Search for Majoron-emitting modes of $^{136}$Xe double beta decay with the complete EXO-200 dataset

A search for Majoron-emitting modes of the neutrinoless double-beta decay of $^{136}$Xe is performed with the full EXO-200 dataset. This dataset consists of a total $^{136}$Xe exposure of 234.1 kg$\cdot$yr, and includes data with detector upgrades that have improved the energy threshold relative to previous searches. A lower limit of T$_{1/2}^{\rm{^{136}Xe}}>$4.3$\cdot$10$^{24}$ yr at 90\% C.L. on the half-life of the spectral index $n=1$ Majoron decay was obtained, a factor of 3.6 more stringent than the previous limit from EXO-200, corresponding to a constraint on the Majoron-neutrino coupling constant of $|\langle g_{ee}^{M}\rangle|$$<(0.4$-$0.9)\cdot10^{-5}$. The lower threshold and the additional data taken resulted in a factor 8.4 improvement for the $n=7$ mode compared to the previous EXO search. This search provides the most stringent limits to-date on the Majoron-emitting decays of $^{136}$Xe with spectral indices $n=1,2,3,$ and 7.