Ñ] K ¡( a • Ö כ  Ç ù p § T “ Ó Þ” X ¢ 40 Ca; c 8 ý” X ¢ K −  ˜ mŠ ˜ m Ž ì ŏ Œ

° Ë

Ñ] K ¡( a • Ö כ  Ç ù p § T  “ Ó Þ” X ¢ ⁴⁰ Ca; c 8 ý” X ¢ K ⁻  ˜ mŠ ˜ m Ž ì ŏ Œ



™ »\ 8 ;T  ^∗ ·  ™ »* > ¬ £

/ B

N Å Ò@ /† < Ɠ § 6 £ x6   xÓ ü t o † < Æõ , / B N Å Ò 314-701 (2004¸   2 Z 4 7{ 9  ~ à Î6 £ §)

ç ß

–é ß –ô  Ç s ³ 1 Ï s  : r \    H   # Œ Ù þ ˜ î ß –\ " f_  H s “ : r F  g† < Æ( J $ ™[ > `  ¦ Ä »• ¸ “ ¦ H s “ : r-Ù þ ˜ í ß –ê ø Í ‰ & ³ © œ õ

 q “ § l  0 AK  ( Ž É Ó'  á Ԗ ÐÕ ªÏ þ ›`  ¦ s 6   x # Œ Klein-Gordon(KG) ~ ½ Ó& ñ d ” `  ¦ Û  ¦ # Q  › ' a8 £ ¤| ¾ ӓ   p ì  r í

ß

–ê ø Í é ß –€  & h `  ¦ > í ß – % i  . 800 MeV/c\ " f_  K

⁻

-

⁴⁰

Ca_  í ß –ê ø Í ‰ & ³ © œ`  ¦ “ ¦ 9 # Œ p ì  r í ß –ê ø Í é ß –€  & h  _  z  ´] j z  ´+ « >u ü < s  x 9  p \  ¦ “ ¦ 9 # Œ % 3 # Q”   F  g† < Æ( J $ ™[ > `  ¦ s 6   x % i `  ¦  â Ä º_  s  : r& h  > í ß –   õ 

\

 ¦ q “ § % i “ ¦ ¢ ¸ô  Ç s  : r u \  ¦ ‰ & ³ © œ : r& h Ü ¼– Ð ˜ Ð& ñ ô  Ç F  g† < Æ ( J $ ™[ > `  ¦ s 6   x % i `  ¦  â Ä ºü < q “ §, ì  r$ 3  

% i  .

PACS numbers: 25.80.N

Keywords: F  g† < Æ( J $ ™[ > , H s “ : r í ß –ê ø Í

I. " e  ] Ø

t

è ß – Y >  K  1 l xî ß – ×  æç ß – ü < Z …o “ : r  © œ  ñ Œ •6   x \  @ /K " f

´ ú

§“ É r ƒ  ½ ¨ ”  ' Ÿ ÷ &% 3  .  s “ : r ×  æç ß – \  @ /K " f  H z  ´ +

« >\  _ K " f % 3 # Q”     õ \  ¦ s  : r Ü ¼– Ð S X ‰ “   “ ¦  Ž ž Ð 



 H ƒ  ½ ¨ | 9 ×  æ& h Ü ¼– Ð s À Ò# Q& ’ Ü ¼
 [1] ì ø ̀  \  l ¬ ¹• ¸

\

 ¦ t   H ×  æç ß – “   H s “ : r õ  Z …o “ : r  s _   © œ  ñ Œ •6   x

\

 @ /K " f  H  f ”  ´ ú §“ É r ƒ  ½ ¨   õ  µ 1 ϳ ð÷ &t  · ú §“ ¦ e ”  .

ô 

Ǽ # , KEK [2]\ " f % 3 # Q”   D h– Ðî  r 8 £ ¤& ñ ° ú כ\   F G~ à Î  œ í Ù þ

˜[ þ t(hypernuclei) \  @ /ô  Ç | 9 ×  æ& h “   ƒ  ½ ¨ s  : r õ  z  ´+ « >,

€

ª œA á ¤ \ " f s À Ò# Qt “ ¦ e ” Ü ¼ 9,   É r ô  Ǽ # Ü ¼– Ѝ  H ×  æç ß – ü <

Z

…o “ : r  © œ  ñ Œ •6   x \ " f / B N" î `  ¦ l Õ ü t l  0 AK  s ³ 1 Ï @ / g A › ¸| [ þ t(Chiral Symmetry Constraints: CSC)`  ¦ s 6   x

  H % ƒ6 £ § Ü ¼– Ð r • ¸÷ &  H ƒ  ½ ¨ [3–5][ þ t s  ”  ' Ÿ ÷ &“ ¦ e ”  .

s

 Qô  Ç ƒ  ½ ¨ â ì2 £ § \  ´ ú Æ Ò# Q þ j  H { 9 ‘ : r Riken ƒ  ½ ¨™ è1 p x \ 

"

f z  ´+ « > ƒ  ½ ¨  Ö ¸µ 1 Ïy  s À Ò# Qt “ ¦ e ”   H œ íÙ þ ˜_  Ò q t$ í õ  Õ

ª $ í | 9  ƒ  ½ ¨\  l œ í | ¨ c à º e ”   H H s “ : r õ  Ù þ ˜   s  _

  © œ  ñ Œ •6   x_  ƒ  ½ ¨\  ¦  „ ½ ÓÜ ¼– Ð Ù þ ˜? / H s “ : r F  g† < Æ ( J $ ™ [ >

`  ¦ > í ß – “ ¦ s \  ¦ s 6   x # Œ H s “ : r õ  Ù þ ˜_  Ø  æ[  t‰ & ³ © œ

`

 ¦ ƒ  ½ ¨ % i  . Ù þ ˜ î ß –\ " f_  H s “ : r F  g† < Æ( J $ ™[ > `  ¦ Ä »• ¸

l  0 AK  ç ß –é ß –ô  Ç s ³ 1 Ï s  : r`  ¦ s 6   x % i “ ¦ [6] z  ´] j z  ´ +

«

>   õ ü < q “ § l  0 AK  ( Ž É Ó'  á Ԗ ÐÕ ªÏ þ ›`  ¦ s 6   x # Œ Klein-Gordon(KG) ~ ½ Ó& ñ d ” `  ¦ Û  ¦ # Q [7] H s “ : r-Ù þ ˜ í ß –ê ø Í ‰ & ³



© œ_  p ì  rí ß –ê ø Í é ß –€  & h `  ¦ > í ß – % i  . s  H s “ : r  © œ  ñ Œ • 6

 

x`  ¦ l Õ ü t l  0 AK " f  H ¸ ú ˜ · ú ˜ 9”   Ericson-Ericson F  g† < Æ

∗

E-mail: [email protected]

(

J $ ™[ >  [8]_  { 9 ì ø Í+ þ A`  ¦ s 6   x % i Ü ¼ 9 s   © œ  ñ Œ •6   x õ  ° ú  

“ É

r l œ í\ " f p   © œ  ñ Œ •6   x_  > í ß –\ " f  H ˜ Ð& ñ  ) a Migdal

† <

Êà º\  ¦ s 6   x % i  . > í ß – ) a F  g† < Æ ( J $ ™[ > `  ¦ ˜ Ð& ñ # Œ z  ´ ]

j H s “ : r-Ù þ ˜ Ø  æ[  t ‰ & ³ © œ_  p ì  r í ß –ê ø Í é ß –€  & h `  ¦ ¸ ú ˜ [ O " î

  H ‰ & ³ © œ : r& h “   F  g† < Æ( J $ ™[ > `  ¦ > í ß –K ˜ Г ¦ s  : r& h Ü ¼– Ð

>

í ß – ) a F  g† < Æ( J $ ™[ > õ  q “ §, ì  r$ 3  % i  .

II. T  Â ] Ø õ m Í + s ÇÊ Ý

Ù þ

˜Ó ü t| 9 \ " f s  H s “ : r  © œ  ñ Œ •6   x`  ¦ l Õ ü t l  0 AK " f

¸ ú

˜ · ú ˜ 9”   Ericson-Ericson_  F  g† < Æ( J $ ™[ >  [8,9]`  ¦  6   xô  Ç



. s \  ¦ ‚ à Г ¦ë  H‰  ³ [9]\  ¦    ç ß –é ß –y  l Õ ü t €    6 £ § õ  ° ú  



. ×  æç ß –  © œ † < Êà º, Ψ(r)“ É r { 9   ü < y Œ •y Œ •_  Ù þ ˜ [ þ t – ÐÂ Ò '

_  í ß –ê ø Í [ þ t_  ½ + ËÜ ¼– Ð ³ ð‰ & ³ ) a  .

Ψ(r) = exp(ik · r) + X

i

exp(iµ |r − r i |)

|r − r i | F i (ω, k)Ψ ^{ef f} _i (r i ) (1)

#

Œl " f ωü < k  H ×  æç ß – _  \  -t ü < î  r1 l x| ¾ Ós “ ¦, µ  H ¨ 8 Š í

ß –| 9 | ¾ Ós  9, í ß –ê ø Í   H i  P :_  Ä »´ ò ×  æç ß –  © œ Ψ ^{ef f} _i (r _i ) ü <

í

ß –ê ø ͔  ; Ÿ ¤ F i (ω, k) x 9 
š ¸  H ½ ¨€   ü <_  Y  L Ü ¼– Ð
 è ß –



. H s “ : r-Ù þ ˜ > \ " f „    “ § ¨ 8 Š G V , `  ¦ Ÿ í† < Êr v l  0 A K

 „  Û ¼— 2 ; I\  ¦ “ ¦ 9 €   i í ß –ê ø Í \  _ ô  Ç í ß –ê ø ͔  ; Ÿ ¤`  ¦



A ü < ° ú  s  ³ ð‰ & ³½ + É Ã º e ”  .

F i (ω, k) = f I=0 (ω, k)P _I=0 ⁱ + f I=1 (ω, k)P _I=1 ⁱ (2)

-333-

#

Œl \ " f P I ⁱ   H Å Ò# Q”   „  Û ¼— 2 ; I\  ¦ ° ú   H H s “ : r-Ù þ ˜ 



© œI \  ¦ _ p ô  Ç . s \  ¦ { 9   H s “ : r õ  Ù þ ˜ _  „  Û ¼— 2 ; '

Ÿ § > =, τ _k , τ _i \  ¦ s 6   x # Œ ³ ð‰ & ³ €  , P ₀ ⁱ = 1 − τ k · τ i

4 , P ₁ ⁱ = 3 + τ k · τ i

4 (3)

s

 9, ¢ ¸ô  Ç „  Û ¼— 2 ; x 9 • ¸  H ρ I (r) = < 0 |P I ⁱ δ ³ (r − r _i ) |0 >_  + þ AI – Ð Å Ò# Qt Ù ¼– Ð s  Qô  Ç ³ ð‰ & ³[ þ t`  ¦ s 6   x # Œ d ”

 (1)`  ¦  r  & ñ o  €    6 £ § õ  ° ú   .

Ψ(r) = exp(ik · r) + X

I=0,1

Z

d ³ r ⁰ exp(iµ |r − r ⁰ |)

|r − r ⁰ |

× f I (ω, k)ρ _I (r ⁰ )Ψ ^{ef f} _I (r ⁰ ) (4) Ψ(r) \  @ /ô  Ç KG ~ ½ Ó& ñ d ” “ É r

(ω ² + ∇ ² − m ² K )Ψ(r) = −4π X

I=0,1

f I ρ I (r)Ψ ^{ef f} _I (r) (5)

{ 9

  כ s “ ¦ s  ~ ½ Ó& ñ d ” `  ¦ Û  ¦ l  0 AK  Ä »´ ò © œ`  ¦ “ ¦ 9K    9 i  P :\  ¦ ] jü @ô  Ç — ¸Ž  H   É r { 9  ü <_   © œ  ñ Œ •6   x`  ¦ “ ¦ 9 ô 

Ç i  P :_  í ß –ê ø Í _  Ä »´ ò © œ“ É r  6 £ § õ  ° ú  s 
 è ß – .

Ψ ^{ef f} _i (r i ) = exp(ik · r i ) + X

j!=i

exp(iµ |r i − r j |)

|r i − r j |

× F j (ω, k)Ψ ^{ef f} _j (r j ) (6) s

M : „  Û ¼— 2 ; ½ ¨› ¸\  ¦ “ ¦ 9 €  , Ψ ^{ef f} _I (r) = exp(ik · r) + X

I

⁰

=0,1

Z

d ³ r ⁰ exp(iµ |r − r ⁰ |)

|r − r ⁰ |

× ρ I

⁰

(r ⁰ )C I,I

⁰

(r, r ⁰ )f I

⁰

Ψ ^{ef f} _I

0

(r ⁰ ) (7) s

 9, # Œl \ " f C ^I,I

⁰

(r, r ⁰ )“ É r Ù þ ˜ -Ù þ ˜   © œ  ñ Œ •6   x(N − N )_   © œ› ' a† < Êà ºs “ ¦ s   H ¿ º { 9   „  Û ¼— 2 ; x 9 • ¸ ρ _I,I

0

(r, r ⁰ ) ü <  6 £ § õ  ° ú  “ É r › ' a > \  ¦ ”   .

ρ I,I

⁰

(r, r ⁰ ) = < 0 | X

i

X

i!=j

P _I ⁱ P _I ^j

0

δ ³ (r − r i )δ ³ (r − r j ) |0 >

= ρ I (r)ρ I

⁰

(r ⁰ )[1 + C I,I

⁰

(r, r ⁰ )] (8) 800 MeV/c_  H s “ : r \  -t \  ¦ “ ¦ 9½ + É M :  © œ› ' a † < Êà º  H ß

¼>  ×  æ כ ¹ t  · ú §Ü ¼ 9 ¢ ¸ô  Ç ‘ : r  7 Hë  H \ " f “ ¦ 9 “ ¦ e ”   H é

ß –í  Hô  Ç — ¸+ þ A\ " f  © œ› ' a ´ òõ  t  “ ¦ 9   H  כ “ É r  H _ p 

 \ O Ü ¼Ù ¼– Ð ‘ : r ƒ  ½ ¨\ " f  H  © œ› ' a  à º ´ òõ \  ¦ “ ¦ 9 t 

· ú

§  H  . s  Qô  Ç   H  ? /\ " f  ^ ‰\  -t  [9]\  ¦ s 6   x # Œ

×

 æç ß –  © œ\  › ' aô  Ç Klein-Gordon ~ ½ Ó& ñ d ” `  ¦ ³ ð‰ & ³ €    6 £ § õ

 ° ú   .

(ω ² + ∇ ² − m ² K )Ψ(r) = Π(ω, k)Ψ(r) (9)

#

Œl \ " f,  ^ ‰\  -t  x 9  F  g† < Æ ( J $ ™[ > “ É r  6 £ § õ  ° ú  s  j þ t Ã

º e ”   [10].

Π(ω, k) = 2ωU opt (ω, k) = −4π(f 0 ρ 0 + f 1 ρ 1 ) (10) Ù þ

˜? /_  € ª œ$ í  (×  æ$ í  ) ï  r0 A ` …Ø Ôp î  r1 l x| ¾ Ó p ^p _F (p ⁿ _F )   t

 e ” “ ¦, Ù þ ˜ “   € ª œ$ í  ü < ×  æ$ í  _   © œI [ þ t“ É r î  r1 l x| ¾ Ó p, Û ¼— 2 ;s  ± ¹ 2 Õ ªo “ ¦ „  Û ¼— 2 ; ± ¹ 2 Ü ¼– Ð ³ ð‰ & ³  9 „   Û

¼— 2 ; ρ _I   H  6 £ § õ  ° ú  s  Å Ò# Q”   .

ρ 0 = 1

4 (ρ p + ρ n ) − τ _K ³

4 (ρ p − ρ n ) ρ 1 = 3

4 (ρ p + ρ n ) + τ _K ³

4 (ρ p − ρ n ) (11)

#

Œl \ " f K ^± \  @ /K  τ K ³ = ±1s “ ¦, ρ ^p = (p ^p _F ) ³ /3π ² , ρ _n = (p ⁿ _F ) ³ /3π ² “ É r € ª œ$ í  ü < ×  æ$ í  \  @ /ô  Ç x 9 • ¸\  ¦
 



· p . # Œl \ " f p ^p F = p ⁿ _F = p _F , ρ _p + ρ _n = ρ s  9 ρ 0 = ρ 1 /3 = ρ/4   ) a  . î  r1 l x“    ^√ _M ^s \  ¦ “ ¦ 9 # Œ d ”  (10)`  ¦



r  & ñ o  €    ^ ‰\  -t  ¢ ¸  H F  g† < Æ( J $ ™[ > “ É r  6 £ § õ  ° ú   s


 è ß – .

2ωU opt = −4π

√ s M

 f ₀ ρ

4 + 3f ₁ ρ 4



(12)

#

Œl " f √

s  H H s “ : r õ  Ù þ ˜_  | 9 | ¾ Ó×  æd ”  \  -t s “ ¦, M“ É r

40 Ca Ù þ ˜_  | 9 | ¾ Ós  . s  í ß –ê ø ͔  ; Ÿ ¤“ É r \  -t \  _ ” > r   H í

ß –ê ø ÍU  ´s  [4,5] a(ω)_  † ½ ÓÜ ¼– Ð ³ ð‰ & ³| ¨ c à º e ” Ü ¼ 9, l ¬ ¹• ¸

 s(strangeness) = − 1{ 9   â Ä º\  @ /K " f í ß –ê ø ͔  ; Ÿ ¤`  ¦ ³ ð

‰ &

³ €    6 £ § õ  ° ú   .

f ₀ ^{s= −1} (ω, k = 0) = 2a(K ⁻ p → K ⁻ p) − a(K ⁻ n → K ⁻ n) f ₁ ^{s= −1} (ω, k = 0) = a(K ⁻ n → K ⁻ n) (13) s(strangeness) = + 1{ 9   â Ä º\  @ /K " f  H í ß –ê ø ͔  ; Ÿ ¤ s    6

£

§ õ  ° ú  s 
 è ß – .

f ₀ ^s=+1 (ω, k = 0) = 2a(K ⁺ n → K ⁺ n) − a(K ⁺ p → K ⁺ p) f ₁ ^s=+1 (ω, k = 0) = 2a(K ⁺ p → K ⁺ p) (14)

#

Œl " f a  H í ß –ê ø ÍU  ´s (off-shell scattering length)\  ¦ z  ´+ « >

u

 [6,11,12]\    H   # Œ @ /{ 9   9 s   H $ \  -t  % ò % i \ 

"

f H s “ : r õ  Ù þ ˜   © œ  ñ Œ •6   x \ " f_  z  ´+ « >u \  ¦ ¸ ú ˜ [ O " î K  ï

 r  . y Œ • G V , \ " f_  í ß –ê ø ÍU  ´s _  ° ú כ“ É r  6 £ § õ  ° ú  s  Å Ò# Q

”

  .

a ^K

⁺

^p = −0.31 ± 0.01 fm, a ^K

⁺

ⁿ = −0.20 ± 0.01fm a ^K

⁻

^p = −0.67 + i0.63 fm,

a ^K

⁻

ⁿ = +0.37 + i0.57 f m (15)

s

[ þ t`  ¦ K ^± \  @ /K " f í ß –ê ø ͔  ; Ÿ ¤_  d ”  (13)õ  (14)\  @ /{ 9 

# Œ & ñ o  €    6 £ § õ  ° ú   .

K ⁺ ; f ₀ ^s=+1 = 2( −0.20) − (−0.31) = −0.09 fm f ₁ ^s=+1 = −0.31 fm

K ⁻ ; f ₀ ^{s= −1} = 2( −0.67 + i0.63) − (0.37 + i0.57)

= ( −1.71 + i0.69) fm

f ₁ ^{s= −1} = (0.37 + i0.57) f m (16) s

] j, 800 MeV/c\ " f_  K ⁻ - ⁴⁰ Ca_  í ß –ê ø Í ‰ & ³ © œ`  ¦ “ ¦ 9K 

˜

Ѐ  ,  © œ@ / : r& h “   \  -t   H  6 £ § d ” Ü ¼– Ð Å Ò# Q| 9   כ s  9, ω =

q

p ² _K c ² + (m _K c ² ) ² , (17)

#

Œl \ " f î  r1 l x| ¾ Ó p _K   H H s “ : r î  r1 l x| ¾ Ó 800 MeV/c_  | 9 

|

¾ Ó×  æd ” \  @ /ô  Ç ° ú כs  . ¢ ¸ô  Ç | 9 | ¾ Ó×  æd ”  \  -t  s  H  6 £ § õ

 ° ú  s  Å Ò# Q”   .

s = p µ p ^µ

= p ⁰ p ⁰ − K · p K

= (M + E K ) ² − p K · p K

= M ² + 2M E K + E _K ² − p ² K

= M ² + 2M E _K + m ² _K (18) H

s “ : r_  & ñ t | 9 | ¾ Ó \  -t   H m K = 494 MeV s “ ¦, ³ ð& h  Ù þ

˜“   ⁴⁰ Ca_  & ñ t | 9 | ¾ Ó \  -t   H M = 37.3 GeV s Ù ¼– Ð Å

Ò# Q”   ° ú כ\  @ /K  | 9 | ¾ Ó×  æd ”  \  -t  √

s\  ¦ ½ ¨½ + É Ã º e ”  .

¢

¸ô  Ç ρ  H „   x 9 • ¸ì  r Ÿ í\  ¦ s 6   x  9  6 £ § õ  ° ú  “ É r 3   Ã

º Woods-Saxon + þ A`  ¦ s 6   xô  Ç .

ρ(r) = ρ ₀ 1 + w _R ^r
2

1 + exp ^{r −R} _z (19) s

 .  6   xô  Ç ⁴⁰ Ca_  Woods-Saxon + þ A_  3  à º ° ú כ“ É r   6

£

§ õ  ° ú    [13,14].

ρ ₀ = 0.17f m ⁻³ , R = 3.697f m, z = 0.587f m, ω = −0.083 (20) ρ 0   H r = 0 \ " f_  Ù þ ˜x 9 • ¸, R“ É r ρ 0 /2 \ " f_  ì ø Í â , z  H

³

ð€   ¿ ºa _  S X ‰í ß –• ¸, w  H B > h  à ºs  .   õ & h Ü ¼– Ð d ”  (12) Ü ¼– Ð Å Ò# Qt   H s   F  g† < Æ( J $ ™[ >  ° ú כ“ É r  6 £ § õ  ° ú   .

U _opt ^s (r) ≈ (6.90 − i27.6)MeV · ρ(r)/ρ 0 (21) s

ü < ° ú  s  > í ß – ) a F  g† < Æ( J $ ™[ > `  ¦  6   x # Œ s \  ¦ KG ~ ½ Ó

&

ñ d ” \  @ /{ 9  # Œ K ⁻ - ⁴⁰ Ca_  í ß –ê ø Í_  p ì  rí ß –ê ø Íé ß –€  & h `  ¦

>

í ß –ô  Ç   õ  [7]\  ¦ D. Malow(1982) [15]_  z  ´+ « >u ü < q “ §

% i   H X < Fig. 1\ " f ^  ¦ à º e ”   H  ü < ° ú  s  z  ´+ « >u \   s `

Fig. 1. The differential cross sections of K ⁻ - ⁴⁰ Ca scatter- ing at 800 MeV are shown. The solid line represents for the calculated results using the optical potentials with s and p wave included, the dotted line for the results with s wave only included, and the dashed line for the results with the phenomenological term added.

›

  3 l w p u   H   õ \  ¦ % 3 “ ¦ e ” 6 £ §`  ¦ · ú ˜ à º e ”  . s \  ¦ ˜ Ð

&

ñ l  0 AK  p _  l # Œ\  ¦ > í ß – % i   H X <, s   ^ ‰\  - t

  H  _  ¢ - a# 4  >  > í ß –s  ÷ &  H  כ \  ì ø ÍK , p “    â Ä º



 H [16] Õ ªX O t  3 l w  9, š ¸f ”  # Œl  © œI  Λ-/ B N" î õ  Σ-/ B N" î _  1  l # Œë ß –`  ¦ Ÿ í† < Ê >  ÷ &  H X <, s   H H s “ : r " é ¶  \ 

@

/K " f ‚ à Г ¦ë  H‰  ³ [17]\ " f% ƒ! 3   Ò  ¦ M : s   ^ ‰\  -t 

\

" f  H Á ºr ÷ &
 p \ " f  H ×  æ כ ¹ô  Ç % i ½ + É`  ¦ >   ) a  .

p    t \  ¦ “ ¦ 9½ + É  â Ä º\  Ä »´ ò í ß –ê ø ͔  ; Ÿ ¤`  ¦ „  > h # Œ

˜

Ѐ    6 £ § õ  ° ú   .

F _{ef f} ^(s) ⇒ F ef f = F _{ef f} ^(s) + c 0 q · q ⁰ (22) p  \  ¦ “ ¦ 9½ + É  â Ä º\  H s “ : r_   ^ ‰\  -t  > í ß –\  Ÿ í† < Ê

÷ &  H H s “ : r õ  Ù þ ˜ _  Ø  æ[  t \  _ K  Ò q t$ í ÷ &  H Λ, Σ_  / B N

"

î “ É r Fig. 2 ü < ° ú  s 
 
 9, # Œl \ " f & h ‚  “ É r H s “ : r s

“ ¦ z  ´‚  “ É r Ù þ ˜ s  .

Ä

º‚   Λ / B N" î `  ¦ “ ¦ 9 €   Migdal † < Êà º  H  A ü < ° ú  s  Å

Ò# Q”   .

φ(w, k) =

Z d ³ p (2π) ³

n(p

w − E Λ (p + k) + E _N (p) (23)

#

Œl " f M Λ = 1115 MeV, p _K ≤780 MeV, w K =

√ 494 ² + 780 ² ∼924 MeVs  9, s  ° ú כ[ þ t“ É r H s “ : r`  ¦  © œ@ /



: r& h Ü ¼– Ð 2 [/ å L % i `  ¦ M :_  ° ú כ[ þ t s  . ¢ ¸ô  Ç N “ É r ` …Ø Ôp 

% ò

% i \  e ” Ü ¼Ù ¼– Ð k F ∼270 MeV,  F ∼40 MeV  M N s 

“

¦ w N ∼ = M N +  F – Ð j þ t à º e ”  . I = 0Ü ¼– Ð Å Ò# Qt   H

Fig. 2. Λ and Σ resonances are formed by the collision of kaon with nucleon. The dotted line represents for the kaon, the solid lines for the nucleon and hyperon.

Λ ü < I = 1{ 9  M :_  Σ s ` … : r / B N" î `  ¦ “ ¦ 9 Ù ¼– Ð \  - t

  H  6 £ § õ  ° ú  s 
 è ß – .

w _Λ = w _N + w _K ∼ 1864MeV = m Λ + E _Λ

= .

= E _Λ ∼ 749MeV (m Λ ∼ = 1115M eV )

= E _Σ ∼ 644MeV (m Σ ∼ = 1220M eV ) (24)

#

Œl " f E  H y Œ • s ` … : r_  î  r1 l x \  -t \  ¦ _ p ô  Ç . \ P 



2 ;> _  & ñ \  _  €  , _m ^p

^Λ

Λ

∼1\ " f  H Λ\  ¦  © œ@ / : r& h Ü ¼– Ð 2

[/ å L½ + É € 9 כ ¹ e ”  . Õ ª Q
 Ä ºo   H  A  ? /6   x_  s Ä »– Ð

“

 K  q  © œ@ / : r& h Ü ¼– Ð  Ò  ¦  כ s  .

d ”

 (23)Ü ¼– Ð Å Ò# Qt   H Migdal † < Êà º  H  A ü < ° ú  s  ˜ Ð

&

ñ | ¨ c à º e ”  . d ”  (23)_  ì  r — ¸ ×  æ \ " f, E Λ (p + k) − E N (p) = p ²

2M _N

 M N

M _Λ − 1

 + k ²

2M _Λ + p · k M _Λ (25)

–

Ð j þ t à º e ” “ ¦ # Œl " f _2M ^k

²_Λ

∼ ₂ ⁷⁸⁰ _·1115

²

∼273 MeVs  . p  H Ù þ

˜  î  r1 l x| ¾ Ó, k  H H s “ : r_  î  r1 l x| ¾ Ós  9, n(p)  H  6 £ §_  F

Gô  Ç\ " f Å Ò  ) a l # Œ\  ¦ % 3 • ¸2 Ÿ ¤ ´ òõ & h Ü ¼– Ð ¸ ú ˜ ? /  H % i 

½ +

É`  ¦ >   ) a  .

0 = E N (p + k) − E N (p) − w

∼ = k ²

2M _Λ − w + p · k M _Λ + p ²

2M _N

 M N

M _Λ − 1

 (26)

0

A_  F Gô  Ç › ¸| `  ¦ ë ß –7 á ¤ r v l  0 AK " f  H |p| ∼800 MeV

&

ñ • ¸ ÷ &# Q  ô  Ç . ` …Ø Ôp  î  r1 l x| ¾ Ә Ð   s `›    H p ∼800 MeV \ " f ß ¼>  l # Œ\  ¦   H X < ì ø ÍK " f, 2M ^p

²_N

= 320 MeV s

Ù ¼– Ð 2M ^p

²_N

 M

_N

M

_Λ

− 1 



 H Á ºr  0 p x  . ¢ ¸ô  Ç,

w − k ²

2M _Λ  kp F

M _Λ , φ 0 ∼ ρ p

w − _2M ^k

²_Λ

(27) s

 ÷ &Ù ¼– Ð  ^ ‰\  -t   H  6 £ § õ  ° ú  s  j þ t à º e ”  .

Π Λ (ω, k) ∼ = 2



− 1

√ 3 (D + 3F )

 2

1 f _K ²

p ² _k

w − _2M ^k

²_Λ

ρ p (28) Ä

» ô  Ç ~ ½ ÓZ O Ü ¼– Ð f ¹

_K

√ 2(D −F )_    ½ + Ë`  ¦ ° ú   H K ⁻ +n → Σ ⁻ / B N" î \  _ ô  Ç  ^ ‰\  -t \  ¦ ½ ¨ €  ,

Π Λ (ω, k) ∼ = 2( − √

2(D − F )) ² 1 f _K ²

p ² _k

ω − _2M ^k

²_Σ

ρ n (29)

–

Ð
 è ß – . # Œl \ " f Λü < Σ_  G V , \ " f ° ú   H   ½ + Ë © œ Ã

º, Λ\  @ /ô  Ç √ ¹

3 õ  Σ\  @ /ô  Ç √

2 ° ú כ[ þ t_    H    H 3   s

³ 1 Ï  Õ ª| ½ Ót î ß – L 3 „  > h\ " f % 3 `  ¦ à º e ”   [6].

L 3 = − 1 f [ √

2(D + F ) ¯ Ψ p σ ^µ Ψ n ∂ µ π ⁺ + √

2(D − F ) ¯ Ψ p σ ^µ Ψ _Σ

+

∂ µ K ⁰ + (h.c.)

− 1

√ 3 (D + 3F ) ¯ Ψ p σ ^µ Ψ Λ ∂ µ K ⁺ + (D − F ) ¯ Ψ n σ ^µ Ψ _Σ

0

∂ µ K ⁺ ]

− 1 f [ √

2(D − F ) ¯ Ψ n σ ^µ Ψ _Σ

−

∂ µ K ⁺ − 1

√ 3 (D + 3F ) ¯ Ψ n σ ^µ Ψ Λ ∂ µ K ⁰ + (h.c.)

−(D − F ) ¯ Ψ _n σ ^µ Ψ _Σ

0

∂ _µ K ⁰ ] − 1

f [(D + F )( ¯ Ψ _p σ ^µ Ψ _n − ¯ Ψ _n σ ^µ Ψ _n )∂ _µ π ⁰

+ ¯ Ψσ ^µ τ ⁰ Ψ∂ _µ π ⁰ ] + · · · (30)

0 Aü < ° ú  “ É r   ½ + Ë\ " f Λü < Σ_  G V , `  ¦ ³ ðr K  ˜ Ѐ    6 £ § õ

 ° ú  s  j þ t à º e ”  .

f _pΛK

+

= f nΛK

⁰

= − 1

√ 3 (D + 3F )

f _pΣ

+

K

⁰

= −f nΣ

⁰

K

⁰

= √

2(D − F ) (31)

#

Œl " f f K = 1.20 ∗ f π = 1.20 ∗ 93 MeVs “ ¦,   ½ + Ë © œÃ º[ þ t

“ É

r  6 £ § õ  ° ú    [16].

D + F = 1.26, D − F = 0.33, D = 0.795, F = 0.465 (32) K ⁻ + p → Λ\  @ /K " f  H _f ¹

K

√ 1

3 (D + 3F )_    ½ + ËÜ ¼– Ð ³ ð

‰ &

³½ + É Ã º e ” Ü ¼Ù ¼– Ð p \  @ /ô  Ç F  g† < Æ( J $ ™[ > `  ¦ ½ ¨K ˜ Ѐ  , U _opt ^p = Π _Λ

2w ∼ = i0.86M eV (33) Ü

¼– Ð Å Ò# Qt  9 0 Aü < Ä » ô  Ç ~ ½ ÓZ O Ü ¼– Ð f ¹

_K

√ 2(D − F )_ 

 

½ + Ë`  ¦ ° ú   H K ⁻ + n → Σ ⁻ \  @ /K " f  ^ ‰\  -t \  ¦ ½ ¨

# Œ  8K ŠҀ   p \  ¦ “ ¦ 9½ + É M : / B N" î \  @ /ô  Ç F  g† < Æ( J $ ™[ > 

° ú כ“ É r,

U _opt ^p = Π Λ + Π Σ

2ω ∼ = i(0.86 + 0.11)M eV = i0.97M eV (34)

–

Ð Å Ò# Q”   . 0 A\ " f ½ ¨ô  Ç ° ú כ[ þ t`  ¦ s 6   x # Œ s ü < p _  l

# Œ\  ¦ — ¸¿ º “ ¦ 9 # Œ K ⁻ _  F  g† < Æ( J $ ™[ > `  ¦ ½ ¨ €   þ j7 á x

&

h Ü ¼– Ð  6 £ § õ  ° ú  “ É r ° ú כ`  ¦ % 3 >   ) a  .

U _opt ^K

⁻

(r) = U _opt ^s (r) + U _opt ^p (r)

= (6.90 − 26.63i) ρ(r)/ρ 0 M eV (35) 0 A\ " f > í ß –ô  Ç F  g† < Æ( J $ ™[ > `  ¦ Woods-Saxon + þ AI _  r-_ 

”

> r$ í `  ¦ s 6   x “ ¦ d ”  (20)\ " f Å Ò# Q”   ° ú כ[ þ t`  ¦ s 6   x # Œ Õ

ªA á Ԗ Ð Õ ªo €   Fig. 3õ  ° ú  s 
 
 9 s \  ¦ s 6   x # Œ ( Ž

É Ó'  á Ԗ ÐÕ ªÏ þ ›`  ¦ s 6   x # Œ KG ~ ½ Ó& ñ d ” `  ¦ Û  ¦ # Q p ì  rí ß – ê

ø Íé ß –€  & h `  ¦ > í ß – # Œ [7] Õ ªA á Ô\  ¦ Õ ªo €   Fig. 1(z  ´‚  )

\

" f ˜ Ðs   H  ü < ° ú  s 
 è ß – . Õ ªA á Ô\ " f > í ß – ) a p  ì

 rí ß –ê ø Í é ß –€  & h  ° ú כ[ þ t“ É r # Œ„  y  z  ´+ « >u [ þ t_   A A á ¤ \  ì  r

Ÿ

í† < Ê`  ¦ ^  ¦ à º e ”  . ¢ ¸ô  Ç p _  l # Œ\  _ ô  Ç ´ òõ   H š ¸ y

 9 z  ´+ « >u \ " f Y O # Qt   H ´ òõ \  ¦ Šғ ¦ e ” 6 £ §`  ¦ › ' a¹ 1 Ͻ + É Ã

º e ” % 3  . d ü < f  _  l # Œ 1 p x_  4 Ÿ ¤¸ ú š “ ¦, ˜ Ð  & ñ S X ‰ ô 

Ç > í ß –`  ¦ r • ¸ l  „  \ , ‚ à Г ¦ë  H‰  ³ [10]\ " f r • ¸  ) a   ü

< ° ú  s , s [ þ t_  ´ òõ \  ¦ · ú ˜ ˜ Ðl  0 AK   6 £ § õ  ° ú  s  ‰ & ³



© œ : r& h “   ˜ Ð& ñ `  ¦ # Œ > í ß –`  ¦ % i  .

U _ph ^K

⁻

(r) = [U _opt ^s (r) + U _opt ^p (r)]b _p

= (6.90 − 26.63i)b p ρ(r)/ρ 0 M eV (36) b p = (1.27 − 1.49i) fm– Ð Å Ò# Q| 9  M : Fig. 1_  = å S|   ‚  \ 

"

fü < ° ú  s  z  ´+ « >u ü < ¸ ú ˜ { 9 u † < Ê`  ¦ ˜ Ð# ŒÅ Ò% 3 Ü ¼ 9 s  כ “ É r F 

g† < Æ( J $ ™[ > _  z  ´Ã ºÂ Òü < ) ‡Ã ºÂ Ò — ¸¿ º 6 £ §{ 9   â Ä º\  K  {

© œ  9 s [ þ t_  [ jl  s ü < p ë ß –`  ¦ “ ¦ 9 # Œ > í ß –ô  Ç

° ú

כ[ þ t õ  q “ § # Œ 2∼3C  9 þ t  â Ä º\  K { © œô  Ç . Fig. 3\ 

"

f ph-real“ É r ‰ & ³ © œ : r& h  F  g† < Æ( J $ ™[ > _  z  ´Ã ºÂ Òs  9, ph- imaginary  H Õ ª ) ‡Ã ºÂ Ò\  ¦
 ? /“ ¦ e ”  . s  Qô  Ç   õ 

Fig. 3. The optical potentials of K ⁻ - ⁴⁰ Ca scattering at 800 MeV are shown. The solid line is the calculated optical potentials when s and p waves are included, the dotted line when s wave is only included, and the dashed line when the phenomenological term is added. The real part of the phenomenological optical potential is labeled as ph-real, the imaginary part as ph-imaginary.



 H q 2 Ÿ ¤ [ jl ü < Õ ª l  † < Æ& h  + þ AI \  e ” # Q  ™ è_  s   H e ”

Ü ¼
 y © œô  Ç f  ¨ à º — ¸+ þ A(strong absorption model)`  ¦ s 6   x

# Œ > í ß –ô  Ç F  g† < Æ ( J $ ™[ > _    õ  [19]ü < { 9 u    H  כ s 



. Garcia-Recio 1 p x“ É r ‚ à Г ¦ë  H‰  ³ [10]\ " f q 2 Ÿ ¤ ] X   H~ ½ ÓZ O 

“ É

r Ä ºo ü <  Ø Ô
, d ü < f  _  l # Œ\  ¦  8† < ÊÜ ¼– Ð+ ‹  © œ{ © œ ô 

Ç ´ òõ \  ¦ % 3 # Q z  ´+ « >u \  ¦ ¸ ú ˜ [ O " î † < Ê`  ¦ ˜ Г     e ” Ü ¼Ù ¼– Ð Ä

ºo • ¸ s ³ 1 Ï Ä »´ ò © œ : r \    H  ô  Ç s  : r& h  > í ß –Ü ¼– Ð · ú ¡ Ü

¼– Ð d ü < f  _  l # Œ\  ¦ > í ß – # Œ F  g† < Æ( J $ ™[ > `  ¦ > h‚  

† <

ÊÜ ¼– Ð+ ‹ s ³ 1 Ï s  : r õ  z  ´] j í ß –ê ø Í ‰ & ³ © œ s _  “ ¦o \  ¦ ]

j/ B N l  0 Aô  Ç ƒ  ½ ¨\  ¦ > 5 Å q½ + É >  S \ ‰ s  .

III. + s Ç Â ] Ø

H

s “ : r õ  Ù þ ˜   s _   © œ  ñ Œ •6   x \    H   # Œ H s “ : r F 

g† < Æ( J $ ™[ > `  ¦ Ä »• ¸ “ ¦ s \  ¦ s 6   x # Œ H s “ : r-Ù þ ˜ç ß – í ß – ê

ø Í ‰ & ³ © œ`  ¦ [ O " î l  0 AK  Klein-Gordon ~ ½ Ó& ñ d ” `  ¦ Û  ¦ # Q p

ì  rí ß –ê ø Íé ß –€  & h `  ¦ > í ß – “ ¦ s \  ¦ z  ´+ « >u ü < q “ § % i  .

‰ &

³F  t _  s ü < p \  ¦ “ ¦ 9 # Œ > í ß –ô  Ç   õ   H é ß –í  Hô  Ç

s ³ 1 Ï — ¸+ þ A\    H  ô  Ç F  g† < Æ( J $ ™[ > `  ¦ s 6   x½ + É  â Ä º p ì  r í

ß –ê ø Íé ß –€  & h  z  ´+ « >u \  3 l w p u   H ° ú כs  % 3 # Qf ” `  ¦ ˜ Ð# ŒÅ Ò

%

3  . ‰ & ³ © œ : r& h Ü ¼– Ð ( J $ ™[ >  y © œ• ¸\  ¦ ˜ Ð& ñ # Œ 2∼3C  K 

×

 ¦  â Ä º z  ´+ « >u ü < { 9 u    H   õ \  ¦ % 3 `  ¦ à º e ” % 3 “ ¦ s   H z 

´+ « >u  % 3 # Q”   \  -t  # 3 0 A 800 MeV/c– Ð “ ¦\  -t 

% ò

% i \  5 Å q Ù ¼– Ð d ü < f   1 p x_  “ ¦ ´ òõ \  ¦ Á ºr ½ + É Ã º

\ O

Ü ¼ 9  8 ´ ú §“ É r / B N" î ´ òõ • ¸ “ ¦ 9K   ô  Ç   H  כ `  ¦ r  

  H  כ Ü ¼– Ð ˜ Г   .

P c

p 8 ý ò k >

‘

: r ƒ  ½ ¨ „  ì ø Í\    5 g ´ ú §“ É r • ¸¹ ¡ §`  ¦ ŠҒ     1 l x€ 9  “ §Ã º_ ”  a

 y Œ ™ × ¼o  9, { 9 Â Ò > í ß –\  › ¸ƒ  `  ¦ K  ŠҒ   s ‚ ½ Ó¨ 8 Š “ § Ã

º_ ” , & ñ <  ª “ §Ã º_ ” a • ¸ y Œ ™ × ¼w n m  .

Y c

p w Š à U Ø ”  ô

[1] T. S. Park, H. Jung, D. P. Min, J. Korean Phys.

Soc. 41, 195 (2002).

[2] H. Tamura, plenary talk given at PANIC02, Osaka, Japan 2002.

[3] N, Kaiser, P. B. Siegel and W. Weise, Nucl. Phys. A 594, 325 (1995).

[4] N, Kaiser, P. B. Siegel and W. Weise, Phys. Lett. B 362, 23 (1995).

[5] N, Kaiser, T. Wass and W. Weise, Nucl. Phys. A 612, 297 (1997).

[6] C.-H. Lee, G. E. Brown, D.-P. Min, M. Rho Nucl.

Phys. A 585, 401 (1995).

[7] B. C. Clark, S. Hama, G. R. Calbermann, R. L.

Mercer, L. Ray, Phys. Rev. Let. 55, 592 (1985).

[8] M. Ericson and T. E. O. Ericson, Ann. Phys. 36, 323 (1966).

[9] T. Wass, M. Rho, W. Weise Nucl. Phys. A 617, 449 (1997).

[10] C. Garcia-Recio, A. J. Melgarejo, J. Nieves, Phys.

Rev. C 67, 047601 (2003).

[11] C.-H. Lee, H. Jung, D.-P. Min, M. Rho Phys. Lett.

B 335, 14 (1994).

[12] T. Barnes and E. S. Swanson, Phys. Rev. C 49, 1166 (1994).

[13] S. M. Wong, Introductory Nuclear Physics, (Prentice Hall, New Jersey, 1995) pp. 348-351.

[14] de. Jager, de. Vries, Atomic Data and Nucl. Data Tables 36, 495 (1987).

[15] D. Malow, P. D. Barnes, N. J. Colella, Phys. Rev.

C 25, 2619 (1982).

[16] A. Ramos, E. Oset, Nucl. Phys. A 671, 481 (2000).

[17] C. Garcia-Recio, J. Nieves, E. Oset and A. Ramos, Nucl. Phys. A 703, 271 (2002).

[18] E. Oset, A. Ramos, Nucl. Phys. A 635, 99 (1998).

[19] Y. J. Kim, M. H. Cha, J. Korean Phys. Soc. 34, 339 (1999).

Study of of K ⁻ - ⁴⁰ Ca Scattering by Using an Optical Potential

Sugie Shim ^∗ and Min-Soo Kim

Department of Physics, Kongju National University, Kongju 314-701 (Received 7 February 2004)

Kaon optical potentials in the nucleus are calculated based on a simple chiral model, and they are used in the Klein-Gordon equation to calculate numerically the differential cross - section of K

⁻

-

⁴⁰

Ca scattering at 800 MeV/c. The calculated differential cross sections are compared with the experimental values and analyzed. Also, the calculated optical potentials are compared with the phenomenologically modified optical potentials that can explain the experimental data pretty well.

PACS numbers: 25.80.N

Keywords: Optical potential, Kaon scattering

∗

E-mail: [email protected]