1

Possible molecular bound state of two charmed baryons

- hadronic molecular state of two Λc s -

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

[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 ( Λc, Σc, Σc

* ) for two reasons.

Λc 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 (JP=0+) in two-body systems of Λc, Σc, Σc

* are considered as follows. Especially, our study is hadronic molecular state of two Λc s

(JP=0+ I=0).

Λc Λc

No open channels

Relevant channels

Λc ΣcOpen channel

Relevant channels

ΣcΣc

Open channel

relevant channels

7

OVERVIEW

[TARGET] : Hadronic molecular state of two Λcs (JP=0+ I=0)

[MODEL] : One-pion exchange potential + short range cutoff

• Long range : one-pion exchange potential

• Short range : phenomenological cutoff

Λc Λc

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 Λc s can not exchange a single pion

8

“POTENTIAL MODEL”

9

FRAMEWORK 　• Λc, Σc, Σc

* : Heavy quark limit (mQ → ∞)

• 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 g2 and g4.

(The g2 and g4 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

→

12

“NUMERICAL CALCULATION”

13

Channel 1 : Λc Λc (1S0)

Channel 2 : Σc Σc (1S0)

Channel 4 : Σc* Σc

* (5D0)

Channel 3 : Σc* Σc

* (1S0)

Channel 5 : Σc Σc* (5D0)

COUPLED CHANNELS Schrödinger equation of coupled channels

: (Transition) Potential of channel i to channel j

: Wave function of channels ie.g.

e.g.

Notation

To solve Schrödinger equation, we use variation method “Gaussian expansion method”. [E. Hiyama et al. Progress 51, (2003)]

14

NUMERICAL RESULTS 　• 3 channels [Λc Λc (1S0), Σc Σc (1S0), Σc

* Σc* (1S0)] (Only S-wave channels)

• 4 channels [(Λc Λc (1S0), Σc Σc (1S0), Σc* Σc

* (1S0), Σc* Σc

* (5D0)]Three S-wave channels + D-wave channel

• 4 channels [Λc Λc (1S0), Σc Σc (1S0), Σc* Σc

* (1S0), Σc Σc* (5D0)]

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.

→ 　 Σc Σc* (5D0) channel is more important for bound states.

15

5 channels [Λc Λc (1S0), Σc Σc (1S0), Σc* Σc

* (1S0), Σc* Σc

* (5D0), Σc Σc* (5D0)]

Λ = 1.3 [GeV]

Λ = 1.0 [GeV]

Radial wave function

← Beyond our model

(Full channels)

16

“SUMMARY”

17

SUMMARY

We get some binding solutions of two Λc s.

D-wave channels (tensor force) especially, Σc Σc*

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 Λc s.

18

“BACKUP SLIDES”

19

(i) i, j = 1~4

(ii) i ≠ 5, j=5

(iii) i = 5, j=5

Potential

e.g. : Pauli matrix: Transition spin

: Spin 3/2 matrix : Spin operator

: Coupling constant: Effective pion mass: Effective cutoff

20

Transition potentials (Λc Λc → another channels)

21

Diagonal potentials

22

Transition potentials (Other transition potentials)

23

Transition potentials (Other transition potentials)

24

5 channels [Λc Λc (1S0), Σc Σc (1S0), Σc* Σc

* (1S0), Σc* Σc

* (5D0), Σc Σc* (5D0)]

Λ = 1.1 [GeV]

Λ = 1.0 [GeV]

Radial wave function

(Full channels)

25

5 channels [Λc Λc (1S0), Σc Σc (1S0), Σc* Σc

* (1S0), Σc* Σc

* (5D0), Σc Σc* (5D0)]

Λ = 1.3 [GeV]

Λ = 1.2 [GeV]

(Full channels)

26

5 channels [Λc Λc (1S0), Σc Σc (1S0), Σc* Σc

* (1S0), Σc* Σc

* (5D0), Σc Σc* (5D0)]

Λ = 1.5 [GeV]

Λ = 1.4 [GeV]

(Full channels)

27

VARIATION METHOD The wave functions ψi (i=1,5) are expanded in term of a

set of Gaussian basis functions.[Prog 51,203]Gaussian expansion method

Nnl : normalization constant

Range parameter {nmax, r1, rmax}

……

[Bas

e fu

nctio

n]

…

28

SPIN MATRIXDEFINE (Transition spin for static limit)

Transition spin :

Define :

(2× ４ )

DEFINE(spin3/2 matrix)

Sin3/2 matrix :

Define :

(4×4)