We present a lattice quantum chromodynamics calculation of the $x$-dependent isovector quark helicity parton distribution function (PDF) of the proton in the large momentum effective theory (LaMET) framework. Through operator product expansion (OPE) we also extract the ${\tilde{d}}_2$ moment of the twist-3 PDF $g_T(x)$ for the first time in the $\overline{\mathrm{MS}}$ scheme, which is proportional to the average color Lorentz force experienced by the quark in the proton. This calculation is performed on a lattice of spacing $a=0.076\mathrm{fm}$ at physical quark masses. The quasi-PDF matrix elements are measured in proton states boosted to momenta $P_z=\{0,0.25,1.02,1.53\}\mathrm{GeV}$. We first extract the lowest few helicity PDF moments from the renormalization-group (RG) invariant ratios of the matrix elements with OPE. Combined with the matrix elements relevant for $g_T(x)$, we obtain ${\tilde{d}}_2^{u-d}(2\mathrm{GeV})=0.0024(46)$ at next-to-leading order in $\overline{\mathrm{MS}}$. Then, the helicity quasi-PDF matrix elements are renormalized in the hybrid scheme with linear renormalon resummation and Fourier transformed to the $x$-space after an asymptotic extrapolation. The quasi-PDF is perturbatively matched to the $\overline{\mathrm{MS}}$ PDF with RG and threshold resummations at next-to-leading power and next-to-next-to-leading logarithmic accuracies. After resummations, we determine the PDF in the region $x\in [0.25,0.75]$. The end-point regions are then parametrized, combined with the LaMET prediction at moderate $x$, and fitted to the short-distance matrix elements in coordinate space.

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• Lattice field theory
• Parton distribution functions
• Nucleons
• Nucleon spin structure