Functional renormalization group study of the critical region of the quark-meson model with vector interactions

The critical region of the two flavour quark-meson model with vector interactions is explored using the Functional Renormalization Group, a non-perturbative method that takes into account quantum and thermal fluctuations. Special attention is given to the low temperature and high density region of the phase diagram, which is very important to construct the equation of state of compact stars. As in previous studies, without repulsive vector interaction, an unphysical region of negative entropy density is found near the first order chiral phase transition. We explore the connection between this unphysical region and the chiral critical region, especially the first order line and spinodal lines, using also different values for vector interactions. We find that the unphysical negative entropy density region appears because the $$s=0$$s=0 isentropic line, near the critical region, is displaced from its $$T=0$$T=0 location. For certain values of vector interactions this region is pushed to lower temperatures and high chemical potentials in such way that the negative entropy density region on the phase diagram can even disappear. In the case of finite vector interactions, the location of the critical end point has a non-trivial behaviour in the $$T-\mu _B$$T-μB plane, which differs from that in mean field calculations.

Singularities in water waves and Rayleigh–Taylor instability

This paper is concerned with singularities in inviscid two-dimensional finite-amplitude water waves and inviscid Rayleigh–Taylor instability. For the deep water gravity waves of permanent form, through a combination of analytical and numerical methods, we present results describing the precise form, number and location of singularities in the unphysical domain as the wave height is increased. We then show how the information on the singularity can be used to calculate water waves numerically in a relatively efficient fashion. We also show that for two-dimensional water waves in a finite depth channel, the nearest singularity in the unphysical region has the form as for deep water waves. However, associated with such a singularity, there is a series of image singularities at increasing distances from the physical plane with possibly different behaviour. Further, for the Rayleigh–Taylor problem of motion of fluid over vacuum, and for the unsteady water wave problem, we derive integro-differential equations valid in the unphysical region and show how these equations can give information on the nature of singularities for arbitrary initial conditions. We give indications to suggest that a one-half point singularity on its approach to the physical domain corresponds to a spike observed in Rayleigh-Taylor experiment.