Gauge Fixing Term Research Articles

NONCOMMUTATIVITY AND NON-ANTICOMMUTATIVITY PERTURBATIVE QUANTUM GRAVITY

In this paper, we will study perturbative quantum gravity on supermanifolds with both noncommutativity and non-anticommutativity of spacetime coordinates. We shall first analyze the BRST and the anti-BRST symmetries of this theory. Then we will also analyze the effect of shifting all the fields of this theory in background field method. We will construct a Lagrangian density which apart from being invariant under the extended BRST transformations is also invariant under on-shell extended anti-BRST transformations. This will be done by using the Batalin–Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this theory can be elegantly written down in superspace with a two Grassmann parameter.

Dual-BRST symmetry: 6D Abelian 3-form gauge theory

Within the framework of the Becchi–Rouet–Stora–Tyutin (BRST) formalism, we demonstrate the existence of the novel off-shell nilpotent (anti-)dual-BRST symmetries in the context of a six (5+1)-dimensional (6D) free Abelian 3-form gauge theory. Under these local and continuous symmetry transformations, the total gauge-fixing term of the Lagrangian density remains invariant. This observation should be contrasted with the off-shell nilpotent (anti-)BRST symmetry transformations, under which, the total kinetic term of the theory remains invariant. The anticommutator of the above nilpotent (anti-)BRST and (anti-)dual-BRST transformations leads to the derivation of a bosonic symmetry in the theory. There exists a discrete symmetry transformation in the theory which provides a thread of connection between the nilpotent (anti-)BRST and (anti-)dual-BRST transformations. This theory is endowed with a ghost-scale symmetry, too. We discuss the algebra of these symmetry transformations and show that the structure of the algebra is reminiscent of the algebra of de Rham cohomological operators of differential geometry.

Mtheory on deformed superspace

In this paper we will analyse a noncommutative deformation of the ABJM theory in N=1 superspace formalism. We will then analyse the BRST and the anti-BRST symmetries for this deformed ABJM theory, in linear as well as non-linear gauges. We will show that the sum of gauge fixing term and the ghost term for this deformed ABJM theory can be expressed as a combination of the total BRST and the total anti-BRST variation, in Landau and non-linear gauges. We will show that in Landau and Curci-Ferrari gauges this deformed ABJM theory is invariant under a additional set of symmetry transformations. We will also discuss the effect that the addition of a bare mass term has on this theory.

Perturbative quantum gravity on complex spacetime

In this Letter we will study non-anticommutative perturbative quantum gravity on spacetime with a complex metric. After analysing the BRST symmetry of this non-anticommutative perturbative quantum gravity, we will also analyse the effect of shifting all the fields. We will construct a Lagrangian density which is invariant under the original BRST transformations and the shift transformations in the Batalin–Vilkovisky (BV) formalism. Finally, we will show that the sum of the gauge-fixing term and the ghost term for this shift symmetry invariant Lagrangian density can be elegantly written down in superspace with a single Grassmann parameter.

Cosmic magnetic fields and dark energy in extended electromagnetism

We discuss an extended version of electromagnetism in which the usual gauge fixing term is promoted into a physical contribution that introduces a new scalar state in the theory. This new state can be generated from vacuum quantum fluctuations during an inflationary era and, on super-Hubble scales, gives rise to an effective cosmological constant. The value of such a cosmological constant coincides with the one inferred from observations as long as inflation took place at the electroweak scale. On the other hand, the new state also generates an effective electric charge density on sub-Hubble scales that produces both vorticity and magnetic fields with coherent lengths as large as the present Hubble horizon.

A Theorem of Equivalence between TransverseDiff Theories and Scalar-Tensor Gravity

Transverse Diffeomorphism (TDiff) theories are well-motivated theories of gravity from the quantum perspective, which are based upon a gauge symmetry principle. The main contribution of this work is to firmly establish a correspondence between TransverseDiff and the better-known scalar-tensor gravity — in its more general form —, a relation which is completely analogous to that between unimodular gravity and General Relativity. We then comment on observational aspects of TDiff. In connection with this proof, we derive a very general rule that determines under what conditions the procedure of fixing a gauge symmetry can be equivalently applied before the variational principle leading to the equations of motion, as opposed to the standard procedure, which takes place afterwards; this rule applies to gauge-fixing terms without derivatives.

Unimodular loop quantum gravity and the problems of time

We develop the quantization of unimodular gravity in the Plebanski and Ashtekar formulations and show that the quantum effective action defined by a formal path integral is unimodular. This means that the effective quantum geometry does not couple to terms in the expectation value of energy proportional to the metric tensor. The path integral takes the same form as is used to define spin foam models, with the additional constraint that the determinant of the four-metric is constrained to be a constant by a gauge fixing term. This extends the results of L. Smolin [Phys. Rev. D 80, 084003 (2009)] to the Hilbert space and path integral of loop quantum gravity. We review the proposal of Unruh, Wald, and Sorkin---that the Hamiltonian quantization yields quantum evolution in a physical time variable equal to elapsed four-volume---and discuss how this may be carried out in loop quantum gravity.

Infrared Behavior and Gauge Artifacts in de Sitter Spacetime: The Photon Field

We study the infrared (long-distance) behavior of the free photon field in de Sitter spacetime. Using a two-parameter family of gauge-fixing terms, we show that the infrared (IR) behavior of the two-point function is highly gauge-dependent and ranges from vanishing to growing. This situation is in disagreement with its counterpart in flat spacetime, where the two-point function vanishes in the IR region for any choice of the gauge-fixing parameters. A criterion to isolate the physical part of the two-point function is given and is shown to lead to a well-behaved two-point function in the IR region.

One-loop effects of extra dimensions on theWWγandWWZvertices

The one-loop contribution of the excited Kaluza-Klein (KK) modes of the $SU_L(2)$ gauge group on the off-shell $WW\gamma$ and $WWZ$ vertices is calculated in the context of a pure Yang-Mills theory in five dimensions and its phenomenological implications discussed. The use of a gauge-fixing procedure for the excited KK modes that is covariant under the standard gauge transformations of the $SU_L(2)$ group is stressed. A gauge-fixing term and the Faddeev-Popov ghost sector for the KK gauge modes that are separately invariant under the standard gauge transformations of $SU_L(2)$ are presented. It is shown that the one-loop contributions of the KK modes to the off-shell $WW\gamma$ and $WWZ$ vertices are free of ultraviolet divergences and well-behaved at high energies. It is found that for a size of the fifth dimension of $R^{-1}\sim 1TeV$, the one-loop contribution of the KK modes to these vertices is about one order of magnitude lower than the corresponding standard model radiative correction. This contribution is similar to the one estimated for new gauge bosons contributions in other contexts. Tree-level effects on these vertices induced by operators of higher canonical dimension are also investigated. It is found that these effects are lower than those generated at the one-loop order by the KK gauge modes.

Automatic calculation of supersymmetric renormalization group equations and loop corrections

SARAH is a Mathematica package for studying supersymmetric models. It calculates for a given model the masses, tadpole equations and all vertices at tree-level. This information can be used by SARAH to write model files for CalcHep/ CompHep or FeynArts/ FormCalc. In addition, the second version of SARAH can derive the renormalization group equations for the gauge couplings, parameters of the superpotential and soft-breaking parameters at one- and two-loop level. Furthermore, it calculates the one-loop self-energies and the one-loop corrections to the tadpoles. SARAH can handle all N = 1 SUSY models whose gauge sector is a direct product of SU ( N ) and U ( 1 ) gauge groups. The particle content of the model can be an arbitrary number of chiral superfields transforming as any irreducible representation with respect to the gauge groups. To implement a new model, the user has just to define the gauge sector, the particle, the superpotential and the field rotations to mass eigenstates. Program summary Program title: SARAH Catalogue identifier: AEIB_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEIB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 97 577 No. of bytes in distributed program, including test data, etc.: 2 009 769 Distribution format: tar.gz Programming language: Mathematica Computer: All systems that Mathematica is available for Operating system: All systems that Mathematica is available for Classification: 11.1, 11.6 Nature of problem: A supersymmetric model is usually characterized by the particle content, the gauge sector and the superpotential. It is a time consuming process to obtain all necessary information for phenomenological studies from these basic ingredients. Solution method: SARAH calculates the complete Lagrangian for a given model whose gauge sector can be any direct product of SU ( N ) gauge groups. The chiral superfields can transform as any, irreducible representation with respect to these gauge groups and it is possible to handle an arbitrary number of symmetry breakings or particle rotations. Also the gauge fixing terms can be specified. Using this information, SARAH derives the mass matrices and Feynman rules at tree-level and generates model files for CalcHep/CompHep and FeynArts/FormCalc. In addition, it can calculate the renormalization group equations at one- and two-loop level and the one-loop corrections to the one- and two-point functions. Unusual features: SARAH just needs the superpotential and gauge sector as input and not the complete Lagrangian. Therefore, the complete implementation of new models is done in some minutes. Running time: Measured CPU time for the evaluation of the MSSM on an Intel Q8200 with 2.33 GHz. Calculating the complete Lagrangian: 12 seconds. Calculating all vertices: 75 seconds. Calculating the one- and two-loop RGEs: 50 seconds. Calculating the one-loop corrections: 7 seconds. Writing a FeynArts file: 1 second. Writing a CalcHep/CompHep file: 6 seconds. Writing the LaTeX output: 1 second.

On the infrared behaviour of 3d Chern-Simons theories in $ \mathcal{N} = 2 $ superspace

We discuss the problem of infrared divergences in the N=2 superspace approach to classically marginal three-dimensional Chern-Simons-matter theories. Considering the specific case of ABJM theory, we describe the origin of such divergences and offer a prescription to eliminate them by introducing non-trivial gauge-fixing terms in the action. We also comment on the extension of our procedure to higher loop order and to general three-dimensional Chern-Simons-matter models.

BRST and Anti-BRST Symmetries in Perturbative Quantum Gravity

In perturbative quantum gravity, the sum of the classical Lagrangian density, a gauge fixing term and a ghost term is invariant under two sets of supersymmetric transformations called the BRST and the anti-BRST transformations. In this paper we will analyse the BRST and the anti-BRST symmetries of perturbative quantum gravity in curved spacetime, in linear as well as non-linear gauges. We will show that even though the sum of ghost term and the gauge fixing term can always be expressed as a total BRST or a total anti-BRST variation, we can express it as a combination of both of them only in certain special gauges. We will also analyse the violation of nilpotency of the BRST and the anti-BRST transformations by introduction of a bare mass term, in the massive Curci-Ferrari gauge.

BRST INVARIANCE AND DE RHAM TYPE COHOMOLOGY OF 't HOOFT–POLYAKOV MONOPOLE

We exploit the 't Hooft–Polyakov monopole to construct closed algebra of the quantum field operators and the BRST charge Q BRST . In the first-class configuration of the Dirac quantization, by including the Q BRST -exact gauge-fixing term and the Faddeev–Popov ghost term, we find the BRST invariant Hamiltonian to investigate the de Rham type cohomology group structure for the monopole system. The Bogomol'nyi bound is also discussed in terms of the first-class topological charge defined on the extended internal two-sphere.

GAUGE FIXING IDENTITY IN THE BACKGROUND FIELD METHOD OF QCD IN PURE GAUGE

In this paper we derive a gauge fixing identity by varying the covariant gauge fixing term in [Formula: see text] in the background field method of QCD in pure gauge. Using this gauge fixing identity, we establish a relation between [Formula: see text] in QCD and [Formula: see text] in background field method of QCD in pure gauge. We show the validity of this gauge fixing identity, in general noncovariant and general Coulomb gauge fixings respectively. This gauge fixing identity is used to prove factorization theorem in QCD at high energy colliders and in nonequilibrium QCD at high energy heavy-ion colliders.

A superspace formulation of Yang–Mills theory on sphere

A superspace approach to the Becchi–Rouet–Stora–Tyutin (BRST) formalism for the Yang–Mills theory on an n-dimensional unit sphere S1n is developed in a manifestly covariant manner based on the rotational supersymmetry characterized by the supergroup OSp(n+1∣2). This is done by employing an (n+2)-dimensional unit supersphere S1n∣2 parametrized by n commutative and two anticommutative coordinate variables so that it includes S1n as a subspace and realizes the OSp(n+1∣2) supersymmetry. In this superspace formulation, referred to as the supersphere formulation, the so-called horizontality condition is concisely expressed in terms of the rank-3 field strength tensor of a Yang–Mills superfield on S1n∣2. The supersphere formulation completely covers the BRST gauge-fixing procedure for the Yang–Mills theory on S1n provided by us [R. Banerjee and S. Deguchi, Phys. Lett. B 632, 579 (2006); arXiv:hep-th/0509161]. Furthermore, this formulation admits the (massive) Curci–Ferrari model defined on S1n, describing the gauge-fixing and mass terms on S1n together as a mass term on S1n∣2.

Rigid rotor as a toy model for Hodge theory

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.

From superpotential to model files for FeynArts and CalcHep/CompHep

SARAH is a Mathematica package for building and studying supersymmetric models. It calculates for a given superpotential and gauge sector the full Lagrangian of a model. With the new version of SARAH it is possible to calculate automatically all interactions for the different eigenstates and write model files for FeynArts and CompHep/CalcHep. In addition, the tadpole equations are calculated, gauge fixing terms can be given and ghost interactions are added, particles can be integrated out and non-supersymmetric limits of the theory can be chosen. CP and flavor violation can easily be switched on or off.

Hamilton-Jacobi Formulation of Green-Schwarz Superstring

The canonical formalism for fermion string ‘superstring’ theory is constructed by following the Hamilton-Jacobi prescription. The Hamiltonian mechanics of the Green-Schwarz superstring are discussed. The complete set of constraints is presented, implies appearance of mixed first and second class fermionic constraints, which are treated with no need to introduce gauge fixing terms. The equations of motion are derived as total differential equations in many variables. Besides, a mechanical model of D=11 superstring is presented.

Bimetric truncations for quantum Einstein gravity and asymptotic safety

In the average action approach to the quantization of gravity the fundamental requirement of "background independence" is met by actually introducing a background metric but leaving it completely arbitrary. The associated Wilsonian renormalization group defines a coarse graining flow on a theory space of functionals which, besides the dynamical metric, depend explicitly on the background metric. All solutions to the truncated flow equations known to date have a trivial background field dependence only, namely via the classical gauge fixing term. In this paper we analyze a number of conceptual issues related to the bimetric character of the gravitational average action and explore a first nontrivial bimetric truncation in the simplified setting of conformally reduced gravity. Possible implications for the Asymptotic Safety program and the cosmological constant problem are discussed in detail.

Nonquadratic gauge fixing and global gauge invariance in the effective action

The possibility of having a gauge fixing term in the effective Lagrangian that is not a quadratic expression has been explored in spin-two theories so as to have a propagator that is both traceless and transverse. We first show how this same approach can be used in spontaneously broken gauge theories as an alternate to the 't Hooft gauge fixing which avoids terms quadratic in the scalar fields. This "non-quadratic" gauge fixing in the effective action results in there being two complex Fermionic and one real Bosonic ghost fields. A global gauge invariance involving a Fermionic gauge parameter, analogous to the usual BRST invariance, is present in this effective action.