Revising the Mass of Light Hybrid Mesons: NLO QCD Sum Rules Point to $φ(2170)$ as a Prime Candidate

TL;DR

The paper addresses the mass of the light vector hybrid meson with $J^{PC}=1^{--}$ by performing a comprehensive next-to-leading order (NLO) QCD sum-rule analysis that includes condensates up to dimension-8. Using both Laplace sum rules (LSR) and Gaussian sum rules (GSR), the authors show that NLO corrections substantially reduce the mass predictions from LO estimates, yielding $m obreak\approx obreak 2.25-2.31$ GeV for the light strange and non-strange configurations and confirming a consistent picture across methods. The results, $m_{sar{s}g} obreak oughly obreak 2.31 ext{ GeV}$ and $m_{uar{u}g} obreak oughly obreak 2.25 ext{ GeV}$ with uncertainties of about $ obreak\pm 0.23$ GeV, align with lattice QCD ranges and flux-tube models, and place the $ ext{φ}(2170)$ resonance as a prime candidate for the light vector hybrid meson (or a state with a large hybrid component). The work also demonstrates that vector-meson mixing has a minor impact on the mass, while the OPE shows good convergence, and it provides robust cross-checks via two-resonance modeling and Gaussian fits. Overall, the study resolves a long-standing discrepancy between LO sum-rule predictions and other approaches, highlighting the importance of NLO and higher-dimension condensates in.hybrid spectroscopy.

Abstract

We present a comprehensive next-to-leading order (NLO) QCD sum rule analysis for light hybrid mesons with $J^{PC}=1^{--}$, incorporating condensates up to dimension-8 and NLO corrections to the perturbative, gluon condensate, and four-quark condensate contributions. These corrections are found to be substantial and reveal the necessity of contributions beyond leading order. Employing both Laplace (LSR) and Gaussian (GSR) sum rules, our analysis predicts a mass in the conservative range of $2.1-2.4\,\text{GeV}$ for the light $1^{--}$ hybrid. These predictions are significantly lower than previous leading-order (LO) estimates (around $2.9\,\text{GeV}$) and bridge the gap between QCD sum rules and other approaches. Our findings establish the $φ(2170)$ resonance as a prime candidate for the light vector hybrid meson.

Figures (20)

• Figure 1: Diagrams of perturbative contributions; the crosses denote the counterterms.
• Figure 2: Diagrams of $\langle GG\rangle$ contributions; the crosses denote the counterterms.
• Figure 3: Diagrams of $\langle qq\rangle^2$ contributions.
• Figure 4: Diagrams involved in hybrid operator renormalization at $O(g^2)$.
• Figure 5: Diagrams corresponding to $m\langle \bar{q}q\rangle$ and $\langle GGG\rangle$ contributions; eq.\ref{['con_d6_2g']} is used for $\langle GGG\rangle$ contribution.