5.1 VMD Contribution

In conventional VMD models, the amplitude for the process is obtained by calculating the Feynman diagrams shown in Fig.(7).

 
Figure 7: Feynman diagrams for the process from Vector Meson contributions  

All the couplings of our amplitudes can be deduced from the two previously presented lagrangians, eqs. (36,71), with . In addition to the couplings (38), which satisfactorily describe the set of data for , for processes like , where or , one needs to introduce a small contamination of non-strange quarks in the meson. The relative amount can be deduced from the experimental [13] decay width =5.80.6 KeV, which can be reproduced by VMD with

where the sign comes from observed interference effects in [13].

With our lagrangians and the quoted values for the coupling constants, let us now turn to predict the intermediate vector-meson contributions to . >From the kinematical point of view these processes involve the two following amplitudes

where are the polarizations of the final photon (initial vector meson), and q () are the corresponding four-momenta; P=p+q, are those for the virtual (intermediate) vector mesons (V and ) of the direct and crossed terms (see Fig.(7)). The total VMD amplitude is then found to be

where is the coupling constant and is the decaying vector meson. The intermediate ones, V and , can be either the or the -mesons, with in and in -decays; for one obviously has and . The coefficient C is the same for both terms (using SU(3)-symmetric couplings) and changes from process to process according to well-known quark-model or nonet-symmetry rules:

and

for the -decays where the Zweig-rule is operative.

>From the above amplitudes, the partial widths are obtained performing a numerical integration of

where the factor (1/2) has to be included only for decays.

Our results using the full VMD amplitudes (51) and the three-body phase space formulae (54) are shown in the two last columns of Table 1. For comparison we also include (first column) the upper limit for the three experimentally studied decay rates [13,20] and the predictions of other authors [22,23,24] who have worked in our same context. Our results are not incompatible with those by Singer [25], who first noticed the simple relation for the VMD part of the rate. This relation allows for a comparison of our results with those by Renard [23], quoted (in parenthesis) in the second column of Table 1. The accompanying values are the original ones [23] corrected by the present-day data for and , and turn out to be in excellent agreement with our predictions. The agreement with ref.[24] is somewhat less satisfactory. Finally, we disagree in the complete list of numerical predictions quoted in ref.[22] even if the initial expressions for the lagrangians are the same, since that our coupling constant g has been defined as of that in ref.[22].

 
Table 1:   Global contribution of intermediate vector mesons to decay rates (in eV) and branching ratios (last column) for different transitions as predicted by several authors. Experimental upper limits are also quoted.

Notice that the branching ratios (BR) appearing in the last column of Table 1, do not always coincide with the simple product of branching ratios for the individual vector dominated diagrams of Fig.(7). This point has been discussed in detail in ref.[26]

Concentrating on -decays one first observes that our vector-meson dominated mechanism predicts a completely negligible , contrasting with the four orders of magnitude larger prediction from ref.[22]. We have carefully analyzed our calculation and, for this channel containing exclusively soft photons ( MeV), an analytic expression for the amplitude in this low-E limit has been obtained. One has

where p, are the pseudoscalar four-momenta, and

accidentally containing the small numerical factor [13]. In other words, the decay is predicted to be exceptionally suppressed not only by the obviously scarce available phase-space but also due to an almost complete destructive interference in the amplitude, as explicitly shown in Fig.(8). (Reversing the sign of the interference term would enlarge the width by 2 orders of magnitude).

 
Figure 8: Photonic spectrum generated by intermediate vector-mesons in (solid line). Dashed and dotted lines correspond to twice the contribution of a single diagram and their interference, respectively.  

Our mechanism also predicts sizable contributions to and decays. The corresponding photonic spectra are shown in Figs.(9) and (10), where the interference effects have again been separated.

 
Figure 9: Photonic spectrum in with conventions as in Fig.(8).  

 
Figure 10: Photonic spectrum in . Dashed and dot-dashed lines are the contributions of each diagram and the dotted line corresponds to their interference.  

These interference effects contribute to enhance the peak at high E in the spectrum. Roughly one-half of this decay contains a photon with an energy E in the narrow range 400 MeV MeV. Alternative mechanisms, such as are expected to produce exclusively lower energy photons, thus minimizing the interferences and allowing for separated analyses, particularly in . We also notice that our predictions for this (and ) decay include events with the and invariant mass on the -peak. As discussed in [26] for the case of , our calculation implicitly contains simpler estimates in terms of two-body branching ratios. The latter imply [13] and =, only in marginal agreement with our tabulated results because important interference and off-mass-shell effects have been neglected. We believe that our calculation should be prefered to these simpler estimate. However, these rough estimates are useful and allow for a numerical check of our predictions. Indeed, by artificially reducing the -width in our complete (three-body) calculation one recovers the expected agreement with the above simpler (two-body) estimates discussed at the end of section 3 of [26].