Clover Term Research Articles

Transversity GPDs of the proton from lattice QCD

We present the first calculation of the $x$-dependence of the isovector transversity generalized parton distributions (GPDs) for the proton within lattice QCD. We compute the matrix elements with non-local operators containing a Wilson line. The calculation implements the Breit symmetric frame. The proton momenta are chosen as $0.83,\,1.25,\,1.67$ GeV, and the values of the momentum transfer squared are $0.69,\,1.02$ GeV$^2$. These combinations include cases with zero and nonzero skewness. The calculation is performed using one ensemble of two degenerate-mass light, a strange and a charm quark of maximally twisted mass fermions with a clover term. The lattice results are renormalized non-perturbatively and finally matched to the light-cone GPDs using one-loop perturbation theory within the framework of large momentum effective theory. The final GPDs are given in the $\overline{\rm MS}$ scheme at a scale of 2 GeV. In addition to the individual GPDs, we form the combination of the transversity GPDs that is related to the transverse spin structure of the proton. Finally, we extract the lowest two moments of GPDs and draw a number of important qualitative conclusions.

The Calculation of Force in Lattice Quantum Chromodynamics

Quantum chromodynamics (QCD) is the fundamental quantum field theory of quarks and gluons. To discuss it in a mathematically well-defined way, the theory has to be regularized by replacing space-time with a Euclidean lattice. This regularized theory, called lattice QCD (LQCD), has proven to be an efficient approach which allows for both theoretical understanding and computational analysis. LQCD has become a standard tool in elementary particle physics, which can be solved by the hybrid Monte Carlo method. The calculation of force is most difficult part in the hybrid Monte Carlo method. This lecture gives the details of the force calculation in one-loop Symanzik improved action, Wilson fermion with clover term, asqtad fermion, HISQ fermion, rooted staggered fermion, smeared fermion, staggered Wilson fermion, overlap fermion and domain wall fermion. The even-odd precondition are also considered in these calculations.

Unpolarized and Helicity Generalized Parton Distributions of the Proton within Lattice QCD.

We present the first calculation of the x dependence of the proton generalized parton distributions (GPDs) within lattice QCD. Results are obtained for the isovector unpolarized and helicity GPDs. We compute the appropriate matrix elements of fast-moving protons coupled to nonlocal operators containing a Wilson line. We present results for proton momenta 0.83,1.25,1.67GeV, and momentum transfer squared 0.69,1.38 GeV^{2}. These combinations include cases with zero and nonzero skewness. The calculation is performed using one ensemble of two degenerate mass light quarks, a strange quark, and a charm quark of maximally twisted mass fermions with a clover term. The lattice results are matched to the light-cone GPDs using one-loop perturbation theory within the framework of large momentum effective theory. The final GPDs are given in the MS[over ¯] scheme at a scale of 2GeV.

Simulating twisted mass fermions at physical light, strange, and charm quark masses

We present the QCD simulation of the first gauge ensemble of two degenerate light quarks, a strange and a charm quark with all quark masses tuned to their physical values within the twisted mass fermion formulation. Results for the pseudoscalar masses and decay constants confirm that the produced ensemble is indeed at the physical parameters of the theory. This conclusion is corroborated by a complementary analysis in the baryon sector. We examine cutoff and isospin breaking effects and demonstrate that they are suppressed through the presence of a clover term in the action.

Pion vector form factor from lattice QCD at the physical point

We present an investigation of the electromagnetic pion form factor, $F_\pi(Q^2)$, at small values of the four-momentum transfer $Q^2$ ($\lesssim 0.25$ GeV$^2$), based on the gauge configurations generated by European Twisted Mass Collaboration with $N_f = 2$ twisted-mass quarks at maximal twist including a clover term. Momentum is injected using non-periodic boundary conditions and the calculations are carried out at a fixed lattice spacing ($a \simeq 0.09$ fm) and with pion masses equal to its physical value, 240 MeV and 340 MeV. Our data are successfully analyzed using Chiral Perturbation Theory at next-to-leading order in the light-quark mass. For each pion mass two different lattice volumes are used to take care of finite size effects. Our final result for the squared charge radius is $\langle r^2 \rangle_\pi = 0.443~(29)$ fm$^2$, where the error includes several sources of systematic errors except the uncertainty related to discretization effects. The corresponding value of the SU(2) chiral low-energy constant $\overline{\ell}_6$ is equal to $\overline{\ell}_6 = 16.2 ~ (1.0)$.

Simulation of an ensemble of Nf = 2 + 1 + 1 twisted mass cloverimproved fermions at physical quark masses

We present a general strategy aimed at generating Nf = 2+1+1 configurations with quarks at their physical mass using maximally twisted mass fermions to ensure automatic O(a) improvement, in the presence of a clover term tuned to reduce the charged to neutral pion mass difference. The target system, for the moment, is a lattice of size 643 × 128 with a lattice spacing a ~ 0:08 fm. We show preliminary results on the pion and kaon mass and decay constants.

Low-lying baryon masses using Nf=2 twisted mass clover-improved fermions directly at the physical pion mass

The masses of the low-lying baryons are evaluated using an ensemble with two degenerate light twisted mass clover-improved quarks with mass tuned to reproduce the physical pion mass. The Iwasaki improved gluonic action is employed. The coupling constant value corresponds to a lattice spacing of $a=0.0938(3)(2)$ fm, determined from the nucleon mass. We find that the clover term supresses isospin symmetry breaking as compared to our previous results using $N_f=2+1+1$ twisted mass fermions. The masses of the hyperons and charmed baryons evaluated using this ensemble are in agreement with the experimental values. We provide predictions for the mass of the doubly charmed $\Xi_{cc}^*$, as well as of the doubly and triply charmed $\Omega$s that have not yet been determined experimentally.

First physics results at the physical pion mass from Nf=2 Wilson twisted mass fermions at maximal twist

We present physics results from simulations of QCD using ${N}_{f}=2$ dynamical Wilson twisted mass fermions at the physical value of the pion mass. These simulations are enabled by the addition of the clover term to the twisted mass quark action. We show evidence that compared to previous simulations without this term, the pion mass splitting due to isospin breaking is almost completely eliminated. Using this new action, we compute the masses and decay constants of pseudoscalar mesons involving the dynamical up and down as well as valence strange and charm quarks at one value of the lattice spacing, $a\ensuremath{\approx}0.09\text{ }\text{ }\mathrm{fm}$. Further, we determine renormalized quark masses as well as their scale-independent ratios, in excellent agreement with other lattice determinations in the continuum limit. In the baryon sector, we show that the nucleon mass is compatible with its physical value and that the masses of the $\mathrm{\ensuremath{\Delta}}$ baryons do not show any sign of isospin breaking. Finally, we compute the electron, muon and tau lepton anomalous magnetic moments and show the results to be consistent with extrapolations of older ETMC data to the continuum and physical pion mass limits. We mostly find remarkably good agreement with phenomenology, even though we cannot take the continuum and thermodynamic limits.

Renormalization functions for Nf=2 and Nf=4 twisted mass fermions

We present results on the renormalization functions of the quark field and fermion bilinears with up to one covariant derivative. For the fermion part of the action we employ the twisted mass formulation with $N_f{=}2$ and $N_f{=}4$ degenerate dynamical quarks, while in the gluon sector we use the Iwasaki improved action. The simulations for $N_f{=}4$ have been performed for pion masses in the range of 390MeV - 760MeV and at three values of the lattice spacing, $a$, corresponding to $\beta{=}1.90,\,1.95,\,2.10$. The $N_f{=}2$ action includes a clover term with $c_{\rm sw}{=}1.57551$ at $\beta{=}2.10$, and three ensembles at different values of $m_\pi$. The evaluation of the renormalization functions is carried out in the RI$'$ scheme using a momentum source. The non-perturbartive evaluation is complemented with a perturbative computation, which is carried out at one-loop level and to all orders in the lattice spacing, $a$. For each renormalization function computed non-perturbatively we subtract the corresponding lattice artifacts to all orders in $a$, so that a large part of the cut-off effects is eliminated. The renormalization functions are converted to the ${\overline{\rm MS}}$ scheme at a reference energy scale of $\mu{=}2$ GeV after taking the chiral limit.

Erratum: Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass [Phys. Rev. D92, 114513 (2015)

We present results on the nucleon scalar, axial and tensor charges as well as on the momentum fraction, and the helicity and transversity moments. The pion momentum fraction is also presented. The computation of these key observables is carried out using lattice QCD simulations at a physical value of the pion mass. The evaluation is based on gauge configurations generated with two degenerate sea quarks of twisted mass fermions with a clover term. We investigate excited states contributions with the nucleon quantum numbers by analyzing three sink-source time separations. We find that, for the scalar charge, excited states contribute significantly and to a less degree to the nucleon momentum fraction and helicity moment. Our analysis yields a value for the nucleon axial charge agrees with the experimental value and we predict a value of 1.027(62) in the $\overline{\text{MS}}$ scheme at 2 GeV for the isovector nucleon tensor charge directly at the physical point. The pion momentum fraction is found to be $\langle x\rangle_{u-d}^{\pi^\pm}=0.214(15)(^{+12}_{-9})$ in the $\overline{\rm MS}$ at 2 GeV.

Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass

We present results on the nucleon scalar, axial, and tensor charges as well as on the momentum fraction, and the helicity and transversity moments. The pion momentum fraction is also presented. The computation of these key observables is carried out using lattice QCD simulations at a physical value of the pion mass. The evaluation is based on gauge configurations generated with two degenerate sea quarks of twisted mass fermions with a clover term. We investigate excited states contributions with the nucleon quantum numbers by analyzing three sink-source time separations. We find that, for the scalar charge, excited states contribute significantly and to a less degree to the nucleon momentum fraction and helicity moment. Our result for the nucleon axial charge agrees with the experimental value. Furthermore, we predict a value of 1.027(62) in the $\overline{\mathrm{MS}}$ scheme at 2 GeV for the isovector nucleon tensor charge directly at the physical point. The pion momentum fraction is found to be $⟨x{⟩}_{u\ensuremath{-}d}^{{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}}=0.214(15){(}_{\ensuremath{-}9}^{+12})$ in the $\overline{\mathrm{MS}}$ at 2 GeV.

Progress in Simulations with Twisted Mass Fermions at the Physical Point

In this contribution, results from Nf = 2 lattice QCD simulations at one lattice spacing using twisted mass fermions with a clover term at the physical pion mass are presented. The mass splitting between charged and neutral pions (including the disconnected contribution) is shown to be around 20(20) MeV. Further, a first measurement using the clover twisted mass action of the average momentum fraction of the pion is given. Finally, an analysis of pseudoscalar meson masses and decay constants is presented involving linear interpolations in strange and charm quark masses. Matching to meson mass ratios allows the calculation of quark mass ratios: ms=ml = 27:63(13), mc=ml = 339:6(2:2) and mc=ms = 12:29(10). From this mass matching the quantities fK = 153:9(7:5) MeV, fD = 219(11) MeV, fDs = 255(12) MeV and MDs = 1894(93) MeV are determined without the application of finite volume or discretization artefact corrections and with errors dominated by a preliminary estimate of the lattice spacing.

Perturbative calculation of the clover term for Wilson fermions in any representation of the gauge group SU(N)

We calculate the Sheikholeslami-Wohlert coefficient of the O(a) improvement term for Wilson fermions in any representation of the gauge group SU(N) perturbatively at the one-loop level. The result applies to QCD with adjoint quarks and to N=1 supersymmetric Yang-Mills theory on the lattice.

Renormalization constants of local operators for Wilson type improved fermions

Perturbative and non-perturbative results are presented on the renormalization constants of the quark field and the vector, axial-vector, pseudoscalar, scalar and tensor currents. The perturbative computation, carried out at one-loop level and up to second order in the lattice spacing, is performed for a fermion action, which includes the clover term and the twisted mass parameter yielding results that are applicable for unimproved Wilson fermions, as well as for improved clover and twisted mass fermions. We consider ten variants of the Symanzik improved gauge action corresponding to ten different values of the plaquette coefficients. Non-perturbative results are obtained using the twisted mass Wilson fermion formulation employing two degenerate dynamical quarks and the tree-level Symanzik improved gluon action. The simulations are performed for pion masses in the range of 480 MeV to 260 MeV and at three values of the lattice spacing, a, corresponding to beta=3.9, 4.05, 4.20. For each renormalization factor computed non-perturbatively we subtract its perturbative O(a^2) terms so that we eliminate part of the cut-off artifacts. The renormalization constants are converted to MS-bar at a scale of mu=2 GeV. The perturbative results depend on a large number of parameters and are made easily accessible to the reader by including them in the distribution package of this paper, as a Mathematica input file.

Perturbative renormalization factors of four-quark operators for improved Wilson fermion action and Iwasaki gauge action

The renormalization factor and O(a) improvement coefficient of four-quark operators are calculated perturbatively for the improved Wilson fermion action with clover term and the Iwasaki gauge action. With an application to the K → ππ decay amplitude in mind, the calculation is restricted to the parity odd operator, for which the operators are multiplicatively renormalized without any mixing to operators that have different chiral structures.

Non-perturbative renormalization of quark mass in N f = 2 + 1 QCD with the Schrödinger functional scheme

We present an evaluation of the quark mass renormalization factor for N f = 2 + 1 QCD. The Schrödinger functional scheme is employed as the intermediate scheme to carry out non-perturbative running from the low energy region, where renormalization of bare mass is performed on the lattice, to deep in the high energy perturbative region, where the conversion to the renormalization group invariant mass or the \( \overline {\text{MS}} \) scheme is safely carried out. For numerical simulations we adopted the Iwasaki gauge action and nonperturbatively improved Wilson fermion action with the clover term. Seven renormalization scales are used to cover from low to high energy regions and three lattice spacings to take the continuum limit at each scale. The regularization independent step scaling function of the quark mass for the N f = 2 + 1 QCD is obtained in the continuum limit. Renormalization factors for the pseudo scalar density and the axial vector current are also evaluated for the same action and the bare couplings as two recent large scale N f = 2 + 1 simulations; previous work of the CP -PACS/JLQCD collaboration, which covered the up-down quark mass range heavier than m π ∼ 500 MeV and that of PACS-CS collaboration for much lighter quark masses down to m π = 155MeV. The quark mass renormalization factor is used to renormalize bare PCAC masses in these simulations.

Precise determination of the strong coupling constant inNf= 2+1 lattice QCD with the Schrödinger functional scheme

We present an evaluation of the running coupling constant for Nf=2+1 QCD. The Schroedinger functional scheme is used as the intermediate scheme to carry out non-perturbative running from the low energy region, where physical scale is introduced, to deep in the high energy perturbative region, where conversion to the MS-bar scheme is safely performed. Possible systematic errors due to the use of perturbation theory occur only in the conversion from three-flavor to four-flavor running coupling constant near the charm mass threshold, where higher order terms beyond 5th order in the $\beta$ function may not be negligible. For numerical simulations we adopted Iwasaki gauge action and non-perturbatively improved Wilson fermion action with the clover term. Seven renormalization scales are used to cover from low to high energy region and three lattice spacings to take the continuum limit at each scale. A physical scale is introduced from the previous Nf=2+1 simulation of the CP-PACS/JL-QCD collaboration, which covered the up-down quark mass range heavier than $m_\pi\sim 500$ MeV.

QuenchedBK-parameter from Osterwalder-Seiler tmQCD quarks and mass-splitting discretization effects

We apply an Osterwalder-Seiler version of twisted mass QCD to a study of the $B_K$ parameter, in which three of the four quark fields making up the relevant $\Delta S =2$ operator are maximally twisted with the same twist angle, while the fourth one has a twist angle of opposite sign. It is known that this setup ensures automatic improvement of the bare $K^0$-$\overline K^0$ operator matrix element and multiplicative renormalization of the $\Delta S =2$ operator, at the price of breaking the $K^0$-$\overline K^0$ mass degeneracy by discretization effects. As a result, two dominant systematic errors of the $B_ K$ determination with Wilson fermions are kept under control. With the Clover term included in the fermion action, we perform a feasibility study and find, in the quenched approximation, a significant improvement of the scaling behaviour of $B_K$, compared to earlier standard tmQCD determinations. Moreover, we study in detail the $K^0$-$\overline K^0$ mass splitting that characterizes this approach and confirm that, in the presence of the Clover term, it is greatly reduced in a maximally twisted theory.

Nonperturbative improvement of stout-smeared three-flavor clover fermions

We discuss a three-flavor lattice QCD action with clover improvement in which the fermion matrix has single level stout smearing for the hopping terms together with unsmeared links for the clover term. With the (tree-level) Symanzik improved gluon action this constitutes the stout link nonperturbative clover or SLiNC action. To cancel $O(a)$ terms the clover term coefficient has to be tuned. We present here results of a nonperturbative determination of this coefficient using the Schr\odinger functional and as a by-product a determination of the critical hopping parameter. Comparisons of the results are made with lowest order perturbation theory.

Scaling analysis of fat-link irrelevant clover fermion actions

The fat-link irrelevant clover fermion action is a variant of the O(a)-improved Wilson action where the irrelevant operators are constructed using smeared links. While the use of such smearing allows for the use of highly improved definitions of the field strength tensor F{sub {mu}}{sub {nu}}, we show that the standard 1-loop clover term with a mean field improved coefficient c{sub sw} is sufficient to remove the O(a) errors, avoiding the need for nonperturbative tuning. This result enables efficient dynamical simulations in QCD with the fat-link irrelevant clover fermion action.