Nuclear modification of high transverse momentum particle production in p+A collisions at RHIC and LHC

We present results and predictions for the nuclear modification of the differential cross sections for inclusive light hadron and prompt photon production in minimum bias d+Au collisions at $\sqrt{s} = 200$ GeV and minimum bias p+Pb collisions at $\sqrt{s} = 5$ TeV at RHIC and LHC, respectively. Our calculations combine the leading order perturbative QCD formalism with cold nuclear matter effects that arise from the elastic, inelastic and coherent multiple scattering of partons in large nuclei. We find that a theoretical approach that includes the isospin effect, Cronin effect, cold nuclear matter energy loss and dynamical shadowing can describe the RHIC d+Au data rather well. The LHC p+Pb predictions will soon be confronted by new experimental results to help clarify the magnitude and origin of cold nuclear matter effects and facilitate precision dense QCD matter tomography.

p+Pb results we present predictions in the framework of this approach to incorporating CNM effects. Specifically, we focus on inclusive particle production in d+Au and p+Pb collisions and show results for the nuclear modification factor of neutral pions, charged hadrons and prompt photons. This paper is organized as follows. In Sec. II, we present our approach to studying CNM effects. We first review the basic perturbative QCD formalism for single inclusive particle production in p+p collisions. We then discuss how various CNM effects are incorporated in this formalism. In Sec. III, we implement these CNM effects and show that the formalism can describe particle production in d+Au collisions at RHIC energies. We then present our theoretical model predictions for the upcoming p+Pb collision results at the LHC. We summarize our paper in Sec. IV.

II. SINGLE INCLUSIVE PARTICLE PRODUCTION
In this section we review the basic perturbative QCD formalism for single inclusive particle production in p+p collisions. In particular, we focus on light hadron and prompt photon production. We then discuss the incorporation of various cold nuclear matter effects within this formalism. Specifically, we comment on the isospin effect, Cronin effect, cold nuclear matter energy loss and dynamical shadowing.

A. Basic pQCD formalism
To leading order in the framework of factorized perturbative QCD, single inclusive hadron production in p+p collisions, p(p 1 ) + p(p 2 ) → h(p h ) + X, can be written as follows [27]: where y and p T are the rapidity and transverse momentum of the produced hadron and a,b,c runs over all parton flavors. In Eq. (1) s = (p 1 + p 2 ) 2 , D h/c (z c ) is the fragmentation function (FF) of parton c into hadron h, H ab→c (ŝ,t,û) are the hard-scattering coefficient functions withŝ,t,û being the usual partonic Mandelstam variables [27,28]. We also include a phenomenological K-factor to account for higher order QCD contributions [17,29]. f a,b/N (x, k 2 T ) are the parton distribution functions with longitudinal momentum fraction x and transverse momentum component k T . We have included this k T -dependence in order to incorporate the Cronin effect in p+A collisions. We assume a Gaussian form in this variable [17,27], where f a/N (x a ) are the usual collinear PDFs in a nucleon. On the other hand, prompt photon production in p+p collisions has two components -the so-called direct and fragmentation contributions [27,30]: The fragmentation contribution is given by: In other words, it is exactly the same as Eq. (1) if one replaces the parton-to-hadron FF D h/c (z c ) by the parton-tophoton FF D γ/c (z c ). On the other hand, the direct contribution can be written as: where H ab→γ are the well-known partonic hard-scattering functions for direct photon production [27,31,32].
In the numerical calculations we choose k 2 T pp = 1.8 GeV 2 in p+p collisions [24,25]. We use CTEQ6L1 PDFs [33], fDSS parametrization for parton-to-hadron FFs [34] and GRV parametrization for parton-to-photon FFs [35]. We fix the factorization and renormalization scales to the transverse momentum of the produced particle, µ f = µ r = p T . We find that a O(1) K-factor can give a good description of hadron production in both RHIC and LHC energies, see Fig. 1. For prompt photons the K-factor values are slightly larger and also √ s-dependent but, importantly, the shape is well-described. In the left panel of Fig. 1, we compare our calculation to RHIC photon and π 0 data at √ s = 200 GeV [36,37]. We also consider LHC photon data at √ s = 7 TeV [38] and charged hadron data at √ s = 2.76 TeV [39], shown in the right panel. The leading order pQCD formalism provides a reasonable baseline description of particle production in p+p collisions. The precise value of the K-factor is not important for the study of the nuclear modification of inclusive particle cross sections in p+A collisions (it cancels in ratios). We will incorporate nuclear effects relevant to p+A reactions in this formalism in the next subsection. GeV and rapidity y = 0 is compared to RHIC photon data (top) [36] and π 0 data (bottom) [37]. We choose K = 1 for π 0 and K = 1.8 for photon production. Right panel: comparison to LHC photon data at √ s = 7 TeV and |y| < 0.9 [38] and charged hadron data at √ s = 2.76 TeV and |y| < 1 [39]. We choose K = 1.5 for photon and K = 1 for charged hadron production.

B. Cold nuclear matter effects
The p+A (e.g. d+Au or p+Pb) nuclear modification factor R pA is usually defined as: where N coll is the average number of binary collisions. The deviation of R pA from unity reveals the presence of CNM effects in p+A collisions. A variety of CNM effects can affect particle production. In this manuscript we discuss the ones that have been theoretically evaluated as arising from the elastic, inelastic and coherent scattering of partons in large nuclei [16]. We also account for the proton and neutron composition of the interacting nucleus. In particular, we implement the isospin effect, Cronin effect, cold nuclear matter energy loss and dynamical shadowing. These effects have been well documented in the literature. We give a brief overview of their implementation in preparation for LHC predictions.
Isospin effect. Isospin effect can be easily accounted for on average in the nPDFs for a nucleus with atomic mass A and Z protons via [17,18]: In Eq. (7), f a/p (x) and f a/n (x) are the PDFs inside a proton and neutron, respectively. The PDFs in the neutron are related to the PDFs in the proton via isospin symmetry.
Cronin effect. Theoretical approaches to understanding the Cronin effect are well documented in [19]. It can be modeled via multiple initial-state scatterings of the partons in cold nuclei and the corresponding induced parton transverse momentum broadening [40,41]. In particular, if the parton distribution function f b/A (x b , k 2 b,T ) has a normalized Gaussian form, the random elastic scattering induces further k T -broadening in the nucleus [17,29]: Here k b,T is the transverse momentum component for the parton prior to the hard scattering, ζ = ln(1 + δp 2 T ) [24,25], and we choose δ = 0.14 GeV −2 , µ 2 = 0.12 GeV 2 , λ g = C F /C A λ q = 1 fm. These parameters can describe reasonably well RHIC data.
Cold nuclear matter energy loss. As the parton from the proton undergoes multiple scattering in the nucleus before the hard collisions, it can lose energy due to medium-induced gluon bremsstrahlung. This effect can be easily implemented as a momentum fraction shift in the PDFs Ideally, Eq. (9) should include a convolution over the probability distribution of cold nuclear matter energy loss [43]. However, concurrent implementation of such distribution together with the Cronin effect and coherent power corrections is computationally very demanding. The main effect of the fluctuations due to multiple gluon emission is an effective reduced fractional energy loss ǫ eff. relative to the mean value ǫ = i ∆Ei E . Here, the sum runs over all medium-induced gluons. In this work we follow Ref. [42] and use ǫ eff. = 0.7 ǫ . The average cold nuclear matter energy loss is obtained by integrating the initial-state medium-induced bremsstrahlung spectrum, first derived in [20]. It also depends on the typical transverse momentum transfer squared µ 2 per interaction between the parton and the medium and the gluon mean free path λ g . For this reason, the parameters are constrained to be the same as in the implementation of the Cronin effect, µ 2 = 0.12 GeV 2 and λ g = 1 fm in our comparison to RHIC data in the next section. This calculation of initial-state cold nuclear matter energy loss has been shown to give a good description of the nuclear modification of Drell-Yan production in fixed target experiments [43].
Dynamical shadowing. Power-suppressed resummed coherent final-sate scattering of the struck partons leads to shadowing effects (suppression of the cross section in the small-x region) [23]. The effect can be interpreted as a generation of dynamical parton mass in the background gluon field of the nucleus [44]. It is included via: where x b is the parton momentum fraction inside the target nucleus, C d = C F (C A ) if the parton d = q(g) in the partonic scattering ab → cd. ξ 2 represents a characteristic scale of the multiple scattering per nucleon. In the RHIC energy range √ s = 200 GeV, ξ 2 q = C F /C A ξ 2 g = 0.12 GeV 2 [22,23] can give a good description for nuclear modification in d+Au collisions for both single hadron and di-hadron production [23,26].

III. THEORETICAL MODEL PREDICTIONS IN P+A COLLISIONS
In this section we compare our theoretical model predictions to existing light hadron and direct photon data in minimum bias d+Au reactions at RHIC at √ s = 200 GeV. We also present predictions for these inclusive final states in minimum bias p+Pb collisions at the LHC at √ s = 5 TeV. It should be understood that in nucleus collisions we quote the center-of-mass energy per nucleon pair.

A. Nuclear modification factor in d+Au reactions at RHIC
We have incorporated the cold nuclear matter effects discussed in the previous section into the leading order pQCD photon and hadron production formalism, Eqs. (1) and (4),(5). Such theoretical approach is able to give a good description of single inclusive particle production in d+Au collisions at RHIC energies. While such calculations have been presented in the past, in this manuscript we extend our results to direct photons at forward rapidity. In Fig. 2, we plot the minimum bias nuclear modification factor R dAu as a function of transverse momentum for both photon (left panel) and π 0 (right panel) production at √ s = 200 GeV. The photon data is from the PHENIX collaboration at rapidity y = 0 [36]. For π 0 s we have included both mid-rapidity y = 0 [45] and forward rapidity 3 < y < 3.8 [46]  Empty and filled circles are low-mss virtual photon and real photon experimental data at mid-rapidity from PHENIX [36]. Filled squares are the minimum bias mid-rapidity π 0 data [45], filled triangles are the forward rapidity y = 4 STAR π 0 data [47], and empty squares is the forward rapidity 3 < y < 3.8 PHENIX π 0 s [46]. We have also presented predictions for rapidity y = 1.5 (dashed curves) and y = 3 (dotted curves) for both photon and hadron production.
PHENIX data, and forward rapidity y = 4 STAR data [47]. We conclude that implementation of cold nuclear matter effects based on the physics of multiple parton scattering in dense QCD matter can describe both photon and hadron data quite well at mid and forward rapidities. With the future RHIC upgrade program in mind, we also show predictions for the nuclear modification factor R dAu for photon and π 0 production in different kinematic regions: the dashed curves are for y = 1.5 and the dotted curves are for y = 3.

B. Predictions for the p+Pb run at the LHC
We are now in good position to present predictions for minimum bias p+Pb reactions at the LHC energy of √ s = 5 TeV. These results will soon be confronted by new experimental data. The main parameters in our approach include: the Cronin momentum broadening parameters µ 2 , λ q,g , which also determine the cold nuclear matter energy loss, and the dynamical shadowing parameter ξ 2 . As emphasized in [31], µ 2 /λ q,g and ξ 2 represent the strength of the multiple scattering between the incoming and outgoing partons and the target nucleus, thus proportional to the number of the soft gluons in the nuclear medium. Following [31], beyond the possibility of fixed ξ 2 , which we take as the lower limit for this parameter, we study a scenario where ξ 2 ∝ σ in , the inelastic nucleon-nucleon scattering cross section. Taking into account that σ in = 42 mb [48] at RHIC energy of √ s = 200 GeV and σ in = 70 mb [49] at the LHC energy √ s = 5 TeV, we consider a potential increase in the ξ 2 parameter by 67%. Specifically, for the upper limit of this parameter we take ξ 2 = 0.20 GeV 2 in our prediction for the LHC p+Pb run at √ s = 5 TeV. We will also consider a correlated enhancement in µ 2 /λ g .
In Fig. 3 we present our theoretical model predictions for the nuclear modification factor R pP b as a function of p T for neutral pion (left panel) and charged hadron (right panel) production in minimum bias p+Pb collisions at the LHC energy of √ s = 5 TeV. Rapidities y = 0 (top), y = 2 (middle), and y = 4 (bottom) were considered. The upper edge of the bands corresponds to the RHIC parameters (µ 2 , ξ 2 ) = (0.12, 0.12) GeV 2 . The lower edge corresponds to a potential enhancement of these parameters as discussed above, (µ 2 , ξ 2 ) = (0.2, 0.2) GeV 2 , to be tested against the experimental data. In Fig. 4, we present similar results for prompt photon production.
The general features of these nuclear modification factors are very similar to the ones observed at RHIC energies. At mid-rapidity y = 0, there is a very small "Cronin peak" in the low p T region. The peak is very close to unity and not as pronounced as the one observed in low energy fixed target experiments. This is because the dynamical shadowing becomes important and strongly suppress the particle production in this region. Furthermore, initial-state energy loss is larger due to the bigger contribution of diagrams with incoming gluons to the cross section. At high p T we still have ∼ 15% suppression, which is also due to the cold nuclear matter energy loss. In general, at forward rapidity all CNM effects are amplified due to the larger values of the momentum fraction x a from the incoming proton (relevant to cold nuclear matter energy loss), the smaller values of the momentum fraction x b of the incoming nucleus (relevant to dynamical shadowing) and the steeper falling spectra (relevant to the Cronin effect). As a result, at forward rapidity and at low p T the dynamic shadowing effect can be most important and lead to stronger suppression of inclusive particle production while at high p T the suppression is a combined effect of cold nuclear matter energy loss and the Cronin effect. Finally, the behavior of the R pP b for π 0 s, charged hadrons and direct photons is qualitatively the same and the quantitative differences are minor.
It is interesting to point out that our predictions for inclusive particle suppression at forward rapidities are for observable effects that are stronger than the effects found in calculations using EPS09 nuclear PDFs. On the other hand, they lead to weaker suppression than the CGC predictions from a hybrid formalism in rc-BK-MC approach [8,14]. In other words, our predictions fall between EPS09 and CGC based approaches.

IV. SUMMARY
We studied the nuclear modification of high transverse momentum particle production in minimum bias d+Au and p+Pb collisions at RHIC and LHC, respectively. Such nuclear modification is manifestation of nontrivial and not yet fully understood QCD dynamics in large nuclei. It is, therefore, critical to further our understanding of high energy nuclear reactions through concurrent theoretical advances and comparison of model predictions to precise experimental data. With this motivation, we presented results from a theoretical approach that combines the leading order perturbative QCD baseline calculation of inclusive light hadron and photon production with cold nuclear matter effects that arise from the elastic, inelastic and coherent parton scattering in large nuclei. Our numerical simulations included the isospin effect, Cronin effect, cold nuclear matter energy loss and resummed QCD power corrections to the leading twist results. We found that this approach can describe quite well the nuclear modification factor in √ s = 200 GeV d+Au collisions at the RHIC for both photon and light hadron production, including mid and forward rapidities. Detailed theoretical model predictions were then presented for inclusive particle production relevant to the recently completed p+Pb run at √ s = 5 TeV at the LHC. Our results for the nuclear modification of light hadrons and photons at forward rapidities fall between the ones based on EPS09 shadowing parameterization and CGC approaches. These predictions will soon be confronted by new experimental data to help constrain the magnitude and clarify the origin of cold nuclear matter effects. Within the framework of our approach, such comparison will also shed light on the transport properties of cold nuclear matter.