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Low energy (0-250 MeV) pion-nucleon phase-shift fits Salomon, M. 1974

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TRIUMFLOW-ENERGY (0 - 250  MeV) PION-NUCLEON PHASE-SHI  FT F IT SM . SaIomonDepartment o f  P h y s i c s  U n i v e r s i t y  o f  B r i t i s h  ColumbiaMESON FAC IL IT Y  OF:UN IVERS ITY  OF ALBERTA  SIMON FRASER UN IVERS ITY  UN IVERS ITY  OF V ICTOR IA  UN IVERS ITY  OF B R IT I SH  COLUMBIATR I - ~ j k - 2TRIUMFLOW ENERGY (0 -250  MeV) PION-NUCLEON PHASE-SHI  FT F IT SM. SalomonDepartment o f  P h y s i c s  U n i v e r s i t y  o f  B r i t i s h  ColumbiaP o s t a l  a d d re s s :TRIUMFU n i v e r s i t y  o f  B r i t i s h  Columbia  Vancouver ,  B.C.Canada V6T 1W5 December 197^1 . INTRODUCTIONThe p i o n - n u c l e o n  i n t e r a c t i o n  has been s t u d i e d  e x p e r im e n t a l l y  ove r  more than 20 y e a r s  and a l t h o u g h  th e re  i s  a v a s t  amount o f  e xpe r imenta l  data a v a i l a b l e ,  the t h e o r e t i c a l  i n t e r p r e t a t i o n  i s  s t i l l  u n c e r t a i n .  T h e re f o r e ,  i t  i s  u s e f u l  to p a r am e t r i z e  the expe r imenta l  i n f o rm a t i o n ,  and a c o n ven ie n t  method to do t h i s  i s  to expand the s c a t t e r i n g  am p l i t u d e 1 o f  the i n t e r a c t i o n  in a p a r t i a l  wave s e r i e s .  The c o e f f i c i e n t s  o f  t h i s  e x p an s i o n  depend on phase  s h i f t s 1 wh ich  a re  ene rgy  dependent.  In  t h i s  work we f i t  a t h e o r e t i c a l  f u n c t i o n  to the expe r imenta l  v a l u e s  o f  the phase  s h i f t s  by means o f  a s t a nda rd  l e a s t  s q u a re s  f i t  code. T h i s  type  o f  f i t  i s  u s u a l l y  r e f e r r e d  to as  ' E n e r g y -D e p e nd e n t '  p h a s e - s h i f t  a n a l y s i s ,  and the re  i s  a v a s t  amount o f  l i t e r a t u r e  d ed i c a ted  to t h i s  type  o f  work.  Most  o f  the recen t  work can be found in  r e f e r e n c e s  2 f o r  low energy  and in r e f e r e n c e s  3 f o r  h i g h  energy  i n t e r a c t  i o n s .The main pu rpo se  o f  t h i s  p a r am e t r i z a t i o n  i s  to p r o v i d e  a s imp le  ana ­l y t i c a l  e x p r e s s i o n  o f  the s c a t t e r i n g  amp l i t u d e .  T h i s  then can be used in impulse  a pp r o x im a t i o n  c a l c u l a t i o n s  o f  complex n u c l e i ,  in  the e v a l u a t i o n  o f  app rox imate  p o t e n t i a l s  o f  the LLO i n t e r a c t i o n ,  and in  the c a l c u l a t i o n  o f  d i s ­p e r s i o n  i n t e g r a l s .Recent expe r iment s  a t  low e n e r g y 4 and new e v a l u a t i o n s  o f  phase s h i f t s  o f  p r e v i o u s l y  e x i s t i n g  d a t a 5 have improved the expe r imenta l  i n f o rm a t i o n  a t  low energy ,  i n d u c i n g  new i n t e r e s t  in t h i s  type  o f  work.2. GENERAL BACKGROUND2. 1 Phase  S h i f t  D e f i n i t i o nThe p i o n - n u c l e o n  s y s t em 1 i s  c h a r a c t e r i z e d  by an i s o s p i n  I  = 1/2 o r  3/2 and t o t a l  a n g u l a r  momentum J = 1/2, 3/2, . . .  e t c .  The d i f f e r e n t i a l  c r o s s - s e c t i o n  can be w r i t t e n  as( 1) (2a)(2b)do sc = |g(©) |2 + |h(e) |2whereandg (6) = I  U + l ) f £ + + P ^ c o s G )£h(e) = - I£f £ + "  f £ - |  P £[ ( c o s 0)-  2 -a re  the n o n s p i n - f l i p  and s p i n - f l i p  am p l i t u d e s .  The s i g n  ± i n d i c a t e s  J = Jl±l/2. These  terms must be s u b d i v i d e d  f u r t h e r  to a l l o w  f o r  the  two i s o ­s p i n  c h a n n e l s .  ThenL 2 I  21 2 i 6 o " S j  f 2 I  T£ ,2J  _ nj  e - 1? 7 11 q 2 i qwhere 6 i s  the phase  s h i f t ,  n the i n e l a s t i c i t y  and both a r e  a f u n c t i o n  o f  the ene rgy  o f  the LLWO system.  The q u a n t i t y  T i s  the p a r t i a l  wave s c a t t e r i n g  ampli  tude.2 .2  K in em a t i c sIn s t a n d a rd  e xpe r imen t s  the p ion  i s  the  p r o j e c t i l e  and the  nu c leon  the t a r g e t .  T h e r e f o r e ,  one o f  the  k i n em a t i c  pa ramete rs  used i s  the k i n e t i c  ene rgy  T^ o f  the  p i o n  in the lab.The two o t h e r  q u a n t i t i e s  u s u a l l y  used a r e  the t o t a l  ene rgy  in the  c e n t r e  o f  mass u = / s ’ ( a l s o  c a l l e d  i n v a r i a n t  mass o f  the  sy stem)  and the  momentum in  the c e n t r e  o f  mass q. In what f o l l o w s  we w i l l  n e g l e c t  the  mass s p l i t t i n g  in n u c le on s  and p i o n s ,  and take  a s  the nuc leon  mass M = 9 3 3 .3  MeV/c2 and the mass o f  the p ion  p = 139-6  MeV/c2 . Then the f o l l o w i n g  k i n em a t i c  r e l a ­t i o n s  a r e  used (c = 1):T =2 -  (M+p)22Ma) = y/p^ + M2 + 2M (T+p) = J q 2 + p 2 + / ^ j 2 + M2 q = J (p2-M2+gj2 ) 2 -  4a)2p 2 .2 .3  Some Expe r imenta l  P r o p e r t i e s  o f  the LLO I n t e r a c t i o nIn the low -ene rg y  r e g i o n  (T < 250 MeV) the LLO i n t e r a c t i o n  has s e v e r a l  f e a t u r e s  wh ich  s i m p l i f y  i t s  a n a l y s i s .  F i r s t ,  the s c a t t e r i n g  i s  p u r e l y  e l a s ­t i c ;  t h e r e f o r e ,  n in  Eq. (3) i s  u n i t y  (w i t h i n  1%).  F u r the rmore ,  o n l y  S -  and P-waves  c o n t r i b u t e  s i g n i f i c a n t l y  in the s e n se  t h a t  D-waves a r e  l e s s  than 0 . 5 %  o f  S -  and P-waves .T h e r e f o r e ,  (3) can be w r i t t e n  as-  3 -T = e ' ^ s i n f i  = c o s 6 s i n 6  + i s i n 26 (A)and the o n l y  s i x  phase  s h i f t s  c o n s i d e r e d  a re6s , r  6s , r  V r  6 - ENR G- ENR G- EY 21where the n o t a t i o n  c o r r e s p o n d s  to 6^ •Ano the r  important  p r o p e r t y  o f  the  LLO i n t e r a c t i o n  i s  t h a t  in a l l  chan ­n e l s  th e re  a r e  s t r o n g  re sonance s  a t  a r e l a t i v e l y  low ene rgy .  As the se  re sonance s  have l a r g e  w i d t h s ,  i t  i s  n e c e s s a r y  to i n c l u d e  them in  the  a n a l y t i c  d e s c r i p t i o n ,  even i f  they  a re  o u t s i d e  the r a n g e o f  ene r g y  f i t t e d  in t h i s  work.The p o s i t i o n  and w id th  o f  th e se  r e sonance s  a s  used in t h i s  work a re :T ab le  IName a j T U)j(MeV)r oi(MeV)S 11 0 1/2 1/2 1535 110S 31 0 1/2 3/2 1650 150p 3 1 l 1/2 3/2 1900 280P 1 3 l 3/2 1/2 1800 250P l l l 1/2 1/2 1470 250P 33 l 3/2 3/2 1231 112These  v a l u e s  were  o b t a i n e d  from s t a nda rd  Baryon  t a b l e s . 6 F u r t h e r  d e t a i l sc o n c e rn i n g  the v a l u e s  o f  to* and T _ : w i l l  be d i s c u s s e d  l a t e r  in S e c t i o n  6.J 1 0  13. ANALYTIC MODEL FOR THE ttN INTERACTIONI t  i s  c l e a r  from the  ttN c r o s s - s e c t i o n s  t h a t  the i n t e r a c t i o n  i s  a m i x tu re  o f  r e s o n an t  and n o n - r e so n a n t  am p l i t u d e s .  To e x p r e s s  such  a m i x t u r e  in a c o n s i s t e n t  way and a t  the same t ime s a t i s f y  the u n i t a r i t y  requ i rement  o f  the t o t a l  amp l i t u d e ,  one  s h ou ld  u se  the  S -m a t r i x  f o rm a l i sm .  We f o l l o w  here a d e r i v a t i o n  by D a l i t z , 7 ’ 8s = s rs b (5)4 -where S i s  the t o t a l  S -m a t r i x ,  r and b s tand  f o r  r e so nan t  and ' b a c k g r o u n d '  o r  n o n - r e s o n a n t ,  r e s p e c t i v e l y .  Both S f and S b a re  u n i t a r y .  Then the  r e l a ­t i o n  between S and the amp l i t u d e s  i sS = 1 + 2 i T = (1 + 2 i T r ) (1 + 2 i T b )from whichT = T b + T r S bi f  the background  i s  p u r e l y  e l a s t i c  S b = e2 i 6b ,  ft D andid i 2 iT = e 1'"' s i n d  = e ’^ 0 sin<5b + e " '  8 T r . (6 )A l t e r n a t i v e  ways to proceed can be found in r e f e r e n c e s  3*3•1 Resonant Amp l i t udeT h i s  p a r t  ha s the u sua l  B r e i t -W i g n e r  shape  w i t h  r e l a t i v i s t i c  m o d i f i c a ­t i o n  a s  d e s c r i b e d  by J a c k s o n . 9 [See a l s o  r e f e r e n c e  10] We then w r i t e  the r e so nan t  amp l i t u d e  asT r = e-  I(7)wherea l  so- IT/2andT e l a s t i cx =r = r,      	     - II       I	  (8)In t h i s  ca se  we found tha t  good a pp r o x im a t i o n .  Then( 025Me Do (w)1 and t h a t  in ou r  ene rgy  r e g i o n  t h i s  i s  a2 2 U)0-UJ qo 2Z+1(9)In the  above e x p r e s s i o n s  w0 i s  the re sonance  ene rgy ,  T0 the t o t a l  w id t h ,  and q0 the c e n t r e  o f  mass momentum c o r r e s p o n d i n g  to u)Q .3 -2  Non -Resonan t  o r  Background Amp l i t udeWe u se  the e f f e c t i v e  range  a p p r o x im a t i o n 1 to d e s c r i b e  the background  amp l i tude .  Thenq2£+1 c o t S b = —  + y -  q 2 + Cq4 = A +  Bq2 + Cq1* (10)where a i s  the s c a t t e r i n g  l e n g th  and r Q i s  the e f f e c t i v e  range.ANALYTIC EXPRESSION OF THE TOTAL AMPLITUDEFrom Eqs.  (6) and (7)o re2+ l  U 2+l.For v a l u e s  o f  e > 1, x < 1 and smal l  a n g l e s  6b (<20 deg) the above e x p r e s s i o n  can be approx imatedT h i s  a p p r o x im a t i o n  i s  b e t t e r  than 2% f o r  a l l  c a s e s  c o n s i d e r e d  here .  Then u s i n g  Eqs. ( 9 ) ,  (10) and (11)tan6 1 x r 0o)0qo^+1Y* (q) = T £ 7T= r Y* ( q ) = = T £7tN   A+Bq2+Cq4 u)2 -u)I t  i s  t h i s  f u n c t i o n  Y^ (q )  wh ich  has been f i t t e d  to the expe r imenta l  v a l u e s  o f  the phase  s h i f t s .  The f o u r  a d j u s t a b l e  pa ramete rs  a re  A, B, C and x.  The c o n s t a n t s  u Qr o and q 0 were taken  from Baryon  t a b l e s  a s  shown in Tab le  I  in S e c t i o n  2 .3 *The r a t i o  t a n 6 /q2^+1 in Eq. (12) i s  the same f u n c t i o n  used by M c K i n l e y 2 to f i t  expe r imenta l  phase  s h i f t s .  I t  has s e v e r a l  a d van ta ge s  o ve r  the use  o f  the phase s  th emse lve s .  F i r s t ,  i t  imposes the  c o n d i t i o n  t h a t  S-*0 f o r  q->-0. S e cond l y ,  i t  a l l ow s  the i n t r o d u c t i o n  o f  the  s c a t t e r i n g  l e n g th  aas  an a d d i t i o n a l  da ta  p o i n t .  Note t h a t  a = Y (0) . F i n a l l y ,  Y v a r i e s  smooth­l y  and a l l o w s  an ea sy  e s t im a te  o f  the range  o f  v a l i d i t y  o f  E q . ( 1 3 ) -  One impor tan t  d i s a d v a n t a g e  i s  t h a t  i t  d i v e r g e s  a t  a re sonance  ( f o r  q = q0 ) . How­eve r ,  t h i s  can be s o l v e d  e a s i l y  by f i t t i n g  Y " 1 wh ich  i s  we l l  behaved. T h i s( IDS(q-K)) = a q 2X,+1 (13)was done here  to  f i t  the P 33 phase  s h i f t s .Note t h a t  the q u a n t i t y  x r e f l e c t s  no t o n l y  the e l a s t i c  p a r t  o f  the re sonance  but  a l s o  the r e l a t i v e  phase  between the background  component and the re s o n an t  amp l i tude .5- LEAST SQUARES F ITT ING  PROCEDUREThe input  da ta  used were taken  from r e f e r e n c e s  h and 5» and from the'CERN T h e o r e t i c a l  phase  s h i f t s ' 10 f o r  T^ up to 300 MeV.The code used was w r i t t e n  in  the Los Alamos  S c i e n t i f i c  L a b o r a t o r y  and a l l owed  to f i t  any  a n a l y t i c a l  f u n c t i o n  w i t h  up to ten f r e e  pa ramete rs .  We used Eq. (12) f o r  f i t t i n g  f u n c t i o n  and the pa ramete rs  A, B, C and x c ou ld  v a r y  i n dependen t l y .In p r a c t i c e  we found i t  c o n v e n ie n t  to f i x  e i t h e r  C o r  x  and l e t  theo t h e r  th ree  v a r y .  (The rea son  f o r  t h i s  p ro cedu re  i s  t h a t  x and A, B and Ca re  s t r o n g l y  c o r r e l a t e d . )  Then B was f i x e d  and the r e s t  v a r i e d  and so  on u n t i l  a minimum in  the x 2 was found.  F i n a l l y ,  a l l  o f  them were a l l owed  to v a r y .  The program p r i n t e d  ou t  the number o f  i t e r a t i o n s ,  the  sum o f  s q ua re s  o f  the d e v i a t i o n s ,  the v a r i a n c e ,  the f i n a l  v a l u e  o f  the f i t t e d  pa ramete rs  w i t h  t h e i r  s t a nda rd  d e v i a t i o n ,  and the m a t r i x  o f  c o r r e l a t i o n s  between the f r e e  pa ramete rs .  The f i n a l  r e s u l t s  a r e  l i s t e d  in T ab le  I I .6. RESULTS AND COMPARISON WITH PREVIOUS DATAThe r e s u l t s  o f  the  f i t t e d  phase  s h i f t s  a r e  shown in F i g s .  1 and 2.The v a l u e s  o f  the f i t t e d  pa ramete rs  in  Eq. (12) a r e  l i s t e d  in  T ab le  I I  wh ich  a l s o  has the v a l u e s  o f  s c a t t e r i n g  l e n g th s  a s  d e f i n e d  in E q . (13 ) *In o rde r  to  e s t im a te  the q u a l i t y  o f  the  f u n c t i o n  used to f i t  the p o i n t s ,  we used the same data  w i t h  the f u n c t i o n  Y = (A + Bq2 + Cq1* + Dq6 ) -1 i n s te ad  o f  Eq. (12 ) .  Comparing the sum o f  s q u a r e s  o f  the  d e v i a t i o n s  f o r  the two e x p r e s s i o n s  th e re  was no s i g n i f i c a n t  improvement w i t h  the new f u n c t i o n ,and in  some c a s e s  ( S 3 3 , P n ,  P 33 ) a c l e a r  i n d i c a t i o n  t h a t  e x p r e s s i o n  (12)was b e t t e r .  These  a re  the c a s e s  in  wh ich  x i s  l a r g e .In some p r e v i o u s  a n a l y s i s 2 the P 33 was f i t t e d  w i t h  a pure  r e s o nan tamp l i t u d e  and th e re  was some d i f f i c u l t y  to rep roduce  the  c o r r e c t  shape  o f  there sonance  in the low momentum pa r t .  An a n a l y s i s  o f  t h i s  problem i s  d e t a i l e d-  6 -Table II-  7 -ShTft A (MeV/c)2Ul B (MeV/c)2^-1 C (MeV/c)2Jl_3 x a • y-<2A+l)Sn (8.098 ± 0.027) x 102 {S.kS ± 0. 25) x 10“3 (-1.42 ± 0.95) x lo-9 0.03 ± 0.02 0.174 ± 0.004S31 (-1.124 ± 0.008) x 103 (1.324 ± 0.041) x io-2 (-9-04 ± 0.44) x io-8 0.73 ± 0.05 -0.095 ± 0.004 P31 (-7.^11 ± 0.076) x io7 (-1.174 ± 0.036) x io3 (-9.4 ± 4.0) x io-9 0.06 ± 0.08 -0.037 ± 0.004P13 (-1.815 ± 0.023) x IO8 (-5.79 ± 0.33) x io3 (3.21 ± 0.30) x io-2 -0.51 ± 0.07 -0.016 ± 0.005Pn (-4.076 ± 0.017) x IO7 (-5.70 ± 0.27) x io2 (-5-72 ± 0.93) x IO-2 0.92 ± 0.02 -0.054 ± 0.008P33 (2.397 ± 0.077) x 107 (3-77 ± 0.25) x io2 (-2.04 ± 0.85) x 10“3 0.99 ± 0.02 0.205 ± 0.016in r e f e r e n ce  6. I t  seems from the se  r e s u l t s  t h a t  a m i x tu re  o f  r e sonan t  and background  amp l i t u d e s  o f  the form o f  Eq. (12) g i v e s  s a t i s f a c t o r y  r e s u l t s .The s c a t t e r i n g  l e n g t h s  as  d e f i n e d  in Eq. (13) a re  l i s t e d  in T ab le  I I .  Compar i son s  o f  the s c a t t e r i n g  l e n g th  f o r  Z=0 w i t h  recent  c a l c u l a t i o n s 11,12 u s i n g  f o rwa rd  d i s p e r s i o n  r e l a t i o n s  a re  l i s t e d  in T ab le  I I I .-  8 -Tab le  I I IOur R e s u l t s Bugg e t  a l . Samaranayake and Woo lcocka l 	0.17** ± 0 . 0 0 6  p_1 - 0 . 0 9 5  ± 0 .0 05  y " 10 .170  ± 0.00*4 y _1- 0 . 0 9 2  ± 0 .0 02  p ' 10.181  ± 0 . 0 0 8  y ” 1 - 0 . 0 8 9  ± 0 . 0 0 5  y _1These r e s u l t s  a g ree  we l l  w i t h i n  the s t a t e d  e r r o r s .Fo r  &=1 i t  i s  p o s s i b l e  to  e v a l u a t e  ( a g a in  u s i n g  f o rward  d i s p e r s i o n  r e l a t i o n s )  l i n e a r  c omb in a t i o n s  o f  the s c a t t e r i n g  l e n g t h s .  R e s u l t s  o b t a i n e d  by Bugg e t  a l . 11 and Samaranayake and Woo lcock13 a re  compared w i t h  o u r  r e s u l t s  in T ab le  IV .Tab le  I VBugg e t  a l . Samaranayake and Woo lcock Our R e s u l t s       	  N  	   N  		         	   	    		a H + 2 a 13+2 a 31+i4a 3 3    N   	   	   N  		- 0 . 5 7 0  ± 0 .004  y -3  0 .202  ± 0.00*4 p -30 .593  ± 0 .027  p-3  -0 .501  ± 0.0*45 P-3- 0 . 5 2 2  ± 0 .0 35   	 0.20*4 ± 0 .035   	 - ( ) ) -  2 - ( - 0 .  570  7 - ( . 0 0  2 - ( - 0 .  ±70A l t h o u gh  the  r e s u l t s  in  T ab le  I V  do not a g re e  v e r y  w e l l ,  the  e r r o r s  a r e  qui te l a r g e .I t  i s  a l s o  not c l e a r  what e f f e c t  the nuc leon  re sonance  has on the a ^  s c a t t e r i n g  l e n g th .CONCLUSIONSFrom the se  r e s u l t s  i t  i s  o b v i o u s  t h a t  in o r d e r  to  improve the knowledge  o f  the l ow -ene rg y  LLO i n t e r a c t i o n  i t  i s  n e c e s s a r y  to  make more mea­su rements  in  the 10 to  100 MeV re g i o n .  H o p e f u l l y ,  t h e se  measurements w i l l  be done in the nea r  f u t u r e  by the new meson f a c i l i t i e s  s t a r t i n g  o p e r a t i o n .I t  seems from the p r e sen t  data a v a i l a b l e  tha t  the a n a l y t i c  e x p r e s s i o n  (12) can be used s u c c e s s f u l l y  to pa rame t r i z e  the l ow -ene rgy  ttN i n t e r a c t i o n  (up to  250 MeV).I t  i s  a common p r a c t i c e  to approx imate  the l ow -ene rgy  ttN phase  s h i f t  by the Eq. (13 ) .  From t h i s  a n a l y s i s  one see s  t h a t  the a pp ro x im a t io n  i s  o n l ygood to  5% f o r  k i n e t i c  ene rgy  o f  l e s s  than 8 MeV.ACKNOWLEDGEMENTSI s h o u ld  l i k e  to  thank  Dr. J.M. M cM i l l a n  f o r  i l l u m i n a t i n g  d i s c u s s i o n sand G. Rowe f o r  v a l u a b l e  h e lp  in r un n in g  the LSF program.REFERENCES1. H. Mu i rhead ,  The P h y s ic s  o f  E lem en ta ry  P a r t i c l e s  (Pergamon, N .Y . ,  1968)2. L.D.  Roper,  R.M. W r i g h t  and B.T.  Fe ld ,  Phys .  Rev. B 1 38 , 190 (1965)J.M. M cK i n l e y ,  Rev. Mod. Phys .  35., 788 (1963)3. A.T.  D a v i e s ,  Nuc l.  Phys .  2 1 B , 359 (1970)B.H. B ransden  e t  a l .  , Phys .  Rev. B13 9 , 1566 (1965)P.W. C o u l t e r ,  Phys .  Rev. L e t t e r s  23_, 450 (1972)4. J .R .  C a r t e r ,  D.V. Bugg and A.A.  C a r t e r ,  Nuc l .  Phys .  58lB, 378 (1973)5. S. Almehed and C. L o ve la c e ,  CERN p r e p r i n t  TH 1408 (1971)6. T.A.  L a s i n s k i  e t  a l .  , Rev. Mod. Phys .  hS_, 525 (1973)7. R.H. D a l i t z ,  S t r a n g e  P a r t i c l e  Resonant S t a t e s  in Ann. Rev. Nuc l .  S c i .  1 3 ,339 (1963)8. See a l s o  Ch. 14 in S tr o n g ly  I n t e r a c t i n g  P a r t i c l e s  by R. Levi  S e t t i  and T.A.  L a s i n s k i  (Un iv .  C h ic a go  P r e s s ,  1973)9. J .D .  J a c k so n ,  Nuovo Cimento  3 4 , 1644 (1964)10. D.J.  Herndon, A. B a r b a r o - G a l t i e r i  and A.H. R o s e n fe l d ,  UCRL-20030 (1970)11. D.V. Bugg,  A.A.  C a r t e r  and J .R .  C a r t e r ,  Phys .  L e t t e r s  4 4 B , 278 (1973)12. V.K.  Samaranayake and W.S. Woo lcock,  Nuc l .  Phys .  4 8 B , 205 (1972)13- V .K.  Samaranayake and W.S. Woo lcock ,  Nuc l .  Phys .  49B , 128 (1972)01OO

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