Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 47 / 3 / c

For Electron-neutrino scattering the weak charged current adds an Electron left-current. The supplied Fierz rearrangement has a minus sign for matrix components; the extra minus from ordering the external fermions cancels it. Accordingly the charged contribution is constructive in the neutrino-current/Electron-current ordering used in part (b):Thus ${\mathcal{M}}_e$ is obtained from the physically consistent ${\mathcal{M}}_{\mu }$ by $c_V\mapsto c_V+1$, $c_A\mapsto c_A+1$. The right-Electron coupling $c_V-c_A$ is unchanged, while the left-Electron coupling $c_V+c_A$ increases by two. This is the charged-current shift in neutrino-electron scattering.

For the antineutrino, choose the external phase convention in whichThe reversed ordering of the antineutrino spinors implements crossing symmetry; the displayed overall sign can be changed by an external-state phase. Its neutrino trace has the opposite antisymmetric part, so the $s^2$ and $u^2$ weights exchange:The phase-space integration already performed in part (b) immediately givesEquivalently, with $a=c_V+1,b=c_A+1$, these are $G_F^{2}s(a^2+ab+b^2)/(3\pi )$ and $G_F^{2}s(a^2-ab+b^2)/(3\pi )$. This is the neutrino-antineutrino interchange of chiral scattering weights. It uses the contact limit, not the full resonant $s$-channel propagator.

If the printed flavour-mismatched interaction is retained literally, its ${\nu }_{\mu }e$ amplitude already equals the displayed ${\nu }_{e}e$ amplitude. There is then no further flavour shift between those two processes, although antineutrino crossing still exchanges the weights. The distinction identified in part (b) is therefore necessary for the Standard Model interpretation of all three processes.