QCD atθ∼πreexamined: Domain walls and spontaneousCPviolation

We consider QCD at $\theta \sim \pi$ with two, one and zero light flavors $N_f,$ using the Di Vecchia–Veneziano–Witten effective Lagrangian. For $N_f=2,$ we show that $\mathrm{CP}$ is spontaneously broken at $\theta =\pi$ for finite quark mass splittings, ${z=m}_d{/m}_u\ne 1.$ In the $z-\theta$ plane, there is a line of first order transitions at $\theta =\pi$ with two critical end points, $z_1^{*}<z<\mathrm{}{z}_2^{*}.$ We compute the tension of the domain walls that relate the two $\mathrm{CP}$ violating vacua. For $m_u{=m}_d,$ the tension of the family of equivalent domain walls agrees with the expression derived by Smilga from chiral perturbation theory at next-to-leading order. For $z_1^{*}<z<\mathrm{}{z}_2^{*},$ $z\ne 1,$ there is only one domain wall and a wall-some sphaleron at $\theta =\pi .$ At the critical points, ${z=z}_{1,2}^{*},$ the domain wall fades away, $\mathrm{CP}$ is restored and the transition becomes of second order. For $N_f=1,$ $\mathrm{CP}$ is spontaneously broken only if the number of colors $N_c$ is large and/or if the quark is sufficiently heavy. Taking the heavy quark limit $(\sim N_f=0)$ provides a simple derivation of the multibranch $\theta$ dependence of the vacuum energy of large $N_c$ pure Yang-Mills theory. In the large $N_c$ limit, there are many quasistable vacua with a decay rate $\Gamma \sim \exp \left(-N_c^4\right).\mathrm{}$