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PHYSIC {I, RCi'IEW 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 16 411q11rl 1976 I( viwd tnot~iistrtpr reccil~ed 179 Yo~~(rt~her 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 ann~unced."~ In this note we will explore the assignment of those mesons to Regge trajectories and the con- sequent predictions of other mesons lying higher on those same trajectories. Our analysis will be independent of any detailed quark model of the new mesons, but will turn out to be reasonably con- sistent with the quark-model calculation of Ref. 3, We begin by assuming that the particle detected in the ICa, Kair, and h'iian modes at 1870 MeV (we take the average mass of the charged and neu- tral states), which is denoted by D, is a pseudo- scalar. The peak in the recoil spectrum to the D observed at 2010 MeV we will assume represents a vector particle denoted by D*. These two as- signments together with the relationship between the vector and pseudoscalar trajectories proposed by Ademollo, Veneziano, and Weinberg4 (AVW) enable us to determine the parameters of the Regge trajectories. The version of the rule proposed by AVW that is relevant to us can be stated as where ay is a vector trajectory and lizp is the mass of the appropriate pseudoscalar particle. This rule is empirically extremely successful, as can be seen in the two exa~nples a, (rnr2) =$ and cy,*(r~z,~) =$. In our case the rule becomes Using ~z, = 1870 MeV, the vector assignment of the D* [~r,*(iii,*~)=l] and a linear trajectory, we determine the slope of D* trajectory to be 0.92 GeV-'. We note that this value is quite consistent with the value of the slope of most well-established Regge trajectories. For example, the p trajector- ies in a recent charge-exchange experiment5 were found to have a slope of 0.93 GeVm2, though differ- ent determinations vary son~ewhat.~ This is con- siderably different than the value of 0.33 GeVW2 that one obtains by placing the +(3100) and the JP =2+ ~(3550) on the same linear trajectory. If we are willing to believe a linear extrapolation of the D* trajectory to vanishing invariant mass we obtain an intercept of -2.72; thus the trajectory takes the form The next highest particle on the D* trajectory is, if we assume weak exchange degeneracy, a tensor particle which we denote by D**. Using the tra- jectory of the D* given in Eq. (3) we find I>/,** =2264 MeV. This value is slightly below the value of 2330 MeV predicted in Ref. 3. By comparison with decays of known tensor niesons we would esti- mate the partial width of D** - Dri at about 1 MeV and that of D** - D*T at about 0.3 MeV. The full width would therefore be in the 2-3 MeV range since there is relatively little phase space for multipion final states. These estimates are sensi- tive to the assumed mass of the D**; for example, pH,** =2.33 GeV implies a partial width of D** - ~n of 2.4 MeV. Even though exchange-degenerate Regge analy- sis is less reliable for unnatural- than for natural- spin-parity trajectories, it is amusing to consider that if the D trajectory were parallel to the D* trajectory given in Eq. (3) it would pass through unity at a mass of 2141 MeV and give rise to an axial-vector meson D: at that mass. A peak at just about this mass is in fact seen in the missing- mass spectrum reported in Ref. 1; however, this peak has an alternative explanation3 as a reflection from D;B$ production. While perhaps both effects are 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 seem to be closely associated with particles or at least with prominant structure in the e'e- annihilation cross section. Compare the DD threshold at 3.74 GeV with the +' at a mass 3.68 GeV, the DD; threshold at 3.88 GeV with the possible structure

Regge spectroscopy of charmed mesons

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Page 1: Regge spectroscopy of charmed mesons

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

Page 2: Regge spectroscopy of charmed mesons

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.