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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

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Related: physics.stackexchange.com/q/21/2451 –  Qmechanic Jul 14 '12 at 19:37

2 Answers 2

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.

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Sorry I am missing something in the 1st paragraph. The curve f(x,y) in the x-y plane, is it something described by an equation like x^2+y^2=a^2 (which is a cricle of radius a)? is that what you mean by f(x,y)? Also in the 2nd line, what square? –  Revo Aug 12 '11 at 11:14
    
Even if it's possible to interpret every Monte Carlo simulation in physics as some kind of numeric integration (which I doubt), it's certainly not in all cases the most useful way to look at it. I would not interpret simulated annealing as integration, to name a simple example. — The method you explain is indeed quite straightforward integration, but is it the only kind of Monte Carlo used in experimental high energy physics? –  leftaroundabout Aug 12 '11 at 11:24
    
@leftaroundabout I am asking about MC in studying detectors efficiency and analyzing data just in the same way anna v did. I have modified the question to be more specific. –  Revo Aug 12 '11 at 11:46
    
@Revo I mean of course y=f(x). Throwing random y and x 's one covers the phase space. I am editing. –  anna v Aug 12 '11 at 13:16
    
@leftaroundabout yes, AFAIK. Though any simulation, imo, is a form of integration, analogue instead of numeric. –  anna v Aug 12 '11 at 13:20

Monte Carlo are very important in almost any particle physics experiment and are used in a variety of ways including

  1. To prototype the experiment without spending many millions of dollars.

    They can be used to

    • show that the proposed physics signal will be detectable among the many known physics effects
    • to test variations on the proposed detector design
    • to estimate the size of the data set and determine what kind of computation resources will be needed to manage the data

    This kind of use is often an input to the proposal.

  2. 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.

  3. 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.

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the formulae after"actually record A' " do not come up correctly in this paragraph. the rest are ok. I am getting a mesage about mathjax that I do not use. "This page seems to use the older default config/MathJax.js file, and so needs to be updated. This is explained further at mathjax.org/help/configuration"; –  anna v Aug 12 '11 at 15:27
    
Thanks anna, that's because I had removed the delimiters in order to keep typing without undue delay and I forgot to put them back. –  dmckee Aug 12 '11 at 15:30
    
very ilustrative answer –  lurscher Jul 14 '12 at 19:45

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