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Nuclear Physics A 697 (2002) 521–545 www.elsevier.com/locate/npe Energy dependence of diffractive production at HERA E. Gotsman a,d,, E. Levin a,b , M. Lublinsky c , U. Maor a , K. Tuchin a a HEP Department, School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science, Tel Aviv University, Tel Aviv, 69978, Israel b DESY Theory Group, 22603, Hamburg, Germany c Department of Physics, Technion — Israel Institute of Technology, Haifa 32000, Israel d Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA Received 31 May 2001; accepted 11 July 2001 Abstract Intrigued by the unexpected recent results from HERA, that the ratio σ diff tot is only weakly dependent on energy, we have attempted to reproduce this result within the framework of perturbative QCD. To this end we generalize the Kovchegov–McLerran formula for the ratio using Glauber–Mueller approach for shadowing corrections and AGK cutting rules. We investigate several phenomenological approaches and also compare with the successful Golec-Biernat–Wüsthoff model. We have not managed to reproduce the data, and conclude that, apparently, there are soft nonperturbative contributions present at short distances which should be included. 2002 Elsevier Science B.V. All rights reserved. 1. Introduction One of the intriguing phenomena, observed at HERA, is the behaviour of the energy dependence of the ratio σ diff tot in deep inelastic scattering (DIS). It appears (see Ref. [1] and Fig. 1) that this ratio, as a function of energy, is almost constant for different mass intervals of the diffractively produced hadrons over a wide range of photon virtualities Q 2 . At present, there is no theoretical explanation for this striking experimental observation. Even though the quasiclassical gluon field approach [2] yields a constant ratio, this is attained using a nonrealistic total as well as diffractive cross sections which do not depend on energy. * Corresponding author. E-mail addresses: [email protected] (E. Gotsman), [email protected], [email protected] (E. Levin), [email protected] (M. Lublinsky), [email protected] (U. Maor), [email protected] (K. Tuchin). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII:S0375-9474(01)01238-6

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Nuclear Physics A 697 (2002) 521–545www.elsevier.com/locate/npe

Energy dependence of diffractive productionat HERA

E. Gotsmana,d,∗, E. Levina,b, M. Lublinskyc, U. Maora, K. Tuchina

a HEP Department, School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science,Tel Aviv University, Tel Aviv, 69978, Israel

b DESY Theory Group, 22603, Hamburg, Germanyc Department of Physics, Technion — Israel Institute of Technology, Haifa 32000, Israel

d Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA

Received 31 May 2001; accepted 11 July 2001

Abstract

Intrigued by the unexpected recent results from HERA, that the ratioσdiff /σtot is only weaklydependent on energy, we have attempted to reproduce this result within the framework ofperturbative QCD. To this end we generalize the Kovchegov–McLerran formula for the ratiousing Glauber–Mueller approach for shadowing corrections and AGK cutting rules. We investigateseveral phenomenological approaches and also compare with the successful Golec-Biernat–Wüsthoffmodel. We have not managed to reproduce the data, and conclude that, apparently, there are softnonperturbative contributions present at short distances which should be included. 2002 ElsevierScience B.V. All rights reserved.

1. Introduction

One of the intriguing phenomena, observed at HERA, is the behaviour of the energydependence of the ratioσdiff/σtot in deep inelastic scattering (DIS). It appears (see Ref. [1]and Fig. 1) that this ratio, as a function of energy, is almost constant for different massintervals of the diffractively produced hadrons over a wide range of photon virtualitiesQ2.At present, there is no theoretical explanation for this striking experimental observation.Even though the quasiclassical gluon field approach [2] yields a constant ratio, this isattained using a nonrealistic total as well as diffractive cross sections which do not dependon energy.

* Corresponding author.E-mail addresses:[email protected] (E. Gotsman), [email protected], [email protected]

(E. Levin), [email protected] (M. Lublinsky), [email protected] (U. Maor), [email protected](K. Tuchin).

0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9474(01)01238-6

522 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

Fig. 1. Experimental data for the ratioσdiff/σtot taken from Ref. [1].

Recently, K. Golec-Biernat and M. Wüsthoff [3] suggested a phenomenological modelwhich incorporates two main theoretical ideas regarding the transition between “hard” and“soft” processes in QCD [4]: (i) the appearance of a new scale which depends on energy,and is related to the average transverse momentum of a parton in the parton cascade;and (ii) the saturation of the parton density at high energies. This model is successfulin describing all the available experimental data on total and diffractive cross sections,including the energy behaviour ofσdiff/σtot [3,5]. A possible theoretical interpretationof this experimental observation, which is also motivated by parton saturation in thehigh energy limit, has been suggested by Yu. Kovchegov and L. McLerran [6]. In theirpublication the constancy ofσdiff/σtot is closely related to strong shadowing corrections(SC) for diffractive production. We note that in this calculation, diffractive dissociationproceeds exclusively through the production of a quark–antiquark pair neglecting the qqGcontribution.

The success of the above mentioned papers prompted us to reexamine the energybehaviour ofσdiff/σtot in perturbative QCD (pQCD) in more detail. Our approach is basedon two main ideas used to describe the total and diffractive cross section for DIS:

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 523

Fig. 2. Diffractive dissociation of the virtual photon into quark–antiquark pair (qq) andquark–antiquark pair plus one extra gluon (qqG) in pQCD.

1. The final state of a virtual photon induced diffractive processes in the HERAkinematic region [7], are described by the diffraction dissociation of the virtualphoton (γ∗) into an intermediate quark–antiquark pair, or a quark–antiquark pair plusone extra gluon (see Fig. 2).

2. The Glauber–Mueller approach [8] for calculating SC in DIS total and diffractivecross sections was successfully used to describe other indications of strong SC inHERA data [7,9], such as theQ2-behaviour of theF2 slope and the energy behaviourof the diffractive cross section [10].

The paper is organized as follows: in Section 2 we give a simple derivation of theKovchegov and McLerran formula based ons-channel unitarity constraints. We generalizethis formula for the case of qqG final state (see Fig. 2) and discuss the relation between thisapproach and the AGK cutting rules [11]. Section 3 is devoted to a numerical calculation ofthe ratioσdiff/σtot as a function of energy for different photon virtualitiesQ2. A summaryof our results as well as a discussion on future HERA experiments are given in Section 4.

In this paper the most important results of our research are included. A detaileddiscussion of some other related issues can be found in the extended version [12].

2. Shadowing corrections in QCD

2.1. Notations and definitions

In this paper we improve our approach for diffractive production in DIS started inRef. [7]. We use the same notation and definitions as in Ref. [7]. In this paper we utilize themain physical ideas of Ref. [13], i.e., that the correct degrees of freedom at high energies(low x) are colour dipoles, rather than quarks and gluons which appear explicitly in theQCD Lagrangian. The consequence of this hypothesis is that a QCD interaction at highenergies does not change the size and energy of a colour dipole. Hence, the majority of ourvariables and observable are related to the distribution and interaction of the colour dipolesin a hadron.

524 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

Fig. 3. Diffractive production of a quark–antiquark pair.

To facilitate reading the paper we list the definitions which we need (see Fig. 3). We usethe same notation as in Ref. [7]:

1. Q2 denotes the virtuality of the photon in DIS,M the produced diffractive massandW the center-of-mass energy;s =W2.

2. Bjorken variable isx =Q2/W2.3. k⊥ denotes the transverse momentum of the quark, andr⊥ the transverse separation

of quark and antiquark, i.e., the size of the colour dipole.4. z is the fraction of the photon energy carried by quark (or antiquark) in the hadron

rest frame, 1− z is the energy fraction carried by the companion antiquark (orquark).

5. l⊥ is the transverse momentum of the gluon emitted by the quark (antiquark).6. b⊥ is the impact parameter of the reaction. It is the Fourier-conjugate ofq⊥, the

momentum transfer from the incoming photon to the recoiled proton;q2⊥ = −t .7. Thes-channel unitarity constraint in the impact parameter representation has the

form

2 Imf el(s, b⊥)=∣∣f el(s, b⊥)

∣∣2 +Gin(s, b⊥), (1)

wheref el is the elastic scattering amplitude of a virtual photon, off the proton andGin denotes the summed contribution of the all inelastic processes to the scatteringamplitude. This implies

σtot = 2∫

d2b⊥ Imf el(s, b⊥), (2)

σel =∫

d2b⊥∣∣f el(s, b⊥)

∣∣2, (3)

σin =∫

d2b⊥Gin(s, b⊥), (4)

where we have introduced the total, elastic and inelastic cross sections forγ∗pscattering.

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 525

8. xGDGLAP(x,Q2) is the gluon structure function which satisfies the DGLAPevolution equation [14].

9. The imaginary part of the elastic amplitude for acolour dipolescattering on anucleon inleading-twistperturbative QCD is given by

ImaelLT(s, r⊥;b⊥)= π2αS

6r2⊥xG

(x,

4

r2⊥;b⊥

), (5)

wherexG(x,4/r2⊥;b⊥) is defined by∫d2b⊥ xG

(x,

4

r2⊥;b⊥

)= xG

(x,

4

r2⊥

). (6)

It has been shown [4] that we can write

xG

(x,

4

r2⊥;b⊥

)= xGDGLAP

(x,

4

r2⊥

)S(b⊥), (7)

where S(b⊥) is the nucleon profile function which is a pure nonperturbativeingredient in our calculations. We assume it to have the Gaussian form

S(b⊥)= 1

πR2 e−b2⊥/R2, (8)

whereR is the effective transverse size of the nucleon. We will take its valuefrom high-energy phenomenology and HERA experimental data (see Section 3 fordetails).

10. σ (x, r⊥) is the total cross section for dipole–nucleon scattering, and, in the leadingtwist (LT), is given by (see Ref. [7] and references therein)

σLT(x, r⊥)= 2∫

d2b ImaelLT = π2αS

3r2⊥xGDGLAP

(x,

4

r2⊥

). (9)

11. To characterize the strength of the colour dipole interaction we introduce aparameter

κ(x, r⊥)= σLT

πR2 = π2αS

3πR2 r2⊥xGDGLAP

(x,

4

r2⊥

). (10)

We illustrate the physical meaning of Eq. (10) by rewriting it in the form:

κ = σ0ρ(x, r⊥), where (11)

ρ(x, r⊥) = xGDGLAP(x,4/r2⊥)πR2 (12)

is the density of colour dipoles of sizer⊥ and energy fixed byx in the transverseplane. In Eq. (11),σ0 denotes the cross section for the interaction of a dipole of sizer⊥ with a point-like probe

σ0 = π2αS

3r2⊥.

Hence,κ is a packing factorfor dipoles of sizer⊥ in a proton. Ifκ is small, wehave a diluted gas of colour dipoles in a proton, but at lowx the gluon density

526 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

increases [10] andκ → 1. In such a kinematic region our parton cascade becomes adense system of colour dipoles which should be treated nonperturbatively.

12. The probability to find a quark–antiquark pair with transverse separationr⊥ andquark energy fractionz inside a virtual photon is [8,15]

Pγ∗ (z, r⊥;Q2) = αemNc

2π2

∑f

Z2f

∣∣Ψγ∗(z, r⊥;Q2)∣∣2

= αemNc

2π2

∑f

Z2f

(z2 + (1− z)2

)a2K2

1(ar⊥)

+ 4Q2z2(1− z)2K20(ar⊥)

. (13)

Here,Ψγ∗is the wave function of the quark–antiquark pair in the virtual photon,

Nc = 3 stands for the number of colours, whileZf is a charge of the quarkf inelectron charge units. In the present paper we neglect the quark masses in whichcasea2 = z(1− z)Q2.

2.2. SC for the penetration of a qq pair through the target

2.2.1. General approachThe physics underlying our approach has been formulated and developed in Refs. [8,16].

During its passage through the target, the distancer⊥ between a quark and an antiquarkcan vary by an amount"r⊥ ∝ Rk⊥/E, whereE is the pair energy andR is the size of thetarget. Since the quark’s transverse momentumk⊥ ∝ 1/r⊥, the relation

"r⊥ ∝Rk⊥E

≈R1

r⊥E r⊥ (14)

holds if

r2s 2mR, (15)

wheres =W2 = 2mE with m being the mass of the hadron.Eq. (15) can be rewritten in terms ofx, namely:

x 2

mR. (16)

From Eq. (16) it follows thatr⊥ is a good degree of freedom [8] for high energy scattering.We can therefore write the total cross section for the interaction of the virtual photon withthe target as follows:

σtot(γ∗p)(x,Q2) =

1∫0

dz∫

d2r⊥ Pγ∗(z, r⊥;Q2)σ (x, r⊥) (17)

= 2∫

d2b⊥1∫

0

dz∫

d2r⊥ Pγ∗(z, r⊥;Q2) Imael(x, r⊥;b⊥). (18)

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 527

The amplitude for diffractive production of a qq-pair is equal to

f diff (γ∗p → qq)=∫

d2r⊥Ψ γ∗(z, r⊥;Q2)ael(x, r⊥;b⊥)Ψ qq(k⊥, z, r⊥), (19)

where Ψ qq(k⊥, z, r⊥) is the wave function of the quark–antiquark pair with fixedmomentumk⊥ and fraction of energyz. To calculate the total cross section of the diffractiveproduction we should integrate over allk⊥ andz. Using the completeness of the qq wavefunction one obtains

σdiff (γ∗p→ qq)=

∫d2b⊥

1∫0

dz∫

d2r⊥ Pγ∗ (z, r⊥;Q2)∣∣ael(x, r⊥;b⊥)

∣∣2. (20)

Utilizing the unitarity constraint we obtain a prediction for the ratio [6]

(x,Q2)= σdiff

σtot=

∫d2b⊥

∫ 10 dz

∫d2r⊥ Pγ∗

(z, r⊥;Q2)|ael(x, r⊥;b⊥)|22∫

d2b⊥∫ 1

0 dz∫

d2r⊥ Pγ∗(z, r⊥;Q2) Imael(x, r⊥;b⊥)

. (21)

Eq. (21) is a prediction for the ratio [6,17]. It shows that the diffractive dissociation andtotal cross sections are related through unitarity. However, Eq. (21) is too general to beused for practical estimates.

One can apply the unitarity constraint of Eq. (1) in the case of dipole scattering, as wellas in the case of photon scattering. Assuming that the dipole–proton amplitude is mainlyimaginary at high energy, the unitarity constraint has the following general solution:

ael(x, r⊥;b⊥) = i(1− e−Ω(x,r⊥;b⊥)/2), (22)

gin(x, r⊥;b⊥) = 1− e−Ω(x,r⊥;b⊥), (23)

whereΩ is an arbitrary real function andgin stands for all inelastic processes.Using Eqs. (22) and (23), Eq. (21) reduces to

(x,Q2)= σdiff

σtot=∫

d2b⊥∫ 1

0 dz∫

d2r⊥ Pγ∗(z, r⊥;Q2)(1− e−Ω(x,r⊥;b⊥)/2)2

2∫

d2b⊥∫ 1

0 dz∫

d2r⊥ Pγ∗(z, r⊥;Q2)(1− e−Ω(x,r⊥;b⊥)/2)

. (24)

The main goal of our microscopic approach, based on pQCD, is to findΩ .

2.2.2. The Glauber–Mueller approachOne way to obtain a more detailed picture of the interaction, is to consider the dipole–

proton interaction in the eikonal model, which is closely related to Glauber–Muellerapproach [8]. The main assumption of this model is to identify the functionΩ in Eq. (24)with the exchange ofone“hard” pomeron (gluon ladder) as shown in Fig. 3 in which casewe have [18]1

Ω(x, r⊥;b⊥)=ΩP(x, r⊥;b⊥)= 2 ImaelLT = κ(x, r⊥)e−b2⊥/R2

. (25)

Thus, Eq. (22) gives the contribution of the multiple pomeron exchange in the eikonalapproximation to the dipole scattering amplitude.

1 We will use the superscript P to emphasize that the quantity relates to one pomeron exchange, whereas thesuperscript MP will specify the multiple pomeron exchange.

528 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

Since the gluon distribution given by the DGLAP evolution equations originates frominelastic processes of gluon emission, we assume an oversimplified structure of thefinal states in the eikonal model. It consists only of a proton and a quark–antiquarkpair (“elastic” scattering), and an inelastic state with a large number of emitted gluons(∝ ln(1/x)). In particular, we neglect the rich structure of the diffraction dissociationprocesses and simplify them to the final state of p+ qq. For example, we neglect thediffractive production of an excited nucleon in DIS. Discussion of those issues can befound in our paper [12].

SubstitutingΩ =ΩP (see Eq. (25)) in Eq. (24), and assuming thatQ2 is large, we obtain

1∫0

dzPγ∗ (z, r⊥;Q2)≈

∑f

Z2f

4αemNc

3π2Q2 × 1

r4⊥, (26)

which upon substitution in Eq. (20) yields the Kovchegov–McLerran formula [6] fordiffractive dissociation.

We make some simple estimates, assuming thatxGDGLAP(x,4/r2⊥) is a slowly varyingfunction of r2⊥, and substitute 4/r2⊥ = Q2 (which is correct at asymptotically highQ2).Using Eqs. (18) and (20) together with Eq. (22) we obtain for the integrals in Eq. (24):

σtot ∝ σdiff ∝ R2Q2/⟨r2⊥⟩,

where the relevant scale〈r2⊥〉 can be evaluated from the saturation condition

κ(x, 〈r⊥〉)= 1, (27)

which leads to

1⟨r2⊥⟩ ∝ xGDGLAP(x,Q2). (28)

Eq. (28) gives the energy independent ratio as well asσdiff ∝ R2Q2 ×xGDGLAP(x,Q2).This simple estimate, given in Ref. [6], illustrates that the SC lead to a constant ratio (upto logarithmic corrections), or, vice versa, the constant ratio can be a strong argumentfor substantial SC.

To examine the influence of SC on we calculate the ratio using Eq. (26) (integratingover z), assuming that 4/r2⊥ =Q2 in the gluon structure functionxGDGLAP (integratingoverr⊥) and, finally, taking the remaining integral overb⊥. We obtain [19]:

= σdiff

σtot=∫∞

0 db2⊥(1− exp

−12κe−b2⊥/R2)2

2∫∞

0 db2⊥(1− exp

−12κe−b2⊥/R2)

= 2E1(κ2)−E1(κ)+ ln κ

4 +C

2[E1(κ2)+ ln κ

2 +C] , (29)

wereC stands for the Euler constant andE1 denotes the exponential integral of the firstorder.

In Fig. 4 the ratio, given by Eq. (29), is plotted as a function ofκ . One can see thatat very largeκ this ratio has a weak dependence onκ . However, the values ofκ that we

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 529

Fig. 4. The ratioσdiff/σtot is calculated using Eq. (29) in the eikonal model for qq diffractiveproduction. Assuming that the impact parameter dependence factorizes and is proportional to thedelta functionδ(b⊥) (as in Golec-Biernat–Wüsthoff model) we obtain the dashed line (σDD standsfor σdiff ).

Fig. 5. The packing factorκ in the HERA kinematic region versus lgx = log10(x) andQ2 in GeV2.Calculation is performed using the LO GRV94 (a) and NLO GRV98 (b) parameterizations [20] forthe gluon structure function.

are dealing with are not very large (κ ≈ 1, see Fig. 5, whereκ is calculated using theGRV94 and GRV98 parameterizations for the gluon distribution [20]). Fig. 4 shows thatfor κ = 0.2–2 we cannot expect Eq. (29) to yield a constant ratio. Nevertheless, wewould like to draw the reader’s attention to the fact that in the kinematical region of ourinterest, is no longer proportional toκ as it would be in the linear regime.

It is important to realize, that neglecting theb⊥ dependence results in a considerableenhancement of shadowing corrections as is shown by the dashed line in Fig. 4. So, thesuccess of the Golec-Biernat and Wüsthoff model [5] in explanation of the energyindependence is, perhaps, an artifact. We will return to this point in the last section.

530 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

These simple estimates indicate that the SC are essential, but they are still not sufficientlystrong to use the asymptotic formulae. We are in the transition region from low densityQCD, described by the DGLAP evolution, to high density QCD where we may use thequasiclassical gluon field approximation [23]. In the transition region the Glauber–Muellerapproach is a natural way to obtain reliable estimates of the SC. However, we need to studycorrections to Eq. (24) more carefully. The most important among them, are the diffractiveproduction of nucleon excitation [12] and of the qqG system, which gives the dominantcontribution to the diffractive cross section [7].

2.2.3. AGK cutting rules and diffractive productionIn this section we derive Eq. (24) in the Glauber–Mueller approach exploiting the AGK

cutting rules [11]. We intend to use the AGK cutting rules, so as to calculate the diffractionproduction of a qqG system.

The AGK cutting rules provide a prescription of how to calculate the cross sectionsfor processes with different multiplicities of produced particles, provided one knows thestructure of the pomeron exchange, and the expression for the total cross section interms of multi pomeron exchanges [11,21]. In the Glauber–Mueller approach we have asimilar expression for the total colour dipole–proton cross section. Namely, one can easilygeneralize the leading-twist formula given in Eq. (9) using Eq. (22):

σ (x, r⊥)= 2∫

d2b⊥1− e−ΩP(x,r⊥;b⊥)/2, (30)

whereΩP is given by Eq. (25) (see Fig. 6). We first need to define the pomeron structure,i.e., to specify the kind of inelastic processes that are described byΩP. From Eq. (25) wesee thatΩP describes the inelastic processes with large average multiplicity,〈nP〉, of theproduced partons, since it is related to the DGLAP parton cascade. For example, at largeQ2 and lowx, 〈nP〉 = 2

√αS lnQ2 ln(1/x) 1, whereαS = αSNc/π . The AGK cutting

rules allow us to calculate cross sections for the processes with average multiplicities 2〈nP〉,3〈nP〉 and so on, as well as the process for diffractive dissociation with multiplicities muchsmaller than〈nP〉.

Fig. 6. Total cross section for the dipole–nucleon interaction in the Glauber–Mueller approach. Thedashed line shows the diffraction dissociation cut.

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 531

Eq. (30) can be rewritten in the form

σ = 2∫

d2b⊥∞∑n=1

(−1)n+1

n!(ΩP

2

)n, (31)

where each term in the sum corresponds to the exchange ofn pomerons. Thus, the totalcross section forn pomeron exchange is

σn = 2∫

d2b⊥(ΩP

2

)n1

n! . (32)

By AGK cutting rules, the contribution of the diagram withk cut pomeronsσkn can beexpressed through thetotal cross section for then pomeron exchangeσn as follows:

σ kn

σn= (−1)n−k

n!k!(n− k)!2

n−1, if k = 0; (33)

= (−1)n(2n−1 − 1

), if k = 0. (34)

Note, that each cut pomeron produces∼ 〈nP〉 particles in the final state.A diffractive cross section corresponds to the elastic dipole scattering, i.e., to the diagram

with no cut pomerons. In other words, using Eqs. (34) and (32) and summing over allpossible diagrams we get

σqqdiff =

∫d2b⊥

∞∑n=1

σ 0n =

∫d2b⊥

∞∑n=1

(−1)n

n! (2n−1 − 1)2

(ΩP

2

)n

=∫

d2b⊥(1− e−ΩP/2)2, (35)

which is simply∫

d2b⊥ |ael|2.

2.3. Cross section for qqG diffractive production with SC

2.3.1. First correction to the Glauber–Mueller formula for the total cross sectionAs shown in Fig. 6, the eikonal approach takes into account only rescatterings of the

fastest colour dipole. In this subsection we will extend the formalism so as to include alsothe rescatterings with the target of the two fastest color dipoles: the initial colour dipoleand the fastest gluon in Fig. 6. Our goal is to include all diagrams of the type shown inFig. 7. Dashed lines in Fig. 7 indicate the diffraction dissociation cuts. These diagramsinclude the diffractive production of qqG system as well as a qq pair.

Since Eqs. (22) and (23) give the general solution to the unitarity constraint, our problemis to find an expression forΩ which will be more general than Eq. (25) withκ defined byEq. (10).

The natural generalization ofκ is to substitute in Eq. (10) the Glauber–Mueller formulafor the gluon structure function [8,18,19], namely:

ΩP = 1

3π2αSr

2⊥xGDGLAP(x,

4

r2⊥

)S(b⊥) →

532 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

Fig. 7. Total cross section for dipole–nucleon interaction in the first iteration of the Glauber–Muellerapproach. The dashed lines show the diffraction dissociation cut.

ΩMP = 1

3π2αSr

2⊥xGMP(x,

4

r2⊥, b⊥

), (36)

where (recall Eq. (6))

xGMP(x,

4

r2⊥, b⊥

)= 4

π3

1∫x

dx ′

x ′

∞∫r2⊥

dr′2⊥

r′4⊥

2(1− e−ΩP

G(x′,r ′⊥;b⊥)/2). (37)

Note that the ladder originating at the gluonΩG gets an additional colour factor ascompared to the ladder originating at the quark–antiquark pair:ΩG = 9

4Ω . Eq. (37) takesinto account the rescattering of the gluon with the target in the eikonal approach (seeFig. 7). However, the question arises of why do we need to include only gluon rescatteringfor the qqG system. To understand the physics of Eq. (37) it is advantageous to consider theequation that describes the rescattering of all partons. Kovchegov [22] using two principleideas suggested by Mueller [13] proved that the GLR nonlinear equation [4] is able todescribe such rescatterings. The principles are:

1. The QCD interaction at high energy does not change the transverse size of theinteracting colour dipoles, and thus they can be considered as the correct degreesof freedom at high energies.

2. The interaction of a dipole with the target has two clear stages:(a) The transition of the dipole into two dipoles, the probability for this is given by

∣∣Ψ ((r⊥)01 → (r⊥)02 + (r⊥)12)∣∣2 = 1

z

(r⊥)201

(r⊥)202(r⊥)212

, (38)

where 0,1 are the coordinates of the quark and antiquark and 2 are the coordinatesof the radiated gluon (in the transverse plane).

(b) The interaction of each dipole with the target is given by the amplitudeael(x, r⊥;b⊥).

The equation is illustrated in Fig. 8 and it has the following analytic form (assuming thatthe amplitude is purely imaginary)

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 533

Fig. 8. Pictorial representation of the nonlinear evolution equation that takes into account therescattering of all partons with the target (x denotesr⊥).

dael((r⊥)01, b⊥, y

)dy

= −2CFαS

πln

((r⊥)201

ρ2

)ael((r⊥)01, b⊥, y

)

+ CFαS

π2

∫ρ

d2(r⊥)2 (r⊥)201

(r⊥)202(r⊥)212

× 2ael((r⊥)02, b⊥ − 1

2(r⊥)12, y)

− ael((r⊥)02, b⊥ − 12(r⊥)12, y

)ael((r⊥)12, b⊥ − 1

2(r⊥)02, y). (39)

ρ is a cutoff parameter,y = ln(1/x) is the rapidity andCF =Nc/2 (similar equations werealso derived using different approaches in Ref. [23]). The first term on the r.h.s. of theequation gives the contribution of virtual corrections, which appear in the equation due tothe normalization of the partonic wave function of the fast colour dipole (see Ref. [13]).The second term describes the decay of the colour dipole(r⊥)01 into two dipoles(r⊥)02

and(r⊥)12, and their interactions with the target in the impulse approximation (notice afactor 2 in Eq. (39)). The third term corresponds to the simultaneous interaction of twoproduced colour dipoles with the target and describes the Glauber-type corrections forscattering of these dipoles. The initial condition for this equation [22] is given by Eq. (22)with Ω =ΩP from Eq. (25) at somex = x0.

In DIS the dominant contribution comes from the decay of a small dipole into two largedipoles. Therefore, we can reduce the kernel of Eq. (39) to [22]

534 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

∫ρ

d2(r⊥)2 (r⊥)201

(r⊥)202(r⊥)212

−→ (r⊥)201π

+−2QCD∫

(r⊥)201

d(r⊥)202

(r⊥)402

. (40)

We make the first iteration of Eq. (39), by substituting the Glauber–Mueller formula fora color dipole rescattering (see Eq. (22) withΩ =ΩP from Eq. (25)), and obtain

ael(first iteration) = CFαS(r⊥)201

2(1− e−ΩP/2)− (

1− e−ΩP/2)2= CFαS(r⊥)201

1− e−2ΩP/2. (41)

Eq. (41) gives the Glauber–Mueller formula for the gluon structure function of Eq. (37).2

Note, that two assumptions have been made in deriving Eq. (41): (i)b⊥ (r⊥)12 or (r⊥)02;and (ii) we neglected the first term in Eq. (39). Both approximations hold in the double logapproximation of pQCD [22].

Eq. (37) describes the passage of the qqG system through the target, as it correspondsto the interaction of two colour dipoles, which are identified with our qqG system. In theparton cascade in the DIS kinematic region, the gluon always corresponds to two colourdipoles of the same size. By changingΩP →ΩMP we take into account the fact that initialqq pair can fluctuate in the qqG system many times during the passage through the target.This is illustrated in Fig. 7.

In order to find an exact solution to Eq. (39) one should continue the iterations. Thisway rescatterings (by multiple pomeron exchange) of all radiated gluons can be taken intoaccount. The main features of the exact solution were studied in Ref. [24]. However, aswe will see in the next section, in the HERA kinematical region it is sufficient to takeinto account rescatterings of the fastest gluon, which itself is a small correction to thesemiclassical Glauber–Mueller formula.

Finally, by including gluon radiation in Eq. (30), the equation for the total cross sectionof dipole–nucleon scattering reads:

σ (x, r⊥)=∫

d2b⊥ 21− e−ΩMP(x,r⊥;b⊥)/2, (42)

whereΩMP(x, r⊥;b⊥) is given by Eqs. (36) and (37).

2.3.2. Cross section for diffractive productionWe would like to obtain the cross section for the diffractive production of both qq and

qqG final states using the AGK cutting rules. In Fig. 7 one can see which cuts in thetotal cross section are related to the diffractive processes. They are shown in Fig. 7 bydashed lines. First, we have to generalize the AGK cutting rules, since in Section 2.2.2 weidentifiedΩ with one-pomeron (gluon “ladder”) exchange settingΩ =ΩP. This fact wasused in the discussion of Section 2.2.3. In Eq. (42),ΩMP itself has a more complicatedstructure which can be recovered by applying the AGK rules of Eqs. (33) and (34) onceagain.

2 Eq. (39) is written for large number of coloursNc 1. For finiteNc, 2ΩP in Eq. (41) should be replacedby 9

4ΩP.

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 535

We derived Eq. (35) by summing all diagrams with no cut pomerons. So, to include thefastest gluon rescatterings we just substituteΩMP instead ofΩP in Eq. (35):

σqqdiff =

∫d2b⊥

(1− e−ΩMP/2)2. (43)

Now, we make one cut. SinceΩMP itself includes pomeron rescatterings, we can cutit in such a way that the final state will contain just one additional (fastest) gluon. Thiscorresponds to the elastic scattering of that gluon and, thus, to the diffraction with a qqGfinal state. Therefore, we replace the cutΩMP by the half (see footnote 2) of the gluonelastic scattering amplitudeΩel

G which is given by:

ΩelG = 4αs

3πr2⊥

1∫x

dx ′

x ′

∞∫r2⊥

dr′2⊥

r′4⊥

(1− e−ΩP

G(x′,r ′⊥,b⊥)/2)2, (44)

where we used Eqs. (36) and (37). Then, using the AGK cutting rules, Eqs. (33) and (31),we obtain once again

σqqGdiff =

∫d2b⊥

∞∑n=1

1

n!2(ΩMP

2

)n−1ΩelG

2 · 2(−1)n−1 n!

(n− 1)!2n−1

= 1

2

∫d2b⊥ e−ΩMP

ΩelG . (45)

Let us note, for future reference, that using this method one can easily calculate thecontribution ofk gluons in the final state:

σqqGk

diff =∫

d2b⊥∞∑n=1

1

n!2(ΩMP

2

)n−k(ΩelG

2 · 2

)k(−1)n−k n!

(n− k)!k!2n−1

=∫

d2b⊥∞∑n=1

(−1)n−k

k!(n− k)!(ΩMP)n−k(Ωel

G

2

)k. (46)

Summing the contribution of all possible numbers of radiated gluons we get

∞∑k=1

σqqGk

diff =∫

d2b⊥ e−ΩMP(eΩ

elG/2 − 1

). (47)

Finally, collecting Eqs. (42), (43) and (45) we obtain the generalization of Eq. (24),which takes into account the diffractive final state production of both a qq pair, and a qqpair plus an extra gluon:

(x,Q2) = σdiff

σtot, where (48)

σdiff(x,Q2) =

∫d2b⊥

∫dz∫

d2r⊥ Pγ∗(z, r⊥;Q2)

×(

1− e−ΩMP(x,r⊥;b⊥)/22

536 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

+ e−ΩMP(x,r⊥;b⊥)r2⊥2αS(r⊥)

1∫x

dx ′

x ′

∞∫r2⊥

dr ′2⊥r ′4⊥

1− e−ΩP

G(x′,r ′⊥;b⊥)/22

),

σtot(x,Q2) = 2

∫d2b⊥

∫dz∫

d2r⊥ Pγ∗(z, r⊥;Q2)

(1− e−ΩMP(x,r⊥;b⊥)/2).

We note that Eq. (48) was derived in the double log approximation of the DGLAPevolution equations, in which the colour dipoles in the produced qqG system are muchlarger than the initial qq dipole.

The factor e−ΩMP(x,r⊥;b⊥) in the second term of Eq. (48) leads to a suppression ofdiffractive production of the the qqG system. Therefore, in the asymptotic limit, at largevalues ofκ , only elastic rescattering of the qq pair survives, leading to the value of the ratioσdiff/σtot → 1/2. It turns out that this factor is important for numerical calculations in theHERA kinematic region (see the next section).

3. Numerical calculation for σdiff/σtot

In this section we present our numerical result for the ratio of the diffractive dissociationcross section to the total cross section in DIS.

The main parameters that determine the value ofκ , are the value ofR2 andthe value of the gluon distribution. We chooseR2 = 10 GeV−2 from “soft” high-energy phenomenology [25,26], and is in agreement with HERA data on J/, photo-production [27]. ForxGDGLAP(x,Q2) we use the GRV94 parameterization and theleading-order solution of the DGLAP evolution equation [20]. We have two reasons forthis choice:

1. Our goal in this paper is to understand the influence of the SC on the energy behaviourof the ratioσdiff/σtot. However, experience shows that we are able to describe almostall HERA data by changing the initial conditions of the DGLAP evolution equations.Unfortunately, we have no theoretical restrictions on this input for any of the pdfparameterizations on the market. On the other hand, we know theoretically [4] thatthe SC work in such a manner that they alter the initial conditions for the DGLAPevolution, making it impossible to apply them at fixedQ2 =Q2

0. With SC we haveto solve the DGLAP equations starting withQ2 =Q2

0(x) whereQ20 is the solution

of the equationκ(x,2/Q0(x)) = 1. Therefore, we are in a quandary: we require aDGLAP input but, if we take it from the pdf global fits, there is a danger that we willincorporate the effects of the SC in the initial condition of these parameterizations.In general, data taken after the 1995 runs are at energies sufficiently high to effectthe lowx behaviour of the initial inputs. GRV94 is based on the experimental dataat rather largex, so we hope that SC are minimal in the initial conditions of thisparameterization.

2. The GRV parameterization starts with rather low virtualities (as low asQ2 ≈0.5 GeV−2). This is a major weakness of this approach, since one cannot guarantee

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 537

that only the leading-twist contribution is dominant in the DGLAP evolutionequations. We agree with this criticism, but the low value ofQ2

0 leads to the solutionof the DGLAP equation which is closer to the leading log approximation, in whichwe can guarantee the accuracy of our master equation (see Eq. (48)).

Before we present our numerical results we have to make an important comment. Itconcerns the substitution in Eq. (37), where the iterated gluon densityxGMP(x,4/r2⊥;b⊥)is defined. Due to technical difficulties, two modifications of the formula were made in theactual numerical calculations. First of all we assumed an exponentialb⊥ factorization ofxGMP(x,4/r2;b⊥):

xGMP(x,4/r2⊥;b⊥)= 1

πR2xGMP(x,4/r2⊥

)e−b2⊥/R2

.

We justify the factorization by the following two arguments. The first one is numerical,we have actually checked that factorization holds numerically with satisfactory accuracy.The second argument is that having assumed factorization forxGDGLAP we can assume itapplies forxGMP. The second modification, made with Eq. (37), is due to the fact that theGRV parameterizations, which we are working with, do not satisfy the DGLAP equationin the DLA (see Ref. [19] for a discussion of this problem). The authors of [19] suggesteda modification of Eq. (37), which we implement in all our numerical calculations. The finalformula forxGMP has the form:

xGMP(x,Q2) = 4

π2

1∫x

dx ′

x ′

∞∫4/Q2

dr ′2⊥r ′4⊥

∫db2⊥ 2

(1− e−ΩGRV(x ′,r ′⊥,b⊥)/2)

+ xGGRV(x,Q2)− αSNc

π

1∫x

Q2∫Q2

0

dx ′

x ′dQ′2

Q′2 x′GGRV(x ′,Q′2). (49)

In all the figures representing our numerical results the upper solid line corresponds tothe sum of all contributions, the widely spaced dashed line to diffractive production of qqpair, while the narrowly spaced dashed line to the diffractive production of qqG.

In Fig. 9 we plot the results of our calculations using Eq. (48). We have not added anycontribution from the nucleon excitations. In Fig. 10 we show our predictions for smallvalues ofQ2. The first observation is that the ratio increases considerably with energy,reaching a value of about 25%. It should be stressed that, even at rather small valuesof Q2, we do not see any sign of saturation of the energy behaviour of this ratio, which iscontinuously increasing in the HERA kinematic region. The calculations were done withGRV94. As noted the main features are also obtained with GRV98.

It is interesting to consider separately the qq diffractive production and the productionof the qqG final state. Both cross sections increase as a function of the energyW . AtsmallQ2 (Fig. 10) the total diffractive cross section is dominated (about 70%) by thediffractive production of the qq pair. The situation changes when higher values ofQ2 areconsidered. Then the qqG final state contributes even more than the qq pair. Indeed, at highenergyW = 200 GeV, qqG is produced in approximately 25% of all diffractive events at

538 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

Q2 = 8 GeV2 Q2 = 14 GeV2

Q2 = 27 GeV2 Q2 = 60 GeV2

Fig. 9. The ratioσdiff /σtot versusW for different values ofQ2 (R2 = 10 GeV−2). See text for details(σDD stands forσdiff ).

Q2 = 1 GeV2 Q2 = 2.5 GeV2

Fig. 10. The ratioσdiff/σtot versusW for small values ofQ2 (R2 = 10 GeV−2). See text for details(σDD stands forσdiff ).

moderateQ2 = 8 GeV2. For smaller values ofQ2 this fraction decreases, being only 10%at Q2 = 1 GeV2. This is an expected behaviour if the SC play an important role. Theresults presented show that the contribution of the extra gluon emission is crucial for thepredictions, and may lead to a doubling the value of the ratio.

At small Q2, the fraction of qqG diffractive production to the total cross sectionincreases sufficiently slowly due to the suppression factor e−ΩMP(x,r⊥;b⊥) in Eq. (48). For

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 539

Q2 = 1 GeV2, R2 = 5 GeV−2 Q2 = 2.5 GeV2, R2 = 5 GeV−2

Q2 = 8 GeV2, R2 = 5 GeV−2 Q2 = 14 GeV2, R2 = 5 GeV−2

Q2 = 27 GeV2, R2 = 5 GeV−2 Q2 = 60 GeV2, R2 = 5 GeV−2

Fig. 11. The ratioσdiff/σtot versusW for R2 = 5 GeV−2. See text for details (σDD stands forσdiff ).

largerQ2, ΩMP becomes small and the fraction of qqG diffractive production increasesfaster with the energy.

In Fig. 11 we illustrate the dependence of our calculation on the value ofR2. Wecompare the calculations with

(1) R2 = 10 GeV−2 which is related to the Glauber–Mueller approach, correspondingto the eikonal model for the nucleon target;

(2) R2 = 5 GeV−2 which is the average radius in a two channel model [12].One can see from Fig. 11 that the value of the ratioσdiff/σtot depends onR2. However, theenergy dependence is still very pronounced.

One can also try to improve this situation by fixing the produced massM (see Fig. 3) andcalculating the total and diffractive cross sections for diffractive production in certain mass

540 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

Q2 = 8 GeV2 Q2 = 27 GeV2

Fig. 12. The ratioσdiff/σtot as a function ofW for different mass bins (R2 = 10 GeV−2). The uppersolid line corresponds to the full calculation, the widely dashed line to diffractive production of a qqpair, while the narrowly dashed line to the diffractive production of qqG (σDD stands forσdiff ).

intervals. The results of this work (see Ref. [12]) are shown in Fig. 12. One clearly observesthat the separation into mass bins does not significantly influence the energy dependenceof .

Therefore, the general conclusion of this section is that the SC fail to reproduce theconstant ratio ofσdiff/σtot for DIS seen in the HERA kinematic region.

4. Discussion

4.1. Comparison with the Golec-Biernat–Wüsthoff model

It is worthwhile to compare our model with the Golec-Biernat–Wüsthoff model [3],which successfully reproduces the experimental data (Fig. 1). In the Golec-Biernat–Wüsthoff model the effective dipole cross sectionσGW(x, r⊥), describing the interactionof the qq pair with a nucleon has the form:

σGW(x, r⊥)= σ0[1− e− 1

4r2⊥/R2

0(x)], (50)

whereR0(x) = (x/x0)λ/2 GeV−1, σ0 = 23.03 mb,λ = 0.288,x0 = 3.04× 10−4. In this

model, the diffractive dissociation cross section is given by

σGWdiff = σ 2

GW/(16πBD), BD = 6 GeV−2. (51)

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 541

The total and diffractive cross sections calculated using the G–W dipole cross sections aredenotedσ tot

GW andσ diffGW. A comparison between G–W model and our model is presented in

Fig. 13. We found a significant difference between the two models . The advantage of G–Wmodel is that this model takes the saturation scale into account in a simple way through therelationQ2

s(x)= 4/R20(x). However, in doing so, this model loses its correspondence with

the DGLAP evolution equation. Our approach has a correct matching with the DGLAP

(a) (b)

(c) (d)

(e) (f)

Fig. 13. Comparison between G–W model and our approach atx = 10−3: (a) the ratioσ /σGW isplotted versusr2⊥; (b) the ratioσdiff/σ

diffGW is plotted versusr2⊥; (c) the ratio of the total cross sections

σtot/σGWtot for Q2 = 8 GeV2 as function of energyW ; (d) the ratio of the diffractive cross sections

σdiff/σdiffGW for Q2 = 8 GeV2 as a function of energyW ; (e) and (f) are the same as (c) and (d) for

Q2 = 60 GeV2.

542 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

evolution and we expected that we would be able to provide a better description of theexperimental data than the G–W model. It turns out (see Fig. 13) that we do not reproducethe behaviour of theσdiff/σtot ratio as well as the G–W model, mostly due to our refinementin the region of smallr2⊥.

There is a substantial difference in the way the two models describe the qqG state. Wefailed to find a correspondence between our formula for qqG production which followsfrom the AGK cutting rules, and the G–W description of this process. However, our failureto fit the experimental data is mostly due to a large difference in the dipole cross section,rather than in the different treatment of the qqG diffractive production.

The first row of Fig. 14 shows our dipole cross sections at different energies. The maindifference with the G–W model is the energy rise of the cross sections which follows fromtheb⊥ dependence included in the eikonal formulae, but neglected in the G–W model (seeFig. 4). The other graphs of Fig. 14 show what the essential distances in our calculationsare. One can see that the main contributions to the diffractive cross sections stem fromlonger distances than in the total cross sections, as have been predicted theoretically [4].The typical distances in diffractive cross section are of the order of the saturation scaleQs(x) (κ(x, r⊥ = 2/Qs(x))= 1) in contrast to the total cross section were typical distancesare much shorter , about 1/Q. Therefore, a possible reason of why we failed to describethe ratio of interest, could be that our model cannot describe the dipole cross sectionin the vicinity of the saturation scale. However, it was demonstrated in the AGL papers(see Ref. [4]) that our eikonal model gives a good approximation to the correct nonlinearevolution equations at the particular distances which are essential accordingly to Fig. 14.

4.2. Summary

In this paper we developed an approach to describe the diffractive dissociation in DIS,based on three main ideas:

(1) the dominant contribution to diffractive dissociation processes in DIS stems fromrather short distances [7] and, therefore, we can use pQCD to describe them;

(2) the final state in diffractive dissociation can be simplified in the HERA kinematicregion by considering only the production of qq and qqG [7];

(3) the shadowing corrections which are essential for the description of the diffractiveprocesses, can be taken into account, using the Glauber–Mueller approach [8].

Using pQCD and the Glauber–Mueller approach we derived a generalization of Kovche-gov–McLerran formula [6] for the ratioσdiff/σtot (see Eq. (48)) which is applicable tothe HERA experimental data on diffractive production at HERA. However, we foundthat Eq. (48) does not yield the approximate energy independence of the ratioσdiff/σtot

observed experimentally. Our attempts to introduce the experimental cuts for diffractiveproduction do not change this pessimistic conclusion.

In conclusion, we believe that this paper provides a strong argument that a nonperturba-tive QCD contribution is essential for the description of diffractive production. Thus, ourapproach, based exclusively on pQCD, is incomplete. However, we showed that the mainsource of the observed energy dependence arises from rather short distances (see Figs. 13

E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545 543

Fig. 14.σ (x, r⊥) (left column) andσdiff (x, r⊥) (right column) versusr⊥ at different values ofx. Allothers figures are the integrands and partial contributions to the cross sections versus dipole sizesr⊥at different values ofx andQ2 (σDD stands forσdiff ).

and Fig. 14), where we did not see any evidence for a nonperturbative correction to the to-tal DIS cross section. In principle, it was pointed out in Ref. [28] that the scale anomaly ofQCD generates a nonperturbative contribution at high energy at sufficiently short distances

544 E. Gotsman et al. / Nuclear Physics A 697 (2002) 521–545

r2⊥ ≈ 1/M20 ∼ 0.25 GeV−2. We plan to examine the influence of such nonperturbative cor-

rections in further publications.We studied in detail the contribution of the excited hadronic states in diffractive

production. We found that the contribution of nucleon excitations should depend onQ2

andx, leading to large cross section at bigger values ofQ2 and higherx.We hope that our paper will draw the attention of the high-energy community to the

beautiful experimental data on the energy dependence of the ratioσdiff/σtot which havestill not received an adequate theoretical explanation, and which can provide a new insightto the importance of nonperturbative corrections even at short distances.

Acknowledgements

E.L. would like to acknowledge the hospitality extended to him at DESY Theory Groupwhere this work was started.

This research was supported in part by BSF # 9800276 and by the Israel ScienceFoundation, founded by the Israeli Academy of Science and Humanities.

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