Past exam of the particle physics course of the University of Oxford / 2018 / 3

The flavour states $K^0=d\bar{s}$ and ${\bar{K}}^0=s\bar{d}$ have definite Strangeness and are distinguished by flavour-sensitive production or decay. The propagation states diagonalize the effective Hamiltonian $H_{\mathrm{eff}}=M-i\Gamma /2$; their complex eigenvalues give a pole mass and a decay width. The common flavour combinations specified in the source assume the usual CPT-symmetric two-state description and are normalized with $|p{|}^2+|q{|}^2=1$.

Choose the phase convention $CP|K^0⟩=|{\bar{K}}^0⟩$. If mixing conserves CP, $p=q$ can be chosen, and the symmetric and antisymmetric combinations are CP even and odd. More generally the equality must be understood with the corresponding flavour-state phase convention; $|p|=|q|$ alone is not a statement fixing an arbitrary written phase.

The almost CP-even state decays rapidly to two pions, whereas the almost CP-odd state is largely restricted to slower channels such as three pions. Their very different lifetimes are easier to distinguish than their tiny mass splitting, motivating the names $K_S^0$ and $K_L^0$. In the source's first notation the subscript $L$ means light, not long-lived: the light state is identified with the short-lived kaon, and the heavy state with the long-lived kaon. Below, $\ell ,h$ label light/heavy to avoid that collision.

The two indistinguishable paths to the same final state must be added as probability amplitudes:Taking the squared modulus gives the time-dependent decay rate, with the common final-state phase-space normalization absorbed into the probability amplitudes:The cross term is the interference between direct decay and decay after flavour conversion. For reference, diagonal propagation gives $g_{\pm }=(e_{\ell }\pm e_h)/2$, where $e_j=e^{-i{m}_{j}t-{\Gamma }_{j}t/2}$ and $t$ is proper time. The unfactored expression remains usable if $A_f=0$, when the ratio $\lambda$ itself is undefined.

Starting with an antikaon instead givesThusThis factored form assumes $A_f,{\bar{A}}_f\ne 0$; otherwise use the preceding probability amplitude. The three mechanisms are separated below. Comparisons for distinct final states must pair $f$ with its CP conjugate $\bar{f}$.

Direct CP violation means unequal magnitudes of CP-conjugate decay probability amplitudes:It changes decay probabilities even at $t=0$, before any mixing. For a common CP eigenstate final state, this reduces to $|A_f|\ne |{\bar{A}}_f|$. Usually it requires interfering decay contributions with different weak phases and different strong phases; it is not equivalent to a nonzero weak phase in one isolated probability amplitude.

CP violation in mixing occurs whenThe flavour-transition probabilities are proportional to $|q/p{|}^2|g_{-}{|}^2$ for $K^0\to {\bar{K}}^0$, and $|p/q{|}^2|g_{-}{|}^2$ for the reverse. Their inequality remains when decay probability amplitudes themselves conserve CP. The coefficients multiplying the mixed-decay terms in the rate equations encode this asymmetry.

Interference CP violation concerns the relative phase between direct decay and decay after mixing. For a common CP eigenstate final state, it can occur with both $|q/p|=1$ and $|{\bar{A}}_f/A_f|=1$, so neither of the first two mechanisms is needed. In that limit $|\lambda |=1$, butproduces different interference terms for $\lambda$ and ${\lambda }^{-1}={\lambda }^{\ast }$. With $\Delta m=m_h-m_{\ell }$,The term proportional to $\mathrm{Im}\lambda \sin (\Delta mt)$ consequently reverses sign. In the CP-conserving limit, $\lambda$ is the real CP parity of the final state in a consistent phase convention. For non-CP-eigenstate final states, compare the full coefficients in CP-conjugate rates rather than imposing this simplified criterion.

Introduce all four physical probability amplitudes: $A_f,{\bar{A}}_f,A_{\bar{f}},{\bar{A}}_{\bar{f}}$. Under exchange of the flavour and final-state labels, defineThis CP-conjugate decay-amplitude ratio gives the conjugate-channel ratesCP is a unitary discrete transformation, so exchanging state labels does not mean complex-conjugating every strong phase by fiat. Nor should one assume that CP-conjugate probability amplitudes are equal while studying CP violation.

The equality $\bar{\lambda }={\lambda }^{-1}$ holds whenin particular when $\bar{f}$ and $f$ are the same CP eigenstate ray, with a consistent state phase. It is not automatic for two distinct CP-conjugate final states. For example, $p=q$, $A_f={\bar{A}}_{\bar{f}}=2$ and ${\bar{A}}_f=A_{\bar{f}}=1$ conserve the paired decay probability amplitudes but give $\lambda =\bar{\lambda }=1/2$, not inverse values.

Assume the usual semileptonic flavour selection rule $\Delta S=\Delta Q$, negligible wrong-sign decay probability amplitudes, and equal magnitudes for CP-conjugate allowed probability amplitudes. Then $f_{+}={\pi }^{-}{\ell }^{+}\nu$ is reached directly only from $K^0$, and $f_{-}={\pi }^{+}{\ell }^{-}\bar{\nu }$ only from ${\bar{K}}^0$. Put $|A_{f_{+}}|=|{\bar{A}}_{f_{-}}|=|A|$. The four rates areThere is only one path to each tagged semileptonic channel, so no direct/mixed decay interference survives. The semileptonic wrong-sign mixing asymmetry isIt isolates CP violation in mixing under the stated assumptions. Allowing direct semileptonic CP violation or wrong-sign probability amplitudes would spoil that isolation.^[ref]