Spin and torsion in general relativity: I. Foundations

From physical arguments space-time is assumed to possess a connection \(\Gamma _{ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} + S_{ij}^{ k} - S_{j i}^{ k} + S_{ ij}^k = \left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\} - K_{ij}^{ k} \) . \(\left\{ {\begin{array}{*{20}c} k \\ {ij} \\ \end{array} } \right\}\) is Christoffel's symbol built up from the metric g _ij and already appearing in General Relativity (GR). Cartan's torsion tensor \(S_{ij} ^k = \tfrac{1}{2}(\Gamma _{ij}^k - \Gamma _{ji}^k )\) and the contortion tensor K _ij ^k , in contrast to the theory presented here, both vanish identically in conventional GR. Using the connection introduced above in this series of articles, we will discuss the consequences for GR in the framework of a consistent formalism. There emerges a theory describing, in a unified way, gravitation and a very weak spin-spin contact interaction. In section 1 we start with the well-known dynamical definition of the energy-momentum tensor σ ^ij ～ δ?/δg _ij , where ? represents the Lagrangian density of matter (section1.1). In sections1.2,3 we will show that due to geometrical reasons, the connection assumed above leads to a dynamical definition of the spin-angular momentum tensor according to τ^k _ji ～ δ?/δK _ij ^k . In section1.4, by an ideal experiment, it will become clear that spin prohibits the introduction of an instantaneous rest system and thereby of a geodesic coordinate system. Among other things in section1.5 there are some remarks about the rôle torsion played in former physical theories. In section 2 we sketch the content of the theory. As in GR, the action function is the sum of the material and the field action function (sections2.1,2). The extension of GR consists in the introduction of torsion S _ij ^k as a new field. By variation of the action function with respect to metric and torsion we obtain the field equations in a general form (section2.3). They are also valid for matter described by spinors; in this case, however, one has to introduce tetrads as anholonomic coordinates and slightly to generalize the dynamical definition of energy-momentum (sections2.4,5).     

Study of f 0(980) and f 0(1500) from B s →f 0(980)π,f 0(1500)π decays

In this paper, we analyze the scalar mesons f ₀(980) and f ₀(1500) from the decays $\bar{B}^{0}_{s}\to f_{0}(980)\pi^{0},\allowbreak f_{0}(1500)\pi^{0}$ within Perturbative QCD approach. From the leading-order calculations, we find that (a) in the allowed mixing angle ranges, the branching ratio of $\bar{B}^{0}_{s}\to f_{0}(980)\pi^{0}$ is about (1.0～1.6)×10^?7, which is smaller than that of $\bar{B}^{0}_{s}\to f_{0}(980)K^{0}$ (the difference is a few times even one order); (b) the decay $\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0}$ is better to distinguish between the lowest lying state or the first excited state for f ₀(1500), because the branching ratios for two scenarios have about one-order difference in most of the mixing angle ranges; and (c) the direct CP asymmetries of $\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0}$ for two scenarios also exists great difference. In scenario II, the variation range of the value ${\mathcal{A}}^{\mathrm{dir}}_{CP}(\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0})$ according to the mixing angle in scenario II is very small, except for the values for mixing angles near 90° or 270°, while the variation range of ${\mathcal{A}}^{\mathrm{dir}}_{CP}(\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0})$ in scenario I is very large. Compared with the future data for the decay $\bar{B}^{0}_{s}\to f_{0}(1500)\pi^{0}$ , it is easy to determine the nature of the scalar meson f ₀(1500).     

Charged Higgs observability through associated production with W ± at a muon collider

The observability of a charged Higgs boson produced in association with a W boson at future muon colliders is studied. The analysis is performed within the MSSM framework. The charged Higgs is assumed to decay to $t\bar{b}We study $B_{s}^{0} \to J/\psi f_{0}(980)$ decays, the quark content of f ₀(980) and the mixing angle of f ₀(980) and ??(600). We calculate not only the factorizable contribution in the QCD factorization scheme but also the nonfactorizable hard spectator corrections in QCDF and pQCD approach. We get a result consistent with the experimental data of $B_{s}^{0} \to J/\psi f_{0}(980)$ and predict the branching ratio of $B_{s}^{0}$ ?CJ/???. We suggest two ways to determine f ₀?C?? mixing angle ??. Using the experimental measured branching ratio of $B_{s}^{0} \to J/\psi f_{0}(980)$ , we can get the f ₀?C?? mixing angle ?? with some theoretical uncertainties. We suggest another way to determine the f ₀?C?? mixing angle ?? using both experimental measured decay branching ratios $B_{s}^{0} \to J/\psi f_{0}(980) (\sigma)$ to avoid theoretical uncertainties.     

Regularity for Eigenfunctions of Schr?dinger Operators

We prove a regularity result in weighted Sobolev (or Babu?ka?CKondratiev) spaces for the eigenfunctions of certain Schr?dinger-type operators. Our results apply, in particular, to a non-relativistic Schr?dinger operator of an N-electron atom in the fixed nucleus approximation. More precisely, let ${\mathcal{K}_{a}^{m}(\mathbb{R}^{3N},r_S)}$ be the weighted Sobolev space obtained by blowing up the set of singular points of the potential ${V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}}$ , ${x \in \mathbb{R}^{3N}}$ , ${b_j, c_{ij} \in \mathbb{R}}$ . If ${u \in L^2(\mathbb{R}^{3N})}$ satisfies ${(-\Delta + V) u = \lambda u}$ in distribution sense, then ${u \in \mathcal{K}_{a}^{m}}$ for all ${m \in \mathbb{Z}_+}$ and all a ?? 0. Our result extends to the case when b _j and c _ij are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a?<?3/2.     

K→π form factors with reduced model dependence

Using partially twisted boundary conditions we compute the K→π semi-leptonic form factors in the range of momentum transfers $0\lesssim q^{2}\leq q^{2}_{\max}=(m_{K}-m_{\pi})^{2}$ in lattice QCD with N _f =2+1 dynamical flavours. In this way we are able to determine $f_{+}^{K\pi}(0)$ without any interpolation in the momentum transfer, thus eliminating one source of systematic error. This study confirms our earlier phenomenological ansatz for the strange quark mass dependence of the scalar form factor. We identify and estimate potentially significant NNLO effects in the chiral expansion that guides the extrapolation of the data to the physical point. Our main result is $f_{+}^{K\pi}(0)=0.9599(34)(^{+31}_{-47})(14)$ , where the first error is statistical, the second error is due to the uncertainties in the chiral extrapolation of the lattice data and the last error is an estimate of potential discretisation effects.     

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p =  2, 3, . . .) of central charge c _{1, p} =1 ? 6(p ? 1)²/p. This family includes the theories corresponding to the singlet algebras ${\mathcal{M}(p)}$ and the triplet algebras ${\mathcal{W}(p)}$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The ${W^{(2)}_n}$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of ${\widehat{\mathfrak{sl}}(n)_k}$ , generalising the Bershadsky–Polyakov algebra ${W^{(2)}_3}$ . Inspired by work of Adamovi? for p = 3, vertex algebras ${\mathcal{B}_p}$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra ${\mathcal{B}_p}$ is a quotient of ${W^{(2)}_{p-1}}$ at level ?(p ? 1)²/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra ${\mathcal{W}(p)}$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra ${\mathcal{M}(p)}$ is similarly realised inside ${\mathcal{B}_p}$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p =  2 and 3.     

Decoupling the NLO coupled DGLAP evolution equations: an analytic solution to pQCD

Using repeated Laplace transforms, we turn coupled, integral-differential singlet DGLAP equations into NLO (next-to-leading) coupled algebraic equations, which we then decouple. After two Laplace inversions we find new tools for pQCD: decoupled NLO analytic solutions $F_{s}(x,Q^{2})={\mathcal{F}}_{s}(F_{s0}(x),G_{0}(x))$ , $G(x,Q^{2})={\mathcal{G}}(F_{s0}(x), G_{0}(x))$ . ${\mathcal{F}}_{s}$ , $\mathcal{G}$ are known NLO functions and $F_{s0}(x)\equiv F_{s}(x,Q_{0}^{2})$ , $G_{0}(x)\equiv G(x,Q_{0}^{2})$ are starting functions for evolution beginning at $Q^{2}=Q_{0}^{2}$ . We successfully compare our u and d non-singlet valence quark distributions with MSTW results (Martin et al., Eur. Phys. J. C 63:189, 2009).     

Recent Results on Hadron Spectroscopy at Belle

Recent activities of the Belle experiment in Hadron spectroscopy are presented. The discovery of two charged bottomonium-like particles, ${Z_b^{\pm}(10610)}$ and ${Z_b^{\pm}(10650)}$ in the ${\Upsilon (nS)\pi^{\pm}}$ (n = 1, 2, 3) and h _b (m P)π ^± (m = 1, 2) final states is followed by the observation of the corresponding peaks in the ${B^*\bar{B}^{(*)}}$ final states that favors the molecular interpretation of Z _b . In addition, a neutral partner candidate is explored at 10,610 MeV in the ${\Upsilon (2S)\pi^0}$ mass spectrum projection. We have got no evidence of an X(3872) partner in J/ψ π ^± π ⁰, J/ψ η and ${\chi_{c1,2}\gamma}$ final states while an evidence of the narrow peak at 3,823 MeV/c ² in ${\chi_{c1}\gamma}$ is thought to be ³ D ₂ charmonium (ψ ₂) candidate. The feasibilities of the search for X(3872) partner in the B → J/ψ π ⁰ π ⁰ K decays, the measurement of light flavored baryons production cross section, and the study of K–p interaction in ${\phi p}$ final state are also discussed.     

Semileptonic form factors D \rightarrow \pi, K and B \rightarrow \pi, K from a fine lattice

We extract the form factors relevant for semileptonic decays of D and B mesons from a relativistic computation on a fine lattice in the quenched approximation. The lattice spacing is a = 0.04 fm (corresponding to a ^-1 = 4.97 GeV), which allows us to run very close to the physical B meson mass, and to reduce the systematic errors associated with the extrapolation in terms of a heavy-quark expansion. For decays of D and D_s mesons, our results for the physical form factors at $\ensuremath q^2 = 0$ are as follows: $\ensuremath f_+^{D\rightarrow\pi}(0) = 0.74(6)(4)$ , $\ensuremath f_+^{D \rightarrow K}(0) = 0.78(5)(4)$ and $\ensuremath f_+^{D_s \rightarrow K} (0) = 0.68(4)(3)$ . Similarly, for B and B_s we find $\ensuremath f_+^{B\rightarrow\pi}(0) = 0.27(7)(5)$ , $\ensuremath f_+^{B\rightarrow K} (0) = 0.32(6)(6)$ and $\ensuremath f_+^{B_s\rightarrow K}(0) = 0.23(5)(4)$ . We compare our results with other quenched and unquenched lattice calculations, as well as with light-cone sum rule predictions, finding good agreement.     

Formation of Light p-shell Ξ − Hypernucleus Via (K −, K +) Reaction

We have calculated ⁷Li(K ^?, K ⁺) reaction spectrum at ${p_{K^-} = 1.65}$ GeV/c and ${\theta_{K}^{+} = 0^\circ}$ within the framework of distorted wave impulse approximation (DWIA) using the ${[\Xi^-{-}^6{\rm He}(0^+)]-[\Xi^-{-}^6{\rm He}^*(2^+)]}$ coupled-channel Green’s function approach, and examined whether the peak corresponding to the Ξ ^? hypernuclear states can be visible in the reaction spectrum, by employing various Ξ ^??⁶He effective potentials.     

Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise

Motivated by the general problem of studying sample-to-sample fluctuations in disorder-generated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model??the ideal periodic 1/f Gaussian noise. Our main object of interest is the number of points $\mathcal{N}_{M}(x)$ above a level $\frac{x}{2}V_{m}$ , with V _m =2lnM standing for the leading-order typical value of the absolute maximum for the sample of M points. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of $\mathcal{N}_{M}(x)$ for 0<x<2. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level x, results in an important difference between the mean and the typical values of $\mathcal{N}_{M}(x)$ . This can be further used to determine the typical threshold x _m of extreme values in the pattern which turns out to be given by $x_{m}^{(\mathit{typ})}=2-c\ln\ln M /\ln M $ with $c=\frac{3}{2}$ . Such observation provides a rather compelling explanation of the mechanism behind universality of c. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum p _max of intensity is to be given by $-\ln p_{\mathit{max}}=\alpha_{-}\ln M +\frac{3}{2f'(\alpha_{-})}\ln\ln M+O(1)$ , where f(??) is the corresponding singularity spectrum positive in the interval ????(?? _?,?? ₊) and vanishing at ??=?? _?>0. For the 1/f noise case we further study asymptotic values of the prefactors in scaling laws for the moments of the counting function. Our numerics shows however that one needs prohibitively large sample sizes to reach such asymptotics even with a moderate precision. This motivates us to derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.     

Few Body Systems Made of Pseudoscalars

Few body systems made of pseudoscalars, like ${K\, K \, \bar K,\,\pi \, K \, \bar K}$ , are studied within a coupled channel approach based on solving the Faddeev equations considering two-body chiral t-matrices as input. As a result, we have found dynamical generation of several states which can be associated with some of the pseudoscalar states listed by the Particle Data Group, like K(1460) or π(1300). The amplitudes obtained have been then used to study systems like f ₀(980) π π and ${f_0(980)K \, \bar K}$ and an evidence for a f ₀ resonance around 1,790 MeV is found.     

Sign of the Singlet (np)-Scattering Length, Neutron Radiative Capture by the Proton and Problem of the Virtual Level of the (np) System

It is shown that, taking into account the process of neutron radiative capture by the proton and the negative sign of the length of singlet (np)-scattering (a _s =? ?f _s (0) <? 0), the singlet (np)-scattering amplitude f _s has a pole at a complex energy ${\widetilde{E}_s}$ , the real part of which is negative ( ${{\rm Re}\,\widetilde{E}_s < 0}$ ) and the imaginary part is positive ( ${{\rm Im}\, \widetilde{E}_s > 0}$ ). This means that a singlet state of the (np) system, which would decay into the deuteron in the ground state and the ?? quantum (??singlet deuteron??) does not exist, and the pole ${\widetilde{E}_s}$ corresponds to the virtual but not true quasistationary level.     

K S 0 , Λ and ≡ production at intermediate to high pT from Au+Au collisions at \sqrt {s_{NN} } = 39, 11.5 and 7.7 GeV

We report on the p _T dependence of nuclear modification factors (R _CP) for K _S ⁰ , ??, ?? and the $\bar NK_S^0 $ ratios at mid-rapidity from Au+Au collisions at $\sqrt {s_{NN} } $ = 39, 11.5 and 7.7 GeV. At $\sqrt {s_{NN} } $ = 39 GeV, the R _CP data show a baryon/meson separation at intermediate p _T and a suppression for K _S ⁰ for p _T up to 4.5 GeV/c; the $\bar \Lambda K_S^0 $ shows baryon enhancement in the most central collisions. However, at $\sqrt {s_{NN} } $ = 11.5 and 7.7 GeV, R _CP shows less baryon/meson separation and $\bar NK_S^0 $ shows almost no baryon enhancement. These observations indicate that the matter created in Au+Au collisions at $\sqrt {s_{NN} } $ = 11.5 or 7.7 GeV might be distinct from that created at $\sqrt {s_{NN} } $ = 39 GeV.     

Study of Light Hypernuclei by the (e,e′K +) Reaction

We have been performing Λ hypernuclear spectroscopic experiments by the (e,e′K ⁺) reaction since 2000 at Thomas Jefferson National Accelerator Facility (JLab). The (e,e′K ⁺) experiment can achieve a few 100 keV (FWHM) energy resolution compared to a few MeV (FWHM) by the (K ^?, π ^?) and (π ⁺, K ⁺) experiments. Therefore, more precise Λ hypernuclear structures can be investigated by the (e,e′K ⁺) experiment. ${^{7}_{\Lambda}{\rm He}}$ , ${^{9}_{\Lambda}{\rm Li}}$ , ${^{10}_{\Lambda}{\rm Be}}$ , ${^{12}_{\Lambda}{\rm B}}$ , ${^{28}_{\Lambda}{\rm Al}}$ , and ${^{52}_{\Lambda}{\rm V}}$ were measured in the experiment at JLab Hall-C. In addition, ${^{9}_{\Lambda}{\rm Li}}$ , ${^{12}_{\Lambda}{\rm B}}$ , and ${^{16}_{\Lambda}{\rm N}}$ were measured in the experiment at JLab Hall-A.     

Effects of Final State Interactions in Pure Annihilation Decay of B^{0}_{s}\rightarrow\pi^{0}\pi^{0}

We investigate the effects of final state interactions (FSI) contributions in the nonleptonic two body $B^{0}_{s} \rightarrow \pi^{0}\pi^{0}$ decay. The short distance interaction amplitude is calculated by using the annihilation diagrams and a tiny branching ratio is obtained, then the long distance amplitude is considered and calculated within FSI effects. For contributions of FSI, the ρ ⁰ ρ ⁰, π ⁺ π ^?(ρ ⁺ ρ ^?), K ⁺ K ^?(K ^+? K ^??) and $K^{0}\bar{K}^{0}(K^{0*}\bar{K}^{0*})$ are produced for intermediate states, in this case the π ⁰, π ^?(ρ ^?), K ^?(?) and $\bar{K}^{0(*)}$ mesons are exchanged. The absorptive part of the diagrams is directly calculated and the dispersive part of the rescattering amplitude can be obtained from the absorptive part via the dispersion relation. The imaginary and real parts of the amplitudes are summed over all intermediate states. The predicted branching ratio of $B^{0}_{s} \rightarrow \pi^{0}\pi^{0}$ is 0.69×10^?8 in the absence of FSI effects and it becomes 1.86×10^?4 when FSI contributions are taken into account, while the experimental result is less than 2.1×10^?4.     

No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis $\alpha _{i}$ chosen by Alice, irrespective of Bob’s axis $\beta _{j}$ (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s and Bob’s spins are identified as $A_{ij}$ and $B_{ij}$ , even though their distributions are determined by, respectively, $\alpha _{i}$ alone and $\beta _{j}$ alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins $A_{ij},B_{ij}$ across all values of $\alpha _{i},\beta _{j}$ are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs $\left( A_{ij},A_{ij'}\right) $ and $\left( B_{ij},B_{i'j}\right) $ with $\alpha _{i}\not =\alpha _{i'}$ and $\beta _{j}\not =\beta _{j'}$ (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of $A_{ij}\not =A_{ij'}$ and $B_{ij}\not =B_{i'j}$ ). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs $\left( A_{ij},B_{ij}\right) $ that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical–mechanical correlations. No-matching theorem says that there are no subsets of the spin variables $A_{ij},B_{ij}$ whose distributions can be fixed to be compatible with and only with QM-compliant correlations.     

Radiative and semileptonic B decays involving higher K-resonances in the final states

We study the radiative and semileptonic B decays involving a spin-J resonant $K_{J}^{(*)}$ with parity (?1) ^J for $K_{J}^{*}$ and (?1) ^J+1 for K _J in the final state. Using large energy effective theory (LEET) techniques, we formulate $B\to K_{J}^{(*)}$ transition form factors in the large recoil region in terms of two independent LEET functions $\zeta_{\perp}^{K_{J}^{(*)}}$ and $\zeta_{\parallel}^{K_{J}^{(*)}}$ , the values of which at zero momentum transfer are estimated in the BSW model. According to the QCD counting rules, $\zeta_{\perp,\parallel}^{K_{J}^{(*)}}$ exhibit a dipole dependence in q ². We predict the decay rates for $B\to K_{J}^{(*)}\gamma$ , $B\to K_{J}^{(*)}\ell^{+}\ell^{-}$ and $B\to K_{J}^{(*)}\nu \bar{\nu}$ . The branching fractions for these decays with higher K-resonances in the final state are suppressed due to the smaller phase spaces and the smaller values of $\zeta^{K_{J}^{(*)}}_{\perp,\parallel}$ . Furthermore, if the spin of $K_{J}^{(*)}$ becomes larger, the branching fractions will be further suppressed due to the smaller Clebsch–Gordan coefficients defined by the polarization tensors of the $K_{J}^{(*)}$ . We also calculate the forward–backward asymmetry of the $B\to K_{J}^{(*)}\ell^{+}\ell^{-}$ decay, for which the zero is highly insensitive to the K-resonances in the LEET parametrization.     

On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of 1D Gaussian band matrices, i.e., of the Hermitian N × N matrices H _N with independent Gaussian entries such that 〈H _ij H _lk 〉 = δ _ik δ _jl J _ij , where ${J=(-W^2\triangle+1)^{-1}}$ . Assuming that ${W^2=N^{1+\theta}}$ , ${0 < \theta \leq 1}$ , we show that the moment’s asymptotic behavior (as ${N\to\infty}$ ) in the bulk of the spectrum coincides with that for the Gaussian Unitary Ensemble.     

K^{-} nuclear quasi-bound states in a chirally motivated coupled-channel approach

K ^?? nuclear optical potentials are constructed from in-medium ${\bar K}N$ scattering amplitudes within a chirally motivated coupled-channel model. The strong energy and density dependence of the scattering amplitudes at and below threshold leads to K ^?? potential depths ?Re $V_{K^-}(\rho_0) \approx 80 -100$ ?MeV. Self consistent calculations of K ^?? nuclear quasi-bound states are discussed.