Soft Contributions Stabilize NNLO QCD Corrections to Quarkonium Production and Decay

Technical Summary: Soft Contributions Stabilize NNLO QCD Corrections to Quarkonium Production and Decay

Problem Statement
The paper addresses a persistent issue in Non-Relativistic QCD (NRQCD) regarding the perturbative stability of Next-to-Next-to-Leading Order (NNLO) corrections for quarkonium production and decay. While NNLO calculations have become feasible for many processes, theoretical predictions frequently exhibit unphysical behavior, such as negative partial decay widths or cross sections, and a significant degradation of perturbative convergence. Specifically, the $\mathcal{O}(\alpha_s^2)$ coefficients in the short-distance coefficients (SDCs) are often large and negative, leading to results that contradict experimental data or exhibit excessive renormalization-scale dependence. This instability has been observed in processes including $\eta_c \to \gamma\gamma$, $J/\psi \to e^+e^-$, and $e^+e^- \to J/\psi J/\psi$.

Methodology and Proposed Solution
The authors identify the origin of this instability as an incomplete treatment of the interplay between SDCs and Long-Distance Matrix Elements (LDMEs) within the standard Modified Minimal Subtraction ($\overline{\text{MS}}$) renormalization scheme. In conventional NRQCD calculations, the $\overline{\text{MS}}$ scheme removes only the infrared (IR) poles ($1/\epsilon$) from the LDMEs, effectively setting the finite soft contributions to the wavefunction at the origin to zero.

The authors propose a remedy inspired by analogous cancellations observed in para-positronium decays and potential NRQCD (pNRQCD) studies. They argue that for S-wave color-singlet states, the perturbative expansion should be reorganized to include the finite soft (non-Coulombic) contributions to the wavefunction at the origin up to $\mathcal{O}(\alpha_s^2)$, while keeping the Coulombic and non-perturbative (NP) parts resummed.

The core of their methodology involves:

1. Redefining the Wavefunction: They derive a fixed-order expansion for the renormalized wavefunction at the origin, $\psi_{s,n}(0)$, which includes specific $\mathcal{O}(\alpha_s^2)$ soft terms (Eq. 2). These terms depend on the NRQCD factorization scale $\mu_\Lambda$, the principal quantum number $n$, and the spin $s$.
2. Cancellation Mechanism: By incorporating these soft contributions into the LDMEs, the large positive terms generated by the wavefunction renormalization are designed to cancel the large negative $\mathcal{O}(\alpha_s^2)$ coefficients found in the SDCs.
3. Scale Independence: This approach ensures that the $\mu_\Lambda$ dependence in the SDCs is exactly canceled at NNLO, eliminating the need for Renormalization Group (RG) evolution of the LDMEs at this order and removing the associated scale uncertainty.

Key Results
The authors applied this framework to nine physical observables measured experimentally, including decay widths ($\eta_c \to gg, \eta_c \to \gamma\gamma, \eta_b \to gg, J/\psi \to \gamma\gamma\gamma, J/\psi \to e^+e^-, \Upsilon(1S) \to e^+e^-$) and exclusive production cross sections ($e^+e^- \to J/\psi \eta_c, e^+e^- \to J/\psi J/\psi$).

• Stabilization of Perturbative Series: The inclusion of soft contributions transforms the NNLO predictions from unphysical (negative values) to physical (positive definite) for all examined processes.
• Improved Convergence: The magnitude of the $\mathcal{O}(\alpha_s^2)$ coefficients is significantly reduced. For example, in $\eta_c \to \gamma\gamma$, the coefficient shifts from $-11.16$ (SDC only) to $+5.17$ after cancellation, resulting in an alternating-sign series behavior rather than a rapidly growing one.
• Reduced Uncertainties: The renormalization-scale uncertainty is substantially reduced (by a factor of 2–3 in some cases) compared to both NLO and the standard NNLO (SDC-only) predictions. The NNLO bands are fully contained within the NLO bands, indicating improved stability.
• Agreement with Data: The revised NNLO predictions show markedly better agreement with experimental data (e.g., BaBar measurements for $\gamma\gamma^* \to \eta_c$ and PDG values for decay widths) compared to previous NNLO calculations.
• Exception: The process $e^+e^- \to J/\psi \eta_c$ remains sensitive to large logarithms ($\log(s/m_c^2)$), which are not addressed by this specific soft-contribution mechanism and likely require separate resummation.

Significance and Claims
The paper claims that the perturbative instability in NNLO quarkonium calculations is not an intrinsic failure of NRQCD but rather a consequence of an suboptimal renormalization scheme that neglects finite soft contributions. By explicitly including these contributions in the LDMEs, the authors demonstrate that:

1. The unphysical negative cross sections and widths are resolved.
2. The theoretical predictions achieve a level of perturbative convergence and scale stability consistent with expectations for a well-behaved effective field theory.
3. The approach provides a simple, systematic remedy that can be extended to S-wave color-octet and P-wave states, provided the corresponding generalizations of the wavefunction expansion are available.

The authors emphasize that their results rely on the combination of SDCs and LDMEs being the physical observable, and that their specific treatment of the wavefunction at the origin restores the expected cancellations between short- and long-distance physics.