Kinetic energy and speed powers vⁿ of a heavy quark inside S wave and P wave heavy-light mesons

The average values of the powers of jq⃗jⁿ ≡ qⁿ and jv⃗jⁿ ≡ vⁿ are commonly used physical quantities, where q⃗ and v⃗ are the three-dimensional momentum and velocity of a quark inside a meson, respectively. For example, the average kinetic energy μ²_π ¼ q² is an important quantity in the heavy quark effective theory, and the expansion of vⁿ is commonly used in the nonrelativistic quantum chromodynamics. In this paper, based on the instantaneous Bethe-Salpeter equation method, we calculate the average values qⁿ and vⁿ (n ¼ 1, 2, 3, 4) of a heavy quark inside S wave and P wave heavy-light mesons. We obtain the kinetic energy μ²_π ¼ 0.455 GeV² for the B meson, which is consistent with the experimental result 0.464 ± 0.076 GeV². For the B_s, D, and D_s mesons, the μ²_π are 0.530, 0.317, and 0.369 GeV², respectively, and v² ¼ 0.0185, 0.0215, 0.121, and 0.140 for B, B_s, D, and D_s, respectively. We obtain some relationships, for example, qⁿ₀−ðmSÞ ≈ qⁿ₁−ðmSÞ, qⁿ₀þðmPÞ ≈ qⁿ₁þ0ðmP⁰Þ > qⁿ₁þðmPÞ ≈ qⁿ₂þðmPÞ, and qⁿðmSÞ < qⁿðmPÞ (m ¼ 1, 2, 3), etc.