405
22.5. Fermion masses
It is clearly possible to go on like this, and arrange for all the fermions,
quarks as well as leptons, to acquire a mass by the same ‘mechanism’. We will
look more closely at the quarks in the next section. But one must admit to
a certain uneasiness concerning the enormous difference in magnitudes represented by the couplings g νe , . . . g e , . . . g t . If m νe < 1 eV then g νe < 10
−11 , while
g t ∼ 1! Besides, whereas the use of the Higgs field ‘mechanism’ in the W–Z
sector is quite economical, in the present case it seems rather unsatisfactory
simply to postulate a different ‘g’ for each fermion–Higgs interaction. This
does appear to indicate that we are dealing here with a ‘phenomenological
model’, once more, rather than a ‘theory’.
As far as the neutrinos are concerned, however, there is another possibility,
already discussed in sections 7.5.2, 20.3 and 21.4.1, which is that they could
be Majorana (not Dirac) fermions. In this case, rather than the four degrees
of freedom (ν eL , ν eR , and their antiparticles) which exist for massive Dirac
particles, only two possibilities exist for neutrinos, which we may take to be
ν eL and ν eR . With these, it is certainly possible to construct a Dirac-type
mass term of the form (22.134). But since, after all, the ν eR component has
been assigned zero quantum members both for SU(2) L W-interactions and for
U(1) B-interactions (see table 22.1), we could consider economically dropping
it altogether, making do with just the ν eL component.
Suppose, then, that we keep only the field ˆ
ν eL . We need to form a mass
term for it. The charge-conjugate field is defined by (see (7.151))
T
†T
¯
(ˆ ν eL ) C = iγ
2 γ 0 ν ˆ eL = iγ
2 ν ˆ ,
(22.135)
L
and we know that the charge-conjugate field transforms under Lorentz transformations in the same way as the original field. So we can use (ˆ ν eL ) C to form
a Lorentz invariant
(ˆ ν eL ) C ν eL
(22.136)
which has mass dimension M
3 . Hence we may write a mass term for ˆ
ν eL in
the form
1
−
¯
m M [(ˆ ν eL ) C ν ˆ eL + ν ˆ eL (ˆ ν eL ) C ]
(22.137)
2
where the
1 is conventional. Written out in more detail, we have
2
ν
T
ν
T
(ˆ ν eL ) C ν ˆ eL = ˆ eL (−iγ
2† γ 0 )ˆ ν eL = ˆ eL iγ
2 γ 0 ν ˆ eL ,
(22.138)
in the representation (20.14). Now
(
)
−iσ 2 0
iγ
2 γ 0 =
.
(22.139)
0
iσ 2
But since ˆ
ν eL is an L-chiral field, only its 2 lower components are present (cf
(20.26)) and (22.138) is effectively
ν
T
(ˆ ν eL ) C ν ˆ eL = ˆ eL (iσ 2 )ˆ ν eL .
(22.140)
22.5. Fermion masses
It is clearly possible to go on like this, and arrange for all the fermions,
quarks as well as leptons, to acquire a mass by the same ‘mechanism’. We will
look more closely at the quarks in the next section. But one must admit to
a certain uneasiness concerning the enormous difference in magnitudes represented by the couplings g νe , . . . g e , . . . g t . If m νe < 1 eV then g νe < 10
−11 , while
g t ∼ 1! Besides, whereas the use of the Higgs field ‘mechanism’ in the W–Z
sector is quite economical, in the present case it seems rather unsatisfactory
simply to postulate a different ‘g’ for each fermion–Higgs interaction. This
does appear to indicate that we are dealing here with a ‘phenomenological
model’, once more, rather than a ‘theory’.
As far as the neutrinos are concerned, however, there is another possibility,
already discussed in sections 7.5.2, 20.3 and 21.4.1, which is that they could
be Majorana (not Dirac) fermions. In this case, rather than the four degrees
of freedom (ν eL , ν eR , and their antiparticles) which exist for massive Dirac
particles, only two possibilities exist for neutrinos, which we may take to be
ν eL and ν eR . With these, it is certainly possible to construct a Dirac-type
mass term of the form (22.134). But since, after all, the ν eR component has
been assigned zero quantum members both for SU(2) L W-interactions and for
U(1) B-interactions (see table 22.1), we could consider economically dropping
it altogether, making do with just the ν eL component.
Suppose, then, that we keep only the field ˆ
ν eL . We need to form a mass
term for it. The charge-conjugate field is defined by (see (7.151))
T
†T
¯
(ˆ ν eL ) C = iγ
2 γ 0 ν ˆ eL = iγ
2 ν ˆ ,
(22.135)
L
and we know that the charge-conjugate field transforms under Lorentz transformations in the same way as the original field. So we can use (ˆ ν eL ) C to form
a Lorentz invariant
(ˆ ν eL ) C ν eL
(22.136)
which has mass dimension M
3 . Hence we may write a mass term for ˆ
ν eL in
the form
1
−
¯
m M [(ˆ ν eL ) C ν ˆ eL + ν ˆ eL (ˆ ν eL ) C ]
(22.137)
2
where the
1 is conventional. Written out in more detail, we have
2
ν
T
ν
T
(ˆ ν eL ) C ν ˆ eL = ˆ eL (−iγ
2† γ 0 )ˆ ν eL = ˆ eL iγ
2 γ 0 ν ˆ eL ,
(22.138)
in the representation (20.14). Now
(
)
−iσ 2 0
iγ
2 γ 0 =
.
(22.139)
0
iσ 2
But since ˆ
ν eL is an L-chiral field, only its 2 lower components are present (cf
(20.26)) and (22.138) is effectively
ν
T
(ˆ ν eL ) C ν ˆ eL = ˆ eL (iσ 2 )ˆ ν eL .
(22.140)