multi-quark hadrons in the quark model

Multi-Quark Hadrons in the Quark Model

Makoto Oka Advanced Science Research Center, JAEA

March 2019, YITP

Makoto Oka (ASRC, JAEA)


Contents1. Introduction

2. Dibaryon d*= DΔ = ΔΔ (I=0, J=3)

3. Pentaquark Pcs (ccbaruds)

4. Pentaquark Pc (ccbaruud)

5. Conclusion

2


IntroductionHadrons are made of quarks bound by gluonic forces. Dynamics of quarks and gluons are highly non-perturbative. Low-lying hadrons are described in terms (only) of valence, or constituent quarks. No hadron with constituent gluon is confirmed. The constituent quark seems to exhibit “non-relativistic” degrees of freedom even for light (chiral) quarks, i.e., orbital motion, spin, flavor and color. The short-range interactions among the quarks are described by spin-color-flavor dependent two-body forces. The color Coulomb interaction and the color-magnetic interaction are dominant. At long distances, the color confinement is to be realized, but no consensus is reached for how the confinement is described by potentials. The pair production/annihilation of quarks are not under control in the potential model.

3

Color Interactions of Quarks


Potential Quark Model

Linear confinement with color Casimir dependence

OgE: Electric Coulomb interaction

OgE: Magnetic spin-spin interaction

Non-relativistic quarks with

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�a

2�µ

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Spin dependenceSpin-spin interaction aka Color-Magnetic Interaction (CMI)

12

C2[SU(g)]([f1, f2, . . . , fg]) =�

i

fi(fi � 2i + g + 1)� N2

g

VCMI = ��

i<j

(⇧⇤i · ⇧⇤j)(⇧⌅i · ⇧⌅j)f(rij) f(rij) ⇥ ⇥(rij)VCMI = ��

i<j

(⇧⇤i · ⇧⇤j)(⇧⌅i · ⇧⌅j)f(rij) f(rij) ⇥ ⇥(rij)

�VCMI⇥(0s)N = � �f(r)⇥0s �cm

prefers symmetric color-spin states

�CM ⇥ ⇤��

i<j

(⇤�i · ⇤�j)(⇤⇥i · ⇤⇥j)⌅color singlet

= 8N � 2C2[SU(6)] +43S(S + 1)�CM = 8N � 2C2[SU(6)cs] +

43S(S + 1) + C2[SU(3)c]

C2[singlet] = 0

�CM = V0�CM


Spin dependenceCMI prefers color-spin symmetric states, i.e. flavor antisymmetric states.

13

�CM(10)��CM(8) = 8� (�8) = 16

�CM(H)� 2�CM(⇥) = �24� 2(�8) = �8

�CM(D�)� 2�CM(�) = 16� 2⇥ 8 = 0

�CM = 8N � 2C2[SU(6)cs] +43S(S + 1) + C2[SU(3)c]

M(�)�M(N) = 16V0 � 300 MeV

V0 � 300/16 � 19 MeV

DΔ (ΔΔ, I=0, S=3)

H (ΛΛ+NΞ+ΣΣ, S=0)

From ABC to DΔ


ABC effectA. Abashian, N.E. Booth, K.M. Crowe Possible anomaly in meson production in p+d collisions, PRL 5, 258 (1960) Anomaly in meson production in p+d collisions, PRL 7, 35 (1961)

15

phase-space volume adjusted to the data

phase space + π-π S-wave enhancement for aI=0= 3.9 fm (exp.~0.2fm)

p + d → 3He + π-π (I=0) Ep=624-743 MeV


ABC effectA. Abashian, N.E. Booth, K.M. Crowe Possible anomaly in meson production in p+d collisions, PRL 5, 258 (1960) Anomaly in meson production in p+d collisions, PRL 7, 35 (1961)

Low mass ππ enhancement observed in the inclusive production,p + d → 3He + X, 3H + X (Ep=624-743 MeV, Berkeley) X = ππ (I=0) for 3He ππ (I=1) for 3He and 3H As the beam energies correspond to ΔΔ excitation in nucleus, the ππ enhancement is attributed to the ΔΔ excitations. The π0π0 enhancement is much larger than estimate in ΔΔ production by Alvarez-Ruso, Oset, Hernandez, NPA 633 (1998) 519. An s-channel resonance at mR~2.36 GeV may explain the results. ABC effect in basic double-pionic fusion: A new resonance? WASA@COSY, PRL 106, 242302 (2011) p+d → d+π0+π0+pspectator at Tp=1.0, 1.2, 1.4 GeV

16


d* resonanceWASA@COSY, PRL 106, 242302 (2011) p + n(d) → d + π0 + π0 (+pspectator) at Tp=1.0, 1.2, 1.4 GeV

A di-baryon resonance, d* (I=0, Jπ=3+) (in pn and ΔΔ) is suggested.

17

ΔΔ contributions

d* : s-channel resonance mR=2.37 GeV and Γ=68 MeV


d* resonanceWASA@COSY+SAID, PRL 112, 202301 (2014)Evidence for a new resonance from polarized n-p scattering d(↑) + p → np + pspectator

np analyzing power, Ay(θ), at Tn=1.108-1.197 GeV A phase shift analysis of 3D3 (3+) amplitudes shows a narrow resonance at M=2380 MeV and Γ~70 MeV.

18


d* resonanceWASA@COSY+SAID, PRL 112, 202301 (2014)Evidence for a new resonance from polarized n-p scattering d(↑) + p → np + pspectator

np analyzing power, Ay(θ), at Tn=1.108-1.197 GeV A phase shift analysis of 3D3 (3+) amplitudes shows a narrow resonance at M=2380 MeV and Γ~70 MeV.

19


DΔ (ΔΔ)I=0 di-baryon

20


Di-baryonby M.O. (1980)

21

Pauli Allowed channels

Fermion statistics for Quarks

�CM � ��

i<j

(�ai �a

j )(�ki �k

j ) = 8n� 2C6 +43S(S + 1)

4C6

C6 � C2[SU(6)cs] =�

i

fi(fi � 2i + 7)� n2

6

R.L. Jaffe, PRL 38 (1977) 195

��(I = 0, S = 3) V = V0 � 0

��(I = 3, S = 0) V = V0 � 32

H = ��(I = S = 0) V = V0 � (�8)

V0 = 300/16 � 18(MeV)

�CM(�) = +8�CM(N) = �8

Color Magnetic Interaction


Spin dependenceCMI prefers color-spin symmetric states, i.e. flavor antisymmetric states.

23

�CM(10)��CM(8) = 8� (�8) = 16

�CM(H)� 2�CM(⇥) = �24� 2(�8) = �8

�CM(D�)� 2�CM(�) = 16� 2⇥ 8 = 0

�CM = 8N � 2C2[SU(6)cs] +43S(S + 1) + C2[SU(3)c]

M(�)�M(N) = 16V0 � 300 MeV

V0 � 300/16 � 19 MeV

DΔ (ΔΔ, I=0, S=3)

H (ΛΛ+NΞ+ΣΣ, S=0)


DΔ (ΔΔ)I=0 di-baryonS=3, I=0 (Δ2) bound state

Relative wave function

No repulsive core100 200 MeV

7S3 phase shift

24


Di-lepton enhancementDLS puzzle: low energy di-lepton enhancement in pA, AA collisions @ E~1GeV/A originally found in DLS detector at Bevalac PRL 79, 1229 (1997), PRC 57, 1865 (1998) confirmed by HADES@Darmstadt PLB 690 (2010) 118

25


Di-lepton enhancement

26

DLS PRC 57, 1865 (1998)

pd/pp ratio


Di-lepton enhancement

27

PLB 690 (2010) 118

Enhancement at Mee ~ 0.3~0.6 GeV


Di-lepton enhancementThe DΔ formation in the pn scattering may be the origin. M. Bashkanov, H. Clement, EPJ A50 (2014) 107 On a possible explanation of the DLS puzzle

The isospin factor for the (ππ)I=0 formation from pp vanishes.

So the enhancement is attributed to the DΔ resonance in p+n, which gives relevant energy dependence.

28

NN � �� NN�� NN�� NNe+e�

(��)I=1 � �0 � ��

��

�

1/2 1 3/21/2 1 3/2INN 1 Itot

��

�

��

�

1/2 1 3/21/2 1 3/21 1 1

��

� = 0

Hidden-Charm Pentaquarks


Exp: Hidden-Charm Pentaquark Pc

Pc → J/ψ+p (ccuud)LHCb (PRL 115 (2015) 07201) found two penta-quark states with hidden cc.

30

Pc(4450) (5/2-)

Pc(4380) (3/2+)


Hidden-Charm PentaquarksConstituent quark model analyses

“Study of qqq cbar c five quark system with three kinds of quark-quark hyperfine interaction”,S.G. Yuan, K.W. Wei, J. He, H.S. Xu, B.S. Zou, Eur. Phys. J. A 48 (2012) 61. “The hidden charm pentaquarks are the hidden color-octet uud baryons?”Sachiko Takeuchi, Makoto Takizawa, Phys. Lett. B 764 (2017) 254. “Flavor-singlet hidden charm pentaquark”Yoya Irie, MO, Shigehiro Yasui, Phys. Rev. D 97 (2018) 034006 “Quark model estimate of hidden-charm pentaquark resonances”E. Hiyama, A. Hosaka, MO, J. M. Richard, Phys. Rev. C98 (2018) 045208.

31


Hidden-Charm Pentaquarkscolor 1 cc56 = (8, 1/2) + (10, 3/2) (8,1/2) ΔCM = -8 Pc= cc uud = ηc or J/ψ + p (10,3/2) ΔCM = 8

color 8 cc70 = (1, 1/2) + (8, 1/2) + (8, 3/2) + (10, 1/2) (1,1/2) ΔCM = -14 Pcs= cc uds = η8/ψ8+ Λ8(singlet) (8,1/2) ΔCM = -2 η8/ψ8+ N8The most favored state with cc by CMI may not be J/ψ + p.

Pcs family (I=0, Str= -1) (cc)8,J=1 + (uds)8, J=1/2 Jπ＝1/2-, 3/2- (cc)8,J=0 + (uds)8, J=1/2 Jπ＝1/2-

32

�CM ⇥ ⇤��

i<j

(⇤�i · ⇤�j)(⇤⇥i · ⇤⇥j)⌅color singlet

= 8N � 2C2[SU(6)] +43S(S + 1)

u du

cc


Flavor Singlet Pentaquark Pcs

Potential Quark Model

Linear confinement with color Casimir dependence

Coulomb electric interaction from one-gluon-exchange

Color magnetic spin-spin interaction from OGE

Non-relativistic quarks with

33

m(u, d) = 313 MeV m(s) = 522 MeV


Instanton-Light quark coupling gives an effective interaction


The3-bodyIIIisrepulsiveinflavorsingletu-d-ssystems

2-bodyIII

3-bodyinteraction

2-bodyinteraction

[3]G. ‘t Hooft, Phys. Rev. Lett 37 (1976) 8　　　　　[4]G. ‘t Hooft, Phys. Rev. D14 (1976) 3432[5]S. Takeuchi, M. Oka, Nuclear Physics A547 (1992) 283c-288c 34



Pcs family (I=0, Str= -1)

35

u ds

cc

8*

Energy Spectrum

Thelowestenergystateis8’.

Theinstantoninducedinteractionlowersthemassesbyabout80MeV.　　

Two1/2-statesmixbytheCMI.8 8’

Y. Irie, MO, S. Yasui.Phys. Rev. D 97 (2018) 034006

Avariationalmethodisusedforaqualitativeevaluationofthespectrum.

8*

Energy Spectrum

Thelowestenergystateis8’.

Theinstantoninducedinteractionlowersthemassesbyabout80MeV.　　

Two1/2-statesmixbytheCMI.

8 8’


Avariationalmethodisusedforaqualitativeevaluationofthespectrum.

8*

Decays

8 8’FlavorSU(3)：suppressed

(barely)allowed8* :D-wavedecay

8* :S-wavedecayWithInstantons→forbidden


nocharge

negativecharge

Production

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Pcs<latexit sha1_base64="batGNA8MHLZAu9BsmBiqKeLRLwM=">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</latexit><latexit sha1_base64="JVBE7hu+5gIXidMhv7mMV+vY+Vk=">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</latexit><latexit sha1_base64="JVBE7hu+5gIXidMhv7mMV+vY+Vk=">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</latexit><latexit sha1_base64="kIIjPuMvJf4OFNorQ3x+Gx4tfdU=">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</latexit>


Pentaquark Pc

E. Hiyama, A. Hosaka, MO, J. M. Richard, “Quark model estimate of hidden-charm pentaquark resonances”, Phys. Rev. C98 (2018) 045208.

Complete calculation of the Pentaquark ccuud (I=1/2, JP= 1/2-, 3/2-) in the NR potential quark model with two-body confining potential,

39

Vconf = �a�

all(ij)pairs

(�(i)c · �(j)

c ) rpij

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B. Silvestre-Brac, Few Body Syst. 20 (1996) 1-25

(p=1 or 2/3)


Variational method scattering channels confined channels

Real scaling method scale the relative coordinate in channels C=1, 2 by R(1,2) → α R(1,2) (α ~ 1.0 – 1.5) As the ranges of the variational basis functions are finite, all the eigenstates are discrete. Under the real scaling, the energy of the scattering states will change (decrease) towards the scattering threshold, but the compact state will stay.

40

r(1)ρ (1)

R (1)s(1)

C=1

q

qq

c

c

1 4

5

r(2 ) ρR (2) s (2)(2)

1

2

3

4 5

q

qc

q

c

C=2

r(3)R (3)

ρ (3)s(3)

1

2

3

45

q

q

q

cc

C=3

r(4) R (4) ρ(4)s(4)

1

2

3

45

q

q

q

cc

C=4

2 3

Pentaquark Pc


Energy levels before coupling to the scattering channels

41

4000

4100

4200

4300

4400

4500

4600

4700

MeV

4119

4236

4497

458145934629

46794708

4221

4577

4617

4700

3900

48004748

4711

4900

48364840

4896

JP =1 2 JP = 3 2

c* +D* (4587)+N (4584)

c +D* (4505)

c +D* (4323)

c +D (4353)

c +D (4171)

J +N (4040)

c +N (3920)

Thresholds

c +N (4555)

Jπ= 1/2- 3/2-

J/� + N<latexit sha1_base64="H0phQDUQOK1/S69w4nhQGI/u53Q=">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</latexit><latexit sha1_base64="TI/SXk9FJ6zWHgEuTsv4a2UUHT8=">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</latexit><latexit sha1_base64="TI/SXk9FJ6zWHgEuTsv4a2UUHT8=">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</latexit><latexit sha1_base64="TYkVxDx1/2RjOFwz9KIHStQbGrk=">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</latexit>

�c + N<latexit sha1_base64="lEm83f5224s8Dxp7tUfOLLGBJUA=">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</latexit><latexit sha1_base64="H9d/OXLkQZyNXjNyyKVj7OxA4yE=">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</latexit><latexit sha1_base64="H9d/OXLkQZyNXjNyyKVj7OxA4yE=">AAACcXicjVE9S8NQFD2N3/WjVRfFwaBUBKHcVq3VqeDiJH5VCyoliU8NpklIXgtanAVHFwddFBzUn+HiLDj4E8SxgouDN6kiDoo3JO+88+45ufc+3bVMXxI9RZSGxqbmlta2aHtHZ1cs3t2z6jtlzxB5w7Ecr6BrvrBMW+SlKS1RcD2hlXRLrOl7s8H5WkV4vunYK3LfFZslbcc2t01Dk0wVNoTUisbYfDE+TMnpbGZ8PKVSkjLpTHoyANM0kZ1SU0kKYzjXf1R0HgZrC078BhvYggMDZZQgYEMytqDB52cdKRBc5jZRZc5jZIbnAoeIsrbMWYIzNGb3+LvDu/VP1uZ94OmHaoP/YvHrsVJFgh7pmmp0T7f0TO/slfjFrRq6BNXs86rX1cItxo77lt/+oSvxKrH7rftDE1QusY1sWLHJHbghE/Ri1B0qB6e15ZmlRHWELumFu7igJ7rjPuzKq3G1KJbOEOVr+Jq1+jtYTSdTjBf5PnKoRysGMIRRnvoUcpjDAvLhtE9whvNITelXVGWonqpEPjW9+BHK2Ad6L5EC</latexit><latexit sha1_base64="UcoeJTr0QodAvva8Co/1pWh2PDw=">AAACcXicjVE9S8NQFD2N3/Wr6qK4BEtFEMptq211Krg4iV/VgkpJ4lODaRKS14IW/4Cji4MuCg7qz3DxDzj4E8SxgouDN2lFHBRvSN55591zcu99umuZviR6jiht7R2dXd090d6+/oHB2NDwhu9UPUMUDcdyvJKu+cIybVGUprREyfWEVtEtsakfLgTnmzXh+aZjr8sjV+xUtH3b3DMNTTJV2hZSKxvTS+VYnJJz+Wwmk1IpSdl0Nj0bgDmayefUVJLCiKMVy07sDtvYhQMDVVQgYEMytqDB52cLKRBc5nZQZ85jZIbnAieIsrbKWYIzNGYP+bvPu60Wa/M+8PRDtcF/sfj1WKkiQU90Sw16pHt6oQ/2SvziVg9dgmqOeNWbauGWB09H197/oavwKnHwrftDE1QusYd8WLHJHbghE/RiNB1qx+eNtfnVRH2SrumVu7iiZ3rgPuzam3GzIlYvEOVr+Jq1+jvYSCdTjFcoXii0LqQb45jAFE89hwIWsYxiOO0zXOAy0lDGFFWZaKYqkZZmBD9Cmf4E3qONcA==</latexit>

many thresholds

Pc(4380)<latexit sha1_base64="KNS5XJ2wlFSQQq3SSSrhOim6HBE=">AAACcnicjVE9T+NAEH0xcAeBu+SOBkFjiHLiJBRNyAUsqkg0lAmQAAIU2WYTLPwl24kEEX+AloICKECiAH7GNfcHKOgoQZREoqFg7IAQBejG8u7bt/Pezuxqrmn4AdF1TOrq7vnytbcv3j/w7Xsi+eNnxXcani7KumM63rKm+sI0bFEOjMAUy64nVEszxZK2NRvuLzWF5xuOvRhsu2LdUuu2UTN0NWBqpVjVx//kFPpdTaYoM52fUqZyMmWyuTAY5BVlMpeXsxmKIlWQS7c3AIpO8gJr2IADHQ1YELARMDahwudvFVkQXObW0WLOY2RE+wK7iLO2wVmCM1Rmt3is82r1hbV5HXr6kVrnU0z+PVbKSNMVndMD/aNLuqMn9kp/4NaKXMJqtnnWOmrhVhN7QwuP/6GzeA6w+ab7RBNWHqAGJarY4A7ciAl70TsOzZ2Dh4WZ+XTrF53SPXdxQtf0l/uwm239rCTmDxHnZ3i9a/ljUJnMZBmX+D0K6EQvRjCGcb71aRQwhyLKfK6FfRzhONaWhqVRKdVJlWIvmkG8C2niGV8Lj1c=</latexit><latexit sha1_base64="zlmtHJM2FvNggbH1N4Q/L4pw+Vo=">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</latexit><latexit sha1_base64="zlmtHJM2FvNggbH1N4Q/L4pw+Vo=">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</latexit><latexit sha1_base64="iOkfQ0CoVuuAKKoMqglXxc3PX/U=">AAACcnicjVFLLwNRFP46XlWvYiNsSlMhkea0VSZWTWws61GPIM3MuJiYV2amTWj8AVsLCyxILPAzbPwBCz9BLElsLJwZI2JBnMnc+93v3u+759yjOobu+USPMampuaW1Ld6e6Ojs6u5J9vYte3bN1URFsw3bXVUVTxi6JSq+7hti1XGFYqqGWFH3ZoP9lbpwPd22lvx9R2yayo6lb+ua4jO1Vq5qY5MFmcaryTRlp4tT8lQhRdlcIQgGRVnOF4qpXJbCSCOKsp28wQa2YENDDSYELPiMDSjw+FtHDgSHuU00mHMZ6eG+wCESrK3xKcEnFGb3eNzh1XrEWrwOPL1QrfEtBv8uK1PI0ANd0wvd0y090Tt7ZX5xa4QuQTb7PKufauFUe44GFt/+oTN59rH7rftDE2TuYxtymLHOFTghE9SifTrUD05eFmcWMo1RuqRnruKCHumO67Dqr9rVvFg4RYLb8PXWqd/Bcj6bYzxP6VIpakgcQxjBGL/6NEqYQxkVvtfEMc5wHnuVBqVhKeqeFIs0/fgR0sQHx8yM3g==</latexit>

Pc(4450)<latexit sha1_base64="Ia2kc5me7nh7k+wc/tm0QDRi2sk=">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</latexit><latexit sha1_base64="wBBcm6SrWf5KApdcIRO+CGxOr/k=">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</latexit><latexit sha1_base64="wBBcm6SrWf5KApdcIRO+CGxOr/k=">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</latexit><latexit sha1_base64="RjhheNh7M//ks9CsOTzHVwHnSU0=">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</latexit>

Pentaquark Pc

(α=1)


Real scaling around 4100-4300 MeV for 1/2-

Coupling of ηcN and J/ψN is weak because of the HQ spin symmetry

42

4000

4100

4200

4300

4400

4500

4600

4700

MeV

4119

4236

4497

458145934629

46794708

4221

4577

4617

4700

3900

48004748

4711

4900

48364840

4896

JP =1 2 JP = 3 2

c* +D* (4587)+N (4584)

c +D* (4505)

c +D* (4323)

c +D (4353)

c +D (4171)

J +N (4040)

c +N (3920)

Thresholds

c +N (4555)

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Pentaquark Pc


At the end, most of the states are gone with the scattering channels and there remains a narrow 1/2- state at E=4690 MeV.

43

Full calculationFree solutions

Pentaquark Pc

Estimated decay width from the level crossing is about 40 MeV.


Pentaquark Pc

Another resonance structure at E=4920 MeV for Jπ=3/2-. No other (sharp) resonance is found for 1/2+-, 3/2+-, 5/2+- channels.

Why do we not reproduce the LHCb pentaquark(s)? Choice 1: This model does not predict loosely-bound hadronic molecules, because it does not induce any long-range interaction between color singlet hadrons, that is, no meson exchange force. Thus the possibility remains that the LHCb pentaquark(s) are Σc + D* molecular resonance state(s). (Not very interesting?) Choice 2: This model does not have a correct (confinement) potential, so that it cannot be applied to the pentaquark systems.

44


ConclusionI have shown three cases of analyses in the constituent quark model. So far, the available spectroscopy data (including lattice QCD results) are consistent with the model whenever the color-spin interaction plays the dominant role. It is not confirmed (or may be even doubtful) that the confinement by the color-dependent two-body potential is correct. If the Pc pentaquark is a compact 5-quark state, then the model is not good enough to reproduce such a state.

45

multi-quark hadrons in the quark model

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