A Comment on the Measurement of Neutrino Masses in Beta-Decay Experiments

We discuss the physics potential of future tritium beta-decay experiments having a sensitivity to a neutrino mass ~\sqrt{|\Delta m^{2}_{23}|}~5x10^{-2} eV. The case of three-neutrino mixing is analised. A negative result of such an experiment would imply that the neutrino mass spectrum is of normal hierarchical type. The interpretation of a positive result would depend on the value of the lightest neutrino mass; if the lightest neutrino mass satisfies the inequality min(m_j)<<\sqrt{|\Delta m^{2}_{23}|}, it would imply that the neutrino mass spectrum is of the inverted hierarchical type.


Introduction
The discovery of neutrino oscillations in the experiments with solar, atmospheric and reactor neutrinos [1,2,3,4,5,6,7,8] is the first particle physics evidence for existence of a new beyond the Standard Model physics. The solar, atmospheric, reactor and K2K neutrino data imply the presence of 3-ν mixing in the weak charged lepton current: Here ν lL (x) is the mixed flavor neutrino field, U is the 3×3 unitary PMNS [9,10] mixing matrix and ν iL (x) is the field of neutrino with mass m i . All currently existing ν-oscillation data, except the data of the LSND experiment 2 [11], can be described perfectly well assuming 3-ν mixing in vacuum and we will consider this possibility in what follows. The minimal 4-ν mixing scheme which could incorporate the LSND indications for ν-oscillations is strongly disfavored by the data. The ν-oscillation explanation of the LSND results is possible assuming 5-ν mixing [13]. The experimental study of neutrino oscillations allowed to determine the values of the two neutrino mass-squared differences ∆m 2 12 and |∆m 2 23 | (∆m 2 ik = m 2 k − m 2 i ) and to obtain The measurement of the absolute values of neutrino masses and the determination of the type of neutrino mass spectrum is one of the highest priority and most difficult problems in neutrino physics (see, e.g., [17]). The solution of this problem is of fundamental importance for the progress in understanding the origin of neutrino masses and mixing.
Information about the absolute values of neutrino masses can be obtained from: a) precision measurements of the β-spectrum in the end-point region [19,20]; b) investigation of neutrinoless double β-decay, if the neutrinos with definite mass are Majorana particles [21,22]; c) measurement of power spectrum of the large scale distribution of galaxies (see, e.g., [23]).
From the data of the Mainz [24] and Troitsk [25] tritium β-decay experiments the following upper bound on the measurable neutrino mass was obtained (95% C.L.): In the future KATRIN experiment [26] a sensitivity to is planned to be achieved 3 .
Using the Cosmic Microwave Background (CMB) data of the WMAP experiment, combined with data from large scale structure surveys (2dFGRS, SDSS), for the sum of neutrino masses upper bounds in the range were found (see, e.g., [28]). Data on weak lensing of galaxies by large scale structure, combined with data from the WMAP and PLANCK experiments, may allow m i to be determined with an uncertainty of δ ∼ 0.04 eV [29].
The sign of ∆m 2 23 , which drives the dominant atmospheric neutrino oscillations, can be determined by studying oscillations of neutrinos and antineutrinos, say, ν µ → ν e and ν µ →ν e , in which matter effects are sufficiently large. This can be done, e.g., in long-baseline ν-oscillation experiments (see, e.g., [30]). Information about sgn(∆m 2 23 ) can be obtained also in atmospheric neutrino experiments by studying the oscillations of the atmospheric ν µ and ν µ which traverse the Earth [31].
In the present article we investigate the physics potential of β-decay experiments having a sensitivity which permits to probe the neutrino mass range corresponding to |∆m 2 23 | ∼ = 5·10 −2 eV 2 . At present such a sensitivity does not seem reachable in any realistic experiment. However, this situation may change in the future. In our analysis we take into account the existing and prospective neutrino oscillation data on the neutrino mass squared differences and neutrino mixing angles. We consider different possible types of neutrino mass spectrum as well 4 .

On the Measurement of Neutrino Mass in β-Decay Experiments
The measurement of β-spectrum in the end-point region in tritium β-decay, is the classical method of direct determination of neutrino mass, which originated from the pioneering articles of Fermi and Perrin on β-decay [19,20]. This decay has many advantages (see, e.g., [24]). First of all, it is a super-allowed decay. Thus, the nuclear matrix element is a constant and the electron spectrum is determined by the relevant phase space factor only.
Other advantages of this decay are the relatively small energy release (E 0 ≃ 18.574.3 ± 1.7 eV) and convenient half-life (12.3 years).
Taking into account the neutrino mixing, for the effective Hamiltonian of the process (15) we have where J α is the hadronic charged current. For the state vector of the final neutrinos and electron we obtain from (16) where Here N is the product of standard normalization factors, p is the momentum of electron, p i is the momentum of antineutrino (right-handed neutrino in the Majorana case) with mass m i , P and P ′ are the total initial and final momenta.
The final state neutrinos are not detected in tritium β-decay experiments. Taking into account the orthogonality of the vectors |ν i and neglecting the recoil of the 3 He nucleus, for the electron spectrum we get the incoherent sum where Here E e ≤ E 0 − m i is the kinetic energy of the electron, E 0 is the energy released in the decay (15), p e is the electron momentum, m e is the mass of the electron, F (E e ) is the Fermi function which takes into account the Coulomb interaction of the final state particles, and C is a constant. In eq. (20) (E 0 − E e ) is the neutrino energy and Neutrino masses enter into the expression for the electron spectrum through neutrino momenta. It is obvious that the maximal distortion of the electron spectrum can be observed in the region (E 0 − E e ) ∼ m i , which is less than of the order of few eV. However, for reasons of luminosity, in tritium experiments relatively large end-point intervals of the electron spectrum are measured: in the Mainz experiment [24] E 0 − E e ≤ 70 eV; in the future KATRIN experiment [26] the region E 0 − E e < ∼ 20 eV will be explored.
Usually the quantity m β defined by is considered as the neutrino mass related observable in β-decay experiments. The expression (21) is obtained (see refs. [33,34]) by developing the neutrino momentum over (20). Let us note that in the region sensitive to the neutrino mass this expansion is not valid, while in the region (E 0 − E e ) ≫ m i the effects of neutrino mass can be neglected.
For the neutrino mass spectrum with normal ordering, the neutrino masses are given (in the standardly used convention) by In the case of spectrum with inverted ordering of neutrino masses we have The neutrino mass-squared differences ∆m 2 12 and |∆m 2 23 | have been measured in the neutrino oscillation experiments. The existing data allow a determination of ∆m 2 12 and |∆m 2 23 | at 3σ with an error of approximately 12% and 50%, respectively. These parameters will be measured with higher accuracy in the future. The highest precision in the determination of |∆m 2 23 | is expected to be achieved from the studies of ν µ -oscillations in the T2K (SK) [35] experiment: the 3σ uncertainty in |∆m 2 23 | is estimated to be reduced in this experiment to ∼ 6%.
The unknown parameter in eqs. (22) and (23) is the lightest neutrino mass m 1 (m 3 ). It follows from (22) and (23) that the minimal value of the heaviest neutrino mass in the cases of normal and inverted mass ordering, m 3 and m 2 , are given by m min 3(2) = ∆m 2 12 + |∆m 2 23 | ≃ |∆m 2 23 | .
As we have discussed earlier, depending on the value of the lightest neutrino mass, three types of neutrino mass spectrum are usually considered: NH (normal hierarchical), IH (inverted hierarchical) and QD (quasi-degenerate), eqs. (9) - (11). The QD spectrum is realised if the value of the lightest neutrino mass is relatively large, min(m j ) > ∼ 0.1 eV. This spectrum requires an approximate symmetry of the neutrino mass matrix (see, e.g., [36,18]). The NH and IH spectra correspond to negligibly small value of min(m j ). The NH spectrum is typically predicted by the GUT models which unify quarks, charged leptons and neutrinos (see, e.g., [18,37]). The IH spectrum can be associated with the existence of a broken L e −L µ −L τ symmetry in the lepton sector [38] (see also, e.g., [39]).
The Mainz, Troitsk and KATRIN experiments can probe only the quasi-degenerate neutrino mass spectrum. If in the KATRIN experiment, which is under preparation at present, a positive effect due to the neutrino mass will be observed, we will have In this case no information on the sgn(∆m 2 23 ), i.e., on the type of ordering of neutrino masses, will be obtained.
A negative result of the KATRIN experiment would imply that the neutrino mass spectrum is either NH or IH, or else is with partial normal or inverted hierarchy [21]. It will be crucial in this case to improve the sensitivity of direct neutrino mass measurement experiments by approximately a factor of 4. If the sensitivity of the β-decay experiments will allow to probe values of neutrino masses m min 3 (m min 2 ) ∼ = |∆m 2 23 | ∼ = (3.9 − 5.8) · 10 −2 eV, these experiments will provide fundamental information on the absolute scale of neutrino masses and on the type of neutrino mass spectrum independently of the nature of massive neutrinos, which, as is well-known, can be Dirac or Majorana particles (see, e.g., [40]).
Indeed, if the neutrino mass spectrum is of the NH type, the contribution of the heaviest neutrino mass m 3 ∼ = |∆m 2 23 | to the distortion of the electron spectrum is suppressed by the factor |U e3 | 2 = sin 2 θ 13 < 5 · 10 −2 and will be unobservable. The distortion of the spectrum due to the mass m 2 ∼ = ∆m 2 12 ∼ = 9 · 10 −3 eV, which is not suppressed by the corresponding mixing matrix element, will also be unobservable. Thus, the electron spectrum that will be observed in the β-decay experiments in the case of NH neutrino mass spectrum will effectively correspond to one zero mass neutrino: In contrast, if the neutrino mass spectrum is of the IH type, the two heaviest neutrino masses m 1 ∼ = m 2 ∼ = |∆m 2 23 | will enter into the expression for the electron spectrum with the coefficient 1 − |U e3 | 2 ≃ 1 . The spectrum will have the form: It follows from the above discussion that the non-observation of the effect of neutrino mass in a β-decay experiment having a sensitivity to |∆m 2 23 | would imply that the neutrino mass spectrum is of the normal hierarchical type, i.e., that ∆m 2 23 > 0 and m 1 ≪ m 2 ≪ m 3 , independently of whether the massive neutrinos are Dirac or Majorana particles. If the spectrum of neutrino masses is of the inverted hierarchical type, the effect of neutrino mass must be observed in such an experiment.
The interpretation of a positive result of a β-decay experiment with a sensitivity to a neutrino mass ∼ |∆m 2 23 |, however, will not be unique in what regards the sgn(∆m 2 23 ) and the value of the lightest neutrino mass (the type of neutrino mass spectrum). Indeed, in the discussion above of the cases of NH and IH spectra it was always assumed that the lightest neutrino mass is negligible, i.e., m 1 ≪ ∆m 2 12 in the NH case and m 3 ≪ |∆m 2 23 | in the IH one. However, this may not necessarily be valid. In principle, for both normal and inverted neutrino mass ordering, we can have min(m j ) < ∼ |∆m 2 23 |, which corresponds to a spectrum with partial hierarchy [21]. Thus, the distortion of the electron spectrum in the case of positive result of a β-decay experiment under discussion could be due either to i) spectrum with inverted neutrino mass ordering, ∆m 2 23 < 0, of two possible types: a) inverted hierarchical, m 3 ≪ m 1 < m 2 , or b) with partial inverted hierarchy, m 3 < m 1 < m 2 [21]; or to ii) spectrum with normal neutrino mass ordering, ∆m 2 23 > 0, but with partial neutrino mass hierarchy, m 1 < m 2 < m 3 [21].
As an example of the possibility ii) consider the following hypothetical spectrum: m 1 = 5.0 · 10 −2 eV, m 2 = m 2 1 + ∆m 2 12 ∼ = 5.1 · 10 −2 eV, m 3 = m 2 1 + ∆m 2 13 ∼ = 6.9 · 10 −2 eV, where we have used the best fit values of ∆m 2 12 and ∆m 2 13 given in eqs. (3) and (5). The sum of neutrino masses is equal to Σm i ∼ = 0.17 eV. Obviously, a β-decay experiment having the precision under discussion will not be sensitive to the difference between the masses m 1 and m 2 ; and it will not be sensitive to the distortion of the electron spectrum due to the mass m 3 since the contribution of the latter is suppressed by the factor sin 2 θ 13 . In this case we will have for the electron spectrum, which practically coincides with the form of the electron spectrum predicted in the case of neutrino mass spectrum of inverted hierarchical type, eq. (27). Let us note that neutrino mass spectrum with partial hierarchy can possibly be probed by future cosmological/astrophysical observations (see, e.g., [29]).

Conclusion
The knowledge of neutrino mass spectrum is decisive for the understanding of the origin of neutrino masses and mixing. From the data of neutrino oscillation experiments the two independent neutrino mass-squared differences in the case of 3-neutrino mixing, which are responsible for the solar and atmospheric neutrino oscillations, were determined. However, the existing data does not allow to determine the sign of the neutrino mass squared difference driving the atmospheric neutrino oscillations, sgn(∆m 2 23 ). Thus, we can have ∆m 2 23 > 0, corresponding (in the standardly used convention) to neutrino mass spectrum with normal ordering of neutrino masses, m 1 < m 2 < m 3 ; or ∆m 2 23 < 0, which is associated with neutrino mass spectrum with inverted ordering of neutrino masses, m 3 < m 1 < m 2 . In both cases the minimal value of the heaviest neutrino mass is given by m min 3(2) ≃ |∆m 2 23 | ∼ 5 · 10 −2 eV. We do not know at present the mass of the lightest neutrino min(m j ) as well. The existing data from β-decay experiments and the cosmological data allow us to obtain only an upper bound on min(m j ).
Depending on the value of the mass of the lightest neutrino, there are three possible characteristic types of neutrino mass spectrum: quasi-degenerate, in which the lightest neutrino mass is relatively large, m 2 1(3) ≫ |∆m 2 23 |; and two spectra in which the lightest neutrino mass is negligibly small: normal hierarchical with ∆m 2 23 > 0 and m 1 ≪ ∆m 2 12 , and inverted hierarchical, with ∆m 2 23 < 0 and m 3 ≪ |∆m 2 23 |. The future KATRIN β-decay experiment will probe the quasi-degenerate neutrino mass spectrum. If effect of neutrino mass will be observed in this experiment, the lightest neutrino mass will be measured. However, the sgn(∆m 2 23 ), and thus the character of neutrino mass spectrum -with normal or inverted ordering of neutrino masses, will not be determined in this case.
If in the KATRIN experiment the effect of nonzero neutrino mass will not be observed, it will be crucial to improve the sensitivity of the β-decay experiments by approximately a factor of four, which would permit to probe values m min 3(2) ∼ = |∆m 2 23 |. As we have shown, the non-observation of the effect of neutrino mass in a β-decay experiment having a sensitivity to |∆m 2 23 | would imply that the neutrino mass spectrum is of the normal hierarchical type, i.e., that ∆m 2 23 > 0 and m 1 ≪ m 2 ≪ m 3 , independently of whether the massive neutrinos are Dirac or Majorana particles. If the spectrum of neutrino masses is of the inverted hierarchical type, the effect of neutrino mass must be observed in such an experiment.
It follows from our analysis, however, that the interpretation of a positive result of a β-decay experiment with a sensitivity to a neutrino mass ∼ |∆m 2 23 | will not be unique in what regards the sgn(∆m 2 23 ) and the type of neutrino mass spectrum. The distortion of the electron spectrum observed in such an experiment could be due either to i) spectrum with inverted neutrino mass ordering (∆m 2 23 < 0) of two possible types, a) inverted hierarchical, m 3 ≪ m 1 < m 2 , or b) with partial inverted hierarchy, m 3 < m 1 < m 2 ; or to ii) spectrum with normal neutrino mass ordering (∆m 2 23 > 0), but with partial neutrino mass hierarchy, m 1 < m 2 < m 3 .