Hongyue Duyang University of South Carolina for the Nova

NOvA Pion Measurements

Hongyue Duyang University of South Carolina

For the NOvA Collaboration

1 The NOvA Near Detector NOνA Near Detector Construction NO�A NO�A Far Detector (Ash River, MN) MINOS Far Detector (Soudan, MN) A broad physics scope • • Detector construction and instrumentation0.3 kton, 4.2mX4.2mX15.8m, completed Aug. Using ��→�e , � ͞ �→� ͞ e … Determine the � mass hierarchy 2014 • 1 km from source, underground Determine at theFermilab. �23 octant Constrain �CP

• PVC cells filled with liquid scintillatorUsing ��→�� ,. � ͞ �→� ͞ � … • Neutrinos observed within seconds of turning on! Precision measurements of sin22� and m2 . • 23 32 Alternating planes of orthogonal (Exclude view. �23=�/4?) Over-constrain the atmos. sector Results (four oscillation channels) Also … Neutrino cross sections at the NO�A Near Detector Bin to bin correlation matrix: Sterile neutrinos Supernova neutrinos Fermilab Other exotica 4 cm ⨯ 6 cm Ryan Patterson, Caltech

• Low-Z, fine-grained (1 plane ~ 0.15X0), highly-active tracking calorimeter. Beam

Mass weight of detector component: C12 Cl35 H1 Near DetectorTi48 O16 Others 0.3 kton 11 Jonathan M. Paley 66.8% 16.4% 10.5% 206 3.3%layers 2.6% 0.4% 2 The measured inclusive cross section from Gargamelle, T2k, and NOvA as shown. There is also shown the predicted cross section for nue on carbon from GENIE. There is large correlation between the energy bins for NOvA results (see Top table). Our detector material is dominant by the carbon, chlorine, and hydrogen.

11/17 NuInt 2015 Xuebing Bu (Fermilab) 28 Pions in The NOvA ND

NOvA ND Data π0π =>0 => γ γ γ γ • Photons from neutral pion decay make EM showers. • Reconstructing both Beam Toppi0 view photons provide constraint on background and energy scale. Side view

3 Simulated Event + - ChargePions Current in The NOvA π ND/π

NOvA ND Data π0π =>0 Simulated=> γ γ γ γ Event• Photons from neutral pion + +decay- make EM showers. NOvA SimulationCharge Currentπ (0.4• ReconstructingGeV) both Beam Toppi0 view π /π photons provide constraintμ - (0.4 GeV) on background and energy scale. Simulated+ NOvA Event with EventID > 0.9 NOvA SimulationSide view π (0.4 GeV) - Top view NOvA Simulationμ (0.4 GeV) π+ (0.4 GeV)

μ- (0.4 GeV) • Charged pions are tracks in Top view Top view NOvA ND. • Developing charged pion tracking algorithm. Muon(1.7 GeV) + Pion(0.4 GeV) - π - μ Side view Side view 4 π+ Fig. Simulation of a NOvA event μ- 7 / 12 Side view

Signal: CC events with at least one charged pion (π+/π-) in the final state

17 Signal: CC events with at least one charged pion (π+/π-) in the final state

27 17

27 NuMI off-axis beam

NO�A detectors are sited 14 mrad off the NuMI NuMI Beam beam axis The Neutrino Flux

With the medium-energy NuMIνµ CC inclusive - Summary of Uncertainties tune, yields a narrow 2-GeV spectrum at the NO�A detectors NOvA Simulation 2.5 0.1

0.08 → Reduces NC and � CC ➔ Detectors2 are installed onby axis being e off beam axis backgrounds in the 14 mrad 0.06

➔ (NO�A) oscillation analyses 1.5 Narrow band beam peaked at 2 GeV 0.04 while maintaining ➔ Near maximum oscillation

1 high � flux at 2 GeV. ➔ � Reduced NC background Statistical Uncertainty 0.02 ➔ Reco Muon Kinetic Energy (GeV) Electron neutrino flux counts ~1% 0.5 0 of0.5 total 0.6flux. 0.7 0.8 0.9 1 Reco Cosθµ Ryan Patterson, Caltech • Statistical7 uncertainties areFermilab typically JETP, August< 2% 6, 2015 • Narrow band neutrino beam 1~3GeV peak at ~2GeV, Dominated by • Systematicsνμ (94%) are still being assessed, but we expect for the differential measurement ~10% highly correlated (normalization) flux uncertainties, and all others systematics • Hadron production uncertainty constraint by external hadron 11/17combined NuInt 2015 to be 5-8%. Xuebing Bu (Fermilab) 5 production data. (See Leo Aliaga’s talk on Monday) • σ(E) measurement systematics will be similar, although systematics from energy scale uncertainties will be larger on the rising5 and falling edges of the spectrum.

14 Jonathan M. Paley Charged Pion production in ⌫µ CC interactions

NOvA Pion Measurements Overview

• Pion production makes background to oscillation analysis. Charged pion production in ⌫µ CC interactions. • We want to measure them• in our own detector! ⌫ + N µ⌥+ N + ⇡± + X • Pion kinematics are sensitive to final-stateµ interaction! (elastic/inelastic scattering, absorption, charge-exchange).I asinglechargedpionproducedcouldmaketheeventmimictheCCQE • We are in 1~3 GeV region: cross-checktopology. with MINERvA, MiniBooNE, T2K. • Working on several pion analysis: • NC COH π0 : reporting preliminary result first time! • Work in progress: • CC π0 • NC π0 • CC π+/π-

Fig. Summary of the current knowledge of ⌫µ charged-current cross sections (Plot courtesy of G. Zeller) and Feynmann diagram for ⌫µ CC resonant single pion production, the dominant channel for pion production.

3 / 12 NOvA Pion Measurements Overview

• Pion production makes background to oscillation analysis. • We want to measure them in our own detector! • Pion kinematics are sensitive to final-state interaction (elastic/inelastic scattering, absorption, charge-exchange). • We are in 1~3 GeV region: cross-check with MINERvA, MiniBooNE, T2K. • Working on several pion analysis: • NC COH π0 : reporting preliminary result first time! • Work in progress: • CC π0 • NC π0 • CC π+/π-

7 NC Coherent π0

• Neutrinos scatter coherently off entire target nucleus with small momentum transfer. • Single forward-going pion, without other pions or nucleons.

• Identify the NC π0 sample • Absence of muon. • Two showers identified as photons by dE/dx-based likelihoods. • Reconstruct invariant mass. • Background dominated by RES and DIS π0s. • Cut on invariant mass further reduces background. • Also serve as a check of photon reconstruction and energy scale. 8 NC Coherent π0

• Divide the NC π0 into two sub-samples: • Signal sample: events with most of their energy in the 2 photon- showers and low vertex energy: it has >90% of the signal. • Control sample: the events with extra energy other than the photons or in the vertex region, dominated by non-coherent π0 s (RES and Control Sample Signal Sample DIS).

9 NC Coherent π0 RES in Control Sample DIS in Control Sample

• Fit the backgrounds to control sample data in π0 energy vs angle 2D space.

10 NC Coherent π0 RES in Control Sample DIS in Control Sample

• Fit the backgrounds to control sample data in π0 energy vs angle 2D space. RES in Signal Sample DIS in Signal Sample

• Apply the background tuning to the signal sample. 11 NC Coherent π0

• Background fit result are applied to the backgrounds in the signal sample. • Coherent signal measurement by subtracting normalized background from data in the coherent region of the energy and angle 2D space.

12 NC Coherent π0 Table 11: List of systematic and statistic uncertainties.

391 N = N N = 987.4 • Sig,raw Data,selected Bkg,norm • Coherent signal measurement by subtracting normalized background from data in 392 The ⌫ ﬂux () has been discussed in Sec. 2. The number of integrated neutrino ﬂux (0 120 energy and angle 2D space. µ ⇠ 393 GeV) we use is

• Measured flux-averaged cross-section: 2 10 394 = 123.2/cm /10 POT σ = 14.0 ± 0.9(stat.) ± 2.1(syst.)•×10⌫ -40cm2/nucleus 395 The eciency of coherent signal selection(✏) and the number of target nucleus in the ﬁducial • Total uncertainty 16.7%, systematic dominant. 396 volume (NTarget) will be discussed in the following subsections.

397 7.1 Eciency13 0 398 The eciency (✏) is defined as the ratio of the final selected ⌫µ coherent ⇡ signal events to 399 the total generated signal events in the fiducial volume. We use the SA ART files to count the 0 400 number of coherent ⇡ signal interactions at generated level. The numbers we get are

401 N = 857.7 • sig,selected

402 N = 20832.9 • sig,generated

403 corresponding to the data pot, which leads to the eciency

404 ✏ = N /N =0.041 • sig,selected sig,generated

405 7.2 Number of Target Nucleus

406 The targets for neutrino coherent interactions are nuclei rather than individual nucleons. The 407 NOvA ND is mainly composed of scintillator oil and PVC [30]. The ﬁducial mass is calculated 408 by scaling from the total detector volume (table 12). The mass of each element is calculated 409 using CAFAna script reading gdml ﬁles [27]. The total number of target nucleus is calculated 410 as

Mi NA NTarget = ⇤ (3) Wmolar,i Xi

37 NC Coherent π0 Table 11: List of systematic and statistic uncertainties.

Source (%) Calorimetric Energy Scale 3.4 Background Modeling 10.0 Control Sample Selection 2.9 EM Shower Modeling 1.1 Coherent Modeling 3.7 Rock Event 2.4 Alignment 2.0 Flux 9.4 Total Systematics 15.3 Measurements scaled to C12 by A2/3 Signal Sample Statistics 5.3 Control Sample Statistics 4.1 Total Uncertainty 16.7 For more details see my poster tonight! 391 N = N N = 987.4 • Sig,raw Data,selected Bkg,norm • Coherent signal measurement by subtracting normalized background from data in 392 The ⌫ ﬂux () has been discussed in Sec. 2. The number of integrated neutrino ﬂux (0 120 energy and angle 2D space. µ ⇠ 393 GeV) we use is

397 7.1 Eciency14 0 398 The eciency (✏) is defined as the ratio of the final selected ⌫µ coherent ⇡ signal events to 399 the total generated signal events in the fiducial volume. We use the SA ART files to count the 0 400 number of coherent ⇡ signal interactions at generated level. The numbers we get are

401 N = 857.7 • sig,selected

402 N = 20832.9 • sig,generated

403 corresponding to the data pot, which leads to the eciency

404 ✏ = N /N =0.041 • sig,selected sig,generated

405 7.2 Number of Target Nucleus

Mi NA NTarget = ⇤ (3) Wmolar,i Xi

37 NOvA Pion Measurements Overview

15 0 νμ-CC π CCPi0ID CCPi0ID Input VariablesSignal: νμ-CC events with at least one primary π0 in the final state. The analysis created its own PID Alluded to for last slides First, a EM ΔLL selector is evaluated for each prong in the event Principle PID for CCPi0 analysis Four input variables to this ΔLL CCPi0ID is the highest EM ΔLL Bragg Peak Identifier – ratio of value for all prongs in the event dE/dx at end of prong to bulk Energy per Hit Doesn’t include muon prong Gap to Vertex Shown here is the PID integrated Number of Missing Planes along prong over the full CCPi0 sample Background decomposed into red,CCPi0ID 0 CCPi0IDCCPi0ID InputInput Variables νμ-CC π TheThe analysis analysis created created its its ownown PIDPID yellow, and blue Alluded to for last slides 5 First,First, a a EM ΔLL Δ selectorLL selector is evaluated is evaluated for eachfor Red is a catch-all for anything• Use that non-muonis not ν CC, showerblue gives variables ν CC that to has form no a π0 Principle PID for CCPi0 analysis μ μ eachprong prong in the in event the event π0 in the event, and yellow givesidentifier: ν CC that has a secondary π0 FourFour input input variables variables to to thisthis ΔΔLLLL CCPi0ID is the highest EM ΔLL μ BraggBragg Peak Peak Identifier Identifier ––ratioratio ofof • Bragg peak identifier. value for all prongs in the event 7 dE/dx at end of prong to bulk dE/dx at end of prong to bulk • Energy per hit. EnergyEnergy per per Hit Hit Doesn’t include muon prong GapGap to to Vertex Vertex Shown here is the PID integrated• Photon gap from vertex. Number of Missing Planes along prong Number of Missing Planes along prong over the full CCPi0 sample • Number of missing planes. Background decomposed into• red,Fit signal and background MC to data in each

CCPi0ID Input Variablesyellow, and blue 6 kinematic bin. 5 16 The analysis created its own PID Red is a catch-all for anything that is not ν CC, blue gives ν CC that has no • μ μ 0 First, a ΔLL selector is evaluated for each 0 Use non-muon shower variables0 to form a π identifier: π in the event, and yellow gives νμ CC that has a secondary π prong in the event • Bragg peak identifier. Four input variables to this ΔLL 7 Bragg Peak Identifier – ratio of • Energy per hit. dE/dx at end of prong to bulk • Photon gap from vertex. Energy per Hit • Gap to Vertex Number of missing planes. 0 0 Number of Missing Planes along prong • Background from νμ-CC no π , νμ-cc secondary π and non-νμ-cc events. • Fit signal and background MC to data in each 6 kinematic bin. 15 0 π0 Kinematics νμ-CC π Wanted to squeeze one plot • Signal is dominantly RES (38.3%) showing signal pion kinematics and DIS (61.3%). into this talk Plot the reconstructed pion • Uncertainty (~15%) is systematic momentum, decomposed into dominant. GENIE truth label

Reco pπ = CalE of prong 38.3% of signal is RES 61.3% of signal is DIS 0.4% of signal is QE / MEC 0% is coherent, by symmetry • Plan to report flux-averaged differential cross section in final state muon and pion kinematics. 8 • Αt final stage of internal review. Preliminary result very soon!

17 NOvA Pion Analysis Overview

18 Neutral Current π0

• Signal: NC with at least one π0.

• Important background to νe appearance. • A event-level Boost Decision Tree (BDT) developed using shower 0 BDTG output (after evaluation)variables as inputs.Reconsructed ! mass using Fiducial + Containment• +Work ReMID in < progress.0.36 + Reco withK.E >Multivariate 0.5 GeV as Technique a pre selection. using pre selection + BDTG > 0.45 • Aiming to report differential cross-section in π0 kinematics.

3 NOvA Simulation ×106 NOvA Simulation ×10 0.015 Signal Signal 0.6 Total background Total background NC background POTs POTs 0.01 NC background 0 20 CC-π background 20 0 0 10 ⋅ 10 CC-π background CC-no background ⋅ 0.4 π CC-noπ0 background

0.005

Events/4.0 0.2 Events/4.0

0 0 −1 −0.5 0 0.5 1 0 0.05 0.1 0.15 0.2 0.25 0.3 BDTG Reconstructed π0 Mass [GeV] 19

11 13 NOvA Pion Measurements Overview

20 Charged Pion production in ⌫µ CC interactions

Fig. Summary of the current knowledge of ⌫µ charged-current cross sections (Plot courtesy of G. Zeller) and selecting a slice having at-least Feynmann diagram for ⌫ CC resonant single pion production, the dominant channel for pion production. µ • Work in progress: one muon and at-least one charged pion in the final state. • Efficiency of pre-selection and 3 / 12 Found the maximum figure of merit CVN selection. at 0.385 and defined the PionCVN • Systematics on CVN. > 0.385 for the signal region. • Sideband background-fitting study. • The first goal is to report differential cross-section in muon kinematics.

5 Charged Pion production in ⌫µ CC interactions

Charged pion production in ⌫ CC interactions. • µ ⌫µ + N µ⌥+ N + ⇡± + X ! + - I asinglechargedpionproducedcouldmaketheeventmimictheCCQECharge Current π /π topology. Signal: CC events with at least one charged pion (π+/π-) in the final state. • Working on improving π+/π- reconstruction. • Using deep-learning technique: A event-level Convolutional Visual NetworkCVN (CVN) for the Charged pion classifier. Maximum FoM at 0.385 • Uses full near detector raw data/MC image as an input. ! Fig. The event classifier, for Fig. Summary of the current knowledge of ⌫µ charged-current cross sections (Plot courtesy of G. Zeller) and selecting a slice having at-least Feynmann diagram for ⌫ CC resonant single pion production, the dominant channel for pion production. µ • Work in progress: one muon and at-least one charged pion in the final state. • Efficiency of pre-selectionby Jyoti and Tripathi For more details 3see / 12 the poster tonight Found the maximum figure of merit CVN selection. at 0.385 and defined the PionCVN • Systematics on CVN. > 0.385 for the signal region. • Sideband background-fitting study. • The first goal is to report differential cross-section in muon kinematics.

5 Summary

• NOvA is entering the game of neutrino pion production measurement. • Fine-grained liquid scintillator detector. • Narrow band neutrino flux at 1~3 GeV. • High statistics neutrino data. Taking anti-neutrino data too. • NC-Coherent π0: preliminary result reported for the first time • CC π0, NC π0,and CC π+/π- in progress. • Stay tuned!

23 Looking Forward and Summary Thank you!

25 Jonathan M. Paley 24 Back up slides

25 NuMI off-axis beam

NO�A detectors are sited NuMI Beam 14 mrad off the NuMI NuMI Beam beam axis The Neutrino Flux

With the medium-energy NuMI tune, yields a narrow 2-GeV spectrum at the NO�A detectors ➔ Detectors are installed by being off beam axis ➔ Detectors are installed onby axis being → Reduces NC and �e CC ➔ Narrow band beam peaked at 2 GeV backgrounds in the off beam14 mrad axis (NO➔ �A)Near maximum oscillation oscillation analyses ➔ Narrow band beam peaked at 2 GeV ➔ Reduced NC background while maintaining ➔ Near maximum oscillation ➔ Electron neutrino flux counts ~1% high �� flux at 2 GeV. ➔ Reducedof total NC flux.background ➔ Electron neutrino flux counts ~1% of total flux. 7 Ryan Patterson, Caltech • Narrow band neutrinoFermilab beam JETP, August 1~3GeV 6, 2015 peak at ~2GeV. 11/17 NuInt 2015 Xuebing Bu (Fermilab) 5 • Dominated by νμ (94%), with small contribution from νe (1%). • Hadron production uncertainty constraint by external hadron production 11/17 NuIntdata. 2015 (See Leo Aliaga’s talkXuebing on Monday) Bu (Fermilab) 5 • Also working on in situ flux measurement by neutrino-electron scattering. 26 NC Coherent π0

• Signal Sample: one pi0 decaying into two photons, both reconstructed in NOvA ND. No other nucleons or pions.

27 0 Eventνμ-CC πDisplay

µ- NOvA Simulation

proton Top view π0 => γ γ 0 νμ CC π Inclusive Analysis

Side view

An example RES event that passes our analysis selection. Both photons are Signal: CC events with at least one primary 0 in the final state reco’d along withνμ- the muon. The proton leavesπ one hit (just below vertex in x- 0 π productionview) vital and forisn’t νreco’de into a prong. Add to world-knowledge 28 3 appearance searches ANL (1981)

NOνA’s fine segmentation BNL (1986) affords EM identification K2K (2009)

νμ CC → lepton reconstruction MiniBooNE (2010) Flux-averaged differential cross MINERvA (2015) (anti-neutrino) sections in final state kinematics

10 Simulated Event Charge Current π+/π-

NOvA Simulation π+ (0.4 GeV) μ- (0.4 GeV)

Top view

π+ μ- Side view

Signal: CC events with at least one charged pion (π+/π-) in the final state

29 NOvA Simulation: ND Neutral Current π0

NOvA Simulation

π0 => γ γ

Top view

Side view

22 Signal: NC events with at least one π0 in the final state

30 NC Coherent π0

• Fit background to control sample data in π0 energy vs angle 2D space.

31 NC Coherent π0

• The control sample is used to fit background to data in π0 energy vs angle 2D space.

32 Neutral Current π0 BDTG Training Result-I using Fiducial + Containment + ReMID < 0.36+ Reco K.E > 0.5 GeV as a pre selection. • Signal: ν -NC with at least one π0. Start with 2-prong events. μ Input Variables • A event-level Boost Decision Tree (BDT) developed using shower variablesBelow listed as variables inputs. are selected as input variables for BDTG. TMVA overtraining check for classifier: BDTG Input variable: CVN_numu Input variable: Prong1 epi0LLL Input variable: Prong1 epiLLL

14 Signal 2.5

0.071 Background5

dx Signal (test sample) Signal (training sample) /

12 0.0527 0.0758

/ /

2 / 10 Background (test4 sample) Background (training sample) 1.5 (1/N) dN 8 3 Kolmogorov-Smirnov(1/N) dN test: signal (background) probability(1/N) dN = 0.182 (0.187) 6 1 4 2 4 1 0.5

2 (1/N) dN

0 U/O-flow (S,B): (0.0, 0.1)% / 0.0)% 0 U/O-flow (S,B): (0.0, 0.0)% / 0 U/O-flow (S,B): (0.0, 0.0)% / −1.5 −1 −0.5 0 0.5 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0 0.5 1 3 CVN_numu Prong1 epi0LLL Prong1 epiLLL Input variable: Prong1 Cont planes Input variable: Prong1 epLLT Input variable: Prong1 Width

4.5 0.06 0.25 2.94 4 0.107 /

0.05 0.774 cm 3.5 0.2 /

0.04 2 3

(1/N) dN 0.15 (1/N) dN 2.5 0.03 2 (1/N) dN 0.1 0.02 1.5 1 0.01 0.05 1 0.5

0 U/O-flow (S,B): (0.0, 0.0)% / 0 U/O-flow (S,B): (0.0, 0.0)% / 0 U/O-flow (S,B): (0.0, 0.0)% / 0 20 40 60 80 100 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 5 10 15 20 25 30 Prong1 Cont planes Prong1 epLLT Prong1 Width [cm]

0 U/O-flow (S,B): (0.0, 0.0)% / −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Separation is zero for identical signal and background and it is one for shapes with no overlap.BDTG response

9 NC Coherent π0

Measurements scaled to C12 by A2/3 Table 11: List of systematic and statistic uncertainties.

Source (%) Calorimetric Energy Scale 3.4 Background Modeling 10.0 Control Sample Selection 2.9 • Coherent signal measurement by subtracting EM Shower Modeling 1.1 normalized background from data in energy and Coherent Modeling 3.7 angle 2D space. Rock Event 2.4 Alignment 2.0 • Measured flux-averaged cross-section: Flux 9.4 -40 2 σ = 14.0 ± 0.9(stat.) ± 2.1(syst.)×10 cm /nucleus Total Systematics 15.3 • Total uncertainty 16.7%, systematic dominant. Signal Sample Statistics 5.3 Control Sample Statistics 4.1 Total Uncertainty 16.7 34

391 N = N N = 987.4 • Sig,raw Data,selected Bkg,norm

392 The ⌫ ﬂux () has been discussed in Sec. 2. The number of integrated neutrino ﬂux (0 120 µ ⇠ 393 GeV) we use is

2 10 394 = 123.2/cm /10 POT • ⌫

395 The eciency of coherent signal selection(✏) and the number of target nucleus in the ﬁducial 396 volume (NTarget) will be discussed in the following subsections.

397 7.1 Eciency 0 398 The eciency (✏) is defined as the ratio of the final selected ⌫µ coherent ⇡ signal events to 399 the total generated signal events in the fiducial volume. We use the SA ART files to count the 0 400 number of coherent ⇡ signal interactions at generated level. The numbers we get are

401 N = 857.7 • sig,selected

402 N = 20832.9 • sig,generated

403 corresponding to the data pot, which leads to the eciency

404 ✏ = N /N =0.041 • sig,selected sig,generated

405 7.2 Number of Target Nucleus

Mi NA NTarget = ⇤ (3) Wmolar,i Xi

37 52

553 The DUET result gives a modern and relatively precise measurement. The collaboration also published the covari- 554 ance matrix with their results[29] while the Asher result does not. Therefore, the simulation was tuned to this DUET 555 result. The cross section curve for ⇡± charge exchange determined by NO⌫A simulation was tested according to the 2 556 DUET data. Using the covariance matrix, a ﬁt is performed to determine what range of scale factors of the 557 simulation curve are compatible with the data[31]. This resulted in the symmetric interval, 1.061 0.146. ± 558 The conﬁdence interval constructed suggests the central value is notably higher than the NO⌫A simulation. As the 559 NO⌫A monte carlo set was generated before the DUET result was published, this analysis weights the nominal monte 560 carlo up by 6.1% to take advantage of the narrowest error band allowed. All numbers and plots in this document, 561 apply this 1.061 scale factor to the ⇡± CEx cross section. This was not mentioned earlier for sake of continuity of the 562 ideas presented. Charge Exchange

FIG. 69. A comparison of the Ashery[28] and DUET[29] measurements of the ⇡+ ⇡0 CEx cross section and the spread in cross section of the NO⌫A simulation consistent with these results. A 68.3% conﬁdence! interval for the range of scale factors 35 applied to the nominal simulation was calculated as 1.061 0.146 when comparing to the DUET data. ±

563 2. Weighting Monte Carlo Events to Scaled CEx Cross Section

564 Weighting monte carlo events to a new value of the CEx cross section is not as simple as weighting events that 0 565 have a ⇡± ⇡ interaction. This would change the number of background neutrino interactions with a ⇡± in the ! 566 ﬁnal state which is clearly wrong for this e↵ect. It is also not as simple as weighting background events with a CEx, 567 and adjusting the remaining background events with a ⇡± to impose unitarity. In the case when the probability for a 568 ⇡± to CEx becomes very close to 1, this system would lead to negative weights applied to the remaining background 569 events with a ⇡±. 570 A model was developed[31] which assumes the number of CEx interactions in an interaction follows a Poisson 571 distribution. Eqn. 7 shows the probability for an event with a ⇡± in the ﬁnal state to have 0 CEx interactions is 572 simply e.

N e P (,N)= = P (, 0) = e (7) N! )

573 To weight monte carlo events to a scaled CEx cross section, the expected number of CEx interactions, in the 574 Poisson distribution, must be scaled by the same scale factor in the CEx cross section. Writing the desired scale factor 575 as 1 + , the weight needed to be applied to background events with a ⇡± in the ﬁnal state but no CEx interaction is

(1+) P ((1 + ), 0) e ln P (,0) = = e = e (8) P (, 0) e

576 Then, enforcing unitarity on the number of background events with ⇡±, the scale for events with a CEx interaction 577 is