The $A_1-\rho -\pi$ system is studied and it is shown that the question of possible subtractions in the dispersion relations for form factors is crucial for understanding the difference between hard- and soft-pion results in the $A_1$ and $\rho$ decays. A double dispersion representation is suggested for three-point functions. Using a truncation approximation, the Bjorken technique is employed to derive several new sum rules for the spectral functions. Assuming that the sum rules are saturated by low-lying meson states, a study is made to evaluate the constant terms in the dispersion representations of the form factors involved in the $A_1\rho \pi$ and $\rho \pi \pi$ vertices. The case of no subtractions, which formally leads to the soft-pion results, is ruled out on the basis of this investigation, and the well-known hard-pion results for $A_1$ and $\rho$ decays are rederived.

• Received 1 May 1968

DOI:https://doi.org/10.1103/PhysRev.174.1743