RI/MOM renormalization of the parton quasidistribution functions in lattice regularization

We analyze the lattice spacing dependence for the pion unpolarized matrix element of a quark bilinear operator with a Wilson link (parton quasidistribution functions operator, quasi-PDFs) in the rest frame, using 13 lattice spacings ranging from 0.032 to 0.121 fm. We compare results for three different fermion actions with or without good chiral symmetry on dynamical gauge ensembles from three collaborations. This investigation is motivated by the fact that the gauge link generates a $1/a$ divergence, the cancellation of which in many ratios can be numerically tricky. Indeed, our results show that this cancellation deteriorates with decreasing lattice spacing, and that the RI/MOM method leaves a linearly divergent residue for quasi-PDFs. We also show that in the Landau gauge the interaction between the Wilson link and the external state results in a linear divergence which depends on the discretized fermion action.

Figure 1

The bare ratios $R_{\pi }(T/2,t,z;a)$ defined in Eq. (2) for the pion using overlap (OV) fermions on HISQ sea ensembles at different lattice spacings, for the cases at $z\simeq 0.36$, 0.72, and 1.08 fm. There are clear plateaus in the region $t-T/4\in [-1,1]\mathrm{fm}$.

Figure 2

The ratios between the pion matrix element $h_{\pi ,{\gamma }_t}(z)$ using clover (CL) fermions and overlap (OV) fermions, on the HISQ sea ensembles at different lattice spacings. The ratios are consistent with 1 within $2\sigma$ at all lattice spacings.

Figure 3

The real part of the relative mixing between $O_{{\gamma }_t}(z)$ and $O_{\mathrm{\Gamma }\ne {\gamma }_t}(z)$, using clover fermions on the HISQ sea ensemble with $a=0.121\mathrm{fm}$. The upper panel shows the $O_{{\gamma }_t}$ matrix element with a $\mathrm{\Gamma }\ne {\gamma }_t$ projector, and the lower panel shows the $O_{\mathrm{\Gamma }\ne {\gamma }_t}$ matrix element with a ${\gamma }_t$ projector. In all cases, the results are consistent with zero within less than 0.1% statistical uncertainty.

Figure 4

The $z$-dependent ratio $R_{\mathrm{RI}/\mathrm{MOM}}(z,\mu ,{\mu }^{\prime })$ defined in Eq. (11) with $\mu \simeq 3\mathrm{GeV}$ and ${\mu }^{\prime }\simeq 1.8\mathrm{GeV}$ (upper panel) and also $R_{\mathrm{RI}/\mathrm{MOM}}(z,\mu ,{\mu }^{\prime \prime })$ with ${\mu }^{\prime \prime }=0\mathrm{GeV}$ (lower panel) using the given valence action and lattice spacing on the HISQ sea ensembles. The results with clover (CL) and overlap (OV) actions converge to the same continuum limit.

Figure 5

The normalization factor $Z_{{\gamma }_t}(0,\mu )$, which contains both ${\alpha }_s$ and also $a^2$ corrections as shown in Ref. [33]. The lines correspond to the renormalization factors with the $\mathcal{O}(a^2)$ correction subtracted.

Figure 6

The RI/MOM renormalized pion matrix element of the vector current, using overlap (OV) fermions on DWF sea ensembles. The data points correspond to the cases with 1-step of HYP smearing, and the bands are the results without HYP smearing of the Wilson link.

Figure 7

The RI/MOM renormalized pion matrix element of the vector current, using overlap (OV) fermions on HISQ sea ensembles and using a Wilson link without HYP smearing (upper panel) and with 1-step of HYP smearing (lower panel). The residual linear divergences in both cases are pronounced and similar.

Figure 8

The same RI/MOM renormalized pion matrix element as in Fig. 7 but using clover (CL) fermions on HISQ sea ensembles. The residual linear divergences are much more obvious than for the overlap case. The colored bands show the results when the clover coefficient are tuned such that the critical quark mass $|m^{\mathrm{cri}}|\le 0.02$. The residual linear divergences are still roughly the same.

Figure 9

The same RI/MOM renormalized pion matrix element as in Fig. 8 but using clover (CL) fermions on CL sea ensembles. The residual linear divergence is still there.

Figure 10

The ratio of gauge links in Coulomb gauge and Landau gauge on MILC ensembles with 1-step of HYP smearing. These curves seem to converge at small lattice spacings, which indicates that the linear divergence of the Wilson link is independent from the gauge fixing condition, at least for the Landau or Coulomb ones.

Figure 11

The value of $C_{US}(z)$ defined in Eq. (13) for the subtracted correlation between the 1-step HYP smeared Wilson link and the clover fermion (upper panel) and overlap fermion (lower panel) external states on MILC ensembles in Landau gauge. The linear divergence is obvious for both fermion actions, while that in the overlap fermion case is much smaller.

Figure 12

The pion quasi-PDF matrix element renormalized by the Coulomb gauge fixed Wilson line, using the overlap fermion on HISQ ensembles.

Figure 13

The ${\tilde{F}}_t(k;z)$ with nonzero momentum along the Wilson link as function of $z$ and normalized by ${\tilde{F}}_t(0;z)$, for overlap or clover fermions on HISQ sea ensembles. The darker data points show the real part, and the lighter ones show the imaginary part.

Figure 14

The ${\tilde{F}}_p(0;z)$ with zero momentum along the Wilson link (upper panels) and that with momentum $k=10\pi /L$ (lower panels), normalized by ${\tilde{F}}_t(0;z)$, using the same simulation setup as Fig. 13.

Figure 15

Similar to Fig. 14 but for the ${\tilde{F}}_p(0;z)$ case.

Figure 16

The RI/MOM renormalized pion matrix elements $h_{\pi ,{\gamma }^t}^r(z)$ using the overlap (OV, upper left panel) or clover (CL, upper right panel) fermion on HISQ sea ensembles. The cases using the minimum projection are also shown in the two lower panels for comparison. The darker data points correspond to the real parts, and lighter ones are the imaginary parts. It is funny that the lattice spacing dependence of $h_{\pi ,{\gamma }^t}^r(z)$ with the p-slash projection in the clover fermion case is mostly moved to the imaginary part.