# Perturbative Corrections to Heavy Quark-Diquark Symmetry Predictions for Doubly Heavy Baryon Hyperfine Splittings

## TL;DR

This paper analyzes perturbative corrections to heavy quark-diquark symmetry predictions for hyperfine splittings in doubly heavy baryons, showing corrections are small and calculable within NRQCD, with nonperturbative effects further suppressed.

## Contribution

It introduces a new operator in NRQCD that accounts for corrections to hyperfine splittings, refining previous symmetry-based predictions.

## Key findings

- Corrections to hyperfine splittings are a few percent or smaller for charm and bottom systems.
- The new operator enhances understanding of heavy quark interactions within baryons.
- Nonperturbative corrections are suppressed by mbda_{
m QCD}^2/m_Q^2, not mbda_{
m QCD}/m_Q.

## Abstract

Doubly heavy baryons $\left(QQq\right)$ and singly heavy antimesons $\left(\bar{Q}q\right)$ are related by the heavy quark-diquark (HQDQ) symmetry because in the $m_Q \to \infty$ limit, the light degrees of freedom in both the hadrons are expected to be in identical configurations. Hyperfine splittings of the ground states in both systems are nonvanishing at $O(1/m_Q)$ in the heavy quark mass expansion and HQDQ symmetry relates the hyperfine splittings in the two sectors. In this paper, working within the framework of Non-Relativistic QCD (NRQCD), we point out the existence of an operator that couples four heavy quark fields to the chromomagnetic field with a coefficient that is enhanced by a factor from Coulomb exchange. This operator gives a correction to doubly heavy baryon hyperfine splittings that scales as $1/m_Q^2 \times \alpha_S/r$, where $r$ is the separation between the heavy quarks in the diquark. This correction can be calculated analytically in the extreme heavy quark limit in which the potential between the quarks in the diquark is Coulombic. In this limit, the correction is $O(\alpha_s^2/m_Q)$ and comes with a small coefficient. For values of $\alpha_s$ relevant to doubly charm and doubly bottom systems, the correction to the hyperfine splittings in doubly heavy baryons is only a few percent or smaller. We also argue that nonperturbative corrections to the prediction for the hyperfine splittings are suppressed by $\Lambda^2_{\rm QCD}/m_Q^2$ rather than $\Lambda_{\rm QCD}/m_Q$. Corrections should be $\approx 10\%$ in the charm sector and smaller in heavier systems.