Quark contribution to the proton spin from $2+1+1$-flavor lattice QCD

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 11 ensembles of $2+1+1$-flavor of highly improved staggered quarks (HISQ) generated by the MILC Collaboration. They cover four lattice spacings, $a\approx \{0.15,0.12,0.09,0.06\}\mathrm{fm}$, and three pion masses, $M_{\pi }\approx \{315,220,135\}\mathrm{MeV}$, 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 $M_{\pi }\approx \{315,220\}\mathrm{MeV}$. 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 $\mathrm{\Delta }u\equiv ⟨1{⟩}_{\mathrm{\Delta }{u}^{+}}=0.777(25)(30)$, $\mathrm{\Delta }d\equiv ⟨1{⟩}_{\mathrm{\Delta }{d}^{+}}=-0.438(18)(30)$, and $\mathrm{\Delta }s\equiv ⟨1{⟩}_{\mathrm{\Delta }{s}^{+}}=-0.053(8)$, adding up to a total quark contribution to proton spin of ${\sum }_{q=u,d,s}(\frac{1}{2}\mathrm{\Delta }q)=0.143(31)(36)$. 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 $0.13<\frac{1}{2}\mathrm{\Delta }\mathrm{\Sigma }<0.18$ and with the $\mathrm{\Delta }q$ obtained from various global analyses of polarized beam or target data.

Figure 1

The connected (left) and disconnected (right) three-point diagrams that contribute to the flavor diagonal matrix elements of the axial operator (labeled by $\otimes$ at time slice $t$) within the nucleon state. The black blobs denote nucleon source and sink, separated by Euclidean time $\tau$.

Figure 2

The data and the two-state fit to the light (top two rows) and strange (bottom two rows) quark disconnected contribution to the bare $g_A^{(l,s),\text{disc}}$. The grey error band and the solid line within it is the $\tau \to \infty$ estimate obtained using the two-state fit to data at different $t$ and $\tau$. The result of the fit for each individual $\tau$ is shown by a solid line in the same color as the data points.

Figure 3

Top: The extrapolation of the renormalized $g_A^{l,\mathrm{disc}}$ and $g_A^{s,\mathrm{disc}}$ data using the chiral-continuum ansatz given in Eq. (6). In each panel, the pink band shows the result of the simultaneous fit plotted versus a single variable with the other variable set to its physical value. The result at the physical point, $M_{\pi }=135\mathrm{MeV}$ and $a=0$, is marked with a red star. The grey band shows the fit versus only $M_{\pi }$, i.e., ignoring the dependence on $a$. It highlights the need for a simultaneous fit in both $a$ and $M_{\pi }$. Bottom: The data in each panel are plotted after extrapolation to the physical point ($a=0$ or $M_{\pi }=135\mathrm{MeV}$) in the nonplotted variable to facilitate comparison with the simultaneous fit.

Figure 4

Chiral-continuum extrapolated PNDME’18 results (this work) for $\mathrm{\Delta }u$, $\mathrm{\Delta }d$, and $\mathrm{\Delta }s$ are compared with ETMC [17, 27] values obtained at a single-lattice spacing and with moments from global fits to polarized PDF (NNPDFpol1.1’13 [28], DSSV’08 [29, 30], Jam’15 [31], and JAM’17 [32]). All PDF results are taken from Ref. [4] and are at 2 GeV in the $\overline{\mathrm{MS}}$ scheme.