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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
aaINFN,SezionediPisa,LargoBrunoPontecorvo3,I-56127Pisa,Italy
bDipartimentodiFisica,UniversitádiPisa,LargoBrunoPontecorvo3,I-56127Pisa,Italy cEuropeanGravitationalObservatory,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 experimentalbind-ingenergiesof3H,3Hebyabout1 MeVandthatof4Hebyabout 4 MeV[4].This missingbinding energycan be accounted forby introducingaTNFintothenuclearHamiltonian[4].
*
Correspondingauthor.E-mailaddress:[email protected](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 the3H,3He and4Hebinding 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 ofamomen-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 originalonesdeterminedin[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 4He ina no-core shell model calculation. In Ref. [36], fixing the
rangeofcD between
−
3 and3,theparametercE wasdeterminedfittingthebindingenergiesof3Hand3He.Inthiswaytheauthors
ofRef.[36]obtainedtwocurvescE
(
cD)
fulfillingthepreviouscon-straints.Asthetwocurvesturnedouttobeveryclose,theauthors ofRef.[36] performedan averagebetweenthem.Finally,foreach setofparameters
(
cE,cD)
,theGamow–Teller(GT)matrixelementof 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 3Heandrequiringtogetthebestsaturationdensity
ρ
0 of
symmet-ric nuclear matter. As we havesaid before,for
=
450 MeV we haveexploredjusttwoparametrizationsoftheN2LOLTNF,namelyN2LOL6 andN2LOL7.ConcerningtheparametrizationN2LOL6,we
havefixedthevaluesofthelowenergyconstantscD andcE inthe
same wayofN2LOL4 inthecase
=
500 MeV;forparametriza-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) wherewehavedefineddx3=
Tr(τ3,σ3)dr3 andn
(
1,
2,
3)
isthedensitydistributionofnucleon3 inrelationtonucleon1 at
r
1andnucleon 2 at r2.The functionn
(
1,
2,
3)
contains the effectoftheNNshort-rangecorrelationssuppressingtheprobabilityoffinding twonucleonsclosetoeachother.Inthefollowing,weassumethat thisdensitydistributioncanbefactorizedas[37,12]
n
(
1,
2,
3)
=
ρ
g2(
1,
3)
g2(
2,
3) ,
(2) whereρ
isthenucleondensity,g(
1,
3)
andg(
2,
3)
arethe correla-tionfunctionsbetweenthenucleons(
1,
3)
and(
2,
3)
respectively.Thelatterquantitiescanbewrittenasg
(
1,
3)
=
1−
η
(
1,
3)
,whereη
(
1,
3)
isthe so-calleddefect function(andsimilarly for g(
2,
3)
). The defect functionη
(
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 the1S0 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,isospinandspaceexchangeoperatorsbetweennucleoni 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)hasbeenneglectedintheBHFcalculationsreportedin[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 neutronmatter 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 symmetricnuclearmatter 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 andanen-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 densityρ
0=
0.
16 fm−3 isE
=
2.
73 MeV for symmetricnuclear matterand
E
=
4.
84 MeV for pureneutron matter.The parametrizationsN2LOL2,N2LOL3 andN2LOL4, allproducea sat-uration densityof 0.
15 fm−3, but the corresponding energy pernucleon ranges between
−
11.23 MeV and−
12.16 MeV (seeTa-ble 1) to be compared withthe empirical value (
−
16±
1) MeV. WenotethattheparametrizationN2LOL5hasbeenconstructedto reproducethe3Hand3Hebindingenergiesandthebestsaturation densityofsymmetric nuclearmatter. However, we wantagainto remarkthatinthiscasetheGTmatrixelementoftritiumβ
-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 areverysimilartotheonesFig. 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 issystematicallysmallerinthe case
=
450 MeV compared to the=
500 MeV one. Thisbehavior 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,theGTmatrixelementoftritiumβ
-decayandthe satura-tionpointofsymmetricnuclearmatter.Howeverthe parametriza-tions N2LOL1 andN2LOL7, which reproduce3H and3He bindingenergies,producealsoasatisfactorysaturationpointforsymmetric nuclearmatter.Suchachievementsrepresentabigimprovementof ourpreviouscalculations[12].
In the case of asymmetric nuclear matter withneutron den-sity
ρ
n,protondensityρ
p,totalnucleondensityρ
=
ρ
n+
ρ
p andasymmetryparameter
β
= (
ρ
n−
ρ
p)/ρ
theenergypernucleonintheBHFapproachcanbeaccuratelyreproduced[50]usingthe so-called parabolicapproximation
E
A
(
ρ
, β)
=
EA
(
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)asobtainedfromrecentcalculations[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)
andtheso-called slopeparameter
L
=
3ρ
0∂
Esym(
ρ
)
∂
ρ
ρ0 (4) Table 2Properties 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 2and 3) for the TNFparametrizations considered in thiswork. As we canseeourcalculated Esym(
ρ
0)
andL areingoodagreementwiththevaluesextractedfromexperimentaldata[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 calculatedsatura-tiondensity,isgenerallyquitelowwhencomparedwiththevalue
deduced from measured isoscalar giant monopole resonances in
medium-heavynuclei K0
=
240±
20 MeV[57] K0.Ourcalculatedvalue 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,intherangeL
= (
16.
0–36.
5)
MeV,fortheslopeparametercalculatedusingthe parabolicapproximationEq.(3)attheempiricalsaturationdensityρ
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))oftheTNFplaya 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 ofρ
0.Finally we haveobservedareduceddependenceonthemomentumcutoff
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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