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. [:MadWeight:Computation of the Weight]
3. AcceptanceTerm

-- Main.OlivierMattelaer - 22 May 2009