363
21.4. Neutrino mixing and oscillations
at least one oscillation length fits into the resonant density region, then it
can be shown that the state stays with state |2› m (‘adiabatic evolution’) until
it reaches the surface of the Sun, when θ m → θ e2 . The probability that the
neutrino will survive to the earth is then (using (21.133)) |‹ν e |2› m |
2 = sin
2 θ e2 ,
which has a value of about 1/3. In the alternative limit, (ii), in which the
oscillation length in matter is relatively large with respect to the scale of
density variation, the state may ‘jump’ to the other mass state |1› m (‘extreme
non-adiabatic evolution’), and then |‹ν e |1› m |
2 = cos
2 θ e2 . These are clearly
extreme cases, and numerical work is required in the general case. However,
the data from SNO and other water Cerenkov detectors are consistent with
the first (adiabatic) alternative, and with the value sin
2 θ e2 ∼ 1/3. Note that
2
2
the solar data imply that (m − m ) cos 2θ e2 > 0.
2
1
By contrast, for the lowest energy neutrinos we can take θ m ≈ θ, so that the
neutrinos are produced in the state cos θ e2 |1›+sin θ e2 |2›, and propagate as in a
vacuum, oscillating with maximum excursion sin
2 2θ e2 . The detectors average
over many oscillations, giving a factor of 1/2, so that the survival probability
for the low energy ν e s is 1 −
1 sin
2 2θ e2 ∼ 5/9. The Gallium experiments are
2
sensitive to the lower energy neutrinos, and indeed record some 60–70% of the
expected flux.
In summary, the solar neutrino data are consistent with the interpretation
in terms of neutrino oscillations, as modified by the Wolfenstein-MikheevSmirnov (MSW) effect. A global solar + KamLAND analysis yields best fit
values (Aharmim et al. 2010)
2
θ e2 = 34.06
+1.16 Δm 21 = 7.59
+0.20
.
(21.142)
−0.21 × 10
−5 eV
2
−0.84
21.4.5 Further developments
Despite the remarkable experimental progress in the studies of neutrino oscillations over the last decade, there still remain some basic gaps in our knowledge. Perhaps the most fundamental is the Dirac/Majorana nature of massive
neutrinos. The most feasible (but very difficult) test is neutrinoless double
β-decay (0νββ-decay), already touched on in section 20.3. As noted there, the
amplitude is proportional to an average Majorana mass parameter ‹m›. Experiments place a lower bound on the half-life for the decay, which translates
into an upper bound on ‹m›. The most stringent lower bounds on the halflives have been obtained with decays of
76
Ge (Klapdor-Kleingrothaus et al.
2001),
130
Te (Andreotti et al. 2011) and
100
Mo (Arnold et al. 2006). Lower
bounds on the half-lives range from 10
24 to 10
25 years, with corresponding
upper bounds on ‹m› of the order of 0.5 eV. It should, however, be noted that
some participants of the Heidelberg-Moscow experiment claimed the observation of 0νββ decay of
76
Ge with half-life of 2.23
+0.44
a
−0.31 ×10
25 years, from which
they deduced ‹m› = 0.32 ± 0.03 eV (Klapdor-Kleingrothaus et al. 2006). The
GERDA experiment (Ur et al. 2011) should be able to check this claim after
one year of running. Other experiments currently running, or planned, will
21.4. Neutrino mixing and oscillations
at least one oscillation length fits into the resonant density region, then it
can be shown that the state stays with state |2› m (‘adiabatic evolution’) until
it reaches the surface of the Sun, when θ m → θ e2 . The probability that the
neutrino will survive to the earth is then (using (21.133)) |‹ν e |2› m |
2 = sin
2 θ e2 ,
which has a value of about 1/3. In the alternative limit, (ii), in which the
oscillation length in matter is relatively large with respect to the scale of
density variation, the state may ‘jump’ to the other mass state |1› m (‘extreme
non-adiabatic evolution’), and then |‹ν e |1› m |
2 = cos
2 θ e2 . These are clearly
extreme cases, and numerical work is required in the general case. However,
the data from SNO and other water Cerenkov detectors are consistent with
the first (adiabatic) alternative, and with the value sin
2 θ e2 ∼ 1/3. Note that
2
2
the solar data imply that (m − m ) cos 2θ e2 > 0.
2
1
By contrast, for the lowest energy neutrinos we can take θ m ≈ θ, so that the
neutrinos are produced in the state cos θ e2 |1›+sin θ e2 |2›, and propagate as in a
vacuum, oscillating with maximum excursion sin
2 2θ e2 . The detectors average
over many oscillations, giving a factor of 1/2, so that the survival probability
for the low energy ν e s is 1 −
1 sin
2 2θ e2 ∼ 5/9. The Gallium experiments are
2
sensitive to the lower energy neutrinos, and indeed record some 60–70% of the
expected flux.
In summary, the solar neutrino data are consistent with the interpretation
in terms of neutrino oscillations, as modified by the Wolfenstein-MikheevSmirnov (MSW) effect. A global solar + KamLAND analysis yields best fit
values (Aharmim et al. 2010)
2
θ e2 = 34.06
+1.16 Δm 21 = 7.59
+0.20
.
(21.142)
−0.21 × 10
−5 eV
2
−0.84
21.4.5 Further developments
Despite the remarkable experimental progress in the studies of neutrino oscillations over the last decade, there still remain some basic gaps in our knowledge. Perhaps the most fundamental is the Dirac/Majorana nature of massive
neutrinos. The most feasible (but very difficult) test is neutrinoless double
β-decay (0νββ-decay), already touched on in section 20.3. As noted there, the
amplitude is proportional to an average Majorana mass parameter ‹m›. Experiments place a lower bound on the half-life for the decay, which translates
into an upper bound on ‹m›. The most stringent lower bounds on the halflives have been obtained with decays of
76
Ge (Klapdor-Kleingrothaus et al.
2001),
130
Te (Andreotti et al. 2011) and
100
Mo (Arnold et al. 2006). Lower
bounds on the half-lives range from 10
24 to 10
25 years, with corresponding
upper bounds on ‹m› of the order of 0.5 eV. It should, however, be noted that
some participants of the Heidelberg-Moscow experiment claimed the observation of 0νββ decay of
76
Ge with half-life of 2.23
+0.44
a
−0.31 ×10
25 years, from which
they deduced ‹m› = 0.32 ± 0.03 eV (Klapdor-Kleingrothaus et al. 2006). The
GERDA experiment (Ur et al. 2011) should be able to check this claim after
one year of running. Other experiments currently running, or planned, will