The Influence of the NN* Channel on Elastic NN Scattering Citation Shepard, Harvey Kenneth (1966) The Influence of the NN* Channel on Elastic NN Scattering. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5HRF-4A20. https://resolver.caltech.edu/CaltechTHESIS:10192015-090318183 Abstract The problem of two channels NN and NN*, coupled through unitarity, is studied to see whether sizable peaks can be produced in elastic nucleon-nucleon scattering due to the opening of a strongly coupled inelastic channel. One-pion-exchange (OPE) interactions are calculated to estimate the NN*→NN* and NN→NN* amplitudes. The OPE production amplitudes are used as the sole dynamical input to drive the multichannel ND -1 equations in the determinental approximation, and the effect on the J = 2+ ( 1 D 2 ) elastic NN scattering amplitude is studied as the width of the unstable N* and strength of coupling to the inelastic channel are varied. A cusp-type enhancement appears in the NN channel near the NN* threshold but for the known value of the N* width the cusp is so “wooly” that any resulting elastic peak is likely to be too broad and diminished in height to be experimentally prominent. A brief survey of current experimental knowledge of the real part of the 1 D 2 NN phase shift near the NN* threshold is given, and the values are found to be much smaller than the nearly “resonant” phase shifts predicted by the coupled channel model. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Frautschi, Steven C. Thesis Committee: Unknown, Unknown Defense Date: 14 December 1965 Funders: Funding Agency Grant Number NSF UNSPECIFIED Atomic Energy Commission UNSPECIFIED Record Number: CaltechTHESIS:10192015-090318183 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10192015-090318183 DOI: 10.7907/5HRF-4A20 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9225 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 19 Oct 2015 20:25 Last Modified: 08 Mar 2024 00:25 Thesis Files Preview PDF - Final Version See Usage Policy. 23MB Repository Staff Only: item control page

THE INFLUENCE OF THE NN* CHANNEL ON ELASTIC NN SCATTERING Thesis by Harvey Kenneth Shepard In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1966 (Submitted December 14, 1965) ii ACKNOWLEDGEMENTS I wish to thank Professor Steven Frautschi for suggesting the study of this topic and for his continuing encouragement and gentle guidance during the course of this work. Among the many others who have been of help in some aspect of this research are E. Abers, L. Balazs, J. Blue, Y. Dothan, R. Gerbracht, R. Griffith, Y. Hara, B. Kayser, R.S. McKean, A. Scotti, W.G. Wagner, and G. Zweig . I am deeply grateful to the National Science Foundation for their financial support as an NSF Predoctoral Fellow during part of this work, and I am also indebted to the U.S. Atomic Energy Commission for providing funds to cover computing costs. Finally, however inadequately, I must thank my wife Nancy for her understanding and loving support. iii ABSTRACT * The problem of two channels, NN and NN , coupled throu gh unitarity, is studied to see whether sizable peaks can be produced in elastic nucleon-nucleon scattering due to the opening ~f a strongly coupled inelastic channel. One-pion-exchange (OPE) interactions are calculated to estimate the NN * ~ NN* and NN ~ NN* amplitudes. The · OPE production amplitudes are used as the sole dynamical input to drive the multichannel ND -1 equations in the determinental approxima- tion , and the effect on the J = 2+ ( 1 D2 ) elastic NN scattering ampli- * and strength of coupling tude is studied as the width of the unstable N to the inelastic channel are varied. the NN channel near the NN A cusp-type enhancement appears in * threshold but for the known value of the * width the cusp · is so "wooly" that any resulting elastic peak is N likely to be too broad and diminished in height to be experimentally prominent. A brief survey of current experimental knowledge of the real part of the 1 D NN phase shift near the NN 2 * threshold is given, and the values are found to be much smaller than the nearly "resonant" phase shifts predicted by the coupled channel model. - -- ·------ --- - --- - - -- · - · - · - - - · - - - -- - . iv TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii. ABSTRACT iii I Introduction 1 II General Features of the NN and NN* Channels 15 III The OPE Amplitudes 28 IV Unitarity of Helicity Amplitudes in the Isobar Model 58 V Removal of Kinematical Singularities and Continuation below the Inelastic Threshold VI 62 Details of the Calculation and General Properties of the Solutions 76 VII Results and Conclusions 89 VIII Implications of su Symmetry 3 113 APPENDICES A Notation and Kinematics B Description of Helicity States and Calculation of the c 120 OPE Partial Wave Helicity Amplitudes 124 The Complex Singularities 137 REFERENCES 142 - 1 - I. · INTRODUCTION After much work in single channel dynamical calculations, there has been increasing effort within the last four years to study the effects of higher channels which are coupled through unitarity to the lower energy elastic scattering processes. The most ambit ious attempts have been in the nN problem where the nnN channel has been included, usually in some approximate way via a pseudo-two-particle pN or nN* channell-B). Although the calculations are not quanti- tatively convincing it does seem clear that inclusion of these higher channels may help to explain the second and third pion-nucleon resonances. From this work and other simple models which have been studied it is evident that the existence of such inelastic channels may act to produce an enhancement in the elastic scattering amplitudes (and cross-section) at energies near and even below the threshold where the inelastic scattering can physically occur. Experimentally, many resonances are found to lie close to the threshold for an inelastic process. Even if this association with thresholds proves accidental, one can speculate that the succession of resonances seen in the nN and nn systems as higher energies are explored can be explained by incl ud ing more and more of the i!lelastic ·Channels which becomes accessible as the energy is increased. Let us list briefly some of the ways a higher e.n ergy channel can 9 cause an enhancement in a lower coupled channel ) : 1. . 3 4 10 11 12) Threshold Effects (Cusps,Ball-Fra zer mechanism) ' ' ' Here - 2 - the effect is primarily kinematical and occurs very near to the energy where the inelastic channel opens up. If the inelastic channel contains an unstable particle (and thus the channel is really a three- or more-particle channel, the cusp can become Founded (a socalled "wooly" cusp) and the enhancement smeared over a broader region. Cusps are most prominent when the particles in the inelastic channel are produced in a low orbital angular momentum state (viz. £ = O). * The prominence of the cusp (i.e.'· .whether it is observable or not) is also dependent on the strength of the coupling between the elastic and inelastic channels. The Ball-Frazer peak is associated with . the rapid ris e of an inelastic cross-section to its unitarity limit. 2. Strong Ine lastic Forces (Cook-Lee Mechanism) P If the c;oupling between channels is strong enough a resonance may appear in the lower elastic channe l below the inelastic threshold. Loosely speaking, there is an attraction produced due to a virtual transition from the elastic to the inelastic channel and back again. in a resonance like configuration . The system oscillates By including unitarity in our calculations we allow such a strongly coupled inelastic channel to produce its effect in the elastic channel below the inelastic threshold. Above the i nelastic threshold unitar i ty acts to severely damp the elastic amplitude. * A good example now see ms to be available: an S-wave cusp in elastic rrN scattering at the ~N threshold. This can be seen in the analysis of P. Auvil and C . Lovelac e , Nuovo Cimento 33, 473 (1964). - 3 - 3, Virtual Bound State Resonance: (Dalitz, Sakurai)~ 3 ,l 4 , 9 ). Here the higher channel, if isolated, contains enough attraction to produce a bound state. But because of the existence of a lower energy channel the "virtual" bound state produced in the higher elastic channel . decays and is seen as a resonance in the lower channel, usually not too far below the ine lastic threshold ·k ) • For this mechanism . to operate a fairly weak coupling between channels seems necessary in order to partially isolate the higher channel. The question naturally arises whether the existence of higher channels in the NN system can produce any effects such as thos e listed above. In the present work we shall consider one specific higher channel, the 11'.NN channel, which we shall treat as a pseudo-two-particle state consisting of an N and an N* , where the N* is the I J = 3/2+ 1238 MeV 11'.N resonance. = 3/2, 1 2 Via this isobar model ' ) we consider only that part of the 11'.NN state in which two of the final particles are produced in a resonant state **) •k . . This model of NN production has been successful in explaining many of the features of pion production in NN collisions (e.g., momentum spectra of the pion and recoil nucleon, angular correlations between pious and nucleons, etc.) below a couple of Be VlS). In our actual dynamical calculations we shall partially include the true three-body nature of the state by use ~f a phase space factor which allows for the width(and hence instability) * An example of this type of resonance is the_interpretation of the *-le 11'.A (Y1 *) resonance as a bound state of the KN channel. See, for example, reference 13. We will ignore the error introduced by not anti-synunetrizing with respect to the two nucleons in the final state. - 4 . .: * 11) of the N Most of the formalism we shall use has been fully discussed in references 1 and 2. i~ Besides the fact that NN production is prominent experi- mentally this particular inelastic channel was chosen for study for * several other reasons. Because of the sizable width of the N , approximately 125 MeV full width at half-maximum, one expects that "force" diagrams such as Figures l(a) and l(b) below N N~ N~ N N N .:rr N (a) (b) One Pion Exchange (OPE) Interactions in * and (b) N + N* ~ N + N* (a) N + N ~ N + N Figure 1 * would make a large contribution due to the large coupling at the :n:NN vertex. (A large coupling between channels might suggest that. mechanism (2) discussed above could . produce an enhancement in the elastic NN channel.) Diagram l(b) also contains a cut in the physical region . i(2 [running from S = 2(M 2 + }1) - µ 2 ~'>'2 to S = (M 2 2 2 - }1) /µ , where * = N* mass = 1238 MeV, µ M = nucleon mass = 938 MeV, M 139 MeV] which, as noted by Peierls in elastic cross-sections*). 16 pion mass =:= ) can sometimes produce peaks However, such a peak is usually not more careful e x amination of specific cases has shown that these * Apeaks may not be present. See, for example, C. Goebel, Phys. Rev. Letters -13, 143 (1964). . ______________.._._........_,;;,;____,_,,_______ =·· -=::.....--"'~===~~~=~== =- ~-=··= · ··--- -- -~-- - -·· - 5 - confined to a single J, P, I state and has a shape and an asymUtetry different than a resonance-like Breit-Wigner distribution. Experimentally, there a r e no known resonances in NN scattering. (There is, of course, a bound state - the deuteron.) Neither the total nor the differential cross-sections show any strong peaking_ below 1 BeV . However, several of those who have attempted to phenomenologically describe the data find some enhancement in the I = 1 state near a lab energy of 600 MeV. For example, in Fig. 2 2 below, we have plotted p d cr (15.c) 2 dn (e CM = n/2) vs. lab kinetic energy for the case of proton-proton scattering (which is pure I= 1). The data is t aken from Table A2 in Wilson's book on the NN interaction 17 ). Although the 90° (CM) scattering does diminish the contribution from some odd £ states both even and odd orbital angular momenta contribute, and we cannot simply isolate the enhancement to a specific state of . ~'c) angular momentum and parity . The proposed enhancement occurs at an energy near or slightly below the threshold for the inelastic process •k N+N -7N+N (a strong reason for studying this channel). This reaction only can occur in the I= 1 state (where the N* is the I= 3/2 isobar). * In spinless scattering since P£(n/2) = 0 for odd £ 1 s, only event 's are present at eCM ~ n/2. But for pp scattering, using the notation of reference 18, although M11 (n/2) = O, ~-l (n/2) O, M00 (n/2) = 0 both M (n/2) and M (n/2) are non zero and contain odd £ states. 01 10 - 6 - Figure 2 p 2 ~~ for proton-proton scattering at 90° c.m. - 7 :... 0 0 " 0 0 "'I"" 0 0 ~ < co ...J - 8 Our ideas on cusps also encourage us to look for states in which the n ew channel is in a low angular momentum configuration, preferably L = 0. The nucl e on has J + p = 1/2 ; the three -three resonanc e treated as an unstable particle has JP= 3/2+. the J Therefore, putting · NN* in a n L = 0 state r e qu i res INN*) be in a P = +, total = 1 or 2 state. To couple to INN) with T = 1, P +,the generalized Pauli principle forces the NN state to have S = O, L = 2 and thus J = 2. with LNN* = 0 the pertinent channel spin-par i ty is = 1 with the NN in a 1 D2 state. examine some of the forces in other J p Although we shall briefly states by looking at the OPE Born ampl i tudes , we shall ultimately pe rform d e tailed dynamica l calculations only in this one state. Above 400 MeV a phase shift analysis of proton-proton scattering i s greatly complicated by the presence of inelastic scattering which causes the phase shifts to become complex; i.e., not pur e ly r e al. Most of the work done in the 600 MeV region has been performed i n the U.S.S . R . using· the 6-meter synchrocyclotron in Dubna. For pp , neglecting ine lasticity and using the requirement of un i tarity , five indepe ndent experiments •k) performed at all angles are sufficie nt * Brie fly , we can l is t the k i nds of experiments from which to choose. 1. Single scatte ring experiments to measure the unpolarized differential cross-section. 2. Double scatte ring experiments to measur e the p olarization P. 3. Triple scatte ring e x p e rime nts to measure R (related to the rotation of the polarization vector in the plane of the s e cond scattering), D (describes the extent to which the second scattering depolarizes an initially polarize d beam) , or spin corre l ation experiments in which one measures the polari z ations o f the recoiling target usually in. correlation with the polarization of the outgoing nucleon. At one energy and angle there are nine independent quantities which can be measured. For more details, see refs. 17, 18, and 19. A, - 9 to determine M the scattering matrix at all angles 18) • This however, is only academic interest even aside from inelasticity, for in practice, one takes a number of pieces of (hopefully reliable) experimental data and attempts to ·,. fit with several (often variable) parameters and then uses some goodness of fit criteria (commonl y chi-square) to search for acceptable solut i ons. Usually several such solutions are found and other types of data (e . g . ' data on p + p~ 1/ + d) or auxiliary requirements must be used to reduce the number of satisfactory solutions. 18) · In the so-called modified phase shift· analysis the use of the one- pion exchange (OPE) contribution for high angular momentum states allows one to get. as good chi - squares using many fewer parameters 2 ( for examp l e at 310 MeV one can get the same X using nine parameters and OPE as using fourteen parameters with no OPE) 19 ). In making a phase shift analysis of pp scattering at 657 MeV, v Azhgirei e t al . 20) assume one meson exchange for £ =::, (~ = They assume the phase shifts are complex 5 states. o + iy) and use absorp- tion coefficients (averaged over J) for the 3P , 1 , 2 , 0 3 F 2 , 3 ; and 1 D2 states which are obtained from the experimental cross-sections for the inelas t ic processes p + p ~ n+ + d, p + p ~ n+ + n + p , . 0 p + p ~ 11'. + p + p . They us e d 45 experimental quantities and determined twelve parameters, nine phase shifts ( £ ~ 4) and three a bsorption coefficients. With a 2 2 X = 36 . 7 (For a good fi t X = number of data points - number of parameters) ·they find . 2 oreal (1D2 ). = 11.7° ± 2.2, r 2 = .668 ( e - y = r). -10- Using new data on the parameter A they modified their values to 5 = 8.7 ± 4.9, r 2 = .678. Similar analyses (Golovin Dzhelepov 23) ) . 21 ), Zul'karneev and Silin also find t ha t Re 51 D ~. 10° for lab . 2 22 ), energ ies near 600 MeV. It thus seems reasonably safe to conclude from thes e analyses) althou gh they are not yet completely satisfactory, that no true . . · resonance exists in t h e lD 2 pp state. The enhancement which may be present seems more likely to be what we have classed as a threshold or cusp effec t. In this paper we shall study the coupled processes * * in the isobar N + N --+ N + N, N 1- N H N + N , N + N~'( --+ N + N * as an unstable particle. approximation treating the N Ultimately in our dynamical calculations (using the determinental approximation to the multichannel ND -1 method) we shall ignore all forces but those * due to the inelastic process N + N -+N + N. And only the longest range part of this interaction, the OPE diagram Fig. l(a) will be kept. Besides providing the longest ran g e interaction, pion exchange l eads to cuts in the partial wave amplitudes lying nearest the physical region . Other particle exchanges were ne g lected for simplicity and because the relevant effective coupling constants * * (e.g. > for the pNN, mNN, we could use the su 6 ~NN * vertices) are not well known, though symmetry scheme to es t imate them·. It is the inelastic OPE amplitudes which shall drive the syste m of coupled equations. Our work is clearly rela ted to the strip approximation diagram for NN scattering shown in Figure 3. This diagram would - 11 - N N _1!_ N'lt N N A Strip Approximation Diagram for Nucleon-Nucleon Scattering Figure 3 correspond to the lowest order term in the solution to the set of coupled unitarity equations. The elastic forces from the processes N + N -7N + N (e.g., forces due to exchange of n, p, m, ~' n, p, 24) * -7N + N* (exchange of . ) and N + N .•. )will be ignored in our dynami~ql calculations for several ~'( * elastic forces, at least reasons. As we shall see in Sec. II the NN -7NN those given by the diagram of Fig. l(b), do not seem to contain either very much attraction or repulsion where this is estimated by studying the sign and (non zero) magnitudes of the amplitudes (with momenta divided out) at or near threshold. For this reason, and because of a need to restrict the size and complexity of the problem these NN elastic forces were omitted. * * As will be discussed in Sec. I, to "/(. describe elastic NN ..; -7 NN : scattering in a state of definite J, P, and I requires in general, a 4 x 4 matrix (i.e., ten independent - 12 amplitudes)*). Since elastic NN scattering in a state of definite J, P, I ·requires a 2 x 2 matrix ( i nvolving two eigenphases and a mixing parameter) for triplet scattering (but only a single amplitude for the S 0 case), ic the most general coupled NN, NN involve 6 x 6 matrices. Thus, the II two II problem would ~'c channel NN, NN system r~ally consists of six coupled channels. Besides the limit on complexity, perhaps the strongest reason for ignoring all elastic forces is just the desire to ·isolate and study the effect of a production process on an elastic channel coupled to it via the unitarity relations. We thus can study the mechanisms (1) and (2) described above in their purest form without any complications . 25) II As Cook and Lee have noted it is likely in many processes that the "elastic" forces contribute very little to the structure of the elasti c amplitude near the thresholds of inelastic channels. It is rather the particular form of the coupling between the amplitudes; II i.e., the unitarity relations, which is responsible for the structure. We shall find that our coupled channel model does predict a cusp type enhancement near the NN ic threshold. However, because of the broad ic width of the N , we find that the cusp in the elastic amplitude becomes rounded and flattened, and diminished in height by the presence of sizable i nelasticity. Thus, the peaks predicted by our model would not be very prominent experimentally. However, our model does predict real phase shifts close to 90° and thus disagrees with .the e xperimental * For small J's (< 2) fewer amplitudes are required . (J = 0 requires 1 and J = 1 requires 6). - 13 - data we have cited. We therefore must conclude that important repulsive forces have been omitted from our model. The influence of the NN* channel on NN s.c attering has also been considered by Leung 26 ) and Coulter et al., 27 ) and we shall compare their work and approach with ours in Sec. VII. The organization of this paper is as follows: In Section II we describe some general features of the NN, NN* system; especially the restriction on the number of independent amplitudes imposed by angular momentum, parity, and isotopic spin conservation and time-reversal invariance. Section III contains the calculation of the OPE amplitudes for the diagrams of Fig. 1. The properties of the spin 3/2 N* state and the nNN* vertex are reviewed. We then project out partial wave helicity amplitudes and form amplitudes of definite parity. The problem of off-mass-shell terms in the partial wave amplitudes is briefly discussed. The high energy and threshold behavior of the .amplitudes are examined. Signs and relative magnitudes of forces in different JP states ~re estimated. ~'( The analytic structure of the NN ~NN partial wave amplitudes is described and the appearance of complex singularities is noted. In Section IV we state the unitarity relations for partial wave amplitudes in the isobar approximation and give the solution in the form of the multichannel ND -1 equations. Next, (in Section V) we choose proper phase space .factors to eliminate kinematical singularities (at least those clos e to the physical region) and guarantee that our resultant "unitarized" amplitude will have the correct threshold behavior. We discuss and - 14 - * illustrate the modified phase space factor that we use for the NN state. This pseudo-three particle phase-space factor is compared · ll) in · t h eir · . f orm suggeste d b y Nauenb erg an d P ais wit h t h e approximate Also in this section we perform the discussion of wooly cusps. * Born amplitudes to the region between the continuation of the NN ~NN ·k NN and NN thresholds since this will be needed for the dynamical calculations. In Section VI, we discuss the approximations to be made in our determinental calculation. Detailed questions concerning the choice of subtraction point,the convergence of integrals, the use of a cutoff, and the handling of principal value integrals are explained. Some general properties of the solution are also discussed. Section VII contains the numerical results of our computer calculations. We examine and plot graphically the dependence of the NN elastic scattering amplitude on the magnitude of the coupling between channels (the size of the inelastic amplitude) , the width of ~~ the N , the choice of subtraction point and the cutoff. of ghosts is briefly discussed. The problem Finally, we sunnnarize the results of our work. In Section VIII we briefly discuss some possible ·su 3 implications of our coupled channel model. Appendix A sunnnarizes the symbols, notations and relevant kinematics we use, and Appendix B gives many details needed for projecting out the OPE partial wave helicity amplitudes. In Appendix C the problem of complex singularities and anomalous thresholds is examined further . · - -- -- . · - - - - 15 •k II. GENERAL FEATURES OF THE NN AND NN CHANNELS In the center of mass the NN threshold is at 1.876 BeV and the NN * threshold at 2.176 BeV. *) . (Later when including the N* width we shall effectively be aliowing the threshold to extend· to lower energies.) In terms of Tlab the lab kinetic energy of the incident nucleon we have ~M - 4~ (II.l) 2M and also = (II.2) where we are using energy units (c = 1). With WCM 2.176 BeV we find * threshold occurs at Tlab ~ 646 MeV. the NN The general kinematical features of nucleon-nucleon sca ttering 28 are Well -known ). Assuming · · d epen d ence, pari•t y conserva t ion ' c h arge in and time reversal invariance, five independent amplitudes are required for a complete characterization of nucleon-nucleon (or NN) scattering. The NN state can have either isotopic spin 0 or 1. For a given value of total angular momentum J the two-nucleon system can be in an S = 0 or S = 1 state. Because of the Pauli principle .there can be no transi- tions b e twe en the two spin states. For the singlet state (S = 0) we have only L = J and one amplitude characterizes the scattering. * In Appendix A we have summarized the notation and conventions we will be using in our calculations and given a brief review of the kinematics of the reactions N + N*--7N + N* and N + N --7N + N*. - 16 - If S = 1 we have L = J or L = J ± 1. (If J = 0 clearly. L = J - 1 is not present.) For L = J, again one amplitude is sufficient since parity conservation forbids transitions to L = J ± 1. If L = J ± 1 we need three ampl°itudes to describe the transitions L = J - 1 ~L = J - 1, J + 1~J+1, J - 1 ~J + 1 (which is equal to the amplitude for J + 1 ~J - 1 by time-reversal invariance). Hence a total of five independent amplitudes (for J > O) are needed. The generalized Pauli principle implies that for allowed transitions in N-N scattering (-l) I + L + S __ _ 1, and thus for example, fixing the parity and isospin determines the allowed S. Using the relat:ions between amplitudes implied by P, T, and I conservation the same kind of counting procedure as above can also be carried . t h e h e l"icity . . 28,29) . out in representation * ~N + N* scattering. Now consider N + N is 1 or 2 and hence only the I state . Here, the total I = 1 part of NNi' can couple to an NN With a fixed J, we have either S = l . or 2, and because the N and N* are not identical particles, there can be transitions between the different spin states. And specifying I places no restrictions on allowed values of L or S as it did for the antisynunetrized NN state. With S = 1 we can have L = J, J ± 1 (assuming J > O), and for S = 2, L = J, J ± 1, J ± 2 (J > 1) are all possible. js 1, L = J}, Hence the states js = 2, L = J}, js = 2, L = J + 2), and js = 2, L = J - 2) all have parity (-l)J and there may be transitions between any of them. Using time-reversal invariance (e.g., amplitude for J + 2 ~ J - 2 equals amplitude for J - 2 ~ J + 2) we find we need - 17 - ten independent amplitudes to describe elastic NN* scattering for p = (-1) J. Similarly 1s = 1, L = J + 1)' is = 1, L = J - 1), \s = 2, L = J + 1) J 1s = 2, L = J - 1) all have parity (-1) J .:.1 and again ten independent amplitudes are needed to describe scattering with this parity. So for J > 1, a total of twenty independent amplitudes would be needed for a complete characterization of elastic NN* scattering in a given isotopic spin state. (It is easy to see that for J = 0 only IS= 1, L = 1) and IS= 2,·L = 2) are allowed, thus requiring one amplitude for each parity; while for J = 1 there are six independent amplitudes for each parity.) Let us perform the same counting of independent amplitudes in terms of a helicity description of each two-particle state (we are talking about . the N* as if it were a stable particle). Define the partial wave helicity amplitudes * ~3(N) * + 4(N) by (A A3 iTJ (W)iAl A2 ) for the process l(N) + 2(N) 4 the relations*) (II. 3) I (2J + 1) <A4 A3 I TJ (W) I Al A2) J (II.4) * See Appendix Band Reference 29 -for more details . on the helicity formalism. - 18 - = 4, 2 = 3) and in the case of elastic scattering (1 (II.5) Parity conservation implies (II.6) where ~i is the intrinsic parity of particle i and Si its intrinsic "/~ spin. Hence, for NN ~'( ~NN we have ~l = ~ 2 = ~ 3 = ~ 4 =+and sl = s4 = 1/2, s2 = s3 = 3/2 and thus ~g = +. Time-reversal invariance requires (II. 7) It is now easy to see that between the eight initial states 1112 3/2), 1112 1/2), 1112 -1/2), 1112 -3/2), l-1/2 3/2), 1-1/2 1/2), l-1/2 -1/2), l-1/2 -3/2) ~nd similar eight final states (just replace the ket by a bra vector) only twenty independent amplitudes may be formed which we have defined in the following matrix. 1112 3/2) 1112 1/2) 1112 -1/2) 1112 -3/2) l-1/2 3/2) l-1/2 1/2) 1-1/2 -1/2} 1-1/2 -3/2) TJ 1 TJ 2 TJ 3 TJ 4 TJ 5 TJ 6 TJ 7 TJ 8 TJ 2 TJ 9 J TlO J Tll J Tl2 J Tl3 J Tl4 TJ 7 TJ 3 J TlO J Tl5 J Tl6 J Tl7 J TlS J T13 TJ 6 I (1/2 3/21 I (1/2 1/21 (1/2 -1/21 (1/2 -3/21 I TJ J Tll J Tl6 J T19 J T20 J T17 J T12 TJ 5 (-1/2 3./21 \ TJ J T12 J T17 J T20 J T19 J T16 J Tll J T4 (-1/2 1121 \ TJ 6 J T13 J T18 J T17 J T16 J Tl5 J TlO TJ 3 TJ 7 J T14 J Tl3 J T12 J Tll J TlO TJ 9 TJ 2 \ T~ TJ 7 TJ 6 TJ 5 TJ 4 TJ 3 TJ 2 TJ (-1/2 -1121 (-1/2 -3/21 4 5 \ 1 \ti - 20 - On the sides of the matrix we have listed the initial and final h e licity states. To form helicity amplitudes with a definite parity we follow reference 29 and use the relation (II. 8) * states J2 1 (jA.a 11,) ± 1-"-a' -11,)) Thus, we construct normali z ed NN which have parity ± (-l)J . Now, we c~n * form scattering matrices for elastic NN scattering in states of definite parity (we have used Eqs. (II.6) and (II.7) and the definitions in the last matrix. }z (!1/2 3/2) + ]z. <1112 1/2) + Jz <ll/2 -1/2) + ~ <1112 -3/2) + \-1/2 -3/2)) l-1/.2 -1/2)) 1-1/2 1/2)) l-1/2 3/2)) 72<<112 3/2\ + (-1/2 -3/21) TJ + TJ 1 ·. 8 TJ +TJ 2 7 . TJ + TJ 3 6 TJ + TJ 4 5 ~2((1/2 1/2\ + (-1/2 -1/2 \) TJ + TJ 2 7 . J J T9 + Tl4 J J TlO + Tl3 .J J Tll + Tl2 /z<<112 -1/21 + (-1/2 1/21) TJ + TJ 3 6 J J T10 + T13 J J . Tl5 + Tl8 J J Tl6 + Tl7 ~<(1/2. -312 \ + (-1/2 3/2 \) ./ TJ + TJ 4 5 J J Tll + Tl2 J J Tl6 + Tl7 J J Tl9 + T20 1 all of which have parity+ (-1) ]2<1112 3/2) \-1/2 -3/2)) ~ q 112 1/2) - 3 N I-' 1 fI (\1/2 -1/2) - ~ (\1/2 -3 /2) - \-1/2 -1/2)) l-1/2 1/2)) 1-1/2 3/2)) TJ - TJ 3 6 TJ - TJ 4 5 ~((1/2 3/2 I - (-1/2 - 3/2 I) TJ - TJ 1 8 TJ - TJ 2 7 5'2<<112 1/2 1 - (-1/2 -1/21) TJ - TJ 2 7 T9 . J - Tl4 J J TlO - Tl3 TJ 72«112 -1/2 I - (-1/2 1/2 I) TJ - TJ 3 6 J . J TlO - Tl3 J J Tl5 - Tl8 J J Tl6 - Tl7 Jz<(i/2 -3/2\ - (-1/2 3/21) TJ - TJ 4 .5 J J Tll - Tl2 J J Tl6 - Tl7 J J Tl9 - T20 J 11 - T ..J 12' - 2:.2 - all of which have parity - (-l)J. So we verify again that there are ten independent amplitudes required to describe scattering in a state of given J, P, and I. ~'( Now consider N + N ~ N + N scattering. Time'.""reversal invari-· ance will relate this process to the reaction N + N* ~N + N but will not affect our counting of the independent amplitudes in * N + N ~N + N. The channels couple only in the I= 1 state. Hence, the NN system must have L even (P = +) and S = 0 or L odd (P = -) and S = 1 to satisfy the Pauli principle. In an I = 1 state of given J we have the following possibilities: -I< With P = +, IS = O, L = J)NN can couple to the NN states IS= 1, L = J), is= 2, L = J), is= 2, L = J + 2) and, IS= 2, L = J - 2) so four independent amplitudes are needed (assuming J > 1). Note that for even parity all odd J amplitudes will vanish. With P = and J odd, the NN states IS = 1, L J ) , can couple to - INN-/() with IS= 1, L = J), 1s = 2, L = J), js = 2, L = J + 2) and js = 2, L = J - 2). This again requires four amplitudes (for J > with P = - and J even, the NN' states 1s = 1, L 1) or J - 1) and 1s = 1, L = J+ 1) will couple to NN* states with \s = 1, L = J - 1), 1s = 1, L = J+ 1), 1s = 2, L ]s = 2, L = J - 1). J + 1) and For this latter case, there will be eight independent amplitudes (for J Hence, a total of sixteen independent amplitudes is required for a * (or N + Ni< ~ N.+ N) complete characterization of N + N ~ N + N scattering (for J ~ 2). > O). - 23 In terms of helicity amplitudes for the process l(N) + 2(N) ~I: + 4(N) conservation ~f parity implies (cf Eq. (II.6) ~3(N) (II. 9) and we can form the matrix of scattering amplitudes (NN~NN) 1112 1/2) (1/2 3121 \1/2 -1/2) t2 J (1/2 1/21 ts (1/2 -1121 t J (1/2 -3/2 I tl3 (-1/2 3 /21 -t16 (-1/2 1/21 -tl2 (-1/2 -1121 -t (-1/2 -3/21 -t4 t6 J J J -tl5 J -tll J tll t8 J J J J J tl2 J J t15 tl6 J -tl4 -tlO -t J 6 -t J -t 3 t J 4 J 7 -t 1-1/2 -1/2) J J t14 J J t7 J J 8 t3 tlO 9 \-1/2 1 /2) J -tl3 -t 9 -t J 2 or in terms of amplitudes of definite parity we have J J 5 J - 24 - A. For P = + (-l)J = - (-l)J-l 'INN) . (NN''f' ]2<1112 1/2) - l-1/2 -1/2)) ~<1112 -1/2 - l-1/2 1/2)) h-«112 3/2 I + (-1/2 -3/2\) ft-«112 1/2 I + ( - 1/2 -1/21) t6 J - t 7 J 1 J2' ( 1/2 -1/2 +. (-1/2 1/2 \) 1«112 -3/2] + (-1/2 3/2]) and B. for P = - (-l)J = + (-l)J-l ~INN) (NN" i"'- }z ((1/2 3/2 I (-1/2 -3/2 ]) ~2 «112 1/2 1 (-1/2 -1/2 \) }z-«112 -1121 (-1/2 1/2]) ~((112 - 3/2 I . (-1/2 3/2\) ~<1112 1/2) + l-1/2 -i/2)) 5z<1112 -1/2) + l-1/2 1/2)) - 25 - For P = + the only allowed NN state with the proper syrrunetry is Fz <\ 1/2 1/2) - l-1/2 -1/2)) and the amplitudes in the second column of case (A) will all vanish leaving only four independent amplitudes. (With I = 1, P = + we must have SNN 0 and it is easy to see physically that l+l/2 -1/2) and l-1/2 1/2) are S = 1 states.) The- syrrunetrization requirement for identical particles also means that for odd J only Al# A is allowedJO). 2 Thus, for P = -, J odd the first column of case (A) vanishes and we are again left with only four amplitudes as we found earlier by considering allowed Land S values. All of these results will be satisfied by the NN -7NN* OPE amplitudes presented in the next section. ~~ Now, we can briefly consider the coupled channel NN, NN problem. For each parity (and J 2:: 2) we, in general, have a 6 x 6 scattering matrix, which contains the 2 x 2 elastic NN scatter'ing submatrix and 4 x 4 submatrix for elastic NN* scattering. For the cases P +, J even and P = -, J odd (L = J) only one NN state is allowed and the coupled channel scatte~ing matrix reduces to 5 x 5. To give a little more feeling for the states which couple between the two channels we have shown,in Table I, the angular - 26 - momentum states of the NN system which couple to the angular momentum * system for some low lying values of the total states of the NN angular momentum. Table I The NN, NN* Coupled Angular Momentum State s for J < 2 * JP NN NN a+ ls SD 0 0 - 3p 0 3p . 0 0 l+. 1 3 - pl ~ 1 s 1 D2 3 s1 , 3 Dl, SD1 s . -s lp 2+ 2- 3 sl , 3D1 p2) 3F 2 s s2 , p2' s s Fl'. D2' s 3 F2 ' 3 . pl G2' p2' 3D 2 3F 2 As discussed i n the Introduct i on we shall ult i mately be specializing to scattering in the J = 2+ state and thus we in principle will be working with S x S matrices, containing four independent ine lastic (NN -7NN) scattering amplitudes. The actual input (of Born amplitudes) to our dynamical problem will look like - 27 - tJP 5 J 8- 0 JP ts, a0 JP t9,12tJP 13Jl6- where JP (II.10) t.1. J k + - * ~NN*) forces equal since we will b e setting all elastic (NN ~NN , NN to zero. - 28 III. THE OPE AMl'LITUDES In this section we shall calculate the OPE amplitudes for the diagrams of Fig. 1: project out partial wave helicity amplitudes for states of definite parity, discuss the threshold and high-energy behavior of these amplitudes and use the threshold val_ues (the "scattering lengths") to estimate the strength of attraction in different JP states, I The amplitudes will be calculated using Feynman s rules 31) · . This will le ad to the appearance of some "off-mass-shell" pieces for low J's, terms which have no dynamical cuts and would not have appeared in a strict S-matrix derivation of the amplitude. These terms lead to some difficulties (for the low J states in which they appear) and one is not certain exactly how to handle them, There seem to be good arguments both for keeping and ignoring them. They make our c onclusions for the J = 0 and J = 1 states somewhat unreliable. Before we can calculate the Feynman Amplitudes we must discus~ *32) the field theoretic treatment of the spin 3/2N • The wave functions for spin 3/2 particles will be written as a o, 1, 2, 3·, set of four four-component spinors ijrµ) µ = being a tensor index in space time. The 1jr µ the index µ satisfy the equations )~ (p - .M ) ljrµ = 0 y µ tµ = 0 . ·cirt .1) (III.2) The second equation is a subsidiary condition which removes the spin 1/2 part in 1jr • µ Th e se equations imply the Lorent z condition - 29 - p '\jr µ In terms of plane wave µ = 0 •. (III.•,3) solutions of the spin 3/2 equation we have '\jr (x) = t) u (p,A.)e-ip · x µ . µ (IIL4) with the normalization u u µ = - 1. µ and (IIL5) uµ and uµ obey the equations -Jc ,,_ (p - M ) uµ (pJ 0 0 (III. 7) -Jc {i ('{> - M ) µ u µ y µ (III. 6) 0 (III·. 8) = 0 . (III'. 9) The state vectors corresponding to different helicity states for the spin 3/2 particle are uµ (3/2) = Eµ (1) U (:{:II ~ lOa) (1/2) uµ (1/2) =~ Eµ (0) u (1/2) +A Eµ (1) u(-1/2) (IIL lOb) u (-1/2) µ =A (III. lOc) uµ (-3/2) = € Eµ(-1) u(l/2) µ (-1) u(-1/2) +fi Eµ(O) u(-1/2) (III.10d) - 30 - where we are considering the spin 3/2 states a direct product of spin 1 and spin 1/2 states. € µ Definitions and details regarding u(± 1/2) and are found in Appendix B. ~'c We write the rtNN and rtNN couplings (omitting the isotopic spin part) as (III.11) and (III.12) respective ly, (~ =pion t =nucleon field). field, In momentum space these become g u (III.13) Y5 u and ~·( Using this coupling we determine the effective rtNN G, by calcu~ating coupling constant, * decay rat e (into N + rt) and equating it to the N the experimental full width at half-maximum, 1 r = 3rt (~)2 r. We finds) 3 PN (~ + M) * (III.15) M where EN = j PN2 + ~· (III.16a) - 31 - p 2 = [ (M*)2 - (M + µ)2][ (M*)2 N ~~ 2 (M (III.16b) 4(M ) "/( Using M = 1238 MeV, r = 125 MeV, we get. (~) 2 61. 7 /Bev (III.17) or 2 . G = 54.3. Now, we are ready to calculate the OPE a~plitudes (for Fig. 1) ~·( using the nNN and nNN couplings given in (III.13) and (III.14) above and. the conventional rules for Feynman diagrams (as given for example, by Schweber, ref. 31). First, consider the process N + N* ~N + N* · The N + N* ~N + N* OPE Interaction Fig. 4 In momentum space - 32 - - -? -? - -? -? u(p4,"-4) (k - P4)µ uµ(p2,A.2) uv(P3,"-3) (pl - k)V u(pl,A.l) (III.18a) (p3 - pl) = 4i (2n) 4 (~) 2 - µ 2 2 (III.18b) u(A.4) (p4)µ uµ(A.2) uV(A.3) (pl)V u(A.l) (p3 - pl) 2 - µ 2 where we have used momentum conservation at the vertices and Eq. (B8) to simplify (III.18a). (p 3 - pl) (p 3 - p ) 1 2 - µ 2 We also have · 2 •k2 + M2 - 2E E - 2q cos e = M 1 3 + M2 - µ 2 - 2E1E3 (M*2 2q2 - cos 2 2l - 2q2 (A - cos B). (III.19) e) (III.20) Using equations (III.18b), (III.20), (III.10), (Bl4,15,16), and (A7 - 12) we may calculate the Feynman amplitude R for any combination of initial and final helicities . (Using parity conservation and time reversal invariance only 20 amplitudes must be calculated. · See Section I.) We next form partial wave helicity amplitudes lf A. jRJjA. A.)= 3 1 2 2 -1 1 (x,w) -"-4"-3i"-1"-2 R. d~µ (e ) dx · (III. 21) - 33 where x = cos e and with our conventions*) 1 (III.22) where we are temporarily omitting the isotopic spin part of the * amplitude (Nk ojk N) - (N ojt Nt"k ). Finally, after much tedious calculation, we combine the partial wave helicity amplitudes into amplitudes of definite parity according to the prescription given in Section II . We have listed the results in Appendix B. expressions for TJP when J > 3 are listed. amplitudes are not present. Only the For smaller J's some of the (We know from Section I that for J =0 there is only one amplitude for each parity and for J = 1 there are six amplitudes for each parity.) And when J < 3 those partial wave ampli- tudes which are present contain extra terms, so called off-mass-shell . '"'<) terms • It wou ld take us too far afield to launch into a thorough of Section I. We may -i (211:)5 In the expressions (writing t = (p3 t ---= 2 t - m 2 m t 2 + 1 m - 4 2 m t - m 2 t - 2 p ) ) 1 m 2 +t+m, etc. Those terms with no denominators on the right-hand side of the e quations will produce off-mass-shell terms in the partial wave amplitudes. - 34 discussion of such terms but a few tentative remarks can be made on the basis of some examination of their properties in the present case (no details will be given). The on-shell (J 2: 3) amplitudes are found to have the expected threshold behavior q q (L 2(J - 2) + 1 p 2 (J - 1) + 1 p = - (-1) J =(~ - ~ are the lowest orbital momenta which can appear for fixed J ~ 2, P = ±(-1) 3 2 1 and at threshold 5r,'"'"' q L + .) Thus, these on-shell amplitudes, if evaluated for J < 2, will not have the ordinary q quite singular at q = O. 2L + 1 min threshold behavior, and can even be But the off-shell amplitudes do turn out to have the "proper" threshold behavior (where "proper" includes the fact that for q 2 = O, parity and angular momentum conservation may forbid L = 0 in the second order diagram). Another argument in favor of keeping these perturbation theory terms is that they are the result of our fieldtheoretic calculation (i.e., they are present in the Feynman amplitudes) and, in addition, they often seem.to express some physics, at least to second order. On the other hand, we may argue against keeping the off~mass- shell terms for the following reasons: 1. These terms frequently turn out to have quite divergent highenergy behavior and thus must be damped out anyway before used in a - 35 dynamical calculation. 2. Single particle exchanges generally are a poor approximation fo·r 3 . 1 waves especia . 11y wh en t h ere is . a b sorption . t h e 1 ow partia present .3) . 3. S-matrix theory does not seem to contain such terms and in a dynamical calculation we want to use as our input just the singular part of the interaction. 4. * vertex Choosing a different field theoretic coupling for the nNN would result in different off-shell terms appearing, so they are ambiguous if nothing else. Fortunately, in our dynamical calculation we will not be considering a J,P state where such terms are present, but on the basis of the previous arguments one is inclined to favor discarding the offshell terms when they are present and a dynamical dispersion-theoretictype calculation is going to be made. In such a calculation, the proper threshold behavior is usually guaranteed by the choice of amplitude or by making suitable subtractions in the dispersion relations. When we estimate the signs and strengths of forces in the ~'( various J, P, I NN states by calculating the theoretical values with momentum factors divided out we . shal l also give the results for the off-shell amplitudes b~t not take them too seriously. * OPE Before doing this, let us consider the N + N ~N + N amplitudes - 36 - -,,,,_ - (b) (a) * OPE Interaction The N + N ~N + N Fig. 5 As the figure symbolically shows, we must form the difference of the amplitudes with particles 1 and 2 interchanged in order to properly * amplitude. antisymmetrize the NN .~ NN 3 ~~ MM g (~)· y5 u(~,~)} (III.23) where 2~ 2E E - 2pp' cos B 1 3 (III .24a) 2~ 2E E + 2pp' cos B 1 3 (III;24b) - 37 and (p3 - pl) 2 - µ 2 =2pp (z - cos e) 1 (III.25a) (III. 25b) with (II!.26) Using Eqs. (B-21 - 24) we can calculate R for any set of initial and final helicities and then form partial wave helicity amplitudes according to equation (III.21). Finally, we combine these into amplitudes of definite parity following the description in Sec. I. The results are: JP a )J(J + 1) tl,4± = 2/2 (2J + 1) 1 . IQC z ) 7 [J+l J even 0 J odd. JP' tl3,16± = (III.2 7a) 212 P' )<J-l)(J)(J+l)(J+l)~ a 1 Q(z) J-2 + (2J+l) (2J-l) = 0 a JP t9,12±= - ,fb ] . [ (2J+3)(2J+l) {+pl) a:) J odd. 2Q(z) J + (2J+3) (2J-l) J even (III.27b) j_Jp + 121 [ Q(z) _ Q(z) (2J + 1) ( z) QJ+2 J+l J-1 J 1 pp' (III.27c) - 38 r J even = 0 (III.27c) J odd • J even =0 JP a t6,7+ = .[b . J odd. 1 /J(J + 1) pp' [ = 0 (III .2 7d) (·2J + 1) [ Q(z) _ Q(z)J . J+l J-1 J even J odd. (III.27e) - 39 tJP = .::.!!. 6,1.f6" _1_ JJ(J + 1) pp' <2J + 1) + 5Jl ( = 0 · (z)l LQJ+1 - QJ-iJ • - Ct21 p J odd (z) JQJ+l 1,. p { [ ·· · (z) +(J + l~J-1 c-~2 + "2 ~ - "2 BJO} = 0 J odd. =~ 1.- 2.rr p. { (z) ( 1r.(z) JQJ+l + J + 1/<J-1 [ J J even J 2J + 1 - "2 ~ BJl + Q~z) [-a2 + ~2 ~} . J = 0 J even • JP 1 = + -14,15+ a 2p -, p • { r~ (J+l) QJ~~ (III.27e) 2J+ 1 + QJ 2,3- t I, (z) ( J even. tJP = a 2 ,3+ "'Zn tJP - odd (III.27f) j (J - 1) (J + 2) • 3QJ(z) . · J QJ+2 (z) (2J-1) (2J+3) + (2J+l) (2J+3) [<2J-l) (2J+l) + J ~1 - 40 - -Q(z) + Q(z) J-1 J+l [ 2J + 1 + = 0 JP = tl4,15- J J odd. 1 + 2./2 L p' j(J -(J + l)Q(z) J-2 { [ (2 J -l) (2J+ l ) . l)(J + 2) + (2J- l) (2J+3) [-QJ"'-1 (z) + QJ+ l ( z ) + (2J + 1) =0 JP J even + a 1 pp I { JP J odd. 1 (III.27g) . [JQ(z) +(J + -nq(z) . J+l J J-1 2J + 1 · . J[ ( .J even 0 J a J odd J even~ tlO, 11+ = + ,f6' tlO, 11- J QJ+2(z) (2J+ l) (2J+3) l 2 P cx2 _ l\P * M 'El)+ - 41 - 2 5 Jl [1 +3 Z p a2 + + Q}z)[ = 0 alpE4] ~-r M 131 +z(.!.p{3 (t p a2 _ i31p'E1) 2 2 - -:;P E4)]} M M ~'r J even • J odd (III.27h) where (III.28a) al,2 !31,2 2E E + M 1 = 2p' ,2 [1 ±(E + M)(E + M)* 3 4 J [± E31+ M + (E + M)~ (E + M*) 1 and an i spin factor of - 2/3 4 (III.28b) J (III.28c) ;-? has been included*) in a. We have chosen the phases so that our inelastic· amplitudes are real (which will make our Born input real and symmetric) and as we shall see later, the results of our dynamical calculation are independent of the sign of the inelastic amplitudes. The · partial wave amplitudes listed above are valid for all J and we note that off-shell pieces only appear for J * = 0 and 1, and thus are not present in J = 2+, - Tk N) (Nj -* 5jk N). The i spin amplitude is (N - 42 - the state for which we make dynamical calculations*). ,'c There is another NN from our discussion * OPE diagram which we have omitted -7NN iri<) N ,'c An N + N -7 N + N* OPE Diagram which · has been Neglected Fig. 6 The reasons for choosing to calculate the amplitude for the diagram of Fig. 4 but not that of Fig. 6 are principally practical ones with only limited theoretical justification. Probably most important is the ,'( ,'c fact that we have almost no experimental knowledge of the ~N N vertex * ,'( coupling constant using su • though we could estimate an effective ~N N 6 Our difficulty with off-shell pieces may be related to . the pheno* menon discussed by S. Mandelstam, Nuovo Cimento 30, 1113 (1963), of extra terms which are present in perturbation amplitudes but which will not be present if sufficient Reggeization, including that of the external particles, is performed. Both Fig. 6 and Fig. 4 represent pieces, perhaps partially overlapping, of the complete NN~ -7 NN~ interaction. - 43 - On the theoretical side, since Fig. 6 contains no unstable vertices, there will be no force cut which runs into the physical . region*) as there was for Fig. 4, and thus, no opportunity for a Peierls type enhancement (see the Introduction). In any case, in our actual dynamical calculation all elastic forces are omitted so naturally Fig. 6 will also not be included. Properties of the OPE Amplitudes Now, let us go back to the OPE partial wave helicity amplitudes we have calculated and briefly discuss their threshold and highenergy behavior. After a great deal of tedious algebra and using the relation t l tn 2 2 x - 1 -I 1 (III.29) n n=l ~'( we find for the large W behavior of the NN* ~NN partial wave amplitudes the results summarized in the following table, which gives the energy dependence of the leading term for W ~ * 2 2 The cut stops at S = (M* + M) - µ /2. 00 (S = w2). - 44 Table II The Energy Dependence of the Elastic NN* OPE Amplitudes in the Limit W ~ oo (8 = if) I ! J = 1 J =2 J>2 T J 1 Const Const Const Tll T J 2 .f8 .fs l/.f8 Tl2 T J 3 8 8 ,enS/8 Tl3 T J 4 ..f8 l/..f8 Tl4 T J ..fs .f8 Tl5 J = 0 5 J = 0 J J Tl6 T J 7 9fs &f8 11.fs Tl7 T J 8 s 8 .enS/S Tl8 Const 8 ,en8/8 Tl9 sfs 9fs lNs T20 s Const sis sf8 l/.f8 s2 s2 .en8/8 82 82 Const sl'8 ..f8 J's l/.f8 8 £n8/S s 8 Const ,tnS/S J J J J s = O, 1, 2 columns correspond to the amplitudes with off-mass-shell terms included. off-shell pieces. Const J Const In this Table the J s2 J s J TlO s J s s J > 2 J T J 6· T J 9 J = 2 J = 1 For J > 2 there are no (Some of the amplitudes in the J > 2 colunm may be missing a factor of .en S.) As mentioned earlier we note that the low J off-shell ampli- - 45 tudes tend to violate the unitarity limit (TJ .....,. const) very badly while most of the on-shell (J > 2) amplitudes are reasonably behaved for large W. ,.,) Before these badly divergent amplitudes could be used in any dynamical calculation one would have to damp them in some way; e.g., by use of a form factor modification at the vertices (although the expressions usually pr oposed do not seem theoretically justified since they imply the existence of low mass intermediate s tat es which do not exist) or by some Regge-type cutoff. Now, let us e x amine the threshold behavior of the elastic * * amplitudes and from this make some crude estimate of the NN ~NN sign and strength of the forces in different JP states. Using the limit x ~0 --=£..;......!{ _xl). (2.P, + 1) ! ! \ P,+l (III.30) We evaluated the amplitudes of the matrices representing elastic * scattering for definite J p states in the limit q 2 ~ O, keeping NN 2 the l owest order term in q , and then diagonalized these real symmetric matrice s (which are 4 x 4 for J ~ 2). Since we do not usually find hig h J resonances appear i n g i n el eme ntary particle It is in t eresting to n o t e that t h e J > 2 h e licity amp litudes ob e y t he h e u r istic rule TJ ,..._, (1/S) (.Tf0 'A + µ for large S (modulo a possibl e factor of P,nS. The behavi or o f the low J off-shell amplitudes may also be predicted by determining the u d e p e ndence (u = cross ed mome ntum transfer = (p3 - p 1 )2) of the helicity amplitudes before a partial wa v e e x pansion is made. - 46 interactions we only examined the OPE forces in lower J states (J :S_ 4). The results are listed below in Table III, where the momentum 1 JP . P dependence of - T is given in parentheses following the J (in this q 2 calculation the value G = 43 corresponding to rN* = 100 MeV was used). Table III "/~ Eigenvalue s of the Elastic NN OPE Born Amplitudes Evaluated at Threshold for States of Definite J, Parity, and I-Spin JP I Eigenvalues =2 I = 1 JP Eigenvalues I = 2 I = 1 + 4 0off(q ) 1080 - 359 2 0 off(q ) - 170 56.4 2 1+ off(q ) 64.7 .00612 - 105 - 21.6 -.00204 34.9 2 1 off(q ) 152 20.7 - 2.86 - 50. 7 - 6,90 . 954 2+ off(q ) 145 .00300 -.0125 - 24.8 - 48.2 -.00100 .00417 8.27 2 2 off(q ) .00474 -.00158 .000628 -.000209 l .osx 10-7 -3.49x 10-8 26.3 - 79.0 2'. ( 0) on q 8.43. .0659 -.00354 -.0380 - 2 . 81 -.0220 .00118 .0127 2 2 on (q ) .841 - .0354 -.342 - 79.3 · -.280 .0119 .114 26.4 3+ (q4) 213 -.866 - 2. 85 - 659 - 70.9 .288 .951 · 220 3- (q2) 79.7 .56 7 -.00452 -.433 - 26.6 -.189 .00150 .144 4+ (q4) 905 6.14 -.0291 - 5.24 - 302 - 2 .05 .00969 1. 75 4- (q6) 3750 325 - 106 - 7600 - 1250 - 108 35.2 2540 2 The subscripts "on" and "off" refer to on- and off-shell amplitudes. · It is interesting to note that for J = 1+ and 2+ the off-shell ampli- - 47 tudes have no L = 0 terms for q 2 = O. These diagonalized threshold values can b e used as a crude qualitative guide to the attraction and repulsion produced by the OPE potential in a way similar to that in which for a single channel case the Born approximation scattering length tells us the sign and approximate magnitude of the phase shift and hence the sign and strength of the potentia1 34 ), (For this connection to be valid, the potential must be relatively weak, at least too weak to produce a bound state,) In our case where we have more than one elastic channel and thus can have both attractive and repulsive "scattering lengths" p for the same J , the interpretation becomes much more ambiguous and because of the difficulties associated with off-shell amplitµdes the results for low J states are somewhat in doubt, and 2:ff values are quite similar.) . (But note that 2 on With all of these qualifications we may still try to . draw some qualitative conclusions from Table III. ~·( For the coupled NN, NN problem we consider only the I = 1 colunm and look for states with strong attractive forces (with our definitions this corresponds to positive entries in the table) , For I = 1 we find possible attraction in the states 0-, l+, 2-, 3+, 4 (note the alternating parities) and repulsion in o+, 1±, 2+, 3±, 4+. (Attraction in I = 1 corresponds to repulsion in I = 2 and vice versa since the signs are opposite*)) . . In those states with sizable * Very crudely, if all attractions indicated a bound state or resonance we might speculate that we have two series of rotational levels; I= 2: o+, 2+, 4+; I= 1: o-, 2-, 4- . - 48 ~~ attraction, the NN OPE elastic force should probably be included in any multichannel calculation. But in states with (mild) repulsion or only weak attraction we might be justified in ignoring this elastic force. This seems to be the case for the JP = 2+ state in which we are especially interested. Also, some further investigation of the J = 2+, I= 1 amplitudes showed that they remained rather small (i.e., much below their unitarity limit) and repulsive in a region above threshold. As discussed in the introduction we tend to concentrate our attention on values of JP which allow the NN* to be in an orbital L = 0 state .since that is when threshold and cusp effects are most prominent. And we look for states in which either or both the elastic * -7NN* force seems very attractive or the NN -7NN . * amplitudes rise NN quickly from threshold to a value close to their unitarity limit. Hence, we next examined the threshold behavior of the inelastic amplitude(s). We first verified that our amplitudes (including off- shell pieces) satisfied the usu~l threshold momentum dependence 1 ::r;pv- . (L___ ) . (LNN*) . min t JP .w ( p )~ min( p ') (III.3la) which for J > 2 becomes 1 ~ JP t {(pp')J - 1 for P = +(-l)J - 1 ~ • • (III.3lb) p J (p I) J - 2 for p = - ( -1) J - l 1he only states for which the rise of the amplitudes from the inelastic threshold was investigated were J 2+andJ=2-. As is evident from Fig. 7 the four J = 2+ amplitudes rise very quickly to values exceeding the unitarity limit (which is .5 for - 49 - Figure 7 The J = z+ N + N . ~N + N* inelastic OPE amplitudes - 50 I ,.--.. (>O I.ti- -....... ...µ 0 t'l \t\ 0 ,........ UJ > co ......._ ,_ ~ c.!J LIJ w ~ '° z Z" u - 51 - inelastic amplitudes ) while the J = 2 amplitudes (not shown on Figure) rise much more slowly and peak (when they do so) at much higher values of W. The steep growth of the OPE J = 2+ inelastic amplitudes suggests that since we seem to have all the prerequisites for a Ball-Frazer or cusp type enhancement, we study in detail their effect on the elastic * threshold. And many reasons including the fact that the elastic NN* OPE potential seems to provide very little NN -7NN amplitude near the NN attraction in this JP state plus our desire for a coupled channel problem of not unreasonable complexity, and finally, an interest in studying the mechanism of inelastic coupling in its purest form, all caused us to decide to ignore the elastic forces and use as our sole dynamical input the J = 2+ inelastic amplitudes. With this input and the requirements of analyticity and unitarity ( expressed via the multi-channel ND-l dispersion equations) we shall study the coupled NN, NN * J = 2+ channels. Now that we have decided on the dynamical problem to be investigated let us redefine and list the four inelastic Born amplitudes. Al3 1 2+ = + ~ Q(z) tS,8$ 2 pp' + z * [ (- 1 2 p 0:2 + (0:1p'E1 + 131 p E4) + z !31p'E1 * M ) 2 p 0:2 + '"1M:E4)] (III. 32a) 2 ( 1 M 2+ Al4 = + tl' 4- = a 13 10 G3 (z) - Ql 1 (131 + o:l z) (z~ p' (III.32b) - 52 - .J:,. [ ( a 2p 'E1 + l\P) a t2+ Al5 = 9,12- = - -5 (Q3 (z) - Q1 (z)) pp + z . 2+ Al6 = t13,16- = - o;2 pE 4 ) ( o;l p 2 + * * 2 M J (III.32c) M a .{3 o;2 p' (- 2Q (z) Q4 ( z) 2 + 35 21 Qo(z) 15 ). (III-. 32d) While we have the amplitudes in front of us let us discuss some of their properties which we shall need later in writing dispersion relations and setting up the ND For W -) oo -1 . equations. we find the asymptotic behavior 1 Al3 w2 A14 w tn W 1 (III.33) A15 . A16 1 w const. In the limitW-70, all theA / s (J= 3, 4, 5, 6) 1 3 2 (W)- / • This kinematical singularity will be removed by a proper choice of phase space factors. The behavior of the NN -7NN* amplitudes under W -7 - Wis rather interesting. We find (III.34a) A14(W) =+--- ~ - 53 - =+ Al5 (W) [;;' (III.34a) . Al6(W) = - )pp' . i.e., in all cases we have (III.34b) * -)NN* OPE amplitudes and the NN -)NN We also examined some of our NN amplitudes of Ref. 28 under the transformation W--) - W. And, for all of these examples of fermion-fermion scattering, we find (III. 35) = Thus, in contrast to the case of boson-fermion (e.g., ~N) scattering where we have the MacDowell synnnetry = '' for 2-fermion scattering )amplitudes of the ~ parity are related. . 35) *) This relation has been noted by Hara who gives a partial proof * Perhaps one may argue for a general rule by the following heuristic reasoning: Consider what intermediate states are allowed in the direct (S) channel for the different combinations of two-pa rticle boson (B) and fermion (F) states. For BF -)BF scattering only F intermediate states are allowed. Since the parity of F = - parity of F (anti-particle) r eversing the sense of W corresponds to changing the parity. But for BB-?BB, FF-)FF, BB-)FF, andFF-)FF the intermediate state must be a boson which has the ~ parity as the corresponding antiboson . Thus, the same parity amplitudes are related by W --)- W. Hence, our general rule would be that if the intermediate state can b e a fermion then opposite-parity amplitudes are related by changing the sign of W. - .54 - Analytic Structure of the Production Amplitudes Next, we must investigate the analytic structure of the * partial wave amplitudes. NN ---1NN A more general discussion of the singularities of production amplitudes has been given by Cook and Lee 1) . 2) , and Ball, Frazer, and Nauenberg and will not be repeated here. As is well-known, because of the presence of an unstable vertex * N~) our amplitudes will have complex singularities which are (N . contin. . u ation . . an externa 1. mass 36 ) treate d .b y an. ana 1ytic in N~" is most convenient). i< (N) mass. ( t h at o f t h e Let a be the square of the variable external The branch points of our production amplitudes are found by setting the argument of the Q functions, z, equal to ± 1. I (The Q.t s contain .tn ( z + 1 z - 1 ) which has these branch points.) We find that the branch point lying farthest to the right is given by 1 2 .. 2 = 2 (3~ + a - µ ) S+(cr) (III.36) As we increase a the point S+ moves to the right and reaches the elastic threshold S = (2M) 2 at a = o1" + 2µ 2 ). This is the value for which an anomalous threshold develops, since if we give a a small imaginary part and increase cr further the branch point circles around the point (2M) 2 and S+ moves to the left having gone from the second sheet through the physical cut and onto the physical sheet. When a = (M + µ) 2 the branch point is at S+ = M(2M + µ) = S , and as a is increased further (the vertex is unstable for a > (M + µ) 2 ) the branch point moves into the complex - 55 - plane. *2 . With a = (M ) , the complex branch points are given by + i With our choice for the Qt functions the branch . cut lies on the locus of W (or S) corresponding to a cut. in the z plane on the real axis between -1 and +l. We have numerically traced the cut in the W plane and find for our NN ~NN* amplitudes the singularity structure shown in Fig. 8. Because the complex cut crosses (at W = 2.033) the physical region for elastic NN scattering and thus intersects the elastic unitarity c~t, certain difficulties appear which will be discussed later. - 56 - Figure 8 Singularities of the OPE production amplitudes in the complex W plane - 57 - 2.0 RE "'N' ( 13 EV) - 58 - IV. UNITARITY OF HELICITY AMPLITUDES IN THE ISOBAR MODEL The unitarity relations and multi-channel ND -1 method with appropriate extensions for three particle states in the isobar approximation have been discussed at length by many authors. Although the details in these treatments do not always agree and some problems remain, we shall not try to give a thorough presentation of the subject here but simply select the pertinent results we shall need from the work of References 1 and 2, who seem to have considered the problem most carefully. Based on Reference 1, Eqs. and Reference 2, Eqs. (19), (26), and (34), (3.8) and (3.9), we may write down the discontinuity equations for partial wave helicity amplitudes (of definite parity). As is usual when fermions are involved, to avoid certain kinematical branch points we choose to work in the W plane. Then, in terms of ,amplitudes Fij from which the kinematical singularities have been removed (see the next section) the unitarity relations (in the isobar approximation) may be expressed for W 2: 2M by FJP(W )] 22 FJP(W ) + FJP(W 22 - F JP ( w_,a I 32 33 - +'a,a I - 59 - FJP (W J CJ J CJ I 33 - r 1 JP( .,) JP(W 2i I F33 w+,cr,cr - F33 _,cr,a ')] . L (IV.l) where w± ::: w ± i €, e is the usual step function (equal to one where the argument is positive and zero otherwise) and a (CJ I) is the energy * square of the 1tN system (the N ) in its center-of-mass. The subscripts 2 and 3 refer to the two- and three-particle channels, NN and NNn respectively, and thus for our case where each "channel" may actually consist of several channels (since for J fixed several L,S states are possible) a summation is implied. (We recall for J == 2 * states). one NN state coupled to four possible NN + there is These equations are identical in form to the partial wave unitarity relations for . stable two-particle channels. entirely contained in * are The properties of the unstable N p (W) which is a generalized phase space 3 integral (the integration being over er, the mass of the unstable particle). Explicit expressions for p (W) and p (W) are given in the 3 2 next section. Since in our approximation we have (pseudo) two- particle channels)the solution to the discontinuity equations may be expressed in terms of the usual multi-channel ND-l method 37 ) without the full generali ty to multiparticle reactions as derived by - 60 - Blankenbecler 38) . Thus writing (IV.2) F .. l.J the constraint imposed by the unitarity conditions becomes (IV .3) where W. is the threshold energy for channel i. l. If F has force cuts with discontinuities disc F, then (in terms · of matrices) (IV.4) disc N = (disc F) D in the region of these cuts. With the matrix B(W) defined as the dispersion integral over the force cuts of the true F J disc F(W')dW' B(W) = _l_ 2ni W' - W force cuts The basic ND -1 N(W} c B(W) + equations are ~ JdW' [B(W') u D(W) = I (IV.5) ofp w- w p(W')N(W'.l W' - W CW') N cw') dW' (IV.6) (W 1 - W)(W' - W0) (IV. 7) where the contour U is the entire real axis except for \Wj < Wi (where - n u the integration for D . . begins depends on the subscript i) and I l. J is the unit matrix. We have assumed that N satisfies an unsubtracted dispersion relation while D requires one subtraction . (We have put the subtraction point at the same place for all channels although this - 61 - is not necessary.) Because the force cut (in the OPE amplitudes) intersects the unitarity cut (see Fig. 21) the integration contour U must be deformed to avoid the protruding singularities. As a result of the anomalous thr eshold and complex singularities additional . will appear in the N and D equations. integral~ This problem is discussed quite thoroughly in Referertces 1 and 2 (by performing an analytic continuation in the N* mass). Although, (for reasons to be given in Section VI) we ignore the presence of the anomalous singularity in our determinental calculations, we shall make some further remarks on the subject in Appendix C. For a more complete discussion of the ND -1 method, including its properties and some examples of how the calculations are performed, the r ea der is referred to references 37 and 34, and other references which are given there. - 62 V. REMOVAL OF KINEMATICAL SINGULARITIES AND CONTINUATION BELOW THE INELASTIC THRESHOLD We must now specify in detail the phase space factors p (W) 2 and p (W). 3 These factors serve several purposes and must meet certain . ot h er requirements 39) • Principally they are chosen to eliminate · "kinematical" singularities (which we have assumed are not contained I II in F) and to guarantee the 'correct threshold behavior **) • partial wave threshold behavior (5t ~ q 2,t+l The . for spinless elastic scattering) will not result from unitarity and analyticity requirements, but does follow from the Mandelstam representation. (In non- relativistic scattering the ordinary threshold behavior can be derived 40) if the potential is suitably restricted. With our approximation for the left hand cut (the "potential") it would be purely accidental if the correct threshold behavior resulted and the amplitude vanished at threshold with the proper momentum dependence. Hence, we divide out the factors of momentum so our new amplitude will just approach a constant at threshold. We have seen that the inelastic amplitudes A (W) (J = 3,4,5,6) 13 2 behave like ...,_ J;;i p at the thresholds (p ~ O, p' ~ 0). And because of their W ~ - W behavior they have the same momentum dependence at the negative W thresholds. Using this and recalling the behavior at W = 0 we choose phase space factors ** There is no a priori guarantee that a suitable phase space factor exists. - 63 - . 5 . (W - 2M)5/2 312 32W(W + 2M) = W(W + 2M) 4 [w2 - (M + M}2]1/2[if - {M* £_ = ,'( I P3 (W) (V .1) w M/l 1/2 2 2W where we have so far assumed the N* is stable. (V.2) We now define (V,3) (S .. - o .. )/2i). (where t.. l.J l.J l.J Because of a desire to have Pz (W) w~ 00 const and p (W) ----+const so 3 w that our dispersion integrals will be convergent , we were forced to include a factor (W + 2M) -2 in Pz which introduces a kinematical zero at the beginning of the left hand unitarity cut. The placement of this zero is rather arbitrary except for the feeling that it should be reasonably distant from the region of physical .i nterest "le (W ....... M+M). Since we decided to ignore the integrations over the negative W unitarity cut (see below) it does not matter that we have disturbed the threshold behavior at the W = - 2M threshold. Aside from this, F . . is expected to have no kinematical zeros and only l.J dynamical singularities throughout the W plane and thus satisfies the dispersion relations given in the previous section, The Born amplitudes which constitute the dynamical input to our coupled channel problem are thus given by (W) J = 3 4 5 6 .B Jl ' ' ) . (V .4) Although we shall shortly discuss the appropriate modification of p 3 - 64 - to include the width of the N* , we note that it is correct when forming the BlJ to use the p 3 given above, assuming a stable N* , since the A J were calculated with this assumption. 1 . 1 approximation . . 41 > wh ere we rep 1 ace I n t h e d eterminenta N . . by the corresponding Born amplitude, those D integrals beginni~g ij at the NN threshold (W ; 2M) require the evaluation of BlJ in the region below the inelastic threshold (W = M + M*). formed are The BlJ we have . at the NN* threshold but they must be smooth .functions rewritten before one can calculate their values for . W < M + M* . Using the relations p'~ + i Ip' I for p •2 < 0 z~ z' -= i 2~ - µ 2 - 2E E i 2pl p' I 1 3 (V.5) and thus, 1 1) -7 i t:an-1 ( tn ( zz + - 1 2 2 Q2 ~ i I [2 (- 3 z '2 we have, for W < M + M* 2 2z' r2 _ ) 1 (V.6) - 65 - • - z'2 =- (V~ 7a) a' fl (V. 7b) = - (V. 7c) , = - (V. 7d) where a prime on any quantity refers to the unprimed quantity evaluated with p' replaced by I~' I. (a, a,~ are defined in Eq. (III . 28)). In Fig. 9 we have graphed the amplitudes B J in the region 1 near the NN* threshold. In Figs. lO(a) and lO(b) we display the continued amplitudes BlJ over the whole region between the elastic and inelastic thresholds (2M ".S, W ".S, M + M*). The rather troubling behavior near W = 2.033 is a result of the fact that the branch cut joining the complex - 66 - Figure 9 The NN ~NN* input Born amplitudes - 67 - 15"00 l&OO w 0 :::> 1.J 0- I: <( s-oo 2..10 2.. IS' 2.20 W (BE:V) - 68 - Figure 10 The NN ~NN* input Born amplitudes in the region between the elastic and inelastic thresholds (a) B (W) and B (W) (b) B (W) and B (W) 14 13 15 16 - 69a - 3 ,0 UI Q '::> 1- ..J 0.. I: <. 1.0 W (BEV) (a) - 69b - 3.0 ~ uJ 0 ::> 1- :J n. r <t 1.0 (b) . · -:- ·---~ - · - -- - - - ~ -- · ------------- - 70 - singularities crosses the real W axis at this point. This strong peaking of the Born amplitudes is reflected in a corresponding peaking in the elastic NN amplitudes which we calculate via the determinental approximation. But for several reasons we felt justified in ignoring such peaks. W = 2.033 corresponds to Tlab~ 330 MeV and thus lies essentially in the purely elastic scattering region. But in our calculation we center our attention on the region close to the inelastic threshold (Tlab~ 646 MeV) and in fact believe the results are only valid for ene.r gies not too far from W = 2 .176 . (When the dynamical calculations were repeated using B J's cut flat for 1 W :5_ 2.176; i.e., BlJ (W < 2.176) = B J (2.176), the results near the 1 inelastic threshold changed very little.) We also believe that a more exact treatment of the NN~ state and a proper continuation of the dynamical equations past the anomalous singularities would tend to eliminate any spurious behavior. * by modifying Now, let us include the instability of the N . 42) p (W) to take into account the width of the resonant N~ state • 3 To do this we replace p' in p (W) by 3 (W - M)2 ~ f c{. . . do 2_;....:;.~-..cr_-_...(w___-_M=)._2.......] )__1_1_2 [;;....;;cr'----_(._W_+_M..._) (V .8) (M + µ)2 * (r/2) 2M where r(cr) = ~~ (~) 2 3 q (EN+ M) F = the width of the N* - 71 - 2 2 2 = [ a - (M + µ) Ha - (M - µ) ] q 4a and a is again the energy square of the Nn state in the Nrc center of mass. To verify this replacement and understand how it is derived note that in the limit r ~o r ~ O .j~p' (s,a)o(a - M*2 )da = p' (V. 9) (for stable N*). Thus, we have just a two-particle momentum distribution weighted by a "finite · width delta-function"*). The limits on the integral for p~ are the smallest and largest "mass" square of an Nrc state which is part of a total NNrc state with c.m. energy W. Following the method used by Nauenberg and Paisll) and Bazll) we can derive an approximate expression for p' the case of u very small width (width/mass of resonance *) << 1). Heuristically, to understand the finite width a-function, one can note that since 1/ (ill - illo - i€) = P/ (m - ill ) + irco(m - ill ) as 0 0 € ~o, then it is r easonable to say - i y/2) ~re o (ill - ru ) or o (ru - ru ) ~ (y/2rc) I 0 0 + y 2 /4] for y finite. We also used the relations Im 1/ (ill - ru [ (ru - ru~) 2 0 5(a - M 2 ) = l* (o(Jcr - M*) + o(./Ci'+ M*)] with J(i'+ M* z 2M 2M*. - 72 - With a very narrow width nearly all the contribution to the integral . *2 comes from a very small region around cr = M • We thus may extend the limits so that the integration runs along the whole Re cr axis and then we can perform a contour integral by closing the contour in the lower-half cr plane enclosing the pole at cr = M*2 - i M* r, where r is *2 the constant obtained by evaluating r (cr) for cr = M . Although the integral is formally divergent it is to be interpreted in the sense that only the contribution pf the N* peak is to be included. Performing the contour integral using Cauchy's Residue Theorem and evaluating for W ~ M* + M we find ic ~ Pu" = ~p •2 + i. M*Mr M +M J 1/2 · . (V .10) The real part of p" gives an approximate description of three-body u phase space when two of the final three particles form a narrow width resonance and we ar e not too far from the resonanc e energy. The imaginary part . of p" results from that part of the extended integration u (W - M) 2 < cr < (W + M) 2 corresponding to Ja' - M < W < /CT+ M (below the inelastic threshold) and is a useful expression when some type of effective range (or K matrix) approximation is being made. However, for widths much larger than a few MeV graphical comparison shows that p 11 i s not a very good representation of the true u p' especia lly if one is interested in the region close to the NN* u threshold. In general, p~ is larger and flatter than p~; while e ven for rather large widths p~ ri ses steeply upwards i n a short range around the inelastic threshold. - 73 - At larger energies (W 2: 2.5 BeV) all expressions for p', very similar numerical values. p~, p~ give For this reason, and because of the computing time (and hence expense) of doing an integral for each different value of W we used p~ in our expressions for p (w) when 3 they occurred in integrals (viz. for Re D.. ), but used p' when no u 1] integrals were involved (e.g., Im D. . ). . 1J I In Fig. 11 we have graphed p (W) 3 ,'r values of the N width. Pu w for several different This shows clearly that the broader the width the slower and flatter the rise of p . 3 As we shall see later this is reflected in the shape and height of the peaks produced by the inelastic amplitudes. - 74 - Figure 11 * phase space factor p 3 , for different values of the N* width The NN 0 - 75 - 0 "' 0 - 76 - VI. DETAILS OF THE CALCULATION AND GENERAL PROPERTIES OF THE SOLUTIONS In this section we shall describe the details of our approximate calculation. The numerical results and conclusions are given in the next section. Rather than solve the set of full ND-l equations we decided to employ the lowest order determinental method 37 ) in which the integral ·equation for N, (IV.6), is simply replaced by the approximation N(W) = B(W) and then D is calculated from N by the dispersion relation Eq. (IV.7). Of course, in our calculation, as in any calculation of finite size, B(W) is approximated by including only that part of the potential due to the e xchange of certain particles, usually the lightest mass ones (corresponding to the longest range part of the potential) or exchanges which have features such as form or strength of coupling, or sing ularitie s , which are of speci al interest . discussed earlier , we chose to keep only the OPE force. As Primarily, we used the determinental method because we did not want our calculation to become exc essive ly involved and costly(= much computer time). But, in fact , considering the other approximations made in the problem (e.g., for B), it is not clear that the full ND -1 calculation would nec essarily b e superior or the results more "accurate", or at least b e that much better to justify the greatly increased complexity of the me thod. In problems which have been solved both by the ND -1 and det erminental methods the gen e ral features of the r e sults usua lly are . · 1 ar 43) • very simi In a sense it is misleading to just call the determine ntal me thod an approx imation to the ND -1 e quatione . Ra t h er - 77 - it should be thought of as a different method, with special properties of its own, which may or may not be a useful or accurate method for any particular problem. Several of its features are undesirable, especially the dependence of results on choice of subtraction point and the lack of symmetry (F . . # F . . ) of the solutions. 1J 1J The ND-l method does not have these troubles and in that sense is superior. We thus have the following set of equations to solve (note that we are changing some of our earlier conventions) (VI. l) F. . 1J 00 D. . j :re .1J ~ p. B . . dW 1 I 1] - w0 ) J (VI.2) - 00 (W - W0 ) :re p.1 B1] . . dW' 1 (W - W)(W 1 - w ) 0 -w.1 where W. is the threshold energy for channel i. 1 With our approximations (neglecting all but the inelastic amplitude s) Bij = 0 except for B13 = B31 , B14 = B , B15 = B51 and 41 B = B , and these are given by the OPE amplitude s of Se ction V. 61 16 Writing these as BlJ' with J = 3,4,5,6 only (subscript 1 refers to "Ir NN state and subscript J = 3,4,5,6 to NN state), we have = - (VI.3) - 78 - !"'°+ J-r:C }-----P3__;;;..;;__BlJd_w'- DJl - - (w : WO){ . ~~ -~\ (W' - W - ie) (W' - W ) 0 (VI.4) where p 2 and p 3 are the phase space factors for the (two-particle)· NN and (three-particle) NN "I< channels, respectively, discussed in Section IV. Allowing for the instability of the N* , WNN"'~ = 2M + µ . The only other non-zero D.. are those for which i and = j l.J then D.. = D .. l.J 5 .. = 1. Thus, our matrix D may be represented in l. l. l.l. the form 1 D = Dl4 DlS Dl6 D31 D13 1 0 0 0 D41 0 1 0 0 DSl 0 0 1 0 D61 0 0 0 1 (VI.5) In writing the equations for DlJ and DJl we have ignored any possible de f ormations o f the contour of int egration along the unitarity cut to avoid intersecting the force cut crossing the physical region . (The intersection occurs between WNN and WNN* and thus only D J (and D ) 1 11 need be modified.) We shall try to discuss this further in Appendix C but we can remark here that because we are using the determinental method our integrands contain BlJ and thus we are - 79 - including some of the effects of the complex singularities. And, of course, with a proper treatment of the three-particle NNn state such anomalous cuts may not even be present 44) • Before we further discuss the calculation of DlJ and DJl let us look at the form of the solutions for F ..• 1.J The elastic NN sea ttering amplitude (VI.6) sum over J = 3,4,5,6. Using the relation (D -1 cof DlJ ) Jl det D (VI. 7) and the matrix for D, Eq. (VI.5), given above, we find (VI.8) with a sum over J implied, The off-diagonal inelastic amplitudes turn out to be BlJ = det D (VI. 9) and (VI.10) In general, F 1 J ~ FJl' but it is interesting to note that if we had only one ine lastic channel we would have the symmetric solutions - 80 - As will be mentioned in the next section, our numerical results were not very symmetric. -1 we had used the full ND. Of course, this could have been avoided if equations .. Several further approximations were made in calculating . DlJ and DJ • 1 - oo . The integral over the left-hand unitarity cut, j' > -W. was neglected because for Win the region of physical interest,i the denominator (W' - W) remains almost constant and is never near zero -oo (as it can be for the integral on the right cut); thus the integral~ -Wi will show little variation with W and is considerably smaller than the similar integral over the right-hand cut. So we are essentially neglecting a probably small constant piece in the integrals for DlJ and DJl' And of course we are neglecting much other singularity structure (e.g., due to other particle exchanges) so the neglect of this cut which is rather distant from the physical region is riot a further drastic approximation. The choice of subtraction point, w0 , is a difficult matter. Usually in the determinental method (recall with the full ND-l that the results are independent of w ) the subtraction point is chosen to 0 41 . . d egree o f crossing . . insure some approximate symmetry ) an d t h us w is 0 placed near or at the right-hand edge of the force cut (near the place of maximum discontinuity) due to the exchanged particle. subtraction point, W = w , we have D .. 0 iJ At the = 5 .. and hence F .. = B ..• iJ iJ iJ We expect the Born approximation to be a fair representation of the true amplitude over that part of the force cut where the diagram being considered provides the only or major discontinuity. In our · case, - 81 - · where the right-most part of the OPE cut lies in the complex plane and intersects the physical region the selection of w is not so easy. 0 The decision was largely made by considering the singularity structure of perhaps the next most important part of the pot·ential which we have omitted, that due to p~exchange (Fig. 12), N N N Rho Meson Exchange Interaction in N + N ~N + N* Fig. 12 where we assume the p is a stable particle with mass = 763 MeV. Now * no vertice.s a re unstable (M<M+m) and thus, no complex singularities p are present. The partial wave amplitudes have a force cut on the right half plane which runs from s_ to s+ where 4~ J 1/2 (VI.11) m p In the W plane the right edge of this cut lies at W ':: 1.84 BeV. - 82 - Since we are omitting this part of the potential, we decided to place our subtraction point between W = 1.84 and W = 1.88, the beginning of the physical region for elastic NN scattering. So we In our numerical work w 0 chose w = 1.86 as a reasonable location. 0 was varied between w = 1.60 and w = 1.86 and we found that the results 0 0 (viz., location and height of peaks) were rather sensitive to the value of w . 0 This seems to be one of the most undesirable features of the determinental method. By making one subtraction in our dispersion relations for D all integrals are convergent since in the worst case pB as W ~ oo. ,.._, 16 const. Nevertheless, the calculation was made with a (variable) cutoff on all integrals. One may ask why use a cutoff .o r make subtractions when they are not needed? Because we 'know that we have omitted much physics from our problem - e.g., other particle exchanges, other channels, possible Reggeistic damping of amplitudes - and by having some parameters which can be varied we can get some feeling for how our solutions depend on the physics we have left out. Using a cutoff, for example, gives us a way to study the possible effect higher energy channels and modified high energy behavior of amplitudes would have on our results. Let us now rewrite the expressions for n 13 and n 31 separating out the real and imaginary parts. J wc = - ~ w R I NN _ DlJ + i DlJ P2 BlJ dW' _----=.;;___ - i (W' - W) (W' - W ) 0 (VI.12) - 83 - w ·C =- (W - w0 ) :rr P j' p 3 BlJ dW' _(_w_'- __w_)__(__w_•___ w__) - i 0 WNN:n: = D~l + i D~l , (VI.13) where P signifies that the principal value is taken when W lies in the range of integration and W is the cutoff energy. c I p u R In the integrals for DJl~P3 Wwas approximated by II Pu since the time for numerical integration w:> uld have been, greatly Re w increased if we had to calculate p' (which involves an integration) u The value of r used in p" was for each mesh point in the Simpson sum. u chosen in each case so that Re p" resembled as closely as possible the u actual p'. u The e~ror made by this approximation is negligible since the shape of the (cusp) peak is determined primarily by the p 1 I DJl and here the true p 3 = ~ -W was used. 3 in Numerically, the principal value integrals were performed by subtracting out the singular point pfp(W') B(W') dW' (W' - W) (W' = p - w0 ) , J[ p (W I ) B (W I ) - p (W) B (W) ] dW I . (W' - W)(W' - + p(W)B(W) pf + w0 ) dW' (W' - W)(W' - w ) 0 ' where the second integral on the RHS can be done analytically. (VI.14) To do the numerical integration, Simpson's rule was used, after a change of variable was made, W' = wthresh y Increased accuracy was achieved by using the Aitken convergence procedure 45 ) in which Simpson sums for 4n, 2n, and n intervals are - 84 - combined according to tne formula SUM(4n) - j [ SUM(4n) - SUM(2n)] 2 } l (SUM(4n) + SUM(n) - 2 SUM(2n)J . = SUM General Properties of the Solutions Let us rewrite our solution for the elastic NN J = 2+ amplitude and note some of its properties. = "e ioNN sin 5 11 NN = We have - p2 BlJ DJl (VI.15a) 1 - DlJ DJl (VI.15b) - i (a sum over J is always implied). If we are above the threshold for inelastic scattering we must remember that 5NN' in general, will be complex and then we have 2i5R P2 Fll = -25 where 11 = e I ( e2i:i- 1) = (n e 2i -1) (VI.16) and 5 = oR + i o . 1 One of the first things to note is that our result for F independent of the sign of the inelastic amplitudes. 11 is Whether positive or negative the effect of the inelastic amplitudes is attractive in the elastic channel. are unimportant. Even the relative signs between the different B J's 1 - 85 - If there were a resonance in the elastic scattering amplitude we would have near the resonance energy (VI.17) where r is the total width (at half-maximum) of the supposed resonance ~'( (not to be confused with the width of the N ) and r . is the fiartl.al 1 width for decay of the resonance into channel 1. <r =I r., 1. where the sum is over all accessible channels). i Defining }~ i(D~J D~ 1 ) - i(D~J D~l (VI.18) we can write = (VI.19a) or d 2 + d 2 R (VI.19b) I Equation (VI.19a) is useful for comparison with the form of Eq. (VI.17) ·a nd (VI.19b) is convenient if we wish to examine the real and imaginary parts of p 2 F11 • Compar ing Eqs. (VI.19a) and (VI . 17) we see that at resonance - 86 - (VI.20a) or (VI.20b) and thus using (VI . 20) (1 +· n) 2 (VI.21) In terms of the elements of D . . l.J (VI.22) (VI.23) Above the threshold for inelastic scattering ~ < 1 , and so 1 + n 2 inelastic scattering will cause the elastic amplitude to decrease above the inelastic threshold. For energies below or close to the ine lastic threshold - 87 - Now, the resonance condition is Re det D = d R = 0 R R 1 - DlJ DJl = det (Re D) (VI.25) and ·the unitarity limit is (VI.26) Clearly, (VI.22) and (VI.23) reduce to (VI.27) 1 (VI.28) as they should for energies below the inelastic threshold. For reasonably narrow N* widths p 3 is small for W < WNN* = * (see Fig. 11) and hence any resonances below the inelastic = M+ M threshold are essentially given by Eq. (VI.25). R R ~'( Since the product DlJ DJl is positive, increasing the N N~ coupling constant, G, will bring us closer to satisfying the resonance condition and thus increase the height of any peaks (or cusps) present. R The factor DJl ' defined by the principal valve integral in Eq. (VI.13), has a cusp (at the NN* threshold) whose shape is controlled by the phase space factor p • 3 p 3 The cusp is sharpest and most pronounced when rises most steeply and quickly, and this occurs when the· width is narrowest and the orbital angula r momentum of the NN* state is lowest. (In our case, for J momenta or broad widths p 3 = 2+ , LNN* = O.) For higher angular rises more slowly and thus shifts the - 88 - effects of the integral to higher energies, where its value is reduced. From the resonan':e condition, Eq. (VI.25), it is also easy to note that the effect of each inelastic channel (J = 3,4,5,6) is additive - each one adds to the attraction and increases the likelihood o f a resonance, We can briefly observe a few other general properties of our solution. From Eq. (VI.15) we s e e that for very large coupling constant the 1 in d e t D may be neglected and Pz F magnitude of G with jp for all J. 2 F 11 becomes independent of the R 11 j near its unitarity limit if jD Jj 1 The inelastic amplitude (see Eq. (VI.9)) FJl -70 like 1 G as G -7 oo • When such strong coupling is present there is a de- coupling between the NN and NN* channels and the transition between the two- and three-particle state s will not be a permanent (asymptotic) one. However , we must be a little cautious in drawing simple general conclusions about the lar ge G l i mit because it is well-known from the study of several soluble models 13 12 ' ) that ghost poles will appear for G sufficiently large . In the weak coupling limit: G 2 G and F11 -7 - BlJ DJl oe G -7 0. -7 0, and both F lJ and FJl -7 - 89 - VII. RESULTS AND CONCLUSIONS Our numerical results are sununarized in Figures 13 - 20. Usually what has been plotted is the square of the elastic NN scattering amplitude, \p 2 F 2 11 j, versus c.m. energy so that the unitarity limit (below the inelastic threshold) for the elastic channel is 1. In Figures 13, 14, 15, we show the dependence of \p 2 F 11 \ 2 on the N*N~ coupling constant for three different values of the N* width r = 1, 25, and 125 MeV. to the coupling constant G. The quantity A is directly proportional For A ~ 1 the curves become rather flat; we seem by then to have saturated the elastic amplitude with inelastic forces. The peaking seems to move to somewhat lower energies and there is a possibility "ghosts" have appeared in our solution (see further comments later in this section). From these figures we note the general features that increasing the coupling constant causes cusps to grow and become more prominent until they become rather flat, and near the unitarity limit below the inelastic threshold but fall off rather sharply above that point. For small widths the peaks display a definite asymmetry as .the lower energy side grows to the unitarity limit with increasing coupling constant but the high energy side approximately maintains its shape. As the width of the N* becomes larger the cusps become less distinct - they are "woolier" (more rounded). Also now the loss of synunetric shape with increasing coupling constant, though still present, is less noticeable. In comparing these figures one should remember that the width of the N* and the - 90 - Figure 13 The elastic NN scattering amplitude squared with an N* width of 1 MeV for different values of the nNN* coupling constant • w0 = 1,86 BeV and Wc = 500 BeV For these curves r eo o >> w UJ U) 00 > w '9 <> --• lJ\ - 91 - . N 11 < 0 0 N - 92 - Figure 14 The elastic NN scattering amplitude squared with an N* width of 25 MeV for different values of the ~NN* coupling constant. = 1.86 and Wc = 500 .. For these curves 0 >>> w UJ UJ ~ ClO 00 ln ~ t; N -: II \J) II L j~ - 93 - . \\ \I\ <. . II t"\ <.. 0 C'i 0 - 94 - Figure 15 The elastic NN scattering amplitude squared with an N* width of 125 MeV for different values of the ~NN* coupling constant. = 1.86 and W = 500 1 c For these curves 0 >>> llJ LLJ LU '<::) 0 l:: ca ct) ~ II II - lrl '9 \.(\ QO II - L :i-; . 00 - 95 - a l..r\ c-4 ('ol 0 <N ('-< 3 - 96 - • Figure 16 The elastic NN scattering amplitude squared with input Born amplitudes assumed constant (equal to their value at the NN* threshold) for enert gies below the inelastic threshold. For these curves r(N*) = 1 MeV, w0 = 1.86 BeV, Wc = 500 BeV. Compare with Figure 13. > > uJ lJJ > QO 00 Ul -.g 0 :E' ()0 0 II _..:. lJ\ II II L._ '30 -3 -.J> r--. II N ~ h ~ ci5 '9 II ,,...... N !:; h " ~ CD - 97 - 0 II> c-i ('4 "' 0 N - 98 - Figure 17 The elastic NN scattering amplitude squared with an N* width of 1 Mev and coupling constant A = .5 for different values of the subtraction point. For these curves W = 500. c 0 > w 0 co 0 > \.I) II ._, \.I) l: If 3 UJ II <t: L C><> _ ... ~ w= ....s> - 99 - 0 ("( Ln . \'l 0 ("'( "' 0 3 - 100 - Figure 18 The elastic NN scattering amplitude squared with an N* width of 1 MeV and coupling constant A= .7 for different values of the subtraction point. For these curves W c = 500. 0 > UJ ~ II I". II L~ II j:o > w ~ Li) g II ; . ....$ - 101 - . N 0 Ln C"' ~ 0 c--4. C"' \.n N 0 - 102 - Figure 19 * width of 125 MeV The elastic NN scattering amplitude squared with an N and coupling constant A point. .7 for different values of the s ubtraction For these curves W = 500. c 0 > UJ l: l.n II II "' "'• t:_ > w ~ 0 Vi 0 II <C 3 . '9 - 103 - . N N Lr\ C"i 0 N ("'( - 0 ~ 0 . - 104 - Figure 20 The elastic NN scattering ampli tude squared with an N* width of 1 MeV for different values of the cutoff energy. and A= .7. For these curves w 0 1.86 q UJ > ta > ....s 00 II 0 ct: U1 l: . r-II H L ~ .. ~ . "1..1!' - ..., - 105 - 0 Ln ~ N 0 ('<( ~ 0 N \J\ ~ 0 - 106 - effective N*Nrr coupling are not independent quantities. Eq. (III.15) we see that r oe (G) 2 From so that it is not perfectly proper to compare curves corresponding to different widths . but the same A(G) . Using the experimental width r = 125, the effective G derived from (III.15) corresponds to A~ 1.7. For a smaller width A must be reduced correspondingly. In Section V we discussed the behavior of the Born amplitude near W = 2.033 where the anomalous cut crosses the physical region. This does produce some erratic points in the elastic scattering ampl i tude near this energy which we have ignored for the reasons given in Se ction V. But, in. order to get some quantitative idea of the size of the effects, we recalculated the elastic scattering amplitude using as input Born amplitudes which are constant for W < M + M* ; i.e., the B 's are cut ~lat below the inelastic threshold with 13 B (W < 2.176) 13 = B1 J(W = 2.176). For large widths most of the differe nces are hidden because the peaks are so broad and flat. But in Fig. 16 we show the results for the r = 1 MeV case. Comparing with Fig. 13 we see that the curves are rather similar in the neighborhood of the NN* threshold for smaller coupling constants,but as A incre ases the corresponding peaks for the flat B J input are slightly 1 higher , more symmetrical,and occur at lower ene rgies. For the calculations represented by Figs. 13 - 16, the subtraction point is at w = 1.86 BeV and the cutoff energy is W = 500 BeV. 0 c. This value for W gives results essentially identical to those for c W c ~ oo but for numerical reasons we could not, in practice, use - 107 higher values for W • c Figures 17, 18, and 19 show the sensitivity of ' our results to the choice of subtraction point. Again for broad widths (Fig. 19) the effect is not so noticeable but for very narrow N* widths (r = 1 MeV) the elastic scattering amplitude is clearly fairly sensitive to a variation in subtra~tion point. In general, the effect of reducing w0 seems to be roughly equivalent to increasing the magnitude of the coupling constant while keeping w constant; for example, the curves 0 for w = 1.60, A = .7 and w = 1.86, A= 1.0 are rather similar. 0 0 Reducing WO tends also to move the location of the peaks to slightly lower energies (see Fig. 17). As discussed in Section VI we have no very convincing argument for the placement of w . 0 Recall that we did however decide a priori that 1.84 S w S 1.88 was perhaps the 0 most reasonable region. One feature of our results seems also to support that choice; viz., with w = 1.86 the cusps in Figs. 17 and 18 occur at the NN* 0 . threshold and not at lower energies as they do for w = 1.60 0 *) . But again we must .admit that the rather strong dependence of the solutions on the value of w is a fairly severe weakness of the 0 determinental method. In Fig. 20 we illustrate the fact that our results are essentially independent of the cutoff, This is no doubt true because 1 of the rapid convergence of the integrals for D~J and D~ • quite low cutoffs, e .g., W c Only for S 4, do we start to see any change in the since lowering Wo seems to be equivalent to increasing G * However, it is not surprising that peaks move to lower energies. - 108 solutions and then just a small reduction in the height of the peaks. We can probably conclude from this that modifying the high energy behavior of the input amplitudes (for example, a la Reggeism) would have very little effect on our results. The inelastic amplitudes are too numerous to graph, but we can make a few general observations (based mainly on the r case). With more than one inelastic amplitude as input)FJl # FlJ' i.e., the determinental solution is not synunetric. (It is most synunetric for small values of the coupling constant.) is that W~ = 1 MeV j.J p 2 p 3 What we find FJl I reach their maximum value ($ .4) for A R:J .5 at 2.20 BeV and decrease for higher energies or larger values of "le the N Nn coupling constant. They are generally well below the r- 'le unitarity limit (\~p p 2 3 FJll < .5) ·and show no peaking at the NN 2 3 F 1 J I's In contrast we find that the j~p p threshold. seem to steadily increase with increasing W (at least up to W = 2.32 BeV) or increasing coupling constant, and do eventually violate the unitarity limit (for A ;;:; .5) although they also do not display any peaking behavior. Based on the work of Reference 3 our results for the inelastic amplitudes suggest the possible presence of "ghosts" for coupling constant corresponding to A~ .5 (at least for very narrow widths). The problem of "ghosts", or spurious singularities which develop to . -1 allow the ND equations to satisfy unitarity, cannot be properly examined without studying the properties of our solutions in a much larger region of the W plane (e.g., for W < w ) than was compu0 tationally feasible for the present problem. As a result of their - 109 work with some simple models and a similar type calculation, Ball and 3 Thurnauer ) suggest that any approximation which ignores the existence of complex singularities may be haunted by "ghosts". Our numerical results also verified the increased effective attraction due to there being more than one inelastic channel. We now may briefly summarize the conclnsions to be drawn from our calculations. The NN ~NN* OPE amplitudes clearly may act to produce a cusp in the elastic NN scattering amplitude near the NN* threshold. We found that the shape and features of this cusp-type enhancement depend in a reasonable way on the assumed width of the N* and. the N*Nn effective coupling constant. But for the known experi- mental value of rN* the curves of Fig. 15 indicate that the cusp is so "wooly" that the resulting "peak"· is likely to be too broad and flat to be experimentally prominent. The rather large width of the N* means that there is much inelasticity present near the N-stable N* threshold (W = 2.176 BeV) and this necessarily reduces the height of the elastic peak even for resonant-like real phase shifts. Naturally, we would be unjustified in drawing quantitative experimental predictions from these approximate calculations since we have omitted so much of the true complexity of the problem (including all elastic NN, NN* forces, other channels, etc.). In fact, if present experimental evidence (cited in Section I) is confirmed, it is likely that the magnitude of the real part of the phase shift in this energy region near the inelastic threshold can be adequately explained solely by the elastic NN one-boson-exchange (OBE) - 110 - . *) • Our model predicts real phase shifts close forces of Scotti and Wong to 90°**) and therefore, we must conclude that we have omitted significant repulsive forces from our calculation, perhaps those due to OPE in the elastic NN* channel and forces due to particles besides the pion being exchanged. However, when more detailed experimental results are available we may find that the detailed shape of the iD 2 NN phase shift near the NN* threshold is influenc ed by the NN ~ NN* production amplitudes. And certa.inly some inelasticity 0 is needed to damp the elastic amplitude at higher energies. The total computer time expended for this project excluding that wasted in programming errors was about one and a half hours. All calculations were done on the CIT IBM 7040-7094 computer system. We should briefly mention two other papers which have also considered the influence of the NN* channel o.n elastic NN scattering. The work of Leung * 26 ) is rather similar to our own but the following This can be seen from the dashed curve in the figures of reference 27 which show that with the Scotti-Wong elastic NN OBE forces (ref ~r­ ence 24) as sole dynamical input, r eal phase shifts on the order of 13° (at 600 MeV lab . energy) are obtained. The author is indebted to Professor Scotti fo r a discussion of this point . (p F )in the form of Eq. (VI.16) one can .show that ** Writing 2 11 2 . 2] . 2 1 2Im(p2F11)-l IP l = 1 - 4[Im(p 2 Fll) and sin BR= ¥1 + T\ ] • 2 F 11 2 Using our numerical results for Im(p2 F11) and \p 2 F 11 \ we can thus extract values for T\ and 5R_. When this is done for values of A 2:: . 6 we find (even for b;oad widths) that BR exceeds 80° for some energies close to the NN threshold. But the present experimental values of oR lie in the range 8° - 15°. 'I) - 111 - differences may be noted: . 1. He calculates no elastic NN* ~NN* forces; 2. His inelastic NN ~NN* OPE amplitudes do not agree with ours; e.g., he has B 16 =O, and some of his amplitudes seem·to have incorrect threshold behavior; 3. In performing the dynamical calculation he combines all of the (four) inelastic amplitudes into one "averaged" amplitude; 4. His calculation consists of a modified pole approximation to the ND-l equations; 5. He does not vary the N* width; 6. From his Fig. 6 it seems that the input averaged inelastic amplitude has been cut flat for W < M + M* . 27 Coulter, Scotti, and Shaw ) use the single channel N/D equations modified to take into account inelastic processes by means of _20 1 the factor Tl ( = e ) • The elastic NN forces due to multi-meson exchanges (calculated by Scotti and Wong 24 )) are included but the inelasticity factor T) is taken from experiment for lab energies < 800 MeV with several different asymptotic forms assumed for T) at high energy. They find that 01D generally peaks at 400 - 500 MeV (with 5 ...J 16°) and goes negative for Tlab ~ 1 BeV. However, we note 2 that recent work by several authors 46 ) indicatesthat one must be cautious in using the one channel N/D method with inelastic unitarity instead of the more correct multi-channel ND-l equations. - 112 - There are many obvious ways in which our calculation could be improved or extended. We could include elastic NN and NN* forces*) and interactions arising from other than pion exchange. Instead of the determinental method we could solve the full ND-l equations. Other channels (e.g., ~+D) might be added. And of course, there are many JP states besides z+ which could be studied 47 )• • * The model calculations of Fra zer and Hendry, Ref. 12 , indicate that an attractive elastic NN* -7NN* force has the reasonable effect of increasing the strength of attraction in the elastic NN channel. - 113 - IMPLICATIONS OF su3 SYMMETRY VIII. The results presented in the last section indicate an enhancement in the I = 1, 1 n2 NN state near the NN* threshold Because of the broad width of the N* , our ( ~ 2175 MeV c .m. energy). model predicts any peaking would not tend to be experimentally prominent and this seems to be in agreement with the data. t~e But aside from detailed predictions of any particular approximate model it is interesting to investigate what the implications of the Eightfold Way . syrrunetry scheme 48 49 1 ' ) are,assuming some such NN n enhancement exists 2 and corresponds to a bound or resonant state. In su we have the following decompositions into irreducible 3 representations: 8 x 8 = 1 + 8 + 8 + 10 + 10 + 27 8 x 10 = 8 + 10 + 2 7 + 35 10 x 10 = ro + 21 + 28 + 35 The nucleon is assumed to belong to an octet (8) and the N* isobar to a decuplet (10). Only the 27 multiplet has a Y = 2, I could correspond to the NN 1 1 piece (which n2 state) and is contained in 8 x 8, 8 x 10, and 10 x 105 0). Thus, if su3 were an exact syrrunetry, we would expect all of . the other members of the 27 to exist in nature as J resonant states of mass ,.., 2J75 MeV. states + = 2 ' B = 2 bound or The 27 contains the following - 114 - y = 2·, I = 1 y = l; I = 1/2, 3/2 y = O; I = 0,1,2 y =-1; I = 1/2' 3/2 y =-2; I =1 However, it is well-known that su will not all be equal. 3 is not exact and thus the masses Assuming octet breaking of su 3 *) , the masses will be given by the Gell-Mann - Okubo mass formula M =a+ bY + c[I(I + 1) or, for J 2 Y /4] 2 states since they are formally bosons 2 M =a+ py + y[I(I + 1) - y2/4] . In Table IV we have listed the predicted octet splitting for each member of the 2 7. * We expect octet breaking to be dominant with maybe a small amount of 27, 64, and other representations present. ~ =a+ ~y + y[I(I + 1) - y2;4] (Y,I) M = a + bY + c[ I(I + 1)- - y2 /4] ( 2, 1) a + 2b + c a+ 2~ + y ( 1, 3/2) a+ b + 7/2 c a+ ~ + 7/2 y ( 1, 1/2) a + b + 1/2 c a+ ~ + 1/2 y ( o, 2) a+ 6 c a+ 6 y ( o, 1) a + 2 c a+2y ( 0' 0) a a (-1, 1/2) a - b + 1/2 c a - ~ + 1/2 y (-1, 3/2) a - b + 7/2 c a - ~ + 7/2 y (-2' 1) a - 2b + c a - 2~ + y Table IV Octet Breaking of the 27 !--' !--' Ul - 116 - . Sl) h as given . . Gerstein t h e expansion o f t h e 27 in terms of baryon-baryon states for each Y, I, I z state. What is the experimental evidence concerning possible manifestations of the proposed 2 7? 52 enhanc ement ) Except for the pp 1 n2 cusp-type the experimental .d ata is rather meager. Besides NN the only two-baryon states which have been studied to any extent are L:N and AN. And here the best evidence seems · to be a possible Ap 3 peak at 2360 Mev5 ). 4 at 2098. Mev5 ). 0 There is also a possible A n peak (Y = 1, I= 1/2) Several other low energy Ap peaks have b een .seen but these are probably S-wave interactions and thus not J On the basis of our dynamical model for NN 1 = 2+ states. D enhancement 2 we expect other members of the 27 to also be associated with thresholds for the inelastic reaction 8 + 8 the following table: ~ 8 + 10. Therefore we make (Y, I) Allowed 8 + 8 states (and c,m, threshold) Coupled 8 + 10 states (c , m, threshold) .. * ( 2J1) NN(l876) NN (2176) ( 1, 3 /2) I:N (2131) NY * (2323) , AN (2353) , I:N (2431) 1 ( 1, 1/2) I:N(2131), AN (2053) ( 0,2) I:I:(2386) ( 0,1) '.:::N (2255), A1:(2308), I:I:(2386) ( o, 0) ::::N(2255), M(2230), I:I:(2386) (-1, 1/2) A::: (2432), E:::: (2510) (-1 , 3/2) I::=: (2510) .., . . * * NY * (2323), .EN* (2431) 1 .'.:::N * (2555), Eyl * (2578) N:::: *(2468), I:yl * (25 78) , AY * (2500), :::: N * (2555) 1 N:::: * (2468), I:yl * (2578) N n- (2613), A::: * (2645) , I::=:* (2723), ::::Y * (2702) 1 * . * r::;;, (2723) J ::::_~l (2702) :=::=: (2634) r:n - (2868), :::::=: *(2847) ~ (-2J1) Table V Allowed B = 2,8 + 8, 8 + 10 states in the 27 I-' I-' -...J - 118 - If we make the assumption that the masses of the members of the 27 will lie near one of the thresholds for the coupled 8 + 10 states, then ·we can use this table to get approximate values for the constants (a,b,c or ex,13,y) appearing in the Gell-Mann - Oktibo (GMO) mass formula. Let us choose three values and then predict the others. We pick the three by looking for (Y,I) states where the allowed 8 + 10 thresholds have a narrow energy spread and then use the average of the inelastic threshold energies. We decided to use the following three (Y,I) states . . ( 2' 1) : ( 0,2): (-2,1): a+ 2b + c = 2176 or ex+ 213 + y = (2176) a + 6c = 2565 or ex + 6y = (2565) a - 2b + c = 2857 or ex - 213 + y 2 2 -(2857/ 6 Solv.ing, we find a= 2507, b = - 170 , c = 9.7; and ex = 6.42 x 10 , 6 6 13 = - .857 x 10 , y = .0261 x 10 With these values the GMO formulae· predict the masses given in Table VI. (Y, I) M(linear GMO formula) M(quadratic GMO formula ( 1, 3/2) 2372 2378 2323, 235 3, 2431 ( 1, 1/2) 2342 2362 2 323, 2431 ( o, 1) 2526 2544 o, 0) 2507 2534 2468, 2578, 2500' 2555 2468, 2578 (-1, 1/2) 2682 2700 (-1, 3/2) 2712 2714 ( Experimental 8 + 10 threshold energ ies 2613, 2645' 2723 ~ 2 702 2723, 2702 Table VI. Predicted octet mass splitting of the 27 assuming three members lie near to inelastic 8 + 8 -7 8 + 10 thresholds - 119 - It is not very surprising that the predicted masses lie close to the 8 + 10 inelastic thresholds since the masses M and M are 8 10 themselves known to follow the pattern of octet splitting. The results of our dynamical calculation suggest that it might be worthwhile to do a similar calculation using other membe~s of the 8 and 10 multiplets; for example , if the experimental evidence for a 2360 Ap peak becomes convincing it would be interesting to look at the AN, NY * , 1 * coupled channel problem. ~N Aside from differences in i-spin factors some of the amplitudes are the same as those for the NN, NN* problem. Among the other work which h a s been done on the two-baryon system and su 3 1 51 is that of Gerstein ) on the s 0 NN virtual state 55) -as part of a 27 and that of Oakes on the deut eron as part of a 10. 56 Dyson and Xuong ) have looked at Y = 2 states in su theory. 6 - 120 - Appendix A. NOTATION AND KINEMATICS We define the scalar product x . Y = xoyo - x . y = x µ oµV Yv = xµy}l ~ ~ (A. l) with metric tensor oµV = 0 forµ:/: V and 0 00 = - 511 = - 522 = ~ 033 = 1. .(A. 2) Then YµYv + YvYµ = 2oµV 2 Yo = - Y1 2 = (A.3) - Y2 2 = - Y3 2 = 1 (A.4) where ~ y = ( 0 ct) -rt 0 (A.5) .with (A .6) We define ' = a µ y·µ for any four vector a µ . and numerical values µ = pion mass = 139 MeV M = nucleon mass = 938 MeV * = N* mass = 1238 MeV M We use the following symbols - 121 - W = total c.m. energy s = w2 2 nNN coupling constant (g /4n = 15.) g ,.J * coupling constant q = c.m. momentum for elastic NN* scattering p,p' = c.m. momentum of NN, NN* states, respectively, for N + N ~N + N* scattering. G = effective nNN E. =Jp. 1. 2 1. + m.1. 2 =energy of particle i in c.m. e = c .m . scattering angle. I. * ~N + N* scattering in the c.m. we have For N + N N x. /_» < 4 . N-t /Nrr 3 P1 = (El' o, o, q) (A. 7a) P2 (E2' o, o, -q) (A. 7b) P3 (E3' -q sin e, o, -q cos B) (A . 7c) P4 = (E4' +q sin e, o, +q cos e) (A. 7d) where El = E4 =Jq2 + ~· = w2 + ~ - M*2 2W (A.8) -· 122 - (A. 9) (A.10) (A .11) q (w2 - (M* + M) 2J[w2 - (M* - M) 2] 2 4w2 = [ (W + M) 2 2 2 2 - M* ][(W - M) - M* ] 4w2 II. (A.12a) (A .12b) * Now consider the reaction N + N -)N + N 3 In the c.m. p 1 = (El, o, o, p) (A .13a) o, o, -p) (A .13b) P2 = (E2' o, -p' cos B) (A .13c) I P4 = (E4' +p' sine, o, +p cos B) (A .13d) P3 = (E3' -p' sin e' -' 123 - (A .14) E .= 3 E 4 = J •2 +ii w2 +ii2W - M*2 = p Jp •2 + M*2 = . __ 2 *2 W- + M 2W (A.15) 2 - ~ (A .16) . (A .17) (A.18) p2 = (~- ~) = (W + 2Mi(W - 2M) °(A .19) 2 2 p'2 = rw2 - (M* + M) l[w2 - (M* - M) ] (A.20) 4w2 (A.21) E * 4 ± M (W ± M*) 2 - ~ 2W = - (A.22) - 124 Appendix B. DESCRIPTION OF HELICITY STATES AND CALCULATION OF THE OPE PARTIAL WAVE HELICITY AMPLITUDES We write for positive energy spinors d·p) where 1 ~ E+M ~ a . p E+M (B.2) 1 and with (B.3) "- = ± 1/2 so u( + - l) 2 =!1! ( 2M 1 ± 112 > 2 "- r I + i12 E+M - J (B.4) and + = u Yo ti< +- l) 2 =~ 2M ( (± -2f...p (+ l f I E+M - 2 1) . (B.5) For a particle moving in the e,~ = 0 direction I 21 + _ ) (cos e12) sin B/2 (B.6a) 1 (-sin B/2) cos B/2 (B.6b) I- 2) = - 125 - and i f moving in the direction 1( - 8' cp I+ t I- (- >= 1 ) = 2 ( + 1( sin e12 cos 8/2 cos 8/2 sin e12 (cp = O) ) (B.7a) ) (B.7b) The spin 1 four-vectors satisfy = 0 (B.8) (B. 9) - 1 Writing € µ (€ , °€), we have 0 1<+ 1) = ( ~ :os ee ) (B.lOa) sin -:co) = ( si~ e ) (B. lOb) cos e °t(-l) (cos 8 = ~ :in e ) for a particle moving in the 8, q> = 0 direction. the 11: - 8, q> = 11: direction the (B. lOc) If it is moving in i's are the same with the ± 1 states interchanged. Using (B.8) and (B.9) we construct four-vectors from the threevectors of Eq. (B.10) - 126 - € € 1 µ µ (+ 1) = - (B. lla) [i 1 (0) = (B.llb) * M 0 € µ cos e ) 1 (-1) = - J2 ( = !in e If we reverse the direction of motion € µ (B. llc) (~ - e, = ~), €µ(+ 1) and ~ (-1) change places and the time component of € µ (O) changes sign. We now list some of the quantities which were used in the calculation of the helicity amplitudes. I. * ~N + N* (see Appendix A for kinematical definitions and N + N conventions) (1'4JAi> = {- where (+ -) sin e/2 (+ +) sin e/2 (- -) cos e/2 (+ -) (B.12) = (- +) 1 = (r..4 = + t1r..2 = - 2 ), etc. I (1'3 1'2 ) = { sin e/2 (+ +) sin e/2 (- -) cos 6/2 (+ -) = (- +) Define ii(f..4) u(t..2) ii ( t..3) u(t..l) = 0\4/..21..3/..1) (B.13) . - 127 Then (+ + + +) = (- - - -) = -(- - + +) _ - - * 2 (El + M) ( E2 + M ) 2M * (+ + - -) . 2 sin 2M B/2 G- (El + M) :E2 + M*)J (B.14a) (+ - + +) = (- + + +) = (- - + -) = (- - - +) = - (+ - - -) = - (- + - -) = - (+ + + - ) = - (+ + - +) * -- - (.E12+M M) ( E22+M*M ) sin e/2 cos e12 .~ (B.14b) (+ - + -) = (- + + -) = (+ - - +) = (- + - +) * = + (E\: M)( E2 +*M 2M 2 ) cos2 B/2 '1_ + . L (El+ M)(E + 2 . 2 . M*)J (B.14c) (B .15a) 1 - ---:;; (q El + q E M = - 1 f2 q sin e 3 cos e) (B.15b) (B.15c) (B.16a) - 128 - II. (p 4 )µ ~µ(O) 1 = - -:; (q E + q E cos 8) 4 2 (p4)µ €µ(-1) = (B.16b) M }r q sin 8. (B.16c) * N + N ~ N + N ={ sin 8/2 sin 8/2 cos 8/2 (- -) (+ -) = (- +) sin 8/2 sin 8/2 cos B/2 (+ +) (- -) (+ -) cos 8/2 sin B/2 sin 8 /2 (+ +) = (- -) (+ -) (+ +) (B.17) (B .18) = (- +) (B.19) (- +) (++)=(--) (+ -) (- +) (B.20) · Define u(A ) u(A ) u(A ) y u(A ) - (A A A A ) and 3 2 1 4 2 3 1 4 5 4 ~(A ) u(A ) u(A ) y 1 3 5 u(A ) ~ ~A A A 3 A ». 2 4 1 2 Then, omitting a common factor of . { i (El : 2 (+ + + +) (+ + - -) = - (- - + +) = 2 - sin B/2 F G M) (- - - -) (B.2la) 129 - (+ - + + ) = (- + + +) = (+ - - -) = (- + - -) sin S/2 cos S/2 F+G- (B .2lb) . (+ + + - ) = (- - - +) = - (+ + - +) = ~ (- - + -) B/2 F_G+ = - sin B/2 cos (B.Zlc) (+ - + -) = (- ~ + -) = - (+ - - +) = - (- + - +) = - cos 2 B/2 F+G+ (B.2ld) ((+ + + +)) = ((- - + +)) = - ((+ + - -)) = = cos 2 (( - - - -)) B/2 F G (B.22a) - (( - + + +)) ((+ - + +)) = ((- + - -)) = - ((+ - - -)) = sin e/2 cos B/2 F+G- (B.22b) . ((+++ -)) = ((+ + - +)) = (( - - + -)) = (( - - - +)) = sin B/2 cos B/2 F_G+ ((+ - + -)) = ((+ - - +)) = - (( - (B. 22c) + + -)) -((- + - +)) (B.22d) wher e ( B.23) G ± p : [ E 1 + M p' ± E 3 +M J (B.24) • - 130 - Some of the relations used in maki ng the partial wave expansions (both for NN ~ NN* and NN* ~ NN~ are: a) The well-known recursion relations and symmetry properties of the d functions (See reference 29). b) Using if 1 2 x n Pix n < J or n - J odd + 2·l·J - 1 for n - J = 2µ (B.25) -1 where (a;b;c) = a(a + b)(a + 2b) ... (a+ (c - l)b), and lf 1 Pk(x)dx = 2 (B.26) B - x -1 we can show that Pk(x)dx B - x = B Qk (B) - 0kO (B .27a) -1 1 lf 2 2 x Pk (x)dx B - x (B .27b) -1 1 1.Jx 3 2 Pk (x)dx = B - x 1 2 3 Bokl - s ok2 . (B.27c) -1 c) (B.28) (B) = 0 for t > 0. (t + l)Q(B) - B(2t + l)Q (B) + tQ t+l t t- 1 (B.29) - 131 - * * partial wave amplitudes of definite parity The resultant NN ~NN (calculated according to the procedure outlined in Section II) will now be listed. Define JP T. J ± T J T1, . k+ - (B.3.0) k 1 (where T . J is defined in Section I) 1 - µ A 2 = (B.31) v = L (g_) 2 q (El + M) (E2 + M*) 2n M 0 Then for J 1) p = + W I= 2 . I .= ) > 3 we have (- l)J - 2 ,• / (B.32b) - 132 - (B.32c) Tll,12+ = - -2_ .f3 4 . • - 2R R { + . T 1 VO J (J-l)(J+l)(J+2) ( E 1 +EA M* 3 )-12 (R2+ - R2)- - ~2 (R2+ + R2)- J (B. 32d) {Q(A) l-(J) (1 - A2) (R2 - R2) - J(l ·- A2)A(R2 + R2) 1 , 8+ = 8 J + 1 l J 2 - (1 + A ) + QJ-l + - + - (~ - R:) - 2A(R! + R=~ ~a!+ R~) (1 + A2 ) + 2A(R! - R~~} ( 3 + l (1 - A2 ) (R+ 2 + R2_)~} + Q(A) - 2A R R El +EA) [ J-1 + M* 2 R+R- {Q~~~ (B . 32e) (B.32f) [<2 _ J2 _ J) + A2(J2 + J + 2 ~ (J+l) )(J-1) (J+2) . + Q§A) t 3 A J (J - 1) + A (!- - J - -4~} (B.32g) 133 - + . t ~ (J + 1) <a! + R=)J 2 (J + 1) (1 + A ) (R= - R!) .- (A) [ . (El + El + QJ-l 2A R+R- * M A) (El + E3A) 2 + * M 2 2 (R+ + R_) (B.32h) VO Tl9,2o+ = 8(J-l)(J+l)(J+2) 2 {4 Q(A) J-1 [<R! + R)~ (i + J + 1) - (J - 1) (J + 2~ - D4 - - 3A (R! - R:)] 3 2 3 2 + Q§A)[(R! + R:) (A J(J - 1) + A(-J +2J -J-121 JJ (-A2(J - 1) 2 (J + 4) + (:i2 + J - 8)(J + 1)) - (R! - R:) (B.32j) 2. p = - (- l)J - 2 (B.33a) T 2,7- = (B.33b) (B.33c) Tl0,13- = O T . 11,12- =+-1- n 8 V J o . (J-l)(J+l)(J+2) • Tl' 8- = . 2 2 2 2 {(R + R ) .+ A(R - R + - + )l · -') {2AQ§~~ - (A2 - A2 J + J + l)Q~A)} (B.33d) ~ V0 ( J ~ 1 ) { - [ Q~~ + A J Q~A~ [ (1 +A2) (R! - R:) .. + (J + .. · + 2A(R2 + R2 + - l)Q~A) [(l + A2 ) (R! + R:) + 2A(R! - R:)]} )J (B.33e) - 135 - R (El+ E3A) ( J + l)Q(A)[2AR J + - * M 1 - 2 (1 R+Rr-2AQ(A) + Q(A)l(J + 1) - A2(J 1 (J+l)Jj-1)(J+2) JJ T t 4,5- L l~l :JJ (B.33g) 22 2 El + E3A)· 2 2 ( R ) + R+ - - ( M * 2j 1 2 A .2 + _; (1 + A ) (R - R ) - - (R + R ) 4 + 2 + - (B.33h) T = - 16 ,l 7 - + VO {Q(A) 14AR R +(El+ E3A) • 4 /3(J+l))(J-l) (J+2) . J-l + M* l Q~A)[- 2R+R- J(l (El+ E3A) + * M 2 2 (R+ + R_ )A(J + 2)(J - 1) + . (A 2 J(J - (E +EA) 1 3 (R2 - R2) • + M* 1) + 2(J + l~J} (B.33i) - 136 - ·v =19,20- T + O 8(J-l)(J+l)(J+2) 2 2 (4Q(A) [3A(R + R ) J -1 + - (~ - R=) ((J - 1) (J + 2) - A2 (J2 + J + l))] 2 2 + QiA>[(R! + R=) ~ (J - 1) (J + 4) + p2 + J - 8)(J + 1))· 3 3 2 + (R! - R=) (A(J - 2:l + J + 12) - A J(J - 1) )]}. (B.33j) Using the obvious relation . that d~µ (B) = 0 for J < \max (A., µ) j J J one can determine that for J = 0 only T and T are non-zero , 14 9 JP J J J J J J J J (T9 , 14±:/: O) and for J = 1: T4 ' TS ' Tll' Tl2' Tl6' Tl7' Tl9' T20 are all zero J=l J=l J=l J=l (T4 ,5± = Tll,12± = T16,17± = Tl9,20± = O) When J 2: 2 all twenty amplitudes are present (except TJP which 10,13vanishe s for all J). - 137 Appendix C. THE COMPLEX SINGULARITIES In this section we add a few details to our discussion in Se ction IV (see also Section VI) of how the ND -1 equations are modified du e to the prese nce of a complex singularity the unitarity cut. intersecting As stated before,the question is an involved and difficult one and our remarks are not to be taken as either complete or absolutely rigorous. In addition, we shall only discuss the modifications appropriate to our determinental method calculation and not those required in the full ND -1 equations. The Deformed Contour of Integration around the Unitarity Cut Beginning at the Elastic NN Threshold Fig. 21 In Fig. 21 we have symbolically indicated how the integration contour around the unitarity cut beginning at WNN must be deformed to avoid the protruding singularities of the NN -7NN* Born amplitudes. (Naturally, a synnnetrical situation exists in the left hand W plane but we are omitting integrals over the left-hand unitarity cut.) - 138 - Note that for integrals beginning at the NN* threshold (viz., D~ ) no 1 modifications are r e quired since . they b e gin to the right of the complex cut. Let us denote the contribution of the contour (call. it C) around the anomalous cuts by !:::. D . . • l.J Furthermpre define 1 • 2 1. (disc B . . (W)) =Abs. B .. (W) =a: .. (W) ' l.J . 1-J (C.l) 1-J and choose the branch cuts of O: (W) and p (W) such that 2 * (W) (C.2) a:* (W) • (C.3) - p * 0: (W ) = - 2 Using some of the results of r e f er ence s 1 and 2, we find the lowest order modification in the de t erminental approximation to be "n(W) __ - i fp 2 (W') a:1J(W') dW' LlJJ lJ (C.4) . w' - w re c where we have omitted the subtraction for brevity . Ex panding (C.4) and using Eqs. (C.2) and (C.3)Wwe can write L'.illlJ(W) = - i { re f R · WR Pz O:lJdW' JP2 a:lJdW'} w' - w W_ w' - w W+ = - i{JWR[P2 (W') a:lJ(W') w' - w re =- ~ wf_ :~ ( re WR P2 w' p (W'*) a: (W'*) 2 lJ J} · w'* - w dW' (C.5) .dW' - 139 - where WR is the point where the anomalous cut crosses the Real W axis (WR= 2.033 BeV). R We note that 6. D j only contributes to n . 1 13 Evaluating Eq. (C.5) for the four inelastic amplitudes would be a formidable undertaking. Instead we shall merely make a few observa- tions and a crude guess at the order of magnitude of 6. n for one 13 value of J. Using the relation (C.6) we find for the inelastic amplitudes l 2 (3z2 - 1) (C.7a) P ) = 5/2 z(z 1 2 - 1) (C.7b) (C.7c) (C. 7d) Recall that the anomalous contour is the locus of points for which -1 < z < 1 with z the point W = WR. 2 = 1 corresponding to the end points W± and Then we see that a 14 and a 15 z = 0 = 0 at the end points of the integrals for /::,, D and/::,, D • 14 15 Let us now look at /::,, n in slightly more detail (only because 16 a 16 has a relativeiy simple form). p4 _ 2 p2 PO) ( 35 21 + 15 (C.8) - 140 where p 3 . * phase space factor. here refers to the stable (rNN* = O)NN (P3 = p I /W). After a good deal of algebra we can write = _-2 ff g Q (z2 _ l) 2 . (16)2 M • (W - 2M) ~~ 2 (M + M) w (C. 9) . [P 2o;l6_l W' _ \{j , takes We then can observe that the integrand of 6 n , Im 16 on its maximum value at or near W' a) the factor (z 2 - 1) 2 = WR. This is plausible because has its maximum at z = 0 (W' = WR) and falls off quickly for larger z; (b) (W' - W) increases as we move up along the anomalous contour away from the real axis and, (c) at W' = WR [p 2 a 16 /(W' - W)l is pure imaginary. Taking W ;;- 2.15 as a typical (interesting) value of the energy we evaluate the integrand at W' =WR= 2.033. .3 • We find (C.10) To make a very crude e stimate for 6 n , replace the integral with one 16 whose integrand= .3 at WR and goes linearly to 0 at IW-1 (Since a 16 = 0 at z = 1): Then we would have ; . 1 • ~ (. 3) ..... - .1 (C .11) 6 (Unf.o rtunately) this value is comparable in magnitude to that for D~ at the same energy. - 141 - The work in this section must .be regarded as inconclusive. But certainly it does not seem obvious as some authors have concluded (Leung 26 ), Meiere 7 j that the anomalous contour contributions to the integrals are negligible. What is needed is a much more satisfactory treatment of the three-particle nature of the NN* state and a more accurate evaluation of the integrals over such complex singularities when they are present*). * Such integrals have been included in the work of Ball and Thurna uer, Ref. 3. - 142 REFERENCES 1. L . F . Cook and B.W. Lee, Phy s. Rev. 127 , 283 and 297 (1962). 2. J.S. 3. J.S. Ball and P. Thurnauer , Phys. Rev. 136 , B529 (1964). 4. J . S. Ball and W.R. Frazer , Phys. Rev. Let.ters l, 204 (1961). 5. P.G. Federbush, M. Grisaru and M. Tausner, Ann. Phys.(N.Y . ) 18, 23 (1962) . 6. S. Mandelstam, J.E . Paton, R.F . Peierls, and A.Q. Sarker, Ann . Phys. (N.Y.) 18, 198 (1962) . 7. F.T. Meiere, Phy s . Rev. 136, Bll96 (1964). 8. K. Mee t z, Phys. Rev. 125, 714 (1962). 9. For more discussion of these mechanisms see references 1 and 2 a nd also, S . Barshay , Ann. Phys. (N . Y. ) 21, 383 (1963); Y. Fujii, Progr .' Theoret. Phys.~Kyo to) 29, 71 (1963; and Y. Fujii and M. Uehara, Progr. 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