Dimensional regularization of gauge theories in spherical spacetime: Free field trace anomalies

The O(n + 1) covariant formulation of massless quantum electrodynamics in spherical spacetime is further developed to allow a calculation of the energy-momentum tensor trace anomalies for the free Dirac, electromagnetic, and SU(2) gauge fields. The principal technical development is the construction of the Faddeev-Popov ghosts for electrodynamics and SU(2) Yang-Mills theory. This construction is unconventional first in that the gauge fixing term in the Lagrangian is not a perfect square, and second because it is necessary to remove radial as well as gauge degrees of freedom from the measure of the functional integral. The ghost fields are shown to satisfy a minimal scalar field equation. The free field effective action is found to be divergent in four dimensions, and is renormalized by the inclusion in the Lagrangian of a counterterm local in the gravitational fields. The energy-momentum tensor calculated from this renormalized effective action is shown to have a trace anomaly.     

Point-splitting regularization for gauge theories: Quantum electrodynamics

As a method of regularization, point splitting has played an essential role in the recent theoretical determination of the masses of the Higgs boson and the top quark. It is the purpose of this paper to put this pointsplitting regularization on a firm basis. The result turns out to be extremely simple: replace the usual vertex factor-ieγ ^µ in quantum electrodynamics by $$ - ie(\gamma _\mu - \frac{{\not p}}{{p \cdot \delta }}\delta _\mu ),$$ wherep is the momentum of the photon line, andδ ^µ is the distance for point splitting. No additional vertices are needed.     

Higher order mean field expansions for lattice gauge theories

The leading contribution to the free energy of lattice gauge theories is evaluated in the mean field expansion to the two-loop level. The methods are general but we only deal with theU(1) case in this paper. The corrections improve the agreement with Monte Carlo calculations. We show that in order to obtain a satisfactory formalism it is necessary to include a new redundant parameter, γ, in the mean field expansion. For γ→0 we recover the usual mean field expansion whereas for γ→∞ we obtain the weak coupling expansion. Thus γ measures the amount of resummation that is done by the mean field formalism.     

Dimensional regularization and finite calculations in gauge theories

The Leading finite corrections to a lowest order relationship between renormalized coupling constants in an SU(2) × U(1) gauge model of weak and electromagnetic interactions are presented. The calculations were carried out in two different gauge invariant dimensional regularization schemes. A brief outline and comparison of the technical aspects of these methods is given. For our particular calculation we find that both yield the same finite results as expected.     

The one-loop Green’s functions of dimensionally reduced gauge theories

We apply the dimensional regularization technique as well as that by dimensional reduction to the calculation of the regularized one-loop Green’s functions ind ₀-dimensional Yang-Mills theory with real massless scalars and spinors in arbitrary (real) representations of a gauge groupG. As a particular example, the super-symmetrically regularized one-loop Green’s functions of theN=4 supersymmetric Yang-Mills model are derived.     

The double dimensional regularization and gauge theories without ghosts

A new version of dimensional regularization is introduced as calculational technique for the recently proposed gauge theory without ghosts. The method is applied to Yang-Mills theory.     

Non-abelian gauge couplings in thermal gauge field theories

Properties of the effective gauge couplings renormalized at finite temperature and density in thermal non-abelian gauge field theories are studied within one-loop approximations. Strong and severe vertex dependence is shown to come out both in the temperature and chemical potential dependences. Difficulties appearing in the perturbative calculation of physical quantities, indicated by the above disaster, are discussed. Also discussed is what insight might be gained from the present analysis into the “magnetic” screening of effective charge.     

Nonlocal regularization and spontaneously broken abelian gauge theories for an arbitrary gauge parameter

We study the nonlocal regularization for the case of a spontaneously broken abelian gauge theory in the R-gauge with an arbitrary gauge parameter . We consider a simple abelian-Higgs model with chiral couplings as an example. We show that if we apply the nonlocal regularization procedure (to construct a nonlocal theory with FINITE mass parameter) to the spontaneously broken R-gauge Lagrangian, using the quadratic forms as appearing in this Lagrangian, we find that a physical observable in this model, an analogue of the muon anomalous magnetic moment, evaluated to order O [g²] does indeed show -dependence. We then apply the modified form of nonlocal regularization that was recently advanced and studied for the unbroken non-abelian gauge theories and discuss the resulting WT identities and -independence of the S-matrix elements.     

Quadratic and quartic one-loop divergences in gauge theories and their cancellation in supergravity

The quadratic and quartic divergences cancel in N = 2 supergravity in photon-photon scattering. They are defined by modified dimensional regularization as the residues of poles at d = 2 and d = 0 dimensions, respectively. Other definitions are shown to violate general coordinate invariance and Yang-Mills invariance. 't Hooft's algorithm for the one-loop divergences is extended to the case of higher-order divergences.     

On general-relativistic and gauge field theories

The fundamental open questions of general relativity theory are the unification of the gravitational field with other fields, aiming at a unified geometrization of physics, as well as the renormalization of relativistic gravitational theory in order to obtain their self-consistent solutions. These solutions are to furnish field-theoretic particle models—a problem first discussed by Einstein. In addition, we are confronted with the issue of a coupling between gravitational and matter fields determined (not only) by Einstein's principle of equivalence, and also with the question of the geometric meaning of a gravitational quantum theory. In our view, all these problems are so closely related that they warrant a general solution. We treat mainly the concepts suggested by Einstein and Weyl.     

Anatomy of one-loop effective action in non-commutative scalar field theories

The one-loop effective action of non-commutative scalar field theory with cubic self-interaction is studied. Utilizing the worldline formulation, both the planar and non-planar parts of the effective action are computed explicitly. We find complete agreement of the result with the Seiberg–Witten limit of a string worldsheet computation and with the standard Feynman diagrammatics. We prove that, in the low-energy and large non-commutativity limit, the non-planar part of the effective action is simplified enormously and is resummable into a quadratic action of scalar open Wilson line operators. Received: 22 July 2001 / Revised version: 19 October 2001 / Published online: 7 December 2001     

Non-abelian gauge theories in a background field

Green functions of on-mass-shell transverse gauge fields in a background gauge are proved to be obtained from Green functions in the perturbative vacuum by a mere translation.     

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL (N, C) and explain the Symplectic Hecke correspondence for these systems. The text was submitted by the author in English.     

Self-consistent normal ordering of gauge field theories

Mean field theories with a real action of unconstrained fields can be self-consistently normal ordered. This leads to a considerable improvement over standard mean-field theory. This concept is applied to lattice gauge theories. We construct first an appropriate real action mean-field theory. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean-field theory are derived.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.