With advances in quantum computing, new opportunities arise to tackle challenging calculations in quantum field theory. We show that trotterized time-evolution operators can be related by analytic continuation to the Euclidean transfer matrix on an anisotropic lattice. In turn, trotterization entails renormalization of the temporal and spatial lattice spacings. Based on the tools of Euclidean lattice field theory, we propose two schemes to determine Minkowski lattice spacings, using Euclidean data and thereby overcoming the demands on quantum resources for scale setting. In addition, we advocate using a fixed-anisotropy approach to the continuum to reduce both circuit depth and number of independent simulations. We demonstrate these methods with Qiskit noiseless simulators for a $2+1$D discrete non-Abelian $D_4$ gauge theory with two spatial plaquettes.
We consider N=4 super Yang-Mills theory on a four-dimensional lattice. The lattice formulation under consideration retains one exact supersymmetry at non-zero lattice spacing. We show that this feature combined with gauge invariance and the large point group symmetry of the lattice theory ensures that the only counterterms that appear at any order in perturbation theory correspond to renormalizations of existing terms in the bare lattice action. In particular we find that no mass terms are generated at any finite order of perturbation theory. We calculate these renormalizations by examining th…
A necessary ingredient for our previous results on the form of the long distance effective action of the twisted lattice N=4 super Yang-Mills theory is the existence of a real space renormalization group which preserves the lattice structure, both the symmetries and the geometric interpretation of the fields. In this brief article we provide an explicit example of such a blocking scheme and illustrate its practicality in the context of a small scale Monte Carlo renormalization group calculation. We also discuss the implications of this result, and the possible ways in which to use it in order …
We propose a lattice model of supersymmetric complex quantum mechanics which realizes the non-renormalization theorem on a lattice. In our lattice model, the Leibniz rule in the continuum, which cannot hold on a lattice due to a no-go theorem, is replaced by the cyclic Leibniz rule (CLR) for difference operators. It is shown that CLR allows two of four supercharges of the continuum theory to preserve while a naive lattice model can realize one supercharge at the most. A striking feature of our lattice model is that there are no quantum corrections to potential terms in any finite order of pert…
We study Euclidean lattice formulations of non-gauge supersymmetric models with up to four supercharges in various dimensions. We formulate the conditions under which the interacting lattice theory can exactly preserve one or more nilpotent anticommuting supersymmetries. We introduce a superfield formalism, which allows the enumeration of all possible lattice supersymmetry invariants. We use it to discuss the formulation of Q-exact lattice actions and their renormalization in a general manner. In some examples, one exact supersymmetry guarantees finiteness of the continuum limit of the lattice…
We discuss fixed point actions for various types of free lattice fermions. The iterated block spin renormalization group transformation yields lines of local but chiral symmetry breaking fixed points. For staggered fermions at least the $U(1) \otimes U(1)$ symmetry can be preserved. This provides a basis for approximating perfect actions for asymptotically free theories far from the critical surface. For a class of lattice fermions that includes Wilson fermions we find in addition one non local but chirally invariant fixed point. Its vicinity is studied in the framework of the Gross Neveu mode…
The renormalization factor relating the bare to the renormalization group invariant quark masses is accurately calculated in quenched lattice QCD using a recursive finite-size technique. The result is presented in the form of a product of a universal factor times another factor, which depends on the details of the lattice theory but is easy to compute, since it does not involve any large scale differences. As a byproduct the Lambda-parameter of the theory is obtained with a total error of 8%.
N=4 supersymmetric quantum mechanical model is formulated on the lattice. Two supercharges, among four, are exactly conserved with the help of the cyclic Leibniz rule without spoiling the locality. In use of the cohomological argument, any possible local terms of the effective action are classified into two categories which we call type-I and type-II, analogous to the D- and F-terms in the supersymmetric field theories. We prove non-renormalization theorem on the type-II terms which include mass and interaction terms with keeping a lattice constant finite, while type-I terms such as the kineti…
We present a method to calculate to very high precision the coefficients of the divergences occuring in two-loop diagrams for a massive scalar field on the lattice. The approach is based on coordinate space techniques and extensive use of the precisely known Green's function.
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,…
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 $β=6.0$.
We present a real space renormalization-group map for probabilities of random walks on a hierarchical lattice. From this, we study the asymptotic behavior of the end-to-end distance of a weakly self- avoiding random walk (SARW) that penalizes the (self-)intersection of two random walks in dimension four on the hierarchical lattice.
Lattice regularization is a standard technique for the nonperturbative definition of a quantum theory of fields. Several approaches to the construction of a quantum theory of gravity adopt this technique either explicitly or implicitly. A crucial complement to lattice regularization is the process of renormalization through which a continuous description of the quantum theory arises. I provide a comprehensive conceptual discussion of the renormalization of lattice-regularized quantum gravity models. I begin with a presentation of the renormalization group from the Wilsonian perspective. I then…
We compute the perturbative renormalization factors required to match to the continuum Isgur-Wise function, calculated using lattice Heavy Quark Effective Theory. The velocity, mass, wavefunction and current renormalizations are calculated for both the forward difference and backward difference actions for a variety of velocities. Subtleties are clarified regarding tadpole improvement, regulating divergences, and variations of techniques used in these renormalizations.
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.
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 …
Supersymmetry on the lattice is explicitly broken by the gluino mass and lattice artifacts. However, it can be restored in the continuum limit by fine tuning the parameters based on the renormalized Ward identities. On the renormalization step not only the mass but also the renormalization of the supercurrent needs to be addressed. Here we present a lattice investigation to obtain the renormalization factors of the supercurrent for $\mathcal{N}$=1 Super-Yang Mills theory in a gauge invariant renormalization scheme. We also provide the conversion factors which are necessary in order to translat…
Lattice regularized Schwinger model with a so-called $θ$ term is studied by using the Grassmann tensor renormalization group. We perform the Lee-Yang and Fisher zero analyses in order to investigate the phase structure at $θ=π$. We find a first order phase transition at larger fermion mass. Both of the Lee-Yang zero and Fisher zero analyses indicate that the critical endpoint at which the first order phase transition terminates belongs to the Ising universality class.
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($a$).
Renormalization constants can be computed by means of Numerical Stochastic Perturbation Theory to two/three loops in lattice perturbation theory, both in the quenched approximation and in the full (unquenched) theory. As a case of study we report on the computation of renormalization constants of the propagator for Wilson fermions. We present our unquenched (N_f=2) computations and compare the results with non perturbative determinations.
This thesis is about new methods of achieving RG transformations, in both a continuum spacetime background and on a lattice discretization thereof. The subject is explored from the point of view of euclidean quantum field theory. As a thesis grounded on the computational method of lattice simulation, I emphasize the role of lattice formulations throughout the work, especially in the first two chapters. In the first, I describe the essential aspects of lattice theory and its symbiosis with RG. In the second, I present a new, continuous approach to RG on the lattice, based on a numerical tool ca…
Using functional techniques, we prove, to all orders of perturbation theory, that lattice vector gauge theories with Ginsparg Wilson fermions are renormalizable. For two or more massless fermions, they satisfy a flavour mixing axial vector Ward identity. It involves a lattice specific part that is quadratic in the vertex functional and classically irrelevant. We show that it stays irrelevant under renormalization. This means that in the continuum limit the (standard) chiral symmetry becomes restored. In particular, the flavour mixing current does not require renormalization.
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…
The renormalization of a general action for massive lattice fermions is discussed. The analysis applies for all $m_q a$. Preliminary results for the self energy at one loop in perturbation theory are presented.
We present results at beta=6.0 and 6.2 for the O(a) improvement and renormalization constants for bilinear operators using axial and vector Ward identities. We discuss the extraction of the mass dependence of the renormalization constants and the coefficients of the equation of motion operators.
We apply the Grassmann tensor renormalization group (GTRG) to the one-flavor lattice Gross-Neveu model in the presence of chemical potential. We compute the fermion number density and its susceptibility and confirm the validity of GTRG for the finite density system. We introduce a method analogous to the reweighting method for Monte Carlo method and test it for some parameters.
The functional integral representation for fermionic observables on the lattice is studied. In particular, Grassmannian representations of the scalar $\hatJ^{(S)}$ and pseudoscalar $\hatJ^{(P)}$ currents and pseudoscalar correlator are derived. It is also discussed the connection between the fermionic Fock space and boundary conditions along the time direction.
In order to extract transport quantities from energy-momentum-tensor (EMT) correlators in Lattice QCD there is a strong need for a non-perturbative renormalization of these operators. This is due to the fact that the lattice regularization explicitly breaks translational invariance, invalidating the non-renormalization-theorem. Here we present a non-perturbative calculation of the renormalization constant of the off-diagonal components of the EMT in SU(3) pure gauge theory using lattices with shifted boundary conditions. This allows us to induce a non-zero momentum in the system controlled by …
We apply the Grassmann tensor renormalization group to the lattice regularized Schwinger model with one-flavor of the Wilson fermion. We study the phase diagram in the $(β,κ)$ plane performing a detailed analysis of the scaling behavior of the Lee-Yang zeros and the peak height of the chiral susceptibility. Our results strongly indicate that the whole range of the phase transition line starting from $(β,κ)=(0.0,0.380665(59))$ and ending at $(\infty,0.25)$ belongs to the two-dimensional Ising universality class similarly to the free fermion case.
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.
In the lattice formulation of the Heavy Quark Effective Theory (LHQET), the classical velocity is renormalized. The origin of this renormalization is the reduction of Lorentz (or O(4)) invariance to (hyper)cubic invariance. The renormalization is finite and depends on the form of the decretization of the reduced heavy quark Dirac equation. For the Forward Time - Centered Space discretization, the renormalization is computed both perturbatively, to one loop, and non-perturbatively using an ensemble of lattices provided by the Fermilab lattice collaboration. The estimates of the leading (linear)…
We discuss the usage of continuous external momenta for computing renormalization factors as needed to renormalize operator matrix elements. These kind of external momenta are encoded in special boundary conditions for the fermion fields. The method allows to compute certain renormalization factors on the lattice that would have been very difficult, if not impossible, to compute with standard methods. As a result we give the renormalization group invariant step scaling function for a twist-2 operator corresponding to the average momentum of non-singlet quark densities.
We study the renormalization group evolution up to the fixed point of the lattice topological susceptibility in the 2-d O(3) non-linear sigma-model. We start with a discretization of the continuum topological charge by a local charge density, polynomial in the lattice fields. Among the different choices we propose also a Symanzik--improved lattice topological charge. We check step by step in the renormalization group iteration the progressive dumping of quantum fluctuations, which are responsible for the additive and multiplicative renormalizations of the lattice topological susceptibility wit…
We investigate the nonperturbative renormalization of composite operators in lattice QCD restricting ourselves to operators that are bilinear in the quark fields. These include operators which are relevant to the calculation of moments of hadronic structure functions. The computations are based on Monte Carlo simulations using quenched Wilson fermions.
The Numerical calculations for tadpole-improved U(1) lattice gauge theory in three-dimensions on anisotropic lattices have been performed using standard path integral Monte Carlo techniques. Using average plaquette tadpole renormalization scheme, simulations were done with temporal lattice spacings much smaller than the spatial ones and results were obtained for the string tension, the renormalized anisotropy and scalar glueball masses. We find, by comparing the `regular' and `sideways' potentials, that tadpole improvement results in very little renormalization of the bare anisotropy and reduc…
In the lattice formulation of the Heavy Quark Effective Theory (LHQET), the classical velocity becomes renormalized. The origin of this renormalization is the reduction of Lorentz (or O(4)) invariance to (hyper)cubic invariance. The renormalization is finite, depends on the form of the discretization of the reduced heavy quark Dirac equation, and can persist down to zero lattice spacing. For the Forward Time - Centered Space discretization, the renormalization is computed both perturbatively, to one loop, and non-perturbatively using an ensemble of lattices provided by the Fermilab ACP-MAPS co…
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.
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.
In the framework of noncompact lattice QED with light fermions, we derive the functional dependence of the average energy per plaquette on the bare parameters using block-spin Renormalization Group arguments and assuming that the renormalized coupling vanishes. Our numerical results for this quantity in $8^4$ and $10^4$ lattices show evidence for triviality in the weak coupling phase and point to a non vanishing value for the renormalized coupling constant in the strong coupling phase.
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.
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.
We study truncation effects in the SU(3) gauge actions obtained by the Monte Carlo renormalization group method. By measuring the heavy quark potential we find that the truncation effects in the actions coarsen the lattice by 40-50 % from the original blocked lattice. On the other hand, we find that rotational symmetry of the heavy quark potentials is well recovered on such coarse lattices, which may indicate that rotational symmetry breaking terms are easily cancelled out by adding a short distance operator. We also discuss the possibility of reducing truncation effects.
The renormalization functions involved in the determination of the topological susceptibility in the SU(2) lattice gauge theory are extracted by direct measurements, without relying on perturbation theory. The determination exploits the phenomenon of critical slowing down to allow the separation of perturbative and non-perturbative effects. The results are in good agreement with perturbative computations.
The determination of renormalization factors is of crucial importance. They relate the observables obtained on finite, discrete lattices to their measured counterparts in the continuum in a suitable renormalization scheme. Therefore, they have to be computed as precisely as possible. A widely used approach is the nonperturbative Rome-Southampton method. It requires, however, a careful treatment of lattice artifacts. They are always present because simulations are done at lattice spacings $a$ and momenta $p$ with $ap$ not necessarily small. In this paper we try to suppress these artifacts by su…
We study a block spin transformation in the SU(3) lattice gauge theory on anisotropic lattices to obtain Iwasaki's renormalization group improved action for anisotropic cases. For the class of actions with plaquette and $1\times2$ rectangular terms, we determine the improvement parameters as functions of the anisotropy $ξ= a_s/a_t$. We find that the program of improvement works well also on anisotropic lattices. From a study of an indicator which estimates the distance to the renormalized trajectory, we show that, for the range of the anisotropy $ξ\approx 1$--4, the coupling parameters previou…
We report on an ongoing non-perturbative computation of RI-MOM scheme renormalization constants for the lattice action with four dynamical flavours currently in use by ETMC. For this goal dedicated simulations with four degenerate sea quark flavours are performed at several values of the standard and twisted quark mass parameters. We discuss a method for removing possible O(a) artifacts at all momenta and extrapolating renormalization constant estimators to the chiral limit. We give preliminary results at one lattice spacing.
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 investigate finite lattice approximations to the Wilson Renormalization Group in models of unconstrained spins. We discuss first the properties of the Renormalization Group Transformation (RGT) that control the accuracy of this type of approximations and explain different methods and techniques to practically identify them. We also discuss how to determine the anomalous dimension of the field. We apply our considerations to a linear sigma model in two dimensions in the domain of attraction of the Ising Fixed Point using a Bell-Wilson RGT. We are able to identify optimal RGTs which allow acc…
A Ginsparg-Wilson based calibration of the topological charge is used to calculate the renormalization constants which appear in the field-theoretical determination of the topological susceptibility on the lattice. A systematic comparison is made with calculations based on cooling. The two methods agree within present statistical errors (3%-4%). We also discuss the independence of the multiplicative renormalization constant Z from the background topological charge used to determine it.
We present a precise non-perturbative determination of the renormalization constants in the mass independent RI'-MOM scheme. The lattice implementation uses the Iwasaki gauge action and four degenerate dynamical twisted mass fermions. The gauge configurations are provided by the ETM Collaboration. Renormalization constants for scalar, pseudo-scalar, vector and axial operators, as well as the quark propagator renormalization, are computed at three different values of the lattice spacing, two volumes and several twisted mass parameters. The method we developed allows for a precise cross-check of…
We present a study of the systematic effects in the nonperturbative evaluation of the renormalization constants which appear in the field-theoretical determination of the topological susceptibility in pure Yang-Mills theory. The study is performed by computing the renormalization constants on configurations that have been calibrated by use of the Ginsparg-Wilson formalism and by cooling.
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.
The QCD light quark mass renormalized at a 1 GeV scale in the $ \overline{MS} $ scheme is obtained from the numerical results of the lattice QCD simulation with staggered fermions. The primary emphasis is given to the connection between the lattice and continuum parameters. The results are compared with those from the QCD sum rule.
We propose a non-perturbative method for computing the renormalization constants of generic composite operators. This method is intended to reduce some systematic errors, which are present when one tries to obtain physical predictions from the matrix elements of lattice operators. We also present the results of a calculation of the renormalization constants of several two-fermion operators, obtained, with our method, by numerical simulation of $QCD$, on a $16^3 \times 32$ lattice, at $β=6.0$. The results of this simulation are encouraging, and further applications to four-fermion operators and…
We propose a new method to renormalize lattice operators. The method is based on the technique to compute the spectral sum appearing in the Shifman-Vainshtein-Zakharov QCD sum rule from lattice correlators. The application of this technique to the light quark system is useful for operator renormalization as well as for the test of perturbative QCD and OPE. We determine the renormalization constant of the vector current and discuss extensions to other current operators.
The determination of renormalization factors is of crucial importance in lattice QCD. They relate the observables obtained on the lattice to their measured counterparts in the continuum in a suitable renormalization scheme. Therefore, they have to be computed as precisely as possible. A widely used approach is the nonperturbative Rome-Southampton method. It requires, however, a careful treatment of lattice artifacts. In this paper we investigate a method to suppress these artifacts by subtracting one-loop contributions to renormalization factors calculated in lattice perturbation theory. We co…
