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The paper establishes that smearing experimental data and lattice QCD predictions with the same finite-width kernel yields model-independent Standard Model tests for observables linear in the spectral density and for interference terms, wit

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2026-08-02 18:06 UTC pith:EQJPXKAL

load-bearing objection A well-argued finite-smearing methods proposal with a solid analytic core, but the central feasibility claim — that a workable epsilon window exists for the targeted channels — is unproven and doubtful for narrow resonances. the 3 major comments →

arxiv 2603.15487 v2 pith:EQJPXKAL submitted 2026-03-16 hep-lat hep-ph

Standard Model tests with smeared experiment and theory

Andreas J\"uttner This is my paper
classification hep-lat hep-ph PACS 12.38.Gc13.20.Fc13.20.He
keywords lattice QCDspectral reconstructionsmeared spectral densitiesinclusive semileptonic decayrare semileptonic decaylong-distance effectsPoisson kernelStandard Model tests
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper argues that lattice QCD and experiment can be compared directly at a finite smearing width, avoiding the costly and potentially model-dependent limit of zero smearing that has held back spectral-reconstruction methods. For observables that are linear in the hadronic spectral density, such as inclusive semileptonic meson decay, the same Poisson-smearing kernel can be applied to both experimental differential rates and lattice-reconstructed correlation functions, giving identical quantities at finite width. For observables that are bilinear in the hadronic amplitude, like rare semileptonic decays, the pure long-distance term cannot be smeared into agreement, but the obstruction is isolated in a positive defect term; observables where that term cancels, such as CP asymmetries, can still be compared model-independently. The paper also constructs a concrete smearing prescription for the interference term of rare decays, using transversality and a once-subtracted dispersion relation to eliminate the subtraction constant. A reader should care because this would make many currently impractical lattice predictions accessible with existing resources and would produce new handles on long-distance effects and possible new physics.

Core claim

The paper claims that for a broad class of Standard Model processes with on-shell intermediate hadronic states, a meaningful theory-experiment comparison can be formulated at finite smearing width. In the linear case (inclusive decays), the smeared differential rate is the harmonic extension of the hadronic amplitude, and the same Euclidean correlation functions that define the lattice computation reproduce this extension at finite width; no epsilon-to-zero limit is needed. In the quadratic case (rare semileptonic decays), a direct finite-width comparison fails for the pure long-distance term, but the paper isolates the failure in a positive defect term and shows that interference terms—wher

What carries the argument

The load-bearing object is the Poisson kernel K_epsilon(x' - x) = (epsilon/pi)/((x' - x)^2 + epsilon^2). Smearing with this kernel turns a dispersive amplitude H(x) into its harmonic extension H(x + i epsilon), so lattice-reconstructed smeared quantities are directly comparable to smeared experiment for observables linear in the spectral density. For bilinear observables the paper defines the defect term Delta_epsilon = <|H|^2>_epsilon - |<H>_epsilon|^2, which is positive for epsilon > 0 and quantifies the irreducible mismatch; the semigroup property Delta_{epsilon1+epsilon2} = <Delta_{epsilon1}>_{epsilon2} supplies an additional cross-check. For rare decays, the reconstruction kernel uses t

Load-bearing premise

Everything hinges on there being a finite smearing width that is simultaneously large enough to make the finite-volume lattice spectrum look continuous and small enough to preserve the physical structures the test is meant to see; the paper states these competing conditions but does not show such a width is attainable with current or near-future lattice ensembles.

What would settle it

Take a fixed lattice spacing and compute the Poisson-smeared inclusive B -> X_c l nu differential rate at two box sizes L1 < L2 with the same smearing width epsilon. The finite-width programme predicts agreement within errors and agreement with smeared experimental data; a volume-dependent drift at fixed epsilon, or a need to push epsilon below the peak separation of the finite-volume spectrum, would falsify the existence of the required smearing window. For the defect-term test, a measured negative defect <|H|^2>_epsilon - |<H>_epsilon|^2 in a channel where only long-distance Standard Model p

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Inclusive semileptonic B and D decays become accessible to lattice QCD with controlled systematics, enabling model-independent determinations of |Vcb|, |Vub|, |Vcs|, and |Vcd| at finite smearing width.
  • CP asymmetries in D -> pi l+l- and B -> K(*) l+l- can be predicted from first principles at finite smearing, because the pure long-distance term cancels and the long-distance amplitude enters only linearly.
  • The positive-definite defect term offers a new Standard Model test: a measured negative defect would signal short-distance contributions not accounted for in the Standard Model.
  • The semigroup relation allows a consistency check between experimental data smeared at two different widths and lattice predictions, sensitive to analysis errors or new physics.
  • Avoiding the epsilon-to-zero extrapolation relaxes the need for very large lattice volumes, removing a major computational bottleneck.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same finite-smearing logic should carry over to processes beyond the two examples, such as hadronic tau decays or hadron scattering, provided experiment and lattice agree on a common kinematic variable and kernel.
  • The two-width semigroup check can be read as a data-driven null test of the experimental smearing procedure itself: a failure before new physics is invoked would flag miscalibrated smearing kernels or underestimated systematics.
  • A natural near-term extension is to apply the transverse-kernel reconstruction to the R-ratio comparison already made in the literature, targeting the rho-omega region where the defect term is largest.
  • If a window of admissible epsilon exists, the method effectively converts the ill-posed spectral reconstruction into a well-posed comparison at finite resolution; if not, the programme's claims about current feasibility would need revision—so testing for such a window on existing ensembles is the cheapest next step.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a framework for comparing Standard Model (SM) predictions with experimental data at finite smearing width, thereby avoiding the costly ε→0 extrapolation of energy-smeared spectral reconstructions. The author distinguishes observables linear in the spectral density (case I) from observables bilinear in the hadronic amplitude (case II). For case I, Poisson-smearing the differential decay rate yields a harmonic extension of the dispersion relation, so smeared experiment and lattice reconstruction can be compared directly. For case II, the smeared norm-squared differs from the norm-squared of the smeared amplitude by a positive 'defect term'; the author shows that model-independent comparisons remain possible for observables in which pure long-distance terms cancel (e.g. CP asymmetries), while pure long-distance contributions require model assumptions or multi-width comparisons. The framework is applied to inclusive semileptonic B decay and to the long-distance part of rare D→πℓℓ decay, with explicit reconstruction kernels and illustrative Chebyshev plots.

Significance. The central idea is attractive and potentially important: if finite-smearing SM tests are feasible, lattice QCD could be compared with experiment without the computationally prohibitive ε→0 limit, opening up inclusive decays and long-distance effects in rare decays to first-principles scrutiny. The analytic identities in Sec. 3 (e.g. Eqs. (3.5)–(3.7), (3.14)–(3.20), (4.33)–(4.35)) are internally consistent, and the defect-term construction is a useful model-independent tool. The paper also makes concrete, testable proposals for kernels that smear data and lattice on the same kinematic trajectory. However, the practical feasibility of the whole programme is not demonstrated: no numerical lattice or synthetic-data test is provided, and the finite-smearing window that must satisfy both ε≫1/L and ε≪|Δ(x)| is not shown to exist for the proposed channels.

major comments (3)
  1. [Eq. (4.36) and Sec. 4.2.5] The reconstruction kernel in Eq. (4.36) has sign errors relative to the derivation in Eq. (4.33). The denominator should be ((ω²−|q|²−s)²+ε²), not minus ε², as written the kernel has real-axis poles. The sine term in the numerator should be +ε(ω²−|q|²) sinφ if the target is Re[Φ*(s+iε)H_T(s+iε)] = |Φ|(cosφ ReF + sinφ ImF). Additionally, the Heaviside factor θ(ω²−|q|²−q²_max) restricts to s>q²_max, i.e. outside the semileptonic region, contrary to the text. Since this kernel is the central construction for the rare-decay example and underlies Fig. 2, these errors must be corrected.
  2. [Sec. 2 and Sec. 3.2.2] The central practical claim—SM tests at finite smearing without ε→0—presupposes a finite ε satisfying both ε≫1/L and ε≪|Δ(x)| (or the analogous q²-space condition), but no quantitative evidence is given. For D→πℓℓ, the narrow ω and φ resonances have widths of a few MeV, while current lattice volumes have 1/L∼70–100 MeV; the illustrative ε=0.3 GeV² in Fig. 2 corresponds to an energy smearing of order 150 MeV at s∼1 GeV², which washes out these structures. Conversely, an ε that resolves them would violate ε≫1/L unless very large volumes (MπL∼10–20) are used, which the paper itself cites as prohibitive. The authors should demonstrate that a viable window exists for the proposed channels, or provide quantitative estimates of statistical and finite-volume uncertainties at the chosen ε.
  3. [General] The paper contains no numerical computation: no lattice data, no synthetic data, and no error budget. The assertions that the Chebyshev approximations in Figs. 1–2 are 'excellent' and that there is 'little leakage' rest on visual inspection. Since the paper's contribution is a feasibility claim for finite-smearing SM tests, a demonstration with synthetic data—or at least explicit errors for the truncation, finite-volume, and statistical effects—is needed to support the central claim.
minor comments (4)
  1. [Sec. 4.1.2, Eqs. (4.6)–(4.7)] The relationship between the proposed constant-q² smearing of experimental data and the fixed-three-momentum lattice correlation functions in Eq. (4.8) is not fully explained. As written, it is not obvious how the lattice reconstruction at fixed q is mapped onto the constant-q² path.
  2. [Sec. 4.2.1, Eqs. (4.19)–(4.22)] The definitions of H+ and H− and the lower integration limits E*± are terse. Please define the intermediate-state content and sign conventions more explicitly, and state how the finite-volume treatment applies to these time orderings.
  3. [Eq. (4.38)] The notation Σ_{t=t±} c^±_{t,μ}(s,q,φ,ε) e^{−ω t} is confusing because t is used both as a summation index (taking values 0 and 1) and as Euclidean time elsewhere. Rename the summation index (e.g. n) for clarity.
  4. [Fig. 2 and Sec. 4.2.5] The caption notes that s_th can be chosen freely between 0 and the lightest on-shell intermediate state, but the reconstruction depends on this choice. Discuss its impact on the leakage and on the final systematic uncertainty.

Circularity Check

0 steps flagged

No significant circularity: the smeared observables are defined as harmonic extensions of the same amplitudes and reconstructed from independent Euclidean correlators; self-citations are contextual.

full rationale

The paper's central claim is that finite-width comparisons between smeared experiment and lattice QCD are possible for observables linear in the spectral density (case I) and for short-distance/short-long-distance interference terms (case II). In both cases the smeared quantity is an exact transform of the underlying amplitude: Eq. (3.5) identifies the Poisson-smeared linear observable with the imaginary part of the harmonic extension of the dispersive integral, and Eqs. (4.28)-(4.33) do the same for the interference terms. The lattice reconstruction in Eq. (4.10) and Eq. (4.39) expresses the same smeared quantity as a linear combination of Euclidean correlation functions, which are independent first-principles inputs; no parameter is fitted to the quantity being 'predicted'. The subtraction constant in the once-subtracted dispersion relation is explicitly eliminated by combinations at two kinematic points or two smearing widths (Eqs. (4.34)-(4.35)), so it is not a fitted input. The defect term of Eq. (3.9) and the semigroup relation of Eq. (3.21) are mathematical identities for positive smearing kernels, and the paper uses them as consistency tests rather than as derived physical predictions. The Breit-Wigner discussion in Sec. 3.2.1 is explicitly labeled a model-dependent study of the defect, not a model-independent prediction. Self-citations to Refs. [6] and [47] are contextual (Chebyshev expansion tools and an exclusive-analysis reference) and are not load-bearing for the finite-smearing comparison proposal. The skeptical concern about the existence of a finite smearing width satisfying both epsilon >> 1/L and epsilon << Delta(x) is a quantitative feasibility and systematic-error issue, not a circularity: the paper does not claim to have demonstrated that window, and its absence does not make the derivation equivalent to its inputs. Therefore no circular step can be exhibited from the paper's own equations.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 0 invented entities

The mathematical core rests on standard dispersion-relation assumptions plus three domain/feasibility assumptions that are plausible but unquantified. The only hand-chosen free parameter is the smearing width epsilon; no constants are fitted to data. No new physical entities are introduced; the defect term is a derived observable.

free parameters (1)
  • smearing width epsilon = not fitted; illustrated as 0.1-0.3 GeV (inclusive) and 0.3 GeV^2 (rare decays)
    Chosen by hand; central to the method. The paper gives no criterion beyond the qualitative inequalities epsilon >> 1/L and epsilon << Delta(x) for selecting or validating the value.
axioms (6)
  • domain assumption Hadronic amplitudes have dispersive representations H(x)=lim_{eta->0} integral dx' rho(x')/(x'-x-i*eta), possibly with subtractions.
    Used in Eq. (2.1) for all processes considered; standard in QCD but assumed.
  • domain assumption H is analytic in the upper half-plane and rho decays sufficiently fast at infinity, so the Poisson average equals H(x+i*epsilon).
    Invoked to derive Eq. (3.7) and Eq. (3.12); physical causality is the justification.
  • domain assumption Local form-factor parametrisations (BGL) converge on and off the real axis in the region of interest, and branch cuts (t_+) are far enough from the semileptonic region that they can be neglected for epsilon < t_+ - q^2_max.
    Needed for off-axis evaluation Phi(s+i*epsilon) in Eq. (4.28); stated but not quantified in Sec. 4.2.2.
  • domain assumption The subtraction constant H_T(0) either is taken from perturbation theory [70,71] or cancels in the differences Eqs. (4.34)-(4.35).
    The once-subtracted dispersion relation Eq. (4.27) requires this constant; the paper does not provide a nonperturbative determination.
  • ad hoc to paper There exists a finite epsilon satisfying epsilon >> 1/L and epsilon << Delta(x) with controlled statistical errors for the channels considered.
    This is the core feasibility premise of the whole proposal; asserted in Secs. 2, 4.1.4 and 4.2.6, not demonstrated.
  • ad hoc to paper Experimental data can be Poisson-smeared at constant q^2, or along the proposed kernel, with controlled systematic uncertainties.
    Case I prescription Eq. (4.7) and case II kernel Eq. (4.36) require data products the experiments do not currently provide; no feasibility study is cited.

pith-pipeline@v1.3.0-alltime-deepseek · 19935 in / 19844 out tokens · 193928 ms · 2026-08-02T18:06:42.809015+00:00 · methodology

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read the original abstract

For Standard Model processes in which on-shell intermediate hadronic states contribute - including inclusive semileptonic decays and long-distance effects in rare exclusive decays such as $D\to \pi \ell\ell$ and $B\to K^{(\ast)}\ell\ell$ - spectral-reconstruction techniques provide a promising route to model-independent lattice QCD predictions for use in phenomenological predictions. The central ingredient is the computation of the energy-smeared spectral density. Following the continuum and infinite-volume limits, the physical amplitude is recovered as the limit of vanishing smearing width. However, achieving sufficiently small smearing for a controlled extrapolation remains a significant challenge for current lattice simulations. In this paper, we therefore propose Standard Model tests, in which both experimental results and theory predictions are smeared with finite width, similar to what has previously been done in the literature for experimental and lattice $R$-ratio data in the context of the muon $(g-2)_\mu$. As concrete examples, we discuss the cases of inclusive meson decay and long-distance contributions to rare semileptonic meson decay.

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Causal Bootstrap: Bounding Smeared Spectral Functions from Non-Perturbative Euclidean Data

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  2. Kernel transformations and bounds for smeared spectral functions

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  3. Inclusive semileptonic $D_s\to X_s\ell\bar\nu$ decays from lattice QCD: continuum and chiral extrapolation

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Reference graph

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