NLO QCD sum rules analysis of $1^{-+}$ tetraquark states

TL;DR

This study applies next-to-leading order (NLO) QCD sum rules to isovector $1^{-+}$ tetraquark and molecular currents to reassess the existence of light exotic states. By constructing a comprehensive set of compact and molecular four-quark operators and performing a complete NLO operator product expansion with renormalized currents, the authors extract resonance masses using a resonance+continuum ansatz and moment ratios. They find no $1^{-+}$ state near $1.4\,\text{GeV}$; predicted masses lie mostly above $1.7\,\text{GeV}$, while states around $1.6$ and $2.0$--$2.5\,\text{GeV}$ (e.g., $\pi_1(1600)$, $\pi_1(2015)$) remain compatible with tetraquark interpretations. The sizable NLO contributions are essential for reliable mass predictions and improve the stability of the sum rules, supporting the experimental view that $\pi_1(1400)$ is unlikely to exist as a tetraquark or hybrid–tetraquark state.

Abstract

We present an NLO QCD sum rules analysis of $J^{PC}=1^{-+}$ light four-quark states, investigated several compact tetraquark and four-quark molecule states, we obtain $1^{-+}$ light four-quark states masses,. Crucially, we have not find four-quark states with mass $\sim 1.4\,\text{GeV}$, which is the interpreted to be $π_1(1400)$ exotic state in previous leading-order studies. This result do not support the existence of $π_1(1400)$ state, agrees with the current experimental observation.

Figures (10)

• Figure 1: Perturbative Feynman diagrams for the tetraquark currents. Permutation diagrams are omitted, and thick cross vertices denote hybrid-like cancellation terms. For compact tetraquark currents, the first four diagrams are not needed.
• Figure 2: Ratios of condensate terms $\langle O^n\rangle$ to the perturbative term for $\eta_1^{\mu}$. $\tau=0.3~\mathrm{GeV}^{-2}$ is taken.
• Figure 3: Moment and mass curves as functions of $s_0$ obtained from NLO calculation for the current $J_2^{\mu\nu}$.
• Figure 4: Ratio of NLO to LO contributions in the moment $\mathcal{M}^0$. Left: $J_2^{\mu\nu}$; right: $\eta_2^\mu$.
• Figure 5: $J_2^{\mu\nu}$: Mass $m$ as a function of $\tau$. Left: LO result; right: with NLO correction.