= Solution
{c}
In the <Weyl representation of the gamma matrices>, write each <Dirac field> as $\psi_i=(\ell_{i\alpha},r_i^{\dot\alpha})^T$, where $\ell_i$ and $r_i$ have opposite <chirality>. With the <Weyl sigma matrices> $\sigma^\mu=(\mathbf1,\boldsymbol\sigma)$ and $\bar\sigma^\mu=(\mathbf1,-\boldsymbol\sigma)$,
$$
\boxed{\mathcal L=\sum_{i=1}^2\left[
i\ell_i^\dagger\bar\sigma^\mu\partial_\mu\ell_i
+ir_i^\dagger\sigma^\mu\partial_\mu r_i
-m(\ell_i^\dagger r_i+r_i^\dagger\ell_i)\right].}
$$
The <Dirac mass term> couples the two <Weyl fields>. Under $\ell\mapsto U_L\ell$, $r\mapsto U_Rr$, its first term acquires $U_L^\dagger U_R$, so a common nonzero <mass> requires $U_L=U_R$. At zero <mass> there is no such constraint:
$$
\boxed{G_{\rm number\ preserving}=U(2)_L\times U(2)_R.}
$$
This is the <chiral flavour symmetry of massless Dirac fields>. Its eight <Noether currents> are $j_{L,A}^\mu=\ell^\dagger\bar\sigma^\mu T^A\ell$ and $j_{R,A}^\mu=r^\dagger\sigma^\mu T^Ar$. The sum $j_R+j_L$ is the <vector current>, and $j_R-j_L$ is the <axial current>, for each flavour generator. This statement concerns the free theory in flat spacetime; an interacting theory can have a <chiral anomaly>.
For completeness, allowing the extra transformations described in part <1/e> gives four left-handed <Weyl fields>, $q=(\ell_1,\ell_2,r_1^c,r_2^c)^T$, where $r_i^c$ is the left-handed <charge-conjugate field> of $r_i$. After <integration by parts> their kinetic term is $iq^\dagger\bar\sigma^\mu\partial_\mu q$, whose maximal linear <unitary group> is $U(4)$. The symmetric <mass matrix> for equal <Dirac masses> is
$$
M=m\begin{pmatrix}0&\mathbf1_2\\\mathbf1_2&0\end{pmatrix},\qquad
\mathcal L_m=-\frac12(q^TMq+\text{Hermitian conjugate}),
$$
with the antisymmetric spinor contraction implicit. A unitary change of basis makes $M=m\mathbf1_4$; preserving both the kinetic term and $M$ requires $V^\dagger V=V^TV=\mathbf1_4$, hence real $V\in O(4)$. Thus the unrestricted free-theory enhancement is $U(4)$ at zero <mass> and $O(4)$ at equal nonzero <mass>; the familiar $U(2)_L\times U(2)_R$ and diagonal $U(2)$ are their common-<fermion number>-preserving subgroups.
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