Physics:Acceptance

From HandWiki


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

Hepb img2.gif

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 Hepb img3.gif 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

Hepb img4.gif

Hepb img5.gif 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 Hepb img6.gif be the total detection efficiency for an event given its physical variables x. The acceptance is then the expectation value of Hepb img6.gif ,

Hepb img7.gif

If, to a sufficiently good approximation,

Hepb img8.gif

where Hepb img9.gif is the purely geometric efficiency Hepb img10.gif if the particles hit the detectors, Hepb img11.gif otherwise) and Hepb img12.gif is a constant overall detection efficiency, then

Hepb img13.gif

Hepb img14.gif

ag 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 Hepb img15.gif according to the probability distribution Hepb img16.gif , then the Monte Carlo estimate for a is

Hepb img17.gif

with the estimated variance

Hepb img18.gif

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

Hepb img19.gif

Hepb img20.gif

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 Hepb img5.gif can be subdivided into non-overlapping regions Hepb img21.gif , that the probabilities

Hepb img22.gif

can be calculated exactly, and that the regions are chosen such that Hepb img11.gif for Hepb img23.gif , Hepb img24.gif for Hepb img25.gif , and Hepb img26.gif for Hepb img27.gif ; in other words, the boundary of the accepted region is contained within Hepb img28.gif . Then by restricting the generation of Monte Carlo events to the region Hepb img28.gif , one obtains the estimates

Hepb img29.gif

Hepb img30.gif

which transforms to

Hepb img31.gif

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.