N-quantum calculation of the Nambu–Jona-Lasinio model with isospin

We use the N-quantum approach to quantum field theory to study the Nambu–Jona-Lasinio model with isospin. Our main purpose is to resolve a paradox found earlier. The paradox is that the form factor of a massless particle diverges as 1/$Q^2$ for small momentum transfer, so that the charge diverges and the amplitude cannot be normalized. We study this problem in one-loop approximation. We show that for a charged boson which is massless by virtue of being a Nambu-Goldstone boson the triangle graph which previously diverged now vanishes identically for all momentum transfers. We exhibit two other graphs which also contribute to the form factor in one-loop approximation and give a finite, nonvanishing contribution to the charge. For a charged boson whose mass is arbitrarily set equal to zero and whose bound-state amplitude does not have the structure of the Nambu-Goldstone amplitude the triangle graph still diverges as 1/$Q^2$. On the basis of this result we raise the question whether the only massless composite spin-zero particles which have finite charge are those which arise through the Nambu-Goldstone mechanism, at least in nonsupersymmetric theories.