DFTT 71/99 hep-ph/9912427
Neutrino oscillations and neutrinoless double-β decay
INFN, Sezione di Torino, and Dipartimento di Fisica Teorica, Universit` di Torino, a Via P. Giuria 1, I–10125 Torino, Italy
arXiv:hep-ph/9912427v1 20 Dec 1999
We consider the scheme with mixing of three neutrinos and a mass hierarchy. We shown that, under the natural assumptions that massive neutrinos are Majorana particles and there are no unlikely ?ne-tuned cancellations among the contributions of the di?erent neutrino masses, the results of solar neutrino experiments imply a lower bound for the e?ective Majorana mass in neutrinoless double-β decay. We also discuss brie?y neutrinoless double-β decay in schemes with mixing of four neutrinos. We show that one of them is favored by the data. Presented at TAUP’99, 6–10 September 1999, College de France, Paris, France.
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Neutrino oscillations  have been observed in solar and atmospheric neutrino experiments. The corresponding neutrino mass-squared di?erences are ?m2 ? 10?6 ? 10?4 eV2 sun in the case of MSW transitions, or ?m2 ? 10?11 ? 10?10 eV2 sun in the case of vacuum oscillations, and ?m2 ? 10?3 ? 10?2 eV2 . atm (0.3) (VO) , (0.2) (MSW) , (0.1)
These values of the neutrino mass-squared di?erences and the mixing required for the observed solar and atmospheric oscillations are compatible with the simplest and most natural scheme with three-neutrino mixing and a mass hierarchy:
m1 ? m2 ? m3 .
This scheme is predicted by the see-saw mechanism , which predicts also that the three light massive neutrinos are Majorana particles. In this case neutrinoless double-β decay (ββ0ν ) is possible and its matrix element is proportional to the e?ective Majorana mass | m |=
k 2 Uek mk ,
where U is the neutrino mixing matrix and the sum is over the contributions of all the mass eigenstate neutrinos νk (k = 1, 2, 3). In principle the e?ective Majorana mass (0.5) can be vanishingly small because of cancellations among the contributions of the di?erent mass eigenstates. However, since the neutrino masses and the elements of the neutrino mixing matrix are independent quantities, if there is a hierarchy of neutrino masses such a cancellation would be the result of an unlikely ?ne-tuning, unless some unknown symmetry is at work. Here we consider the possibility that no such symmetry exist and no unlikely ?ne-tuning operates to suppress the e?ective Majorana mass (0.5) . In this case we have | m | ? max | m |k ,
where | m |k is the absolute value of the contribution of the massive neutrino νk to | m |: | m |k ≡ |Uek |2 mk . (0.7)
In the following we will estimate the value of | m | using the largest | m |k obtained from the results of neutrino oscillation experiments. 2
Let us consider ?rst | m |3, which, taking into account that in the three-neutrino scheme under consideration m3 ? ?m2 = ?m2 , is given by atm 31 | m |3 ? |Ue3 |2 ?m2 . atm (0.8)
Since the results of the CHOOZ experiment  and the Super-Kamiokande atmospheric neutrino data  imply that |Ue3 |2 is small (|Ue3 |2 5 × 10?2 ), the contribution | m |3 to the e?ective Majorana mass in ββ0ν decay is very small [6,2,7]. The upper bounds for | m |3 as functions of ?m2 obtained from the present experimental data are shown in atm Fig. 1. The dash-dotted upper limit has been obtained using the 90% CL exclusion curve of 1 the CHOOZ experiment (taking into account  that |Ue3 |2 = 2 1 ? 1 ? sin2 2?CHOOZ , where ?CHOOZ is the two-neutrino mixing angle measured in the CHOOZ experiment), the dashed upper bound has been obtained using the results presented in Ref.  of the analysis of Super-Kamiokande atmospheric neutrino data (at 90% CL) and the solid upper limit, that surrounds the shadowed allowed region, has been obtained using the results presented in Ref.  of the combined analysis of the CHOOZ and Super-Kamiokande data (at 90% CL). The dotted line in Fig. 1 represents the unitarity limit | m |3 ≤ ?m2 . One can see atm from Fig. 1 that the results of the CHOOZ experiment imply that | m |3 2.7 × 10?2 eV, the results of the Super-Kamiokande experiment imply that | m |3 3.8 × 10?2 eV, and the combination of the results of the two experiments drastically lowers the upper bound to | m |3 2.5 × 10?3 eV . (0.9)
Since there is no lower bound for |Ue3 |2 from experimental data, | m |3 could be much smaller than the upper bound in Eq. (0.9). Hence, the largest contribution to | m | could come from | m |2 ≡ |Ue2 |2 m2 . In ?m2 = ?m2 and, since |Ue3 |2 is very small, |Ue2 |2 ? the scheme (0.4) m2 ? 21 sun 1 ? 1 ? sin2 2?sun , where ?sun is the two-neutrino mixing angle used in the analysis of solar neutrino data. Therefore, | m |2 is given by
| m |2 ?
1 1? 2
1 ? sin2 2?sun
?m2 . sun
Solar neutrino data imply bounds for sin2 2?sun and ?m2 . In particular the large mixing sun angle MSW solution (LMA) of the solar neutrino problem requires a relatively large ?m2 sun and a mixing angle ?sun close to maximal: 1.2 × 10?5 eV2 0.58 ?m2 3.1 × 10?4 eV2 , sun sin2 2?sun 1 , (0.11) (0.12)
at 99% CL . The corresponding allowed range for | m |2 as a function of ?m2 is shown sun in Fig.2 (the shadowed region limited by the solid line). The dashed line in Fig.2 represents the unitarity limit | m |2 ≤ ?m2 . From Fig.2 one can see that the LMA solution of the sun solar neutrino problem implies that 7.4 × 10?4 eV | m |2 3 6.0 × 10?3 eV . (0.13)
Assuming the absence of ?ne-tuned cancellations among the contributions of the three neutrino masses to the e?ective Majorana mass, if |Ue3 |2 is very small and | m |3 ? | m |2, from Eqs.(0.6) and (0.13) we obtain 7 × 10?4 eV |m| 6 × 10?3 eV . (0.14)
Hence, assuming the absence of an unlikely ?ne-tuned suppression of | m |, in the case of the LMA solution of the solar neutrino problem we have obtained a lower bound of about 7 × 10?4 eV for the e?ective Majorana mass in ββ0ν decay. Also the small mixing angle MSW (SMA) and the vacuum oscillation (VO) solutions of the solar neutrino problem imply allowed ranges for | m |2, but their values are much smaller than in the case of the LMA solution. Using the 99% CL allowed regions obtained in  from the analysis of the total rates measured in solar neutrino experiments we have 5 × 10?7 eV | m |2 10?5 eV(SMA) , 10?6 eV | m |2 2 × 10?5 eV(VO) . (0.15) (0.16)
If future ββ0ν experiments will ?nd | m | in the range shown in Fig.2 and future long-baseline experiments will obtain a stronger upper bound for |Ue3 |2 , it would mean that | m |2 gives the largest contribution to the e?ective Majorana mass, favoring the LMA solution of the solar neutrino problem. On the other hand, if future ββ0ν experiments will ?nd | m | in the range shown in Fig.2 and the SMA or VO solutions of the solar neutrino problem will be proved to be correct by future solar neutrino experiments, it would mean that | m |3 gives the largest contribution to the e?ective Majorana mass and there is a lower bound for the value of |Ue3 |2 . Finally, let us consider brie?y the two four-neutrino mixing schemes compatible with all neutrino oscillation data , including the indications in favor of ν? → νe oscillations found in the short-baseline (SBL) LSND experiment :
?m2 atm ?m2 sun
(A)m1 < m2 < m3 < m4 ,
?m2 SBL ?m2 sun ?m2 atm
(B)m1 < m2 < m3 < m4 .
Since the mixing of νe with the two massive neutrinos whose mass-squared di?erence generates atmospheric neutrino oscillations is very small , the contribution of the two “heavy” mass eigenstates ν3 and ν4 to the e?ective Majorana mass (0.5) is large in scheme A and very small in scheme B. Hence, the e?ective Majorana mass is expected to be relatively large in scheme A and strongly suppressed in scheme B. In particular, in the scheme A the SMA solution of the solar neutrino problem implies a value of | m | larger than the the present upper bound obtained in ββ0ν decay experiments  and is, therefore, disfavored. Furthermore, since the measured abundances of primordial elements produced in Big-Bang Nucleosynthesis is compatible only with the SMA solution of the solar neutrino problem , we conclude that the scheme A is disfavored by the present experimental data and there is only one four-neutrino mixing scheme supported by all data: scheme B . 4
             See: S.M. Bilenky, C. Giunti, and W. Grimus, Prog. Part. Nucl. Phys. 43, 1 (1999). C. Giunti, hep-ph/9906275 (Phys. Rev. D). M. Apollonio et al. (CHOOZ Coll.), Phys. Lett. B 466, 415 (1999). M. Nakahata, these proceedings. G.L. Fogli, these proceedings. S.M. Bilenky, C. Giunti, C.W. Kim and M. Monteno, Phys. Rev. D57, 6981 (1998). S.M. Bilenky et al., Phys. Lett. B 465, 193 (1999). S.M. Bilenky and C. Giunti, Phys. Lett. B444, 379 (1998). Y. Fukuda et al., Phys. Rev. Lett. 82, 1810 (1999). J.N. Bahcall, P.I. Krastev and A.Yu. Smirnov, Phys. Rev. D58, 096016 (1998). D.H. White (LSND Coll.), Nucl. Phys. B (Proc. Suppl.) 77, 207 (1999). L. Baudis et al., Phys. Rev. Lett. 83, 41 (1999). S.M. Bilenky, C. Giunti, W. Grimus and T. Schwetz, Astropart. Phys. 11, 413 (1999).
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