First measurement of the muon neutrino charged current quasielastic double differentialcross section

A.A. Aguilar-Arevalo,13 C. E. Anderson,18 A. O. Bazarko,15 S. J. Brice,7 B. C. Brown,7 L. Bugel,5 J. Cao,14 L. Coney,5

J.M. Conrad,12 D. C. Cox,9 A. Curioni,18 Z. Djurcic,5 D. A. Finley,7 B. T. Fleming,18 R. Ford,7 F. G. Garcia,7

G. T. Garvey,10 J. Grange,8 C. Green,7,10 J. A. Green,9,10 T. L. Hart,4 E. Hawker,3,10 R. Imlay,11 R.A. Johnson,3

G. Karagiorgi,12 P. Kasper,7 T. Katori,9,12 T. Kobilarcik,7 I. Kourbanis,7 S. Koutsoliotas,2 E.M. Laird,15 S. K. Linden,18

J.M. Link,17 Y. Liu,14 Y. Liu,1 W.C. Louis,10 K. B.M. Mahn,5 W. Marsh,7 C. Mauger,10 V. T. McGary,12 G. McGregor,10

W. Metcalf,11 P. D. Meyers,15 F. Mills,7 G. B. Mills,10 J. Monroe,5 C. D. Moore,7 J. Mousseau,8 R. H. Nelson,4

P. Nienaber,16 J. A. Nowak,11 B. Osmanov,8 S. Ouedraogo,11 R. B. Patterson,15 Z. Pavlovic,10 D. Perevalov,1 C. C. Polly,7

E. Prebys,7 J. L. Raaf,3 H. Ray,8,10 B. P. Roe,14 A.D. Russell,7 V. Sandberg,10 R. Schirato,10 D. Schmitz,5 M.H. Shaevitz,5

F. C. Shoemaker,15,* D. Smith,6 M. Soderberg,18 M. Sorel,5,† P. Spentzouris,7 J. Spitz,18 I. Stancu,1 R. J. Stefanski,7

M. Sung,11 H.A. Tanaka,15 R. Tayloe,9 M. Tzanov,4 R. G. Van de Water,10 M.O. Wascko,11,‡ D.H. White,10

M. J. Wilking,4 H. J. Yang,14 G. P. Zeller,7 and E.D. Zimmerman4

(MiniBooNE Collaboration)

1University of Alabama, Tuscaloosa, Alabama 35487, USA2Bucknell University, Lewisburg, Pennsylvania 17837, USA3University of Cincinnati, Cincinnati, Ohio 45221, USA4University of Colorado, Boulder, Colorado 80309, USA5Columbia University, New York, New York 10027, USA

6Embry Riddle Aeronautical University, Prescott, Arizona 86301, USA7Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA

8University of Florida, Gainesville, Florida 32611, USA9Indiana University, Bloomington, Indiana 47405, USA

10Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA11Louisiana State University, Baton Rouge, Louisiana 70803, USA

12Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA13Instituto de Ciencias Nucleares, Universidad National Autónoma de México, D.F. 04510, México

14University of Michigan, Ann Arbor, Michigan 48109, USA15Princeton University, Princeton, New Jersey 08544, USA

16Saint Mary’s University of Minnesota, Winona, Minnesota 55987, USA17Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA

18Yale University, New Haven, Connecticut 06520, USA(Received 12 February 2010; published 28 May 2010)

A high-statistics sample of charged-current muon neutrino scattering events collected with the

MiniBooNE experiment is analyzed to extract the first measurement of the double differential cross

section ( d2�

dT�d cos��) for charged-current quasielastic (CCQE) scattering on carbon. This result features

minimal model dependence and provides the most complete information on this process to date. With the

assumption of CCQE scattering, the absolute cross section as a function of neutrino energy (�½E��) andthe single differential cross section ( d�

dQ2) are extracted to facilitate comparison with previous measure-

ments. These quantities may be used to characterize an effective axial-vector form factor of the nucleon

and to improve the modeling of low-energy neutrino interactions on nuclear targets. The results are

relevant for experiments searching for neutrino oscillations.

DOI: 10.1103/PhysRevD.81.092005 PACS numbers: 25.30.Pt, 13.15.+g, 14.60.Lm, 14.60.Pq

I. INTRODUCTION

Neutrino charged-current (CC) scattering without pionsin the final state is important to measure and characterize,and is a critical component in the neutrino oscillationprogram of the MiniBooNE experiment [1–4] atFermilab. Most of these events are charged-current quasi-elastic scattering (CCQE) of the muon neutrino on a bound

*Deceased.†Present address: IFIC, Universidad de Valencia and CSIC,

Valencia 46071, Spain.‡Present address: Imperial College; London SW7 2AZ, United

Kingdom.

PHYSICAL REVIEW D 81, 092005 (2010)

1550-7998=2010=81(9)=092005(22) 092005-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.092005

nucleon (�� þ n ! �� þ p). A robust model of theseinteractions is required to support future experimentssuch as NOvA [5] and T2K [6] that are also searchingfor �� ! �e oscillations. Such experiments will use �e CCinteractions to detect the appearance of any �e resultingfrom oscillations in the large distance between productionand detection. Additional use will be made of �� CC

interactions to normalize the neutrino content at produc-tion using a near detector and to search for the disappear-ance of �� via a far detector. These analyses will require all

available experimental and theoretical insight on theCCQE interaction in the � 1 GeV energy range and onnuclear (carbon, oxygen) targets. While many unknownquantities are eliminated in these experiments by consid-ering ratios of far to near events, the cancellation is notcomplete due to differences in neutrino flux and back-grounds in the near and far detectors. Thus, in order topermit precision oscillation measurements, it is importantto have an accurate characterization of the CCQE differ-ential cross sections over a wide span of neutrino energies.

Historically, it has proven difficult to accurately definethe CCQE cross section and precise measurements havebeen unavailable. The experimental execution and datainterpretation are nontrivial for several reasons. Neutrinobeams typically span a wide energy range thereby prevent-ing an incoming energy constraint on the reaction. Theneutrino flux itself is often poorly known, hampering nor-malization of reaction rates. Background processes arefrequently significant and difficult or impossible to sepa-rate from the CCQE signal, for instance, CC pion produc-tion combined with pion absorption in the nucleus. Furthercomplicating the description, the target nucleon is not freebut bound in a nuclear target and correlations betweennucleons may be important. There are differing detectionstrategies employed by different experiments, forexample, some require detection of the final-state nucleonand some do not. Finally, the nuclear target often differsbetween experiments, thus making comparisons lessstraightforward.

The current data on CCQE scattering come from avariety of experiments operating at differing energies andwith different nuclei [7]. Modeling of this data has beenconsistent from experiment to experiment, yet remainsfairly unsophisticated. Preferred for its simplicity, neutrinoCCQE models typically employ a relativistic Fermi gas(RFG) model (such as that of Ref. [8]) that combines thebare nucleon physics with a model to account for thenucleon binding within the specific nucleus. The structureof the nucleon is parametrized with the three dominantform factors: two vector, F1;2ðQ2Þ, and one axial-vector, FAðQ2Þ. The vector form factors, including the Q2(squared four-momentum transfer) dependence, are well-determined from electron scattering. The axial-vector formfactor at Q2 ¼ 0 is known from neutron beta-decay.Neutrino-based CCQE measurements may then be inter-

preted as a measurement of the axial-vector mass, MA,which controls the Q2 dependence of FA, and ultimately,the normalization of the predicted cross section.This simple, underlying model has led to the situation

where neutrino CCQE measurements typically produce ameasurement of MA independent of neutrino energy andtarget nucleus. The resulting world-average is MA ¼1:03� 0:02 GeV [9] (a recent summary of the variousMA values is provided in Ref. [10]). It should be notedthat the data contributing to this world average are domi-nated by higher precision bubble chamber experimentsusing deuterium as a target. In addition, most (but notall) of the MA values have come from the observed distri-bution of CCQE events in Q2 rather than from an overallnormalization of the event yield.Several experiments have recently reported new results

on CCQE scattering from high-statistics data samples withintense, well-understood neutrino beams. The NOMADexperiment extracted a CCQE cross section and MA fromdata taken on carbon in the energy range, 3

Using this high-statistics and low-background event sam-ple, we report the first measurement of an absolute ��CCQE double differential cross section, the main resultof this work. In addition, CCQE cross sections in severalother conventional forms are provided. The layout of theremainder of this paper is as follows. In Sec. II, we providea summary of the MiniBooNE experiment, including thebooster neutrino beamline (BNB) and the MiniBooNEdetector. We detail the neutrino interaction model used todescribe the signal and background in Sec. III. The CCQEselection and analysis strategy is outlined in Sec. IV.Finally, in Sec. V, we report the MiniBooNE flux-integrated CCQE double differential cross section

( d2�

dT�d cos��), the flux-integrated CCQE single differential

cross section ( d�dQ2QE

), and the flux-unfolded CCQE cross

section as a function of energy (�½EQE;RFG� �). To facilitatecomparison with updated model predictions [16,17], weprovide the predicted MiniBooNE neutrino fluxes andmeasured cross section values in tabular form in theappendix.

II. MINIBOONE EXPERIMENT

A. Neutrino beamline and flux

The Booster Neutrino Beamline (BNB) consists of threemajor components as shown in Fig. 1: a primary protonbeam, a secondary meson beam, and a tertiary neutrinobeam. Protons are accelerated to 8 GeV kinetic energy inthe Fermilab Booster synchrotron and then fast-extractedin 1:6 �s ‘‘spills’’ to the BNB. These primary protonsimpinge on a 1.75 interaction-length beryllium target cen-tered in a magnetic focusing horn. The secondary mesonsthat are produced are then focused by a toroidal magneticfield which serves to direct the resulting beam of tertiaryneutrinos towards the downstream detector. The neutrinoflux is calculated at the detector with a GEANT4-based[18] simulation which takes into account proton transportto the target, interactions of protons in the target, produc-tion of mesons in the p-Be process, and transport ofresulting particles through the horn and decay volume. Afull description of the calculation with associated uncer-tainties is provided in Ref. [19]. MiniBooNE neutrino data

is not used in any way to obtain the flux prediction. Theresulting �� flux is shown as a function of neutrino energy

in Fig. 2 along with its predicted uncertainty. These valuesare tabulated in Table V in the appendix. The �� flux has an

average energy (over 0< E� < 3 GeV) of 788 MeV andcomprises 93.6% of the total flux of neutrinos at theMiniBooNE detector. There is a 5.9% (0.5%) contamina-tion of ��� (�e, ��e); all events from these (non-��) neutrino

types are treated as background in this measurement(Sec. IVD).The largest error on the predicted neutrino flux results

from the uncertainty of pion production in the initial p-Beprocess in the target as the simulation predicts that 96.7%of muon neutrinos in the BNB are produced via �þ decay.The meson production model in the neutrino beam simu-lation [19] relies on external hadron production measure-ments. Those of the HARP experiment [20] are the mostrelevant as they measure the �� differential cross sectionin p-Be interactions at the same proton energy and on thesame target material as MiniBooNE. The uncertainty in�þ production is determined from spline fits to the HARP�þ double differential cross section data [19]. The spline-fit procedure more accurately quantifies the uncertainty inthe underlying data, removing unnecessary sources of errorresulting from an inadequate parameterization [21] of theHARP data. The HARP data used was that from a thin (5%interaction length) beryllium target run [20]. While thatdata provides a valuable constraint on the BNB flux pre-diction, additional uncertainties resulting from thick targeteffects (secondary rescattering of protons and pions) areincluded through the BNB flux simulation.

FIG. 1 (color online). Schematic overview of the MiniBooNEexperiment including the booster neutrino beamline andMiniBooNE detector.

(GeV)νE

0 0.5 1 1.5 2 2.5 3

)2/P

OT

/GeV

/cm

ν)

(ν

(EΦ 0

0.050.1

0.150.2

0.250.3

0.350.4

0.45

-910×

>= 788 MeVν

The resulting �þ production uncertainty is � 5% at thepeak of the flux distribution increasing significantly at highand low neutrino energies. There is a small contribution tothe �� flux error from the uncertainty in kaon production

which is significant only for E� > 2:0 GeV. Other majorcontributions to the flux error include uncertainties on thenumber of protons on target (POT), hadron interactions inthe target, and the horn magnetic field. These are groupedas the ‘‘beam’’ component shown together with theaforementioned pion and kaon production uncertaintiesin Fig. 2(b). All flux errors are modeled through variationsin the simulation and result in a total error of � 9% at thepeak of the flux. In practice, a complete error matrix iscalculated in bins of neutrino energy that includes corre-lations between bins. This matrix is used to propagate theflux uncertainties to the final quantities used to extract thecross section results reported here.

B. MiniBooNE detector

The MiniBooNE detector (shown schematically inFig. 1) is located 541 m downstream of the neutrinoproduction target and consists of a spherical steel tank of610 cm inner radius filled with 818 tons of Marcol7 lightmineral oil (CH2) with a density of 0:845 g=cm

3. Thevolume of the tank is separated into an inner and an outerregion via an optical barrier located at a radius of 574.6 cm.The inner and outer regions are only separated optically,the oil is the same in each. The index of refraction of the oilis 1.47, yielding a Cherenkov threshold for particles with�> 0:68. The mineral oil is undoped, that is, no additionalscintillation solutes were added. However, because of in-trinsic impurities in the oil, it produces a small amount ofscintillation light in addition to Cherenkov light in re-sponse to energy loss by charged particles.

The inner (signal) region is viewed by an array of 1280inward-facing 8-inch photo multiplier tubes (PMTs)mounted on the inside of the optical barrier and providing11.3% photocathode coverage of the surface of the innertank region. The outer (veto) region is monitored by 240pair-mounted PMTs which record the light produced bycharged particles entering or exiting the detector volume.

The PMT signals, in response to the light produced fromcharged particles, are routed to custom-built electronicsmodules where they are amplified, discriminated, anddigitized. These (‘‘QT’’) modules extract the start timeand integrated charge from each PMT pulse that is abovea discriminator threshold of� 0:1 photoelectron. This unitof data is called a PMT ‘‘hit.’’ The data is stored in atemporary buffer until a trigger decision is made. Thetrigger system uses information from the Fermilab accel-erator clock signals and PMTmultiplicities to form physicsand calibration trigger signals. The physics trigger for thisanalysis requires only that beam be sent to the BNBneutrino production target. When this condition is satisfied,

all PMT-hit data in a 19:2 �s window starting 5 �s beforethe 1:6 �s beam spill is extracted from the QT modulesand added to the data stream. This readout strategy collectsall PMT data (with no multiplicity threshold) from beam-induced neutrino events as well as any muon decays thatoccur with a characteristic time of 2 �s after the neutrinointeraction.The data within the 19:2 �s readout window is exam-

ined at the analysis stage to organize the hits into temporalclusters or ‘‘subevents.’’ A subevent is any group of at least10 hits (from inner or outer PMTs) where no two consecu-tive hits are separated in time by more than 10 ns. Thesesubevents may then be analyzed separately to extract fur-ther information such as energy and position. With thisscheme, muon-decay electrons or positrons may be iden-tified and separated from the primary neutrino event.The MiniBooNE detector is calibrated via the light from

a pulsed laser source, cosmic muons, and muon-decayelectrons. The laser calibration system consists of a pulseddiode laser injecting light via optical fibers into four 10 cmdispersion flasks located at various depths in the maindetector volume. In addition, one bare fiber is installed atthe top of the tank. This system is used to quantify thecharge and time response of the PMTs and allows for anin situ measurement of the oil attenuation length and lightscattering properties.The muon calibration system consists of a two-layer

scintillation hodoscope located above the detector tankcombined with seven 5-cm cubic scintillators deployed atvarious locations within the tank (near the vertical axis)and used to tag stopped muons. With this system, stoppingcosmic ray muons of energies ranging from 20 to 800 MeVare tracked through the detector enabling an energy cali-bration via the known range of the stopping muons. Usingthis data, the energy (angular) resolution for reconstructedmuons in MiniBooNE is measured to be 12% (5.4�) at100 MeV, improving to 3.4% (1.0�) at 800 MeV.A separate large sample of stopping cosmic muons is

obtained via a dedicated calibration trigger that requiresthe signature of an incoming muon and its decay electron.This sample allows for the calibration and measurement ofthe detector response to muon-decay electrons.More details about the oil medium, detector structure,

PMTs, electronics, and calibration are available inRef. [22].

C. Detector simulation

A GEANT3-based [23] program is used to simulate theresponse of the detector to neutrino interactions. Thissimulation is used to determine the detection efficiencyfor CCQE events, the probability for accepting backgroundevents, and the error on relevant observables due to un-certainties in the detector response.The entire geometry of the MiniBooNE detector is

modeled including the detector tank and all inner compo-

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nents. The major components of the detector housingstructure are modeled as is the surrounding environment.Standard GEANT3 particle propagation and decay rou-tines are utilized with some changes made to better simu-late �0 decay, � decay, and �� capture on carbon. Thelatter process is especially important for the backgroundestimation in the analysis reported here. The GCALOR [24]hadronic interaction package is used in place of the defaultGFLUKA package in GEANT3 because it better reproduces

known data on �þ absorption and charge exchange in therelevant �þ energy range (100–500 MeV).

Some modifications were made to the standard GCALORcode to better simulate �� processes. Pion radiative cap-ture/decay and photonuclear processes are of concern forthe neutrino oscillation search [2] but have negligibleeffect for this analysis. The elastic scattering of �� oncarbon is important and was not simulated in the standardGCALOR code. A model, guided by the available data

[25,26], was added to the simulation and yields only asmall change to the calculated background from pionproduction processes.

Uncertainties of 35% on �þ absorption and 50% on �þcharge exchange are assigned based on the differencebetween the external data [25] and the GCALOR prediction.Note that these errors are relevant for �þ propagation inthe detector medium not intranuclear processes which areassigned separate uncertainties (Sec. III D).

Charged particles propagating through the detector oilproduce optical photons via Cherenkov radiation and scin-tillation. Optical wavelengths of 250–650 nm are treated.The Cherenkov process is modeled with standardGEANT3. The scintillation process is modeled with aMiniBooNE-specific simulation that creates optical pho-tons at a rate proportional to Birk’s law-corrected energyloss with an emission spectrum determined from dedicatedflorescence measurements. Optical photons resulting fromthese production processes are tracked through the detectoroil with consideration of scattering, fluorescence, absorp-tion, and reflection from detector surfaces (includingPMTs). Photons that intersect the PMT surface (and donot reflect) are modeled with a wavelength and incidentangle-dependent efficiency. The photon signal in eachPMT is used together with the known response of thePMT and readout electronics to generate simulated datathat is then input to the data analysis programs.

The models, associated parameters, and errors imple-mented in the detector simulation are determined throughexternal measurements of the properties of materials aswell as internal measurements using data collected in theMiniBooNE detector. It is particularly important to cor-rectly model the optical photon transport since a typicaloptical photon travels several meters before detection. This‘‘optical model’’ is tuned starting from various external (tothe detector) measurements of MiniBooNE mineral oiloptical properties, such as the refractive index, attenuation

length, and fluorescence/scintillation strength. These quan-tities allow for the implementation of various models todescribe the optical photon propagation. The details of themodels are then further adjusted based on MiniBooNEinternal data sets such as muon-decay electrons, cosmicmuons, and laser pulses. In total, 35 optical model parame-ters are adjusted to obtain a good description of the variousdata sets. Values for the uncertainties of these parameters,including correlations among them, are also extracted fromthe data and the effect on the reported observables deter-mined by running the simulation with adjusted values(Sec. IVE).Additional details about the MiniBooNE detector simu-

lation and supporting measurements of oil properties areavailable in Refs. [22,27].

III. NEUTRINO INTERACTION MODEL

The MiniBooNE experiment employs the NUANCE V3event generator [28] to estimate neutrino interactionrates in the CH2 target medium. The NUANCE generatorconsiders all interaction processes possible in the neutrinoenergy region relevant for MiniBooNE. It also enables thevarious processes to be tuned to match the data via inputparameters or source code changes where necessary. TheNUANCE generator includes the following components:

(1) a relativistic Fermi gas (RFG) model for CCQE (andNC elastic) scattering on carbon [8]; (2) a baryonic reso-nance model for CC/NC single and multipion production[29]; (3) a coherent CC/NC single pion production model[30]; (4) a deep inelastic scattering (DIS) model [31,32];and (5) a final-state interaction model to simulate reinter-action of final state hadrons in the nuclear medium [28].Neutrino interactions on both free (protons) and boundnucleons (in carbon) are considered to model the CH2detector medium. Further details on these models, parame-ters, and uncertainties are provided in the followingsubsections.This event generator is used, after the adjustment of

parameters to adequately describe the MiniBooNE data,to calculate the background contribution to the CCQEsignal. The CCQE model is used only in the extraction ofthe model parameters in a shape-fit to the Q2QE distribution

(Sec. IVC). The CCQE differential cross section measure-ments do not depend on this model (excepting some smalldependence due to detector resolution corrections and ���CCQE backgrounds). In addition, since it is such a largebackground to CCQE, the CC1�þ background is con-strained (outside of the NUANCE model) to reproduceMiniBooNE data (Sec. IVC). A summary of interactionchannels considered and the NUANCE predictions for theneutrino interaction fractions in the MiniBooNE detectorare provided in Table I. The final values for the predictedbackgrounds (after event selection cuts) are provided inTable III.

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A. Charged current quasielastic scattering

CCQE scattering is the dominant neutrino interactionprocess in MiniBooNE and the subject of this analysis.This process is defined as the charge-changing scattering ofa neutrino from a single nucleon with no other particlesproduced and it is simulated with the RFG model [8] withseveral modifications. A dipole form is used for the axialform factor with an adjustable axial mass, MA. An empiri-cal Pauli-blocking parameter, �, is introduced [11] toallow for an extra degree of freedom that is important todescribe the MiniBooNE data at low momentum transfer.This parameter is a simple scaling of the lower bound,

Elo, of the nucleon energy integral of Ref. [8] via Elo ¼�ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2F þM2p

q�!þ EBÞ, where pF, Mp, !, and EB are

the Fermi momentum, nucleon mass, energy transfer, andbinding energy, respectively. When � > 1, the Pauli-blocking of final-state nucleons is increased which reducesthe cross section at low momentum transfer.

A parametrization [33] is used to describe the nondipolebehavior of the Dirac and Pauli form factors. Although thecontribution is small, the pseudoscalar form factor, derivedfrom partial conservation of the axial-vector current(PCAC) is also included [34]. The scalar and axial tensorform factors are set to zero as implied from G-parityconservation. The Fermi momentum and binding energyfor carbon are set to 220� 30 MeV=c and 34� 9 MeV,respectively, as extracted from electron scattering data [35]taking account of the purely isovector character of CCQE.

The parameters MA and � were initially extracted fromMiniBooNE CCQE data in a prior analysis [11], and weredetermined to be MeffA ¼ 1:23� 0:20 GeV and � ¼1:019� 0:011. While not the main result of this paper,this exercise is repeated after explicitly measuring theCC1�þ background from MiniBooNE data and is de-scribed in Sect. IVC. The superscript ‘‘eff’’ on MA wasintroduced to allow for the possibility that the axial massmeasured from scattering on nucleons bound in carbonmay be different from the ‘‘bare-nucleon’’ axial massthat appears within the neutrino model. The use of thisnotation is continued in this work. The uncertainties in

these CCQE model parameters do not propagate to theerrors on the measured cross sections for this channel.Neutral current elastic (NCE) scattering is described withthe same model as that for the CCQE interaction with thereplacement of appropriate form factors [36] to describethe NC coupling to the nucleon. The uncertainty from theNCE model parameters on the CCQE results is negligibledue to the small background contribution from this channel(Table III).

B. Resonance interactions

The primary source of single pion production forMiniBooNE is predicted to be baryonic resonance produc-tion and decay, such as,

�� þ p ! �� þ �þþ,! �þ þ p;

�� þ n ! �� þ �þ,! �þ þ n; �0 þ p:

The NUANCE model employs the relativistic harmonic os-cillator quark model of Refs. [29,37]. The pion angulardistribution due to the spin structure of the resonance statesis additionally taken into account. In total, 18 resonancesand their interferences are simulated in reactions withinvariant mass W < 2 GeV, however, the �ð1232Þ reso-nance dominates at this energy scale. For reactions onbound nucleons, an RFG model is employed with a uni-form Fermi momentum and constant binding energy. In-medium effects on the width of resonances are not consid-ered explicitly, however, final-state interactions can pro-duce an effective change in these widths. An axial mass ofM1�A ¼ 1:10� 0:27 GeV, set by tuning to available data,is used for this channel.Multipion processes are considered in the NUANCE simu-

lation with MN�A ¼ 1:30� 0:52 GeV. This parameter wasset (together with M1�A ) so that the simulation reproducesinclusive CC data. The contribution of this channel to theCCQE background is small and the uncertainty is negli-gible in the final errors.

TABLE I. Event type nomenclature and NUANCE-predicted �� event fractions for MiniBooNE integrated over the predicted flux inneutrino mode before selection cuts. For the pion production channels, indirect production (through resonance states) and directproduction (through coherent processes) are included. (CC ¼ charged-current, NC ¼ neutral-current).neutrino process abbreviation reaction fraction (%)

CC quasielastic CCQE �� þ n ! �� þ p 39NC elastic NCE �� þ pðnÞ ! �� þ pðnÞ 16CC 1�þ production CC1�þ �� þ pðnÞ ! �� þ �þ þ pðnÞ 25CC 1�0 production CC1�0 �� þ n ! �� þ �0 þ p 4NC 1�� production NC1�� �� þ pðnÞ ! �� þ �þð��Þ þ nðpÞ 4NC 1�0 production NC1�0 �� þ pðnÞ ! �� þ �0 þ pðnÞ 8multipion production, DIS, etc. other �� þ pðnÞ ! �� þ N�� þ X, etc. 4

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The CC1�þ channel is the largest background contri-bution to the CCQE signal and the uncertainty from themodel prediction alone is substantial [38]. However, theexperimental signature of the CC1�þ reaction inMiniBooNE is distinct and the efficiency is large. So, inorder to reduce uncertainty stemming from the CC1�þmodel prediction, a measurement of the CC1�þ back-ground is performed as part of the CCQE analysis(Sec. IVC).

C. Coherent pion production

Pions are also produced in the CC and NC coherentinteraction of neutrinos with carbon nuclei,

�� þ A ! �� þ �þ þ A; �� þ A ! �� þ �0 þ A:

In NUANCE, this coherent pion production process is de-scribed using the model of Ref. [30] assuming McohA ¼1:03� 0:28 GeV [28].

Coherent scattering is predicted to have distinct featuresin the angular distributions of both the final-state muonsand pions. Both the K2K [39] and MiniBooNE [40] experi-ments have measured the fraction of pions produced co-herently in� 1 GeV neutrino interactions. K2K measureda rate for coherent CC1�þ production consistent with zeroand set an upper limit. MiniBooNE measured a nonzerovalue for coherent NC1�0 production albeit � 35%smaller than the model prediction [28,30]. The latest resultfrom the SciBooNE experiment is consistent with the K2Kmeasurement for CC1�þ coherent production [41].

Because of the current discrepancy between CC and NCcoherent pion measurements and the variation in modelpredictions at low energy, the prediction is reduced to 50%of the default value in NUANCE [28,30] and assigned a100% uncertainty. This choice spans the current resultsfrom relevant experiments and existing theoreticalpredictions.

D. Final state interactions

In the NUANCE simulation, neutrino interactions on nu-cleons are modeled using the impulse approximationwhich assumes the interaction occurs instantaneously onindependent nucleons. The binding of nucleons withincarbon is treated within the RFG model, however, anynucleon-nucleon correlation effects are not. The final-statehadrons may interact within the nucleus as they exit. Theyare propagated through the 12C nucleus with a known,radially-dependent nucleon density distribution [42] andmay undergo final-state interactions (FSI). These are simu-lated by calculating interaction probabilities for the pos-sible processes in 0.3 fm steps until the particles leave the� 2:5 fm-radius spherical carbon atom. The interactionprobabilities are derived from external �� N, N � Ncross section and angular distribution data [43], as wellas the nuclear density of carbon.

To model � absorption in the nucleus, (�þ N ! N þN), a constant, energy-independent probability for an intra-nuclear interaction of 20% (10%) is assumed for �þ þ N,�0 þ N (�þþ þ N, �� þ N) processes. These valueswere chosen based on comparisons to K2K data [44] andare assigned a 100% uncertainty. After an interaction, thedensity distribution and step size are modified to prevent anoverestimate of these FSI effects [45,46].Of these hadron FSI processes, intranuclear pion absorp-

tion and pion charge exchange, (�þ þ X ! X0, �þ þX ! �0 þ X0) are the most important contributions tothe uncertainty in the CCQE analysis. Pion absorptionand charge exchange in the detector medium are addressedseparately in the detector simulation (Sec. II C).A CC1�þ interaction followed by intranuclear pion

absorption is effectively indistinguishable from theCCQE process in MiniBooNE (they are ‘‘CCQE-like,’’Sec. IVD). An event with intranuclear pion charge ex-change is distinguishable in the detector, albeit not with100% efficiency. These effects, combined with the highrate of CC1�þ events, results in a significant backgroundto the CCQE measurement that must be treated carefully.The model for these intranuclear pion processes has beentuned to match the available data [25] in the relevant pionenergy range. A comparison of the adjusted NUANCE modeland relevant data is shown in Fig. 3. A 25% (30%) system-atic error in the overall interaction cross section is used forthe pion absorption (charge-exchange) process.

FIG. 3. A comparison of relevant data [25] with the NUANCEmodel (solid lines) for intranuclear pion (a) absorption and(b) charge exchange as a function of pion kinetic energy. Thedotted lines show the 25% (30%) systematic error bands as-sumed for the pion absorption (charge-exchange) cross sections.

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IV. CCQE MEASUREMENT

The goal of this measurement is to determine the doubledifferential cross section for the CCQE process on carbon,�� þ n ! �� þ p, where the target neutron is bound in12C.

The identification of CCQE interactions in theMiniBooNE detector relies solely on the detection of theCherenkov light from the primary (prompt) muon and theassociated decay-electron. An illustration of this process isshown in Fig. 4. Scintillation light is produced by thecharged lepton and the recoil proton (or nuclear frag-ments). However, with the reconstruction employed here,this light is not separable from the dominant Cherenkovlight. In addition, the proton is typically below Cherenkovthreshold. These conditions are such that the proton is notseparable from the charged lepton and so no requirement isplaced on the recoil proton in this analysis. This is to becontrasted with some measurements of CCQE interactionsthat do require the observation of a recoil proton for somepart of the event sample [10,12–14]. An advantage of thisinsensitivity to the proton recoil is that the extracted crosssections are less dependent on proton final-state modeluncertainties. However, the disadvantage in not detectingthe recoil nucleon is that contributions to scattering fromother nuclear configurations (such as two-nucleon correla-tions) are inseparable. These contributions are, in thestrictest sense, not CCQE, but counted as such in ourexperimental definition.

A requirement of low veto activity for the CCQE sampleensures that all particles produced in the event stop in themain region of the detector. This allows muons to betagged with high efficiency via their characteristicelectron-decay with � � 2 �s.

The CCQE interaction, including the muon decay, pro-ceeds as

1: �� þ n ! �� þ p2: ,! e� þ ��e þ ��:

where each line in this equation identifies a subevent(Sec. II B). The primary muon is identified with the firstsubevent and the subsequent decay-electron with the sec-ond subevent. At BNB neutrino energies, neutrino interac-tion events that contain a primary muon predominantlyresult from CCQE scattering as can be seen in Table I.The largest background is from CC single-pion produc-

tion (CC1�þ). A CC1�þ interaction in the detector con-sists of (with subevents labeled),

1: �� þ pðnÞ ! �� þ pðnÞ þ �þ,! �þ þ ��

2: ,! e� þ ��e þ ��3: ,! eþ þ �e þ ���:

Note that this interaction results in three subevents: theprimary interaction and two muon decays (the muon de-cays can occur in any order). The �þ decays immediatelyand light from the prompt decay products contribute to thetotal light in the primary event. These events may beremoved from the CCQE sample by requiring exactlytwo subevents. This requirement also reduces the back-ground from NC processes to an almost negligible levelbecause they do not contain muons and thus have only onesubevent. This simple strategy results in a fairly puresample of CCQE events. However, a significant numberof CC1�þ events have only two subevents because one ofthe decay electrons escapes detection: the �� is capturedon 12C in the mineral oil (with 8% probability [47]) or the�þ is absorbed. Additionally, the study of CC1�þ eventsfor this analysis has indicated that the prediction for theCC1�þ channel from the NUANCE event generator is notsufficiently accurate for this measurement [38]. For thesereasons, the CC1�þ rate is measured using a dedicatedevent sample. This differs from our previous strategy [11]where the default NUANCE-predictedCC1�þ fraction (withno adjustments) was used, with generous errors, in fits tothe CCQE sample.The resulting procedure for selecting the CCQE sample

and measuring the CC1�þ background involves the fol-lowing steps:(1) selection of a ‘‘supersample’’ of events with a clean

muon signature to isolate CC events (predominantlyCCQE and CC1�þ) via analysis cuts;

(2) application of a subevent cut to separate the super-sample into CCQE (2-subevents) and CC1�þ (3-subevents) samples;

(3) measurement of the CC1�þ rate from the CC1�þsample;

(4) adjustment of the CC1�þ model in the event simu-lation to reproduce the measured rate; and

FIG. 4 (color online). Schematic illustration of a CCQE inter-action in the MiniBooNE detector. The primary Cherenkov lightfrom the muon (Cherenkov 1, first subevent) and subsequentCherenkov light from the decay electron (Cherenkov 2, secondsubevent) are used to tag the CCQE event. No requirements aremade on the outgoing proton.

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(5) subtraction of this adjusted CC1�þ background(along with other predicted backgrounds) from theCCQE signal to produce a a measurement of theCCQE interaction cross section.

The details of this procedure are provided in the followingsubsections.

In this analysis, the reconstruction of the CC1�þ sampleis for the sole purpose of background estimation.Dedicated measurements of the CC1�þ and CC1�0 chan-nels in MiniBooNE have been reported elsewhere [48–50]including detailed reconstruction of the �þ and �0kinematics.

A. Event reconstruction

For this analysis, it is crucial to identify and measure themuon in the CC interaction. This is accomplished with an‘‘extended-track’’ reconstruction algorithm [51] whichuses the charge and time information from all PMT hitsin the first subevent to form a likelihood that is maximizedto determine the best single track hypothesis quantified bythe track starting point, starting time, direction, and kineticenergy. This is performed with both a muon and electronparticle hypothesis from which a (log) likelihood ratio isformed to enable particle identification.

The muon kinetic energy T� and muon scattering angle

�� are extracted from the track reconstruction assuming a

muon hypothesis. These are used to form the fundamentalobservable reported here, the double differential crosssection. For additional reported observables, the recon-

structed neutrino energy EQE� and reconstructed four-momentum transfer Q2QE are obtained via,

EQE� ¼ 2ðM0nÞE� � ððM0nÞ2 þm2� �M2pÞ

2 � ½ðM0nÞ � E� þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2� �m2�

qcos���

; (1)

Q2QE ¼ �m2� þ 2EQE� ðE� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2� �m2�

qcos��Þ; (2)

where E� ¼ T� þm� is the total muon energy and Mn,Mp, m� are the neutron, proton, and muon masses. The

adjusted neutron mass, M0n ¼ Mn � EB, depends on thebinding energy (or more carefully stated, the separationenergy) in carbon, EB, which for this analysis is set to 34�9 MeV.The subscript, ‘‘QE,’’ on these reconstructed quantities

is to call attention to these specific definitions and todistinguish them from quantities obtained in other wayssuch as fits to the underlying true kinematic quantities.These are kinematic definitions that assume the initialnucleon (neutron) is at rest and the interaction is CCQE(‘‘QE assumption’’). While these quantities certainly differfrom the underlying true quantities, they are well-defined,unambiguous, and easily reproduced by others.

B. CCQE and CC1�þ event selectionThe CCQE andCC1�þ candidate events are selected for

this analysis and separated with a sequence of cuts sum-marized in Table II.The first five cuts are designed to efficiently select a

high-purity sample of CCQE and CC1�þ events. Cut 1rejects events with incoming particles such as cosmic raysor neutrino-induced events produced in the surroundingmaterial. It also eliminates events where any of the neu-trino interaction products escape the main detector volume.This is important for an accurate muon energy measure-ment and to avoid missing muon decays which leads tohigher backgrounds. Cut 1 does reduce the efficiency sub-stantially (Table II), however, it is necessary to reducebackground (together with the subsequent cuts). Cut 2requires that the primary (muon) is in-time with the BNBspill window. Cut 3 ensures that the reconstructed primarymuon vertex is located within a fiducial region in the maindetector volume sufficiently far from the PMTs for accu-rate reconstruction. Cut 4 provides a minimum muon ki-netic energy for reliable reconstruction and reducesbackgrounds from beam-unrelated muon-decay electrons.Cut 5 requires that the candidate primary muon is better

fit as a muon than as an electron. Misreconstructed andmultiparticle events tend to prefer the electron hypothesisso this cut reduces such contamination. This also substan-tially reduces the efficiency for selecting CC1�þ events as

TABLE II. List of cuts for the CCQE and CC1�þ event selections. The predicted efficiency and purity values are for the CCQEsignal normalized to all CCQE events with a reconstructed vertex radius, r < 550 cm.

CCQE

cut # description effic.(%) purity(%)

1 all subevents, # of veto hits 0:0 36.0 62.36 # total subevents ¼ 2 for CCQE ( ¼ 3 for CC1�þ) 29.1 71.07 (CCQE-only) 1st subevent, �� e vertex distance >100 cm and

�� e vertex distance >ð500� T�ðGeVÞ � 100) cm 26.6 77.0

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can be seen in Fig. 5 where the �=e log-likelihood ratiodistribution is shown for each of the 2- and 3-subeventsamples. This bias is intended as it selects a sample ofCC1�þ with muon kinematics more closely matched tothose CC1�þ that are background to the CCQE sample. Asis shown in Fig. 5, data and Monte Carlo simulation (MC)

agree fairly well to within the detector errors. The log-likelihood ratio distribution is quite sensitive to details ofan event such as scintillation from hadron recoil via thePMT charge and time information [51]. The data-MCdifference in the number of events passing Cut 5 in boththe 2- and 3-subevent samples is covered by the fullsystematic errors considered in this analysis.Cut 6 separates the samples into CCQE (2 subevents)

and CC1�þ (3 subevents) candidates. For this analysis, thesecond and third subevents are required to contain at least20 tank hits to reduce the probability of accidental coinci-dences with the initial neutrino interaction (first subevent).This requirement reduces the efficiency for identifying themuon-decay electron by � 3%.Cut 7 utilizes the muon-electron vertex distance, the

measured separation between the reconstructed muon andelectron vertices. This cut requires that the decay-electronis correctly associated with the primary muon and is ap-plied to the CCQE (2-subevent) sample only. This elimi-nates many CC1�þ events where the second subevent is adecay-positron from the �þ decay chain and not the elec-tron from the decay of the primary muon. The distributionsof the muon-electron vertex distance for the majorMonte Carlo channels and for data are shown in Fig. 6,after the application of cuts 1–6 for events with 2subevents.As shown in Table II, the efficiency for finding CCQE

events with a reconstructed vertex radius, r < 550 cm, is26.6%. An r < 550 cm volume is used for normalization in

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FIG. 6 (color online). Scatter plots of muon-electron vertex distance as a function of reconstructed muon kinetic energy for MCsamples: (a) CCQE, (b) CC1�þ, and (c) other channels. The distribution of data is shown in (d). These are events with 2-subevents andwith cuts 1–6 applied. The lines with arrows indicate the region selected by the muon-electron vertex distance cut.

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the cross section calculations to correctly account forevents that pass all cuts but have a true vertex with r >500 cm. Normalizing to events with true vertex of r <500 cm yields an efficiency of 35%.

C. CC1�þ background measurementAfter the selection of the CCQE and CC1�þ candidate

events (2 and 3 subevent samples, respectively), theCC1�þ background to the CCQE signal is measured byadjusting the weights of the simulated CC1�þ events toachieve data-MC agreement in the Q2QE distribution of the

3 subevent sample. The same weighting, applied to allsimulated CC1�þ events, then provides an estimate ofthe CC1�þ background to the CCQE signal. Figure 7shows the data and MC Q2QE distributions for the two

samples before the reweighting of CC1�þ MC events.The 3-subevent sample is predicted to be 90% CC1�þand shows a large data-MC disagreement in both shape andnormalization. The kinematic distribution of muons inCC1�þ events is similar in both the 2-and 3-subeventsamples as can be observed in Fig. 7. This occurs becausethe majority ofCC1�þ events that are background in the 2-subevent sample are due to muon-capture or pion absorp-tion and the reconstruction of the primary event is, to agood approximation, independent of this. In addition, the�=e log-likelihood ratio cut (Table II and Fig. 5) is appliedfor both the 2- and 3-subevent samples, further ensuringthat the CC1�þ events are the same in both samples.

The CC1�þ reweighting function [Fig. 7(b)] is a 4th-order polynomial in Q2QE and is determined from the

ratio of data to MC in this sample. The 2-subeventsample shows good shape agreement between data andMC. This is because the event model for CCQE was al-ready adjusted to match data in a previous analysis [11]that considered only the shape of theQ2QE distribution. That

analysis did not consider the overall normalization ofevents.In practice, this determination of the CC1�þ reweight-

ing is done iteratively as there is some CCQE backgroundin the 3 subevent sample. An overall normalization factoris calculated for the CCQE sample to achieve data-MCagreement in the 2 subevent sample after subtraction of theCC1�þ background. This is then applied to determine theCCQE background in the 3 subevent sample. The back-ground from other channels is determined from the simu-lation and subtracted. This process converges after twoiterations.This method determines a correction to the CC1�þ rate

(as a function Q2QE) using data from the 3-subevent sample

rather than relying strictly on simulation. This reweightingis then applied to all simulated CC1�þ events, in particu-lar, those that are contained in the 2-subevent sample andform most of the background for the CCQE measurement.The error onM1�A within the resonant background model isthen set to zero and the resulting error on the CC1�þbackground to the CCQE signal from CC1�þ productionis determined by the coherent �-production errors and the�þ absorption uncertainty. The statistical errors in thisprocedure are negligible. Most CC1�þ events that endup in the 2-subevent (CCQE) sample are due to intranu-clear �þ absorption. This process is modeled in the eventsimulation as explained in Sec. III D and is assigned a25% uncertainty. The coherent �-production process ismodeled as described in Sec. III C and is assigned a100% uncertainty.With the measured CC1�þ background incorporated, a

shape-only fit to the 2-subevent (CCQE) sample is per-formed to extract values for the CCQE model parameters,MeffA and �. This exercise is required to have a consistentdescription of the MiniBooNE data within the simulationafter adjustment of the background. This procedure has noeffect on the CCQE cross section results reported hereother than very small corrections to the antineutrino back-ground subtraction which uses these parameters. In this fit,all systematic errors and correlations are considered. TheCCQE simulated sample is normalized to have the samenumber of events as data which is the same normalizationas determined in the CC1�þ background determination.The Q2QE distributions of data from the 2 and 3 subevent

samples is shown together with the MC calculation inFig. 8. The MC calculations include all the adjustmentsdescribed in this section and agreement with data is good inboth samples.This shape-only fit to the 2-subevent sample yields the

adjusted CCQE model parameters, MeffA and �,

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FIG. 7 (color online). Distribution of events in Q2QE for the(a) 2 and (b) 3 subevent samples before the application of theCC1�þ background correction. Data and MC samples are shownalong with the individual MC contributions from CCQE,CC1�þ, and other channels. In (b), the dashed line shows theCC1�þ reweighting function (with the y-axis scale on the right)as determined from the background fit procedure.

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MeffA ¼ 1:35� 0:17 GeV=c2; � ¼ 1:007� 0:012;2=dof ¼ 47:0=38:

Figure 9 shows the 1� contour regions of this fit togetherwith the results from the original MiniBooNE analysis[11]. The new fit yields different results for both MeffAand � because of the improved CC1�þ background esti-mation method used in this analysis. Note that the current

result is consistent to within 1� with � ¼ 1, unlike theprevious MiniBooNE result. This fit actually provides nolower bound on � as the 1� contour is not closed for � < 1.The value for � is quite sensitive to the CC1�þ back-ground at the lowestQ2QE and the background in that region

has decreased in this analysis. The increase in the CC1�þbackground at larger Q2QE values has resulted in a larger

value for the extracted MeffA . The previous and currentparameter contours are consistent at the 1� level. Neitherthis nor the prior analysis result is consistent with theworld-average MA of 1:03� 0:02 GeV [9], as can beseen in Fig. 9. The 2=dof assumingMA ¼ 1:03 GeV, � ¼1 is 67:5=40 corresponding to a 2 probability of � 0:5%.The reconstructed four-momentum transfer, Q2QE, de-

pends upon the muon energy as can be seen in Eq. (2).The reconstructed muon energy calibration has beenchecked by comparing the measured range of muons de-termined from the muon-electron vertex distance (Fig. 6)with the expected muon range determined from the energyprovided by the reconstruction algorithm, which does notuse this vertex distance. As an example, a comparison ofthe measured and expected muon ranges for muons of400< T� < 500 MeV is shown in Fig. 10 for both data

and simulation. The agreement is good for all muon en-ergies and verifies the energy calibration to 2%, well withinthe errors calculated by the simulation. This also showsthat any light produced by hadronic particles in the neu-trino interaction (for both CCQE and background chan-nels) is adequately simulated and considered in thereconstruction.A final and more complete check that the simulation

correctly models the data can be made by examining the 2-dimensional muon kinetic energy and angle (T�, cos��)

distributions. While Fig. 8 shows that data is well-described in Q2QE, the adjusted model may not be adequate

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FIG. 9 (color online). The 1� contour plot for theMeffA � � fit.The filled star shows the best fit point and 1� contour extractedfrom this work. The open star indicates the best fit point and 1�contour from the previous work [11]. Two error ellipses areshown for the previous work, the larger reflects the total uncer-tainty ultimately included in Ref. [11]. The circle indicates theworld-average value for MA [9].

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500 MeV. The subtracted offset, (500� T�ðGeVÞ � 100) cm,corresponds to the solid diagonal line in Fig. 6.

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when applied to the (T�, cos��) distribution of events.

This could occur if an adjustment in Q2QE is hiding an

incorrect neutrino energy distribution. The ratio of datato simulation in (T�, cos��) after correction of the CC1�

þ

background and before the new CCQE model parametersare applied is shown in Fig. 11. The ratio after all correc-tions is shown in Fig. 12 and is much closer to unitythroughout the muon phase space. As can be seen in

Fig. 11, the regions of constant ratio mainly follow lines

of constant Q2QE and not EQE� . Also, almost no structure

remains in Fig. 12. The exception is band of � 20% dis-agreement at EQE� � 0:4 GeV where the error on the neu-trino flux is of that order [Fig. 2(b).] This data-simulationagreement provides additional support for our procedure ofadjusting only the Q2QE behavior of the model and not the

energy distribution of the neutrino flux.

D. Extraction of the cross sections

With the CC1�þ interaction background prediction de-termined from experimental data (Sec. IVC) and the re-maining channels predicted from the interaction model(Sec. III), the cross section for the CCQE interaction canbe extracted. A total of 146070 events pass the CCQEselection (Sec. IVB) resulting from 5:58� 1020 protonson target (POT) collected between August 2002 andDecember 2005. The efficiency for CCQE events passingthese cuts is calculated to be 26.6% for CCQE events withtrue vertices within a 550 cm radius from the center of thedetector tank. The sample is estimated to contain 23.0%background events. These numbers, together with a break-down of predicted backgrounds, are summarized inTable III.The background is dominated by CC1�þ interactions,

which are estimated to be 18.4% of the CCQE candidatesample. Their predicted distribution in Q2QE is shown in

Fig. 8(a). As can be seen, this background is a substantialfraction of the sample in the lowest Q2QE region. The

majority (52%) of the CC1�þ background is predicted tobe events in which the �þ is absorbed in the initial targetnucleus. These are defined as ‘‘CCQE-like’’ in that theycontain a muon and no pions in the final state. The remain-ing CC1�þ background consists of CC1�þ events wherethe �þ is absorbed outside of the target nucleus (33%), is

TABLE III. Summary of the final CCQE event sample includ-ing a breakdown of the estimated backgrounds from individualchannels. The fraction is relative to the total measured sample.The channel nomenclature is defined in Table I.

integrated protons on target 5:58� 1020energy-integrated �� flux 2:88� 1011 ��=cm2CCQE candidate events 146070

CCQE efficiency (R < 550 cm) 26.6%background channel events fraction

NCE 45

not identified due to a missed muon decay (11%), orundergoes charge exchange in the nucleus or detectormedium (4%), These last three classes of CC1�þ back-grounds are not considered CCQE-like. All CC1�þ back-ground events, including those that are CCQE-like, aresubtracted from the data to obtain the final CCQE crosssection results. However, to facilitate examination of themodel used for these processes, the effective cross sectionfor CCQE-like background events is separately reported inthe appendix.

In this analysis, the small contamination of ���, �e, and

��e interactions are treated as background and are sub-tracted from the data based on their MC prediction (seeTable III). The majority of these are ��� CCQE interactions.

The sameMeffA and � as measured in the �� CCQE sample

are used to predict non-�� CCQE events. These parameters

have been shown to adequately reproduce the MiniBooNECCQE data collected in antineutrino mode [52].

To extract differential cross sections in muon kinematicvariables, the reconstructed kinematics are corrected fordetector-specific effects. A correction procedure is imple-mented using an ‘‘unfolding’’ process based on the detec-tor simulation. We employ an ‘‘iterative Bayesian’’ method[53] to avoid the problem of amplification of statisticalfluctuations common in the ‘‘inverse response matrix’’method [54]. A disadvantage to the iterative Bayesianmethod is that the result depends on the initial CCQEmodel assumptions (the ‘‘prior’’ probability). However,this problem is addressed by an iterative method thatuses the extracted signal distribution to correct the pre-dicted distributions and repeating this procedure. In prac-tice, the simulation was already tuned to reproduce the databased on previous work [11], and the result from the firstiteration shows satisfactory convergence. The systematicuncertainty in this procedure is determined from the dif-ference between the initial (0th) and final (1st) iterations ofthe algorithm and by examining the dependence of the finalvalues on the initial model assumptions.

The various correction and normalization factors canthen be brought together in a single expression used toextract the CCQE cross section for the ith bin of a particu-lar kinematic variable,

�i ¼P

j Uijðdj � bjÞ

i � T �� ; (3)

where the index j labels the reconstructed bin and i labelsthe unfolded (estimate of the true) bin. In this equation,Uijis the unfolding matrix, i is the efficiency, T is the numberof neutrons in the fiducial volume, and � is the neutrinoflux. This expression is used to obtain the double and single

differential cross sections, d2�

dT�d cos��, and d�

dQ2QE, respec-

tively, after the multiplication of the appropriate bin widthfactors. Note that the choice of normalization yields cross

sections ‘‘per neutron’’. Here, the flux, �, is a singlenumber (2:88� 1011 ��=cm2) and is determined by inte-grating the BNB flux over 0< E� < 3 GeV. Therefore,these differential cross sections are ‘‘flux-integrated.’’An additional quantity, the flux-unfolded CCQE cross

section as a function of neutrino energy, �½EQE;RFG� �, isextracted from this same expression with the replacementof the total flux, �, with the flux in a particular neutrinoenergy bin, �i. The unfolding procedure is used to correct

the data from bins of reconstructed neutrino energy, EQE� ,[using Eq. (1)] to an estimate of the true neutrino energy,

EQE;RFG� . It is important to note that, unlike the differential

cross sections, d2�

dT�d cos��and d�

dQ2QE

, the calculation of this

cross section relies on the interaction model to connect EQE�to EQE;RFG� . The superscript ‘‘RFG’’ indicates the interac-tion model assumed in the unfolding process [8]. Thisprocedure introduces a model dependence into this crosssection, however, this method is consistent with that com-monly used by experiments reporting a CCQE cross sec-tion as a function of neutrino energy. This modeldependence should be considered when comparing mea-surements of this quantity from different experiments.

E. Error analysis

The errors on the measured cross sections result fromuncertainties in the neutrino flux, background estimates,detector response, and unfolding procedure. To propagatethese error sources, a ‘‘multisim’’ method [55] is used tocalculate the errors on the final quantities by varyingparameters in separate simulations. This method producesan error matrix Vij for each reported distribution that can

then be used to extract the error on each bin (�i ¼ffiffiffiffiffiffiVii

p)

and the correlations between quantities of different bins.The error matrix is calculated by generating a large

number of simulated data sets with different parameterexcursions, based on the estimated 1� uncertainties inthose parameters and the correlations between them. Theerror matrix for a particular distribution is then calculatedfrom these M data sets,

Vij ¼ 1MXM

s¼1ðQsi � Q̂iÞðQsj � Q̂jÞ: (4)

Here, Qsi is the quantity of interest in the ith bin from the

sth simulation data set and Q̂i is the ‘‘best’’ estimate of theparameters. The quantities of interest Qi could be, forinstance, the number of events in each bin of Q2QE or the

calculated cross section in each kinematic bin [Eq. (3)].In practice, the errors are classified into four major

contributions: the neutrino flux, background cross sections,detector modeling, and unfolding procedure. The parame-ters within each of these groups are varied independentlyso the resultant error matrices from the individual groups

A.A. AGUILAR-AREVALO et al. PHYSICAL REVIEW D 81, 092005 (2010)

092005-14

can be added to form the total error matrix. For the neutrinoflux and background cross section uncertainties, a re-weighting method is employed which removes the diffi-culty of requiring hundreds of simulations with adequatestatistics. In this method, each neutrino interaction event isgiven a new weight calculated with a particular parameterexcursion. This is performed considering correlations be-tween parameters and allows each generated event to bereused many times saving significant CPU time. The natureof the detector uncertainties does not allow for this methodof error evaluation as parameter uncertainties can only beapplied as each particle or optical photon propagatesthrough the detector. Approximately 100 different simu-lated data sets are generated with the detector parametersvaried according to the estimated 1� errors includingcorrelations. Equation (4) is then used to calculate thedetector error matrix. The error on the unfolding procedureis calculated from the difference in final results when usingdifferent input model assumptions (Sec. IVD). The statis-tical error on data is not added explicitly but is included viathe statistical fluctuations of the simulated data sets (whichhave the same number of events as the data).

The final uncertainties are reported in the followingsections. The breakdown among the various contributionsare summarized and discussed in Sec. VD. For simplicity,the full error matrices are not reported for all distributions.Instead, the errors are separated into a total normalizationerror, which is an error on the overall scale of the crosssection, and a ‘‘shape error’’ which contains the uncer-tainty that does not factor out into a scale error. This allowsfor a distribution of data to be used (e.g. in a model fit) withan overall scale error for uncertainties that are completelycorrelated between bins, together with the remaining bin-dependent shape error.

V. RESULTS AND DISCUSSION

A. CCQE flux-integrated double differentialcross section

The flux-integrated, double differential cross section per

neutron, d2�

dT�d cos��, for the �� CCQE process is extracted as

described in Sec. IVD and is shown in Fig. 13 for thekinematic range, �1< cos��

B. Flux-integrated single differential cross section

The flux-integrated, single differential cross section perneutron, d�

dQ2QE

, has also been measured and is shown in

Fig. 14. The quantityQ2QE is defined in Eq. (2) and depends

only on the (unfolded) quantities T� and cos��. It should

be noted that the efficiency for events with T� < 200 MeV

is not zero because of difference between reconstructedand unfolded T�. The calculation of efficiency for these

(low-Q2QE) events depends only on the model of the detec-

tor response, not on an interaction model and the associ-ated uncertainty is propagated to the reported results.

In addition to the experimental result, Fig. 14 alsoshows the prediction for the CCQE process from theNUANCE simulation with three different sets of parameters

in the underlying RFG model. The predictions are abso-lutely normalized and have been integrated over theMiniBooNE flux. The RFG model is plotted assumingboth the world-averaged CCQE parameters (MA ¼1:03 GeV, � ¼ 1:000) [9] and the CCQE parameters ex-tracted from this analysis (MA ¼ 1:35 GeV, � ¼ 1:007) ina shape-only fit. The model using the world-averagedCCQE parameters underpredicts the measured differentialcross section values by 20%–30%, while the model usingthe CCQE parameters extracted from this shape analysisare within� 8% of the data, consistent within the normal-ization error ( � 10%). To further illustrate this, the modelcalculation with the CCQE parameters from this analysisscaled by 1.08 is also plotted and shown to be in goodagreement with the data.

C. Flux-unfolded CCQE cross section as a function ofneutrino energy

The flux-unfolded CCQE cross section per neutron

�½EQE;RFG� �, as a function of the true neutrino energyEQE;RFG� , is shown in Fig. 15. These numerical values are

tabulated in Table X in the appendix. The quantity EQE;RFG�is a (model-dependent) estimate of the neutrino energyobtained after correcting for both detector and nuclearmodel resolution effects. These results depend on the de-tails of the nuclear model used for the calculation. Thedependence is only weak in the peak of the flux distributionbut becomes strong for E� < 0:5 GeV and E� > 1:2 GeV,i.e., in the ‘‘tails’’ of the flux distribution.

In Fig. 15, the data are compared with the NUANCEimplementation of the RFG model with the world averageparameter values, (MeffA ¼ 1:03 GeV, � ¼ 1:000) and withthe parameters extracted from this work (MeffA ¼1:35 GeV, � ¼ 1:007). These are absolute predictionsfrom the model (not scaled or renormalized). At the aver-age energy of the MiniBooNE flux ( � 800 MeV), theextracted cross section is � 30% larger than the RFGmodel prediction with world average parameter values.The RFG model, with parameter values extracted from

the shape-only fit to this data better reproduces the dataover the entire measured energy range.Figure 15(b) shows these CCQE results together with

those from the LSND [56] and NOMAD [10] experiments.It is interesting to note that the NOMAD results are betterdescribed with the world average MeffA and � values. Alsoshown for comparison in Fig. 15(b) is the predicted crosssection assuming the CCQE interaction occurs on freenucleons with the world average MA value. The crosssections reported here exceed the free nucleon value forE� above 0.7 GeV.

D. Error summary

As described in Sec. IVE, (correlated) systematic andstatistical errors are propagated to the final results. These

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

)2 (

cmσ

02468

10121416

-3910×

MiniBooNE data with shape errorMiniBooNE data with total error

=1.000κ=1.03 GeV, effARFG model with M=1.007κ=1.35 GeV, effARFG model with M

(GeV)QE,RFGνE

(a)

(GeV)QE,RFGνE

-110 1 10

)2 (

cmσ

02468

10121416

-3910×

MiniBooNE data with total error=1.000κ=1.03 GeV, effARFG model with M=1.007κ=1.35 GeV, effARFG model with M

=1.03 GeVAFree nucleon with M

NOMAD data with total errorLSND data with total error(b)

FIG. 15 (color online). Flux-unfolded MiniBooNE �� CCQEcross section per neutron as a function of neutrino energy. In (a),shape errors are shown as shaded boxes along with the totalerrors as bars. In (b), a larger energy range is shown along withresults from the LSND [56] and NOMAD [10] experiments.Also shown are predictions from the NUANCE simulation for anRFG model with two different parameter variations and forscattering from free nucleons with the world-average MA value.Numerical values are provided in Table X in the appendix.

TABLE IV. Contribution to the total normalization uncertaintyfrom each of the various systematic error categories.

source normalization error (%)

neutrino flux prediction 8.66

background cross sections 4.32

detector model 4.60

kinematic unfolding procedure 0.60

statistics 0.26

total 10.7

A. A. AGUILAR-AREVALO et al. PHYSICAL REVIEW D 81, 092005 (2010)

092005-16

errors are separated into normalization and shape uncer-tainties. The contributions from each error source on thetotal normalization uncertainty are summarized inTable IV. As is evident, the neutrino flux uncertainty domi-nates the overall normalization error on the extractedCCQE cross sections. However, the uncertainty on theflux prediction is a smaller contribution to the shape erroron the cross sections. This can be seen in Fig. 16 whichshows the contribution from the four major sources to theshape error on the total (flux-unfolded) cross section.

The detector model uncertainty dominates the shapeerror, especially at low and high energies. This is becauseerrors in the detector response (mainly via uncertainties invisible photon processes) will result in errors on the recon-structed energy. These errors grow in the tails of theneutrino flux distribution due to feed-down from eventsin the flux peak. This type of measurement usually haslarge errors due to non-negligible uncertainties in theCC1�þ background predictions. In this measurement,that error is reduced through direct measurement of theCC1�þ background. However, this error is not completelyeliminated due to the residual uncertainty on the rate ofintranuclear pion absorption that is included. This uncer-tainty is not as important for the measurement of the CCQEcross section measurement as a function of energy but is alarge contribution to the error at lowQ2QE in the differential

distributions.The unfolding error is small in the region of the flux

peak but grows in the high- and low-energy region becauseof the uncertainty in the feed-down from other energy bins,similar to that described for the detector model.

VI. CONCLUSIONS

In this work, we report measurements of absolute crosssections for the CCQE interaction using high-statistics

samples of �� interactions on carbon. These include the

first measurement of the double differential cross section,d2�

dT�d cos��, measurement of the single differential cross

section, d�dQ2

QE

, and the flux-unfolded CCQE cross section,

�½EQE;RFG� �. The double differential cross section containsthe most complete and model-independent informationthat is available from MiniBooNE on the CCQE process.It is the main result from this work and should be used asthe preferred choice for comparison to theoretical modelsof CCQE interactions on nuclear targets.The reported cross section is significantly larger

( � 30% at the flux average energy) than what is com-monly assumed for this process assuming a relativisticFermi Gas model (RFG) and the world-average value forthe axial mass, MA ¼ 1:03 GeV [9]. In addition, the Q2QEdistribution of this data shows a significant excess of eventsover this expectation at higher Q2QE even if the data is

normalized to the prediction over all Q2QE. This leads to

an extracted axial mass from a ‘‘shape-only’’ fit of theQ2QEdistribution of MeffA ¼ 1:35� 0:17 GeV, significantlyhigher than the historical world-average value.These two observations, unexpectedly large values for

the extracted cross section and MeffA , are experimentallyseparate. However, within the model prediction, a largervalue for MA implies a larger cross section because theCCQE cross section increases approximately linearly withMA. The predicted CCQE cross section with this highervalue of MA agrees with the measurement within thenormalization error of the experiment ( � 10%). Whilethis may be simply a coincidence, it is important to note.In recent years, there has been significant effort to

improve the theoretical description of the CCQE interac-tion on nuclear targets [16,17]; however, there seems to beno simple explanation for both the higher cross section andthe harder Q2QE distribution of events (resulting in a larger

MeffA ) as evidenced by the MiniBooNE CCQE data.Nuclear effects can have some impact on the measuredMeffA , but it is not obvious that they are large enough. Also,it is expected that such effects should reduce the crosssection, not increase it. This can be seen in Fig. 15 wherethe cross section for the CCQE interaction on free nucleonsis compared to that from bound nucleons (in an RFGmodel). Note the reduction in cross section from free tobound nucleons. It is interesting that the MiniBooNE mea-surement is also larger than this free nucleon value (at leastat higher energies). This may indicate a significant contri-bution from neglected mechanisms for CCQE-like scatter-ing from a nucleus such as multinucleon processes (forexample, Ref. [17]). This may explain both the higher crosssection and the harder Q2 spectrum, but has not yet beenexplicitly tested. It may also be relevant for the differencebetween these results and those of NOMAD (or otherexperiments) where the observation of recoil nucleonsenter the definition of a CCQE event. An important test

(GeV)QE,RFGνE

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

frac

tion

al s

hape

err

or

0

0.05

0.1

0.15

0.2

0.25Detector errorFlux errorBackground errorUnfolding error

FIG. 16 (color online). Fractional shape error on theMiniBooNE �� CCQE flux-unfolded cross section separated

into major components. The overall normalization error of10.7% is not shown.

FIRST MEASUREMENT OF THE MUON NEUTRINO . . . PHYSICAL REVIEW D 81, 092005 (2010)

092005-17

for such models will be their ability to accurately repro-duce the MiniBooNE double differential cross sections atleast as well as the RFG model assuming a higher axialmass value.

As yet, there is no easily recognized solution to explainthe difference between the CCQE cross sections measuredin MiniBooNE at lower neutrino energy (E� < 2 GeV) andthe NOMAD results at higher neutrino energies (E� >3 GeV). Model-independent measurements of the CCQEcross section anticipated from SciBooNE [57],MicroBooNE [58], and MINERvA [59] as well as theT2K [6] and NOvA [5] near detectors running with 2<E� < 20 GeV, will be important to help resolve theseresults.

ACKNOWLEDGMENTS

This work was conducted with support from Fermilab,the U.S. Department of Energy , and the National ScienceFoundation .

APPENDIX A: TABULATION OF RESULTS

This appendix contains tables of numerical values cor-responding to various plots appearing in the main body ofthe paper. In addition, the effective cross section for theCCQE-like background to the CCQEmeasurement is tabu-lated. These tables are also available via the MiniBooNEwebsite [60].

1. Predicted �� flux

Table V lists the predicted �� flux (Fig. 2) at the

MiniBooNE detector in 50 MeV-wide neutrino energybins. The flux is normalized to protons on target (POT).The mean energy is 788 MeV and the integrated flux overthe energy range (0:0< E� < 3:0 GeV) is 5:16�10�10 ��=POT=cm2. For this analysis, the total POT col-lected is 5:58� 1020 yielding an integrated flux of 2:88�1011 ��=cm

2.

2. CCQE flux-integrated double differentialcross section

Table VI contains the flux-integrated �� CCQE double

differential cross section values ( d2�

dT�d cos��) in bins of

muon energy T� and cosine of the muon scattering angle

with respect to the incoming neutrino direction (in the labframe) cos��. These values correspond to the plot of

Fig. 13. The integrated value over the region (� 1<cos��

CCQE signal in MiniBooNE. These events originate fromthe CC1�þ interaction but contain 1 muon and no pions inthe final state. In the main analysis, this background issubtracted to obtain the CCQE observables. In order tofacilitate comparisons with models (or other analyses) thatconsider all CCQE-like events as CCQE signal, the effec-tive double differential cross section for the CC1�þ inter-action with intranuclear pion absorption is presented in

Table VIII. These values are determined from theNUANCE-event generator corrected to reproduce the

MiniBooNE 3-subevent sample and are calculated usingEq. (3) with (dj � bj) replaced by b0j, the number ofCCQE-like background events. A CCQE-like cross sectionmay be obtained by adding these numbers (Table VIII)with those from Table VI.

TABLE VI. The MiniBooNE �� CCQE flux-integrated double differential cross section in units of 10�41 cm2=GeV in 0.1 GeV bins

of T� (columns) and 0.1 bins of cos�� (rows).

cos��T� (GeV) 0.2,0.3 0.3,0.4 0.4,0.5 0.5,0.6 0.6,0.7 0.7,0.8 0.8,0.9 0.9,1.0 1.0,1.1 1.1,1.2 1.2,1.3 1.3,1.4 1.4,1.5 1.5,1.6 1.6,1.7 1.7,1.8 1.8,1.9 1.9,2.0

þ0:9, þ1:0 190.0 326.5 539.2 901.8 1288 1633 1857 1874 1803 1636 1354 1047 794.0 687.9 494.3 372.5 278.3 227.4þ0:8, þ0:9 401.9 780.6 1258 1714 2084 2100 2035 1620 1118 783.6 451.9 239.4 116.4 73.07 41.67 36.55 � � � � � �þ0:7, þ0:8 553.6 981.1 1501 1884 1847 1629 1203 723.8 359.8 156.2 66.90 26.87 1.527 19.50 � � � � � � � � � � � �þ0:6, þ0:7 681.9 1222 1546 1738 1365 909.6 526.7 222.8 81.65 35.61 11.36 0.131 � � � � � � � � � � � � � � � � � �þ0:5, þ0:6 765.6 1233 1495 1289 872.2 392.3 157.5 49.23 9.241 1.229 4.162 � � � � � � � � � � � � � � � � � � � � �þ0:4, þ0:5 871.9 1279 1301 989.9 469.1 147.4 45.02 12.44 1.012 � � � � � � � � � � � � � � � � � � � � � � � � � � �þ0:3, þ0:4 910.2 1157 1054 628.8 231.0 57.95 10.69 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �þ0:2, þ0:3 992.3 1148 850.0 394.4 105.0 16.96 10.93 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �þ0:1, þ0:2 1007 970.2 547.9 201.5 36.51 0.844 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �0.0, þ0:1 1003 813.1 404.9 92.93 11.63 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:1, 0.0 919.3 686.6 272.3 40.63 2.176 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:2, �0:1 891.8 503.3 134.7 10.92 0.071 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:3, �0:2 857.5 401.6 79.10 1.947 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:4, �0:3 778.1 292.1 33.69 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:5, �0:4 692.3 202.2 17.42 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:6, �0:5 600.2 135.2 3.624 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:7, �0:6 497.6 85.80 0.164 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:8, �0:7 418.3 44.84 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:9, �0:8 348.7 25.82 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��1:0, �0:9 289.2 15.18 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

TABLE VII. Shape uncertainty on the MiniBooNE �� CCQE flux-integrated double differential cross section in units of10�42 cm2=GeV corresponding to Table VI. The total normalization error is 10.7%.

cos��T�(GeV) 0.2,0.3 0.3,0.4 0.4,0.5 0.5,0.6 0.6,0.7 0.7,0.8 0.8,0.9 0.9,1.0 1.0,1.1 1.1,1.2 1.2,1.3 1.3,1.4 1.4,1.5 1.5,1.6 1.6,1.7 1.7,1.8 1.8,1.9 1.9,2.0

þ0:9, þ1:0 684.3 1071 1378 1664 1883 2193 2558 3037 3390 3320 3037 3110 2942 2424 2586 2653 3254 3838þ0:8, þ0:9 905.0 1352 1754 2009 2222 2334 2711 2870 2454 1880 1391 1036 758.7 544.3 505.5 359.6 � � � � � �þ0:7, þ0:8 1134 1557 1781 1845 1769 1823 1873 1464 963.8 601.6 339.6 184.1 170.1 230.6 � � � � � � � � � � � �þ0:6, þ0:7 1435 1455 1581 1648 1791 1513 1068 598.2 267.2 155.1 69.28 89.01 � � � � � � � � � � � � � � � � � �þ0:5, þ0:6 1380 1372 1434 1370 1201 870.2 432.3 162.2 71.88 49.10 54.01 � � � � � � � � � � � � � � � � � � � � �þ0:4, þ0:5 1477 1273 1365 1369 1021 475.5 161.6 55.58 16.32 � � � � � � � � � � � � � � � � � � � � � � � � � � �þ0:3, þ0:4 1267 1154 1155 965.3 574.7 149.2 53.26 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �þ0:2, þ0:3 1293 1105 1041 742.5 250.6 77.66 110.3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �þ0:1, þ0:2 1351 1246 1048 415.1 114.3 41.02 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �0.0, þ0:1 1090 1078 695.5 238.2 45.96 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:1, 0.0 980.4 783.6 515.7 114.6 20.92 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:2, �0:1 917.7 746.9 337.5 50.92 3.422 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:3, �0:2 922.7 586.4 215.6 55.88 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:4, �0:3 698.0 553.3 135.3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��0:5, �0:4 596.9 482.6 57.73 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �