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On The Equivalence of Soft and Zero-Bin Subtractions

Thomas Mehen, Ahmad Idilbi

TL;DR

The paper investigates whether zero-bin subtractions in SCET are equivalent to soft Wilson-line subtractions for avoiding double counting in factorization. Using DR-regulated one-loop quark form factor and threshold DIS, it shows that zero-bin and soft subtractions yield the same IR structure and, at one loop, are equivalent at the integrand level; it then extends the Abelian two-loop analysis to show the equivalence persists beyond leading order. The results establish a consistent endpoint factorization for DIS and confirm that soft contributions subtracted from naive collinear matrix elements align with the soft Wilson-line approach, enabling robust resummation via RG. The work highlights regulator dependencies and provides a pathway to all-orders generalization through field redefinitions, strengthening the theoretical foundation of SCET factorization.

Abstract

Zero-bin subtractions are required to avoid double counting soft contributions in collinear loop integrals in Soft-Collinear Effective Theory (SCET). In traditional approaches to factorization, double counting is avoided by dividing jet functions by matrix elements of soft Wilson lines. In this paper, we compare the two approaches to double counting, studying the quark form factor and deep inelastic scattering (DIS) as x_B \to 1 as examples. We explain how the zero-bin subtractions in SCET are required to reproduce the well-established factorization theorem for DIS as x_B \to 1. We study one-loop virtual contributions to the quark form factor and real gluon emission diagrams in DIS. The two approaches to double counting are equivalent if dimensional regularization (DR) is used to regulate infrared (IR) divergences. We discuss in detail ambiguities in the calculation of one-loop scaleless integrals in DR in SCET and perturbative QCD. We also demonstrate a nontrivial check of the equivalence of the zero-bin subtraction and the soft Wilson line subtraction in the virtual two-loop Abelian contributions to the quark form factor.

On The Equivalence of Soft and Zero-Bin Subtractions

TL;DR

The paper investigates whether zero-bin subtractions in SCET are equivalent to soft Wilson-line subtractions for avoiding double counting in factorization. Using DR-regulated one-loop quark form factor and threshold DIS, it shows that zero-bin and soft subtractions yield the same IR structure and, at one loop, are equivalent at the integrand level; it then extends the Abelian two-loop analysis to show the equivalence persists beyond leading order. The results establish a consistent endpoint factorization for DIS and confirm that soft contributions subtracted from naive collinear matrix elements align with the soft Wilson-line approach, enabling robust resummation via RG. The work highlights regulator dependencies and provides a pathway to all-orders generalization through field redefinitions, strengthening the theoretical foundation of SCET factorization.

Abstract

Zero-bin subtractions are required to avoid double counting soft contributions in collinear loop integrals in Soft-Collinear Effective Theory (SCET). In traditional approaches to factorization, double counting is avoided by dividing jet functions by matrix elements of soft Wilson lines. In this paper, we compare the two approaches to double counting, studying the quark form factor and deep inelastic scattering (DIS) as x_B \to 1 as examples. We explain how the zero-bin subtractions in SCET are required to reproduce the well-established factorization theorem for DIS as x_B \to 1. We study one-loop virtual contributions to the quark form factor and real gluon emission diagrams in DIS. The two approaches to double counting are equivalent if dimensional regularization (DR) is used to regulate infrared (IR) divergences. We discuss in detail ambiguities in the calculation of one-loop scaleless integrals in DR in SCET and perturbative QCD. We also demonstrate a nontrivial check of the equivalence of the zero-bin subtraction and the soft Wilson line subtraction in the virtual two-loop Abelian contributions to the quark form factor.

Paper Structure

This paper contains 6 sections, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Zero-Bin at One-Loop: Quark Form Factor
  3. DIS as $x \to 1$: Soft Subtraction
  4. Zero-Bin At Higher Orders
  5. Conclusions
  6. Appendix

Figures (6)

  • Figure 1: The quark form factor in SCET. In the first two graphs the thick line attached to the fermion line represents a collinear Wilson line, while in the last graph both thick lines are soft Wilson lines.
  • Figure 2: Real gluon contribution to the soft factor.
  • Figure 3: Real gluon contribution to the PDF.
  • Figure 4: Real gluon contribution to the outgoing jet function.
  • Figure 5: Zero-Bin Regions At Two-Loops.
  • ...and 1 more figures