Acceptance

We define the acceptance a of an experiment as the average detection efficiency. Frequently, the word is also used in the more restricted sense of geometric acceptance defined below.

Let N be the total number of events that occurred, out of which n are observed. Then the expectation values of N and n are related by

One may consider the acceptance as a function of one or more variables, or in a small region of phase space, e.g. in one bin of t for a two-body process.

By this general definition, the acceptance includes all effects that cause losses of events: the finite size of detectors, the inefficiencies of detectors and of off-line event reconstruction, dead times, effects of veto counters, etc.

Let be the physical variables that describe an event, such as the momenta of the particles, positions of interaction vertices, and possibly also discrete variables like the number of particles, spin components, etc. These are random variables following a probability distribution

is the allowed region for x, and the integral includes a sum over discrete variables. The non-normalized density F(x) is given by the experimental conditions, i.e. beam, target, etc., and is proportional to the differential cross-section. For a sufficiently small phase space region the differential cross-section is nearly constant and hence drops out from the normalized probability density f(x).

Let be the total detection efficiency for an event given its physical variables x. The acceptance is then the expectation value of ,

If, to a sufficiently good approximation,

where is the purely geometric efficiency if the particles hit the detectors, otherwise) and is a constant overall detection efficiency, then

a_g is called the geometric acceptance .

Acceptances are usually estimated by Monte Carlo integration (see Bock98). If one is able to simulate the experiment by generating M (pseudo-) random events according to the probability distribution , then the Monte Carlo estimate for a is

with the estimated variance

If out of M generated events m events are accepted, then for the geometric acceptance a_g one has the unbiased estimates from a binomial distribution Acceptance

If some part of the integration can be done analytically, then this will reduce the variance; fewer events are necessary, hence the computing load is reduced, sometimes substantially. We will show this by an example: assume that can be subdivided into non-overlapping regions , that the probabilities

can be calculated exactly, and that the regions are chosen such that for , for , and for  ; in other words, the boundary of the accepted region is contained within . Then by restricting the generation of Monte Carlo events to the region , one obtains the estimates

which transforms to

If the acceptance of an experiment varies with time, then the total acceptance will be a weighted average of the acceptances in different periods of time, where the appropriate weight of a period is the number of beam particles, or the integrated luminosity.  also Cross-Section.

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