Neutrino magnetic transition moment in the Zee model

V01ume 216, num6er 3,4

PHY51C5 LE77ER5 8

NEU7R1N0 MA6NE71C 7RAN51710N M0MEN7

12 January 1989

1N 7 H E 2 E E M 0 D E L

J1an9 L1U Phy51c5Department, Carne91e-Me110n Un1ver51ty,P1tt56ur9h, PA 15213, U5A Rece1ved 1 June 1988; rev15ed manu5cr1pt rece1ved 8 Au9u5t 1988

We 5h0w that 1n c0ntra5t t0 the recent 5u99e5t10n 0f 8a6u and Mathur, the 2ee m0de1 actua11ycann0t pr0v1de the re4u1red am0unt 0f Maj0rana neutr1n0 prece5510nf0r the 0kun-V0105h1n-Vy50t5ky501ut10nt0 the 501arneutr1n0 pr061em.

ReCent1y, 8a6U and MathUr have ana1y2ed [1] Maj0rana neutr1n0 tran51t10n m0ment5 ( M N 7 M ) 1n a 9aU9e m0de1 0r191na11y 5U99e5ted 6y 2ee [ 2 ]. 7 h e y ar9Ued that 1f there 15 a heavy f0Urth 9enerat10n 0f 1ept0n, then the m0de1 Can pr0V1de a 5Uff1C1ent1y1ar9e Va1Ue 0 f M N 7 M and thU5 rea112e the 0kun-V0105h1nVy50t5ky ( 0 V V ) 501ut10n [3 ] t0 the 501ar neutr1n0 pr061em. 1n th15 n0te we w0u1d 11ke t0 p01nt 0ut that, 1n fact, th15 15 n0t the ca5e. M0re 5pec1f1ca11y, we w111 f1r5t 5h0w that 1n c0ntra5t t0 the 5u99e5t10n 0f 8a6u and Mathur, the 2ee m0de1 actua11y cann0t pr0v1de a 5uff1c1ent1y 1ar9e va1ue 0 f M N 7 M wh11e at the 5ame t1me keeP1n9 the c0rre5p0nd1n9 neutr1n0 ma55 term van15h1n91y 5ma11. Furtherm0re, we w111p01nt 0ut that 1n d15cu551n9 Maj0rana neutr1n0 5p1n prece5510n5, veL -,v~R, they have m155ed an 1mp0rtant p01nt 1n that 5uch a pr0ce55 15 1n 9enera1 5evere1y 5uppre55ed 6y a prefact0r ar151n9 fr0m the ma55 d1fference 0 f t h e tw0 d1fferent Maj0rana neutr1n0 5tate5. When th15 p01nt 15 c0rrect1y taken 1nt0 acc0unt, we f1nd that the re5u1t1n9 M N 7 M cann0t 6e much 1ar9er than 0 f t h e 0rder 0f 10-15(mc/1 eV) 1n un1t5 0fthe 80hr ma9net0n (e/2m0). 70 111u5trate the 1dea, 1et u5 f1r5t exam1ne the re1at10n 6etween M N 7 M and the neutr1n0 ma55 0f the m0de1. 7he 2ee m0de1 d1ffer5 fr0m the 5tandard 5 U ( 2 ) L X U( 1 ) y 9au9e m0de1 6y the add1t10n 0f ( 1 ) an extra H1995 d0u61et ~2 and (2) a 51n91et char9ed

H1995 h. 7he new c0up11n95 that are re1evant t0 neutr1n0 prece5510n5 are

fa1,QaL1r72Q1,Rh--]-/2~4~1cr~~ -

"

c

-

-

h- +h.c.,

( 1)

where ~ 15 the u5ua1 1ept0n d0u61et, 4~ 15 the H1995 d0u61et 0f the 5tandard m0de1 and f , 6 = - f , a due t0 Ferm1 5tat15t1c5. At tree 1eve1 a11 neutr1n05 are ma551e55. At 0ne-100p 1eve1, rad1at1ve 1nduced neutr1n0 ma55e5 can ar15e: m,1,=6vm0f, t,[m~f(x,,y,)-m~f(xt,,y~,)]

(2)

,

where rna repre5ent5 the ath 9enerat10n char9ed 1ept0n ma55, m0 def1ne5 the (neutr1n0) ma55 5ca1e and 1nx

1ny

f(x, y) = 1-x

1-y

t Addre55after AU9U5t1, 1988:Phy51C5Department, Un1Ver51ty 0f M1Ch19an,Ann Ar60r, M1 48109-1120, U5A. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. ( N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n )

, xa --

m2 ~, M7

Ye =

rn,2 M~

,

(3) w1th M~ and A12 6e1n9 the tw0 char9ed phy51ca1 H19956050n ma55e5. A150, at 0ne-100p 1eve1, 0ff-d1a90na1 tran51t10n m0ment5, wh1ch can 6e e1ther ma9net1c 0r e1ectr1c d1p01e type [4 ], can 6e 1nduced (1n un1t5 0f 8 0 h r ma9net0n): 12~1,=26Fmem0f~6[9(x~,y~)+9(x1,,y1,)] ,

(4)

w1th 9(x,y)=

x ~-x

x1nx 4- ~( 1 - x )

y 1-y

y1ny (1--12) 2"

(5)

E45. (2)--(5) have 6een 91Ven 1n ref. [ 1 ]. 7he 5ame Ca1CU1at10n ha5 a150 6een Carr1ed 0Ut 6y PetC0V 50me t1me a90 [5], and Very 51m11ar re5U1t5 have 6een 06ta1ned. 367

V01ume 216, num6er 3,4

PHY51C5 L E 7 7 E R 5 8

Up t0 th15 p01nt, the d15cu5510n5 0f ref. [ 1 ] 0n M N 7 M rema1n va11d. 1t 15, h0wever, 1nc0rrect t0 c1a1m that m,6(a, 6=e, 9, 2, 2~ ) can 6e van15h1n91y 5ma11 wh11e #,6 rema1n5 1ar9e. 7he rea50n 15 the f0110w1n9. 51nce rn~. >> rn~ >> m~ >> me, 1n c0ntra5t t0 the 5u99e5t10n 0f 8a6u and Mathur, the tw0 term5 1n e4. (2) cann0t cance1.70 make m~h 5ma11, we theref0re have t0 make the c0m61nat10n 0f m0f~1f(x, y) 5uff1c1ent1y 5ma11. 51nce the fact0r rn0f,~ a150 appear5 1n #,6, #,~ w0u1d at the 5ame t1me 6e 512ea61e 0n1y 1f we c0u1d keep 9(x, y) 1ar9e wh11e tun1n9f(x, y) van15h1n91y 5ma11. H0wever, a5 5h0wn 6e10w, th15 can never 6e rea112ed 1n the pre5ent m0de1. 1ndeed, 1f x, y << 1 (th1515 certa1n1y true 1fthere are 0n1y three 9enerat10n5 0f 1ept0n5), then fr0m e45. (3) and (5) we f1nd t0 1ead1n9 0rder

f(x,y)~1nx-1ny,

9(x,y)~x1nx-y1ny.

(6)

51nce f0r 0 < x , y<< 1

1x1nx-y1ny[ <~11nx-1ny1 ,

(7)

we then have (the e4ua11ty 1n e4. (7) ar15e5 0n1y 1f x = y and thu5f(x, y) =9(x, y) = 0)

19(x,y)[ <<.1f(x,y)] •

(8)

0 n the 0ther hand, 1fx, y>> 1 (c0rre5p0nd1n9 t0 the ca5e 1n wh1ch there 15 a heavy f0urth 9enerat10n 0f 1ept0n), 0ne f1nd5 fr0m e45. (3) and (5)

19(x, Y) 1~ 1f1x, y) 1 •

(9)

C1ear1y, 1n ne1ther ca5e 19(x, Y)1 can 6e much 1ar9er than 1f(x, Y) 1. A5 a re5u1t, f0r the5e ch01ce5 0f the parameter5 0f the m0de1, 1t 15 1mp055161e t0 06ta1n a 1ar9e va1ue 0f #~, wh11e at the 5ame t1me keep1n9 m,6 van15h1n91Y 5ma11. F0r 0ther ch01ce5 0f the parameter5, the re1at10n 6etweenf(x, y) and 9(x, y) and thu5 the re1at10n 6etween m~6 and #,6 6ec0me 1e55 tran5parent. H0wever, 1t 5t111 can 6e 5h0wn that 0nce we m a k e f ( x , y) 5ma11, 9(x, y) w1116e 5ma11 a5 we11. 1n part1cu1ar, 1n the 11m1tf ( x , y) =0, 9(x, y) mu5t van15h ~ 51nce the 0n1y 501ut10n f 0 r f ( x , y) = 0 15 x=y. A van15h1n91y5ma11 va1ue 0f rn~, d0e5 n0t h0wever nece55ar11y re4u1re that a11 e1ement5 1n the neutr1n0 ma55 matr1x are 5ma11. F0r 1n5tance, a h1erarch1ca1 ~t 1t 15 certa1n1y 1nc0rrect t0 have any n0n2er0 va1ue5 0 f 9 ( x , y) (5ee f19. 2 0f ref. [ 1 ] ) 0 n c e f ( x , y) = 0 .

368

12 January 1989

re1at10n, rn,~>> me,, rne~, 1n e4. (2) 15 certa1n1y a110wed a5 10n9 a5 we a55ume rnv,,~>>mv~. Acc0rd1n91y, # ~ can 6e much 1ar9er than /2~ and #~. H0wever, 0nce we d1a90na112e the neutr1n0 ma55 matr1x the re5u1t1n9 tran51t10n m0ment 6etween v~ and v~.~w111a9a1n 6e d1rect1ypr0p0rt10na1 t0 mv~ and thu5 van15h1n91Y 5ma11.7h15 f0110w56ecau5e the 2ee m0de1 neutr1n0 ma55e5 are ca1cu1a61e, and any 1nterna1 heavy ma55 1n5ert10n c0ntr16ut1n9 t0 /~ w111 a150 e4ua11y c0ntr16ute t0 the neutr1n0 ma55. 7hu5, the 5ame mechan15m that 5uppre55e5 the neutr1n0 ma55 w111a150 5uppre55 1t5 ma9net1c m0ment. Furtherm0re, 1n th15 ca5e the 5p1n prece5510n, v~L~v~,~R, 15 further 5uppre55ed 6y a fact0r ar151n9 fr0m the ma55 d1fference 0f the tw0 d1fferent Maj0rana neutr1n0 5tate5. A5 5h0wn 6e10w, th15 fact a10ne ha5 a1ready 1nva11dated the 5cenar10 0f 8a6u and Mathur. 7he ev01ut10n e4uat10n 0f tw0 ener9et1c Maj0rana neutr1n05, v~L and v2L w1th a tran51t10n m0ment #12 and ma55e5 m~ and m2 re5pect1ve1y, pr0pa9at1n9 1n the ma9net1c f1e1d ~2 8 1591ven 6y

d/~V~R,] -- ~#~28

m2/2p,1~v~R,1

(10>

~

where p 15 the m0mentum 0f the neutr1n05 and v3R = (v2L) c 15 the CP-c0nju9ate 0f v2L re4u1red 6y CP7-1nvar1ance. 1n the 4ue5t10n 0f1ntere5t p 15 0fthe 0rder 0f 10 MeV. A55um1n9 8 15 a c0n5tant f0r 51mp11c1ty, then e4. (10) can 6e 501ved exact1y. 1n that ca5e, the 5p1n prece5510n pr06a6111ty 0f 5tart1n9 w1th v~L and t = 0 and f1nd1n9 v~R at t1me t 15 91ven 6y [ 6 ] (2#129) 2

P(t) = (2#128)2 + (Am22/2P) 2 ×51n2(~Am~J2p)2t/2)

,

(11)

where Am~2=m 2 - m 2. 1n the 11m1t Am22=0, e4. ( 1 1 ) reduce5 t0 a 5tandard 5p1n-prece5510n f0rmu1a. F0r th15 ca5e the 11m1t c0rre5p0nd5 t0 the mer9er 0f tw0 Maj0rana neutr1n05 1nt0 a D1rac neutr1n0 w1th v~R-~v1R and the tran51t10n m0ment can n0w 6e re1nterpreted a5 a d1a90na1 m0ment f0r th15 D1rac neutr1n0. H0wever, th15 15 n0t the n0rma1 type 0f D1rac neutr1n0 (wh05e r19ht-handed c0mp0nent 15 5ter11e) rather 0ne 0f the unu5ua1 var1et1e5 d15cu55ed ~2 1n wr1t1n9 e4. (10) we have fact0r12ed 0ut an 1rre1evant 0vera11 pha5e fact0r.

V01ume 216, num6er 3,4

PHY51C5 LE77ER5 8

6y W01fen5te1n [7 ]. 1t ha5 a c0n5erved 2e1d0v1chK 0 n 0 p 1 n 5 k y - M a h m 0 u n d [8] char9e L~-L2. 1t 15 read11y 5een fr0m e4. ( 1 1 ) that the prece5510n 15 5evere1y 5uppre55ed 1f (Am22/2p)2>> ( 2 ~ 1 2 8 ) 2. P(t) 15 519n1f1cant 0n1y 1f the 0ff-d1a90na1 e1ement 1n e4. (10) 15 1ar9er than the d1fference 0f the appr0pr1ate d1a90na1 d e m e n t 5 2 / ~ 2 8 > 1Am~21/2p .

(12)

N 0 w 5upp05e v~ L 15 the e1ectr0n-neutr1n0 60rn 1n the center 0 f t h e 5un and v~R 15 a heavy ant1-neutr1n0 v~L prece55ed 1nt0. U51n9 the emp1r1ca1 6 0 u n d [ 9 ] 0n P~2 ( < 10 m), wh1ch 15 a150 the re4u1red va1ue f0r the 0 V V 501ut10n t0 the 501ar neutr1n0 pr061em, 0ne f1nd5 fr0m e4. ( 1 2 ) (the ma9net1c f1e1d 1n the c0nvect1ve 20ne 0 f t h e 5un 15 0 f t h e 0rder 0 f 103 6 ) Arn~2 ~< 10 - 7 ( e V ) 2 .

(13)

M0de1 ca1cu1at10n5 u5ua11y 91ve a much 5ma11er va1ue 0fthe tran51t10n m0ment. A5 a re5u1t, the actua160und 0n the ma55-54uared d1fference can 0n1y 6e 5ma11er than the 0ne 91ven 1n e4. (13), 1t 15 1ntere5t1n9 t0 exp10re the c0n5e4uence 0f e4. (12) and thu5 t0 5ee whether the 2ee m0de1 can pr0v1de a 519n1f1cant neutr1n0 prece5510n t0 501ve the 501ar neutr1n0 pr061em. We 5tart w1th a tw0-9enerat10n m0de1 that c0nta1n5 0n1y v~L and v~D 1t f0110w5 fr0m e45. ( 2 ) - ( 5 ) that the tw0 neutr1n05 are de9enerate 1n ma55 and thu5 can 6e fu5ed 1nt0 a D1rac neutr1n0 w1th a ma55 (a55um1n9 m~ << m~ << M~.2 ) 2 2 m,, ~ 6vfJ2m~2 m0 1n(M2/M1 ),

14)

and a ma9net1c m 0 m e n t

p~2(m~mv/M2)F~,

15)

12 January 1989

chan9e ~3. U51n9 the emp1r1ca1 60und 0n the char9ed H1995-6050n ma55, M~,2 > 30 6 e V , e4. ( 15 ) 1ead5 t0

1•, <10-15(mv/1eV) .

(17)

7hU5, 91Ven a 5ma11 Va1Ue 0f the neutr1n0 ma55, 1t5 ma9net1C m 0 m e n t W1116e t00 5ma11 t0 6e 0f 1ntere5t. 5eC0nd, 1t 15 rea112ed 6y W01fen5te1n [5] that the ma9net1C m 0 m e n t 0f a 2e1d0v1ch-K0n0p1n5kyM a h m 0 u n d type 0f D1rac neutr1n0 5h0u1d 6e further 5uppre55ed 6y an add1t10na1 fact0r m ~2 / M ~, , here M repre5ent5 a heavy ma55 0 f the 0rder 0f Mw. 1f/tv 15 9enerated 501e1y 6y 1ntermed1ate 9au9e 6050n exchan9e5, 0ne can 5h0w that 1n the 11m1t rn,,0 = mv~ th15 fact0r ar15e5 fr0m a 6 1 M type 0f cance11at10n. 1n the tr1p1et m0de1 [ 10], there 15 a new c0ntr16ut10n t0/t~ due t0 the char9ed H1995-6050n exchan9e. A1th0u9h th15 c0ntr16ut10n d0e5 n0t have a 6u11d-1n 6 1 M cance11at10n, 1t 15 a150 e4ua11y 5uppre55ed 6ecau5e the char9ed c 0 m p 0 n e n t 0f the u5ua1 d0u61et c0nta1n5 ma1n1y unphy51ca1 de9ree5 0 f freed0m that end up 1n W •. Wh11e 1n the pre5ent m0de1, th15 char9ed H1995 f1e1d 15 n0t re4u1red t0 6e 50. A5 a re5u1t, the add1t10na1 5uppre5510n fact0r d0e5 n0t appear. We n0w c0n51der the m0de1 w1th three 9enerat10n5 0f 1ept0n5 (v~D v,L and v~L). 1n 9enera1, the d1a90na112at10n 0 f the neutr1n0 ma55 matr1x w111 re5u1t 1n three c0mp1ete1y d1fferent M a j 0 r a n a neutr1n0 ma55 e19en5tate5. 7heref0re, acc0rd1n9 t0 e4. (1 1 ), neutr1n0 prece5510n5 are 5uppre55ed 1fthe c0nd1t10n 91ven 1n e4. (12) 15 n0t 5at15f1ed. N0w we mu5t a5k h0w th15 c0nd1t10n can 6e 5at15f1ed and what the c0n5e4uence5 are. 51nce rn~ << rn~ << m~, we can rea50na61y a55ume fe ~ ( r n ~ / m ~) <
where ×

F~ =

M 2 (M~ - 1 ) 1 n ( m ~ / M 2 ) M--~2+ ~ 1 ~ ,/1n(M~/M~) "

0

(18)

16)

7 w 0 C0mment5 are n0w 1n 0rder. F1r5t, fr0m e45. ( 15 ) and (16) we 5ee that Pv depend5 0n me 0n1y 109ar1thm1ca11y. 7heref0re, f0r a f1xed va1ue 0f neutr1n0 ma55 rnv, re9ard1e55 0f the ch01ce 0 f the heavy 1ept0n ma55 rn~ the 0rder-0f-ma9n1tude 0f ~v w111 n0t

~3 0 f c0ur5e, when mQ>ML20ur appr0x1mat10n f0rmu1a 1n e45. (14), ( 15 ) can n0 10n9er h01d. 1n that ca5e we 5h0u1d u5e the 0r191na1 exact f0rmu1ae e45. (2)-(5). H0wever, 1t 15 ea5y t0 5h0w that tak1n9 m~ 1ar9er than M~2 w111n0t chan9e the 4ua11tat1ve feature 0f 0ur re5u1t and thu5 0ur f1na1 c0nc1u510n rema1n5 the 5ame. 369

v01ume 216, num6er 3,4

PHY51c5 LE77ER5 8

1n th15 11m1t, the tran51t10n m 0 m e n t matr1x 15 91ven 6y

12 January 1989

6vm0m7. 1n(M7/M~)

26vm0m~F~1n(M~/ M 2) ( m~/ M7 ×

X

0

0

,

.

0

0

0

~47here are t0ta1 three d1fferent way5 0f c0n5truct1n9 a D1rac neutr1n0 0ut 0f the 3 × 3 ma55 matr1x, each 0f wh1ch c0rre5p0nd5t0 a c0n5erved1ept0nnum6er (L~- Le- LmL , - L~- L~, Le- L~- L~). H0wever, f0r a f1xedva1ue0f the neutr1n0ma55, the1r ma9net1cm0ment5 are appr0x1mate1ythe 5ame.

L~,~

05.,,1. ol

(19)

D1a90na1121n9 [ 1 1 ] e4. (18) re5u1t5 1n a ma551e55 Maj0rana neutr1n0 and tw0 de9enerate ma551ve 5tate5 wh1ch can theref0re 6e fu5ed 1nt0 a D1rac neutr1n0 ~4 2 2 w1th ma55 m~=6Frn0rn21n(M11M2) f ~2 + f , ,2 . 7 h e ma9net1c m 0 m e n t 0f th15 D1rac neutr1n0 15 a9a1n 91ven 6y e4. (15) w1th the rep1acement 0f F~ 6y Ft. 51nce F~ ~ F~, the 0rder-0f-ma9n1tude 0f ~ 15 thu5 5t111 91ven 6y e4. (17). A150, the tran51t10n m 0 m e n t 6etween the D1rac neutr1n0 a n d the ma551e55 M a j 0 r a n a neutr1n0 15 2er0. 8y keep1n9 thef~(m~/m~) term 1n e4. (18), the D1rac neutr1n0 w111 5p11t 1nt0 tw0 Maj0rana 5tate5. H0wever, 1n 0rder t0 av01d the 5uppre5510n prefact0r 1n e4. (1 1 ) 50 that the 5p1n prece5510n 15 effect1ve, th15 term mu5t 6e ch05en 5uff1c1ent1y 5ma11. F0r 1n5tance, tak1n9 m~ ~ 1 eV we f1nd fr0m e45. (17) a n d ( 12 ) that/2~ < 10- ~5 and Am 2 < 10- ~2 eV 2. A5 a re"~ 2 --6 2 2 5u1t, f~,(m;/m~)<10 Jx/~+f~. 7heref0re, 1nc1ud1n9 the very 5ma11fc~ term w111 n0t chan9e the 4ua11tat1ve feature 0f 0ur re5u1t. F1na11y, we c0n51der the m0de1 w1th f0ur 9enerat10n5 0f 1ept0n5. F0r 51mp11c1ty, we further a55ume that the f0urth-9enerat10n char9ed-1ept0n ma55 me, 15 a60ut ten t1me5 heav1er than rn~ 50 that rn,, 15 5t111 119hter than M~,2 (th15 a55umpt10n 15 n0t e55ent1a1). F0110w1n9 the 5ame ar9ument pre5ented a60ve, we f1nd that a9a1n 1n 0rder t0 av01d the ma55 d1fference 5uppre5510n fact0r, the tran51t10n m 0 m e n t mu5t 6e at m05t 0f the 0rder 0f 10-~5(mv/eV ). 7h15 f0110w5 6ecau5e 1n the 11m1t 0f tw0 Maj0rana neutr1n05 fu51n9 1nt0 a D1rac neutr1n0 the 4ue5t10n can 6e reduced exact1y t0 the ca5e d15cu55ed a60ve. F0r examp1e, t0 c0n5truct a D1rac neutr1n0 w1th a c0n5erved L ~ , L ~ - L , - 1~1ept0n n u m 6 e r , the neutr1n0 ma55 matr1x mu5t 6e 0f the f0rm

370

0

(20)

Acc0rd1n91y, the tran51t10n m 0 m e n t matr1x 15

26vm0mcF~, 1n(M~/M~) ( m~,/M~) 0 ×

0

0 0 0

0 0 0

-fe,

-L,,

-f,~,

(21)

7he

D1rac neutr1n0 ma55 15 n0w f0und t0 6e +f~,. 1t5 ma9net1c m 0 m e n t 15 5t111 91ven 6y e4. ( 15 ) w1th the rep1acement 0f F~ 6y F~,, and hence 6 0 u n d e d 6y e4. (17). 7 h e re5t 0f the tran51t10n m0ment5 are a9a1n 2er0.

6vm0m2,~

1 w15h t0 t h a n k Pr0fe550r L1nc01n W01fen5te1n f0r m a n y very he1pfu1 d15cu5510n5 and c0mment5. 7h15 w0rk wa5 5upp0rted 1n part 6y the U5 D e p a r t m e n t 0f Ener9y.

Reference5

[ 1] K.5. 8a6u and V.5. Mathur, Phy5. Len. 8 218 ( 1987 ) 218. [2 ] A. 2ee, Phy5. Len. 8 90 (1980) 389. [3] L.8.0kun, M.8. V0105h1nand M.1. Vy50t5ky,50v. J. Nuc1. Phy5, 44 (1986) 440; M.8. V0105h1nand M.1. Vy50t5ky, 50v. J. Nuc1. Phy5. 44 (1986) 544. [4] P.8. Pau1 and L. W01fen5te1n,Phy5. Rev. D 25 (1982) 766. [5] 5.P. Pete0v, Phy5. Lett. 8 115 (1982) 401. [6]L.8. 0kun, M.8. V0105h1n and M.1. Vy50t5ky, 17EP (M05c0w) prepr1nt 86-82 ( 1986); C.-5. L1m and W.J. Marc1an0, 7ana5h1 (Japan) prepr1nt 1N5-Rep.-645 (5eptem6er 1987). [7 ] L. W01fen5te1n,Nuc1. Phy5. 8 186 ( 1981 ) 147. [8] Ya.8.2e1d0v1ch, D0k1. Akad. Nauk. 555R 86 (1952) 505; E.J. K0n0p1n5kyand H. Mahm0und, Phy5. Rev. 92 (1953) 1045. [9] A.V. Kyu1dj1ev,Nuc1. Phy5. 8 243 (1984) 387; J. M0r9an, Phy5. Len. 8 102 ( 1981 ) 247; M.A.8.8e9, W.J. Marc1an0and M. Ruderman, Phy5. Rev. D 17 (1978) 1395. [ 10] 7.P. Chen9 and L,-F. L1, Phy5. Rev. D 22 (1980) 2860. [ 11 ] L. W01fen5te1n,Nuc1. Phy5. 8 175 (1980) 93.