I •

Institute of Theoretical and Experimental Physics

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1

~

w \­ . Baru, J. Haidenbauer, A. Kudryavtsev,\~ V. Tarasov, V. Chernyshev

UDK 539.12 IvI-16

THE PERSPECTIVES OF STUDY LIGHTEST SCALAR ao(980) IN THE REACTION pd ~ ppaoPs CLOSE TO THRESHOLD: Preprint ITEP 30-001

V. Baru1,2, J. Haidenbauer2, A. Kudryavtsev1,2, V. Tarasoy2, V. Chernysheyl,2

- NI., 20QO - 2l fp.

We give estimations of total and differential cross sections for the reaction pn ~ ppao close to threshold. The experimental inforrnation OIl this reaction could be extracted from the process pd ~ ppaoPa on a deuterium target. The privilege of this reaction is that all final particles but (10 are protons. That is why the effective mass of an may be reconstructed easily. We discuss how the extracted cross sections of the reaction pn ~ ppuo Illay be used to get information on such fundamental characteristics of ao-meson as its nominal mass in and its coupling constants gao1r1/ and gaoKk.

Fig. - 7, ref. - 22 names

© HHCTHTyT TeopeTH"tJecKoB 11 3KcnepIDvleHTaJIbHOH q>H3HKII, 2000

1Institute of Theoretical and Experimental Physics, 117259, B. Cheremushkinskaya 25, Moscow, Russia

2rnstitut fur Kemphysik, Forschungszentrum Jiilich GmbH, D-52425 JUlich, Germany

1. INTRODUCTION

The study of the lightest scalar mesons 10(980)0+(0++) and ao(980)1-(O++)

is one of the most fundamental problem in physics of hadrons. Both these

mesons are strongly coupled to the Iik-channel and have practically equal

masses.

The standard interpretation of these mesons as qq-states [1] has difficulties

in explaining simultaneously both these facts. A more refined qq-model was

considered recently in ref. [2]. The authors of paper [2] have suggested, that

the 10(980) is an admixture of u,u, dd and s.9-states. At the same time the

ao(980) was build from non strange quarks only. There exists also an attempt

to build the 10(980) from ss-component only [3]. The coupling of 10(980) to

the non strange sector in this model is due to nonperturbative QeD effects.

Note, that in line with the models [1-3] it is difficult to explain the practical

coincidence of the masses for the 10(980)- and ao(980)-mesons. It is also worth

mention here the four-quarks model for lightest scalars proposed in Ref. [4].

This model could explain the mass degeneration for these mesons and their

large 'branching into 1{K-channel.

In Gribov's confinement model [5] the ao and 10 are rather exceptional

mesons and their properties should differ from the predictions of more standard

quark models like [1-4]. In particular in Gribov~s model the interaction cross

sections of the ao- and la-mesons with hadrons should be small.

We are to mention here also a more phenomenological "hadron molecu­

lar" model, in which both mesons are build from K and [{ [6]. The au­

thors of the paper [7] have developed a meson-exchange model to describe

the 10(980)- and ao(980)-mcsons. In their approach both these mesons orig­

inate from the coupling of the 1rw-channel to the I{[(-channel. But in con­

trast with the 10(980), the a.o(980) would not appear to be a 1ik bound

2

state but a dynamically generated threshold effect with a relatively broad pole

(ra[\ ~ 200.AleF, mao ~ 990.itleY). Note, that different approaches give diffe­

rent predictions for the parameters of the 0.0- and fa-mesons. So the precision

measurement. of parameters for the lightest scalars may really solve the prob­

lem of their nature.

Note, that the proposal to study properties of the at-meson in the reaction

pp ---t dat at the ANKE spectrometer of COSY was discussed already in

detail in ref. [8]. In our paper we discuss the possibility to study the reaction

pd ---t ppao ps to get information on properties of the negative ao -meson.

The resonance 0.0 (980) was observed for the first time in the reaction

K-p ---t I;+(1385)-177r- [9]. This meson manifests itself as a sharp peak in the

7I-1]-effective mass distribution. The mass of ao was found to be in = 981AJeF

and the visible width of the peak to be around 50f\.rfeF. The experimental

data of the paper [9] were analyzed in ref. [10J. In that paper a two-channel

parameterization for the S-matrix was used to get information on the pa­

rameters of the 0.0 -meson. It was found, that the visible width of the peak

r vis ~ 50MeF is smaller, than the bare width f '!f"'1 for decay of ao into 711]­

channel (f1f11 ~ 80MeV). The inequality r vis < r '!f11 is a consequence of the

J(k-threshold, which is strongly coupled to the 0.0(980) and is situated only

10MeV above the position of the peak. Because of that reason the form of the

peak is not of Breit-vVigner type bu t is strongly affected by the K R-threshold

(cusp). Note, that in the four-quark model [4J the bare width r 7f'l should be

much larger, namely of the order of 300MeV, see in this connection also ref.

[11]. That is why the form of the peak is sensitive to the quark content of

ao(980)-meson.

Note, that the experimental data [9J were not in contradiction to the fol­

lowing ratio for the coupling constants:

2 ga- K-K[\ 3

d=--T-=i (1) gOo1T-q

Thi~ value d corresponds to the assumption that ao belongs to the 5U(3)­octet of scalars and 1] - 1]/ mixing angle is zero, 87J7J' = O. Note, that in the

3

5U(3) _13 POqq-nonet interpretation of 0.0 [2] the ratio dis:

d= ~ A_ 2cos2( B1J1JI) ,

where A ~ 0.6. That is why the branching of ao-meson into the /i.-R-channel

is to be relatively suppressed in this model in comparison to the predictions of

the SU(3)-octet approach (d ~ 0.8 in [2]). The recent experimental data on 0.0­

production in the 1f-p-collisions [12]give evidence for the ratio d to be close to

unity. In particular, it follows from the analysis of ref. [12J, that the charged ad has a larger mass, ih, = (998±4)J\;IeV/c2

• Itswidthisf(at) = (69±11)MeV/c2

and the ratio d = 0.91 ± 0.11. Note, that from the analysis of the paper [13]

this ratio d is to be larger, d = 1.16 ± 0.18. In the four-quark approach (4]

the coupling to the Iiit-channel is to be large. We conclude, that the ratio

of the coupling constants is also very sensitive to the quark content of the

ao(980)-meson. Note, that in both experiments [9] and (12] only the 1frJ decay

mode of aa was investigated. That is why the ratio of branchings for charged

0.0 's was never measured directly.

In this paper we suggest to extract the parameters of the ao(980)-meson

measuring its production in nucleon-nucleon collisions. To be more concrete,

we suggest to study the reaction pn -t ppau using a deuterium target. To

reconstruct kinematics. one needs to measure all final protons for the reaction

pd -t ppaoPs, where Ps is the proton-spectator momentum. At the same time

both decay channels of the 0.0 contain negative particles and/or could be easily

identified through the photons from rJ-decay. In the case of identification of

both leading decay channels for the aD-meson the study of this reaction at

the ANKE spectrometer of COSY should provide a unique information 011

fundamental characteristics of this meson.

II. TOTAL CROSS SECTION FOR THE REACTION pn -t ppar; .

Consider the process for exclusive production of a resonance in NN­

collisions, i.e. the reaction

NN -t N1VR.

4

As we shall demonstrate, the results for the absolute value of total cross sec­

tion depend very drastically on measured interval of mass for resonance. The

explanation is simple: in case of a threshold reaction 2 -t 3 and under the

constraint that the production amplitude is constant. the total cross section is

proportional to Q2, where Q is energy excess. That is why the production of a

resonance with mass lighter than the nominal mass is getting more preferable

and one needs to introduce some cut-off parameter. Let us illustrate this by

some concrete formulae. In the case of production of stable particle of mass m

in the pp-collision ( pp -t pp + "stable particle") we get the following expres­

sion for the diprotoll IIlass spectrum near threshold (production amplitude is

taken to be constant!):

d(J ,..------,-:::­- ex F(k)kvQrnp - k2 (2)dmpp

Here F(k) is the proton-proton enhancement factor [14--16] and k is the rela­

tive momentum of the final protons, and m pp = 2mp + k2/mpo If we consider

the production of resonance of finite width r, we should convolute the expres­

sion (2) with the spectral density p(m,r). For a narrow resonance it can be

parameterized by a Breit-Wigner function

r/27f (3)p(m, r) = (m _ m)2 + r2/4'

where in is the averaged (nominal) rrU1SS of the unstable particle. Thus, instead

of (2) we get:

d(J ) l ;c r 127f I.. - 2~d- ex F(k k

mma

dm( -)2 r21 vmp(Q + m - m - k Imp), (4)m pp Trtm ;" 'In - m .+ 4

where Q = -IS - 2mp - fit is the nominal threshold~ mmax = Q + in - k2Imp

and the value of the minimal mass mmin will be specified below.

Rewriting the integral in (4) in dimensionless variables we get

d(J dm ex F(k)kl(xo); (5)

pp

where

(6)

5

Here we introduced notations

Q - /;,2 Imp Q + 1'11. - rnmin - k2Imp Xo = r /2 ' Xmaa: = r12 .

Now let us discuss, what is the reasonable value for cut off parameter mmiTl

in the expression (4). As is seen from the BW-formula (3), the probability to

find a very light 0.0 is small. So it seems, that the diproton mass spectrum

should not be sensitive to the choice of nlmin' But this is not a correct state­

ment. Indeed, taking in view that the lea.ding decay channel for ai)-meson is

ai) -+ "rr-11, let us chose 'lnmin = m rr + m1J' Making so, we immediately get

X max ~ 1 and I(xo) ex 11 J-xo for large negative xo. It means, that

da -d-- ex: const

m pp

at k2 ~ mpQ (i.e. for m « in).

So the distribution (5) in this limit has a long constant tail in the region of

large momenta k and the total cross section depends drastically on the chosen

value of mmin ( that is not the case for binary 2 -+ 2 reactioIls). In Figure 1 we

give the dependence of the total cross section for the reaction pn -+ ppu,(j for

some typical values of minimal mass mmin ::::::: in-C. The production amplitude

is calculated in a simple fusion model and is discussed in the next Section.

Figure 1 demonstrates the drastic dependence of the production cross section

on the value of the cut off parameter C. The contribution from the region of

the resonance peak with I m - m I~ r 12 to this cross section corresponds to

the choice of the parameter C ~ r.

III. PRODUCTION AMPLITUDE FOR THE REACTION pn -+ ppa(j

To estimate the production amplitude for the reaction pn -+ ppa.i) close to

threshold we use a simple fusion model with intermediateTJ- and 1T-mesons.

The corresponding Feynman diagram is shown in Figure 2. vVe use the stan­

dard values for the 1TN1V and ']NN coupling constants, g;NNI41r = 14.6 and

g~NN /41r ::::::: 3.0. a0171r-Coupling is determined by the relation

2 r - ga01r7] rrTJ - ~q1f1" (7)

6

where q1iTJ is the 1]JT-relative momentum. For the barE' partial width

r 11'1] = BOAlcV we get from (7) that 9aOTJ1i = 2.5.

The amplitude Af31 , which corresponds to the Feynman graph of Figure 2

IS

q91JNNFTJ(qf) 91TNN F1T (q2) _U(P3),5q-u.(Pl)il(P4),.5(iItt(P2) (8)M- 34 = V .t. 91r11ao m 2 " 2mp , 2mp (q - m;)(qi - m;)

Here in = 983.41\t[eV is the averaged mass of ao and Flr 'T/ are form factors,

2 2 p.(q2) _ Ai - mi z = IT, 1]. (9)

I - At _ q2'

We take Arr = 1300l\!IeV and ATJ = 1500k!eF in correspondence to the Bonn

pot.ential model [17]. The total production amplitude is to be antisymmetrized

with respect to spin and space variables of the final protons (34 ~ 43). Finally

for averaged production matrix element we get:

P2( 2)p2(' 2) 2 - 4 - 12- 2 2 2 rr q 1/ ,ql mpm

I (10)AI - 91rNN9'1NN9ao1rTJ (q2 - m;)2(qr - m;)2"

where q2 = qt = -n1p m. Taking into account the effect of the strong final

state interaction between the final protons ( E'nhancement factor F (k) ), we

get the following expression for the production cross section

(11)

where

4l(Q r) = [kmaxdkF(k)k2 [mmar dm r /27f , I (Q - _ ,_ k2 j ) , 10 '. 1mmin (m _ iii)2 + r2/4Vm p + Tn Tn , m p ,

mmin = iii - C, rnmax = Q + in - k2/mp , kmax = JrHp(Q + m - mmin),

P3 =2mpih./(2mp + m), 1= 2E1 1Pi 1= (2m p + ·m).jm(m + 4m p )/2. The momentum dependence of the pp-enhancement factor F(k) was dis­

cussed in details in refs. [14,15l. For the reaction with high momentum trans­

fer like N N ~ 1'11'1IT ('", 17', etc.) this momentum dependence practically

doesn It depend on any concrete form of productiOIl amplitude and is unive'r­

sal. At the same time the absolute value of the enhancement factor as it was

7

recently clarified in refs. (16,18] depends on the production amplitude and on

sort of the emitted particle. In our treatment we use the standard form of

the pp-cllhancement factor P'va(k), calculated with a separable Yamaguchi

potential [14]. The absolute value of enhancement factor is obtained by int.ro­

ducing the rescaling factor ~, so

The factor ~ reflects the suppression of tIlE' absolute vR,lur of the enhancement

factor in realistic N N-pot.ent.ials ( Bonn and/or Paris). In line with our

estimations in ref. [18] we take t = 1/5.

The resulting cross section versus excess energy Q is shown in Figure 1 for

different values of C. The expression (11) for the cross section ignores the

influence of the IfK -threshold. To get the expression for partial cross sections

pn --+ ppI<- KO and pp --+ pp7r-q we substitute the spectral density in equation

(3) by

(12)

In this formula we put again r 1r11 to be constant, but include the dependence

of r J{ k on mass:

(13)

where mf..--l\-O = ffiK-'ffigo/(mJ\- + mf{o).

In this two-channel approach the total and partial cross sections for the

reaction pn --+ ppao are shown in Figures 3a and 3b for different values of the

ratio d = 0,3/2 and 3- As it is seen from Figure 3a, the total cross section is

getting smaller, when gar; [{OJ.(- increases. This decreasing of the cross section is

a consequence of a destructive interference between the direct decay amplitude

ao --+ 7r'TJ and the amplitude for the decay ao ---t 'lr'TJ through the intermediate ](K-state.

8

IV. HOW TO EXTRACT PARAMETERS OF ao-MESON FROM THE

EXPERIMENT

"Ye are to extract the following pa.rameters for the aD -lllcson: its nominal

mass in. and the coupling constants gao1T1J and Yaol,/{' These are the parameters

which determine the form of the resonance peak.

The ratio of the coupling constants d (1) may be extracted from the direct

measurement of the ratio for branchings at a fixed mass of ao-meson:

-: 2 /Br(0,0 -+ K K) _ gaoK j( V 2rn!..:.- F:a (111, - mKo - mf(- ) (14)

Br( aO -+ 71'1]) - g;O'TrT] q1rT]

After getting the experimental information on the ratio (14) we have still

to determine two parameters in and gaa1rT/. \Ve see two alternative ways how

to extract these parameters.

i) Study of effective mass spectruln. In the case of the reaction

pn -+ ppao close to threshold the strong pp-FSI has to be taken into account.

The distribution over the effective 7f1]-mass is given hy the following expression

where kmax = Jmp(Q - m + m). In addition to the phase volume the

dependence of expression (15) on the mass m comes from the enhancement

factor F( k) in the integral:

rkmCtz F(k) ~7(ik2dk (16)10 V r..:max ­

The form of the resonance IIlasS spectrum given by expression (15), is presented

in Figures 4£1, and 4b for different values of r 1r'I7: ratio of couplings d and mass

'm of an. It illustrates the sensitivity of the mass spectrum to these parameters.

The Figure for dtj / dmf{ f.: at different values of d and r Jr77 is also shown (see

Figure 5).

ii) Binnie's method. In the paper of Binnie et. al. [19] this method

was used to study the production of w -,'1] - and lJ'-mesons in the reaction

1f-p -+ 11,.'\" . The method consists in studying the resonance excitation curve

lversus beam momentum under the CO:dition t.hat. the moment.um of the final

neutron is kept. fixed.

Considering the ease of the reaction pn --? ppar;. let us look at the expression

for the following differential cross section:

d6(J IAl(E. P, k)1 2 PUt') Cr1]

ikdP = 41 (161f)2mpj.l3vsIQ - P'2/2Il3 -1.:2/rnp + ir/2P (17)

where P = P3 + P4, 2k = PS - P4. Let us also fix the momenta P and k in

some small intervals I::i.P* and !\k* . In this case the dependence of t.he double

differential cross section d2(J /dP*dk* is determined by the following equation:

As is seen, the energy dependence for this expression (18) reproduces the form

of resonance. To apply this method, one needs to vary the beam energy and

to study the curve of resonance excitation of ao-meson. Note, that application

of both methods i) and ii) should help getting additional information on the

dependence of the production amplitude M( v'8) on energy ys.

V. SOME REMARKS ON KINEMATICS

In the case of the reaction pn --? ppan in the laboratory system close to

threshold all final fast moving particles are in a narrow cone around the beam

direction. The longitudinal momenta of all these three particles are of the

order of IGeV and the perpendicular momenta are small. For example for

energy excess Q = 50NIeV the maximal laboratory scattering angles for kaons

arc around 200 The scattering angles for T/ and Jr are larger because of larger •

energy release in the decay ao ~ 1fT/.

To measure the reaction pn --? ppao we are to use the deuteron target, i.e.

to study the reaction pd --? ppaoPs. We suggest to study this reaction for small

momenta of the proton-spectator, Ps ~ 2001VIeV/ c. In the Table we give values

of threshold energies for this reaction at some typical values of momentUIJl of

spectator. To estimate the cross section on deuteron, we suggest a simple pole

mechanism for this reaction. The corresponding Feyllman diagram is shown in

10

Figure 6. Usually pole approximation works well up to momenta of spectator

Ps ;S 150 - 200AleV. In this region all re-scattering diagrams give negligible

corrections to this leading pole t.enn.The distribution over the spectator

momentum which corresponds to pole mechanism of Figure 6 , is given by the

expression:

(19)

where 'l/;d(i) is the deuteron wave function in the momentum space. The

relation between sand s' is determined by kinematics. The distribution over

the proton-spectator momenta for y'Si being fixed is shown in Figure 7. Note,

that near the threshold the production cross section for the reaction pn -t ppao

is a rather sharp function of.jSi. That is why the expression (19) at fixed

energy of beam may be used to study the energy dependence for the cross

section pn -t ppar; .

Table 1:

ps(l'·-feV/c) cosPP. Tp{tvleV/c2 ) 1~(Mf;V/C)

0 2454 2159

100 -1 2884 2268 100 0 24tH 2169

100 1 216.5 2080 200 -1 3538 2434

200 0 2577 2198

200 1 1971 2025

300 -1 4581 2678

300 0 27:38 2248

300 1 1846 1987

VI. IS IT POSSIBLE TO EXTRACT THE aoN-AMPLITUDE FROM THE

REACTION pd -+ ppaoPs?

In Section 5 we suggested to use pole approximation to estimate the produc­

tion of ao on deuteron at small spectator momenta PH ~ 200MeVjc. Usually

at small Pa this pole mechanism works well and all the corrections due to the

FSI- effects are small.

11

The produced 0.0 interacts \vit.h the fast protoIls as well as with the slowly

moving spectator. \Nhat about the interaction of the 0.0 with spectator, it is evident.ly negligibly small in this kinematics. The main reason is, that the

produced 00 moves in forward direction with high momentum. That is why

the total energy of the system "ao+ spectator" is also large. In Table we

present values for invariant energy ylsaop. for some typical values of momenta

and angles of the spectators. As is seen from the Table, typical values of .jsaop.

are large, much larger, than mao + rnp, = 1922N[elf. That is why the process

of interaction of the produced mcson with the proton-spectator should be

considered as only a small corrections to the main amplitude. The signal of this

interaction may be definitely found only in the case, when thc spectator moves

along the direction of beam and has momenta > 3001\1eV/ c. Unfortunately

the number of events in this geometry is seriously limited because of small

probability to find in deuteron a proton with a large momentum, see also

Figure 7.

At the same time the interaction of the 0.0 with a fast proton may be ob­

served (if exists or if is not negligibly small; in Gribov's approach it is small,

see [5]). Near the threshold in their rest frame both protons and 00 move slowly.

That is why the interaction between 0.0 and proton is effected and gives serious

influence on aop- and pp-effective mass distributions. In the papers [14,20] we

gave some examples of these distributions for the cases of 7/- and 'fJ'- production

in the pp-collisions. In the case of ao-production these distributions are to be

corrected because of a large width of the produced 0.0 in compari~on to that

\ve have in TJ and TJ' cases.

To conclude, let us stress again, that the measurement of momenta for fast

protons as well as spectator momentum gives possibility to reconstruct the

effective mass of the au and identify the process. Additional measurements

of the final states for the au-meson should provide opportunity to find the

mass and the coupling constants for this lightest negative scalar. The similar

idea for measuring the reaction pn -+ ppTf- on deuteron target at 2 GeV was

realized in the ref. [21]. Recently the reaction pd -+ pprr-Ps was also studied

in the ref. [22].

12

Acknowledgnlents

The authors are very indebted to V.A.Gani for an invaluable help in the

preparation of the manuscript. A.K. acknowledges the hospitality of the Inst,i­

tute fUr Theoretische Kernphysik IKP, Forsdmngzentrum Julich. A.K .. V.B.

and V.T. are also thankful to the Russian Foundation for Basic Research (grant

98-02-17618) for partial support.

13

REFERENCES

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[2] V.V.Anisovich, E-print hep-ph/9712504, Dec. 1997

[3] A.Veinstein et al., EPAN (Elementarnie Chastizi Atomnoe Yadro), 13,542

(1982).

[4] N.N.Achasov and G.N.Shestakov, Sov.Phys.Uspechi, 161, N.6, 53 (1991);

142,361 (1984); Z.Phys., C 41. 309 (1988).

[5] V.~.Gribov, LU-TP-91-7;ORSAY Lectures on Confinernentl, LPT HE­

ORSAY-92-60; V.N.Gribov, Eur, Phys ..l., CI0, 91 (1999); F.Close et aL

Phys.Lett., B319, 291 (1993).

[6] J.\tVeiustein, N.ISgllr., Phys. Rev., D41, 2236 (1990); N.Isgur, KJvlaltman,

J.\Veinstein and T. Barnes, Phys.Rev.Lett., 64, 161 (1990).

[7] G.Janssen, B.C. Pearce, K. Holillde and J.Speth, Phys. Rev., D52, 2690

(1995).

[8] L.A.I\:ondratyuk et al., Preprint ITEP-19-97, :Moscow (1997);

V.Tchernishev et aL COSY proposal N .55: "Study of at-mesons at

ANI\:E" (1997).

[9] J.B.Gay et. a!" Phys. Lett., 63, 220 (1976).

[10] S.l\-f.Flatte, Phys. Lett., 63, 224 (1976).

[11] N.~.Achasov, S.A. Devyanin, G.K.Shestakov., Yad.Fiz., 32, 1098 (1980).

(12] S.Tcigc ct al., Phys. Rev. D, 59, 012001 (1099)

[13] D.Dugg et al.. Phys. Rt"v. D, 50,4412 (1994)

[14] B.L.Druzhinin, A.E.Kudryavtscv and V.E.Tarasov, Z.Phys., A359, 205

(1997)

[15] C. Wilkin, G.Faldt, Phys. Ldt.. B382, 208 (1996)

[16] C.Hanhart and I\:.~akayallla, E-print nucl-th/9809059. Sept. 1998.

14

[17] R..~Iachleidt, K.Holinde and Ch.Elster, Phys.R.ep., 149, Nl (1987)

[18] V.Barn, A.Gasparyan, J.Haidenbauer, A.Kudryavtsev and J.Speth,

E-print nucl-thj0006075, to be published in Phys.At.Nucl., 64 (2001)

[19] D.M.Binnie et al., Phys.Rev. 08, 9, 2789 (1973).

[20] V.Barn, J. Haidenbauer, C. Hanchart, A. Kudryavtsev, P.lIIoskal and

J.Speth, Eur. Phys. J., A6, 445 (1999)

[21] B.Abramov et al., Z.Phys., C69, 409 (1996).

[22] F.Duncan et al., Phys. Rev. Lett., 80, 4390-4393, (1998)

15

100

f

I I75 I

I I

I /

I I I ,I I, I ,I I:0 I, I.s 50

'I I t:> I

I I

I I

//

/I25

I I

/'/

o1....-._-.!~__--J,or--__....I....-__--L- I-_-..-l

-50 0 50 100 Q [MeV]

FIG. 1. Total cross section jor the reaction pn -t ppao versus excess energy Q at different values ofm.min = lh - C. Solid line - C = 290AfeV. dotted line - C = 200;\1d', dashed line ... C = 1OOAleF, long-d(L8hed line - C = 50AfeF . r = 50..:\1eF. The curve for the prvduetion CroS8 section in the case oj stable ao-me.'30n i.s al80 .shown.

16

p p,if; I P4

111 (-)I ao

c:>

IInC-)

P';fJ' ~

I

n p

FIG. 2. Feynman diagram for the production amplitude of the reaction pn -r ppan.

100

17

:c .s t>

50

o

I I

/ /

//

//

//

//

" /

" " ",".,;'-------

-50 o 50 100 Q [MeV]

FIG.3a. Total eTa.';8 section for the reaction pn -7 ppao at different values of d = !l-f'- L'U/g'2- _ . Solid line- d = 0, dotted line - d = 1.5 . dashed line ­

00 \ 1\ U o 7r 7J

d = 3.0. r = 80Jlel', C = lOOl\!clT

fOT all con.~i(h:r>ed cases.

18

8.-------.---'----T""-----.-----r----"T-------,

6

I I I I I I I

I I I

I I

I I

I2 I

I I

I I

I I

I /

/':;.."o L-__.-a. ...1.,.. !:::-""----_--l --L.. -J

-50 0 50 100 Q[MeV]

FIG.3b. Partial cross sections pn -t ppA-- [(0 a.t different vq,lues of d. Solid line - d = 3.0, dashed line ~ d = 1.0. The values of rand C are the sa.me as in Figure 30..

19

0.6

0.4 :> Q> ~.0 -.s ~

E

~ ""0

0.2

a

,r, I I I I I I

I '

,II :I :

I I , I

I : I I

I I

I :I ,

I I

I :I ,

I I

I ' I :

/ / ,I

/ ' / II "

/ ~' / ~

I ,~~

I " I ""

I " ......... ' I "

"

600 700 800 900 1000 1100

mml' [MeV]

FIG. 4a. Differential cross section do- / dm TrTI at different values of d and r. mao is equal to 983.5 MeV. Solid line - r = 80AfcF, d = 1.5. long-dashed line

r = 150A1eF, d = 1.5, dashed line - r = 80ii1eF, d = 3.0. All curves on th.e figures 4 and 5 are pre8cnted at Tfab = 2636Jl.ltV.

0.4 I I I I I I I I I I I

I, I

I I

I /

//

" /.",,;

,..'",..'"

L.--_--L-_---L-_-..1__-L--_.....l----J~_--J

20

0.6

:> Q)

~ -... .0 .s

S:­to:

E ~ t> "0

0.2

.,,'" OL-_......L-_~__600 700 800 900 1000 1100

mml' [MeV}

FIG.4b. Diffcrcntial cross section d(JIdm 1f'f) at different masses of ao-meson but at fixed d and r (d= 0.91.. r = 80 MeV). Solid line - mao = 983.5.l\leV. dashed line- mao = 999l\.1eV.

21

0.1

0.08

:> 0.06 Q)

~.c oS -

I :ll::

o :ll::

E o""0 0.04 "0

0.02

o

,­, , "-" , ,

I ,

, I ,

' , ' , ,,. ", ,I , , I ,

I ' I , " , ' , '

,,"',", , ",\ ",\

~ " ,~,~ ~ ~ ~

990 1000 1010 1020 1030 1040 mK\-[MeV]

FIG..5. Differential CTOSS .section du / dm I-.:i: at difj'erent values of d and r. mao is equal to 98.1. 5 Me V. The notation of the curves see in the caption to Figure 4a.

22

­- -_---- ao

n

FIG. 6. Feynman diagram for the reaction pd --7 ppaoPs.

23

0.01

OL_--'-_-L-_--'­__L---_...I----=:::r:::===:z==:==J 100 300 400

0.005

o 200

PSI MeV/c

FIG. I. The distribution over the p7'Oton-8pcctator' momentum for Vii being fired.