Volume 179, numbcr 3 PHYSICS LETTERS B 23 October 1986

m

T H E OZI R U L E IN C H A R M O N I U M DECAYS ABOVE DD T H R E S H O L D

Harry J L I P K I N 1

Htgh Energy Ph~st~s Dwtston, Argonne National Laborator), Argonne IL 60439, USA

Received 30 July 1986

A drastic change m OZI-wolatmg transitions ~s predicted for charmonlum as one crosses the DD threshold A new source of OZI vmlat lon ts introduced by the opening o f the OZl-allowed DD channel which enables the forbidden process to occur m two OZI-allowed steps with the mtermedmte DD state on shell A smaple est tmate using umtar l ty shows that the sum of the branching ra tms for the OZl-forbldden final states can be the same order o f magmtude as the branching ra tm for the al- lowed DD state It is therefore dangerous to assume that c h a r m o m u m decays jus t above DD threshold are 100% DD, and smadarly for the b o t t o m o m u m decays above the BB threshold

There is still no reliable prescription for calculating the strengths of OZI-forbldden transitions [1 ] because forbidden transitions can go via two-step processes in which each individual transition is OZI allowed [2 -4 ] While stmple perturbatlve QCD calculations seem to be vahd for transitions which go via the annlhtlatlon of a quark-antiquark pair directly into an interme&- ate state containing only gluons, there has been no serious attempt to apply QCD directly to processes in- volving a hadronic Intermediate state, hke q~ ~ KI( -+ p-zr or ~ ~ DD ~ p- r r In these two-step transitions, the quark-antlquark pair in the initial state is first separated into two hadrons by a normal OZI-allowed decay and the pair is annihilated in the second stage by a conventional allowed two-body scattering pro- cess described by "duality diagrams" [5]

One example of the confusion caused by these higher order transitions was the failure to predict the width of the J /~ by extrapolating the OZI-violatlng ~b ~ p-~r amphtude This has since been explained [1 ] as due to the presence of the open OZI-allowed KI( channel in the ~ decay, while there is no open OZI-al- lowed channel in J /$ decay [1 ] In this picture the ~b decay goes via the intermediate KI( state, which is on

~' Work supported by the US Department of Energy Dlws~on of High Energy Physics, Contract W-31-109-ENG-38

i 1985-86 Argonne Fellow, on leave from Welzmann lnstttute of Science, 76 100 Rehovot, Israel

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shell, while the J / ~ decay must go via three gluons The contribution of the off-shell DD states to the J /~ decay is small for reasons that are not fully under- stood They have been described by phenomenologlcal models in which the individual contribution of a given intermediate state is not small, but there are cancella- tions between different contributions resulting from SU(3) flavor symmetry, nonet symmetry and ex- change degeneracy [1 -4 ]

These cancellation mechanisms cannot hold when there is only a single on-shell OZI-allowed open chan- nel, as the contribution from an on-shell Intermediate state cannot be canceled by off-shell contributions This IS particularly noticeable in unltarlty relations, where only on-shell states can contribute, and the OZI-forbldden process is linked by unltarlty to the two-step process It IS therefore dangerous to estmaate OZI-forbidden transitions by extrapolating known data across a threshold for an allowed process

The excited charmonlum states above DD thresh- old should have OZI-vlolatlng decay modes which go via an intermediate on-shell DD state This suggests a drastic change in OZI-vlolatlng transitions in charmo- mum as one crosses the DD threshold A rough esti- mate of this effect is obtainable by writing down the tmitarity relation for the amplitude describing e+e - annlhdatlon into any final state f at an energy just above the DD threshold but below all other naked charm thresholds We can therefore assume that the

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Volume 179, number 3 PHYSICS LETTERS B 23 October 1986

unltanty sum is dominated by the DD intermediate state and obtain

Im(f[TIe+e-)=K(f[T~[DD)(DDITIe+e-) , (la)

where K is the usual kinematic factor This relation holds for any final state [f), and m particular also for DD Thus

Im(DDITIe+ e - )=K(DDIT ~f IDD)(DDITIe+e - ) ( lb)

Dlwdlng these two relatmns, we obtain

Im(flTle+e - ) ( fIT~IDD)

Im (Dfi I Tie+e-) - (DD[ T f [DD) ' (2)

Squaring this relation and summing over all final states f except for DD gives

f@D [ C l~) Im(f lT le+e- ) 2 = (D--~[-T~[~)(flTflD~) 2

= Oln(DD)/Oel(D5) (3)

Thus the ratm of the sum of the cross sections for all the OZI-forbldden transltmns to the OZI-allowed DD transition is given by the ratm of the lnelastm to elastm cross sectmns for DD scattering in the relevant partial wave Although we have no rigorous argument for the value of the inelastic cross section, there are many open OZI-allowed inelastic channels described by the standard allowed connected-quark-hne dia- grams [5 ], m which the charmed quark pair m the ini- tial state anmhflates and light quark pairs are created Thus the right-hand side of eq (3) has no OZI-forbld- denness factor and may very well be of order umty The OZI-forbldden decays into hght quark states may therefore give a slgmficant contnbutmn to the decays of charmonmm states just above the DD threshold Re- cent experunents seem to lndmate the presence of these effects [6]

Note that the dominance of a single OZI-allowed final state is crucial for this derlvatmn Well above charm threshold, where there are contributions from several OZI-allowed transltmns, there can be destruc- tive mterference and cancellatmns on the right-hand side of eq ( la) of the type noted m theoretical argu- ments [1 -4 ] as necessary to explain the success of the OZI rule Below charm threshold there Is no OZI-al- lowed open channel and there is no enhancement of OZI vmlatmn by unltanty Fleld-theoretmal phenom- enology can consider second order OZI-wolatlng tran-

sltions in whmh each of the two steps IS OZI allowed However, here again there are many off-shell interme- diate states and cancellations are expected It is only near thresholds where there is only one allowed inter- mediate state on shell that such cancellations cannot occur Both in the unltarlty sum and in second-order Feynman diagrams there is no way to cancel the con- trlbutlon from a single on-shell intermediate state with off-shell contributions [1]

This effect has been noted and esttmated for the O -~ pzr and f ' ~ nrr OZI-vaolating transitions A sLmple perturbation treatment using the Fermi golden rule expresses the OZI-vlolatlon suppression factor m terms of expertmental masses and widths as follows The transmon matrix element for the transition

O ~ K I ' ( + co ~ prr (4a)

is given by

(p~IMIO)- (pnlMlco) ~ (COlMI~)QIMIO), (4b) M~-M~ , M~,-&

where i denotes all mtermedmte states whmh couple to both the 0 and 60 The sum in (4b) has a pole for the intermediate KI( state at the mass of the 0 The Imaginary part of this pole contribution cannot be canceled by contributions from other intermediate states, and gives a lower bound in the OZI violation Evaluating the sum by converting it to an integral gives

(pTrlMI0!l i m ( P ( E ) dE (COlMIKI()(KI(IMI0)

F(O -~ KI() (colMIKI()

= 2 ( M 0 - M~o) ( 0 1 M I K I ( > ' (5a )

where p (E) is the density of KI( states at the 0 mass and F(0 -+ KI() is the width of the 0 given by Ferml's golden rule

P(0 -+ KI() = 2nl (KI(IMI 0)12 p(E) (5b)

This result shows that the ratio of the OZI-forbldden Oprr coupling to the allowed coprr coupling IS small be- cause of a small quantity on the right-hand side of eq (5b) having no relation to disconnected diagrams or quark-gluon couphngs, namely the ratio of the half- width of the OZI-allowed 0 -+ KI( decay to the Q-co mass difference

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Volume 179 number 3 PHYSICS LETTERS B 23 October 1986

A similar calculation for f ' ~ KI( ~ f -~ 7rrr has given the result [4]

P ( f ' -+ 2rr) 3 (I'(f'--Z_I~)]2[p~]5 P(f'-+KI~)>76\ Mf,-gf ! \~!

= 0 0 3 5 , (6a)

where P~r and PK denote the momenta of the plons and kaons respectwely in f ' decay, d-wave phase space has been used for these final states and the SU(3) relation

(2rr IMI f) = x/~(KI~IMI f) = x / ~ (KK ITI f ' ) (6b)

has been used to eliminate these matr ix elements The result (6a) ls in rough agreement with expertment A similar result ~ 0 05 lS obtained if experimental widths are used to obtain (27r [MID and (KI~ IMlf) with simi- lar phase space factors

Snnllar calculations for the charmonlum and bot- tomonlum systems give the results

( f lMl , I / ' ) P( , I / ' ~ DD) (co'IMIDD) (7a)

( f l g l c o ) ~> 2 ( g , i / ' - M j ) (q / ' IMIDD) '

(f lM[T(4s))]

(f M w ' ) {

F (T(4s ) ~ BB) (w' IMI UB) (7b)

2(gT(4s) - gw') (T(4s) IMIDD) '

where f denotes some OZI-forbldden hght quark state and w' lS some state with co quantum numbers, e g a radially excited w Quantitative evaluation of these ex- pressions lS difficult because the state co' and its cou- phngs to DD and BB are unknown and many possible states If) contribute with large phase space factors However it ls reasonable to assume that Mw' ~< 2 GeV In this case the mass difference factors in (7a) and (7b) are very different and suggest that the OZI viola- tlon for the T(4s) may well be an order of magmtude smaller than for the eg"

It ls therefore of interest to look for OZI-vlolatmg decays m charmonmm just above the DD threshold There should be many decays found, with no reason to expect any particular mode to be dominant All modes coupled to DD scattering in the 1 = 0, odd-G p-wave should be expected to have roughly equal

branching ratios One can look for signatures from states with these quantum numbers and which should be absent or suppressed if the state lS produced from DD decay or from a one-photon background Final states wxth no strange particles and odd G pari ty (odd numbers of plons together with any number of r/'s) would be of interest, since these would be doubly Cablbbo suppressed as DD decays, and also suppressed m one-photon transitions, where the non-strange tso- scalar component of the photon lS also suppressed

An expertmental value for the fraction of OZI-vlo- latmg decays would provide interesting information for theoretical models at tempting to derive the OZI rule from QCD Until these effects are understood, it ls dangerous to assume in the data analyses of complex final states just above DD threshold that these states are nearly 100% DD, and sumlarly for BB states m bo t tomonmm decays [7]

Sttmulatlng dlscusslons with J D Bjorken, D G Hathn, R H Schmdler and E H Thorndnke are grate- fully acknowledged

References

[1] H J Llpkm, Nucl Phys B 244 (1984) 147 [2] C Schmld, D M Webber andC Sorensen, Nucl Phys

B 111 (1976) 317 [3] E L Berger and C Sorensen, Phys Lett B 62 (1976)

303 [4] H J Llpkm, Nucl. Phys B 9 (1969) 349, m New fields

m hadromc physics, Proc Eleventh Rencontre de Monond, Vol 1 (Flame, Haute-Savole, France, 1976) ed J Tran Thanh Van (Rencontre de Morlond, Laboratotre de Phy- sNue Th6orlque et Partlcules Elementatres, Unlverslt6 de Pans-Sud, Orsay, France, 1976) p 169, m Deeper path- ways m high-energy physics, Proc Orbls Sclentme 14th Annual Meeting (Coral Gables, Florida, 1977) eds A Perlmutter and L F Scott (Plenum, New York, 1977) p 567, m Understanding the fundamental constituents of matter, Proc the 1976 Intern School of Subnuclear physics (Ence, Italy), ed A Zlchlchl (European Physi- cal Society, Geneva, 1978) p 179

[5] H Haran, Phys Rev Lett 22 (1969)562, J L Rosner, Phys Rev Lett 22 (1969) 689

[6] R M Baltrusaltlsetal,Phys Rev Lett 56(1986) 2140 [7] E Thorndlke, Ann Rev Nucl Part Scl 35 (1985) 195

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