Divergence of perturbation theory for bosons

Abstract

Perturbation theory is studied in two dimensional space-time. There all non-derivative boson self-interactions are renormalizable and in each order of perturbation theory there are no divergences, that is all renormalizations are finite in perturbation theory. Thus the unrenormalized perturbation series may be studied and it is shown that any interaction of the general form$H_1 (x) = \lambda \sum\limits_{j = 3}^\infty {a_j \times \times :\varphi (x)^j :,a_j \geqslant 0}$leads to Green's functions which are not analytic in λ at λ=0. This result holds in momentum space at a large set of points, enough to show that the Green's functions are not distributions in the momenta which are analytic in λ at λ=0. Furthermore the proper self energy and the two-particle scattering amplitude are shown not to be analytic in λ at λ=0 for certain momenta on or below the bare mass shell. In the course of this analysis we use the integral representations for Feynman graphs to derive a minorization of the form |I)p ₁,...,p _e )|>A B ⁿ for the contribution from alln ^th order connected graphs in a theory with an interaction of the form$H_1 (x) = \lambda \sum\limits_{j = 3}^Q {a_j :\varphi (x)^j :}$. Then the constantsA andB depend only on the momentap _i , and not on the structure of a particular graph.