Is space-time symmetry a suitable generalization of parity-time symmetry?

We discuss space-time symmetric Hamiltonian operators of the form H=H₀+igH^′$H=H_0+ig{H}^{\prime }$, where H₀$H_0$ is Hermitian and g$g$ real. H₀$H_0$ is invariant under the unitary operations of a point group G$G$ while H^′$H^{\prime }$ is invariant under transformation by elements of a subgroup G^′$G^{\prime }$ of G$G$. If G$G$ exhibits irreducible representations of dimension greater than unity, then it is possible that H$H$ has complex eigenvalues for sufficiently small nonzero values of g$g$. In the particular case that H$H$ is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<g_c$0<g<g_c$, where g_c$g_c$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H$H$ may exhibit real or complex eigenvalues for g>0$g>0$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.     

Heavy neutral gauge bosons at the LHC in an extended MSSM

Searching for heavy neutral gauge bosons Z^′$Z^{\prime }$, predicted in extensions of the Standard Model based on a U(1)^′$\mathrm{U}{(1)}^{\prime }$ gauge symmetry, is one of the challenging objectives of the experiments carried out at the Large Hadron Collider. In this paper, we study Z^′$Z^{\prime }$ phenomenology at hadron colliders according to several U(1)^′$\mathrm{U}{(1)}^{\prime }$-based models and in the Sequential Standard Model. In particular, possible Z^′$Z^{\prime }$ decays into supersymmetric particles are included, in addition to the Standard Model modes so far investigated. We point out the impact of the U(1)^′$\mathrm{U}{(1)}^{\prime }$ group on the MSSM spectrum and, for a better understanding, we consider a few benchmarks points in the parameter space. We account for the D-term contribution, due to the breaking of U(1)^′$\mathrm{U}{(1)}^{\prime }$, to slepton and squark masses and investigate its effect on Z^′$Z^{\prime }$ decays into sfermions. Results on branching ratios and cross sections are presented, as a function of the MSSM and U(1)^′$\mathrm{U}{(1)}^{\prime }$ parameters, which are varied within suitable ranges. We pay special attention to final states with leptons and missing energy and make predictions on the number of events with sparticle production in Z^′$Z^{\prime }$ decays, for a few values of integrated luminosity and centre-of-mass energy of the LHC.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.     

Deconfined fractionally charged excitation in any dimensions

An exact incompressible quantum liquid is constructed at the filling factor 1/m²$1/m^2$ in the square lattice. It supports deconfined fractionally charged excitation. At the filling factor 1/m²$1/m^2$, the excitation has fractional charge e/m²$e/m^2$, where e$e$ is the electric charge. This model can be easily generalized to the n$n$-dimensional square lattice (integer lattice), where the charge of excitations becomes e/mⁿ$e/m^n$.     

Constraints on sub-GeV hidden sector gauge bosons from a search for heavy neutrino decays

Several models of dark matter motivate the concept of hidden sectors consisting of SU_(3)C×SU_(2)L×U_(1)Y$\mathit{SU}{(3)}_C\times \mathit{SU}{(2)}_L\times U{(1)}_Y$ singlet fields. The interaction between our and hidden matter could be transmitted by new abelian U^′(1)$U^{\prime }(1)$ gauge bosons A^′$A^{\prime }$ mixing with ordinary photons. If such A^′$A^{\prime }$?s with the mass in the sub-GeV range exist, they would be produced through mixing with photons emitted in decays of η   and η^′${\eta }^{\prime }$ neutral mesons generated by the high energy proton beam in a neutrino target. The A^′$A^{\prime }$?s would then penetrate the downstream shielding and be observed in a neutrino detector via their A^′→e⁺e⁻$A^{\prime }\to e^{+}{e}^{-}$ decays. Using bounds from the CHARM neutrino experiment at CERN that searched for an excess of e⁺e⁻$e^{+}{e}^{-}$ pairs from heavy neutrino decays, the area excluding the γ−A^′$\gamma -A^{\prime }$ mixing range 10⁻⁷???10⁻⁴${10}^{-7}???{10}^{-4}$ for the A^′$A^{\prime }$ mass region 1?_MA′?500 MeV$1?M_{A^{\prime }}?500\text{MeV}$ is derived. The obtained results are also used to constrain models, where a new gauge boson X   interacts with quarks and leptons. New upper limits on the branching ratio as small as Br(η→γX)?10⁻¹⁴$\mathrm{Br}(\eta \to \gamma X)?{10}^{-14}$ and Br(η^′→γX)?10⁻¹²$\mathrm{Br}({\eta }^{\prime }\to \gamma X)?{10}^{-12}$ are obtained, which are several orders of magnitude more restrictive than the previous bounds from the Crystal Barrel experiment.     

A note on embedding non-Abelian finite flavor groups in continuous groups

A non-Abelian finite flavor group G⊂SO(3)$G\subset \mathit{SO}(3)$ can have double covering G^′⊂SU(2)$G^{\prime }\subset \mathit{SU}(2)$ such that G⊄G^′$G\not\subset G^{\prime }$. This situation is not contradictory, but quite natural, and we give explicit examples such as G=Dn$G=D_n$, G^′=Q_2n$G^{\prime }=Q_{2n}$ and G=T$G=T$, G^′=T^′$G^{\prime }=T^{\prime }$. This observation can be crucial in particle theory model building.     

Searching for an extra neutral gauge boson from muon pair production at LHC

We search for signatures of the extra neutral gauge boson Z^′$Z^{\prime }$, predicted in some extensions of the Standard Model, from the analysis of some distributions for p+p→μ⁺+μ⁻+X$p+p\to {\mu }^{+}+{\mu }^{-}+X$, where the only exotic particle involved is Z^′$Z^{\prime }$. In addition to the invariant mass and charge asymmetry distributions, we propose in our search to use the transverse momentum distribution (pT$p_T$) as an observable. We do our calculation for two values of the LHC center of mass energy (7 and 14 TeV), corresponding to 1 and 100 fb⁻¹ of luminosity, in order to compare our findings from some models with the distributions following from the Standard Model. By applying convenient cuts in the invariant mass, we show that the final particles pT$p_T$ distributions can reveal the presence of an extra neutral gauge boson contribution. We also claim that it is possible to disentangle the models considered here and we emphasize that the minimal version of the model, based on SUC₍₃₎×SUL₍₃₎×UX₍₁₎$\mathit{SU}{(3)}_C\times \mathit{SU}{(3)}_L\times U{(1)}_X$ symmetry, presents the more clear signatures for Z^′$Z^{\prime }$ existence.     

Global supersymmetry on curved spaces in various dimensions

We propose methods towards a systematic determination of d  -dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity point of view. In particular, by using appropriate gauged supergravities in various dimensions we show that supersymmetry can be defined in conformally flat spaces, such as non-compact hyperboloids Hⁿ⁺¹${\mathbb{H}}^{n+1}$ and compact spheres Sⁿ${\mathbb{S}}^n$ or – by turning on appropriate Wilson lines corresponding to R-symmetry vector fields – on S¹×Sⁿ${\mathbb{S}}^1\times {\mathbb{S}}^n$, with n<6$n<6$. By group theory arguments we show that Euclidean field theories with rigid supersymmetry cannot be consistently defined on round spheres S^d${\mathbb{S}}^d$ if d>5$d>5$ (despite the existence of Killing spinors). We also show that distorted spheres and certain orbifolds are also allowed by the group theory classification.     

Resolutions of non-regular Ricci-flat Kähler cones

We present explicit constructions of complete Ricci-flat Kähler metrics that are asymptotic to cones over non-regular Sasaki–Einstein manifolds. The metrics are constructed from a complete Kähler–Einstein manifold (V,_gV)$(V,g_V)$ of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Kähler metrics on the total spaces of (i) holomorphic C²/_Zp${\mathbb{C}}^2/{\mathbb{Z}}_p$ orbifold fibrations over V$V$, (ii) holomorphic orbifold fibrations over weighted projective spaces WCP¹${\mathbb{W}\mathbb{C}\mathbb{P}}^1$, with generic fibres being the canonical complex cone over V$V$, and (iii) the canonical orbifold line bundle over a family of Fano orbifolds. As special cases, we also obtain smooth complete Ricci-flat Kähler metrics on the total spaces of (a) rank two holomorphic vector bundles over V$V$, and (b) the canonical line bundle over a family of geometrically ruled Fano manifolds with base V$V$. When V=CP¹$V={\mathbb{C}\mathbb{P}}^1$ our results give Ricci-flat Kähler orbifold metrics on various toric partial resolutions of the cone over the Sasaki–Einstein manifolds Y^p,q$Y^{p,q}$.     

Holomorphic Cartan geometries and Calabi–Yau manifolds

Let M$M$ be a connected complex projective manifold such that _c1(T^(1,0)M)=0$c_1\left(T^{(1,0)}M\right)=0$. If M$M$ admits a holomorphic Cartan geometry, then we show that M$M$ is holomorphically covered by an abelian variety.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

Improvement of initial permeability for Z-type ferrite by Ti and Zn substitution

We have found that the initial permeability μ^′${\mu }^{\prime }$ of _Co2Z${\mathrm{Co}}_2\mathrm{Z}$ ferrite is improved by the substitution of Ti⁴⁺${\mathrm{Ti}}^{4+}$ and Zn²⁺${\mathrm{Zn}}^{2+}$ ions for Fe³⁺${\mathrm{Fe}}^{3+}$ ions. The substituted sample of _{Ba3Co2TixZnxFe24-2xO41}${\mathrm{Ba}}_3{\mathrm{Co}}_2{\mathrm{Ti}}_x{\mathrm{Zn}}_x{\mathrm{Fe}}_{24-2x}{\mathrm{O}}_{41}$ with x=0.85$x=0.85$ has a maximum μ^′${\mu }^{\prime }$ of 24, which is twice as large as that of the non-substituted sample with x=0$x=0$. The particle size and shape are changed by the substitution. This is influential in the densification and the preferential orientation of a toroidal-shape sample, which results in the improvement of μ^′${\mu }^{\prime }$.     

Entanglement entropy and variational methods: Interacting scalar fields

We develop a variational approximation to the entanglement entropy for scalar ?⁴${?}^4$ theory in 1+1$1+1$, 2+1$2+1$, and 3+1$3+1$ dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+1$1+1$ and 2+1$2+1$ dimensions, the entanglement entropy of ?⁴${?}^4$ theory as a function of coupling is monotonically decreasing and convex. While ?⁴${?}^4$ theory with positive bare coupling in 3+1$3+1$ dimensions is thought to lead to a trivial free theory, we analyze a version of ?⁴${?}^4$ with infinitesimal negative bare coupling, an asymptotically free theory known as precarious  ?⁴${?}^4$ theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious ?⁴${?}^4$ theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.     

A rigidity theorem for complete noncompact Bach-flat manifolds

Let (M⁴,g)$(M^4,g)$ be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that (M⁴,g)$(M^4,g)$ has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small _L2$L_2$-norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for (M⁴,g)$(M^4,g)$ with positive scalar curvature.     

Atomic parity violation in the economical 3–3–1 model

The deviation δQW$\delta Q_{\mathrm{W}}$ of the weak charge from its standard model prediction due to the mixing of the W boson with the charged bilepton Y as well as of the Z   boson with the neutral Z^′$Z^{\prime }$ and the real part of the non-Hermitian neutral bilepton X   in the economical 3–3–1 model is established. Additional contributions to the usual δQW$\delta Q_{\mathrm{W}}$ expression in the extra U(1)$\mathrm{U}(1)$ models and the left–right models are obtained. Our calculations are quite different from previous analyzes in this kind of the 3–3–1 models and give the limit on mass of the Z^′$Z^{\prime }$ boson, the Z–Z^′$Z\text{–}{Z}^{\prime }$ and W–Y$W\text{–}Y$ mixing angles with the more appropriate values: MZ^′>564 GeV$M_{Z^{\prime }}>564\text{GeV}$, −0.018<sinφ<0$-0.018<\sin \phi <0$ and |sinθ|<0.043$|\sin \theta |<0.043$.