Average speed and its powers $v^n$ of a heavy quark in quarkonia

Abstract

The typical velocity of a heavy quark in a quarkonium is a widely used quantity, in this paper, based on the relativistic Bethe-Salpeter equation method, we calculate the average values $\overline{|\mathbf{q}{|}^n}$ and $\overline{|\mathbf{v}{|}^n}\equiv v^n$ of a heavy quark in a $S$ wave or $P$ wave quarkonium rest frame, where $\mathbf{q}$ and $\mathbf{v}$ are the three dimensional momentum and velocity, $n=1,2,3,4$. For a charm quark in $J/\psi$, we obtained $v_{J/\psi }=0.46$, $v_{J/\psi }^2=0.26$, $v_{J/\psi }^3=0.18$, and $v_{J/\psi }^4=0.14$, for a bottom quark in ${\rm Y}(1S)$, $v_{{\rm Y}(1S)}=0.24$, $v_{{\rm Y}(1S)}^2=0.072$, $v_{{\rm Y}(1S)}^3=0.025$, and $v_{{\rm Y}(1S)}^4=0.010$. The values indicate that $v^n>v^{n_1}\cdot v^{n_2}$, where $n_1+n_2=n$, which is correct for all the charmonia and bottomonia. Our results also show the poor convergence if we make the {speed} expansion in charmonium system, but good for bottomonium. Based on the $v^n$ values and the following obtained relations $v_{4S}^n>v_{3S}^n>v_{2S}^n>v_{1S}^n$, $v_{4P}^n>v_{3P}^n>v_{2P}^n>v_{1P}^n$ and $v_{mP}^n>v_{mS}^n$ ($n,m=1,2,3,4$), we conclude that highly excited quarkonia have larger relativistic corrections than those of the corresponding low excited and ground states, and there are large relativistic corrections in charmonium system.

arXiv comment13 pages, 4 tables

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