CT or P Problem and Symmetric Gapped Fermion Solution

Abstract

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.

arXiv comment28 pages. Two classes of solutions, without or with the mirror fermion sector. A companion to arXiv:2212.14036. Related models: arXiv:1307.7480, arXiv:1807.05998, arXiv:2202.12355. v3: More misprints corrected. New refinement and clarification upon the Physical Review D version

References in corpus (29)

• Generalized Global Symmetries
• An Improved Experimental Limit on the Electric Dipole Moment of the Neutron
• A Method for Simulating Chiral Fermions on the Lattice
• Topological phases of fermions in one dimension
• Measurement of the permanent electric dipole moment of the neutron
• The effects of interactions on the topological classification of free fermion systems
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Protected edge modes without symmetry
• Topological boundary conditions in abelian Chern-Simons theory
• Reflection positivity and invertible topological phases
• Stability of Chiral Luttinger Liquids and Abelian Quantum Hall States.
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Boundary Degeneracy of Topological Order
• Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory
• Interacting Topological Insulator and Emergent Grand Unified Theory
• A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model
• Comments on Symmetric Mass Generation in 2d and 4d
• Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model
• A Solution to the 1+1D Gauged Chiral Fermion Problem
• A Non-Perturbative Definition of the Standard Models
• P not PQ
• Gapped Chiral Fermions
• Symmetry-protected many-body Aharonov-Bohm effect
• Fermion masses without symmetry breaking in two spacetime dimensions
• The Origin of Families and $SO(18)$ Grand Unification
• On the decoupling of mirror fermions
• Ultra Unification
• Gauge Enhanced Quantum Criticality Beyond the Standard Model
• C-P-T Fractionalization

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