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

CONTENTS  INTRODUCTION  POTENTIAL MODEL  NUMERICAL CALCULATION  SUMMARY 2

“INTRODUCTION” 3

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

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

6 The lowest states (J P =0 + ) in two-body systems of Λ c, Σ c, Σ c * are considered as follows. Especially, our study is hadronic molecular state of two Λ c s (J P =0 + I=0). ΛcΛc ΛcΛc No open channels Relevant channels ΛcΛc ΣcΣc Open channel Relevant channels ΣcΣc ΣcΣc Open channel relevant channels

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

“POTENTIAL MODEL” 8

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

10 Heavy quark spin symmetry reduces 6 coupling constants to 2 independent ones and our choices are g 2 and g 4. (The g 2 and g 4 are estimated from strong decay.) Effective Lagrangian ( ) chirally invariant (NG boson field) ← Quark model [T. Yan et al. PRD 46, (1992)]

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

“NUMERICAL CALCULATION” 12

Channel 1 : Λ c Λ c ( 1 S 0 ) Channel 2 : Σ c Σ c ( 1 S 0 ) Channel 4 : Σ c * Σ c * ( 5 D 0 ) Channel 3 : Σ c * Σ c * ( 1 S 0 ) Channel 5 : Σ c Σ c * ( 5 D 0 ) COUPLED CHANNELS Schrödinger equation of coupled channels 13 : (Transition) Potential of channel i to channel j : Wave function of channels i e.g. Notation To solve Schrödinger equation, we use variation method “Gaussian expansion method”. [E. Hiyama et al. Progress 51, (2003)]

NUMERICAL RESULTS 14 3 channels [Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 )] (Only S-wave channels) 4 channels [(Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 ), Σ c * Σ c * ( 5 D 0 )] Three S-wave channels + D-wave channel 4 channels [Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 ), Σ c Σ c * ( 5 D 0 )] Three S-wave channels + D-wave channel → There is no bound states in three S-wave channels. → D-wave channels (tensor force) are important for bound states. → Σ c Σ c * ( 5 D 0 ) channel is more important for bound states.

15 5 channels [Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 ), Σ c * Σ c * ( 5 D 0 ), Σ c Σ c * ( 5 D 0 )] Λ = 1.3 [GeV] Λ = 1.0 [GeV] Radial wave function ← Beyond our model (Full channels)

“SUMMARY” 16

SUMMARY  We get some binding solutions of two Λ c s.  D-wave channels (tensor force) especially, Σ c Σ c * 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 Λ c s. 17

“BACKUP SLIDES” 18

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

20 Transition potentials (Λ c Λ c → another channels)

21 Diagonal potentials

22 Transition potentials (Other transition potentials)

23 Transition potentials (Other transition potentials)

24 5 channels [Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 ), Σ c * Σ c * ( 5 D 0 ), Σ c Σ c * ( 5 D 0 )] Λ = 1.1 [GeV] Λ = 1.0 [GeV] Radial wave function (Full channels)

25 5 channels [Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 ), Σ c * Σ c * ( 5 D 0 ), Σ c Σ c * ( 5 D 0 )] Λ = 1.3 [GeV] Λ = 1.2 [GeV] (Full channels)

26 5 channels [Λ c Λ c ( 1 S 0 ), Σ c Σ c ( 1 S 0 ), Σ c * Σ c * ( 1 S 0 ), Σ c * Σ c * ( 5 D 0 ), Σ c Σ c * ( 5 D 0 )] Λ = 1.5 [GeV] Λ = 1.4 [GeV] (Full channels)

VARIATION METHOD The wave functions ψ i (i=1,5) are expanded in term of a set of Gaussian basis functions. 27 [Prog 51,203] Gaussian expansion method N nl : normalization constant Range parameter {n max, r 1, r max } … … [Base function] …

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