Lattice QCD Calculations of Transverse-Momentum-Dependent Soft Function through Large-Momentum Effective Theory

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 $a=0.098\mathrm{fm}$. We extract the rapidity-independent (or intrinsic) part of the soft function through a large-momentum-transfer pseudoscalar 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.

Figure 1

Illustration of the pseudoscalar meson form factor $F$ calculated in this Letter. 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 $b_{\perp }$. $t_{\mathrm{sep}}$ 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 $C_3(b_{\perp },P^z,t_{\mathrm{sep}},t)/C_2(0,P^z,0,t_{\mathrm{sep}})$ (data points) that converge to the ground state contribution at $t,t_{\mathrm{sep}}\to \infty$ (gray band) as function of $t_{\mathrm{sep}}$ 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. (9). With different pion momentum $P^z$, the results are consistent with each other. The dashed curve shows the result of the one-loop calculation [see Eq. (7)], 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) and extracted Collins-Soper kernel (lower), 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 three-loop perturbative calculations (at small $b_{\perp }$) with the strong coupling ${\alpha }_s$ at the scale $1/b_{\perp }$ and also the nonperturbative result from the pion quasi-TMDPDF. Results based on quenched lattice calculations, labeled as “Hermite” and “Bernstein” [25], are also shown for comparison. Errors in the lower panel correspond to the statistical and systematic errors from the nonzero imaginary part, as well as the operator mixing effects.