An analytic formula and an upper bound for ε′ ε in the Standard Model

Using the idea of the penguin box expansion we find an analytic expression for ε′ ε in the Standard Model as a function of m_t, m_s (m_c) and two non-perturbative parameters B₆^{( 1 2)} and B₈^{( 3 2)}. This formula includes next-to-leading QCD/QED short distance effects calculated recently by means of the operator product expansion and renormalization group techniques. We also derive an analytic expression for the upper bound on ε′ ε as a function of |V_cb|, |V_ub/V_cb|, B_K and other relevant parameters. Numerical examples of the bound are given.