Direct Measurements of Neutrino Mass
Joseph A. Formaggio, André Luiz C. de Gouvêa, R. G. Hamish Robertson
DDirect Measurements of Neutrino Mass
Joseph A. Formaggio
Laboratory for Nuclear Science and Dept. of Physics,Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Andr´e Luiz C. de Gouvˆea
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 (Dated: February 15, 2021)The turn of the 21st century witnessed a sudden shift in our fundamental understandingof particle physics. While the minimal Standard Model predicts that neutrino masses are ex-actly zero, the discovery of neutrino oscillations proved the Standard Model wrong. Neutrinooscillation measurements, however, do not shed light on the scale of neutrino masses, nor themechanism by which those are generated. The neutrino mass scale is most directly accessedby studying the energy spectrum generated by beta decay or electron capture – a techniquedating back to Enrico Fermi’s formulation of radioactive decay. In this Article, we reviewthe methods and techniques – both past and present – aimed at measuring neutrino masseskinematically. We focus on recent experimental developments that have emerged in the pastdecade, overview the spectral refinements that are essential in the treatment of the mostsensitive 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
I. Introduction II. Neutrino Masses: Current Status a r X i v : . [ nu c l - e x ] F e b E. Direct laboratory measurements 13
III. Neutrino Mass Models and New Physics
IV. Kinematic determination from beta decay
V. Experimental Progress
Ho electron-capture decay 38D.
Re beta decay 40
VI. Other Research
VII. Spectrum Refinements
VIII. Experimental sensitivity
IX. Future
X. Conclusion XI. Acknowledgments References I. INTRODUCTION
The existence of neutrinos was first postulated by Pauli nine decades ago. In his famous ‘DearRadioactive Ladies and Gentlemen’ letter [1, 2], Pauli also made the first non-trivial estimate ofthe mass of the neutrino: “The mass of the neutrons should be of the same order of magnitudeas the electron mass and in any case not larger than 0.01 times the proton mass.” His qualitativeprediction – m ν ∼ m e – based on aesthetics and minimality, turned out to be too large by manyorders of magnitude.Laboratory searches for a nonzero neutrino mass started in the 1930s and have continued inearnest up to the present. Pauli’s neutrino, now called the electron neutrino ν e , was not directlydetected until the work of Reines and Cowan in the 1950s [4, 5]. Two other neutrino flavors werediscovered, ν µ in 1962 [6] and ν τ in 2001 [7], and the searches diversified in order to accommodatethe possibility that the different neutrino species had qualitatively different masses. The moststringent upper bounds to the mass of the electron neutrino evolved from Pauli’s 10 MeV qualitativeupper 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 ofatmospheric neutrino oscillations by the Super-Kamiokande collaboration [8], building on previoushints for atmospheric neutrino oscillations obtained by the Irvine-Michigan-Brookhaven [9] andKamiokande Collaborations [10], and the different collaborations that helped define the solar neu-trino puzzle: Homestake [11], Gallex [12], SAGE [13], and Kamiokande [14]. The solar neutrinopuzzle was definitively resolved by the Sudbury Neutrino Observatory (SNO) collaboration [15] andis also a consequence of nonzero neutrino masses. The 2015 Nobel Prize in Physics was awardedto Takaaki Kajita – from the Super-Kamiokande Collaboration – and Arthur B. McDonald – fromthe SNO collaboration – “for the discovery of neutrino oscillations, which shows that neutrinoshave mass.” “Neutron” was the name attributed to the hypothetical particle by Pauli. In order to avoid confusion with themodern 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. [3]. The Nobel Prize in Physics 2015. . Several twenty-first century oscillation experiments provide precision measurements of the neu-trino oscillation phenomenon [16–23]. These translate into rather precise measurements of neutrinomass-squared differences and reveal that at least two of the three neutrinos are massive and theheaviest neutrino mass is at least 0.05 eV. Oscillation experiments, however, are powerless whenit comes to measuring the individual values of the neutrino masses – they are only sensitive tomass-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 con-nection between measurement and neutrino masses is mediated by a theoretical framework, alongwith other hypotheses. Such observables include the rate for neutrinoless double-beta decay andthe large-scale structure of the universe. Other observables are more directly sensitive to neutrinomasses – the kinematics of the observable are directly established by the fact that neutrinos havenonzero masses in a way that is virtually independent from the nature of the physics responsiblefor 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 amongthese direct probes of nonzero neutrino masses are the subject of this review.A number of reviews of the subject are available [24–27]. As the most recent was seven yearsbefore the present one, it is an opportunity to review the recent progress and to consider wherethe field will go in the future. The KATRIN experiment is now running, tightening the upper limiton neutrino mass by a factor 2 after only a month of operation. The new method of cyclotronradiation emission spectroscopy has passed a crucial proof-of-principle test. Prompted by this suc-cess, the possibility of a neutrino mass experiment based on atomic tritium is once again receivingconsideration. Microcalorimetry is advancing technically to enable studies of isotopes other thantritium.This review is organized as follows. In Section II, we summarize the direct and indirect infor-mation 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 understand-ing of fundamental particle physics, along with the different outstanding questions that we hopewill be informed by the direct observation of nonzero neutrino masses. Nature has provided onlytwo 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 inSection V. Related research on sterile neutrinos and the relic neutrino background is covered inSection VI. In Section VII, we discuss spectrum refinements that are essential in the treatment ofthe most sensitive experiments, and, in Section VIII, provide a simple yet effective protocol forestimating the sensitivity. In Section IX, the new techniques that are emerging to advance the fieldare 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 neutrinomasses. For a detailed review, see, for example, the ‘Neutrino Masses, Mixing, and Oscillations’chapter of the Particle Data Book [28].
A. Neutrino oscillations
Precision measurements of the flux of solar neutrinos reveal that fewer electron-type neutrinosarrive at the Earth than predicted. What came to be known as the solar neutrino problem presenteditself with the first measurements of the solar neutrino flux in the 1960s – see references in Ref. [11]– and persisted until it was definitively resolved by the SNO experiment in the early 2000s [15].Ultimately, solar neutrino data imply that electron neutrinos are, in fact, linear superpositions ofat least two neutrino mass-eigenstates and that at least one of these has a nonzero mass. Thedifference between the neutrino masses-squared is of order ∆ m ≡ m − m ∼ − eV . Here m and m are the masses of two different neutrino mass eigenstates, labeled ν and ν .Precision measurements of the flux of atmospheric neutrinos also reveal that fewer muon-typeneutrinos survive passage through the Earth than expected. The effect depends on the distancebetween the neutrino production and detection points and the neutrino energy. The solution tothis new problem, the atmospheric neutrino problem, first revealed by data from the IMB andKamiokande experiments and later confirmed beyond reasonable doubt by the Super-Kamiokandeexperiment, was the realization that muon neutrinos are linear superpositions of at least twoneutrino mass eigenstates and that at least one of these has a nonzero mass. In this case thedifference between the neutrino masses-squared is of order ∆ m ≡ m − m ∼ − eV . Here m is the mass of the third distinct neutrino mass eigenstate, labeled ν .In the last two decades, multiple experiments with multiple neutrino sources and detector tech-nologies have confirmed the existence of neutrino oscillations and have allowed the construction ofa very robust three-massive-neutrinos paradigm. It asserts that neutrinos interact as prescribedby the Standard Model of particle physics and that the neutrino charged-current-interaction eigen-states – ν e , ν µ and ν τ – are linear superpositions of the neutrino mass eigenstates, ν , ν , ν , withmasses, respectively, m , m , and m : ν α = (cid:88) i U αi ν i , (II.1)where i = 1 , , α = e, µ, τ and U αi are the elements of the 3 × m > m and | m − m | , | m − m | > m − m in sucha way that m > m > m , termed the normal mass-ordering (NMO), or m < m < m ,termed the inverted mass-ordering (IMO). In the NMO, ∆ m , ∆ m are positive while in theIMO ∆ m , ∆ m are negative. The two mass orderings are depicted in Figure 1. The massordering is currently unknown. A slight preference in the world neutrino data for the NMO [29]has recently disappeared with new data [30]. ( D m ) sol ( D m ) sol ( D m ) atm ( D m ) atm n e n m n t (m ) (m ) (m ) (m ) (m ) (m ) normal hierarchy inverted hierarchy FIG. 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) indicatesthe fraction | U αi | of each distinct flavor ν α , α = e, µ, τ contained in each mass eigenstate ν i , i = 1 , , | U e | is equal to the fraction of the ( m ) “bar” that is painted red (shading labeled as “ ν e ”).From [31]. The PMNS matrix is parameterized with three mixing angles θ , θ , θ and one (three) CP-odd phase(s) δ ( δ, η , η ) if the neutrinos are Dirac (Majorana) fermions. Throughout, we will usethe PDG parameterization for the PMNS matrix [28] and will assume the three-massive-neutrinosparadigm is true, unless otherwise noted.Neutrino oscillation experiments can constrain the PMNS matrix and the neutrino mass-squareddifferences ∆ m ij = m i − m j , ij = 21 , ,
32. Only two of the three mass-squared differences areindependent because ∆ m = ∆ m − ∆ m . The world neutrino data translate into robustmeasurements of the neutrino mass-squared differences. According to the Nufit Collaboration[29] ∆ m = (cid:0) . +0 . − . (cid:1) × − eV , (II.2)∆ m = (cid:0) . +0 . − . (cid:1) × − eV (NMO) , (II.3)or∆ m = (cid:0) − . +0 . − . (cid:1) × − eV (IMO) . (II.4)Neutrino oscillation experiments do not inform the values of the individual neutrino masses, onlythe mass-squared differences. For the individual neutrino masses, information outside of neutrinooscillations is required. B. Neutrinoless double beta decay
With the discovery of nonzero neutrino masses, one of the most important outstanding questionsin particle physics is the nature of the neutrinos: are they Majorana or Dirac fermions? If neutrinosare Majorana fermions, lepton-number L conservation is not an exact law of nature and has tobe 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, ∆ L = 2 processes are strictlyforbidden.Searches for neutrinoless double-beta decay (0 νββ ), ∆ L = 2 nuclear-decay processes of the type( Z, A ) → ( 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 νββ , including neutrinoless double-beta-plus decay, ( Z, A ) → ( Z − , A ) + e + e + and lepton-number violating electron capture, ( Z, A ) + e − → ( Z − , A ) + e + , along with µ − → e + conversionin nuclei, forbidden meson decays [28], including K + → µ + µ + π − , and the production of same-signdi-leptons and no missing energy at hadron colliders (e.g., pp → e + µ + + X , where X is a statewith zero lepton number). Here, we concentrate on constraints from 0 νββ . Several phenomenological collaborations regularly collect and analyze the neutrino oscillation data, estimating thevalues of the oscillation parameters. Here, we will use the results from [29], unless otherwise noted. These areconsistent with other global fits, including those presented in Refs. [32, 33].
If Majorana-neutrino exchange is the dominant contribution to 0 νββ , the rate for 0 νββ is afunction of the neutrino masses. If all neutrino masses are small relative to the typical energyscales involved in 0 νββ , which is of order dozens of MeV, in the absence of new physics other thannonzero Majorana neutrino masses, the amplitude for 0 νββ is proportional to a linear combinationof the neutrino masses: m ββ ≡ (cid:88) i ( U ei ) m i = cos θ (cid:0) cos θ m (cid:48) + sin θ m (cid:48) (cid:1) + sin θ m (cid:48) , (II.5)where m (cid:48) i = e iφ i m i , where φ i are combinations of the CP-odd phases in the PMNS matrix, see, forexample, Ref. [28] for a concrete parameterization; m ββ is a complex parameter and experiments aresensitive to its magnitude. θ and θ are two of the mixing angles used to parameterize the PMNSmatrix. Unless otherwise noted, for the mixing angles, we use the PDG parameterization [28].Under these conditions, a measurement of the rate for 0 νββ , combined with input from neu-trino oscillations, provides nontrivial information on the neutrino masses. Using information fromneutrino oscillation experiments, it is possible to parameterize | m ββ | as a function of two rela-tive phases among the m (cid:48) i parameters and the value of the lightest neutrino mass m least . For thedifferent mass orderings m least = m (NMO) or m least = m (IMO) . (II.6)The fact that these so-called Majorana phases are unknown and, in practice, impossible to constrainexperimentally in any other way, renders the information on m least from 0 νββ always imperfect.Agostini, Benato and Detwiler [34] have carried out a Bayesian analysis incorporating existing datato make predictions, under well-defined assumptions, of the discovery probability for true valuesof | m ββ | . The distributions are shown in Fig. 2. It is striking that, barring some particular physicsthat would drive | m ββ | or m least to zero, the discovery probability is not small, especially in theIMO. It can also be seen that a direct mass measurement, essentially a measurement of m 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 ofa particular isotope. That rate depends on the product of | m ββ | , a phase-space factor, and anuclear matrix element for the ( Z, A ) → ( Z + 2 , A ) transition, the most commonly investigatedtype. Currently, the matrix elements are poorly constrained. Estimates performed using differenttechniques differ by a factor of a few. Qualitative and quantitative improvements are expected inthe near future, but it is fair to expect that, for the foreseeable future, the theoretical uncertaintyon extracting | m ββ | from the rate for 0 νββ will be sizable. For detailed, recent reviews see, for [eV] l m - - - - -
10 1 [ e V ] bb m - - - -
10 1 a) NO, QRPA ] - p r obab ili t y den s i t y [ e V - -
101 [eV] l m - - - - -
10 1 it { m _ { e t a e t a }} [ e V ] - - - -
10 1 b) IO, QRPA
FIG. 2: Marginalized posterior distributions for | m ββ | and m least ( m l in the figure) in the NMO (left) andIMO (right). From Agostini et al. [34]. example, [34, 35]. Notwithstanding the matrix-element problem, experimental progress in the pastdecade has been remarkable, limiting the effective Majorana mass to values well below the levelspresently accessible to direct measurements. Table I, updated from that compiled in [35], gives thehalf-life T ν / and Majorana mass | m ββ | limits from the most recent experiments. TABLE I: Published half-life and Majorana mass limits from recent experiments.Isotope T ν / ( × y) | m ββ | (eV) Experiment Ref. Ca > . × − < . −
22 ELEGANT-IV [36] Ge > < . − .
180 GERDA [37] > . < . − . Majorana Demonstrator [38] Se > . × − < . − .
43 NEMO-3 [39] Zr > . × − < . − . Mo > . × − < . − .
62 NEMO-3 [41] Cd > . × − < . − . Te > . < . − .
350 CUORE [43] Xe > . < . − .
165 KamLAND-Zen [44] > . < . − .
286 EXO-200 [45] Nd > . × − < . − . In viewing the experimental results, it must be kept in mind that the rate for 0 νββ is a functiononly of the neutrino masses when light-neutrino exchange is the leading contribution to 0 νββ . Moregenerally, lepton-number violating physics can impact 0 νββ in a way that the connection betweenthe rate for 0 νββ and the neutrino masses is either indirect or, in some cases, non-existent. Foran overview, see, for example, Ref. [47].0The experimental results of the current generation have been obtained with detectors havingisotopic masses in the range of tens to hundreds of kilograms. A new generation of detectors anorder of magnitude larger is now beginning. A comprehensive summary of the plans and statusmay be found in the APPEC Committee Report [48] prepared for the European strategy. The goalof 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 neutrinomasses can be extracted from searches for lepton-number violation. Conversely, an observation ofneutrinoless double beta decay is unambiguous evidence of lepton-number violation, independentof the uncertainties that affect a mass determination therefrom.
C. 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 [49, 50] and the large-scale structure of theuniverse, including precision measurements of the properties of the cosmic microwave background(CMB) [51], are consistent with the existence of a thermal relic-neutrino background. Theseneutrinos played a significant role in the expansion history of the universe even if, today, they arerather 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 ν ∼ × − eV. Hence, as the universe expanded and the relic neutrino background cooled, the behaviorof neutrinos changed from that of ultrarelativistic relics – “radiation” – to that of non-relativisticspecies – “matter” – as long as the neutrino masses are larger than T ν . Given information fromneutrino oscillations, at least two of the three neutrino masses are known to be much larger than T ν . This transition leaves an imprint in the large-scale structure of the universe in such a way thatprecision measurements provide nontrivial information on the neutrino masses.In the absence of other light particles or new neutrino interactions, the relic neutrino backgroundis best described as a homogeneous mixture of the neutrino mass eigenstates ν , ν , and ν and theirantiparticles. , Theoretically, cosmic surveys are sensitive to the values of the individual neutrinomasses, m , m , m . In practice, given the expected sensitivity of next-generation experiments, If neutrinos are Majorana fermions, the situation is very similar, with left-helicity neutrino states playing the roleof particles and right-helicity neutrino states playing the role of antiparticles. There 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 [50] and would not impact the discussionhere. Flavor-dependent effects are subtle and could impact the picture more significantly. These are still the subjectof intense exploration [52–55]. ≡ (cid:88) i m i . (II.7)Σ can be expressed in terms of the known mass-squared differences and the, currently unknown,lightest neutrino mass, m least :Σ = m least + (cid:113) ∆ m + m + (cid:113) ∆ m + m (NMO) , (II.8)orΣ = m least + (cid:113) − ∆ m + m + (cid:113) − ∆ m + m (IMO) . (II.9)If the neutrino mass ordering is normal (inverted), current oscillation data constrain Σ > . × − eV (Σ > . × − eV). The direct laboratory measurement by KATRIN [56] constrainsΣ < . < .
111 eV (95% CL), according to [57].Other recent analyses of cosmic surveys output similar upper bounds including Σ < .
12 eV (95%CL) [51] and Σ < .
16 eV (95% CL) [58]. Bounds obtained before the Planck-2018 data becameavailable were only slightly weaker, Σ < .
19 eV (95% CL) [59]. These bounds do depend on thevalues of the individual neutrino masses and the neutrino mass ordering, but not very strongly(see recent discussion in [58, 60]). Estimates in [60] output, for the NMO, Σ < .
15 eV (95% CL),and Σ < .
17 eV (95% CL) for the IMO. These translate, roughly, into m least < .
05 eV, mostlyindependent from the mass ordering.In the next decade, it is widely anticipated that next-generation experiments, including CMB-S4 [61], will be sensitive to Σ > × − eV. If expectations are realized, cosmic surveys should beable to determine that Σ is nonzero at better than the three-sigma level [61], independent of themass ordering, assuming no new degrees of freedom or interactions beyond those in the StandardModel.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 asa function of time and to the formation of large-scale structure as a function of time. While theevidence that there are relic neutrinos is very compelling (see, for example, [49, 50]), the presenceof neutrinos is only indirectly inferred and so are statements about their properties. New neutrinoproperties can impact the sensitivity to Σ significantly. The authors of Ref. [62], for example,2argued very recently that if neutrinos are unstable but still very long lived (lifetime between 10 − and 10 − times the age of the universe), bounds on Σ can be relaxed by an order of magnitude. Thesame can be said for more new ingredients to the Standard Model of cosmology. For example, thenature of the dark energy – often parameterized by the dark-energy equation-of-state parameter w – impacts the sensitivity to Σ [63]. Allowing for different palatable ingredients in the StandardModel of cosmology loosens the upper bound on Σ by about a factor of three [57, 60] or more (thisis, of course, not guaranteed. For a counter example, see, for example, [64]). The current tensionbetween early-universe and late-universe estimates of the the Hubble parameter [51, 65] has invitedspeculation concerning new neutrino properties and interactions (see, for example, [66]); some ofthese may have a significant impact on extracting constraints on Σ. See, for example, [67] for avery recent discussion.Massive neutrinos are a required ingredient of cosmological models but since the masses arepresently unknown they must be treated as fit parameters. There are few cosmological parameterssusceptible to laboratory measurement, and neutrino mass is one. A measurement would alleviatethe models of a degree of freedom and allow better determinations of those parameters that canonly be determined from cosmology, such as the equation of state of dark energy and the Hubbleconstant [68, 69]. D. Neutrinos from Astrophysical Sources – Time of Flight
Throughout the universe, neutrinos are produced in cataclysmic astronomical events, includingType II Supernova explosions. Since these are short-duration bursts, it is possible to obtaininformation on the neutrino velocity and hence – since the neutrino energy can be measured – theneutrino mass. The time-spread of the neutrinos observed from SN1987A allows one to constrainthe neutrino mass to be less than a few eV. A very detailed analysis was performed in Ref. [70] –which includes references to several other estimates – and the authors constrained what they referto as the “electron neutrino mass” to be less than 5.7 eV at the 95% confidence level. Strictlyspeaking, given what is known about neutrino mixing, the analysis is more involved and shouldinclude the fact that there are three mass eigenstates with different probabilities for interacting viacharged-current interactions with electrons. In practice, given what is known about the neutrinomass-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 with3energy E a distance D away it will arrive at the detector at t ( E, m ) = D (cid:20) m E (cid:21) , (II.10)assuming all three neutrino masses are degenerate and equal to m , and m (cid:28) E . Relative to amassless particle, the time delay of a massive neutrino is∆ t = D (cid:18) m E (cid:19) = 25 . × (cid:18) D50 kpc (cid:19) (cid:16) meV (cid:17) (cid:18)
10 MeVE (cid:19) . (II.11)For supernova neutrinos, t is not known but one can investigate whether neutrinos with differentenergies arrive at different times, leading to, for example, a larger-than-expected spread in theneutrino arrival times.The detection of neutrinos from the next galactic supernova will allow one to perform a similarmeasurement, perhaps with higher statistics and richer data given the existence of bigger andbetter detectors. The JUNO collaboration, for example, estimates that the JUNO experiment [71],currently under construction, is sensitive to m > D (cid:39)
20 kpc were to be detected. The dependence on D is rather mild and the sensitivity worsensas D increases; the larger ∆ t is compensated by the loss of statistics as D increases. Similarsensitivity – down to at most m > . E. Direct laboratory measurements
In processes involving neutrinos where the total energy of the initial state is well known andthe kinematics of the final state can be measured with precision, it is possible to constrain, usingenergy and momentum conservation, the neutrino mass. Such measurements are often referred toas 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 [76]: “Onpeut essayer de d´eduire de la forme des spectres continus d’´emission une indication sur la valeurde cette masse inconnue...” (One could attempt to deduce from the shape of the continuousemission spectra an indication of the value of this unknown mass...). Fermi independently reached4that conclusion quantitatively in his seminal 1934 article [77], which introduces a so-called four-fermion interaction to describe nuclear beta-decay – the Fermi interaction. Fermi suggested thatthe energy spectrum of the β -rays can be used to determine the mass of the neutrino: “[t]he shape ofthe continuous β -spectrum is determined from the transition probability [computed perturbativelyusing the Fermi-interaction Hamiltonian]. We want to discuss first how this shape depends onthe rest mass of the neutrino µ , in order to determine this constant by comparison with empiricalcurves” [78]. The effect of a nonzero neutrino mass is illustrated in Fig. 3, from [77]. Fermiconcluded that “[t]he greatest similarity to the empirical curves is given by the theoretical curvefor µ = 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” [78]. FIG. 3: Fermi’s illustration of the impact of a nonzero neutrino mass µ on the shape of the β -ray energyspectrum, for “large” (groß), “small” (klein) and µ = 0 [77]. Large and small are defined, loosely, relativeto the mass of the electron. If neutrinos are produced or absorbed via charged-current interactions associated with thecharged-lepton (cid:96) α = e, µ, τ the differential rate associated with the process, which is a function of See [78] for a translation of Fermi’s paper to English. Fermi published preliminary work on the theory of β -decayseveral months before Ref. [77] – in “La Ricerca Scientifica” [79] and “Il Nuovo Cimento” [80] – and submitted histheory for publication in Nature. The Nature submission was rejected, famously, because ‘it contained abstractspeculations too remote from physical reality to be of interest to the reader’ [81]. For more details on the historyof Fermi’s contribution to the theory of β -decay, see [82]. We are indebted to David Kaiser for providing us mostof this information. (cid:88) i | U αi | Γ α ( m i ) . (II.12)Here Γ α ( m i ) is the differential rate of the process of interest when a neutrino with mass m i isemitted or absorbed. For the observables under consideration here, neutrino production is bestdescribed as incoherent, under the assumption that the neutrino masses are within the sensitivityof 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 , m , m as long asall | U αi | (cid:54) = 0, which turns out to be the case. In practice, one needs to account for intrinsic andexperimental uncertainties associated with the initial state and the finite resolution of the variousmeasuring apparatuses. Taking these uncertainties into account, in the limit where the neutrinomasses are small enough compared to the various energy scales of the system, one can express Γ α ( m i ) = Γ α (0) + m i dΓ α d m i (0) + O ( m i ) . (II.13)Here, Γ α ( m i ) stands for the differential rate convoluted with the uncertainties associated to themeasurement in question (slightly different from the object in Eq. (II.12)). This object is smootharound m i = 0 in such a way that the series expansion above is meaningful.To leading order in the neutrino masses, taking uncertainties into account, (cid:88) i | U αi | Γ α ( m i ) (cid:39) (cid:20) Γ α (0) + m ν α dΓ α d m i (0) (cid:21) , (II.14)where m ν α ≡ (cid:88) i | U αi | m i , (II.15)is an effective neutrino mass-squared associated with the charged-current processes involving thecharged-lepton (cid:96) α . In the limit where all neutrino masses are very small, all such kinematicalsearches translate into bounds on different m ν α . It is often the case that experiments will quote upper bounds for the square-root of m ν α ,defined to be m ν α ≡ (cid:112) m ν α . In turn, m ν e , m ν µ , m ν τ are sometimes referred to in the literature This discussion is meant to be generic and purely for illustrative and pedagogical purposes. We return to thespecific cases of the charged-lepton energy spectrum of electron-mediated charged-current processes, includingnuclear beta-decay, in more detail in Sec. IV. Different constraints can also be obtained, at least in theory, from neutral current processes. In practice, thereare no low-energy, high-statistics neutral-current processes one can use to extract meaningful information. Theseinclude, for example, ν µ e → ν µ e scattering and the very rare K → πν ¯ ν decay. m ν τ comes from precision measurements of τ -decays into multi-pionfinal states. The strongest such bound was reported by the ALEPH collaboration [83]: m ν τ < . τ − → ν τ π − π + π − – around 3,000 events – and τ − → ν τ π − π + π − ( π ) – around 60 events. It hasbeen estimated that an order-of-magnitude improvement is possible if one were to take advantageof the τ -samples recorded by the B-factories [84] (almost 10 τ − → ν τ π ± π − ).The strongest bound on m ν µ comes from precision measurements of pion decay at rest, π + → µ + ν µ . The authors of Ref. [85], analysing the decay at rest of positively charged pionsat PSI, extracted the upper bound m ν µ < .
17 MeV at the 90% confidence level, along with themeasurement m ν µ = ( − . ± . . From this experiment, taking the new knowledgeof neutrino-flavor oscillations into account, the currently most precise value for the charged pionmass is deduced [86].Given what is known about neutrino masses from neutrino oscillations and constraints on m ν e ,to be discussed momentarily, the constraints on m ν µ and m ν τ discussed above are not especiallyrelevant when it comes to informing the values of the light neutrino masses. Indeed, there are noprocesses involving muons or tau leptons – today or in the foreseeable future – capable of competingwith current and future information from electron-mediated charged-current processes.Before proceeding, we highlight that a variety of alternatives to the notation m ν e are found inthe literature, including m β , m ν , and m . Henceforth, we will make use of m β ≡ (cid:113)(cid:80) i | U ei | m i ,for a few reasons. As already discussed, m ν e should be deprecated because the electron neutrinois not a particle and does not have a mass. The term m ν is better but is used often in manydifferent contexts. The term m is sometimes defined as the mass of the lightest eigenstate ( m or m depending on the ordering) and we want to avoid confusing the two different objects. Thechoice m β is also the one made by the Particle Data Group [28]. The quantity m β is a particularcombination of the masses of real (propagating) neutrinos, as distinct from the virtual or effectivemass m ββ in 0 νββ , introduced in Sec.II B, that does not correspond to propagating neutrinos.As will be shown, m β (cid:39) m . A kinematic measurement of m β simultaneously determines all 3eigenmasses, up to a binary uncertainty in the mass ordering.7Assuming the neutrino mixing matrix is unitary, m β = m + | U e | ∆ m + | U e | ∆ m , (II.16)for either mass ordering, keeping in mind that ∆ m is positive for the NMO and negative for theIMO. Since, it turns out, | U e | and ∆ m are both quite small the approximation m β (cid:39) m workswell unless m is very small. Quantitatively it holds at the percent level or better for m valuesdown to 0.05 eV, and never differs by more than 8 meV even in the limit m = 0. In brief, onecan say that, to a good approximation, beta decay and electron capture measure the mass m ,independent from the mass ordering.The strongest bound on m β comes from precision measurements of tritium beta decay. Thisis the subject of the bulk of this review. The KATRIN experiment, after collecting data forfour weeks, established the strongest bound to date, m β < . m β = − . +0 . − . eV , consistent with zero. The ultimatesensitivity of KATRIN is to m β > . m β , see [87]. To leading order, and for all practical purposes,direct searches for kinematical effects of nonzero neutrino masses also do not depend on the natureof the neutrinos – Majorana or Dirac fermions. The fact that there is a charged lepton with awell defined charge in the final or initial state renders the leading order amplitudes identical forMajorana and Dirac neutrinos. The same is true of all relevant QED corrections. At higher orderin the weak interactions, however, there are unobservably small differences between Majorana andDirac neutrinos. These differences are not only suppressed by the Fermi constant to some powerbut are also proportional to the neutrino masses. A concrete example is the electron spectrumassociated with the five-body final-state neutron decay, n → pe − ¯ ν i ν j ¯ ν k in the Dirac case, n → pe − ν i ν j ν k in the Majorana case, where i, j, k = 1 , ,
3, the different mass eigenstates. In theMajorana case, for example, there are interference effects between ν i and ν j when i = j , if m i isnot zero. These are clearly not present in the Dirac case. For illustrations of this phenomenon, see The second and third terms in Eq. (II.16) are | U e | ∆ m ∼ × − eV , and | U e | ∆ m ∼ ± × − eV . eγ scattering into neutrinos or [89] for a discussion ofthe end point of the bremsstrahlung spectrum of coherent neutrino scattering on nuclei ( ν + N → ν + N + γ ).The presence of new “neutrino” states, however, modifies the interpretation of results from thesetypes of experiments. Indeed, the existence of new, relatively heavy, “neutrino” states is stronglyconstrained by precision measurements of β -decay, meson-decay, tau-decay, etc. For recent reviewssee, for example, [90–96]. 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 robustupper bound on the neutrino masses. The KATRIN bound, combined with results from the currentoscillation data, translates into (two significant digits)0 < m < . , (II.17)7 . × − eV < m < . , (NMO) (II.18)2 . × − eV < m < . , (II.19)or2 . × − eV < m < . , (II.20)2 . × − eV < m < . , (IMO) (II.21)0 < m < . , (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 neutrinomasses. Figure 4 depicts the values of the three light neutrino masses as a function of m least , forboth neutrino-mass orderings.Above, we highlighted the fact that the relation between precision measurements of the β -decay spectrum and the neutrino masses, given what we know from oscillation experiments, is veryrobust. 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 – are “large,” the presence of the extra heavy neutrino will distort the β -decay spectrum. If, instead, the new neutrino masses are “small,” the presence of the extraneutrino will simply add to m β , i.e., the sum in Eq. (II.15) would encompass all mass eigenvalues m i , i = 1 , , , . . . for all m i less than a few eV. For more details see, for example, [97–99]. TheKATRIN collaboration recently made available the results of a search for new neutrino states [100];we discuss it in detail later in this review, see Sec. VI A. As an aside, new neutrino mass states that9 m m m m least ( eV ) m a ss ( e V ) m m m m least ( eV ) m a ss ( e V ) FIG. 4: Current best-fit values of the neutrino masses m , m , m as a function of the lightest neutrinomass, for the normal mass-ordering (top) and the inverted mass ordering (bottom). admix with electron-flavor neutrinos only render the electron-weighted mass-squared parameter m β larger. Hence upper bounds to m β are especially robust and cannot be bypassed by postulatingthe existence of new particles.0 III. NEUTRINO MASS MODELS AND NEW PHYSICS
In this section, we briefly discuss how direct searches for neutrino masses inform our under-standing of the origin of neutrino mass and can be used to discover other new physics.
A. Neutrino mass models
Similar to all fermions in the Standard Model of particle physics, the known neutrinos canacquire nonzero masses only after electroweak symmetry breaking. Unlike charged fermions, thedynamical mechanism behind nonzero neutrino masses is unknown. Identifying the physics respon-sible 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 degreesof 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 neutrinomasses may come from a large range of experimental efforts, from searches for rare muon processes(e.g. µ → ee + e − decays) to the Large Hadron Collider to next-generation neutrino-oscillationexperiments. Direct measurements of the neutrino mass, along with pursuits of lepton-numberviolation, 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 smallestlepton mixing angle is almost as large as the largest quark mixing angle – and the potentialorganizing principles responsible for its observed features may be qualitatively different. Severalof the theoretical approaches to the problem of lepton flavor also make predictions for the valuesof the neutrino masses, which will be informed most straightforwardly by direct searches for thekinematical 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 under-standing of the origin of neutrino masses. For example, if neutrinos are Dirac fermions, m least = 0allows one to contemplate that there are only two right-handed neutrino fields, in stark contrast toall other fermionic degrees of freedom in the Standard Model that come in three flavors. The sameis true if the neutrinos are Majorana fermions and their masses are a consequence of the so-calledType-I seesaw mechanism [101–106]. In this case, m least = 0 translates into the possibility that1there are only two right-handed neutrinos. 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 least is exactly zero since m least = 0and m least (cid:28) ∆ m , | ∆ m | are, in practice, indistinguishable. Instead, one could determine that m least (cid:54) = 0 with some confidence. More quantitatively, m β = m + | U e | ∆ m + | U e | ∆ m , (III.1)= m + 7 . × − eV , (NMO) (III.2)or m β = m + | U e | ( − ∆ m − ∆ m ) + | U e | ( − ∆ m ) , (III.3)= m + 2 . × − eV , (IMO) (III.4)using, in accordance with Ref. [29], | U e | = 0 . | U e | = 0 .
297 and | U e | = 0 . m (cid:54) = 0, one needs to constrain, in a statistically significant way, m β > . × − eV ( m β > . × − eV ) assuming the neutrino mass ordering is known to benormal (inverted).Another qualitatively different hypothesis that, if confirmed, would impact our understanding ofthe 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, stillan option. For example, if m least = 0 . m most − m least ∼ .
12 ( m most + m least )2 , (III.5)where m most is the heaviest neutrino mass, for both mass orderings. Current constraints allowneutrino masses that are almost degenerate – all of the same order of magnitude – especially ifone considers that the cosmology bounds can be significantly alleviated with the introduction ofnew ingredients. Experiments sensitive down to m β ∼ .
01 eV can definitively test the hypothesisthat the neutrino masses are almost degenerate. Strictly speaking, even if there are only two right-handed neutrinos, the lightest neutrino mass is expected to benonzero, generated at the two-loop level even in the absence of new neutrino interactions. In this case, however, m least is expected to be many orders of magnitude lighter than the other neutrino masses [107]. The magnitudes of the mass-squared differences and | U ei | , i = 1 , , B. Sensitivity to new phenomena
As discussed in Sec. II, cosmic surveys, searches for 0 νββ , and direct kinematic measurementsof neutrino masses are all sensitive to the values of the neutrino masses. The first two probesare indirect. They rely on other ingredients that govern the expansion history of the universe, onthe absence of new interactions involving neutrinos, on the Majorana nature of neutrinos, on theabsence of more, directly accessible lepton-number violating interactions, etc. This means thatby combining these different probes of the values of the neutrino masses we can verify whetherthe assumptions that go into relating cosmic surveys and the rate for 0 νββ to the values of theneutrino masses are valid.In the absence of new particle physics and new cosmology-related ingredients, m β , m ββ , and Σare strictly correlated. In particular, assuming the neutrino oscillation parameters are known, it istrivial to express both m ββ and Σ as functions of m β – see Eqs. (III.2, III.4). These are depicted inFigs. 5 and 6, for both mass orderings. Here we assume neutrinos are Majorana fermions; for Diracfermions, the rate for 0 νββ is zero. In the case of m ββ , the bands are a consequence of all possiblevalues of the relative Majorana phases, currently completely unconstrained. For everything else,we use the current best-fit values of the oscillation parameters from [29]. The relevant oscillationparameters – sin θ , sin θ , ∆ m , ∆ m – are all known at better than the 4% level.Figs. 5 and 6 allow one to identify circumstances that would imply the existence of new phe-nomena. For example, independent from the neutrino mass ordering, if precision measurements oftritium β -decay revealed that m β is larger than 0.05 eV, cosmic surveys imply the existence of newcosmology-related ingredients. This is a very robust statement. Even if, ultimately, we find thatnew neutrino mass eigenstates are the dominant contribution to m β , their existence, given thattheir mixing with active neutrinos is rather large, is ruled out by cosmic surveys in the absence ofnew ingredients. See, for example, the bounds on the effective number of neutrinos, N eff , in, forexample, [51].If, on the other hand, m β is constrained to be smaller than 0.1 eV and one finds m ββ to be largerthan 0.1 eV – the sensitivity of current experiments approaches m ββ ∼ . νββ other than neutrino exchange.3 m β ( eV ) m ββ ( e V ) FIG. 5: m ββ as a function of m β , for both the normal (lighter, blue) and inverted (darker, red) massorderings. 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 [29]. The whited-outregion inside the light-blue contour is meant to highlight the values of m β for which m ββ can vanish exactly.We assume the neutrinos are Majorana fermions. If neutrinos are Dirac fermions, m ββ = 0. The grey,horizontal band corresponds to the 95% CL upper bound on m ββ from GERDA [37]. The width of theband is a consequence of uncertainties in the nuclear matrix element for the neutrinoless double-beta decayof Ge. The vertical line corresponds to the current 90% upper bound on m β [56]. IV. KINEMATIC DETERMINATION FROM BETA DECAY
In beta decay the energy available from the nuclear mass difference is carried away by theelectron and the neutrino. The two particles share the energy in a statistical way, determinedquantum mechanically by the available phase space for each. Because the electron cannot abscondwith all the energy if the neutrino has rest mass, that small amount of energy alters the electronspectrum near its endpoint where it would otherwise have taken all the energy. The beta spectrumin 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 ismaximized by choosing isotopes with the smallest Q-values. As we discuss, however, a low Q-valuealone does not guarantee a good basis for an experiment.4 m β ( eV ) Σ ( e V ) FIG. 6: Σ as a function of m β , for both the normal (blue,solid) and inverted (red,dashed) mass orderings.We use the current best-fit values of the oscillation parameters from [29]. The, horizontal band correspondsto the range of 95% CL upper bounds on Σ discussed in [60]. Different upper bounds correspond to differentingredients added to the Standard Model of cosmology. The vertical line corresponds to the current 90%upper bound on m β [56]. A. Beta spectrum
As discussed in Sec. II, the fact that the three eigenmasses m , m , and m are linked byneutrino-oscillation data has simplified the experimental task of determining the mass scale becauseit is now possible to work with beta decay alone; separate determinations involving µ and τ leptons are no longer needed. The most sensitive direct searches for m β to date are based on theinvestigation of the electron spectrum of tritium β -decay. The electron energy spectrum of β -decayfor a neutrino with component masses m , m , and m is the incoherent sum of the contributionsfrom each mass eigenstate: d Γ dE = G F | V ud | π ( G V + 3 G A ) F ( Z, β ) β ( E + m e ) ( E − E ) × (cid:88) i =1 , | U ei | (cid:2) ( E − E ) − m i (cid:3) Θ( E − E − m i ) , (IV.1)where G F is the Fermi coupling constant, V ud is an element of the CKM matrix [28], E ( β ) denotesthe electron’s kinetic energy (velocity), E , the ‘endpoint energy,’ corresponds to the maximumkinetic energy in the absence of neutrino mass, F ( Z, β ) is the Fermi function, taking into account5the Coulomb interaction of the outgoing electron in the final state, and Θ( E − E − m i ) is thestep function that ensures energy conservation. The vector and axial-vector matrix elements are G V = 1 and G A = − . C = G F | V ud | π ( G V + 3 G A ) . (IV.2)The relationships between E , the Q-value, and the atomic mass difference are detailed in [109]and one is given in Eq. (V.1). As both the matrix elements and F ( Z, β ) 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, d Γ dE ≈ r ( E − E ) (cid:88) i =1 , | U ei | [( E − E ) − m i ] / Θ( E − E − m i ) , (IV.3)where r is the detected event rate per atom in the last eV of the spectrum in the absence of mass(if E and E are also in eV). The variables β and E + m e and the function F ( Z, β ) are evaluatedat the endpoint and absorbed in the constant r . r = λC t E (IV.4)where λ = 1 . × − s − is the tritium decay constant. The constant C t is the ratio of theintegral of the spectrum of the form of Eq. (IV.1) to one of the form of Eq. (IV.3), renormalized byany molecular or atomic branching to states populating the final eV of the spectrum. The valueof C t (cid:39) F ( Z, β ) (cid:39) πZα . − . ββ (1 − e − πZα/β ) , (IV.5)where α is the fine-structure constant and Z is the charge on the daughter nucleus. For the barenucleus, the extrapolated endpoint energy in the laboratory is 18522.44 eV [109], and the fractionalintensity in the last eV of the spectrum is 2 . × − . One can also write the constant C in termsof the decay constant, C = 3 λC t (nuclear) E F ( Z, β ) β ( E + m e ) (IV.6)where β is the value of β at the endpoint. If there are N atoms in the source, the total ratein the last eV is r = N r and the total activity is C t rE . Figure 7 shows the shape of thespectrum near the endpoint, where the effects of neutrino mass are most pronounced. Neutrino6 (cid:2)(cid:5)(cid:3)(cid:9) (cid:2)(cid:5)(cid:3)(cid:4) (cid:2)(cid:4)(cid:3)(cid:9) (cid:4)(cid:3)(cid:4)(cid:4)(cid:6)(cid:23)(cid:5)(cid:4) (cid:1)(cid:4)(cid:3) (cid:8)(cid:23)(cid:5)(cid:4) (cid:1)(cid:4)(cid:3) b (cid:2) (cid:7) (cid:8)(cid:6) (cid:5) (cid:14) (cid:1) (cid:11) (cid:5) (cid:13) (cid:8) (cid:1) (cid:7) (cid:1) (cid:3) (cid:7) (cid:1) (cid:1) (cid:9) (cid:10) (cid:1) (cid:12) (cid:1) (cid:2) (cid:8) (cid:4) (cid:1) (cid:2) (cid:11)(cid:17)(cid:14)(cid:13)(cid:22)(cid:21)(cid:20)(cid:19)(cid:1)(cid:14)(cid:19)(cid:14)(cid:21)(cid:15)(cid:24)(cid:1)(cid:1) (cid:1)(cid:2)(cid:1)(cid:1) (cid:2) (cid:1)(cid:16)(cid:19)(cid:1)(cid:14)(cid:12)(cid:1)(cid:2) n (cid:1)(cid:10)(cid:1)(cid:4)(cid:4)(cid:4)(cid:4)(cid:1)(cid:18)(cid:14)(cid:12)(cid:1)(cid:2) n (cid:1)(cid:10)(cid:1)(cid:4)(cid:7)(cid:9)(cid:4)(cid:1)(cid:18)(cid:14)(cid:12)(cid:1)(cid:2) n (cid:1)(cid:10)(cid:1)(cid:5)(cid:4)(cid:4)(cid:4)(cid:1)(cid:18)(cid:14)(cid:12) FIG. 7: Beta spectrum of the decay of atomic tritium near the endpoint, as given by Eq. (IV.9), from [111].In the figure, m ν ≡ m β . mass experiments in beta decay are fundamentally just counting experiments. The neutrino mass m β can in principle be determined or limited from a single measurement of the number of events in asuitably chosen interval ∆ 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 knownwell enough from other information. This is an idealization but not unrealistic for experimentslike Project 8 and calorimetric detectors (described below) where data both on the backgroundabove and the spectrum below the endpoint are automatically taken “for free” because all eventsare recorded as they occur. The principle would also apply to an experiment like KATRIN [112]that collects integral spectral data point–by–point, but with additional time spent to obtain theneeded information. This ‘time expansion ratio’ is discussed in Sec. VIII A. The endpoint energyis not needed in an absolute sense; it need only be determined relative to ∆ E from the shape ofthe spectrum outside that window.The total number of signal events N s in time t in this window is obtained by integratingEq. (IV.3), N s = rt (cid:88) i =1 , | U ei | [(∆ E ) − m i ] / (IV.7) (cid:39) rt (∆ E ) (cid:34) − (cid:80) i =1 , | U ei | m i (∆ E ) (cid:35) . (IV.8)In the last step we invoke the unitarity of the PMNS matrix and the assumption that (∆ E ) (cid:29) (cid:80) i =1 , | U ei | m i to allow a first-order expansion. The latter assumption exploits the fact that7neutrino mass experiments generally explore masses considerably smaller than the instrumentalwidths and backgrounds for which ∆ E is optimized. The summation in Eq. (IV.8) was definedearlier as m β = (cid:88) i =1 , | U ei | m i that is used as a single parameter representing the result of a beta-decay measurement whereinthe individual mass eigenstates are not resolved, and Eq. (IV.8) motivates the replacement. Withthis replacement, the simplified beta spectrum becomes d Γ dE ≈ r ( E − E )[( E − E ) − m β ] / Θ( E − E − m β ) . (IV.9) B. Isotopes of Interest
Neutrino mass affects the shape of the beta spectrum only near the endpoint. Because thespectrum rises quadratically from the endpoint, the fraction of decays that produce events ina region of width m β at the endpoint scales as ( m β /Q ) , which means that a low Q-value isadvantageous, other things being equal. This generalization ignores details of the spectral shapeat lower energies, but is sufficient to guide attention to suitable isotopes. A low Q value is verydesirable [113], but another important factor is the specific activity of the source. In Table II low-Q-value candidates are compared via a benchmark decay rate in the last eV of the spectrum. Even TABLE II: Source mass required to produce 1 event per day in the last eV of the spectrum. Q A is theatomic mass difference.Isotope Spin-Parity Half-life Specific Activity Q A Branching ratio Last eV Source Massy Bq/g eV g H ⁄ + → ⁄ + . × . × − . × − In ⁄ + → ⁄ + . × . × − . × − . × Cs ⁄ + → ⁄ − . × . ×
440 (0 . − × − . × − Re ⁄ + → ⁄ − . × . × . × − Ho ⁄ − → ⁄ − . × ∼ − ∼ . × − though tritium has the highest Q-value of the four, its superallowed beta decay and low atomicmass have made it the isotope of choice through 70 years of direct mass searches. The low Q-valuesof In,
Cs, and
Re are outweighed by the forbidden nature of the beta decays. Calorimetricmeasurements of the
Re decay were carried out successfully down to a neutrino mass limit of 158eV [114], but pressing on much further would require prohibitively large source masses. Similarly,were it not for the source mass, the
In [115–119] and
Cs [120] decays would be compelling fortheir low Q-values. The In transition is accompanied by a prompt gamma that could be used forbackground reduction.A different approach was suggested by De R´ujula [121], who noted that
Ho has a very low Q-value. This isotope decays by electron capture and emits neutrinos instead of antineutrinos. Thevisible energy release is dominated by sharp lines corresponding to vacancies created in variousatomic shells, but the Lorentzian tails of the lines extend to a kinematic endpoint that is sensitiveto neutrino mass, just as in beta decay. There is no simple prescription for the branch to thelast eV, but recent work [122] reports a Q-value of 2858(11) eV. A rough estimate of the sourcemass needed for an equivalent sensitivity of about 1 eV has been extracted from [123], and thisinformation is also included in Table II. Holmium is a viable candidate at this basic level. Recentexperimental work from the ECHo collaboration [124] has yielded a Q-value of 2838(14) eV, and alimit on the neutrino mass of 150 eV.
V. EXPERIMENTAL PROGRESS
Experiments focused specifically on determining the ‘mass of the neutrino’ (as it was thenthought 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 hasthe highest specific activity. By the 1970s it was clear that neutrino mass effects were small enoughthat molecular and atomic effects competed, leading to the “final-state” problem. If anything, thiscemented the role of tritium because of its simple atomic structure. However, a different approachthat could circumvent the final-state problem completely, the microcalorimeter, emerged and hasbeen the scene of intensive technical development. In this section the chronology and status of theexperimental research on the three viable isotopes, tritium,
Ho, and
Re, are presented.The final-state problem takes a different shape for each isotope and has not been completelycircumvented, as we describe in Sec. VII. Henceforth for brevity we replace H with T to denotetritium symbolically.9
A. Tritium beta decay
The first experiments to quantitatively constrain the mass of the neutrino took place in 1948with one in Glasgow and the other in Chalk River. Both made use of gaseous tritium in a propor-tional counter. Over the subsequent half-century, limits on the mass were pushed down thanks toexperimental and conceptual improvements, with a wide variety of instruments. The experimentsare summarized in Table III (some experiments, for example [125], are not included for lack ofinformation). The experiment of Bergkvist [133, 134, 151] ushered in the modern era with an
TABLE III: Neutrino mass experiments with tritium. Units: eV.Group Date Source Spectrometer Limit or mass Ref.Curran et al. Proportional counter < Proportional counter <
500 [127]Curran et al. Proportional counter < <
250 [129]Hamilton et al. <
200 [130]Salgo & Staub 1969 T O Electrostatic <
320 [131]Daris & St. Pierre 1969 T:Al Magnetic <
75 [132]Bergkvist 1972 T:Al Magnetic <
55 [133, 134]R¨ode & Daniel 1972 T:Polystyrene Magnetic <
86 [135]ITEP 1980 T:Valine Magnetic = 30 ±
16 [136, 137]Simpson 1981 T:Si Si(Li) <
65 [110]ITEP 1985 T:Valine Magnetic = 35 +2 − [138]Zurich 1986 T:C Magnetic <
18 [139]ITEP 1987 T:Valine Magnetic = 30 +2 − [140]LANL 1987 T Magnetic <
27 [141]INS Tokyo 1988 T:CdArachidate Magnetic <
29 [142]INS Tokyo 1991 T:CdArachidate Magnetic <
13 [143]LANL 1991 T Magnetic < . <
11 [145]Mainz 1993 T MAC-E < . MAC-E < .
35 [147]Mainz 1998 T MAC-E < . MAC-E < . MAC-E < .
05 [150]KATRIN 2020 T MAC-E < . FIG. 8: Illustration of the electrostatic-magnetic spectrometer used by Bergkvist in 1972 [151]. 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 thecomplicated tritiated molecule used, the amino acid valine, and the energy loss in the source. Twoexperiments, at Los Alamos and Livermore National Laboratories, made use of gaseous molecularT , for which the final-state spectrum could be well calculated. The Los Alamos apparatus is shownin Fig. 9. 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 whenthere are more counts in the endpoint region than expected, and it was eventually shown in 2015[109] that the problem was, once again, the final-state distribution. More recent calculations ofthis distribution [154] resolve this problem, and with the new calculations the Los Alamos andLivermore data are consistent with zero mass. More information on this is given in the sectionon final-state distributions, Sec. VII A. Other experiments also reported negative mass-squaredvalues that were eventually traced to different systematic errors. The early results from the Mainzexperiment, which used as a source a frozen film of tritium [146], were affected by dewetting of thefilm at temperatures near 4K. The microcrystalline ‘frost’ that resulted caused the line broadeningfrom energy loss to be anomalously large. Lowering the temperature to 2K stabilized the films and1 FIG. 9: The LANL tritium beta decay experiment [141, 144, 152]. A windowless gaseous source of molecularT on the left produces beta electrons that are guided by solenoid coils to a focal-point collimator at theentrance to a magnetic spectrometer. The spectrometer is the Tret’yakov cascaded iron-free toroidal type[153] in which electrons cross the axis four times before reaching the silicon multipixel detector at theright-hand end. eliminated the negative mass squared effect [149]. The Troitsk experiments exhibited evidence fora step in the spectrum a few eV below the endpoint. While a specific explanation for the spectralshape has not been found, the effect was found to be associated with runs during which the sourcepressure was not monitored [150]. 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 III are included). The plot reveals a striking Moore’s-Law character over70 years. Also indicated on the plot is the mass that would close the universe with neutrinos(hot dark matter) alone (HDM Ω = 1), the electron-weighted mass corresponding to the smallestpossible 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 He.While neutrino mass can only be deduced from beta decay, the mass difference can be determinedboth from beta decay and from independent mass spectrometry methods. The comparison servesas a uniquely valuable check on possible systematic effects that might influence the beta decayexperiments, with no other symptoms. Indeed, it was mass spectroscopy that initially supported2 m β m a ss li m i t ( e V ) Year HDM Ω = 1 IMO lower limit NMO lower limit FIG. 10: Upper limits on the neutrino mass obtained from tritium beta decay. The point with error bars isa non-zero result (see text). HDM: Hot Dark Matter. and then finally contradicted the ITEP claim of a non-zero neutrino mass [155]. Agreement of thetwo kinds of determination within uncertainties is a necessary, although not sufficient, conditionfor a valid neutrino mass result [155]. As the state of the art in both fields of measurementadvances, however, the power of this comparison is beginning to diminish because of the presenceof work-function differences that can shift the beta endpoint by fractions of an eV. This alsohas the consequence that highly precise atomic mass determinations cannot be used to improvethe sensitivity of neutrino mass measurements by fixing a fit parameter. Table IV gives recentdeterminations of the atomic mass difference. The measured quantity in the beta decay of molecular
TABLE IV: Atomic mass difference between T and He. Units: eV.Group Year Mass Difference Ref.Univ. Washington 1993 18590.1(17) [156]SMILETRAP 2006 18589.8(12) [157]Florida State Univ. 2015 18592.01(7) [158]KATRIN 2019 18591.5(5) [56] tritium is the ground-state to ground-state extrapolated endpoint energy E , which is related to3the atomic mass difference Q A by E = Q A − b + b ( f )0 − E rec , (V.1)where b = 4 .
59 eV is the binding energy of the initial molecular state, b ( f )0 = − .
71 eV is thebinding energy of the final molecular state and E rec = 1 .
705 eV is the recoil energy [109]. Thespecific values are for T beta decay. With these corrections, a measurement of the endpoint fromthe KATRIN experiment translates to the value shown in the table, which is in good agreementwith the mass spectroscopic values. The 0.5-eV uncertainty is dominated by work functions. B. The MAC-E Filter and KATRIN
Until the 1990s the Tret’yakov cascaded toroidal magnetic spectrometer [153], such as thatdepicted in Fig. 9, was the premier instrument for tritium beta decay experiments. It had good ac-ceptance and resolution by the standards of the day but, as experimental groups contemplated thenext steps, a scale-up in size was clearly necessary for increased statistical precision. The Troitskand Mainz groups turned to a concept that had been developed for photoelectron spectroscopy, theretarding-field analyzer. For neutrino mass experiments with tritium, this type of instrument hasa unique advantage: energy conservation eliminates the possibility of a high-energy tail in the re-sponse function. Instrumental tails extending beyond the endpoint are particularly deadly becausethey shift the mass-squared value negatively if not recognized, and even when recognized severelydegrade the sensitivity to neutrino mass. Moreover, the retarding-field analyzer combined with amagnetic 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 β at the endpoint of a beta spectrum of total kinetic energy E , instrumental resolution of order m β E is needed. The spectral fraction per decay that falls in thelast m β of the beta spectrum is approximately ( m β E ) , to within a constant of order unity.For spectrometric experiments in which the source and the detector are physically separated, alimit on source thickness is set by the cross section for inelastic interactions of outgoing electrons,such that one must have σn s ≤
1, where σ is the inelastic cross section and n s the superficialnumber density. More intense sources to reach smaller neutrino masses must therefore have largerareas. On general grounds, the dimensions of the source (radius R src ) and the dimensions of thespectrometer (length or radius R ana ) are related through the resolution needed, specifically,∆ EE (cid:39) (cid:18) R src R ana (cid:19) α . (V.2)4For magnetic spectrometers such as the Tret’yakov type, α = 1. The MAC-E filter, however, hasa different scaling relationship. The source is immersed in a high magnetic field B src , and thespectrometer in a relatively low field B ana . Electrons move from source to analyzer adiabaticallyalong magnetic field lines, and the energy resolution is determined by the field ratio [112]:∆ EE = B ana B src (V.3)= (cid:18) R src R ana (cid:19) . (V.4)Thus, for magnetic-electrostatic retarding-field analyzers (MAC-E filters) α is a more favorable 2.This scaling property was decisive in shifting the focus of the experimental community toward theMAC-E filter, leading ultimately to the KATRIN project.The Magnetic-Adiabatic-Collimation-Electrostatic filter concept was first described in 1976 byHsu and Hirschfield [159] and further developed by Beamson et al. [160] and Kruit and Read in1983 [161], and it soon found adoption in many areas of electron spectroscopy. It combines goodsource acceptance with high resolution. The Mainz and Troitsk experiments were the first to adoptthis new technology for tritium beta decay, with Troitsk mating it to a gaseous T source on thelines of the Los Alamos design, and Mainz mating it to a source of tritium frozen on a substrateof highly oriented pyrolytic graphite. The basic principle of the MAC-E filter is shown in Fig. 11.The MAC-E filter has very high acceptance. All electrons from a cross-section in the high-fieldsource that are emitted with a pitch angle smaller than a selected value are transmitted to theanalyzing plane. The pitch angle (the angle between the momentum and the field direction) isestablished by means of a solenoidal “pinch” magnet located somewhere between the source anddetector. The energy resolution is determined by the ratio of the magnetic fields in source andspectrometer, and the choice of accepted pitch-angle range [111]. It has a simple, analytic form.The KATRIN experiment represents what is likely to be the ultimate realization of the MAC-Etechnology. 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 − mbar. The design magneticfields of 0.3 mT at the analyzing plane of the spectrometer and 4 T in the source lead to anintegral energy resolution-function step of 0.93 eV at the tritium endpoint. An elevation view ofthe KATRIN experiment is shown in Fig. 12. Detailed descriptions of the KATRIN approach andapparatus can be found in [24, 25, 162].KATRIN began commissioning with tritium at low concentrations in 2018, and in 2019 gathered22 live days of data for its first neutrino-mass measurement. The spectrum is shown in Fig. 13.5 FIG. 11: Principle of the MAC-E filter (from [162]). Electrons produced in a high magnetic field regionenter the large, empty high-vacuum space of the main spectrometer. The magnetic field is lower there by afactor ∼ The MAC-E filter is intrinsically both a magnetic trap and a Penning trap, and both modescan be sources of background. In KATRIN, the presence of the prespectrometer forms a secondPenning 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 aplasma and a catastrophic discharge. This mode was a principal motivator for achieving ultra-highvacuum in the spectrometers, and indeed it was found that the trap could ignite at pressures in the10 − mbar range, but not at the operating pressure 1 . × − mbar [56]. The magnetic modein the main spectrometer traps relatively high-energy electrons in the keV to MeV range, and,somewhat surprisingly, Rn 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, slowlylosing energy by ionization to the residual gas [163, 164]. The resulting background of low-energyelectrons is troublesome because it is non-Poissonian. Installation of large liquid-nitrogen cooledchevron baffles in front of the getter chambers largely solved the problem [165], but the baffleefficiency for Rn decreases as water vapor accumulates. Installation of a subcooler to reduce thetemperature 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.6
FIG. 12: Top: View of the KATRIN experiment (from [162]). Tritium gas recirculates continuously throughthe windlowless gaseous tritium source (WGTS). A solenoidal magnetic field guides beta electrons to themain spectrometer (MS) through a differential pumping restriction (DPS) and a cryogenic pumping re-striction (CPS) that prevent tritium from entering the spectrometers. A prespectrometer (PS) blockslower-energy electrons from reaching the MS. Electrons that surpass the MS potential are detected in thefocal-plane detector (FPD). Calibration devices are located in the rear system (RS). Bottom: Photographof KATRIN’s main spectrometer.
This ‘magnetic shielding’ has proven to be highly effective in rejecting radioactive and cosmic raybackgrounds from the walls [166, 167]. In addition, layers of grids totalling 20000 wires spacedfrom the walls and biased to negative voltages in the 100 – 500 V range further reject backgroundsof soft electrons, and also permit a more precise shaping of the electric field inside the spectrometer[168, 169]. A few of these grids became shorted to each other during a bakeout, but with no majorimplications for KATRIN’s performance.In KATRIN a new and unexpected kind of background was discovered, the production of Ryd-berg atoms and their photoionization in the volume of the main spectrometer. Atoms of hydrogenand heavier species are dislodged from the walls of the spectrometer by the decay of radon daugh-7 a) C oun t r a t e ( c p s ) KATRIN data with 1 errorbars 50 Fit resultb) R e s i du a l s () Stat. Stat. and syst.c)
Retarding energy (eV) T i m e ( h ) FIG. 13: Integral beta spectrum of molecular T from the KATRIN experiment [56]. The spectrum contains2 . × events and was taken over a period of 22 live days. The column density of tritium in the WGTSsource 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 thedistribution of measurement times chosen at each retarding potential. ters embedded therein. These atoms are often neutral and in a distribution of excited states, someof which are so close to the ionization edge that they can be ionized by thermal black-body ra-diation as they cross the spectrometer volume. KATRIN is developing strategies to mitigate thissignificant background. It exemplifies a vulnerability of the MAC-E filter method, that signal elec-trons are slowed almost to rest before being reaccelerated into the detector. Therefore, in additionto the Rydberg atoms, any process that makes slow electrons (ionization of the residual gas, forexample, or the decay of errant tritium) becomes a potential background.These backgrounds, together with additional smaller contributions from the FPD, amounted atthe beginning of operations to about 0.5 counts per second (cps), a factor of 50 larger than the target8value in the Design Report [112]. Increasing the magnetic field in the main spectrometer reducedthe background to 0.29 cps at some cost in resolution. This configuration was used for the firstKATRIN neutrino mass measurement [56]. For this background rate, the optimal measurement-time distribution (bottom panel of Fig. 13) peaks 12 eV below the extrapolated endpoint. Thestatistical contribution to neutrino-mass sensitivity, however, depends approximately on the sixthroot of the background (see Eq. VIII.16), and if no further steps were to be taken, the finalsensitivity would decrease from 200 meV to about 300 meV. Of course, KATRIN has movedaggressively to deal with backgrounds. Shifting the analysis plane toward the detector serves tofurther reduce the effective spectrometer volume and therefore the main backgrounds by a factorof 3. Additional measures are under investigation.Another unexpected complication to KATRIN’s neutrino mass extraction has emerged fromplasma potentials. Although the presence of electric and magnetic potentials has no influence onthe molecular dynamics of KATRIN’s tritium source, it certainly affects the diffusion properties ofthe emitted electron and tritium ions created in the decay. Electron-ion recombination and chargedrift eventually neutralize the source; however, neutralization occurs over long time scales, leadingto both spatial and temporal variations of the charge density of the source. These electromagneticpotential fluctuations impact the energy of the decay electron. A campaign is currently focusedon constraining the uncertainties arising from plasma potentials. Plasma potentials can influenceany technique that makes use of magnetic confinement, including cyclotron radiation emissionspectrometers. C. Ho electron-capture decay
Stimulated by the report of non-zero neutrino mass observed in tritium beta decay by Lyubimov et al. [136], a search began for alternative methods as a verification. The low Q-value for theelectron-capture decay of
Ho to
Dy, about 3 keV [170], attracted interest in the possibilityof a neutrino mass measurement with this isotope. The subshell ratios, i.e. the relative intensityof electron capture in each atomic shell, depend on the neutrino mass because the available phasespace is limited for the deeper shells by the neutrino rest mass. This formed the basis of theinitial attack on the problem by Bennett et al. [171], but they found that the theory of subshellratios in heavy nuclei was not adequate to extract a neutrino mass. In the same year, De R´ujulaproposed [172] a different approach, internal bremsstrahlung in electron capture (IBEC), a radiativeprocess producing a continuous spectrum of photons with an endpoint shape that is modified by9neutrino mass, quite analogous to beta decay. He suggested the calorimetric technique that is thebasis of experimental work today. The term IBEC as used by De R´ujula in his unified treatmentcovered processes that are usually considered separately, the radiative process when a real photonattaches to the electron or the W -boson ( ‘innere bremsstrahlung’ ), and the Lorentzian tails of X-raytransitions as determined by the vacancy lifetime. The X-ray widths are complicated by atomicstructure effects, which were treated schematically. In a paper a year later [121], however, thereference to IBEC is dropped and the X-ray widths are used to derive the transition rate near theendpoint. Springer et al. [173] carried out the first Ho neutrino mass experiment in 1987 andincluded a remarkably detailed theoretical study of IBEC complete with the interference effectsthat were found to be substantial. They set an upper limit on the mass of 225 eV, although theiruse of a Q-value that is now discounted may have unduly influenced the derived limit. In 1994,Yasumi et al. [174] set a less stringent limit on the neutrino mass in a study more in the spiritof the original Bennett et al. one [171] in which the subshell ratios were measured, except theycircumvented the theoretical issues via a direct photoionization measurement of the X-ray yield ata synchrotron light source.Thereafter, experimental work converged on the microcalorimetric approach in which the de-excitation of the
Dy, whether in the form of photons or electrons, is entirely captured andconverted to heat. The endpoint of such a spectrum is just the Q-value, apart from a smallbinding-energy correction in the lattice. Neutrino phase space modifies the endpoint region of thespectrum as in beta decay. The experimental efforts to pin down the Q-value and extract a neutrinomass or a limit, are summarized in Table V. It is a testament to the difficulty of this problem that
TABLE V: Neutrino mass and Q-value experiments with
Ho. Units: eVGroup Date Source Spectrometer Limit or mass Q-value Ref.Hopke et al. − < et al. − > et al. − Proportional counter − et al. Si(Li) <
225 2561(20) [173]Yasumi et al. <
460 2710(100) [174]Gatti et al. − − et al. (ECHo) 2017 Ho:Au Microcalorimeter − et al. (ECHo) 2019 Ho:Au Microcalorimeter <
150 2838(14) [124]
Ho microcalorimeter programs have been pursued, HOLMES, ECHo, and NuMECS.The HOLMES apparatus is based on superconducting transition-edge sensors [181] of molybdenum-copper on a silicon nitride substrate. The source pad is gold, which will be implanted with
Ho[182]. 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 simi-larly molybdenum-copper transition-edge sensors, but supported on nanofabricated silicon beams[183]. The
Ho has been incorporated from aqueous solution into nanoporous gold pads at thebeam ends and spectra have been obtained. While the resolution at 35 eV FWHM is not yetsufficient, the purity and specific activity of the Ho material are good. The ECHo collaboration[122, 184–186] makes use of metallic magnetic calorimeters (MMC) to read out the thermal signalsvia SQUIDs. These devices have delivered the fastest risetimes, < ∼
100 ns, and also hold the recordfor the best energy resolution, 1.6 eV FWHM with an X-ray source of Fe. The MMC devicesthemselves are not intrinsically faster than TES’s, but risetime in TES’s is usually kept longer tomatch the dc-SQUID readout better. The speed is determined by heat diffusion in gold in bothdevices. Use of MMC’s with multiplexing will unavoidably require limitations in detector speed.Ultrapure mass-separated
Ho ions were implanted into gold source pads to produce spectra (seeFig. 24) with an instrumental resolution of 9.2 eV FWHM [124]. Only 2 events above the endpointwere observed, evidence that the background is very low. When the
Ho isotope is reactor-produced by irradiation of
Er, the isotopic contaminant m Ho is also produced, necessitatingmass separation. Both ECHo and HOLMES use this method, while the NUMECS material wasproduced by proton bombardment of Dy, which avoids the m Ho byproduct but has a lower crosssection. D. Re beta decay
In a similar vein as for
Ho,
Re also offers a low Q-value, 2.5 keV, in comparison to that oftritium and thus emerged as an attractive isotope for neutrino mass investigation. Despite its lowQ-value, there are inherent difficulties in using
Re as a neutrino mass target. In particular, its1decay process:
Re(J = 5 / + ) → Os(J = 1 / − ) + e − + ¯ ν e (V.5)is a unique first-order forbidden transition (∆ J π = 2 − ), which significantly alters the phase space ofthe decay electron near the endpoint and gives rise to a formidably long lifetime ( τ = 4 . × y).This places severe requirements on the amount of target material necessary to observe a significantnumber of decay events with electron energies near the endpoint of rhenium (Table II).The beta decay of Re was originally observed using proportional counters, and a definitedetermination of the beta decay process was made by Brodzinski and Conway [187] and by Husterand Verbeek [188] using gas proportional counters. From those early measurements, they deter-mined a Q value of 2.6 keV. Inspired by the results by Lubimov indicating a positive value forneutrino mass, McCammon [189] and later Vitale et al. [190] raised the possibility of using mi-crocalorimeters to better measure the neutrino mass signal. Vitale et al. proposed a rhenium-basedcalorimeter to measure the total visible energy produced from the beta decay process, from whicha neutrino mass measurement could be extracted.The first measurements of
Re decay using microcalorimeters were done in Genoa by theMANU project [191]. They used metallic rhenium as a self-contained absorber, taking advantageof the fact that rhenium at temperatures below 1.7 K becomes superconducting. Under theseconditions, the electronic contributions of the heat capacity are exponentially suppressed, leavingonly phonon/lattice contributions to the heat capacity. When cooled well below the transitiontemperature (usually <
100 mK), the heat capacity should be significantly reduced, allowing forhigh energy resolution, as needed for an accurate endpoint measurement. Unfortunately, theenergy released in the decay by the electron at such low temperatures tends to be trapped inthe form of quasi-particles, for which the recombination time to phonons (and thus a detectablesignal) can be as long as several seconds. As such, metallic rhenium proved difficult to realize asa microcalorimeter for the purposes of beta decay detection.A parallel effort, led by the Milan group MiBETA, used a dielectric compound of rhenium toachieve 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 MiBETAgroup eventually settled on silver perrhenate (AgReO ) as yielding the best performance in termsof efficiency and energy resolution ( σ (cid:39)
18 eV FWHM at 6 keV). Both the MANU and MIBETAgroups extracted measurements of the neutrino mass, both limits below 20 eV. The extracted Kurieplot from the decay of
Re is shown in Fig. 15. The MANU experiment also measured for the2
Advances in High Energy Physics (a) C o u n t s Al K 𝛼 Si K 𝛼 Cl K 𝛼 Δ E ≈ eV @ keV 𝜏 R ≈ ms (b) F igure 24 : AgReO crystal glued on the fi rst XRS array of MARE- . Th e usable pixels give Δ𝐸 ≈ eV at . keV, 𝜏 𝑅 ≈ ms (a).Spectrum measured with the best pixel (b). Indeed, also the other experimental e ff orts of MARE- encountered several di ffi culties [ ]. For example, the settingup of arrays of AgReO crystals turned out to be moretroublesome than expected (Figure ). Th e freshly polishedsurfaces of AgReO crystals shaped to cuboids resulted tobe incompatible with the sensor coupling methods usedsuccessfully in MIBETA with as-grown small crystals. Despitethe use of a micromachined array of silicon implanted ther-mistors, the performance of the pixels was irreproducible and,while gradually populating and testing the XRS array withAgReO crystals, the performance of the instrumented pixelsstarted to degrade. Th is made the array fi nally unusable.Given the sorts of the MARE project, also this branch of theMARE- program was thus dropped in . . . Future of Rhenium Experiments. From the MARE expe-rience, it is clear that a large scale neutrino mass experimentbased on
Re beta decay is not foreseeable in the near future.It would require a major step forward in the understandingof the superconductivity of rhenium, but, a ft er more than years of e ff orts, this is not anymore in the priorities ofthe LTD scienti fi c community. Besides the intrinsic problemsof metallic rhenium, there are other considerations whichmake rhenium microcalorimeters not quite an appealingchoice for high statistics measurements. Because of the largehalf-life of Re, the speci fi c activity of metallic rhenium istoo low to design pixels with both high performance andhigh intensity beta sources. Re activity required by a highstatistics experiment must be therefore distributed over alarge number of pixels—of the order of —while the di ffi -culties inherent with the production of high quality metallicrhenium absorbers contrast with the full microfabrication ofthe arrays. MARE- also demonstrated that AgReO is not aviable alternative to metallic rhenium. For these reasons, the new hope for a calorimetric neutrino mass experiment withLTDs is Ho.
7. Current Experiments . . Calorimetric Absorption Spectrum of Ho EC.
De R´ujulaintroduced the idea of a calorimetric measurement of
HoEC decay already in [ ], but it was only one year laterthat this idea was fully exploited in the paper written byLusignoli [ ]. Th e EC decay Ho + 𝑒 − → Dy + ] 𝑒 ( )has the lowest known 𝑄 value, around . keV, and its half-life of about years is much shorter than Re one. In[ ], the authors compute the calorimetric spectrum and givealso an estimate of the statistical sensitivity to the neutrinomass at the spectrum end-point, including the presenceof the pile-up background. Unfortunately, at that time, theexperimental measurements of the 𝑄 value were scatteredbetween keV and keV causing large uncertainties on theachievable statistical sensitivity.A calorimetric EC experiment records all the deexcitationenergy and therefore it measures the escaping neutrinoenergy 𝐸 ] ; see ( ). Th e deexcitation energy 𝐸 𝑐 is the energyreleased by all the atomic radiations emitted in the processof fi lling the vacancy le ft by the EC decay, mostly electronswith energies up to about keV (the fl uorescence yield isless than − ) [ ]. Th e calorimetric spectrum has lines atthe ionization energies 𝐸 𝑖 of the captured electrons. Th eselines have a natural width Γ 𝑖 of a few eV; therefore, theactual spectrum is a continuum with marked peaks withBreit-Wigner shapes (Figure ). Th e spectral end-pointis shaped by the same neutrino phase space factor ( 𝑄 −𝐸 ) √ ( 𝑄 − 𝐸 ) − 𝑚 ] 𝑒 that appears in a beta decay spectrum, FIG. 14: A view of several microcalorimeter pixels using 500 µ g AgReO absorbers for the MARE experi-ment [192]. first time oscillations in the beta energy spectrum due to environmental fine structure from thetarget crystal, although also revealing a potential systematic uncertainty in their neutrino massextraction. The effect was also measured in the AgReO detectors by the MiBETA group [193]. TABLE VI: Neutrino mass and Q-value experiments with
Re. Units: eVGroup Date Source Spectrometer Limit (90 % C.L.) or mass Q-value Ref.Brodzinsky & Conway 1965 (C H ) ReH Proportional counter N/A 2620(90) [187]Huster & Verbeek 1967 (C H ) ReH Proportional counter N/A 2650(40) [188]MANU (Genoa) 1999 Metallic Re Microcalorimeter ≤
19 2470(4) [194]MIBETA (Milano) 2004 AgReO Microcalorimeter ≤
15 2465.3(17) [195]
The MANU and MIBETA groups eventually combined to propose a new calorimetric experimentusing rhenium called Microcalorimeter Arrays for a Rhenium Experiment (MARE), with up to10,000 of such microcalorimeters to reach sub-eV sensitivity to the neutrino mass [196]. TheMARE effort eventually gave way to the holmium efforts of ECHo and HOLMES, as discussedearlier in this section.3
FIG. 15: Kurie plot showing the measured beta spectrum from
Re decay from the Milano group’s exper-iment [195].
VI. OTHER RESEARCH
The kinematic direct mass search method can be applied not only to the mass of the threeknown active neutrinos, but also to other significant physics questions. Are there sterile neutrinosthat mix slightly with the active species and thereby become observable? They may disclose theirexistence in the form of kinks in the otherwise smooth beta spectrum. Can the primordial seaof relic neutrinos created in the big bang be detected? They would produce a peak in the betaspectrum just beyond the endpoint energy.
A. Sterile Neutrinos
The width of the Z -boson fixes the number of active light neutrinos at 3 [28], but the possibilityremains that there might be neutral fermions with no standard-model couplings. If they admixslightly with the active neutrino states they become observable both in oscillation and direct-massexperiments. The discovery of neutrino mass in particular renewed interest in this possibilitybecause light right-handed singlets would be introduced into the Standard Model if neutrinos areDirac fermions, and heavier partners might exist if neutrinos are Majorana fermions. In directmass searches, these ‘sterile neutrinos’ would appear as a kink in the beta spectrum if the mass isin the kinematically allowed range.4Perhaps the most famous example is the “17-keV neutrino” that was first reported by Simpson in1985, from the observation of a kink in the beta spectrum of tritium implanted into a silicon detector[197]. Much excitement ensued when the same signal seemed to show up in many experiments ontritium and other isotopes, but a convincing demonstration against it was carried out by Mortara et al. [198]. Summaries of the saga can be found in Ref. [199–201].More recent interest in sterile neutrinos has largely been motivated by oscillation searches thathave produced results that are either inconsistent with the 3-neutrino picture, or inconsistent withanother observable such as the reactor antineutrino flux. Recent reviews may be found in Refs. [202–204]. Direct measurements give no indication of sterile neutrino admixtures. The most stringentpublished limit contour from tritium beta decay comes from the Troitsk experiment [205, 206], and[205] also includes a summary of limits from other isotopes. Figure 16 displays the existing limitsfrom direct searches in beta decay. The published KATRIN data on active neutrinos [56] have -6 -5 -4 -3 -2 -1 | U e | m N [keV] Troitsk 2013
Mainz 2013
Troitsk expectations -6 -5 -4 -3 -2 -1 | U e | m N [keV] Troitsk 2013
Mainz 2013
Troitsk expectations
FIG. 16: Limits on admixtures | U e | of a single sterile neutrino as a function of the neutrino’s mass.Admixtures above the curves are excluded. The left panel is from [205], and the right from [207]. Sourcesof the data are given in the original publications. been used to derive limits on sterile admixtures, but such analyses are not rigorous, leaving outcorrelation effects. The KATRIN collaboration has released their own analysis [100], depicted inFig. 17, that takes such effects into account. The background in KATRIN precludes disentanglingthe effect of a sterile neutrino and an unconstrained active neutrino for masses below 10 eV. Onthe other hand, the statistical power of KATRIN opens large new regions of parameter space inwhich to search for sterile neutrino admixtures in the mass range >
10 eV. In order to be able to5 -2 -1 | U e4 | m ( e V ) Mainz 95% C.L. : m = 0 eV Troitsk 95% C.L. : m = 0 eV KATRIN sensitivity 95% C.L. : m = 0 eV KATRIN 95% C.L. : m = 0 eV KATRIN 95% C.L. : m freeKATRIN 95% C.L. : m free , ( m ) = 1 eV FIG. 17: Exclusion curves at the 95% confidence level in the ( | U e | , m ) plane obtained from the analysis ofKATRIN data including statistical and systematic uncertainties [100]. The two solid lines show the expectedsensitivity (light grey) and the associated exclusion (blue online) for fixed m β = 0 eV ). The dotted line(dark blue online) illustrates the exclusion curve obtained with a free m β ). The dot-dashed line (turquoiseonline) displays the intermediate exclusion curve with m β introduced as a Gaussian pull term with a centralvalue of 0 and an uncertainty σ ( m β ) = 1 eV . Also shown are the Mainz [208] and Troitsk bounds [206]. Inthe figure, m ν ≡ m β . take advantage of the available tritium source strength, a new, highly segmented, high-resolutionfocal-plane detector for KATRIN is being developed [209–211] by the TRISTAN collaboration.If the technical initiatives are successful, KATRIN would be able to reach unprecedented | U e | levels, potentially as low as 10 − [212]. Tiny admixtures of neutrinos in the keV mass range areparticularly interesting because they are not excluded by astrophysical bounds, could be darkmatter, and could help explain why supernovae explode in nature but rarely in computers [213].It is also possible to search for sterile neutrino admixtures in electron-capture decays. In thatcase, since capture in atomic subshells produces nominally monoenergetic neutrinos, a precisionmeasurement of the recoil energy by calorimetric means can reveal the admixture of massive neu-trinos. Friedrich et al. [207] have carried out such a measurement for Be embedded in a super-6conducting tunnel junction, obtaining the most sensitive limits by the direct method, as low as | U e | < × − , on the admixture of neutrinos in the mass range 100 - 1000 keV. Their resultsare shown as the unshaded curve in the right-hand panel of Fig. 16. B. Relic Neutrinos
As Fisher has commented, “Every neutrino physicist has wondered at some time if there mightbe a way to detect the relic neutrino background” [214]. While not strictly a topic within ourpurview, since Weinberg’s seminal paper in 1962 [215] the method of choice for such considerationshas always been neutrino capture observed as a peak just beyond the endpoint of a tritium betadecay spectrum, separated from the extrapolated endpoint by a an amount of order m β (seeFigure 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 [216], and thereforethe kinetic energy is negligible in comparison to m β . Cosmology gives the mean relic neutrinodensity in the universe, 56 cm − per neutrino flavor and chirality [217]. The contribution Ω ν ofneutrinos to the closure density of the universe is [216]:Ω ν h = (cid:18) Σ92 . (cid:19) (VI.1)where h is the Hubble constant in units of 100 km s − MPc − . The rate of neutrino capture on anuclear target depends on the neutrino density and its flavor content, the nuclear matrix elementand Q-value, and the target mass. Betts et al. [218] find a capture rate of 951(3) events per yearper kg of tritium for the mean universal density modeled as a Fermi-Dirac distribution throughoutspace.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 tobe expected (see, for example, [217, 220, 221]) show that the neutrino contrast scales approximatelyas the square of the mass but is relatively small [217, 220]: n ν n ν − . (cid:16) m ν eV (cid:17) . , (VI.2)where n ν and n ν are the local and universal neutrino densities, respectively.Experimentally the difficulty is that, tritium being a shortlived radioactive isotope, the targetmass is limited. In KATRIN, the largest tritium neutrino mass experiment, the mass under obser-vation is about 100 µ g, and the expected capture rate is roughly 1 every 10,000 y. The PTOLEMY7 FIG. 18: Spectral features from tritium beta decay and neutrino capture on tritium. From [219]. project, whose primary goal is to measure big bang relic neutrinos, is designed to put 0.1 kg, 1MCi, under observation [218, 222]. To achieve sensitivity for such a large target mass, PTOLEMYplans to use large surface area targets with atomic tritium adsorbed on graphene, so as to reducecharge accumulation effects. The feasibility of using tritium-loaded graphene is currently underinvestigation. An experiment with such a tritium source under observation would potentially alsobe 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, repre-sented by Eq. (IV.9) and the corresponding figure (Fig. 7, Fig. 18). In practice, however, one neverdeals with the decay of an isolated nucleus of tritium or any other isotope, and the surroundingelectrons 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 nota result of electron energy loss in a surrounding medium following the decay, an effect that mustalso be considered, but occur simultaneously with the decay because the initial and final states areboth complex quantum systems with many internal degrees of freedom. Thermal excitations inthe initial state may even increase the lepton energy. Nor can one escape these considerations byturning to electron capture, as will be discussed below. Bergkvist, in his pioneering experimentson tritium (see [133, 134, 151] and Fig. 8), was the first to point out that progress beyond the55-eV limit he obtained would demand an understanding of the FSD.8There are also spectrum corrections that arise in the nuclear decay itself. We discuss thosebelow, finding that most are negligible for the purposes of neutrino mass measurement, with theexception of radiative corrections.
A. 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 matrixelement W kj . Omitting small recoil-order corrections, the spectral density for each transitionbecomes [109] (cid:18) d Γ dE e (cid:19) kj = CF ( Z, β ) | W kj | βE e (∆ kj − E e ) ×× (cid:88) i | U ei | (cid:20) (1 − m i (∆ kj − E e ) (cid:21) / Θ(∆ kj − E e ) . (VII.1)This expression is written in terms of the total electron energy, E e = E + m . The energy ∆ kj is themaximum energy available for each transition. An expression for the matrix element W kj is givenby Jonsell, Saenz, and Froelich [223]. The transition matrix element for a final-state molecular ionexcitation k ≡ ( v ( f ) , J ( f ) , M ( f ) , n ( f ) ) from an initial molecule state j ≡ ( v, J, M, n ) may be written, | W kj ( K ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:104) χ n ( f ) v ( f ) J ( f ) M ( f ) ( R ) (cid:105) ∗ S n ( R ) e i K · R ξ nvJM ( R ) d R (cid:12)(cid:12)(cid:12)(cid:12) . (VII.2)In this expression, χ and ξ are the rotational-vibrational wave functions of the final-state molecularions and initial-state molecules, respectively, and S n ( R ) is an electronic overlap integral. Theexponential of the dot product of the recoil momentum K and the nuclear separation R is aconsequence of the recoil motion of the daughter nucleus, He in the case of tritium decay. It maybe seen from the form of Eq. (VII.1) that because each transition has a slightly different endpointenergy, the shape of the spectrum in the region where one seeks evidence of neutrino mass isstrongly modified by the final-state distribution.
1. Tritium
The ITEP result reporting a 30-eV neutrino mass with a source of tritiated valine (an aminoacid) precipitated a substantial theoretical effort to calculate the FSD but in the end it was clearthat such molecules are too complex. The 30-eV result either arose from shortcomings in theFSD theory for valine, or from inexact energy-loss corrections. Evidence emerges from the fact9that these affect both the neutrino mass and the extrapolated endpoint energy, and the latter canbe checked experimentally via mass spectroscopy [155]. An early measurement of the T – Hemass difference indeed seemed to support the FSD and energy-loss theory, but was afflicted withsystematic errors that were controlled in further experiments. The valine experiment did not givethe correct endpoint energy.The Los Alamos group turned to atomic and molecular tritium as nearly ideal sources. Molecularhydrogen had been a topic of theoretical interest since the studies by Cantwell in 1956 [224]. Inprinciple, since the electromagnetic potential is known exactly, the calculations can be carried outto 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 [144] and Livermore[225], a number of new calculations of the molecular FSD were undertaken, and the work of theQuantum Theory Project [226] was used to interpret the results. Both experiments yielded negativefit values for m β , and it was not until 2015 that this was traced [109] to inadequacies in the FSD. R e l a t i v e p r o b a b ili t y ( % / e V ) -250 -200 -150 -100 -50 0 Binding energy (eV)
FIG. 19: Comparison of the FSD spectra for the T molecule calculated by Fackler et al. [226] (dashed blueline) and Saenz [154] (solid red line). From [109]. Theoretical work continued apace in the 1990 – 2005 interval with significant progress. Detailedassessments of the status can be found in [109, 111]. One of the most difficult aspects of the problemis the continuum, wherein one or more of the molecular electrons are ejected during the decay. In0
TABLE VII: Comparison of zeroth, first, and second moments (columns 3, 4, and 5, respectively) of theo-retical final-state distributions (see [27, 109]).Reference Energy range (cid:80) k | W k | (cid:104) b ( f ) k (cid:105) σ b eV eV eV Fackler et al. [226] 0 to 165 0.9949 -17.71 611.04Saenz et al. [154] 0 to 240 0.9988 -18.41 694.50 the model of Fackler et al. [226] continuum states have discrete energies like bound states. Animportant advance by Froelich et al. [154, 227] was the development of ‘complex scaling’ whereinthe radial variable of continuum wave functions is made complex. Another major advance was theinclusion of nuclear motion by Saenz and Froelich in 1997 [228]. The two FSD spectra are shownfor comparison in Fig. 19 and in the figure the discrete states of the Fackler et al. spectrum in thecontinuum have been arbitrarily assigned a 3-eV standard deviation for display and comparisonwith the calculation of Saenz et al. [154]. The complex scaling approach directly gives realisticdistributions in the continuum, joining smoothly (with a modest order-unity normalization) to theLevinger shakeoff distributions [229] that are analytic for hydrogenic atoms. Table VII lists thefirst three moments of the binding-energy distributions for the two theories.
TABLE VIII: Neutrino mass squared extracted from two experiments, in one case with the original 1985theoretical calculations of the FSD and in the second case with a more modern calculation.LANL [144] LLNL [225]
As published.
Theory: Fackler et al. [226] m β -147(79) -130(25) eV Re-evaluated.
Theory: Saenz et al. [154] m β The data for the Los Alamos and Livermore experiments are no longer available, but it ispossible to estimate the changes that would result had the theory of Saenz et al. been used insteadof the Fackler et al. theory, with the aid of the following relationship between an error in thevariance of the FSD or other resolution contribution and the consequent error in the neutrino masssquared [27]: ∆ m β (cid:39) − σ . (VII.3)The results are shown in Table VIII with the results of the LANL and LLNL experiments as1originally reported, both having been analyzed with the theory of Fackler et al. [226] and re-evaluated via Eq. (VII.3) with the theory of Saenz et al. [154]. The large negative value of m β iseliminated in both experiments, subject to the limitations of Eq. (VII.3). These results provide astriking, 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 [230] carried out a calculation in the same geminal basis used by Froelich etal. Their results are the same as those obtained by Saenz et al. [154] up to 40 eV excitation, butshould 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. [231]) also appearanomalously broad in their treatment, and it is therefore recommended to use the calculations ofSaenz et al. whenever the full spectral range is needed [232].Theoretical advances notwithstanding, the most powerful weapon to reduce reliance on the cal-culations is actually the statistical power of state-of-the-art experiments such as KATRIN. If anexperiment can gather sufficient data in the last 25 eV of the spectrum, then most of the uncer-tain aspects of the FSD, particularly the continuum, drop away. Only the rotational-vibrationalmanifold of the electronic ground state remains (Fig. 19, the peak at zero binding energy). Quan-titatively, the variance of the full FSD is 695 eV , but the variance of the ground-state group isonly 0.16 eV . Even that small variance must be known to a precision of 2% for KATRIN to meetits goal of 200-meV neutrino mass sensitivity.There is no known way to measure the FSD directly (other than in beta decay itself), butseveral kinds of experimental verification of the FSD are possible [109]. Interestingly, the LosAlamos and Livermore experiments that covered the full range of the FSD spectrum provide directexperimental confirmation of the correctness of the variance of the Saenz FSD at approximatelythe 2% level of accuracy needed by KATRIN. Since this includes the difficult continuum region, itis 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 theoryand experiment remained. When a tritium atom in the T or HT molecule decays, the daughter Hecan remain bound in an ion, or can be liberated. The electronic ground state of the molecular ionTHe + 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. Twoexperiments carried out in the 1950s [233, 234] reported that the intensity of this ground-statemanifold, 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 experiment2with a novel instrument, TRIMS (Tritium Recoil-Ion Mass Spectrometer) [235]. 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 T . c a d e f b FIG. 20: Experimental data from the TRIMS time-of-flight mass spectrometer experiment showing the richcomplexity 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 [235]) As shown in Table IX, TRIMS finds close agreement with the theoretical prediction, and givesmuch detail about the ionic final states that can be used in further tests of the theory. In the Table,
TABLE IX: Branching ratio to the bound molecular ion for HT and TT.Snell et al.
Wexler Theory TRIMS[233] [234] [154, 223] [235]Molecule Quasibound Bound TotalHT 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) the column ‘Quasibound’ is the part of the ‘Total’ branching ratio to the ground-state manifoldthat lies above the dissociation threshold. Many of these states have high angular momentum andtherefore long lifetimes, and may travel through the TRIMS apparatus without breaking up. Thelifetimes of these states are not known. In the case of the HT molecule, the quasibound fractionis small, and TRIMS provides the most decisive experimental test there. Since the theory of the3FSD is the same for both HT and TT, the close agreement is confirmation of the FSD for bothisotopologues.The FSD for molecular tritium is now very robust and no longer contributes significantly to thesystematic uncertainty budget even in the KATRIN experiment. Nevertheless the broadening ofthe response function caused by the FSD exacts a statistical penalty and makes it more difficultto reach small neutrino masses. The appeal of a source of atomic tritium is illustrated in Fig. 21.The atomic line is very narrow, broadened only by thermal motion, and the first excited state is R e l a t i v e p r o b a b ili t y -10 -8 -6 -4 -2 0 2 Relative Extrapolated Endpoint (eV)T Atomic T
FIG. 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(∆ − m e in [109]) differ by 9.99 eV, but the molecular rotational and vibrational excitations that broadenthe molecular peak make the endpoint energy for the atomic decay effectively about 8 eV smaller than themolecule. Translational Doppler broadening corresponding to a temperature of 1 K is included in the atomicline. The molecular line is from [154]. at -49 eV on this scale. The FSD for atomic tritium in the sudden approximation is simply thesquared overlap of the radial wavefunction R nl ( r, Z ) of the hydrogen ground state and the radialwavefunctions of the ns -states of He + , which are analytic functions. For example, the ground-state4transition probability is (8 / = 0 . T (0) fi = (cid:90) ∞ R n ( r, Z = 2) R ( r, Z = 1) r dr (VII.4)= 8 n (cid:112) n − n ! n − (cid:88) k =0 ( − k k ( k + 2)( n − − k )! k ! ( n + 2) k +3 . (VII.5)Table X lists the transition probabilities as calculated by Williams and Koonin [236]. The third TABLE X: Population of excited final states in the beta decay of atomic tritium [236].He + state Excitation energy (eV) | T (0) fi | | T (0) fi + T (1) fi |
1s 0 70.23% 70.06%2s 40.817079 25.00% 25.17%3s 48.375798 1.27% 1.28%4s 51.021349 0.38% 0.39%5s 52.245862 0.17% 0.17%Continuum 54.422773 2.63% 2.62% column is the zeroth-order sudden approximation value as in Eq. (VII.5), and the fourth columnincludes the interaction of the outgoing beta with the atomic electron. The sudden approximationis well supported.The spectrum in the continuum when the atomic electron is ejected can be calculated followingthe method of Levinger [229], but without making the large- Z approximations he used, where Z is the charge on the daughter nucleus ( He in this case). The transition probability per energyinterval dW is: P (1 s, κ ) dW = 64( Z − Z − (1 − e − πκ ) − κ exp (cid:20) − κ arctan 1 x (cid:21) ( x + 1) − dW. (VII.6)In this expression: κ = (cid:114) E K ( Z ) W (VII.7) x = κ (cid:18) − Z (cid:19) , (VII.8)and E K ( Z ) is the binding energy of an electron in the ground state of the daughter ion.We return to consideration of an atomic source and experimental aspects below.
2. Rhenium
In the beta decay of the isotopes
Re,
In, and
Cs, the low-energy parent-progeny tran-sitions are not allowed or super-allowed: for
Re, it involves a first-order forbidden transition5(∆ J π = 2 − ). Unlike its tritium counterpart, the matrix element inherits an angular momentumdependence that alters the spectral shape, including the region around the endpoint of the spec-trum. The decay spectrum will incorporate contributions from the s / and p / electrons. Thedecay spectrum from these two contributions (ignoring smaller recoil-order effects) can be writtenas [237, 238]: d Γ dE = G F | V ud | π C β ( E ) F ( Z, β ) β ( E + m e ) ( E − E ) × (cid:88) i =1 , | U ei | (cid:2) ( E − E ) − m i (cid:3) Θ( E − E − m i ) , (VII.9)where C β ( E ) is the shape correction factor, C β ( E ) = BR λ p ( Z, β ) p e + p ν ) , (VII.10) B = g A R (cid:104) / − || (cid:88) n τ + n { σ n ⊗ r n }|| / + (cid:105) . (VII.11)The shape factor itself depends on the axial-vector coupling constant g A , the coordinates of the n th nucleon, r n and the nuclear radius R . The Fermi function is also altered, and λ p represents theratio of a generalized Coulomb Fermi function normalized against that used in Eq. IV.1. Althoughthe lifetime and spectral shape well below the endpoint can help constrain the uncertainties thatarise from the theoretical uncertainties, they certainly complicate evaluation of the spectrum at theprecision typically required for neutrino mass evaluation. Drexlin et al. [25] find that the electronterm proportional to p e is 4 orders of magnitude larger than the neutrino one proportional to p ν and also note that in calorimeters the signal is the sum of the beta energy and any electromagneticenergy from excited states. In principle this can modify the shape of the spectrum, but the effectsdo not appear to be at the level of experimental significance.The calorimetric evaluation of the beta decay spectrum from Re suffers from an additionalcomplication, that of environmental alteration of the spectrum due to interactions of the emergingbeta electron with the crystal structure of the absorber. The presence of the lattice itself modifiesthe shape of the beta spectrum by reshaping the final-state phase space for beta electrons, pro-ducing “beta electron fine structure” (BEFS) in the spectrum of tritium or
Re [194, 239, 240]embedded in a solid. The oscillation scale is set by the momentum of the electron and the atomicseparation of the crystal. Figure 22 shows the oscillation spectrum as measured in rhenium crys-tals [194]. Although in good agreement with theoretical predictions, its presence introduces anothercomplexity in a detailed understanding of the endpoint spectrum. The spectral difficulties asso-ciated with rhenium beta decay and the total source masses necessary to achieve the statistical6accuracy for a neutrino mass measurement are among the reasons why the calorimetry groups haveshifted their focus to
Ho, as discussed in the next section.
FIG. 22: Experimental fraction residuals from the beta decay of
Re, fit to the theoretical spectrumdistortion from beta environmental fine structure [194]
3. Holmium
In electron capture, an atomic electron is captured by the nucleus and a neutrino is ejected.Because the electronic binding energies are quantized, the neutrino spectrum nominally consists oflines of energy Q − E b , the difference between the Q-value and the binding energy. Several factorsmodify this simple picture.The atomic vacancies created by electron capture have short lifetimes and refill by X-ray, Auger,and Coster-Kronig transitions. As a result they have widths that impart to the neutrino lines aLorentzian shape modified by phase-space restrictions in the wings. The wings of the lines extendto the Q-value and provide a means for measuring the neutrino mass because a non-zero massmodifies the phase-space distribution near the endpoint just as it does in beta decay [121].De R´ujula and Lusignoli calculate the spectrum to be expected [172, 241, 242]. Expanding the7neutrino flavor eigenstate in the mass basis the spectrum takes the following form [172, 243, 244]: dλ EC dE c = G F | V ud | π ( Q − E c ) × (cid:88) i | U ei | (cid:2) ( Q − E c ) − m i (cid:3) / × (cid:88) j β j C j | M j | Γ j E c − E j ) + Γ j , (VII.12)where β j is the amplitude of the electron wave function at the origin, C j is the nuclear shapefactor, and E j and Γ j are the excitation energy and natural width of atomic configuration j . The‘visible energy’ E c = Q − E ν , where E ν is the total neutrino energy. The quantity M j is an overlap(monopole) electronic matrix element between the ground state of the decaying atom and state j of the daughter atom. Exchange effects [245] and orbital occupancies can be absorbed into M j .A further modification [244] results from the change in the nuclear charge in electron capture.The atomic wavefunctions of Dy are not the same as those of Ho, a final-state effect that requiresexpanding the final state in terms of the complete set of Dy levels, both bound and continuum. Inthis expansion are many multi-vacancy states with electrons in the continuum that appear as rela-tively weak satellite structures in the spectrum (Fig. 23). The calculations have been approached Energy at the bolometer [keV] -2-101234 L og C oun t s p e r b i n [ a r b it r a r y un it s ] experiment1+2 holes1+2 holes + shake off Pm Figure 9: Experimental and theoretical results of the sum of the one- and two-hole deexci-tations and the sum of the one-, two-hole and the shake-off deexcitation for the bolometerspectrum (7). The experimental data are from the ECHo collaboration [4] and [21]. The twotheoretical spectra are adjusted to experiment at the N , s / peak. The nature of the onehole states are indicated. The two-hole peaks are by about two orders of magnitudes smallerthan the one hole peaks. The shake off contributions can hardly been seen in this scale.Some bins contain no experimental counts, thus the log for these experimental values areminus infinity. To fit the 1931 experimental points for the bolometer energy of 0.0 to 2.8keV, the theoretical spectrum of 200 mesh points had to be interpolated to the data pointsfor this figure. Figure 7 contains the 200 original theoretical results without interpolationfor the bolometer spectrum over E c between 0.0 and 2.8 keV. The interpolation is normallyvery good (compare figures 7 and 9) but difficult at some sharp minima and maxima. | K, ( b ) − , Ho > = X L = K ,bound a L · | L, Dy > + Z to ∞ dE ′ · a ( L, E ′ ) | L, E ′ , Dy >a L = < L, Dy | ( b ) − , Ho >,a ( L, E ′′ ) = < L, E ′′ , Dy | ( b ) − , Ho > = Z to ∞ · dE ′ · a ( L, E ′ ) · < L, E ′′ , Dy | L, E ′ , Dy > (23)16 FIG. 23: Calculated final-state spectrum for the electron-capture decay of
Ho by Faessler et al. [246](solid red, blue curves), compared to the experimental data of the ECHO collaboration [184]. in different ways by De R´ujula and Lusignoli [247], the T¨ubingen group [245, 246, 248, 249], andthe Heidelberg group [250] and give quite different spectra (Fig. 24). The Brass et al. calculation8 − − − − − − s s s p d p / p / d / d / d d f d d f } I n t e n s i t y ( c o un t s p e r h a l f - li f e ) Energy (eV) − − − − d Γ / d ω / ( Q − ω ) ( c o un t s / h a l f - li f e / e V ) excitation energy (eV)neutrino energy (eV) experimentlocal orbitalslocal + Auger FIG. 24: Top: Calculated final-state spectrum for the electron-capture decay of
Ho by Brass et al. [250].Bottom: From Brass et al. [251], the same calculation convolved with the instrumental resolution (solidcurve lowest between 500 and 1000 eV excitation, blue online) and compared with the data of Velte etal. [124] (solid histogram, black online). The third curve (red online) is a modified theoretical descriptionincluding Auger emission into the continuum. ( ω ≡ E c in the figure.) [250] appears to give a better account of the experimental features that are emerging as the statis-tical accuracy of the data mounts. In particular, the satellite structure above the 4s peak is wellrepresented. The bottom panel of Fig. 24 displays the experimental and theoretical results withthe phase-space factor divided out so that the comparison near the endpoint is easier to visualize.9At least to this two-vacancy order, both calculations find the spectrum at the endpoint to befree of satellite structures. The previously accepted Q-value near 2500 eV raised concerns thatthe spectrum shape at the endpoint might be so modified by satellites that a reliable extraction ofneutrino mass would be impossible. Even though satellite peaks raise the intensity, their shapes areheavily dependent on theory. The SHIPTRAP measurement [179] of 2833(34) eV, spectroscopicallyconfirmed by ECHo [122, 124], fortunately moves the endpoint to a region that seems to have alocally smooth energy spectrum. However, the additional 300 eV of Q-value reduces the intensityat the endpoint substantially.The theoretical spectrum reproduces to an impressive degree the general features revealed by theincreasingly precise experimental data, but it does not explain the observed spectrum completely.The original concept of IBEC in Ho has been set to one side while these subtle quantum-mechanical aspects of the atomic physics are being worked out, and the inner bremsstrahlungcomponents wait to be included. They interfere coherently (with a phase that is not yet known)with the resonance tails, and can be expected to change the spectral shape between resonancesand near the endpoint.Another consideration is that the decaying atom is located in a potential well in the host latticeand is subject to zero-point energy that broadens the neutrino lines. This is a line-broadeningeffect that is likely small but needs to be accounted for in a neutrino mass analysis.
B. Theoretical corrections to the beta spectrum shape
A number of small effects are known to modify the basic spectrum shape given in Eq. (IV.9).In a 1991 paper [252], Wilkinson enumerates and calculates them: the Fermi function, screening,exchange with atomic electrons, finite nuclear size effects in both the charge and weak-interactiondistributions, radiative corrections, and a collection of 4 recoil-order effects, namely weak mag-netism, V-A interference, three-body (rather than two-body) phase space and the relative motionof the electron and nuclear charge. A recent summary and calculation of these effects has beengiven by Kleesiek et al. [111]. Interestingly, as the field has advanced and experiments have becomemore sensitive, these effects have become even less important rather than more so, because thespectral range for investigation has shrunk to mere tens of eV near the endpoint.The exception is the outer radiative correction, which modifies the spectral shape near theendpoint. Kleesiek et al. [111] use the correction from Repko and Wu [253]. Were it to be neglected,the spectral distortion near the endpoint in the KATRIN experiment would resemble a neutrino0mass of 46 meV.We turn next to a spectral effect that arises not from subtle theoretical corrections, but fromstatistical considerations. The spectrum in Eq. (IV.9) is valid only for positive or zero values of m β .Measurements will produce statistically too many counts in the endpoint region half the time, andtoo few half the time. The analysis of data from an experiment calls for a way to handle smoothlythe transition across this boundary. For that purpose, a function must be defined that describes thespectrum for negative values of the fit parameter, m β . The function is a matter of choice for eachtype of experiment, but the criterion it needs to meet is that the fit parameter (for example, χ )should display the parabolic behavior that is expected of a normally distributed quantity and thatthe form of the parabola matches the positive- m β regime. Experimental groups (except KATRIN)have chosen a variety of different functions that meet this criterion in the circumstances of theirapparatus and data sets, as illustrated in Fig. 25. The functions used by the different groups are K u r i e a m p li t ude ( a r b . ) Electron Energy (eV) mass = 100 eVmass = 100i eVmass = 0 Troitsk LANL &LLNL LLNLMainz
Functional Forms for Negative m KATRIN
FIG. 25: Illustration of the variety of functional forms chosen to continue Eq. (IV.9) into the negative- m β regime. The functions are calculated for m β = − , , and 10 eV , and E = 18576 eV. The Kurieamplitude is the square root of the phase-space density. as given in Table XI. In each case, (cid:15) = E − E (VII.13) k = − m β (VII.14)1and for m < (cid:15) (cid:113) (cid:15) − m β Θ( (cid:15) − m β ) part of the phase-space expression inEq. (IV.9) by the expression in the table. TABLE XI: Empirical functions used by various experimental groups to parameterize the tritium spectrumin the negative-mass-squared regime.Group Function Remarks Ref.LANL ( (cid:15) + k ) Θ( (cid:15) ) [144]LLNL | (cid:15) + k (cid:15) | (cid:15) | | Θ( (cid:15) + k )Mainz (cid:104) (cid:15) + µ exp( − − (cid:15)µ ) (cid:105) √ (cid:15) + k Θ( (cid:15) + µ ) µ = 0 . k [149]Troitsk [2 (cid:15) − (cid:15) √ (cid:15) − k ] Θ( (cid:15) ) Smoothed with [150]bidimensional splines.KATRIN (cid:15) √ (cid:15) + k Θ( (cid:15) ) [56] The functions do not play a major role in the reporting of upper limits from experiments, whichare derived from fits to positive values of m β . The reported central value and its uncertainty,however, clearly depend on the functional parameterization when the central value happens to fallin the negative- m β regime. The central values and their uncertainties are valuable for combiningthe results of different experiments and for assessing the statistical likelihood of a given upper limit.The KATRIN analysis [56] uses the same function as the original phase space, simply allowing m β to go negative, which gives an asymmetric χ function. However, the central value found is the sameas when a more symmetrizing function from the Mainz group [146] is used, and the phase-spaceform was a simplification.For the general analysis of tritium beta decay data, a clear overview of the different method-ologies, classical, unified, and Bayesian, may be found in Ref. [111]. VIII. EXPERIMENTAL SENSITIVITY
It is useful to ask, what is the smallest mass detectable in beta decay, and what experimentalapproach is most likely to be fruitful? The main aspects that enter into the answer are readily iden-tified. They are the statistical accuracy, the energy resolution, the background, and the systematicuncertainties.Three classes of instrument can be distinguished:1. Differential filters: A point in the spectrum is counted,22. Differential spectrometers: The spectrum is counted as a whole with each event sorted byenergy, and3. Integral filters: The intensity above a point in the spectrum is counted.Examples of these instruments are, 1) magnetic ‘spectrometers’ (really filters) of the Tret’yakovtype, 2) microcalorimeters and CRES (cyclotron radiation emission spectrometers, see Sec. IX),and 3) MAC-E filters.In the next section we develop a simple method for estimating the physics reach of a directneutrino mass experiment, taking into account the distinct attributes of the instrument.
A. Estimation method
The statistical sensitivity to neutrino mass is fundamentally determined by the number of eventsin an ‘analysis window’ as given in Eq. (IV.8), reproduced here: N s (cid:39) rt (∆ E ) (cid:34) − (cid:80) i =1 , | U ei | m i (∆ E ) (cid:35) . More realistically, if there is an additional differential background rate b that is energy-independent,the total number of events in an analysis window of width ∆ E is N tot = rt (∆ E ) (cid:34) − m β (∆ E ) (cid:35) + bt ∆ E (VIII.1)This background description is appropriate for differential instruments. For integral filters, theintegral background rate b int does not scale appreciably with the size of the analysis window. Wedeal first with the differential description.The statistical uncertainty σ m β is thus related to the variance in the total number of events: ∂N tot ∂m β = − rt ∆ E σ m β = 23 rt ∆ E (cid:112) N tot (VIII.3) (cid:39) rt (cid:114) rt ∆ E + bt ∆ E , (VIII.4)It is assumed that the neutrino mass is small compared to the width ∆ E . There is an optimumchoice of ∆ E that minimizes the uncertainty,∆ E opt = (cid:114) br . (VIII.5)3For this choice, σ m β (cid:39) / b / t / r / (VIII.6) ≡ σ opt . (VIII.7)The minimum is broad – setting the analysis window width incorrectly by a factor m results in anincrease in the statistical uncertainty by a factor σσ opt = (cid:115) (cid:18) m + 1 m (cid:19) . (VIII.8)For example, if ∆ E is 2 times wider than the optimum, there is a 10% increase in the statisti-cal uncertainty. As a practical matter, the ratio b/r may be very small when rates are high orbackgrounds low. The optimum analysis window ∆ E may then be determined by other factors,for example instrumental broadening with a standard deviation σ instr , because the neutrino-masseffect on the spectrum is now smeared over this larger interval. In turn, improving the instrumen-tal resolution beyond a certain point is not useful if one encounters a limit set by the broadeningcaused by the final-state distribution (FSD). In the decay of molecular T to T He + , the molecularfinal-state distribution of the ground-state rotational and vibrational manifold has a standard de-viation σ FSD (cid:39) . σ trans [109]. The quadrature of the individual contributions forms a quantitative basisfor fixing ∆ E : ∆ E = (cid:114) br + C ( σ + σ + σ + ... ) (VIII.9)where C = √ . σ i of each resolution component is determined experimentally by some meanswith an associated uncertainty u ( σ i ) that propagates into the square of the neutrino mass, leadingto additional, non-statistical, contributions to σ m β of the form σ m β = (cid:118)(cid:117)(cid:117)(cid:116)(cid:88) i (cid:34) ∂ ( m β ) ∂ ( σ i ) u ( σ i ) (cid:35) , (VIII.10)There is a simple relationship [27] between an error in the width of a resolution contribution andthe corresponding error introduced into the neutrino mass: σ m β ≈ − u ( σ i ) (VIII.11)= − u ( σ i ) σ i (VIII.12)4Equation (VIII.10) then becomes, σ m β = 4 (cid:118)(cid:117)(cid:117)(cid:116)(cid:88) i σ i (cid:18) u ( σ i ) σ i (cid:19) , (VIII.13)to be combined with Eq. (VIII.4) to yield the total uncertainty. This approach to estimating thesensitivity of an experiment is approximate but can be quite accurate. Care is needed with thechoice of resolution parameters. For example, the variance of the full FSD in molecular tritiumdecay is very large (695 eV ) but experiments with high sensitivity can focus on the last ∼ ∼ .
16 eV . A similar caveat applies tocontributions from inelastic scattering, which have a large total variance but do not set in untilabout 13 eV below the endpoint. We show later the application of this ansatz to estimating thesensitivity of Project 8 and Ho experiments.The main difference for integral filter instruments is that background events accumulate at anintegral rate b int independent of ∆ E : N tot = rt (∆ E ) (cid:34) − m β (∆ E ) (cid:35) + b int t. (VIII.14)In this case the optimum window is ∆ E opt = (cid:18) b int r (cid:19) / , (VIII.15)and the statistical uncertainty becomes σ m β (cid:39) (cid:18) (cid:19) / b / t / r / . (VIII.16)For the purposes of experiment planning in establishments equipped only with cocktail napkins,Eq. (VIII.4) provides the necessary estimations for differential spectrometers. The 90% CL limitthat could be set on a small neutrino mass in a background-free, systematics-free experiment is m β < ∼ (cid:18) ∆ Ert (cid:19) / . (VIII.17)For example, with ∆ E = 1 eV and a source producing 1 detectable count per day in the last eV,a limit of about 0.23 eV could be set on the mass in a year.The rule is the same for integral filters, except that an adjustment to the running time t isrequired to account for the additional time needed for measuring the background, the spectrumintensity, and the endpoint energy. This adjustment has been done by Monte Carlo for KATRIN5and produces the measurement-time distribution (MTD) that assigns an optimal run time to eachenergy point.With the publication of the first neutrino mass limit from KATRIN [56, 255], it becomes possibleto compare the prediction of Eq. (VIII.16) to the performance of the actual experiment. Since theMAC-E filter is a point-by-point instrument, what emerges is a ‘time-expansion ratio’, the increasein running time needed in order to provide the additional information beyond the number ofevents in the analysis window (i.e. the background, spectrum intensity, and endpoint energy).Table XII gathers the necessary information. The optimum window size of 11 eV agrees well with TABLE XII: The time expansion ratio for KATRIN as derived from its first neutrino mass measurement[56]. Parameter Value Unit Ref.Fraction of nuclear T events in last eV 2 . × − Eq. (IV.1)T branch to electronic ground state 0.57 [154]Maximum pitch angle 50.4 deg [56]Detector area 63.6 cm [256]Detector pixels used 117/148 [56]Detector spectral region of interest 0.95 [56]Source column density ρd . × T cm − [56]Unscattered fraction 0.80 [56]T atoms / total 0.976 [56]Magnetic field at source 2.52 T [56]Magnetic field at detector 2.52 TVisible source activity 2 . × BqAccepted source activity 2 . × BqAccepted endpoint activity r . × − Bq eV − Background b int E m β [56]Effective live time t . × s Eq. (VIII.16)Actual live time 521.7 h [56]Time expansion ratio 110 the spectral emphasis chosen by KATRIN for data taking [56]. The time expansion ratio of 110 atthe bottom of Table XII is a quantitative measure of how much more time is needed to carry outan experiment with a point-by-point MAC-E filter than would be needed simply to accumulate the6statistics in the optimum analysis window. As Kleesiek et al. note [111], the choice of measurementtime distribution for filter instruments cannot be just a numerical optimization exercise becauseit is necessary to obtain a spectrum rather than four numbers at strategically chosen energies.There is information about the quality of the data to be found in the spectrum that cannotbe obtained in any other way. The time expansion ratio is a motivation to seek spectrometricmethods that allow simultaneous acquisition of data across (part of) the spectrum. Of course,spectrometric measurements have their own time-expansion ratios too, because the parametersobtained from regions of the spectrum outside the analysis window contribute statistically, butsimulations indicate that they are typically much closer to unity.The analysis of an actual experiment’s data would be done without the approximations used inthis section for illustration. B. Backgrounds
The decay rate in the last eV of the tritium spectrum is roughly 2 × − of the total rate, andit scales as the cube of the energy interval. Backgrounds become increasingly important as eversmaller neutrino masses are explored. When physical detectors such as semiconductor detectors areused to register events from a spectrometer, natural radioactivity and cosmic-ray backgrounds areoften the dominant source. Every advance in sensitivity and scale brings a new and unanticipatedbackground, and we described above the Rydberg atoms and trapped electrons from radon thathave been discovered in the KATRIN background.The method of Cyclotron Radiation Emission Spectroscopy (CRES), described in Sec. IX B,is expected to have low background because there is no physical detector with which electronsinteract, and at no point are the electrons brought nearly to rest. The registration and energymeasurement of decays is carried out via microwave signals. Sources that can contribute includecosmic-ray and radioactivity interactions with the tritium gas, false events produced by noise inthe receiver chain, and noise from the abundant lower energy electrons that are inevitably presentalong with the few higher-energy ones near the endpoint.A special but important case arises with a putative experiment on atomic tritium, such asthat planned for the CRES experiment Project 8 [257]. The inevitable presence of molecular T represents a background to the atomic spectrum near its endpoint because the Q-value of themolecular decay is 10 eV larger than for the atomic decay. To estimate the influence of this7background, the total rate in the analysis window in the absence of neutrino mass becomes N tot (cid:39) r a t (∆ E ) + bt ∆ E + r m t [( E m + ∆ E ) − E m ] , (VIII.18)where E m (cid:39) r a and r m are the rates in the last eV of theatomic and molecular spectra, respectively. As was shown in Equation VIII.3, σ m β = 23 r a t (cid:114) ( r a + r m ) t ∆ E + 3 r m tE m + ( b + 3 E m r m ) t ∆ E . (VIII.19)The optimum analysis window for the background component similarly becomes,∆ E opt = (cid:115) b + 3 E m r m r a + r m . (VIII.20)If b is small and ∆ E opt is limited to 1 eV by other effects, the implied ratio r m /r a < . Ho in a reactor inevitably leads to co-production of other isotopes, and rigorous chemicalpurification followed by electromagnetic mass separation has been needed to produce essentiallybackground-free spectra [124, 185]. Control of contamination by K is particularly crucial becauseit produces a spectrum of Auger and X-ray lines around the Ho endpoint at 2.8 keV. The airborneRn daughter products must be controlled as in any material detector system.
IX. FUTURE
In the KATRIN experiment now in operation one finds the accumulated knowledge of decades ofdirect mass measurement experiments and the theoretical work on the final-state distribution of thetritium molecule. KATRIN was conceived at the time when neutrino oscillations were discoveredand neutrino mass was shown to be non-zero, with a definite lower limit to the average mass. Itwas designed for an order of magnitude improvement in mass sensitivity, which corresponds to fourorders in simple statistical terms. The sheer scale of KATRIN dwarfs all previous experiments.KATRIN may indeed find the mass within its search range but, if not, the mass range between 0.2and 0.01 eV remains inaccessible to laboratory experiment. In either event, is there any approachthat would allow either confirmation of a result from KATRIN, or exploration of the last remainingwindow?8As has been noted above, MAC-E filters are point-by-point instruments with a time expansionratio of 100 or so. In addition, the need to extract the electrons from the source places a limit onthe source thickness, and hence also its activity. These statistical restrictions may be susceptibleto alleviation.The final-state spectrum has been a limitation and a source of systematic error since 1970, andmotivated the search for methods in which its role is diminished or eliminated, while at the sametime theoretical advances have greatly reduced the systematic uncertainties. The low-temperaturecalorimetric methods used with
Re and
Ho arose from the quest to evade the final-stateuncertainty because all the decay energy not carried away by neutrinos is delivered as heat to thelattice, collapsing the final-state distribution into the desired delta function. But even with thattechnique there are caveats, as has been described, and new kinds of final-state effects becomemanifest.In 2009 a novel idea, cyclotron radiation emission spectroscopy, was proposed by Monreal andFormaggio [258]. As a spectroscopic method it can have a small, order-unity time expansion ratio,and there is no need to extract the electrons, lifting the source-size limitation. In addition, itmay make possible an experiment with atomic tritium, for which the final-state spectrum is freeof uncertainty.We look into these new methods in more detail in this section. Promising though they are, thetechnical challenges are formidable.
A. Calorimetric techniques
One of the difficulties inherent in the electromagnetic techniques discussed thus far, includingthose borne of magnetic adiabatic collimation, is that the electron needs to be transported awayfrom the source before its energy can be measured. In doing so, techniques need to be devisedto minimize (or calibrate away) any energy losses collected in the transport process. Calorimetrictechniques potentially circumvent this fundamental limitation, since they collect all the energy re-leased in the weak decay process (aside from that carried away by the neutrino). As such, they offera system with inherently different systematic uncertainties compared to those in electromagneticmethods. Experimental calorimeters used thus far also make use of primogenitors that are distinctfrom tritium (mainly
Ho and
Re), providing further orthogonal checks on the validity of anypotential positive mass signal.The most common form of calorimeter used to date for direct neutrino mass measurements has9been the cryogenic microcalorimeter, so we will concentrate on devices of this type. A schematicof an idealized version of such a detector is shown in Fig. 26, although not all calorimeters havethis specific configuration. Energy deposited from a radioactive decay (∆ E ) is transmuted into FIG. 26: Left: Schematic of an idealized cryogenic microcalorimeter. Energy deposited from a radioactivedecay (∆ E ) is transmuted into thermal energy (e.g. thermal phonons). The thermal pulse (∆ T ) is in turnconverted to an electrical signal (∆ V ) by coupling to an external system . The timing and response of themicrocalorimeter is determined via its electrothermal link ( G ), and the heat capacity C of the absorber.Right: SEM picture of the central part of the fully microfabricated magnetic microcalorimeters with thefour pixels (A, B, C and D) prepared to be irradiated. Figure reproduced from [186]. thermal energy (e.g. thermal phonons). The conversion from kinetic to thermal energy dependscritically on the heat capacity, C ( T ), of the target absorber and the surrounding system:∆ T = ∆ E/C ( T ) , (IX.1)where ∆ T is the change in temperature induced by a deposition of energy ∆ E . The specific heatcapacity of the system depends strongly on temperature, with superconductors and dielectricshaving a cubic dependence ( C ( T ) ∝ T b ), while normal metals have a linear dependence ( C ( T ) ∝ T b ). As such, these detectors work best when the base temperature T b is extremely low, usuallybelow 100 mK.The change in temperature must eventually be converted into a detectable (electrical) signal.The conversion from a change in temperature δT to a change in voltage ∆ V depends on thespecific technology deployed. Typical methods include using the superconducting transition inthin materials, such as used by transition-edge sensors (TES, e.g. HOLMES), changes in themagnetization using metallic magnetic calorimeters, (MMC’s, e.g.
ECHo), or detecting changesin resistance as registered by neutron transmutation doped detectors (NTD, e.g.
MANU). Which0technology is used often defines the kind of experiment being conducted, as each type of systemoften has trade-offs for its performance and sensitivity.A critical metric for the performance of such microcalorimeters is the energy resolution of thedetected energy deposition. The energy resolution depends greatly on the specific configurationand readout method employed for the detector. As an example, the theoretical lower limit fora cryogenic microcalorimeter read out by a transition edge sensor is given approximately by theformula: σ E = (cid:115) k b T b C tot α T (cid:114) β T + 12 (IX.2)where k b is the Boltzmann constant, C tot is the total heat capacity of the microcalorimeter, β T isthe exponent of the temperature dependence of the thermal conductivity between the calorimeterand the thermal bath (typically β T (cid:39) α T is a dimensionless quantity that represents thesensitivity of the TES to the change in temperature ( α T = TR dRdT , where R is the resistance of theTES. Values for α T around 50-100 are not uncommon). One thing to note is that the theoreticalthermal noise limit is determined by the total heat capacity of the system. Therefore, finer energyresolution scales with both temperature and the mass of each absorber. As a result, such systemsare operated in the millikelvin regime and are constructed as microcalorimeters , so as to reducethe total heat capacity of the system.Another critical parameter that dictates the performance of microcalorimeter systems is thetiming response. This is dictated both by the capacity of the system and by the electrothermallink conductance G th responsible for removing the excess heat from the absorber. τ = CG th (IX.3)The rise and decay of the heat pulse depend critically on the thermal coupling between the absorberand the detector, as well as the coupling between the absorber and the heat bath. The onsetand decay of the signal pulses determine the maximum possible rate that a given absorber cantolerate before multiple pulses appear within a given time window (pile-up). Pile-up leads to amis-identification of the energy deposition. Given that the density of events is far greater below theendpoint of the spectrum, pile-up manifests itself as a background, which scales approximately as A × f pileup × (2 Q ), where A is the source activity and f pileup is the fraction of pileup events withina given absorber. A given calorimetry-based experiment needs to balance the inherent activity,the energy resolution, and the timing response of each absorber. All these factors have moved thefield toward multiplexed systems wherein thousands to hundreds of thousands of detectors need tobe instrumented in order to achieve the desired neutrino mass sensitivity.1One can use the estimation method outlined in Sec. VIII A to provide guidance on the potentialreach of the calorimetric effort. Such is done in Figure 27. For reference, the current first phase ofthe HOLMES and ECHo experiments aim at a total detector mass of roughly 20 µ g. FIG. 27: Uncertainty obtainable as a function of the
Ho mass under observation ( µ g · y). We assumea detector energy resolution of 1 . . × − .We have also assumed that approximately one part in 10 decays occurs in the last eV of the holmiumspectrum. B. Frequency techniques
A new approach was suggested in 2009 by Monreal and Formaggio [258]. Electrons from agaseous beta emitter like tritium, when in a magnetic field, produce cyclotron radiation that canbe detected in a sensitive receiver. Because of a relativistic effect, the frequency of the radiationdepends on the kinetic energy and so makes possible a new spectroscopy, Cyclotron RadiationEmission Spectroscopy (CRES). In a uniform magnetic field B , the frequency f and power P radiated by an electron of kinetic energy E are given by2 πf = 2 πf γ = eBm e + E/c , (IX.4) P = 2 πe f (cid:15) c β sin θ − β . (IX.5)2The pitch angle θ is the angle between the momentum vector and field direction. The zero-energyelectron cyclotron frequency f is a fundamental constant [28], f = 27 . − . (IX.6)The maximum power radiated by an 18-keV electron in a 1 T field is about 1 fW.The first experimental demonstration of CRES was made in 2014 by the Project 8 Collaboration[259] with the isotope m Kr, the decay of which produces sharp internal conversion lines. Theexperimental cell consisted of a section of WR-42 rectangular waveguide having a cross section of10.7 × θ near π/
2. Signals produced by electronswere transmitted by the waveguide to a low-noise cryogenic amplifier, superheterodyne receiver,and digitizer. The first event recorded is shown in the iconic plot reproduced in Fig. 28. For
FIG. 28: Left: First event observed by the CRES method (Project 8 Collaboration [259]). The spectrogramshows RF power in 25-kHz frequency bins and 40- µ s time bins. An electron is created by m Kr decay nearthe lower left corner and forms a track that slopes upward due to radiation loss. The discontinuities fromtrack to track are caused by the electron scattering inelastically from the background gas, which is mainlyhydrogen. The most probable jump size corresponds to about 14 eV. Eventually the electron scatters out ofthe trap and is lost. Right: Waveguide insert used for the CRES technique by the Project 8 collaboration.The inner diameter of the waveguide is 1 cm. each such decay event, the initial electron energy is derived from the frequency at the onset ofpower in the first track. The potential for good energy resolution is visually apparent from the3narrowness, approximately one frequency bin, of the tracks. From the relationship between energyand frequency, dEE = − γγ − dff . (IX.7)A 30-keV electron has γ = 1 . Natural line widths: 1.84 &1.4 eV; Observed FWHM 3.3 eV Separation is 52.8 eV
Region of interest near the 30.4 keV lines (bins are 0.5 eV wide)
Natural line widths: 1.99 &1.66 eV; Observed FWHM 3.6 eV Separation is 7.7 eV
Region of interest near the 32 keV lines (bins are 0.5 eV wide)
FIG. 29: Internal-conversion electron lines in the decay of m Kr measured by the Project 8 collaborationwith the CRES method [257]. Left: The L2 and L3 lines from the 32-keV isomeric decay transition. Right:The M2 and M3 lines from the same transition. The events not in the sharp peaks arise mainly from shakeupand shakeoff processes in the decay [260], and partly from scattering in the residual gas.
The need for a trap leads to complications in the energy spectrum. In the experiment underdiscussion, the electron moves back and forth in the trap along a magnetic field line that is alignedwith the axis of the waveguide. The received signal is frequency modulated by the axial motionbecause of the Doppler effect. Amplitude or frequency modulation of a steady carrier introducessidebands that are spaced from the carrier by multiples of the modulating frequency. When fre-quency modulation causes shifts in the frequency that are greater than the frequency of modulationitself, sidebands proliferate at multiples of the modulation frequency. The power is spread exces-sively and it becomes difficult to detect and identify the many weak sidebands, while the carrier4power also becomes small and can disappear altogether. The critical parameter is the modulationindex, which is defined as h = ∆ ff a (IX.8)where ∆ f is the peak one-sided frequency shift and f a the modulating frequency, the axial frequencyin this application. When h ≤ h = 2 .
405 (the first zero ina related Bessel function) the carrier vanishes completely. The phenomenology of this effect isdescribed in [261], and it strongly restricts the usable axial amplitude and pitch angle in the trap.This effect coupled with the small dimensions of the waveguide result in an effective volume of thegas in the source, the equivalent volume that contributes detectable events, that is only a fractionof a cubic millimeter.Application of CRES to the neutrino mass problem requires much larger effective volumes of tri-tium for statistical accuracy. Two approaches to circumventing these limitations are being exploredby the Project 8 collaboration. One is to collect the cyclotron radiation emitted perpendicular tothe motion of the electron’s guiding center in the trap [257]. The resulting reduction in solid angleplaces severe demands on the noise level of the amplifiers and the temperature of the system. Thesecond approach is the use of a cavity to collect the RF energy emitted. Certain TE modes areDoppler-free but a large overmoded cavity is complex to use and analyze.The ultimate reach of experiments based on molecular tritium is limited by the final-statedistribution, which smears the beta spectrum at the endpoint. The CRES method raises onceagain the long-sought goal of an atomic tritium experiment. Since only microwave photons andnot the beta electrons themselves need to be directly detected, a magnetic trap for atomic tritiumis usable. The fact that molecular tritium levels have magnetic moments at least thousands oftimes smaller than free atomic tritium means that molecules are essentially not trapped. Thisproperty helps in providing the low ratio of molecules to atoms that is necessary for a background-free measurement at the atomic endpoint. Atomic tritium contained within Tesla-scale magneticwalls is necessarily very cold, at a sub-Kelvin temperature, reducing thermal Doppler broadening.Figure 30 shows calculated neutrino mass sensitivities for molecular and atomic tritium withnumber densities chosen to reproduce the mean track durations presently used in Project 8, as afunction of the product of volume, efficiency, and live time. The estimation method of Sec. VIII A isused for these calculations. The cross-section cited by Aseev et al. [262], 3 . × − cm , has beenused for electron scattering by molecules, and for atoms we have used 9 × − cm based on the5 S t a n d a r d d e v i a t i o n i n m ν , e V -6 -5 -4 -3 -2 -1 Volume x Efficiency x Time, m -y % C L m a ss li m i t , e V Atomic T, 7x10 m -3 T , 2x10 m -3 FIG. 30: Uncertainty obtainable as a function of volume under observation for various choices of numberdensity per cm . Systematic uncertainties due to imperfect knowledge of contributions to the resolution areincluded. The densities correspond to mean times between scatters of 200 µ s, as used in current Project 8data. The frequency chosen is 26.5 GHz, the energy resolution is 3 ppm rms, the source temperature formolecular T is 30K while for atomic T it is 1K, and the background is 10 − per second per eV. work of Shah et al. [263]. For concreteness, we assume that the distributions σ i in Eq. (VIII.9) areeach known to 1%, i.e. u ( σ i ) /σ i = 0 .
01. For calculating the ‘sensitivity’ shown here, the expectedvalue for m β is taken to be 0, and, statistically, positive and negative values for this quantity areequally probable. The 90% CL is a one-sided interval derived by setting the 1.28-sigma upperthreshold on m β , which is assumed to be Gaussian distributed. The square root of this number isdisplayed on the right-hand axis.The physics reach of a Project 8 experiment depicted in Fig. 30 is attractive, but should beregarded as an optimistic estimate of what could be done with this type of measurement. Thesystematic uncertainties on resolution-like parameters are assumed to be very small, 1%, andmany effects are omitted.As can be seen, an experiment with gaseous molecular T reaches a limit in sensitivity of order100 meV because of the width of the FSD combined with Doppler broadening associated with the6minimum feasible operating temperature near 30 K. It would be necessary to know the FSD toan accuracy of 0.1% to reach the 40-meV level, and the running time would be 10 times longerthan with an atomic experiment of the same size and efficiency. For these reasons, the Project 8collaboration is exploring the development of an atomic T source in a magnetic configuration thattraps both spin-polarized atoms and the betas. The density required is in an achievable range, oforder 10 m − . The mean energy of the atoms that can be stored depends on the magnetic wallheight, and is about a factor of 10 to 20 below the energy equivalent to the height in order to slowevaporation and obtain lifetimes longer than tens of seconds. A magnetic wall of height 1 T canretain 60-mK atoms for periods of a minute or so.In addition to the resolution contributions from the gaseous source itself, specifically the FSDwidth and the translational Doppler broadening, there are instrumental contributions. The instru-mental resolution has two readily identifiable components, field inhomogeneity and noise. Axialand radial variations of the trapping and background fields mean the average cyclotron frequencydepends on the electron’s position within the trap and on its axial amplitude. Moreover, thepresence of a radial gradient in the magnetic field causes the electron’s guiding center to circulateslowly around the axis, passing through regions that may have slightly different average field. Thedrift velocity in the presence of a magnetic field gradient is given by [24] u ⊥ = (cid:20) (2 E (cid:107) + E ⊥ ) e ∇ ⊥ B B (cid:21) × B , (IX.9)where E (cid:107) and E ⊥ are the parallel and perpendicular energies. In the quasi-uniform field thattypifies a weak trap, a 20-keV electron moves azimuthally at 2 m/s if the field is 1 T and has agradient of 10 − T/m. In stronger traps, the gradient is large and depends on the axial amplitude,leading to more complex behavior. The sign of this “grad-B” motion can even change with radius[264].Noise is fundamental in the performance of a CRES experiment. The signal must be detectableabove receiver and thermal noise, which implies a bandwidth limitation. That in turn implies aminimum observation time for reliable detection of a track, which, finally, sets a limit on the gasdensity n . The mean free path λ = 1 /σ n , where σ is the cross section. For 18.6-keV electronsincident on molecular tritium, the inelastic cross section is [262], 3 . × − cm , while for atoms itis 9 × − cm [263] (the ratio of cross-sections confers almost a factor of 2 advantage on atomicexperiments for a given activity density). It was found in Project 8 [257, 259] that the trackduration that optimized count rate and detection efficiency was about 200 µ s. Noise directly playsa role in the energy resolution by introducing uncertainty into the frequency measurement and the7start-time measurement. Both quantities enter into the energy determination because radiationloss means the frequency is changing with time. Noise also sets the background level for a givendetection threshold because of random alignments of spectrogram pixels having above-average noisepower. C. Atomic tritium
Atomic tritium is an attractive candidate for freedom from final-state effects. The groundstate in the He + daughter is separated by 40 eV from higher excited states and receives 70% ofthe decay intensity. There are no rotational and vibrational excitations, and (in gaseous form)no lattice effects. We mentioned above the low cross section for electron inelastic scattering onatomic tritium compared to molecular. The Los Alamos experiment [144, 265] was designed tooperate with atomic tritium, but the technical challenges proved insurmountable and molecularT data were used. As it turns out, both the Los Alamos and Livermore [225] experiments weresystematically affected by the limitations of the contemporary theory of the molecular final statespectrum, as was subsequently found [109].Atomic hydrogen can be produced from molecular hydrogen either in an RF or DC dischargeor by thermal ‘cracking’ on a hot tungsten surface. That the latter strategy is efficient may besurprising because at 2600K k B T = 0 .
25 eV while the binding energy of the H molecule is 4.5 eV,but once dissociated the atoms cannot easily recombine because it is necessarily a 3-body process.A comprehensive review and compendium of the dissociation of hydrogen and formation of beamsis given by Lucas [266]. Atomic beams of tritium have been produced only three times, by Nelsonand Nafe in 1949 [267], by Prodell and Kusch in 1957 [268], and by Mathur et al. [269] in 1967.A DC discharge was used by Nelson and Nafe and RF discharges by the other two groups, andin each case the tritium was lost from circulation in a matter of hours. Dilution with hydrogen(protium) also occurred quickly.The Los Alamos research into atomic tritium used the RF discharge method, a 50-MHz elec-trodeless discharge in a borosilicate or silica glass tube. Tritium was never used, and it was alwaysfound that the dissociation fraction with protium and deuterium decreased over time periods ofhours to days, and the tube became discolored. Similar experiences were reported in research onBose-Einstein condensation in hydrogen [270], and it was commonly attributed to contaminantsin the gas or vacuum system. Recently we reexamined the Los Alamos logbooks and find possibleevidence for a different mechanism. The RF discharge produces fast electrons, hard UV light, and8reactive atoms, and is capable of attacking the glass envelope. The glass surface was probablyreduced to silicon monoxide, which is brown in color and was incorrectly ascribed to contaminants.At the same time, the hydrogen was converted to water that was subsequently found on vacuum-system cold traps after operation of the discharge, but not otherwise. Slevin and Stirling, however,report operation of a seemingly similar RF discharge tube over thousands of hours [271], so it isevident that not all the relevant factors have been identified. Inventory conservation and equipmentlongevity are important in a tritium experiment. We venture that the RF discharge method willbe unsatisfactory for an atomic tritium experiment where efficient recycling is essential, and thatthermal cracking will be preferable. It is furthermore prudent to avoid ionizing the gas at any pointbecause it leads to ion pumping of the tritium (some thermal crackers use electron bombardmentheating, which can ionize the background gas). Ferrous metals are known to take up and releasehydrogen in high-vacuum systems, and it is of interest to explore the use of other materials suchas copper and aluminum.Another challenge of atomic tritium is that the Q-value is 8 eV less than for the molecule [109](see Fig. 21). The tritium spectrum rises parabolically below the endpoint, and, as described aboveand shown in Eq. (VIII.19), a very small molecular contamination at the 0.01% level will representan important background to the atomic spectrum in future sensitive experiments. Typically thedissociation fraction from a dissociator is not better than 90%, insufficient for an atomic tritiummeasurement if there is no further purification. Adding to the difficulty, the Los Alamos spec-trometer resolution, about 25 eV FWHM, was insufficient to resolve the atomic and molecularcomponents, so the group planned to use frequency doubled and tripled Nd:YAG laser light for anabsorption measurement to determine and monitor the molecular fraction. However, that technol-ogy was in its infancy at the time. These factors played a major role in the retreat to moleculartritium at both Los Alamos and Livermore. Much higher resolution is now the norm in tritiumbeta decay, such that the molecular and atomic components would be readily resolvable in thespectrum itself. This will not mitigate the need for high atomic purity but it does at least allow acontinuous internal calibration of it.As experiments push below the 1-eV level, ever larger activities of tritium must be underobservation to keep measurement times within reason. The KATRIN experiment has approximately1 Ci under observation in the source tube. Will it be possible to develop a similar activity in atomictritium? The answer hinges on the production and containment of atomic tritium. We considercontainment first.Physical bottle containment has long been used in hydrogen masers. Glass surfaces, sometimes9coated with hydrocarbon or fluorocarbon films, are quite effective in inhibiting both recombinationand spin flip. For a tritium experiment, however, insulating surfaces are a danger because of theaccumulation of unknown surface charges that affect the measured spectrum. This motivated theLos Alamos group to carry out an experimental search for a metallic surface that did not encouragerecombination. It was found that aluminum (a standard 6069 alloy) performed as well as glass(Fig. 31). The source tube in the Los Alamos experiment was therefore made of aluminum, 5 cm FIG. 31: Recombination probability per bounce for hydrogen and deuterium atoms on aluminum [265]. Thesolid curve is for borosilicate glass [272]. in diameter and internally polished to reduce declivities where recombination would be enhanced.Today, this approach to an atomic trap is disqualified by the molecular background, which cannotbe reduced to the < ∼ − level that a sensitive atomic experiment requires. On the other hand,aluminum, glass, and possibly other materials are well suited to the accommodation step wheredissociated atoms are first cooled before entering the trap. Fluorocarbons, however, are not suitablewith tritium because they are degraded by radiolysis [273], which leads to the formation of TF, acorrosive gas.Atomic hydrogen has a magnetic moment µ of approximately 1 Bohr magneton, which makes0possible a purely magnetic trap without physical walls. A major advantage is that the magneticmoment of molecular hydrogen is 3 orders of magnitude smaller, and molecules will be able toescape the trap with relative ease, improving the atomic purity. The magnetic energy is U = µ · B . (IX.10)The magnetic energy is displayed in a Breit-Rabi diagram, Fig. 32. If the magnetic moment is -4-20 F r e q u e n c y , G H z Magnetic Field, T a b c d (+1/2,+1/2)(+1/2, -1/2) (-1/2,-1/2)(-1/2,+1/2)F=1F=0 (m s , m i ) Low-field seeking states U = + μ B High-field seeking states U = - μ B FIG. 32: Breit-Rabi diagram for atomic tritium, with parameters from [274]. The effective magnetic momentis proportional to the slope of the hyperfine levels. At low fields the total angular momentum and itsprojection, (
F, m F ), are good quantum numbers, whereas at high field the electronic and nuclear spinsorient independently and their projections ( m s , m i ) become good quantum numbers. in the “low-field-seeking” orientation with respect to the magnetic field, an atom in a local fieldminimum is attracted to the minimum. Maxwell’s equations do not permit a local field maximum,but high-field-seeking atoms can be trapped in a saddle-point region if physical walls can also beused to prevent their escape to higher fields. The reason for using the evidently more complicatedhigh-field trap is related to dipolar spin-flip interactions [275], which are energetically forbidden inthis configuration. They dominate the loss rate from field-minimum traps.It was discovered in the course of experiments to make a Bose-Einstein condensate of hydrogen[276] that atomic hydrogen could be efficiently contained in a high-field bottle trap in which thewalls were coated with a film of superfluid He [274]. Unfortunately the method does not workwith atomic tritium, because the adsorption energy of atomic tritium on superfluid He is too high[274]. From theory, the adsorption energy for hydrogen (protium) is 0.85 K and for tritium 3.21K, a prediction that is experimentally supported at the ∼
15% accuracy level for protium. Thesurface recombination rate depends exponentially on this quantity.Two kinds of field-minimum magnetic trap have been devised, the Ioffe-Pritchard trap [277]and the Halbach array [278]. In both types the magnetic ‘walls’ of the trap are regions of relativelyhigh magnetic field orthogonal to the central field. If an atom approaches these regions not tooquickly, its magnetic moment µ orients with the local direction of the field B . Atoms in low-field-seeking states are repelled by the magnetic walls back to the central region, while atoms inhigh-field-seeking states are ejected from the trap.The simplest form of the Ioffe-Pritchard trap is a quadrupole magnet with pinch coils at eitherend, and it has been successfully used to trap hydrogen atoms [279] and free neutrons [280].Increasing the multipolarity of the trap magnet gives a larger central volume of relatively uniformfield (provided by a separate solenoid), and the ALPHA collaboration use an octupole trap in theirantihydrogen experiments [281, 282]. In principle, a large superconducting Ioffe-Pritchard trapof high multipolarity combined with a solenoid can provide both the magnetic walls for trappingatomic tritium and the uniform central magnetic field for carrying out CRES, although such alarge trap has never been built.Halbach arrays are periodic assemblies of permanent-magnet blocks arranged to produce a fieldnear the surface that is > X. CONCLUSION
In his formulation of the theory of beta decay in 1933-4, Fermi remarked [77] “...we concludethat the rest mass of the neutrino is either zero, or in any case, very small in comparison to themass of the electron.” In the intervening years experimentalists have relentlessly pressed onward2to find the mass, their efforts recorded in a perfect Moore’s Law of technical progress (Fig. 10).We know now that there are not just one but three different kinds of neutrino. The revolutionarydiscovery of neutrino oscillations at the end of the last century showed that neutrinos indeedhave mass, and that the particles with the well-defined flavors – electron, mu, or tau – are actuallylinear superpositions of particles with well-defined mass. Oscillations reveal the differences betweenthe squares of the masses and set a minimum value because no mass can be less than zero, butthey do not yield the masses themselves. Nevertheless, this information enormously simplifies the(still difficult) task of the experimentalist because only the ‘easy’ mass that is coupled mainly toelectrons needs to be measured to determine them all, assuming the mass ordering is also known.And, perhaps even more significant, neutrino mass is finally confined within a window having bothupper and lower bounds, a window that is steadily shrinking with each new experimental idea.At each stage in this odyssey, the next step has seemed insurmountable, but always new ideasand insights have opened a path. The development of magnetic spectrometers, gaseous tritiumsources, the MAC-E filter, microcalorimetry, and cyclotron radiation emission spectroscopy areexamples.The KATRIN experiment is in operation, after 17 years of construction. The ingenuity and carein its design are bearing fruit with a factor of 2 improved limit in only a month’s data taking. Itwill in due course reach its limit of sensitivity, either finding the neutrino mass or setting a limit inthe vicinity of 0.2 eV. Otten and Weinheimer [24] note KATRIN’s singular nature, “This could leadto a somewhat uneasy situation, in particular, if a finite but small mass signal happens to appear.How can the requirement to independently check a new result be fulfilled?” The appearance onthe scene of a novel technology, cyclotron radiation emission spectroscopy (CRES) [258], may offeran answer to this important question. It seems likely that a CRES atomic tritium experiment at some scale could be mounted, but whether it can approach or exceed KATRIN’s reach is at presentunknown. The technical challenges are great.Neutrino mass is the only fermion mass for which the minimal Standard Model makes a firmprediction: zero. That the prediction was incorrect is the first contradiction of the Standard Model,rather than simply something omitted. Neutrino mass seems to arise from a mechanism differentfrom the Standard Model’s Higgs mechanism and finding out what that is will illuminate the wayto a more comprehensive theory. At the same time, we are convinced that the universe is filled withneutrinos from the big bang, and their mass has affected the formation of the largest structures.The cosmological model with a cosmological constant and cold dark matter, like the StandardModel, is an extraordinarily predictive theory, and yet it is assembled from ingredients with which3we have no earthly familiarity. In both theories there are signs of tension. Laboratory measurementof the mass of the neutrino is one of the keys needed to unlock the mysteries.
XI. ACKNOWLEDGMENTS
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