Impacts of dark energy on weighing neutrinos after DESI BAO

Recently, DESI has released baryon acoustic oscillation (BAO) data, and DES has also published its five-year supernova (SN) data. These observations, combined with cosmic microwave background (CMB) data, support a dynamically evolving dark energy at a high confidence level. When using cosmological observations to weigh neutrinos, the results of weighing neutrinos will be significantly affected by the measurement of dark energy due to the degeneracy between neutrino mass and the dark-energy equation of state. Therefore, we need to understand how the dynamical evolution of dark energy in the current situation will affect the measurement of neutrino mass. In this work, we utilize these latest observations and other additional distance measurements to discuss the mutual influence between neutrinos and dark energy, then calculate the Bayes factor to compare models. We consider three neutrino mass hierarchies including degenerate hierarchy (DH), normal hierarchy (NH), and inverted hierarchy (IH), as well as three dark energy models including $\Lambda \rm CDM$, $w\rm CDM$, and $w_0w_a \rm CDM$ models. Cosmological data combined with the prior of particle physics experiments can provide strong to decisive evidence favoring the $w_0w_a {\rm CDM}+\sum m_\nu$ model with NH. In the $w_0w_a \rm CDM$ model, using the CMB+DESI+DESY5 data, we obtain constraints on the total neutrino mass, $\sum m_\nu<0.171\ \rm eV,\ 0.204\ \rm eV,\ 0.220\ \rm eV$, for DH, NH, and IH, respectively. Furthermore, taking into account the neutrino hierarchy or incorporating additional distance measurements results in a more pronounced deviation from the $\Lambda$CDM model for dark energy. The latter, particularly, exhibits a deviation at a confidence level that surpasses $4\sigma$.