Lattice-QCD Calculations of TMD Soft Function Through Large-Momentum Effective Theory

TL;DR

This study presents the first lattice-QCD determination of the intrinsic TMD soft function $S_I(b_\perp,\mu)$ using LaMET on a $2{+}1$ flavor CLS ensemble, extracting it from a large-momentum form factor of a light pseudoscalar meson and its ratio with the quasi-TMDWF. The Collins-Soper kernel $K(b_\perp,\mu)$ is obtained from the momentum evolution of the quasi-TMDWF, with results showing mild $P^z$-dependence and consistency with perturbative expectations at small $b_\perp$ and with previous quenched calculations. A two-state fit framework controls excited-state contamination, and nonperturbative renormalization reveals operator mixing up to a few percent at moderate $b_\perp$, which is incorporated as a systematic uncertainty. Overall, the work demonstrates a viable path to first-principles predictions for Drell-Yan and related processes at small transverse momentum, using lattice QCD to provide nonperturbative inputs for TMD factorization.

Abstract

The transverse-momentum-dependent (TMD) soft function is a key ingredient in QCD factorization of Drell-Yan and other processes with relatively small transverse momentum. We present a lattice QCD study of this function at moderately large rapidity on a 2+1 flavor CLS dynamic ensemble with $a=0.098$ fm. We extract the rapidity-independent (or intrinsic) part of the soft function through a large-momentum-transfer pseudo-scalar meson form factor and its quasi-TMD wave function using leading-order factorization in large-momentum effective theory. We also investigate the rapidity-dependent part of the soft function---the Collins-Soper evolution kernel---based on the large-momentum evolution of the quasi-TMD wave function.

Figures (14)

• Figure 1: Illustration of the pseudo-scalar meson form factor $F$ calculated in this work. The initial and final momenta of the pion are large and opposite. The transition "current" is made of two local operators at a fixed spatial separation $b_\perp$. $t_{\rm sep}$ is the time separation between the source and sink of the pion.
• Figure 2: Results for the $\ell$ dependence of the quasi-TMDWF with $z=0$, and also the square root of the Wilson loop which is used for the subtraction, taking the $\{P^z, b_{\perp}, t\}=\{6\pi/L, 3a, 6a\}$ case as a example. All the results are normalized with their values at $\ell=0$.
• Figure 3: The ratios $C_3(b_\perp,P^z,t_{\rm sep},t)/C_2(0, P^z,0,t_{\rm sep})$ (data points) which converge to the ground state contribution at $t,t_{\rm sep}\rightarrow \infty$ (gray band) as function of $t_{\rm sep}$ and $t$, with $\{P^z, b_{\perp}\}=\{6\pi/L, 3a\}$. As in this figure, our data in general agree with the predicted fit function (colored bands).
• Figure 4: The intrinsic soft factor as a function of $b_{\perp}$ with $b_{\perp,0}=a$ as in Eq. \ref{['eq:S_ratio_z=0']}. With different pion momentum $P^z$, the results are consistent with each other. The dashed curve shows the result of the 1-loop calculation, see Eq. (\ref{['eq:S_ratio_1loop']}), with the strong coupling constant $\alpha_s(1/b_{\perp})$. The shaded band corresponds to the scale uncertainty of $\alpha_s$: $\mu\in [1/\sqrt{2},\sqrt{2}]\times 1/b_\perp$. The systematic uncertainty from the operator mixing has been taken into account.
• Figure 5: Quasi-TMDWF (upper panel) and extracted Collins-Soper kernel (lower panel), as functions of $b_{\perp}$. The visible $P^z$ dependence of the quasi-TMDWF can be primarily understood by that from the Collins-Soper kernel, as the kernel we obtained with tree level matching is consistent with up to 3-loop perturbative calculations (at small $b_{\perp}$) with the strong coupling $\alpha_s$ at the scale $1/b_{\perp}$, and also the non-perturbative result from the pion quasi-TMDPDF. Results based on quenched lattice calculations, labeled as "Hermite" and "Bernstein" Shanahan:2020zxr, are also shown for comparison. Errors in the lower panel correspond to the statistical errors and the systematic errors from the non-zero imaginary part as well as the operator mixing effects.