Nuclear scattering lengths from the spectroscopy of muonic molecules

The shift in energy levels of μ molecules produced by the nuclear interaction energy has been calculated. It is proportional to the probability of coincidence of the two nuclei and to their zero-energy scattering length. In pμp it is 0.01 eV; in pμd it is 0.005 eV. The effect may be detected by measuring the hyperfine splitting of pμd, of which it is a considerable fraction, and a null test for its presence is given. Its magnitude yields the difference of the nuclear scattering lengths in different spin states. As a by-product of our analysis of the pμd hyperfine splitting, we discuss the possibility it presents, as compared to pμp, of studying the μ-p spin-spin interaction, which is a sensitive test of quantum electrodynamics.