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Multiple scattering theory with proper wave function symmetry applied to piondeuteron scttering at threshold Bendix, Peter Bernard 1973

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c MULTIPLE SCATTERING THEORY WITH PROPER WAVE FUNCTION SYMMETRY APPLIED TO PI0N-DEUTERON SCATTERING AT THRESHOLD by Peter Bernard Bendix B.S., California Institute of Technology, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1 9 7 3 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia Vancouver 8, Canada Date A p r i l 2 0 , 1 9 7 3 i Abstract The scattering amplitude for pions on deuterons is calculated in the threshold limit using Watson's multiple scattering theory. Care i s taken to use wave functions with proper symmetry throughout and i t is shown that the results are identical with those obtained using unsymme-trized wave functions in intermediate states. Terms up to second order in the multiple scattering series are calculated, gradually increasing the complexity of the assumptions u n t i l a l l quantitatively relevant features are taken into account. Sp e c i f i c a l l y , we treat single scattering, double elastic scattering, double charge-ex-change scattering, and second order binding correction terms. New quantitative results are obtained which acr, count for non-zero binding energy of the deuteron and nu-cleon excitation i n the propagators, Lorentz-invariant and inelastic scattering kinematic factors in the two-body scattering amplitudes, phase-shift f i t t e d pion-nucleon scattering amplitudes up to P-waves, an S-wave Gartenhaus deuteron wave function, and r e l a t i v i s t i c effects in high-momentum intermediate states. In addition, a general method u t i l i z i n g graphs analogous to Feynman diagrams i s presented which easily reproduces each order contribution of the multiple scattering series (for constant two-body T matrices) and allows ene to sum the whole series in closed form. In particular, we find the sum of the whole series for TT~-deuteron scattering at threshold, including a l l isospin-flipping terms, a result incorrectly obtained in previous l i t e r a t u r e . We also find the series sum for TT~ scattering on an arbitrary nucleus of neutrons and protons,including charge-exchange scattering. (This result does not appear in the literature,). From the series sum we then calculate the higher-order contribution with a Hulthen and then a Gartenhaus S-wave deuteron wave function f i r s t neglecting charge-exchange and then including i t . We find the higher-order contribution to be roughly twenty per cent of the f i r s t and second order terms combined (at threshold). Our best estimate of the pion-deuteron scatter ing amplitude at threshold (the pi-d scattering length) i s F ,= -.0273 fermis. Because pion-deuteron scattering is a three-body problem and because of the simila r i t i e s with multiple scattering theory, we have included a short discussion of the Faddeev equations. We give particular emphasis to wave function symmetry in the Faddeev approach. i i i Table of Contents page INTRODUCTION AND MOTIVATION 1 Chapter 1 MULTIPLE SCATTERING IDEAS PROM 5 BORN TO WATSON 1a The Born Series 8 1b Quantum Electro-dynamics 13 1c Preliminary Scattering Notions 25 1d Watson's Multiple Scattering Series 37 Chapter 2 SIMPLIFIED LOW ORDER PI-DEUTERON 47 SCATTERING USING WATSON'S MULTIPLE SCATTERING THEORY 2a Low Order Multiple Scattering Terms 52 for Tf-d Scattering Neglecting Symmetrization 2b Symmetrization of Intermediate States 63 Chapter 3 SUMMATION OF THE MULTIPLE SCATTERING 72 SERIES TO ALL ORDERS 3a Graph Rules and Order by Order 75 Summation 3b Series Sums for Arbitrary Nuclei 83 Chapter 4 REALISTIC PION-DEUTERON CALCULATIONS 87 4a Pi-d Scattering Complications One 88 Step at a Time 4b Binding Corrections 105 4c Higher Order Corrections 108 4d R e l a t i v i s t i c Corrections 111 4e 4f Chapter 5 5a Chapter 6 BIBLIOGRAPHY Appendix 1 Appendix 2 Appendix 3 Appendix 4 Appendix 5 Appendix 6 Appendix 7 Appendix 8 Appendix 9 Appendix 10 Appendix 11 i v page Summary of Corrections and Best 115 Estimate Review of Other Pi-d Literature and 1 1 7 Discussion of Results THE FADDEEV EQUATIONS AND SYMMETRI- 122 ZATION Symmetrization of the Faddeev Equa- 123 tions CONCLUSIONS 128 130 Field-theoretic Two-fermion Propa- 132 gator Deuteron Wave Function Conventions 136 Expectation Value of 1/r 138 Second Order Unsymmetrized Calcula- 140 tions Conversion of the T Matrix to the 145 Scattering Amplitude for F i r s t and Second Order Terms Second Order Symmetrized Calculations 147 Pi-riucleon Scattering Lengths 150 Relation Between the T Matrix and the 151 Scattering Amplitude for Inelastic Scattering Phase Shift Momentum-dependence for 152 Inelastic Scattering Pi-nucleon S-wave Phase Shift Para- 156 meters Pi-nucleon P-wave Phase Shift Para- 160 meters V Appendix 12 Treatment of Appendix 1 3 R e l a t i v i s t i c page Spin-flipping Mechanism 163 Propagator 170 v i L i s t of Tables Table I Table II Comparison of Hulthen and Gartenhaus_s Results for P. F i r s t and Second Order Pi-d Scatter-ing Amplitudes for Various Complica-tions page 90 104 v i i Acknowledgement The author expresses his indebtedness to his advisor, D.S. Beder, not only for his p r o l i f i c technical advice throughout the course of this work, but also for provid-ing an atmosphere conducive to an informal faculty-student relationship, for always being available for discussions, and for patience extraordinaire with a most captious graduate student. In addition, the author gives his sincere apprecia-tion to the United States government for i t s undying interest in his a c t i v i t i e s and for allowing him to com-plete his graduate l i f e i n Canada. 1, Introduction and Motivation The most important part of any calculation i s not the result but the reason for undertaking the task at a l l . The organization of the finished thesis does not reflect the chronological order of events usually, so the begin-ning of this thesis w i l l be devoted to an explanation of how and why the body of the enclosed work began and devel-oped. It was suggested by D.S. Beder that low energy scatt-ering of pions on deuterons should be re-investigated because early work by various people f a i l e d to account for the Pauli exclusion principle i n intermediate states.^ The low energy range was chosen because here the effects of the exclusion principle would be more dramatic (for a detailed explanation of this statement see chapter 2). But why look at deuterons at a l l ? The looming pres-ence of the TRITJMP meson f a c i l i t y has influenced the course of more than one graduate research proposal and having a straight-forward calculation ready before the machine i s in operation would give experimentalists a chance to make the real world f i t the theory. Besides i t looks good to See for example references (2), ( 1 0 ) , ( 1 1 ) , ( 1 9 ) . (22), (23), (24), and (26). 2 everyone i f a camaraderie between various groups is man-i f e s t . In addition, maintaining an interest in strong interaction physics is not incompatible with investigating pion-deuteron scattering. Very l i t t l e progress has been made in recent years on the pi-nucleon interaction and one might hope for some clues to the two-body problem from the three-body one. In support of this notion, Love-lace has said,^ My opinion is that two-particle systems are now finished. By this I do not mean that we have done everything we hoped to do, but rather that we have done everything we are going to be able to do. I think the future of strong interactions now l i e s with many-particle systems. More s p e c i f i c a l l y , the pi-deuteron scattering amplitude depends on the of f - s h e l l pi-nucleon scattering amplitudes which are not well understood. Many recent strong inter-action theories inter-relate the off-shell and on-shell scattering amplitudes and a knowledge of off-shell behav-ior would lead to better understanding of on-shell pro-perties. Of course off-shell behavior cannot be deter-mined experimentally from the pi-nucleon interaction alone; therefore, the pi-deuteron scattering amplitude i s the simplest candidate to yield off-shell information on the pi-nucleon interaction. F i n a l l y , the methods used in treat-See reference (20), p.437 3 pion-deuteron scattering might shed some l i g h t on the less-understood two and three-particle resonances of the hadrons. With the previous statements in mind, the work was 3 undertaken using well-known techniques, adding-more and more complications in an effort to find the quantitatively correct pion-deuteron scattering amplitude in the thresh-old l i m i t (zero energy pions). But when the exclusion principle was applied by symmetrizing (which we take to mean making symmetric or anti-symmetric) intermediate state wave functions, divergences appeared, terms which looked 4 out of place crept i n , and chaos ensued. People suggested throwing away the nasty terms or cancelling them by adding others with opposite sign, but no suggestion could be jus-t i f i e d . Finally after normalizing the symmetrized propa-gators correctly, a consistent and ju s t i f i a b l e treatment was discovered which eliminated the divergent terms. It was then fo\ind that the fi n a l results are independent of the choice between symmetrized and unsymmetrized intermedi-ate state wave functions (provided i n i t i a l and f i n a l states See the last footnote of chapter 4 , section F. 4 Private communication with members of the University of Br i t i s h Columbia physics department, including Rubin Landau and Leonard Scherk. 4 are properly symmetrized always). Moyer and Koltun (ref-erence (22)) point out that i t is unnecessary to use symmetrized wave functions in intermediate states in the Lippmann-Schwinger equation, T = V + VGT, because V i s already symmetric in a l l target particles. However, i t i s quite another matter to draw the same conclusion?;for the Watson multiple scattering series for T. We demonstrate in this thesis exactly how the symmetrization effects in the usual multiple scattering terms and the binding correction terms cancel in pairs to each order. Even though i t was found that symmetrization of inter-mediate states is unnecessary, the time spent worrying about the divergent terms was not wasted. It was believed that the sum of the whole multiple scattering series, diver-gences and a l l , would be f i n i t e ; therefore, a method was devised which allows one to easily sum a l l the orders of scattering. After the divergence problems were eliminated, the method of summing the whole series s t i l l remained valid and so laborious methods of the past were simplified enor-mously. This in turn f a c i l i t a t e d the correcting of previous 5 ly incorrectly evaluated series sums in the lit e r a t u r e . See the Appendix of reference (28). 5 1.Multiple Scattering Ideas from Born to Watson Although Watson's multiple scattering theory did not appear u n t i l the early 1950's, the basic notions of mul-tiple scattering had been formulated well before. In particular, the earliest interpretation of multiple scatt-ering i s found in the Born series iterative solution of the Schr*6dinger equation.^ The r e l a t i v i s t i c analog of this is manifest in Feynman's diagrammatic approach to quantum electro-dynamics (QED). ; The Born, Feynman, and Watson approaches share the properties of starting with Green's function techniques to solve a d i f f e r e n t i a l equa-tion and iterating the solution ( i n multiples of some relevant scattering parameter). A physical interpretation i s then attached to each type of term in the expansion and a diagram can be drawn to represent each term, thus reducing messy algebraic manipulations to graph problems and associating an integral (which can be written by in-spection) with each graph. Historically and in practice, graphs are not used i n the Born or Watson methods because the number of terms retained is usually small and the expressions relatively simple compared to their r e l a t i v i s t i c See any quantum mechanics text on the Born approximation. What we here c a l l the Born series is perhaps more correctly termed Green's function theory. 6 counterparts i n quantum electro-dynamics. We emphasize the association of graphs with Born and Watson terms here partly for comparison with QED but also with some foresight regarding summation of Watson's series. In what follows we w i l l derive the Born series by formally solving the Schro'dinger equation to obtain an integral equation for the wave function; iteration of the integral equation produces a series expansion for the wave function. The scattering amplitude i s defined and using the series expansion of the wave function we obtain the Born series for the scattering amplitude. Graphs are then associated with each term in the expansion of the scattering amplitude. The Feynman approach i s similar to the Born work except r e l a t i v i s t i c equations (Dirac and Maxwell) for the wave functions replace the Schro'dinger equation and complications arise from the more d i f f i c u l t equations and from invoking particle s t a t i s t i c s (Bose-Einstein and Fermi-Dirac) on the wave functions. The complexity of the solutions i s greatly simplified by the use of Feynman diagrams which we introduce and use to comment upon a few relevant d i f f i c u l t i e s which w i l l appear in the Watson work. We only treat the Feynman approach heuristically owing to the complexity of the problem. 7 In the Watson work, we lay the preliminary groundwork of scattering theory and then derive the Watson multiple scattering series i n terms of the T matrix, closely re-lated to the scattering amplitude. The Watson work i s an extension of the Born work in that the scattering potential is broken up into a sum of ,two-body potentials between the incident particle and each constituent of the scatterer (nucleus), but then the total scattering amplitude is expressed in terms of two-body scattering amplitudes rather than in terms of the total potential. The sim-i l a r i t i e s to the Born and Feynman subsections should be obvious by the end of the Watson derivations and we w i l l avoid lengthy comparisons since they w i l l be undertaken when we try to explain away symmetrization d i f f i c u l t i e s in Watson's theory. 8 1a.The Born Series •We consider the scattering of a particle of mass m and momentum nk by a potential V(r) (where r i s the pos-i t i o n vector from the potential source to the p a r t i c l e ) . The Born series solution begins with the Schrodinger equation re-arranged to > t t ) V (?) = i S ^ V(Jf)^(r; C l a - 2 ) Our choice of Green's function G(r,r') s a t i s f i e s ^ O a - 3 ) & ( £ ? ' ) = &(?:?) This implies that % /5( ? - r " ) <i-« 1 implies derivatives with respect to r . Let ( V 2 + k 2 ) 1 be the appropriate inverse operator of ( v 2 + k 2 ) , (see equation (1a-11)). 6See equations (1d-45) and (1a-11).. 9 2 2 - 1 Then multiplying (1a-3) on the l e f t by (V + k ) (1a-5) Multiplying (1a - 2 ) on the l e f t similarly V ( ? ) = ^ r (^+^)" 'v (? )VC?) (,a-6> Multiplying (1a-5) "by Y ( r / ) V ( r/) and integrating over r ' and using (1a-6) we immediately obtain were C i s a constant of integration ( i . e . , a function of r but not r ) . To determine C, note that as r G(r, r') 0 (see equation (1a-11)).,and i n this l i m i t V (r) must reduce to a plane wave (the i n i t i a l state wave function) travelling along the k direction so that > -3 /x f s . r The complete solution of the problem is thus (1a-9) with the i n i t i a l state (given in (1a-8)). 10 Iterating (1a-9) for V produces the Born series expansion for the wave function in the form V(?Y- Vr(r)+ \ G ( ? , ? ' ) V ( ? ' ) W?9 iV (1a-10) The solution of (1a-3) with appropriate boundary conditions is (1a-11) - 9 - » / For r >> r we can write (1a-12) so that ( 1 a - 1 3 ) and therefore one obtains 5 See reference ( 1 5 ) , p.303, eq..364b. 11 From (1a-H) we can read off the scattering amplitude, f ( 9 ) , defined as the coefficient of C_ , where r the scattering angle, 9, i s defined as follows: Write a f i n a l state plane wave, 4^  , in terms of the f i n a l momentum (more correctly,wave vector) as where = k^ r / r . For elastic scattering we write therefore kf = = k ( i stands for i n i t i a l ) 9 = angle between k^ and k^ Now . <;write f(9=) from (1a-14) by using (1a -15) to obtain 1 ' (1a-16) + . . . j where the 9-dependence on the right side of (1a-16) i s implicit in that • & ~ R COS& and means complex conjugate 12 The equation (1a-16) is. interpreted as follows: The complete scattering amplitude f(9) i s the amplitude to scatter once plus the amplitude to scatter, propagate, and scatter again, J % ( ? ' ) V(r'J> G { r ' } ? ' ) V ( r V V. (r*; JV'JV' etc. Diagrammatically we would represent the single, double, etc. scattering terms, respectively, as Oa-17) where solid lines represent the incoming, prop;agating, and scattered p a r t i c l e , and a dotted line with a '® represents one interaction with the potential V. This summarizes the relevant features of the Born series. 13 1b.Quantum Electro-dynamics No attempt w i l l be made to derive quantum electo-dynamics; rather we w i l l show the way Feynman's diagram-matic approach i s applied to some simple scattering prob-lems after some very intuitive discussion on the origin of the diagrams. The resolution of some d i f f i c u l t i e s in QED which are relevant to Watson's theory w i l l also be A 6 covered. Consider the Dirac equation for an electron where p i s the electron four-momentum operator, m is the electron mass, x is the electron four-vector pos-i t i o n coordinates, and (x) is a four component spinor wave function representing the f i e l d of the electron. The slash through p means where the ^ are 4x4 Dirac matrices. Feynman writes in analogy to (1a-9) the integral equation for the wave For a more detailed account of a l l that follows on QED see references ("1):_and (12). 14 (1b-2) where 4^  ^(x) i s a solution of the f r e e - f i e l d equation (1b-1) before scattering, e is the charge of the electron ( e < 0), Sp(x-y) is the propagator for Dirac particles (corresponding to G(r5,r')), and A is the electro-magnetic four-potential (corresponding to V ( r ) ) . Looking back at section 1a, we see that the equation for G(r,r'),(1a-5), i s h e u ristically obtained from (1a-2) by inverting the operator on the l e f t of V(r) i n (1a-2) (with V=0); i.e., G Q ^ r ' ) ^ ( vVk^"'xl ..U-b-3) realizing that the delta function is the unit operator, 1. , in the coordinate representation. In analogy, S,,(p), the Fourier transorm of S_,(x-y) i s obtained from (lb-1) by inverting the operator to the l e f t of 4^(x), so that in momentum space , j f ~* f - ^ (1b-4) Everything in (1b-2) is determined except A. The electro-magnetic four-potential satis f i e s Maxwell's equations 15 \ \ ^ C * ) ' > W <1b-5> where j (x) is a four-current (the source of the potentials) Again in analogy to (1a-5) and (1b-4) we obtain the photon propagator, Dj,, from (1b-5) by inverting the oper-ator to the l e f t of A as where q is the four-momentum of the photon f i e l d (note that q's Fourier transform is -± )u) . Then in analogy to (1a-9) and (1b-2), p where Dj,(x-y) i s the Fourier transform 63 Dp(q. '). Put (1b-!£) into (1b-2) to get V(x) -- V. (*) + e [ J*.,«le$>!)) PF(r*)/T(j) <"-a) When we obtained our expression for the scattering ampli-tude (1a-16), part of the left-most G ( i n (1a-9)j became the t / r of (1a-16) and the rest of this left-most G became ^ \±n (1a-16).c In analogy the expression for 16 the scattering amplitude in the QED case removes the left-most Sw of (1b-8) to become (aside from kinematic factors) „here (3) = ^ ^ the T means hermitian adjoint, and ]f is one of the Dirac matrices. The iteration of (1b-8) for into (1b-9) gives the analog of (1a-16); namely, (1b-10) Just as the solution of (1a-16) i s determined i n principle once the potential source V i s known, the solution of (1b-10) is determined once the current source j i s known. Let's see how this works with an example. Suppose we want to treat coulomb scattering of an electron by a point charge Q located at the origin. 17 Then the appropriate four-current i s By (1b-10), to f i r s t order the scattering amplitude i s \ - 4 j D F ( r o f e ^ (•j)^"1 ( n " 1 1 ) and corresponds to the diagram (1b-12) where the two soli d lines represent the incoming and out-going electron ( and fjr; )» wavy line represents the propagating photon (Dp), and the ® represents the source of the photon (Q b (a)). The V*s both connect to the point labeled y, the photon propagates from z to y, and the source of the photons i s at point z. The next higher term of (1b-10) i s represented diagrammatically by u) (1b-13) where internal s o l i d lines connecting two photon lines represent a propagating electron (Sj,), and the coordinate 6See reference (1), Vol.1, p.100. The f of (1b-11) as written w i l l contain an energy-conserving & function which should be removed. 18 labels at each vertex correspond to those used in (1b-10). The idea is to f i r s t draw and label the diagrams and then write the corresponding integrals from the diagrams by-inspection rather than start with the more cumberson (1b-10). Now we come to one of the most important aspects of QED that w i l l carry over to multiple scattering theory. In everything done so far we have not symmetrized the terms as required by Bose-Einstein and Permi-Dirac s t a t i s t i c s (we could also symmetrize the Born work but choose to symmetrize only QED here to avoid duplication and because the problems which arise i n QED are relevant to multiple scattering). Consider electron-positron scattering. The relevant current is now the current of the positron as seen by the 7 electron (or vice versa) and is given by j * CO w ? , r J" V / r * ) <1*-H> where the superscripts on the VJ/ *s mean positron states. The choice of j is motivated by the fact that j should be See reference 1, p.135. 19 density which is something on the order of T i A second order graph looks li k e (second order in e, f i r s t order in e^) (1b-15) (where subscript 1 refers to electron, 2 to positron, unprimed p's are i n i t i a l momenta, primed p's are f i n a l momenta) and corresponds to a term Peynman states that electrons going forward in time are equivalent to positrons running backward in time so that we would more correctly draw (1b-15) with arrows indicating the direction of time for an electron as (1b-17) in which we replace a positron with momentum p^ going for-ward i n time by an electron with momentum -p^ going back-ward i n time. Let's try to symmetrize the above diagram 20 by interchanging electron lines where possible. At y an electron entering from the past can proceed into the future as shown with momentum p^ but i t can also go back into the past with momentum ~j>2> a^ z "^ne electron coming fn from the future with momentum -p^ can scatter into the past with momentum -p^ as shown or i t can scatter into the future with momentum p1. We would draw the "exchange diagram" resulting from the above observation as (1b-18) in which lines p1 and -p2 of ( i b - 1 7 ) have been interchanged to obtain (1b-18). In other words, we can interchange electron lines as long as we conserve charge; i.e., some-thing l i k e is not allowed because the electron converts to a positron (the direction of the arrow reverses as we follow the continuous world line of the electron on the l e f t ) . It is more conventional to twist the photon lin e of (1b-18) 21 into Ob-19) so that the exchange diagram of the second order electron-positron scattering corresponds to annihilation into a photon followed by decay into another electron-positron pair. Fermi-Dirac s t a t i s t i c s require that (1b-19) adds to (1b-17) with a relative minus sign (two fermions were exchanged). Now consider fourth order electron-positron scatter-Q ing. The most obvious graph i s and exactly as in the second order case, interchange of f. (1b-20) (1b-21) a Ibid., pp .H8 -151 . 22 where we interchanged the location of the bottom of line -Pg with the top of line p^ in obtaining (1b-21) from (1b-20). The fermion interchange requires that (1b-21) adds to (1b-20) with a relative minus sign. Interchange of the right side of line q^ with the top of line q.1 in (1b-21) yields which must add to (1b-21) with a relative plus sign (boson interchange). The integral corresponding0 to (1b-22) d i r s verges, and i n general a l l graphs with photon bubbles on electron lines l i k e w i l l diverge. To eliminate these divergence problems one proceeds very crudely as follows: The electron propagator (drawn as a solid line and given by (1b-4))is modified by add-ing to i t a l l bubbles; graphically we write r, (1b-22) f. Ob-23) 23 changes t© (1b-24) The i n f i n i t e series of divergent terms can be summed and the result changes the propagator to ) " (1b-25) — rm. + £ where A i s the i n f i n i t e contribution of one bubble. One then says that the mass m i n (1b-4) is the "bare" mass of the electron and that the physically measured mass i s m - A = m n . (1b-26) electron So the theory becomes "renormalized" by changing m to m in a l l electron propagators. The same type of divergences w i l l occur in multiple scattering theory upon symmetrizing, One last exchange graph must be discussed. Appro-priate interchange of photons and electrons in (1b-20) w i l l lead to the graph ( , M W ( 1 b " 2 7 ) whose corresponding integral also diverges. Graphs such 24 as these are called disconnected and are thrown out in QED. The j u s t i f i c a t i o n for throwing them away is this the vacuum is constantly producing such bubbles by spon-taneous emission and reabsorption and since everything is measured relative to the vacuum one ignores such graphs. The ideas discussed for QED w i l l appear again when we use • Watsdrn •"sc^theiory. 0 25 1c.Preliminary Scattering Notions In obtaining the Born series and QED results, we worked with the basic f i e l d equations and interaction potentials. In multiple scattering theory, especially when applied to strong interactions (nuclear physics), the basic interaction potentials and f i e l d equations are not known. Therefore the expansion for the total scatt-ering amplitude of a projectile on a composite system of scatterers (nucleus) i n powers of the two-body potentials between the projectile and scattering constituents i s replaced by an expansion in powers of some more directly measurable quantity. Since the objective i s to treat scattering of a particle by a collection of other par-t i c l e s , the most directly measurable quantities involved are the d i f f e r e n t i a l cross sections for the incoming par-ticle>scattering on each of the constituents of the scatt-erer. Any quantities closely related to the experimentally observed two-particle d i f f e r e n t i a l cross sections are l i k e l y candidates to use as a replacement for the two-body potentials. Historically one chooses to expand in powers of the so-called "T matrix". In the next several pages we develop the groundwork which defines the T matrix and 26 relates i t to the potential V, the scattering amplitude f , and the dif f e r e n t i a l cross section. We imagine a two-body scattering experiment to start with a free particle approaching a scatterer at time t=- ^  , the free particle state unaffected at this time by tine presence of the scatterer. As the particle moves closer to the scatterer, the scatterer starts to modify the free particle wave function and continues to do so unt i l the particle gets sufficie n t l y far away from the scatterer again, with the wave function becoming a new free particle state (or superposition of many free particle states) at t=+ oo . Define the Heisenberg S matrix by describes the system at time t (similarly X i s the free particle state before the scattering event). By our now familiar Green's function work, which comes from (1a-9). Now put (1c-2) into (1c-1) getting <X t (S I X ^ - X^2t)£a*)A* <1-1> 27 9 Now sacrifice rigor for brevity. The time-dependent part of the product of X- ^ and a contributes a factor i(£rOt ^ v • y and the propagator G behaves like i 2 Here the k part brought in the i n i t i a l energy E and fit 2 the y part brought in the energy at t=+oo which i s Eb (the minus sign in front of E^ comes from the fact that E= p /2m o< -7 /2m). Piecing this together transforms the S matrix (1c-3) into d c - 4 ) where the wave functions i n the integral have their time-dependences e x p l i c i t l y removed. Now use the identity Those appalled by what follows can consult reference (15), p.178, eq.16 and preceeding pages. 28 U e — - — r - a n l i f e - E^ N ( i e - 5 ) to write (1c-4) as < M s K > <\(*0> - (vo (l0.6) which is exact (even though derived heuristically here). The T matrix is usually defined in terms of the S matrix by <XJS|X*) = < X b l X . > - J n . J f e - ^ ) ( 1 c . 7 ) As (1c-7) is not a particularly useful form for our pur-poses, we led the reader up to (1c-6) so that we can make the correspondence^ O C j T l X O = ^ J V t K > d o - e ) an expression we w i l l make direct use of in obtaining the Tjippman-Sch winger equation, the cornerstone of Watson's theory. . - — Ibid., p.178, eq.16. 29 We have now defined the new expansion parameter T (which we w i l l use instead of V) and have related T to V by (1c-8). Next we must show how T is directly related to the scattering amplitude f , or equivalently, how i t is related to the d i f f e r e n t i a l cross section, the quantity determined experimentally. As a f i r s t step in obtaining an expression for the d i f f e r e n t i a l cross section in terms of T, we state that (by the definition of S) the transition probability, Pb a» to go from state X to state X- k is P U = \<^ lS|7U.>r CI 0 - 9 ) In most scattering experiments one doesn't measure scatt-ered states which overlap i n i t i a l states X because the detectors would be swamped by the incoming beam. Thus we can say d c - i o ) so that using (1c-7). The square of the & function is tricky; 30 11 to treat i t properly we paraphrase Bjorken and D r e l l . Jf we consider transitions in a time interval from - t / 2 to +t /2 the & function would be smeared out and we o' would replace i t according to ( 1 0 - 1 2 ) 1 E - E . v> We then have, squaring the above equation and integrating both sides over an)2" J (o) 2nt6 This allows us to write (1c-13) (auf [f ^ = c o J ^ , - * 0 ( 1 c - H ) The abjove is the desired result which converts the square of the <j function into a single energy-conserving i function. Combining (1c-H) with (1c -11) , the transition probability See reference ( 1 ) , p.101. 31 per u n i t time i s found to be • ± -- J( e i-«0|<x t |T(;cx>r c i ^ i 5 ) The T matrix conserves momentum so i t i s customary to f a c t o r out the momentum conserving part by de f i n i n g a new T through i h \ r \ t ^ - - J ? ( f V J v ) T u . ( 1 ° - 1 6 ) where p a and p^ are t o t a l momenta of p r o j e c t i l e plus t a r -get f o r i n i t i a l and f i n a l states, r e s p e c t i v e l y . In a manner analogous to our treatment of the squared energy & function, we smear out the squared momentum & function over a f i n i t e volume of space, v , to write the t r a n s i t i o n p r o b a b i l i t y per u n i t space-time as Pi v t The t r a n s i t i o n rate per unit space-time into the i n t e r v a l 3 / 3 / d p.jd p£ is then <l / J (1c-18) 32 where subscript 1 refers to projectile and 2 to target momentum in the f i n a l state. The cross section is the transition rate per incident flux of particles summed 12 over a l l f i n a l states, or (1c-19) where vr e l is the relative velocity of projectile and target. Now work in the center of momentum system of the projectile and target; i.e., P.+ ft" P*.-= p t= p,'+ px'= O V,= P i / E , ' Differentiating both sides of the f i r s t of the above equa< tions and using the second of the above equations we have The (2IT ) comes from flux considerations; see e.g., reference (15) , p.87. 33 where we also used the r e l a t i v i s t i c connection between a l l E's and corresponding p's m Multiplying (1c-21) by p1 and re-arranging gives The expression (1c-19) for the cross section when d i f f -erentiated with respect to solid angle and using (1c-22) becomes an expression for the d i f f e r e n t i a l cross section — dc-23) where p1 i s the projectile momentum i n the f i n a l state, E^ and E^ are the projectile and target f i n a l (total) energies, respectively, vr e l is the relative velocity of the target and projectile given by (1c-24) 34 unprimed p's and E's refer to the i n i t i a l state, and (1c-23) i s the di f f e r e n t i a l cross section i n the center of momentum system of the projectile and scatterer. By definition of the scattering amplitude, f , given 13 hy the-statements following (1a-14) we can write A S L .41 do-25) which establishes the correspondence between j f | and l T bal 5 o n l v "t n e relative phase between f and T^a is l e f t undetermined. Taking the non-relativistic l i m i t of (1c-23) and (1c-24) and let t i n g the scatterer's mass become i n f i n i t e (fixed scatterer) we obtain the limiting case (1c-26) which can be compared with results we w i l l obtain using section 1a since there the non-relativistic limit and fixed scatterer were assumed. If we take equation (1a-16) and write i t i n i t s non-expanded form i t becomes See reference (8), p.57. 35 using our new notation for free and interacting states. But by (1c-8) and (1c-16), Comparing (1c-28) with (1c-27) we get the relative phase between f and from ba V - - ± S . - L . * O o . 2 9 ) By taking the square root of (1c-23) we can define the relation between f and T^a for elastic scattering (primed 14 and unprimed p's and E's equivalent then) (-t;*0 So far this section has been rather long-winded but necessary to establish conventions; l e t us b r i e f l y review See reference (15) , p.223, eq.6. 36 what has been done. We wanted to expand the total scatt-ering amplitude for a projectile on a compositer-^ystem of scatterers i n terms of some new quantity T closely related to the more directly measurable f , the scattering amplitude of the projectile on one constituent of the composite scatterer rather than expand in powers of the harder-to-determine two-body potentials between the pro-je c t i l e and scattering constituents. The f i r s t step was to define T (equations (1c-7) and (1c-1)) .and find the correspondence between T and the scattering;potential V (equation (Vc-8)). F i n a l l y , i t was necessary to relate T to f (and thus to the dif f e r e n t i a l cross section also) to connect T to a real world experiment (equations (1c-30) and (1c-25)). Having accomplished a l l t h i s , we now proceed to develop the Lippmann-Schwinger"equation and Watson*s multiple scattering theory. 37 id.Watson's Multiple Scattering Series Consider a scattering event with a total Hamiltonian H describing the system. Begin by breaking up H into the free Hamilitonian K and the potential V according to H-= K+V <ia-i> and define eigenstates of H and K by fKP= ^ ( 1 d . 2 ) K l - E Note that H, K, and V are operators in the above. Now find an expression for ^ in terms of Substituting (1d-1) for K i n the second equation of (ld-2) Re-write the f i r s t equation of (1d-2) and (1d-3) as ( e - H ) V = o Subtracting the second of the above from the f i r s t gives 38 (E- H)(V- *0 = V X dd-4) Multiplying (1d-4) on the l e f t hy (E-H)"1 and adding X, to both sides one gets To show the analogy of this treatment to section 1a. invert (1d-5) for in terms of Sf^  by f i r s t multi-plying by V yielding vv= [ v + v ( - i ^ ) v ] X (10-6) and now combine the two terms on the right into one by writing Inverting for we find ( i ) ( e - H ) ( e J r | c ) v H ; ( 1 d . 7 ) 39 Now put (1d-7) back into (1d-5) and cancel operators with their inverses to obtain the Lippmann-Schwinger equa-tion for the wave function ^ = + ( i ^ ) v ^ (1d"8) This i s identical to (1a -9 ) of the Born work. In other words, judicious juggling of (1d-5) reproduces (1a - 9 ) . Leaving this digression, we proceed to find an oper-ator expression for the T matrix. Equation (1c-8) states that <*JrK>, < < \ \ \ t \ V K y dee) and (1d-5) into (1c-8) gives which immediately implies T - V 4- V (i^)v dd-9) Invert the above to get V in terms of T by combining the two terms into one, writing 4-0 and invert to get T Putting the above into the rightmost V of (1d-9) leads us to This is the T matrix form of the famous LippmannrSchwinger equation which we w i l l use ad nauseum throughout the: rest 15 of our multiple scattering work. We now proceed with the idea of doing nuclear physics. Consider the scattering of a single projectile on a com-pound system of n scatterers. The total potential V i s then written as the sum of a l l two-body potentials between the projectile and each scatterer; i.e., Let ^ represent the two-body T matrix for the projectile scattering on particle i . Then the two-body Lippmann-Schwinger equations are (1d-10) (1d-11) ^Ibid., p.751, eq.252b. See also p.198, eq.85 for our equation (1d-8) which Goldberger and Watson refer to as the Lippmann-Schwinger equation. 41 (1d-12) where 3e ( 1 d-1 5> in which Ei i s the total energy of the incoming projectile plus'target particle i i n the i n i t i a l state before the scattering, and i s the free energy operator with the property (1d-14) where p and m are the projectile momentum and mass in state |Xj^ a nd p^ and m^  are the target momentum and mass in state The total T matrix for the complete scatt-ering s a t i s f i e s the appropriate Lippmann-Schwinger equation also; namely, T ~ V + V CrT (1d-15) where V i s given in (1d-11) and G- — ~ z d d - 1 6 ) i n which E is the total energy of projectile plus a l l 42 target particles i n the i n i t i a l state before scattering and K-- Z K . c-l (1d-17) Invert (1d-12) by writing so that V . - t and thus (1d-18) Iteration of (1d-15) gives (1d-19) and putting (1d-18) into (1d-l9) we find 1 (1d-20) 4 43 Now expand the terms in parentheses according to dd-21) and re-group terms in powers of the t's to obtain T= I t ("I Z X t.6t : - I t , , . ! -I- . . . Re-arranging the sums in each order to separate out terms with the same t ^ appearing next to i t s e l f (separated by propagators of course) we have1^ T = (1d-22) Terms which contain factors (G-gi) represent binding corrections which are worthy of consideration i n them-17 selves, but one customarily writes (1d-23) 16 17 See reference (22). See the work of Koltun in reference (29), for example. 44 and ignores the contribution of these binding correction terms. We w i l l return to this point when we discuss sym-metrization. This leads us to Watson's formula for the total T matrix in terms of the two-body T matrices, i-i <'=» l-l { J i=j j - i fe-i * ) *• + ... (1d-24) and establishes the f i n a l result associated with the t i t l e of this section. Now we have to state that everything leading to (1d-24) was done so crudely that we are compelled to make some verbal explanations. F i r s t , the way we write the propagators as in (1d-13) or (1d-16) is something li k e shovelling a l l the di r t under the rug. There are problems with the denominators when they become zero. Suffice 17 to say that we should make the replacement G - I; E - K * i n (1d-25) where P means take the principal value of the appropriate 17 - ' - - - ' See reference (15), p.72, eq.52 and p.74, eq.63. 45 integral in which G appears. In summary, (id-16) i s sloppy and should be replaced by (1d-25), and similarly a replacement should be made for (1d-13). Second, the t's that we have written i n (1d-24) are two-body T matri-ces and (1d-24) implies that the projectile scatters off particle i and a l l other n-1 scatterers s i t as spectators; this i s certainly an approximation (the so-called impulse approximation) and the more correct statement of Watson's result i s that a l l the t's in (1d-24) are really TJ • s, where X, is the appropriate T matrix for scattering of the projectile on particle i i n the presence of a l l 18 the n-1 other scatterers. Third, we have not dealt with the problem of symmetrization of the wave functions (nuclei are composed of many identical particles); though we w i l l not go through this section again properly sym-metrizing everything, l e t us just state that a l l states, i n i t i a l , f i n a l , and intermediate, must be properly sym-metrized. One then finds that symmetrized intermediate states give exactly the.' same results as unsymmetrized intermediate states (provided i n i t i a l and fi n a l states are always symmetrized in both cases) when properly handled. We w i l l return to this point in section 2b and related 1 8I b i d . , p.754, eq.265. 46 appendices. Note that our statements do not necessarily apply when the projectile is the 'same as one of the nuclear constituents. The reader should pay particular attention to Goldberger and Watson, reference (15), pages 131 to 133 and the last sentence of page 750 which continues over to page 751. We quote i t as follows: We naturally assume, however, that the target wave functions g are appropriately symmetrized i n the coordinates of identical bound particles. One f i n a l comment i s necessary. Our derivation of Watson's multiple scattering series, (1d-24), is not the derivation of Watson but follows the simpler but less rigorous argu-ments of Moyer and Koltun (reference C'22)). Watson finds coupled integral equations for the wave function ^ which formally solve equation (1d-8). These coupled equations are as follows: Iteration of (1d-26) combined with (1c-8) reproduces (1d-24) except for the replacement of the t's by 't •s (defined on the previous page). 19 Ibid., p.751, eqs.253a and 253b. n (1d-26) 47 2.Simplified Low Order Pi-Deuteron Scattering Using Watson*s Multiple Scattering Theory Our ultimate goal i s to obtain the pi-d1 scattering amplitude using Watson's multiple scattering theory and taking a l l quantitatively relevant effects into account. Naturally this i s a large order to serve, so rather than present the meal in i t s entirety, we choose to offer i t in several courses. To keep the arguments as simple as possible, we w i l l concentrate on TT -& scattering in the threshold l i m i t (the pion stikes the deuteron with zero momentum in the center of mass and lab systems). The incident projectile is chosen to be a pion (rather than a proton or neutron) to avoid extra symmetrization d i f f i c u l t i e s and because the two-body pi-nucleon inter-action i s well known (phenomenologically at l e a s t ) . Also pion beams w i l l be readily available once the TRIUMF meson f a c i l i t y is in operation. We choose a n rather than a TT0 to avoid coulomb effects in intermediate states of the charge exchange process (to be covered in detail shortly) although this is a rather moot point since the advantage is lost in the elastic scattering contribution. A deuteron i s chosen as the target nucleus because i t is the simplest multi-particle aggregate, the deuteron wave 48 function is well-known, and we naively expect symmetri-zation to make a big difference here (see the third para-graph below). F i n a l l y , we consider zero energy scattering because in this limit we can throw away the delta function part of the propagators (to be explained when we take 20 this step) and the expressions become simpler. The menu of the pi-d banquet begins with calculations of low order scattering terms neglecting symmetrization of intermediate state wave functions. Then we discuss how symmetrization of intermediate states is handled and why i t has no effect on the results. We assume constant two-body T matrices throughout this section ( i . e . , the choice (2a-6)). This approximation is a'highly unrealistic one but is introduced nevertheless because i t greatly simplifies the calculations. In momentum space the constant two-body T matrix is just a constant multiplied by an overall momentum conserving delta function. We abandon the constant two-body T matrix assumption in chapter 4. See equation (1d-25). 49 Note that the symmetrization effects we expect to see only occur i n second and higher order scattering terms hut not in f i r s t order (there i s no sum over intermediate states in f i r s t order). Now usually when one does a calculation of a sum of terms one expects the f i r s t order to dominate, and each successive order to contribute much less than the previous one. However, by accident nature has chosen to make the 77 -neutron scattering amplitude almost equal in magnitude and opposite in sign to the T T -proton scattering amplitude at threshold. The f i r s t order contribution in the deuteron is roughly the sum of these two scattering amplitudes so that the f i r s t order contri-bution i s very small, and in fact about equal in magnitude to the second order contributions. For this reason the deuteron i s an unusually good target for studying higher order (second and beyond) effects. Two remarks on symmetrization in the deuteron are now in order. F i r s t , i f one neglects the small D-wave admix-ture i n the deuteron wave function, the state is then pure S-wave (the orbital angular momentum between the neutron and proton i s zero). Now consider double charge-exchange (second order charge-exchange) scattering, to be referred 50 to from here on as DCE, in which a Tf" enters and strikes the proton converting the proton to a neutron and i t s e l f to a TT°; thus one has an intermediate state consisting of two neutrons and a i f0; the proceeds to strike the original neutron converting i t to a proton and the TT 0 back to a TT . Since the original proton and neutron were in a relative S-state, an incoming zero energy pion shouldn't excite the nucleons so we expect the intermediate state nucleons (two neutrons) to s t i l l be predominantly in an S-state. In addition, the deuteron has total angu-l a r momentum J=1 and since L=0 (S-state), we must conclude that the spin i s S=1. So i f we don't allow for any spin-flipping mechanism in the pi-nucleon interaction, the intermediate state of two neutrons must also have S=1 . This means that the spin state of the two neutrons i s symmetric ( t r i p l e t ) and the space part of the wave function i s also symmetric (S-state) so that the tota l wave function for the intermediate state i s symmetric, which i s forbidd-en by the Pauli exclusion principle. Therefore we expect symmetrization of the wave functions to drastically reduce the contribution of the DCE term in the multiple scattering series. We have made two contradictory statements so f a r . 51 In an earlier paragraph we stated that symmetrization of intermediate states has no effect whereas we also showed that we expect a large effect in the DCE term. In fact, there i s a term coming from symmetrization which does reduce the DCE contribution, but there are other terms from the binding corrections which cancel the effects of symmetrization. This w i l l a l l become clearer when we present the detailed calculations. Secondly, we wish to comment on the v a l i d i t y of symmetrizing intermediate states in general. Some might doubt the whole symmetrization requirement on the grounds that intermediate states don't behave i n the same way as external ( i n i t i a l and fi n a l ) states; for example, energy is not conserved in intermediate collisions (par-t i c l e s are not on their mass s h e l l s ) . One is thus tempted to ask i f symmetrization i s required. If one believes in quantum f i e l d theory, the answer is an unequivocal a f f i r -mation of the symmetrization requirement. For details the reader is referred to Appendix 1. 52 2a.Low Order Multiple Scattering Terms for T[ -d Scattering Neglecting Symmetrization We now use Watson's theory and calculate the low order terms of (1d-24) with unsymmetrized intermediate state wave functions. Our aim is to start with the sim-plest cases and build up the work in stages u n t i l we can generalize our conclusions. The crudeness of the approx-imations we make here w i l l be checked in chapter 4 when we do the problem in a l l i t s complexity. Suffice to say that additional complications to be introduced in chapter 4 would only obscure the relevant features and would not affect the general argument here. The deuteron wave function is taken to be We are neglecting the small D-state admixture. In the above, P and R are the momentum and position coordinates of the deuteron center of mass, r i s the relative position coordinate of the two nucleons, r^ and r2 are the nucleon position coordinates in the lab system, and we write is a Hulthen function given in Appendix 2. ->> —•> -9 (2a-2) 53 We assume throughout this section that the mass of the neutron and proton are equal and also take the TT and TT ° masses equal so that i t i s convenient to define the center of mass momentum P and relative momentum k in terms of the nucleon lab momenta.ip^ and pg as It i s also convenient to write the relative position 21 vector part of (2a-1) in terms of i t s Fourier transform As a complete set of intermediate states we take the plane wave states ( ^ n)Va. £ _) 3,v where the subscript n refers to the particular intermediate state and a l l the r ' s , p's, and k's are defined as before —> and in analogy r ^ is the position vector of the pion in 21 See Appendix 2. 54 the lab and p_ i s the pion lab momentum in the intermer-diate state n. We choose a particularly simple form for the two-body T matrix operator in coordinate space V &">Wn-r,) (2a-6) 3 /-> -9 V- < w > t.r(v<0 where t is a constant. This choice is motivated by noticing that That i s , t sandwiched between plane wave pion-nucleon states is a constant multiplied by an overall momentum-conserving delta function. The constant t w i l l vary depending on whether we scatter off a proton e l a s t i c a l l y (tQ=tp), off a neutron e l a s t i c a l l y ("t0='tn)» o r charge-exchange scatter (t =t ).. o ce Let's start by calculating the single scattering terms. From (1d-24), these are 55 where the subscript "b" stands for the f i n a l state, "a" for the i n i t i a l state and V ^ (2a-7) —•5 - 9 (any'2-where p' and p* are the i n i t i a l and fin a l pion momenta which we w i l l set equal to zero in the end (threshold l i m i t ) . If we le t particle 1 be the proton and 2 the />- A neutron the t. term w i l l give a factor t and the t0 term 1 p 2 a factor t but neither w i l l contribute any t factors n w ce because a single charge-exchange cannot leave the f i n a l nucleons in the deuteron state (because there w i l l be two neutrons). Let us just concentrate on the t1 term (the t2 term i s treated analogously). Combining (2a-7), (2a-6), (2a-4), and (2a-3) e (a 3/1- ^ -G3*) Qan)^1 (2a-8) 56 —7 -O where P and P, are the i n i t i a l and f i n a l deuteron center a b of mass momenta. Now use (2a-2) to convert the delta function part of the T matrix into one involving <cm and relative coordinates to obtain The integral over R is easy with the above delta function; the remaining position vector integrals just give delta functions over the appropriate momenta; and we can also integrate over to remove one delta function yielding ^ U O / V (2a-10) Now take the threshold limit setting PQ and P, equal to a o zero and notice that the resulting integral over the V 's just gives a factor of one (because the wave functions are unit normalized). The result i s <WHI>-- tri3(P;k+fw-^-fL) <2.-ii, where the delta function insures momentum conservation. To obtain the scattering amplitude one throws away the 57 delta function (see (1c-16) and (1c-30)). The term is handled exactly the same way and not surprisingly in the threshold l i m i t . Since a l l future calculations in this section are treated along the same lines as the previous one, we put most of the details of succeeding dalculations in the appendices. Now consider second order scattering terms. According to (1d-24) these are expressed by In this case we must insert the plane wave states of (2a-5) so that (2a-13) < K \ K \ t y and similarly for the t^Gt^ term. The calculation of each of the above matrix elements is straightforward and we refer the reader to Appendix 4. Only two comments A A A . are necessary. For elastic scattering the t^Gt2 "f c e r m w i l l contribute a term proportional to t t and so w i l l n p 58 A A A the t 2 G t1 term since they represent scattering off one nucleon and then the other. But for charge-exchange A A A scattering, only the tgGt-j term contributes because i t represents a rf scattering off the proton converting the proton to a neutron and i t s e l f to a TT° and then charge-exchanging back again off the original neutron; however, the t.jGt£ term cannot contribute to charge-exchange because i t represents a n striking a neutron f i r s t and thus cannot convert the neutron to a proton and s t i l l conserve charge. The result is that we get a term proportional to 2% t from elastic scattering and one proportional to i'^n p 2 -t . The minus sign comes from the fact that the neutron ce and proton are interchanged after the scattering by the charge-exchange process. (By choosing this minus sign we are effectively symmetrizing the i n i t i a l and f i n a l states). It is also a simplifying assumption to neglect the kinetic energy of the nucleons in the propagator G as well as taking the binding energy of the deuteron equal to zero in the propagator. Summing (2a-13) over intermediate 2 2 states one then finds for second order scattering See Appendix 4. 59 (2a-H) where we define the expectation value of 1/r by <^>^ \ V,\r) i / r <*-1» Application of (1c-30) to convert the T matrix to the scattering amplitude gives (up to second order terms) J -f - £ (2a-16) n n~n — ^ TT~n " ^ i t ~ j — > -rr"|> the f's being the various two-body scattering amplitudes. In writing (2a-16) we have neglected terms of order m /m TT n compared to unity. Equation (2a-16) is well-known.24 23 See Appendix 5. 2 4See references ( 2 ) , ( 1 0 ) , ( 1 9 ) , ( 2 2 ) , ( 2 6 ) , especially (10), eq.A.3. 60 There i s one curious result l e f t to discuss which occurs in a l l orders of scattering past th i r d . For ex-ample, consider el a s t i c scattering from the p to the n, then'back to the p, and f i n a l l y back to the n once more (a fourth order contribution). The appropriate T matrix element is (from (1d-24)) J^t^ t.&t^ t, ( 2 - 1 7 ) . and treating this just as we did the second order elastic terms of (2a-T6) we find making the same approximations as before. We claim that in general the n**1 order scattering term w i l l be propor-tional to the expectation value of (1/r) " . But the expectation value of ( 1 / r )n diverges for n greater than two i f j^(r=0) 4 0^  ; To see this in a specific case write (2a -19) and r e c a l l that See section 3a 61 - * r - B r y (*) ex. e - e * P 7 — for our Hulthen choice of ^ . Therefore the integral in (2a-19) diverges for n>2 so werconclude that every term in the multiple scattering series past third order w i l l diverge. The divergence results from our bad (un-physical) choice of twe-body T matrix (2a-6) but i t is s t i l l possible to reconcile the d i f f i c u l t y and, get a f i n i t e result for the complete pi-d scattering amplitude without abandoning (2a-6). For example, i f one sums a l l the terms of the multiple scattering series for pi-d scatt-ering the ( l / r )n terms add in a geometric series and the sum of a l l terms can be written in closed form. The re-26 suit neglecting charge-exchange i s V f . 4- 2 f n 4 p / r \ C 2 a r 2 0 ) I- ( H A 1 ) where we neglected terms of order m /m . TT n To summarize, in pi-d scattering a l l terms of the multiple scattering series past third order diverge for 2 6 See reference (2), reference ( 1 9 ) , and chapter 3. 62 our simple choice of constant two-body T matrices. Nev-ertheless, one can s t i l l obtain a f i n i t e result for the complete pi-d scattering amplitude by summing the multiple scattering series to a l l orders without abandoning the constant T matrix approximation. The scattering ampli-tude w i l l always be f i n i t e (even when we include charge-exchange) although this has not been shown in general. We show how to always obtain f i n i t e results in chapter 3. 63 2b.Symmetrization of Intermediate States We wi l l now demonstrate how to incorporate symmet-rized intermediate states into Watson's multiple scatt*-ering theory. We w i l l also show that this procedure leads to no new results so that one is ju s t i f i e d in ne--glecting symmetrization of intermediate states in gen-eral (at least when the projectile i s not the same as one of the scatterers). F i r s t , however, we would like to show how direct insertion of symmetrized intermediate states into Watson's series (1d-24) leads to incorrect results. The point i s that binding corrections which are usually ignored (e.g., r e f . ( T q ) ) become fundamentally important when intermediate-states-are symmetrized. To begin, the i n i t i a l and f i n a l deuteron states must be completely antisymmetric in space, spin, and isospin variables. Since the deuteron wave function is spacially symmetric (S and D-waves) and the spin part is symmetric (spin one), the isospin part must be antisymmetric so we write the isospin part as and particle 2 a proton, etc. The three symmetric isospin means particle 1 i s a neutron 64 states are s i m i l a r l y written l f f > and the above four i s o s p i n states form a complete set which we w i l l sum over i n intermediate s t a t e s . Note that we w i l l not write the s p i n part of the wave functions because we always work with two-body T matrices that don't f l i p spin ( i n this section)«so i t i s unnecessary to p u l l the spin part through a l l the c a l c u l a t i o n s . The intermediate states must a l s o be t o t a l l y a n t i -symmetric and since the s p i n part i s symmetric ( s p i n one does not change) we must have the space part and i s o s p i n part of opposite symmetry (one symmetric, the other a n t i -and ^ be the space symmetric and space antisymmetric parts of the allowed intermediate state wave functions. Then symmetrizing ( 2 a-5) we must have f o r plane wave intermediate states 65 The allowed intermediate states then are nf>-ip»>] l x . " > Uf> (2b-2) We must now decide how to i n s e r t a complete set of states with proper normalization. When states are not symmet-r i z e d , one writes the unit operator a s so that multiplying both sides by ^ X - ^ gives as i t should since f o r unsymmetrized plane wave s t a t e s , But f o r symmetrized plane wave states normalized as above <£uir> ^  66 so the unit operator must be written (2b-3) n where the sum is over a l l states of the forms given i n (2b-2). We are now prepared to calculate the DCE term with symmetrized wave functions. From the previous paragraph's-we write .A , . , ^A <<Cnn)|G|X;Cnn))<X>)(trit(^ )> (2b-4) In the above we have only written those intermediate states which give a non-zero contribution. That i s , a T T scattering off a deuteron and charge-ex changing to a t T ° can only leave behind two neutrons in the intermediate state. First remove the isospin parts in (2b^4) as follows: Since operates on nucleon 2 we have <»Mt"| " J Z p ^ r < n n | i £ - ) t ~ = ^ (2b"5) I f and similarly Ace £ce (2b-6) 67 We can thus use (2b-5) and (2b-6) to reduce (2b-4) to <^i&u:><^itne> The terms in (2b-7) can now be handled in a straight-forward manner as was done in the non-symmetrized case, 27 The result i s where A - ^ j ^ J f (2b-9) Because the deuteron wave function was symmetrized in isospin space, neutron and proton are not always particle 2 and particle 1, repsectively, as they were in the non-A A A symmetrized case. Therefore the t^Gt2 term is not zero A A A this time but is equal to the t^Gt1 term (by symmetry). So the complete DCE contribution to the T matrix is twice the right side of (2b-8). The corresponding DCE scattering 27 See Appendix 6. 68 amplitude i s (2b-10) We see that the A decreases the DCE contribution as we expected. Unfortunately, however, A is i n i f i n i t e and in fact i t numerically is equivalent to the bubble graph (2b-11) ? TT w p n which we expect to be in f i n i t e by analogy with the Feynman QED work. The i n f i n i t y i s really no problem since i t results from an unphysical choice of two-body T matrix, (2a-6), so that a more r e a l i s t i c choice for the t's would make A f i n i t e . But even more perplexing is the fact that when we calculatecthe elastic scattering contri-bution in second order, symmetrizing intermediate states the same way, we get Xt\A <5ri. e l . F - X I T d 2. C2b-12) This just cannot be correct for the following reason. The intermediate state for e l a s t i c scattering consists of a 69 neutron and a proton (not two neutrons as in DCE) so that the particles are not ide n t i c a l . This means that sym-metrizing should not give different results from non-symmetrized calculations. We can resolve the d i f f i c u l t i e s encountered from symmetrization by going back to (1d-22) instead of (1d-24). Then we see that the complete second order contribution i s When we neglected nucleon excitation in the propagators we assumed that but tliis is only true i f the intermediate states are not symmetrized. The important point to remember is that the g's are two-body propagators (one pion and one nucleon) and the G's are three-body propagators (pion and both nuc-leons) so that symmetrized intermediate states are sand-wiched between G and the t's, but unsymmetrized (pion and one nucleon) states must be sandwiched between the g's 70 28 and the t's* When this is done properly we find the following results for DOE: so that the complete second order contribution to the scattering amplitude with intermediate state symmetriza-tion included i s which is the same result we found without symmetrizing intermediate states (see (2a-16)). Similarly, the second order elastic result is the same as the unsymmetrized re-sult when binding corrections are properly handled, and in general, symmetrization of intermediate states is un-necessary since the results are always the same as un-The author is deeply indebted to D.S. Beder for pointing this out and thus resolving the symmetrization d i f f i c u l t i e s . 71 symmetrized ones (providing that the incoming particle is different from the target p a r t i c l e s ) . However, i n i t i a l and f i n a l states must always he symmetrized. In section 2a we did not e x p l i c i t l y symmetrize i n i t i a l and f i n a l deuteron states but the results are correct because we accounted for symmetrization by choosing the minus sign in (2a-16) next to f2. ce Moyer and Koltun, reference (22), mention that inter-mediate states need not be symmetrized, but their argument does not analyze the situation in detail as we have done. Theirs is the only paper to even mention the equivalence of symmetrized and unsymmetrized calculations, and on this account deserves considerable credit. Note that Moyer and Koltun discuss symmetrization based on the Lippmann-Schwinger equation whereas we work with the multiple scattering series. In conclusion, we can ignore symmetrization of inter-mediate state wave functions in a l l subsequent work of this paper. 72 3.Summation of the Multiple Scattering Series To A l l Orders It has been shown that the multiple scattering series diverges in each term past third order; here we want to sum the whole series and obtain f i n i t e results. To accomplish our goal, i t is necessary to write the various scattering amplitudes for each order of scattering. In general this is not possible, when non-constant two-body T matrices are used, for example; i.e., the integrals to each order cannot be evaluated (except numerically). But i f we keep the approximations we have made in the previous sections (constant t's, zero binding and no nucleon exci-tation i n G) i t is,possible to evaluate the integrals and obtain closed analytic expressions for each order of scattering. It is also possible to write the scatter-ing amplitude immediately for each order of scattering by looking at the appropriate graphs. We therefore w i l l state the rules for obtaining the scattering amplitude to each order and then sum the terms to a l l orders. No attempt w i l l be made to derive the rules that w i l l be given because they are easily established by the tech-niques of Appendix 4 although i t is tedious to do so. 7 3 However, we w i l l make reference to previous calculations of second order terms to indicate the origin of our graph rules. We also postpone numerical evaluation of series sums u n t i l chapter 4. The sum of the elastic multiple scattering series for pions on deuterons has been evaluated long ago ( i n 1 9 5 3 by Brueckner, reference ( 2 ) ) but the charge-exchange process was neglected. Not u n t i l 1972 was an attempt made by Kolybasov and Kudryavtsev (reference ( 1 9 ) ) to sum the series including charge-exchange and T T 0 elastic scattering. Unfortunately their result is incorrect, which they state in arnote added in proof. But their later paper (reference (28)) which they claim corrects their previous error is also wrong. In the present section we show how to obtain the series sum including charge-exchange by a new method which "folds" the charge-exchange and T T 0 elastic scattering contributions into the original series sum neglecting these processes. This technique greatly reduces the complexity of the problem and avoids the tedious labor of summing a great many extra graphs. In„ addition'" to our pi-deuteron series sum, we evaluate the series sum neglect-ing charge-ex change and TT ° elastic scattering for pions on an arbitrary nucleus of N neutrons and Z protons. 74 Using our new technique we then f o l d i n the charge-exchange contribution so that we evaluate the s e r i e s sum for an a r b i t r a r y nucleus including e l a s t i c TT and charge-exchange s c a t t e r i n g . We do not include e l a s t i c TT° sc a t t e r i n g i n this more general case because the expressions then become gargantuan. Nevertheless, our technique does allow one to p e r s i s t , i f desired. 75 3a.Graph Rules and Order by Order Summation We begin by looking back at (2a-16), the f i r s t order contribution to the scattering amplitude i s f + f n p (3a-1) For simplicity, l e t graphs corresponding to (3a-1) be drawn as n ' P f f C3a-2) where i t i s understood that deuteron lines 'Should be joined to the beginning and end of the two nucleon lines in (3a-2) and a l l that follows. If we ignore i n i t i a l and f i n a l lines the relevant feature of (3a-2) i s a p i -neutron vertex in the f i r s t graph and a pi-proton vertex in the second. Thus, the amplitude which, we write" by inspection from (3a-2) is just (3a-1), where f is written for the pi-neutron vertex and f for the pi-proton vertex. Now look at second order elastic terms. The diagrams are n f I' / n ' rr (3a-3) 76 and from (2a-16) the corresponding scattering amplitude is where i t is understood that (3a-4) is to be averaged over the deuteron wave function (take the expectation value). That i s , instead of taking the expectation value of each term in the multiple scattering series and then adding a l l the terms, we f i r s t add the whole series and then take the expectation value .(both methods must give the same results); Neglecting external lines in (3a -3 ) , the f i r s t graph has a pi-neutron vertex, a pi-proton vertex, and an internal pion propagating; the two vertices contri-bute the factors f and f and the propagating pion gives the factor 1/r. The second graph contributes the same so that (3a-4) results by inspection from (3a -3 ) . Now look at the DCE graph (3a-4) y TT (3a-5) with the corresponding amplitude from (2a-16) (3a-6) 76a Each pion-nucleon vertex in (3a-5) contributes a factor f because the pion changes i t s charge after each scatt-C c ering, the propagating pion contributes a factor 1/r, and a minus sign i s necessary because the proton and neutron have exchanged places in the fnal state with respect to the i n i t i a l states. We can now state a l l the graph rules. In analogy to QED rules, we choose to abbreviate our rules with the letters MSG (multiple scattering graphs). The rules are as follows: MSG Rules 1. Draw a l l possible graphs with no bubble diagrams allowed (no graphs like (2b-11) , for example). 2. For each vertex associate a factor f , f , f , or f n' p' ce' o corresponding to elastic rr -neutron, rr -proton, charge-exchange, or elastic Tl 0 scattering, repectively. 3. For each internal propagating pion line associate a fac-tor 1/r. 4.If two nucleons^are interchanged in the f i n a l state, multiply by -1. (3a-7) Keep i n mind that the above rules apply only under the assumptions of "page 72; and in the threshold l i m i t . 77 As a check to see i f you can apply the MSG rules to a more complicated case, the amplitude associated with the diagram ? 77' is given by (3a-8) (3a-9) Using the MSG rules we draw the graphs and write the corresponding amplitudes for the next few orders of elastic scattering (neglect the charge-exchange process for now). We obtain for the third order processes n P n f T (3a-10) where n^-*p means the same graph with the pion striking the proton f i r s t . To fourth order we get 78 r n (3a-11) + n r The pattern i s obvious so we write the sum of a l l odd order terms as (3a-12) and the sum of a l l even order terms as (3a-13) and summing (3a-12) and (3a-13) gives the complete p i - d sc a t t e r i n g amplitude neglecting charge-exchange ( 3 a - H ) 79 where i t is understood that the right side of (3a - H ) i s to he averaged over the deuteron wave function (take the expectation value). We have thus shown that the series of divergent terms can be summed to a f i n i t e result, the result (3a-14) having been obtained previously by Brue.ckne r . Now consider the charge-exchange contribution. We could in principle write the graphs to each order as before by'.. including extra factors of f and f . However, the ce o combinations are so many and varied that i t i s d i f f i c u l t to see a pattern to each order so that one has trouble summing the series proceeding as before. Instead, suppose we replace every f in (3a-14) by some factor which takes into account a l l ways to scatter from a Tl on a p to a TT, " off a p with any and a l l possible f and fQ scatterings in between. That i s , replace the vertex s Tl f by the sum of graphs 80 In other words, we have summed a l l paths which start with a "ft on a proton with a neutron as a spectator and f i n i s h with a T f off a proton and a neutron as spectator, and no intermediate elastic scattering (of Tl --*) allowed. There i s no similar replacement for the pi-neutron vertex because the pion would have to f i r s t e l a s t i c a l l y scatter over to the proton before i t couUld charge-exchange, and such contributions are accounted for in (3a-15). Suffice to say, then, that the series sum including charge-exchange and elastic TT ° scattering is obtained by replacing a l l f • s in (3a-14) hy the expression (3a-15). We then find that the sum of the multiple scattering series including 81 charge-exchange and elastic TT scattering is (3a-16) The dubious reader is urged to verify (3a-16) to any order by expanding the denominator. To the authsr's knowledge, (3a-16) is a new result not before obtained. We remark that the result (3a-16) is f i n i t e (as i s seen by multiplying numerator and denominator by r (1+fQ/r); writing the expectation value of the resulting quantity with an ex p l i c i t deuteron wave function makes the finiteness manifest). One curious feature of (3a-16) is the third term in the denominator which is not present in (3a-14). In practice f i s negative so that the denominator of (3a-14) has no zeroes. But the denominator of (3a-16) does have zeroes for positive (but small) r so that F , has a pole T d r when charge-exchange is included. The quantitative ram-ificatio n s of this fact w i l l be discussed in the end of the next chapter. In writing our MSG rules we ngglected terms of order rn^/n^. They are easily accounted for but we choose not to introduce them here because they do not add anything to the discussion. Furthermore, since we w i l l see that the 82 choice of constant two-body T matrices is a poor one, there i s no point in trying to improve the results for the series sum since the sum cannot be so easily eval-uated for non-constant two-body t*s. Although our MSG rules were only written for pions on deuterons, i t i s a simple matter to extend them to include scattering of a projectile on any size system of scatterers, each different from the proje c t i l e . We show how to sum the series for scattering on an arbitrary nucleus i n the next section. 83 3b.Series Sums for Arbitrary Nuclei It i s too cumbersome to try to sum the multiple scattering series for an arbitrary projectile striking an arbitrary nucleus. However, we would like to demonstrate the generality of our method more f u l l y so that the reader should have no d i f f i c u l t y ( i n principle) applying i t to other series sums. Let us br i e f l y review how we summed the multiple scattering series for pi-d scattering. F i r s t we summed the series of elastic scattering graphs neglecting charge-exchange. Let us refer to this sum as the skeleton graph sum. Next we included charge-ex.ehange scattering by re-placing the vertex f by the same vertex plus a l l ways to charge-ex change and finish with a t\ coming off a p. Last, 2 we replaced the vertex product f by the same product plus a l l ways to - charge-exchange off a proton,elastically scatter a IT0 , and f i n i s h with a charge-ex change off a neutron producing a proton. Now consider "fl" scattering on a nucleus consisting of N neutrons, Z protons, where N+Z=A is the total number of nuclear p a r t i c l e s . Let's f i r s t sum the skeleton graphs ( TT -p and ft -n elastic scattering terms only). Assume 84 that the f i r s t scattering occurs on a neutron. There are N ways for this to happen so the amplitude for scattering on a neutron f i r s t is NfQ. Then the pion can succesively stike N-1 other neutrons as many times as i t likes before striking a proton. This brings in the factor jj (H-0^| where r is some average value of r . . , the relative distance * 3 between target particles i and j (see ref.(22)). Next there are Z protons to choose from so we get a factor Zf / r , and then the pion can strike Z-1 protons any number of times before striking a neutron again. This gives a: - -r' factor -J ( £ V r ) [ l - H H (30-D and i n analogy, the next neutron scatterings bring in a factor rt (3b-2) ( N V r ) t ( - W - l ) k ] " Then the factors (3b-1) and (3b-2) come in alternatively, on and on. Therefore:tne total amplitude assuming a neutron was struck f i r s t i s 85 Similarly, the total amplitude assuming a proton was struck f i r s t is obtained from (3b-3) by the replacements N<->Z, f£-+l^, which gives (3b-4) The complete pi-nucleus scattering amplitude at threshold neglecting charge-exchange i s the sum of (3b-3) and (3b-4). If we want to include f , we can do so by the replacement i n (3b-3) and (3b-4) z. fp f," Nice (3b-5) since after charge-exchanging off one proton there are N neutrons to choose from for the next (and f i n a l ) charge-exchange. We could go on to include fQ scattering but the technique should be obvious without doing so. The reader is warned, however, that including f makes the expressions for the series sums very complicated. In i e ' 29 practic there is l i t t l e need to include f scattering because See reference (18) for example. 86 and since f as - f at threshold we have f 0. n p o We f e e l that we have demonstrated our technique of summing the multiple s c a t t e r i n g s e r i e s i n s u f f i c i e n t g e n e r a l i t y . We remark that the r e s u l t s (3b-3), (3b-4), and (3b-5) are new and do not ex i s t i n the l i t e r a t u r e to the author's knowledge. I f one p e r s i s t s i n doing nuclear physics, equations (3b-3), (3b-4), and (3b-5) may be of some use. 87 4.Realistic Pion-Deuteron Calculations The preceeding two chapters analyzed pi-d multiple scattering for constant two-body T matrices so that gen-eral features would not be obscured by the f u l l complexity of the problem. Now we wish to include those complica-tions which w i l l quantitatively change the results of previous work; i.e., we want to find the scattering ampli-tude in the threshold l i m i t which we believe to be the experimentally observed amplitude (to date no reliable experimental value exists but the new meson^facilities now in production should soon provide an answer to compare with our results). Our approach w i l l be to add in one complication at a time, calculating the scattering ampli-tude anew at each successive step. Finally we discuss additional complications and compare our results to pre-vious calculations in the l i t e r a t u r e . 88 4a.Pi-d Scattering Complications One Step at a Time In our previous calculations with constant two-body T matrices, we saw that terms beyond third order diverged (see (2a-18), for example). However, i f more r e a l i s t i c two-body T matrices are used, the integrals associated with each order scattering term w i l l be reduced drastic-a l l y at high intermediate state pion energies due to the energy-dependence of the t's. The integrals w i l l then be f i n i t e and owing to the smallness of the pi-nucleon scattering amplitudes the magnitude:; of each term past second order is small compared to the f i r s t and second order terms. Therefore our main work w i l l not go beyond second order in the multiple scattering series. The results for constant two-body T matrices w i l l also only be retain-ed up to second order for comparison with improved cal-culations of this section but one should keep in mind that for constant two-body T matrices keeping terms only up to second order i s not necessarily a good approximation to summing the whole series. In the following, we present numbers at each stage, but also present a l l stages together at the,end in Table ,11 for comparison. 89 A^ For constant two-body T matrices, zero deuteron binding energy, equal mass nucleons (m^=m^=m^), equal mass pions (m = m ) in the propagator,,the scattering IT - Tf/O amplitude is (to second order) 1st 2nd e l . 2nd ce.  e l . i where 1st _2nd e l . 2nd ce. r , x "\ (4a-1) The above results come from (2a-16) and Appendix 5. For the f's we f i r s t take the scattering lengths (pi-nucleon scattering'iamplitudes in the threshold limit) from refer-ence (7), the Samaranayake and Woolcock data, fn = - .143 fm f = .118 fm ( 4 a"2 ) P f = -.185 fm ce 90 For the masses we use mjj. » 939.0 Mev m^  = (939.6 + 938.3 -2 .2 ) Mev = 1875.7 Mev m = 139.6 Mev (4a-3) and for ^ l A * ^ using a Hulthen wave function (see Appendix 3) we find <1/r^ > = .594 f n f1 (4a-4) and with an S-wave Gartenhaus wave function (see reference (21) and ( 4 a-14))2 9 <1/r^> = .446 fm"1 (4a-5) so that with the above values the numerical results for (4a-1) are the foilowingt Table I: Comparison of Hulthen and Gartenhaus Results for F . - IT d yHulthen ^Gartenhaus \ d -.0268 -.0268 ,,2nd e l . IT d -.0246 -.0185 „2nd ce. -.0250 -.0188 with a l l amplitudes above and from here on in fermis. The Hulthen and Gartenhaus wave functions are simple and accurate enough for our purposes. 91 With the introduction of subsequent complications we do not repeat the calculations with the Gartenhaus wave func-tion except for the l a s t , most complicated case. However, we feel the above results give an indication of the further reduction to be expected in each subsequent complication. B} The f i r s t complication introduced i s to account for a pole in the propagator resulting from the unequal nucleon and pion masses and the non-zero binding energy of the deuteron. That i s , as shown i n Appendix 4,we take the propagator G to be whe re so that instead of Of course this complication has no effect on f i r s t order terms. When the appropriate integral with the new G was 9 2 computed, taking the pole into account, the second order charge exchange term changed negligibly, becoming p2nd ce = _# Q 2 4 8 f m ( 4 a.6 ) H d and the second order elastic term didn't change at a l l to three significant figures. Thus the effect of unequal masses and non-zero binding is insignificant. c 3 The next complication involves the nucleon excitation in the propagator (as shown in Appendix 4) so that now we have E = m. + m - B + m n p n -for charge-exchange and similarly for e l a s t i c scattering. We then find a substantial effect with the results F 2 f 6 1' = -.0166 fm trd (4a-7) _2nd ce. „1 c c, „ F ^ = -.0168 fm Of course the f i r s t order terms are again unaffected. D] The next complication arises when we keep track of the various reference frames of each scattering event. 93 For example, i f we look at a second order term of the multiple scattering series T1 2 = ^ S t g (4a-7) and choose to evaluate 2 ^n "t n e pi~cl center of momentum A A frame, then t-j and t2 must also be evaluated in the pi-d cm frame. In our previous work we made the very crude approximation that the t's are constant in a l l reference frames, an approximation which violates Lorentz-invariance of the theory. Suppose now we say that each two-body t A i s constant in the pi-nucleon cm frame; i.e., t^ i s con-stant in the cm frame of the pion and nucleon i . We want A to convert t^ from the pi-nucleon i cm frame to the pi-d cm frame because the la t t e r i s what must be used in (4a-7) and in general for every two-body t of the multiple scatt-ering series. Consider the matrix element in which state ^n contains a pion with total lab energy (not just kinetic energy) E and nucleon i has total lab n energy EN and similarly for state X-m. Denote the n m corresponding total energies of the pion and nucleon i 94 in their om frame hy putting a (*) over each B. Then by reference (15), page 86, equation 112, the conversion of the two-body t's from the pi-nucleon cm frame;;to the lab frame is given by -(X. | t . I X > \ c* c* (4a-8) In our previous work we used C M which is a constant, but we should have used which is not a constant by (4a-8). (Note that the lab frame and the pi-d cm frame are identical in the threshold l i m i t ) . It is a straightforward matter to find the various energies in (4a-8) from the known momenta of the par t i c l e s . We just mention that for simplicity we choose to evaluate the E's non-re l a t i v i s t i c a l l y so that we write 95 and similarly for the other E's. Let us refer to the application of (4a-8) as the inclusion of Lorentz factors. Since this effectively makes the two-body t's non-constant i t is appropriate to include another complication along with the Lorentz factors. Looking back at (1c-30), i s say the pion total energy and Eg the struck nucleon total energy (in the p i -nucleon cm frame). But should E^ and Eg be the energies of the pion and nucleons before or after the scattering? Actually (1c-30) only holds for e l a s t i c scattering in which the energies of the two particles don't change after the scattering; that is not the case in multiple scattering theory so we have to generalize (1c-30) for inelastic scatt-ering. The details are covered in Appendix 8 and the generalized result is (4a-9) so that f i s effectively no longer constant but is multi-plied by the square root of the term in brackets in (4a-9). 96 The application of (4a-8) and (4a-9) changes the previous results for the pi-d scattering amplitudes to F1® * = -.0267 fm trd ?2nd e l . = _> 0 1 0 5 f m (4a-10) TI d —2nd ce. „ . A. „ P , = -.0104 fm TTd The results (4a-10) hold for unequal pion and nucleon masses, non-zero binding energy, and nucleon excitation i n the propagator, and constant two-body scattering ampli4 tudes but with Lorentz factors and inelastic scattering factors included. E ] Now l e t us drop the assumption of constant two-body scattering amplitudes and rather f i t the two-body scatter-ing amplitudes with partial wave phase sh i f t data. As a f i r s t step we only include S-waves (the f i r s t term in the partial wave expansion of f ) . One usually parameter-izes the two-body scattering amplitudes i n terms of the pion momentum in the pi-nucleon cm frame, but for inelastic scattering the pi momentum is different after the scatter-ing. Therefore to satisfy time reversal invariance the parameterization of the f's must be invariant under the interchange of i n i t i a l and f i n a l pion momenta. If qi and qf are the i n i t i a l and f i n a l pion momenta in the p i -97 nucleon cm frame then we find that the appropriate quan-30 t i t y to use in the parameterization of the f's i s q = { V * f Ua-11) instead of q^. With the parameterization for the S-wave phase shifts given in Appendix 10 we find for the pi-d scattering amplitudes = -.00760 fm TTd ^2nd e l . = _> 0 1 1 5 (4a-12) F2nd ce. = _> 0 1 f m fid In obtaining the result (4a-12) we work with complex scattering amplitudes yet the results written in (4a-12) are r e a l . The reason for this i s that we only keep the real part of the pi-d scattering amplitude because in the threshold limit F ^ must be r e a l . This follows from the optical theorem which states that the total cross section 31 is related to the forward scattering amplitude by 3© See Appendix 9.~j ' " c 31 See reference (27), p.74. 98 In the threshold limit we have q—?>0 so that ImF must also vanish in this l i m i t to keep cr f i n i t e . Thus F i s 32 real in the threshold l i m i t . pj The next complication we introduce is the inclusion of P-waves i n the two-body scattering amplitudes (keeping the second term of the partial wave expansion). It i s unnecessary to go beyond the P-wave terms because the low energy contribution from the higher partial waves is negligible as one can see from the data. In addition to including P-waves we also allow for a spin-flipping term in the pi-nucleon scattering amplitude (see Appendices 1'1 and 12 for our choice of P-wave parameterization and our treatment of the spin-flipping term). The s p i n - f l i p part of the pi-nucleon scattering amplitude complicates the calculations because i t changes the symmetry of the intermediate states depending on the total spin of the two nucleons after each scattering; nevertheless, care-f u l l y accounting f o r the proper symmetrization gives the same results as not symmetrizing the intermediate states This conclusion only follows i f we neglect absorption effects} ->like TT d •—>NN—J»Tid. Nevertheless our retention of only the real part of F ^ calculated from complex *s is admittedly but an ansatz; the appropriate off-shell behavior of the f_,T*s which guarantees a real F , merits ^ ^ . j T N ° rrd further study. 9,9 (as in our previous work). The results for P-waves and spin - f l i p in the pi-nucleon scattering amplitudes are found to he P1^ » .000229 fm nd p2nd e l . = .< 0 1 5 5 f m (4a-13) n/d p2nd ce. = _> o u o f m TT d G] It is well-known that the Hulthen wave function for the deuteron is not a good approximation for small r (relative position coordinate of the two nucleons). A better approximation is obtained with the Gartenhaus S-wave deuteron wave function (see referenee(21)) of the form where c1 , c^, <x , are a l l constants. With the improved Gartenhaus wave function and a l l other previous complica-tions combined we find P 1 ! * = -.00394 fm T f d p2nd e l . = _> 0 1 1 8 f m (4a-15) p2nd ce. = _> 0 1 1 g f m 1G0 The results (4a-15) represent our best estimate of the f i r s t and second order terms of the multiple scattering series. We have collected the results of each successive approximation in Table I I . Before we consider other terms of the multiple scatt-ering series, some remarks are in order concerning the single scattering contribution. Looking at the f i r s t column of Table II we see an erratic fluctuation of the single scattering terms beyond approximation 4 . The impli-cation i s that we have l i t t l e confidence in the f i n a l result (approximation 7) for the single scattering contri-bution. Let us examine the single scattering contribution in each approximation. In approximation 1, each single scattering term ( f ^ and f ^ essentially) is one order of magnitude larger than either double scattering term, but the two single scattering contributions are of opposite sign so that when we add them together there is a partial cancellation; the total single scattering contribution is thus an order of magnitude smaller than either single scattering term. The inclusion of Lorentz and inelastic factors in approximation 4 has l i t t l e effect i n single scattering because there is no integration over interme-diate states and these extra factors are only important 101 at higher momenta where they out-off the two-body T matrices. When we go to approximation 5 there is a con-siderable decrease in the magnitude of the single scatt-ering contribution. This results because at low momenta (but not zero momentum) the S-wave phase sh i f t s are such that the cancellation between fn and f is even greater than at zero momentum. For example, at zero momentum where the a's are the pi-nucleon S-wave scattering lengths. However, at q=68 Mev/c (pion momentum in the pi-nucleon cm frame) In the range 0^q.<68 Mev/c there i s no experimental data for the S-wave phase shifts and we have no idea how to extrapolate the phase shifts in this range. It i s pre-cisely i n this small q range that the major contribution to the single scattering terms occurs. If one assumes a q2^+ 1 phase-shift dependence for the l ^ *1 -wave phase shift for small q then i t is impossible to f i t the S-wave scattering lengths and the known phase shifts at q=68 Mev/c we have fn + f = ( 4 / 3 ) a5 + (2/3)a1 = -.025 fm P 102 21 1 simultaneously. In other words, q * q for S-waves (1=0) so the scattering amplitude i s constant for small q; but fn+ f i s not constant in the range 0< q< 68 Mev/c (as we just showed) and therefore some other (and arbitrary) 21+1 parameterization different from q must be chosen for the S-wave phase s h i f t s . Therefore the extrapolation of the S-wave phase shifts for 0<q<68 Mev/c is; arbitrary and we have l i t t l e confidence in our single scattering calculation u n t i l the S-wave phase shifts are known better. The situation i s not so c r i t i c a l for the P-waves because the q + dependence f i t s the P-wave scattering lengths and P-wave phase shifts at q=68 Mev/c simultaneously. Since the S and P-wave contributions do not interfere in single scattering, we can say with confidence that the P-wave single scattering contribution i s (from approxi-mations 5 and 6, column 1, Table II) ^ d * (p - w a v e) = •000229 - (-.00760) = .00783 fm The S-wave contribution remains an open question u n t i l better (lower energy) pi-nucleon data is available. Note that the second order scattering terms are not sensitive to our choice of the S-wave phase shifts in the range 103 0 < q. <l 68 Mev/c because there is no delicate cancella-. . tion of f + f but rather the contribution looks like n p f f . Therefore we are confident of our second order n p scattering r e s u l t s . 104 Table II: F i r s t and Second Order Pi-d Scattering Amplitudes for Various Complications, ( a l l F's in fermis) 'Complications »1at V w2nd e l . Fttd -2nd ce. PU 2 1.constant two-body f's and t's, m=m "\ n p J m =ni V in G t\- TJO r B=0 J -.0268 -.0246 -.0250 -.0764 2.Same as 1 but m An \ n^ p m /m V in. TT T?° B/0 J -.0268 -.0246 -.0248 -.0762 3.Same as 2 plus nucleon excita-tion in G. -.0268 -.0166 -.0168 -.0602 4.Same as 3 but Lorentz and in-elastic factors i n t's. -.0267 -.0103 -.0104 -.0474 5.Same as 4 but f's not constant and only S-waves in f ' s . -.00760 -.0115 - .0115 -.0306 6.Same as 5 but also P-waves and sp in-d? 1 ipp ing in f ' s . +.000229 - .0135 -.0140 -.0272 7.Same as 6 but Gartenhaus Y ^  instead of Hul-then -.00394 -.0118 - .0119 -.0276 105 4b.Binding Corrections In addition to those terms we have already con-sidered in the multiple scattering series ( f i r s t , second order e l a s t i c , and second order charge-exchange) we must comment on those remaining; the f i r s t of these i s the binding berrections. Looking back at (1d-22), the second order binding correction terms are given by M n d' = t1(G-g1)t1 + t2(G-g2)t2 (4b-1) The three-body propagator G i s straightforward and is given by ( A 4 - 4 ) . The two-body propagators g1 and g2 are a real problem to evaluate in our case, however. One defines the two-body propagator gi (i=1,2) by ( l d - 1 3 ) with E^ the i n i t i a l (before scattering) energy of the incident pion plus nucleon i . Unfortunately the Fermi motion of the nucleons in the deuteron allows a whole range of i n i t i a l state energies for each nucleon. The choice for E^ i n the propagator g^ is therefore completely arbitrary and for lack of any information we take E^sm^+m^ (note that this is consistent with our choice of E in G when we set the binding energy equal to zero). We there-fore write 106 <H,* - [ £ + (4«^ \ ( j W l -I (4b-2) and similarly for G-g2» replacing the p/2-k in the second term by a p/2+k. In G we have summed the kinetic ener- . gies of a l l three particles and in g^ we summed the pion and nucleon 1 kinetic energies; p is the intermediate state pion momentum ( i n the lab frame) and k is the inter-mediate state relative momentum for the two nucleons (see (2a-3)). Note that while the individual terms t^Gt1 and t-jg-jt-j are divergent for constant two-body T matrices, the difference tl(G-gl)t1 is f i n i t e . With the choice (4b-2), Hulthen ^ » a ^ constant two-body T matrices the contribution of second order binding terms to the pi-d scattering amplitude i s ^2nd bind. = _# 0 0 8 1 8 f m ( 4 b_3 ) accurate to two per cent. Including Lorentz and inelas-t i c factors i n the two-body T matrices gives a substan-t i a l reduction due to the cut-off at higher energies 107 ( i n the intermediate state) with the r e s u l t p2nd bind. m 0 0 3 5 5 f m ( 4 b _ 4 ) again accurate to two per cent. Comparing (4b-4) with the l a s t column of approximation 4, Table II we see that the second order binding c o r r e c t i o n i s l e s s than ten per cent of the f i r s t plus second order s c a t t e r i n g terms. For t h i s reason and because we are not sure how to choose g^, i t was considered unprofitable to introduce any further complications i n c a l c u l a t i n g the binding corrections to second order. We are only interested i n an order-of-magnitude estimate. Note that the binding corrections become more important f o r tighter-bound n u c l e i (see r e f -erence (22)). 108 4c.Higher Order Corrections To investigate the contribution of the multiple scattering terms beyond second order i t is necessary to invoke the constant two-body T matrix approximation. If more complicated t's are used (Lorentz factors, phase s h i f t s , etc.) i t becomes impossible to sum the series analytically and worse than that, the evaluation of third and higher order integrals becomes a formidable task. We therefore return to the results we obtained for the series sum using the MSG rules of chapter 3. Fir s t consider Brueckner's well known result (3a-14) which neglects charge-exchange scattering. The sum of a l l terms up to and including second order is (neglect-ing binding corrections) where the subscript H means we used a Hulthen deuteron wave function. Evaluating the series sum (3a-14) for a Hulthen wave function we find (4c-2) 109 This means that the contribution of a l l terms beyond second order is -1.6$ of the f i r s t plus second order terms. Therefore in the approximation of neglecting charge-exchange the contribution of third and higher order terms i s quite small. If we include charge-exchange, the sum of the f i r s t and second order terms is now For the series sum with charge-exchange but neglecting fQ scattering, the appropriate expression to evaluate i s (3a-16) with f =0 and we find In this case the higher order correction is -11.8$ of the f i r s t plus second order terms. Including f i n (3a-16) gives us for the complete multiple scattering series 110 Now the higher order correction is -23.7$ of the f i r s t plus second order terms. As our best estimate of the higher order contribution, we evaluated (3a-16) with the Gartenhaus wave function instead of the Hulthen to find The sum of the f i r s t and second order terms in the Garten-haus case is so that comparing (4c-6) with (4c-7) we find the higher order correction i s -19.3$ of the f i r s t plus second order terms. This represents our best estimate of the contri-bution of the higher order terms of the multiple scattering series. We therefore conclude (and/or guess) that the high-er order scattering terms contribute roughly -20$ of the f i r s t plus second order terms (not including binding corrections) even for non-constant t's. 111 4d.Relativistic Corrections A l l our results up to now have been no n - r e l a t i v i s t i c . We must therefore estimate the corrections arising when the problem is treated r e l a t i v i s t i c a l l y . There is no need to modify the deuteron wave function since i t is accurate' enough for^i our work here." The Lorentz -and inelastic factors in the two-body T matrices could be treated rela-t i v i s t i c a l l y but the kinematics becomes much more compli-cated. In addition, the Lorentz factors are ratios of energies and so an over-estimate of the energy i n the numerator (by treating i t non-relativistically) i s com-pensated by an over-estimate in the denominator. For simplicity and because we do not belive i t to be the dominant r e l a t i v i s t i c contribution, we neglect relativ-i s t i c modifcations i n the Lorentz and inelastic factors. By far the greatest effect i s in the propagator. Instead of using the non-relativistic energy we choose to replace this with the r e l a t i v i s t i c expression for a l l intermediate state energies in the propagator so 112 33 that (A4-4) is replaced by P - fV > ^ = ^ C4d-1) and E is unchanged. For convenience and because i t has l i t t l e effect, we neglect the binding energy and take .." nucleon masses equal (m =m =m„) and pion masses equal n p N (m =m ) in the propagator. Of course the r e l a t i v i s t i c TT o n -corrections we are making have no effect on the single scattering terms. With a Hulthen wave function, constant t's, no nucleon 2 2 % excitation ( i . e . , K = (p +m )u) and no Lorentz or inelas-t i c factors we find the results for our r e l a t i v i s t i c propagator are p2nd ce. = _> 0 5 5 9 f m (4d-2) The increase over the f i r s t row's results of Table II See Appendix 13 for j u s t i f i c a t i o n of this replacement. 113 2 ia obvious since 1/p dies out more rapidly than i t s 2 2 % r e l a t i v i s t i c counterpart 1/(p +m ) . When we include nucleon excitation ( i . e . , use the K of (4d-1)) we find F2?d e l* = -.0224 fm 1 1 ( 1 (4d-3) F2nd ce. _ - > 0 2 2 7 fm ti d Putting more in the denominator of the propagator makes the results smaller than (4d-2). When we now include the Lorentz and inelastic factors (but treat them non-r e l a t i v i s t i c a l l y ) the results decrease to p2nd e l . = _ ^0 1 2 0 f IT d F2nd ce. s _ f m ^ d (4d-4) Comparing (4d-2) with the f i r s t row of Table II we see that the r e l a t i v i s t i c results are 1.44 times larger than the non-relativistic counterparts. Comparing (4d-4) with row 4 of Table I I , the r e l a t i v i s t i c results with Lorentz factors are 1.16 times larger than the non-/relativistic counterparts. In other words, the effect of introducing r e l a t i v i s t i c energies in the propagator is less s i g n i f i -cant when Lorentz and inelastic factors are included. This follows because the r e l a t i v i s t i c effects are larger at higher momenta and the Lorentz and inelastic factors 114 decrease the high-momentum contribution. If we include more detail (phase 3hifts, etc.) we expect the relativ-i s t i c effects to be even less important. To test our b e l i e f s , we calculate approximation 7 of Table II with our r e l a t i v i s t i c propagator ( s t i l l neglecting binding energy in the propagator, however). That i s , we run the most r e a l i s t i c case with our r e l a t i v i s t i c propagator and find —2nd e l . o r. ,. F , = - .0129 im TT d —2nd ce. M 7 1 „ F , = -.013  fm rfd (4d-5) These results are 1.10 times larger in magnitude than the non-relativistic counterparts (row 7, Table I I ) . We therefore conclude that for the most r e a l i s t i c case cal-culated the r e l a t i v i s t i c second order correction is - (-.oil*- 0li<O-t- ( - . 0 l ^ - - * t 3 O : — - 9.1% of the non-relativistic second order terms. 115 4e.Summary of Corrections and Best Estimate We have shown in Table I I , row 7, that the f i r s t and second order scattering terms of the multiple scatt-ering series for pi-d threshold scattering are p1'^ = -.00394 fm p2nd e l . = _> 0 1 1 8 f m (4e-1) p2nd ce. = _ > 0 1 1 9 f m Summing the above and taking 10$ of the sum gives the second order binding correction , p2nd bind. = _< 0 0 2 ? 6 f m ( 4 e_2 ) *nd Taking -20$ of the sum of a l l terms in C4e-1) gives the higher order correction phigh. ord. = + > 0 0 5 5 2 f m (4e-3) n o-P i n a l l y , taking 10$ of the sum of the second order terms in (4e-1) estimates the r e l a t i v i s t i c correction to these FIT r e l* = --0 0 237 fm (4e-4) 116 Adding a l l three terms of (4e-1), (4e-2), (4e-3), and (4e-4) gives our best estimate of the pi-d scattering length _best estimate A O„ -•nd = --0275 fm (4e-5) 117 4f.Review of Other Pi-d Literature and Discussion of Results People have teen estimating the pi-d scattering amplitude since 1950 and possibly earlier (see reference (11) and bibliography therein). In many cases the authors were interested i n comparing their results with then-known pi-d scattering data (above threshold) so their results are not directly comparable with those here. We therefore w i l l only briefly mention the approaches used in such papers. Other papers in which the pi-d scattering amplitude in the threshold limit is calculated w i l l be discussed in more d e t a i l . One of the earliest attempts to obtain the pi-d d i f f -erential cross section was performed by Fernback, Green, and Watson', (reference (11) ) . They did a very crude e s t i -mate by taking the product of the free particle scattering amplitudes with an overlap integral of the i n i t i a l and f i n a l wave functions to get the di f f e r e n t i a l cross section. Thus they neglected double scattering and a l l ramifications thereof. In an effort to find the contribution from higher order scattering terms, Brueckner (reference (2)) evaluated the scattering amplitude by solving the Schro-dinger equation for the scattering of a fast particle by two heavy scatterers. He found the result (3a-14) although 118 his is more general because he allowed arbitrary momentum transfer. Of course his result is not very practical since i t effectively assumes constant two-body T matrices for pi-nucleon scattering. In addition, a serious draw-back of Brueckner's result is the neglect of charge-exchange scattering. It took thirteen years before Wilkin (ref-erence (26)) pointed out " the necessity of including charge-exchange. Wilkin included terms up to second order but assumed constant two-body T matrices and his results apply only to high energy scattering. Since Wilkin, others have evaluated the pi-d 'scattering amplitude (see references (24), (14), (3), (23)) at various energies using multiple scattering approaches taking more details into account, but the results are not comparable with ours since they were performed at higher energies. Dispersion relation approaches were t r i e d by FUldt (reference (10)) and Schiff and Tran Thanh Van (reference (25)) in an effort to treat the problem covariantly and also to tackle the d i f f i c u l -ties of unitarity . Again their results do not apply at threshold so we cannot compare with ours. After Brueckner, a paper which, t r e a t s ^ pion-nucleus scattering in the threshold limit is that of Moyer and Koltun (reference 22)). Unfortunately the lightest nucleus they treat is Helium so their numerical results are not 119 comparable with ours. We mention this paper because i t treats the binding and higher order corrections i n the threshold l i m i t and finds their contribution to be non-negligible. However, their treatment of the binding correc-tions is substantially different from ours in that they write the difference G-g^ of the binding correction as a single operator and solve a Lippmann-Schwinger type equation for i t using separable Yamaguchi potentials. Their method is therefore closely related to Faddeev equation; approaches (see next chapter) as far as binding corrections are con-cerned. Except for the binding corrections, a l l work of Moyer and Koltun assume constant two-body T matrices. Note that the sum of the multiple scattering series in their paper (equation A9) does not allow for charge-exchange scattering of a pion oh a deuteron. The most complete calculation of the pi-d scattering amplitude at threshold to date is by Kolybasov and Kudryav-tsev (reference ( 1 9 ) ) . Because they used different p i -nucleon scattering lengths from ours their results are-: somewhat different but the relative effect of each compli-cation introduced compares favorably with ourivresults. For example, when they introduce nucleon excitation i n the propagator they find a f i f t y to seventy per cent reduction in the magnitude of the double scattering term. 120 We find a sixty seven per cent reduction. When they include P-waves in the pi-nucleon interaction they find the P-wave contribution to be about .thirty per cent of the S-wave in double scattering. We find i t to be about twenty per cent. The reason for the discrepancy i s mostly due to the neglect of Lorentz and inelastic factors in their calculations. In addition, they do not use a phase-shi f t parameterization for their pi-nucleon scattering amplitudes. Their results are obtained with a Hulthen instead of a Gartenhaus wave function and there is no treatment of binding corrections or r e l a t i v i s t i c effects. F i n a l l y , they calculate the series sum for constant two-body T matrices but their final expression is wrong (which they state in a note added in proof). There is l i t t l e point in comparing their best estimate with ours since the result depends on the choice of S and P-wave scattering lengths and their choice d i f f e r s from ours. Nevertheless, Kolybasov and Kudryavtsev find for their particular choice of pi-nucleon scattering lengths P = -.047 fm trd compared to our result P.d = -.0273 fm 121 We expect their results to be higher than ours mainly because Lorentz and inelastic factors were neglected in their calculations and because they used a Hulthen instead of a Gartenhaus wave function for the deuteron. We stated that our goal was to obtain the pi-d scattering amplitude at threshold accounting for a l l quanti-tatively relevant complications. We believe this goal has been achieved within our present knowledge of the pi-nucleon phase s h i f t s . We have neglected the small D-state part of the deuteron wave function and real absorption effects (TT d —>2n). Our crude estimates indicate that these additional complications w i l l introduce much less than a ten per cent correction. We are hoping therefore that this thesis puts the final nails in the coffin of 34 the pi-d scattering length. Attributed to D.S. Beder while groping for the truth. 122 5.The Faddeev Equations and Symmetrization Pion-deuteron scattering is a three-body problem. In 1960 the Russian mathematician Faddeev published an ar t i c l e showing how to solve the three-body problem once the two-body T matrices for a l l pairs of particles (three 35 pairs) are known. The Faddeev equations are closely related to Watson's multiple scattering series since both Faddeev and Watson start with the Lippmann-Schwinger equa-tion. The beauty of Faddeev's approach is that he writes an integral equation for the complete three-body T matrix solely in terms of free two-body T matrices instead of Watson's bound two-body T matrices (the % 's). In addition, the Faddeev equations account for scattering of one nucleon on the other (for the case of pi-d scattering) and Watson's approach only treats scattering of the pion on each nucleon. The purpose of this chapter is to quickly review the Faddeev equations with particular emphasis placed on intermediate state wave function symmetrization. One can consider this brief chapter to be a warning concerning symmetrization and divergent bubble graphs in the s p i r i t of Appendix 6 and section 2b. No attempt w i l l be made to solve the Faddeev equations. See reference ( 9 ) . 123 5a.Symmetrization of the Faddeev Equations Rather than follow Faddeev*s original approach, we choose to paraphrase the approach of Hetherington and Schick (reference (17)) because theirs demonstrates the analogy to Watson's multiple scattering series. Let particle*,2 be the pion and particles 1 and 3 the nucleons. Then we can write the Lippmann-Schwinger equation, (1d-10), for the complete three-particle T matrix as T - - V , 4 - V , + ft + V g ^ T (5a-1) where the three-particle Green's function i s given by (see (1d-16)) ) E - l ^ - l / , , ^ <5a-2) Here E i s the total three-particle energy, K is the sum of a l l three energy operators (pion, nucleon 1, and nucleon 2) and "V\ i s the potential between particles j and k with j / i , k / i , j/k. One also defines the free three-particle Green's function & = E-K + i ^ ( 5 ~ 5 ) By the easy-to-follow steps of Hetherington and Schick 124 one quickly obtains the Faddeev equations for the com-plete three-particle T matrix ( 5 a - 4 ) where ( 5 a - 5 ) ik with t ^ and G given by = G ( I - ^ ( 5 a - 6 ) ( 5 a - 7 ) As Hetherington and Schick point out, iteration of (5a-5- ) gives the multiple scattering series T'J t , i <, t t . + t . C ^ t ^ t . + . . . ( 5 a - 8 ) The point i s that ( 5 a - 8 ) only applies i f the intermediate statee/wave functions are not symmetrized. If one tr i e s to symmetrize them, the same divergences associated with bubble graphs arise that we saw in Appendix 6 . In addition, ( 5 a - 8 ) doe3 not contain binding correction terms. The error i s not in the Faddeev equations as Faddeev writes 125 them hut rather in the way Hetherington and Schick write (5a-6). In (5a-6) they have associated the wrong expres-; sion with the two-body T matrix, t ^ . The correct associa-tion i s to write * 0~ v O c ) ~ ' v« ( 5 a ' 9 ) where a - f c , c u- i< . . (5a-10) is the energy of particle j , etc., and i t is understood that non-symmetrized two-particle states are summed over between g^ and whereas in (5a-6) one sums over three-particle symmetrized states between G and v^. Prom Hetherington and Schick, equation (10), one has T'"' = J"..(l-"f<0"ly- + I-O-^tj'vA^ (5a-,1) The correct Faddeev equations are obtained by writing (5a-11) in terms of the correct two-body t ^ s , (5a-9), instead of the incorrect t ^ s , (5a-6). One then finds with the help of 126 that the appropriate modification to the Faddeev equations as written by Hetherington and Schick i s Of course iteration of (5a-13) with t 2=0 reproduces the multiple scattering series with binding corrections, (1d-22). We urge the reader to remember that (5a-13) is valid for symmetrized intermediate states and (5a-5) is not. There seems to be no regard to this fact in many calculations based on the Faddeev approach. In summary then, people who use equation (5a-5) with symmetrized intermediate states w i l l not get correct numerical results. Worse than that, they w i l l more than l i k e l y not even see their errors when they solve the integral equations numerically. This follows because i t is customary to use separable Yamaguchi potentials in the two-body T matrices, t ^ , which f a l l rapidly with increasing energy; this in turn suppresses the divergences associated 35 with bubble terms. Also, keeping only S-waves i n the two-body interactions^suppresses the divergences and makes them f i n i t e . Therefore, i f Yamaguchi potentials, separable ti' s , and S-wave two-body interactions are the approximations 35 y See reference (17), for example. (5a-13) where (5a-H) 127 employed i n solv i n g the Faddeev equations numerically, the contributions from bubble graphs (which should not be present) w i l l not be overly large compared to other proper scattering terms and one w i l l find i n c o r r e c t r e s u l t s without r e a l i z i n g i t . The point we wish to emphasize i s the following: Use equation (5a-5) but do not symmetrize the intermediate state wave functions. I f you i n s i s t on symmetrizing i n t e r -mediate states, (5a-13) must be used. 128 6. Conclusions We have examined pi-deuteron scattering in the •* threshold l i m i t , including a l l quantitatively relevant features of the problem, we believe. Along the tortuous path we investigated the effects of symmetrizing inter-mediate state wave functions in the multiple scattering series and found the results identical to those obtained with non-symmetrized intermediate states when properly treated. We also found that symmetrizing intermediate state wave functions in the Faddeev equations to solve the three-body problem leads to incorrect results unless one is very careful in interpreting the Faddeev equations. The moral here i s , "Don't symmetrize intermediate states in the Faddeev equations i f the projectile particle differs from the target p a r t i c l e " . We have also demonstrated a technique in chapter 3 (using the MSG rules) that allows one to sum the multiple scattering series i n closed form for any nucleus, taking into account a l l isospin-flipping mechanisms in the two-body scattering amplitudes. In par-t i c u l a r , we applied this technique to TT~-deuteron scattering to find the series sum including a l l isospin-flipping terms. We also evaluated the series sum for TT~ scattering on an arbitrary nucleus of neutrons and protons including charge-exchange scattering terms. 129 The most important question remaining now i s , "What are the pi-nucleon scattering amplitudes at very low v ^  ' energies, especially the S-wave scattering lengths?". Our uncertainty in the pi-nucleon scattering lengths propagates an uncertainty i n the pi-deuteron single scatt-ering contribution and therefore an uncertainty in the pi-deuteron scattering length i t s e l f . We stress, as did Kolybasov and Kudryavstev (reference (19)) that the deter-mination of the pi-nucleon scattering lengths is an imt -* portant experimental task. Once these are known we can give a theoretically determined value for the pi-deuteron scattering length based on ourchoice for the off-shell p i -nucleon T matrix (a re-calculation of the single scattering contribution once the pi-nucleon scattering lengths are better known is simple and can then be added to our second and higher order res u l t s ) . If our theoretically determined pi-deuteron scattering length agrees with the experimental value then we can say with confidence that we understand the off-shell behavior of the pi-nucleon scattering ampli-tude. This would be of fundamental importance in under-standing the complete pi-nucleon interaction and perhaps strong interactions in general. This question is being asked on April 1, 1973. 129a It is well known (see reference (30)) that one can determine the scattering length for pions on l i g h t nuclei by observing the gamma ray emissions of pi-mesic atoms. One produces a pion bound in a high n quantum number shell of an atom and measures the frequency of the emitted gamma ray as the pion f a l l s to the lowest orbit (K s h e l l ) . The pion in the high n shell is not affected by the strong interaction of the nucleus because the pion in this shell is farther from the nucleus than the K shell o r b i t . The strong interaction does s h i f t the energy of the pion in the K s h e l l , however. I f one knows the energy level of the high n shell from which the pion f a l l s , the energy of the emitted gamma ray is the difference in energy between the (unshifted) high n shell and the (shifted) K shell to which the pion f a l l s . The energy level shift (due to the strong interaction of the pion in the K shell with the nucleus) is proportional to the pi-nucleus scattering length (see reference (31) ) . Measurement of the level shift can be used to give quite accurate values of the pi-proton scattering length by applying the above procedure to pi-mesic hydrogen. One then finds a level shift of about 6 ev in magnitude4(to be compared with the n=2 to n=1 transition for pi-mesic hydrogen which is 2.77 kev). The magnitude of the pi-proton scattering length i s about 129b four times our estimate of the pi-deuteron s c a t t e r i n g length so that one expects a l e v e l s h i f t f o r pi-mesic deuterium on the order of 1 ev (magnitude). Therefore the experimentalist's task i n determining the pi-deuteron s c a t t e r i n g length v i a t h i s method i s quite d i f f i c u l t . Note al s o that this method w i l l not allow one to f i n d the p i -neutron sc a t t e r i n g length. 1 30 Bibliography (1)Bjorken, J.D., and D r e l l , S.D. , R e l a t i v i s t i c Quantum Mechanics (Vol.1.) and Re l a t i v i s t i c Quantum Fields ' ' ' -(Vol.II), McGraw-Hill, New York(1964). 2) Brueckner, K.A., Phys.Rev. 8£, 834(1953). 3) Carlson, C., SLAC Pre-print SLAC-PUB-706, Stanford University(1970). 4) Davydov, A.S., Quantum Mechanics, Addison-Wesley, Inc., Reading, Mass.(1965). 5) DeBenedetti, S., Nuclear Interactions, John Wiley & Sons, Inc., New York(1964). 6) Dicke, R.H., and Wittke, J.P., Introduction to Quantum  Mechanics, Addison-Wesley, Inc., Reading, Mass.(1960). 7) Bbel, G., Pilkuhn, H., and Steiner, F., Nucl.>Phys. B 1 7 , No.1(1970). 8) Elton, L.R.B., Introductory Nuclear Theory, Second Ed., W.B. Saunders Co.., Philadelphia( 1966) . 9) Faddeev, L.D., Soviet Physics JETP, Vol.12, No.5, 1014(1961). 10) F*aldt, G., Nucl. Phys. BU), 597(1969). 11) Fernbach, S.,' Green, F.A., and Watson, K., Phys. Rev. 82, 980(1951). 12) Feynman, R.P., Quantum Electro-dynamics. W.A. Benjamin, Inc., New York(1962). 13) Frazer, W.R., Elementary Par t i c l e s . Prentice-Hall, Inc., Englewood C l i f f s , New Jersey(1966). 14) Gibbs, W.R., Phys. Rev. 3C, No.3, 1127(1971). 15) Goldberger, M., and Watson, K., Collision Theory, John Wiley & Sons, Inc., New York(1962). 131 (16) Herndonf D., Barbaro-Galtieri, A., and Rosenfeld, A., TT N Partial-Wave Amplitudes, from Particle Data Group, UCRL-20030 TTN( 1970). (17) Hetherington, J.H., and Schick, L.H., Phys. Rev. 137, No.4B, 939(1965). (18) Jacob, J . , and Chew, G., Strong-Interaction Physics, W.A. Benjamin, Inc., New York(1964). (19) Kolybasov, V.M., and Kudryavtsev, A.E., Nucl. Phys. B41, 510(1972). (20) Lovelace, C , in Strong Interactions and High Energy Physics, ed. by R.G. Moorhouse, Plenum Press, New York(1964). (21) Moravcsik, M., Nucl. Phys. 7, 113(1958). (22) Moyer, L.R., and Koltun, D.S., Phys. Rev. 182, 999 (1969). (23) Pendleton, H., Phys. Rev. V31> 1833(1963). (24) Rockmore, R.M., Phys. Rev. 105, 256(1957). (25) Schiff, D., and Tran Thanh Van, J . , Nucl. Phys. B3_, 671(1967). (26) Wilkin, C., Phys. Rev. Lett, H , 561(1966). (27) Williams, W.S.C., An Introduction to Elementary  Pa r t i c l e s , Second E d . , Academic Press, New York(1971). (28) Kolybasov, V.M., and Kudryavtsev, A.E., Soviet Physics JETP.. 63, 35(1972). (29) Koltun, D.S., in Advances in Nuclear Physics V o l . 3 , ed. by M. Barranger and E. Vogt, Plenum Press, New York(1970). (30) Bailey, J . , Phys. Lett. 33B, 369(1970). (31) Deser, S., Goldberger, M.L., Baumann, K., and Thirring, W., Phys. Rev. 96, 774(1954). 132 Appendix 1: Field-theoretic Two-fermion Propagator We wish to show here using field-theoretic arguments that the propagator for a two-fermion state only contains odd angular momentum states; this means that the Pauli exculsion principle applies in intermediate states (as well as i n i t i a l and f i n a l states). We use the notation of reference (1) throughout. The two-particle fermion propagator is < M T { f y O T V , ) n * O T > ) l l o > where the V 's are Dirac field s ( i . e . , operators) and the T operator is defined by ( o •o X| (A1-1) and the o subscript on the x's means the time component. 37 By Wick's theorem -^See e.g. reference (1), Vol.11, p.181. 133 where the line connecting two ^ ' s means ^ ) ^ * , ) = <o\T{ V o c o V ( * o H 4 > - ' 5 F O ^ " * . ) ( A i - 3 ) which is^the single-particle fermion propagator. The Pauli exclusion principle (or equivalently, Fermi-Dirac s t a t i s t i c s ) for the ^ ' s i s contained in which applied to (A1-2) gives (A1-5) For simplicity we neglect the extra bookkeeping of spin i n what follows. The propagator is written is terms of •50 i t s Fourier transform as"^ (A1-6) •3Q See equation (1b-4). 39 ^See reference (1), p.95, eq.6.46. 134 and (A1-6) together with (A1-3) allows us to express (A1-2) in the form d p d f L (A1-7) The part of (A1-7) in brackets at the end is the desired result only odd angular momentum waves propagate. Perhaps this is more easily appreciated i f we go back to 135 the o r i g i n a l form of the two-particle propagator and draw the diagrammatic correspondence to the above as x. :x4 x 2 i n which one of the fermions s t a r t s at l o c a t i o n x„ and 4 propagates to l o c a t i o n x^ while the other goes from x^ to x 1 . Thus, x3~ x-j i s the r e l a t i v e coordinate between the two nucleons and therefore our i n t e r p r e t a t i o n of (A1-7) as a statement of the P a u l i exclusion p r i n c i p l e i s confirmed, 136 Appendix 2: Deuteron Wave Function Conventions The relative coordinate part of the deuteron wave function, V ^ C r ) , which we use in (2a-1) is chosen to he 40 the Hulthen f ornT (A2-1) where and N is a normalization constant which we now determine. The normalization condition for N is given by ^ i [ C ^ T j V r (A2-2) The evaluation of the integral is elementary and we find N* , tltlSX (A2-3) The Fourier transform of T/JJ(?) is found via simple ^See reference (5), p.46. 137 integrations as follows: frtK> and inverting the Fourier transform gives (A2-4) (A2-5) 138 Appendix 3: Expectation Value of 1/r Our objective is to calculate O / r ^ f o r Hulthen wave function of Appendix 2 so that we want to find (A3-D If we square the term in parentheses above we w i l l have a sum of three separate integrals to evaluate, each of which diverges, although as we w i l l see the sum of a l l three divergent integrals i s f i n i t e . We therefore proceed by writing co r - oir - R f V i i ^  and differentiating with respect to <* + £ (A3-2) Integrating (A3-2) to get I (e( ,p) back including the constant of integration (which we obtain by simple manipulations) we have K ) L ^ U p 1 C A 3 - 5 ) 139 and (A3-3) back into (A3-1) solves the problem with (A3-4) where we used (A2-3). Using the values of o{ and f} given in Appendix 2 we find the numerical value of ^1/r^ to be -I (A3-5) 140 Appendix 4: Second Order TJnsymmetrized Calculations Our goal here i s to obtain (2a-14) starting from (2a - l 3 ) . F i r s t write the separate matrix elements of (2a-13) with the aid of (2a-2) through (2a-7). For the f i r s t term of (2a-13) we can write ^ I S J ^ - ^ V J A r eC^'R e " f ^ (A4-1) where we only consider e l a s t i c scattering (for charge-exchange replace t by t ) . Integrating over R eliminates -> H> the delta function,^integrating over r and r^ produces two more delta functions, and integrating over k^ eliminates one of these two new delta functions (just as we did i n section 2a for single scattering) so that • 3 , (A4-2) where we set p fe = P^ = 0 (threshold l i m i t ) . Similarly for the third term of (2a-13) we find H 1 with p = P =0. The appropriate propagator term of "a (2a-13) i s ( A 4 - 4 ) where £ = % + = + m^- B) + n%-B>0 i s the binding energy of the deuteron, and i t is understood that the integral is evaluated in the limit *\—3>0. Now we make the approximation of neglecting the kinetic energy terms of the two nucleons in K, take mn=nip, and neglect the small binding energy of the deuteron. Then (A4-4) becomes: 142 and integrating over a l l the coordinates immediately pro-duces three delta functions according to Referring back to (2a-13) , we multiply (A4-2) , (A4-3) , and (A4-5) and integrate over the intermediate states to write - —> -»> -5> Integration over the four momenta p , P , k , and P u m m m n eliminates four delta functions and one i s l e f t with (A4-7) H3 i-;It is customary to write the integral i n coordinate space rather than momentum space so we proceed to write the integrand of (A4-7) in terms of i t s Fourier transform. F i r s t write the wave functions in terms of their trans-forms according to (A2-4) to convert (A4-7) to V ( ? ' ) e i ( i f e + k V ? ' ( _ J V ) Integration over k gives a delta function and integration —>/ over r eliminates this delta function so that we get Now observe that Putting (A4-9) hack into (A4-8), integrating over p*to produce a delta function, and integrating over r*/ then H 4 gives?the result A < T k i ^ l ! ^ ) = - u , ) \ t „ t f l ! ( ^ ^ ) (A4-10) Invoking the statements directly preceeding equation (2a-H) we obtain for a l l the double scattering terms ^ (A4-11) which i s just (2a-14). 145 Appendix 5: Conversion of the T matrix to the Scattering Amplitude for F i r s t and Second Order Terms To convert the T matrix (2a-14) to the corresponding scattering amplitude, use (1c-30) and (1c-16) to write and use of (1c-30) once more for the t's gives ( s t ^ ^ y ^ * ^ ) U 5"1 ) and taking m fag a s negligible compared with unity we reduce the above to The single scattering terms (2&-11) and (2a-l2) are converted to their corresponding scattering amplitudes using (1c-30) to immediately obtain P 1 s * ' "L I 1 (A5-3) 146 where in keeping with the approximations of section 2a we have set m^  = m^  = m^ . Again throwing away negli-gible terms, (A5-3) reduces to KV - fn • fp <A5-4> and cominging (A5-2) with (A5-4) we get the desired pi-d scattering amplitude to second order 147 Appendix 6t Second Order Symmetrized Calculations We derive here the DCE contribution to the T matrix with symmetrized wave functions neglecting binding correc-tions. Start with (2b-7) and calculate each term separ-ately. Using (2b-1) write and using the results of Appendix 4 we get Similarly, ( A 6 - 1 ) fl ce r The symmetrized propagator is 148 -9 -9 (A6-3) Combining (A6-1), (A6-2), and (A6-3) with (2b-7) yields and performing the integrals just as i n Appendix 4 we find *u/> tcl | .A . . 1 . 3 (A6-4) 149 The f i r s t integral in brackets i s just the one of (A4-7) so we can replace i t as in Appendix 4. The second inte-gral is reduced by substitution of variables f i r s t to write trin ' and since the wave functions are normalized, 3 Thus, (A6-4) is reduced to < vf c| t it tH " l O - l - ^ ] ( - % ) (2b-8) 150 Appendix 7: Pi-nucleon Scattering Lengths We wish to show here how the results (4a-2) are ob-tained from the data of reference (7). One usually writes the pi-nucleon scattering amplitudes in terms of the iso-spin 3/2 and isospin 1/2 scattering amplitudes, f ^ / 2 fny2 (e.g., reference (13) , p.49) f - f - - f n = TT" n —^ TT n ~ 3/2 *p s ' ^ - p - d / 5 ) ( M1 / 2 • * ,/ 2i J~2 fce = fn - p ~ ^°n = 3 (f3/2 " f1/2) (A7-1) In the notation of reference (7), ^5/2 S a3* f1/2 3 a1 ' and according to their tables* (m^ = 1 . 4 H fm) a1 - a5 = (.277)(1.414 fm) = .392 fm a1 + 2a5 = (-.026)(1.414 fm) = -.0367 fm implying. a_ = - .143 fm 3 (A7-2) = .249 fm which together with (A7-1) immediately gives (4as2). 151 Appendix 8: Relation Between the T Matrix and the Scattering Amplitude for Inelastic Scattering We wish to generalize equation (1c-30) to include inelastic scattering. Combining (1c-23) and (1c-24) we get for inelastic scattering M e * * — » — + 1 I 2. - I where unprimed :p»s and B's denote before scattering and primed denote after scattering. Using (1c-25) and the phase'^established i n section 1c we get the desired result 4 - ' (an)1 T which i s just the result (4a-9) written in s l i g h t l y d i f f e r -ent notation. 152 Appendix 9: Phase Shift Momentum-Dependence for Inelastic Scattering It i s customary to expand the two-body scattering 41 amplitude f i n p a r t i a l waves by writing where q is the momentum of either particle in the twofbody cm frame, the ^ are the phase s h i f t s , P^ are Legendre polynomials, and © is the scattering angle in the cm frame. Unfortunately (A9-1) only applies for ela s t i c scattering since we don't know whether to use q^ or q^ ( i n i t i a l and final)')or what for inelastic scattering. Whatever we use for q in the inelastic case, i t must be some combination of q^ and qf that is invariant under the interchange of both these momenta so that time reversal invariance i s s a t i s f i e d . Our goal i s to find the appropriate combina-tion of qi and qf that replaces q in (A9-1) for inelastic scattering. In the notation of reference (4), section 103, the equation 103.13 states that 4 1See e.g., reference (27), p.69, eq.3.4. 153 * ~ J < C ( 0 e"' b ' r V ( r , ? ) £ ( f ) e 5 ^ < « - « in the Born approximation ( i . e . , to f i r s t order in the Born s e r i e s ) , where 0 is the wave function of the target (subscripts a and b refer to i n i t i a l and f i n a l ) , V is the interaction potential, r and k are the coordinate and momentum vectors of the projectile, and |" are the target coordinates. I f we assume V i s separable so that V ( ? , ? ) - - V ( ? ) Y ( f ) U 9 _ 5 ) equation (A9-2) becomes { - L K f O T W O j ? } y{K~K)%\ (A9.4) Treat the term i n brackets above as a proportionality ".T factor independent of the k's and l e t K - i t - ^ which allows us to re-write (A9-4) as V ^ i . J « " V ( ? ) i \ ~ 1 (A9-5) (A9-6) 154 For simplicity and keeping in mind that we are dealing with strong interactions, take for V the Yukawa potential v ( ? ) = v 8 £ " r ( A 9 . 7 ) r where VQ and <X are constants. Then we can integrate (A9-6) using (A9-7) to obtain where COS 6 - * | For small k& and k^ (low energy scattering) we can neglect 2 2 2 the terms k& and k^ compared to oL (but keep the cos© term to preserve the angle dependence). Then we can write (A9-8) by expanding the denominator as J k j i f * ? + ( * M k £ ^ , . . ] ( A 9 . 9 ) Looking back at (A9-1), the phase shifts are usually par-H-H (A9-10) 42 ameterized by writing 42 See reference (6), p.306, eq.16-98. 155 so that (A9-10) into (A9-1) gives f o r small °° -2>£ o r„-r /s \ (A9-11) 0~o 0 Keeping only the lowest power of q (small q) i n each fac-l t o r of (cos©) we can write (A9-11) as I and comparing each (cos©) term i n (A9-12) and (A9-9) we make thei a s s o c i a t i o n or i n our o r i g i n a l notation l ^ h U (A9-15) But when ^=4^ we must have q=q^ so that the proportion-a l i t y f a c t o r of (A9-13) i s determined and we can write the f i n a l r e s u l t f or i n e l a s t i c s c a t t e r i n g O ^ - P ^ - M ™ = J q T L ( A 9 - H ) 'effective ~ M * j * f Notice that ( A 9 - H ) i s inva r i a n t under the interchange of q^ and q^ as i t must be. 156 Appendix 10: Pi-nucleon S-wave Phase Shift Parameters I f T denotes the total isospin quantum number of a pi-nucleon state (T=3/2 or 1/2) i t i s customary to expand the two-body pi-nucleon scattering amplitude for each 43 value of T by writing r t'° Kf ( A 1 0 - 1 ) where / \ fL = - L f I ^ ' ^ H ] (A10-2) and means j = /±:^. (We are neglecting spin-flipping mechanisms for now). The lj • s are called absorption coefficients and the & •s are the phase s h i f t s . It is also customary to re-label the $*a and r| 's to include the total angular momentum quantum number j as well as the orbital angular momentum quantum number X and the total isospin quantum number T. One uses the spectro-scopic notation S,P,D,F, etc. for JL =0,1,2,3, etc., respectively, and writes I t 43 See reference (18) for example. 157 Keeping S-waves means that we only keep the Z =0 term of (A10-1). The phase s h i f t s and absorption coefficients we use are polynomial f i t s to the TJCEL tables (reference (16)), and in particular we choose the data of Kirsopp. It was found necessary to consider momenta up to 1000 Mev/c to make our integrals convergent„to one per cent accuracy. To be more s p e c i f i c , squaring the momentum-space deuteron wave function (A2-4) and integrating over k, i t i s necessary to integrate out to k=1000 Mev/c to get within one per cent of the integral evaluated out to i n f i n i t e k. We l i s t the parameterization employed as follows: 0^ q-^  100 Mev/c: ( a l l S's in degrees) = Sn = (-5.95x10"6q2 + 5.3x10~4q + 7.25x10"2)q S5 1 = (-1.66x10"6q2 + (1.4lx10~4q -4.15x10~2)q 1 1 1 = ^ 3 1 = 100^ q< 200 Mev/c: Sn = 2.5xlO~2q + 4.1 S5 1 = -7.7x10~2q + 3.3 I n = n 3 1 = L O O 200< <i< 300 Mev/c: Sn = 1. 1x10~2qr-iv 6.9 S3 l = -9.3x10"2q + 6.5 , »| 1 1 = rj ^ = 1.00 158 3 0 0 £ q<4-00 Mev/c: Sn = 1.16x10"1q -24.6 S3 1 = -7.1x10"2q -.1 ^ 1 1 = -9.0x10~4q + 1.27 Vj 5 1 = -7.0x10~4'q + 1 .21 4 0 0 i q ^ 500 Mev/c: S1 1 = 2.93x10"1q -95.0 S3 1 = -3.3x10"2q -15.3 *\ T 1 = -5.2xlO"5q + 2.99 /} 5 1 = -4.0x10"3q + 2.53 500 £ q ^ 600 Mev/c: Sn = 5.59x10"1q -229 S3 1 = -3.22x10"1q + 129 Y\ n = 3.2x10"5q -1 .21 r\ 5 1 = 1 .3x10"\ -.12 600 £ q < 700 Mev/c: Sn = 2.57xiO~1q -48 S5 1 = -6.0x10"5q -60.4 *l n = 3.0x10"4q +.53 H 5 1 = 2.4x10"3q -.78 700£q<800 Mev/c: S1 t = 1.58x10"1q + 22.7 159 S3 1 = 9.4x10 "q -131 ^ n = 1.00x10"4q + .75? 5 1 = -4.0x10"3q +^ 3 n 800^q< 1000 Mev/c: Sn = -3.35x10~1q + 417 S5 1 = -2.48x10"1q + 143 ^n = -7.0x10"4q + 1 .23 ^ 5 1 = -3.5x10"5q + 3.3 160 Appendix 11: Pi-nucleon P-wave Phase Shift Parameters In analogy with Appendix 10, we l i s t here the P-wave =1i) absorption coefficients and phase s h i f t s . The change i n notation appropriate here is 5 S. = P. 3 , + ^ i i * - ta and we again f i t the TJCEL tables (Kirsopp data) with polynomials i n q. The parameterization is as follows: Pn ( i n degrees) = (-1.83x10~6)q5 , 0 £ q ^ 1 0 0 Mev/c 1.00, 1 0 0 £ q < 200 Mev/c 1.74x10"6q3 -6.85x10"2q , 2 0 0 £ q< 400 Mev/c 4.35q. -89 , 4 0 0 £ q< 600 172 , 6 0 0 £ q < 1000 I n = 1 , 0 0 » 0£ t q < 300 Mev/c -7.0x10"5q + 3.1 , 3 0 0 £ q < 4 0 0 Mev/c 1.0x10"5q-.1 , 400 ±q_-^ 600 Mev/c -4.2x10"5q + 3.0 , 6 0 0 £ q < ^ 700 Mev/c 2.74x10"\ -1.84 , 700 ^  q < 1000 Mev/c 161 13 0.0 , 0 £ q ^ 200 Mev/c -3.0 , 200^ q< 300 Mev/c -5.0 , 300£ q< 600 Mev/c -10.0 , 600^ q< 1000 Mev/c 1 1 3 = 1.00 , 0 £ q^ 500 Mev/c -1.69x10"5q + 1 .85 , 5005 q<£ 1000 Mev/c 31 -7.90x10~7q5 , 0^q^100 Mev/c -1.14x10"4q2 + 3.3x10"3q , 100^ q* 500 Mev/c -26.8 , 500£ q< 700 Mev/c - T H 8 q + 77 , 7 0 0 £ q < 1000 Mev/c 1 31 1 33 1 .00 0 £ q < 400 Mev/c , -3 -1.02x10 ''q + 1.41 , 400 <q<^ 1000 Mev/c For P^^ the appropriate momentum dependence i s well-known from Chew-Low theory (see reference (27), page 233, eq.8.22) and i s given by where «j~s is the total center of mass energy. We again have the problem of inelastic scattering so i t i s most convenient to parameterize Js" in terms of q. The choice 162 >fs = 1.5x10"\2 + .32(1 + 1078 gives a reasonable f i t to the data out to q=500 Mev/c and for higher momenta the P ^ phase s h i f t is f a i r l y 44 constant out to 1000 Mev/c so we take p^5 = 172 degrees, 5 0 0 £ q-^ 1000 Mev/c 44 given The inelastic scattering is handled by taking q = q f f in (A9-H) in the <fa parameterization. 163 Appendix 12: Treatment of Spin-Flipping Mechanism We show here how to treat the added complication when spi n - f l i p terms are included in the pi-nucleon scattering amplitudes. Unfortunately we w i l l go through considerable effort to find the s p i n - f l i p contribution (up to P-waves only). The most general pi-nucleon scattering amplitude 45 must have the form oO where the prime on P^ denotes derivative with respect to cos©, qfe and q& are unit vectors in the direction of the i n i t i a l and f i n a l pion ( i n the pi-nucleon cm frame), *~= ^^1'^2'^3^ i s tiie 1 > a u l i s P i n matrices written as.-.a three-component vector, the A's are given by (A10-2), and |a^ and |b^ are the i n i t i a l and f i n a l nucleon spin states. Compare (A12-1) with (A10-1) which neglects the sp i n - f l i p terms. I f we only keep the terms up to P-waves in (A12-1) we have 45 See reference (18), p.28, just below eq.2-24. 164 XT where 4 < t ( A + B ^ - ( | u x ^ ) K > ( A 1 2 . 2 , The spin state | b^ must be the same as | a^ >: ;be-cause i n the threshold limit no energy i s available to " f l i p spins. Therefore there can be no spi n - f l i p contri-bution i n the single scattering terms since i n i t i a l and f i n a l spin states are the same. Now write a typical second order term where in writing the proportionality we do so because we are only concerned with spin and angular variables, not momenta and energies (this w i l l become clearer l a t e r ) . Then substituting (A12-3) i n the above (with the appro-priate conversion from f to t) gives 165 < t i t , Ct J O < x < M A , 4 8 , * f , ) U > where A1 , A2> B1 , are "the same as (AT2-3) hut with the appropriate kinematic factors to convert the scattering amplitudes to the corresponding two-body T matrices, and a r e ^D-e pion directions in the cm of pion and nucleon 1 and 2, respectively, in state ]n^. Now we evaluate the necessary kinematic factors of (A12-4) n o n r e l a t i v i s t i c a l l y . Let us f i r s t l i s t a l l lab frame momenta and velocities (as determined by (2a-3)) as follows: ~*> lab momentum of pion i n state lb> - Ptr = fvv \L -O lab momentum of pion i n state l a ^ ~ Yv ~ M -O ' ' "A, Tl Tl^ lab momentum of pion in state Ln>- PIT - m^V 166 (continued from previous page) lab momentum of nucleon 1 in state (ri^>-|? - — (fnn + ^  ^  lab momentum of nucleon 2 in state |n) - ^x = ~ (EHo--'k^^ 2-(A12-5) where we used |> = -p i n (2a-3). Then the velocity of n the cm of pion and nucleon 1 in state )n> i s 1 - ± - (A12-6) and the velocity of the cm;of pion and nucleon 2 in state fcO i s • rv> + rv,., " _ i (A12-7) so that by definition and using (A12-5) through (A12-7), the q * s in (A12-4) are 167 With (A12-8) we can perform the required cross products of (A12-4), hut note that we want cross-products of unit vectors so that a l l vectors in (A12-8) must be divided by their magnitudes before taking cross products, com-plicating the work. Thus, using (A12-8), - " ( ^ ( t v - i o fo —~> To proceed, define - " f^- kn f f>n (A12-9) so that If we pick an arbitrary direction of p and integrate *n over i t this i s equivalent to picking an arbitrary direc-tion of p xk and integrating over i t . So we write Integration over £ gives zero for the i and j terms and integration over o{ gives zero for the k term. The mag-nitudes q.j , q2, q&, q^ depend on 9 but not on <* or (3 and the same is true for a l l other kinematic factors in (A12-4) Therefore integrating the second and third terms of (A12-4) (A.jB2 and B1A2) over angular variables gives zero just as 168 above, using (A12-9). The f i r s t term of (A12-4) i s the one we have already evaluated when we did the non-spin f l i p case. It remains to evaluate the last term of (A12-4) to get the s p i n - f l i p contribution. Consider the expression -t- (A12-10) and consider also the integral (A12-11.) where the subscripts 1 and 2 mean^the operator only operates on the f i r s t or second part of the two-nucleon spin states, and the notation M i ) means both nucleons have their spins up (along the axis f =0), ( ? ) ( » means nucleon one i s spin down and nucleon 2 spin up, etc. 169 In (A12-11) we have chosen u > u > = ! ( ' o ) f t ) > since the spins of the nucleons are parallel (spin 1) in the deuteron. The only possible intermediate spin state \n.y i s | ( ^ m ^ s i n c e each spin operator in (A12-11) only operates once on each nucleon. Then putting (A12-10) i into (A12-11) we have The contrihution of the sp i n - f l i p part of (A12-4) i s thus whereas the non-spin-flip contribution is 4nA1A2 (the 4T1 comes from integrating over o< and Q ) . Numerical integration of these two contributions over G, p , and k shows the s p i n - f l i p part to be negligible. 1 70 Appendix 13: Re l a t i v i s t i c Propagator We would like to justify the r e l a t i v i s t i c replace-ment (4d-1) in the propagator for the non-relativistic (A4-4). For convenience we w i l l consider a two-particle propagator of spin-zero particles. I f p^ and p2 are the four-momenta of each particle then we want, to show that the non-relativistic propagator is more correctly replaced by where E^ is the i n i t i a l total energy of both particles and for convenience we take the mass of the particles equal. Prom f i e l d theory the propagator for a single scalar spin-zero particle i s pi-mN-ife f x - i w - ( 1 T av r ' ( A 1 3 - 1 ) Por two spin-zero particles the propagator is just the product of the single particle propagators ( A 1 3 - 2 ) 171 A t ^ h i g h e n e r g i e s t h e d e l t a - f u n c t i o n p a r t o f t h e p r o p a g a t o r d o m i n a t e s s o we w r i t e & ( f / j f i ) ^ H f r ^ ) 5 ( f \ - " 0 (A15-5) D e f i n e t h e t o t a l e n e r g y b y „ h e r e E , O f - r + ^ ) ^ E l < | P , l v + ^ ) 4 a n d s i m i l a r l y f o r t h e i n i t i a l s t a t e I n t e r m s o f S ' we h a v e ( A 1 3 - 4 ) s o t h a t ( A 1 3 - 4 ) a l l o w s u s t o r e - w r i t e t h e t w o - p a r t i c l e p r o p a g a t o r ( A 1 3 - 3 ) i n t e r m s o f S 7 U s ' - l E ^ - a ^ E ^ i f , - ^ ( A 1 3 _ 5 , 172 Now use the Cauchy integral formula (write a dispersion relation) ( A 1 3 - 6 ) s ' - s where the contour C i s appropriately chosen within the limits of Cauchy's theorem. Performing the integral over the delta functions i n ( A 1 3 - 5 ) we obtain (A13-7) We can neglect the P-j'P2 term in the denominator since i t s average over angles gives zero. We are then l e f t with o< I - ! i •> ' ( W ^ ) and the right-most factor i s just for normalization so that we obtain our desired result The careful reader w i l l note that the result ( A 1 3 - 8 ) only holds for high energies since ( A 1 3 - 6 ) can only be used i n conjunction with ( A 1 3 - 5 ) for large values of S±; that i s 173 the integrand of (A13-6) is peaked for Senear Si so i f is small the major contribution to (A13-6) may not come-from, large s', i n which case (A13-5) may not be appro-priate. We therefore take (A13-8) as a very heuristic, hand-waving result but believe i t to be close to r e a l i t y , nevertheless. One could test our belief by using the f u l l propagator (A13-2) in (A13-6) but we leave that task to the reader. One can proceed analogously for three particles instead of two and treat two of them as spin % Dirac particles with similar results. The general conclusion i s that non-relativistic energies in propagators get replaced by their r e l a t i v i s t i c analogs. 

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