GEM and the $\Upsilon(1S)$

D. White


The Gluon Emission Model (GEM), first proposed by F. Close in 1979, has been shown to serve very nicely as a basis for calculations of not only the widths of the $\rho$ meson, the $\phi$ meson, the $K^*$(892), and the $J$ meson, but also for the determination of the strong coupling parameter, $\alpha_s$, over essentially the entire range of experimentally reachable energy, leading to an evaluation of $\alpha_s$ at the $Z$ boson energy of $0.121 \pm0.003$. The GEM has built into its framework two precepts of prime importance for the carrying out of the above types of calculations:\ (1)~the specification of a quark spin-flip matrix element as the central determinant of a vector meson resonance and (2)~the virtual photon and the gluon as two aspects of the same entity, viz., the four-momentum propagator. The
prime significance of (1) is that the square of the quark spin-flip matrix elements in vector meson width calculations are proportional to $q_i^4$, where $q_i$ represents the magnitude of the charge of quark type ``$i$''. The significance of (2)~is that the virtual photon and the gluon essentially obtain their identities from what the vertices of origin and termination are in the relevant Feynman Diagram. Close, as a point of fact, represents
the virtual photon as transmuting into a gluon $\ldots$ and vice versa $\ldots$ where necessary, all transmutation couplings being of magnitude,~1. The ramifications of (1)~are that, as $(2/3)^4$ is 16 times $(1/3)^4$, it is quite easy to determine that the
$cc^*$ (charm -- anti-charm) structure of the
$J(3097)$ must transmute to an $ss^*$ (strange --- anti-strange) in point-like manner, such that it is the $ss^*$ structure that undergoes the spin-flip at the
$J(3097)$ resonance. Likewise, the $\Upsilon(1S)$ must transmute in point-like manner from its original
$bb^*$ (bottom -- anti-bottom) structure to a $cc^*$ structure before decaying. The ramification of
(2)~is that the leptonic width to hadronic width ratio associated with the same basic decaying structure must be in the ratio of $\alpha$ to $\alpha_s$, where $\alpha$ represents the fine structure constant $= (1/137.036)$.

At the present juncture in the literature is found that the GEM predicts the hadronic width of the $\Upsilon(1S)$ to be $\sim$41\,{\rm Kev}, whereas the figure for same as stated in the 2008 Meson Table from the Particle Data Group (PDG) is $\sim$50\,{\rm Kev}. The discrepancy noted above (23\%) is extremely important, because, if we were to assume that the GEM was in error by such amount, it turns out that all other GEM calculations, currently essentially exactly on the mark as to the $\rho$, the $\phi$, the $K^*$(892), the $J$, and $\alpha_s$ at the $Z$ mass, would have to be rendered as 23\% too large by bringing the GEM's determination of the $\Upsilon(1S)$ in line with the PDG's determination of same through adjustment of the GEM's determination of $\alpha_s$. Hence, in order to make the GEM as currently constructed fit the PDG as to the hadronic width of the $\Upsilon(1S)$, all other GEM calculations would be discrepant by the same amount, i.e., 23\%, at each diverse point of the energy spectrum where  the GEM has been successfully applied. Clearly, then, what must be addressed in the present work are the details in the GEM's determination of the width of the $\Upsilon(1S)$, with an eye towards any reasonable modifications that might remove the above-mentioned disparity. Unlike the theoretical structures prevalent in the literature that one encounters as to determining the width of the $\Upsilon(1S)$, the GEM theory is about as simple as it gets: One fundamental process is posited for the formation and decay of any spin one meson, i.e., a quark spin-flip; the gluon absorption cross-section for said process is then integrated over energy, and from there, the Feynman Diagram resulting in hadron or lepton pairs is then calculated. We review the development of the GEM and its applications, from its beginnings in 1979 through 2009 $\ldots$ including the 23\% disparity noted above.\ We then postulate the existence of an additional process involved in the decay of the $\Upsilon(1S)$ $\ldots$ one not assumed to be extant in the other, less massive vector mesons to which the GEM has been successfully applied. We find that the ``additional route of decay'' removes completely the noted disparity without affecting the GEM in its other applications. Finally, the GEM ansatz as presented herein is applied to the $\Upsilon(2S)$ with noteable success.


Gluon emission model; $\Upsilon(1S)$; Quark spin-flip matrix element; Four-Momentum propagator

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