160 The very early universe
where
GF≡
1
4
√
2
g^2
MW^2=
π
√
2αw^2
MW^2 1. 166 × 10 −^5 GeV−^2 (4.75)is Fermi coupling constant. The experimental value ofGF,the Weinberg angle
θw 28. 7 ◦
(
sin^2 θw 0. 23)
, determined in the neutral current interactions, and
the measured masses
MW 80 .4 GeV, MZ 91 .2 GeV,are in very good agreement with the theoretical predictions of the standard
electroweak model. From (4.69), “the weak fine structure constant”αw 1 /29 is
seen to be 4.5 times larger than the fine structure constant. As follows from (4.66)
and (4.68), the expectation value for the Higgs field is
χ 0 =2 MWsinθw
e250 GeV. (4.76)However, since the quartic coupling constantλcan be arbitrary, the Higgs mass,
given in (4.65), is not predicted. Higgs particles have not yet been discovered.
The experimental lower bound on their masses ismH>114 GeV.Requiring
self-consistency of the theory, namely, the validity of the perturbative expansion
inλ, one could expect thatλ<1 and hencemHcannot greatly exceed 350 GeV.
Problem 4.14Given that the electric charges ofuanddquarks equal+ 2 /3 and
− 1 /3 respectively, determine their hypercharges. Derive the Lagrangian for the
first quark generation. (Note that bothuRanddRare present in the Lagrangian as
SU( 2 )singlets.) Draw the vertices describing the quark weak interactions.
The neutron decay can be interpreted as the underlying quark process:d→
u+W.As a result the neutron, which is a bound state of three quarksudd,is
converted into the proton consisting ofuudquarks.Because quarks always appear
in bound systems, the calculation of the weak interactions of hadrons is more
complicated and suffers from various uncertainties.
4.3.5 Fermion masses
Until now we have treated all fermions as massless. This is obviously in disagree-
ment with experiment. However, fermion masses, introduced by hand, would spoil
gauge invariance and renormalizability. Therefore, the only way to generate them
is again to use the Higgs mechanism.