Gauge-fixing Parameter Research Articles

On the conformal limit of a QED-inspired model

A conformal invariant QED-inspired model is solved for a general covariant linear gauge using the Dyson–Schwinger equations for the propagators assuming a pure vector like interaction. The leading corrections to the asymptotic solutions and the exponents, that characterize the corrections to each of the two fermion propagator functions, are computed as a function of the coupling and gauge fixing parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. For the scalar component of the fermion propagator our findings generalizes for linear covariant gauges previous results found in the literature and reproduce the outcome of the perturbative analysis of quenched QED in the Landau gauge. Our solution for the exponent associated with vector component of the fermion propagator is new and, in the weak coupling regime, agrees with the estimation based on the perturbative analysis of quenched QED. Of the two critical exponents describing the conformal limit of the vector interaction, one of them is, in QED, associated with the regime where chiral symmetry is broken dynamically, which demands one mass scale, namely the Miransky scaling. A second mass scale has to be introduced at larger coupling constants and is associated with a change on the nature of the fermion wave function. This provides one example, that it is possible to find two interwoven cycles in Quantum Field Theory, albeit in a truncated framework, as it is known in the quantum few-body problem in the limit of a zero-range interaction.

Photon propagator for inflation in the general covariant gauge

Photon propagator for power-law inflation is considered in the general covariant gauges within the canonical quantization formalism. Photon mode functions in covariant gauges are considerably more complicated than their scalar counterparts, except for the special choice of the gauge-fixing parameter we call the simple covariant gauge. We explicitly construct the position space photon propagator in the simple covariant gauge, and find the result considerably more complicated than its scalar counterpart. This is because of the need for explicitly inverting the Laplace operator acting on the scalar propagator, which results in Appell’s fourth function. Our propagator correctly reproduces the de Sitter and flat space limits. We use this propagator to compute two simple observables: the off-coincident field strength-field strength correlator and the energy-momentum tensor, both of which yield consistent results. As a spinoff of our computation we also give the exact expression for the Coulomb gauge propagator in power-law inflation in arbitrary dimensions.

Mass spectrum in a six-dimensional SU(n) gauge theory on a magnetized torus

We examine six-dimensional SU(n) gauge theories compactified on a two-dimensional torus with a constant magnetic flux background to obtain a comprehensive low-energy mass spectrum. We introduce general background configurations including the magnetic flux and continuous Wilson line phases, consistent with classical equations of motion. Under the standard gauge fixing procedure, the complete mass spectrum in low-energy effective theory for the SU(n) case is newly presented without imposing restrictions on the gauge fixing parameter. Our analysis confirms the inevitable existence of tachyonic modes, which neither depend on the background configurations of Wilson line phases nor are affected by the gauge fixing parameter. Masses for some low-energy modes exhibit dependence on the gauge fixing parameter, and these modes are identified as would-be Goldstone bosons that are absorbed by massive four-dimensional vector fields. We discuss the phenomenological implications associated with stabilization or condensation of the tachyonic states. Various mass spectra and symmetry-breaking patterns are expected with flux backgrounds in the SU(n) case. They are helpful for constructing phenomenologically viable models beyond the standard model, such as gauge-Higgs unification and grand unified theories.

FeynGrav 2.0

We present a new version of FeynGrav. The present version supports Feynman rules for matter with non-vanishing mass and SU(N) Yang-Mills model. We revisit the gauge fixing procedure for gravity and derive interaction rules valid for an arbitrary gauge fixing parameter. We provide a few simple examples of calculations to illustrate package usage. Program summaryProgram Title: FeynGravCPC Library link to program files:https://doi.org/10.17632/rwnrbbdkkv.1Developer's repository link:github.com/BorisNLatosh/FeynGravLicensing provisions: GPLv3Programming language: Wolfram Mathematica 10 and higherExternal routines/libraries: FeynCalc [1,2]Nature of problem: FeynGrav is a software package that simplifies calculations involving the Feynman rules for gravity. These rules are complex and consist of numerous terms, but FeynGrav makes it possible to work with a specific set of rules within the FeynCalc environment. The package includes interaction rules for regular matter with different spin values, as well as for the SU(N) Yang-Mills model in curved spacetime and general relativity. FeynGrav provides scripts to generate the relevant interaction rules and libraries of pre-generated expressions, making it a powerful tool for researchers in the field of theoretical physics.Solution method: I propose a framework that provides a straightforward way to obtain expressions for the corresponding Feynman rules by efficiently utilizing perturbative expansions. In particular, within the perturbative approach to quantum gravity, certain quantities admit finite perturbative expansions, while others have infinite perturbative expansions. I derive formulas that allow the generation of terms with infinite expansions for any given order of perturbation theory. In all actions describing gravitational interactions, all derivatives can be placed on terms with finite perturbative expansions via integration by parts. In this way, the proposed framework gives a simple algorithm for deriving Feynman rules.Additional comments including restrictions and unusual features: Scripts responsible for generating interaction rules are placed in the “Rules” folder. They can be used separately and only require FeynCalc to run. The “Libs” folder contains the “FeynGravLibrariesGenerator.wl” script and the libraries it generates. “FeynGrav.wl” is the main package file that imports data from the libraries and uses it to work with the interaction rules. The file “FeynGrav_Examples.nb” contains examples of calculations.

Gauge dependence of the quark gap equation: An exploratory study

We study the gauge dependence of the quark propagator in quantum chromodynamics by solving the gap equation with a nonperturbative quark-gluon vertex which is constrained by longitudinal and transverse Slavnov-Taylor identities, the discrete charge conjugation and parity symmetries and which is free of kinematic singularities in the limit of equal incoming and outgoing quark momenta. We employ gluon propagators in renormalizable $R_\xi$ gauges obtained in lattice QCD studies. We report the dependence of the nonperturbative quark propagator on the gauge parameter, in particular we observe an increase, proportional to the gauge-fixing parameter, of the mass function in the infrared domain, whereas the wave renormalization decreases within the range $0 \leq \xi \leq 1$ considered here. The chiral quark condensate reveals a mild gauge dependence in the region of $\xi$ investigated. We comment on how to build and improve upon this exploratory study in future in conjunction with generalized gauge covariance relations for QCD.

Spectral functions of gauge theories with Banks-Zaks fixed points

We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalization group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a K\"all\'en-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation of complex conjugated branch cuts renders a causal representation impossible. We also derive relations for scaling exponents that determine the presence or absence of propagator nonanalyticities. Further results include spectral functions for all fields up to five loop order, bounds on the conformal window, and an algorithm to find running gauge coupling analytically at higher loops. Implications of our findings and extensions to other theories are discussed.

Gauge dependence of the standard model effective potential at two-loop order

The effective potential of the standard model of electroweak unification is determined at two-loop order in a linear gauge involving two massless and two massive gauge fixing parameters. The divergent and logarithmic terms involving the massive gauge parameters are found to be determined also by the gauge parameter identities. The renormalization group method for summing logarithms is shown to be applicable also at scales below that of the massive gauge parameters.

Yang-Mills propagators in linear covariant gauges from Nielsen identities

We calculate gluon and ghost propagators in Yang-Mills theory in linear covariant gauges. To that end, we utilize Nielsen identities with Landau gauge propagators and vertices as the starting point. We present and discuss numerical results for the gluon and ghost propagators for values of the gauge parameter $0<\xi \le 5$. Extrapolating the propagators to $\xi \to \infty $ we find the expected qualitative behavior. We provide arguments that our results are quantitatively reliable at least for values $\xi\lesssim 1/2$ of the gauge fixing parameter. It is shown that the correlation functions, and in particular the ghost propagator, change significantly with increasing gauge parameter. In turn, the ghost-gluon running coupling as well as the position of the zero crossing of the Schwinger function of the gluon propagator remain within the uncertainties of our calculation unchanged.

Towards Higgs masses and decay widths satisfying the symmetries in the (N)MSSM

In models with an extended Higgs sector, such as the (N)MSSM, scalar states mix with one another. Yet, the concept of Higgs mixing is problematic at the radiative level, since it introduces both a scheme and a gauge dependence. In particular, the definition of Higgs masses and decay amplitudes can be impaired by the presence of gauge-violating pieces of higher order. We discuss in depth the origin and magnitude of such effects and consider two strategies that minimize the dependence on the gauge-fixing parameter and field-renormalization of one-loop order in the definition of the mass and decay observables, both in degenerate and non-degenerate scenarios. In addition, the intuitive concept of mixing and the simplicity of its definition in terms of two-point diagrams can make it tempting to include higher-order corrections on this side of the calculation, irrespectively of the order achieved in vertex diagrams. Using the global SU(2)_{mathrm{L}}-symmetry in the decoupling limit, we show that no improvement can be expected from such an approach at the level of the Higgs decays, but that, on the contrary, the higher-order terms may lead to numerically large spurious effects.

A critical look at the electroweak phase transition

The electroweak phase transition broke the electroweak symmetry. Perturbative methods used to calculate observables related to this phase transition suffer from severe problems such as gauge dependence, infrared divergences, and a breakdown of perturbation theory. In this paper we develop robust perturbative tools for dealing with phase transitions. We argue that gauge and infrared problems are absent in a consistent power-counting. We calculate the finite temperature effective potential to two loops for general gauge-fixing parameters in a generic model. We demonstrate gauge invariance, and perform numerical calculations for the Standard Model in Fermi gauge.

The renormalization structure of [formula omitted] supersymmetric higher-derivative gauge theory

We consider the harmonic superspace formulation of higher-derivative 6D,N=(1,0) supersymmetric gauge theory and its minimal coupling to a hypermultiplet. In components, the kinetic term for the gauge field in such a theory involves four space-time derivatives. The theory is quantized in the framework of the superfield background method ensuring manifest 6D,N=(1,0) supersymmetry and the classical gauge invariance of the quantum effective action. We evaluate the superficial degree of divergence and prove it to be independent of the number of loops. Using the regularization by dimensional reduction, we find possible counterterms and show that they can be removed by the coupling constant renormalization for any number of loops, while the divergences in the hypermultiplet sector are absent at all. Assuming that the deviation of the gauge-fixing term from that in the Feynman gauge is small, we explicitly calculate the divergent part of the one-loop effective action in the lowest order in this deviation. In the approximation considered, the result is independent of the gauge-fixing parameter and agrees with the earlier calculation for the theory without a hypermultiplet.

De Sitter field equations from quadratic curvature gravity: A group theoretical approach

In this paper, the linearized field equations related to the quadratic curvature gravity theory have been obtained in the four-dimensional de Sitter (dS) space–time. The massless spin-2 field equations have been written in terms of the Casimir operators of dS group making use of the ambient space notations. By imposing some simple constraints, arisen from group theoretical interpretation of the field equations, a new four-dimensional Gauss–Bonnet (GB-)like action has been introduced with the related field equations transforming according to the unitary irreducible representations (UIRs) of dS group. Since, the field equations transform according to the UIRs of dS group, the GB-like action, we just obtained, is expected to be a successful model of modified gravity. For more clarity, the gauge invariant field equations have been solved in terms of a gauge-fixing parameter [Formula: see text]. It has been shown that the solution can be written as the multiplication of a symmetric rank-2 polarization tensor and a massless minimally coupled scalar field on dS space. The Krein–Gupta–Bleuler quantization method has been utilized and the covariant two-point function has been calculated in terms of the massless minimally coupled scalar two-point function, using the ambient space notations. It has been written in terms of dS intrinsic coordinates from the ambient space counterpart. The two-point functions are dS invariant and free of any theoretical problems. It means that the proposed model is a successful model of modified gravity and it can produce significant results in the contexts of classical theory of gravity and quantum gravity toy models.

Flows of multicomponent scalar models with U(1) gauge symmetry

We investigate the renormalization group flows of multicomponent scalar theories with $U(1)$ gauge symmetry using the functional renormalization group method. The scalar sector is built up from traces of matrix fields that belong to simple, compact Lie algebras. We find that in general these theories are non-renormalizable even at zero gauge coupling, but if we add a $U(1)$ factor to the Lie algebra structure, then they are consistent. In accordance with our earlier findings, fluctuations introduce anomalous, regulator dependent gauge contributions, which are only consistent with the flow equation for a given set of gauge fixing parameters. We establish connections between regularization procedures in the standard covariant and the $R_\xi$ gauges arguing that one is not tied by introducing regulators at the level of the functional integral, and it is allowed to switch between schemes at different levels of the calculations. We calculate $\beta$ functions, classify fixed points, and clarify compatibility of the flow equation and the Ward-Takahashi identity between the scalar wavefunction renormalization and the charge rescaling factor.

Effective potential in the 3D massive 2-form gauge superfield theory

In the N=1, d=3 superspace, we propose a massive superfield theory formulated in terms of a spinor gauge superfield, whose component content includes a two-form field, and a real scalar matter superfield. For this model, we explicitly calculate the one-loop correction to the superfield effective potential. In particular, we show that the one-loop effective potential is independent the gauge-fixing parameters.

The decay h \u2192 Z\u03b3 in the Standard-Model Effective Field Theory

We calculate the S-matrix element for the Higgs boson decay to a Z-boson and a photon, h → Zγ, at one-loop in the Standard-Model Effective Field Theory (SMEFT) framework and in linear Rξ -gauges. Our SMEFT expansion includes all relevant operators up to dimension-6 considered in Warsaw basis without resorting to any flavour or CP- conservation assumptions. Within this approximation there are 23 dimension-6 operators affecting the amplitude, not including flavour and hermitian conjugation. The result for the on-shell h → Zγ amplitude is gauge invariant, renormalisation-scale invariant and gauge-fixing parameter independent. The calculated ratio of the SMEFT versus the SM expectation for the h → Zγ decay width is then written in a semi-numerical form which is useful for further comparisons with related processes. For example, the h → Zγ amplitude contains 16 operators in common with the h → γγ amplitude and one can draw useful results about its feasibility at current and future LHC data.

Two-loop perturbative corrections to the constrained effective potential in thermal QCD

In this paper, we compute the constrained QCD effective potential up to two-loop order with finite quark mass and chemical potential. We present the explicit calculations by using the double line notation and analytical expressions for massless quarks are obtained in terms of the Bernoulli polynomials or Polyakov loops. Our results explicitly show that the constrained QCD effective potential is independent on the gauge fixing parameter. In addition, as compared to the massless case, the constrained QCD effective potential with massive quarks develops a completely new term which is only absent when the background field vanishes. Furthermore, we discuss the relation between the one- and two- loop constrained effective potential. The surprisingly simple proportionality that exists in the pure gauge theories, however, is in general no longer true when fermions are taken into account. On the other hand, for high baryon density μB and low temperature T, in the massless limit, we do also find a similar proportionality between the one- and two-loop fermionic contributions in the constrained effective potential up to mathcal{O}left(T/{mu}_Bright) .

Gauge-independent approach to resonant dark matter annihilation

In spontaneously broken gauge theories, transition amplitudes describing dark-matter (DM) annihilation processes through a resonance may become highly inaccurate close to a production threshold, if a Breit-Wigner (BW) ansatz with a constant width is used. To partially overcome this problem, the BW propagator needs to be modified by including a momentum dependent decay width. However, such an approach to resonant transition amplitudes generically suffers from gauge artefacts that may also give rise to a bad or ambiguous high-energy behaviour for such amplitudes. We address the two problems of gauge dependence and high-energy unitarity within a gauge-independent framework of resummation implemented by the so-called Pinch Technique. We study DM annihilation via scalar resonances in a gauged U(1)X complex-scalar extension of the Standard Model that features a massive stable gauge field which can play the role of the DM. We find that the predictions for the DM abundance may vary significantly from previous studies based on the naive BW ansatz and propose an alternative simple approximation which leads to the correct DM phenomenology. In particular, our results do not depend on the gauge-fixing parameter and are consistent with considerations from high-energy unitarity.

Two-loop effective potential for generalized gauge fixing

We obtain the two-loop effective potential for general renormalizable theories, using a generalized gauge-fixing scheme that includes as special cases the background-field $R_\xi$ gauges, the Fermi gauges, and the familiar Landau gauge, and using dimensional regularization in the bare and \MSbar renormalization schemes. As examples, the results are then specialized to the Abelian Higgs model and to the Standard Model. In the case of the Standard Model, we study how the vacuum expectation value and the minimum vacuum energy depend numerically on the gauge-fixing parameters. The results at fixed two-loop order exhibit non-convergent behavior for sufficiently large gauge-fixing parameters; this can presumably be addressed by a resummation of higher-order contributions.

Bounds on Higgs and top quark masses from vacuum stability (degeneracy) with gravitational contributions

Based on the weak coupling expansion of gravity, we calculate the gravitational contributions to yukawa coupling, scalar quartic coupling as well as gauge couplings with general Landau-DeWitt gauge-fixing choice and a gauge preserving (of SM gauge group) cut off regularization scheme. We find that the results depend on the Landau-DeWitt gauge-fixing parameter. Based on the two loop RGE of SM couplings with one loop full gravitational contributions in harmonic gauge, we study the constraints on the higgs and top quark mass from the requirement of existing the other degenerate vacua at the Planck-dominated region. Our numerical calculations show that nature will not develop the other degenerate vacua at the Planck-dominated region with current higgs and top quark masses. On the other hand, requiring the existence of the other degenerate vacua at the Planck-dominated region will constrain the higgs and top mass to lie at approximately 130 and 174 GeV, respectively.

Gaugeon formalism for the second-rank antisymmetric tensor gauge fields

We present a BRST symmetric gaugeon formalism for the two-form gauge fields. A set of vector gaugeon fields is introduced as a quantum gauge freedom. One of the gaugeon fields satisfies a higher derivative field equation; this property is necessary to change the gauge-fixing parameter of the two-form gauge field. A naive Lagrangian for the vector gaugeon fields is itself invariant under a gauge transformation for the vector gaugeon field. The Lagrangian of our theory includes the gauge-fixing terms for the gaugeon fields and corresponding Faddeev--Popov ghosts terms.