P H Y S I C A L R E V I E W D V O L U M E 1 5 , N U M B E R 3 1 F E B R U A R Y 1 9 7 7

Regge spectroscopy of charmed baryons*

J. Finkelstein Department of Physics, Columbia University, New York, New York 10027

S. F. Tuan Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, Hawaii 96822

(Received 4 October 1976)

The implication of assigning the newly discovered candidates for charmed baryons to Regge trajectories is discussed. The feasibility of using the Ademollo-Veneziano-Weinberg relation and exchange-degeneracy notion to determine the trajectory parameters is assessed. Finally the universal-slope ansatz is employed to obtain the lowest-Regge-recurrence baryon masses. Their possible decay patterns and production are briefly discussed with an eye to experimental search.

The discovery of new hea ly baryons. with pos- s ible charm quantum numbers , in neutrino' and photon-inducedz react ions has recently been an- nounced. In this note we will explore the assign- ment of those baryons to Regge t ra jec tor ies and the consequent predictions of o ther baryons lying higher on those s a m e trajector ies . The sp i r i t of our analysis i s s i m i l a r to that of an e a r l i e r work3 on Regge spectroscopy of charmed mesons. We will , however, need to draw upon some detai ls of the SU(4) quark model calculations of De Rdjula, Georgi , and Glashow,' a s input ground s ta tes ; t h e s e predictions appear in any c a s e to fit rrlell the cur ren t data t rend on heavy baryons."'

There ex i s t s tantalizing support f r o m experiments fo r assigning the seen baryon state' with total recoiling hadron m a s s (i2°n'n+n'n-) 2426 i 12 MeV a s the (I= 1 , J ~ = $ ' ) Cc s ta te ( 6 ' 6 ~ ) " , and the seen new antibaryon s tatez which decays into (An-n- a' ) with m a s s 2260 i 10 MeV a s the anti- s ta te to the (I= 0, JP= $+) 11, state (6'@3t)* prc - dicted4 to be 160 MeV below C, (and with s t rong decay transition CEt -A,'+ a'). 'The recent photon experimentz a l so repor t s evidence of a higher- m a s s s ta te (-2.5 GeV:cZ) which decays into the 2 . 2 6 - ~ e v / c ' s t a te via the chain Ka'n'n- a- (2.5 Gev:c2) - As-n-n'(2.26 G ~ v / c ' ) + n'. This anti- baryon s ta te is again i n remarkable agreement with the predicted4 s ta te Cd ( I = 1. J'= $ + ) formed f rom (6'3237) and expected a t 2.48 GeV c Z . Evi- dently, if these assignments were to hold up, the photon experiment2 will need to establish the CC(Z.4Z6) state (@'31X)0, nnd the (@'@@)" mem- bers of 5,(2.426) and C,*(2.500). The la t t e r two could be searched f o r via s t rong decays:

2.5 GeV cz A a- a- n- n '

2.426 G e v / c 2

I\ li' n' n' n- 2.5 GeV c2

2.426 GeV c 2

-Aa+a+ nm(2.26 GeV c2) + n' .

We need of course to r e a s s u r e ourselves that the Ac(2.260) i s indeed charmed. Hints along this line can be obtained i f s ignals in invariant-mass dis- tributions a t 2.26 GeV/c2 continue to appear in (A0a'a'n-) and (xon-a-a') hzlt not in (AOn-a-n') and (KOa'a'n-) in the experiment of Ref. 2.

F o r definiteness, we shal l assume henceforth that A,(2.260), 5,(2.426), and Cd(2.5) a r e indeed the low-lying ground s ta tes of the SU(4)-charm classification. In Ref. 3 , the identification of the meson ground s t a t e s , together with the re- lationship between t ra jec tor ies proposed by Ademollo, Veneziano, and Weinberg was shown to determine the p a r a m e t e r s of the charmed meson Regge t rajector ies .

Suppose we t ry to do the s a m e f o r the baryon t ra jec tor ies : The AWV rule would te l l us that

where a,; i s a JP= 5 + trajectory and )?I%, ,c i s the m a s s of the appropriate JP= $+ part ic le . This rule i s empir ical ly successful f o r a A ( , n N z ) = 1 and for a , T ( ~ ~ ~ , Z ) 2 ay*()izAZ), though a s pointed out by A m , ' the 5- ,tr"lT,rajectory does not seem to com- p a r e well with the A t ra jectory (thus rais ing some questions about the validity of exchange degeneracy for baryons). Our choice of assignments l eads to s lopes a 1 = 0 . 4 4 G e V 2 f o r relation (2a) and a' = 1.35 G e Y 2 for relation (2b). Neither appears sensible in t e r m s of the notion of a universal slope f o r hadrons (to be discussed fur ther below) of a '= 0.93 G e V Z . Note that the AVW relations f o r the usual baryons [a,* ( I ~ I ~ ~ ) = aA(11lN2)

= cu + ( r i l A 2 =i?zZ2) = 11 require the (,?I y* - i)z,)/ y1

(ii1,: - )>I,) to be unity; however. fob the assign- ments of De Rdjula of a1 . 4 f o r the corresponding charm s ta tes this constraint i s very poorly sat isf ied, and s o (2a) and (2b) cannot be simultane- ously satisfied. F r o m a theoret ical viewpoint t e r r ib le pathologies a r e involved in putting baryons into a dual theory. Hence we shall not pursue here fur ther the applicationof AVWto charmed baryons.

In Ref. 3 , i t was found that the t ra jec tor ies of the p and of the D* have essentially the s a m e slope. We a r e thus led to consider the ansatz that all l~nrlrorzs respec t Re*-e t m j e c l o r i e s o-f icnz~~ersnl slope. To wit. we require

despite the known much l a r g e r phenomenological m a s s for the charmed quark' ( i i z ~ = r i z ~ = 336 MeV, iu,= 540 MeV, ic'llile inp, = 1660 MeV). The belief in a universal slope i s rooted in the concept that such a slope i s the basic sca le fac tor of s t rong interact ions. In other words , charmed hadrons (D. D*: A,. Cc, c:) a r e a s much strongly interacting par t i c les a s their s t range counterparts (K,I<*. 11, C. Y:). F r o m the viewpoint of a model, an osci l la tor potential fo r quarks (qq o r qif) in nonrelativistic quantum mechanics gives r i s e to l inear level spacings which a r e not dependent on the quark m a s s e s but on the s t rength of the po- tential (e.g., the k of i k r 2 ) .

There a r e exceptions to (3), fo r instance (as- sunling exchange degeneracy f o r the mesons) . 0!$=0.33 G e V 2 i s obtained by placing the r/)(3100) and the JP= 2+ ~ ( 3 5 5 0 ) on the s a m e l inear t ra - jectory? The Regge slope of d(1019) [determined from exchange degeneracy withf" (1514)l yields cuk = 0.8 GeV-?-somewhat s m a l l e r than the uni- v e r s a l slope (3). The s m a l l e r sIopes observed for par t i c les (6, i , X , etc.) may be related dynam- ically to the smal l slope observed f o r the Ponleron and daughters (cub= 0.4 G~v") a s speculated by severa l author^.^

F o r our purposes here , we will adopt the fol- lowing ivrr~eri~onic rule url reduced q u a ~ k i ~ l n s s e s (the dynamic origin of this rule i s a s myster ious now a s the Zweig rule mus t have seemed in 1964). This rule s ta tes that the magnitude of the slope parameter i s determined by the inverse of the two-body reduced m a s s f o r the quark content of the par t i c le under consideration. F o r the (qq) sys tem, the Regge s lopes of p . D* a r e determined (in the l imit of l a rge charm quark m a s s itzw) by light quarks nz, and rihlin the reduced-mass system. F o r the ( J IG, a', X ) c l a s s and to a l e s s extent fo r the 6 , the relevant reduced m a s s e s a r e of o r d e r nz, and in,. Hence, the appropriate slope param- e t e r s a r e correspondingly reduced. The Pomeron

1s belleved to have a (@'Gt) plece, hence, ~ t s slope 1s slnl l lar to that fo r the pslons.' The baryons a r e t reated a s a superposltlon of reduced m a s s e s of two-body p a r s ( ~ n analogy to the Faddeev-type approxlmatlon). TIZZLS l te expect the a Lope p a m ~ i ~ e f e r a o t the s z t z ~ l j cl~ariized (5 - 0, - 1, -2) barjotza to follow the t rend for normal baryons and mesons a s stipulated In Eq. (3). ( F o r s l m p l l c ~ t y we now neglect the differences between ordlnary and s trange quarks.) However, the Regge recur rences of doubly charmed (6 torfzorr f o r t r lply charmed) baryons a r e ex- pected to follow a different (presumably smal le r ) slope.

Uslng Eq. (3), the Regge recur rences of 12,(2.260), Yc(2.426), and SZ(2.500) occur a t

iii, (3') = 2.694 GeV . (4a) C

i i l ( 2 ' ) = 2.835 GeV, C c (4b)

The ground- s ta te assignments of De Rlijula et nl.' f o r singly charmed baryons a r e f o r S= - 1. - 2

JP Mass (GeV)

(3) ( @ ' ~ 3 z ) ~ , (@'x@)+ + 2.470

(6) (S 'XX)~. (@"x@)+ + 2.560

( s t x n ) " ((stxs)+ + 2.610

(PtXX)" 5' 2.680

( 6 ' ~ h ) ~ st 2.720

while the Regge-recurrence s ta tes [using Eq. (3)) a r e given a s follows:

JP Mass (GeV)

Exchange degeneracy for charmed baryons would obviously complicate the spectrum considerably. F o r instance, Eq. (3) p red ic t s a JP= $- par tner to A,(2.260) a t 2.510 GeV-essentially m a s s degener- a t e with the J' = $' ~ct (2 .5) . Since exchange degen- e racy generally r e q u i r e s mjk = m Z c , which is poorly statisfied f o r charmed baryons, we shal l not analyze fur ther the possible exchange-degen- e r a t e baryon spectrum f o r charm.

904 J . F I N K F , L S T E I N 4 N D S . F . T U N

The production of the family of charmed-baryon- of i t s weak decays a r e antibaryon p a i r s can be searched for in e'-em annihilation a t s'I2 2 4.5 GeV. Because of the comparable m a s s e s predicted for charmed mesons and baryons, it i s not ruled out that the baryons may be produced with abundance comparable to that of the mesons , though the yield may a l so be affected by the dynamics of the three-quark sys tem for baryons ( a s opposed to the quark-antiquark configuration for mesons) . A s s t r e s s e d recently by De RGjula e t a l . , if A, (2.260) i s detected, mea- surement 3 f the reco i l -mass spectrum should show s t ruc ture due to kinematical reflections of the heavier s ta tes . Anf (o r Antn-nt of Ref. 2) can re- veal the whole fanlily of singly charrlled nonstrange baryons. The th ree isotopic doublets of charm 1, strangeness -1 baryons and their Regge recur- rences can a l so be identified analogously. Decay modes of the lightest s ta te at 2.470 GeV should be into two-body channels, e .g . , Z- n', C'K-, C ' P , AOps o r multibody channels formed from the two- body ones with the addition of p a i r s of (n+am) , e.g., Z - i - r + ( 2 ~ ) ~ , EmTi'(4n)0, etc. At s"'-5.0- 6.2 GeV, significant yield due to quasi-two-body pro- duction of charmed strange baryons i s possible . and again once a peak i s found, a study of recoi l m a s s e s can reveal the o ther s tates . The S = - 2, spin-+, charmed baryon i s a lso metastable; some

( a f x x ; J =

nq7r+, a - ~ + ( 2 7 1 ) ~ , n-n+(4n)0,. . . \ -

"OK0, Z0P(2n)0, E 0 p ( 4 n ) 0 . . . . (7) + i S Z , - CER(zn)', CiTi7(4;7)0, . . . . Higher-spin s ta tes with J ~ = $+ and Regge- recur - rence s ta tes with JP= %+, :+ would then decay to the lowest JP= 8' (Pfxx) configuration via photon emission. The e'-e- annihilation c r o s s section should be r ich in s t ruc ture owing to the opening of s o many charm baryon-antibaryon threshold^.^

In photon- (and neutrino-) induced react ions, the s e a r c h f o r the Regge-recurrence s ta tes (4b) and (4c) can employ the cascade strong-decay method Z , * ( i + ) - A,(++) + n , Zc(:) - A,($') + 71 (and their antiparticle analogs). The A,(%+) of Eq. (4a) can decay strongly into c,(;') + T, e.g. , Ad(2.694) - Zz(2.426) + n- o r AE(2.694) - Z: (2.426) + n+ .

Finally, the charmed-baryon-antibaryon p a i r can be produced in hadron- hadron collisions. An es t imate of the production c r o s s section in the central region can be made using Hagedorn- Frautschi thermodynamic^.^ F o r pp collision leading to production of a pa i r of mass ive par t i c les MBc and MLc with I spin 1,11, and ordinary spin J, J', the c r o s s section is given by

The tempera ture T i s of o r d e r 200 MeV in o r d e r given charge state. to be consistent with production charac te r i s t i cs f o r (J/$, h f ) a t El,,> 300 GeV. Application of Eq. (8) to the A, & p a i r of m a s s 2.26 GeV each and to The authors have benefited from conversations the highest-mass p a i r delineated in Eq. (6), with many colleagues (and especially with Tom ( ~ ' X X ) ~ and (gfZ)O with JP= a t 3.090 GeV, leads O'Halloran and Steve Pinsky) at the Aspen Center optimistically to pair-production c r o s s sect ions f o r Physics , where this work was done. They (assuming a l l p a i r productions a r e Zweig allowed) would like to thank the Cente r fo r i t s hospitality in the range 1.1 x cm2 to 8.4 x cm2, fo r a and i t s conducive atmosphere fo r research .

*This work i s supported in p a r t by the U . S. Energy 'P. G . 0. Freund and Y . Kambu, Phys . Rev. Le t t . 34, Research and Development Administrat ion. 1645 (1975); G . F . Chew and C . Rosenzweig, Phys . Rev.

'E. G . Cazzoli e t a1 ., Phys . Rev. Le t t . 2, 1125 (1975). D 12, 3907 (1975); Nucl. Phys. M, 290 (1976). 2 ~ . Knapp e t al ., Phys. Rev. Lett . 5, 882 (1976). 7 ~ e r e we need to modify the reduced-mass-slope rule 3 ~ . Finkelstein and S . S . Pinsky, Phys. Rev. D 3, 360 to be that s lope determined by the la rges t (reduced)

-

(1977). quark m a s s p a i r component of the Pomeron . k A. De Rdjula, H . Georgi , and S. L . Glashow, Phys . Rev. 'A. De Rdjula, H . Georgi , and S . L . Glashow, Phys. Rev.

D 12, 147 (1975). Lett . 37, 398 (1976). 'M. Ademollo, G. Veneziano, and S . Weinberg, Phys . $s. C. Frau tsch i , S. Pakvasa, and S. F . Tuan, Caltech

Rev. Lett . 2, 83 (1969). Report No. CALT 68-562, 1976 (unpublished).