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

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.

Put $\tau =\Gamma t$, $r=r_D$ and $\delta ={\delta }_D$. Two indistinguishable paths reach the wrong-sign kaon-pion final state: a direct doubly Cabibbo suppressed decay amplitude $r{e}^{-i\delta }{g}_{+}$ and a neutral meson mixing amplitude followed by the favoured decay, $g_{-}$. They must be added before taking the absolute square:Direct complex-trigonometric multiplication givesThereforeThe three terms are direct decay, quantum interference and decay after mixing. In particular, the minus sign multiplying $\sin \delta \sin x\tau$ follows from the source's $+i\sin$ convention and its definition $r{e}^{-i\delta }$; importing another convention without transforming $\delta$ would change a physical prediction incorrectly. The common exponential accounts for survival before decay. The expression is a decay-rate density up to an overall normalization, rather than a normalized cumulative probability.

The right-sign denominator has its own small mixing contribution. In the same CP symmetry convention its decay amplitude is ${\mathcal{A}}_{\mathrm{RS}}=g_{+}+r{e}^{-i\delta }{g}_{-}$. After cancelling $e^{-\tau }$, the exact numerator and denominator areDefine $u=y\cos \delta$ and $v=x\sin \delta$. Their Taylor expansions are $N=r^2+r(u-v)\tau +\{x^2+y^2+r^2(y^2-x^2)\}{\tau }^2/4+O({\tau }^3)$ and $D=1+r(u+v)\tau +\{y^2-x^2+r^2(x^2+y^2)\}{\tau }^2/4+O({\tau }^3)$. Dividing the series gives the strict second-order answerThe usually quoted approximation additionally drops terms suppressed by powers of $r$ in the right-sign denominator:For $r=0.1$ and $x=y=0.01$, this is $0.01+0.001\tau +0.00005{\tau }^2$ for $\delta =0$, and $0.01-0.001\tau +0.00005{\tau }^2$ for $\delta =\pi /2$. The first increases; the second initially decreases, reaches approximately $0.005$ at $\tau =10$, then rises. The strict series instead has $(C_1,C_2)=(0.00099,0.000049005)$ and $(-0.00101,0.000051005)$. At $\tau =20$ the exact ratios are approximately $0.04913$ and $0.009934$; the leading approximations give $0.05$ and $0.01$.

With no neutral meson mixing, $x=y=0$ and $g_{-}=0$, so $\boxed{R=r^2=0.01}$ at all decay times. The wrong-sign signal then remains through direct doubly Cabibbo suppressed decay.

A charged lepton in a semileptonic decay directly identifies the decaying charm quark flavour: ordinarily $D^0\to K^{-}{\mu }^{+}{\nu }_{\mu }$ and ${\bar{D}}^0\to K^{+}{\mu }^{-}{\bar{\nu }}_{\mu }$. With an independently tagged initial quark flavour, the wrong-sign charged lepton requires neutral meson mixing, rather than a competing doubly Cabibbo suppressed amplitude. Its leading time dependence is $e^{-\tau }(x^2+y^2){\tau }^2/4$, which looks theoretically cleaner.

That method measures only the quadratic combination $x^2+y^2$ at leading order, with no direct access to their signs. The missing neutrino also worsens kinematic and decay-time reconstruction. The fully reconstructed $K\pi$ channel has substantial yield, a clear invariant-mass peak and a well-measured decay interaction vertex. Most significantly, quantum interference with the small direct amplitude produces a term linear in $x$ and $y$, enhancing sensitivity to small mixing. Its cost is the unknown strong phase $\delta$, so separating $x$ and $y$ needs further channels or an independent strong-phase measurement. Neither one unqualified rate measurement determines both parameters by itself.