33
1.4. Renormalization and the Higgs sector of the Standard Model
Apart from giving mass to the W
± and Z
0 , the Higgs fields have more
work to do. The electroweak gauge symmetry is exact only if all the fermion
masses are zero; this is because it is a chiral symmetry (similar to, but not
the same as, the chiral symmetry of QCD mentioned in section 1.2.2). Once
again, this chiral gauge symmetry is essential to the renormalizability of the
theory: if the fermion masses are incorporated in the usual way as parameters
in the Lagrangian, the latter is no longer gauge invariant and the theory is
non-renormalizable. In the SM, this problem is solved by having no fermion
masses in the Lagrangian, and by postulating gauge-invariant Yukawa interactions between the fermions and the Higgs fields, which are arranged in such
a way that, when the Higgs field gets a vacuum expectation value, the interaction terms yield just the fermion masses. So again, the symmetry breaking
is economically blamed on the same property of the vacuum. When the Higgs
field oscillates away from its vacuum value, the result will be residual interactions between the fermions and the Higgs boson, which will have the
defining characteristic that each fermion will interact with the Higgs boson
with a strength proportional to its (i.e. the fermion’s) mass. This is clearly a
testable prediction, once the Higgs boson is found.
We have emphasized the role that the Higgs fields play in the renormalizability of the GSW theory. The all-important proof of that renormalizability
was given by ’t Hooft (1971b), and he also proved the renormalizability of
QCD (1971a); see also ’t Hooft and Veltman (1972).
The SM Higgs sector is the simplest one that will do the job; more complicated versions are possible. Perhaps the Higgs field is a composite formed in
some new heavy fermion-antifermion dynamics, reminiscent of BCS pairing.
In any case, the SM Higgs sector is there to be tested experimentally. In the
following section we shall discuss briefly what is presently known about the
SM Higgs boson, postponing a fuller discussion until we present the GSW
theory in chapter 22 in volume 2.
Before ending this section we must note that modern renormalization theory is concerned with more than perturbative calculability. The renormalization group and related ideas provide powerful tools for ‘improving’ perturbation theory, by systematically resumming terms which (in the particle physics
case) dominate at short distances. Prominent among the results of this analysis (see chapters 15 and 16) are the concepts of energy-dependent (‘running’)
masses and coupling strengths, and the calculation of QCD corrections to
parton-model predictions.
1.4.2 The Higgs boson of the Standard Model
According to the SM, just one neutral spin-0 Higgs boson is expected; its
mass m H is not predicted by the theory. The experimental discovery of the
SM Higgs boson has been a major goal of several generations of accelerators:
−
the LEP e
+ e collider at Cern, the Tevatron p¯ p collider at Fermilab, and
most recently the LHC pp collider at Cern. Experimentally, bounds on the
1.4. Renormalization and the Higgs sector of the Standard Model
Apart from giving mass to the W
± and Z
0 , the Higgs fields have more
work to do. The electroweak gauge symmetry is exact only if all the fermion
masses are zero; this is because it is a chiral symmetry (similar to, but not
the same as, the chiral symmetry of QCD mentioned in section 1.2.2). Once
again, this chiral gauge symmetry is essential to the renormalizability of the
theory: if the fermion masses are incorporated in the usual way as parameters
in the Lagrangian, the latter is no longer gauge invariant and the theory is
non-renormalizable. In the SM, this problem is solved by having no fermion
masses in the Lagrangian, and by postulating gauge-invariant Yukawa interactions between the fermions and the Higgs fields, which are arranged in such
a way that, when the Higgs field gets a vacuum expectation value, the interaction terms yield just the fermion masses. So again, the symmetry breaking
is economically blamed on the same property of the vacuum. When the Higgs
field oscillates away from its vacuum value, the result will be residual interactions between the fermions and the Higgs boson, which will have the
defining characteristic that each fermion will interact with the Higgs boson
with a strength proportional to its (i.e. the fermion’s) mass. This is clearly a
testable prediction, once the Higgs boson is found.
We have emphasized the role that the Higgs fields play in the renormalizability of the GSW theory. The all-important proof of that renormalizability
was given by ’t Hooft (1971b), and he also proved the renormalizability of
QCD (1971a); see also ’t Hooft and Veltman (1972).
The SM Higgs sector is the simplest one that will do the job; more complicated versions are possible. Perhaps the Higgs field is a composite formed in
some new heavy fermion-antifermion dynamics, reminiscent of BCS pairing.
In any case, the SM Higgs sector is there to be tested experimentally. In the
following section we shall discuss briefly what is presently known about the
SM Higgs boson, postponing a fuller discussion until we present the GSW
theory in chapter 22 in volume 2.
Before ending this section we must note that modern renormalization theory is concerned with more than perturbative calculability. The renormalization group and related ideas provide powerful tools for ‘improving’ perturbation theory, by systematically resumming terms which (in the particle physics
case) dominate at short distances. Prominent among the results of this analysis (see chapters 15 and 16) are the concepts of energy-dependent (‘running’)
masses and coupling strengths, and the calculation of QCD corrections to
parton-model predictions.
1.4.2 The Higgs boson of the Standard Model
According to the SM, just one neutral spin-0 Higgs boson is expected; its
mass m H is not predicted by the theory. The experimental discovery of the
SM Higgs boson has been a major goal of several generations of accelerators:
−
the LEP e
+ e collider at Cern, the Tevatron p¯ p collider at Fermilab, and
most recently the LHC pp collider at Cern. Experimentally, bounds on the