Revealing minijet dynamics via centrality dependence of the double parton interactions in proton-nucleus interactions

TL;DR

This paper proposes a method to distinguish double parton scattering from leading twist processes in proton-nucleus collisions by analyzing their different centrality dependencies, supported by a detailed Monte Carlo simulation including realistic nuclear effects.

Contribution

The authors develop a Monte Carlo framework that incorporates nucleon correlations, transverse geometry, and color fluctuations to measure DPS cross sections via centrality dependence in proton-nucleus interactions.

Findings

Numerical results show DPS rate varies with centrality.

A new universal quantity for testing DPS geometry is proposed.

Method can be extended to triple parton interactions.

Abstract

One of the main challenges hampering an accurate measurement of the double parton scattering (DPS) cross sections is the difficulty in separating the DPS from the leading twist (LT) contributions. We argue that such a separation can be achieved, and cross section of DPS measured, by exploiting the different centrality dependence of DPS and LT processes in proton-nucleus scattering. We developed a Monte Carlo implementation of the DPS processes which includes realistic nucleon-nucleon (NN) correlations in nuclei, an accurate description of transverse geometry of both hard and soft NN collisions as well as fluctuations of the strength of interaction of nucleon with nucleus (color fluctuation effects). Our method allows the calculation of probability distributions of single and double dijet events as a function of centrality, also distinguishing double hard scatterings originating from a single target nucleon and from two different nucleons. We present numerical results for the rate of DPS as a function of centrality, which relates the number of wounded nucleons and the sum of transverse energy of hadrons produced at large negative (along the nucleus direction) rapidities, which is experimentally measurable. We suggest a new quantity which allows to test the geometry of DPS and we argue that it is a universal function of centrality for different DPS processes. This quantity can be tested by analyzing existing LHC data. The method developed in this work can be extended to the search for triple parton interactions.

Equations

$${\sigma }^{\mathrm{DPS}}=\frac{{\sigma }_1{\sigma }_2}{{\sigma }_{\mathrm{eff}}},$$

$${\sigma }_{\mathrm{p}A}=A\frac{{\sigma }_1{\sigma }_2}{{\sigma }_{eff}}+{\sigma }_1{\sigma }_2\int d^{2}b{T}^2(b),$$

$$r(A)=\frac{{\sigma }_{pA}^{DPS}}{{\sigma }_{pp}^{\mathrm{DPS}}}=1+1.1\left(\frac{{\sigma }_{\mathrm{eff}}}{\text{mbox}15mb}\right){\left(\frac{A}{40}\right)}^{0.39}(1+R_{\mathrm{corr}}).$$

$$R_{DPS/LT}(b)={\sigma }_{\mathrm{eff}}T(b),$$

$$\frac{d{\sigma }_{pA}^{\mathrm{DPS}}}{d^{2}b}={\sigma }_{pN}{T}_A(b)+{\sigma }_1{\sigma }_2{T}^2(b).$$

$$f_N(x,Q^2,\rho )=\frac{g(x,Q^2,\rho )}{p(x,Q^2)},$$

$$f_N(x,\rho )=\frac{1}{2\pi B}\exp (-{\rho }^2/2B).$$

$$D^{2\otimes any2}(b)$$

$$D^{2\otimes 1}(b)$$

$$D^{2\otimes 2}(b)$$

$$S(b)=\int d\mbox{\boldmath$\rho$}_{1}\,f_{p}(\rho_{1})\,\sum^{A}_{i=1}\,\psi_{A}^{2}(r_{t}^{(i)},z_{i})\,f_{N}\left(\left|\mbox{\boldmath$\rho$}_{1}+\mbox{\boldmath$b$}-\mbox{\boldmath$r$}^{(i)}_{t}\right|\right)\,.$$

$$P_j=\frac{g_N^{(j)}(\rho )}{{\sum }_{k=1}^A{g}_N^{(j)}(\rho )}.$$

$$N^{D/I}=N(\mathrm{dijet}+\mathrm{pion})/N(\mathrm{dijet}).$$

$${\mathcal{R}}_i=\frac{N_i^{D/I}-N_2^{D/I}}{N_3^{D/I}-N_2^{D/I}}.$$

Full text

I Introduction

At the LHC energies a typical proton–proton (pp) collision involves several parton-parton interactions with transverse momentum transfer of a few GeV, leading to the production of several minijets, which are referred to as multiparton interactions (MPI). Successful Monte Carlo (MC) models of pp inelastic interaction at the LHC, such as the models implemented in the event generators Pythia Sjostrand:2014zea and Herwig Bellm:2015jjp , have to tame such parton-parton interactions up to $p_{\rm T}$ $\sim 4$ GeV. Within these models, the taming has to strengthen with the increase of the invariant energy of collision. Minijets give an important contribution to the production of relatively soft hadrons that give a main contribution to the so called underlying event (UE) with respect to the hard processes. It is generally accepted that characteristics of the UE are measured in the direction perpendicular to the momentum of a high-$p_{\rm T}$ jet Cacciari:2009dp . However, a direct observation of minijets is challenging since it is very difficult to separate them. Over the last decade intensive theoretical and experimental studies of double parton scattering (DPS) were performed Blok:2010ge ; Diehl:2011tt ; Blok:2011bu ; Diehl:2011yj ; Blok:2013bpa ; a comprehensive reviews was recently compiled in Ref. Bartalini:2017jkk .

In particular, a number of experimental analyses have been performed, aiming at finding an optimal kinematics where the ratio of the cross sections of DPS to the competing leading twist processes are somewhat enhanced. Except the case of the double charm production Belyaev:2017sws ; Likhoded:2015zna ; Blok:2016lmd , the best kinematics still corresponds to the DPS being a correction to the LT contribution. Hence, the identification of DPS events is rather sensitive to the particular model adopted to describe LT processes, which are usually rather involved. To illustrate this point, Fig. 1 shows the fraction of the total cross section of dijet production within $|\eta|<$ 2 and $p_{\rm T}^{\rm jet}\geq$ 50 GeV plus a charged particle, which originates from the different parton interaction, with an azimuthal angle difference with respect to the leading jet within $80^{o}<\Delta\phi<100^{o}$, as a function of the pseudorapidity interval between the leading jet and the charged particle, $\Delta\eta$, and of the transverse momentum, $p_{\rm T}$, of the charged particle as obtained with Pythia 8 Monash model Skands:2014pea . The fraction of the cross section due to DPS presented in Fig. 1 is computed as a difference between the standard collision simulation and one with MPI mechanism switched-off, divided by former one. One can see from Fig. 1 that the DPS contribution is significant but not dominant, hence a relatively small uncertainty in the calculation of the LT contribution leads to a pretty large uncertainty in the determination of the DPS contribution to the experimental cross section.

Traditionally the DPS cross section is parameterized in the following form:

[TABLE]

where $\sigma_{i}$ are the cross sections of binary pp collisions, and $\sigma_{\mathrm{eff}}$ is the effective cross section, widely used to characterize the effective transverse area of hard partonic interactions in pp collisions Paver:1984ux ; Mekhfi:1983az .

In QCD one expects $\sigma_{\mathrm{eff}}$ to depend on the Bjorken $x$’s of the colliding partons, their flavors, as well as the hardnesses of the subprocesses. We will not write this dependence explicitly in the following.

The LHC data are consistent with $\sigma_{\mathrm{eff}}\sim$ 20 mb for production of two pairs of jets with $p_{\rm T}^{\rm jet}\geq$ 50 GeV Aaboud:2016dea . In this paper we use the formalism for the description of MPI developed in Blok:2010ge ; Blok:2011bu ; Blok:2013bpa ; Blok:2016lmd , see review and references in Blok:2017alw , which takes into account both the mean field contributions as well pQCD–induced parton–parton correlations and small $x$ soft correlations. This formalism allows to describe all existing LHC data except double $J/\psi$ production Aaboud:2016fzt . For smaller virtualities this formalism predicts $\sigma_{\mathrm{eff}}\sim$ 30 mb, which is consistent with the recent Monte Carlo analyses Blok:2015rka . The model also explains an increase of $\sigma_{\mathrm{eff}}$ from $\sim$ 14 mb to 20 mb between the Tevatron and LHC energies for the kinematical ranges in which measurements were performed.

Though the LHC data strongly suggest the presence of the MPI effects in pp scattering, no accurate determinations of the MPI cross were reported so far (a notable exception is the charm production Belyaev:2017sws ; Likhoded:2015zna ; Blok:2016lmd ). To a large extent, this is due to insufficient accuracy of modeling higher order leading twist (LT) contributions to multijet production.

We suggest that a way out is to study MPI in proton–nucleus collisions as a function of the centrality of the collision. The suggested procedure is based on the observation made a long time ago Strikman:2001gz that MPI are enhanced in proton–nucleus collisions, leading to a parametric enhancement of MPI by the factor $\propto A^{1/3}$ as compared to the LT contribution due to hard scattering off two nucleons. The enhancement strongly increases with centrality of the collision. Hence, the study of the rate of the MPI candidate events as a function of centrality would allow to separate DPS and LT processes and provide an unambiguous measurement of DPS.

We study the centrality dependence of the different contributions to DPS in $pA$ collisions at LHC energies, within a high-accuracy implementation of the Glauber Monte Carlo model. Our model makes use of realistic nucleus configurations including NN correlations Alvioli:2009ab and neutron skin in the lead nucleus Alvioli:2018jls . Other implementations of the Monte Carlo Glauber model for soft processes exist, for example the one in Ref. Loizides:2017ack . In the treatment of the individual soft $pN$ collisions, we also include the color fluctuation effect Alvioli:2014eda , which takes into account the possibility for the incoming proton to fluctuate in different quantum states with substantially different $pN$ interaction strength; this effect is important for an accurate description of the dependence of the hadron production on centrality Aad:2016zif , see discussion in Sec.IV. The main effect of smearing of centrality which we take into account is due to the experimental definition of centrality classes, based on the measured transverse energy distribution $\sum E_{\rm T}$. Eventually, we implement a mechanism for a double hard trigger in each Monte Carlo Glauber event, based on the extension to two hard interactions of an existing model for single hard trigger Alvioli:2014sba .

We organized the paper as follows.

In section II we describe the basic idea and summarize the relevant information from the previous studies. Next, in section III we describe the development of a Monte Carlo event generator for calculating the inclusive rate of DPS. In section IV we describe an extension to the case of DPS of the existing Monte Carlo procedure for the calculation of the probability distribution over the number of the wounded nucleons in events with single hard interaction. In section V we include the effect of smearing over impact parameter for the transverse energy of hadrons for centrality characterization. Based on this calculation we outline the proposed procedure for comparing events of different centrality classes in order to measure the DPS cross section.

II Basic idea

In the optical approximation, which does not include NN correlations and considers the nucleon size much smaller than the internucleon distance, the cross section of DPS in $pA$ collisions for large A can be written as follows  Strikman:2001gz :

[TABLE]

where $b$ is the impact parameter of the proton, and $T(b)=\int_{\infty}^{\infty}\rho(b,z)dz$ is the standard nuclear profile function obtained from the nuclear density $\rho(b,z)$. The first term in Eq. (2) is the contribution of the impulse approximation, in which two partons of the proton interact with two partons of a single nucleon of the target nucleus (Fig. 2a). The second term describes the interaction of two partons in the proton with two partons of two different nucleons of the nucleus, neglecting parton–parton correlations in the projectile proton (Fig. 2b).

Using a realistic nuclear density, for $A\geq 40$ one can calculate the ratio of the DPS contributions in pA and pp scattering as follows  Strikman:2001gz :

[TABLE]

In Eq. (3), $R_{\rm corr}=f(x_{1},x_{2})/f(x_{1})f(x_{2})-1$ accounts for the longitudinal correlations of the constituents of the projectile proton due to the pQCD evolution Blok:2012jr . In the pp case, correlation effects leads to a decrease of $\sigma_{\mathrm{eff}}$ by the factor $(1+5R_{\rm corr})$ as compared to the uncorrelated (mean field) model. Numerical calculations were performed under the assumption that the DGPD (Double Generalized Parton Distributions) are factorized at the scale $Q_{0}^{2}$ Blok:2011bu . For different models of double parton correlations at a low resolution scale, see Ref. Rinaldi:2013vpa and references therein. It was found that for large $p_{\rm T}$, the factor $5R_{\rm corr}=0.5\div 1$ allows to reproduce the measured values of $\sigma_{\mathrm{eff}}(NN)$; see Blok:2017alw and references therein. For the kinematics we discuss in this work, a typical value is $R_{\rm corr}\sim 0.15$, see Fig. 3.

Taking $\sigma_{\mathrm{eff}}$ = 20 mb leads to the expectation that the ratio of DPS to LT contributions is enhanced in pPb collisions by a factor $r(200)\sim 4$. For minijets with $p_{\rm T}$ of a few GeV, one expects $\sigma_{\mathrm{eff}}\sim$ 30 mb, leading to $r(200)\sim 5$. However this enhancement is somewhat reduced due to the leading twist shadowing effect which requires a detailed modeling of the particular kinematic domains Frankfurt:2011cs , hence this effect will be considered elsewhere.

One can try to observe the predicted enhancement of DPS in $pA$ scattering at the LHC by comparing pp and pA data. However this would require comparing two different sets of data in a somewhat different kinematics. An alternative strategy we suggest in this paper is to explore the strong dependence of the DPS/LT ratio on the impact parameter of the $pA$ collision. In the mean field approximation (cf. Eq. (2)), we can write:

[TABLE]

which corresponds to a very large enhancement of DPS for central $pA$ collisions. We can write

[TABLE]

In Eq. (5), we removed the superscript (DPS) in the first term on the right hand side to indicate that $\sigma_{pN}$ includes the leading twist contribution to the DPS cross section since it is also linear in $T(b)$. Hence, Eq. (5) gives a model–independent prediction for the $b$–dependence of DPS in terms of the elementary DPS pp cross section, $\sigma_{1}$ and $\sigma_{2}$, and of T(b).

Obviously, one cannot fix the impact parameter of the collision, but one can still define centrality classes, for example using the method adopted by the ATLAS collaboration Aad:2015zza . An evidence of the validity of such a procedure is that it reproduces correctly the rate of jet production in the kinematics where the parton of the proton carries a moderate Bjorken $x$, like $x\leq 0.1$.

To make realistic predictions for the DPS-related observables we perform the calculation in several steps, extending the existing Monte Carlo generator for the production of dijets Alvioli:2014sba ; Alvioli:2014eda ; Alvioli:2017wou , which allows to calculate the interaction probability distribution as a function of the number of wounded nucleons and of the pA centrality. We take into account the finite transverse spread of the parton distribution in nucleons, and correlations between nucleons in the nucleus Alvioli:2009ab .

III Inclusive DPS beyond mean field approximation

The generalized double parton distributions necessary for the calculation of the DPS off nuclei were calculated in Ref. Blok:2012jr as sum of two terms, as in Eq. (3) and as illustrated in Fig. 2.

The first term in Eq. (3) term accounts for the scattering off two partons of the same nucleon. It can be calculated by a convolution of two double nucleon GPDs plus the pQCD induced correlations, and corresponds to the impulse approximation. The second term in Eq. (3) corresponds to scattering of two partons of the projectile off two partons belonging to two different nucleons of the nucleus. Figure 4 shows the notations used for the various quantities used in this work.

Separating the contribution of scattering off the same nucleon is necessary to account in an economic way for the existence of parton-parton correlations in the nucleons. To calculate the DPS cross section accounting for a finite transverse spread of the parton distributions we introduce the quantity $f_{N}(x,Q^{2},\rho)$, describing the transverse distribution of partons in the nucleon, defined as follows:

[TABLE]

where $g(x,Q^{2},\rho)$ is the diagonal generalized parton distribution and $p(x,Q^{2})$ is the parton distribution. The $\rho$ dependence of the generalized parton distribution is given by the Fourier transform of the two gluon form factor of the nucleon, $F_{2g}(t)$, which is determined from the analysis of $J/\psi$ exclusive photoproduction Frankfurt:2010ea . For simplicity we will use an exponential parameterization of $F_{2g}(t)=\exp(Bt/2)$, and will not write explicitly the dependence of $B$ on $x$ and $Q^{2}$, thus the parton distribution takes the form:

[TABLE]

The value of $B$ in Eq. (7) can be extracted from the analysis of the exclusive $J/\psi$ photoproduction.

The geometric factor entering to the DPS cross section can be written as

[TABLE]

which includes both interactions with two different nucleons ($2\otimes 2$) and the same nucleon ($2\otimes 1$) of the target nucleus. The geometric factor for the same nucleon case is given by:

[TABLE]

The factor for the interaction with two different nucleons, which replaces the $T^{2}(b)$ factor in the optical approximation, Eq. (2), is simply given by the difference $D^{2\otimes any2}(b)-D^{2\otimes 1}(b)$:

[TABLE]

For our numerical studies, we choose $B$ = 3 GeV*-2*, which corresponds to $x\sim 0.01$ for $Q^{2}\sim$ a few GeV2. The effective cross section, $\sigma_{\mathrm{eff}}$ in Eq. (1), is expressed through $B$ as $\sigma_{\mathrm{eff}}=8\pi B$, leading to $\sigma_{\mathrm{eff}}$ = 30 mb for $B$ = 3 GeV*-2*. Smaller values of $\sigma_{\mathrm{eff}}$ at large virtualities result in this approach from pQCD induced correlations Blok:2010ge ; Blok:2011bu ; Blok:2013bpa .

The code developed to calculate Eqs. (8-10) thus allows to obtain the separate contributions due to the DPS with one (Eq. (10)) and two (Eq. (9)) nucleons, both as a function of pA centrality and of the number of wounded nucleons.

Our numerical results for the $b$–distributions for DPS off two and single nucleon can be compared with the optical model approximation. In Fig. 5 we compare $D^{2\otimes 2}(b)$ and $T^{2}(b)$.

We find that the $b$–dependent distribution accounting for the finite nucleon size is a bit broader and the total contribution of the $2\otimes 2$ term is somewhat smaller than in the optical approximation. For example, for $pPb$ scattering, $\int d^{2}bD^{2\otimes 2}(b)/\int d^{2}bT^{2}(b)$ = 0.95, accounting for finite size, accurate treatment of the surface region of matter distribution (neutron skin effect, as described in Ref. Alvioli:2018jls ), and NN correlations. This suppression factor is close to the correction found in the mean field approximation for the nucleus wave function accounting for the final nucleon size: $\approx(1-2r_{N}^{2}/R_{A}^{2})$ Blok:2012jr .

The impulse approximation term $\propto D^{2\otimes 1}(b)$ obviously does not introduce any corrections to the cross section integrated over $b$. However, since the elementary cross section corresponds to the interaction of two nucleons at a finite impact distance, the $b$–distribution of $D^{2\otimes 1}(b)$ should be somewhat broader than for $T(b)$.

The distribution over $b$ for the leading twist distribution is given by

[TABLE]

The difference of S(b) and T(b) is very small, so we do not present the corresponding plot. The double scattering in NN interactions corresponds to a smaller average transverse distance than a single scattering Frankfurt:2003td ; Blok:2010ge ; Blok:2011bu ; Blok:2013bpa . So in this case the deviation of the $b$–distribution from $T(b)$ is even smaller. Hence, in the following we will neglect the small difference between $S(b)$ and $D^{2\otimes 1}(b)$.

IV Distribution over the number of wounded nucleons

In order to calculate the distribution over the number of wounded nucleons we need to distinguish events in which the two interacting partons of the nucleus belong either to the same nucleon or to two different nucleons. In the first class of events, which is described by $D^{2\otimes 1}(b)$ in Eq. (9), we need to calculate the distribution over the number of soft interactions excluding the nucleon involved in the hard interaction. Analogously, we exclude two nucleons in the case of hard interactions with partons from two different nucleons in the nucleus. The procedure is a straightforward extension of the one we developed for dijet production Alvioli:2014sba . For each of the two interacting partons of the proton, we assign one particular nucleon as the one involved in a hard interaction, with probabilities given by:

[TABLE]

Now we need to generate the distribution over number of nucleons involved in soft interactions. We do it in two ways. The first approach is based on the standard Glauber model with an accurate treatment of the distribution of the probability of the inelastic NN interaction over the relative impact parameter. Another approach includes in addition effects of fluctuations of the strength of interaction of the projectile proton with the target nucleus fluctuates from event to event. These fluctuations take into account presence of the inelastic diffraction and provide an effective implementation of the high energy Gribov-Glauber picture of hadron - nucleus scattering. We follow closely the procedure discussed in our paper Alvioli:2014sba . We assign to each incoming proton interaction strength $\sigma$ with probability $P(\sigma)$ - for a detailed discussion see Alvioli:2014sba and calculate averages over a large sample of the events. The variance of the distribution over $\sigma$, $\omega_{\sigma}=\left<\sigma^{2}\right>/\left<\sigma\right>^{2}-1$ is given by Miettinen - Pumplin relation Miettinen:1978jb which expresses $\omega_{\sigma}$ through the ratio of inelastic and elastic NN cross sections at $t=0$. For the LHC energies we estimate $\omega_{\sigma}\approx 0.1$. In the analysis of ATLAS Aad:2016zif $R_{pPb}$ was studied as a function of centrality for three values of $\omega_{\sigma}=0,0.11,0.2$. It is remarkable that the best description was found for $\omega_{\sigma}=0.11$. For such $P(\sigma)$ they found $R_{pPb}$ for the charged particle production to be close to one for $p_{\rm T}\geq 2\div 3\,\mbox{GeV}$ and a wide range of rapidities.

Since we count nucleons which were involved in both soft and hard interaction only once, the distribution for the double nucleon term obviously starts at $\nu=2$, with $\nu$ the total number number of interacting nucleons.

The results of the calculation for the distribution over $\nu$ for the no correlation scenario with account of color fluctuations ($\sigma_{\mathrm{eff}}$ = 30 mb) are presented in Fig. 6 for several centrality classes. For large $\nu$ the account of color fluctuations leads to broadening of the distribution over $\nu$.

One can see that for DPS events distribution over $\nu$ is much broader. Parton–parton correlations lead to an enhancement of the impulse approximation $2\otimes 1$ term in Eq. (9) by a factor $\sigma_{\mathrm{eff}}(m.f.)/\sigma_{exp}=1+5R_{\rm corr}$, and of the double nucleon term by a factor $1+R_{\rm corr}$ Blok:2012jr . For the kinematics discussed in this work (presented in Fig. 4), $R_{\rm corr}\sim 0.15$ (Fig. 3). Hence, its effect for the double scattering term is pretty small, and we will neglect its residual dependence on impact parameter.

To take into account parton–parton correlations in the calculation of the distribution over $\nu$, it is sufficient to take the impulse approximation term $D^{2\otimes 1}(b)$ with an additional factor $(1+5R_{\rm corr})/(1+R_{\rm corr})\sim 1.5$ and normalize to the inclusive cross section where the $D^{2\otimes 1}(b)$ term is also enhanced by the same factor.

V Transverse energy distribution and extraction of the DPS signal

Let us consider a process in which DPS contributes: for example production of four jets in a special configuration, or production of two jets and a hadron with a sufficiently large $p_{\rm T}$ from the underlying event. The main challenge is that the LT process can also contribute to this special configuration (cf. Fig. 1), leading to the need to rely on a Monte Carlo simulation for a rather complicated final state.

If we choose a kinematics where soft contributions (including very soft minijets) can be neglected, there are three contributions to the final state: the leading twist contribution, DPS due to the interaction with one nucleon and DPS due to the interaction with two nucleons. The first two contributions are proportional to roughly the number of nucleons along the projectile path. In the events with a dijet trigger they would result in the same multiplicity of a second dijet (hadron) for different centralities. At the same time the DPS due to the interaction with two nucleons should lead to a contribution which grows with centrality much faster (roughly the square of the number of nucleons along the projectile path). Hence, it is convenient to consider the ratio of the multiplicity $N$ of the candidate DPS final state (for example dijet plus a pion) and the multiplicity of the inclusive dijet production in the same kinematics:

[TABLE]

For such a ratio, deviations from linearity in the number of collisions, which were found in Ref. Alvioli:2014sba , practically cancel out. The dependence of $N^{D/I}$ on centrality is only due to the double nucleon interaction term.

We follow the procedure developed by ATLAS to define centrality classes Aad:2015zza . They use the transverse energy $-3.2\geq\eta\geq-4.9$ (i.e. along the nucleus direction) as a measure of centrality. It was shown in Ref. Aad:2015ziq that $\sum E_{\rm T}$ in this kinematics is not sensitive to production of hadrons at forward rapidities. The distribution over $\sum E_{\rm T}$ as a function of $\nu$ is given in Refs. Aad:2015zza ; Aad:2015ziq ; Aad:2016zif .

We define bins in $\sum E_{\rm T}$ as in Refs. Alvioli:2014eda ; Alvioli:2017wou , and use the 10%-20% (second) bin, in which the first term of Eq. (2)(linear in A) dominates, and build the ratio of the differences in multiplicities in the $i^{th}$ centrality bin as follows:

[TABLE]

We use for subtraction the second bin since there are significant uncertainties in modeling the most peripheral bin. In the differences $N^{D/I}_{i}-N^{D/I}_{2}$ the contribution of the terms linear in A cancels out and the $\sum E_{\rm T}$ dependence originates solely from the geometry of the process. Thus, the dependence of $\mathcal{R}_{i}$ on the momenta of the jets (hadrons) is expected to be universal. This would provide a crucial test of the overall picture of the double scattering process. The predicted dependence of $\mathcal{R}_{i}$ on centrality is very strong, as it is illustrated in Fig. 7 for the Color Fluctuation and Glauber models. One can see that color fluctuations somewhat reduce $\mathcal{R}_{i}$ for most central bin due to additional smearing over impact parameter. Anyway, the predicted effect is large and should be pretty straightforward to observe. Note that in our considerations we assumed that both components of DPS events originate from the leading twist QCD processes. So one needs to select the kinematics where for both subprocesses $R_{pPb}$ is close to one. Based on the analysis of ATLAS Aad:2016zif use of the color fluctuation model with $\omega_{\sigma}\sim 0.1$ appears to be preferable. Note also in the kinematics where deviations of $R_{pA}$ from one for both subprocesses are small one can estimate related corrections for $\mathcal{R}_{i}$.

An important test of the picture is that $\mathcal{R}_{i}$ should be a universal function of $\sum E_{\rm T}$, independent on the angle between the dijet and the hadron, and the hadron transverse momentum.

Let us now consider an example of a process which can be studied using this procedure, the production of a dijet at forward rapidities, in the range $y$ = 2 $\div$ 4, and a hadron from an underlying event with a tight cut on the emission angle $\theta=90^{o}\pm 10^{o}$. We performed the calculations using the Pythia model of the contribution of DPS to the underlying multiplicity. The results of the calculation were shown in Fig. 1. One can see that in a wide range of hadron momenta DPS contributes on the scale of 30% $\div$ 40% to the pp cross section. Hence, for central $pA$ collisions, the underlying multiplicity should be enhanced by a factor of at least 2.5 as compared to pp collisions. The the underlying multiplicity enhancement can be increased by suppressing the LT contribution. For instance, imposing additional requirement on the dijet moment imbalance $(p_{\rm T,1}-p_{\rm T,2})/(p_{\rm T,1}+p_{\rm T,2})<$0.1, where $p_{\rm T,1}$ and $p_{\rm T,2}$ are the transverse momenta of the leading and subleading jets respectively, would increase the DPS contribution to 35-50%. Also due to a relatively high rate of the discussed process an accurate subtraction procedure should be possible both for the narrow angle window we discuss, and for a wider range of the angles. The minimal $p_{\rm T}$ of the hadrons for which our calculations are applicable follow from the requirement that $R_{pPb}$ should be close to one. Depending on the rapidity of the hadron it corresponds to $p_{\rm T}(hadron)\geq\mbox{2}\div\mbox{5 GeV}$ Aad:2016zif . Also one has to impose a restriction to the fraction of the momentum of proton, $x_{p}$, carried by the parton involved in the dijet production $x_{p}\leq 0.1$, since for large $x_{p}$ the centrality dependence gradually changes that maybe due to selection of smaller size configurations by a large $x_{p}$ trigger, see discussions in Alvioli:2017wou ; Alvioli:2014eda .

A clean separation of the $2\otimes 2$ contribution would allow to perform a direct measurement of the parton-parton correlations ($R_{\rm corr}$) (cf. Eq. (3)). Knowing the $A^{4/3}$ term it would be possible to measure correlation effects for two partons of the projectile proton involved in the process (cf. Eq. (3)). Also it would make it easier to extract $\sigma_{\mathrm{eff}}$ from the linear term. In this case $\sigma_{\mathrm{eff}}$ is the only parameter which could be adjusted and it could be determined from the condition that the dependence of the hadron emission on the azimuthal angle with respect to the dijet should disappear.

VI Conclusions

We developed an algorithm for the calculation of the DPS cross section in pA scattering as a function of centrality. We suggested a method to use the centrality to determine the cross section of DPS due to scattering off two different nucleons. In the long run this would allow to study parton–parton correlations in nucleons as a function of virtuality and $x$’s. It would be possible also to look for triple parton scattering Strikman:2001gz using a similar strategy.

Acknowledgments

M.A. acknowledges a CINECA award under ISCRA initiative for making high-performance computing resources available. The research of B.B. was supported by Israel Science Foundation under the grant 2025311. M.S.’s research was supported by the US Department of Energy Office of Science, Office of Nuclear Physics under Award No. DE-FG02-93ER40771.