Measurement of double-polarization asymmetries in the quasi-elastic $^{3}\overrightarrow{He}(\vec{e}, {e}'p)$ process

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Physics Letters B

www.elsevier.com/locate/physletb

Measurement of double-polarization asymmetries in the quasi-elastic

3 He  (  ^e , e ^ p ) process

The Jefferson Lab Hall A Collaboration

M. Mihoviloviˇc

^a,b,c

, G. Jin

^d

, E. Long

^e

, Y.-W. Zhang

^f

, K. Allada

^g

, B. Anderson

^h

, J.R.M. Annand

ⁱ

, T. Averett

^j

, W. Bertozzi

^k

, W. Boeglin

^l

, P. Bradshaw

^j

, A. Camsonne

^g

, M. Canan

^m

, G.D. Cates

^d

, C. Chen

ⁿ

, J.P. Chen

^g

, E. Chudakov

^g

, R. De Leo

^o

, X. Deng

^d

, A. Deltuva

^p

, A. Deur

^g

, C. Dutta

^q

, L. El Fassi

^f

, D. Flay

^r

, S. Frullani

^s

, F. Garibaldi

^s

, H. Gao

^t

, S. Gilad

^k

, R. Gilman

^f

, O. Glamazdin

^u

, J. Golak

^v

, S. Golge

^m

, J. Gomez

^g

, O. Hansen

^g

, D.W. Higinbotham

^g

, T. Holmstrom

^w

, J. Huang

^k

, H. Ibrahim

^x

, C.W. de Jager

^g,†

, E. Jensen

^y

, X. Jiang

^z

, M. Jones

^g

, H. Kamada

^aa

, H. Kang

^ab

, J. Katich

^j

, H.P. Khanal

^l

, A. Kievsky

^ac

, P. King

^ad

, W. Korsch

^q

, J. LeRose

^g

, R. Lindgren

^d

, H.-J. Lu

^ae

, W. Luo

^af

, L.E. Marcucci

^ac

, P. Markowitz

^l

, M. Meziane

^j

, R. Michaels

^g

, B. Moffit

^g

, P. Monaghan

ⁿ

, N. Muangma

^k

, S. Nanda

^g

, B.E. Norum

^d

, K. Pan

^k

, D.S. Parno

^ag

, E. Piasetzky

^ah

, M. Posik

^r

, V. Punjabi

^ai

, A.J.R. Puckett

^aj

, X. Qian

^t

, Y. Qiang

^g

, X. Qui

^af

, S. Riordan

^d

, A. Saha

^g,†

, P.U. Sauer

^ak

, B. Sawatzky

^g

, R. Schiavilla

^g,m

, B. Schoenrock

^al

, M. Shabestari

^d

, A. Shahinyan

^am

, S. Širca

^a,b,∗

, R. Skibi ´nski

^v

, J. St. John

^w

, R. Subedi

^an

, V. Sulkosky

^k

, W. Tireman

^al

, W.A. Tobias

^d

, K. Topolnicki

^v

, G.M. Urciuoli

^s

, M. Viviani

^ac

, D. Wang

^d

, K. Wang

^d

, Y. Wang

^ao

, J. Watson

^g

, B. Wojtsekhowski

^g

, H. Witała

^v

, Z. Ye

ⁿ

, X. Zhan

^k

, Y. Zhang

^af

, X. Zheng

^d

, B. Zhao

^j

, L. Zhu

ⁿ

aFacultyofMathematicsandPhysics,UniversityofLjubljana,SI-1000Ljubljana,Slovenia bJožefStefanInstitute,SI-1000Ljubljana,Slovenia

cInstitutfürKernphysik,JohannesGutenberg-UniversitätMainz,DE-55128Mainz,Germany dUniversityofVirginia,Charlottesville,VA22908,USA

eUniversityofNewHampshire,Durham,NH03824,USA fRutgersUniversity,NewBrunswick,NJ08901,USA

gThomasJeffersonNationalAcceleratorFacility,NewportNews,VA23606,USA hKentStateUniversity,Kent,OH44242,USA

iGlasgowUniversity,GlasgowG128QQ,Scotland,UnitedKingdom jTheCollegeofWilliamandMary,Williamsburg,VA23187,USA kMassachusettsInstituteofTechnology,Cambridge,MA02139,USA lFloridaInternationalUniversity,Miami,FL33181,USA

mOldDominionUniversity,Norfolk,VA23529,USA nHamptonUniversity,Hampton,VA23669,USA

oUniversitàdeglistudidiBariAldoMoro,I-70121Bari,Italy

pInstituteforTheoreticalPhysicsandAstronomy,VilniusUniversity,LT-01108Vilnius,Lithuania qUniversityofKentucky,Lexington,KY40506,USA

rTempleUniversity,Philadelphia,PA19122,USA

sIstitutoNazionaleDiFisicaNucleare,INFN/Sanita,Roma,Italy tDukeUniversity,Durham,NC27708,USA

uKharkovInstituteofPhysicsandTechnology,Kharkov61108,Ukraine

vM.SmoluchowskiInstituteofPhysics,JagiellonianUniversity,PL-30348Kraków,Poland wLongwoodCollege,Farmville,VA23909,USA

xCairoUniversity,Cairo,Giza12613,Egypt

*

Correspondingauthorat:FacultyofMathematicsandPhysics,UniversityofLjubljana,SI–1000Ljubljana,Slovenia.

E-mailaddress:simon.sirca@fmf.uni-lj.si(S. Širca).

† Deceased.

https://doi.org/10.1016/j.physletb.2018.10.063

0370-2693/©^{2018TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

yChristopherNewportUniversity,NewportNews,VA23606,USA zLosAlamosNationalLaboratory,LosAlamos,NM87545,USA

aaDepartmentofPhysics,FacultyofEngineering,KyushuInstituteofTechnology,Kitakyushu804-8550,Japan abSeoulNationalUniversity,Seoul,RepublicofKorea

acINFN-Pisa,I-56127Pisa,Italy adOhioUniversity,Athens,OH45701,USA aeHuangshanUniversity,People’sRepublicofChina

afLanzhouUniversity,Lanzhou,Gansu,730000,People’sRepublicofChina agCarnegieMellonUniversity,Pittsburgh,PA15213,USA

ahTelAvivUniversity,TelAviv69978,Israel aiNorfolkStateUniversity,Norfolk,VA23504,USA ajUniversityofConnecticut,Storrs,CT06269,USA

akInstituteforTheoreticalPhysics,UniversityofHannover,D-30167Hannover,Germany alNorthernMichiganUniversity,Marquette,MI49855,USA

amYerevanPhysicsInstitute,Yerevan,Armenia

anGeorgeWashingtonUniversity,Washington,DC20052,USA aoUniversityofIllinoisatUrbana-Champaign,Urbana,IL61801,USA

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received6July2018

Receivedinrevisedform22October2018 Accepted31October2018

Availableonline5November2018 Editor:V.Metag

Keywords:

Double-polarizationasymmetry Helium-3nucleus

Protonknock-out

Wereportonaprecisemeasurementofdouble-polarizationasymmetriesinelectron-inducedbreakupof 3He proceedingtopd andppn finalstates,performedinquasi-elastickinematicsatQ²=⁰.25(GeV/c)² formissingmomentaupto250MeV/c.Theseobservablesrepresenthighlysensitivetoolstoinvestigate the electromagnetic and spin structure of ³He and the relative importance of two- and three-body effects involved inthe breakupreactiondynamics.The measuredasymmetriescannotbesatisfactorily reproducedbystate-of-the-artcalculationsof³He unlesstheirthree-bodysegmentisadjusted,indicating that the spin-dependent part ofthe nuclear interaction governing the three-body breakupprocess is muchsmallerthanpreviouslythought.

©^{2018TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

The³He nucleusrepresentsthekeychallengeofnuclearphysics duetoitspotentialtorevealthebasicfeaturesofnuclearstructure and dynamics in general. In particular, this paradigmatic three- body system offers a unique opportunity to study the interplay of two-nucleon and three-nucleon interactions, an effort at the forefront of nuclear physics research [1–4]. Modern theoretical descriptions of the structure and dynamics of ³He require, first of all, a detailed understanding of the nuclear Hamiltonian (in- cluding the three-nucleon force), which generates the consistent nuclear ground and scatteringstates, while accounting for final- stateinteractions(FSI).Thereactionmechanismcomprisesalsothe electromagneticcurrentoperator,whichtakesintoaccountmeson- exchange currents (MEC).Experiments on ³He, particularlythose involvingpolarization degreesoffreedom,provideessential input totheorieswhichneedtobeperpetuallyimprovedtoyieldbetter understanding of the underlying physics and to match the cur- rentincrease inexperimentalprecision. The qualityoftheoretical modelsiscrucial toall ³He-basedexperiments seekingtoextract neutroninformation by utilizing³He as an effectiveneutron tar- get,an approximationrelyingonasufficientunderstandingofthe protonandneutronpolarizationwithinpolarized³He.

The³He nucleus isbeststudied byelectron-induced knockout ofprotons, deuterons andneutrons, wherethe sensitivityto var- ious aspects of the process can be greatly enhanced by the use of polarized beam and target [2]. The focus of this paper is on thetwo-body(2bbu)andthree-body(3bbu)breakupchannelswith protondetectioninthefinalstate,³He(e,e^p)d and³He(e,e^p)pn, whichwere investigated concurrently withthe alreadypublished 3He(^e,e^d)data[5].

Ina³He(e,e^p)reactionthe virtualphoton emittedbythe in- comingelectron transfersthe energy

ω

andmomentum q tothe 3He nucleus.The process observablesare then analyzedinterms of missing momentum, defined as the difference between the momentum transfer and the detected proton momentum, pm=

|^q−^pp|^{, thus p}m corresponds to themomentum of therecoiled

deuteronin2bbuandthetotalmomentumoftheresidualpn sys- temin3bbu.

The unpolarized ³He(e,e^p) process at low energies has been studied atMAMI,both onthequasi-elastic peak[6] andbelowit [7].The bulkofourpresenthigh-energy informationcomesfrom experiments in quasi-elastic kinematics at Jefferson Lab [8–10], resultinginreactioncross-sectionsathighpm andyieldingimpor- tant insight into nucleon momentum distributions, isospinstruc- tureofthetransitioncurrents,FSI,andMEC[11–15].However,just asinthe (e,e^d) case,experiments that exploitpolarization offer much greater sensitivity to the fine details of these ingredients.

Suchmeasurementshavebeenextremelyscarce.Asingleasymme- trydatapointwithhighuncertaintyexistsfromNIKHEF[16,17].In addition, we havea precise measurementof both transverseand longitudinalasymmetriesseparately forthe2bbuand3bbuchan- nelsinquasi-elastickinematics[18,19],butthemeasurementwas restrictedto(andsummedover)relativelylow pm.

Early theoretical studies [20–22] have shown strong sensitiv- ities of double-polarization asymmetries in ³He breakup to the isospinstructureoftheelectromagneticcurrent,tothesub-leading components ofthe³He ground-statewave-function,aswell asto the tensor component of the nucleon–nucleon interaction. How- ever, while in the deuteron channel these would predominantly manifest themselvesatlow pm,the2bbu and3bbu protonchan- nels shouldgive moreinformationathigh pm,a regionwhichis, however, difficult to explore experimentally. These diagrammatic evaluationsultimatelygavewaytomorerefined,fullFaddeevcal- culations performed independently by the Krakow [23,24] and the Hannover/Lisbon[25–28] groups, whichweuseinthispaper.

The key featureofour experimentistheunmatched precision of theextractedasymmetriestogetherwithabroadkinematicrange, withpm extendingtoasfaras250MeV/c.Thisextendedcoverage representsacrucialadvantage,sinceFaddeevcalculationsindicate that the manifestations of various wave-functioncomponents, as

well asthe potential effects of three-nucleon forces, imply very differentsignaturesasfunctionsofpm.

Ifbothbeamandtargetarefullypolarized,thecross-sectionfor the³He(^e,e^p)reactionhastheform

d

σ (

h

, 

S

)

d

 =

^d

σ

0

d





1

+ 

^S

· 

^A0

+

^h

(

Ae

+ 

^S

· 

^A

)

 ,

whered=^dedEedp isthedifferentialofthephase-spacevol- ume,

σ

0istheunpolarizedcrosssection,S isthespinofthetarget, and h is the helicity of the electrons. Here A⁰ and Ae are the target and beam analyzing powers, respectively, while the spin- correlation parameters A yield the asymmetries when both the beamandthetarget are polarized. Ifthe target ispolarizedonly inthe horizontal plane defined by the beamand scatteredelec- tronmomenta,theterm S· ^{A0 doesnotcontribute[20],while A}e issuppressedandisnegligiblewithrespectto A.

Theorientationofthetargetpolarizationisdefinedbythean- glesθ^∗ andφ^∗ inthe framewherethe z-axis isalong q andthe y-axisisgivenby p_e×p^_e.Themeasured asymmetryatgivenθ^∗ andφ^∗ isthen

A

(θ

^∗

, φ

^∗

) = 

^S

(θ

^∗

, φ

^∗

) · 

^A

= (

d

σ /

d

)

₊

− (

^d

σ /

d

)

₋

(

d

σ /

d

)

₊

+ (

^d

σ /

d

)

₋

,

where the subscript signs represent the beam helicities. In this paper we report on the measurements of these asymmetries in 3He(^e,e^p)d and³He(^e,e^p)pn processes.Themeasurementswere performedduring the E05-102 experiment atthe Thomas Jeffer- sonNationalAcceleratorFacility inexperimental Hall A[29],with a beam energy of 2.425GeV in quasi-elastic kinematics at en- ergytransfer

ω

≈^{140MeV and} four-momentumtransferof Q²= q²−

ω

²_≈0.25(GeV/c)².

In the experiment we utilized a continuous, longitudinally polarized electron beam with an average polarization of Pe= (84.3±².0)%. The beam polarization was measured periodically by a Møller polarimeter [29], and the given uncertainty is pre- dominantlysystematic.Thebeamcurrentswerebetween5 μAand 11 μA,chosen to ensurestableoperation in conjunctionwiththe polarizedtargetsystem.Thetargetwasa40cm-longglasscellcon- taining³He gas atapproximately 9.3bar (0.043g/cm²), polarized continuously by hybrid spin-exchange optical pumping [30–33].

TwopairsofHelmholtzcoils were usedtomaintain the in-plane targetpolarization directionalongthebeamlineandperpendicu- lartoit,asdictatedbyinstrumentalconstraints.Thiscorresponded to the angles 67^◦ and 156^◦ with respect to q, allowing us to measure A(67^◦,0^◦) and A(156^◦,0^◦), respectively. Electron para- magnetic andnuclear magnetic resonance [34–36] were used to monitorthe target polarization, Pt,which was between50% and 60% whencorrectedfordilutionduetonitrogen,closetothemax- imum polarization of 63% achieved without beam. The dilution factorwasdetermined byusinga referencecell filledwithunpo- larized³He anddifferentamountsofN₂,andmeasuringtherates atdifferentrelativepressures.

The scattered electrons were detected by a High-Resolution magneticSpectrometer (HRS) [29] positionedat12.5^◦,while the protonswere detected by thelarge-acceptance spectrometer Big- Bite placed at 75^◦ equipped with a detector package optimized forhadrondetection[37].Thereconstructedprotonmomentawere correctedforenergylossesinall materialsfromthetarget vertex tothedetectorpackage.Furtherdetails oftheexperimental setup and the procedure to extract the very pure sample of electron- protoncoincidenceeventsaregiveninRef. [5].

The experimental asymmetry for each orientation of the tar- get polarization was determined as the relative difference be- tween the number of background-subtracted coincidence events

Fig. 1. Theasymmetries A(67^◦,0^◦)(top)and A(156^◦,0^◦)(bottom)inthe quasi- elastic³He(e,e^p)process(2bbuand3bbucombined)asfunctionsofmissingmo- mentum,comparedtotheoreticalpredictions(green)showingthe2bbu(blue)and 3bbu (red) contributions as wellas the ratioof3bbu and 2bbu cross-sections (grey,rightaxis).FulllinescorrespondtoKrakow(K)calculations[23,24],whilethe dashedlinescorrespondtoHannover/Lisbon(H/L)calculations[25–28].Onlystatis- ticaluncertaintiesareshown.Forsystematical uncertaintiesandthemeaningofthe errorbandsseetext.

corresponding to positive and negative beam helicities, Aexp= (N₊−^N−)/(N₊+^N−), where N₊ and N₋ have been corrected forhelicity-gatedbeamchargeasymmetry,deadtimeandradiative effects.Thecorrespondingfinalvaluesofthephysics asymmetries werecalculatedas A=^Aexp/(PePt).

The resulting asymmetries as functions of pm are shown in Fig. 1. The largest contribution to their systematic error comes fromtherelativeuncertaintyinthe targetpolarization, P_t,which has been estimated at ±^{5%, followed by the} uncertainty in the target dilution factor (±^{2%) and the absolute} uncertainty of the beampolarization,Pe(±^{2%).Thebackgroundrates,determinedby} empty-cellmeasurements,weresmallerthan0.3%ofthetotal,re- sultingina0.1%systematicuncertaintyofthefinalasymmetry.The beam-chargeasymmetrywasdeterminedtobe(−⁰.2±¹.5)·^10−5. Other helicity-correlatedfalse asymmetries were evaluated to be lessthan0.1%,alsomuchsmallerthanthemeasuredphysicsasym- metry.Theuncertaintyinthetargetorientationanglerepresentsa minorcontribution(±^{0.6%) tothetotal}uncertainty,totalling≈^6%

(relative).

Fig.1alsoshowstheresultsofthestate-of-the-artthree-body FaddeevcalculationsoftheKrakow(K)[23,24] andHannover/Lis- bon (H/L) [25–28] groups. The K calculations are based on the

Fig. 2. Theextractedasymmetriesfor2bbu(left)and3bbu(right).Curvenotationas inFig.1,withtheadditionofthePisa2bbucalculation[40] intheleftpanel(blue dottedlineshiddenbeneaththefullanddashedlines).Seetextfordetails.

AV18nucleon–nucleonpotential[38] andinvolveacompletetreat- ment of FSI andthe dominant part of MEC, but do not include three-nucleon forces; the Coulomb interaction is taken into ac- count inthe ³He bound state. The H/L calculationsare basedon thecoupled-channelextension ofthecharge-dependentBonnpo- tential[39] and alsoinclude FSIandMEC, while the isobar is addedasanactive degreeof freedomprovidinga mechanismfor an effective three-nucleonforce and forexchange currents. Point Coulomb interaction is added in the partial wavesinvolving two charged baryons. In contrast to the K and H/L approaches, the Pisa(P)calculations[40] arebasedonavariationalpair-correlated hyper-spherical harmonic expansion that is comparable in preci- sion to the Faddeev methods and is expected to account for all relevant reaction mechanisms. The P calculations are based on the AV18interaction model(augmented by the UrbanaIXthree- nucleon force [41]), in which full inclusion of FSI is taken into account, as well as MEC. At present, the Pisa group only pro- vides 2bbu calculations. Coulomb interaction is included only in the bound state in K calculations, but in both bound and scat- tering statesin H/L and P calculations. All these calculationsre- producesufficientlywell the nuclearbindingenergies andcharge radii[26,27,42,43]. Dueto theextendedexperimental acceptance, alltheoreticalasymmetrieswere appropriatelyaveraged,resulting intheerrorbandsaround thetheoreticalcurvesinFig. 1.Details canbefoundin[5].

NeithertheKnortheH/Lcalculationreproducesthemeasured asymmetriestoasatisfactorylevel.Similarlytoourfindingsinthe deuteron channel, the theories approximatelycapture their over- all functional forms, but exhibit systematic vertical offsets of up to two percent.In all calculationsshown herea strong cancella- tionis involvedinobtaining each total asymmetry fromits2bbu and3bbu contributions,which are typically opposite insign and ofdifferentmagnitudes.Nevertheless,thefailureofthetheoriesto reproducethedatacanbetracedtothe3bbuasymmetryalone,as discussedinthefollowing.

Sincetheenergyresolutionofourmeasurement(about11MeV FWHM)wasinsufficienttodirectlydisentanglethe2bbuand3bbu channels,theindividualasymmetrieswereextractedbyrestricting thedata sample to pm≤^60MeV/c and studyingthe dependence ofA(67^◦,0^◦)andA(156^◦,0^◦)intermsoftheuppercutinmissing energy, E_m=

ω

−^Tp−⁷.7MeV. Thecomparisonofthemeasured Em spectrum (extendingbelow Em=^{0 due toresolution effects)} withthe simulatedone revealedthat in spiteof the overlapbe- tweenthe twochannels, thelowestportionofthedistribution at

Fig. 3. TheA(67^◦,0^◦)(fullsymbols)andA(156^◦,0^◦)(emptysymbols)asymmetries for2bbu(left)and3bbu(right)dividedbythecorrespondingasymmetriesforelas- ticep scattering atthesamevalueofQ²andforEm≤2.5MeV.Inbothpanelsthe data(circles)arecomparedtothecalculations(squares).Thetinyuncertaintieson thetheoreticalpointsareduetotheaveragingprocedure.

Em≤^{0 is dominated by 2bbu. There the 3bbu} contributes only 7% tothe totalcrosssection, thusoffering apossibilitytoextract the 2bbuasymmetry.The extractedasymmetry A_2bbu agreeswell withthe calculations(see Fig. 2(left)). At the sametime, a very smallresidualdifferencebetweentheexperimentalresultandthe- ory (0.5%inallcases)suggeststhatnearthethresholdthesizeof the 3bbuasymmetry is about1%, muchsmaller than thepredic- tions.Tostudythe3bbuasymmetryabovethethreshold,thedata at Em>0 were incrementally added tothe analysis. Considering that the measured asymmetries contain also the 2bbu contribu- tion,the3bbuasymmetry(Fig. 2(right))hasbeenextractedfrom thedataas

A_3bbu

= (

1

+

^R₃₂

)

Aexp

−

^A2bbu

R^₃₂

.

Here R^₃₂ is the 3bbu/2bbu cross-section ratioobtained fromthe R32showninFig.1byconsidering Em uptoaspecificmaximum value, with pm≈^{0, andcorrectingforfinite momentum andan-} gular resolutions aswell asradiativeeffects.Typically R^₃₂ ranges from0.20 to0.33 andisassumedtobewellundercontrolinboth KandH/Lcalculations, withanuncertaintyofabout10%.Theex- tractedasymmetriesareingoodagreementwiththetheoryinthe limit where the whole spectrum (Em≤^{50MeV) is considered in} the analysis, butstrongly deviatefromthe theory nearthreshold (E_m≤².5MeV)forthe3bbureactionchannel.

In an effort to compensate for the effect of spin orientation ofprotons insidethepolarized ³He nucleus,wehavedivided the nuclear asymmetriesby theasymmetries forelastic^ep scattering [44] atthe samevalue offour-momentum transfer;seeFig. 3.In a simplified picture ofthe ³He(^e,e^p)process, one would expect the2bburatioat pm≈^{0 tobe} −¹/3,corresponding totheeffec- tive polarization of the (almost free)proton inside the polarized 3He nucleus, while the 3bbu ratio should vanish because either of thetwo oppositely polarizedprotons could be knocked out in the process. Indeed,inthe 2bbu caseboth theexperimental and the predictedratios coincide almost perfectly, at the anticipated

“naive” value of −¹/3.On the other hand, in the3bbu case the predictionsclusterapproximatelyaroundunity(andapparentlyre- tain a residual dependence on θ^∗), while the two experimental ratiosaremuchsmaller(andmutuallyconsistent).

Inconclusion,wehaveprovidedtheworld-first,high-precision measurement of double-polarization asymmetries for proton knockout frompolarized³He nucleiattwo differentspin settings and over a broad range of momenta. Two state-of-the-art theo- retical approaches to the ³He disintegration process are able to approximately accommodate the main structural features of our data set.Since theasymmetries arerather smallandstrong can- cellations ofthe two-bodyand three-body breakup contributions are involved,the agreement can be deemed satisfactory and the theoreticalframeworkjustified.However,thehighprecisionofour

measurementshas beenableto reveala substantial deficiencyin the calculations of the three-body breakup process, presumably due to a mismatchbetween the true relativistic kinematics and non-relativisticspin-dependentnucleardynamicsemployedinthe calculations.

On the other hand, since the three-body breakup process is moreselectivethan thecorresponding two-bodybreakupof ³He, itwill beinteresting to investigateifan applicationofconsistent chiraltwo-nucleonandthree-nucleoninteractionswithchiraltwo- nucleon and three-nucleon contributions in the electromagnetic currentoperatorcouldalsoshedlightonthisproblem.

Acknowledgements

Wethank theJeffersonLab HallA andAccelerator Operations technicalstaff fortheir outstanding support. Thiswork was sup- ported in part by the National Science Foundation and the U.S.

Department of Energy. Jefferson Science Associates, LLC, oper- ates JeffersonLab forthe U.S. DOE underU.S. DOE contract DE- AC05-06OR23177. Thiswork wassupported in partby theSlove- nian Research Agency (research core funding No. P1–0102). This workisapartoftheLENPICprojectandwassupportedbythePol- ishNationalScienceCentreunderGrantsNo.2016/22/M/ST2/00173 and 2016/21/D/ST2/01120. The numerical calculations of the Krakow group were partially performed on the supercomputer clusteroftheJSC,Jülich,Germany.

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