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

Physics > Data Analysis, Statistics and Probability
[Submitted on 3 Sep 2026]
Title:Theoretical--operational modelling of complex experiments: parameter robustness and degeneracy in muon--electron conversion
View PDF HTML (experimental)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 $\rho$ is $\eta=0.331$, 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.
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* A theoretical-operational model (TOM) is formulated as a typed factorization of forward prediction in complex experiments.
* The model studies how changes in preparation, phenomenon modeling, readout, backgrounds, and analysis project onto the local manifold generated by physics parameters.
* The resulting response map separates model deformation into a parameter-equivalent component and a residual component.
* The application is to charged-lepton-flavor-violating muon-electron conversion in aluminum.
* A one-bin conversion-rate model exhibits an exact normalization degeneracy.
* In a two-template model of the elastic-inelastic spectrum, common signal normalization is absorbed by the fitted conversion rate without changing operator-sensitive nuclear-response ratios.
* A relative change in elastic-inelastic efficiency is exactly parameter-equivalent to a first-order change in that ratio.
* A $100\,\mathrm{keV}/c$ momentum-scale mismatch shows partial parameter-equivalence: the projection onto the local tangent space defined by the elastic normalization ($R0$) and the logarithmic inelastic-to-elastic response ratio ($\rho$) is $\eta=0.331$.

The theoretical-operational model (TOM) is formulated as a typed factorization of forward predictions in complex experiments, examining how variations in preparation, modeling, readout, backgrounds, and analysis affect the local manifold of physics parameters. The model aims to separate model deformations into components that are parameter-equivalent and those that are residual, which cannot be absorbed by changes in fitted physical parameters. This framework is applied to charged-lepton-flavor-violating muon-electron conversion in aluminum. Specific findings include exact normalization degeneracy in a one-bin conversion-rate model. In a two-template model of the elastic-inelastic spectrum, a common signal normalization is absorbed by the fitted conversion rate without altering operator-sensitive nuclear-response ratios. Furthermore, a relative change in elastic-inelastic efficiency is exactly parameter-equivalent to a first-order change in that ratio. A numerical example involving a $100\,\mathrm{keV}/c$ momentum-scale mismatch shows partial parameter-equivalence: the projection onto the local tangent space generated by the elastic normalization ($R0$) and the logarithmic inelastic-to-elastic response ratio ($\rho$) is $\eta=0.331$, with most spectral deformation remaining as a residual shape.

The work establishes a framework for quantifying robustness and degeneracy in inferring physical parameters from complex experimental data by mapping model variations onto the parameter space manifold. The core implication is that not all observed spectral or rate changes are attributable to fundamental physics parameters; a significant portion is systematic artifact stemming from the experimental realization, modeling choices, or analysis pipeline. This moves the focus from fitting to understanding *why* certain inferences are robust or degenerate. The partial equivalence ($\eta=0.331$) for the momentum mismatch suggests that while some information is directly related to fundamental physics parameters ($R0$ and $\rho$), much of the spectral deformation represents non-physical, residual noise or model-specific effects. This pattern underscores a necessary step in experimental physics: explicitly disentangling the intrinsic physical signal from instrumental and theoretical noise before parameter estimation. The challenge lies in systematically defining and constraining the "local manifold" generated by phenomenology such that these residual components are rigorously categorized as irreducible artifacts rather than unknown physics.

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

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

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