6Li(p,Δ++)6He reaction reanalysed

12 March 1998

Physics Letters B 422 Ž1998. 19–25

6

Li žp, Dqq /6 He reaction reanalysed Bijoy Kundu, B.K. Jain

Nuclear Physics DiÕision, Bhabha Atomic Research Centre, Bombay-400 085, India Received 29 January 1997; revised 10 November 1997 Editor: J.-P. Blaizot

Abstract Using our recently developed formalism for the Žp, D . reaction, which includes the decay of D inside the nuclear medium, we study the 6 LiŽp, Dqq. 6 He reaction, and compare the calculated cross sections with those measured some time back at Saturne. In the formalism, the D resonance decays into ppq either inside or outside the nuclear medium. The medium effects on D as well as on p and pq are incorporated through appropriate optical potentials. The transition potential for the elementary process pp ™ n Dqq is taken to be one pion-exchange, whose parameters are constrained to reproduce the measured spin averaged cross sections on the pp ™ n Dqq reaction over a large energy range. The calculated results agree well with the measured cross sections. q 1998 Published by Elsevier Science B.V. PACS: 25.40.Ve; 13.75.-n; 25.55.-e Keywords: D-production; Li; Decay in medium; Distorted waves

In nuclear interaction and in the description of nuclear reactions at intermediate energies, in addition to nucleons and pions, deltas play an important role. Because of this, there has been great interest over the years in the study of the delta-nucleus interaction and delta producing nuclear reactions w1x. In this context, a clean experiment on the 6 LiŽp, Dqq . 6 He reaction was reported sometime back from Saturne w2x. This experiment used protons of 1.04 GeV and detected the recoiling 6 He along with pions in a hodoscope. The measurements were done on the recoil energy spectrum and the differential cross section as a function of the four momentum transfer. The recoil energy spectrum of 6 He exhibited a clear bump in the region of D excitation. These data had generated much interest. They were theoretically analysed by various authors following the Glauber approach or DWBA, and were reproduced well w2,3x. However, all these approaches considered D as a

stable particle. Since the width of D is large Ž; 116 MeV., this approximation had always been a cause of discomfort, and a source of uncertainty in the calculated cross sections. To remedy the situation, recently we have developed a formalism whereafter referred as JKx in which the unstable nature of the delta has been incorporated w4x. The formalism incorporates the decay of D anywhere on its path in propagating-out from its production point r. This decay point could be either inside or outside the nucleus. The distortion due to nuclear medium of the D Žduring its propagation. and its decay products p and pq Žwhile getting out of the nucleus. is incorporated through appropriate optical potentials. In the present paper we apply this formalism to the 6 LiŽp, Dqq . 6 He reaction. We have calculated d srdt for the 6 LiŽp,pXpq . 6 He reaction, where t is the four momentum transfer to the target nucleus from the beam proton, and identified it with the measured four

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 0 0 5 - 7

B. Kundu, B.K. Jain r Physics Letters B 422 (1998) 19–25

20

momentum distribution in the Saturne experiment. The calculated differential cross sections agree well with the measured ones. For the p ™ Dqq excitation in the above description we have used one pion-exchange. We have not included the contribution of r-exchange for this excitation. This is consistant with our earlier studies and those of several other workers on the pp ™ Dqq n reaction. For example, the detailed experimental studies by Wicklund et al. w5x demonstrate that the pion exchange gives a very good fit to the spin averaged pp ™ Dqq n reaction over a wide kinematic region. The work of Dmitriev et al. w6x and Jain et al. w7x corroborate these findings for beam energies from threshold to very high. They also show that any inclusion of rho-exchange yields very unsatisfactory results. In another study, Jain and Santra w8x find that, while the rho-exchange is absolutely essential to account for the pŽn,p.n data, it is not favoured at all by the pŽp,n. Dqq data. The reason for this dis-favour of the rho-exchange in delta excitation, to a certain extent, is provided by a microscopic study of the r ND vertex by Haider et al. w9x, where they find that the microscopically calculated value of the r ND coupling constant, fr ND , is much smaller than the normally assumed value. Following JK, the differential cross-section, d srdt, for the AŽp,pXpq .B reaction is given by ds dt

priate sum and average over the final and initial spins, respectively. According to JK, Tf i is written as X X y) X Tf i s drdrXxy) p Ž r . xp Ž r .

H

= GD Np GD Ž rX ,r . cD Ž r . , Ž 3. where cD Ž r . is the production amplitude for the delta at a point r, and GD Ž rX ,r . is its propagator from r to rX . It includes the distortion of the D by the nuclear medium. At rX , the D decays into p and pq. x ’s are the distorted waves for the outgoing proton and pion. GD Np is the decay operator for the delta. In momentum space and in a non-relativistic static approximation it is given by fp) GD Np s S. k p T. fp , Ž 4. mp where S and T are the spin and iso-spin transition operators respectively for 12 ™ 32 . kp is the pion momentum in the delta rest frame. Because of the final pion being on-shell Žif we neglect the effect of distortion on it., above form for GD Np does not have the usual form factor F ) in it. The value of fp) is taken equal to 2.156. The D production amplitude, cD Ž r ., at a point r is given by

cD Ž r . s d jcb) Ž j . Gp N N ca Ž j .

H

= Gp Ž r , x . Gp ND xq p Ž r. , 2

s d m d f B d Vp w PS x - < Tf i < ) ,

H

Ž 1.

where w PS x, the factor associated with the phase space and the beam current, is given by mD2 m 2 EA EB kp3

m

w PS x s

Ž 2p . =

5

pc2

s 1

kp2 Ž 's y EB . q Ep Ž K B . kp .

Ž 5.

where Gp ND is the operator for the excitation of the beam proton to the delta, and Gp N N is the interaction operator at the p NN vertex in the nucleus. Like GD Np , their forms are fp) F ) Ž t . † Gp ND s i S .qT † . fp , Ž 6. mp and

.

Ž 2.

Here m and mD Ž s 1232 MeV. are the masses of the proton and the centroid of the delta. 's is the available energy in the centre of mass system and pc is the c.m. momentum in the initial state. m is the invariant mass of the pXpq system. Tf i in Eq. Ž1. is the transition amplitude. The angular brackets around its square denote the appro-

Gp N N s i

fp F mp

s .qt . fp ,

Ž 7.

with fp as the coupling constant at the p NN vertex. Its value is 1.008. F ) and F are the form factors associated with the p N D and p NN vertices, respectively. We have taken mono-pole form, with the length parameter Lp equal to 650 MeV, for them. This value of L has been found in the literature to reproduce the spin averaged cross section data on the pp ™ n Dqq reaction w6,7x.

B. Kundu, B.K. Jain r Physics Letters B 422 (1998) 19–25

ca Ž b .Ž j . in Eq. Ž5. are the nuclear wave functions in the initial and final state. In the present case they refer to the ground states of 6 LiŽ1q,0. and 6 HeŽ0q,1., respectively. In the intermediate coupling scheme, using a Ž1p. 2 configuration, the most general wave functions for these states are: < 6 Li )s a A3 S1 q bA1 P1 q gA3 D 1 ,

Ž 8.

< 6 He )s a B1 S0 q bB3 P0 ,

Ž 9.

where, a A s 0.924, bA s 0.2, gA s 0.102, a B s 1.00 and bB s 0.08, are the mixing parameters. These parameters, apart from reproducing the Ž e,eX . form factors for the 3.56 MeV Ž0q,1. and 5.37 MeV Ž2q,1. states in 6 Li, also fit its ground state moments w10x. Since a A and a B are the dominant mixing parameters, in our calculations we restrict the description of 6 Li and 6 He ground states only to the dominant 3 S1 and 1S0 configurations. With this description, the - < Tf i < 2 ) works out as 2

- < Tf i < )s

4 27

ž

f ) 2 fFF ) mp3

2

/

q 2 < FB A < 2

2

= 4 < k p .q < q < q = kp <

2

.

2

1 t y mp2

,

Ž 10 . where X X y) X FB A s drdrXxy) p Ž r . xp Ž r .

H

= GD Ž rX ,r . r Ž r . xq p Ž r. ,

Ž 11 .

with r Ž r . as the radial transition density for 6 Li Ž1q,0. ™ 6 HeŽ0q,1.. We approximate it by the radial density extracted by Bergstrom et al. w10x from their measured experimental data on the inelastic electron scattering on 6 Li from its ground state to the second excited state Ž0q,1. at 3.56 MeV. It is given by yr 2

r Ž r . s exp

b2

= w 0.1063r 2 y 0.05091r 3 q 0.008433r 4 y0.0001126 r 6 q 0.000001407r 8 x ,

Ž 12 . where b s 2.324 fm.

21

For evaluating FB A wEq. Ž11.x, we use eikonal approximation for continuum waves and the delta propagator, and follow the procedure as given in JK w4x. Since the calculation of d srdt includes integration over certain part of the phase space wsee Eq. Ž1.x, a specific value of t involves a range of energies for the continuum particles in the final state. For calculating their distorted waves we, therefore, require the relevant optical potentials at several energies. Since we use eikonal approximation for the distorted waves, only requirement on these potentials is to give correct elastic scattering in the forward direction. Since most of the optical potentials satisfy this requirement, in choosing a particular prescription for the optical potential we are mainly guided by the computational ease. The latter is necessary because in Eq. Ž11. we already have a six dimensional integral to evaluate. The required optical potentials are chosen as follows. For protons, the distorting potentials beyond 300 MeV are obtained using the high energy ansatz, 1 U q iW s y Õs Tp N Ž i q n . r 0 , 2

Ž 13 .

where s Tp N is the total proton-nucleon cross section at the proton speed, Õ, in their c.m., and n is the ratio of the real to imaginary part of the scattering amplitude. The values for the total cross section and n are taken from the experimental data on protonnucleon scattering, and r 0 is taken equal to 0.17 fm y3 . The radial shape of the potential is approximated by r Ž r .rr Ž0., where r Ž r . is taken from the elastic electron scattering analysis of Li et al. on 6 Li w11x. For the low energy protons i.e. below 300 MeV, the potentials are taken from the analyses of the elastic scattering data of protons on 6 Li at various energies w12x. On delta optical potentials, not much information exists. For TD F 100 MeV we make recourse to the delta-hole model of Hirata et al. Žimproved subsequently by Horikawa et al.. w13x and take WD s y45 MeV. For higher energies, as for protons, we use the high energy ansatz as given in Eq. Ž13.. Here, however, for obtaining s TD N , first we write it as a sum of the elastic and the reactive parts, s TD N s selD N q sr D N . Then assuming that the spin averaged elastic dynamics of the proton and delta are not very differ-

B. Kundu, B.K. Jain r Physics Letters B 422 (1998) 19–25

22

ent, we assume selD N f selN N . For the reactive part, since up to about 1.5 GeV the main reactive channel in D N scattering is D N ™ NN, using the reciprocity theorem we write

sr D N f s D N ™ N N s

1 k N2 N 2 2 kD2 N

s Ž pp ™ n Dqq . ,

Ž 14 .

where k N N is the proton c.m. momentum corresponding to the same energy as is available in the D N c.m. An extra factor 1r2 is introduced to account for the identity of two particles in the final state. Eq. Ž14., of course, is valid for stable particles. For unstable particles, like deltas, it needs to be modified w14x. The modification arises because, for a given energy in the pp system only a certain range of delta mass is accessible. At energies near the threshold this restriction could be severe. At higher energies, where we are interested in the present work, it may, however, be insignificant. We have, therefore, ignored its effect. On pionic optical potentials lot of theoretical work has been done. Several prescriptions exit, which have varying degree of success. The latest amongst them is due to Nieves et al. w15x, which has been developed in context with the pionic atoms. It comprehensively includes the effect of s- and p-wave pions. The potential is non-local. For our purpose, as mentioned earlier, we prefer to use a simpler potential, provided it gives the correct forward scattering in the D excitation region. We find that the potential w16x meets these requiredue to Ericson and Hufner ¨ ments. We, therefore, fix the pion optical potential using their method. In this method, the strength of the optical potential is obtained from the refractive index, n, of the pion in the nuclear medium, where nŽ E . s

K Ž E.

Ž 15 .

kŽ E.

and W Ž E. sy

k2 E

ni Ž E . ,

U Ž E . s 1 y n2r E.

Ž 16 . Ž 17 .

Here K is the pion wave number in the nuclear

medium. This is obtained by solving the dispersion relation K 2 s k 2 q 4pr 0 fp N Ž K , E . ,

Ž 18 .

where fp N is the p y N scattering amplitude in the nuclear medium in the forward direction. Considering that the pion scattering is dominated by p-wave scattering, the expression for pq scattering from a nucleus with N neutrons and Z protons is fpq N s

1

Ž Nfp A

q

n q Zfp q p

N q 3Z

.f

3A

f 33

Ž 19 .

A Breit-Wigner resonant form for the amplitude f 33 gives X Gr4

ni Ž E . s

ž

E y ER q

3 4

2

X

/

1 q 4

,

G

Ž 20 .

2

and 1 2

nr Ž E . s 1 y

ž

ž

X E y ER q

E y ER q

3 4

3 4

2

X

/

X 1

q 4

/ G

,

Ž 21 .

2

with X s 4pr 0 c,

Ž 22 .

and csy

N q 3Z 58 Ž MeV . a 3 3A

1 q Ž ka .

2

.

Ž 23 .

Here r 0 is the nuclear density, and a s 1.24 fm. The dispersion relation given in Eq. Ž18. is equivalent to an optical potential approach, if the latter is defined through the folding of the p-N t-matrix with the nuclear density. Tandy et al. w17x have shown that such optical potentials contain nucleon knock-out as the primary reactive content. Therefore, the main contribution to n i Ž E . in Eq. Ž20. is the nucleon knock-out Žpq,pq N. channel. However, in addition to this, the pion flux can also be lost through real absorption of the pion in the nuclear medium. The dominant channel which contributes to such an absorption is D N ™ NN. We have approximated this contribution to the pion optical potential by Wa b s s g Gsr2,

Ž 24 .

B. Kundu, B.K. Jain r Physics Letters B 422 (1998) 19–25

where Gs is the spreading width. We take Gs s 70 MeV from the delta-hole model of pion absorption w13x. The factor g is added to account for the fact that the Žp,pXpq . reaction is a peripheral reaction. This factor represents the fraction of the central nuclear density in the region where this reaction takes place. We have chosen g s 0.7. Though we have used Wa b s to solely represent the real absorption of pions, Salcedo et al. w18x point out that, because of its phenomenological nature, it might also contain some quasielastic corrections. The radial shape of the above potential, like in the case of the protons, is given by the nuclear density. To check the sensitivity of the calculated results to pion distortion, we also employ another equally simple prescription, called effectiÕe radius model, due to Silbar et al. w19x for the pion optical potential. The potential in this model is given by VR ) Ž r , E . s A

b 1 kp2 2 Ep

rR ) Ž r . ,

Ž 25 .

where r R ) Ž r . is the nuclear density profile with its radius changed to an effective energy dependent radius, R ) . It depends upon the pion momentum, and is related to the nuclear r.m.s. radius, R, through R) 2 s R2 q

3 k2

.

Ž 26 .

This dependence of R ) and the form of the potential in Eq. Ž25. comes from the p-wave p N interaction. The parameter b 1 in this equation is complex and is fixed from the p-wave p N scattering data. Radius R for 6 Li is 2.56 fm. Calculated d srdt for 6 Li target nucleus is shown in Fig. 1. It is obtained, as written in Eq. Ž1., by integrating the five-fold differential cross section in the Ž p, pXpq . reaction over the available invariant mass of the pXpq system, the solid angle of the pion and the azimuthal angle of the recoiling nucleus. The experimental data shown in the figure are from Ref. w2x. The figure has three calculated curves. The solid curve represents the full calculations, including distortion of all the continuum particles. We find that this curve is in reasonable accord, in magnitude as well as shape, with the experimental points. The dash-dot curve corresponds to plane wave ŽPW. results. To bring out separately the effect of distor-

23

Fig. 1. Four momentum transfer distribution for the X 6 Ž Li p,p pq . 6 He reaction. Dash-dot curve has no distortion for any continuum wave ŽPW results.. The dashed curve includes distortions for the beam and delta and the solid curve includes distorX tions for beam, delta, p and pq. Lp s650 MeVrc. The PW and X q no p and p distortion curves are normalized to the peak of the DW curve. The normalization factors are 3.2 and 1.6, respectively The experimental points are from w2x.

tion on the shape and the magnitude of the cross section, this curve has been normalized to the peak cross section in the DW calculations Žsolid curve.. The normalization factor, which is 3.2, gives the reduction in the magnitude of the cross section due to distortion. The effect on the shape can be seen directly in the figure. As we see, the distortion has a large effect on the shape of the four momentum transfer distribution too. So much so, the good agreement in shape with the data obtained in the PW results gets marred beyond < t < f 0.2 ŽGeVrc.y2 by the distortion. The dashed curve brings out the relative importance of the distortion of delta and its decay products. It differs from the solid curve in not having the distortion of pX and pq. This curve is also normalized to the peak in the solid curve. The normalization factor is 1.6. This means that the distortion of pX and pq introduces a reduction of the cross section by this factor. This distortion, however, as we see, does not introduce any further modification in the shape of the distribution, except beyond < t < f 0.3 ŽGeVrc.y2 . All the above results use Ericson-Hufner prescrip¨

24

B. Kundu, B.K. Jain r Physics Letters B 422 (1998) 19–25

tion for the pion distortion. However, to see the results due to other prescription Ždiscussed above. as well, in Fig. 2 we show the calculated cross sections due to ‘‘effective radius’’ prescription also. The solid curve is due to Ericson-Hufner prescription and the ¨ dashed curve is due to ‘‘effective radius’’ model. The two sets of cross sections seem same within about 20%. Finally, in Fig. 3 we attempt to identify the factors which may improve the agreement in shape of the DW results in Fig. 1 beyond < t < f 0.2 ŽGeVrc.y2 . An examination of Eq. Ž10. suggests that the transition matrix is determined by two factors, Ži. nuclear structure factor, FB A , and Žii. the interaction term given by - < Tf i < 2 )r< FB A < 2 multiplied by the delta propagator. To see the influence of these factors separately in determining the shape of the four momentum distribution, in Fig. 3 we plot them in PW approximation. The solid curve represents the nuclear transition density squared Žin momentum space. used in above calculations and dashdot curve corresponds the interaction term. The dashed curve represents the multiplication of these two curves. All the curves are normalized to the peak in the measured cross sections. We observe that the

X Fig. 2. Four momentum transfer distribution for 6 LiŽp,p pq . 6 He reaction. Distortion of all continuum particles is included in both the curves. The solid curve is the one calculated using the Ericson and Hufner method and the dashed curve is calculated using the ¨ effectiÕe radius model for the pion distortion. The value of Lp is the same as in Fig. 2. The experimental points are from w2x.

Fig. 3. Nuclear transition density squared for 6 Li ™ 6 He transition Žsolid curve. and the interaction term in the JK model Ždash-dot curve. as a function of the four momentum transfer squared. The dashed curve is the multiplication of these two factors. All the curves are normalized to the peak in the measured cross sections.

shape of the measured four momentum transfer distribution, which has a drop by over two orders of magnitude, is mainly determined by the transition density of the nucleus. The interaction term contributes to the fall of the cross section only to the extent of a factor of 3. This suggests that the too steep fall introduced in the dashed curve by distortion Žsee Fig. 1. may be compansated by enriching the transition density beyond < t < f 0.2 ŽGeVrc.y2 . Alternatively, a different treatment of distortions might also help in the same direction. Since we have chosen both the factors very judiciously in the present calculations, it is not obvious to us immediately as how they can be improved to introduce the desired effect. It needs to be investigated. In conclusion, we find that the inclusion of the delta decay in nuclear medium preserves, in an overall sense, the agreement reached in the earlier stable delta calculations w7x on the 6 LiŽp, Dqq . 6 He reaction with the data. In the JK model a reasonable agreement with the data is obtained using one pionexchange potential for the pp ™ Dqq n transition with the length parameter Lp s 650 MeVrc. It is found that the dominant nuclear distortions are those of the initial proton beam and the intermediate D state. For

B. Kundu, B.K. Jain r Physics Letters B 422 (1998) 19–25

determining the shape of the four momentum transfer distribution, the final proton and pion distortions are less important.

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