An isobar model for the pd → tπ+ reaction

Nuclear Physics A329 (1979) 477-490 Q North-Holland Publishing Co., Amsurdam Not m be reproduced by photoprint or miaofihn without written permission from the publisher

AN ISOBAR MODEL FOR THE pd->tar+ REACTION A . M. GREEN and M. E. SAIIQIO Research Institua for TTttoreNcal Physics, Unitxrsity of Helsinki, Siltanuorcnpenger 20 C, SF-00170 Helsinki 17, Fntand Received 17 May 1979 (Revised 9 July 1979) A6atead: The differential cross section for the pdit-rr + reaction is calculated at the three proton laboratory energies 400, 470 and 600 MeV. The triton wave function is taken to be of cornlated Gaussian form, and the deuteron is one of the a-wave forms due to Hulthén, McGee and Paris . The pion production is considered to take place from intermediate states based on the five types of. two-body coaelations Nd(SS=, sD Z, °Do) and NN(tDZ, t S o) . The final cross sections are not in good agreement with experiment .

1. Introdaction The isobar configuration model for pion production, which has proved so sutxessful in explaining the pp->, dar+ reaction t'Z), hasrecently been extended to the process pd-->t~r + [ref. 3)]. There it was shown that the quasideuteron approach of refs. `'s), which expresses the pd->tar+ cross section as a product of a kinematical factor, a wave function dependent form factor and the pp->dar+ cross section, i.e. d,flA

(pd~

ta+) = CIF 2 d,(1 ~

(pp ~ dar+ )~

(1 .1)

could lead to sizeable errors amounting in some cases to about an order of magnitude. Since the main purpose of ref. 3) was to study the validity of the quasideuteron approach, the wave functions for the deuteron, triton and isobar states were all taken to be as simple as possible . This resulted in an algebraic expression for the pd->tar+ cross section which made clear the approximations introduced in the quasideuteron approach. However, when evaluated, this theoretical cross section was not in good agreement with experiment. In the present paper many ofthe earlier approximations are now removed. In sect. 2 the transition matrix element for the pd-~ tar+ reaction is written in such a form as to enable the use of more realistic deuteron and isobar wave functions and also permit the inclusion of oorrelations into the s-wave component of the triton wave function . l:n sect. 3 the initial wave function is generalized to include pion production from states in addition to the onebased on the sS2 (Nd)isobar configuration, the only state considered in ref. 3). In sect . 4 the numerical results are given and discussed. 477

478

A. M. GREEN and M. E. SAINIO

2. The effect of improved wave functions ü the pd-> ta+ reaction is dominated by the chain of processes -> L(NoNi)zozaNzl~

~

then the transition matrix element can be expressed as s) Tft oc

J

dkf(k)YZ " ( 1~) Jdra(r)é;''''

x era=cx~

w(Kr) ~ ~ eu~ct+Q>a + (s zr)a(s - zr), Kr

(2.2)

where r=ro-rh s =Z(r°+rl)-rz and Q =~ko - 3q with ko and q being the proton and pion moments in the initial pd and final ta+ centre-of-mass systems respectively as depicted in fig. 1 ; K=âko-zk is the relative momentum between the incident proton and the nucleon in the deuteron with which it first interacts; Sz(K) is the real

Fig. 1. The dominant mechanism for the pd-.ta + reaction .

part of the 1Dz(NN) phase shift; f(k) is the s-wave wmponent of the deuteron wave function in momentum space and w(Kr) is the S Sz(Nd) isobar wave function for particles 0 and 1 . The s-wave radial component of the triton wave function is given by +/i. = N,a (r° - rl)a (rl - rz)a (ro - rz).

(2.3)

In ref. 3) f(k), a(x) and w(Kr) were all approximated by Gaussians, so that the nine-dimensional integral of eq . (2.2) could be performed analytically . It is this series of approximations which are now improved in the present paper.

2.1 . TRTTON CORREL ATIONS

The form of the internucleon correlation in the triton is taken to be with a = 0.1013 fmz, b =1 .3823 fm -z and c = 0.925, the values advocated in ref. e) for giving a reasonable fit to the three-body charge form factor. In ref. s) a (x) was the unoorrelated Gaussian given by c = 0. The three-dimensional s-integral and the angular part of the r-integral of eq . (2.2) can still be performed analytically to reduce the integration to oné of only four dimensions ts). In sect. 4 by evaluating this integral with a Monte Carlo procedures, it is found that the inclusion of correlations has a significant effect on the differential cross section. The simplification of eq . (2.2) into a four-dimensional integral depends completely on ôur use of Gaussian correlations . There seems little hope of extending the present work to non-Gaussian forms . z.2. THE 1sosAR WAVE FvNCr1oN

In ref. s) the radial part of the isobar wave function w(Kr) obtained by solving coupled differential equations for the twabody system, as discussed in ref. t), was fitted by a sum of Gausaians. However, since the integral for the transition matrix elementis now evaluated numerically, the exact values of w(Kr) canbe used directly. These are inserted as two (real and imaginary)19 x 80 grids for the points Ep(lab) _ 300(30)840 MeV and r = 0.1(0 .1)8 .0 fm where EP(lab) = 2flZKZ/M. In the' integration linear interpolation in K and r is used to obtain the required values from the two grids. As a check on the accuracy of this procedure, the results of ref. s) were duplicated to a+ithin 1 °~ when these stores were filled with the approximate wave functions of ref. 3). Furthermore, it is found that the approximate forms of w (Kr) used in ref. s) yield results which are in qualitative agreement with the present more accurate treatment. 2.3 . THE DEUTERON WAVE FUNCTION

Previously 3) only a purely s-wave deuteron with Gaussian form could be treated exactly. However, there an attempt was made to estimate the effect of replacing this by the Hulthén wave function . In the present approach any s-wave deuteron can be used once its form f(k) in momentum space is known. For comparison we consider the Hulthén, PartoviMcGee') and Paris s) forms of f(k) and findthat the differential cross section is more or less the same for each of these. However, the results from the Gaussian form of ref.') are significantly different, as can be expected by simply looking at the ' CERN library pro~amme D114 (RIWIAD) .

480

A. M. GREEN and M. E. SAINIO

corresponding f(k). Even so the difference between the Gaussian and Hulthén forms is not as great as that estimated in ref. s) showing that the scaling approximation made in sect . 6 of that paper was poor . The effect of the deuteron d-state has been neglected in the present article. 3. Generalization of the initial wave function In ref. 3), and also in the previous section, pion production was assumed to take place only through the chain of processes containing the SSz isobar configuration as shown in eq. (2.1). However, it is known l'z) that in the pp->dar+ reaction other configurations can play an important role. For example, the all nucleon 1Dz state, through its interference with the SSz isobar configuration, can increase the total cross section by upto 50% at proton laboratory energies around 500 MeV. However, this increase is somewhat reduced on introduction of the SDz isobar configuration. The effect of both of these two states on the pd -~ tar+ reaction will be estimated below. For polarizations z) the situation is even more complex, since many configurations are needed to get an adequate understanding ofthe experimental data. In this paper, since we concentrate only on the differential cross section, we ignore these small components that are so important in polarization . In the pp ~ dar+ reaction one configuration that makes an anomalously small contribution is the all nucleon 1 So state. This is still found to be true in the pd -~ tar+ reaction . 3.1 . THE 1 Dz(NN) AND S D2 (Nd) CONFIGURATIONS

The two configurations'Dz(NN) and SDz(Nd ) contribute to the pd-> tar+ reaction through the chain of processes

(3.1) and i~ ~ [(NoNi)zoz:lNzJ~ ~ -~ L(doNi)zzz :lNzJ~3 -i [(NoNi)

zii :o

NzJ~~~ -" tar+.

(3.2)

As is seen this requires a model for the d-state of the triton since the pair of nucleons involved in the pion production end up in a 3D1 correlation.

pd-~ ta+

481

In this paper we assume that the triton d-state has the form ~d=N~ {8(ro-ri)Soi+B(ro-rz)Soz+B(rz-ri)Szi}~G~(s)

+ 8(ro-rz)L(NoNz)zu:oNi~~~ where ~Ge(s) _ +/~.~ {L(NoNi)°°°aNz]~

~ _ L(NoNi)°i i :oNz] 3 ~~

Hére ~. is the radial part of the s-wave triton (see eq . (2 .3)), g(x) is a correlation function to be discussed below and S,; is the tensor operator acting on the pair of nucleons (ij) . We have chosen g(x) to be of the form and then the contribution from the 1 Dz(NI~ state to the transition matrix in the notation of eq. (2.2) becomes T~(d)oc dkf(k)Yi,~-(~) ~drrz e~a"21o(~9r) J J0 u(Kr) _1 l_2 Kr 8 l a~ x {e -Ik+Qh/9ar z ~ d.~ x e~ca)

r

a l~ - adr~/z(za+d) -Ik+Qh/u2a+d) . ( I k + QI d~ e e + \a + dl ~°l2(2a +d)1 xr-4rz-d-(~~ +(~k+Q~z-(dr)z)/(2(2a+d)]z +

drz + 1 + ~k+Q~ dr cet ~ ~k+Q~ 2(2a+d) 2a+d 2(2a+d) 2(2a+d)

3 _1_ x ((~ +d

1 1l1 2a+dlJJ'

(3 .5)

Here the unoorrelated form for ~ (x) in eq. (2.4) has been used . Leaving out short range oorrelations is expected to be a safe approximation because of the xz factor in the g(x) functions. In eq. (3.5) also the approximation e-~~''~ jo( z9r) has been made since this has been found to be reasonable in earlier works'). Hefore performing the

482

A. M. GREEN and M. E. SAII~TIO

s-integration we have dropped terms proportional to jz(s~k+Q~) and ja(s~k+Q~) since this was shown to be a reasonable approximation leading to errors that were essentially insignificant in absolute value. The contribution from the SDz(Nd) state has exactly the same form. The problem has now been reduced to knowing a suitable value for d in eq. (3.4). This is determined by maximizing the overlap of g(r) e~°'~ with the triton correlation of ref. 1~, the outcome being the value d = 0.46 fm-z. In the limiting case d -~ 0 some of the approximations made in deriving eq . (3.5) can be avoided and it can be seen that the "crossed" terms contribute - â of the "direct" term . For larger d-values the "crossed" contributions are even smaller and so the approximations made above concern only a small fraction of the whole integrand. 3.2 . THE 1 So (NN) CONFIGURATION

In the pp -> d~r+ reaction the contribution from J = 0 states is anomalously small because of the smallness of the radial integral (3S1~!°(i9r)I1S°). However, it is not d priori clear to what extent this anomaly persists in the pd -> tar+ reaction . The effect of the chain of processes (3.6) and the corresponding chain involving the sD°(Nd ) configuration can be incorporated in the same way as the'Dz(NN), SS z (Nd) and SDz(Nd) configurations, but again these J = 0 contributions seem to have an almost negligible effect . 4. Resdts In the pd -> tzr+ reaction, when it is assumed that the pion production is only due to the sSz isobar configuration, and that the triton wave function is purely s-wave and has the form of the correlated Gaussian of eqs. (2.3) and (2 .4), then the transition matrix element can be reduced to a four-dimensional integral as discussed in subsect. 2.1 . In figs . 2 are shown by the solid lines the resulting differential crosssections at the three proton laboratory energies of 400, 470 and 600 MeV when the deuteron wave function is that of Hulthén (s-wave component only). For comparison the cross section for a triton wave function without correlations is also shown by the dashdotted lines. The latter were obtained by simply setting c equal tozero in eq. (2.4). As can be seen, the introduction of oorrelations into the triton s-wave results in cross sections that have at higher energies much more structure than before . In these cross sections the most conspicuous feature which occurs on the introduction of oorrelations is the appearance of a dip at a momentum transfer of Q ~ 1.8 fm' (see eq .

pd-~ trr+

o"

3o

so

so ey,(cM)

48 3

+zo

rso

~eo

Fig. 2a . The differential çroes section for the pd-. to + reaction at the proton laboratory energy 400 MeV . The solid line refers ro the case when the triton wave function hss the correlated G~~asi~n form and the deuteron wave function is that of Hulthén . The dash-dotted line refers to the unwrrelated triton case.

io

o~

3o

so

so

Q,(CM)

>zo

~

ieo

Fig . 2b. The differential cross section for the pd-" ta` reaction at the proton laboratory energy 470 MeV. The solid line refers to the ase when the triton wave function has the eorrelatod Gaussian form a~ the deuteron wave function is that of Hulthén . The dash-dotted line refers ro the unoorrelated triron case.

48 4

A. M. GREEN and M. E. SAINIO

Fig. 2c . The differential cross section for the pd-i ta + reaction at the proton laboratôry energy 600 MeV. The solid line refers to the case when the triton wave function has the wrrelated Gauasian form and the deuteron wave function is that of HulthEn. The dash-dotted line refers to the uncorrelated triton case . (2 .2)) . Presumably this is related to the similar dip that arises in the 3 He charge form factor and for which the present triton wave function was designed . This result is qualitatively in line with the conclusion of Locher and Weber S). However, it disagrees with the work of Fearing 6) who finds in his quasideuteron approach that such a distinct dip does not arise. Furthermore he attributes the result of Locher and Weber to their use of a triton wave function that is not antisymmetric . In the present calculation the dip still appears even though the triton wave function is correctly antisymmetrized . However, as pointed out in ref. 6 ), in addition to the correct antisymmetrization, the d-state of the deuteron, which has been neglected in the present calculation, also has the property of helpingto fill in the type ofdip seen in fig. 2. Unfortunately the experimental situation s does not seem to indicate the dips introduced by the correlations . The overall effect of these correlations is to move the theoretical curve somewhat closer to the experimental data, but still the agreement is poor . In fig. 3 is shown, at 470 MeV, the effect of different s-wave forms for the deuteron wave function . As can be seen the Hulthén, ParlourMcGee') and Paris s) forms all give very similar results whereas the simple Gaussian of ref. 3) is considerably different. Throughout the remainder of this paper only the Hulthén wave function will be used . s The experimental data shown is a collection of those results published since 1970 [ref. ")], scaled to the three energies of interest using the multiplicative factor at the end of sect . 4 in ref. 3).

48 5

pd-" trr+

a'

3o

eo

ao q,ccti+)

tto

~so

ieo

Fg. 3 . The effect of different deuteron wave functions . The solid line refers to the Hulthén form and the dashed to that of Partovi-Mc('~ee') . The Paris s) results an indistinguishable from the Partovi-McGee form.

As described in subsect. 3.1, adding the effect of pion production from the two-body 'D z (NN) and sDz(Nd) states can be incorporated into the theory by modifying the basic expression given in eq . (2.2) to that in eq . (3.5). The result of this is shown in figs . 4 and is in the direction expected from the similar modification in the pp -' dvr+ reaction, i.e. a significant increase in the cross section at lower energies (see fig. 3 of ref. s )). The combined effect of introducing correlations into the triton and the inclusion of pion production from the tDz(NN) and sDz(Nd )states is seen to give a slightly worse overall agreement with experiment. In order to measure the importance of the "crossed" terms in eq. (3.5) they were set to zero with the result that in the forward direction the change was found to be small (-r5% of the total d-state effect). Even though this percentage does increase with angle and energy, it must be remembered that in such situations the total effect of the triton d-state is itself rather small. In the pp i d~r+ reaction a reasonable understanding of the cross section is obtained for energies below -x700 MeV by simply considering the initial state to be tDz(NN) with the J = 0 states tSo and sDo(Nd) having only'a small effect . In the pd-> ta+ the effect of the J = 0 states ts) is again found to be small as can be seen in figs. 5. However, it is unclear how sensitive this result is to the form of the triton s-wave wave function used in these calculations.

486

A. M. GREEN and M. E. SAINIO " Aslanides et al . ,977 " Carroll et al. 1978

400 MeV

o.,~

6 (CM)

,20

,80

Fg. 4a . The effect of the triton d-state. The solid lines are the same as in fig. 2 and refer to the correlated s-wave calculation. The dashed lines correspond to the case when the triton d-state is included.

,a

a,

or

3o

so

9o 6 (CM)

1~
no

487

pd-~ t~r+

,a

o~

3o

so

so

ew, cayu

,so

,50

,eo

Fig . 5 . The contributions from the 1So(NI~ and'Do (Nd )initial two-body states are inducted in the solid tine, with the dashed line as in fig. 4. The circled crosses ere taken tram the calculation of Fearing 6) neglecting the renormaliTStion fadoa he quotes .

488

A. M. GREEN and M. E. SAINIO s Ddlhop( et iL 7973 n Auld 7977 Carrdl et al. 1978

470 MeV

7a

0.7

0'

30

80

90 8~(CM)

Fig. Sb

Fig. Sc

120

1S0

180

pd-" t~i

489

5. Condoeio~ In this paper there is developed for the pd -~ tzr+ reaction a microscopic model that includes the effects of Gaussian like oorrelations in the triton wave function, realistic s-wave deuteron wave functions and the effect of pion production from the five two-body correations Nd (SSz, sDz, SDo) and NN(1Dz, tSo). This is to be compared with the earlier work s) which simply included an uncorrelated triton wave function, a Gaussian deuteron wave function and pion production from only the SSz(Nd) configtuation. Unfortunately the result of introducing these refinements still does not give good agreement with the experimental pd->t~rr+ differential cross section, altlibugh the fit is better than in the earlier work. Originally, the hope was that there would be agreement comparable to that obtained for the pp-> dor+ cross section or for the total inelastic pp cross section t`) . However, it is now clear that this would require some further developments of the presentmodel, A measure ofthe necessary accuracy can be seen by comparing at 600 MeV and 0° the pp->da+ and pd-~t~r+ experimental cross sections . They are respectively 455 t 23 wb/sr and 14.6 t 1.4 Wb/sr (Gabathuler etal. t 1)). This reduction factor of 3 arises from wave function overlaps and momentum transforms, and so requires a very detailed model for a quantitative understanding. In figs . 5 are also shown the results of Fearing 6) without any of the overall renormalization factors quoted in that article. As can be seen, at lower energies his results are in reasonable . agreement with experiment but fall away badly for large angles at the higher energies . Furthermore, it must be remembered that he has the prescription of setting to unity two of his distortion factors F~ and Fd. Their inclusion would mean that the renormalization factors multiplying the theory to get overall agreement with experiment should be about a factor of two larger than those quoted in ref. 6). In view of this we consider that the results of the present calculation are reasonable. The question can now be raised concerning how the above microscopic model can beimproved. Asalready suggested the s-wave component of the triton wave function could be modified by treating it as a symmetrized cluster wave function since this would be more general than the one used in the present article. Such a foim arises naturally from three-body calculations using the Faddeev equations. Unfortunately the effect of such a wave function modification is not easy to foresee. Another correction would be the inclusion of the d-state component of the deuteron. However, according to the quasideuteron model of Fearing lz) the effect of this would be minor compared with the present difference between theory and experiment. Finally there could arise corrections due to the initial and final state interactions which have been so far neglected. In the initial state the interaction between the incident nucleon and one of the nucleons within the deuteron has been treated rather well by the use of coupled Schrcedinger equations for that two-body subsystem. The effect of the spectator nucleon could then be introduced as a one-body

490

A. M. GREEN and M. E. SAINIO

potential in these coupled equations t3). In the final state, part of the rescattering of the pion by the triton has automatically been included. Only the rescattering by the spectator nucleon 2 (see fig. 1) is not in any way built into the present model. In the previous paragraph several suggestions are made on how to improve the agreement between theory and experiment for the pd~ t-rr+ reaction . However, it is not clear to what extent this should be pursued if the long term aim of these studies is to try to understand the more general reaction pA -> (A + 1)a +. The triton is rather a special system and some of its problems are not necessarily relevant to the more general problem. For example, in heavier nuclei, unlike the triton, use of the shell model is necessary and often centre-of-mass effects can be treated as a correction. This could simplify the theory for such reactions with heavier nuclei . On the other hand, initial and final state interactions become more important as the number of nucleons increases. It is possible that the microscopic model developed here for the pd -> tTr+ reaction will then be more successful when extended to heavier nuclei with initial and final state interactions included. An alternative approach could be similar to the philosophy of the quasideuteron model but where the basic ingredients are parametrized two-body matrix elements as opposed to experimental two-body cross sections. These possibilities are now being studied. The authors wish to acknowledge useful discussions and correspondence with E. Maqueda and to thank Ch. Hajduk for use of his three-body wave functions. Also, oneof us (A.M.G .) wishes to thankthe Science Research Council of Great Britain for granting a research fellowship for the period during which this work was initiated. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

12) 13) 14) 15)

A. M. Green and J. A. Niskanen, Nucl. Phys. A271 (1976) 503 J. A. Niskanen, Nucl. Phys. A298 (1978) 417; Phys. Lett. 79B (1978) 190; 82B (1979) 187 A. M. Green and E. Maqueda, Nucl. Phys. A316 (1979) 215 H. W. Fearing, Phys . Rev. Cll (1975) 1210 M. P. Locker and H. J. Weber, Nucl . Phys . BT6 (1974) 400 H. W. Fearing, Phys . Rev. C16 (1977) 313 I. J. McGee, Phys . Rev. 151 (1966) 772 B. Loiseau, private communication A. J. Kallio, P. Toropainen, A. M. Green and T. Kouki, Nucl . Phys. A1.31 (1974) 77 Ch. Hajduh and P. U. Sauer, Nucl. Phys. A322 (1979) 329 K. Gabathuler et al., Nucl . Phys . B40 (1972) 32 ; W. Dollhopf u aL, Nucl. Phys. A217 (1973) 381 ; E. Aslanides et al., Phys . Rev. Lett. 39 (1977) 1654; E. G. Auld, Proc. 7th Int. Conf . on high-energy physicsand nuclear structure, Zürich, Aug. 1977, pp . 21, 26 and subsequent preprint ; J. Carroll et al., Nucl . Phys . A30S (1978) 502 H. W. Fearing, Phys. Rev. Cll (1975) 1493 A. M. Green, in Meson in nuclei, ed . M. Rho and D. H. Wilkinson (North-Holland, Amsterdam, 1979), p. 227 A. M. Green and M. E. Sainio, J. of Phys. GS (1979) 503 A. M. Green and M. E. Sainio, University of Helsinri report HU-TFT-79-18