326
10. Loops and Renormalization I: The ABC Theory
interaction in the original theory (it would have the form λφ ˆ2
A (x)φ ˆ2
B (x)). In
fact, the graph is convergent, as indicated by the usual power-counting (four
powers of k in the numerator, eight in the denominator from the four propagators). And indeed, the ABC theory is renormalizable – or rather, as noted
earlier, ‘super-renormalizable’.
We shall have something more to say about renormalizability and nonrenormalizability (is it fatal?), at the end of the following chapter. The first
and main business, however, will be to apply what we have learned here to
QED.
Problems
10.1 Carry out the indicated change of variables so as to obtain (10.4) from
(10.3).
10.2 Verify the Feynman identity (10.40).
10.3 Obtain (10.42) from (10.41).
10.4 Obtain (10.51) from (10.50), having replaced the upper limit of the uintegral by Λ.
10.5 Obtain the Feynman rule quoted in the text for the sum of the counter
terms appearing in (10.64).
10. Loops and Renormalization I: The ABC Theory
interaction in the original theory (it would have the form λφ ˆ2
A (x)φ ˆ2
B (x)). In
fact, the graph is convergent, as indicated by the usual power-counting (four
powers of k in the numerator, eight in the denominator from the four propagators). And indeed, the ABC theory is renormalizable – or rather, as noted
earlier, ‘super-renormalizable’.
We shall have something more to say about renormalizability and nonrenormalizability (is it fatal?), at the end of the following chapter. The first
and main business, however, will be to apply what we have learned here to
QED.
Problems
10.1 Carry out the indicated change of variables so as to obtain (10.4) from
(10.3).
10.2 Verify the Feynman identity (10.40).
10.3 Obtain (10.42) from (10.41).
10.4 Obtain (10.51) from (10.50), having replaced the upper limit of the uintegral by Λ.
10.5 Obtain the Feynman rule quoted in the text for the sum of the counter
terms appearing in (10.64).