BRS Symmetry Research Articles

Noether Currents and Maxwell-Type Equations of Motion in Higher Derivative Gravity Theories

In general coordinate invariant gravity theories whose Lagrangians contain arbitrarily high order derivative fields, the Noether currents for the global translation and for the Nakanishi’s IOSp(8|8) choral symmetry containing the BRS symmetry as its member are constructed. We generally show that for each of these Noether currents, a suitable linear combination of equations of motion can be brought into the form of a Maxwell-type field equation possessing the Noether current as its source term.

Plane-wave背景上の超弦理論のdouble spinor形式

Plane-wave背景上の超弦理論のdouble spinor形式

BRS structure of simple model of cosmological constant and cosmology

In arXiv:1601.02203, a simple model has been proposed in order to solve one of the problems related with the cosmological constant. The model is given by a topological field theory and the model has an infinite numbers of the BRS symmetries. The BRS symmetries are, however, spontaneously broken in general. In this paper, we investigate the BRS symmetry in more details and show that there is one and only one BRS symmetry which is not broken and the unitarity can be guaranteed. In the model, the quantum problem of the vacuum energy, which may be identified with the cosmological constant, reduces to the classical problem of the initial condition. In this paper, we investigate the cosmology given by the model and specify the region of the initial conditions which could be consistent with the evolution of the universe. We also show that there is a stable solution describing the de Sitter space-time, which may explain the accelerating expansion in the current universe.

Some solutions for one of the cosmological constant problems

We propose several covariant models which may solve one of the problems in the cosmological constant. One of the models can be regarded as an extension of sequestering model. Other models could be regarded as extensions of the covariant formulation of the unimodular gravity. The contributions to the vacuum energy from the quantum corrections from the matters are absorbed into a redefinition of a scalar field and the quantum corrections become irrelevant to the dynamics. In a class of the extended unimodular gravity models, we also consider models which are regarded as topological field theories. The models can be extended and not only the vacuum energy but also any quantum corrections to the gravitational action could become irrelevant for the dynamics. We find, however, that the BRS symmetry in the topological field theories is broken spontaneously and therefore, the models might not be consistent.

EXTENDED BRS SYMMETRY IN TOPOLOGICAL FIELD THEORIES

A class of topological field theories like the BF model and Chern–Simons theory, when quantized in the Landau gauge, enjoys the property of invariance under a vector supersymmetry, which is responsible for their finiteness. We introduce a new type of gauge fixing which makes these theories invariant under an extended BRS symmetry, containing a new type of field, the ghost of diffeomorphisms. The presence of such an extension is naturally related to the vector supersymmetry discussed before.

Physical renormalization condition for the quark-mixing matrix

We investigate the renormalization of the quark-mixing matrix in the Electroweak Standard Model. The corresponding counterterms are gauge independent as can be shown using an extended BRS symmetry. Using rigid SU(2)L symmetry, we prove that the ultraviolet-divergent parts of the invariant counterterms are related to the field renormalization constants of the quark fields. We point out that for a general class of renormalization schemes rigid SU(2)L symmetry cannot be preserved in its classical form, but is renormalized by finite counterterms. Finally, we discuss a genuine physical renormalization condition for the quark-mixing matrix that is gauge independent and does not destroy the symmetry between quark generations.

BRS Symmetry, the Quantum Master Equation and the Wilsonian Renormalization Group

Recently we made a proposal for realization of an effective BRS symmetry along the Wilsonian renormalization group flow. In this paper we show that the idea can be naturally extended for the most general gauge theories. Extensive use of the antifield formalism is made to reveal some remarkable structure of the effective BRS symmetry. The average action defined with a continuum analog of the block spin transformation obeys the quantum master equation (QME), provided that an UV action does so. We show that the RG flow described by the exact flow equations is generated by canonical transformations in the field-antifield space. Using the relation between the average action and the Legendre effective action, we establish the equivalence between the QME for the average action and the modified Ward-Takahashi identity for the Legendre action. The QME remains intact when the regularization is removed.

EXACT SYMMETRIES REALIZED ON THE RG FLOW

We show that the BRS symmetry is present even with the finite IR cutoff in the Wilsonian RG approach. Based on the antifield formalism, we explain how the symmetry is realized on the RG flow.

Pinch technique at two loops: The case of massless Yang-Mills theories

The generalization of the pinch technique beyond one loop is presented. It is shown that the crucial physical principles of gauge-invariance, unitarity, and gauge-fixing-parameter independence single out at two loops exactly the same algorithm which has been used to define the pinch technique at one loop, without any additional assumptions. The two-loop construction of the pinch technique gluon self-energy, and quark-gluon vertex are carried out in detail for the case of mass-less Yang-Mills theories, such as perturbative QCD. We present two different but complementary derivations. First we carry out the construction by directly rearranging two-loop diagrams. The analysis reveals that, quite interestingly, the well-known one-loop correspondence between the pinch technique and the background field method in the Feynman gauge persists also at two-loops. The renormalization is discussed in detail, and is shown to respect the aforementioned correspondence. Second, we present an absorptive derivation, exploiting the unitarity of the $S$-matrix and the underlying BRS symmetry; at this stage we deal only with tree-level and one-loop physical amplitudes. The gauge-invariant sub-amplitudes defined by means of this absorptive construction correspond precisely to the imaginary parts of the $n$-point functions defined in the full two-loop derivation, thus furnishing a highly non-trivial self-consistency check for the entire method. Various future applications are briefly discussed.

Extended BRS symmetry and gauge independence in on-shell renormalization schemes

Extended BRS symmetry is used to prove gauge independence of the fermion renormalization constant Z 2 in on-shell QED renormalization schemes. A necessary condition for gauge independence of Z 2 in on-shell QCD renormalization schemes is formulated. Satisfying this necessary condition appears to be problematic at the three-loop level in QCD.

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.

Action principles, restoration of BRS symmetry and the renormalization group equation for chiral non-Abelian gauge theories in dimensional renormalization with a non-anticommuting γ5

The one-loop renormalization of a general chiral gauge theory without scalar and Majorana fields is fully worked out within Breitenlohner and Maison dimensional renormalization scheme. The coefficients of the anomalous terms introduced in the Slavnov–Taylor equations by the minimal subtraction algorithm are calculated and the asymmetric counterterms needed to restore the BRS symmetry, if the anomaly cancellation conditions are met, are computed. The renormalization group equation and its coefficients are worked out in the anomaly free case. The computations draw heavily from the existence of action principles and BRS cohomology theory.

PERTURBATIVE QUANTUM GAUGE FIELDS ON THE NONCOMMUTATIVE TORUS

Using standard field theoretical techniques, we survey pure Yang–Mills theory on the noncommutative torus, including Feynman rules and BRS symmetry. Although in general free of any infrared singularity, the theory is ultraviolet divergent. Because of an invariant regularization scheme, this theory turns out to be renormalizable and the detailed computation of the one-loop counterterms is given, leading to an asymptotically free theory. Besides, it turns out that nonplanar diagrams are overall convergent when θ is irrational.

Gauge independence of the effective potential revisited

We apply the formalism of extended BRS symmetry to the investigation of the gauge dependence of the effective potential in a spontaneously symmetry broken gauge theory. This formalism, which includes a set of Grassmann parameters defined as the BRS variations of the gauge-fixing parameters, allows us to derive in a quick and unambiguous way the related Nielsen identities, which express the physical gauge independence, in a class of generalized 't Hooft gauges, of the effective potential. We show in particular that the validity of the Nielsen identities does not require any constraint on the gauge-fixing parameters, contrary to some claims found in the literature. We use the method of algebraic renormalization, which leads to results independent of the particular renormalization scheme used.

RENORMALIZATION OF GAUGE THEORIES AND MASTER EQUATION

The evolution of ideas which has led from the first proofs of the renormalizability of non-Abelian gauge theories, based on Slavnov–Taylor identities, to the modern proof based on the BRS symmetry and the master equation is recalled. This lecture has been delivered at the Symposium in Honour of Professor C. N. Yang, Stony Brook, May 21, 22, 1999.

GAUGE DEPENDENCE IN CHERN–SIMONS THEORY

We compute the contribution to the modulus of the one-loop effective action in pure non-Abelian Chern–Simons theory in an arbitrary covariant gauge. We find that the results are dependent on both the gauge parameter (α) and the metric required in the gauge fixing. A contribution arises that has not been previously encountered; it is of the form [Formula: see text]. This is possible as in three dimensions α is dimensionful. A variant of proper time regularization is used to render these integrals well behaved (although no divergences occur when the regularization is turned off at the end of the calculation). Since the original Lagrangian is unaltered in this approach, no symmetries of the classical theory are explicitly broken and ∊μλν is handled unambiguously since the system is three-dimensional at all stages of the calculation. The results are shown to be consistent with the so-called Nielsen identities which predict the explicit gauge parameter dependence using an extension of BRS symmetry. We demonstrate that this α dependence may potentially contribute to the vacuum expectation values of products of Wilson loops.

Topological Symmetry Breaking on Einstein Manifolds

It is known that if gauge conditions have Gribov zero modes, then topological symmetry is broken. In this paper we apply it to the Witten type topological gravitational theory in dimension n ≥ 3. Our choice of the gauge condition for conformal invariance is R + α = 0, where R is the Ricci scalar curvature. We find when α ≠ 0, topological symmetry is not broken, but when α = 0 and solutions of the Einstein equations exist then topological symmetry is broken. These conditions connect to the Yamabe conjecture, namely negative constant scalar curvature exist on manifolds of any topology, but existence of nonnegative constant scalar curvature is restricted by topology. This fact is easily seen in this theory. Topological symmetry breaking means BRS symmetry breaking in cohomological field theory. But it is found that another BRS symmetry can be defined and physical states are redefined. The divergence due to the Gribov zero modes is regularized, and the theory after topological symmetry breaking becomes semiclassical Einstein gravitational theory under a special definition of observables.

Resolution of the BRS singlet-pair problem in quantum Einstein gravity

The possible existence of BRS singlet pairs causes trouble for the proof of unitarity based on the BRS symmetry, especially in quantum gravity. In quantum Einstein gravity, this difficulty is resolved in the framework of its covariant operator formalism by generalizing the proof based on both BRS and anti-BRS symmetries.

Massless Limits of Massive Tensor Fields. II: Infrared Regularizaiton of the Fierz-Pauli Model

Izawa's gauge-fixing procedure based on BRS symmetry is applied twice to the massive tensor field theory of Fierz-Pauli type. It is shown the second application can remove massless singularities which remain after the first application. Massless limit of the theory is discussed.

Background field technique and renormalization in lattice gauge theory

Lattice gauge theory with a background gauge field is shown to be renormalizable to all orders of perturbation theory. No additional counterterms are required besides those already needed in the absence of the background field. The argument closely follows the treatment given earlier for the case of dimensional regularization by Kluberg-Stern and Zuber. It is based on the BRS, background gauge and shift symmetries of the lattice functional integral.