Pion absorption effects in the reaction 3H(π+, π0)3He

Volume 53B, number 5

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6 January 1975

P I O N A B S O R P T I O N E F F E C T S I N T H E R E A C T I O N 3 H 0 r + ' Iro )3 H e ~ J.M. EISENBERG* and V.B. MANDELZWEIG Racah Institute o f Physics, Hebrew University o f Jerusalem, Jerusalem, Israel Received 25 October 1974 We show that in Glauger theory the angle integrated cross section for 3H (~r+, ~r0)3 He has no peak as a function of energy through the 3,3 resonance region, thus suggesting a direct measurement to elucidate the nature of (n+, 7r°) reactions.

A puzzling situation for the understanding of qualitative features of the pion-nucleus optical potential has recently begun to emerge. At issue is pion single charge exchange scattering leading to the isobaric analog state of the target nucleus rt+ + A ~ A'(i.a.s.) + n"° .

(1)

Experimental indications [1-3] persist in showing a rise in the angle-integrated cross section for process (1) on 13 C as one approaches the energy of the 3,3 rtN resonance, whereas virtually all theories*l which embody distortion - and consequently absorption - of the pion wave in an even semi-quantitative manner show a minimum in the cross section at the 3,3 energy. This minimum is produced because, as the pion energy approaches that of the resonance, the pion-nucleon cross section increases, and the mean free path for the pion in the nucleus decreases to a fraction of a fermi. This effect wins out strongly over the resonating charge-exchange amplitude, with a consequent decrease in the cross section for process (1). Of course, the total and the elastic rt±-A cross sections would be expected to show peaks near the 3,3 resonance energy, as would presumably the quasi-elastic (zr, 7r'N) cross section, but other inelastic cross sections need not, and this is in fact what transpires for single charge exchange scattering to the isobaric analog state. The purpose of the present note is to examine this situation for 3H(rt +, n°)3He, with threefold motivation: (i) to consider a case where a fairly complete and reliable treatment of multiple scattering can be given, in the form of Glauber theory [4, 7], (ii) to allow for the possibility of a direct experimental observation .2 of process (1) by detecting the f'mal-state 3 He, as opposed to the activation methods which have otherwise been necessary [1-3], and (iii) to show that, even for nuclei as light as the A = 3 system, the absorptive effects win out over the single-particle transition amplitude resonance to eliminate the 3,3 peak in the cross section. The calculational techniques we shall use here are conventional and simple. We start with the multiple-scattering series of Glauber [7] for the 7r-(A = 3) scattering amplitude in the form Ffi(Q ) =

ik ~fexp ( I"Q - b) d2bf

drldr2dr38(~(r1+r2+r3))

(2) X ~:(rl,r2,r3) {1-[~/=1 [1-2~fexp(-iq'(b-b/))f/(q)d2q]} ~i(rl'r2'r3)' Work supported in part by the National Science Foundation and by the United States-Israel Binational Science Foundation. * Permanent address: Department of Physics, University of Virginia, Charlottesville, Virginia 22901, U.S.A. ,1 A review of the theoretical situation is provided in ref. [4], and a heuristic discussion of the (Tr+, 7r°) process has been given subsequently in ref. [5] ; a recent coupled-channel optical model calculation of charge exchange reactions is that of Miller and Spencer [6]. ,2 We note that the isospin-reflected reaction 3 He Or-, ~r°)3H may provide a more convenient experimental target, and all of our charge-exchange results apply there as well. We have referred mainly to the Or+, n °) version of the reaction since that is the form which has been studied [ 1 - 3 ] to date.

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.[3

E

v

U "t3

2C 100--

1¢

I

100

I

200

I

300 ~(MeV)

Fig. i. Angle-integrated elastic cross sections for n±-3He scattering as a function of pion lab kinetic energy Trr using parameter set (i) as defined under eq. (7). The data points at 100 MeV are from ref. [14].

01: 30"

0~ 0

30" 60" 90"

60"

90"

I

I

I

1 2 0 " 150" O

Fig, 2. Differential cross sections for n±-3He scattering at 100 MeV an energy deemed to be too low for the present theory to apply quantitatively. The data points are from ref. [14].

where k is the incident pion m o m e n t u m , b is the impact parameter variable perpendicular to k, Q ~ 2 k sin½0 is the m o m e n t u m transfer at scattering angle 0, and ~bi(f)(r 1 , r 2 , ?'3) is the three-nucleon wave function for the initial (final)state. The pion-nucleon scattering amplitude at m o m e n t u m transfer q is parametrized by [8]

f ( q ) = (a 0 + a l T . ~ ) exp ( - ½ 7 2 q 2 ) ,

(3)

where the imaginary part o f a s is fixed by the optical theorem and the real part isintroduced in terms of a quantity p in the form ik a s = ~ {(on+ p + (-- 1)so,r-p) (1 - - i p ) ,

(4)

with O~+p the total cross sections for ,r ± scattering on protons. We note that in eq. (3) we have dropped the nucleon spin degree of freedom in the interests of maximum simplicity, an approximation which we believe to be adequate for the present prelimiary exploration; of course, we must retain all the isospin operators in order to calculate the charge exchange process. (We have also ignored the complications which may arise [9] from the presence of such operators when the ,rN potentials overlap appreciably.) Towards further simplification of the calculation, we use the pure S-state Gaussian form of Schiff for the A =3 wave functions with the correctly antisymmetrized spin-isospin factors [ 10]. The above approximations then yield an especially simple result when the full evaluation [11 ] of the series in eq. (2) is carried out; namely for lr± + 3He -+ 3He + 7r± Ffi(Q ) = (3 a 0 -+a 1)I1 (Q) + (3 ao2 -+2 aoa 1 - 2 a 2) 12 (Q) + (a g + a2al - 2aoa21 ~-{ a31)13 (Q) ,

(5)

and for Ir+ + 3He --, 3He + n ° or n - + 3He -~ 3H + n o Ffi(Q) = - x / 2 a l [I1 (Q) + 2aoI2(Q) + (a20 - { a 2 ) I 3 ( Q ) ]

where

406

,

(6)

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I

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I

Fig. 3. Angla-integrated cross sections for SH(~r+, n°)aHe as functions of pion lab kinetic energy T n. T h e solid line is for parameter set (i) defined under eq. (7) while the dot-dash line is for set (iii). The digits next to the curves indicate the number of seatterings retained for the corresponding cross section, " 1 " referring to impulse approximation and " 3 " to the full Glauber result. The dashed curve marked " 2 " is the single- plus double-scattering result for parameter set (i), and already shows the strong absorptive effect of pion rescattering before or after charge exchange.

I

i )3He_

"_,~

1 I

O0

I

20O

I

Tn(MeV)

300

/I(Q) = exp {-½Q2('y2('Y2+ 1/9°t2)}, I3(Q) = - ~ x2 exp ( - Q 2 ~ 2 / 6 ) ,

6 January 1975

I2(Q) =~-ix exp {-½Q2(~'y 2 + 1/9ot2)}, 1

1

x - -k'y2 1 + 1/6~/2c~2 '

(7a, b) (7c, d)

and t~ is the Gaussian parameter for the A =3 wave functions [10]. In the numerical evaluation of eqs. (5) and (6), we took the value for this parameter determined [10] from electron scattering, 0t = 0.384 fm -1 . The other parameters entering are those of eqs. (3) and (4), where the total 7r±-p cross sections are fairly well known, and were here taken from the tabulation of Barashenkov [12], but the quantities 7 and p are more poorly determined. We used three methods for fLxing their values: (i) fitting eqs. (3) and (4) to the differential 7r±-p cross sections, d o/d I2 = if(q)i 2 as listed by K/illdn [ 13], (ii) using the values quoted by Wilkin [8], and ('fii) using values determined by assuming that the 3,3 channel is completely dominant in lrN scattering at these energies. These three approaches yield parameters which differ by about 50% or less near and above the 3,3 energy region (say 180 MeV ~< T~r~ 280 MeV, for pion lab kinetic energy T~r), and by some 100% well below the resonance (T~r~ 120 MeV); we took this as a measure of the uncertainty in these parameters which propagates into an uncertainty in these parameters which propagates into an uncertainty in our cross section results. Angle-integrated elastic cross sections for rc-+-3Hescattering are shown in fig. 1 as a function of energy for the first parameter set indicated under eqs. (7), and exhibit the anticipated peak in the 3,3 region; results for the other parameter sets are similar, to within about 20%. Figure 1 also shows the only two extant experimental data points [14] for these cross sections, at 100 MeV, which is below the energy (~ 120 MeV) at which one could reasonable suppose the Glauber approach to have any validity for pion-nucleus scattering [15], so that we are not greatly dismayed by the lack of agreement there. Indeed, the major part of the discrepancy may merely reflect the very large uncertainty in our parameters at low energy, where they are known only to within about a factor of two. By the same token, the experimental angular distributions [14] at 100 MeV, shown in fig. 2, are also not in agreement with theory, but the general picture is sufficiently good to suggest that at somewhat higher energies, where the Glauber theory is more adequate for our present purposes and where our input parameters are better known, reliable results are obtained. Our main results is in fig. 3, where are shown the single-, double- and triple-scattering results for the angleintegrated charge exchange cross section, using our first parameter set (iii). These different choices of parameters yield the same qualitative (and, very nearly, quantitative) results: in spite of the very different values for them the single-scattering, or impulse approximation *a , cross section shows a strong peak in the 3,3 region, while the inclusion of all the Glauber multiple-scattering terms - through triple scattering here - yields a cross section which is essentially fiat with energy, and perhaps even carries a hint of a dip near the 3,3 position, Thus, even for the A =3 ,3 We note that an earlier impulse approximation calculation for non-change - exchange scattering has also been reported [ 16 ].

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system, pion rescattering before or after charge exchange has a very strongly absorptive effect. We believe that this qualitative feature offers a fairly clear-cut situation in which to carry out an experimental check o f our understanding o f pion-nucleus charge exchange processes. One o f us (JME) wishes to acknowledge useful discussions on pion charge exchange reactions with Drs. J. Alster, B.F. Gibson, A. Kerman, R.C. Minehart, J.E. Spencer and A.I. Yavin, and the kind hospitality o f the Los Alamos Scientific Laboratory, where this work had its inception.

References [1] D.T. Chivers et al., Nucl. Phys. A126 (1969) 129. [2] M. Zaider etal., preprint, Tel-Aviv University (1973) and Prec. Fifth Int. Conf. on High energy physics and nuclear structure, Uppsala (June, 1973). [3] J. Alster, private communication, August, 1974. [4] Prec. LAMPF Summer School on Pion-nucleus scattering, eds. W.R. Gibbs and B.F. Gibson, Los Alamos report, LA-5443-C (October, 1973). [5] D. Tow and J.M. Eisenberg, Nucl. Phys., to be published. [6] G.A. Miller and J.E. Spencer, preprint, Los Alamos Scientific Laboratory (1974). [7] R.J. Glauber, in Lectures in Theoretical Physics, Vol. 1, eds. W.E. Britten and B.W. Downs, (Interscience, N.Y., 1959) p. 315. [8] C. Wiikin, Nucl. Phys. A220 (1974) 621. [9] D.R. Harrington, Nucl. Phys. B59 (1973) 305. [10] L.I. Schiff, Phys. Roy. 133 (1964) B802. [11] V. France, Phys, Roy. C9 (1974) 1690. [12] V.S. Barashenkov, Interaction cross sections of elementary particles (Israel Program for Scientific Translations, Jerusalem, 1968). [13] G. K~dl6n, Elementary particle physics (Addison-Wesley, Reading, Mass., 1964). [14] I.V. Falomkin et al., Lett. Nuovo Cim. 5 (1972) 1121. [15] A.T. Hess and J.M. Eisenberg, Phys. Lett. 47B (1973) 311. [16] G. Ramachandran and K. Ananthanarayanan, Nucl. Phys. 64 (1965) 652.

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