Past exam of the particle physics course of the University of Oxford / 2019 / 4 / a / Solution

The CKM matrix converts down-type quark mass eigenstates into the combinations entering the weak charged current. Its entry $V_{ij}$ controls a charged-current transition between up-type flavour $i$ and down-type flavour $j$. In the three-generation Standard Model, $V=U_{uL}^{†}{U}_{dL}$ is a product of unitary matrices from diagonalizing the quark mass matrices, so $V^{†}V=1$.

The Wolfenstein parametrization displays the hierarchyThus diagonal transitions dominate; first-second generation mixing is larger than second-third generation mixing, and direct first-third generation mixing is smallest. The parameters $A,\rho ,\eta$ refine this order estimate. With nondegenerate quark masses, a nonzero irreducible weak phase permits CP violation; at leading order the Jarlskog invariant is $J\simeq A^2{\lambda }^6\eta$. Individual complex entries are not enough: the phase must survive allowed quark-field rephasings.

The exact physical CKM matrix is unitary, whereas its displayed cubic truncation is only approximately so. For example its first-row norm is $1+{\lambda }^4/4+A^2{\lambda }^6({\rho }^2+{\eta }^2)$, which is not exactly one for nonzero $\lambda$. The omitted higher-order terms repair this; unitarity must be tested to the retained order.