Improving on the Pythia modelling of equal-scale multi-parton distribution functions

TL;DR

This work tackles the problem of constructing equal-scale multi-parton distribution functions (mPDFs) that are symmetric under parton exchange and satisfy the GS sum rules. Building on the PYTHIA MPI framework, the authors introduce the X-ordered mPDF formulation, adding three physically motivated modifications: an $x$-ordering with smoothing, a revised symmetric companion mechanism, and a scale-dependent damping that suppresses configurations with multiple small-$x$ partons. They demonstrate that the resulting $dPDF$s at $Q=M_Z$ satisfy the sum rules to within ~10% across much of phase space, with similar improvements for $tPDF$s, and show that evolution via the inhomogeneous double DGLAP equation further stabilizes these relations. A toy DPS study of same-sign $WW$ and $ZZ$ production indicates that damping and the companion modification can noticeably alter rapidity asymmetries, illustrating practical implications for MPI modeling and potential integration into event generators. The results provide a symmetric, sum-rule-respecting framework for equal-scale mPDFs and offer concrete guidance for incorporating these ideas into phenomenology and future MC implementations, including extensions to $y$-dependence and unequal-scale generalizations.

Abstract

Multi-parton distribution functions (mPDFs) are non-perturbative objects that are important in the prediction of multiple scattering rates at hadron colliders. In the case where the scales associated with all partons in the mPDF are the same, we have two theoretical constraints on the mPDF. These are symmetry in exchange of the parton indices, and the number and momentum sum rules. In a previous publication (arXiv:2208.08197) we found that the equal-scale mPDFs from the Pythia model could not satisfy both of these constraints simultaneously. In this paper we introduce an algorithm for constructing equal-scale mPDFs that is based on the Pythia procedure but has three additional modifications, such that it yields symmetric mPDFs that should satisfy the sum rules to an improved extent. We test the construction for the case of the double and triple parton distribution functions (dPDFs and tPDFs), finding that the sum rules are obeyed to within 10% over the vast majority of the phase space for the scales tested (and deviations only being mildly above this level). We use our dPDFs to compute rapidity asymmetries for same-sign WW and ZZ production via double parton scattering, and compare the results to predictions obtained using Pythia and the GS09 dPDFs of arXiv:0910.4347.

Figures (22)

• Figure 1: Behavior of the Number Sum Rule integrals for all parton flavors in the X-ordered dPDF model at $Q=M_Z$, as defined in Section \ref{['sec:Xord_dPDF_construction']}.
• Figure 2: Behavior of the Momentum Sum Rule integrals for all parton flavors in the X-ordered dPDF model at $Q=M_Z$, as defined in Section \ref{['sec:Xord_dPDF_construction']}.
• Figure 3: Behavior of the $\bar{d}u_v$ Number Sum Rule response function for our X-ordered dPDFs and for the PYTHIA (unsymmetrised) dPDFs, at $x_1=\{10^{-6},10^{-2}, 10^{-1}\}$. The bottom panels give the difference between the X-ordered and PYTHIA response functions, $\Delta_{\bar{d}u_v}$.
• Figure 4: Behavior of the Number Sum Rule response functions for $\bar{u}u_v$.
• Figure 5: Behavior of the Number Sum Rule integrals for all parton flavors in the X-ordered dPDF model at $Q=2$ GeV.