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) putations of the G-matrix were performed using the CENS [14] T.DoiandM.G.Endres,ComputerPhysicsCommunications 184,117 (2013). codes from Ref. [48]. This work was supported in part [15] W.DetmoldandK.Orginos,Phys.Rev.D87,114512(2013). by the United Kingdom Science and Technology Facili- [16] J. Gu¨nther, B. C. To´th, and L. Varnhorst, Phys. Rev. D 87, ties Council (STFC) under Grants No. ST/L005743/1 and 094513(2013). ST/L005816/1,bytheRoyalSocietyInternationalExchanges [17] H. Nemura, Computer Physics Communications 207, 91 GrantNo.IE150305,byJapaneseGrant-in-AidforScientific (2016). Research (JP15K17667and (C)26400281), by MEXT Strate- [18] N. Barnea, L. Contessi, D. Gazit, F. Pederiva, and U. van gic Program for Innovative Research (SPIRE) Field 5, by a Kolck,Phys.Rev.Lett.114,052501(2015). [19] T.Inoue,N.Ishii,S.Aoki,T.Doi,T.Hatsuda,Y.Ikeda,K.Mu- priorityissue(Elucidationofthefundamentallawsandevolu- rano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), tion of the universe) to be tackled by using Post “K” Com- Phys.Rev.Lett.106,162002(2011). puter, and by the Joint Institute for Computational Funda- [20] T.Inoue,S.Aoki,T.Doi,T.Hatsuda,Y.Ikeda,N.Ishii,K.Mu- mental Science (JICFuS). T.H. was supported partially by rano, H. Nemura, and K. Sasaki, Nuclear Physics A 881, 28 the RIKEN iTHEMS Program and iTHES Project. Calcula- (2012). tions were performed at the DiRAC Data Analytic system at [21] T.Inoue,S.Aoki,T.Doi,T.Hatsuda,Y.Ikeda,N.Ishii,K.Mu- the University of Cambridge (BIS National E-infrastructure rano, H. Nemura, and K. Sasaki (HAL QCD Collaboration), capital grant No. ST/K001590/1 and STFC grants No. Phys.Rev.Lett.111,112503(2013). [22] T. Inoue, S. Aoki, B. Charron, T. Doi, T. Hatsuda, Y. Ikeda, ST/H008861/1,ST/H00887X/1,andST/K00333X/1). N. Ishii, K. Murano, H. Nemura, and K. Sasaki (HAL QCD Collaboration),Phys.Rev.C91,011001(2015). [23] V.Soma`,A.Cipollone,C.Barbieri,P.Navra´til, andT.Duguet, Phys.Rev.C89,061301(2014). [24] H. Hergert, S. K. Bogner, T. D. Morris, S. Binder, A. Calci, ∗ [email protected] J.Langhammer, andR.Roth,Phys.Rev.C90,041302(2014). † [email protected] [25] G.Hagen, T.Papenbrock, M.Hjorth-Jensen, andD.J.Dean, [1] S. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, Rep.Prog.Phys.77,096302(2014). S.Krieg,L.Lellouch,T.Lippert,A.Portelli,K.K.Szabo, and [26] W.DickhoffandC.Barbieri,ProgressinParticleandNuclear B.C.Toth,Science347,1452(2015). Physics52,377 (2004). [2] S. R. Beane, E. Chang, W. Detmold, H. W. Lin, T. C. Luu, [27] A.Carbone,A.Cipollone,C.Barbieri,A.Rios, andA.Polls, K.Orginos,A.Parren˜o,M.J.Savage,A.Torok, andA.Walker- Phys.Rev.C88,054326(2013). Loud (NPLQCD Collaboration), Phys. Rev. D 85, 054511 [28] K. Hebeler, J. Holt, J. Mene´ndez, and A. Schwenk, Annual (2012). ReviewofNuclearandParticleScience65,457(2015). [3] T. Yamazaki, K.-i. Ishikawa, Y. Kuramashi, and A. Ukawa, [29] A.Cipollone,C.Barbieri, andP.Navra´til,Phys.Rev.Lett.111, Phys.Rev.D86,074514(2012). 062501(2013). [4] T. Yamazaki, K.-i. Ishikawa, Y. Kuramashi, and A. Ukawa, [30] A. Ekstro¨m, G. R. Jansen, K. A. Wendt, G. Hagen, T. Pa- Phys.Rev.D92,014501(2015). penbrock, B. D. Carlsson, C. Forsse´n, M. Hjorth-Jensen, [5] K.Orginos,A.Parren˜o,M.J.Savage,S.R.Beane,E.Chang, P. Navra´til, and W. Nazarewicz, Phys. Rev. C 91, 051301 and W. Detmold (NPLQCD Collaboration), Phys. Rev. D 92, (2015). 114512(2015). [31] V.Lapoux, V.Soma`, C.Barbieri, H.Hergert, J.D.Holt, and [6] E. Berkowitz, T. Kurth, A. Nicholson, B. Joo, E. Rinaldi, S.R.Stroberg,Phys.Rev.Lett.117,052501(2016). M. Strother, P. M. Vranas, and A. Walker-Loud, (2015), [32] R. F. Garcia Ruiz, M. L. Bissell, K. Blaum, A. Ekstrom, arXiv:1508.00886[hep-lat]. N.Frommgen,G.Hagen,M.Hammen,K.Hebeler,J.D.Holt, [7] T. Iritani, T. Doi, S. Aoki, S. Gongyo, T. Hatsuda, Y. Ikeda, G.R.Jansen,M.Kowalska,K.Kreim,W.Nazarewicz,R.Neu- T.Inoue,N.Ishii,K.Murano,H.Nemura, andK.Sasaki,Jour- gart, G.Neyens, W.Nortershauser, T.Papenbrock, J.Papuga, nalofHighEnergyPhysics2016,101(2016). A.Schwenk,J.Simonis,K.A.Wendt, andD.T.Yordanov,Nat [8] S.Aoki,T.Doi, andT.Iritani,Proceedings,34thInternational Phys12,594(2016). SymposiumonLatticeFieldTheory(Lattice2016):Southamp- [33] C. Barbieri and W. H. Dickhoff, Phys. Rev. C 65, 064313 ton, UK, July 24-30, 2016, PoS LATTICE 2016, 109 (2016), (2002). arXiv:1610.09763[hep-lat]. [34] C.Barbieri,Phys.Lett.B643,268(2006). [9] T.Iritani,Proceedings, 34thInternationalSymposiumonLat- [35] C. Barbieri and M. Hjorth-Jensen, Phys. Rev. C 79, 064313 tice Field Theory (Lattice 2016): Southampton, UK, July 24- (2009). 30,2016,PoSLATTICE2016,107(2016),arXiv:1610.09779 [36] J.Schirmer,L.S.Cederbaum, andO.Walter,Phys.Rev.A28, [hep-lat]. 1237(1983). [10] N.Ishii,S.Aoki, andT.Hatsuda,Phys.Rev.Lett.99,022001 [37] C.Barbieri, D.VanNeck, andW.H.Dickhoff,Phys.Rev.A (2007). 76,052503(2007). [11] S. Aoki, T. Hatsuda, and N. Ishii, Progress of Theoretical [38] C.BarbieriandA.Carbone,“Self-consistentGreen’sfunction Physics123,89(2010). approaches,”inAnadvancedcourseincomputationalnuclear [12] S.Aoki,T.Doi,T.Hatsuda,Y.Ikeda,T.Inoue,N.Ishii,K.Mu- physics:Bridgingthescalesfromquarkstoneutronstars,Lect. rano, H. Nemura, and K. Sasaki (HAL QCD), PTEP 2012, NotesPhys.,editedbyM.Hjorth-Jensen,M.P.Lombardo, and 01A105(2012),arXiv:1206.5088[hep-lat]. U.vanKolck(2017)Chap.11,arXiv:1611.03923[nucl-th]. [13] N.Ishii,S.Aoki,T.Doi,T.Hatsuda,Y.Ikeda,T.Inoue,K.Mu- [39] T. Doi et al., Proceedings, 33rd International Symposium on 6 Lattice Field Theory (Lattice 2015): Kobe, Japan, July 14- a Computational Environment for Nuclear Structure, 18,2015,PoSLATTICE2015,086(2016),arXiv:1512.01610 https://github.com/ManyBodyPhysics/CENS/tree/master/MBPT/ [hep-lat]. . [40] N.Ishiietal.,Proceedings, 33rdInternationalSymposiumon [49] K.GadandH.Mu¨ther,Phys.Rev.C66,044301(2002). LatticeFieldTheory(Lattice2015): Kobe,Japan,July14-18, [50] H.HergertandR.Roth,PhysicsLettersB682,27 (2009). 2015,PoSLATTICE2015,087(2016). [51] H.Nemura,Int.Jour.Mod.Phys.E23,1461006(2014). [41] K.Sasakietal.,Proceedings,33rdInternationalSymposiumon [52] S.N.More,A.Ekstro¨m,R.J.Furnstahl,G.Hagen, andT.Pa- LatticeFieldTheory(Lattice2015): Kobe,Japan,July14-18, penbrock,Phys.Rev.C87,044326(2013). 2015,PoSLATTICE2015,088(2016). [53] A. Cipollone, C. Barbieri, and P. Navra´til, Phys. Rev. C 92, [42] H.Nemuraetal.,in12thInternationalConferenceonHypernu- 014306(2015). clearandStrangeParticlePhysics(HYP2015)Sendai,Japan, [54] H. D. Vries, C. D. Jager, and C. D. Vries, Atomic Data and September7-12,2015(2016)arXiv:1604.08346[hep-lat]. NuclearDataTables36,495 (1987). [43] A.Carbone,privatecommunication(2015). [55] I.AngeliandK.Marinova,AtomicDataandNuclearDataTa- [44] A. Fetter and J. Walecka, Quantum Theory of Many-particle bles99,69 (2013). Systems,DoverBooksonPhysics(DoverPublications,2003). [56] K. Murano, N. Ishii, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, [45] W.H.DickhoffandD.VanNeck,Many-bodytheoryexposed!, T. Inoue, H. Nemura, and K. Sasaki, Physics Letters B 735, 2nded.(WorldScientificPublishing,London,2008). 19 (2014). [46] V.Soma`,C.Barbieri, andT.Duguet,Phys.Rev.C89,024323 [57] T.Doi,S.Aoki,T.Hatsuda,Y.Ikeda,T.Inoue,N.Ishii,K.Mu- (2014). rano, H. Nemura, K. Sasaki, and HAL QCD Collaboration, [47] M. Hjorth-Jensen, T. T. Kuo, and E. Osnes, Physics Reports ProgressofTheoreticalPhysics127,723(2012). 261,125 (1995). [48] T. Engeland, M. Hjorth-Jensen, and G. Jansen, CENS,