- Email: [email protected]
On a parity-violating nucleon-nucleon potential
~
Nuclear Physics B4 (1968) 657-670. North-Holland Publ. Comp., A m s t e r d a m
ON
A PARITY-VIOLATING
NUCLEON-NUCLEON
POTENTIAL
R. L A C A Z E * Centro Brasiliero de Pesquisas Fisicas, Rio de Janeiro, Brazil Received 28 December 1967
Abstract: A static p a r i t y - v i o l a t i n g nucleon-nucleon potential due to the two-pion exchange is constructed, assuming the c u r r e n t - c u r r e n t and CVC hypotheses for weak interactions. Reseattering t e r m s as well as the ~Tr P-wave in the strong 1T-N vertex a r e taken into account. Our r e s u l t s differ from those obtained by Blin-Stoyle in a perturbation c a l c u l a tion for the strong interaction; namely, the range of our potentials is longer and this might modify P a r t o v i ' s r e s u l t s for the a s y m m e t r y coefficients in photodisintegration of the deuteron.
1. ~ T R O D U C T I O N A s s u g g e s t e d by a c e r t a i n n u m b e r of e x p e r i m e n t s [ 1 , 2 ] , t h e r e i s s o m e e v i d e n c e f o r a p a r i t y v i o l a t i o n in n u c l e a r i n t e r a c t i o n , t h e r a t e of t h e p a r i t y n o n - c o n s e r v i n g p a r t s of t h e i n t e r a c t i o n b e i n g of t h e o r d e r 10 -7 . Such a v i o l a t i o n i s a l l o w e d w h e n o n e a s s u m e s t h e c u r r e n t - c u r r e n t h y p o t h e s i s [3] in t h e w e a k p r o c e s s e s b y i n t r o d u c i n g a w e a k 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 . Such a n i n t e r a c t i o n m a y b e o b t a i n e d by c o n s i d e r i n g e i t h e r a d i r e c t interaction or a ~-meson exchange between the nucleons. T h e f i r s t t y p e of i n t e r a c t i o n ( c a l l e d c o n t a c t i n t e r a c t i o n ) h a s b e e n s t u d i e d by M i c h e l [4] b u t it g i v e s r i s e to v e r y s h o r t - r a n g e f o r c e s . T h e s e c o n d t y p e @ - m e s o n e x c h a n g e ) h a s b e e n s t u d i e d b y B l i n - S t o y l e [5] to s e c o n d o r d e r in p e r t u r b a t i o n t h e o r y f o r t h e s t r o n g - i n t e r a c t i o n v e r t e x ( s e e fig. 1). T h i s c a l c u l a t i o n d o e s n o t t a k e into a c c o u n t t h e r e s c a t t e r i n g t e r m s w h i c h a r e k n o w n to b e i m p o r t a n t in t h e ~ - N s c a t t e r i n g , s o t h a t it s e e m s i n t e r e s t i n g to t r e a t m o r e a c c u r a t e l y t h i s t y p e of m e c h a n i s m g i v i n g r i s e to t h e p a r i t y - v i o l a t i n g nucleon-nucleon forces. T h e e f f e c t s of p a r i t y - v i o l a t i n g p o t e n t i a l s h a v e b e e n s t u d i e d by d i f f e r e n t a u t h o r s . T h u s s t a t i c p o t e n t i a l of t h e f o r m ( a l × a 2 ) . r h a s b e e n u s e d f o r c o m p u t i n g t h e e f f e c t s of p a r i t y v i o l a t i o n in t h e d e u t e r o n p h o t o d i s i n t e g r a t i o n a n d a l s o in t h e r a d i a t i v e c a p t u r e of n e u t r o n s by h y d r o g e n a n d by d e u t e r o n [6, 7]. * On leave of absence from Centre National de la Recherche Scientifique, P a r i s .
658
R. L A C A Z E
\
/
\
J
/ \
/ \/
/% Fig. 1. D i a g r a m s t u d i e d by B l i n - S t o y l e .
With the help of this model P a r t o v i [7] has shown that such effects a r e not easily m e a s u r a b l e at p r e s e n t . F o r example, in the photodisintegration of the deuteron, the a s y m m e t r y coefficients of the c r o s s sections and the p o l a r i s a t i o n s a r e expected to be of the o r d e r of 10 -8 to 10 -7. In the s a m e way, a v e l o c i t y - d e p e n d e n t potential (a 1 - a 2) "p has been u s e d for evaluating the n o n - r e g u l a r v - t r a n s i t i o n a m p l i t u d e s of the 181Ta, 175Lu, 57Fe and 203T1 nuclei [8, 9]. The c i r c u l a r p o l a r i s a t i o n s found exp e r i m e n t a l l y (of the o r d e r 10-4) a r e compatible with a r a t e F taken between 9 × 10 -7 and 110 × 10 -7, where F stands [10] for the r a t e of the p a r i t y non c o n s e r v i n g to the p a r i t y c o n s e r v i n g f o r c e s . A p r e c i s e d e t e r m i n a t i o n of the p a r i t y violating n u c l e o n - n u c l e o n potential m a y favour m o r e i m p o r t a n t effects in light nuclei, and on the other hand, m a y provide m o r e a c c u r a t e p r e d i c t i o n s in heavy nuclei. We have thus computed the contribution to the p a r i t y - n o n - c o n s e r v i n g N-N potential due to the two-pion exchange, taking into account the r e s c a t t e r i n g t e r m s , a s well as the ~-~ p - w a v e t e r m in the 7r-N scattering. F u r t h e r m o r e , the effects of f o r m f a c t o r s in the weak c u r r e n t s a r e also included. In the strong i n t e r a c t i o n c a s e , Amati, L e a d e r and Vitale [11] have c o m puted the t w o - p i o n - e x c h a n g e contribution (TPEC) to the elastic s c a t t e r i n g amplitude, using an analytic continuation of the p i o n - n u c l e o n amplitude at low energy. Cottingham and Vinh Mau [12] have shown that, with a c e r t a i n a p p r o x i m a t i o n , one can d e r i v e f r o m the T P E C an equivalent e n e r g y - i n d e pendent potential. We shall see that the c u r r e n t h y p o t h e s i s for weak p r o c e s s e s will allow a p a r i t y - v i o l a t i n g p a r t in the p i o n - n u c l e o n amplitude and, consequently, a p a r i t y - n o n - c o n s e r v i n g n u c l e o n - n u c l e o n interaction. In the following we shall use the method and the notations of refs. [11, 12]. In sect. 2, we expand the n u c l e o n - n u c l e o n s c a t t e r i n g amplitude in a set of s c a l a r and p s e u d o s c a l a r p e r t u r b a t i v e o p e r a t o r s and we g e n e r a l i z e for the coefficients of t h e s e i n v a r i a n t s the d i s p e r s i o n relation postulated in ref. [11]. Unitarity is used for the computation of the s p e c t r a l functions in t e r m s of the elastic p i o n - n u c l e o n s c a t t e r i n g amplitude. The l a t t e r is c a l c u l a t e d in sect. 3 to the f i r s t o r d e r in the weak interaction.
NN PARITY VIOLATION
659
F r o m t h e s c a t t e r i n g a m p l i t u d e w e c o n s t r a c t , in s e c t . 4, t h e p o t e n t i a l s in t h e w a y g i v e n by C o t t i n g h a m a n d V i n h Mau. T h e r e s u l t s a r e p r e s e n t e d a n d d i s c u s s e d in s e c t . 5.
2. T H E S C A T T E R I N G A M P L I T U D E IN F I E L D T H E O R Y W e c o n s i d e r t h e e l a s t i c s c a t t e r i n g of two n u c l e o n s f r o m a s t a t e w i t h f o u r - m o m e n t a n l , P 1, to a s t a t e with f o u r - m o m e n t a n2,P 2 (fig. 2a).
I
a
!
%* ?
b
q:
c
Fig. 2. a - Nucleon-nucleon scattering, b - one pion exchange, c - two-pion exchange. We choose the three linearly independent four-vectors N = ½(n 1 + n2) ,
A = n 1 -n 2 = P2-Pl,
P : ½(Pl + P 2 ) ,
(2.1)
and the three scalars s = -(Pl +nl )2,
t = -(n 1 -n2 )2,
/-= - ( P l - n2 )2 ,
(2.2)
with the constraint
s+ t+ [= 4 m 2 . The scattering amplitude is defined from the S-matrix elements, Sfi = 6fi + i 5 4 ( n l +
by
Pl - n2 - P2)(m/2~E)2 × ff(P2)xp~u(n2)Xn~MU(nl)XnlU(Pl)XPl,
(2.3)
w h e r e E i s t h e n u c l e o n e n e r g y in t h e c e n t e r - o f - m a s s s y s t e m a n d u ( a ) a n d X~ a r e , r e s p e c t i v e l y , t h e s p a c e - t i m e s p i n o r a n d i s o s p i n o r of t h e n u c l e o n with four-momentum a. Now M m a y b e e x p a n d e d in a s e t of s p i n - a n d i s o s p i n - i n v a r i a n t o p e r a t o r s :
M :
~
i, j , k
k
k
k
P i j ( w , t , t-)S i Tj ,
(2.4)
660
R. LACAZE
w h e r e k stands for the symbols + o r -, with S/+ (S/-) and Tj+ (Tf) s y m m e t r i c ( a n t i s y m m e t r i e ) with r e s p e c t to the exchange of the nucleons n and p. If one only r e q u i r e s c h a r g e c o n s e r v a t i o n , the o p e r a t o r s T / t a k e the f o r m [5,13] T/ = lnl p ,
r n • ~p ,
Tj- = T~I p - l n ~ p ,
n p3 , T3~"
T~ 1p + lnT ~, ;
(v n × r P ) 3 ;
j = 1,4, j = 1,2,
(2.5)
w h e r e (Tn × vP) 3 changes sign under t i m e - r e v e r s a l . The S.~ o p e r a t o r s a r e chosen to be: J n
S1+ = l n l p ,
S ~ = Y5~5 (iyn" P + i y p. N),
$2~ = i y n . P l P 4- l n i y P . N ,
S ~ = v ~ i v n ' P 1 p 4- i nT~5 i ~ p - N ) ,
S; = iyn'piTP'N,
S~= (~4-~5) n.vp,
S; = vn.v p ,
$94- = i ~ l P 4- lni~P5 ,
n
S104- = zT, 5 "niyP. N 4- i y n. Pi~5.
s ; = 75 4 ,
(2.6)
It is possible to show that all o t h e r p o s s i b l e spin o p e r a t o r s may be exp r e s s e d in t e r m s of t h e s e ten o p e r a t o r s , by developing vn and 7P in t e r m s of four orthogonal f o u r - v e c t o r s N, P and A, and by using the D i r a c equation. According to the t r a n s f o r m a t i o n s p r o p e r t i e s given in table 1, t h e s e o p e r a t o r s Si + may be c l a s s i f i e d [14] as p r o p e r s c a l a r s for i = 1, 5, t i m e pseudos c a l a r s for i = 6, space p s e u d o s c a l a r s for i = 7, 8, and p r o p e r p s e u d o s c a l a r s for i = 9 , 1 0 .
Table 1 Transformations properties of the S~ i
P
T
1-5 6 7-8
+ + -
+ +
9 -10
T h e s e invariant o p e r a t o r s a r e a g e n e r a l i s a t i o n of those defined by Amati, L e a d e r and Vitale; they a p p e a r in the computation of f o u r t h - o r d e r Feynman g r a p h s (an invariant of the f o r m ( n ± ?/p)i~/n.pip. N also a p p e a r s but one can e x p r e s s it in t e r m s of S7+ and ~84- without introducing any new s i n g u l a r i ty). The Mandelstam r e p r e s e n t a t i o n f o r the s c a l a r , f u n c t i o n s p k ~ may then be c o n s i d e r e d a s valid to the fourth o r d e r . If it is a s s u m e d to b6'~alid for any o r d e r , then one can w r i t e
NN PARITY VIOLATION
+ oo pi;(w, P i ~ ( w ' t ' t - ) = t _aij~ + - ~ 1 f4U2 t ' - t
t')
661
dt' 1
oo
dt'
oo X i ; ( w ' , t ' )
+-~2f9#2t'-tf4m2
w'-w
dw'.
(2.7)
As we a r e i n t e r e s t e d in the l o n g - r a n g e f o r c e s , we shall neglect the c o n t r i bution of the last integral in eq. (2.7) the t-dependence of which c o r r e s p o n d s to a potential due to an exchange of at l e a s t t h r e e pions [12]. In fact we a r e only i n t e r e s t e d in the p a r i t y violating effects and then we r e s t r i c t o u r s e l v e s to the contributions Pi,j + with i = 6, 10. The pole t e r m in eq. (2.7) c o r r e s p o n d s to the one-pion exchange (fig. 2b). It was pointed out some t i m e ago [15] that a p a r i t y violation in the n-N v e r tex i m p l i e s violation u n d e r t i m e r e v e r s a l , but only if c h a r g e independence is assumed. In fact, if we define the f o r m f a c t o r s by:
(n21JV[nl) = i~(n2){fl(t)vp ----~ g
F2(t ) ~pu(nl +i-- a rn~
(0[J~[Tr~
-n2) u
F3(t)(n 1 - n2) p} u(nl)
= -tzFTr(t ) qp dp- ,
T-, (2.8)
w h e r e rn, p, rnp a r e the nucleon, pion and lepton m a s s e s r e s p e c t i v e l y , g is the g y r o m a g n e t i c ratio, a some constant and FI(0 ) = F2(0 ) = F3(0) = 1, we get, with the c u r r e n t - c u r r e n t hypothesis, a non-vanishing pole t e r m cont r i b u t i o n to the i n v a r i a n t s S9+Tf, S~T~, Sg-T ~ defined by eqs. (2.5) and (2.6). T h i s pole contribution to eq. (2.7) is given by: c~9; = ½ E~N GV rn~ ~ a t I m { F : ( t ) F3(t)} , ot93+ = - e 9 2 +,
e92
-
1 P a t R e { F ; ( t ) F3(t)} , = ~gTrN GV mp
(2.9)
w h e r e g~N is the ~-N coupling constant. We note that such a pole t e r m c o r r e s p o n d s to a second kind c u r r e n t a s defined by Weinberg [16] but, if the f o r m f a c t o r s Fi(t) a r e r e a l , this contribution, a s opposed to the lepton c a s e [17], does not imply a t i m e - r e v e r s a l violation on account of p a r t i c u l a r i s o t o p i c - s p i n dependence (contribution to (v n x rP)3 ). However this t e r m i m p l i e s in non c o n s e r v a t i o n of a v e c t o r c u r r e n t and is d i s r e g a r d e d in the actual computation. The s p e c t r a l functions pi?(w, t') a r e obtained by writing the unitarity r e -
662
R. LACAZE
lation in the t-channel f o r the M - m a t r i x . We obtain:
~ o i ) (kw , t') S k Tjk =
ijk
1 2(2~)2 t ~7/g,
(2.10)
with c'~ = ~ (:~1P2 [ T+ [ 2~>(2~. IT [ n1~2> ' 2~
w h e r e ~" stands f o r the annihilation N + N --* 2~ amplitude. The s u m m a t i o n is to be extended to the i s o s p i n indices and to the four m o m e n t a of the p i o n s in the i n t e r m e d i a t e s t a t e (the l a t t e r is e s s e n t i a l l y an a n g u l a r i n t e g r a t i o n [11]). If we put ~- into the f o r m T = r s + Tw w h e r e Ts stands f o r the s t r o n g int e r a c t i o n a m p l i t u d e and z w f o r the w e a k i n t e r a c t i o n amplitude, the t e r m s TS%w and TW+Ts will give u s the c o n t r i b u t i o n we a r e looking for. T h e s e a n nihilation a m p l i t u d e s a r e obtained by a n a l y t i c continuation of the e l a s t i c s c a t t e r i n g ~-N amplitude. Such an a m p l i t u d e is w r i t t e n in the m o r e g e n e r a l f o r m as: F = F 1 - F 2 t " 7 + F 3 t 3 T 3 - ½ F 4 t 3+½ FhT3+ ~ F 6 ~ + ½ F 7 T 3 ~
+ F 8 ( t x 7-)3+ ~1F9(t. T t 3 + t 3 t . 7 ) + F l o ( (
t x 7)3t3-t3(t
× 7)3 ) ,
(2.11)
with [18]: F i = - A i - A i h i 7 5 + i y " OB i + i y " O ~ h B i 5 ,
(2.12)
w h e r e O is one half of the s u m of the pion q u a d r i m o m e n t a , A and B a r e s c a l a r functions of the usual s c a l a r s s, g, t and t and 7 a r e the usual i s o topic spin m a t r i c e s . We t a k e this g e n e r a l f o r m f o r the weak i n t e r a c t i o n a m p l i t u d e F w but the r e s t r i c t i o n s upon the s t r o n g i n t e r a c t i o n a m p l i t u d e r e d u c e F s to: F s = F~-
F,~ t" 7 ,
(2.11')
with F is = -A is + i
. QB s"
(2.12')
If ~ and fi a r e the i s o t o p i c - s p i n i n d i c e s of the two pions, we m u s t c o n s i d e r the p r o d u c t of the m a t r i x e l e m e n t s __F~f~s* F~(~w and take the s u m o v e r the i s o t o p i c - s p i n s t a t e s of the pions. We get
NN P A R I T Y V I O L A T I O N
s*
asFaI3FYa
663
s* w_n p -~s*~w n p =3F~*(F~ v + F ~ v) l n l p + 2F 2 F 2 , .T - z~ 2 ~3T3T3
1
s*
w
w
+ ~ [ F 1 (F 5 + F 7 +
F~)
s*
w
n
p
+ F 2 F 4](T 3+T 3)
1 S* W W F ~ ) S* W n Tp) +~[F1 (F5+FT+ - F 2 F4](~'3 + 2F~*F~(Tn×rP) 3 .
(2.13)
Taking into account the definitions (2.5), we can write the above formula in the f o r m
ot ~
Fa~F~a = ~ ,~l ws* l ijkl "~ijk~i FT Tk'
(2.13)
where T / a r e the isotopic-spin o p e r a t o r s and the coefficients Ciikl can be obtained f r o m (2.13). With the appropriate p h a s e - s p a c e factors, we can then write: ~ t ( t - 4 ~ 2 ) ½ ijkl ~ clij kC~fCijT l + Sy,
9~:
(2.14)
where the t e r m Sy c o m e s f r o m the product F w* F s and can be obtained dir e c t l y hy s y m m e t r i s a t i o n . If we r e s t r i c t o u r s e l v e s to the part of the 9K which violates the spacial parity conservation, 97~ij takes the form:
c~ij = f [ - . 4 s * + i yP" QBS* ] [ - A ~ i~55+ i ~n. Q~55BjS~ d~Q • n
= aij zY5 + bij
iy p
(2.15)
n n • Niy n5 + cij iy n • Py~ + dij yn YPY5 + eij iyn. piyp" N ~5,
where S*
w
S*
w
aij =f[A i Aj + m a S i d j ] d g t q , bij =
f
vii = - f [a a i eis :
f
A7
,
9 j + m(ol 2
--DR26) BS.B~] d~Q ,
,
eij : f (a3 + N_~P 5 ) B S * B 7 d~2o '
(2.16)
664
R. LACAZE
with
D = N2p 2 - (N.p)2, =I(N2Q.P
- P.N Q.N),
=I(p2Q.N
- P.N ©.P),
6 = Q2 _ ( a p + fiN)2. A f t e r s y m m e t r i z a t i o n , the o p e r a t o r s ( i ~ n. P~s i~ p . Y + i~ n. P i y P . N~ p) can be e x p r e s s e d in t e r m s of the o t h e r s :
in NIgh5
:
Then we obtain the expansion of ~ in t e r m s of the invariant o p e r a t o r s and the knowledge of the pion nucleon amplitude in the a p p r o p r i a t e r e g i o n of int e g r a t i o n allows us to compute the s p e c t r a l functions of eq. (2.10).
3. THE y-N A M P L I T U D E We a r e mainly i n t e r e s t e d in the nucleon s c a t t e r i n g n e a r the t h r e s h o l d , i.e. for w ~ 4m 2, and then for small t and t. The m o s t important c o n t r i b u tion to the integral of the s p e c t r a l function pi~(w, t') in eq. (2.7) o c c u r for not too l a r g e v a l u e s of the v a r i a b l e t'. T h e s e ' s p e c t r a l functions can then be obtained by extrapolation of the p i o n - n u c l e o n amplitude at low e n e r g y [12]. F o r the s t r o n g - i n t e r a c t i o n p a r t such an amplitude h a s been c o n s t r u c t e d by Bowcock, Cottingham and Luri~ [19] and used in refs. [11] and [12]. We make use h e r e of an equivalent amplitude which we have built up in a p r e v i ous work [20]. This amplitude t a k e s into account the pole t e r m due to the nucleon exchange, a r e s c a t t e r i n g t e r m a p p r o x i m a t e d by the pole of the N~3 r e s o n a n c e and the n-n i n t e r a c t i o n in the S- and P - w a v e s also a p p r o x i m a t e d by p o l e s (p-pole for the latter). The n-N amplitude to the f i r s t o r d e r of weak i n t e r a c t i o n can be cons t r u c t e d in a s i m i l a r way in the f r a m e w o r k of the c u r r e n t - c u r r e n t hypot h e s e s . However the t e r m s c o r r e s p o n d i n g to the exchange of the nucleon and of the i s o b a r N~3 (fig. 3) do not contribute if the v e c t o r c u r r e n t is c o n served. Indeed the product of the ~ - c u r r e n t with another c u r r e n t is nothing else than the d i v e r g e n c e , as can be seen in the computation of the pole t e r m . Thus t h e s e contribution a r e ruled out in p r a c t i c a l computation. The t e r m equivalent to the n-~ p wave contribution may be obtained either by including directly the p, o r by introducing the p in the f o r m f a c t o r of the two-pion c u r r e n t ; this last p r o c e d u r e was adopted here. We make use of the following c u r r e n t s :
NN P A R I T Y V I O L A T I O N
\
",x9
f l
665
/
F i g . 3. 7/-N s c a t t e r i n g e x c h a n g e in t h e d i r e c t c h a n n e l .
= - 2Fuu(t)Qgt + b
c FMI(t) ~p.vAv}T5U(nl) "r-
+ m
(3.1)
w h e r e X = G A / O , and obtain by c o m p a r i n g with eqs. (2.11) and (2.12): C
*
A25w = 4 GA m N. Q Im{Fu~ (t) FMI(t)} ,
A85w : - 4 GA C---N" Q R e { F u ; ( t ) FMI(t)} , m
B25w = 2 G A R e { F : ( t ) FA(t)} ,
w-- 2J, B 85w = - 2 GA I m { F u ; (t) FA(t) } .
(3.2)
The f o r m f a c t o r s Fu~(t), FA(t ) and FMI(t) a r e n o r m a l i z e d to 1 for t = 0 and in the physical region for t, they may b e c o m e c o m p l e x [21]. The explicit e x p r e s s i o n f o r A i S , B i S , A~ 5W , B45 W in t e r m s of Q allows J us to p e r f o r m the integration (2.13). The spectra~ functions Pij k a r e then d e t e r m i n e d by (2.10) but t h e i r t - d e p e n d e n c e for t >I 4g 2 r e q u i r e s the knowledge of the f o r m f a c t o r s . We a s s u m e that the f o r m f a c t o r s F ~ ( t ) and FA(t) a r e dominated by the exchange of the p and of the A I (ref. [22]) r e s o n a n c e (1080 MeV), r e s p e c t i v e l y (we a s s u m e for the l a t t e r j P G = i + - ) . The FMI(t) f o r m f a c t o r c o r r e s p o n d s to the exchange of a i ++ s y s t e m [21] and could be dominated by the B - r e s o n a n c e (1210 MeV). However t h e r e is no evidence of such a p s e u d o - t e n s o r i a l c u r r e n t and it has not been included in our num e r i c a l computations.
666
R. LACAZE
4. DEFINITION OF THE P O T E N T I A L A n u c l e o n - n u c l e o n potential c o n s t r u c t e d so as to r e p r o d u c e the s c a t t e r i n g amplitude in field t h e o r y has been obtained by Cottingham and Vinh Mau [12] in the c a s e w h e r e p a r i t y c o n s e r v a t i o n is a s s u m e d . In a s i m i l a r way, we can obtain a potential to the f i r s t o r d e r of weak i n t e r a c t i o n s , the main difference being the introduction of s u p p l e m e n t a r y invariants. B e s i d e s that, if we r e s t r i c t o u r s e l v e s to the d e t e r m i n a t i o n of a p a r i t y violating potential, the a b sence of the pion exchange (because of the a b s e n c e of a second kind v e c t o r c u r r e n t ) s i m p l i f i e s this determination. The s c a t t e r i n g amplitude has been expanded in a set of l i n e a r l y independent i n v a r i a n t s (2.4). T h e r e will be then as many types of l i n e a r l y independent potentials as these invariants:
V(r) : ~ Vaj(r) k k jk , ~2aT ajk
(4.1)
w h e r e k = + and the T k a r e isospin i n v a r i a n t s defined in (2.5). We have chosen the fla k to be: J g~pS+ = (an × aP) • r ,
tiPS = a n ' p a P ' L
9 P V+ = (an - a P ) . P ,
~PV- = ( a n + a P ) . P ,
~2pT
= (a n - a P ) . r , +
~2pT v = (a n x a P ) . P ,
+aP'Pan'L,
~2pT S = (a n + a P ) - r , ~ P T V = - ( a n " r a P . L + aP. r a n. L ) / r 2
(4.2)
w h e r e L = r x P and a n, a P a r e the Pauli spin o p e r a t o r s acting on the nucleons n and p. In the m o m e n t u m space the potential is written: (4.3)
ajk with
~
~PS~ + = - i(an × a p ) " A
~PS =i(an'pap'p
N + (a n _ a p ) . p ~2pv =
~PV = (an + a p ) " P '
+
flPTS =
-i(a n-ap).A
V ~ + = (a n × a P ) . P ~2pT
xzl + a P . P a n. p × A ) , (4.4)
5pT S :-i(a -
n+ap).A,
~PTV = ( a n ' A a P ' P x A
,
+ o p A a n. P × A ) / A 2.
In o r d e r to identify the potential s c a t t e r i n g amplitude given by the L i p p * Our ~ ;
are the same as those of ref. [23].
NN
PARITY
VIOLATION
667
m a n - S c h w i n g e r expansion and the field t h e o r y s c a t t e r i n g amplitude, we expand the l a t t e r in t e r m s of t h e s e invariants. By e x p r e s s i n g the D i r a c m a t r i c e s and s p i n o r s with the help of the Pauli o p e r a t o r s and s p i n o r s , we get s+ s+ ~ + s s u(n2) u(P2) S~/U(Pl) u(nl) : ~a X i ~ ×P2 ×n2 fla ×nl XPl"
(4.5)
In the adiabatic a p p r o x i m a t i o n [12, 24], the t r a n s f o r m a t i o n m a t r i x X conc e r n i n g the p a r i t y violating i n v a r i a n t s is
X;o•
PS+
PS-
PV±
PTS ±
7 8 9
+ t/8m 2 + 1/m 0
½rn2 ½rn3 0
2 - 2/m 0
0 0 ½rn
0 0 t/8 m 3
10
0
0
0
½m
- 3t/8m 2
PTV ~
This t r a n s f o r m a t i o n is e n e r g y - i n d e p e n d e n t and i n t r o d u c e s no new singularity. If we then c o n s i d e r the potential
1
(2~) 3 ~jk
i
1 o~ p~j(t') 2 t, - t
zi (t) T f
, ~k
at}as
k T) ,
(4.7)
it will r e p r o d u c e the s c a t t e r i n g amplitude defined in eq. (2.7) for the longest r a n g e , since the one pion exchange t e r m does not contribute. The potential (4.1) is obtained by an i n v e r s e F o u r i e r t r a n s f o r m a t i o n . If the i n t e r c h a n g e of the o r d e r of integration is a s s u m e d to be valid, one obtains: 1
vkj(r)_
1
~i f4~
(2~) 3 "
k
k
k
2 Xi~(t)PiJ(t) O~(r, t)
e -rt2 r dt,
(4.8)
w h e r e the O~(r, t) depend on the potential type ( s c a l a r , v e c t o r , tensor) k
O~(r, t) = 1 = t 2 (1 + 11)_
r =~
1
for
(~,k) = PV ±, PTV +
for
(~,k) = PS +, PTS +
for
(~,k) = PTV
rt 2 (1 +
3
3, i +--j
(4.9)
r t ~ r2t The polynomial f o r m of x i k ( t ) m a k e s the f o r m u l a (4.8) valid as long as we a r e only i n t e r e s t e d to the longest r a n g e f o r c e s .
668
R. LACAZE
5. RESULTS The amplitudes A i ~ and B i ~ v obtained in eq. (3.2) in the same way a s the A s and B s a m p l i t u d e s [20] enable us to obtain through eq. (2.13) the cont r i b u t i o n s to the s p e c t r a l functions; the integral (2.16) exhibits no special difficulty: the i n t e g r a l s (4.8) a r e computed n u m e r i c a l l y , the f o r m f a c t o r s (sect. 3) being chosen a s B r e i t - W i g n e r f o r m u l a e and n o r m a l i z e d to unity at the origin. In the limit w h e r e the r e s o n a n c e s p and A 1 a r e n a r r o w enough (this c o r r e s p o n d s to take the principal value of the integral with extrapolation of the phenomenological f o r m f a c t o r s defined for t < 0 (refs. [22, 25]), one obtains V(r) = (Vps(r)(anx
lr
a p).~+
Y p v ( r ) ( a n - a D).P)(T~T p - T n. vP),
(5.1)
where +
+
V p S (r) = - r V p s ' 2 = r V p s , 3 ,
V p v ( r ) -- - V p v , 2+ : + V p v , 3+ ,
(5.2)
with G = 1.25x 1.01 x 1 0 - 5 / M 2 we obtain the c u r v e s for Vps(r) and VpV(r) drawn in figs. 4 and 5. We have also drawn, in fig. 4, the potentials obtained by Blin-Stoyle [5]: (5.3) Gf 2 2 1 e -2~r c)3pS = - - ~ (1 + /-~+ #---~r2) r3 ' Q2PV = 4 - ~ 5 ( r ) -~ 0 . 8 2 × 1 0 - 8 ~ - 3 6 ( r ) .
,1.,lo -~
~v
!
I
I
',\
L_
t
Fig. 4. The potential Vps(r). The dotted line r e p r e s e n t s the corresponding potential obtained by Blin-Stoyle.
I" Fig. 5. The potential V p v ( r ).
NN P A R I T Y V I O L A T I O N
669
W e n o t i c e on fig. 4 that the s t a t i c p o t e n t i a l t h a t we h a v e o b t a i n e d (solid line) h a s a m u c h l o n g e r r a n g e than the B l i n - S t o y l e potential. One m i g h t then think t h a t the c o r r e s p o n d i n g e f f e c t s in the o b s e r v e d p r o c e s s e s will be m o r e i m p o r t a n t . Note a l s o t h a t the c o m p u t a t i o n s p e r f o r m e d by P a r t o v i [7] c o r r e s p o n d to a w e a k c o u p l i n g c o n s t a n t G, s m a l l e r t h a n the one u s e d in B l i n - S t o y l e p o t e n t i a l (1 x 10-5 e r g . c m 3 i n s t e a d of 1.40 x 10-5 e r g . cm3). F u r t h e r m o r e , we h a v e o b t a i n e d an i m p o r t a n t v e l o c i t y d e p e n d i n g p o t e n t i a l , the c o r r e s p o n d ing e f f e c t s of which m a y m o d i f y P a r t o v i ' s r e s u l t s . T h e p o t e n t i a l (5.1) s e e m s to u s to be m o r e r e a s o n a b l e , but it m a y be c h a n g e d if we i n c l u d e l a r g e width f o r p and A1, s e c o n d kind axial c u r r e n t and vector current non-conservation. With the e x p e r i m e n t a l width of p a n d A1, we o b t a i n a c o n t r i b u t i o n to flpS-(V n x vP)3 and f l p V - ( ~ n x rP)3 but it s e e m s that a m o r e c a r e f u l d e f i nition of the a n a l y t i c c o n t i n u a t i o n r u l e s out such c o n t r i b u t i o n s . T h e r e a l p a r t of the s e c o n d - k i n d a x i a l c u r r e n t g i v e s r i s e , t h r o u g h eq. (3.2), to a c o n t r i b u t i o n of the f o r m ~ P T S ( T n x TP)3 a n d flPTV-(V n x rP)3. T h e n u m e r i c a l c a l c u l a t i o n with such a t e r m of the o r d e r of the v e c t o r c u r r e n t s h o w s that the c o n t r i b u t i o n to f l P T S - ( T n x TP)3 is i m p o r t a n t but such a r e s u l t h a s no p a r t i c u l a r s i g n i f i c a n c e . A n o t h e r c o n t r i b u t i o n of t h i s f o r m m a y a l s o be o b t a i n e d with a s e c o n d kind v e c t o r c u r r e n t (which v i o l a t e s the v e c t o r c u r r e n t c o n s e r v a t i o n ) . T h e effect of s u c h a c o n t r i b u t i o n due to the pion pole h a s b e e n s t u d i e d [26], but it s e e m s not p o s s i b l e to t e s t the e x i s t e n c e of s e c o n d kind w e a k c u r r e n t with such an effect. T h e effect of the n o n - c o n s e r v a t i o n of t h e v e c t o r N - N * is m o r e e a s i l y a p p r e c i a b l e . I n d e e d if we include in the v e c t o r N - N * c u r r e n t [27] a " m a x i m u m v i o l a t i o n " *, we o b t a i n a c o n t r i b u t i o n quite l a r g e but the m a i n r e s u l t is that t h i s c o n t r i b u t i o n t a k e s the f o r m :
V ' ( r ) : ( V ~ s ( r ) ( a n x a p) . r + V p'v ( r ) ( a
n
_ u p ) . p ) ( l n l n - } ( r n . r p + T~3Tp))
and such a p o t e n t i a l would give r i s e to e f f e c t s in pp and nn i n t e r a c t i o n . I a m i n d e b t e d to P r o f e s s o r R. Vinh Mau f o r s u g g e s t i n g the s u b j e c t of t h i s w o r k and f o r h i s kind g u i d a n c e . T h e help of M. Y. N o i r o t at the e a r l y s t a g e of t h i s w o r k is a p p r e c i a t e d . I a l s o thank D r . A. T o u n s i f o r d i s c u s s i o n s .
REFERENCES [1] Yu. G. Abov, P.A. Krupehitsky and Yu. A. Oratovskny, Phys. Letters 12 (1964) 25; Soy. J. Nucl. Phys. I (1965) 341. * The vector current conservation implies, between the four form factors of the N-N* current, the relation =
o
From the photo production analysis [28] FV(o) = -FV(o)= 5,6 and FV(0) = FV(0) = O. Our "maximal violation" stands for FV(t.L2) = +FV(/~2) = 5.6 and FV(/.L2) = FV(~ 2) =0.
670
R. LACAZE
[2] F. Boehm and E. Kankeleit, Proc. Int. Conf. on nuclear physics, P a r i s , Juillet 1964, vol. I I p . 1181. [3] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193; M. Gell-Mann, Revs. Mod. Phys. 31 (1959) 834. [4] F. C. Michel, Phys. Rev. 133 (1964) B329. [5] R.J. Blin-Stoyle, Phys. Rev. 118 (1960) 1605; 120 (1960) 181. [6] R.J. Blin-Stoyle and H. Feshbach, Nucl. Phys. 27 (1961) 395. [7] F. Partovi, Ann. of Phys. 27 (1964) 114. [8] S. Wahlborn, Phys. Rev. 138 (1964) B530. [9] Z. Szymanski, Nucl. Phys. 76 (1966)539. [10] T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 254. [11] D. Amati, E. Leader and B. Vitale, Nuovo Cimento 17 (1960) 68. [12] W. N. Cottingham and R. Vinh Mau, Phys. Rev. 130 (1963) 735. [13] M.J. Moravcsik, The two-nucleon interaction (Clarendon P r e s s , Oxford, 1963); L. Eisenbud and E. P. Wigner, Proc. Nat. Acad. Sc. 27 (1941) 281. [14] P. Roman, Theory of e l e m e n t a r y particles, 2. ed. (North-Holland Publishing Co., Amsterdam, 1961). [15] V. G. Soloviev, Nucl. Phys. 6 (1958) 618. [16] S. Weinberg, Phys. Rev. 112 (1958) 1375. [17] N. Cabibbo and M. Veltman, p r e p r i n t CERN 65-30. [18] S. Fubini and D.Walecka, Phys. Rev. 116 (1959) 194. [19] J. Bowcock, W. N. Cottingham and D. Lurid, Nuovo Cimento 16 (1960) 918. [20] R. Lacaze, Th~se de 3e Cycle, Bordeaux (1964). [21] S. B~ T r e i m a n , Weak interactions and topics in dispersion physics (Bergen 1962 W. A. Benjamin, 1963). [22] G. Cocho, Nuovo Cimento 42A (1966) 421. [23] A.E.Woodruft, Ann. of Phys. 7 (1959) 65. [24] J. M. Charap and M. J. T a u s n e r , Nuovo Cimento 18 (1960) 316. [25] M. Paty, p r e p r i n t CERN 65-12. [26] D.O. Dal'Karov, J E T P (Sov. Phys. )Letters 8 (1965) 124. [27] S. M. Berman and M. Veltman, Nuovo Cimento 38 (1965) 993; C. H. Albright and L. S. Liu, Phys. Rev. 140 (1965) B748. [28] M. Gourdin and Ph. Salin, Nuovo Cimento 27 (1963) 193,309.
Nuclear Physics B4 (1968) 657-670. North-Holland Publ. Comp., A m s t e r d a m
ON
A PARITY-VIOLATING
NUCLEON-NUCLEON
POTENTIAL
R. L A C A Z E * Centro Brasiliero de Pesquisas Fisicas, Rio de Janeiro, Brazil Received 28 December 1967
Abstract: A static p a r i t y - v i o l a t i n g nucleon-nucleon potential due to the two-pion exchange is constructed, assuming the c u r r e n t - c u r r e n t and CVC hypotheses for weak interactions. Reseattering t e r m s as well as the ~Tr P-wave in the strong 1T-N vertex a r e taken into account. Our r e s u l t s differ from those obtained by Blin-Stoyle in a perturbation c a l c u l a tion for the strong interaction; namely, the range of our potentials is longer and this might modify P a r t o v i ' s r e s u l t s for the a s y m m e t r y coefficients in photodisintegration of the deuteron.
1. ~ T R O D U C T I O N A s s u g g e s t e d by a c e r t a i n n u m b e r of e x p e r i m e n t s [ 1 , 2 ] , t h e r e i s s o m e e v i d e n c e f o r a p a r i t y v i o l a t i o n in n u c l e a r i n t e r a c t i o n , t h e r a t e of t h e p a r i t y n o n - c o n s e r v i n g p a r t s of t h e i n t e r a c t i o n b e i n g of t h e o r d e r 10 -7 . Such a v i o l a t i o n i s a l l o w e d w h e n o n e a s s u m e s t h e c u r r e n t - c u r r e n t h y p o t h e s i s [3] in t h e w e a k p r o c e s s e s b y i n t r o d u c i n g a w e a k 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 . Such a n i n t e r a c t i o n m a y b e o b t a i n e d by c o n s i d e r i n g e i t h e r a d i r e c t interaction or a ~-meson exchange between the nucleons. T h e f i r s t t y p e of i n t e r a c t i o n ( c a l l e d c o n t a c t i n t e r a c t i o n ) h a s b e e n s t u d i e d by M i c h e l [4] b u t it g i v e s r i s e to v e r y s h o r t - r a n g e f o r c e s . T h e s e c o n d t y p e @ - m e s o n e x c h a n g e ) h a s b e e n s t u d i e d b y B l i n - S t o y l e [5] to s e c o n d o r d e r in p e r t u r b a t i o n t h e o r y f o r t h e s t r o n g - i n t e r a c t i o n v e r t e x ( s e e fig. 1). T h i s c a l c u l a t i o n d o e s n o t t a k e into a c c o u n t t h e r e s c a t t e r i n g t e r m s w h i c h a r e k n o w n to b e i m p o r t a n t in t h e ~ - N s c a t t e r i n g , s o t h a t it s e e m s i n t e r e s t i n g to t r e a t m o r e a c c u r a t e l y t h i s t y p e of m e c h a n i s m g i v i n g r i s e to t h e p a r i t y - v i o l a t i n g nucleon-nucleon forces. T h e e f f e c t s of p a r i t y - v i o l a t i n g p o t e n t i a l s h a v e b e e n s t u d i e d by d i f f e r e n t a u t h o r s . T h u s s t a t i c p o t e n t i a l of t h e f o r m ( a l × a 2 ) . r h a s b e e n u s e d f o r c o m p u t i n g t h e e f f e c t s of p a r i t y v i o l a t i o n in t h e d e u t e r o n p h o t o d i s i n t e g r a t i o n a n d a l s o in t h e r a d i a t i v e c a p t u r e of n e u t r o n s by h y d r o g e n a n d by d e u t e r o n [6, 7]. * On leave of absence from Centre National de la Recherche Scientifique, P a r i s .
658
R. L A C A Z E
\
/
\
J
/ \
/ \/
/% Fig. 1. D i a g r a m s t u d i e d by B l i n - S t o y l e .
With the help of this model P a r t o v i [7] has shown that such effects a r e not easily m e a s u r a b l e at p r e s e n t . F o r example, in the photodisintegration of the deuteron, the a s y m m e t r y coefficients of the c r o s s sections and the p o l a r i s a t i o n s a r e expected to be of the o r d e r of 10 -8 to 10 -7. In the s a m e way, a v e l o c i t y - d e p e n d e n t potential (a 1 - a 2) "p has been u s e d for evaluating the n o n - r e g u l a r v - t r a n s i t i o n a m p l i t u d e s of the 181Ta, 175Lu, 57Fe and 203T1 nuclei [8, 9]. The c i r c u l a r p o l a r i s a t i o n s found exp e r i m e n t a l l y (of the o r d e r 10-4) a r e compatible with a r a t e F taken between 9 × 10 -7 and 110 × 10 -7, where F stands [10] for the r a t e of the p a r i t y non c o n s e r v i n g to the p a r i t y c o n s e r v i n g f o r c e s . A p r e c i s e d e t e r m i n a t i o n of the p a r i t y violating n u c l e o n - n u c l e o n potential m a y favour m o r e i m p o r t a n t effects in light nuclei, and on the other hand, m a y provide m o r e a c c u r a t e p r e d i c t i o n s in heavy nuclei. We have thus computed the contribution to the p a r i t y - n o n - c o n s e r v i n g N-N potential due to the two-pion exchange, taking into account the r e s c a t t e r i n g t e r m s , a s well as the ~-~ p - w a v e t e r m in the 7r-N scattering. F u r t h e r m o r e , the effects of f o r m f a c t o r s in the weak c u r r e n t s a r e also included. In the strong i n t e r a c t i o n c a s e , Amati, L e a d e r and Vitale [11] have c o m puted the t w o - p i o n - e x c h a n g e contribution (TPEC) to the elastic s c a t t e r i n g amplitude, using an analytic continuation of the p i o n - n u c l e o n amplitude at low energy. Cottingham and Vinh Mau [12] have shown that, with a c e r t a i n a p p r o x i m a t i o n , one can d e r i v e f r o m the T P E C an equivalent e n e r g y - i n d e pendent potential. We shall see that the c u r r e n t h y p o t h e s i s for weak p r o c e s s e s will allow a p a r i t y - v i o l a t i n g p a r t in the p i o n - n u c l e o n amplitude and, consequently, a p a r i t y - n o n - c o n s e r v i n g n u c l e o n - n u c l e o n interaction. In the following we shall use the method and the notations of refs. [11, 12]. In sect. 2, we expand the n u c l e o n - n u c l e o n s c a t t e r i n g amplitude in a set of s c a l a r and p s e u d o s c a l a r p e r t u r b a t i v e o p e r a t o r s and we g e n e r a l i z e for the coefficients of t h e s e i n v a r i a n t s the d i s p e r s i o n relation postulated in ref. [11]. Unitarity is used for the computation of the s p e c t r a l functions in t e r m s of the elastic p i o n - n u c l e o n s c a t t e r i n g amplitude. The l a t t e r is c a l c u l a t e d in sect. 3 to the f i r s t o r d e r in the weak interaction.
NN PARITY VIOLATION
659
F r o m t h e s c a t t e r i n g a m p l i t u d e w e c o n s t r a c t , in s e c t . 4, t h e p o t e n t i a l s in t h e w a y g i v e n by C o t t i n g h a m a n d V i n h Mau. T h e r e s u l t s a r e p r e s e n t e d a n d d i s c u s s e d in s e c t . 5.
2. T H E S C A T T E R I N G A M P L I T U D E IN F I E L D T H E O R Y W e c o n s i d e r t h e e l a s t i c s c a t t e r i n g of two n u c l e o n s f r o m a s t a t e w i t h f o u r - m o m e n t a n l , P 1, to a s t a t e with f o u r - m o m e n t a n2,P 2 (fig. 2a).
I
a
!
%* ?
b
q:
c
Fig. 2. a - Nucleon-nucleon scattering, b - one pion exchange, c - two-pion exchange. We choose the three linearly independent four-vectors N = ½(n 1 + n2) ,
A = n 1 -n 2 = P2-Pl,
P : ½(Pl + P 2 ) ,
(2.1)
and the three scalars s = -(Pl +nl )2,
t = -(n 1 -n2 )2,
/-= - ( P l - n2 )2 ,
(2.2)
with the constraint
s+ t+ [= 4 m 2 . The scattering amplitude is defined from the S-matrix elements, Sfi = 6fi + i 5 4 ( n l +
by
Pl - n2 - P2)(m/2~E)2 × ff(P2)xp~u(n2)Xn~MU(nl)XnlU(Pl)XPl,
(2.3)
w h e r e E i s t h e n u c l e o n e n e r g y in t h e c e n t e r - o f - m a s s s y s t e m a n d u ( a ) a n d X~ a r e , r e s p e c t i v e l y , t h e s p a c e - t i m e s p i n o r a n d i s o s p i n o r of t h e n u c l e o n with four-momentum a. Now M m a y b e e x p a n d e d in a s e t of s p i n - a n d i s o s p i n - i n v a r i a n t o p e r a t o r s :
M :
~
i, j , k
k
k
k
P i j ( w , t , t-)S i Tj ,
(2.4)
660
R. LACAZE
w h e r e k stands for the symbols + o r -, with S/+ (S/-) and Tj+ (Tf) s y m m e t r i c ( a n t i s y m m e t r i e ) with r e s p e c t to the exchange of the nucleons n and p. If one only r e q u i r e s c h a r g e c o n s e r v a t i o n , the o p e r a t o r s T / t a k e the f o r m [5,13] T/ = lnl p ,
r n • ~p ,
Tj- = T~I p - l n ~ p ,
n p3 , T3~"
T~ 1p + lnT ~, ;
(v n × r P ) 3 ;
j = 1,4, j = 1,2,
(2.5)
w h e r e (Tn × vP) 3 changes sign under t i m e - r e v e r s a l . The S.~ o p e r a t o r s a r e chosen to be: J n
S1+ = l n l p ,
S ~ = Y5~5 (iyn" P + i y p. N),
$2~ = i y n . P l P 4- l n i y P . N ,
S ~ = v ~ i v n ' P 1 p 4- i nT~5 i ~ p - N ) ,
S; = iyn'piTP'N,
S~= (~4-~5) n.vp,
S; = vn.v p ,
$94- = i ~ l P 4- lni~P5 ,
n
S104- = zT, 5 "niyP. N 4- i y n. Pi~5.
s ; = 75 4 ,
(2.6)
It is possible to show that all o t h e r p o s s i b l e spin o p e r a t o r s may be exp r e s s e d in t e r m s of t h e s e ten o p e r a t o r s , by developing vn and 7P in t e r m s of four orthogonal f o u r - v e c t o r s N, P and A, and by using the D i r a c equation. According to the t r a n s f o r m a t i o n s p r o p e r t i e s given in table 1, t h e s e o p e r a t o r s Si + may be c l a s s i f i e d [14] as p r o p e r s c a l a r s for i = 1, 5, t i m e pseudos c a l a r s for i = 6, space p s e u d o s c a l a r s for i = 7, 8, and p r o p e r p s e u d o s c a l a r s for i = 9 , 1 0 .
Table 1 Transformations properties of the S~ i
P
T
1-5 6 7-8
+ + -
+ +
9 -10
T h e s e invariant o p e r a t o r s a r e a g e n e r a l i s a t i o n of those defined by Amati, L e a d e r and Vitale; they a p p e a r in the computation of f o u r t h - o r d e r Feynman g r a p h s (an invariant of the f o r m ( n ± ?/p)i~/n.pip. N also a p p e a r s but one can e x p r e s s it in t e r m s of S7+ and ~84- without introducing any new s i n g u l a r i ty). The Mandelstam r e p r e s e n t a t i o n f o r the s c a l a r , f u n c t i o n s p k ~ may then be c o n s i d e r e d a s valid to the fourth o r d e r . If it is a s s u m e d to b6'~alid for any o r d e r , then one can w r i t e
NN PARITY VIOLATION
+ oo pi;(w, P i ~ ( w ' t ' t - ) = t _aij~ + - ~ 1 f4U2 t ' - t
t')
661
dt' 1
oo
dt'
oo X i ; ( w ' , t ' )
+-~2f9#2t'-tf4m2
w'-w
dw'.
(2.7)
As we a r e i n t e r e s t e d in the l o n g - r a n g e f o r c e s , we shall neglect the c o n t r i bution of the last integral in eq. (2.7) the t-dependence of which c o r r e s p o n d s to a potential due to an exchange of at l e a s t t h r e e pions [12]. In fact we a r e only i n t e r e s t e d in the p a r i t y violating effects and then we r e s t r i c t o u r s e l v e s to the contributions Pi,j + with i = 6, 10. The pole t e r m in eq. (2.7) c o r r e s p o n d s to the one-pion exchange (fig. 2b). It was pointed out some t i m e ago [15] that a p a r i t y violation in the n-N v e r tex i m p l i e s violation u n d e r t i m e r e v e r s a l , but only if c h a r g e independence is assumed. In fact, if we define the f o r m f a c t o r s by:
(n21JV[nl) = i~(n2){fl(t)vp ----~ g
F2(t ) ~pu(nl +i-- a rn~
(0[J~[Tr~
-n2) u
F3(t)(n 1 - n2) p} u(nl)
= -tzFTr(t ) qp dp- ,
T-, (2.8)
w h e r e rn, p, rnp a r e the nucleon, pion and lepton m a s s e s r e s p e c t i v e l y , g is the g y r o m a g n e t i c ratio, a some constant and FI(0 ) = F2(0 ) = F3(0) = 1, we get, with the c u r r e n t - c u r r e n t hypothesis, a non-vanishing pole t e r m cont r i b u t i o n to the i n v a r i a n t s S9+Tf, S~T~, Sg-T ~ defined by eqs. (2.5) and (2.6). T h i s pole contribution to eq. (2.7) is given by: c~9; = ½ E~N GV rn~ ~ a t I m { F : ( t ) F3(t)} , ot93+ = - e 9 2 +,
e92
-
1 P a t R e { F ; ( t ) F3(t)} , = ~gTrN GV mp
(2.9)
w h e r e g~N is the ~-N coupling constant. We note that such a pole t e r m c o r r e s p o n d s to a second kind c u r r e n t a s defined by Weinberg [16] but, if the f o r m f a c t o r s Fi(t) a r e r e a l , this contribution, a s opposed to the lepton c a s e [17], does not imply a t i m e - r e v e r s a l violation on account of p a r t i c u l a r i s o t o p i c - s p i n dependence (contribution to (v n x rP)3 ). However this t e r m i m p l i e s in non c o n s e r v a t i o n of a v e c t o r c u r r e n t and is d i s r e g a r d e d in the actual computation. The s p e c t r a l functions pi?(w, t') a r e obtained by writing the unitarity r e -
662
R. LACAZE
lation in the t-channel f o r the M - m a t r i x . We obtain:
~ o i ) (kw , t') S k Tjk =
ijk
1 2(2~)2 t ~7/g,
(2.10)
with c'~ = ~ (:~1P2 [ T+ [ 2~>(2~. IT [ n1~2> ' 2~
w h e r e ~" stands f o r the annihilation N + N --* 2~ amplitude. The s u m m a t i o n is to be extended to the i s o s p i n indices and to the four m o m e n t a of the p i o n s in the i n t e r m e d i a t e s t a t e (the l a t t e r is e s s e n t i a l l y an a n g u l a r i n t e g r a t i o n [11]). If we put ~- into the f o r m T = r s + Tw w h e r e Ts stands f o r the s t r o n g int e r a c t i o n a m p l i t u d e and z w f o r the w e a k i n t e r a c t i o n amplitude, the t e r m s TS%w and TW+Ts will give u s the c o n t r i b u t i o n we a r e looking for. T h e s e a n nihilation a m p l i t u d e s a r e obtained by a n a l y t i c continuation of the e l a s t i c s c a t t e r i n g ~-N amplitude. Such an a m p l i t u d e is w r i t t e n in the m o r e g e n e r a l f o r m as: F = F 1 - F 2 t " 7 + F 3 t 3 T 3 - ½ F 4 t 3+½ FhT3+ ~ F 6 ~ + ½ F 7 T 3 ~
+ F 8 ( t x 7-)3+ ~1F9(t. T t 3 + t 3 t . 7 ) + F l o ( (
t x 7)3t3-t3(t
× 7)3 ) ,
(2.11)
with [18]: F i = - A i - A i h i 7 5 + i y " OB i + i y " O ~ h B i 5 ,
(2.12)
w h e r e O is one half of the s u m of the pion q u a d r i m o m e n t a , A and B a r e s c a l a r functions of the usual s c a l a r s s, g, t and t and 7 a r e the usual i s o topic spin m a t r i c e s . We t a k e this g e n e r a l f o r m f o r the weak i n t e r a c t i o n a m p l i t u d e F w but the r e s t r i c t i o n s upon the s t r o n g i n t e r a c t i o n a m p l i t u d e r e d u c e F s to: F s = F~-
F,~ t" 7 ,
(2.11')
with F is = -A is + i
. QB s"
(2.12')
If ~ and fi a r e the i s o t o p i c - s p i n i n d i c e s of the two pions, we m u s t c o n s i d e r the p r o d u c t of the m a t r i x e l e m e n t s __F~f~s* F~(~w and take the s u m o v e r the i s o t o p i c - s p i n s t a t e s of the pions. We get
NN P A R I T Y V I O L A T I O N
s*
asFaI3FYa
663
s* w_n p -~s*~w n p =3F~*(F~ v + F ~ v) l n l p + 2F 2 F 2 , .T - z~ 2 ~3T3T3
1
s*
w
w
+ ~ [ F 1 (F 5 + F 7 +
F~)
s*
w
n
p
+ F 2 F 4](T 3+T 3)
1 S* W W F ~ ) S* W n Tp) +~[F1 (F5+FT+ - F 2 F4](~'3 + 2F~*F~(Tn×rP) 3 .
(2.13)
Taking into account the definitions (2.5), we can write the above formula in the f o r m
ot ~
Fa~F~a = ~ ,~l ws* l ijkl "~ijk~i FT Tk'
(2.13)
where T / a r e the isotopic-spin o p e r a t o r s and the coefficients Ciikl can be obtained f r o m (2.13). With the appropriate p h a s e - s p a c e factors, we can then write: ~ t ( t - 4 ~ 2 ) ½ ijkl ~ clij kC~fCijT l + Sy,
9~:
(2.14)
where the t e r m Sy c o m e s f r o m the product F w* F s and can be obtained dir e c t l y hy s y m m e t r i s a t i o n . If we r e s t r i c t o u r s e l v e s to the part of the 9K which violates the spacial parity conservation, 97~ij takes the form:
c~ij = f [ - . 4 s * + i yP" QBS* ] [ - A ~ i~55+ i ~n. Q~55BjS~ d~Q • n
= aij zY5 + bij
iy p
(2.15)
n n • Niy n5 + cij iy n • Py~ + dij yn YPY5 + eij iyn. piyp" N ~5,
where S*
w
S*
w
aij =f[A i Aj + m a S i d j ] d g t q , bij =
f
vii = - f [a a i eis :
f
A7
,
9 j + m(ol 2
--DR26) BS.B~] d~Q ,
,
eij : f (a3 + N_~P 5 ) B S * B 7 d~2o '
(2.16)
664
R. LACAZE
with
D = N2p 2 - (N.p)2, =I(N2Q.P
- P.N Q.N),
=I(p2Q.N
- P.N ©.P),
6 = Q2 _ ( a p + fiN)2. A f t e r s y m m e t r i z a t i o n , the o p e r a t o r s ( i ~ n. P~s i~ p . Y + i~ n. P i y P . N~ p) can be e x p r e s s e d in t e r m s of the o t h e r s :
in NIgh5
:
Then we obtain the expansion of ~ in t e r m s of the invariant o p e r a t o r s and the knowledge of the pion nucleon amplitude in the a p p r o p r i a t e r e g i o n of int e g r a t i o n allows us to compute the s p e c t r a l functions of eq. (2.10).
3. THE y-N A M P L I T U D E We a r e mainly i n t e r e s t e d in the nucleon s c a t t e r i n g n e a r the t h r e s h o l d , i.e. for w ~ 4m 2, and then for small t and t. The m o s t important c o n t r i b u tion to the integral of the s p e c t r a l function pi~(w, t') in eq. (2.7) o c c u r for not too l a r g e v a l u e s of the v a r i a b l e t'. T h e s e ' s p e c t r a l functions can then be obtained by extrapolation of the p i o n - n u c l e o n amplitude at low e n e r g y [12]. F o r the s t r o n g - i n t e r a c t i o n p a r t such an amplitude h a s been c o n s t r u c t e d by Bowcock, Cottingham and Luri~ [19] and used in refs. [11] and [12]. We make use h e r e of an equivalent amplitude which we have built up in a p r e v i ous work [20]. This amplitude t a k e s into account the pole t e r m due to the nucleon exchange, a r e s c a t t e r i n g t e r m a p p r o x i m a t e d by the pole of the N~3 r e s o n a n c e and the n-n i n t e r a c t i o n in the S- and P - w a v e s also a p p r o x i m a t e d by p o l e s (p-pole for the latter). The n-N amplitude to the f i r s t o r d e r of weak i n t e r a c t i o n can be cons t r u c t e d in a s i m i l a r way in the f r a m e w o r k of the c u r r e n t - c u r r e n t hypot h e s e s . However the t e r m s c o r r e s p o n d i n g to the exchange of the nucleon and of the i s o b a r N~3 (fig. 3) do not contribute if the v e c t o r c u r r e n t is c o n served. Indeed the product of the ~ - c u r r e n t with another c u r r e n t is nothing else than the d i v e r g e n c e , as can be seen in the computation of the pole t e r m . Thus t h e s e contribution a r e ruled out in p r a c t i c a l computation. The t e r m equivalent to the n-~ p wave contribution may be obtained either by including directly the p, o r by introducing the p in the f o r m f a c t o r of the two-pion c u r r e n t ; this last p r o c e d u r e was adopted here. We make use of the following c u r r e n t s :
NN P A R I T Y V I O L A T I O N
\
",x9
f l
665
/
F i g . 3. 7/-N s c a t t e r i n g e x c h a n g e in t h e d i r e c t c h a n n e l .
c FMI(t) ~p.vAv}T5U(nl) "r-
+ m
(3.1)
w h e r e X = G A / O , and obtain by c o m p a r i n g with eqs. (2.11) and (2.12): C
*
A25w = 4 GA m N. Q Im{Fu~ (t) FMI(t)} ,
A85w : - 4 GA C---N" Q R e { F u ; ( t ) FMI(t)} , m
B25w = 2 G A R e { F : ( t ) FA(t)} ,
w-- 2J, B 85w = - 2 GA I m { F u ; (t) FA(t) } .
(3.2)
The f o r m f a c t o r s Fu~(t), FA(t ) and FMI(t) a r e n o r m a l i z e d to 1 for t = 0 and in the physical region for t, they may b e c o m e c o m p l e x [21]. The explicit e x p r e s s i o n f o r A i S , B i S , A~ 5W , B45 W in t e r m s of Q allows J us to p e r f o r m the integration (2.13). The spectra~ functions Pij k a r e then d e t e r m i n e d by (2.10) but t h e i r t - d e p e n d e n c e for t >I 4g 2 r e q u i r e s the knowledge of the f o r m f a c t o r s . We a s s u m e that the f o r m f a c t o r s F ~ ( t ) and FA(t) a r e dominated by the exchange of the p and of the A I (ref. [22]) r e s o n a n c e (1080 MeV), r e s p e c t i v e l y (we a s s u m e for the l a t t e r j P G = i + - ) . The FMI(t) f o r m f a c t o r c o r r e s p o n d s to the exchange of a i ++ s y s t e m [21] and could be dominated by the B - r e s o n a n c e (1210 MeV). However t h e r e is no evidence of such a p s e u d o - t e n s o r i a l c u r r e n t and it has not been included in our num e r i c a l computations.
666
R. LACAZE
4. DEFINITION OF THE P O T E N T I A L A n u c l e o n - n u c l e o n potential c o n s t r u c t e d so as to r e p r o d u c e the s c a t t e r i n g amplitude in field t h e o r y has been obtained by Cottingham and Vinh Mau [12] in the c a s e w h e r e p a r i t y c o n s e r v a t i o n is a s s u m e d . In a s i m i l a r way, we can obtain a potential to the f i r s t o r d e r of weak i n t e r a c t i o n s , the main difference being the introduction of s u p p l e m e n t a r y invariants. B e s i d e s that, if we r e s t r i c t o u r s e l v e s to the d e t e r m i n a t i o n of a p a r i t y violating potential, the a b sence of the pion exchange (because of the a b s e n c e of a second kind v e c t o r c u r r e n t ) s i m p l i f i e s this determination. The s c a t t e r i n g amplitude has been expanded in a set of l i n e a r l y independent i n v a r i a n t s (2.4). T h e r e will be then as many types of l i n e a r l y independent potentials as these invariants:
V(r) : ~ Vaj(r) k k jk , ~2aT ajk
(4.1)
w h e r e k = + and the T k a r e isospin i n v a r i a n t s defined in (2.5). We have chosen the fla k to be: J g~pS+ = (an × aP) • r ,
tiPS = a n ' p a P ' L
9 P V+ = (an - a P ) . P ,
~PV- = ( a n + a P ) . P ,
~2pT
= (a n - a P ) . r , +
~2pT v = (a n x a P ) . P ,
+aP'Pan'L,
~2pT S = (a n + a P ) - r , ~ P T V = - ( a n " r a P . L + aP. r a n. L ) / r 2
(4.2)
w h e r e L = r x P and a n, a P a r e the Pauli spin o p e r a t o r s acting on the nucleons n and p. In the m o m e n t u m space the potential is written: (4.3)
ajk with
~
~PS~ + = - i(an × a p ) " A
~PS =i(an'pap'p
N + (a n _ a p ) . p ~2pv =
~PV = (an + a p ) " P '
+
flPTS =
-i(a n-ap).A
V ~ + = (a n × a P ) . P ~2pT
xzl + a P . P a n. p × A ) , (4.4)
5pT S :-i(a -
n+ap).A,
~PTV = ( a n ' A a P ' P x A
,
+ o p A a n. P × A ) / A 2.
In o r d e r to identify the potential s c a t t e r i n g amplitude given by the L i p p * Our ~ ;
are the same as those of ref. [23].
NN
PARITY
VIOLATION
667
m a n - S c h w i n g e r expansion and the field t h e o r y s c a t t e r i n g amplitude, we expand the l a t t e r in t e r m s of t h e s e invariants. By e x p r e s s i n g the D i r a c m a t r i c e s and s p i n o r s with the help of the Pauli o p e r a t o r s and s p i n o r s , we get s+ s+ ~ + s s u(n2) u(P2) S~/U(Pl) u(nl) : ~a X i ~ ×P2 ×n2 fla ×nl XPl"
(4.5)
In the adiabatic a p p r o x i m a t i o n [12, 24], the t r a n s f o r m a t i o n m a t r i x X conc e r n i n g the p a r i t y violating i n v a r i a n t s is
X;o•
PS+
PS-
PV±
PTS ±
7 8 9
+ t/8m 2 + 1/m 0
½rn2 ½rn3 0
2 - 2/m 0
0 0 ½rn
0 0 t/8 m 3
10
0
0
0
½m
- 3t/8m 2
PTV ~
This t r a n s f o r m a t i o n is e n e r g y - i n d e p e n d e n t and i n t r o d u c e s no new singularity. If we then c o n s i d e r the potential
1
(2~) 3 ~jk
i
1 o~ p~j(t') 2 t, - t
zi (t) T f
, ~k
at}as
k T) ,
(4.7)
it will r e p r o d u c e the s c a t t e r i n g amplitude defined in eq. (2.7) for the longest r a n g e , since the one pion exchange t e r m does not contribute. The potential (4.1) is obtained by an i n v e r s e F o u r i e r t r a n s f o r m a t i o n . If the i n t e r c h a n g e of the o r d e r of integration is a s s u m e d to be valid, one obtains: 1
vkj(r)_
1
~i f4~
(2~) 3 "
k
k
k
2 Xi~(t)PiJ(t) O~(r, t)
e -rt2 r dt,
(4.8)
w h e r e the O~(r, t) depend on the potential type ( s c a l a r , v e c t o r , tensor) k
O~(r, t) = 1 = t 2 (1 + 11)_
r =~
1
for
(~,k) = PV ±, PTV +
for
(~,k) = PS +, PTS +
for
(~,k) = PTV
rt 2 (1 +
3
3, i +--j
(4.9)
r t ~ r2t The polynomial f o r m of x i k ( t ) m a k e s the f o r m u l a (4.8) valid as long as we a r e only i n t e r e s t e d to the longest r a n g e f o r c e s .
668
R. LACAZE
5. RESULTS The amplitudes A i ~ and B i ~ v obtained in eq. (3.2) in the same way a s the A s and B s a m p l i t u d e s [20] enable us to obtain through eq. (2.13) the cont r i b u t i o n s to the s p e c t r a l functions; the integral (2.16) exhibits no special difficulty: the i n t e g r a l s (4.8) a r e computed n u m e r i c a l l y , the f o r m f a c t o r s (sect. 3) being chosen a s B r e i t - W i g n e r f o r m u l a e and n o r m a l i z e d to unity at the origin. In the limit w h e r e the r e s o n a n c e s p and A 1 a r e n a r r o w enough (this c o r r e s p o n d s to take the principal value of the integral with extrapolation of the phenomenological f o r m f a c t o r s defined for t < 0 (refs. [22, 25]), one obtains V(r) = (Vps(r)(anx
lr
a p).~+
Y p v ( r ) ( a n - a D).P)(T~T p - T n. vP),
(5.1)
where +
+
V p S (r) = - r V p s ' 2 = r V p s , 3 ,
V p v ( r ) -- - V p v , 2+ : + V p v , 3+ ,
(5.2)
with G = 1.25x 1.01 x 1 0 - 5 / M 2 we obtain the c u r v e s for Vps(r) and VpV(r) drawn in figs. 4 and 5. We have also drawn, in fig. 4, the potentials obtained by Blin-Stoyle [5]: (5.3) Gf 2 2 1 e -2~r c)3pS = - - ~ (1 + /-~+ #---~r2) r3 ' Q2PV = 4 - ~ 5 ( r ) -~ 0 . 8 2 × 1 0 - 8 ~ - 3 6 ( r ) .
,1.,lo -~
~v
!
I
I
',\
L_
t
Fig. 4. The potential Vps(r). The dotted line r e p r e s e n t s the corresponding potential obtained by Blin-Stoyle.
I" Fig. 5. The potential V p v ( r ).
NN P A R I T Y V I O L A T I O N
669
W e n o t i c e on fig. 4 that the s t a t i c p o t e n t i a l t h a t we h a v e o b t a i n e d (solid line) h a s a m u c h l o n g e r r a n g e than the B l i n - S t o y l e potential. One m i g h t then think t h a t the c o r r e s p o n d i n g e f f e c t s in the o b s e r v e d p r o c e s s e s will be m o r e i m p o r t a n t . Note a l s o t h a t the c o m p u t a t i o n s p e r f o r m e d by P a r t o v i [7] c o r r e s p o n d to a w e a k c o u p l i n g c o n s t a n t G, s m a l l e r t h a n the one u s e d in B l i n - S t o y l e p o t e n t i a l (1 x 10-5 e r g . c m 3 i n s t e a d of 1.40 x 10-5 e r g . cm3). F u r t h e r m o r e , we h a v e o b t a i n e d an i m p o r t a n t v e l o c i t y d e p e n d i n g p o t e n t i a l , the c o r r e s p o n d ing e f f e c t s of which m a y m o d i f y P a r t o v i ' s r e s u l t s . T h e p o t e n t i a l (5.1) s e e m s to u s to be m o r e r e a s o n a b l e , but it m a y be c h a n g e d if we i n c l u d e l a r g e width f o r p and A1, s e c o n d kind axial c u r r e n t and vector current non-conservation. With the e x p e r i m e n t a l width of p a n d A1, we o b t a i n a c o n t r i b u t i o n to flpS-(V n x vP)3 and f l p V - ( ~ n x rP)3 but it s e e m s that a m o r e c a r e f u l d e f i nition of the a n a l y t i c c o n t i n u a t i o n r u l e s out such c o n t r i b u t i o n s . T h e r e a l p a r t of the s e c o n d - k i n d a x i a l c u r r e n t g i v e s r i s e , t h r o u g h eq. (3.2), to a c o n t r i b u t i o n of the f o r m ~ P T S ( T n x TP)3 a n d flPTV-(V n x rP)3. T h e n u m e r i c a l c a l c u l a t i o n with such a t e r m of the o r d e r of the v e c t o r c u r r e n t s h o w s that the c o n t r i b u t i o n to f l P T S - ( T n x TP)3 is i m p o r t a n t but such a r e s u l t h a s no p a r t i c u l a r s i g n i f i c a n c e . A n o t h e r c o n t r i b u t i o n of t h i s f o r m m a y a l s o be o b t a i n e d with a s e c o n d kind v e c t o r c u r r e n t (which v i o l a t e s the v e c t o r c u r r e n t c o n s e r v a t i o n ) . T h e effect of s u c h a c o n t r i b u t i o n due to the pion pole h a s b e e n s t u d i e d [26], but it s e e m s not p o s s i b l e to t e s t the e x i s t e n c e of s e c o n d kind w e a k c u r r e n t with such an effect. T h e effect of the n o n - c o n s e r v a t i o n of t h e v e c t o r N - N * is m o r e e a s i l y a p p r e c i a b l e . I n d e e d if we include in the v e c t o r N - N * c u r r e n t [27] a " m a x i m u m v i o l a t i o n " *, we o b t a i n a c o n t r i b u t i o n quite l a r g e but the m a i n r e s u l t is that t h i s c o n t r i b u t i o n t a k e s the f o r m :
V ' ( r ) : ( V ~ s ( r ) ( a n x a p) . r + V p'v ( r ) ( a
n
_ u p ) . p ) ( l n l n - } ( r n . r p + T~3Tp))
and such a p o t e n t i a l would give r i s e to e f f e c t s in pp and nn i n t e r a c t i o n . I a m i n d e b t e d to P r o f e s s o r R. Vinh Mau f o r s u g g e s t i n g the s u b j e c t of t h i s w o r k and f o r h i s kind g u i d a n c e . T h e help of M. Y. N o i r o t at the e a r l y s t a g e of t h i s w o r k is a p p r e c i a t e d . I a l s o thank D r . A. T o u n s i f o r d i s c u s s i o n s .
REFERENCES [1] Yu. G. Abov, P.A. Krupehitsky and Yu. A. Oratovskny, Phys. Letters 12 (1964) 25; Soy. J. Nucl. Phys. I (1965) 341. * The vector current conservation implies, between the four form factors of the N-N* current, the relation =
o
From the photo production analysis [28] FV(o) = -FV(o)= 5,6 and FV(0) = FV(0) = O. Our "maximal violation" stands for FV(t.L2) = +FV(/~2) = 5.6 and FV(/.L2) = FV(~ 2) =0.
670
R. LACAZE
[2] F. Boehm and E. Kankeleit, Proc. Int. Conf. on nuclear physics, P a r i s , Juillet 1964, vol. I I p . 1181. [3] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193; M. Gell-Mann, Revs. Mod. Phys. 31 (1959) 834. [4] F. C. Michel, Phys. Rev. 133 (1964) B329. [5] R.J. Blin-Stoyle, Phys. Rev. 118 (1960) 1605; 120 (1960) 181. [6] R.J. Blin-Stoyle and H. Feshbach, Nucl. Phys. 27 (1961) 395. [7] F. Partovi, Ann. of Phys. 27 (1964) 114. [8] S. Wahlborn, Phys. Rev. 138 (1964) B530. [9] Z. Szymanski, Nucl. Phys. 76 (1966)539. [10] T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 254. [11] D. Amati, E. Leader and B. Vitale, Nuovo Cimento 17 (1960) 68. [12] W. N. Cottingham and R. Vinh Mau, Phys. Rev. 130 (1963) 735. [13] M.J. Moravcsik, The two-nucleon interaction (Clarendon P r e s s , Oxford, 1963); L. Eisenbud and E. P. Wigner, Proc. Nat. Acad. Sc. 27 (1941) 281. [14] P. Roman, Theory of e l e m e n t a r y particles, 2. ed. (North-Holland Publishing Co., Amsterdam, 1961). [15] V. G. Soloviev, Nucl. Phys. 6 (1958) 618. [16] S. Weinberg, Phys. Rev. 112 (1958) 1375. [17] N. Cabibbo and M. Veltman, p r e p r i n t CERN 65-30. [18] S. Fubini and D.Walecka, Phys. Rev. 116 (1959) 194. [19] J. Bowcock, W. N. Cottingham and D. Lurid, Nuovo Cimento 16 (1960) 918. [20] R. Lacaze, Th~se de 3e Cycle, Bordeaux (1964). [21] S. B~ T r e i m a n , Weak interactions and topics in dispersion physics (Bergen 1962 W. A. Benjamin, 1963). [22] G. Cocho, Nuovo Cimento 42A (1966) 421. [23] A.E.Woodruft, Ann. of Phys. 7 (1959) 65. [24] J. M. Charap and M. J. T a u s n e r , Nuovo Cimento 18 (1960) 316. [25] M. Paty, p r e p r i n t CERN 65-12. [26] D.O. Dal'Karov, J E T P (Sov. Phys. )Letters 8 (1965) 124. [27] S. M. Berman and M. Veltman, Nuovo Cimento 38 (1965) 993; C. H. Albright and L. S. Liu, Phys. Rev. 140 (1965) B748. [28] M. Gourdin and Ph. Salin, Nuovo Cimento 27 (1963) 193,309.