Machinery for the analysis of field theories with zero-mass particles

Abstract

We establish a machinery for the analysis of field theory models with zero-mass particles with great generality and in a model independent way. To this end a special class of functions called the class A n -functions, introduced earlier by the author, is quite suitable for such a machinery. We show that Feynman integrands and absolutely convergent Feynman integrals, involving zero-mass particles, belong to such a class as functions of external (and internal) momenta and the non-zero masses of the theory. We show, in particular, how one may find real numbers b r , d r′ , which set 《high-energy》 and 《low-energy》 scales, respectively, with b r >1 and 0< d r′ <1, such that for certain parameters η r′ , λ r′ >0, with η r ⩾ b r and λ r′ ⩽ d r ′, the absolute value of a Feynman integrand may be bounded by introducing suitable high- and low-energy asymptotic coefficients. The analysis provides, in particular, explicit forms for the latter coefficients for the Feynman integrals themselves and hence leads, to powerful results for the study of their asymptotic behaviour.