ConstraintEpsK < UTfit

Constraint from
$\varepsilon_K$

Indirect CP violation in the Kaon system is usually expressed in terms of $| \varepsilon_K |$ parameter which is the fraction of the CP violating component in the mass eigenstates and which is usually defined as:
$\varepsilon_K = \frac{e^{i\pi/4}}{\sqrt{2}\Delta M_K} \left( \Im{M_{12}}+2\xi\Re{M_{12}} \right ),$
where
$\xi=\frac{\Im{A_{\pi\pi.~I=0}}}{\Re{A_{\pi\pi.~I=0}}}.$
Top and charm quarks contribute to the expression of the mixing in K⁰-K⁰ system. The calculation of the box diagram gives
$M_{12} = \frac{G_F^2}{12\pi^2} F_K^2 B_K M_K M_W^2 \left [ \lambda_c^{*2} \eta_t S_0 (x_c) + \lambda_t^{*2} \eta_2 S_0 (x_t)+2\lambda_t^* \lambda_c^* \eta_3 S(x_c,~x_t) \right ]$ where $\lambda_i^*=V^*_{is}V_{id},$
which allows one to write
$\varepsilon_K=C_\epsilon B_K A^2 \Im{\lambda_t} \left \{ \Re{\lambda_c} \left [ \eta_1 S_0(x_c) - \eta_3 S_0 (x_c,~x_t) A^2 \lambda^4 \right ] - \Re{\lambda_t} \eta_2 S_0(x_t) \right \} e^{i \pi /4},$
where
$C_\epsilon = \frac{G_F^2 F_K^2 M_K M_W^2}{6\sqrt{2}\pi^2 \Delta M_K } = 3.84 \cdot 10^4.$
The expression actually used in the UT fit is obtained by writing $| \varepsilon_K |$ in terms of $(\bar{\rho},~\bar{\eta})$ and the other elements of CKM matrix:
$| \varepsilon_K | = C_\epsilon B_K A^2 \lambda^6 \bar{\eta} \left \{ -\eta_1 S_0(x_c) (1-\lambda^2/2) + \eta_3 S_0 (x_c,~x_t) + \eta_2 S_0(x_t) A^2 \lambda^4 (1-\bar{\rho}) \right \}.$
The experimental values we use are summarized in the Table of Inputs. The representation of this constraint in the $(\bar{\rho},~\bar{\eta})$ plane is given below.

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