Proca Field Research Articles

Evolution of Lifshitz metric anisotropies in Einstein–Proca theory under the Ricci–DeTurck flow

By starting from a Perelman entropy functional and considering the Ricci–DeTurck flow equations, we analyze the behavior of Einstein–Hilbert and Einstein–Proca theories with Lifshitz geometry as functions of a flow parameter. In the former case, we found one consistent fixed point that represents flat space-time as the flow parameter tends to infinity. Massive vector fields in the latter theory enrich the system under study and have the same fixed point achieved at the same rate as in the former case. The geometric flow is parametrized by the metric coefficients and represents a change in anisotropy of the geometry toward an isotropic flat space-time as the flow parameter evolves. Indeed, the flow of the Proca fields depends on certain coefficients that vanish when the flow parameter increases, rendering these fields constant. We have been able to write down the evolving Lifshitz metric solution with positive, but otherwise arbitrary, critical exponents relevant to geometries with spatially anisotropic holographic duals. We show that both the scalar curvature and matter contributions to the Ricci–DeTurck flow vanish under the flow at a fixed point consistent with flat space-time geometry. Thus, the behavior of the scalar curvature always increases, homogenizing the geometry along the flow. Moreover, the theory under study keeps positive-definite, but decreasing, the entropy functional along the Ricci–DeTurck flow.

Normal modes of Proca fields in AdS spacetime

A normal mode analysis for Proca fields in the anti-de Sitter (AdS) spacetime is given. It is found that the equations for the Proca field can be decoupled analytically. This is performed by changing the basis of the vector spherical harmonics decomposition. The normal modes and the normal mode frequencies of the Proca equation in the AdS spacetime are then analytically determined. It is also shown that the Maxwell field can be recovered by taking the massless limit of the Proca field with care so that the nonphysical gauge modes are eliminated.

Impact of the wavelike nature of Proca stars on their gravitational-wave emission

We present a systematic study of the dynamics and gravitational-wave emission of head-on collisions of spinning vector boson stars, known as Proca stars. To this aim we build a catalog of about 800 numerical-relativity simulations of such systems. We find that the wavelike nature of bosonic stars has a large impact on the gravitational-wave emission. In particular, we show that the initial relative phase $\mathrm{\ensuremath{\Delta}}\ensuremath{\epsilon}={\ensuremath{\epsilon}}_{1}\ensuremath{-}{\ensuremath{\epsilon}}_{2}$ of the two complex fields forming the stars (or, equivalently, the relative phase at merger) strongly impacts both the emitted gravitational-wave energy and the corresponding mode structure. This leads to a nonmonotonic dependence of the emission on the frequency of the secondary star ${\ensuremath{\omega}}_{2}$, for fixed frequency ${\ensuremath{\omega}}_{1}$ of the primary. This phenomenology, which has not been found for the case of black-hole mergers, reflects the distinct ability of the Proca field to interact with itself in both constructive and destructive manners. We postulate this may serve as a smoking gun to shed light on the possible existence of these objects.

Nonminimal non-Abelian quantum vector fields in curved spacetime

The quantum effective action of non-minimal vector fields with Abelian or non-Abelian gauge degrees of freedom in curved spacetime is studied. The Proca or Yang-Mills fields are coupled to a local mass-like term acting in both coordinate and gauge spaces. Pathologies due to gauge invariance in the ultraviolet are avoided through the introduction of a non-Abelian version of the Stueckelberg field. It is found that the breaking of gauge invariance induced by the mass term affects only the tree level part of the effective action. The ultraviolet divergent part of the effective action to one loop is obtained using the method of covariant symbols and dimensional regularization. Formulas are given valid for any spacetime dimension and explicit results are shown for the two-dimensional case. As already happened for a single vector field, the ultraviolet divergences are local but not of polynomial type.

Geometric Proca with matter in metric-Palatini gravity

In the present work, we study linear, torsion-free metric-Palatini gravity, extended by the quadratics of the antisymmetric part of the Ricci tensor and extended also by the presence of the affine connection in the matter sector. We show that this extended metric-Palatini gravity reduces dynamically to the general relativity plus a geometrical massive vector field corresponding to non-metricity of the connection. We also show that this geometric Proca field couples to fermions universally. We derive static, spherically symmetric field equations of this Einstein-geometric Proca theory. We study possibility of black hole solutions by taking into account the presence of a dust distribution that couples to the geometric Proca. Our analytical and numerical analyses show that the presence of this dust worsens the possibility of horizon formation. We briefly discuss possible roles of this universally-coupled geometric Proca in the astrophysical and collider processes.

Resolving the pathologies of self-interacting Proca fields: A case study of Proca stars

It has been argued that a self-interacting massive vector field is pathological due to a dynamical formation of a singular effective metric of the vector field, which is the onset of a gradient or ghost instability. We discuss that this singularity formation is not necessarily a fundamental problem but a breakdown of the effective field theory (EFT) description of the massive vector field. By using a model of ultraviolet (UV) completion of the massive vector field, we demonstrate that a Proca star, a self-gravitating condensate of the vector field, continues to exist even after the EFT suffers from a gradient instability without any pathology at UV, in which the EFT description is still valid and the gradient instability in EFT may be interpreted as a standard dynamical instability of a high-density boson star from the UV perspective. On the other hand, we find that the EFT description is broken before the ghost instability appears. This suggests that a heavy degree of freedom may be spontaneously excited to cure the pathology of the EFT as the EFT dynamically tends to approach the onset of the ghost instability.

Ghost Instabilities in Self-Interacting Vector Fields: The Problem with Proca Fields.

Massive vector fields feature in several areas of particle physics, e.g., as carriers of weak interactions, dark matter candidates, or an effective description of photons in a plasma. Here, we investigate vector fields with self-interactions by replacing the mass term in the Proca equation with a general potential. We show that this seemingly benign modification inevitably introduces ghost instabilities of the same kind as those recently identified for vector-tensor theories of modified gravity (but in this simpler, minimally coupled theory). It has been suggested that nonperturbative dynamics may drive systems away from such instabilities. We demonstrate that this is not the case by evolving a self-interacting Proca field on a Kerr background, where it grows due to the superradiant instability. The system initially evolves as in the massive case, but instabilities are triggered in a finite time once the self-interaction becomes significant. These instabilities have implications for the formation of condensates of massive, self-interacting vector bosons, the possibility of spin-one bosenovae, vector dark matter models, and effective models for interacting photons in a plasma.

AdS3/AdS2 degression of Fronsdal fields

We analyze the Kaluza–Klein type procedure in AdS3 space called the dimensional degression. The topological theory of the Fronsdal field in AdS3 is reformulated in terms of the fields propagating in AdS2. We find that the Fronsdal field in AdS3 leads to finitely many Kaluza–Klein modes. Namely, the obtained spectrum is the massive Klein–Gordon and Proca fields in AdS2. The result is derived by using the specific mode expansion, gauge fixing, and 2–dimensional Schouten identities.

Lagrangian formulation of the de Broglie–Proca for space with noninteger dimension

Fractals are the fundamental characteristics of the universe. In this research, the Lagrangian formulation of the de Broglie–Proca (dBP) field for space with noninteger dimension is examined. Based on the Laplace of the dBP’s equations in noninteger space, the solutions of these equations are found where the dimensionality of the space is [Formula: see text]. Also, we obtain the wave equation for the electric field and the wave equation for the magnetic field in noninteger space.

Pseudogauges and relativistic spin hydrodynamics for interacting Dirac and Proca fields

We present the explicit expressions of different pseudo-gauge transformations for Dirac and Proca fields considering a general interaction term. The particular case of the interaction of Dirac and Proca fields with a background electromagnetic field is also studied. Starting from the quantum kinetic theory with collisions derived from the Wigner-function formalism for massive spin-1/2 and spin-1 particles, we establish a connection between different pseudo-gauges and relativistic spin hydrodynamics. The physical implications of the various decompositions of orbital and spin angular momentum are discussed.

Superradiance in massive vector fields with spatially varying mass

Superradiance is a process by which massive bosonic particles can extract energy from spinning black holes, leading to the build up of a condensate if the particle has a Compton wavelength comparable to the black hole's Schwarzschild radius. One interesting possibility is that superradiance may occur for photons in a diffuse plasma, where they gain a small effective mass. Studies of the spin-0 case have indicated that such a build up is suppressed by a spatially varying effective mass, supposed to mimic the photons' interaction with a physically realistic plasma density profile. We carry out relativistic simulations of a massive Proca field evolving on a Kerr background, with modifications to account for the spatially varying effective mass. This allows us to treat the spin-1 case directly relevant to photons, and to study the effect of thinner disk profiles in the plasma. We find similar qualitative results to the scalar case, and so support the conclusions of that work: either a constant asymptotic mass or a shell-like plasma structure is required for superradiant growth to occur. We study thin disks and find a leakage of the bosonic condensate that suppresses its growth, concluding that thick disks are more likely to support the instability.

Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

We present new results concerning the Proca massive vector field in a Schwarzschild-AdS black hole geometry. We provide a first principles analysis of Proca vector fields in this geometry using both the vector spherical harmonic (VSH) separation method and the Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k-Santos (FKKS) method that separates the relevant equations in spinning geometries. The analysis in the VSH method shows, on one hand, that it is arduous to separate the scalar-type from the vector-type polarizations of the electric sector of the Proca field, and on the other hand, it displays clearly the electric and the magnetic mode sectors. The analysis in the FKKS method is performed by taking the nonrotating limit of the Kerr-AdS spacetime, and shows that the ansatz decouples the polarizations in the electric mode sector even in the nonrotating limit. On the other hand, it captures only two of the three possible polarizations, the magnetic mode sector is missing. The reason for the absence of this polarization is related to the degeneracy of the principal tensor in static spherical symmetric spacetimes. The degrees of freedom and quasinormal modes in both separation methods of the Proca field are found. The frequencies of the quasinormal modes are also computed. For the electric mode sector in the VSH method the frequencies are found through an extension, which substitutes number coefficients by matrix coefficients, of the Horowitz-Hubeny numerical procedure, whereas for the magnetic mode sector in the VSH method and the electric sector of the FKKS method it is shown that a direct use of the procedure can be made. The values of the quasinormal mode frequencies obtained for each method are compared and showed to be in good agreement with each other. This further supports the analytical approaches presented here for the behavior of the Proca field in a Schwarzschild-AdS black hole background.

Proca balls with angular momentum or flux of electric field

Within SU(2) Higgs-Proca theory, we obtain a family of nontopological static solutions describing localized, finite-energy configurations (Proca balls). The gauge symmetry of the theory is explicitly broken by introducing a vector Proca field whose components have different masses. Such solutions describe particlelike systems, the crucial feature of which is that they either possess a nonzero total angular momentum or have a flux of electric field through the plane of symmetry of such objects. It is shown that the angular momentum is provided by static crossed electric and magnetic fields. The existence of the solutions is caused by the fact that we circumvent the conditions of the no-go theorem, according to which there are no stationary and axially symmetric spinning excitations for the 't Hooft-Polyakov monopoles, Julia-Zee dyons, sphalerons, and also vortices. The dependence of some integral physical quantities on the ratio of the Proca-field masses is studied. It is demonstrated that the inclusion of external sources (charges) enables one to obtain solutions with equal Proca-field masses. We also discuss the possibilities of using quarks as sources of the Proca field under investigation and for treating the Proca balls as glueballs in SU(2) Higgs-Proca theory.

Quantum stability of a new Proca theory

The construction of general derivative self-interactions for a massive Proca field relies on the well-known condition for constrained systems of having a degenerate Hessian. The nature of the existing constraints algebra will distinguish among different classes of interactions. Proca-Nuevo interactions enjoy a non-trivial constraint by mixing terms of various order whereas Generalized Proca interactions satisfy the degeneracy condition order by order for each individual Lagrangians. In both cases the vector field propagates at most three degrees of freedom. It has been shown that the scattering amplitudes of Proca-Nuevo arising at the tree level always differ from those of the Generalized Proca, implying their genuinely different nature and a lack of relation by local field redefinitions. In this work, we show the quantum stability of the Proca-Nuevo theory below a specific UV cut-off. Although Proca-Nuevo and Generalized Proca are different inherently in their classical structure, both have the same high energy behaviour when quantum corrections are taken into account. The arising counter terms have the exact same structure and scaling. This might indicate that whatever UV completion they may come from, we expect it to be of similar nature.

On the duality of massive Kalb-Ramond and Proca fields

We compare the massive Kalb-Ramond and Proca fields with a quartic self-interaction and show that the same strong coupling scale is present in both theories. In the Proca theory, the longitudinal mode enters the strongly coupled regime beyond this scale, while the two transverse modes propagate further and survive in the massless limit. In contrast, in case of the massive Kalb-Ramond field, the two transverse modes become strongly coupled beyond the Vainshtein scale, while the pseudo-scalar mode remains in the weak coupling regime and survives in the massless limit. This indicates a contradiction with the numerous claims in the literature (see eg. [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]) that these theories are dual to each other.

Proca theory from the spinning worldline

We obtain Proca field theory from the quantisation of the mathcal{N} = 2 supersymmetric worldline upon supplementing the graded BRST-algebra with an extra multiplet of oscillators. The linearised theory describes the BV-extended spectrum of Proca theory, together with a Stückelberg field. When coupling the theory to background fields we derive the Proca equations, arising as consistency conditions in the BRST procedure. We also explore non-abelian modifications, complexified vector fields as well as coupling to a dilaton field. We propose a cubic action on the space of BRST-operators which reproduces the known Proca action.

Circuit complexity in Proca theory

In this paper, we study circuit complexity in Proca theory with Nielsen's approach and Fubini-Study (FS) metric approach. We place the fields on a lattice to gain a regularized theory, and obtain the ground state by adopting proper coordinates. We calculate complexities of the ground and thermofield double (TFD) states with Nielsen's approach, complexity of the TFD state is found to grows like a logarithmic function. We quantize the Proca fields and give the approximate ground state and TFD state by acting unitary circuit operators on the associated reference states. The circuit lengths are calculated with FS metric, the minimal lengths are given according to the associated geometric spaces. The complexity of TFD state is found to grows linearly with time.

Thermodynamics of Bardeen regular black hole with generalized uncertainty principle

This study explores the emission of massive charged spin-1 particles from the background of Bardeen regular spacetime by the semi-classical method used to study the Hawking radiation spectrum. We employed the Hamilton–Jacobi method and Wentzel–Kramers–Brillouin approximation technique with the suitable form of the wave function to solve the Proca field equation. We calculated the tunneling probability of outgoing spin-1 particles and the corresponding thermodynamic temperature. Furthermore, we obtained the modified thermodynamic quantities like temperature, entropy as well as heat capacity by utilizing the quadratic form of generalized uncertainty principle (GUP) and minimal length. In the end, we investigated the local stability as well as phase transitions of the Bardeen black hole in the context of GUP-modified heat capacity.

Axially symmetric Proca-Higgs boson stars

We consider strongly gravitating configurations consisting of coupled real Higgs scalar field and vector (Proca) field of mass $\mu_P$. For such a system, we find static regular axially symmetric solutions describing asymptotically flat configurations which may be referred to as Proca-Higgs miniboson stars, since their total mass and spatial dimension are of order $M_{\text{Pl}}^2/\mu_P$ and $\mu_P^{-1}$, respectively.The system possesses an axially symmetric dipole field and may be regarded as a Proca dipole.

Destabilization of Black Holes and Stars by Generalized Proca Fields.

We demonstrate that black holes and stars in general relativity can be destabilized by perturbations of nonminimally coupled vector fields. Focusing on static and spherically symmetric backgrounds, our analysis shows that black holes with sufficiently small mass and stars with sufficiently high densities are subject to ghost- or gradient-type instabilities. This holds for a large class of Einstein-Proca theories with nonminimal couplings, including generalized Proca models that have sparked attention for their potential role in cosmology and astrophysics. The stability criteria translate into bounds of relevance for low-scale theories of dark energy and for ultralight dark matter scenarios.