Quark Bilinear Operators Research Articles

Renormalization of asymmetric staple-shaped Wilson-line operators in lattice and continuum perturbation theory

In this work, we study the renormalization of nonlocal quark bilinear operators containing an asymmetric staple-shaped Wilson line at the one-loop level in both lattice and continuum perturbation theory. These operators enter the first-principle calculation of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD using the formulation of large momentum effective theory. We provide appropriate RI′-type conditions that address the power and logarithmic divergences, as well as the mixing among staple operators of different Dirac structures, using a number of different possible projectors. A variant of RI′, including calculations of rectangular Wilson loops, which cancel the pinch-pole singularities of the staple operators at infinite length and reduce residual power divergences, is also employed. We calculate at one-loop order the conversion matrix, which relates the quasi-TMDPDFs in the RI′-type schemes to the reference scheme MS¯ for arbitrary values of the renormalization momentum scale and of the dimensions of the staple. Published by the American Physical Society 2024

Short-flow-time expansion of quark bilinears through next-to-next-to-leading order QCD

The gradient-flow formalism proves to be a useful tool in lattice calculations of quantum chromodynamics. For example, it can be used as a scheme to renormalize composite operators by inverting the short-flow-time expansion of the corresponding flowed operators. In this paper, we consider the short-flow-time expansion of five quark bilinear operators, the scalar, pseudoscalar, vector, axialvector, and tensor currents, and compute the matching coefficients through next-to-next-to-leading order QCD. Among other applications, our results constitute one ingredient for calculating bag parameters of mesons within the gradient-flow formalism on the lattice.

Moments n = 2 and n = 3 of the Wilson twist-two operators at three loops in the RI′/SMOM scheme

We study the renormalization of the matrix elements of the twist-two non-singlet bilinear quark operators, contributing to the n=2 and n=3 moments of the structure functions, at next-to-next-to-next-to-leading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the MS‾ scheme and the regularization-invariant symmetric momentum subtraction (RI/SMOM, RI′/SMOM) schemes. The obtained results can be used to reduce errors in determinations of moments of structure functions from lattice QCD simulations. The results are given in Landau gauge.

Four quark operators for kaon bag parameter with gradient flow

To study the $CP$-violation using the ${K}_{0}\ensuremath{-}{\overline{K}}_{0}$ oscillation, we need the kaon bag parameter which represents QCD corrections in the leading Feynman diagrams. The lattice QCD provides us with the only way to evaluate the kaon bag parameter directly from the first principles of QCD. However, a calculation of relevant four quark operators with theoretically sound Wilson-type lattice quarks had to carry a numerically big burden of extra renormalizations and resolution of extra mixings due to the explicit chiral violation. Recently, the small flow-time expansion ($\text{SF}t\text{X}$) method was proposed as a general method based on the gradient flow to correctly calculate any renormalized observables on the lattice, irrespective of the explicit violations of related symmetries on the lattice. To apply the $\text{SF}t\text{X}$ method, we need matching coefficients, which relate finite operators at small flow times in the gradient flow scheme to renormalized observables in conventional renormalization schemes. In this paper, we calculate the matching coefficients for four quark operators and quark bilinear operators, relevant to the kaon bag parameter.

Improvement, generalization, and scheme conversion of Wilson-line operators on the lattice in the auxiliary field approach

Nonlocal quark bilinear operators connected by link paths are used for studying parton distribution functions (PDFs) and transverse momentum-dependent PDFs of hadrons using lattice QCD. The nonlocality makes it difficult to understand the renormalization and improvement of these operators using standard methods. In previous work, we showed that by introducing an auxiliary field on the lattice, one can understand an on-axis Wilson-line operator as the product of two local operators in an extended theory. In this paper, we provide details about the calculation in perturbation theory of the factor for conversion from our lattice-suitable renormalization scheme to the MS-bar scheme. Extending our work, we study Symanzik improvement of the extended theory to understand the pattern of discretization effects linear in the lattice spacing, $a$, which are present even if the lattice fermion action exactly preserves chiral symmetry. This provides a prospect for an eventual $O(a)$ improvement of lattice calculations of PDFs. We also generalize our approach to apply to Wilson lines along lattice diagonals and to piecewise-straight link paths.

Nonperturbative renormalization of staple-shaped Wilson line operators in lattice QCD

Quark bilinear operators with staple-shaped Wilson lines are used to study transverse-momentum-dependent parton distribution functions (TMDPDFs) from lattice quantum chromodynamics (QCD). Here, the renormalization factors for the isovector operators, including all mixings between operators with different Dirac structures, are computed nonperturbatively in the regularization-independent momentum subtraction scheme for the first time. This study is undertaken in quenched QCD with three different lattice spacings. With Wilson flow applied to the gauge fields in the calculations, the operator mixing pattern due to chiral symmetry breaking with the lattice regularization is found to be significantly different from that predicted by one-loop lattice perturbation theory calculations. These results constitute a critical step towards the systematic extraction of TMDPDFs from lattice QCD.

Bilinear quark operators in the RI/SMOM scheme at three loops

We consider the renormalization of the matrix elements of the bilinear quark operators ψ¯ψ, ψ¯γμψ, and ψ¯σμνψ at next-to-next-to-next-to-leading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the MS‾ scheme and the regularization invariant symmetric momentum subtraction (RI/SMOM) scheme. The obtained results can be used to reduce the errors in determinations of quark masses from lattice QCD simulations. The results are given in Landau gauge.

Lattice finite-volume dependence of the nucleon parton distributions

There have been rapid developments in the direct calculation of the Bjorken-$x$ dependence of parton distribution functions (PDFs) using lattice QCD, and the technique shows promising results. Among various new methods, large-momentum effective theory (LaMET), which calculates boosted hadron matrix elements of multiple spatial displacements on the lattice, shows promising PDFs at physical pion mass by both ${\mathrm{LP}}^{3}$ and ETMC Collaborations. However, the finite-volume systematics have not yet been studied, and it has been suggested that such systematics can be more significant for LaMET-type operators than the traditional bilinear quark operators in charges and form factors. In this work, we present the first study of the finite-volume systematic for both isovector nucleon unpolarized and helicity matrix elements on three lattice volumes (2.88, 3.84, 4.8 fm) with lattice spacing 0.12 fm and 220-MeV pion mass. We perform two-state simultaneous fits using multiple source-sink separations to remove excited-state contamination and obtain reliable ground-state matrix elements. We then implement nonperturbative renormalization and Fourier transform the matrix elements to momentum-space quasi-PDFs. Overall, we do not see significant finite-volume systematics at the studied boost momenta of ${P}_{z}=1.3$ and 2.6 GeV.

Symmetry properties of nonlocal quark bilinear operators on a Lattice (LP3 Collaboration) * * JWC is partly supported by the Ministry of Science and Technology (105-2112-M-002-017-MY3) and the Kenda Foundation. TI is supported by Science and Technology Commission of Shanghai Municipality (16DZ2260200). TI and LCJ are supported by the Department of Energy, Laboratory Directed Research and Development (LDRD) funding of

Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators containing a straight Wilson line along a spatial direction. We confirm the previous study that mixing among the lowest dimensional operators, which have a mass dimension equal to three, can occur if chiral symmetry is broken in the lattice action. For higher dimensional operators, we find that the dimension-three operators will always mix with dimension-four operators, even if chiral symmetry is preserved. Also, the number of dimension-four operators involved in the mixing is large, and hence it is impractical to remove the mixing by the improvement procedure. Our result is important for determining the Bjorken-x dependence of the parton distribution functions using the quasi-distribution method on a Euclidean lattice. The requirement of using large hadron momentum in this approach makes the control of errors from dimension-four operators even more important.

One-loop renormalization of staple-shaped operators in continuum and lattice regularizations

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-bar scheme. Our results can be immediately used for improving recent nonperturbative investigations of transverse momentum-dependent distribution functions (TMDs) 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.

Glueball spectrum from Nf = 2 lattice QCD study on anisotropic lattices**The numerical calculations were carried out on Tianhe-1A at the National Supercomputer Center (NSCC) in Tianjin and the GPU cluster at Hunan Normal University. This work is supported in part by the National Science Foundation of China (NSFC) (11575196, 11575197, 11335001, 11405053, 11405178, 11275169). Y. C., Z. L. and

The lowest-lying glueballs are investigated in lattice QCD using Nf = 2 clover Wilson fermions on anisotropic lattices. We simulate at two different and relatively heavy quark masses, corresponding to physical pion masses of mπ ∼ 938 MeV and 650 MeV. The quark mass dependence of the glueball masses has not been investigated in the present study. Only the gluonic operators built from Wilson loops are utilized in calculating the corresponding correlation functions. In the tensor channel, we obtain the ground state mass to be 2.363(39) GeV and 2.384(67) GeV at mπ ∼ 938 MeV and 650 MeV, respectively. In the pseudoscalar channel, when using the gluonic operator whose continuum limit has the form of εijk TrBiDjBk, we obtain the ground state mass to be 2.573(55) GeV and 2.585(65) GeV at the two pion masses. These results are compatible with the corresponding results in the quenched approximation. In contrast, if we use the topological charge density as field operators for the pseudoscalar, the masses of the lowest state are much lighter (around 1 GeV) and compatible with the expected masses of the flavor singlet meson. This indicates that the operator εijk TrBiDjBk and the topological charge density couple rather differently to the glueball states and mesons. The observation of the light flavor singlet pseudoscalar meson can be viewed as the manifestation of effects of dynamical quarks. In the scalar channel, the ground state masses extracted from the correlation functions of gluonic operators are determined to be around 1.4-1.5 GeV, which is close to the ground state masses from the correlation functions of the quark bilinear operators. In all cases, the mixing between glueballs and conventional mesons remains to be further clarified in the future.

Renormalization of Wilson-line operators in the presence of nonzero quark masses

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 quasi-distribution 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' and MSbar schemes, as well as the conversion factors of these operators between the two renormalization schemes. Our computations have been performed in dimensional regularization at one-loop level, using massive quarks. The conversion factors can be used to convert the corresponding lattice nonperturbative results to the MSbar 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 which occurs in the presence of nonzero quark masses, both on the lattice and in dimensional regularization.

Non-perturbative renormalization of tensor currents: strategy and results for N_f=0 and N_f=2 QCD

Tensor currents are the only quark bilinear operators lacking a non-perturbative determination of their renormalisation group (RG) running between hadronic and electroweak scales. We develop the setup to carry out the computation in lattice QCD via standard recursive finite-size scaling techniques, and provide results for the RG running of tensor currents in N_f=0 and N_f=2 QCD in the continuum for various Schrödinger Functional schemes. The matching factors between bare and renormalisation group invariant currents are also determined for a range of values of the lattice spacing relevant for large-volume simulations, thus enabling a fully non-perturbative renormalization of physical amplitudes mediated by tensor currents.

RI/MOM and RI/SMOM renormalization of overlap quark bilinears on domain wall fermion configurations

Renormalization constants (RCs) of overlap quark bilinear operators on 2+1-flavor domain wall fermion configurations are calculated by using the RI/MOM and RI/SMOM schemes. The scale independent RC for the axial vector current is computed by using a Ward identity. Then the RCs for the quark field and the vector, tensor, scalar and pseudoscalar operators are calculated in both the RI/MOM and RI/SMOM schemes. The RCs are converted to the $\overline{\rm MS}$ scheme and we compare the numerical results from using the two intermediate schemes. The lattice size is $48^3\times96$ and the inverse spacing $1/a = 1.730(4) {\rm~GeV}$.

Scalar, Axial, and Tensor Interactions of Light Nuclei from Lattice QCD.

Complete flavor decompositions of the matrix elements of the scalar, axial, and tensor currents in the proton, deuteron, diproton, and ^{3}He at SU(3)-symmetric values of the quark masses corresponding to a pion mass m_{π}∼806 MeV are determined using lattice quantum chromodynamics. At the physical quark masses, the scalar interactions constrain mean-field models of nuclei and the low-energy interactions of nuclei with potential dark matter candidates. The axial and tensor interactions of nuclei constrain their spin content, integrated transversity, and the quark contributions to their electric dipole moments. External fields are used to directly access the quark-line connected matrix elements of quark bilinear operators, and a combination of stochastic estimation techniques is used to determine the disconnected sea-quark contributions. The calculated matrix elements differ from, and are typically smaller than, naive single-nucleon estimates. Given the particularly large, O(10%), size of nuclear effects in the scalar matrix elements, contributions from correlated multinucleon effects should be quantified in the analysis of dark matter direct-detection experiments using nuclear targets.

Perturbative matching of continuum and lattice quasi-distributions

Matching of the quasi parton distribution functions between continuum and lattice is addressed using lattice perturbation theory specifically withWilson-type fermions. The matching is done for nonlocal quark bilinear operators with a straightWilson line in a spatial direction. We also investigate operator mixing in the renormalization and possible O(a) operators for the nonlocal operators based on a symmetry argument on lattice.

Evolution of chirality-odd twist-3 fragmentation functions

We derive the complete set of evolutions of chirality-odd twist-3 fragmentation functions at one-loop level. There are totally nine real twist-3 fragmentation functions, among which seven are independent. The renormalization-scale dependence of the nine functions has an important implication for studies of single transverse-spin asymmetries. We find that the evolutions of the three complex fragmentation functions defined by quark–gluon–quark operator are mixed with themselves. There is no mixing with the fragmentation functions defined only with bilinear quark field operators. In the large-Nc limit the evolutions of the three complex fragmentation functions are simplified and reduced to six homogeneous equations.

One-loop calculations in Supersymmetric Lattice QCD

We study the self energies of all particles which appear in a lattice regularization of supersymmetric QCD (${\cal N}=1$). We compute, perturbatively to one-loop, the relevant two-point Green's functions using both the dimensional and the lattice regularizations. Our lattice formulation employs the Wilson fermion acrion for the gluino and quark fields. The gauge group that we consider is $SU(N_c)$ while the number of colors, $N_c$ and the number of flavors, $N_f$, are kept as generic parameters. We have also searched for relations among the propagators which are computed from our one-loop results. We have obtained analytic expressions for the renormalization functions of the quark field ($Z_\psi$), gluon field ($Z_u$), gluino field ($Z_\lambda$) and squark field ($Z_{A_\pm}$). We present here results from dimensional regularization, relegating to a forthcoming publication our results along with a more complete list of references. Part of the lattice study regards also the renormalization of quark bilinear operators which, unlike the non-supersymmetric case, exhibit a rich pattern of operator mixing at the quantum level.

Singlet versus nonsinglet perturbative renormalization of fermion bilinears

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.

Non-perturbative running of renormalization constants from correlators in coordinate space using step scaling

Working in a quenched setup with Wilson twisted mass valence fermions, we explore the possibility to compute non-perturbatively the step scaling function using the coordinate (X-space) renormalization scheme. This scheme has the advantage of being on-shell and gauge invariant. The step scaling method allows us to calculate the running of the renormalization constants of quark bilinear operators. We describe here the details of this calculation. The aim of this exploratory study is to identify the feasibility of the X-space scheme when used in small volume simulations required by the step scaling technique. Eventually, we translate our final results to the continuum MS‾ scheme and compare against four-loop analytic formulae finding satisfactory agreement.