Li astrophysical S factors from the no-core shell model with continuum

HAL Id: in2p3-01334698

http://hal.in2p3.fr/in2p3-01334698

Submitted on 21 Jun 2016

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Li astrophysical S factors from the no-core shell model

with continuum

J. Dohet-Eraly, P. Navrátil, S. Quaglioni, W. Horiuchi, G. Hupin, F. Raimondi

To cite this version:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

3

_He

₍

_α

_,

_γ

₎

7

_{Be and}

3

_H

₍

_α

_,

_γ

₎

7

_{Li astrophysical}

_{S factors}

_from

_the

_no-core

shell

model

with

continuum

Jérémy Dohet-Eraly

a

,

∗

,

Petr Navrátil

a

,

Sofia Quaglioni

b

,

Wataru Horiuchi

c

,

Guillaume Hupin

b

,

d

,

1

,

Francesco Raimondi

a

,

2

a_{TRIUMF,4004WesbrookMall,VancouverBCV6T2A3,Canada}

b_{LawrenceLivermoreNationalLaboratory,P.O.Box808,L-414,Livermore,CA 94551,USA} c_{DepartmentofPhysics,HokkaidoUniversity,Sapporo060-0810,Japan}

d_{InstitutdePhysiqueNucléaire,IN2P3-CNRS,UniversitéParis-Sud,F-91406OrsayCedex,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received17October2015

Receivedinrevisedform6April2016 Accepted7April2016

Availableonline12April2016 Editor: W.Haxton

Keywords:

Nuclearphysics

Light-nucleiradiativecaptures Abinitiocalculation

The3_He(

_α

_,

_γ

₎7_{Be and}3_H(

_α

_,

_γ

₎7_{Li astrophysicalS factorsarecalculatedwithintheno-coreshellmodel}

withcontinuumusingarenormalizedchiralnucleon–nucleoninteraction.The3He(

α

,

γ

)7Be astrophysical

S factorsagreereasonablywellwiththeexperimentaldatawhilethe3_H(

_α

_,

_γ

₎7_{Li onesareoverestimated.}

Theseven-nucleonboundandresonancestatesandthe

α

+3_He/3_{H elastic scatteringare alsostudied}

and comparedwithexperiment.Thelow-lyingresonancepropertiesareratherwellreproducedbyour approach.Atlowenergies,thes-wavephaseshift,whichisnon-resonant,isoverestimated.

©2016TheAuthorsandLawrenceLivermoreNationalLaboratory.PublishedbyElsevierB.V.Thisisan openaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

1. Introduction

The 3He(

α

,

γ

)7Be and 3H(

α

,

γ

)7Li radiative-capture processes holdgreat astrophysical significance. Theirreaction rates for col-lision energies between

∼

20 and500 keV in the center-of-mass (c.m.)frameareessentialtocalculatetheprimordial7_Liabundance in the universe [1–3]. In addition, standard solar model predic-tionsforthefractionofpp-chainbranchesresultingin7Beversus 8_{B neutrinosdependcriticallyonthe}3_He(

_α

_,

_γ

₎7_{Be astrophysical S} factoratabout20 keV c.m.energy[4,5].Because oftheCoulomb repulsionbetween thefusing nuclei, thesecapturecross sections are strongly suppressed at such low energies and thus hard to measuredirectlyinalaboratory.

Concerning the 3He(

α

,

γ

)7Be radiative capture, experiments performedbyseveralgroups inthelast decadehaveled toquite accurate cross-section determinations for collision energies

be-*

Correspondingauthor.

E-mailaddresses:jdoheter@triumf.ca(J. Dohet-Eraly),navratil@triumf.ca

(P. Navrátil),quaglioni1@llnl.gov(S. Quaglioni),whoriuchi@nucl.sci.hokudai.ac.jp

(W. Horiuchi),hupin@ipno.in2p3.fr(G. Hupin),f.raimondi@surrey.ac.uk

(F. Raimondi).

1 _{Presentaddress:CEA,DAM,DIF,F-91297Arpajon,France.}

2 _{Presentaddress:DepartmentofPhysics,FacultyofEngineeringandPhysical}

Sci-ences,UniversityofSurrey,Guildford,SurreyGU27XH,UnitedKingdom.

tween about90 keVand3.1 MeVinthec.m. frame[6–13]. How-ever,theoretical modelsorextrapolations arestill neededto pro-vide thecapture cross section atsolar energies [14]. Incontrast, experimentaldataarelesspreciseandalsomuchlessextensivefor the3H(

α

,

γ

)7Li radiativecapture.Themostrecentexperimentwas performedtwentyyearsagoresultinginmeasurementsatcollision energiesbetweenabout50 keVand1.2 MeVinthec.m.frame[15]. Theoretically,theseradiativecaptureshavealsogeneratedmuch interest:fromthedevelopmentofpureexternal-capturemodelsin theearly60’s[16]tothemicroscopicapproachesfromthelate80’s uptonow[17–19,3,20](seeRef.[5]forashortreview).However, no parameter-free approach is able to simultaneously reproduce the latest experimental 3_He(

_α

_,

_γ

₎7_{Be and} 3_H(

_α

_,

_γ

₎7_Li astrophys-ical S factors. To possibly fill thisgap, an abinitio approach, re-lying on a realistic inter-nucleon interaction, is highly desirable. The abinitio no-core shell model with continuum (NCSMC) [21, 22] hasbeensuccessfulinthesimultaneousdescription ofbound andscatteringstatesassociatedwithrealisticHamiltonians

[23,24]

. Thisapproach canthus be naturallyapplied tothe descriptionof radiative-capture reactions, which involve both scattering (in the initialchannels)andboundstates(inthefinalchannels).

In this letter, we present the study of the 3_He(

_α

_,

_γ

₎7_Be and 3_H(

_α

_,

_γ

₎7_{Li radiative-capture reactions with the NCSMC} ap-proach[21,22],usingarenormalizedchiralnucleon–nucleon(N N) interaction. Thisis thefirst NCSMCstudywhere thelightest

col-http://dx.doi.org/10.1016/j.physletb.2016.04.021

lidingnucleushasthreenucleons andthefirst applicationofthe NCSMCto a radiative capture. We outline the NCSMC formalism inSec. 2 andapply the NCSMCapproach to the studyof seven-nucleonsystems inSec. 3.First, propertiesof7_{Be and} 7_Libound statesandresonancestatesare evaluatedandcomparedwiththe experimental data. Then, the

α

+

3_{He and}

_α

₊

3_{H scattering} statesare studied. Elastic cross sections,elastic phase shifts,and scatteringlengths are computed and compared with experimen-tal data or values obtained by other models. Finally, from the 7_{Be and} 7_{Li bound-state wave functions and the}

_α

₊

3_{He and}

α

+

3_{H scatteringwave functions,we evaluate the} 3_He(

_α

_,

_γ

₎7_Be and3H(

α

,

γ

)7Li radiative-capturecrosssections.

2. Formalism

Thepresentstudyofthe7Beand7Linuclei,ofthe

α

+

3_{He and}

α

+

3_{H elasticscattering,andofthe}3_He(

_α

_,

_γ

₎7_{Be and}3_H(

_α

_,

_γ

₎7_Li radiative-capturereactionsisbasedonthesolutionsofthe micro-scopicSchrödingerequation

H

|

J π T

 =

⎡

⎣



7 i=1 ti

−

Tc.m.

+

7



i<j=1 vi j

⎤

⎦ |

J π T

_{ =}

_E

_|

J π T

_,

₍₁₎

atdifferentenergiesfordifferentvaluesofangularmomentum J ,

parity

π

, andisospin T . The quantum numbers associated with theprojectionsof theangular momentumandof theisospin are omittedfor simplifying the notations.Here H isthe translation-invariantmicroscopic Hamiltonian, ti isthe kinetic energyof

nu-cleoni,Tc.m.isthec.m.kineticenergy,vi j isthepotentialbetween

nucleons i and j, E is the total energy in the c.m. frame, and

|

JπT

_

_{is a partial wave function with quantum numbers J π T .}

TheN N potential vi j,whichisspecifiedinthenextsection,is

re-alistic,inthe sense that itreproduces theexperimental deuteron energyandN N phaseshifts.Forcomputationalreasons,no three-bodyforcesareconsideredinthepresentwork.

Forthesake ofbrevity,inthe restofthissection, the formal-ismispresentedonlyforthe7_Be

_/

_α

₊

3_{He system.Thetreatment} ofthe 7_Li

_/

_α

₊

3_{H system isanalogous. In theNCSMCapproach,} thecollidingnuclei,

α

and3He,aredescribedbysquare-integrable eigenstates ofthe 4_{He and}3_{He systemsobtainedwithin the} no-coreshellmodel (NCSM) [25,26]by diagonalizing a largematrix. These NCSM eigenstates are linear combinations of translation-invariant and fully-antisymmetric states of harmonic oscillator (HO)wavefunctionswithfrequency



andupto NmaxHOquanta abovethelowestenergyconfiguration.Thecompoundsystem,the 7_{Be boundandresonancestatesandthe}

_α

₊

3_{He scatteringstates,} aredescribed by a combinationof seven-bodyNCSM eigenstates, denoted by

|

7

λ

J π T



with

λ

the energy label, and 4_He

₊

3_He clusterstates,denotedby

|

νJrπT



,andbuiltfromtheNCSM

eigen-statesof4_{He and}3_{He bymeans oftheresonating-groupmethod} (RGM)[27,28].Thelatterclusterstatesareexplicitlygivenby

|

J π T νr

 =



|

4_He

_λ

4Jπ₄4T4

|

3He

λ

3J₃π3T3



(sT) Y

(

r

ˆ

43

)

(J π T)

×

δ(

r

−

r43

)

rr43

,

(2) where

|

4_He

_λ

4Jπ₄4T4



isaNCSMeigenstateof4Hewithenergy la-bel

λ

4, total angular momentum J4, parity

π

4, and isospin T4,

|

3_He

_λ

3Jπ33T3



isan analogously defined NCSM eigenstateof 3He,

s is the channel spin, r43 is the relative coordinate betweenthe centers of mass of 4He and 3He,

is the relative orbital angu-lar momentum between the clusters, and

ν

is a collective in-dexfor

{λ

4

,

J4

,

π

4

,

T4

,

λ

3

,

J3

,

π

3

,

T3

,

s

,

}

.Thesecluster statesare

translation-invariant. Full antisymmetrization is obtained by ap-plying an operator

A

ˆ

43 that accountsforexchangesbetween the nucleonsbelongingto4_{Heandthosebelongingto}3_He.TheNCSMC partialwavefunctionisthusexpandedas

|

J π T

_{ =}



λ c_λJ π T

|

7

λ

JπT



+



ν



drr2

γ

J π T ν

(

r

)

r

A

ˆ

43

|

J π T νr

,

(3)

wherethecoefficientsc_λJπT andfunctions

γ

νJπT areunknown

dis-creteandcontinuousamplitudes.

Inserting ansatz (3) in Eq. (1) and projecting over the basis states lead to the NCSMCequations [22], written in a schematic block-matricesnotationas



H₇J π T hJ π T hJ π T

H

J π T



cJ π T

γ

J π T

=

E



I₇J π T gJ π T gJ π T

N

J π T



cJ π T

γ

J π T

.

(4)

Theupperdiagonalblocksintheleft- andright-handsidesofthe equationaretheHamiltonianandoverlapmatrixelements onthe square-integrableseven-bodystatesobtainedfromtheNCSM diag-onalization,

(

H₇J π T

)

λλ

= 

7

λ

JπT

|

H

|

7

λ

JπT

 =

Eλ

δ

λλ

,

(5)

(

I₇J π T

)

λλ

= 

7

λ

JπT

|

7

λ

JπT

 = δ

λλ

.

(6)

ThelowerdiagonalblocksaretheHamiltonianandoverlapmatrix elementsonthe4_He

₊

3_{Heclusterstates,}

H

_ννJ π T

(

r

,

r

)

= 

νJ π Tr

| ˆ

A

43H

A

ˆ

43

|

_νJ π T_r

,

(7)

N

J π T νν

(

r

,

r

)

= 

J π T νr

| ˆ

A

243

|

J π T νr

.

(8)

The off-diagonal blocks are the coupling Hamiltonian and over-lapmatrix elements betweensquare-integrablestatesandcluster states,

h_λJ π T_ν

(

r

)

= 

7

λ

JπT

|

H

A

ˆ

43

|

νJ π Tr

,

(9)

g_λJ π T_ν

(

r

)

= 

7

λ

JπT

| ˆ

A

43

|

νJ π Tr

.

(10)

The system of coupled equations (4), transformed first into an orthogonal form [22], is solved by means of the microscopic

R-matrix method on a Lagrange mesh [29–31], which enforces theproper bound-stateorscattering-stateasymptoticbehavior of functions

γ

νJπT.Theasymptoticnormalizationcoefficientsandthe

phase shifts (or thecollision matrix elements ifseveral channels are open)are thencomputedfromthe R-matrix.When onlyone channel is open, the value of the R-matrix at zerocolliding en-ergy enables a simple and accurate evaluation of the scattering lengths[32].Resonances can alsobe studied withthis approach. Indeed,extendingthe microscopic R-matrixapproachto complex energies (see Ref. [33] fora closely relatedapproach) allows the determinationoftheSiegertstates[34],whicharesolutionsofthe Schrödinger equation (1) with purely outgoing waves at infinite relativedistancer43.Thecomplexenergies ofthesestatesare the polesoftheS-matrix,thusprovidingdirectlytheenergyandwidth oftheresonances.

Fig. 1. (Coloronline.) The7_Beand7_{LispectraobtainedfromtheNCSMandNCSMC}

approachesandfromexperiments[47].OnlystateswithisospinT=1/2 are con-sidered.Energiesaregivenwithrespecttotheα+3_He

/3H threshold.Rectangles symbolizethewidthsofresonances.Thequestionmarkindicatesthatthewidthis notexperimentallydetermined.

3. Results

Thenucleon–nucleoninteractionisdescribed by achiralN3_LO

N N potential [37] softened via the similarity-renormalization-group (SRG) method [38–41], which reduces the influence of momenta higher than an SRG resolution scale h

¯



. For com-putational reasons, in the present applications, both chiral and renormalization-induced three- and higher-body forcesare disre-garded.To getthe correcttail ofthe bound-statewave functions andhencesensibleastrophysicalS factors,theSRGresolutionscale ischosen as



=

2

.

15 fm−1 to reproduce,asaccuratelyas possi-ble,theexperimentalseparationenergiesofthe7_Beand7_Linuclei for the largest accessible model space (see below for its speci-fications). An analogous strategy has already been followed in a similarcontext:thestudyofthe7Be(p

,

γ

)8B radiativecapturewith theNCSM

/

RGMapproach[42].

Allpartial waveswithtotalangularmomentum J

∈ {

1

/

2

,

3

/

2

,

5

/

2

,

7

/

2

}

, positive or negative parities, and isospin T

=

1

/

2 are considered. TheHO frequencyis



=

20 MeV

/

h for

_¯

both square-integrablestates andcluster statesincluded inexpansion (3).All NCSM eigenstates up to about 13 MeV with respect to the

α

+

3_He

_/

3_{H threshold are included in the model space, which} cor-responds to one or two states by partial wave. As an exam-ple, the negative-parity NCSM states can be seen in the left-most columnof Fig. 1. The values Nmax

=

11 for positive-parity states and Nmax

=

10 for negative-parity states are considered. The 4_He

₊

3_{He (}4_He

₊

3_{H) cluster states are built by coupling} the NCSM ground states of 4_He

_[(

_{J π T}

₎

_{= (}

₀+₀

₎

_]

_and 3_{He (}3_H)

[(

J π T

)

= (

1

/

2+1

/

2

)

]

obtainedwith Nmax

=

12, exceptif another value is explicitly specified. For studying the convergence prop-erties,smaller valuesof Nmax are used butthe gapbetweenthe valuesofNmaxusedforcomputingthecolliding-nucleiwave

func-Table 1

Ground-stateenergiesandchargeradiiofthe4_He,3_He,and3_{H nucleicalculated}

withtheno-coreshellmodelwithNmax=12 andcomparedwithconvergedvalues

(obtainedwithhighervaluesofNmax)andwithexperimentaldata(Refs.[43,45]for

energiesandRefs.[44,46]forradii).ThechiralN3_{LON N potentialsoftenedviathe}

SRGwith=2.15 fm−1_isused. Nucleus Eg.s.[MeV] rch[fm] 4_{He −27.97 1.68 N} max=12 −28.03 1.68 converged −28.296 1.681(4) Exp.[43,44] 3_{He −7.49 1.95 N} max=12 −7.55 2.00 converged −7.718 1.973(14) Exp.[45,44] 3_{H −8} .24 1.76 Nmax=12 −8.30 1.79 converged −8.482 1.7591(363) Exp.[45,46] Table 2

Propertiesofthe7_Beand7_{LiboundstatescalculatedwithintheNCSMandNCSMC}

approachesandcomparedwithexperimentaldata[47–51].The7_Beand7_Lienergies

aregivenwithrespecttotheα+3_{He andα+}3_{H thresholds,respectively.}

7_{Be NCSM NCSMC Exp. Refs.} E3/2−[MeV] −0.82 −1.52 −1.587 [47] E1/2−[MeV] −0.49 −1.26 −1.157 [47] rch[fm] 2.375 2.62 2.647(17) [48] Q [e fm2] −4.57 −6.14 − μ[μN] −1.14 −1.16 −1.3995(5) [48] 7_{Li NCSM NCSMC Exp. Refs.} E3/2−[MeV] −1.79 −2.43 −2.467 [47] E1/2−[MeV] −1.46 −2.15 −1.989 [47] rch[fm] 2.21 2.42 2.39(3) [49] Q [e fm2] −2.67 −3.72 −4.00(3) [50] μ[μN] 3.00 3.02 3.256 [51]

tions, the seven-body positive-parity states and the seven-body negative-paritystatesisalwaysthesame.Noclusterstates involv-ingexcitedstatesof4_Heor3_He(3_{H)areconsidered.The} orthogo-nalversionofNCSMCequations

(4)

aresolvedbythemicroscopic

R-matrixwithachannel radiusa

=

12 fm and 40 Lagrange-mesh points.

The 4He, 3He, and 3H ground-state energies andcharge radii obtainedwithNmax

=

12 aregivenin

Table 1

.Theyarecompared with theexact values obtainedby increasing Nmax up to conver-genceandwiththeexperimentaldata.AtNmax

=

12,the4He,3He, and3H ground-statepropertiesareclosetotheirconvergedvalues witharelativedifferenceoflessthan1% fortheenergiesandless than3% fortheradii.Theyarealsoclosetotheexperimental val-ues:the energiesatNmax

=

12 differfromtheexperimentalones bylessthan3% whiletheradiiatNmax

=

12 differbylessthan1% fromtheexperimentalonesorareinagreementwiththem.

Inthefollowing,wediscussthe7_Beand7_{Libound-state} prop-ertiesastheyareobtainedwithintheNCSMCapproach.Both nu-clei havea groundstate characterizedby

(

J π T

)

= (

3

/

2−1

/

2

)

and an excited state characterized by

(

J π T

)

= (

1

/

2−1

/

2

)

.Their ener-gies are displayed in Table 2and compared with the values ob-tainedwiththesquare-integrable partofthe basisonlyandwith the experimental values. The charge radii (rch), quadrupole mo-ments ( Q ),andmagneticdipole moments(

μ

)ofthe 7Beand7Li groundstatesarealsogivenin

Table 2

.

Table 3

Energies(Er)andwidths( )inMeVof7Beand7Liresonancestatesupto10 MeV obtainedfromtheNCSMandNCSMCapproachesandfromexperiments[47].Only resonanceswithisospinT=1/2 areconsidered.Resonanceenergiesaregivenwith respecttotheα+3_He

/3H threshold.Experimentaluncertaintiesarenotdisplayed. Widthsmarkedwithanasteriskcannotbeextractedreliablybecausethe6_Li+N

decaychannelisnotconsideredinthesecalculations.

7_{Be NCSM NCSMC Exp.} Jπ Er Er Er 7/2− 4.42 3.61 0.33 2.98 0.175 5/2− 6.16 4.87 1.00 5.14 1.2 5/2− 7.56 7.55 ∗ 5.62 0.40 3/2− 9.11 9.14 0.29 8.31 1.8 1/2− 9.93 9.93 ∗ – – 7/2− 10.05 9.98 0.40 7.68 – 7_{Li NCSM NCSMC Exp.} Jπ Er Er Er 7/2− 3.53 2.79 0.214 2.18 0.069 5/2− 5.24 4.04 0.785 4.14 0.918 5/2− 6.84 6.84 ∗ 4.99 0.080 3/2− 8.47 8.51 0.297 6.28 4.712 1/2− 9.28 9.28 ∗ 6.62 2.752 7/2− 9.41 9.33 0.435 7.10 0.437

presenceofthe

α

+

3_He

_/

3_{H clusterdegreesoffreedom.The} dis-crepancybetweenitstheoreticalandexperimentalvaluesismostly dueto thetwo-bodyelectromagneticcurrents,whicharemissing inourapproach.

An analogous comparison for the 7_{Be and} 7_{Li computed and} measuredspectra isgivenin

Table 3

andin Fig. 1.Notethat the NCSMapproach can onlyprovide theenergies ofresonances, not their widths. Including the

α

+

3_{He (}

_α

₊

3_{H) cluster states in} themodelspacereducesthegapbetweenthetheoreticaland ex-perimentalenergiesofthefirsttworesonanceswhiletheenergies oftheotherresonancesarenearlyunaffected.Thefirst7

/

2− reso-nanceisoverestimatedby600 keVwhilethefirst5

/

2−resonance isunderestimatedby 300 keVforthe7Be systemandby100 keV forthe7Lisystem.Thewidthsofthesecond5

/

2−andofthe1

/

2− resonances obtained with the NCSMC approach are unphysically small(less than 10 keV). The escape widthfor theseresonances ismissingbecausethe corresponding decaychannel(6_Li

₊

_N)is notincludedinthecalculations. Theexplicitinclusionof6Li

+

N

cluster statesin the basis should cure thisproblem andalso af-fecttheother statescloseorabove the6_Li

₊

_{N threshold.While} atenergiesrelevantforastrophysicstheradiativecaptureisclearly non-resonant,thefirst7

/

2− resonanceinthe7Be spectrumplays aminorbutnon-negligiblerole attherelevantenergies for labo-ratorymeasurements,asitisshownfurther.

The

α

+

3_{He and}

_α

₊

3_{H elastic phase shifts are computed} forrelativecollisionenergiesupto

∼

10 MeVandshownin

Fig. 2

. Forthesake ofclarity, the jumpof

+

180◦ in thephase shiftsat thesecond 5

/

2− and7

/

2− resonanceenergiesare not displayed. Asacomplementaryinformationtothephaseshifts,thescattering lengthsforthe1

/

2+,1

/

2−,and3

/

2−partialwavesareprovidedin

Table 4.Forthe1

/

2− and3

/

2− partialwaves,theyarecompared withthevaluesobtainedinRef.[55]fromtheexperimentalphase shiftsandasymptoticnormalizationcoefficients(ANC)byusing re-lationslinkingthe effective-rangeexpansion andthe ANC.Such a methodisnotapplicable forthe 1

/

2+ partialwave sincethereis noboundstatesinthispartialwave.Therefore,forthe1

/

2+partial wave,we cite thevalue obtainedin Ref.[54] froma microscopic clustermodelreproducingtheexperimentalphaseshits.Thisvalue cannotbeconsidered,strictlyspeaking,asanexperimentalonebut providesareasonablepointofcomparison.

Fig. 2. (Coloronline.) Theα+3_{He andα+}3_{H elasticphaseshiftsobtainedfrom}

theNCSMCapproachandfromexperiments[52,53].Energiesaregivenwithrespect totheα+3_He/3_{H threshold.}

Table 4

Scattering lengths aJπ associated with partial waves Jπ _{for the α+}3_{He and}

α+3_{H collisions. Valuesinferred inRefs.[54,55] fromthe experimentalphase}

shiftsaregivenforcomparisonincolumn“Exp.”(seetextformoredetails).

α+3_{He NCSMC “Exp.” Refs.} a1/2+[fm] 7.7 41.06 [54] a1/2−[fm3] 263.9 413±7 [55] a3/2−[fm3] 210.4 301±6 [55] α+3_{H NCSMC “Exp.” Refs.} a1/2+[fm] 5.4 13.05 [54] a1/2−[fm3_{] 82.6 95.13±1.73 [55]} a3/2−[fm3] 70.0 58.10±0.65 [55]

In both systems, the 1

/

2+ theoretical phase shifts overesti-mate the corresponding experimental ones. To analyze this dis-crepancy,we focusesonone system,namely

α

+

3_{He.The 1}

_/

₂+

Fig. 3. (Coloronline.) 1/2+phaseshiftsfordifferentvaluesoftheSRGparameter: =2.1fm−1(dottedlines),=2.15fm−1(solidlines),and=2.2fm−1(dashed lines).For=2.2fm−1_{,differentvaluesofN}

maxareconsidered;theNmax value

usedforcomputingthecolliding-nucleiwavefunctionsisgiven.

Table 5

1/2+ scatteringlengthfortheα+ 3_{He collisionfordifferentvaluesoftheSRG}

parameteranddifferentvaluesofNmax;theNmaxvalueusedforcomputingthe

colliding-nucleiwavefunctionsisgiven.

[fm−1_{] N} max a1/2+[fm] 2.2 8 −2.5 2.2 10 6.5 2.2 12 9.1 2.15 12 7.7 2.1 12 6.2

Fig. 4. Differentialα+3_{He elasticcrosssections(dσ/d)normalizedbythe}

dif-ferentialRutherfordcrosssections(dσR/d)asafunctionofthescatteringangle measuredinthec.m.frame.ExperimentaldatacomefromRef.[56].

Fornegative-paritypartialwaves,thediscrepancybetween the-oreticalandexperimental resonancesseenin

Fig. 1

is alsovisible inthephaseshifts.Moreover,thesplittingbetweenthe1

/

2− and 3

/

2−isunderestimated,asitcanbeseenfromthecomparisonof the phase shiftsand ofthe scatteringlengths. Instead of analyz-ing the phase shifts andthe scattering lengths, we can compare directlytheoretical andexperimentalcross sections.In

Fig. 4

,the differential

α

+

3_{He elasticcrosssectionsaredisplayedfor} differ-entanglesattwoparticularcollidingenergiesandcomparedwith experimentaldatafromRef.[56],forwhichnophase-shiftanalysis exists.Ourapproachreproduces thegeneraltrends ofthe experi-mentaldata.

Toevaluatetheimpactofthediscrepanciesintheelastic scat-teringonthe3He(

α

,

γ

)7Be and3H(

α

,

γ

)7LiastrophysicalS factors,

weadoptaphenomenologicalmodelbasedontheNCSMCresults inthe largest modelspace. The basicidea is to considerthe

en-Fig. 5. (Coloronline.) AstrophysicalS factorforthe3_He(α,γ)7_Beand3_H(α,γ)7_Li

radiative-captureprocessesobtainedfromtheNCSMCapproachandfromits phe-nomenologicalversionandcomparedwithothertheoreticalapproaches[3,20]and withexperiments[57–60,6–13,61–63,15].Recentdataareincolor(online)andold dataareinlightgrey.

Table 6

3_He(α,γ)7_{Be and}3_H(α,γ)7_{Li astrophysicalS factorsextrapolatedatzerocollision}

energy.ExperimentaldatacomefromRefs.[5,15].Forthe3_He(α,γ)7_Bereaction,

thenumbersinparenthesesaretheerrorsintheleastsignificantdigitscomingfrom theexperimentsandfromthetheoreticalextrapolationwhileforthe3_H(α

,γ)7_Li

reaction,theyarethestatisticalandsystematicerrors.

NCSMC Exp. Refs.

S3He(α,γ )7Be(0)[keV b] 0.59 0.56(2)(2) [5] S3H(α,γ )7Li(0)[keV b] 0.13 0.1067(4)(60) [15]

ergies of the square-integrable NCSM basis states Eλ, appearing

in Eq. (5), as adjustable parameters. These new degrees of free-dom are then used to reproduce the experimental 7_{Be and} 7_Li bound-stateandresonanceenergiesandreducingthegapbetween theoreticalandexperimental1

/

2+phaseshifts.

The 3He(

α

,

γ

)7Be and 3H(

α

,

γ

)7Liastrophysical S factors ob-tained with the NCSMC approach and with its phenomenologi-cal version are displayed in Fig. 5 and compared with experi-ment [57–60,6–13,61–63,15]. The astrophysical S factors extrap-olated atzero collidingenergy are given in Table 6. The electric

E1 and E2 transitionsaswellasthemagneticM1 transitionshave beenconsidered.Fortheenergyrangeswhichareconsidered,the contribution of the E1 transitions is dominant while M1

isslightly overestimatedby our theoretical approach,the energy ofthecorrespondingpeakinthe S factorisalsooverestimated.On the contrary, in the phenomenological approach, the experimen-talenergy andwidthof the 7

/

2− resonance are reproduced and hencethe energyand widthof the corresponding peak inthe S

factor.Theadjustmentofthe7Besquare-integrableenergiesinthe phenomenologicalapproachtoreproducetheexperimental bound-stateenergieswithanaccuracyof5 keV orbetterhasanimpactof 1% orlessontheastrophysical S factorfortheenergyrange con-sideredin

Fig. 5

.Asitcanbeexpected,theadjustmentofthe5

/

2− resonanceenergies has a negligible effecton the astrophysical S

factor,about0

.

1%.The 1

/

2+phaseshiftsarenotverysensitiveto theenergies ofthe square-integrableNCSM states,whichare the adjustableparametersofourphenomenologicalmodel.Evenby in-creasingtheseenergiesbyamountsaslargeas10 MeV,the1

/

2+ phaseshiftisonlyreducedby0

.

5◦at1 MeV,1

.

5◦at2 MeV,and2◦ at3 MeV.Theconsequentimpactontheastrophysical S factorsis withinfew percents. A further improvementofthe 3He(

α

,

γ

)7Be astrophysical S factors wouldrequirethe inclusionof three-body forcesandpossiblytheincreaseoftheaccuracyofthebasisstates, i.e.,theincreaseofNmax.Theimportanceofthethree-bodyforces is also highlighted by comparing the results obtained with the NCSMCfor different SRG resolution scales



(not shown in the figures).Increasing(reducing)



by0.05 fm−1 inducesareduction (anincrease)oftheastrophysical S factorbyan amountbetween about30 eV b and 60 eV b over the energy range considered in

Fig. 5.

In Fig. 5, other theoretical results based on two different realistic N N interactions [3,20] are also presented. Nollet’s ap-proach[3]isnotfullymicroscopicbuthybrid,basedonbothab ini-tio variaini-tionalMonteCarlowave functionsandphenomenological potential-modelwavefunctions.Incontrast,Neff’scalculation[20]

is fully microscopic. As our approach, it is based on resonating-group method wave functions and the microscopic R-matrix to enforce the proper boundary conditions, but the model space is built from fermionic molecular dynamics (FMD) wave functions andonlythe E1 transitionsareconsidered.AlthoughtheFMD ap-proachisnotfullyabletodescribetheshort-rangecorrelationsof thewave function,agoodagreementbetweentheoreticaland ex-perimental3_He(

_α

_,

_γ

₎7_{Be astrophysical S factorwasobtained[20].} Letusstress that Nollet’s andNeff’s approachesand the present oneare–fortechnicalreasons–basedonthreedifferentN N

in-teractions. They are thus not supposed to give the same results and,infact,bothabsolutevaluesoftheastrophysical S factorsand their energydependencediffersignificantly. Toget some insights onthesedifferences,the E1 contributionstothe3_He(

_α

_,

_γ

₎7_Be as-trophysical S factors obtained in thiswork are decomposedinto thedifferentpartial wavesandcompared withNeff’sresults [64]

in

Fig. 6

.Becauseoftheabsenceofthecentrifugalbarrier,the tran-sitionsfromthe1

/

2+ partialwave(

=

0)arethemostimportant andthose are the onesfor which we findthe largest difference. Abetterknowledgeoftheempirical

α

+

3_{He s-wavephaseshifts} couldthusprovideausefultestoftheaccuracyofthetheoretical results.

The3_H(

_α

_,

_γ

₎7_{Li astrophysical S factors areoverestimatedover} the full energy range in our calculation. The phenomenologi-cal approach improves only slightly the situation. As shown in

Fig. 5, a similar behavior is present, though less pronounced, inNeff’s calculations [20] while Nollet’s approach[3] reproduces the 3H(

α

,

γ

)7Li astrophysical S factor but underestimates the 3_He(

_α

_,

_γ

₎7_{Be one.Thissuggestsapossibleunderestimation ofthe} experimental systematic uncertainties and underscores the need for new experimental studies of 3H(

α

,

γ

)7Li and a more com-pletemicroscopiccalculation,includingtheeffectofthree-nucleon forces.Again, the dependence of the results on the SRG

param-Fig. 6. (Coloronline.) Partial-wavedecomposition ofthe E1 contributionstothe

3_He(α,γ)7_{Be astrophysicalS factorswiththeNCSMCapproach(solidlines)}

com-paredwithNeff’scalculations(dashedlines)[64].

Fig. 7. (Color online.) Ratio (R) ofthe 3_He(α

,γ)7_{Be and}3_H(α

,γ)7_Li

radiative-capturecrosssectionstotheseven-nucleonexcitedstateandtotheseven-nucleon groundstateobtained fromthe NCSMC,fromitsphenomenological versionand fromexperiments[9,8,10,62,63,15].

eter has been studied. Increasing (reducing) of



by 0

.

05 fm−1 induces areduction (anincrease)oftheastrophysical S factorby about10 eV b over the energy rangeconsidered inFig. 5. It has to benotedthat thecalculationsbased on



=

2

.

15 fm−1 repro-ducemoreaccuratelytheseven-nucleonbound-stateenergiesand therefore, the astrophysical S factors for this



value should be morereliable.

4. Conclusion

In this letter, the 3He(

α

,

γ

)7Be and 3H(

α

,

γ

)7Li radiative-capture processes are described by means of the no-core shell modelwithcontinuumapproach[21,22].Althoughtheapproachis restrictedtotwo-nucleon forces,arathergooddescriptionof 7Be and7_{Linucleiisobtained. Theoreticalandexperimental}

_α

₊

3_He and

α

+

3_{H elasticphaseshiftsdonotagreeperfectlywell.} How-ever, the discrepancy is difficult to characterize because of the lackofknowledge ontheexperimental uncertainties. New exper-imental studies of the

α

+

3_{He and}

_α

₊

3_{H elastic scattering} would be highly desirable to probe more accurately the quality ofthe scatteringwave functions. At low energies, the theoretical

s-wavephase shifts are overestimated by ourapproach. This has a direct impact on the energy dependence of the astrophysical

S factor at low energies, which is therefore not exactly repro-duced. The overestimation of the s-wave phase shifts is mostly duetothe adjustmentofthenucleon–nucleon interactionfor re-producingthe7Beand7Libindingenergieswithoutthree-nucleon forceswiththelargestconsideredmodelspace. Takingthe three-nucleon forces into account could thus reduce this discrepancy. Moreover,theinclusionofthree-bodyforceswouldreducethe de-pendence of renormalizationof the inter-nucleoninteractions on theresults,whichwouldenable ustoconsidersofterinteractions, leading to a faster convergence. Including the three-body inter-actions is thus the next step in the development of an abinitio

approach of radiative-capture reactions. However, this a particu-larly challenging task, which requires analytic developments and large computational efforts.Approximate ways to includethe ef-fectsofthree-bodyforces viathenormalordering technique [65]

arecurrentlyunderstudy.

Acknowledgements

We would like to thank Thomas Neff for useful discussions and, in particular, for sharing with us the results of his cal-culations and his extractions of experimental data. We are also grateful to Barry Davids for fruitful discussions and for point-ing out usefulreferences to us. TRIUMF receives federal funding via a contributionagreement withthe National ResearchCouncil Canada. Thiswork was supported inpart by NSERC underGrant No. 401945-2011, by LLNL under Contract DE-AC52-07NA27344, bythe U.S.DepartmentofEnergy,Office ofScience,Officeof Nu-clear Physics, under Work Proposal Number SCW1158, and by JSPSKAKENHIGrantNumbers 25800121 and15K05072. W.H. ac-knowledges Excellent Young Researcher Overseas Visit Program of JSPS that allowed him to visit LLNL (2009–2010). Computing supportcamefromtheLLNLinstitutionalComputingGrand Chal-lengeProgramandfromanINCITEAward ontheTitan supercom-puter of the Oak Ridge Leadership Computing Facility (OLCF) at ORNL.

References

[1]S. Burles,K.M. Nollett,J.W. Truran,M.S. Turner,Phys.Rev.Lett.82(1999)4176.

[2]K.M. Nollett,S. Burles,Phys.Rev.D61(2000)123505.

[3]K.M. Nollett,Phys.Rev.C63(2001)054002.

[4]E.G. Adelberger,S.M. Austin,J.N. Bahcall,A.B. Balantekin,G. Bogaert,etal.,Rev. Mod.Phys.70(1998)1265.

[5]E.G. Adelberger,A. García,R.G. HamishRobertson,K.A. Snover,A.B. Balantekin, etal.,Rev.Mod.Phys.83(2011)195.

[6]B.S. NaraSingh,M. Hass,Y. Nir-El,G. Haquin,Phys.Rev.Lett.93(2004)262503.

[7]D. Bemmerer,F. Confortola,H. Costantini,A. Formicola,Gy. Gyürky,etal.,Phys. Rev.Lett.97(2006)122502.

[8]F. Confortola,D. Bemmerer,H. Costantini,A. Formicola,Gy. Gyürky,etal.,Phys. Rev.C75(2007)065803.

[9]T.A.D. Brown,C. Bordeanu,K.A. Snover,D.W. Storm,D. Melconian,A.L. Sallaska, S.K.L. Sjue,S. Triambak,Phys.Rev.C76(2007)055801.

[10]A. DiLeva,L. Gialanella,R. Kunz,D. Rogalla,D. Schürmann,etal.,Phys.Rev. Lett.102(2009)232502.

[11]M. Carmona-Gallardo,B.S. NaraSingh,M.J.G. Borge,J.A. Briz,M. Cubero,etal., Phys.Rev.C86(2012)032801.

[12]C. Bordeanu,Gy. Gyürky, Z. Halász,T. Szücs, G.G. Kiss, Z. Elekes,J. Farkas, Zs. Fülöp,E. Somorjai,Nucl.Phys.A908(2013)1.

[13]M. Carmona Gallardo, Ph.D. thesis, Universidad Complutense de Madrid, Madrid,2014.

[14]A. DiLeva,L. Gialanella,F. Strieder,J.Phys.Conf.Ser.665(2016)012002.

[15]C.R. Brune,R.W. Kavanagh,C. Rolfs,Phys.Rev.C50(1994)2205.

[16]T.A. Tombrello,P.D. Parker,Phys.Rev.131(1963)2582.

[17]T. Kajino,Nucl.Phys.A460(1986)559.

[18]T. Mertelmeier,H.M. Hofmann,Nucl.Phys.A459(1986)387.

[19]A. Csótó,K. Langanke,Few-BodySyst.29(2000)121.

[20]T. Neff,Phys.Rev.Lett.106(2011)042502.

[21]S. Baroni,P. Navrátil,S. Quaglioni,Phys.Rev.Lett.110(2013)022505.

[22]S. Baroni,P. Navrátil,S. Quaglioni,Phys.Rev.C87(2013)034326.

[23]G. Hupin,S. Quaglioni,P. Navrátil,Phys.Rev.C90(2014)061601.

[24]G. Hupin,S. Quaglioni,P. Navrátil,Phys.Rev.Lett.114(2015)212502.

[25]P. Navrátil,J.P. Vary,B.R. Barrett,Phys.Rev.Lett.84(2000)5728.

[26]P. Navrátil,S. Quaglioni,I. Stetcu,B.R. Barrett,J.Phys.G36(2009)083101.

[27]S. Quaglioni,P. Navrátil,Phys.Rev.Lett.101(2008)092501.

[28]S. Quaglioni,P. Navrátil,Phys.Rev.C79(2009)044606.

[29]D. Baye,P.-H. Heenen,M. Libert-Heinemann,Nucl.Phys.A291(1977)230.

[30]M. Hesse,J.-M. Sparenberg,F. Van Raemdonck,D. Baye, Nucl. Phys. A640 (1998)37.

[31]P. Descouvemont,D. Baye,Rep.Prog.Phys.73(2010)036301.

[32]D. Baye,M. Hesse,R. Kamouni,Phys.Rev.C63(2000)014605.

[33]A. Csótó,G.M. Hale,Phys.Rev.C55(1997)536.

[34]A.J.F. Siegert,Phys.Rev.56(1939)750.

[35]D. Baye,P. Descouvemont,Nucl.Phys.A407(1983)77.

[36]P. Navrátil,S. Quaglioni,G. Hupin,C. Romero-Redondo,A. Calci,Phys.Scr.91 (2016)053002.

[37]D.R. Entem,R. Machleidt,Phys.Rev.C68(2003)041001; Phys.Rep.503(2011)1.

[38]F. Wegner,Ann.Phys.506(1994)77.

[39]S.K. Bogner,R.J. Furnstahl,R.J. Perry,Phys.Rev.C75(2007)061001.

[40]E.D. Jurgenson,P. Navrátil,R.J. Furnstahl,Phys.Rev.Lett.103(2009)082501.

[41]E.D. Jurgenson,P. Navrátil,R.J. Furnstahl,Phys.Rev.C83(2011)034301.

[42]P. Navrátil,R. Roth,S. Quaglioni,Phys.Lett.B704(2011)379.

[43]D.R. Tilley,H.R. Weller,G.M. Hale,Nucl.Phys.A541(1992)1.

[44]I. Sick,Phys.Rev.C90(2014)064002.

[45]J.E. Purcell,J.H. Kelley,E. Kwan,C.G. Sheu,H.R. Weller,Nucl.Phys.A848(2010) 1.

[46]I. Angeli,K.P. Marinova,At.DataNucl.DataTables99(2013)69.

[47]D.R. Tilley, C.M. Cheves,J.L. Godwin,G.M. Hale, H.M. Hofmann,J.H. Kelley, C.G. Sheu,H.R. Weller,Nucl.Phys.A708(2002)3.

[48]W. Nörtershäuser,D. Tiedemann,M. Žáková,Z. Andjelkovic,K. Blaum,et al., Phys.Rev.Lett.102(2009)062503.

[49]C.W. DeJager,H. DeVries,C. DeVries,At.DataNucl.DataTables14(1974) 479.

[50]H.-G. Voelk,D. Fick,Nucl.Phys.A530(1991)475.

[51]P. Raghavan,At.DataNucl.DataTables42(1989)189.

[52]R.J. Spiger,T.A. Tombrello,Phys.Rev.163(1967)964.

[53]W.R. Boykin,S.D. Baker,D.M. Hardy,Nucl.Phys.A195(1972)241.

[54]R. Kamouni,D. Baye,Nucl.Phys.A791(2007)68.

[55]R. Yarmukhamedov,D. Baye,Phys.Rev.C84(2011)024603.

[56]P. Mohr,H. Abele,R. Zwiebel,G. Staudt,H. Krauss,H. Oberhummer,A. Denker, J.W. Hammer,G. Wolf,Phys.Rev.C48(1993)1420.

[57]P.D. Parker,R.W. Kavanagh,Phys.Rev.131(1963)2578.

[58]H. Kräwinkel,H.W. Becker,L. Buchmann,J. Görres,K.U. Kettner,etal.,Z.Phys. A304(1982)307.

[59]J.L. Osborne, C.A. Barnes, R.W. Kavanagh, R.M. Kremer, G.J. Mathews, J.L. Zyskind,P.D. Parker,A.J. Howard,Phys.Rev.Lett.48(1982)1664.

[60]M. Hilgemeier,H.W. Becker,C. Rolfs,H.P. Trautvetter,J.W. Hammer,Z.Phys.A 329(1988)243.

[61]G.M. Griffiths,R.A. Morrow,P.J. Riley,J.B. Warren,Can.J.Phys.39(1961)1397.

[62]U. Schröder, A. Redder, C. Rolfs, R.E. Azuma, L. Buchmann, C. Campbell, J.D. King,T.R. Donoghue,Phys.Lett.B192(1987)55.

[63]S. Burzy ´nski,K. Czerski,A. Marcinkowski,P. Zupranski,Nucl.Phys.A473(1987) 179.

[64]T. Neff,J.Phys.Conf.Ser.403(2012)012028.