Pion-induced pion production on deuterium: a quasifree process

NUCLEAR PHYSICS A

Nuclear Physics Af~18 (1992) 562-578 North-Holland

Pion-induced pion production on deuterium: a quasifree process V. Sossi, M.J. lqbal, R.R. Johnson, G. Jones, M. Pavan, F.M. Rozon ~, M. Sevior, D. Vettedi and P. Weber: Department of Physics, Unicersity of British Columbia, Vancouver V6T IZI, B.C., Canada

G. Sheffer and G.R. Smith TRIUMF, Vancoucer V6T 2A3, B.C., Canada

P. Camerini, N. Grion and R. Rui Dipartimento di Fisica dell'Universit~ di Trieste, 34127 Trieste, Italy and Istituto Nazionale di Fisica Nucleare, 34127 Trieste, Italy

N.R. Stevenson Department of Physics, University of Saskatchewan, Saskatoon S7N OWO, Saskatchewan, Canada

M.J. Vicente-Vacas Departamento de F{sica Te6rica and IFIC, Centro mixto Universidad de Valencia CSIC, E-46100 Valencia, Spain Received 17 January 1992 (Revised !l May 1992)

Abstract: A detailed experimental analysis of the ¢r+d-, ~r+ ~r-pp in-plane coincidence data first presented by Rui et al. is compared to an expanded version of the Oset and Vicente-Vacas model for pion-induced pion production on a free nucleon. This extended model averages over Fermi motion to describe the assumed quasifree nature of the process occurring on the deuteron and includes nine additional diagrams to account for the N * ~ N(crZr)p_wave reaction channels. Experimental effects such as pion energy loss in the target and in the detectors, pion decay and muon detection are investigated and incorporated into the comparison of experimental data and theory. Inclusion of Fermi motion was found to be essential to provide good agreement between data and model confirming the quasifree nature of the reaction. When compared to the total-cross-section measurements of Manley et al., the free-reaction model yields a model-dependent estimate of the overall strength of the diagram containing the N*--, N('rrfr)s.wave vertex.

NUCLEAR REACTION 2H('rr +, 7r-), E = 2 8 0 M e V ; measured or(0,,+, 0,,-, E,,+, E~._). Comparison with isobar model with Fermi motion.

Correspondence to: Dr. V. Sossi, TRIUMF, 4004 Westbrook Mail, Vancouver, B.C., V6T 2A3 Canada. Present address: Physics Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. : Present affiliation: IMP/ETH, mailing address: EP-Div CERN, 1211 Geneva 23, Switzerland. 0375-9474/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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1. Introduction

Pion-induced pion production is the dominant 7rN strong-inelastic channel for pion energies below 1 GeV and above the pion-production threshold. This feature, together with the fact that the reaction also involves a ww interaction vertex and a high isospin selectivity through the selection of the initial and final pion charge states makes it a very attractive choice for experimental and theoretical work. Isospin conservation limits the number of allowed intermediate states for each reaction channel thus providing a relatively simple and selective environment for the possible determination of the reaction mechanisms and of the constants. The determination of the A~rA (fa) coupling constant is of particular interest since this vertex also contributes to the double charge exchange (DCX) reaction through the deltanucleon interaction mechanism (DINT)~). By its very nature the DCX reaction involves two nucleons and can be used to study nucleon-nucleon correlation functions once the reaction mechanism is thoroughly understood. At present there are no direct measurements of the fa coupling constant. The existing theoretical predictions tend to agree on values close to 1.0 (expressed in units of ~fNN~)2-6), while the analysis of several isospin channels of the free-pion-induced pion-production reaction by Arndt et al. 7) shows a preference for values around 0.4. The other relevant feature of the pion-induced pion-production reaction is that the w~r interaction vertex is related to the isospin-zero and isospin-two w~r s-wave scattering lengths a ° and a °. A very accurate measurement of these quantities would permit the determination of the coefficient of the fourth-order term in the momentum expansion of the quark lagrangian in chiral perturbation theory. Such a result would discriminate amongst different theoretical approaches s,9). A clear understanding of the reaction mechanism is thus important. ~Iheoretical efforts to describe the free-pion-induced pion-production reactions can be roughly divided into two main streams. Both use effective lagrangians derived from current algebra to describe that part of the reaction mechanism which involves only pions and nucleons, while the intermediate isobar states are described either with an isobar model partial-wave analysis 7,~o,~1) or via Feynman diagrams 12,~3). The model presented in this paper is based on the second approach. We have compared a detailed analysis of the 7r+d-* ~r+~r-p p data first presented in ref. ~4) as well as existing total-cross-section data ii) to an extended version of the Oset and Vicente-Vacas model first given in ref. 12). This model describes an effective wN-, ~rTrN t-matrix, whose parameters are fitted to describe low-energy data. Such effective t-matrices do not satisfy unitarity conditions as they are not generated from a generating kernel V through appropriate Lippmann-Schwinger equations. Use of such effective t-matrices is justified at energies far below the three-pion production-energy threshold ( T~ = 362 MeV) and provided that the model contains the correct energy dependence of the widths of the intermediate state isobars. Both of these conditions are satisfied in our case.

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This paper describes briefly the deuterium experiment itself, but focusses primarily on the description of the corrections that were made in order to compare the experimental spectra with theoretical calculations, the inclusion of Fermi motion to make the model applicable to deuterium (quasifree model) and the extension of the theoretical model itself for both the free and quasifree process.

2. Experiment and data analysis The ¢r+d-~ ~r* ¢r--pp experiment performed at T R I U M F at an incident beam energy of 280 MeV has been reported in refs. ~4-~). The following is a brief summary of the experimental details. We chose a deuterium target (or+d--> w+zr-pp) as opposed to a hydrogen target (,-r-p-~ ~r*,-r-n) for the study of the free process to be able to take advantage of the more intense T R | U M F 7r* beam ~7). This choice was justified by earlier experimental results ~'~) that demonstrated that the pion-induced pion production on deuterium was consistent with a quasifree process on a single nucleon. The positive and negative outgoing particles were detected with an in-plane coincidence apparatus which also provided information concerning the particle energies as well as their polar angles. Negative pions were identified by a quadrupolequadrupole-dipole (QQD) spectrometer ~9), while positive pions were selected by the total absorption range telescope CARUZ 20). The apparatus settings are listed in table 1. The four-fold differential cross sections d4or

were extracted from the data, after correcting for the response of the detectors over the allowed range of pion energies and angles and in part for pion decay as described below. The finite size of the target was also taken into account. TABLE I Apparatus settings 0OoD[deg]

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The corrections for or- detection on the QQD side were straightforward. This spectrometer is capable of rejecting electrons and muons so that an unambiguous w- spectrum could be obtained. When the coincidence events were binned to establish cross sections, a weighting factor accounting for the fraction of the pions that decayed while traversing the apparatus was applied on an event by event basis 2~). Similarly the solid angle for each event was normalized to the nominal QQD solid angle (18 msr) with an additional weighting factor rendering the spectrometer angular acceptance independent of the pion momentum. These corrections have been used in a previous experiment 2,). The lowest or- energy limit of the data used in the analysis presented in this paper was 30 MeV. Above this energy the effects of multiple scattering and of the finite target size were found to be negligible. Consequently no correction was necessary for these effects. Multiple-scattering effects could also be neglected on the side of the CARUZ detector, since the mean value of the multiple-scattering angle was within the angular resolution of the detector. Likewise the effects of hadronic interaction were found to be negligible. On the other hand, the CARUZ could not discriminate between pions and those muons resulting from pion decay between the production point and the end point of the CARUZ. Only a fraction of these muons was detected and they were, of course, recorded with a different angle and energy from those of the original pions. The scintillator elements were too thick to give a range energy separation between pions and muons. Consequently, this effect introduced a source of distortion in the yield and shape of the four-fold differential cross section as a function of the kinetic energy of the positive pions ( T~+ spectra). Another source of such distortion was the finite size of the target, since both the detected energy of each pion and the detection efficiency for pions produced with less energy than necessary to traverse the target (12 MeV), depended on the location of the production point within the target. The pion loss due to the finite size of the target was relevant on the CARUZ side, since the low-energy detection threshold of this instrument was 8 MeV for pions. Such a low-energy threshold was desired to cover regions of phase sphace close to the lower kinematic limits of the reaction. Since this was a coincidence experiment the overall normalization of the negative-pion distributions was however corrected to take into account the detection inefficiency of the CARUZ detector. This correction factor, on average, enhanced the data by 25% and is included in the T~- and 0~+ spectra shown here (where 0 is the polar angle with respect to the beam direction). The T~+ spectra presented in this paper are shown without corrections for pion decay and energy distortion. Instead these effects were convoluted with the T.~+ theoretical spectra when comparing theory to experiment. This was achieved by computer simulation of the experimental conditions using Monte Carlo techniques ~6). Theoretical spectra describing the pion-energy distribution at their production point and constrained by the experimental energy and angular detection windows were used as input to the program. The program, operating on an event

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by event basis, selected randomly the pion-production coordinate along the beam direction within the physical dimensions of the target, while the two transverse coordinates were selected according to a gaussian distribution in order to simulate the beam profile. The pion-decay probability was then calculated as a function of position along the pion trajectory from its production point in the cylindrical deuterium target to the CARUZ scintillators. If a Monte Carlo pion decayed along its trajectory, then the energy and trajectory of the resultant muon were obtained from the w-~/~ decay energy and angular distributions. The software veto imposed by a signal from the last scintillator of the CARUZ stack used during the data analysis to guarantee that the particle stopped in the detector was also incorporated as a veto in the Monte Carlo program. For each input spectrum the program generated an output spectrum describing the measured kinetic energy of the particles detected by the CARUZ. This spectrum is compared to the data in figs. 5, 6 and 8. Note that the spectral distortion due to the experimental conditions yields events outside the pure three-body phase space. The inverse of the ratio between the yield of output spectrum of the program and the input spectrum provided the multiplicative correction factor for the yields of the T.~- and 0~.+spectra mentioned earlier. The shapes of these last two spectra were found not to be significantly affected by the distortion of the shape of the T~÷ spectra, since negative pions from the same energy bin have the corresponding 7r+'s spread over a range of energy wide enough to allow for an average correction factor. Likewise the positive pions in the same angular bin have a fairly uniform energy distribution when averaged over all T~-. 3. Theoretical model

The interpretation of the experimental results is based on an extended version of the Oset and Vicente-Vacas Monte Carlo model ~2) that describes the free-pioninduced pion-production process: 7r-p~ 7r*Tr-n. A similar model by the same authors describing (7r, 27r) on complex nuclei has been previously used in analysis of experimental data 2~) yielding good agreement between the experimental differential cross sections and theoretical calculations. The free-process model describes the reaction mechanism with Feynman diagrams involving nucleons, A's and Ropers in the intermediate state. Only that fraction of the N* (½, ½) Roper resonance that decays into two pions in a relative S-state is included in their model. There are three parameters in their model that are not well determined: the chiral symmetry-breaking parameter, ~, which arises from the part of the model derived from the Weinberg lagrangians, the A~-A coupling constant and the overall strength of the diagrams containing the N*-~ N(Tr~)~.,~,e vertex. This last parameter is a product of the N*-~ N(Tr~)~_w~,,e coupling constant, C, and the N*NTr coupling constant gN*NTr [ref. 23)], both uncertain by about 100%. The ~:-parameter is only allowed to assume a discrete set ofvalues (.~,0, -2, - -24~ , . . . ) [ref. 7)]. Theory predicts ~:to be zero 7.24-26),

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and this value is consistent with experimental results 7). In the present model we therefore fixed ~: to be zero, and the value of the coupling constant to 0.02, which is the value used in ref. ~2) (gN*N~ range 0.01 to 0.02). We found that this value of gN*N~- lead to a value of C that is consistent with its estimate obtained from the N* --> N(zrcr),.wave decay branching ratio, as will be shown in subsect. 3.3. The model of ref. m2) is only applicable to a free process. To extend it to the pion-induced pion production on deuteron we introduced Fermi motion in the calculations. We also included nine diagrams describing the N* -> ( ¢rZr)p.w,,vereaction channels that are not negligible at the energy at which our experiment was performed.

3.1. FERMI M O T I O N

The Fermi motion was introduced in the free-process model by assigning an initial momentum to the nucleon involved in the reaction. In the calculation the nucleor, was treated as an off-shell particle with its energy equal to its mass and its momentum randomly chosen according to the Hulthen nucleon-momentum distribution 3o). The model was based on a Monte Carlo program that evaluated the reaction cross section by calculating the reaction amplitudes for those final states that were kinematically allowed for a given total centre-of-mass energy (S) of the system. The introduction of an initial nucleon momentum that varied from event to event yielded a different centre-of-mass energy for each event. The resulting centre-of-mass energy distribution for T~ = 280 MeV is shown in fig. 1. The distribution is centred on the S-value of the free process occurring at the same T~ and is slightly asymmetric with a longer tail on the low-energy side due to the off-shell nature of the nucleon. The variation of S from event to event yielded different kinematic limits for the outgoing particles thus broadening and smearing the kinematic ranges that were allowed in the free

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reaction. Some phase-space areas that were not available to the outgoing particles in the free process case thus became accessible. The above procedure is described in detail in ref. 16).

3.2. NEW DIAGRAMS

As the Oset and Vicente-Vacas model ~2) underestimates the total cross-section measurement at our energies by about 50% [refs. ~~.27)], additional intermediate-state contributions were examined. A contribution that was missing in ref. ~2) arises from diagrams that describe the N* -* N(~-~-)p.,,~,~ decay fraction. The mechanisms containing N*-~ N(~)p.,,,,,~ decay that were found to be relevant are listed in fig. 2 and the respective amplitudes are presented in appendix A. in addition to the coupling constants used in ref. ~2) the new set of diagrams required the N * J ~ coupling constant. The allowed values of this coupling constant were calculated from the N* ~ N( ~-~)p.,,,,~ decay branching ratio 2s) and they ranged between l.l m ~ and 1.5m~ ~. The contribution of the new diagrams to the total cross section at T.~. = 280 MeV with gN*a.~= l-3m~ ~(an intermediate value ofthis coupling constant) and gnarls. 0.02 is about 15%. =

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The C coupling constant has also been recalculated from the N*-> N(TrTr),.w.~,¢ decay branching ratio. The present limits are - 3 . 0 m ~ j and - l . 5 m ~ min contrast with the -1.11m~ ~ to -0.71m~ t range quoted in the earlier Oset and Vicente-Vacas calculations. With the recalculated range of values of C the model overlaps the experimental results in the energy range from reaction threshold to 280 MeV [refs. 29,11)].

3.3. ESTIMATE OF C AND TOTAL CROSS-SECTION SENSITIVITY TO COUPLING-CONSTANT VARIATION We first explored the sensitivity of the theoretical cross section to the parameter variation in the energy range from reaction threshold to 280 MeV. We considered 280 MeV to be the upper energy limit for the applicability of the model, since at higher energies the contribution of other resonances may become significant. The cross section was calculated for the quoted lower and upper limit of the C and gN*.~ coupling constants separately, maintaining the other parameters at a fixed value. We arbitrarily set fa to be 0.5 and 1.0 to explore the cross-section sensitivity to the variation of this parameter. We found a 100% sensitivity to the variation of C and negligible sensitivity to the variation of the other coupling constants throughout the whole energy range. The large sensitivity to the variation of the C parameter reflects the very significant contribution of the N* -* N( ~r~r),.~,.~ diagrams. The importance of these diagrams is also noticeable in fig. 3 where the three curves compare the total cross section calculated with the pion pole term only, with the pion pole term plus these N* diagrams and the full calculation. On the basis of

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Fig. 4. Tetal cross section according to the model with C = -2.08m7,=compared to experimental values of ref. 3). these results we estimated C by fitting the energy dependence of the model to the four total cross-section measurements of the free reaction of ref. ~1) (203, 230, 255 and 280 MeV). With ~ =0.0, gN*a~ = 1.5m~ ~ a n d f a = 1.0 we determined the product of C and g~, ~,. to be - (4.14 ± 0.08) x 10- 2m ~ using the standard X2 technique. Consequently, taking g ~ , ~ : to be 0.02 as in ref. ~2), C falls in the range (-2.07 + 0.04)m ~~. This range of values of C determines the N* ~ N(crcr),.w~,.e decay branching ratio to be about 10%, consistent with the published values -'~). Fig. 4 shows the theoretical cross section as a function of energy with C fixed to -2.08m ~ compared to the experimental values. The value C =-2.08m~. ~ was selected for the analysis described in the next sections. 4. Comparison of the model to the four-fold differential cross-sectlon data As described earlier the (~-, 2~r) reaction occurring on deuteron was assumed to be a quasifree process. To test this assumption Fermi motion was introduced in the free process model. A simultaneous goal was the test of the model itself at a more microscopic level. The effect of Fermi motion, the broadening of the kinematic ranges allowed to the outgoing particles, is particularly evident in those kinematic regions that are close to the free-reaction phase-space limits. Two such regions were covered by our detectors and the corresponding spectra are shown in fig. 5. (The two theoretical curves are calculated with the same set of parameters C = -2.08m7, I , gN*a,, = 1.3m7, ~ and f j = 1.0.) The most significant fraction of the events populating the low-energy region of the T.,+ spectra in the upper two figures and the high-energy region of the T.~+ spectra in the two low figures is due to the nucleon Fermi motion, even though

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there is some contribution to those regions from muons originating from pion decay (see sect. 2). The agreement of the allowed kinematic limits between calculation and data improves remarkably with the introduction of Fermi motion. In particular at 0,, = 80 ° the yield of the experimental cross section is reproduced by the calculation, since the Fermi-motion contribution to the total energy acounts for the larger fraction of the cross section. In ref. ,4) data in these regions were observed to depart from the predictions of the model of ref. ,2). However, nucleon Fermi motion was not included in the previous comparison. Figs. 6-9 compare all experimental spectra with calculations, in all cases the shapes and the yields are reasonably well reproduced although some discrepancies still indicate that the model might not include properly all of the necessary reaction mechanisms. Using therefore the model mainly as an indicator we investigated the differential-cross-section sensitivity to the coupling constants gN,a,, and .i]. The

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curves in figs. 6-9 show the calculations performed with sample-parameter sets. In particular phase-space configurations the differential-cross-section calculations show a larger cross-section sensitivity to the variation of these coupling constants than in the total-cross-section case. A more precise differential-cross-section measurement could possibly discriminate amongst different values of the coupling constants.

5. Discussion 5.1. ~r, 2~r ON D E U T E R I U M

The improvement in the agreement between data and free-process model achieved after the introduction of Fermi motion strongly suggests that ( ~, 2~r) on deuterium is indeed a quasifree process° We also compared the calculated tolal cross see|ion

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contribute differently to the cross section in different isospin channels, a further test of the model may be provided by a more comprehensive experiment that would measure pion-induced pion production in a variety of charge states. Such an extended experiment would also facilitate determination of the coupling constants, since different isospin channels provide different constraints. For example, a larger f.~ value decreases the ~r*p--> 7r+ 7r+n cross section, while it increases the ~--p--, ~r+~'-n cross section. 6. Conclusion We extended the existing model of ref. ,2) to pion production on deuterium in the energy range of interest by introducing Fermi motion in the calculation. We improved the model itself by incorporating more reaction diagrams in the crosssection calculation. We confirmed that the pion-induced pion production on deuterium is compatible with a quasifree process by significantly improving the agreement between differential-cross-section data and a free-process model that included Fermi motion. The comparison between the model calculations corrected

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~._= +80%

0,=-100°+28 °

9 +=-50°±28"

O
O
5°

1.5 1.0

+

0.5 o.o

I

20

40

60

80

i

i

T _(MeV)

[--4 !

+

20

, 40

,

, 60

, ~ ~ ~ _ _ - 80 I00

120

T _(MeV) !

•' "

!

a

|

I

T,,_=45+15 MeV

0 _ = +80°± 5 ° O
I

120

T .=80~20 MeV ~2

I~

,

0 190

® _=+80o+ 5 °

O
<75 MeV

09

b ,,t

O3 O0

0 10

10

38

66

94

122

150

Fig. 9. T,- and 0,,. theoretical distributions compared to data. Q Q D at 80 °. gN.A, = 1 . 3 m , ~. Short.dashed curve: f,, = 0.0; solid curve: fa = 0.1; dotted curve: fA = 0.5; long.dashed curve: f., = 1.0; dash-dotted curve: f,~ = 2.0. Kinetic energy of the incident pion is 280 MeV.

for experimental effects and differential-cross-section data tested the model on a more microscopic level. The results showed that the model is a good description of the reaction although some discrepancies still occur both in the differential cross section at T,, -- 280 MeV and in the total cross section at the same energy. Within the framework of the model we estimated the C-parameter for the N*-~ N(~-~),.w,ve to be approximately -2.1m~' with ~ fixed to 0.0 and gN'N,~ tO 0.02. These results contribute to the understanding of the overall mechanism of the free-pion-induced pion-production reaction, even though we realize that they are not adequate to decouple the N*N~" and C coupling constants nor to provide significantly improved limits on the coupling constants, gN'J, and fj. A modification of the term describing the ~r~"scattering amplitude in the model to include the pion rescattering should also be explored 9). A more accurate description of this amplitude would certainly lead to a more precise and meaningful determination of the coupling constants, in particular C, since there is a strong correlation between the diagrams containing C and the pion pole term. We believe that a more stringent test of the model and consequently a more accurate determination of such parameters would result from comparing the theoretical distributions to a more extensive set of precise

V. Sossi et at / 2H(u ÷, :r-)

576

differential-cross-section measurements of the free process where there is no broadening of the angular and energy ranges due to Fermi motion• For instance measurements that cover in and out of the reaction-plane cross section similar to those performed by Ortner ~t) et al. would test the full angular dependence of this model. Another natural candidate for further investigation is the reaction ~r+p--> ~* ~ n. Due to isospin-conservation constraints there are fewer intermediate states permitted in this reaction than in the ~ r p ~ ~r* ~--n case. It may thus be easier to isolate the contributions of each reaction mechanism. We are currently studying how the fa coupling constant and the pion scattering length can be extracted from the measurement of the cross section of this reaction. Since the cross section for this process is about one order of magnitude smaller than that of ~r-p-, ~r-~r+n the energy and geometrical settings would however have to be carefully planned in order to guarantee the maximum information from the data. We express our thanks to Byron Jennings for all the useful and helpful discussions•

Appendix A AMPLITUDES RELATED TO THE NEW DIAGRAMS

In calculating these amplitudes we followed the convention of ref. ~2) for the A and N* mass distribution. The N* mass (M*) is taken to be 1440 MeV and its width (F*) at any energy is appropriately scaled to the width value at the peak energy [235 ± 115 MeV, ref. 2,~)]. The amplitudes related to the diagrams of fig. 2 are thus: - i T s'N" = - ( f / I.t )( f / I.t )22v/2 {r " p ~ • p6cr . p, x

- i T 2 "~*-

X//~

i -p6- m-p~/2m 0

(fl~)(f*l~)(gN

a~.)~o"

i

×

+ ir 4-g _

:-r3

-r~b"

i ~/-S- M * + ~ i r * '

u

~

o~S.p~s +.p,

i p~/2M*

~

M*+ zr I.

:~

N ~

= --(f /g )(f*/g

) ( g N . a ~ . ) V ~ S . p 6 S * " p~cr . P5

1

× "~

0

pi/2m+m-ps-(pl+ps)2/2M ×

*

M*+~_iF*

i q~-S-p~-p~/Ema - ma +~ir"

--iT2.N*= u ( f / # ) ( f * / f t )(gN*a~-)V~O"" PsS" p,S+ " p6 x

i p~/2m - p 2 - toR- ( p, + pg)2/2ma +'iF v~-p °- M* +½iF*-p~/2M*'

V. Sossi et aL / 2H(rt +, rr-)

577

-iT3d "N* = -- ( f / Iz )( f * / Iz )(gN*.~-.,)~ V~ S " p , S +. P6~ " P5 i

×

p ~ / 2 m - p ~ + m - M * + ~ i r * - ( p, + p s ) 2 / 2 M * X

--

!-,~3 1 e"

N* - - -

.

p~/2m - p s"- p 6 -"f O R - ( p ~ (fll~)(f*/lz)(gN.a.)}V/2~ .

.

.

p~/2m-p°-fOa-(p,

p~/2m

+ps+p6)-/2ma +~iF" p,S

p 6 S + P5

+ps)2/2ma +~ir

p O_ pO + m - ( p, + p5 + P6)2/2M * + ~i t * - M*'

-

-iT~ "N*= - ( f / t . t )(f*/I.t ) ( g N . a . ) v ~ S • p 6 S +- p s o ' " p, i

i

vr-S - M* +~iF* vr-S-p~- ma - p ~ / 2 m a +~ir" •,'~3 N*

-z.~"

=--(flt~)(f*ll~)(gN*a.)~"

P l s " P5 s +

P6

p~/2m --pO--fOa--(p, +p6)2/2mj +~iF

p~/2m - p s"- p 6 +" m - ( p l + p s + p 6 ) 2 / 2 M * + ~ i F

'

*

-

M *"

_iT3h. N* = - ( r i t z ) ( f * l l ~ "(JgN• a . )"s x/2 " S p s S + . p ~ o ' . p, i

i

M* +½it* where

fO R =

ma - p ~ / 2 m a + ir'

m - ma.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

M.B. Johnson et al., Phys. Rev. Lett. 52 (1984) 593 Adler and Dashen, Current algebra and application to particle physics (Benjamin, New York, 1968) G.E. Brown and W. Weise, Phys. Reports 22 (1975) 279 B. Sakita and K.C. Vali, Phys. Rev. BI39 (1965) 1355 D.G. Sutherland, Nuovo Cim. 48 (1967) 188 R.P. Feynman et al., Phys. Rev. D3 (1971) 2706 R.A. Arndt et al., Phys. Rev. D20 (1979) 651 J. Gasser and H. Leutwyler, Phys. Lett. BI25 (1982) 321,325 A.N. lvanov and N.I. Troitskaya, Soy. J. Nucl. Phys. 43 {1986) 260 D.M. Manley, Phys. Rev. D30 (1984) 536

578

V. Sossi et al. / ZH(Tr+, ~r-)

11) D.M. Manley, Ph,D. Thesis, University of Wyoming, 1981 [Los Alamos National Lab. Report No. LA-9101-T, 1981] 12) E. Oset and M.J. Vicente-Vacas, Nucl. Phys. A446 (1985) 584 13) G. Jiikel et al., Nuci. Phys. ASil (1990) 733 14) R. Rui et al., Nucl. Phys. A517 (1990) 455 15) R. Rui et al., The (~r, 2~') reaction on few body systems, Proc. 12th Int. IUPAP Conf. on Few body problems in physics, Vancouver, Canada, July 1989 16) V. Sossi, UBC Ph.D. Thesis 1991, University of British Columbia, Vancouver, unpublished 17) Triumf Users Handbook (1987) 18) J. Lichtenstadt et al., Phys. Rev. C33 (1986) 655 19) R.J. Sobie et al., Nucl. Instr. Meth. 219 (1984) 501 20) F.M. Rozon et al., Nucl. Instr. Meth. A267 (1988) 101 21) N. Grion et al., Nucl. Phys. A492 (1989) 509 22) F. James, FOWL, A general Monte Carlo phase space program, 1977, CERN Computer Centre Program Library 23) T. Ericson and W. Weise, Pions and nuclei (Clarendon, Oxford, 1988) p. 29 24) S. Weinberg, Phys. Rev. Lett. 17 (1966) 616 25) S. Weinberg, Phys. Rev. Lett. 18 (1967) 188 26) M.G. Olsson and L. Turner, Phys. Rev. Lett. 20 (1968) 1127 27) C,W. Bjork et al., Phys. Rev. Lett. 44 (1980) 62 28) Particle properties data booklet, Phys. Lett. B239 (1990) 29) M.E. Sevior et al., Phys. Rev. Lett., to be published 30) L. Hulthen and M. Sugakawa, Encyclopedia of physics, vol. 39 (1957) p. 1 31) H.W. Ortner et al., Phys. Rev. Lett. 64 (1990) 2759