A study for “elementarity” of composite systems

1. Mini workshop on “Structure and production of charmed baryons II” 2014, Aug. 7-9, J-PARC, Tokai A study for “elementarity” of composite systems Hideko Nagahiro1,2, Atsushi Hosaka2 1 Nara Women’s University, Japan 2RCNP, Osaka University, Japan References: H. Nagahiro, and A. Hosaka, e-print arXiv:1406.3684 [hep-ph] H. Nagahiro, and A. Hosaka, PRC88(2013)055203 (published as an Editors’ Suggestion)

2. Introduction & motivation Many candidates for exotic hadrons : not simple (or )state , (980)/(980), …, (1260), …, (1405), (1535), …, (3872), … vs. “elementary” particle (“quasi-particle”) Dynamically generated resonance (~ can be or …) Nature of possible exotic hadrons Physical state must be a mixture of possible quantum states + … physical state The question : How much they contain “elementary” components ?

3. “compositeness” or “elementarity” bare, un-renormalized, field vanishesfor a bound state Weinberg, PR130(63)776 Lurie-Macfarlane, PR136(65)B816 Weinberg, PR137(65)B682 Hyodo-Jido-Hosaka, PRC(12)015201 ... the wave function renormalization Z “compositeness condition for a bound state probability of finding the elementary particle : … … + + + + + “quasi-particle” of infinite masswith Bound state • Weinberg, PR130(63)776 • a bound state can be represented by introducing a “quasi-particle” • with infinite bare mass and hence Z = 0 • Lurie-Macfarlane, PR136(65)B816 • equivalence between a four-Fermi theory and a Yukawa theory the renormalization constant Z for a Yukawa particle is equal to zero • Weinberg, PR136(65)B816 • Z < 0.2 for deuteron system for a resonant state ?? Discussions given for bound states Hyodo-Jido-Hosaka, PRC85(12)015201

4. Contents of this talk • Wave function renormalization constant ( “elementarity”) is zero for any resonant or bound statedynamically generated by WT type interaction • How in a Yukawa model we can introduce the “fictitious” elementary particle which is equivalent to the s-wave dynamical state by • wave function renormalization constant for the fictitious particle • Underlying mechanism of • Model (cut-off & representation) dependence of • Choice of “elementary particle” as a measure • How we should employ the constant to understand the hadron natures. • A special case of zero-energy bound state • Underlying mechanism for can be different from others

5. D. Lurie, A.J.Macfarlane, PR136(64)B816 D. Lurie, Particle and Fields, 1968 A brief review of “compositeness condition” Yukawa theory with constant Bound state (four-point) model … + = + + … + + = = = wave function renormalization Weinberg also uses this eq. by estimating from low energy p-n scattering.

6. Equivalent Yukawa model to a resonant model? Yukawa theory with constant Resonance case (composite model) Bound state model … + = + + … + + = ? = = wave function renormalization

7. For an -wave composite states Interaction kernel : Weinberg-Tomozawa type energy-dependent [1] Olle-Oset, NPA620(97)438 scattering amplitude with on-shell factorization[1] composite pole + = + + … regularize appropriately loop function by dim. regularization / 3dim cut-off bound state case (constant ) (physical) coupling

8. Equivalent Yukawa model shifted amplitude scattering amplitude with on-shell factorization[1] composite pole + = + + …

9. Equivalent Yukawa model shifted amplitude Yukawa term Yukawa term bare mass of the fictitious elementary particle (cf. Hyodo08) “fictitious” particle Energy-dependent Yukawa coupling

10. Equivalent Yukawa model Composite model Yukawa model … + + + … + + Energy-dependent Yukawa coupling

11. Equivalent Yukawa model Composite model Yukawa model … + + + … + + self-energy full propagator of the fictitious elementary particle How about ? bound state case wave function renormalization constant due to energy-dependence of

12. Wave function renormalization constant wave function renormalization constant due to energy-dependence of

13. Wave function renormalization constant zero ! renormalized coupling finite bare mass of the fictitious elementary particle wave function renormalization constant infinite bare coupling infinite due to energy-dependence of

14. The condition Composite model Yukawa model • The composite states can be equivalently represented by a “quasi-particle” with infinite bare massand hence with [Weinberg(63)] • The “elementarity” is zero for any composite state by WT term … + + + + + … hadronic scale [1] Jido-Oller-Oset-Ramos-Meissner, NPA725(03)181. [2] Inoue-Oset-Vicente Vacas, PRC65(02)035204. [3] Hyodo-Jido-Hosaka, PRC78(08)025203. chiral unitary approach un-natural (1405)… bound state[1] (1535) … bound state[2](but large ? [3]) in the composite model = in the Yukawa model our assumption

15. With an explicit pole term Interaction kernel : Weinberg-Tomozawa type + explicit pole term Introduced by an un-natural cut-off [Hyodo (08)] Introduced by an un-natural cut-off [Hyodo (08)] bare mass of “fictitious” particle Equivalent Yukawa term renormalized coupling finite bare coupling finite If there is an explicit pole term, “elementarity” Z is finite. Wave function renormalization constant finite

16. With an explicit pole term Interaction kernel : Weinberg-Tomozawa type + explicit pole term Introduced by an un-natural cut-off [Hyodo (08)] Scattering amplitude Arbitrariness of “elementarity” • Physical observables are invariant under the simultaneous change in and . • Multiple interpretations for a physical state • Z can be any value and cannot be determined in a model-independent manner. Necessary to specify a model (cut-off scale to be used as a “measure”)

17. Representation dependence of cf.) scattering in sigma model Yukawa model (fictitious particle or “quasi-particle”) “quasi-particle” dominates practically zeroin Nonlinear model + Linear model + • They all have the same , but is different

18. Representation dependence of 92.4 MeV = 138 MeV Linear model 0 1.4 1.2 Yukawa nonlinear 1.0 0.8 0 Re 0.6 0.4 0.2 • the pole positions are the same for all representations • Each indicates the “elementarity” measured by different elementary particle: the elementary particle in different models are different 0 0.2 1500 2000 500 1000 2500 3000 bare mass [MeV] Necessary to specify a model ( = representation)

19. Another mechanism of for zero energy bound state for finite finite 0 finite renormalized coupling wave function renormalization constant or “elementarity” Derivative of the loop function for with

20. Another mechanism of for zero energy bound state Example : Interaction kernel : Weinberg-Tomozawa type + explicit pole term [MeV] 260 255 250 245 240 for but ( 480 MeV,MeV) 1 0.8 0.6 0.4 • Different mechanism from • It does not necessarily mean that“infinite bare mass” of quasi-particle • Z=0 for B=0 does not excludean elementary state near B=0 92.4 [MeV] 550 [MeV] 0.2 0 20 15 10 5 0 binding energy [MeV]

21. Summary • Wave function renormalization constant can be zero for any resonant statedynamically generated by WT type interaction • We have shown that the amplitude can be equivalently represented by a Yukawa model with a “quasi-particle” having infinite bare mass and hencewith . • Different from the “renormalization” due to a divergence of G(s) • The underlying mechanism is the same as a bound state (constant interaction) case • Model (cut-off & representation) dependence of • The arbitrariness leads to multiple interpretations for a physical state • Among a number of possible models, we have a model with . • Z cannot be determined from experiments in a model-independent manner. • Specify firstly : “What is an “elementary particle” to be used as a measure ? ” … choice of a model : problem of “economization” • A special case of zero-energybound state • Underlying mechanism for can be different from other cases • does not exclude an elementary state near the physical state

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23. Another mechanism of for zero energy bound state Example : Interaction kernel : Weinberg-Tomozawa type + explicit pole term [MeV] 260 255 250 245 240 for but ( 480 MeV,MeV) • Different mechanism from • It does not necessarily mean that“infinite bare mass” of quasi-particle • Z=0 for B=0 does not excludean elementary state near B=0 92.4 [MeV] 550 [MeV] 20 15 10 5 0 binding energy [MeV]

24. Model dependence of Yukawa model (“quasi-particle”) “quasi-particle” dominates practically zero in Nonlinear model + linear model +

25. Scattering amplitude in the linear model tree amplitude in the linear model + Scattering amplitude + +… + , , +… + + Wave function renormalization constant ,

26. With an explicit pole term 150 Scattering amplitude 100 50 0 50 Pole position 100 1 Interaction kernel 0.5 0 0.5 1 0 0.2 0.4 1 0.6 0.8

27. Constant interaction (bound state) case scattering amplitude with constant (positive constant) Yukawa term fictitious mass and bare coupling Yukawa model bare coupling must be proportional to in the large limit (limit)