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Born-Oppenheimer EFT: a unified description of ordinary and exotic quarkonia

Antonio Vairo

TL;DR

The paper develops a Born-Oppenheimer EFT as a unified QCD-based framework for heavy-quark bound states, including conventional quarkonia and exotica such as hybrids, tetraquarks, and pentaquarks, at the soft scale $m_Q v$. By combining non-relativistic QCD expansions with lattice QCD inputs for BO potentials, it formulates and solves coupled Schrödinger equations across multiple BO sectors, enabling predictions of spectra, compositions, and threshold effects. Notable results include the identification of the $\chi_{c1}(3872)$ and the $T_{cc}^+(3875)$ within a multiplet structure, as well as organized pentaquark and hybrid spectra with dynamical threshold mixing. The approach explains why not all possible multiplets appear, highlights the central role of lattice-determined potentials, and points to the need for ultrasoft-scale completion to fully describe exclusive processes.

Abstract

We show how the Born-Oppenheimer effective field theory (BOEFT) provides a unified description of ordinary and exotic quarkonia grounded on the non-relativistic expansions of QCD and supplemented with lattice QCD inputs. We apply BOEFT to tetraquarks, pentaquarks, quarkonium hybrids and to assess threshold effects in the quarkonium spectrum.

Born-Oppenheimer EFT: a unified description of ordinary and exotic quarkonia

TL;DR

The paper develops a Born-Oppenheimer EFT as a unified QCD-based framework for heavy-quark bound states, including conventional quarkonia and exotica such as hybrids, tetraquarks, and pentaquarks, at the soft scale . By combining non-relativistic QCD expansions with lattice QCD inputs for BO potentials, it formulates and solves coupled Schrödinger equations across multiple BO sectors, enabling predictions of spectra, compositions, and threshold effects. Notable results include the identification of the and the within a multiplet structure, as well as organized pentaquark and hybrid spectra with dynamical threshold mixing. The approach explains why not all possible multiplets appear, highlights the central role of lattice-determined potentials, and points to the need for ultrasoft-scale completion to fully describe exclusive processes.

Abstract

We show how the Born-Oppenheimer effective field theory (BOEFT) provides a unified description of ordinary and exotic quarkonia grounded on the non-relativistic expansions of QCD and supplemented with lattice QCD inputs. We apply BOEFT to tetraquarks, pentaquarks, quarkonium hybrids and to assess threshold effects in the quarkonium spectrum.

Paper Structure

This paper contains 7 sections, 11 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction: XYZ states
  2. BOEFT
  3. Tetraquarks: $\chi_{c1}(3872)$ and $T_{cc}^+(3875)$
  4. Pentaquarks
  5. Quarkonium hybrids
  6. Threshold effects in quarkonium
  7. Conclusions

Figures (7)

  • Figure 1: Isospin $I=0$$\Sigma_g^+$, $\Pi_u$ and $\Sigma_u^-$ potentials at short distance $r$ of the static sources. Energy gaps between states are also displayed.
  • Figure 2: Possible behaviours towards the open flavor thresholds of the isospin $I=1$$\Sigma_g^+$, $\Pi_g$ and $\Sigma_u^-$ potentials Berwein:2024ztx.
  • Figure 3: Avoided level crossing between the quarkonium potential with BO quantum number $\Sigma_g^+$ and the first two tetraquark potentials with BO quantum numbers $\Sigma_g^{+\prime}$, $\Sigma_g^{+\prime \prime}$ in the isospin-singlet $I=0$ case Berwein:2024ztx. The adiabatic static energies are labeled $1\Sigma_g^+$, $2\Sigma_g^+$, $3\Sigma_g^+$, $1\Pi_g$ and $2\Pi_g$.
  • Figure 4: Mass spectrum of states in the spin multiplet of $\chi_{c1}(3872)$ (red lines) vs thresholds (black lines) Brambilla:2024thx.
  • Figure 5: Born--Oppenheimer potentials for the lowest-lying pentaquarks in the scenario of Brambilla:2025xma.
  • ...and 2 more figures