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20. Introduction to the Phenomenology of Weak Interactions
Strictly speaking, neutrino masses and oscillations lie outside the framework
of the original Standard Model, and they are sometimes so regarded. Apart
from anything else, the phenomenology of massive neutrinos has to allow for
the possibility that they are Majorana, rather than Dirac, fermions. For the
moment, we shall continue with a semi-historical path, and proceed with weak
interaction phenomenology on the basis of the original Standard Model, with
massless neutrinos. We return to the question of neutrino mass when we discuss neutrino oscillations (along with analogous oscillations in meson systems)
in chapter 21.
20.4 The universal current × current theory for weak
interactions of leptons
After the breakthroughs of parity violation and V-A theory, the earlier hopes
(Pontecorvo 1947, Klein 1948, Puppi 1948, Lee, Rosenbluth and Yang 1949,
Tiomno and Wheeler 1949) were revived of a universal weak interaction among
the pairs of particles (p,n), (ν e , e
− ), (ν μ , μ
− ), using the V-A modification to
Fermi’s theory. From our modern standpoint, this list has to be changed
by the replacement of (p,n) by the corresponding quarks (u,d), and by the
inclusion of the third lepton pair (ν τ , τ
− ) as well as two other quark pairs
(c,s) and (t,b). It is to these pairs that the ‘V-A’ structure applies, as already
indicated in section 20.2.2, and a certain form of ‘universality’ does hold, as
we now describe.
Because of certain complications which arise, we shall postpone the discussion of the quark currents until section 20.7, concentrating here on the
¯ ˆ γ
μ ˆ
leptonic currents
2 . In this case, Fermi’s original vector-like current ψ
ψ ν
e
becomes modified to a total leptonic charged current
μ
μ
μ
μ
ˆ j (leptons) = ˆ j (e) + ˆ j (μ) + ˆ j (τ )
(20.38)
CC
wk
wk
wk
where, for example,
ˆ j
μ (e) = ν ¯ ˆ e γ
μ (1 − γ 5 )ˆ e.
(20.39)
wk
In (20.39) we are now adopting, for the first time, a useful shorthand whereby
the field operator for the electron field, say, is denoted by ˆ
e(x) rather than
ˆ
ψ e (x), and the ‘x’ argument is suppressed. The ‘charged’ current terminology
μ
refers to the fact that these weak current operators ˆ j carry net charge, in
wk
contrast to an electromagnetic current operator such as ¯ ˆ e which is eleceγ
μ ˆ
trically neutral. We shall see in section 20.6 that there are also electrically
neutral weak currents.
2 Very much the same complications arise for the leptonic currents too, in the case of
massive neutrinos, as we shall see in section 21.4.
20. Introduction to the Phenomenology of Weak Interactions
Strictly speaking, neutrino masses and oscillations lie outside the framework
of the original Standard Model, and they are sometimes so regarded. Apart
from anything else, the phenomenology of massive neutrinos has to allow for
the possibility that they are Majorana, rather than Dirac, fermions. For the
moment, we shall continue with a semi-historical path, and proceed with weak
interaction phenomenology on the basis of the original Standard Model, with
massless neutrinos. We return to the question of neutrino mass when we discuss neutrino oscillations (along with analogous oscillations in meson systems)
in chapter 21.
20.4 The universal current × current theory for weak
interactions of leptons
After the breakthroughs of parity violation and V-A theory, the earlier hopes
(Pontecorvo 1947, Klein 1948, Puppi 1948, Lee, Rosenbluth and Yang 1949,
Tiomno and Wheeler 1949) were revived of a universal weak interaction among
the pairs of particles (p,n), (ν e , e
− ), (ν μ , μ
− ), using the V-A modification to
Fermi’s theory. From our modern standpoint, this list has to be changed
by the replacement of (p,n) by the corresponding quarks (u,d), and by the
inclusion of the third lepton pair (ν τ , τ
− ) as well as two other quark pairs
(c,s) and (t,b). It is to these pairs that the ‘V-A’ structure applies, as already
indicated in section 20.2.2, and a certain form of ‘universality’ does hold, as
we now describe.
Because of certain complications which arise, we shall postpone the discussion of the quark currents until section 20.7, concentrating here on the
¯ ˆ γ
μ ˆ
leptonic currents
2 . In this case, Fermi’s original vector-like current ψ
ψ ν
e
becomes modified to a total leptonic charged current
μ
μ
μ
μ
ˆ j (leptons) = ˆ j (e) + ˆ j (μ) + ˆ j (τ )
(20.38)
CC
wk
wk
wk
where, for example,
ˆ j
μ (e) = ν ¯ ˆ e γ
μ (1 − γ 5 )ˆ e.
(20.39)
wk
In (20.39) we are now adopting, for the first time, a useful shorthand whereby
the field operator for the electron field, say, is denoted by ˆ
e(x) rather than
ˆ
ψ e (x), and the ‘x’ argument is suppressed. The ‘charged’ current terminology
μ
refers to the fact that these weak current operators ˆ j carry net charge, in
wk
contrast to an electromagnetic current operator such as ¯ ˆ e which is eleceγ
μ ˆ
trically neutral. We shall see in section 20.6 that there are also electrically
neutral weak currents.
2 Very much the same complications arise for the leptonic currents too, in the case of
massive neutrinos, as we shall see in section 21.4.