<Emphasis Type="Italic">ρρ</Emphasis>N and <Emphasis Type="Italic">ρρ</Emphasis>Δ molecules with J<Superscript>P</Superscript> = 5/2<Superscript>+</Superscript> and J<Superscript>P</Superscript> = 7/2<Superscript>+</Superscript>

The ρρN and ρρΔ three-body systems have been studied within the framework of the fixed center approximation of Faddeev equation. The ρρ interaction in isospin I = 0 , spin S = 2 is strongly attractive, and so are the N ρ, ρΔ interactions. This leads to bound states of both ρρN and ρρΔ. We find peaks of the modulus squared of the scattering matrix around 2227 MeV for ρρN, and 2372 MeV for ρρΔ. Yet, the strength of the peak for the ρρN amplitude is much smaller than for ρρΔ, weakening the case for a ρρN bound state, or a dominant ρρN component. A discussion is made on how these states can be searched for in present programs looking for multimeson final states in different reactions.     

On the Uniqueness of Gravitational Centre

The dual volume of order α of a convex body A in R ⁿ is a function which assigns to every a ∈ A the mean value of α-power of distances of a from the boundary of A with respect to all directions. We prove that this function is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n (for α = 0 and for α = n the function is constant). It implies that the dual volume of a convex body has the unique minimizer for α > n or α < 0 and has the unique maximizer for 0 < α < n. The gravitational centre of a convex body in R³ coincides with the maximizer of dual volume of order 2, thus it is unique.      

Typicality of Pure States Randomly Sampled According to the Gaussian Adjusted Projected Measure

Consider a mixed quantum mechanical state, describing a statistical ensemble in terms of an arbitrary density operator ρ of low purity, tr ρ ² ≪1, and yielding the ensemble averaged expectation value tr (ρ A) for any observable A. Assuming that the given statistical ensemble ρ is generated by randomly sampling pure states |ψ〉 according to the corresponding so-called Gaussian adjusted projected measure (Goldstein et al. in J. Stat. Phys. 125:1197, 2006), the expectation value 〈ψ|A|ψ〉 is shown to be extremely close to the ensemble average tr (ρ A) for the overwhelming majority of pure states |ψ〉 and any experimentally realistic observable A. In particular, such a ‘typicality’ property holds  whenever the Hilbert space ℋ of the system contains a high dimensional subspace ℋ₊⊂ℋ with the property that all |ψ〉∈ℋ₊ are realized with equal probability and all other |ψ〉∈ℋ are excluded.     

ρ(ω) → π0π0γ, ρ(ω) → ηπ0γ decays in the local quark Nambu-Jona-Lasinio model

The branching ratios and photon spectra of the rare processes ρ(ω) → π ⁰ π ⁰ γ, ρ(ω) → ηπ ⁰ γ are calculated in the framework of the standard local quark Nambu-Jona-Lasinio model. Three types of diagrams are considered: the quark box and the pole diagrams with scalar (σ, α ₀(980)) and vector (ρ, ω) mesons. The obtained estimates for the widths of the processes ρ(ω) → π ⁰ π ⁰ γ are in satisfactory agreement with existing experimental data. Predictions are made for the widths of the processes ρ(ω) → ηπ ⁰ γ. The text was submitted by the authors in English.     

Strong isospin violation andη−π mixing in |ΔS|=1 weak transition

We consider the effects ofη−π mixing on the violation of the |ΔI|=1/2 rule in |ΔS|=1 weak transitions. The processes considered are theK→2π,K→3π, Λ, Ξ and Λ hyperon decays.     

The decay φ→η″γ

A search for the process e ⁺ e ⁻→φ(1020)→η″(958) γ in the decay channel η″→π ⁺ π ⁻ η, η→γγ was made in an experiment at the VEPP-2M e ⁺ e ⁻ collider with the SND detector. Analysis confirms the occurrence of φ→η″γ decay with probability B(φ→η″ γ)=(6.7 _−2.9 ^+3.4 )×10⁻⁵. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 2, 87–91 (25 January 1999)     

Radiative decays,photoproduction and total cross section of theψ particles

It is pointed out that the coupling characterizing theψ-γ vertex must change substantially between the limits,ψ on mass-shell which occurs inψ→e ⁺ e ⁻ and photon on mass-shell which is relevant in radiative decays likeψ→ππγ, ψ→ηγ and photoproduction ofψ. This has the consequence that the value ofψN total cross section must be larger than what is inferred from the use of naive vector dominance in photoproduction.     

η(1475) and <Emphasis Type="Italic">f</Emphasis><Subscript>1</Subscript>(1420) resonances in the decay processes <Emphasis Type="Italic">J</Emphasis>/ψ → γ(ρρ, γρ<Superscript>0</Superscript>, γϕ) and in γγ* collisions

A scenario that removes the contradiction between the suppression of the η(1475) → γγ decay width and the strong coupling of η(1475) to the ρρ, ωω, and γρ⁰ channels and which leads to a nontrivial prediction for the manifestation of η(1475) in γγ*(Q ²) collisions is considered. Data on the dependence of the cross section for the reaction γγ*(Q ²) → K[`(K)]pK\bar K\pi on the photon virtuality in the energy range 1.35–1.55 GeV are explained here by the production of an η(1475) resonance in contrast to their standard interpretation in terms of the f <sub>1</sub>(1420) resonance. Experimental verification of the present explanation requires determining the spin-parity of resonance contributions, R, in the reactions γγ*(Q <sup>2</sup>) R → R → K[`(K)]pK\bar K\pi and J/ψ → γR → γ(γρ⁰, γϕ).     

On bound state computations in three- and four-electron atomic systems

A variational approach is developed for bound state calculations in three- and four-electron atomic systems. This approach can be applied to determine, in principle, an arbitrary bound state in three- and four-electron ions and atoms. Our variational wave functions are constructed from four- and five-body Gaussoids that respectively depend on six (r ₁₂, r ₁₃, r ₁₄, r ₂₃, r ₂₄, r ₃₄) and ten (r ₁₂, r ₁₃, r ₁₄, r ₁₅, r ₂₃, r ₂₄, r ₂₅, r ₃₄, r ₃₅ and r ₄₅) relative coordinates. The approach allows operating with the more than one electron spin functions. In particular, the trial wave functions for the ¹ S states in four-electron atomic systems include the two independent spin functions χ₁ = αβαβ + βαβα − βααβ − αββα and χ₂ = 2ααββ + 2ββαα − βααβ − αββα − βαβα − αβαβ. We also discuss the construction of variational wave functions for the excited 2³ S states in four- electron atomic systems.     

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

Inspired by the work of D.G.Kelly and S.Sherman on general Griffiths inequalities on correlations in Ising ferromagnets, we formulate and prove Griffith–Kelly–Sherman-type inequalities for the ferromagnetic Potts model with a general number q of local states. We take as local state space for the q-state Potts model the set F ^c  = { − l, − l + 1, ⋯ ,l − 1,l},where l=\fracq-12l=\frac{q-1}{2}. The important properties of F ^c for what follows are that |F ^c | = q and F ^c  = − F ^c .     

Eddy diffusivities derived from a numerical model of the convective planetary boundary layer

Summary Functional forms for the vertical eddy diffusivityK _z (z) are sought that optimize the performance of theK-theory diffusion equation. The method developed to determine the optimal diffusivity is first tested by applying it to analytic solution of the diffusion equation in which the functional form of the diffusivity is known precisely. In all test cases performed, the technique yields the correctK _z profile regardless of the initial estimate ofK _z from which the technique’s search procedure begins. When applied to “observed” mean, cross-wind-integrated point source concentration fields derived from Lagrangian diffusion theory and data from a numerical turbulence model jointly, the technique yields optimal diffusivities that make the solution of the diffusion equation agree within ±20% of the “observed” values within the core of point source plumes. Expressed in terms of the convective-velocity scalew _* and the mixed-layer depthz _i , the optimal diffusivity has a quasiuniversal form for atmospheric stabilities in the rangez _i /L⪯-10 whereL is the Monin-Obukhov length. The optimal diffusivity is found to be strongly dependent on the source hight. TheK _z profiles derived for the two source heightsz _s∼-0.025z _i andz _s ∼-0.25z _i are of opposite shape, but they have comparable maximum values ofK _z ∼-0.25w _* z _i .

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Riassunto Si cercano forme funzionali per la diffusibilità turbolenta verticaleK _z (z) che rendano ottimale l’efficacia dell’equazione di diffusione della teoriaK. II metodo sviluppato per determinare la diffusibilità ottimale è dapprima controllato applicandolo a soluzioni analitiche dell’equazione di diffusione in cui la forma funzionale della diffusibilità è nota esattamente. In tutti i casi di prova la tecnica dà il profilo corretto diK _z qualunque sia la stima iniziale diK _z da cui prende inizio il procedimento di ricerca della tecnica. Quando è applicata contemporaneamente ai campi di concentrazione medi “osservati”, a sorgente puntiforme integrati sui venti trasversali, derivati dalla teoria di diffusione lagrangiana ed ai dati presi dal modello di turbolenza numerico, la tecnica dà diffusibilità ottimali per le quali la soluzione dell’equazione di diffusione è in buon accordo entro il ±20% dei valori “osservati” nel nocciolo di pennacchi della sorgente puntiforme. Espressa in termini della scala di velocità di convezionew _* e della profondità dello strato mistoz _i , la diffusibilità ottimale dipende fortemente dall’altezza della sorgente. I profili diK _z derivati per due altezze di sorgentez _s ∼-0.025z _s ez _s ∼-0.25z _i sono di forma opposta ma hanno valori massimi diK _z paragonabili,K _z ∼-0.25w _* z _i .

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Резюме Определяются функциональные выражения для вертикального вихревого козффициента диффузииK _z (Z), которые оптимизируют получение уравнения диффузии вK-теории. Развитый метод для определения оптимального коэффициента диффузии сначала проверяется на аналитических решениях уравнения диффузии, в которых функциональное выражение козффициента диффузии известно точно. Во всех исследованных случаях предложенный метод дает правильный профильK _z . Предложенный метод дает оптимальные козффициенты диффузии, при которых решение уравнения дуффузии согласуется, в пределах ±20%, с ≪наблюдаемыми≫ величинами. Выраженный через масштаб конвекционной скоростиw _* и глубину смешанного слояz _i оптимальный коэффициент диффузии имеет квази-универсальную форму для атмосферных устойчивостей в областиz _i /L≤10, гдеL есть длина Монина-Обухова. Обнаружено, что оптимальный козффнциент диффузии сильно зависит от высоты источника. ПрофилиK _z , выведенные для двух высот источниковz _i ≅0.025z _i иz _i ≅0.25z _i , имеют противоположную форму, но они имеют сравнимые максимальные значенияK _z ≅0.25w _* z _i .

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To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

Electromagnetic decays of radially excited mesons π0′, ρ0′, ω0′, and production of π0′ at lepton colliders

Radiative decays π ⁰(π ^0′) → γ + γ, π ^0′ → ρ ⁰(ω) + γ, ρ ^0′(ω′) → π ⁰ + γ, ρ ^0′(ω′) → π ^0′ + γ, and some processes of π ^0′ production at lepton colliders are considered in the framework of the nonlocal SU(2) × SU(2) Nambu-Jona-Lasinio model. Mixing of the radially excited and the ground meson states is taken into account. Numerical results for the decay and production processes are presented.     

On Ruelle’s Construction of the Thermodynamic Limit for the Classical Microcanonical Entropy

In 1969 Ruelle published his construction of the thermodynamic limit, in the sense of Fisher, for the quasi-microcanonical entropy density of classical Hamiltonian N-body systems with stable and tempered pair interactions. Here, “quasi-microcanonical” refers to the fact that he discussed the entropy defined with a regularized microcanonical measure as ln (N!⁻¹ ∫ χ _{{ℰ−▵ℰ<H<ℰ}}d^6N X) rather than defined with the proper microcanonical measure as ln (N!⁻¹ ∫ δ(ℰ−H) d^6N X). Replacing δ(ℰ−H) by χ _{{ℰ−▵ℰ<H<ℰ}} seems to have become the standard procedure for rigorous treatments of the microcanonical ensemble hence. In this note we make a very elementary technical observation to the effect that Ruelle’s proof (still based on regularization) does establish the thermodynamic limit also for the entropy density defined with the proper microcanonical measure. We also show that with only minor changes in the proof the regularization of δ(ℰ−H) is actually not needed at all.     

Di-pion emission in heavy quarkonia decays

The dipion spectrum for the ϒ(nS) → ϒ(n′S) transition with n < 4 has the form dw/dq ∼ (phase space) |η − x|², with x = q ² − 4m _π² / (ΔM)² − 4m _π² < q ² ≡ M _ππ², and ΔM = M(nS) − M(n′S). The parameter η is calculated and the spectrum is shown to reproduce the experimental data for all three types of decays: 3 → 1, 2 → 1, and 3 → 2 with η ≈ 0.5, 0, and −3, respectively. The text was submitted by the author in English.     

Two photons into π<Superscript>0</Superscript>π<Superscript>0</Superscript>

We perform a theoretical study based on dispersion relations of the reaction γγ → π ⁰ π ⁰ emphasizing the low-energy region. We discuss how the ƒ ⁰(980) signal emerges in γγ → ππ within the dispersive approach and how this fixes to a large extent the phase of the isoscalar S-wave γγ → ππ amplitude above the threshold. This allows us to make sharper predictions for the cross-section at lower energies and our results could then be used to distinguish between different ππ isoscalar S-wave parameterizations with the advent of new precise data on ππ → π ⁰ π ⁰. We compare our dispersive approach with an updated calculation employing the unitary chiral perturbation theory (U _gC PT). We also pay special attention to the role played by the σ-resonance in γγ → ππ and calculate its coupling and width to γγ, for which we obtain Γ(σ → γγ) = (1.68 ± 0.15) keV.     

Husimi Operator for Describing Probability Distribution of Electron States in Uniform Magnetic Field Studied by Virtue of Entangled State Representation

For the first time we introduce an operator Δ _h (γ,ε;κ) for studying Husimi distribution function in phase space (γ,ε) for electron’s states in uniform magnetic field, where κ is the Gaussian spatial width parameter. The marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions. Using the Wigner operator in the entangled state 〈λ | representation we find that Δ _h (γ,ε;κ) is just a pure squeezed coherent state density operator | γ,ε〉 _{κ κ} 〈γ,ε |, which brings much convenience for studying Husimi distribution, so we name Δ _h (γ,ε;κ) the Husimi operator. We then derive Husimi operator’s normally ordered form that provides us with an operator version to examine various properties of the Husimi distribution. Work supported by the National Natural Science Foundation under the grant: 10775097.     

14N and35Cl nuclear quadrupole coupling constants in thiazyl chloride from microwave spectroscopy andab initio molecular orbital computations

Summary The¹⁴N and³⁵Cl nuclear quadrupole coupling constants of the¹⁴N³²S³⁵Cl isotopic species of thiazyl chloride have been determined from an analysis of the hyperfine structure of sixμ _a andμ _b transitions in the microwave spectrum as (in MHz) _χaa (³⁵CL) = −39.00(3), _χbb (³⁵CL) = 23.84(4), _χaa (¹⁴N) = −1.42(6), _χbb (¹⁴N) = −4.16(6).

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Riassunto Le costanti di accoppiamento quadrupolare di¹⁴N e³⁵Cl della specie isotopica¹⁴N³²S³⁵Cl del tiazil cloruro sono state determinate attraverso l'analisi della struttura iperfine di sei transizioni di tipoμ _a eμ _b dello spettro a microonde. Si sono ottenuti i seguenti valori (in MHz): _χaa (³⁵CL) = −39.00(3), _χbb (³⁵CL) = 23.84(4), _χaa (¹⁴N) = −1.42(6), _χbb (¹⁴N) = −4.16(6).

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Резюме Из анализа сверхтонкой структуры шестиμ _a иμ _b переходов в микроволновом спектре определяются ядерные квадрупольные константы связи¹⁴N и³⁵Cl в¹⁴N³²S³⁵Cl. Получены следующие значения (в МГц): _χaa (³⁵CL) = −39.00(3), _χbb (³⁵CL) = 23.84(4), _χaa (¹⁴N) = −1.42(6), _χbb (¹⁴N) = −4.16(6).kg]PACS. 33.20

     

New data for the 197Au(γ, nX) and 197Au(γ, 2nX) reaction cross sections

The results from experimental and theoretical studies of the total and partial cross sections of photoneutron reactions on the ¹⁹⁷Au isotope were analyzed. The cross sections for reactions σ(γ, nX) = σ(γ, n) + σ(γ, np) + … + σ(γ, 2nX) = σ(γ, 2n) + σ(γ, 2np) + … were evaluated in the energy range 7 ≤ E _γ ≤ 30 MeV using an approach free of the shortcomings of experimental photoneutron multiplicity sorting methods. The total photoneutron reaction cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) + … = σ^exp(γ, n) + σ^exp(γ, np) + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + … were used as the initial experimental data. The contributions from the cross sections σ(γ, nX) and σ(γ, 2nX) to the cross sections σ^exp(γ, xn) were separated using the multiplicity transition functions F ₁ ^theor = σ^theor(γ, 1nX)/σ^theor(γ, xn) and F ₂ ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn), calculated within an updated version of the pre-equilibrium model of photonuclear reactions. New evaluated data for both partial reaction cross sections, i.e., σ^eval (γ, 1nX) = F ₁ ^theorσ^exp(γ, xn) and σ^eval(γ, 2nX) = F ₂ ^theorσ^exp(γ, xn), were obtained. The cross sections σeval(γ, nX) and σ^eval.(γ, 2nX) evaluated using the theoretically calculated functions F _1,2^theor are consistent with the Livermore data, but substantially contradict the Saclay data.     

Search for theχ′ c charmonium states as solution to the CDFψ′ puzzle

The efforts of Roy-Sridhar-Close-Cho-Wise-Trivedi to resolve the CDFψ′ anomaly with cascades from above-thresholdχ′ _c states require well defined signatures [a small total width and a large branching fraction forχ′ _cJ →γ+ψ′] for the solution to be viable. Here we estimate the production of such states from BR(B→χ′ _cJ +X)BR(χ′ _cJ →γψ′) andγγ production ofχ′ _c2 at CLEO II, and comment on the feasibility of testing the hypothesis in terms of current experimental capabilities.     

Double beta decays and Majorana neutrinos

Neutrino-less double beta decays (0νββ) are sensitive and realistic probes for studying the Majorana nature of neutrinos (ν), the ν-mass spectrum, the absolute ν-mass scale, the Majorana CP phases and other fundamental properties of neutrinos and weak interactions. Current 0νββ experiments, which use detectors with the mass sensitivity of the order of 300 meV, study the ν-mass in that mass region. Future experiments with higher sensitivities of the orders of 100meV and 30 meV, using different nuclei and methods (calorimetric, spectroscopic), are indispensable for establishing 0νββ in the quasi degenerate and the inverted hierarchy mass regions. R&D for ultra-high sensitivity detectors are encouraged for studying the normal hierarchy mass region. Theoretical and experimental studies for evaluating nuclear matrix elements are important for extracting the sensible ν-mass from the 0νββ rate. Charge exchange reactions by means of nuclear, electromagnetic and ν probes provide valuable data which are used to evaluate the nuclear matrix elenments. International collaborations for 0νββ experiments and for the matrix elements are crucial for next generation 0νββ studies.