Charm rescattering in $B^0\to K^0\bar{\ell}\ell$: an improved analysis

TL;DR

This work extends charm-rescattering analyses in $B^0\to K^0\bar{\ell}\ell$ by incorporating electromagnetic dipole couplings to intermediate charm mesons and refining form factors via HHChPT and resonance-based models. The authors compute dipole-induced long-distance amplitudes for the dominant $D D^*_s$ and $D^* D_s$ topologies, carefully renormalize the ultraviolet behavior, and quantify how these non-local effects interfere with the short-distance $C_9$ amplitude. A multiplicity factor is introduced to account for additional intermediate states, and the vector form factor is reanalyzed with a generalized tower of resonances to better capture the $q^2$-dependence. The combined results show that, while dipole contributions can be comparable to monopole ones, achieving a sizable (up to ~20%) effect requires strong tuning and induces a pronounced $q^2$-dependence, whereas a natural scenario yields only a few percent impact, particularly at high $q^2$, implying that non-local charm effects alone are unlikely to fully mimic a large, $q^2$-independent shift in $C_9$.

Abstract

We improve upon previous explicit estimates of charm rescattering contributions to the decay $B^0\to K^0\bar{\ell}\ell$ by including contributions from dipole interactions with the intermediate charm-meson states, and by further investigating the structure of the electromagnetic form factors. Using a model of fundamental meson fields inspired by heavy-hadron chiral perturbation theory, augmented by form factors motivated by theoretical considerations as well as experimental data, we provide a thorough investigation of rescattering contributions induced by intermediate $D^{(*)}D^{(*)}_s$ states.

Figures (3)

• Figure 1: One-loop topologies considered in our analysis. Solid single lines denote charmed pseudoscalars ($D$ or $D_s$) and solid double lines denote charmed vectors ($D^*$ or $D^*_s$). The last three topologies result in vanishing contributions in the fully symmetric limit (see text).
• Figure 2: Ratio of the real and imaginary parts of the dipole matrix element over the absolute value of the short-distance matrix element in the low-$q^2$ (top) and high-$q^2$ (bottom) regions. The dashed-dotted and the dashed lines correspond to values of $\mu=1, 4$ GeV for the renormalization scale, respectively.
• Figure 3: Combined results of this work and that of Ref. Isidori:2024lng plotted as $|\delta C_9|/|C_9^{\text{eff}}|$, where $\delta C_9$ is the contribution from charm rescattering triangle diagrams. The dark gray bands give the "natural" results, light gray bands give the partially tuned results, i.e. alignment of all considered $c\bar{c} d\bar{s}$ states, and the dashed lines give the fully tuned results, where we assume maximal alignment between all considered intermediate states as well as maximal interference between short-distance and rescattering effects. See text for further details.