Sivers and Boer-Mulders observables from lattice QCD

TL;DR

The paper develops a lattice QCD framework to compute transverse momentum dependent parton distributions using staple-shaped Wilson lines, enabling access to process-dependent T-odd observables like the Sivers and Boer-Mulders shifts in SIDIS and DY. By formulating non-local correlators in terms of Lorentz-invariant amplitudes and employing ratios that cancel soft factors, the authors extract generalized shifts and tensor-charge related quantities from dynamical 2+1 flavor lattices. Their numerical results show nonzero, sign-consistent Sivers and Boer-Mulders effects for the isovector combination, and reveal relatively stable T-even observables under variations of the staple geometry, with a consistent generalized tensor charge near independent of the Collins-Soper parameter within the explored range. The findings constitute a first-principles probe of TMD physics on the lattice and establish a path toward connecting lattice results with experimental SIDIS and DY measurements, while underscoring the need for larger rapidity parameters and enhanced statistics for definitive phenomenological impact.

Abstract

We present a first calculation of transverse momentum dependent nucleon observables in dynamical lattice QCD employing non-local operators with staple-shaped, "process-dependent" Wilson lines. The use of staple-shaped Wilson lines allows us to link lattice simulations to TMD effects determined from experiment, and in particular to access non-universal, naively time-reversal odd TMD observables. We present and discuss results for the generalized Sivers and Boer-Mulders transverse momentum shifts for the SIDIS and DY cases. The effect of staple-shaped Wilson lines on T-even observables is studied for the generalized tensor charge and a generalized transverse shift related to the worm gear function g_1T. We emphasize the dependence of these observables on the staple extent and the Collins-Soper evolution parameter. Our numerical calculations use an n_f = 2+1 mixed action scheme with domain wall valence fermions on an Asqtad sea and pion masses 369 MeV as well as 518 MeV.

Figures (13)

• Figure 1: \ref{['fig-link-straight']} Straight gauge link. \ref{['fig-link-staple']} Staple-shaped gauge link as in SIDIS and DY.
• Figure 2: Illustration of the leading contribution to SIDIS in factorized form.
• Figure 3: Extraction of the generalized Sivers shift on the lattice with $m_\pi = 518 \,\mathrm{MeV}$ using a lattice nucleon momentum $|\boldsymbol{P}^{\text{lat}}| = 2\pi /(a\hat{L}) \approx 500 \,\mathrm{MeV}$ at the corresponding maximal Collins-Soper evolution parameter $\hat{\zeta} = 0.39$. The continuous horizontal lines are obtained from two independent averages of the data points with staple extents in the ranges $\eta |v| = 7a .. 12a$ and $\eta |v| = -12a .. -7a$, respectively. The outer data points shown with empty symbols have been obtained from an anti-symmetrized mean value of these averages, i.e., the expected T-odd behavior of the Sivers shift has been put in explicitly. These outer data points are our estimates for the asymptotic values at $\eta |v| \rightarrow \pm \infty$ and thus represent the generalized Sivers shifts for SIDIS and DY. Error bars show statistical uncertainties only. Figures \ref{['fig-Sivers_lsqr-1_zetasqrlat4']} and \ref{['fig-Sivers_lsqr-4_zetasqrlat4']} have been obtained with rather small quark field separations $|\boldsymbol{b}_{\mathrm T}|=1a$ and $2a$. Therefore, they might be affected by significant lattice cutoff effects.
• Figure 4: Generalized Sivers shift as a function of the quark separation $|\boldsymbol{b}_{\mathrm T}|$ for the SIDIS case ($\eta |v| = \infty$), extracted on the lattice with $m_\pi = 518 \,\mathrm{MeV}$ for $\hat{\zeta} = 0.39$. The data points lying in the shaded area below $|\boldsymbol{b}_{\mathrm T}| \approx 0.25 \,\mathrm{fm}$ might be affected by significant lattice cutoff effects. Error bars show statistical uncertainties only.
• Figure 5: Generalized Sivers shift on the lattice with $m_\pi = 518 \,\mathrm{MeV}$ for a quark separation of three lattice spacings, $|\boldsymbol{b}_{\mathrm T}| = 3a = 0.36 \,\mathrm{fm}$, extracted at $\hat{\zeta} = 0$ and at our highest value of the Collins-Soper evolution parameter, $\hat{\zeta} = 0.78$. Figure \ref{['fig-Sivers_lsqr-9_zetasqrlat16']} has been obtained from nucleons with momentum $|\boldsymbol{P}^{\text{lat}}| = 2 \times 2\pi /(a\hat{L} ) \approx 1 \,\mathrm{GeV}$ on the lattice. Error bars show statistical uncertainties only.