Improved treatment of the T$$_2$$ molecular final-states uncertainties for the KATRIN neutrino-mass measurement

The KArlsruhe TRItium Neutrino experiment (KATRIN) aims to determine the effective mass of the electron antineutrino via a high-precision measurement of the tritium β-decay spectrum in its end-point region. The target
neutrino-mass sensitivity of 0.2 eV/c$^2$ at 90% CL can only be achieved in the case of high statistics and good control of the systematic uncertainties. One key systematic effect originates from the calculation of the molecular final states of T$_2$ β decay. In the first neutrino-mass analyses of KATRIN the contribution of the uncertainty of the molecular finalstates distribution (FSD) was estimated via a conservative phenomenological approach to be 2 × 10$^{-2}$ eV$^2$/c$^4$. In this work a new procedure is presented for estimating the FSD-related uncertainties by considering the details of the finalstates calculation, i.e. the uncertainties of constants, parameters, and functions used in the calculation as well as its convergence itself as a function of the basis-set size used in expanding the molecular wave functions. The calculated uncertainties are directly propagated into the experimental observable, the squared neutrino mass m$^2_v$ , and thus have to be determined individually for each experimental configuration. ... mehrFor the experimental conditions of the first KATRIN measurement campaign the new procedure is presented in detail, allowing for the application of this procedure to other experiments. This specific calculation leads to a constraint of the FSD-related uncertainty of 1.3 × 10$^{-3}$ eV$^2$/c$^4$, well below the design limit of 7.5 × 10$^{-3}$ eV$^2$/c$^4$ for any individual systematic contribution.