12004, Nov. 18th

Θ+ production and two-meson coupling of antidecuplet

Tetsuo Hyodoa,

RCNP, Osakaa Madridb IFIC, Valenciac

A. Hosakaa, F. J. Llanes-Estradab, E. Osetc, J. R. Pelaezb and M. J. Vicente Vacasc

2

Contents

Introduction and motivationsEffective LagrangianSelf-energy of antidecuplet

Mass shiftsDecay widths

Reaction cross sectionSummary and conclusions

3

Introduction : Exotic activities

taken from K. Goeke, et al., hep-ph/0411195

4

Introduction : Flavor SU(3) symmetry

Existence of Θ+ + Flavor SU(3) symmetryExistence of flavor partners of Θ+

Flavor SU(3) symmetry and its breaking bythe strange quark mass

Gell-Man—Okubo mass formulae ~ 3%N(938), Λ(1116), Σ(1192), Ξ(1320),Δ(1232), Σ(1385), Ξ(1530), Ω(1670)

-> Phenomenologically well realized

Magnetic moments of baryon octetColeman-Glashow relation ~ 20%

5

Introduction : determining the quantum numbers

We have proposed

T. H., A. Hosaka, E. Oset, Phys. Lett. B579, 290 (2004)

K+p → π+Θ+ → π+K+n(K0p)

Polarization observablecan be used to determinethe quantum numbers

14

12

10

8

6

4

2

0

d2

/dM

Idco

sb/M

eV]

15801560154015201500

MI

kin(Lab) = 850 MeV/c = 90 deg

((

1/2+

1/2-

Polarized

Unpolarized

Understanding of reaction mechanism

K p n

K

6

Motivations

Total cross section ~ 2 μb

Why so small? What about K+p -> π+Θ+ ?

Results of π-p -> K-Θ+ reaction at KEK

Two-meson coupling effects for Θ+

K. Miwa, talk given at PENTAQUARK04

Possibility of KπN bound stateP. Bicudo, et al., Phys. Rev. C69, 011503 (2004)T. Kishimoto, et al., hep-ex/0312003F. J. Llanes-Estrada, et al., Phys. Rev. C69, 055203 (2004)

π+π−

K+

K−

p Θ+

7

Two-meson coupling

10–20 %Θ+ → KNN(1710) → πN

N(1710) → ππNΘ+ → KπN

40–90 %Forbidden

Very narrow

Effective interactions which account for the decayN(1710) → ππN

Large??π+π−

K+

K−

p Θ+

8

Criteria to construct the Lagrangian

Interaction is SU(3) symmetric

T.D. Cohen, Phys. Rev. D70, 074023 (2004)S. Pakvasa and M. Suzuki, Phys. Rev. D70, 036002 (2004)S. Ceci, et al., nucl-th/0406055

Assumptions for Θ+N(1710) is the S=0 partner of antidecuplet-> JP = 1/2+

No mixing with 8, 27,...

Small number of derivativesChiral symmetric? -> later

low energy : OK

9

SU(3) structure of effective Lagrangian

8M 8M

8B10P

1 ∈ 10 ⊗ 10

10

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Interaction Lagrangians 1

P 333 =√

6Θ+10

P 133 =√

2 N010 P 233 = −√

2 N+10

P 113 =√

2Σ−10 P 123 = −Σ0

10 P 223 = −√2 Σ+

10

P 111 =√

6Ξ−−10 P 112 = −√

2 Ξ−10 P 122 =

√2Ξ0

10 P 222 = −√6Ξ+

10

Antidecuplet field

Meson and baryon fields

B =

1√2Σ0 + 1√

6Λ Σ+ p

Σ− − 1√2Σ0 + 1√

6Λ n

Ξ− Ξ0 − 2√6Λ

Φ =

1√2π0 + 1√

6η π+ K+

π− − 1√2π0 + 1√

6η K0

K− K0 − 2√6η

11

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Interaction Lagrangians 2

Construction of 8s LagrangianDi

j [8sMM ] = φi

aφaj + φi

aφaj − 2

3δi

jφabφb

a

= 2φiaφa

j − 23δi

jφabφb

a

T ijk[10BMM(8s)] = 2φlaφa

iBmjεlmk + (i, j, k symmetrized)

L8s =g8s

2fPijkεlmkφl

aφaiBm

j + h.c.

12

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Interaction Lagrangians 3

N(1710) → ππ(p-wave, I = 1)NN(1710) → ππ(s-wave, I = 0)NExperimental information

L8s =g8s

2fPijkεlmkφl

aφaiBm

j + h.c.

L8a = 0

L10 = 0 <- symmetry under exchange of mesonsL27 =

g27

2f

[4Pijkεlbkφl

iφajBb

a − 45Pijkεlbkφl

aφajBb

i

]+ h.c.

8s

8a

Terms without derivative

L8a = ig8a

4f2Pijkεlmkγµ(∂µφl

aφai − φl

a∂µφai)Bm

j + h.c.

With one derivative

13

Diagrams for self-energy

g8s = 1.88 , g8a = 0.315

N(1710) → ππ(p-wave, I = 1)NN(1710) → ππ(s-wave, I = 0)N 25 MeV

15 MeV

8s 8s 8a 8a

Real partImaginary partSU(3) breaking

: mass shift: decay width: masses of particles Small!!

14

Results of self-energy : Real part (mass shift)

-140

-120

-100

-80

-60

-40

-20

0

Re

[M

eV]

p0 = 1700 MeV

cutoff = 700 MeV

N

8s

8s + 8a

-140

-120

-100

-80

-60

-40

-20

0

Re

[M

eV]

p0 = 1700 MeV

cutoff = 800 MeV

N

8s

8s + 8a

More bound for larger strangeness.All mass shifts are attractive.

Mass difference between Θ and Ξ -> 60 MeV : ~20 % of 320 = 1860–1540

15

Results of self-energy : Imaginary part (decay width)

Decay [MeV] Γ(8s) Γ(8a) Γ(tot)BMM

N(1710) → Nππ (inputs) 25 15 40

N(1710) → Nηπ 0.58 -

Σ(1770) → NKπ 4.7 6.0 24

Σ(1770) → Σππ 10 0.62

Σ(1770) → Λππ - 2.9

Ξ(1860) → ΣKπ 0.57 0.46 2.1

Ξ(1860) → Ξππ - 1.1

16

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Other possible Lagrangians

L8s =g8s

2fPijkεlmkφl

aφaiBm

j + h.c.

Two-meson 27 interactionL27 =

g27

2f

[4Pijkεlbkφl

iφajBb

a − 45Pijkεlbkφl

aφajBb

i]

+ h.c.

Lχ =gχ

2fPijkεlmk(Aµ)l

a(Aµ)aiBm

j + h.c.

Aµ =i

2(ξ†∂µξ − ξ∂µξ†

)= − ∂µφ√

2f+ O(p3)

(Aµ)la(Aµ)a

i → 12f2

∂µφla∂µφa

i

Chiral symmetric interaction

ξ = eiφ/√

2f

LM =gM

2fPijkεlmkSl

iBmj

S = ξMξ + ξ†Mξ† = O(φ0) − 12f2

(2φMφ + φφM + Mφφ) + O(φ4)

SU(3) breaking interaction M = diag(m, m,ms)

17

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Chiral symmetric Lagrangian

L8s =g8s

2fPijkεlmkφl

aφaiBm

j + h.c.

Lχ(2) =gχ

2fPijkεlmk 1

2f2∂µφl

a∂µφaiBm

j + h.c.

SU(3) structure : Identical !Only loop integral is changed<- adjusting the cutoff, we would have the same results

gχ = 0.218N(1710) decay ->

18

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Results of chiral Lagrangian

ReΣ 8s χ(2)Θ -100 -99N -87 -83Σ -79 -70Ξ -63 -57cutoff 800 525

Decay 8s χ(2)N(1710) → Nππ 25 25N(1710) → Nηπ 0.58 0.32Σ(1770) → NKπ 4.7 4.5Σ(1770) → Σππ 10 3.6Ξ(1860) → ΣKπ 0.57 0.40

Almost the same resultsDifference comes from the SU(3) breaking of momenta at the vertex

[MeV]

19

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

27 and mass Lagrangians

Fitting couplings to the N(1710) decay -> large binding energy of 1 GeV : unrealisticTreat them as a small perturbation to the 8s.

L27 =g27

2f

[4Pijkεlbkφl

iφajBb

a − 45Pijkεlbkφl

aφajBb

i]

+ h.c.

Lint = aL8s + b27,ML27,M

Deviation from a = 1 : weight of new terms

g27 = gM = g8s = 1.88 , b27 = −54(1 − a) , bM =

f2

m2π

(1 − a)

LM =gM

2fPijkεlmk(− 1

2f2)(2φMφ + φφM + Mφφ)l

iBmj + h.c.

20

π−p

K+

n

K+

π+

p

K+

n

K+

π+

∆+

ρ0

p

K+

n

π+

K+

Results of 27 and mass Lagrangians

Contributions of these terms are considered as a theoretical uncertainty in the analysis.

-250

-200

-150

-100

-50

0

Re

[M

eV]

p0 = 1700 MeV

cutoff = 800 MeV

N-120

-100

-80

-60

-40

-20

0

Re

[M

eV]

p0 = 1700 MeV

cutoff = 800 MeV

N

27 M0.90 < a < 1.06 0.76 < a < 1.06

21

Conclusion 1 : self-energy

We study the two-meson virtual cloud effect to the self-energy of baryon antidecuplet.

Two types of Lagrangians (8s, 8a) are important among several possible interaction Lagrangians.

L8s =g8s

2fPijkεlmkφl

aφaiBm

j + h.c.

L8a = ig8a

4f2Pijkεlmkγµ(∂µφl

aφai − φl

a∂µφai)Bm

j + h.c.

22

Conclusion 1 : self-energy

Two-meson cloud effects are always attractive, and contribute to the antidecuplet mass splitting, of the order of 20%.Antidecuplet members have relatively small decay widths to MMB channel.

T. H., A. Hosaka, F. J. Llanes-Estrada, E. Oset, J. R. Pelaez,M. J. Vicente Vacas, in preparation

23

Results of reaction : cross sections

20

15

10

5

0

[m

b]

260024002200200018001600

Ecm

8s 8a Total

100

80

60

40

20

0

[m

b]

260024002200200018001600

Ecm

8s 8a Total

(-,K

-) (K

+,

+)

π+π−

K+

K−

p Θ+

Total cross section of

Destructive Constructive

24

Conclusion 2 : reactions

T. H., A. Hosaka, E. Oset, M. J. Vicente Vacas, in preparation

We investigate the Θ production in (π-,K-) and (K+,π+) reactions, with the vertices obtained from the self-energy study.

The small cross section of the order of a few micro barn in (π-,K-) reaction may require some special mechanisms, such as interference of two amplitudes.

25

Future works

Self-energy

Reaction

Chiral symmetric Lagrangian, Possible mixing with the other flavor multiplets (8, 27, ...),

Quantitative analysis (Form factor), background cross section