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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
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CONTENTS
INTRODUCTION
POTENTIAL MODEL
NUMERICAL CALCULATION
SUMMARY
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“INTRODUCTION”
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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*
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(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]
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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
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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
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“POTENTIAL MODEL”
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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.
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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)]
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[PDG, Particle Listings, CHARMED BARYONS]
Strong decay
The ambiguity of their sign is irrelevant to binding solutions.
: Decay amplitude
: Solid angle of pion
→
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“NUMERICAL CALCULATION”
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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)]
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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.
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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)
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“SUMMARY”
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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.
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“BACKUP SLIDES”
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(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
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Transition potentials (Λc Λc → another channels)
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Diagonal potentials
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Transition potentials (Other transition potentials)
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Transition potentials (Other transition potentials)
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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)
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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)
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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)
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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]
…
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SPIN MATRIXDEFINE (Transition spin for static limit)
Transition spin :
Define :
(2× 4 )
DEFINE(spin3/2 matrix)
Sin3/2 matrix :
Define :
(4×4)