More corrections to the sixth-order anomalous magnetic moment of the muon

The contribution to the sixth-order muon anomaly from second-order electron vacuum polarization is determined analytically to orderm _e/m _μ. The result, including the contributions from graphs containing proper and improper fourth-order electron vacuum polarization subgraphs, is $$\begin{gathered} \left( {\frac{\alpha }{\pi }} \right)^3 \left\{ {\frac{2}{9}\log ^2 } \right.\frac{{m_\mu }}{{m_e }} + \left[ {\frac{{31}}{{27}}} \right. + \frac{{\pi ^2 }}{9} - \frac{2}{3}\pi ^2 \log 2 \hfill \\ \left. { + \zeta \left( 3 \right)} \right]\log \frac{{m_\mu }}{{m_e }} + \left[ {\frac{{1075}}{{216}}} \right. - \frac{{25}}{{18}}\pi ^2 + \frac{{5\pi ^2 }}{3}\log 2 \hfill \\ \left. { - 3\zeta \left( 3 \right) + \frac{{11}}{{216}}\pi ^4 - \frac{2}{9}\pi ^2 \log ^2 2 - \frac{1}{9}log^4 2 - \frac{8}{3}a_4 } \right] \hfill \\ + \left[ {\frac{{3199}}{{1080}}\pi ^2 - \frac{{16}}{9}\pi ^2 \log 2 - \frac{{13}}{8}\pi ^3 } \right]\left. {\frac{{m_e }}{{m_\mu }}} \right\} \hfill \\ \end{gathered} $$ . To obtain the total sixth-order contribution toa _μ?a _e, one must add the light-by-light contribution to the above expression.     

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

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.     

Light quark masses in quantum chromodynamics and chiral symmetry breaking

We derive model independent lower bounds for the sums of effective quark masses \(\bar m_u + \bar m_d \) and \(\bar m_u + \bar m_s \) . The bounds follow from the combination of the spectral representation properties of the hadronic axial currents two-point functions and their behavior in the deep euclidean region (known from a perturbative QCD calculation to two loops and the leading non-perturbative contribution). The bounds incorporate PCAC in the Nambu-Goldstone version. If we define the invariant masses \(\hat m\) by $$\bar m_i = \hat m_i \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^{{{\gamma _1 } \mathord{\left/ {\vphantom {{\gamma _1 } {\beta _1 }}} \right. \kern-\nulldelimiterspace} {\beta _1 }}} $$ and <F ²> is the vacuum expectation value of $$F^2 = \Sigma _a F_{(a)}^{\mu v} F_{\mu v(a)} $$ , we find, e.g., $$\hat m_u + \hat m_d \geqq \sqrt {\frac{{2\pi }}{3} \cdot \frac{{8f_\pi m_\pi ^2 }}{{3\left\langle {\alpha _s F^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} $$ ; with the value <α _u F ²?0.04GeV⁴, recently suggested by various analysis, this gives $$\hat m_u + \hat m_d \geqq 35MeV$$ . The corresponding bounds on \(\bar m_u + \bar m_s \) are obtained replacingm _π ² f _π bym _K ² f _K . The PCAC relation can be inverted, and we get upper bounds on the spontaneous masses, \(\hat \mu \) : $$\hat \mu \leqq 170MeV$$ where \(\hat \mu \) is defined by $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

Self-diffusion in α-Fe2O3 natural single crystals

Measurements of¹⁸O self-diffusion in hematite (Fe₂O₃) natural single crystals have been carried out as a function of temperature at constant partial pressure a_{O 2}=6.5·10^?2 in the temperature range 890 to 1227 °C. The a_{O 2} dependence of the oxygen self-diffusion coefficient at fixed temperature T=1150 °C has also been deduced in the a_{O 2} range 4.5·10^?4 - 6.5·10^?1. The concentration profiles were established by secondary-ion mass spectrometry; several profiles exhibit curvatures or long tails; volume diffusion coefficients were computed from the first part of the profiles using a solution taking into account the evaporation and the exchange at the surface. The results are well described by $$D_O \left( {{{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } s}} \right. \kern-\nulldelimiterspace} s}} \right) = 2.7 \cdot 10^8 a_{O_2 }^{ - 0.26} \exp \left( { - \frac{{542\left( {{{kJ} \mathord{\left/ {\vphantom {{kJ} {mol}}} \right. \kern-\nulldelimiterspace} {mol}}} \right)}}{{RT}}} \right)$$ From fitting a grain boundary diffusion solution to the profile tails, the oxygen self-diffusion coefficient in sub-boundaries has been deduced. They are well described by $$D''_O \left( {{{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } s}} \right. \kern-\nulldelimiterspace} s}} \right) = 3.2 \cdot 10^{25} a_{O_2 }^{ - 0.4} \exp \left( { - \frac{{911\left( {{{kJ} \mathord{\left/ {\vphantom {{kJ} {mol}}} \right. \kern-\nulldelimiterspace} {mol}}} \right)}}{{RT}}} \right)$$ Experiments performed introducing simultaneously¹⁸O and⁵⁷Fe provided comparative values of the self-diffusion coefficients in volume: iron is slower than oxygen in this system showing that the concentrations of atomic point defects in the iron sublattice are lower than the concentrations of atomic point defects in the oxygen sublattice. The iron self-diffusion values obtained at T>940 °C can be described by $$D_{Fe} \left( {{{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } s}} \right. \kern-\nulldelimiterspace} s}} \right) = 9.2 \cdot 10^{10} a_{O_2 }^{ - 0.56} \exp \left( { - \frac{{578\left( {{{kJ} \mathord{\left/ {\vphantom {{kJ} {mol}}} \right. \kern-\nulldelimiterspace} {mol}}} \right)}}{{RT}}} \right)$$ The exponent - 1/4 observed for the oxygen activity dependence of the oxygen self-diffusion in the bulk has been interpreted considering that singly charged oxygen vacancies V _O ^? are involved in the oxygen diffusion mechanism. Oxygen activity dependence of iron self-diffusion is not known accurately but the best agreement with the point defect population model is obtained considering that iron self-diffusion occurs both via neutral interstitals Fe _x ⁱ and charged ones.     

Der Einfluß der Wärmebewegung auf die Breite einer Mössbauerlinie über den Dopplereffekt zweiter Ordnung

AtT=0 a perfect Mössbauer line has natural line widthΓ=?/τ _n . However, with rising temperature the width increases. The reason of the line broadening is the second order Doppler effect which causes a stochastic frequency modulation of theγ-radiation, reflecting the thermal motion of the Mössbauer atom. Following Josephson in treating the second order Doppler shift as a mass changeΔM=E _n/c² of theγ-emitting atom caused by the loss of nuclear excitation energy E _n , and using the well known relaxation formalism for calculating theγ-frequency spectrum, the line broadeningΔ Γ is evaluated within the framework of harmonic lattice theory. For a parabolic lattice frequency spectrum with Debye-temperature Θ one obtains $$\Delta {\Gamma \mathord{\left/ {\vphantom {\Gamma \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma } = \left( {{{\tau _n } \mathord{\left/ {\vphantom {{\tau _n } {\tau _c }}} \right. \kern-\nulldelimiterspace} {\tau _c }}} \right) \cdot \left( {{{E_n } \mathord{\left/ {\vphantom {{E_n } {Mc^2 }}} \right. \kern-\nulldelimiterspace} {Mc^2 }}} \right) \cdot F\left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right),where\tau _c = {{\rlap{--} h} \mathord{\left/ {\vphantom {{\rlap{--} h} k}} \right. \kern-\nulldelimiterspace} k}\Theta $$ is the correlation time of the lattice vibrations. The functionF(T/Θ) may be expanded in powers ofT/Θ, yielding $$F\left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right) = 9720\pi \left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right)^7 forT<< \Theta $$ and $$F\left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right) = 2.7\pi \left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right)^2 forT > > \Theta $$ , respectively. Although unavoidable, the line broadening is obviously too small to be observable by means of the present experimental technique.     

Electron paramagnetic resonance and spinlattice relaxation of Cu2+ ions in ZnSeO4·6H2O

In this paper, we present detailed studies of the EPR spectra of Cu²⁺ ions in single crystals of ZnSeO₄·6H₂O. We describe the spectrum with a rhombic spin Hamiltonian with the following parameters: g_z=2.427; g_y=2.095; g_x=2.097; A _z ⁶⁵ =138.4·10^?4 cm^?1; A _x ⁶⁵ =22.3·10^?4 cm^?1. We studied spin-lattice relaxation in the temperature range 4–300 K at the frequency v≈9.3 GHz. The measured spin-lattice relaxation rate for the orientation H∥L₄ is described well at T<5 K by a linear dependence, while at T>5 K it is described by the sum of three exponentials: $$T_1^{ - 1} = 0.27T + 3.3 \cdot 10^{\text{s}} \exp \left( {\frac{{ - 69.5}}{T}} \right) + 2.6 \cdot 10^7 \exp \left( {\frac{{ - 140}}{T}} \right) + 1.36 \cdot 10^{10} \exp \left( {\frac{{ - 735.6}}{T}} \right){\text{ sec}}^{{\text{ - 1}}} $$ .We discuss possible reasons for the exponential dependence of T ₁ ^?1 for the Raman process.     

Some results for SU(N) gauge-fields on the Hypertorus

We show how to prove and to understand the formula for the “Pontryagin” indexP for SU(N) gauge fields on the HypertorusT ⁴, seen as a four-dimensional euclidean box with twisted boundary conditions. These twists are defined as gauge invariant integers moduloN and labelled byN _μv (=?N _μv ). In terms of these we can write (ν∈#x2124;) $$P = \frac{1}{{16\pi ^2 }}\int {Tr(G_{\mu v} \tilde G_{\mu v} )d_4 x = v + \left( {\frac{{N - 1}}{N}} \right) \cdot \frac{{n_{\mu v} \tilde n_{\mu v} }}{4}} $$ . Furthermore we settle the last link in the proof of the existence of zero action solutions with all possible twists satisfying \(\frac{{n_{\mu v} \tilde n_{\mu v} }}{4} = \kappa (n) = 0(\bmod N)\) for arbitraryN.     

Neutrino production of dimuons

Neutrino interactions with two muons in the final state have been studied using the Fermilab narrow band beam. A sample of 18v _μ like sign dimuon events withP _μ>9 GeV/c yields 6.6±4.8 events after backgroud subtraction and a prompt rate of (1.0±0.7)×10^?4 per single muon event. The kinematics of these events are compared with those of the non-prompt sources. A total of 437v _μ and 31 \(\bar v_\mu \) opposite sign dimuon events withP _μ>4.3 GeV/c are used to measure the strange quark content of the nucleon: \(\kappa = {{2s} \mathord{\left/ {\vphantom {{2s} {\left( {\bar u + \bar d} \right) = 0.52_{ - 0.15}^{ + 0.17} \left( {or\eta _s \frac{{2s}}{{u + d}} = 0.075 \pm 0.019} \right) for 100< E_v< 230 GeV\left( {\left\langle {Q^2 } \right\rangle = {{23 GeV^2 } \mathord{\left/ {\vphantom {{23 GeV^2 } {c^2 }}} \right. \kern-0em} {c^2 }}} \right)}}} \right. \kern-0em} {\left( {\bar u + \bar d} \right) = 0.52_{ - 0.15}^{ + 0.17} \left( {or\eta _s \frac{{2s}}{{u + d}} = 0.075 \pm 0.019} \right) for 100< E_v< 230 GeV\left( {\left\langle {Q^2 } \right\rangle = {{23 GeV^2 } \mathord{\left/ {\vphantom {{23 GeV^2 } {c^2 }}} \right. \kern-0em} {c^2 }}} \right)}}\) using a charm semileptonic branching ratio of (10.9±1.4)% extracted from measurements ine ⁺ e ^? collisions and neutrino emulsion data.     

Reanalysis of the mass difference ofB_d^0 - \bar B_d^0 within the Minimal Supersymmetric Standard Modelwithin the Minimal Supersymmetric Standard Model

We present a detailed and complete calculation of the loop corrections to the mass difference $\Delta {{m_{B_d^0 } } \mathord{\left/ {\vphantom {{m_{B_d^0 } } {m_{B_d^0 } }}} \right. \kern-0em} {m_{B_d^0 } }}$ . We include charginos and scalar up quarks as well as gluinos and scalar down quarks on the relevant loop diagrams. We include the mixings of the charginos and of the scalar partners of the left and right handed quarks. We find that the gluino contribution to this quantity is important with respect to the chargino contribution only in a small part of phase space: mainly when the gluino mass is small (～100 GeV) and the symmetry-breaking parameterm _S is below 300 GeV. This contribution is also important for very large values of tanβ (～50) irrespective of the other parameters. Otherwise, the chargino contribution dominates vastly and can be roughly as large as that of the Standard Model. We also present the contribution of the charged Higgs to the mass difference $\Delta {{m_{B_d^0 } } \mathord{\left/ {\vphantom {{m_{B_d^0 } } {m_{B_d^0 } }}} \right. \kern-0em} {m_{B_d^0 } }}$ in the casem _b tanβ?m _t cotβ. This last contribution can be larger than the Standard Model contribution for small values of the Higgs mass and small values of tanβ.     

Secular-free solution up to third order of a model nonlinear equation for dispersive waves

The perturbation method of Lindstedt is applied to study the non linear effect of a nonlinear equation $$\nabla ^2 {\rm E} - \frac{1}{{c^2 }}\frac{{\partial ^2 {\rm E}}}{{\partial t^2 }} - \frac{{\omega _0^2 }}{{c^2 }}{\rm E} + \frac{{2v}}{{c^2 }}\frac{{\partial {\rm E}}}{{\partial t}} + E^2 \left[ {\frac{{\partial {\rm E}}}{{\partial t}} \times A} \right] = 0,$$ where (A. E)=0 andA,c, ω ₀ andν are constants in space and time. Amplitude dependent frequency shifts and the solution up to third order are derived.     

Study of the mechanisms of pre-equilibrium nuclear reactions in the microscopic approach

The mechanisms of pre-equilibrium nuclear reactions are investigated within the Statistical Multistep Direct Process (SMDP) + Statistical Multistep Compound Process (SMCP) formalism. It has been shown that from an analysis of linear part in such dependences as $$\ln \left[ {{{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} \mathord{\left/ {\vphantom {{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} {\varepsilon _b^{1/2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon _b^{1/2} }}} \right]upon\varepsilon _b $$ and $$\ln \left[ {{{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} \mathord{\left/ {\vphantom {{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right. \kern-\nulldelimiterspace} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right]upon{{U_B } \mathord{\left/ {\vphantom {{U_B } {\left( {E_a - B_b } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {E_a - B_b } \right)}}$$ one can extract information about the type of mechanism (SMDP, SMCP, SMDP→SMCP) and the number of stages of the multistep emission of secondary particles. In the above approach, we have discussed the experimental data for a broad class of reactions in various entrance and exit channels.     

Upper bound on the decay of correlations in the plane rotator model with long-range random interaction

We give an upper bound on the decay of correlation function for the plane rotator model with Hamiltonian $$ - \frac{1}{2}\mathop \sum \limits_{xy} \frac{{J_{xy} \cos (\theta _x - \theta _y )}}{{\| {x - y} \|^{({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2} + \varepsilon )^d } }}$$ in dimensiond=1 andd=2 when (J _xy are independent random variables with mean zero.     

Layered structure of superfluid4He at supercritical motion

Landau’s criterion plays an important role in the theory of superfluidity. According to this criterion, superfluid motion is possible if \(\tilde \varepsilon \left( p \right) \equiv \varepsilon \left( p \right) + pV > 0\) along the curve of the spectrum?(p) of excitations. For⁴He it means thatv<v _c,v _c≈60 m/sec.v _s is equal to the tangent of the slope to the roton part of the spectrum. The question of what happens to the liquid when this velocity is exceeded, as far as we know, remains unclear. We shall show that for small excesses abovev _c a one-dimensional periodic structure appears in the helium. A wave vector of this structure oriented opposite to the flow and equal toρ _c/h whereρ _c is the momentum at the tangent point. The quantity \(\tilde \varepsilon \left( p \right)\) is the energy of excitation in the liquid moving with velocity v. Inequality of Landau ensures that \(\tilde \varepsilon \) is positive. If \(\tilde \varepsilon \) becomes negative, then the boson distribution function \(n\left( {\tilde \varepsilon } \right)\) becomes negative, indicating the impossibility of thermodynamic equilibrium of the ideal gas of rotons; therefore the interaction between them must be taken into account. The final form of the energy operator is $$\hat H = \int {\left\{ {\hat \psi + \tilde \varepsilon \left( p \right)\hat \psi + \tfrac{g}{2}\hat \psi + \hat \psi + \hat \psi \hat \psi } \right\}} d^3 x, g \sim 2 \cdot 10^{ - 38} erg.cm.$$ Then we can seek the rotonψ-operator in the formψ=ηexp(i p _c r/h), determiningη from the condition that the energy is minimized. The result is (η)²=(v?v _c)ρ _c/g, forv>v _c. The plane waveψ corresponds to a uniform distribution of rotons. It leads, however, to a spatial modulation of the density of the helium, since the density operator \(\hat n\) contains a term which is linear in the operator \(\psi :\hat n = n_0 + \left( {n_0 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} {A \mathord{\left/ {\vphantom {A {\hat \psi \to \hat \psi ^ + }}} \right. \kern-0em} {\hat \psi \to \hat \psi ^ + }}\) ), where |A|²～ρ _c ² /2m?(ρ _c). Finally we find that the density of helium is modulated according to the law $$\frac{{n - n_0 }}{{n_0 }} = \left[ {\frac{{\left| A \right|^2 \left( {\nu - \nu _c } \right)\rho _c }}{{n_0 g}}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sin \rho _c x \approx 2,6\left[ {\frac{{\nu - \nu _c }}{{\nu _c }}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \sin \rho _c x$$ . This phenomenon can be observed, in principle, in the experiments on scattering ofx-rays in moving helium.     

On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation

This paper is concerned with the Lévy, or stable distribution function defined by the Fourier transform $$Q_\alpha \left( z \right) = \frac{1}{{2\pi }}\int {_{ - \infty }^\infty \exp \left( { - izu - \left| u \right|^\alpha } \right)du} with 0< \alpha \leqslant 2$$ Whenα=2 it becomes the Gauss distribution function and whenα=1, the Cauchy distribution. Whenα≠2 the distribution has a long inverse power tail $$Q_\alpha \left( z \right) \sim \frac{{\Gamma \left( {1 + \alpha } \right)\sin \tfrac{1}{2}\pi \alpha }}{{\pi \left| z \right|^{1 + \alpha } }}$$ In the regime of smallα, ifα¦logz¦?1, the distribution is mimicked by a log normal distribution. We have derived rapidly converging algorithms for the numerical calculation ofQ _α (z) for variousα in the range 0<α<1. The functionQ _α (z) appears naturally in the Williams-Watts model of dielectric relaxation. In that model one expresses the normalized dielectric parameter as $$ \in _n \left( \omega \right) \equiv \in '_n \left( \omega \right) - i \in ''_n \left( \omega \right) = - \int {_0^\infty e^{ - i\omega t} \left[ {{{d\phi \left( t \right)} \mathord{\left/ {\vphantom {{d\phi \left( t \right)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}}} \right]} dt$$ with $$\phi \left( t \right) = \exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha $$ It has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form ofφ(t).ε″ _n (ω) is shown to be directly related toQ _α (z). It is also shown that if the Williams-Watts exponential is expressed as a weighted average of exponential relaxation functions $$\exp - \left( {{t \mathord{\left/ {\vphantom {t \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)^\alpha = \int {_0^\infty } g\left( {\lambda , \alpha } \right)e^{ - \lambda t} dt$$ the weight functiong(λ, α) is expressible as a stable distribution. Some suggestions are made about physical models that might lead to the Williams-Watts form ofφ(t).     

Alternative technique for Standard Model estimation of the rare kaon decay branchings \left. {BR(K \to \pi \nu \bar \nu )} \right|_{SM}

We estimate $BR(K \to \pi \nu \bar \nu )$ in the context of the Standard Model by fitting for λ _t≡V _tdV _ts ^* of the “kaon unitarity triangle” relation. To find the vertex of this triangle, we fit data from |? _K|, the CP-violating parameter describing K mixing, and a _ψ,K , the CP-violating asymmetry in B _d ⁰ → J/ψK ⁰ decays, and obtain the values $\left. {BR(K \to \pi \nu \bar \nu )} \right|_{SM} = (7.07 \pm 1.03) \times 10^{ - 11} $ and $\left. {BR(K_L^0 \to \pi ^0 \nu \bar \nu )} \right|_{SM} = (2.60 \pm 0.52) \times 10^{ - 11} $ . Our estimate is independent of the CKM matrix element V _cb and of the ratio of B-mixing frequencies ${{\Delta m_{B_s } } \mathord{\left/ {\vphantom {{\Delta m_{B_s } } {\Delta m_{B_d } }}} \right. \kern-0em} {\Delta m_{B_d } }}$ . We also use the constraint estimation of λ _t with additional data from $\Delta m_{B_d } $ and |V _ub|. This combined analysis slightly increases the precision of the rate estimation of $K^ + \to \pi ^ + \nu \bar \nu $ and $K_L^0 \to \pi ^0 \nu \bar \nu $ (by ?10 and ?20%, respectively). The measured value of $BR(K^ + \to \pi ^ + \nu \bar \nu )$ can be compared both to this estimate and to predictions made from ${{\Delta m_{B_s } } \mathord{\left/ {\vphantom {{\Delta m_{B_s } } {\Delta m_{B_d } }}} \right. \kern-0em} {\Delta m_{B_d } }}$ .     

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.     

Parity-violating neutron spin rotation in hydrogen and deuterium

We calculate the (parity-violating) spin-rotation angle of a polarized neutron beam through hydrogen and deuterium targets, using pionless effective field theory up to next-to-leading order. Our result is part of a program to obtain the five leading independent low-energy parameters that characterize hadronic parity violation from few-body observables in one systematic and consistent framework. The two spin-rotation angles provide independent constraints on these parameters. Our result for np spin rotation is $\frac{1} {\rho }\frac{{d\varphi _{PV}^{np} }} {{dl}} = \left[ {4.5 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {2g^{\left( {^3 S_1 - ^3 P_1 } \right)} + g^{\left( {^3 S_1 - ^3 P_1 } \right)} } \right) - \left[ {18.5 \pm 1.9} \right] rad MeV^{ - \frac{1} {2}} \left( {g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 2} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right)$\frac{1} {\rho }\frac{{d\varphi _{PV}^{np} }} {{dl}} = \left[ {4.5 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {2g^{\left( {^3 S_1 - ^3 P_1 } \right)} + g^{\left( {^3 S_1 - ^3 P_1 } \right)} } \right) - \left[ {18.5 \pm 1.9} \right] rad MeV^{ - \frac{1} {2}} \left( {g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 2} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right), while for nd spin rotation we obtain $\frac{1} {\rho }\frac{{d\varphi _{PV}^{nd} }} {{dl}} = \left[ {8.0 \pm 0.8} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^1 P_1 } \right)} + \left[ {17.0 \pm 1.7} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^3 P_1 } \right)} + \left[ {2.3 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {3g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 1} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right)$\frac{1} {\rho }\frac{{d\varphi _{PV}^{nd} }} {{dl}} = \left[ {8.0 \pm 0.8} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^1 P_1 } \right)} + \left[ {17.0 \pm 1.7} \right] rad MeV^{ - \frac{1} {2}} g^{\left( {^3 S_1 - ^3 P_1 } \right)} + \left[ {2.3 \pm 0.5} \right] rad MeV^{ - \frac{1} {2}} \left( {3g_{\left( {\Delta {\rm I} = 0} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} - 2g_{\left( {\Delta {\rm I} = 1} \right)}^{\left( {^1 S_0 - ^3 P_0 } \right)} } \right), where the g ^(X-Y), in units of $MeV^{ - \frac{3} {2}}$MeV^{ - \frac{3} {2}}, are the presently unknown parameters in the leading-order parity-violating Lagrangian. Using naıve dimensional analysis to estimate the typical size of the couplings, we expect the signal for standard target densities to be $\left| {\frac{{d\varphi _{PV} }} {{dl}}} \right| \approx \left[ {10^{ - 7} \ldots 10^{ - 6} } \right]\frac{{rad}} {m}$\left| {\frac{{d\varphi _{PV} }} {{dl}}} \right| \approx \left[ {10^{ - 7} \ldots 10^{ - 6} } \right]\frac{{rad}} {m} for both hydrogen and deuterium targets. We find no indication that the nd observable is enhanced compared to the np one. All results are properly renormalized. An estimate of the numerical and systematic uncertainties of our calculations indicates excellent convergence. An appendix contains the relevant partial-wave projectors of the three-nucleon system.