Approaching the Inverse Problem: Toward Lattice QCD Calculations of Inclusive Hadronic Quantities

William Jay

REVIEW 3 major objections 3 minor 3 cited by

Approaching the Inverse Problem: Toward Lattice QCD Calculations of Inclusive Hadronic Quantities

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nevanlinna–Pick interpolation bounds every possible analytic continuation of lattice-QCD Euclidean data in an explicit disk—the Wertevorrat—whose size falls roughly exponentially with the number of data points.

desk verdict A clean proceedings talk: the Wertevorrat method from Ref. [29] plus one genuinely new empirical scaling observation; honest about limits, though it glosses over two technical hypotheses that matter. read the letter →

arxiv 2501.12259 v1 pith:WP6LNWME submitted 2025-01-21 hep-lat

William Jay This is my paper · ORCID

classification hep-lat PACS 12.38.Gc

keywords latticeQCDspectralreconstructionanalyticcontinuationNevanlinna–PickinterpolationWertevorratsmearedfunctionsinverseLaplacetransformR-ratio

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Lattice QCD computes Euclidean correlation functions at a finite set of times, but many inclusive observables—the R-ratio for $e^+e^-\to$ hadrons, inclusive $\tau$ and $B$ decays, neutrino-nucleus scattering, transport coefficients—require the full spectral function, which is an inverse Laplace transform away and notoriously ill-posed. The paper argues that this inverse problem becomes tractable when viewed as analytic continuation: evaluating the momentum-space Green function at $z=\omega+i\epsilon$ is exactly a Poisson-smoothed spectral function, and the known analytic structure (singularities only on the real axis) can be exploited. After conformally mapping both the frequency plane and the Green-function values onto the unit disk, Nevanlinna–Pick interpolation gives, in the Wertevorrat, a disk that provably contains every analytic continuation consistent with the finite data, with no model assumptions. The new result in this talk is empirical: for a realistic R-ratio model, the radius of that disk decreases roughly exponentially with the number of Euclidean points, reaching percent precision at 60 points and permille at 100 for a fixed smearing.

What carries the argument

The load-bearing mechanism is the reduction of the inverse problem to Nevanlinna–Pick interpolation. A Cayley transform maps the upper half plane to the unit disk and a second conformal map sends the Green function's values into the disk, so the physical analytic continuation becomes an analytic function $f:\mathbb{D}\to\mathbb{D}$ (a Schur function) interpolating the transformed data $(\zeta_\ell,w_\ell)$. Nevanlinna's theorem, built by Schur's algorithm with Blaschke factors, rewrites every interpolant as a fractional-linear expression in one arbitrary Schur function $f_N$; the set of values that expression can attain at a fixed $\zeta$ is the Wertevorrat, a disk whose center and radius are explicit rational functions of the data. Pick's theorem supplies the existence condition: the interpolation problem has a solution if and only if the Pick matrix $(1-w_i\bar{w}_j)/(1-\zeta_i\bar{\zeta}_j)$ is positive semidefinite. The key point is that no model for the spectral function enters: only the data and the analyticity structure (singularities on the real line) determine the bounding disk.

What would settle it

Take a known spectral model (e.g., the R-ratio model used in the paper's numerical tests), generate exact Euclidean data, apply the paper's conformal maps, and evaluate the mapped Green function on a dense grid in the unit disk: if any point has modulus greater than 1, the boundedness premise fails and the Wertevorrat is not guaranteed to contain the true smeared spectral function. On the data side, realistic noisy lattice data that yield an indefinite Pick matrix would likewise show the theorem's condition is not met.

Watch

Extended reading notes

Core claim

The central claim is that the systematic uncertainty of analytic continuation from a finite set of Euclidean points can be completely characterized, not just estimated. Given values $G(i\omega_\ell)$ at equally spaced points on the imaginary axis, the paper treats the Green function as an analytic function and applies the Cayley transform to both domain and codomain, so the problem becomes: find an analytic $f:\mathbb{D}\to\mathbb{D}$ with $f(\zeta_\ell)=w_\ell$. Nevanlinna's theorem says all solutions are parametrized by one arbitrary Schur function $f_N$ through explicit Nevanlinna coefficients, and the Wertevorrat—the set of all values $f(\zeta)$ can take at any target point—is a disk with computable center and radius. The paper claims this disk rigorously contains all possible analytic continuations and therefore provides a complete, model-independent bound on the smeared spectral function obtained at $z=\omega+i\epsilon$; the total width of its imaginary part after mapping back to the physical plane is the uncertainty statement. It further reports the new empirical observation that, for the R-ratio model, the fractional uncertainty falls roughly exponentially with the number of interpolation points, so 60 and 100 points give percent and permille precision respectively for a smearing $\epsilon=0.1$ near the $\rho$ peak.

Load-bearing premise

Everything rests on the assumption that, after the conformal maps, the QCD Green function is a bounded analytic function from the unit disk to itself and that the Euclidean data satisfy the Pick positivity condition; if either fails, the Wertevorrat may not contain the true analytic continuation.

Editorial extensions

If this is right

For inclusive observables such as the R-ratio, the method turns 'how many Euclidean times do we need?' from a guess into a computable question: the Wertevorrat gives the systematic error directly, and 60–100 well-chosen points reach percent-to-permille smeared reconstructions in the tested model.
Because the bound relies only on analytic structure, the same machinery applies to $\tau$ decays, inclusive $B$-meson semileptonic decays, neutrino-nucleus structure functions, and transport coefficients whenever the Euclidean correlator's singularities lie on the real axis.
The arbitrary Schur function $f_N$ in Nevanlinna's theorem is a bookkeeping device for missing information: any future constraint on $f_N$, physical or algorithmic, can only shrink the Wertevorrat, so additional input translates directly into a smaller rigorous uncertainty band.
The exponential point-count scaling gives a practical roadmap for lattice calculations: determine how many points are needed for a target smearing and precision first, then optimize the correlator computation accordingly rather than maximizing statistics blindly.
The Wertevorrat's growth as $\epsilon\to 0$ quantifies the cost of going differential: uncertainty grows roughly exponentially as the smearing is removed, making the trade-off between energy resolution and controlled systematic error explicit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exponential scaling is demonstrated on one R-ratio model; if it persists for spectral functions with sharp thresholds, the practical lesson generalizes—the number of points, not the statistical noise, sets the achievable resolution—an inference the paper does not make.
Pick-matrix failure under statistical noise, which the paper flags as an open question, suggests a natural preprocessing step: project noisy Euclidean data onto the nearest set that makes the Pick matrix positive semidefinite, and only then compute the Wertevorrat; the paper points at this problem but does not solve it.
The same disk bound could be read as a diagnostic of the analyticity assumptions themselves: if a lattice correlator cannot be made Schur-bounded after the conformal maps at any reasonable smearing, that would indicate a missing singularity or a finite-volume artifact, turning the method into a consistency test for the Euclidean data.
Used together, the localized-average point estimate and the Wertevorrat envelope would give both a best reconstruction and a rigorous band around it; the paper presents them as alternatives, and the synthesis is left implicit.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a proceedings talk that re-presents the Wertevorrat method from the author's earlier PRD paper and adds one genuinely new observation—that the bound shrinks roughly exponentially with the number of interpolation points, reaching percent and permille precision at 60 and 100 points for the Bernecker–Meyer R-ratio model. If that scaling survives contact with noisy, truncated, finite-volume data, it is a practically useful result for spectral reconstruction.

The talk does several things well. The Nevanlinna–Pick theorem is standard, quoted correctly, and the recursive Schur-algorithm construction is summarized accurately. The worked R-ratio example is concrete and the connection between analytic continuation and Poisson smearing is nicely explained. The author is also honest: the exponential scaling is explicitly flagged as an empirical observation, and the paper concedes that statistical errors can break the Pick-matrix positivity condition.

The soft spots are real but proportionate. The reduction of the physical Green function to a Schur function (disk-to-disk analytic map) is asserted rather than derived. For bosonic correlators that step is nontrivial—the retarded Green function is not Herglotz on the full upper half-plane—and the talk just points to Ref. [29]. That is acceptable for a proceedings summary if the derivation exists, but it leaves the reader unable to check the key hypothesis. Second, Eq. (11) says the interpolation points {ζ_l} lie in D, but the l=0 Matsubara frequency maps to the boundary of the unit disk, so the open-disk hypothesis of Pick's theorem is violated unless that point is explicitly excluded. The talk does not say so. Neither gap is fatal, but both should be stated cleanly.

The new scaling result is based on a single noiseless model, with no code, data, or robustness checks. That is a limitation, but the author does not overclaim: the figures are presented as an example, not a theorem.

For a proceedings paper, this is solid work. It is not a full research article, but it is a fair, readable summary of a mathematically grounded method with one new quantitative observation. I would send it to referees, with the expectation that they ask for a clarification of the boundary-point issue and a more explicit statement about the bosonic codomain map assumptions.

Referee Report

3 major / 3 minor

Summary. This proceedings contribution from LATTICE summarizes a new perspective on the lattice-QCD spectral reconstruction inverse problem, arguing that smeared spectral functions are the practically and conceptually right target (Sections 2–3), and then presents the Nevanlinna–Pick (NP) interpolation framework as a way to bound the systematic uncertainty of analytic continuation from a finite set of Euclidean data (Section 4). The central technical object is the Wertevorrat, the disk of all possible analytic continuations through given interpolation data, whose radius and center are given in Corollary 1. The talk's new contribution is an empirical observation, shown in Fig. 2 for a Bernecker–Meyer model of the R-ratio, that the fractional uncertainty of the Wertevorrat decreases roughly exponentially with the number of interpolation points, reaching percent and permille precision at 60 and 100 points for one fixed extrapolation point.

Significance. If the hypotheses of the NP framework are satisfied by lattice-QCD Green functions, the Wertevorrat is a parameter-free, rigorous bound on the systematic uncertainty of analytic continuation from finitely many Euclidean data points, with no model assumptions and no ad hoc regularization. This would be a genuinely useful tool for inclusive hadronic quantities, where the full spectral function is needed and smeared observables are the natural interface with experiment. The paper is honest about an important limitation: statistical uncertainties can violate the Pick condition, and the paper correctly describes the treatment of such indefinite Pick matrices as an open question. It also cites and connects to the relevant literature, including the HLT method, Bayesian/GP reinterpretations, and prior NP work in condensed matter. The new exponential-scaling observation is potentially important for planning calculations, but in this manuscript it is purely empirical and rests on a narrow set of tests.

major comments (3)

[Section 4, Eq. (11)] The statement that the mapped Euclidean data satisfy {ζ_l} ∈ D is false for bosonic Matsubara frequencies: for ω_0 = 0, ζ_0 = C(0) = -1 lies on the boundary of D rather than in the open unit disk. Theorem 2, as stated, requires interpolation points in the open disk, so Corollary 1 cannot be applied to the bosonic example in Fig. 1 unless that point is excluded or a separate boundary-limiting argument is supplied. The manuscript should state explicitly how the l = 0 Matsubara point is handled.
[Section 4, paragraph after Eq. (10)] The claimed rigor of the Wertevorrat bound depends on the physical Green function becoming a Schur function—a bounded analytic map from D to D—after the Cayley transform of the domain and the codomain map of Ref. [29]. This is asserted but not established in the manuscript: the bosonic retarded Green function is not Herglotz on the entire upper half-plane, so the codomain construction is nontrivial. If that map does not produce a Schur function, Pick's theorem constrains only a transformed object and Corollary 1 does not bound the true spectral reconstruction. Please either give the codomain map explicitly or clearly state that the bound is conditional on the construction in Ref. [29].
[Section 4, Fig. 2 and preceding bullet] The new scaling result is presented as 'roughly exponential' but no derivation or robustness checks are given. The figure uses a single model (Bernecker–Meyer), a single extrapolation point (s = 0.65 GeV², ε = 0.1), and, as in Fig. 1, exact Euclidean data. The statement that '60 and 100 points suffice to determine R(s, ε) with percent and permille precision' is therefore a model- and point-specific empirical observation, not an established general property. The manuscript should either provide a derivation of the scaling or add checks that vary the spectral model, the smearing ε, and the target energy, and should qualify the caption accordingly.

minor comments (3)

[Section 4, text near Eq. (10)] The phrase 'Poisson kernel of Eq. (6a)' is a typo: Eq. (6a) defines the Gaussian kernel, while the Poisson kernel is Eq. (6b).
[Section 2, first paragraph] The sentence 'One might image that a suitably smooth function function could, in fact, be well determined' contains a duplicated word 'function'.
[Section 4, Eq. (9) and Eq. (11)] For clarity, the manuscript should specify that only nonnegative Matsubara frequencies l ≥ 0 are used in the interpolation, since negative frequencies lie on the negative imaginary axis, which is outside the upper half-plane where the retarded Green function is analytic and on which the Cayley transform is defined.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Wertevorrat bound is a theorem-based construction benchmarked on an external model; the main caveats are delegated proof of the bosonic disk map and boundary/noise conditions, not circular reductions.

full rationale

The central claim that the Wertevorrat rigorously contains all possible analytic continuations follows from Nevanlinna-Pick interpolation theory (external, century-old results cited in Refs. [32-37]) and the explicit center/radius formulas in Corollary 1. These formulas are computed from the input Euclidean data and the interpolation theorem; no parameter is fitted to the target spectral function and then renamed a prediction. The new scaling observation in Fig. 2 is explicitly described as empirical ('the Wertevorrat is observed empirically to decrease roughly exponentially'), made on the external Bernecker-Meyer model for the R-ratio, so it is not a fitted input masquerading as a prediction. The paper does rely on Ref. [29], whose authors include the present speaker, for the bosonic disk-to-disk conformal map and for the identity connecting analytic continuation to Poisson-kernel smearing. This citation is load-bearing in the sense that the talk does not re-derive the disk map, but the cited item is a mathematical construction and not a definitional reversal or a fit, so the central bound retains independent mathematical content. The main limitations are rigor conditions rather than circularity: Eq. (11) writes {zeta_l} in D while the bosonic l=0 Matsubara point maps under the Cayley transform to -1 on the boundary of D, so Theorem 2's open-disk hypothesis is not automatically satisfied unless that point is excluded; the paper does not state this. In addition, the talk itself concedes that statistical uncertainties may make the Pick matrix indefinite, so the rigorous bound and the exponential scaling hold only for data satisfying the hypotheses. These caveats affect the range of validity of the claims but do not make the derivation equivalent to its inputs by construction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No particles, forces, or new physical entities are introduced. The Wertevorrat is a standard Nevanlinna-Pick construction. The only chosen input is the smearing width epsilon; the central quantitative scaling claim depends on it but it is not fitted. The main burden on the reader is the analyticity, boundedness, and Pick-positivity assumptions.

free parameters (1)

smearing width epsilon = 0.06 (Fig. 1) and 0.1 (Fig. 2)
Chosen by hand to define the smeared spectral function; the reported precision of the Wertevorrat scaling depends on this value. It is an input parameter, not fitted to data.

assumptions (5)

standard math Nevanlinna-Pick interpolation theorem and Schur algorithm provide the Wertevorrat representation.
Invoked in Section 4 as Theorem 1 and Corollary 1, with the recursive construction of P_N, Q_N, R_N, S_N taken from Ref. [35].
standard math The Cayley transform maps the upper half-plane to the unit disk and preserves analyticity.
Used in Section 4 to place Euclidean data and real-axis singularities at equidistant positions.
domain assumption The QCD Green function is analytic in the upper half-plane with singularities confined to the real line.
Stated in Section 4: 'the singularities are confined to points z = +/- E_n on the real line.'
domain assumption After conformal maps, the Green function is a bounded analytic function from the disk to the disk.
Required for the codomain mapping in Eq. (11) and for the Schur-class interpolation; not verified for QCD Green functions in this talk.
domain assumption The Pick matrix for the available Euclidean data is positive semidefinite.
Theorem 2; the talk concedes that statistical uncertainties can violate this condition, and lists open work in Refs. [46,47].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Approaching the Inverse Problem: Toward Lattice QCD Calculations of Inclusive Hadronic Quantities." pith.science (2026). https://pith.science/paper/WP6LNWME

@misc{pith2026250112259,
  author       = {Pith},
  title        = {Pith review of: Approaching the Inverse Problem: Toward Lattice QCD Calculations of Inclusive Hadronic Quantities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WP6LNWME}},
  note         = {Machine review of arXiv:2501.12259}
}

read the original abstract

In this talk, I describe some recent ideas relating to the spectral reconstruction inverse problem, which arises frequently in lattice QCD calculations of inclusive hadronic quantities, and provide some physical context for this work. Particular emphasis is given to a new method for rigorously bounding uncertainties using techniques from complex analysis.

Figures

Figures reproduced from arXiv: 2501.12259 by the authors.

**Figure 1.** The reconstruction of the smeared 𝑅-ratio 𝑅(𝑠, 𝜖) using exact Euclidean data at 𝛽 = 96 total points generated according to the Bernecker–Meyer model for 𝑅(𝑠) [49]. The left panel shows the line 𝑧 = 𝜔+𝑖0.06 in the complex plane upon which the smeared spectral function is evaluated. The right panel shows the known exact result for 𝑅(𝑠, 𝜖) (blue curve) as well as the result from the Wertevorrat (black points). a spectr… view at source ↗

**Figure 2.** The scaling of the uncertainty in the smeared spectral reconstruction for 𝑅(𝑠, 𝜖) as the number of points in the interpolation is varied. The fractional uncertainty coming from the Wertevorrat decreases roughly exponentially. The result is for a fixed 𝑧 = 𝜔 + 𝑖𝜖 near the 𝜌-meson peak in 𝑅(𝑠). For a smearing of 𝜖 = 0.1 at 𝑠 = 0.65 GeV2 , 60 and 100 points suffice to determine 𝑅(𝑠, 𝜖) with percent and per mille precis… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

Anisotropic excited bottomonia from a basis of smeared operators
hep-lat 2025-02 conditional novelty 6.0 of 10

Excited bottomonia masses and widths are extracted from lattice NRQCD using a smeared-operator GEVP, showing a width hierarchy and a slowly melting ground state.
Finite temperature hadronic spectral properties
hep-lat 2025-05 conditional novelty 5.0 of 10

Doubly charmed baryons stay stable and the bottomonium ground state loses 30 to 40 MeV as quark-gluon plasma temperature rises, according to new lattice QCD ensembles.
The NRQCD $\Upsilon$ spectrum at non-zero temperature using Backus-Gilbert regularisations
hep-lat 2025-02 conditional novelty 4.0 of 10

On FASTSUM Generation 2L ensembles, Backus-Gilbert and Tikhonov reconstructions agree on the Upsilon ground-state mass, but the decay width is dominated by reconstruction smearing and high-temperature broadening is un...

Reference graph

Works this paper leans on

54 extracted references · 11 canonical work pages · cited by 3 Pith papers

[29]

Bergamaschi, W.I

T. Bergamaschi, W.I. Jay and P.R. Oare,Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D108 (2023) 074516 [2305.16190]

arXiv 2023
[1]

Flavour Lattice A veraging Group (FLAG)collaboration, FLAG Review 2024, 2411.04268

arXiv 2024
[2]

Extended Twisted Mass Collaboration (ETMC) collaboration,Probing the Energy-Smeared R Ratio Using Lattice QCD,Phys. Rev. Lett.130 (2023) 241901 [2212.08467]. 9 Approaching the Inverse Problem William Jay

arXiv 2023
[3]

Extended Twisted Mass collaboration, Inclusive Hadronic Decay Rate of the𝜏 Lepton from Lattice QCD: The¯𝑢𝑠 Flavor Channel and the Cabibbo Angle,Phys. Rev. Lett.132 (2024) 261901 [2403.05404]

arXiv 2024
[4]

Extended Twisted Mass collaboration, Inclusive hadronic decay rate of the𝜏 lepton from lattice QCD, Phys. Rev. D108 (2023) 074513 [2308.03125]

arXiv 2023
[5]

Gambino, S

P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula et al.,Lattice QCD study of inclusive semileptonic decays of heavy mesons,JHEP07 (2022) 083 [2203.11762]

arXiv 2022
[6]

Kellermann, A

R. Kellermann, A. Barone, A. Elgaziari, S. Hashimoto, Z. Hu, A. Jüttnerc et al.,Systematic effects in the lattice calculation of inclusive semileptonic decays, in41st International Symposium on Lattice Field Theory, 11, 2024 [2411.18058]

arXiv 2024
[7]

Barone, S

A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko and R. Kellermann,Chebyshev and Backus-Gilbert reconstruction for inclusive semileptonic𝐵(𝑠)-meson decays from Lattice QCD, PoSLATTICE2023(2024) 236 [2312.17401]

arXiv 2024

Show all 54 references

[8]

Kellermann, A

R. Kellermann, A. Barone, S. Hashimoto, A. Jüttnerc and T. Kanekoa,Studies on finite-volume effects in the inclusive semileptonic decays of charmed mesons,PoS LATTICE2023(2024) 272 [2312.16442]

2024 arXiv
[9]

Barone, S

A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko and R. Kellermann,Approaches to inclusive semileptonic B(𝑠)-meson decays from Lattice QCD,JHEP 07(2023) 145 [2305.14092]

2023 arXiv
[10]

Gambino and S

P. Gambino and S. Hashimoto,Inclusive Semileptonic Decays from Lattice QCD,Phys. Rev. Lett.125(2020) 032001 [2005.13730]

2020 arXiv
[11]

Fukaya, S

H. Fukaya, S. Hashimoto, T. Kaneko and H. Ohki,Towards fully nonperturbative computations of inelasticℓ𝑁 scattering cross sections from lattice QCD,Phys. Rev. D102 (2020) 114516 [2010.01253]

2020 arXiv
[12]

Liang, R.S

J. Liang, R.S. Sufian, B. Wang, T. Draper, T. Khan, K.-F. Liu et al.,Elastic and resonance structures of the nucleon from hadronic tensor in lattice QCD: implications for neutrino-nucleon scattering and hadron physics, 2311.04206

arXiv
[13]

XQCDcollaboration, Towards the nucleon hadronic tensor from lattice QCD,Phys. Rev. D 101 (2020) 114503 [1906.05312]

2020 arXiv
[14]

Rothkopf,Inverse problems, real-time dynamics and lattice simulations,EPJ Web Conf

A. Rothkopf,Inverse problems, real-time dynamics and lattice simulations,EPJ Web Conf. 274 (2022) 01004 [2211.10680]

2022 arXiv
[15]

𝜒QCDcollaboration, PDFs and Neutrino-Nucleon Scattering from Hadronic Tensor, PoS LATTICE2019(2020) 046 [2008.12389]

2020 arXiv
[16]

Bulava,The spectral reconstruction of inclusive rates, PoSLATTICE2022(2023) 231 [2301.04072]

J. Bulava,The spectral reconstruction of inclusive rates, PoSLATTICE2022(2023) 231 [2301.04072]. 10 Approaching the Inverse Problem William Jay

2023 arXiv
[17]

M.LuscherandU.Wolff, HowtoCalculatetheElasticScatteringMatrixinTwo-dimensional Quantum Field Theories by Numerical Simulation,Nucl. Phys. B339 (1990) 222

1990
[18]

Luscher,Two particle states on a torus and their relation to the scattering matrix,Nucl

M. Luscher,Two particle states on a torus and their relation to the scattering matrix,Nucl. Phys. B354(1991) 531

1991
[19]

Hansen, H.B

M.T. Hansen, H.B. Meyer and D. Robaina,From deep inelastic scattering to heavy-flavor semileptonic decays: Total rates into multihadron final states from lattice QCD,Phys. Rev. D 96 (2017) 094513 [1704.08993]

2017 arXiv
[20]

Hansen, A

M. Hansen, A. Lupo and N. Tantalo,Extraction of spectral densities from lattice correlators, Phys. Rev. D99(2019) 094508 [1903.06476]

2019 arXiv
[21]

Backus and F

G. Backus and F. Gilbert,The Resolving Power of Gross Earth Data,Geophys. J. Int.16 (1968) 169

1968
[22]

Pĳpers and M

F. Pĳpers and M. Thompson,Faster formulations of the optimally localized averages method for helioseismic inversions, Astronomy and Astrophysics262 (1992) L33

1992
[23]

Lsdensities: Lattice spectral densities

A. Lupo and N. Forzano, “Lsdensities: Lattice spectral densities.” https://github.com/LupoA/lsdensities
[24]

A. Lupo, L. Del Debbio, M. Panero and N. Tantalo,Bayesian interpretation of Backus-Gilbert methods,PoSLATTICE2023(2024) 004 [2311.18125]

2024 arXiv
[25]

Del Debbio, A

L. Del Debbio, A. Lupo, M. Panero and N. Tantalo,Bayesian solution to the inverse problem and its relation to Backus-Gilbert methods, 2409.04413

arXiv
[26]

Pawlowski, C.S

J.M. Pawlowski, C.S. Schneider, J. Turnwald, J.M. Urban and N. Wink,Yang-Mills glueball masses from spectral reconstruction, Phys. Rev. D108 (2023) 076018 [2212.01113]

2023 arXiv
[27]

Aoyama et al.,The anomalous magnetic moment of the muon in the Standard Model,Phys

T. Aoyama et al.,The anomalous magnetic moment of the muon in the Standard Model,Phys. Rept.887 (2020) 1 [2006.04822]

2020 arXiv
[28]

Muon g-2 collaboration, Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm, Phys. Rev. Lett.131 (2023) 161802 [2308.06230]

2023
[30]

Meyer,Transport Properties of the Quark-Gluon Plasma: A Lattice QCD Perspective, Eur

H.B. Meyer,Transport Properties of the Quark-Gluon Plasma: A Lattice QCD Perspective, Eur. Phys. J. A47(2011) 86 [1104.3708]

2011 arXiv
[31]

Poggio, H.R

E.C. Poggio, H.R. Quinn and S. Weinberg,Smearing the Quark Model,Phys. Rev. D13 (1976) 1958

1976
[32]

Nevanlinna,Über beschränkte Funktionen die in gegebenen punkten vorgeschriebene Werte annehmen, Ann

R. Nevanlinna,Über beschränkte Funktionen die in gegebenen punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. Ser. A13 (1919) . 11 Approaching the Inverse Problem William Jay

1919
[33]

Nevanlinna,Über beschränkte analytische Funktionen,Ann

R. Nevanlinna,Über beschränkte analytische Funktionen,Ann. Acad. Sci. Fenn. Ser. A32 (1929)

1929
[34]

Pick,Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math

G. Pick,Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann.77(1915) 7

1915
[35]

Nicolau,The Nevanlinna-Pick Interpolation Problem, inProceedings of the Summer School in Complex and Harmonic analysis, and related topics, J

A. Nicolau,The Nevanlinna-Pick Interpolation Problem, inProceedings of the Summer School in Complex and Harmonic analysis, and related topics, J. Gröhn, J. Heittokangas, R. Korhonen and J. Rättyä, eds., no. 22 in Reports and Studies in Forestry and Natural Sciences, Publicatio...

2016
[36]

Agler and J

J. Agler and J. McCarthy,Pick Interpolation and Hilbert Function Spaces, American Mathematical Society (2002), 10.1090/gsm/044

2002 doi
[37]

Garcia, J

S. Garcia, J. Mashreghi and W. Ross,Finite Blashke Products and Their Connections, Springer, 1 ed. (May, 2018), 10.1007/978-3-319-78247-8

2018 doi
[38]

Fei, C.-N

J. Fei, C.-N. Yeh and E. Gull,Nevanlinna analytical continuation, Phys. Rev. Lett.126 (2021) 056402

2021
[39]

Nogaki and H

K. Nogaki and H. Shinaoka,Bosonic Nevanlinna Analytic Continuation,J. Phys. Soc. Jap. 92 (2023) 035001 [2305.03449]

2023 arXiv
[40]

C.G. Boyd, B. Grinstein and R.F. Lebed,Constraints on form-factors for exclusive semileptonic heavy to light meson decays,Phys. Rev. Lett.74(1995) 4603 [hep-ph/9412324]

1995 arXiv
[41]

C.G. Boyd, B. Grinstein and R.F. Lebed,Model independent extraction of|𝑉𝑐𝑏| using dispersion relations, Phys. Lett. B353 (1995) 306 [hep-ph/9504235]

1995 arXiv
[42]

C.G. Boyd, B. Grinstein and R.F. Lebed,Model independent determinations of¯𝐵→𝐷ℓ ¯𝜈, 𝐷★ℓ ¯𝜈 form factors,Nucl. Phys. B461(1996) 493 [hep-ph/9508211]

1996 arXiv
[43]

C.G. Boyd, B. Grinstein and R.F. Lebed,Precision corrections to dispersive bounds on form-factors, Phys. Rev. D56(1997) 6895 [hep-ph/9705252]

1997 arXiv
[44]

Grinstein and A

B. Grinstein and A. Kobach,Model-Independent Extraction of|𝑉𝑐𝑏| from ¯𝐵→𝐷∗ℓ𝜈, Phys. Lett. B771 (2017) 359 [1703.08170]

2017 arXiv
[45]

Caprini, L

I. Caprini, L. Lellouch and M. Neubert,Dispersive bounds on the shape of¯𝐵→𝐷★ℓ ¯𝜈 form-factors, Nucl. Phys. B530 (1998) 153 [hep-ph/9712417]

1998 arXiv
[46]

Huang, E

Z. Huang, E. Gull and L. Lin,Robust analytic continuation of Green’s functions via projection, pole estimation, and semidefinite relaxation, Phys. Rev. B107 (2023) 075151 [2210.04187]

2023 arXiv
[47]

Y. Yu, A.F. Kemper, C. Yang and E. Gull,Denoising of imaginary time response functions with Hankel projections, Phys. Rev. Res.6 (2024) L032042 [2403.12349]. 12 Approaching the Inverse Problem William Jay

2024 arXiv
[48]

Huang and S

L. Huang and S. Liang,Reconstructing lattice QCD spectral functions with stochastic pole expansion and Nevanlinna analytic continuation, Phys. Rev. D109 (2024) 054508 [2309.11114]

2024 arXiv
[49]

Bernecker and H.B

D. Bernecker and H.B. Meyer,Vector Correlators in Lattice QCD: Methods and applications,Eur. Phys. J. A47(2011) 148 [1107.4388]

2011 arXiv
[50]

Patella and N

A. Patella and N. Tantalo,Scattering Amplitudes from Euclidean Correlators: Haag-Ruelle theory and approximation formulae, 2407.02069

arXiv
[51]

Bruno, L

M. Bruno, L. Giusti and M. Saccardi,Spectral densities from Euclidean lattice correlators via the Mellin transform, 2407.04141

arXiv
[52]

Wagman,Lanczos, the transfer matrix, and the signal-to-noise problem, 2406.20009

M.L. Wagman,Lanczos, the transfer matrix, and the signal-to-noise problem, 2406.20009

arXiv
[53]

Hackett and M.L

D.C. Hackett and M.L. Wagman,Block Lanczos for lattice QCD spectroscopy and matrix elements, 2412.04444

arXiv
[54]

Hackett and M.L

D.C. Hackett and M.L. Wagman,Lanczos for lattice QCD matrix elements, 2407.21777. 13

arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [29]

Bergamaschi, W.I

T. Bergamaschi, W.I. Jay and P.R. Oare,Hadronic structure, conformal maps, and analytic continuation, Phys. Rev. D108 (2023) 074516 [2305.16190]

arXiv 2023

[2] [1]

Flavour Lattice A veraging Group (FLAG)collaboration, FLAG Review 2024, 2411.04268

arXiv 2024

[3] [2]

Extended Twisted Mass Collaboration (ETMC) collaboration,Probing the Energy-Smeared R Ratio Using Lattice QCD,Phys. Rev. Lett.130 (2023) 241901 [2212.08467]. 9 Approaching the Inverse Problem William Jay

arXiv 2023

[4] [3]

Extended Twisted Mass collaboration, Inclusive Hadronic Decay Rate of the𝜏 Lepton from Lattice QCD: The¯𝑢𝑠 Flavor Channel and the Cabibbo Angle,Phys. Rev. Lett.132 (2024) 261901 [2403.05404]

arXiv 2024

[5] [4]

Extended Twisted Mass collaboration, Inclusive hadronic decay rate of the𝜏 lepton from lattice QCD, Phys. Rev. D108 (2023) 074513 [2308.03125]

arXiv 2023

[6] [5]

Gambino, S

P. Gambino, S. Hashimoto, S. Mächler, M. Panero, F. Sanfilippo, S. Simula et al.,Lattice QCD study of inclusive semileptonic decays of heavy mesons,JHEP07 (2022) 083 [2203.11762]

arXiv 2022

[7] [6]

Kellermann, A

R. Kellermann, A. Barone, A. Elgaziari, S. Hashimoto, Z. Hu, A. Jüttnerc et al.,Systematic effects in the lattice calculation of inclusive semileptonic decays, in41st International Symposium on Lattice Field Theory, 11, 2024 [2411.18058]

arXiv 2024

[8] [7]

Barone, S

A. Barone, S. Hashimoto, A. Jüttner, T. Kaneko and R. Kellermann,Chebyshev and Backus-Gilbert reconstruction for inclusive semileptonic𝐵(𝑠)-meson decays from Lattice QCD, PoSLATTICE2023(2024) 236 [2312.17401]

arXiv 2024