Analysis and Model of Low-EnergyΛK0Production

PH YSICAL REVIEW VOLUME 139, NUMBER 3B xK' Analysis and Model of Low-Energy 9 AUGUST j965 Production*f G. T. HOFF$ The Enrico Fermi Institlte for ÃNclear StuChes and the Department of Physics, The University of Chicago, Chicago, Illinois (Received 28 January 1965; revised manuscript received 12 April 1965) The low-energy AEO production data are analyzed under the assumption of a X*-exchange background. A new interesting result is found; namely, that the observed anomaly in the polarization and slight shoulder in the cross section at 829 MeV is apparently due to a narrow Fs/2 resonance located at 1647 MeV. The parameters of this resonance are determined from the data by adding its contribution to a model proposed previously by the author, which gave fairly good results above and below the energy of the anomaly, after modifying the contribution from a P&g2 resonance included in that model by taking into account the proper threshold behavior. Fits to all the data up to 1.03 BeV/c (900 MeV) (the region of validity of the previous model) are given. The result obtained for the product of relative partial widths agrees remarkably well with that predicted by unitary symmetry for a member of an octet with an asymmetry parameter of 0.674 (Carruther's model). It is also found that the P'it2 resonance (if it exists) cannot be a member of an octet, but it has to belong to either a 10 or 27 representation, if the F5/2 resonance belongs to an octet. Experiments which might confirm or deny the existence of these resonances are suggested. Further, our method of analysis is extended to a more general type of background which includes contributions from the nucleon pole and baryon exchange terms. I. INTRODUCTION AND HISTORIC SURVEY HK very low-energy region of the reaction sr +P-+A+E' (from the AE threshold to the neighborhood of the been the subject of intensive experimental study during recent years. The experiments were performed primarily with the purpose of determining the relative ZA parity from the expected observation of a cusp effect. ' As is well known, the effort did not produce any conclusive results„but the workers in the field have now at their disposal a collection of extremely interesting and rather accurate data. This region covers a range of about 80 MeV of centerof-mass energy. Measurements of the angular distribution of the cross section and polarization have been performed at Brookhaven' at incident pion kinetic energies T of 791, 829, and 871 MeV, and at Berkeley' at an incident pion momentum P of 1.03 BeV/c ZE threshold) has *This work is supported by the U. S. Atomic Energy Commission, C00-264-228. t Thesis submitted to the Department of Physics of the University of Chicago in partial fulfillment of the requirements for the Ph. D. degree. f Present address: University of Illinois at Chicago Circle, Chicago, Illinois. ' See R. Adair, Phys. Rev. 632 {1958);and A. Baz and L. Okun, Zh. Eksperim. i Teor. Fiz. 35, 757 (1958) LEnglish transl. Soviet Phys. JETP 8, 526 (1959) 'L. Bertanza, C. L. Connolly, B. B. Culwick, F. R. Kisler, T. Morris, R. Palmer, A. Prodell, and N. P. Samios, Phys. Rev. : — ill, j. Letters 8, 332 (1962). ' M. H. Alston, J. A. Anderson, P. G. Burke, D. D. Carmony, F. S. Crawford, N. Schmitz, and S. E. Wolf, in Proceedings of the Teeth Annual International Rochester, 1960, edited by Physics at Tinlot, and A. C. Melissinos (Interscience Publishers, Inc. , New York, 1960), p. 378. F. S. Crawford, in Proceediegs of the 1962 Anneal International Conference of Bsgh Energy Physics at CL~R/V, -edited by J. Prentki (CERN Scientific Information Service, Geneva, 1962) p. 270. These data are actually the angular distribution of the cross section and of the polarization of the A as averaged through the chamber. Conference on High-Energy E. C. G. Sudarshan, J. H. (T=900 MeV). The angular distribution of the cross section alone has been measured in the pion kinetic914 MeV 904, 904—907, and 907— energy intervals' 900— of 1.02 BeV/c and at an incident pion momentum' (T= 890 MeV). Besides, there are preliminary Berkeley data available which show the variation of the angular distribution of the cross section and polarization with position in the Alvarez bubble chamber, ' the center of the chamber corresponding to an incident pion momentum of about 1.03 BeV/c. The cross section for the process has been measured at seven different energies. The data well above the ZE threshold in the lowenergy region are much older and less accurate, and a complete set of measurements is availablev at only one energy (P=1.12 BeV/c, T=988 MeV). These data exhibit many interesting features. of 1.19 and shows a very slight shoulder (1) The cross section reaches a maximum mb at T=900 MeV at 829 MeV. (2) The angular distribution' of the E (A) exhibits ' F. Eisler, P. Franzini, J. M. Gai]lard, A. Garfinkel, J. Keren, R. Piano, A. Prodell, and M. Schwartz, Rev. Mod. Phys. 33, 436 (1961). ' Joseph Keren, Phys. Rev. 133, B457 (1964). These data are also the angular distribution as averaged through the chamber. ' S. E. Wolf, N. Schmitz, L. J. Lloyd, W. Laskar, F. S. Crawford, Jr., J. Button, J. A. Anderson, and G. Alexander, Rev. Mod. Phys. 33, 439 (1961). 'F. S. Crawford, E. M. Lyman, Jr. , M. Cresti, M. L. Good, K. Gottstein, I"'. T. Solmitz, M. L. Stevenson, and H. K. Ticho in Proceedings of the 1958' Annual International Conference on leigh-Fnergy Physics ut CERE, edited by B. Ferretti (CERN Scienti6c Information Service, Geneva, 1958), p. 323. J. Steinberger, ibid. , p. 147. {The author has received recently a report from Berkeley containing data at 1.17 BeV/c Dared Arnold Anderson, thesis, University of California Radiation Laboratory Report No. VCRL-10838 (unpublished)g. I wish to thank Dr. J. Anderson for his attention. } "From now on when we write simply "angular distribution" ("polarization" ), it should be understood that we mean "angular distribution of the cross section" ("angular distribution of the polarization" ), and when we write "forward" or "backward direction, it should be understood that we are referring to the forward or backward direction of the E, 8 671 " G. T. B 672 a forward. (backward) peak in all this energy region. (3) The angular distribution clearly exhibits an upward (downward) concavity in the forward (backward) direction at and above T=829 MeV. (4) The polarization of the 4 is large and negative for all angles except at T= 988 MeV and T= 829 MeV, where it develops one and two zeros, respectively. During the past few years several attempts have been made to understand these data in terms of simple models. As early as 1959 it became clear that a resonant contribution seemed necessary in order to fit the total cross section. The fact that the A was strongly polarized in the region of the peak pointed to the need for introducing a background contribution in any model. The first model that combined these two features was proposed by Kanazawa, ' who succeeded in giving a satisfactory fit to the experimental data available in 1959, by assuming, in the spirit of the Mandelstam representation, a background due to contributions from the two Born terms (nucleon pole and Z exchange) and a resonance with spin and parity to be determined from the experimental data. K.anazawa found that either a 3/2 resonance could give reasonable fits. g/2 OI Kanazawa obtained a forward peak at the energy of the resonance by giving opposite signs to the contributions from the nucleon pole (symmetric) and the Z exchange (peaked in the backward direction) and adjusting the relative amounts properly. The background obtained in this way develops, however, an increasing backward peak as we go to higher energies. It was suggested at the time that there might be an important contribution from the exchange of either a ' scalar or vector particle of strangeness 1 and isospin —, yet to be discovered. It was called E'. Its exchange would give rise naturally to A particles peaked in the backward direction. It was an appealing idea because of its simplicity. One year after this suggestion was made, a marked peak at 885 MeV was observed by Alston et at. in the plot of the X' — z c.m. total energy in the reaction +p —+E+~ +p, for a E laboratory momentum of 1150 MeV/c. The isotopic spin of this resonance (which was called E*) was found to be —', on the basis of the relative rates observed for its decay processes (Eo,m ) and ~o), and an application of the Adair argument" to the decay I I " " E (E, ' J. Steinberger in Proceedings of the 1959 Annual International Conference on Iligh-energy Physics at Kiev, 1959 (Academy of Science, U. S.S.R. , Moscow, 1960), p. 444. Data from Berkeley at 940 and 1030 MeV/c were reported at the time. "A. Kanazawa, Phys. Rev. 123, 993 (1961). '~ Jayme Tiomno, in Proceedings of the 1960 Annual International Conference on Hi gh-Energy Physi cs at Rochester (Interscience Publishers, Inc. , New York, 1960), pp. 466, 467, and 513; also J. Tiomno, A. L. L. Videira, and N. Zagury, Phys. Rev. Letters 6, 120 (1961). "M. Alston, L. Alvarez, P. Eberhard, M. Good, W. Graziano, H. Ticho, and S. Wojciki, Phys. Rev. Letters 6, 300 (1961).More recent experiments fix the mass of this resonance at 891 MeV. Adair, Phys. Rev. 100, 1540 (1955). "R. FIG. 1. K* exchange diagram. angular distribution of all the E* produced showed that the spin was either zero or one. Some time later the spin-parity assignment of 1 was firmly established by the experiments of Chinowsky et a/. ,'4 and the E* was subsequently identified as a member M of the octet of vector mesons of Gell-Mann's eightfold way. In 1962 the published data on low-energy AE' production had grown in number and accuracy, and the author, following a suggestion of Sakurai, reexamined the problem using the contribution from the E~ exchange (Fig. 1) as the background and found tha, t all the low-energy AK' production data could not possibly be reproduced by just a single resonance and the E* exchange term, but that this model reproduced the observed angular distribution and polarization amazingly well at all the energies at which the polarization did not exhibit anomalous behavior if the assumed resonance was I'g/2. Since the time our paper was written, more accurate data have become available in the angular-distribution very low-energy region showing a persistent, although not very big, discrepancy with our theoretical results at an above T=829 MeV: an upward (downward) concavity in the forward (backward) direction (see Figs. 2—5). We have found that there is just one single effect that can account for this discrepancy as well as for the anomalous behavior of the polarization and slight shoulder observed at 829 MeV, namely, an F5/& resonant contribution. '7 The existence of a high-angular-momentum resonance near 829 MeV was suggested earlier by Bertanza et al. ' as an explanation of the need to include partial waves higher than P in a polynomial fit at this energy, but these authors failed to determine its spin and parity. An F5/2 resonant contribution has been suspected for a long time to be responsible for the peak in the total cross section in AE' production at about 900 MeV. This is the location of the "third resonance" in the cross section of pion-nucleon scattering, to which, as is well known, the assignment F;/2 has been made on the basis of Peierls' work on photoproduction. " " " "W. Chinowsky, G. Goldhaber, S. Goldhaber, %V. Lee, and T. O'Halloran, Phys. Ilev. Letters 9, 330 (1962). "M. Gell-Mann, California Institute of Technology Report No. CTSL-20, 1961 (unpublished). T. Ho6, Phys. Rev. 131, 1302 (1963). From now on we will refer to this paper as I. Preliminary results have been reported already by the author. See G. T. HoB, Phys. Rev. Letters 12, 652 (1964). F. Peierls, Phys. Rev. 118, 323 (1960). Actually this = -'; analysis shows rigorously only that there is an appreciable contribution in the vicinity of the third pion-nucleon peak. For a criticism of this paper, see B. J. Moyer, Rev. Mod. Phys. 33, 367 "G. " "R. (1961). J MODEL OF LOW —ENERGY AE' PRODUCTION ANALYSIS AND 50 4O — I (0) I20 I T~= 829 I T Me V IOO— = 871 Me V v) (b) ~ 673 (a) T~= 871 Me V &0— I 20— bl~ 10 CL -Io -20 +.6 + I.o 8 COS 60— l .2 +.2 -.6 -Lo CO 40— vo eV 60 a(~ 20— 40 50 . 20 I Io— +I —. 2 0 +.8 +.6 I +.4 'I +.2 0 -.2 —.4 —.6 —.8 - I.o COSH Fzo. 2. (a) asymmetry parameter n times polarization F times differential cross section do/dB; (b) differential cross section at an incident pion kinetic energy of 829 MeV. The continuous lines represent the fits obtained in the present model, and the dashed line, the fit to the polarization obtained in our previous model. In all our present calculations we have used a value for the 0.67 which falls within the uncerasymmetry parameter a= — determined value tainties of the most recent experimentally — 16 0.85) more consistent a larger value ( — ( 0.62~0.07). In Ref. in our reaction data was assumed. polarization with the low-energy Data from Ref. 2. But for years many workers in the field have attempted to relate the peak in our reaction cross section to this resonance, achieving at most only transient (Gourdin and Rimpault") or highly questionable (Rimpau]t") success. Our anajysis shows no sign of a broad F»& resonant contribution in the energy region of the peak and thus opposes this long-favored interpretation. This means that if our main assumption is correct (predominantly IC* exchange background), then if there is such an F~i2 resonance in pion-nucleon scattering, it does not decay appreciably into the AE' channel. The present paper is an improvement of our previous work. We have fitted all the data between the AIL and ZE thresholds by adding the contribution from our F~i2 resonance to the contributions from the IC* exchange and the I'i12 resonance. The parameters of the reported) only in the cases of odd ZA parity with spin of the IC* equal to 1, and even ZA. parity with spin of the E"' equal to 0. It is well established at the present time that the E* is a vector particle and the ZA. parity is even. '0 M. Rimpault, Nuovo Cimento BI, 56 (1964). This purely phenomenological attempt which includes partial waves up to F5i2, besides not being very meaningful at the present time, gives a resonant Esi2 amplitude with an "unphysical" threshold behavior. The meaninglessness arises from two diferent sources: (1) the number of parameters to be determined at each particular energy (11) is too large for the degree of accuracy of the data and (2) the number of possible solutions in a purely phenomenological analysis increases exponentially with the number of partial waves retained, being extremely large for an analysis that includes six partial waves. Large D3/2 and I'ji2 amplitudes were obtained in this work and the anomaly at 829 MeV seems to be related to a na, rrow Slit resonance, I I —.6 1 —.8 - I.o have been determined from the experimental data and a slight readjustment of the other parameters has been made after taking into account the proper threshold dependence of the I'iii2 amplitude. In this paper we also give a, somewhat more detailed analysis of our reaction based on the assumed dominance of the E"' exchange term than those given previously; we present some more general arguments against any possible connection of the peak near 900 MeV in the cross section with "the F5i2 resonance, discuss representa, tions of SU3 to which our resonances might belong, and show that the inclusion of possible contributions from the nucleon pole and Z exchange terms to the background does not alter our assignments. There is still, however, an important question that has not been definitely settled. What causes the departure from our model above the ZE threshold? It is possible that there is no such Pcs resonance (in the usual sense) after all, but just an enhancement of the Piis amplitude in our energy region due to, for insta, nce, the opening of an inelastic channel. Ii5i2 resonance " " I20 I " M. Gourdin and M. Rimpault, Nuovo Cimento 24, 414 (1962). The model proposed by these authors, which included contributions from the Z and I'j* exchange, the nucleon pole, and the resonances N&~2~ (1518) and N&ig (1688) besides the IC* exchange, gave agreement with experiment for total and differential cross sections (no fit to the angular distribution of the polarization was -4 cos 8 FIG. 3. Fits to (a) asymmetry parameter o! times polarization I' times differential cross section do. /dO; (b) differential cross section at an incident pion kinetic energy of 871 MeV. Data from Ref. 2. 00 — Q P& = l. 02 Be V(c 80— cn I- z 60— O u 40— 20— +I.O +.8 +.6 +4 +2 0 C0S 8 -2 -.4 ".6 -. 8 -I.O FIG. 4. Fits to the average angular distribution over the length of the Srookhaven bubble chamber for an incident pion momentum at the center of the chamber of 1.02 BeV/c (T=890 MeV). The old fit (dashed line) represents the typical behavior of the angular distribution below the energy of the I'&i2 resonance obtained in our first model. Data from Ref. 5. " A mechanism of this kind was suggested by Sall and Frazer in order to explain the second and third peaks in pion-nucleon scattering. See J. S. Ball and R, Frazer, Phys. Rev. Letters 7, 204 (1961), , G. T. H OF F 700 2kpkp'+M' (b) —p' —p" 600 500 ~ I- 4oo ~ 300 is always greater than 1 in the physical region of our reaction. Here E (E') denotes the o.m. energy of the proton (A particle). The coupling constants and g correspond to an effective interaction-Lagrangian density X f 200 if((r)K/r)x„)~ K~— (r)vr/r)x„))K * vr, +i g K„*Xy„A+H. c. loo +l.o +.8 +.6 +. 4 +.2 0 I -.4 —. 2 cos 8 I -. 6 I —.8 -l.o Fro. 5. Fits to (a), (b) average oP, and (c) average angular distribution over the length of Alvarez bubble chamber for an incident pion momentum interval oi 1.01—1.05 BeV/c. (T=880— 920 MeV. ) The best (given) fit to the angular distribution was obtained for P = 1.01 BeV/c. (The fit obtained using P = 1.03 BeV/c gave a slightly smaller forward-to-backward ratio. ) Data from Ref. 6. The existence of the I'~&2 resonance cannot be definitely established. unless more accurate data are accumulated above the ZK threshold and the presence of one or more effects that mask the typical behavior of the angular distribution when crossing the resonant energy is surely established. We therefore insist upon the importance of those experiments. II. ANALYSIS) MODEL AND NUMERICAL RESULTS fg 1 since the spread. of mass of the E* is not too big in comparison with the distance from this singularity to our physical region. The amplitudes g and h have the following wellknown partial-wave expansions: a=2 (f(~-u+ — f(~+» E=1 h= 2 (f~ /=1 )2't (x) ~ (3) —f~')&~'( )x We have denoted by f~+ the partial-wave amplitude to orbital angular momentum t and corresponding ', and by x the cosine total angular momentum —, of the c.m. production angle e. P~'(x) is the frrst derivative of the Legendre polynomial J'&(x). In our analysis we separate the contributions from the E* exchange diagram from "the rest" and write a= a.+g. , h=h„+h„ (4) with g„and h„given by the partial-wave expansions (3). It is convenient to introduce the amplitudes u and b in terms of which the 3f matrix is written: (2) Here n denotes the normal defined by n= 1 4 4s WIkI (P — cos8)(IkI .+ (1/M') a.V.)/(V'+M') M=a+i(o 8)b sin8. where W2 (b, j=l+ Let us denote the four-momentum and the mass of the proton (A) by p (p') and m (m'), and those of the (K') by k (k') and. p, (p'). Let M be the mass of the K*, q its four-momentum, 0 the c.m. angle between k and k', and W the c.m. energy. The contributions from the E* exchange diagram to the amplitudes g and h of the production matrix" M=Z+h(o P')(e P) were found in I to be given byss g, = —(E+m)'~'(E'+ m')'"L2W —(m+m') + (p,"—p') (m' —m)/M'jC, — h„= —(E m)'"(E' m')'"— I 2W+ (m+m') —(p, "—p,') (m' —m)/M'jC, The calculation was made using Feynman techniques and the propagator of a stable vector particle Ik'I)"' These amplitudes plane (j'x j)/(I j'xlbl). are related to g and A' simply by a=g+h cos8, "We refer the reader to I for a more detailed exposition of the kinematics of reactions of our type. "In this calculation only the term of the type (4+0')„e„ in the expression for the amplitude of the (E'E~m) vertex and the term of the type y„e„ in the expression for the amplitude of the (AK'N) vertex were retained. Therefore, our product of coupling constants actually takes care of any extra contributions. Form factors were also ignored in our (low-energy) approximation. The eGect on our results of a possible contribution from the o „„e„term in the E"'AN vertex is discussed in Appendix A. The other neglected terms give zero contribution in the limit of exact unitary symmetry. to the production b= h. to Explicit expressions for u and P7/2 were obtained in here. I and b for partial waves up are therefore not given Using the formulas do/dQ= ', Tr(M'M) = aI'+ I b-— I I' sin'8 I'(do/dQ)=-', Tr(Mtrr nM)=2 Im(ab*) sino, (g) MODEL OF LOW —ENERGY AX' PRODUCTION ANALYSIS AND we obtain for the differential cross section (da/dQ) and the A. polarization in the p1 direction (P) if the initial proton is unpolarized the following expressions: OC) I + + O I I + O y+ I I I I 0 (da/dQ) (Pda/dQ), and the interference are given by (da/dQ), (Pda/dQ), C4~~ = Ia I'+ Ib I'»n'8, = 2 Im(a, b„*) sing contributions (Pda/dQ), (da/dQ);, l&m O I I +4 + Cd N CV) (10) (p+) ()— ), cd ~ m(m&1 ) —h& Im(fp+)M(i ) (p+) j sin8. The subscripts L and m stand for both subscript and superscript (sign). The matrix elements 3f( are functions of x and are given in Table I up to 67~2 partial waves. In our first publication the assumption was made that low-energy AE production was dominated by the E* exchange contribution and a single resonance in our channel located near the energy of the peak. Since in our energy region g„(x)/h„(x))7, it was justifiable to ignore h„ in a qualitative analysis. Then it was seen (see formulas (10) and (2) and Table I) that most of the low-energy AEP polarization data (no zeros) behave as the interference of a P wave and the E*exchange term. The discrimination between a P1/2 and a P'@2 resonance was made on the grounds of the angular-distribution data at energies at which the polarization did not exhibit anomalous behavior. After approximating this ~ic4 ppIep I I C4 0 ~ cd m 0 g ~ +- g g g ) lE) + + M (p tfJ ( "R lE) CV 0 I I I e = Q g, Im(f„)M„(p+) Q — h„I (f)M I bQ e~ ~ Re (fo+)i' I lC) CD 0bo Cd m(m&1 + 6+ CD Cd sin8, + P — hp Re(f~)M() ) + ri) o 0 («/dQ)'=Z g. Re(f-)~(~)- (Pda/dQ), ~0 (pda/dQ); *)M)„, = P Im(f)f„*)M) tf) Cd cd Introducing the partial-wave expansions for u„and b„ explicitly, we can write further P Re(f)f l&m + + I I = 2 (gp+ h~ cos8) Re (a,) 1 2hpRe (b„)sin'8, = —2I (g~+hp cos8) Im(b, ) —h„ Im(a„)) sin8. (da-/dQ)„= wjw I 0 + 2h, Re(b„)sin'8, I AIc% 0 p „=g„'+h,'+2g„h, cos8 O + uD I I C4~ C4~ cd I R~ g g lE) Cd c &D O PR O ~O + I .+ lE) cD M ~ (da/dQ) lE) ~ III where 8 675 4 ~ G. T. HOF F 8 676 resonant contribution by a Breit-Wigner formula, we found that the P3/~ choice gave a pronounced forward peaking of the A particles at 1.03 GeV/c and below, contrary to what has been observed, and this choice therefore was rejected. We turn now to the analysis of the polarization data at 829 MeV. We assume as suggested by Bertanza et al. ' that there is a resonance very near this energy, and try to find out which are the possible spin-parity assignments on the grounds of the polarization data. The characteristic shape of I'(do/dQ), exhibiting two zeros, requires the contribution of a term with this kind of behavior. This term can come in principle from the interference of this resonance with the K* exchange term or from its interference with the P~~~~ resonance contribution. From Table I and formulas (10) and (2) we see that there are four possibilities: fa or f,+ interfering with the K* exchange term, and f~+ or f4 interfering with the P j~~ resonance. We obtained in I that the P~~~ resonance contributes less than 3 of the cross section at this energy and that the existing shoulder is not too pronounced. This leads us to conclud. e that the contribution to the polarization must come from the interference with the K* exchange term. Otherwise it will be too small to give rise to such a strong effect. Therefore the D5/p and G7/9 possibilities are ruled out '4 Both of the two remaining choices (F5~& and F7/9) are high-angular-momentum states which give contributions with very marked threshold dependences. Therefore, they are expected to influence the differential cross section at higher energies in spite of the narrowness suggested by the polarization data. It has been mentioned previously that there is a persistent discrepancy with our model at and above 829 MeV: an upward (downward) concavity in the angular distribution for positive (negative) values of cos8. We try to find out if there is a possible connection between either of the two possible choices and this persistent effect. The polarization data at 829 MeV require" g~ Imf3 to be negative and g„ Imf3+ to be positive. This and the fact that the real and imaginary part of a resonant contribution have opposite signs above the resonance energy" give a positive (negative) g„Ref& (g„Refa+) above the location of our resonance. Then from formulas (10) and Table I we see that the f~ (f~+) choice a cubic term to the angular distribution sides facing the right (wrong) direction. Therefore, only a narrow Ii5~~ resonance located slightly below 829 MeV can account for this contributes which gives concave effect" We retain in formulas (10) only from the E* exchange and the P~&/~ wave amplitudes. We approximate by Breit-Wigner formulas with the the contributions and Iis&~ partialthese amplitudes proper threshold behavior: with I'&&' (I'~" &) is the partial width corresponding to the decay of the resonant state into the initial (final) channel, t/t/'„&' ' is the energy of the resonance, X is related to the size of the interaction volume, and the allowed subindex i runs for all the kinematically channels. The expressions for the partial widths are the same ones used by Glashow and Rosenfeld, and they where & " follow from simple centrifugal-barrier arguments. At the energies of our resonances there is another channel available, the two-body pseudoscalar-spinor Ãp channel; and at the energy of the P&~& resonance, ZE production is also kinematically possible. If the reduced widths (momentum dependence factored out) are of the same order of magnitude for all the kinematically allowed channels, we expect F&(' ) to be dominant because of the closeness to the threshold of the other reactions. We thus approximate I'&' &=I', ~' &. (If we use unitary symmetry to make estimates this is a valid approximation a, for representations 8 and 10 but not for 27). In the case of the P~~/~ resonance we make the approximation (in order of simplicity) of setting I' equal to a constant. We keep, however, the momentum dependence of I'~ and F~ in the numerator. We keep the values of the parameters I' (64 MeV) and W„&'— (1704 MeV), and of (f~ /G„) LG„=g„ cos0)] at 1.03 BeV/c, from our previous model; &( (P — but we allow a change in the product of coupling constants in order to readjust the fit to the cross section after the contributions from the F~/~ resonance and the " & & ' A perhaps more convincing argument to eliminate the D5/~ possibility is given later. The G7/& possibility can also be eliminated on the grounds of the closeness to the threshold and because the appearance of a G7/o resonance before an F5/~ would violate the orderly way in which resonances appear. "We recall that n ("old" definition) has been determined experimentally to be negative and equal in absolute value to 0.62&0.07. See W. Cronin and O. E. Overseth, Phys. Rev. 129, J. 1795 (1963). "This is a consequence of the principle of causality and is actually rigorously true for a narrow resonance. See, for instance, R. H. Dalitz, Ann. Rev. Xucl. Sci. 13, 339 (1963). '7 It is easy to see that the same kind of argument also rules out possibility. S. L. Glashow and A. H. Rosenfeld, Phys. Rev. Letters 10, 192 (1963). We use the value for X empirically determined by these authors (X = 350 MeV). the D5/& A&AL Ys yS MODEL OF LOW ENERGY AZ D A ence of the» /'2 a litude account. ave been taken into i —relative to m ' an at e determine ' 'butions of the cross o pRODUCT RO I I 0 T~"-79 l.MeV; I ~ ~ gr 1s—1 in terms of I 2 ll eV/& After this is angular-distribu tion data at done we 6 the scale by rea d'usting jus in the produc t of coui g ver low-energy a Tlr (bj 79 = IMeV ~ between the two sets of values given "" Fi g s. 2 —7j. The neww value obtaine b '"'"" ' ' se the resonance within whic h h th e b Set W, (MeV) 2 1647 1647 of '" arameters of the F5/2 d 1 p F1F2/F 8 1/550 1/870 ts is iven by t width of to thee cr c oss section is ' '/(4s. )'=0. 155. From f'g e h lies f'/4s. =0.1 . 't ry symmetry. " onsistent with thee results resu s obtained from A+A. is we as ' ded manner that a the observe articles in th e a g I a' t 'b tion comes from t e E* exchange term. 31 thee contribution of the ~ ti angular "The formula C. I. Izykson . Szepy tycka, ol t Ch. Peyron, vaa g2/4x' I 79f , d A. Leveque, J. Meyer, l~~~~ ~ 1 = P. 14 from the experimenta 0.090 09 m b) of 829 T (Me V) 87I 900 l I I h iv ive 990 I)40 I l.2 I a Berkeley ~ Brookhaven ~ Columbia f Michigan E b 0.6 0.4 0 I l l00 lO ZOO I P Qf1l. ira 300 (Me V/c) 1 400 1 ld a . y. in h. 1 .' --. 500 cross section 0. on the center-ofAnn incident-pion kin in inetic-energy scale H Miller, C. C. Butler, ), P seems to appear erroneous -l 0.2 di F, B. F 1 0.8 e and H. Jaco acob, Nuovo Cimento E. I -.8 -,6 avit for positive es ed at this energy. e order to obtain a much m larger neg ative os0=0. 8 t an a c ~ ~Q/4+=0. 5 we o t g 't de larger than our ur estimate . See J. . a g he NN coupling constan . urai's definition o t b'is cou 1i o t I -4 l.O 4 . From the estimate gn a contribution gu that with the value of o di to th fi t t th ~ I I 0 cts should be stressed: ce is obvious y very eristic shape of t e angu wise of the polarizati ion would have appeare nance is loca e (2) Th M V. The reason is tw twofold. . a e l.4 emp oy I I +2 bad a at 829 MeV. This is due to the fact that the contribution from t ar e to ive rise o a d arge r e aforwardpea k . 0n th oth h d (s — ~ to 'I'2 (s—&/(I (s—1)& 1Z wi'th the value of j. i" t, h fi t to the angu ar is t 829 MeV begins to ge is the of the cross section b egi distribution +4 d0 dO, (bl do/dQ at an incident pion kinetic energy of 791 Me V . Dt a fo Rf. 2. III. DISCUSSION entioned I +6 cos 8 section re a iv ontribution +,6 o, I' {MeV) I I +I us 1 represents the t oobtained in our mo e contri b ution. Its inclusion brings r close to t e ex k d by a cross) very ree he modification at 871 an d 891 M V sents the contribution from t e G. T. HOF F B 678 to reproduce this last effect correctly we need Refs / Imfs = — ~, which locates this resonance at an energy to 829 MeV. 8 F below the c.m. energy corresponding (3) Practically the same fits above 829 MeV are obtained with both sets of parameters. This is due to the fact that even at 871 MeV the fs amplitude is predominantly we can Inake I real, and adjusting the width accordingly it have the desired value for each value of — (s— 11 (s— 1/(I (s 1)s (4) The contribution from the Fs&~ resonance to the data at 791 MeV is practically zero because of the strong suppression caused by the centrifugal-barrier effect. The only effect that can be seen is the flattening of the differential cross section near cose=+1, which of agrees with experiment. The slight improvement do/d0(f&) apart from this effect and the slight worsening of aP(8)do/dQ(0) in comparison with our older results are due entirely to the change in the contribution from the P~~~~ amplitude when the proper threshold behavior is taken into account. " IV. OUTLOOK AT HIGHER ENERGIES Our new model (like the previous one) begins to fail in the neighborhood of the ZK threshold. There is a discrepancy among the experimental data available as to the energy at which the departure begins. The data of Eisler et al. 4 show a departure (larger forward to 907 backward ratio) just above the ZE threshold (904— MeV), which is very significant. On the other hand, the ratio da. ta of %'olf et al. ' show a forward-to-backward at their highest energy measurements (about P = 1.05 BeV/c, T=920 MeV, W=1700 MeV) consistent with our model. In both of our models the forward-to-back. ward ratio should decrease from about an energy -'I'&' to an energy W„" &+-,'I'o ', the angular Wro —— distribution becoming peaked in the backward direction in the neighborhood of the latter energy. This is an effect due to the interference of the P~~2 resonance with the E* exchange term. There are no complete data a,vailable from 920 to 988 MeV (1.12 BeV/c), at which energy the polarization shows a zero and the angular distribution is strongly peaked in the forward direction, contrary to our theoretical results. If the polarization data are correct (the errors are extremely large) this implies that there is a significant contribution from another partial wave at this energy. The analysis at this energy is, however, more complicated than at 829 MeV because the expected P~~~ contribution is too large to be ignored in a qualitative analysis. According to Table I there are four possibilities: D5~2, D3~2 interfering with the L exchange term or P~~2, P5~~ interfering with the P»2 & & "However, at an energy of about F/2 below 1647 MeV, a downward (upward} concavity for positive (negative) values of cos8 is expected to be observed. This can provide an experimental test of signiiicance for our interpretation of the data; not until this eRect is observed will we know for sure that the phase is pg, ssing through 90'. resonance. Even if we eliminate the F5~& possibility on purely aesthetic grounds, we still have three possible choices left. If we assume that the departure in the and polarization are due to a angular distribution single effect it is possible to see from Table I that the D, &2 choice (as well as the F s&,) is excluded. That leaves only two choices; P'&/& or D3/&. However, we should take into account that there might be some complications in the analysis due to a possible effect on our reaction from the opening of the ZE threshold, and that the In P~~/2 resonance might not exist in the usual sense. either case our simple argument used to eliminate the '4 We should also make clear D&&& choice is not valid. K* that the exchange term alone begins to give a contribution to the cross section greater than its experimental value not too far above 1.12 BeV/c (see Fig. 7). This is probably due to the fact that unitarity has been ignored in our simple E* exchange a,pproximation of the background. In future work in this energy region this approximation must be modified in one way or another. Needless to say, Inuch more work, both theoretical and experimental, is necessary. It seems to us that the interval between 1.05 and 1.20 BeV/c is interesting enough to deserve further experimental study. These experiments should be aimed especially (in the author' s opinion) to provide an answer to the f olio wing questions: " (1) Is the anomaly in the polarization at 1.12 BeV/c actually present? (2) At what energy does the departure from our Inodel actually begin? (3) Is there any feature in the angular distribution that seems to be related to the behavior of the polarization at 1.12 BeV/c (if this persists)? V. POSSIBLE 8U3 ASSIGNMENTS It is reassuring that the value for I'i&' &I's&' &/(I'&' ')' obtained from the experimental data in our model agrees remarkably well with the value obtained from unitary symmetry for a member of an octet with a symmetry mixing parameter n= 0.674 (Carruthers's model for higher baryon-meson resonances). " "gee would like to emphasize that in a multichannel situation it is perfectly possible to have a strong departure from the classical Breit-signer form in spite of the existence of a well-de6ned resonant state (i.e., a resonance in the usual sense), particularly in the neighborhood of a new channel. See Ref. 26. "We have a preference for a D wave (rather than a PI~&) because it is in general more strongly suppressed in the very low- energy region where our model works so well. If the anomaly in the polarization is due to the interference of the IC* exchange term with a D wave, the fitting of the data requires an appreciable imaginary D contribution Lace formulas (2) and (10) and Table On the other hand, the production-cross-section data do not seem to be able to tolerate a large extra contribution (see Fig. 7). This suggests that the real contribution is negligible, and therefore the amplitude may be resonant. Carruthers, Phys. Rev. Letters 12. 259 (1964). P. Carruthers, paper presented at the Conference on Particle Physics at the University of Colorado, 1964 (unpublished). Ij. "P. ANALYSIS AND LOVV —ENERGY MODEL OF Using the formula" n)if s 4 =NL+s+ (3/V'5) (1 — ] (12) and the expressions Ps(T = -', , I'= 1) = —(1/2+5) L3 (cVsr)+3 (ZK) + (Nr&)+ (AK)], — =-', Ps (T=-', , I"=1) P (Nsr)+ (ZE) + (N~) —AK)], (13) where (BP) indica, tes the normalized isospin--, state formed from the baryon 8 and pseudoscalar meson P, we obtain P (T = 'F =-1) = N/(20)'"( —3 (Nsr)+3 (1— j — 2n) (3 — Nrf) (3 4n) (— 2n) (ZK) (AK)]. (14) After identifying the relative reduced widths with the — — — probabilities, this gives I't&s &I', &s &/(I'&' &)'= 1/580, to be compared with I't&' &I's&' &/(I'&' &)'=1/550. This result should be contrasted with the results found if we assume that our resonance (S=O, F'=1, I=-', ) is a member of the other irreducible representations of SU3 to which it could in principle belong": 27 and 10. Repeating the calculations, this time writing" lf s7(T=-'„F =1)= (V3/2+15)f —(Nsr) —(ZK) +3 (Nrf)+3 (AK)] QM-(T= (15) '„ I'=1) =-,'(-—(Nw)+2K) we get —(N~)+ (AK)] r (3 )r (3-) (I'&e — &)s 1 1()9 aIld (16) 254 respectively, so our estimated value favors an eightfold representation. (We should not consider, however, the other representations, especially the 10, as excluded on these grounds. ) It should be mentioned that the fitting of the data requires opposite signs for (I't&e &I's&s &)'" and (I', &'—&I's&' &)'~'. It is easy to see )from formulas (14), (15) and (16) after identifying the square root of the rela, tive reduced widths with the amplitudes] that, this implies that, if the F5~2 resonance is a member of an octet and the P&~2 resonance actually exists, the latter cannot be also a member of an octet (with the same or a close value of n), but has to be a member of either an or a 27-piet. We get I't&' &I's&»/ antidecuplet (I'&' &)'=1/14 and 1/31 under these two assumptions (to be compared with 1/21) so it is dificult to make a choice on these grounds. The expected contributions to the pion-nucleon T= rs elastic (total) cross section are, however, quite different in the two cases: about 5 mb (versus 8.4 mb) and te mb (versus 2 mb), respectively. "See, for example, Ref. 26. Carnegie Institute NYO 2290, 2290-A (unpublished). ''7 P. Tarjanne, of Technology Report No. AE' PROD tJCTION 8 679 So this resonance should show up in an ana1ysis of the elastic reaction only in the former case. Layson's result" thus favors an antidecuplet. In either case an appreciable contribution to s) production (2.7 and 1.2 mb, respectively) is expected. [Recent preliminary experimental results" on g production show a hint of a peak of about 0.3 mb in the cross section in this energy region (the errors are very large and the experimental points are still scarce), and only an St~s or Pt~&s wave seems necessary for a fit of the angular distribution (isotropic). We should mention that if this resonance is a member of an octet (with n=0. 65), its contribution to rf production is negligible (0.074 mb) and its contribution to AE production consistent with experiment (0.62 mb). Our Fs&s resonance (if a member of any possible representation of SUe) should contribute appreciably to the pion-nucleon T=rselastic (total) cross section: 49.1 (49.2), 8.8 (20.8), and 33.7 mb (41.0) mb if a member of an octet, a 27-piet, and an antidecuplet, respectively. Therefore, a narrow spike is expected to appear in the elastic and total cross sections near 829 MeV. This energy region has been explored somewhat recently and no appreciable peak has appeared up to the present time. This seems to be against our SU3 assignments, especially the 8 and 10 representations. However, it should not be taken as evidence against the existence of our resonance. ~ On the other hand, there are indications of a shoulder ', pion-nucleon inat the expected energy in the T= — elastic-cross-section data collected by Omnes and Valladas. ' Also our results could provide an explanation for the shift in the position of the peak corresponding ] "B.T. Feld and W. Layson in Proceedimgs of the Iwfernafior&af Physics, Geneva 196Z, edited by J. Prentki (CERN Scienti6c Information Service, Geneva, Switzerland, 1962) p. 147. Also W. M. Layson, Nuovo Cimento 27, 718 (1963).As these authors did not have at their disposal any polarization data at the time, their results are unfortunately not very reliable. '9 F. Bulos, R. E. Lanou, A. E. Pifer, A. M. Shapiro, M. Widgoff, R. Panvini; A. E. Brenner, C. A. Bordner, M. E. Law, E. E. Ronat, K. Strauch, J. J. Szymanski, P. Bastien, B. B. Brabson, Y. Eisenberg, B. T. Feld, V. K. Fischer, I. A. Pless, L. Rosenson, R. K. Yamamoto, G. Calvelli, L. Guerriero, G. A. Salandin, A. Tomasin, L. Ventura, C. Voci, and F. Waldner, Phys. Rev. Letters 13, 486 (1964). "A simple-minded calculation independent of SU3, using our estimated value for FI(' )F~(' )/(F(' ))' and ignoring the widths corresponding to the decay into other possible channels (i.e. , writing F &'-& =F, &e-&+F, &), gives two solutions for F&&' &/F&' &: 0.998 and 0.002. In the 6rst (second) case we obtain a contribution of 49.2 mb (0.002 mb) and 49.3 mb (0.1 mb) to the elastic and total cross section, respectively. LA similar calculation using our value for F&&' &Fs&' &/(F&&' &)' gives for F&&' &/F&' the two values 0.95 and 0.05. In the erst (second) case we obtain a contribution of 14 mb (0.05 mb) and 15 mb (1 mb) for the elastic and total cross section. g It should also be noted that in the SUe calculations of the relative widths of the F5/2 resonance (and also of the PI/2 resonance) we have ignored any possible decay into the two-pionConference on High-Energy & & nucleon channel. + R. Omnes and G. Valladas in Proceedings of the Air-enProvence Conference on Elementary Particles, 1961, edited by E. Cremien-Alcan, P. Falk-Vairant, and O. Lebey (C.E.N. Saclay, France, 1961) p. 472. G. T. HOP F to the third resonance (from 1.65 BeV at I'=4. 74 BeV/c to 1.'70 BeV at /=8. 94 BeV/c) on inelastic proton-proton scattering obtained by Cocconi et al. 4' [More recently a very large concentration of events " + z.++rr +p has been observed in. the reaction y+p — in the neighborhood of the energy of our resonance. 44] It should be mentioned that there are also very slight indications of structure in the region of the third reso—+A+X++K, where nance in the reaction E +p — three diferent peaks at about the expected energies are observed in the plot of the c.m. energy of the AE system. 4' With the improvement of resolution, if our ideas are correct, more than one peak should appear in many reactions where the "third resonance" is observed as a final-state interaction. Only a small contribution (0.010 mb) to r) production is expected if the Ii 5~2 resonance is a member of an octet, but a very large one (about 7 and 12 mb, respectively) if it is a member of an antidecuplet or a 27-piet. (Recent preliminary experimental results on this reaction" show no maximum at this energy, but they do seem to show a slight departure from the isotropic angular distribution obtained at energies below and above. ) VI. COMMENTS ON OTHER WORKS A few words should be said in connection with other works on the subject: (1) According to our calculations, a Kanazawa-type 4' G. Cocconi, E. Lillethun, J. P. Scanlon, C. A. Stahlbrandt, C. C. Ting, J. Walters, and A. M. Witherell, Phys. Letters 8, 134 (1964). 4'It has not been definitely established at the present time whether the third maximum in pion-nucleon scattering is due to a T= 1/2 F5/2 or a T=1/2 Dz~s resonance, or both, or both plus something else. There are, however, strong indications of T= 1/2 Ii5/2 D&~2 interference in this energy region, as reported by F. Bulos, R. E. Lanou, A. E. Pifer, A. M. Shapiro, M. Widgoff, R.. Panvini, A. E. Brenner, C. A. Bordner, M. E. Law, E. E. Ronat, Szymanski, P. Bastien, B. B. Brabson, Y. EisenK. Strauch, A. Pless, L. Rosenson, R. K. berg, B. T. Feld, V. K. Fischer, Yamamoto, G. Calvelli, I.. Guerriero, G. A. Salandin, A. Tomasin, L. Ventura, C. Voci, and F. Waldner, Phys. Rev. Letters 13, 558 {1964).In view of these results and our analysis of A. production we are inclined to believe that this peak is possibly an unresolved mixture of three resonance contributions, 8~~2, P1~/2, and D5/2, not located at the same energy. If this is the case, the polarization of the recoil nucleon is expected to show strong variations in shape in this energy region. The number of zeros at each particular energy can provide us with important information as to the These possible main states contributing as can be seen in Table experiments are therefore quite powerful and should not be neglected. We would like to point out that the x P elastic polarization data at 981 MeV LR. D. Eardi, T. Devlin, R. W. Kenney, P. G. McManigal, and B. J. Moyer, Phys. Rev. 136, B1187 (1964)] seems to have the characteristic behavior of PI/2 Df/2 interference and not of D;/2 F5/2 interference. This is not, however, the only possible interpretation. 44 See H. R. Crouch, Jr. , R. Hargraves, B. Kendall, R. E. Lanou, A. M. Shapiro, M. WidgoQ', A. E. Brenner, M. E. Law, E. E. Ronat, K. Strauch, J. C. Street, J.J. Szymanski, J. D. Teal, P. Bastien, Y. Eisenberg, B. T. Feld, V. K. Fischer, I. A. Pless, A. Rogers, C. Rogers, L. Rosenson, T. L. Watts, R. K. Yamamoto, L. Guerriero, and G. A. Salandin, Phys. Rev. Letters 13, .636 J. I. E I. J. " (1964); Phys. Rev. Letters 13, 640 (1964). See P. L. Connolly, . K. I, . Hart, K. W. Lai, G. London, G. C. Moneti, R. R. Raw, N. P. Samios, I. O. Skillicorn, S. S. Yamamoto, M. Goldberg, M. Gundzik, J. Leitner, and S. Litchran, P hys. Rev. Letters 10, 371 (1963). model including F'5~~ and I'~~~~ resonances gives results not too far away from ours in the low-energy region if we fix the ratios of the coupling constants from unitary symmetry using a D-to-F mixing ratio of 2.5. However, the value of fts/4' Lft —(JV'rrÃ) coupling constant] obtained from the data (2.7) is one order of magnitude smaller than the well-determined value fts/47r=14. 8. Also, this kind of background begins to develop a backward peak (which increases with energy) slightly above 1.5 BeV/c. This backward peak has not been observed in high-energy experiments. (2) It is seen from Table I that the only way a large Ii 5~~ resonant amplitude in the region of the peak can give the observed polarization behavior is by interfering with a predominantly D3~2 background. A large was, in fact, obtained in the analysis D3~& amplitude by Rimpault' and this author suggested that it might come from the resonance Xt~s* (1518 MeV). However, if we make an estimate of the contribution from this resonance (which seems to be a member of an octet) and a symmetry using unitary symmetry mixing parameter" o. =0.65, we obtain a contribution to the cross section in that energy region of less than 10% of the total in the worst of the cases. 4' According to this it seems that Rimpault's suggestion is not correct. Besides, his D3~2 amplitude has a very large imaginary part and a behavior (with energy) of the ratio of the real to imaginary contributions which does not correspond even remotely to what is expected from a resonance below threshold. We consider the presence of this amplitude of unknown origin, which is fundamental (although not sufficient) for obtaining an Iis~s resonant amplitude, a severe handicap of this work. " VII. SUMMARY AND CONCLUDING REMARKS We have analyzed the low-energy A.E production data under the assumption that the contribution from the E* exchange term dominates the background. We have found that if the anomaly observed by Bertanza et a/. in the polarization is actually present there is a ' See, for instance, Ref. 30. V. A. Belyakov, Wang Yung-Chan, V. I. Veksler, N. M. Viryasov, I. Vrana, Du Yuan-cai, Kim Hi In, E. N. Kladnitskaya, A. A. Kuznetsov, A. Mikhul, Nguyen Dinh-Tu, I. Patera, V. N. Penev, E. S. Sokolova, M. I. Soloviev, T. Hofmokl, Tshen Lin-yen and M. Schneeberger in Proce. doings of the 196Z Annual International Conference on High-Energy Physics at CERN, edited by J. Prentki (CERN Scientific Information Service, Geneva, 1962), p. 252. V. A. Belyakov, Wang Yung-Chan, V. I. Veksler, N. M. Viryasov, Du Yuan-cai, Kin Hi In, E. N. Kladnitskaya, A. A. Kuznetsov, A. Mikhul, Nguyen Dinh-Tu, V. N. Penev, E. S. Sokolova, and M. I. Soloviev, ibid. , p. 261. A. Bigi, S. Brandt, R. Carrara, W. A. Cooper, Aurelia de Marco, G. R. MacLeod, Ch. Peyron, R. Sosnowski and A. Wroblewski, ibid. , p. 247. A backward peak is observed in the intermediate-en'ergy data (1.5 to 2.3 BeV/c) from Berkeley. LS. Schwartz, D. H. Miller, G. R. Kalbfhsch, and G. A. Smith, Bull. Am. Phys. Soc. 9, 420 (1964). This peak decreases, however, from 1.9 to 2.3 BeV/c, becoming very small at the last momentum. This peak is probably due to 4' ] (expected) resonance contribution in this energy region. ' This is the value empirically determined by Glashow and Rosenfeld. See Ref. 28. 49 There are two reported widths for this resonance: 56 and 125 MeV. ANAL YSIS AND MODEL OF LO'|A'-ENERG Y ~EO PRODUCTroX narrow Fs~~ resonance very near this energy. Although our detailed fit to the very low-energy data involved the inclusion of a P~~~2 resonance whose existence has not been definitely established, our spin-parity assignment to the resonance of Bertanza et at. did not depend at all upon the existence of that resonant contribution. Even more, this spin-parity assignment does not depend on the detailed nature (E* exchange) of the background; it actually rests on the more general assumption that it is predominantly 5 wave, as can be easily seen from formulas (10) and Table I. However, as it seems that we can be sure that there is a large Pj~2 contribution in the energy region of the peak and that a Breit-Wigner form provides a convenient parametrization in our energy region, we used this parametrization to provide the appropriate background for the Ii 5~2 resonance in order to determine the values of its parameters. The property that we used to find the orbital angular momentum of the resonant partial wave, i.e. , that the number of zeros of the polarization is equal to the orbital angular momentum l, seems to be a general property of the interference of an 5 wave with any other partial-wave amplitude, and it holds in photoproduction. Therefore, measurements of the polarization at several angles and various energies in the + A+E+ probably reaction p+P — would say the last word in connection with the resonances that contribute in the low-energy region of this reaction. Data above the ZE threshold will allow us to identify conwithout ambiguity the additional amplitude tributing at 1.12 BeV/c (988 MeV). II it turns out to be resonant, the extension of our model to that energy will allow the determination of the sign of the square root of the product of the partial widths of the resonance relative to those of the other two, and therefore we will have a constraint on the possible representations of SU3 to which it can belong. The same thing can be done in the region from 1.5 to 2.4 BeV/c, where one bubble-chamber group at Berkeley has been gathering data" with the purpose of 6nding the contribution from the E~~2* (2190) resonance to AE production. These constraints are by no means irrelevant. Once it has been determined to which representation any of the resonances contributing to our reaction belongs, the rest of the resonances can be classified in two groups, those which are members of octets, and those which are members of antidecuplets or 27-plets (if we assume that n does not depart too much from the value 0.65 for the different octets). In this respect reactions in which the initial " " "See, for instance, G. T. Hoff, Phys. Rev. 122, 665 (1961). the present time the most successful models for this reaction include the "F~~/~ resonance" plus Born and E~-exchange terms; see S. Hatsukade and H. J. Schnitzer, Phys. Rev. 132, 1301 (1963) and J. Dufour, Nuovo Cimento 36, 645 (1964). However, except at an incident photon energy of 1000 MeV the polarization has been measured at only one angle and, therefore, more data are needed before any model can be de6nitely established (at 1000 MeV the polarization has been measured at only two angles). S. Schwartz et a/. , Ref. 47. "At " and final states are diHerent have an advantage ovei. the elastic ones. According to our findings the energy region of the third resonance is more complex than was thought just a few years ago; it has structure. If subsequent experiments confirm our suspicions, we are confronting a very interesting situation. The T=-'„5=0, 1 states begin to show a certain regularity. First we have the '+, then the "second resonucleon with spin-parity — nance, which seems to be an admixture of states53 of '+ and ~ —and then the "third resonance, spin-parity —, possibly an admixture of 2+, ~+, and another state. And maybe there is a big surprise in store in the ', 5=0, neighborhood of 2190 MeV where the next T= —, 1 "resonance" is located. However, much more work has to be done in our energy region before the facts that our analysis suggest can become definitely established. First of all, the energy region between 1.05 and 1.20 BeV/c in our reaction should be explored thoroughly to determine the behavior of the P~~~2 amplitude with increasing energy and which additional partial wave (if any) is present. Secondly, it would be desirable for another experimental group to repeat the measurements at 829 MeV to check if the anomaly observed by Bertanza et aL. actually exists, and for the angular distribution and polarization to be determined at a slightly lower energy (T= 815 MeV) to find out if they agree with the predictions of our model. Thirdly, we suggest measurements of the angular distribution of the polarization (probably from five to nine angles would be enough) in pion-nucleon scattering, g production, photoproduction, and associated photoproduction at certain critical wellselected energies, and a thorough experimental study of very low energy ZE production (below 1.20 BeV/c). We expect our method of analysis or a variant of it to be helpful in the interpretation of the data on these reactions. If the existence of some or all of these resonances is definitely established, there will still remain the task of finding their companions in the representations of SU& to which they may belong. As a last remark, we would like to emphasize that the measurement of the angular distribution of the polarization of the recoil baryon is an extremely powerful tool in "resonance spectroscopy, as can be seen from just glancing at Table I. We foresee a time in the near future when classical partial-wave computer analysis will be outmoded, and the spin and parity of baryonmeson resonances will be determined essentially by 8= " , " 8= " ~8 See L. D. Roper, Phys. Rev. Letters 12, 340 (1964) and also P. Auvil, C. Lovelace, A. Donnachie, and A. T. Lea, Phys. Letters 12, 80 (1964). In the last paper two solutions for the phase shifts are obtained: one which corresponds to a normal PI/2 resonant amplitude and one which corresponds to a rapidly varying P1/2 amplitude which departs strongly, however, from the typical Breit-%igner form. It is not clear yet whether in the second case a resonance exists or not. It is interesting to note that the departure from typical behavior begins to occur in the neighborhood of the gn threshold. G. T. HOF F means of certain eGects that can be "seen" in the experimental data. the result ~ r =~~fgr L1/(i&+&f")]{t (k+k'). v. —L1/(m+m')](k+k'), (p+p'), } ACKNOWLEDGMENTS The author is deeply indebted to Professor J. J. Sakurai for suggesting AE production as a research topic, much patient guidance during the initial steps, and sharp criticism, and thankful to Professor Y. Nambu for willingly taking over the responsibilities of sponsorship during Professor Sakurai's leave of absence. She also wishes to take advantage of this opportunity to express her gratitude to Professor R. H. Dalitz and Professor J. J. Sakurai for much-needed encouragement during her stay at The University of Chicago. A conversation with experimentalist thought-provoking Dr. Roger Hill, is acknowledged with pleasure. APPENDIX A In this Appendix we discuss the eGect on our analysis of the inclusion of the contribution from the tensor type of X*AX interaction. The contribution from the tensor term in the X*A~'lt' vertex F, =V 2 fg.,[1/(rf +~ )]{s(k+k')„~„ + $(a+b cos8)/(m+m')]}, where a = W'+ 2kppp' g. =(g.)v+(g. )r=(g.)v 1+ From formulas (5) and (6) of I the contributions to the amplitudes g„and h~ can be found immediately. We obtain (g, ) r= $(a— +b cos8)/(m+ m')]}Cr, m')'"{ 2W+ (m+m') m)' '(E' — (h„)r = —(E— +[(a+b cos8)/(m+m')]}Cr, where Cr = ~~ fgr alld — A calculation shows that both the constant and the cos8 term multiplying g&/gv in the expression for g„ are negative and less than 0.1 in absolute value in our energy region, while the constant (cos8 term) multiplying gr/gv in the expression for h„ is about 1.25 (less than 0.05). If we recall that h~(gr in our energy region we realize that the inclusion of the tensor term cannot influence our results very much unless the ra, tio g&/gr is large. This ratio between the tensor and vector coupling constants is not known. However, it could be estimated form-factor data and unitary from electromagnetic symmetry. From the assumption that the contributions from the vector mesons dominate the form factors, the ratio between the tensor and vector pNE coupling u„—1= 3.7. constants is found to be'4 (gr/gv), ~~ N~ — If we now use the results that the tensor coupling of the vector-meson octet to the baryon octet seems to be predominantly of the D type" (D to F ratio of about "J. I «'-8)(l&l I&'I)"-'' (P Nuovo (A3) y" p' — 2koko'+ M' — os8)/(m+m') gr(a+b c— ig' g m)/— -— (A4) 2W+ ('m+m')+ (a+b cos8)/(m+m') gr- 2W+ (m+m') "See, for instance, J. D. Jackson and H. Pilkuhn, Cimento 33, 906 (1964). J. Sakurai, Nuovo Cimento 34, 1682 (1964). — — — 4~ wll as before. We can now write 2W —(m+m') h„=(h, )v+(h, )r (h„)r 1+ —, —(8+m)'"(8'+m')"'{2W —(m+m') 2W —(m+m')+ (p" p') (m' —— —m" —y', b= 2kk'. I „(p) p, (p') {s[g&/(m+m')]~„„(p p') „.„'}— to the E*-exchange Feynman amplitude can be easily calculated using perturbation techniques. We obtain (A1) which can also be written —(p" ' m)/M—' p') (m— gi 3.5), and that if the ro, p, and q dominate the Dirac form factors of the baryons, the vector-type coupling of the vector meson octet must be pure Ii type, and we find (gr/gr)irons~ — 0.52(gr/gv), iver=1. 9, which is not too large. The ratio between the amplitudes g~ and h„can now be estimated. We obtain g„/h„=4(g„)v/(h„)v. At 829 MeV, we have (g~)v/(h~)i =12, so if the above estimate is correct, g„/h„=3. Therefore, the amplitude g„ still dominates and we should expect that its interference with the resonance contribution should dominate the polarization. We should mention, however, that from our experimental data in the region of the " " 5'M. Gell-Mann and F. Zachariasen, Phys. Rev. 124, 965 (1961). In the (improbable) event that the amplitude h„dominates, " the (qualitative) spin-parity assignment of our resonance would be ~7, a possibility that we have already rejected on the grounds of closeness to the threshold and orderly manner in which the resonances seem to be appearing. Also in this case, the (qualitative) assignment of the (conjectured) 1704-MeV resonance would be —, which would give rise to a pronounced backward peak ifrom the contribution of the 1+3 cos'S term) in the region of the maximum which has not been observed. ', MODEL OF LOW —ENERGY AE0 PRODUCTION ANALYSIS AND peak it seems unlikely that the above estimate for gr/gv gives the correct value: A tensor contribution of that size would decrease the polarization (in absolute value) appreciably (because it increases the amplitude h~, which does not contribute to the polarization). More likely is a ratio g&/gr&1. Also, the fact that we obtained such good quantitative 6ts to all the data while neglecting the tensor-type contribution is a very strong indication of a smaller ratio. " APPENDIX B Although all the available data in AE' production and low-energy) suggest very (high-, intermediate-, strongly the peripheral nature of the background of this reaction, it might be pertinent to discuss the effect on our analysis of the inclusion of possible contributions from the nucleon-pole and Z-exchange terms. We obtain using perturbation techniques g& k ——(E+m)'~'(E'+ m')'i'C W m.— 5C&, — — — = (E m)'"(E' m')"'LW+m„jCip, (81) where W2~G, y, 1 (IkI Il 'I) &P )w[k/ w' —m' m'+— mz]Cz, gz= —(E+m)"'(E'+m')"'j W m — — m)'~'(E' — m') '"LW+ m+ m' — mz]Cz, hz —(E— 2\4 where Cz= and —(Gpfp) V2 4 I & 4~ 1 1 i WlkI (P'+c»~) (IkI Ik'I)'" m — 2ppkp +mz — p 2IkI Ik'I %2) in which fi, Gi, fp, Gp are the usual s)VIV, E1VA, Zhm, and SZE coupling constants. It should be noticed that the nucleon-pole amplitudes are constant and the Z exchange amplitudes increase in absolute value in the backward direction. It is easily verified that g&/h&=10 and gz/hz=10 in our energy region so we should still have dominance of the g background amplitude when these contributions "A larger D-to-F mixing ratio for the tensor-type interaction would decrease the ratio gz/gI. For pure D coupling a value 1.2 is obtained. are included. However, it is no longer true a priori that the background contribution to g (g„) has no zeros. It is not difficult to convince ourselves that there are five different possible types of behavior: (1) g~ has a forward peak and no zeros; a backward peak and no zeros; forward and backward peaks and no (2) g~ has (3) g„has zeros; (4) Ig„I has zero; (5) Ig„I has forward and backward peaks and one forward and backward peaks and two zeros besides the trivial one in which g„ is a constant. From the polarization data in the region of the peak (1.01 to 1.05 BeV/c), where it is safe to assume ths, t the polarization is y.ven by P(do/dQ) = —2g„ Im(b„), possibilities 4 and 5. From here on our analysis (which rested basically on the fact that g, dominates and has no zeros) goes on practically the same way. The only difference is that the forward peaking of the background is required by the data (as concluded by Kanazawa) instead of being assumed. The results are the same. It seems obvious that any linear combination of a of background and (nucleon-pole Kanazawa-type Z-exchange terms combined so as to produce a forward peaking) and E~ exchange background (with small should give good quantitative tensor contribution) results. There is, however, one difhculty involved. If we wish the mÃX coupling constant to be equal to its well-determined value (fiP/4s =14.8), the Kanazawa type of background should give rise to a contribution to g„ five times larger than its experimental value. Therefore, a E* exchange contribution either 6 or 4 times larger than ours (and with opposite sign) is needed in order to give a background of the correct size (coupling constant gr 6 or 4 times larger) The E.* exchange background amplitude is slightly concave upwards and the Kanazawa-type background amplitude slightly concave downwards. It is obvious that the percent concavity is increased by roughly a factor of 10. This would give rise to severely distorted 6ts. Also this type of background develops an increasing backward peak at higher energies which, as mentioned before, has not been observed. we eliminate " ' H instead of Gxing the couplings from unitary symmetry we allow for smaller couplings of the E meson, the amount of distortion can be reduced.