We evaluate the gluon and quark contributions to the spin of the proton using an ensemble of gauge configurations generated at physical pion mass. We compute all valence and sea quark contributions to high accuracy. We perform a nonperturbative renormalization for both quark and gluon matrix elements. We find that the contribution of the up, down, strange, and charm quarks to the proton intrinsic spin is 1 2 P q¼u;d;s;c ΔΣ q þ ¼ 0.191ð15Þ and to the total spin P q¼u;d;s;c J q þ ¼ 0.285ð45Þð10Þ. The gluon contribution to the spin is J g ¼ 0.187ð46Þð10Þ yielding J ¼ J q þ J g ¼ 0.473ð71Þð14Þ confirming the spin sum. The momentum fraction carried by quarks in the proton is found to be 0.618(60) and by gluons 0.427(92), the sum of which gives 1.045(118) confirming the momentum sum rule. All scale and scheme dependent quantities are given in the MS scheme at 2 GeV.

In this paper we present one-loop results for the renormalization of nonlocal quark bilinear operators, containing a staple-shaped Wilson line, in both continuum and lattice regularizations. The continuum calculations were performed in dimensional regularization, and the lattice calculations for the Wilson/clover fermion action and for a variety of Symanzik-improved gauge actions. We extract the strength of the one-loop linear and logarithmic divergences (including cusp divergences), which appear in such nonlocal operators; we identify the mixing pairs which occur among some of these operators on the lattice, and we calculate the corresponding mixing coefficients. We also provide the appropriate RI -like scheme, which disentangles this mixing nonperturbatively from lattice simulation data, as well as the one-loop expressions of the conversion factors, which turn the lattice data to the MS scheme. Our results can be immediately used for improving recent nonperturbative investigations of transverse momentum-dependent distribution functions on the lattice.Finally, extending our perturbative study to general Wilson-line lattice operators with n cusps, we present results for their renormalization factors, including identification of mixing and determination of the corresponding mixing coefficients, based on our results for the staple operators.

address : Department of P hysics, T emple U niversity, P hiladelphia, P A 19122 − 1801, U SA c P resent address : Department of P hysics, U niversity of M ichigan, Ann Arbor, M I 48109, U SA In this paper we present the perturbative evaluation of the difference between the renormalization functions of flavor singlet and nonsinglet bilinear quark operators on the lattice. The computation is performed to two loops and to lowest order in the lattice spacing, for a class of improved lattice actions, including Wilson, tree-level (TL) Symanzik and Iwasaki gluons, twisted mass and SLiNC Wilson fermions, as well as staggered fermions with twice stout-smeared links. In the staggered formalism, the stout smearing procedure is also applied to the definition of bilinear operators.

In this paper, we examine the effect of nonzero quark masses on the renormalization of gauge-invariant nonlocal quark bilinear operators, including a finite-length Wilson line (called Wilson-line operators). These operators are relevant to the definition of parton quasidistribution functions, the calculation on the lattice of which allows the direct nonperturbative study of the corresponding physical parton distribution functions. We present our perturbative calculations of the bare Green's functions, the renormalization factors in RI 0 and MS schemes, as well as the conversion factors of these operators between the two renormalization schemes. Our computations have been performed in dimensional regularization at oneloop level, using massive quarks. The conversion factors can be used to convert the corresponding lattice nonperturbative results to the MS scheme, which is the most widely used renormalization scheme for the analysis of experimental data in high-energy physics. Also, our study is relevant for disentangling the additional operator mixing that occurs in the presence of nonzero quark masses, both on the lattice and in dimensional regularization.

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