Average speed and its powers vn of a heavy quark in quarkonia

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 le and <(vertical bar q vertical bar(n))over bar> and <(vertical bar v vertical bar(n))over bar> (math) v(n) of a heavy quark in a S wave or P wave quarkonium rest frame, where q and 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 Upsilon(1S), v(Upsilon(1s)) = 0.24, v(Upsilon(1s))(2) = 0.072, v(Upsilon)(3) = 0.025, and v(Upsilon(1s))(4) = 0.010. The values indicate that v(n) > v(n1) . v(n2), 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.