Variational method for X(3872)
Aaron Park
Woosung Park, Su Houng Lee(Theoretical Nuclear and Hadron Physics Group)

Yonsei UniversityIntroduction
Recently, a number of new states containing a 𝑐𝑐̅ quark pair have been observed.
In particular, interesting channel is 𝐽𝑃 = 1+ .
𝑋 3872

PRL 91, 262001 (2003)

𝑍𝑐 (4430)+

PRL 100, 142001 (2008)

1𝑍 3900

PRL 110, 252001 (2013)

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Hamiltonian: confinement: hyperfine
𝑚𝑢 = 𝑚𝑑 = 340 MeV
𝑚𝑐 = 1858 MeV
𝜅 = 104.5 MeV fm
𝑎0 = 0.0328 (MeV −1 fm)1/2
𝐷 = 913.5 MeV
𝑟1 = 0.4545 fm
𝛼 = 0.0015

1𝑟0 =𝑚𝑖 𝑚𝑗
1+𝛼𝑚𝑖 + 𝑚𝑗
𝑟12

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Mesons
Mtheory

Mtheory (corrected)

Mexp

Δ

𝜋

250.14135.37134.98

0.39

779.7

780.3

775.26

5.04

𝐷

1912.93

1860.55

1864.84-4.29

𝐷∗2030.142012.26

2006.965.3

3071.58

2990.18

2983.66.58

𝐽/𝜓

3121.63096.573096.92

-0.35

𝜌

𝜂𝑐

Unit : MeV

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Coordinate Systems

c 1 𝑞(2)𝑐̅(3)𝑞� (4)D. M. Brink and F. Stancu,Phys. Rev. D 57, 6778 (1998)Asymptotic form
(direct channel)

Asymptotic form
(exchange channel)

Closed form
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Wave function (orbital part)
The most general Gaussian form with L=0 is a function of six scalar quantities
expressed in either of the coordinate systems.

where5J-PARC Hadron workshop

Wave function (spin-color part)[3] ⊗ [3] = [6] ⊕ [3][3] ⊗ [3] = [6] ⊕ [3][3] ⊗ [3] = [3] ⊗ [3] = [8] ⊕ [1][3] ⊗ [3] ⊗ [3] ⊗ [3] = ([6] ⊕ [3]) ⊗ ([6] ⊕ [3])= ([6] ⊗ [6]) ⊕ ([6] ⊗ [3]) ⊕ ([3] ⊗ [6]) ⊕ ([3] ⊗ [3])[3] ⊗ [3] ⊗ [3] ⊗ [3] = ([8] ⊕ [1]) ⊗ ([8] ⊕ [1])= ([8] ⊗ [8]) ⊕ ([8] ⊗ [1]) ⊕ ([1] ⊗ [8]) ⊕ ([1] ⊗ [1])Coordinate 1 :Coordinate 2 :Coordinate 3 :|113 124 >, |813 824 >|114 123 >, |814 823 >|3� 12 334 >, |612 6� 34 >6J-PARC Hadron workshop

Wave function (spin-color part)
By coupling four fermions of spin 1/2, the total spin is S=0, 1, or 2.
For S=1, there are three spin independent states.

χ1 =| V13 P24 >, χ 2 =| V13V24 >, χ 3 =| P13V24 >𝑉: vector

𝑃: pseudoscalarHence, for S=1, one deals with a six-dimensional color-spin basis formed of

113 124 > 𝜒1, 113 124 > 𝜒2, |113 124 > 𝜒3,|813 824 > 𝜒1, |813 824 > 𝜒2, |813 824 > 𝜒3,7

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Hamiltonian (spin matrix)

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Hamiltonian (color matrix)

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X(3872) in (𝐽/𝜓 + 𝜌) configurationc 1 𝑞(2)𝑐̅(3)𝑞� (4)𝑐C=+113124 V13V24813824 V13V24𝑐̅𝑞C=-𝑞�

10813824 P13V24113124 V13 P24813824 V13 P24113124 P13V24

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Charge conjugation = +11J-PARC Hadron workshop

Ground state in (𝐽/𝜓 + 𝜌) configuration
For C=+, ground state is 113124 V13V24𝑐

A11=10.6, A22=2, A33 ≅ 0, A12=0

E=3876.98 MeVThis is the sum of two mesons(J/ψ + 𝜌).

12𝑞

1A111A331A22𝑐̅𝑞�

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Excited state in (𝐽/𝜓 + 𝜌) configuration

For C=+, excited state is0.333 113124 V13V24 + 0.943 813824 V13V24

A11=2.0755, A22=2.0755, A33=4.15, A12=2.0745

𝑐1A111A33

E=3883.71 MeV
It contains octet-octet state.

𝑞

And A33 is not zero.

13

1A22𝑐̅

𝑞�

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X(3872) in (𝐷 + 𝐷 ∗ ) configurationc 1 𝑞(2)𝑐̅(3)𝑞� (4)

𝑐𝑞�

𝑞

𝑐̅

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c 1 𝑞(2)𝑐̅(3)𝑞� (4)Coordinate Transformation
From coordinate transformation,813824 V13 P24813824 V13V24813824 P13V24113124 V13 P24113124 V13V24113124 P13V24

814823 V14 P23814823 V14V23
814823 P14V23114123 V14 P23114123 V14V23114123 P14V23

1( 114123 P14V23 − 114123 V14 P23 )21( 814823 P14V23 − 814823 V14 P23 )2

C=+1( 114123 P14V23 + 114123 V14 P23 )21( 814823 P14V23 + 814823 V14 P23 )2814823 V14V23

C=-114123 V14V23

But some of them are not charge conjugation eigenstate.
So we need to rearrange the basis.
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Corresponding state in (𝐷 + 𝐷∗ ) configuration
For C=+, local minimum state is𝑐

1( 114123 P14V23 − 114123 V14 P23 )
2

1B111
B33

B11=4.15, B22=4.15, B33 ≅ 0, B12=0𝑞�

E=3883.71 MeV∗This is the sum of two mesons(𝐷 + 𝐷 ).

𝑞

1
B 22𝑐̅

 Excited state in (𝐽/𝜓 + 𝜌) configuration is also the sum of two mesons.
16

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Charge conjugation = -17

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Ground state in (𝐽/𝜓 + 𝜌) configuration
For C=-, ground state is 113124 V13 P24𝑐

A11=10.6, A22=6.1, A33 ≅ 0, A12=0
E=3231.95 MeVThis is the sum of two mesons(J/ψ + 𝜋).

18𝑞

1A111A33

1A22𝑐̅𝑞�

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Excited states in (𝐽/𝜓 + 𝜌) configuration

For C=-, Excited state is 113124 P13V24𝑐

A11=13.9, A22=2, A33 ≅ 0, A12=0

E=3770.5 MeVThis is the sum of two mesons(η𝑐 + 𝜌).

19𝑞

1A111A33

1A22𝑐̅𝑞�

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Summary
1. For 𝑐𝑐𝑐̅𝑞� , ground state is just the sum of two meson states in (𝐽/𝜓 + 𝜌)
configuration .

2. Excited state in (𝐽/𝜓 + 𝜌) configuration corresponds toin (𝐷 + 𝐷 ∗ ) configuration.

1(𝐷𝐷 ∗ − 𝐷 ∗ 𝐷)
2

3. There is no tetraquark state near E=3872MeV or 3900MeV in this model.
4. If we could put the additional interaction term, then we might be able toobtain bounded tetraquark state in this nonrelativistic potential model.

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