Gluonium contaminations in tensor mesons

Z. Phys. C Particles and Fields 28, 573 577 (1985) Part ~Jr Physik e

and O Springer-Verlag 1985

Gluonium Contaminations in Tensor Mesons

A. Bramon, R. Casas, J. Casulleras 1 and F. Cornet Departament de Fisica Te6rica, Universitat Aut6noma de Barcelona, Bellaterra, Barcelona, Spain

Received 21 February 1985

Abstract. The possibility of a small gluonium piece in the f(1270) a n d / o r f ' (1525)-mesons is discussed making use of well tested arguments and a wide set of well established experimental decay widths. Contrary to other authors, we are led to conclude that the conventional and ideally mixed nonet of qc] tensor mesons explains (to the expected accuracy) the data, and there are no compelling reasons to introduce 99 contaminations in f or f ' .

The discovery of gluonic mesons or, more simply, glueballs is crucial for the viability of Quantum Chromodynamics (QCD). Indeed, this widely accepted theory predicts the coexistence in the hadronic world of ordinary (quark composed) hadrons, glueballs [1] (i.e., bound states of two or more gluons) and, possibly, hybrid or mixed states consisting of both types of constituents [21. The lowest lying glueballs are expected to be jPC= 0 § + or 2 § § states carrying no flavour quantum numbers. For these reasons they are allowed to decay into qq pairs (the r/-particle could also have a gluonic component) and have been intensively searched by means of this peculiar decay signature. As a result, at least two glueball candidates seem to exist: the G(1595) and the 0(1690) decaying into r/q and having, respectively [3], jPC = 0 § + and 2 § +. Alter- natively, glueballs can be identified through their appearance in processes which are not allowed for ordinary q~-mesons. Again, several candidates [1,4], with j P c = 2 ++, have been detected as q~4~ resonant states in the OZI-forbidden reaction re- p --* 4) ~b n. To these states a new, lower mass glueball candidate (again, with j P c = 2 + +) has to be added. It has been observed [5] as a peak in the K ~ ~

1 Also in Dept. de Tecnologia de Computadors, Fac. d'Inform~ttica, Universitat Polit6cnica da Catalunya, Barcelona

system around 1410 MeV and cannot be accommo- dated in the already known tensor meson nonet with similar masses. Nonetheless, a definite, direct identification of a glueball seems to be lacking and the situation appears as extremely controversial [11.

Independently, the possibility of detecting a glueball component in a (mainly) ordinary q~-system has also been investigated. In particular the f(1270) meson, having the same quantum numbers as the above mentioned JPC=2++ candidates, seems mostly appropriate and has been claimed to present a fully detectable gluonium admixture by Rosner [63 and other authors [73. These analyses are based on S U(3) and S U(6) arguments relating different strong decay rates of 2 ++ mesons into pseudoscalar and vector mesons, T ~ VP and T~PP ' . According to [6] and [73, SU(3) and SU(6) predict F(f~nTr),~ l l 5 M e V which contrasts with the experimental value [8] F(f~r~rc)-~150MeV. The agreement is restored invoking the presence of a gluonium piece in the up- to-now innocent f-meson. This, in turn, implies the existence of a mainly glueball tensor-state, f r , with a mass of 1.4-1.9 GeV, orthogonal to the f-meson. Even if this type of analyses does not necessarily attribute a glueball nature to the non-qt] piece of I = 0 tensor mesons, this is considered as the most likely explanation and, in this sense, as an indirect but relevant test on the existence of glueballs. It would be then rather attractive to identify (as analyzed in [9]) the mainly glueball tensor meson, f r (mixed with f and f ' ) , with the 0(1690). Indeed, its mass lies in the appropriate range and, more important, the 0(1690) branching ratios into ~ , KK" and q~/ are incompatible with a pure q~ nature for this meson [1, 8,101. However, such an identification leads to several difficulties, and alter- natives, such as the one based on the mixing among f , f ' and the recently observed o o K s K s peak around 1410 MeV [51, have been proposed [111.

The purpose of this paper is to proceed with the

574 A. Bramon et al.: Gluonium Contaminations in Tensor Mesons

Table 1. Experimental data on A 2, K * ~ , f , f ' and r decays coming, mainly, from PDG [8] and different fits to these data. The details of the fits are explained in the text and are based on idea l f - f ' mixing (Fits 1 and 3) or on a t r i p l e f - f ' - f r mixing formulation (Fits 2 and 4). The decay widths are all given in MeV. The relative quality of the different fits indicates that there is no need for the latter, triple mixing phenomena

Partial Fits Global Fits Fits without f ' -da ta EXP. Data Fit T ~ V P Fit T---, PP ' Fit 1 Fit 2 Fit 3 Fit 4

77.1• 73.8• 73.9• 74.1• 73.8_+3.6 73.8• 5.4 • 0.9 6.4 • 0.4 6.5 + 0.4 6.5 • 0.4 6.4 + 0.4 6.1 __+ 0.4

(16.0 • 1.5) (24.6 + 1.5) (24.2 • 1.6) ( < 2.2) (0.54 • 0.03) (0.53 • 0.03)

24.5 • 3.3 29.4 • 1.5 29.4 • 1.4 29.4 • 1.4 8.8 • 1.3 8.9 • 0.4 8.9 • 0.4 8.9 • 0.4 4.2• 1.6 2.4• 2.4• 2.4•

44.8 • 5.0 50.1 +__ 3.1 49.7 + 3.2 (5 • 5) (o) (o)

150.1 • 17.0 126.6 • 7.9 124.4 • 8.0 5.2 + 0.7 4.5 _• 0.3 4.4 • 0.3

(0.9 _+ 0.3) (0.71 • 0.04) (0.69 • 0.04)

13.9 • 0.7 39.9 • 2.5

A2 --* pT~ --+ K K

--* Tc r#

K** ~ K * ~ ~ p K ~ o K --* K ~ --* K q

f -+1~1"~ - -*KK

f ' - - , K * K + K * K ~ K K -~ other -'* all

f -*y~

A2-~Ty

f ' ~ 7 7

A2-~?y

R, (4)

29.5 • 1.4 29.4 • 1.4 9.0 • 0.4 8.9 _+ 0.4 2.4• 2.4 • 0.12

49.7 +_ 2.9 49.6 • 3.0 46.8 • 3.0

124.4 _+ 2.9 127.7 • 9.4 127.6 _+ 12 4.4 • 0.3 4.8 • 0.6 5.3 _+ 1.0

14.1 • 37.2 • 3.3

(6 • 4) 7 0 • 6 0 • 5 7 •

3.41.+0.59 25/9 2.70+_0.11 25/9 2.55_+0.16

0.18 _ 0.10 2/9 0.25 • 0.05

1.92 • 0.75 2.53 _+ 0.06 1.98 ___ 0.76

x2/d.o.f. - - 4.2/3 5.8/3 12.9/10 11.9/8 11.2/7 8.9/5

analyses of these tests involving tensor-meson states along the lines of [6,7, l l ] with the following points in mind:

i) SU(6) arguments (relating T-* VP with T ~ P P ' decays) will not be invoked. This partially contrasts with the procedure of [6, 7] and makes our analysis more reliable. Indeed, S U(3) symmetry is known to be much more precise than S U(6),

ii) the experimental data will be taken directly from the recent PDG compilation [8] and, in particular, those concerning the well established f'(1525) meson will not be ignored. This asseptical attitude should probably be preferred to that of I-6,7],

iii) we will concentrate on the study of decay widths and forget all considerations concerning the masses of the tensor-mesons. This contrasts also with the recent analyses of 1-9, 11], and circumvents the ambiguities associated with mass relations (the use of linear-versus quadratic-mass formulae or the consideration of energy-dependent versus energy-independent mass- matrices are just two examples [9]),

iv) we will not introduce in our fits data corresponding to decay modes involving neither r/, r/' particles, nor any tensor glueball candidate. This seems a safe attitude since the quark and (eventually) gluonium content of these objects is unknown and their inclusion would imply the presence of additional, unfixed para-

meters. However, since a few decay modes involving r/particles have been measured, we will present (as in [6] the corresponding predictions of our scheme assuming a pure quarkonium structure and a reasonable mixing [12] for the q-r/' system.

In other words, we will restrict our analysis to the old-fashioned phenomenology concerning all the best known tensor-meson decays: T ~ VP, PP' and 77, and the recently measured branching ratios [-8,10] for

~ 7f and yf ' decays. The present analysis will be based on simple and well-established quark-model arguments such as the Zweig rule or nonet symmetry. Our initial guess (and to some extend, our hope) was that we would be able to reinforce from these simple and solid grounds the attractive result [6,7, 11] that tensor glueballs are somehow present in the nowaday available data. But it has not been so and we will finally conclude from our analysis that one cannot claim for any substantial departure from the quark-model relations expected to hold for purely q@ (i.e. without gluonium and/or other contaminations) tensor mesons.

Let us first review the T ~ VP decay modes. Four of them have been measured and the corresponding decay widths [8] are shown in the first column of Table 1. These decay widths are strongly correlated by S U(3) symmetry which allows for a one-parameter fit. The

A. Bramon et al.: Gluonium Contaminations in Tensor Mesons

free parameter is the coupling constant gv appearing in the generic expression for the decay width

F ( T ~ VP) = gZlPvlS/4Ort. (1)

Assuming exact S U(3) symmetry for the TVP coupling constant, ideal a) ~b mixing and S U(3) breaking in the phase space factor one obtains the results shown in the second column of Table 1. The quality of the fit is rather high [;~2 = 4.2 for 3 degrees of freedom (d.o.f.)] as one could have anticipated from similar analyses performdd by Rosner [6] and Montanet [13]. The fit fixes quite precisely

g~, = 1078 _+ 53 GeV -4 (2)

and its quality can be considered as representa- tive of the expected accuracy of our approach based on SU(3). Indeed, since there is no gluonium contamination in these T--, VP decays with V= A z or K**, any failure of the fit should be attributed to ordinary SU(3)-breaking in the T V P coupling constants related to g 2 through simple S U(3) factors.

This is not necessarily the case for the T ~ P P ' decays to which we now turn. The experimental data are shown in the first column of Table 1 and the phenomenological analysis has to be performed by means of the decay width expression

F(T--. P P') = #~[PvlS/3Om2T n, (3)

analogous to (1). Assuming, for the moment, ideal f - f ' mixing and proceeding as before one obtains the best fit shown in the third column of Table 1 for the four clearest (i.e., not involving t/, r/' mesons) T ~ PP' decays. The quality of the fit, Z 2 = 5.8 for 3 d.o.f., is certainly lower than in the previous T ~ VP case as could have been expected from the results of [6,7]. Indeed, while the A2---*KK and K**~Krr decay widths are predicted slightly above their experimental values, those for f ~ l r l r and f ~ K R are predicted somewhat below. This observation has essentially been the clue which led the authors of [6,7] to claim that a gg admixture was present in the flavour neutral f - meson.

Up to this point we have followed the lines of the previous analyses [6, 7, 9] and no use has been made of an important and well established information concerning the f ' (1525) meson, namely, its total decay width [8] F ( f ' ~ a l l ) = 7 0 + 10MeV. The dominant contribution to this total width is expected to be due to the K / ( decay mode (the only observed one) and possibly to the f ' ~ K * K + K * K decay [8]. The remaining decay modes of the f ' -meson can be estimated from our knowledge of the corresponding ones for the other members of the tensor-nonet. From the data on F ( T ~ PP') and for reasonable values [12] of the tl-t f mixing angle, S U(3) predicts F ( f ' ~ tlq ) ~- 2-4 MeV, in agreement with the experimental [14] upper bound F ( f ' ~ t l t l ) / F ( f ' ~ K K ) < 1. (The smallness of our S U(3) estimate is a consequence of the small phase-space and the identity of final particles;

575

only an exceptional amount of gluonium in both f ' and t/ particles could modify substantially our estimate). The phase-space factor reduces also other f ' decay modes such as f ' ~ t / t / ' and (the potentially dangerous analogue of the non-negligible [8] f(1270)--,470 f '~Kl~rcrc. G-parity forbids f ' ~ 3 ~ and f ' ~ ~brcrc (similar to K**~K*rcrc) and the most important three-body decay mode, f '--,,KKlt, is expected to proceed through the already considered f ' ~ K*/( + / ( * K ~ K/s decay chain. The energetically favoured f ' -~z tn and f ' ~ 4 ~ decay modes are known to be experimentally suppressed [8]. The suppression of the first one is so drastic that it requires either an almost idea l f - f ' mixing (if no other states are involved; the conventional OZI rule) or an effective cancellation between OZI allowed amplitudes if a three-particle mixing takes place. In any case, the strong suppression of f '~ rcrc suggests a rather low f ' ~ 4 ~ branching ratio, far below the experimental upper bound [8]. Similarly, the stringent upper limit [8] (1~o) for the f~t/rcrc branching ratio should be incompatible with a non-negligible f '~ t / r t rc decay width (for which only a poor upper limit is known [8]). Therefore, it seems reasonable to conclude that a rough estimate for all theft decay modes other than f ' ~ K/s and f ' ~ K * K + K * K could b~ F ( f ' ~ o t h e r ) = 6___ 4MeV, as quoted in Table 1. Notice that this partial decay width is at most of the order of the error affecting the total f ' width, namely [8], __+ 10 MeV. In this sense a more accurate and detailed evaluation for F ( f ' ~ other) (apart from being difficult) is irrelevant and unnecessary for our purposes.

Additional information concerning tensor meson decays can be obtained from the recent results [8,15] on 7Y ~ T transitions. There are two independent data of interest, namely, the ratios between the f and f ' ~ 7Y decay widths to the corresponding one for Az--*TT. These two ratios have been quoted also in the first column of Table 1 and the corresponding phase-space corrections will be neglected due to the similarity among the f , f ' and A 2 masses. Finally, one further piece of relevant information comes from the recent measurement of the branching ratio [10, 14]

R = FOP~Tf)BR(f--*~rO _ = 1.92+0.75 (4) F(~9 ~ 7f')BR (f'--* K K)

also quoted in the first column of Table 1. The rele- vance of ratio (4) is due to the fact that being the ~0 decays dominated by c~-annihilation diagrams, where gluons are copiously produced, any glueball contamination in the f or if-meson could easily be detected. In summary, the global set of experimental data which will be considered in what follows includes this last ratio R(4), two ?7-decay ratios and nine (strong) decay widths into VP or PP final states, i.e., a total number of twelve experimental values which we will try to describe performing different fits.

Let's first consider the most economical and simple possibility of accounting for the whole set of, twelve

'1

576

data in terms of two purely qci and ideally mixed tensor mesons: f ~ ( u ~ + dcl)/x/2 and f ' ~sg. The fit contains two adjustable parameters, 9v and 9p, which (apart from S U(3)-factors) have been defined in (1) and (3). It leads to the predictions quoted in Table 1 (Fit 1). The minimum X 2 is found to be 12.9 for 10 d.o.f., and g2 is fixed to 92v=1081+__50GeV -4 in good agreement with the corresponding determination, using only the four T ~ VP decay widths, (2). This agreement, as well as the similar quality of the two fits, allows us to consider this global Fit 1 as the most natural and equally acceptable extension of that for T--, VP decays. Since the (small) failures of the latter has to be explained in terms of conventional S U(3)- breaking, the same explanation can account for the smaller discrepancies in Fit 1, thus making unnecessary to claim for a (so far neglected) g9 contamination in the f and/or f ' mesons.

In order to reinforce this conclusion, let us consider [6,7,9,11,163, that there exists a third I = 0 tensor meson, fT, which contains a glueball admixture, gg, and mixes with f and f ' . Following the notation of [9], the coefficients of the non-strange [ (mi+ dd)/x/2], strange (ss3 and glueball (g9) components for the f meson will be denoted by x, y and z. Similarly, the corresponding ones for the f ' ( fT) meson will be given by x', y' and z' (x T, Yr and zr). This amounts to the introduction of four independent parameters besides gv and gv: three Euler angles, describing the triple mixing, and the ratio r, measuring the relative strength between the g 9 --* ~ ~z and (u ti + d ~)/x/2 ~ 7r 7r transitions. One has, for instance, z ~ s in0sin0, z ' -=cos0s in0 , zr-=cos0, xr=sinOsinO, Y r - - sin 0 cos ~b, etc. As in [6, 7, 9, 11] this set of parameters is reduced to four independent ones imposing the suppression of the f ' ~ rc~z and fr--* 7zn decay modes. These are two well established experimental facts deducible from the appearence of just one state, the f meson, in the accurate data on rcz interactions. Those suppressions imply r = - tan 0 sin q5 and tan ~, = - tan 4~/cos 0, and we are led to a four adjustable parameter fit: 9v, 9e, tan 4) and cos 0. The results of this global fit have been quoted in the fifth column of Table 1 (Fit 2) and the two relevant parameters of the set are fixed to 92 = 1082_+ 52GeV -4 and cos0=0.987+~176 The first agrees with its

- 0 . 0 4 2 "

two previous determinations showing the stability of our approach, while the angle 0 is found to be compatible with zero. This implies that, f r , decouples from f and f ' which, in turn, appear as ideally mixed, i.e., Fit 2 corresponds to a situation very close to that assumed in Fit 1. Moreover, the quality of Fit 2, X z = 11.9 for 8 d.o.f., has not improved that for Fit 1, )~2 = 12.9 for 10 d.o.f. All these results substantiate our previous conclusion, namely, that a 99 contamination in the f and/or f ' meson is of no help when trying to describe our set of data and, therefore, that these cannot reveal the presence of that 9g piece.


Since this conclusion contrasts with those from [6,7,11, 16] let us investigate the origin of the discre- pancy. To this aim we perform two new fits (Fits 3 and 4) neglecting the three experimental data involving the f '-meson. Firstly, in Fit 3 we assume ideal and uncontaminated f - f ' mesons and obtain the values quoted in Table 1. The corresponding Z 2 is 11.2 for 7 d.o.f., which is somewhat less satisfactory than in its analogue Fit 1 to the whole set of data. The same applies to our final Fit 4, where the triple f - f ' - f r mixing considered in the previous Fit 2 has been introduced. Now one has •2 = 8.9 for 5 d.o.f, which is slightly less satisfactory than in Fit 2. The main difference between these two fits, however, is their predictions for the mixing angle 0 measuring the 99 contaminations in f and f ' . In our final Fit 4 we obtain cos0=aa~+~176 (and 0 . 6 < t a n 0 < 3 . 1 ) , i.e., a non-negligible 99 piece in those mesons; by contrast, in Fit 2 one had c o s 0 = naQ'~+~176 (and v . ~ , o / - 0 . 0 4 2

tan~b= 1.2+_0.2) indicating a negligible gg contamination in f and f ' . Therefore, the discrepancies between our approach and that of [6,7,11] are essentially due to the inclusion or not in the fits of the data corresponding to thef'(1525)-meson. These data tend to restore the viability of a naive q~ and ideally- mixed tensor meson nonet. This is particularly due to the large value, F( f '~KI~)~_5OMeV, one is forced to attribute to this not directly measured decay width. Such a large decay width and SU(3) predict F(f--* tort) ~- 150 MeV, close to the experimental value, thus compensating the effects discussed when performing our partial fit to T ~ P P ' or in [6, 7,11]. In this sense, experimental data on the f ' branching ratios are highly desirable. If they imply F ( f ' ~ K K ) ~ - � 8 9 F ( f ' ~ all) -~ 35 MeV (as used in [11]) we could hardly maintain our conclusion. On the contrary, this would be reinforced if future data turn out to agree with the values implied by our analysis F ( f ' ~ K K ) ~- 0.7F(f ' --. all) -~ 50 MeV. We find this last possibility very likely since all the decay modes of the f ' meson have been carefully analyzed. In particular, the f ' ~ K * K + K * K - * K K r c decay chain, which con- stitutes the most important decay mode after f ' --* K/s has been investigated in detail using the value of 92 quoted in eq. (2) (common to all the fits); one always obtains F(f'---,KKrc)~-14MeV, thus leaving plenty of room for f ' - -*KK.

In conclusion, the whole set of best established data on A2, K**, f ' and f decays has been analyzed in an attempt to uncover the gg contamination in the latter tensor mesons claimed by other authors. Our analysis makes use exclusively of well tested arguments such as the OZI rule and SU(3) symmetry. The whole set of data can be reasonably described in terms of an ideal j v c = 2 + + nonet without gluonium contaminations. Indeed, the accuracy and quality of our simple descrip- tion are compatible with those expected from S U(3) arguments. Moreover, the situation does not improve if one allows for arbitrary gg contaminations. This


con t r a s t s wi th the conc lus ions of p r e v i o u s ana lyses and is a c o n s e q u e n c e of h a v i n g i n t r o d u c e d f ' decay d a t a in o u r fits. F o r tha t r e a s o n a be t t e r e x p e r i m e n t a l k n o w l e d g e of the f ' ( 1 5 2 5 ) m e s o n is desirable .

Acknowledgements. The work has been partially supported by CAICYT. The financial assistance by CIRIT is also acknowledged by one of the authors (A.B.).

References

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Gluonium contaminations in tensor mesons