Doubly magic nuclei from Lattice QCD forces at $M_{PS}=$469 MeV/c$^2$

0.33 MB·

by  C. McIlroy

DoublymagicnucleifromLatticeQCDforcesat M =469MeV/c2
PS
C. McIlroy,1,∗ C. Barbieri,1,† T. Inoue,2 T. Doi,3 and T. Hatsuda3,4
1Department of Physics, University of Surrey, Guildford GU2 7XH, UK
2Nihon University, College of Bioresource Sciences, Kanagawa 252-0880, Japan
3Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan
4iTHEMS Program and iTHES Research Group, RIKEN, Wako 351-0198, Japan
(Dated:January12,2017)
Weperformabinitioself-consistentGreen’sfunctioncalculationsofclosedshellnuclei4He,16Oand40Ca,
based on two-nucleon potentials derived from Lattice QCD simulations, in the flavor SU(3) limit and at the
pseudo-scalarmesonmassof469MeV/c2.ThenuclearinteractionisobtainedusingtheHALQCDmethodand
itsshort-distancenucleon-nucleonrepulsionwastreatedbymeansofladderresummationsoutsidethemodel
7 space. Comparing to previous Brueckner Hartree-Fock calculations, the total binding energies are sensibly
1 improvedbythefullaccountofmany-bodycorrelations. Theresultssuggestaninterestingpossiblebehaviour
0 inwhichislandsofstabilityappearatfirstaroundthetraditionaldoubly-magicnumberswhenthepionmassis
2
loweredtowarditsphysicalvalue. Thecalculatedone-nucleonspectraldistributionsarequalitativelycloseto

n thoseofrealnucleievenfortheheavypseudo-scalarmesonmassconsideredhere.
a
J

1 Introduction. Thequantumchromodynamics(QCD)isex- scalar meson (which corresponds to the pion) as low as
1 pectedtoultimatelyexplainthestructureandtheinteractions M =469 MeV/c2. In these cases, potentials have been ob-
PS
of all hadronic systems, together with the small corrections tainedforthe1S0 andthecoupled3S1-3D1 channels[19,20].
]
h of electroweak origin. At the moment, systematic and non- ExploratorycalculationsbasedontheseHALQCDpotentials
t perturbative calculations of QCD can be carried out only by were performed in the Brueckner Hartree-Fock (BHF) ap-
-
l Lattice QCD (LQCD). Indeed, high-precision studies have proach[21,22]. Thisisquantitativeenoughtogivetheessen-
c
been shown to be possible e.g., in the case of single-hadron tialunderlyingphysicsforinfinitematterbutitislessreliable
u
n masses [1]. ”Direct” calculations of multi-baryon systems infinitesystems. Moresophisticatedcalculationsareneeded
[ havealsobeenattemptedonalatticebyseveralgroups[2–6]. in order to go beyond the mean-field level, which is neces-

However, thetypicalexcitationenergy∆E formulti-baryons sary to properly predict binding energies and to describe the
2
v isonetotwoordersofmagnitudesmallerthanO(ΛQCD). Ac- truly complex structures of nuclei at low energy. Moreover,
7 cordingly, highstatisticdatawithverylargeEuclideantimes BHFbecomesevenmorequestionableforfinitenucleidueto
0 t >∼ (cid:126)/∆E ∼10-100fm/cisrequired. Thisisstillfarbeyond assumptionswiththeunperturbedsingleparticlespectrum—
6
reach due to exponentially increasing errors in t and A (the wherethereisaprobleminthechoicebetweenacontinuous
2
atomic number), as demonstrated theoretically and numeri- oragapform,neitherofwhichiscompletelysatisfactory.
0
. callyinrecentstudies [7–9]. Ab initio theories for medium mass nuclei have advanced
1
0 InthisLetter,wefollowadifferentrouteandperformafirst greatlyinthepastdecadeandmethodssuchascoupledclus-
7 abinitiostudyformediummassatomicnucleidirectlybased terandself-consistentGreen’sfunction(SCGF)arenowrou-
1 onQCDbytakingatwo-stepstrategy. Inthefirststep,weex- tinely employed to study the structure of full isotopic chains
:
v tractthenuclearforcefromabinitioLQCDcalculationswith uptoCaandNi[23–26],withtheinclusionofthree-nucleon
i the HAL QCD method, which generates nucleon-nucleon forces[27]. Theirusewithsoftinteractionsfromchiraleffec-
X
(NN) potentials faithful to the scattering phase shifts [10– tivefieldtheoryhaveledtofirstprinciplepredictionsofexper-
r
a 12]. Inthesecondstep, wecalculatethepropertiesofnuclei imentaltotalbindingenergies[28,29]andnuclearradii[30–
withabinitiomany-bodymethodsusingtheLQCDpotentials 32] with unprecedented accuracies. Interactions like the
as input. The advantage of the HAL QCD approach is that HALQCDpotentialsposeabiggerchallengeforcalculations
one can extract the potential dictating all the elastic scatter- of medium mass isotopes due to their short-distance inter-
ing states below the inelastic threshold from the lattice data nucleon repulsion. However, the SCGF method has the ad-
fort >∼ 1fm/c[13]. ThismakestheLQCDcalculationofpo- vantagethattwo-nucleonscattering(ladder)diagramsatlarge
tentialsaffordablewithreasonablestatistics,togetherwiththe momenta—needed to resolvethe short-range repulsion—can
helpofadvancedcomputationalalgorithms[14–17]. Wecan bedealtwithexplicitlythroughaG-matrix[33–35].
thentaketheadvantageoftherecentdevelopmentsinnuclear
In this Letter, we will employ the SCGF method in the
many-bodytheoriestocalculatevariousinformationonnuclei
third-order algebraic diagrammatic construction [ADC(3)]
suchasbindingenergiesandspectraldistributions. Notethat approximation[36–38]anduseaG-matrixastheeffectivein-
asimilarstrategyhasalsobeentakeninRef.[18]byapplying
teraction, with some slight improvement over the formalism
effectivefieldtheoriestoLQCDdata.
detailed in Ref. [35]. We find that this is accurate enough to
Recent LQCD studies in the flavor SU(3) limit by the makestatementsontheperformanceofthepresentHALQCD
HAL QCD collaboration have led to interactions in both potentials. This work is also a first step toward advancing
the nucleon and hyperon sectors with masses of the pseudo- many-body approaches that can handle hard interactions and
2
a) Σ8 ( ) termsofasimplifiedMFreferencepropagator, gREF(ω),that
Σ* = + 2p1h/ is chosen to best approximate g(ω) through its first two mo-
2h1p
mentsattheFermienergy(seeRef.[35]fordetails). Inprac-
tice,wesolveinsequenceforEqs.(2), (1)andgREF(ω),and
b) repeatuntilconvergencebetweentwosuccessiveiterations.
... ...
= ... For forces with a sizeable short-range repulsion, like the
HAL QCD interactions, usual truncations of the oscillator
space (of up to 12 shells in this case) are not sufficient and
FIG.1.(Coloronline)DiagrammaticcontentoftheADC(3)approx- aresummationofladderdiagramsoutsidethemodelspaceis
imation. (a) The self-energy splits in a static mean field part and required. Ourself-energyiscorrectedtoincludethesecontri-
aenergydependentscontributionaccordingtoEq.(2). (b)Thedy-
butionsintwoways. First,wecalculateaG-matrixaccording
namiccontributionsareobtainedasinfiniteresummationsofladder
to Refs. [47, 48] and add the corresponding diagrams to its
(pp/hh)andthering(ph)diagrams. TheADC(3)approachincludes
MFpart,whichnowbecomesenergydependent[35]:
staticcorrectionstothecouplingofnucleonstointermediateexcita-
tions(matrixDinEq.(2)),anexampleofwhichisshownbythetop (cid:88)(cid:90) dω(cid:48)
portionofthelastGoldstonediagram[38]. Σ(∞)(ω)= G (ω+ω(cid:48))g (ω(cid:48))eiω(cid:48)η
αβ 2πi αγ,βδ γδ
γδ
(cid:88)(cid:88)
= G (ω+ε−)Yk(Yk)∗ , (3)
itwillbeanimportantdirectioninviewofLQCDcalculations αγ,βδ k δ γ
thatareinprogressatnearlythephysicalpionmass[39–42]. γδ k
TheBHFstudyofRef.[21]showedthattheHALQCDin-
whereG (ω)aretheelementsoftheG-matrix. Secondly,
teractionintheSU(3)limitwith MPS =469MeV/c2 leadsto we extraαcγt,βaδstatic effective interaction that we use to calcu-
a saturated nuclear matter and it is confirmed by later SCGF
late the ADC(3) self-energy (last term of Eq. (2)) within the
calculations [43]. Thus, this is a suitable choice to investi-
chosen model space. To do this, we solve the Hartree-Fock
gate possible self-bound nuclei at large pion masses. In this
(HF)equationswiththeMFpotentialofEq.(3):
work, we will focus on this potential and refer to it as the
(cid:40) (cid:41)
‘HAL469SU(3)’interaction,or‘HAL469’forsimplicity. (cid:88) (cid:104)α| p2 |β(cid:105)+Σ(∞)(ω=εHF) ψr =εHFψr , (4)
Formalism. We follow closely the approach of Ref. [35] 2m αβ r β r α
andfocusonthesingle-particlepropagatorgivenby[44,45]: β N
wherelatinindiceslabelHFstates,anddefineastaticinterac-
(cid:88) (Xn)∗Xn (cid:88) Yk(Yk)∗
g (ω)= α β + α β , (1) tioninthisHFbasissimilarlytoRefs.[35,49]:
αβ ω−ε++iη ω−ε−−iη
n n k k 1(cid:104) (cid:105)
V = G (εHF +εHF)+G (εHF +εHF) . (5)
whereXn ≡ (cid:104)ΨA+1|c†|ΨA(cid:105)(Yk ≡ (cid:104)ΨA−1|c |ΨA(cid:105))arethespec- rs,pq 2 rs,pq r s rs,pq p q
α n α 0 α k α 0
troscopicamplitudes, ε+n ≡ EnA+1 −E0A (ε−k ≡ E0A −EkA−1)are The Vrs,pq matrix elements are then transformed back to the
their quasiparticle energies and c†(c ) are the second quan- harmonicoscillatrorspacetobeusedinthecomputations.
α α
tisationcreation(annihilation)operators. Inthesedefinitions, NotethattheexcludedspaceusedtocalculatetheG-matrix
|ΨA+1(cid:105)and|ΨA−1(cid:105)representtheexacteigenstatesofthe(A±1)- isgivenbythefullmodelspace,sothatEq.(3)includesonly
n k
nucleon system, while EA+1 and EA−1 are the corresponding ladderdiagramsthatarenotalreadyresummedintheADC(3)
n k
energies. We perform calculations within a spherical har- computation. Thus,ourG-matrixdependsonlyonthemodel
monic oscillator model space and use Greek indices, α,β,... spaceandcarriesnoambiguitywiththeregardstothechoice
to label its single particle states. Within this space, the one- ofthesingle-particlespectrumattheFermisurface.Likewise,
body propagator is obtained by solving the Dyson equation the HF results from Eq. (4) are different from the standard
withanirreducibleself-energygivenby BHFtheory,whereonlytheoccupiedorbitsareexcluded.
It is well known that short-range repulsion, implicitly ac-
(cid:34) (cid:35)
Σ(cid:63) (ω)=Σ(∞)+(cid:88)D† 1 D . (2) countedbyEq.(3),hasthedoubleeffectofreducingthespec-
αβ αβ αi ω−(K+C)±iη jβ tralstrengthfordominantquasiparticlepeaksandofrelocat-
ij ij
ingittolargemomentaandlargenegativeenergies[26].Since
This is the sum of a mean-field (MF) term, Σ(∞), and the we cannot currently calculate for the location of strength at
contributions from dynamical correlations. The the cou- high momenta, it is not possible to quantify the magnitude
pling (D) and interactions matrices (K and C) are computed of these two effects. However, they contribute to the Koltun
intheADC(3)approximation, byimposingconsistencywith sumrulefortotalbindingenergywithoppositesignsandmust
the perturbative self-energy at third order plus infinite all- canceltoalargeextent.Thus,wechosetoneglectbothcontri-
ordersummationsof2p1hand2h1pconfigurations,asshown butionsandmaintainastaticΣ(∞)tosolvetheDysonequation.
in Fig. 1 [36, 38]. We follow the sc0 approximation of Thisiscurrentlythemajorapproximationinourcalculations
Refs. [35, 46], in which Σ(∞) is calculated exactly from the and its uncertainty is best estimated from the benchmark on
fully-dressedpropagatorg(ω)whileD,KandCarewrittenin 4Hebelow. Resolvingitwillbethesubjectoffuturework.
3
y, E [MeV]0--210 NNNN mmmm aaaa xxxx ==== 15791 4He 1200 NNNN mmmm aaaa xxxx ==== 15791 16O 24000 NNN mmm aaa xxx === 579 40Ca Ground sta
und state energ---543 Exact result -100 ----86420000 0te energy, E [M
Gro 5 10 15 20 25 30 35 40 -20 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40eV]
(cid:15514)Ω [MeV] (cid:15514)Ω [MeV] (cid:15514)Ω [MeV]
FIG.2. (Coloronline)Groundstateenergyof4He,16Oand40Caasafunctionoftheharmonicoscillatorfrequency,(cid:126)Ω,andthemodelspace
size,Nmax.SymbolsmarktheresultsfortheHAL469potentialfromfullself-consistentcalculationsintheG-matrixplusADC(3)approach.
Results. The one-body propagators of 4He, 16O and deviatesfromtheexactsolutionbylessthan10%. Sincethe
40Ca were calculated in spherical harmonic oscillator spaces SCGFapproachresumslinkeddiagrams,andthusissizeex-
of different frequencies, (cid:126)Ω, and increasing sizes up to tensive, one should expect that similar errors will apply for
N =max{2n+(cid:96)}=11(andN ≤9for40Ca). AG-matrix larger isotopes. Fig. 2 also demonstrates that 16O and 40Ca
max max
wascalculatedforeachfrequencyandmodelspaceandthen convergencesimilarlyto4He. Theirextrapolatedgroundstate
itwasusedtoderivethestaticinteractionsofEq.(5). Wesub- energiesarealsogiveninTab.I,wherethefirsterroristheun-
tracted the kinetic energy of the center of mass according to certaintiesinthemodelspaceextrapolation[52]. Thesecond
Ref.[50]andcalculatedtheintrinsicgroundstateenergyfrom error corresponds to many-body truncations and we estimate
g(ω)usingtheKoltunsumrule. Thesamelatticesimulation ittobe10%basedonthefindingfor4He. TheSCGFresults
setupusedtogeneratetheHAL469interactiongivesanucleon are sensibly less bound than our previous BHF results [22].
massofm =1161.1MeV/c2 inadditiontothepseudo-scalar This pattern is completely analogous to the case of 4He and
N
massof M =469MeV/c2. Thus,weemployedthisvalueof weinterpretitasalimitationofBHFtheory.
PS
mN inallthekineticenergyterms. A key feature of our calculations is the use of an har-
The exact binding energy of 4He for HAL469 is known monic oscillator space, which effectively confines all nucle-
to be 5.09 MeV [51] and can be used to benchmark our ap- ons. The last line Tab. I reports the deduced breakup ener-
proach. Fig. 2 displays the ground state energies calculated giesforseparatingthecomputedgroundstatesintoinfinitely
with the G-matrix plus ADC(3) method. The resummation distant 4He clusters. The 16O is unstable with respect to 4-α
of ladder diagrams outside the model space tames ultravio- breakup,by≈2.5MeV.Allowinganerrorinourbindingen-
letcorrectionsandwefindthattheinfraredconvergencedis- ergiesofmorethan10%couldmakeoxygenboundbutonly
cussedinRef.[52]appliesverywellforlargeoscillaltorfre- veryweakly. Thisisincontrasttotheexperimentalresults,at
quencies. Fromcalculations upto (cid:126)Ω=50 MeV,we estimate the physical quarks masses, where the 4-α breakup requires
a converged binding energy of 4.80(3) MeV for 4He, where 14.4 MeV. On the other hand, 40Ca is stable with respect to
theerrorcorrespondstotheuncertaintiesintheextrapolation. breakup in α particles by ≈24 MeV. We expect that these
Allresultsfor4HearesummarisedinTab.Iwherewealsolist observations are rather robust even when we consider the
BHFcalculationsdonewiththesamegapchoiceandmethods (LQCD) statistical errors in the HAL469 interaction. While
ofRef.[22]. ThissuggeststhattheBHFmethodcanoveres- suchstatisticalfluctuationsintroduceadditional∼10%errors
timatethebindingenergyforHAL469evensizeably. Onthe onbindingenergies[22],theyareexpectedtobestronglycor-
otherhand,thefullinclusionoflong-rangeeffectsinADC(3) related among 4He, 16O and 40Ca. Hence, for QCD in the
SU(3)limitat M =469MeV/c2,wefindthatthedeuteronis
PS
unbound[20]and16Oisonlyjustslightlyabovethethreshold
EA[MeV] 4He 16O 40Ca for α breakup, while 4He and 40Ca are instead bound. The
0
HAL469 interaction has the lowest M value among those
BHF[22] -8.1 -34.7 -112.7 PS
G(ω)+ADC(3) -4.80(0.03) -17.9(0.3)(1.8) -75.4(6.7)(7.5) considered in Refs. [19, 20], while from Ref. [21] we know
thatitistheonlyonesaturatingnuclearmatter(althoughnot
ExactResult[51] -5.09 – –
at the physical saturation point). Moreover, we have tested
Separationinto4Heclusters: -2.46(0.3)(1.8) 24.5(6.7)(7.5)
that SCGF attempts at calculating asymmetric isotopes, like
28O,predictstronglyunboundsystemsevenforHAL469. All
TABLE I. Ground state energies of 4He, 16O and 40Ca
at M =469 MeV/c2 obtained from the HAL469 interaction. theseresultstogethersuggestthat,whenloweringofthepion
PS
‘G(ω)+ADC(3)’aretheresultsofthepresentworkandarecompared masstowarditsphysicalvalue, closedshellisotopesarecre-
toBHFandexactresults.Thelastlineisthebreakupenergyforsplit- ated at first around the traditional magic numbers. This hy-
tingthesystemin4Heclusters(oftotalenergyA/4×5.09MeV). pothesis should also be seen in the light of the limitations in
thepresentHAL469Hamiltonian,whichwasbuilttoinclude
4
eV] 40 s p , p d , d 0.4 40 BHF
M 30 1/2 1/2 3/2 3/2 5/2 Ca HF
s [ G(ω) + ADC(3)
ent 20 3] 0.3
m 10 -m
C(3) q.p. frag --21000 16O (r) [f ρ 00..21 16O MPS = 469 Mev/c2
D -30
A
020406080100020406080100020406080100 0
SF [%] SF [%] SF [%] 0 1 2 3 4 5 6
r [fm]
FIG.3.(Coloronline)Singleparticlespectralstrengthdistributionof
16OobtainedfromthedressedpropagatorinthefullG-matrixplus FIG. 4. (Color online) Point-matter distributions of 16O and 40Ca.
ADC(3) approach. Each panel displays partial waves of different TheHFdensitydistributionisobtainedfromthesolutionsofrefer-
angularmomenta. Theverticalaxesgivethequasiparticleenergies encestateofEq.(4),whileADC(3)isthefullyfragmentedspectral
(thatis,thepolesofEq.(1)),whilethelengthofthehorizontalbars function.BHFlabelstheresultsfromRef.[22].
givethecalculatedspectroscopicfactors.
derived from Lattice QCD. For the HAL469 force, the total
onlythe1S ,3S and3D partialwavesandthereforeneglects binding energy agrees with the exact 4He benchmark within
0 1 1
thethree-bodyforcesandspin-orbitinteractions. 10% and shows that ab initio methods can be improved to
Figure3demonstratesthespectralstrengthdistributionsof reachlargenucleievenwithhardnuclearinteractions.
16O obtained for N =11 and (cid:126)Ω = 11MeV. Quasiparticle Thepresentaccuracyissufficienttomakesemi-quantitative
max
fragmentscorrespondingtospin-orbitpartnersdonotsplitdue statementsondoublymagicnuclei,whicharelessboundcom-
totheabsenceofaspin-orbitterminHAL469. Otherwise,all paredtoearlierBHFestimates. FortheHALQCDpotentials
the remaining qualitative features of the experimental spec- we have found that the behaviour when lowering the pion
traldistributionareseenalsoforthe M =469MeV/c2 case. mass towards the physical values is consistent with the idea
PS
A closer look to particle-hole gaps shows that the calculated that nuclei near to the traditional magic numbers are formed
separation between the s1/2 and p1/2 dominant peaks in 16O first. At MPS=469MeV/c2, intheSU(3)limitofQCD,both
is8.0MeV,whiletheempiricalvalueis11.5MeV.Thespec- 4He and 40Ca have bound ground states while the deuteron
tral strength of 40Ca is similar; however we obtain a gap of isunboundand16Odecaysintofourseparatealphaparticles.
10.1MeVbetweenthe f andd states, whiletheexperi- However, 16O is close to become bound. The calculated nu-
7/2 3/2
mentis7.5MeV.Thesefindingsarereflectedinthecalculated clearspectralfunctionsreflectthedifferentnuclearsizes(con-
point-matter distributions and root mean square radii shown sistentlywiththeshorterrangeoftheYukawainteraction)and
in Fig. 4 and Tab. II. Radii are larger (smaller) that the ex- themissingspin-orbitinteractions;theyareotherwisequalita-
periment for 16O (40Ca). The small size of calcium can be tivelyconsistentwithexperimentalobservations.
understoodintermsofthehigherpseudo-scalarmassthatre- Importantfutureworkwillbetheinclusionthespin-orbitas
ducestherangeoftheYukawainteraction. Ontheotherhand, wellasthree-nucleonforces,andLQCDcalculationsforthese
thelargespatialextensioncomputedfor16Oisconsistentwith interactionsareinprogress[56,57]. Inaddition,similarstud-
the suggestion of an unbound state that would expand to in- iesatphysicalquarkmassesareexpectedinnearfuture,since
finityiftheoscillatorwallswereremoved. TheHFapproach LQCD simulations for nuclear and hyperon forces at almost
ofEq.(4)andthestandardBHFapproachesgivesimilarradii physicalquarkmassesarecurrentlyunderway[39–41].
in spite that they predict very different binding energies (see
Tab.II).However,radiiareincreasedbyfullmany-bodycor-
relations. For all nuclei, the full G-matrix plus ADC(3) cal- 16O 40Ca
culationspushesthematterdistributiontolargerradiiandre-
r : BHF[22] 2.35fm 2.78fm
pt−matter
duces the central density. For charge radii in Tab. II we as-
HF 2.39fm 2.78fm
sumedthephysicalchargedistributionsoftheprotonandthe G(ω)+ADC(3) 2.64fm 2.97fm
neutron(seeRef.[53]fordetails).
r : G(ω)+ADC(3) 2.77fm 3.08fm
charge
Summary. We investigated the use of a G-matrix for re-
Experiment[54,55] 2.73fm 3.48fm
summing missing two-nucleon scattering diagrams outside
the usual truncations of the many-body spaces, while a full
abinitioADC(3)approachhasbeenretainedwithinthemodel TABLEII.Computedmatterandchargeradiiof16Oand40Causing
spaceitself. Abenchmarkon4Heshowsthatthepresentim- M =469MeV.Resultsaregivenfordifferentlevelsofapproxima-
PS
plementation works relatively well and it allows to solve the tions and the charge radii from the full G-matrix plus ADC(3) are
comparedtotheexperimentalvalues.
self-consistentGreen’sfunctionfortheHALQCDpotentials
5
Acknowledgements. We thank the HAL QCD Collabo- rano, H. Nemura, and K. Sasaki, Physics Letters B 712, 437
ration for providing the HAL469 interaction. Com- (2012).
SU(3)
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