Assessment of molecular effects on neutrino mass measurements from tritium $\beta$ decay

The $\beta$ decay of molecular tritium currently provides the highest sensitivity in laboratory-based neutrino mass measurements. The upcoming Karlsruhe Tritium Neutrino (KATRIN) experiment will improve the sensitivity to 0.2 eV, making a percent-level quantitative understanding of molecular effects essential. The modern theoretical calculations available for neutrino mass experiments agree with spectroscopic data. Moreover, when neutrino mass experiments performed in the 1980s with gaseous tritium are reevaluated using these modern calculations, the extracted neutrino mass squared values are consistent with zero instead of being significantly negative. However, the calculated molecular final-state branching ratios are in conflict with dissociation experiments performed in the 1950s. We reexamine the theory of the final-state spectrum of molecular-tritium decay and its effect on the determination of the neutrino mass, with an emphasis on the role of the vibrational- and rotational-state distribution in the ground electronic state. General features can be reproduced quantitatively from considerations of kinematics and zero-point motion. We summarize the status of validation efforts and suggest means for resolving the apparent discrepancy in dissociation rates.

Figure 1

Tritium $\beta$ spectrum with three active neutrinos with masses $m_{\nu i}\simeq 1$ eV for the case of no daughter excitation. The left panel shows the full spectrum. The right panel shows the last 5 eV before the end point, with the dotted curve indicating the spectral shape for $m_{\nu i}=0$.

Figure 2

Energy levels relevant to atomic and molecular-tritium decay, patterned after Fig. 5 in Otten and Weinheimer [7]. The mass difference $Q_A$ is taken from Audi, Wapstra, and Thibault [48]. Dissociation energies are derived from calculations by Doss [49]; the ionization energy of ${}^3{\mathrm{HeT}}^{+}$ is from calculations by Kołos et al. [50]. The ionization energies for T [51] and for ${}^3\mathrm{He}$ [52] are taken from recent compilations.

Figure 3

Molecular spectrum excited in the $\beta$ decay of ${\mathrm{T}}_2$ $(J=0)$ as calculated by Saenz et al. [8] (solid curve, red online) and by Fackler et al. [59] (dotted curve, blue online). For the purposes of display and comparison, discrete states in the latter spectrum have been given a Gaussian profile with a standard deviation of 3 eV.

Figure 4

Recoil kinetic energy imparted to a ${}^3\mathrm{He}$ daughter by the $\beta$ decay of an isolated tritium nucleus at rest. The upper boundary of the envelope corresponds to ${\theta }_{e\nu }=0$ and the lower one to ${\theta }_{e\nu }=\pi$.

Figure 5

Distributions of excitation energy in the ground-state rotational and vibrational manifold of ${}^3{\mathrm{HeT}}^{+}$ (left) and ${}^3{\mathrm{HeH}}^{+}$ (right), as calculated by Saenz et al. [8]. The expected value for the excitation energy in each case, based on kinematic considerations, is indicated by a vertical line. An excitation energy of 0 corresponds to a binding energy of 1.897 eV [49].

Figure 6

Calculated recoil excitation energy spectra from zero-point motion in the parent molecule (dotted curves, blue online), compared to the final-state distributions calculated by Saenz et al. [8] at 30 K (solid curves, red online). The curves from zero-point motion are parameter-free except for normalization and have the standard deviations indicated in Table 3. An excitation energy of 0 corresponds to a binding energy of 1.897 eV.

Figure 7

Comparison of the variance of the ground-state-manifold FSD produced in ${\mathrm{T}}_2$ decay as calculated in the semiclassical model, Eq. (30) (solid curve, red online), with variances taken from calculations for states up to $J=3$ described in Ref. [9] (blue dots).