403
22.5. Fermion masses
that the unitarity bound will be saturated at E = E f (TeV) ∼ π/m f (TeV).
Thus for m t ∼ 175 GeV, E t ∼ 18 TeV. This would constitute a serious flaw
in the theory, even though the breakdown occurs at energies beyond those
currently reachable.
However, in a theory with spontaneous symmetry breaking, there is a way
of giving fermion masses without introducing an explicit mass term in the
Lagrangian. Consider the electron, for example, and let us hypothesize a
‘Yukawa’–type coupling between the electron-type SU(2) doublet
(
)
ν ˆ e
ˆ l eL =
,
(22.124)
−
ˆ e
L
the Higgs doublet φ ˆ , and the R-component of the electron field:
¯
ˆ
L ˆ e
ˆ ˆ
¯ φ
† ˆ
Yuk = −g e (l eL φe ˆ R + e ˆ R l eL ).
(22.125)
In each term of (22.125), the two SU(2) L doublets are ‘dotted together’ so as
to form an SU(2) L scalar, which multiplies the SU(2) L scalar R-component.
Thus (22.125) is SU(2) L -invariant, and the symmetry is preserved, at the
Lagrangian level, by such a term. But now insert just the vacuum value
(22.28) of φ ˆ into (22.125): we find the result
v
L ˆ e
( ¯
¯
Yuk (vac) = −g e √ e ˆ L e ˆ R + e ˆ R e ˆ L )
(22.126)
2
which is exactly a (Dirac) mass of the form (22.121), allowing us to make the
identification
√
m e = g e v/ 2.
(22.127)
When oscillations about the vacuum value are considered via the replacement (22.29), the term (22.125) will generate a coupling between the electron
and the Higgs fields of the form
√
¯ ˆ
ˆ
−g e e ˆe ˆH/ 2 = −(m e /v) ¯ e ˆe ˆH
(22.128)
= −(gm e
ˆeH.
(22.129)
/2M W ) ¯ eˆ ˆ
¯
The presence of such a coupling, if present for the process f f → W
+ W
−
0
0
considered earlier, will mean that, in addition to the f -exchange graph analogous to figure 22.4 and the annihilation graphs of figure 22.23, a further graph
shown in figure 22.24, must be included. The presence of the fermion mass in
the coupling to H suggests that this graph might be just what is required to
cancel the ‘bad’ high energy behaviour found in (22.123) – and by this time
the reader will not be surprised to be told that this is indeed the case.
At first sight it might seem that this stratagem will only work for the
= −
1 components of doublets, because of the form of ‹0|φ ˆ |0›. But we
t 3
2
(
)
u
learned in section 12.1.3 that if a pair of states
forming an SU(2)
d
22.5. Fermion masses
that the unitarity bound will be saturated at E = E f (TeV) ∼ π/m f (TeV).
Thus for m t ∼ 175 GeV, E t ∼ 18 TeV. This would constitute a serious flaw
in the theory, even though the breakdown occurs at energies beyond those
currently reachable.
However, in a theory with spontaneous symmetry breaking, there is a way
of giving fermion masses without introducing an explicit mass term in the
Lagrangian. Consider the electron, for example, and let us hypothesize a
‘Yukawa’–type coupling between the electron-type SU(2) doublet
(
)
ν ˆ e
ˆ l eL =
,
(22.124)
−
ˆ e
L
the Higgs doublet φ ˆ , and the R-component of the electron field:
¯
ˆ
L ˆ e
ˆ ˆ
¯ φ
† ˆ
Yuk = −g e (l eL φe ˆ R + e ˆ R l eL ).
(22.125)
In each term of (22.125), the two SU(2) L doublets are ‘dotted together’ so as
to form an SU(2) L scalar, which multiplies the SU(2) L scalar R-component.
Thus (22.125) is SU(2) L -invariant, and the symmetry is preserved, at the
Lagrangian level, by such a term. But now insert just the vacuum value
(22.28) of φ ˆ into (22.125): we find the result
v
L ˆ e
( ¯
¯
Yuk (vac) = −g e √ e ˆ L e ˆ R + e ˆ R e ˆ L )
(22.126)
2
which is exactly a (Dirac) mass of the form (22.121), allowing us to make the
identification
√
m e = g e v/ 2.
(22.127)
When oscillations about the vacuum value are considered via the replacement (22.29), the term (22.125) will generate a coupling between the electron
and the Higgs fields of the form
√
¯ ˆ
ˆ
−g e e ˆe ˆH/ 2 = −(m e /v) ¯ e ˆe ˆH
(22.128)
= −(gm e
ˆeH.
(22.129)
/2M W ) ¯ eˆ ˆ
¯
The presence of such a coupling, if present for the process f f → W
+ W
−
0
0
considered earlier, will mean that, in addition to the f -exchange graph analogous to figure 22.4 and the annihilation graphs of figure 22.23, a further graph
shown in figure 22.24, must be included. The presence of the fermion mass in
the coupling to H suggests that this graph might be just what is required to
cancel the ‘bad’ high energy behaviour found in (22.123) – and by this time
the reader will not be surprised to be told that this is indeed the case.
At first sight it might seem that this stratagem will only work for the
= −
1 components of doublets, because of the form of ‹0|φ ˆ |0›. But we
t 3
2
(
)
u
learned in section 12.1.3 that if a pair of states
forming an SU(2)
d