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 $⟨M_2|{\mathcal{O}}_h|M_1⟩$. These diagrams, usually overlooked under the assumption that $⟨M_2|{\mathcal{O}}_h|M_1⟩\sim ⟨M_2|J_{q_3{q}_4}|0⟩\times ⟨0|J_{q_1{q}_2}|M_1⟩\gg ⟨M_2|J_{q_3{q}_2}\times J_{q_1{q}_4}|M_1⟩$, 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.

Comments: 38 pages, 5 figures