Evidence for Color-by-Color Disengagement from the Process of Lepton Production Associated with the $\Psi$-Series and $\Upsilon$-Series Mesons

Abstract

Form factors associated with the $\Psi(1S)$ and $\Up(2S)$ are calculated directly from relevant experimental data in order to verify that they are given by $f_{1} = (1-q_{s}^{2}) = (8/9)$ in the case of the $\Psi$-Series mesons and $f_{2} = (1-q_{c}^{2}) = (5/9)$ in the case of the $\Upsilon$-series mesons, where $q_{s} = -1/3$ represents the charge of the strange $(s)$ quark and $q_{c}= 2/3$ represents the charge of the charm $(c)$ quark. In two recent articles by the author form factors have been shown to represent the fraction of the originally produced quark/anti-quark $(QQ^*)$ state which makes a transition to a $QQ^*$ state of the next lowest mass $\dots ss^*$ in the case of the $\Psi$-Series mesons and $cc^*$ in the case of the $\Upsilon$-series mesons $\dots$ and thus figure prominently into the calculation of the hadronic and leptonic widths of a given meson via the constructs of the Gluon Emission Model (GEM). We undertake to calculate the form factors of the $\Psi(2S)$, the $\Psi(4040)$, the $\Psi(4160)$, and the $\Psi(4415)$ in order to show that $f_{1} = (8/9)$ is representative of all $\Psi$-states listed above, if and only if, it is assumed that one quark color (in the case of the $\Psi(2S)$) or two quark colors (in all other cases) become disengaged from lepton production. A~similar set of calculations is performed as to the $\Upsilon$-series mesons, such illustrating that all three quark colors are functional in lepton production in $\Upsilon(2S)$ decay, fewer than three functional in $\Upsilon(3S)$ and $\Upsilon(4S)$ decay, with likely only one color functioning in $\Upsilon(10860)$ and $\Upsilon(11020)$ decay. For each meson series, then, lepton decay is characterized by the phenomenon of sequential disengagement of quark color from lepton production as a function of increasing mass. In the $\Upsilon(3S)$ and $\Upsilon(4S)$ decay $\dots$ and in the $\Psi(3770)$ decay $\dots$ it is observed that half-integer color contributions are in force $\dots$ possibly leading to a heretofore unobserved quark property: "shade" of color ("light" and "dark").