# Determining the effective Wilson coefficient $a_2$ in terms of $BR(B_s\to \eta_c\phi)$ and evaluating $BR(B_s\to \eta_cf_0(980))$

## TL;DR

This paper investigates $B_s$ meson decays into $ o \,eta_c$ and other states, extracting the Wilson coefficient $a_2$ using a light-front quark model, and concludes that $f_0(980)$ is likely a tetraquark or molecular state based on experimental data.

## Contribution

The study provides a theoretical calculation of hadronic matrix elements and extracts the Wilson coefficient $a_2$, offering insights into the quark structure of $f_0(980)$ and its implications for $B_s$ decay predictions.

## Key findings

- Theoretical predictions do not match experimental data regardless of mixing adjustments.
- Recent measurements suggest $f_0(980)$ is predominantly a four-quark state.
- The Wilson coefficient $a_2$ is extracted within the light-front quark model framework.

## Abstract

In this work, we investigate decays of $B_s\to \eta_c\phi$ and $B_s\to \eta_cf_0(980)$ in a theoretical framework. The calculation is based on the postulation that $f_0(980)$ and $f_0(500)$ are mixtures of pure quark states ${1\over\sqrt 2} (\bar uu+\bar dd)$ and $\bar ss$. The hadronic matrix elements for $B_s\to \phi$ and $B_s\to f_0(980)$ are calculated in the light-front quark model and the important Wilson coefficient $a_2$ which is closely related to non-perturbative QCD is extracted. However, our numerical results indicate that no matter how to adjust the mixing parameter to reconcile contributions of $f_0(980)$ and $f_0(500)$, one cannot make the theoretical prediction on $B_s\to \eta_c+\pi^+\pi^-$ to meet the data. Moreover, the new measurement of $BR(B_s\to J/\psi+f_0(500))<1.7\times 10^{-6}$ also negates the mixture scenario. Thus, we conclude that the recent data suggest that $f_0(980)$ is a four quark state ( tetraquark or $K\bar K$ molecule ), at least the fraction of its pure quark constituents is small.