Wilson lines - color charge densities correlators and the production of eta' in the CGC for pp and pA collisions
aa r X i v : . [ h e p - ph ] A ug Wilson lines - color charge densities correlators and theproduction of η ′ in the CGC for pp and pA collisions Fran¸cois Fillion-Gourdeau ∗ and Sangyong Jeon † Department of Physics, McGill University,3600 University Street, Montreal, Canada H3A 2T8 (Dated: November 9, 2018)We compute the inclusive differential cross section production of the pseudo-scalarmeson η ′ in high-energy proton-proton ( pp ) and proton-nucleus ( pA ) collisions. Weuse an effective coupling between gluons and η ′ meson to derive a reduction formulathat relates the η ′ production to a field-strength tensor correlator. For pA collisionswe take into account saturation effects on the nucleus side by using the Color GlassCondensate formalism to evaluate this correlator. We derive new results for Wil-son line - color charges correlators in the McLerran-Venugopalan model needed inthe computation of η ′ production. The unintegrated parton distribution functionsare used to characterize the gluon distribution inside protons. We show that in pp collisions, the cross section depends on the parametrization of unintegrated partondistribution functions and thus, it can be used to put constraints on these distri-butions. We also demonstrate that in pA collisions, the cross section is sensitive tosaturation effects so it can be utilized to estimate the value of the saturation scale. I. INTRODUCTION
One of the main challenges in high energy hadronic collisions is the understanding ofparticle production. The theoretical description of these complex phenomena involves bothmany-body physics and the theory of strong interactions. There are many approaches thataim toward a better comprehension of these topics. One of the most successful techniqueis perturbative quantum chromodynamics (pQCD) where one studies the limit where thecoupling constant is small and where the usual loop expansion can be used in principle.Because QCD is an asymptotically free theory, this happens when the exchanged momentaare large compared to the QCD scale Λ
QCD . Even in that regime however, physical observ-ables computed using this machinery suffer from infrared divergences that spoil the naive ∗ Electronic address: ffi[email protected] † Electronic address: [email protected] erturbative expansion. These have to be resummed and this leads to various factorizationformalisms like the collinear factorization and the k ⊥ -factorization. In collinear factoriza-tion, meaningful physical quantities are obtained in terms of parton distribution functions(PDF). These distributions characterize the non-perturbative (large distance) physics andhave to be determined experimentally from a fit to structure functions. This formalismcan be applied on system where the typical exchanged momentum Q is hard, which meansthat it satisfies the inequality given by Λ ≪ Q ∼ s where Λ QCD ≈ . √ s is the center of mass energy. Note that Q ∼ M ⊥ ≡ M + k ⊥ , whichmeans that unless the mass of the produced particle M is of the order of √ s , the collinearfactorization is valid only for very large k ⊥ . This can be relaxed in the k ⊥ -factorization for-malism which considers semihard collisions, meaning that the typical exchanged momentumobeys Λ ≪ Q ≪ s . The resummation implemented in this approach takes care of largecontributions that look like (cid:2) ln( Q / Λ QCD ) α s (cid:3) n , (cid:2) ln( Q / Λ QCD ) ln(1 /x ) α s (cid:3) n and [ln(1 /x ) α s ] n [1, 2, 3, 4]. This technique is used successfully to compute the production of many kindsof particles in high energy proton-proton collisions like heavy quarks [1, 5, 6, 7, 8] and in anumber of other processes (see [9, 10, 11] for reviews of many applications). It is also usedfor predictions of Higgs boson production [12, 13]. In this article, we use this formalism tocompute the η ′ inclusive cross section at the RHIC energy.When there is a nucleus involved in a collision at very high energy, there are new effectsnot included in the previous approaches due to the high density of gluons resulting fromthe emission enhancement at small- x (where x is the momentum fraction). These effectsintroduce a new scale Q s called the saturation scale at which the probability of havinginteractions between gluons of the same nucleus becomes important. At this transversemomentum scale, the gluons recombine and this slows down the growth of partons at smaller x . A naive estimation of Q s shows that it depends on x and the number of nucleons A like Q s ∼ A δ x − λ [14, 15] so at small enough x or large enough A , the saturation scale is hard( Q s ≫ Λ ). When the typical exchanged momentum is smaller than the saturationscale such as Λ ≪ Q ≤ Q s ≪ s , saturation effects have to be taken into accounteven if the system is still in the perturbative regime. This can be achieved in the ColorGlass Condensate (CGC) formalism which is a semi-classical effective theory where the non-linearities are dealt with by solving exactly the Yang-Mills equation of motion. This takescare of gluon recombinations and introduces the effects of saturation in observables.In this article, we are using the CGC to compute the inclusive differential cross section of η ′ meson in pA collisions at the RHIC energy ( √ s = 200 GeV). The main goal of this workis to look at the effect of saturation in η ′ production to validate the CGC approach andestimate the value of the saturation scale by comparing our predictions with experimentaldata. The η ′ is a pseudoscalar meson with a mass of M = 0 .
957 GeV, a decay width of2 .
203 MeV and quantum numbers of I G ( J P C ) = 0 + (0 − + ) [16]. One of the most importantfeatures of η ′ is that it couples to the QCD anomaly [17, 18]. One way to implement andmodel this physics is by introducing an effective interaction between gluons and η ′ mesons.This was done in [19], where the authors are proposing a vertex that couples two gluons and a η ′ meson ( ggη ′ ) to explain B mesons decay ( B → η ′ + X ). This vertex contains a form factorthat depends generally on gluons and η ′ momenta and that can be related to the η ′ wave-function. The structure of this vertex was investigated thoroughly using various techniqueslike the hard scattering and the running coupling approaches. [20, 21, 22, 23, 24, 25]. Weuse these results on the gluons- η ′ coupling to study the η ′ production mechanism based ongluon fusion.The η ′ production in pp collisions at high energy was studied in [26] where the exclusivecross section for the diffractive process p + p → p + p + η ′ is computed. In our study, wefocus on the inclusive production mechanism p + p → η ′ + X which shares similar featureswith this previous analysis. The first attempt to compute η ′ production in high energy pA collisions was done by one of the present author in [27]. In this study, the collinearfactorization is used to compute the cross section at RHIC by including intrinsic transversemomentum in the PDF with a Gaussian distribution. Based on physical arguments, thewidth of the Gaussian, which represents the typical transverse momentum of gluons insidethe nucleus, is chosen to be Q s . The authors show that the η ′ production is sensitive to thesaturation scale implemented in this way. However, they acknowledge that their calculationcan be improved because they use the collinear formalism outside its range of validity. Thegoal of this article is to revisit the η ′ production with a more rigorous approach by doing afull CGC computation that includes recombination effects more realistically.The computation of meson production in the CGC was undertaken in the past usingmostly an “hybrid” approach where the proton and the nucleus are described by the collinearfactorization and the CGC respectively. In this formalism based on pQCD-like techniques,the fragmentation function of collinear factorization is convoluted with the gluon or quarkcross section computed in the CGC formalism. This is suitable for well-known mesons likepions for which a wealth of experimental data have been measured and for which fragmenta-tion functions are well-known. Pion production for pA collisions is computed in [28, 29, 30]using this methodology. Contrary to pions, the data in high energy hadronic collisions for η ′ is scarce, so another approach is required to take care of hadronization effects and internalstructure of the η ′ meson. In [31], an effective theory is used to estimate the tensor mesonproduction in pp collisions. We use a similar approach in this article where the interactionbetween gluons and η ′ is described by an effective theory. As discussed previously, we in-clude these effects in an effective vertex that includes a form factor. As will be shown inthis article, this can be implemented easily in the CGC formalism.3e consider only the case of pp and pA collisions at RHIC. For nucleus-nucleus ( AA )collisions, the total number of η ′ -mesons produced by semihard collisions in the first instants(for t < η ′ mean lifetime, which is about t η ′ ≈ .
93 fm/c, is smaller than the time where the medium exists, which is from 1 fm/c upto 10 fm/c. Moreover, by considering pp and pA , we avoid all the complications that wouldresult from the creation of the medium which include the understanding and modelling ofthe quark-gluon plasma properties. Finally, there are analytical solutions for the gauge fieldin pp and pA collisions, while the analytical solution in AA is still elusive. For these reasons,our present analysis is only applied to pp and pA collisions.This article is organized as follows. In section II we describe the effective vertex usedthroughout the rest of the article. In section III, we show how to compute η ′ production inthe CGC for pA and pp collisions. We start by deriving a reduction formula that relates thecross section to a correlator of field-strength tensors. This correlator is then evaluated toleading order in pA collisions and the result can be interpreted in terms of physical processes.We also show how the k ⊥ -factorized cross section for pp collisions can be recovered in the lowdensity limit of pA cross section. In section IV, we compute the correlators appearing in theexpression of the pA cross section using the McLerran-Venugopalan model. In section V, weevaluate numerically the cross section for pA and pp and discuss the range of validity of ourcomputation. Sections III and IV contain a lot of technical details. The reader interestedin results can jump directly to section V.Throughout the article, we use both light-cone coordinates defined by p + = p + p √ p − = p − p √ g µν = (1 , − , − , − II. EFFECTIVE THEORY
The effective theory used in this article couples gluons and the η ′ meson. In momentumspace, the g ∗ g ∗ η ′ effective vertex (where g ∗ means off-shell gluon) is given by V µν ( M, p, q ) = iF ( p , q , M ) δ ab ǫ µνρα p ρ q α (2)where ǫ µνρα is the Levi-Civitta antisymmetric tensor, M is the η ′ mass, a and b are colorindices, p ρ and q α are gluon momenta and F ( p , q , M ) is the η ′ form factor. The explicit4xpression of the interaction vertex have been studied in a number of articles where differentparametrizations of the form factor can be found [19, 20, 21, 22, 23, 24, 25]. To get afirst approximation of η ′ production and because we are mostly interested in making acomparative study between pp and pA collisions, we use a simple expression given by [26] F ( p , q , M ) = H M ( M − p )( M − q ) (3)where H = F (0 , , M ). To get a better approximations of η ′ production, other parametriza-tions should be used. In the limit of no gluon virtualities ( p , q = 0), the form factoris a constant that can be fixed by looking at the decay of ψ → η ′ + γ . It is given by H = F (0 , , M ) ≈ . B and Υ decay whereprocesses such as g ∗ → g + η ′ and g ∗ + g ∗ → η ′ are considered [19, 32, 33, 34, 35]. Morerecently, gluon fusion was used to compute η ′ production in high energy hadronic collisions[26, 27, 36] and from a thermalized medium [37].It is convenient for our purpose to consider the interaction Lagrangian given by L int ( x ) = 18 Z d yd zF (cid:2) ( x − y ) , ( x − z ) , M (cid:3) G µνa ( y ) e G a,µν ( z ) η ( x ) . (4)that reproduces the vertex Eq. (2) in the perturbative expansion. As seen in the nextsection, this can then be used to derive a reduction formula. Here, G µνa ( x ) is the usualfield-strength tensor given by G µνa ( x ) = ∂ µ A νa ( x ) − ∂ ν A µa ( x ) − gf abc A µb ( x ) A νc ( x ) (5)where A µa is the gauge field of gluons and ˜ G µνa ( x ) = ǫ µνρσ G a,ρσ ( x ) is the dual field-strengthtensor. The Lagrangian is non-local because the vertex includes a form factor. It can beeasily seen that L int ( x ) contains three types of vertices, namely ggη ′ , gggη ′ and ggggη ′ . Atleading order however, only the first one is necessary and considered in this article. III. PRODUCTION OF η ′ FROM THE CGC
In collisions at very high energy, the wave function of nuclei is dominated by gluonsthat have small longitudinal momenta (soft gluons) because of the emission enhancementat small- x . The CGC is a semi-classical formalism that describes the dynamics of thesedegrees of freedom. In this approach, the hard partons, which carry most of the longitudinalmomentum, and soft gluons which have small longitudinal components, are treated differ-ently. Because the occupation number of the soft gluons is large, classical field equationscan be employed to understand their dynamics. The hard partons act as sources for these5lassical field and are no longer interacting with the rest of the system (for reviews of CGC,see [14, 15, 38]).In this formalism, computing a physical quantity involves two main steps. The first oneis to solve the Yang-Mills equation of motion[ D µ , F µν ( x )] = J ν ( x ) (6)where the current J νa ( x ) = δ ν + δ ( x − ) ρ p,a ( x ⊥ ) + δ ν − δ ( x + ) ρ A,a ( x ⊥ ) represents random staticsources localized on the light-cone [14, 15] and D µ = ∂ µ − igA µ is the covariant derivative.The functions ρ p,A ( x ⊥ ) are color charge densities in the transverse plane of the proton andnucleus respectively. The next step is to take the average over the distribution of colorcharge densities in the nuclei with weight functionals W p,A [ ρ p,A ]. For any operator that canbe related to color charge densities, this can be written as h ˆ O i = Z D ρ p D ρ A O [ ρ p , ρ A ] W p [ x p , ρ p ] W A [ x A , ρ A ] . (7)Computing the weight functional is a highly non-perturbative procedure so it usually involvesapproximations based on physical modelling. In the limit of a large nuclei at not too small x ,it can be approximated by the McLerran-Venugopalan (MV) model, which assumes that thepartons are independent sources of color charge [39, 40]. Within this assumption, the weightfunctional W A [ ρ A ] is a x A independent Gaussian distribution and the two point correlatoris simply [14, 15, 39, 40] h ρ A,a ( x ⊥ ) ρ A,b ( y ⊥ ) i = δ ab µ A δ ( x ⊥ − y ⊥ ) (8)where µ A = A/ πR is the average color charge density and R is the radius of the nucleus.It is assumed here that the nucleus has an infinite transverse extent with a constant chargedistribution. Edge effects can be included by changing µ → µ ( x ⊥ ) and by choosing a suit-able transverse profile. Throughout this article, we only consider the constant distributioncase.Within the MV model, the weight functional does not depend on longitudinal coordinatesand therefore, the model is boost invariant. This however can be relaxed by consideringthe quantum version of the CGC. In that theory, quantum radiative corrections becomeimportant below a certain scale x ≈ .
01. These corrections can be resummed by usinga renormalization group technique which leads to the JIMWLK evolution equation [41, 42,43, 44, 45]. In the quantum CGC, the weight functionals W , [ ρ , ] obey this non-linearevolution equation in x . Because the MV model is valid in the range x ≈ . − .
1, itcan be used as an initial condition for the evolution at smaller x . In this article however,we consider only the regime where the MV model is valid and do not consider the small- x evolution although it could be done in principle.6n the proton side, the average computed with W p [ x p , ρ p ] can be related to the uninte-grated parton distribution function (uPDF) φ like g h ρ ∗ p,a ( p ⊥ ) ρ p,b ( q ⊥ ) i = 4 π δ ab ( N c − (cid:20) p ⊥ + q ⊥ (cid:21) × Z d y ⊥ e i ( p ⊥ − q ⊥ ) · y ⊥ dφ (cid:0) x, p ⊥ + q ⊥ | y ⊥ (cid:1) d y ⊥ (9)where N c is the number of color. By construction, the uPDF obeys Z d y ⊥ dφ ( x, p ⊥ | y ⊥ ) d y ⊥ = φ ( x, p ⊥ ) (10)and is normalized such that Z µ φ , ( x, p ⊥ ) ≈ xG ( x, µ ) (11)where xG ( x, µ ) is the collinear parton distribution function and µ is the factorization scale.The uPDF can be obtained from a fit to structure function and evolved to the desired valueof x p , Q and p ⊥ using evolution equations such as the BFKL or the CCFM equations.One important ingredient is missing for the computation of η ′ meson production crosssection. We need a relation between the cross section and a correlator that can be evaluatedusing the CGC formalism. This is done in the next section using a reduction formula andthe effective theory. A. Reduction Formula and the Cross Section
The computation of η ′ mesons from the CGC can be calculated from a reduction formula.The starting point is the expression of the average number of η ′ produced per collisions givenby ¯ n = P ∞ n =1 nP n where P n is the probability to produce n particles. This can be convertedto an equation in terms of creation/annihilation operators that can be evaluated in quantumfield theory. This is given by [46](2 π ) E k d ¯ nd k = h in | ˆ a † out ( k )ˆ a out ( k ) | in i (12)where | in i is the in vacuum. Then, the standard LSZ procedure can be used to write thisas [47] (2 π ) E k d ¯ nd k = 1 Z Z d xd ye ik · ( x − y ) (cid:2) ∂ x + M (cid:3) (cid:2) ∂ y + M (cid:3) h in | ˆ η ( x )ˆ η ( y ) | in i (13)where Z is the wave function normalization and where we assumed the asymptotic conditionslim t →±∞ ˆ η ( x ) = √Z ˆ η out , in ( x ) for the η ′ field operator in Heisenberg representation. It ispossible to use the equation of motion of ˆ η ( x ) given simply by( ∂ + M )ˆ η ( x ) = ˆ T ( x ) (14)7here we defined ˆ T ( x ) ≡ R d yd zF [( x − y ) , ( x − z ) , M ] ˆ G µνa ( y ) ˆ e G a,µν ( z ) to rewrite thereduction formula in a convenient way. We get finally that(2 π ) E k d ¯ nd k = h ˆ T † ( k ) ˆ T ( k ) i (15)where ˆ T ( k ) is the Fourier transform of ˆ T ( x ) evaluated at a η ′ meson on-shell momentumand where we set Z = 1. The angular brackets h ˆ O i here indicates expectation value of ˆ O inthe initial state.The only assumptions used in deriving Eq. (15) are: • There are no η ′ mesons in the initial state. • The η ′ is produced on-shell.The first assumption is justified by the fact that in high-energy collisions, the number of η ′ in a hadron before the collision (in the initial state) is negligible. This allows us touse the in vacuum and the fact that a in ( k ) | in i = 0 to simplify the reduction formula.Using the second assumption, we can treat the η ′ meson as a stable particle which canbe produced on-shell and which is well described by the free spectral density that lookslike ρ ( M ) ∼ δ ( p − M ). Therefore, by making this assumption, it is possible to use theasymptotic conditions described earlier. However, η ′ -mesons are resonances, so the spectraldensity should look rather as a Breit-Wigner function ρ ( M ) ∼ Γ / [( p − M ) + M Γ ]where Γ is the decay width. These effects however are taken into account by the form factor F ( q, p, M ).Eq. (15) is the main result of this section. It relates the average number of η ′ mesonsproduced to a correlator of field strength tensors. This correlator can then be evaluated usingany analytical or numerical methods. The average h ... i depends on the system studied.Looking at a plasma at equilibrium, it could be computed using finite temperature fieldtheory or the AdS/CFT correspondence. These two formalisms are relevant to nucleus-nucleus collisions where a medium at equilibrium is created. We are interested here in pA and pp collisions where no such medium is formed so these techniques are not pursued inthis study. Rather, we use the CGC which describes initial state and saturation effects inhigh-energy hadronic collisions.Having expressed the average number of η ′ produced in terms of a correlator of fieldstrength tensors, it is possible to compute the inclusive cross section in the CGC formalism8hich is given by [46, 48](2 π ) E k dσd k = Z d b ⊥ (2 π ) E k d ¯ n ( b ⊥ ) d k = Z d b ⊥ Z D ρ p D ρ A T ∗ [ ρ p , ρ A ] T [ ρ p , ρ A ] × W p [ x p , ρ p ] W A [ ρ A ; b ⊥ ] . (16)In this expression, b ⊥ is the impact parameter. The fields T are functionals of the sourceonce the Yang-Mills equation of motion of the gauge field is solved (see Eq. (5) for theexpression of the field strength tensor as a function of the gauge field). B. Cross section in pA Collisions In pA collisions, there are two saturation scales (one for the proton ( Q p ) and one forthe nucleus ( Q A )) that satisfy Q p < Q A . When the transverse momentum of the η ′ issmall enough, the nucleus is in a saturation state while the proton is not because we have Q p < Q = M ⊥ < Q A (remember that M ⊥ = M + k ⊥ is the transverse mass of the η ′ ). In that case, the system is semi-dilute, meaning that one of the source is strong (orequivalently, the typical transverse momentum is small) and obeys ρ A,a ( k ⊥ ) /k ⊥ ∼ ρ p,a ( k ⊥ ) /k ⊥ ≪ ρ A,a ( k ⊥ ) /k ⊥ and to first order in ρ p,a ( k ⊥ ) /k ⊥ in different gauges [50, 51, 52, 53]. We use here the solution in the light-conegauge of the proton [52] but in Appendix C, we perform the same calculation in covariantgauge to show that our result is gauge invariant.
1. Gauge Field and Power Counting
In the light-cone gauge of the proton ( A + = 0) with a nucleus in covariant gauge movingin the negative z direction, the solution of the gauge field in pA collisions can be separatedin three parts A µa ( k ) = A µp,a ( k ) + A µA,a ( k ) + A µpA,a where A µp,a ( k ) is the field associated withthe proton (of O ( ρ p )), A µA,a ( k ) is the field associated with the nuclei (of O ( ρ A ) ∼ O (1)) and A µpA,a ( k ) is the field produced by the collision (of O ( ρ p ρ ∞ A ) ∼ O ( ρ p )) [52]. Note that the field9 µA is strong and satisfies A µp , A µpA ≪ A µA . The explicit solution is given by [52, 54] A ip,a ( k ) = 2 πgδ ( k − ) k i k + + iǫ ρ p,a ( k ⊥ ) k ⊥ (17) A − A,a ( k ) = 2 πgδ ( k + ) ρ A,a ( k ⊥ ) k ⊥ (18) A ipA,a ( k ) = − igk + ik + ǫ Z d p ⊥ (2 π ) (cid:20) k i ( k + + iǫ )( k − + iǫ ) − p i p ⊥ (cid:21) ρ p,b ( p ⊥ ) × (cid:2) U ab ( k ⊥ − p ⊥ ) − (2 π ) δ ( k ⊥ − p ⊥ ) δ ab (cid:3) (19)where U ab ( k ⊥ ) is a Wilson line in adjoint representation defined in Eq. (35), g is the usualQCD coupling constant, δ ab is the Kronecker delta in color space and f abc is the structureconstant of the SU ( N c ) group. The component A − pA,a ( k ) is non-zero and is related to A ipA,a ( k )but it does not appear in the final expression of the cross section so it is not needed in thecomputation of η ′ production. All the other components are zero.The production cross section of η ′ mesons is related to a field strength tensor correlatorgiven by B ( k ) ≡ Z d pd q (2 π ) F ( p , p ) F ∗ ( q , q ) h G ∗ µνa ( p ) e G ∗ µνa ( k − p ) G αβb ( q ) e G αβb ( k − q ) i (20)where p , q = k − p, q . In principle, a correlator like this contains contributions from allorders in both sources. Because the proton source is weak, it is possible to simplify thisconsiderably using a power counting argument to isolate the leading order contribution.For the sake of this power counting argument, we use a n ( ... ) which denotes terms having n gauge fields A µ ( ... ) and where a A ∼ O ( ρ A ) and a p , a pA ∼ O ( ρ p ). At first, let us consider onlythe contributions from the abelian part of the field-strength tensor. The terms in thesecontributions have four powers of gauge field such as B abelian ∼ a ∼ ( a A + a p + a pA ) .Naively, one would expect the dominant contribution to come from terms that have manypowers of the nucleus gauge field like a A ∼ O ( ρ A ) and a A a p , a A a pA ∼ O ( ρ A ρ p ). However,these terms vanish because of the Lorentz structure. For example, a typical term in theabelian contribution would look like T abelian ∼ ǫ µνρσ p µ q ν A ρ A σ . When we sum on indices,this kind of term will contain at most one strong gauge field A − ∼ a A . Thus, the dominantcontributions are like a A a p , a A a pA ∼ O ( ρ A ρ p ). Using a similar argument, it is possible toshow that the non-abelian part have no leading order contribution in the sense that it isat least B non − abelian ∼ O ( ρ A ρ p ) ≪ O ( ρ A ρ p ). The possible higher order contributions like a A a p ∼ O ( ρ A ρ p ) for example also vanish because of the Lorentz structure of the correlator.This is because the typical non-abelian contributions look like T ′ non − abelian ∼ ǫ µνρσ p µ A ν A ρ A σ and T ′′ non − abelian ∼ ǫ µνρσ A µ A ν A ρ A σ . Once the Lorentz indices are summed, the second typicalterm T ′′ = 0 because in this gauge, A + = 0. For T ′ , it contains only one strong field A − ∼ a pA A i ∼ a p , a pA . Thus, when it is squared, it givesat most a contribution of B non − abelian ∼ O ( ρ A ρ p ). C. Evaluation of the Correlator
Using the explicit expression of the field strength tensor in terms of gauge field andkeeping only the dominant and non-zero contributions, the correlator can be written as B ( k ) = 64 Z d pd q (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ )( k + − p + )( k + − q + ) ǫ ij ǫ kl p i q k ×h A −∗ A,a ( p ) A − A,b ( q ) A j ∗ s,a ( k − p ) A ls,b ( k − q ) i (21)where ǫ ij is the Levi-Civitta antisymmetric tensor with i, j = 1 , A s,a ≡ A p,a + A pA,a .To obtain the preceding expression we make the assumption that the virtualities in theform factors are due solely to transverse momentum such as F ( p , p ) = F ( − p ⊥ , − p , ⊥ ).This approximation is necessary to recover k ⊥ -factorization in the dilute limit as seen insection III E.It is convenient to separate B ( k ) in four different terms such as B z,z ′ ( k ) = 64 g ( k + ) Z dp − dq − d p ⊥ d q ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) ǫ ij ǫ kl p i q k p ⊥ q ⊥ × h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) A j ∗ z,a ( k − p ) A lz ′ ,b ( k − q ) i (cid:12)(cid:12) p + = q + =0 (22)where z, z ′ = { p, pA } and where we performed the integration on p + and q + using the deltafunctions in Eq. (18). These four terms can be evaluated explicitly by substituting thesolution of gauge fields Eqs. (17), (18) and (19).For the first term, it is a straightforward calculation to show that B p,p ( k ) = 64 g Z d p ⊥ d q ⊥ d r ⊥ d s ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i r j q k s l p ⊥ q ⊥ r ⊥ s ⊥ δ bc (2 π ) δ ( k ⊥ − p ⊥ − r ⊥ ) δ ad (2 π ) δ ( k ⊥ − q ⊥ − s ⊥ ) ×h ρ ∗ p,d ( r ⊥ ) ρ p,c ( s ⊥ ) ih ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) i . (23)The calculation of the second term is similar but requires some more work. By direct11ubstitution of the expression of the gauge fields, we have B pA,p ( k ) = 64 ig ( k + ) Z dp − d p ⊥ d q ⊥ d r ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i q k q l p ⊥ q ⊥ q , ⊥ p − ik + ǫ " p j ( p +2 − iǫ )( p − − iǫ ) − r j r ⊥ × (cid:20) h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) U ∗ ac ( p , ⊥ − r ⊥ ) ih ρ ∗ p,c ( r ⊥ ) ρ p,b ( q , ⊥ ) i− (2 π ) δ ( p , ⊥ − r ⊥ ) δ ac h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) ih ρ ∗ p,c ( r ⊥ ) ρ p,b ( q , ⊥ ) i (cid:21) . (24)The integration on the longitudinal momentum p − can be done by looking at the analyticalstructure of the equation. First, we write a part of the integrand as1( k − p ) − ik + ǫ " p j ( k + − iǫ )( k − − p − − iǫ ) − r j r ⊥ p + =0 =12 k + h k − − p − − ( k − p ) ⊥ k + − iǫ i " p j ( k + − iǫ )( k − − p − − iǫ ) − r j r ⊥ . (25)In the complex plane of p − , the first term in the RHS of Eq. (25) has two poles on thesame side of the real axis and goes like p − ) when p − → ∞ . Thus, closing the integrationcontour in the upper-half plane and using the residue theorem, the integration on p − of thisterm leads to a zero contribution because the contour at infinity has a zero contribution andbecause the contour does not enclose any singularities. For the second term of Eq. (25), weuse the principal part (PP) identity x ± iǫ = PP ∓ i πδ (x). The integration on the principalpart is zero because the integrand does not depend on p − and lim R →∞ R R − R dp PP − a = 0while the delta function integration is trivial. We finally get B pA,p ( k ) = 32 g Z d p ⊥ d q ⊥ d r ⊥ d s ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i q k r j s l p ⊥ q ⊥ r ⊥ s ⊥ h ρ ∗ p,d ( r ⊥ ) ρ p,c ( s ⊥ ) i× (cid:20) −h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) i δ bc (2 π ) δ ( p , ⊥ − r ⊥ ) δ ad (2 π ) δ ( q , ⊥ − s ⊥ )+ h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) U ∗ ad ( p , ⊥ − r ⊥ ) i δ bc (2 π ) δ ( q , ⊥ − s ⊥ ) (cid:21) . (26)The calculations of B pA,p ( k ) and B pA,pA ( k ) are similar. Going through the same steps as for12 p,pA ( k ), we get B p,pA ( k ) = 32 g Z d p ⊥ d q ⊥ d r ⊥ d s ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i q k r j s l p ⊥ q ⊥ r ⊥ s ⊥ h ρ ∗ p,d ( r ⊥ ) ρ p,c ( s ⊥ ) i× (cid:20) −h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) i δ bc (2 π ) δ ( p , ⊥ − r ⊥ ) δ ad (2 π ) δ ( q , ⊥ − s ⊥ )+ h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) U bc ( q , ⊥ − s ⊥ ) i δ ad (2 π ) δ ( p , ⊥ − r ⊥ ) (cid:21) (27)and B pA,pA ( k ) = 16 g Z d p ⊥ d q ⊥ d r ⊥ d s ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i q k r j s l p ⊥ q ⊥ r ⊥ s ⊥ h ρ ∗ p,d ( r ⊥ ) ρ p,c ( s ⊥ ) i× (cid:20) h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) i δ bc (2 π ) δ ( p , ⊥ − r ⊥ ) δ ad (2 π ) δ ( q , ⊥ − s ⊥ ) −h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) U bc ( q , ⊥ − s ⊥ ) i δ ad (2 π ) δ ( p , ⊥ − r ⊥ ) −h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) U ∗ ad ( p , ⊥ − r ⊥ ) i δ bc (2 π ) δ ( q , ⊥ − s ⊥ )+ h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) U ∗ ad ( p , ⊥ − r ⊥ ) U bc ( q , ⊥ − s ⊥ ) i (cid:21) . (28)The final result for the correlator B ( k ) = B p,p ( k ) + B p,pA ( k ) + B pA,p ( k ) + B pA,pA ( k ) can bewritten compactly as B ( k ) = 16 g Z d p ⊥ d q ⊥ d r ⊥ d s ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) ǫ ij ǫ kl p i q k r j s l p ⊥ q ⊥ r ⊥ s ⊥ ×h ρ ∗ p,d ( r ⊥ ) ρ p,c ( s ⊥ ) ih ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) ˜ U ∗ ad ( p , ⊥ − r ⊥ ) ˜ U bc ( q , ⊥ − s ⊥ ) i (29)where we defined ˜ U ab ( k ⊥ ) = U ab ( k ⊥ ) + (2 π ) δ ( k ⊥ ) δ ab . (30)From this equation for B ( k ), we can now evaluate the differential cross section. It is givenby (2 π ) E k dσd k = g Z d b ⊥ Z d p ⊥ d q ⊥ d r ⊥ d s ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i q k r j s l p ⊥ q ⊥ r ⊥ s ⊥ h ρ ∗ p,c ( r ⊥ ) ρ p,d ( s ⊥ ) i×h ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) ˜ U ∗ ac ( p , ⊥ − r ⊥ ) ˜ U bd ( q , ⊥ − s ⊥ ) i . (31)13 ...)+=A (...)+(...)+p =pA = ++ + (...)+ FIG. 1: Diagrams included in the gauge field A µA , A µp and A µpA . The crossed circles ⊗ representinsertions of the strong source ρ A,a ( x ⊥ ) while the crossed squares ⊠ represent insertions of theweak source ρ p,a ( x ⊥ ). By solving the Yang-Mills equation with retarded boundary conditions, itresums all the tree diagrams such as the ones depicted in this figure [55]. Thus, we can relate the η ′ production cross section to a correlator of sources and Wilsonlines. This will be evaluated in the next section in the MV model. Clearly, this expressiondoes not have the k ⊥ -factorization structure but it can be recovered in the dilute limit. First,we can look at the physical interpretation of the different terms in the cross section. D. Diagrammatic Content and Physical Interpretation
To facilitate the physical interpretation, it is convenient to interpret the gauge fieldexpressed in Eqs. (17),(18) and (19) in terms of Feynman diagrams as shown in Fig. 1.The strong gauge field A µA corresponds to a resummation of tree diagrams with any numberof strong source insertions [55]. The weak gauge fields A µp and A µpA also resum an infinitenumber of tree diagrams, the difference being that they contain one weak source insertion[50]. 14 pA + η η pA FIG. 2: Diagrams included in η ′ production at leading order. The thick lines represent insertions ofthe proton ( p ), nucleus ( A ) and produced field ( pA ), and the dashed line is the η ′ meson. The field A µpA contains multi-scattering diagrams shown in Fig. 3. The first figure represents the interactionbetween two gluons producing a η ′ meson that goes through the nucleus without interacting. Thesecond figure corresponds to the situation where the gluons emitted by the proton goes throughthe nucleus before producing the η ′ . (...)+p A=pA Ap+ FIG. 3: Multi-scattering diagrams included in A µpA shown in Fig 2. The last diagram is a typicaldiagram associated with the Wilson line. This field contains the multiscattering effects. Then, the η ′ production and the correlator given in Eq. (21) can be represented diagram-matically in Fig. 2 and 3. These figures show all the diagrams included in the calculation.The first term in the figure corresponds to the part where the η ′ is produced from gluonfusion. The second term contains multiscattering effects and eventually, saturation effects.Overall, this leads to the following physical picture. A gluon inside the proton interactwith the classical background field of the nucleus and gets multiscattered. Once it has gonethrough the nucleus, it combines with a gluon and produce the η ′ . E. Recovering k ⊥ -factorization in the cross section It is possible to recover k ⊥ -factorization from Eq. (31) by looking at the dilute limit ofthe nucleus characterized by a weak source such as ρ A,a ≪
1. In that case, we are allowedto keep only the first term of the Wilson line expansion ˜ U ae ( k ⊥ ) = 2(2 π ) δ ( k ⊥ ) δ ae + O ( ρ A ).15n this low-density limit, the cross section becomes(2 π ) E k dσ low − density d k = g Z d b ⊥ Z d p ⊥ d q ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × ǫ ij ǫ kl p i q k p j q l p ⊥ q ⊥ p , ⊥ q , ⊥ h ρ ∗ p,a ( p , ⊥ ) ρ p,b ( q , ⊥ ) ih ρ ∗ A,a ( p ⊥ ) ρ A,b ( q ⊥ ) i . (32)In terms of Feynman diagrams, this expression corresponds to neglecting all the multi-scattering diagrams shown in Fig. 3. The neglected diagrams are the ones that break k ⊥ -factorization as can be seen from the following argument.In this dilute limit, the correlator of the nucleus can also be related to the uPDF φ likein Eq. (9). Using this results, we find that the cross section is given by(2 π ) E k dσ low − density d k = 4 π ( N c − Z d p ⊥ d q ⊥ | F ( − p ⊥ , − q ⊥ ) | × ǫ ij ǫ kl p i p k q j q l p ⊥ q ⊥ φ ( q ⊥ ) φ ( p ⊥ ) δ ( k ⊥ − p ⊥ − q ⊥ ) . (33)This is the k ⊥ -factorized expression of the cross section and is totally equivalent to Eq.(A7) obtained directly from the k ⊥ -factorization formalism (see Appendix A). Thus, inthe low-density limit of pA collisions, we recover a formalism that describes pp collisions inthe semihard regime. This is very similar to quark and gluon production in pA collisions[48, 50, 56, 57]. Note that to obtain this result, it is necessary to assume from the beginningthat the form factors depend only on transverse momenta. IV. COMPUTATION OF CORRELATION FUNCTIONS IN THE MV MODEL
In this section, we compute the relevant correlators appearing in our expression of thecross section using the McLerran-Venugopalan (MV) model. Throughout this calculation,we use the notation of [58, 59]. We are interested in correlators containing both Wilson linesand color charge densities such as the ones included in Eq. (31). In App. D, we discussthe general case and give more details on the calculation. Note here that to make senseof the ordered path, we start with color charge densities that depend on the longitudinalcoordinate x + . At the end of the calculation, we take ρ a ( x + , x ⊥ ) = δ ( x + ) ρ a ( x ⊥ ) since weuse the MV model which assumes that the nucleus is moving at the speed of light. In thismore general case, the 2-point function is simply h ρ a ( x + , x ⊥ ) ρ b ( y + , y ⊥ ) i = δ ab µ ( x + ) δ ( x + − y + ) δ ( x ⊥ − y ⊥ ) (34)where µ ( x + ) is the average color charge density at point x + . It is related to the averagecolor charge density by µ = R dx + µ ( x + ) = A/ πR where R is the radius of the nucleus.16n this model, W [ ρ ] is still Gaussian, so all even-point functions can be written in terms ofthe 2-point function using Wick theorem and all odd-point functions are zero.The Wilson line is defined as U ab ( b + , a + | x ⊥ ) = P + exp " − ig Z b + a + dz + Z d z ⊥ G ( x ⊥ − z ⊥ ) ρ c ( z + , z ⊥ ) t c ab (35)where t c are the SU ( N c ) generators in adjoint representation, P + is the path ordering inthe light-cone coordinate z + and G is a Green function solution of ∂ ∂x ⊥ G ( x ⊥ ) = δ ( x ⊥ ) . (36)With these definitions, it is possible to compute the needed correlators. The generalstrategy is to express the correlators in terms of the following known results for Wilson linesin adjoint representation [59]: h U ab ( b + , a + | x ⊥ ) i ≡ ¯ U ( b + , a + | x ⊥ ) δ ab = δ ab exp (cid:20) − N c L ( x, x )¯ µ ( b + , a + ) (cid:21) (37) h U ab ( b + , a + | x ⊥ ) U cd ( b + , a + | x ⊥ ) i ≡ δ ac δ bd N c − V ( b + , a + | x ⊥ , x ⊥ )= δ ac δ bd N c − (cid:2) − N c ¯ µ ( b + , a + ) ( L (0 , − L ( x ⊥ , x ⊥ )) (cid:3) (38)where L ( x, y ) = Z d z ⊥ G ( x ⊥ − z ⊥ ) G ( y ⊥ − z ⊥ ) (39)and where we defined the quantity ¯ µ ( b + , a + ) ≡ R b + a + dz + µ ( z + ). In the next subsections,we express the correlators appearing in the η ′ cross section in terms of ¯ U and ¯ V which aredefined in Eqs. (37) and (38). A. 1 Wilson line - 1 color charge density correlator: the main building block
The first correlator does not appear explicitly in the η ′ production cross section but it isuseful to understand the other results. We want to evaluate F , ( b + , a + ) ≡ h U ab ( b + , a + | x ⊥ ) ρ c ( y +1 , y ⊥ ) i . (40)where we assume that b + > y +1 > a + . For clarity, we define the sources included in Wil-son lines as internal sources as opposed to external sources which appear explicitly in thecorrelator (like ρ c ( y +1 , y ⊥ ) in Eq. (40)). 17 = = 00= FIG. 4: These are the different types of possible contractions using Wick theorem and path ordering.The first type is a contraction like h ρ a j − ( z + j − , z j − ⊥ ) ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ih ρ c ( y + , y ⊥ ) ρ a j ( z + j , z j ⊥ ) i where ρ c is an external source. The other ones have only internal sources. The secondone is like h ρ a j ( z + j , z j ⊥ ) ρ a j +2 ( z + j +2 , z j +2 ⊥ ) ih ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ρ a j +3 ( z + j +3 , z j +3 ⊥ ) i , the third one islike h ρ a j ( z + j , z j ⊥ ) ρ a j +3 ( z + j +3 , z j +3 ⊥ ) ih ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ρ a j +2 ( z + j +2 , z j +2 ⊥ ) i and the last one is like h ρ a j ( z + j , z j ⊥ ) ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ih ρ a j +2 ( z + j +2 , z j +2 ⊥ ) ρ a j +3 ( z + j +3 , z j +3 ⊥ ) i . Only the last one has a sup-port and thus, a non-zero contribution. The first step is to expand the Wilson line. The expression can then be written as F , ( b + , a + ) = ∞ X n =0 ( − g ) n n Y i =1 Z d z i ⊥ G ( x ⊥ − z i ⊥ )( f a ...f a n ) ab × Z b + a + dz +1 Z z +1 a + dz +2 ... Z z + n − a + dz + n ×h ρ c ( y +1 , y ⊥ ) ρ a ( z +1 , z ⊥ ) ...ρ a n ( z + n , z n ⊥ ) i (41)where we used t abc = − if abc and where f abc is the antisymmetric SU ( N c ) structure constant.Using Wick theorem, the n-point correlation function can be expressed in terms of 2-pointfunctions as the sum of all possible contractions. At first, we look only at the contractionsof ρ c ( y +1 , y ⊥ ) with ρ a ( z +1 , z ⊥ ) ...ρ a n ( z + n , z n ⊥ ). Using the MV expression for the two pointfunction Eq. (34), we get F , ( b + , a + ) = µ ( y +1 ) G ( x ⊥ − y ⊥ ) ∞ X n =0 ( − g ) n n X j =1 " n Y i =1 ,i = j Z d z i ⊥ G ( x ⊥ − z i ⊥ ) × ( f a ...f a j − f c f a j +1 ...f a n ) ab × Z b + a + dz +1 Z z +1 a + dz +2 ... Z z + j − y +1 dz + j − Z y +1 a + dz + j +1 ... Z z + n − a + dz + n ×h ρ a ( z +1 , z ⊥ ) ...ρ a j − ( z + j − , z j − ⊥ ) ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ...ρ a n ( z + n , z n ⊥ ) i . (42)As argued in [58, 59], only adjacent sources can be contracted due to the ordering in z + . Allthe other nested and overlapping contractions have no support. It can also be shown that the18ontraction h ρ a j − ( z + j − , z j − ⊥ ) ρ a j +1 ( z + j +1 , z j +1 ⊥ ) i (where ρ a j ( z + j , z j ⊥ ) is contracted with theexternal source) does not have support either. These properties are shown diagrammaticallyin Fig. 4. They can be used to split the correlator in two parts like h ρ a ( z +1 , z ⊥ ) ...ρ a j − ( z + j − , z j − ⊥ ) ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ...ρ a n ( z + n , z n ⊥ ) i = h ρ a ( z +1 , z ⊥ ) ...ρ a j − ( z + j − , z j − ⊥ ) ih ρ a j +1 ( z + j +1 , z j +1 ⊥ ) ...ρ a n ( z + n , z n ⊥ ) i . (43)By using these properties and by reorganizing the series, we have F , ( b + , a + ) = µ ( y +1 ) G ( x ⊥ − y ⊥ ) f c dd ′ × (cid:26) ∞ X l =0 ( − g ) l " l Y i =1 Z d z i ⊥ G ( x ⊥ − z i ⊥ ) ( f a ...f a l ) ad × Z b + y +1 dz +1 Z z +1 y +1 dz +2 ... Z z + l − y +1 dz + l h ρ a ( z +1 , z ⊥ ) ...ρ a l ( z + l , z l ⊥ ) i (cid:27) × (cid:26) ∞ X m =0 ( − g ) m " m Y j =1 Z d w j ⊥ G ( x ⊥ − w j ⊥ ) ( f b ...f b m ) d ′ b × Z y +1 a + dw +1 Z w +1 a + dw +2 ... Z w + m − a + dw + m h ρ b ( w +1 , w ⊥ ) ...ρ b m ( w + m , w m ⊥ ) i (cid:27) . (44)This complicated expression is just a combination of Wilson lines that is given more suc-cinctly as F , ( b + , a + ) = µ ( y +1 ) G ( x ⊥ − y ⊥ ) f c dd ′ h U ad ( b + , y +1 | x ⊥ ) ih U d ′ b ( y +1 , a + | x ⊥ ) i . (45)This can be simplified further by using Eq. (37) and the fact that ¯ U is an exponential. Wecan get easily that F , ( b + , a + ) = µ ( y +1 ) G ( x ⊥ − y ⊥ ) f c ab ¯ U ( b + , a + | x ⊥ ) . (46)Specializing to the case of a charge distribution moving at the speed of light we have that ρ a ( x + , x ⊥ ) = δ ( x + ) ρ a ( x ⊥ ). Integrating both sides by y + , we get h U ab ( b + , a + | x ⊥ ) ρ c ( y ⊥ ) i = µ A G ( x ⊥ − y ⊥ ) f c ab ¯ U ( b + , a + | x ⊥ ) . (47) B. 1 Wilson line - 2 color charges densities correlator
In this subsection, we compute the correlator F , ( b + , a + ) ≡ h U a b ( b + , a + | x ⊥ ) ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) i . (48)It is possible to devise diagrammatic rules shown in Fig. 5 that can be used to write F , ( b + , a + ) in terms of known quantities. The rules are discussed in more details and19 +b+ + + (b,a x ) + + (b,a x ) ab c = = + +U ab (z,z ) c ρ + (...) + + (b,a x ) + + (b,a y ) = acbd = + + (...)U cd U ab ab = + + (...)=U ab FIG. 5: Diagrammatic rules shown with some of the first topologies they resum. Note that for thesecond correlator h U ab ( b + , a + | x ⊥ ) U cd ( b + , a + | y ⊥ ) i , only the ladder-like diagram are allowed sinceall other topologies have no support and are zero [58, 59]. Finally, in this notation, the light-conecoordinates are ordered following the arrow, from left (the smallest) to right (the biggest) so that b + > a + . y+, c y+, cb+ a+b a b a y+, c y+, c b a y+, c y+, ca+ a+b+b+d d d’ d’ d d’ d d’ + + FIG. 6: Diagrammatic representation of the 1 Wilson line - 2 color charge densities correlator F , ( b + , a + ). The first figure corresponds to the contraction of two external sources. The otherfigures represent the cases where the external sources are contracted with internal sources. Theindices a , b , c , c , d , d ′ , d , d ′ are color indices while a + , b + , y +1 , y +2 are light-cone coordinates. generalized to all cases in App. D. Using these results, F , ( b + , a + ) can be representeddiagrammatically as in Fig. 6. This includes all possible contractions and topologies thatneed to be resummed. According to the rules of App. D, it can be written as20 , ( b + , a + ) = h ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) ih U a b ( b + , a + | x ⊥ ) i + θ ( y +1 − y +2 ) µ ( y +1 ) µ ( y +2 ) f c d d ′ f c d d ′ G ( y ⊥ − x ⊥ ) G ( y ⊥ − x ⊥ ) (49) ×h U b d ( b + , y +1 | x ⊥ ) ih U d ′ d ( y +1 , y +2 | x ⊥ ) ih U d ′ a ( y +2 , a + | x ⊥ ) i + θ ( y +2 − y +1 ) µ ( y +1 ) µ ( y +2 ) f c d d ′ f c d d ′ G ( y ⊥ − x ⊥ ) G ( y ⊥ − x ⊥ ) (50) ×h U b d ( b + , y +2 | x ⊥ ) ih U d ′ d ( y +2 , y +1 | x ⊥ ) ih U d ′ a ( y +1 , a + | x ⊥ ) i . This can be simplified further by using the explicit expressions given in Eqs. (34), (37) and(46). We get that F , ( b + , a + ) = δ c c δ a b µ ( y +1 ) δ ( y +1 − y +2 ) δ ( y ⊥ − y ⊥ ) ¯ U ( b + , a + | x ⊥ )+ µ ( y +1 ) µ ( y +2 ) G ( x ⊥ − y ⊥ ) G ( x ⊥ − y ⊥ ) ¯ U ( b + , a + | x ⊥ ) × ( f c b d f c da θ ( y +1 − y +2 ) + f c b d f c da θ ( y +2 − y +1 )) (51)Considering that the nuclei is moving at the speed of light and integrating on both sides by y +1 and y +2 , we get h U ab ( b + , a + | x ⊥ ) ρ c ( y ⊥ ) ρ c ( y ⊥ ) i = δ c c δ a b µ A δ ( y ⊥ − y ⊥ ) ¯ U ( b + , a + | x ⊥ )+ µ c G ( x ⊥ − y ⊥ ) G ( x ⊥ − y ⊥ ) ¯ U ( b + , a + | x ⊥ ) × ( f c b d f c da + f c b d f c da ) (52)where we defined µ c = Z ∞−∞ du + Z ∞ u + dv + µ ( u + ) µ ( v + ) . (53) C. 2 Wilson lines - 2 color charges densities correlator
In this subsection, we compute the correlator F , ( b + , a + ) ≡ h U a b ( b + , a + | x ⊥ ) U a b ( b + , a + | x ⊥ ) ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) i . (54)This calculation is similar to the one of the second correlator F , using the diagrammaticrules. Three typical diagrams of F , ( b + , a + ) out of seven are shown in Fig. 7. The first onecorresponds to the contraction of the two external sources and the other ones correspondsthe connected part which consists in all contractions between internal and external sources.The first diagram is straightforward to compute. It is given by F , ( b + , a + ) ≡ h U a b ( b + , a + | x ⊥ ) U a b ( b + , a + | x ⊥ ) ih ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) i . (55)Using Eqs. (34) and (38), we get that F , ( b + , a + ) = µ ( y +1 ) δ ( y +1 − y +2 ) δ ( y ⊥ − y ⊥ ) ¯ V ( b + , a + | x ⊥ , x ⊥ ) δ c c δ a a δ b b N c − . (56)21 +b+ a a b b y+, c y+, cy+, c y+, cd d’ d d’ b b a a b+ a+ β (2) β (2) + y+, cd d’ b b a a b+ a+ β (2) d d’ β (1) y+, c++ (...) FIG. 7: These are the first few diagrams included in the calculation of F , ( b + , a + ). They differessentially by the way the sources are inserted between the blobs. The second diagram shown in Fig. 7 is given by F , ( b + , a + ) = θ ( y +1 − y +2 ) µ ( y +1 ) µ ( y +2 ) f c d d ′ f c d d ′ G ( y ⊥ − x ⊥ ) G ( y ⊥ − x ⊥ ) (57) ×h U b d ( b + , y +1 | x ⊥ ) U b β (2)1 ( b + , y +1 | x ⊥ ) ih U d ′ d ( y +1 , y +2 | x ⊥ ) U β (2)1 β (2)2 ( y +1 , y +2 | x ⊥ ) i×h U d ′ a ( y +2 , a + | x ⊥ ) U β (2)2 a ( y +2 , a + | x ⊥ ) i . (58)Using Eqs. (37) and (38), we get F , ( b + , a + ) = θ ( y +1 − y +2 ) µ ( y +1 ) µ ( y +2 ) f c d d f c d d G ( y ⊥ − x ⊥ ) G ( y ⊥ − x ⊥ ) (59) × ¯ V ( b + , y +1 | x ⊥ , x ⊥ ) ¯ V ( y +1 , y +2 | x ⊥ , x ⊥ ) ¯ V ( y +2 , a + | x ⊥ , x ⊥ )= 0 (60)which is zero because of the color structure. All the other diagrams included in F , ( b + , a + )can be computed in a similar way. There are five other different ways of inserting the source22nd they all vanish because of the color structure. The only non-zero term is the first oneand so we have F , ( b + , a + ) = F , ( b + , a + ). Finally, with a nucleus moving at the speed oflight, we obtain h U a b ( b + , a + | x ⊥ ) U a b ( b + , a + | x ⊥ ) ρ c ( y ⊥ ) ρ c ( y ⊥ ) i = µ A δ ( y ⊥ − y ⊥ ) δ c c δ a a δ b b N c − × ¯ V ( b + , a + | x ⊥ , x ⊥ ) . (61)This concludes the computation of correlators. We are now in a position to evaluate thecross section within the MV model. V. NUMERICAL EVALUATION OF THE CROSS SECTIONA. Proton-proton case
In this section, we evaluate the cross section numerically in pp collisions. This is done byusing the expression of the cross section given by Eq. (33) which can be obtained either fromthe dilute limit of the pA result or from k ⊥ -factorization techniques (see Appendix A). In pp collisions, the cross section is related to uPDF that describe the distribution of gluons insideeach protons. There exist many parametrizations of these distribution functions differingmainly in the way the evolution equation is solved. Among the most successful ones are (thedescription of these parametrizations can be found in [9] ): • DIG (Derivative of the Integrated Gluon distribution function) • CCFM (Catani, Ciafaloni, Fiorani, Marchesini) [8, 60, 61, 62, 63, 64] • KMR (Kimber, Martin, Ryskin) [65]These parametrizations are used to compute the cross section for η ′ production in pp colli-sions and to compare with the result for pA collisions at small saturation scale.The final result is obtained by integrating Eq. (33) using the VEGAS and the CUHREalgorithms implemented in the CUBA package [66]. The number of color is set to N c = 3,the center of mass energy to √ s ≈
200 GeV (RHIC) and the mass of η ′ to M = 0 .
957 GeV.To make a comparison with pA collisions for a wide range of transverse momentum where theMV model is valid (see Fig. (9)), we chose the rapidity y = 1. The results of the numericalcalculation are shown in Fig. (8) where we also present the result for the pA case at smallsaturation scale ( Q s = 1 GeV) and with the proton described by the parametrization CCFMJ2003 set 3. ∼ jung/cascade/updf.html. |k ⊥ | (GeV) d σ / dydk ⊥ | y = ( G e V - ) pp (CCFM J2003 set 3)pp (CCFM set A0 )pp (CCFM set B0)pp (KMR)pp (DIG)pA (Q s2 = 1 GeV ) FIG. 8: Numerical results of the inclusive differential cross section at rapidity ( y = 1) and atRHIC energy ( √ s =200 GeV). The results for pp collisions are scaled by the number of nucleons A to make a comparison with pA . The cross section for pA collisions is evaluated at the saturationscale Q s = 1 GeV. B. Proton-nucleus case using the MV model
We now consider η ′ production for pA collisions for which the cross section is given byEq. (31). This formal expression can be simplified by using the results of section IV wherethe correlators of Wilson lines are evaluated in the MV model. Note here that in the MVmodel, the nucleus is considered as an infinite source of charge in the transverse plane sothere are no edge effects. In this kind of description, translation invariance in the transverseplane is preserved and therefore the correlators have the following property h ρ A,a ( x ⊥ ) ρ A,b ( y ⊥ ) U ce ( z ⊥ ) U c ′ e ′ ( w ⊥ ) i = h ρ A,a ( x ⊥ − w ⊥ ) ρ A,b ( y ⊥ − w ⊥ ) U ce ( z ⊥ − w ⊥ ) U c ′ e ′ (0) i . (62)Then, we can write(2 π ) E k dσd k ≈ g π ( N c − Z d p ⊥ d q ⊥ d r ⊥ (2 π ) F ( − p ⊥ , − p , ⊥ ) F ∗ ( − q ⊥ , − q , ⊥ ) × (cid:20) h ρ ∗ A,a ( p ⊥ ) ρ A,a ( q ⊥ ) i (2 π ) δ ( p , ⊥ − r ⊥ )(2 π ) δ ( q , ⊥ − r ⊥ )+ Z d x ⊥ d y ⊥ d z ⊥ d w ⊥ e ip ⊥ · x ⊥ − iq ⊥ · y ⊥ + i ( p , ⊥ − r ⊥ ) · z ⊥ − i ( q , ⊥ − r ⊥ ) · w ⊥ ×h ρ A,a ( x ⊥ ) ρ A,b ( y ⊥ ) U ac ( z ⊥ ) U bc ( w ⊥ ) i (cid:21) ǫ ij ǫ kl p i q k r j r l p ⊥ q ⊥ r ⊥ φ p ( r ⊥ , x ) . (63)24o obtain this expression, we used Eq. (9) to convert the average on proton sources to anunintegrated distribution function and we Fourier transformed the correlator in the secondterm. We also neglected all terms with only one Wilson line because they are numericallyvery small compared to the other terms. This can be seen as follows. First, using translationinvariance of the correlator, we see that averages with one Wilson line in the cross sectionare proportional to ¯ U (0) (see Eq. (62)). This quantity is small because there is an infraredsingularity appearing in the argument of the exponential in the following way [59]:¯ U (0) = exp (cid:20) − N c L (0 , Q s (cid:21) = exp (cid:20) − N c Q s Z d p ⊥ (2 π ) p ⊥ (cid:21) ≈ . (64)This is not exactly zero because the infrared singularity is regulated by non-perturbativeeffects (confinement) and these induce a cutoff of order Λ QCD . The saturation scale in alarge nuclei at small- x satisfies Q s ≫ Λ so that ¯ U (0) ∼ exp h − Q s Λ i ≪
1. Thus, thecorrelators with one Wilson line can be neglected.It is then a straightforward calculation, involving some change of variables, translationinvariance and Eqs. (34) and (61), to obtain(2 π ) E k dσd k = π g µ A S ⊥ Z d p ⊥ d r ⊥ (2 π ) | F ( − p ⊥ , − p , ⊥ ) | ǫ ij ǫ kl p i p k r j r l p ⊥ ( p ⊥ + Λ ) r ⊥ × φ p ( r ⊥ , x ) (cid:20) (2 π ) δ ( p , ⊥ − r ⊥ ) + Z d z ⊥ e i ( p , ⊥ − r ⊥ ) · z ⊥ ¯ V ( z ⊥ , (cid:21) (65)where S ⊥ = (2 π ) δ (0) is interpreted as the transverse area of the nuclei. We also introducedhere an infrared regulator given by Λ in one of the denominator p ⊥ . This is because gluonsseparated by a distance greater than the size of a nucleon ( ∼ QCD . This is not taken into account explicitly in theMV model, leading to infrared divergent quantities [67] which are regulated by adding theinfrared regulator [56]. This is equivalent to define a gluon distribution function for the MVmodel as φ A ( x, p ⊥ , Q ) = α s ( N c − A π p ⊥ + Λ . (66)We do not introduce a regulator in the other denominator p ⊥ because as seen in Eq. (A7),it is part of the matrix element |M| ∼ ǫ ij ǫ kl p i p k r j r l p ⊥ r ⊥ which is well behaved in the infrared.In the first term of the cross section, one integration can be done easily with the deltafunction. The remaining integrals will be done numerically. The second term can be sim-plified further by doing some integrals analytically as shown in Appendix B. The first termis very similar to the pp cross section and does not involve any saturation scale dependencebecause it does not include any multiscatterings. The Q s dependence occurs in the second25erm through the expression of ¯ V where we define it as Q s = N c µ A g π . (67)
1. Kinematical range
The cross section derived in the previous section is restricted to a certain range of validitybecause we are using the MV model. This model can be used when the radiative correctionswhich goes like α s ln(1 /x ) are not too large, which is when 0 . . x A . .
1. For η ′ production, the momentum fraction of gluons in the proton and the nucleus are given by x p = r M + k ⊥ s e y , (68) x A = r M + k ⊥ s e − y . (69)The kinematic range in terms of the η ′ transverse momentum and rapidity where 0 . . x A . . x A by using the JIMWLKequation [41, 42, 43, 44, 45]. This renormalization group equation resums the large radiativecorrections due to very small x modes. The effect is to change the correlation betweensources and in general, one looses the Gaussian structure of the weight functional W [ ρ ].This complicates the computation of Wilson line correlators and is outside the scope of thisarticle.
2. Numerical Results
We present in this section the numerical results for η ′ production. We integrate numeri-cally Eq. (B6) and the first term of Eq. (65) using the same values for the parameters as in pp collisions. The strong coupling constant appearing in the cross section is evaluated at the η ′ transverse mass scale M ⊥ . The unintegrated distribution functions chosen for the protonsis the CCFM J2003 set 3 because it is very successful in the description of other observ-ables like charm and bottom production at the Tevatron [64]. The results of the numericalintegration are shown in Figs. (10) and (11) for different values for the rapidity.In Fig. (12), we show the result for the inverse nuclear modification factor R pA for y = 1defined as 1 R pA = A dσ pp d k ⊥ dydσ pA d k ⊥ dy . (70)We use this because the pp cross section is zero at | k ⊥ | = 0.26 k ( G e v ) FIG. 9: The grey region is the kinematic range where the calculation using the MV model is validfor RHIC energies in terms of the η ′ transverse momentum k = | k ⊥ | and rapidity y . C. Analysis
For the pp cross section, there is a wide range of variability for the cross section dependingon the parametrization of the uPDF used. This feature can be used to discriminate betweenthe different parametrizations to determine the most accurate one. Thus, by combining thisanalysis with experimental data, the η ′ production becomes another observable that canbe utilized to constrain models of uPDF. A similar conclusion was reached in [31] wherethe f -meson production was studied. Note however that the results of the cross section atsmall transverse momentum should be treated carefully. As stated in the appendix A, thevalidity of the k ⊥ -factorization approach depends on the presence of a large scale comparedto the QCD scale given in our case by Q ∼ M ⊥ . At very small momentum, we havethat M ⊥ Λ ≈
23 for which the semihard inequality Λ ≪ Q ≪ √ s that guaranteesthe accuracy of k ⊥ -factorization is only marginally satisfied. Thus, at small momentum,the cross section should be seen as an extrapolation of the k ⊥ -factorized cross section to aregime where k ⊥ -factorization cannot be rigorously proven.The pA cross section shows a strong dependence on the saturation scale. As Q s becomessmaller, the magnitude of the cross section looks more like the pp cross section. On theother hand, the cross section diminishes as Q s becomes larger. This can be understood in27 .5 1 1.5 2 2.5 3 |k ⊥ | (GeV) d σ / dydk ⊥ | y = ( G e V - ) pppA (Q s2 = 1 GeV )pA (Q s2 = 4 GeV )pA (Q s2 = 10 GeV ) FIG. 10: Numerical results of the inclusive differential cross section at midrapidity ( y = 0) andat RHIC energy ( √ s =200 GeV). The first curve is the result of the cross section for pp collisionsscaled by the number of nucleons A . The other curves are the results for the cross section for pA collisions for different values of the saturation scale ( Q s =1,4 and 10 GeV). the following way. The saturation scale is defined as the momentum at which the probabilityof interaction between different parton cascades is of order one [14, 15]. The partons havinga transverse size satisfying δx ⊥ ∼ Q ⊥ ≥ Q s will have a very high probability to recombine.As the saturation scale is increased, there will be more gluons that will be sensitive to thesenonlinear effects. The final result is that as more and more gluons have a high probability torecombine, the growth of their population will decrease and so is the cross section for a givenset of parameters. This effect is clearly seen in the cross section computed in our model.The fact that the η ′ production cross section is sensitive to the saturation scale can be usedto estimate the numerical value of Q s . The current estimate based on HERA data for deepinelastic scattering shows that for RHIC energy, it is given by Q s ∼ − Q s based on η ′ production and see if it is consistent with the previous estimation.The nuclear modification factor shows clearly the saturation effects for 0 . . | k ⊥ | . . R pA >
1. At large transverse momentum, it approaches the value R pA ≈ .
45. One would expect R pA to be one in that range of momentum because it corre-sponds to the regime where there are no saturation effects and where the k ⊥ -factorized crosssection can be used. This discrepancy can be explained by looking at the approximations28 |k ⊥ | (GeV) d σ / dydk ⊥ | y = ( G e V - ) pppA (Q s2 = 1 GeV )pA (Q s2 = 4 GeV )pA (Q s2 = 10 GeV ) FIG. 11: Numerical results of the inclusive differential cross section at rapidity y = 1 and at RHICenergy ( √ s =200 GeV). The first curve is the result of the cross section for pp collisions scaled bythe number of nucleons A . The other curves are the results for the cross section for pA collisionsfor different values of the saturation scale ( Q s =1,4 and 10 GeV). made during the calculation. We neglect two terms in the cross section that are proportionalto one Wilson line (see Eq. (64)). In the dilute limit (when | k ⊥ | > Q s ), these terms cannotbe neglected and have a non zero contribution that would make R pA ≈
1. In that sense, ourapproximation fails at large transverse momentum and our result should be treated carefullyin that regime.
VI. CONCLUSION
In this article, the inclusive cross section for η ′ production in pp and pA collisions iscomputed. The pp case is analyzed for different reasons. First, it serves as a basis tovalidate our model for η ′ production against experimental data. This model includes twomain ingredients. The first one is the effective theory which is at the foundation of ouranalysis. We use in our study a very simplified form of this effective theory where thevertex is given by Eq. (3). As shown in [19, 20, 21, 22, 23, 24, 25], this is a very crudeapproximation of the real vertex, so there are some improvements that can be done in thisdirection in the future. The second ingredient are the uPDF parametrizations. As seenin Fig. (8), there is still a large variability in the predictions made by different uPDF.Therefore, η ′ production could be used to constrain the models of uPDF once it is compared29 |k ⊥ | (GeV) R p A Q s2 = 1 GeV Q s2 = 4 GeV Q s2 = 10 GeV FIG. 12: Numerical results for the inverse nuclear modification factor at rapidity ( y = 1) andat RHIC energy ( √ s =200 GeV). We show the results for different values of the saturation scale( Q s =1,4 and 10 GeV). to experimental data. The same conclusion was obtained in [26] for the exclusive process p + p → p + p + η ′ . Finally, from the theoretical point of view, we studied pp collisions to see ifthe cross section can be obtained as the low density limit of the pA cross section. There arenow many known examples where this can be seen like gluon production [50, 56, 68], quarkproduction [48, 57] and tensor meson production [31]. We have shown that η ′ productionalso obeys this property and in that sense, it is a consistency check for the approach usedfor the pA case.The pA results for the η ′ inclusive cross section show some very interesting features. First,we show that they are sensitive to the value of the saturation scale. This can also be seen inthe plot of the nuclear modification factor. This property could be used to make an estimateof Q s by comparing with experimental data, which is one of the main goal of our analysis.This information is very important for the study of other particle production such as quarkand gluon production in both pA and AA where saturation effects play an important role.Thus, a measurement of η ′ at RHIC would improve our knowledge of gluon distribution ina nucleus.Throughout the article, we assumed that the gluon fusion process was the dominantmechanism in η ′ production. There is one other production process that could also beimportant. It is the photon fusion where two off-shell photons emitted by the protons or the30ucleus merge to give a η ′ such as γ ∗ + γ ∗ → η ′ + X . This is estimated in [26] and accordingto this analysis, it should be subdominant in the exclusive production. We assume that thisholds also in our study, although a careful analysis of this process should be performed. Ofcourse, our calculation could be made more accurate by investigating this last issue. Acknowledgments
The authors want to thank F. Gelis, R. Venugopalan, T. Lappi, Y. Kovchegov, K. Tuchinand J.-S. Gagnon for interesting and stimulating discussions.
APPENDIX A: CROSS SECTION IN pp COLLISIONS
The calculation of the cross section for η ′ production in proton-proton collisions at RHICcan be performed in the k ⊥ -factorization formalism. This formalism can be used in thesemihard regime where the factorization scale satisfies the inequality Λ ≪ Q ≪ √ s . For η ′ , this inequality is only marginally satisfied at small transverse momentum because its massis relatively low. It is recovered at larger transverse momentum, around | k ⊥ | ∼ − k ⊥ -factorization:(2 π ) E k dσ pp → η ′ X d k = 16 π Z dx x dx x Z d q ⊥ d p ⊥ (2 π ) φ ( x , p ⊥ , Q ) φ ( x , q ⊥ , Q ) × (2 π ) E k dσ g ∗ g ∗ → η ′ X d k (A1)where (2 π ) E k dσ g ∗ g ∗ → H d k = 12ˆ s |M g ∗ g ∗ → η ′ | (2 π ) δ ( p + q − k ) (A2)is the high-energy limit of the cross section for off-shell gluons g ∗ to on-shell η ′ mesons,ˆ s = x x s is the k ⊥ -factorization flux factor [1, 2], x , are momentum fractions of gluonsand φ , ( x , , k ⊥ , Q ) are unintegrated gluon distribution functions of proton 1 and 2. Theunintegrated distribution functions are related to usual parton distribution functions ofgluons (appearing in collinear factorization ) by Z Q dk ⊥ φ ( x, k ⊥ , Q ) ≈ xG ( x, Q ) (A3)where G ( x, Q ) is the usual gluon distribution function in collinear factorization.To compute the production cross section of η ′ mesons, the high energy limit of the lowestorder matrix element between two off-shell gluons and one on-shell η ′ has to be calculated.31his can be evaluated by using Feynman rules where the vertex, evaluated from the in-teraction Lagrangian, is given by Eq. (2). Note also that in k ⊥ -factorization, the sum onpolarization tensors is given by [1, 2] X λ ǫ ∗ µλ ( p ) ǫ νλ ( p ) = p µ ⊥ p ν ⊥ p ⊥ (A4)where p µ ⊥ ≡ (0 , p ⊥ , p = − p ⊥ . The exact form is due tothe coupling of gluons to partons through eikonal vertices as well as gauge invariance andWard identities [2].In the center of mass frame, the 4-momenta of partons inside the proton moving in the ± z direction in Minkowski coordinates can be written,as: P = (cid:18) √ s , , , √ s (cid:19) ; Q = (cid:18) √ s , , , − √ s (cid:19) (A5)Then, the momenta of gluons in the large energy limit ( | p ⊥ | , | q ⊥ | ≪ √ s ) are simply p = (cid:18) x √ s , p ⊥ , x √ s (cid:19) ; q = (cid:18) x √ s , q ⊥ , − x √ s (cid:19) (A6)Using this kinematics, it is possible to compute the high-energy limit of the matrix element M . This can then be inserted in the expression of the cross section which is finally givenby (see [12, 13, 31] for more details and similar calculations)(2 π ) E k dσ pp → η ′ X d k = 4 π ( N c − Z d p ⊥ d q ⊥ φ ( x + , p ⊥ , Q ) φ ( x − , q ⊥ , Q ) × δ ( k ⊥ − p ⊥ − q ⊥ ) | F ( − p ⊥ , − q ⊥ ) | [ ǫ ij p i q j ] p ⊥ q ⊥ (A7)where x ± = M ⊥ √ s e ± y . This expression can be used to study the phenomenology of η ′ produc-tion in pp collisions.
1. Limit of Collinear Factorization
The procedure to recover collinear factorization cross sections from k ⊥ -factorization iswell-known [1, 6, 7, 48] and will serve as a consistency check for Eq. (A7). The limit | p ⊥ | , | q ⊥ | → π ) E k dσ pp → η ′ X coll . d k = π M | F (0 , | s ( N c − G ( x ′ + , Q ) G ( x ′− , Q )(2 π ) δ ( k ⊥ ) (A8)where x ′± = M √ s e ± y . This expression corresponds exactly to the well-known result for leading-order η ′ production in pQCD collinear formalism [27].32 PPENDIX B: SIMPLIFICATION OF EQ. (65)
In this Appendix, we simplify the second term of the cross section and put it in a formthat can be evaluated numerically. First, using the definition of ¯ V given by Eq. (38), wecan perform the integrals in the exponent and we get¯ V ( z ⊥ ,
0) = exp (cid:26) − Q s (cid:20) z ⊥ (cid:18) − γ + ln (cid:20) QCD | z ⊥ | (cid:21)(cid:19) + z ⊥ Λ F (cid:18) ,
1; 2 , , (cid:12)(cid:12)(cid:12)(cid:12) − z ⊥ Λ (cid:19)(cid:21)(cid:27) (B1)where γ is Euler constant and where F (1 ,
1; 2 , , | z ) is the generalized hypergeometricseries. The constant Λ QCD appears in this expression as an infrared regulator.Then, we consider the second term of Eq. (65) given by(2 π ) E k dσ d k = π g µ A S ⊥ Z d z ⊥ d p ⊥ d r ⊥ (2 π ) | F ( − p ⊥ , − p , ⊥ ) | ǫ ij ǫ kl p i p k r j r l p ⊥ ( p ⊥ + Λ ) r ⊥ × φ p ( r ⊥ , x ) e i ( p , ⊥ − r ⊥ ) · z ⊥ ¯ V ( z ⊥ , . (B2)By letting v ⊥ = k ⊥ − p ⊥ − r ⊥ and by using the identity Z π dθe ix cos( θ ) = 2 πJ ( x ) (B3)(where J ( x ) is a Bessel function of the first kind), we find that(2 π ) E k dσ d k = 2 π g µ A S ⊥ Z d p ⊥ d v ⊥ (2 π ) | F ( − p ⊥ , − p , ⊥ ) | ǫ ij ǫ kl p i p k ( k − p − v ) j ( k − p − v ) l p ⊥ ( p ⊥ + Λ )( k − p − v ) ⊥ × φ p [( k − p − v ) ⊥ , x ]Γ( | v ⊥ | ) . (B4)where Γ( | v ⊥ | ) ≡ Z ∞ dzzJ ( | v ⊥ | z ) ¯ V ( z ) . (B5)We let q ⊥ = p ⊥ + v ⊥ and we use polar coordinates to get(2 π ) E k dσ d k = α s ( M ⊥ ) µ A S ⊥ Z ∞ dpdq Z π dθd Θ | F ( − p , − p − k + 2 pk cos( θ )) | × pq [ q sin( θ − Θ) − k sin( θ )] ( p + Λ )[ q + k − qk cos(Θ)] × φ p [ q + k − qk cos(Θ) , x ]Γ( | q + p − pq cos( θ − Θ) | ) . (B6)where α s is the strong coupling constant. This expression is integrated numerically and theresult is shown in section V B 2. 33 PPENDIX C: COMPUTATION OF EQ. (31) IN COVARIANT GAUGE
In this Appendix, we compute the cross section by using the solution of the Yang-Millsequation in covariant gauge. This serves as a check for Eq. (31) and it shows that ourresult is gauge independent. The solution of the gauge field is given by A µa ( k ) = A µp,a ( k ) + A µA,a ( k ) + A µpA,a where A µp,a ( k ) is the field associated with the proton (of O ( ρ p )), A µA,a ( k ) isthe field associated with the nuclei (of O ( ρ A ) ∼ O (1)) and A µpA,a ( k ) is the field produced bythe collision (of O ( ρ p ρ ∞ A ) ∼ O ( ρ p )) [50, 53]. The fields A µp and A µpA are weak and are usedas small parameters to solve the Yang-Mills equation perturbatively. The field A µA is strongand satisfies A µp , A µpA ≪ A µA . The explicit solution is given by [50, 53] A + p,a ( k ) = 2 πgδ ( k − ) ρ p,a ( k ⊥ ) k ⊥ (C1) A − A,a ( k ) = 2 πgδ ( k + ) ρ A,a ( k ⊥ ) k ⊥ (C2) A ipA,a ( k ) = − igk + ik + ǫ Z d p ⊥ (2 π ) ρ p,b ( p ⊥ ) p ⊥ × (cid:26) C µU ( k, p ⊥ ) (cid:2) U ab ( k ⊥ − p ⊥ ) − (2 π ) δ ( k ⊥ − p ⊥ ) δ ab (cid:3) + C µV ( k, p ⊥ ) (cid:2) V ab ( k ⊥ − p ⊥ ) − (2 π ) δ ( k ⊥ − p ⊥ ) δ ab (cid:3)(cid:27) (C3)where V ab ( k ⊥ − p ⊥ ) is a Wilson line with a coefficient in the exponential (see [50]) and C + U ( k, p ⊥ ) ≡ − p ⊥ k − + iǫ , C − U ( k, p ⊥ ) ≡ p ⊥ − p ⊥ · k ⊥ k + , C iU ( k, p ⊥ ) ≡ − p i ,C + V ( k, p ⊥ ) ≡ k + , C − V ( k, p ⊥ ) ≡ k ⊥ k + , C iV ( k, p ⊥ ) ≡ k i . (C4)The power counting is very similar to the light-cone gauge. We get that the leading ordercontribution to the field-strength correlator is given by B cov z,z ′ ( k ) = 64 Z d pd q (2 π ) F ∗ ( − p ⊥ , − p , ⊥ ) F ( − q ⊥ , − q , ⊥ ) ǫ ij ǫ kl ×h (cid:2) q i ( k − q ) j A − A,b ( q ) A + z,b ( k − q ) − q i ( k − q ) + A − A,b ( q ) A jz,b ( k − q ) (cid:3) × (cid:2) p k ( k − p ) l A −∗ A,a ( p ) A + ∗ z ′ ,a ( k − p ) − p k ( k − p ) + A −∗ A,a ( p ) A l ∗ z ′ ,a ( k − p ) (cid:3) i . (C5)Using the expression of the gauge field, it is a straightforward calculation to show that B LCp,p ( k ) = B p,p ( k ). Just like in light-cone gauge, the other terms require more work. We willfirst look at B cov pA,p ( k ) = 64 g Z d p ⊥ d q ⊥ dq − (2 π ) F ∗ ( − p ⊥ , − p , ⊥ ) F ( − q ⊥ , − q , ⊥ ) ǫ ij ǫ kl p k ( k − p ) l q ⊥ p ⊥ ( k − p ) ⊥ ×h (cid:2) q i ( k − q ) j A + pA,b ( k − q ) − q i k + A jpA,b ( k − q ) (cid:3) × ρ A,b ( q ⊥ ) ρ ∗ A,a ( p ⊥ ) ρ ∗ p,a ( k ⊥ − p ⊥ ) i (C6)34hich is obtained by replacing the gauge fields with their explicit expression. Now, we havethat q i ( k − q ) j A + pA,b ( k − q ) − q i k + A jpA,b ( k − q ) = ig ( k − q ) + ik + ǫ Z d r ⊥ (2 π ) ρ p,e ( r ⊥ ) r ⊥ (cid:20) q i ( k − q ) j r ⊥ k − − q − + iǫ + 2 q i k + r j (cid:21) × (cid:2) U be ( k ⊥ − q ⊥ − r ⊥ ) − (2 π ) δ ( k ⊥ − q ⊥ − r ⊥ ) δ be (cid:3) (C7)which does not depend on V ab ( k ⊥ ). This is required because this Wilson line does not appearin the light-cone gauge calculation. Furthermore, the first term is zero when it is integratedon q − because it has two poles on the side of the real axis (similar to the LC calculationin section III C). The integration on q − of the second term can be done with the principalpart identity (see again section III C). The result is that B cov pA,p ( k ) = B pA,p ( k ). A similarcalculation can be performed with the two other terms and we find that B cov ( k ) = B ( k ). APPENDIX D: DIAGRAMMATIC RULES AND n WILSON LINES - m COLORCHARGES DENSITIES CORRELATORS
In this appendix, we discuss the diagrammatic rules used in Sec. IV to compute theWilson lines - color charge densities correlators. These techniques can be used in any repre-sentation of the SU ( N ) generators, as long as the Wilson line correlators such as F j ( b + , a + |{ a } , { b } ) ≡ h U a b ( b + , a + | x ⊥ ) U a b ( b + , a + | x ⊥ ) ...U a j b j ( b + , a + | x j ⊥ ) i (D1)are known, which we assume throughout the following discussion. Here, { a } , { b } are the setsof color indices defined as { a , a , ..., a j } and { b , b , ..., b j } respectively. The most generalcorrelator we study here is F m,n ( b + , a + ) ≡ h ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) ...ρ c m ( y + m , y m ⊥ ) × U a b ( b + , a + | x ⊥ ) U a b ( b + , a + | x ⊥ ) ...U a n b n ( b + , a + | x n ⊥ ) i . (D2)We want to express this correlator in terms of Wilson line correlators shown in Eq. (D1).This can be done in a general way. First, we start by defining a number of new quantitiesnecessary for the computation. We define a quantity that represents a correlator with anumber j of color charge densities such as G j (1 , ,..., { k,k +1 } ,...,j +1 ,j +2) ≡ h ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) ...ρ c k − ( y + k − , y k − ⊥ ) ρ c k +2 ( y + k +2 , y k +2 ⊥ ) ... × ρ c j +2 ( y + j +2 , y j +2 ⊥ ) i . (D3)35n this definition of G j (1 , ,..., { k,k +1 } ,...,j +1 ,j +2) , the upper index counts the number of sourceswhile the lower index indicates in bracket which sources are missing. We also define H j,m in a similar way by H j,n (1 , ,..., { k,k +1 } ,...,j +1 ,j +2) = h ρ c ( y +1 , y ⊥ ) ρ c ( y +2 , y ⊥ ) ...ρ c k − ( y + k − , y k − ⊥ ) ρ c k +2 ( y + k +2 , y k +2 ⊥ ) ... × ρ c j +2 ( y + j +2 , y j +2 ⊥ ) × U a b ( b + , a + | x ⊥ ) U a b ( b + , a + | x ⊥ ) ...U a n b n ( b + , a + | x n ⊥ ) i conn . (D4)where j ≤ m and where the subscript conn . indicates that only the connected part ofthe correlator is considered, which means that all external sources are contracted with an internal source (note that external and internal are defined in Sec. IV A). Now, there aretwo possible cases: m can be odd or even.1. Even m For the case where m is even, we can use Wick theorem to write F m,n ( b + , a + ) ≡ G m H ,n + X i,j,i
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