CP violation in mixing and decay

Consider two decay contributions with real positive magnitudes a one and a two. After removing a common phase, write $A=a_1+a_2e^{i(\delta_s+\phi_w)}$ and its CP-conjugate amplitude $\bar A=a_1+a_2e^{i(\delta_s-\phi_w)}$. The strong rescattering phase difference delta s is unchanged by CP, while the weak coupling phase difference phi w changes sign. This representation assumes the strong dynamics conserve CP. Complex conjugating the entire amplitude would incorrectly reverse the strong phase as well as the weak phase.

Squaring the amplitudes gives a difference proportional to the sine of each phase difference. Direct rate CP violation therefore requires interfering contributions with distinct weak phases and nontrivial relative strong phases. Either phase difference can eliminate this rate asymmetry when its sine vanishes. Equal magnitudes in a particular channel do not prove all CP observables vanish: another channel or an interference measurement can remain sensitive. Also, instrumental charge asymmetries and unequal production rates must be controlled before a measured counting imbalance is interpreted as a decay asymmetry.

CP violation in mixing is diagnosed by $|q/p|\ne1$, which makes the two opposite flavor-conversion probabilities differ under the usual CPT assumptions. For a CP eigenstate f accessible from both neutral flavors, define $A_f=\langle f|H|M\rangle$ and $\bar A_f=\langle f|H|\bar M\rangle$. Their interference depends on $\lambda_f=(q/p)(\bar A_f/A_f)$. Under rephasing of the flavor basis, the change in q over p cancels the change in the amplitude ratio. Its phase consequently has a physical meaning unavailable to either factor separately.

An imaginary part of lambda can produce a CP asymmetry through interference even when its magnitude is one and mixing itself conserves CP. For a CP eigenstate dominated by one weak decay phase, the eigenvalue and mixing phase are included in the full lambda rather than added later by memory. If several weak decay amplitudes matter, extracting a fundamental phase requires their hadronic ratios. Calling every measured sine coefficient simply a weak angle would hide those channel-dependent assumptions. The invariant is a compact bookkeeping device, not a substitute for the amplitude analysis.

At zero proper time the sine vanishes, so the defined rate asymmetry is minus C. This agrees with comparing the unmixed initial decay amplitudes directly when the mixing magnitude is one. At a quarter oscillation, where the full mass phase is pi over two, the cosine vanishes and the asymmetry equals S. These checks catch reversed numerators and inconsistent propagation signs. The coefficient bound $S_f^2+C_f^2\le1$ follows directly from the complex lambda definitions and provides another useful algebraic diagnostic.

Actual meson systems can require unequal-width hyperbolic terms, corrections from a nonunit mixing magnitude, time-resolution convolution, and flavor-tagging calibration. The displayed formula deliberately isolates one common controlled limit. A significant CP asymmetry establishes a symmetry failure only after those experimental and dynamical effects are accounted for; assigning it to new interactions further requires a Standard Model prediction with uncertainties. The same observed number may involve direct and interference contributions, so report the measured coefficients and conventions before attaching a microscopic explanation.