[Paper Review] Transverse-Momentum-Dependent Wave Functions of Pion from Lattice QCD

Min-Huan Chu, Jin-Chen He+12 more|arXiv (Cornell University)|Feb 20, 2023

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper presents the first ab initio lattice QCD calculation of the pion's transverse-momentum-dependent wave functions (TMDWFs) using large-momentum effective theory (LaMET). By combining quasi-TMDWFs from lattice simulations with a nonperturbatively determined soft function and extrapolating to infinite momentum, the authors extract light-cone TMDWFs, achieving consistency across two distinct lattice ensembles and providing a crucial first-principles input for exclusive processes in QCD factorization.

ABSTRACT

We present a first lattice QCD calculation of the transverse-momentum-dependent wave functions (TMDWFs) of the pion using large-momentum effective theory. Numerical simulations are based on one ensemble with 2+1+1 flavors of highly improved staggered quarks action with lattice spacing $a=0.121$~fm from the MILC Collaboration, and one with 2 +1 flavor clover fermions and tree-level Symanzik gauge action generated by the CLS Collaboration with $a=0.098$~fm. As a key ingredient, the soft function is first obtained by incorporating the one-loop perturbative contributions and a proper normalization. Based on this and the equal-time quasi-TMDWFs simulated on the lattice, we extract the light-cone TMDWFs. The results are comparable between the two lattice ensembles and a comparison with phenomenological parametrization is made. Our studies provide a first attempt of $ab$ $initio$ calculation of TMDWFs which will eventually lead to crucial theory inputs for making predictions for exclusive processes under QCD factorization.

Motivation & Objective

• To compute the transverse-momentum-dependent wave functions (TMDWFs) of the pion from first principles in lattice QCD.
• To overcome the challenge of rapidity divergences in TMDWF definitions by employing large-momentum effective theory (LaMET) and nonperturbative soft function matching.
• To provide a reliable, systematic, and controlled theoretical input for exclusive hadronic processes under QCD factorization.
• To reduce phenomenological uncertainties by replacing model-dependent parametrizations with first-principles lattice results.

Proposed method

• Simulate equal-time quasi-TMDWFs on two lattice ensembles: one with 2+1+1 flavors of highly improved staggered quarks (MILC, a=0.121 fm) and one with 2+1 flavor clover fermions (CLS, a=0.098 fm).
• Compute the soft function nonperturbatively by incorporating one-loop perturbative corrections and ensuring proper normalization.
• Perform large-λ extrapolation of the quasi-TMDWFs using a joint fit across multiple b⊥ values to control systematic uncertainties from the regulator.
• Extrapolate the quasi-TMDWFs to infinite momentum (Pz→∞) using a 1/(Pz)² fitting form to extract the light-cone TMDWFs.
• Estimate systematic uncertainties from both large-λ and infinite-Pz extrapolations, combining them quadratically with statistical errors.
• Use the matching kernel (Collins-Soper kernel) to relate the quasi-TMDWFs to the physical light-cone TMDWFs, with the kernel computed nonperturbatively on the CLS ensemble.

Experimental results

Research questions

• RQ1 Can transverse-momentum-dependent wave functions of the pion be computed from first principles in lattice QCD using LaMET?
• RQ2 How do the results for TMDWFs from two distinct lattice ensembles (MILC and CLS) compare in terms of consistency and systematic control?
• RQ3 What is the impact of large-λ and infinite-Pz extrapolations on the final TMDWF results, and how are these uncertainties quantified?
• RQ4 How do the real and imaginary parts of the TMDWFs behave with increasing transverse distance b⊥, and what explains their different convergence patterns?
• RQ5 To what extent do the lattice results agree with phenomenological parametrizations of the pion TMDWFs?

Key findings

• The real parts of the TMDWFs Ψ± decrease with increasing b⊥ on both MILC and CLS ensembles, indicating a suppression of large transverse momentum components.
• The imaginary parts of the TMDWFs increase with b⊥ and stabilize for b⊥ > 0.36 fm, with Ψ− showing convergence at large b⊥ while Ψ+ does not, due to the sign difference in the logarithmic term in the hard kernel.
• The results for Ψ− on the CLS ensemble are smaller than on MILC at small b⊥ (b⊥ < 0.3 fm), likely due to discretization effects from the finer lattice spacing.
• The large-λ extrapolation is well-controlled, with fits matching original data and uncertainties increasing only at large λ, and the extrapolation form is validated via alternative λL choices.
• The infinite-Pz extrapolation using a 1/(Pz)² fit form successfully approaches the physical limit, with the difference between the largest Pz and the extrapolated value used as a systematic uncertainty.
• The final TMDWFs, including statistical and systematic uncertainties, show consistency between the two lattice ensembles, validating the methodological approach for first-principles TMDWF calculations.