Field-dependent BRST-antiBRST Transformations in Yang-Mills and Gribov-Zwanziger Theories

We introduce the notion of finite BRST-antiBRST transformations, both global and field-dependent, with a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in Yang-Mills theories. It turns out that the finite transformations are quadratic in their parameters. At the same time, exactly as in the case of finite field-dependent BRST transformations for the Yang-Mills vacuum functional, special field-dependent BRST-antiBRST transformations, with $s_{a}$-potential parameters $\lambda_{a}=s_{a}\Lambda$ induced by a finite even-valued functional $\Lambda$ and by the anticommuting generators $s_{a}$ of BRST-antiBRST transformations, amount to a precise change of the gauge-fixing functional. This proves the independence of the vacuum functional under such BRST-antiBRST transformations. We present the form of transformation parameters that generates a change of the gauge in the path integral and evaluate it explicitly for connecting two arbitrary $R_{\xi }$-like gauges. For arbitrary differentiable gauges, the finite field-dependent BRST-antiBRST transformations are used to generalize the Gribov horizon functional $h$, given in the Landau gauge, and being an additive extension of the Yang-Mills action by the Gribov horizon functional in the Gribov-Zwanziger model. This generalization is achieved in a manner consistent with the study of gauge independence. We also discuss an extension of finite BRST-antiBRST\ transformations to the case of general gauge theories and present an ansatz for such transformations.Comment: 31 pages, no figures, misprint in the Eq. (5.1) correcte

Field-Dependent BRST-antiBRST Lagrangian Transformations

We continue our study of finite BRST-antiBRST transformations for general gauge theories in Lagrangian formalism, initiated in [arXiv:1405.0790[hep-th] and arXiv:1406.0179[hep-th]], with a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters and prove the correctness of the explicit Jacobian in the partition function announced in [arXiv:1406.0179[hep-th]], which corresponds to a change of variables with functionally-dependent parameters $\lambda_{a}=U_{a}\Lambda$ induced by a finite Bosonic functional $\Lambda(\phi,\pi,\lambda)$ and by the anticommuting generators $U_{a}$ of BRST-antiBRST transformations in the space of fields $\phi$ and auxiliary variables $\pi^{a},\lambda$. We obtain a Ward identity depending on the field-dependent parameters $\lambda_{a}$ and study the problem of gauge dependence, including the case of Yang--Mills theories. We examine a formulation with BRST-antiBRST symmetry breaking terms, additively introduced to the quantum action constructed by the Sp(2)-covariant Lagrangian rules, obtain the Ward identity and investigate the gauge-independence of the corresponding generating functional of Green's functions. A formulation with BRST symmetry breaking terms is developed. It is argued that the gauge independence of the above generating functionals is fulfilled in the BRST and BRST-antiBRST settings. These concepts are applied to the average effective action in Yang--Mills theories within the functional renormalization group approach.Comment: 20+7 pages, no figures, presentation improved, typos corrected, reference added, remarks on composite field approach added in Sec. 4 and App.

Finite BRST-antiBRST Transformations in Generalized Hamiltonian Formalism

We introduce the notion of finite BRST-antiBRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST-antiBRST transformations for the Yang--Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790[hep-th]], special field-dependent BRST-antiBRST transformations with functionally-dependent parameters \lambda_{a}=\int dt\(s_{a}\Lambda) , generated by a finite even-valued function $\Lambda(t)$ and by the anticommuting generators $s_{a}$ of BRST-antiBRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters $\lambda_{a}$, and study the problem of gauge-dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary $R_{\xi}$-like gauges in the Yang--Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790[hep-th]], which justifies the unitarity of the $S$-matrix in the Lagrangian approach.Comment: 23 pages, published version, no figures, 1 table, presentation improved, references [17], [27] updated, ref. [40] and comments added. arXiv admin note: text overlap with arXiv:1405.079

Finite BRST-antiBRST Transformations in Lagrangian Formalism

We continue the study of finite BRST-antiBRST transformations for general gauge theories in Lagrangian formalism initiated in [arXiv:1405.0790[hep-th]], with a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters, and find an explicit Jacobian corresponding to this change of variables for constant $\lambda_{a}$. This makes it possible to derive the Ward identities and their consequences for the generating functional of Green's functions. We announce the form of the Jacobian [proved to be correct in [arXiv:1406.5086[hep-th]] for finite field-dependent BRST-antiBRST transformations with functionally-dependent parameters, $\lambda_{a}=s_{a}\Lambda$, induced by a finite even-valued functional $\Lambda(\phi ,\pi,\lambda)$ and by the generators $s_{a}$ of BRST-antiBRST transformations acting in the space of fields $\phi$, antifields $\phi_{a}^{\ast}$, $\bar {\phi}$ and auxiliary variables $\pi^{a},\lambda$. On the basis of this Jacobian, we solve a compensation equation for $\Lambda$, which is used to achieve a precise change of the gauge-fixing functional for an arbitrary gauge theory. We derive a new form of the Ward identities containing the parameters $\lambda_{a}$ and study the problem of gauge-dependence. The general approach is exemplified by the Freedman--Townsend model of\ a non-Abelian antisymmetric tensor field.Comment: 11 pages, no figures, (5.9) and Discussion extended, section with Freedman-Townsend model, 4+6 references and acknowledgments adde

Finite BRST Mapping in Higher Derivative Models

We continue the study of finite field dependent BRST (FFBRST) symmetry in the quantum theory of gauge fields. An expression for the Jacobian of path integral measure is presented, depending on a finite field-dependent parameter, and the FFBRST symmetry is then applied to a number of well-established quantum gauge theories in a form which includes higher-derivative terms. Specifically, we examine the corresponding versions of the Maxwell theory, non-Abelian vector field theory, and gravitation theory. We present a systematic mapping between different forms of gauge-fixing, including those with higher-derivative terms, for which these theories have better renormalization properties. In doing so, we also provide the independence of the S-matrix from a particular gauge-fixing with higher derivatives. Following this method, a higher-derivative quantum action can be constructed for any gauge theory in the FFBRST framework.Comment: 9 pages, published in Braz. J. Phy