Resonances

".5. RESONANCES

1000 1200 1400 1600 1800 2000

E(MeV)

0.0 0.2 0.4 0.6 0.8 1.0 1.2

degrees

δ

₃⁰

− KK

Monopole Gaussian

Figure 4.15: Phase shiftδ₃⁰. The dashed blue line is the result with conventional monopole type form factors. The solid red line is with the Gaussian type form factor.

poles in the meson-meson scattering in comparison with those in experiments. All the pole positions of the calculated and experimental resonances in ππ−KK¯ −πη−ηη and πK−ηK scatterings are summarized in Tables 4.5 and 4.6, respectively.

There are 3 pole resonances of the coupled ππ−KK¯ −ηη scattering at the I = 0, S-wave state, that is, f = 0(980), σ₁(400) and σ₍600). The f₀(980) have the physical mass m = 1000 MeV and the width Γ = 40 MeV for monopole form factor case (see Table 4.5). For the Gaussian case, we have the pole at the physical mass m = 1075 MeV and the width Γ = 340 MeV. We realize that the width of the resonance in the case of Gaussian form factor is too wide in comparison with the experiment data. In the case of monopole form factor, the ratio of residue matrix elements at the pole is

|R_ππ|² : |R_KK¯|² : |R_ηη|² = 0.41 : 0.59 : 0.00 for f₀ resonance, 0.98 : 0.017 : 0.002 for σ₁ resonance, and 0.48 : 0.33 : 0.18 for σ₂ resonance. These ratio means that there are two pure dynamical resonances at the I =J = 0 in the ππ−KK¯ −ηη scattering, that is, f₀(980) and σ₁(400). The σ₂ resonance corresponds to pole by the s-channel epsilon exchange in the ππ−KK¯

Because of the well-reproduced ππphase shifts of theI = 0,D-wave in theππ−KK¯− ηη scattering, we determine the f₂(1270) resonance rather exactly with the experiment data. This resonance is at m = 1270MeV with the width of 220 MeV for monopole form factor case (see Table 4.5). For the Gaussian case, the resonance is at m = 1250 MeV with the width of180 MeV. For the case ofD-wave in monopole form factor, we have the ratio of residue matrix elements such as |R_ππ|² :|R_KK¯|² :|R_ηη|² = 0.66 : 0.27 : 0.07. This

".5. RESONANCES ratio shows that the f₂ resonance has only small coupling to ηη and considerably large couping (27%) to KK¯ channel.

The agreement oftheoretical results with the experimental data in phase shifts indi-cates that both the position and the width ofthe ρ meson are described very well both in monopole and Gaussian form factors. In fact, the resonance of the I = J = 1 is at m= 800 MeV with the width of 140 MeV for monopole form factor case (see Table 4.5).

For the Gaussian form factor case, the resonance is at m = 800 MeV with the width of 120 MeV. For the case of P-wave with the monopole form factor, we have the ratio of residue matrix elements such as |R_ππ|² : |R_KK¯|² : |R_πη|² = 0.67 : 0.0005 : 0.33. This ratio shows that the ρ resonance has no KK¯ component but 33% comes from the πη component.

Table 4.5: Resonance poles in the ππ−KK¯ −ηπ−ππ scattering.

Experiment Data Monopole Gaussian f₀(980) (970−1010)−i(20−50) 1000−i20 1075−i170

σ₁ (400−500)−i(200−350) 410−i560 390−i500

σ₂ − 580−i360 430−i380

a₀(980) 980−i(25−50) 845−i15 800−i15 ρ(770) 775−i74 800−i60 800−i60 φ(1020) 1019−i2.1 1016.5−i1.6 1022.5−i1.6 f₂(1270) (1275±1.2)−i93 1270−i110 1050−i90

With the strangeness S = 0,I = 1,S-wave, we have the a₀ resonance. This resonance is at m = 845 MeV with the width of 30 MeV for the monopole form factor case (see Table 4.5). For the Gaussian case, the resonance is at m = 800MeV with the width of30 MeV. For the case of monopole form factor, we have the ratio of residue matrix elements such as |R_πη|² :|R_KK¯|² = 0.14 : 0.86. This ratio mean that the a₀ resonance has a large KK¯ component.

Theκresonance withs-channel meson exchange ofthe I = ¹₂,S-wave in the πK−ηK scattering is also reproduced very well. The existence oftwo κpoles in thes-wave is very interesting point ofthe πK−ηK scattering. In the previous calculation ofD.Lohse [10], they have not considered the ηK scattering. The st pole ofκ(700)is a dynamical pole in the πK scattering. In our calculation, we investigate the coupling with the ηK channel, therefore, the second pole of κmaybe come from the interaction by the s-channel meson exchange diagrams. The corresponding physical mass ofthe meson scalar is m_κ = 1450 MeV with the width Γ_κ = 150 MeV for monopole form factor and m_κ = 1440 MeV with the width Γ_κ = 70 MeV in Gaussian form factor. For the case of monopole form factor, we have the ratio ofresidue matrix elements such as |R_πK|² :|R_ηK|² = 0.92 : 0.08for the κ(1450) resonance and |R_πK|² : |R_ηK|² = 0.96 : 0.04 for the κ(700). These ratios show

".5. RESONANCES

that these resonances are dominated by the πK channels.

Table 4.6: Resonance poles position of the πK−ηK scattering.

Experiment Data Monopole Gaussian κ(700) (653−711)−i270 650−i230 650−i190 κ(1450) (1375−1475)−i(95−175) 1450−i75 1440−i35 K^∗ 892−i25 907−i20 910−i18

The pole with s-channel meson exchange of the P-wave of the πK −ηK scattering reproduces well by the K^∗(892)resonance. In Table 4.6, we see that the physical mass of vector meson exchange is m_K∗ = 907 MeV with the width Γ_K∗ = 40 MeV for monopole form factor and m_K∗ = 910 MeV, Γ_K∗ = 36 MeV for Gaussian form factor. In compar-ison with the experiment data, the calculated pole position are a little shifted, but it is reasonable because of the not-well-tting of the phase shifts δ₁¹² at the high energy region (E >950 MeV) as shown in Fig 4.10. For the case of monopole form factor, we have the ratio of residue matrix elements such as |R_πK|² : |R_ηK|² = 0.63 : 0.37. This ratio shows that ηK component is not small in this resonance.

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Chapter 5

Roles of η meson

In this chapter, we discuss the roles of η channels (πη, ηη, ηK channels) in the ππ−KK¯−πη−ηη andπK−ηK scatterings. It should be no doubt that the roles of the η-meson are indispensable in these scatterings to have an adequate view of meson-meson interactions.

We dene two models that we consider in this section, the model with η, which contains the ππ−KK¯ −πη−ηη and πK−ηK interactions, and the model without η, which includes only the ππ−KK¯ and πK interactions.

400 600 800 1000 1200 1400

E(MeV)

0 50 100 150 200 250 300 350

degrees

δ

⁰₀

Propopescu (exp.data) Kaminski (exp.data) K-2pi:Rosselet (exp.data) Grayer Sol.B (exp.data) Grayer Sol.C (exp.data) Monopole

Monopole-NO-eta Gaussian Gaussian-NO-eta

Figure 5.1: Phase shifts δ₀⁰ in the model with η. The solid blue line is the result with conven-tional monopole form factors. The solid red line is with the Gaussian form factors. The dashed lines are those in the model without η. The experimental phase shift analyses are taken from Refs. [1316].

When we keep the same parameter values with those given in Tables 4.1 and 4.2, the phase shifts of the model with η and the model without η are almost the same in both kinds of form factors except the phase shifts in the I = 1, S-wave (see Fig. 5.4).

1000 1020 1040 1060 1080 1100

E(MeV)

0 50 100 150 200

degrees

δ

₁⁰

− K K ¯

Monopole Monopole-NO-eta Gaussian Gaussian-NO-eta

Figure 5.2: Phase shifts δ₁⁰ in the model with η. The solid blue line is the result with conven-tional monopole form factors. The solid red line is with the Gaussian form factors. The dashed lines are those in the model without η.

In the I = 1,S-wave πη−KK¯ scattering, we nd a resonance pole corresponding to a₀(980) resonance, as shown in Fig. 5.4. In the model with η (the πη−KK¯ model), we nd this resonance in the πη phase shifts at the correct energy E = 980MeV.

On the other hand, in the model without η (the single channel KK¯ model), this resonance appears as a resonance in the KK¯ phase shifts at E = 1070MeV (in the case of the monopole factor), which is much higher than the a₀ resonance mass.

On the complex E-plane, this resonance pole is identied at E = 845−i15 MeV (in the model with η) and 890−i35 MeV (in the model without η) as given in Table 5.1.

This means that the resonance position dened by phase shifts on the real E-axis is much dierent from the pole of the S-matrix on the complex E-plane. Such very interesting behavior may be understood by the moving pole with the E-dependent interactions.

800 850 900 950 1000 1050 1100 1150 1200

E(MeV)

0 50 100 150 200

degrees

δ

₀¹

− πη

Monopole Monopole-NO-eta Gaussian Gaussian-NO-eta

Figure 5.3: Phase shifts δ₀¹ in the model with η. The solid blue line is the result with conven-tional monopole form factors. The solid red line is with the Gaussian form factors. The dashed lines are KK¯ phase shifts in the model withoutη.

600 800 1000 1200 1400 1600

E(MeV)

0 10 20 30 40 50

μ b/GeV

dσ

_πη

/dE

_cm

Monopole Monopole-NO-eta Gaussian Gaussia-NO-eta Gay

Figure 5.4: The solid blue line is the result with conventional monopole form factors. The solid red line is with the Gaussian form factors. The dashed lines are the KK¯ cross sections in the model without η. The experimental πη cross sections are taken from Refs. [20].

Table5.1:Resonancepolesintheππ−K¯K−πη−ηηscatterings. ExperimentDataMonopoleMonopole-noηGaussianGaussian-noη f0(980)(970−1010)−i(20−50)1000−i201000−i201075−i1701060−i200 σ1(400−500)−i(200−350)410−i560430−i570390−i500390−i510 σ2−580−i360560−i360430−i380430−i390 a0(980)980−i(25−50)845−i15890−i35860−i15880−i5 ρ(770)775−i74800−i70800−i70800−i60800−i60 φ(1020)1019−i2.11016.5−i1.61016.5−i1.61022.5−i1.61022.5−i1.6 f2(1270)(1275±1.2)−i931270−i1101270−i1101050−i901280−i110 Table5.2:ResonancepolesintheπK−ηKscatterings. ExperimentDataMonopoleMonopole-noηGaussianGaussian-noη κ(700)(653−711)−i270650−i230660−i220650−i190650−i200 κ(1450)(1375−1475)−i(95−175)1450−i751450−i751440−i351440−i30 K∗ 892−i25907−i20−910−i18−

When we compare the phase shifts in the two models for the πK−ηK scattering, we realize that there are a large change in the phase shifts in the I = ¹₂,P-wave (see the Fig.

5.6). This means that the η channel has a large eect in this P- wave. It is reasonable that the pole position of K^∗ do not exist at the same calculated region, it was shifted to the higher energy region. The detail values of pole positions are shown in Table 5.2 to compare its in the two models.

700 800 900 1000 1100 1200 1300 1400

E(MeV)

0 20 40 60 80 100 120 140

degrees

δ

₀^1/2

M.J.Matison(exp.data) Estabrook(exp.data) Aston(exp.data) Monopole Monopole-NO-eta Gaussian Gaussian-NO-eta

Figure 5.5: Phase shiftsδ₀¹² in the model with η. The solid blue line is the result with conven-tional monopole form factors. The solid red line is with the Gaussian form factors. The dashed lines are those in the model without η. The experimental phase shift analyses are taken from Refs. [2325].

As a brief summary, we see that theηchannels play very important and necessary roles in the full treatment of the meson-meson interaction by one-meson-exchange mechanisms.

The ηη channel has a small eect in coupling to the ππ −KK¯ scattering, while the πη channel has a large eect to the KK¯ channel. The η channel contributed to the appearance of a₀ resonance in the ηπ interaction. In the the I = ¹₂, P-wave for the πK scattering, the η channel also has a part in producing the well-tted phase shifts.

700 800 900 1000 1100 1200 1300

E(MeV)

0 50 100 150 200

degrees

δ

₁^1/2

Estabrook (exp.data) R.Mercer (exp.data) Monopole Monopole-NO-eta Gaussian Gaussian-NO-eta

Figure 5.6: Phase shiftsδ₁¹² in the model with η. The solid blue line is the result with conven-tional monopole form factors. The solid red line is with the Gaussian form factors. The dashed lines are those in the model withoutη.The experimental phase shift analyses are taken from Refs.

[24,26].

Chapter 6 Conclusion

In the purpose of description of hadron-hadron interactions in a unied way, we con-structed the one-meson-exchange potential model of the meson-meson interactions based on the interaction Lagrangian which satises the SU(3)-symmetry. In order to deter-mine parameters in the potential, which include two kinds of form factors (monopole and Gaussian) and coupling constants, we used the phase shift data of ππ and πK scatter-ings at low energies. The results of the t of the ππ−KK¯ −πη −ηη and πK−ηK scatterings are reproduced to the most recent energy-dependent phase shift analyses at low and intermediate energy region based on the SU(3)-symmetric one-meson-exchange model. The quantitative phase shifts of the KK interaction are determined by extending the ππ−KK¯ −πη−ηη and πK−ηK scatterings. In the KK interaction, t-channel ρ, ω and φ meson exchange plays an important role.

We know that the resonance is one of the most prominent phenomena in the whole range of scattering experiments. We approach the problem of resonance by the analytic properties of the amplitude scattering of meson-meson interactions. Beside the well-reproduced poles of pure dynamical resonance f₀(980 at the isospin I = 0, S-wave we also found the existence of the other pure dynamical pole position of σ₁ and the σ₂, which come from the s-channel exchange diagram. By calculating the ratio of residue matrix elements at the poles, we understand that the σ₁ and σ₂ have much dierent character. Moreover, by using Haftel-Tabakin method to deal with the singularity of the Green function of the scattering amplitude, we obtained the numerical calculations of the physical masses and the width of the pole positions of ππ−KK¯ −πη −ηη and πK − ηK scattering, that is, ρ(770), a₀(980), φ(1020) f₂(1270), κ(700), κ(1430) and K^∗(892) resonances. We also have two pole position of κ of the πK−ηK scattering at the isopin I = ¹₂, S-wave. In this dissertation, we also give a brief introduction to the compositeness of the two-body resonances.

In addition, the ππ−KK¯ −ηη and πK−ηK interactions also have been discussed to investigate the role of η meson in the ππ and πK scatterings. The ηη channel has a small eect in coupling to the ππ−KK¯ scattering, while the πη channel has a large

eect to the KK¯ channel. The η channel contributed to the appearance of a₀ resonance in the ηπ interaction. In the I = ¹₂, P-wave η channel also has a part in producing the well-tted phase shifts.

In the future, we will rene the meson-meson interaction (ππ−πK¯ −πη−ηη and πK−ηK) by the meson-exchange model. We may make the more well-tted parameter set to well-reproduce the phase shift δ₀⁰ and δ₀¹², especially in the case of Gaussian form factors. We also would like to investigate the eld normalization to understand the quantitative measure of compositeness of the two-body resonances. After that, we would like to advance forward by solving the three-body problems to study the existences and properties of the resonances in the hadronic systems.

Appendix A

The avor SU(3) model of hadrons and hadron interaction

A.1 The avor SU(3) model of hadrons

The extension from SU(2) to SU(3) is immediate if we extend the basic u, d doublet to triplet u, d, s and investigate transformations of

φ ≡

⎛

⎝ u d s

⎞

⎠ of the form

φ =U φ

where U is now a 3×3unitary unimodular matrix. Following the SU(2) case write U ≡exp

1

2iθˆn·λ

where the λ_i are eight dependent Hermitian traceless 3×3 matrices analogous to the σ_i of SU(2). Canonically these are chosen to be [28]

λ₁ =

⎛

⎝ 0 1 · 1 0 ·

· · ·

⎞

⎠ λ₂ =

⎛

⎝ 0 −i · i 0 ·

· · ·

⎞

⎠ λ₃ =

⎛

⎝ 1 0 · 0 −1 ·

· · ·

⎞

⎠

λ₄ =

⎛

⎝ 0 · 1

· · · 1 · 0

⎞

⎠ λ₅ =

⎛

⎝ 0 · −i

· · · i · 0

⎞

⎠ λ₆ =

⎛

⎝ · · ·

· 0 1

· 1 0

⎞

⎠

λ₇ =

⎛

⎝ · · ·

· 0 −i

· i 0

⎞

⎠ λ₈ = 1

√3

⎛

⎝ 1 0 0 0 1 0 0 0 −2

⎞

⎠ (A.1)

where the dots are zeros and we have written them in this fashion to highlight the SU(2) subgroups contained within SU(3). The λ_1,2 have the structure

⎛

⎝ σ_1,2 0 0 0 0 0

⎞

⎠ (A.2)

A.1. THE FLAVOR SU(3) MODEL OF HADRONS and hence exhibit the SU(2) isospin subgroup. The λ_6,7 are

⎛

⎝ 0 0 0 0

0 σ_1,2

⎞

⎠ (A.3)

and exhibit an SU(2) subgroup calledU-spin while theλ_4,5are related toa third subgroup V-spin. In terms of the basic triplet of Fig. these SU(2) doublets are

u,d(I); d,s(U); u,s(V)

The operator F₃ ≡ ¹₂λ₃ is the isospin operator since acting on u, d, s it has eigenvalues

±¹₂,0respectively. The hypercharge operator is Y = 2

√3F₈ ≡ 2

√3· 1

2λ₈ (A.4)

Table A.1: Structure constants of SU(3) f₁₂₃ = 1

f₁₄₇ =f₂₄₆ =f₂₅₇ =f₃₄₅ =f₅₁₆ =f₆₃₇ = ¹₂ f₄₅₈ =f₆₇₈ = ^√₂³

d₁₁₈ =d₂₂₈ =d₃₃₈ =−d₈₈₈ = √¹

d₁₄₆ =d₁₅₇ =d₂₅₆ =d₃₄₄ =d₃₅₅3= ¹₂ d₂₄₇ =d₃₆₆ =d₃₇₇ =−¹₂

d₄₄₈ =d₅₅₈ =d₆₆₈ =d₇₇₈ =−₂^√1₃

The commutation relations of the matrices ¹₂λ_i can be obtained by explicit calculation.

1 2λ_i,1

2λ_j

=if_ijk 1

2λ_k

(A.5) with the structure constants f_ijk having the value in Table A.1 and being antisymmetric under interchange of any pair of indices. The matrices also satisfy anticommutation

relations

1 2λ_i,1

2λ_j

= 1

3δ_ij+d_ijk 1

2λ_k

(A.6) where the d_ijk are symmetric under interchange of indices.

As in the SU(2) case we can generalize these results by dening F_i ≡ ¹₂λ_i satisfying commutation relations

[F_i, F_j] =if_ijkF_k (A.7)

(i = 1· · ·8). A full study of SU(3) then consists of nding N ×N matrices F_i which transform N-dimensional states by

φ →φ = (1 +iθˆn·F)φ (A.8) These states form N-dimensional multiplets of SU(3).

A.2. HADRON INTERACTION

A.2 Hadron interaction

Hadrons are dened as particles that interact by the strong interaction. We can subdivide hadrons to baryons and mesons by their spin: baryons are hadrons with half-integer spins (1/2, 3/2, 5/2, ...), and mesons are hadrons with half-integer spins (0, 1, 2,...).

Thus, there are 3 types of the interaction between hadrons, that is, baryon-baryon, meson-baryon and meson-meson interactions.

Baryon-baryon interactions

S = 0 N N

S =−1 ΛN −ΣN

S =−2 ΞN −ΛΛ−ΛΣ−ΣΣ

S =−3 ΞΛ−ΞΣ

S =−4 ΞΞ

(A.9) Meson-baryon interactions

S = 1 KN

S = 0 πN−ηN −KΛ−KΣ

S =−1 πΛ−πΣ−KN¯ −ηΛ−ηΣ−KΞ S =−2 πΞ−ηΞ−KΛ¯ −KΞ¯

S =−3 KΞ¯

Meson-meson interactions

S=−2 K¯K¯

S=−1 πK¯ −ηK¯

S= 0 ππ−KK¯ −ηπ−ηη

S= 1 πK−ηK

S= 2 KK

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List of Publications by the Author

Journal Publications

1. Ngo Thi Hong Xiem and Shoji Shinmura, "Pion-pion, pion-kaon, and kaon-kaon interactions in the one-meson-exchange model", Progress of Theoretical and Ex-perimental Physics 023D04, doi:10.1093/ptep/ptu001 (2014).

International Conference Publications

1. Ngo Thi Hong Xiem and Shoji Shinmura, "K^barK^bar Interaction by vector- and scalar-meson-exchange mechanisms", The 20th International IUPAP Conference on Few-Body Problems in Physics (FB20) Poster session, Fukuoka, Japan (2012).

2. Ngo Thi Hong Xiem and Shoji Shinmura, "The KK Interaction by Meson-Exchange Model", International Workshop on Strangeness Nuclear Physics 2012 (SNP12), Osaka, Japan, Genshikaku Kenkyu, vol. 57, Suppl. 3 (2012).

3. Shoji Shinmura and Ngo Thi Hong Xiem, "K^bar-Hyperon Interactions and Possible S-Wave Resonances", Few-Body Systems, vol. 54, issue 7-10, pp 1171-1174 (2013).

4. Ngo Thi Hong Xiem and Shoji Shinmura, "Roles of η channels in ππ, πη and πK scatterings", XV International Conference on Hadron Spectroscopy, Nara, Japan, http://pos.sissa.it/archive/conferences/205/145/Hadron%202013_145.pdf (2013).

5. Ngo Thi Hong Xiem and Shoji Shinmura, "Resonances in meson-meson interac-tion by one-meson-exchange model", Workshop on J-PARC hadron physics in 2014 Ibaraki Quantum Beam Research Center, Tokai, Ibaraki, Japan (2014).

Domestic Conference Publications

1. Ngo Thi Hong Xiem and Shoji Shinmura "Kaon-Kaon Interaction by Meson-Exchange Mechanisms", The Physical Society of Japan 2012 Fall Meeting, Kyoto (2012).

ドキュメント内 Meson-Meson Interactions in the One-Meson-Exchange Model (ページ 45-65)