- #1

ChrisVer

Gold Member

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## Main Question or Discussion Point

Hi, let's say in some experiment with ##Z^0## (eg LEP) you are able to determine the "misidentification" of your particles.

Then you can find the efficiency matrix ##M_{eff}## which is given (for ##Z^0## decays to leptons or hadrons):

[itex] \begin{pmatrix} N_e \\ N_\mu \\ N_\tau \\ N_{had} \end{pmatrix}_{detect} = M_{eff} \begin{pmatrix} N_e \\ N_\mu \\ N_\tau \\ N_{had} \end{pmatrix}_{real} \Rightarrow \vec{N}_{det} = M_{eff} \vec{N}_{real} [/itex]

With some errors in its elements, so in general you measure ##M_{eff} \pm \delta M_{eff}##.

My question is then, since I want to find what is actually my real measured events for the decay products:

[itex] \vec{N}_{real} = M_{eff}^{-1} \vec{N}_{det}[/itex]

How can you find the error in the elements of [itex]M_{eff}^{-1}[/itex]?

One example I thought of is quite primitive, and in general very untrustworthy, for example get [itex]M_{eff}+\delta M_{eff}[/itex] find its inverse, and do the same for [itex]M_{eff} - \delta M_{eff}[/itex]. Then I know how much (in the worst case scenario the elements will diverge from the mean valued [itex]M_{eff}^{-1}[/itex].

I find it more plausible to create different matrices [itex] A_i[/itex] which have as elements different values in the range allowed from [itex]M_{eff} \pm \delta M_{eff}[/itex], take their inverse [itex]A_i^{-1}[/itex] and then mean value it and find its statistical error.

However with a ##4 \times 4## matrix (or 16 elements) I don't know how many [itex]A_i[/itex]'s can give a trustworthy result...any help?

Then you can find the efficiency matrix ##M_{eff}## which is given (for ##Z^0## decays to leptons or hadrons):

[itex] \begin{pmatrix} N_e \\ N_\mu \\ N_\tau \\ N_{had} \end{pmatrix}_{detect} = M_{eff} \begin{pmatrix} N_e \\ N_\mu \\ N_\tau \\ N_{had} \end{pmatrix}_{real} \Rightarrow \vec{N}_{det} = M_{eff} \vec{N}_{real} [/itex]

With some errors in its elements, so in general you measure ##M_{eff} \pm \delta M_{eff}##.

My question is then, since I want to find what is actually my real measured events for the decay products:

[itex] \vec{N}_{real} = M_{eff}^{-1} \vec{N}_{det}[/itex]

How can you find the error in the elements of [itex]M_{eff}^{-1}[/itex]?

One example I thought of is quite primitive, and in general very untrustworthy, for example get [itex]M_{eff}+\delta M_{eff}[/itex] find its inverse, and do the same for [itex]M_{eff} - \delta M_{eff}[/itex]. Then I know how much (in the worst case scenario the elements will diverge from the mean valued [itex]M_{eff}^{-1}[/itex].

I find it more plausible to create different matrices [itex] A_i[/itex] which have as elements different values in the range allowed from [itex]M_{eff} \pm \delta M_{eff}[/itex], take their inverse [itex]A_i^{-1}[/itex] and then mean value it and find its statistical error.

However with a ##4 \times 4## matrix (or 16 elements) I don't know how many [itex]A_i[/itex]'s can give a trustworthy result...any help?