Leading large $N_c$ contributions to Lepton Number Violating Meson Decays

Abstract

Lepton number violating meson decays, such as $M_1^- \to M_2^+\ell_1^-\ell_2^-$, provide constraints on $d=9$ $\Delta L = 2$ operators. RGE-improved bounds on the Wilson coefficients of these operators have been presented in the literature, taking into account perturbative QCD one-loop corrections and the corresponding operator mixing. Here, we present for the first time the contribution of connected diagrams to the hadronic matrix elements $\langle M_2 | {\cal O}_h | M_1 \rangle$. These diagrams, usually overlooked under the assumption that $\langle M_2 | {\cal O}_h | M_1 \rangle \sim \langle M_2 | J_{q_3 q_4} | 0 \rangle \times \langle 0 |J_{q_1 q_2} | M_1 \rangle \gg \langle M_2 | J_{q_3 q_2} \times J_{q_1 q_4} | M_1 \rangle$, can give indeed a significant contribution to the matrix element. Including these connected diagrams is but the first step towards a full non-perturbative computation of the long-range QCD effects in these operators, that should be performed using lattice field theory techniques. However, connected diagrams represent the leading order in the $1/N_c$ expansion of the QCD non-perturbative effects and thus our work can be understood as a realistic, first approximation to a complete calculation of the long-range part of the matrix elements.