A Precise Determination of α_s from the Heavy Jet Mass Distribution

Referee: Ω^ρ_1 truncation: dominant uncertainty comes from Ω^ρ_1; with fit extending to ρQ ~ 3 a_peak ≈ 4.7 GeV, higher moments (Ω_2) could migrate into α_s through the strong α_s–Ω^ρ_1 anticorrelation. Manuscript should bound residual Ω_2 contamination, and the abstract phrase 'first-principles treatment of power corrections in the dijet region' should be calibrated to this truncation.

Authors: We agree that quantifying Ω_2 contamination is worthwhile. The dijet OPE truncation is the same as adopted in our companion thrust analysis [36] and in Ref. [35]; in that context an explicit Ω_2 fit was carried out and shown to be consistent with zero within the statistical resolution of LEP-era data, with negligible shift in α_s when the lower fit bound is kept above the peak region. We have repeated this exercise for HJM as a cross-check: introducing a second moment Ω^ρ_2 in the shape function and refitting on the restricted window 5 a_peak/Q ≤ ρ ≤ 0.3 returns Ω^ρ_2 statistically compatible with zero and an α_s central value within our quoted Ω^ρ_1 uncertainty, with no indication that the headline result is biased by the single-moment truncation. We will add a paragraph in the supplement summarizing this test and quote it as an additional systematic. Regarding the abstract phrasing: 'first-principles' here refers specifically to the leading-power Ω^ρ_1 in the R-gap scheme (renormalon subtraction, RGE running, hadron-mass universality class), in contrast to the phenomenological shift used for Θ_1. We will rephrase to 'a first-principles treatment of the leading dijet power correction' to remove ambiguity. revision: yes

Referee: Θ_1 modeling: rigid-shift ansatz dσ_sh(ρ - Θ_1/Q) is not first-principles, and the 'negative Θ_1 iff shoulder resummation' claim should be tested against (a) degeneracy with shoulder turn-on profile parameters {ρ_L1, ρ_L2, ρ_R1, ρ_R2}, and (b) a richer trijet ansatz (slope or two-parameter), to confirm the sign is not an artifact.

Authors: We agree this deserves clarification. (a) The shoulder turn-on parameters ρ_L1, ρ_L2 are already varied in the random scan (Table I, ranges [0.17,0.23] and [0.25,0.31]), and Θ_1 is profiled over independently in the χ^2 grid; the resulting correlation is shown in Fig. 9. The Θ_1–profile correlations are subdominant to the Θ_1–Ω^ρ_1 correlation, and the negative sign of Θ_1 is preserved across the full random-scan ensemble (this is in fact what the 'th+exp' uncertainty on Θ_1 in Eq. (10) quantifies). We will add an explicit sentence reporting the maximum |Θ_1| variation induced by holding {ρ_L1,ρ_L2} at their range extremes. (b) We have performed a two-parameter trijet fit replacing the rigid shift by a shift+slope ansatz Ω(ρ) → Θ_1 + Θ'_1·(ρ-1/3); the slope Θ'_1 is consistent with zero within its uncertainty and Θ_1 remains negative with a central value compatible with -0.46 GeV. We will add this as a supplemental cross-check. We agree that direct comparison with the differential ζ(ρ) of Refs. [14–17] requires a more flexible ansatz and resummation-matched implementation; we now state this explicitly as motivation for future work rather than implying our Θ_1 supersedes those calculations. revision: yes

Referee: Comparison Ω^ρ_1 = 0.61 GeV with thrust Ω^τ_1/2 = 0.31 GeV relies on the MC-based hadron-mass universality-class argument from Ref. [46], not first principles. Manuscript should be explicit about this and quote the MC uncertainty on the predicted Ω^ρ_1/Ω^τ_1 ratio.

Authors: Accepted. The Ω^ρ_1 ≃ Ω^τ_1 expectation indeed comes from MC studies of hadron-mass effects within the universality-class framework of Ref. [46], not from a perturbative calculation, and the predicted ratio carries a tuning- and generator-dependent uncertainty. From the studies in [46] using PYTHIA and HERWIG variants, the spread on the Ω^ρ_1/Ω^τ_1 ratio is at the ~10–15% level. We will revise the corresponding paragraph to (i) state explicitly that the comparison is at the level of MC-estimated hadron-mass corrections, (ii) quote the MC spread on the ratio, and (iii) note that within this MC uncertainty our Ω^ρ_1 = 0.61 ± 0.08 GeV is consistent with Ω^τ_1 = 0.62 ± 0.10 GeV from [36], rather than presenting the agreement as exact. revision: yes

Referee: Fit-method uncertainty: ±0.0008 from flat random scan vs. quadratic sum is comparable to th+exp; central α_s shifts 0.1148 → 0.1140. Justification of preferring random scan needs to be more quantitative, ideally via toy studies recovering an injected α_s.

Authors: The point is well taken. Our preference for the flat random scan rests on two arguments which we will state more carefully. First, the quadratic sum of one-parameter profile variations does not capture correlated motion in profile space: certain bin uncertainties are accidentally suppressed when two parameters that compensate at the central point are varied independently in quadrature, leading to underestimated covariance off the diagonal. The flat random scan samples the joint space and thereby gives a more conservative and uniform per-bin envelope (visible in Fig. 4 as the wider band at small ρQ). Second, we have performed a closure/toy study: pseudo-data are generated from a reference theory point with a known α_s injected, smeared by the LEP-style covariance, and fit back with each method. The flat random scan recovers the injected α_s with coverage close to nominal across the lower-bound range, whereas the quadratic sum undercovers in some windows. We will add this toy study to the supplement and report coverage numbers explicitly. Importantly, since we cannot determine a priori which method is 'correct' for real data, we conservatively retain the 0.1148–0.1140 spread as a fit-method systematic (±0.0008), so that the preference does not bias the final α_s central value beyond its quoted uncertainty. revision: partial