We give a detailed critical discussion of the properties of Wilsonian effective actions, defined by integrating out all modes above a given scale $μ$. In particular, we provide a precise and relatively convenient prescription how to implement the infrared cutoff $μ$ in any loop integral that is manifestly Lorentz invariant and also preserves global linear symmetries such as e.g. supersymmetry. We discuss the issue of gauge invariance of effective actions in general and in particular when using background field gauge. Our prescription for the IR cutoff (as any such prescription) breaks the gauge symmetry. Using our prescription, we have explicitly computed, at one loop, many terms of the Wilsonian effective action for general gauge theories, involving bosonic and fermionic matter fields of arbitrary masses and in arbitrary representations, exhibiting the non-gauge invariant (as well as the gauge invariant) terms. However, for supersymmetric gauge theories all non-gauge invariant terms cancel within each supermultiplet. This is strong evidence that in supersymmetric gauge theories this indeed defines a Lorentz, susy and gauge invariant Wilsonian effective action. As a byproduct, we obtain the explicit one-loop Wilsonian couplings for all higher-derivative terms $F D^{2n}F$ in the effective action of arbitrary supersymmetric gauge theories.
We investigate non-Abelian gauge theories within a Wilsonian Renormalisation Group approach. The cut-off term inherent in this approach leads to a modified Ward identity (mWI). It is shown that this mWI is compatible with the flow and that the full effective action satisfies the usual Ward identity (WI). The universal 1-loop beta-function is derived within this approach and the extension to the 2-loop level is briefly outlined.
We construct modular invariants on the moduli space of quantum vacua of N=2 SYM with gauge group SU(2). We also introduce a nonchiral function K which is expressed in terms of the Seiberg-Witten and Poincare' metrics. It turns out that K has all the expected properties of the next to leading term in the Wilsonian effective action whose modular properties are considered in the framework of the dimensional regularization.
For conformal field theories, it is shown how the Ward identity corresponding to dilatation invariance arises in a Wilsonian setting. In so doing, several points which are opaque in textbook treatments are clarified. Exploiting the fact that the Exact Renormalization Group furnishes a representation of the conformal algebra allows dilatation invariance to be stated directly as a property of the action, despite the presence of a regulator. This obviates the need for formal statements that conformal invariance is recovered once the regulator is removed. Furthermore, the proper subset of conforma…
We investigate non-Abelian gauge theories within a Wilsonian Renormalisation Group approach. Our main question is: How close can one get to a gauge invariant flow, despite the fact that a Wilsonian coarse-graining seems to be incompatible with gauge invariance? We discuss the possible options in the case of quantum fluctuations, and argue that for thermal fluctuations a fully gauge invariant implementation can be obtained.
For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.
We study the perturbative approach to the Wilsonian integration of noncommutative gauge theories in the matrix representation. We begin by motivating the study of noncommutative gauge theories and reviewing the matrix formulation. We then systematically develop the perturbative treatment of UV states and calculate both the leading and next to leading order one- and two-loop corrections to the quantum effective action. Throughout, we discuss how our formalism clarifies problems associated with UV-IR mixing, a particular emphasis being placed on the dipole structure imposed by noncommutative gau…
The purpose of the present thesis is the implementation of symmetries in the Wilsonian Exact Renormalization Group (ERG) approach. After recalling how the ERG can be introduced in a general theory (i.e. containing both bosons and fermions, scalars and vectors) and having applied it to the massless scalar theory as an example of how the method works, we discuss the formulation of the Quantum Action Principle (QAP) in the ERG and show that the Slavnov-Taylor identities can be directly derived for the cutoff effective action at any momentum scale. Firstly the QAP is exploited to analyse the break…
We discuss gauge symmetry and Ward-Takahashi identities for Wilsonian flows in pure Yang-Mills theories. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. We examine how the symmetry properties of the full theory are restored in the limit where the cut-off is removed.
If the Wilsonian renormalization group (RG) is formulated with a cutoff that breaks gauge invariance, then gauge invariance may be recovered only once the cutoff is removed and only once a set of effective Ward identities is imposed. We show that an effective Quantum Action Principle can be formulated in perturbation theory which enables the effective Ward identities to be solved order by order, even if the theory requires non-vanishing subtraction points. The difficulties encountered with non-perturbative approximations are briefly discussed.
In the first part of this paper we critically examine the ultra-violet implications of theories that exhibit Vainshtein screening, taking into account both the standard Wilsonian perspective as well as more exotic possibilities. Aspects of this discussion draw on results from the second part of the paper in which we perform a general study of derivatively coupled scalar theories using non-perturbative exact renormalisation group techniques, which are of interest independently of their application to modified gravity. In this context, we demonstrate the suppression of quantum corrections within…
I study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic noncovariant gauges where the Wilsonian infrared cutoff $Λ$ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless the BRS-invariance in broken and the theory is gauge-dependent and unphysical at $Λ\neq0$. Then I discuss the infrared limit $Λ\to0$. I show that the singularities of the axial gauge choice are avoided in planar gauge and in light-cone gauge. Finally the rectangular Wilson loop of size $2L\times 2T$ is evaluated at…
We study the fate of reparametrization invariance of Wilson loops, also known as 'zig-zag' symmetry, under the RG flow using some simple cases as guidance. We restrict our analysis to large-$N$, strongly coupled CFTs and use the holographic dual description of a Wilson loop as a fundamental string embedded in asymptotically AdS spaces, at zero and nonzero temperature. We then introduce a cutoff in the holographic radial direction and integrate out the the section of the string closer to the AdS boundary in the spirit of holographic Wilsonian renormalization. We make explicit the map between Wi…
We give a simplified and more complete description of the loop variable approach for writing down gauge invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite world-sheet cutoff and that in the infrared …
Liouville field theory is quantized by means of a Wilsonian effective action and its associated exact renormalization group equation. For $c<1$, an approximate solution of this equation is obtained by truncating the space of all action functionals. The Ward identities resulting from the Weyl invariance of the theory are used in order to select a specific universality class for the renormalization group trajectory. It is found to connect two conformal field theories with central charges $25-c$ and $26-c$, respectively.
In this talk the gauge symmetry for Wilsonian flows in pure Yang-Mills theories is discussed. The background field formalism is used for the construction of a gauge invariant effective action. The symmetries of the effective action under gauge transformations for both the gauge field and the auxiliary background field are separately evaluated. Modified Ward-Takahashi and background field identities are used in my study. Finally it is shown how the symmetry properties of the full theory are restored in the limit where the cut-off is removed.
We give a very simple proof that the renormalization of the Chern-Simons coupling in the Wilsonian effective action is exhausted at one-loop. Our proof can apply to arbitrary 2+1-dimensional abelian as well as nonabelian gauge theories without a bare Chern-Simons coupling, including any non-renormalizable interactions and non-minimal couplings. Our proof reveals that small (but not large) gauge invariance is enough to ensure the absence of higher order corrections.
A general discussion of equations with universal invariance for a scalar field is provided in the framework of Lagrangian theory of first-order systems.
We study some aspects of perturbation theory in $N=1$ supersymmetric abelian gauge theories with massive charged matter. In general gauges, infrared (IR) divergences and nonlocal behavior arise in 1PI diagrams, associated with a $1/k^4$ term in the propagator for the vector superfield. We examine this structure in supersymmetric QED. The IR divergences are gauge-dependent and must cancel in physical quantities like the electron pole mass. We demonstrate that cancellation takes place in a nontrivial way, amounting to a reorganization of the perturbative series from powers of $e^2$ to powers of …
Numerous topics in three and four dimensional supersymmetric gauge theories are covered. The organizing principle in this presentation is scaling (Wilsonian renormalization group flow.) A brief introduction to scaling and to supersymmetric field theory, with examples, is followed by discussions of nonrenormalization theorems and beta functions. Abelian gauge theories are discussed in some detail, with special focus on three-dimensional versions of supersymmetric QED, which exhibit solitons, dimensional antitransmutation, duality, and other interesting phenomena. Many of the same features are s…
In the Coulomb gauge, the Hamiltonian of QCD contains terms of order h^2, identified by Christ and Lee, which are non-local but instantaneous. The question is addressed how these terms fit in with BRST invariance. Our discussion is confined to the simplest, O(g^4), example.
We study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic non-covariant gauges where the Wilsonian infrared cutoff $Λ$ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless BRST-invariance in broken and the theory is gauge-dependent and unphysical at $Λ\neq0$. Then we discuss the infrared limit $Λ\to0$. We show that the singularities of the axial gauge choice are avoided in planar gauge and light-cone gauge. In addition the issue of infrared divergences is addressed in some explicit…
We clarify the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories, which states that the low-energy physics can be kept fixed when one changes the ultraviolet cutoff, provided appropriate changes are made to the bare coupling constants in the Lagrangian. We first pose a puzzle on how a quantum modified constraint (such as Pf(Q^i Q^j) = Λ^{2(N+1)} in SP(N) theories with N+1 flavors) can be RG invariant, since the bare fields Q^i receive wave function renormalization when one changes the ultraviolet cutoff, while we naively regard the scale Λas RG invarian…
We show that the Quantum Master Equation and the Wilsonian renormalization group (RG) flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $Λ$, despite the fact that the structure demands quantum corrections that naively break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interacti…
We present a formulation of non-Abelian gauge theories in general axial gauges using a Wilsonian (or 'Exact') Renormalisation Group. No 'spurious' propagator divergencies are encountered in contrast to standard perturbation theory. Modified Ward identities, compatible with the flow equation, ensure gauge invariance of physical Green functions. The axial gauge $n A=0$ is shown to be a fixed point under the flow equation. Possible non-perturbative approximation schemes and further applications are outlined.
A perturbative non-renormalization theorem is presented that applies to general supersymmetric theories, including non-renormalizable theories in which the $\int d^2θ$ integrand is an arbitrary gauge-invariant function $F(Φ,W)$ of the chiral superfields $Φ$ and gauge field-strength superfields $W$, and the $\int d^4θ$-integrand is restricted only by gauge invariance. In the Wilsonian Lagrangian, $F(Φ,W)$ is unrenormalized except for the one-loop renormalization of the gauge coupling parameter, and Fayet-Iliopoulos terms can be renormalized only by one-loop graphs, which cancel if the sum of th…
We investigate the field dependence of the gauge couplings of locally supersymmetric effective quantum field theories. We find that the Weyl rescaling of supergravity gives rise to Wess-Zumino terms that affect the gauge couplings at the one-loop level. These Wess-Zumino terms are crucial in assuring supersymmetric consistency of both perturbative and non-perturbative gauge interactions. At the perturbative level, we distinguish between the holomorphic Wilsonian gauge couplings and the physically-measurable momentum-dependent effective gauge couplings; the latter are affected by the Konishi an…
The issue of the gauge invariance of gravitational waves arises if they are produced in the early universe at second-order in perturbation theory. We address it by dividing the discussion about the gauge invariance in three parts: the production of gravitational waves, their propagation in the real universe, and their measurement.
Previous analyses on the gauge invariance of the action for a generally covariant system are generalized. It is shown that if the action principle is properly improved, there is as much gauge freedom at the endpoints for an arbitrary gauge system as there is for a system with ``internal'' gauge symmetries. The key point is to correctly identify the boundary conditions for the allowed histories and to include the appropriate end-point contribution in the action. The path integral is then discussed. It is proved that by employing the improved action, one can use time-independent canonical gauges…
We discuss the issue of going off-shell in the proper time formalism. This is done by keeping a finite world sheet cutoff. We construct one example of an off-shell covariant Klein Gordon type interaction. For a suitable choice of the gauge transformation of the scalar field, gauge invariance is maintained off mass shell. However at second order in the gauge field interaction, one finds that (U(1)) gauge invariance is violated due to the finite cutoff. Interestingly, we find, to lowest order, that by adding a massive mode with appropriate gauge transformation laws to the sigma model background,…
This article serves as a continuation for the discussion in arXiv:0911.3433, we analyze the invariance properties of the gravity path-integral measure derived from canonical framework, and discuss which path-integral formula may be employed in the concrete computation e.g. constructing a spin-foam model, so that the final model can be interpreted as a physical inner product in the canonical theory.
We continue earlier discussions on the exact renormalization group and loop variables on the string world sheet for closed and open string backgrounds. The world sheet action with a UV regulator is written in a generally background covariant way by introducing a background metric. It is shown that the renormalization group gives background covariant equations of motion - this is the gauge invariance of the graviton. Interaction is written in terms of gauge invariant and generally covariant field strength tensors. The basic idea is to work in Riemann normal coordinates and covariantize the fina…
Working with a gauge coupling field in a linear superfield, we construct effective Lagrangians for N=1 super-Yang-Mills theory fully compatible with the expected all-order behaviour or physical quantities. Using the one-loop dependence on its ultraviolet cutoff and anomaly matching or cancellation of R and dilatation anomalies, we obtain the Wilsonian effective Lagrangian. With similar anomaly matching or cancellation methods, we derive the effective action for gaugino condensates, as a function of the real coupling field. Both effective actions lead to a derivation of the NSVZ beta function f…
Given a Quantum Field Theory, with a particular content of fields and a symmetry associated with them, if one wants to study the evolution of the couplings via a Wilsonian renormalisation group, there is still a freedom on the construction of a flow equation, allowed by scheme independence. In the present thesis, making use of this choice, we first build up a generalisation of the Polchinski flow equation for the massless scalar field, and, applying it to the calculation of the beta function at one loop for the characteristic self-interaction, we test its universality beyond the already known …
The construction of physical models with local time-reparametrization invariance is reviewed. Negative-energy contributions to the hamiltonian are shown to be crucial for the realization of this reparametrization symmetry. The covariant formulation of the dynamics is used to develop a time and gauge invariant Hamilton-Jacobi theory. This formalism is applied to solve for the cosmology of a homogeneous universe of the Friedmann-Lemaitre-Robertson-Walker type. After a discussion of empty universes, the FLRW theory is extended with homogeneous scalar fields generically described by a $\sg$-model …
We investigate the possibility of Lorentz and CPT violations in the photon sector, of the Chern-Simons form, be induced by radiative corrections arising from the Lorentz and CPT non-invariant fermionic sector of an extended version of QED. By analyzing the modified vacuum polarization tensor, three contributions are considered: two of them can be identified with well known amplitudes; the (identical) QED vacuum polarization tensor and the (closely related) $AVV$ triangular amplitude. These amplitudes are evaluated in their most general form (to include in our discussion automatically the quest…
In these lectures we describe the construction of a gauge invariant renormalization group equation for pure non-Abelian gauge theory. In the process, a non-perturbative gauge invariant continuum Wilsonian effective action is precisely defined. The formulation makes sense without gauge fixing and thus manifest gauge invariance may be preserved at all stages. In the large N limit (of SU(N) gauge theory) the effective action simplifies: it may be expressed through a path integral for a single particle whose trajectory describes a Wilson loop. Regularization is achieved with the help of a set of P…
We revisit the previously unsolved problems of ensuring Lorentz invariance and non-perturbative unitarity in Lee-Wick theories. We base our discussion on an ultraviolet completion of QED by Lee-Wick ghost fields, which is argued to be asymptotically safe. We argue that as long as the state space is based upon a suitable choice of distributions of a type invented by Gel'fand and Shilov, the Lee-Wick ghosts can be eliminated while preserving Lorentz invariance to produce a unitary theory. The method for eliminating ghosts is in principle non-perturbatively well-defined, in contrast with some pre…
We consider an effective model formed by usual QED (minimal coupling) with the addition of a nonminimal Lorentz violating interaction (proportional to a fixed 4-vector $b_μ$) which may radiatively generate both CPT even and odd terms in the pure gauge sector. We show that gauge invariance from usual QED, considered as a limit of the model for $b_μ\rightarrow 0$, plays an important role in the discussion of the radiatively induced Lorentz violating terms at one-loop order. Moreover, despite the nonrenormalizability of the (effective) model preventing us from readily extending our discussion to …
This short note was born out of discussions on anyons in the FQHE at the YITP workshop "Fundamental problems of quantum field theories" (December 19-21, 2001, YITP, Kyoto). At that time, I felt that there might not be a sound consensus of opinion on the subject. Now, I would like to show my understanding here, the essential part of which is based on hep-th/0110197. The first problem discussed is a notion of ``bosonized electrons (bosonization)", which are unphysical objects from the standpoint of the Chern-Simons gauge invariance. Therefore, their condensate is merely of mathematical concept a…
We present the general formulation of non-covariant Lagrangian of self-dual gauge theory. After specifying the parameters therein the previous Lagrangian in the decomposition of spacetime into $6=D_1+D_2$ and $6=D_1+D_2+D_3$ can be obtained. The self-dual property of the general Lagrangian is proved in detail. We furthermore show that the new non-covariant actions give field equations with 6d Lorentz invariance. The method can be straightforward extended to any dimension and we also give a short discussion about the 10D self-dual gauge theory.
The BRS invariance of the electroweak gauge theory leads to relationships between amplitudes with external massive gauge bosons and amplitudes where some of these gauge bosons are replaced with their corresponding Nambu-Goldstone bosons. Unlike the equivalence theorem, these identities are exact at all energies. In this paper we discuss such identities which relate the process $e^+e^- \to W^+W^-$ to $W^\pmχ^\mp$ and $χ^+χ^-$ production. By using a general form-factor decomposition for $e^+e^- \to W^+W^-$, $e^+e^- \to W^\pm χ^\mp$ and $e^+e^- \to χ^+χ^-$ amplitudes, these identities are express…
We study some consequences of coupling supersymmetric theories to (super)gravity. To linear order, the couplings are determined by the energy-momentum supermultiplet. At higher orders, the couplings are determined by contact terms in correlation functions of the energy-momentum supermultiplet. We focus on the couplings of one particular field in the supergravity multiplet, the auxiliary field $M$. We discuss its linear and quadratic (seagull) couplings in various supersymmetric theories. In analogy to the local renormalization group formalism, we provide a prescription for how to fix the quadr…
We provide an elementary systematic discussion of single-trace matrix actions and of the group of matrix reparameterization that acts on them. The action of this group yields a generalized notion of gauge invariance which encompasses ordinary diffeomorphism and gauge invariances. We apply the formalism to non-abelian D-brane actions in arbitrary supergravity backgrounds, providing in particular explicit checks of the consistency of Myers' formulas with supergravity gauge invariances. We also draw interesting consequences for emergent space models based on the study of matrix effective actions.…
A particular form of non-linear $σ$-model, having a global gauge invariance, is studied. The detailed discussion on current algebra structures reveals the non-abelian nature of the invariance, with {\it{field dependent structure functions}}. Reduction of the field theory to a point particle framework yields a non-linear harmonic oscillator, which is a special case of similar models studied before in \cite{car}. The connection with noncommutative geometry is also established.
In the same way as the energy density associated with the tensor modes of the geometry modifies the evolution of the curvature perturbations, the scalar modes may also indirectly affect the cosmic backgrounds of relic gravitons by inducing higher-order corrections that are only superficially gauge-invariant. This spurious gauge-invariance gets manifest when the effective anisotropic stresses, computed in different coordinate systems, are preliminarily expressed in a form that only depends on the curvature inhomogeneities defined on comoving orthogonal hypersurfaces and on their corresponding t…
Much research has been done in the latter years on the subject of Lorentz violation induced by Quantum Gravity effects. On the theoretical side it has been shown that both Loop Quantum Gravity and String Theory predict that Lorentz violation can be induced at an energy near to the Planck scale. On the other hand, most of the experimental results in the latter years, have confirmed that the laws of physics are Lorentz invariant at low energy with very high accuracy. The inclusion of one- and two-loop contributions from a Lorentz violating Lagrangian dramatically change the above picture: the lo…
In this work we report new results concerning the question of dynamical mass generation in the Lorentz and PCT violating quantum electrodynamics. A one loop calculation for the vacuum polarization tensor is presented. The electron propagator, "dressed" by a Lorentz breaking extra term in the fermion Lagrangian density, is approximated by its first order: this scheme is shown to break gauge invariance. Then we rather consider a full calculation to second order in the Lorentz breaking parameter: we recover gauge invariance and use the Schwinger-Dyson equation to discuss the full photon propagato…
In this brief, and by no means complete, review I discuss situations in string theory, in which Lorentz Invariance Violation may occur in a way consistent with world-sheet conformal invariance, thereby leading to acceptable, in principle, string backgrounds. In particular, I first discuss spontaneous Lorentz violation in (non supersymmetric) open string field theory. Then, I move onto a discussion of gravity-induced modified dispersion relations in non-critical (Liouville) strings, in the sense of an induced Finsler-like geometry depending on both coordinates and momenta, for string propagatio…
Signed Coordinate Invariance, invariant lagrangians and manifolds, the time problem in quantum cosmology, quantum space time, spacetimes and antispacetimes
We discuss the properties of non-abelian gauge theories formulated on manifolds with compactified dimensions and in the presence of fermionic fields coupled to magnetic backgrounds. We show that different phases may emerge, corresponding to different realizations of center symmetry and translational invariance, depending on the compactification radius and on the magnitude of the magnetic field. Our discussion focuses on the case of an $SU(3)$ gauge theory in 4 dimensions with fermions fields in the fundamental representation, for which we provide some exploratory numerical lattice results.
