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

TL;DR

This work addresses lepton-number-violating $\Delta L=2$ meson decays mediated by dimension-9 operators, focusing on long-range QCD effects. It extends prior analyses by estimating the leading ${\cal O}(1/N_c)$ corrections to hadronic matrix elements via connected diagrams, complementing the vacuum-insertion approximation and RG running. The authors derive the full operator basis for $K^+ \to \pi^- \ell^+ \ell^+$ in both identical and different lepton cases, implement one-loop QCD running to evolve Wilson coefficients from a high scale to a hadronic scale, and compute RG-improved bounds that include connected-diagram contributions. They show that including connected diagrams can dramatically tighten some bounds (e.g., ${C_2}$ and ${C_3}$) due to operator mixing, though the overall bounds remain far from current direct limits; the work also discusses lattice QCD matching and proposes improved $B$-parameter definitions for future nonperturbative calculations.

Abstract

Lepton number violating meson decays, such as $M_1^- \to M_2^+\ell_1^-\ell_2^-$, provide constraints on $d=9$ $ΔL = 2$ 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.

Figures (5)

• Figure 1: Tree-level diagrams responsible for the $K^+ \to \pi^- \, \ell^+ \ell^+$ transition in the type-I see-saw model. Left: s-channel amplitude. Right: t-channel amplitude.
• Figure 2: Disconnected (left) and connected (right) diagrams of the hadronic matrix element for the decay $M^+_1 \to M^-_2 \ell^+ \ell^+$, with effective operator ${\cal O} = {\cal O}_h \times J_{\ell \ell}$. Left diagram is ${\cal O}(N_c^2)$ and right diagram is ${\cal O}(N_c)$, as they contain two and one fermion loops, respectively.
• Figure 3: Effective $d=9$ operator description of the short-range mechanisms (SRM) of the meson decay $M_1^{-} \to M_2^{+} \ell_1^- \ell_2^-$. The tree-level diagram can be obtained, e.g., by integrating out gauge bosons and heavy fermions in Fig. \ref{['fig:seesawdiagrams']}. On the other hand, diagrams (a)-(c) give some representative one-loop perturbative QCD corrections to the amplitude arising by the interchange of a hard gluon.
• Figure 4: Disconnected diagrams with one gluon emission between initial quark lines (top left) or one (top right), two (bottom left) or three (bottom right) gluons emission between initial and final quark lines for the decay $M^+_1 \to M^-_2 \ell^+ \ell^+$, with effective operator ${\cal O} = {\cal O}_h \times J_{\ell \ell}$. The top left diagram is ${\cal O}(N_c^2)$, as it contains three loops and two couplings, $g_s^2 \propto 1/N_c$. On the other hand, the other three diagrams are ${\cal O}(1)$, as they contains one, two and three loops and $g_s^2, g_s^4, g_s^6$ couplings, respectively.
• Figure 5: Connected diagrams with one gluon emission between initial quark lines (left) or between initial and final quark lines (right) for the decay $M^+_1 \to M^-_2 \ell^+ \ell^+$, with effective operator ${\cal O} = {\cal O}_h \times J_{\ell \ell}$. Both diagrams are ${\cal O}(N_c)$, as they contain two loops and two couplings, $g_s^2 \propto 1/N_c$.