Analysis of the Form Factors Present in the Description of Heavy Vector Meson Decay

Abstract

In recently published work by the author, in which the decay rates of all known $1S$ vector meson states have been explained via the Gluon Emission Model ({\rm GEM}), form factors, $f_{j}$ , come into play as associated with the descriptions of the decays of the $\Psi (1S)$ and the $\Upsilon (1S)$, where, in the case of the $\Psi (1S)$, $f_{1} = 1 - q_{s}^{2}$ $(q_{s}$ represents the charge of the strange $(s)$ quark in units of the electron charge), and, in the case of the $\Upsilon (1S)$, $f_{2} = 1$. Specifically, $f_{j}$ represents the fraction of relevant given vector meson states which make a point-like transition to a quark/anti-quark structure of the next lesser mass \dots charm/anti-charm $(cc^*) $ to strange/anti-strange $(ss^*)$ in the case of the $\Psi (1S)$ and bottom/anti-bottom $(bb^*)$ to $cc^*$ in the case of the $\Upsilon(1S)$ \dots the latter respective structures either forming the major portion of the decay scheme (the $\Psi(1S)$), or its entirety (the $\Upsilon (1S)$).\ Investigation of $\Psi (NS)$ and $\Upsilon (NS)$ states, with $N>1$, has revealed three highly interesting eventualities: (1)~$f_{1}$ retains its form noted above as associated with all presently known $\Psi (NS)$, while (2)~$f_{2}$ is seen to be unique to the $\Upsilon (1S)$, as for the known $\Upsilon (NS)$ states, the resulting form factor is seen to be $f_{3} = 1 - q_{c}^{2}$, where $q_{c }$ represents the charm quark charge. In addition to the above it appears convincingly that (3) for a respective given ``$N$'' such that $N \ge 2$, quark color disengagement from lepton production takes place. In the work which follows we attempt to represent the above-mentioned form factors in a logically consistent way as stemming from what we term as ``Reduction Operators''. Necessarily, $f_{2}$ is of a slightly different form than that of $f_{1}$ or $f_{3}$; therefore, we posit a logical reason as to the nature of the difference, viz., the $\Upsilon (1S)$ never does find itself as a $bb^*$ construction. Rather, it starts out as and decays as a $cc^*$ construction. In addition, from a detailed look into the situation pertaining to the $\psi (NS)$ decay, we suggest that ``quark color disengagement'' from the decay of the relevant $N \ge 2$ states is consistent what we denote as ``dimensional reduction'', which is seen to involve reflection-invariant arrays of entangled di-quark structures within an assumed cubic lattice arrangement of same. Investigation of the analogous situation pertaining to the $\Upsilon (NS)$ decay suggests, on the other hand, that ``dimensional reduction'' is its own phenomenon. Nevertheless, we attempt to make the case that the two phenomena are intricately tied together.