Covariant Gauge Fixing Research Articles

BRST quantization of general relativity in unimodular gauge and unimodular gravity

Unimodular gravity (UG) is an important theory which may explain the smallness of the cosmological constant. To get insight into the covariant quantization of UG, we discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the unimodular gauge. This process requires the introduction of an additional pair of BRST doublets which decouple from the physical sector together with the other three pairs of BRST doublets for the TDiff. We show that the physical spectrum is the same as GR in the usual covariant gauge fixing. We then study a theory derived by making "Fourier transform" of GR in the unimodular gauge with respect to the cosmological constant as a candidate of "quantum UG." We clarify the difference from GR and point out problems in this theory.

Gauge-invariant approach to the beta function in Yang-Mills theories with universal extra dimensions

The radiative correction to beta function is comprehensively studied at 1 loop in the context of universal extra dimensions. Instead of using cutoffs to regularize 1-loop divergences, the dimensional regularization scheme is used. Large momenta effects are removed from physical amplitudes by adjusting the parameters of the appropriate counterterms. The use of a SU(N)-covariant gauge-fixing procedure is stressed. 1-loop contributions of KK excitations are characterized by discrete KK sums and continuous momenta sums, which can diverge. Two types of UVs are identified, one arising from poles of the gamma function and associated with short-distance effects in the usual 4-dimensional spacetime manifold, and the other emerging either from poles of the 1-dimensional Epstein function or from the gamma function, and corresponding to short-distance effects in the compact manifold. We address the cases of 5 and 4+n dimensions (n>1) separately. In 5 dimensions the 1-dimensional Epstein function is convergent, so the usual counterterm renormalizes the vacuum polarization function. For 4+n dimensions, the 1-dimensional Epstein function is divergent, so renormalization is implemented by interactions of canonical dimension higher than 4, already present in the effective theory. The polarization function is renormalized using both mass-dependent and mass-independent schemes, with extra-dimensions effects decoupling in the former case but not in the latter. The beta function is calculated for an arbitrary number of extra dimensions. Our main result is that Yang-Mills theory remains perturbative at 1 loop, which is in disagreement with the results obtained in the literature by using a cutoff regulator, which suggest that Yang-Mills theory in more than 4 dimensions ceases to be perturbative. We emphasize the advantages of a mass-dependent scheme in this type of theories, in which decoupling is manifest.

Background Independence in Gauge Theories

Classical field theory is insensitive to the split of the field into a background configuration and a dynamical perturbation. In gauge theories, the situation is complicated by the fact that a covariant (w.r.t. the background field) gauge fixing breaks this split independence of the action. Nevertheless, background independence is preserved on the observables, as defined via the BRST formalism, since the violation term is BRST exact. In quantized gauge theories, however, BRST exactness of the violation term is not sufficient to guarantee background independence, due to potential anomalies. We define background-independent observables in a geometrical formulation as flat sections of the observable algebra bundle over the manifold of background configurations, with respect to a flat connection which implements background variations. A theory is then called background independent if such a flat (Fedosov) connection exists. We analyze the obstructions to preserve background independence at the quantum level for pure Yang–Mills theory and for perturbative gravity. We find that in the former case, all potential obstructions can be removed by finite renormalization. In the latter case, as a consequence of power-counting non-renormalizability, there are infinitely many non-trivial potential obstructions to background independence. We leave open the question whether these obstructions actually occur.

Kac-Moody and Virasoro characters from the perturbative Chern-Simons path integral

We evaluate to one loop the functional integral that computes the partition functions of Chern-Simons theories based on compact groups, using the background field method and a covariant gauge fixing. We compare our computation with the results of other, less direct methods. We find that our method correctly computes the characters of irreducible representations of Kac-Moody algebras. To extend the computation to non-compact groups we need to perform an appropriate analytic continuation of the partition function of the compact group. Non-vacuum characters are found by inserting a Wilson loop in the functional integral. We then extend our method to Euclidean Anti-de Sitter pure gravity in three dimensions. The explicit computation unveils several interesting features and lessons. The most important among them is that the very definition of gravity in the first-order Chern-Simons formalism requires non-trivial analytic continuations of the gauge fields outside their original domains of definition.

Linear gravity from conformal symmetry

We perform a unified systematic analysis of d + 1 dimensional, spin representations of the isometry algebra of the maximally symmetric spacetimes AdSd+1, and dSd+1. This allows us to explicitly construct the effective low-energy bulk equations of motion obeyed by linear fields, as the eigenvalue equation for the quadratic Casimir differential operator. We show that the bulk description of a conformal family is given by the Fierz–Pauli system of equations. For this is a massive gravity theory, while for conserved currents we obtain Einstein gravity and covariant gauge fixing conditions. This analysis provides a direct algebraic derivation of the familiar AdS holographic dictionary at low energies, with analogous results for Minkowski and de Sitter spacetimes.

Minimal gauge invariant and gauge fixed actions for massive higher-spin fields

Inspired by the rich structure of covariant string field theory, we propose a minimal gauge invariant action for general massive integer spin n field. The action consists of four totally symmetric tensor fields of order respectively n, n − 1, n − 2 and n − 3, and is invariant under the gauge transformation represented by two also totally symmetric fields of order n − 1 and n − 2. This action exactly has the same gauge structure as for the string field theory and we discuss general covariant gauge fixing procedure using the knowledge of string field theory. We explicitly construct the corresponding gauge fixed action for each of general covariant gauge fixing conditions.

Eight dimensional QCD at one loop

The Lagrangian for a non-abelian gauge theory with an $SU(N_{\! c})$ symmetry and a linear covariant gauge fixing is constructed in eight dimensions. The renormalization group functions are computed at one loop with the special cases of $N_{\! c}$ $=$ $2$ and $3$ treated separately. By computing the critical exponents derived from these in the large $N_{\! f}$ expansion at the Wilson-Fisher fixed point it is shown that the Lagrangian is in the same universality class as the two dimensional non-abelian Thirring model and Quantum Chromodynamics (QCD). As the eight dimensional Lagrangian contains new quartic gluon operators not present in four dimensional QCD, we compute in parallel the mixing matrix of four dimensional dimension $8$ operators in pure Yang-Mills theory.

Landau-Khalatnikov-Fradkin transformation for the fermion propagator in QED in arbitrary dimensions

We explore the dependence of fermion propagators on the covariant gauge fixing parameter in quantum electrodynamics (QED) with the number of spacetime dimensions kept explicit. Gauge covariance is controlled by the Landau-Khalatnikov-Fradkin transformation (LKFT). Utilizing its group nature, the LKFT for a fermion propagator in Minkowski space is solved exactly. The special scenario of 3D is used to test claims made for general cases. When renormalized correctly, a simplification of the LKFT in 4D has been achieved with the help of fractional calculus.

Tricritical points in a compactU(1)lattice gauge theory at strong coupling

Pure {\it compact} $U(1)$ lattice gauge theory exhibits a phase transition at gauge coupling $g \sim {\cal{O}}(1)$ separating a familiar weak coupling Coulomb phase, having free massless photons, from a strong coupling phase. However, the phase transition was found to be of first order, ruling out any nontrivial theory resulting from a continuum limit from the strong coupling side. In this work, a compact $U(1)$ lattice gauge theory is studied with addition of a dimension-two mass counterterm and a higher derivative (HD) term that ensures a unique vacuum and produces a covariant gauge-fixing term in the naive continuum limit. For a reasonably large coefficient of the HD term, now there exists a continuous transition from a regular ordered phase to a spatially modulated ordered phase. For weak gauge couplings, a continuum limit from the regular ordered phase results in a familiar theory consisting of free massless photons. For strong gauge couplings with $g\ge {\cal{O}}(1)$, this transition changes from first order to continuous as the coefficient of the HD term is increased, resulting in tricritical points which appear to be a candidate in this theory for a possible nontrivial continuum limit.

Hopf-algebraic renormalization of QED in the linear covariant gauge

In the context of massless quantum electrodynamics (QED) with a linear covariant gauge fixing, the connection between the counterterm and the Hopf-algebraic approach to renormalization is examined. The coproduct formula of Green’s functions contains two invariant charges, which give rise to different renormalization group functions. All formulas are tested by explicit computations to third loop order. The possibility of a finite electron self-energy by fixing a generalized linear covariant gauge is discussed. An analysis of subdivergences leads to the conclusion that such a gauge only exists in quenched QED.

Trilinear gauge boson couplings in the standard model with one universal extra dimension

One-loop effects of Standard Model (SM) extensions comprising universal extra dimensions are essential as a consequence of Kaluza-Klein (KK) parity conservation, for they represent the very first presumable virtual effects on low-energy observables. In this paper, we calculate the one-loop $CP$-even contributions to the SM $WW\ensuremath{\gamma}$ and $WWZ$ gauge couplings produced by the KK excited modes that stand for the dynamical variables of the effective theory emerged from a generalization of the SM to five dimensions, in which the extra dimension is assumed to be universal, after compactification. The employment of a covariant gauge-fixing procedure that removes gauge invariance associated to gauge KK excited modes, while keeping electroweak gauge symmetry manifest, is a main feature of this calculation, which is performed in the Feynman-'t Hooft gauge and yields finite results that consistently decouple for a large compactification scale. After numerical evaluation, our results are comparable with the one-loop SM contributions and well within the reach of a next linear collider.

The Standard Model with one universal extra dimension

Effects of universal extra dimensions on Standard Model observables first arise at the one-loop level. The quantization of this class of theories is therefore essential in order to perform predictions. A comprehensive study of the SU C(3) × SU L(2) × U Y(1) Standard Model defined in a space-time manifold with one universal extra dimension, compactified on the oribifold \(S^1/Z_2\), is presented. The fact that the four-dimensional Kaluza–Klein theory is subjected to two types of gauge transformations is stressed and its quantization under the basis of the BRST symmetry discussed. A SU C(3) × SU L(2) × U Y(1)-covariant gauge-fixing procedure for the Kaluza–Klein excitations is introduced. The connection between gauge and mass eigenstate fields is established in an exact way. An exhaustive list of the explicit expressions for all physical couplings induced by the Yang–Mills, Currents, Higgs, and Yukawa sectors is presented. The one-loop renormalizability of the standard Green’s functions, which implies that the Standard Model observables do not depend on a cut-off scale, is stressed.

Covariant gauge fixing and canonical quantization

Theories that contain first class constraints possess gauge invariance which results in the necessity of altering the measure in the associated quantum mechanical path integral. If the path integral is derived from the canonical structure of the theory, then the choice of gauge conditions used in constructing Faddeev's measure cannot be covariant. This shortcoming is normally overcome either by using the "Faddeev-Popov" quantization procedure, or by the approach of Batalin-Fradkin-Fradkina-Vilkovisky, and then demonstrating that these approaches are equivalent to the path integral constructed from the canonical approach with Faddeev's measure. We propose in this paper an alternate way of defining the measure for the path integral when it is constructed using the canonical procedure for theories containing first class constraints and that this new approach can be used in conjunction with covariant gauges. This procedure follows the Faddeev-Popov approach, but rather than working with the form of the gauge transformation in configuration space, it employs the generator of the gauge transformation in phase space. We demonstrate this approach to the path integral by applying it to Yang-Mills theory, a spin-two field and the first order Einstein-Hilbert action in two dimensions. The problems associated with defining the measure for theories containing second-class constraints and ones in which there are fewer secondary first class constraints than primary first class constraints are discussed.

GAUGE FIXING IDENTITY IN THE BACKGROUND FIELD METHOD OF QCD IN PURE GAUGE

In this paper we derive a gauge fixing identity by varying the covariant gauge fixing term in [Formula: see text] in the background field method of QCD in pure gauge. Using this gauge fixing identity, we establish a relation between [Formula: see text] in QCD and [Formula: see text] in background field method of QCD in pure gauge. We show the validity of this gauge fixing identity, in general noncovariant and general Coulomb gauge fixings respectively. This gauge fixing identity is used to prove factorization theorem in QCD at high energy colliders and in nonequilibrium QCD at high energy heavy-ion colliders.

A Lorentz invariant formulation of the Yang-Mills theory with gauge invariant ghost field Lagrangian

A new formulation of the Yang-Mills theory which allows a manifestly covariant gauge fixing accompanied by a gauge invariant ghost field interaction is proposed. The gauge condition selects a unique representative in the class of gauge equivalent configurations.

General covariant gauge fixing for massless spin-two fields

The most general covariant gauge fixing Lagrangian is considered for a spin-two gauge theory in the context of the Faddeev-Popov procedure. In general, five parameters characterize this gauge fixing. Certain limiting values for these parameters give rise to a spin-two propagator that is either traceless or transverse, but for no values of these parameters is this propagator simultaneously traceless and transverse. Having a traceless-transverse propagator ensures that only the physical degrees of freedom associated with the tensor field propagate, and hence it is analogous to the Landau gauge in electrodynamics. To obtain such a traceless-transverse propagator, a gauge fixing Lagrangian which is not quadratic must be employed; this sort of gauge fixing Lagrangian is not encountered in the usual Faddeev-Popov procedure. It is shown that when this nonquadratic gauge fixing Lagrangian is used, two fermionic and one bosonic ghosts arise. As a simple application we discuss the energy-momentum tensor of the gravitational field at finite temperature.

Radiative lifting of flat directions of the MSSM in de Sitter background

The one-loop effective potential for a charged scalar field in de Sitter background is studied. We derive an approximate form for the gauge boson propagator with a general covariant gauge-fixing parameter ξ. This expression is used to derive an effective potential, which agrees with earlier results obtained in Landau gauge, and the independence of ξ explicitly indicates the gauge-invariance of the calculation. Adding the scalar- and fermion-loop contributions, that arise in a supersymmetric Higgs model, we find that the effective potential is vanishing to first order in the Hubble rate H 2 in the absence of additional chiral matter. The contribution due to a chiral multiplet running in the loop is however non-vanishing in the presence of Yukawa couplings to the Higgs field. When applied to a flat direction of the MSSM with VEV w, a logarithmically running correction of order h 2 H 2 w 2 arises, where h is a Yukawa coupling. This is of particular importance for D-term inflation or models endowed with a Heisenberg symmetry, where Hubble scale mass terms for flat directions from supergravity corrections are absent.

Bilepton effects on theWWV*vertex in the 331 model with right-handed neutrinos via aSUL(2)×UY(1)covariant quantization scheme

In a recent paper [J. Montano, F. Ram\'{\i}rez-Zavaleta, G. Tavares-Velasco, and J. J. Toscano, Phys. Rev. D 72, 055023 (2005).], we investigated the effects of the massive charged gauge bosons (bileptons) predicted by the minimal 331 model on the off-shell vertex $WW{V}^{*}$ ($V=\ensuremath{\gamma}$, $Z$) using a $S{U}_{L}(2)\ifmmode\times\else\texttimes\fi{}{U}_{Y}(1)$ covariant gauge-fixing term for the bileptons. We proceed along the same lines and calculate the effects of the gauge bosons predicted by the 331 model with right-handed neutrinos. It is found that the bilepton effects on the $WW{V}^{*}$ vertex are of the same order of magnitude as those arising from the standard model and several of its extensions, provided that the bilepton mass is of the order of a few hundred of GeVs. For heavier bileptons, their effects on the $WW{V}^{*}$ vertex are negligible. The behavior of the form factors at high energies is also discussed as it is a reflection of the gauge invariance and gauge independence of the $WW{V}^{*}$ Green function obtained via our quantization method.

New Covariant Gauges in String Field Theory

A single-parameter family of covariant gauge fixing conditions in bosonic string field theory is proposed. It is a natural string field counterpart of the covariant gauge in the conventional gauge theory, which includes the Landau gauge as well as the Feynman (Siegel) gauge as special cases. The action in the Landau gauge is largely simplified in such a way that numerous component fields have no derivatives in their kinetic terms and appear in at most quadratic in the vertex.

The QCD partition function in the external field in the covariant gauge

The QCD partition function in the external stationary gluomagnetic field is computed in the third order in external field invariants in arbitrary dimension and arbitrary covariant gauge. The contributions proportional to third order invariants in gluon field strength are shown to be dependent on covariant quantum gauge fixing parameter \alpha