Generalized Proca Theories Research Articles

Quantum corrections to the generalized Proca theory via a matter field

We study the quantum corrections to the generalized Proca theory via matter loops. We consider two types of interactions, linear and nonlinear in the vector field. Calculating the one-loop correction to the vector field propagator, three- and four-point functions, we show that the non-linear interactions are harmless, although they renormalize the theory. The linear matter-vector field interactions introduce ghost degrees of freedom to the generalized Proca theory. Treating the theory as an effective theory, we calculate the energy scale up to which the theory remains healthy.

Black holes in the generalized Proca theory

We investigate static and spherically symmetric black hole solutions in the generalized Proca theory which corresponds to the generalization of the shift-symmetric scalar–tensor Horndeski theory to the vector–tensor theory. Any solution obtained in this paper possesses a constant spacetime norm of the vector field, $$X:=-\frac{1}{2}g^{\mu \nu }A_\mu A_\nu =X_0=\mathrm{constant}$$ . The solutions in the theory with generalized quartic coupling $$G_4(X)$$ generalize the stealth Schwarzschild and the Schwarzschild- (anti-) de Sitter solutions obtained in the theory with the nonminimal coupling to the Einstein tensor $$G^{\mu \nu } A_\mu A_\nu $$ . While in the vector–tensor theory with the coupling $$G^{\mu \nu }A_\mu A_\nu $$ the electric charge does not explicitly affect the spacetime geometry, in more general cases with nonzero $$G_{4XX}(X_0)\ne 0$$ this property does not hold in general. The solutions in the theory with generalized cubic coupling $$G_3(X)$$ are given by the Schwarzschild- (anti-) de Sitter spacetime, where the dependence on $$G_3(X)$$ does not appear in the metric function.

Cosmology in beyond-generalized Proca theories

The beyond-generalized Proca theories are the extension of second-order massive vector-tensor theories (dubbed generalized Proca theories) with two transverse vector modes and one longitudinal scalar besides two tensor polarizations. Even with this extension, the propagating degrees of freedom remain unchanged on the isotropic cosmological background without an Ostrogradski instability. We study the cosmology in beyond-generalized Proca theories by paying particular attention to the dynamics of late-time cosmic acceleration and resulting observational consequences. We derive conditions for avoiding ghosts and instabilities of tensor, vector, and scalar perturbations and discuss viable parameter spaces in concrete models allowing the dark energy equation of state smaller than $-1$. The propagation speeds of those perturbations are subject to modifications beyond the domain of generalized Proca theories. There is a mixing between scalar and matter sound speeds, but such a mixing is suppressed during most of the cosmic expansion history without causing a new instability. On the other hand, we find that derivative interactions arising in beyond-generalized Proca theories give rise to important modifications to the cosmic growth history. The growth rate of matter perturbations can be compatible with the redshift-space distortion data due to the realization of gravitational interaction weaker than that in generalized Proca theories. Thus, it is possible to distinguish the dark energy model in beyond-generalized Proca theories from the counterpart in generalized Proca theories as well as from the $\Lambda$CDM model.

Field redefinitions in theories beyond Einstein gravity using the language of differential forms

We study the role of field redefinitions in general scalar-tensor theories. In particular, we first focus on the class of field redefinitions linear in the spin-2 field and involving derivatives of the spin-0 mode, generically known as disformal transformations. We start by defining the action of a disformal transformation in the tangent space. Then, we take advantage of the great economy of means of the language of differential forms to compute the full transformation of Horndeski's theory under general disformal transformations. We obtain that Horndeski's action maps onto itself modulo a reduced set of non-Horndeski Lagrangians. These new Lagrangians are found to be invariant under disformal transformation that depend only in the first derivatives of the scalar. Moreover, these combinations of Lagrangians precisely appear when expressing in our basis the constraints of the recently proposed Extended Scalar-Tensor (EST) theories. These results allow us to classify the different orbits of scalar-tensor theories invariant under particular disformal transformations, namely the special disformal, kinetic disformal and disformal Horndeski orbits. In addition, we consider generalizations of this framework. We find that there are possible well-defined extended disformal transformations that have not been considered in the literature. However, they generically cannot link Horndeski theory with EST theories. Finally, we study further generalizations in which extra fields with different spin are included. These field redefinitions can be used to connect different gravity theories such as multi-scalar-tensor theories, generalized Proca theories and bi-gravity. We discuss how the formalism of differential forms could be useful for future developments in these lines.

Stable solutions of inflation driven by vector fields

Many models of inflation driven by vector fields alone have been known to be plagued by pathological behaviors, namely ghost and/or gradient instabilities. In this work, we seek a new class of vector-driven inflationary models that evade all of the mentioned instabilities. We build our analysis on the Generalized Proca Theory with an extension to three vector fields to realize isotropic expansion. We obtain the conditions required for quasi de-Sitter solutions to be an attractor analogous to the standard slow-roll one and those for their stability at the level of linearized perturbations. Identifying the remedy to the existing unstable models, we provide a simple example and explicitly show its stability. This significantly broadens our knowledge on vector inflationary scenarios, reviving potential phenomenological interests for this class of models.

Extended vector-tensor theories

Recently, several extensions of massive vector theory in curved space-time have been proposed in many literatures. In this paper, we consider the most general vector-tensor theories that contain up to two derivatives with respect to metric and vector field. By imposing a degeneracy condition of the Lagrangian in the context of ADM decomposition of space-time to eliminate an unwanted mode, we construct a new class of massive vector theories where five degrees of freedom can propagate, corresponding to three for massive vector modes and two for massless tensor modes. We find that the generalized Proca and the beyond generalized Proca theories up to the quartic Lagrangian, which should be included in this formulation, are degenerate theories even in curved space-time. Finally, introducing new metric and vector field transformations, we investigate the properties of thus obtained theories under such transformations.

Absence of solid angle deficit singularities in beyond-generalized proca theories

In Gleyzes-Langlois-Piazza-Vernizzi (GLPV) scalar-tensor theories, which are outside the domain of second-order Horndeski theories, it is known that there exists a solid angle deficit singularity in the case where the parameter $\alpha_{\rm H}$ characterizing the deviation from Horndeski theories approaches a non-vanishing constant at the center of a spherically symmetric body. Meanwhile, it was recently shown that second-order generalized Proca theories with a massive vector field $A^{\mu}$ can be consistently extended to beyond-generalized Proca theories, which recover shift-symmetric GLPV theories in the scalar limit $A^{\mu} \to \nabla^{\mu} \chi$. In beyond-generalized Proca theories up to quartic-order Lagrangians, we show that solid angle deficit singularities are generally absent due to the existence of a temporal vector component. We also derive the vector-field profiles around a compact object and show that the success of the Vainshtein mechanism operated by vector Galileons is not prevented by new interactions in beyond generalized Proca theories.

Anisotropic cosmological solutions in massive vector theories

In beyond-generalized Proca theories including the extension to theories higher than second order, we study the role of a spatial component v of a massive vector field on the anisotropic cosmological background. We show that, as in the case of the isotropic cosmological background, there is no additional ghostly degrees of freedom associated with the Ostrogradski instability. In second-order generalized Proca theories we find the existence of anisotropic solutions on which the ratio between the anisotropic expansion rate Σ and the isotropic expansion rate H remains nearly constant in the radiation-dominated epoch. In the regime where Σ/H is constant, the spatial vector component v works as a dark radiation with the equation of state close to 1/3. During the matter era, the ratio Σ/H decreases with the decrease of v. As long as the conditions |Σ| ≪ H and v2 ≪ ϕ2 are satisfied around the onset of late-time cosmic acceleration, where ϕ is the temporal vector component, we find that the solutions approach the isotropic de Sitter fixed point (Σ = 0 = v) in accordance with the cosmic no-hair conjecture. In the presence of v and Σ the early evolution of the dark energy equation of state wDE in the radiation era is different from that in the isotropic case, but the approach to the isotropic value wDE(iso) typically occurs at redshifts z much larger than 1. Thus, apart from the existence of dark radiation, the anisotropic cosmological dynamics at low redshifts is similar to that in isotropic generalized Proca theories. In beyond-generalized Proca theories the only consistent solution to avoid the divergence of a determinant of the dynamical system corresponds to v = 0, so Σ always decreases in time.

Solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor

We investigate the static and spherically symmetric solutions in a class of the generalized Proca theory with the nonminimal coupling to the Einstein tensor. First, we show that the solutions in the scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor can be those in the generalized Proca theory with the vanishing field strength. We then show that when the field strength takes the nonzero value the static and spherically symmetric solutions can be found only for the specific value of the nonminimal coupling constant. Second, we investigate the first-order slow-rotation corrections to the static and spherically symmetric background. We find that for the background with the vanishing electric field strength the slowly rotating solution is identical to the Kerr- (anti-) de Sitter solutions in general relativity. On the other hand, for the background with the nonvanishing electric field strength the stealth property can realized at the first order in the slow-rotation approximation.

Effective gravitational couplings for cosmological perturbations in generalized Proca theories

We consider the finite interactions of the generalized Proca theory including the sixth-order Lagrangian and derive the full linear perturbation equations of motion on the flat Friedmann-Lema\^{i}tre-Robertson-Walker background in the presence of a matter perfect fluid. By construction, the propagating degrees of freedom (besides the matter perfect fluid) are two transverse vector perturbations, one longitudinal scalar, and two tensor polarizations. The Lagrangians associated with intrinsic vector modes neither affect the background equations of motion nor the second-order action of tensor perturbations, but they do give rise to non-trivial modifications to the no-ghost condition of vector perturbations and to the propagation speeds of vector and scalar perturbations. We derive the effective gravitational coupling $G_{\rm eff}$ with matter density perturbations under a quasi-static approximation on scales deep inside the sound horizon. We find that the existence of intrinsic vector modes allows a possibility for reducing $G_{\rm eff}$. In fact, within the parameter space, $G_{\rm eff}$ can be even smaller than the Newton gravitational constant $G$ at the late cosmological epoch, with a peculiar phantom dark energy equation of state (without ghosts). The modifications to the slip parameter $\eta$ and the evolution of growth rate $f\sigma_8$ are discussed as well. Thus, dark energy models in the framework of generalized Proca theories can be observationally distinguished from the $\Lambda$CDM model according to both cosmic growth and expansion history. Furthermore, we study the evolution of vector perturbations and show that outside the vector sound horizon the perturbations are nearly frozen and start to decay with oscillations after the horizon entry.

Cosmology in generalized Proca theories

We consider a massive vector field with derivative interactions that propagates only the 3 desired polarizations (besides two tensor polarizations from gravity) with second-order equations of motion in curved space-time. The cosmological implications of such generalized Proca theories are investigated for both the background and the linear perturbation by taking into account the Lagrangian up to quintic order. In the presence of a matter fluid with a temporal component of the vector field, we derive the background equations of motion and show the existence of de Sitter solutions relevant to the late-time cosmic acceleration. We also obtain conditions for the absence of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations in the small-scale limit. Our results are applied to concrete examples of the general functions in the theory, which encompass vector Galileons as a specific case. In such examples, we show that the de Sitter fixed point is always a stable attractor and study viable parameter spaces in which the no-ghost and stability conditions are satisfied during the cosmic expansion history.

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.

Beyond generalized Proca theories

We consider higher-order derivative interactions beyond second-order generalized Proca theories that propagate only the three desired polarizations of a massive vector field besides the two tensor polarizations from gravity. These new interactions follow the similar construction criteria to those arising in the extension of scalar-tensor Horndeski theories to Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories. On the isotropic cosmological background, we show the existence of a constraint with a vanishing Hamiltonian that removes the would-be Ostrogradski ghost. We study the behavior of linear perturbations on top of the isotropic cosmological background in the presence of a matter perfect fluid and find the same number of propagating degrees of freedom as in generalized Proca theories (two tensor polarizations, two transverse vector modes, and two scalar modes). Moreover, we obtain the conditions for the avoidance of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations. We observe key differences in the scalar sound speed, which is mixed with the matter sound speed outside the domain of generalized Proca theories.

Screening fifth forces in generalized Proca theories

For a massive vector field with derivative self-interactions, the breaking of the gauge invariance allows the propagation of a longitudinal mode in addition to the two transverse modes. We consider generalized Proca theories with second-order equations of motion in a curved space-time and study how the longitudinal scalar mode of the vector field gravitates on a spherically symmetric background. We show explicitly that cubic-order self-interactions lead to the suppression of the longitudinal mode through the Vainshtein mechanism. Provided that the dimensionless coupling of the interaction is not negligible, this screening mechanism is sufficiently efficient to give rise to tiny corrections to gravitational potentials consistent with solar-system tests of gravity. We also study the quartic interactions with the presence of non-minimal derivative coupling with the Ricci scalar and find the existence of solutions where the longitudinal mode completely vanishes. Finally, we discuss the case in which the effect of the quartic interactions dominates over the cubic one and show that local gravity constraints can be satisfied under a mild bound on the parameters of the theory.