Double parton scattering versus jet quenching

A novel observable, the double nuclear modification factor, is proposed to probe simultaneously the initial and final state effects in nucleus-nucleus collisions. An interesting competition between the combinatorial enhancement in the double parton scattering and the suppression due to parton energy loss can be observed in the production rate of two hard particles. In particular, the production of $J/\psi$ mesons in association with a $W$ boson is not suppressed but is enhanced in the region of moderate transverse momenta, unlike the case of unassociated (inclusive) $J/\psi$ production. At the same time, in the region of high enough transverse momenta the nuclear modification factor for associated $J/\psi+W$ production converges to that of unassociated $J/\psi$.


I. INTRODUCTION
A huge number of intriguing and exquisite observations have been made with the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC).Many of them could have never been systematically studied at the accelerators of previous generation.In particular, hard multiparton interactions (MPI) are just among these interesting phenomena.The existence of MPI in hadron-hadron collisions at high energies is a natural consequence of the steep increase in the parton flux at small parton longitudinal momentum fractions, together with the unitarity requirement for the cross sections in perturbative QCD.The inclusive cross section of a hard process is usually calculated under the assumption that, in any collision, along with many soft interactions, there occurs only a single hard interaction because of its relatively low probability.Nevertheless, it is also possible that two (or more) different parton pairs undergo hard scattering in the same hadronic collision.These double parton scatterings (DPS) have been theoretically studied for many years, starting from the early days of the parton model.The current state of the MPI theory and the results of many years of research have been recently reviewed in abook [1] (see also a review [2]), which contains an extensive bibliography.Thus, the investigation of double, triple, and n-parton scatterings [3,4] allows us to extract unique and completely new information about the yet unknown three-dimensional partonic structure of hadrons and about momentum, flavor, and color correlations in their wave function.
The DPS events have been initially observed by the AFS (Axial Field Spectrometer) [5] and UA2 (Underground Area 2) [6] collaborations at CERN and later by the CDF (Collider Detector at Fermilab) [7,8] and D0 [9,10] collaborations at the Tevatron, with sufficiently high statistics for primary analysis and study.As expected, the LHC luminosity and energy provided the observation [1] of events with hard MPI's in numbers that are considerably larger than those in the aforementioned experiments.The DPS contribution has now been reliably separated and measured [1] in a number of processes containing in the final state jets, gauge bosons (γ, W, Z), heavy quarks (c, b) and quarkonia (J/ψ, Υ).For example, here is the list of some recent results from the collaborations ATLAS (A Toroidal LHCApparatuS) [11][12][13], CMS (Compact Muon Solenoid) [14][15][16] and LHCb (Large Hadron Collider beauty) [17][18][19].A triple parton scattering has also been observed very recently [20], in accordance with an early suggestion [3,21].
The DPS is actively discussed for proton-nucleus (p-A) and nucleus-nucleus (A-A) collisions as well, since its relative contribution increases, compared to naive scaling expectation.Unique new options emerge for further studies and measurements of momentum correlations.The latest achievements in and prospects for these studies may be found in a review [4].For nucleus-nucleus collisions, it opens yet a unique possibility to probe the collective properties of a new state of dense matter, the quark-gluon plasma (see, e.g., [22][23][24]) The experiments at RHIC and the LHC have provided clear evidence that the production of hadrons in A-A collisions goes through the formation of a fireball of hot and dense quark-gluon plasma.This follows from the observation of strong suppression of high-p T particle spectra (the so-called jet quenching phenomenon expressed in the nuclear modification factor R AA ) and from the results of hydrodynamic simulations of A-A collisions.
The main purpose of this Letter is to bring reader's attention to an interesting possibility to probe the initial and final state effects in nucleus-nucleus collisions simultaneously, with a single measurement.This can be realised by introducing a novel observable, the double nuclear modification factor.Our note is organized as follows.First, we describe our theoretical approach.Then we show an interesting competition between the combinatorical enhancement in the DPS and the suppression due to parton energy losses.We illustrate it by the example of the associated production of J/ψ mesons and W bosons.Some further possible directions of studies are discussed in Conclusions.

II. THEORETICAL SETUP
Let us consider nucleus-nucleus collisions at the LHC.The parton flux is enhanced by the number A of nucleons in each nucleus, and then -modulo (anti)shadowing effects in the nuclear parton distribution function -the single-parton cross section is simply expected to be that of p-p collisions, or, more exactly, that of nucleon-nucleon collisions (N -N ) scaled by the factor A 2 , i.e. [25]: as long as the final state effects are out of consideration.
Here T A (b) are the nucleus thickness function, which describe the transverse nucleon density of nuclei, the impact parameter vector b connects the centers of the colliding nuclei in the transverse plane, T AA (b) the standard nuclear overlap function normalised to A 2 , and σ SPS (N N →a) the inclusive hard single scattering cross section.
Traditionally the level of the energy losses by the hard scattered parton (the final state effects) is quantitatively characterized by the nuclear modification factor R AA (a): (the second equality is justified if the effects of initial and final states are factorized).R AA (a) = 1 if a = W, Z, γ.We can generalize this nuclear modification factor to the case of two hard scattered partons by introducing a double nuclear modification factor: where σ SPS (N N →ab) is the inclusive cross section to produce two hard particles a and b in a hard single parton scattering (SPS) in N -N collision, and σ SPS (AA→ab) is the inclusive cross section to produce two hard particles a and b in a hard SPS in nucleus-nucleus collision.Again, as in the case of one hard particle production (1) we have a similar relation for two hard particles: if the final state effects are omitted.
The inclusive cross section for the production of two hard particles a and b in a single hard parton scattering can always be represented as a product of the SPS cross sections for the individual production of the corresponding final-state particles, normalized by the effective DPS cross section: (m = 1 if a = b, and m = 2 if a and b are different) [32].
The dimensionless factor K is introduced to recover the correct value of σ SPS (N N →ab) .It mainly depends on the region (the transverse momenta) and on the type of hard particles.This factor determines the relative weight of DPS in comparison with SPS.
The presentation (5) allows us to express the double nuclear modification factor through the ordinary modification factors if the initial and final state effects are factorized, i.e., if each hard parton independently loses energy in the medium; namely: The factor K and the effective DPS cross section σ eff,pp cancel out in this relation.Usually the both hard particles are created in SPS, while the contribution from DPS is negligible in nucleon-nucleon collisions since the K-factor in the relation ( 5) is very large.However, in the case of nucleus-nucleus collisions the relative weight of DPS becomes more significant due to the combinatorial enhancement, which can partly compensate the role of factor K. The inclusive cross section of a DPS process with two hard subprocesses in an A-A collision (as A is large, A−1 ≈ A) can be written as [25]: where the last approximation showing the A-dependence of the DPS cross sections applies to large nuclei and The factor C in Eq. ( 8) indicates the enhancement in the DPS cross sections in A-A collisions compared to the corresponding A 2 -scaled values in nucleon-nucleon collisions.This enhancement amounts to C ∼ 27 (for small A = 40) or C ∼ 215 (for large A = 208).In all the estimates throughout this work we use the value of the effective DPS cross section σ eff, pp = 15 mb.Thus, the double nuclear modification factor taking the SPS and DPS contributions together can finally be presented as: We are going to investigate this novel observable in its dependence on the type of hard particles (a and b) and on the kinematical region (mainly the transverse momenta).We can watch an interesting competition between the combinatorial enhancement C and the K-and R AA -suppressions.

III. NUMERICAL EXAMPLES
A specific interplay between the effects of DPS and jet quenching can be illustrated in the simplest case when one of the two hard particles does not lose its energy when passing through a dense matter.To be more solid, we can employ available experimental results on the associated production of two hard particles.The presentation (5) enables us to extract the needed K-factor directly from the data without appealing to Monte Carlo simulations.
As an example, consider the production of J/ψ mesons in association with a W boson. Relying on the measurements [26] performed by ATLAS collaboration, we obtain the K-factor as a function of the J/ψ transverse momentum as shown in Fig. 1.
Since the W boson passes through the nuclear matter without losing energy, our main theoretical prediction (10)  The combinatorial enhancement C does not depend on the process kinematics and the type of hard particles, but is mainly governed by the atomic number A. For the minimum bias P b-P b collisions this enhancement amounts to C ∼ 215.The measured nuclear J/ψ modification factor at the LHC [27][28][29] amounts to R AA (J/ψ) ≃ 0.5 at low transverse momenta (p T ≃ 2 GeV/c) and to R AA (J/ψ) ≃ 0.3 over a wide interval of higher transverse momenta (p T > 5 GeV/c).The "measured" Kfactor demonstrates strong dependence on the transverse momentum: it changes from ∼ 1.4 at p T lying in the interval [8.5-10] GeV/c to ∼ 374 at p T ∈ [60-150] GeV/c.Thus we can expect that the production of J/ψ mesons in association with a W boson is not suppressed, but is enhanced in the region of moderate transverse momenta, contrary to unassociated (inclusive) J/ψ production.For p T ∈ [8.[5][6][7][8][9][10] GeV/c, we have R AA (J/ψ, W ) ≃ 30 while R AA (J/ψ) ≃ 0.3!In the region of high enough transverse momentum (p T > 60 GeV/c), the behavior of W -associated J/ψ production converges to the unassociated case: R AA (J/ψ, W ) ≃ R AA (J/ψ) since the ratio (C − 1)/(K + 1) becomes small.This example clearly demonstrates the competition between the effects of DPS (initial state effect) and jet quenching (final state effect).
The associated production of D mesons and W bosons shows yet a more intriguing behavior.In this case, there is a notable difference [30,31] between the oppositesign and same-sign production cross sections, and the K-factor is significantly larger for W D configurations of the opposite sign than for configurations of the same sign.The energy loss is independent of the sign of D mesons: R AA (D + ) ≃ R AA (D − ).It means that the double nuclear modification factor for the W D associate production will be notably larger for the same-sign configurations then for the opposite-sign ones: at the same kinematics.

IV. CONCLUSIONS
We propose a novel observable, the double nuclear modification factor, to probe the initial and final state effects in nucleus-nucleus collisions at a time, "in one package".An interesting competition between the combinatorial enhancement due to DPS and the suppression due to the parton energy losses can be observed.As an illustration we demonstrate that the production of J/ψ mesons in association with W bosons is not suppressed but is enhanced in the region of moderate transverse momenta, unlike the case of unassociated J/ψ production.At the same time, in the region of high enough transverse momenta the nuclear modification factor for associated J/ψ + W production converges to that of unassociated J/ψ.In the production of D mesons associated with a W boson, the double nuclear modification factor will be larger for same-sign W D configurations then for opposite-sign ones at the same kinematics.
We come to the conclusion that measurements of the double nuclear modification factor potentially open a wide room for further studies of an interplay between the effects of DPS and jet quenching, extending to vari-ous types of hard final state particles in a wide interval of their transverse momenta.
The results discussed above assumed "minimum bias" A-A collisions without any selection in the reaction centrality.In the future one can also apply this study to different centrality classes.The required cross sections for the SPS and DPS events lying within a certain impactparameter interval corresponding to a given centrality percentile may be found in [25].

FIG. 1 :
FIG.1: K-factor as a function of transverse momentum.The bands show the uncertainties of the cross section determination for J/ψ + W production.