Vector models with spontaneous Lorentz-symmetry breaking

After imposing the Gauss law constraint as an initial condition upon the Hilbert space of the Nambu model, in all its generic realizations, we recover QED in the corresponding non-linear gauge A_{\mu}A^{\mu}=n^{2}M^{2}. Our result is non-perturbative in the parameter M for n^{2}\neq 0 and can be extended to the n^{2}=0 case. This shows that in the Nambu model, spontaneous Lorentz symmetry breaking dynamically generates gauge invariance, provided the Gauss law is imposed as an initial condition. In this way electrodynamics is recovered, with the photon being realized as the Nambu-Goldstone modes of the spontaneously broken symmetry, which finally turns out to be non-observable

We have obtained an exact vacuum solution from a gravity sector contained in the minimal standard-model extension. The theoretical model assumes a Riemann spacetime coupled to the bumblebee field which is responsible for the spontaneous Lorentz symmetry breaking. The solution achieved in a static and spherically symmetric scenario establishes a Schwarzschild-like black hole. In order to study the effects of the spontaneous Lorentz symmetry breaking, we have investigated some classics tests including the advance of the perihelion, bending of light and Shapiro's time-delay. Furthermore, we have computed some upper-bounds from which the most stringent one attains a sensitivity at the $10^{-13}$ level.

It is shown that a spontaneously-broken gauge theory of the Lorentz group contains Ashtekar’s chiral formulation of general relativity accompanied by dust. From this perspective, gravity is described entirely by a connection ω valued in the Lie algebra of the complexified Lorentz group and a Lorentz-charged scalar field t. The model is ‘pre-geometric’ in the sense that the spacetime metric may be constructed only in the symmetry-broken regime. We speculate on the extent to which this dust may account for dark matter and on the behaviour of the theory in regimes where the symmetry is not broken.

In the theory of gravity, the spontaneous breaking of Lorentz symmetry due to the non-minimal coupling between the Kalb-Ramond field and Einstein gravity results in the existence of exactly static and spherically symmetric black hole solutions related to the Lorentz violating parameter. Based on the consideration of the interaction between the black hole and cosmological horizons, this paper studies the thermodynamic properties of Kalb-Ramond-de Sitter (KR-dS) spacetime. The Smarr relation expressed by equivalent thermodynamic quantities is found, and it is proved that the equivalent thermodynamic quantities of the KR-dS spacetime satisfy the universal Euler's theorem. It is discovered that the heat capacity of the KR-dS spacetime with respect to the ratio of the two horizons and the variation curve of the heat capacity with effective temperature possess the characteristics of Schottky specific heat. Moreover, the black hole and the cosmological horizon in the equivalent thermodynamic system are regarded as two different energy levels, and the heat capacity of the KR-dS spacetime is discussed using the general form given by the ordinary two-level system. It is found that the heat capacity of KR-dS spacetime described by equivalent thermodynamic quantities not only conforms to the characteristics of Schottky specific heat but also is consistent with the heat capacity of the ordinary two-level system. This result reflects that when the cosmological constant and the charged carried in the KR-dS spacetime remain unchanged, the heat capacity of the system can be represented by a universal two-level system. By comparing the maximum value of the heat capacity curves, the number of microscopic particles between the two horizons can be estimated, which reflects the quantum properties of the KR-dS spacetime. These studies will open a new perspective to probe the thermodynamics of black holes.

In this chapter we discuss the vector-tensor gravity theories, namely, Einstein-aether gravity and bumblebee gravity, for which we discuss some exact solutions. The special attention is devoted to the problem of spontaneous breaking of Lorentz symmetry in a curved space-time which motivated these models. Besides, we present a general discussion of possibility of developing a consistent Lorentz-breaking extension for gravity.

In our five-dimensional cosmological model, we investigate the role of a Lorentz violating vector "{\ae}ther" field on the moduli stabilization mechanism. We consider the case of a space-like {\ae}ther field on a compact circle with Maxwell-type kinetic term. The Casimir energy of certain combinations of massless and massive bulk fields generates a stabilizing potential for the radius of the compact direction while driving the accelerated expansion in the non-compact directions. It is shown that the {\ae}ther field can reduce the influence of the Casimir force and slow down the oscillation of the radion field. This property proves crucial to the stability of the extra dimension in the universe where non-relativistic matter is present. We speculate that this scenario might reveal a hidden connection between the dimensionality of spacetime and the spontaneous breaking of Lorentz symmetry.

In string or M theories, the spontaneous breaking of 10D or 11D Lorentz symmetry is required to describe our space-time. A direct approach to this issue is provided by the IIB matrix model. We study its 4D version, which corresponds to the zero volume limit of 4D super SU(N) Yang-Mills theory. Based on the moment of inertia as a criterion, spontaneous symmetry breaking (SSB) seems to occur, so that only one extended direction remains, as first observed by Bialas, Burda et al. However, using Wilson loops as probes of space-time we do not observe any sign of SSB in Monte Carlo simulations where N is as large as 48. This agrees with an earlier observation that the phase of the fermionic integral, which is absent in the 4D model, should play a crucial role if SSB of Lorentz symmetry really occurs in the 10D IIB matrix model.

It is shown that an alternative to the standard scalar quantum electrodynamics (QED) is possible. In this new version, there is only global gauge invariance as far as the charged scalar fields are concerned, although local gauge invariance is kept for the electromagnetic field. The electromagnetic coupling has the form jμ(Aμ +∂μB) where B is an auxiliary field and the current jμ is Aμ independent, so that no "sea gull terms" are introduced. As a consequence of the absence of sea gulls, it is seen that no Klein paradox appears in the presence of a strong square well potential. In a model of this kind, spontaneous breaking of symmetry does not lead to photon mass generation, instead the Goldstone boson becomes a massless source for the electromagnetic field. When spontaneous symmetry breaking takes place infrared questions concerning the theory and generalizations to global vector QED are discussed. In this framework, Q-Balls and other nontopological solitons that owe their existence to a global U(1) symmetry can be coupled to electromagnetism and could represent multiply charged particles now in search in the large hadron collider (LHC). Furthermore, we give an example where an "Emergent" Global Scalar QED can appear from an axion–photon system in an external magnetic field. Finally, formulations of Global Scalar QED that allow perturbative expansions without sea gulls are developed.

The structure of annealed and quenched models with localU(1) gauge invariance is studied in terms of the Helmholtz free energy. The first non-trivial, or one-loop, account of fluctuations in the annealed model suggests that spontaneous symmetry breaking occurs in two and three dimensions, through a first-order phase transition. Within the same approximation scheme, the quenched model displays a continuous phase transition. A more complete account of the fluctuations in the annealed model changes the nature of the transition to a continuous one, whereas spontaneous symmetry breaking is then absent with quenched disorder.

A domain-wall pump by an extended cluster model of S=1/2 spins is proposed with local U(1) gauge invariance. In contrast to the conventional pump, the current and the gauge field are defined on sites instead of links. Its snapshot ground state is gapped and doubly degenerated due to Z_{2} invariance, which is broken by an infinitesimal boundary magnetic field. The ground state associated with the spontaneous symmetry breaking is, at the same time, symmetry-protected with additional spatial inversion that is characterized by the Z_{2} Berry phase defined by the twist on the boundary site. We investigate the topological domain-wall pump with and without boundaries. The topological pump associated with the inversion symmetry-breaking path induces a nontrivial Chern number of bulk and a singular behavior of edge states of the domain wall. Generalization to the multispin interaction is also explicitly given. These extended quantum degrees of freedom are prominent qubits that are topologically protected.

An analysis of the spontaneous breaking of chiral symmetry in three dimensional electrodynamics is described. It is argued that this model, when treated in a 1/ N expansion, does exhibit spontaneous chiral symmetry breaking. This is established by finding analytic and numer· ical solutions to the Dyson-Schwinger equation and then computing the composite-operator effective potential. During the fifty years since Y uka wa proposed the J[ meson as the intermediary of nuclear forces, our view of this particle has continued to evolve. We now know it to be a quark-antiquark composite, and in quantum chromodynamics, QCD, we have a strong interaction theory that may well describe its dynamics and that of the other hadrons. Among all the hadrons, however, the 1f meson is believed to play a special role. The underlying theory of strong interactions posseses a near chiral symmetry, SU(2h X SU(2) R, because of the approximate masslessness of the up and down quarks. This symmetry must then break spontaneously in order to explain the effective 300 MeV masses that these quarks appear to possess as the constituents of hadrons. The spontaneous breaking of any continuous symmetry necessarily leads to the existence of massless Goldstone bosons, and in the case of chiral symmetry, the 1f mesons approximately play this role. The small bare mass of the up and down quarks leads to a small mass for 1f mesons 1

We show how trinification models based on the gauge group SU(3)C× SU(3)L× SU(3)R realized near the TeV scale can provide naturally a variety of dark matter (DM) candidates. These models contain a discrete T parity which may remain unbroken even after spontaneous symmetry breaking. The lightest T-odd particle, which could be a fermion, a scalar, or a gauge boson, can constitute the dark matter of the universe. This framework naturally admits a doublet-singlet fermionic DM, a singlet scalar DM, or a vector boson DM. Here we develop the vector boson DM scenario wherein the DM couples off-diagonally with the usual fermions and vector-like fermions present in the theory. We show consistency of this framework with dark matter relic abundance and direct detection limits as well as LHC constraints. We derive upper limits of 900 GeV on the vector gauge boson DM mass and 4.5 TeV on the vector-like quark masses. We also show the consistency of spontaneous gauge symmetry breaking down to Standard Model times an extra U(1) while preserving the T-parity.

Spontaneous Lorentz violation realized through a nonlinear vector field constraint of the type A μ A μ = M 2 ( M is the proposed scale for Lorentz violation) is shown to generate massless vector Goldstone bosons, gauging the starting global internal symmetries in arbitrary relativistically invariant theories. The gauge invariance appears in essence as a necessary condition for these bosons not to be superfluously restricted in degrees of freedom, apart from the constraint due to which the true vacuum in a theory is chosen by the Lorentz violation. In the Abelian symmetry case the only possible theory proves to be QED with a massless vector Goldstone boson naturally associated with the photon, while the non-Abelian symmetry case results in a conventional Yang–Mills theory. These theories, both Abelian and non-Abelian, look essentially nonlinear and contain particular Lorentz (and CPT) violating couplings when expressed in terms of the pure Goldstone vector modes. However, they do not lead to physical Lorentz violation due to the simultaneously generated gauge invariance.

The nature of superfluid, superconducting, and magnetic ordering is elucidated for mesoscopic systems in which the single-particle level spacing is much larger than both the temperature and the critical temperature of ordering. Ordering is defined as a spontaneous breaking of symmetry, the gauge invariance and time reversal being by definition symmetries broken in superfluidity (superconductivity) and magnetism contexts, respectively. Superfluidity and superconductivity are realized in thermodynamic equilibrium states with a nonintegral average number of particles and are accompanied by the spontaneous breaking of time homogeneity. In Fermi systems, two types of superfluidity and superconductivity are possible which are characterized by the presence of pair or single-particle “condensates.” The latter is remarkable in that spontaneous breaking of fundamental symmetries such as spatial 2π rotation and double time reversal takes place. Possible experiments on metallic nanoparticles and ultracold atomic gases in magnetic traps are discussed.

We study NfD6-brane probes in the supergravity background dual to NcD4-branes compactified on a circle with supersymmetry-breaking boundary conditions. In the limit in which the resulting Kaluza-Klein modes decouple, the gauge theory reduces to non-supersymmetric, four-dimensional QCD with Nc colours and Nf << Ncflavours. As expected, this decoupling is not fully realised within the supergravity/Born-Infeld approximation. For Nf = 1 and massless quarks, mq = 0, we exhibit spontaneous chiral symmetry breaking by a quark condensate, ⟨ψ⟩≠0, and find the associated massless `pion' in the spectrum. The latter becomes massive for mq > 0, obeying the Gell-Mann-Oakes-Renner relation: Mπ2 = −mq ⟨ψ⟩/fπ2. In the case Nf>1 we provide a holographic version of the Vafa-Witten theorem, which states that the U(Nf) flavour symmetry cannot be spontaneously broken. Further, we find Nf2−1 unexpectedly light pseudo-scalar mesons in the spectrum. We argue that these are not (pseudo-)Goldstone bosons and speculate on the string mechanism responsible for their lightness. We then study the theory at finite temperature and exhibit a phase transition associated with a discontinuity in ⟨ψ⟩(T). D6/ pairs are also briefly discussed.