Mathematical modeling of the strong coupling parameter based upon the gluon emission model for hadron production associated with vector meson decay

Abstract

F. Close’s Gluon Emission Model (GEM), employed in conjunction with the general form for the absorption cross-section of the gluon involved in hadron production as associated with vector meson decay, has been shown to enable very accurate determinations of the widths of the K* (892) and the J (3097), a reasonably accurate determination of the width of the ϒ (1S), and, by construction, essentially exact determinations of the widths of the ρ (776) and the φ (1019). In determining the theoretical widths of the known vector mesons (and that of the K* (892)) via GEM, it is seen that said widths are proportional to the product of three main factors. The first such factor indicates the very significant electromagnetic component involved in vector meson (and K* (892)) decay, as said first factor involves the fourth power of the quark charge comprising the resonant state just prior to its decay. The second factor, a property of all quantum systems decaying directly to complete dissolution, is proportional to the mass of the resonant state raised to the negative third power. The third factor is the one of primary interest for the present work, as the third main factor involved in calculating the widths of vector mesons via the GEM is the strong coupling parameter, δ5 . As the charges of the quarks making up the ρ (776), the K* (892), the φ (1019), the J (3097), and the ϒ (1S) are all well-known, as are each of their masses, the GEM obviously may be employed to obtain δ s at a number of energies (E) over a wide range, viz., from E ≈ 700Mev ≈ the ρ -meson energy to E ≈ 10, 000 Mev ≈ the ϒ (1S) energy. Notably, we find that over the entire range of energy stated immediately above the strong coupling parameter may be described by one very simple function, viz., δ s ≈ 1.2[1n(E/50Mev)]-1 , interesting for two reasons: (1) the QCD “Scale Factor” (Λ) shows itself as Λ = 50 Mev from considerations of the GEM as evidenced by the behavior of δ s above; such value for Λ is quite small – only about one sixth of the average value of same as determined by the Particle Data Group (PDG) in recent times. Nevertheless, formal gauge theory does consider Λ to be an arbitrary parameter; its usefulness, therefore, no matter the value assumed for it, may be ascertained in the full context in which it is employed. (2) At the low end of the energy scalementioned above (i.e., E ≈ 700 Mev) the strong coupling parameter has a value (as determined via the GEM) of roughly 0.45. Even at the high end (i.e., E ≈ 10, 000 Mev) its associated value is about 0.23. Hence, the entire range of energy overwhich δ s is determined via the GEM cannot be considered as a range of energy over which δ s may be determined accurately via perturbative methods – certainly not any such to only the first order. Yet, the GEM yields a very simple form for the strong coupling parameter, and, as will be shown, it meets the test of experiment to a high degree of accuracy over said range. Beyond the ϒ (1S) energy, δ s as equal to 1.2[1n(E/50Mev)]-1 , becomes no longer viable. A simplemathematical “smoothing” function is derived to “blend” δ s as determined via the GEM in its “low energy range” into its asymptotic, ultra-high energy behavior. We evaluate δ s at the Z -boson mass, such evaluation having become a popular ldquo;standard” over the recent years. We find a statistical match to the experimental findings reported for such in each of two excellent review articles pertaining to the status of the strong coupling parameter – one authored by Michael Schmelling and the other by C. Amsler et al. (available on the PDG’s www pages). Further, we show that unless a given experiment agrees to within 10% of the GEM determination of δ s at a given energy, there is a less than 10% chance that such experiment will yield the presently accepted value of δ s associated with the Z mass.