Relativity symmetries and Lie algebra contractions

We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n)$SO(m,n)$ symmetry as an isometry on an m+n$m+n$ dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n)$G(m,n)$ preserving a symmetry of the same type at dimension m+n−1$m+n-1$, e.g.   a G(m,n−1)$G(m,n-1)$, together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4)$SO(2,4)$, which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3)$G(1,3)$ may be relevant to real physics.     

Geometrical CP violation from non-renormalisable scalar potentials

We consider in detail the non-renormalisable scalar potential of three Higgs doublets transforming as an irreducible triplet of Δ(27)$\mathrm{\Delta }(27)$ or Δ(54)$\mathrm{\Delta }(54)$. We start from a renormalisable potential that spontaneously leads to a vacuum with CP-violating phases independent of arbitrary parameters – geometrical CP violation. Then we analyse to arbitrarily high order non-renormalisable terms that are consistent with the symmetry and we demonstrate that inclusion of non-renormalisable terms in the potential can preserve the geometrical CP-violating vacuum.     

Yangian symmetries and integrability of the Dirac equation with spin symmetry

We show that a Yangian symmetry, namely, Y(su(2))$Y\left(su\left(2\right)\right)$, exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2))$Y\left(su\left(2\right)\right)$ explicitly and get the energy spectrum of this model from the representation theory for Y(su(2))$Y\left(su\left(2\right)\right)$. We also show that this model is integrable, from RTT relations.     

Hidden global conformal symmetry without Virasoro extension in theory of elasticity

The theory of elasticity (a.k.a. Riva–Cardy model) has been regarded as an example of scale invariant but not conformal field theories. We argue that in d=2$d=2$ dimensions, the theory has hidden global conformal symmetry of SL(2,R)×SL(2,R)$SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ without its Virasoro extension. More precisely, we can embed all the correlation functions of the displacement vector into a global conformal field theory with four-derivative action in terms of two scalar potential variables, which necessarily violates the reflection positivity. The energy–momentum tensor for the potential variables cannot be improved to become traceless so that it does not show the Virasoro symmetry even with the existence of global special conformal current.     

Symmetry breaking and restoration for interacting scalar and gauge fields in Lifshitz type theories

We consider the one-loop effective potential at zero and finite temperature in field theories with anisotropic space–time scaling, with critical exponent z=2$z=2$, including both scalar and gauge fields. Depending on the relative strength of the coupling constants for the gauge and scalar interactions, we find that there is a symmetry breaking term induced at one loop at zero temperature and we find symmetry restoration through a first-order phase transition at high temperature.     

Algebraic approach and coherent states for a relativistic quantum particle in cosmic string spacetime

We study a relativistic quantum particle in cosmic string spacetime in the presence of a magnetic field and a Coulomb-type scalar potential. It is shown that the radial part of this problem possesses the su(1,1)$su(1,1)$ symmetry. We obtain the energy spectrum and eigenfunctions of this problem by using two algebraic methods: the Schrödinger factorization and the tilting transformation. Finally, we give the explicit form of the relativistic coherent states for this problem.     

Experimental probes of emergent symmetries in the quantum Hall system

Experiments studying renormalization group flows in the quantum Hall system provide significant evidence for the existence of an emergent holomorphic modular symmetry Γ₀(2)${\Gamma }_0(2)$. We briefly review this evidence and show that, for the lowest temperatures, the experimental determination of the position of the quantum critical points agrees to the parts per mille   level with the prediction from Γ₀(2)${\Gamma }_0(2)$. We present evidence that experiments giving results that deviate substantially from the symmetry predictions are not cold enough to be in the quantum critical domain. We show how the modular symmetry extended by a non-holomorphic particle–hole duality leads to an extensive web of dualities related to those in plateau–insulator transitions, and we derive a formula relating dual pairs (B,Bd)$(B,B_d)$ of magnetic field strengths across any transition. The experimental data obtained for the transition studied so far is in excellent agreement with the duality relations following from this emergent symmetry, and rule out the duality rule derived from the “law of corresponding states”. Comparing these generalized duality predictions with future experiments on other transitions should provide stringent tests of modular duality deep in the non-linear domain far from the quantum critical points.     

Particle-like platonic solutions in scalar gravity

We construct globally regular gravitating solutions, which possess only discrete symmetries. These solutions of Yang–Mills-dilaton theory may be viewed as exact (numerical) solutions of scalar gravity, by considering the dilaton as a kind of scalar graviton, or as approximate solutions of Einstein–Yang–Mills theory. We focus on platonic solutions with cubic symmetry, related to a rational map of degree N=4$N=4$. We present the first two solutions of the cubic N=4$N=4$ sequence, and expect this sequence to converge to an extremal Reissner–Nordström solution with magnetic charge P=4$P=4$.     

Mini little Higgs and dark matter

We construct a little Higgs model with the most minimal extension of the standard model gauge group by an extra U(1)$U(1)$ gauge symmetry. For specific charge assignments of scalars, an approximate U(3)$U(3)$ global symmetry appears in the cutoff-squared scalar mass terms generated from gauge bosons at one-loop level. Hence, the Higgs boson, identified as a pseudo-Goldstone boson of the broken global symmetry, has its mass radiatively protected up to scales of 5–10 TeV. In this model, a Z₂$Z_2$ symmetry, ensuring the two U(1)$U(1)$ gauge groups to be identical, also makes the extra massive neutral gauge boson stable and a viable dark matter candidate with a promising prospect of direct detection.     

Natural inflation and flavor mixing from Peccei–Quinn symmetry breaking

We propose a left–right symmetric model to simultaneously give natural inflation and flavor mixing from a Peccei–Quinn symmetry breaking at the Planck scale. Our model can be embedded into SO(10)$\mathit{SO}(10)$ grand unification theories.     

Low-scale leptogenesis and the domain wall problem in models with discrete flavor symmetries

We propose a new mechanism for leptogenesis, which is naturally realized in models with a flavor symmetry based on the discrete group A₄$A_4$, where the symmetry breaking parameter also controls the Majorana masses for the heavy right-handed (RH) neutrinos. During the early universe, for T?TeV$T?\text{TeV}$, part of the symmetry is restored, due to finite temperature contributions, and the RH neutrinos remain massless and can be produced in thermal equilibrium even at temperatures well below the most conservative gravitino bounds. Below this temperature the phase transition occurs and they become massive, decaying out of equilibrium and producing the necessary lepton asymmetry. Unless the symmetry is broken explicitly by Planck-suppressed terms, the domain walls generated by the symmetry breaking survive till the quark–hadron phase transition, where they disappear due to a small energy splitting between the A₄$A_4$ vacua caused by the QCD anomaly.     

Wormholes with a barotropic equation of state admitting a one-parameter group of conformal motions

The theoretical construction of a traversable wormhole proposed by Morris and Thorne maintains complete control over the geometry by assigning both the shape and redshift functions, thereby leaving open the determination of the stress–energy tensor. This paper examines the effect of introducing the linear barotropic equation of state p_r=ωρ$p_r=\omega \rho$ on the theoretical construction. If either the energy density or the closely related shape function is known, then the Einstein field equations do not ordinarily yield a finite redshift function. If, however, the wormhole admits a one-parameter group of conformal motions, then both the redshift and shape functions exist provided that −3<ω<−1$-3<\omega <-1$. In a cosmological setting, the equation of state p=ωρ$p=\omega \rho$, ω<−1$\omega <-1$, is associated with phantom dark energy, which is known to support traversable wormholes. The restriction −3<ω<−1$-3<\omega <-1$ that arises in the present wormhole setting can be attributed to the assumption of conformal symmetry.     

Brane tachyon dynamics

The dynamics of a tachyon attached to a Dvali, Gabadadze and Porrati (DGP) brane is investigated. Exponential potential and inverse power law potential are explored, respectively. The quasi-attractor behavior, for which the universe will eventually go into a phase similar to the slow-roll inflation, is discovered in both cases of exponential potential and inverse power law potential. The equation of state (EOS) of the virtual dark energy for a single scalar can cross the phantom divide in the branch θ=−1$\theta =-1$ for both potentials, while the EOS of the virtual dark energy for a single scalar cannot cross this divide in the branch θ=1$\theta =1$.     

A model with two inert scalar doublets

We consider an extension of the standard model (SM) with three SU(2)$SU\left(2\right)$ scalar doublets and discrete S₃⊗Z₂$S_3\otimes {\mathbb{Z}}_2$ symmetries. The irreducible representation of S₃$S_3$ has a singlet and a doublet, and here we show that the singlet corresponds to the SM-like Higgs and the two additional SU(2)$SU\left(2\right)$ doublets forming a S₃$S_3$ doublet are inert. In general, in a three scalar doublet model, with or without S₃$S_3$ symmetry, the diagonalization of the mass matrices implies arbitrary unitary matrices. However, we show that in our model these matrices are of the tri-bimaximal type. We also analyzed the scalar mass spectra and the conditions for the scalar potential is bounded from below at the tree level. We also discuss some phenomenological consequences of the model.     

Zero finite-temperature charge stiffness within the half-filled 1D Hubbard model

Even though the one-dimensional (1D) Hubbard model is solvable by the Bethe ansatz, at half-filling its finite-temperature T>0$T>0$ transport properties remain poorly understood. In this paper we combine that solution with symmetry to show that within that prominent T=0$T=0$ 1D insulator the charge stiffness D(T)$D\left(T\right)$ vanishes for T>0$T>0$ and finite values of the on-site repulsion U$U$ in the thermodynamic limit. This result is exact and clarifies a long-standing open problem. It rules out that at half-filling the model is an ideal conductor in the thermodynamic limit. Whether at finite T$T$ and U>0$U>0$ it is an ideal insulator or a normal resistor remains an open question. That at half-filling the charge stiffness is finite at U=0$U=0$ and vanishes for U>0$U>0$ is found to result from a general transition from a conductor to an insulator or resistor occurring at U=U_c=0$U=U_c=0$ for all finite temperatures T>0$T>0$. (At T=0$T=0$ such a transition is the quantum metal to Mott-Hubbard-insulator transition.) The interplay of the η$\eta$-spin SU(2)$SU\left(2\right)$ symmetry with the hidden U(1)$U\left(1\right)$ symmetry beyond SO(4)$SO\left(4\right)$ is found to play a central role in the unusual finite-temperature charge transport properties of the 1D half-filled Hubbard model.     

On a relativistic scalar particle subject to a Coulomb-type potential given by Lorentz symmetry breaking effects

The behaviour of a relativistic scalar particle in a possible scenario that arises from the violation of the Lorentz symmetry is investigated. The background of the Lorentz symmetry violation is defined by a tensor field that governs the Lorentz symmetry violation out of the Standard Model Extension. Thereby, we show that a Coulomb-type potential can be induced by Lorentz symmetry breaking effects and bound states solutions to the Klein–Gordon equation can be obtained. Further, we discuss the effects of this Coulomb-type potential on the confinement of the relativistic scalar particle to a linear confining potential by showing that bound states solutions to the Klein–Gordon equation can also be achieved, and obtain a quantum effect characterized by the dependence of a parameter of the linear confining potential on the quantum numbers {n,l}$\{n,l\}$ of the system.