Table Of ContentEvidence of α particle condensation in 12C and 16O and Nambu-Goldstone boson
S.Ohkuboa
aUniversityofKochi,Kochi780-8515,Japanand
ResearchCenterforNuclearPhysics,OsakaUniversity,Ibaraki,Osaka567-0047,Japan
Abstract
3
ItissuggestedthatdirectevidenceofBose-Einsteincondensationofαparticlesisobtainedbyobservingaphase
1
0 mode (Nambu-Goldstone boson) with long wavelength even when characteristic features such as superfluidity is
2 diffuculttoobserve. Forthe7.65MeV0+ Hoylestatein12Cand15.1MeV0+ statein16O,whicharecandidatesfor
2
n anα particlecondensate,it is suggestedthatthe emergentband head0+ state of the K = 0+2 rotationalbandwith a
a verylargemomentofinertiaisconsideredtobeaNambu-Goldstoneboson.
J
Keywords: αparticlecondensation,Nambu-Goldstoneboson,spontaneoussymmetrybreaking,αclusterstructure,
1
3 12C;16O
PACS:21.60.Gx,27.20.+n,03.75.Nt
]
h
t
- The 0+ state in 12C at excitation energy E =7.65 groundstateandhasadilutematterdistribution.
l 2 x
c MeV,theHoylestate,isakeystatefornucleosynthesis, In orderto get directexperimentalevidencethat the
u
theevolutionofstarsandtheemergenceoflife. Theex- Hoyle state is a condensate, Raduta et al. [10] ob-
n
[ istenceofthisresonantstatejustabovethe αthreshold served three α particles with the same energy emitted
was predicted by Hoyle to explain the enhanced triple simultaneouslyfromtheHoylestateandsuggestedthat
1 α reactions needed to understand the nucleosynthesis thethreeαparticlesweresitting inthelowest0sstate.
v
5 of 12C. In the last 60 years, the understanding of the Ontheotherhand,inasimilarcoincidenceexperiment
8 Hoyle state has been deepened by the study of the α Manfrediet al. [11] skeptically set an upperlimit of a
4 clusterstructureof 12C.However,itsinterpretationhas probability of 0.45%, which confirms the previous re-
7 changedconsiderably. Morinaga[1] proposedthat the sult by Freer et al. [12]. In spite of much effort, firm
.
1 Hoylestateisabandheadstatewithalinearchainstruc- experimental evidence of Bose-Einstein condensation
0 ture of three α particles. The three α structure of 12C suchassuperfluidityoravortexhasnotbeenfoundfor
3 was most thoroughlyinvestigatedby Uegaki et al. [2] theHoylestate.
1
intheirpioneeringwork, whichshowedthattheHoyle
: Thepurposeofthispaperistosuggestthatevidence
v statehasadilutestructureinanew“α-bosongasphase” ofαparticlecondensationcanbeprovidedbytheobser-
Xi andclarifiedthesystematicexistenceofa“newphase” vationoftheNambu-Goldstone(NG)bosoncausedby
ofthethreeαparticlesabovetheαthreshold.
r thespontaneoussymmetrybreakingoftheglobalphase
a InspiredbytheobservationofBose-Einsteinconden- symmetry. Wesuggestthatfor16Oabandhead0+ state
sation of cold atomic gas clusters, much attention has predictedat16.6MeVthataccompaniesaK =0+band
beenpaidtoinvestigatewhetherBose-Einsteinconden- 2
withverylargemomentinertiaisaNGbosoncollective
sationofαparticlesoccursinlightnucleiwhereαclus- zero mode associated with the condensation of four α
terstructurewidelyexists. Thethree αparticlesystem particles of the 0+ state at 15.1 MeV. We also suggest
of12C,especiallythedilutepropertiesoftheHoylestate in 12C that a 0+ state and a rotational band built on it
due to α particle condensation, has been studied ex- 3
should emerge slightly above the Hoyle state as a NG
tensively by many authors [3, 4, 5, 6, 7, 8, 9, 10, 11]. bosoniftheHoylestateisacondensateofthreeαpar-
It is now evident theoretically and experimentally that
ticles. We show an experimentalcandidateforthe NG
the Hoyle state has a larger radius compared with the bosonstatein16Oand12C.
If the concept of α particle condensation persists in
Emailaddress:[email protected](S.Ohkubo) nuclei,thefourαparticlesystemof16Oisanothergood
PreprintsubmittedtoPhysicsLettersB February1,2013
candidate. Tohsaki et al. [3] conjectured that the 0+ and in nature [19]. For example, for infinite systems,
3
state at11.26MeV in16O, whichis located3.18MeV insuperconductorstheNGmode(inwhichtheNGbo-
below the four α threshold energy, may be an α parti- sonhasbeeneatenbytheplasmon[20])andtheHiggs
clecondensatebecauseitiswellrepresentedbyaBose mode[21]havebeenobserved.AsseeninFig.2,inpar-
wavefunctionwithadilutefourαparticlestructure.Fu- ticlephysicsthemasslesspionandtheamplitudemode
nakietal. [13]investigated0+statesin16Ousingafour σ meson emerge as a collective mode from the vac-
α cluster model in the bound state approximation and uumduetothechiralcondensationofthevacuum[17].
suggestedthatthe0+ state at15.1MeV isprobablyan We notethataband(Reggetrajectory)ofthestates, in
α condensed state. Ohkubo and Hirabayashi [14] in- whichorbitalangularmomentumoftherelativemotion
vestigated not only the 0+ states but also the so-called betweenaquarkandananti-quarkisexcited,isbuilton
fourαlinearchainstateswithhigherspins[15,16]from top of the NG boson pion. The spectra including the
theviewpointoftheunifieddescriptionofquasi-bound radially excited states are explained well by the quark
states and α scattering by analyzing the α+12C elas- model[23]. ForfinitesystemsaNambu-Goldstonebo-
tic scattering and α+12C(0+) inelastic scattering using sonaswellasaHiggsmodebosonhavebeenobserved
2
a coupled channel method. They showed that the so- insuperfluidnucleiasapairingrotationandapairingvi-
calledfourαlinearchainstatescanbeunderstoodasa bration,respectively[24].ForBose-Einsteincondensed
K =0+band(Fig.1)withtheα+12C(0+)configuration. coldatomgaseswithfinitenumberofparticlesinafinite
2 2
Theysuggestedthattheobserved15.1MeV0+ stateis volume,theexistenceofaNambu-Goldstonebosonhas
acandidateforthefourαparticlecondensateinacom- beenshowntheoreticallyfromtheWard-Takahasiiden-
pletelydifferentquantumstateunderBosestatistics. tityinquantumfieldtheory[25,26].
Theappearanceofthe K = 0+ bandinFig.1(a)pre-
2
dictedinRef.[14]justabovethe15.1MeV0+statemay
beunderstoodfromtheviewpointofspontaneoussym-
metrybreakingasfollows: Ifthe15.1MeV0+ state is
acondensate,aNGbosonzeroenergycollectivemode
shouldappear. Consideringthat(1)the0+ at16.6MeV
is a first 0+ state above the condensed vacuum (15.1
MeV0+ state)withtheverysmallexcitationenergyof
1.5 MeV (almost massless) and (2) the very large mo-
ment of inertia of the band built on it, which is due to
a collective motion with the largestdimension that the
system allows(long wavelength),this state maybe re-
gardedastheemergenceoftheNGbosoncausedbythe
spontaneoussymmetrybreakingoftheglobalphasedue
Figure1: (a)ThecalculatedK=0+andK=0−rotationalbandswith to the fourα particlecondensation. ThisNG state is a
2 2
theα+12C(0+)clusterstructurein16Onearthefourαthreshold[14]. collective state with enlargedstructure of four α parti-
2
(b)Theexperimental rotational bandstates with αcluster structure cleswithdilutepropertyduetoitslongwavelengthna-
taken from Ref.[14,18]. Thecentroid ofeach ofthespin states is
ture, i.e. deformedorthogonalto thesphericalvacuum
shownbyablacksquare.Thecandidate15.1MeV0+stateoffourα
condensateisdisplayedbyagreensquare. Thelinesaretoguidethe 15.1MeV0+ inthe0sstates.
eye. In fact, in Ref.[14] it was shown that the 0+ reso-
nantstatepredictedat16.6MeVhasalooselycoupled
Ifthe15.1MeV0+ stateisaspecialstateofaBose- α+12C(0+)clusterstructurewithlargedeformation. As
2
Einstein condensate of four α particles, the state can shown in Fig. 1 (a), their calculated resonances with
be described by a macroscopic wave function in the spin2+,4+and6+arelocatedontheJ(J+1)lineforming
Ginzburg-Landautheory,Ψ =|Ψ|eiθ where|Ψ|istheor- a rotational band. The predicted negative parity band
derparameter.Inacondensate,continuousglobalphase K = 0− [14] is considered as a parity-doublet partner
2
symmetryingaugespaceisspontaneouslybrokenwith of the K = 0+ band with almost the same moment of
2
|Ψ|, 0 in the NG phase and two collective modes ap- inertia. The very large moment inertia of the K = 0+
2
pear,i.e.,amassless(zeroenergywithainfinitewave- bandstemsfromtheNGbosonnatureofthebandhead
length) phase (θ) mode (NG boson) and a finite mass 0+statewithamasslesslongwavecollectivemotion.It
amplitude(|Ψ|)mode(Higgsboson)[17]. Spontaneous is noted that the calculated resonances of the K = 0+
2
symmetry breaking is ubiquitous in physical systems band correspond well with the experimental α cluster
2
MeV(Fig.1(a))[14].Thisstronglysupportsthepresent
interpretation. The18.8MeV0+ state maycorrespond
tothe0+ statearound10MeVinFig.3.
4
Nowwecometotheαparticlecondensationofthree
αparticlesin12C.Accordingtotheaboveinterpretation,
the understanding of the K = 0+ rotational band built
2
ontheHoylestateisdrasticallychanged. Morinaga[1]
interpreted that the Hoyle state and the 10.3 MeV 2+
state form a rotational band with a very large moment
ofinertia,whichleadstothethreeαlinearchainmodel.
Recently2+ (9.6MeV)[27,28,29]and4+ (13.3MeV)
2 1
[30] states above the Hoyle state have been observed.
The2+statehasbeenconsideredtobearotationalband
2
statebuiltontheHoylestate[31]. IftheHoylestate is
acondensateofthreeαparticles, aNG bosonzeroen-
ergy collective mode should appear just above it. The
Figure2: Theexperimental spectra oftheNGbosonpion, andthe
NG boson is massless in principle, however, in nature
orbitalandradialexcitedstatesofthepionaredisplayedtogetherwith
theHiggsbosonσmeson. Thelinesaretoguidetheeye. Dataare itsenergyisnotalwaysexactlyzeroasseeninthepion
fromRef.[22]. case.IftheNGstatehaszeroenergy,the2+(9.63MeV)
2
and4+ (13.3MeV) abovestand onthe Hoyle state. In
1
this case it is unlikelythat the 2+ and 4+ states are ro-
states observed in the four α coincidence experiments 2 1
tational member states because the condensate Hoyle
[15,16]. stateissphericalwiththethreeαparticlessittinginthe
Beforeproceedingto the experimentalcandidatefor 0sstate. Ontheotherhand,iftheNGbosonmodeen-
theNGboson0+state,wewouldliketoemphasizethat ergyisnotexactlyzeroasinthecaseof16O,another0+
the15.1MeV0+stateshouldnotconsideredasamem- state emerges just above the Hoyle state. Because the
berstate of the K = 0+ bandbuilt onthe NG 0+ state. NG0+stateisnotspherical,arotationalbandshouldap-
2
Theintrinsicstructureofthe15.1MeV0+stateisquite pear. Theobserved2+ (9.63MeV)and4+ (13.3MeV)
differentinnaturefromthedeformedNGboson0+state states are considered2to be a memberof t1his rotational
and its associated rotational band states. The NG bo- band. ThebandheadNGboson0+ stateandthe2+ and
sonstateisalogicalconsequenceoftheexistenceofthe 4+ states should have the same α cluster configuration
15.1 MeV state that spontaneously violates the global in nature, mostly a loosely coupled α+8Be configura-
phase symmetry due to condensation. While the 15.1 tion created by lifting an α particle from the vacuum
MeV0+ stateissphericalwithBosestatisticsinthe0s (Hoyle state). As shown in Fig. 3, this is an analog of
state,theK = 0+ rotationalbandstatesaredeformedin thelooselycoupledα+12C(0+)structurebuiltontheNG
a local condens2ate state in which the three α particles boson0+ statepredictedat126.6MeVin16O.Theintra-
arecondensedformingtheHoylestate[14]. Ifwecon- bandB(E2)transitionsshouldbeverylargeifmeasured
nectthe15.1MeV0+ stateandthe2+ state(Fig.1(b)), becauseofalargedeformation,i.e. largemomentofin-
the estimatedmomentof inertiaappearsdrastically re- ertia. On the other handthe B(E2)transition from the
duced from that of the K = 0+ rotational band as dis- 2+ totheHoylestate shouldbemuchsmallerthanthat
2 2
cussedinRef.[14]. fromtheintrabandB(E2:2+→0+).
2 3
Accordingto ourinterpretation, the existence of the The results of the ab initio calculation of 12C in the
NG boson 0+ state predicted at 16.6 MeV is logically boundstateapproximationinRef.[32]seemstosupport
veryimportant. However,no0+ statehasbeenreported the present interpretation theoretically. The calculated
aroundhereintheliterature. Itmaynotbeeasytoob- 0+,2+and4+stateshavetheα+8Beintrinsicconfigura-
3 2 1
serve the NG boson because of a large decay width of tionandthe intrabandtransition B(E2)valuesare very
morethan1MeV.VeryrecentlyItohetal. reported[18] large, 600 and 310 (e2fm4) for the transitions 4+→2+,
1 2
that they have observed new broad resonant 0+ states 2+→0+,respectively.OntheotherhandtheB(E2)tran-
2 3
at 16.8 MeV and 18.8 MeV. The 16.8 MeV state with sition from the 0+ to the Hoyle state is one third, 100
3
anαclusterstructureofawidthofabout1MeV(adi- (e2fm4). Kurokawa and Kato¯ emphasized [31, 33] the
amond in Fig. 1(b)) agrees precisely with the theoret- importanceoftreatingtheunboundresonancestatesin
ically predicted energy16.6 MeV and the α width 1.1 thisenergyregioncarefullyunderthecorrectthree-body
3
a third0+ statefromtheexperimentalside, becauseits
existenceslightlylowersthepositionoftheHoylestate
duetoquantumorthogonality.Thereforeathird0+state
isexpectedtoexistslightlyabovethepositionextrapo-
lated from the J(J + 1) line of the rotational band for
the 2+ (9.63 MeV) and 4+ (13.3 MeV) states. If we
2 2
lookintotheexperimentaldataoftheisoscalarstrength
distributionof 12C(α,α’) in Fig. 8(a)of Ref.[28], there
is a peak in the excitation energy at around 8-9 MeV.
Thisseemstosuggesttheexistenceofanew0+ stateat
3
around9MeVjustabovetheHoylestate. Veryrecently
Itoh et al. [18] observed the 0+ and 0+ states with an
3 4
αclusterstructureat9.04MeV (width1.45MeV)and
10.56MeV (width 1.42MeV), respectively. The exis-
tenceofthe0+statejustabovetheHoylestateonwhich
3
a K = 0+ bandisbuilthaslogicallythesamestructure
2
as seen in the 16O case, which gives strong supportto
Figure3:Theexperimentalstateswiththeαclusterstructurein12C: the 0+ state being a NG boson collective mode state.
the0+ state[18](reddiamond)oftheNambu-Goldstoneboson,the 3
3 TheemergenceoftheNGbosonisstrongevidencethat
2+[28]and4+[30]states(redtriangle)oftheK=0+rotationalband.
TheαcondensateHoylestate0+(greensquare)and2the0+state(red theHoylestateisacondensateofthreeαparticles. The
2 4
circle)[18]arealsoshown.Thelineistoguidetheeye. structureofFig.3resemblesthe16OcaseinFig.1(b).It
isinterestingtoexplorewhetherahigherspinstatebuilt
onthe0+,whichcouldbeacandidateforthebandcor-
4
resonanceconditionand foundbroadresonancestates, respondingto π(1300)inFig. 2, exists. Kurokawaand
Γa0b+3=o1avt.e1EtMxh=ee8Vα.9Rt5heMrfe.s[e3hV2o]lwd.)iTthahneΓdy=0s1u+4.4ga8gteMsEteexV=d1t(h1oa.n8tl7tyhM1is.6e0V6+MswtaeittVhe Kwmhaetinoc¯tho[3ffo3irn]meprstrieaadKaicst=thae02+3K+b=satan0tde+wboaintnhtdha.elmoobsstetrhveedsa0m+4e[m18o]-,
3 2
maybeahighernodalstate,inwhichradialmotionbe- We note that a dilute gaseous property with a large
tween 8Be and the α particle is excited [33]. We note radius is not evidence for condensation of α particles
this α cluster 0+ state is located below the newly ob- without firm evidence such as superfluidity (supercon-
3
served2+ (9.63MeV)stateandmaycorrespondtoour ductivity), a vortex or a NG boson. Even a non-
2
NG boson 0+ state because (1) the excitation energy condensate state has a gas-like dilute density distribu-
1.28MeVfromtheHoylestate issmall, whichissim- tion due to the threshold effect. For example, the 3−
1
ilar to the 1.5 MeV of the 16O case (massless) and (2) state in 12C, which is not a well-developed α cluster
this0+stateformsaK =0+rotationalbandwiththe2+ state, has a large radius comparable to the Hoyle state
3 2 2
(9.63MeV) and the 4+ (13.3 MeV) states with a large due to the thresholdeffect [9]. The 0+, which is inter-
moment of inertia sim1ilar to the 16O case (long wave- preted to have a dominant [8Be(2+)⊗ℓ3=2] structure
J=0
length). in Ref.[2] and a linear chain-likestructure in Ref.[32],
Intheliteraturenosucha0+ state witha zero-mode alsohasamuchlargerradiusthantheHoylestate. The
3
nature has been observed just above the Hoyle state. dilutepropertyofagaseousstateofanαparticlesystem
The existence of such a 0+ state will be important in arisesforthefollowingfourreasons:(1)thequantumα
3
the nucleosynthesis of 12C from triple α particle colli- particlecondensationinmomentumspace,whichbrings
sions. From this viewpoint, if we look into the exci- aboutspatialdilutenessduetotheuncertaintyprinciple,
tationenergyoftheobservedthreestatescarefully,the (2)theNGbosonstate,whichcausesdilutenessbecause
Hoyle state seems to be located slightly below the en- of its long wavelength nature, (3) the threshold effect
ergy expected by extrapolating the rotational band of bywhichthewavefunctionextendsnearandabovethe
the 2+ (9.63MeV) and 4+ (13.3 MeV) states. The es- Coulomb barrierand (4) the Higgs boson state, which
2 1
timated moment of inertia connecting the Hoyle state appearsabovetheNGbosonstate,andcanbedilutebe-
andthe2+ (9.63MeV)stateisabout80%ofthatesti- causeofthe collectivityofthe orderparameter |Ψ|and
2
matedfromthe2+ (9.63MeV)andthe4+ (13.3MeV) thethresholdeffect. Thepresentmechanismisgeneral
2 1
states. Thistrendissimilartothefourαcasediscussed andtheobservationofaNGbosoncollectivemodewith
before and seems to suggest the possible existence of longwavelength(largemomentofinertia)willbeuse-
4
fulfortheconfirmationofanαparticlecondensatenear [24] R.A.Broglia, O.Hansen, andC.Riedel, Adv.Nucl. Phys.6
thethresholdenergy. (1973)259.
To summarize, we have investigated α particle con- [25] H.Enomoto,M.Okumura,andY.Yamanaka, Ann.Phys.321
(2006)1892.
densation in 16O and 12C from the viewpoint that evi- [26] M. Mine, T. Koide, M. Okumura, and Y. Yamanaka, Prog.
denceofαparticlecondensationmaybeprovidedbythe Theor.Phys.115(2006)683.
observationoftheNGbosoncausedbythespontaneous [27] W.R.Zimmermanetal.,Phys.Rev.C84(2011)027304.
[28] M.Itohetal.,Phys.Rev.C84(2011)054308.
symmetry breakingof the global phase symmetry of a
[29] M.Freeretal.,Phys.Rev.C86(2012)034320.
condensate.In16Othe0+stateat15.1MeVisshownto [30] M.Freeretal.,Phys.Rev.C83(2011)034314.
beacondensatebecausethebandhead0+ stateat16.6 [31] C.KurokawaandK.Kato¯,Phys.Rev.C71(2005)021301;
MeV just above the 15.1 MeV 0+ state is considered Nucl.Phys.A738(2004)455c.
[32] Y.Kanada-En’yo,Prog.Theor.Phys.117(2007)655.
to be a NG boson zero mode state. The K = 0+ band
2 [33] C.KurokawaandK.Kato¯,Nucl.Phys.A792(2007)87.
builtontopofithasalooselycoupledα+12C(0+)clus-
2
terstructure correspondingwell with experimentalob-
servation.In12Cweshowedthatthenewlyobserved2+
2
(9.63 MeV) and 4+ (13.3 MeV) α cluster states above
2
the Hoyle state are considered to be a rotational band
builtontheNGboson0+ state. Thisemergesasalogi-
calconsequenceoftheHoylestatebeingathreeαcon-
densatestate. Itisinterestingtoinvestigatetheoretically
andexperimentallythe emergenceofthe α condensate
andtheassociatedNambu-Goldstonecollectivemodein
othernuclei.
The author thanks M. Itoh for useful communica-
tions.
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5
