Proton isovector helicity PDF at NNLO and the twist-3 moment $\tilde{d}_2$ from lattice QCD at physical quark masses

Abstract

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 $\overline{\rm MS}$ 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 fm at physical quark masses. The quasi-PDF matrix elements are measured in proton states boosted to momenta $P_z=\{0, 0.25, 1.02, 1.53\}$ 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 $\tilde{d}_2^{u-d}(2\ {\rm GeV})=0.0024(46)$ at next-to-leading order in $\overline{\rm MS}$. 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 $\overline{\rm MS}$ 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 $x\in[0.25,0.75]$ with controlled systematic uncertainties. The end-point regions are then parameterized, combined with the LaMET prediction at moderate $x$, and fitted to the short-distance matrix elements in coordinate space.

Figures (16)

• Figure 1: The fit of the three-point function to two-point function ratio $R_1(\tau,t_{\mathrm{sep}},P_z;z)$ at the largest momentum $P_z=1.53$ GeV with $z=4a$ (upper panels) and $z=8a$ (lower panels). Left: the real part. Right: the imaginary part. The colored bands are reconstructed from two-state fit while the grey band denotes the extracted bare matrix element.
• Figure 2: Similar to Fig. \ref{['fig:fit-ratio']} but for $R_T(\tau,t_{\mathrm{sep}},P_z;z)$ at momentum $P_z=1.02$ GeV with $z=2a$ (upper panels) and $z=4a$ (lower panels).
• Figure 3: Isovector bare matrix elements as functions of $z$ for the helicity channel (top row, $\tilde{h}^{B}_1$) and the transverse twist-3 channel (bottom row, $\tilde{h}^{B}_T$), shown at several nucleon boosts $P_z$ (see legends). Left (right) panels show the real (imaginary) parts.
• Figure 4: The real (left panel) and imaginary (right panel) part of the RG-invariant ratios $\mathcal{M}_1(\lambda,z^2)$ are shown for $z\in[a,5a]$.
• Figure 5: Helicity PDF moments $\langle x^n\rangle(\mu=2\rm{~GeV})$ fitted from the OPE of isovector (left) and isoscalar (right) RG-invariant ratios. The consistency between NLL and NNLL accuracy suggest good perturbative convergence. The leading order (LO) results with $C_n=1$ have been included to demonstrate the importance of perturbative corrections. The combined correlated fit among different $z$ values are shown separately as filled squares on the left, which are consistent with individual fits for lower moments. For comparison, we integrate the quark helicity PDFs from JAM22 Buckley:2014anaCocuzza:2022jye to get the moments and plot as bands.