Axial Vector Current and Induced Pseudoscalar Coupling Constant on $\mu^- p \to n \nu_{\mu} \gamma$ reaction

The recent TRIUMF experiment for $\mu^- p \to n \nu_{\mu} \gamma$ gave a surprising result that the induced pseudoscalar coupling constant $g_P$ was larger than the value obtained from $\mu^- p \to n \nu_{\mu}$ experiment as much as 44 %. Reexamining axial vector current on the gauge theory, we found an additional term to the matrix element of Beder and Fearing which was used to extract the $g_P$ value from the measured photon energy spectrum. This additional term, which is self gauge invariant, plays a key role in restoring the reliability of $g_P (- 0.88 m_{\mu}^2) = 6.77 g_A (0)$. Comparison with conventional approaches is also presented.