Non-singlet conserved charges and anomalies in 3+1 D staggered fermions

Abstract

In this paper, we show that the 3+1 D staggered fermion Hamiltonian possesses, in addition to the conserved charge $Q_0$ that generates the vector $\mathrm{U}(1)_V$ transformation, conserved charges $Q_F$ that generate the $\mathrm{SU}(2)_A$ transformations in the continuum limit, acting simultaneously on left- and right-handed Weyl fermions in opposite directions. Each conserved charge $Q_F$ can be regarded as the generator of a $\mathrm{U}(1)_F$ subgroup of $\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{U}(1)_A$. One of these lattice charges satisfies the Onsager algebra. On the lattice, the charges $Q_F$ do not commute with $Q_0$, and no symmetric mass term exists that commutes with both $Q_0$ and $Q_F$. This signals the presence of a mixed anomaly. Remarkably, however, in the continuum limit, a symmetric mass term commuting with both $Q_0$ and $Q_F$ can be constructed. % This implies that the non-trivial lattice anomaly becomes a trivial anomaly in the continuum QFT. This means that the mixed anomaly that is nontrivial on the lattice becomes trivial in the IR QFT obtained in the continuum limit, which is consistent with the analysis of the Ward--Takahashi (WT) identity on the lattice. Indeed, by evaluating this identity associated with the $\mathrm{U}(1)_F$ transformation on the lattice, we confirm that $\mathrm{U}(1)_F$ symmetry is exactly conserved.