Deconfinement and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="script">C</mml:mi><mml:mi mathvariant="script">P</mml:mi></mml:mrow></mml:math> breaking at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:math> in Yang-Mills theories and a novel phase for SU(2)
Shi Chen, Kenji Fukushima, Híromichi Nishimura, Yuya Tanizaki
Abstract
We discuss the deconfinement and the $\mathcal{C}\mathcal{P}$-breaking phase transitions at $\ensuremath{\theta}=\ensuremath{\pi}$ in Yang-Mills theories. The 't Hooft anomaly matching prohibits the confined phase with $\mathcal{C}\mathcal{P}$ symmetry and requires ${T}_{\mathrm{dec}}(\ensuremath{\theta}=\ensuremath{\pi})\ensuremath{\le}{T}_{\mathrm{CP}}$, where ${T}_{\mathrm{dec}}(\ensuremath{\theta}=\ensuremath{\pi})$ and ${T}_{\mathrm{CP}}$ denote the deconfinement and the $\mathcal{C}\mathcal{P}$-restoration temperatures, respectively, at $\ensuremath{\theta}=\ensuremath{\pi}$. We analytically study these two phase transitions in softly broken $\mathcal{N}=1$ supersymmetric Yang-Mills theories on small ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ with the periodic boundary condition for gluinos. For most gauge groups except SU(2) in this model, we find that the inequality is saturated, so deconfinement and $\mathcal{C}\mathcal{P}$ restoration occur simultaneously. We demonstrate special features of the SU(2) gauge theory: there is a finite window of two temperatures, ${T}_{\mathrm{dec}}(\ensuremath{\pi})<{T}_{\mathrm{CP}}$, which indicates the existence of a novel $\mathcal{C}\mathcal{P}$-broken deconfined phase. We also discuss an implication of the novel phase for domain walls and their junctions.