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 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, $M_\pi \approx \{315,220,135\}$ 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 $M_\pi \approx \{315,220\}$ 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 $\Delta u \equiv \langle 1 \rangle_{\Delta u^+} = 0.777(25)(30)$, $\Delta d \equiv \langle 1 \rangle_{\Delta d^+} = -0.438(18)(30)$, and $\Delta s \equiv \langle 1 \rangle_{\Delta s^+} = -0.053(8)$, adding up to a total quark contribution to proton spin of $\sum_{q=u,d,s} (\frac{1}{2} \Delta q) = 0.143(31)(36)$. 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 $0.13<\frac{1}{2} \Delta \Sigma<0.18$, and with the $\Delta q$ obtained from various global analyses of polarized beam or target data.

Introduction: In 1987, the Europeon Muon Collaboration measured the spin asymmetry in polarized deep inelastic scattering and presented the remarkable result that the sum of the spins of the quarks contributes less than half of the total spin of the proton [1]. This unexpected result was termed the "proton spin crisis". Lattice QCD can unravel the mystery of where the proton gets its spin by measuring the matrix elements of appropriate quark and gluon operators within the nucleon state. In this paper, we present the first lattice calculation of the contribution of the intrinsic spin of the quarks to the proton spin with high-statistics and control over systematic errors. Our result, q=u,d,s 1 2 ∆q = 0.143 (31) (29), is in good agreement with the COMPASS Analysis 0.13 < 1 2 ∆Σ < 0.18 at 3 GeV 2 [2].
To calculate the nucleon spin using lattice QCD, one starts with Ji's sum rule [3] that provides a gauge invariant decomposition of the nucleon's total spin as where ∆q ≡ ∆Σ q ≡ 1 ∆u + ≡ g q A is the contribution of the intrinsic spin of a quark with flavor q; L q is the orbital angular momentum of that quark; and J g is the total angular momentum of the gluons. Thus, to explain the spin of the proton starting from QCD, one needs to calculate the contributions of all three terms. In this paper we present results for the relatively better determined first term, 1 2 ∆Σ ≡ q=u,d,s 1 2 ∆q. On the lattice, the axial charge g q A is given by the matrix element of the flavor diagonal axial current, qγ µ γ 5 q, where Z A is the renormalization constant. g q A is also the first Mellin moment, ∆q, of the polarized parton distribution function (PDF) integrated over the momentum fraction x [4]. The charges, g u,d,s A , also quantify the strength of the spin-dependent interaction of dark matter with nucleons [5,6]. Of these, ∆s is the least well known and current analyses [4] often rely on assumptions such as SU(3) symmetry and ∆s = ∆s.
The Lattice Methodology for the calculation of the flavor diagonal charges is now mature [7,8]. The challenges in the calculation of g q A are to obtain high statistics results for both the connected and disconnected contributions to nucleon three-point functions illustrated in Fig. 1 and address systematics. Here, we show that lattice discretization errors are large, and the extrapolation of renormalized charges evaluated at the physical pion mass, M π 0 = 135 MeV, to the continuum limit is essential.
The calculations of the connected and disconnected contributions to g u,d A were done separately using 2+1+1flavor ensembles of HISQ fermions [9] generated by the MILC Collaboration [10]. The construction of the 2-and 3-point correlation functions used in the analysis was  ) and strange (N s conf ) quarks, the number of random sources (Nsrc) and the ratio NLP/NHP of LP to HP solves used to estimate the quark loop on each configuration. The parameters of the 11 ensembles used for the connected contribution are given in Table 1 in Ref. [8]. carried out using Wilson-clover fermions. We refer to this as the clover-on-HISQ lattice formulation, which in the continuum limit is expected to give results for QCD. All results presented are for degenerate u and d quarks, with the s and c quark masses tuned to their physical values.
Results for the connected contributions have been obtained using eleven HISQ ensembles that cover the range 0.06 a 0.15 fm in the lattice spacing, 135 M π 320 MeV in the pion mass and 3.3 M π L 5.5 in spatial lattice size. The analysis of the connected data, including the simultaneous chiralcontinuum-finite volume fits have been presented in Ref. [8]. We reproduce only the final results in Table III, and refer the reader to that paper for all the details.
The computationally expensive analysis of the disconnected contributions has been carried out on six (for light u and d quarks) and seven (strange quark) HISQ ensembles described in Table I. The calculation of the vacuum polarization loop with the current insertion in the disconnected diagram is carried out stochastically using Gaussian or Z 4 random sources on each background gauge configuration as described in Ref. [7]. The final error is a combination of the error in the stochastic evaluation on each configuration and the error due to the average over the gauge configurations required by the path integral. We have analyzed the disconnected contributions separately because the quality of the statistical signal is weaker, restricting data to smaller values of source-sink separation τ in the 3-point function.
To increase the statistics cost-effectively, the calculations of both the 2-and 3-point nucleon correlation functions were carried out using the truncated solver method with bias correction [11,12]. In this method, correlation functions are constructed using quark propagators inverted with low precision (LP) stopping criteria between r LP ≡ |residue| LP /|source| = 10 −3 and 5 × 10 −4 , high precision (HP) with r HP between 10 −7 and 10 −8 [7,8]. The bias corrected correlation functions on each configuration are given by where C LP and C HP are the 2-or 3-point functions calculated in LP and HP, respectively, and x LP i and x HP i are the source positions for the two kinds of propagator inversion. We found no evidence of possible bias in our data.
Excited-State Contamination (ESC): To obtain the charges, we need to evaluate the matrix elements of the corresponding quark bilinear operators within the ground state of the nucleons. We use the same tool kit (2-state fits to both the operator insertion time t and multiple source-sink separation τ in the 3-point functions) described in Refs. [7,8] to remove the ESC.
The data and the 2-state fits for the disconnected contributions are shown in Fig. 2. The data are noisier compared to the connected part analyzed in Ref. [8]. In many cases there is no clear pattern of convergence towards the τ → ∞ value. We, therefore, first determined the direction of convergence versus τ for both g l,s A by analyzing data at small τ . Data at small τ have smaller statistical errors but larger ESC, and can therefore be used to facilitate the determination of the direction of convergence. We have used the size of the errors and the sign of the curvature shown in Fig. 2 in selecting the best set of τ values to use in the 2-state fits to get the final bare results collected together in Table II.
We emphasize that the disconnected contribution converges from above, while the connected contribution [8] converges from below, i.e., the ESC in the disconnected 3-point function is opposite to that observed in the con-nected 3-point function [8]. In both cases, removing the ESC increases the magnitude of their contribution. A more negative contribution from the sea quarks reduces the fraction of the nucleon spin carried by the quarks.
Analyzing the connected and disconnected contributions separately to remove ESC introduces an approximation. To define connected and disconnected contributions individually, one has to work in a partially quenched theory with an additional quark with flavor u . However, in this theory the Pauli exclusion principle does not apply between the u and u quarks. The upshot of this is that the spectrum of states in the partially quenched theory is larger, for example, an intermediate u ud state would be the analogue of a Λ rather than a nucleon [13]. Thus, the spectral decomposition for this partially quenched theory and QCD is different. In the ESC fits, we use the same QCD spectral decomposition in the 2-state fits for both the 2-and 3-point functions. So we must assume that any additional systematic is well within the quoted uncertainty since the difference between the plateau value in the largest τ data and the τ → ∞ value on each ensemble is a small effect in the individual connected and disconnected contributions, and the extrapolated value is not very sensitive to the precise details of the spectra.
Renormalization of the operators: The renormalization of flavor diagonal light quark operators, qγ µ γ 5 q, requires knowing both non-singlet and singlet factors [14]. The difference starts at two-loops in perturbation theory. For the axial vector operators, explicit calculations have shown that Z non−singlet A and Z singlet A agree to within a percent [15][16][17]. In this work we, therefore, use factors given in Ref. [8] to renormalize both the disconnected and connected contributions in two ways: g ren1 The results for disconnected contributions are given in Table II and we take the average for our final values. The connected contributions are taken from Ref. [8].
The Chiral-Continuum Extrapolation: We include the lowest order corrections to fit the disconnected data given in Table II versus a and M π : and the results are shown in Fig. 3. For discretization effects, we keep the term linear in a since the action and the operators in our clover-on-HISQ formalism are not O(a) improved. The leading dependence on M π , given by finite volume chiral perturbation theory [18][19][20][21][22][23][24], is M 2 π . We neglect finite volume corrections since no significant evidence was found in the connected analysis [8].
The extrapolated values are given in Table III, along with the connected contributions reproduced from Ref. [8]. Our result, 1 2 ∆Σ ≡ q=u,d,s 1 2 ∆q = 0.143 (31), is in good agreement with the COMPASS result [2]. Scaling our value of g s A by 1/m q suggests that the neglected charm contribution could be ≈ 0.005.
In Ref. [8], our result for the isovector axial charge, g u−d A = 1.218 (27) (30), was 0.058 below the experimental value g u−d A = 1.2766 (20). This can be explained if the connected g u A is underestimated by 0.058 or g d A is more negative by this amount or any combination of the two. (The disconnected contributions cancel in g u−d A .) The first case would increase ∆Σ/2 by 0.029 and in the second case reduce it by the same amount. We, therefore, quote an additional systematic error of 0.029 to account for the fact that the same connected data used here underestimates g u−d A . Comparison with previous work and conclusions: In Fig. 4, we compare lattice results, restricted to those from physical pion mass ensembles, with the moments extracted from global fits to polarized PDFs reviewed in Ref. [4]. All results are in the M S scheme at 2 GeV. The ETMC results from a single physical mass ensemble at a = 0.093 fm [16,25] are consistent with ours after correction for the a dependence found in Fig. 3 (40), is also low and consistent with ours. Our results are also compatible with the moments extracted from global PDF fits.
In conclusion, we present first results with chiralcontinuum extrapolation of up, down and strange quark spin contributions. These fits are based on 6 (7) ensembles for the disconnected contribution of light (strange) quarks, and on 11 ensembles for the dominant connected contributions that were analyzed fully in Ref. [8]. We demonstrate in Fig. 3 that a chiral-continuum extrapolation significantly reduces the contribution of the quark spin to the proton spin and is, therefore, essential for getting physical results from lattice calculations. Our final result, 1 2 ∆Σ = 0.143(31) (29), is consistent with the 2015 COMPASS analysis. The ≈ 5% underestimate of g u−d A in our lattice calculation is, at present, the largest systematic that needs to be addressed in future works.