Out of oxygen: Extremely metal-poor galaxy candidates identified at $2.5 < z < 6.5$ with deep JADES medium-band imaging

JWST is beginning to uncover a population of extremely metal-poor galaxies (EMPGs, $Z < 1\%~\mathrm{Z}_\odot$) at $z > 3$, mostly through serendipitous NIRSpec discoveries and blind slitless spectroscopy. To accelerate our understanding of pristine star formation, we further develop a methodology to identify EMPG candidates from photometry, using the extensive deep medium-band imaging from JADES. Our EMPG candidates at $2.5 < z < 6.5$ exhibit strong photometric boosts by H$\alpha$, yet correspondingly weak boosts by [O III] + H$\beta$, likely indicating extremely low metallicity to explain their lack of [O III] emission. We further demand our EMPG candidates to have strong Balmer jumps, as revealed by medium-band imaging, to ensure that they are young starbursts, as opposed to broad-line AGN/LRDs, though contamination by dusty/dense-gas starbursts and highly-obscured AGN remains a concern. SED-fitting with near-pristine models (${\sim}0.1$-$1\%~\mathrm{Z}_\odot$) indicates that our 22 EMPG candidates are low-mass (median $M_* \approx 10^{6.7}~\mathrm{M}_\odot$), faint dwarf galaxies ($M_\mathrm{UV} \approx -16.6$), with high ionizing photon production efficiencies ($\log\, (\xi_\mathrm{ion, obs}/\mathrm{(Hz\ erg^{-1})}) \approx 26.0$). Hence these are plausible sites of near-pristine star formation, comprising ${\sim}0.04$-$0.6\%$ of $2.5 < z < 6.5$ galaxies at $-19 < M_\mathrm{UV} < -16$. We discuss this extremely metal-poor extension to the mass-metallicity relation. We forecast that deep (${\sim}28$ h) NIRCam slitless spectroscopy can identify bright EMPGs through strong H$\beta$ but lack of [O III] emission, or secure the redshifts of fainter systems through H$\alpha$ detections. Highly-multiplexed NIRSpec spectroscopy offers an alternate route to discovering the faintest pristine galaxies out to $z=10$, without requiring deep medium-band/MIRI imaging to identify secure candidates.