We discuss a specific cut-off effect which appears in applying the non-perturbative RI/MOM scheme to compute the renormalization constants. To illustrate the problem a Dirac operator satisfying the Ginsparg-Wilson relation is used, but the arguments are more general. We propose a simple modification of the method which gets rid of the corresponding discretization error. Applying this to full-QCD simulations done at a=0.13 fm with the Fixed Point action we find that the renormalization constants are strongly distorted by the artefacts discussed. We consider also the role of global gauge transformations, a freedom which still remains after the conventional gauge fixing procedure is applied.
Viable candidate theories for electroweak dynamical symmetry breaking are expected to show (near) conformal behaviour in order to accommodate current phenomenological constraints. In principle, renormalisation group studies using finite volume renormalisation schemes are well-suited to verify this property in a given model. The most practical schemes are based on the Schrodinger functional (SF), but suffer from potentially large O(a) effects. Some care has to be taken to remove these effects and to set up a scheme where cutoff effects are small and under control. We here take a step in this di…
In spite of the breakthrough in non-perturbative chiral gauge theories during the last decade, the present formulation has stubborn artefacts. Independently of the fermion representation one is confronted with unwanted CP violation and infinitely many undetermined weight factors. Renormalization group identifies the culprit. We demonstrate the procedure on Weyl fermions in a real representation.
We present our results on the non-perturbative evaluation of the renormalization constant for the quark field, $Z_q$, in Landau gauge within RI-MOM scheme. Using three lattice spacing we are able to isolate lattice artefacts of various origin, both perturbative and non-perturbative. In particular, the existence of the dimension-two gluon-condensate is discussed, and confirmed.
We briefly review and compare three methods (one perturbative, one based on Ward Identities and one non-perturbative) for the calculation of the renormalization constants of lattice operators. The following results are presented: (a) non perturbative renormalization of the operators with light quarks; (b) the renormalization constants with a heavy (charm) quark mass and its KLM improvement; (c) the non perturbative determination of the mixing of the $ΔS = 2$ operator.
We review the gradient flow for gauge and fermion fields and its applications to lattice gauge theory computations. Using specific examples, we discuss the interplay between perturbative and non-perturbative calculations in the context of renormalization with the gradient flow.
Renormalization constants of vector ($Z_V$) and axial-vector ($Z_A$) currents are determined non-perturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole improved clover quark action using the Schrödinger functional method. Non-perturbative values of $Z_V$ and $Z_A$ turn out to be smaller than the one-loop perturbative values by $O(10%)$ at $a^{-1}\approx 1$ GeV. A sizable scaling violation of meson decay constants $f_π$ and $f_ρ$ observed with the one-loop renormalization factors remains even with non-perturbative renormalization.
We present non-perturbative renormalization factors for $ΔS=2$ four-quark operators in quenched domain-wall QCD using the Schroedinger functional method. Non-perturbative renormalization factor for $B_K$ is evaluated at hadronic scale. Combined with the non-perturbative RG running obtained by the Alpha collaboration, our result yields renormalization factor which converts lattice bare $B_K$ to the renormalization group invariant one. We apply the renormalization factor to bare $B_K$ previously obtained by the CP-PACS collaboration with the quenched domain-wall QCD(DWQCD). We compare our result…
We discuss the application of the regularization independent (RI) scheme of Rome/Southampton to determine the normalization of heavy quark operators non-perturbatively using the Fermilab action.
Our objective is to compute the moments of the deep-inelastic structure functions of the nucleon on the lattice. A major source of uncertainty is the renormalization of the lattice operators that enter the calculation. In this talk we compare the renormalization constants of the most relevant twist-two bilinear quark operators which we have computed non-perturbatively and perturbatively to one loop order. Furthermore, we discuss the use of tadpole improved perturbation theory.
We apply a recently introduced non-perturbative renormalization method to two types of lattice operators: the $ΔS=2$ four fermion operator and the heavy-light static axial current, which are relevant for the physics of $K$ and $B$ mesons respectively. The results of the non-perturbative calculations of the renormalization constants are compared with the corresponding perturbative ones.
Recently a block spin renormalization group approach was proposed for the dynamical triangulation formulation of two-dimensional quantum gravity. We use this approach to examine non-perturbatively a particular class of higher derivative actions for pure gravity.
We define, within the Schrödinger functional (SF) scheme, the matrix elements of the twist-2 operators corresponding to the first two moments of non-singlet parton density, and the first moment of singlet parton densities. We perform a lattice one-loop calculation that fixes the relation between the SF scheme and other common schemes and shows the main source of lattice artefacts. Few remarks on the improvement case are added.
We present an exploratory study of a gauge-invariant non-perturbative renormalization technique. The renormalization conditions are imposed on correlation functions of composite operators in coordinate space on the lattice. Numerical results for bilinears obtained with overlap and O(a)-improved Wilson fermions are presented. The measurement of the quark condensate is also discussed.
We discuss the necessity of non-perturbative renormalization in QCD and HQET and explain the general strategy for solving this problem. A few selected topics are discussed in some detail, namely the importance of off-shell improvement in the MOM-scheme on the lattice, recent progress in the implementation of finite volume schemes and then particular emphasis is put on the recent idea to carry out a non-perturbative renormalization of the Heavy Quark Effective Theory.
We compute non-perturbatively the renormalization constants of quark bilinears on the lattice in the quenched approximation at three values of the coupling beta=6/g_0^2=6.0,6.2,6.4 using both the Wilson and the tree-level improved SW-Clover fermion action. We perform a Renormalization Group analysis at the next-to-next-to-leading order and compute Renormalization Group invariant values for the constants. The results are applied to obtain a fully non-perturbative estimate of the vector and pseudoscalar decay constants.
We present renormalization factors for the bilinear operators obtained using the non-perturbative renormalization method (NPR) in the RI-MOM scheme with improved staggered fermions on the MILC asqtad lattices ($N_f = 2+1$). We use the MILC coarse ensembles with $20^3 \times 64$ geometry and $am_{\ell}/am_s = 0.01/0.05$. We obtain the wave function renormalization factor $Z_q$ from the conserved vector current and the mass renormalization factor $Z_m$ from the scalar bilinear operator. We also present preliminary results of renormalization factors for other bilinear operators.
It has been recently proposed to use the operator product expansion to evaluate the expectation values of renormalized operators without the need of a direct computation of the relevant renormalization constants. We test the viability of this idea in the two-dimensional non-linear sigma-model discussing the non-perturbative renormalization of the energy-momentum tensor.
We present our progresses in the use of the non-perturbative renormalization framework based on considering QCD at finite temperature with shifted and twisted (for quarks only) boundary conditions in the compact direction. We report our final results in the application of this method for the non-perturbative renormalization of the flavor-singlet local vector current. We then discuss the more challenging case of the renormalization of the energy-momentum tensor, and show preliminary results on the relevant one-point functions for the computation of the renormalization constants of its non-singl…
Using lattice overlap fermions, we have computed the 1-loop renormalization factors of several operators that measure DIS structure functions and weak amplitudes. Computer codes written in the algebraic manipulation language FORM have been used. The improvement of the operators is also discussed.
Renormalization constants of vector ($Z_V$) and axial-vector ($Z_A$) currents are determined non-perturbatively in quenched QCD for an RG-improved gauge action and a tadpole-improved clover quark action using the Schrödinger functional method. Meson decay constants $f_ρ$ and $f_π$ show much better scaling when $Z_V$ and $Z_A$ estimated for infinite physical volume are used instead of $Z$-factors from tadpole-improved one-loop perturbation theory.
We present the first numerical implementation of a non-perturbative renormalization method for lattice operators, based on the study of correlation functions in coordinate space at short Euclidean distance. The method is applied to compute the renormalization constants of bilinear quark operators for the non-perturbative O(a)-improved Wilson action in the quenched approximation. The matching with perturbative schemes, such as MS-bar, is computed at the next-to-leading order in continuum perturbation theory. A feasibility study of this technique with Neuberger fermions is also presented.
The momentum diffusion coefficient of a heavy quark in a hot QCD plasma can be extracted as a transport coefficient related to the correlator of two colour-electric fields dressing a Polyakov loop. We determine the perturbative renormalization factor for a particular lattice discretization of this correlator within Wilson's SU(3) gauge theory, finding a ~12% NLO correction for values of the bare coupling used in the current generation of simulations. The impact of this result on existing lattice determinations is commented upon, and a possibility for non-perturbative renormalization through th…
We present the results of an extensive non-perturbative calculation of the renormalization constants of bilinear quark operators for the non-perturbatively O(a)-improved Wilson action. The results are obtained at four values of the lattice coupling, by using the RI/MOM and the Ward identities methods. A new non-perturbative renormalization technique, which is based on the study of the lattice correlation functions at short distance in x-space, is also numerically investigated. We then use our non-perturbative determination of the quark mass renormalization constants to compute the values of th…
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 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…
Renormalization factors for bilinear and four-quark operators with the Kogut-Susskind fermion action are perturbatively calculated to one-loop order in the general covariant gauge. Results are presented both for gauge invariant and non-invariant operators. For four-quark operators the full renormalization matrix for a complete set of operators with two types of color contraction structures are worked out and detailed numerical tables are given.
Renormalization constants ($Z$-factors) of vector and axial-vector currents are determined non-perturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole improved clover quark action using the Schrödinger functional method. Non-perturbative values of $Z$-factors turn out to be smaller than one-loop perturbative values by $O(15%)$ at lattice spacing of $a^{-1}\approx$ 1 GeV. The pseudoscalar and vector meson decay constants calculated with the non-perturbative $Z$-factors show a much better scaling behavior compared to previous results obtained with tadpole i…
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 define a family of Schroedinger Functional renormalization schemes for the four-quark multiplicatively renormalizable operators of the $ΔF = 1$ and $ΔF = 2$ effective weak Hamiltonians. Using the lattice regularization with quenched Wilson quarks, we compute non-perturbatively the renormalization group running of these operators in the continuum limit in a large range of renormalization scales. Continuum limit extrapolations are well controlled thanks to the implementation of two fermionic actions (Wilson and Clover). The ratio of the renormalization group invariant operator to its renormal…
We show that the renormalization factor relating the renormalization group invariant quark masses to the bare quark masses computed in lattice QCD can be determined non-perturbatively. The calculation is based on an extension of a finite-size technique previously employed to compute the running coupling in quenched QCD. As a by-product we obtain the $Λ$--parameter in this theory with completely controlled errors.
We propose a general renormalization method, which avoids completely the use of lattice perturbation theory. We present the results from its numerical applications to two-fermion operators on a $16^3 \times 32$ lattice, at $β=6.0$.
I review the strategies which have been developped in recent years to solve the non-perturbative renormalization problem in lattice field theories. Although the techniques are general, the focus will be on applications to lattice QCD. I discuss the momentum subtraction and finite volume schemes, and their application to scale dependent renormalizations. The problem of finite renormalizations is illustrated with the example of explicit chiral symmetry breaking, and I give a short status report concerning Symanzik's improvement programme to O($a$).
We compute non-perturbatively the evolution of the twist-2 operators corresponding to the average momentum of non-singlet quark densities. The calculation is based on a finite-size technique, using the Schrödinger Functional, in quenched QCD. We find that a careful choice of the boundary conditions, is essential, for such operators, to render possible the computation. As a by-product we apply the non-perturbatively computed renormalization constants to available data of bare matrix elements between nucleon states.
We discuss the status and the problems related to the application of the off-shell non-perturbative renormalization method in a fixed gauge to operators relevant to weak decays. In particular, we critically reappraise the method recently proposed for the $ΔI=1/2$ rule. We also present a general analysis of the renormalization for the $ΔI=3/2$ operators, and apply it to the $ΔS=2$ operator.
A general strategy to solve the non-perturbative renormalization problem in lattice QCD, using finite-size techniques and numerical simulations, is described. As an illustration we discuss the computation of the axial current normalization constant, the running coupling at zero quark masses and the scale evolution of the renormalized axial density. The non-perturbative calculation of O(a) correction terms (as they appear in Symanzik's improvement programme) is another important field of application.
We formulate and numerically test the Schrödinger functional scheme for domain-wall QCD. We then apply it to a non-perturbative calculation of the renormalization factors for vector and axial-vector currents in quenched domain-wall QCD with plaquette and renormalization group improved gauge actions at $a^{-1}\simeq 2$ GeV.
We report on our studies of the renormalization of flavor singlet quark bilinear operators in lattice QCD. The renormalization constants are determined non-perturbatively using gauge field ensembles with Nf=2 dynamical clover improved fermions. The renormalization is performed in the RI'-MOM scheme. The difference between flavor singlet and non-singlet quark bilinear operators is a disconnected contribution, which has to be evaluated by stochastic estimators. We compare our results for the running with perturbation theory.
We present results of the wave function renormalization factor $Z_q$ and mass renormalization factor $Z_m$ obtained using non-perturbative renormalization (NPR) method in the RI-MOM scheme with HYP improved staggered quarks. We use fine ensembles of MILC asqtad lattices ($N_f = 2+1$) with $28^3 \times 96$ geometry, $a \approx 0.09$\,fm, and $am_\ell/am_s = 0.0062/0.031 $. We also study on scalability of $Z_q$ and $Z_m$ by comparing the results on the coarse and fine ensembles.
Non-perturbative renormalization of lattice composite operators plays a crucial role in many applications of lattice field theory. We sketch the general problems involved in this task and the methods which are currently used to cope with them. We present a detailed investigation of a new approach based on the operator product expansion. We test the new method on the two-dimensional O(3) sigma-model and discuss its advantages and limitations. Ph.D. thesis at Scuola Normale Superiore, Pisa, Italy.
We present preliminary results of matching factors of the four-fermion operators relevant to $B_K$, which are obtained using the non-perturbative renormalization (NPR) method in the RI-MOM scheme with HYP-smeared improved staggered fermions. We use the MILC asqtad coarse ($a \cong 0.12\,$fm) ensembles with $20^3 \times 64$ geometry and $am_{\ell}/am_s = 0.01/0.05$. We compare NPR results with those of one-loop perturbative matching.
A general non-perturbative analysis of the renormalization properties of $ΔI=3/2$ four-fermion operators in the framework of lattice regularization with Wilson fermions is presented. We discuss the non-perturbative determination of the operator renormalization constants in the lattice Regularization Independent (RI or MOM) scheme. We also discuss the determination of the finite lattice subtraction coefficients from Ward Identities. We prove that, at large external virtualities, the determination of the lattice mixing coefficients, obtained using the RI renormalization scheme, is equivalent to …
We study the non-perturbative determination of the renormalization constants of flavor non-singlet quark bilinear operators on the lattice. The renormalization condition is imposed on correlation functions of bilinear operators in the coordinate space. The results are converted to the value at 2 GeV in the $\rm\overline{MS}$ scheme by a perturbative matching. The calculation is carried out on gauge configurations generated with the Mobius domain-wall fermions at two lattice spacings $a^{-1} = 2.4$ GeV and $a^{-1} = 3.6$ GeV.
We discuss the renormalization of the lattice formulation of the Heavy Quark Effective Theory (LHQET). In addition to wave function and composite operator renormalizations, on the lattice the classical velocity is also renormalized. The origin of this renormalization is the reduction of Lorentz (or O(4)) invariance to (hyper)cubic invariance. We present results of a new, direct lattice simulation of this finite renormalization, and compare the results to the perturbative (one loop) result. The simulation results are obtained with the use of a variationally optimized heavy-light meson operator,…
Renormalization factors relate the observables obtained on the lattice to their measured counterparts in the continuum in a suitable renormalization scheme. They have to be computed very precisely which requires a careful treatment of lattice artifacts. In this work we present a method to suppress these artifacts by subtracting one-loop contributions proportional to the square of the lattice spacing calculated in lattice perturbation theory.
By imposing axial and vector Ward identities for flavour-non-singlet currents, we estimate in the quenched approximation the non-perturbative values of combinations of improvement coefficients, which appear in the expansion around the massless case of the renormalization constants of axial, pseudoscalar, vector, scalar non-singlet currents and of the renormalized mass. These coefficients are relevant for the completion of the improvement programme to O(a) of such operators. The simulations are performed with a clover Wilson action non-perturbatively improved.
We evaluate renormalization factors of the domain-wall fermion system with various improved gauge actions at one loop level. The renormalization factors are calculated for quark wave function, quark mass, bilinear quark operators, three- and four-quark operators in modified minimal subtraction (MS-bar) scheme with the dimensional reduction(DRED) as well as the naive dimensional regularization(NDR). We also present detailed results in the mean field improved perturbation theory.
We calculate one-loop renormalization factors of generic DeltaS=2 four-quark operators for domain-wall QCD with the plaquette gauge action and the Iwasaki gauge action. The renormalization factors are presented in the modified minimal subtraction (MS-bar) scheme with the naive dimensional regularization. As an important application we show how to construct the renormalization factors for the operators contributing to K^0-K^0bar mixing in the supersymmetric models with the use of our results.
We study the heavy quark mass dependence of the perturbative renormalization factors for the heavy-light currents involving Wilson, Clover, and NRQCD heavy quarks. We find that the one-loop Z-factor for the axial-vector current depends significantly on the heavy quark mass commonly for all actions we study, while that for the vector current has smaller dependence.
Renormalization factors for $ΔS =1$ four-quark operators in the effective weak Hamiltonian are perturbatively evaluated in domain-wall QCD. The one-loop corrections of $ΔS=1$ four-quark operators consist of two types of diagrams: one is gluon exchange between quark lines and the other is penguin diagrams containing quark loops. Combining both contributions, our results allow a lattice calculation of the amplitude for $K\to ππ$ decays with $O(g^2)$ corrections included.
We calculate one-loop renormalization factors for heavy-light bilinears as well as four-fermion operators relevant for $B^{0} - \bar{B}^{0}$ mixing calculations on the lattice. We use the static approximation for heavy quarks and the domain-wall formulation for light quarks. We present results for different choices of improved gauge action.
Renormalization factors for three- and four-quark operators, which appear in the low energy effective Lagrangian of the proton decay and the weak interactions, are perturbatively calculated in domain-wall QCD. We find that the operators are multiplicatively renormalizable up to one-loop level without mixing with any other operators that have different chiral structures. As an application, we evaluate a renormalization factor for $B_K$ at the parameters where previous simulations have been performed, and find one-loop corrections to $B_K$ are 1-5% in these cases.
We present a strategy for a non-perturbative determination of the finite renormalization constants of the energy-momentum tensor in the SU(3) Yang-Mills theory. The computation is performed by imposing on the lattice suitable Ward Identites at finite temperature in presence of shifted boundary conditions. We show accurate preliminary numerical data for values of the bare coupling g_0^2 ranging for 0 to 1.
We calculate the 1-loop renormalization of a set of extended fermionic bilinears which form a basis corresponding to moments of the parton distribution functions. We use the overlap action for fermions and Luescher-Weisz (LW) action for gluons. Our results are presented as a function of the overlap parameter rho and the parameters entering the LW action.
We carry out the renormalization and the Symanzik O(a)-improvement programme for the static vector current in quenched lattice QCD. The scale independent ratio of the renormalization constants of the static vector and axial currents is obtained non-perturbatively from an axial Ward identity with Wilson-type light quarks and various lattice discretizations of the static action. The improvement coefficients cVstat and bVstat are obtained up to O(g_0^4)-terms by enforcing improvement conditions respectively on the axial Ward identity and a three-point correlator of the static vector current. A co…
We compute non--perturbatively the renormalization coefficients of scalar and pseudoscalar operators, local vector and axial currents, conserved vector and axial currents, and $O_{LL}^{ΔS=2}$ over a wide range of energy scales using a scaling technique that connects the results of simulations at different values of coupling $β$. We use the domain wall fermion formulation in the quenched approximation at a series of three values of $β, 6.0, 6.45, 7.05$ corresponding to lattice spacing scaling by factors of two.
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 carry out the non-perturbative renormalization of the chromo-magnetic operator in Heavy Quark Effective Theory. At order 1/m of the expansion, the operator is responsible for the mass splitting between the pseudoscalar and vector B mesons. We obtain its two-loop anomalous dimension in a Schr"odinger functional scheme by successive one-loop conversions to the lattice MS scheme and the MS-bar scheme. We then compute the scale evolution of the operator non-perturbatively in the N_f=0 theory between $μ\approx 0.3$ GeV and $μ\approx 100$ GeV, where contact is made with perturbation theory. The o…
We calculate one-loop renormalization factors of bilinear operators made of physical quark fields for domain-wall QCD. We find that finite parts of such renormalization factors have reasonable values at 1-loop except an overlap factor between the physical quark field and the zero mode in the theory. We point out that the 1-loop estimate of overall renormalization factors becomes unreliable at the coupling where numerical simulations are currently performed, due to the presence of this overlap factor. We show that this problem disappears if the mean-field improved perturbation theory is employe…
We present a preliminary result of Nf=2+1 QCD running coupling in Schroedinger functional scheme. We adopted Iwasaki gauge action and non-perturbatively improved Wilson fermion action with clover term. We use seven renormalization scales to cover from low energy to high energy perturbative region and three lattice spacings to take the continuum limit at each scale. A scaling behabior of the step scaling function is discussed together with its renoralization group flow in the continuum. We argue on introduction of the physical scale through the Sommer scale r0.
We present our progress in the non-perturbative O(a) improvement and renormalization of tensor currents in three-flavor lattice QCD with Wilson-clover fermions and tree-level Symanzik improved gauge action. The mass-independent O(a) improvement factor of tensor currents is determined via a Ward identity approach, and their renormalization group running is calculated via recursive finite-size scaling techniques, both implemented within the Schrödinger functional framework. We also address the matching factor between bare and renormalization group invariant currents for a range of lattice spacin…
