Anomalous scaling of linear power corrections

Casey Farren-Colloty; Gavin P. Salam; Jack Helliwell; Rtvik Patel; Silvia Zanoli

REVIEW 2 major objections 4 minor 1 cited by

Anomalous scaling of linear power corrections

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that non-perturbative $1/Q$ corrections to linear jet observables are governed by a simple all-order exponential factor $R(Q)$, identical for thrust, $C$-parameter and energy correlators, and independent of the soft-gluon…

desk verdict Genuinely new analytic derivation of S1 and an all-order exponentiation claim for 1/Q power corrections, backed by thorough numerics, but the exponentiation rests on an unproved longitudinal-recoil factorization that the paper itself only partially supports. read the letter →

arxiv 2507.18696 v2 pith:NQOPKZYI submitted 2025-07-24 hep-ph

Casey Farren-Colloty , Jack Helliwell , Rtvik Patel , Gavin P. Salam , Silvia Zanoli This is my paper · ORCID

classification hep-ph

keywords linearpowercorrectionsrenormalonshadronisationeventshapesthrustC-parameterenergy-energycorrelatorsall-orderresummation

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 argues that the traditional assumption that hadronisation corrections scale simply as $1/Q$ is incomplete: for linear observables like thrust, the $C$-parameter and energy-energy correlators, the actual scaling is $(1/Q)\,R(Q)$, where $R(Q)$ is a known all-order exponential built from the QCD coupling. The authors show that summing over arbitrary numbers of large-angle soft gluons does not produce the complicated non-global structure one might expect; instead, in the large-$N_c$ limit and with linear kinematic recoil maps, the correction reduces to a single exponential factor. If correct, this means the non-perturbative transverse momentum parameter $T_{q\bar q}$ is the only new non-perturbative input, and its scale ambiguity is absorbed into $R(Q)$. The paper further identifies the D-scheme for hadron masses as the one where this simple form survives.

What carries the argument

The paper's central object is the 'gluer': a single non-perturbative gluon inserted at transverse momentum below an infrared scale $\mu_{\rm np}$. Its effect on a linear observable is captured by the transverse momentum per unit rapidity, $T_{q\bar q}$, and by the ratio $\rho_{qg\bar q}(\eta,\eta_g)$ measuring how a soft perturbative gluon modifies that density. The mechanism that carries the argument is the integrated recoil identity $\int d\eta_g\,[\rho_{qg\bar q}(\eta,\eta_g)-1] = -4(1-\ln 2)=-S_1/2$, which turns the first-order correction into the coefficient $C_A S_1/(2\pi)$. Exponentiation follows from a differential equation in $\ln Q$ in which each new scale slab of soft gluons multiplies the non-perturbative density by the same factor $R(Q)$; this step relies on two linear kinematic recoil maps whose longitudinal recoil is local to the emitting dipole.

What would settle it

Compute the coefficient of $(\alpha_s \ln Q/\mu_{\rm np})^2$ in full colour with a second inequivalent linear recoil map: if it is not $\tfrac12(C_A S_1/2\pi)^2$, the exponentiation collapses. Alternatively, a precise measurement of the mean $C$-parameter at high $Q$ (for example at a future Higgs factory) that deviates from $c_C T_{q\bar q} R(Q)/Q$ by more than the stated higher-power uncertainties would falsify the predicted scaling.

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Extended reading notes

Core claim

The central result is Eq. (11): for a $q\bar q$ event dressed by an arbitrary cloud of soft gluons, the non-perturbative correction to a linear observable is $\langle\delta V_{\rm np}\rangle = c_V\, T_{q\bar q}\, R(Q)/Q$, with $R(Q)=\exp[-\lambda(Q,\mu_{\rm np})\,C_A S_1/(2\pi)]$ and $S_1=8(1-\ln 2)$. Equivalently, $R(Q)=(\alpha_s(Q)/\alpha_s(\mu_{\rm np}))^{C_A S_1/\beta_0}$, where $\lambda$ is the logarithmic integral of $\alpha_s$ between the infrared matching scale $\mu_{\rm np}$ and $Q$. The same factor applies to thrust, the $C$-parameter and energy-energy correlators, holds with arbitrary numbers of additional soft gluons, and extends to the three-jet region. In the D-scheme for hadron masses the factor is independent of the gluer mass, so a single $T_{q\bar q}$ suffices; other mass schemes introduce additional non-perturbative parameters and their power corrections can even change sign at high $Q$.

Load-bearing premise

The all-order step assumes that, after averaging over the perturbative gluon cloud, the longitudinal recoil of gluon endpoints carries exactly the same factor $R(Q)$ as the transverse momentum density; this is checked numerically for one smooth rapidity boundary but not proven for arbitrary dipole configurations, and the linear recoil maps themselves remain unproved for gluon endpoints.

Editorial extensions

If this is right

The scale $\mu_{\rm np}$ ambiguity is absorbed into the normalisation of $T_{q\bar q}$, so the all-order prediction depends on a single non-perturbative parameter.
The 3-jet prediction is the bare 3-parton result rescaled by $R(\xi V Q)$ rather than $R(Q)$, adding a calculable $V$-dependence to the event-shape distribution.
In the D-scheme, the anomalous dimension is universal across thrust, $C$-parameter and energy-energy correlators, and independent of the hadron-mass distribution; in E- and P-schemes it is not.
Differences between mass schemes have positive anomalous dimensions, so in the E- and P-schemes the hadronisation correction changes sign at sufficiently high $Q$.
The numerical agreement with a leading Monte Carlo hadronisation model, after fixing one value of $T_{q\bar q}$, suggests the scaling can be used to extrapolate hadronisation corrections between observables and energies.

Reading between the lines

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

If the exponential structure is universal among linear observables, power corrections for new observables could be predicted from a single fitted $T_{q\bar q}$ and the known $R(Q)$, replacing per-observable hadronisation modelling.
A direct test beyond the paper would compute the $\alpha_s^2\ln^2(Q/\mu_{\rm np})$ coefficient in full colour with a second inequivalent linear recoil map; agreement with half the square of the first-order term would confirm exponentiation, while a mismatch would reveal the boundaries of the linear-recoil paradigm.
The D-scheme preference suggests experimental analyses should adopt D-scheme observable definitions if they want hadronisation corrections to be parametrised by one number; this is a practical recommendation the paper states only implicitly.
The same approach may extend to other linear soft-sensitive observables, such as jet broadening or transverse-momentum-like measures in deep inelastic scattering, provided an appropriate linear recoil map can be constructed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, this is genuinely new: Farren-Colloty et al. derive the first-order anomalous coefficient S1 = 8(1-ln2) analytically — Dasgupta and Hounat only had it numerically — and then claim it exponentiates to all orders into R(Q) = (alpha_s(Q)/alpha_s(mu_np))^{C_A S1/beta0} for thrust, C-parameter and EEC, in the two-jet limit, beyond it, and with finite hadron masses in the D-scheme. That is a real claim, not a repackaging. Second, the exponentiation is conditional on an unproved factorization. Eq. (10) assumes the all-order longitudinal recoil from qg and gqbar dipoles gets the same R(Q) factor as the transverse momentum density. The supplement's recursive argument assumes a gluon endpoint responds to gluer recoil like a quark endpoint — which is exactly the gluon-endpoint linearity that the main text says remains to be proved.

Where they earn credit: the analytic derivation of S1 is clean, and the numerical program is unusually thorough. They validate the exponent in the asymptotic alpha_s -> 0 limit, with fixed-lambda runs, with two different shower maps, at finite alpha_s with extrapolation, and against Pythia from 30 GeV to 10 TeV. The agreement with the concurrent operator-based calculation for the EEC in the D-scheme [62] is meaningful evidence. They also state their limitations explicitly; the "paradigm remains to be fully established" sentence is not buried, and the supplement flags the gluon-endpoint open question.

Soft spots, in proportion. The central step, Eq. (10), is not derived from the NLO result. The NLO identity fixes the integral of rho-1; it does not determine the all-order longitudinal factor. The numerical check of longitudinal recoil in Supplement Eq. (40) is first order in alpha_s, uses a single perturbative gluon, and one smooth rapidity boundary. It cannot test arbitrary multi-gluon clouds. The stress-test concern lands: if the longitudinal anomalous dimension differs from the transverse one, Eq. (11) gets a different exponent; if the longitudinal response is not proportional to R at all, Eq. (12) fails. The authors are honest about this, but the abstract and intro present the exponentiation as the result rather than as a conjecture resting on an unproved property. I would want the paper to either prove that property or clearly mark the exponentiated form as conditional.

Minor: the Pythia comparison uses a fitted T_qqbar at one Q value; fine for illustrating scaling, not for predicting normalisation. The thrust shows larger discrepancies than C-parameter, which they acknowledge.

Who this is for: anyone doing alpha_s from event shapes, hadronisation corrections, or renormalon-based power corrections. It deserves a serious referee — the first-order part is solid and the all-order claim is important enough to engage with even while it is incomplete. If I were editor, I'd send it to review with the explicit request that the longitudinal-recoil factorization be proved or sharply delimited, and that the conjecture status be visible at the level of the abstract.

Referee Report

2 major / 4 minor

Summary. This letter proposes an all-order resummation of the linear (Λ/Q) non-perturbative correction to a class of 'linear' observables (thrust, C-parameter, energy correlators). Starting from the standard one-gluer formula, the authors derive the NLO anomalous-dimension coefficient S1=8(1-ln2) for a qqbar system dressed by one soft gluon, and then argue, through the differential equation in Eq. (10), that the all-order correction exponentiates as R(Q)=(α_s(Q)/α_s(µ_np))^{C_A S1/β0}, Eq. (11). They support the result with shower-based numerical tests: an asymptotic-α_s test for the transverse-momentum density (Fig. 3), finite-α_s tests for the C-parameter in the 3-jet region (Fig. 4), mass-scheme studies (Figs. 5 and 11), and a comparison with Pythia8 (Figs. 6 and 12). The authors are transparent that the recoil-map linearity for gluon endpoints is not yet proved. The central claim is the parameter-free exponent S1 and the resulting factorization T_all-order(Q)=T_qqbar R(Q).

Significance. The result is significant if it survives scrutiny: it turns what one might expect to be a non-global-like all-order problem into a single exponential, gives a concrete prediction for the Q-dependence of the leading power correction, and explains the apparent universality across observables. Strengths of the paper include the clean first-order derivation of S1 (Supplement §2), the parameter-free character of the exponent (the only fitted number, T_qqbar, affects the Pythia normalisation and not the exponent), the consistency with the external numerical result of Ref. [46], and the use of the open-source PanScales code for the numerical tests. The agreement with Pythia8 across observables and Q is compelling phenomenological support. The main caveat is the unproved all-order longitudinal-recoil factorization used in Eq. (10).

major comments (2)

[Eq. (10) and Supplement §3] The all-order step in Eq. (10) assumes that the non-perturbative longitudinal recoil from the qg and gqbar dipoles carries the same factor R(Q) as the transverse momentum density. The first-order statement Eq. (7) does not determine this factor, because it concerns only the integral of ρ−1. The recursive argument in Supplement §3 iterates an S_+(Y) relation, but each iteration treats the gluon endpoint on the same footing as a quark endpoint; the text itself states in the main body and in Supplement §1 that the linearity of the recoil maps for gluon endpoints is not yet proved. The numerical check in Eqs. (40a) and (40b) involves a single perturbative soft gluon and one smooth rapidity boundary, so it does not test arbitrary multi-gluon configurations. If the longitudinal factor differs from the transverse one, Eq. (10) would still exponentiate but with a different anomalous-dimension constant; if the longitudinal response is not proportional to R(Q) at all, the factorization Eq. (12) is lost. I ask the authors to either provide a proof of the gluon-endpoint linearity or to add a numerical test that directly measures the longitudinal response for multi-gluon clouds (for example, the S_+(Y) observable with several successive perturbative emissions) rather than relying on the transverse-density test of Fig. 3.
[Fig. 4 and the beyond-two-jet claim] The abstract states that the simplicity holds 'also even beyond the two-jet limit'. The evidence of Fig. 4 is a finite-α_s shower test at fixed λ: the ratio of the cloud-corrected C-parameter shift to the bare 3-parton shift is flat and agrees with the 2-parton semi-analytic result. This is supportive but does not have the asymptotic α_s→0 extraction that the 2-jet Fig. 3 uses to isolate single-logarithmic terms, and the comparison is made at fixed λ and α_s values, so subleading logarithmic effects are not controlled. The statement that the anomalous dimension is universal across the full spectrum would be stronger if the authors provided an α_s→0 version of the 3-jet test, or explicitly marked the beyond-two-jet universality as a conjecture at this stage.

minor comments (4)

[Supplement Eq. (20)] The expression `(lndV/V)−3/4` should be parenthesised as `(ln(dV/V)−3/4)` to avoid ambiguity with the factor multiplying S1 in the denominator.
[Fig. 5 caption] The caption of Fig. 5 does not identify which line style or colour corresponds to the D, E and P schemes; please add this information so that the reader can read the figure without referring to the text.
[Main text, paragraph after Eq. (9)] The phrase 'straightforwardly extended to all orders' is stronger than the subsequent caveats warrant; I suggest rewording to 'we argue' or 'we conjecture' until the factorization property in Eq. (10) is established.
[Fig. 3 caption] The caption says α_s→0 for fixed λ but physically this requires ln(pt,max/pt,min) to diverge; it may be clearer to write 'the limit of asymptotically small α_s at fixed λ'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the exponent is a recursive resummation solution of the first-order response, benchmarked against external results; the all-order recoil equality is an explicitly stated assumption, not a fitted or self-referential input.

full rationale

The central result Eq. (11) is not obtained by fitting or by defining R(Q) to be the exponential. The first-order coefficient S1 comes from the analytic integral in Eq. (7) / Supplement §2, and the paper verifies it against the independent numerical calculation of Ref. [46] and against the 3-jet power-correction results of Refs. [25,35,36]. Eq. (10) is a differential equation for R(Q) whose RHS is proportional to R; exponentiating an ODE of the form dR/dlnQ = c R is not circular. The only step that carries an all-order burden is the sentence in the main text: 'we make use of the property that the non-perturbative all-order longitudinal recoil from the qg or gqbar dipoles gets the same R(Q) factor as the transverse momentum density ([53], §3).' This is presented as a framework assumption, tested numerically in Supplement §3 only for a single smooth rapidity boundary (Eq. (40)); it is a genuine correctness/rigor limitation, but it is not equivalent to the claimed result by construction. Likewise, the Pythia comparison calibrates one overall normalisation T_qq at Q=1000 GeV and then compares Q-dependence and event-shape dependence; that single parameter does not build in the exponent. Self-references to the supplemental material are not load-bearing in a circular way because the S1 value and the all-order benchmark are cross-checked against external, non-overlapping results: Ref. [46], the [25] code, and the independent operator-based calculation of Ref. [62]. I therefore find no qualifying circular step.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central exponent contains no fitted constants: S1 is derived analytically. Two load-bearing assumptions are the linear recoil paradigm from Refs. [35,36] and the factorization property behind Eq. (10). The only numeric fit, T_qqbar = 0.68 GeV, is used for the Pythia comparison.

free parameters (1)

T_qqbar (Pythia effective normalisation) = 0.68 GeV
Set by requiring agreement between gluer insertion and Pythia8 for the average C-parameter at Q = 1000 GeV; determines only the overall normalisation of the Pythia comparison, not the anomalous exponent.

assumptions (5)

domain assumption Observables are linear in soft momenta: V(soft partons) - V(hard) = sum_i (k_ti/Q) f_V(eta_i) + O(k_ti^2/Q^2) (Eq. 2).
Defines the class of observables (thrust, C-parameter, EEC) for which the gluer method applies.
domain assumption Kinematic recoil maps (PanGlobal/PanLocal) are linear in the gluer momentum for gluon dipoles.
Taken from Refs. [35,36]; the text states this is 'yet to be proved for final-states with gluons'.
domain assumption Large-Nc limit: replace 2 C_F by C_A and treat the gluon as equivalent to a quark for dipole emission.
Used in deriving Eqs. (8) and (11); colour-suppressed contributions are dropped.
ad hoc to paper All-order factorization of longitudinal recoil: the non-perturbative effect on any rapidity patch acquires the same R(Q) factor as the transverse momentum density.
Underlies Eq. (10); argued heuristically in supplement section 3 and validated numerically only for one smooth boundary.
domain assumption A single on-shell gluer reproduces the off-shell double-soft result up to a universal Milan factor.
Standard renormalon/gluer paradigm from Refs. [20-23], used in Eq. (1).

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Cite this review

Pith. "Pith review of Anomalous scaling of linear power corrections." pith.science (2026). https://pith.science/paper/NQOPKZYI

@misc{pith2026250718696,
  author       = {Pith},
  title        = {Pith review of: Anomalous scaling of linear power corrections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQOPKZYI}},
  note         = {Machine review of arXiv:2507.18696}
}

abstract

Non-perturbative corrections to hadronic observables represent a critical obstacle to increasing accuracy at colliders. Long taken to scale simply as $1/Q$, where $Q$ is the centre-of-mass scattering energy, recent work has opened the path towards calculating the anomalous dimension that modifies that scaling. A priori, the problem is complex, requiring a resummation involving arbitrary numbers of large-angle and low-energy gluons. Within a specific framework for kinematic recoil, we show that it reduces to a simple exponential for key observables like the thrust, $C$-parameter and energy correlators. This simplicity holds for a specific hadron mass scheme, and also even beyond the two-jet limit.

Figures

Figures reproduced from arXiv: 2507.18696 by the authors.

**Figure 2.** FIG. 2. The effect of gluer emission on transverse momentum [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. All-order evaluation of [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: uses a soft-gluon cloud with pt,max ≪ Q. In practice, when the observable has a value V ≪ 1, the actual pt,max is limited to be ≲ ξV Q, with ξ of order 1. Thus we expect the shift to involve a factor R(ξV Q) rather than R(Q), bringing additional V -dependence relative…

**Figure 5.** Figure 5: FIG. 5. Dependence of the scaling factor [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 7.** Figure 7: FIG. 7. Average non-perturbative shift for thrust (left) and [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Approach to the 2-jet limit for thrust (left) and [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. 2d grid of [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. Numerical all-order results for [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

**Figure 11.** Figure 11: FIG. 11. Calculation of [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: FIG. 12. Non-perturbative shifts in the [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

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Comparison of PanGlobal and PanLocal results to those in the literature The core formula that we use when identifying the coefficient of the non-perturbative power correction with a given kinematic map is ⟨δVnp⟩q¯q= Z µnp 0 dktn ktn dηn 2CFα(eff) s (ktn) π [V(p 1,p 2,kn)−V(˜p ...
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Analytic derivation ofS 1 The essence of our derivation will be to consider an observable that is the sum of transverse momenta inside a rapidity window|η|<YwithY≫1 St(Y) = X i ptiΘ(|ηi|<Y),(22) where thep i in this definition are to be understood as running over all momenta, ...
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Universality of longitudinal recoil The observables that we consider in the main text all have the property thatfV (η) in Eq. (2) vanishes for large soft- particle rapiditiesη i, so that for aq¯qsystem longitudinal recoil of theqand ¯qdoes not affect the observable. However, a...
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Numerical evaluation ofR Given the surprising simplicity of Eq. (11), it is important to verify it numerically. To do so, we start with aq¯q system, and use a perturbative parton shower to add a cloud of soft gluons between scalesp t,min andp t,max≪Q. Summing over all the pert...
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Mass schemes Calculations of non-perturbative power corrections typically use massless partons: indeed, if one has a massive gluer as an intermediate step, a typical full calculation will account for the decay of that massive gluer, according to the well-known double-soft matr...
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Study of Q dependence of shifts in distributions We determine hadronisation corrections in Pythia8 as follows. Firstly, we run Pythia8 including hadronisation and determine the hadron-level value of an observable from the hadrons. We then inspect the event record to identify t...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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