259
19.1. Massive and massless vector particles
the numerator in (19.22) arising from the spin sum (19.19). Thus the dangerous factor k
μ k
ν /M
2 can be traced to the spin sum (19.19): in particular, at
large values of k the longitudinal state ∈
μ (k, λ = 0) is proportional to k
μ , and
this is the origin of the numerator factors k
μ k
ν /M
2 in (19.22).
We shall not give further details here (see also section 22.1.2), but merely
state that theories with massive charged vector bosons are indeed non-renormalizable. Does this matter? In section 11.8 we explained why it is thought
that the relevant theories at presently accessible energy scales should be renormalizable theories. And, apart from anything else, they are much more predictive. Is there, then, any way of getting rid of the offending ‘k
μ k
ν ’ terms
in the X-propagator, so as (perhaps) to render the theory renormalizable?
Consider the photon propagator of chapter 7 repeated here:
i[−g
μν + (1 − ξ)k
μ k
ν /k
2 ] .
(19.23)
k 2 + i∈
This contains somewhat similar factors of k
μ k
ν (admittedly divided by k
2
rather than M
2 ), but they are gauge-dependent, and can in fact be ‘gauged
away’ entirely, by choice of the gauge parameter ξ (namely by taking ξ = 1).
But, as we have seen, such ‘gauging’ – essentially the freedom to make gauge
transformations – seems to be possible only in a massless vector theory.
A closely related point is that, as section 7.3.1 showed, free photons exist
in only two polarization states (electromagnetic waves are purely transverse),
instead of the three we might have expected for a vector (spin-1) particle –
and as do indeed exist for massive vector particles. This gives another way
of seeing in what way a massless vector particle is really very different from
a massive one: the former has only two (spin) degrees of freedom, while the
latter has three, and it is not at all clear how to ‘lose’ the offending longitudinal
state smoothly (certainly not, as we have seen, by letting M → 0 in (19.5)).
These considerations therefore suggest the following line of thought: is it
possible somehow to create a theory involving massive vector bosons, in such
a way that the dangerous k
μ k
ν term can be ‘gauged away’, making the theory
renormalizable? The answer is yes, via the idea of spontaneous breaking of
gauge symmetry. This is the natural generalization of the spontaneous global
symmetry breaking considered in chapter 17. By way of advance notice, the
crucial formula is (19.74) for the propagator in such a theory, which is to be
compared with (19.22).
The first serious challenge to the then widely held view that electromagnetic gauge invariance requires the photon to be massless was made by
Schwinger (1962), as we pointed out in section 11.4. Soon afterwards, Anderson (1963) argued that several situations in solid state physics could be interpreted in terms of an effectively massive electromagnetic field. He outlined a
general framework for treating the phenomenon of the acquisition of mass by
a gauge boson, and discussed its possible relevance to contemporary attempts
(Sakurai 1960) to interpret the recently discovered vector mesons (ρ, ω, φ . . .)
as the gauge quanta associated with a local extension of hadronic flavour sym
19.1. Massive and massless vector particles
the numerator in (19.22) arising from the spin sum (19.19). Thus the dangerous factor k
μ k
ν /M
2 can be traced to the spin sum (19.19): in particular, at
large values of k the longitudinal state ∈
μ (k, λ = 0) is proportional to k
μ , and
this is the origin of the numerator factors k
μ k
ν /M
2 in (19.22).
We shall not give further details here (see also section 22.1.2), but merely
state that theories with massive charged vector bosons are indeed non-renormalizable. Does this matter? In section 11.8 we explained why it is thought
that the relevant theories at presently accessible energy scales should be renormalizable theories. And, apart from anything else, they are much more predictive. Is there, then, any way of getting rid of the offending ‘k
μ k
ν ’ terms
in the X-propagator, so as (perhaps) to render the theory renormalizable?
Consider the photon propagator of chapter 7 repeated here:
i[−g
μν + (1 − ξ)k
μ k
ν /k
2 ] .
(19.23)
k 2 + i∈
This contains somewhat similar factors of k
μ k
ν (admittedly divided by k
2
rather than M
2 ), but they are gauge-dependent, and can in fact be ‘gauged
away’ entirely, by choice of the gauge parameter ξ (namely by taking ξ = 1).
But, as we have seen, such ‘gauging’ – essentially the freedom to make gauge
transformations – seems to be possible only in a massless vector theory.
A closely related point is that, as section 7.3.1 showed, free photons exist
in only two polarization states (electromagnetic waves are purely transverse),
instead of the three we might have expected for a vector (spin-1) particle –
and as do indeed exist for massive vector particles. This gives another way
of seeing in what way a massless vector particle is really very different from
a massive one: the former has only two (spin) degrees of freedom, while the
latter has three, and it is not at all clear how to ‘lose’ the offending longitudinal
state smoothly (certainly not, as we have seen, by letting M → 0 in (19.5)).
These considerations therefore suggest the following line of thought: is it
possible somehow to create a theory involving massive vector bosons, in such
a way that the dangerous k
μ k
ν term can be ‘gauged away’, making the theory
renormalizable? The answer is yes, via the idea of spontaneous breaking of
gauge symmetry. This is the natural generalization of the spontaneous global
symmetry breaking considered in chapter 17. By way of advance notice, the
crucial formula is (19.74) for the propagator in such a theory, which is to be
compared with (19.22).
The first serious challenge to the then widely held view that electromagnetic gauge invariance requires the photon to be massless was made by
Schwinger (1962), as we pointed out in section 11.4. Soon afterwards, Anderson (1963) argued that several situations in solid state physics could be interpreted in terms of an effectively massive electromagnetic field. He outlined a
general framework for treating the phenomenon of the acquisition of mass by
a gauge boson, and discussed its possible relevance to contemporary attempts
(Sakurai 1960) to interpret the recently discovered vector mesons (ρ, ω, φ . . .)
as the gauge quanta associated with a local extension of hadronic flavour sym