On the decoupling of mirror fermions

TL;DR

The paper tackles the problem of formulating chiral gauge theories on the lattice by testing mirror decoupling in the anomaly-free 3-4-5 model. It uses Ginsparg-Wilson fermions and a strong Yukawa Higgs sector to probe the mirror sector via the polarization tensor, leveraging the splitting theorem to isolate mirror dynamics. Nonperturbative lattice simulations reveal a directional discontinuity in the mirror polarization tensor that persists in the continuum limit, consistent with massless mirror modes and suggesting that the mirror sector does not decouple. The results imply that anomaly freedom does not guarantee mirror decoupling and point to two possible infrared structures (a chiral massless mirror or a massless vectorlike pair), highlighting challenges and directions for future work in lattice chiral gauge theories.

Abstract

An approach to the formulation of chiral gauge theories on the lattice is to start with a vector-like theory, but decouple one chirality (the "mirror" fermions) using strong Yukawa interactions with a chirally coupled "Higgs" field. While this is an attractive idea, its viability needs to be tested with nonperturbative studies. The model that we study here, the so-called "3-4-5" model, is anomaly free and the presence of massless states in the mirror sector is not required by anomaly matching arguments, in contrast to the "1-0" model that was studied previously. We have computed the polarization tensor in this theory and find a directional discontinuity that appears to be nonzero in the limit of an infinite lattice, which is equivalent to the continuum limit at fixed physical volume. We show that a similar behavior occurs for the free massless Ginsparg-Wilson fermion, where the polarization tensor is known to have a directional discontinuity in the continuum limit. We thus find support for the conclusion that in the continuum limit of the 3-4-5 model, there are massless charged modes in the mirror sector so that it does not decouple from the light sector. The value of the discontinuity we obtain allows for two interpretations: either a chiral gauge theory does not emerge and mirror-sector fermions in a chiral anomaly free representation remain massless, or a massless vectorlike mirror fermion appears. We end by discussing some questions for future study.

Figures (9)

• Figure 1: A graph representing the gauge invariant mirror interactions in the 3-4-5 model. The vertices denote the Weyl fermion fields, see Table \ref{['fc']}, of the mirror theory or their conjugates. The thick connecting lines denote appropriate powers of the unitary "Higgs" field $\phi$. If a line connects a fermion field on one vertex with the conjugate of the field on the other vertex, then the corresponding interaction is part of $S_{\text{Yuk.,Dirac}}$. If a line connects a fermion field with the field (rather than the conjugate) on the other vertex, then the interaction is part of $S_{\text{Yuk.,Maj.}}$.
• Figure 2: Complex phase distribution for the set of Yukawa coupling constants given in Table \ref{['tb:coupling1']}, on an $8 \times 8$ lattice.
• Figure 3: $\tilde{\Pi}_{11}^{\text{mirror},\prime}(k)$ on a $6\times 6$ lattice. The lines show the extrapolation $k \to 0$ for different angles of approach. A clear discontinuity in the directional limit can be seen.
• Figure 4: $\tilde{\Pi}_{11}^{\text{mirror},\prime}$ on an $8 \times 8$ lattice with the couplings given in Table \ref{['tb:coupling1']}.
• Figure 5: $\tilde{\Pi}_{11}^{\text{mirror},\prime}$ on a $10 \times 10$ lattice with the couplings given in Table \ref{['tb:coupling1']}. Only the smallest values of $k$ were studied and the $0^o$ approach to the origin was omitted because of the expense of the calculation.