Nuclear matter as a liquid phase of spontaneously broken semiclassical SU(2)L×SU(2)R chiral perturbation theory: Static chiral nucleon liquids

Abstract

The standard model of particle physics (SM), augmented with neutrino mixing, is either the complete theory of interactions of known particles at energies naturally accessible on earth, or very nearly so, with a Lagrangian symmetric under the global $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{R}$ symmetry of two-massless-quark QCD, spontaneously broken to $\mathrm{SU}{(2)}_{L+R}$. Using naive dimensional operator power counting that enables perturbation and truncation in inverse powers of ${\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}\ensuremath{\approx}1\phantom{\rule{4.pt}{0ex}}\text{GeV}$, we show that, to $\mathcal{O}({\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB})$ and $\mathcal{O}({\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}^{0})$, SU(2) chiral perturbation theory [SU(2)$\ensuremath{\chi}\mathrm{PT}$] of protons, neutrons, and pions admits a liquid phase, with energy required to increase or decrease the nucleon density. We further show that in the semiclassical approximation---i.e., quantum nucleons and classical pions---``pionless SU(2)$\ensuremath{\chi}$PT'' emerges in that chiral liquid: soft static infrared Nambu-Goldstone-boson pions decouple from ``static chiral nucleon liquids'' (Static$\ensuremath{\chi}$NLs). This vastly simplifies the derivation of saturated nuclear matter (the infinite liquid phase) and of finite microscopic liquid drops (ground-state heavy nuclides). Static$\ensuremath{\chi}$NLs are made entirely of nucleons. They have even parity, total spin zero, even proton number $Z$, and even neutron number $N$. The nucleons are arranged so local expectation values for spin and momentum vanish. We derive the Static$\ensuremath{\chi}$NL effective Lagrangian from semiclassical SU(2)$\ensuremath{\chi}$PT$\phantom{\rule{0.28em}{0ex}}$symmetries to order ${\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}$ and ${\mathrm{\ensuremath{\Lambda}}}_{\ensuremath{\chi}SB}^{0}$, including all relativistic four-nucleon operators that survive Fierz rearrangement in the nonrelativistic limit and SU(2)$\ensuremath{\chi}$PT$\phantom{\rule{0.28em}{0ex}}$fermion exchange operators and isovector exchange operators which are important when $Z\ensuremath{\ne}N$. Mean-field Static$\ensuremath{\chi}$NL nontopological solitons are true solutions of SU(2)$\ensuremath{\chi}$PT$\phantom{\rule{0.28em}{0ex}}$semiclassical symmetries; e.g., they obey all conserved vector current (CVC) and partially conserved axial current (PCAC) conservation laws. They have zero internal and external pressure. The nuclear liquid-drop model and Bethe--von Weizs\"acker semiempirical mass formula emerge---with correct nuclear density and saturation and asymmetry energies---in an explicit Thomas-Fermi construction.