TetraneutronresonanceAndreyShirokov

M.V.Lomonosov MoscowStateUniversity

NTSE-2018,Daejeon,November2,2018

Tetraneutron experiment: history• Morethan50yearsoftetraneutronsearches• ForhistoricalsurveyseeR.Ya.Kezerashvili,arXiv:1608.00169 [nucl-th](2016)

• Earlystudies:4He(𝜋", 𝜋$)4nreaction−noresonanceorboundstate

• Later:7Li(11B,14O)4n;7Li(7Li,10C)4n−noevidencefor4n• F.M.Marquésetal.,Phys.Rev.C65,044006(2002):

14Be→10Be+4n－boundtetraneutron???Not confirmed... Experimental program stopped...

Tetraneutron

K.Kisamori etal.,Phys.Rev.Lett.116,052501(2016):ER =0.83± 0.63(statistical)± 1.25(systematic)MeV;widthΓ ≤ 2.6MeV

MoreexperimentaldataareexpectedOtherexperimentsarestarting

Tetraneutronexperiment:future

Tetraneutronexperiment:future

Tetraneutrontheory:history• ForahistoricalsurveyseeR.Ya.Kezerashvili,arXiv:1608.00169 [nucl-th](2016)• Therewasalotoftheoreticalstudiesoftetraneutronstartingfrom1970’swithvariousNN andNNNinteractionswithinvariousapproaches:democraticdecay(hyperspherical approach),Faddeev−Yakubovsky equations,Gamowshellmodel,complexscaling,analyticcontinuationinthecouplingconstant,variousboundstatetechniques…

• Anundoubtfulconclusion:notetraneutronboundstate• Noindicationinpreviousstudiesofaresonanceatlowenoughenergiesandnarrowenoughtobedetectedexperimentallyfrom numerousstudiesallowingforcontinuum

• Therewere,however,someindicationonapossiblelow-lyingtetraneutronresonancefromsomebound-statecalculations…

Tetraneutron:anexampleofanindicationonapossiblelow-lyingtetraneutronresonancefromGFMCbound-statecalculations

Tetraneutron:anexampleofarecentstudyofapossiblelow-lyingtetraneutronresonance

Tetraneutron theory: history

• So,anundoubtfulconclusion:notetraneutronboundstate• Noindicationinpreviousstudiesofaresonanceatlowenoughenergiesandnarrowenoughtobedetectedexperimentallyfrom numerousstudiesallowingforcontinuum

• We,however,obtainsucharesonancewithinanewlydevelopedSS-HORSE-NCSMapproachwithourJISP16NN interactionfittedtoNNdataandpropertiesoflightnuclei:AMSetal.,Phys.Rev.Lett.117,182502(2016):ER =0.8 MeV;widthΓ =1.4 MeV

Our approach• Ourapproach:No-coreShellModel(NCSM)+SS-HORSEtechniquetocalculateS-matrixattheNCSMeigenstates(moredetailsinA.Mazur’stalktoday):

• Calculatingasetof𝐸, eigenstateswithdifferentℏΩ and𝑁012 withintheNCSM,weobtainasetof𝛿 𝐸, valueswhichwecanapproximatebyasmoothcurveatlowenergies;the𝛿 𝐸, parametrizationincludespoletermsassociatedwithresonances,etc.

• Specificfortetraneutronisthedemocraticdecay

tan �(E�) =SN+1,l(E�)

CN+1,l(E�)

S-matrix at low energiesSymmetryproperty:

Hence

As

Boundstate:

Resonance:

S(�k) =1

S(k)

S(k) = exp 2i�

k ! 0 : �` ⇠ k2`+1 ⇠ (pE)2`+1

S(i)b (k) =

k + ik(i)b

k � ik(i)b

,

�0 ⇥ ⇥ � arctan

sE

|Eb|+ c

⇤E + d(

⇤E)3 + f(

⇤E)5...

S(i)r (k) =

(k + �(i)r )(k � �(i)⇤

r )

(k � �(i)r )(k + �(i)⇤

r )

�1 ' � arctanapE

E � b2+ c

pE + d(

pE)3 + ..., c = � a

b2.

�(�k) = ��(k), k ⇠pE,

� ' CpE +D(

pE)3 + F (

pE)5 + ...

Tetraneutron• Democraticdecay(noboundsubsystems)• Hypersphericalharmonics:

(r1, r2, ..., rA) = �(⇢)Yk⌫(⌦), ⇢ =

vuutAX

i=1

(ri �R)2,

�nK ⌘ �Ln(⇢) = ⇢�(3A�4)/2'nK(⇢), L = K +

3A� 6

2;

~22m

� d2

d2⇢+

L(L+ 1)

⇢2

��L

n(⇢) +X

L0

VL,L0�L0

n (⇢) = E�Ln(⇢).

Approximation:

the only open channel is with L = Lmin = Kmin + 3 = 5.

All possible L (K) values are accounted for in diagonalization of the

NCSM Hamiltonian

TetraneutronS-matrix: S = exp 2i�L

�L = CpE +D(

pE)

3+ F (

pE)

5+ ...

As E ! 0 : �L ⇠ (

pE)

2L+1 ⇠ (

pE)

11– huge power!

� = � arctan

apE

E � b2� �3,6(E),

�3,6(E) =

w1

pE + w3

⇣pE⌘3

+ c⇣p

E⌘5

1 + w2E + w4E2+ w6E3

+ dE4,

�3,6(E) = M9

⇣pE⌘+O

✓⇣pE⌘11

◆.

Tetraneutron, JISP16

Resonanceparameters:Er =186keV,Γ =815keV.

0 5 10 15 20 25 30E [MeV]

0

30

60

90

120

150

180

δ [d

egre

es]

Nmax= 10 12 14 16 18SS HORSERes. term

4n, gs

AresonancearoundEr =850keV withwidtharoundΓ =1.3MeVisexpected!

Canitbeavirtualstate? No.

Tetraneutron, JISP16

Canitbeacombinationofafalsepoleandresonantpole:

� =� arctanapE

E � b2

� arctan

sE

|Ef |� �3,6(E)?

Yes!Resonanceparameters:Er =844keV,Γ =1.378MeV,Efalse =-55keV.

0 5 10 15 20 25 30E [MeV]

-90

-60

-30

0

30

60

90

120

150

180

δ [d

egre

es] Nmax= 10

12 14 16 18SS HORSERes. termFalse term

4n, gs

Res+False

Tetraneutron, JISP16

Options:Resonanceparameters:Er =844keV,Γ =1.378MeV,Efalse =-55keV.

0 5 10 15 20 25 30E [MeV]

0

30

60

90

120

δ [d

egre

es] Nmax= 10

12 14 16 18ResRes+False

4n, gs

Comparison

OrEr =186keV,Γ =815keV ???

The 2018 developmentLargermodelspaces(upto𝑁056 = 26) andsmallerℏΩ values:Wegetphaseshiftsatsmallerenergiesandfindthatitisimpossibletofit𝛿 ∼ 𝑘;; atlowenergies

Origin:Hypersphericalpotentialsarelong-ranged:𝑉 ∼ 𝜌"> for3bodies,for4bodies?

Thelong-range𝑉 ∼ 𝜌">(? ) behaviorofhyperspherical potentialsspoilsthephaseshiftsatlowenergiesandresultsinconvergenceproblemsatlarge𝑁012

Suchaslowdecreaseoftheinteractionspoilsthephaseshiftsatlowenergies

The 2018 results with JISP16

0 5 10 15 20 25 30E [MeV]

0

30

60

90

120

δ [d

egre

es] Nmax= 2

4 6 8 10 12 14 16 18

4n, gs

Before2018:convergenceseemstobeachievedat𝑁012 ≤ 18

At2018:convergenceseemstobenotachievedwhenlarger𝑁012 werecalculated

The 2018 results with JISP16At2018:however,theconvergenceseemstobeachievedatthesmallestenergies

The 2018 development• ToresolvethisproblemweusetheJ-matrixinversescatteringapproach(S.A.Zaytsev,Theor.Math.Phys.115,575(1998);AMSetal,PRC70,044005(2004);PRC79,014610(2009));i.e.,weconstructaninteractionasafinitetridiagonalmatrixintheoscillatorbasisdescribingourSS-HORSEhypersphericalphaseshiftsobtainedwithsome𝑁012 valueandsearchnumericallyfortheS-matrixpoles.

• Ideallyweneedtoconstructtheinfinitepotentialmatrixtoguaranteethedescriptionofthelong-range𝜌"> interactiontail,but...

• So,weconstructasetofinteractionmatricesofincreasingrankN,obtainthepolesandextrapolatetheresonantenergiesandwidthssupposingtheirexponentialconvergencewithN.

The 2018 results:inverse scattering phase shifts

The 2018 results:inverse scattering phase shifts

Withlargermatrixoftheinversescatteringpotential(andlargerℏΩvalue)wedescribephaseshiftsinalargerenergyinterval

The 2018 results: resonanceenergy and width for 𝑵𝐦𝐚𝐱 = 𝟐𝟔

The 2018 results: resonance energy and width for various 𝑵𝐦𝐚𝐱

The 2018 results: surprisingly, we have two resonances

The 2018 JISP16 results: extrapolated resonance energies and widths

𝐸 ≈ 0.29MeV, Γ ≈ 0.85MeV 𝐸 ≈ 0.8MeV, Γ ≈ 1.3MeVBeforewehad:

Er =186keV,Γ =815keVEr =844keV,Γ =1.378MeV,Efalse =-55keV

The 2018: extrapolated resonance energies and widths with various interactions

𝐸 ≈ 0.3MeV, Γ ≈ 0.85MeV 𝐸 ≈ 0.8MeV, Γ ≈ 1.3MeVBeforewehadwithJISP16:

Er =186keV,Γ =815keVEr =844keV,Γ =1.378MeV,Efalse =-55keV

Conclusions• Weobtaintwolow-lyingnarrowenoughresonancesintetraneutronwithvariousmoderninteractionsintheminimaldemocraticapproximation

• WeplantoincludemoreHHwith𝐾 = 𝐾0RS, 𝐾 = 𝐾0RS + 2, 𝐾 = 𝐾0RS + 4, etc.,toverifythevalidityoftheminimaldemocraticapproximation

• Clearly,moreexperimentalinformationisdesiredandawaited• Unfortunately,experimentalistsdonnotmeasureS-matrixpolesbutcrosssections;reactionmechanismmaybeveryimportant

• Thankyou!