CT or P Problem and Symmetric Gapped Fermion Solution

An analogous "Strong CP problem" is identified in a toy model in 2-dimensional spacetime: a general 1+1d abelian U(1) anomaly-free chiral fermion and chiral gauge theory with a generic theta instanton term $\frac{\theta}{2 \pi} \int F$. 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 $\bar{\theta}$ angle (including the effect of $\theta$ 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 content, confinement, and Anderson-Higgs-induced mass by Yukawa-Higgs term. One solution replaces some chiral fermion's Higgs-induced mean-field mass with SMG-induced non-mean-field mass. Another solution enriches this toy model by introducing several new physics beyond the Standard Model: a parity-reflection PR discrete symmetry maps between the chiral and mirror fermions as fermion doubling localized on two domain walls at 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 at lower energy. Without loss of generality, an arguably simplest 1+1d U(1) symmetric anomaly-free chiral fermion/gauge theory (e.g., Weyl fermions with $3_L$-$4_L$-$5_R$-$0_R$ U(1) charges) is demonstrated. As an analogy to the superfluid-insulator or order-disorder quantum phase transition, in contrast to the Peccei-Quinn solution sitting in the (quasi-long-range-order) superfluid phase, our solution is in the SMG insulator disordered phase.