{"id":"1dc72c4c-921a-4307-8caf-6c0fbddab28c","arxiv_id":"1908.04048","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"F2F fits Fourier coefficients of a mass spectrum to the known Fourier transform of a Breit-Wigner propagator, extracting mass and width without modeling the signal or interference.","lead":"A new analysis method, F2F, extracts the mass and width of a resonance by fitting the Fourier coefficients of its invariant-mass distribution, even when the resonance signal interferes with background. It needs no signal model, which could help LHC searches where interference distorts new particle signals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 15's single global amplitude/phase is not proven to absorb a general slowly-varying production envelope; the Sec. III toys scan only one envelope and constant phases, leaving model-independence untested.","rationale":"The paper is a method proposal with a self-contained derivation and internally consistent toy experiments; the toys genuinely recover M and Gamma for the tested envelope and phases, and the detector-resolution treatment is a useful feature. The concern is not that the method is fraudulent or obviously wrong, but that the word 'model-independent' is stronger than the derivation supports: Eq. 15's global B,C is an ansatz, not a theorem, for arbitrary slowly-varying production amplitudes. The reader's weakest assumption (f slow) identifies the right area, but the sharper issue is that even a slowly-varying Ab is convolved with the propagator in Fourier space, and the paper gives no scale-separation argument or envelope-shape scan to show that B,C suffice. The suggested numerical scan would settle whether the ansatz is adequate across realistic slopes and phase gradients. Because the issue is testable and could be addressed by adding a validity condition or a broader scan without changing the method, the existing CONDITIONAL verdict remains appropriate rather than a rejection.","tokens_in":12136,"tokens_out":19710,"duration_ms":233471,"concrete_test":"Using the Sec. III toy (Eq. 18) with M=125 GeV and Gamma=10 GeV, generate 1000 toys for each of several production amplitudes: the original exponential, Ab(x)=1+alpha(x-x0)/L with alpha=0, 0.2, 0.5, 1.0, and Ab(x)=1+gamma(x-x0)^2/L^2 with gamma=0, 0.5, 1.0, while also allowing a phase gradient delta(x)=delta0+beta(x-M)/Gamma with beta=0, 0.5, 1.0. Run F2F setup2 (Eqs. 13-15) and record the pull of fitted M and Gamma. If pulls exceed +/-1 as alpha, gamma, or beta grow, Eq. 15's global B,C cannot absorb the slowly-varying envelope and the model-independence claim needs an explicit validity condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation 15 is the load-bearing template: from Eq. 5 to Eq. 6 the paper drops f(x) with only 'Assuming f(x) is a slow-varying function' (Sec. II), and Eq. 15 then absorbs all remaining model dependence into a global B, a global C, and a low-order polynomial A(k). The exact interference term in Eq. 2 contains a numerator factor (x^2-M^2), which is not slow-varying relative to the squared propagator, so the stated assumption is not even satisfied by the paper's own model. More importantly, for a genuine production amplitude Ab(x), the exact Fourier coefficient is a convolution of the transform of Ab with the propagator transform, producing k-dependent amplitude and phase corrections. A single B and C can absorb only the first few derivative terms of a sufficiently mild variation; no quantitative slow-varying condition, no error bound from the truncation, and no scan over envelope slope, curvature, or phase gradients are given. The Sec. III toy uses one exponential Ab with tau=200 GeV and constant delta, and scans only delta, so the toy evidence does not cover the claimed model-independence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes F2F, a method to extract the mass M and width Gamma of a Breit-Wigner resonance from the Fourier coefficients of an invariant-mass distribution in the presence of coherent signal-background interference. After subtracting a known background, the effective signal is expanded in cosine modes; Eq. (15) models the k-dependence of the coefficients using the exact Fourier transform of the squared propagator, with a polynomial A(k) and a global amplitude B and phase C absorbing model dependence. The method is validated on toy experiments for four constant phase angles and three detector resolutions, and is applied to the SM Higgs width measurement and to a new-resonance search. The contour integration leading from Eq. (5) to Eq. (6) is correct for an idealized constant envelope, and the toy studies are genuine closure tests; the central open question is whether the 'slow-varying f' approximation is quantitatively controlled.","tokens_in":12332,"tokens_out":8928,"duration_ms":102016,"significance":"If the envelope approximation can be justified, F2F is a valuable addition to the hep-ph toolbox: it avoids explicit signal modeling, includes interference at the template level, treats detector smearing analytically, and the Higgs-width application suggests competitive sensitivity. The paper's concrete strengths are the clean contour-integral derivation of Eq. (6), the analytical resolution factor in Eq. (7), and closure tests over several phases and resolutions; no machine-checked proofs or code are provided, but the toy studies are described in sufficient detail to be reproduced. The main limitation is that the central 'model-independent' claim currently rests on an unquantified slow-variation assumption; as written, the significance of the method is conditional on that assumption being satisfied or empirically demonstrated.","major_comments":[{"comment":"The transition from Eq. (5) to Eq. (6) simply drops f(x) after the sentence 'Assuming f(x) is a slow-varying function'. For the paper's own model in Eq. (2), f(x) is not slow-varying in the relevant sense: the interference term contains (x^2-M^2), whose variation over the resonance width is O(2M Gamma), which is not small compared with the denominator scale. The exact Fourier coefficient of the product f(x) G(x;M,Gamma) is a convolution of the transforms of f and the squared propagator; a single global amplitude B and phase C in Eq. (15) can absorb only the first few terms of a Taylor expansion of f. No quantitative condition on f, no truncation error bound, and no scan over envelope slope, curvature, or phase gradient are given. The Sec. III toys use one exponential envelope with tau=200 GeV and constant delta, so they do not exercise the claimed independence from the signal model. This is load-bearing: if f has k-dependent amplitude or phase corrections, the fitted M and Gamma will be biased.","section":"Sec. II, Eqs. (2), (5), (6), (15); Sec. III"},{"comment":"The method assumes that the background Fourier coefficients are known exactly: c_k(seff) is the difference between data and background-sample coefficients. In the Higgs and search applications, the background MC uncertainty is indicated as a band but is explicitly not included in the fit, and the polynomial A(k) in Eq. (15) is asserted to absorb background mismodeling without demonstration. A background shape error with power at high Fourier modes cannot be absorbed by a low-order polynomial and would be misinterpreted as resonance information. A systematic study of background-modeling uncertainties, or an explicit statement that the method requires essentially perfect background knowledge, is needed before the claimed precision can be taken at face value.","section":"Secs. II and IV-V, Eqs. (11)-(12)"},{"comment":"The search application computes local p0 values with 2 degrees of freedom at each point of a mass-width grid but does not address the trials factor from scanning the grid. The background-only toy in the left panel of Fig. 14 shows an apparent feature near (1.3 TeV, 5 GeV); if this is the maximum local p0 over the scanned grid, the global significance should be reported, or the paper should explicitly restrict the claim to local p0. Without this, the statistical interpretation in a search context may be overstated.","section":"Sec. V, Eq. (19), Fig. 14"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors and language issues, including 'reosnant', 'Stardard-Model', 'inteference', 'hardon colliders', 'responce', 'seires', 'indepdent', 'siganl', 'measurment', and 'seaches'. A thorough language editing pass is needed.","section":"Throughout"},{"comment":"Reference [9] cites a Wikipedia page for contour integration; a standard textbook reference would be more appropriate for a journal submission.","section":"Ref. [9]"},{"comment":"The text says 2000 toy data samples are generated, but Figs. 11 and 12 show distributions from 1000 toys; please clarify the sample sizes and state whether the reported upper limits are based on 1000 or 2000 toys.","section":"Sec. IV"},{"comment":"The PDF defined in Eq. (13) is not guaranteed to be positive for all x when the effective signal yield c0 is negative or when the Fourier template dips; the fit should either enforce positivity or restrict the fit range, and this should be stated explicitly.","section":"Eq. (13)"},{"comment":"The estimate Gamma = k pi sigma^2 / L in Eq. (17) is presented as a sensitivity estimate without specifying the mode k at which the two exponential terms are comparable; please state that this is a heuristic order-of-magnitude estimate and define k.","section":"Eq. (17)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the underlying idea is worth pursuing. The main barrier is the unquantified slow-varying envelope approximation that underpins Eq. (15); I would ask the authors to add a quantitative condition or a much broader set of stress tests (varying envelope slope, curvature, and phase gradients), and to address the background-subtraction assumption. The trials-factor issue in the search application should also be clarified. No concerns about novelty or citation fairness, apart from the Wikipedia reference."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my honest read. The paper is worth knowing about: it proposes a genuinely different way to get (M, Γ) when interference distorts the line shape. Instead of fitting a signal model in mass space, you Fourier-transform the data-minus-background distribution and exploit the known Fourier transform of the Breit-Wigner squared propagator. The contour-integral result in Eq. 6 is the core, and it is correct. The detector resolution enters as a simple exponential damping, which is elegant. The toy studies recover the input width and mass with precision comparable to an explicit signal fit, and the method does not need a signal model at all. That is a real idea and a useful one for TeV-scale searches.\n\nNow the soft spots, in proportion. First, the slow-varying assumption on f(x) is the load-bearing wall, and it is not properly supported. The paper says 'Assuming f(x) is a slow-varying function' and then drops it. But in the paper's own model, the interference term contains (x^2-M^2) times the background amplitude, which is not slow-varying over the fit range. The fit still works, which suggests the floating B, C, and polynomial A(k) absorb the leading variation, but that is not demonstrated for general cases. Only one background slope (τ=200 GeV) and constant interference phases are scanned. No scan over envelope slope, curvature, phase gradients, or energy-dependent couplings. So 'model-independent' is too strong; 'insensitive to the specific model in the tested cases' is accurate.\n\nSecond, background subtraction. The method relies on knowing the background shape and normalization precisely; the toys use the MC background with fixed shape and yield, and the quoted uncertainties ignore MC statistical and systematic errors. The polynomial A(k) is a patch for mismodeling, but its bias when the background is slightly wrong is not studied. Real data will have correlated background uncertainties, and that is the main risk to the method.\n\nThird, the Higgs-width application overclaims. The abstract says 'measure the SM Higgs width,' but the results are upper limits, and the width distribution from toys is non-Gaussian with a spike near zero. Also, no comparison with the off-shell method is made, by the author's admission. This section is more a proof-of-principle than a measurement proposal.\n\nMinor points: no code or data released, and reference [9] to Wikipedia for contour integration is below the standard of the rest.\n\nWho should read this: experimentalists doing resonance searches, and pheno people building model-independent tools. It deserves peer review. A revision should quantify the slow-varying truncation error, broaden the toy scan, include background uncertainties, and fix the abstract.","headline":"Clever Fourier trick for extracting resonance parameters under interference, but the claimed model-independence is not yet backed by proof or enough toy diversity.","tokens_in":12846,"tokens_out":4698,"would_cite":false,"duration_ms":50358,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A Fourier-fit method, F2F, recovers a particle's mass and width from its measured mass distribution even when interference hides the resonance, without modeling the signal amplitude or phase.","keywords":["F2F","interference","Breit-Wigner propagator","Fourier coefficients","mass and width measurement","Higgs width","resonance search","model-independent fit"],"falsifier":"Generate a toy sample with a known mass and width but with an interference phase that changes sharply across the fit range, apply F2F, and check whether the fitted M and Gamma deviate from the inputs; a clear deviation would map the boundary of the method's validity, while no deviation would mean the slow-envelope assumption is weaker than stated.","tokens_in":11922,"feed_emoji":"⚛️","tokens_out":6514,"duration_ms":64032,"temperature":0.7,"pith_summary":"Interference between a new particle's signal and Standard-Model background can make the particle appear as a bump, a dip, or a distortion rather than a clean resonance, so its mass and width are hard to fit without knowing the interaction details. This paper proposes F2F, a method that extracts the mass M and width Gamma directly from the Fourier coefficients of the observed mass distribution. The key step is expressing the sum of signal and interference as a cosine Fourier series, whose high-mode coefficients are controlled by the universal Breit-Wigner squared propagator rather than by the unknown interference phase. Toy experiments with four relative phases recover the input mass and width with precision comparable to fits using the true signal model, and the method gives a competitive sensitivity to the Standard-Model Higgs width. Its practical payoff is a way to measure an interfering resonance's properties and to search for new resonances without committing to a signal model.","feed_headline":"Fourier fit pulls mass and width out from under interference","feed_subtitle":"No signal model needed; toy fits match the true mass and width at explicit-model precision.","key_machinery":"The load-bearing object is the cosine Fourier coefficient c_k of the effective signal distribution. The analytic identity obtained by contour integration relates c_k to the squared Breit-Wigner propagator: the coefficient is proportional to a damped cosine whose damping rate is set by Gamma and whose oscillation frequency is set by M. This identity is what lets the fit separate the two parameters without knowing the interference phase; the slow-varying envelope f(x) only contributes to low modes, where a polynomial A(k) absorbs it, while the high modes carry the propagator's signature. In practical fits the working formula is c_k(M,Gamma,A,B,C) = A(k) + B e^{-(k'$\\sigma$)^2/2 - k' $\\sqrt$(gamma_0) sin($\\theta$/2)} cos(k' $\\sqrt$(gamma_0) cos($\\theta$/2) - C), with the detector-resolution factor included.","core_discovery":"The paper's central claim is that the mass and width of a resonance can be determined from the Fourier coefficients of the data-minus-background mass distribution even when the resonance and background amplitudes interfere with an unknown, possibly x-dependent phase. Writing the effective signal as f(x)/(($x^{2}$ - $M^{2}$)^2 + $M^{2}$ $Gamma^{2}$) and assuming f(x) varies slowly, the cosine coefficients c_k satisfy c_k proportional to (pi/$gamma_0^{{3/2}}$) sin($\\theta$) e^{-|k'| $\\sqrt$(gamma_0) sin($\\theta$/2)} cos(|k'| $\\sqrt$(gamma_0) cos($\\theta$/2) - $\\theta$/2), with gamma_0 = $\\sqrt$($M_0^{2}$($M_0^{2}$+$Gamma^{2}$)), tan($\\theta$) = Gamma/M_0, and M_0 = M - x_min. This analytic form carries both an exponential decay set by Gamma and an oscillation set by M, so a fit of this form to the measured coefficients yields M and Gamma directly. Detector smearing enters as a multiplicative Gaussian factor $e^{{-(k'sigma)^2/2}}$ per mode, and slowly-varying background mismodeling is absorbed by a low-order polynomial in mode number. In toy experiments with Gamma = 10 GeV and M = 125 GeV, the F2F fits agree with the inputs within uncertainties for all tested interference phases, and in the Higgs-width application the method improves the 90% confidence upper limit from 1.54 GeV with a nominal on-shell fit to 0.92 GeV with a long-range F2F fit.","pith_inferences":["A natural stress test is to let the interference phase delta(x) vary rapidly across the fitted range, making f(x) no longer slow; the paper's own assumption predicts biased M and Gamma there, so an adaptive range or mode-truncation rule could be developed.","The same coefficient formula may generalize to other observables where a Breit-Wigner-like propagator controls a distribution, such as energy-dependent line shapes or time-dependent oscillation envelopes, as long as the slow-envelope condition holds.","Because background mismodeling enters only through the low-mode polynomial A(k), the method could in principle use a data-driven background estimate rather than Monte Carlo, as long as its high-mode coefficients are negligible.","The periodicity implied by the cosine series creates boundary artifacts; the paper checks that these are confined to edges, which suggests an explicit edge-exclusion or windowing procedure would make the method more robust in low-statistics searches."],"forward_implications":["For a narrow resonance like the Higgs boson, the width sensitivity is set by the balance between the detector-resolution damping and the Gamma-driven damping, giving a crude estimate Gamma approximately k pi sigma^2 / L; the paper shows F2F applied to the four-muon channel in a long mass range yields a smaller 90% C.L. upper limit on Gamma_H than a nominal on-shell fit.","The same Fourier-coefficient fit can be used as a resonance-search statistic: fitting with and without a resonance hypothesis on a grid of (M,Gamma) and taking the chi-square difference gives local p0 values that identify an injected 1 TeV, 50 GeV resonance.","The method requires no signal Monte Carlo or model of the interference, so it can be applied when the new particle's couplings and phase are unknown.","Extending the fitted mass range improves width sensitivity because more modes sample the exponential decay, provided the slow-envelope and MC-statistics conditions are met.","Using more modes than L/sigma is recommended, so the fit includes the region where the resolution factor has not yet suppressed all information."],"supporting_citations":[{"why":"Supplies the contour-integration technique that turns the squared Breit-Wigner propagator into the damped-oscillation formula for c_k.","marker":"[9]"},{"why":"Provide the gluon-fusion Higgs production simulation used in the Higgs width-measurement application.","marker":"[10, 11]"},{"why":"Provides the quark-antiquark Higgs production simulation used for the second signal component in the same application.","marker":"[12]"},{"why":"Define the four-muon selection, background treatment, and comparison basis for the Higgs width study.","marker":"[13, 14]"},{"why":"Supplies the 1.19 GeV four-muon mass resolution used in the Higgs width toy study.","marker":"[15]"}],"fun_headline_variants":["Fourier fit pulls mass and width from interference","Model-free Fourier method measures mass and width","Interference-proof mass-width extraction via Fourier fit","Fourier coefficients reveal resonance mass and width","No-signal-model fit for mass and width under interference"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that f(x), the slowly-varying envelope multiplying the squared Breit-Wigner propagator, does not vary much across the fitted mass range, so its Fourier transform is concentrated at low modes and the high-mode data-minus-background coefficients follow the propagator's analytic form.","fun_headline_variants_meta":{"raw":{"variants":["Fourier fit pulls mass and width from interference","Model-free Fourier method measures mass and width","Interference-proof mass-width extraction via Fourier fit","Fourier coefficients reveal resonance mass and width","No-signal-model fit for mass and width under interference"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00018,"raw_usage":{"total_tokens":1384,"prompt_tokens":1103,"completion_tokens":281,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":208}},"tokens_in":719,"tokens_out":281,"duration_ms":3401,"temperature":1.0,"reasoning_tokens":208,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:53:28.157818+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate a toy sample with a known mass and width but with an interference phase that changes sharply across the fit range, apply F2F, and check whether the fitted M and Gamma deviate from the inputs; a clear deviation would map the boundary of the method's validity, while no deviation would mean the slow-envelope assumption is weaker than stated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the contour-integration technique that turns the squared Breit-Wigner propagator into the damped-oscillation formula for c_k."}],"review_version":1}