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20. Introduction to the Phenomenology of Weak Interactions
m = 0 limit) to the two independent two-component ‘Weyl’ equations.
Eφ 0 = σ · p φ 0
Eχ 0 = −σ · p χ 0 .
(20.32)
Remembering that E = |p| for a massless particle, we see that φ 0 has positive
helicity and χ 0 negative helicity. In this strictly massless case, helicity is
Lorentz invariant, since the direction of p cannot be reversed by a velocity
transformation with v < c. Furthermore, each of the equations in (20.32)
violates parity, since E is clearly a scalar while σ · p is a pseudoscalar (note
that when m = / 0 we can infer from (20.21) that, in this representation, φ ↔ χ
under P, which is consistent with (20.32) and with the form of β in (20.14)).
Thus the (massless) neutrino could be ‘blamed’ for the parity violation. In
this model, neutrinos have one definite helicity, either positive or negative. As
we have seen, the massless limit of the (four-component) V-A theory leads to
the same conclusion.
Which helicity is actually chosen by Nature was determined in a classic
experiment by Goldhaber et al. (1958), involving the K-capture reaction
−
152 Eu → ν +
152 Sm
∗
e +
,
(20.33)
as described by Bettini (2008), for example. They found that the helicity
of the emitted neutrino was (within errors) 100% negative, a result taken as
confirming the ‘2-component’ neutrino theory, and the V-A theory.
We now know that neutrinos are not massless. This information does not
come from studies of nuclear decays, but rather from a completely different
phenomenon – that of neutrino oscillations, which we shall mention again in
the following section, and treat more fully in section 21.4. Neutrino masses
are so small that the existence of the ‘wrong helicity’ component cannot be
detected experimentally in processes such as (20.33), or indeed in any of the
reactions we shall discuss, apart from neutrino oscillations.
In section 4.2.2 we introduced the charge conjugation operation C (see also
section 7.5.2). As we noted there, C is not a good symmetry in weak interactions. The V-A interaction treats a negative helicity fermion very differently
from a negative helicity antifermion, while one is precisely transformed into
the other under C. However, it is clear that the helicity operator itself is
odd under P. Thus the CP conjugate of a negative helicity fermion is positive helicity antifermion, which is what the V-A interaction selects. It may
easily be verified (problem 20.4) that the ‘2-component’ theory of (20.32)
automatically incorporates CP invariance. Elegance notwithstanding, however, there are CP-violating weak interactions, as mentioned in section 4.2.3.
How this is accommodated within the Standard Model we shall discuss in
section 20.7.3.
For charged fermions the distinction between particle and antiparticle is
clear; but is there a conserved quantum number which we can use instead of
charge to distinguish a neutrino from an antineutrino? That is the question
to which we now turn.
20. Introduction to the Phenomenology of Weak Interactions
m = 0 limit) to the two independent two-component ‘Weyl’ equations.
Eφ 0 = σ · p φ 0
Eχ 0 = −σ · p χ 0 .
(20.32)
Remembering that E = |p| for a massless particle, we see that φ 0 has positive
helicity and χ 0 negative helicity. In this strictly massless case, helicity is
Lorentz invariant, since the direction of p cannot be reversed by a velocity
transformation with v < c. Furthermore, each of the equations in (20.32)
violates parity, since E is clearly a scalar while σ · p is a pseudoscalar (note
that when m = / 0 we can infer from (20.21) that, in this representation, φ ↔ χ
under P, which is consistent with (20.32) and with the form of β in (20.14)).
Thus the (massless) neutrino could be ‘blamed’ for the parity violation. In
this model, neutrinos have one definite helicity, either positive or negative. As
we have seen, the massless limit of the (four-component) V-A theory leads to
the same conclusion.
Which helicity is actually chosen by Nature was determined in a classic
experiment by Goldhaber et al. (1958), involving the K-capture reaction
−
152 Eu → ν +
152 Sm
∗
e +
,
(20.33)
as described by Bettini (2008), for example. They found that the helicity
of the emitted neutrino was (within errors) 100% negative, a result taken as
confirming the ‘2-component’ neutrino theory, and the V-A theory.
We now know that neutrinos are not massless. This information does not
come from studies of nuclear decays, but rather from a completely different
phenomenon – that of neutrino oscillations, which we shall mention again in
the following section, and treat more fully in section 21.4. Neutrino masses
are so small that the existence of the ‘wrong helicity’ component cannot be
detected experimentally in processes such as (20.33), or indeed in any of the
reactions we shall discuss, apart from neutrino oscillations.
In section 4.2.2 we introduced the charge conjugation operation C (see also
section 7.5.2). As we noted there, C is not a good symmetry in weak interactions. The V-A interaction treats a negative helicity fermion very differently
from a negative helicity antifermion, while one is precisely transformed into
the other under C. However, it is clear that the helicity operator itself is
odd under P. Thus the CP conjugate of a negative helicity fermion is positive helicity antifermion, which is what the V-A interaction selects. It may
easily be verified (problem 20.4) that the ‘2-component’ theory of (20.32)
automatically incorporates CP invariance. Elegance notwithstanding, however, there are CP-violating weak interactions, as mentioned in section 4.2.3.
How this is accommodated within the Standard Model we shall discuss in
section 20.7.3.
For charged fermions the distinction between particle and antiparticle is
clear; but is there a conserved quantum number which we can use instead of
charge to distinguish a neutrino from an antineutrino? That is the question
to which we now turn.