Radiative Muon Capture and the Giant Dipole Resonance inCa40

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We have calculated the photon spectrum and total rate for radiative muon capture in ${\mathrm{Ca}}^{40}$ using the giant dipole resonance (GDR) model used by Foldy and Walecka to significantly improve theoretical calculations of ordinary muon capture rates. In this model we relate the dipole parts of the nuclear matrix elements relevant for radiative muon capture to integrals over experimental photo-absorption cross sections in the region dominated by the giant dipole resonance. The remaining other multipole and velocity parts of the nuclear matrix elements are evaluated in the closure approximation using harmonic oscillator wave functions. We present numerical results for the photon spectrum as calculated in the GDR model for various values of the weak-interaction coupling constants and of the average maximum photon energy ${k}_{m}$ which is used in the other multipole and velocity parts of the calculation. The most important effect of the GDR calculation is to reduce the dipole contribution to the capture rate, thus giving an absolute radiative rate some 40% lower and a relative rate (ratio of radiative and ordinary rates) some 20% lower than that obtained in the closure-harmonic-oscillator model. As a consequence, the GDR model requires a larger value of the induced pseudoscalar coupling constant ${g}_{P}$ to reproduce a given spectrum than does the closure-harmonic-oscillator model. Finally, we compare our results with the data of Conversi et al., who found by interpreting their data in the closure-harmonic-oscillator model that ${g}_{P}=(13.3\ifmmode\pm\else\textpm\fi{}2.7){g}_{A}$, where ${g}_{A}$ is the axial vector coupling constant. We find that the GDR calculation requires ${g}_{P}=(16.5\ifmmode\pm\else\textpm\fi{}3.1){g}_{A}$ for a fit to the experiment, where we have assumed ${k}_{m}=88$ MeV and have taken currently accepted values for the other coupling constants. Alternatively, by taking the Goldberger-Treiman value ${g}_{P}\ensuremath{\cong}7{g}_{A}$ and varying the induced tensor coupling constant ${g}_{T}$ we obtain a fit to the data for ${g}_{T}\ensuremath{\gtrsim}35{g}_{V}$. As these results are quire sensitive to ${k}_{m}$, we give in addition results for other values of ${k}_{m}$.