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Using the Ly$\alpha$ mass assignment scheme (LyMAS), we make theoreticalpredictions for the 3-dimensional 3-point correlation function (3PCF) of theLy$\alpha$ 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}$$h^3$ Mpc$^{-3}$ placed either in massive halos or randomly. On scales of$10-30$ $h^{-1} \mbox{ Mpc}$, the flux 3PCF displays hierarchical scaling withthe square of the 2PCF, but with an unusual value of $Q \equiv\zeta_{123}/(\xi_{12} \xi_{13} + \xi_{12} \xi_{23} + \xi_{13} \xi_{23}) \approx-4.5$ that reflects the low bias of the Ly$\alpha$ forest and theanti-correlation between mass density and transmitted flux. For halo-basedquasars and an ionizing photon mean free path of $\lambda = 300$ $h^{-1} \mbox{Mpc}$ comoving, UVB fluctuations moderately depress the 2PCF and 3PCF, withcancelling effects on $Q$. For $\lambda = 100$ $h^{-1} \mbox{ Mpc}$ or 50$h^{-1} \mbox{ Mpc}$, UVB fluctuations substantially boost the 2PCF and 3PCF onlarge scales, shifting the hierarchical ratio to $Q \approx -3$. We scale oursimulation results to derive rough estimate of the 3PCF detectability inobservational data sets for the redshift range $z=2.1 - 2.6$. At $r = 10$$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 $r = 40$ $h^{-1} \mbox{ Mpc}$ thepredicted SNR is lower by $\sim$ 3$-$5 times. Measuring the flux 3PCF would bea novel test of the conventional paradigm of the Ly$\alpha$ forest and helpseparate the contributions of UVB fluctuations and density fluctuations toLy$\alpha$ forest clustering.

UV Background Fluctuations and Three-Point Correlations in the Large Scale Clustering of the Lyman-alpha Forest