Parton correlations effects in double parton distribution functions

TL;DR

This paper discusses the calculation of double parton distribution functions (dPDFs) and their potential to reveal three-dimensional partonic structures within protons through phenomenological analysis relevant for upcoming LHC experiments.

Contribution

It presents new calculations of dPDFs and explores their impact on measurable observables, advancing understanding of parton correlations inside protons.

Findings

dPDFs can be calculated and modeled

Observable effects of parton correlations are identified

Potential for experimental estimation at the LHC

Abstract

Double parton distribution functions (dPDFs), measurable in hadron-hadron collisions and encoding information on how partons inside a proton are correlated among each other, could represent a new tool to explore the three dimensional partonic structure of hadrons. In the present contribution, results of the calculations of dPFDs are presented. Phenomenological calculations of experimental observables, sensitive to dPDFs are also discussed showing how double parton correlations could be estimated in the next LHC run.

Equations

$$F_{ij}(x_1,x_2,{\vec{z}}_{\perp },\mu )=f_i(x_1,\mu )f_j(x_2,\mu )T({\vec{z}}_{\perp }),$$

$${\sigma }_{DPS}^{A+B}=\frac{m}{2}\frac{{\sigma }_{SPS}^A{\sigma }_{SPS}^B}{{\sigma }_{eff}},$$

$$rati{o}_{ab}$$

$$rati{o}_{ab}^P$$

$$rati{o}_{ab}^{NP}$$

$$\displaystyle F_{ab}(x_{1},x_{2}=0.01,k_{\perp}=0;Q^{2})|^{P}=\big{[}a(x_{1};Q^{2})b(x_{2}=0.01;Q^{2})\big{]}^{dPDFevolution}~{}.$$

$$d{\sigma }_{DPS}^{AB}=\frac{m}{2}\sum _{i,j,k,l}d{\vec{z}}_{\perp }{F}_{ij}(x_1,x_2,{\vec{z}}_{\perp },\mu )F_{ij}(x_3,x_4,{\vec{z}}_{\perp },\mu )$$

Full text

Chapter 0 Parton correlations effects in double

parton distribution functions

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1 Introduction

Since a long time it has been established that multiple parton interactions (MPI) are a fundamental piece in the investigation on hadron-hadron collisions at the LHC. In these kind of events, more than one parton of a hadron interact with partons of the other colliding hadron. However, the MPI contribution to the total cross section is suppressed with respect to the single partonic interaction. Nevertheless, MPI could represent a background for the search of new Physic at the LHC and the measurement of their cross sections is an important experimental challenge. From a theoretical point of view, one of the main interests on MPI is the possibility of accessing new fundamental information on the partonic proton structure. To this aim, we focused our attention on the double parton scattering (DPS), the most simple case of MPI, which can be observed, in principle, in several processes, e.g., $WW$ with dilepton productions and double Drell-Yan processes (see, Refs. [1, 2, 3, 4, 5] for recent reviews). At the LHC, DPS has been observed some years ago [6] and represents also a background for the Higgs production in several channels. Formally, the DPS cross section depends on the so called double parton distribution functions (dPDFs), $F_{ij}(x_{1},x_{2},{\vec{z}}_{\perp},\mu)$, which describe the joint probability of finding two partons of flavors $i,j=q,\bar{q},g$ with longitudinal momentum fractions $x_{1},x_{2}$ and distance $\vec{z}_{\perp}$ in the transverse plane inside the hadron, see Ref. [7] . Here $\mu$ is the renormalization scale. Since for the moment being no data are available for dPDFs, they are usually approximated in experimental analyses through a fully factorized ansantz:

[TABLE]

being $f_{i}(x,\mu)$ the usual one-body parton distribution functions (PDFs) and $T(\vec{z}_{\perp})$, a distribution encoding the probability of finding two partons at distance $z_{\perp}$. The assumption (1) is to neglects all possible unknown double parton correlations (DPC) between the two interacting partons (e.g. Ref. [8] ). Being dPDFs non perturbative quantities in QCD, they can not be easily evaluated and constituent quark model (CQM) calculations could help to grasp their basic features checking if the factorized approximation is a suitable ansatz for dPDFs see Refs. [9, 10, 11, 12, 13] . Double PDFs are first of all calculated at the low hadronic scale of the model, $\mu_{0}$, then, in order to compare the outcome with future data taken at high momentum transfer, $Q>\mu_{0}$, it is necessary to perform a perturbative QCD (pQCD) evolution by using dPDF evolution equations, see Refs. [14, 15, 16]. In particular, this step is fundamental to understand to what extent DPC survive at the kinematic conditions of experiments. Thanks to this procedure, future data analyses of the DPS processes could be guided, in principle, by model calculations. To this aim in Refs. [11, 17] , DPC in dPDFs have been studied at the energy scale of the experiments and the so called $\sigma_{eff}$, evaluated in Refs. [18, 13, 19] . In fact, DPS cross section, in processes with final state $A+B$, is written through the following ratio (see e.g. Ref. [20]):

[TABLE]

where $m$ is combinatorial factor depending on the final states $A$ and $B$ ($m=1$ for $A=B$ or $m=2$ for $A\neq B$) and $\sigma^{A(B)}_{SPS}$ is the single parton scattering cross section with final state $A(B)$. The present knowledge on DPS cross sections has been condensed in the experimental extraction of $\sigma_{eff}$ [20, 21, 22, 23, 24, 25, 26, 27]. A constant value, $\sigma_{eff}\simeq$ 15 mb, is compatible, within errors, with data: a result obtained within the fully uncorrelated ansatz for dPDFs ( all DPC are neglected). $\sigma_{eff}$ can be calculated within CQM in order to characterize “signals” of DPC. In the next sections the possible dependence of $\sigma_{eff}$ on the longitudinal momentum fraction carried by the interacting partons is presented, see Refs. [18, 13, 19] . Furthermore a phenomenological calculation of DPS cross section for the $WW$ same sign production, is discussed: the main outcome of this last investigation is that DPC could be accessed in the next LHC run.

2 Calculation of dPDFs in CQM and pQCD evoultion

In our first studies, dPDFs have been calculated within CQM, see Refs. [10, 11] , and it has been shown that at the hadronic scale, DPC can not be neglected, in particular the factorization on the $x_{1}$ and $x_{2}$ dependence is never supported. Furthermore, in Ref. [13] , it has been demonstrated that also the factorization in the $\vec{z}_{\perp}-(x_{1},x_{2})$ dependence of dPDFs, evaluated within the relativistic Light-Front (LF) approach [29, 30], does not represent a good approximation for dPDFs at the hadronic scale, independently on the chosen model for the proton wave function. The pQCD evolution of model calculations of dPDFs has been investigated and addressed in Ref. [11, 17] . A deep study on the role of perturbative and non perturbative correlations on dPDFs at high energy scales has been presented in Ref. [17] . To this aim, the following ratios have been defined:

[TABLE]

where here:

[TABLE]

In the above expressions $a(x;Q^{2})$ and $b(x;Q^{2})$ are the single PDFs for two given partons of flavor $a$ and $b$ and $Q^{2}=250$ GeV2. The ratio (3) would be equal to one if the factorized ansatz, Eq. (1) holds. However, since in this quantity the numerator and the denominator evolve by means of different evolution equations, $ratio_{ab}$ is sensitive to non perturbative correlations, encoded in the proton wave function used to calculate dPDFs and PDFs, and to the perturbative ones due to the difference in the evolution equations of dPDFs and PDFs. In order to disentangle these two different contributions, the ratios Eqs. (4, 5) have been introduced and calculated. In addition, the quantity Eq. (6), appearing in Eqs. (4, 5), is evaluated by evolving the product of PDFs through the dPDF evolution scheme. Thanks to this feature, $ratio_{ab}^{P}$ is sensitive to perturbative correlations, in fact, its numerator and denominator evolve differently in pQCD but having the same input at the initial scale. On the other hand, since in $ratio_{ab}^{NP}$, the numerator and denominator evolve within the same scheme, but using a different input at the initial scale, this quantity is sensitive to non perturbative correlations.

An example of these ratios is shown in Fig. 1 (left panel) for, e.g., $a=b=g$, being gluons the highest partonic component at LHC kinematics. In particular, here use has been made of the dPDFs discussed in Ref. [11] and obtained within the CQM described in Ref. [31]. The full ratio $ratio_{ab}$, Eq. (3), (dashed line), influenced by both perturbative (dot-dashed line) and non-perturbative (continuous line) effects, is compared with those where perturbative and non-perturbative correlations are disentangled. As one can see in the left panel of Fig. 1, for the gluon-gluon distribution such components coherently interfere. Furthermore, in Ref. [17] , in order to show how much these conclusions do not depend on the model used, a semi factorized ansatz has been adopted in order to include non perturbative sea quarks and gluons at a given initial scale. For gluon-gluon dPDFs, partonic correlations are still present also in the low-$x$ region. The strength of the correlations seems to become smaller but they are still sizable and should not be neglected.

3 Calculation of $\sigma_{eff}$ by means of CQM

A fundamental quantity, relevant for the experimental analyses of DPS is the so called effective cross section, $\sigma_{eff}$ whose formal expression in terms of PDFs and dPDFs, has been described in Ref. [18] . Here, $\sigma_{eff}$ has been evaluated at zero rapidity through the dPDFs addressed in Ref. [11] and calculated at $Q^{2}=250$ GeV2 for different degrees of freedoms. The range of obtained values of $\sigma_{eff}$ is comparable with the three old experimental extractions of $\sigma_{eff}$ [33, 34, 35] . Moreover, as discussed in Ref. [13] , if Melosh rotations were neglected, the average value of $\sigma_{eff}$ at the hadronic scale would change by a factor 2, making the impact of relativistic effects in $\sigma_{eff}$ very important. The main important result found and discussed in Refs. [18], is the strong $x_{i}$ dependence of $\sigma_{eff}$ in the valence region, a clear sign of DPC. In Ref. [19] , $\sigma_{eff}$ has been studied also through dPDFs calculated within the AdS/QCD correspondence [36, 37, 38] . As one can see, the mean value of $\sigma_{eff}$ and its $x$ dependence, obtained within this different model, is comparable with those found within the LF approach in Ref. [18] . In order to emphasize this $x$ dependence, in Fig. 2, the ratio ${\sigma_{eff}(x_{1},x_{2},\mu_{0}^{2})}/{\sigma_{eff}(x_{1}=10^{-3},x_{2},\mu_{0}^{2})}$, evaluated by means of the AdS/QCD correspondence, has been plotted. One can see that the ratio strongly depends on $x_{2}$ in the valence region and longitudinal correlations cannot be neglected.

4 Parton correlations in same sign $W$ pair production

In this last section the main outcomes discussed in Ref. [28] will be summarised. Due to results described in the previous section, we have studied to what extent double parton correlations could be accessed at the LHC. To this aim we have considered the same sign $W$ pair production, a promising process for the observation of DPS [39, 40, 41, 42] . The differential DPS cross section can be written as follows [7] :

[TABLE]

In order to properly evaluate Eq. (7) by means of the use of dPDFs calculated in Ref. [11] , it is worth to notice that in this scenario the initial hadronic scale lies in the infrared region so that the DPS cross section is extremely sensitive to its value. Therefore, in Ref. [28] , it has been proposed fo fix the initial scale of the model ($Q_{0}^{2}$), used to calculate dPDFs appearing in eq. (7) as follows: $i)$ we evaluated the single parton scattering cross section for $W$ production by using for PDFs the same LF CQM used for dPDFs of Ref. [11], $ii)$ then results found are compared with those obtained by means of DYNNLO code[43] at LO within the PDF parametrization of Ref. [32]. The best match is obtained for $0.24<Q^{2}_{0}<0.28$ GeV2. We therefore choose $Q_{0}^{2}=0.26$ GeV2 as central value. Furthermore, uncertainties due to higher order corrections have been taken into account into the scheme by also varying the final scale value, i.e., $0.5M_{W}<\mu_{F}<2.0M_{W}$, being $M_{W}$ the $W$ mass. Details of the fiducial DPS phase space adopted in the analysis are discussed in Ref. [28] . In particular we found that the total $W$-charge summed DPS cross section (considering both $W^{\prime}$s decays into same sign muons), calculated by means of dPDFs [11] , is found to be $\sigma^{++}+\sigma^{--}[\mbox{fb}]=0.69\pm 0.18(\delta\mu_{F})^{~{}+0.12}_{~{}-0.16}(\delta Q_{0}^{2})$. This result is consistent with those obtained by using for dPDFs the ansatz Eq. (1) with PDFs evaluated with the MSTW parametrization [32] and those obtained with dPDFs of the model described in Ref. [1] . Further results of this analysis are presented in terms of the muon rapidities, i.e., $\eta_{1}\cdot\eta_{2}\simeq 1/4~{}\mbox{ln}(x_{1}/x_{3})\mbox{ln}(x_{2}/x_{4})$. In particular, the DPS cross section converted in per-bin number of events, for a luminosity $\mathcal{L}=300~{}\mbox{fb}^{-1}$, has a maximum for $\eta_{1}\cdot\eta_{2}\sim 0$ where annihilating partons share the momentum fraction in at least one scattering. As one can see in Ref. [28] , results obtained within the LF CQM and with the prescription of Ref. [1] , are compatible within the errors. Moreover, the effects of DPC have been investigated by observing the $\sigma_{eff}$ dependence on $\eta_{1}\cdot\eta_{2}$. Hereafter we call such quantity, evaluated within the LF CQM, $\widetilde{\sigma}_{eff}$. In particular, for this process we found a mean value $\langle\widetilde{\sigma}_{eff}\rangle=21.04\pm 0.07(\delta Q^{2}_{0})^{~{}+0.06}_{~{}-0.07}(\delta\mu_{F})$ mb, consistently with calculations discussed in previous section. However, the most interesting result is shown in Fig. 3. Here, in fact, we appreciate a clear signature of the presence of double correlations by observing the departure of $\widetilde{\sigma}_{eff}$ from a constant. We therefore demonstrated that the experimental analysis of $\sigma_{eff}$ in bins of $\eta_{1}\cdot\eta_{2}$ is a suitable choice for studies on DPC. We have also estimated that, with a luminosity of $\mathcal{L}\sim 1000~{}\mbox{fb}^{-1}$, at 68% confidence level, such a departure of $\sigma_{eff}$ from a constant value can be measured in the next run of the LHC.

5 Conclusions

In this contribution we have shown new results from the calculations of dPDFs evaluated within a LF relativistic quark model. In particular, by using pQCD evolution of dPDFs, we found that longitudinal correlations affect dPDFs at very high momentum transfer and in the small $x$ region, also for gluon-gluon distributions. Moreover, due to the correct treatment of the relativity, thanks to the Light-Front approach, also correlations between $x$ and transverse position $\vec{z}_{\perp}$ are found to be important and largely model independent. The study of correlation effects in $\sigma_{eff}$ shows that in the valence region such a quantity is not constant and depends on $x$ and also using the AdS/QCD correspondence for the calculation of dPDFs the result is confirmed. In order to provide a detailed analysis of $\sigma_{eff}$ in a given process, we studied the same sign $W$ pair production process by calculating its cross section and $\sigma_{eff}$. Results we found are consistent with previous investigations and indicate that in this specific channel, correlations in dPDFs can be observed in the next run of the LHC. This work was supported in part by the Mineco under contract FPA2013-47443-C2-1-P and SEV-2014-0398. This work is also supported though the project “Hadron Physics at the LHC: looking for signatures of multiple parton interactions and quark gluon plasma formation (Gossip project)”, funded by the “Fondo ricerca di base di Ateneo” of the Perugia University. The author thanks the organizers of the conference for the support given for this talk.