406
22. The GSW Gauge Theory of Electroweak Interactions
This is just the form of the mass term for a Majorana field, as we saw in
equation (7.159). The two formalisms are equivalent.
As noted in section 21.4.1, the mass term (22.137) is not invariant under
a global U(1) phase transformation
−iα ˆ
ν ˆ eL → e ν eL
(22.141)
which would correspond to lepton number (if accompanied by a similar transformation for the electron fields): the Majorana mass term violates lepton
number conservation.
There is a further interesting aspect to (22.140) which is that, since two
ν ˆ eL operators appear rather than a ˆ
ν e and a ˆ
ν e
† (which would lead to L e
conservation), the (t, t 3 ) quantum numbers of the term are (1,1). This means
that we cannot form an SU(2) L invariant with it, using only the Standard
1
Model Higgs φ ˆ , since the latter has t = and cannot combine with the (1,1)
2
operator to form a singlet. Thus we cannot make a ‘tree-level’ Majorana
mass by the mechanism of Yukawa coupling to the Higgs field, followed by
symmetry breaking.
However, we could generate suitable ‘effective’ operators via loop corrections, perhaps, much as we generated an effective operator representing an
anomalous magnetic moment interaction in QED (cf section 11.7). But whatever it is, the operator would have to violate lepton number conservation,
which is actually conserved by all the Standard Model interactions. Thus
such an effective operator could not be generated in perturbation theory. It
could arise, however, as a low energy limit of a theory defined at a higher
mass scale, as the current–current model is the low energy limit of the GSW
one. The typical form of such operator we need, in order to generate a term
ν ˆ
T iσ 2 ν ˆ eL , is
eL
g eM ¯ ˆ
† ˆ
−
(l eL φ ˆ C )
T iσ 2 (φ ˆ l eL ).
(22.142)
C
M
Note, most importantly, that the operator ‘(lφ)(φl)’ in (22.142) has mass
dimension five, which is why we introduced the factor M
−1 in the coupling;
it is indeed a non-renormalizable effective interaction, just like the current–
current one. We may interpret M as the mass scale at which ‘new physics’
enters, in the spirit of the discussion in section 11.7. Suppose, for the sake of
argument, this was M ∼ 10
16 GeV (a scale typical of Grand Unified Theories).
After symmetry breaking, then, (22.142) will generate the required Majorana
mass term, with
2
v
m M ∼ g eM
∼ g eM 10
−2 eV.
(22.143)
M
Thus an effective coupling of ‘natural’ size g eM ∼ 0.1 emerges from this
argument, if indeed the mass of the ν e is of order 10
−3 eV.
A more specific model can be constructed in which a relation of the form
(22.143) can arise naturally. Suppose ˆ
ν R is an R-type neutrino field which is
an SU(2) × U(1) singlet, and which has a gauge-invariant Yukawa coupling
22. The GSW Gauge Theory of Electroweak Interactions
This is just the form of the mass term for a Majorana field, as we saw in
equation (7.159). The two formalisms are equivalent.
As noted in section 21.4.1, the mass term (22.137) is not invariant under
a global U(1) phase transformation
−iα ˆ
ν ˆ eL → e ν eL
(22.141)
which would correspond to lepton number (if accompanied by a similar transformation for the electron fields): the Majorana mass term violates lepton
number conservation.
There is a further interesting aspect to (22.140) which is that, since two
ν ˆ eL operators appear rather than a ˆ
ν e and a ˆ
ν e
† (which would lead to L e
conservation), the (t, t 3 ) quantum numbers of the term are (1,1). This means
that we cannot form an SU(2) L invariant with it, using only the Standard
1
Model Higgs φ ˆ , since the latter has t = and cannot combine with the (1,1)
2
operator to form a singlet. Thus we cannot make a ‘tree-level’ Majorana
mass by the mechanism of Yukawa coupling to the Higgs field, followed by
symmetry breaking.
However, we could generate suitable ‘effective’ operators via loop corrections, perhaps, much as we generated an effective operator representing an
anomalous magnetic moment interaction in QED (cf section 11.7). But whatever it is, the operator would have to violate lepton number conservation,
which is actually conserved by all the Standard Model interactions. Thus
such an effective operator could not be generated in perturbation theory. It
could arise, however, as a low energy limit of a theory defined at a higher
mass scale, as the current–current model is the low energy limit of the GSW
one. The typical form of such operator we need, in order to generate a term
ν ˆ
T iσ 2 ν ˆ eL , is
eL
g eM ¯ ˆ
† ˆ
−
(l eL φ ˆ C )
T iσ 2 (φ ˆ l eL ).
(22.142)
C
M
Note, most importantly, that the operator ‘(lφ)(φl)’ in (22.142) has mass
dimension five, which is why we introduced the factor M
−1 in the coupling;
it is indeed a non-renormalizable effective interaction, just like the current–
current one. We may interpret M as the mass scale at which ‘new physics’
enters, in the spirit of the discussion in section 11.7. Suppose, for the sake of
argument, this was M ∼ 10
16 GeV (a scale typical of Grand Unified Theories).
After symmetry breaking, then, (22.142) will generate the required Majorana
mass term, with
2
v
m M ∼ g eM
∼ g eM 10
−2 eV.
(22.143)
M
Thus an effective coupling of ‘natural’ size g eM ∼ 0.1 emerges from this
argument, if indeed the mass of the ν e is of order 10
−3 eV.
A more specific model can be constructed in which a relation of the form
(22.143) can arise naturally. Suppose ˆ
ν R is an R-type neutrino field which is
an SU(2) × U(1) singlet, and which has a gauge-invariant Yukawa coupling