Chiral symmetry Where can I find Information on how electrons have chiral symmetry only if the massless and how it is broken by electron mass correction term? Everything I keep reading seems to be to do with quark's and the strong force
 A: Chiral symmetry is a property of any massless fermion in even-dimensional spacetime. It is defined by the following transformations on spinor fields:
$$
\bar{\psi} \rightarrow \bar{\psi} e^{i \theta\gamma_5} \qquad
\psi \rightarrow e^{i \theta\gamma_5}  \psi 
$$
Note, that the sign in the exponent is the same in both expressions.
Kinetic term is invariant under this transformation:
$$
i \bar{\psi} e^{i \theta\gamma_5} \gamma^{\mu} e^{i \theta\gamma_5} \partial_\mu  \psi \rightarrow i \bar{\psi} e^{i \theta\gamma_5} \gamma^{\mu} e^{i \theta\gamma_5} \partial_\mu  \psi
$$
Here we used anticommutativity $\{\gamma_\mu, \gamma_5\} = 0$.
The mass term, however, is not invariant:
$$
m \bar{\psi} \psi \ e^{i 2 \theta\gamma_5}
$$
The introduction of this term breaks the chiral symmetry explicitly.
For QCD there is somehow a generalized definiton of chiral symmetry. Assuming we have $N_f$ massless species of quarks QCD Lagrangian possesses following symmetry, rotating each of the fields $\bar{q}, q$ independently:
$$
U(N_f) \times U(N_f) = SU(N_f) \times SU(N_f) \times U_V(1) \times U_A(1)
$$
$U_1(A)$ is an axial symmetry broken by the anomaly, and the chiral symmetry is a part $SU(N_f) \times SU(N_f)$. It is a symmetry of Lagrangian on the classical level. However on a quantum level there is an observable - chiral condensate, which violates this symmetry:
$$
\langle \bar{q}_i q_j \rangle = \sigma \delta_{ij}
$$
$\sigma$ developing nonzero expectation value indicates the breaking of chiral symmetry. Above certain temperature $T_c$ chiral symmetry in QCD is restored, but the exact mechanism,explaining how does this exactly happen is unknown so far. There is an evidence from the lattice simulations, and holographic models, where one can account for chiral symmetry breaking, effective model, like NJL.
