CT or P Problem and Symmetric Gapped Fermion Solution

Abstract

An analogous "Strong CP problem" is identified in a toy model in two dimensional spacetime: a general 1+1d abelian U(1) anomaly-free chiral fermion and chiral gauge theory with a generic theta instanton term θ 2π F . The theta term alone violates the charge-conjugation-time-reversal CT and the parity P discrete symmetries. The analogous puzzle here is the CT or P problem in 1+1d: Why can the θ angle (including the effect of θ and the complex phase of a mass matrix) be zero or small for a natural reason? We show that this CT or P problem can be solved by a Symmetric Mass Generation mechanism (SMG, namely generating a mass or energy gap while preserving an anomaly-free symmetry). This 1+1d toy model mimics several features of the 3+1d Standard Model: chiral matter contents, confinement, and Anderson-Higgs mass by Yukawa-Higgs term. One solution enriches this toy model by introducing several new physics beyond the Standard Model: a parity-reflection PR discrete symmetry mapping between the chiral and mirror fermions as fermion doubling localized on two domain walls at a high energy, and SMG dynamically generates mass to the mirror fermion while still preserving the anomaly-free chiral symmetry at an intermediate energy scale, much before the Higgs mechanism generates mass to the chiral fermion in at a lower energy. Another solution no longer requires the mirror fermion sector, but directly replaces the chiral fermion's Higgs mass to the SMG mass. Without loss of generality, an arguably simplest 1+1d U(1) symmetric 3L-4L-5R-0R anomaly-free chiral fermion and chiral gauge theory is demonstrated in detail.

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