Searching for GEMS: TOI-5688 A b, a low-density giant orbiting a high-metallicity early M-dwarf

We present the discovery of a low-density planet transiting TOI-5688 A b, a high-metallicity M2V star. This planet was discovered as part of the search for transiting giant planets ($R \gtrsim8$ M$_\oplus$) through the Searching for GEMS (Giant Exoplanets around M-dwarf Stars) survey. The planet TOI-5688 A b was discovered with the Transiting Exoplanet Survey Satellite (TESS), and characterized with ground-based transits from Red Buttes Observatory (RBO), the Table Mountain Observatory of Pomona College, and radial velocity (RV) measurements with the Habitable-Zone Planet Finder (HPF) on the 10 m Hobby Eberly Telescope (HET) and NEID on the WIYN 3.5 m telescope. From the joint fit of transit and RV data, the mass of the planet is $124\pm24$ M$_\oplus$ and the radius is $10.4\pm0.7$ R$_\oplus$. This planet has a density of $0.61^{+0.20}_{-0.15}$ g/cm${}^3$, and is on a $\sim2.95$ day orbit around its host star. The spectroscopic and photometric analysis of the host star TOI-5688 A shows that it is a high metallicity ([Fe/H] $ = 0.47\pm0.16$ dex) M2V star, favoring the core-accretion formation pathway as the likely formation scenario for this planet. In this paper, we analyze potential mechanisms of planet formation in the context of the formation of TOI-5688 A b. Additionally, observations with Gaia suggest the presence of a wide-separation binary companion, TOI-5688 B, which has a projected separation of $\sim5"$ (1110 AU) and is an M4V. This makes TOI-5688 A b part of a growing number of GEMS in wide-separation binary systems.