1 0. only the strong coupling solution is chosen. From a eOnlptltafional point of’ view. tllell, the nature of the P reduces to tile study of tile ieroes of’ ~(g) and tile evaluation of y(g1 ) and zIg,). Tilis is a task not mucil less complicated tilan solving the full RFT. There are two modes of attack wilich we know. One is likely to he unreliable: tile second is complicated in procedure, hut likely to be reliable. Tile first technique is to use perturbation tileory to learn ~(g). In lowest order we need the graphs shown in f’ig. 4.5 and fig. 4.8 which yields (4----D) g+Kg~.
K> 0,
(444)
when the phase space ilas D space dimensions. This ~(g) Ilas a zero with positive slope at 4---D --
.
(4,45)
4K
It’D 4, tilen g, would be very small and having evaluated only the lowest graphs f’or ~(g) would be extremely reliable. Physics requ.nres D = 2. however and the accuracy of keeping oillv these graphs is at best problematic. Indeed. only the graphs silowil lead to 0.17.
(4.46)
and =
(‘or D
= 2.
1.08
To next order in an expansion in 4- 1)
(4.47> it
is found that 169. 701
H.D.I. Abarbanel eta!., Reggeon field theory: formulation and use
161
+ Fig. 4.8. The lowest order contributions to
4—D
—y(g 1) =
4—D\2 257
[~
4
i~(1 ,i)
371
(~-~)+(~~ log~+~j+
...
(4.48)
and 2 I’155
z(g 1)
1
+
‘4—D\ 1 ‘4—D\ ~—)—+~——“) \l2
-
2
12
L” 24
4
791
log+”j+ 3
...
(4.49)
48
or —~y(g1)
(4.50)
0.38,
and z(g1)~
1.18.
(4.51)
It would seem, therefore, that this series in 4--D or equivalently in g~is unreliable at 4—D = 2. Such series expansions are employed in statistical3)physics for thebecritical second orderto terms might small, indices but we in know no way phase transitions. It is possible that the O((4—.D) be sure except by direct (laborious) calculation. From this procedure, then, one has definite, albeit unreliable, values for ‘y and z. If one reformulates RFT on a lattice, then one can envision using techniques developed in statistical mechanics to evaluate ‘y and z directly in two-dimensions [71—73]. It has been possible to show that if there is an infra-red stable fixed point of the type discussed above, —y ~ z ~ 2; so the Froissart bound is satisfied and a 0tot [731. Numerical calculations of y and z are 51 ~ presently in progress [71--731 Here let us comment on the physical implications of the existence of a zero of p3(g) at g 1 ‘‘ 0. First of all, of course, we have the scaling form which is similar to the strong coupling solution of Gribov and Migdal [63]. Indeed, the weak coupling or quasi-stable P would appear to be out. There is a very significant difference, however, since in the present solution 170 is not equal to but has the value —a, where a is small, as in (4.32). By computation, the problem of negative total cross sections does not arise. Second, even though the bare Pomeron pole is extinguished there is a moving pole, on the trajectory ~
—~,
a(t)
(4.52)
= 1 + ctl/z(gi)
This can be shown either by a perturbative evaluation of Ipii, [25, 341 or by a more accurate evaluation using the full power of the renormalization group [59]. In either evaluation of z(g1) given above, z(g1)> 1, so any trajectory has a cusp at t = 0. Third, when one couples particles back into the theory, one finds a hierarchy of contributions to ar(s) for A + B -~ anything (see fig. 4.9): ar(s)
‘~
g~g~(l0gr)’
—f~8(logrY’+ k~~(logs)_2 (ii)
+
(4.53)
111)1.
162
A harbanel eta!., Rm:geo;m field 1!meori’: formulation amid misc
(a)
s(s)
g
5 g~(icq
-
~-:~S
h~-~Ieg 0
I ii.’. 4.9. ‘The imierareim~of contributions to the total cross ser’tjuIm fur 5 4 u —--‘aim timing v-ommliime ruin lime strulme eoupli me, Sealimme Soiutlofl to time Pummmeromm RI-I, ‘T IS tim,’ imnoimiaiomms dimension ma’ time P field. in an expansiomi of’ the RI-I about seale inv:mrialmt theory
-~is
found (po~sihivmmnreliahiy) tim be LmhOmmt
‘~-
It is iniportant note that the dominance, of’ logs, of the generalited graph of 9a. yieldsto a f~ictori:ed as)’m/)to tic totalb~’powel’s (‘ros,s’ set-lion. Sines’ y(g, 1 > 0. this ‘‘pole’’ cross section rises. fig. 4. 4.5. Secondary trajectories and multi—J’onueron corrections Having in iland a theory of tile P by itself it becomes quite natural o ask wllat will he tile structure of’ partial wave amplitudes with quantum number exchange or, like the P’with a( 0) > I This has direct application to experimental fact in tile ease of the p trajectory with a~(0) since experiments oii irp 71°Il for a Lirge range of’ incident beam niomentum show an s and I dependence consistent with the exchange of a sinlple p Regge pole with trajectory I 7 I —~
a,~(t)=
+
t/(GeV/c)2.
(4.54)
Also in the case of fermion trajectories there is the long staiidi ig problem of parity doublets we will C0il1~ to this. Since tile excilange of any Reggeon witil a-R (0) < I requires an energy gap E( 0) 1 a,~(0) -> 0. the important processes wilichl determine the sillgularity near.! = a 1(( 0) are one Reggeon-multiple P excilanges. All such processes have an infinite number of tilresholds at F = I a,~(0). If’ we restrict ourselves to triple cotipliigs by phase space or simplicity argtulleilts, then only graphs such as fig. 4. 1 0 will occur and we have two couplings: r0 from our earlier work and a Reggeon Reggeon P (RRP) coupling which must be itire imaginary for tile sanie sigilature reason. Because Reggeons with aR(O) < I are to occur only once, tlleir number operator is conserved and we illay solve (‘or the coupled P-Reggeon Green’s functions as all P energies = I --J~--~0 and all shifted Reggeon energies (~= First we consider tile case of boson trajectories (P. A2. p. w, or wilat have you.) lilis was studied ‘
H.D.I. A barbanel et al., Reggeon field theory: formulation and use
163
Fig. 4.10. Reggeon graphs for the propagator of a secondary trajectory, dotted line, coming from its interaction with the P, wiggly line. The P interacts with itself. For the usual reasons only triple couplings are considered,
in the early Soviet work by Gribov, Lenin and Migdal [74].. They rather casually treated the ratio of slopes for the P and Reggeon and neglected the triple P coupling altogether. This latter point is certainly in the spirit of the quasi-stable P. They found that by using a “Ward identity” they were able to determine almost on dimensional grounds that an originally linear Reggeon trajectory aRO(t) = aRO + a~
(4.55)
0t
was strongly modified by the RRP interaction to (4.56)
T11’~,
aR(t)
= aR(O) ±~K’y
or if there were no
RRP
aR(t) = aR(O)
a~t
+
—
coupling but only a RRPP quartic coupling, the resulting trajectory was Ct/(logt)3.
(4.57)
Both these solutions are at best true in the weak coupling case. In the case of strong coupling no solution was presented. The renormalization group approach [75] does not presuppose the absence of a triple P coupling; indeed, it is a crucial element. From a field theory of P’s and Reggeon one finds scaling laws for the Green’s functions as before. There turn out to be three zeros of the beta function for the P-Reggeon vertex which could yield the infra-red behavior of the coupled P-Reggeon vertex functions. Two of these yield Reggeon trajectories which have cusps at t = 0. One gives a linear trajectory, with renormalized intercept and slope, near t = 0 aR(t)
=
aR(O)
+
a’Rt + 0(t2)
(4.58)
which in light of the experimental facts on irN charge exchange is an attractive result. The amplitude, for charge exchange for example, would read in this last case TAB~A~B1(s,t)
gAA’R
(t) gBB’R (t) s’~R(t) (exp {—ilTaR(t) }
+
r~)(logS)’TR,
(4.59)
t small
where TR is the Reggeon signature and aR(t) is given in (4.58). The index “y~is in the expansion about D = 4 described above = 1/12.
(4.60)
/I,D.I. ‘1 barbanel eta!, - R,’,i,’~eommfield the’orm- - /oromulatiomm amid
164
mOo
In any case it is not unreasoilable to expect y14 to he small in niagnitude, i’ltis leaves the result of the boson trajectory witil all P cori’eetions 111 fine shape as far as experitilcu i goes, It is quite sum’-prising that tile collision of’ all tile multiple P en ts has so niild an effect Tile problenl ot’ term ion trajectories is cnorniouslv more contplicated in detail ~tltiiotigli I cutl— ceptually is tile same as the boson secondary trajectory 76 I - The key C\ pert ttletltal fact is that f’ernl ion trajectories are essentially linear in tile Matitlelstani variable u appropriate for backward scattering =
a1,-( 0)
±
u.
t -Lu
It
and 5111CC botll positive and negative parity trajectories in ust be present to tnauit aiil the Mandelstam allalyticity of’ the scattering amplitude. 110111 trajectories would scent to be pic~~’i1 1. Only trajectories carrying eitiler positive or negative parity are ktiown: never, built. A successi uI theory ot’ femniion trajectories ilitist, therefore. avoid these parity doublets. Starting witil the standard P theory described before and positive a/ic! negative parit~-’trajectories of the f’orm ±~~/7+
a01,-il.
(4,02)
finds. tilrougll tile usual renormalitation group procedure. a i’enoriiiali.’ed lrajec-tu/’t’ a/must linear in a. l’urtllerlllore f’or a < 0 in the scattering region both parity poies am’,’ 1)11 tile physietl sheet of’ tile ~ plane. For a > 0 in tile l’egime where pai-ticles lie on tile Regge 1ra~ectories.Oils’ I)! tile parity partners slips onto an unphysical sheet tilrougll the P F cuts, Built ititportant observed properties ot’ ternhioll trajectories are thus achieved. An “unnatural’’ asl-leet of’ ills’ treatment (it fermion trajectories is that the t’ixed points in the space of coupling constants ale onl coilsi itional is’ stable, ‘[his means that the results we have stated univ if tliers’ is a certain i’eiation aillong tilt’ couplings and slopes. ‘file results of the calcuilations of the multiple P corrections to secondary trajectories, by their accord with observed facts give support to tile solution to the P problem by itself. To test any of these in detail is difficult because of the present uileertailties ut (lie actual values of tile indices sucil as y, not even to meiition the experimental problems in diff’erentiating among various variations iii logs. Clearly the whole package ilasan attractiveness and coileretlee \Vilieil is quite pleasilIg. OilC
4.u. Higher p0/lit
P
couplings’
In setting up the RFT as sumnlarized in tile Lagrange 1 nlletioll (4.151 we briefly’ diseusset-l and theil ignored four and more point P couplings as well as derivative couplings An heuristic argunient based on phase space was given for tilis. It is possible to make a stronger case. Within the context of tile weak coupling or quasi—stable P. Gribov and Migdal were able to use ordinar pci’turhation theory to evaluate corrections to their leading P structure [61 I . Here tile pilase space arguments are both correct aild persuasive. Typical of the results obtaiiled are the correeti( us to 2 logt correctile inverse propagator as given in (4.25) where a linear trajectory is modified by a t tion a harnlless addition, In tile case of tile latter day renornialization group strong coupling solution, as discussed above. the problem is again non-perturbative. Migdal. Poiyakov and Ter-Martirosyan gave a iluillber of -
H.D.I. .4 barbanel et al., Reggeon field theory: formulation and use
165
arguments why four point and higher and derivative couplings would become negligible in the
infrared limit [251. Abarbanel and Bronzan showed that when the triple P coupling vanishes, the fixed value of the renormalized four P coupling is zero, so that the P is weak coupling [77] . This is consistent with the notion of a dominant triple P coupling. Finally Brower and Ellis demonstrated that the simplest derivative couplings were harmless in the infrared limit [78]. Basically the task of each of these calculations was to show that the effective coupling constants, like the ~ of the three P case, for higher point or derivative couplings were driven to zero in the infrared limit while the triple P coupling approached the same zero g, of ~3(g).in short, both the stability of the original calculation and the ignorability of higher order couplings
-—
all in tile infrared limit
—
were suggested by these exercises.
A much more complete treatment of this matter was given by Jengo and Calucci [79, 80]. They considered, at once, an infinite set of n P couplings, n = 3, 4, ..., and all derivative couplings. If tile E~,q. phase space integrals were always extended to infinity, then such a theory would certainly be non-renormalizable. However, since we are here involved with an infrared phenomenon it is natural to introduce a cut-off. As we discussed in section 2 such a cut-off is almost certainly required to consistently extract the RFT from an underlying theory. The action of this theory is A =fdrd2x{[~ ~
.
rr0
V~_~o~t~_ (V2~t)(V2~)+...]
r’0 +
~
[~~t2~2+~
...]+ ...
(~t2v2~+~2v2~t)+
(4.63)
with the prescription that behavior in momentum integrals to be cutoffofatthe Iq, cutoff I = A, A. Now 2. The infrared of the space theoryallought to beareindependent lE~ I = ct~,A one scales the cutoffs to lq,l = A/A, IE~I = ct~,A2/Band expresses the scaled theory in terms of new constants ~ ~o, ~‘o,etc. which are functions of A, B, ct~,~ (Scaling the cutoffs, as iildicated above, is equivalent to holding the cutoffs fixed and scaling momenta and energies; thus A or B play the role of ~ in (4.36).) Carrying out this procedure it is found that if one only requires ~o to be stable at zero, that is, = I, then the renormahization group strong coupling solution is reproduced with all couplings except 7~,the three P coupling, going to zero in the infrared limit. If one puts further constraints on the couplings, then it is possible to find other solutions to the RFT including, strikingly enough, the quasi-stable P. To achieve that, however one must essentially require that r 0, be specially chosen so that it maps into zero under the scalings of the cutoff. From the point of view of physics there is no special motivation to further constrain the theory in this way. Indeed, if we do so and launch upon the pathway of the quasi-stable P, then we are plagued with the various decouphings of P. This rather general treatment of all couplings at once provides a posteriori support for the model Lagrangian of (4. 1 5). ~
4.7. The formal status of the RFT
There are two points that are essential to our understanding of the solution of the RFT with the
166
11,1)1, -l barhanc’l it a!,, Re
comm fim’ld thi’orm’, Joromimlatiomm amid mice
Lagrangian of (4. I 5) provided by tile reilornlalisation group. Firstly we tieeul to know that the (Ileory is renorillalisable in tile eotlveiltional sense. so that the forillalisnl of the renornialisation group is applicable. In particular we wouilsl like to he sure that we can safely set ~ = 0. since this is esseiltial to tile dmlensioilal analysis uised. Secondly we need to ktlow how (lie theory can be constructed perturhatively since thus is tIle oilly’ way we can explicitly check Reggeon unitarity, Tilese two points are not uilrelatesi, In the original stuidy of the RFT it was t’m,uisl tisefuil to generalise (lie two space shilliensions to D dimensions. There are two reasons for this. Firstly in a general. non—i ntegs’r number of di— mensions all Feynnlan graphs can he defined by’ analytic continuation from I) < 2 the ultra—violet thvergenees are relegated to poles itl the variable D and a simple regutlarisatioti proceslure is provided in principle by suihtraetillg such poles, with their (real) i’esidues. froni tile Greetl’s f’uuietionx il wilicil tlley occur. However, the RFT is actuall super—renormalisable iti terms of simple power counting (‘or D < 4. This means that (lie subti’aetion oI the poles has to be equuvaleilt to an intercept (mass) renormalisation anti it is not eleai- then that this suihiraction is coinpat ible with the = 0 condition. Tile second reason for varying D is that at D = 4. a clearly uilphysieal situation, the theory possesses a (broken) scale invariance itl separate space. .v. anti “tuue r scalings. Illis ntanil’ests itself in tile fact that tile dii’nensionless coupling constant. g = (r/a’°”4)L’~?~j4 - has no explicit dependence on tile nornlalisation point lay, As a ecnsequence tilere is a iero 0! [3(g) at g, = 0 which governs the infra—red behavior of the theory. It is this latter t’aet which motivates the expansion of tIle theory around D = 4 in powers of 4- -1), The poles fronl tile uilti-a-vioiet divergences occur at .
4
D = 2/n.
ii
and a series expansion
--~
ill
...
(4.64)
powers of (4 D) actually hides (lie poles since
n24-- D)
/1 =
= 1, 2.
---
~----...
(4.65)
‘fileref’ore the expansion of’ tile theory around I) = 4 sidesteps tile problem of the intercept renornlalisation. In f’act forD < 4 tile problem of the ultra-violet divergences of (lie tlleory cannot he separated from the infrared behaviour if we set ~ = 0. The theory cannot he reilornlalised order by order in perturbation tileory. fllis problem has bceil stuidied in detail by Sugar ansi White, both for the present RFT [81, 82] and for conventional Xcp4 tileOries [81]. where tile problems are very similar. Tile essence of the dit’ficulty is that t’or ~ = 0, the iilterccpt renormalizaijomi eotinter-(erill, ö~ ilas tile forni =
~
“Thj~ 4--D).
(4,66)
where f is dimensionless. Tilis follows froill dimensional analysis only. Clearly ö~canilot have a power series expansion in r0, and any perturhative construction of tile propagator must ilecessarily involve a re-ordering of standard pertuirhation theory. Sugar and White provided such a scheme rising renormahzation group apparatuis. Tile derivative renormalization condition on .1> is
H.D.I.
Abarbanel et al., Reggeon field theory: formulation and use 11 Ztx “ N’
—‘—iF~”~~E 01 E’EN /
167
p467
where Z is tile wave function renormahization constant and
4] 4/(4_D)E~,1 . 0/a~,’~ Eq. (4.67) can be integrated to give x~,=
[r
(4.68)
ir~i)(E,0) = _L~xNfdxNx~Z(xNyi with x
=
xNEN-/(’--L).
(4.69)
When this integral representation is combined with the formulae
Z(g) = exp[fdg’~g’)/P(g’)],
(4.70)
(r 0/a’~~)E~ 4)/4
=
g exp[_jdg’(g’ ~ +~(g’)~)]
which are fairly easily derived from the renormalization group equations [821, we obtain a complete set of equations for calêulating F”(E, 0) given y(g) and ~(g). Provided that ~(g) has a zero with positive slope this set of equations solves all of the above problems. ‘Tile infra-red behavior of pO,i) follows immediately and when the representation of p~~’) is compared with perturbation theory we find =
(r0/a~)~4-D)
f
~NXN
[I Z(xN)~]
(4.71)
-
which does indeed contain all of the ultra-violet poles (as divergences at XN = theory 0). When equa2 they can be combined with perturbation forthe p(’2) to tions are generalised to non-zero q provide a complete iterative scheme for constructing p~’)in such a way that it always satisfies = 0 and has the correct infra-red scaling behavior. Finally the representation of F(11)(E, q2) allows us to show that perturbation theory can be used for large (—E) and (—q2) showing that F~”0(E,q2) has a leading P pole and Reggeon unitarity is satisfied. Note also that when the representation of 5z~is modified to allow for the presence of a cut-off it can be shown that S~is positive and so (to the extent that higher P couplings can be ignored) we must have a~,>1 to obtain a renormalised intercept at one. In this case each term of the perturbation expansion will appear to violate the Froissart bound, even though the sum respects it. 4.8.
Ideas about a 0> 1
We have seen that the requirement ~ = 0 is met only if r0 and a’~,are related in a special manner (eq. (4.71)). Since this relationship is not automatically satisfied, the requirement ~ = ~,
0
lID,!, -I harbanel ci a!, -
165
!ie~,’,5’eo/mlie/cl
theorm’ - formmmmm/atiomm
ammo’ mice
be ‘‘unnatural’’ even thougil it is indicated experinlentailv. One alternative to thus situa(ioii is to ask wilat llappens when ~ hecoilles arbitrarily large and negative. In model ealeulatiorls in which all ifltilite set of cuits is sunlnled, tIle F’roissart bound is not violated, even thouigh every iiidividual cuit violates the bound [26. 27] . I nstead. the new “P’’ is a pair of branch points at illigilt
a(t)
= I ± 21a~
ni] m 2
1(a0
tmonieron of (lie Regge-e ikonal mould
)~_
2’). 33
I
lii
sotlie respects.
It
:2t
comits’s
hue “P” reseni bles from a pole with a tile h 11 > I . and in mi pact n~u’anls’terspas’s’ it is a shsk whose t’aslnis nil )ws like Imi s. I-iowever, tile disk is gray. not black. and total doss sections faetoriie in tlis’ It igli cilergy lint ii rather than approaching a common value, ci1 (,s’) (lu,s ~2 has the miiaxuii 11111 rate (if increase ul loss esl by’ the broissart bout 1151, TIus seems to be !‘ortui tons because s--channel uut itarit\ is miot in posesi iii RiTh On the oilier hansl, it nay not he accidental that it is i mu possibls’ to have a rettornialited ~ -- (I (a > I I within (lie Ri”~l’.even thouigh s-channel unitarits’ is 2not ~
(4,73)
where a
11 and ~7oare pi’opoi’tional to ~ ansi sleps’tid in sls’tail ott this’ Lagi’angiait before sluftitig. ‘[his new “f’ree” Lagratigiati has E( q) sped ru itt 2 = (I’ cc(q))2 = (77~ 2)24a1 lf(q)
1+a~1q
where when a(q)
T~=
= I ±2\/a~)a
)47~)
2 0q
for small q2. This, in general f’ea(uires, resemu hhes the results of refs. I 26. The more general problem of’ why ~ = 0 or how one can incorporate the RF’I’ remains open ansi inviting,
27
it
I
in a “nattiral” wa
ni 15)
Other ‘weak coupling’’ ic/eat
4,5),
The idea that tile soiutioil to ouir RF’l’ is stroilg eouiphing in (lie iltrared lint it. namely
[3(g 1)=
g,
0, has certain uinattractive features t’ront an aesthetic Ibm I of’ view even though (lie physics is certainly sounsl. The reasoning is more or less that simlee we had to begm ouir RF’i’ by a choice of’ bare trajectory correct ns’ar .1 = I and I = 0 it is perhaps discomfortmg that we did ilot reproduice that singuilaritv after stimmi ig all graplls. To do so wouiltl requnre that tile f3(g) (‘or tile RFT’ muist hlave a zero at g = 0 wluchi governs the inf’rared behavior. for then all Green’s f’uulctions hecoille tile original Green’s ftuictions plus sillall corrections. Sumch a RFT could he said to bootstrap itself near E1, q1 0. Some examples of such tileories ilave been giveil hy Aharbanel 185 1 . One 0.
H.D.I. A barbanel et al., Reggeon field theory: formulation and use
169
which has the scaling form similar to the Moscow—Batavia strong coupling theory is G~”~(E, q2)” = E~(l+a0q2/E3P~), where the power p is undetermined. This theory bootstraps itself in the sense stated above when a constant triple P coupling is present. A serious fault of these and any other theories which begin with non-linear bare trajectories is that demonstrating that multiparticle t-channeh unitarity is satisfied is a difficult task. it should be recalled from section 3 that t-channel unitarity is nothing less than the foundation of RFT. (4.76)
5. Inelastic processes in
RFT
In this section we consider particle production processes in the RFT. This is a very important problem in its own right, and it is crucial for studying the internal consistency of the RFT. In constructing tile RFT for the elastic amplitude t-channel unitarity was built in from the start, but the constraints of s-channel unitarity were not. One must verify a posteriori that these constraints are satisfied, and in most cases this requires some information about production processes. One approach to the problem is to derive RFT rules for the production amplitudes and then use the techniques of section 4 to study their asymptotic behavior. As indicated in section 2, it is
difficult to apply Reggeon unitarity to 2—N production amplitudes; however, it has been possible to study the asymptotic behavior of these amplitudes in the multi-Regge region of phase space using the hybrid Feynman diagram procedure. Drummond [86] and Campbell [87] have studied hybrid diagrams which head to Reggeon graphs with one closed loop, and recently Bartels [881 has obtained the rules for a general Reggeon diagram. Here we will briefly discuss the results for the two-to-three amplitude. First, one must identify the counterparts of the signatured partial wave amplitudes encountered in the elastic process. We refer to the kinematics of fig. 5.1. In the double Regge limit we consider the five point amplitude A 5i2~ Sf3, ~23 -÷oo and t 5 with 1 = Q~,t2 = Q~,and 77i2 = Si2/Si~,S23fixed. When we have Regge poles ct1(t,) in the t1 channel, study of a hybrid graph as in fig. 5.2 leads to A5(s12, Si3, =
523,
7112, t1, t2)
!31(t~)(32(t2)
{SS~”~i ~cii
~02_01~(t1, t2, h1i2)+s~s~”~2 ~2L1~~52 v2(t1, t2, llu) }
(5.1)
~‘~~~I)sm3
Fig.
5.1.
Kinematics associated
with the
double Regge limit of the five point amplitude. In this limit s12, si 3 Q~ and ~i12 “S12/Si 3S23 held fixed.
~2 3
~
with Q~,
1
/1,1),!, .4 barbaric! ci a!,, Re,m,’g-omm (ie!d theorm’: Jornmulatio,m ammo’
70
N
m/sc’
/
/ fig. 5.2. A ii’ hod l-’eynmnan graph and
its Reggemmn gra pim contribution to time slimmmbie Reese iinm ii ot’ time t’ic’e point t’simmc tim 5i 2 moist ‘5i ~ tue othem. fl “t 2 and ~2 3 distinct partial cease terms enter; one has simultaneous stiseontmnuities ~
‘icc
-
with —
cxp{ i~(ai---a’ 2)}+riT2
~0i02
—
a2)
Si117T(a1
The deconlhbosition ol’ A illdicated ill dtl. 1 5. 1) is consistent with tile Steinmaii relations, which tell us that A5 cannot hlave simultalleouis diseontinuities in ‘~i3 aild s23. The two particle-Reggeon coupling [3is real. The two Reggeon-particle coumplings ~“1 are real and, in models. are analytic at 17i2 = 0. This suggests that the appropriate partial wave amplitudes are tile generalizations of V
1 and V2. it also suggests than one will requure two kimlds of two Reggeon-particle couplings to build a calculus for evaluating Reggeon cuit contributions to the 2 3 process. fhese expectations are borne out by the study of more eonlphicated llyhrid graphs~for example, thioss’ wilich correspond to tile Reggeon graph of fig.5.3. Thus aild fancier graphs yield (lie form for A, —~
A,
=f~ ~‘ ~
1t’ t2.77i2)+~~
~
~
~m ~‘1~.’2,
‘J 2~~
J2 V26’m..J2.
ti
t2,77i2)~.
(5.2) the double Regge limit. The Reggeon graph rules yiehuls V1 ~uid “2’ Bartels has generalized the above results to (lie muitiparticle production amphituides. A4~0,I 881 Tile only new difficulty is that tile decomposition of’ tile amplitudes requured by (lie Steinman relations becomes cuimbersonie. However, for processes in wilicil onlypartial the P singuilarity is impor2 beilavior of tile multiparticie wave amplitudes. In tailt, one only needs the small L’ and k ill
E 1z!-
J~
Fig. 5.3. A Rcggeon graph with branch points in J1 and J2 which contributes to the double Regge himmii of time 2
3 process,
H.D.I. A barbanel et al,, Reggeon field theory: formulation and use
171
~I=I Fig. 5.4. The picture of the basic Pomeron (P) as generalized ladder graph.
this case it sufficies to approximate the bare vertices and coupling functions by constants, and
only one independent amplitude enters for each n. In some cases the analysis of inclusive cross sections is even cleaner than that of exclusive cross
sections. For example, the RFT rules for the single particle inclusive cross section in the triple Regge limit have been obtained both from the hybrid graph approach [59] and from the Reggeon unitarity relations [58]. Further work on inclusive cross sections is presently in progress. Another approach to the s-channel content of a Reggeon calculus has been given by Abramovskii, Gribov and Kanchelhi (AGK) [36]. They bypass RFT by studying the s-channel absorptive part of two-body amplitudes directly. Since such absorptive parts are immediately related to production processes in the s-channel, we gain directly the information desired. However, one must pay a certain price for this splendid efficiency since it is necessary to specify how one cuts through Reggeons in extracting absorptive parts. The formulation of AGK assumes that the
Reggeon, in particular the P is given by a set of generalized ladder graphs as in fig. 5.4. The cut through this P is, up to considerations of signature to be treated in a moment, just as in the fat1iihiar multiperipheral model shown in fig. 5.5. It gives rise to a uniform distribution in produced particle rapidity in the central region for single particle inclusive production; it yields multiplicity of produced particles proportional to log s, etc. With cutting rules in hand, one may cut up any given Reggeon graph to find the contribution to the desired s-channel process. Clearly this method is most attractive when only a small number of Reggeon graphs need be treated. As such it lends itself well to the so-called weak coupling P discussed in the last section. That P is without terribly high regard at the time of this writing and summation of infinite sets of P graphs appears rather necessary. These caveats exposed we turn to a description of the AGK cutting rules and some consequences thereof. Given the rules for cutting a P, AKG argue that the details of the basic production mechanism
Fig. 5.5. The cut across the P reveals a multiperipheral production process in the s-channel.
II,D.J.
172
.4 barbanel eta!,, Rem~y011 Jie!d thc’oru’,’ /ormmmmilation amid mice
UT JP
TO ~
jY
(bl CUT UP
9
9
+
TO STUDY
I ig. 5.6. (a) Absorptive corrections to the basic production process. production t roimm ts5’cu cim~mins.and production with a iarpe rapidity cap these amphtudes are revealed by the cmmt of the tsvuu Reggeon graph contrihimtion to time elastis’’~imnphtmmde.One cuts one, tcvo, or zero Reggeons respectively. (b) ‘the prodmmctiomm iirocessc~revealed by cut tiny ~mpthe Reizgeiun miraph of’ tip. 2.6,
need not be specified. Withl the cuittimlg rumles they study the effects omi ~ie generahlied ladsier P arising from multiple P exchange ails! from P iteractions. Two low order graphs are silown in fig. 5.6 along with tile contributions to produictioil anlphi(uides generated by cumttillg thelll. Tile features revealed in this n~annerare absorptive corrections to the laduler-hike produietiori aillplituiu!e of t’ig. 5.5, produiction l’rom more (Flail one ( lllultiperiphleral like) chaill, aild the possibility of’ large rapidity gaps between produced particles. One cai~already make some quiahita(ive statenlents from fig. 5.6. First, if the average iluniher of particies arising fromli the exchange of’ tile basic P is n1 a log s, (hell ii front the exehailge of’
k non—interacting P’s will he of order ku
~. In the absence of’ P iilteraetiolls then, tile iliuihtiphicitV 5 ~. P interaetiomls will t’ill ill (lie uhistribution will be as shown in f’ig. 5.7a with a,l/uk,l (log S) gaps and smooth ouit to look more like f’ig. 5.7h. The actual possibility of seeing tile peaks in a, 1 at kn rests on model dependent couiphings not estimated by AGK or anyone else. Furthlernlore
a
Ii
I
it~ C
/
~~
3~
~
lip. 5.7. (a) The distribution in number (n) of particles produced by time graphs ~itli nmuitipie non’interaeting exchanged P. Fig. 1.2 is an example of this, (b) The expected smoothing of the o,~distribution of (a) which will coimie froimi P interactions.
1-ID.I. A barbanel eta!., Reggeon field theory: formulation and use
173
/~IIIHIHK
Fig. 5.8. A cut part way through a Reggeon which leaves a hanging chain with rapidity spread v. The particle line carrying momen2 ~ cY, which must be large since a Reggeon is cut. Since large q2 is presumed to be absent, so is the partial tum q will have mass q cut of this Reggcon.
production from k independent P’s is expected to give rise to the order of ka particles per unit rapidity interval. In individual events one expects to encounter long range fluctuations in the
rapidity distribution. Calculation of fluctuation probabilities have been made by AGK for the weak coupling P [361. The detailed cutting rules depend crucially on the observation that for a particular discontinuity to be non-negligible, the cut must go completely through any Reggeon in the graph or hot cut it at all. As an example of this consider the cut of the two Reggeon graph in figs. 2.1 and 2.2 as shown in fig. 5.8. In order for it to be interpreted as a cut through a Reggeon the rapidity spread y across the cut lines must be large. In that case the particle line carrying momentum q will have an enormous mass q2 = e~.However, it is a basic assumption of the hybrid graph approach that any diagram is negligible when a line carries a large mass. For small values of y the cut is interpreted as passing through the Regge vertex functions Mn). This argument applies to any Reggeon diagram, for a partially cut Reggeon leaves a hanging chain and yields an unacceptably large q2 somewhere.
For diagrams involving the exchange of k non-interacting P’s the analysis is straightforward. The contribution to the amplitude T(s, q~)from k P’s is
T~(s, q~)= _i~sffl~~
~2(qj~
qJj)N~~k(qll,
...q~ 1)2fl
~G1(s,q~1),
(5.3)
where G1(s, q~)=
_50/(q~)
~
sin{~lrct1(q1)
}
.
(5.4)
Cutting through N~ should have no effect on them. Recall that N(2) can be written as in eq. (3.41) as an integral over the Reggeon-particle absorptive part. Since such an absorptive part involves only on-shell intermediate states, cutting does nothing. For N~,k> 2, the argument is more formal, requiring an elaborate excursion into Sudakov land. The conclusion is that cut N’s are the same as uncut N’s. The s-channel absorptive part of T~(s,q~)is now reduced to a combinatorial problem in enumerating the ways of and weights associated with cutting 0, 1, ... k P’s in all possible ways. Take T(2)(s, q~)for example:
174
liD.!. .1 harhanc’I et al.. Reggeon tie!d ihcori’: formulation ammo’ use
2~(s, q~)= ~ir Abs T~
s
d2k11 dk (ir)
k
-
k 21 ~
(~ir)~
x {1 G1G~+G2G’~’ I + [AbsG1
( iGs)*+iG’2 .AbsG’1+AhsG2 ( iG )“+iGAbsG-21 +1 2AhsGm AbsG2i
where AbsG’~= 1mG1 =s°i.
(5,6)
‘File ternis in square brackets restilt respectively Ironi cuitting f’irs( 0, then I . ansi finally’ 2 P’s. Note that for identical Reggeons (lie signs and weights of these eontribtitions are +2, 8. ansi +4 yielding an overall negative two P cuit term ui (lie elastic process. Diagranis involving only triple P vertices can also be easily dealt with since s’uitting across thhis vertex leaves it unchanged. However. the diseontinuuty across (lie gemleral nP niP vertex cannot be expressed in terms of tile vertex itself when iitn > I . As a resumit in order to calculate indivisitial contributions to s-chaiinel discontinutities ff0111 Reggeon diagrams cotitaining such vertices, it is necessary to have greater knowledge of tile vertex than is required to evaltiate tile Reggeon graph itself. (A tileory witil only triple P vertices as building blocks for n i-n transitions is thus qtiite attractive.) Tile cutting rules were employed by AGK in stuidying single particle inclusive cross sections in tile central legion. This is the regime of’ tile process a + b e + 1’a X where iii the cc titer of mass = ~ logs or of h, t’b = ~log .c frame of’ a + b, the rapidity i’ of e is finite as the rapidity of a. grows large. Taking tile uiiagrani of fig. 5.9a ails! cuitting it gives the leading eontrihuition to the (histribution in y and P-i , the transverse illolllelltunll of c: -—~
—~
~
da(a+b - c+X) dvd2p = gagh!~(pT)lmG(3’ --
5~--p,0) lmG(j-’
~b’
0).
(5,7)
which is precisely the flat rapidity distrihuition5~h expected 0) =uip1 for (lieproportional P. (‘orreetions and 5.Qe.when Tileya(add to he to to (5.7) result from Reggeon graphs as in figs. 5.
_______________
go
3
Pa
.
‘
P 0
2
/y~,:.))
-
~
(P1) 2b
(a)
~b’0~
>
(b) —-—/*—-
)
(C) - - -~
lip, 5.9. Reggeon graph contributions to the 3 —~ 3 amplitude whmise discontinuity in = (P inclusive distribcmtimun a -f b —~ c + X in time central region,
+ ~h
~)2
gives time single particle
H.DI. A barbanel eta!., Reggeon field theory: formulation and use
1
f~(i4)[
(Ya”Y)
±
1
175
(5.8)
(YYb)
after a variety of marvelous cancellations. It is worth emphasizing once again that results like (5.8) are useful only when a small number of Reggeon diagrams make the major contribution to some process. The signal of the weak coupling P, which allows a small number of graphs, is the appearance of corrections as in (5.8) which are only of order (log s)~or equivalently (rapidity)’ The corresponding calculations with the strong coupling P are far more difficult. In this case one must sum an infinite set of cut Reggeon diagrams, and as yet no convincing method has been presented for identifying the ones which make the leading contributions to the inclusive cross section. A start in this direction has been made by the Moscow group [251, and by Caneschi and Jengo [89], who consider a theory with both cut and uncut P’s. The Russians argue that the leading contribution to the single particle inclusive cross section is again given by eq. (5.7) except that is now to be interpreted as the full P propagator. For large values of y ImG(y, 0) grows like y”t, so in this case the rapidity distribution is not flat. One of the most important open questions concerning the strong coupling solution of the RFT is whether the constraints of s-channel unitarity are in fact satisfied. Although a definitive answer cannot be given at the present time, the preliminary indications are very encouraging. First, the Froissart bound in D dimensions requires that G
u~11~ ( c(logs)’~,
(5.9)
where c is a constant. The bound is satisfied for small values of c (see eqs. (4.46) and (4.53)), and it will be satisfied at D = 2 provided an infra-red stable fixed point exists [73]. The crucial test must await the completion of the direct calculations in two dimensions which are now in progress [71—731. A closely related test comes from the calculation of the exclusive cross sections in the multiRegge region of phase space. Long ago Finkelstein and Kajantie [90] showed that if the P were a simple pole with intercept one and if the P—P-particle coupling did not vanish at zero momentum transfer, then one was led directly to a violation of the Froissart bound. The Moscow group has repeated this calculation for the strong coupling P and found [251 “~
c,~(lns)°~,
(5.10)
where u0~2is the cross section to produce n+2 particles, c~is a constant, and to first order in cs=2—7/12 ~=
1
—/4.
(5.11) (5.12)
So, at least for small values of , there is no Finkeistein—Kajantie disease. The calculation leading to eq. (5.10) starts with a constant bare P—P-particle vertex function, but because of the absorptive nature of the P the renormahized vertex function vanishes when all transverse momenta go to zero. Similarly when one calculates the single particle inclusive cross section in the triple Regge limit [25, 58, 591, a constant bare triple P vertex leads to a renormalized one which vanishes when
176
11,1)1, -i barbanel it a!,, Reggm’omm field tht’orm-: formulation ammo’ mice
the nionientuim transfer does. For small values of there is no violation oh’ the etiergy conservation sumn~rutle. Finally (‘anesehi ansi Jeilgo [89 I have ealetilated tue moments of’ the tiluiltiphici tv u!istrihuitioti in models with cuit ansi uineui( P’s. They limit! no eotltrauhetion with (lie constraints arising from tile positivity of the partial cross sections.
6.
Conclusions,
outlook and critical problems
Up to now our diseussioll has been prilllarilv concerned with (lie theoretical shevelOpillent of Reggeon field theories rather than with their eoi~sequiences [‘oi’ pllenolllenologv. illere are several reasons f’or this. First. unltil tile development of the Moscow Batavia strong coupling solution it was not at all clear tilat a self—consistent theory of (lie Ponteron, its self’ interactions. ansi its illter— actions with other Reggeons was available, even in tile RFT. It is an exceedingly slouihtftil busitiess to do phenomenology witil SO uncertain a f’otmndation. Second, a real feeling t’or the size of P interactions has only been available since the stuidy of higll energy incluisive processes began at (lie CERN—ISR and at the Fermi National Accelerator Laboratory. We are not going to attelllpt here eitbler a review of or a eonstructiot-i of a phenoiliemlology of hadron i’eae(ions at very high energies on the basis of RFT, huit we will indicate wilat iii our opinion are tile lessotis we have learnesi that wihl play a basic role in any sumell uheserrptioii of phiellonlena. Tile first issue concerns the size of tile triple P cotipling and tile energy doniaui in which it is necessary to sum tile lull series of P graphs. We will concentrate on the grapils in fig. 6. 1 which we know will coiitribute to tile dominant term in the total cross section (see fig. 4.8). The order of magnituide of the corrections to tue single P exchange is set by tile dimensioilleSs parameter g0 = r~/a~E’. In terms of s this means that -
(r~/~)logs
(6.1)
sets tile scale for the convergence of the P series. ‘Flie best estinlate of’ the size of the hare triple coupling, r0, comes from the2,single B A + is X small, (see f’ig. s/rn2 particle -÷ withinclusive t fixed).experiment If the tripleA P+ coumpling one6.2) tllavin tile triple Regge region (s, m use the lowest order in the P series to estimate its size. Under this assuimption one finds frolll °°
JIll
-~
r~/
+ X
and pd
-~
d
+ X
P
—~
data tilat
1/50,
(ft2)
-+
)~y~ -I_i,-
6.1. Reggeon gTaphs which contribute to time leading behavior of the total cross section. The expansion parameter imere is (ro/oo) lug a. When it is of order OflC, the whole series must he summed to give time strong coupling scaling solution of RFT. Indications arc that at present energies this parameter is only of order 1/8 or so, Fig.
H.D.I. A barbanel et al., Reggeon field theory: formulation and use
177
btAAtA
~M2~
Fig. 6.2. The triple Rcgge region of the inclusive process A + B
—~
A + anything, where one may estinmate the triple Regge coupling.
and guessing that when (r~/ct~)lns ~ we will need a large number of terms learn that logs
ill
25
the P series, we
(6.3)
is where the whole sum given by the Moscow—Batavia scaling solution, will certainly be necessary. Now the logs values available at the CERN-ISR are only 8 or 8.5 at the most, and at FNAL they range up to about 7. At the energies of these devices the full sum of P graphs would not appear to be necessary. This conclusion is certainly in line with the fashionable phenomenology based on s-channel approaches such as the multiperipheral model, which is motivated by the apparent “short-range rapidity correlation” nature of production processes observed experimentally. We cannot completely rule out the possibility that the scaling solution is applicable at ISR—FNAL energies. If it were, then in the calculation of the inclusive cross section the quantity r~/a~ would be replaced by an effective coupling which is independent of r0, [25]. This alternative seems unlikely because we expect this effective coupling to be of order of magnitude unity; how-
ever, this question cannot be fully resolved until we have reliable calculations of the critical exponents and scaling functions in two-dimensions. Work in this direction is presently in progress. When only a few terms in the P series are required for a good numerical estimate of the process under study then in the small log s regime one must face squarely the matter of secondary contributions from N P coupling with N> 3 and from possible derivative couplings. We know of no way to choose among the alternatives which present themselves except by a trial and error approach. However, we are encouraged by the fact that it has been possible to fit the pp, pp, ~ and K~pelastic scattering data over a wide range of energies with only a few terms in the P series [91, 921. There is a nice feature to having only a finite number of terms to address. In such a case the AGK construction, even with its need for a specific assumption on the result of cutting P’s [40], will be a useful tool in relating specific s-channel production processes to the f-plane physics. Another lesson that we learned in section 4 was that when we required Z~=
l—a(0) = 0, as we desire for the full sum of P graphs, the bare gap, z~= 1 —a0(0) < 0, so the bare P has ao(0)> 1. (Happily the best first in refs. [91, 921 also require ~~(0)> 1, with i~ having the right order of magnitude [93].) In making the estimate in eq. (6.3) we followed the approach where the perturbation series was developed using propagators with the renormalized intercept gap, L~= 0. Since we expect the bare P to be observable at present accelerator energies, it may be more appropriate to rearrange the perturbation series so that the bare gap enters the propagators. In this case each term in the series violates the Froissart bound, but the sum is constructed to satisfy
11,1).!, ,.1 barbanel eta!., Rr’ggco,i field theorm’: formulation and
178
t. Actually tile value of ~ is on tlìe order of’ 5
misc
1 0-2 wInch means a very slow growth in s from my P graph. Taking thus eft’ect into aecoulilt we niay boldly ililagille that tile estimate in eq. ( ~g.3 5 too large by a f’actor of as much as two. ‘filen the iced f’or the (‘till P series might set in at ogs 1 2 5. It’ this were to be so, the use of’ (lie scaling forni f’or tile P propagator might well e an attractive, compact expression to use at t’illite energies. To really be of use, however, we would again need reliable valuies of the sealing inslices and reliable knowledge of’ (lie sealing (tine— Lions, Despite our eatitiouis ahllost pessinl istis’ view towam’sl the ui tihitv of a Reggeon ealculuts phienoni— Inology at tile present stage of theoretical sleveloplllent. thlere have been several at tempts at fitting real data. Serious evaluation of’ these. often attractive. phenolllenological essays is difficuilt. We pass tilis task on to tile reader by providing a f’ew references l’ronl wilicil he or she can begin [91. 92, ~4I Here we (tim oumr attentioll to a resumllC~of (lie ideas covered in this article and to a elleerluil view of tile progress achieved in the slevelopments we have reported. We began by recahhimlg time necessity of braneil points in the .J—plane arising froth (lie combination of moving poles ill .1 with unitarity. We assummed that tile Ollly branch points were those coming from tile presence of mtiltipie moving poles in the 1-channel, fills assumptiomi is actumahly extremely conservative. It resembles almost in detail (he experience of many years in hoeatimlg the position of brailehi p0111(5 in the energy planes arisimlg in conventional f’ield theories. There, of eoumrse. one has a variety of’ dispersion relatiomis and suim rumles wInch allow a direct positive assessnien( of’ the validity of’ this assumption. We are not so fortumnate here. We are forced to f’all back on (lie ( lever entirely convincing, however persuasive and attractive) observation that satisf’yiiig t-eilanllel muiltipartiele unitarity is straightforward and natural wilen branch points arise via moving poles and dil’ficult. if possible at all, otilerwise. Clearly if’ tllere are hralleh points in .1 arising from otiier sources, we have missed them. We next saw how to obtain expressions (‘or (lie discontinumities across the f-plane cuts froni tile inultiparticle 1—channel relations. These Reggeon unitarity relations are cruicial for an understanding of the P near J = 1 and I = U since they lead to strong couphings among tile multi-P channels whose thresholds all collide at tins point. Ill order to insure that the full Reggeon unitarity relations are satisfied, we introduced a field theory to describe (he emission, absorption aild propagation of Reggeons. Tile Reggeon field mp(x. T) operates in a world with two space and one time dimension. The branch points in the .1-plane arise naturally as singularities in the Feynman integrals of tile perturbation expansion of tile field tlleory. These singularities represent, as iii conventional field theories, thresholds f’or tile prosluiction of tile quanta (Reggeons) described by X
-—- 1
ip.
When tile renormahized intercept gap valhisiled, we (‘ouncl that (lie RFT exhibited infra-red behavior analogous to that of conventional fielsi theories with massless particles. En this case oi~e could utse the renormalization group to study the behavior of the theory near J = I and I = 0. Wilen an infra-red stable fixed point exists, the Reggeon Green’s functions amid tile total anti elastic cross sections satisfy scaling laws which are given in eqs. (1.4), (1.5) and (4.41). We saw tilat such a fixed point does exist in D = e’ space dimensions, and attempted to extrapolate to the physically interesting ease, D = 2. How are we to view the f’ield p(x, i-)? Since the Reggeons it describes are composite states of’ the observed hadrons, it represents some averaged or mean behavior of (lie umnderiying iladroils. This mean behavior is likely to be ratiler independent of the constituent liadrons if the distamiees --
H.D.I. A barbanel eta!,, Reggeon field theory: formulation and use
179
IxI and “times” r represented in the field ~o(x, r) are large compared to the scales of the hadrons. The natural scales for hadrons are lxi (rnprotonY1 and r = logs I when s is measured in units of(rnproton)2. So far lxi (mprotoni’ and r~.’I, the average field maybe expected to be a good representative of the collective behavior of the underlying hadronic matter. Now this translates into small momentum, IqI, and energy E for the quanta described by the field. This is just the limit where we employed our RFT to learn in detail about P interactions and amplitudes. in a -“
~.‘
sense this is very attractive, and in another sense this is terribly disappointing. The latter comes because we are saying that in large s, small t processes we will not be learning about the basic
structure of hadrons; indeed, we are averaging over the hadron coordinates in a grand fashion. There is a very persuasive analogy for this point of view. In many body problems near critical points one describes the free energy of the system in terms of mean fields which average over large blocks of local, more fundamental coordinates. These mean fields interact; their quanta are emitted, are absorbed, and propagate. They are the clear analogue of our Reggeon field. One never produces the quanta of the mean field as free states outside the medium (electrons and ion cores) which gives them life. One never produces Reggeons as free states outside the medium (particles acting as sources) which gives them life. At the critical temperature the correlation length for static correlation functions goes to infinity. This as usual, is the signal for a long range interaction mediated by a massless particle. We, too, have a massiess excitation called the Pomeron. It provides for infinite range correlations in rapidity, the time dimension. The universality of scaling functions and critical exponents in the theory of second order phase transitions carries over directly into our Reggeon language. We, when a(O) = 1, sit precisely at the analogue of T= Tcriticai. This marvelous universality means that phase transition phenomena near critical points will not teach one about the detailed dynamics underlying the observed phenomena. So, too, are we not learning aboumt the detailed dynamics underlying the Pomeron by studying very large s, small t elastic amplitudes. This brings us to the first of our critical problems now open for discussion in Reggeon physics; namely, the s-channel content of the theory we have built with quite explicit 1-channel unitarity. Since the Reggeons reflect an infinite number of s-channel production processes, tilere is clearly a rich well of information on the structure of the Reggeons to be plumbed by the detailed study of s-channel phenomena. We can say this in a language rather well adapted to the view of the generation of long range correlations in rapidity by Reggeon interactions. In the elastic amplitudes which have been the primary concern of our report, one probes the amplitude for a source (two particles) to emit Reggeons and then at a later time (rapidity) for another set of two particles acting as a sink to absorb them (fig. 6.3). In an s-channel production process such as single particle inclusive reactions in the triple Regge region there are three times involved log(mpro 2= 1on) logM2, and logs (fig. 6.4). [The energy scale is always mproton I GeV/c~.]We, by studying this inclusive process, are probing intermediate times, which requires the Pomeron to reveal some of its short range (in rapidity) structure. Other examples of intermediate time probes will come directly to mind. Each has its counterpart in an s-channel process which is part of the building up of the Reggeons. Study of these many time correlation functions should provide a systematic method to learn how to put together Reggeons. It has the incidental, highly non-trivial attractiveness of discussing experimentally accessible s-channel processes. The elucidation of these s-channel properties is then an issue of the first importance in Reggeon theories. Another issue, slightly more elusive perhaps, is that of the nature of bare Reggeons. The bare Pomeron is an important example. This is the quantity that is probed at present machine and col-
I,
11.1)1. .4 barhammel ct
180
a!., Rc’ggc’omm field timeort’: Jorrmmulation and use
x e~oqM~
Y
~~:Iog
S
i’ig. 6.3. I’iastic amplitudes study the two time correlation mmnetnmn for sources (called particles) tsm emit Repgcsmns ammd reabsorb tlmenm, Time timm~esare 0 aimd log a for time elastis’ process, I- or long timmmes time scaling solo lion of RI”’F is appl ir’:mble, It averages over enormmmous numnbers of s-channel inelastic prmcesses and over the lmadromm coordinates which are at time imearI nt Reggeon bmmil ding.
~ lip. 6.4, Single particle inclusive anmplitudes (as in fig. 6.21 in’ 2 , and log .r. ‘flme study of many voive t liree times: 0, log M timmmr’ correiatiomm tuns’tions reveals Imow Poimmeromms beha~e at internmediatc t immmes. Sucim correl:m tion fcmnutions’,mrc eqmmiv:mtemm
to learning about a’—cimanrmcl physics in
RI
1’.
hiding beam emiergies. Its strtmeture is stire to reflect the detailesi hadroil slynamiues of wliatevei’ thieoi’y, omle cilooses as an attractive candidate. We havemi’t really mumch of an imif’orniative natuire to say about tile building of’ Reggeons. Despite decades of clever work on the miiatter, (lie issume of’ how hadromls bind to f’orm hadrons amid Reggeons remains open amid enticing. The importance of’ the m’natter clearly transcends the somewllat cireumserihesi set of problems tins report has been able to brimig oum( ails!, insleesi. seems a ehallengillg note oii winch to f’imiish.
Ackno wiedgenients We woumid like to tilank our Reggeon Fielsi Tileory colleagues (‘or countless shscussiomis which clarified oumr thinking on the ideas presented here. Unforttmmla(ely they are too miumnieroums to cite imidividually. We are particularly indebted to Estiler Singer f’or 11cr extraordinary ef’l’orts in P~’~paring this mam’iuscript. Finally, three of ums (J.B.B., R.L.S. amid A.R.W.) woumhd like ts) express our appreciation for the hospitality extended to us at Fermilab, where tins work was begumn.
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[151CE. DeTar, D.Z.
182
11.1).!. ,‘i barbane! eta!,, Rr’ggeomm field theorm’: Jormimu!ation and
I 58~iL, Cardy, R.L. Sugar aimd AR. W’imite, U.C. Bcrkeie~ Sammta Barbara preprint 159] 11.1)1. Abarbancl, J , Bands, lB. llronzaim aimd I). Sidlmmm, to l,e published. ~-
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