As a vector field, an expectation value for ${\omega }_0$ is generated whenever there is net baryon charge in the system and is also why the energy density increases. This is exactly like QED with net charge, except that here the ${\omega }_{\mu }$ meson is massive. In contrast, for the chiral condensate, the energy density is lowered when it has a nonzero expectation value. I thank L. McLerran for discussions on this point.

This is true with either the Stucklelberg [125] or $BF$ [126, 127, 128, 129, 130] formalisms. With the $BF$ formalism, auxiliary two-index gauge potentials $B_{\alpha \beta }^i$ could be introduced and defined to transform under one-form gauge transformations ${\lambda }_{\alpha }^i$ as $B_{\alpha \beta }^i\to B_{\alpha \beta }^i-{\partial }_{\alpha }{\lambda }_{\beta }^i+{\partial }_{\beta }{\lambda }_{\alpha }^i$. Adding to the action a three-index field strength tensor for $B_{\alpha \beta }^i$, integration over $B_{\alpha \beta }^i$ generates the coupling $\sim {\kappa }^2$ if the $BF$ term is chosen as $\sim \kappa {\epsilon }^{\alpha \beta \mu \nu }{\varphi }^i{B}_{\alpha \beta }^i{\mathcal{F}}_{\mu \nu }$. However, such a $BF$ term is gauge invariant only for constant ${\varphi }^i$, and not for a dynamical field, where ${\partial }_{\alpha }{\varphi }^i\ne 0$. This is unremarkable, given that mass terms for non-Abelian fields, such as $\overrightarrow{\rho }$ and ${\overrightarrow{a}}_1$, also cannot be introduced in a form which respects gauge invariance and unitarity [125, 130].

For an arbitrary number of flavors, $N_{\mathrm{fl}}$, under a chiral rotation the chiral field $\mathrm{\Phi }$ transforms as $\mathrm{\Phi }\to {\mathrm{e}}^{i{\theta }_A}{\mathrm{e}}^{i{\alpha }_L}\mathrm{\Phi }{\mathrm{e}}^{-i{\alpha }_R}$, where ${\theta }_A$, ${\alpha }_L$, and ${\alpha }_R$ are elements of the Lie algebra for $U_A(1)\times SU(N_{\mathrm{fl}}{)}_L\times SU(N_{\mathrm{fl}}{)}_R$, respectively. For arbitrary $N_{\mathrm{fl}}$, the coupling $\sim {\kappa }^2$ in Eq. (2) generalizes trivially to ${\omega }_{\mu }^2\mathrm{tr}{\mathrm{\Phi }}^{†}\mathrm{\Phi }$.

This is implicit in the analysis of Ref. [72]: I thank V. Dexheimer for discussions on the difficulty of eliminating the contribution of the ${\omega }_0$ energy in mean field theory.

There are also two other terms with dimensions of inverse mass, Eq. (2) of Ref. [102], but as these only involve two spatial derivatives, they do not have a dramatic effect.

For arbitary $N_{\mathrm{fl}}$, $\mathrm{\Phi }$ is a complex, $N_{\mathrm{fl}}\times N_{\mathrm{fl}}$ matrix. While in general, matrix models are not easily solved at large $N_{\mathrm{fl}}$, for the ansatz where $SU(N_{\mathrm{fl}}{)}_L\times SU(N_{\mathrm{fl}}{)}_R$ breaks to $SU(N_{\mathrm{fl}}{)}_V$, which is $\mathrm{\Phi }={\sigma }_0\mathbf{1}$, the solution follows directly from the vector model, with $N\to 2N_{\mathrm{fl}}^2$.

In Ref. [102], we used the term pionic quantum spin liquid ($\pi \mathrm{QSL}$). In this paper, I adopt the more accurate term quantum pion liquid, as the phenomenon has nothing to do with spin. I thank L. Classen and F. Rennecke for pointing this out and suggesting the term.

My language is common but imprecise: since the fermions carry $Z(3)$ charge, any flux string can always break by the production of fermion-antifermion pairs. By confinement, I mean that there is a mass gap for all particles in the spectrum, which by necessity are gauge singlets.