The strong interaction shift in pionic 3He1 A. W. THOMAS

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TRIUMF, University ofBritis11 Col~lmbici,Vc~'rrnco~nrer, B.C., C N ~ NV6T ~ N I W5 Received January 10, 1978 We calculate the multiple scattering contribution to the strong interaction shift in the n - 3He atomic Is level. The result is in reasonableagreement with recent measurements. However, it is also noticed that this particular system is very sensitive to the inclusion of absorptive effects. In fact, when the dispersive effect of absorption is included in the usual way, only one (relatively old) set of nN scattering lengths gives a result near the data. We stress the importance of a definitive measurement in both this system and then-d system toclarify matters. Nous calculons la contribution de la diffusion multiple au deplacement d'intelxction forte du niveau atomique 1s de n- 3He. Le resultat se trouve en accord raisonnable avec les mesures rkcentes. On remarque aussi, toutefois, que ce systeme particulier est tres sensible B I'inclusion d'effets d'absorption. De fait, lorsqu'on inclut I'effet dispersif de I'absorption selon la methode habituelle, un seul ensemble(relativement ancien) de longueurs dediffusionnN donne un resultat voisin des donnees expirimentales. Nous insistons sur I'importance d'effectuer des mesures definitives, dans ce systtme ainsi que dans n-d, afin de clarifier la situation. [Traduit par lejournal] Can. J . Phys., 56,687(1978)

The n - 3He atom offers a unique example in x-nucleus physics of an attractive, s-wave, strong interaction shift ( I , 2). Next to the n-d case, on which much theoretical attention has been lavished in recent years (see refs. 3 and 4 for a complete list of references), this system is the most readily amenable to accurate calculation in terms of multiple scattering theory. Indeed, we shall see that triple scattering is so small as to be insignificant. Perhaps the most significant fact about n- 3He, however, in terms of the really fundamental questions in intermediate energy physics, is its relatively large width. Indeed two very recent experiments (1,2) suggest a width of between 42 and 68 eV, compared with a shift of +27 to +44 eV. Wherever the true values lie within this range, it is clear that absorption (or rather its so-called "dispersive effect" (5)) could play as important a role in determining the real part of the shift as the normal multiple scattering terms. T o emphasize the comparative sensitivity here, as opposed to xd where most serious efforts to calculate the dispersive effect of absorption have been made so far (6-8), we note that there the width (due to n-d -+ nn) is expected to be of order 0.01 III,-' (or -1 eV) (9, lo), compared to an observed shift of about 5 eV (35). This aspect of the problem will be discussed in more detail in the later section on absorption, but it is clear a priori that n - 3He should be a better testing ground for any theoretical approach. In what follows, we compute the s-wave n - 3He scattering length (and hence the strong interaction shift) by systematic evaluation of each term in the 'Research supported by a grant from the National Research Council of Canada.

appropriate multiple scattering series. In fact, the single and double scattering contributions turn out to be sufficient. Again, as for nd, this leads to a strong dependence on the poorly known isoscalar nN scattering length (i.e., the combination rr, = (a,-,, + a,-,)). This sensitivity to the n N scattering lengths is tested explicitly by using three different analyses (1 1-13).

+

Multiple Scattering Through the s-wave .irN Interaction The three sets of scattering lengths from refs. 11-13 are given in Table 1. In order to evaluate the n~ultiplescattering series we rely on the weak binding theorem proved recently by Fiildt (14), once again for the n~ system. In particular, he showed that one could neglect all binding corrections, without making an error significantly larger than the present uncertainty in the input scattering lengths. Neglecting the binding correction in single scattering (SS) one finds

The double scattering contribution (including double charge exchange) in the same approximation is

We shall neglect the small differences in the (pn) and (pp) separations. Using Schiff's Gaussian and Irving

C A N . J . PHYS. VOL. 56. 1978

TABLE1. The three sets of EN scattering lengths defined in the text (in units of l1ln-l) Value Salonion (1 1)

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Scattering length

Bugg et (12)

[I/.

Samaranayake and Woolcock (13)

TABI.E2. Contributions of various orders of multiple scattering through the 5-wave EN interaction (units rli,-'), and through thep-wave EN interaction Value

Parameter

Salonion (1 1)

Bugg et (11. (12)

Saniaranayake and Woolcock (13)

Single scattering Double scattering Triple scattering Total (EN s-wave) (EN p-waves) Total multiple scattering AEI~

0.080f0.013 -0.044f0.006 -0.002 +0.034k0.014 -0.006f0.002 +0.028+0.014 22f I1 eV

0.081f0.004 -0.042_+0.003 -0.002 +0.037f0.005 -0.006f0.002 +0.031+0.006 2 4 k 5 eV

0.101fO.011 -0.043k0.005 - 0.002 +0.056f0.012 -0.006+0.002 +0.050f0.012 3 9 k 9 eV

wave fiinctions (15), one finds respectively (I/,.) e (0.47, 0.53) fni- ' and (0.50, 0.56) fni- Of course, tlie 3H - 3He Coulo~nbenergy difference gives a direct measure of (I/,.), once we have corrected for exchange currents and finite size effects. For example, a typical estimate of the point Coulonib energy difference of 680 keV leads to ( I li.) equal to 0.47 fin - I . As a reasonable mean vzlue we take

'.

Finally, we note that there are a large number of contributions to tlie triple scattering terni (TS). However, the net result is very sniall, and essentially inde- . pendent of the exact input nN scattering lengths, viz. - 0.002~1,- . The nuinerical results for the Re {a(n- 3He)) are sumniarized in Table 2. (Note that as we have no iiniqi~eprescription for combining errors, the errors in SS and DS have been treated as independent.)

and Kolybasov (I 8). In fact, in the nd case these two terms cancel almost exactly. For n- jHe there is still only one neutron but two (np) pairs. Therefore the s-1) cross term is dominant, albeit sniall, and gives a net repulsive contribution. As a first estimate of this effect we have simply taken the values calculated by Kolybasov and Kudryavtsev (17) (with s-1) cross term niultiplied by two). This gives a reasonable first 2 (-0.007)), estimate of - 0.006 177,- '(= f0.008 which even though it may be in error by 25x, is more than sufficient for our present purposes.

+

Summary of Multiple Scattering Calculation The total niultiple scattering contribution to Re ( ~ ( n -3He)) for the three sets of scattering lengths is shown in Table 2. To express the strong interaction shift of this level (he,,) in ternis of the scattering length, we rely on the results shown in Table 1 of Hiifner et ul. (19). They showed that for n- 4He the perturbation for~nula(16) relating Ae,, to the scatContribution of the T N p-wave tering length worked very well. In particular, a scatAs Ericson (16) has stressed, the p-wave n-p scat- tering length of 0.098 ill,-', corresponds to a shift of tering volume is only about (1/14)th of the n-n value. 75.7 eV. Therefore, multiple pion rescattering through the Clearly the inultiple scattering result for all three p-wave nN interaction is negligible. This leaves only sets of scattering lengths is coinpatible with the the p-wave single scattering terni, and tlie s-1) cross present range of experimental values from 27 t o terms. For the deuteron these terms have been dis- 44 eV (although the smaller (TRIUMF) value is cussed in detail by Kolybasov and Kudryavtsev (17) slightly favoured). Note that the main difference be-

-

'

689

THOMAS

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TABLE3. Results for the Re {o(n- 3He)', (and the Is energy shift), including elfects of absorption through the expression R e {$"(n- 3He)) = -1m jo"bs(n- 3He)]. (Note that the variation inside brackets corresponds to an increase (-) or decrease (+) of the total width by the quoted error. Experimentally AE], is between 27 and 45 eV)

rlol,, (ev)

Reference

R e {rr(n- 3He)] (III,-I)

A E I , (eV)

Salomon (1 1) Bugg et ol. (12) Samaranayake and Woolcock (13) Salomon (1 1) Bugg et 01. (12) Samaranayake and Woolcock (13)

0.010[T0.005]k0.014 0.014[T0.005]k0.005 0.033[T0.005]k0.012

8(~4)?111 IO(T4)k 4 25(T4)+ 9(

3 7 k 10

tween the result of Samaranayake and Woolcock to ass~ri17eRe B, = -1111 B, (29). More recently, [5] (13) and the other two comes froni the single scat- was used to estimate the effect of absorption in a tering term, and particularly the difference in the systematic ~nultiplescattering study of s-wave pionic isoscalar scattering lengths (a, in Table 1). The major atom level shifts for A > 4 (30). improvenient that could be made in this calc~~lation For the present, in view of its widespread acceptwould involve a better determination of a,. ance, we also use [5] to estimate the dispersive effect of absorption. Using Phillips value for the width (cf. Absorption [4]) this leads to Let LIS start with the estimate for the total TC- 3He width calc~~lated by Phillips (20) and Phillips and Roig (21). They find r,,,is 37 f 10 eV,2 of which In the long term, however, in view of the theoretical about 73% (22), or 27 8 eV, is true pion absorp- proble~iisand the strong sensitivity of the TC- 3He tion, calc~~lated in a two-nucleon model. Again using system to this effect, it niay be ~iiostprofitable to turn Table 1 of Hiifner et a/. (19), one finds the problem around, and use the measurement of A&,, to test [5]. Including the result [6] together with the niultiple scattering results in Table 2, one finds the 3He energy Now, as we mentioned in the Introduction, one of shifts given in the top half of Table 3. If one uses the 12eV) measured at the fundamental questions in the theory of n-nucleus much larger width (68 TRIUMF (2), one gets the lower half of Table 3. scattering is the separation of the effect of absorption To conclude this section, it must be said that alfrom the standard multiple scattering contribution. Many years ago Brueckner (5) gave a very qualitative though the sort of phenonienology we are using is necessary at the moment, it is equally important that proof that serious ~nicroscopiccalculations (for example, in the Re {a, -A"b" = - Im {a, -A'bs] two-nucleon model) be undertaken as soon as pos[5I Further theoretical work has onlv occurred verv sible.

+

+

recently (7, 23-25), and has been la&ely confined t i the nNN s y s t e ~ nBy . ~ and large these recent estimates have tended to confirm [5] in the n-d case. Note, however, that the calc~~lations of Hachenberg et a/. (26) and Hachenberg and Pirner (27), using a very different off-shell extrapolation of the TCNinteraction, tend to suggest a small positive value of Re The general result, [5], has found widespread acceptance in the literature. For example, in using an absorptive term Bop2 (28) in the n-nucleus optical potential describing pionic atoms, it is now common ZThe error quoted is sonlewhat optimistic, Including only part of the many uncertainties inherent in such a calculation (A. C. Phillips, private communication). 3Although, within the quasi-deuteron model of pion absorption, one expects the results to carry over to heavier nuclei.

Comparison with the Ericson-Krell Potential Mason et a/. (2) have compared their experimental result with the predictions of the optical model of Krell and Ericson (31). These authors have demonstrated the unphysical effects of the p-wave absorption term in such a light nucleus. In fact, in their final calculations they set C, = 0. (In this respect it w o ~ ~ l d be interesting to see the effect of the similar term introduced by Landau and Thomas (32) in a s t ~ ~ d y of low energy pion scattering, since this term has niucli snioother off-shell behaviour.) They have also dropped the dispersive effect of absorption in the s-wave absorptio~iterm (i.e., Re B, = 0). With these t s directly comparable two assuniptions their r e s ~ ~ lare with our Table 2. In particular the effect of TCN

690

C A N . J. PHYS. VOL. 56. 1978

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p-waves is similar. The uncertainty in their parameter 6, is analogous to our worries over a,, although their results seem slightly more sensitive to it.

4. A. W. THOMAS.Carnegie-Mellon Conference, AIP Conf. Proc. 33,375 (1976). 5. K. BRUECKNER. Phys. Rev. 98,769(1955). 6. 1. R. AFNANand A. W. THOMAS.Phys. Rev. C, 10, 109 (1974). Summary and D. S . KOLTUN.Ann. Phys. (N.Y.), 109, 1 7. T. MIZUTANI It has been shown that the .n- 3He strong interac(1977). and N. MUKHOPADHYAY. Carnegie-Mellon tion shift is very sensitive both to the poorly known 8. T. MIZUTANI Conference, AIP Conf. Proc. 33, 172 (1976). nN scattering lengths, and the method of including 9. C. M. ROSE.Phys. Rev. 154, 1305 (1967). absorption effects. The former uncertainty, in fact, 10. G . FALDT.NucI. Phys. B, 10,597 (1969). prevents us from drawing a firm conclusion. Never- 11. M. SALOMON. TRIUMFreport TRI-74-2. 1974. theless it does seem that either the nN analyses of 12. D. V. BUGG,A. A. CARTER,and J . R. CARTER.Phys. Lett. B, 44,278 (1973). Salomon (1 1) and Bugg et a/. (12) are inaccurate, or V. K. S A M A R A N A Y A and K EW. S. WOOLCOCK. Nucl. Phys. there is a significant deviation from the prescription 13. B, 48,205 (1972). for iiicluding absorption given in [ 5 ] . I11 view of the 14. G . FALDT.University of Lund preprint LUTP 1974.13. University of Lund, Lund, Sweden. 1974. very widespread use of [5] in pionic atom analyses (29), and low energy pion scattering (32, 33), it is 15. L. SCHIFF.Phys. Rev. 133, B802(1964). 16. T. E. 0 . ERICSON.Proceedings of the Banff summer inlportant to resolve this question. school. Ediretl by G. C. Neilson, W. C. Olsen. and S . We stress that the sensitivity of the n - 3He scatVarma. Nuclear Research Centre. University of Alberta, tering length to absorptive effects is far greater than Edmonton, Alta. 1970. p. 102. and A. E. KUDRYAVTSEV. NucI. Phys. for the n-d scattering length. Thus the proposed 17. V. M. KOLYBASOV B, 41,510 (1972). measurement of a(n-d) to an accuracy of 0.2 eV (34) V. M. KOLYBASOV. Carnegie-Mellon Conference, AIP is a crucial step in solving this problem. In particular, 18. Conf. Proc. 33,394 (1976). it should provide a reliable value of the isoscalar nN 19. J. HUFNER,L. TAUSCHER, and C . W I L K I NNucI. . P h y s A, 231, 455 (1974). scattering length (a,). S . Prog. P h ~ s40,905 . (1977). Finally we note that even given the rather large 20. A. C. P H ~ L L ~ PRep. 21. A. C. PHILLIPS, and F . ROIG.Nucl. Phys. B, 60.93 (1973). theoretical errors, the present calculation favours 22. M , D. HASINOFF, F, C O R R I V E A U , D, F, MEASDAY, and M. the TRIUMF value of 27 f 5 eV for A E , , ( ~ 3He). SALOMON. Abstract Volume of 7th International Conference on H~gh-EnergyPhysics and Nucle:uStructure. S I N , (The calculation of the width by Phillips (20) favours Villigen. C19. 1977. the SIN value.) It is obviously very important that the A. W. THOMAS. Ph.D. Thesis. Flinde~sUniversity ofSouth apparent discrepancy between the two experiments 23. Australia, Bedford Park, S o ~ ~Australia. th 1973. Chapt. 3. and if possible, the errors significantly 24. T. M I Z U T A N IPh.D. be . Thesis. Unive~sity of Rochester, reduced. If this can be done, we can look forward Rochester, NY. 1975. to an important new insight into the low energy 2 5 A. S . RINAT.1977. Nucl. Phys. A, 287,399(1977). 26. F. HACHENBERG, J. H U F N E Rand , H. PIRNER. 1977. Phys. n-nucleus interaction. Lett. B, 66,425 (1977). 27. F. HACHENBERG and H. P I R N E RUniversity . of Heidelberg Acknowledgements prepl.int. University of Heidelberg, Heidelberg, W. Germany. 1977. I would like to thank G. Beer, M. Dixit, and G. Mason for discussions concerning their data. It is 28. M. ERICSONand T. E. 0 . ERICSON.Ann. Phys. (N.Y.), 36, 323 (1966). also a pleasure to acknowledge useful discussions 29. J . HWFNER. Phys. Rep. 21C, l(1975). on the calculation of n-nucleus scattering lengths 30. F . MYHRER and D. S. KOLTUN. Z. Phys. A276,29 (1976). 1969. Nucl. Phys. B, 11, 31. M. KRELLand T. E. 0. ERICSON. with F. Myhrer and M. Krell. 521 (1969). 32. R. H. LANDAUand A. W. THOMAS.NucI. Phys. A. In 1. R. ABELA,G. BACKENSTOSS, A. BRANDAO-D'OLIVEIRA, press. and J . CARR.Michigan State H. MCMANUS, M. I ~ Y C KH. I , 0 . MEYER,I. SCHWANNER, L . TAUSCHER, 33. K. STRICKER, University prepl.int. Michigan State University, East LansP. B L U M W. , FETSCHER, D. GOTTA,H. KOCH,H. POTH, ing, MI. 1977. and L. M. SIMONS. Phys. Lett. B,68,429 (1977). 2. G. R. MASON,G. A. BEER,D. A. B R Y M A N M. , S . DIXIT,S . 34. G. BEERef ol. Improved measurement of the strong interaction shift in pionic deuterium. IEP Grant Application, UniA. O L I N ,R. M. PEARCE, M. K. K I M ,J. A. MACDONALD, versity of VictorialTRIUMF. 1977. KRELL,and J. S. VINCENT. Phys. Lett. B. In press. D. V. BUGG,U. GASTALDI, P. HATTERSLEY, D. 3. A. W. THOMAS.Proceedings of the International Confer- 35. J. BAILEY, R. J E R E M I A H E., KLEMPT,K. NEUBECKER, E. POLACCO, ence on Few Body Problems in Nuclear and Particle and J. WARREN. Phys. Lett. B, 50,403 (1974). physics. Laval University, Quebec, P.Q. 1974. p. 271.

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1. G.R. Mason, G.A. Beer, M.S. Dixit, S.K. Kim, J.A. MacDonald, A. Olin, R.M. Pearce, W.C. Sperry, J.S. Vincent. 1980. Pionic K X-rays in liquid 3He. Nuclear Physics A 340:2, 240-248. [CrossRef] 2. F. Myhrer. 1980. Pion-nucleon and pion-few nucleon interactions. Nuclear Physics A 335:1-2, 255-265. [CrossRef] 3. K.P. Lohs. 1978. Scattering lengths of pionic 3He and 4He. Nuclear Physics A 312:3, 297-310. [CrossRef]