θ =π in S U N / ℤ N $$ \mathrm{S}\mathrm{U}(N)/{\mathbb{Z}}_N $$ gauge theories

• Main Authors: Ryuichiro Kitano, Takao Suyama, Norikazu Yamada
• Format: Article
• Language: English
• Published: SpringerOpen 2017-09-01
• Series: Journal of High Energy Physics
• Subjects: Discrete Symmetries; Lattice Quantum Field Theory; Wilson; ’t Hooft and Polyakov loops; Confinement
• Online Access: http://link.springer.com/article/10.1007/JHEP09(2017)137

Abstract In SU(N ) gauge theory, it is argued recently that there exists a “mixed anomaly” between the CP symmetry and the 1-form ℤ N $$ {\mathbb{Z}}_N $$ symmetry at θ = π, and the anomaly matching requires CP to be spontaneously broken at θ = π if the system is in the confining phase. In this paper, we elaborate on this discussion by examining the large volume behavior of the partition functions of the S U N / ℤ N $$ \mathrm{S}\mathrm{U}(N)/{\mathbb{Z}}_N $$ theory on T 4 à la ’t Hooft. The periodicity of the partition function in θ, which is not 2π due to fractional instanton numbers, suggests the presence of a phase transition at θ = π. We propose lattice simulations to study the distribution of the instanton number in S U N / ℤ N $$ \mathrm{S}\mathrm{U}(N)/{\mathbb{Z}}_N $$ theories. A characteristic shape of the distribution is predicted when the system is in the confining phase. The measurements of the distribution may be useful in understanding the phase structure of the theory.