Constraining the GENIE model of neutrino-induced single pion production using reanalyzed bubble chamber data

1.2 MB·

by  Philip Rodrigues

Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
Constraining the GENIE model of neutrino-induced single
pion production using reanalyzed bubble chamber data
Philip Rodrigues1, Callum Wilkinson2, Kevin McFarland1
1UniversityofRochester,DepartmentofPhysicsandAstronomy,Rochester14627,NewYork,USA
2University of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High Energy Physics (LHEP), Bern 3012,
Switzerland
6
1
November9,2016
0
2

v Abstract Thelongstandingdiscrepancybetweenbub- nucleus. Experimental data on nuclear targets presents
o ble chamber measurements of ν -induced single pion aconfusingpicture,withrecentdatafromtheMINERνA[1,
µ
N
production channels has led to large uncertainties in 2] and MiniBooNE [3] experiments in poor agreement
8 pionproductioncrosssectionparametersformanyyears. witheachotherintheframeworkofcurrenttheoretical
We extend the reanalysis of pion production data in models [4, 5]. An additional problem is the disagree-

] deuterium bubble chambers where this discrepancy is ment between measurements of the neutrino-nucleon
x
e solved (Wilkinson et al., PRD 90 (2014) 112017) to in- single pion production cross section in the 100MeV to
- clude the ν n → µ−pπ0 and ν n → µ−nπ+ channels, few-GeV energy range most relevant for current and
p µ µ
e and use the resulting data to fit the parameters of the plannedneutrinooscillationexperiments.Theaxialform
h GENIE pion production model. We find a set of pa- factor for pion production on free nucleons cannot be
[
rameters that can describe the bubble chamber data constrained by electron scattering data, so relies upon

3 better than the GENIE default parameters, and pro- datafromtheArgonneNationalLaboratory’s12ftbub-
v vide updated central values and reduced uncertainties blechamber(ANL)andtheBrookhavenNationalLab-
8
for use in neutrino oscillation and cross section anal- oratory’s 7ft bubble chamber (BNL). However, these
8
yses which use the GENIE model. We find that GE- datasetsdifferinnormalizationby30–40%forthelead-
8
1 NIE’s non-resonant background prediction has to be ingpionproductionprocessν p→µ−pπ+,whichleads
µ
0 significantly reduced to fit the data, which may help to large uncertainties in the predictions for oscillation
.
1 to explain the recent discrepancies between simulation experiments [6–11], as well as in the interpretation of
0 anddataobservedbytheMINERνAcoherentpionand data taken on nuclear targets [12].
6
NOνA oscillation analyses. It has long been suspected that the discrepancy be-
1
: tweenANLandBNLwasduetoanissue withthenor-
v
malization of the flux prediction from one or both ex-
i
X 1 Introduction periments1, and it has been shown by other authors
r that their published results are consistent within the
a
Agoodunderstandingofsinglepionproductionbyneu- experimental uncertainties provided [6, 16]. In Refer-
trinos with few-GeV energies is important for current ence[17],wepresentedamethodforremovingfluxnor-
and future oscillation experiments, where pion produc- malizationuncertaintiesfromtheANLandBNLνµp→
tioniseitherasignalprocess,oralargebackgroundfor
analyseswhichselectquasi-elasticevents.Attheseener- 1The ANL neutrino beam [13] was produced by focusing 12.4
giesthedominantproductionmechanismisviathepro- GeV protons onto a beryllium target. Two magnetic horns were
used to focus the positive pions produced by the primary beam
duction and subsequent decay of hadronic resonances.
inthedirectionofthebubblechamber,thesesecondaryparticles
Completemodelsofneutrino-nucleussinglepionpro- decayed to produce a predominantly νµ beam peaked at ∼0.5
duction interactions are usually factorized into three GeV.TheBNLneutrinobeam[14,15]wasproducedbyfocusing
29 GeV protons on a sapphire target, with a similar two horn
parts:theneutrino-nucleoncrosssection;additionalnu-
designtofocusthesecondaryparticles.TheBNLνµ beamhada
cleareffectswhichaffecttheinitialinteraction;andthe
higherpeakenergyof∼1.2GeV,andwasbroaderthantheANL
“final state interactions” (FSI) of hadrons exiting the beam.
2
µ−pπ+ measurements by taking ratios with charged- (ANL) and [25] (BNL) for this work. To produce flux-
current quasielastic (CCQE) event rates in which the integrated event rate predictions with GENIE, the flux
normalization cancels. Then we obtained a measure- was taken from References [13] (ANL) and [15] (BNL).
ment of νµp → µ−pπ+ by multiplying the ratio by The Eν-dependent distributions for both ANL and
an independent measurement of CCQE (which is well BNL are taken from the reanalysis of ν p → µ−pπ+
µ
known for nucleon targets). Using this technique, we data presented in Reference [17], and the reanalysis of
foundgoodagreementbetweentheANLandBNLνµp→ νµn→µ−pπ0 and νµn→µ−nπ+ using the same tech-
µ−pπ+ datasets. In this work, we extend that method niqueandpresentedinAppendixA.Thesedatasetsare
to include the subdominant νµn → µ−pπ0 and νµn → neutrino-deuterium cross sections, as no correction has
µ−nπ+ channels,andusetheresultingdata,alongwith been applied to account for deuterium nuclear effects.
theQ2-spectra(whereQ2 isthefour-momentumtrans- Additionally, in Appendix B we present reanalyzed re-
fer)fromthesameexperiments,toconstraintheparam- sults for the three pion production channels in which
eters of the GENIE single pion production model [18]. an additional correction has been applied to include
Whilemoresophisticatedsinglepionproductionmodels the effect of the invariant mass cut W ≤ 1.4 GeV on
exist [19–21], the GENIE generator is widely used by theE -dependentdistributions.Thereanalyzedresults
ν
current and planned neutrino oscillation experiments, with W ≤ 1.4 GeV were not used in the present work,
so tuning the generator parameters represents a prag- but we provide it for use in future model comparisons.
matic approach to improving its description of avail- The E -dependent distributions for both ANL and
ν
able data. We find that the reanalyzed data, where the BNL are shown for all three single pion production
normalization discrepancy has been resolved, is able to channels in Figure 1 along with other bubble chamber
significantly reduce the uncertainties on the pion pro- measurements available for these channels. The origi-
ductionparameters.Wealsofindthatthenon-resonant nal ANL and BNL results are also shown so the effect
background prediction from GENIE needs to be signif- of the reanalysis can be seen. It is clear that the re-
icantly reduced to fit the data. analysisaffectsallchannels,althoughtheeffectismore
Reduced uncertainties on pion production param- pronounced for the dominant ν p → µ−pπ+ channel,
µ
eters are vital for current and future neutrino oscilla- where the statistical errors are smaller and biases are
tion experiments, which have very stringent systematic easier to see. The ANL and BNL datasets agree well
uncertainty requirements [22, 23]. We recommend that in all three channels after the reanalysis. In Figures 1a
our new uncertainties should be used by experiments and1b,theBEBCdataonahydrogentargethasanin-
which use the GENIE neutrino interaction generator, variantmasscutofW ≤2GeV[26],whichremovescon-
and the reanalyzed ANL and BNL datasets presented tributions from diffractive processes. The FNAL data
here and in Reference [17] should be used instead of onahydrogentargetisselectedwithaninvariantmass
the published ANL and BNL datasets for future model cut of W ≤ 1.4 GeV, in order to isolate the ∆++ con-
comparisons. tribution to the cross section, which also cuts out any
The paper is organized as follows. In Section 2, we diffractivecontributionsfromthecrosssection[27].Ad-
describe the datasets used in this analysis. In Section 3 ditionally, the FNAL result was scaled by 14% to ac-
we describe the GENIE single pion production model count for ∆++ contributions with W >1.4 GeV.
andcomparethenominalGENIEmodelanderrorbands
Despite the caveats associated with the subdomi-
with the data. The χ2 statistic which is minimized and
nant ν n → µ−pπ0 and ν n → µ−nπ+ channels, we
µ µ
the fit machinery are discussed in Section 4.1; the fit
recommend that all three channels are used for future
results are presented in Section 4.2; and there is a dis-
comparisons with ANL and BNL data. It should be
cussionofthegoodnessoffitinSection4.3.Finally,our
stressed that most of the deficiencies in the reanalyzed
conclusions are presented in Section 5.
results detailed here are also present in the original
ANL and BNL results, and should be borne in mind
2 Datasets used in this analysis when using reanalyzed or published ANL and BNL re-
sults.
In this work, we use the Q2 and E -spectra from ANL A final note of caution regarding the use of these
ν
and BNL for all three charged-current single pion pro- corrected datasets is that there is a hidden correlation
duction modes (ν p → µ−pπ+, ν n → µ−pπ0 and betweenthethreechannelsforeachexperiment.Asthe
µ µ
ν n→µ−nπ+), giving a total of twelve datasets. CCQE events used in the correction are common to
µ
The Q2-dependent distributions used are presented all channels, statistical errors are correlated in a way
asflux-integratedeventrateswithoutanyinvariantmass whichisdifficulttoquantify.Anexampleoftheproblem
cut applied, and were digitized from References [24] can be seen by looking at the three ANL channels for
3
1.4 1.4
) )
n n
o BNL o BNL
cle 1.2 ANL cle 1.2 ANL
u u
n BEBC 86 n BEBC 86
2/m 1 BEBC 90 2/m 1 BEBC 90
c FNAL 78 c FNAL 78
8 0.8 8 0.8
3 SKAT 89 3 SKAT 89
-0 -0
1 1
( 0.6 ( 0.6
+) +)
pp 0.4 pp 0.4
- -
m m
0.2 0.2
fip fip
n(m 0 n(m 0
s 10-1 1 10 102 s 10-1 1 10 102
Neutrino energy (GeV) Neutrino energy (GeV)
(a)νµp→µ−pπ+ published (b)νµp→µ−pπ+ reanalyzed
) )
on 0.7 on 0.7
e BNL e BNL
ucl 0.6 ANL ucl 0.6 ANL
n n
2/m 0.5 BEBC 90 2/m 0.5 BEBC 90
c c
38 0.4 SKAT 89 38 0.4 SKAT 89
-0 -0
1 1
0) ( 0.3 0) ( 0.3
pp pp
- 0.2 - 0.2
m m
0.1 0.1
fin fin
n(m 0 n(m 0
s 10-1 1 10 102 s 10-1 1 10 102
Neutrino energy (GeV) Neutrino energy (GeV)
(c)νµn→µ−pπ0 published (d)νµn→µ−pπ0 reanalyzed
) )
on 0.7 on 0.7
e BNL e BNL
ucl 0.6 ANL ucl 0.6 ANL
n n
2/m 0.5 BEBC 90 2/m 0.5 BEBC 90
c c
38 0.4 SKAT 89 38 0.4 SKAT 89
-0 -0
1 1
+) ( 0.3 +) ( 0.3
p-n 0.2 p-n 0.2
m m
0.1 0.1
fin fin
n(m 0 n(m 0
s 10-1 1 10 102 s 10-1 1 10 102
Neutrino energy (GeV) Neutrino energy (GeV)
(e)νµn→µ−nπ+ published (f)νµn→µ−nπ+ reanalyzed
Fig.1:Thepublishedandextracted ANLandBNLdata arecomparedwithother measurementsof thethreepion
production channels [26–29]. All data is on hydrogen and deuterium targets, except for SKAT 1989 data which
wastakenonheavyfreon(CF Br).NotethatbothpublishedandreanalyzedANLandBNLdatashownherehave
3
no invariant mass cut in the event selection, whereas the other datasets have an invariant mass cut of W ≤2GeV
applied unless otherwise mentioned.
4
1.0≤E ≤1.1GeVwhereanupwardfluctuationofthe gle pion production are not considered further in this
ν
CCQE event rate leads to a decrease in the reanalyzed work3.
cross section. It should be noted that this problem is TheoriginalRein-SehgalmodelinReference[31]in-
also present in the published ANL cross section results cludes non-resonant single pion production as an addi-
because they used the measured CCQE event rate to tional resonance amplitude, while in GENIE, the non-
correct their flux prediction. This problem is not dealt resonant component is implemented as an extension of
withinthefitspresentedinthiswork,wesimplyusethe the deep inelastic scattering model. The non-resonant
statistical errors from the reanalysis without consider- (DIS) contribution to the GENIE single pion produc-
ingcorrelations,butitwillleadtoanincreaseintheχ2 tionmodeliscalculatedusingtheBodek-Yangparametriza-
ofthefits(becausetheχ2 penaltyforalargestatistical tion [36], with other relevant parameters described in
fluctuation is applied three times rather than once). detail in Reference [18]. Hadronization is described by
theAKGYmodel[37],whichusesKNOscaling[38]for
invariantmassesofW ≤2.3GeV,andPYTHIA[39]for
3 GENIE single pion production model invariant masses of W ≤3.0 GeV, with a smooth tran-
sistion between. The low-W KNO model is tuned to
ThesinglepionproductionmodelinGENIEisdescribed data from the Fermilab 15-foot bubble chamber exper-
in Reference [18] and does not change significantly be- iment[40],andthehigh-W PYTHIAmodelistunedto
tween the GENIE major versions 2.6.X and 2.8.X (X BEBCdata[41].WenotethatretuningofthePYTHIA
denotes a the minor version number) investigated in modelhasbeendiscussedelsewhere[42],althoughisnot
this study2. All processes simulated in GENIE use the considered here as it affects larger W values than are
Bodek-Ritchie RFG model to describe the initial state relevant for this study.
nucleonmomentumdistribution[30]forallnucleartar- AmajordifferencebetweentheGENIEversions2.6.X
gets, including deuterium. In GENIE, single pion pro- and 2.8.X is the change to the default Final State In-
duction is separated into resonant and non-resonant teraction (FSI) model, which is applied to all outgoing
terms,withinterferencetermsbetweenthetwoneglected. particles produced at the vertex for both the resonant
Theresonantcomponent(RES)isamodifiedversionof and non-resonant contributions to single pion produc-
the Rein-Sehgal (R-S) model [31]. In the original R-S tion. The default FSI model for both versions of GE-
model, the production and subsequent decay of 18 nu- NIEisthehAintranuclearcascademodel[18],whichis
cleon resonances with invariant masses W ≤ 2 GeV tuned to π+-56Fe and p-56Fe data, then extrapolated
are considered. In GENIE, only 16 resonances are in- to other targets based on A2/3 scaling (where A is the
cluded, based on the recommendation of the Particle atomicnumber).InGENIEv2.6.X,FSIeffectsareneg-
Data Group [32]. The cross section calculation has not ligiblefordeuterium,butthehAmodelwasretunedfor
beenmodifiedtoincludeleptonmassterms,buttheef- GENIE v2.8.X [43], which leads the deuterium FSI to
fectofleptonmasstermsonthephasespaceboundaries reduce the total cross section predictions for all chan-
is taken into account. The cross section is cut off at a nels relevant for this work by 20–30%. In this work we
tunable invariant mass value, which is W ≤ 1.7 GeV have ignored this difference and make the assumption
by default (and in this study). No in-medium modifi- that interactions on deuterium can be treated as in-
cations to resonances are considered, and interferences teractionsonquasi-freenucleonswhichareonlyloosely
betweenresonancesareneglectedinthecalculation.By boundtogether,andsoneglectFSIeffects.Low-Q2bins
default, the resonances decay isotropically in their cen- (Q2 <0.1GeV2)arenotincludedinthefittoavoidthe
terofmassframe.Ingeneral,thereisanadditionalcon- region where FSI effects are expected to have a signifi-
tribution to single pion production from coherent pion canteffectindeuterium.WenotethatinReference[44]
production processes, which are modeled (by default) a careful study of FSI effects for pion production inter-
in GENIE using the Rein-Sehgal coherent model [33] actions on deuterium was carried out. This work found
with lepton mass corrections included [34]. However, thatinteractionsbetweenthefinalstatenucleonssignif-
for the ANL and BNL channels considered here, the icantly modifies the cross section for the ν p→µ−pπ+
µ
selection criteria include requirements on the struck or
spectator nucleon, which effectively excludes coherent 3Additionally,diffractiveprocesses[35],wheretheneutrinointer-
actscoherentlywithafreenucleon(ratherthananentirenucleus)
pionproductionasthedeuteriumisnolongerboundin
can contribute to pion production. GENIE has an implementa-
the final state. As such, coherent contributions to sin- tion of the diffractive pion production model described in Ref-
erence [35], but this is not included by default in the GENIE
2We include both versions as a sanity check which ensures that model. Diffractive processes do not affect the main body of this
our results are consistent between the GENIE major versions workbecauseANLandBNLhaveadeuterium,ratherthanfree
usedbycurrentlyrunningexperiments. proton,target.
5
and ν n → µ−nπ+ channels, the most notable feature total prediction is equal to the measured rate summed
µ
being a suppression of the cross section for very for- over all three channels. The Q2 predictions shown are
ward pions. A more careful treatment of FSI based on therefore shape-only, but with the relative normaliza-
the work presented in Reference [44] would be an im- tionbetweenthedifferentpionproductionchannelspre-
provementtofutureiterationsofthiswork,butthecal- served5.Alldataconsideredhavenoinvariantmasscuts
culationdoesnotcurrentlypredicttheentirefinalstate applied and the event selection in GENIE is based on
of the interaction so is not ready to be implemented in the particles produced at the initial interaction vertex,
a generator. not those surviving GENIE’s FSI model as previously
In GENIE, there are a number of systematic pa- discussed. In Figures 2 and 3, the GENIE prediction is
rameters which can be varied to change the single pion also shown broken down into resonant (RES) and non-
production model [45]. These parameters are summa- resonant (DIS) contributions. Additionally, the domi-
rized in Table 1. Note that although GENIE allows nant ∆ contribution to the RES component is shown
the normalization of charged-current non-resonant sin- separatelyforreference.ThetotalGENIEpredictionis
glepionproductiononprotonsandneutronsseparately, the incoherent sum of the RES and DIS contributions,
we have grouped them here into a single category (la- where interference terms have been neglected. On each
belled“DIS”)intheabsenceofanyreasontotreatthem plotthe1σerrorbandproducedbycombiningthenom-
differently4. inalGENIEuncertaintiesonMRES,RESnormalization
A
The axial form factor used for resonant pion pro- and DIS normalization is also shown for comparison.
duction in GENIE is given by It is clear from Figures 2 and 3 that the nominal
GENIE prediction cannot describe all of the pion pro-
F (0)
F (Q2)= A , (1) duction channels well for the reanalyzed datasets. In
A (cid:18) (cid:19)2
Figure2,itisnoticeablethat,whilethemeasuredcross
1+ Q2
(MARES)2 sectionsforthesubdominantνµn→µ−pπ0 andνµn→
µ−nπ+ channels are similar, there are large differences
whereFA(0)≡ 53Z ≡gA =1.267,Z isarenomalization betweenthenominalGENIEpredictionsforthesechan-
factor for the axial-vector coupling constant of a quark nels. The non-resonant component of the GENIE pre-
obtainedfromdatainReference[46](alsoconsideredin diction, which contributes strongly to these channels,
Reference [47]) and MRES is the resonant axial mass, appears to be too large. It can be seen from Figure 3
A
availableasaparameterinGENIE’sreweightingframe- that the nominal GENIE prediction fails to describe
work. The normalization of the axial form factor is not the low-Q2 data well for some channels. We also note
availableintheGENIEreweightingframework(chang- that the GENIE uncertainties are larger than the data
ing its value requires events to be regenerated with a suggests, and may be reduced by tuning the GENIE
modified value of the “RS-Zeta” parameter), but a sim- model to the ANL and BNL data. These observations
ilar parameter for the overall resonant normalization motivate this work.
is available. Modifying F (0) is the more physically-
A
motivated alternative, but as modifying the resonant
4 Fitting the GENIE model
normalization is more convenient for users, we perform
two fits, each with one of these parameters modified.
In this section, the datasets described in Section 2 are
The nominal GENIE v2.8.2 cross sections for the
used to constrain the GENIE model introduced in Sec-
threesinglepionproductionchannelsconsideredinthis
tion 3. The χ2 statistic which is minimized is given in
work(ν p→µ−pπ+,ν n→µ−pπ0andν n→µ−nπ+)
µ µ µ
Section 4.1, and results are given in Section 4.2. A dis-
are shown as a function of the neutrino energy E in
ν
cussionofthegoodnessoffitisgiveninSection4.3.The
Figure 2, and compared with ANL and BNL data. The
MINUIT package [48] as implemented in the ROOT li-
nominalGENIEv2.8.2eventratepredictionsasafunc-
brary [49] is used to perform all fits.
tion of Q2, produced using the ANL and BNL fluxes,
The fits are performed separately for four GENIE
arecomparedseparatelytoANLandBNLdatainFig-
configurations. Both GENIE v2.6.2 and GENIE v2.8.2
ure3.TheGENIEpredictionfortheQ2 distributionsis
predictions are fit using all parameters available in the
normalized separately for ANL and BNL such that the
standardversionofGENIE(asdescribedinTable1).A
4NotealsothatwhentuningGENIEbasedonthiswork,itwould fit was performed to GENIE v2.8.2 where the normal-
be reasonable to consider these dials to be fully correlated with ization of the resonant axial form factor, F (0), is used
A
thenormalizationoftheneutralcurrentnon-resonantsinglepion
production prediction for interactions on both a target neutron 5Itisnotpossibletomakeacorrectlynormalizedeventratepre-
andtargetproton,andasfullycorrelatedwiththecorresponding diction because neither experiment gives sufficient information
antineutrinodials. onthenumberoftargetnucleonsinthebubblechamber.
6
s 150 s
nit GENIE Nom. nit 300 GENIE Nom.
u u
y y
ar GENIE Nom. RES ar GENIE Nom. RES
Arbitr 100 RES from D Arbitr 200 RES from D
GENIE Nom. DIS GENIE Nom. DIS
ANL BNL
50 100
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0
Q2 (GeV) Q2 (GeV)
(a)ANLνµp→µ−pπ+ (b)BNLνµp→µ−pπ+
s 40 s 150
nit GENIE Nom. nit GENIE Nom.
u u
y y
Arbitrar 30 GREESN IfEro Nmo Dm. RES Arbitrar 100 GREESN IfEro Nmo Dm. RES
GENIE Nom. DIS GENIE Nom. DIS
20
ANL BNL
50
10
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0
Q2 (GeV) Q2 (GeV)
(c)ANLνµn→µ−pπ0 (d)BNLνµn→µ−pπ0
s 40 s 150
nit GENIE Nom. nit GENIE Nom.
u u
y y
Arbitrar 30 GREESN IfEro Nmo Dm. RES Arbitrar 100 GREESN IfEro Nmo Dm. RES
GENIE Nom. DIS GENIE Nom. DIS
20
ANL BNL
50
10
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0
Q2 (GeV) Q2 (GeV)
(e)ANLνµn→µ−nπ+ (f)BNLνµn→µ−nπ+
Fig. 3: The nominal GENIE prediction is shown as a function of Q2 for the three single pion production channels
of interest, and is compared separately to the ANL and BNL data. The total prediction is broken down into the
resonant(RES)andnon-resonant(DIS)contributions,andadditionally,the∆-contributiontotheREScomponent
is shown. The error band shown on the total GENIE prediction shows the 1σ error bands for all GENIE default
parameters given in Table 1 combined in quadrature (note that F (0) is not a default GENIE parameter so is not
A
included in the error band).
7
Parameter GENIE value Parameter name
Resonant axial mass (MRES) 1.12 ± 0.22 GeV [47] GXSec_MaCCRES
A
Resonant normalization
100 ± 20% GXSec_NormCCRES
(RES norm.)
Non-resonant normalization GXSec_RvnCC1pi
100 ± 50%
(DIS norm.) GXSec_RvpCC1pi
Normalization of the axial
100% (no GENIE uncertainty) N/A
form factor (F (0))
A
Table1:VariableparametersintheGENIEsinglepionproductionmodel[45].Thenormalizationoftheaxialform
factor is not a variable parameter in GENIE currently, but is varied in this work as described in the text.
asafitparameterinsteadoftheresonantnormalization, inthefitandtheinnersummationsareovertheN bins
as motivated in Section 3. Finally, a fit was performed ofeachdataset.xalsocontainsnormalizationtermsfor
to GENIE v2.8.2 without the resonant normalization ANL and BNL which affect the Q2 datasets only. As
or F (0) to investigate the effect that correlations be- previouslyremarked,theQ2 datasetsareshape-onlyin
A
tween the axial mass (which has a strong effect on the thefit,buttherelativenormalizationbetweenthethree
normalization of the cross section) and the normaliza- pionproductionmodesispreservedseparatelyforANL
tion parameters have on the results. and BNL.
Note that this statistic is appropriate for minimiza-
tion, but χ2/N is not strictly correct as measure of
DOF
4.1 χ2 definition thegoodnessoffitbecausethePoisson-likelihoodterms
contribute constant terms to the χ2. A more rigorous
No information about systematic uncertainties, corre-
measureofthegoodnessoffitisdiscussedinSection4.3.
lationswithindatasetsorcorrelationsbetweendatasets
is available, so only statistical errors are considered for
all datasets, and the function to be minimized can be
expressed as a sum over the datasets included in the
fit. This is reasonable as the statistical uncertainties
are large. Additionally, the datasets used are efficiency
corrected by the experiments, but are not unfolded, so
the treatment of detector effects is likely to be inade-
quate6. For these reasons, any measure of goodness of
fit should be treated as approximate.
APoisson-likelihoodstatisticisusedforthedatasets
asafunctionofQ2 becausemanyofthehigherQ2 bins
have low event rates. Note that for Q2 datasets, the
sum is over the N bins with Q2 ≥ 0.1 GeV2.
(cid:40) N (cid:20) (cid:21)(cid:41)
χ2 = (cid:88) 2(cid:88) µ (x)−n +n ln ni
i i i µ (x)
i
Q2datasets i=1
+ (cid:88) (cid:40)(cid:88)N [ni−µi(x)]2(cid:41) (2)
σ2
Eν datasets i=1 i
where n and µ (x) are the measured and predicted
i i
number of events in the ith bin, σ is the statistical
i
errorontheithbin,xarethemodelparametersvaried
6Unfortunately,insufficientinformationhasbeenpublishedtodo
amoresophisticatedanalysis,sothisisacaveatwhichappliesto
allanalyseswhichuseANLorBNLdata.
8
4.2 Results
2)m GENIE Nom.
c
-38 0 GENIE Nom. RES Two fake data studies were performed to validate the
+) (1 1.0 RGEESN IfEro Nmo Dm. DIS fifotrtear.llFfiorsutrlyG, AEsNimIEovc[o5n0fi]gfuakraetdioantsa ficotsnswideererepdr.oTduhceesde
pp
- ANL
m provide basic validation that the fitter found the cor-
fip 0.5 BNL rectminimumandgivetheexpectedsizeoftheparam-
m
n( eter uncertainties, which can be used to validate the
s
fit results. Secondly, pull studies were performed for all
GENIEconfigurationstocheckthattheteststatisticis
0.0
0 1 2 3 4 an unbiased estimator of central values and uncertain-
Neutrino energy (GeV) tiesfortheparametersvariedinthefits.Nobiaseswere
(a)νµp→µ−pπ+ observed.
The best fit results to all twelve datasets are shown
2)m 0.8 GENIE Nom. inTable2forthefourGENIEconfigurationsconsidered
c
-380 ) (10 0.6 GRGEEESNN IIfEEro NNmoo Dmm.. RDEISS ififnatkstehadirsaewtcaoosrntksu.isdTtyeh.neTtphwaeirtnahomrtemhtoeasrleizuapntrcieoedrnitcaotifendtthiebesyngotihnvee-rneAsbsoyinmathonvet
pp
- ANL
m 0.4 background was reduced significantly in all fits, as was
fin BNL expected given the nominal model comparisons shown
n(m in Section 3. The resonant axial mass, MRES, was also
s 0.2 A
reducedfromtheGENIEnominalvalueofMRES=1.12
A
GeV in all fits, although we note that there is a strong
0.00 1 2 3 4 anticorrelationbetweenMRES andtheRESnormaliza-
A
Neutrino energy (GeV) tion, and between MRES and F (0) (as can be seen in
A A
(b)νµn→µ−pπ0 Figure 4). The GENIE v2.8.2 fits were also repeated
using GENIE’s free nucleon cross sections in order to
2)m 0.8 GENIE Nom. ensure that the results are not biased by GENIE’s ini-
c
-38+ ) (10 0.6 GRGEEESNN IIfEEro NNmoo Dmm.. RDEISS fitwitaesrl.estcaotnesinsutecnletartomwoidtehli.nIt1σwafsorfoaulnldpatrhaamtettheersreinsulatlsl
pn
m- 0.4 ANL The GENIE v2.6.2 and GENIE v2.8.2 (RES) re-
fin BNL sults, with the same parameters available in the stan-
m
n( dard version of GENIE (as described in Table 1) give
0.2
s
consistentresults.Thefitwhichusesthenormalization
of the axial form factor, F (0), as a fit parameter is
A
0.0
0 1 2 3 4 consistent with the other fits, although as FA(0) has a
Neutrino energy (GeV) smallQ2 dependence,thecorrelationwithMRES isdif-
A
(c)νµn→µ−nπ+ ferent and the value for MARES is less suppressed than
in fits with RES normalization free (which has no Q2
Fig. 2: The nominal GENIE prediction is shown as a
dependence).
functionofE forthethreesinglepionproductionchan-
ν ThebestfitdistributionsfortheGENIEv2.8.2(RES)
nels of interest, and is compared to the corrected ANL fitareshownfortheν p→µ−pπ+ distributionsinFig-
µ
andBNLdata.Thetotalpredictionisbrokendowninto ure 5, for the ν n → µ−pπ0 distributions in Figure 6,
µ
the resonant (RES) and non-resonant (DIS) contribu- and for the ν n → µ−nπ+ distributions in Figure 7.
µ
tions, and additionally, the ∆-contribution to the RES Data points with Q2 ≤ 0.1 GeV2 are shown but not
component is shown. The error band shown on the to- included in the χ2 calculation.
tal GENIE prediction shows the 1σ error bands for all
default GENIE parameters given in Table 1 combined
in quadrature (note that F (0) is not a default GENIE 4.3 Goodness of fit
A
parameter so is not included in the error band).
As has been previously remarked, the χ2/N is not
DOF
anappropriatemeasureofthegoodnessoffitforEqua-
tion 2 as it involves Poisson-likelihood terms with low-
9
GENIE v2.6.2 (RES) v2.8.2 (RES) v2.8.2 (F (0)) v2.8.2 (no norm.)
A
χ2 558.8 615.3 615.3 615.3
nom
χ2 311.2 324.4 327.3 330.3
min
N 157 157 157 158
DOF
MRES (GeV) 0.94 ± 0.05 0.94 ± 0.05 1.00 ± 0.04 1.04 ± 0.02
A
DIS norm. (%) 46 ± 4 43 ± 4 43 ± 4 42 ± 4
RES norm. (%) 115 ± 7 115 ± 7 – –
F (0) norm. (%) – – 107 ± 4 –
A
Table 2: Best fit results for the four GENIE configurations used in this analysis.
nits Best Fit (c 2 = 10.6) nits 1.5 Best Fit (c 2 = 5.6)
u u
y Best Fit DIS y Best Fit DIS
bitrar 1.5 GENIE Nom. (c 2 = 16.3) bitrar GENIE Nom. (c 2 = 15.3)
Ar GENIE Nom. DIS Ar 1.0 GENIE Nom. DIS
Data ANL Data BNL
1.0
0.5
0.5
0.0 0.0
0 2 4 6 0 2 4 6
Neutrino energy (GeV) Neutrino energy (GeV)
(a)ANLEν (b)BNLEν
nits Best Fit (c 2 = 9.5) nits Best Fit (c 2 = 23.0)
u 100 u 300
y Best Fit DIS y Best Fit DIS
bitrar GENIE Nom. (c 2 = 6.6) bitrar GENIE Nom. (c 2 = 15.3)
Ar GENIE Nom. DIS Ar GENIE Nom. DIS
200
Data ANL Data BNL
50
100
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0
Q2 (GeV) Q2 (GeV)
(c)ANLQ2 (d)BNLQ2
Fig. 5: Best fit distributions and post-fit uncertainties for the four ν p→µ−pπ+ datasets included in the GENIE
µ
v2.8.2(RES)fit.Thenominalpredictionisshownforreference,andtheχ2 contributionfromeachdatasetisgiven
in the legend for both the nominal and best fit distributions. The nominal and best fit DIS contribution to the
total GENIE prediction (RES+DIS) are also shown for reference.
statisticsbins.InFigure8,theexpectedχ2 distribution grating the distribution to the right of any given χ2
min
has been produced by making 100,000 toy experiments fit value as the χ2 distribution is independent of the
in which a fake dataset has been produced with statis- fit type. Figure 8 shows the sampling distribution from
tical errors thrown for all twelve datasets included in thismethod,withtheactualfitvalueforGENIEv2.8.2
the fit. The χ2 is calculated between each toy experi- (F (0)),whichhasap-value(cid:28)10−4,indicatingapoor
A
ment and the nominal data (without thrown statistical fit between data and the model, even after the fit pro-
errors).Thep-valueofanyfitcanbecalculatedbyinte- cedure.
10
2)m Best Fit (c 2 = 16.1) 2)m Best Fit (c 2 = 35.6)
-38 c10 0.3 BGeEsNt IFEit NDoISm. (c 2 = 19.8) -38 c10 BGeEsNt IFEit NDoISm. (c 2 = 31.1)
0) ( GENIE Nom. DIS 0) ( 0.4 GENIE Nom. DIS
p-p 0.2 Data ANL p-p Data BNL
m m
fin fin
0.2
m m
n( 0.1 n(
s s
0.0 0.0
0.0 0.5 1.0 1.5 0 1 2 3
Neutrino energy (GeV) Neutrino energy (GeV)
(a)ANLEν (b)BNLEν
0 Best Fit (c 2 = 21.3) 0 150 Best Fit (c 2 = 46.4)
pp pp
- 30 -
m Best Fit DIS m Best Fit DIS
fin GENIE Nom. (c 2 = 24.5) fin GENIE Nom. (c 2 = 46.7)
m GENIE Nom. DIS m100 GENIE Nom. DIS
n2 Q 20 Data ANL n2 Q Data BNL
d d
/ /
sd sd
50
10
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0
Q2 (GeV) Q2 (GeV)
(c)ANLQ2 (d)BNLQ2
Fig. 6: Best fit results and post-fit uncertainties for the four ν n→µ−pπ0 datasets included in the GENIE v2.8.2
µ
(RES) fit. The nominal prediction is shown for reference, and the χ2 contribution from each dataset is given in
the legend for both the nominal and best fit distributions. The nominal and best fit DIS contribution to the total
GENIE prediction (RES+DIS) are also shown for reference.
ν p→µ−pπ+ ν n→µ−pπ0 ν n→µ−nπ+
µ µ µ
Dataset ANL BNL ANL BNL ANL BNL
E Q2 E Q2 E Q2 E Q2 E Q2 E Q2
ν ν ν ν ν ν
Nominal χ2 16.3 6.6 15.3 15.3 19.8 24.5 31.1 46.7 45.3 44.0 265.6 84.6
Best fit χ2 10.6 9.5 5.6 23.0 16.1 21.3 35.6 46.4 33.0 32.2 59.6 31.3
N 7 18 7 19 7 18 10 19 7 18 11 19
DOF
Table 3: Contributions to the nominal χ2 for GENIE v2.8.2 and to the best fit χ2 for the GENIE v2.8.2 (RES)
min
fit from each of the twelve datasets included in the fit.
In Table 3, the contribution that each of the twelve tisticalerrors,whicharedominant,buttherearesignif-
datasets makes to the nominal and best fit χ2 values is icant normalization uncertainties due to detector cor-
given for the GENIE v2.8.2 (RES) fit. The N con- rections and background subtractions which are not
DOF
tributed by each dataset is also shown for comparison. included. These corrections are also likely to have an
It is clear that there is a disproportionate contribution effect on the shape of the distributions, but it is not
fromthereanalyzedE -dependentdatasetsforthesub- possibletocalculatemeaningfulshape-uncertaintiesfor
ν
dominantchannels(ν n→µ−pπ0 andν n→µ−nπ+). theseeffects(noraretheyincludedinthepublishedre-
µ µ
Theuncertaintyonthesedistributionsonlyincludesta- sults from ANL or BNL). It should also be noted that