Slavnov Identities Research Articles

Landau gauge formalism for non-Abelian gauge theories

A local operator formulation of non-Abelian gauge theories in the Landau gauge is presented and discussed. The formalism involves the usual gauge fields ${\stackrel{\ensuremath{\rightarrow}}{A}}_{\ensuremath{\mu}}$, matter fields, unphysical ghost fields, and a further multiplet of unphysical local scalar fields $\stackrel{\ensuremath{\rightarrow}}{B}$. The gauge-fixing term in the Lagrangian is $\stackrel{\ensuremath{\rightarrow}}{B}\ifmmode\cdot\else\textperiodcentered\fi{}(\ensuremath{\partial}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{A})$, which replaces the usual term $(\frac{1}{\ensuremath{\alpha}}){(\ensuremath{\partial}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{A})}^{2}$ characteristic of the generalized Lorentz gauges. The $\stackrel{\ensuremath{\rightarrow}}{B}$ field, formally the limit of $(\frac{1}{\ensuremath{\alpha}})\ensuremath{\partial}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{A}$ for $\ensuremath{\alpha}\ensuremath{\rightarrow}0$, thus provides a local momentum operator which is canonically conjugate to ${\stackrel{\ensuremath{\rightarrow}}{A}}_{0}$, and generates the Landau-gauge relation $\ensuremath{\partial}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{A}=0$ as a field equation. Both operator and functional methods are used to deduce the transversality conditions, Slavnov identities, and renormalization-group equations obeyed by the Green's functions. A functional formalism for vertex functions is presented, and it is shown that these functions are well defined in spite of the fact that the $\mathrm{AA}$ propagator has no inverse and the $\mathrm{BB}$ propagator vanishes. The gauge-field vertex functions are shown to be the $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ limits of those in the Lorentz gauges.

Matter coupling and BRS transformations with auxiliary fields in supergravity

The formulation of supergravity which includes the recently found set of minimal auxiliary fields that close the algebra is coupled to the Maxwell vector multiplet. The BRS transformations and Slavnov identities for this formulation of pure supergravity are also presented. Considerable simplification is obtained with respect to the earlier forms of both these results.

Slavnov–’t Hooft identities in Mandelstam’s formalism

In Mandelstam’s gauge-independent quantization formalism, Slavnov identities are shown to originate from the fact that auxiliary Green’s functions in various gauges (Feynman gauge, Landau gauge, etc.) satisfy the same fundamental equation. Furthermore, we have the practically important result that the infinite number of Slavnov identities are obtained in concise and concrete form. With the help of this form, we are able to easily derive various ’t Hooft identities, which assure us of the gauge independence and the unitarity of the S matrix.

Gauge fixing and mass renormalization in the lattice gauge theory

The lattice gauge theory proposed by Wilson is discussed. Gauge fixing is defined for the lattice theory, and it is shown that gauge fixing is done in this theory solely for calculational purposes. The gauge-fixing method is used to study the mass renormalization of the gauge field quantum. An explicit calculation is done to lowest order which shows that there is no mass renormalization. This same result is proved to all orders in perturbation theory using the Slavnov identity.

Renormalization of higher-derivative quantum gravity

Gravitational actions which include terms quadratic in the curvature tensor are renormalizable. The necessary Slavnov identities are derived from Becchi-Rouet-Stora (BRS) transformations of the gravitational and Faddeev-Popov ghost fields. In general, non-gauge-invariant divergences do arise, but they may be absorbed by nonlinear renormalizations of the gravitational and ghost fields (and of the BRS transformations). Fortunately, these artifactual divergences may be eliminated by letting the coefficient of the harmonic gauge-fixing term tend to infinity, thus considerably simplifying the renormalization procedure. Coupling to other renormalizable fields may then be handled in a straightforward manner.

Spontaneous symmetry breaking of Slavnov symmetry A restriction on the gauge condition

It is shown that the condition of vanishing vacuum expectation value of the gauge operator, using a gauge-fixing term quadratic in this operator, does not necessarily follow from the Slavnov identity, due to a possible spontaneous breakdown of the Slavnov symmetry. For a consistent renormalization such a condition may have to be imposed order by order in perturbation theory, depending on the choice of the gauge. This restriction on non-singular gauges is particularly relevant for the discussion of the spontaneously broken realizations of the gauge symmetry.

Proof of the Slavnov identity in a non-abelian Higgs-Kibble model containing massless fields

An SU(2) Higgs-Kibble model containing massless fields is formulated within the BPHZ framework utilizing the auxiliary mass methods of Lowenstein and Zimmermann. Normal product Green functions are defined and their properties are reviewed. Using these properties the methods of Becchi, Rouet and Stora in the massive case are shown to be applicable in proving the Slavnov identity for this model.

Proof of the Slavnov identities in the renormalized Georgi-Glashow model

Using the version of the quantum action principle for theories involving both massive and massless particles, the proof of the Slavnov identities given by Becchi, Rouet and Stora is extended to the Georgi-Glashow O(3) gauge model involving massless fields.

Renormalization of gauge theories

Gauge theories are characterized by the Slavnov identities which express their invariance under a family of transformations of the supergauge type which involve the Faddeev Popov ghosts. These identities are proved to all orders of renormalized perturbation theory, within the BPHZ framework, when the underlying Lie algebra is semisimple and the gauge function is chosen to be linear in the fields in such a way that all fields are massive. An example, the SU2 Higgs Kibble model is analyzed in detail: the asymptotic theory is formulated in the perturbative sense, and shown to be reasonable, namely, the physical S operator is unitary and independent from the parameters which define the gauge function.

Supersymmetric BPHZ renormalization: (II). Supersymmetric extension of the pure Yang-Mills model

We generalize the supersymmetry way of performing the BPHZ renormalization in introduced in paper (I) to cases including massless fields. This is used to prove the Slavnov identity in the supersymmetric extension of the pure Yang-Mills theory.

Pure massless electrodynamics in Veltman's gauge

To show how the method developed by C. Becchi, R. Stora and the present author to prove Slavnov's identities in gauge theories works in Abelian cases including a nonlinear gauge without any discrete symmetry, a specific example is worked out, exhibiting the details of the technical procedure.

Supersymmetric BPHZ renormalization: (I). Supersymmetric extension of QED

After having introduced a supersymmetric way of performing the BPHZ subtraction we prove the Slavnov identity together with gauge invariance for the case of extended QED, using the Becchi-Rouet-Stora method. This model is worked out in view of a possible generalization to non-Abelian cases.

Auxiliary-mass formulation of the pure Yang-Mills model

Abstract The massless Yang-Mills model is formulated within the BPHZ framework. Normal products are introduced which satisfy normalization conditions at both vanishing and non-vanishing mass values. The Slavnov identity is derived using the convenient properties of these normal products. Slavnov invariant integrated normal products are constructed and used to derive the Callan-Symanzik (renormalization group) equation corresponding to scaling of the auxiliary mass parameter used to normalize the Green functions.

Perturbation theory and renormalization of supersymmetric Yang-Mills theories

We solve the problem of introducing, in supersymmetric Yang-Mills Lagrangians, a supersymmetric gauge breaking term and a Faddeev-Popov ghost interaction term. The resulting Lagrangian turns out to be invariant under a global symmetry transformation which is the supersymmetric extension of the Slavnov symmetry. We show that the complete analysis of all primitively divergent supergraphs ensures, in conjunction with the Slavnov identities, the renormalizability of the theory, once a supersymmetric and gauge invariant regularizing procedure has been introduced. We find that the simplest regularizing procedure is a generalization of the higher covariant derivatives method. In the case of interaction with matter fields we prove that no mass counter term is needed, in exact analogy with the model without gauge fields. Finally we show that, in the Abelian situation, a supersymmetric mass term for the vector multiplet can be introduced without spoiling the renormalizability, thus providing the supersymmetric extension of massive vector bosons theories.

Renormalization of the abelian Higgs-Kibble model

This article is devoted to the perturbative renormalization of the abelian Higgs-Kibble model, within the class of renormalizable gauges which are odd under charge conjugation. The Bogoliubov Parasiuk Hepp-Zimmermann renormalization scheme is used throughout, including the renormalized action principle proved by Lowenstein and Lam. The whole study is based on the fulfillment to all orders of perturbation theory of the Slavnov identities which express the invariance of the Lagrangian under a supergauge type family of non-linear transformations involving the Faddeev-Popov ghosts. Direct combinatorial proofs are given of the gauge independence and unitarity of the physicalS operator. Their simplicity relies both on a systematic use of the Slavnov identities as well as suitable normalization conditions which allow to perform all mass renormalizations, including those pertaining to the ghosts, so that the theory can be given a setting within a fixed Fock space. Some simple gauge independent local operators are constructed.

Ward-Takahashi-Slavnov identities for one-particle irreducible vertices

Ward -- Takahashi -- Slavnov identities were recently formulated by Lee in terms of generating functionals for non-Abelion gauge theories. These results are analyzed and extended to the most general gauge theories. (LBS)