A possible formation scenario of the Gaia ID 3425577610762832384: inner binary merger inside a triple common envelope

Recently, an identified non-interacting black hole (BH) binary, Gaia ID 3425577610762832384 (hereafter G3425), contains a BH ($\sim$3.6 M$_{\odot}$) falling within the mass gap and has a nearly circular orbit, challenging the classical binary evolution and supernova theory. Here, we propose that G3425 originates from a triple through a triple common envelope (TCE) evolution. The G3425 progenitor originally may consist of three stars with masses of 1.49 M$_{\odot}$, 1.05 M$_{\odot}$, and 21.81 M$_{\odot}$, and inner and outer orbital periods of 4.22 days and 1961.78 days, respectively. As evolution proceeds, the tertiary fills its Roche lobe, leading to a TCE. We find that the orbital energy generated by the inspiral of the inner binary serves as an additional energy imparted for ejecting the common envelope (CE), accounting for $\sim$97\% of the binding energy in our calculations. This means that the outer orbit needs to expend only a small amount of the orbital energy to successfully eject CE. The outcome of the TCE is a binary consisting of a 2.54 M$_\odot$ merger produced by the inner binary merger and a 7.67 M$_\odot$ helium star whose CE successfully ejected, with an orbital period of 547.53 days. The resulting post-TCE binary (PTB) has an orbital period that is 1-2 orders of magnitude greater than the orbital period of a successfully ejected classical binary CE. In subsequent simulations, we find that the successfully ejected helium star has a 44.2\% probability of forming a BH. In the case of a non-complete fallback forming a BH, with an ejected mass of 2.6 M$_{\odot}$ and a relatively low natal kick ($11^{+16}_{-5}$ ${\rm km/s}$ to $49^{+39}_{-39}$ ${\rm km/s}$), this PTB can form G3425 in the Milky Way.