Insights from the Interplay of K -> pi nu anti-nu and epsilon_K on the New Physics Flavour Structure

In certain new physics (NP) models, such as the Littlest Higgs model with T-parity, a strict correlation between the K_L -> pi0 nu anti-nu and K+ -> pi+ nu anti-nu branching ratios has been observed, allowing essentially only for two branches of possible points, while in other NP frameworks, such as the general MSSM or warped extra dimensional models, no visible correlation appears. We analyse the origin of the correlation in question and show it to be a direct consequence of the stringent experimental constraint on epsilon_K, provided that the NP enters with comparable strength and a universal weak phase in both Delta S = 2 and Delta S = 1 transitions. This happens in many NP scenarios with either only SM operators, or where the NP induces exclusively right-handed currents while the left-right Delta S = 2 operators are absent. On the other hand, if the NP phases in Delta S = 2 and Delta S = 1 processes are uncorrelated, epsilon_K has no power to put constraints on the K -> pi nu anti-nu system. The latter appears in particular in those NP models where K^0 - anti-K^0 mixing receives contributions from the chirally enhanced left-right operators. We discuss the stability of the correlation in question against small deviations from the assumption of universal Delta S = 2 and Delta S = 1 weak phases, and in the presence of non-negligible NP contributions to epsilon_K.