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 called Gradient Flow (GF). Simulation results from quartic scalar field theory in 2 and 3 dimensions ($ϕ^4_d$) and 4-dimensional 12-flavor SU(3) gauge theory will be presented. In the third and fourth chapters, the focus becomes more analytic. Chapter 3 is an introductory review of Functional Renormalization Group (FRG). In chapter 4, I introduce the concept of Stochastic RG (SRG) by working out the relationship between FRG and stochastic processes.
We describe some properties of Renormalization Group transformations. Especially we show why some of the RG transformations have redundant eigenoperators with eigenvalues that cannot be determined by simple dimensional analysis and give the corresponding formulae.
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 propose a family of renormalization group transformations characterized by free parameters that may be tuned in order to reduce the truncation effects. As a check we test them in the three dimensional XY model. The Schwinger--Dyson equations are used to study the renormalization group flow.
We define a one-particle irreducible (1PI) Wilson action in the gradient flow exact renormalization group (GFERG) formalism as the Legendre transform of a Wilson action. We consider quantum electrodynamics in particular, and show that the GFERG flow equation preserves the invariance of the 1PI Wilson action (excluding the gauge-fixing term) under the \emph{conventional\/} $U(1)$ gauge transformation. This is in contrast to the invariance of the original Wilson action under a modified $U(1)$ gauge transformation. The global chiral transformation also takes the \emph{conventional\/} form for the…
By combining two distinct renormalization group transformations, opposing scale transformations, we obtain a composite transformation which does not rescale the system, and drives it to a "geometrical" fixed point, controlling the effective geometry and locality. The latticized (deconstructed) action for an extra-dimensional field theory becomes a ``perfect action,'' with a linear ladder spectrum for N KK-modes.
The study of nonlinear phenomena in systems with many degrees of freedom often relies on complex numerical simulations. In trying to model realistic situations, these systems may be coupled to an external environment which drives their dynamics. For nonlinear field theories coupled to thermal (or quantum) baths, discrete lattice formulations must be dealt with extreme care if the results of the simulations are to be interpreted in the continuum limit. Using techniques from renormalization theory, a self-consistent method is presented to match lattice results to continuum models. As an applicat…
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…
In this preliminary work, I provide the outline of an argument (leaving the full proof to a future publication) that there exists a valid renormalization group blocking transformation which converts the continuum fermion action into a Ginsparg-Wilson lattice action. I construct the blocking for the massless overlap operator as a specific example, indicating how other Ginsparg-Wilson lattice Dirac operators can be derived in a similar fashion. This renormalization group transformation modifies the gauge action and adds a number of irrelevant terms to the lattice action. The procedure is not val…
We propose a new Real Space Renormalization Group transformation useful for Monte Carlo calculations in theories with global or local symmetries. From relaxation arguments we define the block-spin transformation with two tunable free parameters, adapted to the system's action. Varying them it is possible to place the fixed point very close to the simulation point. We show how the method works in a simple model with global symmetry: the three dimensional XY model.
We propose an exact renormalization group equation for Lattice Gauge Theories, that has no dependence on the lattice spacing. We instead relate the lattice spacing properties directly to the continuum convergence of the support of each local plaquette. Equivalently, this is formulated as a convergence prescription for a characteristic polynomial in the gauge coupling that allows the exact meromorphic continuation of a nonperturbative system arbitrarily close to the continuum limit.
In this paper we present one-loop results for the renormalization of nonlocal quark bilinear operators, containing a staple-shaped Wilson line, in both continuum and lattice regularizations. The continuum calculations were performed in dimensional regularization, and the lattice calculations for the Wilson/clover fermion action and for a variety of Symanzik-improved gauge actions. We extract the strength of the one-loop linear and logarithmic divergences (including cusp divergences), which appear in such nonlocal operators; we identify the mixing pairs which occur among some of these operators…
I develop a renormalization-group blocking framework for lattice QCD with staggered fermions. Under plausible, and testable, assumptions, I then argue that the fourth-root recipe used in numerical simulations is valid in the continuum limit. The taste-symmetry violating terms, which give rise to non-local effects in the fourth-root theory when the lattice spacing is non-zero, vanish in the continuum limit. A key role is played by reweighted theories that are local and renormalizable on the one hand, and that approximate the fourth-root theory better and better as the continuum limit is approac…
We show that for a SU(N) Yang-Mills theory the classical background-quantum splitting is non-trivially deformed at the quantum level by a canonical transformation with respect to the Batalin-Vilkovisky bracket associated with the Slavnov-Taylor identity of the theory. This canonical transformation acts on all the fields (including the ghosts) and antifields; it uniquely fixes the dependence on the background field of all the one-particle irreducible Green's functions of the theory at hand. The approach is valid both at the perturbative and non-perturbative level, being based solely on symmetry…
In a previous publication, we have constructed the Schrödinger functional in Wilson's lattice QCD. It was found that the naive continuum limit leads to a well-defined classical continuum theory. Starting from the latter, a formal continuum definition of the Schrödinger functional is given and its saddle point expansion is carried out to one-loop order of perturbation theory. Dimensional regularization and heat kernel techniques are used to determine the one-loop divergences. These are partly canceled by the usual renormalizations of the quark mass and the coupling constant in QCD. An additiona…
The COntractor REnormalization group (CORE) method, a new approach to solving Hamiltonian lattice systems, is presented. The method defines a systematic and nonperturbative means of implementing Kadanoff-Wilson real-space renormalization group transformations using cluster expansion and contraction techniques. We illustrate the approach and demonstrate its effectiveness using scalar field theory, the Heisenberg antiferromagnetic chain, and the anisotropic Ising chain. Future applications to the Hubbard and t-J models and lattice gauge theory are discussed.
Using the example of configurations generated with the worm algorithm for the two-dimensional Ising model, we propose renormalization group (RG) transformations, inspired by the tensor RG, that can be applied to sets of images. We relate criticality to the logarithmic divergence of the largest principal component. We discuss the changes in link occupation under the RG transformation, suggest ways to obtain data collapse, and compare with the two state tensor RG approximation near the fixed point.
A non-perturbative and continuous definition of RG transformations as stochastic processes is proposed, inspired by the observation that the functional RG equations for effective Boltzmann factors may be interpreted as Fokker-Planck equations. The result implies a new approach to Monte Carlo RG that is amenable to lattice simulation. Long-distance correlations of the effective theory are shown to approach gradient-flowed correlations, which are simpler to measure. The Markov property of the stochastic RG transformation implies an RG scaling formula which allows for the measurement of anomalous…
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 decimation transformations applied to high-$q$ Potts models result in non-Gibbsian measures even for temperatures higher than the transition temperature. We also show that majority transformations applied to the Ising model in a very strong field at low temperatures produce non-Gibbsian measures. This shows that pathological behavior of renormalization-group transformations is even more widespread than previous examples already suggested.
We construct an energy-momentum tensor on the lattice which satisfies the appropriate Ward Identities (WIs) and has the right trace anomaly in the continuum limit. It is defined by imposing suitable WIs associated to the Poincare` invariance of the continuum theory. These relations come forth when the length of the box in the temporal direction is finite, and they take a particularly simple form if the coordinate and the periodicity axes are not aligned. We implement the method for the SU(3) Yang--Mills theory discretized with the standard Wilson action in presence of shifted boundary conditio…
The gradient flow bears a close resemblance to the coarse graining, the guiding principle of the renormalization group (RG). In the case of scalar field theory, a precise connection has been made between the gradient flow and the RG flow of the Wilson action in the exact renormalization group (ERG) formalism. By imitating the structure of this connection, we propose an ERG differential equation that preserves manifest gauge invariance in Yang--Mills theory. Our construction in continuum theory can be extended to lattice gauge theory.
Integral transformations of the QCD invariant (running) coupling and of some related objects are discussed. Special attention is paid to the Fourier transformation, that is to transition from the space-time to the energy--momentum representation. The conclusion is that the condition of possibility of such a transition provides us with one more argument against the real existence of unphysical singularities observed in the perturbative QCD. The second conclusion relates to the way of "translation" of some singular long--range asymptotic behaviors to the infrared momentum region. Such a transiti…
By writing the flow equations for the continuum Legendre effective action (a.k.a. Helmholtz free energy) with respect to a particular form of smooth cutoff, and performing a derivative expansion up to some maximum order, a set of differential equations are obtained which at FPs (Fixed Points) reduce to non-linear eigenvalue equations for the anomalous scaling dimension $η$. Illustrating this by expanding (single component) scalar field theory, in two, three and four dimensions, up to second order in derivatives, we show that the method is a powerful and robust means of discovering and quantify…
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 develop a formalism for performing real space renormalization group transformations of the "decimation type" using low temperature perturbation theory. This type of transformations beyond $d=1$ is highly nontrivial even for free theories. We construct such a solution in arbitrary dimensions and develop a weak coupling perturbation theory for it. The method utilizes Schur formula to convert summation over decorated lattice into summation over either original lattice or sublattice. We check the formalism on solvable case of $O(N)$ symmetric Heisenberg chain. The transformation is particularly…
We introduce a non-perturbative improvement for the renormalization group step scaling function based on the gradient flow running coupling, which may be applied to any lattice gauge theory of interest. Considering first SU(3) gauge theory with $N_f = 4$ massless staggered fermions, we demonstrate that this improvement can remove $O(a^2)$ lattice artifacts, and thereby increases our control over the continuum extrapolation. Turning to the 12-flavor system, we observe an infrared fixed point in the infinite-volume continuum limit. Applying our proposed improvement reinforces this conclusion by …
We present the nonperturbatively renormalized nucleon gluon momentum fraction using ensembles with $2+1+1$ flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. The calculation is done using clover fermions for the valence action with three pion masses, $220$, $310$ and $690$ MeV and three lattice spacings, 0.09, 0.12, and 0.15 fm. The renormalization is done using RI/MOM nonperturbative renormalization and using cluster-decomposition error reduction (CDER) to enhance the signal-to-noise ratio of the renormalization constant. We find the CDER technique is particu…
These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification …
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 discrete $β$ function of the 2-flavor SU(3) sextet model using the finite volume gradient flow scheme. Our results, using clover improved nHYP smeared Wilson fermions, follow the (non-universal) 4-loop $\overline{\textrm{MS}}$ perturbative predictions closely up to $g^2 \approx 5.5$, the strongest coupling reached in our simulation. At strong couplings the results are in tension with a recently published work using the same gradient flow renormalization scheme with staggered fermions. Since these calculations define the discrete $β$ function in the same continuum renormaliza…
We compute non-perturbatively the renormalization constants of composite operators for overlap fermions by using the regularization independent scheme. The scaling behavior of the renormalization constants is investigated using the data from three lattices with similar physical volumes and different lattice spacings. The approach of the renormalization constants to the continuum limit is explored.
We extend the position-space renormalization procedure, where renormalization factors are calculated from Green's functions in position space, by introducing a technique to take the average of Green's functions over spheres. In addition to reducing discretization errors, this technique enables the resulting position-space correlators to be evaluated at any physical distance, making them continuous functions similar to the $O(4)$-symmetric position-space Green's functions in the continuum theory but with a residual dependence on a regularization parameter, the lattice spacing $a$. We can then t…
It is shown that the four dimensional antiferromagnetic lattice phi4 model has the usual non-asymptotically free scaling law in the UV regime around the chiral symmetrical critical point. The theory describes a scalar and a pseudoscalar particle. A continuum effective theory is derived for low energies. A possibility of constructing a model with a single chiral boson is mentioned.
Different phenomenological RG transformations based on scaling relations for the derivatives of the inverse correlation length and singular part of the free-energy density are considered. These transformations are tested on the 2D square Ising and Potts models as well as on the 3D simple-cubic Ising model. Variants of RG equations yielding more accurate results than Nightingale's RG scheme are obtained. In the 2D case the finite-size equations which give the {\it exact} values of the critical point or the critical exponent are found.
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 compute, on the $(λΦ^4)_{1+1}$ model on the lattice, the soliton mass by means of two very different numerical methods. First, we make use of a ``creation operator'' formalism, measuring the decay of a certain correlation function. On the other hand we measure the shift of the vacuum energy between the symmetric and the antiperiodic systems. The obtained results are fully compatible. We compute the continuum limit of the mass from the perturbative Renormalization Group equations. Special attention is paid to ensure that we are working on the scaling region, where physical quantities remain …
We propose inverse renormalization group transformations within the context of quantum field theory that produce the appropriate critical fixed point structure, give rise to inverse flows in parameter space, and evade the critical slowing down effect in calculations pertinent to criticality. Given configurations of the two-dimensional $ϕ^{4}$ scalar field theory on sizes as small as $V=8^{2}$, we apply the inverse transformations to produce rescaled systems of size up to $V'=512^{2}$ which we utilize to extract two critical exponents. We conclude by discussing how the approach is generally app…
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 present a study for the pion decay constant $f_π$ in the quenched approximation to lattice QCD with the Kogut-Susskind (KS) quark action, with the emphasis given to the renormalization problems. Numerical simulations are carried out at the couplings $β= 6.0$ and 6.2 on $32^3\times 64$ and $48^3\times 64$ lattices, respectively. The pion decay constant is evaluated for all KS flavors via gauge invariant and non-invariant axial vector currents with the renormalization constants calculated by both non-perturbative method and perturbation theory. We obtain $f_π= 89(6)$ MeV in the continuum limi…
We present a physical interpretation of machine learning functions, opening up the possibility to control properties of statistical systems via the inclusion of these functions in Hamiltonians. In particular, we include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system. Results in the two-dimensional Ising model evidence that the field can induce an order-disorder phase transition by breaking or restoring the symmetry, in contrast with the field of the conventional order parame…
We propose a renormalization scheme that can be simply implemented on the lattice. It consists of the temporal moments of two-point and three-point functions calculated with finite valence quark mass. The scheme is confirmed to yield a consistent result with another renormalization scheme in the continuum limit for the bilinear operators. We apply a similar renormalization scheme for the non-perturbative renormalization of four-fermion operators appearing in the weak effective Hamiltonian.
Cluster Representation of Renormalization Group Transformations and a Rigorous Proof for Convergence of the RG-Flow of the Ising Model to Trivial Fixed Points away from Criticality
Albert Einstein Center for Fundamental Physics（阿尔伯特·爱因斯坦基础物理中心）
Starting from the QCD Schroedinger functional (SF), we define a family of renormalization schemes for two four-quark operators, which are, in the chiral limit, protected against mixing with other operators. With the appropriate flavour assignments these operators can be interpreted as part of either the $ΔF=1$ or $ΔF=2$ effective weak Hamiltonians. In view of lattice QCD with Wilson-type quarks, we focus on the parity odd components of the operators, since these are multiplicatively renormalized both on the lattice and in continuum schemes. We consider 9 different SF schemes and relate them to…
The continuum limit and scaling properties of an asymptotically free field theory regularized on a random lattice are compared with those on a regular square lattice. We work on random lattices parametrized by a degree of ``randomness'' $κ$. We show that the continuum limit exists and different $κ$ are related by a finite renormalization.
We non-perturbatively calculate the scale dependence of the static axial current in the Schroedinger functional scheme by means of a recursive finite-size scaling technique, taking the continuum limit in each step. The bare current in the O(a) improved theory as well as in the original Wilson regularization is thus connected to the renormalization group invariant one. The latter may then be related to the current at the B-scale defined such that its matrix elements differ from the physical (QCD) ones by O(1/M). At present, a (probably small) perturbative uncertainty enters in this step. As an …
We use perturbation theory to construct perfect lattice actions for fermions and gauge fields by blocking directly from the continuum. When one uses a renormalization group transformation that preserves chiral symmetry the resulting lattice action for massless fermions is chirally symmetric but nonlocal. When the renormalization group transformation breaks chiral symmetry, the lattice action becomes local but chiral symmetry is explicitly broken. In particular, starting with a chiral gauge theory in the continuum one either obtains a lattice theory which is gauge invariant but nonlocal, or a l…
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 reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases…
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.
Inspired by direct and indirect maximal center gauge methods which confirm the existence of vortices in lattice calculations and by using the connection formalism, we show that under some appropriate gauge transformations vortices and chains appear in the QCD vacuum of the continuum limit. In the direct method, by applying center gauge transformation and \textquotedblleft center projection,\textquotedblright QCD is reduced to a gauge theory including vortices, which corresponds to the non-trivial first homotopy group $Π_1\left( \text{SO}(3)\right) =Z_2.$ On the other hand, using the indirect m…
