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Nucleon-antinucleon potential from single-meson exchanges
-~
NuClear Physics B5 (1968) 201-219. North-Holland Publ. Comp., A m s t e r d a m
NUCLEON-ANTINUCLEON FROM
SINGLE-MESON
POTENTIAL EXCHANGES
R. A. B R Y A N * Lawrence Radiation Laboratory, Livermore, California, USA and R. J . N. P H I L L I P S ** A . E . R . E . , Harwell, U.K. Received 15 F e b r u a r y 1968
A b s t r a c t : The static and non-static Bryan-Scott potentials, which a r e based on oneboson-exchange and fit nucleon-nucleon scattering, are adapted to the nucleonantinucleon system. T e r m s corresponding to odd-G exchanges a r e r e v e r s e d in sign, and an e m p i r i c a l absorptive potential is added. Satisfactory fits to nucleonantinucleon data a r e achieved, in the 0-300 MeV range.
1. ~ T R O D U C T I O N M o d e l s b a s e d on o n e - b o s o n - e x c h a n g e (OBE) a p p r o x i m a t i o n s h a v e b e e n v e r y s u c c e s s f u l in t h e N - N p r o b l e m [ 1 - 5 ] . T h i s p a p e r d e s c r i b e s how two OBE potential models - the static and non-static Bryan-Scott potentials [ 1 , 5 ] - c a n b e a d a p t e d to d e s c r i b e n o n - r e l a t i v i s t i c N - N i n t e r a c t i o n s , and s u c c e s s f u l l y fit a v a i l a b l e d a t a . It w a s n o t i c e d l o n g a g o t h a t m e s o n e x c h a n g e c o n t r i b u t i o n s to N - N a n d N - N s c a t t e r i n g a r e t h e s a m e a p a r t f r o m a f a c t o r G, t h i s b e i n g the G - p a r i t y of t h e e x c h a n g e d m e s o n s y s t e m . H e n c e a n y m e s o n e x c h a n g e m o d e l of n u c l e a r f o r c e s at o n c e i m p l i e s a m o d e l f o r t h i s p a r t of a n t i n u c l e a r f o r c e s . T h e N - N i n t e r a c t i o n a l s o i n c l u d e s two o t h e r k i n d s of p r o c e s s with no s i m p l e N - N c o u n t e r p a r t : (i) e x c h a n g e s of b a r y o n n u m b e r B =2, w h i c h h a v e v e r y s h o r t r a n g e a n d a r e u s u a l l y i g n o r e d ; (ii) i n t e r m e d i a t e a n n i h i l a t i o n p r o c e s s e s , w h i c h a r e u s u a l l y r e p r e s e n t e d by s o m e e m p i r i c a l a b s o r p t i o n . T h u s , i f w e r e g a r d t h e m e s o n e x c h a n g e s a s d e t e r m i n e d by N - N s t u d i e s , o n l y the a b s o r p t i o n r e m a i n s to b e f i x e d . B a l l a n d C h e w [6] m a d e t h e f i r s t N - N m o d e l a l o n g t h e s e l i n e s in 1957, u s i n g the b e s t m e s o n - e x c h a n g e p o t e n t i a l of t h e d a y and a b s o r p t i v e b o u n d a r y c o n d i t i o n . F o r l a t e r w o r k , s e e a r e c e n t r e v i e w [7]. Many m o r e e x p e r i m e n t s * P r e s e n t a d d r e s s : Physics Dept., Texas A & M University, College Station, Texas, USA. ** P r e s e n t a d d r e s s : Rutherford Laboratory, Chilton, Berkshire, U.K.
202
R.A. BRYAN and R.J.N. PHILLIPS
have now been made, however, and it s e e m s that the models p r o p o s e d hitherto do not fit all the data well. The p r e s e n t work is intended to bring the Ball-Chew p r o g r a m up to date, by using the latest OBE potentials; at the same time, we find that quantitative a g r e e m e n t with data can be achieved. The only adjustable p a r a m e t e r s a r e in the absorption, r e p r e s e n t e d by a Woods-Saxon potential with no spinor i s o s p i n - d e p e n d e n c e . The g e n e r a l p r o p e r t i e s of N-N s c a t t e r i n g a r e d e s c r i b e d in the l i t e r a t u r e (see ref. [7] and r e f e r e n c e s therein). Here we confine o u r s e l v e s to the OBE models in question, and to a c o m p a r i s o n with data.
2. E X P E R I M E N T A L DATA The p r e s e n t data a r e far f r o m a complete set, so t h e r e is no question of d e t e r m i n i n g any of the amplitudes d i r e c t l y f r o m experiment. However, the data a r e v a r i e d and a c c u r a t e enough to give stringent t e s t s of any fewp a r a m e t e r model. We consider the n o n - r e l a t i v i s t i c region, up to 300 MeV. The p-p data include total c r o s s sections, elastic c r o s s s e c t i o n s and inelastic c r o s s sections (divided into 0-, 2-, 4- and 6 - p r o n g events) [8-13]. T h e r e a r e angular distributions for elastic s c a t t e r i n g [8, 12, 14, 15] and c h a r g e exchange [16]. P o l a r i z a t i o n m e a s u r e m e n t s have only been made at higher e n e r g i e s [17-19]. Z e r o - p r o n g pp events c o n s i s t of c h a r g e - e x c h a n g e and nv ° annihilations. Most e x p e r i m e n t s do not distinguish between these p o s s i b i l i t i e s , and we must allow for this when c o m p a r i n g with t h e o r y * The p r o p o r t i o n of neut r a l annihilations is known [10, 20] to be 3.5% at r e s t (i.e. in S-waves), but h e r e they a r e inhibited b e c a u s e on]z the singlet S-state has the right chargeconjugation n u m b e r C =+1. A r e a s o n a b l e e s t i m a t e for neutral annihilations in flight is t h e r e f o r e about 7%. The t h e o r e t i c a l p r e d i c t i o n s in fig. 1 include not only total annihilation and total c h a r g e - e x c h a n g e c r o s s sections, but also e s t i m a t e s for visible annihilation (93% of total annihilation) and for z e r o - p r o n g modes ( c h a r g e - e x c h a n g e plus 7% of total annihilation). T h e r e is no p - n in flight data "'et, but the r a t i o of ~-p to ~-n annihilations at r e s t in d e u t e r i u m has be~:~l m e a s u r e d [21-23]. Using this, the r e l a tive capture r a t e s , f r o m singlet and triplet s t a t e s of ~ +p and ~ +n, have been e s t i m a t e d [24]. 3. STATIC BRYAN-SCOTT P O T E N T I A L B r y a n and Scott [1] c o n s i d e r e d f i r s t a static potential ** based on the exchange of ~, ~?, w and (~ m e s o n s , plus two s c a l a r m e s o n s ~o and ~1 with T =0 and T=I, r e s p e c t i v e l y . The latter may be r e g a r d e d as p a r a m e t e r i z i n g n o n - r e s o n a n t exchanges with the c o r r e s p o n d i n g quantum n u m b e r s . * Refs. [8, 16] do give pure charge-exchange cross sections, however. Their results lie systematically below other 'zero-prong' data, as we expect, and are distinguished in fig. 1 by an extra circle. ** This form contains errors: see ref. [5]. Fortunately, the numerical error is slight
~
--
,oo LAB
ENERGY
,~o
IN
~o
-L~_ L.~_,_~_,_,_,. ~,~_,,._,_,_,_,_-~
\
MEV
~o
, 300
-
t
)
350
Fig. 1. I n t e g r a t e d ~p c r o s s s e c t i o n data c o m p a r e d with the n o n - s t a t i c m o d e l . Total, v i s i b l e - a n n i h i l a t i o n , e l a s t i c and z e r o - p r o n g data a r e shown by open c i r c l e s , s o l i d c i r c l e s , t r i a n g l e s and c r o s s e s , r e s p e c t i v e l y . C h a r g e - e x c h a n g e (as d i s t i n c t f r o m z e r o prong) m e a s u r e m e n t s a r e d i s t i n g u i s h e d by a c i r c l e r o u n d the c r o s s . F o r r e f e r e n c e s , s e e the text. Solid c u r v e s , f r o m Lop to b o t tom, a r e the n o n - s t a t i c m o d e l p r e d i c t i o n s f o r total, total annihilation, e l a s t i c and c h a r g e e x c h a n g e c r o s s s e c t i o n s . B r o k e n c u r v e s a r e e s t i m a t e s of v i s i b l e - a n n i h i l a t i o n and z e r o - p r o n g c r o s s s e c t i o n s , m a d e by a s s i g n i n g 7% of all a n n i h i l a t i o n s to n e u t r a l modes.
U
0
u'l
ul
z 0
b9
0 H
I
204
R.A. BRYAN and R. J. N. PHILLIPS
For s c a l a r , i s o s c a l a r meson exchange, ~with coupling constant g defined by the effective Lagrangian density 23 =(4n)~g~/q5, the non-relativistic static N-N potential is p2 V(r) = _g2u(1 - 8 ~p_ M 2 / [ F ( / x r ) + 2 - ~ a ( p r ) L "S] .
(1)
Here /~ and M are the meson and nucleon m a s s e s , L is the orbital angular momentum and S =½(a 1 + a 2) is the total spin. The functions F and G are
F(x) = e x p ( - x ) / x , C(x)
(2)
1 ~d F(x)
= - x
•
(3)
For pseudoscalar, i s o s c a l a r meson exchange, with coupling constant defined by ./2 =(4~)½g~75~ , the N-N potential is V(r) = g2
#3 12M2 [ F ( ~ r ) a 1 .a 2 + H(l~r) S12 ] ,
(4)
where
g(x) = (1 + 3 / x + 3 / x 2) F(x) , S12 = 3 a 1 . r e 2 . r / r 2 - a 1 . a 2 .
(5)
(6)
For vector, i s o s c a l a r meson exchange, there are two coupling constants f and g, defined by the Lagrangian density * A2 = (47r)½ ~ [g~/#~b/~ + 4f~ atlv(~vdPll - apqbv) ] ~V .
(7)
In t e r m s of these, the static potential of ref. [1] has the form
V(r) =g2t~{R1F(l~r)+R2[2F(t~r)al " a 2 - H ( # r ) S 1 2 ] - R 3 G(I~r)L "S} ; 1 ~2 1 ~2 R 1 =[1 + ~ - ~ + - - - - f ] 2 4 M2g
1 /~2 [1
R2 =_~2__~
1 ~2
R3 = 2 - ~
'
+f/g]2
[3 + 4 f / g + 3(l~/2M)2(f/g) 2] .
* In ref. [1], the definition of f differs by a factor 2M/~.
(8) (9)
(10)
(11)
N-N POTENTIAL
205
Table 1 Parameters for the Bryan-ScottN-N potentials. Parameter
Static case
Non-static case
g2(~)
11.7
12.66
g2(u)
7.0
3.0
g2 (p)
0.68
2.44
f/g(p) g2 (w)
4.4 21.5
1.13 23.7
g2 ((to)
9.4
9.46
m((~o) g2(gl)
560 6.1
550 1.97
m (0"1)
770
600
For ¢o f / g is assumed zero; rn denotes mass in MeV/c 2.
T h e s e p o t e n t i a l s w e r e m e a n t to be c o r r e c t t h r o u g h o r d e r ( p / M ) 2 only. F o r i s o v e c t o r m e s o n e x c h a n g e s , add the i s o s p i n f a c t o r r 1 "32. B r y a n and Scott [1] a d j u s t e d the c o u p l i n g c o n s t a n t s , and the s c a l a r m e s o n m a s s e s , to fit N - N p h a s e s h i f t s ; the r e s u l t i n g v a l u e s a r e given in table 1. T h e y a l s o put the whole p o t e n t i a l equal to z e r o f o r r < 0.6 fro. F o r the N - N i n t e r a c t i o n , we r e v e r s e d the s i g n s of the p o t e n t i a l s f o r the o d d - G m e s o n s ~, o) and (h" We r e t a i n e d the z e r o c u t - o f f on the m e s o n p o t e n t i a l at r = 0.6 fm. We added an e m p i r i c a l a b s o r p t i v e p o t e n t i a l and a d j u s t e d it to give the b e s t fit to data, a s d e s c r i b e d in s e c t i o n 5.
4. NON-
STATIC
BRYAN-
SCOTT
POTENTIAL
B r y a n and Scott [5] l a t e r i n c l u d e d v e l o c i t y - d e p e n d e n t c e n t r a l t e r m s , of the f o r m [V2U(r) + U(r)V 2 ] / ( 2 M 2 ) , in the N - N potential. At the s a m e t i m e , t h e y slightly a l t e r e d s o m e of the s t a t i c p o t e n t i a l s V(r). F o r s c a l a r i s o s c a l a r m e s o n e x c h a n g e , k e e p i n g the s a m e n o t a t i o n as in s e c t i o n 3, V and U a r e g i v e n by
V(r) = _g2~{(i - 4~2) F(/Ir) U(r) : -g2u F(t~r) .
t~2 +-~-~G(ttr) L.S},
(12) (13)
F o r p s e u d o s c a l a r m e s o n e x c h a n g e s , V(r) is u n c h a n g e d and U(r) = O. F o r v e c t o r , i s o s c a l a r m e s o n e x c h a n g e , the c o e f f i c i e n t s R 1 and R 3 a r e c h a n g e d , and a v e l o c i t y - d e p e n d e n t t e r m i s added:
206
R.A. BRYAN
and R. J. N. PHILLIPS
~2 R 1 = 1 +~-~
n3
(1 + f / g )
,
~2
~-~ (3 ÷4//g),
U(r) = _ g 2 p F ( # r )
.
(14)
(15)
(16)
The v e l o c i t y - d e p e n d e n t p o t e n t i a l s can be i n t e g r a t e d by a d a p t i n g s t a n d a r d m e t h o d s [25]. In effect, they r e n o r m a l i z e the s t a t i c t e r m s and r e d u c e the s i n g u l a r i t y of OBE p o t e n t i a l s at the o r i g i n f r o m r -3 to r -2. The r e m a i n i n g s i n g u l a r i t y can be r e d u c e d a s f o l l o w s . F r o m each m e s o n - e x c h a n g e p o t e n t i a l i s s u b t r a c t e d the p o t e n t i a l for a heavy m e s o n , of m a s s A, with the s a m e quantum n u m b e r s and coupling c o n s t a n t ; this i s e s s e n t i a l l y a P a u l i - V i l l a r s regulator. B r y a n and Scott [5] m a d e two s e p a r a t e s t u d i e s with v e l o c i t y - d e p e n d e n t N - N p o t e n t i a l s . F i r s t , e x p l i c i t l y e x c l u d i n g S - w a v e s , they found that the c e n t r i f u g a l b a r r i e r in h i g h e r p a r t i a l w a v e s w a s s t r o n g enough to o v e r c o m e the 1 / r 2 s i n g u l a r i t y , and no c u t - o f f was n e e d e d . Secondly, they i n c l u d e d S - w a v e s ; a c u t - o f f was then n e e d e d . The o p t i m u m p a r a m e t e r s for the l a t t e r i n v e s t i g a t i o n a r e shown in t a b l e 1. F o r the N - N i n t e r a c t i o n , we r e v e r s e d the sign of o d d - G m e s o n p o t e n t i a l s and added an e m p i r i c a l a b s o r p t i o n ; s e e s e c t i o n 5. The f i r s t n o n - s t a t i c p o t e n t i a l led to a s o m e w h a t u n s a t i s f a c t o r y fit to NN data, at l e a s t with our p a r t i c u l a r a s s u m p t i o n s ; we s h a l l not d i s c u s s this c a s e h e r e . The s e c o n d n o n - s t a t i c p o t e n t i a l (including S - w a v e c r i t e r i a ) gave a good fit to d a t a , howe v e r . T h i s c a s e i s d i s c u s s e d and i l l u s t r a t e d below. In the N - N c a s e , A = 1.5 GeV i s given as the c u t - o f f m a s s , though 1.0 GeV i s a l s o a c c e p t a b l e [5]. F o r the N - N c a s e , we u s e d the l a t t e r value only.
5. F I T T I N G THE DATA To the r e a l OBE p o t e n t i a l s we a d d e d an i m a g i n a r y W o o d s - S a x o n p o t e n t i a l i W , i n d e p e n d e n t of spin and i s o s p i n : W ( r ) : - W o / [ 1 + a exp(br)] .
(17)
The e x t r a f r e e d o m f r o m v a r y i n g a g a v e l i t t l e a d v a n t a g e , so we fixed a = 1. The r e m a i n i n g p a r a m e t e r s Wo and b w e r e then a d j u s t e d to fit the data. S p e c i f i c a l l y , we c h o s e t h e m to m a t c h the t o t a l and e l a s t i c c r o s s s e c t i o n s aT(PP) and Crel(~p) at 100 MeV. The p o t e n t i a l so d e t e r m i n e d p r o v e d to fit a l s o the e n e r g y - d e p e n d e n c e of c r o s s s e c t i o n s , and a n g u l a r d i s t r i b u t i o n s , and c h a r g e - e x c h a n g e . P a r a m e t e r s for the s t a t i c OBE c a s e a r e a = 1, b = 6 fm -1 and Wo = 62 GeV. F o r the n o n - s t a t i c O B E c a s e , a = 1 , b =5 fm -1 and Wo = 8 . 3 GeV.
N-N POTENTIAL
207
C o m p a r i s o n s with d a t a f o r the s t a t i c c a s e have a l r e a d y b e e n shown i n r e f . [7]. F o r the n o n - s t a t i c c a s e , we i l l u s t r a t e the q u a l i t y of the d a t a f i t t i n g i n f i g s . 1-6 and 11. As f o r t e c h n i c a l d e t a i l s , the S c h r 6 d i n g e r e q u a t i o n w a s i n t e g r a t e d n u m e r i c a l l y for u n c o u p l e d p a r t i a l w a v e s with L --<4 a n d c o u p l e d w a v e s with J < 5, f o r e a c h i s o s p i n I = 0 and I = 1. S i n g l e - p i o n e x c h a n g e a m p l i t u d e s w e r e a d d e d f o r the h i g h e r p a r t i a l w a v e s , but m a d e no p e r c e p t i b l e d i f f e r e n c e e x c e p t for
20
o LD F¢nl5
Z
g d 'v
OL 1.0
[
I
0.5
0
(co=O)
IqP -0.5
-I.0
c,m,
Fig. 2. ~p differential cross section at 62.7 MeV [14], compared with the non-static model. The solid line is the p + p ~ ~+p prediction. The broken line is the fi+n -* fi+n prediction, differing by the Coulomb effects.
208
R.A.
BRYAN
and R. J. N. PHILLIPS
25
,4 2 0 LU I-U9
z
15
E u
b
5
O I
i.O
r 0.5
r 0
-0.5
(C o s e) c . m .
F i g . 3. ~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 at 99.8 MeV [14] c o m p a r e d with the n o n - s t a t i c m o d e l .
ol.O
N-N POTENTIAL
209
25
i
A2C
tu u~
m
z
15
d
b
C 1.0
0.5
0
-0.5
( c o s O) ¢.m.
F i g . 4. ~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 at 136.8 MeV [14] c o m p a r e d with the n o n - s t a t i c m o d e l .
-I .o
210
R.A. BRYAN
and R.J.N.
PHILLIPS
30
25
,,=, 2 0 f-U)
~E Z
f5
"o ,.... 0
I0
C
1.0
0.5
0 (co-, e) ¢.m.
-0.5
Fig. 5. ~p differential cross section at 175.0 MeV compared with the non-static model.
-I.0
[14]
N-N POTENTIAL
211 4"0
4"0 5 0 - II S 93
McV
McV
115- 180
DATA
149
THEORY
McV
McV
DATA
THEORY
]
3-C
3"0
uo E 2"0
uE 2 - 0 I¢:
i=
~_)
~D
I'O
I.O
1
]
O t I'0
0 COS
-I-0
I'0
O cm"
-I.O
0 COS
Ocm
Fig. 6. ~ + p ~ ~ + n a n g u l a r d i s t r i b u t i o n s , at a v e r a g e e n e r g i e s 93 a n d 149 MeV [16], c o m p a r e d with the n o n - s t a t i c m o d e l .
+p ~ fi + n. I s o s p i n a m p l i t u d e s A (I) w e r e c o m b i n e d in the u s u a l way: A ( ~ n ~ ~ n ) = A (1) , A ( p p - - * ~ p ) : -12,4(0) + ½A(1) ,
(18)
A(pp--* fin) = ½A(0) - ½A(1) . C h a r g e i n d e p e n d e n c e w a s t h u s a s s u m e d in d e t e r m i n i n g n u c l e a r p a r t i a l w a v e a m p l i t u d e s , but the n - p m a s s d i f f e r e n c e w a s i n c l u d e d in the p h a s e s p a c e f a c t o r f o r ~ + p --* fi + n. Coulomb effects were treated as follows. Integrations were made with no C o u l o m b p o t e n t i a l , to get p u r e n u c l e a r p h a s e s h i f t s ; ~ - p s c a t t e r i n g w a s p r e d i c t e d by a d d i n g C o u l o m b and n u c l e a r p h a s e s h i f t s , w h i c h a m o u n t s to a d d i n g the C o u l o m b a m p l i t u d e to a m o d i f i e d n u c l e a r a m p l i t u d e . S y m b o l i c a l l y , for e a c h p a r t i a l w a v e , exp(2iS+2i~)-
1 = e x p ( 2 i ~ ) - 1 + e x p ( 2 i c r ) [ e x p ( 2 i 6 ) - 1] ,
(19)
212
R.A. BRYAN and R. J. N. PHILLIPS
w h e r e 5 and a a r e the n u c l e a r and Coulomb p h a s e shifts. We calculated OT(pp) f r o m the optical t h e o r e m , u s i n g the p u r e n u c l e a r amplitude. We defined ael(pp) by the p u r e n u c l e a r c r o s s section; thus it equals ~el(fin). Coul o m b e f f e c t s e n t e r ~ +p - fi + n through wave function d i s t o r t i o n in the initial state. T h i s effect w a s c a l c u l a t e d by adding p h a s e f a c t o r s exp(ia) to each p a r t i a l wave, and p r o v e d to be negligible.
6. DISCUSSION The fit to data s e e m s s a t i s f a c t o r y . F u r t h e r questions now a r i s e . What is the significance of this fit? How m u c h is due to the OBE p o t e n t i a l s and how m u c h to the a b s o r p t i o n ? What f u r t h e r p r e d i c t i o n s can be m a d e ? We d i s c u s s t h e s e and other questions below, in s e p a r a t e s u b s e c t i o n s (see a l s o the d i s c u s s i o n in r e f . [7]).
6.1. Role of W(r) The a b s o r p t i v e potential s e e m s to dominate e l a s t i c s c a t t e r i n g . One can fit aT, ael and da/d~2 f o r ~p quite well with p u r e a b s o r p t i o n [7, 14]. Now, W(r) is v e r y s t r o n g at s m a l l r and g r e a t l y a t t e n u a t e s the wave function h e r e . Thus it s u p p r e s s e s s h o r t - r a n g e i n t e r a c t i o n s , and in this r e s p e c t a c t s s o m e w h a t like the r e p u l s i v e c o r e in o r d i n a r y n u c l e a r f o r c e s . In the N - N p r o b l e m , the m e d i u m - r a n g e p o t e n t i a l s m u s t be c a r e f u l l y adjusted to get the c o r r e c t S-wave p h a s e shifts. In the N - N c a s e , S - w a v e s a r e a l m o s t c o m p l e t e l y a b s o r b e d anyway, so the m e d i u m - r a n g e d e t a i l s m a t t e r l e s s . The a b s o r p t i o n g i v e s negative r e a l p a r t s to low p a r t i a l a m p l i t u d e s , hence a negative r e a l p a r t to the s p i n - a v e r a g e d f o r w a r d s c a t t e r i n g amplitude, and hence d e s t r u c t i v e Coulomb i n t e r f e r e n c e in ~ - p s c a t t e r i n g (see below). We a s s u m e W is independent of spin and isospin, b e c a u s e it r e p r e s e n t s the a v e r a g e effect of v e r y m a n y annihilation channels. Thus W alone does not give any p o l a r i z a t i o n , or c h a r g e - e x c h a n g e , or d i f f e r e n c e s between ~ - p and ~ - n s c a t t e r i n g . T h e s e a r e the p l a c e s to look for OBE effects. 6.2. Role of OBE terms T h e s e give all the spin and i s o s p i n dependence. The r e l e v a n t data so f a r a r e ~+p--* fi+n c r o s s s e c t i o n s (figs. 1 and 6); both m a g n i t u d e s and a n g u l a r d i s t r i b u t i o n s a r e fitted s a t i s f a c t o r i l y . It s e e m s that c h a r g e - e x c h a n g e is m a i n l y due to the o n e - p i o n exchange p o t e n t i a l [7]. T h e r e is s o m e i n f o r m a t i o n on r e l a t i v e s t r e n g t h s of p - p and ~ - n a b s o r p tions at r e s t (see below), that is c o n s i s t e n t with <~ur m o d e l s . The OBE p o t e n t i a l s a l s o play a p a r t by drawing in the outer wave function to enhance a b s o r p t i o n in m a n y s t a t e s . Without the OBE t e r m s , W(r) h a s to be w e a k e r and of s u b s t a n t i a l l y longer r a n g e [7]. Since the r a n g e of W(r) is not known a priori, this a s p e c t is h a r d to study, but it a f f e c t s the slope of the e l a s t i c d i f f r a c t i o n p e a k [7]. F u t u r e e x p e r i m e n t s on p o l a r i z a t i o n and ~ - n s c a t t e r i n g will t e s t the OBE terms further.
N-N
POTENTIAL
213
6.3. Measuring the pion coupling It w o u l d b e i n t e r e s t i n g to u s e d a / d ~ l ( ~ p ~ f i n ) to d e t e r m i n e t h e y N c o u p l i n g c o n s t a n t g 2 , by e x t r a p o l a t i n g to t h e p o l e a s in t h e p + n --. n + p c a s e [26]. O n e - p i o n e x c h a n g e i s t h e s a m e in b o t h c a s e s . T h o u g h e x p e r i m e n t a l e r r o r s a r e m u c h b i g g e r f o r YtN, t h e b a c k g r o u n d c o n t r i b u t i o n s f r o m s h o r t e r - r a n g e t e r m s a r e m u c h l e s s . F o r i n s t a n c e , f o r w a r d c h a r g e e x c h a n g e at 150 MeV i s a b o u t 3 m b / s r f o r ~ + p a n d 11 m b / s r f o r p +n. P r e s e n t d a t a w o u l d p e r m i t o n l y a c r u d e d e t e r m i n a t i o n o f g 2, h o w e v e r . I n c i d e n t a l l y , the s o m e w h a t low v a l u e s of g 2 ( ~ ) in t a b l e 1 a r e p a r t l y due to t h e a b s e n c e of a r e l a t i v i s t i c f a c t o r M/E in the p o t e n t i a l eq. (4). 6.4. Scattering lengths T h e z e r o - e n e r g y S - w a v e s c a t t e r i n g l e n g t h s a(I,S ) f o r i s o s p i n I a n d s p i n S a r e s h o w n in t a b l e 2. T h e c o n v e n t i o n u s e d i s a = tg 5 / k , w h e r e k i s t h e r e l a tive momentum. Table 3 Relative ~ capture rates.
Table 2 S-wave s c a t t e r i n g lengths in fm.
Experimental Static case
Non-static case
Static
Non-static I
s
II
a(1,1)
-0.77 + 0.68i
-0.82 + 0.87i
Yn
0.37
0.38
0.19
0.60
a(1,0)
- 1 . 0 1 + 0.51i
- 1 . 0 0 + 0.60i
~
0.62
0.61
0.73
0.73
a(0,1)
- 1 . 4 4 + 0.76i
-1.01 + 0.65i
t Yn
0.50
0.55
0.63
0
a(0,0)
-0.35 + 1.18i
-0.62 + 1.33i
~
0.53
0.48
0.65
0
T h e a n n i h i l a t i o n r a t e at v e r y low e n e r g y i s p r o p o r t i o n a l to I m a . M e a s u r e m e n t s of ~ a n n i h i l a t i o n at r e s t in h y d r o g e n a n d d e u t e r i u m p e r m i t e s t i m a t e s of t h e r e l a t i v e r a t e s f o r c a p t u r e on n e u t r o n s a n d p r o t o n s , in s i n g l e t a n d t r i p l e t s t a t e s , if s o m e f u r t h e r a s s u m p t i o n s a r e m a d e [24]. In t a b l e 3 w e s h o w two s u c h e s t i m a t e s f r o m r e f . [24], b a s e d on two d i f f e r e n t e x t r e m e a s s u m p t i o n s , l a b e l l e d I and II; 7ps d e n o t e s t h e r e l a t i v e c a p t u r e r a t e on p r o t o n s in s i n g l e t s t a t e s , e t c . , and w e u s e t h e a r b i t r a r y n o r m a l i z a t i o n 3 7 ~ + 3~ns + 7 t + 7tn = 4. P r e d i c t i o n s f r o m t h e s t a t i c and n o n - s t a t i c m o d e l s a r e a l s o shown; t h e y s e e m c o n s i s t e n t w i t h t h e e x p e r i m e n t a l p o s s i b i l i t i e s . In p a r t i c u l a r , t h e s i n g l e t c a p t u r e r a t e i s g r e a t e r on p r o t o n s t h a n on n e u t r o n s , while the triplet rates are about equal. 6.5. ~ - n scattering O u r m o d e l s p r e d i c t ~ + n --* ~ + n s c a t t e r i n g . F i g . 7 s h o w s total~ e l a s t i c a n d a n n i h i l a t i o n c r o s s s e c t i o n s , c o m p a r e d w i t h p + p --, p + p p r e d i c t i o n s . In e a c h c a s e , t h e p + p v a l u e s a r e b i g g e r . T h i s e f f e c t , due to t h e O B E p o t e n t i a l s , i s not d o m i n a t e d by o n e - p i o n e x c h a n g e a l o n e [7]. F i g . 8 c o m p a r e s ~ - p a n d ~ - n 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 at 100 MeV. T h e b i g g e r ~ - p c r o s s s e c t i o n g o e s w i t h a s t e e p e r f o r w a r d p e a k , a s we e x p e c t f r o m t h e b l a c k d i s c a n a l o g y . See a l s o s e c t i o n 6.6.
214
R.A. BRYAN and R. J. N. PHILLIPS
I.,,\\~. \\\\~
200
I>,,
z u) z 0 FU
NNIHILATION(~
¢n
0
I00
U Z IZ
ELASTIC
"""
-.,,....
I
I I00 LAB.
I ENERGY
I IN
I
200 McV
300
Fig. 7. Comparison of ~p and On integrated cross section predictions, for the non-static model.
6.6. P o l a r i z a t i o n P r e d i c t i o n s a r e shown in figs. 10 and 11, for ~ - p and ~-n. P o l a r i z a t i o n c o m e s f r o m i n t e r f e r e n c e between the non-flip amplitude (dominated by absorption) and a c e r t a i n spin-flip amplitude that is dominated by the one-pion exchange potential (OPEP) [7]. Note however that O P E P gives no p o l a r i z a tion in Born approximation, nor in damped Born a p p r o x i m a t i o n with L - d e pendent damping; the effect is e s s e n t i a l l y n o n - l i n e a r . This is why p r e d i c t e d p o l a r i z a t i o n does not s i m p l y change sign between ~-p and ~ - n c a s e s , although O P E P does.
N-N POTENTIAL
215
25
~p
20
d Iz uJ lu~
i
5
\
-u
b
0 1.0
I 0.5.
~ 0 ¢o$0
~
: -0.5
-I.0
c.m.
Fig. 8. Comparison of ~p and ~n differential cross section predictions at 100 MeV, for the non-static model.
T h e r e a r e no m e a s u r e m e n t s y e t b e l o w 600 MeV, but the ~ - p p r e d i c t i o n s at f o r w a r d a n g l e s h a v e the s a m e s i g n a n d o r d e r of m a g n i t u d e a s a r e f o u n d at h i g h e r e n e r g i e s .
216
R.A. BRYAN and R. J. N. PHILLIPS
0.4
I00 0.2 Z 0 I,u~
:
o
_1 0 o.
-0.2
-0
.4
1.0
I
I
0.5
0 co$0
I -0.5
-I-O
¢.m.
Fig. 9. ~p polarization predictions at 100, 200 and 300 MeV, for the non-static model.
6.7. Coulomb interference Our models p r e d i c t d e s t r u c t i v e Coulomb i n t e r f e r e n c e in d(x/dft(~p). At m o d e r a t e e n e r g i e s this is quite a s m a l l effect (see fig. 2 at 63 MeV) and at higher e n e r g i e s it is s m a l l e r still. But at the lower e n e r g i e s it is a big effect, r e a c h i n g out to wide angles, a s i l l u s t r a t e d at 15 MeV in fig. 11. This r a i s e s a question about c o m p a r i n g t h e o r y with experiment. We define the i n t e g r a t e d c r o s s section ael(pp) by r e m o v i n g all Coulomb effects. In p r a c t i c e , however, an e x p e r i m e n t e r cannot distinguish the Coulomb effects except at s m a l l angles, and cannot r e m o v e it fully. We conclude that it is not useful to c o m p a r e ael(pp) at low e n e r g i e s ; one should just c o m p a r e d(~/d~(~p). 6.8. Saw-tooth effect In the original Ball-Chew model [6], the onset of absorption in each
N-N POTENTIAL
217
0.4
0.2
300
0
-0
-2
-0.4
i-O
] 0-5
I 0
-0.5
I
-I.0
cos 0 c.m.
Fig. 10. ~n polarization predictions at 100, 200 and 300 MeV, for the non-static model.
partial wave was sudden, giving cross sections a sawtoothed look as funct i o n s of e n e r g y . T h i s c a m e f r o m a p h y s i c a l a s s u m p t i o n a b o u t how an a b sorptive core would act. In o u r m o d e l s , r e p r e s e n t i n g t h e a b s o r p t i o n by a p o t e n t i a l l e a d s to a s m o o t h v a r i a t i o n w i t h e n e r g y . T h e r e i s no s a w - t o o t h e f f e c t w h a t e v e r ; not even smooth bumps are given. 6.9. Static and n o n - s t a t i c c o m p a r i s o n T h e two p o t e n t i a l m o d e l s g i v e r a t h e r s i m i l a r p r e d i c t i o n s . ( F o r f i g u r e s i l l u s t r a t i n g t h e s t a t i c c a s e , s e e r e f . [7].) T h e b i g g e s t d i f f e r e n c e s e e m s to b e in a e l ( P p) b e l o w 50 MeV; at 40 MeV, f o r i n s t a n c e , t h e s t a t i c a n d n o n s t a t i c m o d e l s g i v e 87 m b a n d 81 m b , r e s p e c t i v e l y . This approximate agreement is not surprising, since the OBE potentials a r e r a t h e r s i m i l a r a t m e d i u m a n d l o n g r a n g e . T h e b i g d i f f e r e n c e i s at s h o r t r a n g e , w h e r e in t h e s t a t i c c a s e t h e O B E t e r m s a r e c u t - o f f to z e r o w h i l e in
218
R.A. BRYAN and R. J. N. PHILLIPS
15
.a I O E _= E ,J
I
b
O 1.0
i 0.5
I 0
i -0.5
i -I 0
COS # ¢..m•
Fig. 11. Illustration of Coulomb interference in dff/dfZ(Dp) at 15 MeV. The solid and broken curves are predictions of the non-static model, with and without Coulomb effects. The data are p r e l i m i n a r y [15].
the n o n - s t a t i c c a s e they a r e d a m p e d but r e m a i n s t r o n g . S h o r t r a n g e d i f f e r e n c e s t e n d to be s u p p r e s s e d by the a b s o r p t i o n , h o w e v e r . T h e two a b s o r p t i v e p o t e n t i a l s s e e m r a t h e r d i s s i m i l a r i n s t r e n g t h , but t h i s i s d e c e p t i v e . W h e n W(r) i s v e r y s t r o n g and of s h o r t r a n g e , q u i t e l a r g e c h a n g e s i n s t r e n g t h h a v e l i t t l e effect on the s c a t t e r i n g . An a p p r o x i m a t e fit to d a t a would be a c h i e v e d i n e i t h e r c a s e with p a r a m e t e r s a = 1, b = 5 f m - 1 a n d Wo = 20 GeV: the c h a n g e s f r o m t h e s e v a l u e s to the b e s t fit i n e i t h e r case represent relatively small adjustments• We a r e g r a t e f u l to R. B i z z a r r i , R. G o l d b e r g a n d B. L. Scott for e a r l y i n f o r m a t i o n a b o u t e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s .
REFERENCES [I] R. A. Bryan and B. L. Scott, Phys. Rev. 135 (1964) B434.
[2] S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Progr. Theor. Phys. 32 (1964) 380.
N-N POTENTIAL
219
[3] A. Scotti and D. Y. Wong, Phys. Rev. 138 (1965) B145; J. S. Ball, A. Seotti and D. Y. Wong, Phys. Rev. 142 (1966) 1000. [4] R. A. Arndt, R.A. Bryan and M. H. MacGregor, Phys. L e t t e r s 21 (1966) 314; R.A. Bryan and R . A . A r n d t , Phys. Rev. 150 (1966) 1299. [5] R.A. Bryan and B. L. Scott, L i v e r m o r e preprint UCRL-70409 and California State College, Long Beach, preprint. [6] J. S. Ball and G. F. Chew, Phys. Rev. 109 (1958) 1385. [7] R . J . N . Phillips, Rev. Mod. Phys. 39 (1967) 681. [8] C. A. Coombes, B. Cork, W. Galbraith, G.R. Lambertson and W. A. Wenzel, Phys. Rev. 112 (1958) 1303. [9] B. Cork, O. I. Dahl, D.H. Miller, A. G. Tenner and C. L. Wang, Nuovo Cimento 25 (1962) 497. [10] J. Loken and M. Derrick, Phys. L e t t e r s 3 (1963) 334. [11] U. Amaldi, T. Fazzini, G. Fidecaro, C. Ghesqui~re, M. Legros and H. Steiner, Nuovo Cimento 34 (1964) 825. [12] A. Hossain and M. Shaukat, Nuovo Cimento 38 {1965) 737. [13] U. Amaldi, B. Conforto, G. F i d e e a r o , H. Steiner, G. Baroni, R. B i z z a r r i , P. Guidoni, V. Rossi, G. Brautti, E. Castelli, M. Ceschia, L. Chersovani and M. Sessa, Nuovo Cimento 46 (1966) 171. [14] B. Conforto, G. Fidecaro, H. Steiner, R. B i z z a r r i , P. Guidoni, F. Marcelja, G. Brautti, E. Castelli, M. Cesehia and M. Sessa, Antiproton-proton elastic scattering between 63 and 175 MeV, to be published. [15] R. Goldberg, private communication. [16] R. B i z z a r r i , E. Castelli, M. Ceschia, B. Conforto, G. C. Gialanella, P. Guidoni, F. M a r c e l j a and M. Sessa, Nora Interna no. 136, Rome {1967). [17] J. Button and B. Maglic, Phys. Rev. 127 (1962) 1297. [18] L. Dobrzynski, C. Ghesqui~re, N.H. Xuong and H. Torte, Phys. L e t t e r s 23 {1966) 614. [19] H. Sens, private communication. [20] C. Baltay, P. Franzini, G. Liitjens, J. Severiens, D. Tycko and D. Zanello, Phys. Rev. 145 (1966) 1103. [21] V. Barnes, K.W. Lai, P. Anninos, L. Gray, P. Haggerty, E. Harth, T. Kalogeropoulos, S. Zenone, R. B i z z a r r i , U. Dore, G. Gialanella, G. Monetti and P. Guidoni, Proc. of 1964 Dubna Conf., Vol. I, p. 731. [22] W. Chinowsky and G. Kojoian, Nuovo Cimento 43 (1966) 684. [23] A. Bettini, M. Cresti, S. Limentani, L. Peruzzo, R. Santangelo, S. Sartori, L. Bertanza, A. Bigi, R. C a r r a r a , R. Casali and P. L a r i c c i a , Nuovo Cimenfo 47A (1967) 642. [24] R. B i z z a r r i , CERN preprint (1967). [25] A. M. Green, Nucl. Phys. 33 (1962) 218. [26] P. Cziffra and M . J . Moravcsik, Phys. Rev. 116 (1959) 226; A. Ashmore, W.H. Range, R. T. Taylor, B. M. Townes, L. Castillejo and R. F. P e i e r l s , Nucl. Phys. 36 {1962) 258 and r e f e r e n c e s therein.
NuClear Physics B5 (1968) 201-219. North-Holland Publ. Comp., A m s t e r d a m
NUCLEON-ANTINUCLEON FROM
SINGLE-MESON
POTENTIAL EXCHANGES
R. A. B R Y A N * Lawrence Radiation Laboratory, Livermore, California, USA and R. J . N. P H I L L I P S ** A . E . R . E . , Harwell, U.K. Received 15 F e b r u a r y 1968
A b s t r a c t : The static and non-static Bryan-Scott potentials, which a r e based on oneboson-exchange and fit nucleon-nucleon scattering, are adapted to the nucleonantinucleon system. T e r m s corresponding to odd-G exchanges a r e r e v e r s e d in sign, and an e m p i r i c a l absorptive potential is added. Satisfactory fits to nucleonantinucleon data a r e achieved, in the 0-300 MeV range.
1. ~ T R O D U C T I O N M o d e l s b a s e d on o n e - b o s o n - e x c h a n g e (OBE) a p p r o x i m a t i o n s h a v e b e e n v e r y s u c c e s s f u l in t h e N - N p r o b l e m [ 1 - 5 ] . T h i s p a p e r d e s c r i b e s how two OBE potential models - the static and non-static Bryan-Scott potentials [ 1 , 5 ] - c a n b e a d a p t e d to d e s c r i b e n o n - r e l a t i v i s t i c N - N i n t e r a c t i o n s , and s u c c e s s f u l l y fit a v a i l a b l e d a t a . It w a s n o t i c e d l o n g a g o t h a t m e s o n e x c h a n g e c o n t r i b u t i o n s to N - N a n d N - N s c a t t e r i n g a r e t h e s a m e a p a r t f r o m a f a c t o r G, t h i s b e i n g the G - p a r i t y of t h e e x c h a n g e d m e s o n s y s t e m . H e n c e a n y m e s o n e x c h a n g e m o d e l of n u c l e a r f o r c e s at o n c e i m p l i e s a m o d e l f o r t h i s p a r t of a n t i n u c l e a r f o r c e s . T h e N - N i n t e r a c t i o n a l s o i n c l u d e s two o t h e r k i n d s of p r o c e s s with no s i m p l e N - N c o u n t e r p a r t : (i) e x c h a n g e s of b a r y o n n u m b e r B =2, w h i c h h a v e v e r y s h o r t r a n g e a n d a r e u s u a l l y i g n o r e d ; (ii) i n t e r m e d i a t e a n n i h i l a t i o n p r o c e s s e s , w h i c h a r e u s u a l l y r e p r e s e n t e d by s o m e e m p i r i c a l a b s o r p t i o n . T h u s , i f w e r e g a r d t h e m e s o n e x c h a n g e s a s d e t e r m i n e d by N - N s t u d i e s , o n l y the a b s o r p t i o n r e m a i n s to b e f i x e d . B a l l a n d C h e w [6] m a d e t h e f i r s t N - N m o d e l a l o n g t h e s e l i n e s in 1957, u s i n g the b e s t m e s o n - e x c h a n g e p o t e n t i a l of t h e d a y and a b s o r p t i v e b o u n d a r y c o n d i t i o n . F o r l a t e r w o r k , s e e a r e c e n t r e v i e w [7]. Many m o r e e x p e r i m e n t s * P r e s e n t a d d r e s s : Physics Dept., Texas A & M University, College Station, Texas, USA. ** P r e s e n t a d d r e s s : Rutherford Laboratory, Chilton, Berkshire, U.K.
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have now been made, however, and it s e e m s that the models p r o p o s e d hitherto do not fit all the data well. The p r e s e n t work is intended to bring the Ball-Chew p r o g r a m up to date, by using the latest OBE potentials; at the same time, we find that quantitative a g r e e m e n t with data can be achieved. The only adjustable p a r a m e t e r s a r e in the absorption, r e p r e s e n t e d by a Woods-Saxon potential with no spinor i s o s p i n - d e p e n d e n c e . The g e n e r a l p r o p e r t i e s of N-N s c a t t e r i n g a r e d e s c r i b e d in the l i t e r a t u r e (see ref. [7] and r e f e r e n c e s therein). Here we confine o u r s e l v e s to the OBE models in question, and to a c o m p a r i s o n with data.
2. E X P E R I M E N T A L DATA The p r e s e n t data a r e far f r o m a complete set, so t h e r e is no question of d e t e r m i n i n g any of the amplitudes d i r e c t l y f r o m experiment. However, the data a r e v a r i e d and a c c u r a t e enough to give stringent t e s t s of any fewp a r a m e t e r model. We consider the n o n - r e l a t i v i s t i c region, up to 300 MeV. The p-p data include total c r o s s sections, elastic c r o s s s e c t i o n s and inelastic c r o s s sections (divided into 0-, 2-, 4- and 6 - p r o n g events) [8-13]. T h e r e a r e angular distributions for elastic s c a t t e r i n g [8, 12, 14, 15] and c h a r g e exchange [16]. P o l a r i z a t i o n m e a s u r e m e n t s have only been made at higher e n e r g i e s [17-19]. Z e r o - p r o n g pp events c o n s i s t of c h a r g e - e x c h a n g e and nv ° annihilations. Most e x p e r i m e n t s do not distinguish between these p o s s i b i l i t i e s , and we must allow for this when c o m p a r i n g with t h e o r y * The p r o p o r t i o n of neut r a l annihilations is known [10, 20] to be 3.5% at r e s t (i.e. in S-waves), but h e r e they a r e inhibited b e c a u s e on]z the singlet S-state has the right chargeconjugation n u m b e r C =+1. A r e a s o n a b l e e s t i m a t e for neutral annihilations in flight is t h e r e f o r e about 7%. The t h e o r e t i c a l p r e d i c t i o n s in fig. 1 include not only total annihilation and total c h a r g e - e x c h a n g e c r o s s sections, but also e s t i m a t e s for visible annihilation (93% of total annihilation) and for z e r o - p r o n g modes ( c h a r g e - e x c h a n g e plus 7% of total annihilation). T h e r e is no p - n in flight data "'et, but the r a t i o of ~-p to ~-n annihilations at r e s t in d e u t e r i u m has be~:~l m e a s u r e d [21-23]. Using this, the r e l a tive capture r a t e s , f r o m singlet and triplet s t a t e s of ~ +p and ~ +n, have been e s t i m a t e d [24]. 3. STATIC BRYAN-SCOTT P O T E N T I A L B r y a n and Scott [1] c o n s i d e r e d f i r s t a static potential ** based on the exchange of ~, ~?, w and (~ m e s o n s , plus two s c a l a r m e s o n s ~o and ~1 with T =0 and T=I, r e s p e c t i v e l y . The latter may be r e g a r d e d as p a r a m e t e r i z i n g n o n - r e s o n a n t exchanges with the c o r r e s p o n d i n g quantum n u m b e r s . * Refs. [8, 16] do give pure charge-exchange cross sections, however. Their results lie systematically below other 'zero-prong' data, as we expect, and are distinguished in fig. 1 by an extra circle. ** This form contains errors: see ref. [5]. Fortunately, the numerical error is slight
~
--
,oo LAB
ENERGY
,~o
IN
~o
-L~_ L.~_,_~_,_,_,. ~,~_,,._,_,_,_,_-~
\
MEV
~o
, 300
-
t
)
350
Fig. 1. I n t e g r a t e d ~p c r o s s s e c t i o n data c o m p a r e d with the n o n - s t a t i c m o d e l . Total, v i s i b l e - a n n i h i l a t i o n , e l a s t i c and z e r o - p r o n g data a r e shown by open c i r c l e s , s o l i d c i r c l e s , t r i a n g l e s and c r o s s e s , r e s p e c t i v e l y . C h a r g e - e x c h a n g e (as d i s t i n c t f r o m z e r o prong) m e a s u r e m e n t s a r e d i s t i n g u i s h e d by a c i r c l e r o u n d the c r o s s . F o r r e f e r e n c e s , s e e the text. Solid c u r v e s , f r o m Lop to b o t tom, a r e the n o n - s t a t i c m o d e l p r e d i c t i o n s f o r total, total annihilation, e l a s t i c and c h a r g e e x c h a n g e c r o s s s e c t i o n s . B r o k e n c u r v e s a r e e s t i m a t e s of v i s i b l e - a n n i h i l a t i o n and z e r o - p r o n g c r o s s s e c t i o n s , m a d e by a s s i g n i n g 7% of all a n n i h i l a t i o n s to n e u t r a l modes.
U
0
u'l
ul
z 0
b9
0 H
I
204
R.A. BRYAN and R. J. N. PHILLIPS
For s c a l a r , i s o s c a l a r meson exchange, ~with coupling constant g defined by the effective Lagrangian density 23 =(4n)~g~/q5, the non-relativistic static N-N potential is p2 V(r) = _g2u(1 - 8 ~p_ M 2 / [ F ( / x r ) + 2 - ~ a ( p r ) L "S] .
(1)
Here /~ and M are the meson and nucleon m a s s e s , L is the orbital angular momentum and S =½(a 1 + a 2) is the total spin. The functions F and G are
F(x) = e x p ( - x ) / x , C(x)
(2)
1 ~d F(x)
= - x
•
(3)
For pseudoscalar, i s o s c a l a r meson exchange, with coupling constant defined by ./2 =(4~)½g~75~ , the N-N potential is V(r) = g2
#3 12M2 [ F ( ~ r ) a 1 .a 2 + H(l~r) S12 ] ,
(4)
where
g(x) = (1 + 3 / x + 3 / x 2) F(x) , S12 = 3 a 1 . r e 2 . r / r 2 - a 1 . a 2 .
(5)
(6)
For vector, i s o s c a l a r meson exchange, there are two coupling constants f and g, defined by the Lagrangian density * A2 = (47r)½ ~ [g~/#~b/~ + 4f~ atlv(~vdPll - apqbv) ] ~V .
(7)
In t e r m s of these, the static potential of ref. [1] has the form
V(r) =g2t~{R1F(l~r)+R2[2F(t~r)al " a 2 - H ( # r ) S 1 2 ] - R 3 G(I~r)L "S} ; 1 ~2 1 ~2 R 1 =[1 + ~ - ~ + - - - - f ] 2 4 M2g
1 /~2 [1
R2 =_~2__~
1 ~2
R3 = 2 - ~
'
+f/g]2
[3 + 4 f / g + 3(l~/2M)2(f/g) 2] .
* In ref. [1], the definition of f differs by a factor 2M/~.
(8) (9)
(10)
(11)
N-N POTENTIAL
205
Table 1 Parameters for the Bryan-ScottN-N potentials. Parameter
Static case
Non-static case
g2(~)
11.7
12.66
g2(u)
7.0
3.0
g2 (p)
0.68
2.44
f/g(p) g2 (w)
4.4 21.5
1.13 23.7
g2 ((to)
9.4
9.46
m((~o) g2(gl)
560 6.1
550 1.97
m (0"1)
770
600
For ¢o f / g is assumed zero; rn denotes mass in MeV/c 2.
T h e s e p o t e n t i a l s w e r e m e a n t to be c o r r e c t t h r o u g h o r d e r ( p / M ) 2 only. F o r i s o v e c t o r m e s o n e x c h a n g e s , add the i s o s p i n f a c t o r r 1 "32. B r y a n and Scott [1] a d j u s t e d the c o u p l i n g c o n s t a n t s , and the s c a l a r m e s o n m a s s e s , to fit N - N p h a s e s h i f t s ; the r e s u l t i n g v a l u e s a r e given in table 1. T h e y a l s o put the whole p o t e n t i a l equal to z e r o f o r r < 0.6 fro. F o r the N - N i n t e r a c t i o n , we r e v e r s e d the s i g n s of the p o t e n t i a l s f o r the o d d - G m e s o n s ~, o) and (h" We r e t a i n e d the z e r o c u t - o f f on the m e s o n p o t e n t i a l at r = 0.6 fm. We added an e m p i r i c a l a b s o r p t i v e p o t e n t i a l and a d j u s t e d it to give the b e s t fit to data, a s d e s c r i b e d in s e c t i o n 5.
4. NON-
STATIC
BRYAN-
SCOTT
POTENTIAL
B r y a n and Scott [5] l a t e r i n c l u d e d v e l o c i t y - d e p e n d e n t c e n t r a l t e r m s , of the f o r m [V2U(r) + U(r)V 2 ] / ( 2 M 2 ) , in the N - N potential. At the s a m e t i m e , t h e y slightly a l t e r e d s o m e of the s t a t i c p o t e n t i a l s V(r). F o r s c a l a r i s o s c a l a r m e s o n e x c h a n g e , k e e p i n g the s a m e n o t a t i o n as in s e c t i o n 3, V and U a r e g i v e n by
V(r) = _g2~{(i - 4~2) F(/Ir) U(r) : -g2u F(t~r) .
t~2 +-~-~G(ttr) L.S},
(12) (13)
F o r p s e u d o s c a l a r m e s o n e x c h a n g e s , V(r) is u n c h a n g e d and U(r) = O. F o r v e c t o r , i s o s c a l a r m e s o n e x c h a n g e , the c o e f f i c i e n t s R 1 and R 3 a r e c h a n g e d , and a v e l o c i t y - d e p e n d e n t t e r m i s added:
206
R.A. BRYAN
and R. J. N. PHILLIPS
~2 R 1 = 1 +~-~
n3
(1 + f / g )
,
~2
~-~ (3 ÷4//g),
U(r) = _ g 2 p F ( # r )
.
(14)
(15)
(16)
The v e l o c i t y - d e p e n d e n t p o t e n t i a l s can be i n t e g r a t e d by a d a p t i n g s t a n d a r d m e t h o d s [25]. In effect, they r e n o r m a l i z e the s t a t i c t e r m s and r e d u c e the s i n g u l a r i t y of OBE p o t e n t i a l s at the o r i g i n f r o m r -3 to r -2. The r e m a i n i n g s i n g u l a r i t y can be r e d u c e d a s f o l l o w s . F r o m each m e s o n - e x c h a n g e p o t e n t i a l i s s u b t r a c t e d the p o t e n t i a l for a heavy m e s o n , of m a s s A, with the s a m e quantum n u m b e r s and coupling c o n s t a n t ; this i s e s s e n t i a l l y a P a u l i - V i l l a r s regulator. B r y a n and Scott [5] m a d e two s e p a r a t e s t u d i e s with v e l o c i t y - d e p e n d e n t N - N p o t e n t i a l s . F i r s t , e x p l i c i t l y e x c l u d i n g S - w a v e s , they found that the c e n t r i f u g a l b a r r i e r in h i g h e r p a r t i a l w a v e s w a s s t r o n g enough to o v e r c o m e the 1 / r 2 s i n g u l a r i t y , and no c u t - o f f was n e e d e d . Secondly, they i n c l u d e d S - w a v e s ; a c u t - o f f was then n e e d e d . The o p t i m u m p a r a m e t e r s for the l a t t e r i n v e s t i g a t i o n a r e shown in t a b l e 1. F o r the N - N i n t e r a c t i o n , we r e v e r s e d the sign of o d d - G m e s o n p o t e n t i a l s and added an e m p i r i c a l a b s o r p t i o n ; s e e s e c t i o n 5. The f i r s t n o n - s t a t i c p o t e n t i a l led to a s o m e w h a t u n s a t i s f a c t o r y fit to NN data, at l e a s t with our p a r t i c u l a r a s s u m p t i o n s ; we s h a l l not d i s c u s s this c a s e h e r e . The s e c o n d n o n - s t a t i c p o t e n t i a l (including S - w a v e c r i t e r i a ) gave a good fit to d a t a , howe v e r . T h i s c a s e i s d i s c u s s e d and i l l u s t r a t e d below. In the N - N c a s e , A = 1.5 GeV i s given as the c u t - o f f m a s s , though 1.0 GeV i s a l s o a c c e p t a b l e [5]. F o r the N - N c a s e , we u s e d the l a t t e r value only.
5. F I T T I N G THE DATA To the r e a l OBE p o t e n t i a l s we a d d e d an i m a g i n a r y W o o d s - S a x o n p o t e n t i a l i W , i n d e p e n d e n t of spin and i s o s p i n : W ( r ) : - W o / [ 1 + a exp(br)] .
(17)
The e x t r a f r e e d o m f r o m v a r y i n g a g a v e l i t t l e a d v a n t a g e , so we fixed a = 1. The r e m a i n i n g p a r a m e t e r s Wo and b w e r e then a d j u s t e d to fit the data. S p e c i f i c a l l y , we c h o s e t h e m to m a t c h the t o t a l and e l a s t i c c r o s s s e c t i o n s aT(PP) and Crel(~p) at 100 MeV. The p o t e n t i a l so d e t e r m i n e d p r o v e d to fit a l s o the e n e r g y - d e p e n d e n c e of c r o s s s e c t i o n s , and a n g u l a r d i s t r i b u t i o n s , and c h a r g e - e x c h a n g e . P a r a m e t e r s for the s t a t i c OBE c a s e a r e a = 1, b = 6 fm -1 and Wo = 62 GeV. F o r the n o n - s t a t i c O B E c a s e , a = 1 , b =5 fm -1 and Wo = 8 . 3 GeV.
N-N POTENTIAL
207
C o m p a r i s o n s with d a t a f o r the s t a t i c c a s e have a l r e a d y b e e n shown i n r e f . [7]. F o r the n o n - s t a t i c c a s e , we i l l u s t r a t e the q u a l i t y of the d a t a f i t t i n g i n f i g s . 1-6 and 11. As f o r t e c h n i c a l d e t a i l s , the S c h r 6 d i n g e r e q u a t i o n w a s i n t e g r a t e d n u m e r i c a l l y for u n c o u p l e d p a r t i a l w a v e s with L --<4 a n d c o u p l e d w a v e s with J < 5, f o r e a c h i s o s p i n I = 0 and I = 1. S i n g l e - p i o n e x c h a n g e a m p l i t u d e s w e r e a d d e d f o r the h i g h e r p a r t i a l w a v e s , but m a d e no p e r c e p t i b l e d i f f e r e n c e e x c e p t for
20
o LD F¢nl5
Z
g d 'v
OL 1.0
[
I
0.5
0
(co=O)
IqP -0.5
-I.0
c,m,
Fig. 2. ~p differential cross section at 62.7 MeV [14], compared with the non-static model. The solid line is the p + p ~ ~+p prediction. The broken line is the fi+n -* fi+n prediction, differing by the Coulomb effects.
208
R.A.
BRYAN
and R. J. N. PHILLIPS
25
,4 2 0 LU I-U9
z
15
E u
b
5
O I
i.O
r 0.5
r 0
-0.5
(C o s e) c . m .
F i g . 3. ~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 at 99.8 MeV [14] c o m p a r e d with the n o n - s t a t i c m o d e l .
ol.O
N-N POTENTIAL
209
25
i
A2C
tu u~
m
z
15
d
b
C 1.0
0.5
0
-0.5
( c o s O) ¢.m.
F i g . 4. ~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 at 136.8 MeV [14] c o m p a r e d with the n o n - s t a t i c m o d e l .
-I .o
210
R.A. BRYAN
and R.J.N.
PHILLIPS
30
25
,,=, 2 0 f-U)
~E Z
f5
"o ,.... 0
I0
C
1.0
0.5
0 (co-, e) ¢.m.
-0.5
Fig. 5. ~p differential cross section at 175.0 MeV compared with the non-static model.
-I.0
[14]
N-N POTENTIAL
211 4"0
4"0 5 0 - II S 93
McV
McV
115- 180
DATA
149
THEORY
McV
McV
DATA
THEORY
]
3-C
3"0
uo E 2"0
uE 2 - 0 I¢:
i=
~_)
~D
I'O
I.O
1
]
O t I'0
0 COS
-I-0
I'0
O cm"
-I.O
0 COS
Ocm
Fig. 6. ~ + p ~ ~ + n a n g u l a r d i s t r i b u t i o n s , at a v e r a g e e n e r g i e s 93 a n d 149 MeV [16], c o m p a r e d with the n o n - s t a t i c m o d e l .
+p ~ fi + n. I s o s p i n a m p l i t u d e s A (I) w e r e c o m b i n e d in the u s u a l way: A ( ~ n ~ ~ n ) = A (1) , A ( p p - - * ~ p ) : -12,4(0) + ½A(1) ,
(18)
A(pp--* fin) = ½A(0) - ½A(1) . C h a r g e i n d e p e n d e n c e w a s t h u s a s s u m e d in d e t e r m i n i n g n u c l e a r p a r t i a l w a v e a m p l i t u d e s , but the n - p m a s s d i f f e r e n c e w a s i n c l u d e d in the p h a s e s p a c e f a c t o r f o r ~ + p --* fi + n. Coulomb effects were treated as follows. Integrations were made with no C o u l o m b p o t e n t i a l , to get p u r e n u c l e a r p h a s e s h i f t s ; ~ - p s c a t t e r i n g w a s p r e d i c t e d by a d d i n g C o u l o m b and n u c l e a r p h a s e s h i f t s , w h i c h a m o u n t s to a d d i n g the C o u l o m b a m p l i t u d e to a m o d i f i e d n u c l e a r a m p l i t u d e . S y m b o l i c a l l y , for e a c h p a r t i a l w a v e , exp(2iS+2i~)-
1 = e x p ( 2 i ~ ) - 1 + e x p ( 2 i c r ) [ e x p ( 2 i 6 ) - 1] ,
(19)
212
R.A. BRYAN and R. J. N. PHILLIPS
w h e r e 5 and a a r e the n u c l e a r and Coulomb p h a s e shifts. We calculated OT(pp) f r o m the optical t h e o r e m , u s i n g the p u r e n u c l e a r amplitude. We defined ael(pp) by the p u r e n u c l e a r c r o s s section; thus it equals ~el(fin). Coul o m b e f f e c t s e n t e r ~ +p - fi + n through wave function d i s t o r t i o n in the initial state. T h i s effect w a s c a l c u l a t e d by adding p h a s e f a c t o r s exp(ia) to each p a r t i a l wave, and p r o v e d to be negligible.
6. DISCUSSION The fit to data s e e m s s a t i s f a c t o r y . F u r t h e r questions now a r i s e . What is the significance of this fit? How m u c h is due to the OBE p o t e n t i a l s and how m u c h to the a b s o r p t i o n ? What f u r t h e r p r e d i c t i o n s can be m a d e ? We d i s c u s s t h e s e and other questions below, in s e p a r a t e s u b s e c t i o n s (see a l s o the d i s c u s s i o n in r e f . [7]).
6.1. Role of W(r) The a b s o r p t i v e potential s e e m s to dominate e l a s t i c s c a t t e r i n g . One can fit aT, ael and da/d~2 f o r ~p quite well with p u r e a b s o r p t i o n [7, 14]. Now, W(r) is v e r y s t r o n g at s m a l l r and g r e a t l y a t t e n u a t e s the wave function h e r e . Thus it s u p p r e s s e s s h o r t - r a n g e i n t e r a c t i o n s , and in this r e s p e c t a c t s s o m e w h a t like the r e p u l s i v e c o r e in o r d i n a r y n u c l e a r f o r c e s . In the N - N p r o b l e m , the m e d i u m - r a n g e p o t e n t i a l s m u s t be c a r e f u l l y adjusted to get the c o r r e c t S-wave p h a s e shifts. In the N - N c a s e , S - w a v e s a r e a l m o s t c o m p l e t e l y a b s o r b e d anyway, so the m e d i u m - r a n g e d e t a i l s m a t t e r l e s s . The a b s o r p t i o n g i v e s negative r e a l p a r t s to low p a r t i a l a m p l i t u d e s , hence a negative r e a l p a r t to the s p i n - a v e r a g e d f o r w a r d s c a t t e r i n g amplitude, and hence d e s t r u c t i v e Coulomb i n t e r f e r e n c e in ~ - p s c a t t e r i n g (see below). We a s s u m e W is independent of spin and isospin, b e c a u s e it r e p r e s e n t s the a v e r a g e effect of v e r y m a n y annihilation channels. Thus W alone does not give any p o l a r i z a t i o n , or c h a r g e - e x c h a n g e , or d i f f e r e n c e s between ~ - p and ~ - n s c a t t e r i n g . T h e s e a r e the p l a c e s to look for OBE effects. 6.2. Role of OBE terms T h e s e give all the spin and i s o s p i n dependence. The r e l e v a n t data so f a r a r e ~+p--* fi+n c r o s s s e c t i o n s (figs. 1 and 6); both m a g n i t u d e s and a n g u l a r d i s t r i b u t i o n s a r e fitted s a t i s f a c t o r i l y . It s e e m s that c h a r g e - e x c h a n g e is m a i n l y due to the o n e - p i o n exchange p o t e n t i a l [7]. T h e r e is s o m e i n f o r m a t i o n on r e l a t i v e s t r e n g t h s of p - p and ~ - n a b s o r p tions at r e s t (see below), that is c o n s i s t e n t with <~ur m o d e l s . The OBE p o t e n t i a l s a l s o play a p a r t by drawing in the outer wave function to enhance a b s o r p t i o n in m a n y s t a t e s . Without the OBE t e r m s , W(r) h a s to be w e a k e r and of s u b s t a n t i a l l y longer r a n g e [7]. Since the r a n g e of W(r) is not known a priori, this a s p e c t is h a r d to study, but it a f f e c t s the slope of the e l a s t i c d i f f r a c t i o n p e a k [7]. F u t u r e e x p e r i m e n t s on p o l a r i z a t i o n and ~ - n s c a t t e r i n g will t e s t the OBE terms further.
N-N
POTENTIAL
213
6.3. Measuring the pion coupling It w o u l d b e i n t e r e s t i n g to u s e d a / d ~ l ( ~ p ~ f i n ) to d e t e r m i n e t h e y N c o u p l i n g c o n s t a n t g 2 , by e x t r a p o l a t i n g to t h e p o l e a s in t h e p + n --. n + p c a s e [26]. O n e - p i o n e x c h a n g e i s t h e s a m e in b o t h c a s e s . T h o u g h e x p e r i m e n t a l e r r o r s a r e m u c h b i g g e r f o r YtN, t h e b a c k g r o u n d c o n t r i b u t i o n s f r o m s h o r t e r - r a n g e t e r m s a r e m u c h l e s s . F o r i n s t a n c e , f o r w a r d c h a r g e e x c h a n g e at 150 MeV i s a b o u t 3 m b / s r f o r ~ + p a n d 11 m b / s r f o r p +n. P r e s e n t d a t a w o u l d p e r m i t o n l y a c r u d e d e t e r m i n a t i o n o f g 2, h o w e v e r . I n c i d e n t a l l y , the s o m e w h a t low v a l u e s of g 2 ( ~ ) in t a b l e 1 a r e p a r t l y due to t h e a b s e n c e of a r e l a t i v i s t i c f a c t o r M/E in the p o t e n t i a l eq. (4). 6.4. Scattering lengths T h e z e r o - e n e r g y S - w a v e s c a t t e r i n g l e n g t h s a(I,S ) f o r i s o s p i n I a n d s p i n S a r e s h o w n in t a b l e 2. T h e c o n v e n t i o n u s e d i s a = tg 5 / k , w h e r e k i s t h e r e l a tive momentum. Table 3 Relative ~ capture rates.
Table 2 S-wave s c a t t e r i n g lengths in fm.
Experimental Static case
Non-static case
Static
Non-static I
s
II
a(1,1)
-0.77 + 0.68i
-0.82 + 0.87i
Yn
0.37
0.38
0.19
0.60
a(1,0)
- 1 . 0 1 + 0.51i
- 1 . 0 0 + 0.60i
~
0.62
0.61
0.73
0.73
a(0,1)
- 1 . 4 4 + 0.76i
-1.01 + 0.65i
t Yn
0.50
0.55
0.63
0
a(0,0)
-0.35 + 1.18i
-0.62 + 1.33i
~
0.53
0.48
0.65
0
T h e a n n i h i l a t i o n r a t e at v e r y low e n e r g y i s p r o p o r t i o n a l to I m a . M e a s u r e m e n t s of ~ a n n i h i l a t i o n at r e s t in h y d r o g e n a n d d e u t e r i u m p e r m i t e s t i m a t e s of t h e r e l a t i v e r a t e s f o r c a p t u r e on n e u t r o n s a n d p r o t o n s , in s i n g l e t a n d t r i p l e t s t a t e s , if s o m e f u r t h e r a s s u m p t i o n s a r e m a d e [24]. In t a b l e 3 w e s h o w two s u c h e s t i m a t e s f r o m r e f . [24], b a s e d on two d i f f e r e n t e x t r e m e a s s u m p t i o n s , l a b e l l e d I and II; 7ps d e n o t e s t h e r e l a t i v e c a p t u r e r a t e on p r o t o n s in s i n g l e t s t a t e s , e t c . , and w e u s e t h e a r b i t r a r y n o r m a l i z a t i o n 3 7 ~ + 3~ns + 7 t + 7tn = 4. P r e d i c t i o n s f r o m t h e s t a t i c and n o n - s t a t i c m o d e l s a r e a l s o shown; t h e y s e e m c o n s i s t e n t w i t h t h e e x p e r i m e n t a l p o s s i b i l i t i e s . In p a r t i c u l a r , t h e s i n g l e t c a p t u r e r a t e i s g r e a t e r on p r o t o n s t h a n on n e u t r o n s , while the triplet rates are about equal. 6.5. ~ - n scattering O u r m o d e l s p r e d i c t ~ + n --* ~ + n s c a t t e r i n g . F i g . 7 s h o w s total~ e l a s t i c a n d a n n i h i l a t i o n c r o s s s e c t i o n s , c o m p a r e d w i t h p + p --, p + p p r e d i c t i o n s . In e a c h c a s e , t h e p + p v a l u e s a r e b i g g e r . T h i s e f f e c t , due to t h e O B E p o t e n t i a l s , i s not d o m i n a t e d by o n e - p i o n e x c h a n g e a l o n e [7]. F i g . 8 c o m p a r e s ~ - p a n d ~ - n 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 at 100 MeV. T h e b i g g e r ~ - p c r o s s s e c t i o n g o e s w i t h a s t e e p e r f o r w a r d p e a k , a s we e x p e c t f r o m t h e b l a c k d i s c a n a l o g y . See a l s o s e c t i o n 6.6.
214
R.A. BRYAN and R. J. N. PHILLIPS
I.,,\\~. \\\\~
200
I>,,
z u) z 0 FU
NNIHILATION(~
¢n
0
I00
U Z IZ
ELASTIC
"""
-.,,....
I
I I00 LAB.
I ENERGY
I IN
I
200 McV
300
Fig. 7. Comparison of ~p and On integrated cross section predictions, for the non-static model.
6.6. P o l a r i z a t i o n P r e d i c t i o n s a r e shown in figs. 10 and 11, for ~ - p and ~-n. P o l a r i z a t i o n c o m e s f r o m i n t e r f e r e n c e between the non-flip amplitude (dominated by absorption) and a c e r t a i n spin-flip amplitude that is dominated by the one-pion exchange potential (OPEP) [7]. Note however that O P E P gives no p o l a r i z a tion in Born approximation, nor in damped Born a p p r o x i m a t i o n with L - d e pendent damping; the effect is e s s e n t i a l l y n o n - l i n e a r . This is why p r e d i c t e d p o l a r i z a t i o n does not s i m p l y change sign between ~-p and ~ - n c a s e s , although O P E P does.
N-N POTENTIAL
215
25
~p
20
d Iz uJ lu~
i
5
\
-u
b
0 1.0
I 0.5.
~ 0 ¢o$0
~
: -0.5
-I.0
c.m.
Fig. 8. Comparison of ~p and ~n differential cross section predictions at 100 MeV, for the non-static model.
T h e r e a r e no m e a s u r e m e n t s y e t b e l o w 600 MeV, but the ~ - p p r e d i c t i o n s at f o r w a r d a n g l e s h a v e the s a m e s i g n a n d o r d e r of m a g n i t u d e a s a r e f o u n d at h i g h e r e n e r g i e s .
216
R.A. BRYAN and R. J. N. PHILLIPS
0.4
I00 0.2 Z 0 I,u~
:
o
_1 0 o.
-0.2
-0
.4
1.0
I
I
0.5
0 co$0
I -0.5
-I-O
¢.m.
Fig. 9. ~p polarization predictions at 100, 200 and 300 MeV, for the non-static model.
6.7. Coulomb interference Our models p r e d i c t d e s t r u c t i v e Coulomb i n t e r f e r e n c e in d(x/dft(~p). At m o d e r a t e e n e r g i e s this is quite a s m a l l effect (see fig. 2 at 63 MeV) and at higher e n e r g i e s it is s m a l l e r still. But at the lower e n e r g i e s it is a big effect, r e a c h i n g out to wide angles, a s i l l u s t r a t e d at 15 MeV in fig. 11. This r a i s e s a question about c o m p a r i n g t h e o r y with experiment. We define the i n t e g r a t e d c r o s s section ael(pp) by r e m o v i n g all Coulomb effects. In p r a c t i c e , however, an e x p e r i m e n t e r cannot distinguish the Coulomb effects except at s m a l l angles, and cannot r e m o v e it fully. We conclude that it is not useful to c o m p a r e ael(pp) at low e n e r g i e s ; one should just c o m p a r e d(~/d~(~p). 6.8. Saw-tooth effect In the original Ball-Chew model [6], the onset of absorption in each
N-N POTENTIAL
217
0.4
0.2
300
0
-0
-2
-0.4
i-O
] 0-5
I 0
-0.5
I
-I.0
cos 0 c.m.
Fig. 10. ~n polarization predictions at 100, 200 and 300 MeV, for the non-static model.
partial wave was sudden, giving cross sections a sawtoothed look as funct i o n s of e n e r g y . T h i s c a m e f r o m a p h y s i c a l a s s u m p t i o n a b o u t how an a b sorptive core would act. In o u r m o d e l s , r e p r e s e n t i n g t h e a b s o r p t i o n by a p o t e n t i a l l e a d s to a s m o o t h v a r i a t i o n w i t h e n e r g y . T h e r e i s no s a w - t o o t h e f f e c t w h a t e v e r ; not even smooth bumps are given. 6.9. Static and n o n - s t a t i c c o m p a r i s o n T h e two p o t e n t i a l m o d e l s g i v e r a t h e r s i m i l a r p r e d i c t i o n s . ( F o r f i g u r e s i l l u s t r a t i n g t h e s t a t i c c a s e , s e e r e f . [7].) T h e b i g g e s t d i f f e r e n c e s e e m s to b e in a e l ( P p) b e l o w 50 MeV; at 40 MeV, f o r i n s t a n c e , t h e s t a t i c a n d n o n s t a t i c m o d e l s g i v e 87 m b a n d 81 m b , r e s p e c t i v e l y . This approximate agreement is not surprising, since the OBE potentials a r e r a t h e r s i m i l a r a t m e d i u m a n d l o n g r a n g e . T h e b i g d i f f e r e n c e i s at s h o r t r a n g e , w h e r e in t h e s t a t i c c a s e t h e O B E t e r m s a r e c u t - o f f to z e r o w h i l e in
218
R.A. BRYAN and R. J. N. PHILLIPS
15
.a I O E _= E ,J
I
b
O 1.0
i 0.5
I 0
i -0.5
i -I 0
COS # ¢..m•
Fig. 11. Illustration of Coulomb interference in dff/dfZ(Dp) at 15 MeV. The solid and broken curves are predictions of the non-static model, with and without Coulomb effects. The data are p r e l i m i n a r y [15].
the n o n - s t a t i c c a s e they a r e d a m p e d but r e m a i n s t r o n g . S h o r t r a n g e d i f f e r e n c e s t e n d to be s u p p r e s s e d by the a b s o r p t i o n , h o w e v e r . T h e two a b s o r p t i v e p o t e n t i a l s s e e m r a t h e r d i s s i m i l a r i n s t r e n g t h , but t h i s i s d e c e p t i v e . W h e n W(r) i s v e r y s t r o n g and of s h o r t r a n g e , q u i t e l a r g e c h a n g e s i n s t r e n g t h h a v e l i t t l e effect on the s c a t t e r i n g . An a p p r o x i m a t e fit to d a t a would be a c h i e v e d i n e i t h e r c a s e with p a r a m e t e r s a = 1, b = 5 f m - 1 a n d Wo = 20 GeV: the c h a n g e s f r o m t h e s e v a l u e s to the b e s t fit i n e i t h e r case represent relatively small adjustments• We a r e g r a t e f u l to R. B i z z a r r i , R. G o l d b e r g a n d B. L. Scott for e a r l y i n f o r m a t i o n a b o u t e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s .
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[2] S. Sawada, T. Ueda, W. Watari and M. Yonezawa, Progr. Theor. Phys. 32 (1964) 380.
N-N POTENTIAL
219
[3] A. Scotti and D. Y. Wong, Phys. Rev. 138 (1965) B145; J. S. Ball, A. Seotti and D. Y. Wong, Phys. Rev. 142 (1966) 1000. [4] R. A. Arndt, R.A. Bryan and M. H. MacGregor, Phys. L e t t e r s 21 (1966) 314; R.A. Bryan and R . A . A r n d t , Phys. Rev. 150 (1966) 1299. [5] R.A. Bryan and B. L. Scott, L i v e r m o r e preprint UCRL-70409 and California State College, Long Beach, preprint. [6] J. S. Ball and G. F. Chew, Phys. Rev. 109 (1958) 1385. [7] R . J . N . Phillips, Rev. Mod. Phys. 39 (1967) 681. [8] C. A. Coombes, B. Cork, W. Galbraith, G.R. Lambertson and W. A. Wenzel, Phys. Rev. 112 (1958) 1303. [9] B. Cork, O. I. Dahl, D.H. Miller, A. G. Tenner and C. L. Wang, Nuovo Cimento 25 (1962) 497. [10] J. Loken and M. Derrick, Phys. L e t t e r s 3 (1963) 334. [11] U. Amaldi, T. Fazzini, G. Fidecaro, C. Ghesqui~re, M. Legros and H. Steiner, Nuovo Cimento 34 (1964) 825. [12] A. Hossain and M. Shaukat, Nuovo Cimento 38 {1965) 737. [13] U. Amaldi, B. Conforto, G. F i d e e a r o , H. Steiner, G. Baroni, R. B i z z a r r i , P. Guidoni, V. Rossi, G. Brautti, E. Castelli, M. Ceschia, L. Chersovani and M. Sessa, Nuovo Cimento 46 (1966) 171. [14] B. Conforto, G. Fidecaro, H. Steiner, R. B i z z a r r i , P. Guidoni, F. Marcelja, G. Brautti, E. Castelli, M. Cesehia and M. Sessa, Antiproton-proton elastic scattering between 63 and 175 MeV, to be published. [15] R. Goldberg, private communication. [16] R. B i z z a r r i , E. Castelli, M. Ceschia, B. Conforto, G. C. Gialanella, P. Guidoni, F. M a r c e l j a and M. Sessa, Nora Interna no. 136, Rome {1967). [17] J. Button and B. Maglic, Phys. Rev. 127 (1962) 1297. [18] L. Dobrzynski, C. Ghesqui~re, N.H. Xuong and H. Torte, Phys. L e t t e r s 23 {1966) 614. [19] H. Sens, private communication. [20] C. Baltay, P. Franzini, G. Liitjens, J. Severiens, D. Tycko and D. Zanello, Phys. Rev. 145 (1966) 1103. [21] V. Barnes, K.W. Lai, P. Anninos, L. Gray, P. Haggerty, E. Harth, T. Kalogeropoulos, S. Zenone, R. B i z z a r r i , U. Dore, G. Gialanella, G. Monetti and P. Guidoni, Proc. of 1964 Dubna Conf., Vol. I, p. 731. [22] W. Chinowsky and G. Kojoian, Nuovo Cimento 43 (1966) 684. [23] A. Bettini, M. Cresti, S. Limentani, L. Peruzzo, R. Santangelo, S. Sartori, L. Bertanza, A. Bigi, R. C a r r a r a , R. Casali and P. L a r i c c i a , Nuovo Cimenfo 47A (1967) 642. [24] R. B i z z a r r i , CERN preprint (1967). [25] A. M. Green, Nucl. Phys. 33 (1962) 218. [26] P. Cziffra and M . J . Moravcsik, Phys. Rev. 116 (1959) 226; A. Ashmore, W.H. Range, R. T. Taylor, B. M. Townes, L. Castillejo and R. F. P e i e r l s , Nucl. Phys. 36 {1962) 258 and r e f e r e n c e s therein.