Regge spectroscopy of charmed mesons

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P H Y S I C { I , R C i ' I E W D Regge spectroscopy of charmed mesons* J. Finkelstein Department of Physics. Columbia University. New York, New York 10027 Stephen S. Pinsky Department of Physlcs, Ohlo State Unrverslty, Columbus, Ohlo 43210 (Recrivcd 1 6 411q11rl 1976 I ( viwd tnot~iistrtpr reccil~ed 179 Y o ~ ~ ( r t ~ h e r 19761 The implication of assigning the newly discovered particles to Regge trajectories is discussed. The Ademollo- Veneziano-Weinberg relation is used to determine the trajectory parameters and exchange degeneracy is used to predict a tensor particle at 2264 MeV and an axial-vector particle at 2140 MeV. The discovery of new heavy mesons produced in e'e- annihilation has recently been a n n ~ u n c e d . " ~ In this note we will explore the assignment of those mesons to Regge t ra jec tor ies and the con- sequent predictions of other mesons lying higher on those s a m e trajector ies . Our analysis will be independent of any detailed quark model of the new mesons, but will turn out to be reasonably con- s i s ten t with the quark-model calculation of Ref. 3, We begin by assuming that the part ic le detected in the ICa, Kair, and h'iian modes at 1870 MeV (we take the average m a s s of the charged and neu- t r a l s ta tes ) , which i s denoted by D, i s a pseudo- s c a l a r . The peak in the recoi l spectrum to the D observed at 2010 MeV we will assume represen ts a vector part ic le denoted by D*. These two a s - signments together with the relationship between the vector and pseudoscalar t ra jec tor ies proposed by Ademollo, Veneziano, and Weinberg4 (AVW) enable us to determine the p a r a m e t e r s of the Regge t rajector ies . The version of the rule proposed by AVW that i s relevant t o us can be stated a s where a y i s a vector t rajectory and lizp i s the m a s s of the appropriate pseudoscalar par t ic le . This rule i s empir ical ly extremely successful, a s can be seen in the two e x a ~ n p l e s a , (rnr2) = $ and cy,*(r~z,~) = $ . In our c a s e the rule becomes Using ~ z , = 1870 MeV, the vector assignment of the D* [~r ,* ( i i i ,*~)=l ] and a l inear t rajectory, we determine the slope of D * t rajectory to be 0.92 GeV-'. We note that this value i s quite consistent with the value of the slope of mos t well-established Regge t ra jec tor ies . F o r example, the p t ra jector- ies in a recent charge-exchange experiment5 were found to have a slope of 0.93 GeVm2, though differ- en t determinations vary s o n ~ e w h a t . ~ This i s con- s iderably different than the value of 0.33 GeVW2 that one obtains by placing the +(3100) and the JP = 2 + ~ ( 3 5 5 0 ) on the same l inear t rajectory. If we a r e willing t o believe a l inear extrapolation of the D* trajectory to vanishing invariant m a s s we obtain an intercept of -2.72; thus the t rajectory takes the fo rm The next highest particle on the D* trajectory i s , if we assume weak exchange degeneracy, a tensor part ic le which we denote by D**. Using the t r a - jectory of the D * given in Eq. (3) we find I>/,** =2264 MeV. This value i s slightly below the value of 2330 MeV predicted in Ref. 3. By comparison with decays of known tensor niesons we would e s t i - mate the part ia l width of D** - Dri at about 1 MeV and that of D** - D*T at about 0.3 MeV. The full width would there fore be in the 2-3 MeV range s ince there i s relatively litt le phase space for multipion final s ta tes . These es t imates a r e sensi- tive to the assumed m a s s of the D**; fo r example, pH,** =2.33 GeV implies a part ia l width of D** - ~n of 2.4 MeV. Even though exchange-degenerate Regge analy- s i s is l e s s rel iable fo r unnatural- than for natural- spin-parity t ra jec tor ies , i t i s amusing to consider that if the D t ra jec tory were paral le l to the D* t ra jec tory given in Eq. (3) it would pass through unity at a m a s s of 2141 MeV and give r i s e to an axial-vector meson D: at that m a s s . A peak at just about this m a s s i s in fact seen in the missing- m a s s spec t rum reported in Ref. 1; however, this peak h a s an alternative explanation3 a s a reflection f rom D;B$ production. While perhaps both effects a r e present , the D ~ D has the advantage over the other mode of being an s-wave decay. It can be observed that two-body thresholds s e e m to be closely associated with part ic les o r a t l eas t with prominant s t ruc ture in the e'e- annihilation c r o s s section. Compare the DD threshold a t 3.74 GeV with the +' a t a m a s s 3.68 GeV, the DD; threshold a t 3.88 GeV with the possible s t ruc ture R E G G E S P E C T R O S C O P ' L O F C H A R V E D M E S O N S 36 1 a t 3.95 GeV, and the D;D: threshold a t 4.02 GeV with the s t ruc ture at 4.03 GeV. There is a l so the analogous case of the KR threshold a t 0.99 GeV t o b e compared with the ~5 at a m a s s of 1.019 GeV. With our speculation on the m a s s e s of the D** and Di we note that Ule D**D and D * D ~ thresholds a r e a t 4.13 GeV and 4.15 GeV, respectively, c lose t o the s t ruc ture in the e t e - c r o s s section at 4.1 GeV, and the D**D: threshold i s a t 4.4 GeV, in agreement with the s t ruc ture a t 4.4 GeV. There a l so appears to be a n interesting paral le l between the A, and the Dz since both a r e closely associated with kinematic enhancements. The A, i s s u s - *Work supported in part by the U. S. Energy Research and Development Administration. ' G . Goldhaber el a1 ., Phys. Rev. Lett . 31, 2 5 5 (1976). '1. Peruzzi et a l . , Phys. Rev. Lett . 37, 569 (1976). 3 ~ . De Rlijula, H. Georgi, and S. L . Glashow, Phys. Rev. Lett. 37, 398 (1976); 37, 783 (1976). 'M. Ademollo, G. Veneziano, and S. Weinberg, Phys. pected to be associated with a Deck effect in the pn channel. Similarly, we es t imate the D: t o have about the s a m e m a s s a s a kinematic reflection of D * T . ~ It may in fact be that these a r e dual de- scr ipt ions. Note added in proof. The r a t e f o r D** - Dn w a s calculated f r o m the ra te f o r Kt* - K n assuming that the r a t e v a r i e s a s ~ " ( m , t t ) ~ . The authors have benefited f r o m conversation with G. Goldhaber, and would like t o acknowledge the hospitality of the Aspen Center f o r Physics , where this work was done. Rev. Lett. 22, 83 (1969). 5 ~ . V. Barnes et a l . , Phys. Rev. Lett. 37, 76 (1976). 6V. Barger, in Proceedings o f the XI711 International Conierence on High Energy Phys ics , London, 1974, edited by J . R . Smith (Rutherford Laboratory, Chilton, Didcot, Berkshire, England, 1974), p. 1-210.