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Epsilon_K in the Standard Model and the kaon phase conventions

The parameter epsilon_K, that quantifies CP violation in kaon mixing, is the observable setting the strongest constraints on new physics with a generic flavour and CP structure. While its experimental uncertainty is at the half percent level, the theoretical one is at the level of 15%. One of the largest sources of the latter uncertainty is the poor perturbative behaviour of the short-distance contribution of the box diagram with two charm quarks. In this proceeding, based on arXiv:1602.08494, I summarise how that contribution can be removed, from the imaginary part of the mixing amplitude, by a rephasing of the kaon fields. A first outcome is a mild reduction of the total theoretical uncertainty of epsilon_K: while this might look counterintuitive at first sight, if different "pieces" (i.e. short- and long-distance) of an observable are computed with different techniques, then it is possible to choose a phase convention where the total uncertainty of that observable is optimised. Moreover, it is worthy to discuss if and how this freedom of rephasing, which has been somehow overlooked in the past, can help in making progress in lattice QCD computations of immediate relevance for epsilon_K.