422
2
22. The GSW Gauge Theory of Electroweak Interactions
the Higgs case, the matter is much more delicate. The whole phenomenology
depends on the renormalized coefficient having a negative value, triggering
the spontaneous breaking of the symmetry. This means that the O(Λ
2 ) one2 φ ˆ† ˆ
loop correction must be cancelled by the ‘bare’ mass term
1 m
φ so as to
2 H,0
achieve a negative coefficient of order −v
2 . This cancellation between m H,0
and Λ
2 will have to be very precise indeed if Λ – the scale of ‘new physics’ –
is very high, as is commonly assumed (say 10
16 GeV).
The reader may wonder why attention should now be drawn to this particular piece of renormalization: aren’t all divergences handled this way? In a
sense they are, but the fact is that this is the first case we have had in which
we have to cancel a quadratic divergence. The other mass-corrections have all
been logarithmic, for which there is nothing like such a dramatic ‘fine-tuning’
problem. There is a good reason for this in the case of the electron mass,
which we remarked on in section 11.2. Chiral symmetry forces self-energy
corrections for fermions to be proportional to their mass, and hence to contain only logarithms of the cut-off. Similarly, gauge invariance for the vector
bosons prohibits any O(Λ
2 ) connections in perturbation theory. But there
is no symmetry, within the Standard Model, which ‘protects’ the coefficient
of φ ˆ† φ ˆ in this way. It is hard to understand what can be stopping it from
being of order Λ
2 , if we take the apparently reasonable point of view that the
Standard Model will ultimately fail at some scale Λ where new physics enters.
Thus the difficulty is: why is the empirical parameter v ‘shielded’ from the
presumed high scale of new physics? This ‘problem’ is often referred to as the
‘hierarchy problem’, or the ‘fine-tuning problem’. We stress again that we are
dealing here with an absolutely crucial symmetry-breaking term, which one
would really like to understand far better.
Of course, the problem would go away if the scale Λ were as low as, say
a few TeV. As we shall see in the next section this happens to be, not accidentally, the same scale at which the Standard Model ceases to be a perturbatively calculable theory. Various possibilities have been suggested for
the kind of physics that might enter at energies of a few TeV. For example,
‘technicolour’ models (Peskin 1997) regard the Higgs field as a composite of
some new heavy fermions, rather like the BCS-pairing idea referred to earlier. A second possibility is supersymmetry (Aitchison 2007), in which there
is a ‘protective’ symmetry operating, since scalar fields can be put alongside
fermions in supermultiplets, and benefit from the protection enjoyed by the
fermions. A third possibility is that of ‘large’ extra dimensions (Antoniadis
2002).
These undoubtedly fascinating ideas obviously take us well beyond our
proper subject to which we must now return. Whatever may lie ‘beyond’
it, the Lagrangian of the Higgs sector of the Standard Model leads to many
perfectly definite predictions which may be confronted with experiment, as we
shall briefly discuss in section 22.8.3 (for a full account see Dawson et al. 1990,
and for more compact ones see Ellis, Stirling and Webber 1996, chapter 11,
and the review by Bernardi, Carena and Junk in Beringer et al. 2012). The
2
22. The GSW Gauge Theory of Electroweak Interactions
the Higgs case, the matter is much more delicate. The whole phenomenology
depends on the renormalized coefficient having a negative value, triggering
the spontaneous breaking of the symmetry. This means that the O(Λ
2 ) one2 φ ˆ† ˆ
loop correction must be cancelled by the ‘bare’ mass term
1 m
φ so as to
2 H,0
achieve a negative coefficient of order −v
2 . This cancellation between m H,0
and Λ
2 will have to be very precise indeed if Λ – the scale of ‘new physics’ –
is very high, as is commonly assumed (say 10
16 GeV).
The reader may wonder why attention should now be drawn to this particular piece of renormalization: aren’t all divergences handled this way? In a
sense they are, but the fact is that this is the first case we have had in which
we have to cancel a quadratic divergence. The other mass-corrections have all
been logarithmic, for which there is nothing like such a dramatic ‘fine-tuning’
problem. There is a good reason for this in the case of the electron mass,
which we remarked on in section 11.2. Chiral symmetry forces self-energy
corrections for fermions to be proportional to their mass, and hence to contain only logarithms of the cut-off. Similarly, gauge invariance for the vector
bosons prohibits any O(Λ
2 ) connections in perturbation theory. But there
is no symmetry, within the Standard Model, which ‘protects’ the coefficient
of φ ˆ† φ ˆ in this way. It is hard to understand what can be stopping it from
being of order Λ
2 , if we take the apparently reasonable point of view that the
Standard Model will ultimately fail at some scale Λ where new physics enters.
Thus the difficulty is: why is the empirical parameter v ‘shielded’ from the
presumed high scale of new physics? This ‘problem’ is often referred to as the
‘hierarchy problem’, or the ‘fine-tuning problem’. We stress again that we are
dealing here with an absolutely crucial symmetry-breaking term, which one
would really like to understand far better.
Of course, the problem would go away if the scale Λ were as low as, say
a few TeV. As we shall see in the next section this happens to be, not accidentally, the same scale at which the Standard Model ceases to be a perturbatively calculable theory. Various possibilities have been suggested for
the kind of physics that might enter at energies of a few TeV. For example,
‘technicolour’ models (Peskin 1997) regard the Higgs field as a composite of
some new heavy fermions, rather like the BCS-pairing idea referred to earlier. A second possibility is supersymmetry (Aitchison 2007), in which there
is a ‘protective’ symmetry operating, since scalar fields can be put alongside
fermions in supermultiplets, and benefit from the protection enjoyed by the
fermions. A third possibility is that of ‘large’ extra dimensions (Antoniadis
2002).
These undoubtedly fascinating ideas obviously take us well beyond our
proper subject to which we must now return. Whatever may lie ‘beyond’
it, the Lagrangian of the Higgs sector of the Standard Model leads to many
perfectly definite predictions which may be confronted with experiment, as we
shall briefly discuss in section 22.8.3 (for a full account see Dawson et al. 1990,
and for more compact ones see Ellis, Stirling and Webber 1996, chapter 11,
and the review by Bernardi, Carena and Junk in Beringer et al. 2012). The