4.3 Electroweak theory 161
The gauge-invariant Yukawa coupling between scalars and fermions for the first
lepton generation is
LeY=−fe(
ψ ̄eLφeR+e ̄Rφ†ψeL)
, (4.77)
wherefeis the dimensionless Yukawa coupling constant. This term is obviously
SU( 2 )-invariant and, if the hypercharges satisfy the condition
YL=YR+Yφ, (4.78)it is also invariant with respect toU( 1 )transformations.Substitutingφ=χζφ 0 (see
(4.47)) in (4.77), we can rewrite the Yukawa coupling in terms ofSU( 2 )-invariant
variables as
LeY=−feχΨ ̄eLφ 0 eR+h.c.=−feχ(e ̄LeR+e ̄ReL)=−feχee ̄, (4.79)where h.c. denotes the Hermitian conjugated term. To write the last equality we
have used a well known relation from the theory of Dirac spinors. If the scalar field
takes a nonzero expectation value,so thatχ=χ 0 +φ,the electron acquires the
mass
me=feχ 0. (4.80)Withχ 0 given in (4.76) andfe 2 × 10 −^6 ,we get the correct value for the electron
mass.The appearance of such a small coupling constant has no natural explanation
in the electroweak theory, wherefeis a free parameter. The termfeφ ̄eedescribes
the interaction of Higgs particles with electrons. Note that the particular form of the
Yukawa coupling (4.77) gives mass only to the lower component of the doublet.
The neutrino remains massless.
For quarks some complications arise. First of all, both components of the doublets
should acquire masses. Second, to explain flavor nonconservation in the weak
interactions, we have to assume that the lower components of theSU( 2 )doublets are
superpositions of the lower quark flavors and hence are not quark mass eigenstates.
This suggests we should simultaneously consider all three quark generations. Let
us denote the upper and lower components of theSU( 2 )gauge-invariantquark
doublets by
ui≡(u,c,t)and
d′i≡(
d′,s′,b′