Final state hadrons in inclusive electroproduction processes

ANNALS

OF PHYSICS

104,

54-94 (1977)

Final State Hadrons

in Inclusive NAMIK

Electroproduction

Processes*

K. PAK+

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305 Received May 10, 1976

Inclusive deep inelastic electroproduction processes, in which one (or more) hadrons in the final state are observed in coincidence with the electron, are investigated using Regge-Mueller language, via the inclusive virtual photoproduction processes. The hadron distributions and generalized scaling laws are obtained. Assuming that Pomeron is the dominant Regge trajectory, it is a simple pole, and therefore factorizes at high energies. The contribution of the isospin-carrying secondary trajectories cause charge asymmetries in the central and current fragmentation regions, and these increase with QZ, for tixed total energy. Average multiplicity grows logarithmically with energy, as in the hadronic case. However, in the Bjorken limit, the major contribution comes from the current fragmentation region, contrary to the hadronic case; and the contribution of central region scales, and increases with increasing w- I. Finally, we show that the average transverse momentum of the produced particles is limited, and this limit depends on the density of the particles in the phase space (rate of increase of average multiplicity with In s), and o. All the predictions are consistent with the data.

I. INTRODUCTION The scaling laws proposed by Bjorken [l] for the highly inelastic electron scattering processes are consistent with all the experimental data [2], and at present this phenomena is quite well established and understood [3]. In purely hadronic production processes, another scaling law of rather different character had been proposed, quite a while ago [4]. The interest in this hadronic scaling law has been reviewed by Feynman [5] and Yang [5], quite recently. This scaling law is also consistent with the present experimental data [6]. Recently there has been increasing theoretical [7] and experimental [S] interest in deep inelastic lepton scattering processes in which one or more of the final hadrons are detected in coincidence with the scattered lepton (though the interest in these processes was overshadowed by the very excitting discoveries of heavy narrow vector mesons in the efe- annihilation channel [lo]). The most recent experimental results of SLAC, Cornell, and DESY were reported at the 1975 SLAC conference [9]. The data seem to offer no big surprises. * Work supported in part by the Energy Research and Development Administration. t Partially supported by the Scientific and Technical Research Council of Turkey.

54 Copyright All rights

(g 1977 by Academic Press. Inc. of reproduction in any form reserved.

ISSN

0003-4916

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On the theoretical side [8], the first attempts have been focused on the possibility that the invariant distributions will exhibit some sort of scaling laws similar to those of Bjorken and Feynman and Yang, The exclusive subsets of these processes have been investigated by Lee [8], using an SU, phenomenological Lagrangian technique. The inclusive processes have been analyzed by Drell and Yan [8], Landshoff and Polkingshorne [8], and Feynman [3], using different forms of the parton model. Stack [8] investigated the same problem by using the free-field light-cone commutators, and assumed the dominance of light-cone singularity for certain semiconnected diagrams. Attempts have been made by the present author [ll] and others [Ill to study the problem, by adapting the formalism developed by Mueller [12] for purely hadronic production processes. This approach is based on the very simple and plausible assumption that processes involving highly virtual photons are also dominated by Regge trajectory exchanges, Pomeron being the highest. at high energies. The price paid for not having any fictitious entities like partons or quarks is that the predictions in this approach are not as detailed as those of a parton model [3]. This work is a corrected and updated version of the second reference in [ 1I], which was the first thorough investigation of the problem. 1 All the predictions are in perfect agreement with present data [9]. A final word: Although I will always use the terminology electroproduction, everything I say applies to muon production also (the terminology leptoproduction does not seem to be popular somehow).

II.

DEEP INELASTIC

ELECTROPRODUCTION

For the sake of completeness we shall start with a short review of deep inelastic electron-proton scattering from the Regge point of view. By treating the electromagnetic interaction to lowest order, the process is represented by the Feynman diagram given in Fig. 1. By applying Feynman rules, the differential cross section for fixed incident electron energy and fixed scattering angle is given by do ikxmF=

-=da dQ” dv

9

g [ W2(Q2,v) COST t -+ 2 W,( Q2, v) sin2i], (2.0

1The last reference in [I I] was an attempt to modify the erroneous treatment of the current fragmentation (and some kinematical errors) in the second reference in [ll]. Unfortunately, they failed-first, to notice the w-dependence of the phase space, which is so crucial (especially in proving that no new regimes other than those studied in this work open up at high Qa; in other words, the phase space is completely filled by the secondaries). Second, their scaling variable (and also scaling distribution) for the current fragmentation region is not a scaling variable for high Qp, as is obvious from our list (4.30). So their only two new results being incorrect, they fail to outdate the second reference in (111. In this work, in addition to correctly and thoroughly treating the kinematics, we manage to get a unified scaling variable (which is related to Feynman’s purely hadronic scaling variable) for both fragmentation regions, which is supported beautifully by the data.

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FIG.

K. PAK

1. Optical theorem for deep inelastic electroproduction process.

where E(E’) is the laboratory energy of the incident (final) electron, 0 is the lab scattering angle, m is the nucleon mass, 01is the fine-structure constant, and q is the four momentum transferred to the proton by the photon; we have also defined Q2 = -q2. The invariant structure functions are defined by

(2.2)

where A,, = g,, - (quqv/q2). Both W, and W, are antisymmetric Wi(Q2, -v)

under v -+ -v:

= - Wi(Q2, v).

The total absorption cross section for longitudinal aLsT, is defined by

and transverse virtual photons,

(Flux) uT.L = 47r2cx~*T*L 11 W““e;*L,

(2.3)

where eTsL?are the polarization vectors for virtual photons which satisfy the gauge conditio; 4’. E = 0. In a frame in which q = (q,, , 0, 0, qJ we choose them as

cL = WQN, >f-40,a,), ET* !A = (l/29(0, Substituting

(2.4)

1, fi, 0).

in Eq. (2.3) we get (Flux) O’T = 47~~01 Wl(Q2, I’), (2.5)

(Flux) CJ‘ = 47r20r[( 1 + $)

W2(Q2, v) -

Wl(Q2, v)].

By dimensional arguments, oL - Q2 as Q2 + 0, and (JT (Q2 = 0, v) is the total photoabsorption cross section for real photons of energy v. It is suggestive to write the differential cross section in terms of uT and uL defined above: da dsZdE’

e2 (v” + Q2)li2 E’ uT z1_EP = 271.2 Q2

+ 43

(2.6)

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ELECTROPRODUCTION

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where E=

[

1+2tan+

+ $)]-’

(2.7)

is called the polarization parameter [13], is often small, and R = oL/aT . Bjorken [I] argued that the limits of WI and VI+‘, exist as both v and Qz tend to infinity with their ratio fixed:

here w = (Q2/2mv). Present experimental data [2] are in agreement with this proposal for Q2 ;> 0.5 (GeV/c)2, and we will take this scaling phenomenon. Bjorken scaling, as an experimental fact. Furthermore, experiments indicate that the ratio R is very small 121: R = 0.14 & 0.067. Now consider the forward elastic scattering of a massive photon of spacelike momentum q by a physical nucleon of momentump; so q” :~mm -Qz < 0, p2 = m?. The amplitude averaged over nucleon spins has the form T,JQ2, v) = i 1 d4.ue ia-yp 1 T[Jpyx) = -(4,, The structure functions

J;“(o)]

TdQ2, v> + (l/m2)(n

/ pl ~),(fl

WI,, are related to the absorptive

. p)u T2(Q2, vi

(2.9)

parts of T, and r2 : (2.10)

Wi(Q2, v) = (1/2n-) Im Ti(Q2, v).

It can be shown that [14] TI(Q2, v) IS . a helicity nonflip amplitude for the t-channel process y + y -+ N + ;ir. So we may decompose it into partial waves according to m

T,CQ”, v) = 1 (2J + 1) t1(Q2, J)[P~(cos

J=O

0,) -t PA--OS

01,

(2.11)

appropriately analytically continued from the region t > 4mN2, q? > 0 to the Compton region required for our problem, namely, t == 0, q2 < 0 with 1-j_ p . q physical (Fig. 2). Here 8, is the scattering angle in the center of mass frame of the t channel, and for small t, is given by cos Bt - v/q, where q = (q2)li2. Now let us look at the behavior of cos 9, at various limits: (a) 8, -+ x.

Regge limit. 1’ = p . q/m + co, while q2 is large and fixed. Obviously

cos

(b) Bjorken limit V, q2 --f ~13while w = Q2/2mv fixed. In this case, cos 0, (--q,‘2mw) ---f co. So the conditions for a Regge expansion, or to be more precise, a crossed channel SO(3) expansion, are satisfied in the Bjorken scaling region. The essential point is that the v-dependence in the Bjorken limit is exhibited by the Regeg representation.

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K. PAK

FIG. 2. Regge expansion of the forward virtual Compton scattering.

Dropping the background integral in the Sommerfield-Regge

expansion, we get

TdQ’, 4 - 1 Cb, + 1) 4f”1.) n [P&OS 6,) + P,,(-cos et)]. n

(2.1 la)

To leading order, Pa, (cos 0,) - (cos Ot)anas cos Bt ---f co, so we get (Fig. 2)

wI(Q2, 4 = &

Im 7’dQ2,4 = c & &,np(O) ($,““‘“‘, n

(2.1 lb)

where we have assumed factorization of Regge residues and absorbed the irrelevant factors into the /I’s. Obviously, the validity of this argument requires Q2 to be large. For small Q2 this is rather dubious, for the neglected terms would be comparable to the nonleading Regge contributions of the Regge expansion. There is also the question of the region of validity of expansion (2.11a). Since the sum extends over all leading trajectories, by the use of duality arguments, one may hope that (2.11a) is a good representation of the amplitude TI(Q2, V) even for moderate values of v, and not just for the asymptotic region. Note that we can pass to the Bjorken region from the Regge region, by keeping v large and fixed and letting q2---f co. We observe that the simplest way to obtain the scaling laws (2.8) from the Regge expansion (2.1 lb) is to have [15] (2.12) for the photon-photon-Reggeon vertex for large Q2. Then we see that, if we believe in Bjorken scaling, in some loose sense the photon-Reggeon coupling strength decreases as Q -+ co. This simple observation is very interesting and suggestive in the light of recently popular “asymptotically free field theories” [16], which claim that Bjorken scaling is a consequence of asymptotic freedom. Equation (2.12) is the crucial prescription we shall extensively use in the rest of the paper. Using (2.12) we get

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i.e., (2.13) Similar arguments apply to Wz(Q2, V) and we get a similar result: (2.14) Now if we take only the leading (Pomeron) trajectory, (2.13) and (2.14) yield F,(w) ;

+J?

E2(w) 2 20,

(2.15a) (2.15b)

in accord with the relation urL/ur -+ 0. A warning remark is appropriate here. Actually, the point 01(o)= 1 is a nonsense point for W, , and the Pomeranchuk trajectory decouplesunlessa fixed pole is present which restores its contributions. If we include the secondary Regge trajectory (I”), we get F,(w)

;

;

b

+ w1/2 7

(2.16) F,(w) ;

2a + 2b(w)1’2.

Notice that Fz goes to a constant from above, as l/w = (2~ .4)/Q” limit).

cc (Regge

111. INCLUSIVE ELECTROPRODUCTION III. 1. Definitions and Kinematics We shall consider the processin which an electron scatters from a hadron (nucleon), and one of the hadrons in the final state, in addition to the scattered electron. is detected: e(l) + h( p) - e’U’) + h’(k) + ff( pd. Treating the electromagnetic interaction to lowest order the processis represented by the Feynman diagram given in Fig. 3, where E(E) is the lab energy of the incident (final) electron. Applying Feynman rules, the differential cross section for fixed incident electron energy, fixed electron scattering angle, and fixed hadron scattering angles(0 and I#), and summing over all else,is given by (3.1)

60

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K.

PAK

H(‘H>

FIG.

3. Inclusive deep inelastic electroproduction.

where the masses of target and detected hadron are denoted by m and p. Simple Dirac algebra gives IWY- 2[1,1,’ + IJ, - g,,l *‘l’]

with

me2 -

0,

%v= 4 7 c (PIJ,+(O) IWPH), k)(k,H(PH) IJv(W> x (27713S4(q + p- k - PH) spins

H

(3.2)

where J,, is the hadron electromagnetic current, 1p) is a one-proton state, and j k, H) is a state of the one hadron being detected plus all possible others with quantum numbers summarized by H. Here CH means the sum over all intermediate states H, which is connected to the proton by Jem, and integration over their invariant phase space. Our metric, normalization of states, etc., are the same as in [17]. We denote the average over the initial spin by C. Lorentz invariance tells us that mUV must be a second-rank tensor. Because of the average over spins we have only the vectors qp , pu , k, , and the tensor g,, at our disposal. Tensors of the type eLIvaS q”pB or c,,,pqak@ are excluded because mUy has positive parity (current operator is a polar vector under spatial reflections). Furthermore, the electromagnetic current is conserved, i.e., q,,muv = muvqy = 0. Therefore, the most general form for the symmetric tensor l@@”is

with Lv = g,, - (qwqv/q2).

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ELECTROPRODUCTION

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Note that because of the conservation of the leptonic current, and a special choice of the gauge (E,q@ = 0), the relevant form of the tensor i%‘,,” in the inclusive virtual photoproduction processes is

For the process y + p 4 h + anything, we need 3 .. 4 - IO + 2 = 4 invariant independent variables, and for the process e + p ---+ CJ’ + h -~ anything, we need 3 ,.~ ~’ 5 - 10 + 1 = 6 independent invariant variables to express the cross sections. But because of the single-photon exchange dominance, hadronic and leptonic parts factor in the differential cross section as depicted in (3.1). Because of this fact, which is known as the locality of the lepton vertex, the hadronic tensor mUU cannot depend on the I, and I,‘, except insofar as the p, k. and pH are tied to I and I’ by the overall conservation laws. By virtue of these laws, q == 1 - I’ is, effectively, a hadronic fourvector, so p&v can depend on its components. But (I + I’), == g, is not a hadronic fourvector, and mUy cannot depend on its components to the extent that the latter are unconstrained. There are two such constraints: (a) The magnitude of q, is related to the magnitude of qu : p2 I (/ + /‘)2 = 4m,? - 4’; (b) Projection of p,, on q,, is not free either: qup” = 12 - 1’” = 0. This implies that six variables, which describe the whole process, can be chosen in such a way that mw,, is independent of two of them. The dependence of do on these two variables must, therefore, be explicitly contained in the lepton factor I,, . In the lab frame (rest frame of the target) we shall define the direction of qw as Oz axis. We shall take the Ox axis, in the plane spanned by the vectors I and I’. Then the azimuthal angle of the detected hadron, measured from the Ox axis, is going to be the angle between the planes (or their normals), defined I and I’ and q and k. We shall choose the four invariant variables to describe the hadronic tensor W,,. as: Qz = -q2 > 0, pv’ = k * q, mK =p*k, tan 4 = k,!k,

.

Two new variables are v’, the energy loss of the electrons in the rest frame of the detected hadron, and K, the energy of the detected hadron in the lab frame. Other sets

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K. PAK

of variables which are equally well suited for describing the hadronic part of the dn are (Q2, t, M2, 4) or (Q2, 1k I , cos 0, #)>, where t = (p - k)2 = m2 f p2 -

hK,

M2 = (p + q - k)2 = m2 + p2 + 2mv - 2t.w’ - 2mK - Q2.

(3.4)

Here 8 is the polar angle of the detected hadron, measured from the photon direction (Fig. 4). The invariant phase-space volume element, in terms of these different sets, is given by d3k -= 2ko

$fdIkjdcosBdq5, 0

dv’ dK d&

2(v2 ;Q2)‘i2 1

(3.5)

dt dM2 d#. = 8m(v2 + Q2)1/2

The old and new invariant constraints

variables (Q2, v) and (v’,

o
K)

satisfy the kinematical

2j.d + 2mK < 2mv(l - co),

(3.6)

which are derived from the positivity conditions and

s=(q+p)2>m2

M2=(q+p-k)2>0

(3.7)

in the Bjorken limit. As stated above, due to the summing over the final spins, and averaging over the initial spins (unpolarized lepton scattering over unpolarized proton target), the number of independent helicity amplitudes is four. The differential cross section in terms of these functions is da 012 E’ 1 dCYdE’(d3k/2ko) = G 2mE Q2(l -

l)

X [ W++ f EWoo- EW+-

cos

24 - 2(e + e2)1/2(ReW+O)cos ~$1.

(3.8) f

Y

lab

j

c.m.

Y

FIG. 4. Generalized optical theorem, for three-to-three amplitude.

INCLUSIVE

Here E is the polarization

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parameter defined in (2.7) and Wab are

where (&) stands for two transverse polarizations, and (0) stands for the longitudinal polarization as defined in (2.4). The lengthy relations between Wah and the l%i are given in Appendix 2. It is obvious from (3.8) that if we integrate over the azimuthal angle 4, the last two terms vanish, and we are left with two structure functions, Wf I and Woo. Let us do this more systematically by integrating the hadronic tensor Wuv over the azimuthal angle 4:

x

d4x ezq’x(p j J,+(x) s 47r

1k>(k

j J,(O) I p>.

(3.9)

Now WUv can be written in terms of two structure functions as

Y@” = -VO,v %(Q’,

v, v’, K> +

f

(A

and the differential cross section takes the familiar da

l \ dsZ’ dE’(d3k/2k,)

>

= !k!

Q4

El2

* P),CA

3 p)v

%(Q2,

K),

(3.9a)

form

[K~ cosa4 + 2%> sin2 i],

where ( ) stands for the azimuthal integration. The differential virtual inclusive photoproduction is defined in the usual way: da (F1ux) (d3k/2k,)

v, v',

- 47r2a~,*E,lP~v. -

(3.10)

cross section for

(3.11)

Conventionally, the flux factor for the virtual photon is defined as the flux factor for a real photon, with the same initial invariant mass (q + P)~ as (Flux) = q .p - iQ2 z nw(l

- w).

(3.12)

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PAK

We shall define the flux factor as we do for hadron beams, whenever it is convenient, as (Flux) = Kq *PI” + m2Q211/2. A simple calculation gives

+ $ After integrating

Substituting

Re(c*L . p)(eL * k) I@4.

over the azimuthal

angle we get

these in (3.11) we get

8rr2a Q2 x

2 - (v” + Q2)lj2 Wl(Q2 , v, v’, 4, p

2mv-

8n2a x 2 (v” + Q2)l12 2mv - Q2 p X

-KtQ2,

[

(3.14)

v, v’, K) + (1 + $)

“/y2(Q2,v, v’, K)].

In terms of the normalized inclusive distributions, which are defined as the ratio of the Lorentz invariant distribution to the total cross section (to get rid of all the irrelevant factors), we explicitly get

AQ”, v>v’, K) = 6

E,*E,?v-“~

> = 2(v2 :Q2)li2

E,%, wuv ’

(3.15)

in terms of the structure functions $

=

$

(9

+

Q2)W

$, 1

pW

zz $

(v”

+

Q2)W

[ I;;;;

(3.16) ;z

]. 2

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P T+L is written in this form, because we know that it is (l/m) v W, which scales in the

Bjorken limit. Noting that (v2 + Q 2) l/2 _- v in the Bjorken limit. we get

(3.17) T+L3 [(‘h/d P BI __ F,(w)

‘“%I .

From (3.17) it is obvious that for the inclusive cross sections (integrated over $) there is one more power of v, multiplying the scaling structure functions, compared to the ordinary electroproduction structure functions. Of course, everything we did above is true if the virtual photon vertex factors out. We shall see that this is not true in the photon end of the phase space; it is impossible to make such simple predictions. For Q2 --f 0 we can relate inclusive electroproduction to the inclusive photoproduction as follows (see Appendix 1):

du dQ” dv dv’ dtc >

In discussing purely hadronic inclusive processes it proved to be useful to parametrize the particle momenta in terms of rapidity variables. Therefore we shall use the same parametrization here also: p = m(cosh yZ , 0, 0, sinh JJJ, q = Q(sinh y1 , 0, 0, cash Jo),

k = PL (cash y, 2,

(3.19)

sinh y),

where k, is the transverse momentum of the produced particle, with p and q taken to be colinear in the z direction; and pI = (p2 + k12)lj2 is the transverse mass. The rapidity yi specifies the longitudinal Lorentz transformation that relates the lab frame to the rest frame of the ith hadron. For a space-like photon, the rapidity parameter is the Lorentz boost which relates the lab frame (or whichever frame in which this labeling is done) to the frame where the photon has only a space component and no energy. This frame is called a Breit (or brick-wall) frame, and its usefulness in processes involving highly space-like photons was advertized by Feynman in [3]. The invariants v = ( p . q)/nz, v’ = (k . q)/p, and K = (k . p)/ m, which we have chosen to represent

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the inclusive electroproduction,

K.

can be expressed in terms of these rapidity variables:

v = ;P

. q = Q sinh(y, - yZ),

v’ = $ k * q = 2 K= ;

PAK

Q sinh(y, - y),

k * q = pI cosh(y - yz).

The longitudinal and transverse momenta, k, and k, , of the detected hadron in the c.m. frame of the initial photon-hadron system are related to the rapidity variable as y = In ! k”Lk3)

=iln(

2::).

(3.21)

We put all longitudinally moving frames on an equal footing by the use of rapidity variables, since they are all related by a simple shift of the scale. That is, a longitudinal Lorentz transformation, characterized by /3 = tanh U, merely changes y to y’ = y + u. Let us now find the absolute kinematical limits imposed on y, by energy-momentum conservation. In the Regge region (which is a subregion of the Bjorken region), (3.22)

M2=(p+q-k)2~-Q2+2p*q-2q*k-2p.k~0.

From (3.20) we get, in the lab frame, v = Q Sh Y z $QeY, v’ = $ K =

Q Sh(Y - y),

plShy.

Since it is only K, which does not involve Y, we can neglect 2mK compared to other terms in (3.22), for the minimum value of y, and obtain 2mv(l - w) > 24. Substituting

in the value 2pv’ for y N ymrn , 24

- plQeY-‘min,

we finally obtain ymin

k

In (5)

+

In ($-A).

(3.23a)

INCLUSIVE

ELECTROPRODUCTION

Note that this lower limit is particularly

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simple for nucleon production:

for small w (which of course means the Regge limit). Again, from (3.20) we see that for is small; but because of the factor Q multiplying it, V’ cannot ?‘=ymax, y-yy,, be neglected compared to the other terms. From

2v 2v e”max + - eCumax5 L_ (1 - w). I?1

A careful study of this inequality

I

gives

.+h~ 5 In (+---)

= In (c)

$- In (*I.

Again notice that for nucleon production,

this upper limit gets very simple:

So in summary.

limits on y are

the absolute kinematical In (*)

+ In (A)

And for nucleon production

5 y ~2 In (;k,

t In (+I.

(3.23b)

(3.24)

in the Regge region (w ‘v 0), w 5 y 5 ln(s/m2).

One interesting thing about this result is that the lower limit depends on w. In the deep Bjorken limit this dependence gets more pronounced. Another distinctive feature is that the phase space (of the detected hadron) is not fixed by the rapidities of the target or projectile; in other words it is not y2 ,( -v < vI . Calculating the difference dy = ymax - y1 = In (1 - w) + In

(Q/,u,),

(3.25)

we see that, for Q2 N pL2, this quantity is negative and small (because Q2 N pL2 means w ‘v 0). As Q2 gets larger the second term gets large; but fortunately, the first

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PAK’

term also gets large and negative. Therefore we may think that, since the large pI production is severely suppressed [9, 221 ( see Section V for a detailed study), dy is always finite and never gets large. 111.3. Generalized Optical Theorem In purely hadronic inclusive reactions, Mueller [12] indicated that the inclusive cross section is a piece of the discontinuity of the forward three-to-three amplitude. Stapp’s [12] formal proof goes through for virtual particles also. Therefore, we are going to use the generalized optical theorem in our case as well, to relate the virtual photoinclusive cross section to the discontinuity of the forward three-to-three amplitude yhh’ -+ yhh’ (Fig. 5).

FIG. 5. Definition

of the kinematic variables in the laboratory

Denoting this discontinuity

and barycentric frames.

by A(q, p, h), the optical theorem states that

duTsL ~ (F1ux) d3k/2k, = ATsL(q,P, 4,

(3.26)

where ATsL(q,P, 4

E*T,LET,L LL Y

!

1

s

d*x eiQYp, -k / J,,+(X) J&O) / p, -k)

!

(3.26a) and du=J (F1ux) d3k/2k,,

&.&*T.LET.L ii

Y

!c I$

eiq% I J,,+(x) I k)/ (3.26b)

At least this is not implausible, since the virtual photon at hand is space-like, whereas it is known that the normal threshold branch points and cuts exist only in the right half of the q2 plane. On the other hand, in the production of p+p- pairs by timelike photons, these singularities may be relevant, and the use of Mueller’s results would require further justification.

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IV. 0(1,2) EXPANSIONS In his famous work, Mueller ]12] expanded the absorptive part A into O(1, 2) harmonics to investigate the scaling properties in the purely hadronic inclusive reactions. In doing so his main purpose was to connect the scaling properties of particle production (such as pionization and limiting fragmentation), and average multiplicity with Regge singularities familiar from two-body reactions. He showed that an appropriate single 0( 1, 2) expansion gives limiting fragmentation [5] (slow particles in the rest frame of the one of the initial particles are the fragments of that particular particle), and the appropriate double O(1, 2) expansion gives pionization (pionization products are those pions which maintain a finite momentum in the cm. frame of the initial particles as the energy of the initial particles become very large). Following Mueller, in getting our generalized scaling laws, we shall use O(1, 2) expansions, though our kinematic configuration again has sufficient symmetry to render an O(1, 3) expansion natural. If we use the O(1, 3) group, our expansion parameters would be the rapidity variables which we defined earlier. In both the Regge limit and the Bjorken limit y1 ~ ~9~becomesvery large. There are, as in the purely hadronic case, three distinct types of regions available in a single-particle spectrum: (a) y1 - y large, J’ - y2 finite (=P( 1)): target fragmentation region: (b) both ~1~ ~3 and 1’ - yZ large = central region; (c) .r - l’Z large, .rl - ~3finite (4(l)): current fragmentation region. Despite the fact that we made these definitions in exact analogy to the purely hadronic production processes,it is not at all obvious whether a hadron detected in any of the above regions should show the distinct features of the one detected in purely hadronic production processes.Because of the short-range correlation assumption we make in the rapidity space, we should not be surprised to see that a hadron detected in region (a) above is slow in the target rest frame: but there is no a priori reason why the particles detected in region (b) should be slow in the barycentric frame, or why the particles detected in the region (c) should be slow in the Breit frame. Since A is an invariant function of p q, h- q, and p li (also Q” of course) we may choose any coordinate frame which is convenient for our O(1. 2) parametrization. It turns out that the most convenient frame is the one in which the produced particle is at rest. We shall make the following O(l. 2) parametrization in this particular frame: h- = ~(1, 0, 0, 0) = (k,, , k,, , k,, , k,), q = Q(sinh f1 , cash c1 cos 4, cash f1 sin 4, 0), p r= m(cosh fZ, -sinh t,, 0, 0),

(4.1)

where 0 < 5‘,>52 G a’,

-i? < rf t( 7r.

In terms of these. the three invariant variables become p . q = mQ(sinh f1 cash 5, + cash f, sinh 4, cos I$), q . k = PQ sinh fl, p . k = mp cash tZ .

(4.2)

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Considered as a function of the independent variables & , & , and 4, A can be expanded in 0( 1,2) harmonics [ 181 (neglecting the discrete series)

When t1(e2) becomes large the behavior of (4.3) is governed by the leading singularities in (I&l,). When both t1 and t2 become large the asymptotic behavior is governed by the leading singularities 01~and 01~in (1, and (1, . Before starting the study of different asymptotic regions let us give the relation between the rapidity variables and the 0( 1,2) parameters: p . q = mQ sinh( y1 - y2) = mQ(sinh (I cash f2 + cash 5, sinh 5, cos $), k * q = pLIQ sinh( y1 - y) = PQ sinh & , (4.4) k-p =mplcosh(y -y2) =mpcosht,. IV. 1. Target Fragmentation From (4.3) we see that when y, - y is large (the photon and the produced hadron are well separated in the rapidity plane), t1 is also large, and fa is finite because y - y2 is (as long as k, is small, which seems to be case from the experiments). The 0(1, 2) analysis of [r , for large t1 , yields (Fig. 6)

A(Q”, E2, A&) g

(cash &)6 Bt(Q2,f2 3$1,

(4.5)

where 01is the leading singularity in (1, of A,“1 , which we assumed to be a simple pole. Assuming that only the Pomeranchuk pole dominates we have 01= 1. Also, assuming that the leading singularity is a simple pole probably means its residue factorizes in the usual sense:

FIG.

6.

Single

Regge

exchange

diagram

relevant

for

the

target

fragmentation.

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71

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We used our prescription (2.12) for the photon-photon-Reggeon step. Substituting these in (4.5) we get

coupling in the last

da (&+/2,+,,) ~5

(F’ux) Going back to the invariant

(4.6)

variables v, v’,

K,

k.q

v’

get

we

cos,, (, = ti = “” +; t?lp

-- t ,

cos 4 = [I - $]-liZ[+

- J.

Large fI means large v’/Q, and finiteness of 5, and C$means the finiteness oft and v’~v. For the normalized cross section we finally obtain

AQ”, v, ~“1 ‘d

E -‘dtk Yl--YI IJz

(4.7)

> K>,

r-Qnite

where xt = (li . q)/(p

$

* q) is the new scaling variable. Using (3.17), we obtain

v2Yf12G ; F,(w) ,I~+~(Q~, v, v’, K> =

Recalling that Fl is one power true for 4 and 9YZ : -1 111

I’%;

F,(w)

.Qf;(&

. K).

down in w, compared to F, , we see that the same is

-

F&O, Bj YI-Y-

y-y,finite

X;

K)

=

&(w)

Gt(xt

, K)

(4.8)

72

and _

1

111 p

v2-e2 -

Bj YI-Y+m y-y,finite

Now the question is whether limiting hadron fragmentation

Fzt(W,

X;

K)

=

F,(W)

Gt(Xt

, K).

this particular single Regge limit corresponds to the in the sense of [5]. In order to prove that this really is

72

NAMIK

K.

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the case we have to show that the above limit includes the case where k3 is small in the laboratory frame. Taking the extreme case k, = 0, we have cash 5, = b P’

finite;

So we see that the hadron produced in region (a) has similar features to the target, fragments of the purely hadronic processes and Eq. (4.7) shows that in the kinematical region, which is a combination of the ordinary hadron fragmentation region and the Bjorken scaling regions, the quantity p(Qz, v, v’, K) scales in a general way. The scaling variables are the old w = Qz/(2p . q) and the new xt = (pv’/mv). (Note that in the target fragmentation region we have enough freedom to choose another scaling variable, (2$/Q2), and these two scaling variables are related to each other as 2/d -=

Q2

2pv’ 2mv __zrnv

Q2

=;’

xt

But we shall see below that our choice xt = (pv’/mv) has better features; that it is proportional to the celebrated Feynman scaling variable xF = (2k3*/s1j2).) The meaning of the scaling variable xt is not immediately clear. In order to relate it to a more natural experimental variable let us consider the c.m. system of p and q, in which (dropping the asterisk) P =(P,O,O,P)>

q = K p2 -

Q2Y2, 0, 0, -PI.

Note that, using s ru 2mv(l - o), k . q N k3( p2 - Q2)V2 + p N k2N2.

Therefore, Xf

=

~~(l-CO)~~(l-CO)x,.

(4.9)

Since 0 < 1 - w < 1 and 1XF 1 < 1, we have 1x 1 < 1. We also immediately obtain the following relation, which is well known in purely hadronic inclusive reactions:

$p---

1

k-q

1 -c.0p.q

1 - xp ;

as S/M2 becomes large XF ‘v 1; this corresponds to the boundary of the phase space. When restated in terms of our new variable, xt , this says that the phase-space boundary in the hadronic side corresponds to xt = 1 - w, i.e., it depends on w. We also notice that, to leading order in terms of Regge singularities, the scaling occurs in a factorized form: (Hadronic Feynman scaling) x (Leptonic Bjorken

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73

scaling). Adding secondary Regge trajectories spoils this form of factorization. Including only the next leading trajectories, 9’ and A, , we have

Using factorization of Regge residuesand (2.12), and recalling cash f1 = v’/Q, we get

Arriving at this general scaling law we have usedthe factorization of the Pomeranchuk residue and the prescription (2.12) for the photon-photon-Reggeon vertex (the factorization of the Pomeranchuk residue also implies that the fragments of the hadron are essentially independent of the virtual photon beam). 1V.2.

Central Region

We shall how look at the central region of the single-particle spectrum, where both 2’1- .1’and 1%- ~1~are large, at high energies. From (4.4) we seethat

cash 5, = $

cosh(y - J,.J = ;

So both ,$I and 5, are large. If the leading singularities in (1,, il, are simple poles, we get, from (4.3) (Fig. 7).

From (3.20) we have P’4 (k . q)(k . p) = 5

FIG.

7.

Double

1 Sh(y, - vzl . Sh(,, - J,) Ch( 1’ - ~3~)= z ’

P

k

P

k

4

Reggeexchangediagramrelevant to the central region.

74

NAMIK

K.

PAK

Also from (4.2), (k &k4.p)

=

1 + cos c$ /.2 .

This gives us 1 + cos 4 N pz/p L2. This relation tells us that cos 4 is always finite. The Pomeranchuk trajectory can always contribute, so that the largest 01~is 1. Assuming that the leading singularity is a simple pole probably means that it is factorizable in the usual sense, in which case,

Again using our prescription (2.13) for the photon-photon-Reggeon

vertex, we get (4.12)

or, for the normalized

distribution

dQ”, “, v’, K> 5

YI-Y-tm

x0EC(&),

(4.13)

y--Y2p@=

where x = (k * 4XP * 4 N pL2 c

(P

.4)

*

We see that in the general kinematical region which combines the double Regge and Bjorken scaling regions we get scaling, and the new scaling variable is x, . Recalling the definition cash f2 = [(p . k)/nzp] we notice that if we also make (p . k) finite, which takes us from the double Regge region back to single Regge region, we recover the scaling variable xt again, which we found for the hadron fragmentation region. The function EC($) we found above is a universal function depending only on the type of particle observed; not on the virtual photon beam or the target. The coordinate system described by (4.1) is not a very transparent one in terms of physical quantities In particular, the meaning of 4 is not immediately clear. In order to relate $ to a more natural experimental variable, again let us consider the c.m. frame of p and q(dropping the asterisk): x&w+

I)(ko--k3[(ko--k3-~~0].

For small values of k, ,

This shows that for small k, , the x, dependence is equivalent to the kL2 dependence, which is the transverse momentum of the produced particle in the c.m. frame of the virtual photon-hadron system.

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75

In the purely hadronic production processes, pionization products are those hadrons (pions) which maintain a finite momentum in the c.m. system of the initial particles, as the energy of these initial particles becomes very large. The most characteristic momentum value for the pionization products are those for which k, = 0. Let us see whether the double Regge expansion (4.10) is valid for k, := 0. We have (4.15) In order to have
+ (+j1j2

&s&~,)

$j

or, since the cross terms break the scaling, we can split this into two pieces:

p(Q”, v, v’, K) = xc [ &&,)

+ --$

+ ($)l”

f$f j$j &,e,(x,)]

(4.16)

(

one scaling, and one scale breaking. Equation (4.16) shows that to get the scaling, Pomeron dominance is crucial (the Reggeon-Reggeon contribution is suppressedlike w1i2,but increaseswith Q2 causing charge asymmetries). Note that when the contribution of the secondaries becomes comparable to the Pomeron contribution at the boundary of the central region, i.e., for

these mean mv’

=

7

Q sinh(y, - v) N Q2

or

(4.17a) K =

PL

cosh(y

-

~2)

-

P,

76

NAMIK

K. PAK

cosh(y - yJ N 2.

(4.17b)

Equations (4.17a) and (4.17b) show the passage to the current fragmentation and target fragmentation regions, respectively. By using (3.17) we find the scaling behavior of the structure functions as follow:

1 -q-+ 6Tw,xc)= m4 GW, mV%central region

(4.18)

-!- v”W2---ip %&J, xc) = 4(w) GW. mCL central region

Here the function Gc(x,) is a universal function depending only on the type of particle observed, but not on the virtual photon beam or the target. IV.3.

Phase-SpaceBoundary on the Target End (Triple Regge Limit)

If the particle that is produced is sufficiently near the hadron end of the spectrum, that is if, say y - yz is nearly as small as possible, it becomes possible to calculate ft($, y - yz) in (4.7) explicitly in the Bjorken limit. This is the route followed in the case of purely hadronic inclusive reactions. They show that the phase-space boundary corresponds to the Triple Regge Limit (TR). Here we shall follow the reverse route, by studying the mathematical TR limit, and then looking at what physical region it correspond to. From Fig. 8 we see that in the TR region, ATsL(Q2, v, v’, K) is given by A=*=(Q',

v,

v’,

K)

g

[/?;;(t)]2(COs

6~)2a’0

[$

~‘~M2y

9

“’

“I,

(4.19)

where a’(t) is the leading Regge trajectory in the t channel, ATsL stands for the absorptive part of the virtual photon-Reggeon forward elastic scattering amplitude, M2 is the missing mass, and Bt is the c.m. angle for the process p + q + k + anything k

P

k T%.c

n

FIG.

8. Triple

Regge exchange diagram

for the hadron end of the phase space.

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in the t channel and for small t, and if the detected‘hadron

is a nucleon given by (4.20)

where xt = (k . q)/( p . c). When the detected hadron is a nucleon, then the dominant a’ trajectory is a Pomeron. So cos 8, has a power of 2. A similar analysis gives

ATsLtQ2,M2, 0 M2g2-rm g,*,*,(f, f, 0) ~:;L(Q2>(cosbda(o).

(4.21)

where g,,,,,(t, t, 0) is the triple Reggeon vertex, and eM is the scattering angle in the barycentric frame on the cross channel of the photon-photon channel for the forward processes photon i- Reggeon ---f photon + Reggeon, and is given by

cos e&f rv - 1 (-t)‘l” Pomeranchukon is the dominant Reggeon vertex, we get AT.L(Q2,

442,

Q

a-trajectory;

t)

arm

2, &!ma?i Substituting

(P * 4) - (k . PI

ga’a’bflyy (-ry2

(4.22)

’

using (2.12) for the photon-photon

T*L

( (P

. 4)

-

Q”

(k

. Y,) ,’

(4.23)

this in (4.19) we get

where

The region in which constraints: (a)

our expression

is valid is given by the following

kinematical

t is small and fixed.

(b) (w + x,)/C 1 - x,) is large, which means xt ‘v 1, and this ratio is very sensitive to the variation of xt around xt E 1. In terms of Feynman’s variable this means XF ‘v l/(1 - w). (c) [( p . q) - (k . q)]/Q is large. This condition is automatically satisfied in the Bjorken limit, as long as (b) is satisfied. The constraint (b) implies y1 - y N y1 - 1~~. This shows that indeed the TR region corresponds to the boundary of the phase space avaialable for the single-particle spectrum.

78

NAMIK

K. PAK

So we see that for a nucleon produced near the end of the phase space, the invariant distribution is da

T,L

(FWdsk/2k,~G

2 (a

+ d” w

~1

(4.25)

-3 (At)

i.e., there is a peak at the target end of the phase space in terms of the variable xt . This agrees with the data beautifully (Fig. 9) [9]. The existence of this peak (elastic peak) was also predicted, via quite formal arguments, [ 191. If the detected hadron is a pion, then the dominant (Y’ trajectory is a baryon trajectory with the intercept, ~~(0) = 0.3. In this case, cos ey CT?!1 - *

z A. ’

Substituting

T1:

t

these in (4.19), we get (Px)

duTsL pL-J d3k/2k, z

E

FIG. 9. The normalized

distribution

(1 - XJV

(4.26)

TR

0 5 = 2.2, cl2 = 1.2 s=2.2,Q2=3.9

q

for the process yO + p +p + anything (Cornell data 191).

This distribution vanishes at the boundary. Again, this result agrees with experiment (Fig. 10). The scaling law for the structure function are, using (3.17),

;

1

VW1 $+ .q+J,

xt , K) =

TR

(4.27)

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TT ‘7 T r -r;-r-r2.2<..6<2.8 GeV

r-~ ~

-0.15 I-

b< UY

ro 72

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A 0.5 <$
GeV’ p0 cut

0 Photoproduction 43

-0.10

L #

mi

1 SC0

I

FIG. 10. The normalized longitudinal distribution for the process y,, + p + T- + anything, for 2.2 < W i 2.8 GeV at Q2 = 0, and for Qe intervals 0.3-0.5 and 0.5-I .4 GeV2 CDESY data [9]).

The first ones are for nucleon production, the second for pion production.

IV.4.

Current Fragmentation

From (4.4) we see that when J’ - y, is large, while J’~- 4’ is finite (Fig, 11) the 0( 1, 2) expansion yields

where QIis the leading singularity and is assumedto be a simple pole. Again, assuming k

FIG. 11. Single Regge exchange diagram

q

relevant for the current

fragmentation.

80

NAMIK

K.

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that this leading singularity is the Pomeron trajectory, and that it factorizes, we obtain da (Flux) d3k/2k,

= (+

1 A”

(4.29a)

where the function /3yB stands for the right-hand blob in Fig. 9. Since in this particular region, we do not explicitly have the photon-photonReggeon vertex in a factorized form, this is as far as we can go in our approach, namely, we cannot predict explicit scaling forms. For the normalized distribution we have

p(Q”, v, v’, K)

27

y,-yfinite Y-Y.A+a

+ /%(Q”; YI - Y>.

Ekg

(4.29b)

YY

All we can really do is guess and find plaussibility arguments for the scaling variables, and the distributions. Let us first study the possible candidates to be the scaling variables. Since y, - y E O(l), and y - y, large, we have, in the Bjorken limit,

v’ - O(Q), K -

N

K - 07(Qb),

@(Q-l),

;

N

V

O(Q-7, (4.30)

+

-

U(Q-7,

m'e --;-I CL"

f$

-

O(Q-‘1,

1 fth(Y-Yd w (

2

1*

From this list we see that if we would like to have a scaling variable in the conventional form (conventional in the hadronic sense), the only likely candidate is the last one. But unfortunately, that it is nothing but the Bjorken’s scaling variable can be seen by explicitly calculating it out (as is already obvious from (4.30)) in any frame we like. Doing it in the c.m. frame we find ml-5 -= P*vf

kolk, 1 + &hh)

P- 2w- $1 k&eco*

That eventually we took the limit k, --f 00, was in accord with the definition that the detected hadron has a finite fraction of the photon mass. Since both photon and the detected hadron are well separated from the center of the rapidity space, this means that photon fragments have large longitudinal moments in the c.m. frame. Since the photon fragmentation region is quite a peculiar region for high Q2, we have to give up the conventions we made for the purely hadronic processes.

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To tind the correct scaling variable obtained for the neighboring regions:

let us review

81

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the scaling variables

we have

Calling Sh( .rl -- y) = z1 , Sh( J - yn) = z2 , Sh( y1 -~ yr) :- z. we see that

The scaling variable which meaning would be

would

match with X, and X, and would

carry the same (4.31)

In terms of this new scaling variable,

our normalized

distribution

becomes (4.29c)

As we see, p does not even have Bjorken consider the inclusive sum rule [6]:

scaling, as it is. To guess the scaling form,

We know that the total available energy is &t Y v. We shall show in the next section, quite generally, that the transverse momentum of outgoing particles is limited. From our detailed kinematical analysis above, we see that in the current fragmentation region k, N O(V), and it is at least one power of v*J’! suppressed in the neighboring region (because the boost parameter which relates the lab and the cm. frame is Ch p = [v/2m(l - w)]‘/~). Assuming that cross sections are bounded as Q” ---f cc (i.e., multiplicities do not grow as powers of Q2), then this sum rule is saturated by the contributions in the current fragmentation region. This means that the integral in rapidity extends only a finite region:

Since ki, - lP(v), the simplest way of satisfying this sum rule would be to have p(Q”, Y, v’, K) independent of Q2, i.e., scale in w. (This would mean that VY?; and v”w2 are the scaling structure functions, as in the neighboring region. and is in

82

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PAK

agreement with the parton model predictions [8] also, if this means anything). This is simply achieved if the explicit Q-dependence of the blob /3z(Qz, y - y,) is

P%Q', Y - ~1, PJ = ; P%Y - ~1, pd. If this ansatz is correct, we finally obtain

p;ssL(Q”,v, v’, 4

z ~,fz& Y-Y*+=

9PI),

(4.34)

y,yfinite

where in the last step we also used the fact

WY -

~2)

xy = Sh(y, - y2) - e Adding the next leading trajectory breaks the scaling (4.35a)

p,TsL(Q2, v, v’, K) m x,,

As K gets large we approach scaling from above. Note that for x, - 1, close to the end of the boundary, we have K/p N v/Q. Substituting this in (4.35) we get (4.35b)

py(Q2, v, v’, K> - Xv

For small Q (and fixed value of initial total energy) the Pomeron contribution dominates. But if we increase Q2, the contribution of the isospin-carrying secondary trajectory (Y + A,) becomes comparable to the B contribution. This implies that ?r+/‘rr- asymmetry, which is due to isospin carrying secondary Regge trajectory, increases as a function of Q2, or as a function of w for fixed value of s. This prediction, although not as detailed as that of the parton model [20], is supported by the data (Fig. 12) [9]. Now, let us investigate the questions of whether the hadrons produced in the so-defined current fragmentation region show the same features as in the purely hadronic case. (In the purely hadronic case, particles which are slow in the beam rest frame are the fragments of the beam.) Because it does not make sense to talk about the rest frame of a space-like photon, let us work in the barycentric frame: cash 5%= ‘2

?+ -x,7-+ kt-tm ;

a.

(4.36)

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6-

*

~%r-

Harvard

--I

FIG. 12. The charge asymmetry versus l/w for the process ye + p ---fhi + anything

So our expansion is valid for the fast-particle production. there are any slow particles produced in this region also:

Let us see now whether

(v/m)1/2

cash LJ, r5 JfL. k3-m /t

[2(1 - a~)]~/~-

[9].

(4.37a)

*’

1 1 - 2w sinh [I = ~k-q rv __- large. pQ k,+o 2[w(l - c,J)]~/~ “,+O 2w1/2

(4.37b)

This result shows that particles with k,* N 0 are produced in the central region, and the central region may be called the “pionization region,” as in the hadronic case, without any further reservation. The structure functions scale as -1 V% B1 t %“(w, x, . PJ = 4(w) CTk m Y--Y*+m

. p ,L

y,-uiinite

(4.38) 1 mll.

V2W2

B1 Y-Ya-m

5 FT'(w,

x,

3 PJ

=

F,(m)

G;+Lk

, ~1).

y,-yfinite

The hope that we can find the form of G,(x, , pI) explicitly,

by taking the single-

84

NAMIK

K.

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pion exchange diagram for pion production close to, the boundary, fails when Qz is large, because (q - k)2 c1 -Q2. This is a peculiarity of the highly space-like photon.

V. AVERAGE MULTIPLICITIES V. 1. Contributions of Each Kinematical Region to the Multiplicity

The average multiplicity

of the produced particles is defined ,by

utot(Q2, v) ti(Q2, v) = 1 g

( d"(Q27 " "' K, ,), d3k/2k,

(5.1)

where atot is the total spin averaged virtual photoh-proton cross section at a given v and Q2 and the integral (5.1) extends over the allowed phase space. In our definition, n means the average multiplicity for a specific type of particle, say pion (n+, n-, or no) to which du/(d3k/2ko) refers. To get the average multiplicity for all kinds of pions produced in virtual inclusive photoproduction processes, we have to add all the pions contributions. Clearly, if one does not have information about the transverse momentum distributions, it is difficult to estimate the multiplicity of the pions. However, if k, dependence of the invariant distribution (explicitly ft , f, , and g) falls faster than (k12)-‘, then the leading behavior of $Q2, LJ)can be calculated when the Pomeranchuk pole is the leading singularity. At the end of this section we shall carefully study this point and show that the suppression of transverse momentum comes out naturally as a result of other assumptions already made. The data seem to be in perfect agreement with this (Fig. 13) [9, 221. Making a variable change from (k, , k,) to ( y, k,), we get WQ”, 4 = j- d2k, 14

r&t,

PJ + j dy j d2k, p&J

+ / d2k, j 4 PA-G , pd.

(5.2) Because the pI behavior is damped, the total multiplicity is proportional to the length of the phase space, ln[(l - w)(s/m”)]. Now let us calculate the shares of each region. The first integration is over a finite length of rapidity (-2), and therefore is a constant. The second integral is proportional to the length of the phase space of that region, because the integral of the universal function p&J over k, is constant: I+’

’ dy

s d2k, p&J

= (const) x L, ,

where Lc is the length of the central region, in rapidity space. Even though we assume that the photon fragments very much like a hadron does, we know that certain things are different, because the photon is very highly space-like. To find the contribution of the current fragments, let us keep Q2 fixed and large,

INCLUSIVE

ELECTROPRODUCTION

r

I

I

x

SLAG/MIT

0.4
SLAC

0.4
0.25 l

1 1

DESY /Glasgow

I

0.4 < x’ < 0.8 G> g

85

PROCESSES

0.20

?

0.10 -I-

_]

0

3

I

Q*

(G,vzj2

FIG. 13. (k,*) versus Qz (GeW) for the process yc + p - vm f anything.

and decrease s until the projectile communicates with the target in the rapidity plane (this can be achieved by keeping Q2 fixed and letting w go to 1). The length of the projectile fragmentation region is L, = Y-

Lt ‘v In [s(l

- w)].

Substituting

we get

L., ‘v In [( 1 --- ~0)”-$]. I

(5.3)

If we now let s - cc (keeping Q2 large and fixed), the central region reappears again. The size of the central region is, then

=ln

1

(-.w 1

(5.4)

86

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K. PAK

Finally, the contribution of each region to the total average charged multiplicity is (Fig. 14) &(Qz, V) = constant, Ec(Q2, V) = (const) ln(l/w), (5.5) &(Q2, v) = (const) + (const) 1n(Q2/pA2). V.2. An Upper Bound on the Average Transverse Momentum Now it is well known that the logarithmic growth of the average multiplicity (at least in the purely hadronic case) arises from populating the longitudinal phase space dk,/k, = dy in a statistically independent manner [6]. We would like to point out here that for this type of longitudinal distribution the transverse momentum must be limited for the general case like the one in hand, where we have highly off-shell virtual particles involved [21, 221. We shall make the following assumptions: (a) phase space is completely filled; (b) transverse momentum is independent of rapidity. Since assumption (a) is the most crucial one for the processes involving highly virtual particles, let us give a plausibility argument in support of it, recalling the kinematical study in Section 111.2. The quantity dy = Ymax - yl = ln(l

-

a> + In (*)

would measure the unfilled positions of the longitudinal phase space. Fortunately, it is small (-O(l)) for all values of Q2. If dy were not small, it would mean that there are new regimes opened up in the phase space, other than the so defined target fragmentation, central, and current fragmentation regimes. That dy - O(1) means that phase space actually extends only between limits fixed by the rapidities of the leading particles, although it does not seem to be SO at first sight. Let us calculate the mass, A, of the “Feynman gas” of length Y. Let us first translate the gas along y by a Lorentz transformation so that it is centered at y = 0. Let us call the average density of particles integrated over k, , C; i.e., di’i/dy = C. Since k, = pI sinh y, with the gas centered at y = 0, its total momentum

FIG. 14. The shape of the distribution

in rapidity space.

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ELECTROPRODUCTION

PROCESSES

is zero; therefore its mass is its total energy. Assuming that FL is independent we obtain JH = 2CjiI From (3.24) we immediately

( Y )N CtIi,eY!2.

sinh 2

of y,

(5.7)

see that Y = In [(l - w) --$I.

Substituting

87

(5.8)

this in (5.7), we get .4% = C#G,(l - W)1/2 (-$)li2.

(5.9)

Since &? cannot exceed the total available c.m. energy,

we get

From (5.10) we see that if C is to be of order 1, as it is experimentally (Fig. 15) [9], then for w N 0 (deep Regge region) FL is constrained to be on the order of a typical hadron mass or less, the same behavior we see in hadronic production processes [6]. As Q2 gets very large, or as w -+ 1, (5.10) predicts a large deviation from this typical hadronic behavior. If C is again to be of order 1, then FL/m may get very large with increasing Q2, for fixed s: 1 EL-<-!- -=&(1+%). m -cl--w

(5.11)

This linear increase with Qz seems to be consistent with the data (Fig. 13). From the data (Fig. 15) we get C N 1. So we see that this inequality is well satisfied by the data; for the left-hand side is 0.47, but the right-hand side is 1.08. Notice also that (5.9) and (5.10) set a rough upper limit on the value of C, because p < FL . These upper limits are

and these give

88

NAMIK o

A 5 ,

UCSC/SLAC SLAC I 1

K. PAK n l

r

s

Cornell DESYlGlasgow

I1111,

1

(GeV*)

FIG. 15. Average charged multiplicity versus s for different Q* intervals [9].

Now it is the right place to note that if the central plateau is really two plateaus, the hadronic plateau and the current plateau, lying between the current fragmentation and the hole fragmentation regions [23], our result (5.10) applies again. If they are of equal height C is their common height; if they are of different height it is their average, C = &(C, + C,).

VI. FINAL REMARKS

In the preceding sections, following Mueller’s analysis of purely hadronic processes closely, we calculated the momentum distribution of the final-state hadrons in deep inelastic electroproduction processes, and obtained general scaling laws. Our approach to the problem is fairly model independent. We can list the ingredients that led up to the above result as follows: (a) We assumed that the amplitude for the forward virtual Compton amplitude can be written as a sum over the leading Regge poles over a wide range of values

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ELECTROPRODUCTION

of QZ and 1’. and not just the asymptotic duality).

89

PROCESSES

limit (a plausibility

argument

for this is

(b) The scaling law obtained by Bjorken for the ordinary deep inelastic electroproduction processes is consistent with all the experimental data available at present. Therefore, we take it as an experimental fact, and combining this with (a) above we get a prescription for the Q”-dependence of the virtual photon--virtual photon-Reggeon vertex. (c) We assume that Pomeranchukon singularity is a Regge pole, so that its residue factorizes. This assumption enables us to use the information we obtained in (b) for the ordinary deep inelastic electroproduction processes. (d) We expand the absorptive part of the forward three-to-three amplitude. 7 “virtual photon t nucleon + pion + virtual photon -+ nucleon + pion,” into harmonics of 0( I. 2), as Mueller did in analyzing the purely hadronic inclusive reactions. Because of the existence of a highly space-like virtual photon, this is not a trivial exercise at all. Also, in this analysis, we did not assume the mass of the virtual photon. Q”. to be small compared to the total energy. Although we assumed that the virtual photon fragments very much like a hadron does, we get some interesting features in our case which are absent in the purely hadronic case. For instance, we observe that the size of the virtual photon fragmentation region in the rapidity plane changes with Q”. Factorization of the Pomeronchukon residue ensures that in our case the target nucleon fragments exactly as it does in purely hadronic inclusive processes, i.e., its size in the rapidity plane is a constant. so its contribution to the multiplicity is a constant. The average multiplicity again grows logarithmically with energy, as in the purely hadronic case. But this time the contribution from the photon fragmentation region is really large. If we fix w at a not too small value, we see that the major contribution to the multiplicity comes from the photon fragmentation region, whereas in the purely hadronic case it was coming from the central region. We have, then. shown quite generally that the logarithmic increase of the multiplicity imposes a constraint on the average transverse momentums produced, and this upper limit increases with Qi?. We have shown that, at not too high energies, where the contributions of secondary Regge trajectories are comparable to the Pomeron, the charge asymmetries carried by these isospin carrying trajectories increase with increasing Qz, both in the current fragmentation region and central region. This behavior for the current fragmentation region is beautifully supported by the data. The kinematics relevant to the problem are very thoroughly and carefully investigated throughout. But since energies arc not high enough to develop a plateau, the prediction for the central region is to be tested in the future. We point out, as a final digression on the soft pions (in Appendix 3). that the contributions of the soft pions do not increase with energy; i.e.. in the center of mass frame. it is the soft pions’ contribution which gives early onset of scaling.

90

NAMIK

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APPENDIX 1

For Q2 + 0 we should have [duL/(d3k/2k,)]

-+ 0; this gives us the relation

lim wI(Q2, V, v’, K) = lim [ ” W2(Q2, v, v’, K)] . Q%O Q2

(Al.l)

0%

Also, [du’/(dak/2k0)] (Q2 = 0, v, v’, K) is the inclusive photoproduction. We observe that as Q2 + 0, sin2(8/2) --+ 0, and so cos2(8/2) + 1; therefore da dQ” dv dv’ dK > Comparing

(Al .2)

this with the photoinclusive

cross section

477% N lim WI 0% mv Qe4

(d3$ko)

(Al .3)

we finally obtain da dQ” dv dv’ dK >

(Al .4)

APPENDIX

2

The relations between Wab and Wi are W++

=

E;*+V’V

woo

=


+ j

= =

WI

+ 4

.k

-wL(1+9

12W2

=

W--,

w,

1 EO- k I2 W, + -& Re(EO* - P)(EO - k) W, ,

w+- = B+*E-w” 11 Y

= ~1 c-b* - k)(c P(

Re W+O = 3 (e” . k) W, + $

APPENDIX

1&

. k) W, ,

(8 . p) W, .

3: INCLUSIVE SOFT-PIONELECTROPRODUCTION

We shall now consider the inclusive electroproduction particle is a soft pion: e(l) + N(p) + e’(l’> + &dW

process where the detected

+ anything (PH).

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ELECTROPRODUCTION

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We shall first look at the problem as a special case of the general pion production process we investigated in the preceding chapters. The notion of a soft meson depends on a particular Lorentz frame, just as a soft (infrared) photon does. We are going to assume here that in some Lorentz frame. which we are going to specify explicitly below, all soft-pion momenta. say li,, , are so small, that they satisfy [24] h

(A3.1)


In order to get started we first have to choose a special frame. The special frames at our disposal are the rest frames of the target and projectile, and the center-of-mass frame of the initial virtual photon-nucleon system. If we take the lab frame (target rest frame) as our special Lorentz frame, emitting soft pions correspond to target fragmentation, or, expressed in another way, we say in the lab frame target fragmentation limits and soft-pion limits are kinematically the same (to be more precise. it is not really the whole target fragmentation region, but the end region of it, i.e., the triple Regge limit). We can rephrase this as follows: A soft meson is soft only in the rest frame of the primary particle that emits it (we must be careful at this point, because it may be difficult to identify the rest frame of the primary particle, since, in order to radiate, it must experience an acceleration, and therefore its rest frame changes). The invariant kinematical variables in this case are

and v’ = [(k * 4)/p] z v. Therefore, (A3.2)

d/v ‘v I. For a consistency

check let us calculate the rapidity

variable

in this case:

This tells us that in the soft-pion limit the rapidity difference cannot be large. So the Regge limits which involve large values of rapidity difference y - J+ are inapplicable in this case, and the only Regge limit allowed is the one in which ?‘I - .l’n - co, ?>I - y + 03, and >’ - 1~~finite. So for the soft-pion production in the lab frame we have the scaling law (4.7) with the special values (A3.2) of the scaling variables. The particular value of 1 N 1 GeV2 shows that one nucleon exchange approximation is a fairly good approximation for the soft-pion production in the lab frame. We have shown in Section V that the contribution of the target fragmentation products to the multiplicity is a constant. Since the soft pions in the target frame are the fragments of the target their multiplicity is a constant.

92

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Now we shall choose the c.m. frame of the initial system as our special Lorentz frame, as is usually done [5]. In this case the process of emitting soft real or virtual pions is known as pionization. The invariant variables in this case are

For the soft-pion production in the c.m. frame we have the scaling law (4.13) with the following special value for the scaling variable T: (A3.3) For a consistency check let us calculate the rapidity variables in this case: cosh(y - yZ) = ; (A3.4) k.q

cosh(y, - y) = X

i

1 - 2tL

ok + 2l -lP (m”)w c 1

---f ccl.

Again recalling the last section’s prediction that the contribution of the pionization products to the multiplicity is ln(l/w), i.e., scales, the multiplicity of the soft pions, in the cm. frame, is ln( l/w). So we see that which ever special Lorentz frame we choose for our definition, the multiplicity for the soft pions does not grow with the energy, it is constant, and it scales in the c.m. frame. Lee [8] showed that there is a general relation between the multiplicity and the energy scale (he defines the scale s, , so that when s is bigger than s, the structure functions scale). His prediction is that if the average multiplicity E increases with energy, say,

where K is a number, then the scale s, for a channel with a large multiplicity increase exponentially with n, i.e.,

should

SC- hfN2enfK. In the case of finite average multiplicity, of multiplicity IZ is approximately

he finds that the scale for any channel

s, - (MAI + hMd2, where h is some large factor, say 10. Applying

his predictions to our problem,

we

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claim that for the soft-pion production in the deep inelastic processesthe scaling sets in early, i.e., the overall scale is low, for the multiplicity is finite irrespective of the Lorentz frame we choose. Then the possibility exists that if one excludes all the hard mesonsin the deep inelastic electroproduction processes,the remaining hadron multiplicity at infinite energy may stay finite and not too high. Notice that our prediction is contrary to that of Lee’s. He claims that the growing multiplicity is due to soft mesons, and our prediction is that it is rather due to hard mesons, and it is the soft mesonswhich set the scaling so early. Now integrate the equation (3.10) over V’ around the value V’ - v in an interval of length ~k I/IX in the lab frame. Since the structure functions PY’~and vWZ scale with the special values, K = E, - I”. x N p/m of the variables, we get du 4m2 E’ dQ’ dv dE, ,,p~i,, p” c7 i

e

fr) sin” -2] Ik . (A3.5) and in the c.m. frame, again integrating over v’ along an interval of length I k we get dff 4rrc2 E’ dQ2 dv dE, - T E W2”(x,) cos20 z -C 27$‘(.u,) sin’ 20 k I. c I[ 1

i/m,

(A3.6)

where x,. - p m2in the soft-pion limit, and 13is the electron scattering angle. ACKNOWLEDGMENTS 1 would like to thank J. D. Jackson for his hospitality at the University of California Lawrence Berkeley Laboratory where this work was started, S. Drell for his hospitality at the Stanford Linear Accelerator Center where it was completed, and Turkish Science Research Council for partial financial support.

REFERENCES 1. 2.

J. D. BJORKEN,

Phvs.

Rev. 179 (1969),

1547.

R. E. TAYLOR, Inelastic electron-nucleon scattering experiments at SLAC; L. W. MO, Highenergy muon scattering at Fermi Laboratory, in “Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies” (W. Kirk, Ed.), Stanford Linear Accelerator Center, Stanford, California, 1976. 3. R. P. FEYNMAN, “Photon-Hadron Interactions” Benjamin, New York, 1972. 4. D. AMATIet nl., Nuovo Cimento 26 (1962), 896; K. WILSON, Acta Phys. Austriuca 17 (1963), 37. 5. R. P. FEYNMAN, Phys. Rev. Lett. 23 (1964), 1415; C. N. YANG et al., Phys. Rev. 188 (1969), 2159. 6. K. KANG, Fortschr. Physik 21 (1973), 507; M. JACOB, Hadron physics at IST energies, Lecture notes at Bonn Summer Institute for Theoretical Physics, Ausut 1974:L. STODOLSKY, irr “Proceedings of the Seventh Recontre de Moriond,” Vol. 2, 1972.

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7. SLAC data: M. PERL et al., Phys. Rev. D 10 (1974), 1401; DESY data: V. ECHARDT et al., Nucl. Phys. J3 55 (1973), 45; Cornell data: J. BEBEK et al., Phys. Rev. Lett. 30 (1973), 624. 8. S. DRELL AND T. YAN, Phys. Rev. Lett. 24 (1971), 855; P. LANDSHOFF AND J. POWNGHORNE, Nucl. Phys B 33 (1971), 221; J. STACK, Phys. Rev. Lett. 28 (1972), 57; J. BJORKEN, S. BERMAN, AND J. KOCUT, Phys. Rev. D 4 (1971), 338; T. D. LEE, Ann. Phys. 66 (1971), 857. 9. Harvard-Cornell Collaboration by K. M. HANSON, Inclusive and exclusive virtual photoproduction results at Cornell; SLAC-UCSC Collaboration by R. F. MOZLEY, Muon proton scattering at SLAC; DESY-Glasgow Collaboration by V. ECHARDT, in “Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies” (W. Kirk, Ed.), Stanford Linear Accelerator Center, Stanford, California, 1976. 10. The most recent experimental and theoretical situation is extensively covered by several people in “Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies” (W. Kirk, Ed.), Stanford Linear Accelerator Center, Stanford, California, 1976. 11. K. BARDAKCIAND N. K. PAK, Lett. Nuovo Cimento 4 (1972), 719; N. K. PAKU. C. L. B. L.-773, Ph.D. Thesis, 1972, unpublished; H. ABARBANEL AND D. GROSS, Phys. Rev. D 5 (1972), 699; T. P. CHEN AND A. ZEE, Phys. Rev. D 6 (1972), 885; J. D. BJORKEN, in “Proceedings of the 1971 International Symposium on Electron and Photon Interactions at High Energies” (N. B. Mistry, Ed.), Cornell, Ithaca, New York, 1972; R. CAHN AND W. COLGLAZIER, Phys. Rev. D 8 (1974), 3019. 12. A. MUELLER, Phys. Rev. D 2 (1970), 2963. The formal proofs of the generalized optical theorem are given by H. STAPP, Phys. Rev. D 3 (1971), 2172; C. I. TAN, Phys. Rev. D 4 (1971), 2413; H. STAPP AND K. CAHILL, Phys. Rev. D 6 (1972), 1007. 13. N. DOMBEY, Rev. Mod. Phys. 41 (1969), 236. 14. D. GROIN AND H. PAGELS, Phys. Rev. 172 (1968), 1381. 15. H. ABARBANEL, M. GOLDBERGER, AND S. TREIMAN, Phys. Rev. Lett. 22 (1969), 500. 16. A good review on “asymptotic freedom” is H. D. POLITZER, Phys. Rep. C 14 (1974). 17. J. BJORKEN AND S. DRELL, “Relativistic Quantum Fields,” McGraw-Hill, New York, 1965. 18. M. TOLLER, Nuovo Cimento 32 (1965), 631. 19. J. BJORKEN AND J. Koour, Phys. Rev. D 8 (1973), 1341; see also Ref. [3]. 20. J. T. DAKIN AND G. J. FELDMAN, Phys. Rev. D 8 (1973), 2862. 21. A similar analysis for purely hadronic processes is given by L. STODOLSKY, Phys. Rev. Lett. 31 (1973),

139.

.

For an excellent up-to-date review of large transverse momentum processes, see D. SIVERS, S. BRODSKY, AND R. BLANKENBECLER, Phys. Rep. C 23 (1976). 23. J. D. BORKEN, Phys. Rev. D 7 (1973), 282. 24. S. ADLER AND R. DASHEN, “Current Algebras and Applications to Particle Physics,” Benjamin, New York, 1968.

22.