K → π ν ν decay in the Standard Model In the Standard Model, the master formula for Br(K⁺→ π⁺ ν ν) and BR(K_L→ π⁰ ν ν) correspond to the conventional separation of long- and short-distance parts
where λ_q=V^*_qsV_qd, X_t=1.464 ± 0.041, P_u,c=P_c + δP_u,c=0.41±0.04 for m_c=1.30±0.05, I_ν^+,0 is the phase space integral.
Experimental data on K_l3 branching ratios and form-factor slopes permit to estimate the ϰ coefficients very precisely
Using these values and CKM couplings from CKMfitter and UTfIt collaborations, we get in the SM (see complete note)
The error on CKM matrix elements largely dominates, while those from LD coefficients are negligible. The present experimental situation from E787 - E949 and E391a is the following

These values are expected to be improved by the planned experiments at CERN and JPARC (KEK E391a).

K → π ν ν decay beyond the Standard Model Download Picture in eps format
Some reference of recent analyses about rare K decays beyond the SM; MFV, C-MFV, MSSM (1, 2), LHT (1, 2), 3-3-1 Z´Model, Four-Generation Model (1, 2)

June 2010 by F. Mescia & C. Smith