Derivative expansion of the exact renormalization group

Physical Review D, 2004

We further develop an algorithmic and diagrammatic computational framework for very general exact renormalization groups, where the embedded regularisation scheme, parametrised by a general cutoff function and infinitely many higher point vertices, is left unspecified. Calculations proceed iteratively,by integrating by parts with respect to the effective cutoff, thus introducing effective propagators, and differentials of vertices that can be expanded using

Physical Review D, 2000

We consider scalar field theories in dimensions lower than four in the context of the Wegner-Houghton renormalization group equations (WHRG). The renormalized trajectory makes a non-perturbative interpolation between the ultraviolet and the infrared scaling regimes. Strong indication is found that in two dimensions and below the models with polynomial interaction are always non-perturbative in the infrared scaling regime. Finally we check that these results do not depend on the regularization and we develop a lattice version of the WHRG in two dimensions.

International Journal of Modern Physics A, 2001

After a brief presentation of the exact renormalization group equation, we illustrate how the field theoretical (perturbative) approach to critical phenomena takes place in the more general Wilson (nonperturbative) approach. Notions such as the continuum limit and the renormalizability and the presence of singularities in the perturbative series are discussed. This paper has two parts. In the first part we restrict ourselves to the presentation of some selected issues taken from a recent review on the exact renormalization group (RG) equation (ERGE) in the pure scalar case.

Journal of High Energy Physics, 2002

The requirement that the quantum partition function be invariant under a renormalization group transformation results in a wide class of exact renormalization group equations, differing in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. We demonstrate this scheme independence in four dimensional scalar field theory by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and thus, under very general conditions, the universal result is recovered.

International Journal of Modern Physics A, 2001

We investigate the Polchinski ERG equation for d-dimensional O(N) scalar field theory. In the context of the non-pertubative derivative expansion we find families of regular solutions and establish their relation with the physical fixed points of the theory. Special emphasis is given to the limit N=∞ for which many properties can be studied analytically.

Modern Physics Letters A, 1998

An alternative model to the trivial φ 4 -theory of the standard model of weak interactions is suggested, which embodies the Higgs-mechanism, but is free of the conceptual problems of standard φ 4 -theory. We propose a N -component, O(N)-symmetric scalar field theory, which is originally defined on the lattice. The model can be motivated from SU(2) gauge theory. Thereby the scalar field arises as a gauge invariant degree of freedom. The scalar lattice model is analytically solved in the large N limit. The continuum limit is approached via an asymptotically free scaling. The renormalized theory evades triviality, and furthermore gives rise to a dynamically formed mass of the scalar particle.

Physical Review B

We present a functional renormalization group investigation of an Euclidean three-dimensional matrix Yukawa model with U(N) symmetry, which describes N = 2 Weyl fermions that effectively interact via a short-range repulsive interaction. This system relates to an effective low-energy theory of spinless electrons on the honeycomb lattice and can be seen as a simple model for suspended graphene. We find a continuous phase transition characterized by large anomalous dimensions for the fermions and composite degrees of freedom. The critical exponents define a new universality class distinct from Gross-Neveu type models, typically considered in this context.

Physical Review D, 1996

We develop a field theoretical approach to the cold interstellar medium (ISM). We show that a non-relativistic self-gravitating gas in thermal equilibrium with variable number of atoms or fragments is exactly equivalent to a field theory of a single scalar field φ(x) with exponential selfinteraction. We analyze this field theory perturbatively and non-perturbatively through the renormalization group approach. We show scaling behaviour (critical) for a continuous range of the temperature and of the other physical parameters. We derive in this framework the scaling relation ∆M (R) ∼ R d H for the mass on a region of size R, and ∆v ∼ R q for the velocity dispersion where q = 1 2 (dH − 1). For the density-density correlations we find a power-law behaviour for large distances ∼ | r1 − r2| 2d H −6. The fractal dimension dH turns to be related with the critical exponent ν of the correlation length by dH = 1/ν. The renormalization group approach for a single component scalar field in three dimensions states that the long-distance critical behaviour is governed by the (non-perturbative) Ising fixed point. The corresponding values of the scaling exponents are ν = 0.631..., dH = 1.585... and q = 0.293.... Mean field theory yields for the scaling exponents ν = 1/2, dH = 2 and q = 1/2. Both the Ising and the mean field values are compatible with the present ISM observational data: 1.4 ≤ dH ≤ 2, 0.3 ≤ q ≤ 0.6. As typical in critical phenomena, the scaling behaviour and critical exponents of the ISM can be obtained without dwelling into the dynamical (time-dependent) behaviour. The relevant rôle of selfgravity is stressed by the authors in a Letter to Nature,

Physical Review D, 2001

The type-I region of phase transitions at finite temperature of the U (1)-Higgs theory in 3+1 dimensions is investigated in detail using a Wilsonian renormalisation group. We consider in particular the quantitative effects induced through the cross-over of the scale-dependent Abelian charge from the Gaussian to a non-trivial Abelian fixed point. As a result, the strength of the first-order phase transition is weakened. Analytical solutions to approximate flow equations are obtained, and all characteristics of the phase transition are discussed and compared to the results obtained from perturbation theory. In addition, we present a detailed quantitative study regarding the dependence of the physical observables on the coarse-graining scheme. This results in error-bars for the regularisation scheme (RS) dependence. We find quantitative evidence for an intimate link between the RS dependence and truncations of flow equations.

Physical Review B, 1999

We study the Principal Chiral Ginzburg-Landau-Wilson model around two dimensions within the Local Potential Approximation of an Exact Renormalization Group equation. This model, relevant for the long distance physics of classical frustrated spin systems, exhibits a fixed point of the same universality class that the corresponding Non-Linear Sigma model. This allows to shed light on the long-standing discrepancy between the different perturbative approaches of frustrated spin systems.

Physical Review E, 2013

We present some exact results on the behavior of Branching and Annihilating Random Walks, both in the Directed Percolation and Parity Conserving universality classes. Contrary to usual perturbation theory, we perform an expansion in the branching rate around the non trivial Pure Annihilation model, whose correlation and response function we compute exactly. With this, the non-universal threshold value for having a phase transition in the simplest system belonging to the Directed Percolation universality class is found to coincide with previous Non Perturbative Renormalization Group approximate results. We also show that the Parity Conserving universality class has an unexpected RG fixed point structure, with a PA fixed point which is unstable in all dimensions of physical interest.

Physical Review E, 2012

We present in detail the implementation of the Blaizot-Méndez-Wschebor (BMW) approximation scheme of the nonperturbative renormalization group, which allows for the computation of the full momentum dependence of correlation functions. We discuss its signification and its relation with other schemes, in particular the derivative expansion. Quantitative results are presented for the testground of scalar O(N) theories. Besides critical exponents which are zero-momentum quantities, we compute in three dimensions in the whole momentum range the two-point function at criticality and, in the high temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.

International Journal of Modern Physics A, 2008

In the present paper, a particular form of Slavnov–Taylor identities for the Curci–Ferrari model is deduced. This model consist of Yang–Mills theory in a particular nonlinear covariant gauge, supplemented with mass terms for gluons and ghosts. It can be used as a regularization for the Yang–Mills theory preserving simple Slavnov–Taylor identities. Employing these identities, two nonrenormalization theorems are proved that reduce the number of independent renormalization factors from five to three. These new relations are verified by a comparison to the already known three-loops renormalization factors. These relations include, as a particular case, the corresponding known identities in Yang–Mills theory in Landau gauge.

Physical Review B, 2008

The self-energy of the critical 3-dimensional O(N) model is calculated. The analysis is performed in the context of the Non-Perturbative Renormalization Group, by exploiting an approximation which takes into account contributions of an infinite number of vertices. A very simple calculation yields the 2-point function in the whole range of momenta, from the UV Gaussian regime to the scaling one. Results are in good agreement with best estimates in the literature for any value of N in all momenta regimes. This encourages the use of this simple approximation procedure to calculate correlation functions at finite momenta in other physical situations.

Physical Review E

In the present article we analyze Non-Perturbative Renormalization Group flow equations in the order phase of Z2 and O(N) invariant scalar models in the derivative expansion approximation scheme. We first address the behavior of the leading order approximation (LPA), discussing for which regulators the flow is smooth and gives a convex free energy and when it becomes singular. We improve the exact known solutions in the "internal" region of the potential and exploit this solution in order to implement a numerical algorithm that is much more stable that previous ones for N > 1. After that, we study the flow equations at second order of the Derivative Expansion and analyze how and when LPA results change. We also discuss the evolution of field renormalization factors.

Journal of Statistical Physics

It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension −1. In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the O(N) model. We use three different approximation schemes: ǫ expansion, large N limit and third order of the Derivative Expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than −1. This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the O(N) model with N ∈ {2, 3, 4}.

Physical Review Letters

We provide analytical arguments showing that the non-perturbative approximation scheme to Wilson's renormalisation group known as the derivative expansion has a finite radius of convergence. We also provide guidelines for choosing the regulator function at the heart of the procedure and propose empirical rules for selecting an optimal one, without prior knowledge of the problem at stake. Using the Ising model in three dimensions as a testing ground and the derivative expansion at order six, we find fast convergence of critical exponents to their exact values, irrespective of the well-behaved regulator used, in full agreement with our general arguments. We hope these findings will put an end to disputes regarding this type of non-perturbative methods.

Physical Review E

We compute the critical exponents ν, η and ω of O(N) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-nextto-leading order [usually denoted O(∂ 4)]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter-typically between 1/9 and 1/4-compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field theoretical techniques. We also reach a better precision than Monte-Carlo simulations in some physically relevant situations. In the O(2) case, where there is a longstanding controversy between Monte-Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte-Carlo but clearly exclude experimental values.

2019

We discuss the calculation of one-loop effective actions in Lorentzian spacetimes, based on a very simple application of the method of steepest descent to the integral over the field. We show that for static spacetimes this procedure agrees with the analytic continuation of Euclidean calculations. We also discuss how to calculate the effective action by integrating a renormalization group equation. We show that the result is independent of arbitrary choices in the definition of the coarse-graining and we see again that the Lorentzian and Euclidean calculations agree. When applied to quantum gravity on static backgrounds, our procedure is equivalent to analytically continuing time and the integral over the conformal factor.

Physical Review D, 2021

Working with scalar field theories, we discuss choices of regulator that, inserted in the functional renormalization group equation, reproduce the results of dimensional regularization at one and two loops. The resulting flow equations can be seen as nonperturbative extensions of the MS scheme. We support this claim by recovering all the multicritical models in two dimensions. We discuss a possible generalization to any dimension. Finally, we show that the method also preserves nonlinearly realized symmetries, which is a definite advantage with respect to other regulators.

Physical Review D

We show that the Conformal Standard Model supplemented with asymptotically safe gravity can be valid up to arbitrarily high energies and give a complete description of particle physics phenomena. We restrict the mass of the second scalar particle to ∼ 300 GeV and the masses of heavy neutrinos to ∼ 340 GeV. These predictions can be explicitly tested in the nearby future.

Physical Review D

There are many hints that gravity is asymptotically safe. The inclusion of gravitational corrections can result in the ultraviolet fundamental Standard Model and constrain the Higgs mass to take exactly one value, which can be calculated. Taking into account the current top quark mass measurements this calculation gives Higgs mass ∼ 131 GeV, which is in disagreement with the current experimental measurement. This article considers the predictions of the Higgs mass in two minimal Beyond Standard Model scenarios. One is the sterile quark axion model, while the other is the U (1) B−L gauge symmetry model introducing a new massive Z gauge boson. The inclusion of Z boson gives the prediction much closer to the observed value, while inclusion of sterile quark(s) gives only a slight effect. Also a new, gravitational solution to the strong CP problem is discussed.

Journal of Physics A: Mathematical and Theoretical, 2009

Equations related to the Polchinski version of the exact renormalisation group equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterisation invariance, are postulated. Reparameterisation invariance ensures that the equations determine the anomalous dimension η unambiguously and the equations are such that the result is exact to O(ε 2) in an ε-expansion for any multi-critical fixed point. It is also straightforward to determine η numerically. When the dimension d = 3 numerical results for a wide range of critical exponents are obtained in theories with O(N) symmetry, for various N and for a ranges of η, are obtained within the local potential approximation. The associated η, which follow from the derivative approximation described here, are found for various N. The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in ε.

Molecular Physics, 2011

We consider the equilibrium behavior of fluids imbibed in disordered mesoporous media, including their gas-liquid critical point when present. Our starting points are on the one hand a description of the fluid/solid-matrix system as a quenched-annealed mixture and on the other hand the Hierarchical Reference Theory (HRT) developed by A. Parola and L. Reatto to cope with density fluctuations on all length scales. The formalism combines liquid-state statistical mechanics and the theory of systems in the presence of quenched disorder. A straightforward implementation of the HRT to the quenched-annealed mixture is shown to lead to unsatisfactory results, while indicating that the critical behavior of the system is in the same universality class as that of the random-field Ising model. After a detour via the field-theoretical renormalization group approach of the latter model, we finally lay out the foundations for a proper HRT of fluids in a disordered porous material.

arXiv: High Energy Physics - Theory, 2019

The functional renormalization group method is applied for a scalar theory in Minkowski space-time. It is argued that the appropriate choice of the subtraction point is more important in Minkowski than in Euclidean space-time. The parameters of the cutoff theory, defined by a subtraction point in the quasi-particle domain, are complex due to the mass-shell contributions to the blocking and the renormalization group flow becomes more involved. The Landau poles are avoided when the parameters are complexified. The absence of the UV pole owing to the marginal parameters makes the scalar theory asymptotically free in four dimensions. However the continuation of the trajectory beyond the regularized IR pole at the UV-IR crossover in the phase with spontaneously broken symmetry is possible only if the non-trivial saddle points to the blocking are taken into account.

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011

After a short elementary introduction to the exact renormalization group for the effective action, I discuss a particular truncation of the hierarchy of flow equations that allows for the determination of the full momentum of the n -point functions. Applications are then briefly presented, to critical O ( N ) models, to Bose–Einstein condensation and to finite-temperature field theory.

Physical Review E, 2006

In a companion paper (hep-th/0512317), we have presented an approximation scheme to solve the Non Perturbative Renormalization Group equations that allows the calculation of the n-point functions for arbitrary values of the external momenta. The method was applied in its leading order to the calculation of the self-energy of the O(N) model in the critical regime. The purpose of the present paper is to extend this study to the next-to-leading order of the approximation scheme. This involves the calculation of the 4-point function at leading order, where new features arise, related to the occurrence of exceptional configurations of momenta in the flow equations. These require a special treatment, inviting us to improve the straightforward iteration scheme that we originally proposed. The final result for the self-energy at next-to-leading order exhibits a remarkable improvement as compared to the leading order calculation. This is demonstrated by the calculation of the shift ∆T c , caused by weak interactions, in the temperature of Bose-Einstein condensation. This quantity depends on the self-energy at all momentum scales and can be used as a benchmark of the approximation. The improved next-to-leading order calculation of the selfenergy presented in this paper leads to excellent agreement with lattice data and is within 4% of the exact large N result.

Physical Review D, 1999

We study the derivative expansion for the effective action in the framework of the Exact Renormalization Group for a single component scalar theory. By truncating the expansion to the first two terms, the potential U k and the kinetic coefficient Z k , our analysis suggests that a set of coupled differential equations for these two functions can be established under certain smoothness conditions for the background field and that sharp and smooth cutoff give the same result. In addition we find that, differently from the case of the potential, a further expansion is needed to obtain the differential equation for Z k , according to the relative weight between the kinetic and the potential terms. As a result, two different approximations to the Z k equation are obtained. Finally a numerical analysis of the coupled equations for U k and Z k is performed at the non-gaussian fixed point in D < 4 dimensions to determine the anomalous dimension of the field.

Journal of High Energy Physics

We study an f(R) approximation to asymptotic safety, using a family of non-adaptive cutoffs, kept general to test for universality. Matching solutions on the four-dimensional sphere and hyperboloid, we prove properties of any such global fixed point solution and its eigenoperators. For this family of cutoffs, the scaling dimension at large n of the nth eigenoperator, is λn ∝ b n ln n. The coefficient b is non-universal, a consequence of the single-metric approximation. The large R limit is universal on the hyperboloid, but not on the sphere where cutoff dependence results from certain zero modes. For right-sign conformal mode cutoff, the fixed points form at most a discrete set. The eigenoperator spectrum is quantised. They are square integrable under the Sturm-Liouville weight. For wrong sign cutoff, the fixed points form a continuum, and so do the eigenoperators unless we impose square-integrability. If we do this, we get a discrete tower of operators, infinitely many of which are...

Journal of High Energy Physics, 2019

In this paper we encode the perturbative BFKL leading logarithmic resummation, relevant for the Regge limit behavior of QCD scattering amplitudes, in the IR-regulated effective action which satisfies exact functional renormalization group equations. This is obtained using a truncation with a specific infinite set of non local vertices describing the multi-Regge kinematics (MRK). The goal is to use this framework to study, in the high energy limit and at larger transverse distances the transition to a much simpler effective local reggeon field theory, whose critical properties were recently investigated in the same framework. We perform a numerical analysis of the spectrum of the BFKL Pomeron deformed by the introduction of a Wilsonian infrared regulator to understand the properties of the leading poles (states) contributing to the high energy scattering.

The European Physical Journal C, 2015

We write new functional renormalization group equations for a scalar nonminimally coupled to gravity. Thanks to the choice of the parametrization and of the gauge fixing they are simpler than older equations and avoid some of the difficulties that were previously present. In three dimensions these equations admit, at least for sufficiently small fields, a solution that may be interpreted as a gravitationally dressed Wilson-Fisher fixed point. We also find for any dimension d > 2 additional analytic scaling solutions which we study for d = 3 and d = 4. One of them corresponds to the fixed point of the Einstein-Hilbert truncation, the others involve a nonvanishing minimal coupling.

Physical Review D, 2015

The critical behavior of a relativistic Z 2-symmetric Yukawa model at zero temperature and density is discussed for a continuous number of fermion degrees of freedom and of spacetime dimensions, with emphasis on the role played by multi-meson exchange in the Yukawa sector. We argue that this should be generically taken into account in studies based on the functional renormalization group, either in four-dimensional high-energy models or in lower-dimensional condensed-matter systems. By means of the latter method, we describe the generation of multi-critical models in less then three dimensions, both at infinite and finite number of flavors. We also provide different estimates of the critical exponents of the chiral Ising universality class in three dimensions for various field contents, from a couple of massless Dirac fermions down to the supersymmetric theory with a single Majorana spinor.

Physical Review D, 2017

In this paper we extend our recent non perturbative functional renormalization group analysis of Reggeon Field Theory to the interactions of Pomeron and Odderon fields. We establish the existence of a fixed point and its universal properties, which exhibits a novel symmetry structure in the space of Odderon-Pomeron interactions. As in our previous analysis, this part of our program aims at the investigation of the IR limit of reggeon field theory (the limit of high energies and large transverse distances). It should be seen in the broader context of trying to connect the nonperturbative infrared region (large transverse distances) with the UV region of small transverse distances where the high energy limit of perturbative QCD applies. We briefly discuss the implications of our findings for the existence of an Odderon in high energy scattering.

The European Physical Journal C, 2018

We show how the use of standard perturbative RG in dimensional regularization allows for a renormalization group-based computation of both the spectrum and a family of coefficients of the operator product expansion (OPE) for a given universality class. The task is greatly simplified by a straightforward generalization of perturbation theory to a functional perturbative RG approach. We illustrate our procedure in the-expansion by obtaining the nextto-leading corrections for the spectrum and the leading corrections for the OPE coefficients of Ising and Lee-Yang universality classes and then give several results for the whole family of renormalizable multi-critical models φ 2n. Whenever comparison is possible our RG results explicitly match the ones recently derived in CFT frameworks.

The European Physical Journal C, 2020

We discuss the possibility to define exact RG equations for a UV regulated Wilsonian action based on a proper time (PT) regulator function. We start from a functional mapping which shows how each particular flow equation (and RG scheme) is associated to infinitely many scale dependent field redefinitions, which are related to specific coarse-graining procedures. On specializing to a sub-family of one parameter PT regulators we briefly analyze few results for the Ising Universality class in three dimensions, obtained within a second order truncation in the derivative expansion of the Wilsonian action.

Journal of High Energy Physics, 2022

The critical behavior of infinite families of shift symmetric interacting theories with higher derivative kinetic terms (non unitary) is considered. Single scalar theories with shift symmetry are classified according to their upper critical dimensions and studied at the leading non trivial order in perturbation theory. For two infinite families, one with quartic and one with cubic interactions, beta functions, criticality conditions and universal anomalous dimensions are computed. At the order considered, the cubic theories enjoy a one loop non renormalization of the vertex, so that the beta function depends non trivially only on the anomalous dimension. The trace of the energy momentum tensor is also investigated and it is shown that these two families of QFTs are conformally invariant at the fixed point of the RG flow.

Physical Review B, 1996

The universal behavior of superconductors near the phase transition is described by the three-dimensional field theory of scalar quantum electrodynamics. We approximately solve the model with the help of nonperturbative flow equations. A first-or second-order phase transition is found depending on the relative strength of the scalar versus the gauge coupling. The region of a second-order phase transition is governed by a fixed point of the flow equations with associated critical exponents. We also give an approximate description of the tricritical behavior and briefly discuss the crossover relevant for the onset of scaling near the critical temperature. Final confirmation of a second-order transition for strong type-II superconductors requires further analysis with extended truncations of the flow equations.

Journal of High Energy Physics, 2016

We extend our prescription for the construction of a covariant and backgroundindependent effective action for scalar quantum field theories to the case where momentum modes below a certain scale are suppressed by the presence of an infrared regulator. The key step is an appropriate choice of the infrared cutoff for which the Ward identity, capturing the information from single-field dependence of the ultraviolet action, continues to be exactly solvable, and therefore, in addition to covariance, manifest background independence of the effective action is guaranteed at any scale. A practical consequence is that in this framework one can adopt truncations dependent on the single total field. Furthermore we discuss the necessary and sufficient conditions for the preservation of symmetries along the renormalization group flow.

Journal of High Energy Physics, 2016

In this paper we investigate the possibility whether, in the extreme limit of high energies and large transverse distances, reggeon field theory might serve as an effective theory of high energy scattering for strong interactions. We analyse the functional renormalization group equations (flow equations) of reggeon field theory and search for fixed points in the space of (local) reggeon field theories. We study in complementary ways the candidate for the scaling solution, investigate its main properties and briefly discuss possible physical interpretations.

The European Physical Journal C, 2019

We investigate multi-field multicritical scalar theories using CFT constraints on two-and three-point functions combined with the Schwinger-Dyson equation. This is done in general and without assuming any symmetry for the models, which we just define to admit a Landau-Ginzburg description that includes the most general critical interactions built from monomials of the form φ i 1. .. φ i m. For all such models we analyze to the leading order of the-expansion the anomalous dimensions of the fields and those of the composite quadratic operators. For models with even m we extend the analysis to an infinite tower of composite operators of arbitrary order. The results are supplemented by the computation of some families of structure constants. We also find the equations which constrain the nontrivial critical theories at leading order and show that they coincide with the ones obtained with functional perturbative RG methods. This is done for the case m = 3 as well as for all the even models. We ultimately specialize to S q symmetric models, which are related to the q-state Potts universality class, and focus on three realizations appearing below the upper critical dimensions 6, 4 and 10 3 , which can thus be nontrivial CFTs in three dimensions.

Physical Review E

We revisit the approach to the lower critical dimension d lc in the Ising-like ϕ 4 theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in d 2, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near d lc. We show that the convergence of the fixed-point effective potential is nonuniform in the field when d → d lc with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension d lc and for the behavior of the critical temperature as d → d lc , which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

Physical Review D, 2012

We consider an interacting Lifshitz field with z = 3 in a curved spacetime. We analyze the renormalizability of the theory for interactions of the form λφ n , with arbitrary even n. We compute the running of the coupling constants both in the ultraviolet and infrared regimes. We show that the Lorentz-violating terms generate couplings to the spacetime metric that are not invariant under general coordinate transformations. These couplings are not suppressed by the scale of Lorentz violation and therefore survive at low energies. We point out that in these theories, unless the effective mass of the field is many orders of magnitude below the scale of Lorentz violation, the coupling to the four-dimensional Ricci scalar ξ (4) Rφ 2 does not receive large quantum corrections ξ ≫ 1. We argue that quantum corrections involving spatial derivatives of the lapse function (which appear naturally in the so-called healthy extension of the Hořava-Lifshitz theory of gravity) are not generated unless they are already present in the bare Lagrangian.

Physical Review D, 2020

We outline a general strategy developed for the analysis of critical models, which we apply to obtain a heuristic classification of all universality classes with up to three field-theoretical scalar order parameters in d = 6−ǫ dimensions. As expected by the paradigm of universality, each class is uniquely characterized by its symmetry group and by a set of its scaling properties, neither of which are built-in by the formalism but instead emerge nontrivially as outputs of our computations. For three fields, we find several solutions mostly with discrete symmetries. These are nontrivial conformal field theory candidates in less than six dimensions, one of which is a new perturbatively unitary critical model.

Physical Review D, 2014

There have been some speculations about the existence of critical unitary O(N)-invariant scalar field theories in dimensions 4 < d < 6 and for large N. Using the functional renormalization group equation, we show that in the lowest order of the derivative expansion, and assuming that the anomalous dimension vanishes for large N , the corresponding critical potentials are either unbounded from below or singular for some finite value of the field.

AIP Conference Proceedings, 2017

In the quest for an effective field theory which could help to understand some non perturbative feature of the QCD in the Regge limit, we consider a Reggeon Field Theory (RFT) for both Pomeron and Odderon interactions and perform an analysis of the critical theory using functional renormalization group techniques, unveiling a novel symmetry structure.

Physical Review D, 2003

We uncover a method of calculation that proceeds at every step without fixing the gauge or specifying details of the regularisation scheme. Results are obtained by iterated use of integration by parts and gauge invariance identities. The initial stages can even be computed diagrammatically. The method is formulated within the framework of an exact renormalization group for SU (N) Yang-Mills gauge theory, incorporating an effective cutoff through a manifest spontaneously broken SU (N |N) gauge invariance. We demonstrate the technique with a compact calculation of the one-loop beta function, achieving a manifestly universal result, and without gauge fixing, for the first time at finite N. 10 ignoring the spacetime index and spatial dependence

International Journal of Modern Physics A, 2015

We will study the AdS/CFT correspondence in an intermediate region between the strong form of this correspondence (string theory on AdS being dual to a boundary CFT), and the weak form of this correspondence (supergravity on AdS being dual to a boundary CFT). We will go beyond the supergravity approximation in the AdS by using the fact that strings have an extended structure. We will also calculate the CFT dual to such string corrections in the bulk, and demonstrate that they are consistent with the strong form of the AdS/CFT correspondence. So, even though the conformal dimensions of both the relevant and the irrelevant operators will receive string corrections, the conformal dimension of marginal operators will not receive any such corrections.