288
20. Introduction to the Phenomenology of Weak Interactions
where v, p and E are respectively the electron speed, momentum and energy,
P is the magnitude of the polarization, and θ is the angle of emission of the
electron with respect to ‹J ›.
Why does this indicate parity violation? To see this, we recall from the
discussion of the parity operation P in section 4.2.1 that the angular momentum J is an axial vector such that ‹J › → ‹J › under P, while p is a polar
vector transforming by p → −p. Hence, in the parity-transformed system,
the distribution (20.11) would have the form
I P (θ) = 1 + P v cos θ
(20.12)
The difference between (20.12) and (20.11) implies that, by performing the
measurement, we can determine which of the two coordinate systems we must
in fact be using. The two are inequivalent, in contrast to all the other coordinate system equivalences which we have previously studied (e.g. under threedimensional rotations, and Lorentz transformations). This is an operational
consequence of ‘parity violation’. The crucial point in this example, evidently,
is the appearance of the pseudoscalar quantity ‹J › · p in (20.10), alongside the
obviously scalar quantity ‘1’.
The Fermi theory, employing only vector currents, needs a modification
to accommodate this result. We saw in section 4.2.1 that a combination of
vector (‘V’) and axial vector (‘A’) currents would be parity-violating. Indeed,
after many years of careful experiments, and many false trails, it was eventually established (always, of course, to within some experimental error) that
the currents participating in Fermi’s current–current interaction are, in fact,
certain combinations of V-type and A-type currents, for both nucleons and
leptons.
20.2.2 V-A theory: chirality and helicity
Quite soon after the discovery of parity violation, Sudarshan and Marshak
(1958), and then Feynman and Gell-Mann (1958) and Sakurai (1958), proposed a specific form for the current–current interaction, namely the V-A
(‘V minus A’) structure. For example, in place of the leptonic combination
u ¯ e − γ μ u ν , these authors proposed the form ¯
u e − γ μ (1 − γ 5 )u ν , being the difference (with equal weight) of a V-type and an A-type current. For the part
involving the nucleons the proposal was slightly more complicated, having the
form ¯
u p γ μ (1 − rγ 5 )u n where r had the empirical value r ≈ 1.2. From our
present perspective, of course, the hadronic transition is actually occurring at
the quark level, so that rather than a transition n → p we now think in terms
of a d → u one. In this case, the remarkable fact is that the appropriate cur1
rent to use is, once again, essentially the simple ‘V-A’ one, ¯
u u γ μ (1 − γ 5 )u d .
This V-A structure for quarks and leptons is fundamental to the Standard
Model.
1 We shall see in section 20.7 that a slight modification is necessary.
20. Introduction to the Phenomenology of Weak Interactions
where v, p and E are respectively the electron speed, momentum and energy,
P is the magnitude of the polarization, and θ is the angle of emission of the
electron with respect to ‹J ›.
Why does this indicate parity violation? To see this, we recall from the
discussion of the parity operation P in section 4.2.1 that the angular momentum J is an axial vector such that ‹J › → ‹J › under P, while p is a polar
vector transforming by p → −p. Hence, in the parity-transformed system,
the distribution (20.11) would have the form
I P (θ) = 1 + P v cos θ
(20.12)
The difference between (20.12) and (20.11) implies that, by performing the
measurement, we can determine which of the two coordinate systems we must
in fact be using. The two are inequivalent, in contrast to all the other coordinate system equivalences which we have previously studied (e.g. under threedimensional rotations, and Lorentz transformations). This is an operational
consequence of ‘parity violation’. The crucial point in this example, evidently,
is the appearance of the pseudoscalar quantity ‹J › · p in (20.10), alongside the
obviously scalar quantity ‘1’.
The Fermi theory, employing only vector currents, needs a modification
to accommodate this result. We saw in section 4.2.1 that a combination of
vector (‘V’) and axial vector (‘A’) currents would be parity-violating. Indeed,
after many years of careful experiments, and many false trails, it was eventually established (always, of course, to within some experimental error) that
the currents participating in Fermi’s current–current interaction are, in fact,
certain combinations of V-type and A-type currents, for both nucleons and
leptons.
20.2.2 V-A theory: chirality and helicity
Quite soon after the discovery of parity violation, Sudarshan and Marshak
(1958), and then Feynman and Gell-Mann (1958) and Sakurai (1958), proposed a specific form for the current–current interaction, namely the V-A
(‘V minus A’) structure. For example, in place of the leptonic combination
u ¯ e − γ μ u ν , these authors proposed the form ¯
u e − γ μ (1 − γ 5 )u ν , being the difference (with equal weight) of a V-type and an A-type current. For the part
involving the nucleons the proposal was slightly more complicated, having the
form ¯
u p γ μ (1 − rγ 5 )u n where r had the empirical value r ≈ 1.2. From our
present perspective, of course, the hadronic transition is actually occurring at
the quark level, so that rather than a transition n → p we now think in terms
of a d → u one. In this case, the remarkable fact is that the appropriate cur1
rent to use is, once again, essentially the simple ‘V-A’ one, ¯
u u γ μ (1 − γ 5 )u d .
This V-A structure for quarks and leptons is fundamental to the Standard
Model.
1 We shall see in section 20.7 that a slight modification is necessary.