{"id":"16fa6e20-21ba-4aa7-bfb0-4616a7ffd790","arxiv_id":"2505.10002","paper_version":1,"verdict":"UNVERDICTED","confidence":"HIGH","novelty_score":1.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"A pedagogical chapter explains how lattice QCD eigen-energies are converted into hadron masses, widths, and scattering amplitudes via the Lüscher formalism, using known examples.","lead":"This chapter teaches how lattice QCD simulations compute hadron masses, decay widths, and scattering amplitudes from the strong-interaction Lagrangian. It explains the finite-volume Lüscher formalism and illustrates it with conventional and exotic hadron examples, but presents no new numerical results.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (11) hangs on the geometric series for the finite-volume correlator; a missing check is whether the omitted exponentially suppressed terms that dress the finite-volume correction are uniformly small for the small pion masses and large L used in the cited examples.","rationale":"The reader and this pass converge on the same load-bearing premise: the neglect of exponentially suppressed finite-volume terms in the Luscher relation, flagged before Eq. (9). The chapter is a pedagogical reprint, not a research preprint; its central claim is a methodological one already validated by the cited literature, so the appropriate verdict remains UNVERDICTED. The concern is genuine but not fatal: it is a standard systematic effect that the field controls by varying L and by using parametrizations of T(E), and the concrete examples shown come from refereed papers with their own checks. It would be improved by an explicit statement of the mL values in the examples, but the absence of that quantification does not undermine the pedagogical claim. The stress-test pass therefore agrees with the reader's verdict and identifies no independent, load-bearing, non-standard flaw.","tokens_in":17480,"tokens_out":1297,"duration_ms":12056,"concrete_test":"Re-derive the finite-volume loop difference in Eq. (9) keeping the residue at the second pole k_0=E+omega_k instead of dropping it, and evaluate its size for the lightest example (m_pi=320 MeV, L=3.6 fm, E near the rho resonance) against the retained on-shell term, or quote the corresponding control from the original papers [15,16] if the exercise is meant as exposition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, that finite-volume eigen-energies determine the infinite-volume scattering amplitude via Eq. (12), rests on the derivation of Eq. (11), where the finite-volume correlator is resummed as C_V(E)=C_inf(E)-A' [1/(T+iF)] A. The text explicitly sets aside 'the exponentially suppressed finite-volume corrections e^{-mL}' before Eq. (9), and the subsequent residue argument splits the finite-volume loop correction into a non-exponentially-suppressed on-shell piece and an exponentially suppressed remainder. The geometric series in Eq. (11) then sums an infinite chain of the non-exponential piece F while discarding the exponentially suppressed terms at each insertion. This is a standard and controlled assumption for large mL, and the reader correctly identifies it as the weakest premise. The concern is not that the neglect is wrong in general, but that it is asserted rather than quantified in a chapter whose pedagogical examples (rho at m_pi=320 MeV with L=3.6 fm, T_cc near the DD* threshold) operate near the regime where mL is moderate and where pole extractions are sensitive to parametrization. The cited examples rely on published studies with their own systematic checks, so the chapter itself is not making a new unverified claim; the fragility is in the pedagogical derivation's unquantified truncation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a pedagogical book chapter on lattice QCD hadron spectroscopy. It explains how masses of strongly stable hadrons are extracted from two-point correlation functions, and how resonance masses, widths, and lineshapes are obtained from scattering amplitudes T(E). The central technical thread is a simplified derivation of the one-channel Lüscher relation, which maps finite-volume eigen-energies E(L) to the infinite-volume scattering amplitude at the same energy, followed by explicit examples: the rho and K* resonances, the D*_{s0}(2317) bound state, the T_cc near the DD* threshold, coupled-channel f0/f2 systems, and nucleon-nucleon scattering. The chapter concludes with static-potential and HAL QCD approaches. The text is intentionally not a full review; it delegates technical details to the cited literature.","tokens_in":17820,"tokens_out":14379,"duration_ms":132155,"significance":"The chapter serves as a self-contained introduction to a technically demanding subject. Its main strength is the explicit derivation of the Lüscher relation from the finite-volume correlator (Eqs. (8)–(12)), which is presented in a form accessible to graduate students, with the technical details correctly deferred to Refs. [12,14]. The examples are drawn from published lattice studies, so the pedagogical statements are backed by peer-reviewed results. The chapter does not claim new numerical results; its value is didactic. The claim that lattice QCD is a first-principles, systematically improvable method is supported by the cited calculations of stable hadron masses and scattering phase shifts. The main correctness risk is not circularity—the finite-volume energies are mapped to the infinite-volume amplitude, not fitted circularly—but rather the unquantified neglect of exponentially suppressed finite-volume effects and the simplified treatment of partial waves, both of which are noted in the text.","major_comments":[{"comment":"As typeset, the final expression for C_V(E)=C_∞(E)−A'[1/(T+iF)−1]A does not follow from the preceding geometric series C_∞(E)−A'F Σ_{j=0}^∞(−iTF)^j A, and it is inconsistent with the pole condition in Eq. (12), which requires T^{-1}(E)=−iF(E). The denominator should read T^{-1}+iF (or an equivalent rearrangement of the standard Lüscher form). Please correct this equation; the intended result is standard, but the printed form will mislead a student.","section":"4.1, Eq. (11)"},{"comment":"The derivation in Section 4.1 is restricted to s-wave scattering (l=0, F=F_{00,00}), but the first application is the p-wave rho resonance. The footnote under Figure 6 states that Eq. (12) 'applies here although this is scattering with l=1,' which is misleading because Eq. (12) as derived does not hold for l=1; the relevant Lüscher relation uses the p-wave kinematic function (e.g., from Eq. (22) of Ref. [14]). The chapter should explicitly explain how the p-wave example is connected to the l=0 derivation, or reduce the derivation claim to the s-wave case and refer to the general formalism for the examples.","section":"4.2, Fig. 6 and Eq. (12)"}],"minor_comments":[{"comment":"\"Deutron\" should be \"deuteron\" in the text and in the Figure 12 caption.","section":"4.2, final paragraph; Fig. 12 caption"},{"comment":"\"The so-called so-called time-dependent method\" should be \"the so-called time-dependent method.\"","section":"Footnote 6"},{"comment":"The phrase \"at large enought\" should be \"at large enough t.\"","section":"Section 2, before Eq. (2)"},{"comment":"The neglect of exponentially suppressed e^{-mL} corrections is asserted without quantification; a sentence noting that the cited applications control these corrections through multiple volumes and systematic studies (e.g., Refs. [12,15,16]) would help the reader distinguish the pedagogical simplification from a general proof.","section":"Section 4.1, before Eq. (9)"},{"comment":"The Breit-Wigner form in Eq. (13) is introduced after the phase-shift plot; the text should specify that this is a parametrization of T(E) chosen for fitting, not a direct consequence of Eq. (12), to avoid implying that the pole parameters are extracted without a model assumption.","section":"Section 4.2, Eq. (13)"}],"recommendation":"minor_revision","confidential_remarks":"This is a pedagogical review chapter, so the standards for novelty and completeness differ from a research article. The main risk is that the simplified equations are applied beyond their stated regime of derivation (s-wave), but this is a presentation issue. I recommend minor revision. The author should also verify that Eq. (11) in the arXiv version is not garbled; the typeset formula appears inconsistent with the surrounding derivation, though the intended result is standard."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a textbook chapter, not a research paper. It re-derives the standard Lüscher formalism, GEVP, coupled-channel determinant, and HAL QCD potential method, then illustrates them with published lattice results. There is no new result, and the author does not claim one. What the chapter does well is explain the logic of finite-volume spectroscopy in a mostly self-contained way. The derivation of the Lüscher relation from the finite-volume correlator, Eqs. (8)-(12), is concise and readable, and the selection of examples—rho, K*, Ds0*, Tcc, Tbb, NN—is well chosen to show both what works and where the difficulties lie. The exposition of the coupled-channel case and the pole-extraction strategy is particularly clear.\n\nThe soft spots are minor but real. First, Eq. (12) is derived explicitly for l=0, yet the rho example is a p-wave. The footnote says the relation still applies, but a pedagogical chapter should explain why: the generalized Lüscher matrix F is not the same scalar function for l=1. The reader is left to fill in a nontrivial gap. Second, the neglect of exponentially suppressed finite-volume corrections is asserted before Eq. (9) and never quantified. That is standard and acceptable for an introduction, but since the chapter stresses systematic improvability, a sentence pointing to the original checks in the cited examples would have been honest. The stress-test note worries about this, and I think it lands: the assumption is fine, but it is not stated as an assumption with caveats.\n\nIs the math sound? As a presentation of established results, yes. There is no circularity: finite-volume energies are mapped to infinite-volume amplitudes, and the examples fit parameters to those amplitudes. The citation pattern is appropriate; the chapter draws on the original papers and correctly identifies left-hand cut subtleties in the Tcc case.\n\nWho is this for? Graduate students and researchers entering hadron spectroscopy. It will serve well as a starting point before moving to Briceño, Dudek, and Young's review. It is not something I would cite in a research paper, but I would give it to a student.\n\nMy recommendation: if this is submitted to a journal as a research article, it should be desk rejected; it has no new result. If it is a book chapter for a volume where pedagogical reviews are welcome, a careful referee should still see it, mainly to check the p-wave point and the unquantified e^{-mL} neglect. As a desk editor, I would not block it.","headline":"A clean, conventional pedagogical chapter on lattice QCD spectroscopy: useful for students, no new physics, with two small spots where the simplification is too quick.","tokens_in":18310,"tokens_out":1830,"would_cite":false,"duration_ms":20830,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.Gc"],"model":"deepseek-v4-flash","headline":"Lattice QCD can derive hadron masses, widths, and scattering amplitudes from first principles.","keywords":["lattice QCD","hadron spectroscopy","hadronic resonances","Lüscher formalism","scattering amplitudes","exotic hadrons","finite-volume eigen-energies","coupled-channel scattering"],"falsifier":"Take a single scattering channel at a fixed pion mass, extract the phase shift from two or more lattice volumes differing by roughly a factor of two, and check whether the extracted phase-shift points agree to within the quoted statistical error; a trend with $L$ at the level of $e^{-mL}$ would show the neglected corrections are not negligible. A second check is to refit the same eigen-energy sets with two different parametrizations of $T(E)$ and compare the pole positions.","tokens_in":17317,"feed_emoji":"⚛️","tokens_out":8079,"duration_ms":81261,"temperature":0.7,"pith_summary":"This chapter sets out to show that hadron spectroscopy is accessible from first principles through lattice QCD: strongly stable hadrons get their masses directly from ground-state eigen-energies, while unstable hadrons have to be read off the scattering amplitude for the channel into which they decay. The key step is the finite-volume quantization condition, which turns each discrete two-hadron eigen-energy in a periodic box into a value of the infinite-volume scattering amplitude $T(E)$ at that same energy. Once enough eigen-energies are collected, a resonance's pole in the complex energy plane gives its mass and width, and the amplitude gives the phase shifts and lineshapes. The chapter illustrates the route with one-channel examples ($\\rho$, $K^*$, $D_{s0}^*$), near-threshold exotic states ($T_{cc}$, $T_{bc}$), and a coupled-channel case ($\\pi\\pi$-$K\\bar{K}$-$\\eta\\eta$). If the programme works, then the properties of both ordinary and exotic hadrons can be predicted from QCD rather than inserted by hand.","feed_headline":"Lattice QCD derives hadron masses and widths from QCD itself","feed_subtitle":"A finite-volume relation maps boxed eigen-energies to the amplitudes behind hadron masses, widths, and lineshapes.","key_machinery":"The load-bearing object is the finite-volume quantization condition for two-hadron states, usually called the L\\\"uscher relation. In its simplest s-wave, zero-momentum form it reads $T_{l=0}^{-1}(E) = -iF(E)$, where $F$ is a known kinematic function built from the difference between a continuum loop integral and a discrete sum over the momenta allowed in a periodic box. The derivation keeps only the part of the loop in which both intermediate particles are on shell; the residual difference is exponentially suppressed as $e^{-mL}$ and is neglected. The same $F$ generalizes to moving frames and to several coupled channels, where the condition becomes $\\det(T + iF^{-1}) = 0$. This function is what carries the argument: it converts a set of box energies into phase shifts and pole positions.","core_discovery":"The central claim, stated in the author's own terms, is that the spectroscopic information for any strongly interacting hadron is contained in the discrete eigen-energies $E_n$ of QCD in a finite volume. For a stable hadron, $m = E_1(\\vec P=\\vec 0)$ after continuum and infinite-volume extrapolations. For an unstable hadron, the finite-volume quantization condition $T_{l=0}^{-1}(E) = -iF(E)$ maps each lattice eigen-energy to the corresponding point of the infinite-volume scattering amplitude, and the resonance parameters are read from the pole position $E_p = m - i\\Gamma/2$. Coupled channels generalize this to the determinant condition $\\det(T + iF^{-1})=0$, where the energy dependence of $T$ is parametrized and fitted to the computed levels. The examples presented are offered as demonstrations that this extraction is practical at current lattice sizes and quark masses.","pith_inferences":["If the exponentially suppressed terms $e^{-mL}$ were computed instead of omitted, near-threshold states with tiny binding energies (for example $T_{cc}^+$, bound by under 1 MeV) would get a first-principles systematic error; today that error is only assumed small.","The same mapping from box energies to amplitudes is the template for the emerging three-hadron formalism; the one- and two-channel cases in this chapter are the natural testbed for those extensions.","A direct comparison of the quantization-condition, static-potential, and HAL QCD routes on the same heavy-quark system would isolate the model dependence that currently enters through the parametrization of $T(E)$.","Reanalysis of existing published spectra with alternative parametrizations of the amplitude could reveal whether reported pole positions (especially for broad states) are physical or fit-dependent."],"forward_implications":["Masses of stable hadrons such as the proton, neutron, and heavy-light mesons can be computed from QCD with sub-percent precision once quark masses and lattice spacing are fixed, with no hadronic model input.","For resonances that decay through a single channel, lattice eigen-energies at several volumes and momenta produce phase shifts that fix the resonance mass and width; physical-quark-mass results for $\\rho$ and $K^*$ agree with experiment.","The same machinery assigns masses and binding energies to near-threshold exotic states; the $T_{cc}$ pole trajectory shows how a state can pass from resonance to virtual to bound as quark masses change.","For states above several thresholds, a coupled-channel analysis constrains all elements of the scattering matrix and locates poles such as $f_0$ and $f_2$.","Because every lattice eigen-energy yields one point on the amplitude, increasing the number of volumes, total momenta, and interpolating operators directly sharpens the extracted pole parameters."],"supporting_citations":[{"why":"Supplies the finite-volume quantization condition for two hadrons in moving frames; the derivation of the main relation is based on this reference.","marker":"[12]"},{"why":"Original two-particle-on-a-torus relation between finite-volume energies and the scattering matrix, the foundation of the method.","marker":"[13]"},{"why":"Introduces the generalized eigenvalue approach used to extract eigen-energies from correlation matrices.","marker":"[8]"},{"why":"Review of scattering processes and resonances from lattice QCD, framing the extraction of $T(E)$.","marker":"[4]"},{"why":"Pedagogical $\\rho\\to\\pi\\pi$ example: lattice energies and phase shifts at $m_\\pi\\simeq 320$ MeV.","marker":"[15]"},{"why":"Physical-quark-mass measurement of $\\rho$ and $K^*$ resonance parameters from phase shifts.","marker":"[16]"},{"why":"$T_{cc}$ pole trajectory in $DD^*$ scattering combining the quantization condition with effective field theory.","marker":"[22]"},{"why":"Coupled-channel $\\pi\\pi$-$K\\bar{K}$-$\\eta\\eta$ example that yields $f_0$ and $f_2$ poles via the determinant condition.","marker":"[30]"}],"fun_headline_variants":["Lattice QCD computes hadron masses and scattering amplitudes","From lattice eigen-energies to hadron resonance parameters","Calculating hadron spectra from QCD on a finite grid","Lattice simulations extract hadron masses and widths directly","Discrete QCD grid maps to real hadron scattering amplitudes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that exponentially suppressed finite-volume corrections of order $e^{-mL}$ really are negligible, so that each discrete lattice eigen-energy can be equated with the infinite-volume scattering amplitude at that same energy.","fun_headline_variants_meta":{"raw":{"variants":["Lattice QCD computes hadron masses and scattering amplitudes","From lattice eigen-energies to hadron resonance parameters","Calculating hadron spectra from QCD on a finite grid","Lattice simulations extract hadron masses and widths directly","Discrete QCD grid maps to real hadron scattering amplitudes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1437,"prompt_tokens":865,"completion_tokens":572,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":490}},"tokens_in":481,"tokens_out":572,"duration_ms":5895,"temperature":1.0,"reasoning_tokens":490,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:18:54.000787+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a single scattering channel at a fixed pion mass, extract the phase shift from two or more lattice volumes differing by roughly a factor of two, and check whether the extracted phase-shift points agree to within the quoted statistical error; a trend with $L$ at the level of $e^{-mL}$ would show the neglected corrections are not negligible. A second check is to refit the same eigen-energy sets with two different parametrizations of $T(E)$ and compare the pole positions.","supporting_citations":[{"cited_title":"Two particle states on a torus and their relation to the scattering matrix,","cited_arxiv_id":null,"evidence_quote":"Original two-particle-on-a-torus relation between finite-volume energies and the scattering matrix, the foundation of the method."}],"review_version":1}