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.     

Collisional damping of transverse waves in a plasma

The effect of collisions on transverse waves in a homogeneous, field free plasma is investigated by means of Gross-Krook collision model. The dispersion relation is calculated by assuming the collision frequency to be small andKλ _D ?1. The damping rate ω _I is obtained as $$\omega _I = \frac{{\nu _{ei} }}{2}\frac{{\omega _p^2 }}{{\omega _0^2 }}\left[ {1 + \frac{{3K^2 \lambda _D^2 \omega _p^2 }}{{\omega _0^2 }} - \frac{{K^2 \lambda _D^2 \omega _p^4 }}{{\omega _0^4 }}} \right] + \frac{{\nu _{ee} }}{2}\frac{{\omega _p^2 }}{{\omega _0^2 }}\left( {\frac{{K^2 \lambda _D^2 \omega _p^2 }}{{\omega _0^2 }}} \right)$$ where ω ₀ ² =c ² K ²+ω ₂ ^p , andv _ei andv _ee are electron-ion and electron-electron collision frequency respectively.     

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 $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties

From the path probability density for nonlinear stochastic processes a Lagrangean for classical field dynamics is derived. This formulation provides a convenient approach to the mode coupling equations and the renormalization group theory of critical dynamics. An application is given for the time-dependent isotropic Heisenberg ferromagnet. The dynamical exponent is derived aboveT _c for all dimensionsd>2.     

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.     

On the Essential Spectrum of a Class of Singular Matrix Differential Operators. I: Quasiregularity Conditions and Essential Self-adjointness

The essential spectrum of singular matrix differential operator determined by the operator matrix

Vacuum energy ingϕ d /4 -theory forg→∞

In the nonlocalg? _d ^/4 (d≧1) and localg? ₂ ⁴ theory theS-matrix is obtained in a form of the functional integral which is proved to exist. The density of vacuum energy $$E(g) = - \mathop {\lim }\limits_{V \to \infty } \frac{1}{V}\ln \left\langle {0\left| {S_V (g)} \right|0} \right\rangle $$ is investigated. It is proved to be analytic through the whole complexg-plane except for the negative real axis and pointg=0. Its asymptotic behaviour forg→∞ is found.     

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.     

Surface-induced finite-size effects for first-order phase transitions

We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems such as small magnetic clusters, we consider models with free boundary conditions. For a field-driven transition with two coexisting phases at the infinite-volume transition pointh=h _t , we prove that the low-temperature, finite-volume magnetizationm _free(L, h) per site in a cubic volume of sizeL ^d behaves like $$m_{free} (L,h) = \frac{{m_ + + m_ - }}{2} + \frac{{m_ + - m_ - }}{2}tanh\left[ {\frac{{m_ + - m_ - }}{2}L^d (h - h_\chi (L))} \right] + O\left( {\frac{1}{L}} \right)$$ whereh _x (L) is the position of the maximum of the (finite-volume) susceptibility andm _± are the infinite-volume magnetizations ath=h _t +0 andh=h _t ?0, respectively. We show thath _x (L) is shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boundary conditions, which for an asymmetric transition with two coexisting phases is proportional only to 1/L ^2d . One can consider also other definitions of finite-volume transition points, for example, the positionh _U (L) of the maximum of the so-called Binder cumulantU _free(L,h). Whileh _U (L) is again shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t , its shift with respect toh _χ (L) is of the much smaller order 1/L ^2d . We give explicit formulas for the proportionality factors, and show that, in the leading 1/L ^2d term, the relative shift is the same as that for periodic boundary conditions.     

On the upper critical dimension of Bernoulli percolation

We derive a set of inequalities for thed-dimensional independent percolation problem. Assuming the existence of critical exponents, these inequalities imply: $$\begin{gathered} f + v \geqq 1 + \beta _Q , \hfill \\ \mu + v \geqq 1 + \beta _Q , \hfill \\ \zeta \geqq \min \left\{ {1,\frac{{v^, }}{v}} \right\}, \hfill \\ \end{gathered} $$ where the above exponents aref: the flow constant exponent, ν(ν′): the correlation length exponent below (above) threshold, μ: the surface tension exponent, β _Q : the backbone density exponent and ζ: the chemical distance exponent. Note that all of these inequalities are mean-field bounds, and that they relate the exponentv defined from below the percolation threshold to exponents defined from above threshold. Furthermore, we combine the strategy of the proofs of these inequalities with notions of finite-size scaling to derive: $$\max \{ dv,dv^, \} \geqq 1 + \beta _Q ,$$ whered is the lattice dimension. Since β _Q ≧2β, where β is the percolation density exponent, the final bound implies that, below six dimensions, the standard order parameter and correlation length exponents cannot simultaneously assume their mean-field values; hence an implicit bound on the upper critical dimension:d _c ≧6.     

Spin waves in paramagnetic Boltzmann gases

As the temperature is lowered we get an interesting temperature region? _d?T?? ²/mr ₀ ² (where? _d is the quantum degeneracy temperature,m the mass of a gas molecule,r ₀ the radius of interparticle interaction) in which the thermal de Broglie wavelength Λ of a particle is considerably greater than its sizer ₀ though Λ turns out to be less than the mean interparticle distanceN ^?1/3?Λ?r ₀. Although the gas molecules obey the classical Boltzmann-Maxwell statistics the system as a whole begins to exhibit a larger number of essentially quantum macroscopic collective features. One of the most interesting and dramatic features is the possibility of propagation of weakly damped spin oscillations in spin-polarized gases (external magnetic field, optical pumping). Such oscillations can propagate both in the low-frequencyθτ?1 regime and the high frequencyθτ?1. The last case is highly non-trivial for a Boltzmann gas with a short range interaction between particles. The weakness of relaxation damping of spin modes implies that the degree of polarization is high enough 1>/|α|?|a|/Λ, whereα=(N ₊?N _?)N,a is the two-particles-wave scattering length. Under these conditions the spectrum of magnons has the form (Bashkin 1981, 1984; Lhuillier and Laloe 1982) 1 $$\omega = \Omega _H + \left( {{{K^2 \nu _{\rm T}^2 } \mathord{\left/ {\vphantom {{K^2 \nu _{\rm T}^2 } {\Omega _{int} }}} \right. \kern-\nulldelimiterspace} {\Omega _{int} }}} \right)\left( {{{1 - i} \mathord{\left/ {\vphantom {{1 - i} {\Omega _{int} }}} \right. \kern-\nulldelimiterspace} {\Omega _{int} }}\tau } \right), \Omega _{int} = {{ - 4\pi ahN\alpha } \mathord{\left/ {\vphantom {{ - 4\pi ahN\alpha } m}} \right. \kern-\nulldelimiterspace} m}, \nu _{\rm T}^2 = {T \mathord{\left/ {\vphantom {T m}} \right. \kern-\nulldelimiterspace} m}$$ where Ω _H is the Larmor precession frequency for spins in the magnetic fieldH. Collisionless Landau damping restricts the region of applicability of (1) with not too large wave vectorsKv _T?|Ω_int|. The existence of collective spin waves has been experimentally confirmed in NMR-experiments with gaseous atomic hydrogen H↑ (Johnsonet al 1984). The presence of undamped spin oscillations means automatically the existence of long range correlations for transverse magnetization. Such correlations decrease with the distance according to the power law 2 $$\delta _{ik} \left( r \right) = 2\left| a \right|\frac{{\left( {\beta N\alpha } \right)^2 }}{\gamma }\delta _{ik} $$ . Hereβ is the molecule magnetic moment. Spin waves being even damped can nevertheless reveal themselves atT?? ²/mr ₀ ² or when |α|?r ₀/Λ. The first experimental discovery or damped spin waves in gaseous³He↑ has been done in Nacheret al 1984. Oscillations of magnetization can also propagate in some condensed media such as liquid³He-⁴He solutions, semimagnetic semiconductors etc.     

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 $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

The coupled coherent and incoherent motion of excitons and its influence on the line shape of optical absorption

We treat the coupled coherent and incoherent motion of Frenkel excitons by a model calculation. The model contains the four parametersa (distance of neighbouring atoms),J (exchange interaction integral), γ_o (describing the strength of the local energy fluctuations) and γ₁ (a measure of the fluctuations of the exchange interaction integral, i.e. nonlocal fluctuations). Calculation of the optical absorption of systems with two differently oriented molecules/unit cells results in the Davydov-splitting given by Δ=8J and the linewidth given by Γ=γ_o+γ₁. From the equation of motion of the density matrix we derive a diffusion equation. The diffusion constant is given by \(D = \frac{{a^2 }}{\hbar }\left( {2\gamma _1 + \frac{{J^2 }}{{\gamma _1 + \Gamma }}} \right)\) . Comparison with experiment yields γ_o=70cm^?1, γ₁=0.1 cm^?1 at room temperature and Γ<1 cm^?1 at 4.2 °K. Using these numerical values and the criterium of Haken and Strobl we derive that at room temperature the exciton motion is incoherent and may be described by a hopping process whereas at low temperature it is coherent.     

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.     

Critical Hardy–Lieb–Thirring Inequalities for Fourth-Order Operators in Low Dimensions

This paper considers Hardy–Lieb–Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality

Gauge-invariant on-shellZ 2 in QED,QCD and the effective field theory of a static quark

We calculate theon-shell fermion wave-function renormalization constantZ ₂ of a general gauge theory, to two loops, inD dimensions and in an arbitrary covariant gauge, and find it to be gauge-invariant. In QED this is consistent with the dimensionally regularized version of the Johnson-Zumino relation: d logZ ₂/da ₀=i(2)^–D e ₀ ² d ^D k/k ⁴=0. In QCD it is, we believe, a new result, strongly suggestive of the cancellation of the gauge-dependent parts of non-abelian UV and IR anomalous dimensions to all orders. At the two-loop level, we find that the anomalous dimension _F of the fermion field in minimally subtracted QCD, withN _L light-quark flavours, differs from the corresponding anomalous dimension of the effective field theory of a static quark by the gauge-invariant amount

The integrated density of states for the difference Laplacian on the modified Koch graph

We consider the integrated density of statesN(λ) of the difference Laplacian ?Δ on the modified Koch graph. We show thatN(λ) increases only with jumps and a set of jump points ofN(λ) is the set of eigenvalues of ?Δ with the infinite multiplicity. We establish also that $$0< C_1 \leqslant \mathop {\lim }\limits_{\lambda \to 0} \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }}< \overline {\mathop {\lim }\limits_{\lambda \to 0} } \frac{{N(\lambda )}}{{\lambda ^{d_s /2} }} \leqslant C_2< \infty$$ whered _s =2log5/log(40/3) is the spectral dimension of MKG.     

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.     

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.