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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
12/2/2014 N.T.H Xiem@JPARC-2014 1
Content
1. Introduction
2. Formalization of the model
3. Results
4. Conclusion
12/2/2014 N.T.H Xiem@JPARC-2014 2
Introduction
12/2/2014 N.T.H Xiem@JPARC-2014 3
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.
12/2/2014 N.T.H Xiem@JPARC-2014 4
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)
12/2/2014 N.T.H Xiem@JPARC-2014 5
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
Formalization of the model
12/2/2014 N.T.H Xiem@JPARC-2014 6
Meson exchange potential
• Feynman Diagram
12/2/2014 N.T.H Xiem@JPARC-2014 7
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
• SU(3) Symmetric Lagrangians
12/2/2014 N.T.H Xiem@JPARC-2014 8
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
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
12/2/2014 N.T.H Xiem@JPARC-2014 9
Λ : cut-off parameters
𝑞𝛼: momentum
𝑚𝛼: mass of exchanged mesons
𝜔𝛼 : total energy
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
12/2/2014 N.T.H Xiem@JPARC-2014 10
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
12/2/2014 N.T.H Xiem@JPARC-2014 11
*This resonance renormalization follow: H.Polinder an Th. A. Rijken Phys. Rev. C 72, 065211 (2005)
Pure Dynamical Poles (without s-channel exchange)
Results
12/2/2014 N.T.H Xiem@JPARC-2014 12
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).
12/2/2014 N.T.H Xiem@JPARC-2014 13
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
Phase shift results
12/2/2014 N.T.H Xiem@JPARC-2014 14
𝛿𝐽
𝐼
; 𝐼: isospin. 𝐽: total angular momentum
Phase shift results
12/2/2014 N.T.H Xiem@JPARC-2014 15
𝛿𝐽
𝐼
; 𝐼: isospin. 𝐽: total angular momentum
Resonances results
12/2/2014 N.T.H Xiem@JPARC-2014 16
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)
Resonances results
12/2/2014 N.T.H Xiem@JPARC-2014 17
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)
Summary Table of Pole Positions
12/2/2014 N.T.H Xiem@JPARC-2014 18
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).
𝜎 500 and 𝜅(800)
12/2/2014 N.T.H Xiem@JPARC-2014 19
*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
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.
12/2/2014 N.T.H Xiem@JPARC-2014 20
THANK YOU FOR YOUR ATTENTION
12/2/2014 N.T.H Xiem@JPARC-2014 21

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1202-06-Xiem.pptx

  • 1. RESONANCES IN MESON-MESON INTERACTION BY ONE-MESON-EXCHANGE MODEL Ngo Thi Hong XIEM and Shoji SHINMURA Graduate School of Engineering Gifu University 12/2/2014 N.T.H Xiem@JPARC-2014 1
  • 2. Content 1. Introduction 2. Formalization of the model 3. Results 4. Conclusion 12/2/2014 N.T.H Xiem@JPARC-2014 2
  • 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. 12/2/2014 N.T.H Xiem@JPARC-2014 4
  • 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) 12/2/2014 N.T.H Xiem@JPARC-2014 5 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 12/2/2014 N.T.H Xiem@JPARC-2014 6
  • 7. Meson exchange potential • Feynman Diagram 12/2/2014 N.T.H Xiem@JPARC-2014 7 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 12/2/2014 N.T.H Xiem@JPARC-2014 8 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 12/2/2014 N.T.H Xiem@JPARC-2014 9 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 12/2/2014 N.T.H Xiem@JPARC-2014 10
  • 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 12/2/2014 N.T.H Xiem@JPARC-2014 11 *This resonance renormalization follow: H.Polinder an Th. A. Rijken Phys. Rev. C 72, 065211 (2005) Pure Dynamical Poles (without s-channel exchange)
  • 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). 12/2/2014 N.T.H Xiem@JPARC-2014 13 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 12/2/2014 N.T.H Xiem@JPARC-2014 14 𝛿𝐽 𝐼 ; 𝐼: isospin. 𝐽: total angular momentum
  • 15. Phase shift results 12/2/2014 N.T.H Xiem@JPARC-2014 15 𝛿𝐽 𝐼 ; 𝐼: isospin. 𝐽: total angular momentum
  • 16. Resonances results 12/2/2014 N.T.H Xiem@JPARC-2014 16 800 850 900 950 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 12/2/2014 N.T.H Xiem@JPARC-2014 17 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 12/2/2014 N.T.H Xiem@JPARC-2014 18 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) 12/2/2014 N.T.H Xiem@JPARC-2014 19 *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. 12/2/2014 N.T.H Xiem@JPARC-2014 20
  • 21. THANK YOU FOR YOUR ATTENTION 12/2/2014 N.T.H Xiem@JPARC-2014 21