Radiative tritium β–decay and the neutrino mass

The shape of the electron energy spectrum in H β-decay permits a direct assay of the absolute scale of the neutrino mass; a highly accurate theoretical description of the electron energy spectrum is necessary to the empirical task. We update Sirlin’s calculation of the outer radiative correction to nuclear β-decay to take into account the non-zero energy resolution of the electron detector. In previous H β-decay studies the outer radiative corrections were neglected all together; only Coulomb corrections to the spectrum were included. This neglect artificially pushes m ν < 0 in a potentially significant way. We present a computation of the theoretical spectrum appropriate to the extraction of the neutrino mass in the sub-eV regime. email: [email protected] email: [email protected] email: [email protected] Empirical evidence of neutrino oscillations in atmospheric, solar, and reactor neutrino data [1, 2, 3] compels the existence of non-zero neutrino masses, yet such experiments are insensitive to the absolute scale of a neutrino mass, for the oscillation experiments determine ∆mij ≡ mi −mj , where mi is the mass of neutrino i. To determine the absolute value of the neutrino mass requires different methods. Cosmological constraints on the neutrino mass do exist [4], though our focus shall be on the study of the electron energy spectrum in tritium β-decay near its endpoint, as this represents the most sensitive terrestrial measurement. The spectrum shape constrains the mass of the neutrino, be it of Dirac or Majorana character, and the inferred mass is insensitive to phases in the neutrino mixing matrix — in contradistinction to the constraint on the neutrino mass from neutrinoless double βdecay. An accurate theoretical description of the expected electron energy spectrum is crucial to the determination of the neutrino mass; this demand grows as the sensitivity of the experiments increase. Indeed, future studies expect to probe the neutrino mass at the sub-eV level [5]. It is our purpose to realize a theoretical form of the requisite accuracy, though we shall begin by describing the form used in earlier tritium experiments. With an anti-electron neutrino of mass mν , neglecting neutrino mixing for simplicity, the Fermi form of the electron energy spectrum for tritium β-decay is [6] dΓF dEe = GF 2π3 |M|F (Z,Re, Ee)peEe(E e − Ee) √ (Emax e − Ee) −mν , (1) where GF is the Fermi constant, pe, Ee, and E max e are the momentum, energy, and maximum endpoint energy, respectively, of the electron, and |M| is the absolute square of the nuclear matrix element, with |M| ∼ 5.3. A form of this ilk has been used to bound mν in previous experimental analyses of molecular tritium β-decay [7, 8, 9, 10, 11, 12, 13, 14]. Following the usual practice, we include a non-zero neutrino mass in the phase space contribution only. We set ~ = c = 1 throughout. The Fermi function, F (Z,Re, Ee), captures the correction due to the Coulomb interactions of the electron with the charge Ze of the daughter nucleus [15]. We adopt the usual expression [16], derived from the solutions of the Dirac equation for the point-nucleus potential −Zα/r evaluated at the nuclear radius Re [17]; it differs from unity by a contribution of O(α). The Fermi function includes the dominant electromagnetic effect, though an accurate extraction, or bound, of the neutrino mass does demand the inclusion of the remaining O(α) correction. We shall demonstrate this point explicitly. This last effect, termed the radiative correction, is conventially separated into an “inner” piece ∆R, which is absorbed in |M|, as it is energy independent and thus of no consequence to our current study, and an “outer” piece δR [18]. The outer radiative correction applied in β-decay studies is due to Sirlin [18]: dΓ dEe = dΓ0 dEe ( 1 + α 2π gS(Ee, E max e ) )