Using the Ly$alpha$ mass assignment scheme (LyMAS), we make theoretical predictions for the 3-dimensional 3-point correlation function (3PCF) of the Ly$alpha$ forest at redshift $z=2.3$. We bootstrap results from the (100 $h^{-1} mbox{ Mpc}$)$^3$ Horizon hydrodynamic simulation to a (1 $h^{-1}$ Gpc)$^3$ $N$-body simulation, considering both a uniform UV background (UVB) and a fluctuating UVB sourced by quasars with a comoving $n_q approx 10^{-5}$ $h^3$ Mpc$^{-3}$ placed either in massive halos or randomly. On scales of $10-30$ $h^{-1} mbox{ Mpc}$, the flux 3PCF displays hierarchical scaling with the square of the 2PCF, but with an unusual value of $Q equiv zeta_{123}/(xi_{12} xi_{13} + xi_{12} xi_{23} + xi_{13} xi_{23}) approx -4.5$ that reflects the low bias of the Ly$alpha$ forest and the anti-correlation between mass density and transmitted flux. For halo-based quasars and an ionizing photon mean free path of $lambda = 300$ $h^{-1} mbox{ Mpc}$ comoving, UVB fluctuations moderately depress the 2PCF and 3PCF, with cancelling effects on $Q$. For $lambda = 100$ $h^{-1} mbox{ Mpc}$ or 50 $h^{-1} mbox{ Mpc}$, UVB fluctuations substantially boost the 2PCF and 3PCF on large scales, shifting the hierarchical ratio to $Q approx -3$. We scale our simulation results to derive rough estimate of the 3PCF detectability in observational data sets for the redshift range $z=2.1 - 2.6$. At $r = 10$ $h^{-1} mbox{ Mpc}$ and 20 $h^{-1} mbox{ Mpc}$, we predict a signal-to-noise (SNR) of $sim$ 9 and $sim$ 7, respectively, for both BOSS and eBOSS, and $sim$ 37 and $sim$ 25 for DESI. At $r = 40$ $h^{-1} mbox{ Mpc}$ the predicted SNR is lower by $sim$ 3$-$5 times. Measuring the flux 3PCF would be a novel test of the conventional paradigm of the Ly$alpha$ forest and help separate the contributions of UVB fluctuations and density fluctuations to Ly$alpha$ forest clustering.
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