Unitary scalar-vector mixing in the <Emphasis Type="Italic">ξ</Emphasis> gauge

The effect of the unitary mixing of scalar and vector fields is considered in the ξ gauge. For this effect to emerge, it is necessary that the vector current not be conserved; in the ξ gauge, there arise additional complications because of the presence of an unphysical scalar field. Solutions to the Dyson-Schwinger equations are obtained, and the renormalization of complete propagators is investigated. The use of the Ward identity, which relates a few different Green's functions, is a key point in performing this renormalization. It is shown that the dependence on the gauge parameter ξ disappears in the renormalized matrix element.     

Divergence cancellations in the renormalization of electroweak couplings

Cancellations of nonuniversal ultraviolet divergent contributions to the renormalization of couplingsvlW,vvZ,llZ (in the lowest nontrivial order) in theunitary (U) gauge of the Weinberg-Salam model are reconsidered. The expansion ofU-gauge Feynman diagrams into a set of simpler “secondary” graphs devised some time ago by Kummer and Lane is employed. It is shown that this approach reveals details of the cancellation mechanism which have not been discussed in previous treatments. The connection of cancellations in question with underlying gauge symmetry and the corresponding Higgs mechanism thus becomes more transparent.     

On the renormalization of the charm quartet model

Some aspects of the renormalization program for the charm quartet model are discussed, with special emphasis on the role played by the Cabibbo angle. The cancellation of divergences in the W-quarks and Higgs-quarks sector is examined by explicit calculations at the one-loop level, both in the unitary and 't Hooft-Feyman gauges. The main analysis is based on a renormalization scheme which allows for the most general counter-terms generated by the Yukawa couplings. A second approach, equivalent at the S-matrix level, is briefly discussed. As a byproduct of our work we verify that the standard perturbative analysis leads to the same expressions for the coefficients of the divergent parts in the renormalization of the gauge coupling g₀ as a recent current algebra formulation of the radiative corrections.     

Infrared fixed point of the 12-fermion SU(3) gauge model based on 2-lattice Monte Carlo renomalization-group matching

I investigate an SU(3) gauge model with 12 fundamental fermions. The physically interesting region of this strongly coupled system can be influenced by an ultraviolet fixed point due to lattice artifacts. I suggest to use a gauge action with an additional negative adjoint plaquette term that lessens this problem. I also introduce a new analysis method for the 2-lattice matching Monte Carlo renormalization group technique that significantly reduces finite volume effects. The combination of these two improvements allows me to measure the bare step scaling function in a region of the gauge coupling where it is clearly negative, indicating a positive renormalization group β function and infrared conformality.     

Adler–Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories

We reconsider the Adler–Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin–Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and we identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.     

Gauge coupling renormalisation with several U(1) factors

We examine the renormalisation of gauge coupling constants in theories with a G × U(1)^N gauge group (which appears to be the gauge symmetry of many possible superstring vacua). In general, the abelian gauge bosons mix among themselves, so a correct renormalisation requires including this mixing in the evolution of the gauge couplings. We present general results and note that the mixing is scale independent to all orders if the renormalization group trajectory passes through a unification point. We discuss the cases of one loop and two loops explicitly. An example, based on a possible superstring-inspired model, is given.     

The Structure of Renormalization Hopf Algebras for Gauge Theories I: Representing Feynman Graphs on BV-Algebras

We study the structure of renormalization Hopf algebras of gauge theories. We identify certain Hopf subalgebras in them, whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms. This can be understood physically as wave function renormalization and renormalization of the coupling constants, respectively. After taking into account the Slavnov–Taylor identities for the couplings as generators of a Hopf ideal, we find Hopf subalgebras in the corresponding quotient as well. In the second part of the paper, we explain the origin of these Hopf ideals by considering a coaction of the renormalization Hopf algebras on the Batalin-Vilkovisky (BV) algebras generated by the fields and couplings constants. The so-called classical master equation satisfied by the action in the BV-algebra implies the existence of the above Hopf ideals in the renormalization Hopf algebra. Finally, we exemplify our construction by applying it to Yang–Mills gauge theory.     

A proof of the irreversibility of renormalization group flows in four dimensions

We present a proof of the irreversibility of renormalization group flows, i.e. the c-theorem for unitary, renormalizable theories in four (or generally even) dimensions. Using Ward identities for scale transformations and spectral representation arguments, we show that the c-function based on the trace of the energy-momentum tensor (originally suggested by Cardy) decreases monotonically along renormalization group trajectories. At fixed points this c-function is stationary and coincides with the coefficient of the Euler density in the trace anomaly, while away from fixed points its decrease is due to the decoupling of positive-norm massive modes.     

Renormalization Proof for Spontaneously Broken Yang–Mills Theory with Flow Equations

In this paper we present a renormalizability proof for spontaneously broken SU(2) gauge theory. It is based on Flow Equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU(2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based.     

Migdal renormalization group approach to deconfinement phase transition

The Migdal renormalization group approach is applied to a finite temperature lattice gauge theory. Imposing the periodic boundary condition in the timelike orientation, the phase structure of the finite temperature lattice gauge system with a gauge groupG in (d+1)-dimensional space is determined by two kinds of recursion equations, describing spacelike and timelike correlations, respectively. One is the recursion equation for ad-dimensional gauge system with the gauge groupG, and the other corresponds to ad-dimensional spin system for which the effective theory is described by the nearest neighbor interaction of the Wilson lines. Detailed phase structure is investigated for theSU(2) gauge theory in (3+1)-dimensional space. Deconfinement phase transition is obtained. Using the recursion equation for the three dimensional spin system of the Wilson lines, it is shown that the flow of the renormalization group trajectories leads to a phase transition of the three dimensional Ising model.     

Parity violating Compton amplitude in unified theories

The leading contribution of O(αG_F) to the parity violating piece of the electron Compton amplitude is calculated in the Weinberg Salam unified theory and is shown to vanish. The reason for this is the nonexistence of parity violating charge and electric dipole moment. In the computation, a problem with the usual gauge fixing terms in the Weinberg model was encountered and resolved; the gauge fixing terms made it impossible to satisfy the electromagnetic Ward identity off mass-shell. The resolution of this problem has led to changed Feynman rules resulting in fewer graphs and gauge dependent vertices. Further features are a renormalization of the parity violating infinities, and a discussion of how the γ₅-algebra and n-dimensional regularization can coexist peacefully.     

On gauge independence in quantum gravity

We prove the gauge independence of the one-loop path integral for on-shell quantum gravity obtained in the framework of the modified geometric approach. We use a projector on pure gauge directions constructed via the quadratic form of the action. This enables us to formulate the proof entirely in terms of determinants of non-degenerate elliptic operators without reference to any renormalization procedure. The role of the rotation of the conformal factor in achieving gauge independence is discussed. Direct computations on CP² in a general three-parameter background gauge are presented. We comment on the gauge dependence of previous results by Ichinose.     

General theory of renormalization of gauge invariant operators

We study the question of renormalization of gauge invariant operators in the gauge theories. Our discussion applies to gauge invariant operators of arbitrary dimensions and tensor structure. We show that the gauge noninvariant (and ghost) operators that mix with a given set of gauge invariant operators form a complete set of local solutions of a functional differential equation. We show that this set of gauge noninvariant operators together with the gauge invariant operators close under renormalization to all orders. We obtain a complete set of local solutions of the differential equation. The form of these solutions has recently been conjectured by Kluberg Stern and Zuber. With the help of our solutions, we show that there exists a basis of operators in which the gauge noninvariant operators “decouple” from the gauge invariant operators to all orders in the sense that eigenvalues corresponding to the eigenstates containing gauge invariant operators can be computed without having to compute the full renormalization metrix. We further discuss the substructure of the renormalization matrix.     

Slavnov-Taylor identities in Weinberg's renormalization scheme

Weinberg's renormalization scheme, although more cumbersome from the computational point of view, has a more immediate physical interpretation than 't Hooft's minimal renormalization scheme. It is expected to lead to smaller higher-order coefficients in a perturbative approach to QCD. However, it a priori violates the Slavnov-Taylor identities. A complete study of this problem is performed, both theoretically and for the practitioner's sake. The ambiguities in the choice of the tensorial basis of some of the QCD vertices, as well as the dependence in the gauge parameter are used for substantiating, eventually, the Slavnov-Taylor identities in this renormalization scheme.     

Modified action glueballs

The mass of the 0⁺ glueball in 4-dimensional lattice gauge theory with a mixed SU(2)-SO(3) action is obtained via Monte Carlo. We work in a region far from the critical end point in the phase diagram, with an action partly motivated by renormalization group flows in the Migdal-Kadanoff approximation. A large-N resummation of perturbation theory is used to show that the mass gap scales as predicted by the perturbative renormalization group. Independent of this, our results show that the ratio of the glueball mass to the square root of the string tension, obtained from a previous Monte Carlo, is a renormalization group invariant.     

Superaxial Gauges

Supersymmetric gauge theories can be suitably quantized in non-supersymmetric “superaxial” gauges without abolishing the basic advantages of the superfield technique. In this review the state of the art is presented. It includes the proof of renormalization and the proof of gauge independence and supersymmetry of observable physical quantities.     

A remark on divergence cancellations in the Weinberg-Salam model

An alternative proof of ultraviolet divergence cancellation (at the one-loop level) in certain ratios of renormalized coupling constants in the Weinberg-Salam model is suggested. Working in the unitary gauge, we use a simple coordinate-space method for manipulating Feynman integrals proposed by Kummer and Lane. In a particular example it is shown that this method provides an extremely useful tool for solving the problem.Dedicated to Academician Václav Votruba on the occasion of his seventieth birthday.