{"id":"b2d26998-6fa3-41f8-9379-bf20df04668f","arxiv_id":"hep-ph/0312331","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A new effective theory makes resonant production of unstable particles computable as a systematic double expansion in the coupling and the width-to-mass ratio, demonstrated at next-to-leading order for a toy scalar resonance.","lead":"The authors build an effective field theory that separates the large mass of an unstable particle from its small decay width, so that resonant production cross sections can be computed order by order in the coupling and in width over mass. They demonstrate it by computing the next-to-leading-order line shape of a scalar resonance in a toy model.","discovery_kind":"new_method","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"Eq. (17), the letter's only quantitative demonstration, depends on matching coefficients C^(1) and Delta^(2) that are not given here and are cited to an unpublished companion, so the NLO line shape cannot be checked from the manuscript alone.","rationale":"The reader's weakest_assumption correctly identifies the load-bearing soft spot: the matching coefficients that carry the entire NLO result are taken from Ref. [4], listed as 'in preparation', and the letter neither derives nor independently checks them. I agree with this diagnosis and with the CONDITIONAL verdict. The construction itself is internally coherent: the split into L_HSET, L_SCET and L_int is motivated, the power counting in delta is stated term by term, the need for an external-collinear mode is argued rather than asserted, and the gauge-invariance argument for the minimal-substitution step is standard. The one place where the argument could fail is the matching, and that is exactly where the letter is silent. The figure does not supply the missing check because it compares NLO EFT to LO full theory off resonance, which the authors themselves flag as insufficient for a consistent NLO result. I considered whether a sharper concern exists elsewhere, for example in the power counting of the external-collinear mode or in the sign conventions of Eq. (7), but the apparent discrepancies there resolve as OCR artifacts or standard conventions, and none rises to the level of an internal inconsistency. The Ref. [4] dependence is therefore the single most load-bearing concern, and it is a legitimate one under the rule that an 'in preparation' source may be questioned. The concrete test above would settle whether the concern lands: if C^(1) and Delta^(2) reproduce Eq. (17), the demonstration stands; if not, the letter's quantitative claim fails even though the method claim might survive. No verdict adjustment is warranted beyond the reader's CONDITIONAL.","tokens_in":8737,"tokens_out":10378,"duration_ms":198029,"concrete_test":"Independently recompute the on-shell phi-psi-chi three-point function at O(y alpha) in the full theory, extract C^(1) in the MS scheme including the omega^{-1/2} factor, and check whether it reproduces the a_g and a_y coefficients displayed in Eq. (17). If the extracted coefficients differ, Eq. (17) is wrong and the NLO demonstration fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the EFT organizes resonant amplitudes into a systematic expansion in alpha and Gamma/M, and its demonstration is the NLO line shape Eq. (17). That equation is built from three matching quantities: Delta^(1), Delta^(2) and C^(1). The letter gives Delta^(1) = -i Gamma in the pole scheme and defines Delta^(2) through Eq. (9) in terms of the hard self-energy, but never evaluates it. C^(1) is not written down at all; the text says only that it is obtained by matching the on-shell phi-psi-chi three-point function at O(y alpha) and refers to Ref. [4], which is listed as 'in preparation'. The wave-function normalization factor omega^{-1/2} introduced after Eq. (10) is momentum-dependent and must enter that matching; if it is mishandled, the a_g coefficient in Eq. (17) shifts. Consequently the reader cannot verify that the EFT construction, including the external-collinear mode and the omega factor, actually reproduces the full-theory NLO line shape. Figure 1 does not close this gap: it compares the NLO EFT curve to the LO full-theory off-resonance cross section, a mismatch the authors acknowledge when they say a consistent NLO result would require matching to the NLO off-resonance cross section. The concern is not that the method is wrong; it is that the only quantitative check of the method rests on calculations the letter does not contain and that were not published at the time of writing.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The letter proposes an effective field theory for resonant production and decay of a heavy unstable scalar, organized by the hierarchy Γ ≪ M. Three ingredients are introduced: an HQET-like soft scalar field φ_v obtained by removing the rapid phase e^{iMv·x} (Eqs. (4)–(10)); an SCET-like collinear fermion sector (Eq. (12)); and a new 'external-collinear' mode carrying the fixed large momentum M n₋/2 + k with k ~ δ (Eq. (13)), which allows the production and decay vertices to be written as local operators in L_int (Eq. (15)). The effective Lagrangian is matched to the full theory with the method of regions, and the forward-scattering amplitude is computed through NLO, Eq. (17), for a toy abelian gauge–Yukawa model with a scalar, an 'electron' and a 'neutrino'. Figure 1 shows the LO and NLO EFT line shapes together with the LO full-theory off-resonance cross section. The stated aim is a systematic expansion in α and Γ/M that replaces ad hoc width resummation and that can be carried to NNLO.","tokens_in":9062,"tokens_out":7141,"duration_ms":127257,"significance":"If the construction is correct, this is a genuine methodological advance: it turns the treatment of unstable-particle resonances from a prescription (resum self-energies, hope for the best) into an expansion with an explicit power-counting parameter δ ~ Γ/M, in direct analogy with HQET and SCET. The paper earns credit for several things. The power counting is stated explicitly for every field and operator, so the truncation of Eqs. (10), (12) and (15) can be checked line by line. Gauge invariance is preserved by construction through covariant derivatives rather than restored by hand. The external-collinear mode of Eq. (13) is a real structural addition, not a relabelling, and it is what makes the production vertex local. No parameter is fitted: α_g, α_y, α_λ and M are inputs, and the NLO coefficients are definite numbers, so the framework is falsifiable against a full-theory calculation. The claim that NNLO line shapes become feasible is plausible given the operator basis assembled here.","major_comments":[{"comment":"Eq. (17) is the only quantitative result in the letter, and it cannot be checked from the manuscript. The coefficient C^(1) is never written down; the text says only that it follows from matching the on-shell φψχ three-point function at O(yα) and refers to Ref. [4], which is listed as 'in preparation'. Likewise Δ^(2) is defined only implicitly through Eq. (9) in terms of the hard self-energy and is never evaluated, although it enters Eq. (17) through the terms −[Δ^(1)]²/(8D M̂) + Δ^(2)/(2D). Since the a_g and a_y coefficients of Eq. (17) are the substance of the NLO claim, the reader has no way to verify the matching algebra, the scheme dependence, or the cancellation of μ between C^(1) and Δ^(2). Please give C^(1) explicitly in the MS scheme and give the explicit expression (or numerical value at the reference point) for Δ^(2), so that the letter is self-contained. If space is the constraint, the matching equation alone would already allow an independent check.","section":"Paragraph beginning 'The coefficient C(1) is obtained by matching', and Eq. (17)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The wave-function normalization factor ̟^{-1/2} introduced after Eq. (10) is momentum dependent, ̟^{-1} = (M̂² + M̂Δ − k⊥²)^{1/2}/M̂ = 1 + O(δ, α), and therefore its correction is of the same nominal order as the NLO terms retained in Eq. (17). The text asserts that it 'is taken into account in matching calculations', but never shows how it enters C^(1), nor whether its k⊥ dependence survives the leading-order-in-δ matching. Because this factor multiplies each external φ_v line, an error here would shift the a_g coefficient of Eq. (17) directly. Please display the contribution of ̟^{-1/2} to the matching equation for C^(1), or state explicitly why it can be dropped at the accuracy of Eq. (17).","section":"Paragraph after Eq. (10), the factor ̟^{-1/2}","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The letter claims that Eq. (17) 'describes the line shape near resonance with a relative error of α²', but no comparison between the EFT NLO result and a full-theory NLO calculation appears anywhere in the manuscript. Figure 1 shows the NLO EFT curve next to the LO full-theory off-resonance cross section, a mismatch the authors themselves acknowledge when they note that a consistent NLO result would require matching to the NLO off-resonance cross section. As it stands, the accuracy claim is asserted rather than demonstrated, and the reader cannot tell whether the residual difference is the expected O(α²) or something larger. Please either add the full-theory NLO curve (or the expansion of the full-theory NLO result near resonance, which should reproduce Eq. (17)) or state explicitly that this validation is deferred to Ref. [4] and quantify the expected residual.","section":"Figure 1 and the paragraph beginning 'We can now perform the polarization average'","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The second equality in Eq. (7) appears garbled in the version supplied ('= Δ/2 − Δ² + 4k⊥²/8M̂ + O(δ³)'); the intended expression is presumably Δ/2 − (Δ² + 4k⊥²)/(8M̂). Please check this line carefully at proof stage, since the subsequent power counting of Eq. (8) depends on it.","section":"Eq. (7)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The discussion of scaling refers to 'the first line of (8)', 'the second line' and 'the last line of (8)', but the three-line display being discussed is Eq. (10). Please correct the equation references so the power-counting argument can be followed without ambiguity.","section":"Text following Eq. (10)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The initial-state collinear singularities are said to be 'subtracted minimally' with factorization scale ν, but the subtraction prescription is never specified. The coefficient of the double logarithm ln(−2M̂D/M̂²) ln(−2M̂D/ν²) in the a_g term depends on that prescription, so the ν dependence of Eq. (17) cannot be checked independently. A one-sentence definition of the subtraction would remove the ambiguity.","section":"Eq. (17) and the collinear subtraction","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The four-fermion coefficient is quoted as F^(0) = 1/4 with the remark that it follows from matching the four-point function at tree level including subleading terms in δ. Since the result is a single number, a one-line indication of which subleading terms in the external-collinear projectors produce it would make the operator basis easier to trust.","section":"Eq. (15)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption distinguishes 'light grey/magenta dashed' (LO EFT) from 'light grey/magenta' (NLO EFT) and 'dark grey/blue dashed' (LO full theory), which is hard to decode in print. Please add a legend inside the figure and state in the caption the renormalization and factorization scales used for each curve.","section":"Figure 1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text states that the pole mass M = 100 GeV corresponds to M̂ = 98.8 GeV at LO and M̂ = 99.1 GeV at NLO, but does not explain why the converted MS mass differs between the two orders. A brief clarification of the conversion used would help the reader interpret the horizontal positions of the two curves.","section":"Numerical inputs, paragraph before Figure 1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The introduction dismisses the existing resummation prescriptions of Ref. [1] in a single sentence ('there is no method that allows to systematically improve the accuracy'). A short, concrete statement of what those approaches leave uncontrolled, phrased in terms of the δ counting introduced here, would let the reader judge the advance without consulting the companion paper.","section":"Introduction and Ref. [1]","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision"},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"Worth your time if you work on line shapes or resonance systematics, but read it as a framework paper, not a self-contained calculation. The core idea is that the hierarchy Gamma << M lets you build an EFT for resonant production the way HQET and SCET handle their own hierarchies, and the authors carry it off with the expected competence. The three-part split into L_HSET (soft scalar, HQET-style), L_SCET (collinear fermions, SCET-style) and L_int is clean, and the power counting in delta is stated term by term rather than waved at. The genuinely new piece is the external-collinear mode of Eq. (13): generic collinear momenta don't add up to a near-mass-shell scalar momentum, so they introduce a mode with a fixed large component M n_-/2 plus residual k ~ delta. That's the right fix, and it's argued rather than asserted. The assembly is the contribution here; the ingredients are borrowed and the authors say so. The soft spot is exactly where the stress-test note puts it, and it's real. Eq. (17), the NLO amplitude, is the only quantitative deliverable, and it's built from Delta^(1), Delta^(2) and C^(1). Delta^(1) = -i Gamma in the pole scheme is given; Delta^(2) is defined through Eq. (9) but never evaluated; C^(1) is not written down at all, only described as coming from matching the on-shell phi-psi-chi three-point function at O(y alpha). All of that lives in Ref. [4], listed as 'in preparation'. So a reader cannot check Eq. (17) from this manuscript. That's a normal letter-to-companion arrangement, but it does mean the one number the paper stands on is unverified here, and the momentum-dependent omega^{-1/2} normalization factor they introduce after Eq. (10) has to be handled correctly inside that matching. The figure doesn't close the gap either: it compares the NLO EFT curve to the LO full-theory off-resonance cross section, which the authors themselves flag as needing an NLO off-resonance match. That's acknowledged, not hidden. Nothing here is circular; the coefficients are matched to full-theory diagrams and no parameter is fitted. The toy abelian model is a demonstration, not a claim about the Standard Model, and they say as much. My read: the method is sound and the construction is the real result; the missing loop details are an availability problem, not a soundness one. Send it out. A referee should press on whether the companion's matching is consistent with the omega factor and whether the power counting survives beyond the abelian case, but this deserves referee time.","headline":"A serious, well-constructed EFT framework for resonant unstable-particle production, but the letter's one quantitative result leans on an unpublished companion for its matching coefficients.","tokens_in":9809,"tokens_out":1097,"duration_ms":20929},"pith_extraction":{"msc":[],"pacs":["11.15.Bt","11.80.Cr"],"model":"selfhosted/deepseek-v4.1-flash","headline":"The paper argues that resonant production of an unstable particle can be turned into a controlled expansion in the coupling and in the width-to-mass ratio, using an effective field theory with three kinds of modes, and demonstrates it by a ","keywords":["unstable particles","effective field theory","resonance line shape","soft-collinear effective theory","heavy quark effective theory","method of regions","width resummation","next-to-leading order corrections"],"falsifier":"An independent full-theory calculation of the same toy model's line shape through next-to-leading order, compared against the paper's expression near resonance, would settle it: agreement to the quoted relative accuracy would confirm the expansion, while a discrepancy scaling like the coupling squared would show that the matching coefficients or the external-line normalization are wrong.","tokens_in":9998,"feed_emoji":"⚛️","tokens_out":2505,"duration_ms":47093,"temperature":0.7,"pith_summary":"Near a resonance the propagator of an unstable particle blows up, and the standard fix — resumming self-energy insertions into a width — has long been regarded as an ad hoc patch rather than the first term of a controlled series. This paper argues that the hierarchy $M$ (pole mass) versus $M$ times the coupling, with $s-M^2$ of order $M^2$ times the coupling, is a genuine multi-scale problem of the kind effective field theories are built for, and that the cross section can therefore be expanded systematically in $a_i = a_i/(4)$ and in $s$ shifted from the mass squared. The authors construct such a theory for a toy model with a charged scalar decaying to an electron and a neutrino, and use it to compute the forward scattering amplitude through next-to-leading order, i.e. the line shape including all terms suppressed by one extra power of the coupling or of the width-to-mass ratio. If the construction is right, higher-order line-shape calculations become a matter of computing matching coefficients and soft loops order by order, rather than of guessing which self-energy graphs to resum. The paper is explicit that it is a demonstration in a simplified abelian gauge-Yukawa model, not a Standard Model calculation.","feed_headline":"Resonances get a controlled expansion in width over mass","feed_subtitle":"Splitting the unstable scalar into soft, collinear and external-collinear modes turns width resummation into an order-by-order series.","key_machinery":"The load-bearing object is the three-mode decomposition of the effective Lagrangian, $L = L_{HSET} + L_+ + L_{int}$, together with the power counting in the single small parameter $s - M^2$ over $M^2$, which is the same size as the width over the mass. Each field is assigned a scaling in this parameter: the soft scalar propagates as $1/s$ so the field scales as $s^{3/2}$, soft gauge fields scale as $s$, external fermions scale as $s^{3/2}$, and the external state scales as $s^{-1}$, which is how the leading amplitude ends up of order $a/s$. The matching coefficients — the shift $s$ of the complex pole, the vertex coefficient $C$, and the four-fermion coefficient $F$ — absorb all hard momenta of order $M$ and are computed by the method of regions, expanding integrands in the off-shellness and assigning the difference to the near-mass-shell modes. A wave-function normalization factor is applied to each external scalar line to account for the difference between the full propagator and the one the effective theory reproduces.","core_discovery":"The central claim is that the resonant line shape admits a systematic expansion, and that the machinery which produces it is an effective Lagrangian split into three pieces: a heavy-scalar part built exactly as in heavy quark effective theory by removing the rapid phase $e^{-iMvx}$ from the field, a collinear part for the energetic charged fermions borrowed from soft-collinear effective theory, and a new 'external-collinear' mode whose momentum is pinned to $M n_-/2 + k$ so that the two incoming fermions can actually add up to a nearly on-shell scalar. The last of these is the paper's own device: generic collinear fields do not sum to a resonant scalar momentum, so either the kinematic constraint must be imposed by hand on external states or a dedicated mode must be introduced, and the authors choose the second. With this Lagrangian the leading amplitude is a single diagram with two three-point vertices and one resonant propagator, and the next-to-leading correction assembles from hard propagator and vertex insertions, a four-fermion contact operator, and soft-photon loops that correspond to the non-factorizable corrections. The paper exhibits the resulting next-to-leading expression for the amplitude in closed form, and shows the leading and next-to-leading line shapes numerically for $M = 100$ GeV with $a_y = a_g = 0.1$, noting that the effective-theory curve and the full-theory tree-level off-resonance cross section agree in the intermediate region where both are valid, which is what allows a consistent result across all energies.","pith_inferences":["The external-collinear mode is the piece most likely to generalize: any process where two energetic particles must fuse into a particle of fixed large momentum has the same mismatch between generic collinear momenta and the resonant momentum, so the device may carry over to electroweak vector-boson fusion and to top-quark threshold production.","If the expansion is truly systematic, the residual dependence on the factorization scale used to subtract the initial-state collinear singularity should cancel between the hard coefficient and the collinear sector at each order; checking that cancellation would be a sharp internal test of the construction beyond what the letter shows.","The line shape's sensitivity to the scheme in which the mass is defined is visible in the numbers quoted: the same 100 GeV pole mass corresponds to a different renormalized mass at leading and next-to-leading order, which suggests the matching coefficients, not the mass parameter, are where scheme dependence is really being organized.","A natural extension is to unstable particles carrying spin, where the heavy-mode construction requires a velocity-dependent spinor structure rather than the simple phase removal used for the scalar, and where the width enters the numerator as well as the denominator of the propagator."],"forward_implications":["Line-shape predictions near a resonance acquire a definite error budget: the truncation at next-to-leading order leaves a relative error of order the coupling squared, which can be reduced by going to the next order rather than by choosing among resummation prescriptions.","The split between hard matching coefficients and soft, non-factorizable loop corrections becomes explicit, so factorizable and non-factorizable contributions can be computed separately and combined without double counting.","Because the expansion is order-by-order in the width-to-mass ratio, the same construction can in principle be pushed to next-to-next-to-leading order, which the authors state appears feasible.","The method gives a principled way to join the resonance region to the off-resonance full-theory result, since the two descriptions overlap in an intermediate region where both are valid."],"supporting_citations":[{"why":"Collects the existing width-resummation prescriptions that the paper positions itself against as lacking a systematic improvement scheme.","marker":"[1]"},{"why":"Supplies the earlier step of treating resonant production with an effective-theory-like separation of hard and non-factorizable contributions.","marker":"[2]"},{"why":"Provides the method of regions used to split loop integrals and to define the hard matching coefficients.","marker":"[3]"},{"why":"Supplies the heavy quark effective theory construction, which is copied for the soft scalar field: phase removal, velocity vector, residual momentum.","marker":"[5]"},{"why":"Supplies the soft-collinear effective theory framework from which the collinear fermion Lagrangian and light-cone vector conventions are taken.","marker":"[6]"},{"why":"Provides the collinear Lagrangian worked out to the order in the expansion parameter that the calculation needs.","marker":"[7]"},{"why":"The companion paper cited as in preparation, from which the matching coefficients entering the next-to-leading amplitude and the explicit vertex result are taken.","marker":"[4]"}],"fun_headline_variants":["Resonant line shapes expand order by order in width over mass","A new external-collinear mode pins the resonance kinematics","Width resummation becomes a systematic series for resonances","Line-shape agreement holds only at tree level in the intermediate region","Next-to-leading resonance amplitude assembled from hard and soft pieces"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire next-to-leading result rests on matching coefficients that this letter does not derive but takes from a companion paper listed as in preparation, so the numerical line shape is only as trustworthy as those unpublished computations and the handling of the external-line normalization that goes with them.","fun_headline_variants_meta":{"raw":{"variants":["Resonant line shapes expand order by order in width over mass","A new external-collinear mode pins the resonance kinematics","Width resummation becomes a systematic series for resonances","Line-shape agreement holds only at tree level in the intermediate region","Next-to-leading resonance amplitude assembled from hard and soft pieces"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1798,"prompt_tokens":903,"completion_tokens":895,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":824}},"tokens_in":903,"tokens_out":895,"duration_ms":15464,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":824,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-13T08:56:47.816748+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"pipeline_version":"pith-pipeline@v1.5.0-archive-selfhost","input_kind":null,"input_chars":null,"input_total_chars":null,"passes_missing":null,"published_standing":null,"published_confidence":null,"objection_audit":null,"run_record":null,"basis":{"kind":"not_recorded","source":null,"chars_given":null,"chars_total":null},"receipt":null,"falsifier":"An independent full-theory calculation of the same toy model's line shape through next-to-leading order, compared against the paper's expression near resonance, would settle it: agreement to the quoted relative accuracy would confirm the expansion, while a discrepancy scaling like the coupling squared would show that the matching coefficients or the external-line normalization are wrong.","supporting_citations":[{"cited_title":"Unstable particles in One Loop Calculations","cited_arxiv_id":"hep-ph/9312212","evidence_quote":"Collects the existing width-resummation prescriptions that the paper positions itself against as lacking a systematic improvement scheme."},{"cited_title":"Non-factorizable corrections and effective field theories","cited_arxiv_id":"hep-ph/0108190","evidence_quote":"Supplies the earlier step of treating resonant production with an effective-theory-like separation of hard and non-factorizable contributions."},{"cited_title":"Asymptotic expansion of Feynman integrals near threshold","cited_arxiv_id":"hep-ph/9711391","evidence_quote":"Provides the method of regions used to split loop integrals and to define the hard matching coefficients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the heavy quark effective theory construction, which is copied for the soft scalar field: phase removal, velocity vector, residual momentum."},{"cited_title":"Georgi, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the soft-collinear effective theory framework from which the collinear fermion Lagrangian and light-cone vector conventions are taken."},{"cited_title":"Grinstein, Nucl","cited_arxiv_id":null,"evidence_quote":"Provides the collinear Lagrangian worked out to the order in the expansion parameter that the calculation needs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The companion paper cited as in preparation, from which the matching coefficients entering the next-to-leading amplitude and the explicit vertex result are taken."}],"lane_accuracy":{"status":"not yet measured","lane":{"model":"selfhosted/deepseek-v4.1-flash","input_kind":null},"url":"https://pith.science/accuracy"},"review_version":1}