Fractal wavelet dimensions and localization

In this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ?. It will be proved that for φ, ψ∈? we have $$\mathop {\lim \inf }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ + (2)$$ and that $$\mathop {\lim \sup }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ - (2),$$ wherek ^±(2) are the upper and lower correlation dimensions of the spectral measure associated with ψ and ?. A quantitative version of the RAGE theorem shall also be given.     

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

Absence of long-range order in one-dimensional spin systems

For a one-dimensional Ising model with interaction energy $$E\left\{ \mu \right\} = - \sum\limits_{1 \leqslant i< j \leqslant N} {J(j - i)} \mu _\iota \mu _j \left[ {J(k) \geqslant 0,\mu _\iota = \pm 1} \right]$$ it is proved that there is no long-range order at any temperature when $$S_N = \sum\limits_{k = 1}^N {kJ\left( k \right) = o} \left( {[\log N]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)$$ The same result is shown to hold for the corresponding plane rotator model when $$S_N = o\left( {\left[ {{{\log N} \mathord{\left/ {\vphantom {{\log N} {\log \log N}}} \right. \kern-\nulldelimiterspace} {\log \log N}}} \right]} \right)$$      

Measurement of the\sum {^ - \to ne^ - \bar v} rate and search for ΔQ=−ΔS transitions in Σ+→ne + ν decay

In a bubble chamber experiment about 2×10⁶ Σ ^±-decays have been measured to separateΣ ^±→ne^±¯ν events from the two-body modes. NoΣ ⁺ →ne ⁺ ν event was found whereas 607Σ ^?→ne^?¯ν decays could be identified. The data yield for the ΔQ=?ΔS decay an upper limit: $$\frac{{\Gamma \left( {\sum {^ + } \to ne^ + v} \right)}}{{\Gamma \left( {\sum {^ - } \to ne^ - v} \right)}}< 1.9 x 10^{ - 2} (90\% confidence level)$$ and the branching ratio: $$\frac{{\Gamma \left( {\sum {^ - } \to ne^ - v} \right)}}{{\Gamma \left( {\sum {^ - } \to n\pi ^ - } \right)}} = (1.09 \pm 0.06) x 10^{ - 3} .$$      

A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

Lattice systems with a continuous symmetry

We investigate a continuous Ising system on a lattice, equivalently an anharmonic crystal, with interactions: $$\sum\limits_{\left\langle {x,y} \right\rangle } {\left( {\phi _x - \phi _y } \right)} ^2 + \lambda \left( {\phi _x - \phi _y } \right)^4 , \phi _x \in \mathbb{R}, x \in \mathbb{Z}^d .$$ We prove that the perturbation expansion for the free energy and for the correlation functions is asymptotic about λ=0, despite the fact that the reference system (λ=0) does not cluster exponentially. The results can be extended to more general systems of this type, e.g. an even polynomial semibounded from below instead of a quartic interaction. By a suitable scaling, λ corresponds to the temperature.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

One-dimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion

WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where ε>0 andJ _ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0} $$ , ii) $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC ^∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉 _H denotes the Gibbs average with respect to the HamiltonianH.     

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.     

Bounds on the decay of correlations for λ(∇φ)4 models

We consider models, with an abelian continuous group of symmetry, of the type: $$H = \sum\limits_x {\left[ {\frac{1}{2}(\nabla _x \phi )^2 + \frac{\lambda }{4}(\nabla _x \phi )^4 } \right].}$$ We generalize Brascamp-Lieb inequalities to get (λ-independent) bounds on the low momentum behaviour of general correlation functions when these are truncated into two clusters. We then use this result to derive an asymptotic expansion (up the second order in λ) of the dielectric constant of this system.     

Domain and Range of the Modified Wave Operator for Schrödinger Equations with a Critical Nonlinearity

We study the final problem for the nonlinear Schrödinger equation

$i{\partial }_{t}u+\frac{1}{2}\Delta u=\lambda|u|^{\frac{2}{n}}u,\quad (t,x)\in {\mathbf{R}}\times \mathbf{R}^{n},$

where\(\lambda \in{\bf R},n=1,2,3\). If the final data\(u_{+}\in {\bf H}^{0,\alpha }=\left\{ \phi \in {\bf L}^{2}:\left( 1+\left\vert x\right\vert \right) ^{\alpha }\phi \in {\bf L}^{2}\right\} \) with\(\frac{ n}{2} < \alpha < \min \left( n,2,1+\frac{2}{n}\right) \) and the norm\(\Vert \widehat{u_{+}}\Vert _{{\bf L}^{\infty }}\) is sufficiently small, then we prove the existence of the wave operator in L ². We also construct the modified scattering operator from H ^0,α to H ^0,δ with\(\frac{n}{2} < \delta < \alpha\).

     

On the bound state in weakly coupled λ(ϕ6−ϕ4)2

We consider the λ(?⁶??⁴) quantum field theory in two space-time dimensions. Using the Bethe-Salpeter equation, we show that there is a unique two particle bound state if the coupling constant λ>0 is sufficiently small. Ifm is the mass of single particles then the bound state mass is given by $$_B (\lambda ) = 2m\left( {1 - \frac{9}{8}\left( {\frac{\lambda }{{m^2 }}} \right)} \right)^2 + \mathcal{O}\left( {\lambda ^3 } \right).$$      

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.     

Absence of ballistic motion

For large classes of Schrödinger operators and Jacobi matrices we prove that ifh has only one point spectrum then for φ₀ of compact support $$\mathop {\lim }\limits_{t \to \infty } t^{ - 2} \left\| {xe^{ - ith} \phi _0 } \right\|^2 = 0.$$      

Analyticity properties of the weakly coupled double-well model

In this note we study lattice Φ⁴-models with Hamiltonian $$H = \tfrac{1}{2}(\varphi , - \Delta \varphi ) + \lambda \Sigma \left( {\varphi _i^2 - \frac{{m^2 }}{{8\lambda }}} \right)^2$$ and Gaussian boundary conditions. Using the polymer expansion we obtain analyticity of the pressure and the correlation functions in the infinite volume limit in a region $$\left\{ {\left. \lambda \right| \left| \lambda \right|< \varepsilon ,\left| {arg } \right.\left. \lambda \right|< \frac{\pi }{2} - \delta } \right\}$$ for every δ>0.     

Analysis of the \frac{1} {2}^ - and \frac{3} {2}^ - heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the $\frac{1} {2}^ -$ and $\frac{3} {2}^ -$ heavy and doubly heavy baryon states $\Sigma _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi '_Q \left( {\frac{1} {2}^ - } \right)$ , $\Omega _Q \left( {\frac{1} {2}^ - } \right)$ , $\Xi _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Omega _{QQ} \left( {\frac{1} {2}^ - } \right)$ , $\Sigma _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Omega _Q^* \left( {\frac{3} {2}^ - } \right)$ , $\Xi _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ and $\Omega _{QQ}^* \left( {\frac{3} {2}^ - } \right)$ by subtracting the contributions from the corresponding $\frac{1} {2}^ +$ and $\frac{3} {2}^ +$ heavy and doubly heavy baryon states with the QCD sum rules in a systematic way, and make reasonable predictions for their masses.     

g-factors of rotational states in176Hf,177Hf,and180Hf

g-factors of rotational states in¹⁷⁶Hf and¹⁸⁰Hf were measured with the twelve detector IPAC-apparatus of our laboratory [1]. The natural radioactivity 3.78·10¹⁰y¹⁷⁶Lu and the 5.5 h isomer^180mHf were used which populate the ground-state rotational bands of¹⁷⁶Hf and¹⁸⁰Hf. The integral rotations ofγ-γ directional correlations in strong external magnetic fields and in static hyperfine fields of (Lu→Hf)Fe₂ and HfFe₂ were observed. The following results were obtained: $$\begin{array}{l} ^{176} Hf: g\left( {4_1^ + } \right) = + 0.334\left( {38} \right) \\ ^{180} Hf: g\left( {2_1^ + } \right) = + 0.305\left( {14} \right) \\ g\left( {4_1^ + } \right) = + 0.358\left( {43} \right) \\ {{ g\left( {6_1^ + } \right)} \mathord{\left/ {\vphantom {{ g\left( {6_1^ + } \right)} {g\left( {4_1^ + } \right)}}} \right. \kern-\nulldelimiterspace} {g\left( {4_1^ + } \right)}} = + 0.95\left( {12} \right) \\ \end{array}$$ . The hyperfine field in (Lu→Hf)Fe₂ was calibrated by observing the integral rotation of the 9/2^? first excited state of¹⁷⁷Hf populated in the decay of 6.7d¹⁷⁷Lu. Theg-factor of this state was redetermined in an external magnetic field as $$^{177} Hf: g\left( {{9 \mathord{\left/ {\vphantom {9 {2^ - }}} \right. \kern-\nulldelimiterspace} {2^ - }}} \right) = + 0.228\left( 7 \right)$$ . Finally theg-factor of the 2 ₁ ⁺ state of¹⁷⁶Hf was derived from the measuredg(2 ₁ ⁺ ) of¹⁸⁰Hf by use of the precisely known ratiog(2 ₁ ⁺ ,¹⁷⁶Hf)/g(2 ₁ ⁺ ,¹⁸⁰Hf) [2] as $$^{176} Hf: g\left( {2_1^ + } \right) = + 0.315\left( {30} \right)$$ .     

A Rigorous Upper Bound on the Propagation Speed for the Swift–Hohenberg and Related Equations

We prove that if the initial condition of the Swift–Hohenberg equation $$\partial _t u(x,t) = (\varepsilon ^2 - (1 + \partial _x^2 )^2 ){\text{ }}u(x,t) - u^3 (x,t)$$ is bounded in modulus by Ce ^?βx as x→+∞, the solution cannot propagate to the right with a speed greater than $$\mathop {\sup }\limits_{0 < {\gamma } \leqslant \beta } {\gamma }^{ - 1} (\varepsilon ^2 + 4{\gamma }^2 + 8{\gamma }^4 ).$$ This settles a long-standing conjecture about the possible asymptotic propagation speed of the Swift–Hohenberg equation. The proof does not use the maximum principle and is simple enough to generalize easily to other equations. We illustrate this with an example of a modified Ginzburg–Landau equation, where the critical speed is not determined by the linearization alone.     

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).     

On the absence of breakdown of symmetry for the plane rotator model with long-range random interaction

We study the plane rotator model with hamiltonian $$ - \frac{1}{2}\sum\limits_{x \ne y} {J_{xy} \frac{{\cos (\theta _x - \theta _y )}}{{\left| {\left. {x - y} \right|} \right.^{3 + \in } }}}$$ whereJ _xy for different pair (x, y) are independent symmetric random variables. It is proved that for almost allJ, all the Gibbs statesP(J) are rotation invariant.