411
22.6. Higher-order corrections
are indeed dealing with a renormalizable spontaneously broken gauge theory:
renormalizable, because no extra parameters, not in the original Lagrangian,
have had to be introduced; a gauge theory, because the fermion and gauge
boson couplings obey the relations imposed by the local SU(2) × U(1) symmetry; and spontaneously broken because the same symmetry is not seen in
the particle spectrum (consider the mass separation in the t-b doublet, for
instance).
In fact, one can turn this around, in more than one way. First, one crucially
important element in the theory – the Higgs boson – has a mass m H which
is largely unconstrained by theory (see section 22.8.2), and it is therefore a
parameter in the fits. Some information about m H can therefore be gained by
seeing how the fits vary with m H . Actually, we shall see in equation (22.181)
that the dependence on m H is only logarithmic – it acts rather like a cut-off,
so the fits are not very sensitive to m H . The 90 % central confidence range
from all precision data is given by Erler and Langacker as 55 GeV< m H <
135 GeV. By contrast, some loop corrections are proportional to the square
of the top mass (see (22.180)) and consequently very tight bounds could be
placed on m t via its virtual presence (i.e. in loops, for example as shown
in figure 22.25) before its real presence was confirmed, as we shall discuss
shortly and in section 22.7. Secondly, it is still entirely possible that very
careful analysis of small discrepancies between precision data and electroweak
predictions may indicate the presence of ‘new physics’.
After all this (and earlier) emphasis on the renormalizability of the electroweak theory, and the introduction to one-loop calculations in QED at the
end of volume 1, the reader perhaps has a right to expect, now, an exposition
of loop corrections in the electroweak theory. But the fact is that this is a very
complicated and technical story, requiring quite a bit more formal machinery,
which would be outside the intended scope of this book (suitable references include Altarelli et al. 1989, especially the pedagogical account by Consoli et al.
1989; and the equally approachable lectures by Hollik 1991). Instead, we want
to touch on just a few of the simpler and more important aspects of one-loop
corrections, especially insofar as they have phenomenological implications.
As we have seen, we obtain cut-off independent results from loop corrections in a renormalizable theory by taking the values of certain parameters –
those appearing in the original Lagrangian – from experiment, according to
a well-defined procedure (‘renormalization scheme’). In the electroweak case,
the parameters in the Lagrangian are
′
gauge couplings g, g
(22.167)
Higgs potential parameters λ, μ
2
(22.168)
Higgs–fermion Yukawa couplings g f
(22.169)
CKM angles θ 12 , θ 13 , θ 23 ; phase δ
(22.170)
PMNS angles θ e2 , θ e3 , θ μ3 , phase δ
ν (+ α 21 , α 31 ?).
(22.171)
22.6. Higher-order corrections
are indeed dealing with a renormalizable spontaneously broken gauge theory:
renormalizable, because no extra parameters, not in the original Lagrangian,
have had to be introduced; a gauge theory, because the fermion and gauge
boson couplings obey the relations imposed by the local SU(2) × U(1) symmetry; and spontaneously broken because the same symmetry is not seen in
the particle spectrum (consider the mass separation in the t-b doublet, for
instance).
In fact, one can turn this around, in more than one way. First, one crucially
important element in the theory – the Higgs boson – has a mass m H which
is largely unconstrained by theory (see section 22.8.2), and it is therefore a
parameter in the fits. Some information about m H can therefore be gained by
seeing how the fits vary with m H . Actually, we shall see in equation (22.181)
that the dependence on m H is only logarithmic – it acts rather like a cut-off,
so the fits are not very sensitive to m H . The 90 % central confidence range
from all precision data is given by Erler and Langacker as 55 GeV< m H <
135 GeV. By contrast, some loop corrections are proportional to the square
of the top mass (see (22.180)) and consequently very tight bounds could be
placed on m t via its virtual presence (i.e. in loops, for example as shown
in figure 22.25) before its real presence was confirmed, as we shall discuss
shortly and in section 22.7. Secondly, it is still entirely possible that very
careful analysis of small discrepancies between precision data and electroweak
predictions may indicate the presence of ‘new physics’.
After all this (and earlier) emphasis on the renormalizability of the electroweak theory, and the introduction to one-loop calculations in QED at the
end of volume 1, the reader perhaps has a right to expect, now, an exposition
of loop corrections in the electroweak theory. But the fact is that this is a very
complicated and technical story, requiring quite a bit more formal machinery,
which would be outside the intended scope of this book (suitable references include Altarelli et al. 1989, especially the pedagogical account by Consoli et al.
1989; and the equally approachable lectures by Hollik 1991). Instead, we want
to touch on just a few of the simpler and more important aspects of one-loop
corrections, especially insofar as they have phenomenological implications.
As we have seen, we obtain cut-off independent results from loop corrections in a renormalizable theory by taking the values of certain parameters –
those appearing in the original Lagrangian – from experiment, according to
a well-defined procedure (‘renormalization scheme’). In the electroweak case,
the parameters in the Lagrangian are
′
gauge couplings g, g
(22.167)
Higgs potential parameters λ, μ
2
(22.168)
Higgs–fermion Yukawa couplings g f
(22.169)
CKM angles θ 12 , θ 13 , θ 23 ; phase δ
(22.170)
PMNS angles θ e2 , θ e3 , θ μ3 , phase δ
ν (+ α 21 , α 31 ?).
(22.171)