424
22. The GSW Gauge Theory of Electroweak Interactions
For Λ ∼ 10
16 GeV, this gives m H < 160 GeV. On the other hand, if the
non-perturbative regime sets in at 1TeV, then the bound on m H is weaker,
m H < 750 GeV.
This is an oversimplified argument for various reasons, though the essential
point is correct. An important omission is the contribution of the top quark to
the running of λ(E). A more refined version (Hambye and Riesselmann 1997)
concludes that for m H < 180 GeV the perturbative regime could extend all
the way to the Planck mass, ∼ 10
19 GeV.
There is another, independent, argument which suggests that m H cannot
be too large. We have previously considered violations of unitarity by the
lowest-order diagrams for certain processes (see chapter 21 and section 22.6).
As we saw, in a non-gauge theory with massive vector bosons, such violations
are associated with the longitudinal polarization states of the bosons, which
μ
carry factors proportional to the 4-momentum k (see (22.18)). In a gauge
theory, strong cancellations in the high energy behaviour occur between different lowest-order diagrams. This behaviour is characteristic of gauge theories
(Llewellyn Smith 1973, Cornwall et al. 1974), and is related to their renormalizability. One process of this sort which we did not yet consider, however,
is that in which two longitudinally polarized W’s scatter from each other. A
considerable number of diagrams (7 in all) contribute to this process, in leading order : exchange of γ, Z and Higgs particles, together with the W–W self
interaction. When all these are added up the high-energy behaviour of the
total amplitude turns out to be proportional to λ, the Higgs coupling constant
(see for example Ellis, Stirling and Webber 1996, chapter 8). This at first sight
unexpected result can be understood as follows. The longitudinal components
μ ˆ
of the W’s arise from the ‘∂ φ’ parts in (22.30) (compare equation (19.48) in
the U(1) case), which produce k
μ factors. Thus the scattering of longitudinal
W’s is effectively the scattering of the 3 Goldstone bosons in the complex
Higgs doublet. These bosons have self interactions arising from the λ(φ ˆ † φ ˆ )
2
Higgs potential, for which the Feynman amplitude is just proportional to λ.
Now, although such a constant term obviously cannot violate unitarity as the
energy increases (as happened in the other cases), it can do so if λ itself is too
2
big – and since λ ∝ m , this puts a bound on m H . A constant amplitude is
H
pure J = 0 and so, in order of magnitude, we expect unitarity to imply λ < 1.
In terms of standard quantities,
√
2
λ = m H G F / 2,
(22.205)
and so we expect
−1/2
m H < G
∼ 300 GeV,
(22.206)
F
an energy scale we have seen several times before. A more refined analysis
(Lee et al. 1977a,b) gives
( √ ) 1/2
8 2π
m H <
≈ 1TeV.
(22.207)
3G F
22. The GSW Gauge Theory of Electroweak Interactions
For Λ ∼ 10
16 GeV, this gives m H < 160 GeV. On the other hand, if the
non-perturbative regime sets in at 1TeV, then the bound on m H is weaker,
m H < 750 GeV.
This is an oversimplified argument for various reasons, though the essential
point is correct. An important omission is the contribution of the top quark to
the running of λ(E). A more refined version (Hambye and Riesselmann 1997)
concludes that for m H < 180 GeV the perturbative regime could extend all
the way to the Planck mass, ∼ 10
19 GeV.
There is another, independent, argument which suggests that m H cannot
be too large. We have previously considered violations of unitarity by the
lowest-order diagrams for certain processes (see chapter 21 and section 22.6).
As we saw, in a non-gauge theory with massive vector bosons, such violations
are associated with the longitudinal polarization states of the bosons, which
μ
carry factors proportional to the 4-momentum k (see (22.18)). In a gauge
theory, strong cancellations in the high energy behaviour occur between different lowest-order diagrams. This behaviour is characteristic of gauge theories
(Llewellyn Smith 1973, Cornwall et al. 1974), and is related to their renormalizability. One process of this sort which we did not yet consider, however,
is that in which two longitudinally polarized W’s scatter from each other. A
considerable number of diagrams (7 in all) contribute to this process, in leading order : exchange of γ, Z and Higgs particles, together with the W–W self
interaction. When all these are added up the high-energy behaviour of the
total amplitude turns out to be proportional to λ, the Higgs coupling constant
(see for example Ellis, Stirling and Webber 1996, chapter 8). This at first sight
unexpected result can be understood as follows. The longitudinal components
μ ˆ
of the W’s arise from the ‘∂ φ’ parts in (22.30) (compare equation (19.48) in
the U(1) case), which produce k
μ factors. Thus the scattering of longitudinal
W’s is effectively the scattering of the 3 Goldstone bosons in the complex
Higgs doublet. These bosons have self interactions arising from the λ(φ ˆ † φ ˆ )
2
Higgs potential, for which the Feynman amplitude is just proportional to λ.
Now, although such a constant term obviously cannot violate unitarity as the
energy increases (as happened in the other cases), it can do so if λ itself is too
2
big – and since λ ∝ m , this puts a bound on m H . A constant amplitude is
H
pure J = 0 and so, in order of magnitude, we expect unitarity to imply λ < 1.
In terms of standard quantities,
√
2
λ = m H G F / 2,
(22.205)
and so we expect
−1/2
m H < G
∼ 300 GeV,
(22.206)
F
an energy scale we have seen several times before. A more refined analysis
(Lee et al. 1977a,b) gives
( √ ) 1/2
8 2π
m H <
≈ 1TeV.
(22.207)
3G F