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<ul><li><p>Volume 42B, number 4 PHYSICS LETTERS 25 December 1972 </p><p>DUALITY IN EXOTIC PEAKS% </p><p>K. IGIg The Rockefeller University, New York, New York 10021 I USA </p><p>and </p><p>T. EGUCHI Department of Physics, University of Tokyo, Tokyo, Japan </p><p>Received 6 November 1972 </p><p>A simple s-channel resonance model based on duality is proposed to explain: a) the appearance of distinct exotic peaks, and b) the sharp fall-off of exotic cross sections versus energy, recently observed in K-p and Fp backward scattering. </p><p>Recently, exotic backward peaks have been observed in K-p and pp elastic scattering at 5 GeV/c by the CERN experiments [I]. The prominent features of these experiments are: (i) the existence of distinct backward peaks with qualitatively the same slopes and relative heights as those of the ordinary Regge exchanges, and (ii) the sharp fall-off of exotic cross sections varying as sA8 or sPq [2]. </p><p>The interpretation of these features has usually been attempted either in terms of exotic cuts [3] based on dou- ble Regge exchange mechanism, or by exotic trajectory exchanges such as Z *. Exotic cuts [3], however, cannot explain the s-dependence of the rapid fall-off of exotic cross sections up to 5 GeV/c, while the Z*-exchange model requires an unusual value [4] of so e 0.05 GeV2 in the Regge fit to K-p backward scattering. </p><p>In this note we would like to point out that both the overall decrease of exotic cross sections and the existence of exotic peaks are quite naturally understood from the direct-channel-resonance point of view, based on duality, i.e., by the cooperative work of many direct-channel-resonances. We shall give a simple formula that relates the u- (or t-)channel exotic cross sections to those of the line-reversed s-channel exotic processes at all scattering angles. From this formula we shall show that the occurrence of exotic peaks is a universal phenomenon in two-body ha- dronic collisions and these exotic peaks must have the same slopes and relative heights as those of the allowed or- dinary exchanges. </p><p>In order to present our ideas in an intelligible manner, let us first consider the boson system of Regge trajecto- ries spaced by two units of angular momenta bootstrapping among themselves. This possibility has been advocated by the partial-wave analysis of the Regge exchange amplitude [5] and the low-energy rrrr, Kn phase shift data [6]. </p><p>With odd-daughter trajectories absent in this scheme, only even (or odd) angular momentum states are present in each resonance tower, which give a forward-backward symmetric (or antisymmetric) contribution to the scat- tering amplitudes. Accordingly there naturally arises backward (forward) peaks even in u(t)-channel exotic reac- tions at low energy in accord with experiments [7]. As the energy increases, the total widths of resonances begin to spread and the neighboring resonance towers tend to overlap. Since the neighboring resonance towers have al- ternating signs in the backward (forward) scattering regions in u(t)-channel exotic reactions, they interfere destruc- tively and cause the overall decrease of exotic cross sections with increasing energy [7]. However, the sharp back- ward (forward) peaks made up by the individual resonance towers survive and appear as exotic peaks at high ener- </p><p>gY. In order to give mathematical expressions to the above explained mechanism of exotic amplitudes, we next pro- </p><p>*Work supported in part by the US Atomic Energy Commission under Contract number AT(1 l-11-3232. #Permanent address: Department of Physics, University of Tokyo, Tokyo, Japan. </p><p>449 </p></li><li><p>Volume 42B, number 4 PHYSICS LETTERS 25 December 1972 </p><p>pose a simple resonance model starting from considerations on dual models in the zero-width limit. As we see from the preceding discussions, the most crucial point in the understanding of exotic amplitudes consists in (i) the cor- rect evaluation of the angular dependence of each resonance contribution, and (ii) the resonance-overlapping ef- fects caused by finite hadronic total widths. </p><p>Dual models in their zero-width limit apparently violate the above requirement (ii) which is indispensable for obtaining the overall decrease of exotic cross sections. In the conventional Ima prescription to keep (Y~ or (Y, real and to replace crX with crs with os + iImcu,, the requirement (i) is violated since the t f-7 u crossing property of each resonance contribution is broken by making z complex in the resonance expansion of the dual model, </p><p>(s, t) = r(l-(Y,)r(l--(YJ </p><p>d&(Z), r(l-CX-CKJ = -N,I N-4 1 (1) </p><p>withz = 1 + t/2q2. In order to remedy these defects, we propose to preserve all the elastic widths and angular distributions of indi- </p><p>vidual resonances as they are (in the zero-width limit) and only shift the resonances into the second-sheet of S- plane. That is, our prescription amounts to keeping z real and replacing CY~ with CY, + iImcu, in eq. (1). Thus the on- ly feature of the dual model left in our approach is the determination of r~,l which specifies the relative strengths of the resonances. Evidently the t ** u crossing property and the finite-width effects of individual resonance con- tributions are properly taken into account by our prescription and further, by preserving the elastic widths of reso- nances, the Regge behavior in the forward direction is also assured through FESR. Thus our prescription is ex- pected to give a reasonable description at all anglezf the u(t)-channel exotic cross sections. </p><p>The our new (s, I) term (which we denote as (s, r )) is to be expressed as </p><p>which is obtained technically from eq. (1) by replacing CY~ with (Ye + iImor,, keeping z real. Thus we obtain </p><p>(ST) = sin 7r{ (cr,+iImor,)(( 1 +z)/2) + (a,-+)z t f} </p><p>sin 7T(ost i ImcrJ </p><p>X r{(ol,+iImcu,)((1+z)/2)t(cw0-+)z t+} r{(cY,+iImcu,)((l-z)/2)-((ru-+)z+3} </p><p>r(as+i Imor,) (2) </p><p>where z = ltt/2q2 = -l-u/2q2, Imol, = 0 below threshold and crstatt&U = 1 is used to eliminate odd daughters?. </p><p>It is extremely important to notice here that the (>) term of eq. (2) can be explicitly expanded as </p><p>- Clim(cu,+iIm$ +N)FN,[(q2) N,l </p><p>N-ol,-iImcu, P,(z) </p><p>and the relation lim ((us + iImor, -+N) FN t(q*) = rN I can be easily proved. With some trivial algebra, the above eq. (2) in fact shows the Regge behavior for large values of s and Imol, at </p><p>small t (ItI < 4q2/nIm(r,) </p><p>(~7) - exp (-incu,)r( I-a,)$t ,tt (3) </p><p>iSince the elastic widths of resonances in our prescription are preserved as they are in the narrow-width limit, odd daughters do </p><p>not appear even after the replacement (Y~ -t as + ilmor,. Also no ancestors arise in our amplitude. </p><p>*We assume that Imols does not increase faster than linearly in S. </p><p>450 </p></li><li><p>Volume 42B, number 4 PHYSICS LETTERS 25 December 1972 </p><p>and in the backward region (1~1 < 4q2/7rlmcrS) it also exhibits the exotic behavior, </p><p>(SF) - -2rri exp (-nlmcu,) exp (ino,)r-l ((~~)o,* ,# </p><p>which we introduced previously in ref. [6]. The above eq. (4) gives a nice quantitative expression of our foregoing discussions of exotic amplitudes. It ex- </p><p>hibits (i) the damped oscillation of exotic cross sections caused by the alternating sign and cancellation of neigh- boring resonance towers [7], (ii) the existence of backward exotic peaks with similar slopes and relative heights as those of the allowed ones [ 11, and (iii) it also shows the appearance of the backward fixed-u dips (the so-called Odorico zeros) which have been observed in low-energy K-p + Ken cross sections [8]. Thus our eq. (4) concisely embodies our empirical knowledge of backward exotic amplitudes. </p><p>By taking the absolute square of eq. (2), we have for large values of s and Imcu, ,m </p><p>IKZI - exp {-nlmol,( 1 -z)} I(u, t)12 (except for lmor,( 1 +z) - 0). (5) This is quite an interesting formula relating the u-chancel exotic cross section to its line-reversed s-channel cross section at all angles in a very simple fashion. All the effects of overlap and destructive interference of s-channel resonances are simply factored out into exp { -nIma,( 1 -z)} and aside from this the two cross sections are identi- cal. Especially the (~3) term inherits the allowed backward peaks in the (u, t) term and exhibits them as for- bidden peaks. Therefore the occurrence of these forbidden peaks must be universal phenomenon in two-body hadron collisions and these peaks must have the similar slopes and relative heights as the allowed ones. </p><p>Up to this point we have been working in bosonic systems and used the condition os + (or + oyu = 1 to eliminate odd daughtersf. However, eqs. (2)-(5) show that the elimination of odd daughters is not at all an essential requi- site in our understanding of exotics, though it makes physical interpretations much easier. </p><p>Next, in order to make an experimental check on our understanding of exotics, let us use eq. (5) to determine Imol, at high energy and see if it lies on a reasonable extrapolation from the low-energy data. One way to do this is to compare the s-channel exotic cross section and their line-reversed u-channel exotic cross section, such as </p><p>(daldu)(K p), \ </p><p>(K-P)/$KP) I </p><p>- ln I(Z12it(u, r)i2 - -nlmcu,( 1 -z) </p><p>The energy dependence at z - - 1 can also be derived from eq. (2) as </p><p>(6a) </p><p>(6b) </p><p>In making a realistic fit to experimental data spin and other complications ariseff. However, we can show that the crucial exponential dependence on z always appears in our damping factor exp I--nImor,( I -z)} , so far as the YG contribution is assumed to be dominant over the Y; (this is the case for the solution I in ref.[9]). Therefore we have used eqs. (6a, b) without any change in the data analysis. Determination of Imo, from 5 GeV/c data is shown in fig. 1. Logarithms of the ratio of (do/du)(K-p) and (da/du)(K+p) is consistent with a linear function of z for u 2 -4(GeV)2, as predicted by our formula eq. (6a). Moreover, it is worth-while to stress that Imcu, obtained from 5 GeV/c data lies approximately on a linear extrapolation of Ima, from the low-energy Yi, YT resonances </p><p>TFootnote see foreeoine oaee. mwe use ~~{~,+iIm~s+~r~i~Imo1s)/4~~)f}~(or,+iIm~+~+i((Im~s)/4~~)~}/~{~s+iIm~s}l = Jl+((Imor,)lolt)Z]r(ols+Olt))r(OIS+OIU)/r(OIs)l </p><p>Jl+((Im+/Q I(U, r)l and neglect ((Imorv)/arJ2 since they are of the order of 10m2 in reality. </p><p>fSee footnote & page 247 of ref. [6]. #For example thecondition a$+~~+% = 1 is well satisfied with Yr trajectory but not with Yg (ZY$s)+a,,(t)+ZY:(u) - 0.4, </p><p>where h= a--&). This causes some uncertainty in our data analysis. </p><p>451 </p></li><li><p>Volume 42B, number 4 </p><p>In{$(K-p)/~(~+~j} </p><p>0 -0.5 -1.0 ; I I 1 </p><p>IO- </p><p>0.0 - </p><p>-l.O- </p><p>-2.o- </p><p>-3.o- </p><p>-4.0 - </p><p>- </p><p>I I I I 1 1 0.0 I I I 4.0 5.0 6.0 7.0 8.0 -t (GeVj2 5.0 7.5 10.0 t Geh </p><p>7 </p><p>PHYSICS LETTERS </p><p>Ima, </p><p>25 December 1972 </p><p>0.8 </p><p>0.6 </p><p>0.4 </p><p>Fig. 1. Plot of the logarithm of the ratio of K-p and K+p large angle cross sections at 5 GeV/c. We excluded the data for If1 < 3(GeV)* on account of Pomeron </p><p>Fig. 2. Plot of Imol, determined at 5 GeV/c and Img of Yt, Yy resonances. Solid line Imcus = 0.09s - 0.16 is a fit-by-eye and the dash-dotted line Imas = 0.65 In s- 0.76 is its approximation in the </p><p>contributions. It is approximately linear in z as pre- 2 to 5 GeV/c region. Resonance data (O for Yzs, X for YTs) are dieted by eq. (6a). taken from ref. ( 10). </p><p>taken from ref. [lo] (see fig. 2). Fit-by-eye approximately gives lm@, = 0.09s~0.16 which when approximated by Imol, - 0.65 In s-O.76 in the 2 to 5 GeV/c range gives </p><p>$K-p)/$K+p)l,_-, &s-4,1 m agreement with the experimental value s-4A0.4 [ 11. </p><p>Though these figures cannot be taken very seriously owing to both theoretical and experimental uncertainties, nice consistency of data analysis seems to support our understanding of exotic mechanisms. On the contrary, the conventional Imac prescription to keep CY~ real and to replace as with cu,+iImcu, predicts the absolute values of both (s, t) and (u, t) terms to be equal at all values of z. Therefore, the right-hand side of eq. (6a) is predicted to be zero, which is not consistent with experiments (see fig. 1). </p><p>To summarize, we have shown that a simple s-channel resonance model proposed here successfully explains the essential features of exotic amplitudes, i.e., the overall decrease of cross sections and the appearance of exot- ic peaks. Experimental data now available support our approach to exotic amplitudes. </p><p>One of the authors (K.I.) wishes to thank Professor N.N. Khuri, Dr. Odorico and Dr. V. Rittenberg for useful comments on the manuscript. He would also like to thank Professor A. Pais for the kind hospitality at The Rockefeller University, where this work was completed. </p><p>[l] V. Chabaud et al., Phys. Lett. 38B (1972) 445,449. [2] R. Barloutaud, Amsterdam Intern. Conf. on Elementary particles, Amsterdam, The Netherlands, June 1971. [3] C. Michael, Phys. Lett. 29B (1969) 230; </p><p>C. Quigg, Nucl. Phys. B34 (1971) 77. (41 A.S. Carroll et al., Phys. Rev. Lett. 23 (1969) 887. [5] T. Eguchi and K. Igi, Phys. Rev. Lett. 27 (1971) 1319. [6] T. Eguchi and K. Igi, Phys. Lett. 40B (1972) 245. [7] C. Bricman, E. Pagiola and C. Schmid, Nucl. Phys. B33 (1971) 135. [8] R. Odorico, Nucl. Phys. B37 (1972) 509. [9] E.L. Berger and G.C. Fox, Phys. Rev. 188 (1969) 2120. </p><p>[IO] Particle Data Group, Phys. Lett. 39B (1972) 1. </p><p>452 </p></li></ul>

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