伴随费米子与基础费米子的轴矢反常及剩余离散轴矢对称性

Hey there, let's unpack this question about the discrete axial symmetries that survive after the continuous $U(1)_A$ chiral symmetry is broken by quantum anomalies. The core idea is that while the infinite continuous symmetry is killed by the anomaly, finite discrete subgroups can remain because their transformations don't pick up a non-trivial phase in the path integral. Let's break down the two cases:

1. Fundamental Fermions #

Take fermions in the fundamental representation of a Lie group $G$ (like quarks in QCD, which are fundamental of $SU(3)$):

• The chiral anomaly creates a phase factor in the path integral when we perform a global axial rotation $\psi \to e^{i\alpha \gamma_5}\psi$. For this to be a symmetry, the phase must be trivial (equal to 1) for all topological instanton numbers.
• For the standard case of $SU(N)$ (the group behind QCD), the fundamental representation has a Dynkin index $I_f = 1/2$. The condition for a trivial phase simplifies to:
  $$\frac{N_f \alpha}{2\pi} \in \mathbb{Z}$$
  where $N_f$ is the number of fermion flavors.
• This means the smallest non-trivial rotation angle is $\alpha = \frac{2\pi}{N_f}$, so the largest surviving discrete axial symmetry is the cyclic group $Z_{N_f}$. For example, in QCD with 3 light quark flavors, this gives a $Z_3$ discrete axial symmetry.
• Generalizing to any Lie group: the surviving symmetry is $Z_{2N_f I_f}$, where $I_f$ is the Dynkin index of the fundamental representation of $G$.

2. Adjoint Fermions #

Now consider fermions in the adjoint representation (like gluinos in supersymmetric QCD, which are adjoint of $SU(3)$):

• The adjoint representation of $SU(N)$ has a Dynkin index $I_{\text{adj}} = N$. Applying the same phase-triviality condition, we get:
  $$\frac{N_f N \alpha}{\pi} \in \mathbb{Z}$$
• The smallest non-trivial rotation angle here is $\alpha = \frac{2\pi}{N_f N}$, so the surviving discrete axial symmetry is $Z_{2N_f N}$ for $SU(N)$.
• For a general Lie group $G$, the surviving symmetry is $Z_{2N_f I_{\text{adj}}}$, where $I_{\text{adj}}$ is the Dynkin index of the adjoint representation of $G$.

A quick sanity check: the larger the Dynkin index (which measures how "large" the representation is), the smaller the discrete group—since more rotations will pick up an anomalous phase, leaving fewer valid symmetry transformations.

内容的提问来源于stack exchange，提问作者ann marie cœur