Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field

We consider a large neutral molecule with total nuclear charge Z in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Z α ² ≤ κ ₀ for a sufficiently small κ ₀, where α denotes the fine structure constant. We show that, in the simultaneous limit Z → ∞, α → 0 such that κ = Z α ² is fixed, the ground state energy of the system is given by a two term expansion c ₁ Z ^7/3 + c ₂(κ) Z ² + o(Z ²). The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion.     

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

The Primakoff reaction study for pion polarizability measurement at COMPASS

The electromagnetic structure of charged pions can be described by the electric (α_π) and magnetic (β_π) polarizabilities that depend on the rigidity of pion’s internal structure as a composite particle. It is shown that the values of α_π and β_π can be precisely measured via the Primakoff reaction π⁻ + (A, Z) → π⁻ + (A, Z) + γ in the COMPASS experiment at CERN.     

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

Atoms in Strong Magnetic Fields:¶The High Field Limit at Fixed Nuclear Charge

Let E(B,Z,N) denote the ground state energy of an atom with N electrons and nuclear charge Z in a homogeneous magnetic field B. We study the asymptotics of E(B,Z,N) as B→∞ with N and Z fixed but arbitrary. It is shown that the leading term has the form (ln B)² e(Z,N), where e(Z,N) is the ground state energy of a system of N bosons with delta interactions in one dimension. This extends and refines previously known results for N= 1 on the one hand, and N,Z→∞ with B/Z ³→∞ on the other hand. Received: 9 December 1999 / Accepted: 15 February 200     

e + e − pair production in ultrarelativistic heavy-ion collisions at intermediate impact parameters

Using the semiclassical Green’s function in the Coulomb field, we analyze the probabilities of single and multiple e ⁺ e ⁻ pair production at a fixed impact parameter b between colliding ultrarelativistic heavy nuclei. We perform calculations in the Born approximation with respect to the parameter Z _Bα and exactly in Z _Aα, where Z _A and Z _B are the charge numbers of the corresponding nuclei. We also obtain the approximate formulas for the probabilities valid for Z _Aα, Z _Bα ≲ 1. The text was submitted by the authors in English.     

Electromagnetic polarizabilities of the nucleon and properties of the σ-meson pole contribution

The t-channel contribution to the difference of electromagnetic polarizabilities of the nucleon, (α - β)^t, can be quantitatively understood in terms of a σ-meson pole in the complex t-plane of the invariant scattering amplitude A ₁(s, t) with properties of the σ-meson as given by the quark-level Nambu-Jona-Lasinio model (NJL). Equivalently, this quantity may be understood in terms of a cut in the complex t-plane where the properties of the σ-meson are taken from the ππ → σ → ππ, γγ → σ → ππ and Nˉ → σ → ππ reactions. This equivalence may be understood as a sum rule where the properties of the σ-meson as predicted by the NJL model are related to the f ₀(600) particle observed in the three reactions. In the following, we describe details of the derivation of (α - β)^t making use of predictions of the quark-level NJL model for the σ-meson mass. An erratum to this article is available at .     

κ-generalized statistics in personal income distribution

Starting from the generalized exponential function , with exp ₀(x)=exp (x), proposed in reference [G. Kaniadakis, Physica A 296, 405 (2001)], the survival function P_>(x)=exp _κ(-βx^α), where x∈R⁺, α,β>0, and , is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function P_> ⁰(x)=exp (-βx^α) to which reduces as κ approaches zero behaving in very different way in the x→0 and x→∞ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law P_>(x)∼(2βκ)^-1/κx^-α/κ. This makes the κ-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.     

Search for Z’ → WW at LHC with the ATLAS detector

The potential for measuring the Z‘WW coupling with the ATLAS detector at the LHC is investigated under conditions of low and high luminosity. It is estimated that detection of the decay channel Z’ → WW → lν _l jj is possible with 300 fb ⁻¹, for a Z’ boson with sequential model couplings and with a mass up to 2.2 TeV.     

Associated production of Higgs bosons with Z bosons by charged leptons in strong external fields

The probabilities of the associated production of a Higgs boson with a Z boson by a charged lepton in the field of a plane electromagnetic wave of arbitrary intensity and in a constant crossed field are obtained. The behavior of the cross section of the process as a function of the particle energies and the external field intensity is investigated for various values of the Higgs boson mass. It is shown that there is a logarithmic increase in the photoproduction cross section at superhigh energies up to a value significantly exceeding the cross section of the reaction e ⁺+e ⁻→Z+H, which is presently regarded as the most probable channel for the production of Higgs bosons. Zh. éksp. Teor. Fiz. 113, 1979–1990 (June 1998)     

Light squeezing in optical parametric amplification beyond the slowly-varying amplitude approximation

Optical parametric amplification (OPA) described usually by the coupled-wave equations with the first-order derivatives of the signal and idler waves, is solved under the slowly-varying amplitude approximation (SVA). In this article, by keeping the second-order derivatives in the coupled-wave equations, we obtained an analytical solution for the output signal and idler waves up to the first order of (κ/k)¹; the ratio of coupling constant to the wave number. Furthermore, here the signal and the idler waves are distinguished only by their polarizations with the same frequency. Light squeezing is observed in normally ordered variances of the two quadrature operators of the output combined mode when plotted against κL, where κ is the coupling constant and L the interaction length. The variances have different signs for a range of values of κL and their variations are in opposite directions. We also show that this property is strongly dependent on the relative refractive index of the medium (n). It is worth mentioning that the relative index dependency is not an explicit feature in squeezing of OPA under SVA approximation. Furthermore, the squeezing vanishes when n → 1 and κ/k → 0.     

Double atomic photoeffect in the relativistic domain: Angular and energy distributions of photoelectrons

We study the double ionization of the atomic K-shell by a single photon in the relativistic energy domain. The differential and total cross sections of the process are calculated. It is shown that the ratio of the cross sections of double and single ionization increases with the photon energy, tending to the limit 0.34/Z ², where Z is the atomic number or the nuclear charge. The formulas are found to be valid for Z≫1 and αZ≪1, where α=1/137 is the fine-structure constant. Zh. éksp. Teor. Fiz. 114, 1537–1554 (November 1998)     

On Minimal Eigenvalues¶of Schrödinger Operators on Manifolds

We consider the problem of minimizing the eigenvalues of the Schr?dinger operator H=−Δ+αF(κ) (α>0) on a compact n-manifold subject to the restriction that κ has a given fixed average κ₀. In the one-dimensional case our results imply in particular that for F(κ)=κ² the constant potential fails to minimize the principal eigenvalue for α>α_c=μ₁/(4κ₀ ²), where μ₁ is the first nonzero eigenvalue of −Δ. This complements a result by Exner, Harrell and Loss, showing that the critical value where the constant potential stops being a minimizer for a class of Schr?dinger operators penalized by curvature is given by α _c . Furthermore, we show that the value of μ₁/4 remains the infimum for all α >α _c . Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential. In higher dimensions we prove a (weak) local version of these results for a general class of potentials F(κ), and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace–Beltrami operator and is never attained. Received: 17 July 2000 / Accepted: 11 October 2000     

Alpha-particle condensation in nuclei: Experimental problems

Recently, it was supposed the nα light nuclei may have structures resembling the diluted gas of α particles which could be considered as Bose-Einstein condensate (αBEC) in nuclear systems. At the present time, the only more or less definite candidate for an αBEC state predicted by theory is the second 0⁺ state of ¹²C (E = 7.65 MeV) located near the threshold ¹²C → 3α. We discuss several possible experimental approaches to study the properties of interest of the 7.65-MeV level of ¹²C. Among them are shift of the positions of the rainbow minima in the inelastic scattering to this level, extraction of the empirical inelastic form factor from the α and ³He scattering and its comparison with theoretical predictions, and getting information on the ⁸Be transfer reaction form factor. The conclusion was drawn that new measurements of ¹²C(α, α′) ¹²C(7.65) inelastic scattering at ∼120 MeV in the full angular range would be most adequate to solving the problem. The possibility of observation of αBEC effects in heavier nα nuclei is also discussed. In particular, ¹¹²Ba is of great interest owing to possible existence of an α-particle “halo” outside the Z = N = 50 core. The text was submitted by the authors in English.     

Decays of σ, κ, a0(980) and f0(980)

Ratios of coupling constants for these decays are compared with qq̄ predictions and Jaffe’s q²q̄² model. In both models, the predicted ratio g²(κ→Kπ)/g²(σ→ππ) is much too small. Also, for qq̄, the predicted ratio g²(κ→Kη’)/g²(κ→Kπ) is much larger than observed. Both models fail for g²(f₀→KK)/g²(a₀→KK). This ratio requires that f₀ has a dominant KK component. It arises naturally because the f₀ pole lies very close to the KK threshold, giving its wave function a long KK tail. PACS 13.25.Gv, 14.40.Gx, 13.40.Hq     

Thermal Conductivity of the Toda Lattice with Conservative Noise

We study the thermal conductivity of the one dimensional Toda lattice perturbed by a stochastic dynamics preserving energy and momentum. The strength of the stochastic noise is controlled by a parameter γ. We show that heat transport is anomalous, and that the thermal conductivity diverges with the length n of the chain according to κ(n)∼n ^α , with 0<α≤1/2. In particular, the ballistic heat conduction of the unperturbed Toda chain is destroyed. Besides, the exponent α of the divergence depends on γ.     

Random Matrix Theory and ζ(1/2+it)

We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z ^*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N→∞ asymptotics of the moments of |Z| and Z/Z ^*. These CUE results are then compared with what is known about the Riemann zeta function ζ (s) on its critical line Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2+iT) in the limit T→∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of |ζ(1/2+it) |. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles. Received: 20 December 1999 / Accepted: 24 March 2000     

Rank-Driven Markov Processes

We study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κ _N on {1,2,…,N} ^K . Our main results are that, under appropriate conditions on κ _N , the system exhibits threshold behavior at s ^∗∈[0,1], where s ^∗ is a function of κ _N , and the marginal distribution of a randomly selected element converges to U[s ^∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.     

Re-evaluation of the LHC potential for the measurement of <Emphasis Type="Italic">m</Emphasis><Subscript><Emphasis Type="Italic">W</Emphasis></Subscript>

We present a study of the LHC sensitivity to the W boson mass based on simulation studies. We find that both experimental and phenomenological sources of systematic uncertainties can be strongly constrained with Z measurements: the lineshape, dσ _Z /dm, is robustly predicted, and its analysis provides an accurate measurement of the detector resolution and absolute scale, while the differential cross-section analysis, d² σ _Z /dydp _T , absorbs the strong interaction uncertainties. A sensitivity δ m _W ∼7 MeV for each decay channel (W→e ν, W→μ ν), and for an integrated luminosity of 10 fb⁻¹, appears as a reasonable goal.     

Dynamics of α-clusters in N = Z nuclei

The systematics for binding energies per α-particle in N = Z nuclei, E _Bα/N _α, are studied up to ¹⁶⁴Pb. It is shown that, although a geometrical model can be used to explain the systematics for light nuclei, the binding energy per α-particle exhibits structures which are due to the well-known shells of the mean field of nucleons in nuclei. The overall dependence of E _Bα/N _α on N _α in N = Z nuclei (for the ground-state masses) can be described in a liquid-drop model of α-particles. Conditions for a phase change with the formation of an α-particle condensate, a dilute Bose gas in excited compound nuclei are discussed for E _Bα/N _α = 0, at the thresholds. This is achieved when the binding energy per nucleon in nuclei is equal to or smaller than in the α-cluster. At somewhat smaller excitation energies the appearance of a Bose gas with a closed-shell core (N = Z, e.g. of ⁴⁰Ca) is proposed within the same concept. The experimental observation of the decay of such condensed α-particle states is proposed with the coherent emission of several correlated α-particles not described by the Hauser-Feshbach approach for compound-nucleus decay. This decay will be observed by the emission of unbound resonances in the form of ⁸Be and ¹²C ^* (0⁺ ₂) clusters.