19.1. Massive and massless vector particles
257
M → 0, thus indicating already that a massless vector particle seems to be a
very different kind of thing from a massive one (we can’t just take the massless
limit of the latter).
Now consider the loop integral in figure 19.1. At each vertex we will have
a coupling constant g, associated with an interaction Lagrangian having the
¯ ˆ ˆ ˆ
general form gψγ μ ψX
μ (a γ μ γ 5 coupling could also be present but will not
affect the argument). Just as in QED, this ‘g’ is dimensionless but, as we
warned the reader in section 11.8, this may not guarantee renormalizability,
and indeed this is a case where it does not. To get an idea of why this might
be so, consider the leading divergent behaviour of figure 19.1. This will be
associated with the k
μ k
ν terms in the numerator of (19.5), so that the leading
divergence is effectively
∫
(
) ( ρ )
k
μ k
ν
4
k k
σ
1 1
∼ d k
(19.6)
k 2
k 2
k k
for high k-values (we are not troubling to get all the indices right, we are
omitting the spinors altogether, and we are looking only at the large k part
of the propagators). Now the first two bracketed terms in (19.6) behave like
a constant at large k, so that the divergence is effectively
∫
∼ d
4 1 1
k
(19.7)
k k
which is quadratically divergent, and indeed exactly what we would get in a
‘four-fermion’ theory – see (11.98) for example. This strongly suggests that
the theory is non-renormalizable.
Where have these dangerous powers of k in the numerator of (19.6) come
from? The answer is simple and important. They come from the longitudinal
polarization state of the massive X-particle, as we shall now explain. The
free-particle wave equation is
(❗ + M
2 )X
ν
− ∂
ν (∂ μ X
μ ) = 0
(19.8)
and plane wave solutions have the form
X
ν = ∈
ν e
−ik·x .
(19.9)
Hence the polarization vectors ∈
ν satisfy the condition
(−k
2 + M
2 )∈
ν + k
ν k μ ∈
μ = 0.
(19.10)
Taking the ‘dot’ product of (19.10) with k ν leads to
M
2 k · ∈ = 0,
(19.11)
which implies (for M
2 = 0!)
k · ∈ = 0.
(19.12)
/ /
/ /
/
257
M → 0, thus indicating already that a massless vector particle seems to be a
very different kind of thing from a massive one (we can’t just take the massless
limit of the latter).
Now consider the loop integral in figure 19.1. At each vertex we will have
a coupling constant g, associated with an interaction Lagrangian having the
¯ ˆ ˆ ˆ
general form gψγ μ ψX
μ (a γ μ γ 5 coupling could also be present but will not
affect the argument). Just as in QED, this ‘g’ is dimensionless but, as we
warned the reader in section 11.8, this may not guarantee renormalizability,
and indeed this is a case where it does not. To get an idea of why this might
be so, consider the leading divergent behaviour of figure 19.1. This will be
associated with the k
μ k
ν terms in the numerator of (19.5), so that the leading
divergence is effectively
∫
(
) ( ρ )
k
μ k
ν
4
k k
σ
1 1
∼ d k
(19.6)
k 2
k 2
k k
for high k-values (we are not troubling to get all the indices right, we are
omitting the spinors altogether, and we are looking only at the large k part
of the propagators). Now the first two bracketed terms in (19.6) behave like
a constant at large k, so that the divergence is effectively
∫
∼ d
4 1 1
k
(19.7)
k k
which is quadratically divergent, and indeed exactly what we would get in a
‘four-fermion’ theory – see (11.98) for example. This strongly suggests that
the theory is non-renormalizable.
Where have these dangerous powers of k in the numerator of (19.6) come
from? The answer is simple and important. They come from the longitudinal
polarization state of the massive X-particle, as we shall now explain. The
free-particle wave equation is
(❗ + M
2 )X
ν
− ∂
ν (∂ μ X
μ ) = 0
(19.8)
and plane wave solutions have the form
X
ν = ∈
ν e
−ik·x .
(19.9)
Hence the polarization vectors ∈
ν satisfy the condition
(−k
2 + M
2 )∈
ν + k
ν k μ ∈
μ = 0.
(19.10)
Taking the ‘dot’ product of (19.10) with k ν leads to
M
2 k · ∈ = 0,
(19.11)
which implies (for M
2 = 0!)
k · ∈ = 0.
(19.12)
/ /
/ /
/