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.
Recently, the connections between gradient flow and renormalization group have been explored analytically and numerically. Gradient flow (when modified by a field rescaling) can be characterized as a continuous blocking transformation. In this work, we draw a connection between gradient flow and functional renormalization group by describing how FRG can be represented by a stochastic process, and how the stochastic observables relate to gradient flow observables. The relation implies correlator scaling formulae that can be applied numerically in lattice simulations. We present preliminary resu…
Gradient Flow Exact Renormalization Group (GFERG) is a framework to define the Wilson action via a gradient flow equation. We study the fixed point structure of the GFERG equation associated with a general gradient flow equation for scalar field theories and show that it is the same as that of the conventional Wilson-Polchinski (WP) equation in general. Furthermore, we discuss that the GFERG equation has a similar RG flow structure around a fixed point to the WP equation. We illustrate these results with the $O(N)$ non-linear sigma model in $4-ε$ dimensions and the Wilson-Fisher fixed point.
The gradient flow exact renormalization group (GFERG) is an exact renormalization group motivated by the Yang--Mills gradient flow and its salient feature is a manifest gauge invariance. We generalize this GFERG, originally formulated for the pure Yang--Mills theory, to vector-like gauge theories containing fermion fields, keeping the manifest gauge invariance. For the chiral symmetry we have two options: one possible formulation preserves the conventional form of the chiral symmetry and the other simpler formulation realizes the chiral symmetry in a modified form à la Ginsparg--Wilson. We wor…
The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter~$t$, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then illustrate the Wilsonian RG flow on the basis of the gradient flow in two examples that possess an infrared fixed point, the 4D many-flavor gauge theory and the 3D $O(N)$ linear sigma model.
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…
The gradient flow exact renormalization group (GFERG) is a formulation of the exact renormalization group that keeps exact gauge invariance. GFERG can keep also modified chiral invariance. We will show that this formulation reproduces the correct axial anomaly in four dimensions.
We establish a concrete correspondence between a gradient flow and the renormalization group flow for a generic scalar field theory. We use the exact renormalization group formalism with a particular choice of the cutoff function.
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…
We investigate the renormalization group (RG) structure of the gradient flow. Instead of using the original bare action to generate the flow, we propose to use the effective action at each flow time. We write down the basic equation for scalar field theory that determines the evolution of the action, and argue that the equation can be regarded as a RG equation if one makes a field-variable transformation at every step such that the kinetic term is kept to take the canonical form. We consider a local potential approximation (LPA) to our equation, and show that the result has a natural interpret…
We propose a continuous real space renormalization group transformation based on gradient flow, allowing for a numerical study of renormalization without the need for costly ensemble matching. We apply our technique in a pilot study of SU$(3)$ gauge theory with $N_f = 12$ fermions in the fundamental representation, finding the mass anomalous dimension to be $γ_m = 0.23(6)$, consistent with other perturbative and lattice estimates. We also present the first lattice calculation of the nucleon anomalous dimension in this theory, finding $γ_N = 0.05(5)$.
We formulate quantum electrodynamics on the basis of gauge (or BRST) covariant diffusion equations of fields. This is a particular example of the gradient flow exact renormalization group (GFERG). The resulting Wilson action fulfills a simple gauge Ward--Takahashi identity. We solve the GFERG equation around the Gaussian fixed point to the second order in gauge coupling and obtain the 1-loop beta function and anomalous dimensions. The anomalous dimension of the electron field coincides with that of the fermion field diffused by a gauge covariant flow equation of Lüscher.
Renormalization schemes and cutoff schemes allow for the introduction of various distinct renormalization scales for distinct couplings. We consider the coupled renormalization group flow of several marginal couplings which depend on just as many renormalization scales. The usual $β$ functions describing the flow with respect to a common global scale are assumed to be given. Within this framework one can always construct a metric and a potential in the space of couplings such that the $β$ functions can be expressed as gradients of the potential. Moreover the potential itself can be derived exp…
We analyze the exact behavior of the renormalization group flow in one-dimensional clock-models which undergo first order phase transitions by the presence of complex interactions. The flow, defined by decimation, is shown to be single-valued and continuous throughout its domain of definition, which contains the transition points. This fact is in disagreement with a recently proposed scenario for first order phase transitions claiming the existence of discontinuities of the renormalization group. The results are in partial agreement with the standard scenario. However in the vicinity of some f…
We overview the entire renormalization theory, both perturbative and non-perturbative, by the method of the exact renormalization group (ERG). We emphasize particularly on the perturbative application of the ERG to the phi4 theory and QED in the four dimensional euclidean space.
We comment on a recent puzzle regarding renormalization group invariance of exact results in SUSY theories, and argue that a purported renormalization of the dynamical scale Λdoes not in fact occur.
We perform the exact renormalization of two-dimensional massless gauge theories. Using these exact results we discuss the cluster property and confinement in both the anomalous and chiral Schwinger models.
Through appropriate projections of an exact renormalization group equation, we study fixed points, critical exponents and nontrivial renormalization group flows in scalar field theories in $2<d<4$. The standard upper critical dimensions $d_k={2k\over k-1}$, $k=2,3,4,\ldots$ appear naturally encoded in our formalism, and for dimensions smaller but very close to $d_k$ our results match the $\ee$-expansion. Within the coupling constant subspace of mass and quartic couplings and for any $d$, we find a gradient flow with two fixed points determined by a positive-definite metric and a $c$-function w…
We incorporate running parameters and anomalous dimensions into the framework of the exact renormalization group. We modify the exact renormalization group differential equations for a real scalar field theory, using the anomalous dimensions of the squared mass and the scalar field. Following a previous paper in which an integral equation approach to the exact renormalization group was introduced, we reformulate the modified differential equations as integral equations that define the continuum limit directly in terms of a running squared mass and self-coupling constant. Universality of the co…
We introduce a new prescription for renormalizing Feynman diagrams. The prescription is similar to BPHZ, but it is mass independent, and works in the massless limit as the MS scheme with dimensional regularization. The prescription gives a diagrammatic solution to Wilson's exact renormalization group differential equation.
The Wilsonian exact renormalization group gives a natural framework in which ultraviolet and infrared divergences can be treated separately. In massless QED we introduce, as the only mass parameter, a renormalization scale $Ł_R > 0$. We prove, using the flow equation technique, that infrared convergence is a necessary consequence of any zero-momentum renormalization condition at $Ł_R$ compatible with the effective Ward identities and axial symmetry. The same formalism is applied to renormalize gauge-invariant composite operators and to prove their infrared finiteness; in particular we consider…
We formulate a method of performing non-perturbative calculations in quantum field theory, based upon a derivative expansion of the exact renormalization group. We then proceed to apply this method to the calculation of critical exponents for three dimensional O(N) symmetric theory. Finally we discuss how the approximation scheme manages to reproduce some exactly known solutions in critical phenomena.
We propose a gauge invariant formulation of the exact renormalization group equation for nonsupersymmetric pure U(N) Yang-Mills theory, based on the construction by Tim Morris. In fact we show that our renormalization group equation amounts to a regularized version of the loop equation, thereby providing a direct relation between the exact renormalization group and the Schwinger-Dyson equations. We also discuss a possible implication of our formulation to the holographic correspondence of the bulk gravity and the boundary gauge theory.
Renormalization is one of the deepest ideas in physics, yet its exact implementation in any interesting problem is usually very hard. In the present work, following the approach by Glazek and Maslowski in the flat space, we will study the exact renormalization of the same problem in a nontrivial geometric setting, namely in the two dimensional hyperbolic space. Delta function potential is an asymptotically free quantum mechanical problem which makes it resemble non-abelian gauge theories, yet it can be treated exactly in this nontrivial geometry.
The perturbative $β$-function is known exactly in a number of supersymmetric theories and in the 't Hooft renormalization scheme in the $ϕ_4^4$ model. It is shown how this allows one to compute the effective action exactly for certain background field configurations and to relate bare and renormalized couplings. The relationship between the MS and SUSY subtraction schemes in $N = 1$ super Yang-Mills theory is discussed.
An alternative point of view to exact renormalization equations is discussed, where quantum fluctuations of a theory are controlled by the bare mass of a particle. The procedure is based on an exact evolution equation for the effective action, and recovers usual renormalization results.
We discuss a general method of constructing the products of composite operators using the exact renormalization group formalism. Considering mainly the Wilson action at a generic fixed point of the renormalization group, we give an argument for the validity of short distance expansions of operator products. We show how to compute the expansion coefficients by solving differential equations, and test our method with some simple examples.
Given a Wilson action invariant under global chiral transformations, we can construct current composite operators in terms of the Wilson action. The short distance singularities in the multiple products of the current operators are taken care of by the exact renormalization group. The Ward-Takahashi identity is compatible with the finite momentum cutoff of the Wilson action. The exact renormalization group and the Ward-Takahashi identity together determine the products. As a concrete example, we study the Gaussian fixed-point Wilson action of the chiral fermions to construct the products of cu…
We study exact renormalization group equations in the framework of the effective average action. We present analytical approximate solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second order phase transitions, critical exponents, first order transitions (some of which are radiatively induced) and tricritical behaviour.
The non-linear way the anomalous dimension parameter has been introduced in the historic first version of the exact renormalization group equation is compared to current practice. A simple expression for the exactly marginal redundant operator proceeds from this non-linearity, whereas in the linear case, first order differential equations must be solved to get it. The role of this operator in the construction of the flow equation is highlighted.
Motivated by the construction of the cMERA for interacting field theories, we derive a non-perturbative functional differential equation for wave functionals in scalar field theories from the exact renormalization group equation. We check the validity of the equation using the perturbation theory. We calculate the wave functional up to the first-order perturbation and verify that it satisfies the equation.
In these notes the exact renormalization group formulation of the scalar theory is briefly reviewed. This regularization scheme is then applied to supersymmetric theories. In case of a supersymmetric gauge theory it is also shown how to recover gauge invariance, broken by the introduction of the infrared cutoff.
A new form of the Wilson renormalization group equation is derived, in which the flow equations are, up to linear terms, proportional to a gradient flow. A set of coördinates is found in which the flow of marginal, low-energy, couplings takes a gradient form, if relevant couplings are tuned to vanish.
We study exact renormalization group (RG) in O(4) gauged supergravity using the effective average action formalism. The nonperturbative RG equations for cosmological and newtonian coupling constants are found. It is shown the existence of (nonstable) fixed point of these equations. The solution of RG equation for newtonian coupling constant is qualitatively the same as in Einstein gravity(i.e. it is growing at large distances).
We study how to compute the operator product expansion coefficients in the exact renormalization group formalism. After discussing possible strategies, we consider some examples explicitly, within the $ε$-expansions, for the Wilson-Fisher fixed points of the real scalar theory in $d=4-ε$ dimensions and the Lee-Yang model in $d=6-ε$ dimensions. Finally we discuss how our formalism may be extended beyond perturbation theory.
A manifestly gauge invariant exact renormalization group for pure SU(N) Yang-Mills theory is proposed, allowing gauge invariant calculations, without any gauge fixing or ghosts. The necessary gauge invariant regularisation which implements the effective cutoff, is naturally incorporated by embedding the theory into a spontaneously broken SU(N|N) super-gauge theory. This guarantees finiteness to all orders in perturbation theory.
New proves of decoupling of massive fields in several quantum field theories are derived in the effective Lagrangian approach based on Wilson renormalization group. In the most interesting case of gauge theories with spontaneous symmetry breaking, the approach, combined with the quantum action principle, leads to a rather simple proof to all orders.
We discuss exact renormalization group (RG) in $R^2$-gravity using effective average action formalism. The truncated evolution equation for such a theory on De Sitter background leads to the system of nonperturbative RG equations for cosmological and gravitational coupling constants. Approximate solution of these RG equations shows the appearence of antiscreening and screening behaviour of Newtonian coupling what depends on higher derivative coupling constants.
The Polchinski version of the exact renormalization group equation is discussed and its applications in scalar and fermionic theories are reviewed. Relation between this approach and the standard renormalization group is studied, in particular the relation between the derivative expansion and the perturbation theory expansion is worked out in some detail.
Exact renormalization group techniques are applied to mass deformed N=4 supersymmetric Yang-Mills theory, viewed as a regularised N=2 model. The solution of the flow equation, in the local potential approximation, reproduces the one-loop (perturbatively exact) expression for the effective action of N=2 supersymmetric Yang-Mills theory, when the regularising mass, M, reaches the value of the dynamical cutoff. One speculates about the way in which further non-perturbative contributions (instanton effects) may be accounted for.
The validity of the renormalization group approach for large $N$ is clarified by using the vector model as an example. An exact difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space provides infinitely many identities which reduce the infinite dimensional coupling constant space to that of finite dimensions. The effective beta functions give exact values for the fixed points and the susceptibility exponents.
The holographic correspondence between 2d, N=2 quantum field theories and classical 4d, N=2 supergravity coupled to hypermultiplet matter is proposed. The geometrical constraints on the target space of the 4d, N=2 non-linear sigma-models in N=2 supergravity background are interpreted as the exact renormalization group flow equations in two dimensions. Our geometrical description of the renormalization flow is manifestly covariant under general reparametrization of the 2d coupling constants. An explicit exact solution to the 2d renormalization flow, based on its dual holographic description in …
Wilson's approach to renormalization group is reanalyzed for supersymmetric Yang-Mills theory. Usual demonstration of exact renormalization group equation must be modified due to the presence of the so called Konishi anomaly under the rescaling of superfields. We carry out the explicit computation for N=1 SUSY Yang-Mills theory with the simpler, gauge invariant regularization method, recently proposed by Arkani-Hamed and Murayama. The result is that the Wilsonian action S_M consists of two terms, i.e. the non anomalous term, which obeys Polchinski's flow equation and Fujikawa-Konishi determina…
Using the exact renormalization group (ERG) formalism, we study the gauge invariant composite operators in QED. Gauge invariant composite operators are introduced as infinitesimal changes of the gauge invariant Wilson action. We examine the dependence on the gauge fixing parameter of both the Wilson action and gauge invariant composite operators. After defining ``gauge fixing parameter independence,'' we show that any gauge independent composite operators can be made ``gauge fixing parameter independent'' by appropriate normalization. As an application, we give a concise but careful proof of t…
We discuss the question of inverse symmetry breaking at non-zero temperature using the exact renormalization group. We study a two-scalar theory and concentrate on the nature of the phase transition during which the symmetry is broken. We also examine the persistence of symmetry breaking at temperatures higher than the critical one.
We give a functional derivation of the Ward-Takahashi identity for Yang-Mills theory in the framework of the exact renormalization group. The identity realizes non-abelian gauge symmetry nontrivially despite the presence of a momentum cutoff. The cutoff deforms the gauge transformation by introducing composite operators. In our functional method, which is an extension of the method used in our previous work on QED, these composite operators are expressed in terms of the Wilson action that depends on both a UV cutoff and an IR cutoff.
We review the Exact Renormalization Group equations of Wegner and Houghton in an approximation which permits both numerical and analytical studies of nonperturbative renormalization flows. We obtain critical exponents numerically and with the local polynomial approximation (LPA), and discuss the advantages and shortcomings of these methods, and compare our results with the literature. In particular, convergence of the LPA is discussed in some detail. We finally integrate the flows numerically and find a $c$-function which determines these flows to be gradient in this approximation.
The conventional absence of field renormalization in the local potential approximation (LPA) --implying a zero value of the critical exponent η-- is shown to be incompatible with the logic of the derivative expansion of the exact renormalization group (RG) equation. We present a LPA with η\neq 0 that strictly does not make reference to any momentum dependence. Emphasis is made on the perfect breaking of the reparametrization invariance in that pure LPA (absence of any vestige of invariance) which is compatible with the observation of a progressive smooth restoration of that invariance on imple…
The development of the Exact Renormalization Group for fermionic theories is presented, together with its application to the chiral Gross-Neveu model. We focus on the reliability of various approximations, specifically the derivative expansion and further truncations in the number of fields. The main differences with bosonic theories are discussed.
We study exact renormalization group equations in the framework of the effective average action. We present analytical solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second order phase transitions, critical exponents, first order transitions (some of which are radiatively induced) and tricritical behaviour.
We show that Polchinski equations in the D--dimensional matrix scalar field theory can be reduced at large $N$ to the Hamiltonian equations in a (D+1)-dimensional theory. In the subsector of the $\Tr ϕ^l$ (for all $l$) operators we find the exact form of the corresponding Hamiltonian. The relation to the Holographic renormalization group is discussed.
We consider a scalar field theory in Minkowski spacetime and define a coarse grained Closed Time Path (CTP) effective action by integrating quantum fluctuations of wavelengths shorter than a critical value. We derive an exact CTP renormalization group equation and solve it using a derivative expansion approach. Explicit calculation is performed for the $λϕ^4$ theory. We discuss the relevance of the CTP average action in the study of non-equilibrium aspects of phase transitions in quantum field theory.
Here we demonstrate, firstly, the construction of dualities using the exact renormalization group approach and, secondly, that spatial non-commutativity can emerge as such a duality. This is done in a simple quantum mechanical setting that establishes an exact duality between the commutative and non-commutative quantum Hall systems with harmonic interactions. It is also demonstrated that this link can be understood as a blocking (coarse graining) transformation in time that relates commutative and non-commutative degrees of freedom.
We investigate the non-perturbative renormalization group flow of the scalar Galileon model in flat space. We discuss different expansion schemes of the Galileon truncation, including a heat-kernel based derivative expansion, a vertex expansion in momentum space and a curvature expansion in terms of a covariant geometric formulation. We find that the Galileon symmetry prevents a quantum induced renormalization group running of the Galileon couplings. Consequently, the Galileon truncation only features a trivial Gaussian fixed point.
