Neutral-Meson Mixing and Mixing-Induced CP Violation

A neutral meson and its antimeson form a decaying two-state system governed by $\mathsf H=M-i\Gamma/2$, not by a Hermitian oscillation Hamiltonian. Its complex eigenvalues determine masses and widths, $q/p$ describes the flavor content of the eigenstates, and decay to a common final state depends on the rephasing-invariant combination $\lambda_f=(q/p)(\bar A_f/A_f)$. This structure separates $CP$ violation in mixing, in decay, and in their interference.

Required background. Weak effective Hamiltonians and flavor-changing processes supplies $\Delta F=1,2$ amplitudes; resonances, infraparticles, and limits of particle language supplies unstable-state pole conventions. Helpful background. Linear ODEs and evolution operators supplies matrix exponentiation.

The non-Hermitian flavor Hamiltonian

In the flavor basis $(|P^0\rangle,|\bar P^0\rangle)$, the Wigner–Weisskopf approximation gives

$$i\frac{d}{dt} \begin{pmatrix}a(t)\\ b(t)\end{pmatrix} =\mathsf H \begin{pmatrix}a(t)\\ b(t)\end{pmatrix}, \qquad \mathsf H=M-\frac{i}{2}\Gamma,$$

where $M=M^\dagger$ and $\Gamma=\Gamma^\dagger$. Assuming CPT invariance makes the diagonal entries equal:

$$\mathsf H= \begin{pmatrix} m-\frac{i}{2}\Gamma_0& M_{12}-\frac{i}{2}\Gamma_{12}\\ M_{12}^*-\frac{i}{2}\Gamma_{12}^*& m-\frac{i}{2}\Gamma_0 \end{pmatrix}.$$

$M_{12}$ is dispersive mixing and $\Gamma_{12}$ is absorptive mixing through common on-shell decay channels. The eigenvalues are

$$\lambda_\pm =m-\frac{i}{2}\Gamma_0 \pm \sqrt{ \left(M_{12}-\frac{i}{2}\Gamma_{12}\right) \left(M_{12}^*-\frac{i}{2}\Gamma_{12}^*\right) }.$$

The square-root branch and the assignment of “heavy/light” labels must be stated. Write

$$\lambda_{H,L}=m_{H,L}-\frac{i}{2}\Gamma_{H,L}, \qquad \Delta m=m_H-m_L>0, \qquad \Delta\Gamma=\Gamma_L-\Gamma_H.$$

With this width convention,

$$\left(\Delta m+\frac{i}{2}\Delta\Gamma\right)^2 =4 \left(M_{12}-\frac{i}{2}\Gamma_{12}\right) \left(M_{12}^*-\frac{i}{2}\Gamma_{12}^*\right)$$

after the compatible square-root branch is chosen. This complex identity is safer than importing separate real formulas from a source using $\Gamma_H-\Gamma_L$.

Choose normalized eigenstates

$$|P_L\rangle=p|P^0\rangle+q|\bar P^0\rangle, \qquad |P_H\rangle=p|P^0\rangle-q|\bar P^0\rangle, \qquad |p|^2+|q|^2=1.$$

Then

$$\left(\frac{q}{p}\right)^2 =\frac{M_{12}^*-\tfrac{i}{2}\Gamma_{12}^*} {M_{12}-\tfrac{i}{2}\Gamma_{12}}.$$

The square root has a sign ambiguity correlated with the eigenstate convention. No observable depends on the common sign of $p$ and $q$ or on exchanging the names of the two eigenvectors while transforming all definitions consistently. The general formalism is derived in Nir 2005, § III.B, pp. 23–25.

Time evolution and decaying probabilities

Define

$$g_+(t)=\frac12 \left(e^{-i\lambda_Lt}+e^{-i\lambda_Ht}\right), \qquad g_-(t)=\frac12 \left(e^{-i\lambda_Lt}-e^{-i\lambda_Ht}\right).$$

Inverting the eigenstate definitions gives

$$|P^0(t)\rangle =g_+(t)|P^0\rangle +\frac{q}{p}g_-(t)|\bar P^0\rangle,$$ $$|\bar P^0(t)\rangle =g_+(t)|\bar P^0\rangle +\frac{p}{q}g_-(t)|P^0\rangle.$$

These are amplitudes in the undecayed two-state subspace. Their raw squared norms decrease because probability flows into decay products.

The exact synthetic fixture used by a reproducible calculation is

$$\mathsf H= \begin{pmatrix} -i/2&1/4\\ 1/4&-i/2 \end{pmatrix} =-\frac{i}{2}\mathbf1+\frac14\sigma_x.$$

Its eigenvalues are $\lambda_\pm=\pm1/4-i/2$ and one may choose $q/p=1$. Matrix exponentiation gives

$$e^{-i\mathsf Ht} =e^{-t/2} \left[ \cos(t/4)\mathbf1-i\sin(t/4)\sigma_x \right].$$

For an initially pure $P^0$,

$$P_{\rm surv}(t)=e^{-t}\cos^2(t/4), \qquad P_{\rm trans}(t)=e^{-t}\sin^2(t/4).$$

Thus $P_{\rm surv}+P_{\rm trans}=e^{-t}$, not one. At $t=\pi$ each is $e^{-\pi}/2$. Dividing by the surviving total would define a conditional composition, but it would erase the physical decay probability and is not the raw propagation result.

Rephasing-invariant decay interference

For a final state $f$ accessible from both flavors, define

$$A_f=\langle f|\mathcal H_{\Delta F=1}|P^0\rangle, \qquad \bar A_f=\langle f|\mathcal H_{\Delta F=1}|\bar P^0\rangle,$$ $$\lambda_f=\frac{q}{p}\frac{\bar A_f}{A_f}.$$

Under the flavor-state rephasing

$$|P^0\rangle\to e^{i\xi}|P^0\rangle, \qquad |\bar P^0\rangle\to e^{-i\xi}|\bar P^0\rangle,$$

the factors transform as

$$\frac{q}{p}\to e^{2i\xi}\frac{q}{p}, \qquad \frac{\bar A_f}{A_f}\to e^{-2i\xi}\frac{\bar A_f}{A_f},$$

so $\lambda_f$ is invariant. A claim about the phase of $q/p$ alone is convention dependent.

For $\Delta\Gamma=0$ and the stated asymmetry numerator,

$$A_{CP}(t) \equiv \frac{ \Gamma(\bar P^0(t)\to f)-\Gamma(P^0(t)\to f)} {\Gamma(\bar P^0(t)\to f)+\Gamma(P^0(t)\to f)},$$

define

$$C_f=\frac{1-|\lambda_f|^2}{1+|\lambda_f|^2}, \qquad S_f=\frac{2\,\operatorname{Im}\lambda_f}{1+|\lambda_f|^2}.$$

When $|q/p|=1$ and the common normalization assumptions hold,

$$A_{CP}(t) =S_f\sin(\Delta m\,t)-C_f\cos(\Delta m\,t).$$

Reversing the numerator reverses the entire right-hand side. For the exact check $\lambda_f=i$, $\Delta m=1/2$, and $t=\pi$,

$$C_f=0,\qquad S_f=1,\qquad A_{CP}(\pi)=1.$$

For nonzero $\Delta\Gamma$, the additional invariant

$$A_f^{\Delta\Gamma} =-\frac{2\operatorname{Re}\lambda_f}{1+|\lambda_f|^2}$$

enters hyperbolic time dependences and satisfies

$$C_f^2+S_f^2+(A_f^{\Delta\Gamma})^2=1.$$

The full rate formulas and phase-convention analysis are given in Nir 2005, §§ III.C–III.D, pp. 25–27.

Three kinds of CP violation

Type	Rephasing-invariant diagnostic	Required physical ingredient	Common overclaim
in mixing	$\lvert q/p\rvert\ne1$	mismatch between dispersive and absorptive mixing phases	interpreting $\arg(q/p)$ itself as physical
in decay	unequal magnitudes for CP-conjugate decay amplitudes; for a common state, often $C_f\ne0$	at least two contributions with different weak and strong phases	calling any complex amplitude direct $CP$ violation
in interference of mixing and decay	a CP-odd phase in $\lambda_f$; at $\Delta\Gamma=0$, $S_f\ne0$ under the stated final-state assumptions	common final state reached directly and after mixing	dropping the final-state CP convention or reversing the asymmetry numerator

The categories can coexist. They are mechanisms, not mutually exclusive labels for an entire meson system.

Short-distance and long-distance inputs

The off-diagonal Hamiltonian connects directly to the preceding weak-EFT workflow. In relativistic state normalization, the local short-distance part has the schematic form

$$M_{12}^{\rm SD} =\frac{1}{2m_P} \langle\bar P^0| \mathcal H_{\rm eff}^{\Delta F=2} |P^0\rangle.$$

$\Gamma_{12}$ comes from the absorptive part of a time-ordered product of two $\Delta F=1$ Hamiltonians and sums common on-shell final states. Long-distance dispersive contributions can arise from the corresponding principal-value integral. The local $\Delta F=2$ Hamiltonian has

$$\mathcal H_{\rm eff}^{\Delta F=2} =\sum_i C_i^{\Delta F=2}(\mu)Q_i^{\Delta F=2}(\mu),$$

so its scheme and scale cancel only in

$$\sum_i C_i^{\Delta F=2}(\mu) \langle\bar P^0|Q_i^{\Delta F=2}(\mu)|P^0\rangle.$$

Particle mixing effective Hamiltonians and their QCD evolution are treated in Buchalla, Buras, and Lautenbacher 1996, §§ XII–XIII. Numerical matrix elements, current mixing averages, and detector-dependent time fits require versioned sources and are outside this stable formalism.

Input	Method source	Required qualification
$C_i^{\Delta F=2}$	electroweak matching and QCD/QED RG	operator basis, active flavors, scheme, scale, perturbative order
$\langle\bar P\rvert Q_i\lvert P\rangle$	nonperturbative QCD	same scheme/scale, state normalization, continuum and volume limits, covariance
$\Gamma_{12}$	inclusive expansion or sum over common states	duality or channel truncation, on-shell convention, power corrections
$A_f,\bar A_f$	weak Hamiltonian plus final-state dynamics	strong phases, nonlocal terms, radiative corrections
time-dependent inference	experimental likelihood	tagging, resolution, acceptance, background, covariance, numerator convention

Independent checks and limitations

• Hermiticity inputs: $M$ and $\Gamma$ are Hermitian even though $\mathsf H$ is not. Conjugate both $M_{12}$ and $\Gamma_{12}$ in the lower-left entry.
• Width sign: physical eigenvalues have negative imaginary parts, $\lambda=m-i\Gamma/2$ with $\Gamma>0$. Replacing $-i\Gamma/2$ by $+i\Gamma/2$ produces growing states.
• Flavor rephasing: change $\xi$ and verify eigenvalues, $|q/p|$, $\lambda_f$, and rates remain fixed.
• No-mixing limit: setting $M_{12}=\Gamma_{12}=0$ gives $g_-=0$ and pure exponential survival in each flavor.
• Stable limit: setting every width to zero makes evolution unitary and restores total probability one.
• Raw normalization: for decaying states, survival plus transition is the undecayed probability, not a conditional unit sum.
• Coefficient cancellation: vary $\mu$ only after evolving both $C_i$ and matrix elements in the same scheme.
• Domain: the two-state Wigner–Weisskopf approximation excludes regeneration in matter, open-system decoherence beyond decay, and detector response. Those require additional dynamical and experimental models.

Common pitfalls

Replacing $\mathsf H$ by a Hermitian matrix. That removes decay and absorptive mixing. The lost norm is physical probability in decay products.

Using $p/q$ where the convention defines $q/p$. Their phases and magnitudes invert. Start from the declared eigenvectors and derive the time evolution rather than memorizing a sign.

Comparing $S_f$ across opposite numerator conventions. The asymmetry definition fixes the overall sign. Translate both the numerator and the formula.

Informal self-check

Exponentiate the exact fixture $\mathsf H=-i\mathbf1/2+\sigma_x/4$ and verify its raw probability sum.

Answer

The identity and $\sigma_x$ commute, while $\sigma_x^2=\mathbf1$. Therefore $e^{-i\mathsf Ht}=e^{-t/2}[\cos(t/4)\mathbf1-i\sin(t/4)\sigma_x]$. Acting on $(1,0)^{\mathsf T}$ gives amplitudes $e^{-t/2}\cos(t/4)$ and $-ie^{-t/2}\sin(t/4)$. Their squared magnitudes sum to $e^{-t}$, the undecayed probability.

Handoffs

• Send $\lambda_f$, its final-state convention, and separated mixing/decay inputs to the quark-$CP$ and unitarity-triangle synthesis.
• Send $M_{12}$ or $\Gamma_{12}$ short-distance pieces back to the weak-Hamiltonian workflow with the matrix-element scheme and scale.
• Reproduce the exact non-Hermitian propagation and asymmetry sign checks before fitting data.
• Send current averages, likelihoods, acceptance models, and detector effects to a dated specialist evidence analysis rather than embedding them in the evergreen formalism.

References

• Buchalla, Gerhard, Andrzej J. Buras, and Markus E. Lautenbacher. “Weak Decays Beyond Leading Logarithms.” Reviews of Modern Physics 68 (1996): 1125–1244. DOI · Open PDF
• Nir, Yosef. “CP Violation in Meson Decays.” Lectures at the CERN–CLAF and Les Houches schools, 2005, §§ III.B–III.D. arXiv · Open PDF