Semiclassical expansions up to -order in relativistic nuclear physics

Abstract

We present the first calculation of theh 4 -Wigner-Kirkwood corrections to a relativistic system of fermions in the presence of external scalar and vector potentials. The method we propose allows to compute efficiently semiclassical corrections to one body operators such as mean energies and local densities. It also preserves gauge invariance and produces explicitly convergent results despite some apparent divergencies at the classical turning points. As a byproduct we obtain theh 4 corrections stemming from the polarization of the Dirac sea. We apply our results to the relativistic σ-ω Lagrangian in the Hartree valence approximation. We compare the semiclassical expansion with the exact result and with the Strutinsky average whenever it can be obtained. We find that theh 4 corrections are much smaller than typical shell effects. Our results provide convincing arguments to neglect higher than second orderh effects in the Wigner-Kirkwood scheme to relativistic nuclear physics in the mean field approximation.

Elimination of the vacuum instability for finite nuclei in the relativisticσ-ωmodel

“…The counterterm kernel ∆K s (p 2 ) must be such that cancels the ghost term in the propagator D s (p 2 ) in eq. (18). The subtraction does not modify the position of the physical meson pole nor its residue, but it will change the zero-momentum parameters and also the off-shell behavior.…”

“…We warn, however, that the convergence of the gradient or semiclassical expansion is not the same as converging to the exact mean field result, since there could be shell effects not accounted for by this expansion at any finite order. Such effects, certainly exist in the valence part [18]. Even in a seemingly safe case as infinite nuclear matter, where only the zeroth order has a non vanishing contribution, something is left out by the gradient expansion since the exact mean field solution does not exist due to the Landau ghost instability (of course, the situation may change if the Landau pole is removed).…”

Spectral-shift and scattering-equivalent Hamiltonians on a coarse momentum grid

“…(31) can be regularized as proposed in Ref. [38,39], namely introducing a small imaginary energy iε → i0 + , as follows 1 π…”

Quark-diquark string tension, excited baryonic resonances and thermal fluctuations

“…The comparison clearly shows that there are further states in the RQM spectrum above some scale M > M min that may or may not be confirmed in the future as mesons or hadrons, although they could also be exotic, glueballs or hybrids. A semiclassical expansion of the cumulative number of states [22] can be used to study the high mass spectrum for systems where interactions are dominated by linearly rising potentials with a string tension σ, in the range M √ σ. At leading order in the expansion, the cumulative number takes the form…”