The color dipole picture for prompt photon production in $pp$ and $pPb$ collisions at the CERN-LHC

A study on the prompt photon production within the QCD color dipole picture with emphasis in $pp$ and $pA$ collisions at the LHC energy regimes is performed. We present predictions for the differential cross section as a function of photon transverse momentum at different rapidity bins considering updated phenomenological color dipole models, which take into account the QCD gluon saturation physics. The results are directly compared to the recent experimental measurements provided by CMS and ATLAS Collaborations, showing a reasonable agreement in all rapidity bins with no free parameters. Special attention is given to the IPSAT model given its good description of the data in all rapidity bins from low- to high-$p_{T}$ ranges. As a result, a free-parameter approach has succeeded in describing the LHC data for prompt photon production, while new predictions for the 13-TeV data is presented in view of new data to confirm such prospect.


I. INTRODUCTION
The production of photons in hadronic collisions can be understood as a superposition of different sources of production, and isolation criteria are used to reduce the contamination by photons originating from certain production mechanisms. A photon produced in a hadronic collision is considered prompt (isolated and direct have the same meaning) when it does not originate from the decay of a hadron, such as π 0 or η, or when produced with a large transverse momentum, p T . Several data sets on prompt photon production have been collected over the years, covering a large domain of center-of-mass energy and also a wide range of photon rapidity and transverse momentum spectrum. For instance, inclusive measurements of prompt photons have been made at hadron colliders by ATLAS [1], CMS [2], CDF [3], and D∅ [4] Collaborations, making the comparison between predictions and experimental data a quite meaningful scenario.
In addition, a detailed understanding of prompt photon production is crucial to improve the knowledge both in experimental and theoretical sides. As such, the quantum chromodymamics (QCD) predictions for direct photons constitute an important background in the measurements of diphoton decay channel [5,6]. The study of prompt photons is a subject of investigation for a long time and can be related to the deep inelastic scattering (DIS), the Drell-Yan pair production, and the jet production as an important probe of QCD regimes.
Due the nature of the quark-photon vertex, measurements of their production cross sections have been proposed as a clean source of information about the QCD dynamics [7][8][9][10]. Since photons are colorless probes of the dynamics of quarks and gluons and interact electromagnetically only, they escape unchanged through the colored medium created in a high-energy collision. This becomes possible given that they are not sensitive to the QCD induced final-state interactions and hence leave the system without loss of energy and momentum. Therefore, they are considered a powerful probe to investigate the cold nuclear matter effects in the initial stage of the heavy-ion collisions [11]. Besides, studies about photon production in quark-gluon plasma (QGP), known as thermal photons, are also available in the literature (e.g., see Refs. [12,13]).
From the theoretical side, a treatment in the context of the QCD color dipole approach [14] can describe -within the same framework -both direct photon and Drell-Yan pair production processes. The prompt photon production reaction can be seen in the target rest system, where the production mechanism resembles a bremsstrahlung [15]. Therefore, we can apply the color dipole formalism to describe the radiation processes [16]. Such a formulation includes all perturbative and non-perturbative radiation as well as higher-twist contributions. In the color dipole picture, the phenomenology is based on the universal dipole cross section, fitted to DIS data and successfully describing the DESY-HERA ep data for inclusive and exclusive processes. In high-energy collisions, or very low-x Bjorken variable, nonlinear QCD effects, such as gluon saturation, becomes relevant and should be taken into consideration. The growth of the gluon density at low-x regime can be controlled by gluon recombination effects with a transition region delimited by a x-dependent saturation scale, Q s (x). It is expected that the low-p T region be able to provide access to the saturation regime and allows to study spin-dependent and spin-averaged gluon densities (PDFs) of hadrons in a kinematic regime where the theoretical uncertainties from usual perturbative QCD (pQCD) are huge.
Summarizing the recent results on direct photons within the light-cone dipole picture, their azimuthal anisotropy has been identified with an orientation-dependent dipole cross section and it should contribute to the azimuthal asymmetry of direct photons in pA and AA collisions [17]. The orientation was given by an off-diagonal unintegrated gluon density (UGD) at leading order (LO) and in Ref. [17] has been modeled through an eikonal-inspired UGD. Recently, a next-to-leading order (NLO) calculation has been performed [18,19] in the scope of Color Glass Condensate (CGC) formalism and it was found that the contribution of the NLO channel is significantly larger than the LO one at central rapidities at the LHC energies using an UGD for protons based on CGC effective field theory. The similar case for pA collisions in the very same framework has been addressed in Ref. [20] (similar analysis also done in Ref. [21]). The role played by gluon saturation effects and the value of the anomalous dimension has been analyzed in [22] and authors further shown that Cronin enhancement of direct photons can survive at the LHC energy whether nuclear saturation scale acquires large values [23]. The size of finite coherence length (relevant for low energies as at RHIC) has been investigated in Ref. [24] using the Green function technique which incorporates the color transparency and quantum coherence effects. The seminal work of Ref. [25] treats the azimuthal correlations in photon-hadron production in pA collisions showing the large suppression of the away-side peak in photon-hadron correlations at forward rapidities.
Nuclear modification factor, R pA , and photon-hadron azimuthal correlations are predicted.
That work has promoted a series of further investigations using state-of-art phenomenology concerned to dipole-nucleus interaction (see, e.g., Refs. [26][27][28][29][30]). Additional studies on direct photons that take into account other approaches can be found in Refs. [31][32][33]. In this work, we perform calculations for direct photon production at large and intermediate p T in a wide rapidity range considering pp and pA collisions at the LHC. We update previous studies presented in Ref. [34], where semi-analytical expressions for invariant cross section is given for pp and pA collisions. In this context, the role played by the anomalous dimensions, γ s , in the transition between the saturation regime and large-p T (DGLAP-like regime) is clearly identified. In particular, the anomalous dimension at then saturation limit, γ s ≈ 0.76, is crucial to describe the low and intermediate p T region whereas the DGLAP limit, γ s → 1, describes correctly the large-p T photon spectrum. The situation is similar for the longitudinal structure function [36] and multiplicity of charged hadrons [37]. Here, we consider the state-of-art for the phenomenological models for the dipole-nucleus amplitude including its impact parameter dependence. We investigate the Glauber-Gribov approach for nuclear effects as well as the geometric scaling property. We believe that this quantitatively measures the theoretical uncertainties present in the invariant cross section in pA collisions. The main quantity of interest in this study is the nuclear saturation scale, Q s,A , that defines the onset of unitarity corrections for a nuclear case. There is an uncertainty of the order of 20% by considering different prescriptions for it and we will use the one extracted from DIS data for eA collision in the context of geometric scaling formalism applied to ion targets [38]. Such an approach will be directly compared to the calculation using Glauber-Gribov multiple scattering corrections.
The paper is organized as follows. In Sec. II we start by providing the theoretical information to compute the differential cross section within the QCD dipole color formalism.
Sec. III presents predictions that are compared to the recent measurements focusing in the LHC kinematic regime. Finally, in Sec. IV we summarize the main conclusions and propose future investigations.

II. THEORETICAL FRAMEWORK
In this work we consider the real photon production off protons and nuclear targets at high energies, where the color dipole system is adopted to describe this mechanism.
The emission of real photons is then treated as electromagnetic bremsstrahlung by a quark projectile, which interact with the color field of the target in the single gluon approximation, as seen in Fig. 1, with a photon emitted either before or after the quark-target interaction.
At the high energy limit, each of the diagrams in Fig. 1 is factorized into a vertex of the real photon production associated with the quark-target scattering amplitude, which takes part in the matrix element squared [14,15]. Hence, the real photon radiation process can be interpreted in terms of qq dipole scattering off the target. Considering the target as a proton, in Ref. [15] the differential cross section in terms of the photon transverse momentum p T is presented, taking the form After evaluated the integration over the final quark kinematics, only two radiation amplitudes contribute to the cross section, where r 1 and r 2 are the quark-photon transverse separations entering in σ dip . Moreover, the transverse displacements of the final quarks in the amplitudes are correspondinglyr 1 = αr 1 andr 2 = αr 2 [with ∆r = (r 1 −r 2 )]. The parameter α is the relative fraction of the quark momentum carried by the photon. The Bjorken variable x 1,2 is related to the projectile and target momenta, x 1,2 = p T √ s e ±y γ , where y γ is the photon rapidity and √ s is the collision center-of-mass energy. The light-cone wave function of the photon bremsstrahlung is given by where K 0,1 (x) are the modified Bessel function of the second kind. The auxiliary variable ǫ 2 = α 2 m 2 q depends on the effective quark mass, assumed to be m q = 0.2 GeV in our numerical calculations.
The hadronic cross section is obtained from the convolution of the elementary partonic cross section, Eq. (1), with the projectile structure function F p 2 [14,39], where µ 2 = p 2 T will be considered and a F p 2 parametrization presented in Ref. [40]. The Fourier integrals over r 1 and r 2 can be simplified to a one-dimensional integral over the dipole separation r, which was first derived in Ref. [41], The quantities I 1,2,3 are Hankel integral transforms of order 0 (I 1,2 ) and order 1 (I 3 ) given by: In the color transparency region, σ dip ∝ r 2 , the Hankel integrals can be analytically com-puted, resulting in: where the exact prefactors for GBW model (with γ eff = 1) can be found in Ref. [34]. Notice that in the absence of saturation, the dipole approach can be related to the QCD Compton process as demonstrated in Refs. [42,43].
For our purposes, we consider here some phenomenological models based on the idea of parton saturation in order to investigate the differences and uncertainties among them. In a general form, the dipole-proton cross section can be parametrized as follows where γ eff stands for the effective anomalous dimension and Q s is the saturation scale. For instance, in Golec-Biernat-Wsthoff (GBW) saturation model [47] one has γ eff = 1 and fitting parameters [48] using four quark flavors assuming the values σ 0 = 27.32 mb, x 0 = 0.42×10 −4 , and λ = 0.248. Another model that has the same form as Eq. (12) is the Boer-Utermann-Wessels (BUW) model [49]. In this model the effective anomalous dimension takes the form, where ω ≡ p T /Q s and the free parameters are given by a = 2.82 and b = 168 obtained from a fit to describe the RHIC data on hadron production. One common characteristic is that, for large p T , the dipole cross section in the BUW model reproduces the GBW predictions by using a different set of fitting parameters: γ s = 0.63, σ 0 = 21 mb, x 0 = 3.04 × 10 −4 , and λ = 0.288.
In high-energy collisions (or equivalent low-x regime) the effects of QCD parton evolution are present; in particular the effects coming from multiple parton scattering. In order to analyze the effect of QCD evolution in the dipole cross section, we add to our studies the Impact Parameter Saturation (IPSAT) model [50]. In this case, the dipole cross section depends on a gluon distribution evolved via DGLAP equation: where N is the dipole-nucleon scattering amplitude with a factorized impact-parameter dependence given by a Gaussian profile, T (b), for the proton The initial gluon distribution has the form, xg(x, µ 2 0 ) = A g x −λg (1−x) 6 , which is evolved from a scale µ 2 0 up to µ 2 using the DGLAP evolution equations without quarks, with µ 2 = 4/r 2 +µ 2 0 related to the dipole size r. The parameters are extracted from a fit to high-precision combined HERA data for the reduced cross section (see Ref. [51]).
One of the goals of this work is to estimate the cross section for prompt photon production in proton-nucleus collisions, where A is the nucleus atomic mass number. Within the QCD color dipole picture, there are basically two ways to implement the nuclear effects: geometric scaling property from parton saturation models and Glauber-Gribov formalism for nuclear shadowing. We refer to Ref. [38] as an example of using geometric scaling to include A-dependence in the scattering cross section. There, the authors have studied how experimental data on lepton-nucleon collisions constrain characteristic features of particle production in nuclear collisions, such as their dependence on √ s and on A. They have demonstrated that the cross section for DIS off nuclei, γ * A → X, can be written in terms of the cross section for DIS off nucleons, γ * p → X, assuming a dependence only on the scaling variable τ = Q 2 /Q s (x) instead of x and Q 2 separately. Then, the nuclear effects are absorbed into the saturation scale and on nucleus transverse area, S A = πR 2 A (compared to the nucleon one, S p = σ 0 /2 = πR 2 p ). Here, we assume that geometric scaling is valid in the dipole-nucleus amplitude, N A , and, consequently, this is translated into a A-dependence on prompt photon production cross section, being the saturation scaling in protons, Q s , replaced by a nuclear scaling, Q s,A , in the following way: which grows with the quotient 1/δ. The expression for the nuclear radius is R A = (1.12A 1/3 − 0.86A −1/3 ) fm, while the δ and πR 2 p are parameters determined by data, resulting in δ = 0.79 and πR 2 p = 1.55 fm 2 [38]. The very same ansatz has been considered also to describe data for exclusive vector meson production and DVCS at DESY-HERA as well as photonuclear γA cross section in meson production extracted from ultraperipheral collisions at the LHC [52].
On the other hand, we can use Glauber-Gribov formalism to write the dipole-nucleus scattering cross section in terms of the nuclear profile: with the thickness function, T A , computed from the Woods-Saxon distribution.
In next section we will use these phenomenological models to compute the p T and y γ distributions of direct photon production in pp/pA collisions at the LHC.

III. NUMERICAL RESULTS AND DISCUSSIONS
Let us present the predictions obtained with the QCD color dipole framework for prompt photon production in pp and pA collisions. We estimate the transverse momentum and rapidity distributions focusing at the LHC energies and using three phenomenological models for the dipole cross section discussed in the previous section (GBW, BUW, and IPSAT) with the corresponding introduction of nuclear effects via geometric scaling and Glauber-Gribov shadowing.
First, we present the numerical results for pp collisions at √ s = 13 TeV. Figure 2 shows the predictions for the inclusive prompt photon cross sections compared to the measurements from the CMS Collaboration [53]. The results for the differential cross section as a function of y γ and p T are computed considering four different rapidity bins: |y γ | < 0.8, 0.8 < |y γ | < 1.44, 1.57 < |y γ | < 2.1, and 2.1 < |y γ | < 2.5. The GBW (solid lines) and BUW (dashed lines) models predict slightly different results in p T < 300 GeV. Apparently, the GBW and BUW models improve the data description in these p T domain, however we can not distinguish between the models. On the other hand, taking p T > 300 GeV, the GBW results are similar to the BUW model as expected, since at large p T the effective anomalous dimension is identical in both models, namely γ eff = 1. The IPSAT results (dot-dashed lines) at p T < 300 GeV are in accordance with the GBW and BUW models, however, as p T increases, the IPSAT model is in better agreement with data. This improvement compared to the GBW and BUW models comes from the QCD evolution in µ 2 = p 2 T present in the IPSAT model. It does a better job at forward rapidities where smaller values of x are probed. However, at very forward rapidity, the GBW and BUW models are able to predict the correct shape and normalization of the p T -spectrum. In the calculations using IPSAT, we are using the small-r limit for the dipole-proton amplitude where the Hankel transform can be analytically solved, Eq. (10). This is justified by the fact that the typical dipole sizes being probed in direct photons are r ∝ 1/p T , which is sufficiently small at large p T considered here.
In Fig. 3 the predictions are compared to the experimental data from the ATLAS Collaboration [54]. The corresponding results for the differential cross section in terms of p T are obtained considering four distinct rapidity bins: |y γ | < 0.6, 0.6 < |y γ | < 1.37, 1.56 < |y γ | < 1.81, and 1.81 < |y γ | < 2.37. We have verified that we can not distinguish among the results for the three dipole cross section models at the kinematic range of p T 300 GeV. Furthermore, the GBW and BUW models overshoots the experimental data beyond p T > 300 GeV. As seen in the CMS data, the IPSAT model provides a good description of the ATLAS data in all rapidity bins, especially a better agreement at large p T . The general conclusion is that color dipole models are able to describe the LHC data at forward rapidities, even at the large p T range. integration is probing F 2 (x 1 , Q 2 ) at relatively small x at central rapidities, where x 1 = x 2 .
Hence, it is clear that the proton saturation scale in the p T range measured by the CMS and ATLAS detectors is quite smaller than the transverse momenta, Q 2 s ≪ p 2 T , and the color transparency approximation for the color dipole amplitude is quite well justified. The situation will change only for measurements that reach mild to small ranges in transverse momentum, namely p T 10 GeV. Based on the points raised in the present discussion, we propose a simple parametrization for the invariant cross section assuming color transparency in the dipole-target cross section and a DGLAP-like anomalous dimension, γ eff = 1. This allows to compute analytically the Hankel transforms in Eqs. (5-7) and in the massless quark limit, m 1 → 0, only the second term in Eq. (4) survives. Specifically, the non-vanishing contribution in the second term is proportional to an analytical function: I 1 ∝ σ 0 (αQ s ) 2 /p 4 T . Also, we can write a rough approximation for the nucleon structure function based on the GBW model, Therefore, the integration over α can be done, obtaining f (x 1 ) ≈ 1012 1989 − 4 17 x whereσ ∼ σ 2 0 /(64π 4 ) ≃ 0.31 mb/GeV 2 and f (x 1 ) is a well behaved function of x 1 resulting from α-integration, which is basically a constant for small x 1 , f (x 1 ≪ 1) = 0.509 (using λ = 0.248 ≈ 1/4). That limit occurs, for instance, at central rapidities, This scaling function closely resembles the universal multiplicity scaling for prompt photons investigated in Refs. [57][58][59], in which photon p T -spectra at low transverse momentum are scaled with charged hadron pseudorapidity density at midrapidity. In terms of p T and rapidity y γ , Eq. (21) results dσ/dy γ dp T ∝ ( √ s/p T ) 2λ f (y, p T )/p 3 T . To evaluate our predictions with the color dipole model, it is important to compare our results to recent calculations in the literature for the low-p T region. One of them is the full NLO computation of direct photon cross section in the CGC effective field theory presented in Ref. [18,19] (using UGD obtained from CGC formalism) considering energies of 2.76, 7, and 13 TeV. There, authors estimate a 15% systematic uncertainty in the calculations across several rapidity bins and an overall normalization factor K = 2.4 was used. Here, the kinematic phase-space in the region where the saturation corrections should be very important behaves as p T ∼ Q s (x). In Fig. 4 (left) we present our predictions for the low-p T region compared to the data collected by the CMS and ATLAS experiments at 2.76 and 7 TeV. One can see that all three color dipole models are able to describe the data in the four rapidity bins. Comparing these results to those presented in Fig. 3 of Ref. [19], labeled as CGC in Fig. 4, one finds a similar good description of the data, however Ref. [19] assumes a K-factor while our results are parameter-free in all rapidity bins. Based on this evidence, we also present our predictions for the prompt photon production at 13 TeV in three rapidity bins, which demonstrates the need for more data in order to confirm the good description provided by the color dipole models at a lower p T range.
In the following we present the results for prompt photon production in pP b collisions at √ s = 8.16 TeV, where the differential cross section in terms of p T is shown in Fig. 5. In this case, the experimental results are obtained taking into account three different rapidity bins: 1.09 < y * γ < 1.90, −1.84 < y * γ < 0.91, and −2.83 < y * γ < −2.02. The p T spectrum for the first bin, y γ ≈ 1.5, is probing x 2 ≤ 1.3 × 10 −2 and in the backward rapidity bin (y γ ≈ −2.4) large x is probed, x 2 ∼ 0.5. As a remark, the ATLAS data covers the region between small and large x (where the threshold is taken as x ≃ 10 −2 ). The theoretical predictions are compared to the experimental data from the ATLAS detector [55]. For p T < 50 GeV, the GBW and BUW models give predictions slightly below the experimental data points.
In this case the nuclear effects are introduced by geometric scaling property as discussed in previous section. However, in the kinematic range 50 < p T < 100 GeV such models have a better description of the data. At p T > 100 GeV, the results strongly deviate from   the experimental measurements. Once again, the IPSAT model does a good description at large p T in comparison to the GBW and BUW parametrizations, however IPSAT does not describe data in the negative rapidity bin −2.83 < y * γ < −2.02, for which IPSAT accounts a nuclear correction coming from the Glauber-Gribov shadowing. It is surprising that QCD dipole models still describe part of the p T spectrum correctly despite the large x 2 values involved in the measured kinematic range. In the case of IPSAT, the large x threshold is given by (1 − x 2 ) 6 in the input for the gluon distribution at initial scale.
It is timely to discuss now the uncertainty coming from the model for the nuclear saturation scale used in the geometric scaling predictions. Quantitatively, the nuclear saturation scale obtained from Eq. (18) is Q 2 s,P b ≈ 3Q 2 s,p for a Lead nucleus (A = 208). The value of Q s,A can change whether distinct treatments of the nuclear collision geometry are considered. As an example, using a local saturation scale, , with T A being the nuclear thickness function and a Gaussian b-profile, the relation between Q s,A and Q s,p is found in Ref. [56]. In the hard sphere approximation for the nuclear density ρ A , one has Q 2 s,A = 3A(R p /R A ) 2 Q 2 s,p , which produces Q 2 s,P b ≈ 2.3Q 2 s,p . Therefore, the typical theoretical uncertainty on the determination of the saturation scale compared to the proton one is ∼20%. The ATLAS measurement in pP b collisions in forward rapidities is scanning values of x 2 in the range 10 −3 x 2 10 −2 on the measured p T range. This implies in a   1.4 GeV 2 , which demonstrates that p T ≫ Q s,A as in the proton case. We postpone for a forthcoming publication the comparison of our prediction for the nuclear modification factor, R γ pA (y, p T ), with relevant studies in the literature concerning proton-nucleus collisions. As examples, we quote the calculation in Ref. [21] within the CGC framework using color dipole cross sections solved from the running coupling Balitsky-Kovchegov (BK) evolution equation and predictions in Ref. [20] using CGC formalism at level NLO and nuclear UGD.

IV. SUMMARY
We investigate the prompt photon production at small x in pp and pA collisions at the LHC energies at different rapidity bins. We show that direct photon production can be formulated in the QCD color dipole framework without any free parameter. In particular, we employ three dipole cross section models determined by recent phenomenological analysis of DIS data available from DESY-HERA. The predictions for pp and pA reactions have demonstrated that in the low-p T range we can not completely distinguish between GBW, BUW, and IPSAT models. Nonetheless, the IPSAT results provide a better description of the data at the high-p T range compared to the other color dipole models based on fixed or showing that the predictions undershoot the data as expected.
Therefore, our results encourage for additional improvements that may be taken into account to refine the corresponding phenomenology at the large p T spectrum if new data from the LHC energy regime become available. Furthermore, we propose that future measurements of prompt photon production in pp/pA/AA collisions may be performed at the current/future colliders, since these data could be a valuable tool to analyze the color dipole models as wells as constrain high energy QCD dynamics effects such as saturation physics in kinematic domain not yet explored.