x(3872) as a four-quark state in a dyson-schwinger/bethe ... · elisabetta prencipe, giessen, 2019....

Christian S. Fischer (University of Gießen) X(3872) as a four-quark state / 15 1

X(3872) as a four-quark state in a Dyson-Schwinger/Bethe-Salpeter approach

Wallbott, Eichmann and CF, PRD100 (2019) 014033, [1905.02615]

Christian S. Fischer (University of Gießen) X(3872) as a four-quark state / 15

Tetraquark candidates with cqqc-content

2

Many new unexpected statesfound: Belle, BABAR, BES, LHCb …


Related to details of underlying QCD forces between quarks and gluons

Tetraquark candidates with cqqc-content

2

Internal structure ??

Many new unexpected statesfound: Belle, BABAR, BES, LHCb …


Tetraquark candidates with cssc-content

3

Elisabetta Prencipe, Giessen, 2019


Tetraquark candidates with qqqq-content

Light scalar mesons:

4

wrong level ordering

R. L. Jaffe, Phys. Rev. D 15, 267 (1977)


Tetraquark candidates with qqqq-content

Light scalar mesons:

4

wrong level ordering correct level ordering

R. L. Jaffe, Phys. Rev. D 15, 267 (1977)


Key elements: quark propagator and interaction kernels

Tetraquarks from the four-body equation

5

Exact equation:

Two-body interactions Three- and four-body interactions

Kvinikhidze & Khvedelidze, Theor. Math. Phys. 90 (1992)

Heupel, Eichmann, CF, PLB 718 (2012) 545-549

Eichmann, CF, Heupel, PLB 753 (2016) 282-287

+ perm.


DSE-approach: truncations

6

Rainbow-Ladder (RL):

[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)



6


[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)

Ansatz



6


[S(p)]�1 = [�ip/ +M(p2)]/Zf (p2)

Beyond the rainbow (BRL):

Williams, CF, Heupel, PRD 93 (2016) 034026CF, Williams, PRL 103 (2009) 122001

Ansatz


Quarks: mass from interaction

7

0.0 1.0 2.0 3.0 4.0p [GeV]

0.0

0.1

0.2

0.3

0.4

M(p

) [G

eV]

Lattice: quenched Lattice: unquenched (N

f=2+1)

DSE: quenchedDSE: unquenched (N

f=2)

DSE: CF, Nickel, Williams, EPJ C 60 (2009) 47 Williams, CF, Heupel, PRD 93 (2016) 034026Lattice: P. O. Bowman, et al PRD 71 (2005) 054507

S(p) = Zf (p2)�ip/ +M(p2)

p2 +M2(p2)



7

0.0 1.0 2.0 3.0 4.0p [GeV]

0.0

0.1

0.2

0.3

0.4

M(p

) [G

eV]


f=2+1)


f=2)


S(p) = Zf (p2)�ip/ +M(p2)

p2 +M2(p2)

‘current quark’:- small mass; non-composite



7

0.0 1.0 2.0 3.0 4.0p [GeV]

0.0

0.1

0.2

0.3

0.4

M(p

) [G

eV]


f=2+1)


f=2)


S(p) = Zf (p2)�ip/ +M(p2)

p2 +M2(p2)

‘current quark’:- small mass; non-composite

‘constituent quark’: large mass; very composite


Light meson spectrum

8

good agreement with experiment in most channels

scalars are special …. see later

For pseudoscalar and vector ground states: RL equally good

Williams, CF, Heupel, PRD93 (2016) 034026

1.5

1.0

0.5

0.0+−0 −−1 ++0 −+1 ++1 −+0−−0 +−1

2PI-3L3PI-3LPDG

[ܩ] Ҟ(1600)ߨ

Ҟ(1260)ҝ(1450)(1450)ߩ

(1300)ߨ

ߨ

ߩ

Ҟ(1235)Ҟ(1400)ߨ

0++<latexit sha1_base64="gRUVSjT194MX4VTJhTORt7x00KA=">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</latexit><latexit sha1_base64="gRUVSjT194MX4VTJhTORt7x00KA=">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</latexit><latexit sha1_base64="gRUVSjT194MX4VTJhTORt7x00KA=">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</latexit><latexit sha1_base64="gRUVSjT194MX4VTJhTORt7x00KA=">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</latexit>

JPC<latexit sha1_base64="TQcwFm2R5jjnrLrC2EP80NmUi5g=">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</latexit><latexit sha1_base64="TQcwFm2R5jjnrLrC2EP80NmUi5g=">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</latexit><latexit sha1_base64="TQcwFm2R5jjnrLrC2EP80NmUi5g=">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</latexit><latexit sha1_base64="TQcwFm2R5jjnrLrC2EP80NmUi5g=">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</latexit>

BRL our results



9

Exact equation:





+ perm.


Two-body interactions: allow for internal clusteringuse RL-approximation


9

Exact equation:





+ perm.


Singlet: Doublet:

Four-body equation: permutations

10

meson poles

diquark pole

S0 = (p2 + q2 + k2)/4<latexit sha1_base64="ljoy0RTTrt4bB28seQlynm6QXCM=">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</latexit>

s ⇠ p2 + q2 � 2k2

a ⇠ q2 � p2<latexit sha1_base64="2Amb4sAYSWINIOgqBmlzevNtvq8=">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</latexit>

p,q,k: relative momenta


Singlet: Doublet:

Four-body equation: permutations

11

meson poles

diquark pole

S0 = (p2 + q2 + k2)/4<latexit sha1_base64="ljoy0RTTrt4bB28seQlynm6QXCM=">AAAEYnicfZNfj9JAEMD3DtQT/4EXn/ShSkzOSLiCd9EXkwuS6MMZQeXukitHttuh3XS7rbtbPNL0w/iqn8h3P4gLFIECTrPJZOa382c7Y0eMSmWav3d2C8UbN2/t3S7duXvv/oNy5eGZDGNBoEdCFooLG0tglENPUcXgIhKAA5vBue2/m/jPRyAkDflXNY6gH2CX0yElWGnToPzoy8A03hoH0VXz5Td9/Kvmi8OjQblq1s2pGOtKI1OqKJPOoFI4tpyQxAFwRRiW8rJhRqqfYKEoYZCWrFhChImPXbh0RjSSHAcg+8n1tIXUeK79jjEMhT5cGVPr8qUEB1KOA1uTAVaezPsmxo0+V+DIo0TWGFZwrUOs1hKr4Zt+QnkUK+BkVscwZoYKjclzGQ4VQBQbawUTQXU3BvGwwETpRy1ZHL6TMAgwdxIrkrpxL00i6+nkO7Rq6SrgZ4C/DehkQGcGbELaGdLejnS8jLE8rIz/xOpmXHc7cjoPdapHysFLWMlyYKiHbvqnkjYW/nsBwNNEuHaamDWzflwz03WqxWJYQLXGJNJyRlvobDO8JSjxP4OT5orSUzBHPlKHU9dT06g5zHH/YYvy8oy9wmwKE+BFNj0xXOE84c4az6CNecJFLZ8YHcEc0ovWyK/VunLWrDde1Zvdo+pJK1u5PfQYPUMHqIFeoxP0AXVQDxGUoB/oJ/pV+FMsFSvF/Rm6u5Pd2UcrUnzyFznveAc=</latexit>

s ⇠ p2 + q2 � 2k2

a ⇠ q2 � p2<latexit sha1_base64="2Amb4sAYSWINIOgqBmlzevNtvq8=">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</latexit>

cu

cu

cu u

c

hadro charmonium molecule

p,q,k: relative momenta


Bound state vs resonance: light scalars

12

qq-state: 600

RL



12

qq-state: 600

RL

1100

beyond RL



12

MTetra[MeV ]0 1190

restricted kinematics

Bound state of four massive quarks

qq-state:

qqqq state:

600

RL

1100

beyond RL

�(S0, s, a,...)<latexit sha1_base64="pRVZGvgabWLVw+SMJE9QXZ0y/BQ=">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</latexit>



12

MTetra[MeV ]0 1190

restricted kinematics

Bound state of four massive quarks

full kinematics

400

Two-pion resonance

qq-state:

qqqq state:

600

RL

1100

beyond RL

�(S0, s, a,...)<latexit sha1_base64="pRVZGvgabWLVw+SMJE9QXZ0y/BQ=">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</latexit>

identify with f0(500) (‘σ-meson’)


Resonance becomes bound state for large mq

Dynamical decision: meson clusters, not diquarksResults:

Mass evolution of tetraquark

13

qualitatively similar to two-body frameworkHeupel, Eichmann, CF, PLB 718 (2012) 545-549

mssss ⇠ 1.5GeV

mcccc ⇠ 5.7GeV

m� ⇠ 350MeV

m ⇠ 750MeV

ma0,f0 ⇠ 1080MeV

3

4

6

5

2

1

00.0

0.5

1.0

1.5

0.02 4 6 8 10 0.2 0.4 0.6 0.8

[ܯ] [ܩ]

[ܩ] ܯ

ߪ

ߢ

ҝ/ҝ

strange

4-body BSE

2-body BSE

2

2

charm

MTetra[GeV]



DD* dominate !

X(3872)

14

0 200 400 600 800mq [MeV]

3

4

5

6

7M

cqqc

[GeV

]DDú

DDú + ÊJ/�DDú + ÊJ/� + ASmu, ms, mc

M cqqc1++ = 3916(74)MeV

M cssc1++ = 4068(61)MeV

<latexit sha1_base64="xL3eDwiEdmQHWptkGqHye0aT3Nk=">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</latexit>


mc fixedmq varied

cccccssccuuc

X(3872)

X(4140)relation to ?


Four-quarks states (so far) dominated by meson-meson clusters(diquarks are almost irrelevant)

Dynamical description of σ: π-π resonance

Dynamical description of X(3872): DD* dominated

Dynamical description of cssc: DsDs* dominated

Summary

15

Internal dynamics very important !!



Outlook:

preliminary results for other channels: 0++, 1+�


Backup Slides

16


Scalar 0++ and axial vector 1+- channels

17

PRELIMINARY !!

Wallbott, Eichmann and CF, in preparation

0++

1+�

mass pattern matches molecule picture ofCleven et al. PRD 92 (2015) 014005:


X(3872) and X(4140)

18

0 200 400 600 800mq [MeV]

3

4

5

6

7

Mcq

qc

[GeV

]

ASDDú

ÊJ/�mÊ + mJ/�mD + mDúmA + mSmu, ms, mc

0 200 400 600 800mq [MeV]

3

4

5

6

7

Mcq

qc

[GeV

]

DDú

DDú + ÊJ/�DDú + ÊJ/� + ASmu, ms, mc


good channels: 1--,2++, 3--,...acceptable channels: 0-+,I++

deficiencies in other channels: missing spin-structure

Charmonium spectrum

19

CF, Kubrak, Williams, EPJA 51 (2015)Hilger et al. PRD 91 (2015)

ηc(2.98)

ηc(2S)(3.63)

χc0(3.41)

X(3.918)

J/ψ(3.09)

ψ(2S)(3.68)

ψ(3.77)

ψ(4.04)ψ(4.15)

X(4.26)

X(4.36)ψ(4.42)

ψ(4.66)

χc1(3.51)

X(3.872)

hc(3.52)χc2(3.55)

X(3.927)

M[G

eV

]

3

3.2

3.4

3.6

3.8

4

JPC

0−+ 1−− 0++ 1+− 1++ 2++ 2−− 2−+ 3−− 3+− 3++ 0−− 0+− 1−+

PDGη = 1.157

Contour border

DSE/BSE


Determine gauge invariant spectrum from underlying, gauge dependent quark/gluon dynamicsNeed approximations for dressed quark-gluon vertex

Theoretical Tools II: DSEs and BSEs

20

Kernel K uniquely related to quark-DSE via axWTI

→Pion is bound state and Goldstone bosonMaris, Roberts, Tandy, PLB 420 (1998) 267

−1=

−1−

π=

πK

x(3872) as a four-quark state in a dyson-schwinger/bethe ... · elisabetta prencipe, giessen, 2019....

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