Can the hyperfine mass splitting formula in heavy quarkonia be applied to the $B_c$ system?

TL;DR

This paper investigates whether a known hyperfine mass splitting relation for P-wave quarkonium systems applies to the B_c system, using quark model predictions to test its validity.

Contribution

It proposes an extension of a mass relation from charmonium and bottomonium to the B_c system and tests its accuracy with quark model predictions.

Findings

The mass relation holds with very small deviations for B_c.

The relation is confirmed to be accurate within quark model predictions.

Supports the universality of hyperfine mass splitting formulas across heavy quarkonia.

Abstract

The mass relation ${M_{0^{+}}+3M_{1^{+\prime}}+5M_{2^{+}}= 9M_{1^{+}}}$ miraculously holds for the $P$-wave charmonium $(c\bar{c})$ and bottomonium $(b\bar{b})$ systems with soaring precision. The origin of such relation can be addressed from Quark Models, and have been confirmed experimentally in a limited number of cases. In this connection, we propose $M_{0^{+}}+5M_{2^{+}}=3(M_{1^{+\prime}}+M_{1^{+}})$ as an extension to the $P$-wave $B_{c}$ case. In order to test its applicability, we employ a variety of Quark Model predictions for the $B_c$ mass spectrum. Our numerical analysis confirms such formula is accurate up to very small deviations.