No

1. RESONANCES
IN MESON-MESON INTERACTION
BY ONE-MESON-EXCHANGE MODEL
Ngo Thi Hong XIEM and Shoji SHINMURA
Graduate School of Engineering
Gifu University
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2. Content
1. Introduction
2. Formalization of the model
3. Results
4. Conclusion
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3. Introduction
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4. Introduction
• Meson-meson interaction is treated in
• Quark Model
• Chiral perturbation
• J.A.Oller, E.Oset, Nucl.Phys A620, 438-456(1997).
• J.A.Oller, E.Oset, Phys. Rev. D 59, 074001 (1999).
• A.G.Nicola and J.R.Pelaez arXiv:help-ph/0109056v2 (2001).
• LQCD
• Silas R. Beane, Thomas C. Luu et al, Phys. Rev D 77, 094507 (2008).
• Meson exchange model
• D. Lohse et al, Nuc. Phys. A516, 513-548 (1990).
• Meson-exchanged models keeps its validity as a suitable
effective description of the h-h interactions in the low-energies
region.
• In meson-meson interaction 𝜋𝜋, 𝜋𝐾 scatterings have been
investigated both in experiments and theories.
• Construct a unified hadron-hadron potential model which
appropriate for all BB, MB and MM interactions.
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5. Introduction
• By 𝑆𝑈(3)-symmetric one-meson exchange mechanisms,
we construct:
• K𝐾 （𝑆 = −2)
• 𝜋𝐾 − 𝜂𝐾 (𝑆 = −1)
• 𝜋𝜋 − 𝐾𝐾 − 𝜂𝜋 − 𝜂𝜂 (𝑆 = 0)
• 𝜋𝐾 − 𝜂𝐾 𝑆 = 1
• 𝐾𝐾 (𝑆 = 2)
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Prog. Theor. Exp. Phys. (2014) 023D04
doi: 10.1093/ptep/ptu001
D. Lohse et al, Nuc. Phys. A516, 513-548 (1990).
Only show the phase shift calculation!
NO eta

6. Formalization of the model
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7. Meson exchange potential
• Feynman Diagram
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Exchange-meson
𝜌, 𝐾∗
, 𝜙, 𝜔, 𝑓2, 𝜀1, 𝑎0, 𝜅
Bare mass.
𝑡-channel 𝑠-channel
- Interaction Lagrangians for scalar, vector and tensor field
- Interaction Hamiltonian 𝑊 = − 𝐿𝑑3
𝑥
- Coupling constant
- Form factors
Prog. Theor. Exp. Phys. (2014) 023D04
doi: 10.1093/ptep/ptu001

8. • SU(3) Symmetric Lagrangians
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ℒ𝑝𝑝𝑠 =
𝑓𝑝𝑝𝑠
𝑚𝜋
𝑻𝒓 𝜕𝜇𝑃𝜕𝜇𝑃𝑆
ℒ𝑝𝑝𝑣 = 𝑔𝑝𝑝𝑣𝑻𝒓 ( 𝜕𝜇𝑃 𝑃 − 𝑃 𝜕𝜇𝑃 )𝑉
ℒ𝑝𝑝𝑡 = 𝑔𝑝𝑝𝑡
2
𝑚𝜋
𝑻𝒓 (𝜕𝜇𝑃𝜕𝜈
𝑃)𝑇𝜇𝜈
 𝑓𝑝𝑝𝑠, 𝑔𝑝𝑝𝑣, 𝑔𝑝𝑝𝑡: coupling constants
 P is 3 × 3 matrix representation of the
pseudo-scalar octet.
 S is 3 × 3 matrix representation of the
scalar octet.
 V is 3 × 3 matrix representation of the
vector octet.
• 𝑃8 =
𝜋0
2
+
𝜂8
6
𝜋+
𝐾+
𝜋−
−
𝜋0
2
+
𝜂8
6
𝐾0
𝐾−
𝐾0
−
2
3
𝜂8
;
𝑃1 = 𝜂1
• 𝑉8 =
𝜌0
2
+
𝜙
6
𝜌+
𝐾∗+
𝜌− −
𝜌0
2
+
𝜙
6
𝐾∗0
𝐾∗−
𝐾∗0
−
2
3
𝜙
;
𝑉1 = 𝜔
• 𝑆8 =
𝑎0
2
+
𝑓0
6
𝑎+ 𝜅+
𝑎−
−
𝑎0
2
+
𝜙
6
𝜅0
𝜅− 𝜅0 −
2
3
𝜙
;
𝑆1 = 𝜎
Coupling constants

9. Form Factors
• Monopole Type
• 𝑡-channel
• 𝐹(𝑡) (𝑞𝛼
2 ) =
Λ2−𝑚𝛼
2
(Λ2+𝑞𝛼
2 )
• 𝑠-channel
• 𝐹 𝑠
(𝜔𝑝
2
) =
Λ2+𝑚𝛼
2
(Λ2+𝜔𝑝
2 )
• Forth-order (𝑠-channel)
• 𝐹4 (𝜔𝑝
2
) =
Λ4+𝑚𝛼
4
Λ4+𝜔𝑝
4
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Λ : cut-off parameters
𝑞𝛼: momentum
𝑚𝛼: mass of exchanged mesons
𝜔𝛼 : total energy

10. T-matrix and S-matrix
• Using the quasi-potential V and solving the L-S Equation,
we obtain the 𝑇- matrix :
𝑇 = 𝑉 + 𝑉𝐺𝑉
• S-matrix is given by
𝑆 𝑧 =
𝜂 𝑧 𝑒2𝑖𝛿11(𝑧) 𝑖 1 − 𝜂2(𝑧)𝑒2𝑖𝛿12(𝑧)
𝑖 1 − 𝜂2(𝑧)𝑒2𝑖𝛿21 𝑧
𝜂 𝑧 𝑒2𝑖𝛿22(𝑧)
• 𝜂 𝑧 is the inelasticity; 𝛿𝑖𝑗 the phase shift
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11. Resonances
Poles with s-channel meson exchange
• 𝑉 = 𝑉𝑡 + 𝑉
𝑠
• 𝑉
𝑠 contains s-channel diagrams
• 𝑉𝑡 contains t-channel diagrams
• 𝑉
𝑠 𝑝′, 𝑝 = 𝑖 𝛾𝑖 𝑝′ Δi P 𝛾𝑖(𝑝) with bare mass and bare coupling constant
• 𝑇 = 𝑇𝑡 + 𝑇𝑠
• 𝑇𝑠 = 𝛾∗ 𝑝′ Δ∗ P 𝛾∗(p)
• Δ∗ 𝑃 =
Δ 𝑃
1−Δ 𝑃 Σ(p)
: dressed propagator
• Σ 𝑝 : self-energy
• 𝛾∗(𝑝) : dressed vertex
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*This resonance renormalization follow: H.Polinder an Th. A. Rijken Phys. Rev. C 72, 065211 (2005)
Pure Dynamical Poles (without s-channel exchange)

12. Results
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13. Parameters
• 24 parameters
• All are free
parameters.
• Fitting with a large
of experimental
data is difficult,
special for both
well fitting of (𝐼 =
0, 𝐽 = 0) and (𝐼 =
1
2
, 𝐽 = 0).
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Monopole
𝑔𝜋𝜋𝜌 0.50101
𝑔𝜋𝜋𝑓2
0.02843626
𝑔𝜋𝜋𝜀1
0.01620557
𝑔𝜋𝜋𝜅 0.191683 × 10−3
𝑔𝐾𝐾𝑎0
0.0408802
Λ𝜋𝜋𝜌 2896.41242
Λ𝜋𝐾𝐾∗ 2301.47525
Λ𝐾𝐾𝜌 3923.01907
Λ𝐾𝐾𝜔,𝜙 4520.8249
Λ𝜋𝜋𝜌;𝑠 2757.75065
Λ𝜋𝜋𝑓2;𝑠 1481.43534
Λ𝜋𝜋𝜀1;𝑠 1159.73371
Λ𝜋𝐾𝐾∗;𝑠 4061.81573
Λ𝜋𝐾𝜅;𝑠 3622.67379
𝑴𝟎 Mono.
𝜌 1126.66877
𝜀1 2112.07666
𝐾∗ 1403.13526
𝜅 1522.01264
𝑓2 1367.63563
𝑎𝑜 1235.73076
𝜙 1150.23241
Mono.
Λ𝜂𝐾𝐾∗ 864.0841
Λ𝐾𝐾𝑎0
1233.61523
Λ𝐾𝐾𝑎0
2137.02377

14. Phase shift results
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𝛿𝐽
𝐼
; 𝐼: isospin. 𝐽: total angular momentum

15. Phase shift results
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𝛿𝐽
𝐼
; 𝐼: isospin. 𝐽: total angular momentum

16. Resonances results
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800
850
900
950
1000
-60
-50
-40
-30
-20
-10
0
0
2
4
6
8
10
12
14
16
18
"l-dd-a0" u 1:2:5
900
920
940
960
980
1000
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
2
4
6
8
10
12
14
16
"l-dd-f0" u 1:2:5
1200
1250
1300
1350
1400
-200
-180
-160
-140
-120
-100
-80
-60
-40
0
5
10
15
20
25
30
35
40
45
50
"l-dd-f2" u 1:2:5
600
650
700
750
800
850
900
-300
-250
-200
-150
-100
-50
0
5
10
15
20
25
30
35
"l-dd-rho" u 1:2:5
𝑎0(980)
𝑓0(980)
f2(1270)
𝜌(770)
1010
1015
1020
1025
1030
1035
1040
-6
-5
-4
-3
-2
-1
0
0
2
4
6
8
10
12
14
"l-dd-phi" u 1:2:5
𝜙(1020)

17. Resonances results
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1300
1350
1400
1450
1500
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
0
5
10
15
20
25
30
35
40
45
"l-dd-kap14" u 1:2:5
500
550
600
650
700
750
800
-400 -350 -300 -250 -200 -150 -100 -50
0
20
40
60
80
100
120
"l-dd-kap7" u 1:2:5
840
860
880
900
920
940
960
980
-55
-50
-45
-40
-35
-30
-25
-20
-15
0
5
10
15
20
25
30
35
40
45
50
"l-dd-Kst" u 1:2:5
300
350
400
450
500
550
600
-600
-550
-500
-450
-400
-350
-300
-250
-200
0
20
40
60
80
100
120
"l-dd-sigm" u 1:2:5
𝜅(1430)
𝜅(700)
𝐾∗(892)
𝜎(500)

18. Summary Table of Pole Positions
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Exp. Data (PDG*) Monopole
𝑓0(980) 990 ± 20 − 𝑖(40 to 100) 980 − 𝑖70
𝜎1(500) 400 − 550 − 𝑖(200 − 350) 556 − 𝑖382
𝜎2 ------- 420 − 𝑖567
𝑎0(980) 980 ± 20 − 𝑖(50 to 100) 860, −𝑖20
𝜌(770) 775.26 ± 0.25 − 𝑖(147.8 ± 0.9) 790 − 𝑖50
𝜙(1020) 1019.461 ± 0.019 − 𝑖(4.266 ± 0.031) 1020 − 𝑖2
𝑓2(1270) 1275.1 ± 1.2 − 𝑖(184.2−2.4
+4.0
) 1280 − 𝑖100
𝜅(1430) (1425 ± 50) − 𝑖(270 ± 80) 1440 − 𝑖50
𝜅(700) 682 ± 29 − 𝑖(547 ± 24) 640 − 𝑖200
𝐾∗(892) 891.66 ± 0.26 − 𝑖(50.8 ± 0.9) [910, −20]
※𝑀 − 𝑖
𝚪
𝟐
*K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).

19. 𝜎 500 and 𝜅(800)
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*The experimental data of sigma resonances are from Muramatsu, Aitala,Ishida
(97,00B,01), Alekseev (98,99), Troyan, Svec and Augustin.
* The experimental data of kappa resonances are from Ablikim(06C,10E,11B),
Descotes-g, Aitala, Bugg, Bonvicini, Link, Cawlfield, Zhou, Pelaez,Zheng and Ishida.
𝑅𝑒 𝑠
𝐼𝑚
𝑠
𝑅𝑒 𝑠
𝐼𝑚
𝑠

20. Summary
• Our calculations are well reproduce the phase shift, cross-section and
pole position in the meson-meson interaction by the meson exchange
model.
• 𝜋𝜋 − 𝐾𝐾𝑏𝑎𝑟
− 𝜋𝜂 − 𝜂𝜂
• 𝛿0
0
; 𝛿0
1
; 𝛿1
0
; 𝛿1
1
; 𝛿2
0
• 𝜎 600 , 𝑓0 980 , 𝜌 770 , 𝑎0 980 , 𝜙(1020)
• 𝜋𝐾 − 𝜂𝐾
• 𝛿0
1
2
, 𝛿1
1
2
• 𝜅 700 , 𝜅 1430 , 𝐾∗
(982)
• The shown results can prove the suitable effective of meson-exchange
model on well-produce the interaction of mesons.
• 𝜅 700 ; 𝜎(600) poles with broad width
• FURTURE PLAN
• Refine our model by reasonable bare mass fitting-parameter for (𝛿0
0
; 𝛿0
1
2
, 𝛿1
1
2
)
• Calculation with the Gaussian F.F
• Solve the dynamical 3 body problems to study the existence and properties of
the resonances in the hadronic system.
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21. THANK YOU FOR YOUR ATTENTION
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