Tidal Love numbers of multi-state Boson stars

In this paper, we calculate the tidal Love numbers of multi-state boson stars (MSBSs) composed of ground state and first excited state complex scalar fields. Under synchronized and nonsynchronized frequency conditions, the background solutions of MSBSs are classified into single-branch and double-branch types. The field functions, ADM mass, and binding energy of different solutions are discussed. We then calculate the quadrupolar ($\ell=2$) electric and magnetic tidal Love numbers for branches containing stable solutions. Our results show that the electric tidal Love numbers are initially positive and then suddenly transition to negative values. This phenomenon occurs when the parameters satisfy $\tilde{\mu}_1 > 0.891$ or $\tilde{\omega}_0 > 0.777$; for smaller values of these parameters, the electric Love numbers remain positive. The magnetic tidal Love numbers are always negative, with absolute values smaller than those of the electric tidal Love numbers.