PHYSICAL REVIEW D 101, 016003 (2020)
Effective neutrino masses in KATRIN and future tritium beta-decay experiments
† ‡ Guo-yuan Huang,1,2,* Werner Rodejohann ,3, and Shun Zhou1,2, 1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3Max-Planck-Institut für Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany
(Received 25 October 2019; published 3 January 2020)
Past and current direct neutrino mass experiments set limits on the so-called effective neutrino mass, which is an incoherent sum of neutrino masses and lepton mixing matrix elements. The electron energy spectrum which neglects the relativistic and nuclear recoil effects is often assumed. Alternative definitions of effective masses exist, and an exact relativistic spectrum is calculable. We quantitatively compare the validity of those different approximations as function of energy resolution and exposure in view of tritium beta decays in the KATRIN, Project 8, and PTOLEMY experiments. Furthermore, adopting the Bayesian approach, we present the posterior distributions of the effective neutrino mass by including current experimental information from neutrino oscillations, beta decay, neutrinoless double-beta decay, and cosmological observations. Both linear and logarithmic priors for the smallest neutrino mass are assumed.
DOI: 10.1103/PhysRevD.101.016003
I. INTRODUCTION region close to its end point will be distorted in com- parisontothatinthelimitofzeroneutrinomasses.This Neutrino oscillation experiments have measured with kinematic effect is usually described by the effective very good precision the three leptonic flavor mixing neutrino mass [7] angles fθ12; θ13; θ23g and two independent neutrino 2 2 2 2 mass-squared differences Δm21 ≡ m2 − m1 and jΔm31j≡ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 − 2 2 2 2 2 2 2 jm3 m1j. The absolute scale of neutrino masses, how- mβ ≡ jUe1j m1 þjUe2j m2 þjUe3j m3; ð1Þ ever, has to be determined from nonoscillation approaches, using beta decay [1], neutrinoless double- where U (for i ¼ 1, 2, 3) stand for the first-row elements beta decay [2], or cosmological observations [3].Once ei of the leptonic flavor mixing matrix U, i.e., jU 1j¼ the neutrino mass scale is established, one knows the e cos θ13 cos θ12, jUe2j¼cos θ13 sin θ12, jUe3j¼sin θ13 in lightest neutrino mass, which is m1 in the case of normal the standard parametrization [8],andmi (for i ¼ 1,2,3) neutrino mass ordering (NO) with m1
2470-0010=2020=101(1)=016003(20) 016003-1 Published by the American Physical Society HUANG, RODEJOHANN, and ZHOU PHYS. REV. D 101, 016003 (2020)
χ2 Δχ2 TABLE I. The configurations of tritium beta-decay experiments and the resulting β and true defined in Eqs. (19) and (20) arising from the description of the electron spectrum by using the effective neutrino mass mβ. One year of data taking has been assumed. No background is assumed, and the χ2 values can be further reduced taking into account possible background contributions. 0 Δ χ2 χ2 Δχ2 Δχ2 mL ¼ eV Target mass β,NO β,IO true,NO true,IO KATRIN 2.5 × 10−4 g1eV7.4 × 10−7 5.6 × 10−7 1.3 × 10−7 1.1 × 10−7 Project 8 (Molecular 3H) 5 × 10−4 g 0.36 eV 8.9 × 10−5 6.1 × 10−5 2.0 × 10−5 1.4 × 10−5 Project 8 (Atomic 3H) 5 × 10−4 g 0.05 eV 0.064 0.13 0.032 0.015 PTOLEMY 100 g 0.15 eV 428 331 141 81 experiment for tritium beta decays.1 More explicitly, we main results in Sec. IV. Technical details on the considered shall consider the KATRIN [11], Project 8 [17], and experiments and on the likelihoods used for our Bayesian PTOLEMY [18–20] experiments. Their main features analysis are delegated to Appendices. and projected sensitivities have been summarized in Appendix A and Table I. We compare mβ in Eq. (1) with II. THE EFFECTIVE NEUTRINO MASS other effective neutrino masses proposed in the literature and also consider the exact relativistic spectrum of tritium A. The relativistic electron spectrum beta decays. A measure of the validity of mβ in terms of the Before introducing the effective neutrino mass for beta exposure and energy resolution for a beta-decay experiment decays, we present the exact relativistic energy spectrum can be set. As a result, we find that the standard effective of the outgoing electrons for tritium beta decays (or mass mβ and the classical spectrum form can be used for equivalently the differential decay rate), which can be KATRIN and Project 8 essentially without losing accuracy. calculated within standard electroweak theory [21–24], Furthermore, it is of interest to estimate how likely a the result being signal in upcoming neutrino mass experiments, including X3 those using electron capture, is. Toward this end, we dΓ σ¯ðE Þ rel N e U 2H E ;m : perform a Bayesian analysis to obtain the posterior dis- ¼ T π2 j eij ð e iÞ ð3Þ dKe 1 tributions of mβ. The probability to find the beta-decay i¼ signal depends on the experimental likelihood input one 3 considers, in particular the neutrino mass information from Here NT is the target mass of H, and Ee ¼ Ke þ me is the cosmology and neutrinoless double-beta decays. The cos- electron energy with Ke being its kinetic energy. In Eq. (3), mological constraints on neutrino masses reply on the the reduced cross section is given by datasets one has included in generating the likelihood, 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m3 whereas the constraints from neutrinoless double-beta σ¯ ≡ GF 2 He 2 − 2 ðEeÞ jVudj FðZ; EeÞ Ee Ee me 2π m3 decays are subject to the assumption whether neutrinos H are Dirac or Majorana particles. It is thus quantified what C 2 × hf i2 þ A hg i2 ; ð4Þ the consequences of adding more and more additional F C GT mass information are. Moreover, the prior on the smallest V neutrino mass, which could be linear or logarithmic, is 1 166 10−5 −2 where GF ¼ . × GeV is the Fermi constant, important for the final posteriors. ≈ θ jVudj cos C is determined by the Cabibbo angle The remaining part of our paper is organized as follows. θ ≈ 12.8°, FðZ; E Þ is the ordinary Fermi function with In Sec. II, we make a comparison between the exact C e Z ¼ 1 for tritium taking account of the distortion of the relativistic spectrum of electrons from tritium beta decays electron wave function in the Coulomb potential of the with the ordinary one with an effective neutrino mass mβ. 2 ≈ 1 ≈ 1 2695 decaying nucleus, CV and CA . stand for Then, a quantitative assessment of the validity of the the vector and axial-vector coupling constants of the effective neutrino mass is carried out. The posterior charged-current weak interaction of nucleons, respectively. distributions of the effective neutrino mass are calculated 2 2 In addition, hf i ≈ 0.9987 and hg i ≈ 2.788 are the in Sec. III, where the present experimental information F GT squared nuclear matrix elements of the allowed Fermi and from neutrino oscillations, neutrinoless double-beta decays, Gamow-Teller transitions. The kinematics of the tritium and cosmology are included. Finally, we summarize our beta decays is encoded in the function HðEe;miÞ in Eq. (3), namely, 1In this work, we focus only on tritium beta-decay experi- ments. Similar analyses of the effective neutrino mass can be 2 2πη 1 − −2πη performed for the electron-capture decays of holmium, namely, The Fermi function is given bypFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZ; EeÞ¼ =ð e Þ, − 163 163 2 2 e þ Ho → νe þ Dy, which are and will be investigated in where η ≡ ZαEe=pe with pe ¼ Ee − me being the electron the ECHo [14] and HOLMES [15], NuMECS [16] experiments. momentum and α ≈ 1=137 the fine-structure constant.
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2 −4 1 − m =ðm3 E Þ Furthermore, given 1 − m3 =m3 ≈ 1.89 × 10 and ≡ e H e He H HðEe;miÞ 2 2 2 −4 1 − 2 3 3 ≈ 1 82 10 ð Ee=m H þ me=m3 Þ me=m H . × , the relativistic electron spectrum sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H Γ d rel=dKe approximates to the classical one 2 3 mim He mi × y y þ y þ ðm3He þ miÞ ; 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 Γ σ X m H m H d c1 clðEeÞ 2 2 2 ¼ N jU j ðK 0 − K Þ − m dK T π2 ei end; e i ð5Þ e i¼1 − × ðKend;0 KeÞ; ð8Þ 2 ≡ − 3 − − where y Kend Ke with Kend ¼½ðm H meÞ 2 3 2 3 where σ ðE Þ¼σ¯ðE Þ=ðm3 =m3 Þ and σ¯ðE Þ has been ðm He þ miÞ =ð m HÞ being the end point energy corre- cl e e He H e given in Eq. (4). Comparing the classical spectrum in sponding to the neutrino mass mi. Some comments on the kinematics are in order: Eq. (8) with the relativistic one in Eq. (3), one can observe (i) Given the nuclear masses m3 ≈ 2808920.8205 keV that the end point energy in the former case deviates from H −4 3 ≈ 2808391 2193 the true one by an amount of 1 − m3 =m3 m ≈ 10 m . and m He . keV [23], as well as ð He HÞ i i the electron mass m ≈ 510.9989 keV, one can As the PTOLEMY experiment could achieve a relative e −6 obtain the Q value of tritium beta decay Q≡ precision of 10 for the determination of the lightest 3 − 3 − ≈ 18 6023 m H m He me . keV. In the limit of neutrino mass [20], it would be no longer appropriate to use vanishing neutrino masses, the end point energy the classical spectrum in PTOLEMY. However, it is rather Kend turns out to be safe for KATRIN and Project 8 to neglect the factor 1 − 3 3 m He=m H, as their sensitivities to the neutrino mass 2 2 1σ ≡ 3 − − 2 3 are weaker than for PTOLEMY. To be more specific, the Kend;0 ½ðm H meÞ m3He =ð m HÞ 2 2 sensitivities of KATRIN and Project 8 to mβ are σðmβÞ ≈ ≈ 18.5989 keV; ð6Þ 2 2 2 0.025 eV [11] and σðmβÞ ≈ 0.001 eV [17], respectively, 2 corresponding to σðmβÞ=mβ ≈ 0.05ð0.5 eV=mβÞ and which is lower than the Q value by a small amount of 2 σðmβÞ=mβ ≈ 0.002ð0.5 eV=mβÞ . Both values are much Q − K 0 ≈ 3.4 eV. This difference arises from the end; 10−4 recoil energies of the final-state particles and is larger than the correction of order from the factor naturally included when one considers fully relativ- m3He=m3H. To have an expression for the electron spectrum istic kinematics. Since the electron spectrum near its applicable to experiments beyond KATRIN and Project 8, end point is sensitive to absolute neutrino masses, i.e., leading to PTOLEMY, we can slightly modify the which are much smaller than this energy difference classical energy spectrum as follows: of 3.4 eV, it is not appropriate to treat the Q value as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X3 2 Γ0 σ E m3 the end point energy. d cl clð eÞ 2 − 2 − He ≈ − ¼ NT 2 jUeij ðKend;0 KeÞ mi (ii) It is straightforward to verify that Kend Kend;0 dK π m3 e i¼1 H 3 3 3 3 ≈ − mim He=m H and thus y þ mim He=m H Kend;0 2 − 2 3 1 78 ×ðKend;0 KeÞ: ð9Þ Ke, where a tiny term mi =ð m HÞ < . × 10−10 eV for m < 1 eV can be safely ignored. i Let us now check whether the difference between the exact Taking this approximation on the right-hand side relativistic spectrum dΓ =dK and the modified classical of Eq. (5), we can recast the kinematical function rel e one dΓ0 =dK affects the determination of absolute neutrino into cl e masses in future beta-decay experiments with a target mass 10−4 2 of tritium ranging from g in KATRIN to 100 g in 1 − m =ðm3 E Þ ≈ e H e PTOLEMY. In other words, we examine whether these two HðEe;miÞ 2 2 2 1 − 2 3 ð Ee=m H þ me=m3HÞ spectra are statistically distinguishable in realistic experi- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ments. Consider the ratio of these two energy spectra 2 m3 − 2 − He × ðKend;0 KeÞ mi 2 3 Γ 1 − 3 3 m H d rel=dKe me=ðm HEeÞ m He ¼ 2 2 2 · Γ0 1 − 2 3 3 − d =dKe ð Ee=m H þ me=m3 Þ m H × ðKend;0 KeÞ; ð7Þ cl H K ≈ 1.0036 þ 1.7 × 10−9 e ; ð10Þ from which it is interesting to observe that the eV absolute neutrino mass mi in the square root receives 3 3 ≈ 0 999811 a correction factor m He=m H . . The differ- where an expansion in terms of the electron kinetic ence between Eqs. (7) and (5) is negligibly small, so energy Ke ¼ Ee − me has been carried out. First of all, the former will be used in the following discussions. the constant on the rightmost side of Eq. (10) can be
016003-3 HUANG, RODEJOHANN, and ZHOU PHYS. REV. D 101, 016003 (2020) absorbed into the uncertainty of the overall normalization figure out the expected statistics around the end point factor Aβ in the statistical analysis, so it is irrelevant for our according to Eq. (11). For a conservative experimental discussions.3 Considering the term proportional to the setup, e.g., that is achievable in KATRIN with Δ ¼ 1 eV −4 electron kinetic energy Ke in Eq. (10), it could potentially and E ¼ 10 g · yr, the expected event number within the 5 disturb the determination of the normalization factor of the end point bin is 3.2 × 10 . In the limit of mβ ≪ Δ, a finite spectrum. The distortion amplitude induced by the Ke- neutrino mass will induce a relative deviation of events 2 2 dependent term can be characterized by the specified within the window Δ by 3=2 · mβ=Δ . Therefore, the energy window ΔKe below the end point. For example, 1=4 sensitivity to mβ is roughly scaled as ðΔ=EÞ . The choice we have ΔK ¼ 30 eV for KATRIN while ΔK ¼ 5 eV e e of energy window is limited by the smearing effect of finite for PTOLEMY, which is limited by the detector perfor- energy resolution, and the sensitivity to neutrino masses mance; see Appendix A. In this way, we can obtain −8 will drop with a larger energy resolution. This can be the distortion amplitude of 5.1 × 10 for KATRIN and compensated by increasing the exposure, such that an 8.5 × 10−9 for PTOLEMY, respectively. To examine the efficient event number can be acquired to resolve the impact of this distortion, one can compare it with the overall shift due to finite neutrino masses. statistical fluctuation of the events within the corresponding Unless stated otherwise, we will refer from now on to energy window. The integrated number of beta-decay dΓ0 =dK as the exact spectrum in the remaining discussion. events within the energy window below the end point cl e can be calculated via B. Validity of the effective mass Z K 0 Γ end; d In principle, it is the exact relativistic spectrum that Nint ¼ T dKe K 0−ΔK dK should be confronted with the experimental observation in end; e e 3 order to extract the absolute neutrino masses mi (for i ¼ 1, 11 ΔKe E ≈ 3.2 × 10 · · ; ð11Þ 2, 3), since U 2 (for i 1, 2, 3) can be precisely eV 100 g·yr j eij ¼ measured in neutrino oscillation experiments. However, E ≡ often the effective electron spectrum with only one mass where T is the operation time and NT · T is the total exposure. The statistical fluctuation of the beta-decay parameter is considered (see e.g., [25]), eventsffiffiffiffiffiffiffiffi within the energy window is estimated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p −5 2 ≈ 10 Δ 30 dΓ σ 1 E m3 Nint=Nint for KATRINffiffiffiffiffiffiffiffi with Ke ¼ eV and eff c ð eÞ − 2 − He −4 p −7 ¼ NT 2 ðKend;0 KeÞ mβ E 10 ≈ 10 π 3 ¼ g · yr, while Nint=Nint for PTOLEMY dKe m H with ΔK 5 eV and E 100 g · yr. Both values are e ¼ ¼ − much larger than the corresponding distortion amplitudes. × ðKend;0 KeÞ; ð12Þ It is thus evident that the uncertainty in Aβ will be where the effective neutrino mass mβ is usually defined as dominated by the intrinsic statistical fluctuation of the observed beta-decay events in future experiments. As the in Eq. (1). Note that for consistency we have kept the near- 3 3 Γ0 data fluctuation near the end point is most significant unity factor m He=m H as in d cl=dKe of Eq. (9), which is among the entire spectrum, the influence of the spectral necessary when it comes to experiments beyond KATRIN distortion as indicated in Eq. (10) is not important. and Project 8, i.e., PTOLEMY. However, m3He=m3H appears Hence, we conclude that the classical spectrum with the as an overall factor to all neutrino mass parameters, so the neutrino masses corrected by mi → mi · ðm3 =m3 Þ in quantitative impact on our discussion of the validity of He H 4 Eq. (9) works as well as the exact relativistic spectrum the effective mass is actually negligible, but we keep it in Eq. (3) for future beta-decay experiments. nevertheless in our numerical calculations. Another impor- It is worthwhile to emphasize that because of a finite tant point is that the end point energy in Eq. (6) should be energy resolution Δ, which is normally much larger than corrected if we take account of binding energies as well as the absolute neutrino mass mi, it is difficult to resolve the excitations of the daughter system in an actual experiment. true end point. Hence, the experimental sensitivity to For instance, in KATRIN or Project 8 with the molecular neutrino masses is in fact governed by the integrated tritium target, a correction of 16.29 eV to the end point number of beta-decay events within a specified energy energy should be considered owing to the binding energies window below the end point. Taking this energy window to Δ Δ 4 2 2 2 be the experimental energy resolution, Ke ¼ , we can One can easily check that the relation mβ ≡ jUe1j m1 þ 2 2 2 2 → 3 3 jUe2j m2 þjUe3j m3 is stable under mi mi · ðm He=m HÞ → 3 3 3For instance, in the statistical analysis of the simulated data and mβ mβ · ðm He=m HÞ. Thus, any quantitative conclusion for the PTOLEMYexperiment [20], the prior of the normalization made by considering m3He=m3H corrections can be directly factor Aβ is set to be in the range of (0…2), which is wide enough applied to the case without correction of m3He=m3H by shifting to take account of the difference corresponding to the constant all neutrino masses with a relative fraction as small as 10−4, and term in the ratio in Eq. (10). vice versa.
016003-4 EFFECTIVE NEUTRINO MASSES IN KATRIN AND FUTURE … PHYS. REV. D 101, 016003 (2020) of the mother tritium pair, the daughter tritium-helium three effective masses provide very good fits and their molecule, and the combination of the ionized electron [1]. relative differences are very small. For example, it is easy to 02 002 00 −3 Compared to the atomic case, the recoil energy of the verify that ðmβ − mβ Þ=mβ ≲ 10 . If the chosen energy molecular state will also be reduced by a factor of two due 0 window satisfies ΔKe < 2mβ, then mβ can give a better to the doubled mass, which will boost the end point energy 00 fit than mβ, whereas mβ is still an excellent parameter in of electrons. For PTOLEMY, with a foreseen possibility of fitting the spectrum with an almost negligible difference atomic tritium weakly bounded to a graphene layer [18], m2 − m02 =m002 ≲ 10−5 the ionized electron will inevitably interact with the ð β β Þ β . If neutrino masses are hierarchi- complex graphene binding system. Furthermore, the cal, mβ always fits better to the spectrum than the other two recoiled helium (with a kinetic energy of 3.4 eV) will variants. In case of an extremely small value of the lightest 0 00 escape the graphene binding structure with a sub-eV neutrino mass, both mβ and mβ are unable to offer a good binding energy. All these effects need to be systemically fit to the true spectrum. In Fig. 1, we plot three effective considered in the experiment to evaluate the final end point neutrino masses in terms of the lightest neutrino mass energy Kend;0. However, in our case, the spectrum in which is m1 for NO and m3 for IO. One can observe that Eq. (12) mostly depends on the relative deviation from their differences are significant in NO when m1 is small, but − σ¯ the end point energy Kend;0 Ke, and c1ðEeÞ changes very in IO the differences are always unnoticeable. The situation − slowly as a function of Kend;0 Ke near the end point. of IO can be attributed to the fact that the contribution of m3 2 Hence, our results which will be presented in terms of is suppressed by jUe3j while the remaining two neutrino − Kend;0 Ke are still valid if a different Kend;0 is considered. masses m1 and m2 are always nearly degenerate due to the 2 2 On the other hand, the final-state excitations of molecules relation Δm21 ≪ jΔm31j. will smear the energy of outgoing electrons [26–29], and To be more explicit, we look carefully at the main Γ0 this effect will be taken into account as the irreducible difference between the exact spectrum d cl=dKe and the Γ energy resolution of the experiment. effective one d eff=dKe. The difference stems from the Let us summarize the existing expressions of the electron kinematical functions, namely [32], spectra defined in this work: (i) the exact relativistic beta sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spectrum dΓ =dK without making approximations; see X3 2 rel e m3 Γ Γ0 ∝ 2 − 2 − He Eq. (3); (ii) the classical spectrum d cl=dKe in the limit d =dKe jUeij ðKend;0 KeÞ mi · cl 3 3 3 → 1 3 → 0 1 m H of m He=m H and me=m H ; see Eq. (8); (iii) the i¼ modified classical spectrum dΓ0 =dK by making the − cl e × ðKend;0 KeÞ; ð14Þ → 3 3 Γ replacement mi mi · ðm He=m HÞ in d cl=dKe; see Γ Eq. (9); (iv) the effective electron spectrum d eff=dKe sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m3 definedinEq.(12). We have seen that the difference between Γ ∝ − 2 − ð0;00Þ He 0 d eff=dKe ðKend;0 KeÞ mβ · dΓ =dK in Eq. (9) and the classical spectrum dΓ =dK 3 cl e cl e m H in Eq. (8) plays only a role when PTOLEMY is considered. − In addition, the difference to the relativistic spectrum × ðKend;0 KeÞ; ð15Þ Γ d rel=dKe in Eq. (3) is minuscule and the classical spectra can be considered as the exact ones. It remains to compare where the spectra involving our three effective neutrino Γ0 0 00 the so-defined exact spectrum d cl=dKe in Eq. (9) to the masses mβ, mβ, and mβ are collectively given. To analyze Γ effective one d eff =dKe in Eq. (12). the difference, we take mβ in the NO case, for example, but Moreover, in the literature, two different definitions the other effective masses and the IO case can be studied in of the effective neutrino mass have also been introduced a similar way. Let us start with the end point of the electron [30–32], namely, spectrum and then go to lower energies. For convenience, the factor of m3 =m3 is omitted in the following quali- X3 He H 0 2 00 tative discussion, which of course will not affect the main m ≡ m jU j ;m≡ m1: ð13Þ β i ei β feature of the result as we noted above. i¼1 (1) For the exact spectrum, the end point energy Kend The effective electron spectrum can then be obtained by is set by the smallest neutrino mass, i.e., 0 00 − replacing mβ in Eq. (12) with mβ or mβ. In this subsection, Kend;0 Kend ¼ m1, while it is mβ for the effective Γ 2 2 Δ 2 2 we discuss the difference among those three effective spectrum d eff=dKe. Since mβ ¼ m1 þ m21jUe2j þ 2 2 2 neutrino masses and clarify their validity with future Δm31jUe3j >m1, the end point energy of the Γ beta-decay experiments in mind. effective spectrum d eff=dKe is smaller than that Γ0 As has been observed in Ref. [32], the three effective of the exact one d cl=dKe. Therefore, starting from − neutrino masses have different accuracies in fitting the the electron kinetic energy of Ke ¼ Kend;0 m1 and exact spectrum. If neutrino masses are quasidegenerate, all going to smaller values, the effective spectrum
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FIG. 1. The effective neutrino masses versus m1 for NO (left panel) and m3 for IO (right panel). The 3σ CL uncertainties of oscillation parameters have been considered. A similar plot for absolute neutrino masses was given in Ref. [33].
Γ d eff=dKe is always vanishing and thus should be the red dotted curve for comparison. The effective spectra Γ0 Γ 13 4 0 11 9 00 lying below the exact one d cl=dKe. d eff=dKe for mβ ¼ . meV, mβ ¼ . meV, and mβ ¼ (2) As Ke is decreasing further, we come to the point at 10 meV are given by the dark, medium, and light blue − 2 which Kend;0 Ke ¼ mβ is satisfied. Note that mβ ¼ dashed curves, respectively. Those values are obtained for a 2 2 2 2 2 2 10 m3 − Δm31jUe1j − Δm32jUe2j
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FIG. 2. Illustration of the electron spectrum from tritium beta decays, where the total exposure E ¼ 1 g · yr and the best-fit values of Γ0 10 10 5 neutrino mixing angles and mass-squared differences are assumed. The exact spectra d cl=dKe with m1 ¼ meV and m1 ¼ . meV Γ 13 4 0 11 9 are shown as the gray solid and red dotted curve, respectively. The effective spectra d eff =dKe with mβ ¼ . meV, mβ ¼ . meV, 00 and mβ ¼ 10 meV, corresponding to m1 ¼ 10 meV, are represented by the dark, medium, and light blue dashed curves. In the left panel, the energy smearing is ignored, while an energy resolution of Δ ¼ 100 meV is taken into account in the right panel. In both panels, the real spectra are depicted in the upper subgraph, whereas their deviations from the exact spectrum with m1 ¼ 10 meV are given in the lower subgraph. Z ∞ dΓΔ 1 þ dΓ ffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi 0 0 either positive or negative, the total number of beta- ðKeÞ¼p p dKe 0 ðKeÞ dKe 2πðΔ= 8 ln 2Þ −∞ dKe decay events within a very wide energy window is approximately vanishing. To be more concrete, the − 0 2 ðKe KeÞ Δ Γ × exp − pffiffiffiffiffiffiffiffiffiffiffi ; ð16Þ integration of ðd =dKeÞ over an energy window 2 Δ 8 2 2 ð ln Þ ΔKe scales as ΔðdΓ=dKeÞ ∝ mβ=ΔKe [32], which will be vanishing when ΔKe ≫ mβ. If the energy Γ0 resolution Δ ¼ 100 meV is larger than absolute which has been plotted in the right panel for both d cl=dKe Γ neutrino masses, we can evaluate the difference and d eff=dKe. Note that we have not yet specified any planned experimental configuration so far, because the between the effective spectra in the region of − Δ main purpose here is to understand the behavior of Ke
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An immediate question is whether the effective neutrino nonintegrated mode, we will use the following experimen- mass is still a useful parameter for future beta-decay tal configurations: (i) KATRIN with the target mass Γ0 2 5 10−4 Δ experiments. Put alternatively, does d cl=dKe in Eq. (9) mKATRIN ¼ . × g and energy resolution KATRIN ¼ provide a good description of the effective spectrum 1 eV; (ii) Project 8 loaded with molecular tritium gas with Γ 5 10−4 Δ 0 36 d eff=dKe in Eq. (12)? We will now investigate the validity mP8 ¼ × g and P8m ¼ . eV; (iii) Project 8 5 10−4 of the effective neutrino mass by following a simple loaded with atomic tritium gas with mP8 ¼ × g Δ 0 05 statistical approach. The strategy of our numerical analysis and P8a ¼ . eV; (iv) PTOLEMY with mPTOLEMY ¼ is summarized as follows. 100 Δ 0 15 g and PTOLEMY ¼ . eV. The details can be found Given the target mass and the operation time T (i.e., the in Appendix A. For Project 8 loaded with molecular total exposure E), we simulate the experimental data by tritium, the energy resolution is limited by the irreducible Γ0 using the exact spectrum d cl=dKe and divide the simulated width of the final-state molecular excitations [35]. This data into a number of energy bins with bin width Δ, which limitation can be overcome by switching the target to is taken to be the energy resolution of the detector. atomic tritium. Note that for KATRIN in the MAC-E-TOF In general, the event number in the ith energy bin mode, which is still under development, the penalties of − Δ 2 Δ 2 ½Ei = ;Ei þ = is given by the integration of the the tritium decay rate and energy resolution due to the spectrum over the bin width chopping procedure (see Appendix A) are ignored, so the Z Δ 2 configuration here is somewhat idealized for KATRIN. Eiþ = dΓΔ β Ni ¼ T dKe; ð18Þ Nevertheless, we will find the effect of using m even in −Δ 2 dK Ei = e this ideal KATRIN setup is negligible. We adopt Gaussian distributions as in Eq. (16) for the uncertainties caused by where Ei denotes the mean value of the electron kinetic Γ finite energy resolutions in all experiments. In a more energy in the ith bin and d Δ=dKe is the convolution of a realistic analysis with all experimental details taken into spectrum with a Gaussian smearing function as in Eq. (16). cl account, one should consider a strict shape for the energy The simulated event number Ni in each energy bin is resolution function, e.g., a trianglelike shape for KATRIN Γ0 calculated by using d cl=dKe with a specified value of m1 in the developing MAC-E-TOF mode. The actual shape (i.e., the lightest neutrino mass in the NO case). On the for Project 8 with molecular tritium should also be Γ other hand, to clarify how good d eff=dKe can describe the calculated with detailed consideration of final-state exci- eff true data, the predicted event number Ni in each energy tations. However, a different shape of the energy reso- bin is calculated in the same way but with the effective lution from Gaussian should not affect our results by Γ spectrum d eff=dKe, which will be subsequently sent to fit orders of magnitude. cl the simulated true data Ni . In Fig. 3, we show the difference in the event numbers It should be noted that KATRIN operating in the Γ Γ0 of d eff=dKe and d cl=dKe, together with the statistical ordinary mode with the MAC-E-Filter observes actually fluctuation of the events. In the upper two panels, two the integrated number of beta-decay events and has to nominal experimental setups have been chosen for dem- reconstruct the differential spectrum by adjusting the onstration, and in the remaining four panels, we illustrate retarding potential to scan over a certain energy window the cases of realistic experiments. In all panels, the data are containing the end point. The number of events for the Γ0 simulated with d cl=dKe, for which a true value of the differential spectrum in each energy bin turns out to be true lightest neutrino mass m1 ¼ 10 meV has been input, and int − int int Ni ¼ Ni Ni−1,whereNi is the event number of the the data fluctuations are represented by the filled gray integrated spectrum for the scanning point corresponding histograms. For comparison, the event number difference in to Ei. For this reason, the statistical fluctuation of the each energy bin has been calculated for three effective event number for the reconstructed differential spectrum 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spectra with mβ ¼ 13.4 meV, mβ ¼ 11.9 meV, and can be estimated as Nint þ Nint , which should be m00 ¼ 10 meV, which is denoted as the blue dashed curves. ffiffiffiffiffii i−1 β p In addition, the gray solid curve denotes the exact spectrum compared with that of Ni for the direct measurement. true 10 Meanwhile, a longer time of data taking is also expected. with m1 ¼ meV as in Fig. 2, while the red dotted true Therefore, our result should be taken to be conservative curve is for m1 ¼ 10.5 meV. From Fig. 3, two important when considering KATRIN-like experiments operating in observations can be made. First, for a smaller exposure integrated mode. such as in KATRIN and Project 8, the statistical fluctuation KATRIN can also directly measure the nonintegrated can easily overwhelm the deviations, rendering the effec- beta spectrum in a possible MAC-E-TOF mode, as tive description of the beta spectrum more reliable. Second, described in Appendix A. Since tritium experiments in for mβ, the error caused by using the effective spectrum is the future tend to adopt nonintegrated modes to maximize most significant in the energy bin containing the end point. the neutrino mass sensitivity, we shall focus on this The reason is obvious, namely that the data fluctuation scenario. For those tritium experiments operating in the increases and the deviation decreases, as the energy moves
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Γ Γ0 FIG. 3. The difference of event numbers for the effective spectrum d eff =dKe and the exact one d cl=dKe. The data are simulated by true taking m1 ¼ 10 meV. The blue histograms signify the event number deviations of the effective spectra from the exact one, whereas the gray filled histograms stand for the statistical fluctuations. Two nominal experimental setups have been assumed in the upper two panels, and in the remaining four panels, we illustrate the cases of realistic experiments including KATRIN, Project 8 with molecular tritium and with atomic one, and PTOLEMY. Note that different scales on the axes have been adopted for each plot.
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2 FIG. 4. The contours of χβ, see Eq. (19), arising from the description of the electron spectrum by using the effective neutrino mass mβ, are displayed in the plane of smallest mass versus resolution, i.e., the m1 − Δ plane for the NO case (left panel) and the m3 − Δ plane for the IO case (right panel). away from the end point. For the other two effective masses previously, we have fixed the bin size to be the energy 0 00 mβ and mβ, the deviations are even more significant. resolution Δ. In principle, the bin width can be chosen To quantify the difference between the effective and freely. The smaller the bin width is, the more information Δ ≡ one can acquire in the fit. However, this is limited by the exact spectra in a statistical approach, wep defineffiffiffiffiffiffiffi Ni eff − cl cl energy resolution of an experiment, which will smooth out Ni Ni in each energy bin and take Ni to be the corresponding statistical uncertainty.6 In this way, if ΔN is the information within a comparable bin size, such that pffiffiffiffiffiffiffi i further decreasing the bin size will not improve the result negligible compared to Ncl, one can claim that the error i anymore. We have numerically checked that by choosing due to the use of the effective spectrum is unimportant in 2 a bin width smaller than the energy resolution, e.g., Δ=8, that energy bin. For the whole energy spectrum, the χ 2 the χ -function defined in Eq. (19) will increase only by a function can be constructed as factor of ∼70%. Further reducing the bin width will not X 2 ðΔN Þ alter this result. χ2 ¼ i ; ð19Þ Let us make some remarks on the other input in our β Ncl i i numerical calculations. First, the best-fit values of neutrino χ2 oscillation parameters from Ref. [34] are adopted. Second, where i runs over the number of energy bins. This the energy window for the analysis has been taken to be function measures to what degree the effective spectrum K − K 0 ∈ ð−4…4Þ eV. Third, we have assumed no dΓ =dK deviates from the exact one dΓ0 =dK . Because e end; eff e cl e background contributions. The inclusion of possible back- we have used dΓ0 =dK to generate the true data, from cl e ground events will reduce the value of χ2, leading to a the model selection perspective (i.e., fitting two different β models dΓ0 =dK and dΓ =dK with the same data, smaller statistical deviation of the effective spectrum cl e eff e Γ Γ0 2 d eff=dKe from the classical one d =dKe. Four, we take respectively), χβ defines the statistical significance with cl Γ0 Γ the normalization factor to be one, as it can be precisely which one can favor d cl=dKe over d eff=dKe. If one Γ determined by choosing a wider energy window in realistic insists in using the effective spectrum d eff=dKe to fit 2 2 experiments. the data, χβ also measures the goodness-of-fit χβ=v of In the left panel of Fig. 4, for each pair of m1 in the range Γ d eff=dKe given the degree of freedom v in fitting. Since of ð0…0.1Þ eV and Δ in the range of ð0.02…0.6Þ eV, we most deviations of dΓ =dK from dΓ0 =dK distribute 2 eff e cl e present the value of χβ for the total exposure of E ¼ 1 g·yr only in a few energy bins around the end point, the degree in the NO case, where the effective spectrum with mβ is of freedom can be v ¼ Oð1Þ depending on the number of adopted for illustration. Similar calculations have also been bins we use in the actual fit. As has been mentioned carried out in the IO case and the results are given in the m3 − Δ plane in the right panel. Roughly speaking, for 6 This is true when the event number in each energy bin is large, those values of Δ and m1 in the NO case (or m3 in the IO such that the fluctuation follows approximately a Gaussian χ2 ≲ 0 1 distribution, which turns out to be true for all tritium experiments case) corresponding to β . , the effective spectrum Γ in our consideration as can be noticed in Fig. 3. d eff=dKe with mβ is reasonably good to describe the data,
016003-10 EFFECTIVE NEUTRINO MASSES IN KATRIN AND FUTURE … PHYS. REV. D 101, 016003 (2020) i.e., with negligible and fragile statistical significance to Γ0 Γ discriminate d cl=dKe from d eff=dKe and no noticeable impact on the goodness of fit. In the same sense, we can also conclude that mβ is no longer a safe parameter for 2 those values of Δ and m1 (or m3) corresponding to χβ ≳ 10, Γ0 Γ i.e., the statistical power to favor d cl=dKe over d eff=dKe is more than 3σ and a considerable impact on the goodness- 2 of-fit arises (the p value of fit is 0.001565 for χβ ¼ 10 and 1 Γ v ¼ , and the model d eff=dKe is almost ruled out by the data). Although we have fixed the exposure at E ¼ 1 g · yr, it is 2 straightforward to derive the values of χβ for a different Δ 2 cl exposure by noting the fact that ð NiÞ =Ni is linearly proportional to E. As a consequence, for a different ˜ 2 exposure E, the original values of χβ for E will be modified 2 ˜ to be χβ · E=E. For instance, the original value of the 2 contour χβ ¼ 10 for E ¼ 1 × g · yr should be changed to 2 Δχ2 ≡ χ2 − χ2 χβ ¼ 0.01 for E ¼ 1 mg · yr. Nevertheless, if we insist on FIG. 5. The function β βjmin is shown with respect using dΓ =dK to fit the data regardless of the statistical to the lightest neutrino mass m1 in the NO case. The dark red eff e curve is generated by fitting with the exact spectrum dΓ0 =dK , preference for the true model dΓ0 =dK and a poor good- cl e cl e while the dark blue one is by using the effective spectrum ness of fit, the parameter estimation of mβ can always be dΓ =dK with mβ, for the exposure of E ¼ 100 g · yr. The light Γ eff e performed based on using d eff=dKe. In this case, a large curves are for E ¼ 1 g · yr with all else being the same. For 2 value of χβ does not necessarily mean a large value of illustration, we use an energy resolution of Δ ¼ 0.1 eV. Δχ2 ≡ χ2 − χ2 χ2 β βjmin in parameter estimations, where βjmin χ2 denotes the minimum of β by freely adjusting mβ or m1. bf true mβ with m being the best-fit value. The value of m can To explicitly show the error of fitting, the neutrino β β 2 be directly obtained with Eq. (1) once the input value of m1 mass m1 with the effective spectrum, we calculate Δχ in dΓ0 =dK for simulating the data is given. The difference and present the final result with respect to m1 in Fig. 5.In cl e Δχ2 measures how likely one can recover the true value our calculations, we assume the true value of m1 to be true of the model parameter mβ by fitting with dΓ =dK .We 10 meV, corresponding to mβ ¼ 13.4 meV. The energy eff e χ2 Δχ2 resolution is fixed to 0.1 eV. The dark red curve present β and true in Fig. 6 as a function of the exposure represents the result obtained by fitting with the exact E and the energy resolution Δ. We fix the lightest neutrino Γ0 χ2 spectrum d cl=dKe, while the dark blue one corresponds mass as 0 eV for these plots, as β is maximized in this case Γ to the fit by using the effective spectrum d eff=dKe,given according to Fig. 4. theexposureofE ¼ 100 g · yr. The light curves stand for The experimental configurations of KATRIN, Project 8, the case with E ¼ 1 g · yr. One can observe that if the and PTOLEMY have been indicated in Fig. 6, and their 2 effective spectrum with mβ is used, the best-fit value of corresponding χ values have been explicitly summarized bf m1 is found to be m1 ¼ 9.6 meV, which deviates notably in Table I. For PTOLEMY, the effective beta spectrum can bf from m1 ¼ 10 meV obtained by using the exact spec- no longer be adopted. The use of the effective spectrum trum. Even for the exposure of E ¼ 1 g · yr, the true with mβ would result in a huge error in fitting the neutrino value is outside of Δχ2 ≲ 4 when fitting with the effective mass compared to the precision that is supposed to be Δχ2 141 spectrum. The situation becomes worse if we take a achieved in such an experiment, e.g., true ¼ for NO Δχ2 81 larger exposure. and true ¼ for IO for 1 year of data taking. For To systematically study how far the parameter value KATRIN and Project 8, with 1 year of exposure, the Γ 2 fitted by using d eff=dKe can deviate from the true one, we effective mass mβ is fortunately applicable with χβ, 2 define the following difference of χ : Δχ2 ≲ 0 1 true . . Note that there is a little risk for Project 8 loaded with the atomic tritium. To be more specific, in Δχ2 ≡ χ2 true − χ2 bf true βðmβ ¼ mβ Þ βðmβ ¼ mβ Þjmin; ð20Þ the extreme case that the data taking time is set to 10 years and an improvement on the energy resolution is made to 2 true 2 true where χβðmβ ¼ mβ Þ is the χ value when mβ is set to mβ Δ 0 03 Δχ2 ¼ . eV, true for NO can be as large as 1, indicating Γ χ2 bf when fitting with d eff=dKe, and βðmβ ¼ mβ Þjmin is the that the true value of mβ is out of the 1σ CL region by χ2 Γ minimum value of the curve obtained by freely adjusting fitting with the effective spectrum d eff=dKe; hence, the
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χ2 Δχ2 FIG. 6. The contours of β and true, see Eqs. (19) and (20), arising from the description of the electron spectrum by using the effective neutrino mass mβ, are displayed in the exposure-resolution (E − Δ) plane for the NO case (left two panels) and the IO case (right two panels). The lightest neutrino mass is fixed to 0 eV. description by using the effective spectrum would not be the neutrino sector. For the similar analysis relevant for appropriate anymore. the effective neutrino masses in β and 0νββ decays, see Refs. [37–43]. Here we perform an updated analysis for the III. POSTERIOR DISTRIBUTIONS direct neutrino mass experiments, in light of a good number of experimental achievements. As we have already demonstrated in the previous Γ As usual, two important ingredients for the Bayesian section, the effective spectrum d eff=dKe cannot be used analysis should be specified. First, we have to choose the for PTOLEMY, but is safe to use in the KATRIN and prior distributions for the relevant model parameters Project 8 experiments. Following the Bayesian statistical approach [36], we derive in this section the posterior 2θ 2θ Δ 2 Δ 2 ρ σ M fsin 12; sin 13; msol; matm; ; ;G0ν; j 0νj;mLg; distributions of the effective neutrino mass mβ, based on current experimental information from neutrino oscilla- ð21Þ tions, beta decay, neutrinoless double-beta decay (0νββ), Δ 2 Δ 2 Δ 2 Δ 2 Δ 2 and cosmology. Since the description of the beta spectrum where msol ¼ m21 and matm ¼ m31 (or m32)in via the effective neutrino mass is still valid for KATRIN the NO (or IO) case. For all oscillation parameters 2 θ 2 θ Δ 2 Δ 2 and Project 8, posterior distributions of the effective fsin 12; sin 13; msol; matmg, we assume that they neutrino mass should be very suggestive for future experi- are uniformly distributed in the ranges that are wide enough ments. Our results in this section can also be used for the to cover their experimentally allowed values. For the electron-capture experiments ECHo [14], HOLMES [15], absolute neutrino mass scale, which is represented by and NuMECS [16],ifCPT is assumed to be conserved in the lightest neutrino mass mL (i.e., m1 in the NO case or
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FIG. 7. The posterior distributions of the effective neutrino mass mβ in the NO (upper row) and IO (lower row) cases, given a flat prior on the lightest neutrino mass mL. In each of the four subfigures, the upper subgraph shows the posterior distributions whereas the lower subgraph gives the accumulative distributions. The results for four different combinations of experimental information have ð1Þ ð2Þ been displayed in the left column: (i) Losc þ Lβ (orange curves); (ii) Losc þ Lβ þ Lcosmo (red curves); (iii) Losc þ Lβ þ Lcosmo (blue ð3Þ curves); (iv) Losc þ Lβ þ Lcosmo (green curves), while the data from 0νββ using L0νββ are further included in the right column. The ðiÞ cosmological bounds on the sum of three neutrino masses corresponding to Lcosmo (for i ¼ 1, 2, 3) have been summarized in Eq. (B5). The latest result mβ < 1.1 eV from KATRIN is denoted as the vertical solid line, and future sensitivities of KATRIN and Project 8 are represented by two vertical dashed lines.
m3 in the IO case), we consider the following two and motivated by the approximately constant ratios ∼ ∼ possible priors: [44] of charged fermion masses mu=mc mc=mt λ2 ∼ ∼ λ ∼ ∼ λ2 (i) A flat prior on the logarithm of mL in the range of , md=ms ms=mb , and me=mμ mμ=mτ 10−7 10 ∈ −7 1 λ θ ≈ 0 22 ð Þ eV, namely, Log10ðmL=eVÞ ½ ; , (where ¼ sin C . is the Wolfenstein param- which will be referred to as the log prior in the eter), as well as by the in general exponential following discussion. This prior is scale invariant fermion mass hierarchies. Note that an ad hoc lower
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FIG. 8. The posterior distributions of the effective neutrino mass mβ in the NO (upper row) and IO (lower row) cases, given a log prior on the lightest neutrino mass mL. In each of the four subfigures, the upper subgraph shows the posterior distributions whereas the lower subgraph gives the accumulative distributions. The results for four different combinations of experimental information have been L L L L Lð1Þ L L Lð2Þ displayed in the left column: (i) osc þ β (orange curves); (ii) osc þ β þ cosmo (red curves); (iii) osc þ β þ cosmo (blue L L Lð3Þ 0νββ L curves); (iv) osc þ β þ cosmo (green curves), while the data from using 0νββ are further included in the right column. The ðiÞ cosmological bounds on the sum of three neutrino masses corresponding to Lcosmo (for i ¼ 1, 2, 3) have been summarized in Eq. (B5). The latest result mβ < 1.1 eV from KATRIN is denoted as the vertical solid line, and future sensitivities of KATRIN and Project 8 are represented by two vertical dashed lines.
10−7 cutoff eV for mL has been imposed, which is pheaviestffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi neutrino mass is rather small, at most 2 2 necessary to bound the prior volume from below. Δm31=Δm21 ≈ 5 for NO and essentially 1 for Decreasing this cutoff is equivalent to putting more IO, motivating a moderate and nonexponential and more prior volume to very small and essentially ordering of neutrino masses. vanishing values of mL. Without a complete theory for neutrino mass generation, 0…10 (ii) A flat prior on mL in the range of ð Þ eV. we cannot judge which prior is favorable and therefore shall Note that the ratio of the heaviest to the second- treat both of them on equal footing. The prior dependence
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TABLE II. The volume fraction of the mβ posterior covered by KATRIN with a sensitivity of mβ ≃ 0.2 eV and Project 8 with a sensitivity of mβ ≃ 0.04 eV, respectively. A flat and logarithmic prior on the lightest neutrino mass has been assumed for the upper and lower tables, respectively.
ð1Þ ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ Flat prior Losc þ Lβ þLcosmo þLcosmo þLcosmo þL0νββ þL0νββ þ Lcosmo þL0νββ þ Lcosmo þL0νββ þ Lcosmo KATRIN, NO 73% 4.2% 6.9 × 10−5 <10−11 4.7% 0.23% 5.2 × 10−6 <10−11 KATRIN, IO 74% 4.9% 1.2 × 10−4 <10−11 6.5% 0.45% 1.2 × 10−5 <10−11 Project 8, NO 96% 60% 35% 6% 67% 43% 28% 4.9% Project 8, IO 100% 100% 100% 100% 100% 100% 100% 100% L L Lð1Þ Lð2Þ Lð3Þ L L Lð1Þ L Lð2Þ L Lð3Þ Log prior osc þ β þ cosmo þ cosmo þ cosmo þ 0νββ þ 0νββ þ cosmo þ 0νββ þ cosmo þ 0νββ þ cosmo KATRIN, NO 7.2% 6.8 × 10−4 7.9 × 10−7 <10−11 0.15% 3.7 × 10−5 6.4 × 10−8 <10−11 KATRIN, IO 7.1% 9.3 × 10−4 1.4 × 10−6 <10−11 0.19% 6.6 × 10−5 1.7 × 10−7 <10−11 Project 8, NO 17% 3.5% 1.6% 0.17% 6.0% 2.3% 1.2% 0.15% Project 8, IO 100% 100% 100% 100% 100% 100% 100% 100%
of the final posterior distributions reflects that current shown in Figs. 7 and 8, respectively. A summary of the experimental knowledge on the absolute scale of neutrino volume fractions of mβ posteriors covered by future masses is still very poor. If one attempts to set limits on KATRIN and Project 8 sensitivities has been presented model parameters, a prior-independent approach may be in Table II. Some comments on the numerical results are found in Ref. [45]. in order. The likelihood functions for each type of experiments (1) In Fig. 7, a flat prior on mL is assumed. The plots in can be found in Appendix B. Briefly speaking, the global- the first row are for the NO case, whereas those in fit results of all neutrino oscillation data from Ref. [34] the second row are for the IO case. In each row, the will be used to construct the likelihood function upper subgraph in the left column shows the L 2θ 2θ Δ 2 Δ 2 oscðsin 12; sin 13; msol; matmÞ. The tritium beta- posterior distributions of the effective mass in decay experiments Mainz [12], Troitsk [13], and four different scenarios of adopted experimental L L KATRIN [9] are taken into account and the likelihood information: (i) osc þ β for the neutrino oscilla- 2 i Lβ m m ; L L Lð Þ function ð βÞ involves the model parameters f L tion and beta-decay data and (ii) osc þ β þ cosmo 2 θ 2 θ Δ 2 Δ 2 0νββ 1 sin 12; sin 13; msol; matmg. As for experiments, (for i ¼ , 2, 3) for a further inclusion of cosmo- the likelihood function L0νββðmββ;G0ν; jM0νjÞ actually logical upper bounds on the sum of three neutrino contains all the parameters in Eq. (21). For the two masses. The lower subgraph gives the accumulative Majorana CP phases ρ and σ relevant for 0νββ experi- posterior distributions,R which are defined as 0 ∞ ments, we shall take flat priors in the range of ½0…2πÞ,as Pðmβ >mβÞ¼ 0 ðdP=dmβÞdmβ. In addition, mβ there is currently no experimental constraint on them. For the plots in the right column differ from those in the phase space factor G0ν, a Gaussian prior is assumed the left column only by including the experimental 1σ with the central value and error available from Ref. [46]. information on 0νββ. Hence, if neutrinos are Dirac The nuclear matrix elements M0ν take a flat prior in the j j particles, the results including L0νββ do not apply. range spanned by the predictions from different NME The future sensitivities of KATRIN [11] (0.2 eV) models [39]. Finally, the upper bound on the sum of three and Project 8 [17] (40 meV) are shown as the upper Σ neutrino masses ¼ m1 þ m2 þ m3 from cosmological and lower dashed boundaries of the gray bands. This observations will be implemented, and the corresponding gray region represents the gradual improvement LðiÞ Δ 2 Δ 2 likelihood function cosmo depends on fmL; msol; matmg, of the sensitivities. In Fig. 8, the same computations 1 where i ¼ , 2, 3 refers, respectively, to the Planck data on have been carried out for the log prior on mL, where the cosmic microwave background, its combination with all notations follow those of Fig. 7. gravitational lensing data, and their further combination (2) By comparing among the different scenarios in both with baryon acoustic oscillation data, as explained in Figs. 7 and 8, we can make the following important Appendix B. observations: With the priors of model parameters and the likelihood (a) Let us first focus on the impact of 0νββ. If the functions from the relevant experiments, we can compute cosmological observations, namely, the upper the posterior distribution of mβ, i.e., dP=dmβ, in the bounds on neutrino masses, are not considered, standard way of Bayesian analysis. The sampling is done then one can make a comparison between the L L with the help of the MultiNest routine [47–49]. The orange curves (corresponding to osc þ β)in numerical results for the flat and log priors on mL are the left column and those (corresponding to
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L L L osc þ β þ 0νββ) in the right column. It is relativistic one, and its difference to the effective electron Γ evident that the experimental constraints from spectrum d eff=dKe in Eq. (12) which includes the usually 0νββ decays lead to a significant shift of the considered effective neutrino mass mβ or its variants. posterior distribution to the region of smaller Moreover, based on current experimental information from values of mβ. Even in this case, it is very likely neutrino oscillation data, tritium beta decays, neutrinoless that the future beta-decay experiments can de- double-beta decays, and cosmology, we have computed the termine the absolute neutrino mass, no matter posterior distributions of the effective neutrino mass mβ in whether NO or IO is true. For instance, in Fig. 7, Eq. (1). Our main results are summarized as follows. where the flat prior on mL is assumed, Project 8 First, for tritium beta decays, the classical electron Γ can cover 67% of the posteriors in the NO case. spectrum d cl=dKe can be modified by replacing mi with (b) One should investigate what role is played by the mi · ðm3He=m3HÞ to account for the exact electron spectrum cosmological observations. For this purpose, we including relativistic corrections. In this case, the difference concentrate on the plots in the right columns of between the exact relativistic spectrum and the modified Figs. 7 and 8. When the cosmological observa- Γ0 classical spectrum d cl=dKe can be safely ignored, as the tions are considered, one can see that the dominant uncertainties in the measurements at KATRIN, probability of discovering a nonzero effective Project 8, and PTOLEMY arise from the statistical data neutrino mass in beta-decay experiments drops fluctuations. Furthermore, it is interesting to compare the dramatically. In the worst situation, where the exact spectrum with the effective one containing the usually L L Lð3Þ L likelihood set osc þ β þ cosmo þ 0νββ is considered observable mβ. However, as we have demon- taken in the NO case, even Project 8 can only strated in a quantitative way, the validity of the effective cover 4.9% of the posterior. Therefore, the mass mβ actually depends on the energy resolution and the detection of a positive signal in this case would total exposure of a realistic beta-decay experiment. We imply a tension between the beta-decay experi- show that the use of the standard effective neutrino mass ments and cosmological observations. for KATRIN and Project 8 is justified. For the future 100 (c) For the log prior on mL in Fig. 8, it is easy to PTOLEMY experiment with an exposure of g · yr, recognize that the posterior is remarkably it will be problematic to introduce an effective neutrino reduced in the region of effective neutrino masses mass, and the lightest neutrino mass should be used covered by the KATRIN and Project 8 experi- together with the exact spectrum. While this is known, ments. Compared to the flat prior, this can be we have performed here a general analysis with keeping the interpreted as a consequence of the fact that a exposure and energy resolution as free parameters. large fraction of the prior space has been distrib- Second, as we have mentioned above, it is justified to Γ0 uted in the neighborhood of a nearly vanishing describe the exact electron spectrum d cl=dKe by the mL. For the NO case, the effective neutrino mass effective one with the effective neutrino mass mβ in the −3 mβ takes the minimum value ∼8.9 × 10 eV as KATRIN and Project 8 experiments. Therefore, it does make → 0 mL ¼ m1 eV. The beta-decay experiments sense to derive the posterior distributions of the effective like KATRIN and Project 8 are still far from neutrino mass, given the latest experimental data on neutrino achieving this sensitivity. oscillations, beta decays, neutrinoless double-beta decays, Regardless of the prior and likelihood choices, Project 8 and cosmological observations. Although the cosmological can always cover all the posteriors of the IO case. In this upper bound on the sum of three neutrino masses pushes the connection, the discrimination between NO and IO seems posterior distribution of mβ down to the region almost to be very promising in future beta-decay experiments [50], outside of the sensitivity of Project 8 in the NO case, it e.g., an explicit study of the sensitivity has already been does not affect much the situation in the IO case due to the performed for PTOLEMY in Ref. [20]. lower bound on mβ ≳ 50 meV even in the limit of m3 → 0. This also implies that future tritium beta-decay experiments IV. SUMMARY are able to discriminate between neutrino mass orderings. As KATRIN continues to accumulate more beta-decay The determination of absolute neutrino masses is exper- events and the development of the techniques to be imentally challenging, but scientifically very important. As deployed in Project 8 and PTOLEMY is well in progress, fundamental parameters in nature, absolute neutrino masses it is timely and necessary to revisit the effective neutrino must be precisely measured in order to explore the origin of mass and its validity in future beta-decay experiments. The neutrino masses, which calls for new physics beyond the analysis presented in the present work should be helpful in standard model. Motivated by the latest result from the understanding the approximations made in expressions of KATRIN experiment and upcoming tritium beta-decay the beta spectrum and is suggestive for the improvement on searches, we have performed a detailed study of the exact the usage of the effective masses. In light of the precision Γ0 electron spectrum d cl=dKe in Eq. (9), which is a modified measurement of the beta spectrum already in the first run of
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KATRIN, one may go further to extend the analysis to can be inferred from the formula of the tritium ϵ ρ ≈ 2 518 consider the presence of sterile neutrinos and other new molecule number NðT2Þ¼AS · T · d . × 1019 53 2 physics, and/or to consider the electron-capture decay with the source cross section AS ¼ cm , 163 ϵ 0 95 of Ho. the tritium purity T ¼ . , and the column density ρd ¼ 5 × 1017 cm−2; see Eq. (25) and Table 7 of ACKNOWLEDGMENTS Ref. [11] for details. Given the mass per tritium ∼5 10−24 The authors would like to thank Yu-Feng Li and nucleus × g, we obtain the total target 2 5 10−4 Prof. Zhi-zhong Xing for useful discussions. This work mass of the full KATRIN as mKATRIN ¼ . × g. was supported in part by the National Natural Science The energy resolution of KATRIN in this work is Δ 1 Foundation of China under Grants No. 11775232 and fixed to KATRIN ¼ eV. No. 11835013 and by the CAS Center for Excellence in (ii) Unlike the KATRIN experiment, the Project 8 Particle Physics. W. R. is supported by the DFG with Grant Collaboration will utilize the technique of cyclotron No. RO 2516/7-1 in the Heisenberg program. radiation emission spectroscopy to measure the electron energies [17]. If the magnetic field is APPENDIX A: EXPERIMENTAL SETUPS uniform in the spectrometer, the cyclotron radiation of accelerating electrons can be observed for a few Some experimental details about KATRIN, Project 8 and microseconds and its frequency can be precisely PTOLEMY experiments are as follows: determined, leading to an excellent energy resolu- (i) The KATRIN experiment [11] implements the tion. As has already been shown in Fig. 5 of so-called MAC-E-Filter (Magnetic Adiabatic Colli- Ref. [17], with the deployment of Oð10−4Þ g atomic mation combined with an Electrostatic Filter) to 3H and 1 year of running time, Project 8 is able to select electrons from tritium beta decays that can push the upper limit on the effective neutrino mass pass through the electrostatic barrier with the po- down to mβ < 40 meV at the 90% CL, assuming the tential energy of E . The observable in the MAC-E- V true value of mβ ¼ 0. In this work, we will adopt two Filter is the integrated number of the electrons extreme setups for Project 8: (i) an intermediate that have passed through the energy barrier. The 3 phase with the molecular H, a target mass of mP8 ¼ sharpness of the filter is characterized by the ratio 5 10−4 5 1019 3 10−4 × g corresponding to × tritium mol- between the minimum Bmin ¼ × T and the ecules, and an energy resolution Δ 8 ¼ 0.36 eV maximum B ¼ 6 T of magnetic fields, i.e., Δ ¼ P m max limited by the irreducible width of the final state E B =B ≈ 1 eV, where E ≈ Q ¼ 18.6 keV is e min max e molecular excitations [35]; (ii) an ultimate phase with the electron energy in the range close to the end 3 −4 the atomic H, a target mass of m 8 5×10 g, and ≡ 3 − 3 − P ¼ point and Q m H m He me is the Q value for an energy resolution of Δ 8 ¼ 0.05 eV which is tritium beta decay. Since the filter is insensitive to P a limited by the inhomogeneity of the magnetic field the transverse kinetic energy of electrons, the sharp- Δ ∼ 10−7 7 B=B [17]. The target mass mP8 ¼ ness denotes roughly the maximum of transverse 5 10−4 kinetic energies and thus can be regarded as the × g can be achieved with a gas volume of 100 m2 as required by the phase IVof Project 8 and a energy resolution. Adopting the energy window 12 −3 ∈ − 30 5 gaseous tritium number density of 10 cm . EV ½Q eV;Qþ eV and including all the statistic and systematic uncertainties, the KATRIN (iii) The PTOLEMY experiment has been designed ν experiment [11] with a target tritium mass of to detect the cosmic neutrino background (C B) O 10−4 2 1σ [18–20] via the electron-neutrino capture on tritium ð Þ g can measure mβ with a uncertainty 3 − 3 2 ν þ H → e þ He, as suggested by Steven Wein- of 0.025 eV , corresponding to the sensitivity of e berg in 1962 [51]. Thanks to the large target mass of mβ < 0.2 eV at the 90% CL in the assumption of 0 100 g tritium and the low background rate required mβ ¼ as the true value. KATRIN can also directly for the CνB detection, PTOLEMY would have an measure the nonintegrated beta spectrum by overwhelmingly better sensitivity to the absolute extracting the time of flight information from the neutrino mass than KATRIN does, namely, the source to the detector, operating in the so-called −2 relative uncertainty reaches σðm1Þ=m1 ≲ 10 for MAC-E-TOF mode. Since the emitting time of the m1 ¼ 10 meV with an energy resolution of Δ ¼ electron at source is not directly measurable, a 100 meV. In the PTOLEMY experiment, the energy technique has been devised to infer the emitting time by chopping the source with some high voltage
potential frequently. A lower counting rate and a 7 worse energy resolution will be caused by the The energy resolution of Project 8 with atomic tritium may be roughly obtained by the relation ΔE=me ¼ Δf=f ≈ ΔB=B, additional chopping procedure. The total target mass where f is the frequency of the cyclotron radiation and B is the of 3H planned to be loaded in the full KATRIN setup assumed nearly uniform magnetic field.
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of electrons from tritium beta decays will be for IO. The preference of NO over IO can be measured in three steps. First, the MAC-E-Filter represented by the difference of their χ2 minima, Δχ2 χ2 − χ2 ≃ 9 3 is used to select the electrons close to the end point, i.e., MO ¼ NO IO . , implying a more than preventing the calorimeter from being swamped by 3σ preference of NO. the huge number of events in the energy range below (ii) Lβ—By measuring the end point of the β decay the end point. Second, after passing through the spectrum, the tritium β-decay experiments (e.g., MAC-E-Filter, the electrons are then sent to a long Troitsk [13], Mainz [12], and KATRIN [9,10]) uniform solenoid, undergoing the cyclotron motion can already provide us good constraints on the in the magnetic field of 2 T. Hence, the radio signal absolute neutrinoP mass scale via the effective neu- can be implemented to track each single electron. ≡ 2 2 1=2 trino mass mβ ð i mi jUeij Þ . The limits of the Finally, the electrons are decelerated by the electro- former two are given as static voltage until their kinetic energies are 100 eV or so to match the dynamic range of a cryogenic 2 2 mβ ¼ −0.67 2.53 eV ðTroitskÞ; calorimeter. The energy resolution of these electrons 2 2 can be as low as 50 meV [20]. mβ ¼ −0.6 3.0 eV ðMainzÞ: ðB2Þ
Similar to Eq. (B1) in the case of neutrino oscil- APPENDIX B: EXPERIMENTAL LIKELIHOODS lations, the likelihood function can be constructed with the central values and 1σ errors of m2 in The likelihood functions from the following different β classes of experiments have been used in our analysis: Eq. (B2). For KATRIN, we use the likelihood (i) oscillation experiments L ; (ii) β-decay experiments presented in Fig. 4 of Ref. [9]. We find the likelihood osc can be well approximated by a skewed normal Lβ; (iii) 0νββ experiments L0νββ; (iv) cosmological obser- L distribution vations cosmo. To be specific, we collect all the details of each likelihood function as follows: 1 ðm2 − μÞ2 (i) L —The likelihood information of neutrino oscil- L 2 ∝ ffiffiffiffiffiffi − β osc KATRINðmβÞ p exp 2 lation experiments will be taken from the latest 2πσ 2σ global-fit results of the Nu-Fit group [34]. The 2 αðmβ − μÞ L × erfc − pffiffiffi ; ðB3Þ likelihood function can be obtained as osc ¼ 2σ exp ð−Δχ2=2Þ with Δχ2 defined as where erfcðxÞ is the complementary error function, X Θ − Θbf 2 Δχ2 ≡ ð i i Þ with σ ¼ 1.506 eV2, μ ¼ 0.0162 eV2, and m2 in 2 ; ðB1Þ β σ 2 i i units of eV , as well as α ¼ −2.005. Since the KATRIN experiment has the highest sensitivity to Θ ∈ 2θ 2θ Δ 2 Δ 2 Θbf L ≈ L where i fsin 13; sin 12; msol; matmg, i is mβ, we may have β KATRIN. the best-fit value of the parameter from the global (iii) L0νββ—The constraints on the half-life of 0νββ are σ 1σ analysis, and i is the symmetrized error. We take given by the existing 0νββ searches. The limits 1σ the following central values and symmetrized on the effective neutrino mass jmββj can be derived β errors of oscillation parameters relevant for the by using decays: 2 jmββj 2 −1 0ν −1 M 2 sin θ12 ¼ð3.10 0.12Þ × 10 ; ðT1=2Þ ¼ G0νj 0νj 2 ; ðB4Þ me Δ 2 7 39 0 20 10−5 2 msol ¼ð . . Þ × eV ; 2 −2 where G0ν denotes the phase-space factor, M0ν is sin θ13 ¼ð2.241 0.065Þ × 10 ; the nuclear matrix element (NME), and me ¼ Δ 2 2 525 0 032 10−3 2 0 511 matm ¼ð . . Þ × eV . MeV is the electron mass. In our numerical analysis, we use the likelihood functions from for NO, and Refs. [39,52], which include the experimental in- formation of GERDA [53], KamLAND-Zen [54], 2 −1 sin θ12 ¼ð3.10 0.12Þ × 10 ; EXO [55], and CUORE [52]. (iv) L —The cosmological observations can set Δ 2 7 39 0 20 10−5 2 cosmo msol ¼ð . . Þ × eV ; very strong constraints on the sum of the three 2 −2 Σ ≡ sin θ13 ¼ð2.264 0.066Þ × 10 ; neutrino masses m1 þ m2 þ m3. The Planck Collaboration has recently updated their results Δ 2 −2 512 0 033 10−3 2 matm ¼ð . . Þ × eV in Ref. [56]. For illustration, we will adopt the
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likelihood functions by combining different datasets which yield the following bounds on the sum of the three neutrino masses at the 95% CL:
ð1Þ Σ < 0.54 eVðLcosmo; Planck TT þ lowEÞ; ð2Þ Σ < 0.24 eVðLcosmo; Planck TT; TE; EE þ lowE þ lensingÞ; ð3Þ Σ < 0.12 eVðLcosmo; Planck TT; TE; EE þ lowE þ lensing þ BAOÞ: ðB5Þ The likelihood functions have been obtained by analyzing the Markov chain files available from the Planck Legacy Archive. L With the likelihood functions listed above, the total likelihood relevant for our analysis can be calculated as tot ¼ L L L LðiÞ 1 osc × β × 0νββ × cosmo (for i ¼ , 2, 3).
[1] E. W. Otten and C. Weinheimer, Neutrino mass limit from production options for neutrino mass measurements with tritium beta decay, Rep. Prog. Phys. 71, 086201 (2008). microcalorimeter detectors, Nucl. Instrum. Methods B 311, [2] M. J. Dolinski, A. W. P. Poon, and W. Rodejohann, Neu- 131 (2013). trinoless double-beta decay: Status and prospects, Annu. [17] A. Ashtari Esfahani et al. (Project 8 Collaboration), De- Rev. Nucl. Part. Sci. 69, 219 (2019). termining the neutrino mass with cyclotron radiation emis- [3] M. Lattanzi and M. Gerbino, Status of neutrino properties sion spectroscopy-Project 8, J. Phys. G 44, 054004 (2017). and future prospects—Cosmological and astrophysical con- [18] S. Betts et al., Development of a relic neutrino detection straints, Front. Phys. 5, 70 (2018). experiment at PTOLEMY: Princeton tritium observatory for [4] E. Fermi, An attempt of a theory of beta radiation. 1., light, early-universe, massive-neutrino yield, in Proceed- Z. Phys. 88, 161 (1934). ings, 2013 Community Summer Study on the Future of U.S. [5] E. Fermi, Trends to a theory of beta radiation. (In Italian), Particle Physics: Snowmass on the Mississippi (CSS2013): Nuovo Cimento 11, 1 (1934). Minneapolis, MN, USA, 2013 (2013), https://www.slac [6] F. Perrin, Possibility of emission of neutral particles with .stanford.edu/econf/C1307292/docs/submittedArxivFiles/ zero intrinsic mass in beta radioactivity. (In French), 1307.4738.pdf. Comptes Rendus 197, 1625 (1933). [19] E. Baracchini et al. (PTOLEMY Collaboration), PTOLEMY: [7] R. E. Shrock, New tests for, and bounds on, neutrino masses A proposal for thermal relic detection of massive neutrinos and lepton mixing, Phys. Lett. 96B, 159 (1980). and directional detection of MeV dark matter, arXiv: [8] M. Tanabashi et al. (Particle Data Group), Review of 1808.01892. particle physics, Phys. Rev. D 98, 030001 (2018). [20] M. G. Betti et al. (PTOLEMY Collaboration), Neutrino [9] M. Aker et al. (KATRIN Collaboration), An Improved physics with the PTOLEMY project: Active neutrino Upper Limit on the Neutrino Mass from a Direct Kinematic properties and the light sterile case, J. Cosmol. Astropart. Method by KATRIN, Phys. Rev. Lett. 123, 221802 (2019). Phys. 07 (2019) 047. [10] M. Aker et al. (KATRIN Collaboration), First operation of [21] S. S. Masood, S. Nasri, J. Schechter, M. A. Tortola, J. W. F. the KATRIN experiment with tritium, arXiv:1909.06069. Valle, and C. Weinheimer, Exact relativistic beta decay [11] J. Angrik et al. (KATRIN Collaboration), KATRIN design endpoint spectrum, Phys. Rev. C 76, 045501 (2007). report 2004, https://www.katrin.kit.edu/publikationen/ [22] F. Simkovic, R. Dvornicky, and A. Faessler, Exact relativ- DesignReport2004-12Jan2005.pdf. istic tritium beta-decay endpoint spectrum in a hadron [12] C. Kraus et al., Final results from phase II of the Mainz model, Phys. Rev. C 77, 055502 (2008). neutrino mass search in tritium beta decay, Eur. Phys. J. C [23] A. J. Long, C. Lunardini, and E. Sabancilar, Detecting non- 40, 447 (2005). relativistic cosmic neutrinos by capture on tritium: Phe- [13] V. N. Aseev et al. (Troitsk Collaboration), An upper limit nomenology and physics potential, J. Cosmol. Astropart. on electron antineutrino mass from Troitsk experiment, Phys. 08 (2014) 038. Phys. Rev. D 84, 112003 (2011). [24] P. O. Ludl and W. Rodejohann, Direct neutrino mass [14] L. Gastaldo et al., The electron capture in 163Ho experi- experiments and exotic charged current interactions, J. High ment–ECHo, Eur. Phys. J. ST 226, 1623 (2017). Energy Phys. 06 (2016) 040. [15] B. Alpert et al., HOLMES—the electron capture decay of [25] F. Vissani, Nonoscillation searches of neutrino mass in the age 163Ho to measure the electron neutrino mass with sub-eV of oscillations, Nucl. Phys. B, Proc. Suppl. 100, 273 (2001). sensitivity, Eur. Phys. J. C 75, 112 (2015). [26] A. Saenz, S. Jonsell, and P. Froelich, Improved Molecular [16] J. W. Engle, E. R. Birnbaum, H. R. Trellue, K. D. John, Final-State Distribution of HeT þ for the β-Decay Process M. W. Rabin, and F. M. Nortier, Evaluation of 163Ho of T2, Phys. Rev. Lett. 84, 242 (2000).
016003-19 HUANG, RODEJOHANN, and ZHOU PHYS. REV. D 101, 016003 (2020)
[27] O. Fackler, B. Jeziorski, W. Kolos, H. J. Monkhorst, and K. [42] G.-Y. Huang and S. Zhou, Impact of an eV-mass sterile Szalewicz, Accurate Theoretical Beta Decay Energy Spec- neutrino on the neutrinoless double-beta decays: A Baye- trum of the Tritium Molecule and its Neutrino Mass sian analysis, Nucl. Phys. B945, 114691 (2019). Dependence, Phys. Rev. Lett. 55, 1388 (1985). [43] S. Dell’Oro, S. Marcocci, and F. Vissani, Empirical infer- [28] N. Doss, J. Tennyson, A. Saenz, and S. Jonsell, Molecular ence on the Majorana mass of the ordinary neutrinos, Phys. effects in investigations of tritium molecule beta decay Rev. D 100, 073003 (2019). endpoint experiments, Phys. Rev. C 73, 025502 (2006). [44] Z.-z. Xing, Flavor structures of charged fermions and [29] L. I. Bodine, D. S. Parno, and R. G. H. Robertson, Assess- massive neutrinos, arXiv:1909.09610. ment of molecular effects on neutrino mass measurements [45] S. Gariazzo, Constraining power of open likelihoods, made from tritium β decay, Phys. Rev. C 91, 035505 (2015). prior-independent, arXiv:1910.06646. [30] Y. Farzan, O. L. G. Peres, and A. Yu. Smirnov, Neutrino [46] J. Kotila and F. Iachello, Phase space factors for double-β mass spectrum and future beta decay experiments, Nucl. decay, Phys. Rev. C 85, 034316 (2012). Phys. B612, 59 (2001). [47] F. Feroz and M. P. Hobson, Multimodal nested sampling: [31] J. Studnik and M. Zralek, Parametrization of the energy An efficient and robust alternative to MCMC methods for spectrum in the tritium beta decay, arXiv:hep-ph/0110232. astronomical data analysis, Mon. Not. R. Astron. Soc. 384, [32] Y. Farzan and A. Yu. Smirnov, On the effective mass of the 449 (2008). electron neutrino in beta decay, Phys. Lett. B 557, 224 [48] F. Feroz, M. P. Hobson, and M. Bridges, MultiNest: An (2003). efficient and robust Bayesian inference tool for cosmology [33] J. F. Beacom and N. F. Bell, Do solar neutrinos decay?, and particle physics, Mon. Not. R. Astron. Soc. 398, 1601 Phys. Rev. D 65, 113009 (2002). (2009). [34] I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez- [49] F. Feroz, M. P. Hobson, E. Cameron, and A. N. Pettitt, Cabezudo, M. Maltoni, and T. Schwetz, Global analysis Importance nested sampling and the multinest algorithm, of three-flavour neutrino oscillations: Synergies and ten- arXiv:1306.2144. sions in the determination of θ23, δCP, and the mass [50] S. M. Bilenky, M. D. Mateev, and S. T. Petcov, A comment ordering, J. High Energy Phys. 01 (2019) 106. on the measurement of neutrino masses in beta-decay [35] B. Monreal and J. A. Formaggio, Relativistic cyclotron experiments, Phys. Lett. B 639, 312 (2006). radiation detection of tritium decay electrons as a new [51] S. Weinberg, UnCiversal neutrino degeneracy, Phys. Rev. technique for measuring the neutrino mass, Phys. Rev. D 80, 128, 1457 (1962). 051301 (2009). [52] C. Alduino et al. (CUORE Collaboration), First Results [36] R. Trotta, Bayes in the sky: Bayesian inference and model from CUORE: A Search for Lepton Number Violation selection in cosmology, Contemp. Phys. 49, 71 (2008). via 0νββ Decay of 130Te, Phys. Rev. Lett. 120, 132501 [37] G. Benato, Effective majorana mass and neutrinoless double (2018). beta decay, Eur. Phys. J. C 75, 563 (2015). [53] M. Agostini et al., Background-free search for neutrinoless [38] A. Di Iura and D. Meloni, Probability densities of the double-β decay of 76Ge with GERDA, Nature (London) effective neutrino masses m β and m ββ, Nucl. Phys. B921, 544, 47 (2017). 829 (2017). [54] A. Gando et al. (KamLAND-Zen Collaboration), Search [39] A. Caldwell, A. Merle, O. Schulz, and M. Totzauer, Global for Majorana Neutrinos near the Inverted Mass Hierarchy Bayesian analysis of neutrino mass data, Phys. Rev. D 96, Region with KamLAND-Zen, Phys. Rev. Lett. 117, 082503 073001 (2017). (2016); Addendum, Phys. Rev. Lett.117, 109903 (2016). [40] S.-F. Ge, W. Rodejohann, and K. Zuber, Half-life expect- [55] J. B. Albert et al. (EXO-200 Collaboration), Search for ations for neutrinoless double beta decay in standard and Majorana neutrinos with the first two years of EXO-200 non-standard scenarios, Phys. Rev. D 96, 055019 (2017). data, Nature (London) 510, 229 (2014). [41] M. Agostini, G. Benato, and J. Detwiler, Discovery prob- [56] N. Aghanim et al. (Planck Collaboration), Planck 2018 ability of next-generation neutrinoless double-β decay ex- results. VI. Cosmological parameters, arXiv:1807.06209. periments, Phys. Rev. D 96, 053001 (2017).
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