= Solution
The identities $S_L^\dagger\bar\sigma^\mu S_L=\Lambda^\mu{}_{\nu}\bar\sigma^\nu$ and $S_L^T\sigma^2S_L=\sigma^2$ show respectively that the kinetic term and both mass bilinears are Lorentz invariant. Infinitesimally, the first identity makes $u_L^\dagger\bar\sigma^\mu u_L$ transform by the vector law stated in the question, while the second makes $u_L^T\sigma^2u_L$ a scalar.
Varying the anticommuting components of $u_L^\dagger$ gives
$$
i\bar\sigma^\mu\partial_\mu u_L-i m\sigma^2u_L^*=0.
$$
The factor two from varying the antisymmetric quadratic form cancels the $1/2$ in the mass term. This is a <Majorana mass term>: under $u_L\mapsto e^{iq\alpha}u_L$ it carries charge $2q$, so a nonzero mass is compatible with an unbroken electric charge only for $q=0$. A charged massive fermion instead needs an independent Weyl field of opposite chirality to form a Dirac mass.
Back to article page
