21.4. Neutrino mixing and oscillations
353
2.4 × 10
−3 eV
2 , respectively. The smaller value is associated with oscillations
of solar or reactor neutrinos, and the larger with oscillations of atmospheric
or accelerator neutrinos.
Data on the actual mass values are limited. There is a bound on the
ν ¯ e mass from measurements of the electron spectrum near the end-point in
tritium β-decay, which gives (Lobashev et al. 2003, Eitel et al. 2005)
m ν ¯e < 2.3eV 95%CL.
(21.108)
A weaker limit on m νμ comes from measurements of the muon spectrum in
charged pion decay:
m νμ < 0.19 MeV 95%CL.
(21.109)
The strongest upper bound comes from cosmology, assuming three neutrinos.
The Cosmic Microwave Background data of the WMAP experiment, combined
with supernovae data and data on galaxy clustering, can be used to obtain an
upper limit on the sum of three neutrino masses (Spergel et al. 2007):
∑ 3
m νi < 0.68 eV, 95%CL.
(21.110)
i+1
Taking the squared mass differences as indicative of the actual mass scale,
neutrino masses are evidently very much smaller than the masses of the other
fermions in the Standard Model. We shall return to what this might tell
us about the origin of neutrino mass in section 22.5, where we discuss how
gauge-invariant masses are generated in the Standard Model.
Returning to the question of CP violation, we noted in section 4.2.3 that
the CP violation present in the Standard Model was insufficient to account for
the matter–antimatter asymmetry in the universe. However, we now see that
it is possible to have CP violation in the lepton sector, in an extended Standard Model with massive neutrinos. Leptonic matter–antimatter asymmetries
can be converted into baryon asymmetries in the very hot early universe by a
non-perturbative process predicted by Standard Model dynamics – a process
called leptogenesis (Fukugita and Yanagida 1986, Kuzmin, Rubakov and Shaposhnikov 1985). It has been argued that the Dirac and/or Majorana phases
in the neutrino matrix U or U M can provide the CP violation necessary in
leptogenesis models for the generation of the observed baryon asymmetry of
the universe (Pascoli et al. 2007a, 2007b). If such a proposal should prove
to be the case, the reach of Pauli’s ‘desperate remedy’ will have been vast
indeed.
21.4.2 Neutrino oscillations: formulae
The existence of neutrino oscillations means that if a neutrino of a given
flavour ν α (α = e, μ, τ) with energy E is produced in a charged current weak
interaction process, such as π
+
→ μ
+ ν μ , then at a sufficiently large distance L
353
2.4 × 10
−3 eV
2 , respectively. The smaller value is associated with oscillations
of solar or reactor neutrinos, and the larger with oscillations of atmospheric
or accelerator neutrinos.
Data on the actual mass values are limited. There is a bound on the
ν ¯ e mass from measurements of the electron spectrum near the end-point in
tritium β-decay, which gives (Lobashev et al. 2003, Eitel et al. 2005)
m ν ¯e < 2.3eV 95%CL.
(21.108)
A weaker limit on m νμ comes from measurements of the muon spectrum in
charged pion decay:
m νμ < 0.19 MeV 95%CL.
(21.109)
The strongest upper bound comes from cosmology, assuming three neutrinos.
The Cosmic Microwave Background data of the WMAP experiment, combined
with supernovae data and data on galaxy clustering, can be used to obtain an
upper limit on the sum of three neutrino masses (Spergel et al. 2007):
∑ 3
m νi < 0.68 eV, 95%CL.
(21.110)
i+1
Taking the squared mass differences as indicative of the actual mass scale,
neutrino masses are evidently very much smaller than the masses of the other
fermions in the Standard Model. We shall return to what this might tell
us about the origin of neutrino mass in section 22.5, where we discuss how
gauge-invariant masses are generated in the Standard Model.
Returning to the question of CP violation, we noted in section 4.2.3 that
the CP violation present in the Standard Model was insufficient to account for
the matter–antimatter asymmetry in the universe. However, we now see that
it is possible to have CP violation in the lepton sector, in an extended Standard Model with massive neutrinos. Leptonic matter–antimatter asymmetries
can be converted into baryon asymmetries in the very hot early universe by a
non-perturbative process predicted by Standard Model dynamics – a process
called leptogenesis (Fukugita and Yanagida 1986, Kuzmin, Rubakov and Shaposhnikov 1985). It has been argued that the Dirac and/or Majorana phases
in the neutrino matrix U or U M can provide the CP violation necessary in
leptogenesis models for the generation of the observed baryon asymmetry of
the universe (Pascoli et al. 2007a, 2007b). If such a proposal should prove
to be the case, the reach of Pauli’s ‘desperate remedy’ will have been vast
indeed.
21.4.2 Neutrino oscillations: formulae
The existence of neutrino oscillations means that if a neutrino of a given
flavour ν α (α = e, μ, τ) with energy E is produced in a charged current weak
interaction process, such as π
+
→ μ
+ ν μ , then at a sufficiently large distance L