A FIELD THEORY APPROACH TO PION ..DEUTERON ELASTIC SCATTERING • . by JAMES HARRY ALEXANDER B.Sc., U n i v e r s i t y of V i c t o r i a , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 In p r e s e n t i n g t h i s thesis in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree the L i b r a r y s h a l l make i t f r e e l y that a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s thesis f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l written gain permission. Department o f P-Ttysir.s- The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 1 1 . 1 9 7 5 shall not be a l l o w e d w i t h o u t my ii Abstract Pion-deuteron theory elastic o f p i o n s and scattering nucleons. manner, t h e d o u b l e - c o u n t i n g multiple-seattering The in pion-deuteron problem T-matrix i n the s e r i e s making an i s written a r e examined. compared.- double The be particular field nucleons associated field in this with pion be expressed term resulting are also without theoretic expansion states. first first The term a s a sum and of yields i s similar from four doing potential. the The twenty a static the magnitudes of dominant There as a s e r i e s t o the T-matrix. approximation term evaluated The contribution sensible, scattering approximation. cannot can on-shell where p h y s i c a l l y the usually terms of o p e r a t o r s between o n e - n u c l e o n usual single-scattering are treating using a i s avoided. terms i n the s e r i e s term By i s studied the two the second terms. By approximation twenty terms to the c o n v e n t i o n a l generalized o t h e r terms numerical impulse whose m a g n i t u d e studies with a iii TABLE OF CONTENTS 1 Introduction 2 Definition 2.1 1 of O p e r a t o r s and Physical S t a t e s With Barycn Number Z e r o 2.2 P h y s i c a l S t a t e s With Baryon Number One 2.2.1 The 2 . 2 . 2 The 2.3 Physical The 2.3.2 Baryon Tha The Meson .7 ....7 State .............9 11 ................11 State S t a t e and the Pion-Deuteron ............11 State and ..........13 the T - m a t r i x ....16 Exchange S e r i e s 3.1 Cutkosky Statss 3.2 Meson Operator with Baryon Identities ....16 Number Two and Cutkcsky E l e me n t s Operator Identities 3.2.2 Cutkosky Overlap Cutkosky Operator Meson and ...........21 M a t r i x Elements Matrix Elements Exchange S e r i e s Deutaron 3.3.2 The Pion-Deuteron Scattering 3.3.3 The Pion-Deuteron T-raatrix 4.2 One-Meson the F o r The Exchange C o n t r i b u t i o n .30 30 State Vector .....34 37 T-matrix Exchange C o n t r i b u t i o n .22 Vertex State Vector Exchange S e r i e s Zero-Meson of Diagrams for Picn-Deutercn Scattering The 4.1 and .............25 the H a m i l t o n i a n 3.3.1 4 r h e Meson Matrix ...........21 3.2.1 3.2.3 3.3 5 Number Two Dauteron S-matrix .• State P h y s i c a l Two-Nucleon Scattering 2.4 Ona-Nucleon Pion-Nuclecn Scattering S t a t e s With 2.3.1 3 ......4 States to the T-matrix tc the T - m a t r i x 38 ...38 ....40 <4.3 The S t a t i c On-Shall Approximation 5 Conclusions . . Bibliography A Mason O p e r a t o r APPENDIX B Cutkosky Matrix Unexcited B. 2 U n e x c i t e d Overlap Interaction Overlap B.4 Excited Interaction The Matrix . . . . . . . . . 6 3 ....68 .....75 Elements i n the One-Meson ..77 Exchange ...80 D D. 1 T h e D e u t e r o n Elements Elements Matrix T-matrix ...63 Elements Matrix Approximation APPENDIX ...57 58 Elements Matrix Excited C 55 Identities B.3 APPENDIX » • APPENDIX B.1 48 85 wavefunction 85 D.2 Single-Scattering ..86 D. 3 Double-Scattering ...88 Acknowledgements I to would my research encouragement I and would a l s o constant The Council like to take t h i s supervisor, Dr. to Malcolm express my thanks McMillan, for his Margaret, for her assistance. like moral support financial over the opportunity to thank and assistance-. assistance past two years my wife, provided by the i s gratefully National Research acknowledged. 1 1 IHt£2.duct i o n In the next few years many experiments the meson f a c t o r i e s which w i l l use nuclear is a particularly structure. investigate states, pion pion n u c l e a r s t r u c t u r e . S i n c e i t comes i t can p a r t i c i p a t e exchange scattering understanding absorbed The the w i l l be performed or experiments of which nuclear emitted can i n charge by a probe of u s e f u l t o o l to in hopefully three charge double charge further the A l s o , s i n c e the pion can nucleon, be u s e f u l probes a exchange and will states." as at pion-nucleus be absorption c f h i g h e r momentum components i n n u c l e a r wa vaf u n c t i o n s . S i n c e the i n t e r a c t i o n of rather weak pion with energy pions can s i m i l a r to e l e c t r o n s c a t t e r i n g . be pion s c a t t e r i n g and used as complementary The s i m p l e s t way the s i n g l e - s c a t t e r i n g pion s c a t t e r s from approximation, types of a and forces, be i . e . , to assume that the nucleon i n the n u c l e u s . However, i f details structure ever to be using useful the , i t i s necessary t o o b t a i n an terms neglected by t a k i n g the s i n g l e - s c a t t e r i n g Ihe in to t r e a t p i o n - n u c l e u s s c a t t e r i n g i s t o use is approximation utilised the methods. o n l y one nuclear 100 MeV, is e l e c t r o n s c a t t e r i n g can perhaps e l a s t i c pion s c a t t e r i n g of nucleus However, s i n c e pions e l e c t r o n s i n t e r a c t with nucleons v i a d i f f e r e n t elastic the f o r pion k i n e t i c e n e r g i e s below about e l a s t i c s c a t t e r i n g o f low manner the g e n e r a l i z e d impulse approximation (1952) ] expresses the e l a s t i c scattering in single-scattering estimate of the approximation. [Chew and of a extracting Goldberger projectile on a •2 nucleus as a sum of terms, the f i r s t scattering, d o u b l e - s c a t t s r i n g , and scattering describes from one scatters nucleon, again three of which are s i n g l e - binding c o r r e c t i o n s . Double- the process where the p r o j e c t i l e scatters then propogates before l e a v i n g to another nucleon the nucleus. B i n d i n g c o r r e c t i o n s d e s c r i b e the e f f e c t of the n u c l e a r p o t e n t i a l on the nucleon this approach s c a t t e r i n g , a problem mediating the n u c l e a r f o r c e , nucleons as clear to extent what in the This to describe pion-nucleus thought the p r o j e c t i l e s c a t t e r i n g being of as from the exchanged between a p a r t of the n u c l e a r f o r c e . Thus i t i s not the binding" c o r r e c t i o n s double-scattering impulse approximation. "double-counting used to the p a r t i c l e these included is a r i s e s . Since the pion i s nucleons i s i d e n t i c a l This is term of usually are the already generalized referred to as thesis avoids the double-counting problem nucleons. The a c l o u d of p h y s i c a l p i o n s . By no treating double-counting will occur i n v o l v e d i n the process are accounted The approach Pendleton (1963). s c a t t e r i n g . The simplest corrections nucleons, The nucleus the nucleons since case be in chosen chosen which non-zero. properties of in by this a l l the pions for e x p l i c i t l y . use! i n t h i s t h e s i s f o l l o w s a method l e u t e r o n was can the by nucleons assumed to be composed of a bare nucleon core surrounded manner, the problem". c o n s t r u c t i n g a f i e l d theory of p i c n s and the projactile- scattering. If are and is elastic as the t a r g e t used by picn-deuteron since it is d o u b l e - s c a t t e r i n g and b i n d i n g Since it contains i t s wavefunction only are perhaps two the 3 best known o f In chapter states are describing states are any the as a one-nucleon are compared can the of summary a These series calculated from nucleon along with be states. to pion, scattering. introduced. series the defined which T-natrix 2 nucleus. the of the states in are expansion In chapter and after and impulse results. J basic 3 cf to write the of 4 first making some approximation. pion-deuteron between terms of approximations, double-scattering terms Chapter in operators operators two used two-nucleon one-nucleon terms the scattering operators approximate terms used in pisn-nucleon the chapter written single- generalized In and 5 the are resulting consists of 0 £ B r a t o r s and Definition 2 In be this used duced. in The manner section similar one-nuclaon states to that by in describing states will London (1927) applied to used pion-nucleon be t r e a t e d using to states pion-deuteron s c a t t e r i n g and (1955) and some o f t h e o p e r a t o r s and describing zero- States nucleon-nucleon be Chew and first be The by will introin Low (1956) and a Wick two-nucleon d e v e l o p e d by the hydrogen problems will developed scattering. a method describe will which Heitler m o l e c u l e and later Cutkosky(1958) and P e n d l e t o n (1 963) . The and fundamental nucleons dynamical v a r i a b l e s a r e assumed t o be boson f o r the system annihilation and of p i o n s creation t operators k and k and fermion annihilation and creation + a operators commutation operators and and a . These operators anticommutation rules. satisfy That is, the usual the boson satisfy (2. 1) and the fermion operators s a t i s f y {a,b} = {a ,b } = 0 + + (2.2) In addition delta the boson and fermion operators w i l l f u n c t i o n s above a r e a c t u a l l y products of commute. Kroenecker The delta 5 functions of s p i n and i s c s p i n aomentura. The boson with field Dirac quanta will p h y s i c a l pions and the f e r n i o n f i e l d the fundamental dynamical v a r i a b l e s and having eicenstates to antinucleons. physical nucleccs, pion-nuclecn s t a t e s , deuterons, e t c . These s t a t e s w i l l the baryon number 2.1 P h y s i c a 1 States The kill with constructed be corresponding |k> identified assumed t h a t a H a a i l t c c i a r c a r be will quanta of identified It be ke functions be with bare nucleons and using delta operator also B . correspcndirg t c barycn runber zero mescn s t a t e s or the p h y s i c a l vacuum s t a t e . a he e i c e n s t a t e s c f With Barton Kumber Zero state vectors represents scattering mescn momentum, etc.) l a b e l l e d with The guantum numbers mescn (spin, (2.3) P|k> = kjk> (2.4) B|k> = 0 (2.5) z (2.6) z 1 cperatcr and momentum operator, B is the baryon ruaber i s the mass of the p i o n . The speed c f l i g l t and * a r e taken t o be one vacuum s t a t e iscspin, H|k> = E |k> E, = /|k| +m k '— ir the state by k. I t s a t i s f i e s k where P i s will in the abcve equations. The physical | 0 > i s d e f i n e d ty H|0> = 0 B|O> = 0 i ' ) 2 < * 2 1 8 ) 6 It a l s c satisfies As stated atove quanta are p h y s i c a l can k|0> = 0 {2. 9) a|0> = 0 (2. 10) i t will be assuned that p i o n s . That i s , the p h y s i c a l be thought of as r e s u l t i n g frcm creation operator the action on the p h y s i c a l vacuum s t a t e the boscn nescn s t a t e cf the therefore H = Z E k be w r i t t e n i n the form kk + V + k where V i s an o p e r a t o r d e s c r i b i n g nucleon the ( 2 meson-meson and over the meson momentum. (In d i f f e r e n t s p l i t t i n g of H w i l l be i n t r o d u c e d . be more u s e f u l i n s e p a r a t i n g the . 12) meson- i n t e r a c t i o n . The summation over k i n the above equation denotes a sum over a l l the meson quantum numbers as w e l l integral meson (2. 11) + Hamiltonian w i l l |k> |0> k |0> = |k> The field subsection an 3.2,3 That s p l i t t i n g the nucleon energy p a r t of nucleon i n t e r a c t i o n p a r t of H.) as H a will from 7 2.2 Physical The be state either states. With physical are of Phv.sical The with labelled by It to or baryon will be number pion-nucleon importance One-Nucleon in treated in one scattering discussing some state numbers |A> (spin, pion- detail. describes isospin, a physical momentum etc.) satisfies H|A> = E |A> - A (2.13) P|A> = k |A> (2. 1") B|A> = +l|A> (2.15) A E. where M is A physical a be In operator is core the is as a pions. sum o f present constructed written creation as theory a linear operators The treatment a [Chew pion more meson Low(1956) ] as and operator an of creation cloud creation the product which operators. The products as operator nucleon manner. the the creation physical general defined + written pion the A is and clcud of combination and is operator products in operator operator creation "cloud" of written mass. creation creation nucleon (2. 16) 2 nucleon Chew-Low one-nucleon a 2 one-nucleon the bare produces can In = /|k,| +M observed physical follows. of the will State one-nucleon quantum A. states great they physical nucleon Number One corresponding one-nucleon scattering, The Barton vectors As b o t h deuteron 2.2.1 States creation operator of nucleon operators: 8 A M Z a(AM) The nucleon nucleon number + core c r e a t i o n o p e r a t o r A and antinuclecn creation i s simply a jrccuct cf operators with tctal tare baryon one: 8A |0> = +1 A O b v i o u s l y i f A' an ( 2 . 1 7 ) + odd number i s to s a t i s f y cf bare |0> ( 2 . the above equation nucleon and 1 8 ) i t must c c n t a i r antiruclecn creaticr + cperatcrs. The meson c r e a t i o n The meson c l c u d o p e r a t o r H summation i n e g u a t i c n condition the wave that function anticommutation seen that the i n the c f eguation linear and ( 2 . 1 7 ) cf each i n the term |A>. ccttbiraticn the (2.1) r e l a t i o n s of e g u a t i e n s will be lock space c f commutation ard (2.2) i t can A E , e t c . obey the operators and r be following relations: {A,B} However, core mesons. p h y s i c a l nucleon anticommutation nucleon momentum of the s t a t e of the p h y s i c a l nucleon physical From the form i s ever a l l ( 2 . 1 7 ) the t o t a l charge ard mcffentuns c o e f f i c i e n t s a(AH) the cores and of meson c l o u d c r e a t i o n o p e r a t o r s s u b j e c t t c must be equal to the charge and The cf a product operators. c r e a t i o n o p e r a t o r s and the consists expression because of the n e c e s s i t y = { A ' , B for } = ( 2 . 0 (fl,E ) T w i l l be of anticommuting the verj 1 9 ) complicated products of the 9 annihilation operators with the products c f the c r e a t i o n 4. operators i n the e x p r e s s i o n s f o r necessary i t o evaluate t h i s connutatcr and E • The calculations are dene i n an approximate nanner i n Appendix B. 2.2.2 ThePion-Nucleon The pion-nucleon composed of a asymptotically State scattering state physical labelled asymptotically sign) Scattering nuclecn by |Ak> r e p r e s e n t s a s t a t e + with qcantun A and a mescn * i t h l a b e l l e d by k, obeying either numbers quantum numbers outgoing wave o r incoming wave (- sign) boundary c o n d i t i o n s . Since outgoing wave s t a t e s w i l l normally scattering states s a t i s f y outgoing scattering state without wave be d e a l t only with, pion-nuclecr *ill be assumed to + or - s u b s c r i p t s boundary (+ conditions. The pion-nuclecn |Ak> s a t i s f i e s H|Ak> + = (E +E.)|Ak> A k (2.20) ± P|Ak> = (k +k) |Ak> " ~~ 8|Ak>. = +l|Ak> . .* * + (2.21) ± where E A and E. a r e as given (2.22) previously. K Hick (1955) has shewn that the pion-nucleon can be expressed solely i n terms c r e a t i o n o p e r a t o r s , and p h y s i c a l Sick wrote the pion-nucleon consisting of a scattered state free mescn state c f the E a o i l t c n i a n , meson nucleon scattering and a scattering creation state physical operators. as nuclecn a state plus a 1C I Ak> = k | A > + + An explicit expression for |Ak> the (2.23) s scattered state |Ak> can l e s (2.23) obtained by s u b s t i t u t i n g e q u a t i o n equation f o r H f o r the s c a t t e r i n g (H-E -E )|Ak> A Equation k (2.24) + Hk |A> - + s + = k H|A> + + the s t a t e , equation (E +E )k |A> = A i s s i m p l i f i e d by Hk |A> intc + k eigenvalue (2.20) 0 (2.24) writing (2.25) [H,k ]|A> + (2.12), Using the e x p l i c i t form c f the H a m i l t o n i a n , equation commutator i n the above e x p r e s s i o n becomes [H,k ]|A> = E + By tion k |A> + k (2.24) the f o l l o w i n g (H-E -E )|Ak> A An e x p r e s s i o n f o r the k the + ( (2.26) and intc equa- equation i s obtained [V,k ]lA> + s + scattered operator 2.26) + {V,k ]|A> (2.12), (2.25), s u b s t i t u t i n q equations inverting the state (H-E -E ) A =0 and ( 2 > 2 ? ) |Ak> s is imposing crtainec" outgoing ty wave k boundary c o n d i t i o n s |Ak> s = (E A + E -H+ie)' [V,k ]|A> k 1 + (2.28) 11 u s i n g Pendleton's n o t a t i o n f o r the vertex operator [Vk] = [V,k ] f (2.29) [kV] the outgoing = [Vk] wave pion-nucleon |Ak> scattering 2.3 + k (2.30) i s writtec inccrrectlj the o p e r a t o r [ V k ] being r e p l a c e d by [ k V ] . Number Two The P h y s i c a l Two^Nuclaon S t a t e The p h y s i c a l two-nucleon s t a t e , denoted two (2.30) _1 A S t a t e s With Barton 2.3.1 state i s written = k |A> + (E +E -H+ie) [Vk]|A> I t s h o u l d be noted t h a t equation ty Pendleton, + physical nuclsons with quantum by |AB>, r e p r e s e n t s numbers asymptotically l a b e l l e d by A an! B. I t s a t i s f i e s H|AB> = CE +E )|AB> (2.31) P|AB> = (k +k )|AB> (2.32) B|AB> = +2|AB> (2.33) A A 2.3.2 B fi The Deuteron S t a t e and the Pion-Deutergn The vector p h y s i c a l deuteron Scattering state i s represented State l y the state \V> . I t s a t i s f i e s H|P> = EJP> (2.34) 12 P|P> = kjp> ^ (2.35) B\V> = +2\V> y y where ftp i s the observed The pion-deuteron composed of a (2.36) (2.37) 0 deutercn r e s t mass. scattering state physical deutercn |Pk> r e p r e s e n t s a s t a t e + with guartufc numbers asymptotically labelled by V and a meson with quantum nunbers asymptotically labelled by scattering incominq will The be state,the ± k. As pion-deuteron the pion-nucleon s u b s c r i p t r e f e r s t o e i t h e r outgoing c r wave boundary c o n d i t i o n s assumed with and a s t a t e «ith no to s a t i s f y outgoing wave boundary scattering state p+ k ± Ckp-k) It?k> P|Pk> = ± ± B|Pk> = 2|t?k> ± where and E By fc + are as given ± < w r i t i n g the pion-deuteron equation same procedure (2.23), the analogous deutercn s c a t t e r i n g < 3 8 ( 2 # 3 9 2 < U 0 ) ) ) (2.41) s the 2 s c a t t e r i n g s t a t e as + using e x a c t l y ( previcusly. |Pk> = k |P> + |t?k> and conditions. |Pk> s a t i s f i e s H|Pk> = (E E )|t7k> ± subscript as was used following r e s u l t i s o b t a i n e d f o r the p i c r - state |t?k> = k |p> + + (Ep+E^H+ic)" ^]!^ 1 (2.42) 13 2.4 The S-matrix ana the T-matrix The scattering o p e r a t o r S i s defined i n g e E e r a l as (2.43) where | I K (0)> i s the t i n e independent f a c t o r in c f the limit before cf the satisfying the state vector a long time incoming wave boundary c o n d i t i o n s time independent f a c t o r c f the a s y n p t c t i c vectcr a long and asymptotic scattering, | ^ liirit out (0)> i s c f the s t a t e time a f t e r s c a t t e r i n g , s a t i s f y i n g outgoing wave boundary c o n d i t i o n s . Fcr pion-nuclecn s c a t t e r i n g eguation (2.43) can te writter as (2.44) or, u s i n g the f a c t t h a t S i s unitary <Bm S + The = <Bm (2.45) S-matrix f o r pion-nucleon s c a t t e r i n g of eguation = <Bm|s|Ak> = <Bm|Ak> by use the s i m i l a r e g u a t i o n f o r |Bm>_ i s d e f i r e c as + + (2.45). Using eguation (2.46) (2.30) f c r | E n l a c e " 14 |Bm>_ = m |B> + ( E ^ - I l - i e ) ~ + it ( 2 . 4 7) [Vm] | B> 1 f e l l o w s that (2.48) |Bm>_ = |Bm> + { ( E g + E ^ H - i e ) - ( E g + E ^ H + i e ) ' } [Vm] | B> - 1 1 + Using the i d e n t i t y (E +E -H-ie) B equation - 1 m = (Eg+E^H+ie)" (2.49) + 2TT16 ( E + E - H ) 1 B m (2.48) can be w r i t t e n |Bm>_ = |Bm> + 27riS (Eg+E^-H) [Vm] | B> (2.50) + S u b s t i t u t i n g t h i s i n t o eguation (2.46), the e x p r e s s i o n fcr the £-matrix becomes S Bm,Ak = + <Bn>|Ak> - 2 , i 6 O y E ^ E ^ ) < B | [mV] |Ak> + = Bm,Ak " 2iri«(E E -E -E )<B|[mV]|Ak> + 6 B+ It m A k + ( 2 . should be noted t h a t the Kronecker d e l t a f u n c t i o n 6„ 5 1 ) is a Bm,Ak shorthand cf n o t a t i o n f o r the product a l l the quantum c f Kronecker d e l t a f u n c t i o n s numbers of the s t a t e s |Em> and |flk> as well as a d e l t a f u n c t i o n of momentum. The g e n e r a l e x p r e s s i o n f o r the S-matrix i s S FI = 5 FI " 2Tti6(E F -E )T I F I ( 2 < 5 2 ) 15 where I r e f e r s to the i n i t i a l s t a t e , F r e f e r s tc the f i n a l state li and 1 is the T-matrix. Thus, comparing equations (2.52), the T-matrix f o r pion-nuclecn V A k S i m i l a r i l y , using ponding equation for (2.51) arc s c a t t e r i r g can be w r i t t e n <B|[mV]|Ak> = (2.53) + equation |Pk> , (2.42) f c r |Pk> ace the + the T-uatrix corres- f c r picn-derteron s c a t t e r i n g can be w r i t t e n T t?-m,t?k = < t ? 'ltmV]|Pk> (2.54) + The o b j e c t i v e now i s to c a l c u l a t e the r i g h t eq. hand side cf (2.54). To do t h i s , the deutercn s t a t e and the p i c n - d e t t e r c c scattering state will be expanded i n terms c f a p a r t i c u l a r s e t o f two^nucleon s t a t e s i n t r o d u c e d i n the next chapter. 16 3 SS^QU Excharj2§ Esrias In order t o proceed from t h i s states with baryon number pcint, two is choosing p h y s i c a l two-nuclecn s t a t e s expressed states in is creation constructed introduced. which explicitly and meson nethcd used by Cutkosky(1958) and London(1927). in states the of than readily be cne-nuclecr operators following a and based on the work are will be able between one-nucleon s t a t e s . expand s e t of Bather cf These s t a t e s w i l l be c a l l e d Cutkosky Cutkosky states cannot terms creation written nucleon and meson c r e a t i o n o p e r a t o r s , these ccnplete terms of cne-nuclecn c r e a t i o n o p e r a t o r s , a s e t c f operators Since a pion-deuteron to i n terms c f p h y s i c a l matrix by T-matrix, in states. elements betveen be reduced to matrix elements Thus, express the T-matrix e n t i r e l y Feitler using these i t will terms cf te states tc possible tc cne-nucleon matrix elements. 3.1 Cutkosky, S t a t e s with Barren Nujfcer Two The simplest Cutkosky s t a t e i s that which c o n s i s t s cf twe nucleons with no mesons present. T h i s s t a t e , c a l l e d an unexcited Cutkosky s t a t e and denoted by |AB}, i s d e f i n e d by AB} = A B |0> It (3.1) satisfies P|AB} = 8|AB} (k^+k^lAB} = +2|AB> (3.2) (3.3) 17 However, unlike Cutkosky This state i s the physical JAB} i s shown n o t an eigenstate explicitly definition of the state operators anticommute, two-nucleon in of is the equation |AB} a n d t h e f a c t i t state obvious | AB>, the Hamiltonian. (3.44). Using the that physical nucleon that the state |AB } is antisymmetric |AB} Singly extra excited meson is Cutkosky present, are | ABk} = k A B | 0 > + These states + states, i . e . those in which (Ak)V|0> + A (Bk)*|0> + ( 3 states equation The are general where defined in a excited similar and states (either <3.7) of the H a m i l t o n i a n as is shown Cutkosky state manner a product but i t double-scattering excited or with cf will baryon number two, mescn not be calculations. unexcited) will operators, needed General be d e n o t e d by c a n be in the Cutkosky |U}, |V} etc. One the not eigenstates M represents single- |W } ) (3.46). |ABM }, , 5 (3.6) B|ABk> = +2|ABk> in < satisfy p|ABk> = (k^+k^+jylABk)- These aE d e f i E e d by + f (3.4) - | BA} = set of of the fundamental a l l assumptions Cutkosky s t a t e s form of this method a complete set. is that In other 18 words, i t i s assuaed two In that any p h y s i c a l s t a t e with karyon can be expressed as a l i n e a r c c E b i n a t i c r , p a r t i c u l a r a p h y s i c a l two-nucleon written cf the s t a t e s M s t a t e should te able to the Y« coefficients a r ABM te * e M converge quickly. Since (3.8) > be c c a l c u l a t i o n s to be dene i t i s hoped that deternired. equation obtained j u s t by t a k i n g the n o n i n t e r a c t i n g term summation in equation crthcncrmal defined able «ill bound, a to be and the terms i t states). s t a t e s d e f i n e d abeve Cutkosky (3.8) the (3.8) f c r which M c o n t a i n s only one mescn o p e r a t o r ( i . e . s i n g l y e x c i t e d As the Cutkosky Icr the deuteron i s r a t h e r weakly good approximation t o the deuteron s t a t e should be the |U). as AB> = |AB} + Z X I where cunter s t a t e s , denoted are ret orthogonal, ty |U), |V), |W) e t c . , are ( f o l l o w i n g Pendleton (1963)) by the e q u a t i o n <3.9) The summation Cutkosky in states. aquation The (3.9) orthonormal runs over states are all two-nucleon introduced for computational convenience. Using the orthonormal s t a t e s the unit operator can be w r i t t e n i n the form (3. 10) 1 - Z |u)(u| which will be used in the calculation of matrix elements 19 involved (3.10) i n single- i s not true and d o u b l e - s c a t t e r i n g e x p r e s s i o n s . i f the complete set cf Eguation states is not orthonorma1. Using (3.10) equation (3.9), defined i n equation derived as f o l l o w s . and t h e o r t h c n o r m a l i t y o f the s t a t e s an e x p r e s s i o n for the f matrix is wv F * F = EE Defining {X|U} the matrix G by the equation 6 + G = {X|U} xu xu equation (3.11) can be (3.12) written 6wv = E Fuv Fuw + EE Fxv Gxu Fuw Writing the above equation (3.1u 3 )j I J . i n matrix form y i e l d s the equation 1 = F F + F GF + where the a d j o i n t Equation (3.14) of the can Hermitian a s o l u t i o n can be is defined by F^ = v f o r F F . The s o l u t i o n i s can be solved equation F aatrix + + abcve ( 3 . 14) + F F = (1+G)" The 11) O. be 1 solved written < * > 3 15 f o r F . By choosing F t c be 20 F = a+G)' The expansion = i-^ h using c a l c u l a t e approximations and + jf 2 + (3. 16) ••• tha b i n o m i a l theorem w i l l be used t o to . I t w i l l be shewn that f o r s i n g l e - d o u b l e - s c a t t e r i n g c a l c u l a t i o n s , no more than terms in the f i r s t the expansion the first w i l l be needed. For example, by t a k i n g two terms only, the s t a t e |AB) nay be w r i t t e n |AB) = | | A B } - | E | U ) { U | A B } It w i l l two ( . 3 1 7 ) be u s e f u l t o d e f i n e an o p e r a t o r K.^ as f c l l c w s K | U ) = |uk) + This operator i s not easily ( expressed in teras cf nucleon and meson c r e a t i o n o p e r a t o r s . Taking the f i r s t 3 > 1 8 ) physical two terms only i n eg. (3.16) i t f o l l o w s t h a t K |AB) = |ABk) + = ||ABk} - \ g|u}{u|ABk} Because of the secend r e l a t i o n s h i p between o p e r a t o r s i s very K + term in the alove and p h y s i c a l nucleon complicated. (3.19) equation, and mescn the creaticr 21 3.2 Meson Operator In deuteron order I d e n t i t i e s and C u t k c s k j Matrix Elements tc evaluate the pion-deuteron s t a t e and the p i o n - d e u t e r c c s c a t t e r i n g expanded in 1-matrix will orthonormal T - a a t r i x , the state be w r i t t e n i n terms of matrix elements Cutkosky s t a t e s . Using e g u a t i c n s between nonorthonornal (3.8), meson o p e r a t o r i d e n t i t i e s which e v a l u a t i n g and s i m p l i f y i n g 3.2.1 Operator proofs will (3.12), and of natrix elements and p r e s e r t prove these matrix tc be useful in elements. Identities meson operator i d e n t i t i e s being letweet Cutkosky s t a t e s . The purpose of t h i s s e c t i o n i s t c e v a l u a t e Cutkosky matrix The le terms o f the o r t h c n c r i r a l Cutkosky s t a t e s . Thus the (3.16) these matrix elements can be w r i t t e n i n terms elements hill left tc w i l l o n l y te s t a t e d here, the Appendix A. The f i r s t e x p r e s s i o n f o r the comoutator of a product identity cf nescn g i v e s ar creation + operators H will and a product of meson a n n i h i l a t i o n o p e r a t o r s Q. I t be w r i t t e n i n two forms [M ,Q] + QM or + -4 [[r,M J][[Q,r ]]/n(R)I + + = E [[r,M ]][[Q,r ]]/n(R)'. + R t (3.20) (3.21) where [[r,M ]] = [r ,[r ,[...[r ,M ]...]]] (3.22) [[M,r ]] = [[r,M ]] (3.23) + + 1 2 f n + + n (R)=n=numter of o p e r a t o r s i n R R represents a product of meson a n n i h i l a t i o n o p e r a t o r s . The prime cn the summation i n e q u a t i o n unit operator R=1 is the nested just to be e x c l u d e d commutator of equation the sua ever (3.22) is that the a l l R. defined When to le + operators next three identities from one n u c l e c n L r(H-E)" is E have are given 3.2.2 a meson been B. [AB | CD}, calculate below. r only single- states, A _ 1 k cperatcr from and 1 [rVj ( H - E ) " are (3.25) 1 [rV]|A> a n d %here (3.26) E , X* of these E A , and identities Q states the general result elements double-scattering element | AB} a n d | C D } . n and Diagrams Cutkosky matrix matrix n and Elements between those -Z R,Q R Matrix Aside {AB|CD} = _ Z where (H+E^E)" p r e v i o u s l y . The p r o o f s elements The simplest Cutkosky identities (3.24) (H+E -E -E -ie) annihilation Overlap matrix Appendix element These mescr i n Appendix A . Cutkosky, The - _ 1 r defined elements. i n removing [ r V ] |A> = (H+E -E) r 1 kr r _ 1 be u s e f u l A r|Ak> = 6 | A > - where will matrix r|A> = -(H+E - E - i e ) to frcm indicates M . The in (3.20) is It that c a n be are evaluated f o r the matrix which will be needed will be presented between two u n e x c i t e d written <A|Q R|C><B|R Q|D>/n(R):n(Q)*. + t <B|Q R|c><A|R Q|D>/n(R)!n(Q)! ^ ' t + 1 represent products of meson (3.27) annihilation 23 operators. The summations. unit operator Equation (3.27) n e g l e c t e d above are these particular term cf operator describing the it the the AE cere are egual terra c f the product CD from C . T h i s type c f t e r n can the exchange tc but be in a the cere the core from C c r the thought cf as of one or more p a i r s o f tare n u c l e c t s a n t i n u c l e o n s between nucleons nucleons above cperatcrs from A i s not equal to the core o p e r a t o r core o p e r a t o r and which product operators i n a p a r t i c u l a r included i s not an exact r e s u l t . The t e r n s in the is i n the f i n a l state. i n the i n i t i a l state tc form A more complete e x p l a n a t i o n c f the above i s given i n Appendix B . The called above expansion a meson of a Cutkosky matrix exchange element will s e r i e s . The s e r i e s i s ordered number of mesons exchanged, i . e . , by the t o t a l number of te by the nescc + operators i n the product Q R. The n nescn exchange c o n t r i b u t i o n to a matrix element w i l l be denoted by a bracketed superscript n. Thus the zero meson exchange c o n t r i b u t i o n t c the n a t r i x element (AE|CD } i s {AB|CD} ( 0 ) = <A|C><B|D> - < B | O < A | D > = 6 (3.28) A B , CD This can be r e p r e s e n t e d B d i a g r a m a t i c a l l y as D The one-meson {AB|CD} exchange c o n t r i b u t i o n i s = E (1) <A|[Vq]|c><B|[qV]|D>(E +E -E )~ (E +E -E ) c + This i s represented diagramatically antisymmetric The equation represent mesons. (3.29) make the with respect right (3.29) physical The represented right hand diagrams state emitting a meson i n s t a t e nucleon {AB|CD} then i n state denominators on t h e l e f t . If (3.29) inspection of For for the equation a lines state the nucleon a nucleon first state D B. C becoming a obtaining elements on t h e i n state in i n state lines represent example, diagram removed The s o l i d the i n i t i a l to matrix the have been dotted by a n u c l e o n i s no r u l e a r e those o f A and B o r C and D. (3.25). q and becoming diagram can r e p r e s e n t d i f f e r e n t energy the drawn w i t h absorbed of operators corresponds ( 1 J A. T h e r e from and are for is side t o exchange nucleons and t h e f i n a l meson -1 D by • a n t i s y m . ' ty use of e q u a t i o n diagram The q as meson a n n i h i l a t i o n and c r e a t i o n from B C terms i n e q u a t i o n to 1 D -p- A necessary A antisym. B The q the since with energy t h e same different denominators. one or both of the Cutkosky s t a t e s are excited the r e s u l t s are {AB|CDk} (0) = 0 (3.30) 25 {AB|CDk} (1) = E <A|[Vq]|ck><B|[qV]|D>(E +E +E -E )" (E +E -E )" c k q A 1 B q D 1 + E <A|[qV]|ck><Bi[Vq]|D>(E +E -E -E +i )" (E -E -E )" c k q A e 1 B q D 1 + antisym. B q D B \ Q j . N D A c£ \k ' C .... \ <' > 3 31 \k {AB |CDk} P ( 0 ) = « k p p 5 A B -k B {ABp|CDk} (1) D = 6 k p {ABJCD} p distinct 3.2.3 terms mesons, C Cutkosky i n equation p, k + other (1) JP A other (3.32) terms. -k B The ,CD r (3.33) C (3.33) and q , a r e Matrix D Elements are those in which three present of the Vertex Operator and the HiEliltonia n The states matrix are given elements by of the vertex operator between Cutkosky {AB|[mV]|CD} (0) = <5 <B|[mV]|D> + antisym. AC 'AC \ B C A {AB|[mV]|CDk} 0.34) D = 6 <B|[mV]|Dk>+ antisym. (0) AC m ,k (3.35) D {AB|[mV]|CDk} (1) = Z <A|[qV]|c><B|[Vq](H+E -E )" [mV]|Dk>(E +E -E )" 1 q B A q 1 c + \ <A|[qV]|ck><B|[Vq](H+E -E )-l[mV]|D>(E +E -E -E -i )~ q B A q c k 1 E + Z <A| [Vq] |C><B| [ m V l ( H + E - E - E - i ) [ q V ] | Dk> ( E ^ - E ^ " _1 q D k + 2j <A| [Vq] |Ck><B| [mV] ( H + E - E ) q 1 e [qV] | D> ( E ^ E ^ - E ^ " _1 D 1 + antisym. ^ B ' • D c~-—*J A• C c D B A (/ + C S v B qs«sA> A — - B - D n C k W * D <- A 3 t 2 6 ) C \ sm {ABp|[mV]|cDk} (0) = 6 V n kp p B The heavy s o l i d line {AB|[mV]|CU} 1 + a n t i s y m . Jc D indicates be summed over a complete (0) that (3.37) the i n t e r m e d i a t e s t a t e s e t of s t a t e s with baryon number i s to one. 27 That is, complete into the or set of states the diagrams to e v a l u a t e thus are line. the the For n o t be writing t o be matrix useful to i n equation number one f o r each connected example, i n the must term. has by first no be either write the the diagram for of the o i s defined A relations be B Cutkosky Usinq for multiply will excited Cutkosky states. The w r i t t e n g e n e r a l l y as state equation i t (3.40) k H |u} = E |U} any instant. (3. 39) B HjABk} = (E +E +E ) |ABk} for and by A above e q u a t i o n s c a n meson q (3.38) HjAB} = CE +E )|AB} with s i m i l a r line as H = H +H' o where H if equation connected Hamiltonian, Hamiltonian i n the a dotted drawn as o c c u r i n g a t t h e same of inserted significance meson k a r e n o t d i r e c t l y elements (3.36), a Note t h a t r e p r e s e n t i n g the emission a b s o r p t i o n o f the In baryon not d i r e c t l y vertices t h e y need prove with elements drawn, t h e o r d e r i n g o f v e r t i c e s a solid and matrix seconl matrix element vertices (3.36), the (3.41) |U}. (3.41), i t f o l l o w s t h a t {U|H|W} - \ (E +E ) Q U W {U|W} (3.42) 28 As is shown below, the remainder H' has the very important property of having no zero-meson exchange c o n t r i b u t i o n s matrix elements. This separation of H will to i t s be u s e f u l when e v a l u a t i n g approximations to the pion-deuteron T-matrix. The matrix elements of the Hamiltonian between s t a t e s are g i v e n below. Note t h a t the diagrams Cutkosky are drawn f o r t h e matrix elements o f H• o n l y . {AB|H|CD} {AB|H|CD} ( 1 ) = \ + \\ (0) - \ (E +E +E A B C + (E +E +E +E ){AB|CD} A B C E )6 D A B ) C D ( 1 ) D E A | [[qV] q V ] ||C0>< < B||[[Vq] V q ] ||D > ( _ ( E _^ E ^ ) " I < <A| D> A 1 Q + \ E <A| [Vq] |C><B| [qV] |D> ( E ^ E ^ ) " * 1 + antisym. (3.44) B Q: A ^ D c {AB|H[CDk} / B * A (0) p =0 D — C ( 3 . 45) 29 {AB|H|CDk} =| (1) (E +E +E +E +E ){AB|CDk} A B c D (1) k + Z <A|[Vq]|C><B|[qV]|Dk> a (E) ± + Z <A| [Vq] |0<B| [Vk] ( a ( E ) ( H + E ^ - E ^ i e ) " ^ . ( E ) (H+E Eg+ie)" ) [qV] |D> 1 1 2 k + Z <A|[qV]|C><B|[Vq](H-E -E - i ) £ - 1 ( H + E -E + i ) [ V k ] | D > a (E) -1 £ +. a n t i s y r a . / d A D B C A c*—-p D C / (3.46) + {ABp|H|CDk} q A V C ^—^> = 5 , {AB|CH+E )|CD} pk (1) B functions terms and a (E) ± the exchanged Matrix calculated (3.9), are between the states. From matrix elements equation the p a r t i c l e s those (3.46) represent involved. containing equations (3.38) H matrix One i m p o r t a n t Hamiltonian of orthonormal o f t h e above and ( 3 . 1 6 ) . of in three The other mesons p, k r meson q . i n terms (3.11), of (3.47) C i=1,...,4 (3.47) elements element written —d—^ of the energies i n equation terms D A factors + other k P- The (1) k between and (3.42) between Cutkosky states elements result using is equations the two o r t h o n o r m a l i t i s evident nonorthonormal c a n be matrix Cutkosky that states the may b e 30 {U|H|W} = | where and G is uw (3.16) it (E +E )(6 u defined u E w F* v (E il+G) w is q no z e r o 3.3 Meson The states meson Exchange by from this where Series given |U) deuteron is E ) i t (3.9) (V|H'|X) w '' ( (V Cutkosky contribution to its f o r Pion-Deuteron T-matrix inserting point equations H |X) (3.49) 1 exchange and e v a l u a t i n g expressions u + i n the orthonormal pion-deuteron evaluated Using 1 1 diagonal has (3.11). (3.48) 6 + (VIH |X) vx V H ) +• {U|H'|W> < ^ u w VV W) Thus u w that = iu^w t = E + G u w by e g u a t i o n follows (V|H|X) = | w given complete in sections i s necessary an o r t h o n o r m a l wavefunctions. matrix (2.54) elements and 3.2. to express done H' element. of orthonormal 3.1 i s and Scattering In order (U | C> Cutkosky state, This matrix by e q u a t i o n sets the r e s u l t i n g states in be Cutkosky using to the proceed and i n terms the can (U|Pk>, o f known next two subsections. 3.3.1 The D e u t e r o n The the deuteron orthonormal State state vector Cutkosky states V> t Vector \V> c a n b e e x p a n d e d using = E|U)(U|P> equation i n terms of (3.9) (3.50) 31 where (U|P> i s Cutkosky (U|P> incorporates degrees known the is wavefunction into complete At complete degrees wavefunctions Thus an e x p r e s s i o n in terms this explicitly projection excited |PQ> orthonormal wavefunction freedom implicitly. dc net i n c l u d e of the a l l these exact deuteron an approximate This deuteron expression degrees operators orthonormal operators vector deuteron for t h e meson the should cf freedom i n the wavefunction. point, and in of which h o p e f u l l y i s known. deuteron projection meson needed account unexcited wavefunction The deuteron of freedom. wavefunction state deuteron representation. However, take the are i s denoted defined are defined Cutkosky by P a n d P ' which satisfies states. f o r the These respectively. the A following •aquations PHP.|P > = E | P > 0 P (3.51) 0 P 1P > = 0 1 By writinq the deuteron \V> an expression fact state that may b e state = |P > + Q c a n be o b t a i n e d P+P'=1, ( 3 . 52) Q vector as \V > ( 3 . 53) ± f o r the remainder the eigenvalue equation \V^> . Using f o r H f o r the the deuteron written (P+P')H(P+P')\V> = E J P > (3.54) 32 Using equations ( 3 . 5 1 ) - (3.53) the above equation c a n be written Il|t? > + P'HP|P > = E \V > 1 By inverting fP^. Thus Q the o p e r a t o r from V (H-Ep ) equation (3.53) (3.55) X an e x p r e s s i o n the deuteron is obtained state vector for may b e written (3.56) \V> = {l-(U-E )~h'KP}\V > v Using give equation the f o l l o w i n g \ The above terms expression n equation (3.51) diagonal to 0 [( W" ' of i n the orthonormal deuteron rather might wish arbitrary in o-V" Since terms, meson lp,HP}| state matrix 3 57 state vector elements t h e above vector o f H» d o n o t expression Note that in satisfyinq can since be H is states = P'H'P ( 3 . 58) (U|p > o momentum assumption, to (- > V deuteron exchange. Cutkosky to i d e n t i f y wavefunction ] n ( H deutercn exchange expanded \t> > the exact and.(3.52). P'HP One l H expresses any o r d e r ( H - E )-* may b e for approximate any z e r o - m e s o n evaluated the operator ti- l v> o f an equations have (3.38) Q it with space. does seem the conventional Although this reasonable. is a The 33 conventional of only two explicitly. to deuteron nucleons, approximation, consist properties of |PQ> coupled is not nucleons an the (in eigenstate reduced the the which, of two with of states equation state \V> to the example (CD|P> can be drawn term- contribution exchange and the contributions. \V cf of the mere the be That state is, than and Cutkosky in desired | P two Q Hamiltonian drawn no > stated Q to deuteron V, represents terms of with wavefunction. can + ~D above exchange nucleons orthonormal C first correspond > is in states. relating \QQ> . the For as V The account approximation) eigenvalue approximate into zero-meson consisting diagrams (3.57) composed states correspond unexcited deuteron the exchange correct the Cutkosky deuteron system taken properties thus conventional the a ncninteracting (3.53) to not in The zero-tneson space Using mesons present. and (3.52) describes orthonormal states explicitly aquations the The u n e x c i t e d orthonormal mesons wavefunction neglected The the ... zero-meson represent one-meson exchange multiple-meson exchange contribution vanishes. The drawn D k C one-meson exchange approximation to as (CDk|P> can k^ V V, D D C C D 7I V, V, be 34 3.3.2 The P i o n - D e ute r o n The in terms Scattering pion-deuteron State scattering of the o r t h o n o r m a l Vector state vectcr Cutkosky states c a n ba using expanded equation \Vk> = E | u ) ( u | P k > Since the wavefunction written in obtained the from approximation that it can nucleon. Thus nucleon scattering that form of series, the impulse that the a m p l i t u d e in state not known, the by a takes pion for finding decomposition state in (3.59) c a n be by t h e the of place the i s is rapidly from a free pion-two- pion-deuteron amplitude decompcsiticn ) impulse so particular of 9 which The scattering a 5 (3.59) term approximation. i s approximated two-nucleon equation the f i r s t the scattering approximated state 3 is assumes be <- (u|Pk> a (3.9) of for finding the deuteron state. Thus equation written \Vk> = E|Uk)(u|fl> + |S> where the f i r s t state |Pk> a n d + K defined term represents |S> r e p r e s e n t s i n eg. (3.18), the impulse the remainder. equation (3.60) |Pk> = EK |U)(U|P> + |S> + or i n shortened notation (3.60) approximation Using c a n be the to the operator written (3.61) 35 |Pk> where the presence An = K \V> + f (3.62) \S> of the u n i t o p e r a t o r expression f c r |S> assumed. can be obtained from equation f o r H f o r the pion-deuteron (2.38), has been by s u b s t i t u t i n g f o r \Vk> scattering from the e i g e n v a l u e state, (3,62) equation equation g i v i n g the eqnation (3.63) (H-E )|S> = (-EK +E K )\V> f f s g where E s = E P + E (3.64) k By i n v e r t i n g (H-E ) with outgoing s e x p r e s s i o n f o r |s> becomes wave boundary c o n d i t i o n s , t h e * |S> = (E -H+ie)~ (HK -E K )|t?> 1 (3.38) Using e q u a t i o n (Eg-H+ie)" = 1 the pion-deuteron \Vk> = K \V> i • f + t G t g and ( 3 6 5 ) writing JnUVV^'^'^VV ^" 1 scattering s t a t e may 1 (3.66) be w r i t t e n ? [ ( E -H + i ) " U ] ( E -H + i e ) ' ( H K - E K )|P> n—U s o s o ° 1 n . , n 1 t + e T h i s may be put i n a more convenient form by n o t i n g t h a t (3.67) 3 6 (HK -E K )|P> = H'K |P> +ll K \V> + f t -E < \V> -E K \V> f + f Q s v k -EjK \V> + E (Eu+Ek)K |u)(u|P> -K R\V> = U'K \V> + f + f = H'K |P> + E /C H |U)(U|P> -K H|fl> + + + Q ( 3 . 6 8 ) = (H'K - K H')|P> + + Thus t h e pion-deuteron s c a t t e r i n g |Pk> = K \V> f state may be written E [(E -H +i )" H ] (E -H +i )" (H'K -K H')|t?> + 1 s o , n 1 £ s o t t ( 3 . 6 9 ) e n=0 The 3.3.1 \V> i s known from s e c t i o n state deuteron scattering approximate state deuteron s t a t e can |P > Q be and thus evaluated the pion- i n terms of the whose wavefunction i s assumed to be known. the The above e q u a t i o n appears t o resemble pion-deuteron state scattering However they are q u i t e pion-deuteron scatterinq useful state state. different. into That type o f for obtaining free derived Equation expansion ( 3 . 6 2 ) is the K^|AB> , in a p p r o x i m a t i o n , i s e q u a l to the s t a t e physical (3.5) free net state the and a particularly state. just the s t a t e . I t a l s o c o n t a i n s the i n t e r a c t i o n o f example, equation 2 . 3 . 2 . separates c o n t a i n s more than Cutkosky state i n section to the pion-deuteron the meson k with the nucleons i n the the expression f c r ( 2 . 4 2 ) asymptotically approximations On the o t h e r hand., e q u a t i o n asymptotically an the |ABk} state zero - meson which | U ) . For exchanqe according to c o n t a i n s the i n t e r a c t i o n of the meson k with both nucleons A and B. Thus t h e expansion of equation s h o u l d be u s e f u l i n approximating the pion-deuteron state. ( 3 . 6 9 ) 37 3.3.3 The P i o n - D a u t a r o n Having obtained pion-deuteron the matrix elements wavef u n c t i o n s . matrix expressions scattering pion-deuteron T-matrix state T-matrix between Using f c r thedeuteron i n the previous c a n be w r i t t e n Cutkosky equations states (2.54) , state two the subsections, solely i n terms o f and approximate (3.57) and and deuteron (3.69) t h eT - becomes • »:if'.i"".-v"'W < r a , , 1 , , , 1 J X {K 1,[(E - H + i e ) - H ' ] C E ( n u s o f 1 n + = x The in above writing many terms meson [( expression t h eabove to discussed. t t i < > equation. will take be made. into terms No a p p r o x i m a t i o n s In order n o more I n t h enext of have to evaluate The i n f i n i t e account 3 o 7 lp,H p lH ]n(H i s exact. f o r theT-matrix. exchange 1 <J K!ovv" ' o-v" ' }iv assumptions truncated -H +i£)- (H'»: -tC H')}K s> than chapter the T-matrix series made the T-matrix will a l lbe cne-mescn exchange the zero- will been and one- be c a l c u l a t e d and 38 4 The Meson 4 •1 2ero-Meson The by Exchanqe meson expression then exchange (3.70). It f o r the complete (3.70) There For The T - m a t r i x E x c h i ng_e C o n t r i b u t i o n t o inserting equation Series the T-matrix picn-deuteron T-matrix sets writing of orthonormal is Cutkosky the r e s u l t i n g matrix evaluated states elements i s only c a n be one zero-meson exchange term in (3.52), t h e above normalization factor |AB} 3.1 equation =and equation written «"-> 1 V The as series. $1.1* - £u,w <lmul[*)»<>)<°V| Using into |BA} a n d | C D ] =- of 1/4 equation takes |DC}. U s i n q into the c a n be written account equations the fact of that section 3.2.2, « (AB| [ r a V]|CDk)^ - ZUfM r-* % v {u|[ V]| }<°> = {AB| [mV] | C D k } Using equations T (3.35) £i,(fc - and (4.3), ^A,B,C ol <P AB) m W ( 0 ) equation ( (4.2) <*\W]]^> c a n be (CB|V 4 ' 3 ) written (U . U) 39 The above equation is drawn below. V, °0 Equation (4.4) conventional nucleon the T can deuteron T-matrices. laboratory L QSpt,m; 0,k; 0 ) rewritten momentum This frame be is is space done in expressed in terms of the wavefunction and pion Appendix D. The result in below M ' , M ) 6 (kp,-hn-k) (4.5) +1 i • I The wavefunction^ deuteron wavafunction nucleon spin represent averages and Thus for to over (jO, the £?£ usual elastic of contribution (1974), I . nucleon T component spin prejection conventional M T^^ spin orientations of respectively. are given in contribution scattering A fairly teen the T-matrices exchange single of The and scattering. has the total zero-mason deuteron this with T-matrices, <f>^ the corresponds just projection pion-neutron expressions is OO by total and T™^ pion-proton The explicit A p p e n d i x D. to the ccntributicn extensive performed and T-matrix to numerical McMillan and pionstudy Landau uo U.2 One-Meson The matrix Exchange one-meson given C o n t r i b u t i o n to the T-matrix exchange i n aquation v", = W> ccntributicn (3.70) c a n be w r i t t e n )(u|[nV,Kt|w)a)(w| Pk - I <t''|U)(U|H'|X) L to the picn-deuteron T - < 1 > V (X|[mV]K |w) t ( o ) (M|I' >(E -Ep,r o x 1 excited [ + I <t/|u)(u|[mV]|x) °^X^^ ( n x l " ^u,w^x <t? |u) (u| t m V ^ l + x )( 0 ) ( X I ' H V^x-V l w )( 1 )(w| excited The fourth term K + ( E The to third give (4.6) above X " V _ and f o u r t h c a n be r e w r i t t e n | 1 X ) = terms the following v U + I - jW U ) W " ( E S " Kt E X + by n o t i n g i G i n equation expression vV-V0^"i[" i i - l„ u L >" l K + l ) (4.7) X ( 4 . 6 ) c a n now b e c o m b i n e d f o r the T-matrix w>(1)(w iv VluXulH'IXJ^CxItmVl/C+lHj^CWlP A be equation rewritten x ( o ) (x|H'K |w) + <0>)(u|[mV]K |x) + (3.52) t h s sums o v e r yielding 1 O excited I V|u)(u|[mV]|x) ^U,W ^X >(E - B p , ) " ( o ) ( 1 ) (w|t? >(E -E +ic) (x|H'|w) o ( 1 ) s x ( |P >(E -E ie) W o unexcited Usinq 1 s x + { the states | U} a n d ^ Q ) | w ] may 4 T ^,Pk WO^^'V - = T - J ( 1 ) (x|[mV]|CDk) (AB|[mV]|x) (o) (x|H'|CDk) excited - X x (AB|[mV]|xk) x ( 1 ) m (AB|H'|x) x + J {(AB|[ V]|CDk) (1) ( 0 ) unexcited x x ( 1 ) ,)" I ? (E -E +i ) (x|H'|CD) ( o ) (E -E s 1 1 _ 1 e (E -E +ie)- } s 1 x (4.9) As explained in normalization Cutkosky To the factor previous resulting proceed, states between the the are matrix non-orthonormal elements of non-orthonormal states technique elements of zero-meson elements equation can the matrix it factor of 1/4 antisymmetry states. equation may be as used are was for more The done the is of a the in the terms (3.9) cf the H* tc vertex matrix matrix matrix expressed equation elements exchange in do not zero-meson The matrix have since express terms (4.3). exchange However, have orthonormal zero-meson in of do the one-mescn complicated of cf easily elements operator element orthonormal states. u between terms contributions. is states in ( .9) matrix vertex orthonormal Using be exchange of exchange in H ' , since contributions, operator the elements expressed matrix same from the states. Cutkosky the section, any matrix exchange the one-meson operator between elements element of between the non- vertex written (AB| [mV] |CDk) - I U ) W Fj ) A B {U| [mV] |.W) <li. 10) 42 The is one-meson obtained right by hand writing side contributions meson exchange exchange. each as then contribution a sum taking That of those tc the above the three matrix of zero- and products matrix elements cne-meson which yield a element on the exchange net one- is. ,CDk (4. 11) ,CDk Using equations (3.12) (3.16), and F > U,W = ( o u\w F Thus equation (4.11) (AB | [mV] | CDk) ( 1 = 6 it follows that UW - | { u l w } ( 1 > ( 4 * 1 2 ) becomes ) = {AB | [mV] | CDk} - \ I {AB|U} (1) ( 1 ) {u|[mV]|CDk} (0) (4.13) U {AB|[mV]|w} 2 W (o) { |CDk} W ( 1 ) 43 IW J From the above d i s c u s s i o n , i t f o l l o w s P T P V P k = AB)(CD IV that {fAB|[mV]|CDk} -•|l {AB|u}(1){u[[mV]|CDk} (1) ( 0 ) u - | Z {AB|[mV]|u} ( o ) u {u|CDk} ( 1 ) {AB|H1|u}(1){u|[mV]|CDk} - ( o ) (E -E u I ? I )- 1 excited ^{ABlfmVllul^^ulH'ICDkl^^Ep+E^Ey+ie)" + - {AB|[mV]|uk} ° {u|H'|CD} ( ) (1) 1 (E +E -E +i )" } 1 p k u S unexcited In order will be t o evaluate necessary intermediate terms (4 . IH ) states the T-matrix as to truncate expressed the it over the disregarding all summations |U}. T h i s w i l l be done by i n which more than three above, mesons are p r e s e n t . That i s , the only terms r e t a i n e d i n the above summations w i l l be those contain the initial respectively, and final and one exchanged mesons, labelled to i n c l u d e a l l terms which c o n t a i n second states |U} Cutkosky equation and t h i r d which three must sinqly excited be careful mesons. For example, i n terms, the summation should are by k and m meson, l a b e l l e d by q. Whan t r u n c a t i n q the above summations, one the which as well i n c l u d e those as unexcited s t a t e s . I f | U }= | FGp } i n the second term, then by (3.37) {AB|FGp} (1) i t follows {FGp|[mV]|CDk} using that (o) = {AB|FGk} (1) (FG|[mV]|CDk} (o) (4. 15) + terms w i t h 4 mesons p r e s e n t S i m i l a r i l y , i f |U)=|FGpJ in the third term, then by using 44 equation (3.33) it {AB|[mV]|FGp} follows {FGp|CDk} (o) that = {AB|[raV]|FGk} (1) (o) {FG|CD} (1) (4.16) + terms w i t h 4 mesons When |U}=|FGp] in the fourth present term using equation (3.37) will give {AB|H»|FGp} {FGp|[mV]|CDk} (1) = {AB|H'|FGk} (o) (1) {FG|[mV]|CD} + terms w i t h 4 mesons The fifth {AB|[mV]|FGp} (o) ( 1 ? ) present term w i l l a l s o have some terms coming s i n c e by u s i n g equation ^ (o) from |U )=|FGp } (3.47) {FGp|H'|CDk} = {AB|[raV]|FGk} (1) {FG|H'|CD} (o) (1) (4. 18) + terms w i t h 4 mesons Note that the part c a n c e l s the s i x t h Thus, T-matrix of the fifth term of equation present term w r i t t e n above e x a c t l y (4.14). keepinq only those terms c o n t a i n i n g t h r e e i n equation (4.14) mesons, the can be w r i t t e n \ WO °l V AB)(C T P X P k = " J ^F,G { A B |FG} (1) |FGk} {FG|[mV]|CDk} (o) {FG|[mV]|CD} (o) (1) [raV]|FG} ( o ) [mV]|FGk} -1£F,G + H,G { A B { A B { <AB| [ « V ] H' | F G k } (1) [mV]|FG} {FGJCDk} (o) {FG|CD} |CDk} ( 1 ) ( 1 ) (1) { F G | [mV] | CD} ° ( E p + E ^ - E ^ , ) ( ( o ) {FG|H'|CDk} } ( 1 ) _ 1 ( 4 (E +E -E -E +i )" | 1 p k F G £ ' 19 ) 45 Tha T-matrix matrix elements 3.2.3. The The terms expressed The initial final the by in terms is exchanged initial diff emitted final by ^ T i f is by of tha deuteron, and are then can The dashed of lines and ^ i f corresponds final to the state second on nucleon "diff" before be and or meson is and the absorbed indicates that emitted different by after absorbed where cn T ^iff the the by a E t initial second emitted. One the n e meson nucleon of the and the below. lines is first represented final mesons. diagram the absorbed terms drawn solid denote The the is f single deuteron. left T by processes C. nuclecn emitted terms Appendix other be four in are the subscript the elements equation deuteron, can after above the meson or matrix the 20) of exchanged before (4. nucleon emitted mesons form same of and one the The T the 3.2.2 on four either side one-nucleon subsections in + of represent are and tame terms f There absorbed T hand nucleons. is + in of written one-nuclecn nucleon. and T absorbed meson second + be right of is can diff denoted meson expressed th-3 e q u a t i o n s T the terms now b e result P'm,Pk " on meson the using final T can The denote physical double solid drawn the term with right. in the The equation nucleons lines initial diagram (C.2). denote state drawn the on the above 46 The by T^ by the above, these denoted except ± f f emitted As terms that the second there terms by is T^ff terms nucleon are four drawn are of T and terms similar d i f to have f absorbed the the by represented terms denoted exchanged the by T first <jiff a meson nucleon. n ^ o n e o r " below. V, \ /m The diagram The final corresponds terms meson initial meson nucleon and denoted by is emitted and the by the second T* by by the term the same meson second equation processes nuclecn is which emitted nuclecn. corresponding T"*" of represent exchanged absorbed represented to to the where the absorbs the by There (C.3) the are six first six terms different S3.ni c orderings of the emission of the of is drawn T same absorption exchanged and of the final mesons. The by T As except * same emitted by above, these corresponds terms the there terms is One o f the and six the terms k/ — b - S - r O ^ — »0 diagram meson below, M H The initial denoted that drawn by the second are to the T~ are same terms cf T nucleon six fourth terms and term of similar c same represented below. to have absorbed by by 'k ix- equation V, the the terms denoted exchanged the T (C.4). same first and meson nucleon. one of 47 The diagram It corresponds is the f o u r t h worthwhile at which have been approximations deuteron to T-matrix in the term this equation point made form of in given to (C.5). outline expressing the by e g u a t i o n (4.20) the pionand Appendix C . When in matrix subsections neglected. terms elements 3.2.2 That of Hamiltonian and interaction, V meson c r e a t i o n A major neqlecting This final a l l written (4.20) is One m u s t is terms by such that major when (3.53), space of the not l i n e a r (4,20) three the i n the was mesons given by were the were exchange eguation summations mesons about the one-meson aiatrix it written which with must over present in then be state vector the c o n v e n t i o n a l in equation i f is deuteron to specifying exchange made the p i c n - d e u t e r o n T - the deuteron corresponds without as as be meson-meson the cne-meson three assumption that difficult, than a l l be i d e n t i f i e d equation is only the in T-matrix. conclusions when by t a k i n g were to elements equation i n the T-raatrix for the to were i n w h i c h more truncating i n momentum very in writing firstly assume in wavefunction It states i s another as made to a l l terms expression equation matrix. operators, secondly There neglected or a n n i h i l a t i o n terms assumed f o r the matrix operator evaluated n e t be e x p r e s s e d were i n V which was d o n e intermediate the vertex were exchange elements , and any t e r m s 1 of and the core which c o u l d the expressions assumption approximation (3.67) matrix Cutkosky states a l l a l l terms one-nucleon Also, present. and 3 . 2 . 3 , is, negligible. between V, to approximation (4.20). If draw to any the T- one h a s some 48 potential V then, in principle, equation evaluated numerically. However, this Instead, in section, some be made doinq 4.3 which will enable any n u m e r i c a l The S t a t i c In it the next on-Shell useful various terms to The energy total some be energy of energy. (4.20). deuteron i s In order be t o pion. assumptions also necessary T-matrix. pion-nucleon T-matrix T Also, states set small assume small compared will be Bq,Ak = Using to equations as given " < B I [ q V ] obtain do that without sections, sizes of the this, some the kinetic compared to that kinetic tc the the pion called the element CH-E -E -i ) above of one-nucleon and t r u n c a t i n g an a p p r o x i m a t i o n (2.42) by e q u a t i o n A k k complete to assume is e _ 1 -<B|[Vk](H+E -E +ie) matrix will the total static , pion-nucleon Each conclusions of the relative will is approximation. It here. approximations i n the previous i n the deuteron of the incoming These done idea approximation the some done made. of the nucleons energy draw of the type get will first further be be A£2roximation i n equation approximations not could calculations. a calculation i s one t o will (4.20) B can states the summation be and and (2.53) to (3.24), c a n be the the written [Vk]|A> -1 , [qV]|A> simplified (4.21) by i n s e r t i n g pion-nucleon by n e g l e c t i n g a l l a scattering scattering 49 s t a t e s . With t h i s assumption Bq,Ak T the T-matrix -I l^l l™l ^WV " <B = G><G A> 10 22) 1 can be r e p r e s e n t e d d i a g r a m a t i c a l l y as f o l l o w s /k /k A One = B + T,.,.,. dirt and (Eg+Eq-E^-E^ie) either : further —p ± f f and T * + T same B a m e > conserved in term to of with G i n t e r c h a n g e d with a to assume the f o u r p a r t i c l e s i n the above i . e . assume that that energy energy is scattering. This i s equivalent to neglecting the p r i n c i p a l v a l u e i n t e g r a l when the energy is split the terms factor made here i s place on-shell, the to s i m p l i f y proportional B or D). The approximation denominator takes ^p- Most of the ten terms r e p r e s e n t e d similar scattering involving ct— i s necessary are (or a -1 yk A + c* assumption represented by T * the (4. (E^-E^ie)" G by 1 G -I <B|[Vk]|G><G|[qV]|A> 'his can be w r i t t e n denominator above i n the f o l l o w i n g manner V V V V 1 ^ " 1 = * i 6 < B YW E + + P ( E B q-V k Making use of these approximations T\ a n d T become diff same + + E E ) _ 1 (4 * 2 3 ) the terms r e p r e s e n t e d by 50 diff + TLne 4 T x | W o q ^ ^ l V [mV] |Cq><B| [qV] |Dk> -TTi<A| * Ep-2M 6 (E^+E^-Ep-E^) (2 - — ) V - I ATTI <A | [Vq] + ^| X <A| [Vq] G | G > < G | [mV] |C><B| | C><B | [qV] 6 ( E ^ + E ^ - E ^ - E ^ ) | Dk> [mV] | G > < G | [qV] |Dk> <$ ( V Y W 2 M (— ) x (— ) x ) x 1c x ( E + E - E . + i e )-1 C A q E -2M 0 + I <A| [Vq] G |C><B| [qV] [Gk><G] [mV] |D> 6 (Ej+YW (— q x (E_+E - E . + i e ) " C A " I <A|[Vq]|C><B|[qV]|G><G|[mV]|Dk> G - I <A| [Vq] |C><B1 [mV] |Gk><G| [qV] |D> G 1 (E +E -E +ie)" (E +E -E -ie)" 1 c q A B (E +E - E ^ i e ) " q (Eg+E - E ^ i e ) " 1 c 1 G 1 } (4.2") These terms can be r e p r e s e n t e d d i a g r a m a t i c a l l y as below /k v. *0 q, B -Pm m \ • k /k D >>~^— 0,/ B A .'H i> s B A D r /V. G / D C V m\ r\ o B q/ G • k D 51 The result from binding of first T g it that orders of incident obvious first pion that This When shell of fifth and could only with to some make approximation. terms is (about a l l are of The in with the kinetic G interchanged above energy energies, after absorbing than the static energy it the of approximation with masons the to are Mev o r will not are small if the use form magnitude, about It two is compared one the (assuming greater). by T d i f did an not to the numerical q and f Q JJ B or are D). total energy will be made the when s and - 1 (or a m , e it the on- with most a similar K. the before T associated (E - E - E L - E , - i e ) that be term approximation either nucleon of terms first just making fourth denominator k and is four V. denominator follows other of the static proportional (Ep-2M) elements investigated energy the order Jj term and f MeV). Thus, represented the f same terms potential both and 50 sixth terms i 2.2 the than be ^ T matrix of third energy from approximation smaller simplifying valid these etc. second, kinetic the static assuming magnitude term. not and the result deuteron <G|[mV]|C> calculations is the the (4.22) , terms In of <\|[Vg]|3> follows - a m e energy equation two Since the energies, about of 300 not the just nucleon MeV interaction. dealing energies greater Thus the with these the terms terms. Making represented c by 1 only T~ diff c c the and on-shell T~ c a n be same approximation, written 52 T diff + T sa e = m x 2 W > P ) ( C D l V <Aq|[mV]|C><B|[Vq]|Dk> - ^ I ~ ^2 U G *' 5(Eg-E^-E^-E^) <Al [qV] |C><Bl[mV] | G > < G [V<1] | |Dk> -1 V q~ D" k 6( E E + l , <A|[qV]|c><B|[Vq]|G><G|[mV]|Dk> r - l Q -1 (E^-E^-E^+ie) <A| [qV] |C><G| [raV] | D > < B | [Vq] | Gk> 6 ( E ^ - E ^ ) <A|[qV]|c><B|[mV]|Gk><G|[Vq]|D> E ) ( E C- q" A E E + i £ ) (E^E^+ier^Eg-E^-ie) (E^-E^-E^+ie) ( E , - E - E - i e ) 1 ( q - 1 X D } (4.25) These terms can be r e p r e s e n t e d diagramatically as m k/ »0 ^ G q B qv N A N kx' *0 < B G x q A k' D 4? N sm k s / \ first three conpared to nucleon kinetic energy momentum of 400 since the G terms x • B A The of Q; s ' deuteron of MeV/c D equation the c o r r e s p o n d i n q terms about follows over (4.25) in equation 300 MeV w i l l f o r the nuclecns wavefunction will is very be (4.24) yield in small the small since a the relative deuteron. f o r such large 53 relative te momentum, t h e f i r s t neglected three i n c o m p a r i s o n to the f i r s t As i n e g u a t i o n (4.24) t h e s i z e (4.25) c a n o n l y be e s t i m a t e d Thus making approximation T^P teres cf equaticn an term c f the l a s t numerically on-shell where p h y s i c a l l y (4.25) of eguation t%c terns given (4.24). cf eguation some p o t e n t i a l approximation s e n s i b l e , the can and a dominant V. static tern; cf c a n be w r i t t e n P'm.Pk = " 1 7 1 lABCDq < P o'^ ) ( C IV D < A l [ m V ] l C q > < B l [ q V ] l ( V V V V 6 D k > (4.26) In writing energy this eguation c f the deuteron energy of the represented i t has been a s s u n e d is incoming that negligible compared pion. above The the b i n d i n g tc the equation total can be d i a g r a m a t i c a l l y a s below V, 'q / Asa In a d d i t i o n t o t h e term the last two contributions The the can only above conventional nucleon terms given i n eguaticc of both equations be e v a l u a t e d eguation deuteron T-matrices. can be momentum This i s dene l a b o r a t o r y frame i s e x p r e s s e d .(1) TL QSp.'.JS; 0 , k ; M ' . M ) I fc,Jt'=-l there (4.24) a E d *ill (4.25) te whose numerically. written space i r terns wavefuncticn i t A p p e n d i x D. The cf the and pier result in below <S(k ,+m-k) p <«.27) +1 = -rri (4.26) d^q ,M» $ v (K+a-jQi-m))- ^(£) T £ T £ fiCkp.-hn-k) 54 The T-matrix pion and The Equation obtained ^»£ nucleon tnatrices. E. T by represents isospins explicit (4.27) of a an product expression agrees P e n d l e t o n (1963 with ). average for the spins and pion nucleon T- given in Appendix double-scattering result of over two T^,^ is nucleon 55 5 Conclusions Using a scattering the T-matrix first using In term to (called was shown terms final mesons, each operator In terms as a to 4.2, exchange examined. elastic series and The reason the d o u b l e - c o u n t i n g for problems corrections in pion- order magnitudes Assuming that one-meson i n a form than most the of meson and terms the TThis a as to physically exchange i n the of of the and twenty vertex (4.21)). which o f these was made sensible. With contribution facilitated only initial sum elements (eguation expansion Keeping and the approximation which terms in as matrix states where an i n c i d e n t one of of an o n - s h e l l (4.25) the approximation was e v a l u a t e d . was e x p r e s s e d products of the various (equations term) one exchanged approximation the of was e v a l u a t e d . of terms g a t an i n d i c a t i o n important, term) single-scattering order exchange as i n the expansion exchange the next one-nucleon to term (4.5)). the T-matrix was w r i t t e n greater picn-deuteron scattering the u s u a l containing approximations, shown be written static matrix multiple (equation between were matrix were avoid the f i r s t the one-meson those terms, was t o the i n a meson series the zero-meson section (called this with 4,1, the T-matrix In approach, scattering. section matrix of approach associated deuteron theory was e x p a n d e d two t e r m s this usually field to comparison expression twenty as well these the T of the the T- MeV, i t was for (4.26)). pion kinetic was a b o u t the remaining energy two terms. cf 50 orders This term of magnitude was s h o w n to 56 be similar from the conventional the g e n e r a l i z e d additional the to terms impulse whose double-seattering scattering either could processes before only theoretic term. with or after be e v a l u a t e d in at pion perhaps either four mescn exchanged terms the s c a t t e r i n g . numerically There n o t be e a s i l y These using resulting were between a four compared represented The s i z e to single- the nucleons of these particular Low (1956) ] , be compared 4.2 energies or 4.3. numerical with the the f o r a pion has c a l c u l a t e d be w o r t h w h i l e and calculated (4.28)) kinetic section calculated. has (equation Carlson(1970) would a could term terms field p o t e n t i a l V. contribution butions approximation. magnitude P e n d l e t o n (1963) MeV. double-scattering to kinetic energy the double-scattering ranging from calculate Using results double-scattering other a Chew-Low could 61 t o 142 contri- 300 M e V . terms It contained H a m i l t o n i a n [Chew be o b t a i n e d double-scattering of which contribution could usually BIBLIOGRAPHY B l a t t , J.M. and Weisskopf, V.F. 1952. Theoretical (John Wiley and Sons, New York). Nuelear Physios Carlson, C. 1970. Phys. Rev., C2, 1224. Chew, G.F. and Goldberger, M.L. 1952. Phys. Rev., 87_, 778. Chew, G.F. and Low, F.E. 1956. Phys. Rev., 101, 1570. Cutkosky, R.E. 1958. Phys. Rev., 112, 1027. H e i t l e r , W. and London, F. 1927. Z. Physik, 4 4 ^ 455. McMillan, M. and Landau, R.H. 1974. TRIUMF Report, TRI-74-1. Pendleton, H.N. 1963. Phys..Rev., 131, 1833. Wick, G.C. 1955. Rev. Mod. Phys., 27, 339. 58 APPENDIX A Meson Operator The identity Identities f o r the commutator of a product of mescr t creation operators M and a product cf nescii destruction operators Q i s [M ,Q] = -l' [[r,M J][[Q,r ]]/n(R).' + + + R It can be proved { f l . 1) by i n d u c t i o n on the number of o p e r a t o r s i r the product Q . + Let and M be an a r b i t r a r y c f irescn c r e a t i o n operators l e t Q=g. Then [q,M ] = E [r,M ][q,r ] + + r = Z + (A. 1) i s v a l i d Thus eguation Assume the i d e n t i t y only + f o r n(Q)=1. C . It must now i s t r u e f o r Q k . The p r c c f cf f c r the case k ^ C . The o u t l i n e d only as i t i s c o n c e p t u a l l y although ( f l > 2 ) i s true f c r a r b i t r a r y shown t h a t the i d e n t i t y done + [[r,M ]][[q,r ]]/n(R): R be product this sill proof f o r keQ w i l l similar tc i t i s more c o m p l i c a t e d a l g e b r a i c a l l y . the case be be k^Q l o r k ^ C the p r c c f proceeds as f o l l o w s . Using the r e l a t i o n [M ,Qk] = [M ,Q]k + [M ,k]Q + [[k,M ],Q] + and assuming equation + + (A.1) is + true for (A. 3) Q , the conirutatcr 59 [ M , Q k ] oay be w r i t t e n -[M ,Qk] + = E' [[r,M ]][[Q + t R + by R T Q -[M ,Qk] R T cn single n which de net certain (A.4) can be w r i t t e n )! + ° [k,M ] [ Q k , k ] prime ° + E" [ [ , M ] ] [ [ Q k , s ] ] / n ( S ) ! + S 1 the summation indicates meson cperatcrs are The f a c t o r of n ( S ) ~ summations. are n C S ^ r + + operator and t = E' [ [ r , M ] ] [ [ Q k , r ] ] / n ( R f double T these products R q ]]k/n(R).' T d e f i n i n g S = R k , eguation The + [[r,M ]][[k,r ]]Q/n(R).» . . [[r,[k,M ]]][[Q, ]]/ (R): R + E» R Denoting > r 1 1 t h a t the u n i t ejclucec fron comes from the f a c t that d i s t i n g u i s h a b l e ways o f p l a c i n g k i n a given the there product . o Relabelling the r e s u l t . Thus eguation true two sums i n eguation (*.5) y i e l d s the d e s i r e d (A.1) has teen shewn, b j i r d u c t i c n , to f o r a set Q containing be an a r b i t r a r y number of c p e r a t c r s r c of which are i d e n t i c a l and an a r b i t r a r y s e t M. Tc prove e g u a t i o n (A.1) i n g e n e r a l i t i s assumed that i t is true f o r l M , Q k ] t i+1 where keQ. Then i t i s shewn that i t i s t r t e f o r [M , Q k the f o l l o w i n g i d e n t i t y t J J ] . Using [M ,Qk + j + 1 ] = [M ,Qk ]k+ + j [M ,k]Qk + an eguation s i m i l a r t o equation sums over R are s e p a r a t e d j + (A.1) intc SUBS [[k,M ],Qk ] + (A.6) J can be w r i t t e n . ever B n Then where F n the ccntairs 60 the operator k e x a c t l y n times. operators sums over nested not £ o Denote by S o with the n o p e r a t o r s k e x p l i c i t l y The equations nucleon the i n the cf straightfcrvard 1 Thus equation identities written manipula- (A.1) i s true f o r iM^Qk^" " ] that equation true f o r [ M , Q k ^ ] . + cf e q u a l t c k i n R^ . By w r i t i n g the suns ever R^ as commutators, i t i s a matter t i o n t c prove the product i f i t is 1 <A.1) i s true f c r any M and Q. involving meson (3. 24)- (3.26), can be proved annihilation as f c l l c v s . operators, Since the one s t a t e i s an e i g e n s t a t e of H i t f o l l o w s t h a t r(H-E )|A> = 0 (A. 7) A From equation (2. 12) [r,H] = E r + [rV] r Thus eguation ( J > 8 ) (A.7) can be w r i t t e n (H-E.)r|A> + E r|A> + [rV]|A> = 0 I n v e r t i n g the o p e r a t o r (H-E^+E^) the d e s i r e d i d e c t i t j r|A> = -(H+E -E -ie) r When two or more mescn A _1 [rV]|A> operators are (A•10) tc commutator of a meson a n n i h i l a t i o n o p e r a t o r with needed. Using equation i s obtained be removed, the (H-E)~ (A.8), r (H-I) may be w r i t t e n 1 w i l l be 61 r(H-E) Multiplying on the = right (H-E+E^)- , the d e s i r e d The = 1 third annihilation )r + by (A. 11) [rV] (H-E) - (H+E -E) r - 1 operator operators - (H+E^E)" identity from [rV] 1 is matrix = + cr (E -E )rk r k + - (H-E)" used the left kj r[Vk] + (A. 1 involving picn- (2.12) i t f e l l o w s [rV]k 12) i E relieving meson elements nucleon s c a t t e r i n g s t a t e s . From equation [rk ,H] and 1 r e s u l t i s obtained 1 r(H-E)" (H-E+E that (».13) + Using t h i s i d e n t i t y the eguation rk (H-E )|c> = 0 (A. 14) + c may be written (H+E -E -E )rk |c> r Now, c + k - r [ V k ] | c > + [ r V ] k | c > =0 (A.15) + since (H+E -E -E )6 r the f o l l o w i n g c k k r |0 = (H-E )|C> r e s u l t can be obtained C = 0 freir e q u a t i c c ( (A.14) j . 1 6 ) 62 rk |c> + - (H+E^Eg-E^ie)"^^]!^ .= \ |G> ~ r Outgoing wave (H+E-E-E). 10 state \j and boundary Using the (H+E -E -E -i ) r c k conditions definition _:L E are ( A > 1 7 ) [rV]k |c> + imposed cf the pion ( A . 17) the desired when nucleon invertirg scattering lc equation ( A . 12) and identity may be written r|ck> = 6 |c> kr " (H+E -E -E -ie) r c k 1 [rV]|ck> ( A . 18) 63 APPENDIX E.1 Unexcited The the simplest the Matrix Cverla£ overlap Dsing he B Cutkosky Elements Matrix Cutkosky of two Elejrects matrix element unexcited definition of is Cutkcsky Cutkosky states that states this *hich gives | A B }and |CD ]• matrix element nay written {AB|CD} = < o | B A C D | o > + + Using equation (2.16) the operators A, E, C , and £ may te written A = B I A,K a*(AK)AK I b*(8L)8L = B,L = I c(CM)C M C,M Dt = I d(flN)P N t?,N C Using these matrix {AB|CD} expressions element = I + I can I be I + + for the + + physical nuclecn operators, the written a*(AK)b*(8L)c(CM)d(l7N)<o|8LAKC M P N |o> t + + + ( B # 3 ) A,K 8 , L C,M P,N In matrix order elements separated those into terms in tc express {AB|CD} the terms the four the types. in The products summation first summation i n as type which in cf VC^BA of one eguation term (in nuclecn (E.3) are consists of all statenents 64 in this products cf appendix regarding o f t a r e nucleon the o p e r a t o r s foregoing inequality omitted.) should be w r i t t e n VC±±Bk terms do (cr cf the but s i n c e the they will net c o n t r i b u t e t c the summation + + 8A accurate on the c l a s s i f i c a t i o n o f terms, These versa) inequality c r a n t i n u c l e c n o p e r a t o r s , the o r d e r i n g s i n c e a t l e a s t one of the o p e r a t o r s i n C V vice or w i l l be i g n o r e d . To be completely s i g n s have no e f f e c t be the e q u a l i t y and t h u s w i l l a n t i c c n n u t e + + i s not i n 8 A (cr w i t h a l l the o p e r a t o r s i n ) making the term zero by a n n i h i l a t i n g cn the vacuuir state. The second type of term c o n s i s t s o f those terms i n the summation i n which C=A and V=B. The quadruple these terms w i l l be a b b r e v i a t e d I^Q B D s u n n a t i c n ever a l l ) • The t h i r d type c f t e r a c o n s i s t s o f those terms i n the summation i n which C=8 The quadruple ^(BC AD)" I f summation over a l l these terms w i l l a t € r i B i n €quation and V=A, l e abbreviated (B.3) i s to be ncr-2erc then i t can be shown t h a t no two o p e r a t o r s i n A and 8 or i n C and V c a r be identical. I f , f o r example, the o p e r a t o r s aeA and beB a r e i d e n t i c a l then the matrix element on eguation the r i g h t hand side c f (E.3) can be w r i t t e n <o|...ba...jo> = <o|...ab...|o> (E.4) Eut s i n c e a and b a n t i c o B m u t e <o|...ba...|o> = -<o|...ab...|o> thus p r o v i n g that t h e o p e r a t o r s i n A and 8 c r i n C and (E.5) V must be 65 distinct cf i f the t e r n i s t c be ncn-zerc. Thus f c r r c r - z e r c the second w i l l A and V. The Similarily, fourth type are antinucleon as i . e . those terms f o r describing cr t h i r d the exchange effect f o r two nucleon-antinucleon forces of and 8 pair which will tut rare nuclecrand V • . These terms should reasons. F i r s t l y , mass that these terms can be range CP=A8 t cf the bare nucleons are not known, i t w i l l bare which between the p h y s i c a l nuclecn c e r e s C pairs form the p h y s i c a l nucleon c o r e s A have l i t t l e V, and type. These terms can t to type A f o r ncn-zerc terms c f the t h i r d n e i t h e r of the second be thought cf will thought although the masses be assumed t h a t large of short as describing not be s i g n i f i c a n t f c r medium energy of the nucleon cores c o n s i s t cf three cr acre bare and a n t i n u c l e o n s . I t w i l l be assumed t h a t the physical nucleon will the have a s u f f i c i e n t l y s c a t t e r i n g . Secondly, these terms w i l l only a r i s e when at two net c f term c o n s i s t s c f the reiiainder cf the (B.3), terms i n eguation which 8 w i l l have no o p e r a t o r s i n common type C and have no o p e r a t o r s i n common nor w i l l 8 C will and terms be the s m a l l i n that space r e p r e s e n t i n g t h r e e or more bare nucleons Thus i n a l l c a l c u l a t i o n s of Cutkcsky least nucleons wavefuncticn part cf the and cf Fock antinuclecrs. matrix elements these 'core exchange' terms w i l l be n e g l e c t e d . (B.3) H i t h the above arguments, equation car be written ( i g n o r i n g cere exchange terms) as {AB|CD> - <o|8LAKC M t? N |o> I + + + + (AC.BD) (E.6) + I (BC.AD) where each term <O|BLAKC MVN |O> + i n the susmatiens + i s tc be n u l t i p l i e d by a (AK) 66 etc. Since {AB|CD} = [L,K]=0 and {A,8} = 0 t h i s <o|AK8LC M t? N |o> + £ -I + + t + - I + (AC, BD) second equality the second t y p e {8,C'}=0 resulting (3.21) {AB|CD} = I this I R one + + + + the fact terms c a n be ( B # 7 cf the written that third fcr ) terns type cf {A,C'}=0. as <o|AKC [[r,M ]][[L,r ]]8P N |o>/n(R)! + + + + <o|8LC [[r,M ]][[K,r ]]AP N |o>/n(R)! + + + + + (BC,AD) tc nucleon matrix between <o|BLC KM AP N |o> + (AC,BD) R order from and f c r -11 In | o> V (BC,AD) the eguation as (BC,AD) <o|AKC LM 8pV|o> = I written <o | 8LAKC + (AC.BD) Using c a n be the nested 1 . (£.8) express the above elements, the commutators. ][ matrix unit The u n i t elenert operator cperatcr i n terms is inserted c a n be written E Q |o><o|EQ/n(Q)!n(E)! f cf + ( E g ) E,Q where Q is product of satisfying a product bare the cf nucleon mescn and a n t i n u c l e o n + 8 is cperatcrs and E i s annihilation a operators condition 8E |o> - where annihilation the baryon + lE |c> < ' + number o p e r a t o r . B Terms i n which E is 1 0 > the 67 unit operator are Thus matrix the {AB|CD} = Now, I x n(Q)'.n(E)! element terms cn of case the vanish Q + does it either (B. cf not state). t > < written o| Q[[L r ]]BP N |o>/n(R): E J + + t + + + + + vanish. is the have will has For an example, cperatcr in which it causes anticommutaticn an operator anticcmmute A similar in in the common with the matrix identical common w i t h A with A argument cf and holds 11) (iE annihilate for the seccrc 11). commutes creation will operator operator equation + E/l because it this + I[L,r + vacuum Also, meson which which case + (E.9). (E. then the term (BC,AD) equation may be t + in or {AB|CD ] in <o|BLC [[r,M ]]E Q |o><o|EQ[[K,r ]]AP N |o>/n(R): E ^l to included <olAKC [[r,M ]]E Q |o if operators) which E,Q,R be element (AC.BD) - I to + n(Q)!n(E)! term (in E,Q,R \l x a l l first A I also with operators. [[r.M*]] since Similarily [{r.M^]] Q contains commutes with cnly both + ]J a n d M K , r ]] . Since r|0>=0 for any meson annihalaticr operator, [[r,M ]]|o> RM | o> <o|[[L,r ]] <o|LR <o|[[K,r ]] <o|KR + + + Thus equation (E.11) can be written (E.12) 68 {AB|CD} = t <o|AKQ RC M |o><o|8LR QP N |o>/n(R)!n(Q)! I + Q.R -I will + + can t + be zero as w i l l + + , i n the above e q u a t i o n can (since they w i l l Terms of the second the and summation type i n which A=8=C=P second first summation. can be be will type zero). cancel Eerfcrning these written <A|Q R|c><B|R Q|D>/n(R) .n(Q)! I + R - c r B#? terms c f the second (since they w i l l summations, the o v e r l a p matrix element {AB|CD} = acdP^B type can be added t c the be zero) and and t h i r d first now the r e s u l t . That i s , t e n t s in wticb a l l t e r n s i n which C/A be added to the second between . (BC.AD) Aft). A l s o , terms of the t h i r d summation + <o|BLQ RC M |o><o|AKR QP N |o>/n(R) .n(Q)I I r e s t r i c t i o n s cn the sunmaticns CV AB + ( £ > 1 3 ) te removed without a f f e c t i n g and + + (ACBD) Q,R The + , + , (E.1U) Q <B|Q R|c><A|R Q|D>/n(R) !n(Q)'. I + + R,Q Thus the matrix states of between has been w r i t t e n as a sum p h y s i c a l one result i s that B.2 element matrix Katrix elements in this Elements elenents are natrix or the vertex o p e r a t o r , [mV], Cutkosky states. evaluated first Using o n l y approximation Cutkosky core exchange t e r n s have been i g n o r e d . Interaction matrix elements urescitec of products cf n a t r i x nucleon s t a t e s . The Unexcited I n t e r a c t i o n Hamiltonian, H, two The matrix element e l e n e n t s c f the evaluated between o f the h a m i l t o n i a n w i l l as i t uses a l l the ideas necessary tc te evaluate of the vertex o p e r a t o r . the d e f i n i t i o n of Cutkcsky s t a t e s the n a t r i x element 69 of t h e Hamiltonian between u n e x c i t e d s t a t e s c a r be w r i t t e n {AB|H|CD} = <o|BMIC D |o> i <E. 15) + The g e n e r a l method used i s t c ccmoute H tc the r i g h t or and elements cf then separate i n t o one nucleon the preceding s e c t i o n . symmetric with will resulting eleuents f c l l c w i r g Since the respect t c i n i t i a l final and f i n a l H with e i t h e r £ a. the f a c t aatrix the methods result should te s t a t e s the process or be t h e average of the two r e s u l t s Using Cutkcsky matrix should be repeated commuting result the past e i t h e r E. The final c b t a i n e d above. .j. and E that C anticommute HC D |o> = (C HD -D HC )|o> - [H,D ]C |o> + [H,C ]D |o> - HC D |o> (E.16) + + + + + t + + + + + + + t S i n c e C |0> and D |0> a r e e i g e n s t a t e s o f the H a m i l t o n i a n (E +E )cV|o> (C HD -D HC )|o> = + + + + c < * > E D 17 Defining H the l a s t u = E k k E k (E. 18) k t h r e e terms o f equation (B.16) can be w r i t t e n - [H,D ]C |o> + [H,C ]D |o> - HC D |o> + + + + + + = H C D |o> + D H C |o> - C H D |o> + VC D lo> + D VC Io> - C VD |o> + Ihe + + terms i n v o l v i n g verified + H by w r i t i n g c ' + + i n equation and D + + + (E.19) w i l l explicitly and + + + c a n c e l as can te evaluating those 70 terms. Thus the matrix element notation three matrix elements < O | 8 L A K V C V P N | O > + <o + + t t t <o|BAD VC |o> - <o|BAC VD cf (B.2) equations involving |BLAKP N VeV + E |o> a t y p i c a l term ' 2 0 ) cf the * (E 21) e t c . have teen o m i t t e d . cf f i v e dynamical terms. V variables can be as v(l/,W,S,T)l/ WS T + + l/,W,S,T ( <o| BLAKC + M + VP + N + |o> | o> - + w r i t t e n i n terms c f the fundamental I t V is V i s w r i t t e n as a sum V= written D where the c o e f f i c i e n t s a (AK) To proceed, he T C + the can (E +E ) {AB|CD} + <o|BAVC D | O > {AB|H|CD} = Using of the Hamiltonian ( E .22) + where I/ and W a r e e i t h e r u n i t o p e r a t o r s or are products cf cdd numbers of bare nucleon and are equation 1 and products of meson o p e r a t o r s . The c c e f f i c i e r t s v(l/,W,S,T) depend V, a n t i n u c l e c n o p e r a t o r s anc S upon the (E.22) consists type i s broken of of interaction up i n t o terms. New, The will the sum i t first term, a l l those terms i n which l/=W=l . These w i l l terms d e s c r i b i n g the meson-meson terms are denoted five chosen. interaction. The T ether be four by V", V", V" and V". V" c o n s i s t s c f a l l these D R o C terms i n which W=C and V" c o n s i s t s cf a l l these terms i n which D W=V . V" c o n s i s t s cf those terms i n which W i s p a r t l y ccntainec R i n C and p a r t l y contained i n V while V" c o n s i s t s cf these terms L v o in which W c o n t a i n s some bare nucleon or antinuclecn operators 71 not in C orP . Thus, writing V = some V + V ' + V - + simplifications elements involving equation (B.21) + V' C can te made V . The f i r s t < .23) 1 E immediately matrix element i n the matrix involving v i n becomes <o|BLAKVC M I7 N |o> = < o | B L A K V ' C M P N | o > + + + + T + <O|BLAK(V' +v: )c MVN |o> , + The + matrix element T , + involving V + <O|BLAKV;'C MVN |O> + + will vanish + since at ( E , 2 least 4 ) ere o of the annihilation operators t operators in C in W will articemmute with a l l the t and V a n d a n n i h i l a t e on t h e vacuum s t a t e . V^'C |o> = V^»P |o> = o + the as matrix products element + involving V" w i l l of one-nucleon matrix ( n e t be a b l e elements. t element only appears three more bare cr matrix cores element The second + The matrix + element and be n e g l e c t e d C element involving + Since ) matrix cr P (cr bcth) contain that V" will o 5 this cn the b a s i s + 2 t bare i n equation + . E be e x p r e s s e d operators, <o|BLAKP N VC M |o> + tc antinucleon composed o f s i n g l e matrix <O|8LAKPVVC M |O> = either nucleon will are primarily when Since the nucleon nucleons. (B.21) becomes + <O|BLAKP N V'*C M |O> + vanish this + + + f c r t h e same < E ' 2 6 > reason 72 as in equation vanish have as any a (E.2U) cf equation operators in coamcn as nucleon operators) the By a result matrix Thus the + natrix element (E.25). Since (otherwise the V£ c a n n c t iEvclvinq the natris involving any of terns V^' w i l l will R anc anticommutation contain V" cannot element would identical tare w h i c h W=C iE . vanish. arguments <O|BLAKC M VP N |O> + of element similar + the result vanish Thus and + matrix <O|8LAKCVV*PV|O> = elements involving <O|BLAKCVV^'P N |O> V can be (E.27) + + + written <O|BLAK(V' +V')C MVN |o> + <O|8LAKP N (V'•+V )C M |o> , T + + - the + <O|BLAKV-CWN |O>. . above eguation <o|8LAK(V +-V')r' N cV|o> , T C t + + - in , + + + terms + - <o|8LAKC M (V^'+V*)P N |o> <O|BLAK(V^'+V')C MVN |O; + The + c o n t a i n i n q VL <O|8LAKPV(V + T , C + ( can te E ' 2 8 ) written +V*)C M |O> T T ( E . 29) = - In writing since and the each abcve term antinucleon cperatcrs Now operators, <o|8LAKC [V''+V',N ]C M |o> + which equation in since i n V, each equation + C term = and then (3.20) [N ,V"+V] use V" contains cperatcrs are + C cf is has an + + been made ever since of number W cannct fact tare that nuclecE contain any [P,v"]=0. V'+V contains C used to write nescE -I [[r,N ]][[V'»+V,r ]]/n(R)! R ° , of the + + arnihilation (E.30) 73 Thus the terms involving v ' fceccne c <E.31) - Z <olBLAK[[r,t?V]][[V '+V ,r ]]C M |o>/n(R)! , , + + + c R Using methods overlap i d e n t i c a l to matrix - E' R element, these the atove + t cperatcrs evaluation expression + + c a n be + Q,P being + divided the written + + > < o| A K Q + p [( c V by n ( Q ) ! n ( P ) ! . , + V ' ) R ] c t M + l <- > 0 > E The G e n e r a l i z e d 3 2 vertex [ V R ] a n d , [ R V ] a r e d e f i n e d by [VR] = [[V,r ]]/n(R)! + [RV] - [ V R ] Performing t h e sums over + A,8, ( etc. and u s i n g the fact , above expression - E* + c a n be E # 3 3 ) that (E.3H) [ ( V ' + V ' ) R ] c V | o > = [VR]C M |o> the cf G + term in E <o|AKP QRP N lo><o|8LQ P[(V''+V')R]C M |o> + E' E < o | 8 L P Q R P N | o R Q,P each used + written E <A|P QR|D><B|Q P[VR]|C>/n(Q)!n(P)! f + R Q,P + E* R Similarily E <B|P QR|D><A|Q P[VR] |C>/n(Q)!n(P).» + + Q,P the terms involving v^' y i e l d the expression (E.35) 74 E E 1 R <A|P QR|C><B|Q P[VR]|D>/n(Q)!n(P)! + Q,P (E.36) - E' E R Ey <B|P QR|C><A|Q P[VR]|D>/n(Q)!n(P)! + + Q.P defining H the + AC,BVD y l V|c><B|Q P[VR]|D>/n(Q)!n(P): P A = Q + Hamiltonian matrix {AB|H|CD} 1 = (E +E C 1 - initial and final starting with the Using the to L make + H - states analogue - the the of - + B C . A V D H - B D , A V C whole <o|A[B,H] result B + is A V C f B D # 3 7 ) H A D , B V C (E.38) symmetrical equation = <E E ){AB|CD} + H A+ H derivation E written + term i n v o l v i n g V " R the method c a n be A C , B V D 1 <o|(BHA-AHB) same {AB | H | CD} / { A B | V * | C D } order = ){ABICD} D In <o|BAH element ( with procedure respect is to repeated (B.16) <O|B[A,H] - (E.39) <O|BAH obtained - H ^ { A B | V | C D } + terra i n v o l v i n g V ^ ^ + 1 - (E.40) where H The A V C ' BD final = V R E < A Q,P l[ R V ] expression p + Q|c><B|R Q P|D>/n(P):n(Q): + for the + Hamiltonian matrix (E.41) element is 75 {AB|H|CD} = " | ( E + V C E D + E A + + The BD,AVC H term + ) { A B L C D } H Involving in all V' R the Cutkosky (E.20) w i l l result of expression (B.20) the r e s u l t terms. Thus expression where v and V V BVC,AD I R > ( = included than the f c r the manipulcticts vertex operator natrix elenent cf the is AC.BVD AVC,BD this (3.21) i s used r a t h e r fcr - V AD,BVC " in unit {AB|[mV]|CD} = f ( V A C ) B V D + VBVD.AC BC,AVD " AVD,BC V third (the the energy terms come from final and te present the r e s u l t i n g net i n c l u d e these ~ seccEd The not elenent i n the u n i t operator being (3.20)). Since vertex operator [mV] matrix V r e p l a c e d by [mv]. equation the ^ at i s i n c l u d e d because equation will ^ } f e l l o w s the above d e r i v a t i o n s t a r t i r g summations equation ^ the operator preceding H cf with which w i l l the H H n a t r i x elements i n equation derivation BVD,AC " V,AVD " AVD,BC + f for vertex o p e r a t o r , [mV], (E.20) (H AVC,BD " AD,BVC " BVC,AD> " ^ H expression equation 2 AC,BVD + ) " V [mV ] JCD} BD,AVC V + AVC,BD V term i n v o l v i n g <A|P QR|o<B|Q P[mVR] |D>/n(P) ! n ( Q ) ! + ^ + + [mV^] (E.UU) + p ^ <A|[mRV]P Q|0<B|RVp|D>/n(P):n(Q)! R,Q,P + ( £ ^ 5 ) E. 3 Ejjci ted O v e r l a c Matrix B le ne n t s Matrix elements between some additional complications. expanding these matrix nuclecn The excited elements Cutkosky However ictc states the matrix tasic matrix method c f elenects s t a t e s i s the same as f o r the u n e x c i t e d overlap irvclve e l e n e n t c a l c u l a t e d here cf one Cutkosky s t a t e s . w i l l be between 76 two singly exhibits excited all Cutkosky the c o m p l i c a t i n g elements between e x c i t e d between any Cutkcsky based on the states. features states, states Since this of calculation overlap the c v e r l a p matrix can be c a l c u l a t e d u s i n g matrix element methods following calculations. Using the d e f i n i t i o n c f s i n g l y e x c i t e d Cutkosky s t a t e s the matrix element can be w r i t t e n { ABm| CDk} = <o | B A m k V V | o> + <o |BAm(Ck) V " |o> + <o|BAmC (Dk)*|o> + IB.16) The first matrix element cn the r i g h t hand s i d e c f the atove equation can be w r i t t e n and matrix elements i n v o l v i n g o n l y one meson operator can be w r i t t e n , f o r example. <o|BAm(Ck) V | o > Thus every = <o|BA(Ck)^[m,D ] |o> + <o | BA [m, (Ck) ] D + + matrix element i n e q u a t i o n + | o> (B.48) (E.46) can re w r i t t e n as 4) can a l l be expressed i n the form F, and F and + W + + = I F,M f,(FM)F M s a t i s f y equations + + (2. 18)-(2.18). (B.U9) 77 Thus a l l matrix elements r e s u l t i n g usinq equations expanded in equation (E.46) ry (E.47) and (B.48) and s i m i l a r e g u a t i c n s terms tecniques of from of section one nucleon B . 1 . The matrix resultinq can be elements u s i n g the expression can be written {ABm|CDk} = 6 {AB|CD} + km £ + <Bm|R [k ,Q]|D> + + I + + < B | R + + <Bm|R Q|Dk> + Q | D > ^ + <B|[R ,m]Q|Dk> + <Am|Q R|c> <A|Q R|c>{-6, R , Q + <B|[R ,m][k ,Q]|D>} + + + + <A|[Q ,m]R |c>}{<B|R Q|Dk> + + + + <B|R [k ,Q]|D>} + + R , Q ( E . 50) + antisym. The sign o f t h e summations i n the above e q u a t i o n i s determined by n o t i n g respect elements that to expressions exchange are equations the cf simplified (3. 24) - (3. 26) , A and in will as matrix the above with equation ty use c f the <5-functicn i n the summations (E.50) w i l l elements i n v o l v i n g pion-nuclecn s c a t t e r i n g antisymmetric B c r C and D. when the matrix vanish and a l l the terms i n e q u a t i o n written be te able will to re one nuclecn s t a t e s and/or s t a t e s and vertex o p e r a t o r s . E.4 E x c i t e d I n t e r a c t i o n M a t r i x Elements The interaction singly matrix excited element between a Cutkosky Cutkosky s t a t e . The more c o m p l i c a t e d given here s i n c e they a r e not used pion-deuteron T-matrix. calculated state and here an matrix elements i n the will be unejcited will calculations not be c f the However the methcd of e v a l u a t i n g them i s 78 similar to the Using of the te written techniques the used definition Hamiltonian between below and of Cutkosky an excited in secticr states and an the B.3. matrix uEe>cited eleirett state {AB|H|CDk} = -<o|BAHC D k |o> + <o|BAH(Ck) D |o> + <o|BAHC (Dk) |o> + The second methods and of + + third section + terms B.2. above + t can The r e s u l t s be t evaluated can (E.51) using the are <o|BAH(Ck) D |o> = -|(E +E +E +E +E )<o|BA(Ck) D |o> + + f A B + c D + k I <A|M QR|ck><B|Q M[VR] |D>/n(Q)'.n(M) ! + + R Q,M (E.52) I <A|R Q M|Ck><B|[RV]M Q|D>/n(Q):n(M)! R Q,M. + + + + antisyra. <o|BAHC (Dk) |o> = + ^ + + i 2 E + E A B + E J' I R Q,M + + E D k + E ) < 0 ' B A C + ( D k ) + ' 0 > <A|M QR|C><B|Q M[VR]|Dk>/n(Q)!n(M)! + Z R C + <A|R Q M|C><B| [RV]M Q|Dk>/n(Q):n(M)! + + + (E.53) Q,M + antisym. The first term in eguation (E.51) can be written <o|BAHC D k |o> = <o|B[A,k ]HC D |o> + <o|[B,k ]AHC D |o> + + + + + + + E <o|BAk C D |o> + + + k All the previous in the above terms sections. first two can The o n l y matrix be + + + <o|BA[Vk]C D |o> + evaluated differences elements f akcve using are ( • ") + the that beccnes E nethcds the cf energy 5 the tern i ( E + E -E,+E„+E_) 4 2 A B k C D 79 and t h e r e are e x t r a matrix elements i n v o l v i n g [Vk] since (E.55) <o|[A,k ]H = (E -E ) <o|[A,k ] - <o|A[Vk] + + A The final result fc f c r t h e Hamiltonian matrix element i s {AB|H|CDk} = |(E +E +E +E +E ) {AB]CDk} A + T I' R B c D k I <A|M QR|Ck><B|Q M[VR]|D> + <A|R Q M|Ck><B|[RV]M Q|D> + + + + + Q,M + <A|M QR|C><B|Q M[VR]|Dk> + <A|R Q M|C><B|[RV]M Q|Dk> + + + + + - <A|k M QR]C><B|Q M[VR]|D> - <A|k R Q M|C><B|[RV]M Q|D> + + + + f + + - <A|M QR|c><B|k Q M[VR] |D> - <A|R Q M|o<B|k [RV]M QlD> + + + + + + + - |<A|M QR|C><B]Q M[VkR]|D> - -|<A|R Q M]C><B|[RVk]M Q|D> + + jl 4 I + + + + -<A|M QR|C><B|Q M[VkR] |D> - <A|R Q M|C><B| [RVk]M Q|D> + f + + + R Q,M (E.56) + antisym. Although i t appears as though t h e l a s t two terms w i l l meson exchange terms, cne mescn exchange The these terms w i l l derivation will c a n c e l with seme c f the terms. f o r the operator [mVJ between e x c i t e d states give z e r c matrix elements cf the vertex and u n e x c i t e d or e x c i t e c Cutkcsky not be given here as i t can be e v a l u a t e d using the technigues of s e c t i o n B . 2 . 80 APPENDIX In C The T-matrix section approximation 4.2, (1) terms written the T-matrix was w r i t t e n Vm.flk The i n t h e Qne^Meson on the i n terms of as T diff right - + the Approximation one-meson exchange below _ + " in Ejcchanqe T diff hand one-nucleon + + T same side matrix + T of same (C. 1) the above e q u a t i o n elements i n the following equations. "IIABCD ^llABXCD FGq x V > o x ( <A|[mV]|F><F [Vq] |C><B| [Vk] |G><G| [qV] |D> a ^ E ) +<A| [mV] [ F x F [Vq]|C><B|[qV]|G><G|[Vk]|D> a (E) +<A|[Vq]|F><F [mV]|C><B|[Vk]|G><G|[qV]|D> a,(E) 2 +<A|[Vq]|F><F [mV]|C><B|[qV]|G><G|[Vk]|D> a <E) ) 4 diff = 2 EABCD ^ J ^ ™ FGq (C2) V > x o x ( <A| [mV] | F x F [qV]|CxB|[Vk]|GxG|[Vq]|D> a (E) +<A| [mV] | F x F [qV]|C><B|[Vq]|G><G|[Vk]|D> a (E) +<A|[qV]|F><F [mV]|C><B|[Vk]|G><G|[Vq]|D> a (E) 5 g ? +<A|[qV]|F><F [raV]|C><B|[Vq]|G><G|[Vk]|D> a ( E ) g ) are (C.3) T + = i y „™<p' |AB)(CD|P > x A same 2 ^ABCD o o FGq ( <A|[Vq]|C><B|[Vk] | F x F | [qv] G x G | [mV] D> a ( E ) 1 1 9 +<A|[Vq]]C><B [qV]|F><F [Vk] G><G| [mV] D> a io ( E ) +<A|[Vq]|C><B| [mV]|F><F| [Vk] G><G [qv] D> a ( E ) n +<A|[Vq]|C><B [mV] | F x F [qv] G x G [Vk] D> a +<A|[Vq]|C><B [qV]|F><F [mV] G><G [Vk] |D> a 1 2 (E) 1 3 (E) +<A|[Vq]|C><B [Vk]|F><F [mV] | G><G [qv] |D> a ( E ) u ) same FGq D> a x (<A|[qV]|C><BJ[Vk] F><F|[Vq] G><F|[mV] 1 5 (E) D> a +<A|[qV]|C><B|[Vq] F><F|[Vk] G><G|[mV] lfi (E) G><F|[Vq] D> a +<A|[qV]JC><B|[mV] F><F|[Vk] 1 ? (E) a +<A|[qV]|C><B|[mV] F><F|[Vq] G><G|[Vk] D> l g (E) a +<A|[qV]|C><B|[Vq] F><F|[mV] G><G|[Vk] |D> 1 9 (E) +<A|[qV]|C><BJ[Vk] F x F J [mV] |G><G|[Vq] |D> a 2 Q (E) ) (C.5) where L 1 ( E ) ~ v ^ v ^ )^ v 2v 1 + v " v (v 2 3 1 + (C.6) v +v +v ); 2 3 5 5 (C.7) a (E) = 2 v a (E) 3 l 3 V ( v 2 + V 3 ^ i ) V + V / 1 v (v +v ) 1 2 3 \2 V 2 + V 3 5^ + V V l + V 2 + V 3 \ V V V V V / (C.8) 82 a (E) = 2v.4v. — 1 3^ 2 3^ 1 + 4 V a (E) 5 V V + V V + V 2 + V 3 + V = (CIO) V 1 2 V ( V 2 + V 3 ) / l = v a (E) 7 = / . . v v (v +v ) a (E) = V E ) 7 8 l ( v 1 v )(v 2 + 3 2 2 1 + v 2 + V3 v ) + 2v +v 1 9 l ^2^3^ ^ l V V + V 2 + V 3 + V io ( ( E ) " — v i 3 ( n V a 1 2 (E) 1 3 (E) 1 ( V 3 \ (Cll) / 2 2 + V 3 ^1*4 + V 2 V + V 3 V V ) 2 V v r V 2" V 3 \ CC13) v 3 1 2 3 2v v ( 3 2 V l 2 + 2v -W 3 4 0 5 +v +v +v ) 2 3 (C14) ) V : f V + 5 ) (C15) (C.16) (C17) = v a 2 1 a (E) V l * 4 (V +v ) ( v - W ^ + v 2 v l V 1 V v v .9 + 3 2 - v 4 I, 2 ( 3 2 a r v 3 v ( 2+v ) 1 5 (C.12) V ( E ) v V 5 _ v a (C9) 4^ l 3^ 2 v v + v 1 = V 1 + V 2 + V 3 + V 5^ (C.18) 1 2 3 V 3^( V V 83 a, .(E) = 1 4 (C.19) — 1 2 3 V V V (C.20) a 15 ( E ) v 1 v ( v 2 + v 2 3> 3V +V +Y +2V 1 a (E) l f i l V a 1 7 1 8 2 + V 3 l ) ( v + V 2 + V 3 + V 4 > ) V 2Vj+v 4 V 3 I 7 (C.21) CC22) (E) V a ( v 3 2 1 3 V ( V 2 + V 3 ) 2v +2v +v 2 (E) V 1 ( V 2 + V 3 ) ( V 1" 2" 3 W W + V 5 ^ ) 3 V 5 V r 2~ 3 \ V V (C23) a (E) 19 1 Q = — i v v V l 2 (C.24) 3 CC.25) The energy below. - If meson) is appear similar functions the vertex remove! from to the involving each diagram B . .,v 5 are obtained the mescn diagram, then below. — a | cu m (the a l l as final outlined emitted diagrams will 84 The energy f u n c t i o n i s given by v where the i = E indices i F i ~ E l i ( C a r e as s p e c i f i e d ' 2 6 ) i n the diagram above and where E^j, = sum o f energy of p a r t i c l e s g o i n g i n t o v e r t e x i and E . = sum of energy of p a r t i c l e s coming from v e r t e x i . r i -Note that s i n c e time i s assumed to " be left i n the above diagram, E p a r t i c l e s to the l e f t p i increasing towards the corresponds to the energy c f the of the vertex i and E ^ corresponds t c the energy c f the p a r t i c l e s to tha r i g h t of the vertex i . Since should the f u n c t i o n s v^, v , 2 be added and v^ can v a n i s h , a t c each of these energy denominators. T h i s i n s u r e t h a t the meson at each of these v e r t i c e s appropriate term - i e boundary conditions. will cbey will the 85 APPENDIX D D . 1 The Deuteron The deuteron conventional and Way_ef u n c t i o n wavefunction deuteron Landau(1974) in (AB|t>> = o monientuni the the projection singlet (1952). The isospin It of will functions wavefunction to the x 1 £ ire be W e i s s k o p f (1 952) the the } ( McMillan M) (C * 1) (C2) A -k B ) (D.3) deuteron as spin. given a represent be above by The Blatt the function and isospin is Weisskopf and spin, convenient deuteron to separate wavefunction. the spin Thus the written * ( < > V V where of the nucleons. prove will of (k x and the from wavefunction to + A function quantities respectively, related way 6 ( j c - l the is ^ V V V %>~- * £*W K = k M is space following where and (AB|P > M ) = E * *t triplet ( spin - X l* °A C B , a functions < ' > ) D as given by Blatt 4 and and • J . fe) - U GO Y O O <*) * m + W (<) Y 2 > M _ £ m * (D.5) 86 The H functions and the Y (£) mn state spherical coefficient Clebsch-Gordan D are in coefficient. momentum the The space harmonics of the unit last is the conventional term functions wavefunctions U and W are used by vector the S and McMillan and L a n d a u (1974) . D.2 Single-Seattering_ In is section 4.1 the pion-deuteron single-scattering T-matrix written £tek T Delta functions = ^A,B,C J <P of momentum deuteron and t h e the above summation and an integral <A ab) Ir H mV can be < lV ck> CB extracted pion-nucleon T-matrices. over as a product momentum, +4 of the (D sums ever above -Ms By from * the 6) pion- explicitly writing spin isospin eguation and can be written d k d k d k „ <V »|AB)T(Am,Ck) A B t» o 3 A 3 3 (D.7) x Using rt equations functions be (CB|P >• 55 o to expressed ( D . 1) eliminate in the ( k .+EQ.-k + m - k „-]S:) -k) (k A c and two of laboratory (D.4) the and using integrals, frame as the resulting eguation (D.7) delta can 87 ,(0) +1 d K ^' 3 +4 X X 2 WV" 6(^,+m-k) (jc+|(k-m)) c^OO , V A'WV B T T } 1 T(k ,m; k ^ k ; a ^ ; A where k A (C.8) VVV^) = jc + k-m (C.9) kg --jc The sums written T L 0 ) over in the form 0,k; M',M) % ' » B ; - isospin can (D.10) now be done and the final result 5 (k^.+m-k) (E. 11) +1 I £ A'=-l t where TTtl and where T ^ , It frame is should equation related i be (D. 12) to using the the same quantity noted refers that to pion-nucleon equation (B.7) the with the T A = T McMillan C = ~ 2 pion-nucleon laboratory T-matrix of N frame. T-raatrix This in the and L a n d a u (1974) . can pion-nucleon in be CM 88 D.3 Doubletscat ter ing In is section 4.2 the pion-deuteron double-scattering T-matrix written = P ' m ' —TTi <P A,B,C,D,q t ? k , |AB)<A|[mV]|Cq><B|[qV]|Dk>(CD|P >6(E +E -E -E ) D k ° (E.13) Using can T T the be same as in section D.2 the f above equation written (k^. ( 1 ) techniques L 0,k; M',M) «S(k ,+m-k) V (D.14) V +1 I = -TTi f ' * (K+q-|(k-tn)) d Kd q 3 3 <^(K) T £ t £ 6(k ,-to-k) p A,A»=-1 J where T n = I V V V V V V 2 T A , T B ^ X li' V B ( f f ) X U ( o C'V ! 'X-J, (D.15) A t +1 =-1 T q and k k A = B ~~G As matrices tnatrices McHillan stated in the in and in the £ + £ - m = - J i + k - a = _ ~D = ( D ~— previous laboratory the (E.16) frame 1 7 ) ( E . 18) section, can p i o n - n u c l e o n CM f r a m e L a n i a u (1974). . be the pion-nucleon T- related using to equation the T- (E.7) cf
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A field theory approach to pion deuteron elastic scattering Alexander, James Harry 1975
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Title | A field theory approach to pion deuteron elastic scattering |
Creator |
Alexander, James Harry |
Date Issued | 1975 |
Description | Pion-deuteron elastic scattering is studied using a field theory of pions and nucleons. By treating the nucleons in this manner, the double-counting problem usually associated with pion multiple-seattering is avoided. The pion-deuteron T-matrix is written as a series expansion in terms of operators between one-nucleon states. The first two terms in the series are examined. The first term yields the usual single-scattering contribution to the T-matrix. The second term in the series can be expressed as a sum of twenty terms. By making an on-shell approximation and a static approximation where physically sensible, the magnitudes of the twenty terms are compared. The dominant term is similar to the conventional double scattering term resulting from the generalized impulse approximation. There are also four other terms whose magnitude cannot be evaluated without doing numerical studies with a particular field theoretic potential. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085263 |
URI | http://hdl.handle.net/2429/19473 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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