Physics Letters B 547 (2002) 15–20

www.elsevier.com/locate/npe

Scattering of gluons from the color glass condensate

Adrian Dumitru, Jamal Jalilian-Marian

Nuclear Theory Group, Physics Department, BNL, Upton, NY 11973, USA

Received 30 November 2001; received in revised form 30 July 2002; accepted 19 September 2002

Editor: W. Haxton

Abstract

We prove that the inclusive single-gluon production cross section for a hadron colliding with a high-density target factorizesinto the gluon distribution function of the projectile, defined as usual within the DGLAP collinear approximation, times thecross section for scattering of a single gluon on the strong classical color field of the target. We then derive the gluon–proton(nucleus) inelastic cross section and show that it is (up to logarithms) infrared safe and that it grows slowly with center ofmass energy. Furthermore, we discuss jet transverse momentum broadening for the case of nuclear targets. We show that in thesaturation regime, in contrast to the perturbative regime, the width of the transverse momentum distribution is infrared finiteand grows rapidly with energy and rapidity. In both regimes, however, transverse momentum broadening exhibits the sameA

dependence. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Understanding the behavior of hadronic cross sec-tions at very high energy is one of the major unre-solved problems in QCD. Even though Regge theorycan, in principle, predict the energy dependence ofhadronic cross sections and there are many phenom-enological models, based on Regge theory, which aresomewhat successful in describing the data, the rela-tion of Regge theory to QCD is not well understood.Therefore, it would be very important to be able to cal-culate the high energy behavior of hadronic cross sec-tion from QCD itself. However, it is believed that to-tal cross sections are intrinsically non-perturbative andnot amenable to perturbative QCD methods [1]. In this

E-mail address: dumitru@bnl.gov (A. Dumitru).

note, we consider the simpler problem of (real) gluon–proton total inelastic cross section in the very highenergy (smallx) limit, using the effective action andclassical field method developed recently, and showthat its growth with energy is inhibited as comparedwith that expected from perturbative QCD.

At very smallx, a hadron is a color glass conden-sate due to the condensation of gluons into a coherentstate with characteristic momentum ofQs(x) [2–5].In other words, most of the gluons in the wave func-tion of a hadron have momenta of orderQs(x). Aswe go to higher energies,Qs(x) increases and even-tually will become much larger thanΛQCD so thatαs(Qs)� 1 and weak coupling methods can be used.Even though the theory may be weakly coupled, it isstill non-perturbative in the sense that one has highgluon densities and strong classical color fields associ-ated with them so that the standard perturbative QCD

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)02709-0

16 A. Dumitru, J. Jalilian-Marian / Physics Letters B 547 (2002) 15–20

breaks down. Much progress has been made in devel-oping a formalism which describes this weakly cou-pled, though non-perturbative region of QCD. It gener-alizes the standard collinear factorized (leading twist)expressions for hadronic cross sections and allows oneto calculate hadronic cross sections in an environmentwhere higher twist (high gluon density at smallx) ef-fects are important. We refer the reader to [6] and ref-erences therein for a review of this formalism.

One can use this classical field method to calculategluon production at high energy [7]. In this Letter, weuse the results of [8] to prove (collinear) factorizationof the inclusive cross-section for a “dilute” hadronimpinging on a dense target. Based on that resultwe then derive an expression for the gluon–proton(nucleus) inelastic cross section and for its energydependence at very high energies. Using our resultsfor the gluon–proton (nucleus) inelastic cross section,we consider the transverse momentum broadening ofgluons due to scattering from the strong classical fieldof a nucleus and show that it is infrared finite, energydependent and scales likeA1/3.

2. Gluon–proton inclusive cross section

In the classical field and effective action approachto hadronic (nuclear) collisions at high energy, onesolves the classical Yang–Mills equations of motionin the presence of random color charges created bythe valence quarks and gluons at highx. One thenaverages over these color charges with a Gaussianweight to compute physical quantities. The classicalfields of the colliding hadrons (nuclei) before thecollision are given by the single-hadron (nucleus)solutions which, in light cone gauge, are

A±1,2 = 0,

(1)Ai1,2 = i

gU1,2(x⊥)∂iU†

1,2(x⊥).

These fields serve as the initial conditions for solvingthe Yang–Mills equations of motion in the forwardlight cone; they were solved in [8] for the case ofasymmetric collisions where the classical field of oneof the colliding sources is much stronger than theclassical field of the other source.

Fig. 1. Gluon production in inclusivepp scattering at rapidity farfrom the target proton (i.e., the strong color field).

The produced gluon field in the forward light coneregion is given by (Fig. 1)

Ai(τ, x⊥)=U(x⊥)(βi(τ, x⊥)+ i

g∂i)U†(x⊥),

(2)A±(τ, x⊥)= ±x±U(x⊥)β(τ, x⊥)U†(x⊥).

We have chosen the gauge conditionx+A− +x−A+ = 0. Here,τ = √

2x+x− denotes proper timeandx⊥ is the transverse coordinate. TheU ’s are ro-tation matrices in color space, to be specified shortly.At asymptotic times,τ → ∞, the fieldsβ andβi aregiven by superpositions of plane wave solutions

β(τ → ∞, x⊥)

=∫d2p⊥(2π)2

1√2ωτ3

(3)× {a1(p⊥)eip⊥·x⊥−iωτ + c.c.

},

βi(τ → ∞, x⊥)

=∫d2p⊥(2π)2

1√2ωτ

εilpl⊥ω

(4)× {a2(p⊥)eip⊥·x⊥−iωτ + c.c.

}.

The number distribution of produced gluons at rapidityy and transverse momentump⊥ is given by

(5)dN

d2p⊥dy= 2

(2π)2tr(∣∣a1(p⊥)

∣∣2 + ∣∣a2(p⊥)∣∣2).

One can obtain an analytical solution of the classi-cal Yang–Mills equations if one of the fields (the “pro-jectile”) is much weaker than the other (the “target”).

A. Dumitru, J. Jalilian-Marian / Physics Letters B 547 (2002) 15–20 17

This situation is realized physically when|y − yt | |y − yp|, because of the renormalization group evo-lution of the gluon density in rapidity [5,6,9]. In thatcase, it turns out [8] that theU ’s appearing in Eq. (2)are just theU2’s from Eq. (1); that is, to leading orderin the weak field, the plane waves in the forward lightcone are just gauge rotated by the strong field

(6)U2(x⊥, y)=P exp

(−ig

y∫yt

dy ′Φ2(x⊥, y ′)).

Here, we assumed that the target is moving along thenegativez-axis, i.e., at negative rapidityyt < 0. In thecolor glass condensate model, we now have to averagethe squared amplitudes over the gauge potentialsΦ1,2(x⊥, y) using a Gaussian weight [2,4,5]:⟨|a1,2|2

⟩=∫

DΦ1DΦ2∣∣a1,2(Φ1,Φ2)

∣∣2

× exp

[−

yp∫y

dy ′∫d2x⊥

tr(∇2⊥Φ1(x⊥, y ′))2

g2µ21(x⊥, y ′)

(7)

−y∫

yt

dy ′∫d2x⊥

tr(∇2⊥Φ2(x⊥, y ′))2

g2µ22(x⊥, y ′)

].

Here,µ denotes the density of color charge in thesources per unit of transverse area and rapidity. Theradiation number distribution (5) turns out to dependonly on theintegrated color charge densities of thesources

χ1(y,p2⊥

)=yp∫y

dy ′µ21

(y ′,p2⊥

),

(8)χ2(y,p2⊥

)=y∫

yt

dy ′µ22

(y ′,p2⊥

).

Due to the evolution in rapidity,χ1(y) � χ2(y) if|y − yt | |y − yp|. In terms of the gluon distributionfunction in the projectile or target proton [10], respec-tively,

(9)χ1,2(y,p2⊥

)= Nc

N2c − 1

1

πR2

1∫x0

dx gp,t(x,p2⊥

),

wherex0 is on the order of∼ p⊥ cosh(y)/√s. In the

weak-projectile limit the averaging (7) over the gaugepotentials with Gaussian weight can be done [8]. Athigh transverse momentum,p⊥ Qt (whereQt isthe saturation momentum of the target), one recoversthe standard result from perturbation theory [7,10–12],

dN

d2p⊥ dy= 4α2

s

πR2

N2c

N2c − 1

αsNc

p4⊥

×∫d2k⊥π2

p2⊥k2⊥(p⊥ − k⊥)2

×∫dx ′ gp

(x ′, k2⊥

)(10)×

∫dx ′′ gt

(x ′′, (p⊥ − k⊥)2

).

This is to be expected, of course, because the gluonoccupation numbers in the target proton become smallatp⊥ Qt , i.e., that field becomes weak as well andso can be treated perturbatively.

In the collinear limit(k⊥/p⊥ → 0), the first factorin (10) is the DGLAP [13] splitting function forgluons. It evolves the gluon distribution function of theprojectile from the scalek2⊥ to the scalep2⊥,

(11)

xgp(x,p2⊥

)= αsNc

p⊥∫d2k⊥π2

1

k2⊥

∫dx ′ gp

(x ′, k2⊥

).

Thus, in the collinear limit Eq. (10) simply turns into

dN

d2p⊥dy= 4α2

s

πR2

N2c

N2c − 1

1

p4⊥xgp

(x,p2⊥

)(12)×

∫dx ′′ gt

(x ′′,p2⊥

).

Integrating over the impact parameter space (simply afactor of πR2) and dividing by the flux of incominggluons in the projectile proton at the hard scalep2⊥,which is simplyxgp(x,p2⊥), we obtain the inclusivegluon–proton differential cross section atlarge mo-mentum transfer

(13)dσ

pertgp

d2p⊥= 4α2

s

N2c

N2c − 1

1

p4⊥

1∫x0

dx gt(x,p2⊥

).

This is the standard expression for the gluon–protoninclusive cross section [10,14] in the one-gluon ex-change approximation which diverges like 1/t2 ≡

18 A. Dumitru, J. Jalilian-Marian / Physics Letters B 547 (2002) 15–20

1/p4⊥ at small momentum transfer.σ pertgp grows like a

power of energy ifxg(x)∼ 1/xδ with δ > 0.Next, we consider the radiation distribution (5)

at small transverse momentum,p⊥ � Qt , but largeenough so that the projectile proton is still in the weak-field regimep⊥ ΛQCD. In this regime [8]

dN

d2p⊥ dy= 1

4π

1

p2⊥αsNc

∫d2k⊥π2

p2⊥k2⊥(p⊥ − k⊥)2

(14)×1∫

x0

dx gp(x, k2⊥

)

(15)� 1

4π

1

p2⊥x gp

(x,p2⊥

).

This result is valid to leading order inα2s χ1, but to all

orders inα2s χ2. The second line applies in the limit of

nearly collinear splitting(k⊥/p⊥ → 0), where we canemploy Eq. (11). Thus, the inclusive gluon productionagain factorizes into the gluon distribution function ofthe projectile at the scalep2⊥ times the cross sectionfor scattering of a gluon on the high-density target.

To obtain the gluon–proton cross section, we againintegrate over the impact parameter and divide by theflux of incoming collinear gluons in the projectile atthe scalep2⊥ , which isxgp(x,p2⊥). We get

(16)dσ sat

gp

d2p⊥= 1

4π

1

p2⊥πR2.

In the saturation regime, the cross section is of order1 rather thanα2

s as in (13) because the occupationnumber of the target∼ 1/αs cancels one power ofthe coupling in both the amplitude and the complexconjugate amplitude. Note that the above result holdsfor p⊥ ΛQCD only. To get the gluon–proton totalinclusive cross section, we integrate (16) which gives

(17)σ satgp � 1

4πR2 log

(Q2t /Λ

2QCD

)+O(α2s

).

Parameterization of the saturation momentum like apower of energy,Q2

t ∼ sγ as done in [15] then leadsto a logarithmically growing cross section

(18)σgp ∼ πR2 logs.

On the other hand, if we use a DLA DGLAP typeparameterizationQt ∼ exp

√log1/x, the cross section

would grow like a square root of energy. Therefore,

it is clear that different parameterization of the targetsaturation scaleQt would lead to a different growthof the cross section with energy. Curiously enough,assumingQt to grow like s logs would lead to growthof the gluon–proton cross section like log2 s. In orderto determine the energy dependence ofQt rigorously,one would need to formally define it in terms of agluonic two-point function and solve the non-linearevolution equation thatQt follows. This is howeverbeyond the scope of this work.

Strictly speaking, our results are correct for thecross section per unit area,dσ/d2b. What we haveshown here is that the growth of this cross section at afixed impact parameter is inhibited due to high gluondensity effects. In principle, as we go to larger im-pact parametersb, the gluon density becomes smallerand smaller until our classical formalism reduces tothe standard pQCD where the gluon density is not thatlarge. Since large impact parameters (where the gluondensity is low and so the classical approach is notvalid) are believed to give the dominant contribution tothe total cross section, one should understand the areaappearing in (18) as a parameter which we cannot pre-cisely determine. A genuine non-perturbative (strongcoupling vs. our weak coupling methods) calculationwould presumably enable one to determine this factorand its energy dependence.

The above equations hold equally well for a nu-clear target. In Eq. (13) one just has to replacethe gluon distribution function of the target protonby that for the nucleus; in the absence of shadow-ing [3,16], gA(x,Q2) = Ag(x,Q2). On the otherhand, in Eqs. (16), (17) one substitutes the radius ofthe proton by that of the nucleus,RA �A1/3R.

It is now straightforward to compute the transversemomentum broadening of the incoming gluon jettraversing the target nucleus [14,17]. The transversemomentum broadening is given by [14]

(19)⟨p2⊥⟩= 1

A

⟨tA(�b)

⟩ ∫d2p⊥ p2⊥

dσgA

d2p⊥,

where〈tA(�b)〉 is the nuclear thickness function aver-aged over impact parameters:

(20)⟨tA(�b)

⟩= A

πR2A

= ρL,

with L the average thickness of the nucleus, and withρ � 0.15 fm−3 the density of nucleons in the nucleus.

A. Dumitru, J. Jalilian-Marian / Physics Letters B 547 (2002) 15–20 19

In perturbation theory, using (13) and DGLAP [13]evolution,

(21)αsNc

π

∫dx gA

(x,p2⊥

)= d

d logp2⊥xgA

(x,p2⊥

),

one obtains⟨p2⊥⟩= 4π2αs

Nc

N2c − 1

(22)× [xg(x,Q2

max

)− xg(x,Q2

min

)]ρL.

Thus, 〈p2⊥〉 grows proportional toA1/3 if nuclearshadowing is disregarded [14,17]. Appearance ofxg(x,Q2

min) signifies sensitivity to physics at smallmomentum transfer. For a discussion of what value ofx is to be used in (22), see [14,17].

Transverse momentum broadening in the saturationregime is quite different. Using (16) and (19), (20), wefind that

(23)⟨p2⊥⟩� Q2

A

4+O(αs ),

whereQ2A denotes the nuclear saturation scale which,

from the definition ofχ2 ∼ Q2A/α

2s in (9), is A1/3

times larger than that for a proton. Thus, in the satura-tion regime jet broadening grows with thesame powerof the nuclear mass numberA as in the perturbativeregime(p2⊥) ∼ A1/3. Also, our derivation shows thatthe jet transverse momentum broadening is energy de-pendent even though we cannot determine its energydependence in a model independent way. However, forQ2A ∼ sγ [15], we obtain a strong power-law increase

of transverse momentum broadening. We would liketo emphasize [18] that since the saturation scaleQ2

t

of the target is expected to be much larger in the for-ward rapidity (projectile fragmentation) region, the jettransverse momentum broadening will be much largerin the forward region than in the central rapidity re-gion, contrary to the prediction (22) from perturbationtheory.

3. Conclusion

In summary, we have shown that inclusive gluonproduction from a hadron scattering off a high-densitytarget (with “saturated” gluon density) factorizes intothe gluon distribution of the projectile times the cross

section of the beam of incoming collinear gluons onthe dense target. We derived the gluon–proton inclu-sive cross section in the high energy limit. We haveshown that the cross section grows, with reasonable,HERA compatible parameterizations [15] of the sat-uration momentum, only logarithmically with energyrather than power like as expected from perturbationtheory. This “unitarization” of the cross section is dueto the strong classical fields of the target generated bythe high gluon density (higher twist) effects. We havealso considered the transverse momentum broadeningof the gluon jet passing through a nuclear target. Wehave shown that it scales likeA1/3 in both perturbativeand saturation regimes and that it is infrared finite. Wepredict that the jet transverse momentum broadeningwill be larger in the forward rapidity region and that itincreases with energy.

Acknowledgements

We thank Yu. Dokshitzer, D. Kharzeev, and Yu.Kovchegov for helpful comments and L. McLerran formany stimulating discussions and a critical reading ofthis manuscript. This manuscript has been authoredunder contract No. DE-AC02-98CH10886 with theUS Department of Energy. J.J.-M. is also supportedin part by a LDRD from BSA.

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