The MAMMOTH-Grism slitless spectroscopic survey is a Hubble Space Telescope (HST) cycle-28 medium program, which is obtaining 45 orbits of WFC3/IR grism spectroscopy in the density peak regions of three massive galaxy protoclusters at $z=2-3$ discovered using the MAMMOTH technique. We introduce this survey by presenting the first measurement of mass-metallicity relation (MZR) at high redshift in overdense environments via grism spectroscopy. From the completed MAMMOTH-Grism observation in the field of the BOSS1244 protocluster at $z=2.24pm0.02$, We secure a sample of 36 protocluster member galaxies at $zsim2.24$, showing strong nebular emission lines ([O III], H$beta$ and [O II]) in their G141 spectra. Using the multi-wavelength broad-band deep imaging from HST and ground-based telescopes, we measure their stellar masses in the range of $[10^{9},10^{10.4}]M_odot$, instantaneous star formation rates (SFR) from 10 to 240$M_odot yr^{-1}$, and global gas-phase metallicities [1/3,1] of solar. Compared with similarly selected field galaxy sample at the same redshift, our galaxies show on average increased SFRs by ~0.06dex and ~0.18dex at ~10$^{10.1}M_odot$ and ~10$^{9.8}M_odot$, respectively. Using the stacked spectra of our sample galaxies, we derive the MZR in the BOSS1244 protocluster core as $12+log({rm O/H})=(0.136pm0.018)timeslog(M_ast/M_odot)+(7.082pm0.175)$, showing significantly shallower slope than that in the field. This shallow MZR slope is likely caused by the combined effects of efficient recycling of feedback-driven winds and cold-mode gas accretion in protocluster environments. The former effect helps low-mass galaxies residing in overdensities retain their metal production, whereas the latter effect dilutes the metal content of high-mass galaxies, making them more metal poor than their coeval field counterparts.
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