Volume 78B, number 1 PHYSICS LETTERS 11 September 1978

THE OZI RULE VIOLATING RADIATIVE DECAYS OF THE HEAVY PSEUDOSCALARS

A. BILLOIRE, R. LACAZE 1, A. MOREL and H. NAVELET C E.N. Saclay, Service de Physique Th~orique, 91190 Gif-sur- Yvette, France

Received 8 June 1978

In lowest order of QCD, the rates for radiative transitions violating the OZI rule of heavy pseudoscalars are found to be extremely small. The ~c ~ 3̀ + P, co, ~ width is predicted two orders of magnitude below present experimental bounds on X(2.8) ~ 3,+ n+rr -. As a by product, we get for the 43, parapositronium decay width F = me aT. 1.352 × 10 -2.

The 2.8 GeV state, probably the 0 - partner r/c of the J/~/, has been seen both in J /~/decay products [1] and in n - p collisions [2], but only in the 3'3' decay mode. There is an obvious need for its confirmation in other channels. Hadronic modes, expected to be domi- nant (total width in the MeV region), are however difficult to detect experimentally due to high multi- plicities [3]. Radiative channels, such as 3'0 or 7~0 have better signatures, and to lowest order in QCD, we could hope to get the ratio P(r/c-+3` + anything)/P(r/c -+ 3'3') as large as a 3 / a ~ 1 for a s = 0.2, not in contradiction with the experimental bound 1~ (r~c -+ 3"n+rr - ) /P( r /c~ 3'3') ~< 2.3 [4,11]. This motivates us to present quanti- tative estimates of the pseudoscalar OZI violating radia- tive widths. This study has applications not only to the 2.8 problem but as well to the 3.45 GeV state and the pseudoscalar members % of the upsilonium family near 10 GeV.

We base our calculations in QCD in the same spirit as it has been already done for J/ff radiative decays [5]. The hope is that a perturbative expansion in the effective quark-g luon coupling a s makes sense also in this context, at least to predict orders of magnitude. We thus compute the inclusive rates for r/c,b -+ 3 ̀+ W, W being any hadronic system with mass W, to lowest order in a s. We are interested in the spectrum dF/dW, not in view of actually comparing it to inclusive data, certainly obscured by a large 3' background from 7r 0 decays, but rather as representative, at low W, of a few

1 C.N.R.S.

140

-t "0000000000 000000~0'~"

[ 00000'00000 _ _ I 0000000"0"~"

(a) (bl

Fig. 1. Leading order contributions to r~ ~ ,,/+ W, (a) n 3, + 3g; (b) r /~ 3`q~g. The two diagrams have to be symmetrized.

exclusive rates into 3' plus (C = - 1 ) resonances such as p, co, ~ (~c case) and also J/O (%) .

Two classes of Feynman diagrams (figs. 1 a, b) con- tribute to the lowest order g3 (a s =g2/47r). According to the standard non relativistic framework, the corre- sponding rates Pa,b are given by

dFa b = ~[~ (27r)464(en - Pf)lff(0)121Ta bl2d fa, b " (1) f

~(r) is the non relativistic wave function describing r/c (r/b) as a cg (bb) bound state, T~, b the cg (bb) anni- hilation amplitude at threshold corresponding to either graph, and dfa, b the relevant phase space volume ele- ment. In diagram 1 a, only systems containing a flavour singlet part may be produced. Diagram lb involves the production of a light qTq pair and a gluon, which now materialize into isospin states 0 or 1 as well (p, w, % ...). An infrared divergence appears at the upper end of the W spectrum (soft photon); this divergence is irrelevant for this work which concentrates on the lower W region

Volume 78B, number 1 PHYSICS LETTERS 11 September 1978

Bk0

.01 / ;z//

.001

.0001

/ 2 / Bu, d

I / /

/

i/ii/ / / // / /

01

001

1 N [Gev]-i

~ , d / / '

/ " . 1

J / /

/ /

/ /

/ /

/ /

/ / /

/ /

/ /

11 11J

Bs

Be

[

t , 2 p, to q)

Fig. 2. W spect rum for r / ~ 3` + W relative to F3`7(r~) for (a) r/c, (b) r/b. Curves A and B refer to contr ibut ions f rom figs. l a and lb , • • 1 2 respectively. The hght quark mass value are: (a) 20 MeV (Bu,d) 330 MeV (Bu,d) , 400 MeV (Bs), (b) 18 MeV (Bu,d), 350 MeV (Bs) ,

1250 MeV (Bc). In bo th figures Bu, d corresponds to zero mass light quarks and a 300 MeV 3" transverse m o m e n t u m cut-off.

only. Summation over spin states have been carried out using Reduce [6]. Technical details will be given elsewhere [7].

As a by product of the graph l a calculation, we obtain the following parapositronium partial width

into 47's: (2)

I'43"(pps ) = mea7" 1.352 X 10 - 2 = 11.57 X 103 s - 1 ,

which is larger by a factor of about four than a pre- vious result given by Mc Coyd [8]. The latter was in fact computed with two cuts in the photon energies co i. A first cut for all coi less than me/9 was made for "exper imenta l" purposes. For such a cut we obtain 9.03 X 103s - 1 . The second cut, based on an assump- tion about the phase space populat ion, required two photons to have energies less than me/3 * '. By doing

2cuts both cuts we find ['43 , (pps) = 2.6 × 103 s -1 in agreement with the result of ref. [8], which shows that

4-1 The two cuts allowed an expansion of the cross section in powers o f the wi's, leading to an analytic integration over phase space.

the assumption made there is wrong. Since the 2.8 state is seen in the 77 mode, we nor-

malize the rate ( d F / d W) (r/c, b -+7 + W) to the r/c, b -> 77 width F c'b

73' '

V73'c,b = 12rroz2e~[~(O)12/m2,b . (3)

eQ is the charge, mc, b the mass, of the annihilating quark (me, b = Mr/c,b/2 ). In so doing, we eliminate un- certainties connected to the wave function at the origin. Our results for 77c are computed with c~ s = 0.19, which accounts for the experimental value of F(J/I~ ---> hadrons) / I ' ( J /O -+ e+e- ) . They are shown in fig. 2a. The contr ibution of diagram la is given by curve A. The quanti ty plot ted is equal to me ~3 times a universal function of W/m e. The three curves

1 2 Bu,d, Bu,d, B s are the contributions of diagram lb with the following specifications. B s is for a strange final quark with effective mass #s = 400 MeV. The two others are for the incoherent sum of the u and d quark cont r ibut ions , Bl ,d corresponding to a u or d mass /l 1 = 20 MeV and B2,d to/12 = 330 MeV. The effective values/11 and/1 s are those proposed by Georgi and

141

Volume 78B, number 1 PHYSICS LETTERS 11 September 1978

Politzer [9~ in case the scale is set by the J/@ mass. The value/J2 is chosen for reference to the nai~,e quark model. Let us now comment on these curves.

(i) In contrast with the case J /~ -+ 7 + W [5], the spectrum is depressed at low W values, which unfavours low multiplicity exclusive channels. The total rate in the lower half of the W region, relevant to the 7 + P, co, ~p channels, is indeed very small, a few percent of

c PT"~ at most, that is lower by two orders of magnitude than the present experimental limit (2.3P77) on F(2.8 -+7 + n+n-) [4].

(ii) In the zero mass limit for the light quark q, a singularity occurs in diagram lb when a quark and the photon are collinear, which is related to the im- possibility of separating out a photon parallel to a massless particle. This is a reason, together with diffe- rent threshold positions, why B2,d(/J = 330) is well below Bl, d(~ = 20 MeV). This sensitivity to the / l value is not of real importance phenomenologically since for any sensible ~ we get contributions well below present experimental bounds.

However, this mass singularity is of theoretical in- terest since, in QCD, effective masses/J(M) go to zero according to [9]

u(M)/I.t(Mo) = [log (MO/a)/log (M/A)] 12/25 , (4)

in the large M limit (4 flavours). The parameter A sets the strength of the coupling as(M ) = 12n/(25 log × (M2/A2)). One can show that in this limit, the singular part of the spectrum takes on the form

F 7 7 d W M 2 ~ F , (5)

where the W/M 2 factor comes from phase space, F being a regular function.

3 goes Thus, for any If, the above rate, to order a s , to zero in the large M limit, despite the logarithmic mass singularity. As a side remark, we note that if it was experimentally relevant to consider large W's, we would discuss the final states in terms of jets, and then apply a cut-off in the photon jet angular separation.

t

As an illustration, the curve Bu, d in fig. 2a shows the result obtained with/~ = 0 and a 300 MeV cut-off in the 3' momentum transverse to the 3 jets. Within a

,

factor of about 2, the results B ,d and Bu, d are com- parable.

Fig. 2b shows curves corresponding to the r/b case (m b = Mnb/2 = 4.75 GeV), assuming a charge 1/3 for

the new quark. The curves Bu,d, B s and Bc(r/b -+ cE'yg) are respectively for/~ = 18 ,350 and 1250 MeV, values which follow from the previous ones 20 ,400 and 1410 in the r/c case and eq. (4) with A = 70 MeV (in corre- spondence with as(Mj/~ ) = 0.19, consistent with the

!

j/~j +2). Bu,d again corresponds to/J = 0 with a 300 MeV transverse 7 momentum cut-off.

To summarize and conclude, we have found very small radiative decay rates for the heavy pseudoscalars.. This is due on one hand to selection rules which make these decays to occur at relatively high order in a s

2 (the branching ratio B(r/c ~ 7W) is of order a s as com- pared to B(J/~J -+ 7W)). On the other hand, the shape of the W spectrum disfavours the low multiplici ty channels, supposedly represented by the low W tail of this spectrum. Both facts are typical '~, ?D effects in the minimum gluon number scheme, and the predicted smallness of the radiative rates thus deserves experi- mental verification.

Due to the respective quark contents of p and co, the ratio 7P/')'co is 9 for curve B 1 (which represents u,d the total non strange contribution). From fig. 2a, we then infer that the 7P channel is probably the best one. Using B(J/~ -+ 7r/c)B(r/c -+ 3'7) = 0.14 X 10 - 3 [11], we predict a few 7P events in a 10 6 J/t~ sample (a few tens if the effective a s to be used is 0.4). Actual meas- urements, or at least better bounds, for this transition or similar ones leading to relatively clean final states, would clarify the pseudoscalar status and would pro- vide especially valuable tests of such applications of

QCD.

We thank F. Hayot and H. Kluberg-Stern for fruit- ful discussions and comments.

:1:2 A = 500 MeV (a s = 0.4), a more realistic value for other purposes including charmonium spectroscopy [10], would not lead to results qualitatively different.

References

[1] W. Braunschweig et al., Phys. Lett. 67B (1977) 243. [2] W.D. Apel et al., Phys. Lett. 72B (1978) 500. [3] C. Quigg and J.L. Rosner, Phys. Rev. D16 (1977) 1497. [4] J.E. Olsson, Proc. Lepton-photon Hamburg Symp. (1977)

p. 117. [5 ] T. Applequist, A. De Rfijula, H.D. Politzer and S.L. Glashow,

Phys. Rev. Lett. 34 (1975) 365; M.S. Chanowitz, Phys. Rev. 12 (1975) 918;

142

Volume 78B, number 1 PHYSICS LETTERS 11 September 1978

T. Applequist, A. De Rfijula, H.D. Politzer and S.L. Glashow, Phys. Rev. Lett. 34 (1975) 365; M.S. Chanowitz, Phys. Rev. 12 (1975) 918; L. Okun and M. Voloshin, Moscow preprint ITEP 95 (1976); K. KoUer and T.F. Walsh, Phys. Lett. 72B (1978) 227; S.J. Brodsky, T.A. De Grand, R.R. Horgan and D.G. Coyne, Phys. Lett. 73B (1978) 203; H. Fritzsch and K.H. Streng, Phys. Lett. 74B (1978) 90.

[6] A.C. Hearn, Reduce 2 User's Manual, Univ. of Utah, report UCP 19 (1973).

[7] A. Billoire, Thesis, in preparation. [8] G.C. Mc Coyd, Ph.D. Thesis, St. John's Univ. (1965). [9] H. Georgi and H.D. Politzer, Phys. Rev. D14 (1976) 1829.

[10] A. Billoire and A. Morel, Nucl. Phys. B135 (1978) 131. [11] S. Yamada, Proc. Lepton-photon Hamburg Symp.

(1977) p. 69.

143