Acharya, Anjana

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jnJinn J. Phys. 8 0  ( 4 ) ,  4 0 9 - 4 1  1 ( 2 0 0 6 ) oteComplex shape parameter for .v-wave scattering from zero-energy wave functionsA n jR n a A c h u ry aI^ c p a ilm c n t o f  Phys ics . 4 j iu v c is i iy  C 'o lloge tiC E in g in cc iin g , B u ila -7 6K  01 S am ba lpm .O nssa . Ind iaB -m a il |anjanaC«)K)ph ic s . inRet c iv r c l / y  iJt'f 200^, t ir i  cptci/  7 f ’ch n u u 'x  200ft\ l)s t ra c l . U s ing  an cxlc*nsion o l ih c  m ethod o l K erm ode and Van I>i]k. ( /V/v.v Rev <M2 1S91 (1990 ) ] \vc show  how  a c i>m p lc \ shape pa iam eter lot c(’>mplev lo ca l p o te n tia l can he d c te i m ined  in ic im s  o f  /e io -e n c ig y  wave Inn c tio ns  fo r  \ -wave sca ttenngKeyw o rds E lT e c liv c  la ng e  the o ry , shape p a ia in e te i. /e io -e n e ig y  wave fim e tio n sPV(’S Nos. 21 10 .-k .2 1  9 0  + rIt has re c e n tly  b e e n  s h o w n  th a t  th e  s h a p e  p a r a m e te r  ( P )  a n d  c o - d tic ie n ts  o f  h ig h e r  p o w e r s  o f  in  th e  e f fe c tiv e  ra n g e  e x p a n s io n  (P.RI: ) fu n e tk ^ n s  m a y  b e  o b t a in e d  fro m  th e  z e r o - e n e r g y  w a v e  tiinctions fo r  a re a l lo c a l p o te n tia l  w h ic h  s u p p o r ts  n o  m o re  th a n  one b o u n d  s ta tc [  I ]. T h e  u s e f u ln e s s  o f  th is  a p p r o a c h  lie s  in  th e  lact that c a lc u la tio n  o f f *  is n o t g e n e r a lly  s im p le  i f  th e  p o te n tia l  IS nt)t o f  a s im p le  f o rm  a n d  th e  s h a p e  p a r a m e te r  is s m a ll  in m agnitude.In th is  n o te , w e  e x te n d  th e  a p p r o a c h  o f  K e r m o d e  a n d  V an n i j k |2 | fo r  c o m p le x  lo c a l p o te n t ia ls  a n d  tr y  to  s e e  w h e th e r  th e  shape p a r a m e t e r  in  s u c h  a s i t u a t i o n  c a n  a ls o  b e  o b ta in e d  reasonably s im p ly  to  p e r m it  its  a p p lic a t io n  to  a n tip r o to n -p r o to n  scatterings. H e n c e , w e  h a v e  d e r iv e d  a n  e x p r e s s io n  f o r  c o m p le x  shape p a r a m e te r  in te r m s  o f  i n te g r a ls  o f  th e  z e r o - e n e r g y  w a v e  tu n clio n s.In v ie w  o f  th e  p r e s e n t  e x p e r im e n ta l  w o r k s  w ith  lo w  e n e r g y  iin tip ro to n s s c a t t e r i n g  o f f  p r o t o n s  b o t h  e l a s t i c a l l y  a n d  via ^annihilation c h a n n e l ,  th e  d e t e r m i n a t i o n  o f  e f f e c t i v e  r a n g e  param eters h a s  b e e n  o f  c o n s i d e r a b le  i n te r e s t  I 3 - 6 ] .  K ro ll  a n d  Schw eiger [ 7 |  h a v e  f o u n d  a  s e t  o f  c o m p le x  s q u a r e  w e ll p o te n tia l  param eters fo r  v a rio u s  p a rtia l  w a v e s  w h ic h  s e e m  to  fit th e  a n g u la r  d is trib u tio n s a n d  th e  p  p a r a m e t e r  r e a s o n a b l y  s a t is f a c to r i ly , fliey u se  d i f f e r e n t  r a n g e s  f o r  re a l  a n d  im a g in a r y  p a r ts  o f  th e  potential in  th e  prCvSence o f  c o u lo m b  i n te r a c t io n .W h ile  K ro ll  a n d  S c h w e i g e r  u s e d  d i f f e r e n t  r a n g e s  f o r  re al iind im a g in a ry  p a r ts  o f  th e  s q u a r e  w e ll  p o te n tia l  f o r  r e a lis t ic  d a tafittin g  , w e  h a v e  s im p lif ie d  th e  p ro b le m  by ta k in g  id en tica l ra n g e s  fo r  b o th , s in c e  at th is  s ta g e , o u r  in te n tio n  is o n ly  to  d e m o n s tr a te  th e  u s e f u ln e s s  o f  th is  m e th o d  a n d  n o  c la im  is m a d e  a b o u t th e  a c tu a l d e te r m in a tio n  o f  s h a p e  p a r a m e te r  fo r  a n tip ro to n -  p ix n o n  s c a tte r in g . W e w o u ld  n e e d  to  in c lu d e  c o u lo m b  m o d if ic a tio n  in th is  a p p r o a c h . A t th is  s ta g e  o f  a v a ila b le  in f o r m a tio n , it w ill b e  p r e m a tu r e  to  la y  a n y  c la im  to  th a t e ffe c t s in c e  e v e n  s c a tte r in g  le n g th s  a n d  e f f e c t i v e  r a n g e s  a r c  s till  to  b e  u n a m b ig u o u s ly  id e n tif ie d . In v ie w  t^f th is , fo r  d e m o n s tia tio n  p u r p o s e s , w e  h a v e  u s e d  p o te n tia l  p a r a m e te r s  w ith  id e n tic a l ra n g e s . W e th u s  s h o w  th a t K e r m o d e  a n d  V an  D i j k ’s a p p r o a c h  c a n  b e  e x te n d e d  to  c o m p le x  p o te n tia ls  a ls o  w ith  s o m e  m o d ilic a tio n s .Com plex shape param eiet :H e r e , w e  h a v e  ta k e n  a s p h e r ic a lly  s y m m e tr ic  lo c a l c o m p le x  p o te n tia l ( V (/) 4- iW {r ))  in s te a d  o f  a  real p o te n tia l. W e d e f in e  th e  re a l a n d  im a g in a ry  p a rts  o f  th e  .v-state w a v e  fu n c tio n s  a s  (////  ) + i a y ) )  f o r  e n e r g y  k - a n d  f o r  z e r o  e n e r g yre s p e c tiv e ly  a n d  a s s u m e  th e  s a m e  b o u n d a r y  c o n d itio n s  as g iv e n  b y  K e rm o d e  a n d  V an D ijk  fo r  b o th  re al a n d  im a g in a r y  p a r ts  s e p a ra te ly  i /c./, ( 0 )  =: = « ( 0 )  = / / „ / 0 )  = 0 ,  ( l a )u ,(R )  =  « „ ,(/ ? ) .« ,(/ ? )  =  « ,„ (/? ). ( l b )u ,(0 ) =  « „ / 0 )  =  1, « , ( 0 )  =  « o ,(0 )  =  0 ,  ( lc>© 2 (X X iIA C S410 A njana Acharyaw h e r e  R  is th e  d is ta n c e  b e y o n d  w h ic h  b o th  re a l a n d  im a g in a r y  p a r ts  o f  th e  p o te n tia l  a re  n e g lig ib le . A s  m e n tio n e d  a b o v e , w e  a re  a s s u m in g  f o r  c o n v e n ie n c e , th a t  R  is id e n tic a l  in b o th  c a s e s . D e f in in g  a c o m p le x  s h a p e  p a r a m e te r  P  =  P  +  iP  ^ in th e  s ta n d a r d  e f fe c tiv e  ra n g e  e x p a n s io n  fo r  th e  c o m p le x  p o te n tia l .w ithK  c o t {Sr +  iS,) =  "  \ /  {a,. + ia , ) + 1/2  + />o, )k  *- { P ,  + i P i ) { 'a ,  +//•(,, (2)t7^(r)= 1-|/-«,./ («; +«,?)} + r/(,r<:‘ /2 +  ^ k - r ^ a , l { f j { a ;  +  k ' l - - / l ,  ,//,.( r )  =  | r  < 7 ,/ ( « ;  +  a f  ) }  +  r r ^ i k - / 2  +  { ^ j b [ a f .  +  a ' )]w ith  ^ '\s  a s p h a s e  s h if ts , a ’s a s  s c a tte r in g  le n g th s  a n d  a s  e f fe c tiv e  ra n g e s . It c a n  b e  s h o w n  th a t  P  s a t is f ie s( P, + iP, )  (/•„,. +  ;/j,, y ' k - - k - * { ........ )  =  +  (•/<„, )■- (" r  + i" i ) («or + '"o;) -  (“0/ + iiiQi) '  + (“r + '“i ) (“o/- + '■''o,)] (if(3 )a n d  h e n c e  in  th e  lim it —> 0 ,w e  <.)btain e x p r e s s io n s  fo r  P^ a n dP  ^ in  te rm s  o f  z e r o - e n e r g y  w a v e  f u n c tio n s  a s  f o l lo w s( +  if", ) in)r +  i 'b , f  =  ''T l*  2 - .0  > / +  '« 0 , ) ’(“0/ + '“o<)-("()-- +'«<»/)' + (“i- + '» , ) (“()/•(4)H o w e v e r , th e  a b o v e  e x p r e s s io n s  s till  c o n ta in  ii^ , a n d  ii^ w h ic h  a rc  n o t z e r o - e n e r g y  o b je c ts . In  o r d e r  to  g e t r id  o f  th e m , w e  d o  a  T a y lo r  e x p a n s io n  in  k^ a n d  w rite  th e  w a v e  fu n c tio n s  in te rm s  o f  c o m p le x  o b je c ts  (3  ^ +//?, a n d  + Piu, (r) + iUj(r) = Jj + (r) + //?, (/))*  '  (r) + /«(,, (/•))] (5 a)(i7r(r) + /«;(r)) = |l + (/9^  +/?)A:-J(/7o,(r) + H7o,(/■))]. (5b)T h is  le a d s  to(P r+ iP ,)(ro ^+ ir(h f=  +  'A ) ( " O r  +  " 'O i ) "  “  (Ar +A ,)  ( “ o r  +  '" O l ) " ]  ( 6 )w h ic h  a f te r  s e p a r a t io n , g iv e s  th e  r e a l  a n d  im a g in a r y  v a lu e s  o f  th e  s h a p e  p a r a m e te r .However, (P^ + ) ^nd (P^ + /P ) are still to be determined.Determ ination o f  functions p { r )  and P { r )  *Since we have,{Jif (r) -f- iw/ (r ))  = sin(A:r) cot (S^ + iS i) + cos(^r)(7h,to  o r d e r  o f  k ~ , a n d  it  ^ \s  b e in g  th e  f o l lo w in g'■'0r('-) = ! - { ' •  ( I r / { c r  + «/‘  ) }  , (Ku«0. ('•) = {'■ ^'//(«," + o f  ) }  . (M.iU s in g  e q u a t io n s  ( 5 ) , ( 7 )  a n d  ( 8 ) ,  w e  o b ta inAr (f") =  '  [{ 3 (« f  -  of - '■  ) ('b , -  '■ ) + ( " ,  -  '■ ) + 3/- j j6 ( ( « , . - r ) ' , ,1).,,A, ('•) = '|{3/b, { ‘ i f  + d f  -  r  Hr ) + 2 r-a , -  3/- a,i„,  ^ !+ « , ' ) j  . (%iT h e s e  a p p e a r  to  b e  c o r r e c t  e x p r e s s i o n s  b e c a u s e  / / (/  r e d u c e s  to  K e r m o d e - V a n  D ijk 's  P ^ ir )  w h e n  im a g in a ry  part is s w itc h e d  o f f  w h ile  p j { r )  b e c o m e s  z e r o  u n d e r  th a t condition P r^ r)  a n d  P ,{ r )  c a n  n o w  b e  d e t e r m i n e d  b y  su b stiu itin y  ( )  fro m  e q  (."S) in to  th e  a p p r o p r ia te  S c h r d d in g e r  equationg iv in g{A r + i p  )" ( " O r  + '"0/ ) + 2(/?,. + i p  )'• («0r + <"(), )"+(• +(A - ("Or + '"o;) = 0. (lOlL e tt in g  > 0 ,  w e  g e t a  d if f e r e n tia l  e q u a tio n  fo r  c o m p le x  p '  a s("or + '«0/f (A- +'V5j)'= -(«or +'«0/)'r=  -  J  ("o» (-^) + '" o . .The constant of integration is taken to be zero convenience.Complex shape parameter for s-wave scattering from zero-energy wave functions 411f\pplication to complex sc/uarc-well potential o f identical expressed in terms of the zero energy wave function. This featureis especially useful in the determination of the complex shapei hc appropriate informations for a com plex square-well potential Parameter, which is calculated i n a straightforward manner byThe ot depthK-=V+ iW and range R are,K =  K + i K ^ ,K  = ( V + (  W ’ -t-Mihintegrating simple functions made up of the square of the zero- energy wave functions. We have thus shown that Kermode and Van Dijk method can be conveniently extended to complex local ptitentials as well and a complex shape parameter can be defined.H„, (f) =  Ar sinCA: /^') c o sh (A '/)-  4, co.s(A',r)sinh(A-,/) |I»oi ('■) =  cos(A^/-) sinh(A,r) -  /\, sin(A,.f)cosh(A,r) •!" O r (r )  =  I +« , ■ ) } ,(1, = /? - [/f, tan ( /f, /?) (l -  tanh' ( /f, /^)) + /f, tanh(K, R) ^ { \U im \K ,R ))y (K ;-^K ^ )( l  + \m h\K ,R )\;xn^K ,R )),a, =[Af, tan(/r,/?)(l-tanh-(A;',/?))-A:,tanh(A',/?)(I + tan-(A:,/?))y(/r,- -t- /:f )(l  +tanh-(/f,/?)tan-(A:,/?)).With the squarc-wcll potential for which R is taken to be the langeoflhe potential, we used R = 2.02 fm and K = 0.9343 fm”* and we found P = -0 .0 4 0  and P = -0 .0 0 6 . The finiteness to Pr  I IIS interesting because the imaginary part of the potential is very large. We show in Figure 1 the // - functions and we note that while the behaviour of and is very similar to that obtained for purely real case, Pj and P, show a distinctly diticrent behaviour including some oscillations. Further, their magnitude is an order less than P,. 's. Again, we have shown •hat the coefficients of the effective range expansion can all beFigure 1. The real and imaginaiy pans of the P  and /y functions foi the complex square well potentialReferences11 ] L S RcKlherg and R M Thaler hitrodm turn to ihv Quantum Uwory of SratU'nni> (New Yoik : Academic) (1%7)121 M W  Kermode and W Van Dijk /V/v.s. Kev C42 1S91 (IWO)[31 J Mahalanabic, H J Pirner and T A Shihata Nurl Pliys. A485 546(1988)[4] H J Pirnei, H Kervikov and J Mahalanabic Z Phys, A338 I I I(1991)[5] A V Lagu in Pliy\ir\ at U'ar with hnv Enci\{>v Anttproums (eds) ( ’ Amsler et al (New York . Hat wood) p25l (1988)(6| BO Karvikov Nurl. Phys. (Pror. Suppl) B8 273 (1989)[7] P Kroll and W Schweigei Niul Phys. A503 865 (1989)

Complex Shape Parameter for S-Wave Scattering from Zero-Energy Wave Functions

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Complex Shape Parameter for S-Wave Scattering from Zero-Energy Wave Functions

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