Non-perturbative determination of improvement coefficients b_m and b_A-b_P and normalisation factor Z_m*Z_P/Z_A with N_f=3 Wilson fermions
Giulia Maria de Divitiis, Patrick Fritzsch, Jochen Heitger, Carl Christian Koester, Simon Kuberski, Anastassios Vladikas
备注：
36 pages including figures and tables, latex2e; version published in Eur. Phys. J. C; small additions to the text and reference added; results unchanged
We determine non-perturbatively the normalisation constant Z_m*Z_P/Z_A as well as the Symanzik coefficients b_m and b_A-b_P, required in O(a) improved quark mass renormalisation with Wilson fermions. The strategy underlying their computation involves simulations in N_f=3 QCD with O(a) improved massless sea and non-degenerate valence quarks in the finite-volume Schroedinger functional scheme. Our results, which cover the typical gauge coupling range of large-volume N_f=2+1 QCD simulations with Wilson fermions at lattice spacings below 0.1 fm, are of particular use for the non-perturbative calculation of O(a) improved renormalised quark masses.
The coefficient of the O(a)-improved Sheikholeslami-Wohlert action for Wilson fermions are perturbatively determined at one-loop level and estimated at two-loop level.
We introduce an approach to expand gauge-invariant Wilson operators on lattice. This approach is based on non-abelian Stokes theorem and overcomes some shortage of some former methods. It is also suitable for expanding any Wilson operators on lattice.
A completely non-perturbative estimate is given for gA using both quenched and unquenched O(a) improved Wilson fermions. Particular attention is paid to the determination of the axial renormalisation constant, ZA, using the Ward identity for the propagator. For the quenched case, we have results at three lattice spacings allowing a continuum extrapolation.
Using Schroedinger functional techniques, we determine the coefficient of the clover term necessary for non-perturbative O(a) improvement of hypercubic smeared Wilson fermions on a quenched plaquette action background. Unlike for unsmeared Wilson fermions, the resulting clover coefficients are close to the tree-level value even at coarse lattice spacings, indicating the absence of large cutoff effects. A number of exploratory tests are also performed with the improved 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 $K\toππ$ 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.
We present results on light hadron masses from simulations of full QCD and report on experiences in running such simulations on a Hitachi SR8000-F1 supercomputer.
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.
A non-perturbative determination of the axial current improvement coefficient cA is performed with two flavors of dynamical improved Wilson fermions and plaquette gauge action. The improvement condition is formulated with Schroedinger functional boundary conditions and enforced at constant physical volume. Large sensitivity is obtained by using two different pseudo-scalar states in the PCAC relation. We estimate the resulting correction to F_PS at beta=5.2 to be around 10%.
Several Wilson loops on several lattice sizes are computed in Perturbation Theory via a stochastic method. Applications include: Renormalons, the Mass Term in Heavy Quark Effective Theory and (possibly) the beta-function.
We present a perturbative calculation of the improvement coefficients of SU(2) gauge theory with adjoint representation Wilson-clover fermions and using Schrodinger functional boundary conditions. The computation of the boundary improvement terms is necessary for the full O(a) improvement. With two flavours of adjoint representation fermions this theory is called Minimal Walking Technicolor model.
We propose a non-perturbative method to determine the mixing coefficients of $Δs=2$ four-quark operators for the Wilson quark action using chiral Ward identities. The method is applied to calculate B_K in quenched QCD.
We compute the improvement coefficient $c_{sw}$ that multiplies the Sheikholeslami-Wohlert term as a function of the bare gauge coupling for two flavour QCD. We discuss several aspects concerning simulations with improved dynamical Wilson fermions.
We describe the first steps in the extension of the Symanzik O($a$) improvement program for Wilson-type quark actions to anisotropic lattices, with a temporal lattice spacing smaller than the spatial one. This provides a fully relativistic and computationally efficient framework for the study of heavy quarks. We illustrate our method with accurate results for the quenched charmonium spectrum.
We present results for the light quark masses in lattice QCD with two degenerate flavours of dynamical fermions. We used configurations generated by the UKQCD and QCDSF collaborations at six different combinations of beta and kappa_sea.
We calculate the critical value of the hopping parameter, $κ_c$, in Lattice QCD with Wilson fermions, to two loops in perturbation theory. This quantity is an additive renormalization; as such, it is characterized not only by the standard caveats regarding the asymptotic nature of perturbative results, but also by a linear divergence in the lattice spacing. Consequently, our calculation tests rather stringently the limits of applicability of perturbation theory. We compare our results to non perturbative evaluations of $κ_c$ coming from Monte Carlo simulations. Finally, we apply a tadpole impr…
We present a method to compute non-perturbatively the renormalization constant of the scalar density for Ginsparg-Wilson fermions. It relies on chiral symmetry and is based on a matching of renormalization group invariant masses at fixed pseudoscalar meson mass, making use of results previously obtained by the ALPHA Collaboration for O(a)-improved Wilson fermions. Our approach is quite general and enables the renormalization of scalar and pseudoscalar densities in lattice regularizations that preserve chiral symmetry and of fermion masses in any regularization. As an application we compute the…
By considering the local vector current between nucleon states and imposing charge conservation we determine, for $O(a)$ improved Wilson fermions, its renormalisation constant and quark mass improvement coefficient. The computation is performed for both quenched and two flavour unquenched fermions.
Many observables of interest in lattice QCD are extracted from correlation functions involving the vector current. If Wilson fermions are used, it is therefore of practical importance that, besides the action, the current be O($a$) improved in order to remove the leading discretization errors from the observables. Here we introduce and apply a new method to determine the improvement coefficient for the two most widely used discretizations of the current.
We present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one loop level in lattice perturbation theory. Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions. The extended nature of such `long-link' operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors. We present nonperturbative prescriptions to extract the linearly divergent contributio…
By considering the local vector current between nucleon states and imposing charge conservation, we determine its renormalisation constant and quark mass improvement coefficient for Symanzik $O(a)$ improved Wilson fermions. The computation is first performed for quenched fermions (and for completeness also with unimproved fermions) and compared against known results. The two-flavour unquenched case is then considered.
Renormalization factors for local vector and axial vector currents for the Wilson quark action are perturbatively calculated to one loop order including finite quark masses from the ratio of the on-shell quark matrix elements in the Feynman gauge defined on the lattice and in the continuum. For large quark masses of order unity in lattice units, we find that finite quark mass effects are quite large: one-loop coefficients of the renormalization factors differ by 100% compared to those in the massless limit.
We study QCD with two flavors of non-perturbatively improved Wilson fermions at finite temperature on the $16^3 8$ lattice. We determine the transition temperature at lattice spacings as small as $a \sim 0.12$ fm, and study string breaking below the finite temperature transition. We find that the static potential can be fitted by a two-state ansatz, including a string state and a two-meson state. We investigate the role of Abelian monopoles at finite temperature.
Perturbative coefficients for Wilson loops and the static-quark self-energy are extracted from Monte Carlo simulations at weak coupling. The lattice volumes and couplings are chosen to ensure that the lattice momenta are all perturbative. Twisted boundary conditions are used to eliminate the effects of lattice zero modes and to suppress nonperturbative finite-volume effects due to Z(3) phases. Simulations of the Wilson gluon action are done with both periodic and twisted boundary conditions, and over a wide range of lattice volumes (from $3^4$ to $16^4$) and couplings (from $β\approx 9$ to $β\…
We present a new normalization condition for the axial current, derived from the PCAC relation with non-vanishing quark mass. This condition is expected to reduce mass effects in the chiral extrapolation of the results for the normalization factor Z_A. The application to the two-flavor theory with improved Wilson fermions shows that this expectation is indeed fulfilled. Using the Schroedinger functional setup we calculate Z_A(g_0^2) as well as the vector current normalization factor Z_V(g_0^2) for beta = 6/g_0^2 >= 5.2.
We present a new normalization condition for the axial current, which is derived from the PCAC relation with non-vanishing mass. Using this condition reduces the O(r_0 m) corrections to the axial current normalization constant Z_A for an easier chiral extrapolation in the cases, where simulations at zero quark-mass are not possible. The method described here also serves as a preparation for a determination of Z_A in the full two-flavor theory.
We present the results of an extensive lattice calculation of the renormalization constants of bilinear and four-quark operators for the non-perturbatively O(a)-improved Wilson action. The results are obtained in the quenched approximation at four values of the lattice coupling by using the non-perturbative RI/MOM renormalization method. Several sources of systematic uncertainties, including discretization errors and final volume effects, are examined. The contribution of the Goldstone pole, which in some cases may affect the extrapolation of the renormalization constants to the chiral limit, …
We propose a strategy to non-perturbatively match the Wilson coefficients in the three- and four-flavor theories, which uses two-point Green's functions of the corresponding four-quark operators at long distances. The idea is refined by combining with the spherical averaging technique, which enables us to convert two-point functions calculated on the lattice into continuous functions of the distance $|x-y|$ between two operators. We also show the result for an exploratory calculation of two-point functions of the $ΔS=1$ operators $Q_7$ and $Q_8$ that are in the $(8_L,8_R)$ representation of ${…
We present first results from the QCDSF collaboration for the kaon semileptonic decay form factors at zero momentum transfer, using two flavours of non-perturbatively O(a)-improved Wilson quarks. A lattice determination of these form factors is of particular interest to improve the accuracy on the CKM matrix element |V_us|. Calculations are performed on lattices with lattice spacing of about 0.08 fm with different values of light and strange quark masses, which allows us to extrapolate to chiral limit. Employing double ratio techniques, we are able to get small statistical errors.
We verify the operator product expansion (OPE) of deep inelastic scattering (DIS) on the lattice and present first results of a non-perturbative calculation of the Wilson coefficients.
We non-perturbatively determine the renormalization factor of the axial vector current in lattice QCD with $N_f=3$ flavors of Wilson-clover fermions and the tree-level Symanzik-improved gauge action. The (by now standard) renormalization condition is derived from the massive axial Ward identity and it is imposed among Schrödinger functional states with large overlap on the lowest lying hadronic state in the pseudoscalar channel, in order to reduce kinematically enhanced cutoff effects. We explore a range of couplings relevant for simulations at lattice spacings of $\approx 0.09$ fm and below. …
We describe a new set of gauge configurations generated within the CLS effort. These ensembles have N_f=2+1 flavors of non-perturbatively improved Wilson fermions in the sea with the Luescher-Weisz action used for the gluons. Open boundary conditions in time are used to address the problem of topological freezing at small lattice spacings and twisted-mass reweighting for improved stability of the simulations. We give the bare parameters at which the ensembles have been generated and how these parameters have been chosen. Details of the algorithmic setup and its performance are presented as wel…
We calculate perturbative Wilson loops of various sizes up to loop order $n=20$ at different lattice sizes for pure plaquette and tree-level improved Symanzik gauge theories using the technique of Numerical Stochastic Perturbation Theory. This allows us to investigate the behavior of the perturbative series at high orders. We observe differences in the behavior of perturbative coefficients as a function of the loop order. Up to $n=20$ we do not see evidence for the often assumed factorial growth of the coefficients. Based on the observed behavior we sum this series in a model with hypergeometr…
We propose a method to improve lattice operators composed of Wilson fermions which allows the removal of all corrections of $O(a)$, including those proportional to the quark mass, leaving only errors of $O(a^2)$. The method exploits the fact that chiral symmetry is restored at short distances. By imposing this requirement on correlation functions of improved lattice operators at short distances, the coefficients which appear in these operators can be determined. The method is an extension of the improvement program of the ALPHA collaboration, which, up to now, has only been applicable in the c…
We calculate Wilson loops of various sizes up to loop order $n=20$ for lattice sizes of $L^4 (L=4, 6, 8, 12)$ using the technique of Numerical Stochastic Perturbation Theory in quenched QCD. This allows to investigate the behaviour of the perturbative series at high orders. We discuss three models to estimate the perturbative series: a renormalon inspired fit, a heuristic fit based on an assumed power-law singularity and boosted perturbation theory. We have found differences in the behavior of the perturbative series for smaller and larger Wilson loops at moderate $n$. A factorial growth of th…
We compute the light hadron mass spectrum in quenched lattice QCD at $β= 6.0$ using the Sheikholeslami-Wohlert fermionic action. The calculation is done for several choices of the coefficient $c_{SW}$, including $c_{SW} = 0$ and the recently proposed optimal value $c_{SW} = 1.769$. We find that the individual masses change by up to 30\% under $O(a)$ improvement. The spectrum calculation suggests $c_{SW} \approx 1.4$ for the optimal value of the coefficient.
The coefficient c_A required for O(a) improvement of the axial current in lattice QCD with N_f=3 flavors of Wilson fermions and the tree-level Symanzik-improved gauge action is determined non-perturbatively. The standard improvement condition using Schroedinger functional boundary conditions is employed at constant physics for a range of couplings relevant for simulations at lattice spacings of ~ 0.09 fm and below. We define the improvement condition projected onto the zero topological charge sector of the theory, in order to avoid the problem of possibly insufficient tunneling between topolog…
We present the results for light hadron spectrum, decay constants and the quark masses obtained with non-perturbatively improved Wilson fermions. We also give our preliminary results for the heavy-light decay constants.
We investigate the renormalization of a class of gauge-invariant nonlocal quark bilinear operators, including a finite-length Wilson-line (called Wilson-line operators). The matrix elements of these operators are involved in the recent "quasi-distribution" approach for computing light-cone distributions of Hadronic Physics on the lattice. We consider two classes of Wilson-line operators: straight-line and staple-shaped operators, which are related to the parton distribution functions (PDFs) and transverse momentum-dependent distributions (TMDs), respectively. We present our one-loop results fo…
Numerical Stochastic Perturbation Theory was able to get three- (and even four-) loop results for finite Lattice QCD renormalization constants. More recently, a conceptual and technical framework has been devised to tame finite size effects, which had been reported to be significant for (logarithmically) divergent renormalization constants. In this work we present three-loop results for fermion bilinears in the Lattice QCD regularization defined by tree-level Symanzik improved gauge action and n_f=2 Wilson fermions. We discuss both finite and divergent renormalization constants in the RI'-MOM …
We compute the hadron mass spectrum, the quark masses and the meson decay constants in quenched lattice QCD with non-perturbatively $O(a)$ improved Wilson fermions. The calculations are done for two values of the coupling constant, $β= 6.0$ and 6.2, and the results are compared with the predictions of ordinary Wilson fermions. We find that the improved action reduces lattice artifacts as expected.
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 diffe…
This is the third of a series of papers on three-loop computation of renormalization constants for Lattice QCD. Our main point of interest are results for the regularization defined by Iwasaki gauge action and n_f=4 Wilson fermions. Our results for quark bilinears renormalized according to the RI'-MOM scheme can be compared to non-perturbative results. The latter are available for Twisted Mass QCD: being defined in the chiral limit, renormalization constants must be the same. We also address more general problems. In particular, we discuss a few methodological issues connected to summing the p…
Perturbative expansions of several small Wilson loops are computed through next-to-next-to-leading order in unquenched lattice QCD, from Monte Carlo simulations at weak couplings. This approach provides a much simpler alternative to conventional diagrammatic perturbation theory, and is applied here for the first time to full QCD. Two different sets of lattice actions are considered: one set uses the unimproved plaquette gluon action together with the unimproved staggered-quark action; the other set uses the one-loop-improved Symanzik gauge-field action together with the so-called ``asqtad'' im…
We present the status of our study on the equation of state in 2+1 flavor QCD with non-perturbatively improved Wilson quarks coupled with the RG improved glue. We apply the T-integration method to non-perturbatively calculate the equation of state by the fixed-scale approach.
We present a high order perturbative computation of the renormalization constants Z_V, Z_A and of the ratio Z_P/Z_S for Wilson fermions. The computational setup is the one provided by the RI'-MOM scheme. Three- and four-loop expansions are made possible by Numerical Stochastic Perturbation Theory. Results are given for various numbers of flavours and/or (within a finite accuracy) for generic n_f up to three loops. For the case n_f=2 we also present four-loop results. Finite size effects are well under control and the continuum limit is taken by means of hypercubic symmetric Taylor expansions. …
We propose a method to compute the Wilson coefficients of the weak effective Hamiltonian to all orders in the strong coupling constant using Lattice QCD simulations. We perform our calculations adopting an unphysically light weak boson mass of around $2~\mathrm{GeV}$. We demonstrate that systematic errors for the Wilson coefficients $C_1$ and $C_2$, related to the current-current four-quark operators, can be controlled and present a path towards precise determinations in subsequent works.
We explore different branches of the fermion doublers with Wilson fermion in perturbation theory, in the context of additive mass renormalization and chiral anomaly, and show that by appropriately averaging over suitably chosen branches one can reduce cut-off artifacts. Comparing the central branch with all other branches, we find that the central branch, among all the avatars of the Wilson fermion, is the most suitable candidate for exploring near conformal lattice field theories.
We computed the static potential and Wilson loops to $O(α^2)$ in perturbation theory for different lattice quark and gluon actions. In general, we find short distance lattice data to be well described by ``boosted perturbation theory''. For Wilson-type fermions at present-day quark masses and lattice spacings agreement within 10% between measured ``$β$-shifts'' and those predicted by perturbation theory is found. We comment on prospects for a determination of the real world QCD running coupling.
We present new results for the renormalisation and subtraction constants for the four fermion Delta F=2 operators, computed non-perturbatively in the RI-MOM scheme (in the Landau gauge). From our preliminary analysis of the lattice data at beta=6.45, for the B-bar B mixing bag-parameter we obtain B_B^{RGI} = 1.46(7)(1).
I report on cost estimates and algorithmic performance in simulations using 2 flavours of non-perturbatively O(a) improved Wilson quarks together with the Wilson plaquette action.
A Yukawa-Higgs model with Ginsparg-Wilson (GW) fermions, proposed recently by Bhattacharya, Martin and Poppitz as a possible lattice formulation of chiral gauge theories, is studied. A simple argument shows that the gauge boson always acquires mass by the Stückelberg (or, in a broad sense, Higgs) mechanism, regardless of strength of interactions. The gauge symmetry is spontaneously broken. When the gauge coupling constant is small, the physical spectrum of the model consists of massless fermions, massive fermions and \emph{massive} vector bosons.
We compute the perturbative expression of Wilson loops up to order $g^4$ for SU($N$) lattice gauge theories with Wilson action on a finite box with twisted boundary conditions. Our formulas are valid for any dimension and any irreducible twist. They contain as a special case that of the 4-dimensional Twisted Eguchi-Kawai model for a symmetric twist with flux $k$. Our results allow us to analyze the finite volume corrections as a function of the flux. In particular, one can quantify the approach to volume independence at large $N$ as a function of flux $k$. The contribution of fermion fields in…
In calculating hadronic contributions to precision observables for tests of the Standard Model in lattice QCD, the electromagnetic current plays a central role. Using a Wilson action with O($a$) improvement in QCD with $N_{\mathrm{f}}$ flavors, a counterterm must be added to the vector current in order for its on-shell matrix elements to be O($a$) improved. In addition, the local vector current, which has support on one lattice site, must be renormalized. At O($a$), the breaking of the SU($N_{\mathrm{f}}$) symmetry by the quark mass matrix leads to a mixing between the local currents of differ…
We present the results of a non-perturbative determination of the improvement coefficient $c_\mathrm{V}$ and the renormalisation factor $Z_\mathrm{V}$, which define the renormalised vector current in three-flavour $\mathrm{O}(a)$ improved lattice QCD with Wilson quarks and tree-level Symanzik-improved gauge action. In case of the improvement coefficient, we consider both lattice descriptions of the vector current, the local as well as the conserved (i.e., point-split) one. Our improvement and normalisation conditions are based on massive chiral Ward identities and numerically evaluated in the …
We study the finite temperature phase structure for three-flavor QCD with a focus on locating the critical point which separates crossover and first order phase transition region in the chiral regime of the Columbia plot. In this study, we employ the Iwasaki gauge action and the non-perturvatively O($a$) improved Wilson-Clover fermion action. We discuss the finite size scaling analysis including the mixing of magnetization-like and energy-like observables. We carry out the continuum extrapolation of the critical point using newly generated data at $N_{\rm t}=8$, $10$ and estimate the upper bou…
We derive the perturbative expansion of Wilson loops to order g^4 in a SU(N) lattice gauge theory with twisted boundary conditions. Our expressions show that the thermodynamic limit is attained at infinite N for any number of lattice sites and allow to quantify the deviations from volume independence at finite large N as a function of the twist.
