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An on-shell approach to factorization

Ilya Feige, Matthew D. Schwartz

TL;DR

The paper develops a transparent, on-shell, tree-level factorization for QCD amplitudes in soft and collinear limits using spinor-helicity methods, showing that matrix elements factorize into products of gauge-invariant collinear sector operators and a soft Wilson-line factor, with the leading power controlled by a small parameter $\lambda$. The method relies on momentum power counting, reference-vector free spinor-helicity techniques, and Wilson lines to encode soft and collinear emissions, enabling a clean gauge-invariant factorization without assigning unphysical field power counting. It extends progressively from scalar theories to scalar QED, spinor QED, and finally QCD, deriving both collinear, soft, and soft-collinear factorizations and illustrating with the QCD splitting functions. The results connect naturally to Soft-Collinear Effective Theory (SCET) and offer a pathway to all-order factorization proofs and practical resummation strategies through a clean amplitude-level factorization framework.

Abstract

Factorization is possible due to the universal behavior of Yang-Mills theories in soft and collinear limits. Here, we take a small step towards a more transparent understanding of these limits by proving a form of perturbative factorization at tree- level using on-shell spinor helicity methods. We present a concrete and self-contained expression of factorization in which matrix elements in QCD are related to products of other matrix elements in QCD up to leading order in a power-counting parameter determined by the momenta of certain physical on-shell states. Our approach uses only the scaling of momenta in soft and collinear limits, avoiding any assignment of scaling behavior to unphysical (and gauge-dependent) fields. The proof of factorization exploits many advantages of helicity spinors, such as the freedom to choose different reference vectors for polarizations in different collinear sectors. An advantage of this approach is that once factorization is shown to hold in QCD, the transition to Soft-Collinear Effective Theory is effortless.

An on-shell approach to factorization

TL;DR

The paper develops a transparent, on-shell, tree-level factorization for QCD amplitudes in soft and collinear limits using spinor-helicity methods, showing that matrix elements factorize into products of gauge-invariant collinear sector operators and a soft Wilson-line factor, with the leading power controlled by a small parameter . The method relies on momentum power counting, reference-vector free spinor-helicity techniques, and Wilson lines to encode soft and collinear emissions, enabling a clean gauge-invariant factorization without assigning unphysical field power counting. It extends progressively from scalar theories to scalar QED, spinor QED, and finally QCD, deriving both collinear, soft, and soft-collinear factorizations and illustrating with the QCD splitting functions. The results connect naturally to Soft-Collinear Effective Theory (SCET) and offer a pathway to all-order factorization proofs and practical resummation strategies through a clean amplitude-level factorization framework.

Abstract

Factorization is possible due to the universal behavior of Yang-Mills theories in soft and collinear limits. Here, we take a small step towards a more transparent understanding of these limits by proving a form of perturbative factorization at tree- level using on-shell spinor helicity methods. We present a concrete and self-contained expression of factorization in which matrix elements in QCD are related to products of other matrix elements in QCD up to leading order in a power-counting parameter determined by the momenta of certain physical on-shell states. Our approach uses only the scaling of momenta in soft and collinear limits, avoiding any assignment of scaling behavior to unphysical (and gauge-dependent) fields. The proof of factorization exploits many advantages of helicity spinors, such as the freedom to choose different reference vectors for polarizations in different collinear sectors. An advantage of this approach is that once factorization is shown to hold in QCD, the transition to Soft-Collinear Effective Theory is effortless.

Paper Structure

This paper contains 27 sections, 1 theorem, 148 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Orientation
  3. Power counting momenta
  4. Spinor helicities and power counting
  5. Wilson lines
  6. Scalar field theory
  7. Scalar QED
  8. Collinear Factorization
  9. Soft Factorization
  10. Simultaneous Soft-Collinear Factorization
  11. Soft coherence
  12. General soft-collinear factorization
  13. The position-space picture
  14. Factorization for general $S$-matrix elements
  15. Spinor QED
  16. ...and 12 more sections

Key Result

Lemma 1

Two expressions that are independent of the choice of reference vectors, $r_i$, and that agree at leading power for particular $r_i$, must agree at leading power for any $r_i$.

Figures (2)

  • Figure 1: The physical picture of soft photons emitted by a classical source, $Y_j^\dagger$. The blue oval represents magnification that shows the substructure of the collinear sector which is invisible to the soft photons.
  • Figure 2: The physical picture of collinear photons emitted by classical source, $W_j$. The picture on the left is in the frame of the hard scattering and that on the right is in the frame where each component of the $j$-collinear momenta are the same size, $\lambda$.

Theorems & Definitions (1)

  • Lemma