Implicit Regularisation Technique : Calculation of the Two - loop φ 44 - theory β - function

We propose an implicit regularisation scheme. The main advantage is that since no explicit use of a regulator is made, one can in principle avoid undesirable symmetry violations related to its choice. The divergent amplitudes are split into basic divergent integrals which depend only on the loop momenta and finite integrals. The former can be absorbed by a renormalisation procedure whereas the latter can be evaluated without restrictions. We illustrate with the calculation of the QED and φ44-theory β-function to one and two-loop order, respectively. PACS: 11.10.Gh, 12.38 Bx. In dealing with ultraviolet divergences in perturbative calculations of Quantum Field Theories (QFT), one is led to adopt a regularisation scheme (RS) to handle the divergent integrals. A vast arsenal of such schemes is presently available, viz. Dimensional Regularisation (DR), Pauli-Villars (PV), Zeta-function Regularisation, Lattice Regularisation, etc. . The choice of a particular scheme is generally based on its adequacy to a particular computational task or compatibility with the underlying theory in the sense of preserving its vital symmetries. For example, DR is usually employed in particle physics since it preserves unitarity and gauge invariance. However, care must be exercised in DR when parity-violating objects (γ-matrices, ǫμ1...μn tensors) occur in the theory [3]. The properties of such objects depend very much on the space-time dimension and this clashes with the idea of analytic continuation on the dimension of the space-time D. The issue of finding an ambiguity free RS so that the theory in consideration is not plagued by RS-dependent amplitudes is most important, particularly in chiral and nonrenormalisable models. In the latter, the RS is frequently defined as a part of the model. Consequently, any parameters introduced by a specific choice must be adjusted phenomenologically [6], [8]. Recently a step in this direction has been taken. A technique was proposed for the manipulation and calculation of divergent amplitudes in a way that a regularisation need only to be assumed implicitly [1],[2]. The main idea is to manipulate the integrands of the divergent amplitudes by means of algebraic identities until the physical content, i.e. the external momentum dependent part, is isolated and displayed solely in terms of finite integrals . On the other hand, the divergent content is automatically reduced to a set of basic divergent objects which can be organised according to their degree of divergence. Throughout this process, it is assumed that the ultraviolet divergent integrals in the momentum (say, k) are regulated by the multiplication of the integrand by a regularising function G(k2,Λi),