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.

13 pages, 4 tables

References in corpus (14)

• Heavy quarkonium: progress, puzzles, and opportunities
• Improved determination of color-singlet nonrelativistic QCD matrix elements for S-wave charmonium
• Relativistic corrections to $J/\psi$ exclusive and inclusive double charm production at B factories
• Potential-model calculation of an order-v^2 NRQCD matrix element
• Spectrum for Heavy Quankonia and Mixture of the Relevant Wave Functions within the Framework of Bethe-Salpeter Equation
• Decay constants of $P$-wave mesons
• NRQCD matrix elements for S-wave bottomonia and Gamma[eta_b(nS) -> gamma gamma] with relativistic corrections
• Relativistic effects in the production of pseudoscalar and vector doubly heavy mesons from e^+e^- annihilation
• $O({\alpha }_s{v}^2)$ Corrections to Hadronic and Electromagnetic Decays of ${}^1{S}_0$ Heavy Quarkonium
• Next-to-next-to-leading-order QCD corrections to hadronic width of pseudoscalar quarkonium
• Leading twist distribution amplitudes of P-wave nonrelativistic mesons
• Relativistic corrections to light-cone distribution amplitudes of S-wave $B_c$ mesons and heavy quarkonia
• Relativistic description of the double charmonium production in $e^{+}{e}^{-}$ annihilation
• Leading-twist light cone distribution amplitudes for p-wave heavy quarkonium states

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