Zero - Mass Approach to Counterterms

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 Z g and Z m 2 and of quadratically divergent diagrams without tadpole parts for Z φ. 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 IR-divergences 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. In many Feynman integrals, the single nonzero external momentum is still an obstacle to an analytical calculation. In this case one employs a more drastic IR-rearrangement by admitting a final nonzero external momentum which does generate unphysical IR-divergences. These must be properly identified and subtracted to arrive at the desired UV-counterterms. The minimal subtraction scheme which is so convenient for regularizing the theory has, unfortunately, an unpleasant feature as far as the new unphysical IR-divergences are concerned. The new divergences have the same 1/ε i-pole form as the UV-divergences. The identification and subtraction of the infrared parts in the total counterterms are therefore nontrivial. These parts will be called IR-counterterms, for brevity, and we shall develop a diagrammatic method for calculating them for each Feynman diagram. We shall construct so-called IR-diagrams which must, of course, be such that no new UV-divergences arise. This procedure leads, unfortunately, to a proliferation of diagrams, but these have the advantage of containing only massless lines, which greatly simplifies the associated Feynman integrals. In some cases, this is the only way to find an analytic result for the UV-counterterms. In this chapter we describe a recursive diagrammatic construction …