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Both ($e$,$e'p$) and ($p$,$2p$) reactions have been performed to study theproton single-particle character of nuclear states with its relatedspectroscopic factor. Recently, the dispersive optical model (DOM) was appliedto the ($e$,$e'p$) analysis revealing that the traditional treatment of thesingle-particle overlap function, distorted waves, and nonlocality must befurther improved to achieve quantitative nuclear spectroscopy. We apply the DOMwave functions to the traditional ($p$,$2p$) analysis and investigate theconsistency of the DOM spectroscopic factor that describes the ($e$,$e'p$)cross section with the result of the ($p$,$2p$) analysis. Additionally, we makea comparison with a phenomenological single-particle wave function and opticalpotential. Uncertainty arising from a choice of $p$-$p$ interaction is alsoinvestigated. We implement the DOM wave functions to the distorted wave impulseapproximation (DWIA) framework for ($p$,$2p$) reactions. DOM + DWIA analysis on$^{40}$Ca($p$,$2p$)$^{39}$K data generates a proton $0d_{3/2}$ spectroscopicfactor of 0.560, which is meaningfully smaller than the DOM value of 0.71 shownto be consistent with the ($e$,$e'p$) analysis. Uncertainties arising fromchoices of single-particle wave function, optical potential, and $p$-$p$interaction do not explain this inconsistency. The inconsistency in thespectroscopic factor suggests there is urgent need for improving thedescription of $p$-$p$ scattering in a nucleus and the resulting in-mediuminteraction with corresponding implications for the analysis of this reactionin inverse kinematics.

First application of dispersive optical model to ($p$,$2p$) analysis within distorted wave impulse approximation framework