Is there a relationship between axial $U(1)$ symmetry and parity transformations?

Question

Some books, such as Maggiore, write that the axial $U(1)$ transformation is the chiral transformation. A Dirac field $\Psi \in (\frac12, \frac12)$ undergoes a chiral transformation in the following manner.

$$ \Psi \mapsto e^{i\beta\gamma_5}\Psi $$

In a chiral basis, we can write $\Psi \equiv \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix}$ where $\psi_L \in (\frac12,0)$ and $\xi_R \in (0,\frac12)$. Then,

$$ \psi_L \mapsto e^{-i\beta}\psi_L \,,\qquad \xi_R \mapsto e^{i\beta}\xi_R \,. $$

Maggiore defines chiral fields to be ones that violate the axial $U(1)$ symmetry.

Whereas books such as Srednicki define chiral fields to be ones that violate parity. A parity transformation does the following.

$$ \begin{pmatrix} \psi_L\\ \xi_R\end{pmatrix} \mapsto \begin{pmatrix} \xi_R\\\psi_L \end{pmatrix} $$

Srednicki discusses the axial $U(1)$ symmetry in a different chapter as a global symmetry in consistent gauge theories.

So, which of the definitions of chirality is true? And if they are equivalent, how so?

Answer 1

They are both manifestly true. Any confusion is traceable to notational overkill. In your 2d Weyl basis where superfluous spinor indices are ignored, an axial rotation is but $$ e^{-i\beta \sigma_3} =1\!\!1 \cos \beta -i \sigma_3 \sin \beta , $$ where, of course, the finite rotation $\beta=\pi/2$ amounts to the above reducing to $-i\sigma_3= -\tfrac{i}{2}(1\!\!1+i\sigma_2)\sigma_1 (1\!\!1-i\sigma_2)$, so unitarily equivalent to $-i\sigma_1$.

Likewise, $$ P=\sigma_1 $$ which has the same eigenvalues as the π/2 rotation above, and provides its conjugation matrix, $$ P e^{-i\beta \sigma_3} P= e^{i\beta \sigma_3}, $$ since $\sigma_1 \sigma_3\sigma_1=-\sigma_3$.

So, yes, axial rotations differentiate between chiral eigenstates and parity interchanges them. You cannot have an invariant of axial rotations which is not parity invariant; and, conversely, for fermions, you won't be able to construct an parity invariant which violates the axial U(1).

We are talking about fermions here, but, parity-odd spinless states like pseudoscalars (which might be used to violate parity in an action) can be associated with fermion spinless bilinears which comport with this axial rotation scheme: so you might as well consider the two bases connected by $(1\!\!1-i\sigma_2)/\sqrt2 $ equivalent.