Momentum fraction and hard scale dependence of double parton scattering

TL;DR

The paper addresses the nonuniversal nature of the DPS effective cross section $\sigma_{\text{eff}}$ by modeling the double parton distribution as a product of single PDFs and a transverse distribution with width $B(x,\mu)$. A Gaussian transverse profile is used with $B(x,\mu)=\beta+\gamma_1 H(x_0-x)\ln(x_0/x)+\gamma_2 H(x-x_v)+\kappa\ln(\mu/\mu_0)$, and the scale dependence is incorporated through $\mu_{AB}=\sqrt{\mu_A\mu_B}$, leading to a nonuniversal $\sigma_{\text{eff}}(AB)$ via the overlap factor $\Theta$. A global fit to 26 DPS measurements across collider experiments yields best-fit parameters that imply large parton separations at small $x$, reduced separation near $x\sim10^{-2}$, and valence-like separation at large $x$, with a logarithmic rise with the hard scale $\mu$. The results significantly improve the description of DPS data over universal or constant-$\sigma_{\text{eff}}$ approaches and provide testable predictions for unmeasured final states and forward kinematics, deepening our understanding of the proton's transverse structure.

Abstract

The effective cross section of double parton scattering in high-energy hadron collisions has been measured in proton--proton collisions, with significant variation among final-state observables, contrary to the idea of a universal value. Building upon our previous work, we incorporate the dependence on both the parton longitudinal momentum fraction $x$ and the process energy hard scale $μ$ into the transverse part of the double parton distributions, using a Gaussian profile. Employing the experimental data from the LHC and Tevatron experiments (covering different processes, kinematic configurations, and center--of--mass energies), we perform a global fit of the model, extracting the parameters that describe the proton structure. With this result, it becomes possible to calculate the effective cross section for others observables, and we provide predictions for future measurements at the LHC.

Figures (3)

• Figure 1: Double parton scattering effective cross section $\sigma_{\text{eff}}(AB)$ measurements CDF:1993sbjCDF:1997yfaATLAS:2013aphD0:2014owyD0:2014vqlCMS:2013huwD0:2015rpoCMS:2015wcfATLAS:2016rndLHCb:2015wvuATLAS:2016ydtLHCb:2016wuoLansberg:2017chqLansberg:2019adrCMS:2019jcbCMS:2021lxiCMS:2022pioALICE:2023lsnLeontsinis:2022cyiLHCb:2023qguLHCb:2023ybt compared to the results from our fitted model.
• Figure 2: Parameterization of the variance function $B(x,\mu)$ with scales fixed by the masses $\mu_{AB} \equiv m_{J/\psi} = 3.096$ GeV (blue line) and $\mu_{AB} \equiv m_{W} = 80.377$ GeV (red line). At small and large values of the longitudinal momentum fraction $x$, the variance of the Gaussian distribution increases, indicating that the partons within the pair are more widely separated in the transverse plane.
• Figure 3: Predictions of $\sigma_{\text{eff, DPS}}$ in $pp$ collisions for various final states from the global fit of our model. The kinematic cuts are the same as those used by the experiments in DPS analyses (when available) or in corresponding SPS analyses. The purple arrows indicate upper or lower experimental limits for D0 Ref. Shao:2016wor, ATLAS Ref. Lansberg:2016muq, and UA2 Ref. Alitti1991ASO.