= Solution
Adopt the phase convention $CP|K^0\rangle=-|\bar K^0\rangle$ and $CP|\bar K^0\rangle=-|K^0\rangle$. Changing both phases changes the convention for the mixing parameter, not physical predictions. The flavor contents are $K^0=d\bar s$ and $\bar K^0=s\bar d$. A pair of <charged weak currents> changes <strangeness> by two units through the following box contractions, with $u_i,u_j=u,c,t$ summed over.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-53-kaon-boxes.png]
{title=Charged-current box contractions for neutral-kaon mixing; solid arrows carry quark number and wavy lines denote W bosons}
The drawing uses <unitary gauge>: the <W boson> <gauge-boson propagator> contains its longitudinal part. In a gauge of a <renormalizable quantum field theory>, charged <Goldstone boson> boxes must also be included to obtain the same gauge-independent result. The external lines represent the valence <quarks>; the <meson> <matrix element> also contains <strong interaction> binding effects.
Put $\lambda_i=V_{is}^*V_{id}$, $x_i=m_i^2/m_W^2$ and $P_L=(1-\gamma^5)/2$. The <charged weak box contribution to kaon mixing> has the effective <four-fermion interaction> structure
$$
H_{\Delta S=2}=C(\bar s\gamma_\mu P_Ld)(\bar s\gamma^\mu P_Ld)+\text{h.c.},\qquad
C\sim\frac{G_F^2m_W^2}{16\pi^2}\sum_{i,j}\lambda_i\lambda_j S(x_i,x_j).
$$
Here $G_F$ is the <Fermi constant>, $V$ is the <CKM matrix>, and $S$ is a dimensionless loop function from a <Feynman integral>; the estimate suppresses convention-dependent numerical factors. Four <weak interaction> <Feynman vertices> and a one-loop <Feynman integral> give the coupling and <loop order> suppression. <Unitarity> of the <CKM matrix> gives $\sum_i\lambda_i=0$, so the <GIM mechanism> cancels the <mass>-independent part. With a negligible <up quark> <mass>, the leading <charm quark> term has $S(x_c,x_c)\sim x_c$. Define $f_K$ by $\langle0|\bar s\gamma_\mu\gamma^5d|K^0(p)\rangle=if_Kp_\mu$. Estimating the hadronic <matrix element> by $f_K^2m_K B_K$, with the dimensionless <kaon bag parameter> $B_K$ of order unity, gives
$$
\boxed{|H'_{21}|_{\rm charm}\sim
\frac{G_F^2m_c^2}{16\pi^2}|V_{cs}^*V_{cd}|^2 f_K^2m_K B_K}.
$$
The hadronic normalization includes division by $2m_K$ when relativistically normalized states are converted to the <effective Hamiltonian for neutral-meson mixing>. Since $|V_{cs}^*V_{cd}|\simeq\sin\theta_C\cos\theta_C$, the <Cabibbo suppression> is explicit. Illustrative scales $G_F\sim10^{-5}\,\mathrm{GeV}^{-2}$, $m_c\sim1\,\mathrm{GeV}$, $f_K\sim0.1\,\mathrm{GeV}$ and $m_K\sim0.5\,\mathrm{GeV}$ give an order $10^{-16}$ to $10^{-15}\,\mathrm{GeV}$ estimate. The full sum includes <charm quark>-<top quark> and <top quark>-<top quark> terms, important for <CP violation>; long-distance <strong interaction> contributions prevent this dimensional estimate from being a precision prediction. A <meson> <matrix element> necessarily needs hadronic input in addition to electroweak parameters.
For the decaying <kaon> system write $H'=M-i\Gamma/2$, where $M$ and $\Gamma$ are <Hermitian matrices>. The entries denoted $M_{ij}$ in the question are entries of this <effective Hamiltonian for neutral-meson mixing>; they need not themselves form a <Hermitian matrix>. <CPT> invariance equates the diagonal <masses> and diagonal <decay widths>, giving
$$
\boxed{H'_{11}=H'_{22}}.
$$
One can see why there is no <complex conjugation> on the right as follows. In the flavor subspace let $J=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and represent the <antiunitary> <CPT> operation by $J$ followed by <complex conjugation>. <CPT> changes a projected forward propagator into its adjoint. Indeed, if $P$ projects onto the flavor subspace and $H_{\rm full}$ is the Hermitian microscopic generator, its invariance and the <antiunitary> reversal of $i$ give $\Theta P e^{-iH_{\rm full}t}P\Theta^{-1}=P e^{+iH_{\rm full}t}P=(P e^{-iH_{\rm full}t}P)^\dagger$. In the time-independent two-state approximation, the effective generator this means $J{H'}^*J={H'}^\dagger$, whose diagonal entries give $(H'_{22})^*=(H'_{11})^*$; the off-diagonal entries impose no further condition. Equivalently, apply <CPT> separately to the dispersive and absorptive <Hermitian matrices>. Treating the decaying <effective Hamiltonian for neutral-meson mixing> as a <Hermitian matrix> would incorrectly discard the absorptive part.
In the chosen flavor phase convention <CP> is represented by $-J$. Its invariance requires $JH'J=H'$. Thus it gives diagonal equality and additionally
$$
\boxed{H'_{12}=H'_{21}}.
$$
The normalized <CP eigenstates> are
$$
\boxed{|K_1^0\rangle=\frac{|K^0\rangle-|\bar K^0\rangle}{\sqrt2},\qquad
|K_2^0\rangle=\frac{|K^0\rangle+|\bar K^0\rangle}{\sqrt2}}.
$$
Applying <CP> directly gives <eigenvalues> $+1$ and $-1$ respectively.
Now use <CPT> to put $H'=\begin{pmatrix}a&b\\c&a\end{pmatrix}$ and take correlated <square roots> $\alpha=\sqrt b$, $\beta=\sqrt c$. Its <characteristic polynomial> equation is $(a-\mu)^2-bc=0$, and direct multiplication shows
$$
H'\binom{\alpha}{-\beta}=(a-\alpha\beta)\binom{\alpha}{-\beta},\qquad
H'\binom{\alpha}{\beta}=(a+\alpha\beta)\binom{\alpha}{\beta}.
$$
Expressing these two <eigenvectors> in the <CP eigenstate> basis gives, respectively, coefficients proportional to $(\alpha+\beta,\alpha-\beta)$ and $(\alpha-\beta,\alpha+\beta)$. Therefore the <kaon CP mixing parameter> is
$$
\epsilon=\frac{\alpha-\beta}{\alpha+\beta}
=\frac{\sqrt{M_{12}}-\sqrt{M_{21}}}{\sqrt{M_{12}}+\sqrt{M_{21}}},\qquad
\frac qp=\frac\beta\alpha=\frac{1-\epsilon}{1+\epsilon},
$$
and, because the <CP eigenstate> basis is <orthonormal>, the normalized states at production are
$$
\boxed{|K_S^0\rangle=\frac{|K_1^0\rangle+\epsilon|K_2^0\rangle}{\sqrt{1+|\epsilon|^2}},\qquad
|K_L^0\rangle=\frac{|K_2^0\rangle+\epsilon|K_1^0\rangle}{\sqrt{1+|\epsilon|^2}}}.
$$
The <kaon mixing square-root branch convention> correlates the roots continuously with the limit of conserved <CP> $b=c$, where $\alpha=\beta$ and $\epsilon=0$. Reversing one root interchanges the two <eigenstates>. The labels short and long are fixed by their <decay widths>: their complex <eigenvalues> are $\mu_{S,L}=m_{S,L}-i\Gamma_{S,L}/2$, and subsequent evolution supplies $e^{-i\mu_{S,L}t}$. The displayed formula assumes nonzero mixing and $\alpha+\beta\ne0$, as in the physical <kaon> system; degenerate or defective matrices require a separate limiting treatment.
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