Acceptance, efficiency, and fiducial cross sections

A fiducial region is a rule applied to truth-level or particle-level objects with declared definitions: momentum and angular cuts, stable-particle convention, photon dressing where relevant, and treatment of decays or isolation. These rules make the target reproducible for a theoretical prediction. Reconstructed objects approximate those definitions through an imperfect detector, so an offline selection and a fiducial region should be distinguished even when their numerical thresholds resemble one another.

Define acceptance A as the fraction of a specified full signal population lying in the fiducial region. Define efficiency as the probability that a fiducial event is reconstructed and selected, under the stated trigger and exposure conventions. Acceptance concerns the chosen physics population and geometry definition; efficiency concerns response conditional on that population. Their numerical values can depend on a generator’s within-region spectrum, so a fiducial result reduces extrapolation without becoming completely model independent.

Detector resolution can move a true fiducial event out of selection and can admit a signal event whose truth is outside the fiducial region. One convention subtracts outside-fiducial feed-in separately; another includes it in C, defined as all selected signal divided by true fiducial signal. In the second convention C need not lie below one, so calling it a pure efficiency is misleading. Every numerator and denominator must follow the same signal and decay definitions.

If a full production cross section excludes the decay branching fraction while a fiducial final-state cross section includes that decay, the extrapolation requires the branching factor as well as acceptance. Here we define full and fiducial cross sections for the same already specified final-state channel, so no extra branching factor appears. Likewise, luminosity may already include particular acquisition losses. Recording that convention prevents a trigger or live-time factor from being applied both in exposure and in C.

With a known background and fixed correction and luminosity, the observed selected count is still Poisson distributed with mean signal plus background. Consequently the counting variance in the background-subtracted count is the original count variance, not the subtracted signal expectation alone. If background is estimated from another sample, its uncertainty and correlations enter too. Negative background-subtracted estimates can occur; clipping them before fitting changes the estimator and can affect coverage.

Luminosity, response, and acceptance uncertainties often correlate bins and channels. Extrapolation to an unobserved region can be especially sensitive to theoretical assumptions even if the fiducial count is precise. Differential measurements require a migration treatment rather than an automatic independent correction for each reconstructed bin. Publishing the fiducial definitions, correction convention, covariance or likelihood, and theory dependence makes it possible to compare a later prediction without silently changing the measured target.