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REVIEW 5 minor 111 references

From scattering towards multi-hadron weak decays

T0 review · 0 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This review establishes that lattice QCD scattering calculations have matured to the point where they can compute multi-hadron weak-decay amplitudes and feed flavour physics.

desk verdict A solid, honest plenary review that maps lattice scattering's role in flavour physics; no new results, but a useful snapshot with transparent limits. read the letter →

arxiv 2501.19302 v1 pith:GYMXUKUW submitted 2025-01-31 hep-lat

Felix Erben This is my paper · ORCID
classification hep-lat MSC 81V0581T25 PACS 12.38.Gc13.20.-v11.15.Ha
keywords latticeQCDfinite-volumequantizationscatteringamplitudeshadronicresonancesweakdecaysflavourphysicsthree-particleformalismmuong-2
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

This review argues that lattice QCD scattering calculations have reached a turning point: they now feed directly into flavour physics through weak decays whose final or intermediate states contain two or three hadrons. The author surveys a set of recent calculations—kaon decay to two pions, rho and K* resonances at physical quark masses, the Tcc tetraquark, the omega meson, B to rho l nu, and the long-distance muon g-2—that use finite-volume spectra and matrix elements as input. The common thread is that finite-volume formalisms, originally built for two-particle scattering, are being extended to three particles and to electroweak currents. If the picture holds, lattice QCD becomes a decisive non-perturbative source of Standard-Model predictions for processes that mix strong and weak interactions.

What carries the argument

The load-bearing object is the finite-volume quantization condition: Lüscher's relation between the discrete energy levels of a finite periodic box and the infinite-volume scattering phase shift, together with the Lellouch-Lüscher extension that connects finite-volume matrix elements to infinite-volume transition amplitudes. Lattice correlation functions are first analysed with a generalized eigenvalue problem to extract finite-volume energies and optimized interpolators; those energies are then fed through the quantization condition to produce phase shifts, resonance poles, or decay amplitudes. For three-particle systems the review relies on generalizations of these conditions that introduce intermediate K-matrices and account for left-hand cuts, as in the Tcc and omega calculations. A data-driven weighted-histogram procedure converts the many single-level fits into final resonance parameters with combined statistical and systematic uncertainties.

What would settle it

A lattice computation of the omega(782) pole with two lattice spacings, two volumes, and physical pion masses would settle the three-particle formalism: if the extrapolated pole disagrees with the experimental value by more than the quoted uncertainties, the treatment of the rho-pi inelastic channel is incomplete.

Watch

Extended reading notes

Core claim

The central claim is that the combination of mature finite-volume quantization conditions and new numerical techniques has made multi-hadron weak decays computable from first principles. Concretely, the finite-volume energy spectrum obtained from lattice correlation functions is related by the Lüscher quantization condition to infinite-volume scattering phase shifts, and the Lellouch-Lüscher formalism extends this relation to weak transition matrix elements. With these tools, the author reports, a lattice calculation at the physical pion mass reproduces the $\Delta$ I = 1/2 rule in K to pi pi, resonance poles for the rho and K* agree with experiment, and the long-distance part of the muon g-2 can be reconstructed from pi pi scattering states rather than suffering an exponential signal-to-noise problem. Three-particle formalisms, though still under active development, have opened channels such as omega to 3 pi and DD pi (Tcc). The review concludes that scattering calculations are now positioned to make meaningful contributions to flavour physics.

Load-bearing premise

Everything rests on the finite-volume quantization conditions—from Lüscher's two-particle relation to the newer three-particle variants—correctly encoding the infinite-volume physics, with left-hand cuts and inelastic channels fully accounted for.

Editorial extensions

If this is right

  • Hadronic K to pi pi decays can now be computed at the physical pion mass, giving a first-principles value for Re(A0)/Re(A2) consistent with the experimental Delta I = 1/2 rule.
  • The B to rho l nu form factors extracted from B to pi pi l nu pave the way for B to K* l+ l- calculations that can be confronted with the flavour anomalies.
  • Long-distance muon g-2 can be determined by reconstructing the vector-vector correlator from pi pi scattering states, bypassing the exponential signal-to-noise problem at large times.
  • Three-particle resonances such as the omega and the doubly charmed tetraquark Tcc become accessible, with the omega pole extrapolated to the physical pion mass in agreement with experiment.
  • Resonance parameters at physical quark masses can now be reported with both statistical and systematic uncertainties, using Akaike-weighted averages over fit choices.

Reading between the lines

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

  • The same machinery could be turned on other semileptonic decays with unstable final states, such as B to D* l nu or Bs to K* l+ l-, where the final vector meson decays into a multi-hadron state; nothing in the reviewed formalism is specific to pi pi.
  • The left-hand cut problem encountered for Tcc is likely to recur for any weakly bound hadronic molecule near a one-pion exchange threshold; the modified quantization conditions treated in this review may become the standard tool for such systems.
  • The split-even variance-reduction technique, which the review reports cuts rare-kaon-decay errors by factors of 4 to 10, could plausibly be adopted by other long-range matrix-element calculations including g-2 and related precision tests.
  • A natural next step, which the review hints at, is the creation of shared quality criteria for scattering observables analogous to those used for other lattice quantities, so that results from different groups can be compared and averaged.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. This manuscript is the written version of the author's plenary talk at LATTICE2024. It surveys lattice QCD calculations of hadron scattering and of weak decays involving multi-hadron states. The first part covers recent two- and three-particle scattering computations, including the Tcc(3875) and the omega meson, and the author's own physical-point rho(770) and K*(892) calculation with a data-driven AIC-based error analysis. The second part reviews finite-volume formalisms and computations for hadronic K->pi pi decays, 1+J->2 transitions such as B->rho l nu, long-range electroweak matrix elements (rare kaon decays, epsilon_K, B->mu mu gamma, K_L -> mu mu), and the long-distance hadronic vacuum polarization contribution to the muon g-2. The central claim is that scattering calculations are now mature enough to impact flavour physics and that lattice QCD is becoming a key non-perturbative tool.

Significance. The paper is a conference proceedings review, so its significance lies in providing a timely, organized overview of a rapidly moving subfield rather than in a new technical result. It is careful to flag the main open issues: the left-hand cut in the Tcc analysis (Section 2.2), the ongoing development of three-particle formalisms (Section 2.3), and the single-lattice-spacing limitation of the physical-point rho/K* calculation (Section 2.4). The author also highlights reproducible artifacts, including the public release of a 760 GB correlator dataset [58] and open-source software [61,62], and gives a balanced account of both achievements and outstanding uncertainties in long-range electroweak matrix elements. If the qualitative assessment is correct, the review serves as a useful entry point for non-experts and a status record for practitioners.

minor comments (5)
  1. [2.4] The reported number of phase-shift fits, written as 'n_ideal ~ 10020' on page 8, is ambiguous and should be typeset as 100^{20} to convey the intended exponent.
  2. [3.3] In the sentence after Eq. (15), the phrase 'for some L′' is confusing because L is the fixed volume of the finite-volume estimator; the pole condition is that a discrete energy level E_n(L) crosses E_Sigma.
  3. [3.3] The ellipsis in Eq. (15) is not defined; please state which terms are being omitted in the sum over finite-volume states.
  4. [2] There are several typographical errors throughout the text: 'refered' for 'referred', 'statsitical' for 'statistical', 'Lippman-Schwinger' for 'Lippmann-Schwinger', 'comapre' for 'compare' in the caption of Fig. 3(a), and 'transtion' for 'transition' in the caption of Fig. 6.
  5. [1] In Eq. (1), the notation 'finite-volume energy states E_n' would be clearer as 'finite-volume energy levels E_n', and it would help to state explicitly that the spectral decomposition is taken on a finite Euclidean-time interval with |n> denoting energy eigenstates.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is a review/survey; its central claim is a qualitative assessment, and the only author-derived results are benchmarked against experimental data rather than reduced to fitted inputs.

full rationale

The manuscript is a Lattice 2024 review article, so its central claim — that multi-hadron weak decays are making progress and lattice QCD is becoming a key non-perturbative tool — is a qualitative synthesis of the cited literature. No derivation in the paper reduces to its inputs. Section 2.4 describes the author's own rho(770)/K*(892) calculation [56,57], but the chain there is standard and non-circular: lattice correlation functions are computed from a publicly released data set [58], energy levels are extracted by GEVP, phase-shift models are fitted to those levels, and resonance poles are obtained by solving Eq. (2). The poles are not used to define the input energy levels, and the final results are compared with experimental PDG values [67]. The finite-volume formalisms (Luescher, Lellouch-Luescher, three-particle extensions) are imported from external literature, and the review explicitly flags their limitations, e.g. left-hand-cut effects in the Tcc discussion and ongoing three-particle formalism development in Sections 2.2 and 2.3. Self-citations to [56,57] and to software [61,62] are descriptive rather than load-bearing: they point to publicly checkable data and open-source tools, not to an unverified premise on which the review's conclusion uniquely depends. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is smuggled in via self-citation. Accordingly, no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a review; it introduces no free parameters or new entities. It relies on established lattice QCD formalisms (Lüscher, Lellouch-Lüscher, three-particle quantization conditions) as standard domain assumptions.

assumptions (4)
  • domain assumption Finite-volume correlation functions computed in Euclidean time determine finite-volume energies and matrix elements that are independent of metric signature.
    Invoked in Section 1, Eq. (1), as the basis for all lattice extraction methods.
  • domain assumption Lüscher's formalism relates finite-volume energy shifts to infinite-volume scattering phase shifts.
    Core assumption throughout Section 2, introduced via Refs. [4-15].
  • domain assumption Lellouch-Lüscher formalism extends the relation to transition matrix elements for weak decays.
    Foundation for the electroweak matrix elements reviewed in Section 3.
  • domain assumption Three-particle finite-volume formalisms correctly describe systems with three hadrons and resonant subprocesses.
    Assumed in Section 2.3; the paper notes these formalisms are still being developed and mutually compared.

how reviews work

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Cite this review

Pith. "Pith review of From scattering towards multi-hadron weak decays." pith.science (2026). https://pith.science/paper/GYMXUKUW

@misc{pith2026250119302,
  author       = {Pith},
  title        = {Pith review of: From scattering towards multi-hadron weak decays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYMXUKUW}},
  note         = {Machine review of arXiv:2501.19302}
}
read the original abstract

In this article I provide an overview of the current state of scattering within lattice QCD, along with ongoing projects that examine weak decays involving scattering states as either final or intermediate states. Significant progress has been made in the study of multi-hadron weak decays, opening the door for scattering calculations to make meaningful contributions to flavour physics and further establishing lattice QCD as the key non-perturbative tool for QCD predictions. In addition to discussing new calculations, I also highlight recent advancements in finite-volume formalisms, which enable the exploration of previously inaccessible channels.

Figures

Figures reproduced from arXiv: 2501.19302 by the authors.

Figure 1
Figure 1. Complex plane of the Mandelstam variable 𝑠, illustrating possible pole singularities of the scattering amplitude. Sheets are distinguished by the imaginary part of 𝑘. The left plot shows the physical sheet where bound states occur on the real axis. The right plot shows the unphysical sheet where resonances appear above the branch cut and bound states may manifest as virtual bound states. continuous description of th… view at source ↗
Figure 2
Figure 2. Scattering amplitudes for the charmonium resonances computed in [25, 26]. The left plot shows the 𝐽 𝑃𝐶 = 0 ++ and the right plot shows the 𝐽 𝑃𝐶 = 2 ++ channel. Shaded bands are the envelope over fit variations to the energy levels shown as black dots at the bottom of the plots. of 2510 − 2610 MeV. While direct conclusions regarding physical-point scattering cannot be easily drawn from this analysis, such studies off… view at source ↗
Figure 3
Figure 3. Plots from the 𝑇𝑐𝑐 and 𝜔 works involving 3-particle formalisms. 2.3 Three-particle scattering Finite-volume scattering formalisms were initially developed for systems involving two parti￾cles. However, several formalisms for three-particle scattering have been available for some time now [43–48]. For a detailed discussion and comparison of the three available approaches, which are mutually compatible where a compari… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Effective masses of the two lowest GEVP eigenvectors in the P = [110] frame, 𝐵1 irrep in the 𝐾𝜋 scattering system. The overlaid bands correspond to our fit result for the combined systematic and statistical error, estimated from the weighted histogram plotted at the le…
Figure 5
Figure 5. Figure 5: Histograms of the final results of the 𝐾 ∗ and 𝜌 scattering study. Figures taken from [56, 57] associated AIC score, AICPS. Repeating this process across multiple samples allows us to explore the fit space systematically. A histogram of phase shift parameters, such as …
Figure 6
Figure 6. Figure 6: The four form factors describing the 𝐵 → 𝜌ℓ𝜈 transtion as shown in [86]. Lattice data is available at the high-𝑞 2 region, between the black vertical lines. The lighter shaded region outside these lines is an extrapolation from the lattice data on the first-order 𝑧-exp…
Figure 7
Figure 7. Figure 7: Plots displaying the reconstructed 𝑔 − 2 integrand from Mainz [108] and RBC/UKQCD [109], both computed on ensembles with physical pion masses. In both cases, the reconstructed integrand closely matches the direct lattice calculation of the vector-vector correlator, sho…

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