Compositeness of Dynamically Generated Resonances

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1 Compositeness of Dynamically Generated Resonances Takayasu SEKIHARA (Japan Atomic Energy Agency) [1] T. S., Phys. Rev. C95 (2017) [2] T. S., T. Hyodo and D. Jido, PTEP D04. [3] T. S., T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) [4] F. S. Navarra, M. Nielsen, E. Oset and T. S., Phys. Rev. D92 (2015) The Charm and Beauty of Strong ECT* (July 17-28, 2017)

Contents 1. Introduction Hadronic molecules are unique! Thanks to the uniqueness, one can use quantum mechanics. 2. Physical meaning of the compositeness Two-body wave function.

2 Contents 1. Introduction Hadronic molecules are unique! Thanks to the uniqueness, one can use quantum mechanics. 2. Physical meaning of the compositeness Two-body wave function. --> Compositeness = its norm! How to evaluate the two-body wave function? 3. Some exercises for dynamically generated resonances Is the Ds0*(2317) resonance is a KD molecular state???? 4. Summary The Charm and Beauty of Strong ECT* (July 17-28, 2017) 2

3 1. Introduction The Charm and Beauty of Strong ECT* (July 17-28, 2017)

.. --- Actually some hadrons cannot be described by the quark model.

4 1. Introduction ++ Exotic hadrons and their structure ++ Exotic hadrons --- not same quark component as ordinary hadrons = not qqq nor qq. Penta-quarks Tetra-quarks Hybrids Glueballs Hadronic molecules Actually some hadrons cannot be described by the quark model. In particular, the recent experimental condition allows us to study in the heavy Ordinary hadrons quark sector (charm & bottom)! Pc(4450) & Pc(4380). X(3872) & other XYZs. Ds0*(2317).... The Charm and Beauty of Strong ECT* (July 17-28, 2017) 4

5 1. Introduction ++ Exotic hadrons and their structure ++ Penta-quarks Tetra-quarks Hybrids Glueballs Hadronic molecules... Q.1: Do exotic hadrons really exist? Q.2: If they do exist, how are their structure and properties? --- Re-confirmation of quark models. --- Constituent quarks in multi-quarks? Constituent gluons? Q.3: If they do not exist, what mechanism forbids their existence? Q.4: What is the connection between exotic hadrons and QCD? The Charm and Beauty of Strong ECT* (July 17-28, 2017) 5

6 1. Introduction ++ Uniqueness of hadronic molecules ++ Hadronic molecules should be unique, because they are composed of hadrons themselves, which are color singlet. Hadronic molecules (cf. deuteron)... --> Various quantitative / qualitative Diff. from other compact states. Large spatial size due to the absence of strong confining force. Hadron masses are observable, in contrast to quark masses. --> We expect that a hadronic molecule will exist around a two-body (or few- / many-body) threshold. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 6

1. Introduction ++ Hadronic molecules and quantum mechanics ++ An example of hadronic molecule: deuteron. NN phase shift Machleidt, Phys. Rev. C63 (2001) 024001.

7 1. Introduction ++ Hadronic molecules and quantum mechanics ++ An example of hadronic molecule: deuteron. NN phase shift Machleidt, Phys. Rev. C63 (2001) s-wave d-wave Wave function of deuteron Deuteron is a proton-neutron bound state. --- Weinberg proved this fact by using general wave equations in quantum mechanics in the weak binding limit (BE << Etypical). Introduced field renormalization constant Z: --> Bare component B0 > in the total wave function B >. Weinberg (1965). --> Consistent with Z ~ 0! The Charm and Beauty of Strong ECT* (July 17-28, 2017) 7

8 1. Introduction ++ Hadronic molecules and quantum mechanics ++ An example of hadronic molecule: deuteron. NN phase shift Machleidt, Phys. Rev. C63 (2001) s-wave d-wave Wave function of deuteron Lesson: In a similar manner, we can study the structure of general hadronic molecules. --- We can use quantum mechanics (not QCD itself directly) to investigate the hadronic molecular component: Two-body wave function, its norm = compositeness, Scatt. Amp. <--> In contrast, for hadrons of other configurations, we have to treat color multiplet states explicitly and appropriately. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 8

9 1. Introduction ++ Motivation ++ We evaluate the wave function of hadron-hadron composite contribution. --- cf. Wave function for relative motion of two nucleons inside deuteron. For this purpose, we solve the Lippmann- Schwinger equation. <-- We employ a fact that the two-body wave function appears in the residue of the scattering amplitude of the two particles at the resonance pole. The WF and compositeness (= norm) are automatically scaled. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 9

10 2. Physical meaning of the compositeness The Charm and Beauty of Strong ECT* (July 17-28, 2017)

11 2. Physical meaning of compositeness ++ Setup of our model ++ We consider the following system in quantum mechanics. Full Hamiltonian: --- Composed of free part H0 and interaction V. --- The interaction V, determined in some theory, may depend on the energy of the system E. The free Hamiltonian has eigenstates of free two-body state: where or --- The two-body state with relative momentum q. The full Hamiltonian has an eigenstate of a bound state: --- The eigenvalue Epole is real (stable bound state) or complex (resonance). The Charm and Beauty of Strong ECT* (July 17-28, 2017) 11

12 2. Physical meaning of compositeness ++ Compositeness as a norm ++ Definition: Compositeness is defined as the norm of the two-body part of the bound-state wave function. Two-body bound-state wave function in momentum space: n The norm of the two-body wave function is: However, for the moment we have not normalized the boundstate wave function. --> To interpret the compositeness, we have to normalize it. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 12

13 2. Physical meaning of compositeness ++ Wave function from Lippmann-Schwinger Eq. ++ Statement 1: To obtain the correct normalization of bound-state wave function it is better to solve the Lippmann- Schwinger Eq. than the Schrödinger Eq.! Schrödinger Eq. in momentum space: --- Homogeneous integral Eq., so we have to normalize it by hand! Lippmann-Schwinger Eq. in momentum space: --- Inhomogeneous integral Eq., so we need not take care of the normalization of the scattering amplitude! Where is the wave function in Lippmann-Schwinger Eq.? The Charm and Beauty of Strong ECT* (July 17-28, 2017) 13

14 2. Physical meaning of compositeness ++ Wave function from Lippmann-Schwinger Eq. ++ Solve the Lippmann-Schwinger equation at the pole position of the bound state. --- Near the resonance pole position Epole, amplitude is dominated by the pole term in the expansion by the eigenstates of H as --- The residue of the amplitude at the pole position has information on the wave function! or The Charm and Beauty of Strong ECT* (July 17-28, 2017) 14

15 2. Physical meaning of compositeness The idea of the renormalization for: Solve the Lippmann-Schwinger equation at the pole position of --- the We bound (re-)normalize state. the total wave function as ++ Wave function from Lippmann-Schwinger Eq. ++ cf. --- Near the resonance pole position Epole, amplitude is dominated by the pole term in the expansion by the eigenstates of H as --- The residue of the amplitude at the pole position has information on the wave function! or The Charm and Beauty of Strong ECT* (July 17-28, 2017) 15

16 2. Physical meaning of compositeness ++ Wave function from Lippmann-Schwinger Eq. ++ Solve the Lippmann-Schwinger equation at the pole position of the bound state. --- The wave function can be extracted from the residue of the amplitude at the pole position: <-- Off-shell Amp.! --> Because the scattering amplitude cannot be freely scaled (Lippmann-Schwinger Eq. is inhomogeneous!), the WF from the residue of the amplitude is automatically scaled as well! If purely molecule --> <-- We obtain! E. Hernandez and A. Mondragon, Phys. Rev. C29 (1984) 722. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 16

17 2. Physical meaning of compositeness ++ Example 1: Stable bound state ++ A Λ hyperon in A ~ 40 nucleus. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψ by hand! Woods-Saxon potential 2. Solve Lippmann-Schwinger equation: --> Extract WF from the residue: --> --- Without normalizing by hand! The Charm and Beauty of Strong ECT* (July 17-28, 2017) 17

2. Physical meaning of compositeness ++ Example 1: Stable bound state ++ A Λ hyperon in A ~ 40 nucleus. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψ by hand!

18 2. Physical meaning of compositeness ++ Example 1: Stable bound state ++ A Λ hyperon in A ~ 40 nucleus. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψ by hand! Woods-Saxon potential 2. Solve Lippmann-Schwinger equation: --> Extract WF from the residue: --> --- Without normalizing by hand! In 1st way: Points. 2nd way: Lines. Exact coincidence! --- We obtain automatically normalized WF from the Amp.! The Charm and Beauty of Strong ECT* (July 17-28, 2017) 18

19 2. Physical meaning of compositeness ++ Example 1: Stable bound state ++ We define the compositeness X as the norm of the wave function: --- In the following, we calculate X from the scattering amplitude. The compositeness is unity for energy independent interaction. 0s, from Scatt. Amp. X = 1 (v1 = 0) Hernandez and Mondragon (1984). An interesting thing happens when we introduce the energy dependence for V. If the interaction depends on the energy, the compositeness from the scattering amplitude deviates from unity. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 19

20 2. Physical meaning of compositeness ++ Example 1: Stable bound state ++ We define the compositeness X as the norm of the wave function: --- In the following, we calculate X from the scattering amplitude. The compositeness is unity for energy independent interaction. 0s, from Scatt. Amp. X = 1 (v1 = 0) Lines: X from Amp. Points: X = X V/ E Hernandez and Mondragon (1984). Consistent with the norm with energy-dep. interaction. Formanek, Lombard and Mares (2004); Miyahara and Hyodo (2016). The Charm and Beauty of Strong ECT* (July 17-28, 2017) 20

21 2. Physical meaning of compositeness ++ Example 1: Stable bound state ++ We define the compositeness X as the norm of the wave function: e.g. --- In the following, we calculate X from the scattering amplitude. The compositeness is unity for energy independent interaction. 0s, from Scatt. Amp. X = 1 (v1 = 0) Hernandez and Mondragon (1984). Deviation of compositeness from unity can be interpreted as a missing-channel part. T. S., Hyodo and Jido, PTEP D04. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 21

2. Physical meaning of compositeness ++ Example 2: Unstable resonance state ++ Unstable resonance in KN-πΣ system. --> Calculate wave functions in 2 ways. 1.

22 2. Physical meaning of compositeness ++ Example 2: Unstable resonance state ++ Unstable resonance in KN-πΣ system. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψj by hand! Gaussian potential Coupling strength is controlled by x. 2. Solve Lippmann-Schwinger equation: Aoyama et al. (2006). --> Extract WF from the residue: --> --- Without normalizing by hand! Complex scaling method. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 22

23 2. Physical meaning of compositeness ++ Example 2: Unstable resonance state ++ Unstable resonance in KN-πΣ system. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψj by hand! Gaussian potential Coupling strength is controlled by x. 2. Solve Lippmann-Schwinger equation: θ = 20 o --> Extract WF from the residue: --> --- Without normalizing by hand! In 1st way: Points. 2nd way: Lines. Coincidence again! --- Our method is valid even for resonances! The Charm and Beauty of Strong ECT* (July 17-28, 2017) 23

24 2. Physical meaning of compositeness ++ Example 2: Unstable resonance state ++ We define the compositeness X as the norm of the wave function: X Z d 3 q (2 ) 3 h qihq i = Z In the following, we calculate X from the scattering amplitude. <-- The compositeness is unity for energy independent interaction. When we consider the energy dependence of the interaction, the compositeness from the scattering amplitude deviates from unity because of missing channel contribution. --- e.g.: g0 2 V miss = E M 0 0 dq P(q) --- θ Indep.! Hernandez and Mondragon (1984). Lines: X from Amp. Points: X = X V/ E The Charm and Beauty of Strong ECT* (July 17-28, 2017) 24

25 2. Physical meaning of compositeness ++ Model (in)dependence of compositeness ++ Statement 2: Compositeness is a model dependent quantity, in general. Because the wave function and interaction are not observable, they are model dependent quantities. --- In the present study, to calculate the residue γ(q) and wave function, we need the off-shell amplitude, which is not observable and model dependent quantity. --> (E and q are independent as the off-shell Amp.) Furthermore, the compositeness is also not observable and model dependent quantity. --- The field renormalization constant is not observable as well. cf. Deuteron d-wave probability PD ~ 5% is not observable. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 25

26 2. Physical meaning of compositeness ++ Model (in)dependence of compositeness ++ Statement 2: Compositeness is a model dependent quantity, in general. Note: We can uniquely determine the compositeness once we fix the interaction (including its energy dependence). cf. In the pioneering studies, they fixed the interaction first and discussed the compositeness from the scattering amplitude. --- Separable interaction. Gamermann, Nieves, Oset and Ruiz Arriola, Phys. Rev. D81 (2010) ; Yamagata-Sekihara, Nieves and Oset, Phys. Rev. D83 (2011) ; Aceti and Oset, Phys. Rev. D86 (2012) ; Interaction with the Yukawa coupling to a bare state. Hyodo, Jido and Hosaka, Phys. Rev. C85 (2012) The Charm and Beauty of Strong ECT* (July 17-28, 2017) 26

2. Physical meaning of compositeness ++ Model (in)dependence of compositeness ++ Statement 3: If the pole exists near threshold, compositeness becomes a model independent quantity.

27 2. Physical meaning of compositeness ++ Model (in)dependence of compositeness ++ Statement 3: If the pole exists near threshold, compositeness becomes a model independent quantity. --- Compositeness can be expressed with threshold parameters such as scattering length and effective range, which are observable. Deuteron. Weinberg ( 65). f0(980) and a0(980). Baru et al. ( 04), Kamiya-Hyodo, Phys. Rev. C93 (2016) Λ(1405). Kamiya-Hyodo, Phys. Rev. C93 (2016) Model dependent part is a minor contribution to X! observables The Charm and Beauty of Strong ECT* (July 17-28, 2017) 27

28 2. Physical meaning of compositeness ++ Summary of the physical meaning ++ Compositeness is defined as the norm of the two-body part of the bound-state wave function. To obtain the correct normalization of bound-state wave function it is better to solve the Lippmann-Schwinger Eq. than the Schrödinger Eq.! Automatically normalized. Energy Dep. interaction is interpreted as bare-state contribution. Compositeness is a model dependent quantity, in general. If the pole exists near threshold, compositeness becomes a model independent quantity. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 28

29 3. Some exercises for dynamically generated resonances The Charm and Beauty of Strong ECT* (July 17-28, 2017) 29

30 3. Exercises for DGRs ++ What we have done is ++ For a given interaction, we can calculate two-body wave functions from the scattering amplitude. --- NOT solving the Schrödinger Eq. in a usual manner. --- In particular, compositeness (= the norm of the wave function) is automatically normalized in this approach! Normalized! The compositeness is unity for a bound / resonance state generated by an energy independent interaction. For an energy dependent interaction, the compositeness deviates from unity, reflecting a missing channel contribution. e.g. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 30

31 3. Exercises for DGRs ++ Wave functions for hadrons ++ By using the two-body wave function and compositeness (norm), we can distinguish a certain configuration of hadrons in a model. Hadronic molecules as a bound state of hadrons (cf. deuteron) Ordinary hadrons In the previous studies, we have investigated: Λ(1405). Ξ(1690). N(1535) & N(1650).... T. S., Hyodo and Jido, PTEP 2015, 063D04; T. S., PTEP D01; T. S. T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) Evaluated X for these dynamically generated resonances. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 31

32 3. Exercises for DGRs ++ Compositeness for Ds0*(2317) ++ Particle Data Group.??? Measured mass and width do not match the predictions from potential-based quark models. --> Ds0*(2317) may be a DK molecule (or other exotic configuration) rather than a cs system. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 32

33 3. Exercises for DGRs ++ Compositeness for Ds0*(2317) ++ We employ the scattering amplitude for Ds0*(2317) group by Valencia group with KD and coupled channels. Gamermann, Oset, Strottman, and Vicente Vacas, Phys. Rev. D76 (2007) ). --- KD and ηds for Ds0*(2317), The separable interaction V is fixed by a phenomenological Lagrangian, in which the flavor SU(4) symmetry is explicitly broken to SU(3) by the contribution of heavy meson exchanges. Parameters are fixed as: --> Dynamically generates the pole of Ds0*(2317). The Charm and Beauty of Strong ECT* (July 17-28, 2017) 33

3. Exercises for DGRs ++ Compositeness for Ds0*(2317) ++ With the separable interaction, the compositeness Xj (channel j) n and missing channel contribution Z are: T. S.

34 3. Exercises for DGRs ++ Compositeness for Ds0*(2317) ++ With the separable interaction, the compositeness Xj (channel j) n and missing channel contribution Z are: T. S., Hyodo and Jido, PTEP 2015, 063D04.!!! --- The result shows the large KD component for Ds0*(2317), since XKD dominates the sum rule: --- Really KD molecule in this model! The Charm and Beauty of Strong ECT* (July 17-28, 2017) 34

3. Exercises for DGRs ++ Compositeness for Ds0*(2317) ++ In the case of Ds0*(2317), additional support to the molecular interpretation came from lattice QCD (re-)analysis.

35 3. Exercises for DGRs ++ Compositeness for Ds0*(2317) ++ In the case of Ds0*(2317), additional support to the molecular interpretation came from lattice QCD (re-)analysis. Martínez Torres, Oset, Prelovsek, and Ramos, JHEP 1505 (2015) Assuming the separable interaction between KD. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 35

36 4. Summary The Charm and Beauty of Strong ECT* (July 17-28, 2017) 36

37 4. Summary We can extract the two-body WF from the residue of the scattering amplitude at the pole position, both stable and unstable states. Scattering amplitude: The WF from the scattering amplitude is automatically scaled. The compositeness (= norm of the two-body WF) is unity for a bound state in an energy independent interaction. For an energy dependent interaction, the compositeness deviates from unity, reflecting a missing channel contribution. The compositeness is model dependent in general, but it becomes model Indep. when the pole exists near the threshold. From the hadron-hadron amplitudes in appropriate models, we can evaluate compositeness for dynamically generated resonances. The Ds0*(2317) resonance has large KD component in a model. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 37

38 Thank you very much for your kind attention! The Charm and Beauty of Strong ECT* (July 17-28, 2017)

39 Appendix The Charm and Beauty of Strong ECT* (July 17-28, 2017) 39

40 Appendix ++ Compositeness for Λ(1405) ++ Compositeness X for Λ(1405) in the chiral unitary approach. Amplitude taken from: Ikeda, Hyodo and Weise, Phys. Lett. B706, (2011) 63; Nucl. Phys. A881 (2012) 98.!!! Hyodo and Jido ( 12). --- Large KN component for (higher pole) Λ(1405), since XKN is almost unity with small imaginary parts. T. S., Hyodo and Jido, PTEP 2015, 063D04. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 40

!!! K -- Σ + --- Large KΣ component for Ξ(1690), since XKΣ is almost unity

41 Appendix ++ Compositeness for Ξ(1690) ++ Compositeness X for Ξ(1690) in the chiral unitary approach. K 0 Λ T. S., PTEP D01.!!! K -- Σ Large KΣ component for Ξ(1690), since XKΣ is almost unity with small imaginary parts. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 41

Appendix ++ Compositeness for N(1535) and N(1650) ++ Compositeness X for N(1535) & N(1650) in chiral unitary approach. T. S. T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) 035204.

42 Appendix ++ Compositeness for N(1535) and N(1650) ++ Compositeness X for N(1535) & N(1650) in chiral unitary approach. T. S. T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) For both N* resonances, the missing-channel part Z is dominant. --> N(1535) and N(1650) have large components originating from contributions other than πn, ηn, KΛ, and KΣ. The Charm and Beauty of Strong ECT* (July 17-28, 2017) 42

43 Appendix ++ Compositeness for Δ(1232) ++ Compositeness X for Δ(1232) in chiral unitary approach.!? The πn compositeness XπN takes large real part! But non-negligible imaginary part as well. --> Large πn component in the Δ(1232) resonance!? The Charm and Beauty of Strong ECT* (July 17-28, 2017) 43

Compositeness for the N* and Δ* resonances from the πn scattering amplitude

Compositeness for the N* and Δ* resonances from the πn scattering amplitude [1] T. S., Phys. Rev. C95 (2017) 025206. [2] T. S., in preparation. Takayasu SEKIHARA (Japan Atomic Energy Agency) 1. Introduction