Nuclear matter saturation with chiral three-nucleon interactions fitted to light nuclei properties

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Nuclear

matter

saturation

with

chiral

three-nucleon

interactions

fitted

to

light

nuclei

properties

Domenico Logoteta

a

,

Ignazio Bombaci

b

,

a

,

c

,

∗

,

Alejandro Kievsky

a

a_{INFN,SezionediPisa,LargoBrunoPontecorvo3,I-56127Pisa,Italy}

b_{DipartimentodiFisica,UniversitádiPisa,LargoBrunoPontecorvo3,I-56127Pisa,Italy} c_{EuropeanGravitationalObservatory,ViaE.Amaldi,I-56021S.StefanoaMacerata,Cascina, Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received19November2015 Receivedinrevisedform19April2016 Accepted15May2016

Availableonline18May2016 Editor:J.-P.Blaizot

Keywords: Nuclearmatter

Chiralnuclearinteractions Three-bodyforce

The energyperparticleofsymmetricnuclear matterand pureneutronmatteris calculatedusingthe many-bodyBrueckner–Hartree–FockapproachandemployingtheChiralNext-to-next-to-next-toleading order(N3LO)nucleon–nucleon(NN)potential,supplementedwithvariousparametrizationsoftheChiral Next-to-next-to leadingorder(N2LO)three-nucleoninteraction.Suchcombinationisabletoreproduce severalobservablesofthephysicsoflightnucleiforsuitablechoicesoftheparametersenteringinthe three-nucleoninteraction.Wefindthatsomeoftheseparametrizationsprovideasatisfactorysaturation pointofsymmetricnuclearmatterandvaluesofthesymmetryenergyanditsslopeparameter

L in

very goodagreementwiththoseextractedfromvariousnuclearexperimentaldata.Thus,ourresultsrepresent asignificantsteptowardaunifieddescriptionoffew- andmany-bodynuclearsystemsstartingfrom two-andthree-nucleoninteractionsbasedonthesymmetriesofQCD.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The adventofinteractions derived in theframework of chiral

perturbation theory (ChPT) [1–3] opened a new and systematic

waytoinvestigatethepropertiesoffinitenuclearsystemsand nu-clearmatter. Thebig advantage ofusingsuch methodlies inthe factthattwo-bodyaswell asmany-bodyforcescanbecalculated orderbyorderaccordingtoawelldefinedschemebasedona low-energyeffectiveQCDLagrangian whichretainsthesymmetriesof QCD, andin particular the approximate chiral symmetry. Within thisapproach the details of the QCD dynamics are contained in parameters, theso-called low-energy constants (LECs),which are fixedbylow-energyexperimental data.Thissystematicprocedure isparticularlyusefulfornucleonicsystems wheretheimportance ofthethree-nucleonforce(TNF)isawell establishedfeature.Itis indeedwell knownthathighprecision nucleon–nucleon(NN) po-tentials,fittingNNscatteringdatauptoenergyof350 MeV,with a

χ

2 _{per datumnext to 1, underestimatethe experimental}

bind-ingenergiesof3H,3Hebyabout1 MeVandthatof4Hebyabout 4 MeV[4].This missingbinding energycan be accounted forby introducingaTNFintothenuclearHamiltonian[4].

*

Correspondingauthor.

E-mailaddress:ignazio.bombaci@unipi.it(I. Bombaci).

NNpotentialsplusTNFsbasedonChPThavebeenrecentlyused to investigate propertiesof medium-mass nuclei [5,6] andheavy nuclei[7].A veryimportant taskinthisline is the evaluationof the uncertainties originating in the nuclear Hamiltonian [8] and in particular on the LECs and to establish which should be the best fitting procedure to fix them [9]. For example,in Ref. [5]a simultaneousoptimizationoftheNNinteractionplusaTNFinlight

andmedium-massnucleihasbeenperformed.

Inthepresentwork,weusenuclearinteractionsderivedwithin the framework of ChPT to calculate the equation of state (EoS) of symmetric nuclear matter andpure neutron matter using the Brueckner–Hartree–Fock(BHF) approach[10,11].In particular,we intend to study nuclear matter properties using different sets of TNFs constructed to reproduce specific observables in the three-andfour-nucleonsystems.Inthiswaywecanestimate the sensi-tivityintroducedbythedifferentLECs’values.

In a previous paper[12],we have investigatedwhether using thesameinteractionsattwo- andthree-bodylevel,itwaspossible toconcurrentlyreproducepropertiesoffinitelightnucleiand nu-clearmatter. Fixing theparameters of theTNF tosimultaneously describe the3_H,3_{He and}4_{Hebinding energiesandthe neutron–}

deuteron(n-d)doubletscatteringlength[13],wefoundthatnone ofthe considered interactions was able to reproducea good sat-uration point of symmetric nuclear matter. In [12] we used the ArgonneV18 NNpotential [14] supplemented withtwo different

http://dx.doi.org/10.1016/j.physletb.2016.05.042

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

Table 1

Five (two) different parametrizations of the N2LOL three-bodyforce with  =

500 MeV ( =450 MeV).Thevaluesc1= −0.00081 MeV−1,c3= −0.0032 MeV−1

(c3= −0.0034 MeV−1),c4= −0.0054 MeV−1 (c4= −0.0034 MeV−1)and



=

500 MeV ( =450 MeV)havebeen keptfixinallthefive(two)cases.Seetext fordetails. (MeV) cD cE N2LOL1 500 1.00 −0.029 N2LOL2 500 −0.20 −0.208 N2LOL3 500 −0.04 −0.184 N2LOL4 500 0.00 −0.180 N2LOL5 500 0.25 −0.135 N2LOL6 450 −0.24 −0.110 N2LOL7 450 0.40 −0.011

TNFs. In the first casewe employed various parametrizations of theTucson–MelbourneTNF[15],whileinthesecondcase,weused the local version of the chiral N2LO [16] TNF (hereafter N2LOL)

[17].TheN2LOLdiffersfromtheusualN2LOTNFonlyfortheuse ofa localcutoff(see Refs. [13] and[12])forother details. Inthe presentwork,weconsideraninteractionfullybasedonChPTboth forthetwo- andthree- nucleonsectors.Weuseindeedthechiral N3LO[18]NNpotentialinconjunctionwithdifferent parametriza-tions of the N2LOL three-nucleon interaction [17]. As the N3LO NNinteractionhasbeenproducedinversionswithdifferentcutoff values,weintendtostudythesensitivityofourresultsonthe cut-off.Partiallythisdependencecouldarise fromthe factthatwhile the NN potential is calculated at order N3LO ofchiral perturba-tionexpansion,thethree-nucleononeiscalculatedatorderN2LO. Chiral TNFs havealso been calculated atorder N3LO [19]. Their effectinthedescription oflightnucleiisatpresentunder inves-tigation[20].It should be noticed that no additionallow energy constant (LEC) appears at this order. New LECs appear in short rangesubleadingcontributionstotheTNFatorderN4LO[21]and theircontributionseemstobepotentiallyimportant[22].Thelong range part ofTNF at the same order contains no additional LEC

[23].In thiswork welimit ourstudyconsideringTNFscalculated atorderN2LO.

Nuclearinteractions basedonChPThavebeenusedby several groupsforcalculatingtheEoSofpureneutronmatter[24–30]and

symmetric nuclear matter [31–35]. A comparison with some of

thesecalculationswillbeperformedinlastpartofthiswork. Thepaperis organizedasfollows: insection 2we reviewthe parameters oftheN2LOL three-nucleonforce andthe determina-tion ofthe low energyconstants;in section 3 we briefly discuss

how to include a TNF in the Brueckner–Hartree–Fock (BHF)

ap-proach;section4isdevotedtoshowanddiscusstheresultsofour calculations;finally insection 5 weoutline themain conclusions ofthepresentstudy.

2. The N2LOL three nucleon force

FollowingRef.[17],theN2LOLpotentialcanbewrittenas func-tionofthetransferredmomentaq andqoftheexternalnucleons. We adopted[17] acut offfunction ofthe form F

=

e−

q2n 2n _and

a similar one forq. We employ n

=

2 inthe case ofa

momen-tum cutoff parameter



=

500 MeV and n

=

3 in the case



=

450 MeV.For



=

500 MeV,we havethenconsidered five differ-entparametrizationsoftheN2LOLTNF(hereafterN2LOL1,N2LOL2, N2LOL3,N2LOL4,N2LOL5). Forall thesefiveparametrizations,we havekept the values ofparameters c1, c3 andc4 asthe original

onesdeterminedin[18].Forthecase



=

450 MeV,wehave con-sidered two additional parametrizations denoted as N2LOL6 and N2LOL7.Inthiscasec1,c3 andc4 havebeentakenfromRef.[34].

Thevaluesofci arereportedinthecaptionofTable 1forthetwo

considered values of themomentum cutoff



. The valuesof the remainingLECs cE andcD areshowninTable 1.The

parametriza-tion N2LOL1, is the original one proposed in Ref. [17],where cE

and cD were fitted to reproduce the binding energies of 3Hand 4_{He ina no-core shell model calculation. In Ref. [36], fixing the}

rangeofcD between

−

3 and3,theparametercE wasdetermined

fittingthebindingenergiesof3_Hand3_{He.Inthiswaytheauthors}

ofRef.[36]obtainedtwocurvescE

(

cD

)

fulfillingtheprevious

con-straints.Asthetwocurvesturnedouttobeveryclose,theauthors ofRef.[36] performedan averagebetweenthem.Finally,foreach setofparameters

(

cE,cD

)

,theGamow–Teller(GT)matrixelement

of tritium

β

-decay was calculated and, using the corresponding experimental value andits error-bars, theminimaanda maxima valuesforcD andcE satisfyingthisrequirementweredetermined.

IntheparametrizationsN2LOL2andN2LOL3,wehaveadoptedthe minima andthemaximavalues allowed forcD andcE according

totheconstructiondescribed above.TheparametrizationN2LOL4, takenagainfromRef. [34],was obtainedinthe samewayasthe N2LOL2andN2LOL3onesbutallowingthattheGTmatrixelement of tritium

β

-decay to be reproduced with a slight larger uncer-tainty (however lessthan 1%).The lastparametrization (N2LOL5) thatwehaveconsidered,hasbeenobtainedfixingthecouple(cD,

cE) on thetrajectory reproducingthebinding energies of3Hand 3_{Heandrequiringtogetthebestsaturationdensity}

_ρ

0 of

symmet-ric nuclear matter. As we havesaid before,for



=

450 MeV we haveexploredjusttwoparametrizationsoftheN2LOLTNF,namely

N2LOL6 andN2LOL7.ConcerningtheparametrizationN2LOL6,we

havefixedthevaluesofthelowenergyconstantscD andcE inthe

same wayofN2LOL4 inthecase



=

500 MeV;for

parametriza-tion N2LOL7, we have optimized the symmetric nuclear matter

saturation pointwiththeconstrainttoreproducethebinding en-ergies of3Hand3He. Finally,we underline thatin thetwo-body N3LOinteractionwehaveadoptedthesamevalueofthecutoff



usedfortheN2LOL TNF.ThustheLECs(seeTable 1caption)have beenconsistentlycalculatedforthetwodifferentvaluesof



[34].

3. Inclusion of three body forces in the BHF approach

Asdiscussedintheliterature[37–40],three-nucleonforces can-not be included in the BHF formalism [11,41,42] in their origi-nal form. This would require the solution of three-body scatter-ing equations in the nuclear medium (Bethe–Faddeev equations) whichisanextremelydemanding taskfromatechnicaland com-putational point of view. A possible simplification is to build an effectivedensitydependenttwo-bodyforcestartingfromthe orig-inalthree-bodyonebyaveragingoverthecoordinates(spatial,spin andisospin)ofoneofthenucleons.TheeffectiveNNforcedueto the generalthree-nucleonforce W

(

1

,

2

,

3

)

can be writtenas[37, 43]: W

(

1

,

2

)

=

1 3



dx3



c yc W

(

1

,

2

,

3

)

n

(

1

,

2

,

3

)(

1

−

P13

−

P23

) ,

(1) wherewehavedefined



dx3

=

Tr(τ3,σ3)



dr3 andn

(

1

,

2

,

3

)

isthe

densitydistributionofnucleon3 inrelationtonucleon1 at

r

1and

nucleon 2 at r2.The functionn

(

1

,

2

,

3

)

contains the effectofthe

NNshort-rangecorrelationssuppressingtheprobabilityoffinding twonucleonsclosetoeachother.Inthefollowing,weassumethat thisdensitydistributioncanbefactorizedas[37,12]

n

(

1

,

2

,

3

)

=

ρ

g2

(

1

,

3

)

g2

(

2

,

3

) ,

(2) where

ρ

isthenucleondensity,g

(

1

,

3

)

andg

(

2

,

3

)

arethe correla-tionfunctionsbetweenthenucleons

(

1

,

3

)

and

(

2

,

3

)

respectively.

Thelatterquantitiescanbewrittenasg

(

1

,

3

)

=

1

−

η

(

1

,

3

)

,where

η

(

1

,

3

)

isthe so-calleddefect function(andsimilarly for g

(

2

,

3

)

). The defect function

η

(

i

,

j

)

is the difference between the corre-latedandthe uncorrelatedwave functionsandthus represents a measure of short range correlations [44,45]. To simplify the nu-mericalcalculationsandfollowing[37,12],inthepresentworkwe usecentral correlation functions g

(

i

,

j

)

independent on spin and isospin.Moreover,ithasbeenshown[37–39]thatthiscentral cor-relationfunctions,inwhichareincludedthemaincontributionsof the1_S

0 and3S1 channels, are weaklydependent on thedensity,

andinprinciplecanbeapproximatedbyaHeavisidestepfunction

θ (

ri j

−

rc), withrc

=

0

.

6 fm in all the considered density range.

Howeverinthepresentwork

η

(

i

,

j

)

hasbeencalculatedina self-consistent way at each step of the iterative procedure from the BruecknerG-matrix[44].Finally Pi jarethespin,isospinandspace

exchangeoperatorsbetweennucleoni and j.

Adifferentaverageprocedure,withrespecttotheonediscussed above,hasbeenperformedinRef.[46],whereaninmedium effec-tiveNNpotentialwas derivedfromtheN2LOthree-nucleonforce. In[46]justtheon-shellcontributionshavebeendeterminedwhile theoff-shellcounterpartshavebeenobtainedbyextrapolation.

We want to stress that the presence of contributions arising fromthecyclicpermutationsofW

(

1

,

2

,

3

)

andfromtheexchange operators Pi j intheaverageEq.(1)hasbeenneglectedintheBHF

calculationsreportedin[32],wherenosaturationofnuclear mat-terhasbeenobtainedusingtheN3LONNpotentialplustheN2LOL threenucleoninteraction.Thepresenceoftheexchangeoperators in Eq.(1) has been neglected in our previous work [12]. As we will show in the following, these contributions to the TNF play a very important role for the nuclear matter saturation mecha-nism.Inour presentcalculationscheme,we take intoaccount of all internal and external permutations of the TNF. The resulting effectivedensitydependent potential contains in particular some purelyrepulsivecontributionswhicharemissingwhenneglecting the Pi j exchange operators. Such repulsionis needed tocontrast

thestrongattractionprovidedattwo-bodylevelby theN3LO po-tentialaswewillshowinnextsection.

4. Results and discussions

We nowpresent the resultsof our calculationsof theenergy per particle E

/

A of symmetric nuclear matter and pure neutron

matter using N3LO NN potential supplemented with the N2LOL

three-body force. These calculations have been performed using

the BHF approximation of the Brueckner–Bethe–Goldstone

hole-lineexpansion[10,11]withtheBHFcontinuous choice[47,42]for the auxiliary single particle potential U

(

k

)

entering in the self-consistentprocedure todetermine E

/

A.As suggestedinRef. [47]

andlater onnumerically confirmedinRef. [48,49],the BHF con-tinuouschoiceforU

(

k

)

optimizestheconvergenceofthehole-line expansion thus illustrating the accuracy of thecalculated energy perbaryonatthetwo-holelineapproximationwiththis prescrip-tionforthauxiliarypotential.

InFig. 1weshowtheenergyperparticleofpureneutron mat-ter(leftpanel)andsymmetric nuclearmatter (right panel)using thecutoff



=

500 MeV.The dottedlinesin bothpanelsrefer to thecalculationperformedemployingtheN3LOtwo-bodypotential withoutanyTNF.First wenotethesizeablecontributionprovided bythe TNFto theenergyper particle ofboth symmetricnuclear

matter and pure neutron matter. In the case of symmetric

nu-clearmatter the saturation point moves indeed from0

.

41 fm−3 tovaluesbetween0

.

16 fm−3 and0

.

185 fm−3,depending on em-ployed TNF parametrization (see Table 1). More specifically, the parametrizationN2LOL1predictsasatisfactorysaturationpointfor symmetricnuclearmatter atadensityof0

.

185 fm−3 andan

en-Fig. 1. Energy pernucleon for pureneutron matter(left panel) and symmetric nuclearmatter(rightpanel)versusthenucleonicdensityforthefive parametriza-tionsoftheN2LOLmodelwith



=500 MeV.Thegreen-dottedline,inboth pan-els,representstheenergypernucleonwithnothree-bodyforcecontributionand usingthe N3LO NN potential.The empirical saturationpoint of nuclearmatter ρ0=0.16±0.01 fm−3,E/A|ρ0= −16.0±1.0 MeV isdenotedbytheblackboxin

therightpanel.

Fig. 2. Energypernucleonforpureneutronmatter(leftpanel)andsymmetric nu-clearmatter(rightpanel)versusthenucleonicdensityforthetwoparametrizations oftheN2LOLmodelwith



=450 MeV.

ergy per nucleon equal to

−

15

.

48 MeV.The shift introduced by the N2LOL1 TNFto the energyper nucleon atthe empirical sat-uration density

ρ

0

=

0

.

16 fm−3 is



E

=

2

.

73 MeV for symmetric

nuclear matterand



E

=

4

.

84 MeV for pureneutron matter.The parametrizationsN2LOL2,N2LOL3 andN2LOL4, allproducea sat-uration densityof 0

.

15 fm−3, but the corresponding energy per

nucleon ranges between

−

11.23 MeV and

−

12.16 MeV (see

Ta-ble 1) to be compared withthe empirical value (

−

16

±

1) MeV. WenotethattheparametrizationN2LOL5hasbeenconstructedto reproducethe3Hand3Hebindingenergiesandthebestsaturation densityofsymmetric nuclearmatter. However, we wantagainto remarkthatinthiscasetheGTmatrixelementoftritium

β

-decay isnot reproduced.Thecurvesfortheenergyper nucleonofpure neutron matter (right panel in Fig. 1) are very similar inall the densityrangeconsidered.

In Fig. 2, we show the same quantities of Fig. 1, but adopt-ing thecutoff



=

450 MeV.In thiscasewe haveconsideredthe twoparametrizationsN2LOL6andN2LOL7(seeTable 1)oftheTNF. Theresultsobtainedfor



=

450 MeV areverysimilartotheones

Fig. 3. Symmetryenergyversusnucleonicdensityfortheparametrizationsofthe N2LOLwith



=500 MeV (leftpanel)andwith



=450 MeV (rightpanel).The triangleslabeledDSSrepresenttheresultsofRef.[51].

discussed above for the case



=

500 MeV at least fordensities around the empiricalsaturation densityone or slightlylarger. At densitieslargerthan

∼

0

.

25 fm−3 E

/

A issystematicallysmallerin

the case



=

450 MeV compared to the



=

500 MeV one. This

behavior ismainlyduetothestrongattractionintroduced bythe NNpotentialN3LOforthechoice



=

450 MeV ofthecutoff.Itis apparent that none of thepresent interactions can fulfill the re-quirement to reproduce simultaneously the 3H and 3He binding energies,theGTmatrixelementoftritium

β

-decayandthe satura-tionpointofsymmetricnuclearmatter.Howeverthe parametriza-tions N2LOL1 andN2LOL7, which reproduce3_{H and}3_{He binding}

energies,producealsoasatisfactorysaturationpointforsymmetric nuclearmatter.Suchachievementsrepresentabigimprovementof ourpreviouscalculations[12].

In the case of asymmetric nuclear matter withneutron den-sity

ρ

n,protondensity

ρ

p,totalnucleondensity

ρ

=

ρ

n

+

ρ

p and

asymmetryparameter

β

= (

ρ

n

−

ρ

p)/

ρ

theenergypernucleonin

theBHFapproachcanbeaccuratelyreproduced[50]usingthe so-called parabolicapproximation

E

A

(

ρ

, β)

=

E

A

(

ρ

,

0

)

+

Esym

(

ρ

)β

2

_,

₍₃₎

whereEsym(

ρ

)

isthenuclearsymmetryenergy.Thusthe symme-tryenergycanbecalculatedasthedifferencebetweentheenergy perparticleofpureneutronmatter(

β

=

1)andsymmetricnuclear matter(

β

=

0).

Thenuclearsymmetryenergy,calculatedwiththisprescription, isplottedinFig. 3fortheparametrizationsoftheN2LOLTNFwith



=

500 MeV (leftpanel)and



=

450 MeV (rightpanel)usedin thepresentwork.Inthesamefigure,weshow Esym (triangles)as

obtainedfromrecentcalculations[51]ofasymmetricneutron-rich matter with two- and three-body chiral interactions. The results ofRef. [51] haveconfirmedthevalidity ofthequadratic approxi-mation(Eq.(3))fordescribingtheEOShighly asymmetricmatter. However, ithasbeenrecentlyshown[52,53] thatthe

β

4 termin theenergypernucleonofasymmetricnuclearmattercouldnotbe negligible, especially at supranuclear densities, thus having size-ableinfluencee.g.onneutronstarcooling[54].

Tocompareourresultswiththevalueofthesymmetryenergy extractedfromvarious nuclear experimental data [55,56],we re-port in Table 2 and Table 3 the symmetry energy Esym(

ρ

0

)

and

theso-called slopeparameter

L

=

3

ρ

0

∂

Esym

(

ρ

)

∂

ρ





ρ0 (4) Table 2

Properties of nuclear matterat saturation point for  =500 MeV. In the five columnsareshown:parametrizationoftheN2LOLthree-bodyforce,saturation den-sity,correspondingvalueofenergyperparticle E/A,symmetryenergyEsym(ρ0)

andslopeL ofthesymmetryenergyatthecalculatedsaturationdensity.Thelast rowreferstothecalculationinwhichjusttheN3LONNforcehasbeenused.See textfordetails.

ρ0(fm−3) E/A (MeV) Esym(MeV) L (MeV)

N2LOL1 0.185 −15.48 35.5 58.5 N2LOL2 0.15 −11.23 29.0 44.1 N2LOL3 0.15 −11.96 29.3 45.0 N2LOL4 0.15 −12.16 29.4 45.2 N2LOL5 0.16 −13.04 31.3 48.7 N3LO 0.41 −24.25 55.0 108.2 Table 3

ThesameofTable 2butwith



=450 MeV.Seetextfordetails.

ρ0(fm−3) E/A (MeV) Esym(MeV) L (MeV)

N2LOL6 0.15 −13.04 29.4 40.2

N2LOL7 0.18 −15.01 33.6 44.3

atthecalculatedsaturationdensity

ρ

0 (secondcolumninTable 2

and 3) for the TNFparametrizations considered in thiswork. As we canseeourcalculated Esym(

ρ

0

)

andL areingoodagreement

withthevaluesextractedfromexperimentaldata[55]:Esym(

ρ

0

)

=

29

.

0–32

.

7 MeV, and L

=

40

.

5–61

.

9 MeV. This agreement is lost whentheTNFisnotincludedinthecalculations(seee.g.thelast rowinTable 2). Noticethat thevalue ofthenuclear incompress-ibility K0 of symmetric nuclear matter, at the calculated

satura-tiondensity,isgenerallyquitelowwhencomparedwiththevalue

deduced from measured isoscalar giant monopole resonances in

medium-heavynuclei K0

=

240

±

20 MeV[57] K0.Ourcalculated

value of K0 ranges between150 and 170 MeV,depending onthe

TNFmodel.

Let us now compare our results with those of similar calcu-lations presentin literature based on chiral nuclear interactions. Ourpresentresultsareingoodagreementwiththerecent nonper-turbative quantumMonte Carlocalculations[26] ofpure neutron matter. The authors ofRef. [34] performednuclear matter calcu-lations up to third order many-body perturbation theory, adopt-ing theeffectivedensitydependentNNforcederived inRef. [46]. These authors found results in acceptable agreement with ours both fornuclearmatter andpure neutronmatter. Severalnuclear matter calculations based on chiral interactions have been per-formed usingother many-bodytechniques.In Ref.[31],usingthe

Vlowk-approachandthesimilarityrenormalizationgroup(SRG), it

was shownthat fora suitable choiceofthe cutoff

V lowk

itwas possible to reproduce a good saturation point of symmetric nu-clearmatterkeepingfortheparametersoftheTNFthevaluesfixed in few-body calculations. In Ref. [33], using the self-consistent Green’s functionsmethod, it has been found that the saturation point ofsymmetric nuclearmatter isstronglyimproved withthe help oftheN2LOL three-nucleonforcealthough,alsointhiscase, thesaturationenergyresultsunderestimated.

In Fig. 4 we compare our results (using the parametrizations N2LOL1, N2LOL4andN2LOL6)fortheenergyperparticle ofpure neutronmatterwiththoseobtainedbyotherresearchersusing dif-ferent many-bodyapproaches. The black- andred-dashedregions inthe leftpanel ofFig. 4 representthe resultsofthemany-body perturbative calculations ofRef. [25] using completetwo-, three-and four-body interactions at the N3LO of the ChPT. In particu-lar,theregionbetweenthetwoshort-dashedblackcurves (black-dashedbandpartially overlappedbythered-dashedband)is rela-tivetotheN3LOEntem–Machleidt[18] NNpotential,whereasthe

Fig. 4. Comparisonofpureneutronmatterenergyperparticle withother many-bodymethods(seetextfordetails).

potentials[58].The widthofthe bandsrepresentsthe uncertain-tiesrelatedtothe valuesoftheLECsandthecutoffofthe three-andfour-bodyforces.Thedash-dotted(blue)curvesinbothpanels inFig. 4, representtheresults [29] ofan auxiliary field diffusion Monte Carlo (AFDMC) calculation of neutron matter using a lo-cal formof two- andthree-body chiral interactions atN2LO and twodifferent valuesofthe NNcutoff (see [29] formore details). Thegreen-dottedlineinbothpanelsofFig. 4,corresponds tothe resultsofRef. [27] withthe auxiliary field quantumMonte Carlo

(AFQMC) obtained with chiral N3LO two-body force plus N2LO

TNF. Finally, the red-dashed band in the right panel of Fig. 4

represents the results of Ref. [30] for the energy per particle of neutron matter after a renormalization group (RG) evolution of

theN3LOEntem–Machleidt[18]NNinteractiontolow-momentum

scale(



=

394

.

6 MeV)andincludingN2LOTBF(the widthofthe band reflectingthe uncertainties ofthe LECs inthe TNF).As one cansee,ourresultsareinverygoodagreementwithalltheother calculationsconsidered in Fig. 4, except with the calculationsof Ref.[29]fordensitiesclosetotheempiricalsaturationdensity.In fact,inthisdensityregiontheAFDMCcurvesare “flat”compared toother calculations[29], thusimplying low values,intherange

L

= (

16

.

0–36

.

5

)

MeV,fortheslopeparametercalculatedusingthe parabolicapproximationEq.(3)attheempiricalsaturationdensity

ρ

0

=

0

.

16 fm−3.

5. Conclusions

WehaveperformedBHFcalculationsoftheEoSsymmetric

nu-clearmatter andpure neutron matter considering the N3LO NN

potential plus the N2LOL three nucleon interaction. The last one hasbeenreducedtoan effectivedensitydependent two-body in-teractionaveragingoverthecoordinates(spatial,spinandisospin) ofoneofthenucleons.Wehaveshownthatthecontributions aris-ing fromthecyclic permutationsof W

(

1

,

2

,

3

)

andfromthe nu-cleonexchangeoperators Pi j intheaverage(seeEq.(1))oftheTNF

playa decisiveroleforthenuclear mattersaturation mechanism. Infact,nosaturationisobtained[32]whenthesecontributionsare nottakenintoaccount.We wanttostress thatthe parametersof theTNFfittedincalculationsoflight nucleihavebeenkept fixed inournuclearmattercalculations.

Fromone sidewefound thatit wasnot possibletoreproduce simultaneouslythe binding energy of3H, 3He, theGT matrix el-ement of tritium

β

-decay and the empirical saturation point of symmetricnuclearmatter.Howeversometrendscanbeextracted,

in particular, those parametrizations that reproduce the binding energyof3H,3He andtheGTmatrixelement oftritium

β

-decay predict a reasonable value (0

.

15 fm−3) of the saturation density, butthecorrespondingbindingenergypernucleon(B

/

A

= −

E

/

A)

isunderestimated.Thetwoparametrizations,N2LOL1andN2LOL7, that are not constrained to reproduce the GTmatrix element of tritium

β

-decay are able to reproduce quite well the empirical saturation energybutata toolarge value of

ρ

0.Finally we have

observedareduceddependenceonthemomentumcutoff

param-eter. We have found in addition a good agreement with recent

AFQMC[27]andMBPT[25]calculationsofneutronmatter. Theseencouragingresultsrepresentasignificantsteptowarda unifieddescriptionoffew- andmany-bodynuclearsystems start-ingfromtwo- andthree-nucleoninteractionsderivedinthe frame-workofchiralperturbationtheory.

Acknowledgements

This work has been partially supported by “NewCompstar”,

COSTActionMP1304.

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