Kenji Muneyuki, Nobuyoshi Ohta
It has been suggested that higher-derivative gravity theories coupled to a scalar field with shift symmetry may be an important candidate for a quantum gravity. We show that this class of gravity theories are renormalizable in D = 3 and 4 dimensions.
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We study the renormalization group flow of higher derivative gravity, utilizing the functional renormalization group equation for the average action. Employing a recently proposed algorithm, termed the universal renormalization group machine, for solving the flow equation, all the universal features of the one-loop beta-functions are recovered. While the universal part of the beta-functions admits two fixed points, we explicitly show that the existence of one of them depends on the choice of regularization scheme, indicating that it is most probably unphysical.
This work explores an alternative solution to the problem of renormalizability in Einstein gravity. In the proposed approach, Einstein gravity is transformed into the renormalizable theory of four-derivative gravity by applying a field redefinition containing an infinite number of higher derivatives. It is also shown that the current-current amplitude is invariant with the field redefinition, and thus the unitarity of Einstein gravity is preserved.
The second alternative conformal limit of the recently proposed general higher derivative dilaton quantum theory in curved spacetime is explored. In this version of the theory the dilaton is transformed, along with the metric, to provide the conformal invariance of the classical action. We find the corresponding quantum theory to be renormalizable at one loop, and the renormalization constants for the dimensionless parameters are explicitly shown to be universal for an arbitrary parametrization of the quantum field. The renormalization group equations indicate an asymptotic freedom in the IR l…
The one loop UV divergences of Hilbert-Einstein gravity with a cosmological constant and spin 0, 1/2 and 1 matter are computed making use of a covariant derivative expansion and functional methods. For this purpose the transformation that yields the covariant derivative is extended to include a dynamical metric and the expansion in the fields themselves is made covariant which is relevant for the effective action due to the non-linear character of gravity
We study higher derivative extension of the functional renormalization group (FRG). We consider FRG equations for a scalar field that consist of terms with higher functional derivatives of the effective action and arbitrary cutoff functions. We show that the epsilon expansion around the Wilson-Fisher fixed point is indeed reproduced by the local potential approximation of the FRG equations.
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.
This paper has been withdrawn to address an omission. It will be resubmitted in the near future.
The perturbative evaluation of the effective action can be expanded in powers of derivatives of the external field. We apply the renormalization group equation to the term in the effective action that is second order in the derivatives of the external field and all orders in a constant external field, considering both massless scalar $ϕ_4^4$ model and massless scalar electrodynamics. A so-called ``on shell'' renormalization scheme permits one to express this ``kinetic term'' for the scalar field entirely in terms of the renormalization group functions appropriate for this scheme. These renorma…
We generalize the proof of the non-renormalization of the four derivative operators in ${\cal N}=4$ Yang Mills theory with gauge group SU(2) to show that certain terms with 2N derivatives are not renormalized in the theory with gauge group SU(N). These terms may be determined exactly by a simple perturbative computation. Similar results hold for finite ${\cal N}=2$ theories. We comment on the implications of these results.
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.
A renormalization group study of a scalar theory coupled to gravity through a general functional dependence on the Ricci scalar in the action is discussed. A set of non-perturbative flow equations governing the evolution of the new interaction terms generated in both local potential and wavefunction renomalization is derived. It is shown for a specific model that these new terms play an important role in determining the scaling behavior of the system above the mass of the inflaton field.
We consider the softly broken ${\cal N}=1$ supersymmetric electrodynamics, regularized by higher derivatives. For this theory we demonstrate that the renormalization of the photino mass is determined by integrals of double total derivatives in the momentum space in all orders. Consequently, it is possible to derive the NSVZ-like exact relation between the photino mass anomalous dimension and the anomalous dimension of the matter superfields in the rigid theory by direct summation of supergraphs. It is important that both these renormalization group functions are defined in terms of the bare co…
We study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques. The non-perturbative contributions to the $β$-functions shift the known perturbative ultraviolet fixed point into a non-trivial fixed point with three UV-attractive and one UV-repulsive eigendirections, consistent with the asymptotic safety conjecture of gravity. The implication of this transition on the unitarity problem, typically haunting higher-derivative gravity theories, is discussed.
It has been argued that a certain large $N$ matrix model may provide a non-perturbative definition of $M$-theory. This model is the truncation to $0+1$ dimensions of ten-dimensional supersymmetric Yang-Mills theory. It is crucial to this identification that terms with four derivatives in the effective action for the quantum mechanics should not be renormalized. We offer a perturbative proof of this result.
Two-loop $β$-function and anomalous dimension are calculated for N=1 supersymmetric quantum electrodynamics, regularized by higher derivatives in the minimal subtraction scheme. The result for two-loop contribution to the $β$-function appears to be equal to 0, does not depend on the form of regularizing term and does not lead to anomaly puzzle. Two-loop anomalous dimension can be also made independent on parameters of higher derivative regularization by a special choice of subtraction scheme.
A theory of higher-derivative 2D dilaton gravity which has its roots in the massive higher-spin mode dynamics of string theory is suggested. The divergences of the effective action to one-loop are calculated, both in the covariant and in the conformal gauge. Some technical problems which appear in the calculations are discussed. An interpretation of the theory as a particular D=2 higher-derivative $σ$-model is given. For a specific case of higher-derivative 2D dilaton gravity, which is one loop multiplicatively renormalizable, static configurations corresponding to black holes are shown to exi…
On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order $\partial^4$ of the derivative expansion leads to $ν=0.632$ and to an anomalous dimension $η=0.033$ which is significantly improved compared with lower orders calculations.
An efficient way to calculate one-loop counterterms within the Feynman diagrammatic approach and dimensional regularization is to expand the propagators in the integrands of the Feynman integrals around vanishing external momentum. In this way, a generic one-loop diagram is reduced to a sum of vacuum diagrams. The logarithmically divergent part can be extracted by power counting arguments. In case of higher derivative theories, the standard implementation of this procedure on a computer algebra system can become quickly inefficient due to a high proliferation of terms coming from the intermedi…
Amplitudes and Renormalization Group Techniques: A Case Study
本研究以含高阶导数动能项与四次导数耦合的可重整化平移对称模型为对象，通过振幅分析提取参数物理跑动规律，发现其与其他方法结果仅在极限下一致，为高阶导数引力理论研究提供参考。
The renormalization algorithm based on regularization methods with two regulators is analyzed by means of explicit computations. We show in particular that regularization by higher covariant derivative terms can be complemented with dimensional regularization to obtain a consistent renormalized 4-dimensional Yang-Mills theory at the one-loop level. This shows that hybrid regularization methods can be applied not only to finite theories, like \eg\ Chern-Simons, but also to divergent theories.
Wetterich's equation provides a powerful tool for investigating the existence and universal properties of renormalization group fixed points exhibiting quantum scale invariance. Motivated by recent works on asymptotically safe scalar-tensor theories, we develop a novel approximation scheme which projects the functional renormalization group equation onto functions of the kinetic term. Applying this projection to scalars and gauge fields, our analysis identifies a new universality class with a very special spectrum of stability coefficients. The implications of our findings in the context of as…
For the general renormalizable ${\cal N}=1$ supersymmetric gauge theory we investigate renormalization of the Faddeev--Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general $ξ$-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev--Popov ghosts defined in terms of the renormalized coupling constant and examin…
$M$-theory is believed to be described in various dimensions by large $N$ field theories. It has been further conjectured that at finite $N$, these theories describe the discrete light cone quantization (DLCQ) of $M$ theory. Even at low energies, this is not necessarily the same thing as the DLCQ of supergravity. It is believed that this is only the case for quantities which are protected by non-renormalization theorems. In 0+1 and 1+1 dimensions, we provide further evidence of a non-renormalization theorem for the $v^4$ terms, but also give evidence that there are not such theorems at order $…
The fourth derivative models for two dimensional gravity are shown to be equivalent to the special version of the nonlinear sigma models coupled to 2d quantum gravity. The reduction consists in the introduction of the auxiliary scalar fields and can be performed in an explicit way for both metric and general metric-dilaton cases. In view of this we can evaluate the structure of possible counterterms and show that they contains second derivative structures only. The calculations in the theory with an auxiliary fields require some special procedure to be applied. We perform the explicit calculat…
We investigate the convergence of the derivative expansion of the exact renormalization group, by using it to compute the beta function of scalar field theory. We show that the derivative expansion of the Polchinski flow equation converges at one loop for certain fast falling smooth cutoffs. The derivative expansion of the Legendre flow equation trivially converges at one loop, but also at two loops: slowly with sharp cutoff (as a momentum-scale expansion), and rapidly in the case of a smooth exponential cutoff. Finally, we show that the two loop contributions to certain higher derivative oper…
In linearized quantum gravity, a shift of the average energy-momentum can be compensated by a shift of the average gravitational field. This allows a renormalization scheme that naturally removes the contribution of quantum vacuum fluctuations to the cosmological constant, solving the old cosmological-constant problem for weak gravitational fields.
We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order $\partial^2$ of the derivative expansion. This approach allows us to select optimized cut-off functions and to improve the accuracy of the critical exponents $ν$ and $η$. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the cri…
We study the renormalization group flow of gravity coupled to scalar matter using functional renormalization group techniques. The novel feature is the inclusion of higher-derivative terms in the scalar propagator. Such terms give rise to Ostrogradski ghosts which signal an instability of the system and are therefore dangerous for the consistency of the theory. Since it is expected that such terms are generated dynamically by the renormalization group flow they provide a potential threat when constructing a theory of quantum gravity based on Asymptotic Safety. Our work then establishes the fol…
We evaluate the one-loop $β$ functions of all dimension 6 parity-preserving operators in the Abelian Higgs-Kibble model. No on-shell restrictions are imposed; and the (generalized) non-polynomial field redefinitions arising at one-loop order are fully taken into account. The operator mixing matrix is also computed, and its cancellation patterns explained as a consequence of the functional identities of the theory and power-counting conditions.
We investigate the phase structure and the infrared properties of higher-derivative quantum gravity (QG) with matter, in $4-\varepsilon$ dimensions. The renormalization group (RG) equations in $4-\varepsilon$ dimensions are analysed for the following types of matter: the $fφ^4$-theory, the $O(N) φ^4$-theory, scalar electrodynamics, and the $SU(2)$ model with scalars. New fixed points for the scalar coupling appear ---one of which is IR stable--- being some of them induced by QG. The IR stable fixed point perturbed by QG leads to a second-order phase transition for the theory at non-zero temper…
A higher-derivative, interacting, scalar field theory in curved spacetime with the most general action of sigma-model type is studied. The one-loop counterterms of the general theory are found. The renormalization group equations corresponding to two different, multiplicatively renormalizable variants of the same are derived. The analysis of their asymptotic solutions shows that, depending on the sign of one of the coupling constants, we can construct an asymptotically free theory which is also asymptotically conformal invariant at strong (or small) curvature. The connection that can be establ…
The exact NSVZ relation between a $β$-function of ${\cal N}=1$ SQED and an anomalous dimension of the matter superfields is studied within the Slavnov higher derivative regularization approach. It is shown that if the renormalization group functions are defined in terms of the bare coupling constant, this relation is always valid. In the renormalized theory the NSVZ relation is obtained in the momentum subtraction scheme supplemented by a special finite renormalization. Unlike the dimensional reduction, the higher derivative regularization allows to fix this finite renormalization. This is mad…
Approximation only by derivative (or more generally momentum) expansions, combined with reparametrization invariance, turns the continuous renormalization group for quantum field theory into a set of partial differential equations which at fixed points become non-linear eigenvalue equations for the anomalous scaling dimension $η$. We review how these equations provide a powerful and robust means of discovering and approximating non-perturbative continuum limits. Gauge fields are briefly discussed. Particular emphasis is placed on the rôle of reparametrization invariance, and the convergence of…
We consider supersymmetric gauge theories coupled to hyper multiplets on five and six dimensional orbifolds and determine the bulk and local fixed point renormalizations of the gauge couplings. We infer from a component analysis that the hyper multiplet does not induce renormalization of the brane gauge couplings on the five dimensional orbifold S^1/Z_2. This is not due to supersymmetry, since the bosonic and fermionic contributions cancel separately. We extend this investigation to T^2/Z_N orbifolds using supergraph techniques in six dimensions. On general Z_N orbifolds the gauge couplings do…
We apply a derivative expansion to the Legendre effective action flow equations of O(N) symmetric scalar field theory, making no other approximation. We calculate the critical exponents eta, nu, and omega at the both the leading and second order of the expansion, associated to the three dimensional Wilson-Fisher fixed points, at various values of N. In addition, we show how the derivative expansion reproduces exactly known results, at special values N=infinity,-2,-4, ... .
We show explicitly that the sum over punctures for the three times derivative for the Penner free energy $F_{0}^{3}$, given by D.A. Johnston, Phys.Rev.D 51 (1995) is not correct. As a consequence, Eq.(21), the differentiated version for the renormalization group (RG) equation, is wrong. Also, his conclusion that the differentiated version of the (RG) equation for the three-times derivative of the free energy can be obtained from the higher genus (RG) equation can not be true. Finally, the differentiated version of the (RG) equation is extended to any $s$ derivative of the free energy $F_{0}$.
The two-loop anomalous dimension of the chiral matter superfields is calculated for a general ${\cal N}=1$ supersymmetric gauge theory regularized by higher covariant derivatives. We obtain both the anomalous dimension defined in terms of the bare couplings, and the one defined in terms of the renormalized couplings for an arbitrary renormalization prescription. For the one-loop finite theories we find a simple relation between the higher derivative regulators under which the anomalous dimension defined in terms of the bare couplings vanishes in the considered approximation. In this case the o…
The full off-shell one loop renormalization for all divergent amplitudes up to dimension 6 in the Abelian Higgs-Kibble model, supplemented with a maximally power counting violating higher-dimensional gauge-invariant derivative interaction $\sim g ~ ϕ^\dagger ϕ(D^μϕ)^\dagger D_μϕ$, is presented. This allows one to perform the complete renormalization of radiatively generated dimension 6 operators in the model at hand. We describe in details the technical tools required in order to disentangle the contribution to UV divergences parameterized by (generalized) non-polynomial field redefinitions. W…
In the case of using the higher derivative regularization we construct the subtraction scheme which gives the NSVZ-like relation for the anomalous dimension of the photino mass in softly broken ${\cal N}=1$ SQED with $N_f$ flavors in all loops. The corresponding renormalization prescription is determined by simple boundary conditions imposed on the renormalization constants. It allows fixing an arbitrariness of choosing finite counterterms in every order of the perturbation theory in such a way that the renormalization group functions defined in terms of the renormalized coupling constant sati…
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 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.
We consider the harmonic superspace formulation of higher-derivative $6D$, ${\cal N}=(1,0)$ supersymmetric gauge theory and its minimal coupling to a hypermultiplet. In components, the kinetic term for the gauge field in such a theory involves four space-time derivatives.The theory is quantized in the framework of the superfield background method ensuring manifest $6D$, ${\cal N}=(1,0)$ supersymmetry and the classical gauge invariance of the quantum effective action. We evaluate the superficial degree of divergence and prove it to be independent of the number of loops. Using the regularization…
We consider a higher-derivative extension of QED modified by the addition of a gauge-invariant dimension-6 kinetic operator in the U(1) gauge sector. The Feynman diagrams at one-loop level are then computed. The modification in the spin-1 sector leads the electron self-energy and vertex corrections diagrams finite in the ultraviolet regime. Indeed, no regularization prescription is used to calculate these diagrams because the modified propagator always occurs coupled to conserved currents. Moreover, besides the usual massless pole in the spin-1 sector, there is the emergence of a massive one, …
I study some aspects of the renormalization of quantum field theories with infinitely many couplings in arbitrary space-time dimensions. I prove that when the space-time manifold admits a metric of constant curvature the propagator is not affected by terms with higher derivatives. More generally, certain lagrangian terms are not turned on by renormalization, if they are absent at the tree level. This restricts the form of the action of a non-renormalizable theory, and has applications to quantum gravity. The new action contains infinitely many couplings, but not all of the ones that might have…
The consistent recursive subtraction of UV divergences order by order in the loop expansion for spontaneously broken effective field theories with dimension-6 derivative operators is presented for an Abelian gauge group. We solve the Slavnov-Taylor identity to all orders in the loop expansion by homotopy techniques and a suitable choice of invariant field coordinates (named bleached variables) for the linearly realized gauge group. This allows one to disentangle the gauge-invariant contributions to off-shell 1-PI amplitudes from those associated with the gauge-fixing and (generalized) non-poly…
Classical higher-derivative gravity is investigated in the context of the holographic renormalization group (RG). We parametrize the Euclidean time such that one step of time evolution in (d+1)-dimensional bulk gravity can be directly interpreted as that of block spin transformation of the d-dimensional boundary field theory. This parametrization simplifies the analysis of the holographic RG structure in gravity systems, and conformal fixed points are always described by AdS geometry. We find that higher-derivative gravity generically induces extra degrees of freedom which acquire huge mass ar…
We classify the unitary, renormalizable, Lorentz violating quantum field theories of interacting scalars and fermions, obtained improving the behavior of Feynman diagrams by means of higher space derivatives. Higher time derivatives are not generated by renormalization. Renormalizability is ensured by a "weighted power counting" criterion. The theories contain a dimensionful parameter, yet a set of models are classically invariant under a weighted scale transformation, which is anomalous at the quantum level. Formulas for the weighted trace anomaly are derived. The renormalization-group proper…
We construct higher derivative supervertices in an effective theory of maximal supergravity in various dimensions, in the super spinor helicity formalism, and derive non-renormalization conditions on up to 14-derivative order couplings from supersymmetry. These non-renormalization conditions include Laplace type equations on the coefficients of $R^4$, $D^4R^4$, and $D^6R^4$ couplings. We also find additional constraining equations, which are consistent with previously known results in the effective action of toroidally compactified type II string theory, and elucidate many features thereof.
With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson-Polchinski equation in the case of the $N$-vector model with the symmetry $\mathrm{O}(N) $. As a test, the critical exponents $% η$ and $ν$ as well as the subcritical exponent $ω$ (and higher ones) are estimated in three dimensions for values of $N$ ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of ot…
The conditions for fully supersymmetric backgrounds of general N=2 locally supersymmetric theories are derived based on the off-shell superconformal multiplet calculus. This enables the derivation of a non-renormalization theorem for a large class of supersymmetric invariants with higher-derivative couplings. The theorem implies that the invariant and its first order variation must vanish in a fully supersymmetric background. The conjectured relation of one particular higher-derivative invariant with a specific five-dimensional invariant containing the mixed gauge-gravitational Chern-Simons te…
The effective action in quantum general relativity is strongly dependent on the gauge-fixing and parametrization of the quantum metric. As a consequence, in the effective approach to quantum gravity, there is no possibility to introduce the renormalization-group framework in a consistent way. On the other hand, the version of effective action proposed by Vilkovisky and DeWitt does not depend on the gauge-fixing and parametrization off-shell, opening the way to explore the running of the cosmological and Newton constants as well as the coefficients of the higher-derivative terms of the total ac…
The exact one-loop beta functions for the four-derivative terms (Weyl tensor squared, Ricci scalar squared and the Gauss-Bonnet) are derived for the minimal six-derivative quantum gravity (QG) theory in four spacetime dimensions. The calculation is performed by means of the Barvinsky and Vilkovisky generalized Schwinger-DeWitt technique. With this result we gain, for the first time, the full set of the relevant beta functions in a super-renormalizable model of QG. The complete set of renormalization group (RG) equations, including also those for the Newton and the cosmological constant, is sol…
Handbook of derivative AdS amplitudes
本文是此前反德西特空间标量算符树级Witten图研究的延伸，拓展至含时空导数的体相互作用情形，给出d=3边界下两类标量对应振幅的显式公式、配套Mathematica程序包，并揭示有限关联函数也存在非平凡重整化效应。
We show that non-perturbative fixed points of the exact renormalization group, their perturbations and corresponding massive field theories can all be determined directly in the continuum -- without using bare actions or any tuning procedure. As an example, we estimate the universal couplings of the non-perturbative three-dimensional one-component massive scalar field theory in the Ising model universality class, by using a derivative expansion (and no other approximation). These are compared to the recent results from other methods. At order derivative-squared approximation, the four-point co…
The consistency of Matrix theory with supergravity requires that in the large N_c limit terms of order v^4 in the SU(N_c) Matrix effective potential are not renormalized beyond one loop in perturbation theory. For SU(2) gauge group, the required non-renormalization theorem was proven recently by Paban, Sethi and Stern. In this paper we consider the constraints supersymmetry imposes on these terms for groups SU(N_c) with N_c>2. Non-renormalization theorems are proven for certain tensor structures, including the structures that appear in the one-loop effective action. However it is expected othe…
We show that the instabilities of higher derivative gravity models with quadratic curvature invariant $αR^2+βR_{μν}R^{μν}$ can be removed by judicious addition of constraints at the quadratic level of metric fluctuations around Minkowski/de Sitter background. With a suitable parameter choice, we find that the instabilities of helicity-0, 1, 2 modes can be removed while reducing the dimensionality of the original phase space. To retain the renormalization properties of higher derivative gravity, Lorentz symmetry in the constrained theory is explicitly broken.
The Chern-Simons coupling shift is calculated within the framework of the hybrid regularization based on a local higher covariant derivative regulator. When the Yang-Mills term is present in the theory the well-know integer-shift is obtained, but is absent, the shift value is non-integer. These results show a possibility that a non-integer-shift can be derived using a local higher covariant derivative and also suggest that the Yang-Mills term plays an important role in the integer-shift of the Chern-Simons coupling.
We systematically analyze the effective action on the moduli space of (2,0) superconformal field theories in six dimensions, as well as their toroidal compactification to maximally supersymmetric Yang-Mills theories in five and four dimensions. We present a streamlined approach to non-renormalization theorems that constrain this effective action. The first several orders in its derivative expansion are determined by a one-loop calculation in five-dimensional Yang-Mills theory. This fixes the leading higher-derivative operators that describe the renormalization group flow into theories residing…
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…
We give a review of the exact renormalization group (ERG) approach and illustrate its applications in scalar and fermionic theories. The derivative expansion and approximations based on the derivative expansion with further truncation in the number of fields (mixed approximation) are discussed. We analyse the mixed approximation for a three-dimensional scalar theory and show that it is less effective than the pure derivative expansion. For pure fermionic theories analytical solutions for the pure derivative expansion and mixed approximation in the limit $N \to \infty $, where $N$ is the number…
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