� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
An Instanton Induced Interactionin a relativistic constituent quark model of baryons
Bernard Metsch
Helmholtz-Institut fur Strahlen– und Kernphysik
Universitat Bonn
Nußallee 14-16, D-53115 Bonn, GERMANY
e-mail: [email protected]
N*2002 Pittsburgh, 08-13.10.2002 – p.1
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
Outline
!
Instantons in QCD:! "SB! #%$ &' (
-problem!
instanton induced quark interaction:! ) *)-interaction: (pseudo)scalars! ) )-interaction: octet-decuplet-splitting!
a relativistic constituent quark model:!
basic assumptions and approximations: BSE + SE!
parameterisation of confinement!
results:!
mass spectra: Regge trajectories, parity doublets!
e.m properties; form factors and helicity amplitudes!
strong decays
N*2002 Pittsburgh, 08-13.10.2002 – p.2
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
Instantons
Yang-Mills lattice simulationsar
Xiv
:hep
-lat
/931
2071
v2
23
Dec
199
3ar
Xiv
:hep
-lat
/931
2071
v2
23
Dec
199
3
t
z
(a)
5
10
15
20
5
10
15
2
3
4
5
10
15
20
5
10
15
2
3
4
t
z
(b)
5
10
15
20
5
10
15
-0.05
0
0.05
5
10
15
20
5
10
15
-0.05
0
0.05
t
z
(c)
5
10
15
20
5
10
15
0
0.01
0.02
0.03
5
10
15
20
5
10
15
0
0.01
0.02
0.03
t
z
(d)
5
10
15
20
5
10
15
-0.001
0
0.001
0.002
5
10
15
20
5
10
15
-0.001
0
0.001
0.002
t
z
(e)
5
10
15
20
5
10
15
0
0.01
0.02
0.03
5
10
15
20
5
10
15
0
0.01
0.02
0.03
t
z
(f)
5
10
15
20
5
10
15
-0.001
0
0.001
0.002
5
10
15
20
5
10
15
-0.001
0
0.001
0.002
Action densityTopological
charge density
u
u
d_
d_
d_
uL
R
L
R
L
R
ππ
π I
A
!
Gas of instantons:,size
- . 0.3-0.4 fm;,separation
- . 0.9-1.0 fm;! /-symmetry breaking!
effective interaction! ,1032 2 - 465 7
!
constituent quark masses
N*2002 Pittsburgh, 08-13.10.2002 – p.3
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
effectivemassesand interaction
!
after normal ordering:89 :; <
eff
= 89 :> <
eff
= 89 :? <
eff
= 89 :@ <
effeffective masses effective interaction
dd
ud
s
ud
ss_
_uu
_uu
s
dd_
m(d)
m(s)
ds
g
g’
ss_
_
d
_ss_
uu
+ + +
uu_
η
η’
π
Κ∗ρ,ω
Mass
m(n)+m(s)
2m(n)
m(n)+m(s)
2m(s)Κ
2m(s)
2m(n)
N*2002 Pittsburgh, 08-13.10.2002 – p.4
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
(pseudo)scalarmesons
0
200
400
600
800
1000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Mas
s M
[MeV
]
A
g[GeV^(-2)]
g’ = 0.0 GeV^(-2)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
g’[GeV^(-2)]
g = 1.73 GeV^(-2)
pionkaon
etaeta’
uu/dd_
ss_
_
g’g
_
dd_
uu
+
BDC C C mixes E * E%F G *GF H *H orIKJ F ' J for (pseudo)scalars
of’
aoKo
*Ko*of’
ao
fo fo
Mass [MeV]
1000
1500 1460
1350
1240nn
’1500’1450
1430
1470
1430
1320
980
ns
ss
’1000’
η’980
530
495
ηΚ
π140
Mass [MeV]
1000
500
0
910
810
690
ss
ns
nn
η’
495
ηΚ
π140
960
550
LNM = LPO O O Exp.
N*2002 Pittsburgh, 08-13.10.2002 – p.5
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
The instanton-induced Q Q-interaction
In 2 2SR T 5 0VU -channel W instantaneous potential:’t Hooft’s interaction (induced by instantons):
L :? <
’t Hooft
XZYV[ \ YV] ^ 5 _ :@ < XZYV[ \ YV] ^
` ab c
point-interaction
d
\ e fgih h j kml Xon n ^ = gh p j kml X nq ^ r
` a b c
flavour-dependent coupling
sut vw t v =yx z w x z{ j |~}%�� �;
G. ’t Hooft, Phys. Rev. D 14, 3432 (1976)
M. A. Shifman, A. I. Vainstein, V. I. Zakharov, Nucl. Phys. B 163, 46 (1980)
!
flavour-dependent:
j k�l: flavour-antisymmetric quark pairs.
!
spin-dependent:
j |}%� � �; : antisymmetric in spin
�> ? 5 7
.
!
point-interaction:
_ :@ < XZYy[ \ Y] ^ \ W [� ��� �� exp
� \ ���� �� � � �� ��
� does not act on: flavour-decuplet, spin-symmetric states;
� no
��� d ��
, no tensor forces.
N*2002 Pittsburgh, 08-13.10.2002 – p.6
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
a relativistic quark model: assumptions
Pretend to describe baryons in the framework of quantum fieldtheory: Basic quantities: Bethe-Salpeter amplitudes
/��� Xo� >R � ?R � @ ^�� 5 , 7 �� � X � > ^ � Xo� ? ^ � Xo� @ ^ � 01� -(with
*y� � ¡¢ £ ¡¤ £ ¡¥ ;
*� ¤ � ¦ ¤
baryon mass;
§ &%¨© (
quark field operator)
which fulfill the Bethe-Salpeter Equation:
χ(3)- iK χΣ
zykl. Perm.1,2,3
+(2)
iK-χ =
Full quark propagators:
� ª Xo� \ � « ^ 5 , 7 � � � Xo� ^ 0 � Xo� « ^ � 7 -
++ + + ...=
Irreducible 2- and 3 -body interaction kernels:
+ + + + ...-iK(2) =
+ + + + ...=-iK(3)
N*2002 Pittsburgh, 08-13.10.2002 – p.7
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
momentumspace
From Poincaré invariance:
/� � Xo� >R � ?R � @ ^ 5 ¬ � ® �� ¯ °±²X³ ´ ^ °
¯ °±iµX³ ´ ^ ° ¬ � ¶· ² ¬ � ¶u¸ µ /��� X ±² R ± µ ^
� 8-dimensional integral equation
"¹º & ¡» F ¡½¼ ( � ¾J¢ & '¿ *� £ ¡» £ ' À ¡½¼ (�Á ¾J¤ & ' ¿ *Â�Äà ¡» £ ' À ¡¼ (Á ¾J¥ & '¿ *Â� à ¡½¼ (
Å &Ã Æ ( G Ç ¡ È»& À½É ( ÇG Ç ¡ ȼ& À½É ( Ç Ê & *� F ¡» F ¡Ë¼ F ¡ È» F ¡ ȼ ( " ¹º & ¡ È» F ¡ ȼ (
with integral kernel:
Ê� Ê Ì ¥ Í £ * Ê Ì ¤ Í,
* Ê Ì ¤ Í & *� F ¡» F ¡¼ F ¡ È» F ¡ ȼ (ÏÎ �(123)
Ê Ì ¤ Í & �¢ ¤ F ¡» F ¡ È» (�Á Ð ¾J¥ & ' ¿ *� à ¡½¼ (ÑÒ ¢ &À É ( ÇÔÓ Ì Ç Í & ¡½¼ à ¡ ȼ (
N*2002 Pittsburgh, 08-13.10.2002 – p.8
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
Bethe-SalpeterEquation, approximations...
Model assumptions (inspired by NonRelativistic Constituent Quark Model):
!
Free quark propagators replace full propagators:
� ª X ± ^ . Õ
x X ± ^ \ Ö = Õ ×
W Ö effective constituent quark masses
!
Instantaneous approximation: No dependence of interactionkernels on ± ;² und ± ;µ (in the restframe of the baryon):
Ø :@ < X 0Ô� R ±² R ± µ R ± «² R ± «µ ^ � � � � :ÙÛÚ Ü; < 5 L :@ < XÞÝ ² R Ý µ R Ý «² R Ý «µ ^
Ø :? < X � > ?R ±² R ± «² ^ � � � � :ÙÛÚ Ü; < 5 L :? < XÞÝ ² R Ý «² ^
ß ’Retardation effects are neglected’
� Salpeter amplitudeà Ù XÝ ² R Ý µ ^� 5 á d¶ â·? � d¶ â¸? � /���
� X ± ;² R Ý ² ^R X ± ;µ R Ý µ ^ � ãäãäãäã �� � : ÙÚ å <
satisfies the Salpeter equation
æ à Ù 5 ç à Ù è , à Ù � à Ù - 5 ³ ç
N*2002 Pittsburgh, 08-13.10.2002 – p.9
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
SalpeterEquation, SalpeterHamiltonian
é é é approximate treatment of
L :? < é é é :
&ê ëKì ( & í » F í ¼ (�¥
© î ¢ï© ë ì & í » F í ¼ (
£ ð ñ ò¢ Á ñ ò¤ Á ñ ò¥ £ ñ Ò ¢ Á ñ Ò ¤ Á ñ Ò ¥ ó
ô õ Á ô õ Á ô õG¥ ¡ È»& À½É ( ¥G¥ ¡ ȼ&À É ( ¥ B Ì ¥ Í & í » F í ¼ F í È» F í ȼ ( ë ì & í È» F í ȼ (
£ ð ñ ò¢ Á ñ ò¤ Á ñ ò¥ Ã ñ Ò ¢ Á ñ Ò ¤ Á ñ Ò ¥ ó
ô õ Á ô õ Á ' ö G¥ ¡ È»&À É ( ¥÷ B Ì ¤ Í & í » F í È» (Á ' ö ø ë ì & í È» F í ¼ (
£
cycl. perm. (123)
! ñ ù© & íú (Î � ûü ùý ü¤ ûü Energy projectors
! ï© & íú (Î � ô õ & þÿ íú £�� © (Dirac Hamiltonian
N*2002 Pittsburgh, 08-13.10.2002 – p.10
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
Confinement
Quark confinement is realized by a phenomenological string potentialfor 3 quarks (Ansatz similar to NRCQM):
L :@ <
Conf
XZY[ R Y] R Y � ^ 5 A � =
B �� � �� Y � \ Y � �
3-Quark-String
In contrast to the nonrelativistic quark model the relativistic
quasi-potential
L :@ <��� depends on the Dirac structure for three quarks.We choose:A@ 5 � @ ° �t vw t vw t v = x ; w x ; w t v =yx ; w t vw x ; = t vw x ; w x ;
B@ 5 � >? � \ t vw t v w t v =yx ; w x ; w t v =yx ; w t vw x ; = t v w x ; w x ;
W Spin-orbit effects are small (compatible with exp.)
W Regge trajectories are quantitatively correct.
N*2002 Pittsburgh, 08-13.10.2002 – p.11
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
Model parameters
parameter value
quark- ’nonstrange’ � � 330 Mev
masses ’strange’ � � 670 Mev
confinement offset � -744 MeV
slope
�
470 MeV fmÒ ¢
’t Hooft’s nn-coupling � � � 136.0 MeV fm
¥
force ns-coupling � � � 94.0 MeV fm
¥
effective range�
0.4 fm
Parameters are fixed by
!
the
8
-Regge trajectory
\ W Confinement parameters �, �
and Öh
!
baryon ground-states (octet und decuplet)
\ W g�h h , gh p , �and Ö p
N*2002 Pittsburgh, 08-13.10.2002 – p.12
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
-Reggetrajectory
Model
�PDG 20008� ���� � X ³ �� 7 ^
8� � � �� �� � X³ e³ 7 ^
8� � � ���� � X t �� 7 ^
8 � � � ���� � X t ³ U ³ ^
�
!" [GeV
" ]> z?#> >?#$? #@?#
10
8
6
4
2
0
N*2002 Pittsburgh, 08-13.10.2002 – p.13
� �� �� � � �� � � � �� � �� � ���� � � �� ��� � �� �� �� �
-resonances
πP P33 S D31 F F D GH G II3 1135 37 39 H3 11 3 13 3 15 3 1333 35 37 3931KKL2T 2J
1232
20001950
1700
2150
1940 1930
2200
2750
1620
23502400
1900
2300
2420
2950
2390
19051920
1600
1750
1910
J 1/2+ 3/2+ 5/2+ 7/2+ 9/2+ 11/2+ 13/2+ 15/2+ 1/2- 3/2- 5/2- 7/2- 9/2- 11/2- 13/2- 15/2-
Mas
s [M
eV]
1000
1500
2000
2500
3000
*
********
*****
****
**
**
****
***
**
**
****
**
**
*
*
*
*
*******
***
N*2002 Pittsburgh, 08-13.10.2002 – p.14
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Instanton-inducedeffectsin the
F G
-spectrum
π
g [MeV fm ]nn3
conf
inem
ent o
nly
expe
rim
ent
+ ’
t Hoo
ft’s
for
ceco
nfin
emen
t
Roper-resonance
∆−Ν
split
ting
N
J 1/2+ 3/2+ 5/2+ 7/2+ 9/2+ 11/2+ 13/2+
Mas
s [M
eV]
1000
1500
2000
2500
3000
*
**
** **
**
S
****
****
*** **** ****
****
0 136 0 136 1360 1360 0 136 1360 0 136
N*2002 Pittsburgh, 08-13.10.2002 – p.15
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Instanton-inducedeffectsin the
FH -spectrum
π
conf
inem
ent o
nly
+ ’
t Hoo
ft’s
for
ceex
peri
men
t
conf
inem
ent
?
?
g [MeV fm ]nn3
J 1/2- 3/2- 5/2- 7/2- 9/2- 11/2- 13/2-
Mas
s [M
eV]
1000
1500
2000
2500
3000
*
***
**
**
***
****
****
S S
****
****
****
****
0 136 0 136 0 136 0 136 0 136 0 136 0 136
N*2002 Pittsburgh, 08-13.10.2002 – p.16
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-Reggetrajectory
Model
IPDG 2000
and ’t Hoofts forceConfinement
without ’t Hoofts forceonly Confinement,
J K K�LMN O PQ RS S T
J K K K KU N O PQ Q Q S T
J K K K KV N O PWX Y S T
J K K K KLN O PZ[ Z T
\]
^_ [GeV
_ ]`ab cdb ceb c`b c
10
8
6
4
2
0
H f ’t Hooft’s force produces a constant shift in
g b
Furthermore, parity doublets in the nucleon spectrum
N*2002 Pittsburgh, 08-13.10.2002 – p.17
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Parity doublets(high spin)
ω2 h ω2 h
ω3 h
ω4 h
ω5 h
ω
ω
ω
ω
1 h
3 h
4 h
5 h
?
?
g [MeV fm ]nn3
π
ω6 h
J
2000
2500
**
****
**
****
****
****
****
***
1360 1360 0 136 1360
**
0 136
7/2 9/25/2 11/2 13/2
+ + + -+ - - -+ -M
ass
[MeV
]0 1361360 0 136 0 1361360
3000
ω2 h ω2 h ω2 h
ω1 h ω1 h ω1 h
ω3 h ω3 h ω3 h
g [MeV fm ]nn3
π
J
Roper-resonance
∆−Ν
split
ting
N1000
1500
2000
Mas
s [M
eV]
5/21/2 3/2
*
S
****
****
***
0 136
+
*
****
****
S
-
**
****
0 136
+
***
**
S
****
-
**
****
1360
+
**
****
-0 136 0 136 0 136
N*2002 Pittsburgh, 08-13.10.2002 – p.18
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BSE NRCQM (
h
conf.
i hkj j j )
π 1/2+
�
3/2+
�
5/2+
�
7/2+
�
9/2+
�
11/2+
�
13/2+
�
1/2-
�
3/2-
�
5/2-
�
7/2-
�
9/2-
�
11/2-
�
13/2-
�
J
Mas
s [M
eV]
1000
1500
2000
2500
3000
****
****
***
*
*
****
**
****
** **
****
**
****
****
*
*
****
***
**
*
****
** ********
***
N*2002 Pittsburgh, 08-13.10.2002 – p.19
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Relativistic effects
Effects of ’t Hooft’s force in two different confinement models
h lm nopq r sutvxw tyxw tz { | }~ �� � � �� � �� � � i�� � � � � � � � icycl. perm.
�
Modell A (discussed so far)
i ��� � � t� � t� � �� � � � �� � �� � � i�� � � � � � � � i
cycl. perm.
�
h lm n op q r s tvxw ty w tz { |�
� }~ i � �� � � t� � t� � �� �� � � �� � �� � � i� � � � � � � � i
cycl. perm.
�
Modell B (alternative)
Although both models give exactly the same results in thenonrelativistic limit, large differences appear in a fully relativistictreatment (Salpeter equation) !
Example: Roper resonance position � � �
N*2002 Pittsburgh, 08-13.10.2002 – p.20
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interplay of confinementand instanton effects...
2
2
8 [56]2
48 [70]
8 [70]
8 [20]
∆−Ν
split
ting
Roper-resonance
Roper-resonance
nn3g [MeV fm ]
∆−Ν
split
ting
11PN
21+
11PN
21+
nn3g [MeV fm ]
8 [56]2
228 [70]8 [56]
8 [56]
8 [56]
8 [70]
8 [56]8 [20]
8 [70]
8 [70]
2
2
4
22
2
2 8 [56]
8 [70]
8 [70]
8 [70]8 [56]
8 [56]
8 [20]
8 [56]
8 [56]
8 [70]
8 [70]
8 [20]
8 [56]
8 [70]
2
2
2
2
2
2
2
4
2
2
2
4
2
... ...
...
...
Exp.Exp.Confinement Confinement
N
only only+ ’t Hooft + ’t Hooft
N2
Model A Model B
*
S
****
****
***
*
S
****
****
***
Mas
s [M
eV]
1000
1500
2000
2500
89.601360
�
“Correct” masssplittings result froma specific interplayof relativistic effects,the Dirac structure ofthe confinementpotential and ’tHooft’s interaction�
A reliable test ofa residual interactionin fact requires a fullyrelativistic treatment
N*2002 Pittsburgh, 08-13.10.2002 – p.21
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Electromagneticand strongcoupling amplitudes
Mandelstam formalism: Matrix element in initial state rest frame (IA):
�� ���k� �� � s� { � � �� | � } ¡ ¢¤£¥s �§¦ { ¢ ¡ ¢¨£ª©s �§¦ { ¢
«¬® s£ ¯¥w £ ¯© {° ±³² s£ ± { � ° ´² s£ ´ { � ° m ² µ£ ¯m ¶ ·¹¸� �° m ² s£ m { «kº s»£¥w »£x© {
�
Reconstruction of the vertex function in the rest frame
«kº s»£¥w »£x© { | � ¼ ¡m £ ¯¥s �¦ { m ¡m £ ¯©s �§¦ { ¢ h s»£¥w »£x©w »£ ¯¥w »£ ¯© {½ º s»£ ¯¥w »£ ¯© {
�
Boost
s�w » � {ª¾ s ¿� ´ i » � ´w » � {
�� ���k�w ¦ � �ÁÀ �° � � � � | � } ¡ ¢¤£¥s �§¦ { ¢ ¡ ¢ £©s �§¦ { ¢
«¬® s£ ¯¥w £ ¯© {° ±Â² s£ ± { � ° ´² s£ ´ { � ° m ² µ£ ¯m ¶ � « s£ {° m ² s£ m { «kº s»£¥w »£x© {
mesonic Bethe-Salpeter amplitudes from
(M.Koll et al., Eur. Phys. J. A 9, 73 (2000))
N*2002 Pittsburgh, 08-13.10.2002 – p.22
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Static electromagneticproperties
from
à b f S -limit of form factors:
Calc. Exp.ĪŠ�ÇÆ È � ÄÉ �ÇÆ ÈÊ } ÄÉ (PDG)ÄxË � �Æ È� ÄÉ � �Æ Ê � } ÄÉ (PDG)�ÍÌ ´ ÅÏÎ � Æ Ð �
fm
� Æ Ð � È
fm (MMD)�ÍÌ ´ ËÎ � Æ � �
fm
´ � Æ � � } Ñ � Æ � � �fm
´(MMD)� Ì ´ ź � Æ Ê �
fm
� Æ Ð }Òfm (MMD)� Ì ´ ˺ � Æ ÐÒ
fm
� Æ Ð Ð Êfm (MMD)
ÓÔ �ÇÆ � � �ÇÆ � Ò È� Ñ � Æ � � }Õ
(PDG)¿ �ÍÌ ´ Ô � Æ Ò �fm
� Æ Ò � Ñ � Æ � �
fm (Bernard)
MMD: Mergell, Meißner, DrechselBernard: Bernard, Elouadrhiri, Meißner
N*2002 Pittsburgh, 08-13.10.2002 – p.23
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Octet magneticmoments
from “expectation values” of local operators with S.A.:
ÖØ× Ö ÙÚÛ Ù
Û ×Û cnp
octet magnetic moments
1/N
ÜÜÝ
0.70.60.50.40.30.20.10
3
2
1
0
-1
-2
exp. Þ basis size (calc.)
N*2002 Pittsburgh, 08-13.10.2002 – p.24
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Electric form factor of proton and neutron
proton neutron
MMD
¸ ´ß
GeV
´à
á â ãäå æ
32.521.510.50
1
0.8
0.6
0.4
0.2
0
SchiavillaEden
PasschierHerberg, Ostrick
Argonne V18
¸ ´ß
GeV
´ à 32.521.510.50
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
MMD: Mergell, Meißner, Drechsel
N*2002 Pittsburgh, 08-13.10.2002 – p.25
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Electric form factor of proton and neutron
Isoscalar/isovector contributions withoutçéè è è
isovectorisoscalar
dipole
¸ ´ß
GeV
´ à
áê ëì âãäå æ
32.521.510.50
1
0.8
0.6
0.4
0.2
0
-0.2
íËÎíÅîÎdipole
¸ ´ß
GeV
´à
áï ëð âãäå æ
32.521.510.50
1
0.8
0.6
0.4
0.2
0
-0.2
Almost perfectdipole shape
wrong asymptotics ?
N*2002 Pittsburgh, 08-13.10.2002 – p.26
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Magnetic form factor of proton and neutron
proton neutron
MMD
¸ ´ß
GeV
´à
á ñ òóá ô
ãäåæ
10.10.01
1.4
1.2
1
0.8
0.6
0.4
0.2
0KubonAnklinMMD
¸ ´ß
GeV
´à 10.10.01
1.4
1.2
1
0.8
0.6
0.4
0.2
0
N*2002 Pittsburgh, 08-13.10.2002 – p.27
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Electroweaknucleonform factors
õÁö ÷ õÁø axial form factor
Jones
¸ ´ß
GeV
´à
ó ïáï âòá
ï ñãäå æ
32.521.510.50
1
0.8
0.6
0.4
0.2
0
BloomJoos
AmaldiBrauel
delGuerra¸ ´ß
GeV
´à
á ù ãäå æ
10.80.60.40.20
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
N*2002 Pittsburgh, 08-13.10.2002 – p.28
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Photoncouplings
úØû üH [ GeVH `b ý
state Calc PDG Calc PDG�m m s � � } � { þÉ ± ÿ ´ � Ð Ê � � }Õ Ñ ÒþÉm ÿ ´ � � Õ � � � Õ Õ Ñ Ð
° ± ± s � Õ }Õ { þÅ ± ÿ ´ � � } Ê � Ñ }� þ Ë ± ÿ ´ � ÈÕ � � Ò Ñ � È
° ± ± s � Ò Õ � { þÅ ± ÿ ´ Ò Õ } Ñ � Ò þ Ë ± ÿ ´ � � � � � Õ Ñ � �
� ± m s � Õ � � { þÅ ± ÿ ´ � Õ } � � � Ñ Ê þ Ë ± ÿ ´ � � Õ Ê Ñ ÊþÅm ÿ ´ Õ � � Ò Ò Ñ Õ þ Ëm ÿ ´ � Õ � � � } Ê Ñ � �
� ± m s � È� � { þÅ ± ÿ ´ � � Ð � � Ð Ñ � } þ Ë ± ÿ ´ � } � Ñ Õ �þÅm ÿ ´ � � � � � Ñ � � þ Ëm ÿ ´ � Ò � � } Ñ � �
� ±� s � Ò ÈÕ { þÅ ± ÿ ´ Õ � Ê Ñ Ð þ Ë ± ÿ ´ � }Õ � � } Ñ � �þÅm ÿ ´ È � Õ Ñ Ê þ Ëm ÿ ´ � � È � Õ Ð Ñ � }
� ± ± s � � � � { þÅ ± ÿ ´ � � Ð � Ò Õ Ñ � þ Ë ± ÿ ´ � È � � Ñ � �
° m ± s � Ò � � { þÉ ± ÿ ´ � Ð � È Ñ � �
�m m s � È� � { þÉ ± ÿ ´ Ò } � � � Ñ � ÕþÉm ÿ ´ Ò Ð ÐÕ Ñ � �
N*2002 Pittsburgh, 08-13.10.2002 – p.29
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Transition form factors
Nucleon Delta transition
FrolovStein
FosterBartel
¸ ´ß
GeV
´à
á� ñãäå æ
43.532.521.510.50
3
2.5
2
1.5
1
0.5
0
FrolovStein
FosterBartel
¸ ´ß
GeV
´à
á� ñòá� ñã� æ
á ô ãäå æ
43.532.521.510.50
1
0.8
0.6
0.4
0.2
0
N*2002 Pittsburgh, 08-13.10.2002 – p.30
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Helicity amplitudes
F15(1680) Nucleon transition D15(1675) Nucleon transition
þÅm ÿ ´þÅ ± ÿ ´
þÅm ÿ ´ Burkert
þÅ ± ÿ ´ Burkert
¸ ´ß
GeV
´ à
�ï ����� GeV
� ëå �
32.521.510.50
150
100
50
0
-50
-100
þÅm ÿ ´þÅ ± ÿ ´
¸ ´ß
GeV
´à
�ï ����� GeV
� ëå �
32.521.510.50
10
8
6
4
2
0
N*2002 Pittsburgh, 08-13.10.2002 – p.31
%&' ()'* +,-./ 0. 12 - '.3 +45 6789 :; <= :>?@ ABC > D>E >
Strongdecaywidths
J � decay widths
� �
MeV
� � � decay widths
� �
MeV�
Decay Calc
m � � PDG Decay Calc
m � � PDG° ± ± � � Õ }Õ { ¾ �¦ } } � � Ò � Ò Ð Ñ � Õ { � ¢�� ´m ¾ �¦ � � � �
° ± ± � � Ò Õ � { ¾ �¦ } � � Ê � � � Ê Ñ � Ò { � ´�� ¢ ¾ �¦ Õ � } � Ò Ñ Õ { � ´�
� ± m � � Õ � � { ¾ �¦ } Ð È � � Ò Ò Ñ Ò { � ��� ¾ �¦ }Õ }Õ � � � Ñ Ò { �m� ´
� ± m � � È� � { ¾ �¦ � Æ � } � � � � Ñ Õ { ���� ¾ �¦ Ð Ð È È Ð
seen
� ±� � � Ò ÈÕ { ¾ �¦ � � Ð � Ò Ð Ñ È { � ± ¢�� ¾ �¦ }� } � � Ð } Ñ È { � ± �� �� ± ± � � � � � { ¾ �¦ } Ð � � � � � � Ð Ñ � Ð { � ��
� �� ¾ �¦ }Õ � � � Ð Ð Ñ � Ð { � ´�� ´��m m � � � } � { ¾ �¦ Ò � � � Ð � � � Ê Ñ � { ��
��° m ± � � Ò � � { ¾ �¦ � � Ò � } Ð Ñ È { � �� � ¾ �¦ È � � Ð � Ò Ð Ñ � } { � ± ¢� ± ¢
�m m � � È� � { ¾ �¦ � � � � � Õ Ñ � Õ { � ±�� ±� ¾ �¦ Õ � � Ò � � � }Õ Ñ � Õ { � ¢�� ¢�m � � : S. Capstick, W. Roberts, Phys.Rev. D49 (1994) 4570-4586
N*2002 Pittsburgh, 08-13.10.2002 – p.32