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HADRON SPECTROSCOPY
I. Butterworth
To cite this version:
I. Butterworth. HADRON SPECTROSCOPY. Journal de Physique Colloques, 1973, 34 (C1), pp.C1-
173-C1-210. �10.1051/jphyscol:1973119�. �jpa-00215199�
JOURNAL DE PHYSIQUE Colloque C1, supplement au nOIO, Tome 34, Octobre 1973, pagecl-173
HADRON SPECTROSCOPY I. BUTTERWORTH Imperial College, London
1.
-
SU(3) TESTS AND SU(6) MULTIPLETSMany comparisons with the first Aix Conference Table I have been made. Standing here, I am mainly conscious
of the reduction in time allocated to Hadron Spec- troscopy. I trust I can make my report also parallel the changes in what the girls on the Cours Mirabeau wear : that nowadays there's almost nothing left but
it's the interesting areas that are still covered.
Let me start with something reassuring, namely SU(3) looks good as regards branching ratios once one is certain of the classification of a particle.
Two recent checks have been made ; that on baryon states by Barbaro-Galtieri at Batavia [I] and that by Sanios et al.[2]. The following table shows how good the fits to the decay rates are in the latter analysis (see Table I).
To prove it isn't just chance, they perform 50 random permutations of the SU(3) Clebsch-Gordan coefficients to produce 50 nonsense-SU(3)'s and refit to those. If the SU(3) fit were fortuitous,
Multiplet N.D.
B -, 8x8
3/ '2 7.8 3 1/ 2- 3.2 3
312- .02 1
51 2- 5.1 4 51 2' 6.2 4 712' 7.6 4
712- 2.3 3
M
-
8x81- -r 0-0- 0.6 2 2
'
-
1-0- 1.5 22'-0-0- 6.4 4
half these nonsense fits should be better than
2 Moreover the emerging parameters are those expec- SU(3). Figure 1 compares the
x
-values obtained.ted ; where full meson nonets are known, mixing In the lower multiplets where branching ratios are
angles are reasonable - : best known true SU(3) always gives the best fit
-
only in high lying multiplets can you find the odd few rubbish-fits that do as well.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973119
-
= PERMUTATIONx
F i g . 1 . - F i t X2 v a l u e s on permuting SU (3) Clebsch-Gordan c o e f f i c i e n t s .
Table I1
Nonet Mix a n g l e from decay Mix a n g l e from m a s s
- - -
1 - 0 0 31'
+
3' 40'2
lo2+
-
0-0- 48'+
45' 30°+ lo0
2+
-
1-0- 3 4 . 5 O + goFor baryon o c t e t s we can check D/F r a t i o s a g a i n s t t h o s e expected on t h e S U ( 6 ) ~ 0 ( 3 ) quark
Table 111
P D
J a =
,
experimental a,
quark model1/ 2- 1.28 0.06 1.5 (70
,
L = l,
Sq=3/2)3/ 2- 0.28
+
0.15 0.375(70,
L=l,
Sq=1/2)51 2- 1.16
+
0.01 1 . 5 (70,
L=l,
Sq=3/2)512' 0.46
t
0.005 0.6 (56,
L=2,
Sq=1/2)712- 0.17 2 0.02 0.375(70
,
L-3,
Sq=1/2)model.
One does p r e t t y w e l l i g n o r i n g c o n f i g u r a t i o n mixing between s t a t e s of t h e same 'J b u t d i f f e - r e n t quark s p i n o r o r b i t a l a n g u l a r momentum ( t h e o n l y s u r p r i s e i s t h e c l o s e n e s s of t h e w e l l - e s t a b l i - shed 112- o c t e t , t h a t w i t h A (1405) and A (1675)
,
HADRON SPECTROSCOPY C1-175 t o t h e Sq=3/2 p r e d i c t i o n r a t h e r t h a n t o t h a t f o r
Sq=1/2)
.
Not o n l y i s SU(3) good, t h e r e f o r e , b u t we a r e d o i n g r a t h e r w e l l w i t h h i g h e r symmetry schemes ; and i n d e e d t h e emphasis f o r s u c h schemes i s s h i f t i n g away from p u r e c l a s s i f i c a t i o n towards b r a n c h i n g r a t i o p r e d i c t i o n s .
A minor c u r i o s i t y of such SU(3) c h e c k s , o r when one s e e k s t o e n s u r e mass c o m p a t i b i l i t y of d i f f e r e n t d e c a y s of t h e same s t a t e n e a r t h r e s h o l d , i s t h a t t h e Breit-Wigner b a r r i e r f a c t o r r a d i u s w a n t s t o go t o z e r o . ( D e s p i t e t h e f a c t t h a t t h e p a r t i a l wave ana- l y s e s t h a t y i e l d t h e b r a n c h i n g r a t i o s i n v a r i a b l y u s e 1 f e r m i
-
and indeed t h e r e l a t i v i s t i c BW used f o r broad mesons g i v e s t h e b e s t s h a p e w i t h a r a d i u s o f 1 f e r m i o r s o ) .The need f o r S U ( 6 ) ~ 0 ( 3 ) c l a s s i f i c a t i o n i s ob- v i o u s . Meson s y s t e m a t i c s a r e s t i l l messy w i t h o n l y t h e 0-1- and 2' n o n e t s i n r e a s o n a b l e s h a p e , b u t t h e b a r y o n s a r e much h e a l t h i e r . Assuming, f o r unk- nown r e a s o n s , t h e need f o r symmetry u n d e r q u a r k i n - t e r c h a n g e , t h e f u l l y s-wave ground s t a t e must be i n t h e SU(6) s y m e t r i c (56) which i s f u l l y k n a m . P u t t i n g a q u a r k p a i r i n t o a p-wave t o g i v e L=l y i e l d s mixed s p a t i a l s y m e t r y depending which quark p a i r i s i n t e r c h a n g e d , t h e r e b y demanding t h e mixed s y m e t r y SU(6) ( 7 0 ) .
F i g u r e 2 shows o u r knowledge o f t h i s m u l t i p l e t . (Masses i n b r a c k e t s a r e f o r s t a t e s n o t f u l l y e s t a - b l i s h e d o r o f u n c e r t a i n a t t r i b u t i o n . For p u r p o s e s of d i s p l a y , c o n f i g u r a t i o n mixing i s i g n o r e d ) . A l l s=O and most s=-1 s t a t e s a r e k n a m . P u t t i n g a p a i r o f q u a r k s i n a d-wave t o g i v e L=2 we e x p e c t a
symmetric ( 5 6 ) , (Fig. 3)
.
A l l S=O and S = l s t a t e s a r e t e n t a t i v e l y a t t r i b u t e d . The new P33(1850)though shown h e r e i s , however, more l i k e l y t o be a r a d i a l s t a t e a s we s h a l l s e e .
At t h i s p o i n t t h e r e i s a n i m p o r t a n t d i s t i n c t i o n between t h e q u a r k modeland a s i m p l e r symmetry which would p u t a l l p o s i t i v e p a r i t i e s i n ( 5 6 ) ' s a l l nega-
F i g . 3 . - The (56) L=2
.
t i v e s i n ( 7 0 ) ' s . S y m b o l i c a l l y ( F i g . 4) i f p a i r (1,2) i s i n a d-wave p a i r ( 1 , 3 ) must have a p-wave component, hence on t h e q-model we a l s o e x p e c t a (70) o f p o s i t i v e p a r i t y L=2 b a r y o n s i n t h e same mass r a n g e . T h e r e i s now good e v i d e n c e t h a t t h i s i s
MlSSlNG S = -1 No of doubtful cond~dales
2 x Sll 2 511
013 3 013
s o ( F i g . 5 ) . The S a c l a y XN a n a l y s i s , updated h e r e 13) s e e b o t h t h e c r i t i c a l FI5 and F1, s t a t e s . The former i s s e e n , a l b e i t d i s p l a c e d upwards i n mass by Almehed and L o v e l a c e [4
1,
and e x a m i n a t i o n o f t h e i r Argand p l o t s l e a d s t o p o s s i b l e s u p p o r t f o r t h e F17 t o o . (The P-waves o f F i g . 5 a r e p r o b a b l y r e a l b u t a r e b r a c k e t e d s i n c e o f c o u r s e t h e y might r e p r e s e n t r a d i a l r a t h e r t h a n o r b i t a l e x c i t a t i o n s ) .F i g . 2 . - The (70) L = l
.
F i g . 5 . - The (70) L=2
.
Moving t o t h e (70) ~ = 3 - ( F i g . 6 ) , t h e D13 (2040) a l r e a d y r e p o r t e d by Almehed and L o v e l a c e i s now sup- p o r t e d by S a c l a y , i n d e e d you c o u l d a l r e a d y s e e i t i n t h e i r 1972 Argand p l o t s . By t h e t i m e we g e t t o t h e
1
70 3-1
F i g . 6 . - The (70) L=3
.
(56) ~ = 4 + of F i g . 7 t h e s t a t e s a r e g e t t i n g r a t h e r t h i n .
F o r r a d i a l e x c i t a t i o n s , b o t h S=O members o f t h e f i r s t r a d i a l e x c i t a t i o n of t h e (56) L=O h a v e , a s we s h a l l s e e , p r o b a b l y now been found.
2
.-
S=O BARYONSOne o f t h e main s o u r c e s o f i n f o r m a t i o n f o r hadron s p e c t r o s c o p y , h a s been nN e l a s t i c p a r t i a l wave a n a l y s i s dominated i n r e c e n t y e a r s , by t h e CERN and S a c l a y g r o u p s . The l a t e s t "CERN" s e t i s t h a t o f Almehed and L o v e l a c e , 1972 [ 4 ] . Even a t t h e t i m e o f B a t a v i a comparison w i t h t h e new Berkeley c h a r g e exchange d i f f e r e n t i a l c r o s s - s e c t i o n s [5] f a v o u r e d t h e S a c l a y 1972 a n a l y s i s which f o l l o w e d t h e d a t a
F i g . 7 . - The (56) L=4
r e a s o n a b l y w e l l , ( F i g . 8) and looked a s i f w i t h a l i t t l e trimming c o u l d be made t o f i t i t v e r y c l o - s e l y . S a c l a y h a v e now u p d a t e d t h e i r f i t [ 6 ] . New d a t a i n c l u d e d a r e :
i ) Charge exchange DCS's from LBL and Saclay [7
1.
F i g . 9 shows t h e i r c l o s e agreement and t h e d i s a - greement o f t h e MIT d a t a [ 8 ] which i s t h e r e f o r e e x c l u d e d
.
i i ) The p r e c i s e low e n e r g y e l a s t i c DCS's of Bussey e t a 1 . 19 ] (Eg. F i g . 10)
.
As b e f o r e , e n e r g y i n d e p e n d e n t f i t s a r e performed a t some 45 e n e r g i e s which a r e t h e n j o i n e d by a mi- nimum p a t h program w e i g h t e d a g a i n s t s h a r p a n g l e s and h i g h i n d i v i d u a l X2
.
The s e l e c t e d p a t h a p p e a r s u n i q u e i n t h a t , a p p l i c a t i o n of t h e program a f t e r removal o f t h e b e s t X2 p o i n t a t e a c h e n e r g y s t i l l g i v e s e s s e n t i a l l y t h e same, b u t jagged, p a t h which i s t h e n smoot.hed by b e i n g i n p u t i n t o a n e n e r g y de- pendent program ( w i t h s o much d a t a t h e y d o n ' t h a v e enough computer power t o p e r f o r m a n e n e r g y depen- d e n t f i t on raw experimenLa1 d a t a ) . I n t h e f i t t e di @
T=TW e
+
TBG,
t h e background now a l s o c a r r i e s a b a r r i e r f a c t o r which s h o u l d make t h e i r low energy b e h a v i o u r b e t t e r . One t e s t t h a t t h e y have a good f i t i s t o u s e t h e deduced p h a s e s t o f i n d t h e B ( t ) a m p l i t u d e and t o a p p l y forward d i s p e r s i o n r e l a t i o n s t o t h e l a t t e r a t d i f f e r e n t e n e r g i e s t o y i e l d t h e nN c o u p l i n g c o n s t a n t [ l o ] . (The Barger and P h i l i p s [ l l ] Regge f i t i s used a s i n p u t above 2 . 8 GeV/c where t h e p a r t i a l wave a n a l y s i s s t o p s ) . I f a l l i s w e l l F i g . 11 s h o u l d be l i n e a r and h a v e a c o m o n i n t e r c e p t f o r n p and+
namely t h e e x t r a p o l a t e d v a l u e o f f 2.
I t d o e s , and y i e l d s f 2 a s shown. F i g s . 12 and 1 3 show t h e Argands ( t h e d o t t e d l i n e i s t h e non r e s o n a n t background). Commenting on i n d i v i d u a l waves S31 h a s acquired a second l o o p s i n c e 1972 ( F i g . 17-
HADRON SPECTROSCOPY
Coslne of ns Scattering Angle Cos~ne of n' Scattering Angle
lu 7s-UI
F i g . 8 . - LBL charge exchange c r o s s - s e c t i o n s compared w i t h Almehed and Lovelace ( S o l i d curve) and Ayed e t a l . , 1972 (dashed c u r v e ) .
& ]
forward dispersion rekitions0.
+
Fig.lOa.- n p d i f f e r e n t i a l c r o s s - s e c t i o n s from F i g . 9 . - charge exchange d i f f e r e n t i a l c r o s s - s e c t i o n s . Bussey e t a l . (Ref .9).
C1-178 I. BUTTERWORTH
-
L 4<
1 6 -n E
-
0 c
-Iv
1 20 8
0 4 -
0
-
-
I I I I I
- 1 0 - 0 6 - 0 2 0 2 0 6 1 0 Cos 8 Fig.lOb.- i p d i f f e r e n t i e l c r o s s - s e c t i o n s from
Bussey e t a l . ( R e f . 9 ) .
Fig.13.- I = 1 / 2 Argand diagrams of Saclay nN phase s h i f t a n a l y s i s .
-3 0 3 0
of r e f e r e n c e 12 shows t h e 72 s o l u t i o n s ) Langbein and Wagner [13 ] i n t h e i r d -t M a n a l y s i s found a s t a t e w i t h M=1870-1900
,
p 1 4 0 - 1 6 0.
The S a r l a y loop y i e l d s M=2000,
P-307, x N .08.
I f t r u e i t remxnds u s we always f l n d a n S-wave resonance above each new t h r e s h o l d , h e r e t h e ZK.
P33 (1900) Unchanged i n t h e f l t s i n c e l a s t y e a r ~t Fig.11.- D i s p e r s i o n r e l a t i o n s t e s t of t h e Saclay
phases. was n o t s e e n by Almehed and Lovelace who r a t h e r
b r a c k e t e d i t by two weak P33 c l a i m s a t 1690 and
-_
2150. One o f Langbein and Wagner's two EK s o l u -t i o n s had t h e s t a t e a t 1890 w i t h p 1 9 0
.
The s t a t e i s r e p o r t e d i n t h e new LBL-SLAC a n a l y s i s (14) where i t s a m p l i t u d e s i g n s u g g e s t s t h a t i t i s a r a & a l L=O s t a t e r a t h e r t h a n t h e remaining [56] 2' mem- b e r .I
D35 (1890) I showed t h l s I n (70) 3-.
I t h a s a// "
\ I r a t h e r bad h i s t o r y . I t s t a r t e d a t
-
1950 way backo m
"b i n t h e CERN 1 s o l u t i o n and w i t h each s u c c e s s i v e CERN f i t h a s had d i m i n i s h i n g a c c e p t e n c e by t h e a u t h o r s . On t h e o t h e r hand Saclay steadrly r a t e i t more h l g h l y . Lovelace [12] h a s p o i n t e d o u t t h e danger of c o n f u s i o n
8 rn w i t h a c u s p a t t h e x Flg (1688) t h r e s h o l d , and
Almehed and Lovelace have no resonance h e r e b u t s h i f t t h e i r Dg5 c l a i m up t o 2200 a s Regge r e c u r r e n c e of t h e Sgl (1650)
.
Saclay r e p o r t no such h l g h massed s t a t e .- - l l D m O m a o
G39 (2170) A c o n c e i v a b l e low e l a s t ~ c i t y a r r i v a l Fig.12.- I = 3 / 2 Argand diagrams o f Saclay ,N which t h e a u t h o r s do n o t s t r e s s . Turning t o I = 1 / 2
,
phase s h i f t a n a l y s i s . t h e s and P waves a r e l i t t l e changed s i n c e t h e
HADRON SPECTROSCOPY 1972 s o l u t i o n .
Sll (2280) Used t o b e a t 2190 (2100 i n Almehed)
.
--
Onewonders a b o u t a n Nq' t h r e s h o l d r e s o n a n c e though t h a t c h a n n e l could s c a r c e l y s a t u r a t e t h e i n e l a s t i c i t y . Pll They s t i l l s p l i t t h e Roper r e s o n a n c e-
n o t e t h e-
s t r e t c h e d Argand loop y i e l d i n g M1=1413 and M2=1530,
a c l a i m no
-
one e l s e h a s made.D13 (2020) A welcome newcomer f o r t h e [20] L=3 a l r e a d y r e p o r t e d by Almehed and Lovelace (Note t h a t
t h e Argand h a s n o t changed much s i n c e l a s t y e a r when t h e s t a t e a l r e a d y looked p o s s i b l e ) .
D13 (1700) which completed t h e s = O s t a t e s o f L=l (70) was n o t s e e n i n t h e CERN s o l u t i o n b u t a s w e l l a s b e i n g s e e n h e r e i s i n t h e K A a n a l y s i s ofWagner and Lovelace [15], t h e LK a n a l y s i s o f Langbein and Wagner [13
1,
t h e rtA a n a l y s i s o f LBL-SLAC [14]and t h e p h o t o p r o d u c t i o n s t u d y o f Moorehouse and Oberlack [16] ( s u r p r i s i n g l y i t s t i l l h a s n o t made
t h e PDG main- t a b l e [17
1) .
D15 (2076) Though r a t e d p o o r l y by PDG [17
1,
i tseems p r e t t y s o l i d , b e i n g h e r e and i n t h e CERN ana- l y s i s and b e i n g a t t h e r i g h t mass t o e n t e r t h e (70) 3- a s a r e c u r r e n c e o f t h e 1520.
F15 and F17 e n t r i e s t o t h e (70) 2' I have men- t i o n e d . The h i g h e r masses go o u t o f t h e r a n g e o f Almehed and ' l o v e l a c e b u t t h e Gr7
,
G19 and H19were a l s o s e e n i n Wagner's a n a l y s i s r e p o r t e d a t B a t a v i a [18].
T a b l e s IV and V g i v e t h e o v e r a l l p a r a m e t e r s from t h e new Saclay a n a l y s i s . New d a t a n o t y e t i n c l u d e d b u t which should t i g h t e n up t h e p h a s e s f u r t h e r a r e t h e c h a r g e exchange p o l a r i z a t i o n d a t a from LBL [19]
which a l r e a d y f i t s q u i t e n i c e l y on t h e Saclay s o l u - t i o n s d i s c r i m i n a t i n g r a t h e r s t r o n g l y a g a i n s t t h e CERN s o l u t i o n s a t t h e h i g h e r e n e r g i e s ( F i g . 1 4 ) ,
BERKELEY PRELIMINARY VALUES OF
t K-p-.non
PoLaRIzAnou PHASE SHIFT ANALYSES PREDICTIONS-
CERN (1971)---
SACLAY (1972)SACLAY (1973),Smoothed phase-shifts.
&/BOUNDS COMPUTED FROM ISOSPIN INVARIANCE (With the SACLAY11973) smoothed phose-shifts)
F i g . 1 4 . - LBL c h a r g e exchange p o l a r i z a t i o n a g a i n s t p h a s e s h i f t s o l u t i o n (Re*. I Y ) .
and t h e p r e c i s e 180' c r o s s - s e c t i o n s o f Binnie e t a l . [20
1.
F i g . 1 5 from t h i s experiment shows u n i t a - r i t y n i c e l y a t work a s t h e N q c r o s s - s e c t i o n , r i s i n g from t h r e s h o l d t o peak a t Sll (1520),
punches a h o l e i n t h e backward T C - ~ e l a s t i c c r o s s - s e c t i o n .n- +p-n- +p
i t
;rl THRESHOLDF i g . 1 5
.-
Backward rc-p e l a s t i c c r o s s - s e c t i o n (ref.@0 BULOS ET AL.
(ERRORS- 0.08 mb.)
.
THIS EXPT.F i g . 1 6 . - Backward c h a r g e exchange c r o s s - s e c t i o n ( r e f . 20)
T h e i r backward c h a r g e exchange d a t a i s much b e t t e r t h a n t h a t p r e v i o u s l y p u b l i s h e d ( F i g . 16) and though, t o - d a t e , t h e r e i s a 10-20 % n o r m a l i z a t i o n u n c e r t a i n t y t h e n i s no obvious s u p p o r t f o r t h e i d e a t h a t I - s p i n i s v i o l a t e d i n t h e backward d i r e c t i o n [21] ( F i g . 1 7 ) .
~ i g . 1 7 . - I - s p i n bounds fo; backward r r p - c h a r g e exchange compared w i t h experiment ( 2 0 ) .
T a b l e I V
X
0.32 0.08 0.16 1.
0.19 0.17 0.08 0.14 0 . 4 1 0.04 0.09 P a r t i a l
wave S3 1 1 P31 P33 P;3 D33 D35 F35 F37 G39 H311
Mass 1.623 2.001 1.789 1.231 1.90 1.723 1.894 1.869 1.928 2.170 2.392
r
0.161 0.307 0.221 0.109 0.204 0.192 0.121 0.255 0.237 0.205 0.289
HADRON SPECTROSCOPY C1-181
Table V A l l assume SU(6) x O(3) s t a t e c l a s s i f i c a t i o n
P a r t i a l
wave Mass I- X
S~~ 1.519 0.075 0.34
1.673 0.150 0 . 5 4
S';l 2.280 0.320 0.15
1.413 0.182 0.54
pi^
1.532 0.075 0.16pY
1 1.730 0.164 0.17P13 1.696 0.117 0 . 1 4
D13 1.525 0.122 0.56
D i 3 1.710 0.100 0.09
D'i3 2.029 0.116 0 . 1 0
D15 1.660 0.146 0 . 4 1
2.076 0.206 0.09
F15 1.679 0.126 0.59
F i 5 2.025 0.157 0.08
F17 2.049 0.119 0.06
C- 17 2 . 1 4 1 0.243 0.16
;19 2.133 0.193 0.09
;'19 2.249 0.347 0 . 2 0
;he second t r a d i t i o n a l method of baryon formation i s :-ia photoproduction. \.fore a p p r o p r i a t e t o Bonn, 7 x e n t i o n i t h e r e a s a n i n t r o d u c t i o n t o h i g h e r sym- x e t r y checks. We saw SU(3) i s good and, now t o
t e s t h i g h e r s y m e t r i e s we must look, more e x c i t i n g l y , a t s U ( 3 ) - i n e l a s t i c s c a t t e r i n g . With t h e photon jetavin: a s a v e c t o r meson i n i t s s t r o n g c o u p l i n g ,
y!; -. XU , l i k e n!; -. p?; o r ruU + n A , i s such a pro- c e s s . I n each of t+.ese c a s e s a resonance h a s more than one couplin:. Kg. F5 -.An has two l e a d i n g t o f i n a l f o r p waves, F5 * N p has f o u r f o r p vaves w i t h S=1!2 o r 312
.
S i m i l a r l y h e l i c i t y 112 o r 3 ' 2 yN c o u p l i n g s a r e p o s s i b l e ( a l t e r n a t i v e l y e l e c t r i c and magnetic m u l t i p o l a r i t y ) .Three c l o s e l y r e l a t e d approaches have been used f o r c o n s i d e r i n g SU(3) i n e l a s t i c c o u p l i n g s .
i ) /-broken SU(6)W [22].
i i ) I t s modern j u s t i f i c a t i o n from Melosh t r a n s - form21:ions [23 ].
i i i ) E x p l i c i t quark model c a l c u l a t i o n s [24].
and t h u s g i v e t h e same p r e d i c t i o n s f o r r e l a t i v e s i g n s f o r amplitudes w i t h e q u a l i n i t i a l and f i n a l o r b i t a l a n g u l a r momentum (f=#' ; p + p f + f e t c . ) . Now pure SU(6)W a l s o f i x e d , i n c o r r e c t l y , t h e f ' and f ' + 2 r e l a t i o n s h i p
-
i t s c l a s s i c mistake was on t h e D/S r a t i o f o r t h e B-meson p r e d i c t i n g lon- g i t u d i n a l r a t h e r t h a n t r a n s v e r s a l decay. I n f - b r o - ken SU(6)W f ' and f ' + 2 were t h e r e f o r e decoupled by t h e argument t h a t quarks might have t r a n s v e r s e momentum and s o a l o n g t h e c o l i n e a r Z-axis being c o n s i d e r e dlaz I
= 1 might be p o s s i b l e i n a d d i - t i o n t o t h eaZ
= 0 of pure SU(6)W.
[The spe- c i a l c a s e of IALZI
= 1 terms b e i n g dominant i s c a l l e d a11ti-SU(6)~,
and y i e l d s t h e same f ' t o f 1 + 2 r e l a t i o n a s pure SU(6)W b u t w i t h r e v e r s e d-1.
A l l t h i s i s g i v e n r e s p e c t a b i l i t y by t h e Melosh t r a n s f o r m a t i o n .
A u n i t a r y t r a n s f o r m a t i o n i s assumed p o s s i b l e b e t - ween e - s t a t e s of c o n s t i t u e n t and c u r r e n t quarks :
Iconstituent)=Vlcurrent)
.
Thus i f , say, one c a r r i e s o u t PCAC c a l c u l a t i o n s f o r n emission by a hadron s t a t e,
which i s what we want f o r rdJ -.An say, t h e n (hf 1QYlhi)=(constituentIQ 5" 1
c o n s t i t u e n t ) = ( c u r r e n t \v-'Q> ( c n r r e n t ).
The break-through i s t h a t t h e behaviour of V - ' Q ~ under c h i r a l SU(3) x SU(3) i s made up of a v e r y
l i m i t e d number of terms. Thus w i t h t h e a x i a l v e c t o r c u r r e n t d i s c u s s e d h e r e , Melosh shows i t i s made up of two terms : (8,1)0-(1,8)0 and (3,5)1-(5,3)1
.
Theformer l e a d s t o t h e l o g i c a l e q u i v a l e n t of pure SU(6)W i f i t dominates, t h e l a t t e r of a t 1 t i - S U ( 6 ) ~
.
( I t would, of c o u r s e , be n i c e i f one o r o t h e r term dominated f o r a given SU(6) x O(3) m u l t i p l e t )
.
I n e x p l i c i t q-models t h e e q u i v a l e n t of 2 c o u p l i n g s appears through t h e r e c o i l term. But c o n s t r a i n e d by e x p l i c i t wave f u n c t i o n s they t y p i c a l l y have no f r e e - dom t o a d j u s t t h e i r r a t i o i n going between
SU(6) x O(3) m u l t i p l e t s .
For y c o u p l i n g s Gillman e t a l . w h i l s t s t i l l having two c o u p l i n g s f o r t h e (70) have t h r e e f o r t h e (56), though t h e i r f i t shows t h a t one of t h e s e i s , i n p r a c t i c e , n e g l i g i b l e .
Photoproduction experiments a r e w e l l cross-checked.
Two r e c e n t a n a l y s e s both u s i n g f i x e d - t d i s p e r s i o n r e l a t i o n s e x t r a c t t h e yN c o u p l i n g s of w e l l e s t a b l i s hed baryon s t a t e s . That o f Moorehouse, Oberlack and
Rosenfeld [16'] and that of Devenish et al. [25]. With Table VI 17 resonances below 2 GeV to consider they are in
remarkable agreement. Table VI gives, for each re- sonance, the y = 112 and 312 amplitudes for the positive and neutral baryon (for a A these are di- rectly related because the y can only be iso- vector and only A+ is given). The top figure in each row is the experimental result of Moorehouse and Oberlack, the second that of Devenish et al., the third is the q-model prediction of Moorehouse and Oberlack, the fourth is the sign, when calcula-
ted, given by Gilman et al. [26 ] (which in fact always agrees with the sign of the q-model predic- tion). When a q-model prediction bears an asterisk it should be particulary stable as it does no in- volve significant cancellations. This is a highly non-trivial set of agreements
-
very encouraging for higher symmetry schemes. We are far from zoological classification. The quark model with its definite wave functions can give a prediction for radial ex- citations-
and apparently gets them wrong. That is probably a message.Turning to SU(3) inelastic strong couplings a single analysis (14) has dominated our thinking and there are important changes since Batavia. The ana-
-
0m
lysis, by LBL-SLAC, is of < p -tp K R in the E
+ -
range 1310-1990 MeV with a - + n n n
+
0- p ITn
bad gap between I560 and 1630, which allowing for binning means between bin centres at 1540 and 1650 MeV. Maximum likelihood analysis uses the full va- riable set to describe simultaneously the reactions on the simplifying assumption that they are trea- table as a sum of quasi two-body systems. Let me stress the limitations and problems :
i) Only Np, AK and No (shorthand for rn s-wave inserted as Berkeley 6; solution) are assu- med, i.e. ~\i;/~'s are ignored though mass plots
+ +
show such states. They do not fit the, n n K
channel because it shows this omission most clearly having no I=0,1 KIT and supressed A
.
ii) Post hoc, some amplitudes are, almost essen- tially, put to zero. This can, while reducing com- plexity, affect the waves left in, as we shall see.
iii) An energy independent analysis not giving phases, a K-matrix energy-dependent fit through the energy independent points is used to give phase con- tinuity. They anchor to Pll in the range 1310-1540 MeV ; DL5/F15 in the range 1650-1680 MeV and Fg5 in the range 1810-1970 MeV. Note it is Pll that
Top figure : Analysis of Devenish et al. [25]
Second figure : Analysis of Moorehouse and Oberlack [16a]. [The update of 16b is in close agreement
1.
Third figure : q-model prediction of M & 0
.
Sign : Sign when given by Gilman et al. [26 ] (or input)
State A:/~ A:/*
P33(1236) -144116 -262t15
-142+6 -259j;16 (56)ot
(input) (input) S31(1620) 4i-33
9 W 7 6
FI5(1688) 15i-23 +146+31
+
3 5 g 9-
18i-39 6@ 3@i n u t i n u t in ut F35(1890)
-
37+ ?-
22t ?-
6% ? -lo& 7carries the phase through the data gap.
The'72 solution as we shall see in a moment gave amplitude signs grossly at variance with .!-broken SU(6)W
,
but would look better if one could rotate all signs on one side of the gap (which side is ar- bitrary in view of the overall phase ambiguity). Here we see the importance of the new Saclay analysis ofHADRON SPEC TROSCOPY C1-183 Dolbeau and T r i a n t i s [27 ] u s i n g t h e i r own rr-p d a t a
i n t h e below gap r e g i o n a t f o u r CM e n e r g i e s between 1385 and 1535 GeV where t h e y h a v e 2-4 t i m e s t h e p r e - v i o u s s t a t i s t i c s . Adopting a n e a r i d e n t i c a l a p p r o a c h
t h e y d i f f e r from LBL-SLAC p a r t i c u l a r l y i n Pll
.
F i g . 1 8 g i v e s c o n t r i b u t i o n s , open c i r c l e s a r e t h e new S a c l a y r e s u l t s , b l a c k d o t s LBL-SLAC, c r o s s e s a r e o l d d i s c a r d e d S a c l a y s o l u t i o n . F i g . 19 compares Saclay and LBL-SLAC '72 Argands. (For S a c l a y , p h a s e i s a g a i n Pll-locked b u t by a r b i t r a r i l y p u t t i n g t h e PPll An p h a s e e q u a l t o t h e e l a s t i c p h a s e
-
whicht h e LBL-SLAC a n a l y s i s would s a y was r e a s o n a b l e ) . T h e r e a r e s i g n i f i c a n t d i f f e r e n c e s , p a r t i c u l a r l y i n Pll where SLAC s e e Np which LBL-SLAC had p u t e q u a l t o z e r o . I n t h e meantime LBL-SUC had u n d e r p r e s s u r e from Faiman i n s e r t e d Pll Np
-
and i t com- p l e t e l y changes t h e i r c o n t i n u a t i o n t h r o u g h , t h e gap r o t a t i n g i n t h e d e s i r e d way a l l s i g n s above r e l a t i v e t o a l l s i g n s below. H e n c e f o r t h I w i l l d i s c u s s o n l y t h i s ' 7 3 (Faiman) s o l u t i o n . I n a d d i t i o n t o b e i n g r o t a t e d a t t h e gap i t h a s f o u r e x t r a waves. D e t a i l e d K-matrix f i t t i n g h a s n o t y e t been c a r r i e d o u t . Par- t i a l w i d t h s w i l l d i f f e r from t h e ' 7 2 s o l u t i o n b u t r e l a t i v e s i g n s w i l l s t a y t h e same on each s i d e o f t h e gap. Note however F37 h a s s w i t c h e d s i n c e B a t a v i a s i n c e t h e K-matrix f i t g i v e s t h e o p p o s i t e s i g n t o t h a t s e e n a t f i r s t g l a n c e-
a r e p u l s i v e back- ground r o t a t i n g t h e r e s o n a n t p h a s e t h r o u g h 90' ( F i g . 20 shows t h e e f f e c t i n t h e '72 s o l u t i o n ) . F i g . 21 shows t h e new waves. F i g s . 22 & 23 compare some 72 and 73 Argands. Two new r e s o n a n t waves a p p e a r : t h e SDll n A o f t h e Sll (1700) and t h e pp33 n A of t h e P33 (1900) ( t h u s c o n f i r m i n g t h e e x i s t e n c e o f t h i s s t a t e s e e n i n t h e S a c l a y~
e l a s - t i c a n a l y s i s 161. F i g . 24 g i v e s t h e a m p l i t u d e s i g n s o f t h e ' 7 3 s o l u t i o n . Checking a g a i n s t t h e h i g h e r symmetries,/-broken SU(6)W ( o r Melosh),assuming t h e (70) 1-,
c h o o s e s a n t i - S U ( 6 ) W ( o r ( 3 , ? ) ( 3 , 3 ) and t h e (56) 2' c h o o s e s p u r e SU(6)W( o r ( 8 , l ) - ( 1 , 8 ) 0 ) t h e n a l l t h e An s i g n s a r e c o r - 0
r e c t f o r t h e s e m u l t i p l e t s . Moreover, Buccela e t a l . 1281 h a v e p r e d i c t e d t h a t even L m u l t i p l e t s would c h o o s e SU(6)W and odd L a11ti-SU(6)~
.
We a g a i n h a v e h e r e a n i m p r e s s i v e s e t of a g r e e m e n t s . I t makes u s b e l i e v e we c a n deduce t h i n g s f o r o t h e r m u l t i p l e t s and b o t h t h e Pll (1470) and t h e new P33 (1850) have s i g n s a p p r o p r i a t e f o r L=O ( 5 6 ) ' s.
[Since/=e"
t h e two c o n t r i b u t i o n s SU(6) - l i k e o r a n t i - S U ( 6 ) W W
-
l i k e y i e l d t h e same p r e d i c t i o n ] . They would have t h e o p p o s i t e s i g n f o r L=2
.
Hence t h e e a r l i e r s t a t e m e n tt h a t P33 (1850) i s a l m o s t c e r t a i n l y a r a d i a l e x c i - t a t i o n .
SU(6)W o r Melosh c a n n o t c u r r e n t l y p r e d i c t Np s i g n s b u t we have t h e q-model o f Moorehouse and P a r s o n s [29]. Of 5 p r e d i c t e d s i g n s o n l y one, t h a t on which t h e y a r e l e a s t c o n f i d e n t F15 (1690) d i s a -
g r e e s w i t h t h e e x p e r i m e n t .
I t h i n k i t i s s e l f e v i d e n t t h a t t h e d e t a i l e d s u c c e s s o f t h e s e symmetry schemes i s a n i m p o r t a n t development. The c u r r e n t i m p o r t a n c e o f SU(3) i n e - l a s t i c s c a t t e r i n g e x p e r i m e n t c a n n o t be o v e r s t r e s s e d .
3 , - S = -1 BARYONS
I n t h e c a s e of hyperon r e s o n a n c e s , d i s c o u r a g e d no d o u b t by low f l u x e s c o u n t e r p h y s i c i s t s h a v e , more o r l e s s , l e f t t h e f i e l d t o t h e bubble chamber w i t h t h e r e s u l t t h a t t h e b e s t u n d e r s t o o d c h a n n e l i s K - ~
-
Artw i t h s i n g l e I - s p i n and decay p o l a r i z a t i o n and t h e n Err w i t h p a r t i a l p o l a r i z a t i o n w h e r e a s e l a s t i c i t y v a - r i a t i o n s by a f a c t o r o f two a r e n o t uncommon even f o r w e l l e s t a b l i s h e d s t a t e s . Of c o u r s e s e n s i b l y one s h o u l d p e r f o r m a m u l t i c h a n n e l a n a l y s i s . The f i r s t , K-matrix, a t t e m p t by Kim 1301 a l m o s t c e r t a i n l y i n - v o l v e s d i s c o n t i n u o u s l e a p s between ambiguous s o l u - t i o n s , a p r o p e r t y p r o b a b l y s h a r e d by t h e a t t e m p t e d e n e r g y independent a n a l y s i s of Langbein and Wagner [31]. Even u s i n g l e a s t p a t h methods t h e d a t a i s i n - s u f f i c i e n t s e n s i b l y t o c o n s t r a i n t h e a n a l y s i s - we s h a l l s e e below however t h a t t h i s a n a l y s i s i s n o t beyond r e s c u e .
The n i c e s t m u l t i c h a n n e l a n a l y s i s i s t h a t o f Lea e t a l . [32], h e r e a f t e r c a l l e d LMMO) u s i n g a K-matrix
0 0
p a r a m e t r i z e d a s K . .(w) 1 3 = r y.y.(wo-a) 1 J
+
( i , j r e f e r t o
KN,
Err, Art and " t h egrcund, d i s t a n t r e s o n a n c e s , c u t s below t h r e s h o l d , e t c . . . a l l go i n t o t h e l i n e a r t e r m o r i n t o p o l e s o u t s i d e t h e a n a l y s e d r e g i o n (4 p o l e s p e r p a r t i a l wave a l l o w e d ) . The r e s i d u e s y a r e assumed f a c t o r i -
z a b l e , g u a r a n t e e i n g one t o one c o r r e s p o n d e n c e of K- and T - p o l e s of p o s i t i v e w i d t h . They, o f c o u r s e , u s e a minimal number o f p o l e s and F i g . 2 5 i l l u s t r a t e s t h e i r Argands. L i k e s o many o t h e r s t h e y o n l y go up t o 1.2 GeV/c, t h e gap between p u b l i s h e d CHS and CRS d a t a ( a gap now b e i n g c l o s e d a s we saw from a n a l y s e s h e r e by CHS [33
I) .
The b e s t p u b l i s h e d s i n g l e c h a n n e l a n a l y s i s , i n Art, i s p r o b a b l y Van ~ o r n ' s [34] s p a n n i n g 1540 5 E* 5 2215 compared w i t h t h e L i t c h f i e l d a n a l y s i s [35
1,
w i t h which i t c l o s e l y a g r e e s , t h e LBL d a t a i n1865 5 E' 5 2106 MeV i s added. He u s e s BW
+
q b ei%
I. BUTTERWORTH
0.04
aoo
F i g . 1 8 . - c o m p a r i s o n of some a m p l i t u d e s i n t h e m N a n a l y s i s o f LBL-SLAC a n d S a c l a y ( o p e n c i r c l e s : n e w S a c l a y ; crosses : o l d S a c l a y ; d o t s : LBL-SLAC).
SACLAY 73
PI1
@
I8 ' -S
q .I .3 'O
310'
2IV 00.
210'
F i g . 1 9 .
-
C o m p a r i s o n o f A r g a n d s b e t w e e n S a c l a y a n d LBL-SLAC.HADRON SPECTROSCOPY
I = $ T N F37
OLD NEW
A 1127 B 1217 C 1255 0 1310 E 1340 F 1370 G 1400 H 1440 1 1470 J 1490 K 1520 L 1540 No N""
Data for M.N.0 M 1560 N 1585 0 1617 P 1650 0 1670 R 1690 S 1730 T 1770 U 1810
v 1850 W 1890 X 1930 Y 1970
z 2010
Fig.20.- F37 waves in the 72 LBL-SLAC analysis.
Fig.22.- The Pll waves in the 72 and 73 solutions.
OLD A 1310 B 13LO C 1370 D 1400 E lLLO F 1470 G 1490 H 1520 I 15LO
-
- - - - - -J 1650 K 1690 L 1730 M 1770 N 1810 0 1850 P 1890 0 1 9 3 0 R 1 9 7 0 NEW
Fig.21.- New waves added in the 73 LBL-SLAC solution. Fig.23.- Comparison of waves in the new and old solutions.
C1-186 I. BUTTE
Fig.24.- Amplitudes signs of 73 LBL-SLAC solution (Note convention is such that arrows point opposite to loops in Argands of previous figures).
n A
n A L+I
N
?
"pl L
Nf.
with varying polynomial forms for
%
and b ' Fig. 26 shows, in three close alternatives, his Argands and Table VII his parameters. At the Confe-rence, [36] CERN-Heidelberg added their new data at 10 momenta, 1.4-1.8 GeV/c
,
to what is basically the Van Horn set, discarding some old U.S. points and perform an energy independent analysis in the old CHS style, i.e. using A 's and BP's polynomialP
smoothed with energy. D15 (1765) F17 (2030) were fully imposed. Their Argands (Fig. 27) are really very like those of Van Horn, though their El=--04
for FO5 (1915) is the lowest reported, their At smoothing has probably reduced its size.
CHS have presented [37] analyses of K-p + An and
+ +
K-p + C K- including their new data between 1134 and 1462 MeV/c but over a very narrow E* range, 1850-2000 MeV
.
They find only F15 (1915),
giving a large tzx of-
.22+.02 (previous figures-
0.1)with a standard th of - . O ~ . O L
.
They do not support Pog (1860), Sol (1870),
which does not surprise as these are elastic channel claims, nor D13 (1940),
more worrying as it is seen in other An analyses (see below) -however their 'E range is very narrow. College de France-Saclay [38] report a An analysis over 2070 5 E* 5 2370 MeV for thefive lowest momenta of their seven points up to E
'
= 2550.
Clear structure at 2250 needs a peir of resonances, but their J' assignments are still un- certain (Table VIII). TheE*
(1530) K cross-section is structured, a bump in $OKo absent in E*-K+PII m3 SN ms FS 033 sn
ma
~ 3 3 PI^ FS FZshowing that both I-spins are active (A negative A4 gives some support to D5 wave(s) as the lowest
1470 1530
1
Bl3I
PPH DO13
1
OS13
I
PSI1
I
possibility for giving such structure)
.
1650 1670 1690 1670 1700 1700 1850 1850 1850 1920
I t INl" I
1 I
FPlS 0s"1
DSl3 PP33 FF37SD31 DO15 SDll
New
FF35
I
FP15
I
FP35 FF37I I
5531
1 "" 1
PP13I I
F015 spn DPQ
I I
New data presented include K-p and K-d total cross-sections from BNL in the range 410-1060 MeV/c
[39]. Fig. 28 shaws the results and the possible new 1=1 bump(s) should be noted. Chu et al. [40 ] use 50 : 50 propane-freon in the CERN heavy liquid
0 0
chamber to study K-p + E rc in the 1660 region, their cross-section is apparently low compared with that published by CHS [41] for Cox0
.
However the CHS EOKO measurement agrees with the cross-section deduced from charged X's in the CHS K-p and K-n experiments. The discrepancy reduces to a single 2SD low point in the new data, (Fig. 29). There is no evidence from SU(3) checks of any trouble with branchings in this region [1,21.
Van Horn [34] opened a Pandora's box by asking what were the fundamental ambiguities of his solution giving the same data fit and still being physical.
Suppose we had precise scattering amplitudes f(e) and g(8)
,
the transformation f(e) + e i&(e)f(e)
g(e) -, e i6(e) g(e) gives the same
a
and polari- anzation. (In elastic channels 6(0°)=0 by optical theorem but that is the only place it is constrained
-
for inelastic channels even that limitation is removed). The proper way to remove the ambiguity is to have a trustworthy way of calculating the high partial waves as Alcock and Cottingham seem to be able to do in elastic channels [42]. This continuous phase ambiguity is in practise normally removed by sharp truncation of partial waves at some maximum J , P values Jm ; 8, when it turns into a set of dis- crete ambiguities 143,441 most readily treated by Barrelet zeros [44]. Instead of the two observables*
aa and P-
a0 consideran
2(I f
pIa an = l/k ~f (cose)?ig(cose)sinel2.
The ampli- tude combinations f+ig sine are known as transversity amplitudes). Using the variable w = eie(Iw!
= 1gives the physical region) cons'der the analytic
i
function F(w) = f (cost3)
+
g(cos8) which inthe physical region equals f+ig sine
,
then(l*P) =
an k2 IF(w)
l2
on the unit circle, Iwl = 1,
i.e. the physical region.
HADRON SPECTROSCOPY
Fig.25.- Amplitudes of LMMO analysis (ref. 32).
Fit H Fit R Fit K
1500 1600 1700 1800
TOTAL crn ENERGY ( M e V )
Fig.28.- Preliminary K-Nucleon total cross-sections for 1=0 and 1=1 from (ref. 39).
Fit H Fit R Fit K
Real T
I 1 1
Real T
Figs.26a-b.- Argands from An analysis of Van Horn.
C1-188 I. BUTTERWORTH
REAL PART
Fig.27.- Results of semi-energy-independent An analysis (ref. 36).
Table VII
Best values for energy dependent resonant parameters from Van Horn
Wave y/2(GeV) ER (MeV) r (MeV) t @(radians) Status
.16 .63 2042
+
11 178+
13 +.20+
.01 5:
Established .22 .70 1774+
10 146+
18-
.28-
'04 .05 3.14 fixed Established1 1::
-08 .44 1920 +-
20 l5 1 0 2 t 1 8 -.09+.02 3.67 ::
Established .20 1659fig
32+
11 +.09+
.02 0.0+
.1 ~stablished .20 1697$. 21:
66.:; -.13*.04 3.01 : ;
Probable1949
+ :g +
'03.43 160 -.05
-
.02 3.7 +-
Oe5 1.0 Probable1 .O 1668
+
25 2 3 1 -+.I2+
1.82
.3 Probable.25 2004 -j- 40 116
+
40 +.07+ :::
-0.4+
.3 Possible.43 2251
zE
192*
30-
.16 -j- .03 2.8+
.2 Possible.51 1985 f 50 220 *I40 +.05 + - '07 ,02 -0.7
*
.3 Possible.17 1925
+
200 65+ z:
+.06+
.04 .1 t .2 Suggested*
Averaged from best fits with+
indicating the range of values in good fits.a t = amplitude at resonance = f
m / r =
e r+
q1q3 dl(ER)/q3d2(ER).
Sign indicates Su(3) coupling with baryon first convention.y = full width = 2 q3d2 (ER)
.
q1 =
qr
= energy independent branching fraction. q3 = ~ / 2 = reduced width.These parameters are directly related to the SU(3) couplings
HADRON SPECTROSCOPY C1-189 Table V I I I
I
Resonant wavesI
P3 G9I
D5 G9I
P 3 H1lI
D5 H1lI
F i t t e d P3 o r D5 r e s o -
K N
-
(1=0) P,, ( G e V / c )0.700 0.800 0.900 1.000 1.100 1.200 nance G o r Hll
parame- 9 t e r s F7 f i x e d
1 1 I 11
.ooo
1.600 1 700 1.800 1.900
K N CENTRE OF MASS ENERGY ( G e V )
0 0
F i g . 2 9
.-
K-p -. E n.
Crosses g i v e measurements by Chu e t a l . ( r e f . 4 0 ) , open c i r c l e s t h e c r o s s - s e c t i o n deduced from CHS charged X measurements. Line shows r e s u l t of CHS p a r t i a l wave a n a l y s i s .2215+20 [2030]
Normally we w r i t e f(B)
-
T ~ " P!(cosO) J , Pgee)
-
T~~~ P;(COSB)J , P hence F(w) i s a l s o a polynomial :
1 0 9 2 0 [I711
r e - w r i t a b l e i n terms of i t s z e r o e s [44] :
1 1
where n=J +L - - (even o r odd a s J =L
+ -
o rl m m 2 m m 2
Jm=Lm-
7 ) .
To know t h e z e r o e s w. i s t o know t h e a m p l i t u d e . Since t h e d a t a (l+P) only g i v eas2
IF(")
l 2
on t h e u n i t c i r c l e-
they a r e u n a f f e c t e d by w. +-
w. where G ~ = ( < ' ) * (geometric i n v e r s e w . r . t .1 1
-0.10K).02 [M.21]
u n i t c i r c l e ) ; "w. i s a n e q u a l l y good z e r o s u b s t i - t u t a b l e f o r wi t o g i v e a new amplitude. By p i c k i n g and choosing between w. and
a
a t each z e r o we c a n , t h e r e f o r e g e n e r a t e a 2 - f o l d ambiguity of amplitu- n d e s t h a t g i v e t h e same c r o s s - s e c t i o n s and p o l a r i z a - t i o n a t a given energy. Not a l l need by p h y s i c a l-
they may push a p a r t i a l wave o u t s i d e t h e u n i t a r y c i r c l e f o r example, e i t h e r a t some given energy, o r c l e a r l y j o i n t o a s o l u t i o n t h a t has t h i s s i c k n e s s a t some a d j a c e n t energy.
Thus Berends [45] found t h e z e r o e s f o r t h e ampli- tudes corresponding t o t h e CERN EN e l a s t i c p a r t i a l waves a t each energy, and t r i e d s u b s t i t u t i n g a l l t h e p o s s i b l e mirror-zero combinations. At a given energy he found new s o l u t i o n s
-
b u t t h e r e was no new c o n t i - nuous p a t h f o r E*
< 2025 showing t h a t t h e o r i g i n a l random h u n t s had probably found a l l s o l u t i o n s , t h a t t h e goodness of t h e d a t a p r e v e n t s one jumping b e t - ween s o l u t i o n s corresponding t o d i f f e r e n t z e r o e s , and t h a t t h e t i g h t n e s s of e l a s t i c u n i t a r i t y y i e l d s a unique amplitude.Van Horn d i d n o t f i n d t h i s f o r h i s h s o l u t i o n , he could by i n t e r c h a n g i n g z e r o e s g e n e r a t e 6 a l t e r n a - t i v e s o l u t i o n s t h a t were p h y s i c a l , f i t t e d t h e d a t a and showed t h e dominant D15(1765) and F17(2040).
A l l showed D13 (1660) and F15 (1915) b u t p e r m i t t e d v i c i o u s changes i n branchings. For example,
@'
f o r F17 (2030) v a r i e d between .06 and .19
.
They a l s o y i e l d e d a mass of new low l y i n g s t a t e s . F i g . 30 shows t h e Argands-
they a r e by no s t r e t c h of t h e imagination minimal s t r u c t u r e s o l u t i o n s - b u t they would f i t t h e d a t a i n a n i d e n t i c a l way (A and R would remove t h e ambiguity c o m p l e t e l y ) .De B e l l e f o n and Berthon [46] adopt a somewhat d i f - f e r e n t approach. They look a t e s s e n t i a l l y t h e same An d a t a s e t , ( t h e new College d e France-Saclay d a t a i s added [38]), b u t go s t r a i g h t f o r t h e z e r o e s i n t h e hope t h a t by o b s e r v i n g t h e movement of t h e ze- r o e s w i t h energy they w i l l be a b l e t o i d e n t i f y sen- s i b l e amplitudes. They d e r i v e them d i r e c t l y from A 's and B ' s smoothed a s a f u n c t i o n of energy t o
e e
g i v e a n i c e continuous behaviour t o t h e d a t a . I t s t i l l sometimes happens t h a t (1*P) goes nega-
aa
t i v e because of e r r o r s , when
IF(^) I*
p a s s i n g from i t s p h y s i c a l p o s i t i v e v a l u e t o t h i s non-sense nega- t i v e v a l u e and back a g a i n throws up two r e a l r o o t s r a t h e r t h a n two r e s p e c t a b l e z e r o e s i n e q u a t i o n ( 1 ) . That i s n o t t o o much of a problem-
i t simply means t h e d a t a i s f l a g g i n g a danger a r e a when a z e r o i s c l o s e t o t h e u n i t c i r c l e and s o you a r e l i k e l y t o have a n ambiguity a s t o how t h e amplitude changes 0.1k0.2[O]
C1-190 I. BUTTERWORTH
Fig.30.- Some of Van Horn's ambiguous solutions.
with eneygy, since at this point you can cross to the mirror-zero (Fig. 31).
A
more serious problem is the introduction of an extra zero with increasing energy. If one waits un- til its position is certain the new high A 's come&'
in with a discontinuity frightening to the conven- tional phase shift analyst (e.g. Fig. 32). It is a pity that you cannot see where a zero first appears.
If one examines equation (I), demanding that the first introduction of an extra zero shall leave F(w) unchanged, then one sees that for n odd the zero must appear at w = 0
,
i.e. inside the circle.Likewise for n even it must appear off at infinity outside the circle. Ambiguities would only develop under those circumstances where a zero gets too close to the unit circle. However if we do not see a zero until it is really well established we cannot be sure the solution has not had the zero cross the circle while we still could not see it. (Actually where the zero first appears is out of the conver- gence region of the expansion).
De Bellefon and Berthon find basically only two solution sets for the zero trajectories which give both F17 (2030) and D15 (1765). They probably em- brace Van Horn's 7. Surprisingly it is the one that gives the largest couplings of these two states and which seems to correspond to the usual An phase
shift solution which at first sight has the greater problems. The pnd, 3rd, 4th, 5th and 6th zeroes
have all to be inside the unit circle, despite the Fig.31 fact that :
a) even zeroes should start outside
b) zero 5 aims normal to the circle at
-
1915 MeVand it looks suspicious that you do not let it cross outside (Fig. 33 shows the zeroes. Only those in the circle are plotted
-
each has a mirror outside).The second solution is free of these ills. It keeps tD high (-.23) but drops i+ to about .13
-
17
boil? solutions have the FI5 (1915) with t
-
.1.
.-
Zero trajectory continuation ambiguity.0..
l b 8 I 2.0 2.Z i l 1 . 1 2 . 0 1.7 1.. t.0 2.0 2.2
. " <"c.m ," M" c n C.E.O. 1. -1 C." t"C"S. .0M. I.. l d I.D ) . I
c n C l t l l , I* a. I - Ic n & rrrlor 2.0 I" 2.2 M"
8.8 I . 2.0 1.2
c n l r l o r I" M"
,
0 3 W A V E,
0 5 WRVE,
F 5 U R V E lo.rl 0 3 W R V E,
05 U R V E - 1 F 5 WRVE2.0
::: 11! !,
d. 0 . 3, :::
0,sbill, 14 I\, ill'
Ill, ---j*!--L.>
0.0-4--1-
1 b I . 2 . 0 2.1
c " C"C"*, ," M. I.. I . c n CmClOr I" 2 . 0 I . ? M V , . a ,. 7 . 0 z.2 L n C"l"0. I" a*
c " CHC"". 8. M " c " C.C%O. 8. w, c " E"C"*. 8" M" < " C"C"0, ," M" c n r"t"0, ," a" c " C"$"CT 1" a"
Fig.32.- Projection back of waves from amplitude obtained from zero trajectory. a) First solution
ZERO NR 1 ZERO NR 2 b) Second solution.
7 -
The analysis is not the last word, they do not know yet how correctly to introduce phases, and the D13 (1660) region looks a mess (Fig. 32)
-
but wehave here an interesting development in the phase shift business.
One can also use the zeroes in the interesting po- sitive sense of taking a ragged partial wave analysis that you suspect of jumping between solutions and
ZERO NR 3 ZERP NR U
I I I smooth it by judicious replacement of the zeroes by
I
their mirrors. Thus Litchfield [47] showed that the very discontinuous Langbein and Wagner energy inde- pendent analysis 1311 had An zeroes that frequently crossed the circle
-
by reflecting them all inside(like the first solution of de Bellefon and Berthon) the solution comes more or less into step with eve-
I ,
ryone else's-
thus the jaggedD15 then does a res-
2tRO NR 5 ZERO NR 6
- I
pectable Breit-Wigner loop (see Fig. 2 of Litchfield'sZERO NR ? ZERO HA 8
mini-rapport) (the original analysis is multichannel but unitarity puts poor constraints on hrc
-
indeed the size of the hrc signal on average goes down in the variant solution so unitarity is still obeyed).Indeed if you take minimal structure solutions more or less everyone is agreed on at least the An Argands and that they need D13 (1670), Sll (1750), D15 (1765), F15 (1915) and F17 (2030)
.
Moreover [47 ] all have resonant like structure in Pll at-
1900 MeV and DL3 at-
1950.
SU(3) inelastic channels are in poorer shape than for nN
.
CHS [48] report K-p + E (1385)~ in the range 600-1150 MeV/c in an analysis subject to seve- ral limitations :i) the A X'TC- channel is assumed pure ~(1385) x (Fig. 34 shows the Dalitz plot).
ii) The variables to be used in the analysis are Fig.33.- Zero trajectory from ref. 46.
