Nucleon transverse momentum-dependent parton distributions in lattice QCD: Renormalization patterns and discretization effects

Lattice QCD calculations of transverse momentum-dependent parton distribution functions (TMDs) in nucleons are presented, based on the evaluation of nucleon matrix elements of quark bilocal operators with a staple-shaped gauge connection. Both time-reversal odd effects, namely, the generalized Sivers and Boer-Mulders transverse momentum shifts, as well as time-reversal even effects, namely, the generalized transversity and one of the generalized worm-gear shifts, are studied. Results are obtained on two different $n_f=2+1$ flavor ensembles with approximately matching pion masses but very different discretization schemes: domain-wall fermions (DWF) with lattice spacing $a=0.084\mathrm{fm}$ and pion mass 297 MeV, and Wilson-clover fermions with $a=0.114\mathrm{fm}$ and pion mass 317 MeV. Comparison of the results on the two ensembles yields insight into the length scales at which lattice discretization errors are small, and into the extent to which the renormalization pattern obeyed by the continuum QCD TMD operator continues to apply in the lattice formulation. For the studied TMD observables, the results are found to be consistent between the two ensembles at sufficiently large separation of the quark fields within the operator, whereas deviations are observed in the local limit and in the case of a straight link gauge connection, which is relevant to the studies of parton distribution functions. Furthermore, the lattice estimates of the generalized Sivers shift obtained here are confronted with, and are seen to tend towards, a phenomenological estimate extracted from experimental data.

Figure 1

Illustration of the TMD operator with staple-shaped gauge connection. The four-vectors $v$ and $P$ give the direction of the staple and the momentum, while $b$ defines the separation between the quark operators. The values of these variables used in the lattice calculation are given in Table 2. In the present calculation, $b·P=b·v=0$ is chosen, corresponding to evaluating the first moment of the TMDs with respect to the quark momentum fraction $x$.

Figure 2

Dependence of the generalized Sivers shift on the staple extent $\eta |\mathit{v}|$ for the clover (left) and the DWF (right) ensembles at $|{\mathit{b}}_{\mathrm{T}}|=3a$ (top) and $4a$ (bottom). The Collins-Soper parameter is fixed at the highest value for which data are available, $\widehat{\zeta }=0.41$ and 0.32 for the DWF and the clover ensembles, respectively. The line shows the central value obtained using a constant fit to the $\pm \eta |\mathit{v}|$ data analyzed separately. The asymptotic estimate (shown using the diamond symbol in the margins on the left and right) is obtained by averaging the $\pm \eta |\mathit{v}|$ fit results and the error is obtained by a jackknife procedure.

Figure 3

Dependence of the generalized Sivers shift on $|{\mathit{b}}_{\mathrm{T}}|$. In the left panel we compare DWF and clover results for $\widehat{\zeta }\approx 0.3$ and in the right panel we show the higher precision clover data for three values of $\widehat{\zeta }$. The shaded area, $|{\mathit{b}}_{\mathrm{T}}|\le 3a_{\mathrm{DWF}}\approx 0.25\mathrm{fm}$, marks the region in which discretization effects could be expected.

Figure 4

Dependence of the generalized Sivers shift on $\widehat{\zeta }$ for the two different ensembles and for two values of $|{\mathit{b}}_{\mathrm{T}}|=0.34\mathrm{fm}$ (left) and 0.68 fm (right).

Figure 5

Dependence of the generalized Boer-Mulders shift on the staple extent $\eta |\mathit{v}|$ for the clover (left) and the DWF (right) ensembles. The rest is the same as in Fig. 2.

Figure 6

Dependence of the generalized Boer-Mulders shift on $|{\mathit{b}}_{\mathrm{T}}|$ for the two ensembles (left), and for three different values of $\widehat{\zeta }$ analyzed on the clover ensemble (right). The rest is the same as in Fig. 3.

Figure 7

Dependence of the generalized Boer-Mulders shift on $\widehat{\zeta }$ for two values of $|{\mathit{b}}_{\mathrm{T}}|=0.34\mathrm{fm}$ (left) and 0.68 fm (right).

Figure 8

Dependence of the transversity ratio ${\tilde{h}}_1^{[1](0)}/{\tilde{f}}_1^{[1](0)}$ on the staple extent $\eta |\mathit{v}|$ for the clover (left) and the DWF (right) ensembles. The rest is the same as in Fig. 2.

Figure 9

Dependence of the transversity ratio ${\tilde{h}}_1^{[1](0)}/{\tilde{f}}_1^{[1](0)}$ on $|{\mathit{b}}_{\mathrm{T}}|$ for the two ensembles (left), and for three different values of $\widehat{\zeta }$ analyzed on the clover ensemble (right). The rest is the same as in Fig. 3.

Figure 10

Dependence of the transversity ratio ${\tilde{h}}_1^{[1](0)}/{\tilde{f}}_1^{[1](0)}$ on $\widehat{\zeta }$ for two values of $|{\mathit{b}}_{\mathrm{T}}|=0.34\mathrm{fm}$ (left) and 0.68 fm (right).

Figure 11

Dependence of the generalized $g_{1T}$ worm-gear shift on the staple extent $\eta |\mathit{v}|$ for the clover (left) and the DWF (right) ensembles. The rest is the same as in Fig. 2.

Figure 12

Dependence of the generalized $g_{1T}$ worm-gear shift on $|{\mathit{b}}_{\mathrm{T}}|$ for the two ensembles (left), and for three different values of $\widehat{\zeta }$ analyzed on the clover ensemble (right). The rest is the same as in Fig. 3.

Figure 13

Dependence of the generalized $g_{1T}$ worm-gear shift on $\widehat{\zeta }$ for two values of $|{\mathit{b}}_{\mathrm{T}}|=0.34\mathrm{fm}$ (left) and 0.68 fm (right).

Figure 14

Dependence of the transversity (left) and generalized $g_{1T}$ worm-gear shift (right) on the length of the straight-link paths, $|{\mathit{b}}_{\mathrm{T}}|$, for the two different ensembles. The striking observation is that the difference between the DWF and clover data for the worm-gear shift persists for all $|{\mathit{b}}_{\mathrm{T}}|$. The data shown are for nucleon momentum $|\mathit{P}|=2\pi /(aL)$; results for $\mathit{P}=0$ coincide with these data within the uncertainties shown.

Figure 15

Experimental extraction of the SIDIS generalized Sivers shift at $\widehat{\zeta }=0.83$, together with lattice QCD data in the SIDIS limit, $\eta \to \infty$, as a function of the Collins-Soper parameter $\widehat{\zeta }$. Lattice data for $|{\mathit{b}}_{\mathrm{T}}|\approx 0.35\mathrm{fm}$ are given in the left panel where we have included results from an earlier DWF-on-Asqtad study given in Ref. [9]. Results for $|{\mathit{b}}_{\mathrm{T}}|\approx 0.68\mathrm{fm}$ are given in the right panel.