423
22.8. The Higgs sector
elucidation of the mechanism of gauge symmetry breaking is undoubtedly of
the greatest importance to particle physics: quite apart from the SU(2) L ×U(1)
theory, very many of the proposed theories which go ‘beyond the Standard
Model’ face a similar ‘mass problem’, and generally appeal to some variant of
the ‘Higgs mechanism’ to deal with it.
As Higgs noted in the final paragraph of his 2-page Letter (Higgs 1964), an
essential feature of the spontaneous symmetry breaking mechanism, in a gauge
theory, is the appearance of incomplete multiplets of both scalar and vector
bosons. Let us just rehearse this once more, in the SU(2) × U(1) case. We
started with 4 massless gauge fields, belonging to an SU(2) triplet and a U(1)
singlet; and, in addition, 4 scalar fields of equal mass, in an SU(2) doublet.
After symmetry breaking, three massive vector bosons emerged, leaving the
photon massless. In the scalar sector, three of the scalars became the longitudinal components of the three massive vector bosons, and one lone massive
scalar field survived, all that remained of the original scalar doublet. Its mass
√
√
√
is a free parameter of the theory, being given by m H = 2μ = λv/ 2. The
discovery – or otherwise – of this Higgs boson has therefore been a vital goal in
particle physics for over forty years. Before turning to experiment, however,
we want to mention some theoretical considerations concerning m H by way of
orientation.
22.8.2 Theoretical considerations concerning m H
The coupling constant λ, which determines m H given the known value of v,
is unfortunately undetermined in the Standard Model. However, some quite
strong theoretical arguments suggest that m H cannot be arbitrarily large.
Like all coupling constants in a renormalizable theory, λ must ‘run’. For
the (φ ˆ † φ ˆ )
2 interaction of (22.30), a one-loop calculation of the β-function leads
to
λ(v)
λ(E) =
.
(22.202)
1 −
3λ(v)
8π 2 ln(E/v)
Like QED, this theory is not asymptotically free: the coupling increases with
the scale E. In fact, the theory becomes non-perturbative at the scale E
∗ such
that
(
)
8π
2
E
∗
∼ v exp
.
(22.203)
3λ(v)
Note that this is exponentially sensitive to the ‘low-energy’ coupling constant
λ(v) – and that E
∗ decreases rapidly as λ(v) increases. But (see (22.40)) m H is
essentially proportional to λ
1/2 (v). Hence as m H increases, non-perturbative
behaviour sets in increasingly early. Suppose we say that we should like perturbative behaviour to be maintained up to an energy scale Λ. Then we
require
[
] 1/2
4π
2
m H < v
.
(22.204)
3 ln(Λ/v)
22.8. The Higgs sector
elucidation of the mechanism of gauge symmetry breaking is undoubtedly of
the greatest importance to particle physics: quite apart from the SU(2) L ×U(1)
theory, very many of the proposed theories which go ‘beyond the Standard
Model’ face a similar ‘mass problem’, and generally appeal to some variant of
the ‘Higgs mechanism’ to deal with it.
As Higgs noted in the final paragraph of his 2-page Letter (Higgs 1964), an
essential feature of the spontaneous symmetry breaking mechanism, in a gauge
theory, is the appearance of incomplete multiplets of both scalar and vector
bosons. Let us just rehearse this once more, in the SU(2) × U(1) case. We
started with 4 massless gauge fields, belonging to an SU(2) triplet and a U(1)
singlet; and, in addition, 4 scalar fields of equal mass, in an SU(2) doublet.
After symmetry breaking, three massive vector bosons emerged, leaving the
photon massless. In the scalar sector, three of the scalars became the longitudinal components of the three massive vector bosons, and one lone massive
scalar field survived, all that remained of the original scalar doublet. Its mass
√
√
√
is a free parameter of the theory, being given by m H = 2μ = λv/ 2. The
discovery – or otherwise – of this Higgs boson has therefore been a vital goal in
particle physics for over forty years. Before turning to experiment, however,
we want to mention some theoretical considerations concerning m H by way of
orientation.
22.8.2 Theoretical considerations concerning m H
The coupling constant λ, which determines m H given the known value of v,
is unfortunately undetermined in the Standard Model. However, some quite
strong theoretical arguments suggest that m H cannot be arbitrarily large.
Like all coupling constants in a renormalizable theory, λ must ‘run’. For
the (φ ˆ † φ ˆ )
2 interaction of (22.30), a one-loop calculation of the β-function leads
to
λ(v)
λ(E) =
.
(22.202)
1 −
3λ(v)
8π 2 ln(E/v)
Like QED, this theory is not asymptotically free: the coupling increases with
the scale E. In fact, the theory becomes non-perturbative at the scale E
∗ such
that
(
)
8π
2
E
∗
∼ v exp
.
(22.203)
3λ(v)
Note that this is exponentially sensitive to the ‘low-energy’ coupling constant
λ(v) – and that E
∗ decreases rapidly as λ(v) increases. But (see (22.40)) m H is
essentially proportional to λ
1/2 (v). Hence as m H increases, non-perturbative
behaviour sets in increasingly early. Suppose we say that we should like perturbative behaviour to be maintained up to an energy scale Λ. Then we
require
[
] 1/2
4π
2
m H < v
.
(22.204)
3 ln(Λ/v)