1. Dibaryon with S=-1 in the Bound Kaon Approach to the Skyrme Model
T. NISHIKAWA (Tokyo Institute of Technology)
Y. KONDO (Kokugakuin univ.)
2018/11/23
KEK研究会「 現代の原子核物理 」

2. Discovery (?) of strange multibaryons
Strange
T. Suzuki et al., 2004
stopped K- reaction in 4 He
deeply bound ppnK- and pnnK- states
Strange
M. Agnello et al., 2005
strong Back-to-back correlation
between Λ and p from K- absorption
Evidence for ppK- bound state ?
B.E.=115MeV, Γ=67MeV

3. Purpose of this work appropriate for the study of strange multibaryons
To explore the possibility
of deeply bound strange dibaryon
in the bound kaon approach to the Skyrme model
structure, mechanism responsible for the large B.E. (>100MeV)
The bound kaon approach naturally incorporates
KN interaction.
well describes Λ(1405) as well as positive parity
hyperons
appropriate for the study of strange multibaryons

4. Skyrme model Nucleons = topological solitons of pion field
Skyrme’s ansatz (“Hedgehog” ansatz):
Mapping from S3(phys.) to S3(int.)
F(r): profile function
Winding number (conserved) = baryon number
n (integer):
“winding number”

5. Skyrme model
Skyrme, 1961
　　 Nucleons as topological solitons of pion field
t’Hooft, 1974
In NC →∞, QCD → weakly coupled meson theory.
Witten, 1979
Baryons appear as solitons of pion field
Adkins & Nappi, 1984
Collective coordinate quantization
The model qualitatively (~30%) reproduces static properties of the nucleon

6. Bound kaon approach to the Skyrme model
(Callan and Klevanov, 1985)
Basic idea:
SU(2) Skyrmion has fluctuating kaon field.
The kaon field has bound states !
The lowest state: L=1
the next state: L=0
Wess-Zumino-Witten term split S=±1 states
S=-1 states: bound states hyperons
S=+1 states (ex. pentaquark): continuum

7. Spectrum of hyperons P-wave bound states : positive parity hyperons
S-wave bound state: negative parity hyperons
I=0 state: Λ(1405)
M=1325 MeV
(Rho, Riska and Scoccola, 1992)
Λ(1116)
Σ(1193)
Ξ(1318)
M(MeV)
1105
1203
1332
(mπ=0, fπ and e are fitted to MN and MΔ.)
(Schat, Scoccola and Gobbi, 1995)

8. Our approach to strange dibaryon
Place two rotating solitons
Derive the equation of motion for the kaon under the diskyrmions y x z
Skyrmion
-R/2
R/2
kaon energy and wave function
as functions of relative distance
of the 2-skyrmions, R

9. Possible R-dependence of kaon energyEK
If EK<mK in the intermediate region
or EKmK in the short range,
suggest NNK bound states R

10. Ansatz for chiral field
Ansatz for U
where
U(1) and U(2):
Skyrmions centered at z=R/2 and -R/2
UK: kaon field
C: relative orientation of the 2-Skyrmions

11. Relative orientation of the 2-Skyrmions
SS interaction strongly depends on the relative orientation, C. (Oka&Hosaka, 1992)
Kaon-diskyrmion interaction may be so.
We examine 2-cases.
C=1 (“HH channel”)
Unrotated hedgehogs
C=iσ2 (“ATR channel”)
One hedgehog is rotated by π around y-axis.
For ATR channel, 2-Skyrmion system has the lowest energy configuration.

12. Derivation of kaon eq. of motion
Substitute the ansatz into the Lagrangian
and expand up to O(K2) terms
Collective coordinate quantization
Projection of the 2-skyrmions onto IJ=10 state

13. How to solve the eq. of motion
First, solve the eq. of motion for R=0
Starting from the configuration for R=0,
gradually increase R. ΔR
2ΔR R
Numerical relaxation method
(Veerbarschot, 1987)

14. Eq. of motion for K- (R=0) Eq. of motion
The last term comes from Wess-Zumino-Witten term and play a role of potential.
For small r,
Lowest state: l=0 (cf. lowest state is l=1 for B=1)

15. Results of kaon energy (R=0)
B=1
(Callan&Krevanof)
B=2
(HH-channel)
(ATR-channel)
L=0
367MeV ≅0
188MeV
L=1
153MeV
309MeV
unbound
(2-Skyrmions are projected onto IJ=10 state.)

16. Summary
We have applied the bound kaon soliton model to study the strange dibaryon.
The equation of motion for K- coupled with diskyrmions was derived.
The interaction of K- with SS seems to strongly depend on the relative orientation of SS.
When the 2-skyrmions overlap, the s-wave kaon is bound much deeper than that for baryon#=1.