4.4 Energy resolution

4.4.1 Introduction - relevant terms and origin of them

Energy resolution

There are 3 terms in the general energy resolution of calorimeters.

Generally it is fitted like that:

Implementation in code (for fitting):

Double_t res = par[0]*par[0]/x[0] + par[1]*par[1]/(x[0]*x[0]) + par[2]*par[2];
//sigma/e = sqrt(a**2/E + b**2/E**2 + c**2)
return TMath::Sqrt(res);

The individual terms can be traced back to

• a) sampling and statistical fluctuations (a^2/E), which is dominating over most of the useful range of calorimeters
• b) noise, which dominates the low-energy performance of calorimeters (b^2/E^2)
  
• c) effects coming from calibration errors, non-uniformities and non-linearities
  e.g. inherent sampling fluctuations depending on the impact point of the incident particle The later limit the performance at very high energies. In the case of the ATLAS calorimeters, the constant term comes from the accordian structure, see Wigmans, p.211, section 4.4.2.2.
  

reference - nice link: http://rkb.home.cern.ch/rkb/PH14pp/node63.html

How do you determine the energy resolution for jets?

Genereally, one is interested in:

sigma(Erec)/Erec

so that the resolution does not depend on the scale with which you are reconstructing Erec. Since the plots we are producing are Erec/Etruth with fit method (but RMS method is similare) we will have:

mean of gaussian = <Erec/Etruth>
sigma = sigma(Erec/Etruth) = sigma(Erec)/Etruth considering that the variation of Etruth in the bin we are considering is << of the variation of sigma(Erec)

thus we can obtain the percentual resolution as:
sigma/mean = sigma(Erec)/Etruth * <Etruth/Erec> = sigma(Erec)/Erec

So the plot we would like to have is the percentual resolution
(sigma(Erec)/Erec) as a function of Etruth. Do you agree on this ?
Is this quantity what you are plotting ?
Similar considerations can be done for RMS and average.

Clearly the RMS method is more dependent on the behaviour of the tails.
Both methods give information on the resolution we had decided to have the fit version for the moment to understand the behaviour of the bulk of the ditribution.

How are the fits performed?

• Plot Erec/Etrue in bins of eta and energy (these energy bins can be created as a function of - assumed - resolution, e.g.

                                                                      double s = sqrt(0.1/el + 0.01)*el;
                                                                       eh = ceil(el+2*s);
  

  where s is the assumed resolution and the energy bins ( el is lower bin edge, eh is higher bin) are designed to be 2*sigma of the assumed resolution

• Fit a Gaussian to the distributions in the individual bins
• Plots the derived Sigma/Mean against the energy
• Fit the above described energy resolution function to this plots with the fit parameters
  
  • a - sqrt(E) term
  • b - E term
  • c - constant term
    

4.4.2 Results

Fig. 1: Resolution for different Jet Algorithms and calibration methods for the central barrel region of 0.0<|eta|<0.7

Fig. 2: Resolution for the same algorithms for the region of 0.7<|eta|<1.5

Fig 3. Resolution fits for Cone4 Topo Jets (local Hadron Calibration)

Fig 4.: Resolution Fits for Cone 4 Tower Jets (global Hadron Calibration)

-- KristinLohwasser - 19 Jun 2007