Neutral meson mixing and time evolution

On the meson-antimeson subspace use $H_{\mathrm{eff}}=M-i\Gamma/2$, with M and Gamma Hermitian matrices. This is the usual exponential-decay approximation after eliminating the decay products, not an assertion that fundamental quantum mechanics is nonunitary. CPT invariance makes the two diagonal entries equal. Off-diagonal dispersive and absorptive terms describe virtual conversion and shared on-shell decay channels. The eigenvalues can be written mass minus i times width over two. Natural units are used until numerical time units are introduced explicitly.

Define $|M_L\rangle=p|M\rangle+q|\bar M\rangle$ and $|M_H\rangle=p|M\rangle-q|\bar M\rangle$, normalized by $|p|^2+|q|^2=1$. Here H labels the larger mass and L the smaller, so $\Delta m=m_H-m_L>0$. The ratio q over p depends on flavor-state phases, whereas its magnitude does not. When CP is violated in mixing these propagation eigenstates need not be orthogonal in the reduced description. The general formalism handles that feature; our probability example additionally sets the ratio's magnitude to one.

Adding and subtracting the eigenstates expresses the initial flavor M as their equal-amplitude combination divided by twice p. Evolve each component and transform back. Our definition of g minus is e L minus e H over two, so the converted component carries a plus sign multiplying $(q/p)g_-$. Some references define the opposite g minus and compensate by placing a minus sign in the flavor evolution. Either convention works; mixing one convention's definition with the other's evolution changes interference signs and is incorrect.

For equal widths and $|q/p|=1$, the two flavor probabilities share an exponential survival factor. The unmixed component has cosine squared and the mixed component sine squared of half the mass phase. Their sum is the survival probability, not one. Dividing by that sum gives probabilities conditioned on the meson still being present. This distinction matters experimentally because flavor-tagged decay rates include both the oscillation and the decay density. A large converted fraction among surviving mesons need not mean a large number of surviving events.

For flavor-specific decays with equal tagging efficiencies and the same partial-width normalization, integrate the mixed decay density over time. In the equal-width, CP-conserving limit its fraction among all such decays is $\chi=x^2/[2(1+x^2)]$, where x is the mass-oscillation frequency divided by the decay rate. This follows by integrating gamma times the two probabilities; the sum integrates to one. Very slow mixing gives a small fraction, while rapid unresolved mixing tends to one half rather than complete conversion.

Mixing itself is not evidence for CP violation. A real, symmetric effective mixing interaction can already produce mass splitting and oscillation. CP tests compare particle and antiparticle behavior or their interference with decay amplitudes, which the next lesson develops. Unequal widths, detector time acceptance, imperfect initial tagging, and experimental resolution modify observable time distributions. The simple integrated fraction should therefore be used only after stating its assumptions. A laboratory decay time must also be converted to proper time when the meson is moving relativistically.