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Polarization in meson-baryon elastic scattering at high energies
Volume 20, number 3
PHYSICS LETTERS
The m o d e l r e s u l t s f r o m a p a r a m e t r i z e d r e f l e c t i o n c o e f f i c i e n t which is e s s e n t i a l f o r the a b s o r p t i o n m o d e l [7] and the p h e n o m e n o l o g i c a l m o d e l [8].
15 February 1966
4. W.Tobocman, private communication; R. Serber, Phys. Rev. Lett. l0 (1963) 357; Rev.Mod. Phys. 36 (1964) 649; A.D. Kitsch, Phys. Rev. 135 (1964) B1456; L. Marshall and T. Oliphant, Phys. Letters 16 (1965) 83 5. G. Fast and R. Hagerdon, NuovoCimento27 (1963)208; G. Fast, R. Hagerdon and L. W. Jones, Nuovo Cimento 27 (1963) 856. 6. V. Cook, B.Cork, W.R. Holley and M . L . P e r l , Phys. Rev. 130 (1963) 762; R. M. Heinz and M. H. Ross, Phys. Rev. Letters 14
The a u t h o r s would like to thank C. Robinson f o r useful d i s c u s s i o n s .
References 1. W.E. Frahn and R. H. Venter, Ann. of Phys. 24 (1963) 243. 2. R.H.Venter, Ann. of Phys.25 (1963) 405. 3. M . L . P e r l , L.W. Jones and C.C.Ting, Phys. Rev. 132 (1963) 762; L. M. Simmons, Phys. Rev. Letters 12 (1964) 229. A. Dar, M. Kugler, Y. Dothan and S. Nnssinov, Physics Letters 11 (1964) 265.
(1965) 1091. 7. K.Gott~ried and J.D.Jackson, Nuovo Cimento 34 (1964) 735; L.Durand and Y. T. Chiu, Phys. Rev. 199 (1965) B646. 8. A.Dar and B. Kozlowsky, to be published. 9. D.E.Damouth, L.W. Jones andM.L.Perl, Phys. Rev. Letters 11 (1963) 287. 10. M.L. Perl, Y.Y. Lee and E. Marquit, Phys. Rev. 138 (1965) B707.
11. C.Coffin, N.Dikmen, L.Ettlinger, D.Meyer, A. Saulys, K. Ter-williger and D. Williams, private communication. 12. J. Orear et al., Phys. Rev. Letters 15 (1965) 309; D. Harting et al., Nuovo Cimento 13 (1965) 60. K. J. Foley et al., Phys. Rev. Letters 11 (1963) 425.
* * * * *
POLARIZATION
IN MESON-BARYON ELASTIC AT HIGH ENERGIES
SCATTERING
A . DAR and B . K O Z L O W S K Y * The Wei~mann Institute of Science, Rehovoth, Israel
Received 7 January 1966
The strong absorption model is extended to describe the polarization angular distribution in high energy meson-baryon elastic scattering.
As a r e s u l t of t h e i r p o l a r i z a t i o n m e a s u r e m e n t s of ~-p e l a s t i c s c a t t e r i n g in the 1.70-2.50 G e V / c e n e r g y r e g i o n , Suwa et al. [I] concluded that spin e f f e c t s m u s t be included in any meaningful a n a l y s i s of m e s o n - b a r y o n e l a s t i c s c a t t e r i n g in t h i s e n e r g y r eg i o n . In a p r e c e d i n g l e t t e r [2] we have d e m o n s t r a t e d that the m a i n f e a t u r e s of the d i f f e r e n t i a l c r o s s s e c t i o n f o r m e s o n - b a r y o n e l a s t i c s c a t t e r i n g at high e n e r g y m a y be well r e p r o d u c e d by the " s t r o n g a b s o r p t i o n m o d e l " [3] with the n e g l e c t of spin e f f e c t s . In the p r e s e n t note we d e m o n s t r a t e * The research reported in this document has been sponsored in part by the National Bureau of Standards, USA. 314
that a n a t u r a l e x t e n s i o n of t h i s m o d el m a y a c count a l s o f o r f i n e r d e t a i l s of the d i f f e r e n t i a l c r o s s s e c t i o n and f o r the m a i n f e a t u r e s of the p o l a r i z a t i o n an g u l ar d i s t r i b u t i o n . Th e HtifnerDe Shalit a s s u m p t i o n [4] which was u t i l i z e d by A l e x a n d e r et al. [5] a p p e a r s as a s p e c i a l c a s e of our t r e a t m e n t . Our m o d e l is b a s e d on a p o t en t i al p i c t u r e f i r s t s u g g e s t e d by F e r m i [6] which can be g e n e r a l i z e d at once to the r e l a t i v i s t i c domain w h e r e the p o t e n t i a l has at b e s t a p h e n o m e n o l o g i c a l s i g n i f i c a n c e . F o l l o w i n g F e r m i we a s s u m e that the nonc e n t r a l p a r t of the potential i s a s p i n - o r b i t coupling t e r m , while the c e n t r a l p a r t i s r e p r e s e n t e d by a c o m p l e x potential
Volume 20, n u m b e r 3
PHYSICS
L E T T E RS
15 F e b r u a r y 1966
ImS~
==
/
VR >
Vz
Re~¢ Re~; f+ t,~ =t -kb
j =f,+,zz e
ReV
J
R~
,," / j=r,-,,z
B
/ Ir.,7;
>
~/~*
n-"
Fig. 1. a) The r a d i a l shape of the r e a l and i m a g i n a r y c e n t r a l potentials and of the spin o r b i t potential, b) The r a d i a l s h a p e of the effective r e a l potentials in the j = l + ½, j = l - ~ s t a t e s (solid lines) c o m p a r e d with the c e n t r a l r e a l potential (dashed line).
V = i Vi(r) + VR(r) + V.o(r)(s.l) •
(1)
The spin orbit potential Vs.o(r ) is assumed to be real and its radial shape is proportional to the gradient of the "interaction density", while the imaginary and real parts are represented approximately by step functions (see fig. la). For s and I antiparallel, R e V becomes stronger in the surface region and extends to a larger radius than the Re V(r) for s and I parallel. (see fig. lb). Utilizing the W K B approximation [7], one easily sees that the phase shifts (either in the partial wave or impact parameter representation) have similar shapes to the corresponding potentials, (see fig. 2a). The reflection coefficients are given by ~?l = e x p 2 i S / ~ ~ g ( l ) + 2i Re(5
;)
g(1)
(2a)
Fig. 2. a) The shape of the r e a l and i m a g i n a r y p h a s e shifts in the j = l + ½, j = l - ½ s t a t e s , as function of i m p a c t p a r a m e t e r , b) The shape of the complex r e f l e c tion coefficient in the j = l + ½, ] = l - ½ s t a t e s (solid lines) and t h e i r different {heavy line) as function of i m pact p a r a m e t e r . Rey/
= g l ( t ) + E(1 - g l ( t ) ) ,
+ • d g 2~ I m 7/l = U d t
and Re6
<< 1.
Representing the reflection coefficients by contin u o u s f u n c t i o n s of t = l + ½ [3]
'
(2b)
(2c)
a s j u s t i f i e d a b o v e , we f i n d * +
+
~/l -
ni
= i
-
d2g 3
Ira0/l -Z/l) ~
--
(3)
[g(o)l 2 ,
(4a)
iu
dt 2 The gi(t) approximate unit step functions at T ~ kR where R is the s u m of radii of the colliding particles. The differential cross section and the polarization angular distribution dc~/d~ :
where
wz =t-kb
I/(o)I2 +
• The l a s t a s s u m p t i o n was suggested independently by W. E. F r a h n (private communication) [see also 8] and by N. A u s t e r n , p r i v a t e communication through W. E. Fgahn. However, this was introduced together with the a s s u m p t i o n Im ~/~ = #± d2g2/dt2.
315
Volume 20, number 3
PHYSICS LETTERS
p(O) d(x/d~2 = 2 Re [f(O)g*(0)] ,
~Z3 = ~,~
(4b)
m a y be e x p r e s s e d in t e r m s of the reflection c o -
(~/l+ -~/l- )/~ 1(COS 0) ,
(5b)
with 2~ = ~/+ + ~/-. T h e analytic t r e a t m e n t of the strong a b s o r p tion model [3] is independent of the specific functional f o r m of gi(t). The only p r a c t i c a l r e q u i r e ment is that dgiTdt p o s s e s simple F o u r i e r t r a n s forms.
efficients
¢(0) ~ i / 2 k ~ (2/+ 1)(1 - ~-l) P/(cos O) , (5a) Polorization in 1T-P
15 February 1966
elostic scattering
(~I 2.08 Gevlc
+~ dgi -i(t-T)O Fi(aiO) = f di- e dt , E
(6)
-oo
o
•~ T / o
w h e r e Ai is the "rounding" p a r a m e t e r of g/. F o r brevity we have r e s t r i c t e d o u r s e l v e s to the case g l = g2 = g3' (No additional difficulty a r i s e s in the general case g l ¢ g 2 ¢ g3") Eqs. (4a), (4b) reduce then to
-~/-~
-2 ] ' N _
g, n
da
T2
(F[ A0])2 x
.2
P(0) ~d(x = - ~-~(1T2- E)v S -0~ (FLAOj)2JI(TO)~o(TO) ] r (Tb) f o r 0 -.< 0 << 7/- (4T) -1 . I0
i
_.•i
I
I
~ ~
I
I
I
I
I
I
I
[
i
I
[
I
I
I
I
"tr- P ELASTIC SCATTERING P LAB
=
2
GeV/c
I
v
~
oJ
0.01
i/l I ?,L) I
. . . .
t
I.o
I
0.8
I
]
0.6
I
I
0.4
]
]
02
I
I
0
L
I
-0.2
I
I
-0.4
I
I
-0.6
I
I
-0.8
I
-I.o
cos 8 Fig. 3. a) Comparison between experimental results [1, 9] and the theoretical predictions for ~r-p elastic scattering at 2.01-2.08 G e V / c . The broken line is the prediction of ref. 2 neglecting spin orbit effects. The solid line was calculated from eq. (7) with the s a m e param_ eters, b) Comparison between e x p e r i m e n tsl results and theoretical prediction for the polarization in u p elastic scattering at 2.08 G e V / c . The experimental points w e r e taken from ref. 1. The parameters for the theoretical line were those of ref. 2. 316
Volume 20, number 3
PHYSICS LETTERS
F o r b a c k w a r d a n g l e s (~ >/ 0>> (4T) -1)
f(o) = i (1 - ~ ) f l ( o ) - ~ ~ f l ( o ) g(O)
'
= - ~vT
,
(Sa)
a2 ~ T 2- -f-l ( O ) ~ 0
'
(8b)
where
(Sc)
15 February 1966
a g r e e m e n t with e x p e r i m e n t up to b a c k w a r d a n g l e s , where c o n t r i b u t i o n of the 2190 MeV r e s onance p r o b a b l y d o m i n a t e s the diffraction m e c h a n i s m . F r o m the fact that the p o l a r i z a t i o n m e a s u r e d at t h e s e a n g l e s is s y s t e m a t i c a l l y below our p r e d i c t i o n , one may conclude that the d i s c r e p a n c y is p r i n c i p a l l y due to a r e s o n a n t having spin and o r b i t a l a n g u l a r m o m e n t u m a n t i p a r a l l e l . F u r t h e r v e r i f i c a t i o n of the p r e s e n t model awaits p o l a r i z a t i o n m e a s u r e m e n t s at high e n e r gies. The a u t h o r s would like to acknowledge d i s c u s s i o n s with G. A l e x a n d e r and C. Robinson.
with ~ = v - O . One s e e s that the Htifner-De Shalit and the A l e x a n d e r - D a r - K a r s h o n r e s u l t s a r e obtained by taking the l i m i t A-+ 0 in the s m a l l angle f o r m u l a (Tb). F o r s m a l l T and b a c k w a r d angle it i s m o r e p r a c t i c a l to evaluate g(O) d i r e c t l y f r o m e x p r e s sion (5b). In the following a n a l y s i s we have c h o s e n the Woods-Saxon shape f o r the a b s o r p t i o n amplitude g i = {1 + exp[(t - T)/A]} -1
(9)
F o r t h i s choice F [ AO] = , A O / s i n h (TrAO) .
(10)
We have applied eqs. (7) and (8) to the r e c e n t r e s u l t s of Suwa et al. [1] on p o l a r i z a t i o n in y - p e l a s t i c s c a t t e r i n g at 2.08 G e V / c (see fig. 3). The p a r a m e t e r s were t a k e n f r o m our p r e v i o u s a n a l y s i s of m e s o n - b a r y o n e l a s t i c s c a t t e r i n g . One s e e s that the i n c l u s i o n of spin effects i s sufficient for r e p r o d u c i n g the f i n e r d e t a i l s of v p e l a s t i c s c a t t e r i n g . The p o l a r i z a t i o n p r e d i c t e d i s in good
References
1. S. Suwa, A. Yokosawa, N. E. Booth, R.J. Esterling and R.E.Hill, Phys. Rev. Letters 15 {1965} 560. 2. B.Kozlowsky and A.Dar, Physics Letters 20 (1966) 311. 3. W.E. Frahn and R.H.Venter, Ann. Phys. 24 (1963) 243; R.H.Venter, Ann. Phys. 25 11963) 405. W.E.Frahn and R.H.Venter, Ann.Phys. 27 (1964) 135; R.H.Venter and W.E.Fralm, Ann. Phys. 27 (1964) 385. 4. J. Hfffner and A.De-Shalit, Physics Letters 15(1965) 52. 5. G.Alexander, A.Dar and U.Karshon, Phys. Rev. Letters 14 (1965) 918. 6. E . F e r m i , Nuovo Cimento II, Suppl. No. 1 (1955} 84. 7. R.J. Glauber, High energy collision theo~¢ {Boulder Lectures in Theoretical Physics, Vol. 1 (1958}). 8. W.E. Frahn, Intern. Conf. on Polarization phenomena of nucleons, Karlsruhe, September 1965). 9. D.E.Damouth> L.W.Jones and M.L.Perl, Phys. Rev. Letters 11 (1963) 287.
* * * * *
317
PHYSICS LETTERS
The m o d e l r e s u l t s f r o m a p a r a m e t r i z e d r e f l e c t i o n c o e f f i c i e n t which is e s s e n t i a l f o r the a b s o r p t i o n m o d e l [7] and the p h e n o m e n o l o g i c a l m o d e l [8].
15 February 1966
4. W.Tobocman, private communication; R. Serber, Phys. Rev. Lett. l0 (1963) 357; Rev.Mod. Phys. 36 (1964) 649; A.D. Kitsch, Phys. Rev. 135 (1964) B1456; L. Marshall and T. Oliphant, Phys. Letters 16 (1965) 83 5. G. Fast and R. Hagerdon, NuovoCimento27 (1963)208; G. Fast, R. Hagerdon and L. W. Jones, Nuovo Cimento 27 (1963) 856. 6. V. Cook, B.Cork, W.R. Holley and M . L . P e r l , Phys. Rev. 130 (1963) 762; R. M. Heinz and M. H. Ross, Phys. Rev. Letters 14
The a u t h o r s would like to thank C. Robinson f o r useful d i s c u s s i o n s .
References 1. W.E. Frahn and R. H. Venter, Ann. of Phys. 24 (1963) 243. 2. R.H.Venter, Ann. of Phys.25 (1963) 405. 3. M . L . P e r l , L.W. Jones and C.C.Ting, Phys. Rev. 132 (1963) 762; L. M. Simmons, Phys. Rev. Letters 12 (1964) 229. A. Dar, M. Kugler, Y. Dothan and S. Nnssinov, Physics Letters 11 (1964) 265.
(1965) 1091. 7. K.Gott~ried and J.D.Jackson, Nuovo Cimento 34 (1964) 735; L.Durand and Y. T. Chiu, Phys. Rev. 199 (1965) B646. 8. A.Dar and B. Kozlowsky, to be published. 9. D.E.Damouth, L.W. Jones andM.L.Perl, Phys. Rev. Letters 11 (1963) 287. 10. M.L. Perl, Y.Y. Lee and E. Marquit, Phys. Rev. 138 (1965) B707.
11. C.Coffin, N.Dikmen, L.Ettlinger, D.Meyer, A. Saulys, K. Ter-williger and D. Williams, private communication. 12. J. Orear et al., Phys. Rev. Letters 15 (1965) 309; D. Harting et al., Nuovo Cimento 13 (1965) 60. K. J. Foley et al., Phys. Rev. Letters 11 (1963) 425.
* * * * *
POLARIZATION
IN MESON-BARYON ELASTIC AT HIGH ENERGIES
SCATTERING
A . DAR and B . K O Z L O W S K Y * The Wei~mann Institute of Science, Rehovoth, Israel
Received 7 January 1966
The strong absorption model is extended to describe the polarization angular distribution in high energy meson-baryon elastic scattering.
As a r e s u l t of t h e i r p o l a r i z a t i o n m e a s u r e m e n t s of ~-p e l a s t i c s c a t t e r i n g in the 1.70-2.50 G e V / c e n e r g y r e g i o n , Suwa et al. [I] concluded that spin e f f e c t s m u s t be included in any meaningful a n a l y s i s of m e s o n - b a r y o n e l a s t i c s c a t t e r i n g in t h i s e n e r g y r eg i o n . In a p r e c e d i n g l e t t e r [2] we have d e m o n s t r a t e d that the m a i n f e a t u r e s of the d i f f e r e n t i a l c r o s s s e c t i o n f o r m e s o n - b a r y o n e l a s t i c s c a t t e r i n g at high e n e r g y m a y be well r e p r o d u c e d by the " s t r o n g a b s o r p t i o n m o d e l " [3] with the n e g l e c t of spin e f f e c t s . In the p r e s e n t note we d e m o n s t r a t e * The research reported in this document has been sponsored in part by the National Bureau of Standards, USA. 314
that a n a t u r a l e x t e n s i o n of t h i s m o d el m a y a c count a l s o f o r f i n e r d e t a i l s of the d i f f e r e n t i a l c r o s s s e c t i o n and f o r the m a i n f e a t u r e s of the p o l a r i z a t i o n an g u l ar d i s t r i b u t i o n . Th e HtifnerDe Shalit a s s u m p t i o n [4] which was u t i l i z e d by A l e x a n d e r et al. [5] a p p e a r s as a s p e c i a l c a s e of our t r e a t m e n t . Our m o d e l is b a s e d on a p o t en t i al p i c t u r e f i r s t s u g g e s t e d by F e r m i [6] which can be g e n e r a l i z e d at once to the r e l a t i v i s t i c domain w h e r e the p o t e n t i a l has at b e s t a p h e n o m e n o l o g i c a l s i g n i f i c a n c e . F o l l o w i n g F e r m i we a s s u m e that the nonc e n t r a l p a r t of the potential i s a s p i n - o r b i t coupling t e r m , while the c e n t r a l p a r t i s r e p r e s e n t e d by a c o m p l e x potential
Volume 20, n u m b e r 3
PHYSICS
L E T T E RS
15 F e b r u a r y 1966
ImS~
==
/
VR >
Vz
Re~¢ Re~; f+ t,~ =t -kb
j =f,+,zz e
ReV
J
R~
,," / j=r,-,,z
B
/ Ir.,7;
>
~/~*
n-"
Fig. 1. a) The r a d i a l shape of the r e a l and i m a g i n a r y c e n t r a l potentials and of the spin o r b i t potential, b) The r a d i a l s h a p e of the effective r e a l potentials in the j = l + ½, j = l - ~ s t a t e s (solid lines) c o m p a r e d with the c e n t r a l r e a l potential (dashed line).
V = i Vi(r) + VR(r) + V.o(r)(s.l) •
(1)
The spin orbit potential Vs.o(r ) is assumed to be real and its radial shape is proportional to the gradient of the "interaction density", while the imaginary and real parts are represented approximately by step functions (see fig. la). For s and I antiparallel, R e V becomes stronger in the surface region and extends to a larger radius than the Re V(r) for s and I parallel. (see fig. lb). Utilizing the W K B approximation [7], one easily sees that the phase shifts (either in the partial wave or impact parameter representation) have similar shapes to the corresponding potentials, (see fig. 2a). The reflection coefficients are given by ~?l = e x p 2 i S / ~ ~ g ( l ) + 2i Re(5
;)
g(1)
(2a)
Fig. 2. a) The shape of the r e a l and i m a g i n a r y p h a s e shifts in the j = l + ½, j = l - ½ s t a t e s , as function of i m p a c t p a r a m e t e r , b) The shape of the complex r e f l e c tion coefficient in the j = l + ½, ] = l - ½ s t a t e s (solid lines) and t h e i r different {heavy line) as function of i m pact p a r a m e t e r . Rey/
= g l ( t ) + E(1 - g l ( t ) ) ,
+ • d g 2~ I m 7/l = U d t
and Re6
<< 1.
Representing the reflection coefficients by contin u o u s f u n c t i o n s of t = l + ½ [3]
'
(2b)
(2c)
a s j u s t i f i e d a b o v e , we f i n d * +
+
~/l -
ni
= i
-
d2g 3
Ira0/l -Z/l) ~
--
(3)
[g(o)l 2 ,
(4a)
iu
dt 2 The gi(t) approximate unit step functions at T ~ kR where R is the s u m of radii of the colliding particles. The differential cross section and the polarization angular distribution dc~/d~ :
where
wz =t-kb
I/(o)I2 +
• The l a s t a s s u m p t i o n was suggested independently by W. E. F r a h n (private communication) [see also 8] and by N. A u s t e r n , p r i v a t e communication through W. E. Fgahn. However, this was introduced together with the a s s u m p t i o n Im ~/~ = #± d2g2/dt2.
315
Volume 20, number 3
PHYSICS LETTERS
p(O) d(x/d~2 = 2 Re [f(O)g*(0)] ,
~Z3 = ~,~
(4b)
m a y be e x p r e s s e d in t e r m s of the reflection c o -
(~/l+ -~/l- )/~ 1(COS 0) ,
(5b)
with 2~ = ~/+ + ~/-. T h e analytic t r e a t m e n t of the strong a b s o r p tion model [3] is independent of the specific functional f o r m of gi(t). The only p r a c t i c a l r e q u i r e ment is that dgiTdt p o s s e s simple F o u r i e r t r a n s forms.
efficients
¢(0) ~ i / 2 k ~ (2/+ 1)(1 - ~-l) P/(cos O) , (5a) Polorization in 1T-P
15 February 1966
elostic scattering
(~I 2.08 Gevlc
+~ dgi -i(t-T)O Fi(aiO) = f di- e dt , E
(6)
-oo
o
•~ T / o
w h e r e Ai is the "rounding" p a r a m e t e r of g/. F o r brevity we have r e s t r i c t e d o u r s e l v e s to the case g l = g2 = g3' (No additional difficulty a r i s e s in the general case g l ¢ g 2 ¢ g3") Eqs. (4a), (4b) reduce then to
-~/-~
-2 ] ' N _
g, n
da
T2
(F[ A0])2 x
.2
P(0) ~d(x = - ~-~(1T2- E)v S -0~ (FLAOj)2JI(TO)~o(TO) ] r (Tb) f o r 0 -.< 0 << 7/- (4T) -1 . I0
i
_.•i
I
I
~ ~
I
I
I
I
I
I
I
[
i
I
[
I
I
I
I
"tr- P ELASTIC SCATTERING P LAB
=
2
GeV/c
I
v
~
oJ
0.01
i/l I ?,L) I
. . . .
t
I.o
I
0.8
I
]
0.6
I
I
0.4
]
]
02
I
I
0
L
I
-0.2
I
I
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cos 8 Fig. 3. a) Comparison between experimental results [1, 9] and the theoretical predictions for ~r-p elastic scattering at 2.01-2.08 G e V / c . The broken line is the prediction of ref. 2 neglecting spin orbit effects. The solid line was calculated from eq. (7) with the s a m e param_ eters, b) Comparison between e x p e r i m e n tsl results and theoretical prediction for the polarization in u p elastic scattering at 2.08 G e V / c . The experimental points w e r e taken from ref. 1. The parameters for the theoretical line were those of ref. 2. 316
Volume 20, number 3
PHYSICS LETTERS
F o r b a c k w a r d a n g l e s (~ >/ 0>> (4T) -1)
f(o) = i (1 - ~ ) f l ( o ) - ~ ~ f l ( o ) g(O)
'
= - ~vT
,
(Sa)
a2 ~ T 2- -f-l ( O ) ~ 0
'
(8b)
where
(Sc)
15 February 1966
a g r e e m e n t with e x p e r i m e n t up to b a c k w a r d a n g l e s , where c o n t r i b u t i o n of the 2190 MeV r e s onance p r o b a b l y d o m i n a t e s the diffraction m e c h a n i s m . F r o m the fact that the p o l a r i z a t i o n m e a s u r e d at t h e s e a n g l e s is s y s t e m a t i c a l l y below our p r e d i c t i o n , one may conclude that the d i s c r e p a n c y is p r i n c i p a l l y due to a r e s o n a n t having spin and o r b i t a l a n g u l a r m o m e n t u m a n t i p a r a l l e l . F u r t h e r v e r i f i c a t i o n of the p r e s e n t model awaits p o l a r i z a t i o n m e a s u r e m e n t s at high e n e r gies. The a u t h o r s would like to acknowledge d i s c u s s i o n s with G. A l e x a n d e r and C. Robinson.
with ~ = v - O . One s e e s that the Htifner-De Shalit and the A l e x a n d e r - D a r - K a r s h o n r e s u l t s a r e obtained by taking the l i m i t A-+ 0 in the s m a l l angle f o r m u l a (Tb). F o r s m a l l T and b a c k w a r d angle it i s m o r e p r a c t i c a l to evaluate g(O) d i r e c t l y f r o m e x p r e s sion (5b). In the following a n a l y s i s we have c h o s e n the Woods-Saxon shape f o r the a b s o r p t i o n amplitude g i = {1 + exp[(t - T)/A]} -1
(9)
F o r t h i s choice F [ AO] = , A O / s i n h (TrAO) .
(10)
We have applied eqs. (7) and (8) to the r e c e n t r e s u l t s of Suwa et al. [1] on p o l a r i z a t i o n in y - p e l a s t i c s c a t t e r i n g at 2.08 G e V / c (see fig. 3). The p a r a m e t e r s were t a k e n f r o m our p r e v i o u s a n a l y s i s of m e s o n - b a r y o n e l a s t i c s c a t t e r i n g . One s e e s that the i n c l u s i o n of spin effects i s sufficient for r e p r o d u c i n g the f i n e r d e t a i l s of v p e l a s t i c s c a t t e r i n g . The p o l a r i z a t i o n p r e d i c t e d i s in good
References
1. S. Suwa, A. Yokosawa, N. E. Booth, R.J. Esterling and R.E.Hill, Phys. Rev. Letters 15 {1965} 560. 2. B.Kozlowsky and A.Dar, Physics Letters 20 (1966) 311. 3. W.E. Frahn and R.H.Venter, Ann. Phys. 24 (1963) 243; R.H.Venter, Ann. Phys. 25 11963) 405. W.E.Frahn and R.H.Venter, Ann.Phys. 27 (1964) 135; R.H.Venter and W.E.Fralm, Ann. Phys. 27 (1964) 385. 4. J. Hfffner and A.De-Shalit, Physics Letters 15(1965) 52. 5. G.Alexander, A.Dar and U.Karshon, Phys. Rev. Letters 14 (1965) 918. 6. E . F e r m i , Nuovo Cimento II, Suppl. No. 1 (1955} 84. 7. R.J. Glauber, High energy collision theo~¢ {Boulder Lectures in Theoretical Physics, Vol. 1 (1958}). 8. W.E. Frahn, Intern. Conf. on Polarization phenomena of nucleons, Karlsruhe, September 1965). 9. D.E.Damouth> L.W.Jones and M.L.Perl, Phys. Rev. Letters 11 (1963) 287.
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