Poincaré recurrence theorem and the strong $CP$ problem

The existence in the physical QCD vacuum of nonzero gluon condensates, such as $⟨g^2{F}^2⟩$, requires dominance of gluon fields with finite mean action density. This naturally allows any real number value for the unit “topological charge” $q$ characterizing the fields approximating the gluon configurations which should dominate the QCD partition function. If $q$ is an irrational number then the critical values of the $\theta$ parameter for which $CP$ is spontaneously broken are dense in $\mathbb{R}$, which provides for a mechanism of resolving the strong $CP$ problem simultaneously with a correct implementation of $U_{\mathrm{A}}(1)$ symmetry. We present an explicit realization of this mechanism within a QCD motivated domain model. Some model independent arguments are given that suggest the relevance of this mechanism also to genuine QCD.

Figure 1

The scalar (A and B) and pseudoscalar (C and D) quark condensates as functions of $\theta$ for $N_f=3$ in units of $\aleph$. The plots A and C are for rational $q=0.15$, while B and D correspond to any irrational $q$, for instance to $q=\frac{3}{2.02{\pi }^2}=0.15047\dots$ which is numerically only slightly different from $0.15$. The dashed lines in A and C correspond to discrete minima of the free energy density which are degenerate for $m\equiv 0$. The solid bold lines denote the minimum which is chosen by an infinitesimally small mass term for a given $\theta$. Points on the solid line in A, where two dashed lines cross each other, correspond to critical values of $\theta$, at which $CP$ is broken spontaneously. This is signalled by the discontinuity in the pseudoscalar condensate in C. For irrational $q$, as illustrated in B and D, the dashed lines densely cover the strip between $1$ and $-1$, and the set of critical values of $\theta$ turns out to be dense in $\mathbb{R}$. As will be discussed in the last section, an analogy can be seen between A (C) and motion on a torus over a closed trajectory, while a manifestation of the Poincaré recurrence theorem resulting in dense winding of a torus (for example, see 16) can be recognized in B (D).

Figure 2

Values of ${\theta }_l=g^l\theta$ as defined in Eq. (26): boxes occur for rational $q=q_1/q_2$ with $q_1{N}_f=3$, while circles represent some of the points densely distributed, in particular, in the neighborhood of $\theta =0$ for irrational $q$.

Figure 3

Coordinates on a torus, as discussed in the text.