Theoretical--operational modelling of complex experiments: parameter robustness and degeneracy in muon--electron conversion

Abstract

Complex experiments infer theory parameters through a coupled chain of physical models and data reduction. We formulate the theoretical--operational model (TOM) as a typed factorization of this forward prediction and study how changes of preparation, phenomenon modelling, readout, backgrounds, and analysis project onto the local manifold generated by the physics parameters. For a smooth prediction and a locally identifiable weighted least-squares estimate, the resulting response map separates each model deformation into a parameter-equivalent component and a residual component that cannot be absorbed by a change of the fitted physics parameters. This gives local criteria for robustness, exact degeneracy, and partial degeneracy of parameter inference. The construction is applied to charged-lepton-flavour-violating muon--electron conversion in aluminium. A one-bin conversion-rate model exhibits an exact normalization degeneracy. In a two-template model of the elastic--inelastic spectrum, a common signal normalization is absorbed by the fitted conversion rate without changing an operator-sensitive nuclear-response ratio, whereas a relative elastic--inelastic efficiency change is exactly parameter-equivalent to a change of that ratio at first order. A numerical Run-I example based on published Mu2e spectra shows that a \(100\,\mathrm{keV}/c\) momentum-scale mismatch is only partially parameter-equivalent: its projection ratio onto the local tangent space generated by the elastic normalization \(R_0\) and the logarithmic inelastic-to-elastic response ratio \(ρ\) is \(η=0.331\), while most of the weighted spectral deformation remains as a residual shape. TOM thereby provides a local theoretical description of parameter robustness and degeneracy at the interface of particle/nuclear phenomenology and experimental realization.