Prompt photon production in proton collisions as a probe of parton scattering in high energy limit

We study the prompt photon hadroproduction at the LHC with the $k_T$-factorization approach and the $qg^* \to q\gamma$ and $g^*g^* \to q\bar q\gamma$ partonic channels, using three unintegrated gluon distributions which depend on gluon transverse momentum. They represent three different theoretical schemes which are usually considered in the $k_T$-factorization approach, known under the acronyms: KMR, CCFM and GBW gluon distributions. We find sensitivity of the calculated prompt photon transverse momentum distribution to the gluon transverse momentum distribution. The predictions obtained with the three approaches are compared to data, that allows to differentiate between them. We also discuss the significance of the two partonic channels, confronted with the expectations which are based on the applicability of the $k_T$-factorization scheme in the high energy approximation.

with recent data from the LHC. In addition, this analysis is an important step in our ongoing program to constrain the transverse momentum parton distributions with the LHC data, see Ref. [5] for the analysis of the Lam-Tung relation breaking in Z 0 hadroproduction at the LHC.
We improve on the previously known results by carefully comparing two possible realizations of the prompt photon production at the parton level with off-shell gluons g * : the 2 → 2 processes qg * → qγ and qg * → qγ, and the 2 → 3 process g * g * → qqγ. We impose in our analysis the photon isolation criteria of Frixione [31] and use the data from the LHC [32][33][34].
Hence, besides constraining the gluon distribution F g (x, k T , µ F ), we also focus on the comparison of two different partonic channels that should be close to each other at the leading logarithmic approximation for small and moderate values of parton momentum fractions x. These are already mentioned, the qg * → qγ and g * g * → qqγ channels with off-shell gluons. In general, the gluon distribution is larger than the sea quark distributions, and due to larger anomalous dimensions, the gluon distribution is more rapid and drives the evolution of the sea quark sea distributions in scale µ F and in parton x. Hence, one may approximate the sea quarks as coming from the gluon in the last splitting of the parton evolution. This approximation is a basis of a very successful dipole picture of high energy scattering, see e.g. [35,36]. Thus, one expects that the contribution with sea quarks, q sea g * → qγ, should be well represented as a part of the g * g * → qqγ contribution. This should hold true because the amplitude for g * g * → qqγ contains diagrams describing the g * → q sea splitting in the last step, and in addition subleading terms in the collinear limit. So, in principle one could expect an improved theoretical precision of the hard matrix element in the latter approach. Jumping to the conclusions, to some surprise, we find the opposite to be true. We view this result as an interesting theoretical puzzle and a strong inspiration to perform the complete NLO analysis of the process in the k T -factorization framework.
The prompt photon production at the LHC has been thoroughly analyzed within the collinear factorization framework up to the NNLO accuracy [37]. Using this approach, good agreement with the data was found. We do not expect better agreement in our analysis with the k Tfactorization approach since the process has been treated so far only at the tree level. We rather apply more phenomenologically minded logic to use the theoretically clean observables and precision data in order to refine details of the transverse momentum distribution of partons within the k T -factorization framework. Nevertheless, even at the tree level, we find a rather good description of the LHC data in one of the considered scenarios.
The paper is organized as follows. In section II we present the formalism for qg * → qγ and g * g * → qqγ channels. In section III we discuss the transverse momentum dependent gluon distributions used in the paper. In section IV we discuss the photon isolation criteria and present our results. We conclude in section V.

II. OVERVIEW OF THE PROCESS DESCRIPTION
We consider the prompt photon production in the pp scattering in the k T -factorization approach, depicted in Fig. 1. The incoming proton beam momenta, P 1 and P 2 , are consider to be light-like: P 2 1 = P 2 2 = 0 in the high energy approximation: S = (P 1 + P 2 ) 2 4m 2 p . The photon kinematics is parameterized with the help of Feynman variable x F and the transverse momentum q T . In the light-cone variables, the real photon momentum in the pp center-of-mass frame reads where q ± γ = q 0 γ ± q 3 γ . At the leading order (LO), the real photon may be produced in the 2 → 2 partonic channels: qg → qγ, qg → qγ and qq → gγ. At the NLO, these 2 → 2 processes receive one loop correction and the tree level 2 → 3 processes appear: gg → qqγ, qg → qgγ,qg →qgγ, qq → qqγ, qq → qqγ, qq →qqγ. In our approach, we apply the high energy approximation for all the incoming partons.
Since in this regime the gluon density is strongly dominant, it drives evolution of the sea quark densities. Therefore, we shall keep only the contributions with the maximal number of incoming gluons, i.e. qg → qγ andqg →qγ for the 2 → 2 processes and gg → qqγ for the 2 → 3 processes.
We apply the k T -factorization approach for these processes in which gluons carry non-zero transverse momentum and are off-shell: g → g * . For the incoming quarks and antiquarks, we neglect the transverse momentum and we use the collinear approximation. This setup is often called a hybrid factorization approach. The main motivation for this approximation scheme is our focus on the gluon transverse momentum distribution at small and moderate x.
It should be stressed that the available data on the prompt photon hadroproduction at the LHC extend from the kinematic region where the values of parton x are small, x ∼ 0.002, to larger x > 0.1. It is well known that the high energy approximation is best motivated in the former region. It was however argued in detail in Refs. [4,5] that the high energy factorization scheme may provide a good approximation of the high energy amplitudes also for moderate parton x.
The key argument used there is based on a detailed analysis of the values of exchanged gluon kinematics in the Sudakov decomposition. In general, the momentum k of the off-shell gluon exchanged in the t-channel between the proton with momentum P 1 and the hard interaction vertex may be written as k = xP 1 + βP 2 + k ⊥ , where the protons' momenta P 1 and P 2 are approximated to be light-like, and k ⊥ belongs to the plane orthogonal to P 1 and P 2 . Then, the quality of the high energy approximation crucially depends on the value of the Sudakov parameter β [4,5], which is typically much smaller than the value of parton x. These two values are correlated because they both are inversely proportional to the collision energy. The gluon x, however, is driven by the large invariant mass of the state produced in the hard collision, while the Sudakov parameter β is driven by a usually much smaller mass of the proton remnant.
Hence, β x, and the high energy approximation is still applicable at moderate x. For a detailed discussion of this issue and the gauge invariance problem in this approximation scheme for related Drell-Yan and Z 0 hadroproduction processes, see Ref. [5].
A. Photon production in the qg * partonic channel We start the description of the prompt photon hadroproduction in the k T -factorization approach with the 2 → 2 approximation, i.e. with the qg * → qγ andqg * →qγ channels. In what follows, we use the light cone coordinates of the photon polarization vectors corresponding to helicities σ = ±: where (±) The quark-gluon channel contribution to the photon production with rapidity y and transverse momentum q T , derived in [13,14,38], reads where k T = |k T |, e i are quark charges in units of the elementary charge e, q i are collinear distributions of quarks and antiquarks with flavors f andf (not multiplied by x). In addition, the photon Feynman variable reads while the gluon momentum fraction x g is obtained from kinematics of the parton scattering in the high energy limit where z = x F /x q . From now on, we keep the renormalization and factorization scales to be The function F g is the transverse momentum dependent (or unintegrated) gluon density, in the form fixed by the leading logarithmic relation to the collinear gluon density g(x, µ F ): All quarks are assumed to be massless when compared to the photon transverse momentum q T .
The diagonal in photon helicity impact factors are given bỹ where λ 1 and λ 2 are helicities of the incoming and outgoing quark, respectively. With the chosen photon polarization vectors (2), the functions are proportional to the photon emission amplitudes. After summation over quark and photon helicities, the cross-section for the real photon hadroproduction reads where the symmetrization y → −y since both the initial states, q(k 1 ) g * (k 2 ) and g * (k 1 ) q(k 2 ), contribute.
The integrand in (9) is singular for q T = zk T . This is a collinear singularity coming from the emitted photon momentum parallel to the outgoing quark momentum. In experiments, however, the photon measurement requires a separation from the outgoing hadrons (or jets).
Hence, the experimental data assume certain isolation cut, Θ Rγ (q γ , {p H i }), which depends on the photon momentum q γ , the hadron momenta {p H i } and the isolation cone parameter R γ . In the parton level formulas, the isolation criterion is implemented with outgoing parton momenta {p i } instead of the hadronic momenta {p H i }. Thus, in order to obtain a physically meaningful result, Eq. (9) must be complemented by a suitable cut Θ Rγ (q T , z, k T ). The detailed discussion of our implementation is given section (IV A).
FIG. 2: Partonic channels in γ hadroproduction: a) the q val g * → qγ channel and b) the g * g * → qqγ channel. The crossed blobs represent channel dependent hard scattering amplitudes.
The qg * → qγ channel described in the high energy limit within the k T -factorization framework was shown [13,14] to be equivalent to the dipole picture of high energy scattering, in close analogy to the dipole picture of the Drell-Yan process [13,14,27], see also [38,39] for recent developments. This picture is obtained by the Fourier transform of the impact factors to the impact parameter space, and the color dipole scattering amplitude emerges as a result of an interference between the initial and final state photon emission amplitudes. This approach has an advantage to be capable to efficiently include the effects of multiple scattering and it was explored for instance in recent analyses [25,26].
B. Photon production in the g * g * partonic channel In an alternative approach to the prompt photon production in the high energy approximation, one generates sea quarks from the gluon in the last splitting. With this assumption, the sea quark contributions q sea can be absorbed into the g * g * → qqγ hard matrix elements. The valence quark contribution q val , however, is not included in the g * g * channel, and these two contributions enter the cross section additively, see Fig. 2, The g * g * → qqγ hard subprocesses cross section is calculated in the k T -factorization framework. The scheme and details of the calculations follow closely more general calculations performed in the same setup for the Drell-Yan structure functions [5]. In fact, the formulas for the photon production may be recovered from those derived in Ref. [5] by removing leptonic part with boson propagator, taking the limit M → 0 for the Drell-Yan intermediate boson mass and considering only the diagonal in helicity structure functions for the transverse polarizations.
For completeness, we shortly repeat the main steps of these calculations adjusted to the photon hadroproduction case.
In the k T -factorization gluons are virtual, k 2 i −k 2 iT < 0, and quarks are taken to be massless, p 2 3 = p 4 4 = 0. The standard gluon momenta decomposition in the high energy limit is applied: k 1 = x 1 P 1 + k 1 ⊥ and k 2 = x 2 P 2 + k 2 ⊥ , see also the discussion in Sec. II, preceding Sec. II A. The parton level scattering amplitude is a sum of eight diagrams shown in Fig. 3. The high energy limit for virtual gluon polarizations is used, in which the virtual gluon polarization vectors π g * (k i ) are approximated by the so-called "nonsense polarizations", using the so-called Collins-Ellis trick [40] in the derivations, The impact factors T (i) where M (i) µ,αβ are the amplitudes for the diagrams shown in Fig. 3 with amputated polarization vectors of the incoming gluons and outgoing photon. The explicit expressions for the impact factors are given in Appendix of Ref. [5] .
The g * g * → qqγ impact factor, given by µ , is used to calculate the g * g * channel contributions cross-sections for photon production with helicity σ, with F g being the unintegrated gluon distribution introduced in the previous section and where summations are performed over quark flavors f (present through the charge e f in the , color indices of the gluons (a, b) and quarks (i 3 , i 4 ), and over the quark helicities (r 3 , r 4 ). The latter summation leads to traces over Dirac spinors which are evaluated with the FORM program for symbolic manipulations [41]. The resulting expressions, obtained in two independent calculations, are lengthy and do not need to be explicitly displayed. It was also checked that H σ ∼ k 2 iT when the gluon transverse momentum k 2 iT → 0, as required by the gauge invariance condition in the high energy limit.
The phase space for the final state particles of partonic scattering is parameterized in terms of the photon variables (y, q T ) and the variables (z, φ κ ) describing the qq kinematic configuration, where the variables (z, κ) are implicitly defined by the parameterization of the quark and antiquark momenta in which For the comparison with data it is necessary to integrate over the final state quark/antiquark kinematical variables and sum over the photon polarizations, .
Several parameterizations of the transverse momentum dependent gluon distribution in the proton, F g (x, k T , µ F ), were proposed. Many of them were derived in the regime of small-x dynamics, including solutions of the BFKL equation [6][7][8] or the Balitsky-Kovchegov (BK) [42][43][44] equation. The data for the prompt photon production, however, extend out of the small-x domain. Therefore, we consider parameterizations that may be used also for moderate values of x > 0.01. The two most widely used approaches having this feature are: the Kimber-Martin-Ryskin (KMR) approach [45,46] that permits to recover the unintegrated parton distributions from the collinear parton density functions and the approach based on the solutions of the CCFM equation [9][10][11][12], implemented by Jung and Hansson (JH) [47]. We also consider the gluon distribution from the Golec-Biernat-Wüsthoff (GBW) model [35], extended phenomenologically to the moderate values of x. All of these parameterizations were optimized to describe the HERA data on the proton structure function F 2 (for the GBW gluon distribution, the optimization was performed for x < 0.01 but not for its phenomenological extension).

A. KMR-AO gluon distribution
The KMR scheme was used in several variations. Our choice is the integral version of the KMR prescription [48] in which the unintegrated gluon distribution is given by where the function T a is called the Sudakov form factor In the above, f /f denote quark/antiquark flavors, P aa (z, µ) are the Altarelli-Parisi splitting functions given in terms of the expansion in α s (µ)/(2π) and D a (x, µ) are collinear parton distribution functions (PDFs). We choose the leading order splitting functions and the CT10 PDFs [49]. We consider the angular ordering version of the KMR distribution (KMR-AO) in which the function ∆ in the upper integration limits equals This prescription imposes angular ordering in the last step of the evolution [46]. We prefer to use the integral version of the KMR distribution over the differential one since it allows to avoid problems for large values of k T > Q, discussed in detail in [48].
The unintegrated gluon distribution in the all-loop CCFM scheme [12,50,51] takes into account small-x coherence effects which are reflected in angular ordering in gluon cascade branching.
This leads to a non-Sudakov form factor which screens the 1/z singularity in the P gg splitting function. The CCFM scheme was extended for gluons to the region of large x by taking into account terms with finite z in P gg , and the Sudakov form factor with angular ordering of the final state emissions. Infrared parameters in such a scheme were fitted to the HERA data on F 2 by Jung and Hansson in [47], which analysis offers the JH-2003 unintegrated gluon distribution.
The selection of the two above schemes is motivated by both their theoretical consistency for small and moderate x, and good description of data within the k T -factorization approach.

C. Extended GBW gluon distribution
The unintegrated gluon density F g from the GBW saturation model is given by where the the saturation scale Q 2 s = (x/x 0 ) −λ GeV 2 . We use the parameters from the recent refit of the GBW model done in [36]: We also compare the x-dependence of the gluon distributions. We choose to illustrate this dependence by evaluating the 'collinear integral' of the unintegrated gluon distributions This integral should reproduce the true collinear gluon distribution xg(x, µ 2 F ) in the leading logarithmic approximation. In our approach, however, the effects beyond the leading logarithmic approximation are also present, so differences between xg(x, µ 2 F ) and xg(x, µ 2 F ) should appear. In Fig. 5, the gluon distribution (23) computed for the three unintegrated gluon distributions are compared to the collinear NLO gluon distribution CT10 [49] from which the KMR-AO distribution is extracted. In general, the integrated KMR-AO gluon is closest to the collinear gluon distribution, reproducing quite well the x-dependence but having somewhat higher normaliza- by the postulated extrapolation factor (1 − x) 7 , that clearly is too mild for x ∈ (0.01, 0.1).

A. Isolation of γ
In order to perform the comparison of our numerical results with the LHC data, we need to perform photon isolation from jets in similar manner as it was done by the ATLAS and CMS experiments. To this end, we implement the Frixione cone cut at the parton level [31]. For the final state qqγ, we define the distances R i between the outgoing quarks, labeled by i = 3, 4, and the photon: where φ i are parton azimuthal angles and y i are their rapidity. We take R 0 = 0.4. The Frixione cone isolation procedure leads then to the following conditions which have to be satisfied by the kinematic configuration: • no constrains when R 0 < R 3 < R 4 , • if R 4 > R 3 then conditions (25) and (26) are applied with the indices 3 ↔ 4, • for simpler partonic final states, qγ andqγ, only condition (26) is imposed.

B. LHC data sets
In the comparison, we focus on the available sets of data that probe the smallest values of gluon x variable. Hence we choose the following measurements of the prompt photon production from the ATLAS and CMS experiments: • ATLAS data at √ S = 7 TeV with 15 GeV < q T < 100 GeV (ATLAS@7TeV) [32] • CMS data at √ S = 7 TeV with 25 GeV < q T < 400 GeV (CMS@7TeV) [33] • ATLAS data at √ S = 8 TeV with 25 GeV < q T < 1500 GeV (ATLAS@8TeV) [34] where q T is the photon transverse momentum. In these experiments the cross sections were measured as a function of q T in several rapidity intervals. In the following figures, the experimental data are compared to theoretical predictions obtained within the k T -factorization formalism in the two schemes described in the previous sections: the qg * scheme based on the qg * → qγ and qg * →qγ partonic channels, and the g * g * scheme based on a combination of the q val g * → qγ and g * g * → qqγ channels. For each theoretical scheme, we display the results obtained with the three unintegrated gluon distributions: KMR-AO, JH-2003 and GBW.

C. Comparison to data
In Fig. 6, we show the theory curves against the ATLAS@8TeV data for the central rapidity, |y| < 0.6, in the log-log scale. This is the way the experimental results are usually presented.
In the upper plot, the predictions are obtained in the qg * scheme while in the lower plot the predictions are computed in the g * g * scheme. Clearly, the qg * scheme with the KMR-AO gluon distribution gives the best description of the data. The same gluon distribution used in the g * g * scheme leads to a reasonable description of the data, but the cross section is somewhat schemes with the indicated unintegrated gluon distributions, compared to the ATLAS@8TeV data [34].
obtained within both the qg * and g * g * approaches and the KMR-AO and JH-2003 unintegrated gluon distributions.
In order to perform more detailed studies of the description quality, we change the way the data and theoretical results are presented. Namely, in Fig. 7 and Fig. 8 we display the ratios of the theoretical results to the experimental data. For the sake of the optimal comparison, the theoretical result are integrated over the bins used in the experiments. The theoretical uncertainties are due to the variation of the factorization scales, µ F = µ R , between q T /2 and The theory to data ratios obtained within the qg * scheme are shown in Fig. 7 for the three  considered LHC data sets. The absolute value of the photon rapidities |y| grows from the top row down. Clearly, the best overall description of the data is found with the KMR-AO gluon (green shaded bands). It is seen at best for the data set with the smallest errors, that is the ATLAS@8TeV data (right column). For most bins, these theoretical results are below the data  σ( q v a l g * → q γ) / σ( q g * → q γ) σ( q s e a g * → q γ) / σ( q g * → q γ) σ( g * g * → q q γ) / σ( q g * → q γ) and g * g * schemes. Plotted ratios of the partonic cross sections: σ(q val g * )/σ(qg * ), σ(q sea g * )/σ(qg * ), and σ(g * g * )/σ(qg * ).
scale uncertainty of these results is very small and does not allow to overlap with the data. The description of the CMS@7TeV data with q T < 400 GeV (middle column) with the KMR-AO gluon is also very good for all rapidities. The predictions for ATLAS@7TeV (left column) are above the central data points, but stays consistent with the data within the experimental errors, somewhat larger in this measurement.
The data description in Fig. 7 with the JH-2003 gluon (red hatched bands) is reasonable, but the application of this gluon is limited to q T < 100 GeV. The results calculated with JH-2003 are somewhat below the KMR-AO results and more away from the CMS@7TeV and ATLAS@8TeV data. This is interesting in the context of Fig. 5, where one sees a larger integrated gluon from JH-2003 than from KMR-AO distributions. This suggests that the lower prompt photon production cross sections from the JH-2003 distribution are due to the cut-off of larger transverse momenta, k T > µ F , that is not present in the KMR-AO case, see Fig. 4. This shows that the prompt photon data are sensitive to the shape of the transverse momentum gluon distribution, and that the longer k T -tail of the KMR-AO distribution is preferred by the data.
We also see that the uncertainty of the cross section due to scale variation is much larger for JH-2003 than for KMR-AO. The gluon from the GBW model does not reproduce well the q T -dependence of the data. The predictions are significantly below the data at the lower limits of q T and much above data at very large q T . One has to remember, however, that in our study the GBW model is used far outside the range of very small x values and moderate scales, where it was fitted and should work well. The deviations from the data reflect the tendency visible already in the integrated GBW gluon, see Fig. 5 and the corresponding discussion of its content.
Let us move now to the other approach, the g * g * scheme, which results are shown in Fig. 8.
Here the cross sections are obtained by adding the g * g * → qqγ and q val g * → qγ contributions.
Due to different dependencies on x of the valence quark and gluon distributions, the g * g * → qqγ is dominant at lower q T while q val g * → qγ takes over at larger q T . In the region where the gluon x is small or moderate, i.e. for q T < 50 GeV, the g * g * scheme gives a rather good description of all the data sets with the KMR-AO gluon. The scale uncertainty is significantly larger than it was for the qg * scheme. This approach, however, is consistently less successful for q T > 50 GeV. The biggest deviation occurs in the region of q T ∼ 300 − 500 GeV, where the theory to experiment ratio is at the level of 0.6 − 0.7. For larger q T , the theory results become dominated by the q val g * → qγ channel, and the results approach the results of the qg * scheme, see Fig. 9. Within its limits of applicability, the JH-2003 gluon gives rather similar results to the KMR-AO gluon, hence well consistent with the data. The approach based on the GBW gluon experiences strengthened problems due to inaccuracy of the used extrapolation beyond x > 0.01 and too mild scale dependence since the gluon distribution enters the hadronic cross sections twice in the g * g * scheme.
There is an interesting message coming from the comparison of the results in the qg * anf g * g * schemes. On the diagrammatic level, the g * g * contribution incorporates the q sea g * contribution, where the sea quark is produced in the last splitting. Therefore, one expects that the g * g * scheme should lead to results that are close to the qg * scheme. This is, however, not confirmed by the data at larger q T . In order to provide more insight into this problem, in Fig. 9 we show the partonic components of the cross sections in both the schemes. We plot the ratios: σ(q val g * )/σ(qg * ), σ(q sea g * )/σ(qg * ) and σ(g * g * )/σ(qg * ), where the reference cross section σ(qg * ) = σ(q sea g * ) + σ(q val g * ) is obtained in the qg * scheme. Note that σ(q val g * ) is the same in both schemes.
Clearly, at lower q T , that is for smaller values of the parton x, the q sea g * or g * g * contributions strongly dominate over the q val g * channel. For q T ∼ 200 − 600 GeV, depending on the scheme, the q val g * channel is leading. Interestingly enough, both for the KMR-AO and JH-2003 gluons, the g * g * channel is larger at the lowest q T = 10 GeV, but it decreases faster with q T than the q sea g * channel, and at q T ∼ 1000 GeV the difference is already very pronounced. At this stage, we do not have good understanding of this behavior. One could attempt to connect this deviation to going out of the small x domain, but this simple explanation is undermined by the fact that the results obtained in the qg * scheme stay close to data even at q T ∼ 1000 GeV, where the gluon x is not small. Hence we consider this problem to be an interesting theoretical puzzle that calls for explanation.

V. CONCLUSIONS
In this paper, we have analyzed the prompt photon hadroproduction at the LHC using the k T -factorization approach with the qg * → qγ and g * g * → qqγ partonic channels.The data from the ATLAS and CMS collaboration were considered, obtained at With the qg * partonic channel and the KMR-AO gluon distribution, the best overall description of the data was obtained with typical accuracy of 10 − 20%. The results obtained with the JH-2003 gluon were found to be slightly less accurate in the region of their applicability with q T < 100 GeV. The GBW gluon does not provide satisfactory description beyond this region and for central values of the photon rapidity, which shows the expected region of its validity due to small-x nature of this distribution.
We point out that our findings indicate that the precision prompt photon data allow for constraining the transverse momentum distribution of the gluons in the proton. At larger photon rapidities, the JH-2003 description of the data deteriorates for increasing photon transverse momentum q T , underestimating the data while the KMR-AO description stays close to the data.
This effect may be traced back to a much steeper decrease of the JH-2003 distribution than the KMR-AO one for gluon transverse momenta k T larger than the factorization scale µ F .
The description obtained with the g * g * channel was shown to provide a good description of the data at lower photon transverse momentum q T , where also the values of the partons' x are lower. At larger q T , however, the data are significantly underestimated in this approach.
We find this result rather puzzling as the g * g * channel should partially include the NLO effects besides the sea quark contributions to the qg * channel from the last splitting, which is the main contribution to the total qg * channel. This puzzle calls for a complete NLO analysis of the prompt photon hadroproduction in the k T -factorization framework.