Solution of very Singular N/D Equations—An Approach to Nonrenormalizable Field Theory

Abstract

In this paper, we find closed form solutions to unsubtracted and once‐subtracted N/D equations, both nonrelativistic and relativistic, with pathological left‐hand absorptive parts that increase at large ν like α(−ν) ∼ −λν^m, m not necessarily integral. The solutions allow dispersive, unitary and regulator‐free calculation in nonrenormalizable field theory. For example, N/D equations with exchange of a higher spin particle (spin $\frac{3}{2}$ etc.) can be solved. In general, the technique allows the use of a large class of divergent graphs as input, that is, those whose left‐hand absorptive parts are finite, although asymptotically ill behaved. For example, in the W theory of leptonic weak interactions, one of the possible inputs is any number of ladder graphs cut so that all the bosons are on the mass shell. Many nonplanar graphs can be included as well. In much the same way, we can also calculate in the Fermi theory, (multiple exchanges in) theories of higher spin in general, and theories with derivative coupling. There are an infinite number of solutions to these singular integral equations, each of which is characterized by a branch point at g² = 0, and a branch point of oscillatory nature at infinite |ν|. Out of these, we pick the solution which sums the iterative expansion of the equations as most meaningful. It is seen explicitly that the unitary requirement generates its own regulation in the form of rapid oscillations at large unphysical energies, thus eliminating the need for any regulator limiting process. The oscillations are associated with an infinite number of ghosts, but these stay very far from the physical region. In addition, the solutions violate unitarity in the cross channels, so we expect the program to be useful at most in and near the physical region. As shorter range forces are systematically included, there is some indication that the program may converge rapidly for small physical energies.