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Resummation of Double-Differential Cross Sections and Fully-Unintegrated Parton Distribution Functions

Massimiliano Procura, Wouter J. Waalewijn, Lisa Zeune

TL;DR

The paper develops a unified SCET-based framework (SCET_+) to resum double-differential cross sections across multiple phase-space regions, connecting SCET_I and SCET_II via intermediate collinear-soft modes. It provides NNLL ingredients including fully-unintegrated PDFs, TMD beam functions, FU and collinear-soft soft functions, and their rapidity- and virtuality-evolution, and validates the approach with an NLO cross section for Z+0 jets and a cross-check against angularities on a single jet. The work demonstrates continuous interpolation across regions, ensuring consistency with known boundary theories and improving predictive power for multi-variable LHC observables. It also corrects prior NLL cross-section conjectures and lays groundwork for broader multi-variable resummations and more robust MC implementations.

Abstract

LHC measurements involve cuts on several observables, but resummed calculations are mostly restricted to single variables. We show how the resummation of a class of double-differential measurements can be achieved through an extension of Soft-Collinear Effective Theory (SCET). A prototypical application is $pp \to Z + 0$ jets, where the jet veto is imposed through the beam thrust event shape ${\mathcal T}$, and the transverse momentum $p_T$ of the $Z$ boson is measured. A standard SCET analysis suffices for $p_T \sim m_Z^{1/2} {\mathcal T}^{1/2}$ and $p_T \sim {\mathcal T}$, but additional collinear-soft modes are needed in the intermediate regime. We show how to match the factorization theorems that describe these three different regions of phase space, and discuss the corresponding relations between fully-unintegrated parton distribution functions, soft functions and the newly defined collinear-soft functions. The missing ingredients needed at NNLL/NLO accuracy are calculated, providing a check of our formalism. We also revisit the calculation of the measurement of two angularities on a single jet in JHEP 1409 (2014) 046, finding a correction to their conjecture for the NLL cross section at ${\mathcal O}(α_s^2)$.

Resummation of Double-Differential Cross Sections and Fully-Unintegrated Parton Distribution Functions

TL;DR

The paper develops a unified SCET-based framework (SCET_+) to resum double-differential cross sections across multiple phase-space regions, connecting SCET_I and SCET_II via intermediate collinear-soft modes. It provides NNLL ingredients including fully-unintegrated PDFs, TMD beam functions, FU and collinear-soft soft functions, and their rapidity- and virtuality-evolution, and validates the approach with an NLO cross section for Z+0 jets and a cross-check against angularities on a single jet. The work demonstrates continuous interpolation across regions, ensuring consistency with known boundary theories and improving predictive power for multi-variable LHC observables. It also corrects prior NLL cross-section conjectures and lays groundwork for broader multi-variable resummations and more robust MC implementations.

Abstract

LHC measurements involve cuts on several observables, but resummed calculations are mostly restricted to single variables. We show how the resummation of a class of double-differential measurements can be achieved through an extension of Soft-Collinear Effective Theory (SCET). A prototypical application is jets, where the jet veto is imposed through the beam thrust event shape , and the transverse momentum of the boson is measured. A standard SCET analysis suffices for and , but additional collinear-soft modes are needed in the intermediate regime. We show how to match the factorization theorems that describe these three different regions of phase space, and discuss the corresponding relations between fully-unintegrated parton distribution functions, soft functions and the newly defined collinear-soft functions. The missing ingredients needed at NNLL/NLO accuracy are calculated, providing a check of our formalism. We also revisit the calculation of the measurement of two angularities on a single jet in JHEP 1409 (2014) 046, finding a correction to their conjecture for the NLL cross section at .

Paper Structure

This paper contains 21 sections, 113 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Factorization
  3. Effective Theory for the Region between ${\rm SCET}_{\rm I}$ and ${\rm SCET}_{\rm II}$ Boundaries
  4. Factorization Formulae
  5. Ingredients at NNLL
  6. Hard Function
  7. Beam Functions
  8. Soft Functions
  9. Collinear-Soft Function
  10. Renormalization and Anomalous Dimensions
  11. NLL Cross Section
  12. Matching the Effective Theories
  13. NLO Cross Section
  14. Ingredients
  15. Cancellation of IR divergences
  16. ...and 6 more sections

Figures (3)

  • Figure 1: The different regions for the double measurement of $p_T$ and beam thrust ${\mathcal{T}}$ in $Z$-boson production from $pp$ collisions.
  • Figure 2: The modes in ${\rm SCET}_{\rm I}$, ${\rm SCET}_+$ and ${\rm SCET}_{\rm II}$: collinear (green), collinear-soft (blue) and soft (orange). Interactions between modes in the effective theory are shown with wiggly lines. These are removed by the decoupling transformations in eq. \ref{['eq:BPS']}.
  • Figure 3: The $\mu$-evolution resums double logarithms from separations in virtuality (between hyperbolae), while the $\nu$-evolution resums single logarithms related to separations in rapidity (along hyperbolae). The collinear, collinear-soft and soft modes are depicted in green, blue and orange, respectively.