Re-Study on the wave functions of $\Upsilon(nS)$ states in LFQM and the radiative decays of $\Upsilon(nS)\to \eta_b+\gamma$

Understand

The Light-front quark model (LFQM) has been applied to calculate the transition matrix elements of heavy hadron decays.

• However, it is noted that using the traditional wave functions of the LFQM given in literature, the theoretically determined decay constants of the $\Upsilon(nS)$ obviously contradict to the data.
• It implies that the wave functions must be modified.
• Keeping the orthogonality among the $nS$ states and fitting their decay constants we obtain a series of the wave functions for $\Upsilon(nS)$.