A condensed description of the results of the present paper was given by one of us (B. Z.) in Proceedings of the Fifth Coral Gables Conference on Symmetry Principles at High Energy (W. A Inc., New York,). Partial results were presented in Proceedings of the Heidelberg International Conference on Elementary Particles, H. Filthuth (eds.) Publishers, Inc., New York,)

Properties of tree diagrams have been studied by Boulder Lectures in Theoretical Physics Publishers, Inc., New York,)

Y. Nambu, University of Chicago Report,(unpublished)

A discussion of the connection between the structure of diagrams and the power in an expansion in the coupling constant can be found in the paper by Lee and Nieh quoted in Ref. 1

For a discussion in the context of Lagrangian perturbation theory, see the contribution by one of us (S. C.) in IIadrons and Their Interactions, A. Zichichi (eds.) Press inc., New York, 1968)

This decomposition amounts to a particular parametrization of the left cosets G/II by means of the parameters $~. Any other parametrization would give rise to a treatment completely equivalent from the abstract group-theoretic point of view and would furnish equivalent results. However, the particular parametrization we use here is very convenient because of the simple and wellknown properties of exponentials. Alternative parametrizations have been used in concrete applications and can be found, for instance,. in the papers quoted in Ref. 1

This case has been considered in detail by (private communication)

As a special case of this result, we recall that, in nonlinear phenomenological theories, the pion is described as a threedimensional multiplet and it is not necessary to introduce the e 6eld which would be needed to complete the four-dimensional linear representation of SU(2)&(SU{2).This does not mean, however, that one cannot introduce the cr into the theory, if experimental evidence requires one to do so. The 0, being an isoscalar, would be described as an invariant under the full group SU(2) XSV(2). The strength of its couplings to other fields will be unrelated to that of the couplings of the pion to the same Gelds. Nevertheless, invariance under the group imposes well-de6ncd restrictions on the a couplings