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

## TL;DR

This paper calculates the mass shifts of $^3P_J$ heavy quarkonia caused by two-gluon annihilation effects, providing numerical estimates for different states and coupling constants.

## Contribution

It introduces a method to estimate heavy quarkonia mass shifts due to two-gluon annihilation, including IR divergence handling in a nonrelativistic QCD framework.

## Key findings

- Mass shifts for $ ext{chi}_{c0,c1,c2}$ are approximately 1.23-1.58 MeV, 1.57-1.86 MeV, and 5.92-5.45 MeV.
- Results depend on the strong coupling constant $eta_s$ in the range 0.25 to 0.35.
- The approach effectively absorbs IR divergences through color octet channel considerations.

## Abstract

In this work, we calculate the nonrelativistic asymptotic behavior of the amplitudes of $q\overline{q}\rightarrow 2g \rightarrow q\overline{q}$ in the leading 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 and antiquarks on-shell and expand the amplitudes on the three-momentum of the quarks and antiquarks to order 6 and get three nonzero terms. The imaginary parts of the first term and the second term are the old. The real parts of the results have IR divergence. When applying the results to the heavy quarkonia, the corresponding amplitude of $q\overline{q}\rightarrow1g\rightarrow q\overline{q}$ with $q\overline{q}$ in the color octet $^3S_1$ channel is considered 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. The numerical 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 $\alpha_s\approx 0.25\sim0.35$.