Electroweak Single Pion Production and Form Factors of the \Delta(1232) Resonance

0.16 MB·English

by  Jakub Zmuda

Electroweak Single Pion Production and Form Factors of the
D
(1232) Resonance
Jakub Z˙muda, Krzysztof M. Graczyk
5
InstituteforTheoreticalPhysics,UniversityofWrocław,pl.M.Borna9,50-204,Wrocław,Poland
1
0
2 Abstract. Weextendandreviewouranalysisofthenucleon D (1232)transitionelectroweakformfactorsfromRef.[1].
r NewfitoftheD (1232)vectorformfactorstoelectron-protons→catteringF2structurefunctionisintroducedaswell,leadingto
a resultsdifferentfromthepopularparametrizationofRef.[2].Aclearmodeldependenceoftheextractedparametersemerges.
M Fittoneutrinoscatteringdataisperformedinallavailableisospinchannels.TheresultingaxialmassisMAD =0.85+00..0089(GeV)
andC5A(0)=1.10−+00..1145.ThelattervalueisinaccordancewithGoldberger-Treimanrelationaslongasthedeuteron−effectsare
6 included.
2
Keywords: Pion,electron,neutrino,deuteron
PACS: 13.15.+g,13.60.Le
]
h
p
- INTRODUCTION
p
e
h Theproblemofsinglepionneutrinoproduction(SPP)hasbeenstudiedformanydecades.Itsimportancehasbecome
[ clearwiththedevelopmentofneutrinoacceleratorexperiments,suchasMINOS[3],T2K[4],NOvA[5],MiniBooNE

[6],andLBNE[7].Inthefew-GeVenergyregionchartacteristicfortheabovementionedexperimentsthisinteraction
4
channelcontributesalargefractionofthetotalcrosssection.Oneestimates,thatforanisoscalartargetandneutrino
v
energyofaround1GeVSPPaccountsforabout1/3oftheinteractions.
6
8 These SPP events give rise to the background in measurements of quasi-elastic neutrino scattering off nuclear
0 targets if subsequent pion absorptionoccurs. For experimentsaiming at electron neutrino appearancemeasurement
3 neutral currentp 0 productionprocess adds to the backgroundin water Cherenkovdetectors. Correct understanding
0
andmodelingofthe cross-sectionsforthe SPPis crucialforprecise extractionof neutrinooscillationparametersin
.
1 longbaselineexperiments.
0 TheoreticalmodellingoftheSPPprocessesonnucleartargetsisbiasedbysystematicerrorscomingfromnuclear
5 model uncertainties. They are driven by the strong nature of hadron interactions inside the nucleus, which do not
1
allow for a feasable, exact solution of the problem. Experimental measurements also suffer from these efects. An
:
v apparenttensionbetweentheMiniBooNEandveryrecentMINERn ASPPdataon(mostly)carbontarget(Ref.[8,9])
i is one of the key examples. For the purpose of analysis of Nucleon (N) to D (1232)resonance transition vertex one
X
desiresmeasurementsoftheneutrino-productiononfreeoralmostfreetargets.Suchdataexistonlyfor 30yearsold
r ∼
a ArgonneNationalLaboratory(ANL)[10,11]andBrookhavenNationalLaboratory(BNL)[12,13]bubblechamber
experiments,wheredeuteronandhydrogentargetswereutilized.Inthiscaseonemayhopetoreducethemany-body
biasinareasonablemannerwithasimpletheoreticalansatz[14].
One can not understand the neutrino SPP data without introducing an appropriate nonresonant background, see
Ref. [15]. More recent studies of weak SPP fit the N D transition axial form factors utilizing only the neutrino-
protonchanneln m +p m −+p ++p[16,17, 18,19→, 20,21].Simpletotalcrosssectionratioanalysisshows,that
→
the backgroundcontribution is much larger in neutrino-neutronchannels. The neutrino-protonSPP channel can be
described well within a model that contains the D (1232) resonance contribution only, see e. g. Ref. [22]. Thus a
quantitative, statistical, validation of any pion neutrinoproductionmodel should be done using all available isospin
channels,whereone getsmore informationaboutthe backgroundcontribution.In Ref. [22] a consistentfit forboth
ANL and BNL data sets with deuteron effects included yielded CA(0)=1.19 0.08 and M =0.94 0.03 GeV.
5 ± A ±
The attempt to extract the leadingCA(Q2) N D form factor parameters in a model containing both nonresonant
5 →
backgroundand deuteroneffects has been done in Ref. [19]. The results gave the value ofCA(0)=1.00 0.11far
5 ±
fromtheGoldberger-TreimanrelationestimateofCA(0) 1.15[23,24].Lackofthedeuteroneffectspushesthefit
5 ≈
evenfurtherawayfromtheoreticalprediction,givingCA(0)=0.867 0.075inRef.[17].Fromtheabovementioned
5 ±
modelsonlythoseinRefs.[16,18]havebeendirectlytestedagainsttheelectroproductiondata.Authorsof[17,19,20]
usevectorformfactorparametrizationfromRef.[2],basedontheMAIDanalysis[25].TheD formfactorshavebeen
fitteddirectlytotheD helicityamplitudesfromMAID.ThustheapproachfromRef.[2]reliedonlyontheD resonance
excitation without any nonresonant background. The problem is that the resonance helicity amplitudes extraction
procedureismodel-dependent.OnehastomakesomeassumptionshowtoseparateD andbackgroundcontributions
fromthedataandD –backgroundinterferenceeffectsarestrong.Weshowlaterthatifoneusesform-factorsobtained
within one description of pion electroproductionin a model with different physical componentsthe resulting cross
sectionsmaybecomeimprecise.
Theabovementionedcaveatsofpreviousanalyseshavemotivatedustoproposeanimprovedapproach.Weadapt
and develop the statistical framework of Ref. [22] in order to fit both vector and axial form factors of the D (1232)
resonance.Weuseinclusiveelectron-protonscatteringdatafortheelectromagneticinteractionintheD (1232)region,
addinganewfitoftheformfactorsfromRef.[2].ThedeuteronbubblechamberdataofANLandBNLexperiments
areusedfortheweakinteraction.Forthisanalysiswearethefirstauthorstoincorporatetheneutronchannels.Inthis
mannerweincludethedatasets,thatareverysensitivetothenonresonantbackground.
FORMALISM
Intheneutrinosinglepionproductionofffreenucleontargetsonedistinguishesthreeisospinchannels:
n m (l)+p(p) m −(l′)+p +(k)+p(p′) (1)
→
n m (l)+n(p) m −(l′)+p 0(k)+p(p′) (2)
→
n m (l)+n(p) m −(l′)+p +(k)+n(p′) (3)
→
withl,l, p, p andkbeingtheneutrino,muon,initialnucleon,finalnucleonandpionfourmomentarespectively.The
′ ′
definitionoffourmomentumtransferisfollowing:
q=l l =p +k p, Q2= q2, qm =(q0,q) (4)
′ ′
− − −
andthesquareofhadronicinvariantmassis:
W2=(p+q)2=(p +k)2. (5)
′
mn
Metricconventiong =diag(+, , , )isusedthroughoutthispaper.
− − −
Forthepionelectroproductionweareinterestedinprotontargetreactions;
e (l)+p(p) e (l )+p +(k)+n(p) (6)
− − ′ ′
→
e (l)+p(p) e (l )+p 0(k)+p(p). (7)
− − ′ ′
→
Theprotoninteractionchannel(1)isdominatedbythe intermediateD ++ resonanceexcitation,whichmakesit very
sensitive to the properties of this resonance. Neutron channels (Eqs. (2) and (3)) contain a large contribution of
nonresonantpionproduction,thustheypresentmorechallengesfortheorists.Theyare crucialtoverificationofany
consistentSPPmodel.
N D (1232)transition
→
We treatthe D (1232)resonanceexcitationwithinthe isobarframework.The mostgeneralformof positiveparity
spin-3 particleselectroweakexcitationvertexcanbeexpressedas:
2
G am (p,q)= Vam +Aam g 5 (8)
3/2 3/2
h i
where
Vam = C3V(Q2)(gam (cid:30)q qa g m )+C4V(Q2)(gam q (p+q) qa (p+q)m )+C5V(Q2)(gam q p qa pm )+gam CV(Q2)
3/2 M − M2 · − M2 · − 6
(9)
Aam = C3A(Q2)(gam (cid:30)q qa g m )+C4A(Q2)(gam q (p+q) qa (p+q)m )+CA(Q2)gam +C6A(Q2)qa qm g 5. (10)
3/2 M − M2 · − 5 M2
(cid:20) (cid:21)
ArelevantinformationabouttheinnerstructureoftheD (1232)resonanceiscontainedinasetofvectorandaxial
formfactorsCV,A.InthispapertheyareassumedtobefunctionsofQ2only(withtheexceptionofCV whichdepends
j 4
alsoonW).
Vectorcontribution
Theconservedvectorcurrent(CVC)hypothesisgivestherelationbetweenweakandelectromagneticvectorform
factors. In the case of D (1232) resonance and hereby used conventionboth sets are exactly the same. The size and
excellentaccuracyoftheelectromagneticdatasetallowsforanintroductionofmultiplefitparameters.
We explore two models of vector form factors. The first parametrization, refered to as “Model I”, has the same
functionalform,asinRef.[2]:
CV(0) 1
CV(Q2) = 3 (11)
3 Q2 2 Q2
1+D 1+A
(cid:18) ·Mv2(cid:19) Mv2
CV(0) 1
CV(Q2) = 4 (12)
4 Q2 2 Q2
1+D 1+A
(cid:18) · Mv2(cid:19) Mv2
CV(0) 1
CV(Q2) = 5 . (13)
5 Q2 2 Q2
1+D 1+B
(cid:18) ·Mv2(cid:19) Mv2
IntheaboveequationsM =0.84GeVisthestandardvectormass.Everythingelseistreatedasafitparameter.
V
We also propose our own model of electromagnetic form factors. We assume that the N D transition form
factorshavethesamelargeQ2 behaviourastheelectromagneticelasticnucleonformfactors.T→hereexisttheoretical
arguments[26]suggestingthatatQ2 ¥ thenucleonformfactorsfalldownas1/Q4.Followingtheseassumptions
→
we adopt appropriate Padé type parametrization used previously to parametrize the electromagnetic form factors
of the nucleon [27]. In this manner we allow for a deviation from the SU(6)-symmetry quark model relations
CV(Q2)= (M/W)CV(Q2)andCV =0betweentheformfactors[28].Finally,weassumethedipolerepresentation
4 − 3 5
ofCV(Q2)toreducethenumberofparameters.Altogether,ourparametrizationhasthefollowingform:
5
CV(0)
CV(Q2) = 3 (1+K Q2) (14)
3 1+AQ2+BQ4+CQ6 · 1
M 1+K Q2
CV(Q2) = pCV(Q2) 2 (15)
4 −W 3 ·1+K Q2
1
CV(0)
CV(Q2) = 5 . (16)
5 Q2 2
1+D
M2
(cid:18) V(cid:19)
This parametrization reproducesquark model relation betweenCV andCV at Q2 =0. It is also allows for nonzero
3 4
valueofS helicityamplitude.Wecallit“ModelII”.
1/2
Axialcontribution
HeretheleadingcontributioncomesfromCA(Q2)whichisananalogueoftheisovectornucleonaxialformfactor.
5
Partiallyconservedaxialcurrent(PCAC)hypothesisrelatesthevalueofCA(0)withthestrongcouplingconstant f
5 ∗
throughoff-diagonalGoldberger-Treimanrelation[23,24]:
f
CA(0)= ∗ 1.15, (17)
5 √2 ≈
butwewilltreatCA(0)asafreeparameter.Weassume,thatCA hasadipoleQ2 dependence:
5 5
CA(0)
CA(Q2)= 5 (18)
5 Q2 2
1+
M2
(cid:18) AD (cid:19)
TheaxialmassparameterMAD canberelatedto“resonanceaxialchargeradius”.Itisalsosubjecttofit,butweexpect
ittobeoftheorderof1GeV.
TheCA formfactorisananalogueofthenucleoninducedpseudoscalarformfactor.OnecanusePCACtorelateit
6
toCA:
5
M2
CA(Q2)= CA(Q2), (19)
6 m2p +Q2 5
where mp is average pion mass. The CA(Q2) is the axial counterpart of the very small electric quadrupole (E2)
3
transition form factor. Unfortunately, bubble chamber data set is too inacurate to precisely measure its effect. Due
toexpectedsimilaritiesbetweenD andnucleonpropertieswesetCA=0.FortheCA weusetheAdlermodelrelation
3 4
[29]:
CA(Q2)= CA(Q2)/4. (20)
4 − 5
InthiswaytheaxialcontributionisfullydeterminedbyCA(Q2).Altogethertherearetwofreeparameters:CA(0)and
5 5
MAD . If therewere enoughexperimentaldata onecoulddropthe Adlerrelation andtreatC4A(Q2) asan independent
formfactororevendetermine,whetherCA(Q2)hasnonzerovalue.However,theANLandBNLexperimentaldatado
3
nothavesufficientstatisticseventoobtainseparatefitsofCAandCA[30],seealsothediscussioninRef.[19].
5 4
Cross section
a) b) c)
l q l′ l q l′ l q l′
N(p) ∆(p+q) N′(p′) N(p) ∆(p−k) N′(p′) N(p) N˜(p+q) N′(p′)
π(k) π(k) π(k)
d) e) f) g)
l l l q l l q l
q ′ ′ l l′ ′
q
π˜(k q) π(k) π˜(q)
−
N(p) N˜(p−k) N′(p′) N(p) N′(p′) N(p) N′(p′) N(p) N′(p′)
π(k) π(k)
π(k)
FIGURE1. (Coloronline)Basicpionproductiondiagramsfrom[17]:a)Deltapole(D P),b)crossedDeltapole(CD P),c)nucleon
pole(NP),d)crossednucleonpole(CNP),e)pion-in-flight(PIF),f)contactterm(CT)andg)pionpole(PP).
WeexpresstheinclusivedoubledifferentialSPPcrosssectionforneutrinoscatteringoffnucleonsatrestas:
d2s 1 W d3k
dQ2dW = 32p E2G2Fcos2q CM2Z (2p )32Ep (k)E(p′)Lmn Amn d (E(p′)+Ep (k)−M−q0)
Lmn = lm l′n +ln l′m gmn l l′+ie mnab la′ lb
− ·
Amn = (cid:229) p N′|jcmc(0)|N p N′|jcnc(0)|N ∗. (21)
spins
(cid:10) (cid:11)(cid:10) (cid:11)
whereE istheincidentneutrinoenergy,Mistheaveragednucleonmass,Ep (k)andE(p′)arethefinalstatepionand
nucleonenergies,G =1.1664 10 11MeV 2istheFermiconstant,Lmn -theleptonicandAmn -thehadronictensors.
F − −
TheCabibboangle,cos(q )=0·.974,wasfactoredoutoftheweakchargedcurrentdefinition.
C
The informationaboutdynamicsof SPP is contained in transition matrix elements, p N jm (0) N , between an
′ cc
initialnucleonstate N andafinalnucleon-pionstate p N .
′
| i | i (cid:10) (cid:12) (cid:12) (cid:11)
In the model of this paper the dynamics of SPP process is defined by a set of Feynman(cid:12)diagra(cid:12)ms (Fig. 1) with
vertices determined by the effective chiral field theory. They are discussed in Ref. [17], where one can find exact
m
expressionsfor j . The same set of diagramsdescribes also pion electroproduction,with the exceptionof the pion
polediagram,whichispurelyaxial.Wecallthisapproach"HNVmodel"afterthenamesoftheauthorsofRef.[17].
Deuteroneffects
In this paper we consider a deuteron model based on phenomenologicalnucleon momentum distribution, f(p),
taken from the Paris potential [31]. It is assumed that the spectator nucleon does not participate in the interaction
andthattherearenofinalstateinteractions(FSI).ThisassumptionisbasedontheresultsofRef.[32]whereforthe
quasielasticscatteringcaseFSI wereshownto benegiligibleaslongastheneutrinoenergyis largerthan500MeV.
Theveryrecentstudy(newer,thantheherebyanalysis)ofRef.[33]provedtheFSItobeimportantinthenp +channel,
wheretheyleadtoasubstantialreductionofthecrosssectionforforward-goingpions.Furtherstudiesregardingthe
impactof FSI effectsare needed,also in the ANL andBNL experimentaldata analysiswherethe eventselection is
basedonspectatorapproachaswell.Weintroducealsotheeffectivebindingenergy:
B(p)=2E(p) M , (22)
D
−
whereM isdeuteronmass.Theexpressionforthecrosssectionbecomes:
D
dQd2sdW =Z d3pfv(repl.)1G62Fp cEonsE2((QpC)|)J|lll′′′||Z (2p )d323Ekp (k)Z (2p )d332pE′(p′)Lmn Amn (p,q˜,k)d 4(p+q˜−k−p′). (23)
withq˜m =(q0 B(p),~q)andvrel.= (l p)2/EE(p).WealsodefineaJacobianJ:
− ·
p ¶ Q2 ¶ Q2
J = Det ¶ cos(Q ) ¶ q0 (24)
¶W ¶W
¶ cos(Q ) ¶ q0 !
whoseexplicitformiscomplicatedbecausetheinvariantmassW dependsbothontheenergytransferq0andthelepton
scatteringangleQ .
RESULTS OF THEANALYSIS
Ourmaingoalistohaveareliablemodelofweakpionproduction.BecausetheneutrinoSPPdataaresufficientonly
toobtaininformationaboutleadingaxialcouplingoftheD (1232)resonance,weassumethattheextractionofvector
andaxialformfactorscanbedoneindependentlyusingfirstrespectiveelectronscatteringandthenneutrinoSPPdata.
Inthenextparagraphswedescribedetailsofourprocedure.
VectorContribution to WeakSPP andElectroproduction
Our aim is (due to a poor quality of the neutrino SPP data) to reproduce correctly only the most important
characteristics of the neutrino SPP reactions. These include overall cross sections and distributions in Q2. Detailed
analysis of the electroproduction data bases on pion angular distributions. Such a task is beyond the scope of this
paper.
We use the informationcontained in electron-protonF data from [34]. We include 37 separate series of F data
2 2
pointsfromthelowestvalueofQ2(0.225GeV2)upto2.025GeV2.ThisQ2rangeoverlapswiththeoneinANLdata.
Thedatadesribetheinclusivestructurefunction,thuswelimitourfittovaluesofinvariantmassW uptoMp+2mp .
Beyond that value the experimental data contain two pion production and then more inelastic channels. With this
limitationforQ2 2.025andW <Mp+2mp wearestillleftwith603datapoints.
Inordertoensu≤rethattheresultswillreproducewellthedataattheD (1232)peakweexpandedourfittohighervalue
ofinvariantmassW =1.27GeV.TherearenoexclusiveelectronSPPdataintheregionW (M+2mp ,1.27GeV).
∈
Thus we chose to add to our fit a term in which MAID 2007 model predictions are taken as 228 fake data points
with errors identicalto respective Osipenko et al. [34] points. We could not apply the MAID model directly in our
fits sincethe exactformulasfortheirSPP amplitudeshaveneverbeenpublished.Theseadditionalpointshavebeen
generatedusingtheon-lineversionofMAID(http://wwwkph.kph.uni-mainz.de/MAID//).Wehavealso
includedtheinformationaboutMAID2007modelhelicityamplitudes.Thecaveatisthatthepionelectroproduction
experimental results contain both resonant and nonresonant contributions (see e.g. Ref. [35] ). Thus the extracted
helicity amplitudesdependon how one definesthe "Delta" and "background".The HNV modeldifferswith MAID
in the treatment of both. One can not expect the resulting helicity amplitudesto be the same. From that reason the
informationabouthelicityamplitudeshasbeengivenalargeadhocerrorassumptioninourestimator.
Results
The best fit results of our vector form factor parametrization given by Eqs. (11-13) and Eqs. (14-16) are shown
in Table 1. We also present there the values from Ref. [2] in order to compare directly with our model I. In both
modelsthebestfitvalueofCV(0)isclosetotheonefromRef.[2]andwegetaclearbeyond-dipoleQ2 dependence
3
ofCV(Q2) andCV(Q2). Surprisingly, the Q2 dependence ofCV(Q2) is exactly dipole (1+Q2/M2) 2 in model II.
3 4 5 V −
Most importantly, we have shown, that extracted form factors are model-dependent. This clearly follows from the
differenceof best-fitparametersbetweenourmodelI and their counterpartfromRef. [2].One can see that, besides
thesimilarityintheleadingformfactorvalueCV(0),bothfitsdifferbylarge.
3
TABLE1. BestfitcoefficientsforvectorformfactorsgivenbyEqs.(11-13)(“ModelI”)
andEqs.(14-16)(“Model II”).Wedonotreport1s errorsbecauseofhybridcharacterof
ourestimator,seeexplanationsinthetext.
C3V(0) C4V(0) A B C K1 K2 C5V(0) D
Ref.[2] 2.13 -1.51 0.25 1.289 - - - 0.48 1.00
MODELI 2.00 -6.77 0.68 1.40 - - - 5.95 1.15
MODELII 2.10 - 4.73 -0.39 5.59 0.13 1.68 0.62 1.00
Fig.2showsthatqualitativelyintheregionbelowtwopionproductionthresholdourfitreproducesthedatarather
well.OurformfactorsleadtobetteragreementwiththeFp electronscatteringdatathantheformfactorsconsidered
2
inRef.[2].ThesametrendisclearlyseeninFig.3,whereourbestfitresultsarecomparedtotheinclusiveelectron-
proton scattering cross section data. Inspection of Fig. 2 shows that biggest disagreement with data is exhibited in
regionoflowW.Ourfitsaregoingtobeusedintheanalysisofneutrinoscatteringdataandsomediscrepancyatlow
W isofnopracticalimportance.
Anotherconclusionis that each physicalmodelof single pion productionneedsits own separate resonance form
factor analysis. Any change of description of one of the elements such as D propagator and width, background
amplitudes, unitarity constraint etc. will affect the results. In other words, the HNV modelshould be used together
withvectorformfactorsfitedusingtheHNVmodelinordertoincreasetheaccuracyofitspredictions.
Becausebothproposedformfactorsetsleadtoverysimilarresults,wechoosetouse“ModelII”intheaxialfits.
AxialContribution to weakSPP:fits to bubble chamber data
We consider a statistical framework,proposedin Ref. [22]. ANL used a neutrino beam with mean energybelow
1GeVandalargefluxnormalizationuncertaintyD p 20%thatwasnotincludedinthepublishedds /dQ2cross
ANL
∼
sectionforthereactioninEq.(1)[22].ANLreportedthedatawiththeinvariantmasscutW <1.4GeV,whichallows
ustoconfinetotheD (1232)region.Wecanneglectcontributionsfromheavierresonances,whoseaxialcouplingsare
0.5 Osipenko et al. Osipenko et al.
2 2 2 2
Q =0.225 [GeV ] best fit Ref. 2 Q =0.425 [GeV ] best fit Ref. 2
0.4 best fit I best fit I
best fit II best fit II
0.3
P
2
F
0.2
0.1
0
0.4 Osipenko et al. Osipenko et al.
2 2 2 2
Q =0.625 [GeV ] best fit Ref. 2 Q =0.825 [GeV ] best fit Ref. 2
best fit I best fit I
0.3 best fit II best fit II
P
20.2
F
0.1
0
Osipenko et al. Osipenko et al.
2 2 2 2
Q =1.025 [GeV ] best fit Ref. 2 Q =1.225 [GeV ] best fit Ref. 2
best fit I best fit I
0.2
best fit II best fit II
P
2
F
0.1
0
1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.1 1.15 1.2 1.25 1.3 1.35 1.4
W [GeV] W [GeV]
FIGURE2. (Coloronline)BestfitresultsforvectorformfactorsgivenbyEqs.(11-13)(“ModelI”)andEqs.(14-16)(“Model
II”)plottedagainstexperimentaldatafromRef.[34]andpredictionsofHNVmodelwithoriginalLalakulich-Paschosformfactors
ofRef.[2].Verticallinesshowthe2p productionthreshold.
bylargeunknown.OuranalysisusesinformationfromallavailableSPPisospinchannels.Thedetileddescriptionof
thestatisticalapproachcanbefoundinRef.[1].
We treatC5A(0), MAD and normalization factor pANL as free fit parameters in the analysis of axial N →D (1232)
transition.We presentourresultsinTab.2.wherefitsto allthreechannelsseparatelyaswellasthejointfittothree
TABLE2. BestfitfortheD (1232) axialformfactorsondeuterontarget.Errors
forC5A(0)andMAD wereobtainedaftermarginalizationof pANL.
Channel C5A(0) MAD (GeV) pANL c 2/NDF NDF
n m +p m −+p+p + 1.11+00..3342 0.97+00..1177 1.04 0.20 6
n m +n→m −+p+p 0 1.31−+00..7479 1.00−+00..2257 0.93 1.52 9
n m +n→m −+n+p + 2.83−+00..6602 0.76−+00..1133 0.94 1.47 9
→ − −
Jointfit 1.10+0.15 0.85+0.09 0.90 2.06 30
0.14 0.08
− −
channelsarelisted.Ineachcasethenumberofdegreeoffreedomiscalculatedas:
NDF =No.Q2bins No.fittedparameters.
We see, that takense−paratelythe pp + (A1)and pp 0 (A2)channelsare statistically consistent,albeittheir predicted
scale parametersdiffer by around10%. The latter channelseems to carry less informationon the N D transition
→
data 70 data
Sr) 1.2 best fit Ref. 2 Sr) best fit Ref. 2
eV 1 bbeesst tf ifti tI II eV 60 bbeesstt ffitit III
G G 50
n/ 0.8 n/
ar ar 40
b b
0.6 n
0m ( 0 ( 30
q q
d 0.4 d
20
Wd Wd
sd/ 0.2 E=0.730 (GeV) Q l=37.1o sd/ 10 E=2.238 (GeV) cos(Q l)=0.8487
0 0
1.1 1.2 1.3 1.4 1.1 1.2 1.3 1.4
W (GeV) W (GeV)
FIGURE3. (Coloronline)ComparisonofourbestfitresultsandHNVmodelwithLalakulich-PaschosformfactorsofRef.[2]
plottedagainstinclusive p(e,e)data(notincludedinthefit)fromRef.[36](leftpanel)andRef.[37](rightpanel).TheQ2values
′
atpeakarefromlefttoright:0.1(GeV2)and0.95(GeV2)respectively.
axial current than the first one. This fact is reflected in larger uncertainties. We explain it by a bigger background
contributiontothatchannel,whichmakesitlesssensitivetochangesintheD resonanceformfactors.
Thenp +(A3)channelgivesresultsinconsistentwiththeothertwo.CA(0)isobtainedtwiceaslargeasforthe pp +
5
and pp 0 channelsandMAD significantlysmaller. Herethe numberof eventsreportedbyANL iscomparableto pp 0
channel,buttheoreticalcrosssectionpredictedbyourmodelaresmaller.ThisresultsintheoverestimationofCA(0).
5
Surprisingly,thefitstoseparateisospinchannelsgiveacceptablevaluesofc 2 forbothneutronchannels.
min
DeuteroneffectsaffectmostlythevalueofCA(0)byupto20%.Inthejointfitonfreeprotonandneutrontargetswe
5
obtainedC5A(0)=0.93−+00..1133andMAD =0.81−+00..0099GeVcomparedto1.10and0.85GeVinthedeuteroncase.Asignificant
improvementwithrespecttopreviousfitstoHNVmodeldoneinRefs.[17,19]isthatwithdeuterontargeteffectswe
getthebestfitvalueofCA(0)within1s rangefromthetheoreticalGoldberger-Treimanrelation.Thejointfitagrees
5
alsoonthe1s levelwithseparatefitson pp 0and pp +channels.
8 Best fit of Ref. 19 Best fit of Ref. 19
our best fit 1 our best fit
2] 7
2/GeV 6 En m=p0-.>7m(G-pep V+), W<1.4(GeV) 0.8 En m=n0-.>7m(G-nep V+), W<1.4(GeV)
m 5
c 0.6
9
-30 4
1
2 [ 3 0.4
Q
d 2
/ 0.2
sd
1
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Q2 [GeV] Q2 [GeV]
FIGURE4. (Coloronline)DifferentialcrosssectionsonafreenucleontargetforourbestfitformfactorsandforfitsofRef.[19].
We have compared our best fit form factors with the results of Ref. [19] for two different neutrino energies
characteristic for Minosand MINERvA experimentsin Figs. 4 and Fig. 5. Our form factorspredict differentshape
andmagnitudeofds /dQ2.ThesizeofeffectisbiggerforthelowervaluesufQ2andforprotonchannel,wheretheD
contributiondominates.Inthelatteronehasupto10%differenceinthecrosssection.
Normalization factors p fitted separately for each channel are different for neutrons and protons. The proton
ANL
channelprefersthedatatobescaledupandbothneutronchannelspreferthedatatobescaleddown.Thejointfituses
the same p parameterforallchannelsandseemsto preferthe datato bescaled downevenmore(p 0.90).
ANL ANL
Thevaluesof p areallwithintheassumedfluxnormalizadionerrorD p . ≈
ANL ANL
10 Best fit of Ref. 19 1.4 Best fit of Ref. 19
our best fit our best fit
2GeV] 8 En m=p3-.>0m(G-pep V+), W<1.4(GeV) 1.2 En m=n3-.>0m(G-nep V+), W<1.4(GeV)
2/ 1
m
9c 6 0.8
3
-0
2 [1 4 0.6
Q 0.4
d
/ 2
sd 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Q2 [GeV] Q2 [GeV]
FIGURE5. (Coloronline)SameasFig.4butforEn =3GeV.
Finally, we noticed that the best fit values forC5A(0) and MAD in the pp + (A1) channel are different from those
obtainedinRef.[22].Weexplainitbytheinclusionofnonrenonantbackgroundinthecurrentanalysis.
CONCLUSIONS
InthispaperwereviewanewattempttogetaninformationaboutweakN D transitionmatrixelementspreviously
→
presented in Ref. [1] as well as we extent this analysis by presenting the newest fits of the vector form factors and
discussionof the axialformfactors.The fitto electromagneticFp hasclearlyshown,thatthe extractedvectorform
2
factors of the D resonance are model-dependent,e. g. the HNV model gives the best results with the form factors
extractedusingthefullHNVmodel,asithasbeendoneinthispaper.
WediscussedaxialformfactorfitsobtainedbasedontheanalysisofallthreeneutrinoSPPchannels,includingthe
neutrino-neutronchannels. In previousworks usually only neutrino-protonchannelwas utilized to extract the axial
form factors. A critical analysis of neutrino-neutron channel, on qualitative level, appears also in other papers see
e.g.[15,17],adetaileddiscussioncanbefoundinRef.[20]aswell.TheobtainedvalueofCA(0)agrees,onthe1s
5
level,withtheGoldberger-Treimanrelationifthedeuteroneffectsaretakenintoaccount,whichisanimportantresult.
If one neglects the nuclear effects the resultingCA(0) value is lower. Also, there is a strong tension between np +
5
andremainingtwochannels(seealso[17]and[20]).Thesametheoreticalmodeldoesnotseemtogiveaconsistent
reproductionofdatainallchannels.
Therecanbevariousreasonsforthat.FirstlytheexistingbubblechamberdataonneutronSPPchannelsareofpoor
statistics. Secondly,the chiralmodelforthe backgroundis well justified onlynearthe pion productionthreshold.It
may be not reliable in the D (1232)peak region.Last, butnot least, the np + channelis subject to large FSI effects,
asshownin Ref. [38]. Thusthe spectatormodelusedbothin experimentalanalysesofANL andBNL aswellasin
ourcalculationsmaygiveinvalidresultsinthischannel.TheplotsinRef.[38]suggestareductionofthenp + cross
sectionduetoFSI.Furtherstudiesareneeded.
Still another reason of these difficulties may come from a missing unitarization of the model. This constraint,
following the Watson theorem [39], imposes a relation between phases in lepton-nucleon and pion-nucleon elastic
scatteringamplitudes.Unitarityis notsatisfied in ourapproach.In a recentstudy Nieves,Alvarez-Ruso,Hernandez
and Vicente-Vacas [40] tried to correct the HNV model by introducing phenomenological phases. They obtained
betteragreementofthe bestfit valueofCA(0)with the Goldberger-Treimanrelation1. Thisis a strongindicationof
5
theimportanceofpropermodelunitarizationforthepionneutrinoproductioncase.Morestudiesarenecessary.
Relatively large values of experimental errors and our inability to extract independentCA and CA form factors
3 4
implies, that better statistics SPP measurements in the D region on proton or deuteron targets are badly needed.
Keeping in mind difficulties in the treatment of nuclear effects on heavier targets it is the only way to get precise
informationabouttheN D axialtransitionmatrixelements.
→
1 SeealsothecontributionofE.HernandezpresentedduringtheCETUP*2014workshop.
ACKNOWLEDGMENTS
We thank U. Mosel for his remarks on the previous version of the paper. JZ was supported by Grant UMO-
2011/M/ST2/02578.NumericalcalculationswerecarriedoutinWroclawCentreforNetworkingandSupercomputing
(http://www.wcss.wroc.pl),grantNo.268.
REFERENCES
1. K.M.Graczyk,J.Z˙muda,andJ.T.Sobczyk,Phys.Rev.D90,093001(2014),1407.5445.
2. O.Lalakulich,E.A.Paschos,andG.Piranishvili,Phys.Rev.D74,014009(2006).
3. J.Evans,Adv.HighEnergyPhys.2013,182537(2013),1307.0721.
4. K.Abe,etal.,Nucl.Instrum.Meth.A659,106–135(2011),1106.1238.
5. D.Ayres,etal.,NOvA:Proposaltobuilda30kilotonoff-axisdetectortostudynu(mu)-nu(e)oscillationsintheNuMI
beamline,FERMILAB-PROPOSAL-0929(2004),arXiv:0503053[hep-ex],hep-ex/0503053.
6. A.Aguilar-Arevalo,etal.,Phys.Rev.Lett.98,231801(2007),0704.1500.
7. C.Adams,etal.,TheLong-BaselineNeutrinoExperiment:ExploringFundamental SymmetriesoftheUniverse(2013),
arXiv:1307.7335[hep-ex],1307.7335.
8. B. Eberly, et al., Charged Pion Production in n m Interactions on Hydrocarbon at En = 4.0 GeV (2014), arXiv:hep-
h i
ex/1406.6415,hep-ex/1406.6415.
9. J.Sobczyk,andJ.Z˙muda,OnRecentWeakSinglePionProductionData(2014),arXiv:1410.7788[nucl-th],1410.7788.
10. S.J.Barish,M.Derrick,T.Dombeck,L.G.Hyman,K.Jaeger,B.Musgrave,P.Schreiner,andR.Singeretal.,Phys.Rev.D
19,2521(1979).
11. G.M.Radecky,V.E.Barnes,D.D.Carmony,A.F.Garfinkel,M.Derrick,E.Fernandez,L.Hyman,andG.Levmanetal.,
Phys.Rev.D26,3297(1982),[Erratum-ibid.D26(1982)3297].
12. T.Kitagaki,H.Yuta,S.Tanaka,A.Yamaguchi,K.Abe,etal.,Phys.Rev.D34,2554–2565(1986).
13. T.Kitagaki,H.Yuta,S.Tanaka,A.Yamaguchi,K.Abe,K.Hasegawa,K.Tamai,andH.Sagawaetal.,Phys.Rev.D42,1331
(1990).
14. L.Alvarez-Ruso,S.Singh,andM.VicenteVacas,Phys.Rev.C59,3386–3392(1999),nucl-th/9804007.
15. G.L.Fogli,andG.Nardulli,Nucl.Phys.B160,116(1979).
16. T.Sato,D.Uno,andT.Lee,Phys.Rev.C67,065201(2003),nucl-th/0303050.
17. E.Hernandez,J.Nieves,andM.Valverde,Phys.Rev.D76,033005(2007).
18. C.Barbero,G.LopezCastro,andA.Mariano,Phys.Lett.B664,70(2008).
19. E.Hernandez,J.Nieves,M.Valverde,andM.VicenteVacas,Phys.Rev.D81,085046(2010),1001.4416.
20. O.Lalakulich,T.Leitner,O.Buss,andU.Mosel,Phys.Rev.D82,093001(2010).
21. B.D.Serot,andX.Zhang,Phys.Rev.C86,015501(2012),1206.3812.
22. K.M.Graczyk,D.Kielczewska,P.Przewlocki,andJ.T.Sobczyk,Phys.Rev.D80,093001(2009).
23. M.L.Goldberger,andS.B.Treiman,Phys.Rev.110,1178(1958).
24. D.Barquilla-Cano,A.Buchmann,andE.Hernandez,Phys.Rev.C75,065203(2007),0705.3297.
25. D.Drechsel,S.S.Kamalov,andL.Tiator,Eur.Phys.J.A34,69(2007).
26. S.J.Brodsky,andG.R.Farrar,Phys.Rev.D11,1309(1975).
27. J.Kelly,Phys.Rev.C70,068202(2004).
28. J.Liu,N.C.Mukhopadhyay,andL.-s.Zhang,Phys.Rev.C52,1630–1647(1995),hep-ph/9506389.
29. S.L.Adler,AnnalsPhys.50,189(1968).
30. K.M.Graczyk,PoSEPS-HEP2009,286(2009),0909.5084.
31. M.Lacombe,B.Loiseau,R.VinhMau,J.Cote,P.Pires,andR.deTourreil,Phys.Lett.Bp.139(1981).
32. G.Shen,L.Marcucci,J.Carlson,S.Gandolfi,andR.Schiavilla,Phys.Rev.C86,035503(2012),1205.4337.
33. J.-J.Wu,T.Sato,andT.S.H.Lee,Phys.Rev.C91,035203(2015),1412.2415.
34. M.Osipenko,G.Ricco,M.Taiuti,M.Anghinolfi,M.Battaglieri,etal.(2003),hep-ex/0309052.
35. R.Davidson,N.Mukhopadhyay,andR.Wittman,Phys.Rev.D43,71–94(1991).
36. J.S.O’Connell,W.R.Dodge,J.W.Lightbody,X.K.Maruyama,J.O.Adler,K.Hansen,B.Schroder,andA.M.Bernsteinet
al.,Phys.Rev.Lett.53,1627(1984).
37. M.Christy,andP.E.Bosted,Phys.Rev.C81,055213(2010),0712.3731.
38. J.-J.Wu,T.Sato,andT.-S.H.Lee,IncoherentPionProductioninNeutrino-DeuteronReactions(2014),arXiv:1412.2415
[nucl-th],1412.2415.
39. K.M.Watson,Phys.Rev.88,1163–1171(1952).
40. L.Alvarez-Ruso,E.Hernandez,M.J.Vicente-Vacas,andJ.Nieves,Watson’stheorem,Goldberger-Treimanrelationandthe
p ND axialCA(0)couplingconstant(May19-242014),talkatNuInt14,London.
5