294
20. Introduction to the Phenomenology of Weak Interactions
produced e
− via (20.34), where ‘ν’ is the helicity −1 state (or, on the other
interpretation, the carrier of lepton number +1).
The situation may therefore be summarized as follows. In the case of e
−
and e
+ , all four ‘modes’ – e
− (λ = +1), e
− (λ = −1), e
+ (λ = +1), e
+ (λ = −1)
– are experimentally accessible via electromagnetic interactions, even though
only two generally dominate in weak interactions (e
− (λ = −1) and e
+ (λ =
+1)). Neutrinos, on the other hand, seem to interact only weakly. In their
case, we may if we wish say that the participating states are (in association
with e
− or e
+ ) ¯
ν e (λ = +1) and ν e (λ = −1), to a very good approximation.
But we may also regard these two states as simply two different helicity states
of one particle, rather than of a particle and its antiparticle. As we have seen,
the helicity rules do the job required just as well as the lepton number rules. In
short, the question is: are these ‘neutrinos’ distinguished only by their helicity,
or is there an additional distinguishing characteristic (‘electron number’)?
In the latter case we should expect the ‘other’ two states ¯
ν e (λ = −1) and
ν e (λ = +1) to exist as well as the ones known from weak interactions.
If, in fact, no quantum number – other than the helicity – exists which
distinguishes the neutrino states, then we would have to say that the Cconjugate of a neutrino state is a neutrino, not an antineutrino – that is,
‘neutrinos are their own antiparticles’. A neutrino would be a fermionic state
somewhat like a photon, which is of course also its own antiparticle. Such
‘C-self-conjugate’ fermions are called Majorana fermions (Majorana 1937), in
contrast to the Dirac variety, which have all four possible modes present (2
helicities, 2 particle/antiparticle). We discussed Majorana fermions in sections
4.2.2 and 7.5.2.
The distinction between the ‘Dirac’ and ‘Majorana’ neutrino possibilities
becomes an essentially ‘metaphysical’ one in the limit of strictly massless neutrinos, since then (as we have seen) a given helicity state cannot be flipped
by going to a suitably moving Lorentz frame, nor by any weak (or electromagnetic) interaction, since they both conserve chirality which is the same as
helicity in the massless limit. We would have just the two states ν e (λ = −1)
and ¯
ν e (λ = +1), and no way of creating ν e (λ = +1) or ¯
ν e (λ = −1). The
‘
− ’ label then becomes superfluous. Unfortunately, the massless limit is approached smoothly, and neutrino masses are, in fact, so small that the ‘wrong
helicity’ supression factors will make it very difficult to see the presence of the
possible states ν e (λ = +1), ν ¯ e (λ = −1).
One much-discussed experimental test case (see, for example, the review
by Vogel and Piepke in Nakamura et al. 2010) concerns ‘neutrinoless double
β-decay’, which is the process A → A
′ + e
− + e
− , where A, A
′ are nuclei. If
the neutrino emitted in the first β-decay carries no electron-type conserved
quantum number, then in principle it can initiate a second weak interaction,
exactly as in Davis’ original argument, via the diagram shown in figure 20.2.
Note that this is a second-order weak process, so that the amplitude contains
the very small factor G
2
Furthermore, the ν emitted along with the e
−
F .
at the first vertex will be predominately λ = +1, but in the second vertex
20. Introduction to the Phenomenology of Weak Interactions
produced e
− via (20.34), where ‘ν’ is the helicity −1 state (or, on the other
interpretation, the carrier of lepton number +1).
The situation may therefore be summarized as follows. In the case of e
−
and e
+ , all four ‘modes’ – e
− (λ = +1), e
− (λ = −1), e
+ (λ = +1), e
+ (λ = −1)
– are experimentally accessible via electromagnetic interactions, even though
only two generally dominate in weak interactions (e
− (λ = −1) and e
+ (λ =
+1)). Neutrinos, on the other hand, seem to interact only weakly. In their
case, we may if we wish say that the participating states are (in association
with e
− or e
+ ) ¯
ν e (λ = +1) and ν e (λ = −1), to a very good approximation.
But we may also regard these two states as simply two different helicity states
of one particle, rather than of a particle and its antiparticle. As we have seen,
the helicity rules do the job required just as well as the lepton number rules. In
short, the question is: are these ‘neutrinos’ distinguished only by their helicity,
or is there an additional distinguishing characteristic (‘electron number’)?
In the latter case we should expect the ‘other’ two states ¯
ν e (λ = −1) and
ν e (λ = +1) to exist as well as the ones known from weak interactions.
If, in fact, no quantum number – other than the helicity – exists which
distinguishes the neutrino states, then we would have to say that the Cconjugate of a neutrino state is a neutrino, not an antineutrino – that is,
‘neutrinos are their own antiparticles’. A neutrino would be a fermionic state
somewhat like a photon, which is of course also its own antiparticle. Such
‘C-self-conjugate’ fermions are called Majorana fermions (Majorana 1937), in
contrast to the Dirac variety, which have all four possible modes present (2
helicities, 2 particle/antiparticle). We discussed Majorana fermions in sections
4.2.2 and 7.5.2.
The distinction between the ‘Dirac’ and ‘Majorana’ neutrino possibilities
becomes an essentially ‘metaphysical’ one in the limit of strictly massless neutrinos, since then (as we have seen) a given helicity state cannot be flipped
by going to a suitably moving Lorentz frame, nor by any weak (or electromagnetic) interaction, since they both conserve chirality which is the same as
helicity in the massless limit. We would have just the two states ν e (λ = −1)
and ¯
ν e (λ = +1), and no way of creating ν e (λ = +1) or ¯
ν e (λ = −1). The
‘
− ’ label then becomes superfluous. Unfortunately, the massless limit is approached smoothly, and neutrino masses are, in fact, so small that the ‘wrong
helicity’ supression factors will make it very difficult to see the presence of the
possible states ν e (λ = +1), ν ¯ e (λ = −1).
One much-discussed experimental test case (see, for example, the review
by Vogel and Piepke in Nakamura et al. 2010) concerns ‘neutrinoless double
β-decay’, which is the process A → A
′ + e
− + e
− , where A, A
′ are nuclei. If
the neutrino emitted in the first β-decay carries no electron-type conserved
quantum number, then in principle it can initiate a second weak interaction,
exactly as in Davis’ original argument, via the diagram shown in figure 20.2.
Note that this is a second-order weak process, so that the amplitude contains
the very small factor G
2
Furthermore, the ν emitted along with the e
−
F .
at the first vertex will be predominately λ = +1, but in the second vertex