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The K− nucleon interaction in K mesic atoms
Volume 38B, n u m b e r 3
THE
PHISICS
K-
NUCLEON
LETTERS
INTERACTION
7 F e b r u a r y 1972
IN
K MESIC
ATOMS $
W. A. B A R D E E N :~ CERN - Geneva and Institule of Theoretical Physics Stanford University, Stanford, Calif., USA and E. W A Y N E T O R I G O E Institute of Theoretical P h y s i c s , Stanford University, Stanford, Calif., USA
Received 28 December
1971
A study of the K- nucleon interaction is made as it pertains to the absorption of K- mesons in nuclei. Effective interaction strenghts are derived which yield good agreement with recent X ray experiments that determine both the strong interaction energy shifts and widths.
K m e s i c a t o m s c a n b e u s e d to p r o b e o u r u n d e r s t a n d i n g of t h e K - n u c l e o n i n t e r a c t i o n a s well as the nuclear surface. The theoretical a n a l y s i s of K m e s o n a b s o r p t i o n in n u c l e i i s b a s e d u p o n t h e c o n s t r u c t i o n of a K n u c l e u s p s e u d o p o t e n t i a l t h a t c a n b e r e l a t e d to t h e f r e e K nucleon scattering amplitudes. However, there are difficulties associated with the simple applic a t i o n of t h e p s e u d o p o t e n t i a l t e c h n i q u e . T h e e f fective K- nucleus interaction is strongly abs o r p t i v e w h i c h c a n c a u s e d i s t o r t i o n s of t h e K meson wave-function making perturbation techniques unreliable. Also the K nucleon interaction i s e x p e c t e d to b e s t r o n g l y e n e r g y d e p e n d e n t in t h e r a n g e of e n e r g i e s r e a c h e d d u r i n g t h e a b s o r p t i o n p r o c e s s . In t h i s p a p e r , w e p r e s e n t a p s e u d o potential method which can incorporate these effects. Pseudopotential calculations by Ericson and Scheck [I], by Bloom et al. [2], by the present authors [3] give results which are in good agreement with the initial studies of X ray yields by Wiegand et al. [4]. While the experiments confirm our general picture of K meson absorption, such as absorption from predominantly circular
$ R e s e a r c h s p o n s o r e d in part by the Air F o r c e Office of Scientific R e s e a r c h , U.S. Air F o r c e Contract No F44620-71-C-0044. Alfred P. Sloan Foundation Fellow. N. S.F. P r e d o c t o r a l Fellow.
orbits, they are not sufficiently precise as to distinguish the different treatments of the absorption process or to indicate possible modifications of nuclear surface densities. The recent experiments of Backenstoss et al. [5, 6] which determine the strong interaction energy shifts and line broadenings may make such a distinction possible. Indeed, on the basis of the initial studies on sulphur [5], Krell [7] has argued that there must be strong distortion effects occurring in the lower electronic orbits due to the strongly absorptive potential obtained from the free K meson scattering lengths. Hence perturbation treatments of the lower levels are not reliable. Krell suggests that the distortion effects may be incorporated by solving the eigenvalue problem for the K meson with strong interaction pseudopotential included as a part of the potential in the Klein-Gordon equation describing the dynamics of the K meson. Using this approach, Krell finds that agreement with the sulphur data can be attained with a pseudopotential whose absorptive part is in agreement with that expected from the free scattering lengths but whose real part is attractive rather than repulsive as expected from the free scattering lengths. Similar results have been obtained by Seki [8] in fitting the sulphur data. In our previous paper [3], we have argued that the presence of the Yo(1405) resonance in the K- proton channel can strongly influence 135
Volume 38B, number 3
PHYSICS
t h e a b s o r p t i o n p r o c e s s as t h e e n e r g i e s a v a i l a b l e to t h e K - p r o t o n s y s t e m d u r i n g a b s o r p t i o n a r e w e l l b e l o w the f r e e p a r t i c l e t h r e s h o l d and n e a r t h e r e s o n a n c e e n e r g y . One q u a l i t a t i v e r e s u l t of t h e r e s o n a n c e is an e n h a n c e m e n t of t h e a b s o r p t i o n r a t e s on p r o t o n s r e l a t i v e to n e u t r o n s a s c a n b e s e e n in o u r e s t i m a t e s of the a b s o r p t i v e p a r t s of the p s e u d o p o t e n t i a l s t r e n g h t s in r e f . [3]. A n o t h e r q u a l i t a t i v e e f f e c t that is p o s s i b l e if t h e Yo r e s o n a n c e d o m i n a t e s in t h e K - p r o t o n c h a n nel is a c h a n g e in s i g n of the r e a l p a r t of the s c a t t e r i n g a m p l i t u d e n e a r the r e s o n a n c e e n e r g y . T o s e e w h e t h e r t h i s e f f e c t can a c c o u n t for t h e r e s u l t s of K r e l l ' s a n a l y s i s of the s u l p h u r data, we determine energy averaged pseudopotential p a r a m e t e r s by t h e m e t h o d s u s e d in r e f . [3]. W e g i v e a b r i e f r e v i e w of the p s e u d o p o t e n t i a l m e t h o d s u s e d in r e f . [3]. We c o n s t r u c t a K - n u c l e o n p s e u d o p o t e n t i a l w h i c h , in B o r n a p p r o x i m a t i o n , g i v e s the c o r r e c t K- n u c l e o n s w a v e s c a t t e r i n g a m p l i t u d e s . As the K- n u c l e o n i n t e r a c t i o n is s h o r t r a n g e , we u s e a p o t e n t i a l of t h e form V K N ( X - y ) = ( U - iW) 5 ( x - y )
(1)
where -
L E T T ER S
7 February
reduced mass, p. = rnKmN/(rnK+mN). nucleus pseudopotential is constructed of two-body potentials
1972
The Kas a sum
VKA = j~. ( % - i W j ) 5 ( x - y j )
(3)
w h e r e the sum (j) r a n g e s o v e r all n u c l e o n s in the nucleus. The effective strong interaction p o t e n t i a l s e e n by t h e K m e s o n is t h e n VKA(X) = (Up-iWp)pp(X)+(Un-iWn)Pn(X)
(4)
w h e r e p p ( x ) and Pn(X) a r e t h e r e s p e c t i v e p r o t o n and n e u t r o n d e n s i t i e s . As a r g u e d in r e f . [3], t h e p s e u d o p o t e n t i a l in eq. (4) cannot b e u s e d due to the s t r o n g e n e r g y d e p e n d e n c e of f r e e s c a t t e r i n g a m p l i t u d e s which determine the pseudopotential parameters through eq. (2). T h e e n e r g y d e p e n d e n c e of f i t s to the s c a t t e r i n g data by Kim [9] and by M a r t i n and Sakitt [10] is shown in fig. 1. H o w e v e r , as shown in ref. [3], e f f e c t i v e p s e u d o p o t e n t i a l p a r a m e t e r s c o u l d b e d e r i v e d as a s u i t a b l e e n e r g y a v e r a g i n g of t h e s c a t t e r i n g a m p l i t u d e s in fig. 1. H e n c e t h e s c a t t e r i n g a m p l i t u d e s in eq. (2) m u s t b e r e p l a c e d by the e n e r g y a v e r a g e d s c a t t e r i n g a m p l i t u d e s , fKN,
U+ iW = 4~(t/2/2 ~ ) / K N
(2)
J K N is the f r e e s c a t t e r i n g a m p l i t u d e and p is t h e .
.
.
.
.
.
KIM
f K N ~ f K N : f d W f K N ( W ) P(W) .
(5)
T h e f u n c t i o n P(W) is the p r o b a b i l i t y of finding t h e K - a b s o r b e d n u c l e o n s y s t e m with t o t a l c e n t r e o f - m a s s e n e r g y W. If we n e g l e c t i n t e r a c t i o n s b e t w e e n t h e K a b s o r b e d n u c l e o n s y s t e m and the r e s i d u a l nuc l e u s , t h e n t h e c e n t r e - o f - m a s s e n e r g y is the i n i t i a l e n e r g y l e s s t h e k i n e t i c e n e r g y of the m o t i o n of t h e c e n t r e - o f - m a s s ,
0
Wi = r n K - C K +
-1
/I 3'
MARTIN
AND SAKITT
--
0
ii
...--------
-1 13~0 13L70 13l~ 13[~" 1~0 1410i-- 14~
1430 14L~0 1/450
W (MeV)
Fig. 1. The energy dependence of the free K- nucleon scattering amplitudes for the fits of Kim and of Martin and Sakitt.
136
r ni- e . -~P 2cm / 2 W .z
(6)
w h e r e ~K and Ei a r e the K m e s o n and a b s o r b e d n u c l e o n b i n d i n g e n e r g i e s r e s p e c t i v e l y and P c m is t h e m o m e n t u m of the c e n t r e - o f - m a s s . T h e e n e r g y d i s t r i b u t i o n s a r e then s i m p l y r e l a t e d to the centre-of-mass momentum distributions. T h e s e a r e g i v e n by t h e F o u r i e r t r a n s f o r m of t h e p r o d u c t of t h e K m e s o n and n u c l e o n w a v e - f u n c t i o n s u n d e r the a s s u m p t i o n of a s i n g l e p a r t i c l e d e s c r i p t i o n of t h e n u c l e u s and the s h o r t r a n g e n a t u r e of t h e f o r c e in eq. (1). F o r n u c l e o n s t a t e s of d e f i n i t e a n g u l a r m o m e n t u m we can d e r i v e t h e e x p l i c i t f o r m u l a f o r P(W), a n a l o g o u s to eq. (26) in r e f . [3]
PHYSICS
Volume 38B, n u m b e r 3
x
/
dpp2izij(P) l2 6 ( W - Wi(P))
(7)
0 w h e r e Zij(P) i s t h e o v e r l a p of t h e r a d i a l w a v e functions
Zij(P) = / drr2 jj(Prl UiL (r) Rnl(r) . (8) 0 We n o t e t h a t jj(Pr) i s t h e s p h e r i c a l B e s s e l f u n c t i o n , UiL(r) t h e n u c l e o n w a v e - f u n c t i o n of o r b i t a l a n g u l a r m o m e n t u m L , a n d Rnl(r) the r a d i a l wave-function for the K meson; the sum over i is over all nucleon states. U s i n g t h e s h e l l m o d e l d e s c r i p t i o n of t h e n u c l e u s a s g i v e n in r e f . [3] a n d u n d i s t o r t e d K meson wave-functions, the energy distributions w e r e c a l c u l a t e d u s i n g eqs. (7) a n d (8). T h e r e s u i t s of t h e s e c a l c u l a t i o n s f o r s e l e c t e d n u c l e i and indicated circular orbital angular momentum s t a t e s f o r t h e K m e s o n a r e s h o w n in fig. 2 f o r
OO~-- P(W) 004.~003 96~,o(1=4) 002 001
,,~f "-?s!
'°'°<"'> " - A / 'G0(1:2)
0
1.
%
ik
/" 1.
13"/0
1380 1390 1400 IZ+lO 1420 W MeV
Fig. 2. The probability function for K- nucleon s y s t e m to have an energy, W, in the c e n t r e - o f - m a s s .
LETTERS
7 F e b r u a r y 1972
1 6 0 ( / = 2), 4 0 C a ( l = 3), 9 6 M o ( / = 4). In c o m p a r i s o n w i t h fig. 1, the e n e r g y d i s t r i b u t i o n s s h o w a s t r o n g p e a k in t h e r e g i o n of the z e r o of the r e a l p a r t of the K- p r o t o n a m p l i t u d e with a t a i l e x t e n d i n g to l o w e r e n e r g i e s o r m o r e a t t r a c t i v e s c a t t e r ing a m p l i t u d e s . U s i n g t h e e n e r g y d i s t r i b u t i o n s of fig. 2 and t h e two s e t s of s c a t t e r i n g a m p l i t u d e s s h o w n in fig. 1, we c a n c o m p u t e t h e e n e r g y a v e r a g e d s c a t t e r i n g a m p l i t u d e s d e f i n e d in eq. (5). T h e r e s u t l s of t h e s e c a l c u l a t i o n s a r e p r e s e n t e d in t a b l e 1 a l o n g w i t h t h e t h r e s h o l d v a l u e s of the s c a t t e r i n g a m p l i t u d e s or the scattering lenghts. W e s e e t h a t t h e c o n j e c t u r e of K r e l l i s v e r i f i e d f o r the r e s o n a n c e d o m i n a t e d a m p l i t u d e s of fig. 1. The energies available during the absorption p r o c e s s a r e s u f f i c i e n t l y b e l o w t h r e s h o l d a s to r e f l e c t t h e c h a n g e in s i g n of the r e a l p a r t of t h e s c a t t e r i n g a m p l i t u d e s . By c o m p a r i n g t h e a b s o r p t i r e p a r t s of the a v e r a g e d a m p l i t u d e s , we s e e the l a r g e e n h a n c e m e n t of the a b s o r p t i o n s t r e n g h t on p r o t o n s . A s s u g g e s t e d in r e f . [3], t h i s e n h a n c e merit i m p l i e s t h a t X r a y s t u d i e s in K m e s i c a t o m s a r e not s e n s i t i v e to t h e n e u t r o n d i s t r i b u tions. T h e e n e r g i e s a v a i l a b l e d u r i n g a b s o r p t i o n will of c o u r s e v a r y w i t h the n u c l e u s a n d a l s o w i t h the i n i t i a l s t a t e of t h e K m e s i e a t o m . H o e v e r , f r o m t a b l e 1, we o b s e r v e the e f f e c t i v e n e u t r o n p a r a m e t e r s a r e e s s e n t i a l l y c o n s t a n t and a p p r o x i m a t e l y t h e s a m e f o r the two f i t s to t h e s c a t t e r i n g d a t a . T h e a b s o r p t i v e p a r t of t h e p r o t o n a m p l i t u d e s a r e a l s o e s s e n t i a l l y c o n s t a n t but d i f f e r c o n s i d e r a b l y b e t w e e n t h e two f i t s r e f l e c t i n g the m u c h b r o a d e r r e s o n a n c e p e a k f o r t h e K i m fit. T h e r e a l p a r t of t h e p r o t o n a m p l i t u d e i s the m o s t s e n s i t i v e to t h e e n e r g y a v e r a g i n g a l t h o u g h t h e s i g n will c l e a r l y r e m a i n o p p o s i t e to t h e t h r e s h o l d v a l u e f o r t h e r e s o n a n c e d o m i n a t e d a m p l i t u d e s i n fig. 1. T h e e n e r g y a v e r a g e d a m p l i t u d e s of t a b l e 1 c a n b e u s e d to m a k e a d e t a i l e d c o m p a r i s o n w i t h t h e r e c e n t e x p e r i m e n t s of B a c k e n s t o s s et al. [6] which observe with some precision the strong i n t e r a c t i o n e n e r g y s h i f t s a n d l i n e b r o a d e n i n g s of t h e K m e s i c X r a y s f o r a n u m b e r of l i g h t n u c l e i .
Table 1 The energy a v e r a g e d s c a t t e r i n g amplitudes.
i- -p-Kim-/i<-n fm
fm
160(/ = 2)
0.93+i2.27
0.10+i0.47
1 . 1 2 + i 1.48
0.14+i0.40
40Ca(/ = 3)
0.73 ÷i 2.24
0.09+ i0.46
0 . 9 4 + t 1.60
0.13+ i0.39
96Mo(/ = 4)
0 . 7 9 + i 2.32
0.10+ i 0.48
1.01 + i 1.62
0.13 + i 0.42
Threshold
-0.89 +i 0.62
-0.13 +i 0.51
- 0 . 8 8 + i 0.62
- 0 . 0 9 + i 0.54
137
Volume 38B. n u m b e r 3
PHYSICS
LETTERS
7 F e b r u a r y 1972
Table 2 Pseudopotential calculation of the energy shifts and widths. T h r e s h o l d p a r a m e t e r calculations in b r a c k e t s . Atom t r a n s i t i o n
r r . m . s . (fm)
Kim
r0.5(fm ) /(fro)
M.S.
Experimental
EL(keV)
I'L(keV)
~L(keV)
FL(keV)
EL(keV)
FL(keV)
~O~3 0
(-0.23)
O.65 (0.24)
-O.25 (-0.22)
0.72 (0.25)
-0.208 ± 0.035 0.81±0.10
10B((3,2) ~ (2,1))
2.45
-
l l B ( ( 3 , 2 ) -~ (2,1))
2.42
-
-0.30 (-0.23)
0.64 (0.24)
-0.26 (-0.23)
0.70 (0.26)
-0.167±0.035
0.70±0.08
12C((3,2) ~ (2,1))
2.42
-
-0.80 (-0.59)
1.44 (0.55)
-0.67 (-0.5S)
1.58 (0.58)
-0.59±0.08
1.73±0.15
31p((4,3)" ~ (3,2))
3.188
3.34
2.45
-0.52 (-0.49)
1.65 (0.58)
-0.36 (-0.48)
1.68 (0.60)
-0.33 ±0.08
1.44±0.12
32S~,4,3,"' -~(3,2))
3.244
3.33
2.60
-0.88 (-0.82)
2.73 (0.94)
-0.61 (-0.80)
2.78 (0.98)
-0.55 ±0.06
2.33 ±0.06
35C1((4, 3) ~ (3,2))
3.335
3.42
2.60
-1.44 (-1.29)
4.06 (1.45)
-1.05 (-1.27)
4.28 (1.51)
-0.77±0.40
3.8±1.0
W e h a v e u s e d e n e r g y a v e r a g e d a m p l i t u d e s for 1 6 0 ( l = 2) g i v e n in t a b l e 1 to c o n s t r u c t the e f f e c t i v e p s e u d o p o t e n t i a l d e t e r m i n e d t h r o u g h e q s . (4) a n d (2). T h e s t r o n g i n t e r a c t i o n e n e r g y s h i f t s and widths were computed using a program d e v e l o p e d by K r e l l [11] w h i c h s o l v e s t h e e i g e n v a l u e p r o b l e m for t h e K l e i n - G o r d o n e q u a t i o n w i t h t h e p o t e n t i a l i n c l u d i n g the s t r o n g i n t e r a c t i o n p s e u d o p o t e n t i a l . T h e r e s u l t s of t h e s e c a l c u l a t i o n s f o r the i s o t o p e s 10B, l l B , 12C, 3 1 p , 32S, 35C1 a r e g i v e n in t a b l e 2 a n d c o m p a r e d w i t h t h e e x p e r i m e n t a l d a t a [6]. T h e p s e u d o p o t e n t i a l s h a p e s f o r the lighter isotopes were harmonic wells with root mean square radii determined from electron s c a t t e r i n g [12]. T w o p a r a m e t e r F e r m i w e l l s were used for the heavier isotopes with root mean square radii determined from the muonic a t o m s t u d i e s of B a c k e n s t o s s et al. [13]. F r o m t a b l e 2. we s e e t h a t t h e t h e o r e t i c a l p r e d i c t i o n s b a s e d on the r e s o n a n c e d o m i n a t e d a m p l i t u d e s a r e g e n e r a l l y in q u i t e good a g r e e m e n t w i t h t h e d a t a . C a l c u l a t i o n s b a s e d on t h e t h r e s h o l d s c a t t e r i n g l e n g t h s ( s h o w n in b r a c k e t s ) g e n e r a l l y g i v e w i d t h s a r e a f a c t o r of two to t h r e e too s m a l l . W e n o t e t h a t in t h i s r a n g e of p s e u d o p o t e n t i a l parameters the energy shift varies most strongly w i t h t h e i m a g i n a r y p a r t of the p s e u d o p o t e n t i a l , w h i l e the w i d t h d e p e n d s m o s t s t r o n g l y on the r e a l p a r t . c o n t r a r y to p e r t u r b a t i o n t h e o r y . W e o b s e r v e t h a t the M a r t i n - S a k i t t fit p a r a m e t e r s s e e m to g i v e a b e t t e r fit to t h e e n e r g y s h i f t s in t h e h e a v i e r n u c l e i t h a n t h e K i m fit p a r a m e t e r s . T h i s m a y i n d i c a t e t h a t t h e K i m fit g i v e s too large an imaginary part for the proton amplitude in the r e s o n a n c e r e g i o n . D e f i n i t e c o n c l u s i o n s concerning this point must await a more complete 138
a n a l y s i s of d a t a p a r t i c u l a r l y a s t u d y of the r a d i u s a n d a m p l i t u d e d e p e n d e n c e s of t h e m e a s u r e d quantities. T h e r e a r e of c o u r s e m a n y w a y s t h e s e c a l c u l a tions could be improved. For example, distorted K m e s o n w a v e - f u n c t i o n s s h o u l d be u s e d in c o m p u t i n g t h e e n e r g y a v e r a g i n g f u n c t i o n s . In p r i n c i p l e t h e e n e r g y a v e r a g i n g will d i f f e r d e p e n d i n g upon the K meson orbit as well as the isotope involved. T h e s e e f f e c t s a r e b e i n g s t u d i e d a n d will be r e ported elsewhere. T h e r e a r e a l s o b a s i c l i m i t a t i o n s to t h e p s e u d o p o t e n t i a l a p p r o a c h . T h e u s e of the f r e e s c a t t e r i n g a m p l i t u d e s i n v o l v e s the n e g l e c t of i n t e r a c t i o n s of t h e K m e s o n - n u c l e o n s y s t e m w i t h the r e s i d u a l n u c l e u s . W e e x p e c t t h i s a p p r o x i m a t i o n to b e r e a s o n a b l y good a s a b s o r p t i o n o c c u r s in the n u cleon periphery, but this assumption should be s t u d i e d in r e a l i s t i c a t o m s i t u a t i o n s . W e c o n c l u d e t h a t {he p s e u d o p o t e n t i a l m e t h o d w o r k s q u i t e w e l l w h e n s u i t a b l y m o d i f i e d to the k i n e m a t i c a l s i t u a t i o n found in K m e s i c a t o m s . T h e f i t s to the B a c k e n s t o s s d a t a in t a b l e 2 i m p l y t h a t the K- p r o t o n s c a t t e r i n g a m p l i t u d e m u s t b e resonance dominated without a large repulsive b a c k g r o u n d [14]. As o u r u n d e r s t a n d i n g of t h e a b s o r p t i o n p r o c e s s i m p r o v e s , we m a y b e a b l e to u s e K m e s i c a t o m s a s a p r o b e of the K- p r o t o n s c a t t e r i n g a m p l i t u d e s in the r e s o n a n c e r e g i o n a s w e l l a s a p r o b e of t h e n u c l e a r s u r f a c e . O n e of u s (W. A. B.) would l i k e to t h a n k t h e m e m b e r s of t h e B a c k e n s t o s s g r o u p a t C E R N f o r many useful discussions concerning the data and t h e t h e o r y . W . A . B . would a l s o l i k e to t h a n k
Volume 38B, number
3
PHYSICS
T . E . O . E r i c s o n , M. Kre[1 and F. S c h e c k f o r s t i m u l a t i n g d i s c u s s i o n s of the t h e o r y .
Refere~wes [1] T. E. (3. Ericson and F. Scheck, Nucl. Phys. 1319 (1970) 450. [2] S. D. Bloom, M. H. Johnson and E. T e l l e r , Phys. Rev. Letters 23 (1969) 28; UCRL-72807 (1970). [3] W. A. Bardeen and E,W. Torigoe, Phys. Rev. C3 (1971) 1785, [4] C.E. Wiegand. Phys. Rev. L e t t e r s 22 (1969) 1235: C. E. Wiegand and D. A. Mack, Phys. Rev. Letters 18 (1967) 685. [5] G. Backenstoss et al., Phys. L e t t e r s 32B (1970) 399,
L E T T E RS
7 February
1972
[6] G. Backenstoss et al., Phys. Letters 38B (1972) 181. [7] M.Krell. Phys. Rev. Letters 26 (1:971) 584. [8] R. Seki, Submitted to Phys. Rev. [9] J.K.Kim, Phys. Rev. Letters 19 (1967) 1074: F. Von IIippel and J. K. Kim, Phys..Rev. Letters 20 (1968) 1303. [i0] B. R. Martin and M. Sakitt, Phys. Rev. 183 (1969) 1345:183 (1969) 1352. [ii] M. Krell, Submitted to Computer Physics Communications. [12] R. Hofstadter and H. R. Collard, in LandoltB6rnstein, ed. K.H. Hellwege, (Springer Verlag, New York, 1967), groupI, Vol. 2, p. 32. ]13] G. Backenstoss et al., Phys. Letters 25B (1967) 547. [14] D. Cline, R. Laumann and J. Mapp, Phys. Rev. Letters 26 (1971) 1194.
139
THE
PHISICS
K-
NUCLEON
LETTERS
INTERACTION
7 F e b r u a r y 1972
IN
K MESIC
ATOMS $
W. A. B A R D E E N :~ CERN - Geneva and Institule of Theoretical Physics Stanford University, Stanford, Calif., USA and E. W A Y N E T O R I G O E Institute of Theoretical P h y s i c s , Stanford University, Stanford, Calif., USA
Received 28 December
1971
A study of the K- nucleon interaction is made as it pertains to the absorption of K- mesons in nuclei. Effective interaction strenghts are derived which yield good agreement with recent X ray experiments that determine both the strong interaction energy shifts and widths.
K m e s i c a t o m s c a n b e u s e d to p r o b e o u r u n d e r s t a n d i n g of t h e K - n u c l e o n i n t e r a c t i o n a s well as the nuclear surface. The theoretical a n a l y s i s of K m e s o n a b s o r p t i o n in n u c l e i i s b a s e d u p o n t h e c o n s t r u c t i o n of a K n u c l e u s p s e u d o p o t e n t i a l t h a t c a n b e r e l a t e d to t h e f r e e K nucleon scattering amplitudes. However, there are difficulties associated with the simple applic a t i o n of t h e p s e u d o p o t e n t i a l t e c h n i q u e . T h e e f fective K- nucleus interaction is strongly abs o r p t i v e w h i c h c a n c a u s e d i s t o r t i o n s of t h e K meson wave-function making perturbation techniques unreliable. Also the K nucleon interaction i s e x p e c t e d to b e s t r o n g l y e n e r g y d e p e n d e n t in t h e r a n g e of e n e r g i e s r e a c h e d d u r i n g t h e a b s o r p t i o n p r o c e s s . In t h i s p a p e r , w e p r e s e n t a p s e u d o potential method which can incorporate these effects. Pseudopotential calculations by Ericson and Scheck [I], by Bloom et al. [2], by the present authors [3] give results which are in good agreement with the initial studies of X ray yields by Wiegand et al. [4]. While the experiments confirm our general picture of K meson absorption, such as absorption from predominantly circular
$ R e s e a r c h s p o n s o r e d in part by the Air F o r c e Office of Scientific R e s e a r c h , U.S. Air F o r c e Contract No F44620-71-C-0044. Alfred P. Sloan Foundation Fellow. N. S.F. P r e d o c t o r a l Fellow.
orbits, they are not sufficiently precise as to distinguish the different treatments of the absorption process or to indicate possible modifications of nuclear surface densities. The recent experiments of Backenstoss et al. [5, 6] which determine the strong interaction energy shifts and line broadenings may make such a distinction possible. Indeed, on the basis of the initial studies on sulphur [5], Krell [7] has argued that there must be strong distortion effects occurring in the lower electronic orbits due to the strongly absorptive potential obtained from the free K meson scattering lengths. Hence perturbation treatments of the lower levels are not reliable. Krell suggests that the distortion effects may be incorporated by solving the eigenvalue problem for the K meson with strong interaction pseudopotential included as a part of the potential in the Klein-Gordon equation describing the dynamics of the K meson. Using this approach, Krell finds that agreement with the sulphur data can be attained with a pseudopotential whose absorptive part is in agreement with that expected from the free scattering lengths but whose real part is attractive rather than repulsive as expected from the free scattering lengths. Similar results have been obtained by Seki [8] in fitting the sulphur data. In our previous paper [3], we have argued that the presence of the Yo(1405) resonance in the K- proton channel can strongly influence 135
Volume 38B, number 3
PHYSICS
t h e a b s o r p t i o n p r o c e s s as t h e e n e r g i e s a v a i l a b l e to t h e K - p r o t o n s y s t e m d u r i n g a b s o r p t i o n a r e w e l l b e l o w the f r e e p a r t i c l e t h r e s h o l d and n e a r t h e r e s o n a n c e e n e r g y . One q u a l i t a t i v e r e s u l t of t h e r e s o n a n c e is an e n h a n c e m e n t of t h e a b s o r p t i o n r a t e s on p r o t o n s r e l a t i v e to n e u t r o n s a s c a n b e s e e n in o u r e s t i m a t e s of the a b s o r p t i v e p a r t s of the p s e u d o p o t e n t i a l s t r e n g h t s in r e f . [3]. A n o t h e r q u a l i t a t i v e e f f e c t that is p o s s i b l e if t h e Yo r e s o n a n c e d o m i n a t e s in t h e K - p r o t o n c h a n nel is a c h a n g e in s i g n of the r e a l p a r t of the s c a t t e r i n g a m p l i t u d e n e a r the r e s o n a n c e e n e r g y . T o s e e w h e t h e r t h i s e f f e c t can a c c o u n t for t h e r e s u l t s of K r e l l ' s a n a l y s i s of the s u l p h u r data, we determine energy averaged pseudopotential p a r a m e t e r s by t h e m e t h o d s u s e d in r e f . [3]. W e g i v e a b r i e f r e v i e w of the p s e u d o p o t e n t i a l m e t h o d s u s e d in r e f . [3]. We c o n s t r u c t a K - n u c l e o n p s e u d o p o t e n t i a l w h i c h , in B o r n a p p r o x i m a t i o n , g i v e s the c o r r e c t K- n u c l e o n s w a v e s c a t t e r i n g a m p l i t u d e s . As the K- n u c l e o n i n t e r a c t i o n is s h o r t r a n g e , we u s e a p o t e n t i a l of t h e form V K N ( X - y ) = ( U - iW) 5 ( x - y )
(1)
where -
L E T T ER S
7 February
reduced mass, p. = rnKmN/(rnK+mN). nucleus pseudopotential is constructed of two-body potentials
1972
The Kas a sum
VKA = j~. ( % - i W j ) 5 ( x - y j )
(3)
w h e r e the sum (j) r a n g e s o v e r all n u c l e o n s in the nucleus. The effective strong interaction p o t e n t i a l s e e n by t h e K m e s o n is t h e n VKA(X) = (Up-iWp)pp(X)+(Un-iWn)Pn(X)
(4)
w h e r e p p ( x ) and Pn(X) a r e t h e r e s p e c t i v e p r o t o n and n e u t r o n d e n s i t i e s . As a r g u e d in r e f . [3], t h e p s e u d o p o t e n t i a l in eq. (4) cannot b e u s e d due to the s t r o n g e n e r g y d e p e n d e n c e of f r e e s c a t t e r i n g a m p l i t u d e s which determine the pseudopotential parameters through eq. (2). T h e e n e r g y d e p e n d e n c e of f i t s to the s c a t t e r i n g data by Kim [9] and by M a r t i n and Sakitt [10] is shown in fig. 1. H o w e v e r , as shown in ref. [3], e f f e c t i v e p s e u d o p o t e n t i a l p a r a m e t e r s c o u l d b e d e r i v e d as a s u i t a b l e e n e r g y a v e r a g i n g of t h e s c a t t e r i n g a m p l i t u d e s in fig. 1. H e n c e t h e s c a t t e r i n g a m p l i t u d e s in eq. (2) m u s t b e r e p l a c e d by the e n e r g y a v e r a g e d s c a t t e r i n g a m p l i t u d e s , fKN,
U+ iW = 4~(t/2/2 ~ ) / K N
(2)
J K N is the f r e e s c a t t e r i n g a m p l i t u d e and p is t h e .
.
.
.
.
.
KIM
f K N ~ f K N : f d W f K N ( W ) P(W) .
(5)
T h e f u n c t i o n P(W) is the p r o b a b i l i t y of finding t h e K - a b s o r b e d n u c l e o n s y s t e m with t o t a l c e n t r e o f - m a s s e n e r g y W. If we n e g l e c t i n t e r a c t i o n s b e t w e e n t h e K a b s o r b e d n u c l e o n s y s t e m and the r e s i d u a l nuc l e u s , t h e n t h e c e n t r e - o f - m a s s e n e r g y is the i n i t i a l e n e r g y l e s s t h e k i n e t i c e n e r g y of the m o t i o n of t h e c e n t r e - o f - m a s s ,
0
Wi = r n K - C K +
-1
/I 3'
MARTIN
AND SAKITT
--
0
ii
...--------
-1 13~0 13L70 13l~ 13[~" 1~0 1410i-- 14~
1430 14L~0 1/450
W (MeV)
Fig. 1. The energy dependence of the free K- nucleon scattering amplitudes for the fits of Kim and of Martin and Sakitt.
136
r ni- e . -~P 2cm / 2 W .z
(6)
w h e r e ~K and Ei a r e the K m e s o n and a b s o r b e d n u c l e o n b i n d i n g e n e r g i e s r e s p e c t i v e l y and P c m is t h e m o m e n t u m of the c e n t r e - o f - m a s s . T h e e n e r g y d i s t r i b u t i o n s a r e then s i m p l y r e l a t e d to the centre-of-mass momentum distributions. T h e s e a r e g i v e n by t h e F o u r i e r t r a n s f o r m of t h e p r o d u c t of t h e K m e s o n and n u c l e o n w a v e - f u n c t i o n s u n d e r the a s s u m p t i o n of a s i n g l e p a r t i c l e d e s c r i p t i o n of t h e n u c l e u s and the s h o r t r a n g e n a t u r e of t h e f o r c e in eq. (1). F o r n u c l e o n s t a t e s of d e f i n i t e a n g u l a r m o m e n t u m we can d e r i v e t h e e x p l i c i t f o r m u l a f o r P(W), a n a l o g o u s to eq. (26) in r e f . [3]
PHYSICS
Volume 38B, n u m b e r 3
x
/
dpp2izij(P) l2 6 ( W - Wi(P))
(7)
0 w h e r e Zij(P) i s t h e o v e r l a p of t h e r a d i a l w a v e functions
Zij(P) = / drr2 jj(Prl UiL (r) Rnl(r) . (8) 0 We n o t e t h a t jj(Pr) i s t h e s p h e r i c a l B e s s e l f u n c t i o n , UiL(r) t h e n u c l e o n w a v e - f u n c t i o n of o r b i t a l a n g u l a r m o m e n t u m L , a n d Rnl(r) the r a d i a l wave-function for the K meson; the sum over i is over all nucleon states. U s i n g t h e s h e l l m o d e l d e s c r i p t i o n of t h e n u c l e u s a s g i v e n in r e f . [3] a n d u n d i s t o r t e d K meson wave-functions, the energy distributions w e r e c a l c u l a t e d u s i n g eqs. (7) a n d (8). T h e r e s u i t s of t h e s e c a l c u l a t i o n s f o r s e l e c t e d n u c l e i and indicated circular orbital angular momentum s t a t e s f o r t h e K m e s o n a r e s h o w n in fig. 2 f o r
OO~-- P(W) 004.~003 96~,o(1=4) 002 001
,,~f "-?s!
'°'°<"'> " - A / 'G0(1:2)
0
1.
%
ik
/" 1.
13"/0
1380 1390 1400 IZ+lO 1420 W MeV
Fig. 2. The probability function for K- nucleon s y s t e m to have an energy, W, in the c e n t r e - o f - m a s s .
LETTERS
7 F e b r u a r y 1972
1 6 0 ( / = 2), 4 0 C a ( l = 3), 9 6 M o ( / = 4). In c o m p a r i s o n w i t h fig. 1, the e n e r g y d i s t r i b u t i o n s s h o w a s t r o n g p e a k in t h e r e g i o n of the z e r o of the r e a l p a r t of the K- p r o t o n a m p l i t u d e with a t a i l e x t e n d i n g to l o w e r e n e r g i e s o r m o r e a t t r a c t i v e s c a t t e r ing a m p l i t u d e s . U s i n g t h e e n e r g y d i s t r i b u t i o n s of fig. 2 and t h e two s e t s of s c a t t e r i n g a m p l i t u d e s s h o w n in fig. 1, we c a n c o m p u t e t h e e n e r g y a v e r a g e d s c a t t e r i n g a m p l i t u d e s d e f i n e d in eq. (5). T h e r e s u t l s of t h e s e c a l c u l a t i o n s a r e p r e s e n t e d in t a b l e 1 a l o n g w i t h t h e t h r e s h o l d v a l u e s of the s c a t t e r i n g a m p l i t u d e s or the scattering lenghts. W e s e e t h a t t h e c o n j e c t u r e of K r e l l i s v e r i f i e d f o r the r e s o n a n c e d o m i n a t e d a m p l i t u d e s of fig. 1. The energies available during the absorption p r o c e s s a r e s u f f i c i e n t l y b e l o w t h r e s h o l d a s to r e f l e c t t h e c h a n g e in s i g n of the r e a l p a r t of t h e s c a t t e r i n g a m p l i t u d e s . By c o m p a r i n g t h e a b s o r p t i r e p a r t s of the a v e r a g e d a m p l i t u d e s , we s e e the l a r g e e n h a n c e m e n t of the a b s o r p t i o n s t r e n g h t on p r o t o n s . A s s u g g e s t e d in r e f . [3], t h i s e n h a n c e merit i m p l i e s t h a t X r a y s t u d i e s in K m e s i c a t o m s a r e not s e n s i t i v e to t h e n e u t r o n d i s t r i b u tions. T h e e n e r g i e s a v a i l a b l e d u r i n g a b s o r p t i o n will of c o u r s e v a r y w i t h the n u c l e u s a n d a l s o w i t h the i n i t i a l s t a t e of t h e K m e s i e a t o m . H o e v e r , f r o m t a b l e 1, we o b s e r v e the e f f e c t i v e n e u t r o n p a r a m e t e r s a r e e s s e n t i a l l y c o n s t a n t and a p p r o x i m a t e l y t h e s a m e f o r the two f i t s to t h e s c a t t e r i n g d a t a . T h e a b s o r p t i v e p a r t of t h e p r o t o n a m p l i t u d e s a r e a l s o e s s e n t i a l l y c o n s t a n t but d i f f e r c o n s i d e r a b l y b e t w e e n t h e two f i t s r e f l e c t i n g the m u c h b r o a d e r r e s o n a n c e p e a k f o r t h e K i m fit. T h e r e a l p a r t of t h e p r o t o n a m p l i t u d e i s the m o s t s e n s i t i v e to t h e e n e r g y a v e r a g i n g a l t h o u g h t h e s i g n will c l e a r l y r e m a i n o p p o s i t e to t h e t h r e s h o l d v a l u e f o r t h e r e s o n a n c e d o m i n a t e d a m p l i t u d e s i n fig. 1. T h e e n e r g y a v e r a g e d a m p l i t u d e s of t a b l e 1 c a n b e u s e d to m a k e a d e t a i l e d c o m p a r i s o n w i t h t h e r e c e n t e x p e r i m e n t s of B a c k e n s t o s s et al. [6] which observe with some precision the strong i n t e r a c t i o n e n e r g y s h i f t s a n d l i n e b r o a d e n i n g s of t h e K m e s i c X r a y s f o r a n u m b e r of l i g h t n u c l e i .
Table 1 The energy a v e r a g e d s c a t t e r i n g amplitudes.
i- -p-Kim-/i<-n fm
fm
160(/ = 2)
0.93+i2.27
0.10+i0.47
1 . 1 2 + i 1.48
0.14+i0.40
40Ca(/ = 3)
0.73 ÷i 2.24
0.09+ i0.46
0 . 9 4 + t 1.60
0.13+ i0.39
96Mo(/ = 4)
0 . 7 9 + i 2.32
0.10+ i 0.48
1.01 + i 1.62
0.13 + i 0.42
Threshold
-0.89 +i 0.62
-0.13 +i 0.51
- 0 . 8 8 + i 0.62
- 0 . 0 9 + i 0.54
137
Volume 38B. n u m b e r 3
PHYSICS
LETTERS
7 F e b r u a r y 1972
Table 2 Pseudopotential calculation of the energy shifts and widths. T h r e s h o l d p a r a m e t e r calculations in b r a c k e t s . Atom t r a n s i t i o n
r r . m . s . (fm)
Kim
r0.5(fm ) /(fro)
M.S.
Experimental
EL(keV)
I'L(keV)
~L(keV)
FL(keV)
EL(keV)
FL(keV)
~O~3 0
(-0.23)
O.65 (0.24)
-O.25 (-0.22)
0.72 (0.25)
-0.208 ± 0.035 0.81±0.10
10B((3,2) ~ (2,1))
2.45
-
l l B ( ( 3 , 2 ) -~ (2,1))
2.42
-
-0.30 (-0.23)
0.64 (0.24)
-0.26 (-0.23)
0.70 (0.26)
-0.167±0.035
0.70±0.08
12C((3,2) ~ (2,1))
2.42
-
-0.80 (-0.59)
1.44 (0.55)
-0.67 (-0.5S)
1.58 (0.58)
-0.59±0.08
1.73±0.15
31p((4,3)" ~ (3,2))
3.188
3.34
2.45
-0.52 (-0.49)
1.65 (0.58)
-0.36 (-0.48)
1.68 (0.60)
-0.33 ±0.08
1.44±0.12
32S~,4,3,"' -~(3,2))
3.244
3.33
2.60
-0.88 (-0.82)
2.73 (0.94)
-0.61 (-0.80)
2.78 (0.98)
-0.55 ±0.06
2.33 ±0.06
35C1((4, 3) ~ (3,2))
3.335
3.42
2.60
-1.44 (-1.29)
4.06 (1.45)
-1.05 (-1.27)
4.28 (1.51)
-0.77±0.40
3.8±1.0
W e h a v e u s e d e n e r g y a v e r a g e d a m p l i t u d e s for 1 6 0 ( l = 2) g i v e n in t a b l e 1 to c o n s t r u c t the e f f e c t i v e p s e u d o p o t e n t i a l d e t e r m i n e d t h r o u g h e q s . (4) a n d (2). T h e s t r o n g i n t e r a c t i o n e n e r g y s h i f t s and widths were computed using a program d e v e l o p e d by K r e l l [11] w h i c h s o l v e s t h e e i g e n v a l u e p r o b l e m for t h e K l e i n - G o r d o n e q u a t i o n w i t h t h e p o t e n t i a l i n c l u d i n g the s t r o n g i n t e r a c t i o n p s e u d o p o t e n t i a l . T h e r e s u l t s of t h e s e c a l c u l a t i o n s f o r the i s o t o p e s 10B, l l B , 12C, 3 1 p , 32S, 35C1 a r e g i v e n in t a b l e 2 a n d c o m p a r e d w i t h t h e e x p e r i m e n t a l d a t a [6]. T h e p s e u d o p o t e n t i a l s h a p e s f o r the lighter isotopes were harmonic wells with root mean square radii determined from electron s c a t t e r i n g [12]. T w o p a r a m e t e r F e r m i w e l l s were used for the heavier isotopes with root mean square radii determined from the muonic a t o m s t u d i e s of B a c k e n s t o s s et al. [13]. F r o m t a b l e 2. we s e e t h a t t h e t h e o r e t i c a l p r e d i c t i o n s b a s e d on the r e s o n a n c e d o m i n a t e d a m p l i t u d e s a r e g e n e r a l l y in q u i t e good a g r e e m e n t w i t h t h e d a t a . C a l c u l a t i o n s b a s e d on t h e t h r e s h o l d s c a t t e r i n g l e n g t h s ( s h o w n in b r a c k e t s ) g e n e r a l l y g i v e w i d t h s a r e a f a c t o r of two to t h r e e too s m a l l . W e n o t e t h a t in t h i s r a n g e of p s e u d o p o t e n t i a l parameters the energy shift varies most strongly w i t h t h e i m a g i n a r y p a r t of the p s e u d o p o t e n t i a l , w h i l e the w i d t h d e p e n d s m o s t s t r o n g l y on the r e a l p a r t . c o n t r a r y to p e r t u r b a t i o n t h e o r y . W e o b s e r v e t h a t the M a r t i n - S a k i t t fit p a r a m e t e r s s e e m to g i v e a b e t t e r fit to t h e e n e r g y s h i f t s in t h e h e a v i e r n u c l e i t h a n t h e K i m fit p a r a m e t e r s . T h i s m a y i n d i c a t e t h a t t h e K i m fit g i v e s too large an imaginary part for the proton amplitude in the r e s o n a n c e r e g i o n . D e f i n i t e c o n c l u s i o n s concerning this point must await a more complete 138
a n a l y s i s of d a t a p a r t i c u l a r l y a s t u d y of the r a d i u s a n d a m p l i t u d e d e p e n d e n c e s of t h e m e a s u r e d quantities. T h e r e a r e of c o u r s e m a n y w a y s t h e s e c a l c u l a tions could be improved. For example, distorted K m e s o n w a v e - f u n c t i o n s s h o u l d be u s e d in c o m p u t i n g t h e e n e r g y a v e r a g i n g f u n c t i o n s . In p r i n c i p l e t h e e n e r g y a v e r a g i n g will d i f f e r d e p e n d i n g upon the K meson orbit as well as the isotope involved. T h e s e e f f e c t s a r e b e i n g s t u d i e d a n d will be r e ported elsewhere. T h e r e a r e a l s o b a s i c l i m i t a t i o n s to t h e p s e u d o p o t e n t i a l a p p r o a c h . T h e u s e of the f r e e s c a t t e r i n g a m p l i t u d e s i n v o l v e s the n e g l e c t of i n t e r a c t i o n s of t h e K m e s o n - n u c l e o n s y s t e m w i t h the r e s i d u a l n u c l e u s . W e e x p e c t t h i s a p p r o x i m a t i o n to b e r e a s o n a b l y good a s a b s o r p t i o n o c c u r s in the n u cleon periphery, but this assumption should be s t u d i e d in r e a l i s t i c a t o m s i t u a t i o n s . W e c o n c l u d e t h a t {he p s e u d o p o t e n t i a l m e t h o d w o r k s q u i t e w e l l w h e n s u i t a b l y m o d i f i e d to the k i n e m a t i c a l s i t u a t i o n found in K m e s i c a t o m s . T h e f i t s to the B a c k e n s t o s s d a t a in t a b l e 2 i m p l y t h a t the K- p r o t o n s c a t t e r i n g a m p l i t u d e m u s t b e resonance dominated without a large repulsive b a c k g r o u n d [14]. As o u r u n d e r s t a n d i n g of t h e a b s o r p t i o n p r o c e s s i m p r o v e s , we m a y b e a b l e to u s e K m e s i c a t o m s a s a p r o b e of the K- p r o t o n s c a t t e r i n g a m p l i t u d e s in the r e s o n a n c e r e g i o n a s w e l l a s a p r o b e of t h e n u c l e a r s u r f a c e . O n e of u s (W. A. B.) would l i k e to t h a n k t h e m e m b e r s of t h e B a c k e n s t o s s g r o u p a t C E R N f o r many useful discussions concerning the data and t h e t h e o r y . W . A . B . would a l s o l i k e to t h a n k
Volume 38B, number
3
PHYSICS
T . E . O . E r i c s o n , M. Kre[1 and F. S c h e c k f o r s t i m u l a t i n g d i s c u s s i o n s of the t h e o r y .
Refere~wes [1] T. E. (3. Ericson and F. Scheck, Nucl. Phys. 1319 (1970) 450. [2] S. D. Bloom, M. H. Johnson and E. T e l l e r , Phys. Rev. Letters 23 (1969) 28; UCRL-72807 (1970). [3] W. A. Bardeen and E,W. Torigoe, Phys. Rev. C3 (1971) 1785, [4] C.E. Wiegand. Phys. Rev. L e t t e r s 22 (1969) 1235: C. E. Wiegand and D. A. Mack, Phys. Rev. Letters 18 (1967) 685. [5] G. Backenstoss et al., Phys. L e t t e r s 32B (1970) 399,
L E T T E RS
7 February
1972
[6] G. Backenstoss et al., Phys. Letters 38B (1972) 181. [7] M.Krell. Phys. Rev. Letters 26 (1:971) 584. [8] R. Seki, Submitted to Phys. Rev. [9] J.K.Kim, Phys. Rev. Letters 19 (1967) 1074: F. Von IIippel and J. K. Kim, Phys..Rev. Letters 20 (1968) 1303. [i0] B. R. Martin and M. Sakitt, Phys. Rev. 183 (1969) 1345:183 (1969) 1352. [ii] M. Krell, Submitted to Computer Physics Communications. [12] R. Hofstadter and H. R. Collard, in LandoltB6rnstein, ed. K.H. Hellwege, (Springer Verlag, New York, 1967), groupI, Vol. 2, p. 32. ]13] G. Backenstoss et al., Phys. Letters 25B (1967) 547. [14] D. Cline, R. Laumann and J. Mapp, Phys. Rev. Letters 26 (1971) 1194.
139