Neutral meson mixing

Transitions between a neutral meson carrying flavour and its antiparticle. Decays are described on this two-state subspace by an effective Hamiltonian operator $M-i\Gamma /2$ with Hermitian matrices $M,\Gamma$.

If a wrong-sign amplitude is $r{e}^{-i\delta }{g}_{+}+g_{-}$, the same final state's right-sign amplitude is $g_{+}+r{e}^{-i\delta }{g}_{-}$. Their ratio is not exactly obtained by replacing the right-sign rate with a pure exponential. For series $N=n_0+n_{1}t+n_2{t}^2$ and $D=1+d_{1}t+d_2{t}^2$, division gives $N/D=n_0+(n_1-n_0{d}_1)t+[n_2-n_1{d}_1+n_0(d_1^2-d_2)]t^2+O(t^3)$.

For ${\lambda }_f=(q/p){\bar{A}}_f/A_f$, exchanging both initial flavour and final-state labels defines ${\bar{\lambda }}_{\bar{f}}=(p/q)A_{\bar{f}}/{\bar{A}}_{\bar{f}}$. It equals ${\lambda }_f^{-1}$ exactly when $A_{\bar{f}}/{\bar{A}}_{\bar{f}}=A_f/{\bar{A}}_f$. This holds for a common CP eigenstate final ray. CP symmetry of paired amplitudes alone does not imply it for distinct final states: $p=q$, $A_f={\bar{A}}_{\bar{f}}=2$, ${\bar{A}}_f=A_{\bar{f}}=1$ gives both ratios $1/2$.

For mass eigenstates $p|M⟩\pm q|\bar{M}⟩$, unequal forward and reverse flavour transition probabilities occur when $|q/p|\ne 1$. They are proportional to $|q/p{|}^2|g_{-}{|}^2$ and $|p/q{|}^2|g_{-}{|}^2$. This diagnostic is invariant under rephasing the two flavour eigenstates.

If semileptonic decay tags the decaying flavour, wrong-sign decays require neutral meson mixing. With no direct CP violation and no wrong-sign direct decay amplitudes, their rate asymmetry is $(|p/q{|}^2-|q/p{|}^2)/(|p/q{|}^2+|q/p{|}^2)=(1-|q/p{|}^4)/(1+|q/p{|}^4)$. The common time dependence cancels; contaminating direct amplitudes invalidate that isolation.

Neutral B meson mixing converts the flavor eigenstates $B^0=\bar{b}d$ and ${\bar{B}}^0=b\bar{d}$ into one another through the weak interaction. The propagating mass eigenstates are $|B_L⟩=p|B^0⟩+q|{\bar{B}}^0⟩$ and $|B_H⟩=p|B^0⟩-q|{\bar{B}}^0⟩$, with $|p{|}^2+|q{|}^2=1$.

In units $\hslash =c=1$, neglect their width difference and take $|q/p|=1$. Evolution of an initial $B^0$ at proper time $t$ gives the undecayed flavour probabilitieswhere $\Delta m=m_H-m_L$ and $\Gamma$ is the common decay width. Their sum is the survival probability, not one.

For a common final state, mixing and direct decay interfere through ${\lambda }_f=(q/p)({\bar{A}}_f/A_f)$. In the Standard Model, $f=J/\psi K_S$ combines a J/psi meson with a short-lived neutral kaon. Neglecting subleading weak amplitudes and small kaon CP violation,Here the $V_{ij}$ are CKM matrix elements; the minus sign reflects the approximate negative CP parity of this specified final state. A different charmonium-kaon final state need not have that same sign.^[ref]

A decaying particle-antiparticle pair is described by a two-dimensional effective Hamiltonian for neutral-meson mixing with dispersive matrix $M$ and decay matrix $\Gamma$, both Hermitian matrices. The effective Hamiltonian for neutral-meson mixing itself is generally not a Hermitian matrix. CPT symmetry equates its diagonal entries, by separately equating diagonal masses and decay widths. For $H_{\mathrm{eff}}=\left(\begin{array}{cc}a& b\\ c& a\end{array}\right)$ the eigenvalues are $a\pm \sqrt{bc}$ and generic eigenvectors are $(\sqrt{b},\pm \sqrt{c}{)}^T$, with correlated square-root branches. Complex eigenvalues give masses and decay widths through $\mu =m-i\Gamma /2$.

CPT symmetry equates the diagonal entries of the decay-effective Hamiltonian operator. It does not make this effective matrix Hermitian operator; its imaginary diagonal part describes a total decay width.

Neutral D-meson mixing is the time evolution between the flavor states $D^0=c\bar{u}$ and ${\bar{D}}^0=\bar{c}u$. The states with definite propagation masses and decay widths are generally superpositions of these flavor states, so an initially tagged flavor need not remain unchanged before decay.

A measurement tags the production flavor and compares time-dependent rates into selected final states. Some final states can be reached both directly and after mixing, producing interference; a wrong-flavor candidate is therefore not automatically a pure mixing event. The effective two-state Hamiltonian includes both dispersive and absorptive parts. Separating mixing, direct suppressed decays and detector charge asymmetries is part of the inference.^[ref]