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¦ “¦9 %iܼ 9 Λ(1405)_ ÅÒכ¹ ½+Ë©œÃº[þt`¦ 0ArôÇ {9[þt_ ½+Ë©œÃº[þt“Ér 4Ÿ¤ Ôæõ\"f •¸Ø¦

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a °úכ[þt`¦ 6 x %i. éߖ€&h“Ér ÅÒכ¹ ½+Ë©œÃº gKpΛ<sub>1405</sub>_ o\ Bĺ yŒ™ #Œ FgÒqt$í ìøÍ6£x\"f Λ1405†½Ó_ l#Œ tC&he”`¦ ˜Ð#Œïr.

PACS numbers: 25.20.Lj

Keywords: Λ(1405) /BN"î, ~½Ó$í yŒ™ûZ, FgÒqt$í, íߖêøÍéߖ€&h

I. "e Â]Ø

Λ(1405) /BN"î{9_ Û¼—2; Jowü< l¬¹Ãº yŒ•yŒ• J^P = (¹₂)⁻ õ S = −1–Ð ·ú˜94R e”tëߖ Õª ?/ÂÒ ½¨

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$ß¼
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H š¸A1lxîߖ 7HêøÍ ×æ\ e” [2,3]. Λ(1405)H K⁻p >

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 €• 30 MeV A\ e”l M:ëH\ s > 4Ÿ¤ Ôæõ 





H K⁻p → Λγü< K⁻p → Σ⁰γ õ&ñs Λ(1405)_ $í|9

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x_ qÖ¦ (branching ratio)`¦ ›'a8£¤ H כ s r•¸÷&%3

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¦ [4,5], s õ–Ð ÂÒ' Burkhardt [þt [6,7]s
 Work- man [þt [8]“Ér e¦ —¸+þA`¦ 6 x #Œ Λ(1405)_ …;s —¸F'pàÔ κΛΛ1405ü< κ_Σ⁰_Λ1405\¦ s:r&hܼ–Ð Æҏ:rK ˜Ð€Œ¤. ëߖ{9 s X

O

> &ñ)a °úכ[þts &ñSX‰ € 0A\ ƒ/åLôÇ Λ(1405)\

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/ôÇ #ŒQt —¸+þA[þt–Ð s[þt`¦ >íߖK 4Ÿ§Ü¼–Ð+‹ #Q‹"

—

¸+þAs Λ(1405)\¦ ¸ú˜ lÕüt Ht µ1߁n= ú e”`¦ כ s.

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Ǽ#, Ramos [þt“Ér chiral unitaty —¸+þA`¦ 6 xôÇ FgÒqt$í ì

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Í6£x γp → K⁺Λ(1405) >íߖ\"f éߖ€&h`¦ |9|¾ÓoÖ¦

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Ð >íߖ #Œ Λ(1405) /BN"î{9 πΣü< ¯KN ¿º t ½¨ 5

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q ©œI_ ×æ^o?)a õe”`¦ \V8£¤ %i [9]. e¦ —¸+þAܼ

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Ð FgÒqt$í ìøÍ6£x`¦ >íߖôÇ Williams[þt (WCC)“Ér éߖ€&h _

 ß¼l
 yŒ••¸ ìrŸí\¦ \V8£¤ “¦ e” [10]. ÕªQ
 WCC

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¦ crossing @/gAõ duality›'a>\¦ s6 x #Œ t-GV, l#Œ

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H “¦9 t ·ú§l M:ëH\ ÕªXO> &ñ)a ½+Ë©œÃº[þt“Ér



A\ ]jrôÇ Table 1\"f ˜Ðrx 4Ÿ¤ Ôæõ–Ð Æҏ:r

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a ëH‰³ [7]s
 [8]_ õü<H ©œ{©œôÇ s e”. 8

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¨
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 FgÒqt$í“Ér ëH)3 H~½Ó\"f œíl γp ©œI_ JowH ´ú˜ l

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H)3H~½Ó\"f t-GV, l#Œ ×æכ¹ H כ `¦ _pôÇ [11]. "f e¦ —¸+þA\"f Λ(1405) FgÒqt$í“Ér Dh–Ð >íߖK

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¦ €9כ¹ e”.

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r 7HëH“Ér 4Ÿ¤ Ôæõ_ ”;Ÿ¤`¦ >íߖ HX< 6 xôÇ e¦

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¸+þA [6–8,10] ܼ–Ð FgÒqt$í ìøÍ6£x_ éߖ€&h[þt`¦ >íߖ 9 ô



Ç. >íߖ\ €9כ¹ôÇ {9[þt_ ©œÃº°úכ[þt“Ér WCC —¸+þAõ

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H ²ú˜o 4Ÿ¤ Ôæõ\"f &ñ)a °úכ`¦ 6 x½+É כ s. ¢¸ ô



Ç ëH‰³ [7]\"f y©œ›¸ôÇ כ %ƒ!3 s ìøÍ6£x[þt`¦ e¦ —¸+þAܼ

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Ð >íߖ½+É M: ×æכ¹ôÇ %i½+É`¦ H y©œ{9_ +þAI“\¦

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Œl"f•¸ “¦9½+É כ s [11].

-373-

Κ

+

+ (b)

Λ 1405 Κ

, + Λ 0 γ

Κ

(c) Λ

(d) (a)

Σ

1405

Κ

Λ p

1405 +

p Λ

1405 p

Κ

+ γ

p

γ γ

p

Fig. 1. Born diagrams for γp → K⁺Λ(1405) process.

Diagrams (a), (b) and (c) denote the s-, u- and t-channel pole terms with the hadronic form factor depicted as the blob at each vertex. The diagram (d) corresponds to ∆M which is necessary for gauge invariance of the elementary Born terms (a), (b), and (c). It is depicted as a large blob.

II. T Â]Ø

F 

gÒqt$í ìøÍ6£x γp → K⁺Λ(1405)\ @/ôÇ éߖ€&h“Ér dσ

dΩ= 1 4

M MΛ1405

(4πW )²

|q|

|k|Σs,s⁰|M|² (1) Ü

¼–Ð ÅÒ#Qt 9 qü< kH yŒ•yŒ• H“:rõ Fg_ îr1lx|¾Ós“¦

œ

íl{9[þt\ @/ôÇ Û¼—2; ¨îçH`¦ ôÇ כ s. ³ðïr e¦ —¸ +

þ

A\"f …;s ”;Ÿ¤ M“Ér Fig. 1õ 2\"f ˜Ð“ כ %ƒ!3 tree

½

¨›¸_ Born H\¦ 6 x #Œ l‘:r&h“ Born†½Óõ /BN"î {

9

†½Ó_ l#Œ–Ð ½¨$í)a.

M = MB+ MR. (2)

Λ(1405) {9_ FgÒqt$í ”;Ÿ¤“Ér Jow 6£§Ãº“ &h`¦ yŒ™ î

ß

– #Œ hyeron FgÒqt$í ìøÍ6£x γp → K⁺Y [12]\"f Dirac spinor\¦ u(p) → γ5u(p)–Ð ¨8Š € çߖéߖy ½¨½+É Ãº e”



. "f y©œ{9 +þAI“\¦ “¦9ôÇ Λ(1405)_ FgÒqt$í ì

ø

Í6£x\ @/ôÇ PS ½+Ë Born ”;Ÿ¤“Ér Aü< °ú s ÅÒ#Q”



.

MB = −e ¯uΛ₁₄₀₅(p⁰)

½

gKpΛ₁₄₀₅F (s)(/p + /k + M ) s − M²

h /

² − κp

2M² //ki +gKpΛF (u)κΛΛ1405

2M ² //k(/p⁰− /k − MΛ) u − MΛ2

+gKpΣF (u)κΣΛ1405

2M ² //k(/p⁰− /k − MY)

u − MΣ2 + gKpΛ₁₄₀₅F (t)(2q − k) · ² t − m²_K

¾

u(p). (3)

s

p ·ú˜9” ü< °ú s d”(3)\"f F (s), F (u)ü< F (t)

y

Œ

•yŒ• s-, u-, Õªo“¦ t-GV,_ y©œ{9 +þAI“–Ð •¸{9÷&

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 Born ”;Ÿ¤_ >st Ô¦“Ér L:#Qt> )a. sכ “Ér



6£§õ °ú s ÅÒ#QtH †½Ó,

∆M = eg_KpΛ1405u¯_Λ1405(p⁰)

½

(F (s) − bF )(2p + k) · ²

s − M² + (F (t) − bF )(2q − k) · ² t − m²_K

¾

u(p) , (4)

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¦ 8 “¦9†<Êܼ–Ð+‹ 4Ÿ¤"鶝)a [13]. sM: ˜Ð&ñ†<Êú bF H Born H\"f l‘:r&hܼ–Ð &ñ H כ %ƒ!3 on-shell {9 M

: „†½Ó_ FG°úכs d”(3)\"fëߖ l#ŒK l M:ëH\ d

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(4)_ „†½Ós on-shell{9 M: ÕªXO ú e”H 0px$í`¦ C

]j l 0A #Œ (F (s) − bF )ü< (F (t) − bF )H on-shell {9 M

: yŒ•yŒ• 0s ÷&H ›¸| `¦ ôÇ [11].

Fig. 2H ìøÍ6£x\ l#Œ½+É Ãº e”H #ŒQt /BN"î{9†½Ó [

þ

t`¦
·p כ s. #Œl"fH >íߖ_ õ —¸+þA\ _

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r H כ `¦ þj™èo l 0A #Œ 4Ÿ¤ Ôæõ\¦ :Ÿx #Œ Õª



½+Ë©œÃº\¦ Æҏ:r ½+É Ãº e”H כ [þtëߖ /BN"î{9[þt_ l#Œ MR–Ð “¦9ôÇ. Fig. 2_ (a), (b), (c)†½Ó“Ér s-GV, “§¨8Š{9



–Ð N^∗(1650)(¹₂⁻)ü< N^∗(1710)(¹₂⁺)\¦, u-GV, {9–Ð Λ(1405)(¹₂⁻)\¦ Õªo“¦ K^∗(892)(1⁺)ü< K1(1270)(1⁻)\¦ t-GV, “§¨8Š{9–Ð yŒ•yŒ•
·p כ s 9 Õª ”;Ÿ¤[þt“Ér  6

£

§õ °ú s ÅÒ#Q”.

M^N(±) = e g_KN(±)Λ1405F^∗(s)κ_N(±)p

2M u¯_Λ1405(p⁰) (/p + /k ± M_N(±)) s − M_N(±)2+ iΓ_N(±)M_N(±)

/

² /k u(p), (5)

M^Y(±) = −e g_KpY(±)F^∗(u)κ_Y(±)Λ1405

2M u¯Λ1405(p⁰) /² /k (/p⁰− /k ∓ M_Y(±))

u − M_Y^(±)2+ iΓ_Y^(±)M_Y^(±)u(p), (6)

MK^∗= G^K_V^∗F^∗(t) ²αβτ σk^α²^βq^0τ(−g^σµ+ q^0σq^0µ/M_K²∗) t − M_K²∗+ iΓK^∗MK^∗

¯

uΛ1405(p⁰) µ

γµ+ iκK^∗pΛ1405

2M σνµq^0ν

¶ γ5u(p), MK1= G^K_V¹F^∗(t)(k · q⁰²µ− ² · q⁰kµ)(−g^µν+ q^0µq^0ν/M_K²₁)

t − M_K²₁+ iΓK1MK1

¯

uΛ1405(p⁰) µ

γν+ iκ_K1pΛ1405 2M σανq^0α

¶ u(p).

(7)

(f) (g)

Λ

Κ Λ γ Κ

*

, Λ

N

Κ

1405

*

γ

* +

*

Λ 1405

Κ

p (e)

+

p

1

1405 γ

1405 p

Κ +

Fig. 2. Diagrams for resonance exchanges in the γp → K⁺Λ(1405) process.

d

”

(5)ü< (6)\"f Jow ± “ /BN"î{9\¦ N^(±)ü< Y^(±)–Ð y

Œ

•yŒ• ³ðl %iܼ 9 d”(6)\"f Λ(1405)\¦ Y⁽⁻⁾–Ð ³ðl

%i. d”(7)\"f ½+Ë©œÃº G^K_V^∗ = gγKK^∗gK^∗pΛ1405ü<

G^K_V¹ = gγKK1gK1pΛ1405\¦
?/ 9 7˜'Bj”>r_ t-GV, î

r1lx|¾Ó „s q⁰= (q − k) s. l#Œ ß¼t ·ú§`¦ כ ܼ–Ð

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V8£¤÷&H J$™"f ½+ˆ½Ó•¸ ½+Ë©œÃº ¸ú˜ ·ú˜94R e”t ·ú§ l

 M:ëH\ #Œl"fH “¦9 t ·ú§H. s[þt /BN"î{9_ +

þ

AI“H Fig. 1_ (a), (b), (c)\ e”H €ªœ$í, Λ(Σ⁰) Õ

ªo“¦ K⁺„†½Ó[þtõH ½¨Z> l 0A #Œ yŒ• GV,Z>–Ð F^∗–Ð ³ðl %iܼ 9 /BN"î{9[þt_ âĺ\H sQôÇ +þA I

“ [þt#Qü<•¸ Fg l —¸F'pàԖРJ$™"f ½+Ë`¦

l M:ëH\ >st Ô¦“Ér %ò†¾Ó`¦ ~ÃÎt ·ú§H.

0

A_ d”[þt\ 6 x|¨c ½+Ë©œÃº[þt`¦ Table 1\ ú2Ÿ¤ %i



. Table 1\"f BLR(1) [6], BL(2) [7] Õªo“¦ WF [8]–Ð

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a כ [þts 9 ‘:r >íߖ\H s[þt_ °úכ`¦ “¦9 ’x. WCC

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¸+þA_ ½+Ë©œÃº[þt“Ér FgÒqt$í γp → K⁺Λü< γp → K⁺Σ z



´+«>õ\ &h½+Ë •¸2Ÿ¤ fit`¦ :Ÿx #Œ ½¨ôÇ כ [þts [10].

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r 7HëH\"fH 0A_ d”[þt\"f •¸{9ôÇ +þAI“\¦  6

£

§õ °ú s ¿º 9,

F (x) = Λ⁴

Λ⁴+ (x − M_x²)², (8)

€ ª

œ$íü< s(:r Λ, Σ⁰ Õªo“¦ K⁺ „†½Ó[þt\ @/K

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fH cut-off Λ = 1.2 GeV–Ð éH. /BN"î{9 +þAI“ F^∗•¸ (8)d”õ °ú “Ér +þAd”`¦ 2[ ’xtëߖ cut-off Λ = 1.8 GeV–Ð ØÔ> ‚×þ˜ ’x. ˜Ð&ñ†<Êú bF H ·ú¡+‹ ƒ/åLôÇ on-shell ›¸| `¦ ëߖ7ᤠ•¸2Ÿ¤

F = F (s) + F (t) − F (s)F (t)b (9)

–

Ð ¸úšH [12,14].

III. +sÇÊÝ õmÍ ÀXØ8ý

Fig. 3“Ér Table 1\ ÅÒ#Q” °úכ[þt–Ð >íߖôÇ γp → K⁺Λ(1405)_ „ éߖ€&h`¦ Fg_ {9 \-t–Ð




·

p כ s. Fig. 3\"f —¸ŽH ‚[þt“Ér d”(2)ü< s  d”[þt–Ð Å

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