Regularization of Feynman Integrals 8.1 Regularization

For dimensions close to D = 4, the Feynman integrals in momentum space derived in Chapter 4 do not converge since their integrands fall off too slowly at large momenta. Divergences arising from this short-wavelength region of the integrals are called ultraviolet (UV)-divergences. For massive fields, these are the only divergences of the integrals. In the zero-mass limit relevant for critical phenomena, there exists further divergences at small momenta and long wavelength. These so-called infrared (IR)-divergences will be discussed in Chapter 12. In this chapter we shall consider only UV-divergences. We shall therefore often omit specifying their UV character. The divergences can be controlled by various mathematical methods whose advantages and disadvantages will be pointed out, and from which we shall select the best method for our purposes. In principle, all masses and coupling constants occurring here ought to carry a subscript B indicating that the perturbative calculations are done starting from the bare energy functional E B [φ B ] with bare mass m B and bare field φ B , introduced earlier in Section 7.3.1: E B [φ B ] = d D x 1 2 (∂φ B) 2 + m 2 B 2 φ 2 B , (8.1) and perturbing it with the bare interaction E int B [φ B ] ≡ d D x λ B 4! φ 4 B (x). (8.2) However, in Chapter 9 we shall see that the renormalized quantities can eventually be calculated from the same Feynman integrals with the experimentally observable mass m and coupling constant λ. For this reason, the subscripts B will be omitted in all integrals. In four dimensions, the integrals of two-and four-point functions diverge. With the help of so-called regularization procedures they can be made finite. A regularization parameter is introduced, so that all divergences of the integrals appear as singularities in this parameter. There are various possible regularization procedures: (a) Momentum cutoff Λ regularization In field descriptions of condensed matter systems, Feynman diagrams are regularized naturally at length scales a, where the field description breaks down. All momentum integrals are limited naturally to a region |p| < Λ = π/a, so that no UV-divergences 102