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Matrix Element Method

The Matrix Element Method consist in minimizing a likelihood.

The likelihood for N events is defined as $ L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)$

The best estimate of the parameter $\alpha$ is obtained through a maximisation of the likelihood. It is common practice to minimize $-ln(L(\alpha))$ with respect to $\alpha$, $-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$

In general, the probability that an event is accepted depends on the characteristics of the measured event, and not on the process that produced it. The measured probability density $\bar{P}(x,\alpha)$ can be related to the produced probability density $P(x,\alpha)$: $\bar{P}(x,\alpha){{{ Acc(x) P(x,\alpha)$ where $ Acc(x)$ is the detector acceptance, which depends only on $ x $. So the quantity that we have to minimize is $-ln (\tilde{L}) }}}-\sum_{i=1}{N} ln(P(x_i;\alpha)) + N \int Acc(x) P(x,\alpha)dx$ where the term $-\sum_{i=1}N ln(Acc(x_i))$ has been omitted since it does not depend on $\alpha$.

Definition of the Weight

The Matrix Element Method associates a weight to each experimental event $ P( x || \alpha)=\frac{1}{\sigma_{ \alpha}} \int d \phi( y) ||M_{ \alpha}||^2 ( y) dw_1 dw_2 f_1(w_1) f_2(w_2) W(x, y) $ where

  1. $ x $ is the set of information describing the events in the detector (momenta,tag,...)
  2. $ \alpha $ describe a theoretical hyppothesis
  3. $\sigma_{ \alpha}$ is the cross section of this theoretical hyppothesis
  4. $M_{ \alpha}$ is the aplitude linked to this theoretical framework
  5. $f_i(w_i)$ is the parton distribution function associate to the initial parton
  6. $W(x, y)$ is the TransferFunction

Computation of those elements

The MadWeight has created a series of tool to compute the transfer function, the weight, the cross-section, the likelihood,... Some of these tools have their own specific page/

  1. TransferFunction
  2. Computation of the Weight
  3. AcceptanceTerm

-- Main.OlivierMattelaer - 22 May 2009

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