The word “Axial” indicates the presence of \gamma_{5} in the transformation law.
Consider the following axial transformation of a fermion field with TWO FLAVOURS;
\Psi \rightarrow U_{5}\Psi \equiv \exp (i \frac{\epsilon . \tau}{2}\gamma_{5}) \Psi . \ \ (1)
In this, \gamma_{5} operates on the Dirac components of \Psi, while Pauli’s matrices \tau^{i} operate on the internal 2-dimensional flavour space of the fermions. We may call this group SU(2)_{5}. This transformation takes on a simple form when expressed in terms of the chiral components of \Psi. That is
<br />
\Psi_{R}\rightarrow V \Psi_{R} \equiv \exp( i \frac{\epsilon . \tau}{2}) \Psi_{R},<br />
<br />
\Psi_{L}\rightarrow V^{\dagger}\Psi_{L} \equiv \exp( - i \frac{\epsilon . \tau}{2}) \Psi_{L}.<br />
This can be shown by expanding eq(1) in a power series, and using the following
\Psi = \Psi_{R} + \Psi_{L},
\gamma_{5}\Psi = \Psi_{R} - \Psi_{L}.
So, the 8 \times 8 transformation matrix U_{5} can be written as
U_{5} = VP^{+} + V^{\dagger}P^{-}\ \ \ (2),
where
P^{\pm} = \frac{1}{2}(1 \pm \gamma_{5}),
are projection operators.
From eq(2), it follows that
\gamma^{\mu}U_{5}= U^{\dagger}_{5}\gamma^{\mu}.
This means that \Psi and \bar{\Psi} transform in the same way,
\Psi \rightarrow U_{5}\Psi,
\bar{\Psi}\rightarrow \bar{\Psi}U_{5}.
Thus, chirally invariant Lagrangian must be constructed out of massless fermions; the presence of a small fermion mass term provides a mechanism for breaking chiral symmetry.
Since our fermion field has two flavours, the theory must also be invariant under the global group SU(2). So, the total symmetry group is SU(2) \times SU(2)_{5}. This group is equivalent to SU(2)_{L}\times SU(2)_{R} with element
<br />
U_{L}U_{R}= \exp(i\frac{\epsilon_{L}.\tau}{2}P^{-}) \exp( i \frac{\epsilon_{R}.\tau}{2}P^{+}),<br />
where \epsilon_{L} and \epsilon_{R} are the independent parameters. To see the equivalence, let \Psi_{L}= P^{-}\Psi, and \Psi_{R}= P^{+}\Psi be associated with the (finite dimensional) irreducible representations of Lorentz group (1/2,0) \oplus (1/2,0) and (0,1/2) \oplus (0,1/2), respectively. The transformations under two commuting SU(2) groups are
(2,1) \in SU(2)_{L}:
\Psi_{L}\rightarrow U_{L}\Psi_{L}\equiv \exp( i\frac{\epsilon_{L}.\tau}{2})\Psi_{L},
\Psi_{R}\rightarrow \Psi_{R},
which can be combined into
<br />
\Psi \rightarrow e^{i\frac{\epsilon_{L}.\tau}{2}} \frac{1}{2}(1 - \gamma_{5})\Psi + \frac{1}{2}(1 + \gamma_{5})\Psi,<br />
or
\Psi \rightarrow \exp(i\frac{\epsilon_{L}.\tau}{2}P_{-})\Psi .
(1,2) \in SU(2)_{R}:
\Psi_{L}\rightarrow \Psi_{L},
\Psi_{R}\rightarrow U_{R}\Psi_{R} = \exp( i\frac{\epsilon_{R}.\tau}{2})\Psi_{R},
which we can write as
\Psi \rightarrow \exp(i\frac{\epsilon_{R}.\tau}{2}P^{+})\Psi
Therefore, the combined SU(2)_{L}\times SU(2)_{R} transformation is given by
<br />
\Psi \rightarrow U_{L}U_{R}\Psi = \exp[i(\epsilon_{L}P^{-} + \epsilon_{R}P^{+}) . \frac{\tau}{2}] \Psi . \ \ \ (3)<br />
If we define
\alpha = \frac{1}{2}(\epsilon_{R} + \epsilon_{L}),
\epsilon = \frac{1}{2}(\epsilon_{R} - \epsilon_{L}),
then, the SU(2)_{L}\times SU(2)_{R} element U_{L}U_{R} becomes
<br />
U_{L}U_{R} = \exp(i\frac{\alpha . \tau}{2}) \exp(i\frac{\epsilon . \tau}{2}\gamma_{5}),<br />
which belongs to the original symmetry group SU(2) \times SU(2)_{5}.
The meaning of chiral symmetry is, according to eq(3), the statement that an SU(2) symmetry can be INDEPENDENTLY realized on the two subspaces projected out by the P^{\pm} operators, i.e., the transformations on these two spaces can have different parameters.
Regards
Sam
