# Direct Measurements of Neutrino Mass

Joseph A. Formaggio  Laboratory for Nuclear Science and Dept. of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  André Luiz C.
de Gouvêa  Department of Physics and Astronomy, Northwestern University,
Evanston, IL 60208, USA  R. G. Hamish Robertson  Center for Experimental
Nuclear Physics and Astrophysics, and Dept. of Physics, University of
Washington, Seattle, WA 98195, USA

###### Abstract

The turn of the 21st century witnessed a sudden shift in our fundamental
understanding of particle physics. While the minimal Standard Model predicts
that neutrino masses are exactly zero, the discovery of neutrino oscillations
proved the Standard Model wrong. Neutrino oscillation measurements, however,
do not shed light on the scale of neutrino masses, nor the mechanism by which
those are generated. The neutrino mass scale is most directly accessed by
studying the energy spectrum generated by beta decay or electron capture – a
technique dating back to Enrico Fermi’s formulation of radioactive decay. In
this Article, we review the methods and techniques – both past and present –
aimed at measuring neutrino masses kinematically. We focus on recent
experimental developments that have emerged in the past decade, overview the
spectral refinements that are essential in the treatment of the most sensitive
experiments, and give a simple yet effective protocol for estimating the
sensitivity. Finally, we provide an outlook of what future experiments might
be able to achieve.

###### Contents

  1. I Introduction
  2. II Neutrino Masses: Current Status
    1. II.1 Neutrino oscillations
    2. II.2 Neutrinoless double beta decay
    3. II.3 Cosmology
    4. II.4 Neutrinos from Astrophysical Sources – Time of Flight
    5. II.5 Direct laboratory measurements
  3. III Neutrino Mass Models and New Physics
    1. III.1 Neutrino mass models
    2. III.2 Sensitivity to new phenomena
  4. IV Kinematic determination from beta decay
    1. IV.1 Beta spectrum
    2. IV.2 Isotopes of Interest
  5. V Experimental Progress
    1. V.1 Tritium beta decay
    2. V.2 The MAC-E Filter and KATRIN
    3. V.3 163Ho electron-capture decay
    4. V.4 187Re beta decay
  6. VI Other Research
    1. VI.1 Sterile Neutrinos
    2. VI.2 Relic Neutrinos
  7. VII Spectrum Refinements
    1. VII.1 Final State Distributions
      1. VII.1.1 Tritium
      2. VII.1.2 Rhenium
      3. VII.1.3 Holmium
    2. VII.2 Theoretical corrections to the beta spectrum shape
  8. VIII Experimental sensitivity
    1. VIII.1 Estimation method
    2. VIII.2 Backgrounds
  9. IX Future
    1. IX.1 Calorimetric techniques
    2. IX.2 Frequency techniques
    3. IX.3 Atomic tritium
  10. X Conclusion
  11. XI Acknowledgments

##  I Introduction

The existence of neutrinos was first postulated by Pauli nine decades ago. In
his famous ‘Dear Radioactive Ladies and Gentlemen’ letter Pauli (1930); Brown
(1978), Pauli also made the first non-trivial estimate of the mass of the
neutrino: “The mass of the neutrons111“Neutron” was the name attributed to the
hypothetical particle by Pauli. In order to avoid confusion with the modern
neutron, discovered a few years later, the diminutive form ‘neutrino’ was
famously introduced by Fermi. For information on the history of the neutrino,
see, for example, Ref. Franklin (2000). should be of the same order of
magnitude as the electron mass and in any case not larger than 0.01 times the
proton mass.” His qualitative prediction – $m_{\nu}\sim m_{e}$ – based on
aesthetics and minimality, turned out to be too large by many orders of
magnitude.

Laboratory searches for a nonzero neutrino mass started in the 1930s and have
continued in earnest up to the present. Pauli’s neutrino, now called the
electron neutrino $\nu_{e}$, was not directly detected until the work of
Reines and Cowan in the 1950s Reines and Cowan (1953); Cowan et al. (1956).
Two other neutrino flavors were discovered, $\nu_{\mu}$ in 1962 Danby et al.
(1962) and $\nu_{\tau}$ in 2001 Kodama et al. (2001), and the searches
diversified in order to accommodate the possibility that the different
neutrino species had qualitatively different masses. The most stringent upper
bounds to the mass of the electron neutrino evolved from Pauli’s 10 MeV
qualitative upper bound – 1% of the proton mass – to several electron-volts by
the late 1990s.

Conclusive evidence for nonzero neutrino masses was revealed in 1998 with the
discovery of atmospheric neutrino oscillations by the Super-Kamiokande
collaboration Fukuda et al. (1998), building on previous hints for atmospheric
neutrino oscillations obtained by the Irvine-Michigan-Brookhaven Casper et al.
(1991) and Kamiokande Collaborations Hirata et al. (1992), and the different
collaborations that helped define the solar neutrino puzzle: Homestake
Cleveland et al. (1998), Gallex Hampel et al. (1999), SAGE Abdurashitov et al.
(1999), and Kamiokande Fukuda et al. (1996). The solar neutrino puzzle was
definitively resolved by the Sudbury Neutrino Observatory (SNO) collaboration
Ahmad et al. (2002) and is also a consequence of nonzero neutrino masses. The
2015 Nobel Prize in Physics was awarded to Takaaki Kajita – from the Super-
Kamiokande Collaboration – and Arthur B. McDonald – from the SNO collaboration
– “for the discovery of neutrino oscillations, which shows that neutrinos have
mass.”222 The Nobel Prize in Physics 2015.
https://www.nobelprize.org/prizes/physics/2015/summary/.

Several twenty-first century oscillation experiments provide precision
measurements of the neutrino oscillation phenomenon Abe et al. (2008); Adamson
et al. (2008); Abe et al. (2014); Choi et al. (2016); An et al. (2017); Abe et
al. (2018a, 2017); Acero et al. (2018). These translate into rather precise
measurements of neutrino mass-squared differences and reveal that at least two
of the three neutrinos are massive and the heaviest neutrino mass is at least
0.05 eV. Oscillation experiments, however, are powerless when it comes to
measuring the individual values of the neutrino masses – they are only
sensitive to mass-squared differences. Other laboratory observables are
sensitive to nonzero neutrino masses. Some of these observables are only
indirectly sensitive to the masses. In those cases, the connection between
measurement and neutrino masses is mediated by a theoretical framework, along
with other hypotheses. Such observables include the rate for neutrinoless
double-beta decay and the large-scale structure of the universe. Other
observables are more directly sensitive to neutrino masses – the kinematics of
the observable are directly established by the fact that neutrinos have
nonzero masses in a way that is virtually independent from the nature of the
physics responsible for the observable. Among these are precision measurements
of nuclear beta decay, meson decay, charged-lepton decays, and electron and
neutrino capture in nuclei. The most sensitive among these direct probes of
nonzero neutrino masses are the subject of this review.

A number of reviews of the subject are available Otten and Weinheimer (2008);
Drexlin et al. (2013); Holzschuh (1992); Robertson and Knapp (1988). As the
most recent was seven years before the present one, it is an opportunity to
review the recent progress and to consider where the field will go in the
future. The KATRIN experiment is now running, tightening the upper limit on
neutrino mass by a factor 2 after only a month of operation. The new method of
cyclotron radiation emission spectroscopy has passed a crucial proof-of-
principle test. Prompted by this success, the possibility of a neutrino mass
experiment based on atomic tritium is once again receiving consideration.
Microcalorimetry is advancing technically to enable studies of isotopes other
than tritium.

This review is organized as follows. In Section II, we summarize the direct
and indirect information on neutrino masses that is currently available, along
with some near-future expectations. In Section III, we discuss how the
discovery of nonzero neutrino masses impacted our understanding of fundamental
particle physics, along with the different outstanding questions that we hope
will be informed by the direct observation of nonzero neutrino masses. Nature
has provided only two isotopes that continue to offer prospects for gains in
sensitivity, as we describe in Section IV. The decades of progress that have
brought the field to the 1-eV sensitivity level are reviewed in Section V.
Related research on sterile neutrinos and the relic neutrino background is
covered in Section VI. In Section VII, we discuss spectrum refinements that
are essential in the treatment of the most sensitive experiments, and, in
Section VIII, provide a simple yet effective protocol for estimating the
sensitivity. In Section IX, the new techniques that are emerging to advance
the field are introduced, and in Section X, we conclude.

##  II Neutrino Masses: Current Status

In this section we provide an overview of the current understanding of the
values of the neutrino masses. For a detailed review, see, for example, the
‘Neutrino Masses, Mixing, and Oscillations’ chapter of the Particle Data Book
Zyla et al. (2020).

###  II.1 Neutrino oscillations

Precision measurements of the flux of solar neutrinos reveal that fewer
electron-type neutrinos arrive at the Earth than predicted. What came to be
known as the solar neutrino problem presented itself with the first
measurements of the solar neutrino flux in the 1960s – see references in Ref.
Cleveland et al. (1998) – and persisted until it was definitively resolved by
the SNO experiment in the early 2000s Ahmad et al. (2002). Ultimately, solar
neutrino data imply that electron neutrinos are, in fact, linear
superpositions of at least two neutrino mass-eigenstates and that at least one
of these has a nonzero mass. The difference between the neutrino masses-
squared is of order $\Delta m^{2}_{21}\equiv m_{2}^{2}-m_{1}^{2}\sim 10^{-4}$
eV2. Here $m_{1}$ and $m_{2}$ are the masses of two different neutrino mass
eigenstates, labeled $\nu_{1}$ and $\nu_{2}$.

Precision measurements of the flux of atmospheric neutrinos also reveal that
fewer muon-type neutrinos survive passage through the Earth than expected. The
effect depends on the distance between the neutrino production and detection
points and the neutrino energy. The solution to this new problem, the
atmospheric neutrino problem, first revealed by data from the IMB and
Kamiokande experiments and later confirmed beyond reasonable doubt by the
Super-Kamiokande experiment, was the realization that muon neutrinos are
linear superpositions of at least two neutrino mass eigenstates and that at
least one of these has a nonzero mass. In this case the difference between the
neutrino masses-squared is of order $\Delta m^{2}_{31}\equiv
m_{3}^{2}-m_{1}^{2}\sim 10^{-3}$ eV2. Here $m_{3}$ is the mass of the third
distinct neutrino mass eigenstate, labeled $\nu_{3}$.

In the last two decades, multiple experiments with multiple neutrino sources
and detector technologies have confirmed the existence of neutrino
oscillations and have allowed the construction of a very robust three-massive-
neutrinos paradigm. It asserts that neutrinos interact as prescribed by the
Standard Model of particle physics and that the neutrino charged-current-
interaction eigenstates – $\nu_{e}$, $\nu_{\mu}$ and $\nu_{\tau}$ – are linear
superpositions of the neutrino mass eigenstates, $\nu_{1}$, $\nu_{2}$,
$\nu_{3}$, with masses, respectively, $m_{1}$, $m_{2}$, and $m_{3}$:

$\nu_{\alpha}=\sum_{i}U_{\alpha i}\nu_{i},$ (II.1)

where $i=1,2,3$, $\alpha=e,\mu,\tau$ and $U_{\alpha i}$ are the elements of
the $3\times 3$ unitary leptonic mixing matrix, also referred to as the
neutrino mixing matrix or the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.
The neutrino mass eigenstates are defined through the relative values of the
neutrino masses-squared, as follows: $m_{2}^{2}>m_{1}^{2}$ and
$|m_{3}^{2}-m_{1}^{2}|,|m_{3}^{2}-m_{2}^{2}|>m_{2}^{2}-m_{1}^{2}$ in such a
way that $m_{3}^{2}>m_{2}^{2}>m_{1}^{2}$, termed the normal mass-ordering
(NMO), or $m_{3}^{2}<m_{1}^{2}<m_{2}^{2}$, termed the inverted mass-ordering
(IMO). In the NMO, $\Delta m^{2}_{31},\Delta m^{2}_{32}$ are positive while in
the IMO $\Delta m^{2}_{31},\Delta m^{2}_{32}$ are negative. The two mass
orderings are depicted in Figure 1. The mass ordering is currently unknown. A
slight preference in the world neutrino data for the NMO Esteban et al. (2020)
has recently disappeared with new data Kelly et al. (2021).

Figure 1: Illustration of the two distinct neutrino mass orderings that fit
nearly all of the current neutrino data, for typical values of all mixing
angles and mass-squared differences. The color coding (shading) indicates the
fraction $|U_{\alpha i}|^{2}$ of each distinct flavor $\nu_{\alpha}$,
$\alpha=e,\mu,\tau$ contained in each mass eigenstate $\nu_{i}$, $i=1,2,3$.
For example, $|U_{e2}|^{2}$ is equal to the fraction of the $(m_{2})^{2}$
“bar” that is painted red (shading labeled as “$\nu_{e}$”). From de Gouvêa et
al. (2013).

The PMNS matrix is parameterized with three mixing angles
$\theta_{12},\theta_{13},\theta_{23}$ and one (three) CP-odd phase(s) $\delta$
($\delta,\eta_{1},\eta_{2}$) if the neutrinos are Dirac (Majorana) fermions.
Throughout, we will use the PDG parameterization for the PMNS matrix Zyla et
al. (2020) and will assume the three-massive-neutrinos paradigm is true,
unless otherwise noted.

Neutrino oscillation experiments can constrain the PMNS matrix and the
neutrino mass-squared differences $\Delta m^{2}_{ij}=m_{i}^{2}-m_{j}^{2}$,
$ij=21,31,32$. Only two of the three mass-squared differences are independent
because $\Delta m^{2}_{32}=\Delta m^{2}_{31}-\Delta m^{2}_{21}$. The world
neutrino data translate into robust measurements of the neutrino mass-squared
differences. According to the Nufit Collaboration Esteban et al.
(2020)333Several phenomenological collaborations regularly collect and analyze
the neutrino oscillation data, estimating the values of the oscillation
parameters. Here, we will use the results from Esteban et al. (2020), unless
otherwise noted. These are consistent with other global fits, including those
presented in Refs. Capozzi et al. (2018); de Salas et al. (2020).

$\displaystyle\Delta m^{2}_{21}=\left(7.42^{+0.21}_{-0.20}\right)\times
10^{-5}~{}{\rm eV}^{2},$ (II.2) $\displaystyle\Delta
m^{2}_{31}=\left(2.517^{+0.026}_{-0.028}\right)\times 10^{-3}~{}{\rm
eV}^{2}~{}(\rm NMO),$ (II.3) $\displaystyle{\rm or}$ $\displaystyle\Delta
m^{2}_{32}=\left(-2.498^{+0.028}_{-0.028}\right)\times 10^{-3}~{}{\rm
eV}^{2}~{}(\rm IMO).$ (II.4)

Neutrino oscillation experiments do not inform the values of the individual
neutrino masses, only the mass-squared differences. For the individual
neutrino masses, information outside of neutrino oscillations is required.

###  II.2 Neutrinoless double beta decay

With the discovery of nonzero neutrino masses, one of the most important
outstanding questions in particle physics is the nature of the neutrinos: are
they Majorana or Dirac fermions? If neutrinos are Majorana fermions, lepton-
number $L$ conservation is not an exact law of nature and has to be violated,
even if only very feebly. On the other hand, if neutrinos are Dirac fermions,
lepton-number conservation is an exact law of nature or, at the very least,
$\Delta L=2$ processes are strictly forbidden.

Searches for neutrinoless double-beta decay ($0\nu\beta\beta$), $\Delta L=2$
nuclear-decay processes of the type $(Z,A)\to(Z+2,A)+e^{-}e^{-}$, where
$(Z,A)$ is a nucleus with atomic number $Z$ and mass number $A$, are the most
powerful probes of lepton-number conservation. Other searches include
“variations” on $0\nu\beta\beta$, including neutrinoless double-beta-plus
decay, $(Z,A)\to(Z-2,A)+e^{+}e^{+}$ and lepton-number violating electron
capture, $(Z,A)+e^{-}\to(Z-2,A)+e^{+}$, along with $\mu^{-}\to e^{+}$
conversion in nuclei, forbidden meson decays Zyla et al. (2020), including
$K^{+}\to\mu^{+}\mu^{+}\pi^{-}$, and the production of same-sign di-leptons
and no missing energy at hadron colliders (e.g., $pp\to e^{+}\mu^{+}+X$, where
$X$ is a state with zero lepton number). Here, we concentrate on constraints
from $0\nu\beta\beta$.

If Majorana-neutrino exchange is the dominant contribution to
$0\nu\beta\beta$, the rate for $0\nu\beta\beta$ is a function of the neutrino
masses. If all neutrino masses are small relative to the typical energy scales
involved in $0\nu\beta\beta$, which is of order dozens of MeV, in the absence
of new physics other than nonzero Majorana neutrino masses, the amplitude for
$0\nu\beta\beta$ is proportional to a linear combination of the neutrino
masses:

$m_{\beta\beta}\equiv\sum_{i}(U_{ei})^{2}m_{i}=\cos^{2}\theta_{13}\left(\cos^{2}\theta_{12}m_{1}^{\prime}+\sin^{2}\theta_{12}m_{2}^{\prime}\right)+\sin^{2}\theta_{13}m_{3}^{\prime},$
(II.5)

where $m_{i}^{\prime}=e^{i\phi_{i}}m_{i}$, where $\phi_{i}$ are combinations
of the CP-odd phases in the PMNS matrix, see, for example, Ref. Zyla et al.
(2020) for a concrete parameterization; $m_{\beta\beta}$ is a complex
parameter and experiments are sensitive to its magnitude. $\theta_{13}$ and
$\theta_{12}$ are two of the mixing angles used to parameterize the PMNS
matrix. Unless otherwise noted, for the mixing angles, we use the PDG
parameterization Zyla et al. (2020).

Under these conditions, a measurement of the rate for $0\nu\beta\beta$,
combined with input from neutrino oscillations, provides nontrivial
information on the neutrino masses. Using information from neutrino
oscillation experiments, it is possible to parameterize $|m_{\beta\beta}|$ as
a function of two relative phases among the $m_{i}^{\prime}$ parameters and
the value of the lightest neutrino mass $m_{\rm least}$. For the different
mass orderings

$m_{\rm least}=m_{1}~{}~{}({\rm NMO})~{}~{}~{}~{}{\rm or}~{}~{}~{}~{}m_{\rm
least}=m_{3}~{}~{}({\rm IMO}).$ (II.6)

The fact that these so-called Majorana phases are unknown and, in practice,
impossible to constrain experimentally in any other way, renders the
information on $m_{\rm least}$ from $0\nu\beta\beta$ always imperfect.
Agostini, Benato and Detwiler Agostini et al. (2017) have carried out a
Bayesian analysis incorporating existing data to make predictions, under well-
defined assumptions, of the discovery probability for true values of
$|m_{\beta\beta}|$. The distributions are shown in Fig. 2.

Figure 2: Marginalized posterior distributions for $|m_{\beta\beta}|$ and
$m_{\rm least}$ ($m_{\rm l}$ in the figure) in the NMO (left) and IMO (right).
From Agostini et al. Agostini et al. (2017).

It is striking that, barring some particular physics that would drive
$|m_{\beta\beta}|$ or $m_{\rm least}$ to zero, the discovery probability is
not small, especially in the IMO. It can also be seen that a direct mass
measurement, essentially a measurement of $m_{\rm least}$, below 100 meV
becomes highly informative in the search for neutrinoless double beta decay.

Experimental searches for neutrinoless double beta decay measure or limit the
decay rate of a particular isotope. That rate depends on the product of
$|m_{\beta\beta}|$, a phase-space factor, and a nuclear matrix element for the
$(Z,A)\to(Z+2,A)$ transition, the most commonly investigated type. Currently,
the matrix elements are poorly constrained. Estimates performed using
different techniques differ by a factor of a few. Qualitative and quantitative
improvements are expected in the near future, but it is fair to expect that,
for the foreseeable future, the theoretical uncertainty on extracting
$|m_{\beta\beta}|$ from the rate for $0\nu\beta\beta$ will be sizable. For
detailed, recent reviews see, for example, Agostini et al. (2017); Dolinski et
al. (2019). Notwithstanding the matrix-element problem, experimental progress
in the past decade has been remarkable, limiting the effective Majorana mass
to values well below the levels presently accessible to direct measurements.
Table 1, updated from that compiled in Dolinski et al. (2019), gives the half-
life $T_{1/2}^{0\nu}$ and Majorana mass $|m_{\beta\beta}|$ limits from the
most recent experiments.

Table 1: Published half-life and Majorana mass limits from recent experiments. Isotope aa |  $T_{1/2}^{0\nu}\ (\times 10^{25}$ y) aa |  $|m_{\beta\beta}|$ (eV) aaaa | Experiment |  aa Ref.
---|---|---|---|---
48Ca | $>5.8\times 10^{-3}$ | $<3.5-22$ | ELEGANT-IV | Umehara et al. (2008)
76Ge | $>18$ | $<0.079-0.180$ | GERDA | Agostini et al. (2020)
| $>2.7$ | $<0.200-0.433$ | Majorana Demonstrator | Alvis et al. (2019)
82Se | $>3.6\times 10^{-2}$ | $<0.89-2.43$ | NEMO-3 | Barabash and Brudanin (2011)
96Zr | $>9.2\times 10^{-4}$ | $<7.2-19.5$ | NEMO-3 | Argyriades et al. (2010)
100Mo | $>1.1\times 10^{-1}$ | $<0.33-0.62$ | NEMO-3 | Arnold et al. (2015)
116Cd | $>1.0\times 10^{-2}$ | $<1.4-2.5$ | NEMO-3 | Arnold et al. (2017)
130Te | $>3.2$ | $<0.075-0.350$ | CUORE | Adams et al. (2020)
136Xe | $>10.7$ | $<0.061-0.165$ | KamLAND-Zen | Gando et al. (2016)
| $>3.5$ | $<0.093-0.286$ | EXO-200 | Anton et al. (2019)
150Nd | $>2.0\times 10^{-3}$ | $<1.6-5.3$ | NEMO-3 | Arnold et al. (2016)

In viewing the experimental results, it must be kept in mind that the rate for
$0\nu\beta\beta$ is a function only of the neutrino masses when light-neutrino
exchange is the leading contribution to $0\nu\beta\beta$. More generally,
lepton-number violating physics can impact $0\nu\beta\beta$ in a way that the
connection between the rate for $0\nu\beta\beta$ and the neutrino masses is
either indirect or, in some cases, non-existent. For an overview, see, for
example, Ref. Pas and Rodejohann (2015).

The experimental results of the current generation have been obtained with
detectors having isotopic masses in the range of tens to hundreds of
kilograms. A new generation of detectors an order of magnitude larger is now
beginning. A comprehensive summary of the plans and status may be found in the
APPEC Committee Report Giuliani et al. (2019) prepared for the European
strategy. The goal of the next generation is sensitivity in the range of the
IMO.

Finally, we highlight that, of course, if neutrinos are Dirac fermions, no
information on neutrino masses can be extracted from searches for lepton-
number violation. Conversely, an observation of neutrinoless double beta decay
is unambiguous evidence of lepton-number violation, independent of the
uncertainties that affect a mass determination therefrom.

###  II.3 Cosmology

In the Standard Model of cosmology, neutrinos are predicted to be relics of
the big bang. Measurements of the relic abundance of light elements Steigman
(2012); Pitrou et al. (2018) and the large-scale structure of the universe,
including precision measurements of the properties of the cosmic microwave
background (CMB) Aghanim et al. (2020), are consistent with the existence of a
thermal relic-neutrino background. These neutrinos played a significant role
in the expansion history of the universe even if, today, they are rather cold
and make up only a tiny fraction of the universe’s matter and energy budget.

The temperature of the relic neutrino background today is predicted to be of
order $T^{0}_{\nu}\sim 2\times 10^{-4}$ eV. Hence, as the universe expanded
and the relic neutrino background cooled, the behavior of neutrinos changed
from that of ultrarelativistic relics – “radiation” – to that of non-
relativistic species – “matter” – as long as the neutrino masses are larger
than $T^{0}_{\nu}$. Given information from neutrino oscillations, at least two
of the three neutrino masses are known to be much larger than $T^{0}_{\nu}$.
This transition leaves an imprint in the large-scale structure of the universe
in such a way that precision measurements provide nontrivial information on
the neutrino masses.

In the absence of other light particles or new neutrino interactions, the
relic neutrino background is best described as a homogeneous mixture of the
neutrino mass eigenstates $\nu_{1}$, $\nu_{2}$, and $\nu_{3}$ and their
antiparticles.444If neutrinos are Majorana fermions, the situation is very
similar, with left-helicity neutrino states playing the role of particles and
right-helicity neutrino states playing the role of antiparticles.,555There is
the possibility that the asymmetry between neutrinos and antineutrinos is
relatively large. A flavor-universal asymmetry is constrained to be
significantly less than one percent Pitrou et al. (2018) and would not impact
the discussion here. Flavor-dependent effects are subtle and could impact the
picture more significantly. These are still the subject of intense exploration
Dolgov et al. (2002); Wong (2002); Abazajian et al. (2002); Hansen et al.
(2020a). Theoretically, cosmic surveys are sensitive to the values of the
individual neutrino masses, $m_{1},m_{2},m_{3}$. In practice, given the
expected sensitivity of next-generation experiments, data from cosmic surveys
constrain the sum of the neutrino masses, assuming all neutrino masses are
light (say, all masses below a few eV), labelled here for convenience as

$\Sigma\equiv\sum_{i}m_{i}.$ (II.7)

$\Sigma$ can be expressed in terms of the known mass-squared differences and
the, currently unknown, lightest neutrino mass, $m_{\rm least}$:

$\displaystyle\Sigma=m_{\rm least}+\sqrt{\Delta m^{2}_{21}+m_{\rm
least}^{2}}+\sqrt{\Delta m^{2}_{31}+m_{\rm least}^{2}}~{}~{}~{}({\rm NMO}),$
(II.8) $\displaystyle{\rm or}$ $\displaystyle\Sigma=m_{\rm
least}+\sqrt{-\Delta m^{2}_{32}+m_{\rm least}^{2}}+\sqrt{-\Delta
m^{2}_{31}+m_{\rm least}^{2}}~{}~{}~{}({\rm IMO}).$ (II.9)

If the neutrino mass ordering is normal (inverted), current oscillation data
constrain $\Sigma>5.87\times 10^{-2}$ eV ($\Sigma>9.92\times 10^{-2}$ eV). The
direct laboratory measurement by KATRIN Aker et al. (2019) constrains
$\Sigma<3.3$ eV in either mass ordering.

Data from large scale structure, including the CMB, the distribution of
clusters of galaxies, and the Lyman-alpha forest, most recently constrain
$\Sigma<0.111$ eV (95% CL), according to Alam et al. (2020). Other recent
analyses of cosmic surveys output similar upper bounds including $\Sigma<0.12$
eV (95% CL) Aghanim et al. (2020) and $\Sigma<0.16$ eV (95% CL) Ivanov et al.
(2020). Bounds obtained before the Planck-2018 data became available were only
slightly weaker, $\Sigma<0.19$ eV (95% CL) Giusarma et al. (2018). These
bounds do depend on the values of the individual neutrino masses and the
neutrino mass ordering, but not very strongly (see recent discussion in Ivanov
et al. (2020); Roy Choudhury and Hannestad (2020)). Estimates in Roy Choudhury
and Hannestad (2020) output, for the NMO, $\Sigma<0.15$ eV (95% CL), and
$\Sigma<0.17$ eV (95% CL) for the IMO. These translate, roughly, into $m_{\rm
least}<0.05$ eV, mostly independent from the mass ordering.

In the next decade, it is widely anticipated that next-generation experiments,
including CMB-S4 Abazajian et al. (2019), will be sensitive to $\Sigma>2\times
10^{-2}$ eV. If expectations are realized, cosmic surveys should be able to
determine that $\Sigma$ is nonzero at better than the three-sigma level
Abazajian et al. (2019), independent of the mass ordering, assuming no new
degrees of freedom or interactions beyond those in the Standard Model.

The extraction of the sum of light neutrino masses from cosmic surveys is
model dependent. Cosmic surveys are sensitive, in an over-simplified way, to
the expansion rate of the universe as a function of time and to the formation
of large-scale structure as a function of time. While the evidence that there
are relic neutrinos is very compelling (see, for example, Steigman (2012);
Pitrou et al. (2018)), the presence of neutrinos is only indirectly inferred
and so are statements about their properties. New neutrino properties can
impact the sensitivity to $\Sigma$ significantly. The authors of Ref. Escudero
et al. (2020), for example, argued very recently that if neutrinos are
unstable but still very long lived (lifetime between $10^{-8}$ and $10^{-1}$
times the age of the universe), bounds on $\Sigma$ can be relaxed by an order
of magnitude. The same can be said for more new ingredients to the Standard
Model of cosmology. For example, the nature of the dark energy – often
parameterized by the dark-energy equation-of-state parameter $w$ – impacts the
sensitivity to $\Sigma$ Hannestad (2005). Allowing for different palatable
ingredients in the Standard Model of cosmology loosens the upper bound on
$\Sigma$ by about a factor of three Alam et al. (2020); Roy Choudhury and
Hannestad (2020) or more (this is, of course, not guaranteed. For a counter
example, see, for example, Vagnozzi et al. (2018)). The current tension
between early-universe and late-universe estimates of the the Hubble parameter
Aghanim et al. (2020); Riess et al. (2019) has invited speculation concerning
new neutrino properties and interactions (see, for example, Kreisch et al.
(2020)); some of these may have a significant impact on extracting constraints
on $\Sigma$. See, for example, Sekiguchi and Takahashi (2020) for a very
recent discussion.

Massive neutrinos are a required ingredient of cosmological models but since
the masses are presently unknown they must be treated as fit parameters. There
are few cosmological parameters susceptible to laboratory measurement, and
neutrino mass is one. A measurement would alleviate the models of a degree of
freedom and allow better determinations of those parameters that can only be
determined from cosmology, such as the equation of state of dark energy and
the Hubble constant Di Valentino et al. (2016); Di Valentino et al. (2016).

###  II.4 Neutrinos from Astrophysical Sources – Time of Flight

Throughout the universe, neutrinos are produced in cataclysmic astronomical
events, including Type II Supernova explosions. Since these are short-duration
bursts, it is possible to obtain information on the neutrino velocity and
hence – since the neutrino energy can be measured – the neutrino mass. The
time-spread of the neutrinos observed from SN1987A allows one to constrain the
neutrino mass to be less than a few eV. A very detailed analysis was performed
in Ref. Loredo and Lamb (2002) – which includes references to several other
estimates – and the authors constrained what they refer to as the “electron
neutrino mass” to be less than 5.7 eV at the 95% confidence level. Strictly
speaking, given what is known about neutrino mixing, the analysis is more
involved and should include the fact that there are three mass eigenstates
with different probabilities for interacting via charged-current interactions
with electrons. In practice, given what is known about the neutrino mass-
squared differences, the 5.7 eV upper bound applies to all mass eigenvalues.

In a nutshell, the measurement works as follows. If a neutrino is produced at
some $t_{0}=0$ with energy $E$ a distance $D$ away it will arrive at the
detector at

$t(E,m)=D\left[1+\frac{m^{2}}{2E^{2}}\right],$ (II.10)

assuming all three neutrino masses are degenerate and equal to $m$, and $m\ll
E$. Relative to a massless particle, the time delay of a massive neutrino is

$\Delta t=D\left(\frac{m^{2}}{2E^{2}}\right)=25.7~{}\rm
ms\times\left(\frac{D}{50~{}\rm kpc}\right)\left(\frac{m}{\rm
eV}\right)^{2}\left(\frac{\rm 10~{}MeV}{E}\right)^{2}.$ (II.11)

For supernova neutrinos, $t_{0}$ is not known but one can investigate whether
neutrinos with different energies arrive at different times, leading to, for
example, a larger-than-expected spread in the neutrino arrival times.

The detection of neutrinos from the next galactic supernova will allow one to
perform a similar measurement, perhaps with higher statistics and richer data
given the existence of bigger and better detectors. The JUNO collaboration,
for example, estimates that the JUNO experiment An et al. (2016), currently
under construction, is sensitive to $m>1$ eV if neutrinos from a supernova
explosion at $D\simeq 20$ kpc were to be detected. The dependence on $D$ is
rather mild and the sensitivity worsens as $D$ increases; the larger $\Delta
t$ is compensated by the loss of statistics as $D$ increases. Similar
sensitivity – down to at most $m>0.5$ eV – has been estimated for the future
DUNE Rossi-Torres et al. (2015); Abi et al. (2020) and Hyper-Kamiokande
experiments Abe et al. (2018b). Very recently, the authors of Ref. Hansen et
al. (2020b) explored in detail the sensitivity of different next-generation
experiments and different sources. In particular, they discuss the possibility
of comparing the arrival time of the neutrinos with the potential detection of
gravitational waves from the same source and estimate sensitivity to neutrino
masses of order 1 eV.

###  II.5 Direct laboratory measurements

In processes involving neutrinos where the total energy of the initial state
is well known and the kinematics of the final state can be measured with
precision, it is possible to constrain, using energy and momentum
conservation, the neutrino mass. Such measurements are often referred to as
direct measurements of the neutrino mass and are the main subject of this
review.

The first direct laboratory probe of neutrino mass was suggested by Perrin in
1933 Perrin (1933): “On peut essayer de déduire de la forme des spectres
continus d’émission une indication sur la valeur de cette masse inconnue…”
(One could attempt to deduce from the shape of the continuous emission spectra
an indication of the value of this unknown mass…). Fermi independently reached
that conclusion quantitatively in his seminal 1934 article Fermi
(1934a),666See Wilson (1968) for a translation of Fermi’s paper to English.
Fermi published preliminary work on the theory of $\beta$-decay several months
before Ref. Fermi (1934a) – in “La Ricerca Scientifica” Fermi (1933) and “Il
Nuovo Cimento” Fermi (1934b) – and submitted his theory for publication in
Nature. The Nature submission was rejected, famously, because ‘it contained
abstract speculations too remote from physical reality to be of interest to
the reader’ Fermi and Segrè (1962). For more details on the history of Fermi’s
contribution to the theory of $\beta$-decay, see Guerra and Robotti (2018). We
are indebted to David Kaiser for providing us most of this information. which
introduces a so-called four-fermion interaction to describe nuclear beta-decay
– the Fermi interaction. Fermi suggested that the energy spectrum of the
$\beta$-rays can be used to determine the mass of the neutrino: “[t]he shape
of the continuous $\beta$-spectrum is determined from the transition
probability [computed perturbatively using the Fermi-interaction Hamiltonian].
We want to discuss first how this shape depends on the rest mass of the
neutrino $\mu$, in order to determine this constant by comparison with
empirical curves” Wilson (1968). The effect of a nonzero neutrino mass is
illustrated in Fig. 3, from Fermi (1934a). Fermi concluded that “[t]he
greatest similarity to the empirical curves is given by the theoretical curve
for $\mu=0$. …Hence we conclude that the rest mass of the neutrino is either
zero or, in any case, very small in comparison to the mass of the electron”
Wilson (1968).

Figure 3: Fermi’s illustration of the impact of a nonzero neutrino mass $\mu$
on the shape of the $\beta$-ray energy spectrum, for “large” (groß), “small”
(klein) and $\mu=0$ Fermi (1934a). Large and small are defined, loosely,
relative to the mass of the electron.

If neutrinos are produced or absorbed via charged-current interactions
associated with the charged-lepton $\ell_{\alpha}=e,\mu,\tau$ the differential
rate associated with the process, which is a function of the neutrino masses-
squared, can be written as

$\sum_{i}|U_{\alpha i}|^{2}\Gamma_{\alpha}(m_{i}^{2}).$ (II.12)

Here $\Gamma_{\alpha}(m_{i}^{2})$ is the differential rate of the process of
interest when a neutrino with mass $m_{i}$ is emitted or absorbed. For the
observables under consideration here, neutrino production is best described as
incoherent, under the assumption that the neutrino masses are within the
sensitivity of the experimental setup in question. As long as the final-state
neutrino is not measured, however, its coherence, or lack thereof, is
immaterial to our discussion.

Eq. (II.12) reveals that one is sensitive to the individual neutrino masses
$m_{1},m_{2},m_{3}$ as long as all $|U_{\alpha i}|\neq 0$, which turns out to
be the case. In practice, one needs to account for intrinsic and experimental
uncertainties associated with the initial state and the finite resolution of
the various measuring apparatuses. Taking these uncertainties into account, in
the limit where the neutrino masses are small enough compared to the various
energy scales of the system, one can express777This discussion is meant to be
generic and purely for illustrative and pedagogical purposes. We return to the
specific cases of the charged-lepton energy spectrum of electron-mediated
charged-current processes, including nuclear beta-decay, in more detail in
Sec. IV.

$\Gamma_{\alpha}(m_{i}^{2})=\Gamma_{\alpha}(0)+m_{i}^{2}\frac{{\rm
d}\Gamma_{\alpha}}{{\rm d}m_{i}^{2}}(0)+{\cal O}(m_{i}^{4}).$ (II.13)

Here, $\Gamma_{\alpha}(m_{i}^{2})$ stands for the differential rate convoluted
with the uncertainties associated to the measurement in question (slightly
different from the object in Eq. (II.12)). This object is smooth around
$m_{i}^{2}=0$ in such a way that the series expansion above is meaningful.

To leading order in the neutrino masses, taking uncertainties into account,

$\sum_{i}|U_{\alpha
i}|^{2}\Gamma_{\alpha}(m_{i}^{2})\simeq\left[\Gamma_{\alpha}(0)+m_{\nu_{\alpha}}^{2}\frac{{\rm
d}\Gamma_{\alpha}}{{\rm d}m_{i}^{2}}(0)\right],$ (II.14)

where

$m_{\nu_{\alpha}}^{2}\equiv\sum_{i}|U_{\alpha i}|^{2}m_{i}^{2},$ (II.15)

is an effective neutrino mass-squared associated with the charged-current
processes involving the charged-lepton $\ell_{\alpha}$. In the limit where all
neutrino masses are very small, all such kinematical searches translate into
bounds on different $m^{2}_{\nu_{\alpha}}$.888Different constraints can also
be obtained, at least in theory, from neutral current processes. In practice,
there are no low-energy, high-statistics neutral-current processes one can use
to extract meaningful information. These include, for example,
$\nu_{\mu}e\to\nu_{\mu}e$ scattering and the very rare $K\to\pi\nu\bar{\nu}$
decay.

It is often the case that experiments will quote upper bounds for the square-
root of $m^{2}_{\nu_{\alpha}}$, defined to be
$m_{\nu_{\alpha}}\equiv\sqrt{m^{2}_{\nu_{\alpha}}}$. In turn,
$m_{\nu_{e}},m_{\nu_{\mu}},m_{\nu_{\tau}}$ are sometimes referred to in the
literature as the electron-neutrino, muon-neutrino, and tau-neutrino masses.
This is a practice we would like to strongly discourage since the electron
neutrino, muon neutrino, and tau neutrino are not particles in the technical
sense of the word and do not have well defined masses. Instead, they are
interaction eigenstates and linear superpositions of the neutrino mass
eigenstates. The latter are propagating particles in the strict sense of the
word.

The strongest bound on $m_{\nu_{\tau}}^{2}$ comes from precision measurements
of $\tau$-decays into multi-pion final states. The strongest such bound was
reported by the ALEPH collaboration Barate et al. (1998):
$m_{\nu_{\tau}}<18.2$ MeV at the 95% confidence level. The result is obtained
by combining precision measurements of
$\tau^{-}\to\nu_{\tau}\pi^{-}\pi^{+}\pi^{-}$ – around 3,000 events – and
$\tau^{-}\to\nu_{\tau}2\pi^{-}2\pi^{+}\pi^{-}(\pi^{0})$ – around 60 events. It
has been estimated that an order-of-magnitude improvement is possible if one
were to take advantage of the $\tau$-samples recorded by the B-factories
Kobach and Dobbs (2015) (almost $10^{7}$
$\tau^{-}\to\nu_{\tau}\pi^{\pm}\pi^{-}$).

The strongest bound on $m_{\nu_{\mu}}^{2}$ comes from precision measurements
of pion decay at rest, $\pi^{+}\to\mu^{+}\nu_{\mu}$. The authors of Ref.
Assamagan et al. (1996), analysing the decay at rest of positively charged
pions at PSI, extracted the upper bound $m_{\nu_{\mu}}<0.17$ MeV at the 90%
confidence level, along with the measurement $m_{\nu_{\mu}}^{2}=(-0.016\pm
0.023)$ MeV2. From this experiment, taking the new knowledge of neutrino-
flavor oscillations into account, the currently most precise value for the
charged pion mass is deduced Daum et al. (2019).

Given what is known about neutrino masses from neutrino oscillations and
constraints on $m_{\nu_{e}}^{2}$, to be discussed momentarily, the constraints
on $m_{\nu_{\mu}}^{2}$ and $m_{\nu_{\tau}}^{2}$ discussed above are not
especially relevant when it comes to informing the values of the light
neutrino masses. Indeed, there are no processes involving muons or tau leptons
– today or in the foreseeable future – capable of competing with current and
future information from electron-mediated charged-current processes.

Before proceeding, we highlight that a variety of alternatives to the notation
$m_{\nu_{e}}$ are found in the literature, including $m_{\beta}$, $m_{\nu}$,
and $m_{0}$. Henceforth, we will make use of
$m_{\beta}\equiv\sqrt{\sum_{i}|U_{ei}|^{2}m_{i}^{2}}$, for a few reasons. As
already discussed, $m_{\nu_{e}}$ should be deprecated because the electron
neutrino is not a particle and does not have a mass. The term $m_{\nu}$ is
better but is used often in many different contexts. The term $m_{0}$ is
sometimes defined as the mass of the lightest eigenstate ($m_{1}$ or $m_{3}$
depending on the ordering) and we want to avoid confusing the two different
objects. The choice $m_{\beta}$ is also the one made by the Particle Data
Group Zyla et al. (2020). The quantity $m_{\beta}$ is a particular combination
of the masses of real (propagating) neutrinos, as distinct from the virtual or
effective mass $m_{\beta\beta}$ in $0\nu\beta\beta$, introduced in Sec.II.2,
that does not correspond to propagating neutrinos. As will be shown,
$m_{\beta}\simeq m_{1}$. A kinematic measurement of $m_{\beta}$ simultaneously
determines all 3 eigenmasses, up to a binary uncertainty in the mass ordering.

Assuming the neutrino mixing matrix is unitary,

$m_{\beta}^{2}=m_{1}^{2}+|U_{e2}|^{2}\Delta m_{21}^{2}+|U_{e3}|^{2}\Delta
m^{2}_{31},$ (II.16)

for either mass ordering, keeping in mind that $\Delta m^{2}_{31}$ is positive
for the NMO and negative for the IMO. Since, it turns out, $|U_{e3}|^{2}$ and
$\Delta m^{2}_{21}$ are both quite small the approximation $m_{\beta}\simeq
m_{1}$ works well unless $m_{1}$ is very small. Quantitatively it holds at the
percent level or better for $m_{1}$ values down to 0.05 eV, and never differs
by more than 8 meV even in the limit $m_{1}=0$.999The second and third terms
in Eq. (II.16) are $|U_{e2}|^{2}\Delta m_{21}^{2}\sim 3\times 10^{-5}$ eV2,
and $|U_{e3}|^{2}\Delta m_{31}^{2}\sim\pm 6\times 10^{-5}$ eV2. In brief, one
can say that, to a good approximation, beta decay and electron capture measure
the mass $m_{1}$, independent from the mass ordering.

The strongest bound on $m_{\beta}$ comes from precision measurements of
tritium beta decay. This is the subject of the bulk of this review. The KATRIN
experiment, after collecting data for four weeks, established the strongest
bound to date, $m_{\beta}<1.1$ eV at the 90% confidence level Aker et al.
(2019), associated with the measurement $m_{\beta}^{2}=-1.0^{+0.9}_{-1.1}$
eV2, consistent with zero. The ultimate sensitivity of KATRIN is to
$m_{\beta}>0.2$ eV (90% confidence level). KATRIN is discussed in more detail
in Sec. V.2.

Direct searches for kinematic effects of nonzero neutrino masses are, for the
most part, model independent and rely only on the conservation of energy and
momentum to measure the neutrino mass. They do not depend, for example,
significantly on whether neutrino scattering is exactly described by the
Standard Model of particle physics. For a careful, recent exploration of new-
interaction effects on the extraction of $m_{\beta}$, see Ludl and Rodejohann
(2016). To leading order, and for all practical purposes, direct searches for
kinematical effects of nonzero neutrino masses also do not depend on the
nature of the neutrinos – Majorana or Dirac fermions. The fact that there is a
charged lepton with a well defined charge in the final or initial state
renders the leading order amplitudes identical for Majorana and Dirac
neutrinos. The same is true of all relevant QED corrections. At higher order
in the weak interactions, however, there are unobservably small differences
between Majorana and Dirac neutrinos. These differences are not only
suppressed by the Fermi constant to some power but are also proportional to
the neutrino masses. A concrete example is the electron spectrum associated
with the five-body final-state neutron decay, $n\to
pe^{-}{\bar{\nu}}_{i}\nu_{j}\bar{\nu}_{k}$ in the Dirac case, $n\to
pe^{-}\nu_{i}\nu_{j}\nu_{k}$ in the Majorana case, where $i,j,k=1,2,3$, the
different mass eigenstates. In the Majorana case, for example, there are
interference effects between $\nu_{i}$ and $\nu_{j}$ when $i=j$, if $m_{i}$ is
not zero. These are clearly not present in the Dirac case. For illustrations
of this phenomenon, see Berryman et al. (2018) for a recent discussion of very
low-energy $e\gamma$ scattering into neutrinos or Millar et al. (2018) for a
discussion of the end point of the bremsstrahlung spectrum of coherent
neutrino scattering on nuclei ($\nu+N\to\nu+N+\gamma$).

The presence of new “neutrino” states, however, modifies the interpretation of
results from these types of experiments. Indeed, the existence of new,
relatively heavy, “neutrino” states is strongly constrained by precision
measurements of $\beta$-decay, meson-decay, tau-decay, etc. For recent reviews
see, for example, de Gouvêa and Kobach (2016); Drewes and Garbrecht (2017);
Fernandez-Martinez et al. (2016); Drewes et al. (2017); Bryman and Shrock
(2019a, b); Bolton et al. (2020). We return to this issue later in this
subsection and in Sec. VI.

In the absence of new, light degrees of freedom, the KATRIN result can be used
to set a robust upper bound on the neutrino masses. The KATRIN bound, combined
with results from the current oscillation data, translates into (two
significant digits)

$\displaystyle 0<$ $\displaystyle~{}m_{1}^{2}~{}$ $\displaystyle<1.2~{}{\rm
eV}^{2},$ (II.17) $\displaystyle 7.4\times 10^{-5}~{}{\rm eV}^{2}<$
$\displaystyle~{}m_{2}^{2}~{}$ $\displaystyle<1.2~{}{\rm
eV}^{2},~{}~{}~{}~{}~{}~{}~{}(\rm NMO)$ (II.18) $\displaystyle 2.5\times
10^{-3}~{}{\rm eV^{2}}<$ $\displaystyle~{}m_{3}^{2}~{}$
$\displaystyle<1.2~{}{\rm eV}^{2},$ $\displaystyle\rm or$ $\displaystyle
2.4\times 10^{-3}~{}{\rm eV}^{2}<$ $\displaystyle~{}m_{1}^{2}~{}$
$\displaystyle<1.2~{}{\rm eV}^{2},$ (II.20) $\displaystyle 2.5\times
10^{-3}~{}{\rm eV}^{2}<$ $\displaystyle~{}m_{2}^{2}~{}$
$\displaystyle<1.2~{}{\rm eV}^{2},~{}~{}~{}~{}~{}~{}~{}(\rm IMO)$ (II.21)
$\displaystyle 0<$ $\displaystyle~{}m_{3}^{2}~{}$ $\displaystyle<1.2~{}{\rm
eV}^{2},$ (II.22)

where the bounds are heavily correlated given the constraints on the mass-
squared differences. These are, arguably, the most robust, model-independent
upper bounds on all three neutrino masses. Figure 4 depicts the values of the
three light neutrino masses as a function of $m_{\rm least}$, for both
neutrino-mass orderings.

Figure 4: Current best-fit values of the neutrino masses $m_{1},m_{2},m_{3}$
as a function of the lightest neutrino mass, for the normal mass-ordering
(top) and the inverted mass ordering (bottom).

Above, we highlighted the fact that the relation between precision
measurements of the $\beta$-decay spectrum and the neutrino masses, given what
we know from oscillation experiments, is very robust. The main exception to
this robustness is the presence of new neutrino mass eigenstates. In a
nutshell, these manifest themselves in two different ways. If the new neutrino
masses – referred to, here, as $m_{4}$ – are “large,” the presence of the
extra heavy neutrino will distort the $\beta$-decay spectrum. If, instead, the
new neutrino masses are “small,” the presence of the extra neutrino will
simply add to $m_{\beta}^{2}$, i.e., the sum in Eq. (II.15) would encompass
all mass eigenvalues $m_{i}$, $i=1,2,3,\ldots$ for all $m_{i}$ less than a few
eV. For more details see, for example, de Gouvêa et al. (2007); Formaggio and
Barrett (2011); Esmaili and Peres (2012). The KATRIN collaboration recently
made available the results of a search for new neutrino states Aker et al.
(2020); we discuss it in detail later in this review, see Sec. VI.1. As an
aside, new neutrino mass states that admix with electron-flavor neutrinos only
render the electron-weighted mass-squared parameter $m^{2}_{\beta}$ larger.
Hence upper bounds to $m^{2}_{\beta}$ are especially robust and cannot be
bypassed by postulating the existence of new particles.

##  III Neutrino Mass Models and New Physics

In this section, we briefly discuss how direct searches for neutrino masses
inform our understanding of the origin of neutrino mass and can be used to
discover other new physics.

###  III.1 Neutrino mass models

Similar to all fermions in the Standard Model of particle physics, the known
neutrinos can acquire nonzero masses only after electroweak symmetry breaking.
Unlike charged fermions, the dynamical mechanism behind nonzero neutrino
masses is unknown. Identifying the physics responsible for neutrino masses is
among the most important questions in particle physics today.

There are several qualitatively different models capable of explaining why
neutrinos have mass. While all of them require the existence of new degrees of
freedom, the nature of the new degrees of freedom – one or several new states,
fermions or bosons, light or heavy new states, etc – varies dramatically.
Given the high degree of uncertainty, information on the origin of neutrino
masses may come from a large range of experimental efforts, from searches for
rare muon processes (e.g. $\mu\to ee^{+}e^{-}$ decays) to the Large Hadron
Collider to next-generation neutrino-oscillation experiments. Direct
measurements of the neutrino mass, along with pursuits of lepton-number
violation, are guaranteed to provide nontrivial information.

Parallel to the origin of neutrino mass, there is the issue of the pattern of
lepton mixing. Unlike quarks, the mixing angles that parameterize the PMNS
matrix are all large – the smallest lepton mixing angle is almost as large as
the largest quark mixing angle – and the potential organizing principles
responsible for its observed features may be qualitatively different. Several
of the theoretical approaches to the problem of lepton flavor also make
predictions for the values of the neutrino masses, which will be informed most
straightforwardly by direct searches for the kinematical effects of masses.

Direct measurements of the neutrino masses can help reveal if the lightest
neutrino is massless. Knowledge of the masslessness of the lightest neutrino
would impact, very significantly, our understanding of the origin of neutrino
masses. For example, if neutrinos are Dirac fermions, $m_{\rm least}=0$ allows
one to contemplate that there are only two right-handed neutrino fields, in
stark contrast to all other fermionic degrees of freedom in the Standard Model
that come in three flavors. The same is true if the neutrinos are Majorana
fermions and their masses are a consequence of the so-called Type-I seesaw
mechanism Minkowski (1977); Gell-Mann et al. (1979); Yanagida (1979); Glashow
(1980); Mohapatra and Senjanovic (1980); Schechter and Valle (1980). In this
case, $m_{\rm least}=0$ translates into the possibility that there are only
two right-handed neutrinos.101010Strictly speaking, even if there are only two
right-handed neutrinos, the lightest neutrino mass is expected to be nonzero,
generated at the two-loop level even in the absence of new neutrino
interactions. In this case, however, $m_{\rm least}$ is expected to be many
orders of magnitude lighter than the other neutrino masses Davidson et al.
(2007). In many other scenarios, there is no natural way to “explain” why the
lightest neutrino should be massless or much lighter than the other two.

Experimentally, of course, it is impossible to determine that $m_{\rm least}$
is exactly zero since $m_{\rm least}=0$ and $m_{\rm least}\ll\Delta
m^{2}_{21},|\Delta m^{2}_{31}|$ are, in practice, indistinguishable. Instead,
one could determine that $m_{\rm least}\neq 0$ with some confidence. More
quantitatively,

$\displaystyle m_{\beta}^{2}$ $\displaystyle=$ $\displaystyle m^{2}_{\rm
least}+|U_{e2}|^{2}\Delta m^{2}_{21}+|U_{e3}|^{2}\Delta m^{2}_{31},$ (III.1)
$\displaystyle=$ $\displaystyle m^{2}_{\rm least}+7.74\times 10^{-5}~{}{\rm
eV}^{2},\hskip 28.45274pt\rm(NMO)$ $\displaystyle\rm or$ $\displaystyle
m_{\beta}^{2}$ $\displaystyle=$ $\displaystyle m^{2}_{\rm
least}+|U_{e1}|^{2}(-\Delta m^{2}_{32}-\Delta m^{2}_{21})+|U_{e2}|^{2}(-\Delta
m^{2}_{32}),$ (III.3) $\displaystyle=$ $\displaystyle m^{2}_{\rm
least}+2.47\times 10^{-3}~{}{\rm eV}^{2},\hskip 31.2982pt\rm(IMO)$ (III.4)

using, in accordance with Ref. Esteban et al. (2020), $|U_{e1}|^{2}=0.681$,
$|U_{e2}|^{2}=0.297$ and $|U_{e3}|^{2}=0.022$. Hence, to establish that
$m^{2}_{\rm least}\neq 0$, one needs to constrain, in a statistically
significant way, $m_{\beta}^{2}>7.74\times 10^{-5}$ eV2
($m_{\beta}^{2}>2.47\times 10^{-3}$ eV2)111111The magnitudes of the mass-
squared differences and $|U_{ei}|^{2}$, $i=1,2,3$ entries of the PMNS matrix
are currently known at the 4% level or better. Taking uncertainties into
account, the upper bounds quoted here are known at the 5% level (one sigma).
assuming the neutrino mass ordering is known to be normal (inverted).

Another qualitatively different hypothesis that, if confirmed, would impact
our understanding of the origin of neutrino masses is the possibility that all
three neutrino masses are quasi-degenerate. Even if all current bounds on
neutrino masses are taken at face value, this is, experimentally, still an
option. For example, if $m_{\rm least}=0.1$ eV, only in slight tension with
the most stringent indirect constraints from cosmic surveys Aghanim et al.
(2020),

$m_{\rm most}-m_{\rm least}\sim 0.12{\frac{(m_{\rm most}+m_{\rm least})}{2}},$
(III.5)

where $m_{\rm most}$ is the heaviest neutrino mass, for both mass orderings.
Current constraints allow neutrino masses that are almost degenerate – all of
the same order of magnitude – especially if one considers that the cosmology
bounds can be significantly alleviated with the introduction of new
ingredients. Experiments sensitive down to $m_{\beta}^{2}\sim 0.01$ eV2 can
definitively test the hypothesis that the neutrino masses are almost
degenerate.

###  III.2 Sensitivity to new phenomena

As discussed in Sec. II, cosmic surveys, searches for $0\nu\beta\beta$, and
direct kinematic measurements of neutrino masses are all sensitive to the
values of the neutrino masses. The first two probes are indirect. They rely on
other ingredients that govern the expansion history of the universe, on the
absence of new interactions involving neutrinos, on the Majorana nature of
neutrinos, on the absence of more, directly accessible lepton-number violating
interactions, etc. This means that by combining these different probes of the
values of the neutrino masses we can verify whether the assumptions that go
into relating cosmic surveys and the rate for $0\nu\beta\beta$ to the values
of the neutrino masses are valid.

In the absence of new particle physics and new cosmology-related ingredients,
$m_{\beta}$, $m_{\beta\beta}$, and $\Sigma$ are strictly correlated. In
particular, assuming the neutrino oscillation parameters are known, it is
trivial to express both $m_{\beta\beta}$ and $\Sigma$ as functions of
$m_{\beta}$ – see Eqs. (III.1, III.4). These are depicted in Figs. 5 and 6,
for both mass orderings. Here we assume neutrinos are Majorana fermions; for
Dirac fermions, the rate for $0\nu\beta\beta$ is zero. In the case of
$m_{\beta\beta}$, the bands are a consequence of all possible values of the
relative Majorana phases, currently completely unconstrained. For everything
else, we use the current best-fit values of the oscillation parameters from
Esteban et al. (2020). The relevant oscillation parameters –
$\sin^{2}\theta_{13},\sin^{2}\theta_{12},\Delta m^{2}_{21},\Delta m^{2}_{31}$
– are all known at better than the 4% level.

Figure 5: $m_{\beta\beta}$ as a function of $m_{\beta}$, for both the normal
(lighter, blue) and inverted (darker, red) mass orderings. The bands are a
consequence of allowing for all possible values of the relative Majorana
phases. For everything else, we use the current best-fit values of the
oscillation parameters from Esteban et al. (2020). The whited-out region
inside the light-blue contour is meant to highlight the values of $m_{\beta}$
for which $m_{\beta\beta}$ can vanish exactly. We assume the neutrinos are
Majorana fermions. If neutrinos are Dirac fermions, $m_{\beta\beta}=0$. The
grey, horizontal band corresponds to the 95% CL upper bound on
$m_{\beta\beta}$ from GERDA Agostini et al. (2020). The width of the band is a
consequence of uncertainties in the nuclear matrix element for the
neutrinoless double-beta decay of 76Ge. The vertical line corresponds to the
current 90% upper bound on $m_{\beta}$ Aker et al. (2019).  Figure 6: $\Sigma$
as a function of $m_{\beta}$, for both the normal (blue,solid) and inverted
(red,dashed) mass orderings. We use the current best-fit values of the
oscillation parameters from Esteban et al. (2020). The, horizontal band
corresponds to the range of 95% CL upper bounds on $\Sigma$ discussed in Roy
Choudhury and Hannestad (2020). Different upper bounds correspond to different
ingredients added to the Standard Model of cosmology. The vertical line
corresponds to the current 90% upper bound on $m_{\beta}$ Aker et al. (2019).

Figs. 5 and 6 allow one to identify circumstances that would imply the
existence of new phenomena. For example, independent from the neutrino mass
ordering, if precision measurements of tritium $\beta$-decay revealed that
$m_{\beta}$ is larger than 0.05 eV, cosmic surveys imply the existence of new
cosmology-related ingredients. This is a very robust statement. Even if,
ultimately, we find that new neutrino mass eigenstates are the dominant
contribution to $m_{\beta}$, their existence, given that their mixing with
active neutrinos is rather large, is ruled out by cosmic surveys in the
absence of new ingredients. See, for example, the bounds on the effective
number of neutrinos, $N_{\rm eff}$, in, for example, Aghanim et al. (2020).

If, on the other hand, $m_{\beta}$ is constrained to be smaller than 0.1 eV
and one finds $m_{\beta\beta}$ to be larger than 0.1 eV – the sensitivity of
current experiments approaches $m_{\beta\beta}\sim 0.1$ eV (Table 1) – one
will be required to conclude, again independent from the mass ordering, that
there are contributions to $0\nu\beta\beta$ other than neutrino exchange.

##  IV Kinematic determination from beta decay

In beta decay the energy available from the nuclear mass difference is carried
away by the electron and the neutrino. The two particles share the energy in a
statistical way, determined quantum mechanically by the available phase space
for each. Because the electron cannot abscond with all the energy if the
neutrino has rest mass, that small amount of energy alters the electron
spectrum near its endpoint where it would otherwise have taken all the energy.
The beta spectrum in the presence of neutrino mass has a simple analytic form
that reflects the available phase space.

The relative influence of neutrino mass on the spectrum compared to the
available energy is maximized by choosing isotopes with the smallest Q-values.
As we discuss, however, a low Q-value alone does not guarantee a good basis
for an experiment.

###  IV.1 Beta spectrum

As discussed in Sec. II, the fact that the three eigenmasses $m_{1}$, $m_{2}$,
and $m_{3}$ are linked by neutrino-oscillation data has simplified the
experimental task of determining the mass scale because it is now possible to
work with beta decay alone; separate determinations involving $\mu$ and $\tau$
leptons are no longer needed. The most sensitive direct searches for
$m_{\beta}$ to date are based on the investigation of the electron spectrum of
tritium $\beta$-decay. The electron energy spectrum of $\beta$-decay for a
neutrino with component masses $m_{1},m_{2},$ and $m_{3}$ is the incoherent
sum of the contributions from each mass eigenstate:

$\displaystyle\frac{d\Gamma}{dE}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}|V_{ud}|^{2}}{2\pi^{3}}(G_{V}^{2}+3G_{A}^{2})F(Z,\beta)\beta(E+m_{e})^{2}(E_{0}-E)$
(IV.1)
$\displaystyle\times\sum_{i=1,3}|U_{ei}|^{2}\left[(E_{0}-E)^{2}-m_{i}^{2}\right]^{\frac{1}{2}}\Theta(E_{0}-E-m_{i}),$

where $G_{F}$ is the Fermi coupling constant, $V_{ud}$ is an element of the
CKM matrix Zyla et al. (2020), $E$ ($\beta$) denotes the electron’s kinetic
energy (velocity), $E_{0}$, the ‘endpoint energy,’ corresponds to the maximum
kinetic energy in the absence of neutrino mass, $F(Z,\beta)$ is the Fermi
function, taking into account the Coulomb interaction of the outgoing electron
in the final state, and $\Theta(E_{0}-E-m_{i})$ is the step function that
ensures energy conservation. The vector and axial-vector matrix elements are
$G_{V}=1$ and $G_{A}=-1.2646(35)$ for tritium, respectively Akulov and Mamyrin
(2005). Henceforth we gather the leading constants into a single one,

$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}|V_{ud}|^{2}}{2\pi^{3}}(G_{V}^{2}+3G_{A}^{2}).$
(IV.2)

The relationships between $E_{0}$, the Q-value, and the atomic mass difference
are detailed in Bodine et al. (2015) and one is given in Eq. (V.1). As both
the matrix elements and $F(Z,\beta)$ are independent of $m_{i}$, the
dependence of the spectral shape on $m_{i}$ is given by the phase space factor
only.

The beta spectrum near the endpoint can be written in a simplified form for
discussion,

$\displaystyle\frac{d\Gamma}{dE}$ $\displaystyle\approx$ $\displaystyle
3r_{0}(E_{0}-E)\sum_{i=1,3}|U_{ei}|^{2}[(E_{0}-E)^{2}-m_{i}^{2}]^{1/2}\Theta(E_{0}-E-m_{i}),$
(IV.3)

where $r_{0}$ is the detected event rate per atom in the last eV of the
spectrum in the absence of mass (if $E$ and $E_{0}$ are also in eV). The
variables $\beta$ and $E+m_{e}$ and the function $F(Z,\beta)$ are evaluated at
the endpoint and absorbed in the constant $r_{0}$.

$\displaystyle r_{0}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{C_{t}E_{0}^{3}}$ (IV.4)

where $\lambda=1.784\times 10^{-9}$ s-1 is the tritium decay constant. The
constant $C_{t}$ is the ratio of the integral of the spectrum of the form of
Eq. (IV.1) to one of the form of Eq. (IV.3), renormalized by any molecular or
atomic branching to states populating the final eV of the spectrum. The value
of $C_{t}\simeq 1$ is more precisely 0.943 for molecular tritium, 0.767 for
atomic tritium, and 0.537 for a bare tritium nucleus, when Simpson’s form
Simpson (1981) of the Fermi function is used:

$\displaystyle F(Z,\beta)$ $\displaystyle\simeq$ $\displaystyle 2\pi
Z\alpha\frac{1.002037-0.001427\beta}{\beta(1-e^{-2\pi Z\alpha/\beta})},$
(IV.5)

where $\alpha$ is the fine-structure constant and $Z$ is the charge on the
daughter nucleus. For the bare nucleus, the extrapolated endpoint energy in
the laboratory is 18522.44 eV Bodine et al. (2015), and the fractional
intensity in the last eV of the spectrum is $2.93\times 10^{-13}$. One can
also write the constant $C$ in terms of the decay constant,

$\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{3\lambda}{C_{t({\rm
nuclear})}E_{0}^{3}F(Z,\beta_{0})\beta_{0}(E_{0}+m_{e})^{2}}$ (IV.6)

where $\beta_{0}$ is the value of $\beta$ at the endpoint. If there are $N$
atoms in the source, the total rate in the last eV is $r=Nr_{0}$ and the total
activity is $C_{t}rE_{0}^{3}$. Figure 7 shows the shape of the spectrum near
the endpoint, where the effects of neutrino mass are most pronounced.

Figure 7: Beta spectrum of the decay of atomic tritium near the endpoint, as
given by Eq. (IV.9), from Kleesiek et al. (2019). In the figure,
$m_{\nu}\equiv m_{\beta}$.

Neutrino mass experiments in beta decay are fundamentally just counting
experiments. The neutrino mass $m_{\beta}$ can in principle be determined or
limited from a single measurement of the number of events in a suitably chosen
interval $\Delta E$, the ‘analysis window’, that ends at the extrapolated
endpoint energy, as long as other parameters, namely the rate, time, endpoint
energy, and background, are known well enough from other information. This is
an idealization but not unrealistic for experiments like Project 8 and
calorimetric detectors (described below) where data both on the background
above and the spectrum below the endpoint are automatically taken “for free”
because all events are recorded as they occur. The principle would also apply
to an experiment like KATRIN Angrik et al. (2004) that collects integral
spectral data point–by–point, but with additional time spent to obtain the
needed information. This ‘time expansion ratio’ is discussed in Sec. VIII.1.
The endpoint energy is not needed in an absolute sense; it need only be
determined relative to $\Delta E$ from the shape of the spectrum outside that
window.

The total number of signal events $N_{s}$ in time $t$ in this window is
obtained by integrating Eq. (IV.3),

$\displaystyle N_{s}$ $\displaystyle=$ $\displaystyle
rt\sum_{i=1,3}|U_{ei}|^{2}[(\Delta E)^{2}-m_{i}^{2}]^{3/2}$ (IV.7)
$\displaystyle\simeq$ $\displaystyle rt(\Delta
E)^{3}\left[1-\frac{3}{2}\frac{\sum_{i=1,3}|U_{ei}|^{2}m_{i}^{2}}{(\Delta
E)^{2}}\right].$ (IV.8)

In the last step we invoke the unitarity of the PMNS matrix and the assumption
that $(\Delta E)^{2}\gg\sum_{i=1,3}|U_{ei}|^{2}m_{i}^{2}$ to allow a first-
order expansion. The latter assumption exploits the fact that neutrino mass
experiments generally explore masses considerably smaller than the
instrumental widths and backgrounds for which $\Delta E$ is optimized. The
summation in Eq. (IV.8) was defined earlier as

$\displaystyle m_{\beta}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i=1,3}|U_{ei}|^{2}m_{i}^{2}$

that is used as a single parameter representing the result of a beta-decay
measurement wherein the individual mass eigenstates are not resolved, and Eq.
(IV.8) motivates the replacement. With this replacement, the simplified beta
spectrum becomes

$\displaystyle\frac{d\Gamma}{dE}$ $\displaystyle\approx$ $\displaystyle
3r_{0}(E_{0}-E)[(E_{0}-E)^{2}-m_{\beta}^{2}]^{1/2}\Theta(E_{0}-E-m_{\beta}).$
(IV.9)

###  IV.2 Isotopes of Interest

Neutrino mass affects the shape of the beta spectrum only near the endpoint.
Because the spectrum rises quadratically from the endpoint, the fraction of
decays that produce events in a region of width $m_{\beta}$ at the endpoint
scales as $(m_{\beta}/Q)^{3}$, which means that a low Q-value is advantageous,
other things being equal. This generalization ignores details of the spectral
shape at lower energies, but is sufficient to guide attention to suitable
isotopes. A low Q value is very desirable Kopp and Merle (2010), but another
important factor is the specific activity of the source. In Table 2 low-Q-
value candidates are compared via a benchmark decay rate in the last eV of the
spectrum.

Table 2: Source mass required to produce 1 event per day in the last eV of the spectrum. $Q_{A}$ is the atomic mass difference. Isotope | Spin-Parity | Half-life | Specific Activity | $Q_{A}$ | Branching ratio | Last eV | Source Mass
---|---|---|---|---|---|---|---
|  | y | Bq/g | eV |  |  | g
3H2 |  $\nicefrac{{1}}{{2}}$${}^{+}\rightarrow\ $$\nicefrac{{1}}{{2}}$+ | 12.3 | $3.6\times 10^{14}$ | 18591 | 0.57 | $2.9\times 10^{-13}$ | $2.0\times 10^{-7}$
115In |  $\nicefrac{{9}}{{2}}$${}^{+}\rightarrow\ $$\nicefrac{{3}}{{2}}$+ | $4.4\times 10^{14}$ | 0.26 | 147 | $1.2\times 10^{-6}$ | $5.0\times 10^{-7}$ | $7.5\times 10^{7}$
135Cs |  $\nicefrac{{7}}{{2}}$${}^{+}\rightarrow\ $$\nicefrac{{11}}{{2}}$- | $1.5\times 10^{6}$ | $6.8\times 10^{7}$ | 440 | $(0.04-16)\times 10^{-6}$ | $2.2\times 10^{-8}$ | 0.4 - 217
187Re |  $\nicefrac{{5}}{{2}}$${}^{+}\rightarrow\ $$\nicefrac{{1}}{{2}}$- | $4.3\times 10^{10}$ | $1.6\times 10^{3}$ | 2470 | 1.0 | $1.2\times 10^{-10}$ | 57
163Ho |  $\nicefrac{{7}}{{2}}$${}^{-}\rightarrow\ $$\nicefrac{{5}}{{2}}$- | $4750$ | $1.8\times 10^{10}$ | 2858 |  | $\sim 10^{-12}$ | $\sim 1.0\times 10^{-5}$

Even though tritium has the highest Q-value of the four, its superallowed beta
decay and low atomic mass have made it the isotope of choice through 70 years
of direct mass searches. The low Q-values of 115In, 135Cs, and 187Re are
outweighed by the forbidden nature of the beta decays. Calorimetric
measurements of the 187Re decay were carried out successfully down to a
neutrino mass limit of 15 eV Ferri et al. (2015), but pressing on much further
would require prohibitively large source masses. Similarly, were it not for
the source mass, the 115In Cattadori et al. (2005, 2007); Mount et al. (2009);
Urban et al. (2016); Zheltonozhsky et al. (2018) and 135Cs de Roubin et al.
(2020) decays would be compelling for their low Q-values. The In transition is
accompanied by a prompt gamma that could be used for background reduction.

A different approach was suggested by De Rújula De Rújula and Lusignoli
(1982), who noted that 163Ho has a very low Q-value. This isotope decays by
electron capture and emits neutrinos instead of antineutrinos. The visible
energy release is dominated by sharp lines corresponding to vacancies created
in various atomic shells, but the Lorentzian tails of the lines extend to a
kinematic endpoint that is sensitive to neutrino mass, just as in beta decay.
There is no simple prescription for the branch to the last eV, but recent work
Ranitzsch et al. (2017) reports a Q-value of 2858(11) eV. A rough estimate of
the source mass needed for an equivalent sensitivity of about 1 eV has been
extracted from Nucciotti et al. (2018), and this information is also included
in Table 2. Holmium is a viable candidate at this basic level. Recent
experimental work from the ECHo collaboration Velte et al. (2019) has yielded
a Q-value of 2838(14) eV, and a limit on the neutrino mass of 150 eV.

##  V Experimental Progress

Experiments focused specifically on determining the ‘mass of the neutrino’ (as
it was then thought to be) began in 1948 with two contemporaneous experiments
on the beta decay of tritium, which was known to have a low decay energy.
Tritium is the simplest radioactive isotope and has the highest specific
activity. By the 1970s it was clear that neutrino mass effects were small
enough that molecular and atomic effects competed, leading to the “final-
state” problem. If anything, this cemented the role of tritium because of its
simple atomic structure. However, a different approach that could circumvent
the final-state problem completely, the microcalorimeter, emerged and has been
the scene of intensive technical development. In this section the chronology
and status of the experimental research on the three viable isotopes, tritium,
163Ho, and 187Re, are presented.

The final-state problem takes a different shape for each isotope and has not
been completely circumvented, as we describe in Sec. VII. Henceforth for
brevity we replace 3H with T to denote tritium symbolically.

###  V.1 Tritium beta decay

The first experiments to quantitatively constrain the mass of the neutrino
took place in 1948 with one in Glasgow and the other in Chalk River. Both made
use of gaseous tritium in a proportional counter. Over the subsequent half-
century, limits on the mass were pushed down thanks to experimental and
conceptual improvements, with a wide variety of instruments. The experiments
are summarized in Table 3 (some experiments, for example Sun et al. (1993),
are not included for lack of information).

Table 3: Neutrino mass experiments with tritium. Units: eV. Group | Date | Source | Spectrometer | Limit or mass |  | Ref.
---|---|---|---|---|---|---
Curran et al. | 1948 | T2 | Proportional counter | $<1700$ |  | Curran et al. (1948)
Hanna & Pontecorvo | 1949 | T2 | Proportional counter | $<500$ |  | Hanna and Pontecorvo (1949)
Curran et al. | 1949 | T2 | Proportional counter | $<1000$ |  | Curran et al. (1949)
Langer & Moffat | 1952 | T:Succinic acid | Magnetic | $<250$ |  | Langer and Moffat (1952)
Hamilton et al. | 1953 | T:Zr | Electrostatic | $<200$ |  | Hamilton et al. (1953)
Salgo & Staub | 1969 | T2O | Electrostatic | $<320$ |  | Salgo and Staub (1969)
Daris & St. Pierre | 1969 | T:Al | Magnetic | $<75$ |  | Daris and St-Pierre (1969)
Bergkvist | 1972 | T:Al | Magnetic | $<55$ |  | Bergkvist (1972a, b)
Röde & Daniel | 1972 | T:Polystyrene | Magnetic | $<86$ |  | Röde and Daniel (1972)
ITEP | 1980 | T:Valine | Magnetic | $=30\pm 16$ |  | Lyubimov et al. (1980, 1981)
Simpson | 1981 | T:Si | Si(Li) | $<65$ |  | Simpson (1981)
ITEP | 1985 | T:Valine | Magnetic | $=35^{+2}_{-15}$ |  | Boris et al. (1985)
Zurich | 1986 | T:C | Magnetic | $<18$ |  | Fritschi et al. (1986)
ITEP | 1987 | T:Valine | Magnetic | $=30^{+2}_{-13}$ |  | Boris et al. (1987)
LANL | 1987 | T2 | Magnetic | $<27$ |  | Wilkerson et al. (1987)
INS Tokyo | 1988 | T:CdArachidate | Magnetic | $<29$ |  | Kawakami et al. (1988)
INS Tokyo | 1991 | T:CdArachidate | Magnetic | $<13$ |  | Kawakami et al. (1991)
LANL | 1991 | T2 | Magnetic | $<9.3$ |  | Robertson et al. (1991)
Zurich | 1992 | T:C | Magnetic | $<11$ |  | Holzschuh et al. (1992)
Mainz | 1993 | T2 | MAC-E | $<7.2$ |  | Weinheimer et al. (1993)
Troitsk | 1994 | T2 | MAC-E | $<4.35$ |  | Belesev et al. (1995)
Mainz | 1998 | T2 | MAC-E | $<2.8$ |  | Weinheimer et al. (1999)
Mainz | 2005 | T2 | MAC-E | $<2.3$ |  | Kraus et al. (2005)
Troitsk | 2011 | T2 | MAC-E | $<2.05$ |  | Aseev et al. (2011)
KATRIN | 2020 | T2 | MAC-E | $<1.1$ |  | Aker et al. (2019)

The experiment of Bergkvist Bergkvist (1971, 1972a, 1972b) ushered in the
modern era with an advanced spectrometer shown in Fig. 8. With this device, he
reached a sensitivity that was limited by atomic and molecular effects that
modify the shape of the spectrum near the endpoint.

Figure 8: Illustration of the electrostatic-magnetic spectrometer used by
Bergkvist in 1972 Bergkvist (1971).

There was a flurry of excitement in the early 1980s when a group in the Soviet
Union reported a non-zero mass of 30 eV, the right size to close the universe
gravitationally. The result was erroneous, probably owing to some combination
of limited understanding of the final state spectrum of the complicated
tritiated molecule used, the amino acid valine, and the energy loss in the
source. Two experiments, at Los Alamos and Livermore National Laboratories,
made use of gaseous molecular T2, for which the final-state spectrum could be
well calculated. The Los Alamos apparatus is shown in Fig. 9.

Figure 9: The LANL tritium beta decay experiment Knapp (1986); Wilkerson et
al. (1987); Robertson et al. (1991). A windowless gaseous source of molecular
T2 on the left produces beta electrons that are guided by solenoid coils to a
focal-point collimator at the entrance to a magnetic spectrometer. The
spectrometer is the Tret’yakov cascaded iron-free toroidal type Tret’yakov
(1975) in which electrons cross the axis four times before reaching the
silicon multipixel detector at the right-hand end.

While those experiments could rule out the Soviet result with much greater
sensitivity, they ultimately produced mass-squared values that were apparently
negative. That happens when there are more counts in the endpoint region than
expected, and it was eventually shown in 2015 Bodine et al. (2015) that the
problem was, once again, the final-state distribution. More recent
calculations of this distribution Saenz et al. (2000a) resolve this problem,
and with the new calculations the Los Alamos and Livermore data are consistent
with zero mass. More information on this is given in the section on final-
state distributions, Sec. VII.1. Other experiments also reported negative
mass-squared values that were eventually traced to different systematic
errors. The early results from the Mainz experiment, which used as a source a
frozen film of tritium Weinheimer et al. (1993), were affected by dewetting of
the film at temperatures near 4K. The microcrystalline ‘frost’ that resulted
caused the line broadening from energy loss to be anomalously large. Lowering
the temperature to 2K stabilized the films and eliminated the negative mass
squared effect Kraus et al. (2005). The Troitsk experiments exhibited evidence
for a step in the spectrum a few eV below the endpoint. While a specific
explanation for the spectral shape has not been found, the effect was found to
be associated with runs during which the source pressure was not monitored
Aseev et al. (2011). Excluding those runs from analysis eliminated the step.

The limits on neutrino mass from tritium beta decay as a function of time are
shown in Fig. 10 (not all results in Table 3 are included).

Figure 10: Upper limits on the neutrino mass obtained from tritium beta decay.
The point with error bars is a non-zero result (see text). HDM: Hot Dark
Matter.

The plot reveals a striking Moore’s-Law character over 70 years. Also
indicated on the plot is the mass that would close the universe with neutrinos
(hot dark matter) alone (HDM $\Omega=1$), the electron-weighted mass
corresponding to the smallest possible value in the inverted mass ordering,
and similarly for the normal mass ordering.

A closely related experimental quantity is the atomic mass difference between
T and 3He. While neutrino mass can only be deduced from beta decay, the mass
difference can be determined both from beta decay and from independent mass
spectrometry methods. The comparison serves as a uniquely valuable check on
possible systematic effects that might influence the beta decay experiments,
with no other symptoms. Indeed, it was mass spectroscopy that initially
supported and then finally contradicted the ITEP claim of a non-zero neutrino
mass Staggs et al. (1989). Agreement of the two kinds of determination within
uncertainties is a necessary, although not sufficient, condition for a valid
neutrino mass result Staggs et al. (1989). As the state of the art in both
fields of measurement advances, however, the power of this comparison is
beginning to diminish because of the presence of work-function differences
that can shift the beta endpoint by fractions of an eV. This also has the
consequence that highly precise atomic mass determinations cannot be used to
improve the sensitivity of neutrino mass measurements by fixing a fit
parameter. Table 4 gives recent determinations of the atomic mass difference.

Table 4: Atomic mass difference between T and 3He. Units: eV. Group | Year | Mass Difference | Ref.
---|---|---|---
Univ. Washington | 1993 | 18590.1(17) | Van Dyck et al. (1993)
SMILETRAP | 2006 | 18589.8(12) | Nagy et al. (2006)
Florida State Univ. | 2015 | 18592.01(7) | Myers et al. (2015)
KATRIN | 2019 | 18591.5(5) | Aker et al. (2019)

The measured quantity in the beta decay of molecular tritium is the ground-
state to ground-state extrapolated endpoint energy $E_{0}$, which is related
to the atomic mass difference $Q_{A}$ by

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle
Q_{A}-b_{0}+b_{(f)0}-E_{\rm rec},$ (V.1)

where $b_{0}=4.59$ eV is the binding energy of the initial molecular state,
$b_{(f)0}=-11.71$ eV is the binding energy of the final molecular state and
$E_{\rm rec}=1.705$ eV is the recoil energy Bodine et al. (2015). The specific
values are for T2 beta decay. With these corrections, a measurement of the
endpoint from the KATRIN experiment translates to the value shown in the
table, which is in good agreement with the mass spectroscopic values. The
0.5-eV uncertainty is dominated by work functions.

###  V.2 The MAC-E Filter and KATRIN

Until the 1990s the Tret’yakov cascaded toroidal magnetic spectrometer
Tret’yakov (1975), such as that depicted in Fig. 9, was the premier instrument
for tritium beta decay experiments. It had good acceptance and resolution by
the standards of the day but, as experimental groups contemplated the next
steps, a scale-up in size was clearly necessary for increased statistical
precision. The Troitsk and Mainz groups turned to a concept that had been
developed for photoelectron spectroscopy, the retarding-field analyzer. For
neutrino mass experiments with tritium, this type of instrument has a unique
advantage: energy conservation eliminates the possibility of a high-energy
tail in the response function. Instrumental tails extending beyond the
endpoint are particularly deadly because they shift the mass-squared value
negatively if not recognized, and even when recognized severely degrade the
sensitivity to neutrino mass. Moreover, the retarding-field analyzer combined
with a magnetic field for collimation, the ‘MAC-E’ filter (Magnetic Adiabatic
Collimation – Electrostatic), had another major advantage in the way it scaled
in size, as we show next.

In order to detect the effect of $m_{\beta}$ at the endpoint of a beta
spectrum of total kinetic energy $E_{0}$, instrumental resolution of order
$\frac{m_{\beta}}{E_{0}}$ is needed. The spectral fraction per decay that
falls in the last $m_{\beta}$ of the beta spectrum is approximately
$(\frac{m_{\beta}}{E_{0}})^{3}$, to within a constant of order unity.

For spectrometric experiments in which the source and the detector are
physically separated, a limit on source thickness is set by the cross section
for inelastic interactions of outgoing electrons, such that one must have
$\sigma n_{s}\leq 1$, where $\sigma$ is the inelastic cross section and
$n_{s}$ the superficial number density. More intense sources to reach smaller
neutrino masses must therefore have larger areas. On general grounds, the
dimensions of the source (radius $R_{\rm src}$) and the dimensions of the
spectrometer (length or radius $R_{\rm ana}$) are related through the
resolution needed, specifically,

$\displaystyle\frac{\Delta E}{E}\simeq\left(\frac{R_{\rm src}}{R_{\rm
ana}}\right)^{\alpha}.$ (V.2)

For magnetic spectrometers such as the Tret’yakov type, $\alpha=1$. The MAC-E
filter, however, has a different scaling relationship. The source is immersed
in a high magnetic field $B_{\rm src}$, and the spectrometer in a relatively
low field $B_{\rm ana}$. Electrons move from source to analyzer adiabatically
along magnetic field lines, and the energy resolution is determined by the
field ratio Angrik et al. (2004):

$\displaystyle\frac{\Delta E}{E}$ $\displaystyle=$ $\displaystyle\frac{B_{\rm
ana}}{B_{\rm src}}$ (V.3) $\displaystyle=$ $\displaystyle\left(\frac{R_{\rm
src}}{R_{\rm ana}}\right)^{2}.$ (V.4)

Thus, for magnetic-electrostatic retarding-field analyzers (MAC-E filters)
$\alpha$ is a more favorable $2$. This scaling property was decisive in
shifting the focus of the experimental community toward the MAC-E filter,
leading ultimately to the KATRIN project.

The Magnetic-Adiabatic-Collimation-Electrostatic filter concept was first
described in 1976 by Hsu and Hirschfield Hsu and Hirshfield (1976) and further
developed by Beamson et al. Beamson et al. (1980) and Kruit and Read in 1983
Kruit and Read (1983), and it soon found adoption in many areas of electron
spectroscopy. It combines good source acceptance with high resolution. The
Mainz and Troitsk experiments were the first to adopt this new technology for
tritium beta decay, with Troitsk mating it to a gaseous T2 source on the lines
of the Los Alamos design, and Mainz mating it to a source of tritium frozen on
a substrate of highly oriented pyrolytic graphite. The basic principle of the
MAC-E filter is shown in Fig. 11.

Figure 11: Principle of the MAC-E filter (from Arenz et al. (2016)). Electrons
produced in a high magnetic field region enter the large, empty high-vacuum
space of the main spectrometer. The magnetic field is lower there by a factor
$\sim 10000$, and the momentum vector swings forward along the direction of
the field. At the same time a potential gradient is applied between the entry
point and the analyzing plane in the center by floating the entire
spectrometer shell at the (negative) analyzing potential. Electrons with
sufficient energy to surmount the barrier are re-accelerated, re-enter a high
magnetic field, and are detected.

The MAC-E filter has very high acceptance. All electrons from a cross-section
in the high-field source that are emitted with a pitch angle smaller than a
selected value are transmitted to the analyzing plane. The pitch angle (the
angle between the momentum and the field direction) is established by means of
a solenoidal “pinch” magnet located somewhere between the source and detector.
The energy resolution is determined by the ratio of the magnetic fields in
source and spectrometer, and the choice of accepted pitch-angle range Kleesiek
et al. (2019). It has a simple, analytic form.

The KATRIN experiment represents what is likely to be the ultimate realization
of the MAC-E technology. Its main spectrometer, 9.8 m in diameter and 23.3 m
in length, is the largest ultra-high-vacuum vessel in the world and operates
at a base pressure of $10^{-11}$ mbar. The design magnetic fields of 0.3 mT at
the analyzing plane of the spectrometer and 4 T in the source lead to an
integral energy resolution-function step of 0.93 eV at the tritium endpoint.
An elevation view of the KATRIN experiment is shown in Fig. 12.

---
Figure 12: Top: View of the KATRIN experiment (from Arenz et al. (2016)).
Tritium gas recirculates continuously through the windlowless gaseous tritium
source (WGTS). A solenoidal magnetic field guides beta electrons to the main
spectrometer (MS) through a differential pumping restriction (DPS) and a
cryogenic pumping restriction (CPS) that prevent tritium from entering the
spectrometers. A prespectrometer (PS) blocks lower-energy electrons from
reaching the MS. Electrons that surpass the MS potential are detected in the
focal-plane detector (FPD). Calibration devices are located in the rear system
(RS). Bottom: Photograph of KATRIN’s main spectrometer.

Detailed descriptions of the KATRIN approach and apparatus can be found in
Otten and Weinheimer (2008); Drexlin et al. (2013); Arenz et al. (2016).

KATRIN began commissioning with tritium at low concentrations in 2018, and in
2019 gathered 22 live days of data for its first neutrino-mass measurement.
The spectrum is shown in Fig. 13.

Figure 13: Integral beta spectrum of molecular T2 from the KATRIN experiment
Aker et al. (2019). The spectrum contains $2.03\times 10^{6}$ events and was
taken over a period of 22 live days. The column density of tritium in the WGTS
source tube was 20% of the nominal design value. From these data, an upper
limit of 1.1 eV (90% CL) on the neutrino mass was obtained, a factor 2 below
the previous world limit. The lowest panel shows the distribution of
measurement times chosen at each retarding potential.

The MAC-E filter is intrinsically both a magnetic trap and a Penning trap, and
both modes can be sources of background. In KATRIN, the presence of the
prespectrometer forms a second Penning trap, of the opposite sign. The latter
trap was a worry during the design phase because, as an electron trap, the
repeated passage of electrons through the residual gas can produce a plasma
and a catastrophic discharge. This mode was a principal motivator for
achieving ultra-high vacuum in the spectrometers, and indeed it was found that
the trap could ignite at pressures in the $10^{-10}$ mbar range, but not at
the operating pressure $1.2\times 10^{-11}$ mbar Aker et al. (2019). The
magnetic mode in the main spectrometer traps relatively high-energy electrons
in the keV to MeV range, and, somewhat surprisingly, 219Rn emanating from the
getters was found to be the main contributor. In this mode, the electrons
circulate in the magnetic trap for times as long as an hour, slowly losing
energy by ionization to the residual gas F. M. Fränkle et al. (2011); Mertens
et al. (2013). The resulting background of low-energy electrons is troublesome
because it is non-Poissonian. Installation of large liquid-nitrogen cooled
chevron baffles in front of the getter chambers largely solved the problem S.
Görhardt et al. (2018), but the baffle efficiency for Rn decreases as water
vapor accumulates. Installation of a subcooler to reduce the temperature
another 10 K is expected to reduce this background greatly.

Although particles cannot easily get out of a magnetic trap, neither can they
easily get in. This ‘magnetic shielding’ has proven to be highly effective in
rejecting radioactive and cosmic ray backgrounds from the walls K. Altenmüller
et al. (2019a, b). In addition, layers of grids totalling 20000 wires spaced
from the walls and biased to negative voltages in the 100 – 500 V range
further reject backgrounds of soft electrons, and also permit a more precise
shaping of the electric field inside the spectrometer Valerius (2010); Prall
et al. (2008). A few of these grids became shorted to each other during a
bakeout, but with no major implications for KATRIN’s performance.

In KATRIN a new and unexpected kind of background was discovered, the
production of Rydberg atoms and their photoionization in the volume of the
main spectrometer. Atoms of hydrogen and heavier species are dislodged from
the walls of the spectrometer by the decay of radon daughters embedded
therein. These atoms are often neutral and in a distribution of excited
states, some of which are so close to the ionization edge that they can be
ionized by thermal black-body radiation as they cross the spectrometer volume.
KATRIN is developing strategies to mitigate this significant background. It
exemplifies a vulnerability of the MAC-E filter method, that signal electrons
are slowed almost to rest before being reaccelerated into the detector.
Therefore, in addition to the Rydberg atoms, any process that makes slow
electrons (ionization of the residual gas, for example, or the decay of errant
tritium) becomes a potential background.

These backgrounds, together with additional smaller contributions from the
FPD, amounted at the beginning of operations to about 0.5 counts per second
(cps), a factor of 50 larger than the target value in the Design Report Angrik
et al. (2004). Increasing the magnetic field in the main spectrometer reduced
the background to 0.29 cps at some cost in resolution. This configuration was
used for the first KATRIN neutrino mass measurement Aker et al. (2019). For
this background rate, the optimal measurement-time distribution (bottom panel
of Fig. 13) peaks 12 eV below the extrapolated endpoint. The statistical
contribution to neutrino-mass sensitivity, however, depends approximately on
the sixth root of the background (see Eq. VIII.16), and if no further steps
were to be taken, the final sensitivity would decrease from 200 meV to about
300 meV. Of course, KATRIN has moved aggressively to deal with backgrounds.
Shifting the analysis plane toward the detector serves to further reduce the
effective spectrometer volume and therefore the main backgrounds by a factor
of 3. Additional measures are under investigation.

Another unexpected complication to KATRIN’s neutrino mass extraction has
emerged from plasma potentials. Although the presence of electric and magnetic
potentials has no influence on the molecular dynamics of KATRIN’s tritium
source, it certainly affects the diffusion properties of the emitted electron
and tritium ions created in the decay. Electron-ion recombination and charge
drift eventually neutralize the source; however, neutralization occurs over
long time scales, leading to both spatial and temporal variations of the
charge density of the source. These electromagnetic potential fluctuations
impact the energy of the decay electron. A campaign is currently focused on
constraining the uncertainties arising from plasma potentials. Plasma
potentials can influence any technique that makes use of magnetic confinement,
including cyclotron radiation emission spectrometers.

###  V.3 163Ho electron-capture decay

Stimulated by the report of non-zero neutrino mass observed in tritium beta
decay by Lyubimov et al. Lyubimov et al. (1980), a search began for
alternative methods as a verification. The low Q-value for the electron-
capture decay of 163Ho to 163Dy, about 3 keV Hopke et al. (1968), attracted
interest in the possibility of a neutrino mass measurement with this isotope.
The subshell ratios, i.e. the relative intensity of electron capture in each
atomic shell, depend on the neutrino mass because the available phase space is
limited for the deeper shells by the neutrino rest mass. This formed the basis
of the initial attack on the problem by Bennett et al. Bennett et al. (1981),
but they found that the theory of subshell ratios in heavy nuclei was not
adequate to extract a neutrino mass. In the same year, De Rújula proposed A.
De Rújula (1981) a different approach, internal bremsstrahlung in electron
capture (IBEC), a radiative process producing a continuous spectrum of photons
with an endpoint shape that is modified by neutrino mass, quite analogous to
beta decay. He suggested the calorimetric technique that is the basis of
experimental work today. The term IBEC as used by De Rújula in his unified
treatment covered processes that are usually considered separately, the
radiative process when a real photon attaches to the electron or the $W$-boson
(‘innere bremsstrahlung’), and the Lorentzian tails of X-ray transitions as
determined by the vacancy lifetime. The X-ray widths are complicated by atomic
structure effects, which were treated schematically. In a paper a year later
De Rújula and Lusignoli (1982), however, the reference to IBEC is dropped and
the X-ray widths are used to derive the transition rate near the endpoint.
Springer et al. Springer et al. (1987) carried out the first 163Ho neutrino
mass experiment in 1987 and included a remarkably detailed theoretical study
of IBEC complete with the interference effects that were found to be
substantial. They set an upper limit on the mass of 225 eV, although their use
of a Q-value that is now discounted may have unduly influenced the derived
limit. In 1994, Yasumi et al. Yasumi et al. (1994) set a less stringent limit
on the neutrino mass in a study more in the spirit of the original Bennett et
al. one Bennett et al. (1981) in which the subshell ratios were measured,
except they circumvented the theoretical issues via a direct photoionization
measurement of the X-ray yield at a synchrotron light source.

Thereafter, experimental work converged on the microcalorimetric approach in
which the de-excitation of the 163Dy, whether in the form of photons or
electrons, is entirely captured and converted to heat. The endpoint of such a
spectrum is just the Q-value, apart from a small binding-energy correction in
the lattice. Neutrino phase space modifies the endpoint region of the spectrum
as in beta decay. The experimental efforts to pin down the Q-value and extract
a neutrino mass or a limit, are summarized in Table 5.

Table 5: Neutrino mass and Q-value experiments with 163Ho. Units: eV Group | Date | Source | Spectrometer | Limit or mass | Q-value | Ref.
---|---|---|---|---|---|---
Hopke et al. | 1968 | Ho:Al | Proportional counter | $-$ | $<9100$ | Hopke et al. (1968)
Bennett et al. | 1981 | Ho:Al | Si(Li) | $-$ | $>2047$ | Bennett et al. (1981)
Baisden et al. | 1983 | Ho+ | Mass spectrometer | $-$ | 2650(250) | Baisden et al. (1983)
Hartmann & Naumann | 1985 | Ho(fod)3 | Proportional counter | $-$ | 2600(30) | Hartmann and Naumann (1985); Hartmann and Naumann (1992)
Springer et al. | 1987 | HoF3 | Si(Li) | $<225$ | 2561(20) | Springer et al. (1987)
Yasumi et al. | 1994 | Ho metal | Si(Li) | $<460$ | 2710(100) | Yasumi et al. (1994)
Gatti et al. | 1997 | Ho salt | Microcalorimeter | $-$ | 2800(50) | Gatti et al. (1997)
SHIPTRAP | 2015 |  | ICR | $-$ | 2833(34) | Eliseev et al. (2015)
Ranitzsch et al. (ECHo) | 2017 | Ho:Au | Microcalorimeter | $-$ | 2858(11) | Ranitzsch et al. (2017)
Velte et al. (ECHo) | 2019 | Ho:Au | Microcalorimeter | $<150$ | 2838(14) | Velte et al. (2019)

It is a testament to the difficulty of this problem that in the intervening 40
years, the mass limit has come down from 225 eV to 150 eV. Even the Q-value
has been resistant to accurate measurement, with current numbers some ten
standard deviations from earlier measurements. The development of ion
cyclotron resonance (ICR) and Penning-trap mass spectrometers has made
possible dramatic improvements in the accuracy of isotope mass measurements. A
review of the methods and of the progress made can be found in Dilling et al.
(2018).

Three 163Ho microcalorimeter programs have been pursued, HOLMES, ECHo, and
NuMECS. The HOLMES apparatus is based on superconducting transition-edge
sensors Becker et al. (2019) of molybdenum-copper on a silicon nitride
substrate. The source pad is gold, which will be implanted with 163Ho Gallucci
et al. (2020). A total of 1024 sources is planned, which will be multiplexed
for readout by RF SQUIDs (radiofrequency superconducting quantum interference
devices). The NuMECS sensors are similarly molybdenum-copper transition-edge
sensors, but supported on nanofabricated silicon beams Croce et al. (2016).
The 163Ho has been incorporated from aqueous solution into nanoporous gold
pads at the beam ends and spectra have been obtained. While the resolution at
35 eV FWHM is not yet sufficient, the purity and specific activity of the Ho
material are good. The ECHo collaboration Gastaldo et al. (2014, 2017, 2013);
Ranitzsch et al. (2017) makes use of metallic magnetic calorimeters (MMC) to
read out the thermal signals via SQUIDs. These devices have delivered the
fastest risetimes, $\lesssim 100$ ns, and also hold the record for the best
energy resolution, 1.6 eV FWHM with an X-ray source of 55Fe. The MMC devices
themselves are not intrinsically faster than TES’s, but risetime in TES’s is
usually kept longer to match the dc-SQUID readout better. The speed is
determined by heat diffusion in gold in both devices. Use of MMC’s with
multiplexing will unavoidably require limitations in detector speed. Ultrapure
mass-separated 163Ho ions were implanted into gold source pads to produce
spectra (see Fig. 24) with an instrumental resolution of 9.2 eV FWHM Velte et
al. (2019). Only 2 events above the endpoint were observed, evidence that the
background is very low. When the 163Ho isotope is reactor-produced by
irradiation of 162Er, the isotopic contaminant 166mHo is also produced,
necessitating mass separation. Both ECHo and HOLMES use this method, while the
NUMECS material was produced by proton bombardment of Dy, which avoids the
166mHo byproduct but has a lower cross section.

###  V.4 187Re beta decay

In a similar vein as for 163Ho, 187Re also offers a low Q-value, 2.5 keV, in
comparison to that of tritium and thus emerged as an attractive isotope for
neutrino mass investigation. Despite its low Q-value, there are inherent
difficulties in using 187Re as a neutrino mass target. In particular, its
decay process:

$~{}^{187}{\rm Re}({\rm J}={5/2}^{+})\rightarrow~{}^{187}{\rm Os}({\rm
J}={1/2}^{-})~{}+e^{-}~{}+\bar{\nu}_{e}$ (V.5)

is a unique first-order forbidden transition ($\Delta J^{\pi}=2^{-}$), which
significantly alters the phase space of the decay electron near the endpoint
and gives rise to a formidably long lifetime ($\tau=4.12\times 10^{10}$ y).
This places severe requirements on the amount of target material necessary to
observe a significant number of decay events with electron energies near the
endpoint of rhenium (Table 2).

The beta decay of 187Re was originally observed using proportional counters,
and a definite determination of the beta decay process was made by Brodzinski
and Conway Brodzinski and Conway (1965) and by Huster and Verbeek Huster and
Verbeek (1967) using gas proportional counters. From those early measurements,
they determined a Q value of 2.6 keV. Inspired by the results by Lubimov
indicating a positive value for neutrino mass, McCammon McCammon et al. (1984)
and later Vitale et al. Vitale et al. (1985) raised the possibility of using
microcalorimeters to better measure the neutrino mass signal. Vitale et al.
proposed a rhenium-based calorimeter to measure the total visible energy
produced from the beta decay process, from which a neutrino mass measurement
could be extracted.

The first measurements of 187Re decay using microcalorimeters were done in
Genoa by the MANU project Cosulich et al. (1992). They used metallic rhenium
as a self-contained absorber, taking advantage of the fact that rhenium at
temperatures below 1.7 K becomes superconducting. Under these conditions, the
electronic contributions of the heat capacity are exponentially suppressed,
leaving only phonon/lattice contributions to the heat capacity. When cooled
well below the transition temperature (usually $<100$ mK), the heat capacity
should be significantly reduced, allowing for high energy resolution, as
needed for an accurate endpoint measurement. Unfortunately, the energy
released in the decay by the electron at such low temperatures tends to be
trapped in the form of quasi-particles, for which the recombination time to
phonons (and thus a detectable signal) can be as long as several seconds. As
such, metallic rhenium proved difficult to realize as a microcalorimeter for
the purposes of beta decay detection.

Figure 14: A view of several microcalorimeter pixels using 500 $\mu$g AgReO4
absorbers for the MARE experiment Nucciotti (2016).

A parallel effort, led by the Milan group MiBETA, used a dielectric compound
of rhenium to achieve a similar suppression of electron-based noise while
avoiding issues associated with quasi-particle trapping in superconductors
(see Fig. 14). Several compounds were tried and the MiBETA group eventually
settled on silver perrhenate (AgReO4) as yielding the best performance in
terms of efficiency and energy resolution ($\sigma\simeq 18$ eV FWHM at 6
keV). Both the MANU and MIBETA groups extracted measurements of the neutrino
mass, both limits below 20 eV. The extracted Kurie plot from the decay of
187Re is shown in Fig. 15. The MANU experiment also measured for the first
time oscillations in the beta energy spectrum due to environmental fine
structure from the target crystal, although also revealing a potential
systematic uncertainty in their neutrino mass extraction. The effect was also
measured in the AgReO4 detectors by the MiBETA group Arnaboldi et al. (2006).

Table 6: Neutrino mass and Q-value experiments with 187Re. Units: eV Group | Date | Source | Spectrometer | Limit (90 % C.L.) or mass | Q-value | Ref.
---|---|---|---|---|---|---
Brodzinsky & Conway | 1965 | (C5H${}_{5})_{2}$ReH | Proportional counter | N/A | 2620(90) | Brodzinski and Conway (1965)
Huster & Verbeek | 1967 | (C5H${}_{5})_{2}$ReH | Proportional counter | N/A | 2650(40) | Huster and Verbeek (1967)
MANU (Genoa) | 1999 | Metallic Re | Microcalorimeter | $\leq 19$ | 2470(4) | Gatti et al. (1999)
MIBETA (Milano) | 2004 | AgReO4 | Microcalorimeter | $\leq 15$ | 2465.3(17) | Sisti et al. (2004)
Figure 15: Kurie plot showing the measured beta spectrum from 187Re decay from
the Milano group’s experiment Sisti et al. (2004).

The MANU and MIBETA groups eventually combined to propose a new calorimetric
experiment using rhenium called Microcalorimeter Arrays for a Rhenium
Experiment (MARE), with up to 10,000 of such microcalorimeters to reach sub-eV
sensitivity to the neutrino mass Nucciotti (2008). The MARE effort eventually
gave way to the holmium efforts of ECHo and HOLMES, as discussed earlier in
this section.

##  VI Other Research

The kinematic direct mass search method can be applied not only to the mass of
the three known active neutrinos, but also to other significant physics
questions. Are there sterile neutrinos that mix slightly with the active
species and thereby become observable? They may disclose their existence in
the form of kinks in the otherwise smooth beta spectrum. Can the primordial
sea of relic neutrinos created in the big bang be detected? They would produce
a peak in the beta spectrum just beyond the endpoint energy.

###  VI.1 Sterile Neutrinos

The width of the $Z$-boson fixes the number of active light neutrinos at 3
Zyla et al. (2020), but the possibility remains that there might be neutral
fermions with no standard-model couplings. If they admix slightly with the
active neutrino states they become observable both in oscillation and direct-
mass experiments. The discovery of neutrino mass in particular renewed
interest in this possibility because light right-handed singlets would be
introduced into the Standard Model if neutrinos are Dirac fermions, and
heavier partners might exist if neutrinos are Majorana fermions. In direct
mass searches, these ‘sterile neutrinos’ would appear as a kink in the beta
spectrum if the mass is in the kinematically allowed range.

Perhaps the most famous example is the “17-keV neutrino” that was first
reported by Simpson in 1985, from the observation of a kink in the beta
spectrum of tritium implanted into a silicon detector Simpson (1985). Much
excitement ensued when the same signal seemed to show up in many experiments
on tritium and other isotopes, but a convincing demonstration against it was
carried out by Mortara et al. Mortara et al. (1993). Summaries of the saga can
be found in Ref. Wietfeldt and Norman (1996); Robertson (1992); Franklin
(1995).

More recent interest in sterile neutrinos has largely been motivated by
oscillation searches that have produced results that are either inconsistent
with the 3-neutrino picture, or inconsistent with another observable such as
the reactor antineutrino flux. Recent reviews may be found in Refs. Hagstotz
et al. (2020); Böser et al. (2020); Giunti and Lasserre (2019). Direct
measurements give no indication of sterile neutrino admixtures. The most
stringent published limit contour from tritium beta decay comes from the
Troitsk experiment Abdurashitov et al. (2015); Belesev et al. (2013), and
Abdurashitov et al. (2015) also includes a summary of limits from other
isotopes. Figure 16 displays the existing limits from direct searches in beta
decay.

Figure 16: Limits on admixtures $|U_{e4}|^{2}$ of a single sterile neutrino as
a function of the neutrino’s mass. Admixtures above the curves are excluded.
The left panel is from Abdurashitov et al. (2015), and the right from
Friedrich et al. (2020). Sources of the data are given in the original
publications.

The published KATRIN data on active neutrinos Aker et al. (2019) have been
used to derive limits on sterile admixtures, but such analyses are not
rigorous, leaving out correlation effects. The KATRIN collaboration has
released their own analysis Aker et al. (2020), depicted in Fig. 17, that
takes such effects into account.

Figure 17: Exclusion curves at the 95% confidence level in the
($|U_{e4}|^{2}$, $m^{2}_{4}$) plane obtained from the analysis of KATRIN data
including statistical and systematic uncertainties Aker et al. (2020). The two
solid lines show the expected sensitivity (light grey) and the associated
exclusion (blue online) for fixed $m_{\beta}^{2}=0$ eV2). The dotted line
(dark blue online) illustrates the exclusion curve obtained with a free
$m_{\beta}^{2}$). The dot-dashed line (turquoise online) displays the
intermediate exclusion curve with $m_{\beta}^{2}$ introduced as a Gaussian
pull term with a central value of 0 and an uncertainty $\sigma(m_{\beta}^{2})$
= 1 eV2. Also shown are the Mainz Kraus et al. (2013) and Troitsk bounds
Belesev et al. (2013). In the figure, $m_{\nu}^{2}\equiv m_{\beta}^{2}$.

The background in KATRIN precludes disentangling the effect of a sterile
neutrino and an unconstrained active neutrino for masses below 10 eV. On the
other hand, the statistical power of KATRIN opens large new regions of
parameter space in which to search for sterile neutrino admixtures in the mass
range $>10$ eV. In order to be able to take advantage of the available tritium
source strength, a new, highly segmented, high-resolution focal-plane detector
for KATRIN is being developed Brunst et al. (2019); K. Altenmüller et al.
(2018); Mertens et al. (2019) by the TRISTAN collaboration. If the technical
initiatives are successful, KATRIN would be able to reach unprecedented
$|U_{e4}|^{2}$ levels, potentially as low as $10^{-8}$ Mertens et al. (2015).
Tiny admixtures of neutrinos in the keV mass range are particularly
interesting because they are not excluded by astrophysical bounds, could be
dark matter, and could help explain why supernovae explode in nature but
rarely in computers Hidaka and Fuller (2007).

It is also possible to search for sterile neutrino admixtures in electron-
capture decays. In that case, since capture in atomic subshells produces
nominally monoenergetic neutrinos, a precision measurement of the recoil
energy by calorimetric means can reveal the admixture of massive neutrinos.
Friedrich et al. Friedrich et al. (2020) have carried out such a measurement
for 7Be embedded in a superconducting tunnel junction, obtaining the most
sensitive limits by the direct method, as low as $|U_{e4}|^{2}<2\times
10^{-4}$, on the admixture of neutrinos in the mass range 100 - 1000 keV.
Their results are shown as the unshaded curve in the right-hand panel of Fig.
16.

###  VI.2 Relic Neutrinos

As Fisher has commented, “Every neutrino physicist has wondered at some time
if there might be a way to detect the relic neutrino background” Fisher
(1998). While not strictly a topic within our purview, since Weinberg’s
seminal paper in 1962 Weinberg (1962) the method of choice for such
considerations has always been neutrino capture observed as a peak just beyond
the endpoint of a tritium beta decay spectrum, separated from the extrapolated
endpoint by a an amount of order $m_{\beta}$ (see Figure 18). It is a
potential byproduct of every direct mass measurement via beta decay.

The relic neutrino background is cold today, with a temperature of 1.95 K
Hannestad (2004), and therefore the kinetic energy is negligible in comparison
to $m_{\beta}$. Cosmology gives the mean relic neutrino density in the
universe, 56 cm-3 per neutrino flavor and chirality Ringwald and Wong (2004).
The contribution $\Omega_{\nu}$ of neutrinos to the closure density of the
universe is Hannestad (2004):

$\displaystyle\Omega_{\nu}h^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{\Sigma}{92.5\rm\ eV}\right)$ (VI.1)

where $h$ is the Hubble constant in units of 100 km s-1 MPc-1. The rate of
neutrino capture on a nuclear target depends on the neutrino density and its
flavor content, the nuclear matrix element and Q-value, and the target mass.
Betts et al. Betts et al. (2013) find a capture rate of 951(3) events per year
per kg of tritium for the mean universal density modeled as a Fermi-Dirac
distribution throughout space.

Figure 18: Spectral features from tritium beta decay and neutrino capture on
tritium. From Irvine and Humphreys (1983).

The local neutrino density at earth will be larger than the mean density in
the universe because, having mass, neutrinos will be drawn gravitationally
into galaxies. Simulations of the overdensity to be expected (see, for
example, Ringwald and Wong (2004); Zhang and Zhang (2018); de Salas (2020))
show that the neutrino contrast scales approximately as the square of the mass
but is relatively small Ringwald and Wong (2004); Zhang and Zhang (2018):

$\displaystyle\frac{n_{\nu}}{\overline{n}_{\nu}}-1$ $\displaystyle=$
$\displaystyle 76.5\left(\frac{m_{\nu}}{\rm eV}\right)^{2.21},$ (VI.2)

where $n_{\nu}$ and $\overline{n}_{\nu}$ are the local and universal neutrino
densities, respectively.

Experimentally the difficulty is that, tritium being a shortlived radioactive
isotope, the target mass is limited. In KATRIN, the largest tritium neutrino
mass experiment, the mass under observation is about 100 $\mu$g, and the
expected capture rate is roughly 1 every 10,000 y. The PTOLEMY project, whose
primary goal is to measure big bang relic neutrinos, is designed to put 0.1
kg, 1 MCi, under observation Betts et al. (2013); Betti et al. (2019). To
achieve sensitivity for such a large target mass, PTOLEMY plans to use large
surface area targets with atomic tritium adsorbed on graphene, so as to reduce
charge accumulation effects. The feasibility of using tritium-loaded graphene
is currently under investigation. An experiment with such a tritium source
under observation would potentially also be capable of a sensitive measurement
of neutrino mass.

##  VII Spectrum Refinements

We are accustomed to the idealized appearance of the beta spectrum near the
endpoint, represented by Eq. (IV.9) and the corresponding figure (Fig. 7, Fig.
18). In practice, however, one never deals with the decay of an isolated
nucleus of tritium or any other isotope, and the surrounding electrons and
atoms can be excited in the decay. These excitations, referred to loosely as
the “final-state distribution” (FSD) reduce the energy available to the
leptons. The excitations are not a result of electron energy loss in a
surrounding medium following the decay, an effect that must also be
considered, but occur simultaneously with the decay because the initial and
final states are both complex quantum systems with many internal degrees of
freedom. Thermal excitations in the initial state may even increase the lepton
energy. Nor can one escape these considerations by turning to electron
capture, as will be discussed below. Bergkvist, in his pioneering experiments
on tritium (see Bergkvist (1971, 1972a, 1972b) and Fig. 8), was the first to
point out that progress beyond the 55-eV limit he obtained would demand an
understanding of the FSD.

There are also spectrum corrections that arise in the nuclear decay itself. We
discuss those below, finding that most are negligible for the purposes of
neutrino mass measurement, with the exception of radiative corrections.

###  VII.1 Final State Distributions

To include the final (and initial) states, a spectrum in the form of Eq.
(IV.1) for each initial-state $j$ to each final-state $k$ is represented in
the total spectrum, weighting each transition by a matrix element $W_{kj}$.
Omitting small recoil-order corrections, the spectral density for each
transition becomes Bodine et al. (2015)

$\displaystyle\left(\frac{d\Gamma}{dE_{e}}\right)_{kj}$ $\displaystyle=$
$\displaystyle CF(Z,\beta)\left|W_{kj}\right|^{2}\beta
E_{e}^{2}(\Delta_{kj}-E_{e})^{2}\times$ (VII.1)
$\displaystyle\times\sum_{i}|U_{ei}|^{2}\left[(1-\frac{m_{i}^{2}}{(\Delta_{kj}-E_{e})^{2}}\right]^{1/2}\Theta(\Delta_{kj}-E_{e}).$

This expression is written in terms of the total electron energy, $E_{e}=E+m$.
The energy $\Delta_{kj}$ is the maximum energy available for each transition.
An expression for the matrix element $W_{kj}$ is given by Jonsell, Saenz, and
Froelich Jonsell et al. (1999). The transition matrix element for a final-
state molecular ion excitation $k\equiv(v_{(f)},J_{(f)},M_{(f)},n_{(f)})$ from
an initial molecule state $j\equiv(v,J,M,n)$ may be written,

$\displaystyle\left|W_{kj}(K)\right|^{2}$ $\displaystyle=$
$\displaystyle\left|\int\left[\chi_{v_{(f)}J_{(f)}M_{(f)}}^{n_{(f)}}(\bf
R)\right]^{*}S_{n}(R)e^{i{\bf K\cdot R}}\xi_{vJM}^{n}({\bf
R})d^{3}R\right|^{2}.$ (VII.2)

In this expression, $\chi$ and $\xi$ are the rotational-vibrational wave
functions of the final-state molecular ions and initial-state molecules,
respectively, and $S_{n}(R)$ is an electronic overlap integral. The
exponential of the dot product of the recoil momentum ${\bf K}$ and the
nuclear separation ${\bf R}$ is a consequence of the recoil motion of the
daughter nucleus, 3He in the case of tritium decay. It may be seen from the
form of Eq. (VII.1) that because each transition has a slightly different
endpoint energy, the shape of the spectrum in the region where one seeks
evidence of neutrino mass is strongly modified by the final-state
distribution.

####  VII.1.1 Tritium

The ITEP result reporting a 30-eV neutrino mass with a source of tritiated
valine (an amino acid) precipitated a substantial theoretical effort to
calculate the FSD but in the end it was clear that such molecules are too
complex. The 30-eV result either arose from shortcomings in the FSD theory for
valine, or from inexact energy-loss corrections. Evidence emerges from the
fact that these affect both the neutrino mass and the extrapolated endpoint
energy, and the latter can be checked experimentally via mass spectroscopy
Staggs et al. (1989). An early measurement of the T – 3He mass difference
indeed seemed to support the FSD and energy-loss theory, but was afflicted
with systematic errors that were controlled in further experiments. The valine
experiment did not give the correct endpoint energy.

The Los Alamos group turned to atomic and molecular tritium as nearly ideal
sources. Molecular hydrogen had been a topic of theoretical interest since the
studies by Cantwell in 1956 Cantwell (1956). In principle, since the
electromagnetic potential is known exactly, the calculations can be carried
out to any desired accuracy, but in practice even this simple molecule is very
challenging theoretically. In parallel with the development of gaseous tritium
experiments at Los Alamos Robertson et al. (1991) and Livermore Stoeffl and
Decman (1995), a number of new calculations of the molecular FSD were
undertaken, and the work of the Quantum Theory Project Fackler et al. (1985)
was used to interpret the results. Both experiments yielded negative fit
values for $m_{\beta}^{2}$, and it was not until 2015 that this was traced
Bodine et al. (2015) to inadequacies in the FSD.

Figure 19: Comparison of the FSD spectra for the T2 molecule calculated by
Fackler et al. Fackler et al. (1985) (dashed blue line) and Saenz Saenz et al.
(2000a) (solid red line). From Bodine et al. (2015).

Theoretical work continued apace in the 1990 – 2005 interval with significant
progress. Detailed assessments of the status can be found in Bodine et al.
(2015); Kleesiek et al. (2019). One of the most difficult aspects of the
problem is the continuum, wherein one or more of the molecular electrons are
ejected during the decay. In the model of Fackler et al. Fackler et al. (1985)
continuum states have discrete energies like bound states. An important
advance by Froelich et al. Froelich et al. (1993); Saenz et al. (2000a) was
the development of ‘complex scaling’ wherein the radial variable of continuum
wave functions is made complex. Another major advance was the inclusion of
nuclear motion by Saenz and Froelich in 1997 Saenz and Froelich (1997). The
two FSD spectra are shown for comparison in Fig. 19 and in the figure the
discrete states of the Fackler et al. spectrum in the continuum have been
arbitrarily assigned a 3-eV standard deviation for display and comparison with
the calculation of Saenz et al. Saenz et al. (2000a). The complex scaling
approach directly gives realistic distributions in the continuum, joining
smoothly (with a modest order-unity normalization) to the Levinger shakeoff
distributions Levinger (1953) that are analytic for hydrogenic atoms. Table 7
lists the first three moments of the binding-energy distributions for the two
theories.

Table 7: Comparison of zeroth, first, and second moments (columns 3, 4, and 5, respectively) of theoretical final-state distributions (see Robertson and Knapp (1988); Bodine et al. (2015)). Reference |  aa Energy range aa |  aa $\sum_{k}\left|W_{k0}\right|^{2}$ aa |  aaa $\langle b_{(f)k}\rangle$ aaa |  aaa $\sigma_{b}^{2}$ aaa
---|---|---|---|---
| eV |  | eV | eV2
Fackler et al. Fackler et al. (1985) | 0 to 165 | 0.9949 | -17.71 | 611.04
Saenz et al. Saenz et al. (2000a) | 0 to 240 | 0.9988 | -18.41 | 694.50
Table 8: Neutrino mass squared extracted from two experiments, in one case with the original 1985 theoretical calculations of the FSD and in the second case with a more modern calculation. |  aaa LANL Robertson et al. (1991) | LLNL Stoeffl and Decman (1995) |
---|---|---|---
As published. Theory: Fackler et al. Fackler et al. (1985) |
aaa $m_{\beta}^{2}$ |  aaa -147(79) | -130(25) | eV2
Re-evaluated. Theory: Saenz et al. Saenz et al. (2000a) |
aaa $m_{\beta}^{2}$ |  aaa 20(79) | 37(25) | eV2

The data for the Los Alamos and Livermore experiments are no longer available,
but it is possible to estimate the changes that would result had the theory of
Saenz et al. been used instead of the Fackler et al. theory, with the aid of
the following relationship between an error in the variance of the FSD or
other resolution contribution and the consequent error in the neutrino mass
squared Robertson and Knapp (1988):

$\displaystyle\Delta m_{\beta}^{2}$ $\displaystyle\simeq$
$\displaystyle-2\Delta\sigma_{\rm FSD}^{2}.$ (VII.3)

The results are shown in Table 8 with the results of the LANL and LLNL
experiments as originally reported, both having been analyzed with the theory
of Fackler et al. Fackler et al. (1985) and re-evaluated via Eq. (VII.3) with
the theory of Saenz et al. Saenz et al. (2000a). The large negative value of
$m_{\beta}^{2}$ is eliminated in both experiments, subject to the limitations
of Eq. (VII.3). These results provide a striking, and essentially ‘blind’,
measure of experimental confirmation of the calculations of Saenz et al.,
especially in the difficult regime of electronic excited states.

Doss et al. in 2006 Doss et al. (2006) carried out a calculation in the same
geminal basis used by Froelich et al. Their results are the same as those
obtained by Saenz et al. Saenz et al. (2000a) up to 40 eV excitation, but
should not be used above that point owing to inclusion of only the 6 lowest
electronic bound states. The effects of nuclear motion above 45 eV in a 2008
publication (Doss et al. Doss and Tennyson (2008)) also appear anomalously
broad in their treatment, and it is therefore recommended to use the
calculations of Saenz et al. whenever the full spectral range is needed
Sibille (2020).

Theoretical advances notwithstanding, the most powerful weapon to reduce
reliance on the calculations is actually the statistical power of state-of-
the-art experiments such as KATRIN. If an experiment can gather sufficient
data in the last 25 eV of the spectrum, then most of the uncertain aspects of
the FSD, particularly the continuum, drop away. Only the rotational-
vibrational manifold of the electronic ground state remains (Fig. 19, the peak
at zero binding energy). Quantitatively, the variance of the full FSD is 695
eV2, but the variance of the ground-state group is only 0.16 eV2. Even that
small variance must be known to a precision of 2% for KATRIN to meet its goal
of 200-meV neutrino mass sensitivity.

There is no known way to measure the FSD directly (other than in beta decay
itself), but several kinds of experimental verification of the FSD are
possible Bodine et al. (2015). Interestingly, the Los Alamos and Livermore
experiments that covered the full range of the FSD spectrum provide direct
experimental confirmation of the correctness of the variance of the Saenz FSD
at approximately the 2% level of accuracy needed by KATRIN. Since this
includes the difficult continuum region, it is likely that the rotation-
vibration manifold of the electronic ground state is even more reliable.

With the negative neutrino mass-squared problem resolved, one other conflict
between theory and experiment remained. When a tritium atom in the T2 or HT
molecule decays, the daughter 3He can remain bound in an ion, or can be
liberated. The electronic ground state of the molecular ion THe+ is bound by
1.9 eV, and corresponds to the peak near zero binding energy shown in Fig. 19.
Rotation and vibration spread the distribution partly into the energetically
unbound regime. Two experiments carried out in the 1950s Snell et al. (1957);
Wexler (1959) reported that the intensity of this ground-state manifold, as
measured by the bound-ion fraction, was 90 – 95%. The theoretical prediction,
however, is only 57%, a serious disagreement. This has now been resolved in a
new experiment with a novel instrument, TRIMS (Tritium Recoil-Ion Mass
Spectrometer) Lin et al. (2020). TRIMS is a time-of-flight mass spectrometer,
and Fig. 20 shows data from runs with gas that is predominantly HT, with a
small admixture of T2.

Figure 20: Experimental data from the TRIMS time-of-flight mass spectrometer
experiment showing the rich complexity of ionic final states populated in the
beta decay of HT. The bands originate from a) protons, b) He++, c)
H${}^{+}+$He+, d) He+ and T+, e) HHe+, and THe+. (From Lin et al. (2020))

As shown in Table 9, TRIMS finds close agreement with the theoretical
prediction, and gives much detail about the ionic final states that can be
used in further tests of the theory.

Table 9:  Branching ratio to the bound molecular ion for HT and TT. | Snell et al. | Wexler |  aTheory a | TRIMS
---|---|---|---|---
| Snell et al. (1957) | Wexler (1959) | Jonsell et al. (1999); Saenz et al. (2000a) | Lin et al. (2020)
Molecule |  |  | Quasibound | Bound | Total |
HT | 0.932(19) | 0.895(11) | 0.02 | 0.55 | 0.57 | 0.565(6)
TT | – | 0.945(6) | 0.18 | 0.39 | 0.57 | 0.503(15)

In the Table, the column ‘Quasibound’ is the part of the ‘Total’ branching
ratio to the ground-state manifold that lies above the dissociation threshold.
Many of these states have high angular momentum and therefore long lifetimes,
and may travel through the TRIMS apparatus without breaking up. The lifetimes
of these states are not known. In the case of the HT molecule, the quasibound
fraction is small, and TRIMS provides the most decisive experimental test
there. Since the theory of the FSD is the same for both HT and TT, the close
agreement is confirmation of the FSD for both isotopologues.

The FSD for molecular tritium is now very robust and no longer contributes
significantly to the systematic uncertainty budget even in the KATRIN
experiment. Nevertheless the broadening of the response function caused by the
FSD exacts a statistical penalty and makes it more difficult to reach small
neutrino masses. The appeal of a source of atomic tritium is illustrated in
Fig. 21.

Figure 21: The ground-state manifold of molecular tritium compared to the
ground state of atomic tritium. The atomic line is here placed on the same
scale as the molecular binding energy of Fig. 19. The ground-state Q-values
differ by 8.29 eV. Because of recoil effects, the ground-state extrapolated
endpoint values ($\Delta_{00}-m_{e}$ in Bodine et al. (2015)) differ by 9.99
eV, but the molecular rotational and vibrational excitations that broaden the
molecular peak make the endpoint energy for the atomic decay effectively about
8 eV smaller than the molecule. Translational Doppler broadening corresponding
to a temperature of 1 K is included in the atomic line. The molecular line is
from Saenz et al. (2000a).

The atomic line is very narrow, broadened only by thermal motion, and the