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Precision $e^+e^-$ Hemisphere Masses in the Dijet Region with Power Corrections

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that the heavy jet mass distribution's first moment at order 1/Q carries a non-perturbative parameter absent from its tail shift, unlike thrust.

desk verdict The polar expansion and N3LL machinery are real advances, but the 'proof' of an extra 1/Q parameter for HJM moments rests on an expansion the authors admit is unjustified. read the letter →

arxiv 2506.09130 v2 pith:73NLKDDW submitted 2025-06-10 hep-ph

Andre H. Hoang , Vicent Mateu , Matthew D. Schwartz , Iain W. Stewart This is my paper
classification hep-ph
keywords heavyjetmassdihemisphereeventshapespowercorrectionsresummationsoftfunctionshapefunctionsrenormalon
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

This paper establishes that the e+e- heavy jet mass (HJM) and dihemisphere mass (DHM) distributions can be predicted in the dijet region at N3LL' plus O($alpha_s^{3}$) accuracy, including non-perturbative power corrections controlled by a two-dimensional shape function. Its central structural claim is that at order 1/Q the first moment of the HJM distribution depends on an additional non-perturbative parameter compared with the parameter that shifts the tail of the spectrum. This differs from thrust, where a single non-perturbative parameter at order 1/Q describes both the first moment and the tail. The paper therefore disfavors hadronization models that use a single universal 1/Q parameter, such as the low-scale effective coupling model, and provides a complete perturbative and non-perturbative framework for future alpha_s fits from HJM data.

What carries the argument

The central object is the two-dimensional dihemisphere soft function $S(\ell_1,\ell_2)$ together with the two-dimensional non-perturbative shape function $F(k_1,k_2)$. The key technical move is the polar expansion: writing the non-global part of the soft function, $\tilde s_{2f}(L)$, as a rapidly convergent Taylor series in $L=\ln(y_1/y_2)$ around $L=0$, valid on the integration contour $|\theta|<\pi$. This reduces the resummation combined with a general two-dimensional shape function to polynomial operations, making the computation numerically tractable to better than $10^{-6}$. The paper also builds a complete basis for the two-dimensional shape function, implements renormalon subtractions in the R-gap scheme with a running gap parameter, and uses the resulting moments $\Omega_{i,j}$ and $\Upsilon_{i,j}$ to organize power corrections.

What would settle it

Compute or extract the exact three-loop non-global hemisphere soft function and check whether its polar-expansion coefficients lie within the assumed ranges; alternatively, measure the first moment and the tail shift of the HJM distribution at several center-of-mass energies and test whether a single non-perturbative parameter describes both, as opposed to the two-parameter structure predicted here.

Watch

Extended reading notes

Core claim

The paper's central claim is that HJM and DHM can be resummed and matched with power corrections to N3LL' + O($alpha_s^{3}$) accuracy, and that the OPE for HJM moments structurally differs from the OPE for the HJM tail. In the tail, the leading power correction is the single shape-function moment $\Omega_{1,0}$ and produces a shift of the distribution. But the first moment of the full HJM distribution involves the parameter $\widehat{\Upsilon}_1$, defined through integrals with $\theta(k_1-k_2)\,k_1^i$ weightings of the two-hemisphere soft momentum distribution, which is not determined by $\Omega_{1,0}$. Thus the author derives that perturbative and non-perturbative physics do not factor in the simple product form that holds for thrust, and concludes that single-parameter hadronization models are disfavored.

Load-bearing premise

The three-loop non-global piece of the two-hemisphere soft function is unknown, and the paper approximates it by the first three terms of its expansion around equal-hemisphere logarithms, assuming the coefficients $s_{3,2}$ and $s_{3,4}$ are no larger than the ranges $[-250,250]$ and $[-7,7]$. If the true function has larger coefficients or significant structure beyond $L^4$, the claimed N3LL accuracy in the peak region would be compromised.

Editorial extensions

If this is right

  • The HJM and DHM distributions can now be predicted in the dijet region at N3LL' + O(alpha_s^3), with non-perturbative corrections parametrized by a two-dimensional shape function.
  • In the tail region the OPE truncates at leading power with a single new parameter $\Omega_{1,0}$, so high-precision alpha_s fits from HJM tail data become feasible in the same way as for thrust.
  • At order 1/Q the first moment of HJM requires the extra parameter $\widehat{\Upsilon}_1$, so moment-based determinations must account for two hadronic parameters rather than one.
  • The peak region of HJM is controlled by both $\Omega_{1,0}$ and $\widehat{\Upsilon}_1$, which affect the peak position and the peak height respectively; models with a single universal 1/Q parameter are disfavored.
  • The polar expansion reduces the two-dimensional soft-function resummation to polynomial operations, making convolution with a general two-dimensional shape function numerically tractable to high precision.

Reading between the lines

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

  • If the two-parameter structure is confirmed, global fits that combine HJM peak and tail data will need to fit both parameters simultaneously, and the extracted value of alpha_s may shift compared to fits that assume a single universal parameter.
  • The same polar-expansion technique should extend to other doubly differential dijet observables whose soft functions are not sums of single-variable products, once their three-loop soft functions become known.
  • A decisive test could come from comparing Monte Carlo hadronization models to HJM moment data at several center-of-mass energies: single-parameter models predict a specific relation between the first moment and the tail shift, while the paper's structure leaves them independent.
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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 / 4 minor

Summary. This paper develops a high-precision theoretical framework for the e+e- heavy jet mass (HJM) and dihemisphere mass (DHM) distributions in the dijet region. It presents N3LL' resummation based on a polar expansion of the two-loop non-global hemisphere soft function, a two-dimensional shape-function basis, renormalon subtraction with R-evolution, profile functions for peak/tail transitions, and non-singular extractions at O(alpha_s^2) and O(alpha_s^3). The paper also claims to prove that at order 1/Q the first moment of the HJM distribution requires an additional non-perturbative parameter bUpsilon1 beyond the tail-shift parameter Omega1,0, in contrast to thrust, and concludes that single-parameter hadronization models are disfavored.

Significance. If fully established, the bUpsilon1 result would be a novel qualitative distinction: HJM moments would not be governed by the same non-perturbative matrix elements as the tail OPE, ruling out a class of universal power-correction models for this observable. The technical contributions are substantial: the polar expansion convergence is demonstrated to 1e-6 in Fig. 5, the two-dimensional shape-function basis is a useful new tool, the R-gap subtractions are implemented in a documented scheme, and the non-singular extractions are described with explicit fit forms and uncertainties. These ingredients will be valuable for future global fits of alpha_s from HJM data. However, the central proof claim rests on an explicitly unjustified expansion, which tempers the significance until that step is either established or the claim is appropriately weakened.

major comments (2)
  1. [Sec. 5.2, Eq. (5.14), footnote 10] The abstract states that the first moment of the HJM distribution is proven to involve an additional non-perturbative parameter at order 1/Q, but the derivation in Sec. 5.2 is conditional. The step from the exact expression in Eq. (5.14), which contains the non-factorizable kernel bXi[Qell,Q(ell+k1-k2)] under theta(k1-k2), to the conclusion that the non-perturbative dependence is through the combinations theta(k1-k2)(k1-k2)^j k1^i F(k1,k2) is precisely the Taylor expansion that footnote 10 admits is 'not fully justified' because the region ell ~ k1 ~ k2 is equally important. The subsequent definition of bUpsilon1 and the bound in Eq. (5.17) follow only under this expansion. The conclusion that single-parameter hadronization models are disfavored is therefore not proven by the arguments in the manuscript. The claim should either be established by a rigorous treatment of the neglected region, or by a numerical check of the exact convolution, or the abstract and conclusions should be rephrased as a conjecture.
  2. [Sec. 3.3, Eq. (3.29), Table 1] The announced N3LL' precision in the peak region is contingent on the unknown three-loop non-global dihemisphere soft function s3f(L) being accurately represented by the first three terms of its polar expansion, Eq. (3.29), with coefficients s3,2 and s3,4 varied in the ad hoc ranges |s3,2| <= 250 and |s3,4| <= 7. This is an assumption about the size and analytic structure of an uncomputed function; the theory scan can quantify the sensitivity within that family, but not the uncertainty relative to the true s3f. The paper should state explicitly that the N3LL' peak-region predictions are conditional on this assumption, and should avoid presenting the resulting bands as complete perturbative uncertainties.
minor comments (4)
  1. [Sec. 9 / Abstract] The conclusions repeat the word 'proved' and 'we showed' for the bUpsilon1 result without the caveat given in footnote 10; please align the abstract and conclusions with the conditional status of the derivation.
  2. [Sec. 7, Table 1] The text says 'ten free parameters', but Table 1 lists twelve parameters with variation ranges (n0, n1, t2, ts, rs, eJ, eH, ns, s32, s34, epsilon2, epsilon3). Please reconcile the count.
  3. [Sec. 4.1, after Eq. (4.8)] The fit function is written as theta(rho0-rho) f_log_4(rho) + theta(rho-rho0) f_pol_3(rho), but Eq. (4.7) defines f_log_3; this appears to be a typo for f_log_3.
  4. [Fig. 3] The caption would be clearer if it stated explicitly that the 'number of terms' counts the number of nonzero coefficients in the polar expansion, and whether the left-panel label n refers to the highest power theta^n retained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the new HJM claims are derived self-consistently; the main caveat is an admitted Taylor-expansion gap (footnote 10) that is a rigor limitation, not a reduction to inputs.

full rationale

The paper's central factorization and OPE derivations (Secs. 2-5) are self-contained given standard SCET inputs and external fixed-order/numerical results. The new bUpsilon1 parameter is defined as a shape-function moment (Eqs. (1.5)-(1.6), (5.16)) and its appearance in HJM moments is obtained by direct manipulation of the convolution (Eq. (5.14)), not by fitting or by defining it as the residual of the calculation. The tail OPE (Eq. (5.9)) uses Omega1,0; the moment calculation shows a theta(k1-k2)-dependent kernel that does not factorize, so the qualitative distinction between the tail parameter and the moment parameter is not built into the definitions. External benchmarks are used (event2, CoLoRFulNNLO, the exact two-loop soft function), and self-citations to Refs. [22,75,128] provide framework but are not the only support for the new claims. The main caveat is footnote 10: the Taylor expansion used to organize the moments in terms of Omega_i,j and Upsilon_i,j is admitted to be 'not fully justified' because the region ell ~ k1 ~ k2 is equally important. The exact starting point, Eq. (5.14), already contains a non-factorizable theta(k1-k2) kernel, so the existence of an additional moment-dependence is not an artifact of that expansion; however, the specific identification of bUpsilon1 and the bound in Eq. (5.17) rely on the disputed expansion, making the 'proof' in Sec. 5.2 less rigorous than the abstract's wording suggests. This is an internal rigor gap and a correctness risk, but it is not a circular reduction: no fitted parameter is renamed as a prediction, no definition is constructed in terms of the target result, and no load-bearing conclusion is forced by a self-citation chain. Score 2 reflects standard, non-load-bearing self-citations; no circular step was found.

Assumptions & free parameters 12 free parameters · 6 assumptions · 1 invented entities

The main free parameters are the unknown 3-loop non-global soft function coefficients and the profile function parameters used for uncertainty estimation. The axioms are the standard SCET factorization, the assumed convergence of the polar expansion, and the properties of the shape function. The bY_i moments are new non-perturbative parameters introduced by the paper.

free parameters (12)
  • s32 = 0 (varied +/-250)
    Unknown 3-loop non-global dihemisphere soft function coefficient; parametrizes the L^2 term of the polar expansion (Eq 3.29).
  • s34 = 0 (varied +/-7)
    Unknown 3-loop non-global soft function coefficient; parametrizes the L^4 term (Eq 3.29).
  • epsilon2 = 0 (varied [-1,1])
    One-sigma error parameter for the O(alpha_s^2) non-singular fit (Eq 4.6).
  • epsilon3 = 0 (varied [-1,1])
    One-sigma error parameter for the O(alpha_s^3) non-singular fit (Eq 4.8).
  • eH = 0.5 (varied 0.25 to 1.0)
    Hard scale profile parameter (Eq 7.2).
  • rs = 2 (varied 1.33 to 3)
    Soft scale slope in the tail (Eq 7.3).
  • n0 = 2 (varied 1.5 to 2.5)
    Peak-to-tail transition boundary in units of Q/(1 GeV) (Eq 7.4).
  • n1 = 10 (varied 8.5 to 11.5)
    Peak-to-tail transition boundary in units of Q/(1 GeV) (Eq 7.4).
  • t2 = 0.25 (varied 0.225 to 0.275)
    Tail-to-far-tail transition point in rho (Eq 7.3).
  • ts = 0.4 (varied 0.375 to 0.425)
    Tail-to-far-tail transition endpoint (Eq 7.3).
  • eJ = 0 (varied -1 to 1)
    Trumpet factor for the jet scale (Eq 7.5).
  • ns = 0 (varied -1 to 1)
    Non-singular scale parameter (Eq 7.7).
assumptions (6)
  • domain assumption SCET factorization theorem for dijet event shapes (hard x jet x jet x soft) at N3LL
    Invoked in Sec 2.2, Eq (2.9); standard in the literature (Refs [11,13,73]).
  • standard math Non-Abelian exponentiation structure of the soft function
    Used in Sec 3.1, Eq (3.3); constrains the form of the non-global function.
  • domain assumption Polar expansion of the non-global soft function converges on |L| < pi
    Shown numerically in Sec 3.2 (Fig 5) but not proven analytically; used to make resummation tractable.
  • domain assumption Shape function F(k1,k2) is positive, symmetric, and exponentially suppressed
    Stated in Sec 6, Eq (6.1); needed for the OPE and the moment bounds.
  • domain assumption R-gap scheme subtraction removes the u=1/2 renormalon
    Taken from Ref [128], used in Sec 5.3; underpins the power-correction definition.
  • domain assumption Truncation of the tail OPE at leading power is sufficient for rho >> Lambda_QCD/Q
    Used in Sec 5.1, Eq (5.9); validated by the scale hierarchy but not by a rigorous error estimate.
invented entities (1)
  • bY_i (also Y_i,j) independent evidence
    purpose: Non-perturbative parameters needed for the moments of the HJM distribution at order 1/Q; encode the dependence on which hemisphere is heavier.
    Defined in Eqs (1.6) and (5.16); derived from the shape function moments. The paper provides a falsifiable bound: Omega1,0/2 < bY1 < Omega1,0, and predicts that a single-parameter description of HJM moments fails. This can be tested with HJM moment data.

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Pith. "Pith review of Precision $e^+e^-$ Hemisphere Masses in the Dijet Region with Power Corrections." pith.science (2026). https://pith.science/paper/73NLKDDW

@misc{pith2026250609130,
  author       = {Pith},
  title        = {Pith review of: Precision $e^+e^-$ Hemisphere Masses in the Dijet Region with Power Corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73NLKDDW}},
  note         = {Machine review of arXiv:2506.09130}
}
abstract

We derive high-precision results for the $e^+e^-$ heavy jet mass (HJM) $d \sigma/d \rho$ and dihemisphere mass (DHM) $d^2\sigma/(d s_1 d s_2)$ distributions, for $s_1\sim s_2$, in the dijet region. New results include: i) the N$^3$LL resummation for HJM of large logarithms $\ln^n(\rho)$ at small $\rho$ including the exact two-loop non-global hemisphere soft function, the 4-loop cusp anomalous dimension and the 3-loop hard and jet functions, ii) N$^3$LL results for DHM with resummation of logarithms $\ln(s_{1,2}/Q^2)$ when there is no large separation between $s_1$ and $s_2$, iii) profile functions for HJM to give results simultaneously valid in the peak and tail regions, iv) a complete two-dimensional basis of non-perturbative functions which can be used for double differential observables, that are needed for both HJM and DHM in the peak region, and v) an implementation of renormalon subtractions for large-angle soft radiation to ${\cal O}(\alpha_s^3)$ together with a resummation of the additional large $\ln(Q\rho/\Lambda_{QCD})$ logarithms. Here $Q$ is the $e^+e^-$ center-of-mass energy. Our resummation results are combined with known fixed-order ${\cal O}(\alpha_s^3)$ results and we discuss the convergence and remaining perturbative uncertainty in the cross section. We also prove that, at order $1/Q$, the first moment of the HJM distribution involves an additional non-perturbative parameter compared to the power correction that shifts the tail of the spectrum (where $1\gg \rho\gg \Lambda_{QCD}/Q$). This differs from thrust where a single non-perturbative parameter at order $1/Q$ describes both the first moment and the tail, and it disfavors models of power corrections employing a single non-perturbative parameter, such as the low-scale effective coupling model. In this paper we focus only on the dijet region, not the far-tail distribution for $\rho \gtrsim 0.2$.

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Cited by 1 Pith paper

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