Q2-x bin migration

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This page presents an analysis summary for studies into Q2-x bin migration due to track momentum resolution.


The eic-smear package was used to smear the momentum of tracks in PYTHIA events. Event-wise kinematic quantities y (inelasticity), x and Q2 were then recalculated using the smeared track momenta. Events were binned in (Q2, x) with five bins per decade in x and four per decade in Q2. For each event it was determined whether the (Q2, x) bin containing the event was the same or different following smearing.

The following methods are available to calculate the event kinematics:

  • Electron method: kinematics are calculated from the energy and polar angle of the scattered beam electron
  • Jacquet-Blondel method: kinematics are calculated from the hadronic final-state
  • Double-angle method: kinematics are calculated from a combination of scattered electron and hadronic final-state information.

The methods and associated formulae are described in F.D. Aaron et al., JHEP01 (2010) 109. Wherever the energy of the particle was required it was calculated from momentum and mass using E2 = p2 + m2, as the momentum resolution of the detector is expected to be better than the energy resolution except at high momentum. Smearing of the particle type was not performed (i.e. perfect particle identification was assumed). Angular smearing was not performed, as this can be expected to be correlated with the momentum resolution. Momentum resolution was varied as a function of pseudorapidity (η), with the best resolution at midrapidity and the worst near the beam. The following two sets of momentum resolutions were used to study the sensitivity of bin migration to momentum resolution:

η range "Good" set "Bad" set
-1 to +1 ("central") 0.2%P 0.6%P
-3 to -1 ("backward") 1%P 3%P
1 to 3 ("forward")
-5 to -3 ("far backward") 3%P 10%P
3 to 5 ("far forward")

Simple parameterisations were chosen over the generic tracking functions for simplicity and reproducibility. Studies by Will Foreman suggest that the resolutions in the far-forward/backward direction using the tracking formulae give exceedingly poor resolution. Therefore consider these results indicators of what level of migration we can expect if we achieve a given level of momentum resolution with the final detector, rather than what migration we get with a particular detector design.

Comparison of methods

The following figures are for 5 x 100 GeV events with Q2 > 0.1 GeV2.

(a) Calculation of y, x and Q2 using the electron method
(b) Calculation of y, x and Q2 using the Jacquet-Blondel (JB) method
(c) Calculation of y, x and Q2 using the double-angle (DA) method

The electron method gives fair resolution in all quantities. The JB method gives good resolution in y (better than the electron method) but poor resolution in x and Q2. The DA method gives good resolution in Q2 but is worse for y and x. If we use a mixed method, using y from the JB method, Q2 from the DA method and calculate x using Q2 = sxy, we obtain better results.

(a) Calculation of y, x and Q2 using the mixed JB/DA method

Looking at bin migration, we see the following.

(a) Electron method
(b) JB method
(c) DA method
(d) Mixed JB/DA method

The electron method works well at large y but fails as y decreases. The DA method works better than the JB method, with both being more uniform that the electron method and giving better results at large Q2. The mixed method is also more uniform, but gives better performance overall than either the JB or DA methods individually. With the mixed method the bin migration is typically less than 25% for the chosen binning and resolutions (~few %P in the forward direction).

To get an idea of the sensitivity to momentum resolution, compare results for the electron and mixed methods with the "bad" resolution set.

(a) Electron method
(b) Mixed method

With reduced momentum resolution, particularly in the forward and backward directions, the electron method is significantly reduced in effectiveness. The performance of the mixed method is also degraded, but remains more robust, with ~50% bin migration over most of the Q2-x plane for the chosen binning.

Charged-current events with DJANGOH

Charged-current events lack a detectable final-state lepton. Therefore the only available method of kinematic reconstruction is the Jacquet-Blondel method i.e. using the hadronic final state alone. The feasibility of measuring such events with an EIC detector was investigated using smeared events from the DJANGOH event generator. Events were generated for Q2 greater than 1,000 GeV2. Smaller Q2 were not studied due to the rapid drop-off of cross section with low Q2. Radiative corrections were also applied. Two electron beam energies were studied: 10 and 20 GeV. A 250 GeV hadron beam energy was used in each case to maximise the cross section (which increases with centre-of-mass energy). 5 x 250 GeV collisions were not studied due to the limited kinematic span available at Q2 > 1,000. The following detector performance was used when smearing events with eic-smear:

Quantity Parameterisation η range
σ(P) 0.0069 + 0.0014*P + 0.00041*P2 0.00 < abs(η) < 0.25
σ(P) 0.0049 + 0.0021*P + 0.00030*P2 0.25 < abs(η) < 0.75
σ(P) -0.00013 + 0.0040*P + 0.00014*P2 0.75 < abs(η) < 1.25
σ(P) 0.0038 + 0.0068*P + 0.00016*P2 1.25 < abs(η) < 1.75
σ(P) 0.0022 + 0.0061*P + 0.0002*P2 1.75 < abs(η) < 2.25
σ(P) 0.0082 + 0.0079*P + 0.00019*P2 2.25 < abs(η) < 2.75
σ(P) 0.0200 + 0.0120*P + 0.00061*P2 2.75 < abs(η) < 3.25
σ(E) [electromagnetic] 0.122 * √E -1 < η < 4.5
σ(E) [electromagnetic] 0.0178 * √E + 0.0069 * E -4.5 < η < -1
σ(E) [hadronic] 0.38 * √E 1 < η < 4.5

The momentum performance parameterisations are based upon fits to resolutions obtained from the EicRoot Geant simulation of the EIC tracking detectors. Calorimeter resolutions are based on current target resolutions for those detectors. Imperfect particle identification and angular smearing were assumed to give small effects compared to acceptance and resolution in p and E, and were therefore not considered.

The figures below show bin-by-bin purity for the two energy configurations. Purity in a bin is defined as (Ngenerated - Nsmeared out) / (Ngenerated - Nsmeared out + Nsmeared in) i.e. the number of events remaining in the correct bin divided by the total number of events in that bin (including those migrating in) after smearing.

(a) Purity for 20 x 250 collisions
(b) Purity for 10 x 250 collisions

For both energies the typical purity is in the range 75-95%, which should be sufficient to allow unfolding. This is due to the good resolution in both x and Q2 achieved with projected EIC detector performance, as shown by the figures for 20 x 250 GeV, below. Resolutions for 10 x 250 are comparable. The binning used above follows the "standard" EIC binning (4 bins per decade in Q2, 5 per decade in x). Binning could be optimised to give better purity if necessary.

(a) Resolution in y
(b) Resolution in x
(c) Resolution in Q2

Resolution in y is very good, as is typical for the JB method. While resolution in Q2 (and hence x) is poor at low Q2, at high Q2 the method gives good results. The reason for this can be seen by the figure below, where particle counts as a function of pseudorapidity are weighted by particle pT.


Q2 in the JB method is calculated as a sum over particle pT. At high Q2 the hadronic system recoils strongly, so particles are generally produced within the main detector acceptance. The distribution for 10 x 250 collisions is similar, albeit with particle production shifted to larger (forward) η on average. Solid lines indicate nominal detector acceptance (-4.5 < η < 4.5), while the dashed line shows the lower limit of hadronic calorimetry (1 < η < 4.5).

Further steps

  • Effect of imperfect particle identification (causes miscalculation of energy for hadronic methods).
  • Introduce angular resolution also?


Analysis code is located at /eic/u/tpb/analyses/migration. Look in the README for instructions on running the code.