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Single and Double Perturbative Splitting Diagrams in Double Parton Scattering

Jonathan R. Gaunt, W. James Stirling

TL;DR

The paper scrutinizes how perturbative parton splitting diagrams contribute to double parton scattering (DPS) cross sections, focusing on single (2v1) and double (1v1) perturbative splitting graphs. It demonstrates that the established dPDF (2pGPD) framework struggles to cleanly separate a genuine DPS piece for 1v1 diagrams, often requiring arbitrary scales and yielding questionable interpretations, and shows that 2v1 graphs introduce LO logarithmic terms with distinct transverse structure. By deriving explicit expressions for 1v2 and analyzing their relation to 2v2, the authors propose including 2v1 contributions alongside 2v2 while cautioning that simple σ_eff rescalings are unreliable and that a consistent framework must account for different b-dependences and nonperturbative input. Overall, the work highlights significant conceptual and technical challenges in formulating a robust, double-counting-free DPS theory that accurately incorporates perturbative splittings and their interplay with nonperturbative correlations. This has important implications for precise DPS background predictions and for understanding parton-parton correlations inside the proton.

Abstract

We discuss the role of two different types of diagram in the proton-proton double parton scattering (DPS) cross section - single and double perturbative splitting graphs. Using explicit calculations of simple graphs from these classes we show that the treatment of these graphs by the 'double PDF' framework for describing the DPS cross section, introduced a number of years ago by Snigirev and collaborators, is unsatisfactory. We suggest that a contribution from single perturbative splitting graphs should be included in the DPS cross section, albeit with a different geometrical prefactor to the contribution from 'zero perturbative splitting' graphs.

Single and Double Perturbative Splitting Diagrams in Double Parton Scattering

TL;DR

The paper scrutinizes how perturbative parton splitting diagrams contribute to double parton scattering (DPS) cross sections, focusing on single (2v1) and double (1v1) perturbative splitting graphs. It demonstrates that the established dPDF (2pGPD) framework struggles to cleanly separate a genuine DPS piece for 1v1 diagrams, often requiring arbitrary scales and yielding questionable interpretations, and shows that 2v1 graphs introduce LO logarithmic terms with distinct transverse structure. By deriving explicit expressions for 1v2 and analyzing their relation to 2v2, the authors propose including 2v1 contributions alongside 2v2 while cautioning that simple σ_eff rescalings are unreliable and that a consistent framework must account for different b-dependences and nonperturbative input. Overall, the work highlights significant conceptual and technical challenges in formulating a robust, double-counting-free DPS theory that accurately incorporates perturbative splittings and their interplay with nonperturbative correlations. This has important implications for precise DPS background predictions and for understanding parton-parton correlations inside the proton.

Abstract

We discuss the role of two different types of diagram in the proton-proton double parton scattering (DPS) cross section - single and double perturbative splitting graphs. Using explicit calculations of simple graphs from these classes we show that the treatment of these graphs by the 'double PDF' framework for describing the DPS cross section, introduced a number of years ago by Snigirev and collaborators, is unsatisfactory. We suggest that a contribution from single perturbative splitting graphs should be included in the DPS cross section, albeit with a different geometrical prefactor to the contribution from 'zero perturbative splitting' graphs.

Paper Structure

This paper contains 3 sections, 10 equations, 2 figures.

Table of Contents

  1. 'Double Perturbative Splitting' Diagrams in Double Parton Scattering
  2. 'Single Perturbative Splitting' Diagrams
  3. The Total Cross Section for DPS

Figures (2)

  • Figure 1: (a) A diagram that apparently contributes to the leading order DPS cross section according to the framework of Snigirev:2003cq. The partons emerging from the grey proton blobs in the figure are nonperturbatively generated partons -- i.e. ones existing at a low scale $\sim \Lambda_{QCD}$. (b) The 'crossed box' graph. In this part of the figure, A and B are arbitrary single particle final states with $Q_1^2 = Q_2^2 = Q^2 > 0$.
  • Figure 2: (a) A generic graph from the '2v1' class. (b) A simple '2v1' graph in which a gluon splits into a $q\bar{q}$ pair, and these partons then interact in two separate Drell-Yan interactions with a 'nonperturbatively generated' $q\bar{q}$ pair from a proton.