Emission of neutron–proton and proton–proton pairs in neutrino scattering

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Emission

of

neutron–proton

and

proton–proton

pairs

in

neutrino

scattering

I. Ruiz Simo

a

,

J.E. Amaro

a

,

∗

,

M.B. Barbaro

b

,

c

,

A. De

Pace

c

,

J.A. Caballero

d

,

G.D. Megias

d

,

T.W. Donnelly

e

a_{DepartamentodeFísicaAtómica,MolecularyNuclear,andInstitutodeFísicaTeóricayComputacionalCarlosI,UniversidaddeGranada,Granada18071,Spain} b_{DipartimentodiFisica,UniversitàdiTorino,ViaP.Giuria1,10125 Torino,Italy}

c_{INFN,SezionediTorino,ViaP.Giuria1,10125Torino,Italy}

d_{DepartamentodeFísicaAtómica,MolecularyNuclear,UniversidaddeSevilla,Apdo. 1065,41080Sevilla,Spain}

e_{CenterforTheoreticalPhysics,LaboratoryforNuclearScienceandDepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received29July2016

Receivedinrevisedform12September 2016

Accepted13September2016 Availableonline16September2016 Editor: W.Haxton

We use arecently developed modelofrelativistic meson-exchangecurrentsto computetheneutron– protonandproton–protonyieldsin(

νμ

,

μ

−)scatteringfrom12Cinthe2p–2hchannel.Wecomputethe responsefunctionsandcrosssectionswiththerelativisticFermigasmodelfordifferentkinematicsfrom intermediatetohighmomentumtransfers.Wefindalargecontributionofneutron–protonconfigurations intheinitialstate,ascomparedtoproton–protonpairs.Inthecaseofcharge-changingneutrinoscattering the2p–2hcrosssectionofproton–protonemission(i.e., npintheinitialstate)ismuchlargerthanfor neutron–protonemission(i.e., twoneutronsintheinitialstate)bya(

ω

,q)-dependentfactor.Thedifferent emission probabilities ofdistinctspeciesofnucleonpairs areproduced inourmodelonlyby meson-exchange currents,mainly bythe isobar current. We alsoanalyzeothereffects includingexchange contributionsandtheeffectoftheaxialandvectorcurrents.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

1. Introduction

The identification of nuclear effects in neutrino scattering is essential for modern neutrino oscillation experiments [1–7]. In particularthesensitivityofthe neutrinoenergyreconstructionto multi-nucleoneventshasbeenstressedinrecentdataanalyses

[8]

. In theMINERvA neutrino experimentan enhanced population of multi-proton states has been observed between the quasielastic and



peaks.Ontheotherhand,observationofeventswithapair of energetic protons at the interaction vertex accompanying the muonin40_Ar(

_ν

μ

,

μ

−)reactionhasbeenreportedintheArgoNeuT

experiment [9]. From these events several back-to-back nucleon configurations have been identified and associated with nuclear mechanismsinvolvingshort-rangecorrelated(SRC)neutron–proton (np) pairs in the nucleus [10]. However in [11] these “hammer events”havebeenmodeledbyasimplepionproductionand reab-sorption model withoutnucleon–nucleon correlations, suggesting that the distribution of pp pairs in the final state is less

sensi-*

Correspondingauthor.

E-mailaddress:amaro@ugr.es(J.E. Amaro).

tive to details of the initial pair configuration.In the opinion of theauthorsof

[11]

,theeventscannotteachusanythingsignificant aboutSRC. The NUWRO eventgeneratorsupports thatthe excess ofback-to-backeventsinArgoNeuThasakinematicoriginandis notdirectlyrelatedtoSRC

[12]

.

SRCwithback-to-backconfigurationshavebeenalsoidentified in two-nucleonknock-out electronscatteringexperiments on12_C for highmomentum transfer andmissing momentum [13,14]. In this case one expects an excess of np pairs over pp pairs [15,

16]. The experiment reported a number of np pairs 18 times

larger than their pp counterparts. The analysis of these experi-ments is compatible withtheoretical single-nucleon and nucleon pair momentum distributions in variationalMonte Carlo calcula-tions, wherethe importance of the tensor forces inthe ground-statecorrelationsofnucleihasbeenemphasized

[17,18]

.Whilethe kinematicsoftheexperimentshavebeenselectedtominimizethe contributionfromother mechanismsthatcaninduce two-particle emission,suchasmeson-exchangecurrents(MEC)andisobar exci-tations [13],thecontribution ofMECcannot beruled out apriori

[19].

In this work we investigate therelative effectof MEC on the separateppandnpemissionchannelsintheinclusive2p–2h

neu-http://dx.doi.org/10.1016/j.physletb.2016.09.021

0370-2693/©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

trino cross section. Through this work we use the convention that, unlessspecified otherwise, the charge label xy for the pair concerns the final state pair. It has been emphasized that the separate charge distributions of 2p–2h events are useful. One of thereasons isfor their usein MonteCarloeventgenerators

[12,

20].For instancein NUWRO configurations the MEC 2p–2h exci-tations are assumed to occur 95% of the time for events where theinteractionoccursininitialnp pairs

[12,21]

(or finalpp pairs forchargedcurrentneutrinoscattering).Thisvaluewasestimated basedontheassumption,claimedalsoin

[22]

,thatneutrinos inter-actmostly withcorrelatednppairs.From anaive calculationthis value agrees with a factor 18/19,corresponding to the extracted valueofnp

/(

np

+

pp

)

inthe12_C(e

_,

_e_{Np)experimentof}

_[13]

_. How-ever this neutrino generator uses a 2p–2h model that does not give separate pp and np contributions, andtherefore this choice isnotfullyconsistentfromthetheoretical pointofview [12].On theotherhanditisexpectedthattheratiobetweennpandpp in-teractionsshouldbe kinematics dependentand notonly aglobal factor.Thus atheoreticalquantification ofthenp/ppratioandits dependenceon thetypical kinematics wouldbe desired foreach implementationof 2p–2h cross sections.Results for the separate ppandnpcontributionsduetoshort-rangecorrelationshavebeen presentedin

[23]

,forthe RT and RC C responsefunctions,andfor

q

=

400 MeV/c,butnotforthedifferentialcrosssection.The con-tributionofinitialnppairstotheTresponsefoundin

[23]

isabout twicethatoftheinitialnnpairs.

Wehaverecentlydevelopedafullyrelativisticmodelof meson-exchangecurrentsinthe2p–2hchannelforelectronandneutrino scattering

[24]

.Thismodelisanextension oftherelativisticMEC modelof

[25]

totheweaksector.Ithasbeenrecentlyvalidatedby comparingtothe12_C(e

_,

_e_{)inclusivecrosssectiondataforawide} kinematicrangewithintheSuperScalingapproach(SuSA)

[26]

.This modeldescribesjointlythequasielasticandinelasticregionsusing twoscalingfunctionsfittedtoreproducethedata,whilethe2p–2h MECcontributionproperlyfillsthe dipregioninbetween, result-ing in excellent global agreement with the data. The model has been recently extended to the description of neutrino scattering reactionsforavarietyofexperimentsprovidinganexcellent agree-mentwithdata

[27]

.Withthisbenchmark modelwe are ableto studytheseparate npandpp channelsin theresponse functions andcrosssectionforthethree

(

e

,

e

)

,

(

ν

l

,

l−

)

and

(

ν

¯

l

,

l+

)

reactions.

While this analysis is performed in [19] for electron scattering, inthiswork we consider neutrinoreactions. Our modelincludes thecontributionsofpion-in-flight,seagull,pion-poleand



(1232) excitationdiagramsoftheMEC.Thetwo-bodymatrixelements be-tweenrelativisticspinors werepresentedinourrecentwork

[24]

,

where they have been deduced from the weak pion production

amplitudesof

[28]

.

2. Formalismforneutrinoscattering

Theformalismof2p–2hcrosssectionincludingMECinthe rel-ativisticFermigaswasgivenin

[24]

.Wewritethechargedcurrent (CC) crosssectionas d

σ

d



d





=

σ

0





VC CRC C

+

2V



C LRC L

+ 

VLLRLL

+ 

VTRT

±

2



VTRT 



,

(1)

where

σ

0 isa kinematicfactor includingtheweak couplings de-finedin

[29,30]

.Notethatthereisalinearcombinationoffive re-sponsefunctions,labeledasC C

,

C L

,

LL

,

T andT.The Tresponse functioncontributes differentlyfor neutrinos(plus sign)than for antineutrinos(minussign).The



VK factorsarekinematicfunctions

thatweredefinedin

[29,30]

.

TheweakresponsefunctionsRK

₍

_ω

_,

_q

₎

_{—nottobeconfusedwith}

the electromagnetic response functionsused in previous works— dependontheenergyandmomentumtransfer.Theyarecomputed hereina relativisticFermigas (RFG)model,withFermi momen-tumkF,where they canbe expanded asthesumofone-particle

one-hole (1p–1h), two-particle two-hole (2p–2h), plus additional channels.Hereweareinterestedinthe2p–2hchannel,wheretwo nucleons with momenta p₁ and p₂ are ejected out of the Fermi sea,p_i

>

kF,leavingtwoholestatesinthedaughternucleus,with

momentah1andh2 (withhi

<

kF).

The2p–2hresponsefunctionsarecomputedas

R_2pK_−2h

=

V

(

2

π

)

9



d3p₁d3h1d3h2 m4_N E1E2E₁E₂ rK

(

p₁

,

p₂

,

h1

,

h2

)δ(

E1

+

E2

−

E1

−

E2

−

ω

)

× θ(

p₂

−

kF

)θ (

p1

−

kF

)

× θ(

kF

−

h1

)θ (

kF

−

h2

),

(2)

wherethemomentumofthesecond nucleonisfixedby

momen-tumconservationinside the integralsign, p₂

=

h1

+

h2

+

q

−

p1,

V isthevolumeofthesystem,mN isthenucleonmass,while Ei

andE_iaretheenergiesoftheholesandparticles,respectively. Using energy conservation, the calculation of the inclusive 2p–2hresponsesofEq.(2)forgivenenergyandmomentum trans-fer

(

ω

,

q

)

,isreducedtoaseven-dimensionalintegralthatis com-putednumericallyfollowingthemethodsdevelopedin

[31,32]

.The mainingredientofthecalculationisthesetoffiveresponse func-tionsrK

₍

_p

1

,

p2

,

h1

,

h2

)

,fortheelementary2p–2htransition.These elementary response functions are written in terms of the

two-body MEC antisymmetrized matrix elements, summed over spin.

Weseparate thecontributions ofthedifferentchargechannelsto the response functions. These can be

(

np

,

pp

)

forneutrinos, and

(

np

,

nn

)

forantineutrinos.In

[24]

wederivedgeneralformulaefor theseparatenpandppresponsefunctions.

ThetotalCCMECforneutrinoscatteringcanbewrittenas

jμ_MEC

=

τ

+

(

1

)

Jμ₁

(

12

;

1 2

)

+

τ

+

(

2

)

J₂μ

(

12

;

1 2

)

+ (

IV

)

+ J3μ

(

12

;

1 2

),

(3)

where

τ

+

=

τ

x

+

i

τ

yandwehavedefinedtheisospinoperators

(

IV

)

±

= (

IV

)

x

±

i

(

IV

)

y (4)

thatstandsforthe

±

-componentofthetwo-bodyisovector opera-tor

IV

=

i [

τ

(

1

)

×

τ

(

2

)

]

.

(5)

The isospin-independenttwo-body currents Jμ₁, Jμ₂,and J₃μ, fol-low fromthe amplitudesof weakpion productionmodel of[28]

andarewrittenin

[24]

.

Inother models ofneutrinoscattering[22,33],onlythe direct diagrams (a, b) of Fig. 1 are included, while the direct-exchange contributioncorrespondingtothediagrams(c,d)aredisregarded. Inour model,onthecontrary, bothcontributions are considered. Theelementary2p–2htransverseresponsefunctionisgivenforpp emissionby rT_pp

=

4 2



μ=1



s1s2s1s2



_

Jμpp

(

12

;

12

)



2

−

Re Jμpp

(

12

;

12

)

∗Jμpp

(

21

;

12

)

,

(6)

where Jμpp

(

12

;

12

)

istheeffectivetwo-bodycurrentforpp

Jμpp

=

Jμ1

+

J

μ

3

.

(7)

Thefirstterminthetransverse responseisthe“direct” contribu-tion,and thesecond one is the “exchange” contribution,actually

Fig. 1. (Coloronline.) Somecontributionsof2p–2hstatestotheresponsefunctions consideredinthiswork.Thecirclestandsfor theelementary W+N→πN am-plitude.Diagrams(a,b)representthedirectcontribution.Diagrams(c,d)arethe exchangecontributions.

beingtheinterferencebetweenthedirectandexchangematrix el-ements.

Theelementarytransverseresponsefornpemissionhasa sim-ilarexpression r_npT

=

4 2



μ=1



s1s2s1s2



_

Jnpμ

(

12

;

12

)



2

−

Re J_npμ

(

12

;

12

)

∗Jμ_np

(

12

;

21

)

,

(8)

butwiththeeffectivecurrent Jμnp fornpemission

J_npμ

=

Jμ₂

+

Jμ₃

,

(9)

whichisingeneraldifferentfromtheppone becauseofthe dis-tinct isospinmatrixelements.Thedirectcontributioncorresponds toneglectingthesecondtermsinEqs.(6),

(8)

.

3. Results

In the following we present results for the semi-inclusive 12_C(

_ν

μ

,

μ

−pp)and12C(

ν

μ

,

μ

−np) reactions,corresponding tothe

2p–2h channel of the 12C(

ν

μ

,

μ

−) reaction in the two separate

charge channelsinthefinal hadronicstate correspondingto

two-nucleon knockout. Our MEC model and its parameters were

ob-tained from the pion production amplitudes of [28]. This is an extension ofthe electromagneticMEC modelof

[25]

tothe weak sector.

Fig. 2. (Coloronline.) Separateppandnp2p–2hresponsefunctionsof12_{Cforthreevaluesofthemomentumtransfer.Inallthefiguresthechargelabelsrefertothefinal}

Fig. 3. (Coloronline.) SeparateppandnpcontributionstotheT andT2p–2h re-sponsefunctionsof12_{C,forq=600 MeV/c,comparedtothedirectcontributions}

obtainedbyneglectingthedirect-exchangeinterferences.

The five MEC-induced 2p–2h responses for CC neutrino

in-teractionsat fixed momentum transfer are shown in Fig. 2. The Coulomb RC C and transverse RT responses are also present in

electronscattering. Ingeneralthe

ω

dependenceofthe2p–2h re-sponses shows a broad peak coming from the



excitation. The

strength of the MEC peak weakens with q due to the decrease

of the electroweak form factor with Q2_{, especially the}

_

_form factors.Notealsothatthemostimportantcontributiontothe neu-trino crosssection comes formthe two transverse responses RT

andRT.Inthefigureweonlyshowtheseparateppandnp2p–2h

responsefunctions,thetotalresponses beingthesumofthetwo. Forallthe casesin

Fig. 2

we observethatthe pp response func-tionsare much largerthan the nponesby a factor6or less de-pendingonthekinematics.

Thepp/npratiointhepresentneutrinocalculationcanbe com-pared to the np/pp ratio in the

(

e

,

e

)

reaction studied in [19], becausethey correspondtothesamepairsintheinitialstate.For thetransverseresponsethatratioforneutrinoscatteringisroughly afactoroftwosmallerthanfortheelectroncase.

The origin of the dominance of initial np pairs is related to theisospinstructureoftheMECoperators,thatinvolveexchanges ofdifferentcombinationsofchargedandneutralpions depending on the initial andfinal chargesof the nucleon pairs. Due to the large number of Feynman diagrams contributing to each one of theseterms,withdifferentcouplingconstants andisospinfactors, itisnotpossibletoobtainasimpleconstantfactorbetweenboth pp andnp contributionresponses, even neglecting the exchange. Infactour resultsshow thatthe factordependsonthe kinemat-ics.

Forq

=

400 MeV/c ourresultsforRT canbecomparedtothose

oftheSRC modelof

[23]

.Our MECresponse atthe maximumis one order of magnitude larger than that of the SRC one. In our calculation,the pp pair emission transverse response induced by

Fig. 4. (Coloronline.) ComparisonoftheseparateppandnpcontributionstotheT , TandC C 2p–2hresponsefunctionsof12_{C,forq=100,200,...,2000 MeV/c.}

MEC is about a factor of 6 larger than the np one. In contrast,

the mentioned SRC model shows atmost a factor of 3 between

thetwocontributions.TheorderofmagnitudeoftheRC C,onthe

other hand, is smallin both MEC andSRC 2p–2h responses, but stilltheMECresultsareabouttwicethoseof

[23]

.Thepppairsin thefinalstatecontinuetodominatethe RC C MECresponse,while

intheSRCcase,bothpairscontributesimilarly.

In a previous work [24] we showedthat the interference di-agrams (c, d)of Fig. 1 can amount to

∼

25% of the total 2p–2h responses. But for the separate charge channels the interference influence can be truly different, aswe show in Fig. 3. While the interference forpp emission produces areduction of 20%,for np emission the reduction factoris about 1/2. Thus the ratio pp/np criticallydependson the treatmenton theinterference contribu-tions. The sameconclusion was found for theelectron scattering responses in[19].Theeffectfromtheinterferencecontributionis ofthesamesizefortheT response,asshowninthelowerpanel of

Fig. 3

.

Notethattheorderofmagnitudeoftheinterference contribu-tionissimilarfornpandppchannels,butbecauseofthesmallness ofthenpresponses,therelativeimportanceismuchlargerinthis

Fig. 5. (Coloronline.) ComparisonoftheseparateppandnpcontributionstotheT 2p–2hresponsefunctionof12_{C,forq=100,200,...,2000 MeV/c,includingonlythe}

axialMEC.

Fig. 6. (Coloronline.) Doubledifferentialneutrinocrosssectionperneutronof12_C,

forfixedmuonscatteringangleandforthreeneutrinoenergies,asafunctionofthe muonkineticenergy.Theseparate1p–1hand2p–2hcrosssectionsaredisplayed forcomparison.

case.Wehaveestimatedthattheimportanceoftheseinterferences islessthan 25%;however,thisisapproximatelytrueforthetotal responses,butnotfortheseparatecontributions.Duetothelarge numberof diagramsinvolved, implicitin Eqs.(6)–(9),this isthe firsttimethatthisinterferencehasbeencalculatedforneutrinos.

Fig. 7. (Coloronline.) Doubledifferential2p–2hneutrinocrosssectionperneutron of12_{C,forfixedmuonscatteringangleandforthreeneutrinoenergies,asafunction}

ofthemuonkineticenergy.Theseparatenpandppchannelsareshown. In Fig. 4we comparetheseparate np andppcontributions to theresponse functionsRT, RT,RC C,inthefullrangeof

momen-tumtransferfromq

=

100 to2000MeV/c.Thiscorrespondstothe typical kinematic rangeofthe neutrinoexperiments operatingin the few-GeV region. The T response is more than twice the T

both fornp andforpp channels. This seems toindicate that the axial MECcontribution islargerthan thevector one.Indeed,this canbe trulyobservedin

Fig. 5

,whereonlytheaxialMECcurrent isincludedinthecalculation.Notethatthe C C responseismuch smallerthantheothertwo,exceptfortheq

=

ω

point,wherethe

C andL contributionsareapproximatelycancelled.Theywouldbe exactlycancelledifthetotalcurrentwasconserved.Asamatterof fact,nearthephotonpointtheseagullcurrentdominatestheMEC for high q when the



resonance is far away.The axial seagull contribution ismainly longitudinal.Thatiswhy theC C response

in

Fig. 4

islargeatthephotonpointforhighq andtheT response

in

Fig. 5

issosmall.

ToappreciatethesizeoftheMEC2p–2hcontribution,in

Fig. 6

weplotthedoubledifferentialneutrinocrosssectionperneutron,

d2

_σ

_/

_dcos

_θ

μ

/

dTμ

/

N,of12C,asafunctionofthemuonkinetic

en-ergy,forcos

θ

μ

=

0

.

85 andforthreevaluesoftheincidentneutrino

energy. Weshow theseparate contributions of1p–1h and2p–2h channelsinthe RFG.Therelative contributionof2p–2hincreases withtheneutrinoenergy,andtheMECandquasielasticpeaksget closer.Heretheneutrinoenergyisfixed,whileintheexperiments

Fig. 8. (Coloronline.) Doubledifferential2p–2hneutrinocrosssectionperneutron of12_C,d2_σ/dcosθ

μdTμ/N,inunitsof10−39cm2/GeV,asafunctionofcosθμ,Tμ forfixedneutrinoenergyEν=1 GeV.Theseparateppandnpchannelsareshown inthemiddleandbottompanels,respectively.

the neutrino energyis not fixed and the flux produces an aver-ageof all the contributions around the meanenergy. For typical peak energies of 1 GeV, the results of Fig. 6 indicate that one canexpectacontributionofroughly20%tothecrosssectionfrom 2p–2h.

The separate pp and np channels in the differential neutrino crosssection areshownin

Fig. 7

forthesamekinematics.Thepp channelclearly dominatesthe2p–2hcross section.Thepp/np ra-tioisaround5–6nearthemaximum,butitsprecisevaluedepends onthekinematics.Notethatthenpdistributionisshiftedtowards highermuon energiescomparedwiththeppcase.Thiseffectcan be further observed in Fig. 8, where we show the

(

cos

θ

μ

,

Tμ

)

dependence of the 2p–2h double differential cross section, for

Eν

=

1 GeV. Indeed the second and third panels show the sep-arate pp and np distributions. The np is much smaller than the pp one, andit is clearly shifted towards higher Tμ and smaller angles.It can be seenthat for thisneutrinoenergy, theabsolute maximum of the cross section is located around cos

θ

μ

∼

0.85

and Tμ

∼

600 MeV, andcorresponds approximatelyto the

max-imum shownin the middle panel of Fig. 7. The 2p–2h strength isconcentratedin thetop-rightcorner ofFig. 8corresponding to smallanglesandlargemuonkineticenergies,meaninglowenergy transfer, around

ω

=

300 MeV. This corresponds to the excita-tionenergyofthe

(

1232

)

,whichgivesthemaincontributionto theMEC.Ourcalculationpredictsthat,whentheleptonscattering

angleincreases,twoparticleemissionimpliesadecreaseofthe ki-neticenergyofthemuonorlarger

ω

.

4. Conclusions

In this work we have studied the separate charge channels

(

ν

μ

,

μ

−pp

)

and

(

ν

μ

,

μ

−np

)

, from 12C, integrated over the two

emitted nucleons, that contribute to the 2p–2h cross section in quasielastic-likeCCneutrinoscattering.Wehavecomputedthe re-sponsefunctionsanddoubledifferentialcrosssectionsforseveral kinematics.Theppchanneldominatesoverthenpcontributionin thewholedomain.Thepp/npratioisabout5–6fora widerange of neutrino energies. Future plans are to fold the cross section with the neutrino fluxes for the various neutrino oscillation ex-periments.Havingtheseparateisospincontributionswillallowus toapply thisformalism toasymmetricnuclei N

=

Z .Thiswillbe of interest forneutrinoexperiments based, forinstance, on40Ar, 56_Feor208_Pb.

Acknowledgements

This work was supported by Spanish Direccion General de

Investigacion Cientifica y Tecnica and FEDER funds (grants No.

FIS2014-59386-P and No. FIS2014-53448-C2-1), by the Agencia

de Innovación y Desarrollo de Andalucía (grants No. FQM225,

FQM160), by INFN grant No. IS-MANYBODY, and part (TWD)

by U.S. Department of Energy under cooperative agreement

DE-FC02-94ER40818. IRS acknowledges support from a Juan de la

CiervafellowshipfromSpanishMINECO(grantNo.IJCI-2014-20038). GDMacknowledges supportfromaJuntadeAndaluciafellowship (FQM7632,ProyectosdeExcelencia2011).

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