Total Gluon Helicity Contribution to Proton Spin from Lattice QCD

TL;DR

This work delivers a first-principles lattice QCD determination of the total gluon helicity contribution to the proton spin, $\Delta G$, using the topological current in Coulomb gauge in boosted proton states. The calculation combines high-precision lattice techniques (distillation, momentum smearing, and CDER) with nonperturbative RI/MOM renormalization and a three-loop MSbar matching, followed by continuum and infinite-momentum extrapolations. The final result, $\Delta G^{\overline{\mathrm{MS}}}(\mu^2=10\ \mathrm{GeV}^2)=0.231(17)^{\mathrm{sta.}}(33)^{\mathrm{sym.}}$, corresponds to $46(7)\%$ of the proton spin, and the temporal and spatial components of $K^\mu$ yield consistent $\Delta G$ in the IMF, validating Lorentz covariance in this regime. The study provides a precise, first-principles benchmark for the gluon spin contribution and lays groundwork for future investigations into quark and gluon orbital angular momentum and a complete proton-spin decomposition.

Abstract

We report a state-of-the-art lattice QCD calculation of the total gluon helicity contribution to proton spin, $ΔG$. The calculation is done on ensembles at three different lattice spacings $a=\{0.08, 0.09, 0.11\}$ fm. By employing distillation + momentum smearing for proton external states, we extract the bare matrix elements of the topological current $K^μ$ under the 5-HYP smeared Coulomb gauge fixing configurations. Furthermore, we apply a non-perturbative $\mathrm{RI/MOM}$ renormalization scheme augmented with the Cluster Decomposition Error Reduction (CDER) technique to determine the renormalization constants of $K^μ$. The results obtained from different components $K^{t,i}$ (with $i$ being the direction of proton momentum or polarization) are consistent with Lorentz covariance within uncertainties. After extrapolating to the continuum limit, $ΔG$ is found to be $ΔG = 0.231(17)^{\mathrm{sta.}}(33)^{\mathrm{sym.}}$ at the $\overline{\mathrm{MS}}$ scale $μ^2=10\ \mathrm{GeV}^2$, which constitutes approximately $46(7)\%$ of the proton spin.

Figures (8)

• Figure 1: The ratio $R_{K^\mu}(t_f, t_i)$ at $t_f\in[0.63,1.16]\ \mathrm{fm}$ and the fitted BME $\langle K^\mu\rangle$ (the gray error band) for proton external states with $p=0.98\ \mathrm{GeV}$ under ensemble C24P29, utilizing the fitting procedure described in \ref{['eq:ratio_fitting']}.
• Figure 2: The removal of ${\cal O}(a^2p^2)$ discretization effect in $\bar{Z}_{11}^{\mathrm{lat}}\equiv R_{11}Z_{11}^{\mathrm{RI}}$ for the ensemble C48P23. The red, green, and blue data points and their error bands represent the CDER results and their $a^2p^2\rightarrow 0$ extrapolated results using momentum modes $(0,0,p,p)$, $(0,p,p,p)$, and $(p,p,p,p)$ in \ref{['eq:Z11_final']}, where $p=(2\pi/(aL))p_{\mathrm{lat}}$ with $p_{\mathrm{lat}}$ represents momentum in the lattice unit.
• Figure 3: The continuum and infinite momentum extrapolation of the total gluon helicity results using lattice spacings ranging from $0.08\ \mathrm{fm}$ to $0.11\ \mathrm{fm}$ at the $\overline{\mathrm{MS}}$ scale $\mu^2=10\ \mathrm{GeV}^2$. The uncertainties are shown as separate bands: statistical uncertainty (dark gray) and all uncertainties that arise during our analysis (light gray).
• Figure 4: Proton effective mass with different momentum under different smearing schemes under ensemble C24P29. Left figure represents the distillation smearing scheme alone, and the right figure represents the distillation + momentum smearing. The red, green and blue points represent $p=0,0.98,1.96\ \mathrm{GeV}$, respectively.
• Figure 5: Ratio $R_{K^\mu}(t_f, t_i)$ at $t_f\in[0.63,1.16]\ \mathrm{fm}$ and the BMEs of $K^\mu/(2E)$ (the gray error band) for proton external states with varying momenta under ensemble C24P29, utilizing the fitting procedure described in \ref{['eq:ratio_fitting']}. The results are organized such that the first column corresponds to spatial components $K^i$, while the second column represents temporal component $K^t$. Rows one through three display results for proton momenta of $p=0,0.98,1.96\ \mathrm{GeV}$, respectively.