Sensitivities and correlations of nuclear structure observables emerging from chiral interactions

0.23 MB·

by  Angelo Calci

Sensitivities andcorrelations ofnuclear structure observables emerging
from chiral interactions
Angelo Calci
TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3, Canada∗
Robert Roth
Institut fu¨rKernphysik, Technische Universita¨t Darmstadt, 64289 Darmstadt, Germany†
(Dated:August8,2016)
Startingfromasetofdifferenttwo-andthree-nucleoninteractionsfromchiraleffectivefieldtheory,weuse
theimportance-truncatedno-coreshellmodelforabinitiocalculationsofexcitationenergiesaswellaselectric
quadrupole (E2)and magnetic dipole (M1) moments and transitionstrengths for selectedp-shell nuclei. We
6 explorethesensitivityoftheexcitationenergiestothechiralinteractionsasafirststeptowardsandsystematic
1 uncertaintypropagationfromchiralinputstonuclearstructureobservables.Theuncertaintybandspannedbythe
0 differentchiralinteractionsistypicallyinagreementwithexperimentalexcitationenergies,butwealsoidentify
2 observableswithnotablediscrepanciesbeyondthetheoreticaluncertaintythatrevealinsufficienciesinthechiral

g interactions. Forelectromagnetic observables weidentifycorrelations among pairsof E2or M1 observables
u basedontheabinitiocalculationsforthedifferentinteractions. WefindextremelyrobustcorrelationsforE2
A observablesandillustratehowthesecorrelationscanbeusedtopredictoneobservablebasedonanexperimental
datumforthesecondobservable. Inthiswaywecircumventconvergenceissuesandarriveatfarmoreaccurate

5 resultsthananydirectabinitiocalculation.Aprimeexampleforthisapproachisthequadrupolemomentofthe
first2+statein12C,whichispredictedwithandrasticallyimprovedaccuracy.

]
h
PACSnumbers:21.60.De,21.30.-x,21.10.Ky,23.20.-g,21.10.-k,27.20.+n
t
-
l
c
u I. INTRODUCTION of freedom to accelerate the convergenceof the chiral order
n expansion[18].
[
Overthepastdecadetherehasbeensubstantialprogressin These developments on chiral interactions enable numer-

2 the construction of nuclear forces from chiral effective field ous applications in nuclear structure physics. To probe
v theory (EFT), both, on the formal level and on practical as- the predictive power of chiral interactions without introduc-
9 pects[1–3]. Recentlyseveraldifferentregularizationschemes ing uncontrolled approximations, ab initio many-body ap-
0
havebeenimplementedandarebeingexploredinmany-body proaches are the methods of choice. In addition to the tra-
2
calculations. An example are coordinate-space regulators ditional ab initio many-body methods such as the no-core
7
0 leading to fully local two-nucleon (NN) and three-nucleon shellmodel(NCSM)[19–21]andtheGreen’sfunctionMonte
. (3N) interactions up to N2LO that can be used in quantum Carlo (GFMC) method [22–24] there are also recent devel-
1
Monte Carlo calculations[4]. Using a mixedlocal and non- opmentslike the coupled-cluster(CC) methods[25–31], the
0
6 localregularizationschemeEpelbaumetal. havepresenteda self-consistentGreen’sfunctionmethods[32–34],andthein-
1 newfamilyofimprovedchiralNNinteractionsrangingfrom mediumsimilarityrenormalizationgroup(IM-SRG)[35–37]
: leading-order(LO)to next-to-next-to-next-to-next-toleading that extend the range of ab initio nuclear structure calcula-
v
order (N4LO) with five different cutoff values [5, 6]. This tions to medium-massand heavy nucleiregime up to the tin
i
X familyofinteractionsallowsforasystematicstudyoforder- isotopes. The importancetruncatedno-coreshell model(IT-
r by-orderconvergenceandcutoffdependenceofnuclearstruc- NCSM) [38, 39] bridgesthe gapbetween the traditionaland
a ture observables, a critical aspect that was often ignored in novelmany-bodymethods, it can include the 3N interaction
previousnuclearstructureapplications. TheLENPICcollab- explicitlyandcanprobeground-stateandexcitationenergies
oration[7] is exploringthese interactionsin few- and many- aswellasspectroscopicobservablesinp-andlowersd-shell
body calculations [8] and is developingthe consistent chiral nuclei. Theseobservablesconstituteacomprehensivetestbed
3N interactions. In a complementary development, new fit- forthetheoreticalpredictionsofchiralEFT.
tingstrategiesforchiralNN+3NinteractionsatN2LOareex-
Typically, these ab initio approaches attempt to estimate
ploited to quantify the statistical uncertainties related to the
uncertaintiesresulting from truncationsand incomplete con-
parameterfits[9]. Moreover,novelfitproceduresareutilized
vergencewith respect to the many-bodyspace. However, in
thatimprovethedescriptionofbound-statepropertiesfornu-
manyapplicationsof nuclearstructure theory,such as the p-
clei beyond the few-body domain [10, 11] compared to the
shellspectroscopy,theuncertaintiesenteringthroughthechi-
previousgenerationofchiralinteractions[12–17]. Thereare
ral inputs have not been explored, so far. The combination
also efforts to include the ∆ resonance as an explicit degree
ofreliablemany-bodyapproachesandnewchiralinteractions
will allow for a systematic propagation of theory uncertain-
ties to the nuclear structure observables. As a preparatory
∗[email protected] step, we study sensitivity and correlations of different spec-
†[email protected] troscopicobservablesforasetofchiralNN+3Ninteractions.
2
We explore the excitation spectra of 6Li, 10B, and 12C and lators, physical observablesshow generally a small sensitiv-
quantify the sensitivity of excitation energies on the choice ity to variations in the cutoff [4, 5, 44]. Thus, novelstudies
oftheunderlyingchiralinteractions. Thevariationoftheun- with a semi-localregularizationmust includeinformationof
derlying interactions also provides an opportunity to detect thechiralorderconvergencetoextracttheuncertaintiesofthe
andmap-outcorrelationsamongpairsofnuclearstructureob- currentlyavailableNNforces[8].
servables, particularly electromagnetic moments and transi- TheaboveNNforcesareaugmentedby3NforcesatN2LO.
tionstrengths. TheEMandN2LO NNforcesarecombinedwiththelocal
opt
This paper is structured as follows. In Sec. II we intro- 3N force usinga cutoff of 500MeV/c [13]. The LECs c
1,3,4
ducethedifferentchiralinteractionsandthetechnicalaspects of the two-pion exchangeterm are adoptedfrom the NN in-
of the many-body treatment. The sensitivity analysis of the teraction.Theparameterc isfittedtothetritonβ-decayhalf-
D
excitation spectra for selected p-shell nuclei is presented in life[45]andc isfixedbytheA=3and4Hebindingenergy,
E
Sec. III. In Sec. IV correlations in electromagnetic observ- for the EM and N2LO NN interaction, respectively. This
opt
ablesarestudiedandweconcludeinSec.V. yields (c ,c ) = (−0.2,−0.205) for the EM interaction and
D E
(−0.39,−0.398) for the N2LO interaction. Although, the
opt
N2LO interactionsis originallya NN force for brevity we
opt
II. FROMCHIRALHAMILTONIANSTOOBSERVABLES usethisexpressionalsotorefertothecorrespondingNN+3N
interaction introducedin this work. The EGM NN forces at
A. NN+3NinteractionsfromchiralEFT N2LOaretypicallycombinedwithaconsistentnon-local3N
forceatN2LO.WhilethisNN+3Nforceisusedinseveralap-
plicationstoneutronmatter[46–48],nuclearstructurephysics
In this work we investigate interactions from three differ-
beyondthelightestnucleiisfairlyunknown.TheLECsofthe
entchiralschemesthatareobtainedatN2LOorN3LOusing
3N force are fitted to the triton ground-state energy and the
different regularization and fit procedures. The first NN in-
neutron-deuterondoubletscatteringlength[16]. Thepartial-
teractionweintroduceisdevelopedbyEntemandMachleidt
wave decomposed 3N matrix elements at N2LO can be de-
(EM) [12] at N3LO. The nonlocal regulator function uses a
rived explicitly [13, 16, 49] or via a numerical partial-wave
fixed cutoff of 500MeV/c. This potentialprovidesan accu-
decomposition[50, 51]. The latter approachis also applica-
ratedescriptionof NN phaseshiftswitha comparablepreci-
bletocomputethecomplicated3NcontributionsatN3LOfor
sionasmorephenomenologicalhigh-precisionpotentialslike
futureinvestigations.
Argonne V18 [40] and CD-Bonn [41]. The EM potential is
widelyusedinnuclearstructurephysicsandhasbeenastan-
dardchoiceintheabinitiofield.
B. SRGevolutionandbasistransformations
The N2LO interaction by Ekstro¨m et al. [10] is a re-
opt
cently developed chiral interaction at N2LO. The fits of
Although non-local chiral NN interactions are rather soft
the low-energy constants (LECs) have been performed with
duetothemomentumcutoffintheregularization,itisstilldif-
the practical optimization using no derivatives (for squares)
ficultto convergeNCSM-type calculationsbeyondthe light-
(POUNDerS) algorithm [42]. This potentialalso uses a cut-
estnuclei.Alsotheinclusionoftherelevant3Ncontributions
off of Λ = 500MeV/c and an additional spectral function
χ
regularization(SFR)withacutoffofΛ˜ =700MeV/c. can be problematic for ab initio methods when a bare chi-
χ
ral interaction is used. The similarity renormalizationgroup
The NN interaction by Epelbaum, Glo¨ckle, Meißner
(EGM) [14] at N2LO uses the same non-local regular- (SRG)[52–54]isaunitarytransformationthatsoftensthenu-
clear interaction and can be applied consistently in the two-
ization with an additional SFR cutoff. This potential
and three-body space. Therefore, this approach is used in a
is constructed for a sequence of five cutoff combinations
(Λ /Λ˜ ) = {(450/500),(600/500),(550/600),(450/700), variety of nuclear structure applications to soften the chiral
χ χ
NN+3Ninteractions[29,34,36,55–57].
(600/700)}MeV/candprovidesaslightlylesspreciserepro-
TheSRGflowequationfortheHamiltonianH isgivenby
duction of the NN data than the other two NN interactions.
Thepion-nucleonLECsciarefittedindependentlyofthereg- d
ularizationtothepion-nucleonscatteringdata[43]anddiffer Hα =[ηα,Hα], (1)
dα
from the values used for the EM and N2LO interactions.
opt
With the sequence of cutoff parameters it offers the unique with the continuous flow-parameter α, which is related to a
opportunitytostudytheeffectoftheregularizationonnuclear momentumscaleλSRG =α−1/4andthedynamicgenerator
structureobservablesand,thus,todrawconclusionsaboutthe η =(2µ)2 [T ,H ], (2)
α int α
theoreticaluncertaintiesoriginatingfromthechiralinputs.
For chiral interactions with non-local regulators the cut- where µ is the reduced nucleon mass and T is the intrin-
int
offvariationprovidesalegitimatediagnostictooltoestimate sic kinetic-energy operator. It is important to note that the
the uncertainties at an individualchiral order. Nevertheless, SRG evolution induces irreducible many-body contributions
a variation of the chiral order is crucial to study the conver- beyond the particle rank of the initial interaction. With the
gence of the interactions and future works will combine a canonical generator (2) it has been found [55–57], that it
chiral order and cutoff variation for a more elaborate uncer- is indispensable to include the induced three-body contribu-
tainty analysis. Note, for chiralinteractionswith localregu- tions. Therefore,forallresultspresentedinthispaperweuse
3
an initial NN and NN+3N interactionand includeall contri- Westartwiththesimplenucleus6Li.Figure1showstheex-
butions up to the three-body level, which correspond to the citationenergiesofthefirstfourpositiveparitystatesobtained
NN+3N-inducedand NN+3N-fullnomenclatureof previous withthedifferentchiralNNandNN+3Ninteractions.Toindi-
works[55,56,58]. catetheuncertaintyduetotheconvergencewithrespecttothe
We aim at many-body calculations performed in the modelspacewecomparetheresultsatN =8(dashedbars)
max
harmonic-oscillator (HO) representation. Thus the NN and and N = 10 (solid bars). The calculationsare carriedout
max
3N interactions that are obtained in a partial-wave momen- at~Ω = 16MeVandthe SRG evolutionup to α = 0.08fm4
tum representation need to be transformed to the HO space. isperformedatthethree-bodylevel.Asillustratedforthe12C
Therearetechniquestoperformtheevolutionequationofthe spectrum in Ref. [65] once the SRG evolution is performed
SRG in the three-bodymomentumrepresentation[59], how- consistentlyatthethree-bodylevel,excitationenergiesshow
ever,forapplicationsinlocalizedsystems,suchasnuclei,the a negligible flow-parameter dependence. This remains true
discrete HO Jacobibasis isthe mostefficientscheme forthe even for heavier systems, where the absolute energies show
SRGevolution. Therefore,weimmediatelytransformthe3N a sizable flow-parameterdependence[31, 55, 58]. Since the
momentummatrixelementstotheHOJacobirepresentation, SRG induced beyond-3N contributionspredominantly cause
performingtheSRGtransformationsubsequently. an overall shift of all energies, their impact cancels out for
The HO machinery, comprehensively described in [58], theexcitationenergies. Inadditiontothespectrafortheindi-
is utilized to perform the Moshinsky transformation to the vidualinteractionsincludingtheinitialNNandNN+3Npart,
particle-basisrepresentation. Eventually,thematrixelements respectively,Fig.1alsoshowsacombinedspectrumforallthe
arestoredintheso-calledJT-coupledscheme[55,58],which EGMinteractions,wherethebandsindicatethespreadofthe
isthestartingpointforanumberofabinitiomany-bodymeth- excitationenergiesobtainedwiththedifferentcutoffs.Weob-
ods[30,33,34,36,37,55,60–63]. servethattheexcitationenergiesobtainedwiththeEMandthe
N2LO Hamiltonianstypically fall into the bandsextracted
opt
fortheEGMinteractions.
III. EXCITATIONSPECTRA A firstinspectionofthespectra revealsthatthe sensitivity
ofthe differentexcitedstatestotheHamiltonianisquitedif-
Inafirststepwe studytheexcitationspectraofvariousp- ferent. Whereas the excitation energy of the first 0+ state is
shellnucleiusingthedifferentchiralHamiltoniansintroduced largely unaffected by the different choices of chiral interac-
inSec.IIA. Thefocuswillbeonthesensitivityofthediffer- tionsortheinclusionofthechiral3Nforce,theexcitationen-
entexcitedstatesonthechiralinputs,givingrisetosystematic ergiesofthe3+andthe1+statesshowasizablevariation.The
theoryuncertaintiesthatresultfromthevariouschoicesmade inclusionofthechiral3Ninteractioncausesashiftoftheen-
during the construction of the chiral interactions. This in- ergiesinthesamedirectionforallHamiltonians,leadingtoa
cludes,forinstance,thedifferentregularizationschemes,chi- higher2+andalower3+excitationenergyforthefullNN+3N
ralordersaswellasthefitproceduresusedfortheLECs. An interactions. In the case of the EGM interactions, the band
additional source of uncertainty are the statistical uncertain- constructed from the cutoff dependence of the 3+ excitation
tiesoftheLECsresultingfromthefits. Thelatteruncertain- energynicelyoverlapswiththeexperimentalenergy. Forthe
tieshavebeenexploitedrecentlyforfew-bodyscatteringand 2+excitationenergythereisacleardiscrepancyandthechiral
ground-stateobservables[9],butremaintobeinvestigatedfor 3N interaction shifts the state further away from the experi-
p-shellspectroscopy. mentalenergyin all cases. However,the first 2+ state in the
experimentalspectrumisabroadresonanceandthereisanar-
WeemploytheIT-NCSMwiththeSRGevolvedHamiltoni-
ansbasedonthechiralNNorNN+3Ninteractionsdiscussed rowsecond2+stateabout1MeVabove.Thustheinclusionof
continuumdegreesoffreedom,e.g.,throughtheNCSMwith
in Sec. IIB. For all Hamiltonians the IT-NCSM calculation
continuum[60, 66, 67], will be importantto understandand
includesexplicit3NtermsoftheSRG-evolvedHamiltonians.
We perform a full NCSM calculations up to N = 6 for disentanglethese2+ states. Theslowconvergenceofthecal-
6Liand N = 4for12Cand10B. We efficientlmyapxroceedto culated2+statemightserveasafirstindicationforthesecon-
max tinuum effects. For the 0+ excitation energy, the EGM band
largerN withanimportancetruncationincludingathresh-
max
is closer to experimentalthoughit does not overlapeither—
old extrapolationtowards the full NCSM space. The details
experimentallythisstateisanarrowresonance.
oftheIT-NCSMandthethresholdextrapolationarediscussed
in [38, 39]. On should note that the threshold extrapolation Figure2showsasimilaranalysisoftheexcitationspectrum
itself induces a theoretical uncertainty at the level of the fi- of 12C. The excitation energiesof the lowest positive-parity
nal observables, which is quantified systematically through statesobtainedinIT-NCSMcalculationsareshownforallin-
the extrapolation protocol [38]. For excitation energies this teractions. Obviously,thestructureofexcitationspectrumof
uncertainty is of the order of 50keV for the largest spaces. 12Cisricherthanfor6Li. Previousinvestigationshaveshown
The IT extrapolationis very robust for the discussed excita- thatsomeoftheexcitationenergies,e.g.,forthefirst1+and4+
tion spectra in this work. The only exception is the first ex- states, are verysensitive to the 3N interaction. Furthermore,
cited0+ state in12CforasingleEGMinteraction,wherethe in comparisonto experimentthere are clear discrepanciesof
degeneracywiththefirst4+statecausesinaccurateITextrap- the1+excitationenergyobtainedfortheEMinteractionwhen
olations, resultingin an unreliableassessment ofthe angular includingthe 3N interaction [55, 65]. The behaviorof these
momentumofthisstate. statesfordifferentchiralinteractionsis, therefore,highlyin-
4
EGM EGM
EM N2LO (450/500) (600/500) (550/600) (450/700) (600/700) Bands
opt
5
2+0
4
0+1
]
V
e 3
M
[ 3+0
x 2
E
1
1+0
0
.
NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N exp
FIG. 1. (color online) Excitation spectrum of 6Li for the EM, the N2LO and all five cutoff combinations of theEGM NN and NN+3N
opt
interactions. TheparametersoftheIT-NCSMcalculations are N = 10, ~Ω = 16MeV, andα = 0.08fm4. Thedashedbarscorrespond
max
toN = 8calculations. BandsinthelastbutonecolumnontherightindicatethecutoffdependenceoftheEGMpotential. Experimental
max
excitationenergiesaretakenfrom[64].
EGM EGM
EM N2LO (450/500) (600/500) (550/600) (450/700) (600/700) Bands
opt
16
12++10
14 4+0
1+0
12
V]10 0+0
e
M 8
[ 0+0
x 6
E
4 2+0
2
0 0+0
.
NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N exp
FIG. 2. (color online) Excitation spectrum of 12C. IT-NCSM calculations are performed for N = 8 (solid bars) and 6 (dashed bars),
max
remainingparametersasinFig.1.Experimentalexcitationenergiesaretakenfrom[64].
teresting. Note, the excited 0+ states are expected to have a large sensitivity. For all these states the sensitivity, as sum-
distinct cluster structure that cannot be described accurately marizedby thebandsforthe EGMinteractions, reducessig-
in tractable HO modelspaces[68]. Therefore,itis notclear nificantlywiththeinclusionofthechiral3Ninteraction.This
whether the 0+ state obtained in the IT-NCSM corresponds might be interpreted as indication that the theoretical uncer-
thefirstexcited0+ state (Hoylestate) orthesecondone(see tainties are reduced when going from an incomplete chiral
Ref. [65] fora moredetailed discussion). For these reasons, NN interaction to a complete and consistent chiral NN+3N
wewillnotincludethis0+ stateintothefollowingdiscussion HamiltonianatN2LO.Forallinteractions,thechiral3Ncom-
onthesensitivitytotheHamiltonian. ponent shifts the 1+ states to lower excitation energies. As
a result, all interactions underestimate the 1+ excitation en-
Comparing the spectra for the different interactions con-
ergybymorethan2MeV—evenconsideringthe uncertainty
firmsthesensitivityofthe1+and4+excitationenergiestothe
band,thereis acleardiscrepancywith experiment. Sinceall
underlyinginteraction,alsothehigher-lying2+ stateshowsa
5
EGM EGM
EM N2LO (450/500) (600/500) (550/600) (450/700) (600/700) Bands
opt
5
4
2+0
3
]
V
e 2 1+0
M
0+1
[
x 1
E
1+0
0 3+0
-1
.
NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N exp
FIG. 3. (color online) Excitation spectrum of 10B. IT-NCSM calculations are performed for N = 8 (solid bars) and 6 (dashed bars),
max
remainingparametersasinFig.1.Experimentalexcitationenergiesaretakenfrom[64].
Hamiltoniansemployedhereuselocalornon-local3Ninter- 16Oground-stateenergy.
actions at N2LO, it will be very interesting the see whether As the final case, we discuss the excitation spectrum of
next-generation chiral 3N interactions at N3LO can resolve 10B as shown in Fig. 3. The typical excitation energies for
this discrepancy. In contrast to the 1+ and 4+ states, the en- thisodd-oddnucleusaremuchsmallerthanin12C,therefore,
ergy of the first excited 2+ state shows very little sensitivity shiftsoftheexcitationenergiesofindividualstatesby1MeV
to the starting interaction and to the chiral 3N contribution. can change the spectrum drastically. Furthermore, full con-
ThebandsextractedfromtheEGMinteractionsaresmalland vergenceof the excitation energies is more difficult to reach
thebandswithandwithoutthechiral3Ninteractionoverlap. thanforthe12Cspectrum. Particularlytheresultswithchiral
Interestingly,allinteractionstendtounderestimatethe2+ ex- 3NinteractionsshowaresidualN dependence,i.e.,theex-
max
citationenergyslightly. citation energies for N = 8 and 6, indicated in Fig. 3 by
max
the solid and dashed levels, respectively, are slightly differ-
Itisalsointerestingtostudytheabsolute12Cground-state
ent. ThisresidualN dependenceismuchsmallerthanthe
energiesresultingfromthedifferentNN+3Ninteractions.The max
variationsduetodifferentHamiltoniansand,therefore,donot
energies calculated with the EGM interactions span a range
affectthepresentdiscussion.
of−96.8to−80.5MeV. Thisrangecontainstheexperimental
energyof−92.16MeV.Alsotheground-stateenergyobtained Already the first ab initio calculations of 10B with 3N in-
withN2LO interactioniswithintheEGMrange,whilethe teractionshaveshownthattheorderingofthefirst3+ and1+
opt
EM interactions predicts an energy of −97.8MeV and thus statesdependsonthe3Ninteraction[69]. Manyoftherealis-
the largest bindingenergy. However, it is importantto note, ticNN interactionsincorrectlypredictthe1+ asgroundstate
that the energiesare extrapolatedfromNCSM modelspaces andonlythe3Ninteractionrestoresthecorrectlevelordering.
uptoN =8causinganestimatedextrapolationuncertainty This is also observed in Fig. 3 for the chiral interactions—
max
ofabout1−2MeV. Whatismoreimportantistheimpactof with one exception all chiral NN interactions predict the 1+
omittedSRG-induced4Ncontributionsthataresizableforthe below or degenerate with the 3+ state. In all these cases,
SRG flow-parameterα = 0.08fm4 usedhere. Fromananal- thechiral3Ninteractionshiftsthe1+ upwardsrelativetothe
ysis of the flow-parameter dependence we find that the 4N 3+,thus,restoringthecorrectlevelordering. Anexceptionis
contributions are repulsive and, thus, will reduce the above theEGMinteractionwithcutoffs(450/500)MeV/c,whichal-
binding energies. For instance, changing α for the EM in- readygivesthecorrectlevelorderingwiththeNNinteraction,
teraction to α = 0.04fm4, i.e., towards the bare interaction thechiral3Ninteractiononlyleadstoaslightreductionofall
reducesthe bindingenergybyabout2.3MeV. Based on the excitationenergies.TheEGMuncertaintybandforthe1+ex-
flow-parameter dependence we cannot reliably estimate the citationenergyisreducedbyincludingthechiral3Ninterac-
binding energy expected for the bare interaction. Neverthe- tionandrobustlyindicatesthe3+ asthegroundstate. Within
less, we can conclude that the absolute binding energies for the cutoff-uncertainty bands all excitation energies obtained
thebareinteractionswillexhibitaspreadofseveralMeVand withthechiralNN+3Narecompatiblewithexperiment.
tendtounderestimatetheexperimentalbindingenergies.This Our uncertaintyanalysis for the p-shellspectra providesa
sensitivity of the absolute binding energies to the details of crucial verification of the predictive power of the chiral in-
the interactions is consistent with the finding in [9] for the teractions. Besidesdistinctsensitivitiesofexcitationenergies
6
to the 3N force, also systematic deviations from experiment
8.5
beyondthetheoreticaluncertaintycanbeidentified.Thesein-
vestigationsidentifythefirst1+statein12Casanidealbench-
8
markforthenextgenerationofchiralNN+3Ninteractions.
4]7.5
m
f
2
IV. ELECTROMAGNETICTRANSITIONSAND [e 7
MOMENTS +)
0
→ 6.5
Electromagneticobservablesprovideanotherwindowinto +
2
6
structure of nuclei with differentsensitivities and addressing 2,
E
complementary information. Therefore, we extend the dis- (
B5.5
cussionofsensitivitiesofnuclearobservablestoelectromag-
netic moments and transition strengths, focusing on electric
5
quadrupole(E2)andmagneticdipole(M1)observables.
Coming from the discussion of excitation energies, sev-
4.5
eral comments are in order: First, the convergence rate of
.
electromagnetic observables, particularly of E2 observables,
3 4 5 6 7 8 9
issignificantlyslowerthantheconvergenceofexcitationen-
Q(2+)[efm2]
ergies. Owing to the sensitivity of the E2 operator on the
long-rangebehaviorofthewavefunction,largeNCSMbasis
FIG.4. (coloronline)Correlationofquadrupoleobservablesforthe
spacesarerequiredtoobtainthecorrectasymptoticbehavior
first 2+ state in 12C. Plotted is the reduced quadrupole transition
ofthewavefunctionsandtoconvergeE2observables. Even-
strength B(E2,2+ → 0+)totheground stateversusthequadrupole
tually, one mightstill rely on extrapolations,e.g., within the moment Q(2+) obtained with different chiral NN (open symbols)
novelschemesconstructedinaneffectivefieldtheoryframe- andNN+3Ninteractions(solidsymbols): EM(box), N2LO (cir-
opt
work[70],toextractarobustresult. cle), and EGM with cutoffs (Λ /Λ˜ ) = {(450/500),(600/500),
χ χ
Second,theE2orM1operatorsshouldbetransformedcon- (550/600),(450/700),(600/700)}MeV/c(diamond,triangleup,tri-
sistently with the Hamiltonian when using SRG transforma- angle down, hexagon, cross). The IT-NCSM calculations are per-
tionstoimprovetheconvergencebehavior. Theeffectofthis formedat~Ω = 16MeVandα = 0.08fm4 usingamodelspaceof
consistent SRG transformation of electromagnetic operators Nmax = 2(blue),4(green),6(violet),and8(redsymbols). Theer-
rorbarsindicatetheuncertaintiesofthethresholdextrapolationsin
wasonlystudiedinafewselectedcases. Thesestudiesindi-
theIT-NCSM.Thedashedcurvescorrespondstothecorrelationob-
catedthattheconsistentSRGtransformationchangeselectro-
tainedfromformula(5)withaquotient of theintrinsicquadrupole
magneticobservablesonlybyafewpercent[71,72],whichis
momentsettoone(grey)orfittedtotheoreticaldatapoints(black).
whyNCSMapplicationshavenotincludedtheseeffectsofar.
The grey shaded area indicates the error band of the experimental
Finally, chiral EFT also predicts the electromagnetic two- B(E2)[76]and Q[77]value. Theblueshadedareacorresponds to
bodycurrentcontributionsconsistentlywith theinteractions. apredictionforQconsistentwiththetheoreticalcorrelationandthe
Also these contributions should be included for a complete B(E2)measurement.
treatment of electromagnetic observables. Pioneering calcu-
lation in a hybrid frameworkusing chiral EFT currents with
anArgonneinteractionhaveindicateda significantinfluence present these two observablesfor the same set of chiral NN
ofcurrentcontributiontoelectromagneticmomentsandtran- andNN+3Ninteractionsusedforthestudyofexcitationspec-
sitionstrengths[73]. tra. Inadditionweshowtheresultsfordifferentmodel-space
Addressingalltheseeffectsinacomprehensivefashionwill truncationsfromN =2to8,whichisimportantbecauseof
max
betheaimofourfuturestudiesofelectromagneticproperties the slow convergenceof these observables. Thus each sym-
starting from consistent chiral EFT inputs. In a preparatory bol in the figure corresponds to a specific Hamiltonian at a
step towards the complete calculations, we study the impact specific value of N . The grey rectangle indicates the ex-
max
oftheHamiltonianonE2andM1observables.Asanovelas- perimentalvalues for the B(E2) and the quadrupolemoment
pectintheabinitiocontext,weexplorecorrelationsbetween includingtheirexperimentaluncertainty. Theuncertaintyfor
pairsofE2or M1observablesinvolvingthesame states. As thequadrupolemomentisparticularlylarge[77],butnewex-
we will see in the following, the study of such correlations perimentsareplannedtoreducethisuncertainty[78].
in an ab initio framework can be extremely beneficial. Re- The picture that emerges from Fig. 4 is remarkable. All
cently, correlations of E1 observables in closed-shell nuclei data pointsfall ontothe same line, irrespectiveof the under-
havebeenexploitedforimpressivepredictionsofobservables lying chiral NN or NN+3N interactions and of N . There
max
sensitivetothechargeandneutrondistribution[74,75]. isastrongandrobustcorrelationbetweenthetwoE2observ-
We start with the discussion of E2 observables involving ablesemergingfromourabinitiocalculations. Thevaluesof
the first excited 2+ state and the 0+ groundstate in 12C, i.e., theindividualobservablesshowasizable dependenceonthe
the B(E2) transition strengthform the 2+ state to the ground underlyinginteractionand N , buttheyalways stay on the
max
stateandthequadrupolemomentofthe2+ state. InFig.4we correlationline. Asageneraltrend,withincreasingN the
max
7
quadrupolemomentandtheB(E2)continuetoincrease,indi- 11
catingtheslowconvergenceoftheselong-rangeobservables.
Therobustcorrelationbetweenthispairofquadrupoleob-
10
servables emerging from ab initio calculations can be in-
terpreted in terms of the simple rotational model by Bohr ]
4
andMottelson[79],wherebothobservablesinthelaboratory m 9
f
frame are connected to the intrinsic quadrupole moment Q0 2[e
viatheformulas +) 8
2
→
3K2−J(J+1)
Q(J)= (J+1)(2J+3)Q0,s , (3) +4 7
2,
E
and (B
6
5 J 2 J
B(E2,J → J )= Q2 i f . (4)
i f 16π 0,t K0(cid:12)(cid:12)K! 5
(cid:12)
(cid:12)
Here J is the angular momentum with the(cid:12)index i and f re-
ferringtotheinitialandfinalstate, K istheprojectionofthe . 4
total angular momentumon the symmetry axis of the intrin- 4.5 5 5.5 6 6.5
sicallydeformednucleus. Fortheinvestigatednuclei12Cand Q(2+)[efm2]
6Li,K correspondstotheangularmomentumoftheirground
states. Theindicessandtoftheintrinsicquadrupolemoment FIG.5. (coloronline)Correlationofthe B(E2,4+ → 2+)valueand
indicatethe”static”and”transition”observableQandB(E2), the quadrupole moment Q(2+) in 12C. The parameters and defini-
tion of the symbols are as in Fig. 4. The black dashed lines mark
respectively.Onecancombinebothformulassuchthatthera-
theregimeofthecorrelatedtheoreticaldatapoints. Theblueshaded
tio of the intrinsic quadrupolemoments Q /Q is the only
0,t 0,s areacorrespondstoapredictionfortheB(E2)transitionstrengthcon-
parameterthatrelatesthetwoobservables
sistentwiththetheoreticalcorrelationandthepredictedquadrupole
momentfromFig.4.
5 (J+1)(2J+3) 2 J 2 J
B(E2,J → J )= i f
i f 16π(cid:0)3K2−J(J+1)(cid:1)2 K0(cid:12)K!
(cid:12) (5)
(cid:12)
× Q0,t(cid:0)2Q(J)2 . (cid:1) (cid:12)(cid:12) a substantialtheoryuncertainty,which is eliminated through
(cid:16)Q0,s(cid:17) theuseofthecorrelationtogetherwithoneexperimentalob-
servable.Thequadrupolemomentisalsoconsistent,butmore
Ina rigidrotormodeltheintrinsicquadrupolemomentsQ
0,s precise than direct predictionsby nuclear lattice simulations
and Q0,t are expected to be equal. The correlation result- ofQ(2+)=(6±2)efm2obtainedatLO[80].
ing from this assumption is represented by the grey dashed
Due to the stability of the correlation in 12C one can also
lineinFig.4,whichslightlymissesthecorrelationpredicted
address higher excited states of the yrast band, as shown in
in the ab initio calculations. Using the ratio of the intrinsic
Fig. 5. In analogy to the previous correlation analysis we
quadrupolemomentsasaparametertofittheaboverelations
totheabinitioresultsleadstoQ /Q = 0.964andacorre- plot the B(E2,4+ → 2+) as function of the quadrupole mo-
0,t 0,s ment Q(2+). The correlation motivated by the rotor model
lationlinethatmatchestheabinitioresultsperfectly,asseen
is present, but less clean. In particular, for the larger model
fromtheblackdashedlineinFig.4.
spaces the theoretical data points start to spread around the
Afterhavingestablishedthiscorrelationinabinitiocalcu-
fittedquadraticcorrelationcurve. Thisindicatesamorecom-
lations,wecanexploitittomakepredictionsononeofthetwo
plicated structure of the 4+ state in large model spaces, de-
observablesbasedonexperimentaldatafortheotherobserv-
able. Inthisparticularcase,thequadrupolemomentofthe2+ viating from the simple rotor model. Already the excitation
energyof the 4+ state was much more sensitive to the inter-
state is poorly known, whereas the B(E2) has a much lower
actionthanthefirst2+ state(cf. Fig.2),indicatingadifferent
relativeuncertainty. Thuswecanusetheexperimentalvalue
anduncertaintyB(E2)=7.94±0.66e2fm4[76]andtranslateit andmoreintricatestructureforthe4+state.
viatheabinitiocorrelationlineintoanvalueanduncertainty Still, we can identify a correlation band indicated by
for the quadrupole moment of Q(2+) = (5.91± 0.25)efm2. the black curves covering almost all ab initio results and
Theuncertaintyofthisvalueisoneorderofmagnitudesmaller parametrizeitthroughtherotormodelusingarangeforthera-
thantheuncertaintyofthedirectmeasurement. tio Q /Q from0.795to 0.905, which differs significantly
0,t 0,s
Itisalsomuchbetterthanthetheoryuncertaintyforadirect from the rigid rotor. Still, we can use this correlation band
calculationofthequadrupolemoment. ForN = 8thedif- to predict the B(E2,4+ → 2+) in the range from 7.05 to
max
ferentchiralNN+3Ninteractionpredictquadrupolemoments 10.82e2fm4,basedonourpreviousextractionforQ(2+).This
intherangefrom4.5to6.2efm2(redfilledsymbolsinFig.4) B(E2)isnotknownexperimentally. Clustermodelspredicta
and these values still increase with increasing N . So the value around15e2fm4 [81] and a rigid rotor model with the
max
sensitivitytotheinteractionandtheslowconvergenceleadto intrinsicquadrupoledeformationobtainedfromthe3αmodel
8
12 12
11 11
]10 ]10
4 4
m m
2f 9 2f 9
e e
[ [
) )
+1 8 +1 8
→ →
+ 7 + 7
3 3
2, 2,
E 6 E 6
( (
B B
5 5
4 4
3 3
. .
-6.5 -6 -5.5 -5 -4.5 -4 -3.5 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Q(3+)[efm2] Q(1+)[efm2]
FIG. 6. (color online) Correlation of quadrupole observables for FIG. 7. (color online) Correlation of quadrupole observables for
the first 3+ state in 6Li. Plotted is the reduced quadrupole tran- the first 1+ state in 6Li. Plotted is the reduced quadrupole transi-
sition strength B(E2,3+ → 1+) to the ground state as function of tionstrength B(E2,3+ → 1+)tothegroundstateasfunctionofthe
thequadrupolemoment Q(3+). TheIT-NCSMcalculationsareper- quadrupolemomentQ(1+).Theremainingparametersanddefinition
formedfor N = 4(blue),6(green),8(violet),and10(redsym- ofthesymbolshapesareasinFig.6.Thegreyshadedareaindicates
max
bols).Theremainingparametersanddefinitionofthesymbolshapes theerrorbandoftheexperimentalB(E2)[83]andQ(1+)[85].
are as in Fig. 4. The grey shaded area indicates the error band of
theexperimental B(E2)[83]. Note,thereisnomeasurementforthe
spectroscopicquadrupolemoment. nectionwiththequadrupolemomentofthe1+groundstatein-
steadoftheexcited3+state.AsshowninFig.7thispairofE2
observablesdoesnotexhibitarobustcorrelation.Becausethe
appliedinRef.[82]gives25e2fm4[81]. measuredquadrupolemomentof−0.08178(164)efm2 [85]is
Wenowmovetothelighternucleus6Liandrepeatthecor- so close to zero, the experimental data point in the correla-
relationanalysis. ThelowestE2transitionisbetweenthefirst tionplotisincompatiblewiththerigidrotormodel. Thevery
excited 3+ state and the 1+ ground state. As we remarked smallquadrupolemomentoftheground-stateisgovernedby
earlier, the 3+ state is a narrowresonanceand, therefore,the more subtle structural effects rather than the robust effects
definition of the quadrupolemomentis nontrivial. Since we fromtherotormodel.Notetheimpactoftheimportancetrun-
are workingin a bound-stateapproach,we can computethis cationalso becomesnoticeable, becauseof the smallmagni-
observablenevertheless.InFig.6weshowthecorrelationplot tudeofthequadrupolemoment.Moreover,severaltheoretical
forthe B(E2,3+ → 1+)strengthandthequadrupolemoment quadrupolemomentshavea positivesign, i.e., predicta pro-
of the first 3+ state in 6Li. Again we find a tight correlation latedeformation,butmovewithincreasing N towardsthe
max
betweenthesetwoobservablesforallchiralNNandNN+3N slightlyoblatedeformationasexperimentallymeasured.This
interactions and model space truncations. A fit of the rotor exampledemonstrates,thattheE2correlationsin12Cand6Li
modelwithQ0,t/Q0,s =0.961againdescribesthiscorrelation that we identified from our ab initio calculations and inter-
verywell. pretedbythesimplerotormodelarenon-trivialfindings.
Unlike the previous cases, all IT-NCSM calculations un- Finally, we discussan examplefora differentelectromag-
derestimate the B(E2) value—we only get about half of the netic operator, the magnetic dipole or M1 operator. Unlike
experimentaltransition strengths. At the same time, there is theE2observableswediscussedsofar,theM1operatordoes
asystematicdependenceonthemodel-spacetruncationNmax. not depend on the spatial distance, but only probes the spin
Forallinteractionstheabsolutevaluesofthequadrupolemo- and orbital angular-momentum structure of the state. This
mentandtheB(E2)increasemonotonicallywithNmaxwithno leadstoadifferentconvergencebehaviorofM1observablesin
indicationofconvergence.Thisgeneralbehaviorisconsistent NCSM-type calculations. Furthermore, from a macroscopic
withthefindingsinRef.[84]usingtheCD-Bonnpotentialand modelbuildonanintrinsicstatethemagneticdipolemoment
hintsatmissingcontinuumeffectsforthedescriptionofthe3+ andtheB(M1)transitionstrengthshowalesstrivialbutagain
resonance.Still,wecanusetherotormodeltoextractavalue quadratic relation depending on an effective and intrinsic g-
of the quadrupolemomentof Q = −6.59(26)efm2 based on factor[79]. Therefore,itisinterestingtoexplorethespinand
themeasuredB(E2). orbitalstructurethatdeterminespairsofM1observablesinab
We canconsiderthesame B(E2,3+ → 1+) for6Liin con- initiocalculationsandtoidentifycorrelations.
9
N whichmightberelatedtotheresonancenatureofthe0+
max
state.
15.8
Comparingthecalculationstoexperiment,indicatedbythe
narrow grey rectangle in Fig. 8, provides an interesting per-
spective. Theground-statemagneticdipolemomentof6Liis
]
2N15.6 knownwith anexcellentaccuracyfromatomicphysicsmea-
µ
)[ surements[86, 87]. Thecalculationswith chiralNN+3Nin-
+
1 teractionsdeviatefromexperimentbyabout2%—thoughthis
→
isasmalldeviationbyourstandards,itisverysystematic.The
+ 15.4
0 experimentaluncertaintyontheB(M1)arelargerandmostof
M1, the calculations fall within the error bar of the experiment.
( However, the systematic N dependenceof the calculation
B max
15.2 suggeststhattheconvergedB(M1)willbeoutsidetheexperi-
mentalerrorbarforallHamiltonians.
This is clearly a case where precision studies, both in ex-
periment and in ab initio theory will be very valuable. As
15.0
mentionedinthebeginningofthissection,ourstudiesofelec-
. tromagneticobservablesarenotfullycompleteyet. Wehave
0.82 0.83 0.84 0.85 0.86 notincludedtheconsistentSRGevolutionoftheelectromag-
µ(1+)[µN] netic operators and we have not included consistent electro-
magnetic two-body currents from chiral EFT. Both correc-
FIG.8.(coloronline)Correlationofmagnetic-dipoleobservablesfor tionsenterasadditivetwo-bodypiecesintheelectromagnetic
the1+groundstateandthefirst0+excitedstatein6Li.Plottedisthe operators and they will affect both observables in the pairs
magnetic dipole transition strength B(M1,0+ → 1+) to the ground
of E2 andM1 observablesdiscussed here. In case of the E2
stateversusthemagnetic-dipolemomentµ(1+). TheIT-NCSMcal-
observables, we expect the correlation line to be largely un-
culationsareperformedforN =4(blue),6(green),8(violet),and
max affectedbythese corrections. However,theywillplaya role
10 (red symbols). The remaining parameters and definition of the
for the specific values of the observables, particularly at the
symbolshapesareasinFig.4.Thegreyrectangleindicatestheerror
band of theexperimental µ(1+) [86,87] and B(M1,0+ → 1+) [83] precisionlevelof the M1 observablesdiscussed above. This
value. aspectwillbeafocusofourfuturestudies.
In Fig. 8 we plot the B(M1,0+ → 1+) transition strength V. CONCLUSIONS
fromtheexcited0+statetothe1+groundversusthemagnetic
dipolemomentµ(1+)ofthegroundstatein6Li.Asbefore,we WehavepresentedabinitioIT-NCSMcalculationsforthe
considerthefullsetofinteractionsforarangeofmodelspace spectroscopy of p-shell nuclei using a large set of different
truncations Nmax = 4,6,8,and 10. One should note that the chiralNN+3Ninteractions.Inthiswayweaddressedthesen-
rangeof B(M1)andµpresentedintheplot, coveringaround sitivityofexcitationspectraandelectromagneticobservables
7% relative change in both observables, is very small com- to the input interactions, which constitutes a significant and
paredtothetypicalvariationsoftheE2observablesdiscussed previouslyunexploredcontributiontothetheoryuncertainties
before. This alreadyindicatesthatM1 observablesare more ofstate-of-the-artabinitiocalculations. Thevariationofthe
robustwithrespecttointeractionandmodel-spacechoices. input interactions also provides yet unexplored insights into
Thepictureregardingcorrelationsisalsodifferentfromthe thedetailsofnuclearstructure.
E2observables,thecalculationsdonotcollapseona univer- We discussed the sensitivities of individual excitation en-
sal correlation line. There are distinct groupsof points with ergiesandcomparedtheresultingtheoryuncertaintiestoex-
very systematic trends. First, the calculations using chiral periment.Thisprovidesanimportantdiagnosticforthechiral
NN+3N interactions (full symbols) are separated from cal- interactions,particularlyincaseofasystematicdisagreement
culations with only chiral NN forces (open symbols). The with experiment beyond the uncertainties obtained from the
inclusionof thechiral3N interactionreducesthe dipolemo- differentinteractions. Anexampleistheexcitationenergyof
mentsystematicallybyabout1%, simultaneouslythe B(M1) the first 1+ state in 12C, wherethe sensitivity to the different
isreducedbyasimilaramount.Second,withincreasingN chiralNN+3Ninteractionsismuchsmallerthanthedeviation
max
theB(M1)strengthissystematicallyreduced,whilethedipole fromexperiment.Thiswillbeanimportanttestcasefornext-
momentremainspracticallyconstant. Third,thedifferentin- generationchiralinteractions.
putHamiltoniansforfixedN giveverysimilarresultsasin- Forelectromagneticobservablesthevariationoftheunder-
max
dicatedbythegroupsofsame-coloredopenorfullsymbolsin lyinginteractionandofthemodel-spacetruncationallowedus
Fig.8. Insummary,bothobservablesarerobustwithrespect toidentifyrobustcorrelationsbetweenpairsofE2observables
tothechoiceofchiralNN+3Ninteractionbuttheyareinflu- mergingfromabinitiocalculations,whichcanbeinterpreted
encedbythechiral3Nforce.Thedipolemomentisconverged intermsofarigidrotormodel.Thesecorrelationsofferanew
whiletheB(M1)showsasystematicdecreasewithincreasing toolto extractaccurate predictionsfor ab initio calculations.
10
By combiningthe theoreticallypredictedcorrelationsof two thatexploitthefullpotentialofchiralEFT.
observableswithasingleexperimentaldatumforoneobserv-
able we can extracta value for the second observablethat is
far more accurate and robustthan any directab initio result. ACKNOWLEDGMENTS
Anexampleisthequadrupolemomentofthefirstexcited2+
statein12C,thatwepredicttobeQ(2+) = (5.91±0.25)efm2 We thank Marina Petri and Alfredo Povesfor helpfuldis-
basedontheexperimentalvalueoftheassociatedB(E2). cussions. This work is supported by the DFG throughgrant
SFB1245,theHelmholtzInternationalCenterforFAIR(HIC
Thisworkisapreparatorysteptowardsafullquantification forFAIR),theBMBFthroughcontracts05P15RDFN1(NuS-
oftheoryuncertaintiesbasedonconsistentinputsfromchiral TAR.DA) and 05P2015(NuSTAR R&D). TRIUMF receives
EFT.Forexample,usingthenewfamilyofchiralinteractions fundingviaacontributionthroughtheCanadianNationalRe-
from LO to N4LO developed within the LENPIC collabora- searchCouncil. Numericalcalculationshavebeenperformed
tion[5,6,8],wewillbeabletostudytheorder-by-ordersys- at the computing center of the TU Darmstadt (LICHTEN-
tematics of nuclear observables and thus propagate the EFT BERG),attheJu¨lichSupercomputingCentre,attheLOEWE-
uncertaintiesconsistentlytothelevelofnuclearstructureob- CSCFrankfurt,andattheNationalEnergyResearchScientific
servables. Includingtwo-bodycurrentsand consistentSRG- ComputingCenter supportedby the Office of Science of the
evolutionfortheelectromagneticobservableswillbeanother U.S. Department of Energy under Contract No. DE-AC02-
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