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UBC Theses and Dissertations

Coulomb corrections to low energy |N phase shifts Bankes, Stephen Alexander 1982

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COULOMB CORRECTIONS TO LOW ENERGY »rN PHASE SHIFTS by STEPHEN ALEXANDER BANKES B . S c , SIMON FRASER UNIVERSITY, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE / / i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE 1982 © Stephen Alexander Bankes, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood th a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) i i A b s t r a c t In t h i s t h e s i s a phenomenology i s given w i t h the view to determining the electromagnetic c o n t r i b u t i o n s to the low energy irN S-wave phase s h i f t s . The approach we take i s to p a r t i t i o n the Coulomb p o t e n t i a l i n t o a short and a long ranged part at some a r b i t r a r y c u t o f f r a d i u s . Subsequently we deduce expressions which all o w us to estimate, i n two d i s t i n c t stages, f i r s t the short and then the long range electromagnetic m o d i f i c a t i o n s to the purely hadronic p h y s i c a l observable q u a n t i t i e s . The a n a l y s i s f o r m a l l y begins with a d i s c u s s i o n of the s i n g l e channel n + p—> i r*p e l a s t i c s c a t t e r i n g process. We then turn our a t t e n t i o n to the two channel process n'p—»• n~p,ir°n. Although a more complete d i s c u s s i o n of ir'p low energy s c a t t e r i n g should n e c e s s a r i l y e n t a i l the i n c l u s i o n of the t h i r d channel i n e l a s t i c process i r'p—*Yn, we have chosen to ignore t h i s i n order that we may e l u c i d a t e some of the am b i g u i t i e s that have e x i s t e d with the two channel a n a l y s i s . E x p l i c i t c a l c u l a t i o n s are done, for the energy range 11.6 Mev < TJJ,, £ 44.0 Mev. For input we use energy dependent expressions f o r the hadronic phase s h i f t s . Although our estimates t o the Coulomb m o d i f i c a t i o n s t o these q u a n t i t i e s do d i s p l a y the same general f e a t u r e s with those found i n the l i t e r a t u r e , the numbers we get f o r the ir'p phase s h i f t s are con s i d e r a b l y s m a l l e r . This we a t t r i b u t e to our omission of the V n channel. The encouraging feature we do f i n d however i s that the procedure developed f o r t r e a t i n g both the one and the two channel problems are nearly i d e n t i c a l . The only d i f f e r e n c e i s that the l a t t e r n e c e s s i t a t e s the use of the u n i t a r i t y c o n s t r a i n t on the s c a t t e r i n g matrix. This we hope w i l l serve as a u s e f u l guide f o r a f u t u r e d i s c u s s i o n of the three channel problem. i v . Table of Contents Abstract i i Table of Contents i v L i s t of Tables v i Acknowledgements v i i In t r o d u c t i o n 1 Chapter 1. P r e l i m i n a r y M a t e r i a l ...8 1.1 Free F i e l d S o l u t i o n 10 1.2 The Short Ranged P o t e n t i a l 11 1.3 The Coulomb P o t e n t i a l 14 1.4 P a r t i a l Wave S o l u t i o n f o r Coulomb P o t e n t i a l 19 1.5 The Coulomb Mo d i f i e d P o t e n t i a l 21 Chapter 2. Coulomb C o r r e c t i o n s to Low Energy 7T+p Phase S h i f t s 27 2.1 E x t e r i o r C o r r e c t i o n s 31 2.2 I n t e r i o r C o r r e c t i o n s 35 V Chapter 3. Coulomb C o r r e c t i o n s to Low Energy Trp Phase S h i f t s 37 3.1 E x t e r i o r C o r r e c t i o n s 45 3.2 Coulomb M o d i f i e d S c a t t e r i n g Lengths 55 3.3 I n t e r i o r C o r r e c t i o n s 56 Chapter 4. Numerical Considerations 59 4.1 Coulomb M o d i f i c a t i o n s to 7T+p Phase S h i f t s 61 4.2 Coulomb M o d i f i c a t i o n s to Tf p Phase S h i f t s 66 References 79 Appendix A S c a t t e r i n g Amplitude f o r E l a s t i c C o l l i s i o n s 80 Appendix B Threshold Behaviour of P a r t i a l Wave S c a t t e r i n g Amplitude 84 Appendix C Coulomb Wave Functions.^ 87 Appendix D P o t e n t i a l due to Uniform Charge Density 90 Appendix E Isospin Conventions 92 v i L i s t of Tables Tables I and I I Co r r e c t i o n s to n*P S-wave Phase S h i f t s ..65 Table I I I Coulomb C o r r e c t i o n s to 7T p S-wave Phase S h i f t s 73 Table IV Phase S h i f t s C o r r e c t i o n s of Tromborg ... 74 v i i Acknowledgements I would l i k e to thank my s u p e r v i s o r , Dr.D.S.Beder, for h i s guidance and extensive input of ideas i n t o t h i s p r o j e c t , and h i s encouragement and i n t e r e s t throughout i t s developement. 1 INTRODUCTION Over the past few years, a renewal of i n t e r e s t has a r i s e n i n pion-nucleon s c a t t e r i n g , p a r t i c u l a r l y at low energies. The primary aim of t h i s study i s t o provide t h r e s h o l d parameters which can be used to t e s t c e r t a i n aspects of s t r o n g - i n t e r a c t i o n dynamics. One i n p a r t i c u l a r i s the i s o s p i n hypothesis, the v a l i d i t y of which has r e c e n t l y become a renewed subject of debate [12]. For s e v e r a l decades i t has been known that the experime n t a l l y observable low energy s c a t t e r i n g of charged pions from protons i s dominated by the four basic processes n*p —» n*p ir'p —* n'p ir'p •—» tr° n ir"p —» V n ( i ) ( i i ) ( i i i ) ( i v ) I t can be shown t h a t , i f time r e v e r s a l symmetry holds f o r electromagnetic i n t e r a c t i o n s , then the cross s e c t i o n s f o r r a d i a t i v e capture ( i v ) can be deduced. When t h i s i s e x t r a p o l a t e d to zero energy, i t i s p o s s i b l e t o r e l a t e the c r o s s s e c t i o n f o r charge exchange ( i i i ) at t h r e s h o l d v i a a (J'\ *Tr°n~) = z?R [ k v / k / f cr(Y[> -» ir*n) ( v ) . where fik, and hk v are the cm. momenta f o r the processes ( i i i ) and ( i v ) r e s p e c t i v e l y . This presupposes that the q u a n t i t i e s P = cr ( rr\> —i> 7r°n) /o~(7rf> - > Y r O ( v i ) 2 and R = <y ( V n -» jr-f)) / 0" (Vf) -h>7T +n) < v i i ) are w e l l known experimentally at t h r e s h o l d . Furthermore we know t h a t , i f i s o s p i n i s a good quantum number f o r the strong (hadronic) i n t e r a c t i o n , then i t i s a l s o p o s s i b l e to - deduce the charge exchange cross s e c t i o n s d i r e c t l y from a knowledge of the c r o s s s e c t i o n s f o r the e l a s t i c processes ( i ) and ( i i ) . A comparison of t h i s , at t h r e s h o l d , w i t h equation (v) should thus provide us with an a d d i t i o n a l t e s t i n g ground f o r the i s o s p i n hypothesis. U n t i l r e c e n t l y , t h i s study has been confined almost e n t i r e l y to higher energy c o l l i s i o n s [12]. One reason for t h i s i s due t o the r e l a t i v e l y short l i f e t i m e s of the i n c i d e n t p a r t i c l e s . At low energies the i n c i d e n t p a r t i c l e beam i s s i g n i f i c a n t l y contaminated with i t s decay products. This d i f f i c u l t y has been l a r g e l y a l l e v i a t e d , at Triumf at l e a s t , by u t i l i z i n g s h o r t e r pion beams. I t i s hoped that s u c c e s s f u l experiments w i l l soon be c a r r i e d out there with pion l a b o r a t o r y k i n e t i c energies as low as 20 Mev. The other long standing problem of o b t a i n i n g more accurate measurements of the photons from the r a d i a t i v e capture process and those from the decay v" —»\Y has been overcome by the recent use, again at Triumf, of high energy r e s o l u t i o n Nal c r y s t a l s . Hence the renewed i n t e r e s t i n low energy nN s c a t t e r i n g . A fundamental i s s u e connected with the four processes mentioned above, i s the question of to what extent d i f f e r e n t f o r c e s c o n t r i b u t e to the s c a t t e r i n g process. I t i s b e l i e v e d that 3 only the hadronic and electromagnetic forces are releva n t here. Hence i t i s n a t u r a l to ask, to what degree w i l l the hadronic phase s h i f t s or s c a t t e r i n g amplitudes be modified when the Coulomb e f f e c t s are taken i n t o account. In attempting to provide an answer to t h i s query, two basic phenomenological approaches have been taken. One i s to analyse the irN i n t e r a c t i o n using f i e l d theory and subsequently e x t r a p o l a t e the r e s u l t s to th r e s h o l d v i a d i s p e r s i o n theory techniques [8,9]. This approach i s the most fundamental one known. I t s p r i n c i p a l advantage i s to provide c l u e s to the p o s s i b l e mechanisms i n v o l v e d . However, because the motive f o r t h i s t h e s i s i s to determine what the Coulomb m o d i f i c a t i o n s to the supposedly known hadronic p h y s i c a l observables are, t h i s approach seems to overextend the scope of the problem. A n a t u r a l a l t e r n a t i v e , which we s h a l l take, i s to u t i l i z e o r dinary quantum mechanics. This approach i s the more common one taken f o r studying low energy irN s c a t t e r i n g . U n f o r t u n a t e l y , there does not appear to be any c l e a r l y d e f i n e d quantum mechanical p r e s c r i p t i o n for d e a l i n g with the r a d i a t i v e capture process. The most s u c c e s s f u l attempt, to our knowledge, to d e l i v e r a systematic, quantum mechanical treatment of the above four processes i s that given by Rasche and Woolcock [4,5,6]. What they do i s e s s e n t i a l l y d i v i d e the problem i n t o three p a r t s . They begin by p r o v i d i n g an e f f e c t i v e range expansion f o r the phase s h i f t s for the s i n g l e channel process n + p—*it*p. This i s re-presented, i n our own n o t a t i o n , i n chapter four eqn(12). Next, they a r r i v e at an analogous expression, see eqn(4.!3), for 4 the two channel process ir'p ir'p , ir°n. F i n a l l y , they expand t h i s e f f e c t i v e range expression from a 2x2 matrix equation to a . 3x3 matrix equation to encompass the r a d i a t i v e capture process i r"p-»Yn. From t h i s they manage to show, by a n a l y t i c c o n t i n u a t i o n of the s c a t t e r i n g amplitude below the ir'p t h r e s h o l d , how to obtain a parametric expression f o r the Panofsky r a t i o , d efined by e q n ( v i ) . As input f o r t h e i r numerical work f o r both the two and three channel problem, they use the charge independent (hadronic) s c a t t e r i n g lengths a* and a i . These they take from the two channel a n a l y s i s of Samaranayake and Woolcock [10,11], I t seems that the b a s i s of t h e i r work r e s t s l a r g e l y on t h e i r treatment of the two channel problem [ 5 ] , Although they do appear to attempt a complete p r e s e n t a t i o n , there are s e v e r a l t h i n g s which are l a c k i n g . F i r s t and foremost i s the lack of c l a r i t y throughout. For example, they are not too e x p l i c i t i n s t a t i n g how one obtains the f u l l y Coulomb modified s c a t t e r i n g amplitudes from the purely hadronic ones. Furthermore t h e i r method of c a l c u l a t i n g the Coulomb m o d i f i c a t i o n s to the t h r e s h o l d parameters seems not only very cumbersome but a l s o obscure. In the pages which f o l l o w , we w i l l attempt to c l a r i f y some of these i s s u e s . Rather than using the energy independent hadronic s c a t t e r i n g lengths as input which Rasche and Woolcock do, we w i l l s t a r t with low energy expressions f o r the hadronic phase s h i f t s . With these we w i l l show how t o o b t a i n , i n two d i s t i n c t steps, t h e i r f u l l Coulomb m o d i f i c a t i o n s . This i s accomplished by assuming that the Coulomb p o t e n t i a l can be p a r t i t i o n e d i n t o a short and a long ranged p a r t . Subsequently i t b i s shown how one can obtain the short ranged Coulomb m o d i f i c a t i o n s to the hadronic phase s h i f t s . With these l a t t e r q u a n t i t i e s , i t i s then shown how to c a l c u l a t e the long ranged m o d i f i c a t i o n s . As i s mentioned i n chapter two, because our primary concern i s w i t h very low energy »rN c o l l i o n s , we w i l l c o n f i n e most of our a t t e n t i o n to S-waves only. Because i t i s n a t u r a l to b e l i e v e that the three channel problem i s merely an extension of the two channel one, we w i l l omit a d i s c u s s i o n of the r a d i a t i v e capture process. Hence there w i l l be expected e r r o r s i n our f i n a l r e s u l t s . Although the magnitudes of these e r r o r s are d i f f i c u l t to a s c e r t a i n , i t i s b e l i e v e d by some [7] that they cannot be any worse than 30%, which i s a t h r e s h o l d estimate. In chapter one, we s t a r t with a p r e l i m i n a r y d i s c u s s i o n of the p a r t i a l wave s o l u t i o n s to the Schroedinger equation. The s c a t t e r i n g amplitudes are then examined for the v a r i o u s s c a l a r p o t e n t i a l s of i n t e r e s t . This i s followed i n the next chapter with a d i s c u s s i o n of the ir*p problem. A model p o t e n t i a l , r e p r e s e n t a t i v e of the ' p h y s i c a l s i t u a t i o n ' , i s then discussed. Using the m a t e r i a l developed i n chapter one, we then proceed to show how one can o b t a i n the Coulomb m o d i f i c a t i o n s t o the various energy dependent q u a n t i t i e s such as the s c a t t e r i n g amplitudes and t h e i r corresponding phase s h i f t s . This procedure i s c a r r i e d out i n two stages. F i r s t we show how to obtain the long ranged Coulomb modified phase s h i f t s from the short ranged Coulomb modified phase s h i f t s . Subsequently, we show how these l a t t e r q u a n t i t i e s can be obtained from the purely hadronic phase 6 s h i f t s . In chapter three, we examine the two channel problem i r'p— • i r'p, ir°n. There i t i s shown that the Coulomb m o d i f i c a t i o n s to the hadronic q u a n t i t i e s can a l s o be obtained i n an almost i d e n t i c a l f a s h i o n . The only d i f f e r e n c e i s that because the s c a t t e r i n g amplitude i n now a matrix, we are forced to u t i l i z e the u n i t a r i t y c o n d i t i o n of the s c a t t e r i n g m a t r i x . Included a l s o i s a separate d i s c u s s i o n of the Coulomb modified s c a t t e r i n g lengths f o r the two channel problem. This turns out to be simply an extension of the ideas presented i n chapter two for s i n g l e channel s c a t t e r i n g . The f i n a l chapter i s devoted p r i m a r i l y to numerical c a l c u l a t i o n s . As i s mentioned above, we are using as input, a n a l y t i c expressions f o r the low energy hadronic phase s h i f t s . These were taken from M. Salomon e t . a l . [ 7 ] who obtained them by performing a l e a s t square f i t t i n g of parametric expressions with rrN data for T i ^ < 400 Mev. The f u l l Coulomb m o d i f i c a t i o n s to these phase s h i f t s were c a l c u l a t e d by assuming that the hadronic p o t e n t i a l s were s p a t i a l l y constant. Tabulated r e s u l t s are presented f o r the energy range 50 Mev < 'hck < 100 Mev which i s roughly equivalent to 11.6 Mev < Tt"6 < 44.0 Mev. Although these r e s u l t s do e x h i b i t the same basic features as those obtained v i a the d i s p e r s i o n techniques of Tromborg e t . a l . [ 8 ] , the m o d i f i c a t i o n s obtained f o r the two channel case are somewhat s m a l l e r . This perhaps can be a t t r i b u t e d to our negligence of the N-n channel. We conclude the chapter with a d i s c u s s i o n of the two channel treatment of Rasche and Woolcock. In p a r t i c u l a r we show 7 how to de r i v e the e f f e c t i v e range expansion f o r the two channel problem, which i s what they expand upon to incorporate the r a d i a t i v e capture process. 8 I PRELIMINARY MATERIAL An appropiate s t a r t i n g p o i n t f o r the study of the s c a t t e r i n g of two s p i n l e s s p a r t i c l e s v i a a s c a l a r p o t e n t i a l would be with the Klein-Gordon equation. However because our i n t e r e s t i s p r i m a r i l y with low energies, we are j u s t i f i e d i n beginning with i t s low energy counterpart, namely the Schroedinger equation. In the center of momentum frame, we have f o r energy e i g e n s t a t e s [ v l * k i - ( J e n ] ^ ( n = ° where and where and (1) i It = -J 27TJ E m = reduced mass of two body system E = cm energy of each p a r t i c l e V(0= i n t e r a c t i o n p o t e n t i a l r = space vector separating the two p a r t i c l e s . We are supposing that the p r o j e c t i l e o r i g i n a t e s at an i n f i n i t e d i s t a n c e along the negative z - a x i s and that the target i s l o c a t e d at the o r i g i n . \ y The general s o l u t i o n t o eqn(1) f o r any s c a l a r p o t e n t i a l can be expressed i n s p h e r i c a l c o o r d i n a t e s as J / - ( r , f i , r t - 1 t R.M SmCe) ( 2 ) where r . ( X* .+ y 1 • * ] 0 = arctan y/x 0 = arccos z / r The f u n c t i o n s $ u , 6L„ and must s a t i s f y ( r - » 3 , ( r - ^ • ( > - 0(0 - - o ( 3; + M1 } $„(*) « o The s o l u t i o n s to the l a t t e r two equations are the ass o c i a t e d Legendre polynomials, P l M(cos 6 ) ,and e l M * r e s p e c t i v e l y . I f we now assume that our problem e x h i b i t s azimuthal symmetry with respect to the z - a x i s , then i t f o l l o w s that M = 0, hence 1 and consequently eqn(2) becomes the p a r t i a l wave expansion 4 ( r , G ) = 11 RJIcO R (cose") which may be r e w r i t t e n as where f> - kr 10 (3) a L= some f u n c t i o n of L cos & = k • r k = z = z/z = d i r e c t i o n of incidence and TjIlCp") - PRL = which i s the s o l u t i o n to • [ k* - U C O - - t ^ l l ] ) = o Note th a t the requirement that our theory be normalizable makes i t necessary that R L(kr) be f i n i t e over a l l space and consequently y L ( r = o ) = o s (4) For t h i s reason i t i s common to r e f e r t o i t as the re g u l a r s o l u t i o n . The object of t h i s chapter w i l l be t o determine the p r e c i s e form of u u ( ? ) f o r the var i o u s p o t e n t i a l s considered. 1) Free F i e l d S o l u t i o n I f V = 0 everywhere, then the s o l u t i o n s to eqn(3) are the reg u l a r and i r r e g u l a r f u n c t i o n s JL(P) and rUCO , defined r e s p e c t i v e l y by 11 (5) The l a r g e and small argument approximations to these f u n c t i o n s a r C f sin ( p - , p » | and f COS ( P - ) f p » 1 The f u n c t i o n ) i s termed i r r e g u l a r because i t has a s i n g u l a r i t y at r = 0. In l i g h t of t h i s and eqn(4), i t f o l l o w s that only Jl(P) i s a p p l i c a b l e here. Consequently our s o l u t i o n w i l l look l i k e L - O The q u a n t i t y a u i s determined by observing that the corresponding momentum e i g e n s o l u t i o n may a l s o be w r i t t e n as iMM) = IW L C^+'V" J.Lp) V. (cose) Comparison of eqns(7) and (8) r e v e a l s that aL = AA L i i + ') iL ( 9 ) 2) The Short Ranged P o t e n t i a l Suppose we have a s c a l a r p o t e n t i a l which vanishes beyond (8) 12 some di s t a n c e r = R, that i s f V , . (0 , r i R Y ( 0 r > R Because the i n t e r i o r s o l u t i o n to the corresponding r a d i a l equation depends e x p l i c i t l y on the behaviour of V fo r r ^  R, we w i l l ignore t h i s f o r the present and consider only the e x t e r i o r s o l u t i o n . Since t h i s must be a l i n e a r combination of JLC?) an<3 PlC^O defined above, we may w r i t e U L ( r > d) = ft A ILP) + knL hurt _ The q u a n t i t y t a n & i , which i s c h a r a c t e r i s t i c of the p o t e n t i a l , i s a f u n c t i o n of k,L and R. Hence from the manner i n which i t i s presented we see that f o r f i n i t e Vs* l i m tan6 L= 0 (10) R->o To determine what At i s , we observe that f o r any f i n i t e ranged s p h e r i c a l l y symmetric p o t e n t i a l , the wave f u n c t i o n 0 ( r , e ) Iflt P" [ i W + tanLndP)} R lease) L-0 should a s y m p t o t i c a l l y look l i k e a s u p e r p o s i t i o n of a f r e e plane wave e'k'c with a s p h e r i c a l outgoing wave -jr e.'1"- . This means that the asymptotic form f o r the above w i l l be where ¥(k,0)is some f u n c t i o n , c a l l e d the s c a t t e r i n g amplitude, which i s a measure of the extent to which s c a t t e r i n g occurs. 1 3 (11) Hence i f we w r i t e and then make use of eqns(5,6 and 8 ) , i t f o l l o w s that and The q u a n t i t y fj.CL') , which w i l l be of cons i d e r a b l e i n t e r e s t i n l a t e r chapters, i s termed the p a r t i a l wave s c a t t e r i n g amplitude. As a po i n t of i n t e r e s t i t should be noted that eqns(9,l0 and 11) imply that lino ¥ L - ° Consequently, a f t e r some rearrangement our e x t e r i o r s o l u t i o n f o r the short ranged p o t e n t i a l w i l l look l i k e where Note that r • for the fre e f i e l d and 1 4 f o r the short ranged p o t e n t i a l . Comparison of t h e i r r e s p e c t i v e asymptotic forms r e v e a l why the q u a n t i t y 6lC0 i s termed the "phase s h i f t " . In the chapters which f o l l o w we s h a l l i n v e s t i g a t e the behaviour of these phase s h i f t s f o r s e v e r a l d i f f e r e n t p o t e n t i a l s . 3) The Coulomb P o t e n t i a l The r e s u l t s of the past two s e c t i o n s have i n d i c a t e d that although i t i s s u f f i c i e n t to w r i t e the p a r t i a l wave expansion /lA Z- + ') z.L p"x Ui(p) R (cose) as a p o s s i b l e s o l u t i o n t o the Schroedinger equation, i t i s necessary to compare i t w i t h the asymptotic form of the f u l l s o l u t i o n i n order that the p r e c i s e form of the f u n c t i o n UL(P) be known. Hence, before d i s c u s s i n g the p a r t i a l wave s o l u t i o n s f o r the purely p o i n t l i k e Coulomb p o t e n t i a l ( e 2 / r ) , we w i l l f i r s t o u t l i n e a d e r i v a t i o n f o r i t s f u l l s o l u t i o n . P a r t i c u l a r d e t a i l s can be found i n most standard quantum mechanics t e x t s , some of which are l i s t e d i n the refe r e n c e s . S u b s t i t u t i o n of V = e 2 / r i n t o eqn(1) gives where and - yr\C <* /u * - e V h c Rather than using s p h e r i c a l c o o r d i n a t e s , which i s the standard p r a c t i s e when d e a l i n g w i t h the bound s t a t e problem 1 5 p a r a b o l i c coordinates w i l l be used i n order to f a c i l i t a t e the c o n s t r u c t i o n of an asymptotic form which w i l l be made to look l i k e the usual s u p e r p o s i t i o n of a f r e e plane wave with a s p h e r i c a l outgoing wave as c l o s e l y as p o s s i b l e . The wave equation i n p a r a b o l i c coordinates ( S , 3i ^) where 5 - r + 0 = arctart y / x and g i v e s , when we set Y m = /(^ 3(3) the three equations ' 3 ,(33,) + ( i ^ - ? 5 M l 4 / 5 ' ^ l 9 ( 3 ) ° where M and |S are separation constants. Because we are d e a l i n g w i t h a c e n t r a l f o r c e problem which i s symmetric w i t h respect t o the azimuth <j> , we set M = 0 which i m p l i e s that 1. This leaves us w i t h where / and ^ s a t i s f y the above w i t h M = 0. Since we would l i k e the i n c i d e n t wave which o r i g i n a t e s f a r 16 away along the negative z-axis to resemble a fre e plane wave, we w i l l impose the c o n d i t i o n 4 s This suggests that we set and , ; t where w ( 3 ) e 1 3 l i m w(j) = a f i n i t e constant (12) X -oo When these are s u b s t i t u t e d i n t o t h e i r r e s p e c t i v e equations with M = 0 we get jS = k and A change of v a r i a b l e x = i t j transforms the l a t t e r equation i n t o which i s the confluent hypergeometric equation, whose s o l u t i o n s are the regular and i r r e g u l a r c onfluent f u n c t i o n s denoted r e s p e c t i v e l y by M(-i^ r i , i t j ) and U(-^ , i , ^ 3 ). 1 1 / More g e n e r a l l y , the s o l u t i o n to i s M ( Q,b, x ) ( b - x ) c^ x " * ] V J ( x } - o = 2_ (a-O! ( b + M-0! N ' (13) N-o which converges for a l l x provided b * negative i n t e g e r , and Further treatments of these f u n c t i o n s can be found i n references 2 and 3. Although the general s o l u t i o n to eqn(13) i s we must set B = 0 to ensure t h a t the c o n d i t i o n (12) be met. Since ckj s i k ( r - i ) = 2\ p sinxQ/z where Q i s the s c a t t e r i n g angle i n the center on mass frame, the f u l l wave f u n c t i o n f o r the mutual s c a t t e r i n g v i a a p o i n t l i k e Coulomb p o t e n t i a l can be w r i t t e n as lp(r,Q) = A elk* ,1 , *ipsM*e/z) (14) I f we set . e2*i - i j where Nuis some no r m a l i z a t i o n constant, then the asymptotic form to eqn(14) w i l l be r (15) -A 1 o where and ( 1 6 ) Go = a r P ( I + i i ) For l a r g e energies, 1«\ , we have SORN where "I 2 k sin* 9/2 i s the f a m i l i a r Rutherford s c a t t e r i n g amplitude discussed i n appendix A. The above expression f o r i / a l s o lends i t s e l f to an immediate p h y s i c a l i n t e r p r e t a t i o n , which agrees w e l l w i t h what we would i n t u i t i v e l y expect. To see t h i s , observe that * ZTT t\ | ^(o,e)| 2 = \(\\ = I Ait I which i s e s s e n t i a l l y the p r o b a b i l i t y of observing the p r o j e c t i l e i n the near v i c i n i t y of the t a r g e t . This means that f o r very low energy s c a t t e r i n g from a r e p u l s i v e f i e l d 1 •m c pt » i and hence implying a very small l i k e l y h o o d of observing the two p a r t i c l e s c l o s e together. 19 On the other hand, fo r very low energy s c a t t e r i n g from an a t t r a c t i v e f i e l d we have | ^ ( o ^ | 2 ~ ATTlr^l » 1 i n d i c a t i n g a higher p r o b a b i l i t y of the p r o j e c t i l e and target c o e x i s t i n g c l o s e together. 4) P a r t i a l Wave S o l u t i o n f o r Coulomb P o t e n t i a l Had we w r i t t e n the s o l u t i o n to as the p a r t i a l wave expansion r l = o we would f i n d that u J O must s a t i s f y and hence where F L and $ L are commonly known as the regular and i r r e g u l a r Coulomb f u n c t i o n s r e s p e c t i v e l y . I f the choice of norm a l i z a t i o n f o r these two fu n c t i o n s i s made to ensure that \\m[kh.P) ) ^(T'Oj = [jL(0;n,.(^] (1B) e ->o and U ( Kh,<>) •> qJn.")} = {*«*• ' } ( i 9 ) (20) (21 ) where ft = p - *r ~ i h * p + °"-and Oi = ar<$ T ( L + I + / ") ) then i t can be shown that where the confluent f u n c t i o n M i s des c r i b e d above and Cdn) - I r d o ^ D l The i r r e g u l a r f u n c t i o n w i l l have the form 4. - - i - c:ci) p-% Ah.') ZL + 1 where the functions q L(*7), p t (7) and given e x p l i c i t l y i n reference 1, are uniquely determined from e q n s ( l 8 and 19) and the a d d i t i o n a l c o n s t r a i n t £Jl ^P Fl ~ R. ^ p £Jl = 1 where 9, - a / a a o The presence of the l o g a r i t h m i c s i n g u l a r i t y i n C^L at r = 0 re q u i r e s that B =0 i n eqn(!7) and consequently the p a r t i a l wave 2.1 s o l u t i o n f o r the,Coulomb p o t e n t i a l w i l l look l i k e cose) When t h i s i s compared wi t h the equivalent s o l u t i o n which was d e r i v e d i n the previous s e c t i o n , i t i s s t r a i g h t f o r w a r d t o show v i a eqn ( 2 l ) that where ai i s given by e q n ( 2 0 ) . Thus we have shown that the s o l u t i o n to the Schroedinger equation for the p o i n t l i k e Coulomb p o t e n t i a l may be w r i t t e n as where Vu(e) -- e ' 8 1 Kin,'') ( 2 2 ) 5) The Coulomb Modi f i e d P o t e n t i a l The object of t h i s study i s to determine what e f f e c t electromagnetism has on low energy irN phase s h i f t s . Hence i t i s e s s e n t i a l that we consider some p o t e n t i a l which most adequately d e s c r i b e s the p h y s i c a l s i t u a t i o n . Lets begin by assuming that t h i s p o t e n t i a l w i l l c o n s i s t of two p a r t s , namely a long ranged part due to the e 2 / r p o t e n t i a l and a short ranged part due t o the strong p o t e n t i a l modified by short ranged electromagnetic e f f e c t s . I f we d e f i n e R as the r e l a t i v e d i s t a n c e between the two p a r t i c l e s beyond which the p o t e n t i a l becomes the s t r i c t l y p o i n t l i k e Coulomb p o t e n t i a l , then 22 we may w r i t e our p h y s i c a l p o t e n t i a l as V M C (0 r where the s u b s c r i p t s "MC" and "SR" r e f e r to "modified Coulomb" and "short ranged" r e s p e c t i v e l y . As i n s e c t i o n 2 we w i l l concern ourselves f o r the present only with the s o l u t i o n f o r r £ R. Now we know from the previous s e c t i o n that the e x t e r i o r s o l u t i o n to t h i s problem w i l l look l i k e l- 0 We a l s o kmow that i n the R = 0 l i m i t , the p o t e n t i a l becomes the pure p o i n t l i k e Coulomb p o t e n t i a l , and so we have from eqn(22) l i m a u = e ; <^ and l i m bt = 0 This suggests that we set b L= a u t a n VL where U. = U(*> i s what we w i l l c a l l the phase s h i f t f o r V M C . So f a r , a l l we know about VL i s that i t i s c h a r a c t e r i z e d by the behaviour l i m t a n v t = 0 R-> o (24) To determine what the c o e f f i c i e n t a L i s , we impose the c o n d i t i o n that the asymptotic form of our s o l u t i o n equal a su p e r p o s i t i o n of a reg u l a r Coulomb wave </'^ £g. with a d i s t o r t e d s p h e r i c a l s c a t t e r e d wave \J,st . Hence by making use of the asymptotic forms of F t and (\L(R),P) we have 0(r-M»,0) - /OkLC^1-**) i L p ' i - ~ ^ : *'"te + l>0 PLCCOSO) (25) From the l a s t s e c t i o n we know that the regular Coulomb wave i s which has the asymptotic form - /Oktj>L*t)iLf'"^rink Z(coso) ( 2 6 ) where Furthermore, i n analogy w i t h eqn(l5) we have J,lr—) - / U Z L ^ i + O f l ' K W - — j : Hence by demanding that eqn(25) equal the sum of eqns(26) and (27), we ob t a i n a f t e r rearranging and equating the c o e f f i c i e n t s of s i n P and cos P , the r e s u l t s d L = L - sin K exl^ (27) (28) 24 This i m p l i e s that the sum of eqns(26) and (27) i s merely the asymptotic form of oo - M ) •" ^  L-J-2L*'}P" U"C(P) R t e e ) (29) where (30) and _ e i ( o i * ^ 0 pt + sin 14. £ L The q u a n t i t y IS the p a r t i a l wave s c a t t e r i n g amplitude for the modified Coulomb p o t e n t i a l . Observe that eqns(23,24 and 28) imply that l i m VL = 0 (31 ) I t should be emphasized that the asymptotic form for eqn(29) can a l s o be expressed as (32) where ^ ( l e . ^ i s t n e s c a t t e r i n g amplitude for the purely p o i n t l i k e Coulomb p o t e n t i a l given by eqn(16). This completes our i n t r o d u c t o r y p r e s e n t a t i o n upon which much of t h i s paper w i l l depend. However before coming to a 25 c l o s e , a b r i e f summary of the r e l e v a n t r e s u l t s obtained w i l l be given. I t has been shown above that the s o l u t i o n to the Schroedinger equation f o r any s c a l a r p o t e n t i a l with azimuthal symmetry wi t h respect to the d i r e c t i o n of incidence may be w r i t t e n as a p a r t i a l wave expansion where the constant N tdepends on our choice of n o r m a l i z a t i o n . The regular s o l u t i o n s U L(P) are l i s t e d below for the p o t e n t i a l s considered. OO + 0 iLp-*UL(p) ?L fase) 1) V = 0, f o r a l l r 2) V = where and l i m 6L = 0 3) V = e 2 / r , -Tor all r ; UdP) = £l°L Euh.P^ 4) V„c= VL where and l i m V L = 0 I I COULOMB CORRECTIONS FOR LOW ENERGY w*D PHASE SHIFTS In t h i s chapter a procedure i s given which w i l l enable us to determine, to lowest order i n the Coulomb p o t e n t i a l , the electromagnetic c o n t r i b u t i o n to the t o t a l or " p h y s i c a l " phase s h i f t s for low energy i r*N s c a t t e r i n g . This i n turn should allow us t o estimate the energy dependence of the purely non—electromagnetic, or hadronic, phase s h i f t s and hence the s c a t t e r i n g amplitude. As i s mentioned i n the i n t r o d u c t i o n , because only S—waves are of i n t e r e s t here, the proton w i l l be t r e a t e d as a s p i n l e s s p a r t i c l e . Leaving aside f o r the moment why only S-waves are considered, we may p a r t i a l l y j u s t i f y t h i s by observing that the s c a t t e r i n g amplitude f o r the i n t e r a c t i o n of spin 0 and spin 1/2 p a r t i c l e s may be w r i t t e n as where a = P a u l i s p i n matrix n, = u n i t vector normal to the s c a t t e r i n g plane and £,j = amplitude f o r s c a t t e r i n g i n t o the | . L , J = L ± T ) s t a t e Because the only s u r v i v i n g term i n the s c a t t e r i n g amplitude f o r 28 L=0 w i l l be ^ 0 j i , we may t r e a t the s c a t t e r i n g p o t e n t i a l as a purely s c a l a r p o t e n t i a l which i s tantamount t o viewing the proton as a s p i n l e s s p a r t i c l e . We w i l l subsequently w r i t e the S-wave s c a t t e r i n g amplitude as simply ^"(k) with the "L = 0" and " J = 1/2" s u b s c r i p t s suppressed. To understand why only S-waves are releva n t f o r very low energy s c a t t e r i n g , r e c a l l that f o r any f i n i t e ranged s c a l a r p o t e n t i a l , the p a r t i a l wave s c a t t e r i n g amplitude w i l l e x h i b i t the low energy behaviour where a L i s some constant c h a r a c t e r i s t i c of the s c a t t e r i n g p o t e n t i a l . A proof of t h i s i s given i n appendix B. Consequently the only s u r v i v i n g terms i n the s c a t t e r i n g amplitude CO L-o i n the k = 0 l i m i t , w i l l be that due to S-waves. This means that l i m ¥ ( k , 6 ) = a, (1) implying that the cross s e c t i o n acn = J da | ^(k.^r and hence the r e a c t i o n r a t e , w i l l be dominated by the S-wave term for, very low energies. The q u a n t i t y a. d e f i n e d by e q n ( l ) , c a l l e d the s c a t t e r i n g l e n g t h , i s of prime i n t e r e s t i n low energy irN s c a t t e r i n g . Because 29 t h i s i s the zero energy l i m i t of the L = 0 term i n the s c a t t e r i n g amplitude, then since ^ - o C M " , . i fan W O we may a l s o d e f i n e the s c a t t e r i n g length v i a lim-r-tan O^k) = a k-*o k Thi s l a t t e r expression i s j u s t i f i e d s ince we know (see appendix B) that f o r any f i n i t e ranged s c a l a r p o t e n t i a l the low energy phase s h i f t s may be expressed as t a n 6 0 ( k ) = ak + bk 3 + c k 5 + ... where the c o e f f i c i e n t s a,b,c... are energy independent q u a n t i t i e s which are r e f l e c t i v e of the s p a t i a l behaviour of the s c a t t e r i n g p o t e n t i a l . Because the basic c l a s s of p o t e n t i a l s that w i l l be considered are those discussed i n the previous chapter, we know that the f u l l s o l u t i o n to the Schroedinger equation can be w r i t t e n as a p a r t i a l wave expansion L-o For S—waves, t h i s means that our s o l u t i o n w i l l look l i k e At^o) = M P" U (p) (2) L-O To commence wit h our study of ir*p s c a t t e r i n g , we must f i r s t decide on a model p o t e n t i a l which w i l l provide a good r e p r e s e n t a t i o n of the p h y s i c a l s i t u a t i o n . To begin with we know that when no Coulomb f o r c e s are a c t i n g , the p o t e n t i a l w i l l be pur e l y hadronic. This means that i n the absence of 30 electromagnetism the s c a t t e r i n g p o t e n t i a l w i l l have the form o , r > where R H i s the range of the i n t e r a c t i o n . Futhermore we can s a f e l y assume that the electromagnetic p o t e n t i a l , f o r very low energies and s p i n l e s s p a r t i c l e s , can be w r i t t e n as V r ' ' K Here we are d e f i n i n g R as some di s t a n c e beyond which only the i n f i n i t e ranged Coulomb p o t e n t i a l i s a c t i v e whereas w i t h i n which i s f e l t only some f i n i t e ranged electromagnetic e f f e c t . I t thus seems reasonable to suppose that the p h y s i c a l p o t e n t i a l r e s p o n s i b l e f o r low energy n*p s c a t t e r i n g may be w r i t t e n as f VSRC^ , r £ £ 4r . r > d V * ( 0 --where VSR = VH + V E n ;r < R The s u b s c r i p t s "MC" and "SR" r e f e r to "modified Coulomb" and "short ranged" r e s p e c t i v e l y . In order that we may compare the phase s h i f t s , or e q u i v a l e n t l y the s c a t t e r i n g amplitude, a s s o c i a t e d w i t h t h i s p o t e n t i a l with that f o r the hadronic p o t e n t i a l , i t i s u s e f u l to proceed i n two stages. F i r s t we ask ourselves what e f f e c t the long ranged Coulomb p o t e n t i a l w i l l have on the phase s h i f t s f o r 31 the short ranged p o t e n t i a l , and secondly we compare how the l a t t e r w i l l d i f f e r from the purely hadronic phase s h i f t s . We w i l l begin w i t h the former. 1) E x t e r i o r C o r r e c t i o n s I f we l e t and 6 ( k ) represent the phase s h i f t s f o r the r e s p e c t i v e p o t e n t i a l s Vrtc and V S R , then we know from the previous chapter that the corresponding e x t e r n a l S-wave s o l u t i o n s w i l l be r and 'll„ (c>n) = e i S an ( p + 6 ) where P = kr and mc Because both of these s o l u t i o n s must n e c e s s a r i l y s a t i s f y the same i n t e r i o r equation, we are j u s t i f i e d i n equating t h e i r l o g a r i t h m i c d e r i v a t i v e s at r = R. 32 Hence which g i v e s us Fo + Ian y v7o or e q u i v a l e n t l y r = * = fan ( k £ + M = - tin M r*<L Sp f~L ^ and r e w r i t e i t as a cot k Co t. fen (kr + ^ (3) I t i s worth noting that the c o n d i t i o n (see eqn(1-3l)) l i m V - 0 i s s a t i s f i e d here. To see what eqn(3) looks l i k e at very low energies, say hck £ l5Mev, we make use of * 1 (4) r= ft. where c e i 7 T r i - i Now sin c e i t i s reasonable to assume that the c u t o f f radius R w i l l be i n the neighbourhood of 1 fermi ( 1 0 " 1 5 m e t e r s ) , then ( f o r very low energies we may make the approximations (see appendix C) 3d C^KnM) - cos tie. ~ i and ~ 2. where and /\ = J l + l o ^ ^ l ^ l k ^ + £ Y - 1 CO 1 I \ y = =- - /OQI^ I - V m- i The symbol V denotes the E u l e r constant. I t i s sometimes def i n e d by and i s approximately equal to 0.5772. From t h i s we see that at t h r e s h o l d (k = 0,|^| = oo ), 0 and hence A - i s a constant, namely Ao = l i m A = logR - 4.5573 With these approximations eqn(4) becomes Co I c o i v + zi± Ac ~ a where a = lim-£tan& i s the s c a t t e r i n g length for the short ranged p o t e n t i a l . 34 To determine what the Coulomb modified s c a t t e r i n g length i s , we w r i t e the above as -7- "fori V - 7 7 1 C z 0 I t w i l l become apparent i n chapter four that because the short ranged electromagnetic e f f e c t s are almost n e g l i g i b l e compared with the long range e f f e c t s f o r low energies, the above equation gives a reasonably accurate d e s c r i p t i o n of V (k) f o r hck £ 50 Mev. Now when the Coulomb force i s r e p u l s i v e , we may make the approximation and hence — Tan v — c, L from which we obta i n k ^ o k On the other hand, f o r an a t t r a c t i v e force ( f[ - ) one f i n d s that In t h i s case because we have 35 and t h e r e f o r e l i m -]-tan V = co lc-»o k The f a c t that the s c a t t e r i n g l e n g t h , and hence the r e a c t i o n r a t e , at t h r e s h o l d , should vanish f o r r e p u l s i v e Coulomb forces and yet be i n f i n i t e f o r a t t r a c t i v e f o r c e s should come as no s u r p r i s e . T his i s what we would n a t u r a l l y expect when we separate two s t a t i o n a r y charged p a r t i c l e s by an i n f i n i t e d i s t a n c e . 2) I n t e r i o r C o r r e c t i o n s To determine what e f f e c t the short ranged electromagnetic f o r c e w i l l have on the phase s h i f t s , we w i l l begin by c o n s i d e r i n g the two S—wave equations [ a' + ^ - ^ V M M ] U H ( ^ 0 (5) [ & + k l - V«C')) Uu(r* o (6) where V H = hadronic p o t e n t i a l V E M = short ranged electromagnetic p o t e n t i a l I f we m u l t i p l y eqns(5) and (6) on the l e f t by Um and UH r e s p e c t i v e l y and then take t h e i r d i f f e r e n c e , we obtain I n t e g r a t i o n from r - 0 to r = R gives Jo Now i f we r e c a l l that 36 and we can reduce eqn(7) to To lowest order i n the Coulomb p o t e n t i a l we have In chapter four use i s made of eqns(3) and (8) to determine what e f f e c t both the short and the long ranged Coulomb p o t e n t i a l has on the hadronic phase s h i f t s , which are entered i n t o the above equations as presumably known q u a n t i t i e s . This i n turn should enable one to determine what the hadronic phase s h i f t s w i l l be from a knowledge of the " p h y s i c a l " phase s h i f t s l ^ ( k ) . Although the r e s u l t s w i l l be h i g h l y s e n s i t i v e to the p a r t i c u l a r choice of the matching r a d i u s , we have chosen to perform a sample c a l c u l a t i o n f o r one p a r t i c u l a r value, namely R = 1 fermi. The energy range considered i s 50 Mev £ hck £ 100 MEV for cm. momentum which i s roughly eq u i v a l e n t to a l a b k i n e t i c energy of the pion ranging from 11 Mev t o 44 Mev. 37 I I I COULOMB CORRECTIONS TO LOW ENERGY t T P PHASE SHIFTS In the l a s t chapter, a procedure was given that would enable one to estimate, to lowest order i n the Coulomb p o t e n t i a l , the electromagnetic c o r r e c t i o n s to low energy tr*p phase s h i f t s . The present chapter proposes to do the same f o r the rr"p phase s h i f t s . What makes the l a t t e r process so d i f f e r e n t as to n e c e s s i t a t e a separate d i s c u s s i o n i s that u n l i k e n*p system, at low energies the s c a t t e r e d p a r t i c l e s are not always the same as the i n c i d e n t p a r t i c l e s . As was mentioned i n the i n t r o d u c t i o n , a f u l l d i s c u s s i o n of low energy n"p s c a t t e r i n g w i l l n e c e s s a r i l y e n t a i l an examination of the three competing channels rr'p * r"p , ir°n , rn We have chosen however to ignore the l a t t e r process i n order that we may develope more c l e a r l y , the steps required for an a n a l y s i s of the corresponding two channel problem. This i s a dangerous compromise c o n s i d e r i n g that the near t h r e s h o l d c o n t r i b u t i o n of the n"f> —» rn channel t o the t o t a l c r o s s s e c t i o n may very w e l l be as high as 30%. However because i t i s n a t u r a l to assume that the f u l l problem w i l l i n v o l v e merely an extension of the two channel a n a l y s i s , t h i s step seems j u s t i f i e d . As i n the l a s t chapter, the object i s to develop a method which w i l l a l l o w us t o determine how much the S—wave phase 38 s h i f t s corresponding to the " p h y s i c a l " p o t e n t i a l C V 5 R ( r ) • , r i £ v. T , r > ft w i l l d i f f e r from the S—wave phase s h i f t s f o r the hadronic p o t e n t i a l which i s the short ranged p o r t i o n of Vflc minus i n t e r n a l electromagnetic e f f e c t s . As before, the route we s h a l l take w i l l be to f i r s t r e l a t e the s c a t t e r i n g amplitude h£ m c for the p o t e n t i a l V£|C to the corresponding amplitude T$R f o r a short ranged p o t e n t i a l , and subsequently compare t h i s with the purely hadronic s c a t t e r i n g amplitude ¥*„ . In order to accomplish t h i s we must f i r s t develop some more n o t a t i o n . Lets begin by de s i g n a t i n g by channels "1" and "2" the n"p and n°n systems r e s p e c t i v e l y . The cm. momenta and reduced masses of these systems w i l l be l a b e l e d r e s p e c t i v e l y as "hk ,m and tiq ,m. . Consequently fDn- i D p (1) -m - • • — KDTr-jOQn (2) 39 and (3) where the l a t t e r e q u a l i t y i s a statement of energy co n s e r v a t i o n . Now f o r s i n g l e channel s c a t t e r i n g we know that the energy e i g e n s o l u t i o n s a t i s f y i n g the Schroedinger equation f o r any s c a l a r p o t e n t i a l , with azimuthal symmetry with respect to the z — a x i s , may be w r i t t e n i n the form Hence the basic q u a n t i t y of i n t e r e s t f o r S—waves i s r \l>L.(fS> or e q u i v a l e n t l y k" lXL-.o(p) . L a b e l i n g t h i s l a t t e r q u a n t i t y as X(P^ » w e k n o w that f o r s c a t t e r i n g confined to one channel, the S—wave Schroedinger equation may be w r i t t e n as [ £ • k1 - (JM } X (kO - ° where (J(0 = ^ V(0 We w i l l now p o s t u l a t e that the two channel S—wave 40 Schroedinger equation w i l l look l i k e £ 1 + Q L " U X - o where and where I V Q -r k o i n 0 1 J 0 U - - T T M V(0 Vi,(0 V»(0 J The s o l u t i o n to the above equation w i l l be w r i t t e n as X where the i t h column r e f e r s to incidence i n channel i . The e x t e r i o r two channel S-wave equation f o r a short 41 ranged p o t e n t i a l 0 i s s a t i s f i e d by X ( ^ 0 - ° 1 sindr + ^ t : -4 i where (no ^wnO (4) denotes the S-wave amplitude f o r s c a t t e r i n g from channel i i n t o channel j wi t h corresponding phase s h i f t s 6 ^ . In eqn(4), " L = 0 " s u b s c r i p t s have been suppressed and kj = k,q for j = 1,2 r e s p e c t i v e l y . I t i s important to observe that the multichannel phase s h i f t s d iscussed throughout t h i s chapter are complex numbers. This w i l l become apparent l a t e r when a d i s c u s s i o n of the K-matrix i s given. The corresponding equations f o r the modified Coulomb p o t e n t i a l are k 3* + kl <- ^ o 9* * "T 42 and MC 12. He 11 1 smdr f t'^ 4 where and where ^.("j,fcr) , £«,(-?,kf>) a r e t h e i r r e g u l a r , regular Coulomb fu n c t i o n s d e f i n e d i n chapter one and i n the appendix. Observe that MC /A which represents the s c a t t e r e d ' s p h e r i c a l ' wave i n the n _p channel when the i n c i d e n t wave i s i n the ir°n channel, i s i n accord with eqn(30) of chapter one. Furthermore, i f Vjj, - Vjitkj) represents the phase s h i f t f or s c a t t e r i n g v i a V n e from channel i i n t o channel j , then the corresponding s c a t t e r i n g amplitude i s i n matrix n o t a t i o n , the e x t e r i o r S-wave equation and i t s s o l u t i o n f o r V,R can be w r i t t e n as and X^ - Q 1 where f r s/nlcr sm<\r o cos<\r-h : t if and S i m i l a r expressions f o r V are a I • V - V*]xJ»-n and r F + where i F 6 k- i f fo r sincjr J cos<|r J z o m< r i l k r i o H 45 and MC q1 Q 1 1) E x t e r i o r C o r r e c t i o n s To determine a r e l a t i o n between the S-wave s c a t t e r i n g amplitudes f o r the short ranged p o t e n t i a l and the Coulomb modified p o t e n t i a l , we w i l l begin by assuming that Vsa. i s a square w e l l and make use of the e q u a l i t y Jf-i. (7) where and XMc a r e 9 i v e n bY eqns(5) and (6) r e s p e c t i v e l y . Before doing so however, we must f i r s t prove the v a l i d i t y of t h i s equation. To t h i s end we w r i t e the s o l u t i o n to - o (8) where [mVn mVn [/n.Vu Wo Vxx J which i s some s p a t i a l l y constant matrix, i n terms of the c h a r a c t e r i s t i c frequencies 6J and A as r a, jmu>r X + a i ^ r s i n X r 46 When t h i s i s s u b s t i t u t e d i n t o eqn(8) we f i n d that and where and I P. 21 2. mn V a R* - X 1 m V, Although the parameters aj and bj , which c h a r a c t e r i z e the boundary c o n d i t i o n s of the s o l u t i o n and i t s d e r i v a t i v e at r = 0 and r = R , are dependent on the nature of the e x t e r i o r p o t e n t i a l , namely e 2 / r , i t i s important to observe that the l o g a r i t h m i c d e r i v a t i v e of X which we c a l l d, does not have any such dependence. More e x p l i c i t l y d : I X e ^ l * ^ ' s ° l J r . iJ - 1 60 fan X ft X which i s the same with or without long range Coulomb e f f e c t s 47 Hence i f the i n t e r i o r p o t e n t i a l i s t r e a t e d as a square w e l l then eqn(7) n a t u r a l l y f o l l o w s . When eqn(6) i s s u b s t i t u t e d i n t o eqn(7) or e q u i v a l e n t l y we o b t a i n a f t e r some rearrangement (9) TVic (10) where a l l s p a t i a l values are taken at r = R. Towards the end of t h i s chapter we w i l l use t h i s expression to o b t a i n more e x p l i c i t l y both the Coulomb modified s c a t t e r i n g amplitudes and t h e i r r e s p e c t i v e s c a t t e r i n g l e n g t h s . Now s i n c e the i n t e r i o r s o l u t i o n w i l l i n v o l v e coupled equations, the question n a t u r a l l y a r i s e s whether t h i s c o u p l i n g i s preserved or at l e a s t r e f l e c t e d by eqn(lO). Although i t does not seem p o s s i b l e t o answer t h i s question d i r e c t l y here, we can at l e a s t check whether or not one consequence of the c o u p l i n g , namely the u n i t a r i t y c o n d i t i o n i s s a t i s f i e d . What i s meant by t h i s i s developed as f o l l o w s . Consider f o r the moment the e x t e r i o r s o l u t i o n f o r V$R (5) Because t h i s may a l s o be w r i t t e n as where C = I + 2 i ' ¥ « .<"> then i n view of the f a c t that the matrices h + and h" d e f i n e d 48 above are e s s e n t i a l l y outgoing and incoming s p h e r i c a l waves r e s p e c t i v e l y and that the r a t i o between the two i s given by the s c a t t e r i n g matrix S S A , then the statement of p r o b a b i l i t y c o n s e r v a t i o n i s e q u i v a l e n t to the u n i t a r i t y c o n d i t i o n where the dagger " f " means hermit i a n conjugate. Therefore when e q n ( n ) i s s u b s t i t u t e d i n t o t h i s , we f i n d that the s c a t t e r i n g a m p l i i t u d e s must s a t i s f y the c o n s t r a i n t Because t h i s expression i s rather cumbersome to deal w i t h , i t has become the custom when d e a l i n g w i t h multichannel s c a t t e r i n g processes to u t i l i z e what i s known as the K—matrix, sometimes r e f e r r e d to as the r e a c t i o n matrix. Our choice f o r using t h i s formalism i s twofold. Aside from f a c i l i t a t i n g a check on the u n i t a r i t y c o n d i t i o n , i t w i l l a l s o a l l o w us to compare our approach w i t h that taken by Rasche and Woolcock [ 4 ] . The K—matrix formalism can be developed by observing that eqn(5) may a l s o be w r i t t e n as I (12) where - i (13) 49 Since t h i s i m p l i e s that y l y ? s . = [ I - 1 Ks« J N«* then i t f o l l o w s from eqn(11) that 5 « - t i - i K I • I (14) This means that the u n i t a r i t y c o n d i t i o n i s immediately s a t i s f i e d provided KSR. i s h e r m i t i a n C = K. SR. implying that we may parametrize K $ f c as L P where ot ,\ are r e a l numbers and i s complex. When t h i s i s s u b s t i t u t e d i n t o eqn(14) we obtai n T SB. A , d-ib ( 1 5 ) where and - z (<* + > ) - A Two p o i n t s of i n t e r e s t immediatly f o l l o w from t h i s l a t t e r 50 ex p r e s s i o n . The f i r s t i s that i f time r e v e r s a l i n v a r i a n c e holds f o r irN s c a t t e r i n g then t,x - ,from which i t f o l l o w s that K must be a r e a l symmetric m a t r i x . Secondly, we can now show q u i t e e a s i l y that the phase s h i f t s f o r Vice, are n e c e s s a r i l y complex q u a n t i t i e s . For example, we know from eqn(4) that Therefore when use i s made of eqn(l5) we ob t a i n f a n 6 n | + i k ¥n* a - i A 51 S i m i l a r l y "fan & 2.7. \ I - tOC and tan 8 7?/* l-i(cc + V V A + t/E /S Because i t i s q u i t e reasonable to assume that p r o b a b i l i t y c o n s e r v a t i o n a l s o holds i n the presence of long range Coulomb f o r c e s , we may c l a i m that an analogous expression to eqn(l3) a l s o holds f o r -f>u , that i s I * i I ~ r KMC Hence we need only check that K n c i s hermitian to ensure u n i t a r i t y . With t h i s goal i n mind we r e w r i t e eqn(6) as I t i s s t r a i g h t f o r w a r d t o show with the use of t h i s and eqn(9) that (16) where 52 To see and w r i t e d what t h i s looks l i k e when expanded out we r e c a l l eqn(!2) IT? IS (17a) (17b) (17c) where A, = cos^ ( cosk£ - etsmU) - ( Ycosk£ - <*y/nU) A 2 = cosqR. (smU +*cosU) - smtf (ysinkd + Acosk&) A 3 - s/ncfjj (coske - otsmkR) + cosc}£. (V<rosk£ - A J w k ) A . A 3 - A.A. = When use i s made of the above we f i n d that eqnd6) may be w r i t t e n as e-^js* A , _ A | ( o ( s i n k G - c o s k ^ F . where 4- Ai (A s m k £ - V c o s k O . F . C ^ k O 53 Since the only complex q u a n t i t y appearing i n the above i s pe i o» i t f o l l o w s that K*c i s h e r m i t i a n . This guarantees that the u n i t a r i t y c o n d i t i o n i s s a t i s f i e d . When eqn(l7c) i s now s u b s t i t u t e d i n t o eqnOO) i t can be shown, a f t e r c o n s i d e r a b l e a l g e b r a , t h a t and > [(i-iV) cosk£ - ( o t-i^s/nkfc] fa j (18a) (18b) , s/>MC fa Ta, - A;1 p* ^ (18c) - [ V s ; n k e + A c o s U ] ^ W. J where a l l the Coulomb f u n c t i o n s are evaluated at r = R and A e = j^ O-iV") coskg. - (o(-i&) sinkk] l\0(") ,k£) ^ J 54 To express the Coulomb modified amplitudes d i r e c t l y i n terms of the amplitudes and phase s h i f t s f o r V , we observe that • SR. IP vj St This enables us to r e w r i t e the above as vnAC _ i _ ton f k£. -I- hn) ^ p F g EJ (19) vv)MC "121 and i n = i n U n f o r t u n a t e l y i t does not seem p o s s i b l e to re—express ¥ the same manner. I t might i n t e r e s t the reader t o observe that e qnd9) which i s tantamount to + t a n f k £ + <U>F» - Fc In v„ = : 55 i s i d e n t i c a l i n form to i t s s i n g l e channel analogue, eqn(3) c i t e d i n chapter two. As a f i n a l remark we observe that i n the e 2 = 0 l i m i t , one obtains , 2) Coulomb Mo d i f i e d S c a t t e r i n g Lengths I f we adopt the d e f i n i t i o n of s c a t t e r i n g l e n g t h , given i n the l a s t chapter, as being the t h r e s h o l d value of the s c a t t e r i n g amplitude, then we might be tempted to w r i t e for the two channel problem SR and fie as being the s c a t t e r i n g lengths f o r s c a t t e r i n g , v i a VS(L and V n c r e s p e c t i v e l y from channel i i n t o channel j . Therefore by making use of the low energy approximations f o r the Coulomb f u n c t i o n s , F . ( - i , t O ~ where (see page 33) /\o = I03R " 4-5573 56 we o b t a i n m Q~ *  a " - a i V ( z T T i n i ) , / l and 'xt Now since —* oo as I c o then the only f i n i t e term according to the above format w i l l be the ir°n -* ir°n s c a t t e r i n g l e n g t h . Because an i n f i n i t e s c a t t e r i n g length f o r s c a t t e r i n g from channel 2 to channel 1 goes against our i n t u i t i o n , i t would seem more r e a l i s t i c to define a s c a t t e r i n g length f o r each channel of i n c i d e n c e , that i s a : = l i m t\ where k, = k and k x= q. This would keep the l a t t e r two of the above equations constant while l e a v i n g the former two u n a l t e r e d . I t should be emphasized that i n order to show limCL^j =0-1* ,we must f i r s t take the e 2 = 0 l i m i t then e x t r a p o l a t e the r e s u l t to t h r e s h o l d . 3) I n t e r i o r C o r r e c t i o n s Having shown how to obtain from and hence t h e i r corresponding phase s h i f t s , i t remains to determine how can be obtained from the purely hadronic s c a t t e r i n g amplitude 57 We w i l l begin by w r i t i n g the corresponding i n t e r i o r S-wave Schroedinger equations as £ 1 * Q 1 ' U * X*(^<0 -o (20) and & \ + Q 1 - a ] x « ( ' * o - o I f we take the transpose of the above, r i g h t m u l t i p l i e d by Xs&\ > and subt r a c t from i t eqn(20), l e f t m u l t i p l i e d by , we obtain ^r[X H T 3rX i R ' ( d ' X i ) X * * ) = X H ( J S * X 5 * " X H UH X * g In t e g r a t i n g from r = 0 to r = R and r e c a l l i n g that X « ( ^ 0 = q ' 4 [ * • Q " * with a s i m i l a r expression f o r * R ) ' gives us - ? H - //* x ; [ u « T - u « ] x « d r x ; f u e - u « ] x s , +.prx;[u T -a]x* (21 ) Since i t i s reasonable t'o suppose that the i n t e r i o r electromagnetic e f f e c t s w i l l be most s t r o n g l y f e l t i n channel 1 we w i l l w r i t e 2 2m ]/tn f l o " . 0 0 • 0 . (22) 58 To lowest order i n the Coulomb p o t e n t i a l we may set When t h i s along with eqn(22) are s u b s t i t u t e d i n t o eqn(2l) we obtai n (x » r x : x X|| XlZ { X 12 } z + 1? f O 1. •1 o (23) Because the long range Coulomb m o d i f i c a t i o n s to the s c a t t e r i n g amplitude and phase s h i f t s are to be determined d i r e c t l y from the short range Coulomb modified phase s h i f t s v i a eqn(lO), i t i s apparent that any numerical c o n s i d e r a t i o n s must u l t i m a t e l y begin w i t h eqns(2l) or (23). 59 IV NUMERICAL CONSIDERATIONS In t h i s chapter we w i l l use some of the r e s u l t s obtained above to determine the Coulomb m o d i f i c a t i o n s to the supposedly known S-wave hadronic phase s h i f t s f o r trN s c a t t e r i n g . I t i s assumed that the numbers generated w i l l serve as a c t u a l should be able t o do the converse, namely, to give an estimate of the hadronic phase s h i f t s and other r e l e v a n t q u a n t i t i e s . Because of t h i s , i t i s not e s s e n t i a l that exact values of the hadronic phase s h i f t s be used. Since n e a r l y a l l of the e x i s t i n g l i t e r a t u r e on low energy irN s c a t t e r i n g assumes charge independence when expressing the hadronic phase s h i f t s , we are forced to adhere to the usual i s o s p i n conventions (see appendix E ) f o r our i n i t i a l set of numbers. These we w i l l take from M. Salomon et . a l . ( ! 9 7 8 ) as re p r e s e n t a t i o n s of the " p h y s i c a l " phase s h i f t s . With these, one and + where C>"(k) = phase s h i f t f or i s o s p i n a/2 and fok = cm. momentum (Mev/c) 60 Because low energy trN s c a t t e r i n g experiments have been performed f o r pion l a b o r a t o r y k i n e t i c energies as low as 25 Mev, we have chosen to keep our c a l c u l a t i o n s w i t h i n the range 50 Mev £ iick £ 100 Mev which i s roughly equivalent to 11.6 Mev £ T£, £ 44.0 Mev. To determine what the hadronic s c a t t e r i n g amplitudes are, we observe that "fen ba where 4-a as the S—wave s c a t t e r i n g amplitude f o r i s o s p i n a/2 and 6a i s the corresponding phase s h i f t . Therefore, assuming that i s o s p i n i s a good quantum number f o r the hadronic f o r c e , we can o b t a i n the s c a t t e r i n g amplitudes from and In the work that f o l l o w s , we w i l l e x p l i c i t l y show how to o b t a i n the Coulomb modified phase s h i f t s f o r the s c a t t e r i n g of 61 p o s i t i v e and negative pions o f f protons. The procedure taken f o r both cases w i l l be e s s e n t i a l l y the same. Tabulated r e s u l t s f o r the energy range c i t e d above w i l l a l s o be presented. We s h a l l begin by c o n s i d e r i n g the former. 1) Coulomb M o d i f i c a t i o n s to ir*p Phase S h i f t s In chapter two we saw that we could express the phase s h i f t s , which had both short and long range Coulomb m o d i f i c a t i o n s , i n terms of those with only short range m o d i f i c a t i o n s as We a l s o found that we could o b t a i n these l a t t e r phase s h i f t s , to lowest order i n the Coulomb p o t e n t i a l , from J a £M (2. 8) where SR. and Now i f we regard the hadronic p o t e n t i a l as a square w e l l , then u H ( r £ R) «= kA sinhBkr (1) The q u a n t i t i e s A and B, which are f u n c t i o n s of k and R, can be 62 determined by equating t h i s and i t s d e r i v a t i v e to (r £ R) = e ^ - s i n U r + £H) and i t s d e r i v a t i v e r e s p e c t i v e l y at r = R. This gives us and •-i- tanhBkR = tan(kR + &H ) D A = e'^ Hsin(kR +6H)/ksinhBkR (2) (3) To f i r s t order i n e 2 , we expect the short ranged electromagnetic p o t e n t i a l to be of order e 2/R. This prompts us to "try e. 3 ' R (4) which i s the p o t e n t i a l due to a charge uniformly d i s t r i b u t e d over a sphere of radius R. See appendix D f o r a d e r i v a t i o n of t h i s . As a check, to see how s e n s i t i v e our model i s to the p a r t i c u l a r choice of l4n , we w i l l a l s o do a c a l c u l a t i o n with V > 0 G_ 5 ft (5) which i s the space average of the above over the same volume. S u b s t i t u t i o n of these two expressions i n t o eqn(2. 8) gives us the r e s p e c t i v e amplitudes (6) 63 where £ = BkR and The phase s h i f t s are r e a d i l y obtained from these by making use of eqn(3) and r e w r i t i n g the above as where and These equations enable us to obtain b d i r e c t l y from Two sets of phase s h i f t s are t a b u l a t e d below, one f o r each of the p o t e n t i a l s s p e c i f i e d above. The matching radius was chosen to be 1 fermi and the p h y s i c a l constants were taken as = 139.58 Mev m^c2 - 938.21 Mev <x - 1/137 197.32 Mev-Fm rjk /• mcec/h = 0.0045 Fm" 64 The Coulomb f u n c t i o n s (see appendix C) are evaluated at R = ifm. For fcck < 100 Mev, they -can be w r i t t e n to four s i g n i f i c a n t f i g u r e s as o.oo^s k. - o . o o o s k* c o s k + o.oo^o - o.ooiok l f o.oooi oo 1  0H(dty«s) h(deyea) \) (degrees) i - OH AM 5 o - JZ.Ogfrl - 3.12 2X - I . l i n - 0.05I0 - 0 . OI68 O. 106 5 6 0 -2. - 0.0^ X1 -0. Ol6<f 0.10 XT. 7 0 - 3.022.5 -.2.1/4-7 - 0.O3S7 -0. O/S-*} 0.1012 So "3.5.233 - 3.5781 -3.41*2 -0. 0 3 0 8 -0.OI54 0. 1026 -4.0512 - 4 .(I 13 -0. 02.70 -0.OI42 0.1016 1 0 0 -0.026I -0.0143 0. /20€ TABLE I CORKECTIOJOS TO 7T*J> PHASE. SHIFTS V£in% £QNJ( T). I) (deyeet) •y) (decreet) v- 6h 0- OH V- OH 5 0 -2. osez - 3.1234 - /.<)?/? -0. 016? 0. 1063 6 0 -2.5*62 - 0.042.0 - 0.0164 0. 1070 70 - 3.0Z25 - 3.07O? - 0.0 3 56 -0.OIS1 0.1076 eo -3.57*3 - 3.^/55" - 0.0308 - 0.0IS4 0. I0S4 - 4.0SIS -3.142-4 - 0.0570 - 0 .0*41 0 .1014 too -4-602f -4.6741 -4*4**1 -0.02.6 1 - e.oi4' 0.12.06 TABLE II CORRECTIONS TO fT*b PHASE. SHIFTS USING £ Q M ( S ) . 66 The s t r i k i n g t h i n g about these t a b l e s i s that the Coulomb m o d i f i c a t i o n s f o r both are almost i d e n t i c a l . I t i s f o r t h i s reason t h a t , i n d e a l i n g with the two channel problem, we w i l l assume that the short ranged electromagnetic p o t e n t i a l i s that due t o a s p a t i a l averaging of a p o t e n t i a l due t o a uniform charge d i s t r i b u t i o n . I t i s i n t e r e s t i n g to observe that the f r a c t i o n a l d i f f e r e n c e s , f o r both the e x t e r i o r and the i n t e r i o r Coulomb m o d i f i c a t i o n s , decrease uniformly i n magnitude with energy whereas the d i f f e r e n c e s |&-AH( and |y-u«( both increase uniformly w i t h energy. In t a b l e IV (page 74) we have reproduced some of the r e s u l t s of Tromborg [ 8 ] , f o r both the n*p and the u'p problem. Because t h e i r a n a l y s i s covers the energy range of approximately 70 Mev < hck £ 420 Mev, we have f e l t i t appropiate to present only t h e i r lower energy phase s h i f t c o r r e c t i o n s . 2) Coulomb M o d i f i c a t i o n s to ir'p Phase S h i f t s In the l a s t chapter i t was shown that the phase s h i f t s \)yL , f o r s c a t t e r i n g v i a V « ( ^ = j - e r £ r o m channel i i n t o channel j . could be expressed i n terms of the corresponding phase s h i f t s 6* f o r V„ . This «as accomplished wi t h TMC 67 where Q k o O ^ J 9 + ev«r and Sft A , re? (3. 17) which have a l l been defined i n the previous chapter. I t was a l s o shown that the short range q u a n t i t i e s could be obtained d i r e c t l y i n terms of the purely hadronic ones v i a 2<m IF .Xh Xix ii A ix Cxi? -1 (3. 22) In t h i s s e c t i o n we w i l l show how, under c e r t a i n basic assumptions about the hadronic p o t e n t i a l , i t i s p o s s i b l e to 68 c a l c u l a t e e x p l i c i t l y the f u l l Coulomb m o d i f i c a t i o n s t o the hadronic phase s h i f t s . We w i l l begin by making use of eqn(3. 22). To t h i s end we w i l l s t a r t with the assumption that i s o s p i n i s a good quantum number for the hadronic p o t e n t i a l . This means that the hadronic s c a t t e r i n g amplitude as w e l l as the s c a t t e r i n g p o t e n t i a l w i l l be symmetric. Since t h i s i m p l i e s that the pion and nucleon mass d i f f e r e n c e s are zero, we have as a consequence -zm d r V E „ ( 0 An A12 wp SR. vJ7 IK Observe that i n t h i s approximation -h 12. — +a-» , We know from the n*p problem that A ¥ w i l l not be very s e n s i t i v e to VEH provided i t be of. the order e 2/R for f i x e d R. Therefore i t seems reasonable to use the ir~p analogue of eqn(5.) and set II A l l (At (8) which i s a r e l a t i v e l y shallow square w e l l p e r t u r b a t i o n . I f we now suppose that the hadronic p o t e n t i a l i s a l s o a square w e l l , then we know from chapter three that s<r> tor * A X a, 5, a* b,. where the q u a n t i t i e s u , A , i and are d e f i n e d v i a eqn(3. 8 ) . 69 S u b s t i t u t i o n of these expressions i n t o the above gives us (9) Replacement of bj by a; and v i c e - v e r s a gives us A^„ and A -fn r e s p e c t i v e l y . I t remains to show how we may obtain the q u a n t i t i e s q. fkj r to r X ' 5 a n d ^ from the i s o s p i n phase s h i f t s o l • To t h i s end we begin by equating X^r - R ) wi t h at r = R. This gives us ¥,i e.!l" fQ, b , l r to o X (10) which a l l o w s us to determine a} and bj from a knowledge of the eigenfrequencies tO , \ and 5 ,^ . These we f i n d as f o l l o w s . Since the absence of mass s p l i t t i n g s w i l l a l s o imply that k = q, then s u b s t i t u t i o n of H where '= t a n 70 i n t o eqn(3. 13) f o r the K-matrix, t e l l s us that 3 Consequently when we set we o b t a i n 23 Li • H o x / (11) 71 where By l e t t i n g d Cj designate the i j t h element of d H , we see that da = di, ,which i m p l i e s that J = -S and t h e r e f o r e and • X = 5* - | These along with eqn(10) now enable us to e x p l i c i t l y c a l c u l a t e £ f t . From t h i s we next c a l c u l a t e dsn and hence . The next step i s to transform the s c a t t e r i n g amplitudes ¥*f or e q u i v a l e n t l y . r i ? ^ ; r i 7 ? i n t o the amplitudes f o r s c a t t e r i n g , v i a \Z„C , from i s o s p i n B i n t o \j>A8 i s o s p i n A. These l a t t e r q u a n t i t i e s , which we designate as , 72 are obtained v i a Observe that the presence of 2 ' X IMC 3 "_1 i 1 2. and ^ j j . , where the ket | l , I 3 ^ designates the i s o s p i n s t a t e , gives allowance f o r i s o s p i n breaking. Further d e t a i l s are presented i n the appendix. As a f i n a l step, to see what the Coulomb m o d i f i c a t i o n s to the i s o s p i n phase s h i f t s 6" are, we l a b e l them as V^e and proceed to c a l c u l a t e them with the a i d of k Re - i k (U V„° £ and e In the t a b l e given below, we are s e t t i n g Vs - V,} and V, = Vu f o r s c a t t e r i n g w i t h i n the i s o s p i n channels 3/2 and 1/2 r e s p e c t i v e l y . Furthermore, because the q u a n t i t i e s Vo, have such a small imaginary p a r t , at l e a s t f o r the energy range we have chosen, we have tabul a t e d only the r e a l part of Va - 61 which we are d e s i g n a t i n g as A a . « O ft 1 O o * 1 r-O • 1 o 1 IN H O * * \ VO 0 • 1 o <3 • rJ « 00 o-o • *o OO cr O O ^-OO O * °. o ^ 1 -t-H ? o o 0 o 1 *+ - 5 o ^ O • •+ /—• 0 to o o ^ + a <o O o ; -<j> >o o .w • 1 o-»0 o 6 ° • 1 5 1 o ^ * i to to 0 O RO ft RO ^ XT M -6O ° 00 * cJ ~ V* -»• <r — 0 0O o • • ^  o to o o- o s ° * -J 1 1 A o o — o cr o N •: I cr in o N o =F- ^ F"> 4-1 0° O vo o cr O I 0 O »o o f ~ 0 I _ ~0 O r- <r « •D-o O r-<-o VO OO vo 0 -O 00 OO Q 1 • I H FSL O 1 •3-H in no \ o 1 cr o VO * 1 > ul r >—• O O o o O O OO o O o T A B L E J J X 7 4 Although the numbers A 7 and A, appear to be c o n s i d e r a b l y smaller than those given i n the l i t e r a t u r e (see t a b l e I V ) , they do e x h i b i t the same general f e a t u r e s . F i r s t we observe that the q u a n t i t i e s Aa. decrease monotonically with energy. Secondly, they are of opposite signs and that A3 i s c o n s i d e r a b l y l a r g e r i n magnitude than A i . A l s o t h e i r n*p m o d i f i c a t i o n s A appear to decrease with energy f o r -nek £ 112 Mev while our values increase (see page 65). The orders of magnitude are the same however. Given below are some of the Coulomb c o r r e c t i o n s for ir + p, ir'p S-wave phase s h i f t s taken from Tromborg [ 8 ] . The q u a n t i t i e s A and A a a r e the ir + p and i s o s p i n a/2 c o r r e c t i o n s ( i n degrees). J A A, A 3 A,3 0.5 0. / 0 - 0.3 1 0.52 0 . O 6 O. 8 0.0<=\ -D.21 0 . 35 O . 0 6 \-o 0.10 - 0. 15 0.30 O.Ol 1.1 0. 1 0 -0.13 O .2.2 O . 07 0. 1 0 - O.10 0 . z 7 O . O 8 T#BLEJB£ Phase Shift Corrections, of TTombor^ (1 75 Before coming to a c l o s e , i t might prove u s e f u l to say a few words concerning the approach t o the two channel ir"p problem taken by Rasche and Woolcock [4,5,6]. The reason f o r t h i s seemingly l i m i t e d choice amongst such a wide s e l e c t i o n of l i t e r a t u r e on t h i s t o p i c i s t w o f o l d . To begin with,, they provide an e x c e l l e n t review of the l i t e r a t u r e and consequently serve as a good reference source. Secondly, t h e i r approach, which i s r e l a t i v e l y easy to re-present since i t has a few s i m i l a r i t i e s to our own, i n p a r t i c u l a r t h e i r use of eqn(3. 7) which however they make no attempt to j u s t i f y , i s done with the view to extending i t to a three channel formalism to incorporate the r a d i a t i v e capture process. Because t h e i r n o t a t i o n appears rather cumbersome, we w i l l adhere to our own. In t h e i r paper, rather than d i s c u s s i n g the low energy phase s h i f t s as we do, they concentrate on the Coulomb m o d i f i c a t i o n s to the s c a t t e r i n g lengths, presumably a f t e r the i n f i n i t e ranged e f f e c t s have been c a r r i e d out. Although t h e i r a c t u a l mode of c a l c u l a t i o n i s not too c l e a r , the general features of t h e i r paper[4] can be presented as f o l l o w s . In determining the Coulomb c o r r e c t i o n s to the ir'p s c a t t e r i n g l e n g t h s , what they do i s e s s e n t i a l l y look f o r the two channel analogue to I CdpF. W U + o) - C." F 0 r-.R I (12) where / 76 and Ji - Z_ i m t , + ( W T ^ , J - ' ^ ' T " v and a = s c a t t e r i n g l e n g t h f o r V$« . This can be accomplished, a f t e r some rearrangement of eqn(3. 16) v i a c ( ? ' C q 4 c 2r\ k A o 1 o where r C. o (13) (14) J - Co"' F . C - T , kO LEAF 77 OMITTED IN PAGE NUMBERING 78 and r e w r i t e eqn(l3) as where / U + ^ A - i k C ) ^ / ! ( 1 5 ) with A;j being the i j t h element of A. As input f o r t h e i r numerical work, they use estimates of the hadronic s c a t t e r i n g l e n g t h s , or t h e i r i s o s p i n e q u i v a l e n t s a and a , taken from several authors, arguing that the c o r r e c t i o n s are h i g h l y i n s e n s i t i v e to the p a r t i c u l a r values chosen. Although they a l s o are using a square w e l l electromagnetic i n t e r i o r p e r t u r b a t i o n to a square w e l l hadronic p o t e n t i a l , the main advantage of our approach, aside from c l a r i t y , i s that we e x p l i c i t l y c a r r y out the numerical c a l c u l a t i o n i n two d i s t i n c t stages, namely the i n t e r i o r c o r r e c t i o n s and subsequently the e x t e r i o r ones. 79 References 1. M.Abramowitz and I.Stegun, Handbook of Mathematical  Functions , Dover, New York, 1965. 2. N.Mott and H.Massey, The Theory of Atomic C o l l i s i o n s , 3rd ed., Oxford, Claredon Press, 1965. 3. J.R.Taylor, S c a t t e r i n g Theory , New York, Wiley, 1972. 4. G.Rasche and W.Woolcock, Helv.Phys.Acta, Vol.49, 435 ,1976. 5. G.Rasche and W.Woolcock, Helv.Phys.Acta, Vol.49, 455 ,1976. 6. G.Rasche and W.Woolcock, Helv.Phys.Acta, Vol.49, 557 ,1976. 7. M.Salomon e t . a l . , Phys.Rev. CI 8 , 584, 1978. 8. B.Tromborg e t . a l . , Phys.Rev. D15 , 725, 1977. 9. B.Tromborg e t . a l . , Nucl.Phys. B60 , 443, 1973. 10. V.Samaranayake e t . a l . , Nucl.Phys. B48 , 205, 1972. 11. V.Samaranayake e t . a l . , Nucl.Phys. B49 , 128, 1972. 12. B.Robertson, D.Measday, e t . a l . , Paper presented to the Ninth I n t e r n a t i o n a l Conference of High Energy Physics and Nuclear S t r u c t u r e , held at V e r s a i l l e s , Fronce, J u l y 1981. Appendix A S c a t t e r i n g Amplitude for E l a s t i c C o l l i s i o n s Suppose we demand that the s o l u t i o n to the equation v 1 + - U ( 0 ] h { L ) ~- ° ( a , ) where (J( r ) (jim/fr) VU) s a t i s f y the asymptotic c o n d i t i o n (a2) Since we know that (a2) cannot be true for any a r b i t r a r y p o t e n t i a l , we n a t u r a l l y ask what i s the advantage of using i t assuming i t were true? Secondly, f o r what c l a s s of p o t e n t i a l s w i l l t h i s asymptotic c o n d i t i o n be v a l i d ? The answer to the f i r s t question i s that i f i t were t r u e , then i t would be p o s s i b l e to c a l c u l a t e the d i f f e r e n t i a l - cross s e c t i o n by the use of d. ( a 3 ) To see t h i s , l e t Jsc and J, N C represent the s c a t t e r e d and i n c i d e n t f l u x r e s p e c t i v e l y . The number of p a r t i c l e s s c a t t e r e d per second i n t o the element of area da - r r l d£l i s J J C * da . Since t h i s must equal the number i n c i d e n t per second through some element of area d<J -Z-dff s i t u a t e d normal to the d i r e c t i o n of incidence z , then J s c • r d a = J ( N C - i d ( r Now i f <//JC = ¥ M t a n d = e i f e t represent the s c a t t e r e d and i n c i d e n t waves r e s p e c t i v e l y , then J S c = tic V -> «r where b : J? - )> • This i m p l i e s that and hence d a To answer the second q u e s t i o n , we convert the Schroedinger equation i n t o an i n t e g r a l equation with the asymptotic boundary c o n d i t i o n (a2) b u i l t i n , and w r i t e fdV5(t,r)V(£')s((<:') (.4) k i t ) - M & i k i + Since the f i r s t term represents an i n c i d e n t f r e e plane wave, t h i s expression cannot be v a l i d f o r any p o t e n t i a l such as the Coulomb p o t e n t i a l which has an i n f i n i t e range. In order f o r t h i s t o s a t i s f y ( a l ) , i t i s necessary that 2-rYl ( a 5 ) Although the f u n c t i o n J s a t i s f y i n g (a5) i s not unique, the demand that the second term i n (a4) be an outgoing s p h e r i c a l wave r e q u i r e s that (a6) and hence Ml) = A)k 7^  2 | r - r ' | k (a7) To obtain the asymptotic form f o r the i t e r a t i v e s o l u t i o n (a7), we observe that i f f»r' then |r-r'| =r r-r-r' and ther e f o r e | r - f'J ~~ c r ° These approximations give us (a8) where (a9) I t i s important to observe that the above asymptotic expression i s v a l i d only f o r those p o t e n t i a l s which d i e o f f f a s t e r than 1/r 2. If we suppose that the p o t e n t i a l i s purely Coulombic, then to lowest order i n pt the s c a t t e r i n g amplitude w i l l e x h i b i t the asymptotic form where q = |k - k r | = 2ksin6 /2 i s the momentum t r a n s f e r viewed i n the center of momentum frame. Because the above i n t e g r a l i s poorly d e f i n e d , i t i s convenient to adopt some form of a r t i f i c i a l c u t o f f f o r the 83 Coulomb p o t e n t i a l . This can be n a i v e l y accomplished i n one of two ways. We can e i t h e r demand that i t vanish suddenly beyond some a r b i t r a r y c u t o f f radius R c or have i t decrease e x p o n e n t i a l l y . For the c u t o f f p o t e n t i a l V we obtain whereas for e 1 V = ^ e we have For the e x p o n e n t i a l l y decaying p o t e n t i a l , we have for large R the f a m i l i a r Rutherford s c a t t e r i n g amplitude 1 For the c u t o f f p o t e n t i a l , we have f o r la r g e qRc, a r a p i d l y o s c i l l a t i n g f u n c t i o n appended to the Rutherford amplitude. Appendix B Threshold Behaviour of P a r t i a l Wave Amplitude The p a r t i a l wave s c a t t e r i n g amplitude f o r a s p h e r i c a l l y symmetric f i n i t e ranged p o t e n t i a l can be obtained d i r e c t l y from eqn(a9). This i s accomplished by w r i t i n g the wave f u n c t i o n as vhC*'") = & L ( l L + | } i L T P u d k r > ) and u t i l i z i n g the Rayleigh expansion S u b s t i t u t i n g these two expressions i n t o (a9) and making use of the i d e n t i t i e s gives us L-O where (bi ) To determine the low energy behaviour of (b1) we must f i r s t know the corresponding behaviour for U^kr^. To t h i s end, we fo l l o w the procedure taken i n appendix A and s u b s t i t u t e the i t e r a t i v e equation ^0 1 i n t o the d i f f e r e n t i a l equation f o r UL • Doing so w i l l give us CO f - k^  or e q u i v a l e n t l y where = ra t + i 3L a n d For k ~ 0 we know that and -2*" L J^. + t p L+ V (2.L+ 0! Therefore - rr_£L - I ^ 1 k l L t ,'(r<rO which i m p l i e s that UL. must have the same low energy behaviour as J L . Hence i t f o l l o w s from and where Qt, b L , c L,...are a l l s c a t t e r i n g p o t e n t i a l . >1) that energy independent functions of 87 Appendix C Coulomb Wave Functions The regular and i r r e g u l a r s o l u t i o n s to designated $-u(<],p) and ^.fo,?") r e s p e c t i v e l y , form what are known as the Coulomb wave f u n c t i o n s . A complete d e s c r i p t i o n of these f u n c t i o n s i s given by Abramowitz and Stegun [ 1 ] . For S-waves, i t can be shown that i f C o represents the charge b a r r i e r p enetration f a c t o r (sometimes c a l l e d the Gamow f a c t o r ) , defined by then + - - • + 88 4 4-/ r 75 4- -and C 0 ^ 0 - *1 C F . [ 10321*11 p + where and 2V - 1 + A f + y£ 0 0 - loc^l q\ - V 4.' 75 3 3 75 4-89 The q u a n t i t y C ooV^ 0 i s o b t a i n e d from the i d e n t i t y I f we s e t m^c 2 = 938.21 Mev n y c 2 = 139.58 Mev Oi = 1/137 he = 197.32 Mev-Fm then rjk = mccx/ft = 0.00449467 Fm"' T h e r e f o r e i f R = 1 Fm, then t o f i v e s i g n i f i c a n t f i g u r e s we may w r i t e ( v a l i d f o r tick < lOOMev) C'Fo = ^,r>k + (.oo^stOk - (.oooso)k\ * (.ooooz) ks C^pFo c o s ^ + -oo°,oz - (.oozoo)kx +(.0OOil)kH {js0 = cos k - .OOOOG + (.ooosOk 1 - (.00004') kl and ^ - 4 . 5 5 7 3 0 1 ] +• V^o 90 Appendix D P o t e n t i a l due t o U n i f o r m Charge D e n s i t y We know t h a t the p o t e n t i a l a t r , due t o the i n t e r a c t i o n of the p o i n t charge e and the charge p(r)d3r may be e x p r e s s e d as V ( - 0 p c £Q d ' r 1 I r. - c' I I f the charge d e n s i t y P(r) were due t o a charge e d i s t r i b u t e d u n i f o r m l y over a sphere of r a d i u s R, then V ( 0 = & r * d 3 r ' To e v a l u a t e t h i s i n t e g r a l , we make use of i r - r i CO /\ L y — L= O and fdx (xO and r e w r i t e ( c 1 ) as V ( 0 -For r < R we have -t- I 3 £ l r," r I I d r r 1 - ^ - + / dr C r 2 & whereas ( d ) (c2) 91 Hence when (c2) i s averaged over a sphere of radius R we get V r 1 £ l 5 R 92 Appendix E I s o s p i n C o n v e n t i o n s The r u l e s f o r the c o u p l i n g of i s o s p i n 1 w i t h i s o s p i n 1/2 are the same as those f o r s p i n . Hence i f we r e g a r d the nucleons (n,p) and mesons (tr", rr6 , rr*) as the components of the d o u b l e t {\j>j),\{ , + i > ] and t r i p l e t { | I, -1> , I 1, 0 > , | I , H > } r e s p e c t i v e l y then j T + n 1 1 2. ' 2L 3 1_ "I ' 2. (3 3 +1 7_ ' 2. r 1 [T»f)) = | i , o ) | i ' t ) ^ -[=- f T | i - * i ) " j m ) - |i,o>|i.-i) - jf[iT|i.'t> + |*."i> rp) - |..-.)li.t> -^[li.-i>.-n:|i."i> r n ) - i ^ > l t . - i > = 93 T h i s i m p l i e s t h a t i f (I.I, * T- ' TII' ta.i; ( e l ) then the rrN s c a t t e r i n g a m p l i t u d e s f o r the p r o c e s s e s d i s c u s s e d i n the t e x t may be w r i t t e n as I t i s from t h e s e e x p r e s s i o n s t h a t the i s o s p i n a m p l i t u d e s , ( e l ) used i n c h a p t e r f o u r , are o b t a i n e d . 

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