Neutrino Emissivity in $\mathrm{\Delta }$-direct-Urca Processes Inside Neutron Stars

Observed massive neutron stars may contain heavier baryons such as $\mathrm{\Delta }$ isobars and hyperons in their interiors. In the presence of $\mathrm{\Delta }$ isobars, the neutrino emitting nucleonic Urca processes can be accompanied by $\mathrm{\Delta }$-direct-Urca processes involving nucleons and hyperons. Over the past three decades, significant theoretical effort has been dedicated to understanding the thermal evolution of neutron stars with the nucleonic and hyperonic Urca processes emerging as dominant cooling mechanisms. While recent studies have proposed the presence of $\mathrm{\Delta }$ isobars in the dense core of neutron stars, the specific contribution of the $\mathrm{\Delta }$-direct-Urca process to neutrino emissivity has not been explicitly quantified. In this work, we address this gap by deriving the neutrino emissivity associated with $\mathrm{\Delta }$-direct-Urca processes in dense matter inside neutron stars. We show that once $\mathrm{\Delta }$-direct-Urca processes become kinematically allowed, their emissivities are comparable to those of nucleonic-direct-Urca processes.