# Presentation: 207445

## Λ(1405) in a hybrid quark model

- Introduction
- Formulation
- Results
- Summary

- C. Nakamoto (Suzuka College of Technology)
- H. Nemura (Advanced Meson Science Laboratory, DRI, RIKEN)

- PANIC05    24OCT2005

- Introduction

- Λ(1405) ⇒ *q*3 ?		   *K* *N* ?		   or ***q*****4*****q*** ?

- If we introduce the minimum realistic *q*-*q* interaction in a quark model, how is Λ(1405) represented?

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## What is ‘the minimum realistic’?

- Confinement mechanism between quarks

- If the quark-Hamiltonian includes  these effects and reproduces the baryon ground states, how is Λ(1405) represented?

- (2) Gluon-exchange effect

- (3) Meson-exchange effect to describe 	the nuclear force

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## (1) Confinement mechanism between quarks
	⇒ confinement potential : - (λ・λ) a r
	 A free parameter a mainly determines the energy shift of the baryon ground states. 
	 It is determined to reproduce the overall baryon ground states.
a = 43.5 MeV・fm-1

**	****⇒**** ****confinement potential : - (λ****・****λ) *****a***** *****r***
- 	 A free parameter ***a*** mainly determines the energy shift of the baryon ground states.
- 	 It is determined to reproduce the overall baryon ground states.
***a**** *= 43.5 MeV・fm-1

- 2.Formulation

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## (2) Gluon-exchange effect
	⇒	one-gluon exchange potential
	1/4 (λ・λ) αS 
×[1/r-π/2{1/mi2+1/mj2+4/(3mimj)(σ・σ)}δ(r)]
・ the finite spatial extent of the constituent quarks
		δ(r) ⇒ Λ3/(8π3/2)exp[-1/2(Λr)2]

	αS = 1.38 (free parameter)
	mud = 340 MeV, ms = 580 MeV (free parameter)
	Λ= 5 fm-1

- 	⇒	one-gluon exchange potential
- 	1/4 **(λ****・****λ) **αS
- ×[1/r-π/2{1/mi2+1/mj2+4/(3mimj)(σ・σ)}δ(r)]
- ・ the finite spatial extent of the constituent quarks
- 		δ(r) ⇒ Λ3/(8π3/2)exp[-1/2(Λr)2]
- 	αS = 1.38 (free parameter)
- 	*m**ud* = 340 MeV, *m**s* = 580 MeV (free parameter)
- 	Λ= 5 fm-1

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## (3) Meson-exchange effect
	⇒	NG-boson exchange potential
・ πand K-exchange potential with each form factor

	g82/(4π)μ2/(12mimj)
	×[ exp(-μr)/r – (Λ/μ)2 exp(-Λr)/r ]
	×(τ・τ or 2PF)(σ・σ)

	 g82/(4π) = (3/5)2gπNN/(4π)mq2/mN2 = 0.67
	Λπ= 1.8 fm-1 ( free parameter)
	ΛK = 2.3 fm-1 ( free parameter)

- 	⇒	NG-boson exchange potential
- ・ πand *K*-exchange potential with each form factor
- 	g82/(4π)μ2/(12mimj)
- 	×[ exp(-μr)/r – (Λ/μ)2 exp(-Λr)/r ]
- 	×(τ・τ or 2*P**F*)(σ・σ)
- 	 g82/(4π) = (3/5)2gπNN/(4π)*m**q**2**/m**N**2** *= 0.67
- 	Λπ= 1.8 fm-1 ( free parameter)
- 	Λ*K* = 2.3 fm-1 ( free parameter)

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## ・ σ- exchange potential as the chiral partner of π

	- gσ2/(4π) [ exp(-μr)/r –  exp(-Λr)/r ]

	 gσ2/(4π) = g82/(4π) = 0.67
	μσ= √{(2mq)2+mπ2} = 694 MeV
	Λσ= 5 fm-1 

- 	- gσ2/(4π) [ exp(-μr)/r –  exp(-Λr)/r ]
- 	 gσ2/(4π) = g82/(4π) = 0.67
- 	μσ= √{(2*m**q*)2+*m*π2} = 694 MeV
- 	Λσ= 5 fm-1

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## Quark-Hamiltonian

***H = **** ****   *****(*****m******i******+p******i******2******/2m******i*****)****-*****K******cm****** ***

***+        *****(*****U******ij*****Conf*****+U******ij*****OGEP*****+U******ij*****π*****+U******ij******K ******+U******ij*****σ****)**

- ・***U******ij*****Conf ****: ****confinement potential**
- ・***U******ij*****OGEP ****: one-gluon exchange potential **
- ・***U******ij*****π ****: π exchange **
**・*****U******ij******K***** ****: *****K*****-meson exchange **
**・*****U******ij*****σ ****: σ-meson exchange**

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## Slide 9

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## Many-body calculation	⇒ Stochastic Variational Method 	   ( K.Varga and Y. Suzuki, PRC52 (1995) 2885)

- We solve the five-body problem dynamically.

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## Results

- Baryon ground states
- N(939) : 949 MeV
- Δ(1232) : 1245 MeV
- Λ(1116) : 1116 MeV
- Σ(1193) : 1235 MeV
- Ξ(1318) : 1320 MeV

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## Results

- Baryon ground states
- N(939) : 949 MeV
- Δ(1232) : 1245 MeV
- Λ(1116) : 1116 MeV
- Σ(1193) : 1235 MeV
- Ξ(1318) : 1320 MeV

- Meson spectrum
- π(140) : 250 MeV
- K(496) : 526 MeV
- K*(892) : 905 MeV

- PANIC05    24OCT2005

## Results

- Baryon ground states
- N(939) : 949 MeV
- Δ(1232) : 1245 MeV
- Λ(1116) : 1116 MeV
- Σ(1193) : 1235 MeV
- Ξ(1318) : 1320 MeV

- Baryon excited states
- N(1440) : 1399 MeV
- Δ(1600) : 1609 MeV
- Λ(1600) : 1568 MeV
- Σ(1660) : 1648 MeV
- Ξ(unknown) : 1750 MeV

- Meson spectrum
- π(140) : 250 MeV
- K(496) : 526 MeV
- K*(892) : 905 MeV

- PANIC05    24OCT2005

## Contribution from each term

N(949) K:1745 Vconf:291 Vcc:-911 Vcm:-13
		  Vπ:(36 –148)	Vσ:(–148 +97)

N(1398) K:1711 Vconf:489 Vcc:-659 Vcm:-30
		  Vπ:(25 –105)	Vσ:(–99 +67)

- N(949) K:1745 Vconf:291 Vcc:-911 Vcm:-13
- 		  Vπ:(36 –148)	Vσ:(–148 +97)
- N(1398) K:1711 Vconf:489 Vcc:-659 Vcm:-30
- 		  Vπ:(25 –105)	Vσ:(–99 +67)

- g82/(4π)μ2/(12mimj)
- ×[ exp(-μr)/r – (Λ/μ)2 exp(-Λr)/r ](τ・τ)(σ・σ)

- - gσ2/(4π) [ exp(-μr)/r –  exp(-Λr)/r ]

- PANIC05    24OCT2005

## Results

- Baryon ground states
- N(939) : 949 MeV
- Δ(1232) : 1245 MeV
- Λ(1116) : 1116 MeV
- Σ(1193) : 1235 MeV
- Ξ(1318) : 1320 MeV

- Baryon excited states
- N(1440) : 1399 MeV
- Δ(1600) : 1609 MeV
- Λ(1600) : 1568 MeV
- Σ(1660) : 1648 MeV
- Ξ(unknown) : 1750 MeV

- Meson spectrum
- π(140) : 250 MeV
- K(496) : 526 MeV
- K*(892) : 905 MeV

- Λ(1405)
*q*3  : 1421 MeV
*q*4*q* : 1388 MeV

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- PANIC05    24OCT2005

## Summary

- We investigated the baryon spectrum and the Λ(1405) in the minimum realistic quark model, involving a confinement potential, a one-gluon exchange potential and π, K and σ-exchange potentials.
- Free parameters were determined to reproduce the baryon ground states.
- Light mesons and some positive parity excited baryons including Roper resonance were reproduced reasonably.
- Reproduction of Roper resonance may be due to the large contribution from short-range π-exchange potential because owing to the small cut-off parameter, and the introduction of σ-exchange potential between quarks.
- The energy of *q*4*q *– state as Λ(1405) become lower than that of *q*3-state in the present model.
- We will investigate further ; other set of parameters, contribution from the non-central force, channel-coupling effect...

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