How are Monte Carlo simulations used in experimental high energy physics? Particularly in studying detectors limitations (efficiencies?) and data analysis.
I will appreciate giving a simple example to clarify how MC is used if the question is too technical to have a simplified answer
One has to realize that a Monte Carlo simulation is an integration tool. Suppose you have a curve in an xy plot, y=f(x). If you throw random (x,y) pairs in the square containing the f(x) and count the number where y is less than f(x) versus the number y larger than f(x) you get an estimate of the area under f(x), i.e. the integral of the function.
In elementary particle physics, the phase space ( equivalent to the square in the simple example) is known. Theoretical functions are used as a weight to a random number generator, to generate "events" according to their parameters and checked against the real event data. If the fit is bad, the parameters are changed to improve it.
The advantage is
a)Detector limitations can be programmed in the phase space and events generated with the limits of the detector included
2) The method is much more efficient in computer time than the numerical integrations necessary over the innumerable functions entering the problems, detector and theory.
3) Once a Monte Carlo event sample is generated it can be used "as if it is data" over and over again to get plots not thought up beforehand.
In the recent LHC experiments the Monte Carlo events were generated way before the real data, according to the detector limitations and to the theoretical expectations from the Standard Model. The existence of these Monte Carlo data set allowed fast checks on whether new physics is appearing. New physics will appear as statistically significant deviations from the Monte Carlo curves.
Monte Carlo are very important in almost any particle physics experiment and are used in a variety of ways including
To prototype the experiment without spending many millions of dollars.
They can be used to
This kind of use is often an input to the proposal.
To test a proposed analysis, and to evaluate the expected systematic uncertainty associated with it.
In many experiments (especially those at colliders) the data contains a vast number of over-lapping physics channels, and must be filtered (sometimes quite extensively) before a particular observable can be pulled out of the data.
By generating a known amount of the (simulated) desired signal and hiding it among a bunch of (simulated) backgrounds, one can test if the analysis works (so this point could just as well have gone in item #1), but you can do more than that: you can also take note of how many false positives and negatives are introduced in each filtering step and use that data to (a) optimize the filters and (b) evaluate the systematic uncertainly of the analysis.
These applications are integral to the data analysis.
Un-fold complicated detector behavior.
Many times we have a theoretical or phenomenological entity (call it $A(\mathbf{x})$), that we would like to get at, but the data actually record $$A'(\mathbf{x_n}) = \int_n dx_{n-1} \rho(\mathbf{x_n};\mathbf{x_n-1}) \cdots \int_2 dx_2 \rho(\mathbf{x_2};\mathbf{x_1})\int_1 dx \rho(\mathbf{x_1};\mathbf{x}) A(\mathbf{x}) .$$
If your MC is good enough (big if, as that is tough) You can use it to produce an approximate backwards mapping function $f'(A'(\mathbf{x_n})) \to \tilde{A}(\mathbf{x})$. This introduces another uncertainty often called the "model dependent error" (though the term is overloaded for errors dependent on theoretical models as well).
This application is rather less common, but is part of the data analysis.
