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.
Highlights
The first doubly charm baryon Ξþccþ with mass 3621.40 Æ 0.72 Æ 0.27 Æ 0.14 MeV was recently observed by the LHCb Collaboration in the exclusive decay modes Ξþccþ → Λþc K−πþπþ and Ξþccþ → Ξþc πþ [1,2]
Even though the SELEX Collaboration [3,4,5] had earlier reported the observation of doubly charmed baryons years ago, those observations were not confirmed by other experiments such as FOCUS [6], Belle [7,8], and BABAR [9]
The recent experimental observation of the Ξþccþ baryon has greatly revived the interest in the physics of doubly heavy baryons
Summary
The first doubly charm baryon Ξþccþ with mass 3621.40 Æ 0.72 Æ 0.27 Æ 0.14 MeV was recently observed by the LHCb Collaboration in the exclusive decay modes Ξþccþ → Λþc K−πþπþ and Ξþccþ → Ξþc πþ [1,2]. One of the implications of the HQDQ symmetry is the relation between the hyperfine mass splittings of the doubly heavy baryons and heavy antimesons. The chromomagnetic interactions of the diquark and quark are responsible for the hyperfine splittings in the doubly heavy baryons and antimesons. The heavy quark-diquark symmetry implies the relation between the hyperfine splittings is [25,26,27]. The finite size effects were due to operators coupling the light quarks and the diquarks that contribute to the mass of the double heavy baryon. In the body of this paper we review the effective action for heavy diquark fields, introduce the operator, and compute its effect on the prediction for doubly heavy baryon hyperfine splittings. QCD diagrams for QQg → QQ scattering onto NRQCD to Oð1=m2QÞ
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