Neutral kaon mixing ($K^0\leftrightarrow {\bar{K}}^0$)

The neutral flavour states $d\bar{s}$ and $s\bar{d}$ can mix through second-order weak interactions. Box diagrams with W bosons and internal up-type quarks induce a strangeness-changing-by-two four-quark operator. The GIM mechanism removes its flavour-independent part.

The short-lived neutral kaon $K_S$ is one of the two propagating eigenstates of the effective neutral kaon mixing problem. It is a coherent combination $p|K^0⟩+q|{\overline{K}}^0⟩$, with coefficients depending on the phase convention and normalized so $|p{|}^2+|q{|}^2=1$.

In the limit of exact CP symmetry, it is approximately the CP-even combination and readily decays to two pions; the corresponding long-lived state has different allowed decay channels. Small CP violation makes the exact identification with a CP eigenstate imperfect. The observation of two-pion decays in the nominally long-lived beam was therefore evidence for CP violation, not a redefinition of the short-lived state.^[ref]

In the exchange-positive CP convention for neutral kaon mixing, a coherent source with decay amplitudes $A$ and $B$ for $K^0$ and ${\bar{K}}^0$ has projected amplitudes $(A+B)/\sqrt{2}$ and $(A-B)/\sqrt{2}$. For equal phase space, their decay-rate asymmetry is $\mathcal{R}=2\mathrm{Re}r/(1+|r{|}^2)$, $r=B/A$. Expanding the squared moduli proves the identity. Knowledge of $|r|$ alone cannot fix the asymmetry: $r=i|r|$ gives zero, whereas $r=|r|$ gives constructive interference in the symmetric channel.

The CP eigenstates of neutral kaons depend on a flavor-state phase convention. Using $CP|K^0⟩=-|{\bar{K}}^0⟩$, set $|K_1⟩=(|K^0⟩-|{\bar{K}}^0⟩)/\sqrt{2}$ and $|K_2⟩=(|K^0⟩+|{\bar{K}}^0⟩)/\sqrt{2}$. They are respectively CP even and odd. Reversing the flavor-exchange sign reverses which written sum is even; it does not change the CP eigenvalues. With CP violation, the physical short- and long-lived states need not equal these CP eigenstates; kaon CP mixing parameter and neutral-kaon CP impurity and two-pion decay describe the mixtures.

In the flavor convention $K_1=(K^0-{\bar{K}}^0)/\sqrt{2}$ and $K_2=(K^0+{\bar{K}}^0)/\sqrt{2}$, the parameter $ϵ=(a-b)/(a+b)$ writes normalized mass states as $(K_1+ϵK_2)/\sqrt{1+|ϵ{|}^2}$ and $(K_2+ϵK_1)/\sqrt{1+|ϵ{|}^2}$. In the Hermitian mass-only approximation with correlated roots, $ϵ=i\tan (\phi /2)$ and the states are orthogonal. For the full non-Hermitian decay matrix the real part need not vanish. The parameter depends on the flavor-state phase convention; observable CP violation involves convention-invariant combinations with decay amplitudes.

In the phase convention $CP|K^0⟩=|{\bar{K}}^0⟩$, the symmetric flavor combination is CP even and the antisymmetric one CP odd. For an equal-diagonal effective matrix with entries b,c off diagonal, choose correlated roots r,s with $r^2=b$, $s^2=c$ and eigenvalues $a\pm rs$. Its eigenvectors are $(r,\pm s{)}^T$. Writing $ϵ=(r-s)/(r+s)$ gives the displayed CP-basis mixtures, up to an overall phase. Their deviation is small only if $|ϵ|\ll 1$; noninvariance alone does not guarantee that. This convention must not be mixed with one in which flavor exchange carries a minus sign.

With CPT symmetry, a Hermitian kaon mass matrix is $M=\left(\begin{array}{cc}{r}_0& z\\ z^{\ast }& r_0\end{array}\right)$. Its eigenvalues are $r_0\pm |z|$. For correlated roots $a^2=z$, $b^2=z^{\ast }$ and $ab=|z|$, the normalized lower and upper eigenvectors are $(a,-b)/\sqrt{2|z|}$ and $(a,b)/\sqrt{2|z|}$. They are orthogonal. This is a mass-only result; including decay requires the generally non-Hermitian effective matrix $M-i\Gamma /2$.

In a Hermitian neutral-kaon mass matrix, independently choosing square-root signs of $z$ and $z^{\ast }$ can interchange the eigenvalue labels. Choose $z=r{e}^{i\phi }$, $a=\sqrt{r}{e}^{i\phi /2}$ and $b=\sqrt{r}{e}^{-i\phi /2}$, so $ab=r>0$, continuously from the desired CP-conserving limit. Then $(a,-b)$ is the lower-mass eigenvector. At a degenerate off-diagonal zero the mass eigenbasis is not fixed by diagonalization alone.

A Standard Model charged-current box diagram converts $d\bar{s}$ into $s\bar{d}$. Internal quarks run over $u,c,t$, and the flavor sum contains ${\lambda }_i{\lambda }_j$ with ${\lambda }_i=V_{is}^{\ast }{V}_{id}$. The loop function depends on internal masses; CKM matrix unitarity gives ${\sum }_i{\lambda }_i=0$, producing the GIM mechanism cancellation of flavor-independent terms. The amplitude is of order $g^4$ and second order in the weak interaction.

For relativistically normalized kaon states and $Q=(\bar{s}{\gamma }_{\mu }(1-{\gamma }^5)d)(\bar{s}{\gamma }^{\mu }(1-{\gamma }^5)d)$, define $⟨{\bar{K}}^0|Q|K^0⟩=(8/3)f_K^2{m}_K^2{B}_K$. The dimensionless kaon bag parameter measures strong interaction corrections to the vacuum-insertion normalization. The coefficient and matrix element must be defined in the same renormalization scheme and at the same scale. Dividing by $2m_K$ gives the corresponding two-state mixing-Hamiltonian normalization.