Unsubtracted Dispersion Relations in Weak Interactions and a Partially Conserved Axial-Vector Current

Abstract

It is shown that a combined use of unsubtracted dispersion relations (UDR) in weak interactions with a partially conserved axial-vector current (PCAC) due to Nambu leads to a very small wave-function renormalization constant of the pion field (${Z}_{3}\ensuremath{\ll}1$), where an additional condition for PCAC is necessary to obtain the case ${Z}_{3}=0$. The polology version of PCAC due to Gell-Mann and others does not generally lead to such a conclusion. A model of PCAC (${\ensuremath{\partial}}_{\ensuremath{\mu}}{a}_{\ensuremath{\mu}}=\mathrm{const}{\ensuremath{\varphi}}_{\ensuremath{\pi}}$) proposed by Gell-Mann and L\'evy contradicts UDR for the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\mu}$ decay amplitude. But their model can always be modified so as to be equivalent to the polology version or Nambu's version of PCAC.