Electroweak fit inputs and correlated tests

At leading order a conventional electroweak input set contains the electromagnetic coupling, the Fermi constant and the Z mass. These determine combinations of the gauge couplings and vacuum scale, allowing a W-mass prediction. At higher order additional parameters such as top and Higgs masses, strong coupling and hadronic vacuum polarization enter. The chosen renormalization and unstable-particle mass conventions determine how those measured inputs map to Lagrangian parameters.

If the measured W mass is instead included as an input, reproducing that same number cannot be counted as an independent successful W-mass test. One can still test other observables or examine an indirect prediction obtained by excluding that datum. A leave-one-out result and a full-fit residual answer related but different questions. Experimental combinations also need their shared calibration and modeling systematics retained, rather than replacing several correlated analyses with an artificially small independent error.

For two observables, divide each residual by its own standard deviation and denote their correlation by c. Inverting the two-by-two correlation matrix yields a quadratic form with a cross term. Equal-sign standardized residuals lie along a common direction whose variance grows for positive correlation. Opposite-sign residuals lie along a difference direction whose variance shrinks. Correlation can therefore reduce one discrepancy and strengthen another; ignoring it has no uniformly conservative effect.

A covariance built from experimental errors is not automatically a complete theory-error model. Common missing corrections may be correlated across observables, and some theory uncertainties are better represented by explicit nuisance parameters or non-Gaussian likelihoods. If input uncertainties have already been propagated into a covariance, also fitting the same fluctuations independently can double count them. The fit specification should make every uncertainty source and dependence reproducible rather than hiding it inside a single quoted total error.

A nuisance parameter can represent a shared calibration shift, a hadronic input, or a theoretical normalization. Profiling minimizes the objective over its allowed values, including any external constraint term; marginalization instead integrates over it with specified probability distributions. These procedures need not agree beyond simple Gaussian settings. A change in chi-squared should identify which parameters were reoptimized and which external constraints were retained.

For a regular Gaussian model with known covariance and a correctly specified linear mean, a minimized chi-squared may be compared with a distribution whose degrees of freedom subtract the independently fitted parameters. Boundaries, nonlinearities, estimated covariance and model selection can change that calibration. A p-value measures compatibility with a specified test construction, not the probability that the Standard Model is true. Precision information is most useful when a published input list and likelihood allow other researchers to reproduce the claimed constraints and test alternative explanations.