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Photon-nucleus elastic and total cross sections
8.-~.
Nuclear Physics B10 (1969)329-338. North-Holland Publ. Comp., A m s t e r d a m
P H O T O N - N U C L E U S ELASTIC AND TOTAL CROSS SECTIONS B. M A R G O L I S * and C. L. T A N G * Institute of Theoretical Physics, McGill University, Montreal 110, Canada
Received 3 F e b r u a r y 1969 Abstract: The forward s c a t t e r i n g amplitude and hence the total cr o ss section for photons incident on nuclei is calculated in the eikonal approximation. The s c a t tering amplitude is taken to be the sum of two types of coherent amplitudes; (a) el as t i c scattering from each nucleon and (b) a r e g e n e r a t i v e amplitude c o r r e sponding to the production of the v e c t o r mesons p, W and (p on one nucleon and radiative capture on another. A considerable effect due to the finite mass of the v e c t o r mesons is seen at e n e r g i e s up to 10 GeV in heavy nuclei. This manifests itself as an apparent violation of v e c t o r dominance for the photon-nucleus i n t e r action which disappears at v e r y high e n e r g i e s if v e c t o r dominance is valid for the photon-nucleon interaction.
1. I N T R O D U C T I O N W e c a l c u l a t e t h e f o r w a r d a m p l i t u d e f o r e l a s t i c s c a t t e r i n g of p h o t o n s f r o m a t o m i c n u c l e i at h i g h e n e r g i e s in t h e e i k o n a l a p p r o x i m a t i o n . T h i s a m p l i t u d e , to l o w e s t o r d e r in t h e f i n e s t r u c t u r e c o n s t a n t , c o n s i s t s of t w o p a r t s : (a) t h e c o h e r e n t s u m of e l a s t i c s c a t t e r i n g a m p l i t u d e s f r o m e a c h nuc l e o n (a ' o n e - s t e p ' p r o c e s s ) and (b) t h e r e g e n e r a t e d a m p l i t u d e w h i c h c o m e s f r o m p h o t o - p r o d u c t i o n of a h a d r o n on one n u c l e o n of t h e n u c l e u s , f o l l o w e d by r a d i a t i v e c a p t u r e on a n o t h e r n u c l e o n (a ' t w o - s t e p ' p r o c e s s ) . We a s s u m e t h a t a l l t w o - b o d y p r o c e s s e s h a v e a m p l i t u d e s w h i c h a r e i n d e p e n d e n t of s p i n and i s o s p i n . We c o m m e n t on d e v i a t i o n s f u r t h e r on. T h e r e s u l t s s h o w in d e t a i l , c h a r a c t e r i s t i c s d i s c u s s e d q u a l i t a t i v e l y by S t o d o l s k y [1]. T h e r e g e n e r a t e d a m p l i t u d e i n t e r f e r e s d e s t r u c t i v e l y w i t h t h e ' o n e - s t e p ' a m p l i t u d e (a). At i n f i n i t e e n e r g y , a s s u m i n g t h a t t h e only i m p o r t a n t i n t e r m e d i a t e h a d r o n s a r e t h e v e c t o r m e s o n s p, w and (P an d a s s u m i n g v e c t o r d o m i n a n c e in p h o t o n - n u c l e o n i n t e r a c t i o n s , one f i n d s t h a t p h o t o n s s c a t t e r f r o m n u c l e i w i t h a d e p e n d e n c e on m a s s n u m b e r A, t h e s a m e a s f o r s c a t t e r i n g of v e c t o r m e s o n s . At f i n i t e e n e r g i e s h o w e v e r t h e n o n - z e r o m a s s of t h e v e c t o r m e s o n s a c t s to d e c r e a s e t h e r e g e n e r a t i v e a m p l i t u d e an d o n e g e t s a n A - d e p e n d e n c e c l o s e r to t h a t f o r t h e p u r e ' o n e - s t e p ' p r o c e s s . T h i s i s d u e to a p h a s e m i s - m a t c h b e t w e e n t h e photon an d t h e i n t e r m e d i a t e h a d r o n w a v e s in t h e ' t w o - s t e p ' p r o c e s s e s . * Supported in part by the National H e s e a r c h Council of Canada.
:330
B. M A R G O L I S a n d C. L. T A N G
2. CALCULATION OF PHOTON SCATTERING According to the eikonal a p p r o x i m a t i o n [2-4] we w r i t e the o n e - s t e p a m plitude
w h e r e f~7(O) is the f o r w a r d a m p l i t u d e for e l a s t i c photon-nucleon s c a t t e r i n g , and the t w o - s t e p a m p l i t u d e
F (b),Yy(O) ' -~27r ~x fyx(0)fx)'(0) Ix (A, qx; 0, ½Crx)
(2)
Ix(A, qx; 0,½ ~rx) Z 'T •
A2/d2b fide" e-qx
_!
,,
f
--OO
Z'
Z'~
f
×
1
dz' Px(b, z') e ~crx A
f
px(b, z')dz' eiqx z' .
(2a)
--OO
H e r e x is any p a r t i c l e that can b e p r o d u c e d c o h e r e n t l y by photons i n c i d e n t on the nucleus, k is the photon wave n u m b e r and ~x is the total c r o s s s e c tion f o r p a r t i c l e x incident on a nucleon. It is a s s u m e d h e r e that the x - n u cleon s m a l h - a n g l e s c a t t e r i n g a m p l i t u d e is p u r e i m a g i n a r y . O t h e r w i s e ex is to be r e p l a c e d by Crx(1 - i~x) w h e r e ~x is the r a t i o of r e a l to i m a g i n a r y p a r t of the f o r w a r d s c a t t e r i n g amplitude. T h e longitudinal m o m e n t u m t r a n s f e r qx = m x 2 / ( k + P x ) • The n u c l e a r c e n t r e - o f - m a s s motion is neglected. T h e quantities m x and Px a r e the i n t e r m e d i a t e hadron m a s s and m o m e n t u m r e s p e c t i v e l y . The nucleus is taken as a ' p i e c e of n u c l e a r m a t t e r ' of single p a r t i c l e density Po
Po(r) = Po(b, z) = 1+ e + ( r - a ) / c
(3)
T h e justification f o r this has been d i s c u s s e d s e v e r a l t i m e s [3-5]. The Saxon-Woods or F e r m i f o r m (3) has two p a r a m e t e r s a and c which a r e t a k e n f r o m e l e c t r o n s c a t t e r i n g data (6) and given in t a b l e 1. The neutron and proton d e n s i t i e s a r e a s s u m e d to be the s a m e . Due to the m o m e n t u m t r a n s f e r dependence of the two-body a m p l i t u d e s f ~ x ( q 2) and f x x ( q 2) (the l a t t e r c o r r e s p o n d i n g to e l a s t i c s c a t t e r i n g of x by a nucleon) the ' e f f e c t i v e d e n s i t y ' that a p p e a r s in eq. (2a) is Px(r) where
= _ _1 f F(;~) e - ~ 02x" - eiX" r d3 X (2~) 3
(4)
331
I~HOTON-NUCLEUS CROSS SECTION Table 1 The Saxon-Woods parameters a and c of formula (3) from ref. [6].
Target
12 C
28Si
58Ni
115in
181 T a
208pb
a (fro)
0.42
0.637
0.57
0.523
0.637
0.523
c (fro)
2.29
2.946
4.258
5.25
6.448
6.458
Table 2 The function ~-In(R)(A;qp'0,½~p) given by f o r m u l a (2a). ffp = 26mb
~'----.~ k(GeV/c)
3
5
12 C
1.68
28Si
3.64
Target
10
15
20
2.26
2.55
2.61
2.63
2.66
5.85
7.15
7.43
7.52
7.62
~
58Ni
14.4
18.9
19.8
20.2
20.6
115in
18.7
7.82
35.6
47.9
50.7
51.7
53.0
181Ta
21.5
49.4
74.8
81.0
83.3
86.4
208pb
33.6
68.6
97.7
104
107
111
~p = 3 1 m b 12C
1.96
28Si
4.31
58Ni
9.43
2.61 6.73
2.93 8.15
3.00 8.45
3.02 8.56
3.05 8.66
16.6
21.3
22.3
22.7
23.1
115In
22.8
40.8
53.3
56.1
57.1
58.4
181Ta
27.3
57.4
83.4
89.6
91.9
95.0
41.9
78.9
208pb
108
115
117
120
ffp = 37 mb 12C 28Si
2.29 5.08
3.00 7.72
3.35 9.24
3.42 9.56
58Ni
11.3
19.0
23.9
24.9
115In
27.7
46.3
58.8
61.5
181Ta
34.2
66.2
92.2
98.3
208pb
51.7
89.9
118
124
3.45 9.67 25.3 62.5
3.48 9.79 25.8 63.8
100
103
127
130
The r a n g e [7] in f o r m u l a (6),bp = 8 (GeV/c) -2. The Saxon-Woods p a r a m e t e r s a and c in f o r m u l a (3) a r e taken f r o m ref. [6]. The m a s s m p = 765 MeV.
B. MARGOLIS and C. L. TANG
332
Table 3 The f u n c t i o n I(R)(A;qw, 0,½~o)) g i v e n by f o r m u l a (2a). ffW = 2 6 m b ~k(GeV/c) Target
3
5
12C
1.61
2.22
28Si
3.40
58Ni
7.16
5.70
10
15
20
2.54
2.61
2.62
7.11
14.0
18.7
7.40
7.51
2.66 7.62
19.7
20.1
20.6
l151n
17.1
34.3
47.5
50.5
51.6
53.0
181Ta
19.4
46.9
73.8
80.5
83.0
86.4
208pb
30.7
65.6
96.6
104
107
111
(~¢o = 31 mb 12C
1.88
2.57
2.92
3.00
3.02
3.05
28Si
4.04
6.57
8.10
8.43
8.54
8.66
58Ni
16.1
21.1
22.2
22.6
23.1
115In
21.1
8.70
39.4
52.8
55.9
57.0
58.4
181Ta
24.8
54.8
208pb
38.7
75.9
82.3
89.1
107
91.6
114
117
95.0 120
ff¢o = 37 m b 12C
2.20
28Si
4.78
2.95
3.34
7.54
3.42
9.19
3.45
9.53
58Ni
11.5
18.4
23.7
24.8
115In
25.8
45.0
58.4
61.3
181Ta
31.5
63.6
91.1
97.8
208pb
48.1
86.9
117
124
9.66 25.2 62.4
3.48 9.79 25.8 63.8
100
103
127
130
The r a n g e in f o r m u l a (6), bw = 8 ( G e V / c ) - 2 , o b t a i n e d f r o m r e f . [7]. T h e S a x o n Woods p a r a m e t e r s a and c in f o r m u l a (3) a r e t a k e n f r o m ref. [6]. The m a s s m W = = 783 MeV. where
F(X) = f po(r) We take the same
This is not generally meson.
(5)
-i~. r d3r.
s l o p e f o r f v x ( q 2) a n d f x x ( q 2 ) ,
fvx(q2)/f~x(O)
vector
e
= fxx(q2)/fxx(O)
true but it follows from
i.e.
(6)
= e-½bxq 2 . vector
dominance
if x is a
PHOTON-NUCLEUS CROSS SECTION
333
Table 4
The function
~ - ~ G~e V / c )k~ Target
i~orQ( A m ;q(fl' O, ½tYq~)given
3
5
by formula (2a).
10
15
20
oo
1.26
1.35
1.39
1.44
~
12C
0.30
0.83
28Si
0.33
1.63
3.31
58Ni
0.50
3.04
8.32
1151n
1.42
6.45
21.0
181Ta
1.48
6.06
28.5
208pb
2.65
40.2
10.1
3.77
3.95
i0.0
4.14
10.7
11.6
26.4
28.6
31.8
39.8
47.8
52.1
54.2
60.3
69.2
The range in formula (6), b~0 = 5 (GeV/c)-2, obtained from ref. [71]2 The SaxonWoods parameters a and c in formula (3) are from ref. [6]. (rcp mb. The mass m@ = 1.019 GeV.
At infinite e n e r g y qx = 0 and eq. (2) b e c o m e s
F(b)(o) = -k-2~i~ f ~ x ( 0 ) f x ~ ( 0 YY
) ~
[A
-N(A;
0,½~x) ]
x
(7)
where
N(A; 0, ~ax)l
1
= 2~x
f[1 - e - F a x Tx(b))d2b
(8)
with Tx(b )
: A J Px(b,z)dz .
(9)
--oO
We now a s s u m e that the i m p o r t a n t i n t e r m e d i a t e h a d r o n s a r e the v e c t o r m e s o n s p, ¢o and q~ and that v e c t o r d o m i n a n c e is v a l i d f o r the p h o t o n - n u c l e o n i n t e r a c t i o n . T h e x = V w h e r e the V a r e the v e c t o r m e s o n s p, ¢o and ~p and f y v ( q 21 = ~¼
[\ -Y2V'-I ~-) f v v ( q 21 •
(101
H e r e a is the f i n e - s t r u c t u r e c o n s t a n t and the VV a r e the v e c t o r d o m i n a n c e c o u p l i n g c o n s t a n t s . U s i n g the o p t i c a l t h e o r e m , I m f v v ( 0 ) = (Pv/47r)aV, we h a v e at infinite e n e r g y 2 -1 FVV(0) =
Y7 (0)
V7 (0) =
V
r 42
~V
-1
(11) l
334
B. MARGOLIS and C. L. TANG
I
I
I
l
Ill
17
80
IN
I 2
I S
J 10
I IS
J 20
Fig. 1. The ratio a ( y A ) / f f ( y / ~ as a function of the photon lab momentum k for 208Pb and 181Ta. The v e c t o r dominance coupling constants a r e as given in the text. ffq~ = 12 mb, ffp = cr¢o=26, 31, 3 7 m b ,
This infinite energy result can be derived as well using Glauber theory. The methods are similar to those for deriving vector meson photo-product i o n c r o s s s e c t i o n s [2] ( o n e - s t e p p r o c e s s e s ) . One t h e n h a s , u s i n g t h e o p t i c a l t h e o r e m , t h e t o t a l c r o s s s e c t i o n f o r p h o t o n s i n c i d e n t on n u c l e i . ~ 2 -1 ~(~A) = ~(vN) • v
(12) - ~ 2V - 1
A t f i n i t e e n e r g i e s one h a s , u s i n g e q s . (2), (2a) a n d (10)
PHOTON-NUCLEUS CROSS SECTION
1
I
I
I
335
I
IIII
in 11~
26 mb 31 3"/
Ni
s! 26 31 37
I
I 3
t 5
I 10
I 20
l
lS
Fig. 2. The ratio cr('yA)/~(%N) as a function of the photon lab momentum k for l15In and 58Ni. The v e c t o r dominance coupling constants are as given in the text. qq) = =12 rob, ~p =(Y¢o = 26, 31, 37rob.
a(TA) = Aa(]/N)
-
, v ItS> IVR)
= Relv
T2
•
-1
!t
i '
(13)
(13a)
In eqs. (12) and (13) cr(TN) is the photon-nucleon total cross section assumed independent of the spin and charge of the nucleon. The situation becomes more complex if there are appreciable spin and iso-spin dependent parts to fTy(0). We note that the nucleus acts to average over spin dependent a m p l i t u d e s , a n d f o r n u c l e i w h e r e N ~ Z , o v e r i s o - s p i n a s w e l l . In t h e heaviest nuclei where there are considerably more neutrons than protons s o m e r e s i d u a l i s o - s p i n d e p e n d e n c e w i l l r e m a i n in FNN if t h e r e is a s i z e a b l e a m o u n t in fT~.
336
B. MARGOLIS and C. L. TANG [
O",(r~ O"(r~
I
I
24
Si •11
26 rn b
31
37
C 11
7
26 31 37
I
I
1
1
I
3
5
10
15
20
kL.k (Gev/¢)
Fig. 3. The ratio cr(yA)~/(r(yN) as a function of the photon lab moment~tm k for 28Si and 1~C. crq~ = 12 rob, (Tp= (7co =26, 31, 3 7 m b .
R e t u r n i n g to the c a s e of i n f i n i t e e n e r g y we h a v e [2] FTV(0 ) = f y v ( O ) Y ( A ; O, ½aV) oc f v v ( O ) N ( A ;
O, ½aV) = F v v ( O ) ,
(14) (14a)
s o t h a t e l a s t i c s c a t t e r i n g of p h o t o n s at e x t r e m e l y high e n e r g y h a s a d e p e n d e n c e on a t o m i c n u m b e r l i k e t h a t f o r the s c a t t e r i n g of v e c t o r m e s o n s o r f o r p h o t o - p r o d u c t i o n of v e c t o r m e s o n s . In fact f o r m u l a (12) o b t a i n s if one a s s u m e s v e c t o r d o m i n a n c e f r o m the b e g i n n i n g r a t h e r t h a n c o n s i d e r i n g the s u m of a m p l i t u d e s f o r p r o c e s s e s (a) a n d (b), i.e. if one a s s u m e s v e c t o r d o m i n a n c e in p h o t o n - n u c l e u s i n t e r a c t i o n . We now e x a m i n e n u m e r i c a l l y the e n e r g y d e p e n d e n c e of the t o t a l p h o t o n n u c l e u s c r o s s s e c t i o n f o r e n e r g i e s f r o m a few GeV up, u s i n g eqs. (2a) a n d (13). T a b l e s 2 - 4 g i v e Iv(R) f o r a r a n g e of m a s s n u m b e r s a n d i n c i d e n t p h o t o n e n e r g i e s f o r crq~ = 12 m b a n d ~p =o"w = 26, 31 a n d 37 m b . U s i n g the v a l u e s
PHOTON-NUCLEUS CROSS SECTION
337
[7] (1/47r)y 2 = 0.52, 4.69 a n d 3.04 f o r V = p, ¢o a n d q), r e s p e c t i v e l y , we c a l c u l a t e a(yA)/a(yN) a n d t h e r e s u l t s a r e g i v e n in f i g s . 1, 2 a n d 3. One s e e s c o n s i d e r a b l e v a r i a t i o n with e n e r g y in the r a n g e a t t a i n a b l e with p r e s e n t a c c e l e r a t o r s , e s p e c i a l l y f o r the h e a v y e l e m e n t s .
3. DISCUSSION T h e r e s u l t s o b t a i n e d h e r e d e p e n d on the a s s u m p t i o n s that (i) the only i m p o r t a n t i n t e r m e d i a t e h a d r o n s t a t e s a r e the v e c t o r m e s o n s p, ~o and •, a n d (ii) v e c t o r d o m i n a n c e is v a l i d in p h o t o n - h a d r o n i n t e r a c t i o n . One would s t i l l h a v e a d e v i a t i o n f r o m the r e l a t i o n s h i p otyA) cc A if the a s s u m p t i o n s w e r e v i o l a t e d a s l o n g a s i n t e r m e d i a t e h a d r o n s c o n t r i b u t e s i g n i f i c a n t l y , but the r e s u l t s w o u l d d i f f e r i n d e t a i l f r o m t h o s e found h e r e . T h e m o s t s t r i k i n g f e a t u r e of the c a l c u l a t i o n s p e r f o r m e d a b o v e i s the s t r o n g effect due to the n o n - z e r o m a s s of the v e c t o r m e s o n s at e n e r g i e s of a few GeV, on the r a t i o (r(yA)/~(yN) f o r h e a v y n u c l e i . We m i g h t t e r m t h i s a n a p p a r e n t v i o l a t i o n of v e c t o r d o m i n a n c e . G i v e n t h a t v e c t o r d o m i n a n c e is v a l i d , it i s m e r e l y a n o f f - s h e l l effect due to t h e m i s - m a t c h of p h o t o n a n d v e c t o r m e s o n w a v e n u m b e r due to the v e c t o r m e s o n m a s s . T h i s d i f f e r e n c e of wave n u m b e r h a r d l y a f f e c t s the r e s u l t s f o r a n u c l e u s a s s m a l l a s C a r b o n but is i m p o r tant for larger nuclei. We s e e f r o m eq. (13) t h a t s i n c e y2 is s e v e r a l t i m e s s m a l l e r t h a n ~o- or ¢1 y ~ , t h e t e r m s c o r r e s p o n d i n g to V = ¢o o r ~ c o n t r i b u t e w e a k l y . It is to b e "F n o t e d t h a t a c c o r d i n g to the q u a r k m o d e l one e x p e c t s (rco = ap. T h e r a t i o a(yA)/cr(yN) f o r g i v e n A d e p e n d s t h e n m a i n l y on the e f f e c t i v e n u c l e o n d e n s i t y g i v e n by eq. (4), on crp a n d on the l o n g i t u d i n a l m o m e n t u m t r a n s f e r qp. We c o m m e n t on t h e r e l i a b i l i t y of the r e s u l t s o b t a i n e d h e r e . T h e a p p r o a c h u s e d i s of a n o p t i c a l m o d e l c h a r a c t e r . S i m i l a r c o n s i d e r a t i o n s h a v e b e e n f o u n d to g i v e c o n s i s t e n t r e s u l t s i n p h o t o p r o d u c t i o n of v e c t o r m e s o n s [2, 7] f o r a r a n g e of n u c l e i o v e r the p e r i o d i c t a b l e . It h a s a l s o b e e n u s e d s u c c e s s f u l l y f o r e l a s t i c s c a t t e r i n g of p r o t o n s at 20 GeV (ref. [8]). It i s e x p e c t e d t h a t , a l t h o u g h one i s not u s i n g a d e t a i l e d n u c l e a r m o d e l , one will h a v e the same success here. We wish to thank Professor D. G. Stairs for raising a question which led to this investigation. One of us (B.M.) also thank Professor K. Gottfried for a useful discussion.
REFERENCES [i] L. Stodolsky, Phys. Rev. Letters 18 (1967) 135. [2] K. S. KSlbig and B. Margolis, Nucl. Phys. B6 (1968) 85. [3] R. J. Glauber, Boulder lectures in theoretical physics, Vol. I (1958) (Interscience Publ. Inc., New York, 1959). [4] J. S. Bell, CERN preprint TH.887 (1968). [5] R. J. Glauber, High-energy physics and nuclear structure (North-Holland Publ. Comp., Amsterdam, 1967) p. 311.
338
B. MARGOLIS and C. L. TANG
[6] R. Hofstadter, Revs. Mod. P h y s . 103 (1956) 1454; L. R. B. Elton, Nuclear sizes (Oxford University Press, 1961). [7] S. C. C. Ting, Electromagnetic interactions, rapporteur's summary at the 14th Int. Conf. on high-energy physics, Vienna, September 1968, DESYInternal Report F31/4. [8] R. J. Glauber, private communication.
Nuclear Physics B10 (1969)329-338. North-Holland Publ. Comp., A m s t e r d a m
P H O T O N - N U C L E U S ELASTIC AND TOTAL CROSS SECTIONS B. M A R G O L I S * and C. L. T A N G * Institute of Theoretical Physics, McGill University, Montreal 110, Canada
Received 3 F e b r u a r y 1969 Abstract: The forward s c a t t e r i n g amplitude and hence the total cr o ss section for photons incident on nuclei is calculated in the eikonal approximation. The s c a t tering amplitude is taken to be the sum of two types of coherent amplitudes; (a) el as t i c scattering from each nucleon and (b) a r e g e n e r a t i v e amplitude c o r r e sponding to the production of the v e c t o r mesons p, W and (p on one nucleon and radiative capture on another. A considerable effect due to the finite mass of the v e c t o r mesons is seen at e n e r g i e s up to 10 GeV in heavy nuclei. This manifests itself as an apparent violation of v e c t o r dominance for the photon-nucleus i n t e r action which disappears at v e r y high e n e r g i e s if v e c t o r dominance is valid for the photon-nucleon interaction.
1. I N T R O D U C T I O N W e c a l c u l a t e t h e f o r w a r d a m p l i t u d e f o r e l a s t i c s c a t t e r i n g of p h o t o n s f r o m a t o m i c n u c l e i at h i g h e n e r g i e s in t h e e i k o n a l a p p r o x i m a t i o n . T h i s a m p l i t u d e , to l o w e s t o r d e r in t h e f i n e s t r u c t u r e c o n s t a n t , c o n s i s t s of t w o p a r t s : (a) t h e c o h e r e n t s u m of e l a s t i c s c a t t e r i n g a m p l i t u d e s f r o m e a c h nuc l e o n (a ' o n e - s t e p ' p r o c e s s ) and (b) t h e r e g e n e r a t e d a m p l i t u d e w h i c h c o m e s f r o m p h o t o - p r o d u c t i o n of a h a d r o n on one n u c l e o n of t h e n u c l e u s , f o l l o w e d by r a d i a t i v e c a p t u r e on a n o t h e r n u c l e o n (a ' t w o - s t e p ' p r o c e s s ) . We a s s u m e t h a t a l l t w o - b o d y p r o c e s s e s h a v e a m p l i t u d e s w h i c h a r e i n d e p e n d e n t of s p i n and i s o s p i n . We c o m m e n t on d e v i a t i o n s f u r t h e r on. T h e r e s u l t s s h o w in d e t a i l , c h a r a c t e r i s t i c s d i s c u s s e d q u a l i t a t i v e l y by S t o d o l s k y [1]. T h e r e g e n e r a t e d a m p l i t u d e i n t e r f e r e s d e s t r u c t i v e l y w i t h t h e ' o n e - s t e p ' a m p l i t u d e (a). At i n f i n i t e e n e r g y , a s s u m i n g t h a t t h e only i m p o r t a n t i n t e r m e d i a t e h a d r o n s a r e t h e v e c t o r m e s o n s p, w and (P an d a s s u m i n g v e c t o r d o m i n a n c e in p h o t o n - n u c l e o n i n t e r a c t i o n s , one f i n d s t h a t p h o t o n s s c a t t e r f r o m n u c l e i w i t h a d e p e n d e n c e on m a s s n u m b e r A, t h e s a m e a s f o r s c a t t e r i n g of v e c t o r m e s o n s . At f i n i t e e n e r g i e s h o w e v e r t h e n o n - z e r o m a s s of t h e v e c t o r m e s o n s a c t s to d e c r e a s e t h e r e g e n e r a t i v e a m p l i t u d e an d o n e g e t s a n A - d e p e n d e n c e c l o s e r to t h a t f o r t h e p u r e ' o n e - s t e p ' p r o c e s s . T h i s i s d u e to a p h a s e m i s - m a t c h b e t w e e n t h e photon an d t h e i n t e r m e d i a t e h a d r o n w a v e s in t h e ' t w o - s t e p ' p r o c e s s e s . * Supported in part by the National H e s e a r c h Council of Canada.
:330
B. M A R G O L I S a n d C. L. T A N G
2. CALCULATION OF PHOTON SCATTERING According to the eikonal a p p r o x i m a t i o n [2-4] we w r i t e the o n e - s t e p a m plitude
w h e r e f~7(O) is the f o r w a r d a m p l i t u d e for e l a s t i c photon-nucleon s c a t t e r i n g , and the t w o - s t e p a m p l i t u d e
F (b),Yy(O) ' -~27r ~x fyx(0)fx)'(0) Ix (A, qx; 0, ½Crx)
(2)
Ix(A, qx; 0,½ ~rx) Z 'T •
A2/d2b fide" e-qx
_!
,,
f
--OO
Z'
Z'~
f
×
1
dz' Px(b, z') e ~crx A
f
px(b, z')dz' eiqx z' .
(2a)
--OO
H e r e x is any p a r t i c l e that can b e p r o d u c e d c o h e r e n t l y by photons i n c i d e n t on the nucleus, k is the photon wave n u m b e r and ~x is the total c r o s s s e c tion f o r p a r t i c l e x incident on a nucleon. It is a s s u m e d h e r e that the x - n u cleon s m a l h - a n g l e s c a t t e r i n g a m p l i t u d e is p u r e i m a g i n a r y . O t h e r w i s e ex is to be r e p l a c e d by Crx(1 - i~x) w h e r e ~x is the r a t i o of r e a l to i m a g i n a r y p a r t of the f o r w a r d s c a t t e r i n g amplitude. T h e longitudinal m o m e n t u m t r a n s f e r qx = m x 2 / ( k + P x ) • The n u c l e a r c e n t r e - o f - m a s s motion is neglected. T h e quantities m x and Px a r e the i n t e r m e d i a t e hadron m a s s and m o m e n t u m r e s p e c t i v e l y . The nucleus is taken as a ' p i e c e of n u c l e a r m a t t e r ' of single p a r t i c l e density Po
Po(r) = Po(b, z) = 1+ e + ( r - a ) / c
(3)
T h e justification f o r this has been d i s c u s s e d s e v e r a l t i m e s [3-5]. The Saxon-Woods or F e r m i f o r m (3) has two p a r a m e t e r s a and c which a r e t a k e n f r o m e l e c t r o n s c a t t e r i n g data (6) and given in t a b l e 1. The neutron and proton d e n s i t i e s a r e a s s u m e d to be the s a m e . Due to the m o m e n t u m t r a n s f e r dependence of the two-body a m p l i t u d e s f ~ x ( q 2) and f x x ( q 2) (the l a t t e r c o r r e s p o n d i n g to e l a s t i c s c a t t e r i n g of x by a nucleon) the ' e f f e c t i v e d e n s i t y ' that a p p e a r s in eq. (2a) is Px(r) where
= _ _1 f F(;~) e - ~ 02x" - eiX" r d3 X (2~) 3
(4)
331
I~HOTON-NUCLEUS CROSS SECTION Table 1 The Saxon-Woods parameters a and c of formula (3) from ref. [6].
Target
12 C
28Si
58Ni
115in
181 T a
208pb
a (fro)
0.42
0.637
0.57
0.523
0.637
0.523
c (fro)
2.29
2.946
4.258
5.25
6.448
6.458
Table 2 The function ~-In(R)(A;qp'0,½~p) given by f o r m u l a (2a). ffp = 26mb
~'----.~ k(GeV/c)
3
5
12 C
1.68
28Si
3.64
Target
10
15
20
2.26
2.55
2.61
2.63
2.66
5.85
7.15
7.43
7.52
7.62
~
58Ni
14.4
18.9
19.8
20.2
20.6
115in
18.7
7.82
35.6
47.9
50.7
51.7
53.0
181Ta
21.5
49.4
74.8
81.0
83.3
86.4
208pb
33.6
68.6
97.7
104
107
111
~p = 3 1 m b 12C
1.96
28Si
4.31
58Ni
9.43
2.61 6.73
2.93 8.15
3.00 8.45
3.02 8.56
3.05 8.66
16.6
21.3
22.3
22.7
23.1
115In
22.8
40.8
53.3
56.1
57.1
58.4
181Ta
27.3
57.4
83.4
89.6
91.9
95.0
41.9
78.9
208pb
108
115
117
120
ffp = 37 mb 12C 28Si
2.29 5.08
3.00 7.72
3.35 9.24
3.42 9.56
58Ni
11.3
19.0
23.9
24.9
115In
27.7
46.3
58.8
61.5
181Ta
34.2
66.2
92.2
98.3
208pb
51.7
89.9
118
124
3.45 9.67 25.3 62.5
3.48 9.79 25.8 63.8
100
103
127
130
The r a n g e [7] in f o r m u l a (6),bp = 8 (GeV/c) -2. The Saxon-Woods p a r a m e t e r s a and c in f o r m u l a (3) a r e taken f r o m ref. [6]. The m a s s m p = 765 MeV.
B. MARGOLIS and C. L. TANG
332
Table 3 The f u n c t i o n I(R)(A;qw, 0,½~o)) g i v e n by f o r m u l a (2a). ffW = 2 6 m b ~k(GeV/c) Target
3
5
12C
1.61
2.22
28Si
3.40
58Ni
7.16
5.70
10
15
20
2.54
2.61
2.62
7.11
14.0
18.7
7.40
7.51
2.66 7.62
19.7
20.1
20.6
l151n
17.1
34.3
47.5
50.5
51.6
53.0
181Ta
19.4
46.9
73.8
80.5
83.0
86.4
208pb
30.7
65.6
96.6
104
107
111
(~¢o = 31 mb 12C
1.88
2.57
2.92
3.00
3.02
3.05
28Si
4.04
6.57
8.10
8.43
8.54
8.66
58Ni
16.1
21.1
22.2
22.6
23.1
115In
21.1
8.70
39.4
52.8
55.9
57.0
58.4
181Ta
24.8
54.8
208pb
38.7
75.9
82.3
89.1
107
91.6
114
117
95.0 120
ff¢o = 37 m b 12C
2.20
28Si
4.78
2.95
3.34
7.54
3.42
9.19
3.45
9.53
58Ni
11.5
18.4
23.7
24.8
115In
25.8
45.0
58.4
61.3
181Ta
31.5
63.6
91.1
97.8
208pb
48.1
86.9
117
124
9.66 25.2 62.4
3.48 9.79 25.8 63.8
100
103
127
130
The r a n g e in f o r m u l a (6), bw = 8 ( G e V / c ) - 2 , o b t a i n e d f r o m r e f . [7]. T h e S a x o n Woods p a r a m e t e r s a and c in f o r m u l a (3) a r e t a k e n f r o m ref. [6]. The m a s s m W = = 783 MeV. where
F(X) = f po(r) We take the same
This is not generally meson.
(5)
-i~. r d3r.
s l o p e f o r f v x ( q 2) a n d f x x ( q 2 ) ,
fvx(q2)/f~x(O)
vector
e
= fxx(q2)/fxx(O)
true but it follows from
i.e.
(6)
= e-½bxq 2 . vector
dominance
if x is a
PHOTON-NUCLEUS CROSS SECTION
333
Table 4
The function
~ - ~ G~e V / c )k~ Target
i~orQ( A m ;q(fl' O, ½tYq~)given
3
5
by formula (2a).
10
15
20
oo
1.26
1.35
1.39
1.44
~
12C
0.30
0.83
28Si
0.33
1.63
3.31
58Ni
0.50
3.04
8.32
1151n
1.42
6.45
21.0
181Ta
1.48
6.06
28.5
208pb
2.65
40.2
10.1
3.77
3.95
i0.0
4.14
10.7
11.6
26.4
28.6
31.8
39.8
47.8
52.1
54.2
60.3
69.2
The range in formula (6), b~0 = 5 (GeV/c)-2, obtained from ref. [71]2 The SaxonWoods parameters a and c in formula (3) are from ref. [6]. (rcp mb. The mass m@ = 1.019 GeV.
At infinite e n e r g y qx = 0 and eq. (2) b e c o m e s
F(b)(o) = -k-2~i~ f ~ x ( 0 ) f x ~ ( 0 YY
) ~
[A
-N(A;
0,½~x) ]
x
(7)
where
N(A; 0, ~ax)l
1
= 2~x
f[1 - e - F a x Tx(b))d2b
(8)
with Tx(b )
: A J Px(b,z)dz .
(9)
--oO
We now a s s u m e that the i m p o r t a n t i n t e r m e d i a t e h a d r o n s a r e the v e c t o r m e s o n s p, ¢o and q~ and that v e c t o r d o m i n a n c e is v a l i d f o r the p h o t o n - n u c l e o n i n t e r a c t i o n . T h e x = V w h e r e the V a r e the v e c t o r m e s o n s p, ¢o and ~p and f y v ( q 21 = ~¼
[\ -Y2V'-I ~-) f v v ( q 21 •
(101
H e r e a is the f i n e - s t r u c t u r e c o n s t a n t and the VV a r e the v e c t o r d o m i n a n c e c o u p l i n g c o n s t a n t s . U s i n g the o p t i c a l t h e o r e m , I m f v v ( 0 ) = (Pv/47r)aV, we h a v e at infinite e n e r g y 2 -1 FVV(0) =
Y7 (0)
V7 (0) =
V
r 42
~V
-1
(11) l
334
B. MARGOLIS and C. L. TANG
I
I
I
l
Ill
17
80
IN
I 2
I S
J 10
I IS
J 20
Fig. 1. The ratio a ( y A ) / f f ( y / ~ as a function of the photon lab momentum k for 208Pb and 181Ta. The v e c t o r dominance coupling constants a r e as given in the text. ffq~ = 12 mb, ffp = cr¢o=26, 31, 3 7 m b ,
This infinite energy result can be derived as well using Glauber theory. The methods are similar to those for deriving vector meson photo-product i o n c r o s s s e c t i o n s [2] ( o n e - s t e p p r o c e s s e s ) . One t h e n h a s , u s i n g t h e o p t i c a l t h e o r e m , t h e t o t a l c r o s s s e c t i o n f o r p h o t o n s i n c i d e n t on n u c l e i . ~ 2 -1 ~(~A) = ~(vN) • v
(12) - ~ 2V - 1
A t f i n i t e e n e r g i e s one h a s , u s i n g e q s . (2), (2a) a n d (10)
PHOTON-NUCLEUS CROSS SECTION
1
I
I
I
335
I
IIII
in 11~
26 mb 31 3"/
Ni
s! 26 31 37
I
I 3
t 5
I 10
I 20
l
lS
Fig. 2. The ratio cr('yA)/~(%N) as a function of the photon lab momentum k for l15In and 58Ni. The v e c t o r dominance coupling constants are as given in the text. qq) = =12 rob, ~p =(Y¢o = 26, 31, 37rob.
a(TA) = Aa(]/N)
-
, v ItS> IVR)
= Relv
T2
•
-1
!t
i '
(13)
(13a)
In eqs. (12) and (13) cr(TN) is the photon-nucleon total cross section assumed independent of the spin and charge of the nucleon. The situation becomes more complex if there are appreciable spin and iso-spin dependent parts to fTy(0). We note that the nucleus acts to average over spin dependent a m p l i t u d e s , a n d f o r n u c l e i w h e r e N ~ Z , o v e r i s o - s p i n a s w e l l . In t h e heaviest nuclei where there are considerably more neutrons than protons s o m e r e s i d u a l i s o - s p i n d e p e n d e n c e w i l l r e m a i n in FNN if t h e r e is a s i z e a b l e a m o u n t in fT~.
336
B. MARGOLIS and C. L. TANG [
O",(r~ O"(r~
I
I
24
Si •11
26 rn b
31
37
C 11
7
26 31 37
I
I
1
1
I
3
5
10
15
20
kL.k (Gev/¢)
Fig. 3. The ratio cr(yA)~/(r(yN) as a function of the photon lab moment~tm k for 28Si and 1~C. crq~ = 12 rob, (Tp= (7co =26, 31, 3 7 m b .
R e t u r n i n g to the c a s e of i n f i n i t e e n e r g y we h a v e [2] FTV(0 ) = f y v ( O ) Y ( A ; O, ½aV) oc f v v ( O ) N ( A ;
O, ½aV) = F v v ( O ) ,
(14) (14a)
s o t h a t e l a s t i c s c a t t e r i n g of p h o t o n s at e x t r e m e l y high e n e r g y h a s a d e p e n d e n c e on a t o m i c n u m b e r l i k e t h a t f o r the s c a t t e r i n g of v e c t o r m e s o n s o r f o r p h o t o - p r o d u c t i o n of v e c t o r m e s o n s . In fact f o r m u l a (12) o b t a i n s if one a s s u m e s v e c t o r d o m i n a n c e f r o m the b e g i n n i n g r a t h e r t h a n c o n s i d e r i n g the s u m of a m p l i t u d e s f o r p r o c e s s e s (a) a n d (b), i.e. if one a s s u m e s v e c t o r d o m i n a n c e in p h o t o n - n u c l e u s i n t e r a c t i o n . We now e x a m i n e n u m e r i c a l l y the e n e r g y d e p e n d e n c e of the t o t a l p h o t o n n u c l e u s c r o s s s e c t i o n f o r e n e r g i e s f r o m a few GeV up, u s i n g eqs. (2a) a n d (13). T a b l e s 2 - 4 g i v e Iv(R) f o r a r a n g e of m a s s n u m b e r s a n d i n c i d e n t p h o t o n e n e r g i e s f o r crq~ = 12 m b a n d ~p =o"w = 26, 31 a n d 37 m b . U s i n g the v a l u e s
PHOTON-NUCLEUS CROSS SECTION
337
[7] (1/47r)y 2 = 0.52, 4.69 a n d 3.04 f o r V = p, ¢o a n d q), r e s p e c t i v e l y , we c a l c u l a t e a(yA)/a(yN) a n d t h e r e s u l t s a r e g i v e n in f i g s . 1, 2 a n d 3. One s e e s c o n s i d e r a b l e v a r i a t i o n with e n e r g y in the r a n g e a t t a i n a b l e with p r e s e n t a c c e l e r a t o r s , e s p e c i a l l y f o r the h e a v y e l e m e n t s .
3. DISCUSSION T h e r e s u l t s o b t a i n e d h e r e d e p e n d on the a s s u m p t i o n s that (i) the only i m p o r t a n t i n t e r m e d i a t e h a d r o n s t a t e s a r e the v e c t o r m e s o n s p, ~o and •, a n d (ii) v e c t o r d o m i n a n c e is v a l i d in p h o t o n - h a d r o n i n t e r a c t i o n . One would s t i l l h a v e a d e v i a t i o n f r o m the r e l a t i o n s h i p otyA) cc A if the a s s u m p t i o n s w e r e v i o l a t e d a s l o n g a s i n t e r m e d i a t e h a d r o n s c o n t r i b u t e s i g n i f i c a n t l y , but the r e s u l t s w o u l d d i f f e r i n d e t a i l f r o m t h o s e found h e r e . T h e m o s t s t r i k i n g f e a t u r e of the c a l c u l a t i o n s p e r f o r m e d a b o v e i s the s t r o n g effect due to the n o n - z e r o m a s s of the v e c t o r m e s o n s at e n e r g i e s of a few GeV, on the r a t i o (r(yA)/~(yN) f o r h e a v y n u c l e i . We m i g h t t e r m t h i s a n a p p a r e n t v i o l a t i o n of v e c t o r d o m i n a n c e . G i v e n t h a t v e c t o r d o m i n a n c e is v a l i d , it i s m e r e l y a n o f f - s h e l l effect due to t h e m i s - m a t c h of p h o t o n a n d v e c t o r m e s o n w a v e n u m b e r due to the v e c t o r m e s o n m a s s . T h i s d i f f e r e n c e of wave n u m b e r h a r d l y a f f e c t s the r e s u l t s f o r a n u c l e u s a s s m a l l a s C a r b o n but is i m p o r tant for larger nuclei. We s e e f r o m eq. (13) t h a t s i n c e y2 is s e v e r a l t i m e s s m a l l e r t h a n ~o- or ¢1 y ~ , t h e t e r m s c o r r e s p o n d i n g to V = ¢o o r ~ c o n t r i b u t e w e a k l y . It is to b e "F n o t e d t h a t a c c o r d i n g to the q u a r k m o d e l one e x p e c t s (rco = ap. T h e r a t i o a(yA)/cr(yN) f o r g i v e n A d e p e n d s t h e n m a i n l y on the e f f e c t i v e n u c l e o n d e n s i t y g i v e n by eq. (4), on crp a n d on the l o n g i t u d i n a l m o m e n t u m t r a n s f e r qp. We c o m m e n t on t h e r e l i a b i l i t y of the r e s u l t s o b t a i n e d h e r e . T h e a p p r o a c h u s e d i s of a n o p t i c a l m o d e l c h a r a c t e r . S i m i l a r c o n s i d e r a t i o n s h a v e b e e n f o u n d to g i v e c o n s i s t e n t r e s u l t s i n p h o t o p r o d u c t i o n of v e c t o r m e s o n s [2, 7] f o r a r a n g e of n u c l e i o v e r the p e r i o d i c t a b l e . It h a s a l s o b e e n u s e d s u c c e s s f u l l y f o r e l a s t i c s c a t t e r i n g of p r o t o n s at 20 GeV (ref. [8]). It i s e x p e c t e d t h a t , a l t h o u g h one i s not u s i n g a d e t a i l e d n u c l e a r m o d e l , one will h a v e the same success here. We wish to thank Professor D. G. Stairs for raising a question which led to this investigation. One of us (B.M.) also thank Professor K. Gottfried for a useful discussion.
REFERENCES [i] L. Stodolsky, Phys. Rev. Letters 18 (1967) 135. [2] K. S. KSlbig and B. Margolis, Nucl. Phys. B6 (1968) 85. [3] R. J. Glauber, Boulder lectures in theoretical physics, Vol. I (1958) (Interscience Publ. Inc., New York, 1959). [4] J. S. Bell, CERN preprint TH.887 (1968). [5] R. J. Glauber, High-energy physics and nuclear structure (North-Holland Publ. Comp., Amsterdam, 1967) p. 311.
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[6] R. Hofstadter, Revs. Mod. P h y s . 103 (1956) 1454; L. R. B. Elton, Nuclear sizes (Oxford University Press, 1961). [7] S. C. C. Ting, Electromagnetic interactions, rapporteur's summary at the 14th Int. Conf. on high-energy physics, Vienna, September 1968, DESYInternal Report F31/4. [8] R. J. Glauber, private communication.