QCD atθ∼πreexamined: Domain walls and spontaneousCPviolation

Abstract

We consider QCD at $\ensuremath{\theta}\ensuremath{\sim}\ensuremath{\pi}$ 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 $\ensuremath{\theta}=\ensuremath{\pi}$ for finite quark mass splittings, ${z=m}_{d}{/m}_{u}\ensuremath{\ne}1.$ In the $z\ensuremath{-}\ensuremath{\theta}$ plane, there is a line of first order transitions at $\ensuremath{\theta}=\ensuremath{\pi}$ with two critical end points, ${z}_{1}^{*}<z<{z}_{2}^{*}.$ 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 ${z}_{1}^{*}<z<{z}_{2}^{*},$ $z\ensuremath{\ne}1,$ there is only one domain wall and a wall-some sphaleron at $\ensuremath{\theta}=\ensuremath{\pi}.$ 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 $(\ensuremath{\sim}{N}_{f}=0)$ provides a simple derivation of the multibranch $\ensuremath{\theta}$ 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 $\ensuremath{\Gamma}\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}{N}_{c}^{4}).$