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The vertex bootstrap conditions in the dual unitarisation scheme
Nuclear Physics B125 (1977) 264-274 © North-Holland Publishing Company
THE VERTEX BOOTSTRAP CONDITIONS IN THE DUAL UNITARISATION SCHEME Ken-ichi KONISHI Rutherford Laboratory, Chilton, Didcot, Oxon, England
J. KWIECINSKI Institute of Nuclear Physics, Krakow, Poland
Received 17 January 1977
The bootstrap conditions for the triple- and double-Reggevertices are derived utilising the planar unitarity condition for six- and five-pointamplitudes. The problem of symmetry breaking is considered within the one-dimensional approximation. It is shown that the vertex and propagator equations together predict the equal spacing rule for Reggetrajectory intercepts and a well defined pattern of symmetry breaking for vertices.
1. Introduction Much effort has recently been devoted to the exploration of the dual unitarisation (or topological expansion) [1-5], a new perturbative scheme for computing the strong interaction S-matrix which combines duality with unitarity equations. In this approach one starts with the planar amplitudes which are asymptotically controlled by the (exchange degenerate and ideally mixed) planar Regge poles. Their properties are the basis of all calculations of non-planar effects (such as the pomeron and violations of the Okubo-Zweig-lizuka rule). There exists a set of bootstrap conditions [6-11] which constrain the properties of planar Regge poles. For instance, requiring that the Regge propagators should not further be renormalised by planar loop insertions, one formulates the (propagator) bootstrap condition, which essentially fixes the scale of strong interactions. The purpose of this paper is to derive and to discuss similar bootstrap conditions for the triple Regge vertex and the reggeon-reggeon-particle vertex functions (sect. 2). Combining these with the corresponding conditions for propagators in the one dimensional approximation we find a simple solution for the possible pattern of SU(3) symmetry breaking for the trajectories and vertices (sect. 3). 264
265
K. Konishg J. Kwiecinski / Dual unitarisation scheme
2. Propagator and vertex bootstrap conditions The propagator bootstrap conditions are obtained from the planar unitarity equation for the two body amplitude. The detailed form of the planar bootstrap equation depends on the details of the production mechanism assumed for the evaluation of the unitarity integrals. In this note we follow for simplicity the formulation of Rosenzweig and Veneziano [7]. We adopt the following rule for writing the planar diagrams: (i) associate with the cut Regge propagator (apart from numerical constants and factors responsible for energy dependences) 1/P(cQ; (ii) the uncut Regge propagator, -P(1 - a) exp(-ina); (iii) the cut triple-Regge vertex, G. P(a); (iv) the suitably cut double-Regge-particle vertex, V- P(a) (see below eq. (5)). Consider now the s-channel discontinuity of the Regge amplitude (fig. la). The intermediate state in the unitarity summation can be divided into two sets of particles, with widths in rapidity A 1 and A 2 satisfying O~AI~Y-yo,
0<~A2~
where Yo is an arbitrarily chosen point between ~ Y and - ~ Y. Assuming that the two sets of particles are produced via Regge exchanges (fig. lb), and performing the summation over intermediate state in the two sets separately, one arrives at the diagram of fig. lc. Consistency of both sides of fig. 1 leads to [7]
(r(~[_~:l)} -I
X P(1
G2(q, ~)r(1 - a[-(q + ~ 5)2]) AZflvjd2q (a(-62) -c~ ( - [ q + 1 6 ]2) --a (--[q 1 6]2) + 1}2
-a[-(q-15)2])cosrr[o~(-[q+~Sl2)-oL(-[q-15]2)],
(1)
where G is the triple-Regge vertex function, a is the Regge trajectory and N corre-
1
Y/2 )At =
, i
-V12
=
q÷
% 6 /
) zx2
-¥/2 Fig. 1. The planar propagator bootstrap and the kinematics.
Fig. 2. The quark diagram corresponding to the propagator bootstrap.
K. KonishL J. Kwiecinski / Dual unitarisation scheme
266
S2
-v/2 ,-'4-'/
--
(a)
(hi I
-----
~
(c)
/ /
Fig. 3. Discontinuity in s t channel of the six point amplitude and derivation of the triple Regge vertex bootstrap. sponds to the (flavour) symmetry SU(N). The two-dimensional vectors q and 6 are defined in fig. 1c. This equation can be generalised to the case of broken SU(N) symmetry:
{r( ij[-8
-'=
U
{" d2 q G~.(q, 6)G~.(q, 6)P(] - e~ig[-(q + ~-5)2])
k=l J ~aij(--f 2)--OLik( - [q + ~ fi ]~--ajk~-- [q ~ ~ 1 )
× F(1 - ajk[-(q - ~5)21) cos lr[c~ik(-[q + ~6]2)-a/k(-[q
- 1612)1,
2 (2)
where indices i, / and k refer to the quark lines as shown in fig. 2 * An argument similar to that which led to eq. (2), can be used in order to derive the vertex bootstrap conditions. First, let us take a six-point amplitude in the triple Regge region, and consider its discontinuity in the sl channel (fig. 3a). We assume that this discontinuity is built up through the production of two sets of particles via Regge exchanges (fig. 3b). Taking advantage o f the arbitrariness of such a separation, we require the upper and lower sets to be limited in such a way that Al~ly-yl,
A2<~Yl + I Y ,
(3)
where A1 and A 2 are the widths in rapidity of the two sets of particle and Yl is the point in rapidity scale corresponding to the triple Regge vertex **. Replacing the sum* We choose the convention of labelling the vertices by following the quark arrows drawn in the clockwise direction. ** A different choice of this separation would lead to a consistency condition identical to eq. (2).
K. KonishL J. Kwiecinski / Dual unitarisation scheme iIdI H
267
'' t/
s1
8
'
8
/
f
8
]
Fig. 4. The triple Regge vertex bootstrap and the kinematics.
mations over the upper and lower sets of intermediate states by the Regge pole exchanges one arrives at fig. 3c. In order to avoid certain ambiguities which occur in the real part of the triple Regge amplitude, we find it convenient to consider the M 2 discontinuity of both sides of the equation presented in fig. 3 (see fig. 4). This guarantees that the same triple Regge vertex appears on both sides. After integration over A~ and A2 in rapidity, one obtains the factor s~ik(M2)°ti]
- t~ik - C~]k(__S2)Otjk
on both sides, and the consistency of the coefficients leads to the following vertex bootstrap condition:
a~(q,5)
{F(~/k[--(q + ~1 ~ 2) 1 }) - 1
N =/=~1 f d 2 q '
X
l
1
q'
1 ,l ' -lq;q+~g)Gq(q,8) {1 + aq(-8 2 )-c~iR-[q , + 1612)-~lj(-[q' - i612))
Gki(46+
'
'
½6).r(1-a,[-(q
+~
{1 + otik(-iq + l s ] 2 ) - O q l ( - [ q ' + ½ 6 1 2 ) - ~ t k ( - [ q
x F(1 - akl[--(q'
-
q)2])
COS
1)
- q'12)}
rr[oqt(--[q' + ½6]2)--~lk(--[q ' -- q ] 2 ) ] ,
where the corresponding quark indices are defined in fig. 5.
=
E L
Fig. 5. The quark diagram corresponding to the triple Regge vertex bootstrap.
(4)
K. KonishL J. Kwiecinski /Dual unitarisation scheme
268
It
t----
ljr
Sl (, ,-, i "-4
,,
- -
/5 -
(a)
q~_q
Fig. 6. (a) The bootstrap for the Regge-Regge-particle vertex and the kinematics. (b) The quark diagram corresponding to the bootstrap of (a).
A parallel argument applied to the five point amplitude in the double Regge region, leads to the vertex b o o t s t r a p condition for the reggeon-reggeon-particle vertex function (see fig. 6)
V~(q, 5) {r(a/k[--(q + 15)2])}-1 =
~
l X
d2q '
Gli(q, lq __ 1 6 . q __ ½ 3 ) Gik(q l ' -- ½q + ~fi;q + 16) (1 +O~ik(--[q -- 1612)--0./i/(--[q' -- 1612)--Ollk(--[q -- q']2)}
l i Vh(q, 6)r(1 - ,~/~[ - ( q ' + ~6)2]) {1 + ~/k(-[q + ½512)-a/t(-[q ' + ~6]2)--ulk(--[q - q,]2)}
X r(1 - akt[-(q' - q)2]) cos 7 r [ ~ k t ( - [ q - q ] 2 ) - a q ( - [ q '
+ ~-612)] .
(5)
In eq. (5), Vi~(q, 6) is the a p p r o p r i a t e reggeon-reggeon-particle vertex corresponding to disc s discslT s of fig. 6a. Eq. (4) and eq. (5) should be s u p p l e m e n t e d b y two conditions which guarantee the reality o f the vertex functions G and V. They are obtained from eq. (4) and eq. (5), substituting sine instead o f cosine functions in the integrands and setting zero on the left hand sides. The eqs. (2), (4) and (5) are close in spirit to the b o o t s t r a p conditions discussed earlier in the literature [12] *. There are however several i m p o r t a n t new features in eqs. (2), (4) and (5). First, the quantities appearing in those equations are Regge tra* From the formal point of view, conditions eqs. (2), (4) and (5) resemble the field theoretical conditions for vanishing of the wave function and vertex renormalisation constants Z 1 and Z 3 [ 121. The underlying field theory here is the (planar) Reggeon field theory suitably formulated to take into account the planar unitarity signature factor cos rr(a 1 - a2) in the Reggeon loop.
K. Konishi. J. Kwiecinski /Dual unitarisation scheme
269
jectory functions, triple and double Regge vertices, which are directly related to the asymptotic behaviour of hadronic amplitudes. Secondly, the quantities under consideration are all planar ones. The leading meson trajectories correspond to the planar ones. The leading meson trajectories corrrespond to the planar combination of vector and tensor trajectories (which have the phase exp4ns). The vertices are oriented, and together with the fact that the upper and lower legs of the triple Regge vertices are not equivalent, this means that Giki possesses no obvious symmetry in the indices k, i and j *. This last point is important in the discussion of symmetry breaking. Another notable aspect of these bootstrap equations is that they are obtained as the consistency conditions between dual dynamics and unitarity, and thus have a clearer meaning than those in the earlier formulations.
3. SU(3) Symmetry breaking in the one-dimensional approximation In this section, we study the propagator and vertex bootstrap conditions (eqs. (2), (4) and (5)) in the one dimensional approximation with the aim of finding the possible pattern of symmetry breaking, both for trajectories and vertices. The bootstrap conditions take the following form in the one dimensional approximation:
F(si/)- 1 = ~ Gq Gji F(1 - Sik)l-'(1 -- Sjk ) , k ( a i i - aik - sik + 1 )2
t
(Sij
G~LGt). I'(1 - si~)r(1 - s,k) - - SIj + 1) (aik -- Sit -- Stk + 1) '
-- Sil
l (sik -- sit -- ~lk + 1 ) (sik -- sjl -- sik + 1 ) '
(6)
(7)
(8)
where Gi/~kand V~ are the vertices corresponding to G/~ and V~ respectively, suitably averaged over the transverse momentum q, and sij are the effective trajectory intercepts. In broken SU(3) symmetry, one has in general, = ao
(9)
+
O,
(10)
~q = a0 (1 + 3ii} ,
(11)
= Vo{1 +
where g~j, vijk and 36 are all small. Substituting eqs. ( 9 ) - ( 1 1 ) into eqs. ( 6 ) - ( 8 ) (considered in first order with respect to g~., vii and ~3i/), one finds after some algebra * An exception is the relation: (;~1= Gijk* = (;~. which expresses the mutual equality of vertices with particles and antiparticles.
270
K. KonishL J. Kwieeinski / D u a l unitarisation scheme
J V V
E m I ~
i
k
=
k
i
k
(a)
v~v
m2-.. i
V V
=
(b)
I
Fig. 7. (a) Multiparticle production via vertices 1/in a reggeon-reggeon-scattering. (b) A Reggepole amplitude in a reggeon-reggeon scattering.
(see appendix): 13i] = e i + e/ ,
(12)
g/~. = g k ,
(13)
oilk = vi i ,
(14)
and moreover Ot0 - ~
(15)
as the consistency condition for having non-trivial solution satisfying eqs. ( 1 2 ) - ( 1 4 ) . Thus the direction the symmetry could be broken in the SU(3) space is restricted to that given in eqs. ( 1 2 ) - ( 1 4 ) b y the bootstrap conditions. Eq. (12) is the well known equal spacing rule for trajectories, ~P "- C~K* = OtK* -- ~0 "
Let us notice that this result follows without making any assumptions except that the deviations from the symmetric limit is small *. Assuming exact SU(2) symmetry, one gets from eq. (13), Gp;pp = GK,;K,p = G K , ; p K ,
= G¢;K,F** ,
1" The propagator bootstrap (eq. (6)), alone does not lead to any specific results concerning the
breaking pattern of SU(3) symmetry.
K. Konishi, J. Kwiecinski / Dual unitarisation scheme
271
where Go_ KO,K_ , --7-GnPA ,
etc.
The meaning of the solution given by the formulae (12)-(14) may become clearer by the following argument. It is easy to see that with the trajectories and vertices given by eqs. (9)-(14), each term from those shown in fig. 7a has the same structure with respect to the dependence on indices i, j, k and l, as the Regge exchange amplitude shown in fig. 7b *. Eqs. (12)-(14) may be interpreted as a generalisation of the Chan-Paton rule [13] for writing the dual coupling, to the case of broken SU(3). One should keep in mind, however, that strictly speaking this result applies to the effective couplings of the one dimensional approximation. Eqs. (12)-(14) have been assumed in the recent discussion of symmetry properties of the pomeron [10].
4. Summary To sum up, we have formulated the vertex bootstrap conditions based on the planar unitarity equation for six and five point amplitudes. Those conditions strongly constrain both Regge trajectory and vertex functions. In the one dimensional approximation a very simple and physically appealing pattern of SU(3) symmetry breaking emerges. The resulting coupling scheme may be interpreted as a generalization of the Chan-Paton factors. The authors are much indebted to Drs. R.C. Arnold, Chan Hong-Mo and J. Paton for several discussions, and to Dr. R.G. Roberts for reading the manuscripts. One of us (J.K.) thanks Dr. R.J.N. Phillips for his hospitality at the Theory Division of the Rutherford Laboratory.
Appendix In this appendix we give the derivation of eqs. (12)-(15). k °ijk and[3i/. Eqs. (6)-(8) take the following form in first order with respect to gij, defined in eqs. (9)-(11), d C~o~3q ~ {l-'(c@-' Ic~=~o = 2 a ~ f 2 {1-'(1 - ao)} 2 ~ g~. k
* Actually one can construct a multiperipheral model for the planar amplitude, using eqs. (12) and (14) as an input, and consistently producing output Reggeamplitudes with the properties given by eq. (12) and (13).
K. Konishi, J. Kwiecinski /Dual unitarisation scheme
272
+ soGoEFZr {(1 - So) ~
+ s o G o 2 F ~dF 2 d
Gosot3ik ~
d
r(1 - So) ~k (~ik + ~]k) (AJ)
"IV(J- - - 0~0)} 2 2 k (~i] -- nik -- n ] k ) ,
(P(So)} -1 + Go { r ( s o ) } - ' g~ = a3oF2 { r o - so)} 2
X~(g/ki+ ~-+-,3,~2 rv 1 d l gqt + gl]) 001" s°'t -S°)d-soP(1 --So) G(~il+nlk)l dF
3
+GosoF-d~ (F(1 - So)} 2 ~ ~i/- nit- nl/+ nik -nill
(A.2)
nlk} ,
d
V°s° d~ao (1-'(a°)}-~ ~/k + Vo (ESo)} -1 v~. = G2VoF2 (r(l - So)} 2
X Gt (gl. +~a +@}, + GgVosoF2 r o
d
-
+GgVoso {I"(1 -So)}2F~dF ~1 ~3ik- - ~il - - ~ l k
r(1 - so) G ,
+ ~ j k - - ~]l - - ~ l k }
+
,
(A.3)
where Z = --~0
,
F(Z) = 1/(1 + Z).
(A.4)
The SU(3) symmetric part of the bootstrap conditions, eqs. (6)-(8), is 1
(A.5)
o
G°~ = 3C(so) {r(1 - ,~o))2F ~
Using eq. (A.5) in eqs. (A.1)-(A.3), one finds
Cl~ij=~ ~ j
* ~C2 G~i,+f3fl} * ~C3 ~ {-/5il- nil} + 2c3f3i/, (A.6) 1
l
t
k
G{~.+ntk}
+ ~C3 ~ { - 213il-flq-ntk} +c3(nii +{3ik), 1
(A.7)
K. Konishi, J. Kwiecinski / Dual unitarisation scheme
+ 4--'
+
Jr 1 C3 E l
+,
+
273
l
{ - 2~l k - f l i l - ~/l} + C3(~ik -t- ~]k) ,
(A.8)
where
d
C l -z ~ o
~ 0 In (P(ao)} -1
c2 --o
--
d
aao
,
in {CO - ~ o ) } ,
d C3 -= ao ~ In F ( Z ) .
(A.9)
Performing summation over the index k in (A.7) we obtain (Jq = ei + el + (C1 -
C2) ~ (/3il),
3Ca
(A.10)
l
where C 3 -- C 2
1 E g l k i + ~_E ~a} + _ _ ei = - 9C-~ k,i ' l 18C
3
,
(/3l,k) •
(A.11)
From the requirement of having non-trivial solution for J3i/s which is symmetric in the indices, we obtain C! = C2,
(A.12)
~ij = ei + el .
(A.13)
It is seen from eqs. (A.9) that the condition (A.12) is equivalent to c o t 7rot 0 = 0 ,
(A.14)
which leads to eq. (15) of the main text. From (A.6), (A.7) and (A.13) one gets g~.- ~ ~ g ~ . = 0 ,
(A.15)
which implies g~]. = gk .
A similar treatment of eq. (A.8) leads to eq. (14) of the text.
(A.16)
274
K. KonishL J. Kwiecinski / Dual unitarisation scheme
References [1] [2] [3] [4] [5] [6] [7] [81 [9] [10[ [ 11[
[12]
[13]
G. Veneziano, Phys. Letters 52B (1974) 220; Nucl. Phys. B74 (1974) 365. M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. N98 (1975) 472,493. Chart Hong-Mo, J.E. Paton and S.T. Tsou, Nucl. Phys. B86 (1975) 479. Chart Hong-Mo, J.E. Paton, Tsou S.T. and Ng Sing Wai, Nucl. Phys. B92 (1975) 13. Chan Hong-Mo and Tsou S.T., Rutherford Lab. report RL-76-080; K. Konishi, Rutherford Lab. report RL-76-095. Chan Hong-Mo et al., ref. [41; J. Kwiecinski and N. Sakai, Nucl. Phys. B106 (1976) 44. C. Rosenzweig and G. Veneziano, Phys. Letters 52B (1974) 335. J.R. Freeman and Y. Zarmi, Nucl. Phys. Bl12 (1976) 303; J.R. Freeman, Y. Zarmi and G. Veneziano, Nucl. Phys. 120 (1977) 477. S. Feinberg and D. Horn, Tel Aviv Univ. preprint, TAUP-511-76. N. Papadopoulos, C. Schmid, C. Sorensen and D.M. Webber, Nucl. Phys. B101 (1975) 189. J. Dias de Deus and J. Uschersohn, Physica Scripta, to be published, Rutherford Lab. report, RL-76-042: K. Konishi, Nucl. Phys. B l I 6 (1976) 356. R.E. Cutkosky, Phys. Rev. 131 (1963) 1888; R.E. Cutkosky and P. Tarjanne, Phys. Rev. 132 (1963) 1354; R.H. Capps, Phys. Rev. Letters 10 (1963) 312; Chan Hong-Mo, P.C. De Celles and J.E. Paton, Phys. Rev. Letters 11 (1963) 521; J.S. Dowker and J.E. Paton, Nuovo Cimento 15 (1963) 450. Chan Hong-Mo and J. Paton, Nucl. Phys. B10 (1969) 516.
THE VERTEX BOOTSTRAP CONDITIONS IN THE DUAL UNITARISATION SCHEME Ken-ichi KONISHI Rutherford Laboratory, Chilton, Didcot, Oxon, England
J. KWIECINSKI Institute of Nuclear Physics, Krakow, Poland
Received 17 January 1977
The bootstrap conditions for the triple- and double-Reggevertices are derived utilising the planar unitarity condition for six- and five-pointamplitudes. The problem of symmetry breaking is considered within the one-dimensional approximation. It is shown that the vertex and propagator equations together predict the equal spacing rule for Reggetrajectory intercepts and a well defined pattern of symmetry breaking for vertices.
1. Introduction Much effort has recently been devoted to the exploration of the dual unitarisation (or topological expansion) [1-5], a new perturbative scheme for computing the strong interaction S-matrix which combines duality with unitarity equations. In this approach one starts with the planar amplitudes which are asymptotically controlled by the (exchange degenerate and ideally mixed) planar Regge poles. Their properties are the basis of all calculations of non-planar effects (such as the pomeron and violations of the Okubo-Zweig-lizuka rule). There exists a set of bootstrap conditions [6-11] which constrain the properties of planar Regge poles. For instance, requiring that the Regge propagators should not further be renormalised by planar loop insertions, one formulates the (propagator) bootstrap condition, which essentially fixes the scale of strong interactions. The purpose of this paper is to derive and to discuss similar bootstrap conditions for the triple Regge vertex and the reggeon-reggeon-particle vertex functions (sect. 2). Combining these with the corresponding conditions for propagators in the one dimensional approximation we find a simple solution for the possible pattern of SU(3) symmetry breaking for the trajectories and vertices (sect. 3). 264
265
K. Konishg J. Kwiecinski / Dual unitarisation scheme
2. Propagator and vertex bootstrap conditions The propagator bootstrap conditions are obtained from the planar unitarity equation for the two body amplitude. The detailed form of the planar bootstrap equation depends on the details of the production mechanism assumed for the evaluation of the unitarity integrals. In this note we follow for simplicity the formulation of Rosenzweig and Veneziano [7]. We adopt the following rule for writing the planar diagrams: (i) associate with the cut Regge propagator (apart from numerical constants and factors responsible for energy dependences) 1/P(cQ; (ii) the uncut Regge propagator, -P(1 - a) exp(-ina); (iii) the cut triple-Regge vertex, G. P(a); (iv) the suitably cut double-Regge-particle vertex, V- P(a) (see below eq. (5)). Consider now the s-channel discontinuity of the Regge amplitude (fig. la). The intermediate state in the unitarity summation can be divided into two sets of particles, with widths in rapidity A 1 and A 2 satisfying O~AI~Y-yo,
0<~A2~
where Yo is an arbitrarily chosen point between ~ Y and - ~ Y. Assuming that the two sets of particles are produced via Regge exchanges (fig. lb), and performing the summation over intermediate state in the two sets separately, one arrives at the diagram of fig. lc. Consistency of both sides of fig. 1 leads to [7]
(r(~[_~:l)} -I
X P(1
G2(q, ~)r(1 - a[-(q + ~ 5)2]) AZflvjd2q (a(-62) -c~ ( - [ q + 1 6 ]2) --a (--[q 1 6]2) + 1}2
-a[-(q-15)2])cosrr[o~(-[q+~Sl2)-oL(-[q-15]2)],
(1)
where G is the triple-Regge vertex function, a is the Regge trajectory and N corre-
1
Y/2 )At =
, i
-V12
=
q÷
% 6 /
) zx2
-¥/2 Fig. 1. The planar propagator bootstrap and the kinematics.
Fig. 2. The quark diagram corresponding to the propagator bootstrap.
K. KonishL J. Kwiecinski / Dual unitarisation scheme
266
S2
-v/2 ,-'4-'/
--
(a)
(hi I
-----
~
(c)
/ /
Fig. 3. Discontinuity in s t channel of the six point amplitude and derivation of the triple Regge vertex bootstrap. sponds to the (flavour) symmetry SU(N). The two-dimensional vectors q and 6 are defined in fig. 1c. This equation can be generalised to the case of broken SU(N) symmetry:
{r( ij[-8
-'=
U
{" d2 q G~.(q, 6)G~.(q, 6)P(] - e~ig[-(q + ~-5)2])
k=l J ~aij(--f 2)--OLik( - [q + ~ fi ]~--ajk~-- [q ~ ~ 1 )
× F(1 - ajk[-(q - ~5)21) cos lr[c~ik(-[q + ~6]2)-a/k(-[q
- 1612)1,
2 (2)
where indices i, / and k refer to the quark lines as shown in fig. 2 * An argument similar to that which led to eq. (2), can be used in order to derive the vertex bootstrap conditions. First, let us take a six-point amplitude in the triple Regge region, and consider its discontinuity in the sl channel (fig. 3a). We assume that this discontinuity is built up through the production of two sets of particles via Regge exchanges (fig. 3b). Taking advantage o f the arbitrariness of such a separation, we require the upper and lower sets to be limited in such a way that Al~ly-yl,
A2<~Yl + I Y ,
(3)
where A1 and A 2 are the widths in rapidity of the two sets of particle and Yl is the point in rapidity scale corresponding to the triple Regge vertex **. Replacing the sum* We choose the convention of labelling the vertices by following the quark arrows drawn in the clockwise direction. ** A different choice of this separation would lead to a consistency condition identical to eq. (2).
K. KonishL J. Kwiecinski / Dual unitarisation scheme iIdI H
267
'' t/
s1
8
'
8
/
f
8
]
Fig. 4. The triple Regge vertex bootstrap and the kinematics.
mations over the upper and lower sets of intermediate states by the Regge pole exchanges one arrives at fig. 3c. In order to avoid certain ambiguities which occur in the real part of the triple Regge amplitude, we find it convenient to consider the M 2 discontinuity of both sides of the equation presented in fig. 3 (see fig. 4). This guarantees that the same triple Regge vertex appears on both sides. After integration over A~ and A2 in rapidity, one obtains the factor s~ik(M2)°ti]
- t~ik - C~]k(__S2)Otjk
on both sides, and the consistency of the coefficients leads to the following vertex bootstrap condition:
a~(q,5)
{F(~/k[--(q + ~1 ~ 2) 1 }) - 1
N =/=~1 f d 2 q '
X
l
1
q'
1 ,l ' -lq;q+~g)Gq(q,8) {1 + aq(-8 2 )-c~iR-[q , + 1612)-~lj(-[q' - i612))
Gki(46+
'
'
½6).r(1-a,[-(q
+~
{1 + otik(-iq + l s ] 2 ) - O q l ( - [ q ' + ½ 6 1 2 ) - ~ t k ( - [ q
x F(1 - akl[--(q'
-
q)2])
COS
1)
- q'12)}
rr[oqt(--[q' + ½6]2)--~lk(--[q ' -- q ] 2 ) ] ,
where the corresponding quark indices are defined in fig. 5.
=
E L
Fig. 5. The quark diagram corresponding to the triple Regge vertex bootstrap.
(4)
K. KonishL J. Kwiecinski /Dual unitarisation scheme
268
It
t----
ljr
Sl (, ,-, i "-4
,,
- -
/5 -
(a)
q~_q
Fig. 6. (a) The bootstrap for the Regge-Regge-particle vertex and the kinematics. (b) The quark diagram corresponding to the bootstrap of (a).
A parallel argument applied to the five point amplitude in the double Regge region, leads to the vertex b o o t s t r a p condition for the reggeon-reggeon-particle vertex function (see fig. 6)
V~(q, 5) {r(a/k[--(q + 15)2])}-1 =
~
l X
d2q '
Gli(q, lq __ 1 6 . q __ ½ 3 ) Gik(q l ' -- ½q + ~fi;q + 16) (1 +O~ik(--[q -- 1612)--0./i/(--[q' -- 1612)--Ollk(--[q -- q']2)}
l i Vh(q, 6)r(1 - ,~/~[ - ( q ' + ~6)2]) {1 + ~/k(-[q + ½512)-a/t(-[q ' + ~6]2)--ulk(--[q - q,]2)}
X r(1 - akt[-(q' - q)2]) cos 7 r [ ~ k t ( - [ q - q ] 2 ) - a q ( - [ q '
+ ~-612)] .
(5)
In eq. (5), Vi~(q, 6) is the a p p r o p r i a t e reggeon-reggeon-particle vertex corresponding to disc s discslT s of fig. 6a. Eq. (4) and eq. (5) should be s u p p l e m e n t e d b y two conditions which guarantee the reality o f the vertex functions G and V. They are obtained from eq. (4) and eq. (5), substituting sine instead o f cosine functions in the integrands and setting zero on the left hand sides. The eqs. (2), (4) and (5) are close in spirit to the b o o t s t r a p conditions discussed earlier in the literature [12] *. There are however several i m p o r t a n t new features in eqs. (2), (4) and (5). First, the quantities appearing in those equations are Regge tra* From the formal point of view, conditions eqs. (2), (4) and (5) resemble the field theoretical conditions for vanishing of the wave function and vertex renormalisation constants Z 1 and Z 3 [ 121. The underlying field theory here is the (planar) Reggeon field theory suitably formulated to take into account the planar unitarity signature factor cos rr(a 1 - a2) in the Reggeon loop.
K. Konishi. J. Kwiecinski /Dual unitarisation scheme
269
jectory functions, triple and double Regge vertices, which are directly related to the asymptotic behaviour of hadronic amplitudes. Secondly, the quantities under consideration are all planar ones. The leading meson trajectories correspond to the planar ones. The leading meson trajectories corrrespond to the planar combination of vector and tensor trajectories (which have the phase exp4ns). The vertices are oriented, and together with the fact that the upper and lower legs of the triple Regge vertices are not equivalent, this means that Giki possesses no obvious symmetry in the indices k, i and j *. This last point is important in the discussion of symmetry breaking. Another notable aspect of these bootstrap equations is that they are obtained as the consistency conditions between dual dynamics and unitarity, and thus have a clearer meaning than those in the earlier formulations.
3. SU(3) Symmetry breaking in the one-dimensional approximation In this section, we study the propagator and vertex bootstrap conditions (eqs. (2), (4) and (5)) in the one dimensional approximation with the aim of finding the possible pattern of symmetry breaking, both for trajectories and vertices. The bootstrap conditions take the following form in the one dimensional approximation:
F(si/)- 1 = ~ Gq Gji F(1 - Sik)l-'(1 -- Sjk ) , k ( a i i - aik - sik + 1 )2
t
(Sij
G~LGt). I'(1 - si~)r(1 - s,k) - - SIj + 1) (aik -- Sit -- Stk + 1) '
-- Sil
l (sik -- sit -- ~lk + 1 ) (sik -- sjl -- sik + 1 ) '
(6)
(7)
(8)
where Gi/~kand V~ are the vertices corresponding to G/~ and V~ respectively, suitably averaged over the transverse momentum q, and sij are the effective trajectory intercepts. In broken SU(3) symmetry, one has in general, = ao
(9)
+
O,
(10)
~q = a0 (1 + 3ii} ,
(11)
= Vo{1 +
where g~j, vijk and 36 are all small. Substituting eqs. ( 9 ) - ( 1 1 ) into eqs. ( 6 ) - ( 8 ) (considered in first order with respect to g~., vii and ~3i/), one finds after some algebra * An exception is the relation: (;~1= Gijk* = (;~. which expresses the mutual equality of vertices with particles and antiparticles.
270
K. KonishL J. Kwieeinski / D u a l unitarisation scheme
J V V
E m I ~
i
k
=
k
i
k
(a)
v~v
m2-.. i
V V
=
(b)
I
Fig. 7. (a) Multiparticle production via vertices 1/in a reggeon-reggeon-scattering. (b) A Reggepole amplitude in a reggeon-reggeon scattering.
(see appendix): 13i] = e i + e/ ,
(12)
g/~. = g k ,
(13)
oilk = vi i ,
(14)
and moreover Ot0 - ~
(15)
as the consistency condition for having non-trivial solution satisfying eqs. ( 1 2 ) - ( 1 4 ) . Thus the direction the symmetry could be broken in the SU(3) space is restricted to that given in eqs. ( 1 2 ) - ( 1 4 ) b y the bootstrap conditions. Eq. (12) is the well known equal spacing rule for trajectories, ~P "- C~K* = OtK* -- ~0 "
Let us notice that this result follows without making any assumptions except that the deviations from the symmetric limit is small *. Assuming exact SU(2) symmetry, one gets from eq. (13), Gp;pp = GK,;K,p = G K , ; p K ,
= G¢;K,F** ,
1" The propagator bootstrap (eq. (6)), alone does not lead to any specific results concerning the
breaking pattern of SU(3) symmetry.
K. Konishi, J. Kwiecinski / Dual unitarisation scheme
271
where Go_ KO,K_ , --7-GnPA ,
etc.
The meaning of the solution given by the formulae (12)-(14) may become clearer by the following argument. It is easy to see that with the trajectories and vertices given by eqs. (9)-(14), each term from those shown in fig. 7a has the same structure with respect to the dependence on indices i, j, k and l, as the Regge exchange amplitude shown in fig. 7b *. Eqs. (12)-(14) may be interpreted as a generalisation of the Chan-Paton rule [13] for writing the dual coupling, to the case of broken SU(3). One should keep in mind, however, that strictly speaking this result applies to the effective couplings of the one dimensional approximation. Eqs. (12)-(14) have been assumed in the recent discussion of symmetry properties of the pomeron [10].
4. Summary To sum up, we have formulated the vertex bootstrap conditions based on the planar unitarity equation for six and five point amplitudes. Those conditions strongly constrain both Regge trajectory and vertex functions. In the one dimensional approximation a very simple and physically appealing pattern of SU(3) symmetry breaking emerges. The resulting coupling scheme may be interpreted as a generalization of the Chan-Paton factors. The authors are much indebted to Drs. R.C. Arnold, Chan Hong-Mo and J. Paton for several discussions, and to Dr. R.G. Roberts for reading the manuscripts. One of us (J.K.) thanks Dr. R.J.N. Phillips for his hospitality at the Theory Division of the Rutherford Laboratory.
Appendix In this appendix we give the derivation of eqs. (12)-(15). k °ijk and[3i/. Eqs. (6)-(8) take the following form in first order with respect to gij, defined in eqs. (9)-(11), d C~o~3q ~ {l-'(c@-' Ic~=~o = 2 a ~ f 2 {1-'(1 - ao)} 2 ~ g~. k
* Actually one can construct a multiperipheral model for the planar amplitude, using eqs. (12) and (14) as an input, and consistently producing output Reggeamplitudes with the properties given by eq. (12) and (13).
K. Konishi, J. Kwiecinski /Dual unitarisation scheme
272
+ soGoEFZr {(1 - So) ~
+ s o G o 2 F ~dF 2 d
Gosot3ik ~
d
r(1 - So) ~k (~ik + ~]k) (AJ)
"IV(J- - - 0~0)} 2 2 k (~i] -- nik -- n ] k ) ,
(P(So)} -1 + Go { r ( s o ) } - ' g~ = a3oF2 { r o - so)} 2
X~(g/ki+ ~-+-,3,~2 rv 1 d l gqt + gl]) 001" s°'t -S°)d-soP(1 --So) G(~il+nlk)l dF
3
+GosoF-d~ (F(1 - So)} 2 ~ ~i/- nit- nl/+ nik -nill
(A.2)
nlk} ,
d
V°s° d~ao (1-'(a°)}-~ ~/k + Vo (ESo)} -1 v~. = G2VoF2 (r(l - So)} 2
X Gt (gl. +~a +@}, + GgVosoF2 r o
d
-
+GgVoso {I"(1 -So)}2F~dF ~1 ~3ik- - ~il - - ~ l k
r(1 - so) G ,
+ ~ j k - - ~]l - - ~ l k }
+
,
(A.3)
where Z = --~0
,
F(Z) = 1/(1 + Z).
(A.4)
The SU(3) symmetric part of the bootstrap conditions, eqs. (6)-(8), is 1
(A.5)
o
G°~ = 3C(so) {r(1 - ,~o))2F ~
Using eq. (A.5) in eqs. (A.1)-(A.3), one finds
Cl~ij=~ ~ j
* ~C2 G~i,+f3fl} * ~C3 ~ {-/5il- nil} + 2c3f3i/, (A.6) 1
l
t
k
G{~.+ntk}
+ ~C3 ~ { - 213il-flq-ntk} +c3(nii +{3ik), 1
(A.7)
K. Konishi, J. Kwiecinski / Dual unitarisation scheme
+ 4--'
+
Jr 1 C3 E l
+,
+
273
l
{ - 2~l k - f l i l - ~/l} + C3(~ik -t- ~]k) ,
(A.8)
where
d
C l -z ~ o
~ 0 In (P(ao)} -1
c2 --o
--
d
aao
,
in {CO - ~ o ) } ,
d C3 -= ao ~ In F ( Z ) .
(A.9)
Performing summation over the index k in (A.7) we obtain (Jq = ei + el + (C1 -
C2) ~ (/3il),
3Ca
(A.10)
l
where C 3 -- C 2
1 E g l k i + ~_E ~a} + _ _ ei = - 9C-~ k,i ' l 18C
3
,
(/3l,k) •
(A.11)
From the requirement of having non-trivial solution for J3i/s which is symmetric in the indices, we obtain C! = C2,
(A.12)
~ij = ei + el .
(A.13)
It is seen from eqs. (A.9) that the condition (A.12) is equivalent to c o t 7rot 0 = 0 ,
(A.14)
which leads to eq. (15) of the main text. From (A.6), (A.7) and (A.13) one gets g~.- ~ ~ g ~ . = 0 ,
(A.15)
which implies g~]. = gk .
A similar treatment of eq. (A.8) leads to eq. (14) of the text.
(A.16)
274
K. KonishL J. Kwiecinski / Dual unitarisation scheme
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[12]
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