Wakafumi Meguro, Yan- Rui Liu, Makoto Oka (Tokyo Institute of Technology)

1. Possible molecular bound state of two charmed baryons- hadronic molecular state of two Λcs - Wakafumi Meguro, Yan-Rui Liu, Makoto Oka (Tokyo Institute of Technology) BARYONS’10 Dec. 8, 2010, Osaka, Japan

2. CONTENTS • INTRODUCTION • POTENTIAL MODEL • NUMERICAL CALCULATION • SUMMARY

3. “INTRODUCTION”

4. INTRODUCTION Hadronic molecule : Bound state of hadrons in hadron dynamics We consider there might be hadronic (exotic) molecular statesin charmed baryons ( Λc, Σc, Σc* ) for two reasons. [inter-hadron distance] > [confinement size] e.g.deuteron(NN), triton(NNN), hypertriton(Λpn) Σc* Σc Λc N N

5. e.g. Two body systems (ii) Heavy quark spin symmetry [Kinetic Energy] vs [Potential] The effect of heavy quark spin is suppressed in heavy baryons (i) Kinematics Because the reduced mass becomes larger in heavy baryons, the kinetic term is suppressed. → The coupled channels effects in heavy baryons become larger [PDG, Particle Listings, CHARMED BARYONS]

6. The lowest states (JP=0+) in two-body systems of Λc, Σc, Σc* are considered as follows. Especially, our study is hadronic molecular state of two Λcs (JP=0+ I=0). Relevant channels Λc No open channels Relevant channels Open channel Λc Σc Λc relevant channels Open channel Σc Σc

7. OVERVIEW [TARGET] : Hadronic molecular state of two Λcs (JP=0+ I=0) [MODEL] : One-pion exchange potential + short range cutoff Λc • Long range : one-pion exchange potential 5 channels • Short range : phenomenological cutoff The longest-range interactions is important for molecular state. Λc → one-pion exchange potential Two Λcs can not exchange a single pion →coupled channels [METHOD] : Variation method (Gaussian expansion method) [E. Hiyama et al. Progress 51, (2003)]

8. “POTENTIAL MODEL”

9. FRAMEWORK • Λc, Σc, Σc* : Heavy quark limit (mQ→ ∞) • One-pion exchange potential NG boson (pion) → Couplings between pion and charmed baryons are related with heavy quark spin symmetry. Charmed baryon Charmed baryon [T. Yan et al. PRD 46, (1992)] • Form factor → Monopole form factor : cutoff To simplify the calculation, all cutoffs are put as same value.

10. chirally invariant Effective Lagrangian ( ) [T. Yan et al. PRD 46, (1992)] • (NG boson field) • • Heavy quark spin symmetry reduces 6 coupling constants to 2 independent ones and our choices are g2 and g4. ← Quark model (The g2 and g4 are estimated from strong decay.)

11. Strong decay : Decay amplitude : Solid angle of pion → The ambiguity oftheir sign is irrelevant to binding solutions. [PDG, Particle Listings, CHARMED BARYONS]

12. “NUMERICAL CALCULATION”

13. COUPLED CHANNELS Schrödinger equation of coupled channels Notation Channel 1 : ΛcΛc (1S0) Channel 2 : ΣcΣc (1S0) : Wave function of channels i : (Transition) Potential of channel i to channel j Channel 3 : Σc*Σc* (1S0) e.g. Channel 4 : Σc*Σc* (5D0) To solve Schrödinger equation, we use variation method “Gaussian expansion method”. Channel 5 : ΣcΣc* (5D0) [E. Hiyama et al. Progress 51, (2003)] e.g.

14. NUMERICAL RESULTS • 3 channels [ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0)] (Only S-wave channels) ThreeS-wave channels + D-wave channel ThreeS-wave channels + D-wave channel → There is no bound states in three S-wave channels. • 4channels [(ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0), Σc*Σc* (5D0)] • 4channels [ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0), ΣcΣc* (5D0)] →D-wave channels (tensor force) are important for bound states. →ΣcΣc* (5D0) channel is more important for bound states.

15. (Full channels) Λ = 1.0 [GeV] Radial wave function Λ = 1.3 [GeV] 5channels [ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0), Σc*Σc* (5D0), ΣcΣc* (5D0)] ← Beyond our model

16. “SUMMARY”

17. SUMMARY • We get some binding solutions of two Λcs. • D-wave channels (tensor force) especially, ΣcΣc* channel is important for bound states. • In case of Λ=1.0, result is molecule-like and in case of Λ=1.3, result is beyond our model. • It is possible to have a hadronic molecular state of two Λcs.

18. “BACKUP SLIDES”

19. Potential (i) i, j = 1~4 (ii) i≠ 5, j=5 (iii) i= 5, j=5 : Coupling constant : Effective pion mass : Effective cutoff : Spin operator : Pauli matrix : Transition spin e.g. : Spin 3/2 matrix

20. Transition potentials (ΛcΛc → another channels)

21. Diagonal potentials

22. Transition potentials (Other transition potentials)

23. Transition potentials (Other transition potentials)

24. (Full channels) Λ = 1.0 [GeV] Radial wave function Λ = 1.1 [GeV] 5channels [ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0), Σc*Σc* (5D0), ΣcΣc* (5D0)]

25. (Full channels) Λ = 1.2 [GeV] Λ = 1.3 [GeV] 5channels [ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0), Σc*Σc* (5D0), ΣcΣc* (5D0)]

26. (Full channels) Λ = 1.4 [GeV] Λ = 1.5 [GeV] 5channels [ΛcΛc (1S0), ΣcΣc (1S0), Σc*Σc* (1S0), Σc*Σc* (5D0), ΣcΣc* (5D0)]

27. VARIATION METHOD The wave functions ψi (i=1,5) are expanded in term of a set of Gaussian basis functions. Gaussian expansion method [Prog 51,203] Nnl: normalization constant … … [Base function] … Range parameter {nmax, r1, rmax}

28. SPIN MATRIX DEFINE (Transition spin for static limit) (2×４) Sin3/2 matrix : Transition spin : Define : Define : DEFINE(spin3/2 matrix) (4×4)