The calculations address the low-energy failure of conventional perturbative QCD, where perturbative series coefficients grow factorially at high orders and the usual running coupling has Landau singularities at low spacelike momentum. The pole-mass series is affected especially by the infrared renormalon at Borel parameter u = 1/2, which makes the mass ambiguous at order O(ΛQCD). Consequently, many precision analyses use short-distance or renormalon-subtracted mass definitions rather than the pole mass.

Ayala and Cvetič instead used a previously developed resummation method for spacelike and timelike QCD observables. It reorganizes a divergent perturbation series as an integral involving a characteristic function and a running coupling. The method uses the known first four perturbative coefficients of the relation between the pole mass and the MS-bar mass, together with the known Borel-plane structure associated with the u = 1/2 and u = −1 renormalons. A modified Borel transform is parameterized with four quantities fixed by those expansion coefficients, and its inverse Mellin transform supplies the characteristic function used in the resummation.

To avoid ambiguities from Landau poles, the authors used the 3δAQCD spacelike coupling, which is analytic in the complex Q² plane except on the negative timelike axis beyond a threshold scale of about the pion mass. The corresponding timelike coupling is obtained by a contour average of the spacelike coupling. This construction practically coincides with perturbative QCD above 1 GeV² and behaves as A(Q²) proportional to Q² in the deep infrared below 0.1 GeV², consistent with the cited large-volume lattice-QCD results. The main infrared input is varied over M1 = 0.150 with an uncertainty of +0.100 and −0.050 GeV.

The study used three active quark flavours for both the charm and bottom calculations, including the bottom-quark mass and chromomagnetic coefficient. The authors argued that this treatment makes the charm-mass contribution effectively negligible for quantities dominated by the u = 1/2 infrared renormalon. The resummed procedure was also shown to be exactly independent of changes in the renormalization scale. Input values were the Particle Data Group MS-bar masses, 4.183 ± 0.007 GeV for the bottom quark and 1.2730 ± 0.0046 GeV for the charm quark, together with αs^MS-bar(MZ²) = 0.1180 ± 0.0009.

For comparison, the authors also applied truncated perturbation series to the pole-mass relation. Truncating near the smallest term gave 5.004 ± 0.157 GeV for the bottom mass and 1.585 ± 0.202 GeV for the charm mass, with additional uncertainties from αs and the MS-bar masses. These truncation uncertainties were substantially larger than the corresponding infrared-threshold uncertainties in the resummed results. Variation of the modified Borel ansatz produced only a small change in the bottom mass, while the charm mass increased by 0.039 GeV under one alternative ansatz.

Using the resummed pole masses, the authors reevaluated the renormalization-group-invariant chromomagnetic Wilson coefficient. They obtained C-hat(mb²) = 0.510(17) and C-hat(mc²) = 0.636(40), giving a ratio of 0.802(24). Combining this ratio with measured B- and D-meson hyperfine splittings yielded Λeff = 0.236(91) GeV. The detailed renormalon structure beyond the u = 1/2 singularities was not known, so the authors did not estimate an additional ansatz uncertainty for the chromomagnetic coefficient or the quantities derived from it.

The chromomagnetic matrix element provided a cross-check between heavy-quark systems. Its B-meson value was μ-hatG² = 0.507(9) GeV², compared with 0.505(13) GeV² from the D-meson system; the authors preferred the B result because the omitted higher-order corrections are less influential there. The same B-meson value was also reported by an independent hyperasymptotic Principal Value approach. From spin-averaged meson masses, the study obtained Λ-bar = 0.308(104) GeV. The kinetic parameter μπ² was poorly constrained and compatible with zero because of a near cancellation in the extraction formula.

These are model-dependent theoretical estimates rather than direct measurements. The principal limitations are uncertainty in the infrared behavior of the coupling, represented by M1 and alternative 2δAQCD calculations, as well as incomplete knowledge of higher renormalon structures. Charm decoupling and the use of three active flavours also influence the numerical analysis. The results therefore depend on the chosen infrared regulator and resummation framework, even though the framework formally eliminates explicit Landau singularities and renormalization-scale dependence.

Subject of Research: Renormalon-based QCD resummation of heavy-quark pole masses and HQET parameters

Article Title: Renormalon-based resummation for B(D) mesons

Article References: Ayala, C., & Cvetič, G. (2026). Renormalon-based resummation for B(D) mesons. The European Physical Journal C, 86(10), Article 1186. https://doi.org/10.1140/epjc/s10052-026-16480-8

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DOI: 10.1140/epjc/s10052-026-16480-8

Keywords: QCD, renormalon resummation, bottom quark mass, charm quark mass, pole mass, Heavy Quark Effective Theory, B mesons, D mesons, chromomagnetic Wilson coefficient, nonperturbative parameters, Renormalon-based, resummation