Re-study on the contribution of scalar potential and spectra of $c\bar c$, $b\bar b$ and $b\bar c(\bar b c)$ families

TL;DR

This paper investigates the impact of scalar potentials on the spectra of heavy quarkonium and $bar c$ families in QCD, highlighting differences from QED and fitting parameters to experimental data.

Contribution

It provides a phenomenological analysis of scalar potential contributions to heavy quarkonium spectra, incorporating both vector and scalar potentials and testing against experimental data.

Findings

Scalar potential effects are significant in QCD but negligible in QED.

Fitted parameters successfully reproduce known spectra of charmonia and bottomonia.

Predictions for $bar c$ states are consistent with existing measurements.

Abstract

We indicated in our previous work that for QED the role of the scalar potential which appears at the loop level is much smaller than that of the vector potential and in fact negligible. But the situation is different for QCD, one reason is that the loop effects are more significant because $\alpha_s$ is much larger than $\alpha$, and secondly the non-perturbative QCD effects may induce a sizable scalar potential. In this work, we phenomenologically study the contribution of the scalar potential to the spectra of charmonia, bottomonia and $b\bar c(\bar b c)$ family. Taking into account both vector and scalar potentials, by fitting the well measured charmonia and bottomonia spectra, we re-fix the relevant parameters and test them by calculating other states of not only the charmonia, bottomonia, but also further the $b\bar c$ family. We also consider the Lamb shift of the spectra.