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. Renormalization certainly fails for the pseudovector interaction and the possibility of defining a finite S-matrix, formally, with the use of regulators, is considered. It is shown that it is not possible to do this in a manner which is both self-consistent and consistent with the renormalization of the pseudoscalar coupling. This result is related to the "ambiguities" found in the calculation of three-field problems with the use of regulators only, and is illustrated in the appendix by a detailed discussion of the two-photon decay of a neutral spin zero meson. It is concluded that cross sections for interactions, for which renormalization is inadequate, can certainly not be calculated in this manner, and that if, in fact, such theories have any meaning, it must be possible to remove the divergences in a more fundamental way.