Estimate of $\Gamma(H \to gg)$ at $O(\alpha_s^7)$ from Padé approximants with constraints from large-$n_f$

Technical Summary of "Estimate of $\Gamma(H \to gg)$ at $O(\alpha_s^7)$ from Padé approximants with constraints from large-$n_f$"

Problem Statement

Precision measurements of Higgs boson decays at the LHC, the High-Luminosity LHC, and future facilities like the FCC-ee are critical for probing physics beyond the Standard Model. However, theoretical predictions for hadronic Higgs decays, particularly $H \to gg$ and $H \to b\bar{b}$, face two primary challenges: the need for reduced uncertainties in QCD parameters ($\alpha_s$ and quark masses) and the necessity of computing decay widths at higher orders in perturbative QCD (pQCD). While some observables are known up to five loops, calculations at even higher orders are computationally prohibitive. Consequently, there is a demand for reliable, model-independent estimates of the first unknown higher-order coefficients to assess missing higher-order (MHO) uncertainties and residual renormalization-scale dependence.

This work addresses the estimation of the first unknown QCD correction to the decay of the Higgs boson into gluons, $\Gamma(H \to gg)$, which corresponds to the order $O(\alpha_s^7)$ (N5LO) in the heavy-top limit. Additionally, the authors provide updated estimates for the first unknown coefficients of the QCD $\beta$ function, the QCD decoupling relation, the gluonium correlator, and the $H \to b\bar{b}$ decay width.

Methodology

The authors employ Padé approximants (PAs) and D-log Padé approximants, constructed from both the perturbative series in $\alpha_s$ and their Borel transforms (Padé-Borel method). The core strategy involves the following steps:

1. Decomposition of the Decay Rate: Instead of applying approximants directly to the decay rate $\Gamma(H \to gg)$, the authors decompose the problem into its constituent parts: the imaginary part of the gluonium correlator ($\text{Im}\,\Pi_{G^2}$), the Wilson coefficient $C_1$ (derived from the decoupling relation $\zeta_g^2$), and the QCD $\beta$ function. The decay rate is reconstructed using the low-energy theorem and renormalization group (RG) equations. This approach allows for predictions valid for arbitrary renormalization scales and top-quark mass schemes.
2. Large-$n_f$ Constraints: A distinguishing feature of this work is the systematic use of exact large-$n_f$ results (where $n_f$ is the number of light-quark flavors) as constraints. The authors demonstrate that the highest power of $n_f$ in the perturbative coefficients of $\Gamma(H \to gg)$ is entirely driven by the gluonium correlator, for which large-$n_f$ results are known to all orders.
3. Approximant Selection and Filtering: The authors construct various PAs and D-log PAs for $n_f = 1, \dots, 9$. They apply rigorous filtering criteria to discard unreliable approximants:
   • Exclusion of approximants exhibiting Froissart doublets (poles close to zeros).
   • Exclusion of approximants with real poles far from the origin that contribute negligibly to the predicted coefficient.
   • Exclusion of approximants with spurious complex poles near the origin.
   • Large-$n_f$ Consistency Test: A least-squares fit is performed to determine the polynomial dependence of the predicted coefficients on $n_f$. Approximants whose predicted coefficient for the highest power of $n_f$ deviates significantly from the exact large-$n_f$ result are discarded.
4. Validation: The method is validated through "postdiction"—predicting the last known coefficient of each series (e.g., $g_{4,1}$, $d_{4,0}$, $\beta_5$, $p_4$) using only lower-order terms. The authors show that their estimates generally reproduce exact results within quoted uncertainties, provided sufficient non-trivial coefficients are available as input.

Key Contributions and Results

1. Estimation of $\Gamma(H \to gg)$ at $O(\alpha_s^7)$:
The primary result is the model-independent estimate for the first unknown coefficient of the decay width, $\gamma_5$, in the $\overline{\text{MS}}$ scheme with $\mu = m_H$:
$$\gamma_5(5) = -312 \pm 15$$
This coefficient is derived from the estimated fifth-order coefficient of the gluonium correlator, $g_{5,1}(5) = -187 \pm 14$, the decoupling constant $d_{5,0}(5) = -8 \pm 11$, and the six-loop $\beta$-function coefficient $\beta_6(5) = 264 \pm 226$.

2. Impact on Scale Dependence:
Incorporating the estimated $\gamma_5$ into the perturbative expansion of $\Gamma(H \to gg)$ significantly improves the stability of the prediction. The residual renormalization-scale dependence (varying $\mu$ in the range $m_H/2 \leq \mu \leq 2m_H$) is reduced by a factor of almost 1.5 compared to the result truncated at N4LO ($O(\alpha_s^6)$).

3. Updated Estimates for Other Quantities:

• Wilson Coefficient $C_1$: The first unknown coefficient is estimated as $c_5(5) = 985 \pm 677$.
• $H \to b\bar{b}$ Decay: The first unknown coefficient of the imaginary part of the scalar correlator is updated to $p_5(5) = -27.0 \pm 0.5$, showing excellent agreement with previous estimates but with reduced uncertainty.
• $\beta$-function: The sixth-order coefficient is estimated as $\beta_6(5) = 264 \pm 226$.

4. Validation of the Strategy:
The postdiction of the last known coefficient $g_{4,1}$ yielded $g_{4,1}(5) = -73 \pm 20$, which agrees with the exact value ($-75.26$) within 3%. Similarly, the postdiction of $\beta_5$ and $p_4$ showed consistency with exact results within uncertainties. The authors note that the method is less effective for the decoupling constant $d_{4,0}$ (and consequently $c_4$) due to the scarcity of available input coefficients (only two non-trivial terms), highlighting the importance of having at least three known coefficients for reliable predictions.

Significance and Claims

The authors claim that their approach provides a model-independent estimate of unknown higher-order corrections. Unlike previous works that relied on the large-$\beta_0$ limit to guide approximations (which introduces a degree of model dependence), this work relies strictly on exact large-$n_f$ results as constraints.

The significance of the work lies in:

• Reducing Theoretical Uncertainty: By providing a concrete estimate for the N5LO coefficient, the authors reduce the truncation uncertainty and the associated scale dependence of the $H \to gg$ decay width.
• Future Readiness: The authors note that while current uncertainties are dominated by $\alpha_s$, future facilities like the FCC-ee aim for a 0.1% precision on $\alpha_s(m_Z)$ and a 4 MeV precision on $m_H$. At this level, the uncertainty from missing higher orders (measured by residual scale variation) would become comparable to parametric uncertainties. Therefore, such estimates are essential for assessing the theoretical error budget in advance of exact multi-loop calculations.
• Methodological Robustness: The paper demonstrates that combining Padé approximants with exact large-$n_f$ constraints is a reliable tool for estimating missing higher-order corrections in pQCD series where sufficient information (at least three non-trivial coefficients) is available.

The authors conclude that their results are consistent with other estimates in the literature (e.g., Asymptotic Padé Approximants) and that the method serves as a useful tool for phenomenology and error assessment in the absence of direct higher-order computations.