31
1.4. Renormalization and the Higgs sector of the Standard Model
renormalization, in such a way that all multi-particle (higher-order) processes
become finite and calculable – a quite remarkable fact, and one that is of
course an absolutely crucial requirement in the case of the Standard Model
interactions, where the relevant data are precise enough to test the accuracy
of the theory well beyond lowest order, particularly in the case of QED (see
chapter 11). The price to be paid for this taming of the divergences is just
that the basic parameters of the theory, such as masses and coupling constants, have to be treated as parameters to be determined by comparison to
the data, and cannot themselves be calculated.
But some theories cannot be reformulated in this way – they are nonrenormalizable. A simple test for whether a theory is renormalizable or not
will be discussed in section 11.8: if the coupling constant has dimensions of
a mass to an inverse power, the theory is non-renormalizable. An example of
such a theory is the original four-Fermi theory of weak interactions, where the
coupling constant G F has the dimensions of an inverse square mass (or energy)
as we saw in (1.31). We will look at this theory again in section 11.8, but the
essential point for our purpose now is that the dimensionful coupling constant
−1/2
introduces an energy scale into the problem, namely G F
∼ 300 GeV.
It seems reasonable to infer that a more relevant measure of the interaction
1/2
strength will be given by the dimensionless number EG , where E is a
F
characteristic physical energy scale of any weak process under consideration
– for example, the energy in the centre of momentum frame in a two-particle
scattering process, at least at energies much greater than the particle masses.
−1/2
Then, for energies very much less than G F
the effective strength will be
very weak, and the lowest order term in perturbation theory will work fine;
this is how the Fermi theory was used, for many years. But as the energy
increases, what happens is that more and more parameters have to be taken
from experiment, in order to control the divergences; as the energy approaches
−1/2
G
, the theory becomes totally non-predictive and breaks down. Thus
F
renormalizability is regarded as highly desirable in a theory.
One might hope to come up with a renormalizable theory of weak interactions by replacing the four-fermion interaction by a Yukawa-like mechanism,
with exchange of a quantum of mass M and dimensionless coupling y, say.
Then just as in (1.32) we would identify G F ∼ y
2 /M
2 at low energies. However, as we have seen, phenomenology implies that the massive exchanged
quantum must have spin 1. Unfortunately, this type of straightforward massive spin-1 theory is not renormalizable either, as we shall discuss in chapter
22 (in volume 2). The trouble can be traced directly to the existence of the
longitudinal polarization state which, as noted previously, is present for a
massive spin-1 particle. If the exchanged spin-1 quantum were massless, as
in QED, it would lack that third polarization state, and the theory would be
renormalizable. But weak interaction facts dictate both non-zero mass and
spin-1.
In the case of QED, there is a symmetry principle behind both the zero
mass of the photon and the absence of the longitudinal polarization state:
1.4. Renormalization and the Higgs sector of the Standard Model
renormalization, in such a way that all multi-particle (higher-order) processes
become finite and calculable – a quite remarkable fact, and one that is of
course an absolutely crucial requirement in the case of the Standard Model
interactions, where the relevant data are precise enough to test the accuracy
of the theory well beyond lowest order, particularly in the case of QED (see
chapter 11). The price to be paid for this taming of the divergences is just
that the basic parameters of the theory, such as masses and coupling constants, have to be treated as parameters to be determined by comparison to
the data, and cannot themselves be calculated.
But some theories cannot be reformulated in this way – they are nonrenormalizable. A simple test for whether a theory is renormalizable or not
will be discussed in section 11.8: if the coupling constant has dimensions of
a mass to an inverse power, the theory is non-renormalizable. An example of
such a theory is the original four-Fermi theory of weak interactions, where the
coupling constant G F has the dimensions of an inverse square mass (or energy)
as we saw in (1.31). We will look at this theory again in section 11.8, but the
essential point for our purpose now is that the dimensionful coupling constant
−1/2
introduces an energy scale into the problem, namely G F
∼ 300 GeV.
It seems reasonable to infer that a more relevant measure of the interaction
1/2
strength will be given by the dimensionless number EG , where E is a
F
characteristic physical energy scale of any weak process under consideration
– for example, the energy in the centre of momentum frame in a two-particle
scattering process, at least at energies much greater than the particle masses.
−1/2
Then, for energies very much less than G F
the effective strength will be
very weak, and the lowest order term in perturbation theory will work fine;
this is how the Fermi theory was used, for many years. But as the energy
increases, what happens is that more and more parameters have to be taken
from experiment, in order to control the divergences; as the energy approaches
−1/2
G
, the theory becomes totally non-predictive and breaks down. Thus
F
renormalizability is regarded as highly desirable in a theory.
One might hope to come up with a renormalizable theory of weak interactions by replacing the four-fermion interaction by a Yukawa-like mechanism,
with exchange of a quantum of mass M and dimensionless coupling y, say.
Then just as in (1.32) we would identify G F ∼ y
2 /M
2 at low energies. However, as we have seen, phenomenology implies that the massive exchanged
quantum must have spin 1. Unfortunately, this type of straightforward massive spin-1 theory is not renormalizable either, as we shall discuss in chapter
22 (in volume 2). The trouble can be traced directly to the existence of the
longitudinal polarization state which, as noted previously, is present for a
massive spin-1 particle. If the exchanged spin-1 quantum were massless, as
in QED, it would lack that third polarization state, and the theory would be
renormalizable. But weak interaction facts dictate both non-zero mass and
spin-1.
In the case of QED, there is a symmetry principle behind both the zero
mass of the photon and the absence of the longitudinal polarization state: