Chiral Modulations in Curved Space I: Formalism

The goal of this paper is to present a formalism that allows to handle four-fermion effective theories at finite temperature and density in curved space. The formalism is based on the use of the effective action and zeta function regularization, supports the inclusion of inhomogeneous and anisotropic phases. One of the key points of the method is the use of a non-perturbative ansatz for the heat-kernel that returns the effective action in partially resummed form, providing a way to go beyond the approximations based on the Ginzburg-Landau expansion for the partition function. The effective action for the case of ultra-static Riemannian spacetimes with compact spatial section is discussed in general and a series representation, valid when the chemical potential satisfies a certain constraint, is derived. To see the formalism at work, we consider the case of static Einstein spaces at zero chemical potential. Although in this case we expect inhomogeneous phases to occur only as meta-stable states, the problem is complex enough and allows to illustrate how to implement numerical studies of inhomogeneous phases in curved space. Finally, we extend the formalism to include arbitrary chemical potentials and obtain the analytical continuation of the effective action in curved space.Comment: 22 pages, 3 figures; version to appear in JHE

ABOUT THE ORIGIN OF HIGH-ENERGY ELECTRONS IN THE INNER RADIATION BELT

A large flux of > 100 MeV electrons were registered in the inner radiation belt on low-altitude satellites. The origin of that flux is discussed. It appears that slow radial diffusion (D(o) = 10(-13) 1/s) gives a low probability for penetration of these electrons to small L from the boundary of magnetosphere because of synchrotron radiation energy losses. It is found that they can enter to the inner belt region without such losses after great magnetic storms when high speed radial diffusion sometimes takes place. Two great storms on 8-9 Feb.1986 and 24 March 1991 are examples when one can directly observe a penetration of energetic electron fluxes into magnetosphere. The assumption about their Jovian origin is discussed.57220120