Model of Pion Excitation

We study the vector and axial-vector excitation of a pion to an arbitrary single-particle state. The main inputs to our calculation are a hard-pion representation for the relevant off-shell matrix elements, and use of partial conservation of axial-vector current, the Bjorken-Johnson-Low theorem, and current algebra as constraints. We find that for a model incorporating a finite number of poles, the pion cannot be excited to arbitrarily high-spin states. In a particular model we find a cutoff of $J=4\mathrm{}$, and also obtain predictions involving branching ratios for the decay of $f\left(1268\right)\mathrm{}$ and $g\left(1680\right)\mathrm{}$ mesons into $A_1\mathrm{}\left(1070\right)\mathrm{}\pi$ and $\pi \pi$ composites. The results of our model are checked by comparing with sum rules based on local commutation relations.