$CP$-violating theta parameter in the domain model of the QCD vacuum

A nonzero $CP$-violating $\theta$ parameter is treated in the domain model which assumes a clusterlike vacuum structure whose units are characterized, in particular, by a topological charge which is not necessarily an integer number. In the present paper we restrict consideration to rational values of the charge. The model has previously been shown to manifest confinement, spontaneous chiral symmetry breaking and the absence of an axial $U(1)$ Goldstone boson. We find that the specific structure of the minima of the free energy density of the domain ensemble forces a $2\pi$ periodicity of observables in $\theta$ for any number of light quarks, that vacuum doubling occurs at $\theta =\pi$ for any $N_f>1$ and any value of topological charge $q$. These features are in agreement with expectations based on anomalous Ward identities and large $N_c$ effective theories. We find also additional values of $\theta$ depending on $q$ for which vacuum doubling occurs.

Figure 1

The scalar quark condensate as a function of the parameter $\theta$ for $N_f=1$ and $q=0.15$ in units of $\aleph$. The dashed lines correspond to discrete minima of the free energy density which are degenerate for $m\equiv 0$. The solid line denotes the minimum which becomes global for a given $\theta$ in the presence of an infinitesimally small mass term.

Figure 2

The scalar quark condensate as a function of $\theta$ for $N_f=3$ and $q=0.15$ in units of $\aleph$. The meaning of dashed and solid lines is the same as in Fig. 1. Qualitatively the picture is the same as for $N_f=1$: the same periodicity and critical values of $\theta$, which is achieved by an increased number of different discrete minima.

Figure 3

The pseudoscalar quark condensate as a function of $\theta$ for $N_f=1$ and $q=0.15$ in units of $\aleph$. The meaning of dashed and solid lines is the same as in Fig. 1.

Figure 4

The pseudoscalar quark condensate as a function of $\theta$ for $N_f=3$ and $q=0.15$ in units of $\aleph$. The meaning of dashed and solid lines is the same as in Fig. 2.

Figure 5

The scalar quark condensate as a function of $\theta$ for $N_f=3$ and $q=1/N_f$ in units of $\aleph$, which is equivalent to a corresponding plot from 5. The meaning of dashed and solid lines is the same as in Fig. 1.

Figure 6

The pseudoscalar quark condensate as a function of $\theta$ for $N_f=3$ and $q=1/N_f$ in units of $\aleph$. The meaning of dashed and solid lines is the same as in Fig. 1.