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In this work, we calculate the nonrelativistic asymptotic behavior of theamplitudes of $q\overline{q}\rightarrow 2g \rightarrow q\overline{q}$ in theleading order of $\alpha_s$ (LO-$\alpha_s$) with $q\overline{q}$ in the $^3P_J$channels. In the practical calculation we take the momenta of quarks andantiquarks on-shell and expand the amplitudes on the three-momentum of thequarks and antiquarks to order 6 and get three nonzero terms. The imaginaryparts of the first term and the second term are the old. The real parts of theresults have IR divergence. When applying the results to the heavy quarkonia,the corresponding amplitude of $q\overline{q}\rightarrow1g\rightarrowq\overline{q}$ with $q\overline{q}$ in the color octet $^3S_1$ channel isconsidered to absorb the IR divergence in a unitary way in the leading order of$v$ (LO-$v$). The finial results can be used to estimate the mass shifts of the$^3P_J$ heavy quarkonia due to the effect of two-gluon annihilation. Thenumerical estimation shows that the contributions to the mass shifts of$\chi_{c0,c1,c2}$ are about $1.23\sim1.58$ MeV, $1.57\sim1.86$ MeV and$5.92\sim5.45$ MeV when taking $\alpha_s\approx 0.25\sim0.35$.

Mass shifts of $^3P_J$ heavy quarkonia due to the effect of two-gluon annihilation