ELSEVIER

9 June 1994

Physics Letters B 329 (1994) 143-148

PHYSICS LETTERS B

On the role of K*K intermediate states in OZI-rule violating reactions of antiproton annihilation

D. Buzatu, F.M. Lev Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia

Received 25 January 1994; revised manuscript received 28 March 1994 Editor: L. Montanet

Abstract

We explicitly calculate in the on-shell approximation the contribution of the triangle mechanism with K* and K mesons in the intermediate states to the branching ratios of the reactions pn --* cMr-, 15p ~ ~bzr ° and pp --* qb3,, the experimental values of which strongly disagree with the OZI-rule predictions. The calculated branching ratios for the first two reactions are in qualitative agreement with experimental data while even the upper bound for the branching ratio of the latter reaction is much less than the corresponding experimental value. We also discuss the contribution of the above mechanism to the reactions/~p -~ ~MT,/~p ~ ~bp ° and/~p -~ ~bto.

1. The recent experimental data on the pp and pn annihilation at rest obtained by the ASTERIX, CRYSTAL BARREL and OBELIX groups [ 1-3] at LEAR, have shown that the branching ratios of the reactions pp ---, thTr °, pn ~ dp~r- and pp ~ q53, are much bigger than expected from naive OZI-rule estimations. For example, as follows from the present data on the thto mixing angle, a th/to production ratio should be 4 × 10 -3 or less while in practice [1-3]

Br(pp ---, ~bTr ° ) /Br (pp ---, to~.o) = 0.14 + 0.004,

Br (/in ---, thrr- ) / B r (/~n ---, toTr- ) = 0.16 _ 0.04,

Br (/~p ---, th'y)/Br (/Yp ---, toT) = 0.33 _ 0.15.

(1)

(2)

(3)

The interest to the reactions violating the OZI-rule is connected with expectations that these reactions may give evidence of the existence of exotic states such as hybrids and glueballs and of the considerable admixture of strange quarks in the nucleon (see for example the discussion in Refs. [4,5] ). For this reason it is important to investigate whether some of the reactions violating the OZI rule can be explained by usual mechanisms. In particular, the authors of Ref. [6] have shown that the data on the reaction pp ~ 4,4, can be explained by the triangle mechanism with K mesons in the intermediate state. These authors also claim that their estimate for the branching ratio of ~bTr using the same mechanism with K* and K mesons in the intermediate state is of the same order as the experimental values [ 1,2]. The concrete results concerning the contribution of the K * K intermediate states to the reactions /~p ~ ~bTr °,/~n ~ ~bTr - and/~p ~ th'y assuming that the initial antiproton and nucleon are at rest and their orbital angular momentum are equal to zero were independently obtained in Refs. [7,8]. In the present paper we revise our work [7] taking into account referee's recommendations and remarks by the authors of Ref. [ 8].

Elsevier Science B.V. SSD10370-2693 (94)00401-R

144 D. Buzatu, F.M. Lev / Physics Letters B 329 (1994) 143-148

"k "k

(a) (b] (c) (d)

Fig. 1.

It is easy to show that to conserve the C, G and P parities in this case, the antiproton-nucleon system should annihilate into ~b~- only from the state with the isospin I = 1, spin S = 1 and the final ~bTr system is in the state with l = 1, while the pp annihilation into ~by occurs only from the state with S = 0.

The triangle mechanism with K* and K mesons in the intermediate state is described by the four diagrams in Fig. 1 and all these diagrams give equal contributions to the amplitude with I = S = i. As follows from the isotopic invariance, the amplitude of the reaction pn --* chqr- is by ¢2 times bigger than the amplitude of the reaction

pp --, 4,~ °. The contribution of the triangle mechanism with K* and K mesons in the intermediate state to the reaction

pp ~ qby is described by the same diagrams as in Fig. 1 but ~r ° is replaced by y. In this case the contribution of the graph (a) is equal to that of the graph (d) and the contributions of the graphs (b) and (c) are also equal to each other. However since the isotopic invariance does not take place in the process K* ~ KT, we cannot relate directly the diagrams (a) and (b).

2. At the conditions specified above there exists only one relativistically invariant amplitude for the reaction j0p ~ t~7"r°:

Affp~4~n.o =fp-p~ ~b~O( ~ - yU~)eu~p~e ~*p~p~= (2m)Zfp~p~4j~o( qgtTo~2o'i~o)e~ eiklPl . (4)

Here fa,_~ ~ o is some constant, ~ i s the Dirac bispinor describing the initial proton, ~ _ is the Dirac bispinor with the negative energy describing the initial antiproton, e ~ is the polarization vector of the q~ meson, p~ is the 4- momentum of ~r °, P2 is the 4-momentum of ~b, m is the nucleon mass, q~ is the usual spinor describing the proton, o~; ( i = 1, 2, 3) are the Pauli matrices, ~p' is the charge conjugated spinor describing the antiproton, the index T means the transposed spinor, p is the momentum of 7r in the CM frame of the ~blr system, e~o,~ and e~k~ are the absolutely antisymmetric tensors and the sum over repeated indices is assumed. The Green indices take the values 0, I, 2, 3 and the Latin indices take the values 1, 2, 3. A standard calculation using Eq. (4) gives

(2m) 3 Br(/~P~thTr°)= 4~r IL-~-'~°I2p3C~" (5)

where p = IP I and Cap is some constant which is fully determined by the total cross section of the pp annihilation from the state with l = 0.

The pp system can annihilate into K* + K - from both states with S= 1 and S = 0. The amplitude corresponding to S= 1 can be written by analogy with Eq. (4). Letp~ be the momentum of K *÷ andp~ be the momentum of K - . The amplitude corresponding to S = 0 can be proportional only to ( ~'_ Tsq t) (e *P) where e is the polarization vector of K* ÷ and P =p~ +p~. Therefore, as follows from Eq. (4)

* r * (0) AFp~IC*+K- ( 2m)2[f ap~--~l) ~., +K_ (q~,a-o.2~.~o)ek eikzPs +v/2m fap-~K*+K-(¢'T~2~)e *] , (6)

where m* is the mass of K*, eo is the time component of e , , f ' s are some constants and p ' is the momentum of K* +. A standard calculation gives then

(2m) 3 _ _ p ~ 3 ( r ( 1 ) ~(o) -~*+ Br(fiP~K*+K-) 4~" j~p~x.+K-12+ 3" ~p-~^ x - [2)C~p, (7)

D. Buzatu, F.M. Lev / Physics Letters B 329 (1994) 143-148 145

wherep ' = IP'I and Cpp is the same constant as in Eq. (5). Then it follows from Eq. (5) that

Br (/~p ___> 497rO) = ~ _ ) 3 if#_.,~o I Br(f fp__.K.+K-) ~t) ~ co) ~ .+ [2" (8) I f a , - . r*+r -12+ J a p - . ^ r -

It is well known that if the vector meson with the mass M decays into pseudoscalar mesons with the 4-momenta ql and q2 then the amplitude of such a decay can be written as

A = g(ql - q2) •e ~, (9)

and the width of the vector meson relative to this decay is

F = [ g I---'-~2 k 3 (10) 6~M 2 '

where k is the magnitude of the momenta of the pseudoscalar mesons in their CM frame. Ifgi is the constant entering into the amplitude K* ÷ ~ 7r°K + and g2 is the constant entering into the amplitude 49~ K + K - then using the facts that K* in almost 100% cases decays into K~ and 49 in 87% cases decays into K/~ we get

[gl I 2 p ~ Ig2lZp3xe 6rtm.2 = ½ F * , 61rm~ =½.0.87F,~, (11)

where P,~x is the momentum in the CM frame of the K~r system created in the decay of K*, Pxx is the momentum in the CM frame of the KK system created in the decay of 49, F * is the full width of K* and F,~ is the full width of

49. The amplitude of the decay K* ~ KT has the form

Ax* --,r~, =fr* ---,r~,e ~,o~e "*E "PPk ° , (12)

where e ~ is the polarization vector of the photon, E is the polarization vector of the K* meson, P is the 4-momentum of the K* meson and k is the 4-momentum of the photon. The amplitude (12) is gauge invariant and automatically ensures the gauge invariance of the amplitude pp ~ 493'. Let Pr:, be the magnitude of the momentum in the CM frame of KT, and F * be the radiative width of K*. Then, as follows from Eq. (12)

P3r I f r ._~rr l 2 . (13) r~* = 12~r

3. We have done all the preparatory work for calculating the diagrams in Fig. 1. However it is easy to see that these diagrams diverge if no form factors are introduced into the vertices since K* is the vector meson. At the same time the imaginary part of these diagrams can be calculated rather reliably. Consider for example the imaginary part of the diagram in Fig. la. The amplitudepp ~ K* + K - enters into this diagram when all the particles are on their mass shells, and in the amplitudes K *+ ~ 7r°K + and K+K - ---> 49 only K + is not on the mass shell. Let us first consider the approximation when the last two amplitudes are written in the same form as for the case when K ÷ is real. Then, as can be shown by a straightforward calculation, the integrals defining the imaginary part of the diagrams in Fig. 1 can be calculated analytically.

In particular, using Eqs. (6) and (9) it can be shown that the imaginary part of the diagrams in Fig. 1 contributes to the amplitude pp ~ 497r ° as follows:

mp '2 [ (a+ 11] - - 11 t T * iA~p__,¢,,,~- 2.trpf glg2(q~ tr2trlqOeiktpket 2a+(1- -a2) ln[ ,a_- - -~ j / , (14)

where

a = ( 2 E * E , + m 2 - m 2 - m * 2 ) / 2 p p ' , E*=(m*E+p'2) 1/2, E~r=(m2 +p2) 1/2

146 D. Buzatu, F.M. Lev / Physics Letters B 329 (1994) 143-148

a n d f ~s is the part of the constantf ~s) corresponding to the contribution of the isospin I. It is important f fp~K* +K-- that the kinematical conditions are such that a > 1 and therefore the integrals defining the imaginary part of the amplitudes,On ~ ~bTr- and,Op --* ~bTr ° do not contain singularities (the same is true for the case,Op ~ ST). Comparing Eqs. (4) and (14) we find that

_i , 2 [ (a+li] glg2P f ll (15) fp:o ~ ~,~ ~ 2a+ (1 --a 2) In ~a------~]]

and therefore, as follows from Eqs. (8) and ( I 1 ):

Br(fip ~ ~b~ °) =0.87 3pp 'm*2m2~F*F~ If '112 B r ( f p ~ K * + K - ) 32 mE(p~KPKK) 3 [ f lO+fOO]2+[fl~+f01[2

l n ( a+ 1)] 2 " X ,[2a+ (1 - a 2) ~a ----~)J (16)

The result of the analogous calculation for Br(,Op ~ th3') is

a r ( f p ~ q~y) = __9 prp'm*2mZe, F~ dffP ~r*+K-'rr(O) 12 ¢ (1) * (0) [2 Br(~p--*K*+K -) 256 m2(prrpKK) 3 J~p~K + r - [ 2 + l f ap-K*+K -

[ (a'+ql+ × + \ a t - 1/_1

where ct is the (unknown) difference of phases between the diagrams (a) and (b) of Fig. 1 for the case,Op--* ST, pv is the CM frame momentum in the thT system and av is defined by analogy with a with p replaced by pv and m~ replaced by m v = 0.

4. Let us now discuss the obtained results. As follows from Eqs. (16) and (17), even in the on-shell approximation the branching ratios under consideration are not expressed only in terms of observable quantities since we should know additionally the quantitiesf is. The data on B r ( p p ~ K * K ) given in Refs. [9-12] are not in full agreement with each other and therefore the corresponding quantities

glS = If ls12 If ~°+f °°12+ If ll + f O1 [2 Br(tYp--*K*K)

also will differ from each other. Using the fact that Br(,op ~ K ' K ) = 4 Br(pp ~ K* + K - ) and taking the values obtained in the analysis of Ref. [9]:

gll = (23.4___ 2.0) × 10-4, g l °= (6.6+2.7) × 10-4,

g ° l = ( 1 . 7 + l . 0 ) × 1 0 -4 , g ° ° = ( 8 . 4 + 1 . 4 ) × 1 0 - 4 , (18)

we obtain for Br(,op ~ ~bTr °) the value given in Table 1. The quantities Br(pp ~ ~b~r °) and B r ( p n ~ ~brr-) can be related to each other as follows. Since Br(/~p~

th 7r° ) = trap ~ ~ / t r a p , Br (fin ~ qSrr- ) = trfn-, ~,r- /tr~. where O-pp and tran are the total cross sections of the,op and ,On annihilations, and the amplitude ,On ~ qblr- is by X/2 times bigger than the amplitude,Op ~ th~ -°, then

Br(/ in~qbTr-) = 2O'fp (19) Br (/ip ~ qS~ -°) o'p-n

According to the analysis of data made in Ref. [ 13 ], the ratio trap/tra, for the S wave annihilation near threshold is 1.25. Therefore Br(,On ~ ~bzr - ) = 2.5 Br(,Op --* ~brr°), and we obtain for Br(,On --* ~bTr - ) the result given in Table I.

D. Buzatu, F.M. Lev /Physics Letters B 329 (1994) 143-148 147

Table 1 Comparison of our results obtained by calculating the contribution of the triangle mechanism with K*K intermediate states to the branching ratios of different reactions with the experimental data of Refs. [ 1-3].

Reaction

pp--* thlr ° pn ~ thor- P P ~ thY P P ~ the7 p p ~ thpo pp-~ thto

I ,S 1,1 1,1 S=0 0,1 1,0 0,0 theory 2.9+0.2 7.2 +0.5 0.014 0.3 +0.1 0.063 0.08 experiment 7.8 -I- 1.0 12.41 + 0.88 0.22 + 0.06 0.94 4- 0.28 3.4 4- 1.0 5.3 -I- 2.2

The theoretical value for the Br(/~p--* th'F) is the upper bound for Eq. (17), the value of Br(pp--* th'0) is calculated assuming the SU(3) symmetry relation between the constants gx*-~x- ,rO and g x * - ~ r - ~ , and the values for Br(pp--* CpO) and Br(pp~ the) are rough estimations. All the branching ratios are given in units of 10 -4.

The on-shell approximation as evaluated and presented in Table 1 is an upper limit since there are necessary damping factors corresponding to the virtual K-meson in the exchange channel and neglected in the present evaluation. The authors of Ref. [ 8] argue that introducing the off shell factor for the K-meson and taking into account the contributions of the real part of the amplitudes in Fig. 1 and p p intermediate states one can enhance the above value of Br(pp ~ th~ -°) by a factor ranging from 1.5 to 4.3. Therefore we conclude that our results for Br(pp ~ qbTr °) and Br(pn ~ thTr- ) are in qualitative agreement with the experimental data.

At the same time, even the upper bound in Eq. (17) (see Table 1) is much less than the experimental value of Br(pp ~ @y). As shown in Ref. [ 8], this value can be explained taking into account the contribution from thp via the vector meson dominance.

Some deviation from the OZI rule (though not so strong as in the above cases) has been also observed in the reactions p p --* dpr I, p p ~ qbp ° and p p ~ thto [ 1,2]. The contribution of the considered mechanism to these reactions is described by the diagrams of Fig. 1 where the ~.o meson is replaced by the r/, pO and to mesons respectively. The decay constants gK* + -~ K + 7' gK* + ~ K + po and gK* + ~ K +~, are not known since the corresponding processes can

be only virtual. Assuming the SU(3) symmetry we can connect the quantities g K * + ~ K + ~ and g K * ÷ ~r+,rO:

gK* + ~ X ÷ ~ = Vr3gK* + ~ K + ~o. Then our result for Br(pp ~ th'r/) appears to be in qualitative agreement with the

experimental quantity (see Table 1 ). At the same time, even assuming the SU(3) symmetry, we cannot determine the constants g x * ÷ -~ K + po and gK* + ~ K +~" Assuming that these constants are equal to the constant of the decay

K* + ~ K + ~o in the hypothetical case when 7r ° is a vector particle we get for Br(pp --0 thp °) and Br(pp ~ qbto) the

estimations given in Table 1. Since the violation of the OZ! rule in the reactions pp--* dpp ° and p p ~ ~bto are not so drastic as in the reactions

p p --* qb~ ° and p p --* dp),, one might think that the main contribution in the cases of ~bp ° and thto is given not by the mechanism considered above but by the thto mixing. Let us also note that the thTr, thr/, thp and thto channels in the nucleon-antinucleon annihilation are independent of each other since they correspond to different quantum numbers (L S) (see Table 1 ). Therefore there may exist mechanisms of annihilation which contribute to some channels and do not contribute to others.

To better understand the problem of violation of the OZI rule, it is necessary to carry out both the experimental and theoretical study of this problem. For example, the conclusion about the dominant role of the channel with I = S = 1 in the annihilation into K * K was made only using the indirect data while the direct measurement of the reaction p n ~ K * ° K - has not been made. An interesting theoretical problem is to explain the experimental facts [ 14] that when thepp system annihilates from the l = 1 state of the hydrogen-like atom then Br(/Sp ~ thTr °) is small while Br(pp --* the), Br(pp ~ ~bp °) and Br(pp --* thto) are of the same order of magnitude as in the case l = 0, In this connection it is important to measure the branching ratios of the p p annihilation into K * K states from the state of the hydrogen-like atom with l = 1.

148 D. Buzatu, F.M. Lev / Physics Letters B 329 (1994) 143-148

The authors are grateful to M.G. Sapozhn ikov for the s ta tement of the problem, supervis ion and numerous

discussions, to B,Z. Kope l iov ich and M. Locher for valuable remarks and to M. Faessler for sending us the latest

exper imenta l data on Br (thrr), Br(tb 'g) and Br ( ~br/).

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