Vol 190, No 2 (2017)

We consider the processes in compact stars that arise during the potential formation of a pseudoscalar condensate in finite volumes. We do not propose specific hypotheses about the nature of this condensate. Considering that in the regions with a changing pseudoscalar density, photon propagation can be described in the framework of Maxwell–Chern–Simons electrodynamics, we find the reflection/transmission coefficients for regions with different densities. We study the fermion spectrum in the presence of an axial field considering the pseudoscalar condensate gradient. We also study the influence of the modified photon and fermion spectra on the cooling process of compact stars.

We consider the critical behavior of the O(n)-symmetric model of the ϕ⁴ type with an antisymmetric tensor order parameter. According to a previous study of the one-loop approximation in the quantum field theory renormalization group, there is an IR-attractive fixed point in the model, and IR scaling with universal indices hence applies. Using a more specific analysis based on three-loop calculations of the renormalization-group functions and Borel conformal summation, we show that the IR behavior is in fact governed by another fixed point of the renormalization-group equations and the model therefore belongs to a different universality class than the one suggested by the simplest one-loop approximation. Nevertheless, the validity of the obtained results remains a subject for discussion.

We consider the families of polynomials P = { P_n(x)}_n=0^∞ and Q = { Q_n(x)}_n=0^∞ orthogonal on the real line with respect to the respective probability measures μ and ν. We assume that { Q_n(x)}_n=0^∞ and {P_n(x)}_n=0^∞ are connected by linear relations. In the case k = 2, we describe all pairs (P,Q) for which the algebras A_P and A_Q of generalized oscillators generated by { Qn(x)}_n=0^∞ and { Pn(x)}_n=0^∞ coincide. We construct generalized oscillators corresponding to pairs (P,Q) for arbitrary k ≥ 1.

In the framework of the model with fusion of quark–gluon strings on the transverse lattice, we find the asymptotic behavior of the correlation coefficients between observables in separated rapidity intervals with a high string density in a realistic case with an inhomogeneous distribution of strings in the impact parameter plane. We calculate the asymptotic forms for three types of correlations: between the average transverse momenta of particles with rapidity in these intervals, between the average transverse momentum of particles in one rapidity interval and the multiplicity of particles in another, and also between the multiplicities of charged particles in these intervals. We show that the previously found independence of the asymptotic form of the correlation coefficient between the average transverse momenta from the variance in the number of particles produced in string fragmentation holds only in the case of a uniform distribution of strings in the transverse plane. We also show that the found general expressions for the long-range correlation coefficients in the particular case with a uniform distribution of strings in the transverse plane become the formulas previously obtained by another method applicable only in this simple case.

We briefly review recent results of exact calculations of critical Casimir forces of the O(n) ϕ⁴ model as n→∞ on a three-dimensional strip bounded by two planar free surfaces at a distance L. This model has long-range order below the critical temperature Tc of the bulk phase transition only in the limit L→∞ but remains disordered for all T > 0 for an arbitrary finite strip width L < ∞. A proper description of the system scaling behavior near T_c turns out to be a quite challenging problem because in addition to bulk, boundary, and finite-size critical behaviors, a nontrivial dimensional crossover must be handled. The model admits an exact solution in the limit n → ∞ in terms of the eigenvalues and eigenenergies of a self-consistent Schrödinger equation. This solution contains a potential v(z) with the near-boundary singular behavior v(z → 0+) ≈ −1/(4z²)+ 4m/(π²z), where m = 1/ξ+(|t|) is the inverse bulk correlation length and t ~ (T − T_c)/T_c, and a corresponding singularity at the second boundary plane. In recent joint work with colleagues, the potential v(z), the excess free energy, and the Casimir force were obtained numerically with high precision. We explain how these numerical results can be complemented by exact analytic ones for several quantities (series expansion coefficients of v(z), the scattering data of v(z) in the semi-infinite case L = ∞ for all m >/< 0, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force) by a combination of boundary-operator and short-distance expansions, proper extensions of the inverse scattering theory, new trace formulas, and semiclassical expansions.

We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B₃, C₃, and D₃. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D₃.