Past exam of the particle physics course of the University of Oxford / 2016 / 4 / 1

In neutral meson mixing, the mass eigenstates and flavor eigenstates differ. A state prepared as $D^0=c\bar{u}$ develops a ${\bar{D}}^0=u\bar{c}$ component because the effective Hamiltonian operator has off-diagonal mass and decay width entries. In the CP symmetry limit implicit in the source, the propagating states are equal superpositions, with average mass $M=(M_1+M_2)/2$ and average decay width $\Gamma =({\Gamma }_1+{\Gamma }_2)/2$.

To retain the source's positive $i\sin$ convention for $g_{-}$, take $M_2-M_1=\Delta M$ and ${\Gamma }_2-{\Gamma }_1=\Delta \Gamma$, with $g_{\pm }=(e_1\pm e_2)/2$, where $e_j=\exp (-i{M}_{j}t-{\Gamma }_{j}t/2)$. Then $x=\Delta M/\Gamma$ and $y=\Delta \Gamma /(2\Gamma )$ reproduce the given functions. Reversing an eigenstate label changes signs in intermediate formulae; it must also change the phase conventions consistently.

For equal widths, the mixed probability is proportional to $e^{-\Gamma t}{\sin }^2(\Delta Mt/2)$. Thus the angular angular frequency is $|\Delta M|$ in natural units, or $|\Delta M|/\hslash$ with units restored; the period is $2\pi \hslash /|\Delta M|$. A nonzero difference of decay widths adds hyperbolic time dependence.

The ordinary neutral flavour-carrying mesons that mix are $K^0$, $D^0$, $B_d^0$ and $B_s^0$ with their antiparticles. A self-conjugate ${\pi }^0$ has no distinct antiparticle flavour to oscillate into. There are no long-lived top-flavoured mesons, since the top quark decays before hadronization.