Proca Field Research Articles

The decoupling problem of the Proca equation; and treatment of Dirac, Maxwell and Proca fields on the resulting pp-wave spacetimes

In this work we take a formal approach to the problem of decoupling Proca equations in curved space-times. We use Newman-Penrose (NP) two-spinor formalism to represent the Proca vector by one complex and two real scalars. We show that a decoupled second order differential equation for one of the real scalars can be derived if and only if the background space-time admits a covariantly constant null vector. Thus, the background space-time must be a pp-wave vacuum. We evaluate the separability of Proca, Maxwell and Dirac equations on the resulting pp-wave background.

One-loop corrections to vector Galileon theory

The effective action of the recently proposed vector Galileon theory is considered. Using the background field method, we obtain the one-loop correction to the propagator of the Proca field from vector Galileon self-interactions. Contrary to the so-called scalar Galileon interactions, the two-point function of the vector field gets renormalized at the one-loop level, indicating that there is no non-renormalization theorem in the vector Galileon theory. Using dimensional regularization, we remove the divergences and obtain the counterterms of the theory. The finite term is analytically calculated, which modifies the propagator and the mass term and generates some new terms also.

Generalized nonlinear Proca equation and its free-particle solutions

We introduce a non-linear extension of Proca's field theory for massive vector (spin $1$) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter $q$ (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit $q \rightarrow 1$. We derive the nonlinear Proca equation from a Lagrangian that, besides the usual vectorial field $\Psi^{\mu}(\vec{x},t)$, involves an additional field $\Phi^{\mu}(\vec{x},t)$. We obtain exact time dependent soliton-like solutions for these fields having the form of a $q$-plane wave, and show that both field equations lead to the relativistic energy-momentum relation $E^{2} = p^{2}c^{2} + m^{2}c^{4}$ for all values of $q$. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present $q$-generalized Proca theory reduces to Maxwell electromagnetism, and the $q$-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed.

Derivative self-interactions for a massive vector field

In this work we revisit the construction of theories for a massive vector field with derivative self-interactions such that only the 3 desired polarizations corresponding to a Proca field propagate. We start from the decoupling limit by constructing healthy interactions containing second derivatives of the Stueckelberg field with itself and also with the transverse modes. The resulting interactions can then be straightforwardly generalized beyond the decoupling limit. We then proceed to a systematic construction of the interactions by using the Levi-Civita tensors. Both approaches lead to a finite family of allowed derivative self-interactions for the Proca field. This construction allows us to show that some higher order terms recently introduced as new interactions trivialize in 4 dimensions by virtue of the Cayley-Hamilton theorem. Moreover, we discuss how the resulting derivative interactions can be written in a compact determinantal form, which can also be regarded as a generalization of the Born-Infeld lagrangian for electromagnetism. Finally, we generalize our results for a curved background and give the necessary non-minimal couplings guaranteeing that no additional polarizations propagate even in the presence of gravity.

Two-Component Theory of Classical Proca Fields in Curved Spacetimes with Torsionless Affinities

The world formulation of the full theory of classical Proca fields in generally relativistic spacetimes is reviewed. Subsequently the entire set of field equations is transcribed in a straightforward way into the framework of one of the Infeld-van der Waerden formalisms. Some well-known calculational techniques are then utilized for deriving the wave equations that control the propagation of the fields allowed for. It appears that no interaction couplings between such fields and electromagnetic curvatures are ultimately carried by the wave equations at issue. What results is, in effect, that the only interactions which occur in the theoretical context under consideration involve strictly Proca fields and wave functions for gravitons.

Massive Vector Particles Tunneling from the Neutral Rotating Anti-de Sitter Black Holes in Conformal Gravity**Supported by the National Natural Science Foundation of China under Grant No. 11205048, and the Foundation for Young Key Teacher of Henan Normal University

We investigate the massive vector particles' Hawking radiation from the neutral rotating Anti-de Sitter (AdS) black holes in conformal gravity by using the tunneling method. It is well known that the dynamics of massive vector particles are governed by the Proca field equation. Applying WKB approximation to the Proca equation, the tunneling probabilities and radiation spectrums of the emitted particles are derived. Hawking temperature of the neutral rotating AdS black holes in conformal gravity is recovered, which is consistent with the previous result in the literature.

Proca stars: Gravitating Bose–Einstein condensates of massive spin 1 particles

We establish that massive complex Abelian vector fields (mass μ) can form gravitating solitons, when minimally coupled to Einstein's gravity. Such Proca stars (PSs) have a stationary, everywhere regular and asymptotically flat geometry. The Proca field, however, possesses a harmonic time dependence (frequency w), realizing Wheeler's concept of geons for an Abelian spin 1 field. We obtain PSs with both a spherically symmetric (static) and an axially symmetric (stationary) line element. The latter form a countable number of families labelled by an integer m∈Z+. PSs, like (scalar) boson stars, carry a conserved Noether charge, and are akin to the latter in many ways. In particular, both types of stars exist for a limited range of frequencies and there is a maximal ADM mass, Mmax, attained for an intermediate frequency. For spherically symmetric PSs (rotating PSs with m=1,2,3), Mmax≃1.058MPl2/μ (Mmax≃1.568,2.337,3.247MPl2/μ), slightly larger values than those for (mini-)boson stars. We establish perturbative stability for a subset of solutions in the spherical case and anticipate a similar conclusion for fundamental modes in the rotating case. The discovery of PSs opens many avenues of research, reconsidering five decades of work on (scalar) boson stars, in particular as possible dark matter candidates.

Nonlinear interactions between black holes and Proca fields

Physics beyond the standard model is an important candidate for dark matter, and an interesting testing ground for strong-field gravity: the equivalence principle ‘forces’ all forms of matter to fall in the same way, and it is therefore natural to look for imprints of these fields in regions with strong gravitational fields, such as compact stars or black holes (BHs). Here we study general relativity minimally coupled to a massive vector field, and how BHs in this theory lose ‘hair’. Our results indicate that BHs can sustain Proca field condensates for extremely long time-scales.

Tachyonic Cherenkov radiation from supernova remnants

The subexponential decay observed in the γ-ray spectral maps of supernova remnants is explained in terms of tachyonic Cherenkov emission from a relativistic electron population. The tachyonic radiation densities of an electronic spinor current are derived, the total density as well as the transversal and longitudinal polarization components, taking account of electron recoil. Tachyonic flux quantization subject to dispersive and dissipative permeabilities is discussed, the matrix elements of the transversal and longitudinal Poynting vectors of the Maxwell–Proca field are obtained, Cherenkov emission angles and radiation conditions are derived. The spectral energy flux of an ultra-relativistic electron plasma is calculated, a tachyonic Cherenkov fit to the high-energy (1 GeV to 30 TeV) γ-ray spectrum of the Crab Nebula is performed, and estimates of the linear polarization degree are given. The spectral tail shows subexponential Weibull decay, which can be modeled with a frequency-dependent tachyon mass in the dispersion relations. Tachyonic flux densities interpolate between exponential and power-law spectral decay, which is further illustrated by Cherenkov fits to the γ-ray spectra of the supernova remnants IC 443 and W44. Subexponential spectral decay is manifested in double-logarithmic spectral maps as curved Weibull or straight power-law slope.

Topologically massive spin-1 particles and spin-dependent potentials

We investigate the role played by particular field representations of an intermediate massive spin-1 boson in the context of spin-dependent interparticle potentials between fermionic sources in the limit of low momentum transfer. The comparison between the well-known case of the Proca field and that of an exchanged spin-1 boson (with gauge-invariant mass) described by a 2-form potential mixed with a 4-vector gauge field is established in order to pursue an analysis of spin- as well as velocity-dependent profiles of the interparticle potentials. We discuss possible applications and derive an upper bound on the product of vector and pseudo-tensor coupling constants.

Marginal scalar and Proca clouds around Reissner-Nordström black holes

Massive scalar test fields around Kerr black holes can form quasi-bound states with complex frequencies. Some of these states decay in time, but some others -- in the superradiant regime -- grow, causing a superradiant instability. Precisely at the threshold between decaying and growing modes, there exist bound states with real frequency, known as scalar clouds. Fully non-linear counterparts of these clouds have been shown to yield Kerr black holes with scalar hair. Here, we consider massive, (electrically) charged, test scalar and Proca fields on the background of Reissner-Nordstr\"om black holes. By computing the quasi-bound state frequencies, we establish that no such states exist in the superradiant regime for the Proca field -- a similar behavior to that known for scalar fields. But decaying quasi-bound states with an arbitrarily small imaginary part of the frequency exist and thus which are arbitrarily long lived. In the limit of vanishing imaginary part of the frequency, the field does not trivialize and we dub the corresponding configurations as marginal (charged) scalar or Proca clouds, since they are only marginally bound. We comment on the possible existence of non-linear counterparts to these marginal clouds.

Renormalized stress tensor for massive fields in Kerr-Newman spacetime

In a four-dimensional spacetime, the DeWitt-Schwinger expansion of the effective action associated with a massive quantum field reduces, after renormalization and in the large mass limit, to a single term constructed from the purely geometrical Gilkey-DeWitt coefficient $a_3$ and its metric variation provides a good analytical approximation for the renormalized stress-energy tensor of the quantum field. Here, from the general expression of this tensor, we obtain analytically the renormalized stress-energy tensors of the massive scalar field, the massive Dirac field and the Proca field in Kerr-Newman spacetime. It should be noted that, even if, at first sight, the expressions obtained are complicated, their structure is in fact rather simple, involving naturally spacetime coordinates as well as the mass $M$, the charge $Q$ and the rotation parameter $a$ of the Kerr-Newman black hole and permitting us to recover rapidly the results already existing in the literature for the Schwarzschild, Reissner-Nordstr\"om and Kerr black holes (and to correct them in the latter case). In the absence of exact results in Kerr-Newman spacetime, our approximate renormalized stress-energy tensors could be very helpful, in particular to study the backreaction of massive quantum fields on this spacetime or on its quasinormal modes.

Majorana-Oppenheimer Approach to Proca Field Equations

A Dirac-like equation for a massive field obeying the classical Proca equations of motion (PMO) is proposed in close analogy with Majorana’s construct for Maxwell electrodynamics. Its underlying algebraic structure is examined and a plausible physical interpretation is discussed. The behavior of the PMO equations in the presence of an external electromagnetic field is also investigated in the low energy limit, via unitary transformations similar to the Foldy-Wouthuysen canonical transformation for a Dirac fermion.

Extended Gauss–Bonnet gravities in Weyl geometry

In this paper we consider an extended Gauss–Bonnet gravity theory in arbitrary dimensions and in a space provided with a Weyl connection, which is torsion-free but non-metric-compatible, the non-metricity tensor being determined by a vector field. The action considered consists of the usual Einstein–Hilbert action plus all the terms quadratic in the curvature that reduce to the usual Gauss–Bonnet term for vanishing Weyl connection, i.e., when only the Levi-Civita part of the connection is present. We expand the action in terms of Riemannian quantities and obtain vector–tensor theories. We find that all the free parameters only appear in the kinetic term of the vector field, so two branches are possible: one with a propagating vector field and another one where the vector field does not propagate. We focus on the propagating case. We find that in four dimensions, the theory is equivalent to Einstein's gravity plus a Proca field. This field is naturally decoupled from matter, so it represents a natural dark matter candidate. Also for d = 4, we discuss a non-trivial cubic term in the curvature that can be constructed without spoiling the second-order nature of the field equations, because it leads to the vector–tensor Horndeski interaction. In arbitrary dimensions, the theory becomes more involved. We show that, even though the vector field presents kinetic interactions which do not have symmetry, there are no additional propagating degrees of freedom with respect to the usual massive case. We show that, interestingly, this relies on the fact that the corresponding Stückelberg field belongs to a specific class within the general Horndeski theories. Finally, since Weyl geometries provide the natural ground on which to build scale invariant theories, we apply the usual Weyl gauging in order to make the Horndeski action locally scale invariant, and discuss new terms that can be added.

Tachyonic quantum densities of relativistic electron plasmas: Cherenkov spectra of γ-ray pulsars

Abstract Tachyonic Cherenkov radiation in second quantization can explain the subexponential spectral tails of GeV γ-ray pulsars (Crab pulsar, PSR J1836+5925, PSR J0007+7303, PSR J2021+4026) recently observed with the Fermi-LAT, VERITAS and MAGIC telescopes. The radiation is emitted by a thermal ultra-relativistic electron plasma. The Cherenkov effect is derived from a Maxwell–Proca field with negative mass-square in a dispersive spacetime. The frequency variation of the tachyon mass results in exp ( − β ˆ ω 1 − ρ ) attenuation of the asymptotic Cherenkov energy flux, where β ˆ is a decay constant related to the electron temperature and ρ is the frequency scaling exponent of the tachyon mass. An exponent in the range 0 ρ 1 can reproduce the observed subexponential decay of the energy flux. For the Crab pulsar, we find ρ = 0.81 ± 0.02 , inferred from the substantially weaker-than-exponential decay of its spectral tail measured by MAGIC over an extended energy range. The scaling exponent ρ determines whether the group velocity of the tachyonic γ-rays is sub- or superluminal.

Mode-sum construction of the two-point functions for the Stueckelberg vector fields in the Poincaré patch of de Sitter space

We perform canonical quantization of the Stueckelberg Lagrangian for massive vector fields in the conformally flat patch of de Sitter space in the Bunch-Davies vacuum and find their Wightman two-point functions by the mode-sum method. We discuss the zero-mass limit of these two-point functions and their limits where the Stueckelberg parameter ξ tends to zero or infinity. It is shown that our results reproduce the standard flat-space propagator in the appropriate limit. We also point out that the classic work of Allen and Jacobson [“Vector two-point functions in maximally symmetric spaces,” Commun. Math. Phys. 103, 669 (1986)] for the two-point function of the Proca field and a recent work by Tsamis and Woodard [“Maximally symmetric vector propagator,” J. Math. Phys. 48, 052306 (2007)] for that of the transverse vector field are two limits of our two-point function, one for ξ → ∞ and the other for ξ → 0. Thus, these two works are consistent with each other, contrary to the claim by the latter authors.

Generalization of the Proca Action

We consider the Lagrangian of a vector field with derivative self-interactions with a priori arbitrary coefficients. Starting with a flat space-time we show that for a specialchoice of the coefficients of the self-interactions the ghost-like pathologies disappear. This constitutes the Galileon-type generalization of the Proca action with only three propagating physical degrees of freedom. The longitudinal mode of the vector field is associated to the usual Galileon interactions for a specific choice of the overall functions. In difference to a scalar Galileon theory, the generalized Proca field has more free parameters. We then extend this analysis to a curved background. The resulting theory is the Horndeski Proca action with second order equations of motion on curved space-times.

Resonant excitation of black holes by massive bosonic fields and giant ringings

We consider the massive scalar field, the Proca field and the Fierz-Pauli field in the Schwarzschild spacetime and we focus more particularly on their long-lived quasinormal modes. We show numerically that the associated excitation factors have a strong resonant behavior and we confirm this result analytically from semiclassical considerations based on the properties of the unstable circular geodesics on which a massive particle can orbit the black hole. The conspiracy of (i) the long-lived behavior of the quasinormal modes and (ii) the resonant behavior of their excitation factors induces intrinsic giant ringings, i.e., ringings of a huge amplitude. Such ringings, which are moreover slowly decaying, are directly constructed from the retarded Green function. If we describe the source of the black hole perturbation by an initial value problem with Gaussian initial data, i.e., if we consider the excitation of the black hole from an extrinsic point of view, we can show that these extraordinary ringings are still present. This suggests that physically realistic sources of perturbations should generate giant and slowly decaying ringings and that their existence could be used to constrain ultralight bosonic field theory interacting with black holes.

An Analytic Approximation for Researching Tunneling Rate from Black Hole in Proca Field

The analytic solution of a static spherical symmetrical Proca black hole is discussed in this paper. As in the massive vector field, Proca black hole can be considered as the analogy of RN background plus a perturbation with the same order as μ2 due to the mass of vector particle μ satisfying μ2 ≪ 1. Through the action of Proca field, we find the analytic form with the first and arbitrary order approximation. Furthermore, we divide the results into 3 groups according to the real zero solutions of the background (i.e., spacetime in massless vector field). Finally we analyze the Hawking radiation of such black hole, which is significant for constructing black hole thermodynamic.

Hawking Radiation of Black Hole in Einstein-Proca Theory

The Hawking radiation of black hole in Einstein-Proca theory is discussed in this paper. The Einstein-Proca black hole is more general than Reissner-Nordstrom black hole, because Proca field is massive vector field. We calculate several quantum perturbations in this spacetime, and obtain the Hawking radiation at the horizon in Einstein-Proca theory.