The Zero-Bin and Mode Factorization in Quantum Field Theory

TL;DR

The paper develops a zero-bin subtraction framework to systematically avoid double counting across overlapping infrared momentum regions in NRQCD and SCET. It demonstrates how pinch and endpoint singularities arise from improper mode counting and shows that proper zero-bin subtractions convert many IR divergences into UV ones, enabling consistent renormalization and RG evolution. Through extensive NRQCD and SCET_I/II examples, including both cutoff and dimensional regulators, it introduces ø-distributions and rapidity scales μ_± to achieve finite convolutions and genuine factorization in rapidity space. The results provide regulator-independent, complete tiling of IR regions and resolve longstanding puzzles around convolutions in QCD factorization, with clear implications for exclusive and inclusive processes in high-energy QCD.

Abstract

We study a Lagrangian formalism that avoids double counting in effective field theories where distinct fields are used to describe different infrared momentum regions for the same particle. The formalism leads to extra subtractions in certain diagrams and to a new way of thinking about factorization of modes in quantum field theory. In non-relativistic field theories, the subtractions remove unphysical pinch singularities in box type diagrams, and give a derivation of the known pull-up mechanism between soft and ultrasoft fields which is required by the renormalization group evolution. In a field theory for energetic particles, the soft-collinear effective theory (SCET), the subtractions allow the theory to be defined with different infrared and ultraviolet regulators, remove double counting between soft, ultrasoft, and collinear modes, and give results which reproduce the infrared divergences of the full theory. Our analysis shows that convolution divergences in factorization formulæoccur due to an overlap of momentum regions. We propose a method that avoids this double counting, which helps to resolve a long standing puzzle with singularities in collinear factorization in QCD. The analysis gives evidence for a factorization in rapidity space in exclusive decays.

Figures (17)

• Figure 1: Comparison of two setups for the soft and usoft contributions to an NRQCD Feynman graph. In a) the $1/\epsilon$ divergences at the intermediate scale $mv$ cancel between the soft and usoft contributions. In b) there are no IR divergences at this intermediate scale.
• Figure 2: Toy model to illustrate the scales captured by an effective theory with multiple low energy modes (here $q_a$, $q_b$, and $q_c$).
• Figure 3: a) Scales and momentum modes for nonrelativistic field theories like NRQCD. Here s, p, and u denote soft, potential, and usoft respectively.
• Figure 4: a) Scales and momentum modes for ${\rm SCET}_{\rm I}$. Here $cn$, $c{\bar{n}}$, and $u$ denote collinear-$n$, collinear-${\bar{n}}$, and usoft modes respectively and $\lambda\sim \sqrt{\Lambda_{\rm QCD}/Q}$. b) Scales and momentum modes for ${\rm SCET}_{\rm II}$. Here $cn$, $c{\bar{n}}$, and $s$ denote collinear-$n$, collinear-${\bar{n}}$, and soft modes respectively and $\eta\sim \Lambda_{\rm QCD}/Q$.
• Figure 5: Label and Residual momenta for a) NRQCD quarks and b) ${\rm SCET}_{\rm I}$. In both cases $p$ denotes a large momentum, and labels a particular box, whereas $k$ is a small momentum, and gives the final momentum location relative to the reference momentum point in the box labeled by $p$.