Pith. sign in

REVIEW 2 major objections 5 minor 59 references

Hyperasymptotic approximation to the top, bottom and charm pole mass

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the heavy-quark pole mass, when regulated by the principal-value prescription, is a well-defined mass, and that the MS-bar-to-PV conversion for the top quark carries a theoretical uncertainty of 28 MeV.

desk verdict A careful extension of the authors' hyperasymptotic method to the pole mass, with a plausible 28 MeV theory error for the top PV mass once you accept that the u=1 renormalon is absent; the abstract's headline omits the larger alpha_s error and the conditionality. read the letter →

arxiv 1909.01370 v2 pith:GX4RJW7E submitted 2019-09-03 hep-ph hep-lat

Cesar Ayala , Xabier Lobregat , Antonio Pineda This is my paper · ORCID
classification hep-phhep-lat
keywords heavyquarkpolemasshyperasymptoticexpansionprincipalvalueprescriptionrenormalonstopMS-barLambda-barBmeson
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the longstanding 'ambiguity' of the heavy-quark pole mass is not an ambiguity at all: once the divergent perturbative series is resummed with the principal-value (PV) prescription, quantities such as the PV pole mass are well defined with arbitrary achievable accuracy, and the practical question is only how well the conversion between the MS-bar mass and a specific pole-mass definition is known. For the top quark with MS-bar mass 163 GeV, the paper finds this conversion to have a theoretical uncertainty of 28 MeV, with a central value $m_{t,\rm PV} = 173033$ MeV. The argument runs through hyperasymptotic expansions, in which the truncated perturbative sum is supplemented by 'terminants' that encode factorial divergences (renormalons) in the Borel plane. The method is tested in the large-$\beta_0$ approximation, applied to lattice and $B/D$-meson determinations of $\bar\Lambda$, and then used for the top quark.

What carries the argument

The central object is the hyperasymptotic expansion of the pole mass, a Borel-resummed principal-value version of the divergent perturbative series. The workhorse is the terminant $\Omega_d$ (and its mass analogue $m\Omega_m$), the completion that restores the part of the series associated with a singularity at $u=d/2$ in the Borel plane; for the leading infrared renormalon at $u=1/2$ it has the form $\Omega_m \sim \sqrt{\alpha(\mu)}\, K_X^{(P)}\, (\mu/m)\, e^{-2\pi/(\beta_0\alpha(\mu))}\, (\beta_0\alpha(\mu)/4\pi)^{-b}$, with $K_X^{(P)}$ proportional to the normalization $Z_m$. The argument is carried by matching this terminant to the known asymptotic behaviour of the perturbative coefficients $r_n$, and by a renormalon-free running function $F(m,n_f)$ that lowers the top quark mass to scales around 5 GeV where the hyperasymptotic expansion can be evaluated, plus explicit decoupling of bottom and charm finite-mass effects.

What would settle it

Decisive tests: compute the renormalon normalization $Z_m$ independently, for instance from the static potential with reduced uncertainty or from a sum rule free of the leading renormalon, since $m\Omega_m$ is proportional to $Z_m$ and a shift of $Z_m$ by its current uncertainty changes the $173033$ MeV central value by more than 20 MeV. Also determine whether the $u=1$ infrared renormalon normalization is nonzero: if it is, the missing terminant contributes an additional $O(\Lambda_{\rm QCD})$ shift in the conversion, and the 28 MeV theoretical error estimate understates the uncertainty.

Watch

Extended reading notes

Core claim

The central claim is that the PV-regulated pole mass $m_{\rm PV}(m)$ can be computed from the MS-bar mass $m$ by a hyperasymptotic expansion whose leading non-perturbative term is the terminant $m\Omega_m$, proportional to $\sqrt{\alpha}\,(\Lambda_{\rm QCD}/m)$ times the renormalon normalization $Z_m$, and that this expansion is accurate enough that for the top quark the relation $\bar m_t = 163$ GeV to $m_{t,\rm PV}$ has a theoretical error of only 28 MeV (plus a much larger $+119/-123$ MeV from the strong coupling). A corollary is that the often-quoted pole-mass ambiguity of order $\Lambda_{\rm QCD}$ is replaced by a computable, definition-dependent difference: different legitimate pole-mass definitions differ by $O(\Lambda_{\rm QCD})$ with an arbitrary coefficient, so statements about 'the' pole mass ambiguity are ill-posed without a specific definition. The paper also finds evidence for the ultraviolet renormalon at $u=-1$ in the sign-alternating coefficients of the renormalon-free running function $F(m,n_f)$, while finding no evidence for an infrared renormalon at $u=1$.

Load-bearing premise

The load-bearing premise is that the size of the leading 'runaway growth' of the perturbative series—the renormalon normalization—is known to the accuracy of the cited determination, and that there is no competing divergence at twice that location; if either assumption fails, the central value and the 28 MeV error both move.

Editorial extensions

If this is right

  • The top quark pole mass can be defined and used in a scheme-and-scale controlled way: with current perturbative input, $m_{t,\rm PV}(163\,{\rm GeV}) = 173033^{+25}_{-28}({\rm th})^{+119}_{-123}(\alpha)$ MeV, so the purely theoretical uncertainty is well below the typical $\Lambda_{\rm QCD}$ ambiguity quoted in the literature.
  • The same hyperasymptotic machinery gives $m_{b,\rm PV} = 4836^{+8}_{-17}(Z_m)^{+12}_{-11}(\alpha)^{+8}_{-9}$ MeV for the bottom quark and $\bar\Lambda_{\rm PV} = 477\pm 46$ MeV from the $B$ meson mass, allowing cross-checks between lattice and $B$-physics determinations.
  • The renormalon-free running function $F(m,n_f)$ shows a sign-alternating pattern consistent with an ultraviolet renormalon at $u=-1$ and no sign of an infrared renormalon at $u=1$; if the latter is truly absent, the heavy-quarkonium mass analysis closes consistently.
  • The error in the conversion is dominated by the strong coupling, not by renormalon ambiguity, and it is systematically improvable order by order in perturbation theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the $u=1$ infrared renormalon is truly absent, the PV prescription becomes a particularly clean mass definition for electroweak-precision fits, because the dominant theoretical error would then come from $\alpha_s$ and higher-order coefficients, both improvable.
  • Because the large-$\beta_0$ tests show the OPE picture holding down to scales around 667 MeV, the same machinery might give charm quark mass determinations with controlled errors despite the lower scale.
  • The observed sign-alternating coefficients in $F(m,n_f)$ imply that the $u=-1$ ultraviolet renormalon, usually thought to be subleading, may set the practical truncation point in the MS scheme; a lattice-scheme computation with a high effective scale could push the expansion further.
  • Relating the PV mass to experimentally measured top cross sections rather than assuming the measured mass is the pole mass would remove an $O(\Lambda_{\rm QCD})$ theoretical ambiguity from top-quark mass extractions; the paper makes this concrete by giving the PV-to-MS-bar shift with a 28 MeV error.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper constructs hyperasymptotic expansions for the heavy-quark pole mass regulated with the principal-value (PV) prescription, extending earlier work by the same group to include ultraviolet renormalons. After establishing the general formalism, the authors test it in the large-β0 approximation, where exact results are available, and demonstrate that successive hyperasymptotic corrections (mP, mΩm, subleading sums, mΩ−2) produce the expected convergence in both lattice and MS schemes. They then use the method to extract Λ̄_PV from quenched lattice data and from the B-meson mass, and compare with an independent lattice result. Finally, they construct a renormalon-free running function F(m,nf) to evolve the top mass from 163 GeV down to μc=5 GeV, decoupling bottom and charm quarks, and evaluate m_PV(μc)+μcΩm there. The central result is Eq. (66): m_{t,PV}(163 GeV)=173033 (+25/−28)(th) (+119/−123)(α) MeV, i.e., a 28 MeV theoretical uncertainty for fixed α_s, which the abstract quotes.

Significance. The paper is a serious, technically careful contribution. The large-β0 toy model is a genuine check of the hyperasymptotic expansion, and the consistency between lattice-scheme and MS-scheme determinations of Λ̄_PV is encouraging. The decoupling/renormalon-free-running strategy for the top mass is well motivated, and the bottom/charm finite-mass effects are treated in detail. If the error assessment is correct, the result would substantially reduce the often-quoted O(Λ_QCD) pole-mass ambiguity for the top. The main caveat is that the headline uncertainty is conditional on an unquantified assumption about the u=1 renormalon and on a heuristic truncation-error estimate; because the paper's central claim is precisely about the size of an uncertainty, these assumptions are load-bearing rather than cosmetic.

major comments (2)
  1. [§V.B, §V.C (Eqs. (59), (66))] The abstract's δm_{t,PV}=28 MeV and the central value in Eq. (66) are obtained from Eq. (59), which omits the mΩ_2 terminant associated with a possible u=1 infrared renormalon and estimates the remainder as O(μc e^{−2π/(β0α)(1+ln2)}). At μc=5 GeV this remainder is numerically comparable to the quoted 28 MeV, so the quoted uncertainty is not a bound on this contribution but an assumption that its coefficient is small. Section V.C offers only plausibility arguments—the sign pattern of f_n in Table V and the lattice analysis of Ref. [38]—and no quantitative bound on the normalization Z_2. If Z_2 were of order one, the omitted mΩ_2 would shift the central value by an amount comparable to the quoted error and would not be covered by the μc-variation error in Eq. (62), which tests the already-subtracted series. Please provide a quantitative estimate or bound on mΩ_2, or explicitly present the 28 MeV as conditional on Z_2 being zero or negligible.
  2. [§V.B, Eq. (61)] A major component of the theoretical error is the 22 MeV assigned in Eq. (61) to the truncation of F(m,nf), estimated as half the last computed term. This is a standard rule for sign-alternating asymptotic series, but the available series for nf=3 has only four coefficients, and the sign pattern is itself used in Section V.C as evidence about renormalons. Applying the 'half the last term' rule to such a short series is an assumption; it would be more convincing to test the stability of the error estimate under different truncation orders (e.g., excluding or including the last coefficient) and to compare with the large-β0 prediction, where the exact remainder is known.
minor comments (5)
  1. [Abstract and Eq. (66)] The 28 MeV is the theory error for fixed α_s; the α_s contribution is +119/−123 MeV. Please state this explicitly in the abstract, since as written the abstract may be read as a total uncertainty.
  2. [Eqs. (63) and (66)] Please clarify how the quoted +25/−28 MeV theoretical error is obtained from the components +22/−22 (h.o.), +22/−22 (μ), +7/−15 (Z_m), and +9/−9 (μ_c) listed in Eq. (63); a direct quadrature of these components gives a somewhat larger interval.
  3. [Eqs. (63) and (66)] The unit '163MeV' in Eqs. (63) and (66) should read '163 GeV'.
  4. [§IV.A and §IV.E] The word 'Elucubrative' after Eq. (42) is nonstandard; 'Speculatively' or 'As a speculative ansatz' is preferable. Similarly, 'aminorates' in Sec. IV.E should be 'ameliorates'.
  5. [Figure 11] The labels (a)–(d) in the lower panel are not legible at the printed size; consider enlarging the font or using a legend.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the top-mass conversion is computed from externally determined perturbative coefficients and renormalon normalizations, not fitted to the target observable.

full rationale

The central quantity m_{t,PV}(163 GeV) is obtained in Sec. V.B by evaluating Eq. (58) with the MS mass as an input; the perturbative coefficients r_n are taken from Refs. [6-9] and the renormalon normalization Z_m from Ref. [13] (and [23] in the lattice case), both of which are determinations from observables other than the top mass. No top-quark datum is used to adjust the result, and no equation defining the final error budget is equivalent to a fitted parameter. The self-cited formalism of Ref. [1] is a prior theoretical derivation for a different observable (the static potential) and does not assume the pole-mass result, so it is independent support rather than a circular premise. The caveat that the O(μc e^{-2π/(β0 α)(1+ln 2)}) term in Eq. (59) is 'not known' and that the u=1 renormalon is only argued to be absent from the sign pattern of F and from the lattice analysis of Ref. [38] is a genuine limitation of the error estimate—the quoted 28 MeV may undercount an omitted Λ^2/μc effect—but this is an accuracy/validity concern, not a reduction of the derivation to its inputs. Accordingly no circular step can be exhibited, and the score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central top-mass result rests on three classes of inputs: (i) perturbative coefficients r_n up to n=3 from the literature, (ii) renormalon normalizations Z_m from previous fits by the same group and others, and (iii) heuristic assumptions about subleading renormalons and truncation errors. No new particles or forces are introduced.

free parameters (5)
  • Z_m (MS, nf=3) = 0.5626(260)
    Renormalon normalization from [13], used in m_Omega_m for the top and bottom mass conversions; carries the largest theory-driven uncertainty after alpha_s.
  • Z_m (lattice, nf=0) = 17.9(1.0)
    Renormalon normalization from [23] used in the quenched lattice Lambda-bar_PV determination.
  • Lattice beta-function coefficients beta_3..beta_6 = -1.16(3)e6, -1.35(10)e8, -1.44(28)e10, -1.41(60)e12
    Estimated in Table III under renormalon dominance with Z_m = 0.62; used to convert lattice-scheme to MS-scheme results.
  • Decoupling scales mu_b, mu_c = 20 GeV, 5 GeV
    Chosen in the top analysis as central values; residual dependence is small but not zero.
  • K (O(a) coefficient in Lambda-bar_PV lattice fit) = not quoted
    A linear lattice-spacing term is fitted to E_MC lattice data to absorb O(a Lambda_QCD^2) effects; the intercept gives Lambda-bar_PV = 1.42 r0^-1.
assumptions (5)
  • domain assumption The perturbative coefficients of the pole mass and related observables are renormalon dominated, with the asymptotic behavior given by the leading renormalons (Eq. (8) and the c_n^(as) construction).
    Used throughout to construct the hyperasymptotic expansion and to estimate unknown higher-order coefficients; supported by large-beta0 models and numerical coefficient behavior, but not proven in general.
  • domain assumption The OPE factorization in Eq. (1): Observable(Q) = S_PV + K ... Lambda^d/Q^d + ...
    The starting point that separates perturbative and nonperturbative contributions; standard for renormalon-based analyses.
  • domain assumption The pole mass perturbative series is asymptotic and divergent with IR renormalons at positive u and UV renormalons at negative u, as in Eq. (35) (Borel transform).
    Needed for the PV prescription and terminants; proven only in the large-beta0 limit, as the authors note in Section V.A.
  • ad hoc to paper For the top, the renormalon-free running function F(m,nf) can be truncated at NNNLO and its error estimated as half the last term (sign-alternating asymptotic series rule).
    Used in Eq. (61) to assign the h.o. error; a heuristic rule from Dingle [5], not a rigorous bound.
  • ad hoc to paper The u = 1 infrared renormalon in the pole mass is zero or negligible, so only the u = 1/2 and u = -1 singularities drive the asymptotic behavior.
    The paper's sign analysis of f_n and the lattice result [38] suggest this, but the authors explicitly state that higher-order coefficients are needed to confirm; the top error budget leans on this assumption.

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Cite this review

Pith. "Pith review of Hyperasymptotic approximation to the top, bottom and charm pole mass." pith.science (2026). https://pith.science/paper/GX4RJW7E

@misc{pith2026190901370,
  author       = {Pith},
  title        = {Pith review of: Hyperasymptotic approximation to the top, bottom and charm pole mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GX4RJW7E}},
  note         = {Machine review of arXiv:1909.01370}
}
abstract

We construct hyperasymptotic expansions for the heavy quark pole mass regulated using the principal value (PV) prescription. We apply such hyperasymptotic expansions to the $B/D$ meson masses, and $\bar \Lambda $ computed in the lattice. The issue of the uncertainty of the (top) pole mass is critically reexamined. The present theoretical uncertainty in the relation between ${\bar m}_t$, the $\bar{\rm MS}$ top mass, and $m_{t, \rm PV}$, the top pole mass regulated using the PV prescription, is numerically assessed to be $\delta m_{t,\rm PV}= 28\;{\rm MeV}$ for ${\bar m}_t =163$ GeV.

Figures

Figures reproduced from arXiv: 1909.01370 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: As in Fig. 1 but in the [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Comparison of lattice and [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4: As in Fig. 1 but with [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: As in Fig. 1 but with [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of lattice and [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Same caption as in Fig. 1 of [29] including more terms in the perturbative expansion [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p033_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p043_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Plots of Eq. (58) in terms of [PITH_FULL_IMAGE:figures/full_fig_p045_14.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.