Gauge Fixing Term Research Articles

Gauge-invariant critical exponents for the Ginzburg-Landau model.

The critical behavior of the Ginzburg-Landau model is described in a manifestly gauge-invariant manner. The gauge-invariant correlation-function exponent is computed to first order in the 4-d and 1/n expansion, and found to agree with the ordinary exponent obtained in the covariant gauge, with the parameter alpha=1-d in the gauge-fixing term ( partial differential (mu)A(mu))(2)/2alpha.

The relationship between four-dimensional $\theta$ = $\pi$ Yang-Mills theory and the two-dimensional Wess-Zumino-Novikov-Witten model

Used the dimensional reduction in the sense of Parisi and Sourlas, the gauge fixing term of the four-dimensional Yang-Mills field without the theta term is reduced to a two-dimensional principal chiral model. By adding the θ term (θ = π), the two-dimensional principal chiral model changes into the two-dimensional level 1 Wess-Zumino-Novikov-Witten model. The non-trivial fixed point indicates that Yang-Mills theory at θ = π is a critical theory without mass gap and confinement.

Remarks on monopoles in Abelian projected continuum Yang-Mills theories

A possible mechanism accounting for monopole configurations in continuum Yang-Mills theories is discussed. The presence of the gauge fixing term is taken into account.

Renormalizing a Becchi-Rouet-Stora-Tyutin-invariant composite operator of mass dimension 2 in Yang-Mills theory

We discuss the renormalization of a BRST and anti-BRST invariant composite operator of mass dimension 2 in Yang-Mills theory with the general BRST and anti-BRST invariant gauge fixing term of the Lorentz type. The interest of this study stems from a recent claim that the non-vanishing vacuum condensate of the composite operator in question can be an origin of mass gap and quark confinement in any manifestly covariant gauge, as proposed by one of the authors. First, we obtain the renormalization group flow of the Yang-Mills theory. Next, we show the multiplicative renormalizability of the composite operator and that the BRST and anti-BRST invariance of the bare composite operator is preserved under the renormalization. Third, we perform the operator product expansion of the gluon and ghost propagators and obtain the Wilson coefficient corresponding to the vacuum condensate of mass dimension 2. Finally, we discuss the connection of this work with the previous works and argue the physical implications of the obtained results.

SYMPLECTIC EMBEDDING AND HAMILTON–JACOBI ANALYSIS OF PROCA MODEL

Following the symplectic approach we show how to embed the Abelian Proca model into a first-class system by extending the configuration space to include an additional pair of scalar fields, and compare it with the improved Dirac scheme. We obtain in this way the desired Wess–Zumino and gauge fixing terms of BRST-invariant Lagrangian. Furthermore, the integrability properties of the second-class system described by the Abelian Proca model are investigated using the Hamilton–Jacobi formalism, where we construct the closed Lie algebra by introducing operators associated with the generalized Poisson brackets.

Two-loop renormalization of tanβ and its gauge dependence

Renormalization of two-loop divergent corrections to the vacuum expectation values (v_1, v_2) of the two Higgs doublets in the minimal supersymmetric standard model, and their ratio tan beta=v_2/v_1, is discussed for general R_xi gauge fixings. When the renormalized (v_1, v_2) are defined to give the minimum of the loop-corrected effective potential, it is shown that, beyond the one-loop level, the dimensionful parameters in the R_xi gauge fixing term generate gauge dependence of the renormalized tan beta. Additional shifts of the Higgs fields are necessary to realize the gauge-independent renormalization of tan beta.

OBTAINING A LIGHT-LIKE PLANAR GAUGE

In the usual and current understanding of planar gauge choices for Abelian and non-Abelian gauge fields, the external defining vector nμ can either be space-like (n2 < 0) or time-like (n2>0) but not light-like (n2=0). In this work we propose a light-like planar gauge that consists of defining a modified gauge-fixing term, ℒ GF , whose main characteristic is a two-degree violation of Lorentz covariance arising from the fact that four-dimensional space–time spanned entirely by null vectors as basis necessitates two light-like vectors, namely nμ and its dual mμ, with n2=m2=0, n·m≠0, say, e.g. normalized to n·m=2.

Ward identities for N = 2 rigid and local supersymmetry in Euclidean space

We construct and check by explicit Feynman diagram calculations the BRST Ward identities for N = 2 rigid super Yang–Mills theory and N = 2 extended supergravity in four-dimensional Euclidean space without auxiliary fields. We use the Batalin–Vilkovisky formalism. In the supergravity case we need one new contractible pair of complex spinor fields to obtain the usual gauge-fixing term and corresponding Nielsen–Kallosh ghosts.

Algebraic Aspects of the Background Field Method

The background field method allows the evaluation of the effective action by exploiting the (background) gauge invariance, which in general yields Ward identities, i.e., linear relations among the vertex functions. In the present approach an extra gauge fixing term is introduced right at the beginning in the action and it is chosen in such a way that BRST invariance is preserved. The background effective action is considered and it is shown to satisfy both the Slavnov–Taylor (ST) identities and the Ward identities. This allows the proof of the background equivalence theorem with the standard techniques. In particular we consider a BRST doublet where the background field enters with a non-zero BRST transformation. The rationale behind the introduction of an extra gauge fixing term is that of removing the singularity of the Legendre transform of the background effective action, thus allowing the construction of the connected amplitudes generating functional Wbg. By using the relevant ST identities we show that the functional Wbg gives the same physical amplitudes as the original one we started with. Moreover we show that Wbg cannot in general be derived from a classical action by the Gell–Mann–Low formula. As a final point of the paper we show that the BRST doublet generated from the background field does not modify the anomaly of the original underlying gauge theory. The proof is algebraic and makes no use of arguments based on power-counting.

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.

QUANTUM AND CLASSICAL GAUGE SYMMETRIES IN A MODIFIED QUANTIZATION SCHEME

The use of the mass term as a gauge fixing term has been studied by Zwanziger, Parrinello and Jona-Lasinio, which is related to the nonlinear gauge [Formula: see text] of Dirac and Nambu in the large mass limit. We have recently shown that this modified quantization scheme is in fact identical to the conventional local Faddeev–Popov formula without taking the large mass limit, if one takes into account the variation of the gauge field along the entire gauge orbit and if the Gribov complications can be ignored. This suggests that the classical massive vector theory, for example, is interpreted in a more flexible manner either as a gauge invariant theory with a gauge fixing term added, or as a conventional massive nongauge theory. As for massive gauge particles, the Higgs mechanics, where the mass term is gauge-invariant, has a more intrinsic meaning. It is suggested that we extend the notion of quantum gauge symmetry (BRST symmetry) not only to classical gauge theory but also to a wider class of theories whose gauge symmetry is broken by some extra terms in the classical action. We comment on the implications of this extended notion of quantum gauge symmetry.

A FORMULATION OF THE YANG–MILLS THEORY AS A DEFORMATION OF A TOPOLOGICAL FIELD THEORY BASED ON THE BACKGROUND FIELD METHOD AND QUARK CONFINEMENT PROBLEM

By making use of the background field method, we derive a novel reformulation of the Yang–Mills theory which was proposed recently by the author to derive quark confinement in QCD. This reformulation identifies the Yang–Mills theory with a deformation of a topological quantum field theory. The relevant background is given by the topologically nontrivial field configuration, especially, the topological soliton which can be identified with the magnetic monopole current in four dimensions. We argue that the gauge fixing term becomes dynamical and that the gluon mass generation takes place by a spontaneous breakdown of the hidden supersymmetry caused by the dimensional reduction. We also propose a numerical simulation to confirm the validity of the scheme we have proposed. Finally we point out that the gauge fixing part may have a geometric meaning from the viewpoint of global topology where the magnetic monopole solution represents the critical point of a Morse function in the space of field configurations.

DUAL BRST SYMMETRY FOR QED

We show the existence of a co(dual)-BRST symmetry for the usual BRST invariant Lagrangian density of an Abelian gauge theory in two dimensions of space–time where a U(1) gauge field is coupled to the Noether conserved current (constructed by the Dirac fields). Under this new symmetry, it is the gauge-fixing term that remains invariant and the symmetry transformations on the Dirac fields are analogous to the chiral transformations. This interacting theory is shown to provide a tractable field theoretical model for the Hodge theory. The Hodge dual operation is shown to correspond to a discrete symmetry in the theory and the extended BRST algebra for the generators of the underlying symmetries turns out to be reminiscent of the algebra obeyed by the de Rham cohomology operators of differential geometry.

Quark-confinement mechanism for SU(2) Yang–Mills theory in abelian gauge

By dimensional reduction in the sense of Parisi and Sourlas (PS), the gauge fixing term in the abelian gauge of the SU(2) Yang–Mills field is reduced to a two-dimensional O(3) nonlinear \(\sigma \) model. The confinement potential is obtained from magnetic monopoles and frame fluctuations. But the source of quark confinement is frame fluctuations and not magnetic monopoles. Because the frame \(T^a\) cannot be regarded as a fixed one, the abelian projected SU(2) Yang–Mills field turns into a \({\mathrm{U(1)}} \times{\mathrm{U(1)}}\) gauge field – one group element being \(\exp (\mathrm i\varphi ^3T^3)\) with fixed frame \(T^3\), another group gauging the frame \(T^3\). The nonperturbative part \(\varpi _\mu (x) \) becomes a dynamical gauge field in two dimensions, giving rise to the short range linear potential.

Regularisation: many recipes, but a unique principle: Ward identities and normalisation conditions. The case of CPT violation in QED

We analyse the recent controversy on a possible Chern–Simons like term generated through radiative corrections in QED with a CPT violating term: we prove that, if the theory is correctly defined through Ward identities and normalisation conditions, no Chern–Simons term appears, without any ambiguity. This is related to the fact that such a term is a kind of minor modification of the gauge fixing term, and then no renormalised. The past year literature on that subject is discussed, and we insist on the fact that any absence of an a priori divergence should be explained by some symmetry or some non-renormalisation theorem.

NEW LOCAL SYMMETRY FOR QED IN TWO DIMENSIONS

A new local, covariant and nilpotent symmetry is shown to exist for the interacting BRST invariant U(1) gauge theory in two dimensions of space–time. Under this new symmetry, it is the gauge-fixing term that remains invariant and the corresponding transformations on the Dirac fields turn out to be the analogue of chiral transformations. The extended BRST algebra is derived for the generators of all the underlying symmetries, present in the theory. This algebra turns out to be the analogue of the algebra obeyed by the de Rham cohomology operators of differential geometry. Possible interpretations and implications of this symmetry are pointed out in the context of BRST cohomology and Hodge decomposition theorem.

Hodge decomposition theorem for Abelian two-form gauge theory

We show that the BRST/anti-BRST invariant 3+1 dimensional 2-form gauge theory has further nilpotent symmetries (dual BRST /anti-dual BRST) that leave the gauge fixing term invariant. The generator for the dual BRST symmetry is analogous to the co-exterior derivative of differential geometry. There exists a bosonic symmetry which keeps the ghost terms invariant and it turns out to be the analogue of the Laplacian operator. The Hodge duality operation is shown to correspond to a discrete symmetry in the theory. The generators of all these continuous symmetries are shown to obey the algebra of the de Rham cohomology operators of differential geometry. We derive the extended BRST algebra constituted by six conserved charges and discuss the Hodge decomposition theorem in the quantum Hilbert space of states.

Quantum gauge symmetry from finite field dependent BRST transformations

Using the technique of finite field dependent BRST transformations we show that the classical massive Yang–Mills theory and the pure Yang–Mills theory whose gauge symmetry is broken by a gauge fixing term are identical from the view point of quantum gauge symmetry. The explicit infinitesimal transformations which leave the massive Yang–Mills theory BRST invariant are given.

BRST COHOMOLOGY AND HODGE DECOMPOSITION THEOREM IN ABELIAN GAUGE THEORY

We discuss the Becchi–Rouet–Stora–Tyutin (BRST) cohomology and Hodge decomposition theorem for the two-dimensional free U(1) gauge theory. In addition to the usual BRST charge, we derive a local, conserved and nilpotent co(dual)-BRST charge under which the gauge-fixing term remains invariant. We express the Hodge decomposition theorem in terms of these charges and the Laplacian operator. We take a single photon state in the quantum Hilbert space and demonstrate the notion of gauge invariance, no-(anti)ghost theorem, transversality of photon and establish the topological nature of this theory by exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In fact, the topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. On the two-dimensional compact manifold, we derive two sets of topological invariants with respect to the conserved and nilpotent BRST- and co-BRST charges and express the Lagrangian density of the theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of both Witten- and Schwarz-type of topological field theories.

Becchi–Rouet–Stora–Tyutin invariant formulation of spontaneously broken gauge theory in a generalized differential geometry

Noncommutative geometry (NCG) on a discrete space successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory, in which the Higgs boson field is regarded as a kind of gauge field on the discrete space. People also know how to construct the generalized differential geometry (a GDG) as one version of NCG on a discrete space M4×ZN where ZN denotes the N points discrete space. A GDG is a direct generalization of the differential geometry on the ordinary manifold into the discrete one. In this paper, we attempt to construct the Becchi–Rouet–Stora–Tyutin (BSRT) invariant formulation of spontaneously broken gauge theory based on a GDG and obtain the BSRT invariant Lagrangian with the ’t Hooft–Feynman gauge fixing term.