[Paper Review] Quark contribution to the proton spin from 2+1+1-flavor lattice QCD

Huey-Wen Lin, Gupta, Rajan+3 more|arXiv (Cornell University)|Jun 27, 2018

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This study presents the first chiral-continuum extrapolated lattice QCD calculation of the up, down, and strange quark spin contributions to the proton spin using 11 ensembles of 2+1+1-flavor HISQ fermions. The results, $Δu = 0.777(25)(30)$, $Δd = -0.438(18)(30)$, $Δs = -0.053(8)$, yield a total quark spin contribution of $\frac{1}{2}\Delta\Sigma = 0.143(31)(36)$, consistent with global fits and COMPASS data.

ABSTRACT

We present the first chiral-continuum extrapolated up, down and strange quark spin contribution to the proton spin using lattice QCD. For the connected contributions, we use eleven ensembles of 2+1+1-flavor of Highly Improved Staggered Quarks (HISQ) generated by the MILC Collaboration. They cover four lattice spacings $a \approx \{0.15,0.12,0.09,0.06\}$ fm and three pion masses, $M_π\approx \{315,220,135\}$ MeV, of which two are at the physical pion mass. The disconnected strange calculations are done on seven of these ensembles, covering the four lattice spacings but only one with the physical pion mass. The disconnected light quark calculation was done on six ensembles at two values of $M_π\approx \{315,220\}$ MeV. High-statistics estimates on each ensemble for all three quantities allow us to quantify systematic uncertainties and perform a simultaneous chiral-continuum extrapolation in the lattice spacing and the light-quark mass. Our final results are $Δu \equiv \langle 1 angle_{Δu^+} = 0.777(25)(30)$, $Δd \equiv \langle 1 angle_{Δd^+} = -0.438(18)(30)$, and $Δs \equiv \langle 1 angle_{Δs^+} = -0.053(8)$, adding up to a total quark contribution to proton spin of $\sum_{q=u,d,s} (\frac{1}{2} Δq) = 0.143(31)(36)$. The second error is the systematic uncertainty associated with the chiral-continuum extrapolation. These results are obtained without model assumptions and are in good agreement with the recent COMPASS analysis $0.13 < \frac{1}{2} ΔΣ< 0.18$, and with the $Δq$ obtained from various global analyses of polarized beam or target data.

Motivation & Objective

• To compute the intrinsic spin contributions of up, down, and strange quarks to the proton using lattice QCD with controlled systematic uncertainties.
• To perform a simultaneous chiral-continuum extrapolation of the axial charges $\Delta q$ using ensembles with varying lattice spacing and pion mass.
• To reduce systematic errors by including high-statistics data across multiple ensembles and eliminating model-dependent assumptions.
• To provide a first-principles determination of $\Delta\Sigma = \sum_{q=u,d,s} \frac{1}{2}\Delta q$ without relying on SU(3) symmetry or other theoretical assumptions.
• To compare the results with global polarized PDF fits and experimental data from COMPASS to validate the lattice computation.

Proposed method

• Utilized 11 ensembles of 2+1+1-flavor Highly Improved Staggered Quarks (HISQ) from the MILC Collaboration with four lattice spacings ($a \approx \{0.15, 0.12, 0.09, 0.06\}$ fm) and three pion masses ($M_\pi \approx \{315, 220, 135\}$ MeV), including physical pion mass ensembles.
• Computed both connected and disconnected contributions to the nucleon three-point functions for the axial current matrix elements using high-statistics measurements on each ensemble.
• Applied a simultaneous chiral-continuum extrapolation in both the light-quark mass and lattice spacing to extract the physical values at $a=0$ and $M_\pi = 135$ MeV.
• Used the renormalized matrix elements of the flavor-diagonal axial current $\langle N|Z_A \bar{q}\gamma_\mu\gamma_5 q|N \rangle$ to extract $\Delta q = g_A^q$, with $Z_A$ determined non-perturbatively.
• Quantified systematic uncertainties from the chiral-continuum extrapolation, including discretization errors and finite-volume effects.
• Validated results against global fits and COMPASS data, with a focus on consistency in the $\overline{MS}$ scheme at 2 GeV.

Experimental results

Research questions

• RQ1 What is the intrinsic spin contribution of the up, down, and strange quarks to the proton’s total spin, as computed from first-principles lattice QCD?
• RQ2 How do discretization and chiral extrapolation errors affect the determination of $\Delta q$ in lattice QCD, and can they be systematically controlled?
• RQ3 How does the disconnected quark loop contribution to the proton spin evolve with lattice spacing and pion mass, and what is its physical value at the continuum and physical point?
• RQ4 To what extent do the lattice results for $\Delta u$, $\Delta d$, and $\Delta s$ agree with global fits to polarized parton distribution functions and experimental data from COMPASS?
• RQ5 What are the dominant remaining systematic uncertainties in the current lattice determination of the proton spin structure?

Key findings

• The chiral-continuum extrapolated value for the up quark spin contribution is $\Delta u = 0.777(25)(30)$, with the second error representing systematic uncertainty from the extrapolation.
• The down quark spin contribution is $\Delta d = -0.438(18)(30)$, indicating a net negative contribution to the proton’s spin from the down quark.
• The strange quark spin contribution is $\Delta s = -0.053(8)$, which is small and consistent with zero within errors, and does not require SU(3) symmetry assumptions.
• The total quark spin contribution to the proton is $\frac{1}{2}\Delta\Sigma = 0.143(31)(36)$, with the second error due to chiral-continuum extrapolation uncertainty.
• The results are in good agreement with the 2015 COMPASS analysis, which reports $0.13 < \frac{1}{2}\Delta\Sigma < 0.18$ at 3 GeV.
• The ETMC lattice result at $a = 0.0938$ fm is consistent with this work, with small differences attributed to unaccounted discretization effects in their analysis.