Vilkovisky-DeWitt Effective Action Research Articles

Gravitational contributions to gauge Green’s functions and asymptotic free power-law running of gauge coupling

We perform an explicit one-loop calculation for the gravitational contributions to the two-, three- and four-point gauge Green's functions with paying attention to the quadratic divergences. It is shown for the first time in the diagrammatic calculation that the Slavnov-Taylor identities are preserved even if the quantum graviton effects are included at one-loop level, such a conclusion is independent of the choice of regularization schemes. We also present a regularization scheme independent calculation based on the gauge condition independent background field framework of Vilkovisky-DeWitt's effective action with focusing on both the quadratic divergence and quartic divergence that is not discussed before. With the harmonic gauge condition, the results computed by using the traditional background field method can consistently be recovered from the Vilkovisky-DeWitt's effective action approach by simply taking a limiting case, and are found to be the same as the ones yielded by the diagrammatic calculation. As a consequence, in all the calculations, the symmetry-preserving and divergent-behavior-preserving loop regularization method can consistently lead to a nontrivial gravitational contribution to the gauge coupling constant with an asymptotic free power-law running at one loop near the Planck scale.

Quantum gravity, gauge coupling constants, and the cosmological constant

The quantization of Einstein-Maxwell theory with a cosmological constant is considered. We obtain all logarithmically divergent terms in the one-loop effective action that involve only the background electromagnetic field. This includes Lee-Wick--type terms, as well as those responsible for the renormalization group behavior of the electric charge (or fine structure constant). Of particular interest is the possible gauge condition dependence of the results, and we study this in some detail. We show that the traditional background-field method, that is equivalent to a more traditional Feynman diagram calculation, does result in gauge condition dependent results in general. One resolution of this is to use the Vilkovisky-DeWitt effective action method, and this is presented here. Quantum gravity is shown to lead to a contribution to the running charge not present when the cosmological constant vanishes. This reopens the possibility, suggested by Robinson and Wilczek, of altering the scaling behavior of gauge theories at high energies although our result differs. We show the possibility of an ultraviolet fixed point that is linked directly to the cosmological constant.

Vilkovisky-DeWitt effective action for Einstein gravity on Kaluza-Klein spacetimesM4×SN

We evaluate the divergent part of the Vilkovisky-DeWitt effective action for Einstein gravity on even-dimensional Kaluza-Klein spacetimes of the form $M^{4}\times S^{N}$. Explicit results are given for $N$=2, 4, and 6. Trace anomalies for gravitons are also given for these cases. Stable Kaluza-Klein configurations are sought, unsuccessfully, assuming the divergent part of the effective action dominates the dynamics.

Gauge-independent trace anomaly for gravitons.

We show that the trace anomaly for gravitons calculated using the usual effective action formalism depends on the choice of gauge when the background spacetime is not a solution of the classical equation of motion, that is, when off shell. We then use the gauge-independent Vilkovisky-DeWitt effective action to restore gauge independence to the off-shell case. Additionally we explicitly evaluate trace anomalies for some N-sphere background spacetimes.

Superspace dependence of the Vilkovisky-DeWitt effective action for quantum gravity.

The definition of the Vilkovisky-DeWitt effective action for gravity theories permits the introduction of a metric in the field space that depends on a parameter \ensuremath{\lambda}. We study the consequences in the quantum theory of keeping this parameter free in ordinary and in higher derivative gravity. We do this by computing the renormalization group equations for an arbitrary \ensuremath{\lambda}. In both cases we see that it is possible to have a decay mechanism for the cosmological constant while maintaining a phenomenologically interesting behavior for Newton's constant.

One-loop Vilkovisky-DeWitt counterterms for two-dimensional gravity plus scalar field theory.

The divergent part of the one-loop off-shell effective action is computed for a single scalar field coupled to the Ricci curvature of 2D gravity ($c \phi R$), and self interacting by an arbitrary potential term $V(\phi)$. The Vilkovisky-DeWitt effective action is used to compute gauge-fixing independent results. In our background field/covariant gauge we find that the Liouville theory is finite on shell. Off-shell, we find a large class of renormalizable potentials which include the Liouville potential. We also find that for backgrounds satisfying $R=0$, the Liouville theory is finite off shell, as well.

The path integral for gauge theories: a geometrical approach

The author reviews some recent developments in the geometrical approach to the path integral quantisation of gauge theories. In particular the geometrical content of the Faddeev-Popov ansatz is elucidated and used to motivate the introduction of a particular affine structure on the space of gauge fields. This affine structure leads directly to the Vilkovisky-DeWitt effective action, which is manifestly gauge invariant, gauge fixing independent and reparametrisation invariant off-shell.

Examples of the Vilkovisky-DeWitt effective action in one-loop quantum gravity.

We present two examples where the contribution of one-loop gravitons to the renormalized stress-energy tensor may be calculated explicitly from both the standard and the Vilkovisky-DeWitt effective action. The renormalization is carried out using the Hadamard renormalization procedure outlined by Allen, Folacci, and Ottewill [Phys. Rev. D 38, 1069 (1988)]. The examples show that the standard and Vilkovisky-DeWitt formulations of quantum field theory lead to different physical predictions at one loop.

General relativity as the low-energy limit in higher derivative quantum gravity

The curvature induced phase transition in quantum R2-gravity with matter is investigated. The renormalization group approach is used to calculate the effective potential with accuracy to linear curvature terms. The universal expressions for induced gravitational and cosmological constants are obtained. The effective potential, induced cosmological and gravitational constants depend on the coupling constants of the original theory and on the gauge parameters. When matter is described by a scalar field, coupling constants subject to asymptotic freedom are chosen. Then a curvature induced phase transition is possible. If the Vilkovisky-De Witt effective action formalism is used then the induced gravitational and cosmological constants do not depend on gauge parameters.

The Vilkovisky-De Witt effective action in BF-type topological field theories

The one-loop off-shell effective action is reexamined for the case of BF theories in three dimensions. Within the context of the Vilkovisky-DeWitt reparametrization invariant framework, it is shown how the choice of an acceptable field space metric is crucial for obtaining gauge invariant results, even when this metric is field independent. It is found that the phase contribution to the one-loop effective action is proportional to the pure Chern-Simons term. The possible dependence of these results on the choice of field metric is briefly discussed.

Does the Vilkovisky-De Witt effective action in quantum gravity depend on the configuration space metric?

The dependence on the configuration space metric of the Vilkovisky-De Witt effective action in quantum gravity is discussed. As an example d-dimensional Einstein gravity on the flat background R d or R k× T d−k is considered, where the one-loop Vilkovisky-De Witt effective action with the one-parameter dependent configuration space metric is calculated. The dependence on the configuration space metric of the Vilkovisky-De Witt EA in gravity on the space R 4× S 1 has been investigated for the first time by Huggins, Kunstatter, Leivo and Toms.

Vilkovisky-de Witt effective action in quantum field theory in curved spacetime

The potentialities of the application of the Vilkovisky-de Witt effective action in quantum field theory in curved spacetime are discussed. A number of examples is given, in which this effective action eliminates problems related to the standard effective action.

Quantum gravity at large distances and the cosmological constant

The large-distance behaviour of quantum corrections to the gravitational field equations with a cosmological constant term is studied within the formalism of the gauge-invariant Vilkovisky-DeWitt effective action. Implications for a recently proposed Coleman-Weinberg type suppression of the cosmological constant are discussed.

Covariant off-shell effective action for scalars coupled to gravity

The covariant Vilkovisky-DeWitt effective action is applied to Einstein gravity coupled to scalars in a general non-trivial space-time metric. The one-loop off-shell, and on-shell, divergences are then obtained, also for pure gravitation.

The Parametrization Invariant and Gauge Invariant Effective Actions in Quantum Field Theory

The review of formulation and methods of calculation of the parametrization and gauge invariant effective actions in quantum field theory is given. As an example the Vilkovisky-De Witt Effective action (EA) is studied (this EA is a natural representative of gauge and parametrization invariant EA's). The examples where the use of the standard EA leads to the ambiguity are demonstrated. This happens as the result of dependence of the standard EA upon the choice of gauge condition. These examples are as follows: Coleman-Weinberg potential in the finite theories and symmetry breaking, EA in quantum gravity with matter and d = 5 gauged supergravity, the possibility of spontaneous supersymmetry breaking in N = 1 supergravity and the spontaneous compactification in the multidimensional R2-gravity. In all these cases the one-loop Vilkovisky-De Witt EA is found and therefore the problem of gauge dependence of EA is solved. The dependence of standard EA of composite fields upon the gauge is studied for the general gauge theories. The class of gauge and parametrization invariant EA's of the composite fields is offered.

A gauge-independent coupling constant in thermal QCD

We employ the theory of the Vilkovisky-DeWitt effective action, trivially extendable to finite temperatures, to define a manifestly gauge- and vertex-independent coupling constant in the high-temperature phase of quantum chromodynamics. The renormalization- group flow of this effective interaction strength is determined on the basis of a recently constructed thermal generalization of the MOM subtraction scheme (which is briefly reviewed from a rigorous point of view). This scheme intends to minimize the effects of higher orders, and appears to be the most “physical” renormalization prescription at finite T. It is found that the one-loop β-function for the renormalized effective expansion parameter thus defined is positive at high T. It follows that, in contradiction to conventional wisdom, asymptotic freedom at high temperature (and/or density) cannot be derived from bare one-loop renormalization-group arguments.

Linear response of hot gluons

We reexamine the various schemes for calculating the linear response (the retarded Green's function) of a hot gluon plasma. The problems related to gauge invariance are discussed in detail, and results in different gauges are compared. We also point out some issues related to the very definition of a thermal ensemble in the presence of unphysical degrees of freedom. By culculating the retarded Green's function directly in real time, we explicitly study the effects of unphysical degrees of freedom in different gauges. Although there appears to be no unique way to define the response function, we find that several schemes can be questioned on formal grounds and that use of the background-field gauge (BFG) is the most satisfactory in this respect. We discuss two proposals to fix the gauge parameter (α) dependence in the BFG response function, the Vilkovisky-DeWitt effective action corresponding to the choice α = 0 (background Landau gauge), and the “gauge-invariant propagator” of Cornwall et al. corresponding to α = 1 (background Feynman gauge).

The Vilkovisky-De Witt effective action in multidimensional higher derivative gravity

The one-loop gauge and parametrization invariant Vilkovisky-De Witt effective action (VDEA) in multidimensional R 2-gravity on a background R 4×T d−4 is calculated. (Here R 4 is four-dimensional flat space and T d−4 is the ( d−4)-dimensional torus). It is shown that in the Einstein limit the VDEA under consideration leads to the VDAE in Einstein gravity on the background R 4×T d−4. The question of the stability of quantum spontaneous compactication d=5 R 2-gravity is also considered.

Gauge dependence of the effective action in multidimensional supergravities

Abstract The problem of gauge dependence of the effective action in multidimensional supergravities (SG) is investigated. As an example d =5 SGs are considered. The convenient gauge-dependent effective action (EA) and gauge-invariant Vilkovisky-De Witt EA in d =5, N =2, 4, 6, 8 SGs on the background R 4 ×S 1 with accuracy to linear curvature terms are obtained. The convenient three-gauge-parameter-dependent EA in d =5, N =2 gauged SG on the background R 4 ×S 1 is calculated. The Vilkovisky-De Witt EA in d =5 SGs on the background R 4 ×S 1 at non-zero temperature is also obtained.

CASIMIR EFFECT IN MULTIDIMENSIONAL QUANTUM SUPER-GRAVITIES AND SUPERSYMMETRY BREAKING

The one-loop effective action (the Casimir gravitational energy) of the aribitrary Einstein supergravity on the background [Formula: see text], where [Formula: see text] is the Minkowski space with non-zero temperature, Td is the d-dimensional torus, is calculated. The problem of quantum breaking of supersymmetry is discussed. The Vilkovisky-De Witt effective action in the D-dimensional Einstein gravity with the Λ-term on the background [Formula: see text] is found. An idea is expressed that a temperature phase transition in Kaluza-Klein theories is possible. For d=5 gravity, the Vilkovisky-De Witt effective action on the [Formula: see text], where [Formula: see text] is four-dimensional curved space-time with non-zero temperature, is found.