Optimise Reconstruction

Overview

Teaching: 15 min
Exercises: 5 min

Questions

How can I improve the reconstruction when using standard methods?

Objectives

Compare kinematic resolutions for electron reconstruction based on tracks alone and for tracks+calorimetry

Use realistic scattered electron ID in the reconstruction

Optimising the reconstruction

There are four quantities that are used to reconstruct the inclusive kinematics: the scattered electron energy and polar angle, the hadronic final state (HFS) transverse momentum, and the E-pz sum of all particles in the HFS. The values of these quantities depends on the information used in the reconstruction of the scattered electron and the HFS. In some regions of the phase space, the calorimeters may do a much better job of reconstructing the scattered electron compared to the tracking system.

We can investigate this using the script below, which should be copied into a file called OptimiseReconstruction.C

// PODIO
#include "podio/Frame.h"
#include "podio/ROOTFrameReader.h"

// DATA MODEL
#include "edm4eic/InclusiveKinematicsCollection.h"
#include "edm4hep/MCParticleCollection.h"
#include "edm4eic/ReconstructedParticleCollection.h"
#include "edm4eic/MCRecoClusterParticleAssociationCollection.h"
#include "edm4eic/MCRecoParticleAssociationCollection.h"
#include "edm4eic/HadronicFinalStateCollection.h"
#include "edm4eic/ClusterCollection.h"
#include "edm4hep/Vector3f.h"

std::vector<float> calc_elec_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam);
std::vector<float> calc_jb_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam);
std::vector<float> calc_da_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam);
std::vector<float> calc_sig_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam);
std::vector<float> calc_esig_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam);

void OptimiseReconstruction(std::string filename) {

  // Settings
  Float_t E_ebeam = 18;
  Float_t E_pbeam = 275;
  Float_t m_e = 0.000511;

  std::vector<std::string> inFiles = {filename};

  auto reader = podio::ROOTFrameReader();
  reader.openFiles(inFiles);

  // Declare benchmark histograms
  TH1F *hResoX_electron = new TH1F("hResoX_electron","Electron method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_jb = new TH1F("hResoX_jb","JB method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_da = new TH1F("hResoX_da","Double Angle method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_sigma = new TH1F("hResoX_sigma","#Sigma method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_esigma = new TH1F("hResoX_esigma","e-#Sigma method;#Deltax/x;Counts",100,-1,1);

  TH1F *hResoY_electron = new TH1F("hResoY_electron","Electron method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_jb = new TH1F("hResoY_jb","JB method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_da = new TH1F("hResoY_da","Double Angle method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_sigma = new TH1F("hResoY_sigma","#Sigma method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_esigma = new TH1F("hResoY_esigma","e-#Sigma method;#Deltay/y;Counts",100,-1,1);

  TH1F *hResoQ2_electron = new TH1F("hResoQ2_electron","Electron method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_jb = new TH1F("hResoQ2_jb","JB method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_da = new TH1F("hResoQ2_da","Double Angle method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_sigma = new TH1F("hResoQ2_sigma","#Sigma method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_esigma = new TH1F("hResoQ2_esigma","e-#Sigma method;#DeltaQ2/Q2;Counts",100,-1,1);

  TH1F *hResoX_calo_electron = new TH1F("hResoX_calo_electron","Electron method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_calo_jb = new TH1F("hResoX_calo_jb","JB method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_calo_da = new TH1F("hResoX_calo_da","Double Angle method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_calo_sigma = new TH1F("hResoX_calo_sigma","#Sigma method;#Deltax/x;Counts",100,-1,1);
  TH1F *hResoX_calo_esigma = new TH1F("hResoX_calo_esigma","e-#Sigma method;#Deltax/x;Counts",100,-1,1);

  TH1F *hResoY_calo_electron = new TH1F("hResoY_calo_electron","Electron method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_calo_jb = new TH1F("hResoY_calo_jb","JB method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_calo_da = new TH1F("hResoY_calo_da","Double Angle method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_calo_sigma = new TH1F("hResoY_calo_sigma","#Sigma method;#Deltay/y;Counts",100,-1,1);
  TH1F *hResoY_calo_esigma = new TH1F("hResoY_calo_esigma","e-#Sigma method;#Deltay/y;Counts",100,-1,1);

  TH1F *hResoQ2_calo_electron = new TH1F("hResoQ2_calo_electron","Electron method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_calo_jb = new TH1F("hResoQ2_calo_jb","JB method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_calo_da = new TH1F("hResoQ2_calo_da","Double Angle method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_calo_sigma = new TH1F("hResoQ2_calo_sigma","#Sigma method;#DeltaQ2/Q2;Counts",100,-1,1);
  TH1F *hResoQ2_calo_esigma = new TH1F("hResoQ2_calo_esigma","e-#Sigma method;#DeltaQ2/Q2;Counts",100,-1,1);

  Float_t x_truth, x_electron, x_jb, x_da, x_sigma, x_esigma;
  Float_t y_truth, y_electron, y_jb, y_da, y_sigma, y_esigma;
  Float_t Q2_truth, Q2_electron, Q2_jb, Q2_da, Q2_sigma, Q2_esigma;
  Float_t E, theta, sigma_h, pt_had;

  cout << reader.getEntries("events") << " events found" << endl;
  for (size_t i = 0; i < reader.getEntries("events"); i++) {// begin event loop
    const auto event = podio::Frame(reader.readNextEntry("events"));
    if (i%100==0) cout << i << " events processed" << endl;

    // Retrieve Inclusive Kinematics Collections
    auto& kin_truth = event.get<edm4eic::InclusiveKinematicsCollection>("InclusiveKinematicsTruth");
    auto& kin_electron = event.get<edm4eic::InclusiveKinematicsCollection>("InclusiveKinematicsElectron");
    auto& kin_jb = event.get<edm4eic::InclusiveKinematicsCollection>("InclusiveKinematicsJB");
    auto& kin_da = event.get<edm4eic::InclusiveKinematicsCollection>("InclusiveKinematicsDA");
    auto& kin_sigma = event.get<edm4eic::InclusiveKinematicsCollection>("InclusiveKinematicsSigma");
    auto& kin_esigma = event.get<edm4eic::InclusiveKinematicsCollection>("InclusiveKinematicsESigma");

    // Retrieve Scattered electron and HFS
    auto& eleCollection = event.get<edm4eic::ReconstructedParticleCollection>("ScatteredElectronsEMinusPz");
    auto& hfsCollection = event.get<edm4eic::HadronicFinalStateCollection>("HadronicFinalState");

    auto& ecalClusters = event.get<edm4eic::ClusterCollection>("EcalClusters");

    // Store kinematics from InclusiveKinematics branches
    if (kin_truth.empty() || kin_electron.empty() || kin_jb.empty()) continue;
    if (eleCollection.empty()) continue;

    TLorentzVector scat_ele;
    E = eleCollection[0].getEnergy();
    auto& ele_momentum = eleCollection[0].getMomentum();
    scat_ele.SetPxPyPzE(ele_momentum.x, ele_momentum.y, ele_momentum.z, E);
    theta = scat_ele.Theta();
    sigma_h = hfsCollection[0].getSigma();
    pt_had = hfsCollection[0].getPT();

    // Calculate kinematics manually
    std::vector<float> elec_reco = calc_elec_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> jb_reco = calc_jb_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> da_reco = calc_da_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> sigma_reco = calc_sig_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> esigma_reco = calc_esig_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);

    x_truth = kin_truth.x()[0];
    y_truth = kin_truth.y()[0];
    Q2_truth = kin_truth.Q2()[0];

    x_electron = elec_reco[0];
    x_jb = jb_reco[0];
    x_da = da_reco[0];
    x_sigma = sigma_reco[0];
    x_esigma = esigma_reco[0];

    y_electron = elec_reco[1];
    y_jb = jb_reco[1];
    y_da = da_reco[1];
    y_sigma = sigma_reco[1];
    y_esigma = esigma_reco[1];

    Q2_electron = elec_reco[2];
    Q2_jb = jb_reco[2];
    Q2_da = da_reco[2];
    Q2_sigma = sigma_reco[2];
    Q2_esigma = esigma_reco[2];

    // Some example cuts
    bool cuts = true;
    cuts = cuts && (y_truth < 0.95);
    cuts = cuts && (y_truth > 0.01);
    cuts = cuts && (Q2_truth > 1);

    if (!cuts) continue;

    hResoX_electron->Fill((x_electron-x_truth)/x_truth);
    hResoX_jb->Fill((x_jb-x_truth)/x_truth);
    hResoX_da->Fill((x_da-x_truth)/x_truth);
    hResoX_sigma->Fill((x_sigma-x_truth)/x_truth);
    hResoX_esigma->Fill((x_esigma-x_truth)/x_truth);

    hResoY_electron->Fill((y_electron-y_truth)/y_truth);
    hResoY_jb->Fill((y_jb-y_truth)/y_truth);
    hResoY_da->Fill((y_da-y_truth)/y_truth);
    hResoY_sigma->Fill((y_sigma-y_truth)/y_truth);
    hResoY_esigma->Fill((y_esigma-y_truth)/y_truth);

    hResoQ2_electron->Fill((Q2_electron-Q2_truth)/Q2_truth);
    hResoQ2_jb->Fill((Q2_jb-Q2_truth)/Q2_truth);
    hResoQ2_da->Fill((Q2_da-Q2_truth)/Q2_truth);
    hResoQ2_sigma->Fill((Q2_sigma-Q2_truth)/Q2_truth);
    hResoQ2_esigma->Fill((Q2_esigma-Q2_truth)/Q2_truth);

    // Replace electron momentum with energy from calo cluster
    E = eleCollection[0].getClusters()[0].getEnergy();

    // Recalculate kinematics
    std::vector<float> calo_elec_reco = calc_elec_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> calo_jb_reco = calc_jb_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> calo_da_reco = calc_da_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> calo_sigma_reco = calc_sig_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);
    std::vector<float> calo_esigma_reco = calc_esig_method(E, theta, pt_had, sigma_h, E_ebeam, E_pbeam);

    x_electron = calo_elec_reco[0];
    x_jb = calo_jb_reco[0];
    x_da = calo_da_reco[0];
    x_sigma = calo_sigma_reco[0];
    x_esigma = calo_esigma_reco[0];

    y_electron = calo_elec_reco[1];
    y_jb = calo_jb_reco[1];
    y_da = calo_da_reco[1];
    y_sigma = calo_sigma_reco[1];
    y_esigma = calo_esigma_reco[1];

    Q2_electron = calo_elec_reco[2];
    Q2_jb = calo_jb_reco[2];
    Q2_da = calo_da_reco[2];
    Q2_sigma = calo_sigma_reco[2];
    Q2_esigma = calo_esigma_reco[2];

    hResoX_calo_electron->Fill((x_electron-x_truth)/x_truth);
    hResoX_calo_jb->Fill((x_jb-x_truth)/x_truth);
    hResoX_calo_da->Fill((x_da-x_truth)/x_truth);
    hResoX_calo_sigma->Fill((x_sigma-x_truth)/x_truth);
    hResoX_calo_esigma->Fill((x_esigma-x_truth)/x_truth);

    hResoY_calo_electron->Fill((y_electron-y_truth)/y_truth);
    hResoY_calo_jb->Fill((y_jb-y_truth)/y_truth);
    hResoY_calo_da->Fill((y_da-y_truth)/y_truth);
    hResoY_calo_sigma->Fill((y_sigma-y_truth)/y_truth);
    hResoY_calo_esigma->Fill((y_esigma-y_truth)/y_truth);

    hResoQ2_calo_electron->Fill((Q2_electron-Q2_truth)/Q2_truth);
    hResoQ2_calo_jb->Fill((Q2_jb-Q2_truth)/Q2_truth);
    hResoQ2_calo_da->Fill((Q2_da-Q2_truth)/Q2_truth);
    hResoQ2_calo_sigma->Fill((Q2_sigma-Q2_truth)/Q2_truth);
    hResoQ2_calo_esigma->Fill((Q2_esigma-Q2_truth)/Q2_truth);
  }
  // Drawing the histograms
  auto canvas_x_1D = new TCanvas();
  canvas_x_1D->Divide(3,2);
  canvas_x_1D->cd(1); hResoX_electron->Draw("hist");
  hResoX_calo_electron->SetLineStyle(2); hResoX_calo_electron->SetLineColor(kRed); hResoX_calo_electron->Draw("hist same");
  canvas_x_1D->cd(2); hResoX_jb->Draw("hist");
  hResoX_calo_jb->SetLineStyle(2); hResoX_calo_jb->SetLineColor(kRed); hResoX_calo_jb->Draw("hist same");
  canvas_x_1D->cd(3); hResoX_da->Draw("hist");
  hResoX_calo_da->SetLineStyle(2); hResoX_calo_da->SetLineColor(kRed); hResoX_calo_da->Draw("hist same");
  canvas_x_1D->cd(4); hResoX_sigma->Draw("hist");
  hResoX_calo_sigma->SetLineStyle(2); hResoX_calo_sigma->SetLineColor(kRed); hResoX_calo_sigma->Draw("hist same");
  canvas_x_1D->cd(5); hResoX_esigma->Draw("hist");
  hResoX_calo_esigma->SetLineStyle(2); hResoX_calo_esigma->SetLineColor(kRed); hResoX_calo_esigma->Draw("hist same");
  canvas_x_1D->cd(6);
  auto legend_x = new TLegend(0.1,0.1,0.9,0.9);
  legend_x->AddEntry(hResoX_electron, "E_{e} from tracker", "l");
  legend_x->AddEntry(hResoX_calo_electron, "E_{e} from ECAL", "l");
  legend_x->Draw();
  
  auto canvas_y_1D = new TCanvas();
  canvas_y_1D->Divide(3,2);
  canvas_y_1D->cd(1); hResoY_electron->Draw("hist");
  hResoY_calo_electron->SetLineStyle(2); hResoY_calo_electron->SetLineColor(kRed); hResoY_calo_electron->Draw("hist same");
  canvas_y_1D->cd(2); hResoY_jb->Draw("hist");
  hResoY_calo_jb->SetLineStyle(2); hResoY_calo_jb->SetLineColor(kRed); hResoY_calo_jb->Draw("hist same");
  canvas_y_1D->cd(3); hResoY_da->Draw("hist");
  hResoY_calo_da->SetLineStyle(2); hResoY_calo_da->SetLineColor(kRed); hResoY_calo_da->Draw("hist same");
  canvas_y_1D->cd(4); hResoY_sigma->Draw("hist");
  hResoY_calo_sigma->SetLineStyle(2); hResoY_calo_sigma->SetLineColor(kRed); hResoY_calo_sigma->Draw("hist same");
  canvas_y_1D->cd(5); hResoY_esigma->Draw("hist");
  hResoY_calo_esigma->SetLineStyle(2); hResoY_calo_esigma->SetLineColor(kRed); hResoY_calo_esigma->Draw("hist same");
  canvas_y_1D->cd(6);
  auto legend_y = new TLegend(0.1,0.1,0.9,0.9);
  legend_y->AddEntry(hResoY_electron, "E_{e} from tracker", "l");
  legend_y->AddEntry(hResoY_calo_electron, "E_{e} from ECAL", "l");
  legend_y->Draw();
  
  auto canvas_Q2_1D = new TCanvas();
  canvas_Q2_1D->Divide(3,2);
  canvas_Q2_1D->cd(1); hResoQ2_electron->Draw("hist");
  hResoQ2_calo_electron->SetLineStyle(2); hResoQ2_calo_electron->SetLineColor(kRed); hResoQ2_calo_electron->Draw("hist same");
  canvas_Q2_1D->cd(2); hResoQ2_jb->Draw("hist");
  hResoQ2_calo_jb->SetLineStyle(2); hResoQ2_calo_jb->SetLineColor(kRed); hResoQ2_calo_jb->Draw("hist same");
  canvas_Q2_1D->cd(3); hResoQ2_da->Draw("hist");
  hResoQ2_calo_da->SetLineStyle(2); hResoQ2_calo_da->SetLineColor(kRed); hResoQ2_calo_da->Draw("hist same");
  canvas_Q2_1D->cd(4); hResoQ2_sigma->Draw("hist");
  hResoQ2_calo_sigma->SetLineStyle(2); hResoQ2_calo_sigma->SetLineColor(kRed); hResoQ2_calo_sigma->Draw("hist same");
  canvas_Q2_1D->cd(5); hResoQ2_esigma->Draw("hist");
  hResoQ2_calo_esigma->SetLineStyle(2); hResoQ2_calo_esigma->SetLineColor(kRed); hResoQ2_calo_esigma->Draw("hist same");
  canvas_Q2_1D->cd(6);
  auto legend_Q2 = new TLegend(0.1,0.1,0.9,0.9);
  legend_Q2->AddEntry(hResoQ2_electron, "E_{e} from tracker", "l");
  legend_Q2->AddEntry(hResoQ2_calo_electron, "E_{e} from ECAL", "l");
  legend_Q2->Draw();
  

  cout << "Done!" << endl;
}

// electron method
std::vector<float> calc_elec_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam) {
  float Q2  = 2.*E_ebeam*E*(1+TMath::Cos(theta));
  float y = 1. - (E/E_ebeam)*TMath::Sin(theta/2)*TMath::Sin(theta/2);
  float x = Q2/(4*E_ebeam*E_pbeam*y);
  return {x, y, Q2};
}

// jb method
std::vector<float> calc_jb_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam) {
  float y = sigma_h/(2*E_ebeam);
  float Q2 = pt_had*pt_had / (1-y);
  float x = Q2/(4*E_ebeam*E_pbeam*y);
  return {x, y, Q2};
}

// float angle method
std::vector<float> calc_da_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam) {
  float alpha_h = sigma_h/pt_had;
  float alpha_e = TMath::Tan(theta/2);
  float y = alpha_h / (alpha_e + alpha_h);
  float Q2 = 4*E_ebeam*E_ebeam / (alpha_e * (alpha_h + alpha_e));
  float x = Q2/(4*E_ebeam*E_pbeam*y);
  return {x, y, Q2};
}

// sigma method
std::vector<float> calc_sig_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam) {
  float y = sigma_h/(sigma_h + E*(1 - TMath::Cos(theta))); 
  float Q2 = E*E*TMath::Sin(theta)*TMath::Sin(theta) / (1-y);
  float x = Q2/(4*E_ebeam*E_pbeam*y);
  return {x, y, Q2};
}

// e-sigma method
std::vector<float> calc_esig_method(float E, float theta, float pt_had, float sigma_h, float E_ebeam, float E_pbeam) {
  float Q2  = 2.*E_ebeam*E*(1+TMath::Cos(theta));
  float x = calc_sig_method(E,theta,pt_had,sigma_h,E_ebeam,E_pbeam)[0];
  float y = Q2/(4*E_ebeam*E_pbeam*x);
  return {x, y, Q2};
}

This script combines features of the scripts shown in the previous two sections. The plots that are produced when you run

root -l OptimiseReconstruction.C\(\"your_file.root\"\)

show the (reco-true)/true distributions as before, with the reconstructed values coming from the manual calculations.

The output of the basic electron-finder implemented in EICrecon is found in the ScatteredElectronsEMinusPz branch, which is accessed as

auto& eleCollection = event.get<edm4eic::ReconstructedParticleCollection>("ScatteredElectronsEMinusPz");

This returns a list of all particles in ReconstructedParticles that have matched tracks and ECAL clusters that pass an E/p cut, ordered by momentum. In this code, we take the first element (largest momentum) of the list - be aware that this assumption causes a drop in efficiency larger values of inelasticity.

In this script we compare the quality of the reconstruction methods for two different scenarios: the first where the electron energy is found from the track momentum

E = eleCollection[0].getEnergy();

and the second where the electron energy comes from the energy of the associated ECAL cluster

E = eleCollection[0].getClusters()[0].getEnergy();

The benchmark plots for these two scenarios are overlaid on the same canvas. For the larger Q2 file, the two scenarios perform similarly (for March 2025 files considered here) but when looking at the low Q2 file, some differences become apparent. As one might expect, the Double Angle and JB methods are unaffected by this change, as they do not use the scattered electron energy in their calculation. However, the electron method, and to a lesser extent the Sigma methods see an improvement when using the energy value from the ECAL. The takeaway here is to check which approach gives you a better resolution for your analysis - at low Q2 it’s generally better to rely on the calorimeters, as tracking momentum resolutions are worse at shallow angles.

There are many ways in which the reconstruction of the hadronic final state, which has not been discussed much in this tutorial, could be improved. The reader is invited to look through the current HFS reconstruction code, which constructs the HFS as the sum of particles in the ReconstructedParticles branch, excluding the scattered electron. Note that this code uses boosts to correct for the crossing angle at ePIC - this is something to keep in mind if you are reconstructing the HFS manually, as it strongly impacts the HFS inputs to the reconstruction methods (Pt and E-pz sum). Regardless of the difficulties, it is highly recommended to try your own HFS reconstruction, as there are many improvements to be offered by e.g. particle flow algorithms, or kinematic fitting of exclusive final states, that will hopefully be the subject of a future tutorial.

Key Points

Some kinematics may favour a calorimeter based electron energy over a tracking based determination - check which is better for your analysis

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EIC Tutorial: Kinematic Reconstruction

lesson home

Optimise Reconstruction

Overview

Optimising the reconstruction

Key Points

previous episode

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