[Paper Review] QCD Soft Function from Large-Momentum Effective Theory on Lattice

Xiangdong Ji, Yusheng Liu+1 more|arXiv (Cornell University)|Oct 24, 2019

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper proposes a lattice QCD method to calculate the QCD soft function using large-momentum effective theory (LMEFT), formulating the transverse momentum-dependent (TMD) soft function as a form factor of color sources with nearly-lightlike velocities. The approach enables extraction of the soft function via quasi-TMD wave functions and factorization in large-momentum light-meson form factors, offering a non-perturbative lattice framework for TMD factorization.

ABSTRACT

We study Euclidean lattice calculation of the QCD soft function, which involves two conjugate lightcone directions in the framework of large-momentum effective theory. We find that the transverse momentum dependent (TMD) soft function required by TMD factorization can be formulated as the form factor of a pair of color sources traveling with nearly-lightlike velocities, and thus can be calculated using lattice heavy-quark effective theory. A simple generalization shows that the factorization of a large-momentum light-meson form factor combining with quasi-TMD wave function can also be used to extract the soft function on lattice.

Motivation & Objective

• To develop a non-perturbative lattice method for computing the QCD soft function, essential for transverse momentum-dependent (TMD) factorization in high-energy processes.
• To address the challenge of formulating the TMD soft function in Euclidean lattice field theory, where standard Minkowski space techniques are inapplicable.
• To establish a connection between the soft function and form factors of color sources moving at nearly-lightlike velocities, enabling lattice computation.
• To generalize the framework to extract the soft function from large-momentum light-meson form factors combined with quasi-TMD wave functions.

Proposed method

• Formulate the TMD soft function as the matrix element of a pair of color sources with nearly-lightlike velocities, enabling its representation as a form factor in lattice QCD.
• Apply techniques from lattice heavy-quark effective theory (HQET) to compute the form factor of these color sources, leveraging the heavy-quark limit for numerical stability.
• Utilize large-momentum effective theory (LMEFT) to relate the soft function to the factorization of large-momentum light-meson form factors.
• Generalize the approach to include quasi-TMD wave functions, allowing the soft function to be extracted from the factorized form of the light-meson form factor.
• Ensure Lorentz invariance and Euclidean signature compatibility by embedding the soft function in a large-momentum frame with boosted color sources.
• Construct the relevant correlation functions in Euclidean space that encode the soft function through matrix elements of local operators.

Experimental results

Research questions

• RQ1 How can the TMD soft function be formulated in a way compatible with Euclidean lattice field theory?
• RQ2 Can the soft function be extracted from the factorization of large-momentum light-meson form factors using quasi-TMD wave functions?
• RQ3 What is the role of nearly-lightlike color sources in defining the soft function within the framework of large-momentum effective theory?
• RQ4 How does the lattice implementation of the soft function via HQET techniques ensure consistency with TMD factorization theorems?
• RQ5 What is the generalization potential of this method to other hadronic processes requiring non-perturbative soft functions?

Key findings

• The TMD soft function is successfully formulated as a form factor of two color sources with nearly-lightlike velocities, enabling its lattice computation.
• The method leverages lattice HQET techniques to compute the soft function in a stable and controlled manner, avoiding issues in Minkowski space.
• The soft function can be extracted from the factorization of large-momentum light-meson form factors when combined with quasi-TMD wave functions.
• The framework provides a non-perturbative lattice access to the soft function, a key ingredient in TMD factorization theorems.
• The approach generalizes to other processes involving soft gluon radiation, offering a systematic path to non-perturbative TMD functions.
• The formulation ensures compatibility with the factorization structure required by TMD factorization, preserving gauge invariance and Lorentz covariance in the lattice setup.