Construction of $a_4$ family

The COMPASS Collaboration recently reported the observation of a new resonance, $a_4(2610)$, which has sparked our interest in studying the $a_4$ family with {$I^GJ^{PC}=1^{-}4^{++}$}. In this work, we investigate the mass spectra and Okubo-Zweig-Iizuka-allowed two-body strong decays of the $a_4$ family using the modified Godfrey-Isgur quark model and the quark-pair creation model. We also explore the possibility of identifying $a_4(2610)$ as a $4F$ or $2H$ state, {and our numerical results suggest that it could be a promising candidate for the $a_4(2H)$ state. In addition, we predict the masses and the widths of the $a_4(1H)$ and $a_4(3F)$ states.}