Investigation into the Constrained Kinematics of QE Events

Introduction

• The over constrained kinematics of QE events allow for the possibility of predicting the expected proton 4-momentum for a given event (assuming that it is QE).
• Variables can then be constructed to quantify how well the hits in a given event correspond to the hypotheses that that event is either QE or non-QE.
• These variables could then be used as PDFs for the liklihood-based QE event identification.
• The purpose of this page is to investigate the level of accuracy of my predicted proton 4-momentum and to test some potential PDF variables.

How well am I predicting the proton 4-momentum?

• The following series of plots show the reconstructed values of the proton and muon 4-momenta and compare them to the associated StdHep information. The muon 4-momenta components are derived assuming the muon mass and using the reconstructed muon energy and vertex direction cosines. The proton 4-momenta are derived using this knowledge about the muon 4-momenta, the proton mass and the reconstructed neutrino energies for those events. The plots show events from the stardard LE-10 MC reconstructed with release R1_18_2 subject to the following selection:
  • event with one track the passes standard reconstruction and fiducial criteria
  • if muon momentum is measured from curvature that the error on this is not too large (30%)
  • negative or zero track q/p
  • reconstructed track vertex at least 0.5m from the front face of the detector
  • no more than 20% of an event's PH or hits re-used in another event in the same snarl (runt event cut)

Figure 1. Resolution for Muon 4-Momentum Components

• The errors on the muon 4-momentum components must be taken into account when considering the following plot for the proton 4-momentum components and predicted track angle in the U and V planes:

Figure 2. Resolution for Proton 4-Momentum Components

Figure 3. Resolution for Proton Angle in U and V Planes

• If you were to only consider muons for which you knew all components to within 10% you would get the following plots for the proton 4-momentum components:

Figure 4. Resolution for Proton 4-Momentum Components

Figure 5. Resolution for Proton Angle in U and V Planes

• It can be seen from the above figures that although knowing the muon direction cosines and energy well (10%) there is still a reasonably large spread in the proton 4-momentum components. There seems to be a upper limit to the resolution that is limited by nuclear effects and re-scattering.

Using the Kinematic Prediction to Select QE Events

• The bottom 2 pads on figure 3 show that 1 sigma for the difference between the predicted and truth proton track angle in the U and V planes is about PI/6 radians. I have constructed sectors in the (u,z) and (v,z) planes with angles of plus or minus 0.5, 1.0, 1.5 and 2.0 times this sigma from the predicted proton direction in that plane in which to look for vertex hadronic shower hits.
• The idea is to tag the vertex hadronic shower hits and then to see what fraction of these hits and what fraction of their PH lies within a given sector. For true QE events I would expect that this fraction would be high in the smaller sectors compared to the other event types.
• The following plot shows the fraction of PH for true QE events and all background events with the QE and BG histograms indivually normalised to unity ('PDF style').

Figure 6. Fraction of PH in Sectors

• One thing to check is if this gets any better for events where the disagreement between calculated angles and truth angles (for the proton direction) are less that 0.25 radians in both the (u,z) and (v,z) planes:

Figure 7. Fraction of PH in Sectors

• You could consider the PH fraction within the sectors exclusively - that is do not include overlapping regions. In this case the above figure looks as follows:

Figure 8. Fraction of PH in Sectors

• Again if you only consider events with an angular difference of less than 0.25 radians you get the following:

Figure 9. Fraction of PH in Sectors

• There is no real separating power to be had from these variables. I think that a combination of a coarse detector, INTRANUKE and the Fermi momentum of the struck nucleon are degrading the resolution in the QE events' proton 4-momentum components such that they are indistinguishable from a large number of background events. In addition to this there is considerable overlap between the sectors in which you expect to see hits corresponding to the proton (assuming a QE event) and where you would expect to see hits from particles in background events.