BRST Doublets Research Articles

BRST quantization of general relativity in unimodular gauge and unimodular gravity

Unimodular gravity (UG) is an important theory which may explain the smallness of the cosmological constant. To get insight into the covariant quantization of UG, we discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the unimodular gauge. This process requires the introduction of an additional pair of BRST doublets which decouple from the physical sector together with the other three pairs of BRST doublets for the TDiff. We show that the physical spectrum is the same as GR in the usual covariant gauge fixing. We then study a theory derived by making "Fourier transform" of GR in the unimodular gauge with respect to the cosmological constant as a candidate of "quantum UG." We clarify the difference from GR and point out problems in this theory.

On non-commutative U⋆(1) gauge models and renormalizability

Based on our recent findings regarding (non-)renormalizability of non-commutative U⋆(1) gauge theories [1, 2] we present the construction of a new type of model. By introducing a soft-breaking term in such a way that only the bilinear part of the action is modified, no interaction between the gauge sector and auxiliary fields occurs. Demanding in addition that the latter form BRST doublet structures, this leads to a minimally altered non-commutative U⋆(1) gauge model featuring an IR damping behavior. Moreover, the new breaking term is shown to provide the necessary structure in order to absorb the inevitable quadratic IR divergences appearing at one-loop level in theories of this kind. In this paper we compute Feynman rules, symmetries and results for the vacuum polarization together with the one-loop renormalization of the gauge boson propagator and three-point functions.

On the renormalizability of noncommutative U(1) gauge theory—an algebraic approach

We investigate the quantum effects of the nonlocal gauge invariant operator in the noncommutative U(1) action and its consequences to the infrared sector of the theory. Nonlocal operators of such kind were proposed to solve the infrared problem of the noncommutative gauge theories evading the questions on the explicit breaking of the Lorentz invariance. More recently, a first step in the localization of this operator was accomplished by means of the introduction of an extra tensorial matter field, and the first loop analysis was carried out (Blaschke et al (2009 Eur. Phys. J. C 62 433–43)). We will complete this localization avoiding the introduction of new degrees of freedom beyond those of the original action by using only BRST doublets. This will allow us to conduct a complete BRST algebraic study of the renormalizability of the theory, following Zwanziger's method of localization of nonlocal operators in QFT.

Soft breaking of BRST invariance for introducing non-perturbative infrared effects in a local and renormalizable way

The possibility of introducing non-perturbative infrared effects leading to a modification of the long distance behavior of gauge theories through a soft breaking of the BRST invariance is investigated. The method reproduces the Gribov–Zwanziger action describing the restriction of the domain of integration in the Feynman path integral to the Gribov region and a model for the dynamical quark mass generation is presented. The soft symmetry breaking relies on the introduction of BRST doublets and massive physical parameters, which allow one to distinguish the infrared region from the ultraviolet one, within the same theory.

Slavnov-Taylor parameterization for the quantum restoration of BRST symmetries in anomaly-free gauge theories

It is shown that the problem of the recursive restoration of the Slavnov-Taylor (ST) identities at the quantum level for anomaly-free gauge theories is equivalent to the problem of parameterizing the local approximation to the quantum effective action in terms of ST functionals, associated with the cohomology classes of the classical linearized ST operator. The ST functionals of dimension <=4 correspond to the invariant counterterms, those of dimension >4 generate the non-symmetric counterterms upon projection on the action-like sector. At orders higher than one in the loop expansion there are additional contributions to the non-invariant counterterms, arising from known lower order terms. They can also be parameterized by using the ST functionals. We apply the method to Yang-Mills theory in the Landau gauge with an explicit mass term introduced in a BRST-invariant way via a BRST doublet. Despite being non-unitary, this model provides a good example where the method devised in the paper can be applied to derive the most general solution for the action-like part of the quantum effective action, compatible with the fulfillment of the ST identities and the other relevant symmetries of the model, to all orders in the loop expansion. The full dependence of the solution on the normalization conditions is given.

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.

THE BRST SYMMETRY OF AFFINE LIE SUPERALGEBRAS AND NONCRITICAL STRINGS

The topological field theories associated with affine Lie superalgebras are constructed. Their BRST symmetry is characterized by a Kazama algebra containing spin 1, 2 and 3 operators and closes linearly. Under this symmetry all operators are grouped into BRST doublets. The relation between the models constructed and noncritical string theories is explored.

Solving general gauge theories on inner product spaces

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form ph〉 = exp [ Q, ψ]ph〉 0 where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge-fixing operator, and ph〉 o BRST invariant states determined by a hermitian set of BRST doublets in involution. ph〉 0 does not belong to an inner product space although ph〉 does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for ph〉 0 and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that ph〉 0 are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set ph〉 0 are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.

Auxiliary conditions for solving BRST cohomologies

In a previous paper it was shown that a set of BRST doublets may always be used to restrict the BRST-invariant states provided they satisfy a closed algebra. Here we discuss the completeness of such sets and their use in solving operator cohomologies. The resulting formulation involves a generalized form of the quartet mechanism of Kugo and Ojima.