{"id":"64c79541-f577-4b1c-a2f3-b4a4cf42543c","arxiv_id":"2507.18696","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For thrust, C-parameter, and energy correlators, the 1/Q hadronization correction is multiplied by R(Q) = (alpha_s(Q)/alpha_s(mu_np))^{C_A S_1/beta_0} with S_1 = 8(1-ln2), an all-order exponential derived under the linear-recoil assumption.","lead":"This paper derives a simple exponential rule for how non-perturbative hadronization corrections to particle-collision observables change with collision energy. The result can sharpen predictions of the strong coupling constant from event-shape data.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (10) assumes the all-order longitudinal recoil factor equals the transverse R(Q); this is only verified at first order for one smooth boundary, so the central exponentiation is conditional.","rationale":"The reader's weakest assumption identifies exactly the load-bearing point: the all-order step in Eq. (10) requires a universality between longitudinal and transverse non-perturbative recoil that is argued heuristically and checked only in a first-order, single-boundary setting. My independent reading of the manuscript confirms this. The paper itself flags the underlying gluon-endpoint linear-recoil paradigm as unproved, both in the main text ('this is yet to be proved for final-states with gluons') and in Suppl. Sec. 1 ('there remain open questions as to the validity of this general approach when one of the dipole ends is a gluon'). The recursive argument in Suppl. Sec. 3 is a plausible sketch but not a proof: each recursion step assumes the gluon end behaves like a quark end, which is the very point at issue. The numerical support is real, including the direct-insertion checks in Fig. 10 and the consistency with Ref. [62] for the EEC in the D-scheme, but those tests do not independently verify the all-order longitudinal-recoil universality for arbitrary multi-gluon clouds. Therefore the correct disposition is conditional: the result is well-supported but not fully established, and a targeted numerical test can settle the remaining assumption. Since my concern does not change the reader's verdict, no adjustment is needed.","tokens_in":23294,"tokens_out":5216,"duration_ms":62846,"concrete_test":"Run the direct gluer-insertion probe of Eq. (13b) on PanScales PG sdf beta=0 showers at asymptotically small alpha_s with fixed lambda, measuring the change in a rapidity-window plus-momentum observable S+(Y), or its smoothed version Eq. (38), for several boundary choices Y. Extract the all-order factor R_long(Q) defined by <delta S+> = -T_qqbar R_long(Q) e^Y (or the smoothed analogue) and compare it with the transverse R(Q) obtained in Fig. 3 / Eq. (11). A statistically significant difference (beyond the few x 10^-4 level) would falsify the universality assumption in Eq. (10); agreement would close the main open gap in the derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central exponentiation rests on the step in Eq. (10), where the all-order longitudinal recoil from qg and gqbar dipoles is assigned the same factor R(Q) as the transverse momentum density. This is not established by the NLO result: Eq. (7) fixes only the first-order integral of rho-1, while R(Q) is defined via the transverse density. Supplement Sec. 3 attempts a recursive justification using S+(Y), but the recursion assumes that a gluon endpoint of a dipole responds to gluer recoil exactly like a quark endpoint. That assumption is precisely the unproved gluon-endpoint linearity admitted in the main text and in Suppl. Sec. 1. The numerical check in Eq. (40) is first order in alpha_s, involves a single perturbative gluon, and uses one smooth rapidity boundary; it does not test the all-order longitudinal factor for arbitrary multi-gluon clouds. If the longitudinal factor differs from the transverse R, Eq. (10) should read dR/dlnQ = (2 C_F alpha_s/pi) (integral [rho_long - 1]) R with a possibly different anomalous-dimension constant; the exponential form could survive but Eq. (11) would have a different exponent. If the longitudinal response is not proportional to R at all, the factorization in Eq. (12) fails outright.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":23698,"tokens_out":10425,"duration_ms":106591,"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":[{"comment":"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.","section":"Eq. (10) and Supplement §3"},{"comment":"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.","section":"Fig. 4 and the beyond-two-jet claim"}],"minor_comments":[{"comment":"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.","section":"Supplement Eq. (20)"},{"comment":"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.","section":"Fig. 5 caption"},{"comment":"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.","section":"Main text, paragraph after Eq. (9)"},{"comment":"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'.","section":"Fig. 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its main assumption, and the numerical work is of high quality. My main reservation is that Eq. (10) is a genuine gap between the first-order result and the all-order claim; this is exactly the sort of assumption that a journal referee should force the authors to either prove or label explicitly as a conjecture. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhere 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.\n\nSoft 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.\n\nMinor: 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.\n\nWho 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.","headline":"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.","tokens_in":24124,"tokens_out":3547,"would_cite":true,"duration_ms":36846,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["linear power corrections","renormalons","hadronisation","event shapes","thrust","C-parameter","energy-energy correlators","all-order resummation"],"falsifier":"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.","tokens_in":23071,"feed_emoji":"⚛️","tokens_out":5772,"duration_ms":61026,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hadronisation shifts reduce to a single exponential","feed_subtitle":"A new formula gives all-order non-perturbative corrections for thrust, C-parameter and energy correlators.","key_machinery":"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.","core_discovery":"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 $Q$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplied the first-order anomalous dimension $S_1$ for thrust and $C$-parameter, which this paper exponentiates.","marker":"[46]"},{"why":"Established the linear kinematic recoil maps for gluer insertion and the 3-parton power-correction results against which this paper validates.","marker":"[35, 36]"},{"why":"Provided the reference code and results for 3-parton non-perturbative shifts used to validate the recoil maps.","marker":"[25]"},{"why":"Defined the dipole-shower recoil maps used for gluer insertion and linearity properties.","marker":"[51]"},{"why":"Provided the transverse-momentum-ordered shower used for the all-order numerical test of Eq. (11).","marker":"[52]"},{"why":"Supplied the 3-jet matched map used in the validation of non-perturbative shifts.","marker":"[50]"},{"why":"Defined the E- and P-schemes for hadron masses and their anomalous scaling, which the paper compares with the D-scheme.","marker":"[57]"},{"why":"An operator-based calculation of the same anomalous scaling for the energy-energy correlator, quoted as consistent with this paper's result.","marker":"[62]"}],"fun_headline_variants":["Linear power corrections collapse to a single exponential","Anomalous scaling tamed: one exponent for thrust and more","One exponential formula for non-perturbative QCD shifts","All-order power corrections: one exponent for key observables"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Linear power corrections collapse to a single exponential","Anomalous scaling tamed: one exponent for thrust and more","One exponential formula for non-perturbative QCD shifts","All-order power corrections: one exponent for key observables"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000441,"raw_usage":{"total_tokens":2215,"prompt_tokens":904,"completion_tokens":1311,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":1244}},"tokens_in":520,"tokens_out":1311,"duration_ms":12857,"temperature":1.0,"reasoning_tokens":1244,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:08:46.741713+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}