Astrophysical S-factor for the 3 He( α , γ ) 7 Be reaction via the asymptotic normalization coeﬃcient (ANC) method

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Physics Letters B

www.elsevier.com/locate/physletb

Astrophysical S-factor for the ³ He( α , γ ) ⁷ Be reaction via the asymptotic normalization coeﬃcient (ANC) method

G.G. Kiss

^a

, M. La Cognata

^b^,∗

, C. Spitaleri

^b^,^c

, R. Yarmukhamedov

^d

, I. Wiedenhöver

^e

, L.T. Baby

^e

, S. Cherubini

^b^,^c

, A. Cvetinovi ´c

^b

, G. D’Agata

^b^,^c^,^f

, P. Figuera

^b

, G.L. Guardo

^b^,^c

, M. Gulino

^b^,^g

, S. Hayakawa

^b^,^h

, I. Indelicato

^b^,^c

, L. Lamia

^b^,^c^,ⁱ

, M. Lattuada

^b^,^c

, F. Mudò

^b^,^c

, S. Palmerini

^j^,^k

, R.G. Pizzone

^b

, G.G. Rapisarda

^b^,^c

, S. Romano

^b^,^c^,ⁱ

, M.L. Sergi

^b^,^c

, R. Spartà

^b^,^c

, O. Trippella

^j^,^k

, A. Tumino

^b^,^g

, M. Anastasiou

^e

, S.A. Kuvin

^e

, N. Rijal

^e

, B. Schmidt

^e

,

S.B. Igamov

^d

, S.B. Sakuta

^l

, K.I. Tursunmakhatov

^d^,^m

, Zs. Fülöp

^a

, Gy. Gyürky

^a

, T. Szücs

^a

, Z. Halász

^a

, E. Somorjai

^a

, Z. Hons

^f

, J. Mrázek

^f

, R.E. Tribble

ⁿ

, A.M. Mukhamedzhanov

ⁿ

aInstituteforNuclearResearch(ATOMKI),H-4001Debrecen,POB.51,Hungary bIstitutoNazionalediFisicaNucleare,LaboratoriNazionalidelSud,95123Catania,Italy cDipartimentodiFisicaeAstronomia“E.Majorana”,UniversitàdiCatania,95123Catania,Italy dInstituteofNuclearPhysics,UzbekistanAcademyofSciences,100214Tashkent,Uzbekistan eDepartmentofPhysics,FloridaStateUniversity,Tallahassee,FL 32306,USA

fNuclearPhysicsInstituteoftheCzechAcademyofSciences,25068ˇRež,CzechRepublic gFacoltàdiIngegneriaeArchitettura,UniversitàdiEnna“Kore”,94100,Enna,Italy

hCenterforNuclearStudy(CNS),UniversityofTokyo,RIKENcampus,Saitama351-0198,Japan iCentroSicilianodiFisicaNucleareeStrutturadellaMateria(CSFNSM),95123Catania,Italy jDipartimentodiFisicaeGeologia,UniversitàdiPerugia,06123Perugia,Italy

kIstitutoNazionalediFisicaNucleare,SezionediPerugia,06123Perugia,Italy lNationalResearchCenter“KurchatovInstitute”,Moscow123182,Russia

mPhysicalandMathematicalDepartment,GulistanStateUniversity,120100Gulistan,Uzbekistan nCyclotronInstitute,TexasA&MUniversity,CollegeStation,TX 77843,USA

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

Articlehistory:

Received9April2020

Receivedinrevisedform11June2020 Accepted1July2020

Availableonline7July2020 Editor: B.Blank

Keywords:

Nuclearastrophysics Nucleosynthesis

Thedetectionofthe neutrinosproducedinthe p−^p^chainândⁱⁿ^the^CNO^cycle^can^beûsed ^to^test theStandardSolarModel.The³He(α,γ)⁷Bereactionisthefirstreactionofthe2nd and3rd branchof thep−^p^chain,^therefore,^theuncertaintyofitscrosssectionsensitivelyinfluencesthepredictionofthe 7Beand⁸Bneutrinofluxes.Despiteitsimportanceandthelargenumberofexperimentalandtheoretical works devotedtothisreaction,theknowledgeonthereactioncrosssectionatenergiescharacterizing the core of the Sun (15 keV - 30 keV) is limited and further experimental efforts are needed to reachthe desired(≈^3%)âccuracy. ^The^precise^knowledgeôn^theêxternal^capturecontributiontothe 3He(α,γ)⁷Bereactioncrosssectioniscrucialforthetheoreticaldescriptionofthereactionmechanism.

InthepresentworktheindirectmeasurementofthisexternalcapturecontributionusingtheAsymptotic NormalizationCoeﬃcient(ANC) techniqueisreported.To extracttheANC,theangulardistributionsof deuteronsemittedinthe⁶Li(³He,d)⁷Beα-transferreactionweremeasuredwithhighprecisionatE3He= 3.0MeVand E3He=5.0MeV.TheANCswerethenextractedfromcomparisonofDWBAcalculationsto themeasureddataandthezeroenergyastrophysicalS-factorfor³He(α,γ)⁷Bereactionwasfoundtobe 0.534±^0.025^keVb.

©²⁰²⁰^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

*

Correspondingauthor.

E-mailaddresses:ggkiss@atomki.mta.hu(G.G. Kiss),lacognata@lns.infn.it (M. La Cognata),rakhim@inp.uz(R. Yarmukhamedov).

1. Introduction

The ³He(

α

,

γ

)⁷Be is oneof thekey reactionsin nuclearastrophysics, which remained critical after decades, despite the large numberofexperimentalandtheoreticalstudiesdevotedtoit.This ispredominantly duetothefact that theastrophysically relevant

https://doi.org/10.1016/j.physletb.2020.135606

0370-2693/©²⁰²⁰^TheÂuthor(s).^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

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energyregion, the so-calledGamowwindow, lies betweenabout 15 keV and 30 keV for a temperature of 15 MK, characterizing thecoreof theSun,andatthesetemperatures the³He(

α

,

γ

)⁷Be reaction cross section is far too small to be measured directly.

Theory-basedextrapolations arethereforenecessarytoobtainthe reactionrate[1–3].Thereactionisalsoimportantforunderstand- ingthelithiumproblemoftheBigBangNucleosynthesis,atener- giesaround100keV(see[4] andreferencestherein).

Thedetectionoftheneutrinoscomingdirectlyfromthecoreof theSunbecamemore andmorepreciseafter theconstruction of larger andmoreefficient neutrinodetectors, sensitive toa wider neutrinoenergyrangearoundtheturnofthecentury.Theseneu- trinosarereleasedintheβdecayofthe⁷Be,⁸B,¹³N,¹⁵Oisotopes produced inthe p−^p ^chain ândⁱⁿ^the ^CNO^cycle. ^Recently,^the fluxofthep−^p^neutrinos^was^measured^withâ^precisionôfâbout 3.4%bytheBOREXINO,SNOandSuper-Kamiokandecollaborations [5–7]. The precise neutrinoflux measurements canconstrain the StandardSolarModel (SSM)andprovideinformationonthe core temperatureoftheSunifthe relevantnuclearreactioncrosssec- tionsareknownwithmatchingaccuracy.However,atpresentthe uncertainties oftheseinput parameters arefartoo high,typically oftheorderof5-8%[8] (orevenhigher,seebelow)contrarytothe 3%precisionrequired[9,10].

The³He(

α

,

γ

)⁷Be reactionis thefirstreaction ofthe2nd and 3rd p−^p ^chain^branch ând^therefore ^the uncertaintyof its rate stronglyinfluencestheprecisionofthepredictedfluxoftheafore- mentioned⁷Be,⁸Bneutrinos.Thus,animprovementontheknowl- edgeofthelow-energycrosssectionofthisreactionwouldresult inasubstantialreductionoftheuncertaintiesofthesolarneutrino fluxandmighthaveimportantconsequencesfortheSSM.

Not surprisingly the ³He(

α

,

γ

)⁷Be is among those reactions whichwerethemostintensivelystudiedinthepastandtheresults wereextensivelydiscussedinreviewpapers[1–3] (andreferences therein). Because of insuﬃcient experimental information to as- sesstheir systematicerrors,inthemostrecentcompilationsonly datacollectedafter2004aretakenintoaccount[2,3].Theexperi- mentalmethodsused inthe“modern”studiescan besortedinto three groups: thedetection of prompt

γ

rays [11–14], the mea- surementofthe⁷Beactivity[15–19],andthecountingofthe⁷Be recoilswitharecoilmassseparator[20].Regardingthetheoretical description, severaldifferent models - including external capture models (e.g. [21]), potential models (e.g. [22,23]), modiﬁed two- body potential approach [24], resonating group calculation (e.g.

[25]), abinitio models (e.g. [26,27]) andR-matrix theory [28,29]

- wereusedtodescribethereaction.

The recommendedzero energyastrophysical S-factorvalue of [2], derived using the microscopic calculations of [27,30] and rescaledtoﬁtthedataatE≤^{1 MeV}^is^S34(0)=0.56±^0.02^(exp)

±^0.02 ^(theory)^{keV b.}^The^sameexperimentaldatasetwas fitted usingthemodified two-bodypotential approachandsignificantly larger S34(0) =0.613⁺₋⁰₀^._.⁰²⁶₀₆₃ keV b factorwas found [24]. Exclud- ingdatasetII[12,20] wouldleadto S₃₄(0)=0.562±⁰.008 keVb (see[24] formoredetails),whichisthevalueshowninFig.1.The comprehensiveR-matrixextrapolation[29],includingnotonlythe datafittedin [2] and [24] but alsothe recentlymeasured higher energycrosssections[14,17,18],resultedinanS34(0)value(0.542

±^0.011^(Monte^Carlo^ﬁt)±^0.006^(model)⁺₋⁰₀^._.⁰¹⁹₀₁₁^(phase^shift)^keV b)3.2% lowerthanthe onerecommendedin[2]. Furthermore,ab initiono-core shellmodelwithcontinuum approachwas usedto predict S34(0)anda value (S34(0)=0.59keVb) 5.3%largerthan the one of [2] was found [31], almost the same aspredicted by [26] (S₃₄(0)=0.593keVb). Finally,recentlyeffectiveﬁeld theory wasalsousedtoperformextrapolationandavalue(S34(0)=0.578 keVb)3.2%largerthantheoneof[2] wasfound[32].

Whiletheprecisionoftheextrapolationsis oftheorderof6-7%, the difference between the S34(0) values exceeds 10%. The pre-

Fig. 1.Summaryofthemostrecent³He(α,γ)⁷BeS34(0)factorresults:derivedfrom theanalysisofelasticscatteringangulardistributions[23] (pinkstar),theoretical calculations[26,31] (darkredtriangle),extrapolationsofexperimentaldatasets[14, 24,29,32] (bluestar),predictionbasedonneutrinoyieldmeasurement[56] (green box)andderivedusingtheANCtechnique(presentwork,reddiamond)Thesolid centrallinerepresentstherecommendedvalueof[2],withitsuncertaintyindicated withtheshadedarea.ForTursunmakhatov etal.[24],theS34(0)valueobtainedby ﬁtting[11,13,15,16] isshown.

dicted S34(0)factorsareshownFig.1.Itisclearthatthecalculated S34(0)factorsdependstronglyonthemodelusedintheextrapola- tions andhighprecisionexperimentaldataisneededtoconstrain thetheoreticalmodels.

Here we present the resultsof a new approach, proposed by A.M.Mukhamedzhanov,wheretheS34(0)factorofthe³He(

α

,

γ

)⁷Be reaction was derived without extrapolation,usingthe asymptotic normalizationcoeﬃcient (ANC)technique [33].Namely, since the 3He(

α

,

γ

)⁷Be reaction atstellar energies isa pure external direct captureprocess[2],itessentiallyproceedsthroughthetailofthe nuclearoverlapfunction.Therefore,theshapeoftheoverlapfunc- tion inthe tailregion is determinedby the Coulomb interaction, thustheamplitudeoftheoverlapfunctiondetermines therateof the capture reaction [34,35]. Since the direct capture cross sections are proportional to the squares of the ANCs - which are found from transfer reactions - with the study of the near bar- rier ⁶Li(³He,d)⁷Be

α

particle transfer reaction the ANCs for the 3He(

α

,

γ

)⁷Be reaction can be obtained. This independent experimental approach, improving gradually our understanding on the low energybehavior of this reaction,was up-to-now never used to study the ³He(

α

,

γ

)⁷Be reaction. Furthermore,the ANC values are also neededfor the R-matrix calculations. In [29] these val- ueswere deducedfromexperimental crosssectionsandfound to be between3-5.5 fm⁻¹. Accordingly,the independentdetermina- tionoftheANCvaluesalsoincreasestheprecisionoftheR-matrix extrapolations.

2. Experimentaltechnique

The angular distributions of the deuterons emitted in the 6Li(³He,d)⁷Be reaction were measured in two experiments performed using the 3.1 MV single ended coaxial singletron accel- erator of theDepartment ofPhysics andAstronomy (DFA)of the University of Cataniaandthe FNtandem acceleratoratthe John.

D.FoxSuperconductingAcceleratorLaboratoryattheFloridaState University (FSU), Tallahassee, USA. The energy of the ³He beam was E_Lab =3MeVandE_Lab =5MeV,withbeamcurrentstypically between20enAand30enA,respectively.Thesetupinbothexper- imentsconsistedofseveralE−^E telescopes,placedonrotatable turntables anda monitordetector ﬁxed at 165^◦ (DFA) and150^◦

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Fig. 2.E−Ê ^spectrum^measured^with â^silicon^telescope ^positionedât ϑlab = 124.64^◦atElab=5MeVbeamenergy.Thepeaks(markedwithd0andd1)usedfor theanalysisareindicated.Theinsetshowsthedeuteronspectrumdeducedfrom thisidentificationplot.IntegrationisperformedusingGaussianfittingtoremove thresholdproblems.Theredboxisusedtohighlighttheregionofinterestonly.

(FSU)withrespect to thebeam direction. The thicknesses ofthe E detectorswerebetween8 μmand16 μmandthethicknesses of the E detectors were 500 μm. In the experiment performed atDFAa99% compoundpurity,134 μg/cm² thick ⁶LiFtarget(en- richedin95%of⁶Li)wasused.IntheexperimentperformedatFSU the99%compoundpurity,57 μg/cm²thicklithiumtarget(enriched in98%of⁶Li)waspreparedonaFormvarbackingandtransferred tothescatteringchamber inasealedcontainer undervacuumto preventoxidation.Furthermore,asitwillbe discussedinthefol- lowingin thisexperimenttwo additional detectorswere used to monitorthetargetthicknessandforabsolutenormalization.Inthe two experiments, the yield of the emitted deuterons were mea- suredbetween23.0^◦ ≤ϑc.m.≤^172.5^◦^at^ELab =3MeVand23.2^◦

≤ϑc.m.≤^168.5^◦^at^ELab =5MeV,usingtypically6^◦ - 10^◦steps.

Inboth experiments theparticle identificationwas performed usingthestandard E−Ê ^technique ând^the^peakâreas^{- corre-} spondingto the⁷Be groundand1st excited state- were derived byfittingGaussianfunctions. Anexampleofthetwo-dimensional particle identification plots is shown in Fig. 2, and the one di- mensionaldeuteron spectrumispresented initsinset (thepeaks correspondingtothe⁷Begroundand1st excitedstatearemarked withd0 andd1, respectively). It can be seen that the separation ofthedifferentisotopesissufficientforreliableidentification.The sameprocedure was used for absolutenormalization in the two measurements. Namely, atfirst the solid angles of the detectors were derived from the knowngeometry andwere cross-checked using radioactive sources with known activity. Furthermore, the cross section as a function of the angle of the outgoing particle inthe ⁶Li(³He,p) reaction (Q =16.79 MeV)is well known [36], thustherateofthe highenergyprotonswas measuredusingthe monitordetectorplaced ata fixed positionat backwardangle to reconstructthenumberofimpinging³Heparticlesandthetarget thickness. At the experiment performedat FSU two further nor- malizationtechniqueswere used. Following theapproach of [35]

the yield of the ³He induced reactions and elastic ³He scatter- ingon⁶Liweremeasuredwiththeforwardmonitordetectors(M1 andM2).Moreover,by placingeach telescopeat95^◦ withrespect to the beam axis and measuring the ⁶Li(p,p)⁶Li elastic scattering, thetarget thicknessand thetelescope solid angles could be determined, since the ⁶Li(p,p) elastic scattering cross section at 95^◦ with E_p =6.868 MeVprotonbeamwas previously measured with a 3% precision [37]. This approach was used in the previ- ousexperimentsperformedatFSU,seee.g.[38,39].Asaresult,the uncertainty of the absolute normalization was found to be 5.7%

whichcontainsuncertainties fromthetargetthicknessdetermina-

Fig. 3.Angulardistributionsofthe⁶Li(³He,d)⁷Be reactionpopulatingtheground ((a)and(c))andﬁrst(0.429MeV)excited((b)and(d))statesof⁷Be attheprojectile 3He energiesof3((a)and (b))and5((c)and (d))MeV.Error barsaresmaller thanthesizeofthepoints.Graylinesarethecalculatedangulardistributionsas describedinthetext,for p−andα−transfer(forwardandbackwardhemisphere, respectively).

tion,thecurrentmeasurementandthesolid angledetermination.

ExperimentalangulardistributionsareshowninFig.3.

3. DataanalysisandextractionoftheANCforthe

α

+³^He→⁷^Be system

The theoretical analysis of the data was carried out in the framework of the modiﬁed Distorted Wave Born Approximation (DWBA) [40] assuming one step proton and

α

particle transfer [41].

Accordingly — assuming ³He = (d+^p), ⁷^{Be =} ⁽⁶^Li+^p), ⁶^{Li =} (d+

α

)and⁷Be =(³He+

α

)—forﬁxedvaluesofl_{d p}, j_{d p},l_dα and jdα,thedifferentialcrosssection(DCS)fortheperipheraltransfer ofan“e-particle”(whereestandsforpor

α

)inthe⁶Li(³He,d)⁷Be reactioncanbewrittenintheform:

d

σ

d

=

jAe

C²_Ae_;_j

AeR⁽_e^DWBA_;_j ⁾

Ae

(

Ei

, θ ;

^bye;^jye

,

bAe;^jAe

),

(1) R⁽_e^DWBA_;_j ⁾

Ae

(

E_i

, θ ;

^bye;jye

,

b_Ae_;_j_Ae

) =

C²_ye_;_j

ye

σ

_e⁽_;^DWBA_j ⁾

Ae

(

E_i

, θ;

^bye;^jye

,

b_Ae_;_j_Ae

)

b²_ye_;_j

yeb²_Ae_;_j

Ae

,

(2)

where B= A+e and x= y+e;

σ

_e⁽_;^DWBA_j ⁾

Ae is the single-particle DWBA cross section [42], lAe and jAe are the orbital andtotal angular momenta of the transferred particles, Cs are the ANCs for A+ e→^Bând ^y+ê→^x,^which^determine^theâmplitudesôf^the^tails ofthe radial B andx nucleuswave functionsin the(A+ê) ând (y+ê)^channels^[43];^b^sâre^thesingle-particleANCsfortheshell- modelwavefunctionsforthetwo-body[B=(A+ê)ând^x=⁽^y+ê)]

boundstates,whichdeterminetheamplitudesoftheir tails; Ei is the relative kinetic energy ofthe collidingparticles and θ is the center-of-mass scatteringangle. The negligible contributionof d- waves (l_{d p}= 2 and l_dα= 2) is ignored owning to their smallness [43,44].

Eqs. (1) and(2) areusedseparatelyfortheonestep

α

particle exchange reaction andforproton transferreaction (the latter re-

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sultswillbepublishedelsewhere).The swaveANCvaluesforthe d+^p→³^{He and}^the ^d+

α

_→⁶Li are 4.20±^{0.32 fm}⁻¹ ^{[45] and} 5.43±^{0.37 fm}⁻¹ ^[46],respectively.Accordingto[40,47],thevalues ofthreeparametersb_{d p}_;_j_{d p} forl_{d p}=0and j_{d p}=1/2aswell asof b_dα;jdα forl_dα=0and j_dα=0wereﬁxedbyreproducingthecor- respondingANCvaluesenteringtheR⁽_e^DWBA_;_j ⁾

Ae (E_i,θ;^bye;^jye,b_Ae_;_j_Ae) functioncalculatedseparatelyfortheonestepprotontransferand

α

particleexchangemechanisms.

Atthebackwardhemispheretheexperimentaldifferentialcross sectionincreaseswithincreasinganglesandthisﬁndingconﬁrms the presence of a dominant one-step

α

-particle exchange mechanism. Similarly, theone-step proton transfer isdominant inthe forwards hemisphere. Thus, the interference of the two mecha- nismsatsmall(forward)andlarge(backward)anglesisnegligible.

Accordingly, the ANCs for ³He+

α

_→⁷Be and for ⁶Li+^p→⁷^Be were extracted separately within the post form of the modiﬁed DWBA[40] usingtheLOLAcode[42].

First,eight sets ofoptical potentials,obtainedfromthe global parametersets of[48,49],inthe input andoutput channelswere testedandthe one,providingthebestdescription forthe experi- mentaldata,was usedforthefurtheranalysis. Then,thegeomet- rical parameters r0 and a of the Woods-Saxon potential (having the Thomas spin-orbit term) of the two-body ⁷Be [(⁶Li+^p) ^or (³He+

α

)]bound state wave function were varied inthe ranges of1.13≤^r0≤^1.40 ^fm ând^0.59≤â≤^0.72^fm ând^the ^depthôf^the potentialwell was adjustedtofitthecorresponding experimental bindingenergyforeach(r0,a)pair.

To test the peripheral nature of the reaction, the geometrical parameters r₀ anda ofthe Woods-Saxon potential ofthe bound statewave functionwerevaried withintherangesasabove(similarly to [50]) and the resulting R⁽_p^DWBA_;_j ⁾

6 Lip and R⁽α^DWBA;^j_{3 He}⁾_α functions were found to change within about ±^{7% at} ^varying ^the (r0,

α

) pair in the intervals above, for each chosen experimental point of center-of-mass scattering angle θ. By normalizing the calcu- latedDCSs to theexperimental onesforeach experimental point (θ=θ^exp) separatelyfortheforwardandbackwardangleregions, the“indirectlydetermined”valuesoftheANCsfor³He+

α

→⁷^Be andfor⁶Li+^p→⁷^{Be without} ând^with ^takingîntoâccount ^the channelscouplingeffects(CCE)werederived.

The CCE contributions to the DWBA cross sections for each experimental pointof θ^exp – belongingto thebackward andfor- ward peak regions – were determined using the FRESCO code [51] by taking intoaccount onlyone stepprocesses with proton stripping ⁶Li(³He,d)⁷Be and exchange mechanism with the

α

- particleclustertransfer⁶Li(³He,⁷Be)d.Theninenucleons,present in the entrance channel, were replaced by three subsystems: i) 3He+⁶^Li(g.s., ^J^π⁼¹⁺^; ^E^∗⁼^{2.185 MeV,} ^J^π⁼ ³⁺^); ⁱⁱ⁾^d+⁷^Be(g.s., Jπ = 3/2⁻; E^∗= 0.429 MeV, Jπ= 1/2⁻) — p−^transfer ^— ^andⁱⁱⁱ⁾

7Be(g.s., Jπ =3/2⁻; E^∗=0.429 MeV, Jπ=1/2⁻)+d—

α

−^transfer.

Allstatesofthesubsystemsii)andiii) arecoupledwiththesub- systemi)by thereactions withprotons and

α

-particlestransfers.

Couplingsbetweengroundandexcitedstatesofnuclei⁶Li and⁷Be were calculated using the rotational model with the form factor Vλ(r)=(δλ/√

4

π

)dU(r)/drforquadrupoletransitions(λ=2).Here, δλisadeformationlength,whichisdeterminedbyδλ=βλR,where R and βλ are the radiusof thenucleusand thedeformation pa- rameter,respectively.Thereorientationeffects,determinedbythe matrixelement<E, Jπ|^V2|^E,Jπ>[51],werealsoincludedinthe couplingscheme.Thedeformationlengthsδ2 weretakenequalto 3.0fmfor⁶Li,and2.0for⁷Be,whichcorrespondtoβ2 =0.73and β2 =1.0,respectively[50,52,53].

The spectroscopic factors for the ³He and ⁶Li nuclei in the (d+^p)^and⁽

α

₊d)conﬁgurations,respectively,areﬁxedusingthe correspondingANCvaluesmentionedabove.Theyarefoundtobe 1.16and0.94,respectively.Whereas,thespectroscopicamplitudes

forthe ⁷Be nucleusinthe (⁶Li+^p) ^and⁽

α

+³^He)conﬁgurations aretakenfrom[54].Nevertheless,theratiooftheDCSscalculated with and without the CCE contribution (deﬁning the CCE renor- malization factor for the ANCs from Eq. (1), calculated for each scatteringanglebelongingtothemainpeakoftheangulardistri- butions, asdonein[50]),doesnotdepend onthesespectroscopic factors.

ThevaluesofthegeometricparametersoftheWoods-Saxonpo- tential,usedtocalculatethetwo-bodyboundstatewavefunctions, weretakenasin[52].Forthed−⁶^{Li and}^d−³He core-coreinterac- tions intheprotontransferand

α

-particleexchangemechanisms, theopticalpotentialsadoptedfortheentrance(⁶Li+³^He)^channel andtheCoulomb componentforthed−³He potentialwereused, respectively.

TheCCEcontributionenhancestheANCvaluesfrom22%to47%

andupto10.9%for³He+

α

→⁷^Be(g.s)^and³^He+

α

→⁷^Be(0.429 MeV),respectively,withrespecttotheDWBAcalculation,andfrom 1.0%to6.0%andfrom1.6%to12%for⁶Li+^p→⁷^Be(g.s.)ând⁶^Li+ p→⁷^Be(0.429^MeV), respectively. Inparticular, Fig. 3 showsthe calculatedDCSs,normalizedtothecorrespondingmainpeakofthe angular distributions at θ=θ_peakêxp,compared to the experimental results. Foreach experimental angulardistribution, labelledfrom (a)to(d),twocurvesareshown,fortheforwardandthebackward angles,correspondingtop−ând

α

−^particle^transfer,respectively.

The weighed mean values of the square of the ANCs for the 3He+

α

_→⁷Be(g.s.) and ³He+

α

_→⁷Be(0.429 MeV) are equal to C²= 20.84 ± ^1.12 ^[0.82; ^0.77] ^fm⁻¹ ^and ^C²⁼ ^12.86 ± ^0.50 [0.35; 0.36] fm⁻¹, respectively, which are in an excellent agree- ment with thoseof [24] derived fromthe analysisof the experimental S−^factor ^data ^of [11,13,15,16]. The overall uncertainties given herecorrespond to the errors combined in quadrature,including both experimental uncertainties in the d

σ

êxp/d (first terminsquare parentheses)andtheuncertaintycorresponding to theANCford+⁴^He→⁶^Li,âs^wellâs^theuncertaintiescharacter- izingtheR⁽α^DWBA;^j_{3 He}⁾_α function(secondterminsquareparentheses).

4. Summary

Thedirectcapturecontributiontotheastrophysicallyimportant 3He(⁴He,

γ

)⁷Be reaction cross section at energies corresponding to the core temperature of the Sun was derived using the ANC technique. The angular distributions of deuterons emitted in the 6Li(³He,d)⁷Be

α

-transfer reaction were measured with high precision at E3He = 3.0MeV and E3He = 5.0 MeV and the weighed means of the ANCs were used to calculate the total astrophysical S−^factorât^stellarênergies^(including Ê ⁼^0).^Thecalculations wereperformedwithinthemodifiedtwo-bodypotentialapproach framework [46,55],andtheresulting S3 4(0)andS3 4(23keV)fac- torswerefoundtobe S3 4(0)=0.534±^0.025^[0.015;^0.019]^keVb andS3 4(23keV)=0.525± ^0.022^[0.016;^0.016]^keVb.

While theANC approachis well established sincedecades (as discussedinthe recentreview [58]),additionalworkisnecessary toaddressspeciﬁcissuesandimprovetheaccuracyofthepresent paper. Among others, the uncertainty introduced by the use of one-stepprocess inmodellingthetransfer,thecouplingsbetween groundandexcitedstatesof⁶Liand⁷Be,andtheneedofcoupled- channel analysisto derive the ³He+⁴^{He and} ^the ^p+⁶^{Li ANCs.}

The present result provides a completely independent conﬁrma- tionofthecrosssection-basedextrapolationof[14,24,29],andthe deduced ANCsforthe p+⁶^{Li system} ^further^support^the^present result. Moreover, the indirectly derived ANC values can also be used in future R-matrix extrapolations to increase the precision andthereliability,sincethey supplyadditionalconstraintsonthe R-matrixanalysis[57].

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Table 1

SummaryofalluncertaintiesenteringtheevaluationoftheANCoftheα+³He→⁷Be system.Moredetails aregivenin AppendixA.

α+³^He→⁷^Be C_α²[fm⁻¹] E3He

[MeV]

E^∗ [MeV]

θ [deg]

without CCE

with CCE

⁽exp¹⁾

%

⁽exp²⁾

%

exp

% th

% tot

%

1 2 3 4 5 6 7 8 9 10

3.0 0.0 158.4 14.66±1.57[1.18;1.03] 17.87±1.91[1.44;1.25] 4.3 6.8 8.0 7.0 10.7 162.0 17.36±1.85[1.39;1.22] 21.49±2.29[1.72;1.50] 4.2 6.8 8.0 7.0 10.6 164.9 17.68±1.91[1.46;1.24] 22.17±2.40[1.83;1.55] 4.6 6.8 8.0 7.0 10.8 5.0 154.7 14.99±1.57[1.17;1.05] 20.40±2.14[1.59;1.43] 3.8 6.8 7.8 7.0 10.5 158.1 14.69±1.55[1.16;1.03] 21.63±2.29[1.71;1.51] 4.2 6.8 7.9 7.0 10.6 161.7 15.88±1.59[1.14;1.11] 23.03±2.31[1.65;1.61] 2.2 6.8 7.2 7.0 10.0 3.0 0.429 160.0 10.71±1.10[0.80;0.75] 11.60±1.19[0.87;0.81] 3.2 6.8 7.5 7.0 10.2 163.3 11.67±1.20[0.88;0.82] 12.70±1.31[0.96;0.89] 3.2 6.8 7.5 7.0 10.3 166.3 10.81±1.25[0.83;0.76] 11.83±1.23[0.91;0.83] 3.6 6.8 7.5 7.0 10.4 169.4 12.61±1.32[0.98;0.88] 13.90±1.45[1.08;0.97] 3.7 6.8 7.7 7.0 10.4 172.5 12.31±1.26[0.92;0.86] 13.65±1.40[1.02;0.96] 3.1 6.8 7.5 7.0 10.2 5.0 155.6 13.80±1.56[1.22;0.97] 12.86±1.45[1.14;0.90] 5.7 6.8 8.9 7.0 11.3 158.9 13.34±1.39[1.06;0.90] 12.87±1.34[1.02;0.90] 4.0 6.8 7.9 6.8 10.4 162.0 13.74±1.39[1.00;0.96] 13.72±1.39[1.00;0.96] 2.6 6.8 7.3 7.0 10.1 165.4 13.32±1.46[1.12;0.93] 13.74±1.51[1.16;0.96] 5.0 6.8 8.4 7.0 11.0 168.5 14.73±1.84[1.32;1.03] 15.69±1.96[1.40;1.10] 5.8 6.8 9.0 7.0 12.5

weighted mean values

3.0+5.0 0.0 15.68±0.74[0.51;0.53] 20.84±1.12[0.82;0.77] 3.9 3.7 5.4

3.0 16.32±1.41[1.00;1.00] 20.13±1.97[1.39;1.39] 6.9 6.9 9.8

5.0 15.18±0.91[0.67;0.61] 21.66±1.10[0.95;0.87] 4.3 4.0 5.0

3.0+5.0 0.429 12.12±0.62[0.43;0.44] 12.86±0.50[0.35;0.36] 2.7 2.8 3.9

3.0 10.84±0.60[0.36;0.36] 11.80±0.62[0.44;0.44] 3.7 3.7 5.2

5.0 13.74±0.66[0.50;0.43] 13.62±0.69[0.50;0.48] 3.7 3.5 5.1

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Acknowledgements

Thiswork wassupported by INFN (IstitutoNazionale di Fisica Nucleare), by NKFIH (NN128072, K120666), and by the ÚNKP- 19-4-DE-65New NationalExcellence Program of the Ministry of HumanCapacities ofHungary, andsupported in partby theNa- tionalScience Foundation, GrantNo. PHY-1712953 (USA), andby the University of Catania (Finanziamenti di linea 2 and Starting grant 2020). G.G. Kiss acknowledges the support fromthe János Bolyairesearch fellowship oftheHungarianAcademy ofSciences.

R.YarmukhamedovandK.I.Tursunmakhatovacknowledgethesup- port from the Academy of Sciences of the Republic of Uzbek- istan. J. Mrázek and G. D’Agata acknowledge the support from MEYS Czech Republic under the project EF16_013/0001679.A.M.

MukhamedzhanovacknowledgessupportfromtheU.S.DOEGrant No.DE-FG02-93ER40773 andNNSAGrant No. DENA0003841.The authorsacknowledgethesupportofprof.M.G.Grimaldiandofthe technicalstaffoftheDFA.

Appendix A

In Table 1 we show the squared ANCs and their uncertainties(Cα²) forthe

α

+³^He→⁷^{Be system,}ôbtained ⁱⁿ^the ^present work without and with the CCE contributions, for each experimental point of center-of-mass angle θ, at E3He = 3.0 and 5.0 MeV, andtheir weighed meanvalues, forboth ⁷Be ground state (E^∗=0.0 MeV; Jπ= ³₂⁻⁾ ând ^first êxcited ^state ^(E^∗=0.429 MeV;

Jπ=¹₂⁻^).

Thenumbers insquare bracketsare the experimental(C²_exp) and theoretical (C_th²) uncertainties, respectively. They are cal-

culated as follows: the experimental uncertainty is the sum of two contributions, ⁽exp^tot⁾= [(⁽exp¹⁾)²+(⁽exp²⁾)²]¹^/²^,^ﬁrst ^one ^being theuncertaintyontheexperimentalangulardistributions:⁽_exp¹⁾ = [(^d

σ

^exp/d)]/[^d

σ

êxp/d]^, ând ^the ^second ône îs ^linked ^to ^the uncertainty on the

α

+^d→⁶^{Li ANC:} ⁽exp²⁾ =(C^exp)²/(C^exp)². The theoretical uncertainty, corresponding to the effects of the non-peripherality, is calculated from the R-functions (2): th= R^DWBA/R^DWBA (for ease of reading, all subscripts are neglected here). In detail, the uncertainty on the R-function is calculated varying the geometrical parameters (r0 and a) of the adopted Woods-Saxonpotentialwithintheintervalsmentionedinthetext.

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