スカーム模型における束縛K中間子法による

Bound kaon approach
for the ppK- system
in the Skyrme model
Tetsuo Nishikawa (Tokyo Ins. Tech.)
Yoshihiko Kondo (Kokugakuin U.)
2015/10/1
Nuclear Physics at J-PARC 01-02/06/07
ppK- bound state
Lightest K-nucleus proposed by Dote, Akaishi &
Yamazaki.
 Its existence is theoretically still unclear.
 The interpretation of the experiment by FINUDA
collab. is still controversial.
 More clear evidence is expected to be seen in
future exp.@J-PARC.
Our interests
 Does the Skyrme model suggest deeply bound ppKstates?
⇒ Can the energy of K- coupled with pp be
considerably small ?
If yes, we want to know
 mechanism responsible for the strong binding
 possible structure of ppK-
Skyrme model
Skyrme model
Skyrme, 1961
Nucleons = topological solitons of the pion field
 Skyrme’s ansatz for chiral (pion) field:
r
ˆ
U  exp(i  rF(r))
“Hedgehog” ansatz
 The isospin is oriented to the radial direction, rˆ .
Skyrme model
r
ˆ
 Hedgehog ansatz: U  exp(i  rF(r))
mapping: S3(phys.) a S3(int.)
 Soliton profile:
n
F(r)
minimizing the energy
r
 n : “winding number” classifies the mapping
S3
S3
n=1
or
S3
,…
n=2
 Winding number (conserved) = Baryon number
The action of the Skyrme model
Skyrme term
Wess-Zumino-Witten term
 Skyrme term
stabilizes solitons
E
total
Skyrme term
NL-sigma term
min
 (size)
 Wess-Zumino-Witten term
remove an extra symmetry of chiral Lagrangian
ensure the Skyrmion to behave like a fermion (E.Witten, 1983)
Bound kaon approach
to the Skyrme model
Bound kaon approach
to the Skyrme model
(Callan and Klevanov, 1985)
 A kaon field fluctuates around the SU(2) Skyrmion.
 The kaon field has bound states.
 lowest-lying mode: T ( =L+I )=1/2, L=1
 next mode: T=1/2, L=0
 Wess-Zumino-Witten term
split S=±1 states
 S=-1 states: bound states
hyperons
 S=+1 states (e.g. pentaquark)
continuum
Why bound kaon approach?
 No parameter, once we adjust Fπ and e to fit N and
Δ masses in SU(2)f sector.
 KN interaction, which is a key ingredient for the
study of K-nuclei, is unambiguously determined.
 It reproduces the mass of Λ(1405) as well as Λ, Σ,
Ξ etc..
Description of the ppK- system
Description of the ppK- system
(2-Skyrmion)
+ (kaon field fluctuating around the Skyrmion)
bound kaon
Description of the ppK- system
 Derive the kaon’s EoM for Skyrmions at fixed positions
kaon’s energy  K (R)
 Solve the NN dynamics
the binding energy of the ppK- system
VNN
R
ωK?
R
Derivation of the kaon’s EoM
 Substitute the ansatz
r
r
r r r
U  U(r1 )UKU(r2 ), r1  r2  R
into the action
 Expand up to O(K2) and neglect O(1/Nc) terms
Lagrangian of the kaon field
under the background of B=2 Skyrmion
Derivation of the kaon’s EoM
 Collective coordinate quantization
projection onto (pp)S=0
L[K ]   d N1 N 2 L N1 N 2 ,

Average the orientation of R
N 
1
(1)I 3 1/2 D1/2I 3 J 3 ()
2
Spherical partial wave analysis is allowed:
R/2
Equation of motion for the kaon
Results
Energy eigenvalue of K-
P-wave
S-wave
R=2.0 fm
BK=220 MeV
R=1.5 fm
BK=340 MeV
K- (s-wave) distribution
ωK=156MeV
Proton position
R=1.5fm
K- distribution and baryon# density
ωK=272MeV
Proton position
R=2.0fm
K- distribution and baryon# density
mK=495MeV
ωK=349MeV
K- distribution and baryon# density
Molecular
states ?
mK=495MeV
ωK=397MeV
Effects of the Wess-Zumino-Witten
term
Switching off the WZW term,…
Roles of the Wess-Zumino-Witten
term in the Skyrme model
 Remove an extra symmetry the chiral Lagrangian
possesses
 Ensure the Skyrmion behaves as a fermion
(E.Witten, 1983)
 Gives an attractive contribution to S=-1 state
correct values of hyperon masses
 Λ(1405) is bound owing to the WZW term alone !
Roles of the WZW term
 Interaction from the WZW term,
LWZW  
iN c  †
†

B
[K
D
K

(D
K)
K],
(B
: baryon # current)


2
F
attractive contribution VWZW to K-.
 VWZW for B=2 is stronger than that for B=1,
since LWZW ∝(baryon#).
makes K- bound to pp very deeply
 This scenario is very similar to that proposed by AY.
Strong binding of K- to pp seems to be natural
within the Skyrme model.
Summary
 Within the product ansatz Skyrme approach, the
energy of K- bound to pp can be considerably small.
 K- is centered between pp for small R, while for
larger separation it shows molecular nature.
 Only the attractive interaction arising from the
Wess-Zumino-Witten term binds Λ(1405).
 This interaction plays an important role also
for the strong binding of K- to pp .
 Solving the p-p radial motion to obtain the total
energy of ppK is now in progress.

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