BRS Transformations Research Articles

Supersymmetry, chiral symmetry and the generalized BRS transformation in lattice formulations of 4D [formula omitted] SYM

In the context of the lattice regularization of the four-dimensional N=1 supersymmetric Yang–Mills theory (4D N=1 SYM), we formulate a generalized BRS transformation that treats the gauge, supersymmetry (SUSY), translation and axial U(1) (U(1)A) transformations in a unified way. A resultant Slavnov–Taylor identity or the Zinn-Justin equation gives rise to a strong constraint on the quantum continuum limit of symmetry breaking terms with the lattice regularization. By analyzing the implications of the constraint on operator-mixing coefficients in the SUSY and the U(1)A Ward–Takahashi (WT) identities, we prove to all orders of perturbation theory in the continuum limit that, (i) the chiral symmetric limit implies the supersymmetric limit and, (ii) a three-fermion operator that might potentially give rise to an exotic breaking of the SUSY WT identity does not emerge. In previous literature, only a naive or incomplete treatment on these points can be found. Our results provide a solid theoretical basis for lattice formulations of the 4D N=1 SYM.

Pure spinor approach to type IIA superstring sigma models and free differential algebras

This paper considers the Free Differential Algebra and rheonomic parametrization of type IIA Supergravity, extended to include the BRS differential and the ghosts. We consider not only the ghosts lambda's of supersymmetry but also the ghosts corresponding to gauge and Lorentz transformations. In this way we can derive not only the BRS transformations of fields and ghosts but also the standard pure spinor constraints on lambda's. Moreover the formalism allows to derive the action for the pure spinor formulation of type IIA superstrings in a general background, recovering the action first obtained by Berkovits and Howe.

Quantum Master Equation for QED in Exact Renormalization Group

Recently, one of us (H. S.) gave an explicit form of the Ward-Takahashi identity for the Wilson action of QED. We first rederive the identity using a functional method. The identity makes it possible to realize the gauge symmetry even in the presence of a momentum cutoff. In the cutoff dependent realization, the nilpotency of the BRS transformation is lost. Using the Batalin-Vilkovisky formalism, we extend the Wilson action by including the antifield contributions. Then, the Ward-Takahashi identity for the Wilson action is lifted to a quantum master equation, and the modified BRS transformation regains nilpotency. We also obtain a flow equation for the extended Wilson action.

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.

Slavnov-Taylor1.0: A Mathematica package for computation in BRST formalism

Slavnov-Taylor is a Mathematica package which allows us to perform automatic symbolic computation in BRST formalism. This article serves as a self-contained guide to prospective users, and indicates the conventions and approximations used. Program summary Title of program: Slavnov-Taylor Catalogue identifier:ADSS Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSS Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Programming Language: Mathematica 4.0 Platform: Any platform supporting Mathematica 4.0 Computers tested on: Pentium PC Operating systems under which the program has been tested: Linux Memory required to execute: Minimal: 1.254.784 bytes, Standard: 1.281.248 bytes No. of bytes in distributed program, including test data, etc.: 8368 bytes Distribution format: tar gzip file Keywords: Slavnov–Taylor, BRST, Mathematica Nature of physical problem: Symbolic computation in the Slavnov–Taylor formalism for gauge theories in 4-dimensional space-time based on a semi-simple compact Lie group for a general BRS transformations. Restrictions on the complexity of the problem: Only matter in the adjoint is allowed. Typical running time: less than one second

APPLICATION OF FINITE FIELD-DEPENDENT BRS TRANSFORMATIONS TO PROBLEMS OF THE COULOMB GAUGE

We discuss the Coulomb propagator in the formalism developed recently in which we construct the Coulomb gauge path-integral by correlating it with the well-defined Lorentz gauge path-integrals through a finite field-dependent BRS transformation. We discover several features of the Coulomb gauge from it. We find that the singular Coulomb gauge has to be treated as the gauge parameter λ → 0 limit. We further find that the propagator so obtained has good high energy behavior[Formula: see text] for λ ≠ 0 and ∊ ≠ 0. We also find that the behavior of the propagator so obtained is sensitive to the order of limits k0→ ∞, λ → 0 and ∊ → 0; so that these have to be handled carefully in a higher loop calculation. We show that we can arrive at the result of Cheng and Tsai for the ambiguous two-loop Feynman integrals without the need for an extra ad hoc regularization and within the path integral formulation.

CONNECTING GREEN FUNCTIONS IN AN ARBITRARY PAIR OF GAUGES AND AN APPLICATION TO PLANAR GAUGES

We establish a finite field-dependent BRS transformation that connects the Yang–Mills path integrals with the Faddeev–Popov effective actions for an arbitrary pair of gauges F and F'. We establish a result that relates an arbitrary Green function (either a primary one or that of an operator) in an arbitrary gauge F' to those in gauge F that are compatible to the ones in gauge Fby its construction. This is because the construction preserves expectation values of gauge-invariant observables. We establish parallel results also for the planar gauge-Lorentz gauge connection.

Supersymmetry transformation of quantum fields

If supersymmetry is realized non-linearly as is the case when auxiliary fields are eliminated and/or one works in the Wess–Zumino gauge it is usually incorporated in terms of BRS transformations and Slavnov–Taylor identities. On the vertex functional susy transformations act even non-locally. Furthermore, the gauge fixing term breaks supersymmetry. In the present paper we clarify in which sense supersymmetry is still a symmetry of the system and how it is realized on the level of quantum fields. We treat the Wess–Zumino model as an example for chiral models, SQED and massless SYM as prototypes of gauge theories.

Interpolating gauges and the importance of a careful treatment of ?-term

We consider the use of interpolating gauges (with a gauge function F[A;α]) in gauge theories to connect the results in a set of different gauges in the path-integral formulation. We point out that the results for physical observables are very sensitive to the epsilon term that we have to add to deal with singularities and thus it cannot be left out of a discussion of gauge-independence generally. We further point out, with reasons, that the fact that we can ignore this term in the discussion of gauge independence while varying of the gauge parameter in Lorentz-type covariant gauges is an exception rather than a rule. We show that generally preserving gauge-independence as α is varied requires that the e-term has to be varied with α. We further show that if we make a naive use of the (fixed) epsilon term $$-i\in \int d^{4}x[\frac{1}{2}A^2-{\bar c}c]$$ (that is appropriate for the Feynman gauge) for general interpolating gauges with arbitrary parameter values [i.e. α], we cannot preserve gauge independence [except when we happen to be in the infinitesimal neighborhood of the Lorentz-type gauges]. We show with an explicit example that for such a naive use of an e-term, we develop serious pathological behavior in the path-integral as α is/are varied. We point out that correct way to fix the e-term in a path-integral in a non-Lorentz gauge is by connecting the path-integral to the Lorentz-gauge path-integral with correct e-term as has been done using the finite field-dependent BRS transformations in recent years.

On the gauge independence of the S-matrix

The S-matrix is invariant with respect to the variation of any (global) parameter involved in the gauge fixing conditions, if that variation is accompanied by a certain redefinition of the basis of polarization vectors. Renormalizability of the underlying gauge theory is not required. The proof is nonperturbative and, using the `extended' BRS transformation, quite simple.

Finite field-dependent BRS transformation and axial gauges

Finite field-dependent BRS transformation and axial gauges

Exact BRS Symmetry Realized along the Renormalization Group Flow

Using the average action defined with a continuum analog of the block spin transformation, we show the presence of gauge symmetry along the Wilsonian renormalization group flow. As a reflection of the gauge symmetry, the average action satisfies the quantum master equation(QME). We show that the quantum part of the master equation is naturally understood once the measure contribution under the BRS transformation is taken into account. Furthermore an effective BRS transformation acting on macroscopic fields may be defined from the QME. The average action is explicitly evaluated in terms of the saddle point approximation up to one-loop order. It is confirmed that the action satisfies the QME and the flow equation.

A SUPERSPACE FORMULATION OF ABELIAN ANTISYMMETRIC TENSOR GAUGE THEORY

We apply a superspace formulation to the four-dimensional gauge theory of a massless Abelian antisymmetric tensor field of rank 2. The theory is formulated in a six-dimensional superspace using rank-2 tensor, vector and scalar superfields and their associated supersources. It is shown that BRS transformation rules of fields are realized as Euler–Lagrange equations without assuming the so-called horizontality condition in an ad hoc manner and that a generating functional [Formula: see text] constructed in the superspace reduces to that of the ordinary gauge theory of Abelian rank-2 antisymmetric tensor field. The WT identity for this theory is derived by making use of the superspace formulation and is expressed in a neat and compact form [Formula: see text].

CORRECT TREATMENT OF $\bm{{1\over (\eta\cdot{\lc{k}})^{\lc{p}}}}$ SINGULARITIES IN THE AXIAL GAUGE PROPAGATOR

The propagators in axial-type, light-cone and planar gauges contain [Formula: see text]-type singularities. These singularities have generally been treated by inventing prescriptions for them. In this work, we propose an alternative procedure for treating these singularities in the path integral formalism using the known way of treating the singularities in Lorentz gauges. To this end, we use a finite field-dependent BRS transformation that interpolates between Lorentz-type and the axial-type gauges. We arrive at the ε-dependent tree propagator in the axial-type gauges. We examine the singularity structure of the propagator and find that the axial gauge propagator so constructed has no spurious poles (for real k). It however has a complicated structure in a small region near η·k=0. We show how this complicated structure can effectively be replaced by a much simpler propagator.

Relating Green’s functions in axial and Lorentz gauges using finite field-dependent BRS transformations

We use finite field-dependent BRS transformations (FFBRS) to connect the Green functions in a set of two otherwise unrelated gauge choices. We choose the Lorentz and the axial gauges as examples. We show how the Green functions in axial gauge can be written as a series in terms of those in Lorentz gauges. Our method also applies to operator Green’s functions. We show that this process involves another set of related FFBRS transformations that is derivable from infinitesimal FFBRS. We suggest possible applications.

Exact symmetries realized on the renormalization group flow

We show that symmetries are preserved exactly along the (Wilsonian) renormalization group flow, though the IR cutoff deforms concrete forms of the transformations. For a gauge theory the cutoff dependent Ward–Takahashi identity is written as the master equation in the antifield formalism: one may read off the renormalized BRS transformation from the master equation. The Maxwell theory is studied explicitly to see how it works. The renormalized BRS transformation becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a generic global symmetry. The master equation considered for the chiral symmetry provides us with the continuum analog of the Ginsparg–Wilson relation and the Lüscher's symmetry.

WILSON LOOP AND THE TREATMENT OF AXIAL GAUGE POLES

We consider the question of gauge invariance of the Wilson loop in the light of a new treatment of axial gauge propagator proposed recently based on a finite field-dependent BRS (FFBRS) transformation. We remark that under the FFBRS transformation as the vacuum expectation value of a gauge-invariant observable remains unchanged, our prescription automatically satisfies the Wilson loop criterion. Furthermore, we give an argument for direct verification of the invariance of Wilson loop to O(g4) using the earlier work by Cheng and Tsai. We also note that our prescription preserves the thermal Wilson loop to O(g2).

GREEN'S FUNCTIONS IN AXIAL- AND LORENTZ-TYPE GAUGES AND BRS TRANSFORMATIONS

A formula that relates the Green functions in axial-type gauges and Lorentz type gauges is established in a simpler manner. This result sheds further light on some of our earlier results using field transformations (a finite field-dependent BRS transformation) that relates these gauges. An indirect way of looking at these field transformations is presented. An example of such results is given and applications are indicated.

A DERIVATION OF THE CORRECT TREATMENT OF ${1\over (\eta\cdot {\lc k})^{\lc p}}$ SINGULARITIES IN AXIAL GAUGES

We use the earlier results on the correlations of axial gauge Green's functions and the Lorentz gauge Green's functions obtained via finite field-dependent BRS transformations to study the question of the correct treatment of [Formula: see text]-type singularities in the axial gauge boson propagator. We show how the known treatment of the [Formula: see text]-type singularity in the Lorentz-type gauges can be used to write down the axial propagator via field transformation. We examine the singularity structure of the latter and find that the axial propagator so constructed has no spurious poles, but a complex structure near [Formula: see text]. We also give the form of the much simpler propagator which can effectively replace the complicated structure near the region [Formula: see text].

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 ξ 0 of the gauge parameter ξ. The dependence of Green functions on ξ 0 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 ξ 0 of the gauge parameter, and, hence, is not ξ 0-independent as required.