Calculating statistical distributions from operator relations: The statistical distributions of various intermediate statistics

In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as a^†b=Λ(N)$a^{†}b=\Lambda \left(N\right)$ or N=Λ⁻¹(a^†b)$N={\Lambda }^{-1}\left(a^{†}b\right)$, where N$N$, a^†$a^{†}$, and b$b$ denote the number, creation, and annihilation operators, i.e., N$N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical distribution is the Gentile distribution, a statistical distribution in which the maximum occupation number is an arbitrary integer. As examples, we discuss the statistical distributions corresponding to various operator relations. In particular, besides the Bose–Einstein and Fermi–Dirac cases, we discuss the statistical distributions for various schemes of intermediate statistics, especially various q$q$-deformation schemes. Our result shows that the statistical distributions corresponding to various q$q$-deformation schemes are various Gentile distributions with different maximum occupation numbers which are determined by the deformation parameter q$q$. This result shows that the results given in much literature on the q$q$-deformation distribution are inaccurate or incomplete.     

B to light tensor meson form factors derived from light-cone sum rules

Using the recent results for the two-parton light-cone distribution amplitudes of the tensor meson, we calculate the form factors for the decays of B_u,d,s$B_{u,d,s}$ into the light JPC=²⁺⁺$J^{\mathit{PC}}=2^{++}$ tensor mesons via the vector/axial–vector/tensor current with the light-cone sum rules. We also obtain the q²$q^2$-dependence of the form factors.     

A separation property for magnetic Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression L=_ΔA+q$L={\Delta }_A+q$ on a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$, where _ΔA${\Delta }_A$ is the magnetic Laplacian on M$M$ and q≥0$q\ge 0$ is a locally square integrable function on M$M$. In the terminology of W.N. Everitt and M. Giertz, the differential expression L$L$ is said to be separated in L²(M)$L^2\left(M\right)$ if for all u∈L²(M)$u\in L^2\left(M\right)$ such that Lu∈L²(M)$Lu\in L^2\left(M\right)$, we have qu∈L²(M)$qu\in L^2\left(M\right)$. We give sufficient conditions for L$L$ to be separated in L²(M)$L^2\left(M\right)$.     

Comment on noncommutative deformation of instantons

It was argued in [Y. Maeda, A. Sako, Noncommutative deformation of instantons, J. Geom. Phys. 58 (2008) 1784] that the noncommutative deformation of instantons on a 4-torus T⁴$T^4$ should alter the instanton numbers for arbitrary noncommutativity parameter θ$\theta$. We show that this is not the case for the U(N²)$U\left(N^2\right)$ theory discussed there. And we discuss the instanton numbers in general gauge theories on the noncommutative T⁴$T^4$.     

Sneutrino identification in lepton pair production at ILC with polarized beams

Numerous non-standard dynamics are described by contact-like effective interactions that can manifest themselves in electron–positron collisions only through deviations of the observables (cross sections, asymmetries) from the Standard Model predictions. If such a deviation were observed, it would be important to identify the actual source among the possible non-standard interactions as many different new physics scenarios may lead to very similar experimental signatures. We study the possibility of uniquely identifying the indirect effects of s-channel sneutrino exchange, as predicted by supersymmetric theories with R  -parity violation, against other new physics scenarios in high-energy e⁺e⁻$e^{+}{e}^{-}$ annihilation into lepton pairs at the International Linear Collider. These competitive models are interactions based on gravity in large and in TeV-scale extra dimensions, anomalous gauge couplings, Z^′$Z^{\prime }$ vector bosons and compositeness-inspired four-fermion contact interactions. To evaluate the identification reach on sneutrino exchange, we use as basic observable a double polarization asymmetry, that is particularly suitable to directly test for such s-channel sneutrino exchange effects in the data analysis. The availability of both beams being polarized plays a crucial rôle in identifying the new physics scenario.     

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$Y^{p,q}$.     

Space inversion of spinors revisited: A possible explanation of chiral behavior in weak interactions

We investigate a model in which spinors are considered as being embedded within the Clifford algebra that operates on them. In Minkowski space M_1,3$M_{1,3}$, we have four independent 4-component spinors, each living in a different minimal left ideal of Cl(1,3)$\mathit{Cl}(1,3)$. We show that under space inversion, a spinor of one left ideal transforms into a spinor of another left ideal. This brings novel insight to the role of chirality in weak interactions. We demonstrate the latter role by considering an action for a generalized spinor field ψαi${\psi }^{\alpha i}$ that has not only a spinor index α but also an extra index i   running over four ideals. The covariant derivative of ψαi${\psi }^{\alpha i}$ contains the generalized spin connection, the extra components of which are interpreted as the SU(2) gauge fields of weak interactions and their generalization. We thus arrive at a system that is left–right symmetric due to the presence of a “parallel sector”, postulated a long time ago, that contains mirror particles coupled to mirror SU(2) gauge fields.