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Calculation of meson-exchange corrections to triton beta decay using realistic nuclear wave functions
Volume 38B, n u m b e r 1
PHYSICS
CALCULATION TO
OF
TRITON
LETTERS
MESON-EXCHANGE
BETA
NUCLEAR
DECAY WAVE
USING
10 January 1972
CORRECTIONS REALISTIC
FUNCTIONS
E. F I S C H B A C H . E. P . H A R P E R ,
J
Y. E. KIM a n d A. TUBIS
Physics Department, Purdue Uni~,ersity, Lafayette, Indiana 47907, USA and W.K. CHENG
Physics Department, Stevens Institute of Technology, Hoboken, N.J. 07030, USA Received
I0 November
1971
The one-pion-exchange contribution to the a x i a l - v e c t o r m a t r i x element for 3 H ~ 3He + e- + ~e is calculated. The model of Cheng, based on CVC and PCAC, is used to d e t e r m i n e the c o r r e c t i o n s to the a x i a l - v e c t o r operator, and trinucleon wave functions are calculated by solving the Faddeev equations for the Reid soft-core nucleon-nucleon potential. The S-D transition m a t r i x e l e m e n t s give the dominant contributions in a g r e e m e n t with previous r e s u l t s of Blomqvist and of Riska and Brown.
T h e r e h a s b e e n r e n e w e d i n t e r e s t of l a t e [ 1 - 7 ] in t h e p r o b l e m of m e s o n - e x c h a n g e c o r r e c t i o n s to t r i ton ~ - d e c a y , i . e . , t h e c o r r e c t i o n s a r i s i n g f r o m t h e / t - d e c a y of v i r t u a l i s o b a r s a n d m e s o n s in t h e t r i t o n . T h e s e c o r r e c t i o n s r e f l e c t t h e m s e l v e s in a d i f f e r e n c e b e t w e e n t h e f t v a l u e s f o r 3H ~ 3He + e - + Pe a n d n ~ p + e + u e. T h e i n f o r m a t i o n w h i c h c a n b e o b t a i n e d f r o m a s t u d y of t h e s e c o r r e c t i o n s c a n b e s t b e u n d e r s t o o d if t h e p r o b l e m i s a n a l y z e d into i t s t h r e e m o s t e s s e n t i a l c o m p o n e n t s : (i) A n e x p e r i m e n t a l d e t e r m i n a t i o n of f t ( n ~ p) a n d f t ( 3 H ~ 3He). (ii) A t h e o r e t i c a l c a l c u l a t i o n of a n e f f e c t i v e t w o - b o d y e x c h a n g e o p e r a t o r . (iii) A t h e o r e t i c a l c a l c u l a t i o n of t h e 3H a n d 3He w a v e f u n c t i o n s . A n a c c u r a t e k n o w l e d g e of a n y two of t h e t h r e e c o m p o n e n t s c a n t h e n b e u s e d to s t u d y t h e t h i r d . If we a s s u m e , f o r e x a m p l e , t h a t (i) a n d (ii) a r e w e l l k n o w n , t h e n a s t u d y of m e s o n - e x c h a n g e e f f e c t s c a n b e u s e d to p r o b e s u c h q u e s t i o n s a s t h e s h o r t r a n g e b e h a v i o r of t h e t r i t o n w a v e f u n c t i o n . A l t e r n a t i v e l y a d e t a i l e d k n o w l e d g e of t h e t r i t o n w a v e f u n c t i o n c a n b e u s e d to t e s t v a r i o u s e x t r a p o l a t i o n p r o c e d u r e s u s e d in c a l culating the two-body exchange operator. W e p r e s e n t in t h i s n o t e t h e p r e l i m i n a r y r e s u l t s of a c a l c u l a t i o n , b a s e d on r e a l i s t i c n u c l e a r w a v e f u n c t i o n s , of t h e m e s o n e x c h a n g e c o r r e c t i o n 5, 5 ~ <3Hel A ( 2 ) t 3 H y < 3 H e l A ~ I ) I 3H> •
AB ' are,
respectively,
(1)
t h e o n e - b o d y a n d t w o - b o d y c o n t r i b u t i o n s to t h e e f f e c t i v e a x i a l - v e c t o r
operator
d2fl = ½ G'/-2 cos O A~" I ,
A fl = A ~1 ) + A ~2 ),
A [(1) 3 = _ G A ~i a i . r ! +z ) "
(2)
In eq. (2), I = i C ~ / y ( l + Y 5 ) ~ p u i s t h e l e p t o n c u r r e n t , G i s t h e F e r m i c o n s t a n t , GA = 1.23 is t h e u s u a l n u c l e o n a x i a l - v e c t o r f o r m f a c t o r a t z e r o m o m e n t u m t r a n s f e r , a n d 0 i s t h e C a b i b b o a n g l e . (By v i r t u e of t h e c o n s e r v e d v e c t o r c u r r e n t h y p o t h e s i s (CVC), t h e m a t r i x e l e m e n t of t h e p o l a r - v e c t o r c u r r e n t is n o t r e n o r m a l i z e d by m e s o n e x c h a n g e e f f e c t s [2, 3].) A p p r o x i m a t e e x p r e s s i o n s f o r A ~ 2) w e r e o b t a i n e d f r o m t h e o n e - p i o n - e x c h a n g e m o d e l of C h e n g [2, 3] in w h i c h N ~ N~ c u r r e n t m a t r i x e l e m e n t s a r e a p p r o x i m a t e d b y p o l e t e r m s c o r r e s p o n d i n g to i n t e r m e R e s e a r c h supported by the US National Science Foundation Grant No. GP-29522 and the US Atomic Energy Comm i s s i o n Contract No. AT 4997-54-13965.
PHYSICS
Volume 38B, number 1
10 January 1972
LETTERS
diate N* and p r e s o n a n c e s , and P C A C is used to r e l a t e NN* and np c u r r e n t m a t r i x e l e m e n t s to e m p i r i cal 7NN* and ~zrp coupling constants. The explicit f o r m of the o p e r a t o r A ~o~) in c o o r d i n a t e space is given in refs. [2] and [3]. Since the p r e s e n t calculation was c a r r i e d out entirely,~ in m o m e n t u m s p a c e , we give the two-nucleon (relative) m o m e n t u m - s p a c e r e p r e s e n t a t i o n for A~ ~). =
=
=
~J,~, t2 ,~),N (5 , ½),p,
(3)
i<7 " w h e r e the m a t r i x e l e m e n t s of the
"(2)(X zj) a r e given as follows:
,,tfi
,
( p ' [A~2)(N* (~ +, ~), zj) [ p) = F(3, 3)(k 2 +/z 2) -1 {2k[ (ai + oj)" k( r i + rj) (+) + (a i - aj)" !( (r i - r j )(+)] +
+ [-k2(ai×aj) + k(ai×aj)'k](Ti×
rj)(+)},
F(3, 3) = -{4n GA/9~2(m* - m ) } ( f 2 N N , / 4 = ) ~ -0.29 U-3 ; ( p ' IA (fl2)(N*(½+, ½), ~j) ]p> : C(1,1)(/< 2 + 2 ) - l { k [ ( a i
(4a)
+ aj)" k('r i + ~j) (+) + (a i - aj)" k ( r i - rj) (+)] +
+ (1 + m~*') k × [ k × ( a i × o j ) ] ( r i × rj)(+)}, iv/,/ -
(4b)
V(1, 1) = {-27 aA/U2(rn* - m)}(f2NN,/4=) ~ -0.065 ~z-3",
(p' ]A~2)(N*(~3- , ½),ij)]p) = r(3, 1)(k 2 +tx 2 ) -1 { k [ ( a i + a j ) ' k ( r i + r )) (+)+ (a i - a j ) ' k ( r i - r j ) (+)] + - k × [k×(ai×aj)](ri×rj)(+)}, r(3, 1) = -{4~ GA/31x2(m * +m)}(f2NN,/47) ( p ' IA ~2)(p, zj)IP) = F(P)[ k2 + tz2) -1 - (k2 + m 2 ) - l ] [ k ( o i × a j ) "
~ -0.037 U -3 ;
(4c)
k - k2(ai × a j ) l ( r i × rj~, (+) ,
F(p) = {-n CA(1 +/~p - ~n)/m (rn 2 -/.L2)}(f2p77/4=) ~ -0.24 /z -3 .
(4d)
In eqs. (4a)-(4d), [p) denotes a r e l a t i v e m o m e n t u m state of the nucleon pair ij with n o r m a l i z a t i o n < p ' ] p ) = (27)36(p ' - p ) , k = p ' - p , m , m* and /~ a r e the m a s s e s of the nucleon, i s o b a r a n d p i o n r e s p e c t i v e l y , and /~p, /~n a r e r e s p e c t i v e l y the a n o m a l o u s m a g n e t i c m o m e n t s of the proton and neutron. Due to present--day u n c e r t a i n t i e s in the e x p e r i m e n t a l values of the m a s s e s and widths of the v a r i o u s N* r e s o n a n c e s , the values of the coupling constants F(N*) defined in eqs. (4) a r e also uncertain. Consequently, it is of s o m e i n t e r e s t to exhibit explicit f o r m u l a s [3] for calculating the c o n s t a n t s f = N N , which a p p e a r in the e x p r e s s i o n s for F(N*). F o r each r e s o n a n c e we give both the definition o f f = N N , in t e r m s of the d e c a y L a g r a n g i a n d e n s i t y ~O(=NN*) and the f o r m u l a f o r f 2 N N , / 4 = in t e r m s of m*, m, /~ and F w h e r e m*, rn and /~ a r e the m a s s e s of the r e s o n a n c e , nucleon and pion r e s p e c t i v e l y , and F is the width for the d e c a y N* ~ N~. x~,t£+ 3~ fTNN*"" ~2 , 2 J : ~ ( ~ N N * ) = - - - NIxN x .
N , ,~ t • +, ~,1 ]
-
~ *
+n//
, Ox.."r + H . c . ,
: Z ( ~ N N * / : - ~N t _ _ =7~,- - .5 ~ J 4 n( N ,~
--
-
N* = -A N-4n
6/~2 rn*2F
/2NN* "-IN*
+ H.c.,
(pn) -3 •
(5a)
[(m*+m) 2-~2 l 47
2
U2m*2F
3 (m, + m ) 2 [ ( m , _ r n ) 2 _ # 2 ]
(pn)-I
. ,
(5b)
Volume 38H, n u m b e r 1
:PHYSICS
N*(~, I) : J2vNN* = -i f~NN* iV~y5(r. ~ )
LETTERS
N + H.c. '
f2NN* 4v
10 J a n u a r y 1972
2~2m*2F [(m* - m) 2 ~ 2 ]
(Pv)-3 "
(5c)
In eqs. (5) pv = lPv ! is the magnitude of the pion m o m e n t u m in the N* r e s t f r a m e and is given by Ip~I : ~1/2(m*, m, ~)/2m*
(6)
w h e r e ;~(x,y, z) = x 2 + y2 + z 2 _ 2xy - 2xz - 2yz is the usual Kffll~n function. The v a l u e s of the v a r i o u s coupling c o n s t a n t s used in our calculation w e r e : 2 N*(3+, ~):f~NN./47T = 0.36 ~= 0.02,
~,T.~-~+,eJ!~ : f 2 NN,/4v
• , ~2
N*(~, I) :f~NN,/4~ 2 = 0.19+0.00,
p
=
0.0075+0.0032,
:f2v~/4~= 2.4~:0.4.
(7)
The wave functions for 3H and 3He were determined by solving approximately the Faddeev equations [8] for the case in which Coulomb effects are ignored and only the IS 0 -IS0, 3S1 -3SI, 3S1 -3DI, and 3DI -3D 1 elements of the nucleon-nucleont-matrix are taken into account. The Reid soft-core potential [9] was used for the nucleon-nucleoninteraction. For our model of the nucleon-nucleoninteraction, the exact Faddeev formalism gives eight coupled integral equations in two continuous variables [i0]. We have followed the procedure of Malfliet and Tjon [ i i ] in which the Faddeev equations are approximately reduced to three coupled integral equations in two continuous variables. The channel states included in our calculation were (in the notation of ref. [i0]):
[[pq(L1)~2, (Ss)d] 9 ~ z ; ( T t ) ~ z } 1 : l [pq(00)0, (11)½]½'5,'(01) Igz} 1 ,
(8a)
t[Pq (00) 0, (0~)1 ~] ~ ~1 ,, (1½) Igz}l
(85)
l[pq(20)
2, (11) ~] ' ~' ; (0~) ' ~' z } l "
(8c)
It should be noted that the solution of the " t h r e e channel" F a d d e e v equations y i e l d s bound s t a t e p r o b a b i l i t i e s not only for the s t a t e s (8) but a l s o for o t h e r s t a t e s such a s (8a,b) with pq(O0)0 ~ pq(LL)O, L = 2,4...,
I[[q(LL) O, (0~)' ~]1 22,11 , (01) ½~Z}1' I[pq(LL) O, (1½) I ] ' ' ~ ; (11)
½~>I .
.
(L=I ,3,5, .. .),
(8d)
( .L =.I . 3, 5,
(8e)
.)
and
I [ p q ( l l ) 2, (11) ~] ~},1 ' ' (11) 1~z} 1 .
(8f)
The p r o b a b i l i t i e s for s t a t e s (8d) and (Be) with L >/ 3, and f o r s t a t e s (8a,b) with pq(O0) 0 -~ pq(LL) O, L >~ 2, w e r e found to be negligible. The homogeneous F a d d e e v equations w e r e s o l v e d by the i t e r a t e d k e r n e l technique [1I] to y i e l d t r i nucleon b o u n d - s t a t e e n e r g i e s and wave functions. Our c a l c u l a t i o n gives a binding e n e r g y E B = 6.39 MeV, and S' and D s t a t e p r o b a b i l i t i e s P(S') = 1.9% and P(D) = 5.8%. M a l f l i e t and Tjon [11] find E B = 6 . 5 + 0 . 2 5 MeV, P(S') = 1.8%, P(D) = 8.1%. A r e c e n t v a r i a t i o n a l c a l c u l a t i o n [12] b a s e d on n u c l e o n - n u c l e o n i n t e r a c t i o n s (given by the Reid s o f t - c o r e potential) in the 1S0, 3S 1 - 3 D 1, 1D 2 and 3D 2 channels y i e l d s E B = 6.06 MeV, P(S') = 0.43% and P(D) = 8.52%. In table 1, we give the c o n t r i b u t i o n s to 6 a s s o c i a t e d with the v a r i o u s A ~) ~) (oX and components of the 3H, 3He wave functions. The n e a r c a n c e l l a t i o n between the pion and the rho~propagator t e r m s from A ~2)(p) has been p r e v i o u s l y noted [5, 7], a s has the vanishing of 6ss(N*( 3+, ~)). The d o m i n a n c e of the S-~D contribution f r o m A ~ ) ( N * ( ~ +, ~)) is in a g r e e m e n t with the r e s u l t s of B l o m q v i s t [6] and of R i s k a and Brown [7], which w e r e obtained with s i m p l i f i e d 3H, 3He wave functions. The " e x p e r i m e n t a l " value of 6 is given a p p r o x i m a t e l y by 10
Volume 38B, number 1
PHYSICS
LETTERS
10 January 1972
Table 1 Contributions to 5 (eq. (1)) associated with the various A(2)(X) and components of the trinucleon wave functions. 5AB(X, /2) and 5AB(X, rnp) (A, B = S, S', D) denote the contributions p to 5 from terms in A(~)tX) fnvolving the pion and rho propagators respectively. The values of the 6, s are given as percentages. The un'-certainties in 5 are calculated from the uncertainties in the coupling constants given in eq. (7). X
5ss(X , it)
5ss(X, n~p)
5S,s,(X, tz)
5S,s,(X, ~ p )
5 sD(X, tz)
5sD(X, rap)
Total
N*(~ + ' 2-3i
0
-0.00
7.53
7.53 ± 0.41~;{:
N*(~+-,})
0.05
-0.00
0.23
0.28 ±0.12%
N*(~-, ~)
-0.18
0.00
-O.18'Y~
9
0.80
-0.81
-0.01
0.01
2.08
-0.58
1.49 ~ 0.25~)~
Total
0.67
-0.81
-0.00
0.01
9.84
-0.58
9.1 ± 0.5%
5exp(%) = ( ½ + l / 6 G 2 ) ( - l + ( f t ) n / ( f t ) 3 H )
x 100 + 5R + 34-P(S') . 2 p ( D ) .
(9)
5 R is o b t a i n e d by c o n s i d e r i n g the " m i n i m a l " v 2 / c 2 c o r r e c t i o n to the G a m o w - T e l l e r o p e r a t o r [13]
~" ~ a i - o i p 2 / 2 m 2
+ (ai'Pi/2m2)p
i ,
(10)
w h e r e Pi is the m o m e n t u m of n u c l e o n i. We find 5 R v 0.5% which c o r r e s p o n d s to a m e a n v a l u e of v / c , f o r a n u c l e o n in 3H o r 3He, of v 0.12. Recently determinedft values are (ft)n : 1108 ~= 16.45 s e a [14],
(ft)3 H = 1143 ± 3 s e c [15].
(11)
If we use t h e s e f t v a l u e s and P ( S ' ) = 0.43%, P(D) = 8.52% f r o m the c a l c u l a t i o n of J a c k s o n et al. [12] ( w h o s e e s t i m a t e s of t h e s e q u a n t i t i e s s h o u l d be s o m e w h a t m o r e r e l i a b l e than o u r s ) , we find 5 exp ~ 4.9 + 1.0%,
(12)
w h i c h is to be c o m p a r e d with o u r c a l c u l a t e d v a l u e 5 = 9.1 + 0.5%.
(13)
T h e a g r e e m e n t b e t w e e n 5exp and o u r c a l c u l a t e d v a l u e is s a t i s f a c t o r y in v i e w of: 1) our n e g l e c t of s o m e of the D - s t a t e c o m p o n e n t s of the t r i n u c l e o n w a v e f u n c t i o n s ; 2) o u r n e g l e c t of c o n t r i b u t i o n s to 5 f r o m m u l t i p l e m e s o n e x c h a n g e and w a v e f u n c t i o n n o r m a l i z a t i o n c o r r e c t i o n s [5] and 3) u n c e r t a i n t i e s in the v a l u e of P ( S ' ) . We a r e c u r r e n t l y i m p r o v i n g o u r d e t e r m i n a t i o n of the t r i n u c l e o n w a v e f u n c t i o n s by i n c l u d i n g c o n t r i b u t i o n s f r o m the c h a n n e l s t a t e s [[)q(02)2,
(1~) ~ 3] ~ ~' '~' , (0½) 513 z } 1 ,
(14)
which should be next in importance after the states (8a), (Sb) and (8c), on the basis of centrifugal b a r r i e r considerations. Our new calculation will probably lead to an increased P(D), and thus reduce the discrepancy between our results and those of Jackson et al. [12]. It should be noted that the components of the wave functions involving the states (14) do not contribute to the S-D matrix element of A (2)(N*(3+, 23-)). T h u s , although P(D) m i g h t i n c r e a s e , the 3H and 3He p r o b a b i l i t i e s f o r the D s t a t e s (8c) m i g h t d e c r e a s e and r e s u l t in a l o w e r c a l c u l a t e d v a l u e f o r 6. T h e v a l u e of P ( S ' ) is a m a j o r u n c e r t a i n t y in our a n a l y s i s . T h e c a l c u l a t e d v a l u e of 5 is not v e r y s e n s i t i v e to the a m o u n t s of S' in 3H and 3He, but the " e x p e r i m e n t a l " v a l u e of 5 ( s e e eq. (9)) i n c r e a s e s by 107 l i ~ if P ( S ' ) is i n c r e a s e d by 1%. T h e r e a l i s t i c n u c l e o n - n u c l e o n i n t e r a c t i o n s of R e i d [9] and of H a m a d a and J o h n s t o n [16] g i v e v e r y s i m i l a r v a l u e s f o r P ( D ) (8.54% [12] and 9% [17] r e s p e c t i v e l y ) but d i f f e r in a s i g n i f i c a n t way (with r e s p e c t to o u r p r e s e n t a n a l y s i s ) in the v a l u e s they g i v e f o r P(S') (0.429 [12] and 2% [17] r e s p e c t i v e l y ) . P e r h a p s m o r e i m p o r t a n t is the o b s e r v a t i o n that the a d d i t i o n a l i n t e r a c t i o n (inv o l v i n g t h r e e - n u c l e o n f o r c e s ) r e q u i r e d to g i v e the e x p e r i m e n t a l binding e n e r g y 8.49 M e V , m a y w e l l i n c r e a s e P ( S ' ) by s e v e r a l p e r c e n t . We a r e p r e s e n t l y e s t i m a t i n g the c h a n g e s in P(S') and P(D) which r e s u l t f r o m the a d d i t i o n of r e a s o n a b l e t h r e e - b o d y f o r c e s [18] to the n u c l e a r i n t e r a c t i o n .
11
Volume
38B, number
I
PHYSICS
LETTERS
i0 January 1972
T w o of u s (Y. E. K. a n d A. T.) w i s h to t h a n k P r o f e s s o r s H. P . N o y e s a n d S. D. D r e l l f o r h o s p i t a l i t y a n d i n t e r e s t i n g d i s c u s s i o n s a t t h e S t a n f o r d L i n e a r A c c e l e r a t o r C e n t e r ( J u n e - A u g u s t , 1971) w h e r e p a r t of t h i s r e s e a r c h w a s d o n e . One of us (E. F.) w i s h e s to t h a n k P r o f e s s o r s K. E. B e r g k v i s t a n d G. E. B r o w n a n d D r . M. C h e m t o b f o r s e v e r a l s t i m u l a t i n g d i s c u s s i o n s on t h e p r o b l e m of m e s o n - e x c h a n g e e f f e c t s .
References [1] R.J. Blin-Stoyle and M. Tint, Phys. Rev. 160 (1967) 803. [2] W.K. C h e n g a n d E . F i s c h b a c h , Phys. Rev. 188 (1969) 1530. [3] W.K. Cheng, Ph. D. Thesis, University of Pennsylvania (1966), unpublished. In calculating the F, s in eqs. (4), s e v e r a l e r r o r s in [2] and [3 l have been c o r r e c t e d , in these r e f e r e n c e s , A/~(2)(N*( 3+, ½)) should be multiplied by and AB(2)(N*(½~ , ½)), A/3(2)(N*(~-, ½)) and Aj3(2)(p) by -¼, ~ and ½ respe~'tively. [4] R.J. Blifi-Stoyle, Phys. R~v. 188 (1969) 1 5 4 0 . [5] M. Chemtob and M. Rho, Phys. L e t t e r s 29B (1969) 540; Nucl. Phys. A163 (1971) 1. [6] J. Blomqvist, Phys, L e t t e r s 32B (1970) 1. [7] D.O. Riska and G . E . B r o w n , Phys. L e t t e r s 32B (1970) 662. [8] L.O. Faddeev, Zh. Eksper. Teor. Fiz. 39 (1960) 1459, Soviet Phys. J E T P 12 (1961) 1014; Mathematica! aspects of the three-body problem in quantum s c a t t e r i n g theory (Daniel Davey and Company, inc., New York, 1965). [9] R.V. Reid, Ann. of Phys. 50 (1968) 411. [10] E . P . Harper, Y . E . K i m and A.Tubis, Phys. Rev. C2 (1970) 877, e r r a t u m C2 (1970) 2455. [11] R.A. Malfliet and J . A . Tjon, Ann. of Phys. 61 (1970) 425. [12] A.D. Jaekson, A. Lande and P.U. Sauer, Phys. Letters 35B (1971) 3651. [13] J.S. BeLL and R.J. Blin-Stoyle, Nucl. Phys. 6 (1958) 87: R. J. Blin-Stoyle and S. Papageorgiou, Nucl. Phys. 64 (1965) i. [14] R.C. Saigo and N.H. Staub, Nucl. Phys, A138 (1969) 417. [15] K.E.Bergkvist, private communication. [16] T. Hamada and I.D. Johnston, Nucl. Phys. 34 (1962) 382: T. Hamada. Y. Makamura and R. Tamagaki, Progr. Theoret. Phys. (Kyoto) 33 (1965) 769(L). [17] L.M. Delves, J.M. BLatt, C. Pask and B. Davies, Phys. Letters 28B (1969) 472. [18] J. Fujita and H. Miyazawa, Progr. Theoret. Phys. (Kyoto) 17 (1957) 360: C. Pask, Phys. Letters 25B (1967) 78: B.A. Loiseau and Y. Nogami, Nuel. Phys. B2 (1967) 470.
12
PHYSICS
CALCULATION TO
OF
TRITON
LETTERS
MESON-EXCHANGE
BETA
NUCLEAR
DECAY WAVE
USING
10 January 1972
CORRECTIONS REALISTIC
FUNCTIONS
E. F I S C H B A C H . E. P . H A R P E R ,
J
Y. E. KIM a n d A. TUBIS
Physics Department, Purdue Uni~,ersity, Lafayette, Indiana 47907, USA and W.K. CHENG
Physics Department, Stevens Institute of Technology, Hoboken, N.J. 07030, USA Received
I0 November
1971
The one-pion-exchange contribution to the a x i a l - v e c t o r m a t r i x element for 3 H ~ 3He + e- + ~e is calculated. The model of Cheng, based on CVC and PCAC, is used to d e t e r m i n e the c o r r e c t i o n s to the a x i a l - v e c t o r operator, and trinucleon wave functions are calculated by solving the Faddeev equations for the Reid soft-core nucleon-nucleon potential. The S-D transition m a t r i x e l e m e n t s give the dominant contributions in a g r e e m e n t with previous r e s u l t s of Blomqvist and of Riska and Brown.
T h e r e h a s b e e n r e n e w e d i n t e r e s t of l a t e [ 1 - 7 ] in t h e p r o b l e m of m e s o n - e x c h a n g e c o r r e c t i o n s to t r i ton ~ - d e c a y , i . e . , t h e c o r r e c t i o n s a r i s i n g f r o m t h e / t - d e c a y of v i r t u a l i s o b a r s a n d m e s o n s in t h e t r i t o n . T h e s e c o r r e c t i o n s r e f l e c t t h e m s e l v e s in a d i f f e r e n c e b e t w e e n t h e f t v a l u e s f o r 3H ~ 3He + e - + Pe a n d n ~ p + e + u e. T h e i n f o r m a t i o n w h i c h c a n b e o b t a i n e d f r o m a s t u d y of t h e s e c o r r e c t i o n s c a n b e s t b e u n d e r s t o o d if t h e p r o b l e m i s a n a l y z e d into i t s t h r e e m o s t e s s e n t i a l c o m p o n e n t s : (i) A n e x p e r i m e n t a l d e t e r m i n a t i o n of f t ( n ~ p) a n d f t ( 3 H ~ 3He). (ii) A t h e o r e t i c a l c a l c u l a t i o n of a n e f f e c t i v e t w o - b o d y e x c h a n g e o p e r a t o r . (iii) A t h e o r e t i c a l c a l c u l a t i o n of t h e 3H a n d 3He w a v e f u n c t i o n s . A n a c c u r a t e k n o w l e d g e of a n y two of t h e t h r e e c o m p o n e n t s c a n t h e n b e u s e d to s t u d y t h e t h i r d . If we a s s u m e , f o r e x a m p l e , t h a t (i) a n d (ii) a r e w e l l k n o w n , t h e n a s t u d y of m e s o n - e x c h a n g e e f f e c t s c a n b e u s e d to p r o b e s u c h q u e s t i o n s a s t h e s h o r t r a n g e b e h a v i o r of t h e t r i t o n w a v e f u n c t i o n . A l t e r n a t i v e l y a d e t a i l e d k n o w l e d g e of t h e t r i t o n w a v e f u n c t i o n c a n b e u s e d to t e s t v a r i o u s e x t r a p o l a t i o n p r o c e d u r e s u s e d in c a l culating the two-body exchange operator. W e p r e s e n t in t h i s n o t e t h e p r e l i m i n a r y r e s u l t s of a c a l c u l a t i o n , b a s e d on r e a l i s t i c n u c l e a r w a v e f u n c t i o n s , of t h e m e s o n e x c h a n g e c o r r e c t i o n 5, 5 ~ <3Hel A ( 2 ) t 3 H y < 3 H e l A ~ I ) I 3H> •
AB ' are,
respectively,
(1)
t h e o n e - b o d y a n d t w o - b o d y c o n t r i b u t i o n s to t h e e f f e c t i v e a x i a l - v e c t o r
operator
d2fl = ½ G'/-2 cos O A~" I ,
A fl = A ~1 ) + A ~2 ),
A [(1) 3 = _ G A ~i a i . r ! +z ) "
(2)
In eq. (2), I = i C ~ / y ( l + Y 5 ) ~ p u i s t h e l e p t o n c u r r e n t , G i s t h e F e r m i c o n s t a n t , GA = 1.23 is t h e u s u a l n u c l e o n a x i a l - v e c t o r f o r m f a c t o r a t z e r o m o m e n t u m t r a n s f e r , a n d 0 i s t h e C a b i b b o a n g l e . (By v i r t u e of t h e c o n s e r v e d v e c t o r c u r r e n t h y p o t h e s i s (CVC), t h e m a t r i x e l e m e n t of t h e p o l a r - v e c t o r c u r r e n t is n o t r e n o r m a l i z e d by m e s o n e x c h a n g e e f f e c t s [2, 3].) A p p r o x i m a t e e x p r e s s i o n s f o r A ~ 2) w e r e o b t a i n e d f r o m t h e o n e - p i o n - e x c h a n g e m o d e l of C h e n g [2, 3] in w h i c h N ~ N~ c u r r e n t m a t r i x e l e m e n t s a r e a p p r o x i m a t e d b y p o l e t e r m s c o r r e s p o n d i n g to i n t e r m e R e s e a r c h supported by the US National Science Foundation Grant No. GP-29522 and the US Atomic Energy Comm i s s i o n Contract No. AT 4997-54-13965.
PHYSICS
Volume 38B, number 1
10 January 1972
LETTERS
diate N* and p r e s o n a n c e s , and P C A C is used to r e l a t e NN* and np c u r r e n t m a t r i x e l e m e n t s to e m p i r i cal 7NN* and ~zrp coupling constants. The explicit f o r m of the o p e r a t o r A ~o~) in c o o r d i n a t e space is given in refs. [2] and [3]. Since the p r e s e n t calculation was c a r r i e d out entirely,~ in m o m e n t u m s p a c e , we give the two-nucleon (relative) m o m e n t u m - s p a c e r e p r e s e n t a t i o n for A~ ~). =
=
=
~J,~, t2 ,~),N (5 , ½),p,
(3)
i<7 " w h e r e the m a t r i x e l e m e n t s of the
"(2)(X zj) a r e given as follows:
,,tfi
,
( p ' [A~2)(N* (~ +, ~), zj) [ p) = F(3, 3)(k 2 +/z 2) -1 {2k[ (ai + oj)" k( r i + rj) (+) + (a i - aj)" !( (r i - r j )(+)] +
+ [-k2(ai×aj) + k(ai×aj)'k](Ti×
rj)(+)},
F(3, 3) = -{4n GA/9~2(m* - m ) } ( f 2 N N , / 4 = ) ~ -0.29 U-3 ; ( p ' IA (fl2)(N*(½+, ½), ~j) ]p> : C(1,1)(/< 2 + 2 ) - l { k [ ( a i
(4a)
+ aj)" k('r i + ~j) (+) + (a i - aj)" k ( r i - rj) (+)] +
+ (1 + m~*') k × [ k × ( a i × o j ) ] ( r i × rj)(+)}, iv/,/ -
(4b)
V(1, 1) = {-27 aA/U2(rn* - m)}(f2NN,/4=) ~ -0.065 ~z-3",
(p' ]A~2)(N*(~3- , ½),ij)]p) = r(3, 1)(k 2 +tx 2 ) -1 { k [ ( a i + a j ) ' k ( r i + r )) (+)+ (a i - a j ) ' k ( r i - r j ) (+)] + - k × [k×(ai×aj)](ri×rj)(+)}, r(3, 1) = -{4~ GA/31x2(m * +m)}(f2NN,/47) ( p ' IA ~2)(p, zj)IP) = F(P)[ k2 + tz2) -1 - (k2 + m 2 ) - l ] [ k ( o i × a j ) "
~ -0.037 U -3 ;
(4c)
k - k2(ai × a j ) l ( r i × rj~, (+) ,
F(p) = {-n CA(1 +/~p - ~n)/m (rn 2 -/.L2)}(f2p77/4=) ~ -0.24 /z -3 .
(4d)
In eqs. (4a)-(4d), [p) denotes a r e l a t i v e m o m e n t u m state of the nucleon pair ij with n o r m a l i z a t i o n < p ' ] p ) = (27)36(p ' - p ) , k = p ' - p , m , m* and /~ a r e the m a s s e s of the nucleon, i s o b a r a n d p i o n r e s p e c t i v e l y , and /~p, /~n a r e r e s p e c t i v e l y the a n o m a l o u s m a g n e t i c m o m e n t s of the proton and neutron. Due to present--day u n c e r t a i n t i e s in the e x p e r i m e n t a l values of the m a s s e s and widths of the v a r i o u s N* r e s o n a n c e s , the values of the coupling constants F(N*) defined in eqs. (4) a r e also uncertain. Consequently, it is of s o m e i n t e r e s t to exhibit explicit f o r m u l a s [3] for calculating the c o n s t a n t s f = N N , which a p p e a r in the e x p r e s s i o n s for F(N*). F o r each r e s o n a n c e we give both the definition o f f = N N , in t e r m s of the d e c a y L a g r a n g i a n d e n s i t y ~O(=NN*) and the f o r m u l a f o r f 2 N N , / 4 = in t e r m s of m*, m, /~ and F w h e r e m*, rn and /~ a r e the m a s s e s of the r e s o n a n c e , nucleon and pion r e s p e c t i v e l y , and F is the width for the d e c a y N* ~ N~. x~,t£+ 3~ fTNN*"" ~2 , 2 J : ~ ( ~ N N * ) = - - - NIxN x .
N , ,~ t • +, ~,1 ]
-
~ *
+n//
, Ox.."r + H . c . ,
: Z ( ~ N N * / : - ~N t _ _ =7~,- - .5 ~ J 4 n( N ,~
--
-
N* = -A N-4n
6/~2 rn*2F
/2NN* "-IN*
+ H.c.,
(pn) -3 •
(5a)
[(m*+m) 2-~2 l 47
2
U2m*2F
3 (m, + m ) 2 [ ( m , _ r n ) 2 _ # 2 ]
(pn)-I
. ,
(5b)
Volume 38H, n u m b e r 1
:PHYSICS
N*(~, I) : J2vNN* = -i f~NN* iV~y5(r. ~ )
LETTERS
N + H.c. '
f2NN* 4v
10 J a n u a r y 1972
2~2m*2F [(m* - m) 2 ~ 2 ]
(Pv)-3 "
(5c)
In eqs. (5) pv = lPv ! is the magnitude of the pion m o m e n t u m in the N* r e s t f r a m e and is given by Ip~I : ~1/2(m*, m, ~)/2m*
(6)
w h e r e ;~(x,y, z) = x 2 + y2 + z 2 _ 2xy - 2xz - 2yz is the usual Kffll~n function. The v a l u e s of the v a r i o u s coupling c o n s t a n t s used in our calculation w e r e : 2 N*(3+, ~):f~NN./47T = 0.36 ~= 0.02,
~,T.~-~+,eJ!~ : f 2 NN,/4v
• , ~2
N*(~, I) :f~NN,/4~ 2 = 0.19+0.00,
p
=
0.0075+0.0032,
:f2v~/4~= 2.4~:0.4.
(7)
The wave functions for 3H and 3He were determined by solving approximately the Faddeev equations [8] for the case in which Coulomb effects are ignored and only the IS 0 -IS0, 3S1 -3SI, 3S1 -3DI, and 3DI -3D 1 elements of the nucleon-nucleont-matrix are taken into account. The Reid soft-core potential [9] was used for the nucleon-nucleoninteraction. For our model of the nucleon-nucleoninteraction, the exact Faddeev formalism gives eight coupled integral equations in two continuous variables [i0]. We have followed the procedure of Malfliet and Tjon [ i i ] in which the Faddeev equations are approximately reduced to three coupled integral equations in two continuous variables. The channel states included in our calculation were (in the notation of ref. [i0]):
[[pq(L1)~2, (Ss)d] 9 ~ z ; ( T t ) ~ z } 1 : l [pq(00)0, (11)½]½'5,'(01) Igz} 1 ,
(8a)
t[Pq (00) 0, (0~)1 ~] ~ ~1 ,, (1½) Igz}l
(85)
l[pq(20)
2, (11) ~] ' ~' ; (0~) ' ~' z } l "
(8c)
It should be noted that the solution of the " t h r e e channel" F a d d e e v equations y i e l d s bound s t a t e p r o b a b i l i t i e s not only for the s t a t e s (8) but a l s o for o t h e r s t a t e s such a s (8a,b) with pq(O0)0 ~ pq(LL)O, L = 2,4...,
I[[q(LL) O, (0~)' ~]1 22,11 , (01) ½~Z}1' I[pq(LL) O, (1½) I ] ' ' ~ ; (11)
½~>I .
.
(L=I ,3,5, .. .),
(8d)
( .L =.I . 3, 5,
(8e)
.)
and
I [ p q ( l l ) 2, (11) ~] ~},1 ' ' (11) 1~z} 1 .
(8f)
The p r o b a b i l i t i e s for s t a t e s (8d) and (Be) with L >/ 3, and f o r s t a t e s (8a,b) with pq(O0) 0 -~ pq(LL) O, L >~ 2, w e r e found to be negligible. The homogeneous F a d d e e v equations w e r e s o l v e d by the i t e r a t e d k e r n e l technique [1I] to y i e l d t r i nucleon b o u n d - s t a t e e n e r g i e s and wave functions. Our c a l c u l a t i o n gives a binding e n e r g y E B = 6.39 MeV, and S' and D s t a t e p r o b a b i l i t i e s P(S') = 1.9% and P(D) = 5.8%. M a l f l i e t and Tjon [11] find E B = 6 . 5 + 0 . 2 5 MeV, P(S') = 1.8%, P(D) = 8.1%. A r e c e n t v a r i a t i o n a l c a l c u l a t i o n [12] b a s e d on n u c l e o n - n u c l e o n i n t e r a c t i o n s (given by the Reid s o f t - c o r e potential) in the 1S0, 3S 1 - 3 D 1, 1D 2 and 3D 2 channels y i e l d s E B = 6.06 MeV, P(S') = 0.43% and P(D) = 8.52%. In table 1, we give the c o n t r i b u t i o n s to 6 a s s o c i a t e d with the v a r i o u s A ~) ~) (oX and components of the 3H, 3He wave functions. The n e a r c a n c e l l a t i o n between the pion and the rho~propagator t e r m s from A ~2)(p) has been p r e v i o u s l y noted [5, 7], a s has the vanishing of 6ss(N*( 3+, ~)). The d o m i n a n c e of the S-~D contribution f r o m A ~ ) ( N * ( ~ +, ~)) is in a g r e e m e n t with the r e s u l t s of B l o m q v i s t [6] and of R i s k a and Brown [7], which w e r e obtained with s i m p l i f i e d 3H, 3He wave functions. The " e x p e r i m e n t a l " value of 6 is given a p p r o x i m a t e l y by 10
Volume 38B, number 1
PHYSICS
LETTERS
10 January 1972
Table 1 Contributions to 5 (eq. (1)) associated with the various A(2)(X) and components of the trinucleon wave functions. 5AB(X, /2) and 5AB(X, rnp) (A, B = S, S', D) denote the contributions p to 5 from terms in A(~)tX) fnvolving the pion and rho propagators respectively. The values of the 6, s are given as percentages. The un'-certainties in 5 are calculated from the uncertainties in the coupling constants given in eq. (7). X
5ss(X , it)
5ss(X, n~p)
5S,s,(X, tz)
5S,s,(X, ~ p )
5 sD(X, tz)
5sD(X, rap)
Total
N*(~ + ' 2-3i
0
-0.00
7.53
7.53 ± 0.41~;{:
N*(~+-,})
0.05
-0.00
0.23
0.28 ±0.12%
N*(~-, ~)
-0.18
0.00
-O.18'Y~
9
0.80
-0.81
-0.01
0.01
2.08
-0.58
1.49 ~ 0.25~)~
Total
0.67
-0.81
-0.00
0.01
9.84
-0.58
9.1 ± 0.5%
5exp(%) = ( ½ + l / 6 G 2 ) ( - l + ( f t ) n / ( f t ) 3 H )
x 100 + 5R + 34-P(S') . 2 p ( D ) .
(9)
5 R is o b t a i n e d by c o n s i d e r i n g the " m i n i m a l " v 2 / c 2 c o r r e c t i o n to the G a m o w - T e l l e r o p e r a t o r [13]
~" ~ a i - o i p 2 / 2 m 2
+ (ai'Pi/2m2)p
i ,
(10)
w h e r e Pi is the m o m e n t u m of n u c l e o n i. We find 5 R v 0.5% which c o r r e s p o n d s to a m e a n v a l u e of v / c , f o r a n u c l e o n in 3H o r 3He, of v 0.12. Recently determinedft values are (ft)n : 1108 ~= 16.45 s e a [14],
(ft)3 H = 1143 ± 3 s e c [15].
(11)
If we use t h e s e f t v a l u e s and P ( S ' ) = 0.43%, P(D) = 8.52% f r o m the c a l c u l a t i o n of J a c k s o n et al. [12] ( w h o s e e s t i m a t e s of t h e s e q u a n t i t i e s s h o u l d be s o m e w h a t m o r e r e l i a b l e than o u r s ) , we find 5 exp ~ 4.9 + 1.0%,
(12)
w h i c h is to be c o m p a r e d with o u r c a l c u l a t e d v a l u e 5 = 9.1 + 0.5%.
(13)
T h e a g r e e m e n t b e t w e e n 5exp and o u r c a l c u l a t e d v a l u e is s a t i s f a c t o r y in v i e w of: 1) our n e g l e c t of s o m e of the D - s t a t e c o m p o n e n t s of the t r i n u c l e o n w a v e f u n c t i o n s ; 2) o u r n e g l e c t of c o n t r i b u t i o n s to 5 f r o m m u l t i p l e m e s o n e x c h a n g e and w a v e f u n c t i o n n o r m a l i z a t i o n c o r r e c t i o n s [5] and 3) u n c e r t a i n t i e s in the v a l u e of P ( S ' ) . We a r e c u r r e n t l y i m p r o v i n g o u r d e t e r m i n a t i o n of the t r i n u c l e o n w a v e f u n c t i o n s by i n c l u d i n g c o n t r i b u t i o n s f r o m the c h a n n e l s t a t e s [[)q(02)2,
(1~) ~ 3] ~ ~' '~' , (0½) 513 z } 1 ,
(14)
which should be next in importance after the states (8a), (Sb) and (8c), on the basis of centrifugal b a r r i e r considerations. Our new calculation will probably lead to an increased P(D), and thus reduce the discrepancy between our results and those of Jackson et al. [12]. It should be noted that the components of the wave functions involving the states (14) do not contribute to the S-D matrix element of A (2)(N*(3+, 23-)). T h u s , although P(D) m i g h t i n c r e a s e , the 3H and 3He p r o b a b i l i t i e s f o r the D s t a t e s (8c) m i g h t d e c r e a s e and r e s u l t in a l o w e r c a l c u l a t e d v a l u e f o r 6. T h e v a l u e of P ( S ' ) is a m a j o r u n c e r t a i n t y in our a n a l y s i s . T h e c a l c u l a t e d v a l u e of 5 is not v e r y s e n s i t i v e to the a m o u n t s of S' in 3H and 3He, but the " e x p e r i m e n t a l " v a l u e of 5 ( s e e eq. (9)) i n c r e a s e s by 107 l i ~ if P ( S ' ) is i n c r e a s e d by 1%. T h e r e a l i s t i c n u c l e o n - n u c l e o n i n t e r a c t i o n s of R e i d [9] and of H a m a d a and J o h n s t o n [16] g i v e v e r y s i m i l a r v a l u e s f o r P ( D ) (8.54% [12] and 9% [17] r e s p e c t i v e l y ) but d i f f e r in a s i g n i f i c a n t way (with r e s p e c t to o u r p r e s e n t a n a l y s i s ) in the v a l u e s they g i v e f o r P(S') (0.429 [12] and 2% [17] r e s p e c t i v e l y ) . P e r h a p s m o r e i m p o r t a n t is the o b s e r v a t i o n that the a d d i t i o n a l i n t e r a c t i o n (inv o l v i n g t h r e e - n u c l e o n f o r c e s ) r e q u i r e d to g i v e the e x p e r i m e n t a l binding e n e r g y 8.49 M e V , m a y w e l l i n c r e a s e P ( S ' ) by s e v e r a l p e r c e n t . We a r e p r e s e n t l y e s t i m a t i n g the c h a n g e s in P(S') and P(D) which r e s u l t f r o m the a d d i t i o n of r e a s o n a b l e t h r e e - b o d y f o r c e s [18] to the n u c l e a r i n t e r a c t i o n .
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Volume
38B, number
I
PHYSICS
LETTERS
i0 January 1972
T w o of u s (Y. E. K. a n d A. T.) w i s h to t h a n k P r o f e s s o r s H. P . N o y e s a n d S. D. D r e l l f o r h o s p i t a l i t y a n d i n t e r e s t i n g d i s c u s s i o n s a t t h e S t a n f o r d L i n e a r A c c e l e r a t o r C e n t e r ( J u n e - A u g u s t , 1971) w h e r e p a r t of t h i s r e s e a r c h w a s d o n e . One of us (E. F.) w i s h e s to t h a n k P r o f e s s o r s K. E. B e r g k v i s t a n d G. E. B r o w n a n d D r . M. C h e m t o b f o r s e v e r a l s t i m u l a t i n g d i s c u s s i o n s on t h e p r o b l e m of m e s o n - e x c h a n g e e f f e c t s .
References [1] R.J. Blin-Stoyle and M. Tint, Phys. Rev. 160 (1967) 803. [2] W.K. C h e n g a n d E . F i s c h b a c h , Phys. Rev. 188 (1969) 1530. [3] W.K. Cheng, Ph. D. Thesis, University of Pennsylvania (1966), unpublished. In calculating the F, s in eqs. (4), s e v e r a l e r r o r s in [2] and [3 l have been c o r r e c t e d , in these r e f e r e n c e s , A/~(2)(N*( 3+, ½)) should be multiplied by and AB(2)(N*(½~ , ½)), A/3(2)(N*(~-, ½)) and Aj3(2)(p) by -¼, ~ and ½ respe~'tively. [4] R.J. Blifi-Stoyle, Phys. R~v. 188 (1969) 1 5 4 0 . [5] M. Chemtob and M. Rho, Phys. L e t t e r s 29B (1969) 540; Nucl. Phys. A163 (1971) 1. [6] J. Blomqvist, Phys, L e t t e r s 32B (1970) 1. [7] D.O. Riska and G . E . B r o w n , Phys. L e t t e r s 32B (1970) 662. [8] L.O. Faddeev, Zh. Eksper. Teor. Fiz. 39 (1960) 1459, Soviet Phys. J E T P 12 (1961) 1014; Mathematica! aspects of the three-body problem in quantum s c a t t e r i n g theory (Daniel Davey and Company, inc., New York, 1965). [9] R.V. Reid, Ann. of Phys. 50 (1968) 411. [10] E . P . Harper, Y . E . K i m and A.Tubis, Phys. Rev. C2 (1970) 877, e r r a t u m C2 (1970) 2455. [11] R.A. Malfliet and J . A . Tjon, Ann. of Phys. 61 (1970) 425. [12] A.D. Jaekson, A. Lande and P.U. Sauer, Phys. Letters 35B (1971) 3651. [13] J.S. BeLL and R.J. Blin-Stoyle, Nucl. Phys. 6 (1958) 87: R. J. Blin-Stoyle and S. Papageorgiou, Nucl. Phys. 64 (1965) i. [14] R.C. Saigo and N.H. Staub, Nucl. Phys, A138 (1969) 417. [15] K.E.Bergkvist, private communication. [16] T. Hamada and I.D. Johnston, Nucl. Phys. 34 (1962) 382: T. Hamada. Y. Makamura and R. Tamagaki, Progr. Theoret. Phys. (Kyoto) 33 (1965) 769(L). [17] L.M. Delves, J.M. BLatt, C. Pask and B. Davies, Phys. Letters 28B (1969) 472. [18] J. Fujita and H. Miyazawa, Progr. Theoret. Phys. (Kyoto) 17 (1957) 360: C. Pask, Phys. Letters 25B (1967) 78: B.A. Loiseau and Y. Nogami, Nuel. Phys. B2 (1967) 470.
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