Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 305 / 3

Forbidden at tree level. A photon, gluon, $Z$ boson, or neutral Higgs interaction is flavor diagonal in the quark mass basis, while a $W$ boson connects an up-type quark to a down-type quark. The process $u\bar{c}\to c\bar{u}$ would therefore require a flavor-changing neutral current, which first appears through loops in the Standard Model.

Allowed. A single $t$-channel W boson mediates the weak charged current transitions $u\to d$ and $\bar{c}\to \bar{s}$. The amplitude is proportional to the Cabibbo-Kobayashi-Maskawa matrix product $V_{ud}{V}_{cs}^{\ast }$.

Forbidden. The initial state has total lepton number two and the final state has total lepton number minus two. No renormalizable Standard Model vertex changes total lepton number by four.

Allowed. There are two tree-level Feynman diagrams: $t$-channel Z boson exchange through the weak neutral current, and $s$-channel $W^{+}$ formation through the weak charged current.

Allowed. There are again two tree-level Feynman diagrams: $t$-channel Z boson exchange and the crossed charged-current diagram with a W boson exchanged between the neutrino and electron lines.

The complete tree-level list for (ii), (iv), and (v) is:

The first four-fermion operator is the low-energy Fermi interaction obtained by replacing the crossed $W$ propagator in the charged-current diagram by $-g_{\mu \nu }/M_W^2$. The second operator is obtained analogously from the $Z$-exchange weak neutral current; $c_V$ and $c_A$ are its electron vector and axial-vector couplings. Thus the two terms represent respectively the charged-current and neutral-current diagrams in part (v), with their interference retained when the amplitude is squared.

A Fierz rearrangement puts the charged-current operator in the same current ordering as the neutral-current operator. Accounting for the interchange of fermionic fields, the combined amplitude isSum over final spins and average over the initial electron spin. The fermion spin sum and gamma-matrix trace identities giveFor massless two-body scattering in the centre-of-momentum frame, $d\sigma /dt=\overline{|\mathcal{M}{|}^2}/(16\pi s^2)$ and $u=-s-t$. Integrating $-s\le t\le 0$ therefore yieldsConsequently

The result grows linearly with the squared centre-of-momentum energy, $\sigma \propto G_F^{2}s$. This growth eventually violates partial-wave unitarity and signals the breakdown of the pointlike Fermi interaction. At energies comparable to $M_W$ or $M_Z$, the full gauge-boson propagators must replace the contact interaction; the renormalizable electroweak theory then softens the high-energy behavior.