Relativistic non-linear Walecka model

Question

What is meant by a relativistic non-linear Walecka model?

What are some various sources to study it?

[And why cannot Google show a satisfactory result to such a simple question?]

Answer 1

The non-linear Walecka model is a reference to a system composed of nucleons, which interact strongly with each other via the exchange of the $\sigma$, $\omega$ and $\rho$ mesons in the process called one-boson exchange (OBE). This is described by the following Lagrangian consisting of the Dirac particles, mesons, which are appropriately coupled to nucleons as well as containing terms concerning the self-interaction of $\sigma$ and $\omega$ mesons: $$ \begin{aligned} \mathcal{L}=&\bar\psi\left[i\gamma^\mu\partial_\mu-m-g_\sigma\sigma-g_\omega\gamma^\mu\omega_\mu-g_\rho\gamma^\mu\tau^a\rho^a_\mu\right]\psi \\ &+\frac{1}{2}(\partial_\mu\sigma\partial^\mu\sigma-m_\sigma^2\sigma^2)-\frac{1}{3}g_3\sigma^3-\frac{1}{4}g_4\sigma^4 \\ &-\frac{1}{4}\omega_{\mu\nu}\omega^{\mu\nu}+\frac{1}{2}m_\omega^2\omega_\mu\omega^\mu+\frac{1}{4}c_4(\omega_\mu\omega^\mu)^2 \\ &-\frac{1}{4}\rho^a_{\mu\nu}\rho^{a\mu\nu}+\frac{1}{2}m_\rho^2\rho_\mu^a\rho^{a\mu} \ , \end{aligned} $$

where $\psi$ is the Dirac field with the mass $m$; $\sigma$ is the isoscalar, Lorentz scalar field with the mass $m_\sigma$; $\omega$ is the isoscalar, Lorentz vector field with the mass $m_\omega$; $\boldsymbol{\rho}_\mu=\{\rho^0_\mu,\rho^+_\mu,\rho^-_\mu\}$ is the isovector, Lorentz vector field with the mass $m_\rho$; $\omega_{\mu\nu}=\partial_\mu\omega_\nu-\partial_\nu\omega_\mu$; $\rho^a_{\mu\nu}=\partial_\mu\rho^a_\nu-\partial_\nu\rho_\mu^a+g_\rho\epsilon^{abc}\rho^b_\mu\rho^c_\nu$; $\tau^a$ are the standard Pauli matrices; $g_\sigma$, $g_\omega$, $g_\rho$ are the coupling constants for the respective mesons, whereas $g_3$, $g_4$ are the coupling constants of the self-interacting $\sigma$ mesons and $c_4$ is the one for the $\omega$ mesons.

The papers regarding the non-linear Walecka model I like are the following:

[1] Y. K. Gambhir, P. Ring, and A. Thimet. Relativistic mean field theory for finite nuclei. Annals of Physics, 198(1):132–179, February 1990.

[2] Y. Sugahara and H. Toki. Relativistic mean-field theory for unstable nuclei with non-linear σ and ω terms. Nuclear Physics, Section A, 579(3):557–572, October 1994.

[3] H. Toki, D. Hirata, Y. Sugahara, K. Sumiyoshi, and I. Tanihata. Relativistic many body approach for unstable nuclei and supernova. Nuclear Physics, Section A, 588:357–363, February 1995.

[4] Norman K. Glendenning. Compact stars. Nuclear physics, particle physics, and general relativity. Springer, 1997.

[5] G. A. Lalazissis, J. König, and P. Ring. New parametrization for the Lagrangian density of relativistic mean field theory. Physical Review C, 55(1):540–543, January 1997.

[6] Ilona Bednarek. Relativistic mean field models of neutron stars. Katowice: Wydawnictwo Uniwersytetu Śląskiego, 2007.