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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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Content
Introduction
2. Formalization of the model
3. Results
4. Conclusion
1.
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Introduction
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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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Introduction
• By 𝑆𝑈(3)-symmetric one-meson exchange mechanisms,
we construct:
• K𝐾 （𝑆 = −2)
• 𝜋𝐾 − 𝜂 𝐾 (𝑆 = −1)
• 𝜋𝜋 − 𝐾 𝐾 − 𝜂𝜋 − 𝜂𝜂 (𝑆 = 0)
• 𝜋𝐾 − 𝜂𝐾 𝑆 = 1
• 𝐾𝐾 (𝑆 = 2)
D. Lohse et al, Nuc. Phys. A516, 513-548 (1990).
Only show the phase shift calculation!
NO eta
Prog. Theor. Exp. Phys. (2014) 023D04
doi: 10.1093/ptep/ptu001
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Formalization of the model
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Meson exchange potential
• Feynman Diagram
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 = − 𝐿𝑑 3 𝑥
Coupling constant
Prog. Theor. Exp. Phys. (2014) 023D04
Form factors
doi: 10.1093/ptep/ptu001
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Coupling constants
• SU(3) Symmetric Lagrangians
ℒ𝑝𝑝𝑠 =
𝑓𝑝𝑝𝑠
𝑚𝜋
𝑻𝒓 𝜕𝜇 𝑃𝜕𝜇 𝑃𝑆
ℒ𝑝𝑝𝑣 = 𝑔𝑝𝑝𝑣 𝑻𝒓 ( 𝜕𝜇 𝑃 𝑃 − 𝑃 𝜕𝜇 𝑃 )𝑉
2
ℒ𝑝𝑝𝑡 = 𝑔𝑝𝑝𝑡
𝑻𝒓 (𝜕𝜇 𝑃𝜕 𝜈 𝑃)𝑇𝜇𝜈
𝑚𝜋
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
𝜂8
+
2
6
𝜋−
𝜋+
−
𝐾+
𝜋0
𝜂8
+
2
6
𝐾0
𝐾0
𝜌0
𝜙
+
2
6
𝜌+
𝐾 ∗+
𝜌−
𝜌0
𝜙
− +
2
6
𝐾 ∗0
𝐾 ∗−
𝐾 ∗0
𝑎0
𝑓
+ 0
2
6
𝑎+
𝑃1 = 𝜂1
• 𝑉8 =
−
𝑉1 = 𝜔
• 𝑆8 =
𝑎−
𝜅−
𝑆1 = 𝜎
2
𝜂
3 8
𝐾−
−
−
;
2
𝜙
3
𝜅+
𝑎0
𝜙
+
2
6
𝜅0
;
𝜅0
−
2
𝜙
3
;
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Form Factors
• Monopole Type
• 𝑡-channel
•
𝐹 (𝑡)
(𝑞𝛼2
)=
2
Λ2 −𝑚𝛼
2)
(Λ2 +𝑞𝛼
• 𝑠-channel
• 𝐹
𝑠
(𝜔𝑝2
)=
2
Λ2 +𝑚𝛼
2)
(Λ2 +𝜔𝑝
• Forth-order (𝑠-channel)
• 𝐹4 (𝜔𝑝2 ) =
4
Λ4 +𝑚𝛼
4
Λ4 +𝜔𝑝
Λ : cut-off parameters
𝑞𝛼 : momentum
𝑚𝛼 : mass of exchanged mesons
𝜔𝛼 : total energy
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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
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12
𝜂 𝑧 𝑒 2𝑖𝛿 (𝑧)
𝑖 1 − 𝜂 2 (𝑧)𝑒 2𝑖𝛿 (𝑧)
𝑆 𝑧 =
2𝑖𝛿 21 𝑧
2𝑖𝛿 22 (𝑧)
2
𝑖 1 − 𝜂 (𝑧)𝑒
𝜂 𝑧 𝑒
• 𝜂 𝑧 is the inelasticity; 𝛿 𝑖𝑗 the phase shift
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Resonances
Poles with s-channel meson exchange
• 𝑉 = 𝑉𝑡 + 𝑉𝑠
• 𝑉𝑠 contains s-channel diagrams
• 𝑉𝑡 contains t-channel diagrams
• 𝑉𝑠 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
Pure Dynamical Poles (without s-channel exchange)
*This resonance renormalization follow: H.Polinder an Th. A. Rijken Phys. Rev. C 72, 065211 (2005)
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Results
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Parameters
Monopole
𝑴𝟎
Mono.
𝜌
1126.66877
𝑔𝜋𝜋𝜌
0.50101
𝜀1
2112.07666
𝑔𝜋𝜋𝑓2
0.02843626
• All are free
𝐾∗
1403.13526
𝑔𝜋𝜋𝜀1
0.01620557
𝜅
1522.01264
𝑔𝜋𝜋𝜅
0.191683 × 10−3
𝑓2
1367.63563
𝑔𝐾𝐾𝑎0
0.0408802
𝑎𝑜
1235.73076
Λ𝜋𝜋𝜌
2896.41242
𝜙
1150.23241
Λ𝜋𝐾𝐾∗
2301.47525
Λ𝐾𝐾𝜌
3923.01907
Λ𝐾𝐾𝜔,𝜙
4520.8249
parameters.
• Fitting with a large
of experimental
data is difficult,
special for both
well fitting of (𝐼 =
0, 𝐽 = 0) and (𝐼 =
1
, 𝐽 = 0).
Mono.
Λ𝜂𝐾𝐾∗
864.0841
Λ𝜋𝜋𝜌;𝑠
2757.75065
Λ𝐾𝐾𝑎0
1233.61523
Λ𝜋𝜋𝑓2;𝑠
1481.43534
Λ𝐾𝐾𝑎0
2137.02377
Λ𝜋𝜋𝜀1 ;𝑠
1159.73371
Λ𝜋𝐾𝐾∗ ;𝑠
4061.81573
Λ𝜋𝐾𝜅;𝑠
3622.67379
• 24 parameters
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Phase shift results
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𝛿𝐽𝐼 ; 𝐼: isospin. 𝐽: total angular momentum
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Phase shift results
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𝛿𝐽𝐼 ; 𝐼: isospin. 𝐽: total angular momentum
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Resonances results
"l-dd-f0" u 1:2:5
"l-dd-a0" u 1:2:5
18
16
14
12
10
8
6
4
2
0
-60
16
14
12
10
8
6
4
2
0
𝑎0 (980)
-100
-90
-80
-70
-60
-50
-40
-30
-20
-101000
800
-50
850
-40
-30
900
-20
950
-10
𝑓0 (980)
960
980
940
920
900
"l-dd-f2" u 1:2:5
0
1000
"l-dd-rho" u 1:2:5
"l-dd-phi" u 1:2:5
35
14
12
10
𝜙(1020)
8
6
30
50
45
40
35
30
25
20
15
10
5
0
4
2
0
f2 (1270)
25
15
10
5
0
-300
1200 -250
-6
-5
-200
-4
-3
-2
-1
01040
1035
1030
1025
1020
1015
1010
𝜌(770)
20
-180
-160
1250
-140
-120
1300
-100
-80
1350
-60
1400
-40
-200
-150
-100
-50 900
850
800
750
700
650
600
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Resonances results
"l-dd-kap7" u 1:2:5
"l-dd-kap14" u 1:2:5
𝜅(700)
120
45
40
35
30
25
20
15
10
5
0
100
𝜅(1430)
80
60
40
20
0
500
550
600
650
-100
-90
-80
1350
-70
-60
-50
1400
-40
-30
-20
700
-400 -350
-300 -250
-200 -150
-100
1300
750
"l-dd-sigm" u 1:2:5
-50 800
"l-dd-Kst" u 1:2:5
1450
-10
120
100
80
60
40
20
0
1500
0
50
45
40
35
30
25
20
15
10
5
0
𝐾 ∗ (892)
𝜎(500)
-600
-550
-500
-450
300
-400
-55
-50
-45
-40
-35
-30
-25
-20
-15
980
960
940
920
900
880
860
840
350
-350
400
450
-300
-250
-200
600
500
550
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Summary Table of Pole Positions
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+4.0
−2.4 )
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
*K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).
𝚪
𝟐
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𝐼𝑚 𝑠
𝐼𝑚 𝑠
𝜎 500 and 𝜅(800)
𝑅𝑒 𝑠
𝑅𝑒 𝑠
*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.
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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 − 𝐾𝐾 𝑏𝑎𝑟 − 𝜋𝜂 − 𝜂𝜂
• 𝛿00 ; 𝛿01 ; 𝛿10 ; 𝛿11 ; 𝛿20
• 𝜎 600 , 𝑓0 980 , 𝜌 770 , 𝑎0 980 , 𝜙(1020)
• 𝜋𝐾 − 𝜂𝐾
1
2
1
2
• 𝛿0 , 𝛿1
• 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
1
(𝛿00 ; 𝛿02 ,
1
2
𝛿1 )
• 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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THANK YOU FOR YOUR ATTENTION
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