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Interaction current contributions to parity non-conserving nuclear γ- transitions
Volume 36B, number 5
PHYSICS LETTERS
CONTRIBUTIONS TO PARITY INTERACTION C U R R E N T NUCLEAR y-TRANSITIONS *
4 October 1971
NON-CONSERVING
M. GARI** and A. H. HUFFMAN
California Institute of Technology, Pasadena, California, USA Received 25 July 1971
Parity non-conserving interaction current contributions to nuclear y-transitions required by gauge invariance are calculated and discussed. Arguments are given that the strangeness conserving currents should contribute to the one pion-exchange parts of the interaction.
P a r i t y n o n - c o n s e r v i n g effects in n u c l e a r ~ - and r - t r a n s i t i o n s have now b e e n detected in many e x p e r i m e n t s [1, 2]. T h e s e effects have been e s t i m a t e d t h e o r e t i c a l l y by c a l c u l a t i n g a n u c l e o n - n u c l e o n p a r i t y - n o n c o n s e r v i n g potential (VpNc) , u s u a l l y b a s e d on one = or one p exchange (fig. 1). In the conv e n t i o n a l model [3] the AS = 0 weak H a m i l t o n i a n is given by: HAS=0 G [ 2 ÷ + sin20{S~,S+~}+] weak - - ~8-L c ° s O{J~, Jp }+
(1)
where J ~ , Sg a r e s t r a n g e n e s s c o n s e r v i n g (AS = 0) and s t r a n g e n e s s changing (AS = 1) hadronic c u r r e n t s , r e s p e c t i v e l y . The p - c o n t r i b u t i o n to VpN C using either the f a c t o r i z a t i o n a p p r o x i m a t i o n or c u r r e n t a l g e b r a [4] is:
vP(AI=0,2) = Gp [ ( l + ~ v ) ( a 1 x a2).[Pl2,fp(rl2)] where T (12) = r l+~ -2 + ~-1~2 , - + hA(0) : Gcos20(m2pgA/f) ,
_
_
+ (a 1
f p (r12) = e x p ( - m p r l 2 ) / r l 2
,
a 2 ) - P l 2 , f p ( r l 2 ) +1J -i(1+ )2 ,
Gp = hA(O)f/8y~'-2M,
(2)
f 2 / 4 ~ ~ 2.4 ,
gA(0) ~ 1.2 .
Fig. 1. One 7rand p contributions to VPNC. The open circle represents a strong vertex and the circle-with-cross a weak vertex. The ~ - c o n t r i b u t i o n to VpN C can be obtained from the cos20 piece of Hweak by a s s u m i n g i s o s p i n b r e a k i n g [5] and f r o m sin20 piece by r e l a t i n g the weak NN~ v e r t e x to s t r a n g e b a r y o n decays [6]. In either c a s e the potential has the f o r m [5]: V;,II(AI= 1) = o~I,II, . ~Ox + 02)'[P12, f~(rl2)]_ T~2) ,
(3)
but G I and G II o r i g i n a t e f r o m the AS = 0 and AS = 1 c u r r e n t s , r e s p e c t i v e l y , and a r e given by * This work was supported in part by Deutsche Forschungsgemeinschaft, in part by National Science Foundation [GP-28027, GP-19887], and in part by the U.S. Atomic Energy Commision. Prepared under contract AT[04-3]-63 for the San Francisco Operations Office, U.S. Atomic Energy Commission. ** Permanent address: Ruhr-Universit~tt, Bochum, West Germany. 442
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GI'Ilrr : KfI'II/81r~-2-M ,
4October1971
/~(r12) : exp(- m r12)/r12 ,
(4)
w h e r e g i s the s t r o n g ~NN coupling constant (g2/4~ ~ 15) and /I
= C/~2-cos2Oc ( m
_ M ) ~ 0.14 × 10 - 8
(4a)
--/i = 4.8 x i 0 - 8
(4b)
Note that G I v a n i s h e s in the SU(2) l i m i t when the n u c l e o n s a r e taken to be o n - s h e l l . This is not n e c e s s a r i l y t r u e if one t a k e s into account the o f f - s h e l l b e h a v i o u r of the nucleons in c a l c u l a t i n g the weak ~NN v e r t e x [5, 7]. In t h i s c a s e one could g e t f I c o m p a r a b l e to fII. T h e s e p o t e n t i a l s a d m i x s t a t e s of irregular p a r i t y Id(-~)Ei>to the s t a t e s of n o r m a l p a r i t y IJ (~)E) p r o d u c e d by the s t r o n g N-N f o r c e s :
[~J) = ]J(y)E)+ ~.$ F i ~ (-Y)Ei)
(5)
with
E - Ei Thus, the a m p l i t u d e for an i r r e g u l a r y - t r a n s i t i o n i s given by:
= ~
A
EK(~, ) - E A
B
E B - Ej0r)
(6) '
w h e r e E L (ML) d e n o t e s the e l e c t r i c (magnetic) m u l t i p o l e o p e r a t o r of the i r r e g u l a r t r a n s i t i o n . While c a l c u l a t i o n s b a s e d on the above d e s c r i p t i o n a r e in good a g r e e m e n t with e x p e r i m e n t a l r e s u l t s c o n c e r n i n g p a r i t y n o n - c o n s e r v i n g s - d e c a y [8], they u n d e r e s t i m a t e the p a r i t y n o n - c o n s e r v i n g e f f e c t s in ~ - t r a n s i t i o n s by two o r d e r s of magnitude [9]. T h e s e l a t e s t c a l c u l a t i o n s [9] t a k e c a r e f u l account of n u c l e a r s t r u c t u r e and c o r r e l a t i o n s between the nucleons. We wish to point out in this l e t t e r that the i n t e r a c t i o n c u r r e n t c o r r e c t i o n s of fig. 2 r e q u i r e d by gauge i n v a r i a n c e m a y be l a r g e . T h e r e a r e two a t t e m p t s [10, 11] known to us which t a k e into account additional c o n t r i b u t i o n s by i n t r o d u c i n g the e.m. f i e l d into the n o n - r e l a t i v i s t i c p.v. N - N potential, a method we do not follow. We think the b e s t way to p e r f o r m a gauge i n v a r i a n t d e s c r i p t i o n i s a s y s t e m a t i c c a l c u l a t i o n of all r e l e v a n t t i m e - o r d e r e d F e y n m a n d i a g r a m s (fig. 3 b - d and additional) using e i t h e r the gauge i n v a r i ant i n t r o d u c t i o n of the m i n i m a l e.m. i n t e r a c t i o n into the total L a g r a n g i a n , o r the full e l e c t r o m a g n e t i c v e r t e x . F o r the b a r e i n t e r a c t i o n t h e s e two m e t h o d s a r e equivalent.
(a)
(b)
(¢)
(d)
(e)
Fig. 2. Interaction current contributions to parity-nonconservation. (Plus terms with weak and strong interchanged). These diagrams produce a parity n0n-conserving two-body electromagnetic operator c'tgpNE.
(a)
(b)
(¢)
(d)
F i g . 3. T i m e o r d e r e d c o n t r i b u t i o n s of o n e - m e s o n - e x c h a n g e . (Plus d i a g r a m s with weak and s t r o n g interchanged).
A gauge i n v a r i a n t d e s c r i p t i o n m u s t include the p r o c e s s e s of fig. 2 b - e . T h e s e c o n t r i b u t i o n s a r e not included t h r o u g h the wavefunctions eq. (5) and t h e r e f o r e m u s t b e s e p a r a t e l y introduced. T h e s e p a r t s of the i n t e r a c t i o n then act l/k¢ a two body p a r i t y n o n - c o n s e r v i n g e l e c t r o m a g n e t i c (FNE) o p e r a t o r . In c a l c u 443
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4October 1971
lating such PNE contributions one has to be w a r y of overcounting. F o r example, the covariant diagram fig. 2a containts some p a r t s like fig. 3a which have already been included through the wavefunction admixtures, and other p a r t s like fig. 3b, c, d which must be included in the PNE contribution. In this paper we obtain the leading PNE t e r m s to get an estimation of the o r d e r of magnitude of their effects and to get a feeling of where such contributions could be important. A s t r a i g h t f o r w a r d calculation of the PNE o p e r a t o r s is to s t a r t f r o m a gauge invariant Lagrangian which can be obtained by the minimal substitution 0 / ~ + - + (a~ ¥ i e A g ) ~ +
(7)
in the total Lagrangian ~ t o t a l = ( ~ N + .~n + ~ P ) f r e e + "/~WNN+ ~S, NN + .~WpNN+ ~SpNN
(8)
_W,S denote the appropriate coupling of the m e s o n s to the nucleon: where MMNN
S
~TrNN
-
g a ~Kr~g~5~aN ,
(9a)
2M
_W(I) = i - ~ cos 2 0 c J~ 7rNN
0g ~a~(yg +
~
av) - ~ N
(9b)
'
./~W(II) / I ~((7 × V)zN , ~rNN = 23pN NS
= ifp~.~(ybt + ~gv
(9c)
agVOv) ~2 g ,
(9d) (9e)
.CoN N W = __P~lV(YghA(0)- i a g h P (0))5 ~2 g , S
The pion-nucleon interaction .~ 7rNN has been taken to be of the gradient coupling f o r m [12] which is known to satisfy the PCAC relation and can be shown to d e s c r i b e the s o f t - p i o n - l i m i t of photoproduction properly. It is easily verified that the r e p l a c e m e n t (7) in .~SNN gives the K r o l l - R u d e r m a n t h e o r e m [14]. This p r o p e r t y of the Lagrangian is important because without this connection to experimental pion photoproduction the calculations would r e m a i n r a t h e r doubtful concerning the right o r d e r of magnitude of these effects. Wb~ii) The weak pion L a g r a n g i a n s .C originate f r o m the AS = 0 and AS = 1 c u r r e n t s respectively. F o r weak photoproduction the AS = 0 p a r t gives a non-vanishing contribution even in ~U(2) limit. This is v e r y important because it enters into the o p e r a t o r s with the cos20 factor. After making the substitution (7) in the total Lagrangian and retaining t e r m s of lowest o r d e r in P/M, the following PNE contributions a r e obtained: = Qg;.II(AI = 0, 2) + Qg;(AI = I) + Q;P(AI = 0, I, 2 ) , where 7r 2) = GI ' I I e " CPI,II(AI= 0, - n ~a 1 + a2)"
(+) Af~(rl2) T12 ,
• I x a2)J[(s~J(Pl2)- 52P 122 6ij),f.(rx2) ]_ +Ai(o1_o2~l{p~2,[PJ2,f.(r12)] CPI(AI=I) : 2F el(l+.v)A'(io
(10) (lOa) }+]T~2)
(10b)
q~P(AI=O, I,2) = -Cpef p(rl2)12~v(ial X a2).AT~2) + (al + a2).AT~+2)l , where
S~2~p)-- = pipJ_ ~p 1 2 8ij ,
(10c)
F = Gcgcos2O/~/2(2/~247r , bLv = ~p _ / l n ' A is the v e c t o r potential.
A =k~,~ ( ~ V 1/2 (Ak,e+ A:k,~)~ exp(ik-r) . The t e r m Q~;,II(AI= 0, 2) c o m e s f r o m photoproduction at the strong v e r t e x (fig. tribution. This is just t e r m coming f r o m the and 3d) [15]. In lowest 444
2b). We found that the K r o l l - R u d e r m a n (or seagull-) t e r m gave the l a r g e s t conthe p r o c e s s d e s c r i b e d in d i a g r a m 2b, which can be shown to be equivalent to the negative e n e r g y p a r t in the nucleon p r o p a g a t o r in the p s e u d o s c a i a r t h e o r y (fig. 3c o r d e r , photoproduction of a neutral pion vanishes in a c c o r d a n c e with the Kro11-
Volume36B, number 5
PHYSICS LETTERS
4October1971
R u d e r m a n t h e o r e m . T e r m s f r o m figs. 2 c - e a r e of at l e a s t f i r s t o r d e r in P / M and w e r e not w r i t t e n down h e r e . The termC~(Al = 1) d e s c r i b e s the c o n t r i b u t i o n of weak photoproduction, fig. 2a and 2b, with weak and s t r o n g v e r t e x i n t e r c h a n g e d . Fig. 2a was c a l c u l a t e d by taking into account o f f - s h e l l b e h a v i o u r of the i n n e r n u c l e o n - l i n e . T h i s p i e c e was not taken into account through VpN C. AlthoughC~ii i s of second o r d e r in P/M, it has been included h e r e b e c a u s e it o r i g i n a t e s f r o m the s t r a n g e n e s s c o n s e r v i n g (AS - 0) c u r r e n t s and does not v a n i s h in the SU (2) l i m i t . T h i s t e r m should give e f f e c t s of the s a m e o r d e r of m a g n i tude asC)9~i which was d e r i v e d f r o m the s t r a n g e n e s s changing (AS = 1) c u r r e n t s . (Recall that V ~ v a n i s h e s in the SU(2) l i m i t by taking the nucleons o n - s h e l l . ) B e c a u s e the m i n i m a l substitution i s m a d e m an effective L a g r a n g i a n , and o f f - s h e l l b e h a v i o u r u s e d this t e r m is p r o b a b l y l e s s r e l i a b l e than the o t h e r s . O t h e r c o n t r i b u t i o n s of weak photoproduction beyond z e r o o r d e r in P / M w e r e o m i t t e d , as t h e r e a r e no compensating factors. The t e r m c)gO d e s c r i b e s the contribution of s t r o n g p photoproduction (fig. 2b and fig. 3c and d). Weak p - p h o t o p r o d u c t i o n and the p r o c e s s e s of fig. 2 c - e a r e of higher o r d e r in P / M . It should be noted that the PNE contributionsQYP andC~ 7T l e a d the different i s o s p i n s e l e c t i o n r u l e s a s c o m p a r e d to v P N and VI~ . C C As m e n t i o n e d above, the t e r m s C ~ P N S eq. (10) e n t e r in the ~ - t r a n s i t i o n a m p l i t u d e a s ~ i r e c t c o n t r ~ u tions. The total a m p l i t u d e for i r r e g u l a r ~ - t r a n s i t i o n including such effects is thus given by Mid~ - - K ~' ]total = M[J• --+K~ i] admixture +M[J" ~ K y'] d i r e c t ' with: ?T' M[J ~ - * g ]direct = {KTr' ~)~PNE I J ~ ) "
(11)
(12)
In the long wavelength l i m i t (exp ( J R - r ~ 1) Q ~ a n d c ~ P a r e i r r e g u l a r E1 o p e r a t o r s ; that i s , they give E1 r a d i a t i o n with no p a r i t y change. In the c a s e of a r e g u l a r M1 t r a n s i t i o n t h i s r a d i a t i o n p l u s i r r e g u l a r E1 r a d i a t i o n f r o m s t a t e s a d m i x e d by VpN C will i n t e r f e r e with the r e g u l a r M1. Higher o r d e r t e r m s of the PNE o p e r a t o r s in P/M, and the l o w e s t o r d e r t e r m s taken beyond the long wavelength l i m i t , give i r r e g u l a r M1 r a d i a t i o n ; this is M1 r a d i a t i o n with p a r i t y change. The o p e r a t o r s Q 2 p N E connect r e l a t i v e S - s t a t e s to S - s t a t e s , which is v e r y i m p o r t a n t b e c a u s e of the shoft ~ange of the r a d i a l dependence. By c o n t r a s t the a d m i x i n g p o t e n t i a l s VpN C eq. (2) and (3) connect r e l a t i v e S- to P - s t a t e s . Thus, for e x a m ple, in n - p c a p t u r e the C})pNE t e r m s should c o n t r i b u t e a l a r g e i r r e g u l a r E1 t r a n s i t i o n between 1S 0 s c a t t e r i n g s t a t e and the deuteron ground s t a t e i n t e r f e r i n g with the r e g u l a r M1 t r a n s i t i o n to p r o d u c e circular polarization. P r e l i m i n a r y c a l c u l a t i o n s in the c a s e of the 482 k e y line of 181Ta show that the s t r e n g t h of the d i r e c t t r a n s i t i o n M d i r e c t due toQ2pN E (eq. (10)) will be l a r g e r than the s t r e n g t h of the t r a n s f t i o n Mad m due to a d m i x t u r e s of o p p o s i t e p a r i t y s t a t e s , a s given in an e a r l i e r p a p e r [9]. At f i r s t sight one could g u e s s that the c o n t r i b u t i o n s M d i r e c t a r e only dominant in e x p e r i m e n t a l s i t u a t i o n s w h e r e s t a t e s a r e a d m i x e d through a big AE, however the situation m a y not be a s s i m p l e b e c a u s e of n u c l e a r s t r u c t u r e effects. B e c a u s e the PNE o p e r a t o r s have a d i f f e r e n t i s o s p i n dependence than the PNC p o t e n t i a l s , it will be v e r y difficult to get any i n f o r m a t i o n about the o r i g i n of m e a s u r e d effects in ~ - t r a n s i t i o n s , by looking at isopin selection rules. However, b e f o r e s t a r t i n g lengthy c a l c u l a t i o n s of the above d e s c r i b e d effects, a thorough i n v e s t i g a t i o n h a s to be m a d e of the p r o c e s s e s not included in Q2PNE, to e n s u r e a gauge i n v a r i a n t d e s c r i p t i o n . Also the weak photoproduction v e r t e x has to be e x a m i n e d f u r t h e r . I n v e s t i g a t i o n s of the c o m p l e t e d i r e c t c o n t r i b u tions including t e r m s of higher o r d e r in P / M are in p r o g r e s s for n - p c a p t u r e and o t h e r c a s e s of e x p e r imental interest. The authors would like to thank Professors G. E. Brown and L. SchUlke for valuable suggestions. Discussions with our Caltech colleqgues were helpful. One of us (M.G.) would like to thank Professor F. B o e h m and his group at Caltech for their hospitality and for excellent working conditions.
References [1] H. Hlittig, K. HUnchen and H. Wltffler, Phys. Rev. Lett. 25 (1970) 941. [2] F. Boehm, Intern. Conf. in Delft, 1969 (CALT-63-142). [3] R. Feynman and M. GeH-Mann, Phys. Rev. 109 (1958) 1993. [4] E. Fiscbbach, D. Tadi6 and K. Trabert, Phys. Rev. 186 (1969) 1688. [5] L. Schulke, to be published. 445
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[6] B. H. J. McKellar, Phys. L e t t e r s 26B (1967) 107. [7] E.M, Henley, to be published. [8] E.M, Henley, T . E . K e L i h e r and D . U . L . Yu, Phys. Rev. Lett. 23 {1969) 941; M. Gari and H. Kllmmel, Phys. Rev. Lett. 23 (1969) 26; M. Gari, H.K•mmel and J . G . Zabolitzky, Nucl. Phys. A161 (1970) 625. [9] B. Desplanques and N. Vinh-Mau, Phys. L e t t e r s 35B (1971) 28. M. Gari, O. Dumitrescu, J . G . Zabolitzky and H. KilmmeL, Phys. L e t t e r s 35B (1971) 19. [10] F. C. Michel, Phys. Rev. 133B {1964} 329. [11] D. Tadl~, private communication (1970). [12] M. Gell-Mann and M. L~vy, Nuovo Cimento Vol. XVI (1960) 705. [13] J. J. Sakurai, Currents and m e s o n s (The University of Chicago P r e s s , Chicago 1969). [14] N. KrolL and M. A. Ruderman, Phys. Rev. 93 (1954) 233. [15] M. Chemtob and M. Rho, p r e p r i n t T.H. l179-CERN.
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4 O c t o b e r 1971
PHYSICS LETTERS
CONTRIBUTIONS TO PARITY INTERACTION C U R R E N T NUCLEAR y-TRANSITIONS *
4 October 1971
NON-CONSERVING
M. GARI** and A. H. HUFFMAN
California Institute of Technology, Pasadena, California, USA Received 25 July 1971
Parity non-conserving interaction current contributions to nuclear y-transitions required by gauge invariance are calculated and discussed. Arguments are given that the strangeness conserving currents should contribute to the one pion-exchange parts of the interaction.
P a r i t y n o n - c o n s e r v i n g effects in n u c l e a r ~ - and r - t r a n s i t i o n s have now b e e n detected in many e x p e r i m e n t s [1, 2]. T h e s e effects have been e s t i m a t e d t h e o r e t i c a l l y by c a l c u l a t i n g a n u c l e o n - n u c l e o n p a r i t y - n o n c o n s e r v i n g potential (VpNc) , u s u a l l y b a s e d on one = or one p exchange (fig. 1). In the conv e n t i o n a l model [3] the AS = 0 weak H a m i l t o n i a n is given by: HAS=0 G [ 2 ÷ + sin20{S~,S+~}+] weak - - ~8-L c ° s O{J~, Jp }+
(1)
where J ~ , Sg a r e s t r a n g e n e s s c o n s e r v i n g (AS = 0) and s t r a n g e n e s s changing (AS = 1) hadronic c u r r e n t s , r e s p e c t i v e l y . The p - c o n t r i b u t i o n to VpN C using either the f a c t o r i z a t i o n a p p r o x i m a t i o n or c u r r e n t a l g e b r a [4] is:
vP(AI=0,2) = Gp [ ( l + ~ v ) ( a 1 x a2).[Pl2,fp(rl2)] where T (12) = r l+~ -2 + ~-1~2 , - + hA(0) : Gcos20(m2pgA/f) ,
_
_
+ (a 1
f p (r12) = e x p ( - m p r l 2 ) / r l 2
,
a 2 ) - P l 2 , f p ( r l 2 ) +1J -i(1+ )2 ,
Gp = hA(O)f/8y~'-2M,
(2)
f 2 / 4 ~ ~ 2.4 ,
gA(0) ~ 1.2 .
Fig. 1. One 7rand p contributions to VPNC. The open circle represents a strong vertex and the circle-with-cross a weak vertex. The ~ - c o n t r i b u t i o n to VpN C can be obtained from the cos20 piece of Hweak by a s s u m i n g i s o s p i n b r e a k i n g [5] and f r o m sin20 piece by r e l a t i n g the weak NN~ v e r t e x to s t r a n g e b a r y o n decays [6]. In either c a s e the potential has the f o r m [5]: V;,II(AI= 1) = o~I,II, . ~Ox + 02)'[P12, f~(rl2)]_ T~2) ,
(3)
but G I and G II o r i g i n a t e f r o m the AS = 0 and AS = 1 c u r r e n t s , r e s p e c t i v e l y , and a r e given by * This work was supported in part by Deutsche Forschungsgemeinschaft, in part by National Science Foundation [GP-28027, GP-19887], and in part by the U.S. Atomic Energy Commision. Prepared under contract AT[04-3]-63 for the San Francisco Operations Office, U.S. Atomic Energy Commission. ** Permanent address: Ruhr-Universit~tt, Bochum, West Germany. 442
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PHYSICS LETTERS
GI'Ilrr : KfI'II/81r~-2-M ,
4October1971
/~(r12) : exp(- m r12)/r12 ,
(4)
w h e r e g i s the s t r o n g ~NN coupling constant (g2/4~ ~ 15) and /I
= C/~2-cos2Oc ( m
_ M ) ~ 0.14 × 10 - 8
(4a)
--/i = 4.8 x i 0 - 8
(4b)
Note that G I v a n i s h e s in the SU(2) l i m i t when the n u c l e o n s a r e taken to be o n - s h e l l . This is not n e c e s s a r i l y t r u e if one t a k e s into account the o f f - s h e l l b e h a v i o u r of the nucleons in c a l c u l a t i n g the weak ~NN v e r t e x [5, 7]. In t h i s c a s e one could g e t f I c o m p a r a b l e to fII. T h e s e p o t e n t i a l s a d m i x s t a t e s of irregular p a r i t y Id(-~)Ei>to the s t a t e s of n o r m a l p a r i t y IJ (~)E) p r o d u c e d by the s t r o n g N-N f o r c e s :
[~J) = ]J(y)E)+ ~.$ F i ~ (-Y)Ei)
(5)
with
= ~
A
EK(~, ) - E A
B
E B - Ej0r)
(6) '
w h e r e E L (ML) d e n o t e s the e l e c t r i c (magnetic) m u l t i p o l e o p e r a t o r of the i r r e g u l a r t r a n s i t i o n . While c a l c u l a t i o n s b a s e d on the above d e s c r i p t i o n a r e in good a g r e e m e n t with e x p e r i m e n t a l r e s u l t s c o n c e r n i n g p a r i t y n o n - c o n s e r v i n g s - d e c a y [8], they u n d e r e s t i m a t e the p a r i t y n o n - c o n s e r v i n g e f f e c t s in ~ - t r a n s i t i o n s by two o r d e r s of magnitude [9]. T h e s e l a t e s t c a l c u l a t i o n s [9] t a k e c a r e f u l account of n u c l e a r s t r u c t u r e and c o r r e l a t i o n s between the nucleons. We wish to point out in this l e t t e r that the i n t e r a c t i o n c u r r e n t c o r r e c t i o n s of fig. 2 r e q u i r e d by gauge i n v a r i a n c e m a y be l a r g e . T h e r e a r e two a t t e m p t s [10, 11] known to us which t a k e into account additional c o n t r i b u t i o n s by i n t r o d u c i n g the e.m. f i e l d into the n o n - r e l a t i v i s t i c p.v. N - N potential, a method we do not follow. We think the b e s t way to p e r f o r m a gauge i n v a r i a n t d e s c r i p t i o n i s a s y s t e m a t i c c a l c u l a t i o n of all r e l e v a n t t i m e - o r d e r e d F e y n m a n d i a g r a m s (fig. 3 b - d and additional) using e i t h e r the gauge i n v a r i ant i n t r o d u c t i o n of the m i n i m a l e.m. i n t e r a c t i o n into the total L a g r a n g i a n , o r the full e l e c t r o m a g n e t i c v e r t e x . F o r the b a r e i n t e r a c t i o n t h e s e two m e t h o d s a r e equivalent.
(a)
(b)
(¢)
(d)
(e)
Fig. 2. Interaction current contributions to parity-nonconservation. (Plus terms with weak and strong interchanged). These diagrams produce a parity n0n-conserving two-body electromagnetic operator c'tgpNE.
(a)
(b)
(¢)
(d)
F i g . 3. T i m e o r d e r e d c o n t r i b u t i o n s of o n e - m e s o n - e x c h a n g e . (Plus d i a g r a m s with weak and s t r o n g interchanged).
A gauge i n v a r i a n t d e s c r i p t i o n m u s t include the p r o c e s s e s of fig. 2 b - e . T h e s e c o n t r i b u t i o n s a r e not included t h r o u g h the wavefunctions eq. (5) and t h e r e f o r e m u s t b e s e p a r a t e l y introduced. T h e s e p a r t s of the i n t e r a c t i o n then act l/k¢ a two body p a r i t y n o n - c o n s e r v i n g e l e c t r o m a g n e t i c (FNE) o p e r a t o r . In c a l c u 443
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lating such PNE contributions one has to be w a r y of overcounting. F o r example, the covariant diagram fig. 2a containts some p a r t s like fig. 3a which have already been included through the wavefunction admixtures, and other p a r t s like fig. 3b, c, d which must be included in the PNE contribution. In this paper we obtain the leading PNE t e r m s to get an estimation of the o r d e r of magnitude of their effects and to get a feeling of where such contributions could be important. A s t r a i g h t f o r w a r d calculation of the PNE o p e r a t o r s is to s t a r t f r o m a gauge invariant Lagrangian which can be obtained by the minimal substitution 0 / ~ + - + (a~ ¥ i e A g ) ~ +
(7)
in the total Lagrangian ~ t o t a l = ( ~ N + .~n + ~ P ) f r e e + "/~WNN+ ~S, NN + .~WpNN+ ~SpNN
(8)
_W,S denote the appropriate coupling of the m e s o n s to the nucleon: where MMNN
S
~TrNN
-
g a ~Kr~g~5~aN ,
(9a)
2M
_W(I) = i - ~ cos 2 0 c J~ 7rNN
0g ~a~(yg +
~
av) - ~ N
(9b)
'
./~W(II) / I ~((7 × V)zN , ~rNN = 23pN NS
= ifp~.~(ybt + ~gv
(9c)
agVOv) ~2 g ,
(9d) (9e)
.CoN N W = __P~lV(YghA(0)- i a g h P (0))5 ~2 g , S
The pion-nucleon interaction .~ 7rNN has been taken to be of the gradient coupling f o r m [12] which is known to satisfy the PCAC relation and can be shown to d e s c r i b e the s o f t - p i o n - l i m i t of photoproduction properly. It is easily verified that the r e p l a c e m e n t (7) in .~SNN gives the K r o l l - R u d e r m a n t h e o r e m [14]. This p r o p e r t y of the Lagrangian is important because without this connection to experimental pion photoproduction the calculations would r e m a i n r a t h e r doubtful concerning the right o r d e r of magnitude of these effects. Wb~ii) The weak pion L a g r a n g i a n s .C originate f r o m the AS = 0 and AS = 1 c u r r e n t s respectively. F o r weak photoproduction the AS = 0 p a r t gives a non-vanishing contribution even in ~U(2) limit. This is v e r y important because it enters into the o p e r a t o r s with the cos20 factor. After making the substitution (7) in the total Lagrangian and retaining t e r m s of lowest o r d e r in P/M, the following PNE contributions a r e obtained: = Qg;.II(AI = 0, 2) + Qg;(AI = I) + Q;P(AI = 0, I, 2 ) , where 7r 2) = GI ' I I e " CPI,II(AI= 0, - n ~a 1 + a2)"
(+) Af~(rl2) T12 ,
• I x a2)J[(s~J(Pl2)- 52P 122 6ij),f.(rx2) ]_ +Ai(o1_o2~l{p~2,[PJ2,f.(r12)] CPI(AI=I) : 2F el(l+.v)A'(io
(10) (lOa) }+]T~2)
(10b)
q~P(AI=O, I,2) = -Cpef p(rl2)12~v(ial X a2).AT~2) + (al + a2).AT~+2)l , where
S~2~p)-- = pipJ_ ~p 1 2 8ij ,
(10c)
F = Gcgcos2O/~/2(2/~247r , bLv = ~p _ / l n ' A is the v e c t o r potential.
A =k~,~ ( ~ V 1/2 (Ak,e+ A:k,~)~ exp(ik-r) . The t e r m Q~;,II(AI= 0, 2) c o m e s f r o m photoproduction at the strong v e r t e x (fig. tribution. This is just t e r m coming f r o m the and 3d) [15]. In lowest 444
2b). We found that the K r o l l - R u d e r m a n (or seagull-) t e r m gave the l a r g e s t conthe p r o c e s s d e s c r i b e d in d i a g r a m 2b, which can be shown to be equivalent to the negative e n e r g y p a r t in the nucleon p r o p a g a t o r in the p s e u d o s c a i a r t h e o r y (fig. 3c o r d e r , photoproduction of a neutral pion vanishes in a c c o r d a n c e with the Kro11-
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R u d e r m a n t h e o r e m . T e r m s f r o m figs. 2 c - e a r e of at l e a s t f i r s t o r d e r in P / M and w e r e not w r i t t e n down h e r e . The termC~(Al = 1) d e s c r i b e s the c o n t r i b u t i o n of weak photoproduction, fig. 2a and 2b, with weak and s t r o n g v e r t e x i n t e r c h a n g e d . Fig. 2a was c a l c u l a t e d by taking into account o f f - s h e l l b e h a v i o u r of the i n n e r n u c l e o n - l i n e . T h i s p i e c e was not taken into account through VpN C. AlthoughC~ii i s of second o r d e r in P/M, it has been included h e r e b e c a u s e it o r i g i n a t e s f r o m the s t r a n g e n e s s c o n s e r v i n g (AS - 0) c u r r e n t s and does not v a n i s h in the SU (2) l i m i t . T h i s t e r m should give e f f e c t s of the s a m e o r d e r of m a g n i tude asC)9~i which was d e r i v e d f r o m the s t r a n g e n e s s changing (AS = 1) c u r r e n t s . (Recall that V ~ v a n i s h e s in the SU(2) l i m i t by taking the nucleons o n - s h e l l . ) B e c a u s e the m i n i m a l substitution i s m a d e m an effective L a g r a n g i a n , and o f f - s h e l l b e h a v i o u r u s e d this t e r m is p r o b a b l y l e s s r e l i a b l e than the o t h e r s . O t h e r c o n t r i b u t i o n s of weak photoproduction beyond z e r o o r d e r in P / M w e r e o m i t t e d , as t h e r e a r e no compensating factors. The t e r m c)gO d e s c r i b e s the contribution of s t r o n g p photoproduction (fig. 2b and fig. 3c and d). Weak p - p h o t o p r o d u c t i o n and the p r o c e s s e s of fig. 2 c - e a r e of higher o r d e r in P / M . It should be noted that the PNE contributionsQYP andC~ 7T l e a d the different i s o s p i n s e l e c t i o n r u l e s a s c o m p a r e d to v P N and VI~ . C C As m e n t i o n e d above, the t e r m s C ~ P N S eq. (10) e n t e r in the ~ - t r a n s i t i o n a m p l i t u d e a s ~ i r e c t c o n t r ~ u tions. The total a m p l i t u d e for i r r e g u l a r ~ - t r a n s i t i o n including such effects is thus given by Mid~ - - K ~' ]total = M[J• --+K~ i] admixture +M[J" ~ K y'] d i r e c t ' with: ?T' M[J ~ - * g ]direct = {KTr' ~)~PNE I J ~ ) "
(11)
(12)
In the long wavelength l i m i t (exp ( J R - r ~ 1) Q ~ a n d c ~ P a r e i r r e g u l a r E1 o p e r a t o r s ; that i s , they give E1 r a d i a t i o n with no p a r i t y change. In the c a s e of a r e g u l a r M1 t r a n s i t i o n t h i s r a d i a t i o n p l u s i r r e g u l a r E1 r a d i a t i o n f r o m s t a t e s a d m i x e d by VpN C will i n t e r f e r e with the r e g u l a r M1. Higher o r d e r t e r m s of the PNE o p e r a t o r s in P/M, and the l o w e s t o r d e r t e r m s taken beyond the long wavelength l i m i t , give i r r e g u l a r M1 r a d i a t i o n ; this is M1 r a d i a t i o n with p a r i t y change. The o p e r a t o r s Q 2 p N E connect r e l a t i v e S - s t a t e s to S - s t a t e s , which is v e r y i m p o r t a n t b e c a u s e of the shoft ~ange of the r a d i a l dependence. By c o n t r a s t the a d m i x i n g p o t e n t i a l s VpN C eq. (2) and (3) connect r e l a t i v e S- to P - s t a t e s . Thus, for e x a m ple, in n - p c a p t u r e the C})pNE t e r m s should c o n t r i b u t e a l a r g e i r r e g u l a r E1 t r a n s i t i o n between 1S 0 s c a t t e r i n g s t a t e and the deuteron ground s t a t e i n t e r f e r i n g with the r e g u l a r M1 t r a n s i t i o n to p r o d u c e circular polarization. P r e l i m i n a r y c a l c u l a t i o n s in the c a s e of the 482 k e y line of 181Ta show that the s t r e n g t h of the d i r e c t t r a n s i t i o n M d i r e c t due toQ2pN E (eq. (10)) will be l a r g e r than the s t r e n g t h of the t r a n s f t i o n Mad m due to a d m i x t u r e s of o p p o s i t e p a r i t y s t a t e s , a s given in an e a r l i e r p a p e r [9]. At f i r s t sight one could g u e s s that the c o n t r i b u t i o n s M d i r e c t a r e only dominant in e x p e r i m e n t a l s i t u a t i o n s w h e r e s t a t e s a r e a d m i x e d through a big AE, however the situation m a y not be a s s i m p l e b e c a u s e of n u c l e a r s t r u c t u r e effects. B e c a u s e the PNE o p e r a t o r s have a d i f f e r e n t i s o s p i n dependence than the PNC p o t e n t i a l s , it will be v e r y difficult to get any i n f o r m a t i o n about the o r i g i n of m e a s u r e d effects in ~ - t r a n s i t i o n s , by looking at isopin selection rules. However, b e f o r e s t a r t i n g lengthy c a l c u l a t i o n s of the above d e s c r i b e d effects, a thorough i n v e s t i g a t i o n h a s to be m a d e of the p r o c e s s e s not included in Q2PNE, to e n s u r e a gauge i n v a r i a n t d e s c r i p t i o n . Also the weak photoproduction v e r t e x has to be e x a m i n e d f u r t h e r . I n v e s t i g a t i o n s of the c o m p l e t e d i r e c t c o n t r i b u tions including t e r m s of higher o r d e r in P / M are in p r o g r e s s for n - p c a p t u r e and o t h e r c a s e s of e x p e r imental interest. The authors would like to thank Professors G. E. Brown and L. SchUlke for valuable suggestions. Discussions with our Caltech colleqgues were helpful. One of us (M.G.) would like to thank Professor F. B o e h m and his group at Caltech for their hospitality and for excellent working conditions.
References [1] H. Hlittig, K. HUnchen and H. Wltffler, Phys. Rev. Lett. 25 (1970) 941. [2] F. Boehm, Intern. Conf. in Delft, 1969 (CALT-63-142). [3] R. Feynman and M. GeH-Mann, Phys. Rev. 109 (1958) 1993. [4] E. Fiscbbach, D. Tadi6 and K. Trabert, Phys. Rev. 186 (1969) 1688. [5] L. Schulke, to be published. 445
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[6] B. H. J. McKellar, Phys. L e t t e r s 26B (1967) 107. [7] E.M, Henley, to be published. [8] E.M, Henley, T . E . K e L i h e r and D . U . L . Yu, Phys. Rev. Lett. 23 {1969) 941; M. Gari and H. Kllmmel, Phys. Rev. Lett. 23 (1969) 26; M. Gari, H.K•mmel and J . G . Zabolitzky, Nucl. Phys. A161 (1970) 625. [9] B. Desplanques and N. Vinh-Mau, Phys. L e t t e r s 35B (1971) 28. M. Gari, O. Dumitrescu, J . G . Zabolitzky and H. KilmmeL, Phys. L e t t e r s 35B (1971) 19. [10] F. C. Michel, Phys. Rev. 133B {1964} 329. [11] D. Tadl~, private communication (1970). [12] M. Gell-Mann and M. L~vy, Nuovo Cimento Vol. XVI (1960) 705. [13] J. J. Sakurai, Currents and m e s o n s (The University of Chicago P r e s s , Chicago 1969). [14] N. KrolL and M. A. Ruderman, Phys. Rev. 93 (1954) 233. [15] M. Chemtob and M. Rho, p r e p r i n t T.H. l179-CERN.
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