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

D. White


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.


Reduction operators; Heavy vector meson decay; Quark pair vacuum lattice arrays; Dimensional reduction; Quark color disengagement

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