A three-body calculation of the πd → πnp reaction in the Δ-resonance region

Nuclear Physics A379 (1982) 41528 Q North-Holland Publishing Company

A THREE-BODY CALCULATION OF THE ard-~ wrap REACTION IN THE 0-RESONANCE REGION A. MATSUYAMA Department of Physics, Faculty of Science, University of Tokyo, Bunkyd-ku, Tokyo 113, Japan Received 20 July 1981 (Revised 26 October 1981) Abstract : The reaction d(~r* , per* )n at incident pica momentum 340 MeV/c is analyzed based on a relativistic three-body fonmalism . The contributions of the various reaction mechanisms such as impulse processes, pica multiple scattering and nucleon-nucleon final-state interactions are investigated for several cases of typical kinematics. The impulse tenor is dominant when the recoil neutron momentum is small . On the other hand, the NN final-state interaction is found to be relatively important when the recoil neutron momentum is large . The effects of dibaryon resonances which Gave been suggested in ~rd elastic scattering are estimated using a phenomenological model. Comparisons with other work are also made.

1. Inirodaclion In recent years, great efforts have been devoted to the study of the dynamics of a NN-~rNN system, both theoretically and experimentally . The three-body i-6), formalism which is very popular at present, can in principle treat all the relevant processes such as ~rdH~rd, ardHNN, NNHaNN and NNHNN simultaneously. Nevertheless, theoretical works have been concentrated mainly on the two-body channels such as rrdH ~rd, rrdHNN and NNHNN. This is because the experimental data for the breakup reactions are not so plentiful as those for the two-body reactions. ïn addition, the kinematics of three-body final states is complicated for the analysis. Recently, beautiful experimental data for the ~rd-> ~rnp reaction were reported by Hoftiezer et al.'), in which kinematically complete measurements of the differential cross sections were made fqr the incident pica momentum 340 MeV/c. These authors also calculated the cross sections and suggested the possible existence of the J p = 2 + dibaryon resonance. Their conclusion is closely related to our previous work a) where we investigated the dibaryon resonance effects on rrd elasticscattering and concluded that the Jp = 2 + dibaryon resonance may be important. Stimulated by their works, we have calculated the differential cross section for the rr*d-s a *np breakup reaction based on the relativistic three-body formalism, and investigated the dibaryon resonance effects. In sect. 2, we review briefly the basic equations in the three-body theory and describe the practical method of computation. Numerical results are presented in 415

A. Matsuyama / ~rd -~ anp

41 6

sect . 3, where we discuss the importance of various reaction mechanisms such as impulse processes, pion multiple scattering, NN final-state interactions and dibaryon resonance effects. A summary of our analysis is given in sect . 4. 2. Three-body equations and practical calculations 2.1 . BASIC EQUATIONS For the case of a arNN system, there are two kinds of interacting pairs, namely (TrN) and (NN) pairs. Hereafter we denote N(~rN) three-body channels by Greek labels a and ß, and ~(NN) three-body channels by Latin labels m and n. The coupled integral equations for rearrangement T-matrices are written symbolically as follows''Z): Tn .m =~ B~ .~G~Ta .m Ta.m = Ba.~ +~ Ba.aGaTa.m +~ B~ .nGnT~.m n a where B~,Q, BQ, and BQ,ß are the Born terms which represent one-particle (~r or N) exchange interactions . Their explicit expressions are Ba .m(RP'+S)° B~.a(RP'+S)°

g°`(q)(Er+~r'+Er+r')Bm(q~) g°`(q)(Er+Er'+tvr+r')Sa(4')

where q and q' aie the relative momenta of the interacting pairs, and the two-body interactions are assumed to be separable. In eq . (2), ga and g,  are the vertex functions for a -> ~rN and m -> NN transitions respectively, and Er and mr are the energies of the nucleon and the pion respectively . GQ and G, are the propagators of N(~rN) and -rr(NN) systems, with a general expression for both G~ and G g G- (~) = A

-

1

~

d3k

2(k)(Et +Ek)

where ./~ is the total energy of the interacting pair . We use the energy-dependent coupling strength A n~ _ ~ - mn for the P33 (d) channel of the aN interaction . Forthe 3S,-3D1 channel of the NN interaction, we use the following type of propagator which has a correct deuteron bound-state pole : G~1(Q)=(~-Mâ)

8â(k)2 1 _d3 k (2~r) 3 ~ Ek (Q + - 4Ek)(Ma -4Ek)

(4)

A. Matsuyama / ad -" anp

41 7

The relation between the rearrangement T-matrices, T°,, and breakup T-matrix To.m, is

T~ .m,

and a

(s)

To .m = ~ B~GaTa .m +~ BnGnTR.M . a n

Therefore, if the two-body interaction parameters are known, we can calculate the three-body breakup reaction cross section unambiguously within the framework of three-body theory . In the c.m . system, the differential cross section is expressed in terms of the breakup T-matrix ToP,a as d3 v _ 1 (8Ef _ 9sPs -1IT~.na.~a~z~ dq 3 dflq, d.fl~ 128~rs 8po ~ElE2E3 \âp3~

(6)

where El , Ez and E3 are the energies of the outgoing proton, neutron and pion ; p3 is the momentum of the outgoing pion; qs the relative momentum between the outgoing proton and neutron; po the incident pion momentum, and "~S the total energy in the c.m . system . The energy of the final state Ef is expressed in terms of ps and q3 as Et(Ps, 43) = Lfgs-zPs)z+Mz ]' n+L(qs+iPs)z+Mz]i~z+(93+N

.z)'~z

where M and ~ are the nucleon and pion masses respectively . The transformation from the c.m. system to the laboratory system is straightforward . 2.2 . TWO-BODY INPUTS AND PRAC'T'ICAL CALCULATIONS

In the practical calculation, we take into account Sll, S31, P13, P3i and Pss(4) channels for the ~rN interaction, and 3 S1 3Di, 1 So, 1 P1, 3 P1 and 3 Pz channels for the NN interaction. The effect of true pion absorption cannot be included in the present formalism and is ignored. We will make some remarks on the true pion absorption in subsect. 3.3. The relevant parameters in these two-body interactions are taken from the works of Rinat and Thomasl8), and Rinat et al. lb). The D-state probability of the deuteron is fixed to be 6.7% . Calculations are done also with the deuteron wave function of Giraud et al.z) (SF6 .7 in their notation) in order to investigate how the detail of the wave function affects the cross section. We consider as many diagrams as possible by solving the relevant integral equations, because it is crucial that we calculate accurately the multiple-scattering processes in order to make a definite statement on anomalous effects such as dibaryon resonances . Fig. l shows diagrams considered in our calculation. Diagrams (1) and (2). are the impulse terms and diagrams (3) to (7) represent the higher-order processes. We calculate the dominant impulse term (1) for the total angular momentum J ,10, while the other diagrams (2) to (7) only for J , 3 . We have estimated the contributions of diagrams (2) to (7) for the higher partial waves and found them to be very small (~5%).

41 8

A. Mabuyama / ~rd-~ anp

2

d Fg. 1 . Diagrams of the reaction ~rd-~ ~rnp. The box means the full process generated through the iteration of the d-channel and the d-channel . S .P. means 5 11 , 531, P13 and P 31 channels for the ~rN interaction, and 1 So,'P1, 3P 1 and 3Ps channels for the NN interaction .

In order to solve eq . (1) numerically, we use the method of contour deformation to avoid the singularities of kernels. The singularity structure in the variable p' of the Born term Bm,~ ( p, p' ; S) is dependent on the outgoing pion momentum p. Three different possibilities are shown in figs . 2 together with the method of contour deformation 9). In the actual kinematics of the experiment considered here, po = 287.6 MeV/c, p,=286 MeV/c and p 2 =268 MeV/c. Fortunately the momentum range of the pica corresponding to figs . 2(1), (2) is very small and most of the experimental kinematics correspond to fig. 2(3) . We calculate only for the case of fig. 2(3), because the kinematics of fig. 2(2) is much more complicated for the calculation than those of fig. 2(3) . In fact, there are fewexperimental points in thesmall outgoing proton momentum region which correspond to the kinematics of fig. 2(2) . For those kinematics, we extrapolate the calculated values from fig. 2(3) . 3. Results and discussion The ~r~d-> rr }np experiment') was performed at two extreme kinematics . In one kinematics, the outgoing neutron momentum is small and the impulse term is expected to be dominant. (Hereafter we call this kinematics "region 1" .) In the other

R>P>R

R>P>R

ß>P

Fig . 2. Singularity structure of the Born tenor B,,Q (p, p' ; S) in the brealnlp reaction .

41 9

A. Mabuyama / ~rd -~ ~rnp

kinematics, the outgoing neutron momentum is large compared to the deuteron internal momentum, and processes other than the impulse term are also expected to be important. (Hereafter we call this "region 2".) For the present calculation, we choose six typical cases, three cases for each region, out of the eleven cases reported in the work of Hoftiezer et al. They are listed in table 1. Tns~ 1 Kinematics of the experiment

Bp 9~

Case 1

Case 2

Caae 3

Case 4

Case s

Case 6

30° 80°

30° 105°

40° 95°

50° 95°

50° 105°

60° 100°

Angles are in the apposite side and are measured relative to the incident pion beam .

3 .1 . THE IMPULSE APPRO}üMATION

Fig. 3 shows the experimental data and the results of our calculations . Cases 1 to 3 correspond to region 1, and 4 to 6 correspond to region 2. In table 2, column A gives the relative momentum of a pn pair in the initial deuteron of the impulse diagram (1) in the case of a neutron spectator . The neutron spectator process gives many moré contributions to the impulse term than the proton spectator process in the present kinematics. It can be seen from this table that in region 1 the impulse term is dominant due to the small relative momentum of the pn pair, and reproduces the data qualitatively, but not always quantitatively . In contrast, the relative momentum of the pn pair is large in region 2, and the impulse term gives only small contributions. 3 .2. HIGHER-ORDER CONTRIBUTIONS

It is convenient to classify processes (2) to (7) into two groups . One group is composed of processes (2) and (3) where the ~rN pairinteracts finally, and the other is composed of processes (4) to (7) where the NN pair interâcts finally. We call the former ~rN-FSI (final state interaction) and the latter NN-FSI. Fig. 3 shows also these higher-order contributions. In region 1, although the impulse term is dominant, the higher-order processes contribute significantly, in particular for .the ~-d-> ~-np reaction . All included, the theoretical results still disagree with-the experiment by a factor of 2 in some cases. In region 2 the higher-order processes produce large effects on the cross sections. In this region the contribution of the -rrN-FSI is relatively small and that of the NN-FSI is very large . In table 2, column B gives the relative momentum between the pn pair in the final state, which ranges from 100 to 400 MeV/c. In this momentum range, the NN-FSI shoulsi have a considerable effect . In fact, in region 2, the NN-FSI contributes much more than the impulse term . In our analysis, the NN-FSI increases the results by a

420

A . Matsuyama / ~rd -~ ~rnp

CQSe 2

v

350

400

450

500

Pp Q~IeV/c)

550

350

Fig . 3 . Experimental data and the results of three-body calculations. O and ~ are the data points for a' and ~r- respectively. Dashed lines show the impulse term (1), dot-dashed lines show the contributions of the sum of diagrams (1), (2) and (3), and solid lines show the contributions of all diagrams (1) to (7) .

A. Matsuyama / ~rd-" arnp

421

Case 5

Case 6 -to

-10'

350

500 550 350 Pp(MeV/c) Figure 3 (cont)

450

500 Rp(Me1/~c)

550

factor of 2 to 5, and improves the agreement. The contribution of the NN-FSI is almost solely due to diagram (4). 3.3 . SENSITIVTTY TO THE DEUTERON WAVE FUNCTION

We have performed similar calculations using two different kinds of deuteron wave functions, because the NN-FSI is found to be important, especially in region 2 as mentioned before . Fig. 4 shows the results calculated by using the deuteron wave TABLE 2

Column A: Relative momentum of the pn pair in the initial deuteron contributing to the impulse term (neutron spectator case); Column B: Relative momentum of the pn pair in the final state Case

A

B

1 2 3 4 5 6

105-. 65 -. 212 156-. 14 i202 128-. 52-.189 154-+ 124i276 196-.165 -+ 357 222-.215-.347

136-370 83-370 99-354 120-439 126-448 162~20

The values correspond to the data points of minimum PP (outgoing proton momentum) to maximum Pp in units of MeV/c.

422

A. Mattuyama / ~rd i ~rnp r

350

4

--

r

Case 1

450

500

Pp (MeV/c)

550

350

400

450

Pp

500 (Me4/c)

~0

Case 5 ~~- Rinnt etal . --:-- Giraud et al

Rinnt etal . --_.- Giraud et

~to

a

u

~t6'

500

400

450

900 Pp(MeV~c)

550

Fig . 4. Sensitivity to the deuteron wave function . Dashed line and dotted line correspond to the impulse term (1) . Solid line and dot-dashed line correspond to the sum of diagrams (1), (3) and (4) .

A. Mauuyama / ~rd-" arnp

423

functions given by Rinat et al.') and Giraud et al. Z). We calculate the contributions of the impulse term (1) and those of the sum of processes (1), (3) and (4) ; for the purpose of estimating the sensitivity to the deuteron wave function, diagrams (2), (5), (6) and (7) may be dropped since the typical contribution from these diagrams is 20% at most . Agreement between these two cases is good for the impulse term, because it reflects only the low-momentum component of the deuteron wave function . However for the sum of processes (1), (3) and (4) there is a considerable difference, especially for region 2. This is because process (4) involving a loop integral is strongly affected by the behavior of the high-momentum component of the deuteron wave function . It might be noted that we have obtained a better agreement with the data in region 2 by using the parametrization SF6.7 which, according to ref. Z), is more realistic than that of Rinat etal. In region 1, there is no remarkable improvement and the quantitative discrepancy still remains. Thus the disagreement between theory and experiment in region 1 seems to be quite persistent . We will make in subsect. 3.5 a comment on the "improvement" which Hoftiezer et al. obtained by multiplying the impulse approximation result with an attenuation factor. One ingredient missing in the present framework of the three-body theory is an effect of true pion absorption . For the ~rd elastic scattering the true pion absorption effect was estimated by Rinat et al. l~ and Fayard et a1. 1°) who found that it is appreciable in the backward angles but not at forward angles where the impulse term is dominant. We may expect that a similar situation holds in the breakup process. Thus the discrepancy in region 1 probably will not be resolved by this mechanism. We have tried to calculate the effects of true pion absorption by including the Pll channel for the aN interaction in the lowest order with a separable parametrization by Schwarz et a1. 11). The results are shown in fig. 5 for cases 1 and 4. In this unsymmetrical treatment of two nucleons in the intermediate state (namely the true nucleon N and the Pl , channel of the zrN pair), however, there are serious problems such as violation of the Pauli principle and double counting, which is pointed out by Avishai and Mizutai ~) . In order to treat the true pion absorption we must e` incorporate the NN channel consistently in the theory of the ~rNN system .b) . which is beyond our present calculation. Although fig. 5 should be regarded only for demonstration purposes, the effects of true pion absorption are expected not to affect the qualitative feature of the results. Another point to be checked is the off-shéll effects of the aNd vertex . We have examined them by changing the cut-off mass from 355 MeV to 1000 MeV for the impulse term, and found that the effects are only of a few percent. 3.4. DIBARYON RESONANCE EFFECT

We next investigate the possible dibaryon resonance effects on the ~rd breakup reaction which was suggested by Hoftiezer etal. Possible dibaryon resonance effects

42 4

A. Matsuyama / ~rd -~ ~rnp

Fig. 5. Effects of true pion absorption through the P ll channel of aN pair. The upper and lower lines correspond to a + and ~- case, and solid line and dot-dashed line show the resultswithoutP l1 channel and with P l1 channel respectively. Only the solid line is drawn where the two lines coincide .

on the elastic scattering have been pointed out by Kubodera et al. tz) and Kanai et al. t3). According to the estimate of the decay width of the dibaryon resonances by Duck et al. l4) and Grein et al. ts), they predominantly decay into ~rNN channels. Therefore the aNN channel is most suitable for the search of their signals 16). In our previous work a) we assumed that the dibaryon resonances are quasi-bound states of the Nd system and parametrized the Nd interaction so as to reproduce the NN scattering data and at the same time improve the agreement with the ~rd elastic scattering data . We use the same model and parametrization in the present analysis. In this model, we replace the Born term Bxn.xa in eq . (1) by BN,.x, =Bxe .xe+ VNe,Ne, where VrRra,N, is an attractive interaction of a separable type between N and d which generates the dibaryon resonance. The explicit form is VNd,Nd (PW,

9v) = Avw(P)v~(4)

Uw(P)=awvz, .(P) ~ 1

(g) 2

4

m,. ~=i ßi The range ßi, the strength A and the decay ratios a 's are chosen to reproduce the observed dibaryon resonances in the NN scattering data. At this energy of the experiment (~/S ~ 2217 MeV), the dibaryon resonances tDZ( ~ 2170 MeV) and 3F3(=2220 MeV) [ref. t')] seem to contribute . But we take

A. Matsuyama / ad -" ~rnp

42 5

into account only the'DZ resonance here, which is consistent with the suggestion by Hoftiezer et al., since we failed to find consistent interaction parameters for the 3F3 resonance in the previous work 8). Fig. 6 shows the calculational results for the cases 1, 4, 5 and 6. The effects of the dibaryon resonance are not so dramatic as suggested by Hoftiezer et al., but they seem to shift slightly the result in the right direction. It may be possible to improve the agreement by varying the parameters of the interaction VN,,Nd, but we do not make such an attempt here, since those parameters were chosen to be consistent with the NN and ad scattering data . As mentioned in subsect. 3.3, the choice of the deuteron wave function by Giraud et al. makes a significant improvement in region 2. Hence a more improved agreement may be expected if their parameters are used in the calculation of the background processes. This is however not a consistent procedure, since our parameters of the dibaryon resonance are determined relative to the background calculated with the parameters of Rinat et al. Therefore, fig. 6 should be regarded as a qualitative indication of the effects of the dibaryon resonance relative to the background processes. Kubodera et al. lZ) pointed out also that the dibaryon resonances are expected to give stronger signals on spin-dependent observables. We calculate the polarization of the recoil proton in the reaction ~r~-> anli to study its sensitivity to the dibaryon resonances . The result of a typical case is shown in fig. 7, where the polarization is along the incident pion momentum . The signals of the dibaryon resonance are more evident for the polarization than for the cross section as is expected. 3 .5. DISCUSSION

Now let us compare our results with the analysis of Hoftiezer et al. As far as the impulse term is concerned, our results are in good agreement with theirs. As for the higher-order corrections, however, there are disagreements in two points, that is, in the estimations of the effects of the pion multiple scattering and the treatment of the NN-FSI. Firstly we discuss the effects of the pion multiple scattering . In ref.'), the pion multiple scattering correction is evaluatedby summing up the iteration of rrd-> Nd -> rrd in the entrance channel, and it is concluded that this correction reduces the cross section by about a factor of 2 for small outgoing neutron moments. In contrast, our calculation of the pion multiple scattering (diagram 3 in fig. 1) increases the cross section over the wide range of the recoil proton momentum Pp. (Here we note that, although diagram 3 does not exactly coincide with the DWIA diagram in their notation, its contribution should be qualitatively the same as that of their DWIA calculation.) This discrepancy arises in the following manner. Hoftiezer et al. take into account only the on-shell propagation of the ad system in the intermediate state. It was however found in our previous calculation of the -rrd elastic scattering that the

426

A . Matsuyama / ad i ~rnp

450~ (M500~ ) ey

550

350

Case Ô D, R. Effect

Case 5 D.R, Effect

~,b

_70-i

3

4

450

500 Pp( MeV/c)

550

350

400

450

500 Pp (MeV/c)

550

Fig. 6. Dibaryon resonance effects on the cross sections. Solid line shows the background process. Dashed, dotted and dot-dashed lines represent the effects of dibaryon resonance with paraméter sets 1, 2 and 3, in ref . s), respectively .

427

A. Matsuyama / ad -~ m~p

350

400

450

500

550 PpiMeVi~

Fig. 7 . Dibaryon resonance effects on the polarization of the recoil proton . Solid line shows the background process and dashed line repreaents the effects of dibaryon resonance with parameter set 1 in ref . s ) .

off-shell propagation of the ~rd system is as important as the on-shell propagation, and that the two terms approximately cancel each other. If we consider only the on-shell propagation of the rrd system, it would cause a fictitious reduction of the theoretical elastic cross sections by a factor of 2 -~- 3. Judging from this, the reduction of the cross section obtained in ref.') seems to requ~e more investigation. Secondly we investigate the contributions of the NN-FSI. The authors of ref.') calculated the NN-FSI with a model of v-meson exchange potential. This somewhat simplified treatment of the NN-FSI seems to underestimate its contribution . In our analysis the discrepancy of the impulse term and the data in region 2 is almost explained in terms of the NN-FSI, and the effects ofthe dibaryon resonance need not to be so large as in their analysis. Thus, at present, itseems difficult to make a definite statement about effects of the dibaryon resonance on the -rrd->-rrnp cross sections . 4. Concloeion

We have calculated the cross section of the reaction d(a~, prr~)n at the incident pion momentum 340 MeV/c based on the relativistic three-body formalism. In region 1 which corresponds to the small neutron recoil kinematics, the impulse wntribution is dominant and determines the gross structure of the cross section, but the higher-order processes also contribute appreciably in some cases. Although the calculation reproduces the experimental data within a factor of 2, the agreement is not so satisfactory. This is a serious problem in our analysis . On the other hand, in region 2 which corresponds to the large neutron recoil kinematics, the impulse term is relatively small and the NN-hSI contributes significantly . This NN-FSI is sensitive to the deuteron wave function and the agreement with the experimental data is

428

A. Matsuyausa / ord-~ ~rnp

improved by using the more realistic wave function . The effects of the dibaryon resonance are estimated based on our phenomenological model of ref. 8). They are found to be not so dramatic as in the analysis of Hoftiezer et al., but to improve the agreement slightly . I would like to thank K. Yazaki, K. Kubodera and H. Hyuga for useful discussions and comments. The computer calculations of this work have been financially supported by the Institute of Nuclear Study, University of Tokyo . References la) A.S. Rinat and A.W . Thomas, Nucl . Phys . A282 (1977) 365 ; lb) A.S. Rinat, E. Hammel, Y. Starkand and A.W. Thomas, Nucl . Phys . A329 (1979) 285; A.S . Rinat, Y. Starkand and E. Hammel, Nucl. Phys. A364 (1981) 486 2) N. Graud, Y. Avishai, C. Fayard and G.H. Lamot, Phys . Rev. C19 (1979) 465 ; N. Giraud, C. Fayard and G.H . Lamot, Phys. Rev. C21 (1980) 1959 3) H. Garcilazo, Phys . Rev. Lett. 45 (1980) 780; Phys . Rev. C23 (1981) 2632 4) W.M. Klcet and R.R . Silbar, Nucl. Phys . A338 (1980) 281, 317 ; Nucl . Phys . A364 (1981) 346; Phys . Rev. Lett . 45 (1980) 970 5) M. Araki, Y. Koike and T. Ueda, Prog . Theor. Phys . 63 (1980) 335 6a) Y. Avishai and T. Mizutani, Nucl. Phys. A326 (1979) 352; Nucl . Phys . A338 (1980) 377 6b) LR. Afnan and B. Blankleider, Phys. Rev. C22 (1980) 1638 7) J.H . Hoftieur et al., Phys . Lett . 8ßB (1979) 73 ; Phys . Rev. C23 (1981) 407 8) A. Matsuyama and K. Yazaki, Nucl. Phys. A364 (1981) 477 9) R. Aaron and R.D. Amado, Phys. Rev. 150 (1966) 857 10) C. Fayard, G.H . Lamot and T. Mizutani, Phys . Rev. Lett . 45 (1980) 524 11) K. Schwarz, H.F.K . Zingl and L. Mathelitsch, Phys . Lett . 83B (1979) 297 12) K. Kubodera and M.P . Lceher, Phys. Lett . 87B (1979) 169 K. Kubodera, M.P . Locher, F. Myhrer and A.W. Thomas, J. of Phys . G6 (1980) 171 13) K. Kanai, A. Minaka, A. Nakamura and H. Sumiyoshi, Prog . Theor. Phys . 62 (1979) 153 14) I. Duck and E. Umland, Phys . Lett. 96B (1980) 230 15) W. Grain, K. Kubodera and M.P. Lceher, Nucl . Phys . A356 (1981) 269 16) A. Kônig and P. Kroll, Nucl . Phys. A356 (1981) 345 17) N. Hoshizaki, Prog . Theor. Phys. 60 (1978) 1796 ; 61 (1979) 129