12 December 1996

PHYSICS LETTERS B

ELSFXIER Physics Letters B 389 (1996) 374-378

Gluon recombination in the GLR model Wei Zhu

CCAST (World Laboratory), PO. Box 8730, Beijing 100 080, People’s Republic of China Department of Physics, East China Normal University, Shanghai 200 062, People’s Republic of China 1

Received 26 March 1996; revised manuscript received 5 August 1996 Editor: M. Dine

Abstract

The recombination of gluons with different Bjorken variables, x, in the nonlinear evolution equations, is discussed. We find that this modification unreasonably enhances the shadowing effect in the GLR equation, while it does not change the predictions of the modified (momentum conserving) GLR equation. The results show that momentum conservation plays an important role in gluon recombination.

Gluon recombination is one of the most interesting phenomena in small x physics. Perturbative QCD predicts a rapid rise of parton multiplicity inside the proton in the small-x region. At sufficiently large gluon density, the gluons overlap with each other and can no longer be regarded as free. In this case, fusion or recombination of gluon pairs will start to compete with the parton decay processes included in the standard linear evolution equations. Gribov-Levin-Ryskin [l] and Mueller-Qiu [ 21 first studied the recombination effect of gluons in a nonlinear evolution equation (the GLR equation) using perturbative QCD. Recently this dynamical equation has been generalized for restoring momentum conservation in a modified GLR equation [ 31.

We should note that the deductions of the above mentioned evolution equations include additional hypotheses. For example, in the leading twist approximation, the two-gluon distribution Cc*) is simply taken to be the product of two conventional one-gluon distributions as

($2) = &GO. Q%(x*, ~22) N

(lb)

where G(x, Q*) = xg(x, Q*) is the gluon momentum density and RN characterizes the area of a nucleon which the gluons are populated. The first identity (la) means that the different branches of the gluon cascade evolve independently. Levin, Ryskin and Shuvaev [4] and Bartels [5] have discussed the contribution of twist four operator in perturbative QCD and pointed out that interaction between the gluon cascades is small and the factorization (la) is valid in the limit of a large number of colors (N, -+ co). Recently Laenen and Levin

1 Permanent address.

0370-2693/96/$12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII 30370-2693(96)01182-3

W. Zhu/Physics Let?ers B 389 (1996) 374-378 375

prove ‘that the corrections of multigluon correlations on the structure function are small [6]. However, the second identity (lb) implies that the recombination only occurs among gluons with the same value of x; this cannot be derived from QCD but has to be taken as an assumption.

The gluon recombination is expected to be large when the fusing gluons have the same value of rapidity y = In{ 1 lx), since the gluonic wave functions can overlap in the longitudinal direction, given enough time. However, there is apriori no reason to forbid the recombination of two gluons with different values of x. Obviously, the restriction (lb) is unreasonable.

In this paper we generalize the above mentioned nonlinear evolution equations to involve the recombination of gluons with different values of x. We find that this modification enhances the shadowing effect in the GLR equation but does not change the predictions of the modified GLR equation.

The gluon recombination modifies the linear QCD evolution equation as the GLR equation [ 1,2] at small x in the leading shadowing approximation

dG(x, Q2> = 3as(Q2) d,InQ2 ?7

x x

(2)

The nonlinear term in (2) suppresses the rapidly growing gluon density as x becomes small and hence is called the shadowing effect.

Now we consider that the gluon with rapidity y can combine with a gluon with rapidity (y + q) . One such process is shown in Fig. Ia, where y = In(l/x>, y + 77 = ln(l/x*) = ln(l/(Sx)>, and 5 = e-n. A natural generalization of the GLR equation has the following form:

dG(x,Q2) = 3dQ2) ’ dz

s

81&Q’> TG(z>Q2) - 16R2 Q2 dv

X0 dz

dInQ2 ?r N J S IG(z,Q2)G(e-n,,Q2)f(l~l) 7 (3) x x

where S( (r7 I) is the normalized correlation function of fussing gluons. Eq. (3) reduces to the GLR equation if f is the delta function 6(q). For convenience, we rewrite Eq. (3) in the differential form

d&G(y,t) =c-/\exp[-r-exp(t)l J d?lG(y,t)G(y+17,t)f(lrll)

n

=c - A I

dqe ‘7’(‘,‘)f()v])exp[-t-exp(t)]G2(y,t), (4)

where t = ln[ ln(Q2/A2) ] and we represent generally the distribution G(x, Q2) in the small-x region in the form

h(Q2)x-“(&. (5)

The coefficients in (4) are

c = 12/( 11 - 2nf/3) , A = 9r2c3/16R&i2, (6)

where A is the QCD parameter. Obviously, the ratio of the nonlinear terms of (4) and of the GLR equation

R = s drle7”f(lrll)~ (7)

shows the corrections from Fig. la, where S is regarded as an independent parameter. For an example, we take

(8)

376 W. Zhu / Physics Letters B 389 (1996) 374-378

a b Fig. 1. One of the possible recombinations of gluons with different

values of x, where (a) leads to shadowing and (b) to antishad-

owing.

0 0.2 0.4 6 0.6 9:8 1

Fig. 2. The ratio R (which parametrises the effect of recombination of gluons with different momenta) as a function of 6.

with a = 0.2, which means lvrnax] = 5. Fig. 2 gives the ratio R as a function of 6. We find that the process corresponding to Fig. la will enhance

the shadowing effect if the gluon distribution is steep. Unfortunately, this result is incompatible with the HERA data [ 7,8], since they have not seen the expected shadowing effect of the GLR equation for x < 1O-2 [ 91.

The GLR equation does not conserve the momentum of gluons. In our previous work [3], we introduced antishadowing terms to restore the momentum sum rule in a modified GLR equation:

aGtx,Q2) = 3a,(Q2) 1

s

8lcy2(Q2> Xo dz 81a;(Q2> ’ dz

dlnQ2 V $G(zt Q2) - l&2 Q2

N J T[~(z,~2)~~t 16R2Q2 J~c~~Q~u~. N

x n 42

(9)

The last term of (9) arises from the contributions of the vertex 4G --) 2G, in which the cut passes through none of the gluon ladders, or equivalently, from expanding the lower limit of the integration from x to x/2. The contribution of the last term is called the antishadowing effect since it compensates the momentum lost in shadowing.

The differential form of (9) is

~3,,d,G(y,t) =~--h[2-2~~(~,‘)]exp[-t--exp(t)]G~(y,t). (10)

There are some significant features in the modified GLR equation as we have elaborated in [ lo]. For simplicity, we define K1 as the coefficient of the nonlinear term in ( 10) :

K1 = ‘-J _ 22”h’) , (11)

and show this as a function of S in Fig. 2 (solid curve). We see that the nonlinear effect of the gluon recombination is weaker in the region where the gluon shows Lipatov behavior (where 6 N l/2 [ 1 I] ) .

Now we include the processes corresponding to Figs. la and lb in the modified GLR equation, where lb corresponds to the fact that the modification of the gluon distribution at x can come from the enhancement (antishadowing) effect of the fusion of two gluons with different momenta, xn and x:. Note x, + xz = x and xz = 5~~. Therefore, xn = X/ ( 1 + 5) and X: = 5x( 1 + 6). In consequence, we have

W. Zhu/Physics Letters B 389 (19961374-378 377

. 1

K .o

-1

-2 -0 0.2 0.4 0.6 0.8 1 --0 0.2. 0.4

6 6 0.6 0.8 1

Fig. 3. The coefficients of the nonlinear term, KI (solid carve)

and Kz (dashed curve) (I?+. ( 11) and (15) respectively) are

plotted as functions of 6.

Fig. 4. The individual contributions from shadowing (S) and an-

tishadowing (AS), respectively, in the factor KZ in I$. ( 15).

The corresponding differential form is

J2G(x,Q2) = 3as(Q2)

--x dxdlnQ2 z- G(z,Q2) - 8&!;$) J d~Gtx,Q2)G(e-~~,Q2)f([?ll)

N

+ =:(Q? 16R$Q2 J dqG(x/( 1 + eMV >,Q2>G(e-7x/(l + e-~>~Q2>f(l~l> 2

(12)

(13)

or, using (5) we get

d&G(y, t) = c - A J [ &.) 2p%YJ) _ ( eFv -3 Yd)

(1 + e-s>2 > 1 f(I~I)exp[-t - exp(OlG2(y,0. (14)

For comparison with the primal modified GLR equation ( lo), we plot the factor

versus 6 in Fig. 3 (dashed curve), in which f( 17)) is as defined in (8). The result shows that Kr N K2. The factors Kt and KZ control the strength of the gluon recombination in the evolution equation. Thus we conclude that the modifred GLR equation approximately retains its primal form (9) and its original predictions even when the recombination of gluons with different values of x is involved. The above mentioned interesting properties of the modified GLR equation arise from the balance of the shadowing and antishadowing effects. In order to clearly demonstrate this, we plot the contributions to the factor K2 from shadowing (S) and antishadowing

378 W. Zhu/Physics Letters B 389 (1996) 374-378

(AS) respectively in Fig. 4. We see that the contributions from shadowing and antishadowing are opposite and weakened by each other. The net effects of Figs. 1 are small and they can be neglected.

In conclusion, we have investigated the recombination of gluons with differing momenta, i.e., with different values of x. We find that this modification unreasonably enhances the shadowing effect in the GLR equation, while it does not change the primal form and the predictions of the modified (that is, the momentum conserv- ing) GLR equation. The results thus show that momentum conservation is an important consideration in the description of gluon recombination.

This work was supported by the National Natural Science Foundation of China.

References

[I] L.V. Gribov, E.M. Levitt and M.G. Ryskin, Phys. Rep. 100 (1983) 1.

[2] A.H. Mueller and J. Qiu, Nucl. Phys. B 268 (1986) 427.

[3] W. Zhu, D.L. Xue, K.M. Chai and 2.X. Xu, Phys. Lett. B 317 (1993) 200.

[4] E.M. Levin, M.G. Ryskin and A.G. Shuvaev, Nucl. Phys. B 387 (1992) 589.

[5] J. Bartels, Phys. Lett. B 298 (1993) 204. [6] E. Laenen and E. Levin, Nucl. Phys. B 451 (1995) 207.

[7] Hl Collab., S. Aid et al., Nucl. Phys. B 345 (1995) 494.

[S] ZEUS Collab., M. Derrick et al., Phys. L&t. B 345 (1995) 576.

[9] J. Kwiecinski, A.D. Martin, W.J. Stirling and R.G. Roberts, Phys. Rev. 42B (1990) 3645.

[lo] W. Zhu, K.M. Chai and B. He, Nucl. Phys. B 427 (1994) 525.

[ 111 L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904.