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

Technical Summary: Revising the Mass of Light Hybrid Mesons via NLO QCD Sum Rules

Problem Statement
The nature and mass of light hybrid mesons ($q\bar{q}g$), particularly those with conventional quantum numbers $J^{PC} = 1^{--}$, have long been a subject of theoretical inconsistency. While flux-tube models and lattice QCD simulations predict masses in the range of 1.8–2.5 GeV, previous Leading Order (LO) QCD sum rule calculations predicted significantly heavier states, approximately 2.9 GeV. This discrepancy creates a conflict with experimental observations of the $\phi(2170)$ resonance, which has a mass of $\approx 2.16$ GeV and decays into hidden strangeness channels, suggesting a strangeonium-like hybrid nature. The central problem addressed is whether the LO sum rule predictions can be reconciled with experimental data and other theoretical approaches by incorporating higher-order corrections.

Methodology
The authors perform a comprehensive Next-to-Leading Order (NLO) QCD sum rule analysis for the $1^{--}$ hybrid correlator. The methodology involves several key technical components:

1. Correlator Construction: The analysis utilizes an interpolating current $J_\mu = g\Psi^T \sigma_{\mu\nu} \tilde{G}^{\mu\nu} \gamma_5 \Psi$ to describe the hybrid state. The two-point correlation function is expanded using the Operator Product Expansion (OPE) up to dimension-8 condensates.
2. NLO Corrections: Unlike previous LO studies, this work calculates NLO corrections for the perturbative contribution, the gluon condensate $\langle G^2 \rangle$, and the four-quark condensate $\langle \bar{q}q \rangle^2$.
3. Renormalization: A critical part of the formalism involves the renormalization of the generic hybrid current at $O(g^2)$. The authors derive counterterms to handle ultraviolet divergences and ensure the cancellation of infrared (IR) poles, specifically addressing the cancellation of $\log/\epsilon$ poles in the diagrams involving the gluon condensate.
4. Sum Rule Techniques: Two distinct sum rule methods are employed to extract the mass:
   • Laplace Sum Rules (LSR): Applying the Borel transformation to suppress higher states and continuum contributions. The analysis considers both single-resonance and two-resonance (hybrid + vector meson) spectral ansätze.
   • Gaussian Sum Rules (GSR): Utilizing a Gaussian kernel to probe the spectral function more locally, allowing for a fit of the resonance mass and coupling strength while minimizing bias from the continuum threshold $s_0$.
5. Vector Meson Mixing: To address potential contamination from conventional vector mesons (like $\rho$ and $\phi$), the authors calculate the off-diagonal correlator between the vector meson current and the hybrid current to estimate the mixing coupling $f_1$.
6. Error Analysis: A Monte Carlo uncertainty analysis is performed, varying input QCD parameters (quark masses, condensates) and testing the robustness of results against deviations from the vacuum saturation hypothesis for dimension-6 and dimension-8 four-quark condensates (introducing deviation factors $\kappa_6$ and $\kappa_8$).

Key Contributions

• First NLO Calculation: This paper presents the first NLO QCD sum rule calculation for light $1^{--}$ hybrid mesons, explicitly including corrections to the perturbative, $\langle G^2 \rangle$, and $\langle \bar{q}q \rangle^2$ terms.
• Renormalization Formalism: The authors provide a detailed derivation of the renormalization constants and counterterms required for the hybrid current, resolving specific issues regarding IR pole cancellation and gauge invariance in the context of hybrid operators.
• Resolution of Discrepancy: The study demonstrates that the inclusion of NLO corrections is not a minor adjustment but a substantial revision that fundamentally alters the mass prediction, bridging the gap between QCD sum rules and other theoretical frameworks.

Results
The NLO analysis yields mass predictions significantly lower than previous LO estimates:

• Strange Hybrid ($s\bar{s}g$): $2.31 \pm 0.23$ GeV.
• Non-strange Hybrid ($u\bar{u}g$): $2.25 \pm 0.23$ GeV.

These results are derived consistently across both LSR and GSR methods. The inclusion of NLO corrections reduces the predicted mass by approximately 0.7 GeV compared to LO results. The analysis confirms that the NLO perturbative contribution is substantial, amounting to roughly 70% of the LO contribution, indicating slow convergence of the perturbative series at LO.

Regarding the vector meson influence, the coupling strength between the vector meson and the hybrid current is found to be small ($f_1 \approx 0.018 - 0.0195$ GeV$^4$), and the ratio of coupling strengths $f_1^2/f_2^2$ is approximately $1/200$. This suggests that the vector meson contribution to the spectral function is negligible, validating the single-resonance approximation used in the mass extraction.

Furthermore, the results remain robust under variations of the factorization deviation factors ($\kappa_6$ and $\kappa_8$). Even with large deviations, the predicted masses remain in the 2.2–2.4 GeV range, consistent with the baseline NLO results and significantly lower than LO predictions.

Significance and Claims
The paper claims that its findings resolve a long-standing tension between QCD sum rule predictions and other theoretical approaches (flux-tube models and lattice QCD) regarding the mass of light hybrid mesons. By establishing a mass range of 2.1–2.4 GeV, the authors assert that their results are compatible with the experimentally observed $\phi(2170)$ resonance.

The primary conclusion is that the $\phi(2170)$ is a prime candidate for the light vector hybrid meson, or at least a state with a large hybrid component. The study emphasizes that previous LO estimates were insufficient due to the substantial nature of NLO corrections, and that a comprehensive NLO analysis is necessary for accurate mass predictions in this sector. The work does not propose new experimental searches but rather reinterprets existing experimental data ($\phi(2170)$) through a refined theoretical lens.