$\mathrm{SL}(6,C)\mathrm{}$ and Meson Decays

The relativistic extension of $\mathrm{SU}\left(6\right)\mathrm{}$ proposed by Budini and Fronsdal based on $\mathrm{SL}(6,C)\mathrm{}$ has been moderately successful in its application to baryon decays and to electromagnetic and hadron form factors. Work thus far on the three-meson interaction has been limited to the examination of an $\mathrm{SL}(2,C)\mathrm{}$ model calculation by Rühl. The present work is an examination of the $\mathrm{SL}(6,C)\mathrm{}$-invariant three-meson vertex. It is shown that the meson representation adopted by Fronsdal does not admit a three-meson vertex which is invariant under charge conjugation. An alternative representation first proposed by Rühl is examined and shown not to possess a coupling between three 35 multiplets. We conclude that $\mathrm{SL}(6,C)\mathrm{}$ forbids all meson decays, and that a modification of the theory must be made in order to describe these processes.