Measurement of the Inclusive Neutrino Charged-Current Interaction Double Differential Cross Sections with the MINERνA Detector

Transcription

1 Measurement of the Inclusive Neutrino Charged-Current Interaction Double Differential Cross Sections with the MINERνA Detector A thesis submitted in partial fulfillment of the requirement for the degree of Bachelor of Science in Physics from the College of William and Mary in Virginia, by Yiyang Liu Advisor: Jeffrey K. Nelson Williamsburg, Virginia May 18

2 Contents List of Figures iv List of Tables v Abstract v 1 Introduction The Neutrinos and the Neutrino interactions The Cross Sections The MINERνA Detector 5.1 Nuclear Targets and Tracker Region Calorimetry MINOS Near Detector Cross Section Extraction 9.1 Overview Event Selection and Background Subtraction Unfolding Acceptance and Efficiency Correction Results i

3 4.1 Inclusive Double Differential Cross Sections Uncertainties Summary and Future Work A Bin by Bin Data and Simulations After Each Step of Analysis A.1 Background-Subtracted Data A. Unfolded Data A. Efficiency Corrected Data A.4 Cross Sections with Uncertainties ii

4 List of Figures 1.1 Neutrino Charged-Current Interaction Neutrino Neutral-Current Interaction The Front View of the MINERvA Detector Module The Elevation View of the MINERvA Detector Module An Example of a MINERvA Event Selected Events per.5 Gev versus Reconstructed Parallel Muon Momentum The Ratio of selected data events versus Monte Carlo events in each P // bin Selected Events per.5 Gev versus Reconstructed Transverse Muon Momentum The Ratio of Selected Data Events versus Simulated Events in Each P T Bin Background Subtracted Data per P T Bin versus P // The Migration Matrix Unfolded Data per P T Bin versus P // Unfolded Data P // and P T projection The Overall Efficiency Efficiency Corrected Data per P T bin versus P // iii

5 .11 Unfolded Data P // and P T Projection Double-Differential Cross Sections per P T Bin versus P // Double-Differential Cross Sections per P T Bin versus P // with Nominal and Modified GENIE Cross Section P T and P // Projection Plot Comparison with Global Data Bin by Bin Uncertainty Summary iv

6 List of Tables.1 Number of POT in Each Playlist Upper Edges of Momentum Bins A.1 Bin by Bin Background Subtracted Data A. Bin by Bin Unfolded Data A. Bin by Bin Efficiency Corrected Data A.4 Bin by Bin Cross Sections with Uncertainty v

7 Abstract Neutrinos in few-gev range have not been studied since the 7s. The existing cross-section measurements have large uncertainties due to low statistics and poor knowledge of the incoming neutrino flux. This thesis will present the measurement of the neutrino inclusive charge-current cross sections using the MINERvA detector in Fermilab. We obtain double-differential cross sections as functions of the transverse and paralleled muon momentum. The detector provides fine measurement of the cross sections and the NuMI beam line provides neutrinos flux with energy ranging from GeV to 5 GeV. We will present the major steps in data analysis, including event selection, unfolding, and detector efficiency and acceptance correction. We also compare our data with a modified version of GENIE neutrino interaction simulation. We find that the data and the simulation are in agreement, as they are within the uncertainty of each other. The total cross section also indicates that our result is in agreement with other measurements.

8 Chapter 1 Introduction 1.1 The Neutrinos and the Neutrino interactions Neutrinos, denoted as ν (and ν for antiparticles) are a category of electrically neutral leptons with half-integer spins and extremely small masses in the Standard Model of particle physics. Neutrinos have three lepton flavors, as they are paired up with the three charged leptons, the electrons, the muons and the taus. These elusive and small particles are important subjects of research because they can help us understand some fundamental questions, like the dominance of matter over antimatter in the Universe. Neutrinos are not directly observable because they do not participate in strong or electromagnetic interactions, and they interact with matter only via the weak nucleus force, which is mediated by the exchange of either a charged W boson or a neutral boson. Weak interactions involving the exchange of a W ± boson is called charged-current (CC) interactions because the W boson carries a unit of charge. The other type of weak interactions that involve the exchange of a boson is called the neutral-current (NC) interactions because the boson carries zero charge. Figure 1.1 and 1. shows the two types of interactions. In the charged current interaction, the neutrino transforms into the corresponding lepton and the quark 1

9 changes its charge and flavor. In the neutral current interaction, the charge and flavors of the interacting particles remain unchanged. This thesis focuses on the CC scattering of the muon neutrinos on hydrocarbon target. Therefore, the NC interactions will be treated as the background in the analysis (Chapter ). The charged-current interactions can be further categorized based on the states of interacting nucleon. CC scattering includes the quasi-elastic (QE) scattering, resonance production (RES) and deep inelastic scattering (DIS). [5] QE scattering happens when the momentum exchange is relatively small. A QE scattering produces one proton, a muon and zero pions. The interactions for a neutrino and an anti-neutrino are, respectively, written as: n + ν µ µ + p (1.1) n + ν µ µ + + p (1.) The resonance interaction typically involve larger momentum transfer than QE scattering. When it happens, the product is in resonance state. The resonance particle exists only over very short duration and decays. We will only find the result of decays in the detector, which typically includes one pion. The most typical resonance production is the Delta, which has the lowest mass. The Delta Baryon ( ) will decay into a pion and a nucleon. DIS occur when the momentum transfer is relatively large. It refers to the situation in which the target nucleon breaks up. In DIS, the neutrino interacts with quarks within the target and the ejected quarks get hadronized and form complex final states. The products often include multiple pions. We also add the two-particles-two-holes excitations (ph) in our analysis. ph is the interaction in which a neutrino interacts with two nucleons and both nucleons

10 Figure 1.1: Charged-current interaction of a muon neutrino with a down quark: A muon neutrino (zero charge) is converted to a muon (-1 charge) and a down quark (-1/ charge) to an up quark (/ charge) by the exchange of a W boson. Figure 1.: Neutral-current interaction of a muon neutrino and a down quark: The initial and final state particles are the same. The exchanged boson carries charge.

11 are ejected from the target nucleus. We may see two protons in the detector, but one is often of very low energy and may not be detectable. No pion is produced in this interaction, and events of this type most often look like QE events in the detector. The momentum transfer in ph scattering is between that of QE and RES scattering. 1. The Cross Sections The cross section is the probability of interaction between two particles. The cross section is expressed as: σ(p //, P T ) = N(P //, P T ) ΦT (1.) [5] where P // and P T are the parallel and traverse muon momentum, N is the rate of interaction, Φ is the particle flux, and T is the number of targets. In this study, we measure the double differential inclusive cross section of muon neutrino interactions as a function of the parallel and traverse muon momentum. We use hydrocarbon as the target. The neutrino flux is from NuMI beamline, which has neutrino energy from to 5 GeV and a peak at about GeV. Since we are measuring the inclusive cross sections, we will take into account all types of chargedcurrent scatterings mentioned in Section 1.1. The double differential cross section as a function of P T and P // can be understood as the likelihood of neutrino chargedcurrent interaction within a range of P T and P //. We attain the double differential cross section by dividing the crossection σ by the bin width of P T and P //, called y and y, respectively. That is dσ(p //, P T ) dxdy = N(P //, P T ) ΦT ( x)( y) (1.4) The code we use to calculate the cross sections is a modified version of Daniel Ruterbories code for MINERvA QE cross section analysis. [] We remove the QE selections in that code so that it can calculate the inclusive cross sections. 4

12 Chapter The MINERνA Detector The MINERνA detector is a dedicated cross section measuring device, designed to measure neutrino cross sections on different materials simultaneously. [1] It is able to track and identify final state particles and do calorimetric energy measurements. The device consists of a target region in front, followed by a scintillator tracker region and electromagnetic calorimeters (ECAL) and hadronic calorimeter (HCAL) surrounding the scintillator. The detector is built in hexagonal shape on the axis of the NuMI beam. [] Figures.1 and. shows two views of the MINERνA detector..1 Nuclear Targets and Tracker Region The target region contains various liquid and solid nuclear targets. The hydrocarbon inside the tracker region can also be used as target, and as it is this most massive target, we use for this measurement. The active-tracker region is located downstream of the target region. It is a solid scintillator tracker consists of stacked planes of triangular scintillator strips. The triangular configuration is used to let particles pass through more than one strip to enhance the precision of the position measurement. The strips are arranged in three different directions, the vertical direction, and ±6 degrees from the vertical direction. The scintillator emits light when the charged particles pass the material. 5

13 The wavelength-shifting (WLS) fiber inside the triangular strip will capture the light and transmit the signal to the photomultiplier tube (PMT). The region in which the detector has the best resolution is called the fiducial region. The fiducial region in the MINERvA detector from the front view is a hexagon with 85cm edge length.. Calorimetry The electromagnetic calorimeter and hadronic calorimeters are on the sides and downstream of the active-tracker region to contain electrical or hadronic particles and to allow calorimetry.. MINOS Near Detector The magnetized MINOS near detector is immediately downstream of the MINERνA detector. It measures the momentum and charge of the muons. Particles exiting MINERνA will enter the MINOS. The detector has two scintillator orientations that are perpendicular to each other. If the muon stops before exiting the MINOS detector, we measure the momentum by the range it travels in the detector. If the muon exits the detector, we measure the curvature of the particle in the magnetic field. Figure. shows an event in MINERνA detector. This is a candidate for quasielastic scattering. The colors in the figure indicates the amount of energy deposited in each scintillator strip. 6

14 Figure.1: The front view of the detector module. wrapped by the side ECAL and HCAL. The active-tracker region is 7

15 Figure.: An elevation view of the MINERνA detector. The target region, activetracker, ECAL, HCAL, and MINOS locate from upstream to downstream. Figure.: This plot shows a quasi-elastic event in the MINERνA detector. The colors of the little triangles indicate the energy at the point. 8

16 Chapter Cross Section Extraction.1 Overview In order to extract the final cross section, we need go through procedures including event selection, background subtraction, unfolding, and efficiency correction. Then, as introduced in Sec. 1., we divide the number of signal events by the widths of the corresponding transverse and paralleled momentum bins, the flux, and the number of scattering targets. Based on Equation 1.4, the cross section in our analysis is written as: [7] ( dσ dxdy ) ij = αβ U αβij(n data,αβ N bkgd data,αβ ) ɛ ij (ΦT )( x i )( y j ) (.1) The left hand side is the double differential cross section of the i th bin in paralleled momentum and j th bin in transverse momentum. On the right hand side, N data,αβ is the signal read by the detector, and N bkgd data,αβ is the background, so N data,αβ N bkgd data,αβ is the background-subtracted signal. Due to the limitation of detector resolution and inevitable detector bias, the events can be reconstructed to the incorrect bin. In order to reduce the effects of reconstruction errors, we go through a procedure called unfolding (Section.) by multiplying the background-subtracted data with a migration matrix denoted U αβij in Equation.1. ɛ ij on the right hand side is for acceptance and efficiency correction. Neither the 9

17 detector nor our selection algorithm is perfect, so a fraction of the signal will be lost during reconstruction and event selection. We need to divide the number of events in each bin by a efficiency so that the final cross section can represent every event in our sample, not just those we detect. The details will be introduced in Section.4. For this thesis, we analyze data from MINERvA playlist 1 and 1. These playlists cover about eighty percent of the data in the run period 1-1. The POT and simulated data for of each playlist is listed in Table.1. The upper edges of momentum bins when we do analysis are listed in Table.. Playlists Data POT MC POT minerva 1 9.6e e+ minerva 1.e+ 1.1e+1 Table.1: The number of Data POT and Monte Carlo POT we use are listed in this table. Bin Number P // GeV P T GeV Table.: This table shows the upper edges of each momentum bin. 1

18 . Event Selection and Background Subtraction We measure the inclusive cross section of muon neutrino interactions. That is, we will take into account all of the QE, ph, RES and DIS scattering. Therefore, our event selection will mainly be based on the limitations of Minos-Minerνa acceptance. The selected signals will have to pass the following selection requirements: 1. The event must have a reconstructed vertex within the fiducial volume of the detector tracker region.. The muon must enter the MINOS near detector so that there can be a momentum measurement for the muon. The reconstructed track in the MINOS detector must match the reconstructed track in the MINERvA detector. This limits the muon angles θ µ to be no greater than about degrees.. The muon momentum is greater than 1.5 GeV to ensure good detection efficiency in MINOS. 4. The muon should have enough curvature for us to determine its charge and the charge with 5 standard deviation significance. The charge-momentum ratio should be negative, so the particle is a muon instead of an antimuon which carries positive charge. 5. The detector electronics has a ns dead time after reading out data. To avoid reconstruction errors caused by the dead time in portion of the detector, a signal event cannot have more than one dead discriminator along the backward projection of the muon track to ensure the interaction was actually in the fiducial volume. [5] The backgrounds for charged-current inclusive event selection includes neutralcurrent events, anti-neutrino events, and ν e and ν τ events. The ν τ events are not 11

19 simulated by the Monte Carlo because we are below threshold for τ production. The NC and antineutrino events will not pass our charge significance cuts. So, our background subtraction is done with the muon selection. Our event selection results are presented in Figure Figure.1 and. shows the number of muon events per.5 GeV versus reconstructed paralleled and transverse muon momentum, respectively. Figure. and.4 shows the ratio of selected events in data versus that in the Monte Carlo with uncertainty. Figure.5 is the background subtracted data in each P T bin versus P T. Events /.5 GeV MINERνA Preliminary POT Normalized Data POT:.4E+ MC POT:.4E+ DATA ν µ CC QE ν µ CC RES ν µ CC DIS other Reconstructed Figure.1: The vertical axis of this plot shows the number of selected events per.5 GeV, and the horizontal axis is the reconstructed parallel muon momentum. The data points are selected events from MINERvA data, and the colored regions represent the Monte Carlo simulation. The green, yellow, and purple color each represent a type of interaction in the overall sample. The other category mainly consists of ph.. Unfolding As is mentioned in Section.1, the limits in detector resolution can cause the muon momentum to be reconstructed in adjacent bins of the true bin. We reduce this smearing effect through a Bayesian unfolding technique. This procedure utilizes 1

20 Data / MC Figure.: This plot shows the ratio of selected events in the data per muon P // bin versus the those in Monte Carlo. The pink color represents the uncertainty of the Monte Carlo. The data are all within the uncertainty of the simulation. In 5 GeV to GeV region, the model slightly over-predicts the number of signal events. the Monte Carlo simulation to estimate the most probable true muon momentum distribution from a reconstructed distribution. The estimation is done through generating a migration matrix from the simulation that shows how the events migrate to adjacent bins after reconstruction. The migration matrix is a histogram of true muon paralleled momentum (P // ) per P T (transverse momentum) bin versus the reconstructed P // per P T bin. If an event is in the x th P // bin and the y th P T bin, let us say it is in bin (x, y). The migration matrix is normalized to be unitary, so each of its entry is the conditional probability of an Monte Carlo event being reconstructed to bin (i, j), given that it is generated in bin (α, β). The migration matrix for our data analysis is in Figure.6. Denote the number of true P // per P T bin in bin (α, β) as T αβ. Similarly, write the number of reconstructed P // per P T bin in bin (i, j) as R ij. T αβ and R ij will have the following relation: R ij = α,β U ijαβ T αβ (.) 1

21 Events /.5 GeV MINERνA Preliminary POT Normalized Data POT:.4E+ MC POT:.4E+ DATA ν µ CC QE ν µ CC RES ν µ CC DIS other Reconstructed T Figure.: The vertical axis of this plot shows the number of selected events per.5 GeV, and the horizontal axis is the reconstructed transverse muon momentum. The data points are selected events from MINERvA data, and the colored regions represent the Monte Carlo simulation. The green, yellow, and purple color each represent a type of interaction in the overall sample. The other category mainly consists of ph. If U ijαβ is invertible, we can retrieve the true distribution from the reconstructed distribution by: T ij = α,β U 1 ijαβ R αβ (.) In the actual situation, matrix inversion can potentially bring large variances. The unsmearing matrix M ijαβ in our analysis is constructed iteratively through Bayesian unfolding, in which the Bayesian theorem is applied repetitively. In the n th iteration, the unsmearing matrix is calculated from the folded and unfolded distribution in the n 1 th iteration: where ɛ ij is the efficiency in bin (i, j). [] M ijαβ,n = P (R αβ T ij )T ij,n 1 ɛ ij R αβ,n 1 (.4) Figure.7 is the event distribution in each P T vs. the paralleled momentum after unfolding iterations. We have background subtracted data in red and unfolded data in blue. We can see that distribution becomes sharper than it is in the background 14

22 Data / MC T Figure.4: This plot shows the ratio of selected events in the data per muon P T bin versus the those in simulation. The pink color represents the uncertainty of the Monte Carlo. The data are all within the uncertainty of the simulation except low P T region the data show deficit. subtracted version of the same figures. The effect is most obvious in the low parallel momentum region. Figure.8 is the unfolded version of Figure The unfolding slightly changes the shape of distribution around its peak..4 Acceptance and Efficiency Correction The detector acceptance is defined as the fraction of signal seen by the detector. A event needs to enter the MINOS detector to be reconstructed and this accounts for the majority of lost acceptance. The selection efficiency is the fraction of simulated signal events selected with our algorithm. Certain signal events will not be selected if we apply too aggressive cuts in the selection. The efficiency term we use in efficiency and acceptance correction procedure is the product of detector acceptance and selection efficiency. So the efficiency in the (i, j) bin is expressed as ɛ ij = Selected Signal Interaction Rate True Signal Interaction Rate (.5) 15

23 <Pt<.75 GeV.75<Pt<.15 GeV.15<Pt<.5 GeV.5<Pt<.5 GeV <Pt<.4 GeV.4<Pt<.475 GeV.475<Pt<.55 GeV.55<Pt<.7 GeV <Pt<.85 GeV.85<Pt<1. GeV 1.<Pt<1.5 GeV 1.5<Pt<1.5 GeV 1.5<Pt<.5 GeV Figure.5: This figure show the background subtracted data in each P T bin versus the paralleled muon momentum P //. Each histogram shows the events in a P T bin. The bins and the numerical value of each data point are listed in Table A.1 16

24 bin True Muon P per Muon P T # Normalized Entries Reconstructed Muon P per Muon P T bin 1 Figure.6: The histogram in this figure is the migration matrix. The entries in the matrix are normalized to 1. We can see that although there are some off diagonal entries, the migration matrix is still diagonal dominant. 17

25 .<Pt<.75 GeV.75<Pt<.15 GeV.5<Pt<.4 GeV.4<Pt<.475 GeV.7<Pt<.85 GeV.15<Pt<.5 GeV.475<Pt<.55 GeV.85<Pt<1. GeV 1.<Pt<1.5 GeV.5<Pt<.5 GeV.55<Pt<.7 GeV 1.5<Pt<1.5 GeV 1.5<Pt<.5 GeV Figure.7: This figure show the data distribution in each PT bin versus the parallel muon momentum P// after unfolding. The data points in light red are the background subtracted data, and blue data points are the unfolded data. We use iterations of unfolding. The numbers of data and simulated events in each bin are listed in Table A.. 18

26 Events /.5 GeV 5 4 Unfolded Muon P Data MC Data/MC P Data/MC P Events /.7 GeV Unfolded Muon P T Data MC T Data/MC P T Data/MC P T T Figure.8: The two plots on the left are the 1D projections of the distribution. The plots on the right are the ratio of data vs. Monte Carlo in each P // or P T bin. 19

27 Events /.5 GeV MINERνA Preliminary Overall Efficiency True Muon P µ +FV+tdead MINOS match MINOS q/p < 1 Figure.9: This plot shows the overall efficiency-acceptance correction factor as a function of P //. This plot also shows how the efficiency changes as each set of cuts is applied to the data. The first set of selections (colored in brown) requires the signal events to be in the fiducial volume, to have at most one dead discriminator, and to have a MINOS reconstructed track that matches the track in MINERvA Detector. The second selection calculates the charge-momentum ratio to opt out anti-neutrinos events.

28 <Pt<.75 GeV.75<Pt<.15 GeV.15<Pt<.5 GeV.5<Pt<.5 GeV <Pt<.4 GeV.4<Pt<.475 GeV.475<Pt<.55 GeV.55<Pt<.7 GeV <Pt<.85 GeV.85<Pt<1. GeV 1.<Pt<1.5 GeV 1.5<Pt<1.5 GeV <Pt<.5 GeV Figure.1: This figure show the data distribution in each P T bin versus the parallel muon momentum P // after the efficiency correction. The efficiency correction rescales the number of events and changes the distribution in the low p // regions. Figure.9 shows our efficiency-acceptance correction factor. Fig..1 and.11 are the results of the efficiency corrections. We can make direct comparison between Fig and Fig The efficiency rescales the numbers of events to be roughly three times the unfolded ones. When P // is low, the efficiency correction also the shape of distribution significantly. 1

29 Events /.5 GeV Events /.7 GeV Eff. Cor. Muon P Data MC Eff. Cor. Muon P T Data MC T Data/MC P Data/MC P T Data/MC P Data/MC P T T Figure.11: This is the efficiency corrected version of figure.8. Also, the uncertainty in high momentum and large muon angle region became significantly larger after the efficiency correction. The simulation is within the uncertainty of the data except for the lowest P T bins.

30 Chapter 4 Results 4.1 Inclusive Double Differential Cross Sections We compute the final double differential cross sections by dividing the efficiency corrected sample by bin widths, flux and the number of nuclear targets. The bin by bin inclusive cross sections of muon neutrino and the data uncertainties are presented in Table A.4 and Fig Fig. 4. is a neater version of Fig. 4.1 without the last three transverse momentum bins. The figure also includes comparisons between the nominal GENIE and modified GENIE. The nominal GENIE fails to predict the neutrino charged-current interaction within the error bands in large transverse momentum region. The modified GENIE prediction is within all the error bands except in the first bin. The 1D projection plot of the distribution and the data-mc ratio are shown in Figure 4.. We integrate the differential cross sections of all bins, and the total inclusive cross section is: σ cc =. ± (cm /neutron) The TK experiment in Kamioka, Japan has measured the muon neutrino inclusive charged-current cross section in the energy range of 1- GeV with the TK INGRID detector and attained σ cc =.9 ± (cm /neucleon). [6] Our

31 <Pt<.75 GeV.75<Pt<.15 GeV.15<Pt<.5 GeV.5<Pt<.5 GeV <Pt<.4 GeV 9.4<Pt<.475 GeV 9.475<Pt<.55 GeV 9.55<Pt<.7 GeV <Pt<.85 GeV.85<Pt<1. GeV 1.<Pt<1.5 GeV 1.5<Pt<1.5 GeV <Pt<.5 GeV dσ Figure 4.1: This is the double differential cross sections (dp // )(dp T of neutrino inclusive ) charged-current interaction vs. the parallel muon momentum. The cross sections are plotted bin by bin (P T bin). 4

32 dσ Figure 4.: This is the double differential cross sections (dp // )(dp T of neutrino inclusive charged-current interaction vs. the parallel muon momentum. The cross sections ) are plotted bin by bin (P T bin). The last three transverse momentum bins are not included. The red line is the modified GENIE. The blue line is the nominal GENIE. The modified GENIE successfully predicts the neutrino charged-current interactions within uncertainty except in the P T (,.7) GeV bin. The nominal GENIE underpredicts the neutrino interactions in high transverse momentum bins. 5

33 /neutron) /GeV (cm dµ dσ /dµ P T P dσ /dµ dµ P P T Data MC Data/MC P Data/MC P /neutron) /GeV (cm dµ dσ /dµ P T P dσ /dµ dµ P P T Data MC T Data/MC P T P T Data/MC T Figure 4.: This is the measured cross section. The systematic uncertainty is extraordinary large in the bin with highest transverse momentum. MC is within the uncertainty of data in every bin. 6

34 Figure 4.4: This figure presents our measurement in a global data summary. The horizontal axis is the neutrino energy, and the vertical axis the total cross sections divided by the neutrino energy. neutrino flux have an average neutrino energy of E ν =.5 GeV, while the TK result is for neutrino flux at. GeV. The two results are comparable. Figure 4.4 compares the total cross section we get to global data from 1979 to 17. [8] In Figure 4.4, we can see that our measurement is in agreement with several other measurements. 4. Uncertainties Our uncertainties are divided into two major categories, systematic uncertainties and statistical uncertainties. Systematic uncertainties come from muon reconstruction, flux extraction, the GENIE neutrino cross section and final state interaction models, and some smaller other effects. The muon reconstruction uncertainty 7

35 contribute to lateral error bands, while the statistical uncertainty, and the flux uncertainty contribute to the vertical error bands. The error bands are computed and summed by MINERvA s PlotUtils software package. Systematic uncertainties in the simulation are calculated using the many universe method. [5] That is, shifting one parameter within their uncertainty, which creates an alternative universe, and comparing the resulting Monte Carlo with the central value. The uncertainty will be root mean square of the differences between the Monte Carlo in each universe with the Central Value. The statistical uncertainty is calculated by dividing the propagated error by the square root of the number of events in each bin. Figure 4.5 is the error summaries for the cross section results. From the fractional error summary plots, we can see that the total uncertainty is dominated by muon reconstruction uncertainty, especially in bins with larger transverse muon momentum. Flux uncertainties are also significant. The flux fractional uncertainty is almost constant in each bin at about.1, and the statistical uncertainty is at a few percents except in the lowest P T bins. The muon reconstruction uncertainty is exceptionally large in the bin with the highest transverse momentum because these events are at the edge of the range where the muon can enter MINOS detector. Many small systematic errors are likely to change the measurement in this region. The absolute systematic and statistical uncertainties of calculated data in each bin are listed in Table A.4 along with the cross sections. 4. Summary and Future Work We have obtained the double differential cross section for inclusive muon neutrino charged-current interactions as a function of muon parallel and transverse momentum with neutrino flux of GeV to 5 GeV. (The Result of bin summation.) We used 8

36 Fractional Uncertainty <Pt<.75 GeV.75<Pt<.15 GeV.15<Pt<.5 GeV.5<Pt<.5 GeV.5.5 Total Uncertainty Total Uncertainty Total Uncertainty Total Uncertainty Statistical Statistical.45 Statistical Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty FSI Models Flux Muon Reconstruction Others XSection Models.5<Pt<.4 GeV.4<Pt<.475 GeV.475<Pt<.55 GeV.55<Pt<.7 GeV Fractional Uncertainty.5 Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty.45 Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty.45 Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty <Pt<.85 GeV.85<Pt<1. GeV 1.<Pt<1.5 GeV 1.5<Pt<1.5 GeV Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models 1.5<Pt<.5 GeV Fractional Uncertainty.45 Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty Total Uncertainty.6 Statistical FSI Models Flux Muon Reconstruction Others XSection Models Fractional Uncertainty Total Uncertainty Statistical FSI Models Flux Muon Reconstruction Others XSection Models Figure 4.5: The figure show the fractional uncertainty as a funcion of P // in each P T bin. The figure includes information about the statistical uncertainties, the break down of systematic uncertainties and their contribution to total uncertainty 9

37 MINERvA playlist 1 and 1 for data analysis. The total POT number is.e+ for data and.e+1 for Monte Carlo Simulation. The data and simulated cross sections are within the uncertainty of each other. Our uncertainty is dominated by muon reconstruction and flux uncertainties. The muon reconstruction uncertainty gets exceptionally large in bins with larger muon transverse momentum. Our future work will include a Chi-Squared compatibility comparison between the simulation and the result. We will also want to run our code with the RPA (Random Phase Approximation) and resonance simulation model improvements. Acknowledgments I would like to thank Professor Nelson and Amy Filkins for introducing me to this study, guiding me throughout the process, and teaching me to use the tools and software. I also want to thank my parents and friends for their constant support.

38 Bibliography [1] Norrick, Anna (16). Detector Talk, Minerva 11 [PowerPoints slide]. Retrieved from [] L. Aliaga et al. [MINERvA Collaboration], Design, Calibration, and Performance of the MINERvA Detector, Nucl. Instrum. Meth. A 74, 1 (14) doi:1.116/j.nima [arxiv: [physics.ins-det]]. [] Ruterbories, Daniel (17). MINERvA (n)n-ccpi Results [PowerPoints slide]. Retrieved from layouts/15/wopiframe.aspx?sourcedoc=/ organization/theory/jetp/16/wc Ruterbories LEDoubleDiff MINERvA v.pdf [4] Garcia, Alfonso; Sanchez, Federico (17). inclusive CC cross-section measurement on C at TK [PowerPoints slide]. Retrieved from /8/material/slides/.pdf [5] Devan, Joshua. Measurement of Neutrino and Antineutrino Charged-Current Inclusive Cross Sections with the MINERvA Detector, Fermilab Thesis, 9 (15). 1

39 [6] Abe, K. et al. [TK Collaboration], Measurement of the muon neutrino inclusive charged-current cross section in the energy range of 1- GeV with the TK INGRID detector, Phys. Rev. D. 9, 7 (16) [7] E. Patrick, Cheryl. Measurement of the Antineutrino Double-Differential Charged-Current Quasi-Elastic Scattering Cross Section at MINERvA. 1.17/ (18). [8] C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 4, 11 (16) and 17 update.

40 Appendix A Bin by Bin Data and Simulations After Each Step of Analysis A.1 Background-Subtracted Data P // P T N Data (Events/GeV ) N MC (Events/GeV ) (1.5, ) (,.75) (,.5) (,.75) (.5, ) (,.75) (,.5) (,.75) (.5, 4) (,.75) (4, 4.5) (,.75) (4.5, 5) (,.75) (5, 6) (,.75) (6, 8) (,.75) (8, 1) (,.75) (1, 15) (,.75) (15, ) (,.75) (1.5, ) (.75,.15) (,.5) (.75,.15) (.5, ) (.75,.15) (,.5) (.75,.15) (.5, 4) (.75,.15) (4, 4.5) (.75,.15) (4.5, 5) (.75,.15) 5.79 (5, 6) (.75,.15) (6, 8) (.75,.15) (8, 1) (.75,.15) (1, 15) (.75,.15)

41 (15, ) (.75,.15) (1.5, ) (.15,.5) (,.5) (.15,.5) (.5, ) (.15,.5) (,.5) (.15,.5) (.5, 4) (.15,.5) (4, 4.5) (.15,.5) (4.5, 5) (.15,.5) (5, 6) (.15,.5) (6, 8) (.15,.5) (8, 1) (.15,.5) (1, 15) (.15,.5) (15, ) (.15,.5) (1.5, ) (.5,.5) (,.5) (.5,.5) (.5, ) (.5,.5) 66.7 (,.5) (.5,.5) (.5, 4) (.5,.5) (4, 4.5) (.5,.5) (4.5, 5) (.5,.5) (5, 6) (.5,.5) (6, 8) (.5,.5) (8, 1) (.5,.5) (1, 15) (.5,.5) (15, ) (.5,.5) (1.5, ) (.5,.4) (,.5) (.5,.4) (.5, ) (.5,.4) (,.5) (.5,.4) (.5, 4) (.5,.4) (4, 4.5) (.5,.4) (4.5, 5) (.5,.4) (5, 6) (.5,.4) (6, 8) (.5,.4) (8, 1) (.5,.4) (1, 15) (.5,.4) (15, ) (.5,.4) (1.5, ) (.4,.475) (,.5) (.4,.475) (.5, ) (.4,.475) (,.5) (.4,.475) (.5, 4) (.4,.475)

42 (4, 4.5) (.4,.475) (4.5, 5) (.4,.475) (5, 6) (.4,.475) (6, 8) (.4,.475) (8, 1) (.4,.475) (1, 15) (.4,.475) (15, ) (.4,.475) (1.5, ) (.475,.55) (,.5) (.475,.55) (.5, ) (.475,.55) (,.5) (.475,.55) (.5, 4) (.475,.55) (4, 4.5) (.475,.55) (4.5, 5) (.475,.55) (5, 6) (.475,.55) (6, 8) (.475,.55) (8, 1) (.475,.55) (1, 15) (.475,.55) (15, ) (.475,.55) (1.5, ) (.55,.7) 9 16 (,.5) (.55,.7) (.5, ) (.55,.7) (,.5) (.55,.7) (.5, 4) (.55,.7) (4, 4.5) (.55,.7) (4.5, 5) (.55,.7) (5, 6) (.55,.7) (6, 8) (.55,.7) (8, 1) (.55,.7) (1, 15) (.55,.7) (15, ) (.55,.7) (1.5, ) (.7,.85) (,.5) (.7,.85) (.5, ) (.7,.85) (,.5) (.7,.85) (.5, 4) (.7,.85) (4, 4.5) (.7,.85) (4.5, 5) (.7,.85) (5, 6) (.7,.85) (6, 8) (.7,.85) (8, 1) (.7,.85) (1, 15) (.7,.85)

43 (15, ) (.7,.85) (,.5) (.85, 1) (.5, ) (.85, 1) (,.5) (.85, 1) (.5, 4) (.85, 1) (4, 4.5) (.85, 1) (4.5, 5) (.85, 1) (5, 6) (.85, 1) (6, 8) (.85, 1) (8, 1) (.85, 1) (1, 15) (.85, 1) (15, ) (.85, 1) (.5, ) (1, 1.5) (,.5) (1, 1.5) (.5, 4) (1, 1.5) (4, 4.5) (1, 1.5) 6. (4.5, 5) (1, 1.5) (5, 6) (1, 1.5) (6, 8) (1, 1.5) (8, 1) (1, 1.5) (1, 15) (1, 1.5) (15, ) (1, 1.5) (,.5) (1.5, 1.5) (.5, 4) (1.5, 1.5) (4, 4.5) (1.5, 1.5) (4.5, 5) (1.5, 1.5) (5, 6) (1.5, 1.5) (6, 8) (1.5, 1.5) (8, 1) (1.5, 1.5) (1, 15) (1.5, 1.5) (15, ) (1.5, 1.5) (4, 4.5) (1.5,.5) (4.5, 5) (1.5,.5) (5, 6) (1.5,.5) (6, 8) (1.5,.5) (8, 1) (1.5,.5) (1, 15) (1.5,.5) (15, ) (1.5,.5) Table A.1: This table shows the data and simulation after event selections. 6

44 A. Unfolded Data P // P T N Data (Events/GeV ) N MC (Events/GeV ) (1.5, ) (,.75) (,.5) (,.75) (.5, ) (,.75) (,.5) (,.75) (.5, 4) (,.75) (4, 4.5) (,.75) (4.5, 5) (,.75) (5, 6) (,.75) (6, 8) (,.75) (8, 1) (,.75) (1, 15) (,.75) (15, ) (,.75) (1.5, ) (.75,.15) (,.5) (.75,.15) (.5, ) (.75,.15) (,.5) (.75,.15) (.5, 4) (.75,.15) (4, 4.5) (.75,.15) (4.5, 5) (.75,.15) (5, 6) (.75,.15) (6, 8) (.75,.15) (8, 1) (.75,.15) (1, 15) (.75,.15) (15, ) (.75,.15) (1.5, ) (.15,.5) (,.5) (.15,.5) (.5, ) (.15,.5) (,.5) (.15,.5) (.5, 4) (.15,.5) (4, 4.5) (.15,.5) (4.5, 5) (.15,.5) (5, 6) (.15,.5) (6, 8) (.15,.5) (8, 1) (.15,.5) (1, 15) (.15,.5) (15, ) (.15,.5) (1.5, ) (.5,.5) (,.5) (.5,.5) (.5, ) (.5,.5)

45 (,.5) (.5,.5) (.5, 4) (.5,.5) (4, 4.5) (.5,.5) (4.5, 5) (.5,.5) (5, 6) (.5,.5) (6, 8) (.5,.5) (8, 1) (.5,.5) (1, 15) (.5,.5) (15, ) (.5,.5) (1.5, ) (.5,.4) (,.5) (.5,.4) (.5, ) (.5,.4) (,.5) (.5,.4) (.5, 4) (.5,.4) (4, 4.5) (.5,.4) (4.5, 5) (.5,.4) (5, 6) (.5,.4) (6, 8) (.5,.4) (8, 1) (.5,.4) (1, 15) (.5,.4) (15, ) (.5,.4) (1.5, ) (.4,.475) (,.5) (.4,.475) (.5, ) (.4,.475) (,.5) (.4,.475) (.5, 4) (.4,.475) (4, 4.5) (.4,.475) (4.5, 5) (.4,.475) (5, 6) (.4,.475) (6, 8) (.4,.475) (8, 1) (.4,.475) (1, 15) (.4,.475) (15, ) (.4,.475) (1.5, ) (.475,.55) (,.5) (.475,.55) (.5, ) (.475,.55) (,.5) (.475,.55) (.5, 4) (.475,.55) (4, 4.5) (.475,.55) (4.5, 5) (.475,.55) (5, 6) (.475,.55) (6, 8) (.475,.55)

46 (8, 1) (.475,.55) (1, 15) (.475,.55) (15, ) (.475,.55) (1.5, ) (.55,.7) (,.5) (.55,.7) (.5, ) (.55,.7) (,.5) (.55,.7) (.5, 4) (.55,.7) (4, 4.5) (.55,.7) (4.5, 5) (.55,.7) (5, 6) (.55,.7) (6, 8) (.55,.7) (8, 1) (.55,.7) (1, 15) (.55,.7) (15, ) (.55,.7) (1.5, ) (.7,.85) (,.5) (.7,.85) (.5, ) (.7,.85) (,.5) (.7,.85) (.5, 4) (.7,.85) (4, 4.5) (.7,.85) (4.5, 5) (.7,.85) (5, 6) (.7,.85) (6, 8) (.7,.85) (8, 1) (.7,.85) (1, 15) (.7,.85) (15, ) (.7,.85) (,.5) (.85, 1) (.5, ) (.85, 1) (,.5) (.85, 1) (.5, 4) (.85, 1) (4, 4.5) (.85, 1) (4.5, 5) (.85, 1) (5, 6) (.85, 1) (6, 8) (.85, 1) (8, 1) (.85, 1) (1, 15) (.85, 1) (15, ) (.85, 1) (1.5, ) (1, 1.5) (.5, ) (1, 1.5) (,.5) (1, 1.5) (.5, 4) (1, 1.5)

47 (4, 4.5) (1, 1.5) (4.5, 5) (1, 1.5) (5, 6) (1, 1.5) (6, 8) (1, 1.5) (8, 1) (1, 1.5) (1, 15) (1, 1.5) (15, ) (1, 1.5) (.5, ) (1.5, 1.5) (,.5) (1.5, 1.5) (.5, 4) (1.5, 1.5) (4, 4.5) (1.5, 1.5) (4.5, 5) (1.5, 1.5) (5, 6) (1.5, 1.5) (6, 8) (1.5, 1.5) (8, 1) (1.5, 1.5) (1, 15) (1.5, 1.5) (15, ) (1.5, 1.5) (4, 4.5) (1.5,.5) (4.5, 5) (1.5,.5) (5, 6) (1.5,.5) (6, 8) (1.5,.5) (8, 1) (1.5,.5) (1, 15) (1.5,.5) (15, ) (1.5,.5) Table A.: This table shows the unfolded data and simulation in each muon momentum bin. A. Efficiency Corrected Data P // P T N Data (Events/GeV ) N MC (Events/GeV ) (1.5, ) (,.75) (,.5) (,.75) (.5, ) (,.75) (,.5) (,.75) (.5, 4) (,.75) (4, 4.5) (,.75) (4.5, 5) (,.75) (5, 6) (,.75) (6, 8) (,.75) (8, 1) (,.75)

48 (1, 15) (,.75) (15, ) (,.75) (1.5, ) (.75,.15) (,.5) (.75,.15) (.5, ) (.75,.15) (,.5) (.75,.15) (.5, 4) (.75,.15) (4, 4.5) (.75,.15) (4.5, 5) (.75,.15) (5, 6) (.75,.15) (6, 8) (.75,.15) (8, 1) (.75,.15) (1, 15) (.75,.15) (15, ) (.75,.15) (1.5, ) (.15,.5) (,.5) (.15,.5) (.5, ) (.15,.5) (,.5) (.15,.5) (.5, 4) (.15,.5) (4, 4.5) (.15,.5) (4.5, 5) (.15,.5) (5, 6) (.15,.5) (6, 8) (.15,.5) (8, 1) (.15,.5) (1, 15) (.15,.5) (15, ) (.15,.5) (1.5, ) (.5,.5) (,.5) (.5,.5) (.5, ) (.5,.5) (,.5) (.5,.5) (.5, 4) (.5,.5) (4, 4.5) (.5,.5) (4.5, 5) (.5,.5) (5, 6) (.5,.5) (6, 8) (.5,.5) (8, 1) (.5,.5) (1, 15) (.5,.5) (15, ) (.5,.5) (1.5, ) (.5,.4) (,.5) (.5,.4) (.5, ) (.5,.4) (,.5) (.5,.4)

49 (.5, 4) (.5,.4) (4, 4.5) (.5,.4) (4.5, 5) (.5,.4) (5, 6) (.5,.4) (6, 8) (.5,.4) (8, 1) (.5,.4) (1, 15) (.5,.4) (15, ) (.5,.4) (1.5, ) (.4,.475) (,.5) (.4,.475) (.5, ) (.4,.475) (,.5) (.4,.475) (.5, 4) (.4,.475) (4, 4.5) (.4,.475) (4.5, 5) (.4,.475) (5, 6) (.4,.475) (6, 8) (.4,.475) (8, 1) (.4,.475) (1, 15) (.4,.475) (15, ) (.4,.475) (1.5, ) (.475,.55) (,.5) (.475,.55) (.5, ) (.475,.55) (,.5) (.475,.55) (.5, 4) (.475,.55) (4, 4.5) (.475,.55) (4.5, 5) (.475,.55) (5, 6) (.475,.55) (6, 8) (.475,.55) (8, 1) (.475,.55). 6. (1, 15) (.475,.55) (15, ) (.475,.55) (1.5, ) (.55,.7) (,.5) (.55,.7) (.5, ) (.55,.7) (,.5) (.55,.7) (.5, 4) (.55,.7) (4, 4.5) (.55,.7) (4.5, 5) (.55,.7) (5, 6) (.55,.7) (6, 8) (.55,.7) (8, 1) (.55,.7)

50 (1, 15) (.55,.7) (15, ) (.55,.7) (1.5, ) (.7,.85) (,.5) (.7,.85) (.5, ) (.7,.85) (,.5) (.7,.85) (.5, 4) (.7,.85) (4, 4.5) (.7,.85) (4.5, 5) (.7,.85) (5, 6) (.7,.85) (6, 8) (.7,.85) (8, 1) (.7,.85) (1, 15) (.7,.85) (15, ) (.7,.85) (,.5) (.85, 1) (.5, ) (.85, 1) (,.5) (.85, 1) (.5, 4) (.85, 1) (4, 4.5) (.85, 1) (4.5, 5) (.85, 1) (5, 6) (.85, 1) (6, 8) (.85, 1) (8, 1) (.85, 1) (1, 15) (.85, 1) (15, ) (.85, 1) (.5, ) (1, 1.5) (,.5) (1, 1.5) (.5, 4) (1, 1.5) (4, 4.5) (1, 1.5) (4.5, 5) (1, 1.5) (5, 6) (1, 1.5) (6, 8) (1, 1.5) (8, 1) (1, 1.5) (1, 15) (1, 1.5) (15, ) (1, 1.5) (,.5) (1.5, 1.5) (.5, 4) (1.5, 1.5) (4, 4.5) (1.5, 1.5) (4.5, 5) (1.5, 1.5) (5, 6) (1.5, 1.5) (6, 8) (1.5, 1.5) (8, 1) (1.5, 1.5)

51 (1, 15) (1.5, 1.5) (15, ) (1.5, 1.5) (4, 4.5) (1.5,.5) (4.5, 5) (1.5,.5) (5, 6) (1.5,.5) (6, 8) (1.5,.5) (8, 1) (1.5,.5) (1, 15) (1.5,.5) (15, ) (1.5,.5) Table A.: This table shows the efficiency corrected data and simulation in each bin. A.4 Cross Sections with Uncertainties P // P T ( dσ data dxdy sys ± ɛ stat ) 1 4 dσ MC dxdy 14 (cm /GeV /neutron) (cm /GeV /neut.) (1.5, ) (,.75).6956±.77717± (,.5) (,.75).1889±.1476± (.5, ) (,.75).1745±.4947± (,.5) (,.75).175±.1811± (.5, 4) (,.75).95899±.9985± (4, 4.5) (,.75).44148±.556± (4.5, 5) (,.75).8175±.47918± (5, 6) (,.75).5889±.5911± (6, 8) (,.75).14868±.15671± (8, 1) (,.75) ±.1949± (1, 15) (,.75).94787± ± (15, ) (,.75).4987± ± (1.5, ) (.75,.15).749±.5181± (,.5) (.75,.15).6789±.984± (.5, ) (.75,.15).6795±.74618± (,.5) (.75,.15) ±.58186± (.5, 4) (.75,.15).56489±.4876± (4, 4.5) (.75,.15).1411±.6± (4.5, 5) (.75,.15).188±.14877± (5, 6) (.75,.15).4581±.558± (6, 8) (.75,.15).18565±.4771± (8, 1) (.75,.15).1±.58861± (1, 15) (.75,.15).6774±.781± (15, ) (.75,.15).1761± ±

52 (1.5, ) (.15,.5).4418±.79765± (,.5) (.15,.5).4581±.5195± (.5, ) (.15,.5) ±.51± (,.5) (.15,.5) ±.1476± (.5, 4) (.15,.5).97476±.9174± (4, 4.5) (.15,.5).6585±.615± (4.5, 5) (.15,.5).41971±.94± (5, 6) (.15,.5).68554±.66165± (6, 8) (.15,.5).89471± ± (8, 1) (.15,.5).55999±.64748± (1, 15) (.15,.5).75595±.16568± (15, ) (.15,.5).67685±.46± (1.5, ) (.5,.5).91614±.979± (,.5) (.5,.5).5175±.4159± (.5, ) (.5,.5).6±.6414± (,.5) (.5,.5) ±.1864± (.5, 4) (.5,.5) 1.519±.1178± (4, 4.5) (.5,.5).7566±.76545± (4.5, 5) (.5,.5).48846±.4485± (5, 6) (.5,.5).785±.7478± (6, 8) (.5,.5) 1.579± ± (8, 1) (.5,.5).71198±.196± (1, 15) (.5,.5).9168±.1875± (15, ) (.5,.5).4971±.4714± (1.5, ) (.5,.4).8589±1.48± (,.5) (.5,.4).44±.4459± (.5, ) (.5,.4).9647±.54918± (,.5) (.5,.4).497±.886± (.5, 4) (.5,.4) ±.15994± (4, 4.5) (.5,.4).986±.9769± (4.5, 5) (.5,.4).6878±.64589± (5, 6) (.5,.4) 1.57±.9978± (6, 8) (.5,.4) ±.1478± (8, 1) (.5,.4).89517±.15554± (1, 15) (.5,.4) 1.66± ± (15, ) (.5,.4).59159±.8591± (1.5, ) (.4,.475) 4.794±1.469± (,.5) (.4,.475) ±.56987± (.5, ) (.4,.475).5675±.7818± (,.5) (.4,.475).7651±.67681± (.5, 4) (.4,.475) ±.17849± (4, 4.5) (.4,.475) ± ±