Radiative Corrections to Non-Abelian Gauge Theory in Homogeneous Self-Dual Backgrounds

We study the Euclidean version of non-Abelian gauge theories in homogeneous background fields with due regard of negative and zero modes of the gauge field kernel. For general gauge group and general background but without external currents we prove a relation between the propagators belonging to different gauge fixings. Specializing to a self-dual homogeneous background we derive nonperturbatively the propagators. Next we calculate the matter field contribution to the renormalized polarization tensor using a modified dimensional regularization scheme. We prove the validity of a modified Ward identity.     

The background scale Ward identity in quantum gravity

We show that with suitable choices of parametrization, gauge fixing and cutoff, the anomalous variation of the effective action under global rescalings of the background metric is identical to the derivative with respect to the cutoff, i.e. to the beta functional, as defined by the exact RG equation. The Ward identity and the RG equation can be combined, resulting in a modified flow equation that is manifestly invariant under global background rescalings.     

Infinite order radiative corrections and renormalization

By means of the Ward identity and the correlation expansion derived earlier, we calculate virtual radiative corrections to an infinite order. Partial factorization of the correlation effects, as in the simple exchange processes, makes infinite summation possible. Furthermore, because the Ward identity is satisfied order by order and because we are able to carry out the infinite order summation and obtain a closed form, renormalization turns out to be very simple and transparent. Here the calculations are performed for a scalar: ?²: φ model, but are easily generalized to other similar models. We also indicate why this rearrangement of the ordinary perturbation expansion is suitable for strong coupling theories, ordinary local field theories as well as dual models.     

Six-Dimensional Superconformal Field Theories from Principal 3-Bundles over Twistor Space

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose–Ward transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2-bundles in that the so-called Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3-bundles such as the relevant gauge transformations. We thus arrive at the non-Abelian differential cohomology that describes principal 3-bundles with connective structure.     

The axial gauge and supersymmetry

We examine the supersymmetry Ward identity for supersymmetric Yang-Mills theories in the axial gauge. In the pure N = 1 (no matter) case the Ward identity leads to supersymmetric counterterms to all orders. This result does not survive the introduction of matter fields, however, and we therefore conclude that the gauge is not useful in the context of supersymmetry.     

Symmetries in a Constrained System with a Singular Higher-Order Lagrangian

A simple algorithm to construct the generator of gauge transformation for a constrained canonical system with a singular higher-order Lagrangian in field theories is developed. Based on phase-space generating functional of Green function for such a system, the generalized canonical Ward identities under the non-local transformation have been deduced. For the gauge-invariant system, based on configuration-space generating functional, the generalized Ward identities under the non-local transformation have been also derived.The conservation laws are deduced at the quantum level. The applications of the above results to the gauge invariance massive vector field and non-Abelian Chern–Simons(CS) theories with higher-order derivatives are given, a new form of gauge-ghost proper vertices, and Ward–Takahashi identity under BRS transformation and BRS charge at the quantum level are obtained. In the canonical formulation one does not need to carry out the integration over canonical momenta in phase-space path integral as usually performed.     

Gauge fixing,BRS invariance and Ward identities for randomly stirred flows

The Galilean invariance of the Navier–Stokes equation is shown to be akin to a global gauge symmetry familiar from quantum field theory. This symmetry leads to a multiple counting of infinitely many inertial reference frames in the path integral approach to randomly stirred fluids. This problem is solved by fixing the gauge, i.e., singling out one reference frame. The gauge fixed theory has an underlying Becchi–Rouet–Stora (BRS) symmetry which leads to the Ward identity relating the exact inverse response and vertex functions. This identification of Galilean invariance as a gauge symmetry is explored in detail, for different gauge choices and by performing a rigorous examination of a discretized version of the theory. The Navier–Stokes equation is also invariant under arbitrary rectilinear frame accelerations, known as extended Galilean invariance (EGI). We gauge fix this extended symmetry and derive the generalized Ward identity that follows from the BRS invariance of the gauge-fixed theory. This new Ward identity reduces to the standard one in the limit of zero acceleration. This gauge-fixing approach unambiguously shows that Galilean invariance and EGI constrain only the zero mode of the vertex but none of the higher wavenumber modes.     

On dynamical gluon mass generation

The effective gluon propagator constructed with the pinch technique is governed by a Schwinger-Dyson equation with special structure and gauge properties, that can be deduced from the correspondence with the background field method. Most importantly the non-perturbative gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions, a property which allows for a meanigfull truncation. A linearized version of the truncated Schwinger-Dyson equation is derived, using a vertex that satisfies the required Ward identity and contains massless poles. The resulting integral equation, subject to a properly regularized constraint, is solved numerically, and the main features of the solutions are briefly discussed.     

Dilatation invariance and spontaneous symmetry breakdown in dimensional renormalization

This paper completes a systematic study of the renormalization of scale invariant models with spontaneous symmetry breakdown discussing an effective scheme for the construction of Feynman amplitudes. From a general point of view the original scale invariance of the models is accounted for by constraining the dilation anomalies with a minimality criterion analyzed in a previous paper. This criterion is basd on a local Ward identity where the anomalies appear oupled to a spurion field. Here the general framework is adapted to a dimensional renormalization scheme and it is shown to be equivalent to a scale invariant choice of all counterterms and as it happens in minimal schemes, to the mass independence of the renormalization constants, thereby allowing a derivation of a renormalization group equation with standard (mass independent) coefficients. The whole analysis refers as an example to the simplest bosonic model containing all the features common to scale invariant theories with spontaneous symmetry breakdown.     

Gauge dependence of effective potential and self-consistent dimensional reduction

The gauge dependence of the effective potential determining the Kaluza-Klein radius self-consistently is contrasted to that belonging to a gauge theory with spontaneous symmetry breaking at finite temperature. Formally the two quantities are computed in the same way and obey the same Ward identities, but a gauge-independent approximation scheme is possible only in the latter case because of the coupling between the Goldstone field and the longitudinal vector field. The connection to the Ward identities of the two theories is also obtained.     

Computation of the chiral anomaly in the bulk quantization

The bulk quantization method is used for regularizing a conventional four dimensional theory of massless fermions coupled to an external non‐Abelian gauge field and for subsequently evaluating the associated Ward identity. As a result one obtains the well‐known chiral anomaly.     

Arbitrary parameters in implicit regularization and democracy within perturbative description of 2-dimensional gravitational anomalies

We show that the Implicit Regularization Technique is useful to display quantum symmetry breaking in a complete regularization independent fashion. Arbitrary parameters are expressed by finite differences between integrals of the same superficial degree of divergence whose value is fixed on physical grounds (symmetry requirements or phenomenology). We study Weyl fermions on a classical gravitational background in two dimensions and show that, assuming Lorentz symmetry, the Weyl and Einstein Ward identities reduce to a set of algebraic equations for the arbitrary parameters which allows us to study the Ward identities on equal footing. We conclude in a renormalization independent way that the axial part of the Einstein Ward identity is always violated. Moreover whereas we can preserve the pure tensor part of the Einstein Ward identity at the expense of violating the Weyl Ward identities we may as well violate the former and preserve the latter.     

Rigid invariance as derived from BRS invariance: the abelian Higgs model

Consequences of a symmetry, e.g. relations amongst Green functions, are renormalization scheme independently expressed in terms of a rigid Ward identity. The corresponding local version yields information on the respective current. In the case of spontaneous breakdown one has to define the theory via the BRS invariance and thus to construct rigid and current Ward identity nontrivially in accordance with it. We performed this construction to all orders of perturbation theory in the abelian Higgs model as a prelude to the standard model. A technical tool of interest in itself is the use of a doublet of external scalar “background” fields. The Callan-Symanzik equation has an interesting form and follows easily once the rigid invariance is established.     

Background field formalism and construction of effective action for N = 2, d = 3 supersymmetric gauge theories

We review the background field method for three-dimensional Yang-Mills and Chern-Simons models in N = 2 superspace. Superfield proper time (heat kernel) techniques are developed and exact expressions of heat kernels for constant backgrounds are presented. The background field method and heat kernel techniques are applied for evaluating the low-energy effective actions in N = 2 supersymmetric Yang-Mills and Chern-Simons models as well as in N = 4 and N = 8 SYM theories.     

Gauge parameter dependence in the background field gauge and the construction of an invariant charge

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable — also if we consider the local Ward identity — to introduce the normalization value ξ₀ of the gauge parameter ξ. The dependence of Green functions on ξ₀ is governed by a partial differential equation in a similar manner as the dependence on the normalization point κ is governed by the RG equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization value ξ₀ of the gauge parameter, and, hence, is not ξ₀-independent as required.     

An Abelian Ward identity and the vertex corrections to the color superconducting gap

We derive an Abelian-like Ward identity in the color superconducting phase and calculate vertex corrections to the color superconducting gap. Making use of the Ward identity, we show that subleading order contributions to the gap from vertices are absent for gapped excitations.     

An Abelian Ward identity and the vertex corrections to the color superconducting gap

We derive an Abelian-like Ward identity in the color superconducting phase and calculate vertex corrections to the color superconducting gap. Making use of the Ward identity, we show that subleading order contributions to the gap from vertices are absent for gapped excitations.     

Quantal global symmetry for a gauge-invariant system

Based on the configuration-space generating functional obtained by using the Faddeev-Popov trick for a gauge-invariant system, the Ward identities for global transformation are derived. The conservation laws at the quantum level for global symmetry transformation are also deduced. A preliminary application of the present formulation to non-Abelian Chern-Simons (CS) theory is given. The Ward identity and quantal BRS charge under the BRS transformation are deduced. The quantal conserved angular momentum is obtained and the fractional spin for CS theories is discussed.