# Comparing Two Different Initial Conditions AAMQS and Quartic Action in Illustrating the Effect of Valence Color Charges in High-EnergypA Collisions at Forward Rapidity

Authors: Ye-Yin Zhao; Ya-Hui Chen; Ya-Qin Gao; Fu-Hu Liu
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

The inclusive particle productions in
proton-proton (pp) and deuton-gold (d+Au) collisions at
forward rapidity at the Relativistic Heavy Ion Collider (RHIC)
energy are studied in the framework of the color glass condensate
(CGC) theory by using two different initial conditions: AAMQS
(Albacete-Armesto-Milhano-Quiroga-Salgado) and quartic action.
Then, the results obtained by the two different initial conditions
in illustrating the effect of valence color charges in high-energy
proton-nucleus (pA) collisions at forward energy are compared.
Meanwhile, the inclusive particle productions in pA collisions
at forward rapidity at the Large Hadron Collider (LHC) energies
are predicted. The main dynamical input in our calculations is the
use of solutions of the running coupling Balitsky-Kovchegov
equation tested in electron-proton (ep) collision data. Particle
production is computed via the hybrid formalisms to obtain spectra
and yields. These baseline predictions are useful for testing the
current understanding of the dynamics of very strong color fields
against the upcoming LHC data.


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## Body

## 1. Introduction

As a self-consistent effective perturbative quantum chromodynamics (QCD) theory at high energy (or small momentum fractionx), the color glass condensate (CGC) framework has been successfully applied to many QCD processes in high-energy collisions [1, 2]. Generally, in the initial state, the collisions involve at least one hadron or nucleus. Including the needed unitarity corrections, the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) renormalization group equations [3–8] capture the dynamical features of the CGC, which provides the tools to perform such resummation, although the precise prescription for the resummation may vary from process to process or colliding system.In the framework of the CGC, as an instrumental tool, the dipole model formulation of deep inelastic scattering (DIS) allows for a relatively simple implementation of saturation effects in the description of the scattering process [9, 10]. In fact, in the small-x range, many experimental data with a number of features are successfully described by the dipole model. By requiring higher-order corrections to the Balitsky-Kovchegov-JIMWLK (BK-JIMWLK) equations [3–8, 11–13], an important progress was achieved in the past years. Recently, through the calculation of higher-order corrections to the BK-JIMWLK equations, an important step towards promoting the CGC to a practical phenomenological tool was performed [14–17]. In particular, the calculation of running coupling (αs) corrections to the evolution kernel of the Balitsky-Kovchegov (BK) equation [18, 19] renders the possibility to describe various data at high energies. In particular, the BK equation with running coupling (referred to as rcBK equation [18, 19]) grasps most of the higher-order effects and incorporates nonlinear high gluon density effects. These processes close the gap between the first principle theory calculations and the experimental data.In the CGC formalism, by using varying degrees of approximations and models, many researchers investigated single inclusive hadron productions in proton-nucleus (pA) collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) [20–31]. The CGC framework describes well single-particle distributions in both proton-proton (pp) and deuton-gold (d+Au) collisions, as well as the depletion of nuclear modification factors RdAu [27, 29]. In deuton-nucleus (dA) collisions at the RHIC, the observed suppression of single inclusive hadron production and the disappearance of the away-side peak in double-hadron production in the forward rapidity region are perhaps the strongest evidence for the importance and possibly dominance of saturation effects. The CGC calculations firstly predicted these effects [32] and later on quantitatively accounted for such suppression [33, 34]. This will be tested further at the LHC where, due to the larger energy of the collisions, the CGC dynamics in a much larger kinematic region can be probed. In the theoretical framework of the CGC, the single inclusive hadron production cross-section captures the saturation dynamics. The most important ingredient of the cross-section is the fundamental (or adjoint) dipole cross-section, that is, the imaginary part of the quark-antiquark (qq-) scattering amplitude on a proton or nucleus target. This dipole cross-section satisfies the BK-JIMWLK evolution equations [3–8, 11–13]. It also resums the small-x and high gluon density effects. With next-to-leading-order (NLO) accuracy [14–17, 35], one now knows the evolution equation for the dipole cross-section.Because of the larger valence charge density in a nucleus, the CGC effect in a nucleus is expected to be more enhanced than that in a proton. However, due to a large amount of high-quality DIS data on protons at small-x, the most exhaustive searches of the saturation phenomenon are performed using data on proton reactions. The global fits performed by the Albacete-Armesto-Milhano-Quiroga-Salgado (AAMQS) group [36–38], in ep collision at the HERA [39], show that the rcBK equation successfully accounts for the x-dependence of inclusive structure functions at small-x. We would like to point out that the AAMQS initial condition is not derived from an effective action (Gaussian action) for the large-x valence charges. Based on a non-Gaussian action (quartic action), which involves the subleading correction of valence color charges, Dumitru and Petreska derived a new dipole scattering amplitude [40]. Thus, we may compare the two different initial conditions (AAMQS and quartic action), which could illustrate the effect of valence color charges in pA collisions at high energy. In particular, as the number of nucleons is finite, the process of scattering then involves the correction of valence color charges effects.Due to the scientific significances and interests, in this paper, we study the inclusive particle productions inpp and d+Au collisions at the RHIC energy in the framework of the CGC theory by using the previously mentioned two initial conditions. Then, the inclusive particle productions in pA collisions at the LHC energies are predicted by us. As we know, the situations at central rapidity and forward rapidity are different. To focus our attention on fragmentation region, where the leading nucleons and cold spectators are affected by each other, we study only the inclusive particle productions at forward rapidity in this paper. In Sections 2 and 3, we introduce and review the high-energy evolution equation and give single inclusive hadron production, respectively. The conclusions are given in Section 4.


## 2. High-Energy Evolution Equation

### 2.1. The BK Equation with Running Coupling

In the CGC framework, one needs the BK-JIMWLK equations to describe the quantum evolution of hadron structure towards small-x. The BK-JIMWLK equations are equivalent to an infinite set of coupled nonlinear equations for the different Wilson line correlators, where these correlators systematically incorporate small-x gluon emission to all orders [3–8]. In the large number of colors (large-Nc) limit, the JIMWLK evolution equations are reduced to the BK equation [11–13], while the later one is a closed-form equation for the evolution of the dipole amplitude.According to [11–14, 18], the rcBK equation can be given in the following form:
(1)∂𝒩F(r,x)∂ln(x0/x)=∫d2r→1Krun(r→,r→1,r→2)×[𝒩F(r1,x)+𝒩F(r2,x)-𝒩F(r,x)-𝒩F(r1,x)𝒩F(r2,x)],
where 𝒩F, x0, and r→ (=r→1+r→2) denote the forward dipole-target scattering amplitude in the fundamental (F) representation, the value of x where the evolution starts, and the transverse dipole size, respectively. In order to give some numerical solutions and related applications of (1), we can use a kernel validity to running coupling accuracy. According to Balitsky’s prescription [14], the kernel Krun in (1) can be presented as follows:
(2)Krun(r→,r→1,r→2)=Ncαs(r2)2π2[1r12(αs(r12)αs(r22)-1)Ncαs(r2)2π2+r2r12r22+1r22(αs(r22)αs(r12)-1)].
For the running coupling in the above equation, we may use the scheme proposed in [37] at one-loop level.In particular, in the large-Nc limit, one has the following relation between the adjoint (A) and fundamental dipoles:
(3)𝒩A(r,Y)=2𝒩F(r,Y)-𝒩F2(r,Y).
The amplitude 𝒩A(F) incorporates all multiscattering between a projectile color dipole and the target and encodes the small-x dynamics. In the CGC framework, it can be obtained from the solution of rcBK evolution with a given initial condition.


### 2.2. Initial Condition: Beyond the McLerran-Venugopalan Model

The initial conditions required by the high-energy evolution equations are at a rapidity ofY=ln(x0/x)=0. Generally, one takes x0=10-2. The dynamical equations were derived by McLerran and Venugopalan [41, 42] in the limit of an infinitely large nucleus. Let ρa(x⊥,x-) denote the classical color charge density per unit transverse area x⊥ and longitudinal length x-, where kinetic terms ρ are neglected because the transverse momenta are assumed to be small. In the McLerran-Venugopalan (MV) model [41, 42], the large-x valence color charges are then described as recoilless sources on the light cone with ρa(x⊥,x-). As A→∞, the variance of the valence color charge density ~A1/3 grows large corresponding to a very high-density ρ of charges. According to the central limit theorem, the fluctuations of the color charges are described by a Gaussian effective action. Generally, the high multiplicity pp collision corresponds to an unusually high valence color charge density [43]. Then, in this case, we can use the Gaussian effective action to describe it. However, for the real case with the mass number A, the number of valence color charges is finite. Particularly in pp and peripheral nucleus-nucleus (AA) collisions, the considered number is very limited. Therefore, one needs to consider extensions of the MV model action to involve higher powers of the color charge density.In fact, Jeon and Venugopalan [44, 45] introduced an “odderon” operator contribution -dabcρaρbρc/κ3 to the effective action with κ3~g3(A/πR2)2, where R denotes the radius of the nucleus. Dumitru et al. [46] introduced the quartic order contribution δabδcdρaρbρcρd/κ4 with κ4~g4(A/πR2)3. Finally, to go beyond the limit of infinite valence charge density, the effective action is written as follows [46]:
(4)S[ρ(x)]≃∫d2x[δabρaρb2μ2-dabcρaρbρcκ3∫d2x+δabδcd+δacδbd+δadδbcκ4ρaρbρcρd].Based on the new effective action, including the first subleading correction in the semiclassical approximation, Dumitru and Petreska [40] introduced the dipole scattering amplitude 𝒩(r) on a dense target. This dipole scattering amplitude can be matched with a phenomenological proton fit by Albacete et al. [36, 37] over a broad range of dipole size r and can provide a definite prediction for the A-dependence for heavy-ion targets. According to [40], this dipole scattering amplitude has the following form:
(5)𝒩(r)=Qs2r24ln(1rΛ)-βQs2r2ln3(1rΛ),(r2Qs2<1),
where Qs and Λ denote the initial saturation momentum scale and the infrared scale, respectively, and
(6)β≡CF26π3g8Qs2κ4[∫-∞∞‍dz-μ4(z-)]2≃1100,(A=1).
For the nucleus with a mass A, we have βA~A-2/3.As discussed in [40], the amplitude decreases sharply at very short distances. This phenomenon may not be reasonable. As we note, that the new term of the valence color charge has an extremely larger contribution than the other one at very small dipole size. This term corresponds to strong interaction caused by higher-order correlations of valence color charges. More carefully, we have 𝒩(r)~r2 in the limit of r→0. Based on this behavior, the valence color charge term is assumed to be controlled by a weight function γ2(r) to go smoothly to the scattering amplitude. The first term is assumed to be controlled by a weight function γ1(r). The new scattering amplitude is given by the following:
(7)𝒩(r)=γ1(r)Qs2r24ln(1rΛ)-γ2(r)βQs2r2ln3(1rΛ),(r2Qs2<1).
Following this procedure, the unreasonable phenomena can be avoided. It should be noted that taking the weight function involved is just an experiential treatment. We also assume experientially that the weight functions have the following form:
(8)γ1(r)=0.001+0.999exp(-0.0002r2Qs2),γ2(r)=exp(-0.00021r2Qs2).
The parameters above are obtained by balanceing the two terms in the amplitude. That means the contribution of the new term is vanished while the first term is domainal at very short distance. Then, the scattering amplitude for a dipole in the adjoint representation with a given β can be obtained. We would like to point out that to regularize the behavior at large r in the calculation, we have replaced ln(1/rΛ) by ln(e+(1/rΛ)) in (7) and assumed an exponentiation of the 𝒪(r2) expression. This does not affect the behavior at rQs≤1.The dipole scattering amplitude for a proton target was fitted by the AAMQS model [36–38] which gives the amplitude, in the initial condition for a small-x evolution, to be(9)𝒩AAMQS(r,x0=0.01)=1-exp[-14(r2Qs2(x0))γln(e+1rΛ)],
where the infrared scale Λ is taken to be 0.241 GeV. We notice that the case with γ=1 corresponds to the MV model which was derived from an effective action (Gaussian action) for the large-x valence charges. The free parameters are parameter γ and the initial saturation scale Qs, which is probed by quarks. For the proton and nuclear targets, we use respectively, the notations Q0p and Q0A for clearness. From a global fit to proton structure functions in DIS in the small-x region and single inclusive hadron data in pp collisions at the RHIC and the LHC, the proton initial saturation scale gives Q0p2≃0.168GeV2 and γ≃1.119. Simultaneously, this model provides a good description of charged hadron transverse momentum distributions in pp collision at the center-of-mass energy per nucleon pair sNN=7TeV [47, 48].Solving the rcBK equation related to the impact-parameter is still an open question. This renders that in (1) the impact parameter dependence of the collisions is ignored. In fact, this question may not be important for the minimum-bias analysis because the initial saturation scale Q0A of a nucleus can be considered as an averaged value and extracted from the minimum-bias data. In the minimum-bias collisions, one may assume that Q0A2=CA1/3Q0p2, where the parameter C is fixed from a fit to data. In [49], it was shown that C≈0.5 if the DIS data is used for the description of heavy nuclear targets. This result is consistent with the inclusive hadron production data in minimum-bias d+Au collisions at the RHIC energy, which gives an initial saturation scale within Q0A2≈3÷4Q0p2. The uncertainties associated to the variation of initial saturation scale in the rcBK evolution equation should be taken into account.The dependence of dipole scattering amplitudes in the adjoint representation(𝒩A=2𝒩F-𝒩F2) on rQs(x0) are shown in Figure 1 with three different models: MV (the solid curves), AAMQS with γ=1.119 (the dashed curves), and quartic action (the dashed-dotted curves). Figure 1(a) is for a proton assuming that Qs2=0.168GeV2, Λ2=0.058GeV2, and β=0.01. Figure 1(b) is for a nucleus with A=200 (which is between Au and Pb), Qs2~A1/3, and βA~A-2/3 (or β=0.0003). One can see that both the dipole scattering amplitudes resulted from the quartic action and AAMQS models are beyond the MV model. The result of the quartic action is similar to the AAMQS model over a broad rQs(x0) range. At a small size of dipole, the result of the quartic action is much smaller than that of AAMQS. Dumitru and Petreska [40] argued that the suppression of the bremsstrahlung tails by the subleading term possibly explains the difference between the AAMQS and MV models within relevant region 0.05<rQs<1. At very small r, however, the subleading correction starts to dominate the leading one. This implies the breakdown of the simple perturbative expansion.(a) Dependence of dipole scattering amplitude in an adjoint representation (𝒩A=2𝒩F-𝒩F2) on rQs(x0) with three different models indicated in the figure. (a) is for a proton target assuming Qs2=0.168GeV2 and Λ2=0.058GeV2. (b) is for a nucleus target with A=200 (which is between Au and Pb), Qs2~A1/3, and βA~A-2/3 (or β=0.0003).
(a)
(b)To understand more easily the above phenomenon, the scattering amplitude𝒩(r,y) for a dipole in coordinate space at different rapidities is given in Figure 2. The solid and dashed-dotted curves represent the results of the AAMQS with γ=1.119 and quartic action, respectively. From the right curve to the left one, the corresponding rapidities are y=0, 5, and 10, respectively. One can see that, at y=0, there is a very little difference between the results obtained using the two different initial conditions. At y=5 and 10, the two types of results are almost the same. These phenomena are caused by the gluon distribution of the proton itself, which is independent of any particular reactions and renders nearly the same result.Figure 2
Dipole scattering amplitude𝒩(r,y) in coordinate space at different rapidities with two different initial conditions indicated in the figure. From the right curve to the left one, the corresponding rapidities are y=0, 5, and 10, respectively.


### 2.3. Structure Function and Comparison with Experimental Data

To check the validity of these calculations, let us calculate the proton structure functionF2 at small-x. At x≪1, the inclusive structure function obtained from DIS process is given by [50]
(10)F2(x,Q2)=Q24π2αem(σTγ*p+σLγ*p),
where αem, Q2, and σT,Lγ*p denote the electromagnetic coupling, the transverse resolution scale, and the virtual photon-proton (γ*p) cross-section for transverse (T) and longitudinal (L) polarization states of the virtual photon, respectively. In the dipole model, the total γ*p cross-section is given by
(11)σT,Lγ*p=∑f∫d2rT∫01dz[Ψ*Ψ]T,Lf(z,Q2)σqq-(x,rT),
where z, rT, Ψ, f, and σqq-(x,rT) denote the fraction of longitudinal momentum of the virtual photon carried by the quark, the transverse separation between the quark and the antiquark, the light-cone wavefunction for the photon splitting into a qq- pair [51, 52], the flavor of the quark, and the dipole-proton cross-section, respectively. By using the optical theorem, σqq-(x,rT) can be obtained from the elastic dipole-proton scattering amplitude 𝒜(x,rT,Δ),
(12)σqq-(x,rT)=2Im𝒜(x,rT,Δ)=2∫d2bT𝒩(x,rT,bT)=σ0𝒩(x,rT),
where 𝒩(x,rT,bT), bT, and σ0 denote the imaginary part of the forward elastic dipole-proton scattering amplitude, the impact parameter of the dipole-target collisions, and the normalization constant obtained by the bT integration, respectively.To see the dependence of structure functionF2 on x, in Figure 3 we show the comparisons between the experimental data (squares) in ep collision [53–55] and the results by solving rcBK equation with the two different initial conditions, AAMQS (the solid curves) and quartic action (the dashed-dotted curves), by using (2) and (3). Figures 3(a), 3(b), 3(c), and 3(d) are the results for different values of Q2. In the calculation, we take σ0=30.265mb, Q0s2=0.168GeV2, C=1.715, and αfr=1 in the running coupling in (2) for the four Q2 values in Figure 4. One can see that the two different initial conditions describe well the F2 data, although their behaviors in Figure 1 on 𝒩A(r,x0) at small rQs(x0) are different.Proton structure functionF2(x,Q2) versus x at different Q2. The squares represent the experimental data of the HERA Collaboration [53–55]. The solid and dashed-dotted curves represent the results calculated by the AAMQS and quartic action initial conditions, respectively.
(a)
(b)
(c)
(d)Dependence of the dipole scattering amplitudeNA(F)(kT,x), the Fourier transform of 1-𝒩A(F)(r,Y) (15), on kT for (a) a proton target and (b) a nucleus target with A=200 at different rapidities with two different initial conditions indicated in the figure. From the upper curve to the lowest one, the corresponding rapidities are y=2, 4, and 8, respectively.
(a)
(b)
(c)
(d)


## 2.1. The BK Equation with Running Coupling

In the CGC framework, one needs the BK-JIMWLK equations to describe the quantum evolution of hadron structure towards small-x. The BK-JIMWLK equations are equivalent to an infinite set of coupled nonlinear equations for the different Wilson line correlators, where these correlators systematically incorporate small-x gluon emission to all orders [3–8]. In the large number of colors (large-Nc) limit, the JIMWLK evolution equations are reduced to the BK equation [11–13], while the later one is a closed-form equation for the evolution of the dipole amplitude.According to [11–14, 18], the rcBK equation can be given in the following form:
(1)∂𝒩F(r,x)∂ln(x0/x)=∫d2r→1Krun(r→,r→1,r→2)×[𝒩F(r1,x)+𝒩F(r2,x)-𝒩F(r,x)-𝒩F(r1,x)𝒩F(r2,x)],
where 𝒩F, x0, and r→ (=r→1+r→2) denote the forward dipole-target scattering amplitude in the fundamental (F) representation, the value of x where the evolution starts, and the transverse dipole size, respectively. In order to give some numerical solutions and related applications of (1), we can use a kernel validity to running coupling accuracy. According to Balitsky’s prescription [14], the kernel Krun in (1) can be presented as follows:
(2)Krun(r→,r→1,r→2)=Ncαs(r2)2π2[1r12(αs(r12)αs(r22)-1)Ncαs(r2)2π2+r2r12r22+1r22(αs(r22)αs(r12)-1)].
For the running coupling in the above equation, we may use the scheme proposed in [37] at one-loop level.In particular, in the large-Nc limit, one has the following relation between the adjoint (A) and fundamental dipoles:
(3)𝒩A(r,Y)=2𝒩F(r,Y)-𝒩F2(r,Y).
The amplitude 𝒩A(F) incorporates all multiscattering between a projectile color dipole and the target and encodes the small-x dynamics. In the CGC framework, it can be obtained from the solution of rcBK evolution with a given initial condition.


## 2.2. Initial Condition: Beyond the McLerran-Venugopalan Model

The initial conditions required by the high-energy evolution equations are at a rapidity ofY=ln(x0/x)=0. Generally, one takes x0=10-2. The dynamical equations were derived by McLerran and Venugopalan [41, 42] in the limit of an infinitely large nucleus. Let ρa(x⊥,x-) denote the classical color charge density per unit transverse area x⊥ and longitudinal length x-, where kinetic terms ρ are neglected because the transverse momenta are assumed to be small. In the McLerran-Venugopalan (MV) model [41, 42], the large-x valence color charges are then described as recoilless sources on the light cone with ρa(x⊥,x-). As A→∞, the variance of the valence color charge density ~A1/3 grows large corresponding to a very high-density ρ of charges. According to the central limit theorem, the fluctuations of the color charges are described by a Gaussian effective action. Generally, the high multiplicity pp collision corresponds to an unusually high valence color charge density [43]. Then, in this case, we can use the Gaussian effective action to describe it. However, for the real case with the mass number A, the number of valence color charges is finite. Particularly in pp and peripheral nucleus-nucleus (AA) collisions, the considered number is very limited. Therefore, one needs to consider extensions of the MV model action to involve higher powers of the color charge density.In fact, Jeon and Venugopalan [44, 45] introduced an “odderon” operator contribution -dabcρaρbρc/κ3 to the effective action with κ3~g3(A/πR2)2, where R denotes the radius of the nucleus. Dumitru et al. [46] introduced the quartic order contribution δabδcdρaρbρcρd/κ4 with κ4~g4(A/πR2)3. Finally, to go beyond the limit of infinite valence charge density, the effective action is written as follows [46]:
(4)S[ρ(x)]≃∫d2x[δabρaρb2μ2-dabcρaρbρcκ3∫d2x+δabδcd+δacδbd+δadδbcκ4ρaρbρcρd].Based on the new effective action, including the first subleading correction in the semiclassical approximation, Dumitru and Petreska [40] introduced the dipole scattering amplitude 𝒩(r) on a dense target. This dipole scattering amplitude can be matched with a phenomenological proton fit by Albacete et al. [36, 37] over a broad range of dipole size r and can provide a definite prediction for the A-dependence for heavy-ion targets. According to [40], this dipole scattering amplitude has the following form:
(5)𝒩(r)=Qs2r24ln(1rΛ)-βQs2r2ln3(1rΛ),(r2Qs2<1),
where Qs and Λ denote the initial saturation momentum scale and the infrared scale, respectively, and
(6)β≡CF26π3g8Qs2κ4[∫-∞∞‍dz-μ4(z-)]2≃1100,(A=1).
For the nucleus with a mass A, we have βA~A-2/3.As discussed in [40], the amplitude decreases sharply at very short distances. This phenomenon may not be reasonable. As we note, that the new term of the valence color charge has an extremely larger contribution than the other one at very small dipole size. This term corresponds to strong interaction caused by higher-order correlations of valence color charges. More carefully, we have 𝒩(r)~r2 in the limit of r→0. Based on this behavior, the valence color charge term is assumed to be controlled by a weight function γ2(r) to go smoothly to the scattering amplitude. The first term is assumed to be controlled by a weight function γ1(r). The new scattering amplitude is given by the following:
(7)𝒩(r)=γ1(r)Qs2r24ln(1rΛ)-γ2(r)βQs2r2ln3(1rΛ),(r2Qs2<1).
Following this procedure, the unreasonable phenomena can be avoided. It should be noted that taking the weight function involved is just an experiential treatment. We also assume experientially that the weight functions have the following form:
(8)γ1(r)=0.001+0.999exp(-0.0002r2Qs2),γ2(r)=exp(-0.00021r2Qs2).
The parameters above are obtained by balanceing the two terms in the amplitude. That means the contribution of the new term is vanished while the first term is domainal at very short distance. Then, the scattering amplitude for a dipole in the adjoint representation with a given β can be obtained. We would like to point out that to regularize the behavior at large r in the calculation, we have replaced ln(1/rΛ) by ln(e+(1/rΛ)) in (7) and assumed an exponentiation of the 𝒪(r2) expression. This does not affect the behavior at rQs≤1.The dipole scattering amplitude for a proton target was fitted by the AAMQS model [36–38] which gives the amplitude, in the initial condition for a small-x evolution, to be(9)𝒩AAMQS(r,x0=0.01)=1-exp[-14(r2Qs2(x0))γln(e+1rΛ)],
where the infrared scale Λ is taken to be 0.241 GeV. We notice that the case with γ=1 corresponds to the MV model which was derived from an effective action (Gaussian action) for the large-x valence charges. The free parameters are parameter γ and the initial saturation scale Qs, which is probed by quarks. For the proton and nuclear targets, we use respectively, the notations Q0p and Q0A for clearness. From a global fit to proton structure functions in DIS in the small-x region and single inclusive hadron data in pp collisions at the RHIC and the LHC, the proton initial saturation scale gives Q0p2≃0.168GeV2 and γ≃1.119. Simultaneously, this model provides a good description of charged hadron transverse momentum distributions in pp collision at the center-of-mass energy per nucleon pair sNN=7TeV [47, 48].Solving the rcBK equation related to the impact-parameter is still an open question. This renders that in (1) the impact parameter dependence of the collisions is ignored. In fact, this question may not be important for the minimum-bias analysis because the initial saturation scale Q0A of a nucleus can be considered as an averaged value and extracted from the minimum-bias data. In the minimum-bias collisions, one may assume that Q0A2=CA1/3Q0p2, where the parameter C is fixed from a fit to data. In [49], it was shown that C≈0.5 if the DIS data is used for the description of heavy nuclear targets. This result is consistent with the inclusive hadron production data in minimum-bias d+Au collisions at the RHIC energy, which gives an initial saturation scale within Q0A2≈3÷4Q0p2. The uncertainties associated to the variation of initial saturation scale in the rcBK evolution equation should be taken into account.The dependence of dipole scattering amplitudes in the adjoint representation(𝒩A=2𝒩F-𝒩F2) on rQs(x0) are shown in Figure 1 with three different models: MV (the solid curves), AAMQS with γ=1.119 (the dashed curves), and quartic action (the dashed-dotted curves). Figure 1(a) is for a proton assuming that Qs2=0.168GeV2, Λ2=0.058GeV2, and β=0.01. Figure 1(b) is for a nucleus with A=200 (which is between Au and Pb), Qs2~A1/3, and βA~A-2/3 (or β=0.0003). One can see that both the dipole scattering amplitudes resulted from the quartic action and AAMQS models are beyond the MV model. The result of the quartic action is similar to the AAMQS model over a broad rQs(x0) range. At a small size of dipole, the result of the quartic action is much smaller than that of AAMQS. Dumitru and Petreska [40] argued that the suppression of the bremsstrahlung tails by the subleading term possibly explains the difference between the AAMQS and MV models within relevant region 0.05<rQs<1. At very small r, however, the subleading correction starts to dominate the leading one. This implies the breakdown of the simple perturbative expansion.(a) Dependence of dipole scattering amplitude in an adjoint representation (𝒩A=2𝒩F-𝒩F2) on rQs(x0) with three different models indicated in the figure. (a) is for a proton target assuming Qs2=0.168GeV2 and Λ2=0.058GeV2. (b) is for a nucleus target with A=200 (which is between Au and Pb), Qs2~A1/3, and βA~A-2/3 (or β=0.0003).
(a)
(b)To understand more easily the above phenomenon, the scattering amplitude𝒩(r,y) for a dipole in coordinate space at different rapidities is given in Figure 2. The solid and dashed-dotted curves represent the results of the AAMQS with γ=1.119 and quartic action, respectively. From the right curve to the left one, the corresponding rapidities are y=0, 5, and 10, respectively. One can see that, at y=0, there is a very little difference between the results obtained using the two different initial conditions. At y=5 and 10, the two types of results are almost the same. These phenomena are caused by the gluon distribution of the proton itself, which is independent of any particular reactions and renders nearly the same result.Figure 2
Dipole scattering amplitude𝒩(r,y) in coordinate space at different rapidities with two different initial conditions indicated in the figure. From the right curve to the left one, the corresponding rapidities are y=0, 5, and 10, respectively.


## 2.3. Structure Function and Comparison with Experimental Data

To check the validity of these calculations, let us calculate the proton structure functionF2 at small-x. At x≪1, the inclusive structure function obtained from DIS process is given by [50]
(10)F2(x,Q2)=Q24π2αem(σTγ*p+σLγ*p),
where αem, Q2, and σT,Lγ*p denote the electromagnetic coupling, the transverse resolution scale, and the virtual photon-proton (γ*p) cross-section for transverse (T) and longitudinal (L) polarization states of the virtual photon, respectively. In the dipole model, the total γ*p cross-section is given by
(11)σT,Lγ*p=∑f∫d2rT∫01dz[Ψ*Ψ]T,Lf(z,Q2)σqq-(x,rT),
where z, rT, Ψ, f, and σqq-(x,rT) denote the fraction of longitudinal momentum of the virtual photon carried by the quark, the transverse separation between the quark and the antiquark, the light-cone wavefunction for the photon splitting into a qq- pair [51, 52], the flavor of the quark, and the dipole-proton cross-section, respectively. By using the optical theorem, σqq-(x,rT) can be obtained from the elastic dipole-proton scattering amplitude 𝒜(x,rT,Δ),
(12)σqq-(x,rT)=2Im𝒜(x,rT,Δ)=2∫d2bT𝒩(x,rT,bT)=σ0𝒩(x,rT),
where 𝒩(x,rT,bT), bT, and σ0 denote the imaginary part of the forward elastic dipole-proton scattering amplitude, the impact parameter of the dipole-target collisions, and the normalization constant obtained by the bT integration, respectively.To see the dependence of structure functionF2 on x, in Figure 3 we show the comparisons between the experimental data (squares) in ep collision [53–55] and the results by solving rcBK equation with the two different initial conditions, AAMQS (the solid curves) and quartic action (the dashed-dotted curves), by using (2) and (3). Figures 3(a), 3(b), 3(c), and 3(d) are the results for different values of Q2. In the calculation, we take σ0=30.265mb, Q0s2=0.168GeV2, C=1.715, and αfr=1 in the running coupling in (2) for the four Q2 values in Figure 4. One can see that the two different initial conditions describe well the F2 data, although their behaviors in Figure 1 on 𝒩A(r,x0) at small rQs(x0) are different.Proton structure functionF2(x,Q2) versus x at different Q2. The squares represent the experimental data of the HERA Collaboration [53–55]. The solid and dashed-dotted curves represent the results calculated by the AAMQS and quartic action initial conditions, respectively.
(a)
(b)
(c)
(d)Dependence of the dipole scattering amplitudeNA(F)(kT,x), the Fourier transform of 1-𝒩A(F)(r,Y) (15), on kT for (a) a proton target and (b) a nucleus target with A=200 at different rapidities with two different initial conditions indicated in the figure. From the upper curve to the lowest one, the corresponding rapidities are y=2, 4, and 8, respectively.
(a)
(b)
(c)
(d)


## 3. Single Inclusive Hadron Production

### 3.1. Hybrid Formalisms

In asymmetric collisions such aspA to hadron (h) and others (X) at high energy, the scattering occurs between a dilute system of parton and a dense one of nucleus [56, 57]. The cross section for single inclusive hadron production at forward rapidity is given by [58]
(13)dNpA→hXd2pTdη=K(2π)2[(x1ξ,Q)∫xF1dzz2×[x1fg(x1,Q2)NA(x2,pTz)Dh/g(z,Q)+Σqx1fq(x1,Q2)×NF(x2,pTz)Dh/q(z,Q)]+∫xF1dzz2αs2π2z4pT4×∫kT2<Q2d2kTkT2NF(kT,x2)×∫x11dξξ×∑i,j=q,q-,gwi/j(ξ)Pi/j(ξ)x1×fj(x1ξ,Q)Dh/i(z,Q){∫xF1dzz2}],
where the quantities K, ξ, kT, η, and pT denote the normalization constant, the splitting fraction for a given parton splitting into two others, the small transverse momentum which is strongly affected by target saturation effects, the pseudorapidity of the produced hadron, and the transverse momentum of the considered hadron, respectively; the functions fi(x,Q2), Dh/i(z,Q), wi/j(ξ), and Pi/j(ξ) are the parton distribution functions of the incoming proton which depends on the light-cone momentum fraction x and the hard scale Q, the hadron fragmentation function of the ith parton to the final hadron h with a momentum fraction z, the inelastic weight function, and the Altarelli and Parisi [59] splitting function, respectively. The last two functions can be found in equations (74–77) of [58] and equations (16.26–16.29) of [60] for more details. The longitudinal momentum fractions x1 and x2 are defined as follows:
(14)x1=xFz≈pTzsNNeη,x2=x1e-2η≈pTzsNNe-η,
where xF is the Feynman variable in another form. In the above calculation, the hadron masses are neglected since we are only interested in the light hadron production at high pT, which results the rapidity to equal approximately to the pseudorapidity.  As in the two-dimensional Fourier transform, the amplitude NA(NF) in (13) can be given by
(15)NA(F)(kT,x)=∫d2r→e-ik→T·r→(1-𝒩A(F)(r,Y=ln(x0x))).The first integral in (13) denotes the elastic term which was derived in [23] and corresponds to the scattering of collinear partons from the projectile on the target. In the process corresponding to the elastic term, a quark from the projectile scatters on the target and radiates a gluon either before or after the scattering. The incoming quark, the outgoing quark, and the radiated gluon can all multiply scatters on the target. Because the incoming parton initially has zero transverse momentum, but picks up transverse momentum of order Qs after multiply scattering on the target, the elastic term is the most important one. The other part in (13) denotes the inelastic term which was derived in [58] and is important only when the produced hadron has a much higher transverse momentum than Qs. The inelastic term corresponds to high transverse momentum partons radiated from the incoming partons in the projectile wave function. The partons with high transverse momentum may scatter off the target with only a small momentum transfer, and finally they are fragmentated into high-pT hadrons.We show the dipole scattering amplitudeNA(F)(kT,x) and the Fourier transform of 1-𝒩A(F)(r,Y) (15), for a proton target and a nucleus target with A=200 in Figures 4(a) and 4(c), as well as Figures 4(b) and 4(d), respectively. The solid and dashed-dotted curves represents the results of the AAMQS with γ=1.119 and quartic action, respectively. In Figures 4(a) and 4(c), we take Q0s2=0.168GeV2; in Figures 4(b) and 4(d), we take Q0s2=0.5GeV2. From the top curve to the lowest one, the corresponding rapidities are y=2, 4, and 8, respectively.  One can see that in most of the cases the differences between the two types of results obtained using the two initial conditions are small. The value of NA(F)(kT,x) for a proton target is much larger than that of NA(F)(kT,x)/A1/3 for a nucleus with A=200.


### 3.2. Numerical Results and Predictions

To give the numerical results of the hybrid formula, we can use the CT10 NLO [61] parton distribution functions and the DSS NLO [56, 62] fragmentation functions. The only input for the rcBK equation is the initial conditions for the evolution of the dipole amplitude. Our numerical results will be given by using the AAMQS and the quartic action, respectively. As already mentioned previously, the inelastic term in (13) involves an additional coupling factor αs at NLO or named a running coupling scale which is considered as a free parameter in [57]. In this work, we take αs=0.1 which is the same as that in [57].Our aim is not to fit the experimental data but to use the CGC theoretical tools available in low-x physics to highlight the uncertainties involved in making robust predictions for the upcoming pA collisions at the LHC. In all the calculations, we take K=1 in (13). However, the STAR data at very forward pseudorapidity (η=4) shows that a K-factor of ~0.4÷0.6 may be needed (see [27] for more details). The necessity of such a small value of K-factor may correspond to a large-x phenomena, which is not included in the CGC at very forward (pseudo)rapidity region.In Figure5 we show the single inclusive hadron production yields (transverse momentum distributions) in sNN=0.2TeV pp collisions at different pseudorapidities using the rcBK dipole solution with two different initial conditions, AAMQS (the solid curves) and quartic action (the dashed-dotted curves). The symbols (triangles, circles, and squares) represent the experimental data of the STAR [63] and BRAHMS collaborations [64] for different hadrons (π0 [63], h- [64], and h- [64]) at different pseudorapidities (4.0, 3.2, and 2.2) with different magnifying multiples (1, 50, and 200) in the distributions as shown in the figures. As we have already pointed out previously, the value of Q0s at x0=0.01 for proton target in the range 0.168÷0.336GeV2 gives a consistent description of the experimental data from the RHIC. We then choose Q0s2=0.168GeV2 (Figure 5(a)) and Q0s2=0.336GeV2 (Figure 5(b)) as input parameters for rcBK equation with the two different initial conditions. From  Figure 5 we see that the results of the two initial conditions are similar to each other and approximately in agreement with the data.Neutral pion and negatively charged hadron yields inpp collisions at sNN=0.2TeV. The symbols represent the experimental data at different pseudorapidities measured by the STAR [56] and BRAHMS Collaborations [57]. The solid and dashed-dotted curves represent the results calculated by the AAMQS and quartic action, respectively, with Q0s2=0.168GeV2 (a) and 0.336GeV2 (b) in the calculations.
(a)
(b)In Figure6, we compare our calculated results of pT distributions for π0 and h- produced in minimum-bias d+Au collisions at sNN=0.2TeV with the experimental data of the STAR [63] and BRAHMS collaborations [64] at the RHIC. Except the interacting system and the values of Q0s2, other illuminations are the same as those for Figure 5. According to [27, 57], the value of saturation scale for gold within Q0s2=0.5÷0.67GeV2 is consistent with the RHIC data. It is difficult to pin down the exact value of Q0s for nuclei since the DIS data for nuclear targets are limited and have large experimental uncertainties. In this case, we choose Q0s2=0.5GeV2 for Figure 6(a) and Q0s2=0.67GeV2 for Figure 6(b) as input parameters for rcBK equation with the two different initial conditions. We see that at Q0s2=0.5GeV2 the quartic action seems better and at Q0s2=0.67GeV2 the AAMQS seems better when we compare them with the experimental data of the STAR [63] and BRAHMS collaborations [64].Same as Figure3, but for minimum-bias d+Au collisions with Q0s2=0.5GeV2 (a) and 0.67GeV2 (b) in the calculations.
(a)
(b)As shown in Figures5 and 6, the single inclusive hadron production yields at different forward pseudorapidities in pp and minimum-bias d+Au collisions at the RHIC energy are approximately described by the rcBK dipole solution with the two different initial conditions. We notice that the two different initial conditions are matched together at very forward pseudorapidity (η=4). However, the discrepancy goes large when the pseudorapidity goes small, which would be caused by the dynamical evolution with increasing the nuclear mass number.We now present our predictions for single inclusive hadron production at the LHC in terms of the nuclear modification factorRpA and hope that some of the theoretical uncertainties, such as the sensitivity to K-factor, will be reduced. The nuclear modification factor RpA is defined as
(16)RpA=1NcolldNpA→hX/d2pTdηdNpp→hX/d2pTdη,
where Ncoll is the number of binary nucleon-nucleon collisions for a given centrality class in pA collisions, and dNpA→hX/d2pTdη and dNpp→hX/d2pTdη denote the single inclusive hadron productions in pA and pp collisions, respectively. We take Ncoll=6.5 and 7.4 for minimum-bias p+Pb collisions at sNN=4.4 and 8.8 TeV, respectively, according to a standard Monte Carlo Glauber model [65].In Figure7 we show the nuclear modification factor RpAch for inclusive charged hadrons h++h- production in minimum-bias p+Pb (A=Pb in pA) collisions at sNN=4.4TeV (Figure 7(a) with Ncoll=6.5) and sNN=8.8TeV (Figure 7(b) with Ncoll=7.4) with η=4, 5, 6, and 7 (from the upper curve to the lower one). All the results are obtained by using the solution of the rcBK dipole evolution equation with two different initial conditions, AAMQS (the solid curves) and quartic action (the dashed-dotted curves). In both panels, we assume the initial saturation scales of proton and nucleus to be Q0s2=0.168GeV2 and Q0s2=0.5GeV2, respectively. From Figure 7 one can see that RpAch decreases with increasing the pseudorapidity. This is explained from nonlinear evolution which suppresses additional gluon emissions in dense wave functions as compared to the dilute limit. The relationship between RpAch and sNN is weak in the LHC energy region. We note also from Figure 7 that the quartic action as the solution of rcBK equation leads to a faster rise of RpAch with transverse momentum, and the corresponding behavior is obviously different from AAMQS. This is because the particle production at high pT in pA collisions is mainly dominated by a high valence color charge density in the dense target.Nuclear modification factorRpAch versus pT for inclusive charged hadrons h++h- production in minimum-bias p+Pb (A=Pb in pA) collisions at the LHC energies, with the pseudorapidity being 4, 5, 6, and 7 from the upper curve to lower one. The solid and dashed-dotted curves represent the results calculated by the AAMQS and quartic action, respectively, with Q0s2=0.168GeV2 for p and 0.5GeV2 for Pb in the calculations. (a) corresponds to sNN=4.4TeV with Ncoll=6.5, and (b) corresponds to sNN=8.8TeV with Ncoll=7.4.
(a)
(b)Because the DIS data for nuclear targets are limited and have large experimental uncertainties, we calculate the possible ranges ofRpAch for forward rapidities in minimum-bias p+Pb collisions at two energies using two initial saturation momentum scales with two initial conditions, AAMQS (the double solid curves) and quartic action (the double dashed-dotted curves), in Figure 8 (sNN=4.4TeV) and Figure 9 (sNN=8.8TeV), respectively. The lower curves correspond to Q0s2=0.5GeV2 and the upper curves correspond to Q0s2=0.67GeV2 for nuclear target. We note that there are large uncertainties in RpAch at small pseudorapidity at the LHC due to the choice of the initial saturation scale for the rcBK evolution equation. In particular, if we choose the special non-Gaussian action (quartic action) which involves the subleading correction of the valence color charges, the discrepancy will be clearly larger compared to the other one (AAMQS).Nuclear modification factorRpAch versus pT for inclusive charged hadrons h++h- production in minimum-bias p+Pb collisions at sNN=4.4TeV with different η. The double solid curves and the double dashed-dotted curves are calculated, respectively, by the AAMQS and quartic action, which represents the uncertainty ranges in RpAch when Q0s2=0.5 to 0.67GeV2.
(a)
(b)
(c)
(d)Same as Figure8, but for the results at sNN=8.8TeV.
(a)
(b)
(c)
(d)It should be noted that the particle production cross-section given by (13) is right at forward rapidity. If we calculate the particle production at central rapidity, we need another formula, the kt-factorization formalism [66–68]. This will dilute our comparison on the two initial conditions. To give prominence to the comparison, we have used only the hybrid formalism (13) at forward rapidity in this paper.


## 3.1. Hybrid Formalisms

In asymmetric collisions such aspA to hadron (h) and others (X) at high energy, the scattering occurs between a dilute system of parton and a dense one of nucleus [56, 57]. The cross section for single inclusive hadron production at forward rapidity is given by [58]
(13)dNpA→hXd2pTdη=K(2π)2[(x1ξ,Q)∫xF1dzz2×[x1fg(x1,Q2)NA(x2,pTz)Dh/g(z,Q)+Σqx1fq(x1,Q2)×NF(x2,pTz)Dh/q(z,Q)]+∫xF1dzz2αs2π2z4pT4×∫kT2<Q2d2kTkT2NF(kT,x2)×∫x11dξξ×∑i,j=q,q-,gwi/j(ξ)Pi/j(ξ)x1×fj(x1ξ,Q)Dh/i(z,Q){∫xF1dzz2}],
where the quantities K, ξ, kT, η, and pT denote the normalization constant, the splitting fraction for a given parton splitting into two others, the small transverse momentum which is strongly affected by target saturation effects, the pseudorapidity of the produced hadron, and the transverse momentum of the considered hadron, respectively; the functions fi(x,Q2), Dh/i(z,Q), wi/j(ξ), and Pi/j(ξ) are the parton distribution functions of the incoming proton which depends on the light-cone momentum fraction x and the hard scale Q, the hadron fragmentation function of the ith parton to the final hadron h with a momentum fraction z, the inelastic weight function, and the Altarelli and Parisi [59] splitting function, respectively. The last two functions can be found in equations (74–77) of [58] and equations (16.26–16.29) of [60] for more details. The longitudinal momentum fractions x1 and x2 are defined as follows:
(14)x1=xFz≈pTzsNNeη,x2=x1e-2η≈pTzsNNe-η,
where xF is the Feynman variable in another form. In the above calculation, the hadron masses are neglected since we are only interested in the light hadron production at high pT, which results the rapidity to equal approximately to the pseudorapidity.  As in the two-dimensional Fourier transform, the amplitude NA(NF) in (13) can be given by
(15)NA(F)(kT,x)=∫d2r→e-ik→T·r→(1-𝒩A(F)(r,Y=ln(x0x))).The first integral in (13) denotes the elastic term which was derived in [23] and corresponds to the scattering of collinear partons from the projectile on the target. In the process corresponding to the elastic term, a quark from the projectile scatters on the target and radiates a gluon either before or after the scattering. The incoming quark, the outgoing quark, and the radiated gluon can all multiply scatters on the target. Because the incoming parton initially has zero transverse momentum, but picks up transverse momentum of order Qs after multiply scattering on the target, the elastic term is the most important one. The other part in (13) denotes the inelastic term which was derived in [58] and is important only when the produced hadron has a much higher transverse momentum than Qs. The inelastic term corresponds to high transverse momentum partons radiated from the incoming partons in the projectile wave function. The partons with high transverse momentum may scatter off the target with only a small momentum transfer, and finally they are fragmentated into high-pT hadrons.We show the dipole scattering amplitudeNA(F)(kT,x) and the Fourier transform of 1-𝒩A(F)(r,Y) (15), for a proton target and a nucleus target with A=200 in Figures 4(a) and 4(c), as well as Figures 4(b) and 4(d), respectively. The solid and dashed-dotted curves represents the results of the AAMQS with γ=1.119 and quartic action, respectively. In Figures 4(a) and 4(c), we take Q0s2=0.168GeV2; in Figures 4(b) and 4(d), we take Q0s2=0.5GeV2. From the top curve to the lowest one, the corresponding rapidities are y=2, 4, and 8, respectively.  One can see that in most of the cases the differences between the two types of results obtained using the two initial conditions are small. The value of NA(F)(kT,x) for a proton target is much larger than that of NA(F)(kT,x)/A1/3 for a nucleus with A=200.


## 3.2. Numerical Results and Predictions

To give the numerical results of the hybrid formula, we can use the CT10 NLO [61] parton distribution functions and the DSS NLO [56, 62] fragmentation functions. The only input for the rcBK equation is the initial conditions for the evolution of the dipole amplitude. Our numerical results will be given by using the AAMQS and the quartic action, respectively. As already mentioned previously, the inelastic term in (13) involves an additional coupling factor αs at NLO or named a running coupling scale which is considered as a free parameter in [57]. In this work, we take αs=0.1 which is the same as that in [57].Our aim is not to fit the experimental data but to use the CGC theoretical tools available in low-x physics to highlight the uncertainties involved in making robust predictions for the upcoming pA collisions at the LHC. In all the calculations, we take K=1 in (13). However, the STAR data at very forward pseudorapidity (η=4) shows that a K-factor of ~0.4÷0.6 may be needed (see [27] for more details). The necessity of such a small value of K-factor may correspond to a large-x phenomena, which is not included in the CGC at very forward (pseudo)rapidity region.In Figure5 we show the single inclusive hadron production yields (transverse momentum distributions) in sNN=0.2TeV pp collisions at different pseudorapidities using the rcBK dipole solution with two different initial conditions, AAMQS (the solid curves) and quartic action (the dashed-dotted curves). The symbols (triangles, circles, and squares) represent the experimental data of the STAR [63] and BRAHMS collaborations [64] for different hadrons (π0 [63], h- [64], and h- [64]) at different pseudorapidities (4.0, 3.2, and 2.2) with different magnifying multiples (1, 50, and 200) in the distributions as shown in the figures. As we have already pointed out previously, the value of Q0s at x0=0.01 for proton target in the range 0.168÷0.336GeV2 gives a consistent description of the experimental data from the RHIC. We then choose Q0s2=0.168GeV2 (Figure 5(a)) and Q0s2=0.336GeV2 (Figure 5(b)) as input parameters for rcBK equation with the two different initial conditions. From  Figure 5 we see that the results of the two initial conditions are similar to each other and approximately in agreement with the data.Neutral pion and negatively charged hadron yields inpp collisions at sNN=0.2TeV. The symbols represent the experimental data at different pseudorapidities measured by the STAR [56] and BRAHMS Collaborations [57]. The solid and dashed-dotted curves represent the results calculated by the AAMQS and quartic action, respectively, with Q0s2=0.168GeV2 (a) and 0.336GeV2 (b) in the calculations.
(a)
(b)In Figure6, we compare our calculated results of pT distributions for π0 and h- produced in minimum-bias d+Au collisions at sNN=0.2TeV with the experimental data of the STAR [63] and BRAHMS collaborations [64] at the RHIC. Except the interacting system and the values of Q0s2, other illuminations are the same as those for Figure 5. According to [27, 57], the value of saturation scale for gold within Q0s2=0.5÷0.67GeV2 is consistent with the RHIC data. It is difficult to pin down the exact value of Q0s for nuclei since the DIS data for nuclear targets are limited and have large experimental uncertainties. In this case, we choose Q0s2=0.5GeV2 for Figure 6(a) and Q0s2=0.67GeV2 for Figure 6(b) as input parameters for rcBK equation with the two different initial conditions. We see that at Q0s2=0.5GeV2 the quartic action seems better and at Q0s2=0.67GeV2 the AAMQS seems better when we compare them with the experimental data of the STAR [63] and BRAHMS collaborations [64].Same as Figure3, but for minimum-bias d+Au collisions with Q0s2=0.5GeV2 (a) and 0.67GeV2 (b) in the calculations.
(a)
(b)As shown in Figures5 and 6, the single inclusive hadron production yields at different forward pseudorapidities in pp and minimum-bias d+Au collisions at the RHIC energy are approximately described by the rcBK dipole solution with the two different initial conditions. We notice that the two different initial conditions are matched together at very forward pseudorapidity (η=4). However, the discrepancy goes large when the pseudorapidity goes small, which would be caused by the dynamical evolution with increasing the nuclear mass number.We now present our predictions for single inclusive hadron production at the LHC in terms of the nuclear modification factorRpA and hope that some of the theoretical uncertainties, such as the sensitivity to K-factor, will be reduced. The nuclear modification factor RpA is defined as
(16)RpA=1NcolldNpA→hX/d2pTdηdNpp→hX/d2pTdη,
where Ncoll is the number of binary nucleon-nucleon collisions for a given centrality class in pA collisions, and dNpA→hX/d2pTdη and dNpp→hX/d2pTdη denote the single inclusive hadron productions in pA and pp collisions, respectively. We take Ncoll=6.5 and 7.4 for minimum-bias p+Pb collisions at sNN=4.4 and 8.8 TeV, respectively, according to a standard Monte Carlo Glauber model [65].In Figure7 we show the nuclear modification factor RpAch for inclusive charged hadrons h++h- production in minimum-bias p+Pb (A=Pb in pA) collisions at sNN=4.4TeV (Figure 7(a) with Ncoll=6.5) and sNN=8.8TeV (Figure 7(b) with Ncoll=7.4) with η=4, 5, 6, and 7 (from the upper curve to the lower one). All the results are obtained by using the solution of the rcBK dipole evolution equation with two different initial conditions, AAMQS (the solid curves) and quartic action (the dashed-dotted curves). In both panels, we assume the initial saturation scales of proton and nucleus to be Q0s2=0.168GeV2 and Q0s2=0.5GeV2, respectively. From Figure 7 one can see that RpAch decreases with increasing the pseudorapidity. This is explained from nonlinear evolution which suppresses additional gluon emissions in dense wave functions as compared to the dilute limit. The relationship between RpAch and sNN is weak in the LHC energy region. We note also from Figure 7 that the quartic action as the solution of rcBK equation leads to a faster rise of RpAch with transverse momentum, and the corresponding behavior is obviously different from AAMQS. This is because the particle production at high pT in pA collisions is mainly dominated by a high valence color charge density in the dense target.Nuclear modification factorRpAch versus pT for inclusive charged hadrons h++h- production in minimum-bias p+Pb (A=Pb in pA) collisions at the LHC energies, with the pseudorapidity being 4, 5, 6, and 7 from the upper curve to lower one. The solid and dashed-dotted curves represent the results calculated by the AAMQS and quartic action, respectively, with Q0s2=0.168GeV2 for p and 0.5GeV2 for Pb in the calculations. (a) corresponds to sNN=4.4TeV with Ncoll=6.5, and (b) corresponds to sNN=8.8TeV with Ncoll=7.4.
(a)
(b)Because the DIS data for nuclear targets are limited and have large experimental uncertainties, we calculate the possible ranges ofRpAch for forward rapidities in minimum-bias p+Pb collisions at two energies using two initial saturation momentum scales with two initial conditions, AAMQS (the double solid curves) and quartic action (the double dashed-dotted curves), in Figure 8 (sNN=4.4TeV) and Figure 9 (sNN=8.8TeV), respectively. The lower curves correspond to Q0s2=0.5GeV2 and the upper curves correspond to Q0s2=0.67GeV2 for nuclear target. We note that there are large uncertainties in RpAch at small pseudorapidity at the LHC due to the choice of the initial saturation scale for the rcBK evolution equation. In particular, if we choose the special non-Gaussian action (quartic action) which involves the subleading correction of the valence color charges, the discrepancy will be clearly larger compared to the other one (AAMQS).Nuclear modification factorRpAch versus pT for inclusive charged hadrons h++h- production in minimum-bias p+Pb collisions at sNN=4.4TeV with different η. The double solid curves and the double dashed-dotted curves are calculated, respectively, by the AAMQS and quartic action, which represents the uncertainty ranges in RpAch when Q0s2=0.5 to 0.67GeV2.
(a)
(b)
(c)
(d)Same as Figure8, but for the results at sNN=8.8TeV.
(a)
(b)
(c)
(d)It should be noted that the particle production cross-section given by (13) is right at forward rapidity. If we calculate the particle production at central rapidity, we need another formula, the kt-factorization formalism [66–68]. This will dilute our comparison on the two initial conditions. To give prominence to the comparison, we have used only the hybrid formalism (13) at forward rapidity in this paper.


## 4. Conclusions

In this work we have used quantitatively a new initial condition (the quartic action initial condition) as input for the rcBK equation [18, 19] and presented a description of the hadron yields measured at three different pseudorapidities in pp and d+Au collisions at the RHIC by using the hybrid formalism which is proposed in [58]. The results calculated by the quartic action and AAMQS models [36–38] have been compared to each other and some interesting results have been obtained.At medium and largerQs(x0), the quartic action gives a comparable result of the scattering amplitude for an adjoint dipole with the MV [41, 42] and AAMQS models. At small rQs(x0), the quartic action gives a much lower scattering amplitude which is beyond the MV and AAMQS models. The results on the proton structure function F2(x,Q2) calculated by the quartic action and AAMQS models are very similar and describe approximately the HERA data [53–55]. In a few cases, for example, Q2=8.5GeV2, the two models present almost the same results on F2(x,Q2).In the description of hadron transverse momentum distributions inpp collisions at the RHIC energy, both quartic action and AAMQS models present almost the same results at forward pseudorapidity. At the closely mid-pseudorapidity, the results calculated by the two models are similar. In d+Au collisions at the RHIC energy, both of the two models present almost the same results at forward pseudorapidity on the pT distribution. At the closely mid-pseudorapidity, the two modeling results have an obvious difference in the high pT region, which is caused by the dynamical evolution with increasing the nuclear mass number.Based on the comparisons with the experimental data at the RHIC energy, the nuclear modification factorRpAch for charged hadrons produced in pA collisions at the LHC energies is predicted by using the previously mentioned two models. The calculated results show that RpAch decreases with increasing the pseudorapidity, and a large uncertainty in RpAch at small pseudorapidity is obtained. The quartic action as the solution of rcBK equation leads to a faster rise of RpAch with transverse momentum, and the quartic action shows a larger uncertainty than the AAMQS. These behaviors are related to the correlation and density fluctuation of the valence color charges. Our calculated results in the present work are consistent with the results given in recent works [25, 69].



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