Constraining possible $\gamma$-ray burst emission from GW230529 using Swift-BAT and Fermi-GBM

GW230529 is the first compact binary coalescence detected by the LIGO-Virgo-KAGRA collaboration with at least one component mass confidently in the lower mass-gap, corresponding to the range 3-5$M_{\odot}$. If interpreted as a neutron star-black hole merger, this event has the most symmetric mass ratio detected so far and therefore has a relatively high probability of producing electromagnetic (EM) emission. However, no EM counterpart has been reported. At the merger time $t_0$, Swift-BAT and Fermi-GBM together covered 100$\%$ of the sky. Performing a targeted search in a time window $[t_0-20 \text{s},t_0+20 \text{s}]$, we report no detection by the Swift-BAT and the Fermi-GBM instruments. Combining the position-dependent $\gamma-$ray flux upper limits and the gravitational-wave posterior distribution of luminosity distance, sky localization and inclination angle of the binary, we derive constraints on the characteristic luminosity and structure of the jet possibly launched during the merger. Assuming a top-hat jet structure, we exclude at 90$\%$ credibility the presence of a jet which has at the same time an on-axis isotropic luminosity $\gtrsim 10^{48}$ erg s$^{-1}$, in the bolometric band 1 keV-10 MeV, and a jet opening angle $\gtrsim 15$ deg. Similar constraints are derived testing other assumptions about the jet structure profile. Excluding GRB 170817A, the luminosity upper limits derived here are below the luminosity of any GRB observed so far.