Zero-Mass Approach to Counterterms
WORLD SCIENTIFIC eBooks · 2001
Massive Feynman integrals containing more than one loop momentum are hard to evaluate. Fortunately, the understanding of the critical behavior of the field theory requires only knowledge of the divergent counterterms. In the last chapter we have seen that their calculation reduces to the calculation of logarithmically divergent diagrams without tadpole parts for Zg and Z m 2 and of quadratically divergent diagrams without tadpole parts for Zφ. From the superficial divergence of the latter. only the mass-independent part is needed. In addition, the superficial divergences of the logarithmically divergent diagrams are independent of the mass and of the external momenta. These properties have the important consequence that masses and external momenta of Feynman integrals may be modified in a variety of ways without changing the counterterms. In particular, masses and external momenta may be set equal to zero as long as this does not produce unphysical IR-divergences. We shall see that overall IRdivergences do not occur if at least one external momentum is kept nonzero. There are different ways of choosing the nonzero momentum, and the corresponding mathematical modifications of Feynman integrals are called infrared rearrangement (IRR) [1]. A suitable rearrangement allows us to simplify considerably the calculation of counterterms in a massive theory.