12008, Mar. 18th

Λ(1405) in chiral dynamics

TU Münchena YITP, Kyotob

Tetsuo Hyodoa,b

KN int. below threshold

KN

πΣ

KN scatt.?

energy2

Introduction : (well) known facts on Λ(1405)

Mass : 1406.5 ± 4.0 MeVWidth : 50 ± 2 MeVDecay mode : 100%

Introduction : Λ(1405)

Coupled channel multi-scattering

R.H. Dalitz, T.C. Wong and G. Rajasekaran, PR153, 1617 (1967)

“naive” quark model : p-wave ~1600 MeV?

N. Isgur and G. Karl, PRD18, 4187 (1978)

Λ(1405)

Two poles?

KNπΣ

reim

3

Contents

Phenomenology of KN interaction

Structure of the Λ(1405)

Contents

T. Hyodo, W. Weise, 0712,1613 [nucl-th], Phys. Rev. C, in press.

T. Hyodo, D. Jido, L. Roca, 0712.3347 [hep-ph], Phys. Rev. D, in press.

T. Hyodo, D. Jido, A. Hosaka, 0803.2550 [nucl-th]

A. Doté, T. Hyodo, W. Weise, 0802.0238 [nucl-th], Nucl. Phys. A, in press

・Construction of local KN potential by chiral dynamics

・Application to three-body KNN system

・Nc Behavior and quark structure

・Dynamical or CDD (genuine quark state) ?

・Introduction to the chiral unitary approach

4

Chiral unitary approach and Λ(1405)

N. Kaiser, P. B. Siegel, W. Weise, Nucl. Phys. A594, 325 (1995)E. Oset, A. Ramos, Nucl. Phys. A635, 99 (1998)J. A. Oller, U. G. Meissner, Phys. Lett. B500, 263 (2001)M.F.M. Lutz, E. E. Kolomeitsev, Nucl. Phys. A700, 193 (2002), .... many others

S = -1, KN s-wave scattering : Λ(1405) in I=0

strong attraction (

5T. Hyodo, S.I. Nam, D. Jido, A. Hosaka, Phys. Rev. C68, 018201 (2003)

200

150

100

50

0

!T [

mb

]

300200100

K-p

70

60

50

40

30

20

10

0300200100

"0#

200

150

100

50

0300200100

"+$%

60

50

40

30

20

10

0

!T [

mb

]

300200100Plab [MeV/c]

K0n

60

50

40

30

20

10

0300200100

Plab [MeV/c]

"0$0

80

60

40

20

0300200100

Plab [MeV/c]

"%$+

Total cross sections of K-p scatteringChiral unitary approach and Λ(1405)

1660168017001720 -20

-40-60-800.05

0.100.150.200.25

0.050.100.15

0.200.25

Λ(1670)

z=1690-22i

Re[z]

Im[z]

|T|

6

Description of the resonancesPoles of the amplitude : resonance

∼

Tij(√

s) ∼ gigj√s − MR + iΓR/2

1360

1380

1400

1420

1440

-20

-40

-60

-80

0.5

1.0

1.5

0.5

1.0

1.5

!(1405)

Re[z]Im[z]

|T|

Chiral unitary approach and Λ(1405)

Real part MassImaginary part Width/2

Residues Couplings

♦Successful description of KN scattering♦Two poles for the Λ(1405)

7

Deeply bound (few-body) kaonic nuclei?Motivation

K- + p

MeV-27K=E

MeV04=Γ

1 2 3 r fm0

-50

-200

-300

-400

-500

nucl

KUMeV

Λ(1405)Σ+π

Λ+π

K- + pp

MeV-48K=E

MeV61=Γ

1 2 3 r fm0

-50

-200

-300

-400

-500

nucl

KUMeV

H2

KΣ+π

Λ+π

K- + 3He

MeV-108K=E

MeV02=Γ

1 2 3 r fm0

-50

-200

-300

-400

-500

nucl

KU

MeV

H3

K

Σ+π

Λ+π

Y. Akaishi & T. Yamazaki, Phys. Rev. C 65 (2002) 044005

T. Yamazaki & Y. Akaishi, Phys. Lett. B 535 (2002) 70

Potential is purely phenomenological.What does chiral dynamics tell us about it?

8

Construction of the effective interaction

Effective interaction based on chiral SU(3) dynamics

Coupled-channel BS + real interaction

Result of chiral dynamics --> single channel potential

few-body kaonic nuclei(exact)

Single-channel BS + complex interaction

(approximate)Schrödinger equation + local potentialcomplex, energy-dependent

9

Construction of the single channel interactionChannels 1 and 2 --> effective int. in 1

Construction of the effective interaction

tree effect of channel 2

Equivalent to the coupled-channel equations

single channel

-15

-10

-5

0

Vef

f [fm

]

1440140013601320!s [MeV]

Im Veff, full

I=0

-14

-13

-12

-11

-10

-9

Vef

f [fm

]

1440140013601320!s [MeV]

V (tree) Re Veff, full

I=0

10

Single channel KN interaction with πΣ dynamicsConstruction of the effective interaction

effect of πΣ

Strength : comparable with the WT term~ 1/2 of phenomenological (Akaishi-Yamazaki) potentialeffect of πΣ resummation in KN channel is not large

4

2

0

-2

F KN

[fm

]

1440140013601320!s [MeV]

Re F, full

KN(I=0)

1.5

1.0

0.5

0.0

-0.5

-1.0

F !" [f

m]

1440140013601320!s [MeV]

!"(I=0)

Im F, full

11

Scattering amplitude in KN and πΣConstruction of the effective interaction

Binding energy : B = 15 MeV 30 MeV

Resonance in KN : around 1420 MeV

Very strong attraction in KN (higher energy) --> bound stateStrong attraction in πΣ (lower energy) --> resonance

-120

-100

-80

-60

-40

-20

0

Im z

[MeV

]1440140013601320

Re z [MeV]

!" resonance

KN bound stateWithout off-diagonal

KNπΣ

12

Origin of the two-pole structure

KN πΣ

Two poles : natural consequence of chiral interaction

Construction of the effective interaction

Chiral interaction

-120

-100

-80

-60

-40

-20

0

Im z

[MeV

]1440140013601320

Re z [MeV]

!" resonance

z2(full)

z1(full)

KN bound state

KNπΣ

higher order correction? --> theoretical uncertainty (later)B. Borasoy, R. Nissler, W. Weise, Eur. Phys. J. A25, 79-96 (2005)

13

Comparison with phenomenological potential

KN πΣ

Construction of the effective interaction

Chiral interaction phenomenologicalT. Yamazaki, Y. Akaishi, Phys. Rev. C76, 045201 (2007)

Absence of πΣ diagonal coupling--> absence of πΣ dynamics, resonance--> strong (×2) attractive interaction in KN

KN πΣ

πΣ -> πΣ attraction : flavor SU(3) symmetryenergy dependence : derivative coupling

4

2

0

-2

F KN

[fm

]

1440140013601320!s [MeV]

Re (original)

I=0

0.8

0.6

0.4

0.2

0.0

F KN

[fm

]

1440140013601320!s [MeV]

Im (original)I=1

14

KN amplitude with local potentialConstruction of the effective interaction

: to reproduce the resonance

4

2

0

-2

F KN

[fm

]

1440140013601320!s [MeV]

Re (local) Re (original)

I=0

0.8

0.6

0.4

0.2

0.0

F KN

[fm

]

1440140013601320!s [MeV]

Im (local) Im (original)

I=1

Deviation at lower energy : BS eq. local potential + Schrödinger eq.

agreement around threshold : OK

15

Correction of the strength of the potentialConstruction of the effective interaction

4

2

0

-2

F KN

[fm

]

1440140013601320s1/2 [MeV]

Re (corrected) Re (original)

I=0

-1000

-800

-600

-400

-200

0

U(0

) [M

eV]

1440140013601320s1/2 [MeV]

I=0

0.8

0.6

0.4

0.2

0.0

F KN

[fm

]

1440140013601320s1/2 [MeV]

Im (corrected) Im (original)

I=1

-500

-400

-300

-200

-100

0

U(0

) [M

eV]

1440140013601320s1/2 [MeV]

I=1

Pote

ntia

lAm

plitu

de

16

Summary 1 : KN interactionWe derive the single-channel local potential based on chiral SU(3) dynamics.

Resonance structure in KN appears at around 1420 MeV

17

Application to three-body K-pp systemApplication to the few-body anti-K system

Hamiltonian : Realistic interactions

Model wave functionSelf-consistency of kaon energy and KN interaction

JP = 0- , T = 1/2, T3 = 1/2

TN = 1, dominant, used in Faddeev

TN = 0

Realistic NN potential (Av18)KN potential based on chiral SU(3) dynamics (real part)dispersive effect from imaginary part~ 3-4 MeV in two-body KN system

18

Theoretical uncertaintiesApplication to the few-body anti-K system

Different models of chiral dynamics

-100

-80

-60

-40

-20

0

Im z

[MeV

]

14401420140013801360Re z [MeV]

z2(BMN)z2(HNJH)

z2(ORB)

z1(HNJH)z1(ORB)

z1(BMN)z2(BNW)

z1(BNW)

Two options for two-body energy

Define antikaon “binding energy”Energy dependence of KN interaction

4

2

0

-2

F KN

[fm

]

1440140013601320s1/2 [MeV]

I=0

4

2

0

-2

F KN

[fm

]

1440140013601320s1/2 [MeV]

I=0

~1420 MeV-1200

-1000

-800

-600

-400

-200

0

U(0

) [M

eV]

1440140013601320s1/2 [MeV]

I=0, corrected

HNJH

ORBBNW

BMN

19

Summary 2 : K-pp systemWe study the K-pp system with chiral SU(3) potentials in a variational approach.

With theoretical uncertainties,

Application to the few-body anti-K system

B.E.Γ(πYN)

= 19 ± 3 MeV= 40 ~ 70 MeV

Phenomenological potential (~ 2 times stronger than ours)

B.E.Γ

~ 48 MeV~ 60 MeV

No two-nucleon absorption : KNN -> YN ... small?

Faddeev with chiral interaction(separable, non-rel, ...?)

B.E.Γ

~ 79 MeV~ 74 MeV

Y. Ikeda, T. Sato, Phys. Rev. C76, 035203 (2007)

T. Yamazaki, Y. Akaishi, Phys. Rev. C76, 045201 (2007)

A. Doté, T. Hyodo, W. Weise, 0802.0238 [nucl-th], Nucl. Phys. A, in press

20

Structure of dynamically generated resonancesQuark structure of resonances? analysis of Nc scaling

J.A. Oller, E. Oset and J.R. Pelaez, Phys. Rev. D59, 074001 (1999)G. Ecker, J. Gasser, A. Pich, and E. de Rafael, Nucl. Phys. B321, 311 (1989)

c.f. ρ meson in ππ scattering ρ ~ qq

Nc behavior and quark structure

can be used to distinguish qq from others

21

Introduce the Nc scaling into the model and study the behavior of resonance.

Nc scaling in the model

Leading order WT interaction

c.f. meson-meson scattering : = trivialNontrivial Nc dependence of the interaction is in NLO.

T. Hyodo, D. Jido and A. Hosaka, Phys. Rev. Lett. 97, 192002 (2006)T. Hyodo, D. Jido and A. Hosaka, Phys. Rev. D75, 034002 (2007)

(for baryon and Nf > 2)

has Nc dep.

Nc behavior and quark structure

22

S = -1, I = 0 channel in SU(3) basis

1 8 8 27

Coupling strengths with Nc dependence

C ∝ Nc : finite interaction at Nc -> ∞Attractive interaction in singlet channel

Nc behavior and quark structure

23

S = -1, I = 0 channel in Isospin basis

KN πΣ ηΛ KΞ

Coupling strengths with Nc dependence

Attractive interaction in KN -> KNKΞ -> KΞ : attractive -> repulsive for Nc > 9

Off-diagonal couplings vanish at Nc -> ∞--> single-channel problem @ large Nc limit

Nc behavior and quark structure

24

In the large Nc limitNc behavior and quark structure

Attractive interaction in KN(singlet) channels

Critical coupling strength (with Nc dep)

Ccrit(Nc) =2[f(Nc)]2

m[−G(MT (Nc) + m)]

-20

-10

0

10

20

Coup

ling

stren

gths

(dim

ensio

nles

s)

20151050

Nc

Ccrit 1 8 27

Nc = 3

Bound state in “1” or KN channels

25

With SU(3) breaking : Pole trajectories around Nc = 3

1 bound state and 1 dissolving resonance

~ non-qqq (i.e. dynamical) structure

Nc scaling of (excited) qqq baryon

T.D. Cohen, D.C. Dakin, A. Nellore, Phys. Rev. D69, 056001 (2004)

-250-200-150-100

-500

Im W

[MeV

]

2001000-100Re W - MN - mK [MeV]

z1(Nc=3)

z2(Nc=3)

z2(12)

z1(12)

Nc behavior and quark structure

26

Isospin conponents of the polesResidues in the isospin basis

dissolvingother components

bound stateKN dominant

1.0

0.5

0.0

|g_i

| / |g

_KN

|

119753Nc

i = !" i = #$ i = K%

z1 pole4

3

2

1

0|g

_i| /

|g_K

N|

119753Nc

z2 pole

Nc behavior and quark structure

27

SU(3) components of the polesResidues in the SU(3) basis

bound state1 dominant

dissolvingother components

2.0

1.5

1.0

0.5

0.0

|g_i

| / |g

_1|

1210864Nc

i = 8 i = 8' i = 27

z1 pole7

6

5

4

3

2

1

0

|g_i

| / |g

_1|

1210864Nc

i = 8 i = 8' i = 27

z2 pole

Nc behavior and quark structure

28

Summary 3 : Nc behavior of Λ(1405)

Large Nc limit

Behavior around Nc = 3

Existence of a bound state in “1” or KN channel even in the large Nc limit

1 bound state and 1 dissolving pole: signal of the non-qqq state.Residues of the would-be-bound-state : dominated by “1” or KN: consistent with large Nc limit.

We study the Nc scaling of the Λ(1405)

T. Hyodo, D. Jido, L. Roca, 0712.3347 [hep-ph], Phys. Rev. D, in press.

Nc behavior and quark structure

29

Structure of dynamically generated resonancesResonances ~ quasi-bound two-body states

in some case, CDD pole (genuine state).

Dynamical or CDD

Renormalization change of loop function ~ change of interaction kernel

Formulation of the N/D method and the structure of low energy interaction

30

Renormalization schemesDynamical or CDD

Scattering amplitude in N/D method

G : unitarity cutV : other contribution (e.g. CDD pole)

For meson-baryon scattering・Identify G as loop function・Matching with ChPT order by order・V is given by ChPT (interaction kernel)

--> equivalent to solving BS equation

31

Renormalization schemesDynamical or CDD

Renormalization procedure

c. f. K. Igi, and K. Hikasa, Phys. Rev. D59, 034005 (1999) M.F.M. Lutz, and E. Kolomeitsev, Nucl. Phys. A700, 193-308 (2002)

G(µ) = 0, ⇔ T (µ) = V (µ) at µ = MT

Natural renormalization scheme: determine G to exclude CDD pole contribution, V is to be determined

Phenomenological scheme: V is given by ChPT, fit cutoff in G to data

Same physics (scattering amplitude)

32

Pole in the effective interaction

↑data fit ↑given

pole!

Physically meaningful pole :

Dynamical or CDD

** energy scale of the effective pole **

Interaction kernel in natural scheme↑ChPT

0.03

0.02

0.01

0.00

-0.01

-0.02

!V

[MeV

-1]

1800160014001200

s1/2 [MeV]

!V11 !V22 !V33 !V44

4

3

2

1

0!

V [M

eV-1

]

1800160014001200

s1/2 [MeV]

!V11 !V22 !V33 !V44

33

Example : Λ(1405) and N(1535)

Origin of resonances?

Dynamical or CDD

1.7 GeV8 GeV

34

Summary 4 : dynamical or CDD?

Natural renormalization

Analysis of Λ(1405) and N(1535)Λ(1405) : CDD pole would be smallN(1535) : appreciable contribution from CDD pole

We study the origin of the resonances in the chiral unitary approach

Dynamical or CDD

T. Hyodo, D. Jido, A. Hosaka, 0803.2550 [nucl-th]

Exclude CDD pole contribution from the loop function, consistent with N/D

35

Summary 5 : Structure of Λ(1405)

Analysis of Nc behavior

Analysis of natural renormalizationNMB dominates

Schematic decomposition of Λ(1405)

N3