Ground-state properties of 4He and 16O extrapolated from lattice QCD with pionless EFT

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Ground-state

properties

of

4

He

and

16

O

extrapolated

from

lattice

QCD

with

pionless

EFT

L. Contessi

a,b

,

A. Lovato

c

,

F. Pederiva

a,b,

∗

,

A. Roggero

d

,

J. Kirscher

e

,

U. van Kolck

f,g

a_Physics_Department,_University_of_Trento,_Via_Sommarive_14,_I-38123_Trento,_Italy

b_INFN-TIFPA_Trento_Institute_of_Fundamental_Physics_and_{Applications,}_Via_Sommarive,_14,₃₈₁₂₃_Povo_TN,_Italy c_Physics_Division,_Argonne_National_Laboratory,_Argonne,_{IL 60439,}_USA

d_Institute_for_Nuclear_Theory,_University_of_Washington,_Seattle,_WA_98195,_USA e_Department_of_Physics,_The_City_College_of_New_York,_New_York,_NY_10031,_USA

f_Institut_de_Physique_Nucléaire,_CNRS/IN2P3,_Univ._Paris-Sud,_Université_{Paris-Saclay,}_F-91406_Orsay,_France g_Department_of_Physics,_University_of_Arizona,_Tucson,_AZ_85721,_USA

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received25January2017

Receivedinrevisedform8May2017 Accepted24July2017

Availableonline26July2017 Editor:W.Haxton

WeextendthepredictionrangeofPionlessEffectiveFieldTheorywithananalysisofthegroundstateof 16_O_in_leading_order._To_renormalize_the_theory,_we_use_as_input_both_experimental_data_and_lattice_QCD predictionsofnuclearobservables,whichprobethesensitivityofnucleitoincreasedquarkmasses.The nuclearmany-bodySchrödingerequationissolvedwiththeAuxiliaryFieldDiffusionMonteCarlomethod. FortheﬁrsttimeinanuclearquantumMonteCarlocalculation,alinearoptimizationprocedure,which allowsus todevisean accuratetrial wavefunctionwith alarge numberofvariational parameters,is adopted.The methodyields abinding energyof4_He_which_is_in_good_agreement_with _experiment_at physicalpionmassandwithlatticecalculationsatlargerpionmasses.Atleadingorderwedonotﬁnd anyevidenceofa16_O_state_which_is_stable_against_breakup_into_four4_He,_although_higher-order_terms couldbind16_O.

1. Introduction

Establishinga clearpathleading fromthefundamental theory of strong interactions, namely Quantum Chromodynamics (QCD), to nuclear observables, such as nuclear masses and electroweak transitions,isoneofthemaingoalsofmodernnucleartheory.At present, the most reliable numerical technique to perform QCD calculations is Lattice QCD (LQCD). It combines recent advances in high-performance computing, innovative algorithms, and con-ceptualbreakthroughsinnucleartheorytoproducepredictionsof nucleon–nucleon scattering, and the binding energies and mag-neticmomentsoflight nuclei.However, thereare technical prob-lems,whichhavesofarlimitedtheapplicabilityofLQCDto A

≤

4 baryonsystemsandtoartiﬁciallylarge quarkmasses.Then, LQCD calculationsrequire signiﬁcantly smaller computational resources to yieldmeaningful signal-to-noise ratios. In thispaper, we

con-sider as examples LQCD data sets comprised of binding

ener-*

Correspondingauthor.

E-mailaddress:francesco.pederiva@unitn.it(F. Pederiva).

gies obtained at pion masses of mπ

805 MeV [1] and mπ

510 MeV[2].

The link between QCD and the entire nuclear landscape is

a Hamiltonian whose systematic derivation was developed in

the framework of effective ﬁeld theory (EFT) in the last two decades[3–5].Thisisachievedbyexploitingaseparationbetween “hard” (M)and“soft” ( Q )momentumscales.Theactivedegreesof freedom atsoftscales are hadronswhoseinteractions are consis-tent withQCD.Effective potentialsandcurrents arederived from themostgeneralLagrangianconstrainedby theQCD symmetries,

and employed with standard few- and many-body techniques to

make predictions fornuclear observablesin a systematic expan-sion in Q

/

M. The interaction strengths carry information about thedetailsoftheQCDdynamics,andcanbeobtainedbymatching observablescalculatedinEFTandLQCD.

The aimofthiswork is theﬁrst extension ofthisprogram to therealmofmedium-heavynuclei.ByusingPionlessEFT(EFT(

π

/

)) coupled to the Auxiliary Field Diffusion Monte Carlo (AFDMC) method

[6]

we analyze theconnection betweenthegroundstate of16_O_and_its_nucleon_{constituents.}_Beside_physical_data,_the

con-siderationofhigherquark-massinputallowsustoinvestigatethe

http://dx.doi.org/10.1016/j.physletb.2017.07.048

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sensitivity of 16_O _stability _to _the _pion _mass. _The _usefulness _of

EFT(

π

/

) [3,4] for the analysis of LQCD calculationshas been dis-cussedpreviously[7–9].

WhetherEFT(

π

/

) canbe extended to realandlattice nucleiin the medium-mass region is an open question. For physical pion mass, convergence has been demonstrated in leading orders for thelow-energypropertiesofA

=

2

,

3 systems

[10–14]

. Counterin-tuitively,thebindingenergyofthe A

=

4 groundstate wasfound ingoodagreementwithexperimentatleadingorder(LO)[15],and eventhe A

=

6 groundstatecomesoutreasonablywellatthis or-der[16].Asimilarbindingenergypernucleonfor4He(

7 MeV) and16_O₍

_{8 MeV)}_suggests_that_EFT(

_π

_/

_{) might}_converge_for

heav-ier systems. However, the difference in total binding energy be-tweenthetwosystemsisquitelarge.Moreover,many-bodyeffects become stronger, and quantum correlations might substantially changethepicture.Wechose16_O_for_mainly_two_reasons:_First,

be-causeitisadoublymagicnucleus,therebyreducing thetechnical diﬃcultiesrelatedtothe constructionofwave functionswiththe correctquantumnumbersandsymmetries.Second,itscentral den-sityissuﬃcientlyhightoprobesaturationpropertiesandthereby serveasamodelforevenheaviernuclei.

Inthisﬁrstcalculation ofan A

>

6 systeminEFT(

π

/

) wework atLO.Whiletheerrorofan LOcalculation isrelativelylarge (ex-pectedtobeabout30%),itstillallowsustoexcludeacatastrophic failure ofsaturation inthisEFT.Ourpredictions for4Heand16O useasinputonlythepropertiesofthedeuteron,dineutronand tri-ton,andareconsistentwithrenormalization-groupinvariance.We ﬁndthatthegroundstateenergyof16nucleonsis indistinguish-ablefromthefour-4_He_threshold_at_all_values_of_quark_masses_we

consider.Atphysicalpionmasstheexperimental 16_O_binding

en-ergyiswithintheEFTtruncationerror,beingpotentiallyreachable in higherorders. Thus, we can concludethat EFT(

π

/

) has the el-ementsneededfor saturation,andwe provide a baseline against whichthe convergenceof EFT(

π

/

) inmedium-mass nucleican be judged infuture higher-order calculations. At thispoint, because of their complexity, strictly perturbative NLO and N2_LO

calcula-tionshavebeenlimitedto A

=

2

,

3[10–14].

UsingLQCDdataasinputenablesthestudyofthenuclearchart asafunctionofthequarkmasses.Weemploythestandardmethod ofdeterminingEFTparameterstheoretically,thatof“matching” ob-servablescalculated inEFT to thesame observablescalculatedin the underlying theory [17,18]. Operationally, fitting EFT parame-terstoLQCDdataisnotdifferentthanfittingthemtoexperimental data.Incontrast,conceptuallyitmeansthattheresultingEFT pre-dictionsare consequencesof QCDitself. When experimentaldata are fitted instead, the corresponding EFT predictions are conse-quences ofany underlying theory that shares the symmetries of QCD.

At present there is no LQCD data for light nuclei at physical quark masses,so inthiscasewe useexperimental dataasinput. Thereisalsonoconsensusabouthowbindingenergies behaveas quark masses increase, with different lattice approaches yielding contradictoryresults.Weusedata

[1,2]

obtainedfromthe identi-ﬁcationofplateausintheimaginary-time evolutionwithina box. Thisidentiﬁcationhasbeensubjecttocriticism

[19]

.For A

=

2,an independentanalysisofthesamedata

[1]

yields differentbinding energies [20]. Fordifferentgauge configurations,a methodbased ontheextraction(undercertain assumptions)ofa potentialfinds no bound states [21]. Ourinput should thus be seen as illustra-tive.However,asshownbefore[7,9] andconfirmedhere,the4He bindingenergypredictedbyLOEFT(

π

/

) doesagreewiththatfound inRefs. [1,2].Although thelarge errors do not allowa deﬁnitive conclusion, this coincidence is suggestive of a consistent plateau identiﬁcation.ItalsosupportsthevalidityofEFT(

π

/

).

Inpractice,ourcalculationiscarriedoutasfollows.Inorderto show renormalizability,weusepotentialscharacterizedbycutoffs up to

1

.

5 GeV. This introduces non-trivial diﬃculties in solv-ingtheSchrödingerequationduetotherapidlychangingbehavior ofthewave functions. Tothisaim,we developedan eﬃcient lin-ear optimization scheme to devise high-quality variational wave functions. Those have beenemployed asa starting point for the imaginary-timeprojection ofAFDMCwhichenhancesthe ground-state component of the trial wave function. Finally, to alleviate the sign problem, we have also performed unconstrained prop-agations and studied their convergence pattern. We show that, thankstothesedevelopments,theerrorsfromtheAFDMC calcula-tion arenowmuch smallerthantheuncertaintyoriginatingfrom the EFT(

π

/

) truncationandthe LQCD input. The dooris open for higher-ordercalculationswithfuture,morepreciseLQCDinput.

Therestofthepaperisorganizedasfollows:inSec.2wewill brieﬂy review the properties of EFT(

π

/

) that are relevant for our discussion; inSec.3themethodologicalaspectofthecalculations willbediscussed;inSec.4wewillpresentanddiscussourresults; andﬁnallySec.5isdevotedtoconclusions.Anappendixdescribes thewayweestimateerrors.

2. Pionless effective ﬁeld theory

Therelativistic,underlyingtheory,whichpresumablyallowsfor thedescriptionofnucleifromﬁrstprinciples, isQCD. Low-energy processes in nuclear physics involve small enough momenta to justify the use of a nonrelativistic approach. Consequently, nu-cleonnumberisconservedandnucleardynamicscanbedescribed within nonrelativisticmany-bodytheory,whilethestrongnuclear potentialneedstoincludeonlyparityandtime-reversalconserving operators.Allrelativisticcorrectionsaresub-leading.

In thispaperweare interested inthegroundstatesofnuclei. Thecharacteristicmomentuminatwo-bodyboundstateof bind-ingenergy B2isgivenbythelocationofthepoleoftheS matrix

inthecomplexmomentumplane, Q2

=

√

mNB2,wheremN isthe

nucleonmass.Toourknowledge,thereisnoconsensus fora def-inition ofananalogouscharacteristicmomentumforlargernuclei bound by BA;asan estimateone canuseageneralization where

eachnucleoncontributesequally,

QA

=

2mN

BA

A

.

(1)

For lattice 4He at mπ

=

805 MeV (mπ

=

510 MeV), where B4

110 MeV [1] (B4

40 MeV [2]) with mN

1600 MeV (mN

1300 MeV), thisestimategives Q4/mπ

0

.

4

(

0

.

3

)

.Thus, the typ-ical momentum is small in comparison not only with mN, but

alsowithmπ ,allowingforadescriptionwherepionexchangesare treatedasunresolvedcontactinteractions.Thecaseislessclear-cut intherealworld,where Q4/mπ

0

.

8,butLOresults[15]suggest that Eq.

(1)

overestimatesthetypical momentum.Infact, a simi-larinferencecanbemadefromresultsoftheanalogofEFT(

π

/

) for

4_He _atomic_clusters

_[22]

_._At _physical_{mπ ,}_the _binding_energy _per

particlein16Oissimilartothatin4He,soEFT(

π

/

) mightconverge forthisnucleusaswell.

With pions integrated out, mπ gives an upper bound on the

breakdown scale M of the EFT. The momentum associated with

nucleon excitationsofmassmR is givenby Eq.(1)withBA

/

A

→

mR

−

mN.Forthelightestexcitation,theDeltaisobar,m

−

mN

de-creasesasmπ increases.Formπ

=

805 MeV,

√

2mN

(

m

−

mN

)

be-comes comparabletomπ . Thus,throughouttheconsidered range of pionmasses nucleonresonances can also be treatedas short-rangeeffects,andnucleonsareindeedtheonlyrelevantdegreesof freedom.

(3)

Themost generalLagrangian compatiblewiththe symmetries ofQCD consists, atleading orderin 1

/

M, of the nucleon kinetic term,twotwo-nucleoncontactinteractions,andonethree-nucleon contactinteraction. The singularity of these interactions leads to divergencesthatneedtobedealtwithbyregularizationand renor-malization. Here, as in Refs. [7,9], we use a Gaussian regulator thatsuppressestransferredmomentaaboveanultravioletcutoff

. ThischoiceensuresthattheLagrangiancanbetransformedintoa Hamiltoniancontaining only local potentials,suitable to be used withinAFDMC.TheHamiltonianincoordinatespacereads[7,9]

HL O

= −

i

∇

2 i 2mN

+

i<j

C1

+

C2

σ

i

· σ

j

e−r2i j2/4

+

D0

i<j<k

cyc e− r2 ik+r2i j 2/4

,

(2)

wherethesumsareover,respectively,nucleons,nucleonpairs,and nucleontriplets, and

_cyc stands forthecyclicpermutation ofi, j, and k. Dependence on the arbitrary regulator choice is elim-inated by allowing the interaction strengths, orlow-energy con-stants(LECs),C1(),C2()andD0()todependon

.

Tosolvethetwo-nucleonsystem,inprincipleoneiterates inter-actionsonlyinthechannelscontaining S-matrixpoleswithinthe convergencerangeofthe theory [23]. Sincetwo nucleons havea boundstateinthe3S1 channelandashallowvirtualstate (which

becomes a bound state as mπ increases [1,2]) in the 1S0

chan-nel,one needs to includetwo interactions atLO andtreat them non-perturbatively.In Eq.

(2)

we chose the operator basis 1 and

σ

i

· σ

j, butit can by replaced by any other formequivalent

un-der Fierztransformations in SU(2). All thesepossible choicesare equallyconvenientforanAFDMCcalculation.

Whenthethree-bodyproblemissolvedwiththeseinteractions, renormalizabilityrequiresacontactthree-nucleonforceatLO[24]. Asforthetwo-bodyinteractions,thereissomefreedomin choos-ingtheoperatortoincludeintheHamiltonianformulationofthe three-bodyforce. Forsimplicitywe useacentral potential,which makesobvious theWignerspin–isospin symmetry(SU(4)) ofthis force.

Forrenormalization at LO, we ensure that three uncorrelated observablesare

-independent.HerewefollowRef.[9]andchoose theseobservablesasthedeuteronandtritonbindingenergiesand, forphysical (unphysical)pion mass(es) the 1S0 scattering length

(dineutron binding energy). The LECs’ dependence on the cutoff canbefoundinRef.[9].Inparticular,becauseC1

()

C2(),the

LOHamiltonianhasanapproximateSU(4)symmetry.

Interactionswithmorederivativesrepresenthigherorders.For example,atNLO the ﬁrst two-bodyenergycorrections appearin the form of two-derivative contact interactions [23]. For ground stateselectromagneticinteractionsarealsosub-leading,startingat NLOwiththeCoulombinteraction

[13]

.Sincesub-leading interac-tionsaresuppressedby powersof M,they shouldbe includedas perturbations.Treatingthemnon-perturbatively,liketheLOterms, is problematic as the iteration of sub-leading terms usually de-stroys renormalizability. NLO interactions have been dealt with fullyperturbativelyonlyforA

=

2[10,11]andA

=

3[12,13].Given thechallengesposedbysuchcalculations,welimitourselvestoLO inthisﬁrstforayintomedium-massnuclei.

Onefeature of the EFT approach is a better understanding of the systematic uncertainties which reduce the accuracy of pre-dicted nuclear observables. Apart from the errors germane to AFDMC, the EFT at LO is expected to be affected by systematic, relativeerrorsof

O(

QA

/

M

,

QA

/)

plus“measurement”

uncertain-tiesintheLECs.

Forobservablesthat werenot usedasinput, regularization in-troduces an error proportional to the inverse of the cutoff. For example,differentFierz-reorderedformsofthepotentialonlygive thesame resultsforlargecutoffs.In ordertominimize the regu-larizationerror,weﬁtﬁnite-cutoffresultswith

O

=

O

+

C

0

+

C

1

2

+ · · · ,

(3)

where O is the observable at

→ ∞

, while the parameters

C0,

C1,

. . .

, are speciﬁc for each observable. The numberof pow-ers of

needed to perform a meaningful extrapolation is not knownapriori.Thestandardprescriptionconsistsintruncatingthe expansionwhenaddingadditionalpowersof1

/

nolonger inﬂu-ences O . Ina renormalizable theory,observablesconvergein the

→ ∞

limit to a value that must not be confused witha pre-cise physical result. Observables are unavoidably plagued by the truncation error, which cannot be reduced without a next-order calculation.

ThetruncationofanaturalEFTexpansionatordern allowsfor aresidualerrorproportionalto

(

Q

/

M

)

n+1,wheretheconstantof proportionalitydependsonthespeciﬁcobservable.Truncation er-rors are more diﬃcult to assess here because the scales M and QA are not well known. Assuming M

∼

mπ and QA

∼

Q3,4, as

given inEq. (1),one estimates QA

/

M

∼

1

/

3 forphysical quarks.

Analternativethatdoesnotrelyonanestimate forQA uses

cut-offvariation toplace alower bound onthe truncationerror.The residual cutoff dependence cannot be distinguished from higher-order contributions. Assuming that for theobservable ofinterest the leading missing power of 1

/

M is the same as the leading powerof1

/

,varying

fromM to muchlargervaluesgivesan estimateofthetruncationerror.Foranothertechniquetoestimate theEFTtruncationerror,seeforexampleRef.[25].

Finally, experimental and numerical LQCD uncertainties are transcribedthroughtherenormalizationoftheLECs.Whilethisis notanimportantissueforthephysicaldata,LQCD“measurements” carryasigniﬁcantuncertaintywhichcoulddramaticallyaffectEFT predictions.Estimatingtheireffectswouldrequireahuge compu-tational effort, as the calculation would have to be repeated for various combinations of the extreme values the LECs can take. Since the pertinent errors [1,2] are comparable to the LO trun-cationerror,thiseffortisnotyetjustiﬁed.Wewilllimitourselves to show that the Monte Carlo errors discussed in the next sec-tionhavereachedapointwheretheyarenotanobstacletofuture higher-order calculations. At that point, a more detailed analysis of the propagation of “measurement” errors at unphysical pion masseswillberequired.

3. Monte Carlo method

Quantum Monte Carlo (QMC) methods allow for solving the

time-independent Schrödinger equation of a many-body system, providing an accurate estimate ofthe statisticalerror ofthe cal-culation.Forlightnuclei,QMCand,inparticular,Green’sFunction

Monte Carlo (GFMC) methods have been successfully exploited

to carry out calculations of nuclear properties, based on realis-ticHamiltoniansincludingtwo- andthree-nucleonpotentials,and consistentone- andtwo-bodymeson-exchangecurrents

[26]

.

Because the GFMC method involves a sum over spin and

isospin, its computational requirements grow exponentially with thenumberofparticles.OverthepasttwodecadesAFDMC

[6]

has emergedasamoreeﬃcientalgorithmfordealing withlarger nu-clearsystems

[27]

,butonlyforsomewhat simpliﬁedinteractions. WithinAFDMC,thespin–isospindegreesoffreedomaredescribed bysingle-particlespinors,theamplitudesofwhicharesampled us-ingMonteCarlotechniques,andthecoordinate-spacediffusionin

(4)

GFMCisextendedtoincludediffusioninspinandisospinspaces. BothGFMCandAFDMChavenodiﬃcultiesinusingrealistic two-and three-body forces; the interactions are not required to be softandhencecangeneratewavefunctionswithhigh-momentum components.Thisisparticularlyrelevanttoanalyzethecutoff de-pendenceofobservables, asrelativelylargevaluesfor

aretobe consideredinordertoconﬁrmrenormalizability.

QMCmethodsemployanimaginary-time(

τ

)propagationin or-dertoextractthelowestmany-bodystate

0

froma giveninitial trialwavefunction

T:

|

0

=

lim

τ→∞e−(

H−ET)τ

|

T

.

(4)

In the above equation ET is a parameter that controls the

nor-malizationofthewave function andH is theHamiltonianofthe system. In order to eﬃciently deal with spin–isospin dependent Hamiltonians,theHubbard–Stratonovichtransformationisapplied tothequadraticspinandisospinoperatorsenteringthe imaginary-timepropagatortomakethemlinear.Asaconsequence,the com-putationalcost of the calculation isreduced fromexponential to polynomial in the numberof particles, allowing forthe study of many-nucleonsystems.

Thestandard formofthewave functionused inQMC calcula-tionsoflightnucleireads

X

|

T

=

X

|

i<j<k Ui jk

i<j Fi j

| ,

(5)

whereX

= {

x1

. . .

xA

}

andthegeneralizedcoordinatexi

= {

ri

,

σ

i

,

τ

i

}

representstheposition,spin,andisospinvariables ofthei-th nu-cleon. The long-rangebehavior ofthewave function is described bytheSlaterdeterminant

X

| = A{φ

α1

(

x1

), . . . , φ

αA

(

xA

)

} .

(6)

Thesymbol

A

denotestheantisymmetrizationoperatorand

α

de-notes thequantum numbers of the single-particle orbitals, given by

φ

α

(

x

)

=

Rnl

(

r

)

Yz

(

ˆ

r

)

χ

ssz

(σ

)

χ

τ τz

(τ

) ,

(7)

where Rnl

(

r

)

is the radial function, Yz

(

ˆ

r

)

is the spherical

har-monic,and

χ

ssz

(

σ

)

and

χ

τ τz

(

τ

)

arethecomplexspinorsdescribing

thespinandisospinofthesingle-particlestate.

In both the GFMC and the latest AFDMC calculations spin– isospin dependent correlations Fi j and Ui jk are usually adopted.

However, theseare notnecessary forthiswork. Infact, the two-body LO pionless nuclear potential considered in this work does notcontaintensororspin-orbitoperators.Inaddition,theLEC pro-portional to the spin-dependent component of the interaction is muchsmallerthantheoneofthecentralchannel,C2

C1.Finally,

Fierztransformationsallowustoconsiderthepurelycentral three-bodyforceinEq.

(2)

.Asaconsequence,wecanlimit ourselvesto spin–isospinindependenttwo- andthree-bodycorrelationsonly,

Fi j

=

f

(

ri j

) ,

(8) Ui jk

=

1

−

cyc

u

(

ri j

)

u

(

rjk

)

+

u

(

ri j

)

u

(

rik

)

+

u

(

rik

)

u

(

rjk

)

,

(9)

where f

(

r

)

andu

(

r

)

arefunctionsoftheradiusonly.

Theradialfunctionsoftheorbitalsaswellasthoseenteringthe two- andthree-bodyJastrowcorrelationsaredetermined minimiz-ingtheground-stateexpectationvalueoftheHamiltonian,

EV

=

T

|

H

|

T

|

T

.

(10)

In standard nuclear Variational Monte Carlo (VMC) and GFMC

calculations the minimization is usually done adopting a “hand-waving”procedure, whileinmore recentAFDMCcalculationsthe stochastic reconfiguration (SR)method [28] hasbeen adopted.In bothcasesthenumberofvariationalparametersisreducedbyfirst minimizing the two-body cluster contribution to the energy per particle, asdescribed in Refs. [29,30].In thiswork we adopt, for thefirsttimeinanuclearQMCcalculation,themoreadvanced lin-earmethod (LM)[31],whichallows ustodealwithamuchlarger numberofvariationalparameters.

Withinthe LM,ateach optimizationstep we expandthe nor-malizedtrialwavefunction

| ¯

T

(

p

)

=

|

T

(

p

)

√

T

(

p

)

|

T

(

p

)

(11)

atﬁrstorderaroundthecurrentsetofvariationalparameters

p

0

₌

{

p0 1

,

. . . ,

p0Np

}

,

| ¯

lin T

(

p

)

= | ¯

T

(

p0

)

+

Np

i=1

pi

| ¯

iT

(

p0

)

.

(12)

Byimposing

T

(

p0

)

| ¯

T

(

p0

)

=

1,weensurethat

| ¯

i T

(

p0

)

=

∂

| ¯

T

(

p

)

∂

pi

p=p0

= |

i T

(

p0

)

−

S0i

|

T

(

p0

)

(13)

are orthogonal to

|

T

(

p0

)

. In the last equation we have

intro-duced

|

i T

(

p0

)

=

∂

|

T

(

p

)

∂

pi

p=p0 (14)

fortheﬁrstderivative withrespecttothei-thparameter,andthe overlapmatrixisdeﬁnedby S0i

=

T

(

p0

)

|

Ti

(

p0

)

.The

expecta-tionvalueoftheenergyonthelinearwavefunctionisdeﬁnedas

Elin

(

p

)

≡

¯

lin T

(

p

)

|

H

| ¯

Tlin

(

p

)

¯

lin T

(

p

)

| ¯

linT

(

p

)

.

(15)

The variation

p of

¯

the parameters that minimizes the energy,

∇

pElin(p

)

=

0,correspondstothelowesteigenvaluesolutionofthe generalizedeigenvalueequation

¯

H

p

=

ES

¯

p

,

(16)

where H and

¯

S are

¯

the Hamiltonian and overlap matrices in the

(

Np

+

1

)

-dimensionalbasisdeﬁnedby

{| ¯

T

(

p0

)

,

| ¯

1T

(

p0

)

,

. . . ,

| ¯

Np

T

(

p0

)

}

.The authorsofRef. [32]haveshownthat writingthe

expectation values of these matrix elements in terms of covari-ances allows ustokeep their statisticalerror undercontrol even whentheyareestimatedoverarelativelysmallMonteCarlo sam-ple.However,sinceinAFDMCthederivativesofthewavefunction with respect to the orbital variational parameters are in general complex, we generalized the expressions for the estimators re-portedintheappendixofRef.[32].

For a ﬁnite sample size the matrix H can

¯

be ill-conditioned, spoiling therefore the numerical inversion needed to solve the eigenvalue problem. A practical procedure to stabilize the algo-rithmistoaddasmallpositiveconstant

tothediagonalmatrix elements of H except

¯

fortheﬁrst one, H

¯

i j

→ ¯

Hi j

+

(

1

− δ

i0

)δ

i j.

This procedure reduces the length of

p and

¯

rotates it towards thesteepest-descentdirection.

It has to be noted that ifthe wave function dependslinearly upon the variationalparameters, the algorithm convergesin just one iteration. However, in our case strong nonlinearities in the

(5)

variationalparameters make, in some instances,

| ¯

lin

T

(

p

)

signiﬁ-cantlydifferentfrom

| ¯

T

(

p0

+

p

)

.Accountingforthequadratic

termintheexpansionasintheNewtonmethod[33,32]would al-leviatetheproblem,attheexpenseofhavingtoestimatealsothe Hessianof the wave function withrespect to the variational pa-rameters. An alternative strategy consists in taking advantage of the arbitrariness of the wave-function normalization to improve ontheconvergenceby asuitable rescalingoftheparameter vari-ation[32,31]. Wefound that thisprocedure was notsuﬃcient to guaranteethestabilityoftheminimizationprocedure.Forthis rea-sonwehaveimplementedthefollowingheuristicprocedure.Fora givenvalue of

, Eq.(16)is solved. Ifthe linear variation ofthe wavefunctionfor

p

=

p0

+

p is small,

| ¯

lin T

(

p

)

|

2

| ¯

T

(

p0

)

|

2

=

1

+

Np

i,j=1

¯

Si j

pi

pj

≤ δ ,

(17)

ashortcorrelatedrunisperformedinwhichtheenergy expecta-tionvalue

E

(

p

)

≡

¯

T

(

p

)

|

H

| ¯

T

(

p

)

¯

T

(

p

)

| ¯

T

(

p

)

(18)

isestimatedalongwiththefullvariationofthewavefunctionfor a set ofpossible values of

(in our case

≈

100 values are con-sidered).Theoptimal

ischosensoastominimize E

(

p

¯

)

provided that

| ¯

T

(

p

¯

)

|

2

| ¯

T

(

p0

)

|

2

≤ δ .

(19)

Note that, at variance withthe previous expression, here in the numeratorwehavethefullwavefunctioninsteadofitslinearized approximation.Inthe(rare)caseswherenoacceptablevalueof

isfound due to possibly large statisticalﬂuctuationsin the VMC estimators, we perform an additional run adopting the previous parametersetandanewoptimizationisattempted.Inour experi-ence,thisprocedureprovedextremelyrobust.

Thechiefadvantageoftheadditionalconstraintisthatit sup-pressesthe potential instabilitiescausedby the nonlinear depen-denceofthe wave function on thevariational parameters. When using the “standard” version of the LM, there were instances in which,despitethevariationofthelinearwavefunctionbeingwell belowthethresholdofEq.(17),thefull wave functionﬂuctuated signiﬁcantly more, preventing the convergence of the minimiza-tionalgorithm.As forthewave-functionvariation,we found that choosing

δ

=

0

.

2 guaranteesafastandstableconvergence.

Thetwo-bodyJastrow correlation f

(

ri j

)

iswritten intermsof

cubicsplines, characterized bya smooth ﬁrstderivative and con-tinuoussecond derivative. The adjustableparameters to be opti-mizedare the “knots” ofthespline, whichare simply thevalues of the Jastrow function at the grid points, and the value of the ﬁrstderivative atri j

=

0.Analogousparametrizations areadopted

foru

(

ri j

)

and Rn

(

ri

)

.Inthe4Hecaseweusedsixvariational

pa-rametersfor f

(

r

)

,u

(

r

)

,andfortheradialorbitalfunctions Rn

(

r

)

. Thisallowed enough ﬂexibility for the variationalenergies to be veryclosetotheoneobtainedperformingtheimaginary-time dif-fusion forall values of the cutoff andof the pion mass. On the otherhand,toallowforanemergingclusterstructure,forthe16O wavefunctionweused30parametersforthetwo- andthree-body Jastrowcorrelationsand15parametersforeachoftheRn

(

r

)

.

The LM exhibits a much faster convergence pattern than the SR, previouslyused inAFDMC.In Fig. 1,we show the4_He

varia-tionalenergyobtainedforphysicalpionmassand

=

4 fm−1_as_a

functionofthenumberofoptimizationstepsforbothSRandLM. WhiletheLM takesonly

15 stepsto converge,the SRismuch

Fig. 1. (Coloronline)Convergencepatternofthe4_He_variational_energy_at_physical pionmassand=4 fm−1_as_a_function_of_the_number_of_optimization_steps_for theLMmethod(blacksquares)andtheSR(bluecircles).Forcomparison,thered lineindicatestheAFDMCresult.

Table 1

4_He_energy_for_different_values_of_the_pion_mass_m

π andthecutoff,comparedto

experimentandLQCDcalculations[1,2].Seemaintextandappendixfordetailson errorsandextrapolations.

mπ=140 MeV mπ=510 MeV mπ=805 MeV

2 fm−1 ₋₂₃ .17±0.02 −31.15±0.02 −88.09±0.01 4 fm−1 ₋₂₃_.₆₃_±₀_.₀₃ ₋₃₄_.₈₈_±₀_.₀₃ ₋₉₁_.₄₀_±₀_.₀₃ 6 fm−1 ₋₂₅_.₀₆_±₀_.₀₂ ₋₃₆_.₈₉_±₀_.₀₂ ₋₉₆_.₉₇_±₀_.₀₁ 8 fm−1 ₋₂₆_.₀₄_±₀_.₀₅ ₋₃₇_.₆₅_±₀_.₀₃ ₋₁₀₁_.₇₂_±₀_.₀₃ → ∞ −30±_±02 (stat).3 (sys) −39 ±1 (sys) ±2 (stat) −124 ±3 (sys) ±1 (stat) Exp. −28.30 – – LQCD – −43.0±14.4 −107.0±24.2

slower;after 50steps theenergyis still muchabove the asymp-toticlimit.Wehaveobservedanalogousbehavior forother values ofthecutoffandthepionmass.Inthe16Ocase,theimprovement oftheLMwithrespecttotheSRisevenmoredramaticduetothe clustering ofthewave function,whichwill bediscussedin detail inthefollowing.

4. Results

WithLOEFT(

π

/

) LECsdeterminedfromexperimentorLQCD cal-culations, predictions can be made with AFDMC for the binding energiesof4Heand16O.

In Table 1 we report results of 4_He _energies _for _all _the

val-uesofthecutoffandofthepionmassweconsidered.Despitethe differentparametrizationofthevariationalwavefunctions,the re-sultsare invery goodagreement withthosereportedin Ref.[7], where a simpliﬁed version of the variational wave function was usedbecause theLMhadnot beenintroducedyet.Formost cut-offvalues,ourresultsalsoagreewiththoseofRef.[9],whichwere obtainedwiththeResonating-GroupandHyperspherical-Harmonic methods. For

≤

6 fm−1, the QMC results differ by less than 0.1 MeV from Ref. [9], while for

=

8 fm−1 the QMC method binds4He moredeeplybymorethan1 MeV.Inconsequence,the extrapolatedasymptoticvaluesdiffer. Ourresultsdisplay abetter convergencepatternwiththecutoff.Atthephysicalpionmassand with the same input observables, our highest-cutoff result is in goodagreementwiththehighest-cutoffresult(cutoffvaluesinthe range8–10 fm−1_,_but_a_different_regulator_function)_of_Ref._[15]_.

We found that an expansion ofthe type (3)up to 1

/

2 suf-ﬁcestoextrapolatethe4Heenergiesformπ

=

140 MeV,sincethe addition of a cubic term changes neither the extrapolated value nor the best-ﬁt coeﬃcients. For unphysical pion masses, the

(6)

us-ageofthesmallestcutoffisquestionablebecause

=

2 fm−1 _cuts

offmomentummodesbelowthe pionmass.Wethus extrapolate

the values appearing in the tables with the quadratic expansion inEq.(3) butwithoutthe resultat

=

2 fm−1_. _In_all _cases,_we

performﬁtswithandwithoutthe

=

2 fm−1 _results_to_estimate

thesystematicextrapolationerror.Theprocedureadopted forthe systematic andstatistical errors quoted throughout thispaper is detailedintheappendix.

Ithastoberemarkedthatthiscutoffsensitivitystudydoesnot account for the EFT truncation error. Using cutoff variation from cutoff values somewhat larger than the pion mass, for example from

=

2 fm−1,4 fm−1,and6 fm−1 formπ

=

140

,

510

,

and 805 MeV, we mightestimate the error as

±

7,

±

4,and

±

30, re-spectively.Exceptfortheintermediatepionmass,thisisconsistent with the rougher dimensional-analysis estimate QA

/

M

∼

0

.

3. In

anycase,weexpectthetruncationerrortodominateoverthe sta-tisticalandextrapolationerrors.

Giventheconvergenceofthe4_He_binding_energy_with

increas-ingcutoff,we conﬁrmthat, forboth physical

[15]

andunphysical

[7,9] pion masses, LO EFT(

π

/

) is renormalized correctly without the needfor a four-nucleon interaction.In the physicalcase, the binding energy isunderestimated forall values of the cutoff we considered, but the extrapolated value is in agreement with ex-perimentevenifweneglectthetruncationerror.Ofcoursewhen the latter is taken into account we must conclude that such a good agreement is somewhat fortuitous. We expect NLO correc-tions, includingCoulomb andtwo-nucleoneffective-range correc-tions,tochangetheresultbya fewMeV.Formπ

=

510 MeVand

mπ

=

805 MeV, our results reproduce LQCD predictions (where

Coulombisabsent)withinthemeasurementerrorovertheentire cutoffrange.AspointedoutinRefs.[7,9],thisisanon-trivial con-sistencycheck:ifeitherLQCDdataorEFT(

π

/

) weretoowrong,one wouldexpectnosuchagreement.However,LQCDuncertaintiesare toolargeatthispointforustodrawaverystrongconclusion.

It is interesting to study the cutoff dependence of the root-mean-square (rms) point-nucleon radius

√

r_pt2

and the single-nucleon point density

ρ

pt(r

)

. These quantities are related to the chargedensity,whichcanbeextractedfromelectron–nucleus scat-teringdata,butarenotobservablethemselves:few-bodycurrents and single-nucleon electromagnetic form factors have to be ac-countedfor.Still,onecangainsomeinsightintothefeaturesofthe ground-statewavefunctionbycomparingresultsatdifferentpion massesandcutoffs.Sinceneither

√

r2

pt

nor

ρ

pt(r

)

commutewith

the Hamiltonian, the desired expectation values on the

ground-state wave function are computed by means of “mixed” matrix

elements

0

|O|

0

≈

2

T

|O|

0

−

T

|O|

T

.

(20)

Intheaboveequation

|

0

istheimaginary-timeevolvedstate of

Eq. (4), while

|

T

is the trial wave function constructed as in

Eq.(5).

Theresults forthepoint-proton radiusof 4_He _are_reported _in Table 2. (Since Coulomb is absent in our calculation, the point-nucleonandpoint-protonradiiarethesame.)Inthephysicalcase, thecalculatedradius ismuchsmaller thantheempirical value— thatis,thevalueextractedfromtheexperimentaldataofRef.[34]

accounting for the nucleon size, but neglecting meson-exchange currents.Asimilar result,

√

r2_pt

≈

1 fmwas obtainedby the au-thorsofRef.[35]usingalocalformofachiralinteraction.NLOand N2_LO_potentials_in_a_chiral_expansion_based_on_naive_dimensional

analysis[3–5] bring theory intomuch closeragreementwiththe empiricalvalue.Hence,sub-leadingtermsintheEFT(

π

/

) expansion couldplay arelevantrole, atleastforphysicalvaluesofthepion mass.

Table 2

4_He_point-proton_radius_for_different_values_of_the_pion_mass_m

π andthecutoff,

comparedtotheempiricalvalueextractedfromRef.[34]accountingfortheﬁnite nucleonsize.Seemaintextandappendixfordetailsonerrorsandextrapolations.

2 fm−1 ₁_.₃₇₄_±₀_.₀₀₄ ₁_.₄₈₂_±₀_.₀₀₃ ₀_.₈₉₈_±₀_.₀₀₁ 4 fm−1 ₁_.₂₀₃_±₀_.₀₀₄ ₁_.₁₃₃_±₀_.₀₀₃ ₀_.₆₉₉_±₀_.₀₀₁ 6 fm−1 ₁ .109±0.003 1.035±0.002 0.609±0.001 8 fm−1 ₁ .054±0.003 0.976±0.001 0.542±0.001 → ∞ 0.9_±±0₀_..008 (sys)_{2 (stat)} 0.8±_±₀0_..04 (sys)_{1 (stat)} 0.25±_±0₀._.05 (sys)_{06 (stat)}

“Exp.” 1.45 – –

Fig. 2. (Coloronline)4_He_{single-nucleon}_point_density_for_m

π=140 MeV(upper

panel),mπ=510 MeV(middlepanel),andmπ=805 MeV(lowerpanel),at

differ-entvaluesofthecutoff.

Forunphysicallylarge pionmasses, whereEFT(

π

/

) issupposed toexhibit afasterconvergence,thepoint-protonradiusissmaller thanatmπ

=

140 MeV.Thevalueobtainedformπ

=

510 MeV in-dicates aspatial extent similar to thephysical one,while 4_He _at

mπ

=

805 MeV,incomparison,seemstobeamuchmorecompact object. Thisisconsistent withthe behaviorof thesingle-nucleon point density,

ρ

pt, displayed in Fig. 2. For all cutoff values, the

density corresponding tomπ

=

805 MeV isappreciably narrower

thanthat computedformπ

=

510 MeVormπ

=

140 MeV.

Focus-ing on

=

8 fm−1_,

_ρ

pt has a maximumvalue of 11

.

0 fm−3 for

(7)

Table 3

16_O_energy_for_different_values_of_the_pion_mass_m

π andthecutoff,compared

withexperiment.(NoLQCDdataexistforthisnucleus.)Seemaintextandappendix fordetailsonerrorsandextrapolations.

2 fm−1 ₋₉₇_.₁₉_±₀_.₀₆ ₋₁₁₆_.₅₉_±₀_.₀₈ ₋₃₅₀_.₆₉_±₀_.₀₅ 4 fm−1 ₋₉₂_.₂₃_±₀_.₁₄ ₋₁₃₇_.₁₅_±₀_.₁₅ ₋₃₆₂_.₉₂_±₀_.₀₇ 6 fm−1 ₋₉₇ .51±0.14 −143.84±0.17 −382.17±0.25 8 fm−1 ₋₁₀₀ .97±0.20 −146.37±0.27 −402.24±0.39 → ∞ −115±_±1 (sys)_{8 (stat)} −151_±±2 (sys)_{10 (stat)} −504±_±20 (sys)_{12 (stat)}

Exp. −127.62 – –

cases the maximum values are 2

.

1 fm−3 and 2

.

2 fm−3, respec-tively.

The similarity between 4He ground-state properties at mπ

=

510 MeVandthose at thephysical pionmass exists despite dif-ferencesinthestructureoflightersystems.Ifconﬁrmedforother propertiesof4_He _and_heavier_nuclei,_this_semblance_would_mean

thatsimulationsatintermediatepionmassescouldprovideuseful insightsintothephysicalworldwhilesavingsubstantial computa-tionalresources.

InTable 3 the 16Oground-state energies are reported forthe same pion-mass and cutoff values considered for 4_He. _A

strik-ing feature is that 16_O _is _not _stable _against _breakup _into _four 4_He _clusters _in _almost _all _the _cases, _the _only _exception

occur-ringfor mπ

=

140 MeVand

=

2 fm−1_, _where16_O_is ₄

_.

_{5 MeV}

morebound thanfour4Henuclei.Intheother caseswe missthe four-4_He_threshold_by_about_{5 MeV,}_which_is_beyond_our

statisti-calerrorsandrevealsalowerboundonthesystematicerrorofour

QMCmethod.

Even considering only statistical and extrapolation errors the asymptotic values of the 16O energy cannot be separated from the four-4_He _threshold. _The _proximity _of _the _threshold _suggests

thatthestructure ofour 16Oshouldbe clustered.Indeed,despite no explicit clustering being enforced in the trial wave function, thehighlyeﬃcientoptimizationprocedurearrangesthetwo- and three-bodyJastrowcorrelations,aswellastheorbital radial func-tions, in such a way as to favor conﬁgurations characterized by fourindependent4Heclusters.

The single-proton density proﬁles displayed in Fig. 3 indicate thatonlyfor

=

2 fm−1_with_mπ

₌

_{140 MeV}_and_mπ

₌

_{510 MeV}

are the nucleons distributed according to the classic picture of a bound wave function. For all the other combinations of pion massesandcutoffs,nucleonsare pushedaway fromthecenterof the nucleus, which is basically empty — the density at the ori-gin isa minuscule fractionofthe peak —until

2 fmfromthe centerof mass. The erratic behavior of the peak position of the densityprofilesasafunctionofthecutoffhastobeascribedtothe factthattherelativepositionofthefour4Heclustersispractically unaffectedby thecutoffvalue.Infact,oncetheclustersare suffi-cientlyapart,alandscapeofdegenerateminimainthevariational energyemerges.Hence, thesingle-proton densities correspondto wavefunctionsthat,despitepotentiallysignificantlydifferent,lead toalmost identicalvariational energies. In contrast,the widthof thepeaksdecreaseswithincreasingcutoffinstepwiththe shrink-ingoftheindividual4_He_clusters_reported_in

_{Table 2}

_.

The analysisof the proton densities alone doesnot suﬃce to support the claim of clustering. Another indication of clusteriza-tioncomesfromcomparingtheexpectationvaluesofthenuclear potentialsevaluatedin thegroundstatesof16_O_and4_He._For

in-stance,in themπ

=

140 MeV and

=

8 fm−1 caseit turns out thattheexpectationvaluesofthe16Otwo- andthree-body poten-tials are

4

.

05 and

4

.

16 times larger than the corresponding values for 4He. The same pattern is observed for all the com-binations of pion mass and cutoff, except for

=

2 fm−1 _with

Fig. 3. (Coloronline)16_O_{single-nucleon}_point_density_for_m

π=140 MeV(upper

panel),mπ=510 MeV(middlepanel),andmπ=805 MeV(lowerpanel),at

differ-entvaluesofthecutoff.

mπ

=

140 MeVandmπ

=

510 MeV.Inparticular,for

=

2 fm−1 andmπ

=

140 MeV,theexpectationvaluesofthetwo- and three-bodypotentialsin16_O_are

₄

_.

_{65 and}

₆

_.

_{14 times}_larger_than_in 4_He._This_difference_is_a_consequence_of_the_fact_that _the_number

ofinteractingpairsandtripletsislargerwhenclusterizationdoes nottakeplace.

To better visualize the clusterization of the wave function, in

Fig. 4wedisplaythepositionofthenucleonsfollowingthe prop-agation of a single walker for 5000 imaginary-time steps, corre-sponding to

τ

=

0

.

125 MeV−1, printed every 10 steps. In the upperpanel,concerningmπ

=

140 MeVand

=

2 fm−1_,_nucleons

are not organized in clusters. In fact, during the imaginary-time propagation they diffuse in the region in which the correspond-ingsingle-nucleon densityof

Fig. 3

doesnotvanish.Acompletely different scenario takesplace at the samepion mass when

=

8 fm−1_:_the _nucleons_forming_the _four4_He _clusters_remain_close

tothecorrespondingcentersofmassduringtheentire imaginary-timepropagation. Thisisclearevidenceofclustering.Ithastobe notedthattherelativepositionofthefourclustersisnot a tetra-hedron. To prove this, for each conﬁguration we computed the moment-of-inertiamatrixasinRef. [36].Ifthe4_He _clusters_were

positioned attheverticesofatetrahedron, diagonalizationwould yield only two independent eigenvalues. Instead,we found three distinct eigenvalues, corresponding to an ellipsoid — yet another indication oftheabsence ofinteractionsamongnucleons belong-ingtodifferent4_He_clusters.

(8)

Fig. 4. (Coloronline)Imaginary-timediffusionwithtimestepτ=0.125 MeV−1_of asinglewalkerformπ=140 MeV,at=2 fm−1(upperpanel)and=8 fm−1

(lowerpanel).

The non-clustered states at

=

2 fm−1 for mπ

=

510 MeV

andmπ

=

140 MeVdeservefurthercomment. Thestate atmπ

=

510 MeVstandsincontrasttotheotherstatesfoundabove thresh-old whose structure is clustered. We interpret thisas an artifact ofthe numerical method,since a perfect optimizationprocedure shouldhaveproducedaclusteredstructureresemblingthe lower-energystate withfour free 4_He. _While _there _is _no_signal _of16_O

stabilityabovethephysicalpionmass,thestateatmπ

=

140 MeV is certainly stable at the lowest cutoff, that is, when the inter-action has the longest range.On this basis, one might speculate that atsome pionmassabove thephysicalone a transitionfrom a non-clustered to a clustered state is expected. However, such a conclusion cannot be drawn until higher-order calculations in EFT(

π

/

) —whichwillcaptureﬁnereffectsfrompionexchangesuch asthetensorforceatN2_LO_—_are_available.

The smaller relative size of the model space leads to more modestsignsofcutoffconvergencefor16Othan4He,whichare re-ﬂectedinlargerextrapolationerrors,especiallyatmπ

=

805 MeV. Atphysical pionmass,the centralvalue of theextrapolatedtotal energyisonly 10% offexperiment, whichcan be bridged by sta-tisticalandextrapolationerrors.Thisdifferenceissmallcompared totheexpectedtruncationerror,

∼

30%.Ifthereisalow-lying res-onantor virtual state of4_He _nuclei_at _LO_in_EFT(

_π

_/

_{) —}_note _that

ouranalysis doesneither preclude noridentify such a state —it ispossiblethat the(perturbative) inclusionofhigher-orderterms uptoN2LOwillmovethe16Oenergysuﬃcientlyforstabilitywith respecttofour4Heclusters.

Forunphysicalpionmass,ourresultscanbeseenasan exten-sionofLQCDtomedium-massnuclei,withnofurtherassumptions abouttheQCDdynamics.Inthiscase,adeterminationofthe

rela-tivepositionofthefour-4_He_threshold_would_further_require_much

increasedaccuracyinthe A

=

2

,

3 LQCDdatathatweuseasinput.

5. Conclusions

Thispaperrepresentstheﬁrstapplicationofthe effective-ﬁeld-theory formalism, as developed forsmall nuclei without explicit pions,to arelativelyheavy object,16O. Weemployedcontact po-tentials whichrepresentthe leadingorderofa systematic expan-sionofQCD.Thisenabledustoanalyzephysicalnucleonsaswell assimulatedscenarioswithincreasedquarkmasses.

Toovercome thepeculiarchallengesassociated withthe

solu-tion of the Schrödinger equation, we have improved AFDMC by

introducing a new optimizationprotocolof the many-bodywave functiontobeemployedinthevariationalstageofthecalculation. Theschemewe proposeisanextensionofthelinearmethodand providesamuchfasterconvergenceinparameterspacecompared to stochastic reconﬁguration, previously adopted in nuclear QMC calculations.Suchaccuratetrialwavefunctionisthestartingpoint ofthe imaginary-timeprojection inAFDMC,which ﬁltersout the “exact” groundstateoftheHamiltonian. Thisalgorithmwas used topredictnotonlyground-stateenergies,butalsoradii,densities, andparticledistributions.

Ourresultsforthe 4He bindingenergyare inagreementwith previous ﬁndings, including the renormalizability of the four-nucleon system in EFT(

π

/

) withouta LO four-body force. In par-ticular, atphysicalpion massthe energyagrees withexperiment within theoretical uncertainties. Moreover, the calculated point-nucleon radii and single-particle densities reveal a 4_He _structure

atmπ

=

510 MeVsimilartothatatphysicalpionmass.

Withthis successfulbenchmark,we extended thecalculations to16O,obtainingextrapolatedvaluesforthe16Oenergyatallpion masses which are indistinguishable fromthe respective four-4_He

threshold, even considering only the smaller statistical and ex-trapolation errors. Infact, foralmost all cutoffs and pionmasses we considered,16_O_is_unstable_with_respect_to_break-up_into_four 4_He _nuclei. _Our _calculation _of _the 16_O _energy _is _the _ﬁrst _time

LQCD dataare extendedto themedium-mass regionina model-independentway.1

Interestingly, mπ

=

140 MeVand

=

2 fm−1 _is_the _only

pa-rametersetwhichyieldsastable16O.Thissuggeststhatthe long-rangestructureoftheinteractionisdeﬁcientatlargercutoffvalues and might have to be corrected, e.g. via one-pion exchange, to guaranteethebindingofheaviernucleiatLO.Alternatively,within a pionless framework, higher-order terms could act as perturba-tionstomove16_O_with_respect_to_the_four-4_He_threshold._At

phys-ical pionmass thecentral value ofthe totalenergy isjustabout 10% offexperiment. This is only slightlylarger than the statisti-calandextrapolationerrors,andwellwithinthe

∼

30% truncation error estimate.We cannot exclude thepossibility that agreement withdatawillimprovewithorder.Acomprehensivestudyofthe various subsystemsof 16O—forexample,12C, 8Be,and4He–4He scattering—could determine whetheraresonant orvirtual shal-low state atLO is transformed intoa bound state by subleading interactions, thus elucidating the relation between clusterization andQCD.

Note that the tensor force does not appear to be indispens-able for the deuteron or other light nuclei to be bound, or for

1 _As_this _manuscript_was _being _concluded,_a_calculation_of_doubly _magic nu-clei appeared[37], whereatwo-bodypotentialmodelobtainedfromLQCDdata at mπ=469 MeVwassolved withtheSelf-ConsistentGreen’sFunction method.

Thewidelydifferentinputdataandmethodtranslateintomuchsmaller4_He_and 16_O_energies_than_our_results._No_clear_sign_is_found_for_a16_O_state_below_the4_He threshold.

(9)

16_O _to _be _stable _against _the _breakup _into _four 4_He _nuclei. _For

instance, the Argonne v₄ interaction [38], which does not con-tain tensorterms,yields energies of

−

32

.

83

±

0

.

05 MeVfor 4_He

and

−

180

.

1

±

0

.

4 MeV for 16_O_[39]_. _In _EFT(

_π

_/

_{) the} _tensor _force

is only an N2_LO _(typically _10%) _effect, _which _is _small _for _the

deuteron[10].Atlargepionmassesthetensorcomponentshould remainperturbative.Beforeincludingatensortermweneedto ac-countforNLOterms,whichcouldplayaroleinthestabilityof16_O.

Oneof the goals ofcalculating 16_O_in_EFT(

_π

_/

_{) is} _to _assess_which

featuresofthenuclearinteractionareessentialforadescriptionof nuclei,andwhichcanbeconsideredsmalleffects.

In order to better appreciate the cluster nature of our solu-tionfor16_O, _we_have _studied_the _radial_nucleon _density_and_the

sampled probability density for the nucleons. In both cases the occurrenceof clusterizationis evident. From our resultsit is not possibleto infer anysigniﬁcant correlation between the clusters, whichoncemoreconﬁrmstheextremelyweakinteractionamong themwithinEFT(

π

/

).We wouldlike topointout thatlocalization was not imposed in the wave function used to project out the ground state; rather, it spontaneously arises from the optimiza-tionprocedure(despitethecorrelationsbeingfullytranslationally invariant) andit is preserved by the subsequent imaginary-time projection.

Current QMC (AFDMC)results have now reached an accuracy levelthat allowsfordiscussingthefew-MeV energies involvedin thisclass ofphenomena, whichare relevant fora deeper under-standing of how the systematics in nuclear physics arises from QCD.Starting fromLQCD data obtainedforvaluesofmπ smaller than the ones employed in this work, and yet larger than the physicalone, wouldallowusto establishthe thresholdforwhich nucleiaslargeas16Oarestableagainstthebreakupintofour4He clusters, ifsuch a threshold exists.To perform thisanalysis, it is essentialto includehigher-order termsinthe EFT(

π

/

) interaction, possiblyuptoN2LO, wheretensorcontributionsappear.Thisalso requiresa substantial improvementofthe existing LQCD dataon light nuclei, which, even for large mπ , are currently affected by statisticalerrorsthatdonotallowforaneffectiveconstraintofthe interactionparameters.

Acknowledgements

Wewouldlike to thankN.Barnea,D. Gazit, G. Orlandini,and W.Leidemannforusefuldiscussions aboutthesubjectofthis pa-per.Thisresearch was conducted inthe scope ofthe Laboratoire internationalassocié (LIA) COLL-AGAIN andsupported in part by

the U.S. Department of Energy, Oﬃce of Science, Oﬃce of

Nu-clearPhysics,undercontractsDE-AC02-06CH11357(A.L.)and DE-FG02-04ER41338(U.v.K.),andbytheEuropeanUnionResearchand InnovationprogramHorizon2020undergrantNo.654002(U.v.K.). The work of A.R. was supported by NSF Grant No. AST-1333607. J.K.acknowledgessupportbytheNSFGrantNo.PHY15-15738. Un-deran awardof computertime provided by theINCITEprogram, thisresearchusedresourcesoftheArgonneLeadershipComputing FacilityatArgonneNationalLaboratory,whichissupportedbythe OﬃceofScienceoftheU.S.DepartmentofEnergyundercontract DE-AC02-06CH11357.

Appendix A. Statistical and systematic error estimation

The procedure we adopted in order to estimate the error in the extrapolations performedin this work is as follows. We can distinguishbetweentwosourcesoferrors.Theﬁrstisasystematic error corresponding to the choice of neglecting the next (cubic) order in the expansion Eq. (3) and of removing the initial data

pointat

=

2 fm−1_._The_second_is_a_statistical_error_coming_from

theuncertaintiesinthedatausedfortheextrapolation.

The ﬁrst kindof error is estimated by considering the maxi-mum spreadin three different extrapolations: two quadratic ex-trapolations obtained by either neglecting the results at

=

2 fm−1 orby usingall available data(the latter isincludedonly ifthe reduced

χ

2 _is

_≈

₁₎ _and_a _cubic_{extrapolation} _that _uses_all

data.

Forthesecondtypeoferror,itisconvenienttowriteEq.

(3)

as asimplequadraticform,

O

=

O

+ C

0

+ C

1

2

+ · · ·

(21)

where

=

1

/

.Giventhatwehaveonlythreepairs

(,

O

)

,itis straightforwardtoseethat

C

1

=

1

2

−

3

O

2

− O

1

2

−

1

−

O

3

− O

1

3

−

1

(22) togetherwith

C

0

=

O

3

− O

1

3

−

1

− (

3

+

1

)

C

1 (23) and O

= O

1

− C

0

1

− C

1

21

.

(24)

Atthispointitissimpletoestimate theerrorsbypropagationof themeasurementuncertainty.Wehave

δC

1

=

1

2

−

3

δO

₂2

+ δO

2₁

(

2

−

1

)

2

+

δO

32

+ δO

21

(

3

−

1

)

2

(25) and

δC

0

=

δO

₃2

+ δO

₁2

(

3

−

1

)

2

+ (

3

+

1

)

2

δC

₁2

,

(26)

andthenﬁnally

δ

O

=

δO

₁2

+ δC

₀2

₁2

+ δC

₁2

₁4

.

(27)

Botherrorestimatesappearintheresultsreportedinthemain text.

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