[Paper Review] Generalized Threshold Factorization with Full Collinear Dynamics

Gillian Lustermans, Johannes K. L. Michel+1 more|arXiv (Cornell University)|Jan 1, 2019

Particle physics theoretical and experimental studies11 citations

TL;DR

This paper introduces a generalized threshold factorization theorem for color-singlet processes in hadronic collisions, valid when one parton momentum fraction xa →1 while the other xb remains generic, corresponding to large rapidity but finite invariant mass. It captures all soft and collinear singularities to all orders in perturbation theory, including flavor-nondiagonal channels, and enables resummation of large-x logarithms crucial for precision PDF fits. The key innovation is a new beam function related to the N-jettiness beam function, which allows accurate prediction of N3LO contributions to Z and Higgs rapidity spectra beyond the classic soft threshold limit.

ABSTRACT

Soft threshold factorization has been used extensively to study hadronic collisions. It is derived in the limit where the momentum fractions $x_{a,b}$ of both incoming partons approach $x_{a,b} o 1$. We present a generalized threshold factorization theorem for color-singlet processes, which holds in the weaker limit of only $x_a o 1$ for generic $x_b$ (or vice versa), corresponding to the limit of large rapidity but generic invariant mass of the produced color singlet. It encodes the complete soft and/or collinear singular structure in the partonic momentum fractions to all orders in perturbation theory, including in particular flavor-nondiagonal partonic channels at leading power. It provides a more powerful approximation than the classic soft threshold limit, capturing a much larger set of contributions. We demonstrate this explicitly for the Z and Higgs rapidity spectrum to NNLO, and we use it to predict a nontrivial set of its N3LO contributions. Our factorization theorem provides the relevant resummation of large-$x$ logarithms in the rapidity spectrum required for resummation-improved PDF fits. One of our factorization ingredients is a new beam function closely related to the N-jettiness beam function. As a byproduct, we identify the correct soft threshold factorization for rapidity spectra among the differing results in the literature.

Motivation & Objective

• To extend soft threshold factorization to the weaker limit where only one parton momentum fraction xa →1, while xb remains generic, corresponding to large rapidity but finite invariant mass.
• To capture the complete soft and collinear singular structure in partonic momentum fractions to all orders in perturbation theory, including flavor-nondiagonal channels at leading power.
• To provide a more powerful approximation than the classic soft threshold limit, enabling improved resummation of large-x logarithms in rapidity spectra.
• To derive a new beam function closely related to the N-jettiness beam function, which is essential for the generalized factorization structure.
• To validate the factorization at NNLO and predict nontrivial N3LO contributions for Z and Higgs rapidity spectra, supporting resummation-improved PDF fits.

Proposed method

• Derive a factorization theorem in Soft-Collinear Effective Theory (SCET) for the generalized threshold limit λ² ∼ 1 − xa ≪ 1 with generic xb, using light-cone coordinates and power counting in λ.
• Identify the relevant SCET modes: p̄n-collinear (for final-state radiation), Pn, P̄n-collinear (for PDFs), and Ps (soft modes), with ultra-soft modes canceling due to identical measurement.
• Construct the factorized cross section as dσ/dxadxb d⃗qT = Hij(Q²) ∫ dt f_thr^i[xa(1 + t/Q² + qT²/(2Q²))] ˜Bj(˜t, xb, ⃗qT), where ˜t = t + qT²/2.
• Introduce a modified beam function ˜Bj(˜t, x) via convolution over transverse momentum, which inherits the µ-evolution of the standard beam function but has different matching coefficients.
• Use known results for beam function matching coefficients up to O(α²s) for quark jets and O(αs) for gluon jets to compute the new ˜Ijk(˜t, z) matching coefficients.
• Validate the factorization numerically at NNLO by comparing predictions to full fixed-order calculations across partonic channels, confirming power-suppressed convergence as 1 − xa → 0.

Experimental results

Research questions

• RQ1 Can a factorization theorem be derived that generalizes soft threshold factorization to the case where only one parton momentum fraction xa →1 while xb remains generic, corresponding to large rapidity but finite invariant mass?
• RQ2 What is the complete structure of soft and collinear singularities in this generalized limit, including flavor-nondiagonal partonic channels at leading power?
• RQ3 How can a new beam function be defined that captures the collinear dynamics in the generalized threshold limit, and how does it relate to the N-jettiness beam function?
• RQ4 To what extent can this generalized factorization predict higher-order contributions, such as N3LO terms, for Z and Higgs rapidity spectra?
• RQ5 Does the new factorization provide a more accurate approximation than the classic soft threshold limit, especially in regions of large rapidity?

Key findings

• The generalized threshold factorization theorem successfully captures all soft and collinear singularities to all orders in perturbation theory, including flavor-nondiagonal channels such as qg →Lq, which vanish in the classic soft threshold limit.
• The new beam function ˜Bj(˜t, x) is derived via convolution over transverse momentum and has the same µ-evolution as the standard beam function, but with distinct matching coefficients that include polylogarithms with fractional weights.
• Numerical validation at NNLO shows excellent agreement between the generalized factorization prediction and the full fixed-order calculation across all partonic channels, with the difference vanishing as a power of 1 − xa.
• For the Drell-Yan process at µ = Q/2, the generalized factorization performs comparably to the standard scale choice, significantly outperforming the soft threshold expansion.
• For gluon-fusion Higgs production, the leading-power generalized approximation captures ~20% of the full NLO cross section, comparable to the missing piece, indicating its importance for processes dominated by central radiation.
• The paper resolves discrepancies in the literature by identifying the correct soft threshold factorization for rapidity spectra, providing a consistent framework for resummation-improved PDF fits.