Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians

We consider the integrated density of states (IDS) ρ_∞(λ) of random Hamiltonian H_ω=?Δ+V_ω, V_ω being a random field on ? ^d which satisfies a mixing condition. We prove that the probability of large fluctuations of the finite volume IDS |Λ|^?1ρ(λ, H_Λ(ω)), Λ ? ? ^d , around the thermodynamic limit ρ_∞(λ) is bounded from above by exp {?k|Λ|},k>0. In this case ρ_∞(λ) can be recovered from a variational principle. Furthermore we show the existence of a Lifshitztype of singularity of ρ_∞(λ) as λ → 0⁺ in the case where V_ω is non-negative. More precisely we prove the following bound: ρ_∞(λ)≦exp(?kλ^?d/2) as λ → 0⁺ k>0. This last result is then discussed in some examples.     

Nuclear core dynamics and rearrangement energy for hypernuclei

The effect of the nuclear core dynamics on the binding energies of Λ hypernuclei is studied in the framework of variational correlated wave functions. In addition to the core energy and to a Λ single-particle contribution ϵ_ν, we exhibit a rearrangement energy E_R which in particular depends on the ΛN correlations F and on the difference δφ_N between the core wave function and its exact ground-state value. E_R is nonzero only if there are both ΛN correlations (F ≠ 1) and if δφ_N ≠ 0. The latter can result from core polarization by the Λ or from errors in the ground-state variational wave function of the core. E_R varies linearly with δφ_N for small δφ_N. Detailed numerical results for central ΛN and NN potentials with repulsive cores confirm these general features. For a give core wave function ΛN correlations reduce the radius of the core nucleons relative to their c.m.; however E_R leads to an expansion of the core rather than to a contraction with only ϵ_ν. The core polarization eneegy is quite small ⋍ 0.4 MeV. We study the effects of errors in the variational core wave function by detuning this as a function of one of the variational parameters.     

A-dependence of ΛΛ bond energies in double-Λ hypernuclei

The A-dependence of the bond energy ΔB_ΛΛ of the ΛΛ hypernuclear ground states is calculated in a three-body Λ + Λ + ^A Z model and in the Skyrme-Hartree-Fock approach. Various ΛΛ and Λ-nucleus or ΛN potentials are used and the sensitivity of ΔB_ΛΛ to the interactions is discussed. It is shown that in medium and heavy ΛΛ hypernuclei, ΔB_ΛΛ is a linear function of r _Λ ^{? 3} , where r _Λ is rms radius of the hyperon orbital. It looks unlikely that it will be possible to extract ΛΛ interaction from the double-Λ hypernuclear energies only, the additional information about the Λ- core interaction, in particular, on r _Λ is needed.     

Binding energies of hypernuclei and ΛN interaction

A central ΛN potential V _ΛN is found on the basis of an analysis of the binding energies of 1s-shell hypernuclei and Λp scattering. Within the exprerimental errors, this potential makes it possible to reproduce the binding energies of three-, four-, and five-particle ground and excited states of hypernuclei and the angular and energy dependences of the cross sections for Λp scattering. The accuracy and reliability of the calculation of three-, four-, and five-particle hypernuclear and the respective nuclear systems is ensured by determining both upper and lower bounds on the energy. Within the Λ plus core model, the potential V _ΛN is matched with binding energies of heavy hypernuclei.     

Λ and Λ production in K+p interactions at 32 GeV/c

A study of the inclusive and semi-inclusive Λ and $\text{Λ}$ production in K⁺p interactions at 32 GeV/c is presented. The inclusive cross sections for Λ and $\text{Λ}$ amount to 0.78 ± 0.05 and 0.42 ± 0.04 mb thus showing a remarkable growth between 16 and 32 GeV/c with a factor of 1.7 for Λ and 2.8 for $\text{Λ}$. Target and beam fragmentation processes are found to be dominant for Λ and $\text{Λ}$ production respectively with the following lower limits for the corresponding cross sections: $\text{σ(p}\text{→}\text{K}{}^{\mathrm{+}}\text{?) > 0.5}\text{mb}$ and $\text{σ(K}{}^{\mathrm{+}}\text{→}\text{p}\text{?}\text{) > 0.3}\text{mb}$. Although the early scaling conditions are fulfilled for the Λ production in the target fragmentation region, and $\text{Λ}$ production in the beam fragmentation region, scaling is not observed between 16 and 32 GeV/c in the x and p_T² Feynman variables. The Λ production is found to be very similar in the K⁺p inclusive reaction at 32 GeV/c and in the semi-inclusive reaction $\text{K}{}^{\mathrm{?}}\text{p}\text{→ Λ}\text{K}\text{K}\text{X}$ at the same energy. The $\text{Λ}\text{Λ}$ pair production cross section increases significantly in K⁺p interactions from 16 to 32 GeV/c where it reaches the value $\text{σ}{}_{\mathrm{<mtext>\Lambda </mtext><mtext>\Lambda </mtext>}}\text{= 47 ± 11 μ}\text{b}$. The cross sections for Λ or $\text{Λ}$ produced in association with an identified proton are also given and discussed.     

Self-consistent calculations of hypernuclear properties with the Skyrme interaction

An extension of the Skyrme-Hartree-Fock formalism is proposed for calculating Λ binding energies (B_Λ) in hypernuclei throughout the periodic table. The calculation includes the self-consistent nuclear polarization by the Λ hyperon. Realistic estimates for ground state B_Λ's are obtained with a simple effective ΛN force and are compared with a Λ-nucleus potential model. The Λ particle excitation spectra have also been calculated. They are discussed in the context of strangeness analog state formation and the emphasis is put on the symmetry between Λ and nuclear states.     

A shell-model analysis of Λ binding energies for the p-shell hypernuclei III. Further analysis and predictions

A simplified phenomenological analysis of the Λ binding energies for the p-shell hypernuclei is given on the basis of the shell-model configuration [(1s)_N⁴(1p)_N^A?5(1s)_Λ], using the intermediate-coupling formalism set up in Part I, assuming the ΛNN interaction Q₀₀⁰ of central form (as motivated by the double-OPE mechanism), and taking into account the new data recently available about several $\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{-shell}$ hypernuclei. The systematics of the mass matrix for p-shell Λ hypernuclei are discussed in some detail. The fit obtained has only moderate strengths for the (ΛN) spin-orbit and (ΛNN) three-body interactions and requires rather little mixing of different nuclear core states into the hypernuclear wavefunctions, except for the special case _Λ⁸Li. The implications of this fit are spelled out for the well-known hypernuclear species. A discrepancy between the wavefunction calculated here for _Λ⁸Li and the characteristics of the decay _Λ⁸Li → ααπ^? is emphasized, as is also the need for Λ hypernuclear γ-ray data.     

Infrared and 1H nuclear magnetic resonance studies of solvent effect on hydrogen bond in some pyridine n-oxide trifluoroacetates

The IR and ¹H NNR spectra are reported for the trifluoroacetates of five substituted pyridine N-oxides in five dry aprotic solvents of different polarity and studied with respect to the influence of the reaction field. Both the centre of gravity ($\text{V}$_H) of continuous absorption and the chemical shift (δ) show weak dependence upon solvent polarity. In the range of ε from 2.27 to 10.36 the continuous absorption is very similar. In acetonitrile (ε = 37.5) $\text{V}$_H is shifted. The magnitude and sign of the $\text{V}$_H shift depend on the proton acceptor properties of N-oxides and can be correlated with the position of the proton in the complex. The influence of the reaction field on OHO bonds is compared with the influence on OH?N?O^??HN⁺ bonds, and the differences in behavior of these complexes are related to their energy surface. The weak dependence of $\text{V}$_H and δ upon the reaction field in pyridine N-oxide trifluoroacetates is consistent with the single minimum energy surface postulated in the literature for the strong OHO bonds.     

Exact four-body calculation of the 4ΛHe-4ΛH binding energy difference

The coupled, two-variable integral equations that determine the ⁴_ΛHe and ⁴_ΛH bound states, when the NN and ΛN interactions are represented by separable potentials, are derived from the Schrödinger equation. The integral equations are solved numerically for simple s-wave potentials and for tensor potentials in the truncated t-matrix approximation without resort to separable expansion of the kernels. The Λ-separation energy difference ΔB_Λ resulting from the genuine four-body model is shown to be approximately twice as large as that coming from an “effective two-body” model calculation, when identical central potentials are used. The four-body model estimate of ΔB_Λ made with tensor forces is consistent with the experimental value, indicating that charge symmetry breaking implied by the low energy Λ N scattering parameters is compatible with that suggested by the known binding energy difference in the A = 4 hypernuclear isodoublet.     

A mechanical definition of the thermodynamic pressure

A dynamical definition of pressure for grand-canonical Gibbs measures in bounded regions Λ is rigorously discussed: It measures the momentum transferred to the walls of the container by the elastically colliding particles. The local pressureP(r, δΛ) so obtained is proportional to the temperature and the local density at the boundaries of Λ. This allows us to obtain a rigorous proof of the virial theorem of Clausius. In this picture the thermodynamic pressureP ^d (Λ) is obtained as the average ofP(r, δΛ) onδΛ. Its relationship with the usual equilibrium pressureP _eq(Λ) = (βsΛ¦)^?1lnZ _Λ (Z _Λ is the grand-canonical partition function) is then discussed. In the particular case in which the regions A are spheres, it is shown that P_d(Λ) converges in average so that, if the limit of P_d(Λ) exists, it equals P_eq, the thermodynamic limit of the equilibrium pressure P_eq(Λ). Finally, convergence ofP _d(Λ) is proven to hold in the particular case of one-dimensional hard cores in the absence of phase transitions.     

On a Semiclassical Formula for Non-Diagonal Matrix Elements

Let H(?)=?? ²d²/dx ²+V(x) be a Schrödinger operator on the real line, W(x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V(x) in exactly two turning points and lies below V _∞=lim?inf?_|x|→∞ V(x). We consider the semiclassical limit n→∞, ?=? _n →0 and E _n =E where E _n is the nth eigenenergy of H(?). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(?), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.     

The spontaneous breaking of supersymmetry in curved space-time

The spontaneous breaking of supersymmetry in the presence of a cosmological constant Λ is discussed in a class of theories that includes gauged supergravity and the recently constructed model of N = 1 supergravity coupled to supermatter. The stability of de Sitter, anti-de Sitter and Minkowski vacua in these theories is investigated. Positivity of energy is demonstrated in a model independent way for supersymmetric vacua, even if the scalar potential is unbounded below, and for global minima of the potential for Λ ? 0.Free fields in anti-de Sitter space are considered and the distinction made between the coefficients of quadratic terms in the lagrangian, which vanish for Goldstone scalars, and the physical masses, which give the frequencies and total energies of modes. The number of degrees of freedom depends on gauge invariance, not on the vanishing of mass.The one-loop corrections to the cosmological constant are given for Λ ? 0 and they vanish if the physical masses obey certain sum rules. It is, however, the bilinear coefficients in the N = 1 supergravity-supermatter lagrangian, rather than the physical masses, that satisfy a quadratic sum rule. This sum rule depends on Λ so that a given mass splitting can be obtained for arbitrarily large supersymmetry breaking scales if Λ is sufficiently large and negative.     

Torsion of the nitro group in nitroethylene

Eight NO₂-torsional transition frequencies obtained from interferometric far infrared measurements are reported. These data provide accurate information on the potential function for torsional angles up to 50°. The potential parameters are V₂ = 1642.7 ± 3.9 cm^?1 and V₄ = ?88.0 ± 1.7 cm^?1 for a rigid frame/rigid top model and V₂ = 1690.6 ± 4.2 cm^?1 and V₄ = ?90.8 ± 1.8 cm^?1 for a refined model allowing for distortions of the NO₂-group upon torsion. V₆ and V₈ terms are not significant.     

The quark-nucleon phase diagram and quantum chromodynamics

The temperature dependence of a conjectured first-order phase transition between nuclear matter and quark-gluon matter is calculated for temperatures below T = 200 MeV. On the nuclear side a rather successful meson-nucleon mean field theory is applied while quark-gluon matter at large densities and finite temperatures is described perturbatively by quantum chromodynamics. Outside the finite volume of hot and dense quark-gluon matter the physical vacuum is characterized, by the newly determined bag parameter Λ_B = 235 MeV. We observe a dramatic drop in the density of nuclear matter at the phase transition point as the temperature increases, if the scale parameter Λ of QCD is chosen as Λ = 100 MeV. For larger values of Λ the effect is less pronounced. Further work is required to settle this problem.     

Measurements of hyperon semileptonic decays at the CERN Super Proton Synchrotron

The charged hyperon beam at the CERN Super Proton Synchrotron (SPS) has been used to collect data on semileptonic decays ofΣ ^?,Ξ ^? andΛ. A magnetic channel selects 100 GeV/c negatively charged particles produced in the forward direction by interaction of the 200 GeV/c SPS proton beam on a BeO target. TheΣ ^? andΞ ^? hyperons are concurrently identified in a DISC ?erenkov counter, and their decay products are analysed by a magnetic spectrometer. Electron-hadron discrimination is achieved by the combined use of lead-glass and lead/scintillator counters, transition radiation detectors, and a ?erenkov counter. In this article we report results on the \(\Sigma ^ - \to \Lambda e^ - \bar v\) decay mode. Measurements of the Λ polarization and of the centre-of-mass distributions (baryon kinetic energy, electron-neutrino correlation, and Dalitz plot distributions) yield the vector to axialvector form factor ratiof ₁/g ₁=+0.034±0.080, in agreement with the value expected from the conserved vector current hypothesis (f ₁/g ₁=0). TheΣ ^?→Λe ^? v→ branching ratio measurement gives a value of (5.41±0.30)×10^?5. The effects of radiative corrections are not included in these results. They are discussed in the text. Results on the otherΞ ^?,Σ ^?, andΛ semileptonic decays are reported in separate articles.     

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.     

Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ? for the classical Hamiltonian H = H₀ + ?H₁, where H₀ gives rise to a known motion and ? is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ? and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator (s ? L)^?1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ?, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ?ⁿ⁺¹ when the accuracy of the theory is order ?ⁿ.     

The problem of confinement: From two to four dimensions

The Coulomb law of “electrodynamics” in two space time dimensions in extended to four dimensions. The result is that antisymmetric tensor potentials A_μν? subject to the gauge transformation δA_μν? = ?_μΛ_ν? + ?_?Λ_?μ + ?_?Λ_μν can be effectively used as confining potentials.     

On the definition of the pressure of an infinite system

A microcanonical system $\text{L}$^o_Λ is considered together with a manometer $\text{M}$_Λ. The thermodynamic limit Λ → ∞ is taken for the system $\text{L}$_Λ composed of $\text{L}$^o_Λ and $\text{M}$_Λ. This yields a definition of the pressure P(v, ε) of $\text{{}\text{L}{}^{\mathrm{o}}{}_{\mathrm{\Lambda }}\text{; Λ → ∞}}$ for given values ε and v of the energy and particle densities. P(v, ε) is shown to be equal to the thermodynamic function p(z, β) derived from the grand canonical ensemble for almost all values of ε and v provided the appropriate equilibrium values of the temperature β^?1 and the chemical potential μ = β^?1 log z are inserted.