Upper Bound for the Induced Pseudoscalar Form Factor in Muon Capture

It is shown, on very general grounds, that the Goldberger-Treiman (GT) value for the induced pseudoscalar form factor, $h_A$, in muon-capture reactions represents an upper bound for any realistic $h_A$ that takes into account corrections to (i) nuclear GT relations and (ii) the hypothesis of partial conservation of axial-vector current. The proof is independent of the particular model one might choose for these corrections. An application of this bound to muon capture on ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ makes it difficult to understand the sign of the experimentally reported induced pseudotensor form factor in ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$ $\beta$ decay.