dynamically generated resonances from baryon decuplet...
TRANSCRIPT
NSTAR05, Oct 12-15, Tallahassee
Dynamically Generated
Resonances from BaryonDecuplet-Meson Octet Interaction
Sourav Sarkar
Universidad de Valencia-IFIC, SpainCollaborators:
L. Roca, E. Oset, M. J. Vicente Vacas, V. Magas
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Plan
I. General discussion on dynamically generated resonances S.S., E. Oset, M.J. Vicente Vacas, NPA 750 (2005) 294 andEur.Phys.J.A24 (2005) 287
IntroductionUnitary coupled channel formalismResonances observed in different channels
II. Detailed study of the resonance S.S., E. Oset,
M.J.Vicente Vacas, PRC72 (2005) 015206 and L.Roca, S.S., V.K.Magas, E Oset (inpreparation)
Improvements; addition of extra channels, etc.Determination of unknown parameters and couplingsPredictions and comparison with experimental data
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Introduction
Chiral Dynamics + Unitary Coupled Channels Successful prediction of s-wave baryon resonances.s-wave scattering of the pion octet with the nucleonoctet
Dynamical generation of baryonresonances
!" #$ !&% " ' ($ !&% ) '* $ !+ '* $ !
D. Jido et al, NPA 725(03)181;
C. Garcia Recio et al, PLB 582(04)49;
M.F.M. Lutz et al, NPA 700(02)193;
E. Oset et al, NPA 635(98)99
N. Kaiser et al; T. Inoue et al ; etc...
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IntroductionWhat about d-wave baryon resonances?,- , in d-wave has too many unknown terms in .PT
No predictivity!But, these d-wave resonances couple in s-wave to the
baryons decuplet and the mesons octet. Ifthese s-wave channels are dominant the resonancescould appear in the s-wave scattering of the baryonsdecuplet and the mesons octet.
s-wave implies .PT is more predictive!
In fact some of these d-wave resonances( / 0 1 / 0 2 1 3 2 ) have large decaybranching ratios to /54 4 channels even when 4 / isfavoured by phase space!
Kolomeitsev and Lutz, PLB 585(04)243
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Formulation
Lowest order chiral Lagrangian (Decuplet - Octet interaction)6 7 8 9;: <= : <
: <> ?A@ the baryons decuplet and B C the covariant derivative
= C : < > ?@ D C : <> ?A@ E F C G > : <G ?@ E F C G? : <> G@ E F C G@ : <> ? G
F C H I J D C J K E J K D C J ML J I N O P Q I R ST
with a single coupling constant U
E. Jenkins et al, PLB 259 (91) 353
We write : < :WV < whereXZY[\] ^ _ `a ba c dfe eg !ih Yij % d ! Xij % e !
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Formulation
We will consider only the k-wave part of the interaction andthe non-relativistic limit, so thatl Xj m% e m !in o Xj % e ! [ p o q p ^ ^ r s t uv j uwx !
\ r] ^ r\] ^ _ `a ba c d m e m eg !h Yij my% d m ! _ ` a ba c de eg !h Yj % d ! p ^ ^ r [z s t u v j u awx a !&
6 8 :W| 9~: : F f
l ! [] l c o [ # a o z o !
` ` ` [ , ` ` a [ ` b
, ` a a [ ` b q , a a a [ , ` ` b [ ` b )
, ` a b [ ` ) q , a a b [ ` b ) , ` b b [ ` b + q , a b b [ ` b + , b b b [ .
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Formulation
For a meson of incoming (outgoing) momenta we getfor the k- O transition amplitudes,
P 7 U I P E This is used as kernel of a coupled channels BetheSalpeter equation : 7 ¡ H £¢
= +T V G TV
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Formulation
¡¥¤ 8 ¤ ¦ §©¨ 4 § ª 7 ¨ I7 I¤ E 8¬«
¨ I7 I¤ E 8¬«
Using dimensional regularization
¡¥¤ HH®°¯ a ¤ ²± E ³µ´ ¶ a¸·< a E ¹ a · ¶ a · »ºI º ³ ´ ¹ a ·¶ a · E¼ ·Qº ½ ³ ´ k7 I¤ 7 I¤ E ¨ ¤ k E ³ ´ k E I¤ 7 I¤ E ¨ ¤ k 7³µ´ 7 k E I¤ 7 I¤ E ¨ ¤ k 7 ³ ´ 7 k7 I¤ 7 I¤ E ¨ ¤ k ¾¿ 1
with unknown parameters ÀÁ for which the ‘natural size‘ is ÂÃ Ä
corresponding to a cut-off of  700 MeV.
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Formulation
Å N decomposition: , - ,Æ Æ 2Æ Projection on to Å N basis: ÈÇ É PËÊ
Ì 81 Í Î P ÌÏ 1 Ð Î
Strength is proportional to: Ç É ¦ 8Ñ Ò 1 1 1 7
strong attraction in octet, followed by decupletweak attraction in 27, repulsion in 35
We then solve the BS equation and look for poles in thecomplex plane. In the Å N limit we get two poles, oneeach for the octet and decuplet representations
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Results: Trajectories of Poles in the Complex Plane
Two bound states in the Å N limit
1350 1500 1650 1800 1950 2100 Re zR (MeV)
0
50
100
150
200
−Im
zR (M
eV)
S=0, I=1/2S=−1, I=1S=−1, I=0S=−2, I=1/2
Octet
? (1673)
M0=1382 MeVm0=368 MeV
1350 1500 1650 1800 1950 2100 Re zR (MeV)
0
50
100
150
200
−Im
zR (M
eV)
S=0, I=3/2S=−1, I=1S=−2, I=1/2S=−3, I=0
Decuplet
?
(1738)
M0=1382 MeVm0=368 MeV
Break ÓÔ !
symmetry graduallyx Õ iÖ ! [ x q s Ö x Õz x q !× aÕ iÖ ! [ × a q s Ö × aÕ z × a q !$Ø Ö Ø Close to the pole
iÙÚ !
ÕÛ Ù ! [ Ü Õ ÜÛÙ z ÙÚ
residue Ý couplings Ü Õ
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Poles in the Complex Plane
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200-50
0
50
100
150
200
S=0, I=1/2Þ ßàáâãäæå çè é
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200-50
0
50
100
150
200
S=0, I=3/2êàá çëìå áí è éàá ìî ëå áä ì é
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200-50
0
50
100
150
200
S=-1, I=0ïàá è èäæå í ë é
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200-50
0
50
100
150
200
S=-1, I=1ðàá í â îå á è é
àá í ì ëå á ë ì éàîä î áå çè é
1300 1400 1500 1600 1700 1800 1900 2000 2100 2200-50
0
50
100
150
200
S=-2, I=1/2ñ àá ì â îå á ìî é àáã îäæå áâ ë éàî á í îå áã é
1400 1500 1600 1700 1800 1900 2000 2100 2200-50
0
50
100
150
200
S=-3, I=0ò
àî á çáå â ì é
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ª ó 7 Resonances
1300 1500 1700 1900 2100ECM (MeV)
0
0.01
0.02
0.03
0.04
0.05
0.06
|T|2 (1
/MeV
2 )
∆πΣ*K∆π (with width)Σ*K (with width)
(S,I)=(0,1/2)
N*(1520)
1300 1500 1700 1900 2100ECM (MeV)
0
0.001
0.002
0.003
0.004
0.005
|T|2 (1
/MeV
2 )
∆πΣ*K∆η
(S,I)=(0,3/2)
∆(1700)
∆ (?)
1300 1500 1700 1900 2100ECM (MeV)
0
0.005
0.01
0.015
0.02
0.025
|T|2 (1
/MeV
2 )
Σ∗πΞ*K
(S,I)=(−1,0)
Λ(1520)
1300 1500 1700 1900 2100ECM (MeV)
0
0.2
0.4
0.6
0.8
1
|T|2 (1
/MeV
2 )
∆KΣ∗πΣ∗ηΞ*K
(S,I)=(−1,1) −
Σ(1670)
Σ(1940)
1300 1500 1700 1900 2100ECM (MeV)
0
0.05
0.1
0.15
|T|2 (1
/MeV
2 )
Σ*KΞ∗πΞ∗ηΩK
(S,I)=(−2,1/2)
−
Ξ(1820)
Ξ (?)
1300 1500 1700 1900 2100ECM (MeV)
0
0.02
0.04
0.06
0.08
|T|2 (1
/MeV
2 )
Ξ*KΩη
(S,I)=(−3,0)
−
Ω (?)
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ôöõ÷ øù ú
In our treatment so far with the 4 û 0 and üý 0 in coupledchannels the is generated dynamically with alarge coupling to the 4 û 0 channel.
It appears about MeV higher in mass and is about
times broader than the nominal width which is ¢ Ò MeV.
This large width is a consequence of the fact that thepole appears above the 4 û 0 threshold.
However, the width of the comes essentiallyfrom the decay into
9 ü / ( %) and 4 û ( %)
It is mandatory to add these channels to our scheme
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ôöõ÷ øù ú
Other channels like þ and üý could also contributebut their influence will be in the mass and not the width.In any case the position (mass) can be fine tunedthrough the subtraction constants.
The lowest partial wave in which the channels
9 ü / and4 û can couple to spin parity is ÿ .
We will couple these channels in ¦-wave to the 4 û 0
channel and not to the ü ý 0 which is far away from theregion of influenceWe will also introduce the û 0 width in the meson baryonloop function since the 4 û 0 threshold is very close tothe peak of the
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Coupling of -wave channels:K
Σ∗
π
d−wave s−wavek
N (1/2,m) (3/2,M)
à à ! " # $% %
where
&' ( Ä ) * + ( Ä, - . !+ ( Ä Ä ' ( Ä0/ , 1 ) # & ' ( Ä + ( Ä -
so that
à à . ! 2 Ä 3 / , 4 )à , # 65 78 9 ! " # !à + # 9 8 7 :;=<
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K
Σ∗ Σ∗
ππ
N(3/2,M) (1/2,m) (3/2,M´)
a [ > ? @BA* C ! @ DE FHGI # C
n GI E uv A u aJ _ w* * w* c ×%x mz × !K a] J L r MA ! z ! L r J
n GI E uv A u a _ w* * w* c ×%x z × ! K a] J L MA ! z ! L J
Angular integration of the two K z Ý p L L r
Orthogonality of the Clebsch Gordan coefficientsOn-shell factorization of vertex a [ Nn GI E APOQ R aTS D GI E [ UWV X Y GI E DGI E UGI E Y V X
Dynamically Generated
Z []\ ^_ `
Resonances from Baryon Decuplet-Meson Octet Interaction – p.16/45
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a
With the b matrix given by
b cdcecfcdcecfcdcecfcdcg 2 2 ! % 2 h % 2 # g 2 ! % 2 h % # 2 3 i 3 2 j i jg 2 ! % h % 2 # g ! % h % # k k
2 3 i 3 k 3 3 i j3 3 j i 3 i j
2 j i j k 3 j i 3 i j j j i jjcdcecfcdcecfcdcecfcdc
where g 2 2 ; , g ' and g 2 g 2 Ã :l
A Õ [ mn oe Ne z px Õ s × Õ q a R Ne z px Õz × Õ q a R
r qÕ [ e z x aÕ s × aÕn oe
we can continue with the formalism as in ordinary s-wave scattering.
Dynamically Generated
Z []\ ^_ `
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a
To take the t û u width into account we fold ¡ with thespectral function of the û u :
D V X p oe=v x X v × V q Ý Lxw a yzL|w a yz ? oe m z mC Im
moe mz x X s > X pe m qw n~
S D V X p oe=v oe mv × V qwhere
X pe m q [ q A p= v x v × V q
A p Xv v × V q p o z z × V q
m n A p v Xv × V q
A p Xv Xv × V q p o z X z × V q v
Dynamically Generated
Z []\ ^_ `
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a
Using b as the kernel and the loop function we use the coupled channelBS equation to get the amplitudes for
and <
From a fit to the experimental amplitudes
we find the unknownparameters 2 3 , 2 j, 3 3 , 3 j, j j in the b matrix and the subtractionconstants % , in
P p o q z AC o A C o p o q
¡ ¡ ¢£ p o ¤¥¦ § ¨ q
We get the following values
©§ © ª¥ p
MeV « q ª¥ @ pMeV « q ª p
MeV «¦ q ª @ @ p
MeV «¦ q ª @ p
MeV «¦ qz m z m ¬ S m « m mS m « z m ®S m « ¥ z ®S m « ¥ z m mS m « ¥
Dynamically Generated
Z []\ ^_ `
Resonances from Baryon Decuplet-Meson Octet Interaction – p.19/45
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¯ a 1450 1475 1500 1525 1550 1575 1600
-0.2
-0.1
0
0.1
0.2
0.3
Re T~
1450 1475 1500 1525 1550 1575 1600
-0.2
-0.1
0
0.1
0.2
0.3
1450 1475 1500 1525 1550 1575 1600√__ s (MeV)
0
0.1
0.2
0.3
0.4
0.5
Im T~
1450 1475 1500 1525 1550 1575 1600√__ s (MeV)
0
0.1
0.2
0.3
0.4
0.5
K_
N K_
N K_
N πΣ
Fit to the experimental amplitudes. Left column:
°²±³ ´ °²±³ ; right column:
°²±³ ´ C µ.Experimental data from G. P. Gopal et al. NPB119 (1977) 362 and M. Alston-Garnjost et al.PRD18 (1978) 182
Dynamically Generated
Z []\ ^_ `
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¯·¶ a ¸ ¹
1450 1500 1550 1600
-0.06
-0.04
-0.02
0
0.02Re
T (
MeV
-1)
1450 1500 1550 1600-0.03-0.02-0.01
00.010.020.03
1450 1500 1550 1600-0.03-0.02-0.01
00.010.020.03
1450 1500 1550 1600-0.1
-0.08
-0.06
-0.04
-0.02
0
Im T
(M
eV-1
)
1450 1500 1550 16000
0.02
0.04
0.06
0.08
0.1
1450 1500 1550 16000
0.02
0.04
0.06
0.08
0.1
1450 1500 1550 16000
0.002
0.004
0.006
0.008
0.01
|T|2 (
MeV
-2)
1450 1500 1550 1600
√_ s (MeV)
0
0.002
0.004
0.006
0.008
0.01
1450 1500 1550 16000
0.002
0.004
0.006
0.008
0.01
πΣ∗ πΣ∗ πΣ∗ Κ_
Ν πΣ∗ πΣ
From left to right: Unitary amplitudes for º µ ´ º µ , º µ ´ °²±³ and º µ ´ º µ.Dynamically Generated
Z []\ ^_ `
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¯·¶ a ¸ ¹
Close to the peak of the » ¼½¾ ¿À Á the amplitudes are givenby p o q  Âo z ¤¥ ¦ § ¨ à ¡ ¤¥ ¦ § ¨ Ä nfrom where we calculate the couplings of the » ¼½¾ ¿À Á to thedifferent channels:
  z ¡ ¤¥ ¦ § ¨n Å p ¤¥ ¦ § ¨ q Å ¢£ Æ p ¤¥ ¦ § ¨ qÇv
We get Â¥    @ ¬ m z n¬ z È z È
The partial decay widths can be obtained from¡ Â nPº
¤¥ ¦ § ¨ É
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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¯·¶ a ¸ ¹
Predictions:
The reaction ÐÒÑ Ó Ô t Õ×Ö u Õ ¼ ½Ø Ù¾ Á Ô t Õ t Õ » ¼ ½ ½ ½Ú ÁK−
πo p1( ) p
2( )πo
Σ∗op
(k)
(1/2,m) (3/2,M) Λ (1/2,m´)
z ÃÛ pÜÞÝ ¥v ÜÝ q z Ã=ßà áâã äåæ çè é äåã Ä £ã êz äå à ¡ äå ë ê ì Ä èíïî
ðz è Ý Òñ £ ò Ä è
Ý Òó ÃÝ Òô £ z ò Ä èõïö
÷
Symmetrize the amplitude three-body phase space cross-section
Dynamically Generated
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¯·¶ ø ¸ ¹
We also add the following conventional diagrams
p1( )πo
p2( )
K− πo
(1/2,m)p Λ (1/2,m´)p
(k)πo p
2( )
Σ∗ο
p1( )πo
K−
Λ (1/2,m´)p(1/2,m) p
(k)
(a) (b)
K− πo p2( )
πo p1( )
K−πo p
2( )
πo p1( )
(k)
(1/2,m´)Λp(1/2,m) (1/2,m)pΛ (1/2,m´)
(a) (b)
K
(k)
Dynamically Generated
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¯·¶ ø ¸ ¹
350 400 450 500 550 600 650 700q (MeV)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
σ(K
- p
π0 π0 Λ)
(mb)
ÐÒÑ Ó Ôù Õù Õ » cross section (mb) vs ÓúÞû ü (MeV) of Ð Ñ
Experimental data from S. Prakhov et al., PRC69 (2004) 042202
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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¯·¶ ø ¸ ¹
300 320 340 360 380 400 420 440 460q (MeV)
0
0.5
1
1.5
2
2.5
3
3.5
4
σ(K
- p
π+ π− Λ)
(mb)
ÐÒÑ Ó Ôù ýù Ñ » cross section (mb) vs ÓúÞû ü (MeV) of Ð Ñ
Experimental data from T. S. Mast et al., PRD7 (1973) 5
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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¯·¶ ø ¸ ¹1460 1480 1500 1520 1540 1560 1580 1600 1620 1640
MK-p (MeV)
0
100
200
300
400
500
600
700
dσ/d
MK
- p (a
rbitr
ary
units
)
ÐÒÑ Ó invariant mass distribution for the þ Ó Ô Ð ý Ð Ñ Ó
reaction with photons in the range ÿ ¿ Ù Ù
GeV.Experimental data from D. P. Barber et al., Z. Phys. C7 (1980) 17
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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¯·¶ ø ¸ ¹1450 1500 1550 1600 1650
MK-p (MeV)0
10
20
30
40
50dσ
/dM
K- p
(arb
itrar
y un
its)
1450 1500 1550 1600 1650MK-p (MeV)
0
5
10
15
20
25
30pπ− = 1.6-2.4 GeV/c pπ− = 2.9-3.3 GeV/c
ÐÒÑ Ó invariant mass distribution for theù Ñ Ó Ô Ð Õ ÐÒÑ Ó
reaction. Left: pions in the range Ó « ½ Ú ¿
GeV/c; right:Ó « ¿ Ø Ø GeV/c.Experimental data from O. I. Dahl et al., Phys. Rev. 163 (1967) 1377
Dynamically Generated
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Summary: resonances
Systematic study of the interaction of the baryondecuplet with the meson octet taking the lowest orderchiral Lagrangian (Weinberg Tomozawa term) Ø ¿ Ñ resonances
Poles associated to established resonances: ¼ ½¾ ¿ À Á , ¼ ½ À À Á , » ¼ ½¾ ¿À Á ,Ö ¼ ½Ú À Á ,Ö ¼ ½ À Á , ¼ ½ Ù ¿À Á , ¼ ¿ ¿¾ À Á
Prediction of couplings of these resonances to themeson baryon decay channels partial widthsPrediction of extra resonances not yet observed: ¼ ¿½Ú À Á with a width of 40 MeV, and some others toobroad e.g. 1550 MeV.
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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Summary:
Phenomenological introduction of the !channels
" Ð andù Ö into the coupled channel scheme
Introduction of the Ö width in theù Ö loop functionWe find that the » ¼½¾ ¿À Á couples strongly to theù Ö
channel though the branching ratios to the
" Ð andù Ö
channels are much larger
Prediction of amplitudes and couplings of the » ¼ ½¾ ¿ À Á toall the channelsWe obtain good agreement with experimental data inthe reactions Ð Ñ Ó Ô »ù ù , þ Ó Ô Ð ý Ð Ñ Ó andù Ñ Ó Ô Ð Õ Ð Ñ Ó at energies close to and above the» ¼ ½¾ ¿À Á region.
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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Extras
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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Results: # $ # ( %)
States: ù and Ö Ð.
1300 1500 1700 1900 2100ECM (MeV)
0
0.01
0.02
0.03
0.04
0.05
0.06
|T|2 (1
/MeV
2 )
∆πΣ*K∆π (with width)Σ*K (with width)
(S,I)=(0,1/2)
N*(1520)
Pole at ½Ø ¿ & ¿À MeVwith strong coupling to ù
channel.PDG ¼½¾ ¿À Á , ' ½ ¿À
MeVThe peak at 1877 MeV isa threshold effect.
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
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Results: # $ # ( )
States: ù , Ö Ð and )( .Two poles in the complex plane
1300 1500 1700 1900 2100ECM (MeV)
0
0.001
0.002
0.003
0.004
0.005
|T|2 (1
/MeV
2 )
∆πΣ*K∆η
(S,I)=(0,3/2)
∆(1700)
∆ (?)
Pole at ½ Ù & ½Ú ¾ MeVwith strong coupling to ù
channel.No counterpart in PDG.Missing resonance ?
Pole at ½ Ù ¿ & ½ À Ù MeVwith strong coupling toÖ Ð channel.PDG ¼ ½ À À Á , ' Ø À À
MeV
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.33/45
NSTAR05, Oct 12-15, Tallahassee
Couplings of to various channels
*,+ ½ Ù & ½Ú ¾ ½ Ù ¿ & ½ À Ù
-. / -. / -. / -. /
ù ¿ À & ½ ¿ Ù À ¾ 0 & À Ù ½ ÀÖ Ð ½ Ú & ½ Ú ¿Ø Ø Ø 0 & À Ø
( À Ø & À ½ À Ø ½ & ½ ¿ ¿
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.34/45
NSTAR05, Oct 12-15, Tallahassee
Results: # 1 $ # ( )
States: Ö ù and Ð.
1300 1500 1700 1900 2100ECM (MeV)
0
0.005
0.01
0.015
0.02
0.025
|T|2 (1
/MeV
2 )
Σ∗πΞ*K
(S,I)=(−1,0)
Λ(1520)
Pole at ½¾ ¾ À &Ú MeVcouples strongly to Ö ù
channel.PDG » ¼ ½¾ ¿ À Á
The peak around 2000MeV is a threshold effect.
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.35/45
NSTAR05, Oct 12-15, Tallahassee
Couplings of various to channels
*,+ ½¾ ¾ À &Ú
-. / -. /
Ö ù ¿ À & ½ ¾ ¿¾ Ð À & À Ù ½ ¿
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.36/45
NSTAR05, Oct 12-15, Tallahassee
Results: # 1 $ # ( )
States: " Ð, Ö ù , Ö ( and Ð.We find three poles in the complex energy plane
1300 1500 1700 1900 2100ECM (MeV)
0
0.2
0.4
0.6
0.8
1
|T|2 (1
/MeV
2 )
∆KΣ∗πΣ∗ηΞ*K
(S,I)=(−1,1) −
Σ(1670)
Σ(1940)
Pole at ½Ú Ø ¿ & ½¾ MeVwith strong coupling to " Ð channel.PDG: Ö ¼ ½Ú À Á , ' Ú À
MeVPole at ò23 ®54 Ã ò ®3 MeV. Too broad !
Pole at ¿À ¿½ & ¾ is asso-ciated to Ö ¼ ½ À Á with ' ¿ ¿À MeV in the PDG
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.37/45
NSTAR05, Oct 12-15, Tallahassee
Couplings of to various channels
*+ ½Ú Ø ¿ & ½¾ ½Ú Ù & ½ Ù ¿ À ¿½ & ¾
-. / -. / -. / -. / -. / -. /
Ð Ø & À ÀØ Ø À & ½ ½ Ù À & À ¾ À ÚÖ ù ½ ½ 0 & À ½ ½ ¿ ¿ & ¿ À Ø À À Ø 0 & À Ù À Ù
Ö ( ½ Ù & À Ø ½ ½ 0 & À Ú ½ ½ À & À ½ ¿
Ð À Ø 0 & À ¾ À Ú ¿ & ½ Ø À ¿¾ 0 & ½ À ¿
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.38/45
NSTAR05, Oct 12-15, Tallahassee
Results: # 1 $ # ( )
States:Ö " Ð, ù , ( and Ð.We find four poles in the complex energy plane
1300 1500 1700 1900 2100ECM (MeV)
0
0.05
0.1
0.15
|T|2 (1
/MeV
2 )
Σ*KΞ∗πΞ∗ηΩK
(S,I)=(−2,1/2)
−
Ξ(1820)
Ξ (?)
The pole at ½ Ù & ½¾
MeV is associated withthe ¼ ½ Ù ¿À Á which has ' ¿ ý 67Ñ 6 Õ MeVwidth (on real axis) appears reduced dueto Flatt 8:9 effect
Poles at ò3 æ è 4 Ã ò3 è and ò ¬ è; 4 Ã ò æ ®
MeV are too broad to show up
Pole at ¿½Ú ¿ & ½ MeV couples strongly to Ð Ô
quasibound state !Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.39/45
NSTAR05, Oct 12-15, Tallahassee
Couplings of to various channels
< ê ò3 2 æ 4 Ã ò ë>= ; ¬ ì ò3 æ è 4 Ã ò3 è ò ¬ è; 4 Ã ò æ ® è ò 2 è 4 Ã ò ¬
 Š Š Š Š Š Š Š Å
µ ? ± ò ¬@ Ã; ® è ; ò 3 4 Ã ò ò è ò ò ò @ Ã; ò ò ò ; æ 4 Ã; ; È
A ? º ; È @ Ã; ¬ ò ò è æ 4 Ã ò 3 è ¬ ò ò 4 Ã ò ® è ; ; è @ Ã; ® ; ®
A ?:B è È @ Ã; è è 2 ò @ Ã ò æ ò ¬ æ È @ Ã ò ® æ 3 ; 4 Ã; æ ; È
C ± ; ò 4 Ã; ® ; ® è æ 4 Ã; ¬ è ò 2 4 Ã; ò ® è ò @ Ã; ¬ è æ
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.40/45
NSTAR05, Oct 12-15, Tallahassee
Results: # 1 , # ( )
States: " Ð and ( .
1300 1500 1700 1900 2100ECM (MeV)
0
0.02
0.04
0.06
0.08
|T|2 (1
/MeV
2 )
Ξ*KΩη
(S,I)=(−3,0)
−
Ω (?)
We find a pole at ¿½ ½ &Ø Ù MeVCould be associated tothe ¼ ¿ ¿¾ À Á (***), ' ¾ ¾ D
½ Ù MeV in PDG
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.41/45
NSTAR05, Oct 12-15, Tallahassee
Couplings of to various channels
*,+ ¿½ ½ &Ø Ù
-. / -. /
Ð ½ ½ & À Ù ½
( Ø Ø 0 & À Ø
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.42/45
NSTAR05, Oct 12-15, Tallahassee
Formulation
Example: E 4 ò , F ;
G H ± H G I ±J µ ?J º I µ ? H º H µ ? HB µ ? I ºJ A ?J ± I A ? H ± H
G H ± H è è 4 ò ò 4 ç æ ; ; ;
G I ±J è ; 4 ò 4 ç æ 4 ò ; ;
µ ?J º I è è ; ; è ;
µ ? H º H ; ; 4 è ò 4 ò
µ ? HB ; ; ç æ ç æ
µ ? I ºJ è ; è
A ?J ± I è 4 ò
A ? H ± H è
Å µ ? º KML ; N L OP Å µ ? I ºJ N 4 O P Å µ ? H º H N 4 O P Å µ ?J º I N
;Å A ?Q KML ; N L OR Å A ? HQ H N 4 OR Å A ?J Q I N
to finally get for KML ;
µ ? º A ? Q
µ ? º S ç2
A ? Q æ
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.43/45
NSTAR05, Oct 12-15, Tallahassee
N/D Method
Unitarity states that, above threshold,
TU V W4 ò ¼X ÁY ÃZ [ Ã Ãù X
\ ÃZ U V ] ¼X Á
Using a subtracted dispersion relation
W4 ò ¼X Á ] ¼X Á 0 ^4 ò ¼X Á
where ] ¼_ Á
contains an arbitrary subtraction constant and^4 ò
accounts for contact terms which remain at tree levelwhen ] À .The above equation can be cast as
W T½ ^ ]Y 4 ò ^ 0 ^ ] W
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.44/45
NSTAR05, Oct 12-15, Tallahassee
`·¶ a ø ¸ ¹
The actual amplitudes are given by
Ûã äå âã äå L bã äå âã äåÛà cå â à cå L bà cå â à cåÛßà áâã äå L bßà áâã äå d ë O R è PR e f>g h 4 f ìi R)j kJ l ë m:n ì ë4 ò ì lJ k ç S º
Ûã ä â ã äå L bã ä âã äå d ë O R è PR e fg h 4 f ìi R)j kJ l ë m:n ì ë4 ò ì lJ k ç S º
Ûß à áâ ßà á L bßà áâ ßà á ld ë OR è PR e f>g h 4 f ìi R j kJ l ë m n ìpo
o d ë O R è PR e f q g h 4 f q ìi ?R)j k rJ l ë msn q ì ë4 ò ì k rJ k S º
Dynamically Generated
Ê Ë]Ì ÍÎ Ï
Resonances from Baryon Decuplet-Meson Octet Interaction – p.45/45