Spontaneous $CP$ breaking in QCD and the axion potential: an effective Lagrangian approach

TL;DR

The work investigates spontaneous CP violation in QCD at $\theta=\\pi$ using a chiral, large-$N$ effective Lagrangian, revealing a CP-conserving region separated from a CP-breaking region by a hypersurface where a PNGB becomes massless and the topological susceptibility diverges. It demonstrates that, in the presence of an axion, the axion potential can be substantially altered near the CP-critical surface due to mixing with light PNGBs, sometimes requiring a two-field description rather than the conventional single-axion picture. The analysis provides detailed phase diagrams for $N_f=1,2,3$, showing first-order CP-breaking lines and a second-order endpoint whose location depends on quark masses and the YM topological susceptibility parameter $a$. These results have potential implications for axion cosmology and dark matter, especially at finite temperature where changes in $\chi_{YM}$ could modify the axion mass and dynamics; the authors also emphasize the need for lattice studies to test these predictions and discuss Ward-Takahashi identities validating the effective theory.

Abstract

Using the well-known low-energy effective Lagrangian of QCD --valid for small (non-vanishing) quark masses and a large number of colors-- we study in detail the regions of parameter space where $CP$ is spontaneously broken/unbroken for a vacuum angle $θ= π$. In the $CP$-broken region there are first order phase transitions as one crosses $θ=π$, while on the (hyper)surface separating the two regions, there are second order phase transitions signaled by the vanishing of the mass of a pseudo Nambu-Goldstone boson and by a divergent QCD topological susceptibility. The second order point sits at the end of a first order line associated with the $CP$ spontaneous breaking, in the appropriate complex parameter plane. When the effective Lagrangian is extended by the inclusion of an axion these features of QCD imply that standard calculations of the axion potential have to be revised when the QCD parameters fall in the above mentioned $CP$-broken region, in spite of the fact that the axion solves the strong-$CP$ problem. These latter results could be of interest for axionic dark matter calculations if the topological susceptibility of pure Yang-Mills theory falls off sufficiently fast when temperature is increased towards the QCD deconfining transition.

Figures (7)

• Figure 4: Solutions of the stationarity conditions for $N_f=2$, $\mu_d^2 = 2 \mu_u^2$ and $\theta = \pi$ are given by the intersections of the curves shown in different color. The two situations with one or three solutions are shown together with the limiting case corresponding to a second order phase transition.
• Figure 5: Same as Fig \ref{['fig:fig4']} in the $CP$ broken situation, but for two values of $\theta$ on opposite sides of $\theta = \pi$: (a): $\theta < \pi$, (b): $\theta > \pi$. The true minimum (corresponding to the intersection which is farther away from the middle one) swaps abruptly as one goes through $\theta = \pi$.
• Figure 6: $CP$ conserving (filled) and $CP$ breaking (empty) regions for $N_f = 2$ and $N_f= 3$. The vertical axis is $D$, the horizontal is (are) the mass ratio(s).
• Figure 7: $N_f=1$. (a) Evolution of the two eigenvalues of (\ref{['ANf=1']}) for $b = 0.1$ as one varies $\mu^2/a$. The lower eigenvalue is tachyonic. (b) Projections of the two corresponding eigenvectors along the PNGB direction. Maximal mixing occurs in the vicinity of the critical point $\mu^2/a =1$.
• Figure 8: $N_f=2$. (a) Evolution of the three eigenvalues as one varies $\mu^2/a$ for $b = 0.1$ and $\mu_2^2 = 2 \mu_1^2$. One of the three eigenvalue always lies much higher than the other two and is not much affected by the axion. (b) Blow up of the lower part of the figure showing the repulsion (and mixing) of the two lower eigenvalues.