Shapiro, Joel

Yellin, Joel.

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Lawrence Berkeley National LaboratoryRecent WorkTitleA MODEL FOR nn SCATTERINGPermalinkhttps://escholarship.org/uc/item/6v883410AuthorsShapiro, JoelYellin, Joel.Publication Date1968-09-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaRESEIVEO University of California lAWREiKE IlAD\AllON lASURA10RV (leI z 3 \~b~ E ·0 Y AND rnest . L\BRAR lONR d· . oocuMENTS sECT - alation Lawrence Laboratory A MODEL FOR 1T1T SCATTERING Joel Shapiro and Joel Yellin TWO-WEEK LOAN COpy This is a library Circulating Copy which may be borrowed for two weeks. For a personal retention copy. call Tech. -Info. Dioision. Ext. 5545 Berkeley, California UCRL-18500 ~>~ DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. 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Subrriitted to Physical Review Letters UNIVERSIT Y OF CALIFORNIA Lawrence Radiation Laboratory Berkeley,· California AEC Contract No. W-7405-eng-48 A MODEL FOR lTlT SCATTERING Joel Shapiro and Joel Yellin September 1968 ( UCRL-18500 Preprint "'-, ''\: ) " c --A MODEL FOR :n::n: SCATTERING * Joel Shapiro University of California Department of·Physics Berkeley, California and Joel Yellin Lawrence Radiation Laboratory University of California Berkeley, California ABSTRACT We propose ~ model for :n::n: scattering, using . fu.r).ctions of the type recently suggested by Veneziano. We discuss the resulting threshold phenomena, and also the predicted widths and positions of the rr:n: resonances. We discuss here a model for :n::n: elastic scattering, constructing our amplitude ~y using functions of the type recently proposed by V .. 1 eneZlano. If we insist the amplitude be a linear combination of symmetric function's F(x,y), then the most general isospin amplitudes satisfying crossing symmetry and Bose statistics, are cl C2[3F(t,u) + F(s,u) + F(s,t)J + 2[3F(S,u) + 3F(s,t) - F(t,u)J ; (la) (lb) C1F(t,u) + C2 [F(S,U) + F(s,t)J , (lc) -2- UCRL-18S00 where the c. l are constants and where I . A (s,t,u) is the amplitude for x isospin I in the x channel. We choose F(s,t) g r[l - a(s)] r[l - aCt)] r[l - a(s) - a(t)] (2) and a(s) a + bs (3) We assume there exist no I 2 J]'1 resonances. This can be assured by setting c2 = 0 in (1). The properties of F(s,t) guarantee the correct Regge asymptotic behaviorland the saturation of the finite energy sum rules (FESR).2 The three parameters a,b, and g are determined as follows. (There is no constraint on external mass here, so we take 2 I-l = m lL 2 from experiment.) We set r = 112 MeV and m = 764 MeV, leaving one· free parameter pnn P which we take to be a. Further, in order to agree with the I =0, shift,3 cPo(m2 ), form <;; 400MeV, we choose nn nn a= 0.48. -L = 0 . n.n phas e This then gives a 4 . -2 slope of b = 0.90 GeV . Our amplitude is now completely determined. It suffers from the following diseases: (1) The narrow resonance approximation violates unitarity.5 (2) The Pomeranchon is ignored. (3) The amplitude is not unique. (4) Causality, in the form of pos iti vity of the resonance widths, is not necessarily satisfied. The narrow resonance approximation has been discussed by Mandelstam. 5 An admittedly rough argument giving the order of magnitude of the Pomeranchon contribution goes as follows: The asymptotic rr:rr elastic scattering cross-section of . .::::1 mb, estimated using factorization, can be compared ·wJth t.he 1(11 cross-section in the resonance region ,assuming , (/ • -3- UCRL-18500 the Pomeranchon contribution to the elastic scattering is roughly energy independent. This gives an effect, of the order ofa few percent. 6 As pointed out by veneziano,l (2) is not unique.? We have chosen what seems to us the simplest possibility consistent with the absence of I = 2 resonances. We discuss the positivity condition below. Our ampli'tude gives: (A). The masses and widths of the rOT .resonances. (B) The low energy IT IT phase shifts (c) A prediction of the angular and energy dependence of high energy IT IT charge-exchange scattering. A. IT IT Resonances The model predicts the existence 'of families of + even , I =0) and (~= odd-, I = 1) nIT resonances, with fixed relative widths. The resonances lie on exchange degenerate parallel trajectories. For example, the p has a degenerate 0+ partner, ,which can be identified with the broad enhancement sometimes called3 E. The f(1260) has degerier~te 1- (p ,) and 0 + (E ,) partners. Also there should be a state at ~670 MeV with degenerate - + 1 , and 0 . 8 companlons. Our predicted masses and widths are shown in Ta.ble 1. The widths of the resonances lying on the first daughter trajectory are rather large. The most difficult case to deal with on comparison with experiment is the pl.9 The" experimental data of Crennel et al., 10 can be combined with estimates ,of the p and p' production cross sections to give an upper limi t to the p I width of i r I ; /r < 0.13 p -7 rTIT p-7 rTrT (4) ,-4- UCRL-18500 assuming the p' 11 ' is totally elastic. Our simple picture can be preserved only if the p' 12 broad. is highly inelastic, so that the resonance is extremely The requirement that all resonance widths be positive gives a bound on a. ,For the ,first daughter trajectory the condition is 3a + 4J..L2b ~ 1.13 The condition for positivity of all widths seems to be that the 0+ partner of the f, the E,', have a positive width. We have , reached this conclusion by numerical computation and are as yet unable to construct a proof. As can be seen in Table 1, our choice of ,a and b leads to a small negative width for E'. Since rE, changes sign at a ~ 0.496 ,we do not consider this serious, and ascribe the negative width to uncertainties in o 50 (s) plus the inaccuracy of the model. B. Threshold Properties We confine ourselves here to a discussion of the s wave phase '0 shifts 50(s), Since our amplitude is pure real we can only com-pare our results with other analyses for 5 small, say ~35° at best. 'V There have been several semiphenomenological attempts to compute3 Using double subtracted dispersion relations, and an input of Breit-Wigner forms for the f, p, and f', Tryon3 has calculated and for's ~ 950 MeV. He also assumed the s-wave scattering lengths ar satisfy - ~< aO/a2~ - ~ and L = ~(2ao - '5a2) ~ 0.10/~. Tryon's results seem relat'ively insensitive to the'model dependent ratio aO/a2 • We are ! 0 able to get 'a reasonable fit to Tryon's 50 (s) for s "- 400 MeV. His is then also in fair agreement with ours. Near 1 a - -- 2' the change sign because we are near a pole in one of our r f'unctions,'and we can therefore say nothing about aO/a2 . Our L ~ O.ll/J..L, and is relatively insensitive to 14 a. 1 U .' , ;'\f ., \~ , -5-C. Charge Exchange Diffraction The analog of - 0 rr p ~ rr n here, is have the usual form .en (~~) = c + dt , where we compute d. = 1.80 .en (0.90 s) + 6.3 + - 0 0 n:rr ~rrrr Near -2 in units of GeV . UCRL-18500 t o we For s ~ 30 GeV 2, d ~12.3~WhiCh is roughly the same as in j,N and NN ; 16 charge exchange. We have presented here a model which incorporates Regge behavior, pole structure, crossing, and analyticity. It seems to have some predic-tive success. On the other hand, besides suffering from the theoretical difficulties outlined above, it predicts two seemingly nonexistent experi-i mental effects: the pI at the f mass, and f3 (t p 2 15 -0.54 GeV ) = 0.-Whether these features are inescapable remains to be seen. The authors wish to express their thanks to Professor Geoffrey Chew for many helpful suggestions during the course of this work. -6- UCRL-18500 TABLE 1 Partial widths for llJ( decay predicted by the present model. The widths are normalized to r = 112 MeV, and our p trajectory is l prrrr ., a (t) = 0.48 +0.90 t. p 8 +' 7 6+ 5 4+ ;3 2+ 1 112 .+ 0 565 764 34 38 30 96 82 27 112 14 36 -13 77 12 1300 1670 1980 (;' 5 6 ~. 13 10 12 14 11 10 10 27 20 21 1!~ 16 19 10 13 39 16 25 12 10 20 8 14 39 12 25 11 2240 2480 2690 2890' , I \) . /) " -7~ UCRL-18500 FOOTNOTES' AND REFERENCES * This work was supported in part by the U.S. Atomic Energy Commission. 1. G. Veneziano, Nuovo Cimento 21, 190 (1968). 2~ The ~~ problem in the context of finite energy sum rules has been studied by C. Schmid, Phys. Rev. Letters 20, 628 (1968). For discussions of ~,he general problem of infinite families of Regge trajectories see N. N. Khuri, Phys. Rev. Letters 18, 1094 (1967); M. 'Kugler ibid, 21, 570 (1968). 3. E. P. Tryon, Phys. Rev. Letters 20, 769 (1968); E. Malamud and P. Schlein, Phys. Rev. Letters 19,1056 (1967); W. D. Walker et a1., Phys. Rev. Letters 18, 630 (1967). Tryon's phase shifts seem to agree with Walker's. The energy dependence of his seems to be rather different from that of Schlein and Malamud. Walker, and Schlein and Malamud find solutions for 5gC s) which resonate near 750 MeV. These would be consistent with our broad E(O+) resonance. We wish to thank Dr. Tryon for a helpful discussion. 4. This slope is on the low end of the commonly accepted range of values -2 0·9 - 1.1 GeV . For example, F. Arbab and C. Chiu, Phys. Rev. 147, 1045 (1966) give ex (t)= 0.56(±O.01) + 1.08(±O.03)t from an analysis p of nN charge exchange scattering. B. Svensson, Proc. CERN School, Rattvik, (1967), p.153, gives -2 b = 0.91 GeV . 5· S. Mandelstam, Phys. Rev. 166, 1539 (1968). See also H. Cheng and D. Sharp, Ann. Phys.(N.Y.) 22, 481 (1963); Phys. Rev. 132, 1854 (1963) . 6~; This procedure would be in accord with the general philosophy of H. Harari, Phys. Rev. Letters 20, 1395 (1968). We thank Geoffrey Chew for em~hasizing this point to us. 7. We have chosen what seems to us to be the simplest possibility consis-tent with our requirements, considering the rur system in isolation. 8. -8- UCRL-18500 On considering the internal resonances as external states and studying, e.g., n:p scattering, one may very well discover new terms must be added. For an alternate choice of the form of the amplitude seeM. Virasoro, University of Wisconsin preprint (1968), unpublished. One of us (J.Y.), wishes to thank Dr. Virasoro for an interesting discussion. The + -n: n: I = 0 object at 1.05 BeV recently reported by D. H. Miller et al., Purdue preprint (1968) unpublished, would 'not fi t';into this scheme. Though the p' at ~1300 MeV is the most difficult of these objects experimentally, the 1670 MeV 3 object also presents diffi-culties, as the g(16so) is somewhat lower in mass and is supposed to decay predominantly into 2:n:. The general situation in this mass region is deeply confused. (See N. Barash-Schmidt et'al., UCRL-8030 Rev., August, 1968, unpublished.) More investigation into the dip ion spectrum in the region above m n:n: 1 GeV would be highly desirable. The harmonic oscillator quark model of G. Zweig (Report to Philadelphia Meson Conference, April, 1968, unpublished) does not give the odd daughters. 9. Any fit to high energy data employing a p' trajectory runs into the problem of the existence of a resonant 1 I = 1 state between ~1000 and 1300 MeV, unless one assumes the p' represents an effective cut contribution, or unless one puts an arbitrary zero into t3 ,et). p See, for example, V. Barger and R. Phillips, Phys. Rev Letters 21, 865 (1968). 10. D. Crennel et al., Phys. Rev Letters 18, 323 (1967); R. Eisner et aL, Phys. Rev. 164, 1699 (1967). i ,l , -9- UCRL-18500 11. J. D. Jackson and C. Quigg (private ,communication). Fig. 2(b) of Ref. 10 allows a generous upper limit for (p , / p) of 1/10. The pro-. duct ion cross section for p and p' is estimated using the absorp-tive OPE model.which is known to work well in the energy range for p production. At 6 GeV/c the result is d(p')/cr(p) ~ 3/4. We thank Professor J. D. Jackson for numerous helpful comments and for the estimate above. 12. If we take the point of view of Schmid, Phys. Rev. Letters 20, 689 (1968), a given partial wave will be represented by an inward spiral on the Argand plot with the resonance becoming more and more inelastic as we go up in energy. We are now investigating this behavior in our model, and the result will be contained in a more detailed report. 13. This is related to the Veneziano supplementary condition. (See Ref. 1.) At the .point a(s) + aCt) + a(u) = 4[1~ + 3a = 1, the odd daughter trajectories vanish. The condition 1 32 4 2 A 6 3 '" a> 2 - ~ [1 b + ~([1 b ) = 0.496. r , > 0 can be written as € In this note " second daughter trajectory" means second trajectory below leading trajectory. 14. The sum rule for the I = 1 amplitudes is saturated only if we keep 15· the €(760) contribution, since L is close to the current algebra result of ;;; 0.1/[1' This is in accord with Adler's original remarks inthisconilection. See S. L. Adler, Phys. Rev. 140, B736 (1965), Section IV. Factorization.implies that the zero we have in ~ (t) p at a (t) = 0 p also causes. a dip in the nN charge exchange differential cross sec-tion. This does not seem to appear experimentall. Virasoro (see Ref; 7) attempts to circumvent this problem. We thank Haim Harari for pointing out thl2~importance of this zero.to us in a private ~\ .. communication. . 'J,': -10- UCRL-18500 16. R. K. Logan et al., Phys. Rev. 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A MODEL FOR nn SCATTERING

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