Pion-Pion Scattering
So interconnected is hadronic physics in all its parts (as is implied, for example, by the unitarity relations), that no one hadronic process can be thought to be under full theoretical control without an under standing of all. ft is a mor modest goal to settle instead for correlation of phenom ua, introclncing for the description of one process the empiri 'al iwop rties of others. Th· possibilities are limited, however, by th oir umstimc that only a small part of hadronic physics is ac ssibl to clir ct XJ. erim nta.l study. In particular, the inaccessi bility of diT ct I ion-piou scatt ring information has been especially grievonf'i . Although on th 11otion of nuclear democracy1 all particles ar snpposcd to h q ual t h · pi n- which is by a substantial margin th· li ghtest of he hachons-i p rhaps the most equal of all. This near ma. sl s u ,· gives to the ion it prominence in the analysis of the accessible processes (one-pion-exchange models, the PCAC hypothesis, etc.). It has also suggested the idea that the two-pion state, at least for small energies, is perhaps the simplest scattering system of hadrons that one can contemplate theoretically. Some years ago in their di persion r lati n trcatm nt of I.ow -energy pion-pion scattering, Chew and Mand lstam2 inde d sugge ted that one might get away with a elf- fo :;ing approximati n with a tention restricted solely to the two-pion tat in t h cm. sncl hannel . ~'h results, quantitatively, proved to be inconclusive. But in connection with that outstanding empirical feature of the two-pion system, the p-wave p-meson resonance, this scheme gave rise to the bootstrap approximation to self-closing. 3 Th· idea hero is to ace pt the existence of the resonance, relying on it in th rossod hann ls fo provide the forces which in turn produce the resonanc u.cJ{ again in the direct channel. The mass and width of the resonance are supposed to emerge from the demand of self-consistency. Qualitatively, a p-wave resonance does emerge self-consistently; but in the straightforward version of the model, consistency with the actually observed p-meson parameters is not impressive. It may be too much, even for the almost massless mesons, to ignore the rest of hadronic physics. In any case, the impor-
68 tance of pion-pion scattering effects for the understanding of other processes is brought out when one recalls that the p-meson resonance was itself first foreseen theoretically from a dispersion relation analysis of nucleon electromagnetic structure.4 During the past few years, a renewed and intensive interest in pion pion scattering has again developed in the literature. It arises for (at least) two reasons. (I) The pion phase shifts are wanted, more than ev r be£ r , in th udy f oth ~: r acti u. ; notably in connection with tb p1·ocesses K ~ 21T. Insofar as the LJJ = ! rule for w ak, non- eptonio d •cays i. ·lated, as it surely is to som small e ·t nt if only eoau f electromagne io ·ff cts, th -wav pha.i -shift differ nee
S0 - S2 (the subscript deno · i ot pie spin) figi:u·es in th para.met ri7-a tion of the bro.n hing ratios of K + --;.- 7T+ + 7T0, 1 i -4 7T+ + 'IT-, and Ki --? n° + 11°, and in th analy js of th· OP- iolating d ays KJ, -+ ~'IT. (2) n addition, th id .a.s f urrent a lg bra and I AO ar especiaIJy rilcing in th ir apJ>li ation to pion- pi n , cattering. It ha.a been_argu d that the sue ·sses a hiev d elsewhere for these id as can b · 1md rst <.l only if th pion- pion interactions are (imexp ctedly1) mild ll' M' t lrr • ·I old; and onsistent with this, W inb rg' 5 cm·r nt a.l 'B • " t i,1 =Te'Js i.1s1no1,r· A number f attempts have been made recently to ext nd th res111ts some distano awa from threshold, with unitarity effi cts taken into 0 accom1t • Of course, consideration of Wlital'ity do s not by it elf serve to specify uniquely tlie continuation away fr m the t lu·eshold T gion. But one conjecture. simple unitarized :£ Tms for the amplitudes, con straining th·m near thr sh Jd, by th urrent algebra results, and, for th p-wav amplitude, imp sing the xtra demand that it resonate at 69 the empirical p-meson mass. In the analyses of Brown and Goble, and of Arnowitt et al., the s-wave phase shifts turn out to vary smoothly with energy up to the region of the K-meson mass (and beyond), S0 rising there to the value 80 R:J 20°, 82 falling to the value S2 R::; - 10°. The results in this region are not so different from what one would obtain for the s-waves from the simple current algebra expressions, 6 with the approximation ei sin 8 R::; S. There is enough arbitrariness in these procedures, however, that no obvious disaster would befall if experiment were some day to reveal a very different behavior of the phase shifts away from threshold. A failure of the current algebra results close to threshold would, on the other hand, be a greater affront; but even here, as emphasized in the papers of Sucher and Woo, 7 and Fulco and Wong, 8 delicacies in giving a precise meaning to the extrapolations involved in PCAC are by no means absent. One would prefer in these matters to turn from pure theory to pure experiment; but it will be some time before this is possible. A com promise, for the present era, consists in inverting the interest one has in pion-pion scattering for its effects in other processes, with the aim of extracting the pion information from the effects produced. A classical idea along these lines was proposed some time ago by Chew and Low, 9 for the process 7T + N -+7T + 7T + N. The scheme is as follows. Let ..1 2 be the invariant momentum transfer between the nucleons, m the invariant mass of the outgoing di pion system, and ethe angle between the incident and one of the outgoing pions, in the dipion rest frame. Extrapolated in the variable ..1 2 to the vicinity of the unphysical point ..1 2 = - µ 2, the production amplitude should be dominated by the one-pion-exchange term, the residue of the pole at ..1 2 = - µ 2 being expressible in terms of the pion-pion scattering amplitude. Extrapolated to this region, the production differential cross section is then given by the expression da = _ (f2) m(!m2 _ µ2)1/2 ( -..12/µ2) da.,,.,, ... 2 2 2 2 dL1 dm dQ 27T kfab (..1 +µ ) 2 dQ + ' wheref2 C'.'.:'. 0.081 and ktab is the incident pion momentum. The remainder 2 2 term has a part with a first-order pole at ..1 = - µ , plus a part regular 2 at the pole. The factor ( -..1 2/µ ) which appears above could as well be set equal to unity; retained, it reflects the fact that the one-pion exchange amplitude vanishes at ..1 2 = 0 (at which point, however, it is not certain that the remainder term can be neglected). The idea, in any case, is to attempt an extrapolation in the variable ..1 2, from the 2 physical region where there are data, to the unphysical point ..1 2 = - µ • Evidently, this is a difficult and delicate matter. A cruder procedure 70 involves neglecting the remainder term even in the physical region, at least for small values of Ll 2, effecting thereby a direct extraction of the pion-pion cross section. Experimentally, the observed dependence on the Ll 2 'ariable is u t ·well fitted by the pure one-pion-exchange xJ r ssi n, so one introduces an empirical Ll 2-dependent form factor, 2 snp o. ed o a1 proach mrity at Ll 0 = - p-2. A recent attempt along these approximate lines has been made by Walker et al.10 They extract a picture in which the 8-wave phase shift o0 rises steadily with increasing energy out to the K-meson mass and beyond, with o0 reaching the value ~35° at the K-meson mass and passing through 90° (broad 8-wave resonance) somewhere in the region around 900 Me V. The 8-wave phase shift o2 decreases smoothly with energy, reaching the value I=:::! -15° at the K-meson mass. The more elaborate procedure called for by extrapolation in the Ll 2 variable has been carried out most recently by Baton et al., 11 for the process TT- +p -'>'TT-+ TTo + p (this involves the I= 1 and I= 2 dipion states). Retaining only 8- arid p-wave amplitudes in the analysis, they obtain for the p-wave a picture dominated by the p-meson resonance, con viction being lent by the fact that a,,,, is nearly equal to the theoreti cally expected value 12TTA2 at the resonance peak. In the I= 2 8-wave, the phase shift is pictured as decreasing smoothly with energy, the value at the K-meson mass being given by o2 ~ - 8°. The best fit to an 1 effective-range formula yields a scattering length a 2 = - 0.052µ- , not so different from the simple current algebra result. A different procedure for getting at the pion shifts is based on con sideration of backward (180°) pion-nucleon scattering,12 the amplitude being regarded as a function of v = k2, where k is the center-of-mass momentum. It follows from the Mandelstam representation that the backward amplitude is analytic in the whole complex v plane, except for th.~ nucleon pole and the cuts (0, oo ), ( - oo, - µ 2). The physical region ~~or pion-nucleon scattering corresponds to v > 0, whereas along the left-hand cut the continuation describes the annihilation process N + N ___,, 21T at barycentric (squared) energy t = - 4v. The annihila tion amplitude is physical in the t variable only for t > 4M2 (though unphysical in the annihilation-channel scattering angle). In a partial wave expansion for the region 4,u 2 < t < 16µ 2 each partial-wave amplitude must have the phase of the corresponding pion-pion scattering amplitude; and one can suppose that four-pion effects are unimportant even well beyond 16µ 2• Suppose, further, that only the 8- and p-wave amplitudes are important over this range. Then con tinuation in the v variable for backward pion-nucleon scattering to v = - t/4, t > 4µ 2 must reveal the pion-pion phases: 8-wave for the 71 I= 0 crossed-channel amplitude, p-wave for I= 1. The principle, once again, is clear, but the implementation is as difficult and uncertain as it always is when one is attempting extrapolations. Lovelace et al. 13 have recently carried out the attempt, proceeding in a wide variety of ways; and they offer a large choice of curves describing the phase shift 00 as a function of energy. One can infer, so wide is the choice, either that this particular technique is unsuitable as compared to others, or else that the authors are singularly candid. Qualitatively, in all of the various fits, one finds that o0 is positive and rising with energy once well above threshold, and that it inevitably passes through 90° (broads-wave resonance) somewhere or other! Near threshold, fits are available with both signs of the scattering length, the positive choices tending to be rather large. Still further information, this time on the s-wave phase-shift difference o0 - Sa jw~t n.t th /! -me ·on mass, com s fr m compa.r.i on of the de ay 0 rate. for IJ + - 71'+ + 7T 11 KO.. - 7T + + 71' -, K~ -> 7T + 7T0 • Th v ·y cm·- 1· nee of the fir t process represents a l r akdown f the iJI = i rnJ , ancl in gen ral transitions corresponclin to b th iJJ = -jl- an i %must be cont mplated, in adc:Jiti n to the dominant LH = 2 • On the a"m.wip tion that th bs r -d departur s fr m the LJI = { rule can be 11ttl"ibuted solely to L11 =-!transitions, one can extract from the branching ratios 14 information on the phase-shift difference o0 - 02• The data admit of several solutions, the one lying closest to the range of results described abov oonesponding to S0 - o2 ~ 66°. Ther is room fol' fmth · 'P ri mental imprnv ment in th K~ -7 ;...,,. branching l'atios; but in any eas -, t i negle t f th L1I = -~transition . till n ed ind pend nt jui;tilication. It is p ssible, on all f the above evidence to take an ptimisti vi w concemj1J 72 References I. G. F. Chew, The Analytics Matrix (W. A. Benjamin, New York, 1966). 2. G. F. Chew and S. Mandelstam, Phys. Rev. 119, 567 (1960). 3. F. Zachariasen, Phys. Rev. Letters 7, 112 (1961). 4. W.R. Frazer and J. R. Fulco, Phys. Rev. 117, 1609 (1960). 5. S. Weinberg, Phys. Rev. Letters 17, 616 (1966). 6. L. S. Brown and R. L. Goble, Phys. Rev. Letters 20, 346 (1968). R. Arnowitt, M. H. Friedman, P. Nath, and R. Suitor, Phys. Rev. Letters 20, 475 (1968). Further references can be traced from these papers. 7. J. Sucher and C.H. Woo, Phys. Rev. Letters 18, 723 (1967). 8. J. R. Fulco and D. Y. Wong, Phys. Rev. Letters 19, 1399 (1967). 9. G. F. Chew and F. Low, Phys. Rev. 113, 1640 (1959). 10. ~W. D. Walker, J. Carroll, A. Garfinkel, and B. Y. Oh, Phys. Rev. Letters 18, 630 (1967). 11. J.P. Baton, G. Laurens, and J. Reignier, Phys. Letters 25B, 419 (1967). 12. D. Atkinson, Phys. Rev. 128, 1908 (1962). 13. C. Lovelace, R. M. Heinz, and A. Donnachie, Phys. Letters 22, 332 (1966). 14. T. D. Lee and C. S. Wu, Ann. Rev. Nucl. Sci. 16, 511 (1966). 15. N. Cabibbo and A. Maksymowicz, Phys. Rev. 137, B438 (1965). F. A. Berends, A. Donnachie, and G. C. Oades (to be published). A. Pais and S. B. Treiman (to be published). See also references therein. For the experimental data, see R. W. Birge et al., Phys. Rev. 139, Bl600 (1965). AJ 73