The final statesl ± K ± K ± X in jets as signatures ofB_s^0 - \bar B_s^0 mixings

Significant mixing is expected between the neutral bottom mesons \(B_s^0 - \bar B_s^0 \) in the standard model of weak interactions. We propose measurements of the processes \(\left\{ {\begin{array}{*{20}c} {e^ + e^ - } \\ {p\bar p} \\ \end{array} } \right\} \to \begin{array}{*{20}c} {b\bar b} \\ {} \\ \end{array} \to l^ + K^ - K^ - X\) as a measure of such mixing. Rates are presented for energetic bottom quark jets, produced ine ⁺ e ^? annihilation.     

A limit law for the ground state of Hill's equation

It is proved that the ground state Λ(L) of (?1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions, satisfies the law $$\mathop {\lim }\limits_{L \uparrow \infty } P[(L/\pi )\Lambda ^{1/2} \exp ( - \tfrac{8}{3}\Lambda ^{3/2} ) > x] = \left\{ {\begin{array}{*{20}c} {1forx< 0} \\ {e^{ - x} forx \geqslant 0} \\ \end{array} } \right.$$      

A rigorous upper bound in electrostatics on a random lattice ensemble

We consider a Kirchhoff network on a random two-dimensional lattice with links and weights as previously specified, and a circular boundary of radiusR. We show rigorously that the resistance between the central point and the boundary, averaged over all placements of the remaining sites with site density ?, is bounded above by $$\begin{array}{*{20}c} {(4\pi )^{ - 1} [\ln (4\pi \rho R^2 ) + 1] + 16[\tan ^{ - 1} 5^{ - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} + 5^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} /(\sqrt 5 + 1)^2 ]} \\ { \simeq (4\pi )^{ - 1} \ln (4\pi \rho R^2 ) + 12.0.} \\ \end{array} $$      

Lifetimes andg J factors of 6s7p and 5d6p levels of Ba I

Excited states of Ba have been investigated with optical double resonance and Hanle effect. The followingg _J factors and natural lifetimes (in 10^?9 sec) have been measured $$\begin{gathered} 6s7p\left\{ {\begin{array}{*{20}c} {^1 P_1 :g_J = 1.003(2)\tau = 13.5(6)} \\ {^3 P_1 :g_J = 1.4971(8)\tau = 85.0(8.0)} \\ \end{array} } \right. \hfill \\ 5d6p\left\{ {\begin{array}{*{20}c} {^1 P_1 :g_J = 1.004(2)\tau = 12.4(9)} \\ {^3 P_1 :g_J = 1.4847(15)\tau = 11.7(9)} \\ {^3 D_1 :g_J = 0.5064(3)\tau = 17.0(5).} \\ \end{array} } \right. \hfill \\ \end{gathered}$$ g _J is utilized to test the mixing coefficients of the wave functions in the intermediate coupling model. The lifetimes are converted into absolute transition probabilities for all the decays originating from the states investigated under the assumption that their branching ratios obtained elsewhere are correct. This assumption is not unquestionable, however.     

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Exponential lower bounds to solutions of the Schrödinger equation: Lower bounds for the spherical average

For a large class of generalizedN-body-Schrödinger operators,H, we show that ifE<Σ=infσ_ess(H) and ψ is an eigenfunction ofH with eigenvalueE, then $$\begin{array}{*{20}c} {\lim } \\ {R \to \infty } \\ \end{array} R^{ - 1} \ln \left( {\int\limits_{S^{n - 1} } {|\psi (R\omega )|} ^2 d\omega } \right)^{1/2} = - \alpha _0 ,$$ with α ₀ ² +E a threshold. Similar results are given forE≧Σ.     

Effekte höherer Ordnung beim erlaubten β-Zerfall von Li8

Theβ-α angular correlation of Li⁸ has been measured at electron energies of 3·5 and 7·0 Mev. Theβ-energies were selected by a magnetic lens spectrometer. Subtracting the contribution by kinematic effects we find for the coefficientb of the cos²Θ term\(b = \left( {1 \cdot 9\begin{array}{*{20}c} { + 2 \cdot 6} \\ { - 2 \cdot 3} \\ \end{array} } \right)\%\) at 3·5 Mev and\(b = \left( {4 \cdot 0\begin{array}{*{20}c} { + 2 \cdot 0} \\ { - 1 \cdot 3} \\ \end{array} } \right)\%\) at 7·0 Mev. This result is in reasonable agreement with theoretical predictions.     

Eine scheinbare Abschwächung der Lokalitätsbedingung. II

If for a relativistic field theory the expectation values of the commutator (Ω|[A (x),A(y)]|Ω) vanish in space-like direction like exp {? const|(x-y ²|^α/2#x007D; with α>1 for sufficiently many vectors Ω, it follows thatA(x) is a local field. Or more precisely: For a hermitean, scalar, tempered fieldA(x) the locality axiom can be replaced by the following conditions 1. For any natural numbern there exist a) a configurationX(n): $$X_1 ,...,X_{n - 1} X_1^i = \cdot \cdot \cdot = X_{n - 1}^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^1 - X_{i + 1}^1 )} \right]^2 + \left[ {\sum\limits_{i = 1}^{n - 2} {\lambda _i } (X_i^2 - X_{i + 1}^2 )} \right]^2 > 0\) for all λ _i ≧0i=1,...,n?2, \(\sum\limits_{i = 1}^{n - 2} {\lambda _i > 0} \) , b) neighbourhoods of theX _i 's:U _i (X _i )?R ⁴ i=1,...,n?1 (in the euclidean topology ofR ⁴) and c) a real number α>1 such that for all points (x):x ₁, ...,x _n?1:x _i ∈U _i (X _r ) there are positive constantsC ⁽ⁿ⁾{(x)},h ⁽ⁿ⁾{(x)} with: $$\left| {\left\langle {\left[ {A(x_1 )...A(x_{n - 1} ),A(x_n )} \right]} \right\rangle } \right|< C^{(n)} \left\{ {(x)} \right\}\exp \left\{ { - h^{(n)} \left\{ {(x)} \right\}r^\alpha } \right\}forx_n = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ r \\ \end{array} } \right),r > 1.$$ 2. For any natural numbern there exist a) a configurationY(n): $$Y_2 ,Y_3 ,...,Y_n Y_3^i = \cdot \cdot \cdot = Y_n^i = 0i = 0,3$$ with \(\left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^1 - Y_{i{\text{ + 1}}}^{\text{1}} } )} \right]^2 + \left[ {\sum\limits_{i = 3}^{n - 1} {\mu _i (Y_i^2 - Y_{i{\text{ + 1}}}^{\text{2}} } )} \right]^2 > 0\) for all μ _i ≧0,i=3, ...,n?1, \(\sum\limits_{i = 3}^{n - 1} {\mu _i > 0} \) , b) neighbourhoods of theY _i 's:V _i(Y _i )?R ⁴ i=2, ...,n (in the euclidean topology ofR ⁴) and c) a real number β>1 such that for all points (y):y ₂, ...,y _n y _i ∈V _i (Y _i there are positive constantsC _(n){(y)},h _(n){(y)} and a real number γ_(n){(y)∈a closed subset ofR?{0}?{1} with: γ_(n){(y)}\y ₂,y ₃, ...,y _n totally space-like in the order 2, 3, ...,n and $$\left| {\left\langle {\left[ {A(x_1 ),A(x_2 )} \right]A(y_3 )...A(y_n )} \right\rangle } \right|< C_{(n)} \left\{ {(y)} \right\}\exp \left\{ { - h_{(n)} \left\{ {(y)} \right\}r^\beta } \right\}$$ for \(x_1 = \gamma _{(n)} \left\{ {(y)} \right\}r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right),x_2 = y_2 - [1 - \gamma _{(n)} \{ (y)\} ]r\left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right)\) and for sufficiently large values ofr.     

Energy spectrum of photons produced by neutrino (antineutrino) scattering on electrons. Effect of possible neutrino oscillations

The energy spectrum of photons emitted in neutrino (antineutrino) scattering on electrons at E_v ? m_v is calculated with the assumption that the neutral electron flow has an arbitrary (V, A) structure. The result obtained is generalized to the case of possible neutrino oscillations, \(v_e \begin{array}{*{20}c} { - \to } \\ { \leftarrow - } \\ \end{array} v_\mu , \overline v _e \begin{array}{*{20}c} { - \to } \\ { \leftarrow - } \\ \end{array} \overline v _\mu \) , at an arbitrary neutrino mixing angle. Using the Weinberg-Salam model (sin²θ_W = 0.23) estimates of the sections dσ_γ/dω and σ_γ are obtained with consideration of the reactor antineutrino flux \(\bar v_e \) . The contributions from charged and neutral lepton fluxes and their interference to dσ_γ/dω are compared.     

Neutral strange particle exclusive production in charged current high-energy antineutrino interactions

The cross section of the quasi-elastic reactions \(\bar v_\mu p \to \mu ^ + \Lambda (\Sigma ^0 )\) in the energy range 5–100 GeV is determined from Fermilab 15′ bubble chamber antineutrino data. TheQ ² analysis of quasi-elastic Λ events yieldsM _A=1.0±0.3 GeV/c² for the axial mass value. With zero µΛ K ⁰ events observed, the 90% confidence level upper limit \(\sigma (\bar v_\mu p \to \mu ^ + \Lambda {\rm K}^0 )< 2.0 \cdot 10^{ - 40} cm^2 \) is obtained. At the same time, we found that the cross section of reaction \(\bar v_\mu p \to \mu ^ + \Lambda {\rm K}^0 + m\pi ^0 \) is equal to \(\left( {3.9\begin{array}{*{20}c} { + 1.6} \\ { - 1.3} \\ \end{array} } \right) \cdot 10^{ - 40} cm^2 \) .     

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).     

On the problem of quantum tunneling of a singular potential

The problem of quantum tunneling through the singular potential barrier \(V(x) = \left\{ {\begin{array}{*{20}c} {V_0 (b/x - x/a)^2 ,} & {0 < \left| x \right| \leqslant \sqrt {ab} } \\ {0,} & {\left| x \right| > \sqrt {ab} } \\ \end{array} } \right.\) is discussed on the subject of the possibility to replace the singular behavior of the problem at the point x = 0 by a limiting process at the top of the truncated potential. The validity of such a replacement and, on this basis, the zero transparency of the quantum potential barrier are shown.     

Coherent pion production in neutrino and antineutrino interactions on nuclei of heavy freon molecules

Studying the coherent diffractive production of pions in neutrino and antineutrino scattering off the nuclei of freon molecules we have observed for the first time in one experiment all three states of the isospin triplet of the axial part of the weak charged and neutral currents. For the corresponding cross sections we derive $$\begin{array}{*{20}c} {\sigma _{coh}^v (\pi ^ + ) = (106 \pm 16) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ {\sigma _{coh}^{\bar v} (\pi ^ - ) = (113 \pm 35) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}and} \\ {\sigma _{coh}^v (\pi ^0 ) = (52 \pm 19) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ \end{array} $$ . Comparing our data with theoretical predictions based on the standard model of weak interactions we find reasonable agreement. Independently from any model of coherent pion production we determine the isovector axial vector coupling constant to be |β|=0.99±0.20.     

Three-loop on-shell charge renormalization without integration:\Lambda _{QED}^{\overline {MS} } to four loops

Using algebraic methods, we find the three-loop relation between the bare and physical couplings of one-flavourD-dimensional QED, in terms of Γ functions and a singleF ₃₂ series, whose expansion nearD=4 is obtained, by wreath-product transformations, to the order required for five-loop calculations. Taking the limitD→4, we find that the \(\overline {MS} \) coupling \(\bar \alpha (\mu )\) satisfies the boundary condition $$\begin{gathered} \frac{{\bar \alpha (m)}}{\pi } = \frac{\alpha }{\pi } + \frac{{15}}{{16}}\frac{{\alpha ^3 }}{{\pi ^3 }} + \left\{ {\frac{{11}}{{96}}\zeta (3) - \frac{1}{3}\pi ^2 \log 2} \right. \hfill \\ \left. { + \frac{{23}}{{72}}\pi ^2 - \frac{{4867}}{{5184}}} \right\}\frac{{\alpha ^4 }}{{\pi ^4 }} + \mathcal{O}(\alpha ^5 ), \hfill \\ \end{gathered} $$ wherem is the physical lepton mass and α is the physical fine structure constant. Combining this new result for the finite part of three-loop on-shell charge renormalization with the recently revised four-loop term in the \(\overline {MS} \) β-function, we obtain $$\begin{gathered} \Lambda _{QED}^{\overline {MS} } \approx \frac{{me^{3\pi /2\alpha } }}{{(3\pi /\alpha )^{9/8} }}\left( {1 - \frac{{175}}{{64}}\frac{\alpha }{\pi } + \left\{ { - \frac{{63}}{{64}}\zeta (3)} \right.} \right. \hfill \\ \left. { + \frac{1}{2}\pi ^2 \log 2 - \frac{{23}}{{48}}\pi ^2 + \frac{{492473}}{{73728}}} \right\}\left. {\frac{{\alpha ^2 }}{{\pi ^2 }}} \right), \hfill \\ \end{gathered} $$ at the four-loop level of one-flavour QED.     

Nuclear quadrupole interaction at99Tc in lithium-intercalated molybdenum disulphide

A highly crystalline form of lithium intercalated MoS₂ was investigated by performing TDPAC measurements on the 740 — (44) 141 keV γ?γ cascade in⁹⁹Tc. Analysis of the data reveals the presence of two static efg interactions with the following parameters: $$\begin{array}{*{20}c} {\begin{array}{*{20}c} {v_q = 114(3) MHz,} & {\eta _1 = 0.57(5),} & {\delta _1 = 0.48(5);} \\\end{array}} \\ {\begin{array}{*{20}c} {v_{q2} = 645(19) MHz,} & {\eta = 0.45(5),} & {\delta _2 = 0.11(2).} \\\end{array}} \\\end{array}$$      

Dobrushin states for classical spin systems with complex interactions

We consider a classical spin system on the hypercubic lattice with a general interaction of the form $$ H = \frac{\beta } {4}\sum\limits_{\begin{array}{*{20}c} {x,y:} \\ {|x - y| = 1} \\ \end{array} } {|s_x - s_y | - h} \sum\limits_x {x{}_x + } \sum\limits_A {\lambda _A \prod\limits_{y \in A} {S_y } } $$ are the spin variables, Β is the inverse temperature,h is the magnetic field, andλ _A are translation-invariant coupling constants satisfyingλ _A = 0 if diamA > l. No symmetry relating the configurationss ={sinx} and-s=-s _x is assumed. In dimension d-3, we construct low-temperature States which break the translation invariance of the system by introducing so-called Dobrushin boundary conditions which force a horizontal interface into the system. In contrast to previous constructions, our methods work equally well for complex interactions, and should therefore be generalizable to quantum spin systems.     

Entangled Fourier Transformation and its Application in Weyl-Wigner Operator Ordering and Fractional Squeezing

In order to entangle the functions to be transformed, we proposed the entangled. Fourier integration transformation (EFIT) which has the property of keeping modulus-invariant for its inverse transformation. Then we then studied Wigner operator’s EFIT and found that a function’s EFIT is just related to its Weyl-corresponding operator’s matrix element, in so doing we also derived new operator re-ordering formulas \( \delta \left(x-P\right)\left(y-Q\right)=\frac{1}{\pi }{\displaystyle \begin{array}{c}:\\ {}:\end{array}}{e}^{-2i\left(P-x\right)\left(Q-y\right)}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} \);\( \delta \left(y-Q\right)\left(x-P\right)=\frac{1}{\pi }{\displaystyle \begin{array}{c}:\\ {}:\end{array}}{e}^{2i\left(P-x\right)\left(Q-y\right)}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} \), where P, Q are momentum and coordinate operator respectively, the symbol \( {\displaystyle \begin{array}{c}:\\ {}:\end{array}}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} \) denotes Weyl ordering. By virtue of EFIT we also found the operator which can generate fractional squeezing transformation.

     

Nonlinear Schrödinger equations and sharp interpolation estimates

A sharp sufficient condition for global existence is obtained for the nonlinear Schrödinger equation $$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma } \phi = 0,} & {x \in \mathbb{R}^N } & {t \in \mathbb{R}^ + } \\ \end{array} $$ in the case σ=2/N. This condition is in terms of an exact stationary solution (nonlinear ground state) of (NLS). It is derived by solving a variational problem to obtain the “best constant” for classical interpolation estimates of Nirenberg and Gagliardo.