abstract

It is conventionally assumed that the negative mass squared term in the linear sigma model version of the pion Lagrangian is \(M^2 \sim {\mit \Lambda }_{\rm QCD}^2\) in powers of \(N_{\rm c}\). We consider the case where \(M^2 \sim {\mit \Lambda }^2_{\rm QCD}/N_{\rm c}\) so that to leading order in \(N_{\rm c}\), this symmetry breaking term vanishes. We present some arguments why this might be plausible. One might think that such a radical assumption would contradict lattice Monte Carlo data on QCD as a function of \(N_{\rm c}\). We show that the linear sigma model gives a fair description of the data of DeGrand and Liu both for \(N_{\rm c} = 3\) and for variable \(N_{\rm c}\). The values of quark masses considered by DeGrand and Liu, and by Bali et al. turn out to be too large to resolve the case we consider from that of the conventional large-\(N_{\rm c}\) limit. We argue that for quark masses \(m_{q} \ll {\mit \Lambda }_{\rm QCD}/N_{\rm c}^{3/2}\), both the baryon mass and nucleon size scale as \(\sqrt {N_{\rm c}}\). For \(m_{q} \gg {\mit \Lambda }_{\rm QCD}/N_{\rm c}^{3/2}\), the conventional large-\(N_{\rm c}\) counting holds. The physical values of quark masses for QCD (\(N_{\rm c} = 3\)) correspond to the small quark-mass limit. We find pion nucleon coupling strengths are reduced to the order \({\cal O}(1)\) rather than \({\cal O}(N_{\rm c})\). Under the assumption that in the large-\(N_{\rm c}\) limit, the sigma meson mass is larger than that of the omega, and that the omega–nucleon coupling constant is larger than that of the sigma, we argue that the nucleon–nucleon large-range potential is weakly attractive and admits an interaction energy of the order of \({\mit \Lambda }_{\rm QCD}/N_{\rm c}^{5/2} \sim 10\) MeV. With these assumptions on coupling and masses, there is no strong long-range attractive channel for nucleon–nucleon interactions, so that nuclear matter at densities much smaller than that where nucleons strongly interact is a weakly interacting configuration of nucleons with strongly interacting localized cores. This situation is unlike the case in the conventional large-\(N_{\rm c}\) limit, where nuclear matter is bound with binding energies of the order of the nucleon mass and forms a Skyrme crystal.

abstract

This study essentially consists of two separate investigations where we study neutron-rich isotopes of copper. Two different nuclear models were selected to perform the two investigations. In the first part of this paper, the nuclear ground-state properties of neutron-rich copper isotopes (\(72 \leq A \leq 82\)) have been studied with the help of the relativistic mean field (RMF) model. The second portion of this paper is dedicated to calculation of lepton capture rates in stellar environment. Ground and excited states of GT and U1F strength functions were calculated in a microscopic way employing the deformed proton–neutron quasiparticle random phase approximation (pn-QRPA) model. The lepton capture rates were computed on a wide temperature range of (0.01–30)\(\times 10^{9}\) K and stellar density range of (10–10\(^{11}\)) g/cm\(^{3}\). We compared our computed half-lives (GT + U1F) with previous theoretical and measured results. Our calculated terrestrial half-lives agree well with the measured ones. Our study shows that, at high stellar temperatures, allowed GT and, specially, U1F positron capture rates dominate the competing \(\beta \)-decay rates. For a better description of presupernova evolutionary phases of massive stars, simulators are recommended to take into account lepton capture rates on neutron-rich copper isotopes presented in this work.

abstract

The 1D Schrödinger equation for an electron with the potential energy defined as a pair of equal delta-wells is analyzed. If the distance between the wells exceeds a critical value, there exist two negative eigenenergies. We focused attention on the non-stationary motion when the electron alternates periodically its position in the left and right well. The delta-wells are considered as models of quantum dots (QDs) in a quantum wire embedded in a semiconductor structure. To apprehend the motion of an electron between the QDs at low temperatures (which is evidently a tunnelling phenomenon), we employ a density-matrix formalism. We derive the solution of the Liouville (von Neumann) equation in the approximation of a relaxation time. The solution suggests that the oscillatory motion of the electron between the QDs undergoes a damping. The main cause of this damping is the electron–phonon interaction. At very low temperatures, the damping, corroborating a decoherence process, can also be effected artificially when a voltage source with a noise component is employed.

abstract

The problem of averaging the kinetics of two-stage decaying system subject to dichotomous fluctuations in the forward rate is solved exactly. It is shown that the temporal behavior of system’s populations is four-exponential, given finite frequency and amplitude of fluctuations. For frequent fluctuations, this behavior is bimodal typical of deterministic decay, but oppositely, it reduces to three-exponential and bimodal forms, specific of low and resonance amplitude fluctuations. There is an immobilization of initial state at a stochastic resonance point, where forward rate coincides with fluctuation amplitude, whereas backward, decay and fluctuation rates are all negligible.

abstract

We study the effect of confining potentials, generated by different equilibrium (long-time asymptotic or terminal) probability densities, on non-Gaussian stochastic processes, described by Lévy–Schrödinger semigroup dynamics. The former densities belong to the family of so-called \(\mathbb {M}\)-Wright functions of index \(\nu \). Using analytical and numerical arguments, we demonstrate that properly tailored confining potentials can generate the Gaussian distribution (which is also a member of \(\mathbb {M}\)-Wright family at \(\nu =1/2)\) at final stages of time evolution. This means that the Gaussian distribution (and other sufficiently fast decaying distributions like exponential one) can emerge in the differential equation with fractional derivatives, which normally produces the heavy-tailed, slow-decaying probability densities. We discuss the physical implications of the results obtained, for instance, in the evolution of magnetic resonanse lineshapes for complex, multi-peaked resonant lines.