Renormalization of Neutral Mesons in Three-Field Problems

It is shown that a finite $S$-matrix can be obtained to low orders by renormalization if neutral spin zero mesons with (pseudo) scalar coupling are added to the mixture of nucleons, photons, and charged spin zero mesons previously considered, provided that a limited number of contact interactions are introduced. A general proof for any order is not given; but it is shown that at no stage can divergences of a new form appear, requiring the introduction of additional contact terms.