PHYSICAL REVIEW D 73, 034019 (2006)

Gluon propagation inside a high-energy nucleus

Francois Gelis1 and Yacine Mehtar-Tani21Service de Physique Theorique (URA 2306 du CNRS), CEA/DSM/Saclay, Batiment 774 91191, Gif-sur-Yvette Cedex, France

2Laboratoire de Physique Theorique, Universite Paris Sud, Batiment 210 91405, Orsay Cedex, France(Received 8 December 2005; published 15 February 2006)

1550-7998=20

We show that, in the light-cone gauge of the proton, it is possible to derive in a very simple way thesolution of the classical Yang-Mills equations for the collision between a nucleus and a proton. Oneimportant step of the calculation is the derivation of a formula that describes the propagation of a gluon inthe background color field of the nucleus. This allows us to calculate observables in pA collisions in amore straightforward fashion than already proposed. We discuss also the comparison between light-conegauge and covariant gauge in view of further investigations involving higher order corrections.

DOI: 10.1103/PhysRevD.73.034019 PACS numbers: 12.38.Mh, 12.38.Bx, 13.60.Hb

I. INTRODUCTION

The study of semihard particle production in high-energy hadronic interaction is dominated by interactionsbetween partons having a small fraction x of the longitu-dinal momentum of the colliding nucleons. Since thephase-space density of such partons in the nucleon wavefunction is large, one expects that the physics of partonsaturation [1–3] plays an important role in such studies.This saturation generally has the effect of reducing thenumber of produced particles compared to what one wouldhave predicted on the basis of perturbative QCD calcula-tion with parton densities that depend on x according to thelinear BFKL [4,5] evolution equation.

It was proposed by McLerran and Venugopalan [6–8]that one could take advantage of this large phase-spacedensity in order to describe the small x partons by aclassical color field rather than as particles. More precisely,the McLerran-Venugopalan (MV) model proposes a dualdescription, in which the small x partons are described as aclassical field and the large x partons act as color sourcesfor the classical field. In their original model, they had inmind a large nucleus, for which there would be a largenumber of large x partons (at least 3Awhere A is the atomicnumber of the nucleus, from just counting the valencequarks) and therefore they would produce a strong colorsource. This meant that one has to solve the full classicalYang-Mills equations in order to find the classical field.But that procedure, on the other hand, would properlyincorporate the recombination interactions that are respon-sible for gluon saturation. In the MV model, the large xcolor sources are described by a statistical distribution,which they argued could be taken to be a Gaussian for alarge nucleus at moderately small x (see also [9] for a moremodern perspective on that).

Since then, this model has evolved into a full fledgedeffective theory, the so-called ‘‘color glass condensate’’[10–12]. It was soon recognized that the separation be-tween what one calls large x and small x, inherent to thedual description of the MV model, is somewhat arbitrary,and that the Gaussian nature of the distribution of sources

06=73(3)=034019(6)$23.00 034019

would not survive upon changes of this separation scale.This arbitrariness has been exploited to derive a renormal-ization group equation, the so-called JIMWLK equation[10–19], that describes how the statistical distribution ofcolor sources changes as one moves the boundary betweenlarge x and small x. This functional evolution equation canalso be expressed as an infinite hierarchy of evolutionequations for correlators [20], and has a quite useful (andtremendously simpler) large Nc mean-field approximation[21], known as the Balitsky-Kovchegov equation.

In high-energy hadronic collisions, gluon production isdominated by the classical field approximation, and calcu-lating it requires to solve the classical Yang-Mills equa-tions for two color sources moving at the speed of light inopposite directions. This is a problem that has been solvednumerically in [22–26] for the boost-invariant case, andthe stability of this solution against rapidity dependentperturbations has been investigated in [27,28]. In termsof analytical solutions, much less is known. The onlysituation for which gluon production has been calculatedanalytically is the case where one of the two sources isweak and can thus be treated at lowest order (this situationis often referred to as ‘‘proton-nucleus’’ collisions in theliterature, but it can also be encountered at forward rap-idities in the collision of two identical objects). This wasdone in a number of approaches [29–32]. The last tworeferences provide the solution of the Yang-Mills equationin this asymmetrical situation, in the Schwinger gauge(x�A� � x�A� � 0) and Lorenz gauge (@�A� � 0), re-spectively. More recently, Balitsky has proposed an expan-sion in commutators of Wilson lines, where at each orderone treats the two projectiles symmetrically [33].

Although the solution in the Lorenz gauge was fairlycompact, it turned out that it displayed some oddities likethe appearance of a Wilson line containing the couplingconstant g=2 instead of g (this of course disappeared atlater stages from physical quantities). When applied to theproduction of quark-antiquark production in [34], it alsoled to a contribution in which the vertex producing the q �qpair was located inside the nucleus, which is quitecounterintuitive.

-1 © 2006 The American Physical Society

x+

0 ε

M µν

A1ν(0) A1

µ(ε)

FIG. 1. Gluon passing through a nucleus. The region shaded ingray is the region where the color source representing thenucleus is nonzero.

FRANCOIS GELIS AND YACINE MEHTAR-TANI PHYSICAL REVIEW D 73, 034019 (2006)

In this paper, we derive the solution of the Yang-Millsequations in the light-cone gauge A� � 0 (with the nucleusmoving in the negative z direction), and we find a muchsimpler solution (not only the solution is simpler, but it isalso much easier to obtain). In particular, the solution in theA� � 0 gauge presents none of the odd features encoun-tered in the Lorenz gauge. Our central result, derived inSec. II, is in fact a formula that tells how a color fieldpropagates on top of the gauge field of the nucleus.Equations (12) in fact contain all the information whichis needed in order to derive the solution of Yang-Millsequations and compute gluon production, which we per-form as a verification in Sec. III. Beyond the study of ‘‘-proton-nucleus’’ collisions themselves, Eqs. (12) are animportant building block for calculating higher order cor-rections in the weak source to the solution of Yang-Millsequations. And to a large extent, the complexity of thisobject finds its way into the solution of Yang-Mills equa-tions. Therefore, it is important to determine this object,and to find a gauge in which it is particularly simple. Forthe sake of comparison, we derive in the appendix theanalogue of Eqs. (12) for the Lorenz gauge, and they aremuch more complicated, as one expects.

II. GLUON PROPAGATION IN A NUCLEUS

We assume that the nucleus is moving with a velocityvery close to the speed of light, in the negative z direction.Thus, it can be described by the following density of colorsources:

��x���a�x?�: (1)

Some intermediate calculations may require to regularizethe delta function by giving it a small width. When thisis necessary, we replace ��x�� by ���x��, where ���x��is a positive definite function normalized byR�1�1 dx

����x�� � 1 and whose support is �0; ��.

In this paper, we address the following question: know-ing the gauge fields at x� � 0, what are the gauge fields atx� � �, i.e. after having propagated through the nucleus?Finding the gluon propagator inside the nucleus amounts tosolving this problem to linear order in the incoming gaugefields. In practice, we add to the gauge field A�0 of thenucleus a small perturbation A�1 ,

A� � A�0 � A�1 ; (2)

and we wish to find a linear relationship for this perturba-tion before and after the region where the nucleus lives,such as (see Fig. 1):

A�1 �x� � �� � M�

�A�1�x� � 0�: (3)

1In this and in the following equations, we do not writeexplicitly the color indices.

034019

We work in the light-cone gauge of the proton A� � 0.The covariant conservation of the color current reads:1

@�J� � �D�; J�� � �Di; Ji� � 0: (4)

This equation can be solved by

J� � Ji � 0; @�J� � 0: (5)

The first equation is allowed because all the color chargesin our problem are moving in the negative z direction. Thesecond equation means that the color current J� associatedto the nucleus is not affected by the incoming field,2 andtherefore this current J� can be ‘‘hidden’’ once and for allin the gauge field A�0 of the nucleus.

The Yang-Mills equations in this gauge read

@��@�A�� � ig�Ai; @�Ai� � 0

�D�; @�A�� � �Di; Fi�� � J�

@�F�i � �D�; @�Ai� � �Dj; Fji� � 0: (6)

One can see that the first of these equations does notcontain any time derivative (@�). Therefore, it can beseen as a constraint that relates the various field compo-nents at the same time.

The gauge field of the nucleus alone is found by con-sidering only the order zero in A�1 in the above equations.One can readily check that the following is a solution:

Ai0 � 0; A�0 � �1

@2?

J� � �g��x��1

@2?

��x?�: (7)

In order to find the linear relationship between theperturbation A�1 before and after the region where the colorsources of the nucleus live, we must linearize the Yang-Mills equations in A�1 . One obtains

2In other gauges, the incoming A�1 would induce a colorprecession of J�.

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S PHYSICAL REVIEW D 73, 034019 (2006)

@�A�1 �@iAi1�0

�Ai1�2ig�A�0 ;@�Ai��0

�A�1 �2ig�A�0 ;@�A�1 ��2ig�@iA�0 ;A

i1��2ig�@iA�0 �T�A

i1:

(8)

The method for solving the second and third equations wasexplained in [32]. The solution of the second equationinside the nucleus, i.e. for x� 2 �0; ��, reads

Ai1�x�; x�; x?� � U�x�; 0; x?�Ai1�0; x

�; x?�; (9)

whereU is a Wilson line in the adjoint representation of the

GLUON PROPAGATION INSIDE A HIGH-ENERGY NUCLEU

034019

gauge group:

U�x�; 0; x?� T � exp�igZ x�

0dz�A�0a�z

�; x?�Ta�:

(10)

At this point, we could find A�1 simply by solving theconstraint equation. But as a verification, it is interestingto solve explicitly the third equation for A�1 , and at the endto verify that the fields Ai1 and A�1 do obey the constraint.The third equation leads to

A�1 �x�; x�; x?� � U�x�; 0; x?�A�1 �0; x

�; x?�

� igZ x�

0dz�U�x�; z�; x?��@

iA�0 �z�; x?� � T�U�z

�; 0; x?�1

@�Ai1�0; x

�; x?�

� U�x�; 0; x?�A�1 �0; x

�; x?� � �@iU�x�; 0; x?��

1

@�Ai1�0; x

�; x?�: (11)

It is trivial to verify that the constraint that relates A�1 to Ai1would have given the same answer.

Therefore, if we denote U U��; 0; x?� and if we donot write explicitly the x� and x? dependence, we have thefollowing relations:

A�1 ��� � 0; Ai1��� � UAi1�0�;

A�1 ��� � UA�1 �0� � �@iU�

1

@�Ai1�0�:

(12)

These equations are the light-cone gauge expression of thelinear relation we were looking for. Analogous relationswill be derived for the Lorenz gauge in the appendix.

III. GLUON PRODUCTION IN pA COLLISIONS

A. Gauge field

From this linear relation, it is easy to calculate the gaugefield that describes proton-nucleus collisions. In this case,the incoming field A�1 is the field produced by the currentassociated to the proton

J� � g��x���p�x?�: (13)

For x� 0, i.e. before the collision with the nucleus, thecurrent J� remains constant, and we simply have, to linearorder in the proton source �p,

A�1 � A�1 � 0; �Ai1 � ���x��@i�p�x?�: (14)

Solving the latter equation gives the following field atx� � 0:

A�1 �0� � A�1 �0� � 0; Ai1�0� � ��x��@i

@2?

�p�x?�:

(15)

The next step is to use Eqs. (12) in order to find the gaugefield A�1 immediately after the collision with the nucleus,i.e. at x� � �,

A�1 ��� � 0; Ai1��� � U��x��@i

@2?

�p�x?�;

A�1 ��� � �@iU�

1

@���x��

@i

@2?

�p�x?�:

(16)

The final step is to find the gauge field for x� > �. Theequation that governs the evolution of Ai1 is

�Ai1 � 2ig�A�0 ; @�Ai1� � �

@i

@�J�; (17)

and the current J� at x� > � is modified by color preces-sion in the nuclear field A�0 :

J��x� > �� � U��x���p�x?�: (18)

This is a direct consequence of current conservationD�J� � 0. We must solve Eq. (17) with an initial condi-tion at x� � � given by Eq. (16). Using the techniques of[32], we obtain

Ai1�x��Zy���

dy�d2y?G0R�x;y�2@

�y U�y?�

���y��@iy@2y?

�p�y?�

�Zy�>�

d4yG0R�x;y���y

��@iy�U�y?��p�y?��; (19)

where G0R�x; y� is the free retarded propagator obeying

�xG0R�x; y� � ��x� y�. The two terms in this solution

are illustrated in Fig. 2.

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x+

0 ε x+

0 ε

FIG. 2. The two contributions to gluon production in pAcollisions. Left: the proton source emits the gluon before thecollision with the nucleus. Right: the proton source goes throughthe nucleus before emitting the gluon. The thick solid linerepresents the proton color current.

FRANCOIS GELIS AND YACINE MEHTAR-TANI PHYSICAL REVIEW D 73, 034019 (2006)

B. Gluon production

The amplitude for the production of a gluon of momen-tum p and polarization � is given by the Fourier transformof the amputated gauge field, contracted in the relevantpolarization vector,

M ��p� �Zd4xeip�x�A�1 �x��

���� �p�: (20)

In light-cone gauge, the sum over the physical polariza-tions reads X

�

����i �p������j �p� � �gij; (21)

and for this reason we need only the transverse componentsof the gauge field when calculating gluon production. Notethat since the Dalembertian �Ai1�x� is bounded inside thenucleus, the region 0< x� < � does not contribute in thelimit �! 0. Therefore, we can take �! 0 and disregardthe interior of the nucleus when we calculate the ampli-tude. We first obtain:3

�Ai1�x� � 2��x����x���U� 1�@i

@2?

�p�x?�

� ��x�����x��@i�p�x?�

� ��x����x��@i�U�p�x?��; (22)

and then the Fourier transform gives

�p2Ai1�p� � �p2Aiproton�p� � 2i

Z d2k1?

�2��2

�

�pi

2�p� � i"��p� � i"��

ki1k2

1?

��p�k1?�

� �U�k2?� � �2��2��k2?��: (23)

In this equation, k2? p? � k1? and Aiproton�p� is the

3When we evaluate the Dalembertian of the gauge field in theregion x� < 0, we must multiply Eq. (15) by ���x�� in order torestrict this term to the region x� < 0. Equivalently, we couldsimply subtract it from Eq. (19) in order not to overcount it in theregion x� > 0.

034019

Fourier transform of the gauge field of a proton alone,i.e. the Fourier transform of Eq. (15). It is easy to verifythat this expression leads to the standard result for gluonproduction in proton-nucleus collisions.4

IV. CONCLUSIONS

In this paper, we have obtained the solution of theclassical Yang-Mills equations for proton-nucleus colli-sions in the light-cone gauge of the proton. An importantintermediate step is the ‘‘transfer matrix,’’ given inEqs. (12), which tells how a color field propagates throughthe nucleus on top of the color field of the nucleus. It turnsout that this object takes an extremely simple form in thelight-cone gauge, especially when compared to what oneobtains in the (covariant) Lorenz gauge (see the appendix).This transfer matrix is central in deriving the solution ofYang-Mills equations for pA collisions, and will be acrucial building block for calculating higher order correc-tions in the weak source to this solution.

ACKNOWLEDGMENTS

We would like to thank R. Baier, I. Balitsky, J.-P.Blaizot, D. Dietrich, E. Iancu, A. H. Mueller, D. Schiff,and R. Venugopalan for useful discussions on the issuesdiscussed in this paper.

APPENDIX: GLUON PROPAGATOR INCOVARIANT GAUGE

It is sometimes useful to have expressions for the propa-gation of a gluon field inside the nucleus in a covariantgauge (@�A� � 0). We derive in this appendix the ana-logue of Eqs. (12) for the Lorenz gauge. In this gauge, thecolor field of the nucleus is the same as its field in the A� �0 gauge:

A�0 � Ai0 � 0; A�0 � �g��x��

1

@2?

��x?�; (A1)

thanks to the independence of the nuclear sources on x�.The Yang-Mills equation that controls the evolution of A�1is very simple and reads

�A�1 � ig�A�0 ; @

�A�1 �; (A2)

and its solution can be written as

A�1 ��� � VA�1 �0�; (A3)

where we use the same compact notations as in Eq. (12)and where V is a Wilson line,

4In [29,35], the same gauge was used but in a diagrammaticapproach which deals with quark and gluon interactions with thenucleus, so that the derivation of gluon production is moreinvolved.

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S PHYSICAL REVIEW D 73, 034019 (2006)

V T � exp�ig2

Z �

0dz�A�0a�z

�; x?�Ta�; (A4)

that differs from U only in the factor 1=2 in theexponential.

The Yang-Mills equation for Ai1 reads

�Ai1�2ig�A�0 ;@�Ai1�� ig�@

iA�0 �T�A�1 � ig�A

�0 �T�@

iA�1 :

(A5)

The solution of this equation reads

Ai1��� � UAi1�0� � ig2U �@iA�0 � T� V

1

@�A�1 �0�

� ig2U �A�0 � T�

@i

@��VA�1 �0��; (A6)

where we have used the following compact notation:

U1 A U2

Z �

0dx�U1��; x

�; x?�A�x�; x?�U2�x

�; 0; x?�;

(A7)

with U1;2 any pair of Wilson lines evaluated at the sametransverse coordinate, and A the nuclear field or one of itsderivatives. With this notation, one has

@iU � igU �@iA�0 � T� U;

@iV � ig2V �@iA�0 � T� V;

(A8)

and the slightly less obvious relation

U� V � ig2U �A�0 � T� V � i

g2V �A�0 � T� U:

(A9)

A proof of the latter formula was given in [32]. Thanks to

GLUON PROPAGATION INSIDE A HIGH-ENERGY NUCLEU

034019

these relations, it is straightforward to simplify Eq. (A6)into

Ai1��� � UAi1�0� � @i�V

1

@�A�1 �0�

��U

@i

@�A�1 �0�:

(A10)

One can see that all the convolutions between U’s and V’shave disappeared from this expression.

Let us consider finally the Yang-Mills equation for A�1 .This equation involves the current J�, and in the Lorenzgauge the field A�1 induces a color precession of J�, whichcan be seen as a correction J�1 of order O�A�1 � to thiscomponent of the current. The equation that determinesthis correction is determined from the covariant currentconservation, which can be satisfied by

J�1 � Ji1 � 0; @�J�1 � ig�A�1 ; J�0 �; (A11)

where J�0 is the minus component of the nuclear current inthe absence of any extra field. If we recall that J�0 ��@2

?A�0 , we can solve the last equation as

J�1 � ig�@2?A�0 � T�V

1

@�A�1 �0�: (A12)

The Yang-Mills equation for A�1 reads

�A�1 � ig�A�0 ; @

�A�1 �

� J�1 � �ig�2�A�0 � T�

2A�1

� 2ig�A�0 � T�@�A�1 � ig�@

�A�0 � T�A�1

� 2ig�@iA�0 � T�Ai1 � ig�A

�0 � T�@

iAi1: (A13)

It is straightforward to solve this equation and write itssolution as

A�1 ��� � VA�1 �0� � ig2V �@2

?A�0 � T� V

1

@�2 A�1 �0� �

�ig�2

2V �A�0 � T�

2 V1

@�A�1 �0� � igV �A

�0 � T�

@��V

1

@�A�1 �0�

�� i

g2V �@�A�0 � T� V

1

@�A�1 �0� � igV �@

iA�0 � T� �U

1

@�Ai1�0� � @

i�V

1

@�2 A�1 �0�

�

�U@i

@�2 A�1 �0�

�� i

g2V �A�0 � T� @

i�U

1

@�Ai1�0� � @

i�V

1

@�2 A�1 �0�

��U

@i

@�2 A�1 �0�

�: (A14)

After some lengthy manipulations based on Eqs. (A8) and(A9), one can eliminate all the convolution products andobtain

A�1 ��� � VA�1 �0� ��@iU

1

@�� V

@i

@�

�Ai1�0� �

�1

2�@2?U�

�1

@�2 �1

2U@2?

@�2 �1

2@2?U

1

@�2 � �@�V�

1

@�

� @2?V

1

@�2

�A�1 �0�: (A15)

Equations (A3), (A10), and (A15) thus constitute theequivalent of Eqs. (12) in the Lorenz gauge. One can seein these expressions the advantage of working in the light-cone gauge: not only the formulas are much more compact,but in addition they do involve only the Wilson line U, andnot V. As a side note, one can check that

@�A�1 ��� � V@�A

�1 �0�; (A16)

i.e. the Lorenz gauge condition is satisfied at x� � �provided that it was satisfied by the field incoming at x� �0.

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FRANCOIS GELIS AND YACINE MEHTAR-TANI PHYSICAL REVIEW D 73, 034019 (2006)

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