303
10.1. The propagator correction in ABC theory
where we have defined the function −iΠ
[2] (q
2 ) as the loop (or ‘bubble’) amC
2
plitude appearing in figure 10.1. It is a function of q , as follows from Lorentz
invariance. The
[2] refers to the two powers of g, as will be explained shortly,
after (10.15).
Careful consideration of the equivalences among the various contractions
shows that the amplitude corresponding to figure 10.1 is, in fact, just the
simple expression
′
′
[2]
(−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B )
i
2
(−iΠ C (q
2 ))
i
2
q 2 − m + i∈
q 2 − m + i∈
C
C
(10.9)
where Π
[2] (q
2 ) is given in (10.8). We see that whereas the ‘single-particle’
C
pieces, involving one C-exchange, do not involve any integral in momentum–
space, the loop (which involves both A and B particles) does involve a momentum integral. This can be simply understood in terms of 4-momentum conservation, which holds at every vertex of a Feynman graph. At the top (or bottom) vertex of figure 10.1, the 4-momentum q of the C-particle is fully deter′
′
mined by that of the incoming and outgoing particles (q = p A −p = p −p B ).
B
A
This same 4-momentum q flows in (and out) of the loop in figure 10.1, but
nothing determines how it is to be shared between the A- and B-particles;
all that can be said is that if the 4-momentum of A is k (as in (10.7)) then
that of B is q − k, so that their sum is q. The ‘free’ variable k then has to be
integrated over, and this is the physical origin of rule (iii) of section 6.3.5.
We have devoted some time to the steps leading to expression (10.7), not
only in order to follow the emergence of rule (iii) mathematically, but so as to
lend some plausibility to a very important statement: the Feynman rules for
associating factors with vertices and propagators, which we learned for tree
graphs in chapters 6 and 8, also work, with the addition of rule (iii), for all
more complicated graphs as well! Having seen most of just one fairly short
calculation of a higher-order amplitude, the reader may perhaps now begin to
appreciate just how powerful is the precise correspondence between ‘diagrams
and amplitudes’, given by the Feynman rules.
Having arrived at the expression for our first one-loop graph, we must
at once draw the reader’s attention to the bad news: the integral in (10.7) is
divergent at large values of k. We shall postpone a more detailed mathematical
analysis until section 10.3.1, but the divergence can be plausibly inferred just
from a simple counting of powers: there are four powers of k in the numerator
and four in the denominator, and the likelihood is that the integral diverges
∫ Λ
as
k
3 dk/k
4
∼ ln Λ, as Λ → ∞. This is plainly a disaster: a quantity
which was supposed to be a small correction in perturbation theory is actually
infinite! Such divergences, occurring as loop momenta go to infinity, are called
‘ultraviolet divergences’, and they are ubiquitous in quantum field theory.
Only after a long struggle with these infinities was it understood how to obtain
physically sensible results from such perturbation expansions. Depending on
the type of field theory involved, the infinities can often be ‘tamed’ through a
0
10.1. The propagator correction in ABC theory
where we have defined the function −iΠ
[2] (q
2 ) as the loop (or ‘bubble’) amC
2
plitude appearing in figure 10.1. It is a function of q , as follows from Lorentz
invariance. The
[2] refers to the two powers of g, as will be explained shortly,
after (10.15).
Careful consideration of the equivalences among the various contractions
shows that the amplitude corresponding to figure 10.1 is, in fact, just the
simple expression
′
′
[2]
(−ig)
2 (2π)
4 δ
4 (p A + p B − p A − p B )
i
2
(−iΠ C (q
2 ))
i
2
q 2 − m + i∈
q 2 − m + i∈
C
C
(10.9)
where Π
[2] (q
2 ) is given in (10.8). We see that whereas the ‘single-particle’
C
pieces, involving one C-exchange, do not involve any integral in momentum–
space, the loop (which involves both A and B particles) does involve a momentum integral. This can be simply understood in terms of 4-momentum conservation, which holds at every vertex of a Feynman graph. At the top (or bottom) vertex of figure 10.1, the 4-momentum q of the C-particle is fully deter′
′
mined by that of the incoming and outgoing particles (q = p A −p = p −p B ).
B
A
This same 4-momentum q flows in (and out) of the loop in figure 10.1, but
nothing determines how it is to be shared between the A- and B-particles;
all that can be said is that if the 4-momentum of A is k (as in (10.7)) then
that of B is q − k, so that their sum is q. The ‘free’ variable k then has to be
integrated over, and this is the physical origin of rule (iii) of section 6.3.5.
We have devoted some time to the steps leading to expression (10.7), not
only in order to follow the emergence of rule (iii) mathematically, but so as to
lend some plausibility to a very important statement: the Feynman rules for
associating factors with vertices and propagators, which we learned for tree
graphs in chapters 6 and 8, also work, with the addition of rule (iii), for all
more complicated graphs as well! Having seen most of just one fairly short
calculation of a higher-order amplitude, the reader may perhaps now begin to
appreciate just how powerful is the precise correspondence between ‘diagrams
and amplitudes’, given by the Feynman rules.
Having arrived at the expression for our first one-loop graph, we must
at once draw the reader’s attention to the bad news: the integral in (10.7) is
divergent at large values of k. We shall postpone a more detailed mathematical
analysis until section 10.3.1, but the divergence can be plausibly inferred just
from a simple counting of powers: there are four powers of k in the numerator
and four in the denominator, and the likelihood is that the integral diverges
∫ Λ
as
k
3 dk/k
4
∼ ln Λ, as Λ → ∞. This is plainly a disaster: a quantity
which was supposed to be a small correction in perturbation theory is actually
infinite! Such divergences, occurring as loop momenta go to infinity, are called
‘ultraviolet divergences’, and they are ubiquitous in quantum field theory.
Only after a long struggle with these infinities was it understood how to obtain
physically sensible results from such perturbation expansions. Depending on
the type of field theory involved, the infinities can often be ‘tamed’ through a
0