Ground-State Properties of $^{4}$He and $^{16}$O Extrapolated from Lattice QCD with Pionless EFT

2.5 MB·

by  L. Contessi

Ground-State Properties of 4He and 16O
Extrapolated from Lattice QCD with Pionless EFT
L.Contessia,b,A.Lovatoc,F.Pederivaa,b,A.Roggerod,J.Kirschere,U.vanKolckf,g
aPhysicsDepartment,UniversityofTrento,ViaSommarive14,I-38123Trento,Italy
bINFN-TIFPATrentoInstituteofFundamentalPhysicsandApplications,ViaSommarive,14,38123PovoTN,Italy
cPhysicsDivision,ArgonneNationalLaboratory,Argonne,Illinois60439,USA
dInstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195,USA
eDepartmentofPhysics,TheCityCollegeofNewYork,NewYork,NY10031,USA
fInstitutdePhysiqueNucle´aire,CNRS/IN2P3,Univ.Paris-Sud,Universite´Paris-Saclay,F-91406Orsay,France
gDepartmentofPhysics,UniversityofArizona,Tucson,AZ85721,USA
7
1Abstract
0
WeextendthepredictionrangeofPionlessEffectiveFieldTheorywithananalysisofthegroundstateof16Oinleadingorder.
2
Torenormalizethetheory,weuseasinputbothexperimentaldataandlatticeQCDpredictionsofnuclearobservables,whichprobe
n
thesensitivityofnucleitoincreasedquarkmasses. Thenuclearmany-bodySchro¨dingerequationissolvedwiththeAuxiliaryField
a
JDiffusionMonteCarlomethod. ForthefirsttimeinanuclearquantumMonteCarlocalculation,alinearoptimizationprocedure,
3 whichallowsustodeviseanaccuratetrialwavefunctionwithalargenumberofvariationalparameters, isadopted. Themethod
2yields a binding energy of 4He which is in good agreement with experiment at physical pion mass and with lattice calculations
atlargerpionmasses. Atleadingorderwedonotfindanyevidenceofa16Ostatewhichisstableagainstbreakupintofour4He,

h]althoughhigher-ordertermscouldbind16O.
t
-
l
c
1. Introduction carryinformationaboutthedetailsoftheQCDdynamics, and
u
canbeobtainedbymatchingobservablescalculatedinEFTand
n Establishing a clear path leading from the fundamental the-
[ LQCD.
ory of strong interactions, namely Quantum Chromodynam-
The aim of this work is the first extension of this program
1ics(QCD),tonuclearobservables,suchasnuclearmassesand
velectroweaktransitions,isoneofthemaingoalsofmodernnu- to the realm of medium-heavy nuclei. By using Pionless EFT
6 (EFT(π/))coupledtotheAuxiliaryFieldDiffusionMonteCarlo
cleartheory. Atpresent,themostreliablenumericaltechnique
1 (AFDMC) method [6] we analyze the connection between the
toperformQCDcalculationsisLatticeQCD(LQCD).Itcom-
5 groundstateof16Oanditsnucleonconstituents. Besidephys-
6binesrecentadvancesinhigh-performancecomputing,innova-
ical data, the consideration of higher quark-mass input allows
0tivealgorithms,andconceptualbreakthroughsinnucleartheory
.to produce predictions of nucleon-nucleon scattering, and the ustoinvestigatethesensitivityof16Ostabilitytothepionmass.
1
TheusefulnessofEFT(π/)[3,4]fortheanalysisofLQCDcal-
0bindingenergiesandmagneticmomentsoflightnuclei. How-
culationshasbeendiscussedpreviously[7,8,9].
7ever, there are technical problems, which have so far limited
1theapplicabilityofLQCDto A ≤ 4baryonsystemsandtoar- WhetherEFT(π/)canbeextendedtorealandlatticenucleiin
v:tificiallylargequarkmasses. Then,LQCDcalculationsrequire themedium-massregionisanopenquestion. Forphysicalpion
isignificantlysmallercomputationalresourcestoyieldmeaning- mass,convergencehasbeendemonstratedinleadingordersfor
X
ful signal-to-noise ratios. In this paper, we consider as exam- thelow-energypropertiesofA = 2,3systems[10,11,12,13].
arplesLQCDdatasetscomprisedofbindingenergiesobtainedat Counterintuitively,thebindingenergyoftheA=4groundstate
pionmassesofm (cid:39)805MeV[1]andm (cid:39)510MeV[2]. wasfoundingoodagreementwithexperimentatleadingorder
π π
The link between QCD and the entire nuclear landscape (LO)[14],andeventhe A = 6groundstatecomesoutreason-
is a potential whose systematic derivation was developed in ablywellatthisorder[15].
the framework of effective field theory (EFT) in the last two Asimilarbindingenergypernucleonfor4He((cid:39)7MeV)and
decades[3,4,5].Thisisachievedbyexploitingaseparationbe- 16O((cid:39) 8MeV)suggeststhatEFT(π/)mightconvergeforheav-
tween“hard”(M)and“soft”(Q)momentumscales. Theactive ier systems. However, the difference in total binding energy
degreesoffreedomatsoftscalesarehadronswhoseinteractions betweenthetwosystemsisquitelarge. Moreover,many-body
are consistent with QCD. Effective potentials and currents are effects become stronger, and quantum correlations might sub-
derived from the most general Lagrangian constrained by the stantiallychangethepicture. Wechose16Oformainlytworea-
QCDsymmetries,andemployedwithstandardfew-andmany- sons:First,becauseitisadoublymagicnucleus,therebyreduc-
body techniques to make predictions for nuclear observables ingthetechnicaldifficultiesrelatedtotheconstructionofwave
in a systematic expansion in Q/M. The interactions strengths functions with the correct quantum numbers and symmetries.
PreprintsubmittedtoPhysicsLettersB January24,2017
Second, its central density is sufficiently high to probe satura- (m (cid:39) 1300 MeV), this estimate gives Q /m (cid:39) 0.4 (0.3).
N 4 π
tion properties and thereby serve as a model for even heavier Thus, the typical momentum is small in comparison not only
nuclei. with m , but also with m , allowing for a description where
N π
In practice, this first calculation of an A > 6 system in pion exchanges are treated as unresolved contact interactions.
EFT(π/)atLOiscarriedoutasfollows. Inordertoshowrenor- Thecaseislessclear-cutintherealworld,whereQ /m (cid:39)0.8,
4 π
malizability, we use potentials characterized by cutoffs up to but the results at LO [14] suggest that Eq. (1) overestimates
(cid:39) 1.5 GeV. This introduces non-trivial difficulties in solving thetypicalmomentum. Infact,asimilarinferencecanbemade
the Schro¨dinger equation due to the rapidly changing behav- from results of the analog of EFT(π/) for 4He atomic clusters
ior of the wave functions. To this aim, we developed an effi- [16]. Atphysicalm , thebindingenergyperparticlein16Ois
π
cient linear optimization scheme to devise high-quality varia- similar to that in 4He, so EFT(π/) might converge for this nu-
tional wave functions. Those have been employed as a start- cleusaswell.
ing point for the imaginary-time projection of AFDMC which With pions integrated out, m gives an upper bound on the
π
enhances the ground-state component of the trial wave func- breakdown scale M of the EFT. The momentum associated
tion. Finally, to alleviate the sign problem, we have also per- with nucleon excitations of mass m is given by Eq. (1) with
R
formed unconstrained propagations and studied their conver- B /A → m −m . For the lightest excitation, the Delta iso-
A R N
gencepattern. Weshowthat,thankstothesedevelopments,the √bar, m∆ −mN decreases as mπ increases. For mπ = 805 MeV,
errorsfromtheAFDMCcalculationarenowmuchsmallerthan 2mN(m∆−mN)becomescomparabletomπ. Thus,throughout
theuncertaintyoriginatingfromtheEFT(π/)truncationandthe the considered range of pion masses nucleon resonances can
LQCD input. The door is open for higher-order calculations also be treated as short-range effects, and nucleons are indeed
withfuture,morepreciseLQCDinput. theonlyrelevantdegreesoffreedom.
The rest of the paper is organized as follows: in Sec. 2 we Atleadingorderin1/M,theEFT(π/)Lagrangian[17,10,18]
will briefly review the properties of EFT(π/) that are relevant consists of the nucleon kinetic term, two two-nucleon contact
for our discussion; in Sec. 3 the methodological aspect of the interactions, and one three-nucleon contact interaction. The
calculations will be discussed; in Sec. 4 we will present and singularity of these interactions leads to divergences that need
discussourresults;andfinallySec. 5isdevotedtoconclusions. to be dealt with by regularization and renormalization. Here,
Anappendixdescribesthewayweestimateerrors. asinRefs. [7,9],weuseaGaussianregulatorthatsuppresses
transferredmomentaaboveanultravioletcutoffΛ. Thischoice
2. PionlessEffectiveFieldTheory ensuresthattheLagrangiancanbetransformedintoaHamilto-
niancontainingonlylocalpotentials,suitabletobeusedwithin
AnEFTisareformulationofanunderlyingtheoryinterms
AFDMC.TheHamiltonianincoordinatespacereads[7,9]
of degrees of freedom relevant to the problem at hand, which
interactviaoperatorsthatobeysymmetriescompatiblewiththe (cid:88) ∇(cid:126)2 (cid:88)(cid:16) (cid:17)
original interactions. These operators are part of a controlled HLO = − 2mi + C1+C2σ(cid:126)i·σ(cid:126)j e−ri2jΛ2/4
expansion in a suitable small parameter which encapsulates a i N i<j
(cid:88) (cid:88) (cid:16) (cid:17)
separationofscalesinthesystem. +D0 e−ri2k+ri2jΛ2/4, (2)
Therelativistic,underlyingtheory,whichpresumablyallows
i<j<k cyc
forthedescriptionofnucleifromfirstprinciples,isQCD.Low-
energyprocessesinnuclearphysicsinvolvesmallenoughmo- wherethesumsareover,respectively,nucleons,nucleonpairs,
(cid:80)
menta to justify the use of a nonrelativistic approach. Con- andnucleontriplets,and standsforthecyclicpermutation
cyc
sequently, nucleon number is conserved and nuclear dynam- of i, j, and k. Dependence on the arbitrary regulator choice is
ics can be described within nonrelativistic many-body theory, eliminatedbyallowingtheinteractionstrengths,orlow-energy
whilethestrongnuclearpotentialneedstoincludeonlyparity constants(LECs),C (Λ),C (Λ)andD (Λ)todependonΛ.
1 2 0
andtime-reversalconservingoperators. Allrelativisticcorrec- Tosolvethetwo-nucleonsystem,inprincipleoneiteratesin-
tionsaresub-leading. teractionsonlyinthechannelscontainingS-matrixpoleswithin
In this paper we are interested in the ground states of nu- the convergence range of the theory [19]. Since two nucleons
clei. The characteristic momentum in a two-body bound state haveaboundstateinthe3S channelandashallowvirtualstate
1
of binding energy B is given by the location of the pole of (whichbecomesaboundstateasm increases[1,2])inthe1S
2 √ π 0
theS matrixinthecomplexmomentumplane, Q2 = mNB2, channel, oneneedstoincludetwointeractionsatLOandtreat
where mN is the nucleon mass. To our knowledge, there is no themnon-perturbatively. InEq. (2)wechosetheoperatorbasis
consensus for a definition of an analogous characteristic mo- 1 and σ(cid:126) ·σ(cid:126) , but it can by replaced by any other form equiv-
i j
mentumforlargernucleiboundby BA; asanestimateonecan alentunderFierztransformationsinSU(2). Allthesepossible
useageneralizationwhereeachnucleoncontributesequally, choicesareequallyconvenientforanAFDMCcalculation.
(cid:114) When the three-body problem is solved with these interac-
B
Q = 2m A . (1) tions, renormalizability requires a contact three-nucleon force
A N A
at LO [20]. As for the two-body interactions, there is some
For lattice 4He at m = 805 MeV (m = 510 MeV), where freedominchoosingtheoperatortoincludeintheHamiltonian
π π
B (cid:39) 110MeV[1](B (cid:39) 40MeV[2])withm (cid:39) 1600MeV formulation of the three-body force. For simplicity we use a
4 4 N
2
centralpotential,whichmakesobvioustheWignerspin-isospin distinguished from higher-order contributions. Assuming that
symmetry(SU(4))ofthisforce. fortheobservableofinteresttheleadingmissingpowerof1/M
ForrenormalizationatLO,weensurethatthreeuncorrelated isthesameastheleadingpowerof1/Λ,varyingΛfrom M to
observables are Λ-independent. Here we follow Ref. [9] and muchlargervaluesgivesanestimateofthetruncationerror.For
choose these observables as the deuteron and triton binding anothertechniquetoestimatetheEFTtruncationerror,seefor
energies and, for physical (unphysical) pion mass(es) the 1S exampleRef. [21].
0
scattering length (dineutron binding energy). The LECs’ de- Finally,experimentalandnumericalLQCDuncertaintiesare
pendenceonthecutoffcanbefoundinRef. [9]. Inparticular, transcribed through the renormalization of the LECs. While
becauseC (Λ) (cid:29) C (Λ),theLOHamiltonianhasanapproxi- thisisnotanimportantissueforthephysicaldata,LQCD“mea-
1 2
mateSU(4)symmetry. surements”carryasignificantuncertaintywhichcoulddramat-
Interactions with more derivatives represent higher orders. ically affect EFT predictions. Estimating their effects would
Forexample,atNLOthefirsttwo-bodyenergycorrectionsap- require a huge computational effort, as the calculation would
pear in the form of two-derivative contact interactions [19]. have to be repeated for various combinations of the extreme
For ground states electromagnetic interactions are also sub- valuestheLECscantake. Sincethepertinenterrors[1,2]are
leading, starting at NLO with the Coulomb interaction [13]. comparabletotheLOtruncationerror,thiseffortisnotyetjus-
Sincesub-leadinginteractionsaresuppressedbypowersof M, tified. We will limit ourselves to show that the Monte Carlo
they should be included as perturbations. Treating them non- errorsdiscussedinthenextsectionhavereachedapointwhere
perturbatively,liketheLOterms,isproblematicastheiteration theyarenotanobstacletofuturehigher-ordercalculations. At
of sub-leading terms usually destroys renormalizability. NLO thatpoint,amoredetailedanalysisofthepropagationof“mea-
interactionshavebeendealtwithfullyperturbativelyforA=2 surement”errorsatunphysicalpionmasseswillberequired.
[10,11]and A = 3[12,13]. Sofar, noperturbativeNLOcal-
culationhasbeenconductedinEFT(π/)forA≥4. Wetherefore
limit ourselves to LO in this first foray into medium-mass nu- 3. MonteCarloMethod
clei.
One feature of the EFT approach is a better understanding Quantum Monte Carlo (QMC) methods allow for solving
of the systematic uncertainties which reduce the accuracy of the time-independent Schro¨dinger equation of a many-body
predicted nuclear observables. Apart from the errors germane system, providing an accurate estimate of the statistical error
to AFDMC, the EFT at LO is expected to be affected by sys- of the calculation. For light nuclei, QMC and, in particular,
tematic,relativeerrorsofO(Q /M,Q /Λ)plus“measurement” Green’s Function Monte Carlo (GFMC) methods have been
A A
uncertaintiesintheLECs. successfullyexploitedtocarryoutcalculationsofnuclearprop-
For observables that were not used as input, regularization erties,basedonrealisticHamiltoniansincludingtwo-andthree-
introducesanerrorproportionaltotheinverseofthecutoff. For nucleon potentials, and consistent one- and two-body meson-
example, different Fierz-reordered forms of the potential only exchangecurrents[22].
givethesameresultsforlargecutoffs. Inordertominimizethe Because the GFMC method involves a sum over spin and
regularizationerror,wefitfinite-cutoffresultswith isospin, its computational requirements grow exponentially
with the number of particles. Over the past two decades
C C
OΛ =O+ Λ0 + Λ12 +··· , (3) AFDMC [6] has emerged as a more efficient algorithm for
dealing with larger nuclear systems [23], but only for some-
where O is the observable at Λ → ∞, while the parameters whatsimplifiedinteractions. WithinAFDMC,thespin-isospin
C , C , ..., are specific for each observable. The number of degrees of freedom are described by single-particle spinors,
0 1
powers of Λ needed to perform a meaningful extrapolation is the amplitudes of which are sampled using Monte Carlo tech-
notknownapriori. Thestandardprescriptionconsistsintrun- niques, and the coordinate-space diffusion in GFMC is ex-
catingtheexpansionwhenaddingadditionalpowersof1/Λno tended to include diffusion in spin and isospin spaces. Both
longer influences O. In a renormalizable theory, observables GFMC and AFDMC have no difficulties in using realistic
converge in the Λ → ∞ limit to a value that must not be con- two- and three-body forces; the interactions are not required
fusedwithaprecisephysicalresult. Observablesareunavoid- to be soft and hence can generate wave functions with high-
ablyplaguedbythetruncationerror, whichcannotbereduced momentumcomponents.Thisisparticularlyrelevanttoanalyze
withoutanext-ordercalculation. thecutoffdependenceofobservables,asrelativelylargevalues
The truncation of a natural EFT expansion at order n al- forΛaretobeconsideredinordertoconfirmrenormalizability.
lows for a residual error proportional to (Q/M)n+1, where the QMCmethodsemployanimaginary-time(τ)propagationin
constantofproportionalitydependsonthespecificobservable. order to extract the lowest many-body state Ψ from a given
0
Truncation errors are more difficult to assess here because the initialtrialwavefunctionΨT:
scales M and Q arenotwellknown. Assuming M ∼ m and
A π
QA ∼ Q3,4, as given in Eq. (1), one estimates QA/M ∼ 1/3 |Ψ0(cid:105)=τl→im∞e−(H−ET)τ|ΨT(cid:105). (4)
for physical pions. An alternative that does not rely on an es-
timate for Q uses cutoff variation to place a lower bound on In the above equation E is a parameter that controls the nor-
A T
thetruncationerror. Theresidualcutoffdependencecannotbe malization of the wave function and H is the Hamiltonian of
3
thesystem. Inordertoefficientlydealwithspin-isospindepen- workweadopt,forthefirsttimeinanuclearQMCcalculation,
dentHamiltonians,theHubbard-Stratonovichtransformationis the more advanced linear method (LM) [27], which allows us
appliedtothequadraticspinandisospinoperatorsenteringthe todealwithamuchlargernumberofvariationalparameters.
imaginary-time propagator to make them linear. As a conse- WithintheLM,ateachoptimizationstepweexpandthenor-
quence, the computational cost of the calculation is reduced malizedtrialwavefunction
fromexponentialtopolynomialinthenumberofparticles, al-
|Ψ (p)(cid:105)
lowingforthestudyofmany-nucleonsystems. |Ψ¯T(p)(cid:105)= (cid:112)(cid:104)Ψ (Tp)|Ψ (p)(cid:105) (10)
ThestandardformofthewavefunctionusedinQMCcalcu- T T
lationsoflightnucleireads
at first order around the current set of variational parameters
(cid:16) (cid:89) (cid:17)(cid:16)(cid:89) (cid:17) p0 ={p0,...,p0 },
(cid:104)X|ΨT(cid:105)=(cid:104)X| Uijk Fij |Φ(cid:105), (5) 1 Np
i<j<k i<j (cid:88)Np
where X = {x ...x } and the generalized coordinate x = |Ψ¯lTin(p)(cid:105)=|Ψ¯T(p0)(cid:105)+ ∆pi|Ψ¯iT(p0)(cid:105). (11)
1 A i
{r,σ,τ}representstheposition,spin,andisospinvariablesof i=1
i i i
thei-thnucleon. Thelong-rangebehaviorofthewavefunction Byimposing(cid:104)Ψ (p0)|Ψ¯ (p0)(cid:105)=1,weensurethat
T T
isdescribedbytheSlaterdeterminant
(cid:104)X|Φ(cid:105)=A{φα1(x1),...,φαA(xA)}. (6) |Ψ¯iT(p0)(cid:105)= ∂|Ψ¯∂Tp(ip)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12)p=p0
=|Ψi (p0)(cid:105)−S |Ψ (p0)(cid:105) (12)
The symbol A denotes the antisymmetrization operator and α T 0i T
denotes the quantum numbers of the single-particle orbitals, areorthogonalto|Ψ (p0)(cid:105). Inthelastequationwehaveintro-
T
givenby
duced
φα(x)=Rnl(r)Y(cid:96)(cid:96)z(rˆ) χssz(σ)χττz(τ), (7) |ΨiT(p0)(cid:105)= ∂|Ψ∂Tp(ip)(cid:105)(cid:12)(cid:12)(cid:12)(cid:12)p=p0 (13)
for the first derivative with respect to the i-th parameter, and
where R (r) is the radial function, Y (rˆ) is the spherical har-
nl (cid:96)(cid:96)z the overlap matrix is defined by S = (cid:104)Ψ (p0)|Ψi (p0)(cid:105). The
monic,andχssz(σ)andχττz(τ)arethecomplexspinorsdescrib- expectation value of the energy on0tihe lineTar waveTfunction is
ingthespinandisospinofthesingle-particlestate.
definedas
InboththeGFMCandthelatestAFDMCcalculationsspin-
(cid:104)Ψ¯lin(p)|H|Ψ¯lin(p)(cid:105)
iHsooswpeinvedre,ptheensdeeanrtecnoorrtenlaetcieosnssarFyijfoarndthUisiwjkoarrke.uIsnufaalclyt,athdeoptwteod-. Elin(p)≡ (cid:104)Ψ¯Tlin(p)|Ψ¯linT(p)(cid:105) . (14)
T T
bodyLOpionlessnuclearpotentialconsideredinthisworkdoes Thevariation∆p¯ oftheparametersthatminimizestheenergy,
notcontaintensororspin-orbitoperators. Inaddition,theLEC ∇ E (p)=0,correspondstothelowesteigenvaluesolutionof
p lin
proportionaltothespin-dependentcomponentoftheinteraction thegeneralizedeigenvalueequation
ismuchsmallerthantheoneofthecentralchannel,C (cid:28) C .
2 1
Finally, Fierz transformations allow us to consider the purely H¯∆p=∆ES¯ ∆p, (15)
centralthree-bodyforceinEq. (2). Asaconsequence,wecan
where H¯ and S¯ are the Hamiltonian and overlap ma-
limitourselvestospin-isospinindependenttwo-andthree-body
trices in the (N + 1)-dimensional basis defined by
correlationsonly, p
{|Ψ¯ (p0)(cid:105),|Ψ¯1(p0)(cid:105),...,|Ψ¯Np(p0)(cid:105)}. The authors of Ref. [28]
(cid:88) T T T
F = f(r ), U =1− u(r )u(r )u(r ), (8) haveshownthatwritingtheexpectationvaluesofthesematrix
ij ij ijk ij ik jk
cyc elements in terms of covariances allows us to keep their
statistical error under control even when they are estimated
where f(r)andu(r)arefunctionsoftheradiusonly.
overarelativelysmallMonteCarlosample. However,sincein
The radial functions of the orbitals as well as those enter-
AFDMC the derivatives of the wave function with respect to
ingthetwo-andthree-bodyJastrowcorrelationsaredetermined
the orbital variational parameters are in general complex, we
minimizingtheground-stateexpectationvalueoftheHamilto-
generalized the expressions for the estimators reported in the
nian,
appendixofRef.[28].
(cid:104)Ψ |H|Ψ (cid:105)
E = T T . (9) Forafinitesamplesizethematrix H¯ canbeill-conditioned,
V (cid:104)Ψ |Ψ (cid:105)
T T spoiling therefore the numerical inversion needed to solve the
InstandardnuclearVariationalMonteCarlo(VMC)andGFMC eigenvalueproblem.Apracticalproceduretostabilizethealgo-
calculationstheminimizationisusuallydoneadoptinga“hand- rithmistoaddasmallpositiveconstant(cid:15)tothediagonalmatrix
waving” procedure, while in more recent AFDMC calcula- elementsofH¯ exceptforthefirstone,H¯ → H¯ +(cid:15)(1−δ )δ .
ij ij i0 ij
tionsthestochasticreconfiguration(SR)method[24]hasbeen Thisprocedurereducesthelengthof∆p¯ androtatesittowards
adopted. Inbothcasesthenumberofvariationalparametersis thesteepest-descentdirection.
reduced by first minimizing the two-body cluster contribution It has to be noted that if the wave function depends lin-
totheenergyperparticle,asdescribedinRefs.[25,26]. Inthis earlyuponthevariationalparameters,thealgorithmconverges
4
in just one iteration. However, in our case strong nonlinear- 15
ities in the variational parameters make, in some instances, LM
|Ψ¯lin(p)(cid:105) significantly different from |Ψ¯ (p0 +∆p)(cid:105). Account- 10 SR
T T 5 AFDMC
ing for the quadratic term in the expansion as in the Newton
method[29,28]wouldalleviatetheproblem,attheexpenseof 0
V]
having to estimate also the Hessian of the wave function with e
M −5
respect to the variational parameters. An alternative strategy [
𝐸
−10
consists in taking advantage of the arbitrariness of the wave-
functionnormalizationtoimproveontheconvergencebyasuit- −15
able rescaling of the parameter variation [28, 27]. We found −20
thatthisprocedurewasnotsufficienttoguaranteethestability
−25
oftheminimizationprocedure. Forthisreasonwehaveimple- 0 5 10 15 20 25 30 35 40 45 50
mentedthefollowingheuristicprocedure. Foragivenvalueof step
(cid:15),Eq.(15)issolved. Ifthelinearvariationofthewavefunction
forp=p0+∆pissmall, Figure1: (Coloronline)Convergencepatternofthe4Hevariationalenergyat
physicalpionmassandΛ=4fm−1asafunctionofthenumberofoptimiza-
tionstepsfortheSRmethod(blacksquares)andtheLM(bluescircles). For
|Ψ¯lin(p)|2 (cid:88)Np comparison,theredlineindicatestheAFDMCresult.
T =1+ S¯ ∆pi∆pj ≤δ, (16)
|Ψ¯ (p0)|2 ij
T i,j=1
theimaginary-timediffusionforallvaluesofthecutoffandof
ashortcorrelatedrunisperformedinwhichtheenergyexpec-
the pion mass. On the other hand, to allow for an emerging
tationvalue
(cid:104)Ψ¯ (p)|H|Ψ¯ (p)(cid:105) clusterstructure,forthe16Owavefunctionweused30param-
E(p)≡ T T (17) eters for the two- and three-body Jastrow correlations and 15
(cid:104)Ψ¯ (p)|Ψ¯ (p)(cid:105)
T T parametersforeachoftheR (r).
n(cid:96)
is estimated along with the full variation of the wave function TheLMexhibitsamuchfasterconvergencepatternthanthe
for a set of possible values of (cid:15) (in our case ≈ 100 values are SR, previously used in AFDMC. In Fig. 1, we show the 4He
considered). The optimal (cid:15) is chosen so as to minimize E(p¯) variational energy obtained for physical pion mass and Λ = 4
providedthat fm−1asafunctionofthenumberofoptimizationstepsforboth
|Ψ¯T(p¯)|2 ≤δ. (18) SRandLM.WhiletheLMtakesonly(cid:39) 15stepstoconverge,
|Ψ¯T(p0)|2 the SR is much slower; after 50 steps the energy is still much
Notethat,atvariancewiththepreviousexpression,hereinthe above the asymptotic limit. We have observed analogous be-
numerator we have the full wave function instead of its lin- havior for other values of the cutoff and the pion mass. In the
earizedapproximation. Inthe(rare)caseswherenoacceptable 16Ocase,theimprovementoftheLMwithrespecttotheSRis
valueof(cid:15) isfoundduetopossiblylargestatisticalfluctuations evenmoredramaticduetotheclusteringofthewavefunction,
intheVMCestimators,weperformanadditionalrunadopting whichwillbediscussedindetailinthefollowing.
thepreviousparametersetandanewoptimizationisattempted.
Inourexperience,thisprocedureprovedextremelyrobust.
Thechiefadvantageoftheadditionalconstraintisthatitsup-
pressesthepotentialinstabilitiescausedbythenonlineardepen-
denceofthewavefunctiononthevariationalparameters.When 4. Results
usingthe“standard”versionoftheLM,therewereinstancesin
which, despite the variation of the linear wave function being With LO EFT(π/) LECs determined from experiment or
well below the threshold of Eq. (16), the full wave function LQCDcalculations,predictionscanbemadewithAFDMCfor
fluctuatedsignificantlymore,preventingtheconvergenceofthe thebindingenergiesof4Heand16O.
minimizationalgorithm.Asforthewave-functionvariation,we In Table 1 we report results of 4He energies for all the val-
found that choosing δ = 0.2 guarantees a fast and stable con- uesofthecutoffandofthepionmassweconsidered. Despite
vergence. thedifferentparametrizationofthevariationalwavefunctions,
The two-body Jastrow correlation f(r ) is written in terms the results are in very good agreement with those reported in
ij
ofcubicsplines,characterizedbyasmoothfirstderivativeand Ref. [7], where a simplified version of the variational wave
continuoussecondderivative. Theadjustableparameterstobe function was used because the LM had not been introduced
optimized are the “knots” of the spline, which are simply the yet. For most cutoff values, our results also agree with those
valuesoftheJastrowfunctionatthegridpoints,andthevalue of Ref. [9], which were obtained with the Resonating-Group
of the first derivative at r = 0. Analogous parametrizations and Hyperspherical-Harmonic methods. For Λ ≤ 6 fm−1, the
ij
areadoptedforu(r )andR (r). Inthe4Hecaseweusedsix QMC results differ by less than 0.1 MeV from Ref. [9], while
ij n(cid:96) i
variational parameters for f(r), u(r), and for the radial orbital for Λ = 8 fm−1 the QMC method binds 4He more deeply by
functionsR (r). Thisallowedenoughflexibilityforthevaria- morethan1MeV.Inconsequence,theextrapolatedasymptotic
n(cid:96)
tionalenergiestobeveryclosetotheoneobtainedperforming values differ. Our results display a better convergence pattern
5
Λ m =140MeV m =510MeV m =805MeV atthispointforustodrawaverystrongconclusion.
π π π
It is interesting to study the cutoff dependence of the root-
2fm−1 −23.17±0.02 −31.15±0.02 −88.09±0.01 √
4fm−1 −23.63±0.03 −34.88±0.03 −91.40±0.03 mean-square(rms)point-nucleonradius (cid:104)rp2t(cid:105)andthesingle-
nucleonpointdensityρ (r). Thesequantitiesarerelatedtothe
6fm−1 −25.06±0.02 −36.89±0.02 −96.97±0.01 pt
charge density, which can be extracted from electron-nucleus
8fm−1 −26.04±0.05 −37.65±0.03 −101.72±0.03
scattering data, but are not observable themselves: few-body
→∞ −30±0.3(sys) −39±1(sys) −124±3(sys)
±2(stat) ±2(stat) ±1(stat) currentsandsingle-nucleonelectromagneticformfactorshave
Exp. −28.30 – – to be accounted for. Still, one can gain some insight into the
LQCD – −43.0±14.4 −107.0±24.2 features of the ground-state wave function by comparing re-
√
sultsatdifferentpionmassesandcutoffs. Sinceneither (cid:104)r2(cid:105)
Table1: 4Heenergyfordifferentvaluesofthepionmassmπ andthecutoff norρ (r)commutewiththeHamiltonian,thedesiredexpectpat-
Λ,comparedtoexperimentandLQCDcalculations[1,2]. Seemaintextand pt
tionvaluesontheground-statewavefunctionarecomputedby
appendixfordetailsonerrorsandextrapolations.
meansof“mixed”matrixelements
(cid:104)Ψ |O|Ψ (cid:105)≈2(cid:104)Ψ |O|Ψ (cid:105)−(cid:104)Ψ |O|Ψ (cid:105). (19)
0 0 T 0 T T
withthecutoff.Atthephysicalpionmassandwiththesamein-
putobservables,ourhighest-cutoffresultisingoodagreement Intheaboveequation|Ψ0(cid:105)istheimaginary-timeevolvedstate
withthehighest-cutoffresult(cutoffvaluesintherange8−10 ofEq. (4),while|ΨT(cid:105)isthetrialwavefunctionconstructedas
fm−1,butadifferentregulatorfunction)ofRef. [14]. inEq. (5).
Wefoundthatanexpansionofthetype(3)upto1/Λ2suffices Theresultsforthepoint-protonradiusof4Hearereportedin
to extrapolate the 4He energies for m = 140 MeV, since the Table2. (SinceCoulombisabsentinourcalculation,thepoint-
π
nucleon and point-proton radii are the same.) In the physical
additionofacubictermchangesneithertheextrapolatedvalue
northebest-fitcoefficients.Fortheunphysicalpionmasses,the case, the calculated radius is much smaller than the empirical
usageofthesmallestcutoffisquestionablebecauseΛ=2fm−1 value—thatis,thevalueextractedfromtheexperimentaldata
cuts off momentum modes below the pion mass. We thus ex- of Ref. [30] accounting for the nucleon si√ze, but neglecting
meson-exchangecurrents. Asimilarresult, (cid:104)r2(cid:105) ≈ 1fmwas
trapolate the values appearing in the tables with the quadratic pt
expansion in Eq. (3) but without the result at Λ = 2 fm−1. In obtainedbytheauthorsofRef. [31]usingalocalformofachi-
all cases, we perform fits with and without the Λ = 2 fm−1 ralinteraction. NLOandN2LOpotentialsinachiralexpansion
basedonnaivedimensionalanalysis[3,4,5]bringtheoryinto
resultstoestimatethesystematicextrapolationerror. Thepro-
much closer agreement with the empirical value. Hence, sub-
cedure adopted for the systematic and statistical errors quoted
throughoutthispaperisdetailedintheappendix. leading terms in the EFT(π/) expansion could play a relevant
It has to be remarked that this cutoff sensitivity study does role,atleastforphysicalvaluesofthepionmass.
notaccountfortheEFTtruncationerror. Usingcutoffvariation For unphysically large pion masses, where EFT(π/) is sup-
fromcutoffvaluessomewhatlargerthanthepionmass,forex- posed to exhibit a faster convergence, the point-proton radius
ample from Λ = 2 fm−1, 4 fm−1, and 6 fm−1 for m = 140, is smaller than at mπ = 140 MeV. The value obtained for
510,and805MeV,wemightestimatetheerroras±7π,±4,and mπ = 510 MeV indicates a spatial extent similar to the phys-
±30, respectively. Except for the intermediate pion mass, this icalone,while4Heatmπ =805MeV,incomparison,seemsto
beamuchmorecompactobject. Thisisconsistentwiththebe-
is consistent with the rougher dimensional-analysis estimate
Q /M ∼ 0.3. In any case, we expect the truncation error to haviorofthesingle-nucleonpointdensity,ρpt,displayedinFig.
A 2. Forallcutoffvalues,thedensitycorrespondingtom = 805
dominateoverthestatisticalandextrapolationerrors. π
Given the convergence of the 4He binding energy with in- MeVisappreciablynarrowerthanthatcomputedformπ =510
creasingcutoff,weconfirmthat,forbothphysical[14]andun- MeV or mπ = 140 MeV. Focusing on Λ = 8 fm−1, ρpt has a
maximumvalueof11.0fm−3 form = 805MeV,whileinthe
physical [7, 9] pion masses, LO EFT(π/) is renormalized cor- π
m =510MeVandm =140MeVcasesthemaximumvalues
rectly without the need for a four-nucleon interaction. In the π π
are2.1fm−3and2.2fm−3,respectively.
physicalcase,thebindingenergyisunderestimatedforallval-
ues of the cutoff we considered, but the extrapolated value is
in agreement with experiment even if we neglect the trunca-
tion error. Of course when the latter is taken into account we Thesimilaritybetween4Heground-statepropertiesatm =
π
must conclude that such a good agreement is somewhat for- 510 MeV and those at the physical pion mass exists despite
tuitous. We expect NLO corrections, including Coulomb and differences in the structure of lighter systems. If confirmed
two-nucleoneffective-rangecorrections,tochangetheresultby for other properties of 4He and heavier nuclei, this semblance
afewMeV.Form =510MeVandm =805MeV,ourresults wouldmeanthatsimulationsatintermediatepionmassescould
π π
reproduceLQCDpredictions(whereCoulombisabsent)within provide useful insights into the physical world while saving
themeasurementerrorovertheentirecutoffrange. Aspointed substantialcomputationalresources.
outinRefs. [7,9],thisisanon-trivialconsistencycheck: ifei- InTable3the16Oground-stateenergiesarereportedforthe
therLQCDdataorEFT(π/)weretoowrong,onewouldexpect samepion-massandcutoffvaluesconsideredfor4He. Astrik-
nosuchagreement.However,LQCDuncertaintiesaretoolarge ing feature is that 16O is not stable against breakup into four
6
Λ m =140MeV m =510MeV m =805MeV 2.5
π π π Λ=2fm−1
2fm−1 1.374±.0.004 1.482±0.003 0.898±0.001 Λ=4fm−1
4fm−1 1.203±0.004 1.133±0.003 0.699±0.001 2.0 Λ=6fm−1
Λ=8fm−1
6fm−1 1.109±0.003 1.035±0.002 0.609±0.001 −3] 1.5
8fm−1 1.054±0.003 0.976±0.001 0.542±0.001 m
[f
→∞ 0.9±0.008(sys) 0.8±0.04(sys) 0.25±0.05(sys) pt 1.0
±0.2(stat) ±0.1(stat) ±0.06(stat) 𝜌
“Exp.” 1.45 – –
0.5
Tanadbleth2e:cut4oHffeΛp,ocinotm-ppraorteodntroadtihuesefmorpidriiffcaelrevnatluvealeuxetsraocftetdhefrpoimonRmefa.ss[3m0π] 0.0
accountingforthefinitenucleonsize. Seemaintextandappendixfordetails 2.5
onerrorsandextrapolations.
2.0
Λ mπ=140MeV mπ=510MeV mπ=805MeV −3m] 1.5
2fm−1 −97.19±0.06 −116.59±0.08 −350.69±0.05 [fpt 1.0
𝜌
4fm−1 −92.23±0.14 −137.15±0.15 −362.92±0.07
6fm−1 −97.51±0.14 −143.84±0.17 −382.17±0.25 0.5
8fm−1 −100.97±0.20 −146.37±0.27 −402.24±0.39
0.0
→∞ −115±1(sys) −151±2(sys) −504±20(sys)
±8(stat) ±10(stat) ±12(stat) 12.0
Exp. −127.62 – –
10.0
Table3: 16OenergyfordifferentvaluesofthepionmassmπandthecutoffΛ, 8.0
comparedwithexperiment. (NoLQCDdataexistforthisnucleus.) Seemain −3]
textandappendixfordetailsonerrorsandextrapolations. m 6.0
[f
pt
𝜌 4.0
2.0
4He clusters in almost all the cases, the only exception occur-
ringformπ =140MeVandΛ=2fm−1,where16Ois4.5MeV 0.0
more bound than four 4He nuclei. In the other cases we miss 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
𝑟[fm]
the four-4He threshold by about 5 MeV, which is beyond our
statistical errors and reveals a lower bound on the systematic Figure2: (Coloronline)4Hesingle-nucleonpointdensityformπ =140MeV
errorofourQMCmethod. (upper panel), mπ = 510 MeV (middle panel), and mπ = 805 MeV (lower
Evenconsideringonlystatisticalandextrapolationerrorsthe panel),atdifferentvaluesofthecutoffΛ.
asymptotic values of the 16O energy cannot be separated from
thefour-4Hethreshold.Theproximityofthethresholdsuggests despite potentially significantly different, lead to almost iden-
thatthestructureofour16Oshouldbeclustered.Indeed,despite
tical variational energies. In contrast, the width of the peaks
noexplicitclusteringbeingenforcedinthetrialwavefunction, decreases with increasing cutoff in step with the shrinking of
the highly efficient optimization procedure arranges the two-
theindividual4HeclustersreportedinTable2.
andthree-bodyJastrowcorrelations,aswellastheorbitalradial
functions,insuchawayastofavorconfigurationscharacterized
byfourindependent4Heclusters.
The single-proton density profiles displayed in Fig. 3 in- The analysis of the proton densities alone does not suffice
dicate that only for Λ = 2 fm−1 with m = 140 MeV and to support the claim of clustering. Another indication of clus-
π
m = 510 MeV are the nucleons distributed according to the terizationcomesfromcomparingtheexpectationvaluesofthe
π
classicpictureofaboundwavefunction. Foralltheothercom- nuclearpotentialsevaluatedinthegroundstatesof16Oand4He.
binationsofpionmassesandcutoffs,nucleonsarepushedaway Forinstance,inthem =140MeVandΛ=8fm−1caseitturns
π
fromthecenterofthenucleus,whichisbasicallyempty—the outthattheexpectationvaluesofthe16Otwo-andthree-body
densityattheoriginisaminusculefractionofthepeak—until potentials are (cid:39) 4.05 and (cid:39) 4.16 times larger than the corre-
(cid:39) 2 fm from the center of mass. The erratic behavior of the sponding values for 4He. The same pattern is observed for all
peakpositionofthedensityprofilesasafunctionofthecutoff the combinations of pion mass and cutoff, except for Λ = 2
has to be ascribed to the fact that the relative position of the fm−1 with m = 140 MeV and m = 510 MeV. In particular,
π π
four 4He clusters is practically unaffected by the cutoff value. forΛ = 2fm−1 andm = 140MeV,theexpectationvaluesof
π
In fact, once the clusters are sufficiently apart, a landscape of thetwo-andthree-bodypotentialsin16Oare(cid:39)4.65and(cid:39)6.14
degenerate minima in the variational energy emerges. Hence, times larger than in 4He. This difference is a consequence of
the single-proton densities correspond to wave functions that, thefactthatthenumberofinteractingpairsandtripletsislarger
7
0.010
Λ=2fm−1
0.008 ×25 ΛΛ==46ffmm−−11
Λ=8fm−1
−3] 0.006
m
[f
pt0.004
𝜌
0.002
0.000
0.012
0.010 ×25
0.008
3]
−
m
0.006
[f
pt
𝜌 0.004
0.002
0.000
0.030
0.025
0.020
3]
−
m
0.015
[f
pt
𝜌 0.010
0.005
0.000
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 Figure4: (Coloronline)Imaginary-timediffusionwithtimestep∆τ = 0.125
𝑟[fm] MeV−1ofasinglewalkerformπ=140MeV,atΛ=2fm−1(upperpanel)and
Λ=8fm−1(lowerpanel).
Figure3: (Coloronline)16Osingle-nucleonpointdensityformπ =140MeV
(puapnpeel)r,patandeiffl)e,remnπtv=alu5e1s0oMftheVec(umtoiffddΛle.panel), and mπ = 805 MeV (lower absenceofinteractionsamongnucleonsbelongingtodifferent
4Heclusters.
whenclusterizationdoesnottakeplace.
Tobettervisualizetheclusterizationofthewavefunction,in The non-clustered states at Λ = 2 fm−1 for m = 510 MeV
π
Fig. 4 we display the position of the nucleons following the and m = 140 MeV deserve further comment. The state at
π
propagation of a single walker for 5000 imaginary-time steps, m =510MeVstandsincontrasttotheotherstatesfoundabove
π
corresponding to ∆τ = 0.125 MeV−1, printed every 10 steps. threshold whose structure is clustered. We interpret this as an
In the upper panel, concerning m = 140 MeV and Λ = 2 artifact of the numerical method, since a perfect optimization
π
fm−1,nucleonsarenotorganizedinclusters. Infact,duringthe procedure should have produced a clustered structure resem-
imaginary-timepropagationtheydiffuseintheregioninwhich blingthelower-energystatewithfourfree4He. Whilethereis
the corresponding single-nucleon density of Fig. 3 does not nosignalof16Ostabilityabovethephysicalpionmass,thestate
vanish. Acompletelydifferentscenariotakesplaceatthesame at m = 140 MeV is certainly stable at the lowest cutoff, that
π
pion mass when Λ = 8 fm−1: the nucleons forming the four is, when the interaction has the longest range. On this basis,
4Heclustersremainclosetothecorrespondingcentersofmass one might speculate that at some pion mass above the physi-
duringtheentireimaginary-timepropagation. Thisisclearev- caloneatransitionfromanon-clusteredtoaclusteredstateis
idenceofclustering. Ithastobenotedthattherelativeposition expected. However, such a conclusion cannot be drawn until
ofthefourclustersisnotatetrahedron. Toprovethis,foreach higher-ordercalculationsinEFT(π/)—whichwillcapturefiner
configurationwecomputedthemoment-of-inertiamatrixasin effectsfrompionexchangesuchasthetensorforceatN2LO—
Ref.[32]. Ifthe4Heclusterswerepositionedattheverticesof areavailable.
a tetrahedron, diagonalization would yield only two indepen- The smaller relative size of the model space leads to more
denteigenvalues. Instead,wefoundthreedistincteigenvalues, modestsignsofcutoffconvergencefor16Othan4He,whichare
corresponding to an ellipsoid — yet another indication of the reflectedinlargerextrapolationerrors, especiallyatm = 805
π
8
MeV. At physical pion mass, the central value of the extrap- Interestingly, m = 140 MeV and Λ = 2 fm−1 is the only
π
olated total energy is only 10% off experiment, which can be parametersetwhichyieldsastable16O.Thissuggeststhatthe
bridged by statistical and extrapolation errors. This difference long-rangestructureoftheinteractionisdeficientatlargercut-
is small compared to the expected truncation error, ∼ 30%. If off values and might have to be corrected, e.g. via one-pion
there is a low-lying resonant or virtual state of 4He nuclei at exchange, to guarantee the binding of heavier nuclei at LO.
LO in EFT(π/) — note that our analysis does neither preclude Alternatively,withinapionlessframework,higher-orderterms
noridentifysuchastate—itispossiblethatthe(perturbative) couldactasperturbationstomove16Owithrespecttothefour-
inclusionofhigher-ordertermsuptoN2LOwillmovethe16O 4He threshold. At physical pion mass the central value of the
energysufficientlyforstabilitywithrespecttofour4Heclusters. total energy is just about 10% off experiment. This is only
Forunphysicalpionmass,ourresultscanbeseenasanexten- slightly larger than the statistical and extrapolation errors, and
sionofLQCDtomedium-massnuclei,withnofurtherassump- well within the ∼ 30% truncation error estimate. We cannot
tionsabouttheQCDdynamics. Inthiscase,adeterminationof exclude the possibility that agreement with data will improve
therelativepositionofthefour-4Hethresholdwouldfurtherre- with order. A comprehensive study of the various subsystems
quiremuchincreasedaccuracyintheA = 2,3LQCDdatathat of 16O — for example, 12C, 8Be, and 4He-4He scattering —
weuseasinput. could determine whether a resonant or virtual shallow state at
LOistransformedintoaboundstatebysubleadinginteractions,
thuselucidatingtherelationbetweenclusterizationandQCD.
5. Conclusions
In order to better appreciate the cluster nature of our solu-
This paper represents the first application of the effective- tion for 16O, we have studied the radial nucleon density and
field-theory formalism, as developed for small nuclei without thesampledprobabilitydensityforthenucleons. Inbothcases
explicit pions, to a relatively heavy object, 16O. We employed theoccurrenceofclusterizationisevident. Fromourresultsit
contact potentials which represent the leading order of a sys- is not possible to infer any significant correlation between the
tematicexpansionofQCD.Thisenabledustoanalyzephysical clusters, which once more confirms the extremely weak inter-
nucleons as well as simulated scenarios with increased quark actionamongthemwithinEFT(π/). Wewouldliketopointout
masses. thatlocalizationwasnotimposedinthewavefunctionusedto
Toovercomethepeculiarchallengesassociatedwiththeso- projectoutthegroundstate;rather,itspontaneouslyarisesfrom
lutionoftheSchro¨dingerequation,wehaveimprovedAFDMC theoptimizationprocedure(despitethecorrelationsbeingfully
by introducing a new optimization protocol of the many-body translationally invariant) and it is preserved by the subsequent
wave function to be employed in the variational stage of the imaginary-timeprojection.
calculation. The scheme we propose is an extension of the Current QMC (AFDMC) results have now reached an ac-
linear method and provides a much faster convergence in pa- curacy level that allows for discussing the few-MeV energies
rameter space compared to stochastic reconfiguration, previ- involved in this class of phenomena, which are relevant for a
ouslyadoptedinnuclearQMCcalculations. Suchaccuratetrial deeperunderstandingofhowthesystematicsinnuclearphysics
wave function is the starting point of the imaginary-time pro- arises from QCD. Starting from LCQD data obtained for val-
jectioninAFDMC,whichfiltersoutthe“exact”groundstateof ues of mπ smaller than the ones employed in this work, and
the Hamiltonian. This algorithm was used to predict not only yet larger than the physical one, would allow us to establish
ground-stateenergies,butalsoradii,densities,andparticledis- thethresholdforwhichnucleiaslargeas16Oarestableagainst
tributions. the breakup into four 4He clusters, if such a threshold exists.
Ourresultsforthe4Hebindingenergyareinagreementwith Toperformthisanalysis, itisessentialtoincludehigher-order
previous findings, including the renormalizability of the four- terms in the EFT(π/) interaction, possibly up to N2LO, where
nucleon system in EFT(π/) without a LO four-body force. In tensorcontributionsappear. Thisalsorequiresasubstantialim-
particular,atphysicalpionmasstheenergyagreeswithexperi- provement of the existing LQCD data on light nuclei, which,
mentwithintheoreticaluncertainties. Moreover,thecalculated evenforlargemπ,arecurrentlyaffectedbystatisticalerrorsthat
point-nucleon radii and single-particle densities reveal a 4He donotallowforaneffectiveconstraintoftheinteractionparam-
structureatm =510MeVsimilartothatatphysicalpionmass. eters.
π
With this successful benchmark, we extended the calcula-
tionsto16O,obtainingextrapolatedvaluesforthe16Oenergyat
Acknowledgements
allpionmasseswhichareindistinguishablefromtherespective
four-4Hethreshold,evenconsideringonlythesmallerstatistical
andextrapolationerrors. Infact,foralmostallcutoffsandpion We would like to thank N. Barnea, D. Gazit, G. Orlandini,
massesweconsidered,16Oisunstablewithrespecttobreak-up and W. Leidemann for useful discussions about the subject of
into four 4He nuclei. Our calculation of the 16O energy is the
firsttimeLQCDdataareextendedtothemedium-massregion
inamodel-independentway1. nucleiappeared[33],whereatwo-bodypotentialmodelobtainedfromLQCD
dataatmπ =469MeVwassolvedwiththeSelf-ConsistentGreen’sFunction
method.Thewidelydifferentinputdataandmethodtranslateintomuchsmaller
4Heand16Oenergiesthanourresults. Noclearsignisfoundfora16Ostate
1Asthismanuscriptwasbeingconcluded, acalculationofdoublymagic belowthe4Hethreshold.
9
thispaper.ThisresearchwasconductedinthescopeoftheLab- andthenfinally
oratoire international associe´ (LIA) COLL-AGAIN and sup-
(cid:113)
ported in part by the U.S. Department of Energy, Office of δO= δO2+δC2Γ2+δC2Γ4 . (26)
Science,OfficeofNuclearPhysics,undercontractsDE-AC02- 1 0 1 1 1
06CH11357(A.L.)andDE-FG02-04ER41338(U.v.K.),andby
Botherrorestimatesappearintheresultsreportedinthemain
theEuropeanUnionResearchandInnovationprogramHorizon
text.
2020undergrantNo. 654002(U.v.K.). TheworkofA.R.was
supportedbyNSFGrantNo.AST-1333607.J.K.acknowledges
supportbytheNSFGrantNo. PHY15-15738. Underanaward References
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10