10
Loops and Renormalization I: The ABC
Theory
We have seen how Feynman diagrams represent terms in a perturbation theory
expansion of physical amplitudes, namely the Dyson expansion of section 6.2.
Terms of a given order all involve the same power of a ‘coupling constant’,
which is the multiplicative constant appearing in the interaction Hamiltonian
– for example, ‘g’ in the ABC theory, or the charge ‘e’ in electrodynamics. In
practice, it often turns out that the relevant parameter is actually the square
of the coupling constant, and factors of 4π have a habit of appearing on a
regular basis; so, for QED, the perturbation series is conveniently ordered
according to powers of the fine structure constant α = e
2 /4π ≈ 1/137.
Equivalently, this is an expansion in terms of the number of vertices appearing in the diagrams, since one power of the coupling constant is associated
with each vertex. For a given physical process, the lowest-order diagrams (the
ones with the fewest vertices) are those in which each vertex is connected
to every other vertex by just one internal line; these are called tree diagrams.
The Yukawa (u-channel) exchange process of figure 6.4, and the s-channel process of figure 6.5, are both examples of tree diagrams, and indeed all of our
calculations so far have not gone further than this lowest-order (‘tree’) level.
Admittedly, since α is after all pretty small, tree diagrams in QED are likely
to give us a good approximation to compare with experiment. Nevertheless, a
long history of beautiful and ingenious experiments has resulted in observables
in QED being determined to an accuracy far better than the O(1%) represented by the leading (tree) terms. More generally, precision experiments at
LEP and other laboratories have an accuracy sensitive to higher-order corrections in the Standard Model. Hence, some understanding of the physics
beyond the tree approximation is now essential for phenomenology.
All higher-order processes beyond the tree approximation involve loops, a
concept easier to recognize visually than to define in words. In section 6.3.5
we already met (figure 6.8) one example of an O(g
4 ) correction to the O(g
2 )
C-exchange tree diagram of figure 6.4, which contains one loop. The crucial
point is that whereas a tree diagram can be cut into two separate pieces by
severing just one internal line, to cut a loop diagram into two separate pieces
requires the severing of at least two internal lines.
In these last two chapters of volume 1, we aim to provide an introduction to higher-order processes, confining ourselves to ‘one-loop’ order. In the
299
Loops and Renormalization I: The ABC
Theory
We have seen how Feynman diagrams represent terms in a perturbation theory
expansion of physical amplitudes, namely the Dyson expansion of section 6.2.
Terms of a given order all involve the same power of a ‘coupling constant’,
which is the multiplicative constant appearing in the interaction Hamiltonian
– for example, ‘g’ in the ABC theory, or the charge ‘e’ in electrodynamics. In
practice, it often turns out that the relevant parameter is actually the square
of the coupling constant, and factors of 4π have a habit of appearing on a
regular basis; so, for QED, the perturbation series is conveniently ordered
according to powers of the fine structure constant α = e
2 /4π ≈ 1/137.
Equivalently, this is an expansion in terms of the number of vertices appearing in the diagrams, since one power of the coupling constant is associated
with each vertex. For a given physical process, the lowest-order diagrams (the
ones with the fewest vertices) are those in which each vertex is connected
to every other vertex by just one internal line; these are called tree diagrams.
The Yukawa (u-channel) exchange process of figure 6.4, and the s-channel process of figure 6.5, are both examples of tree diagrams, and indeed all of our
calculations so far have not gone further than this lowest-order (‘tree’) level.
Admittedly, since α is after all pretty small, tree diagrams in QED are likely
to give us a good approximation to compare with experiment. Nevertheless, a
long history of beautiful and ingenious experiments has resulted in observables
in QED being determined to an accuracy far better than the O(1%) represented by the leading (tree) terms. More generally, precision experiments at
LEP and other laboratories have an accuracy sensitive to higher-order corrections in the Standard Model. Hence, some understanding of the physics
beyond the tree approximation is now essential for phenomenology.
All higher-order processes beyond the tree approximation involve loops, a
concept easier to recognize visually than to define in words. In section 6.3.5
we already met (figure 6.8) one example of an O(g
4 ) correction to the O(g
2 )
C-exchange tree diagram of figure 6.4, which contains one loop. The crucial
point is that whereas a tree diagram can be cut into two separate pieces by
severing just one internal line, to cut a loop diagram into two separate pieces
requires the severing of at least two internal lines.
In these last two chapters of volume 1, we aim to provide an introduction to higher-order processes, confining ourselves to ‘one-loop’ order. In the
299