{"id":"971360ff-e82e-4f6e-996f-af2f97feabbb","arxiv_id":"2610.40390","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Meson decay widths are extracted from Bethe-Salpeter kernels built from the 4PI effective action of QCD, after the rainbow-ladder kernel is tuned to the experimental masses.","lead":"Using a four-loop 4PI effective action of QCD, the authors derive the quark-antiquark scattering diagrams that produce two-body decays and compute the rho, K* and phi resonance poles in the complex plane. The resulting decay widths are within 24-33 percent of experiment only after the binding kernel is tuned to reproduce the experimental masses.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Width extraction depends on the unproven assertion that omitted diagrams leave the discontinuity of lambda unchanged; the SPM errors in Table II do not cover this source.","rationale":"The reader's weakest assumption and the concern I would press are the same: the width is fixed by the discontinuity of the BSE eigenvalue, and the paper asserts, without a cutting-rule proof or a numerical estimate, that all omitted physics changes only Re lambda. I do not see an internal inconsistency in the 4PI construction, and the agreement between the triangle couplings and the first-order eigenvalue perturbation in Table III is genuine independent support for the implementation. However, that cross-check tests the retained diagrams; it does not test the omitted ones. The 25-33% deviations from PDG could plausibly be Maris-Tandy model dependence as the authors argue, but the possibility that a missing diagram contributes to the discontinuity itself remains open, and the quoted SPM errors exclude this channel. Since the reader already made this the basis of a CONDITIONAL verdict, my reading does not move the verdict. If the proposed cut analysis shows zero omitted discontinuity, ACCEPT would be warranted subject to the acknowledged model uncertainties; if it shows a nonzero contribution, the widths in Table II would need to be revised or accompanied by a systematic error estimate for the omitted discontinuities.","tokens_in":37585,"tokens_out":15714,"duration_ms":112279,"concrete_test":"Apply Cutkosky cutting rules in P^2 to the full one-loop kernel built from the exact four-quark vertex of Eq. (15) and to the dropped diagrams of Fig. 8, namely the vertical t-channel meson exchange, the rest-rest terms, and the two-quark-two-gluon insertions, using the same dressed propagators and BSAs as in Sec. IV. Enumerate every s-channel cut whose intermediate state has the quantum numbers of an open two-body channel for the rho, K* and phi, and evaluate the resulting contribution to disc lambda in the sampling windows of Sec. V A. Compare with the discontinuity obtained from the retained box and crossed-box diagrams alone. If the extra cut contributions vanish identically, the conditional is discharged; if they are nonzero at the level of the SPM errors, Table II is not the full two-body width and a quantitative missing-diagram error term must be added.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for the numerical width claim is the sentence preceding Eq. (39): the sum of all missing diagrams only shifts the real part of the eigenvalue curve on the time-like axis, while the discontinuities are determined by the decay diagrams we compute. Since the reported widths are read from the SPM-continued pole of lambda(P^2)=1, any omitted diagram that has an s-channel discontinuity in an open two-body channel changes Gamma directly, not just M. The paper does not supply a cutting-rule analysis of the omitted terms. The rest term of the on-shell decomposition (15) is not a purely real background: in the time-like region it contains the non-pole part of the pi pi amplitude, and the dropped vertical t-channel meson exchange and two-quark-two-gluon diagrams can in principle acquire s-channel discontinuities after the loop integrations in Fig. 8 are performed. The authors themselves acknowledge that the phi to pi pi pi channel carries roughly 15% of the width and is absent (b363), which is an admitted counterexample to the only-Re-lambda statement; no analogous quantitative check is presented for the rho and K*, where the 4pi and K pi pi channels are argued, not shown, to be sub-permille. The quoted SPM errors therefore do not cover the dominant uncertainty of the claim. The framework is internally consistent and the triangle cross-check in Table III supports the implementation, but the identification of the width with the box and crossed-box diagrams is an assertion about the discontinuity rather than a derivation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper develops a framework for computing two-body hadronic decay widths inside the Bethe-Salpeter equation, starting from the 4PI effective action of QCD. The authors present the four-loop 4PI effective action of QCD in linear covariant gauges (Sec. II B, Figs. 1-3), derive the two-body scattering kernels from it, and use an on-shell decomposition of the four-quark vertex (Eq. 15) into meson poles plus a rest term to identify the box and crossed-box diagrams (Fig. 8) that carry the physical two-body decays. The framework is applied to the rho, K*, and phi mesons: the homogeneous BSE (23) is solved for complex total momenta with optimized contour deformations (Sec. IV C), the resonance pole lambda(P^2)=1 is located by Schlessinger-point analytic continuation, and the rainbow-ladder kernel strength c is tuned per meson (Eq. 40) so that the mass reproduces the PDG value. With this tuning, Table II reports Gamma_rho=109.1+-1.2 MeV, Gamma_K*=31.7+-0.2 MeV, and Gamma_phi(KK-bar)=2.7+-0.3 MeV, against experimental values of 147.4, 47.1 and 3.556 MeV. The triangle couplings computed in Sec. V B are cross-checked against the first-order perturbation of the eigenvalue (Table III) and against the published rho result of Mader et al.","tokens_in":37921,"tokens_out":10160,"duration_ms":70712,"significance":"If the central identification holds, this is a valuable systematic derivation of two-body hadronic decays in the nPI formalism, with a diagrammatic interpretation that extends beyond the vector mesons treated here. The strengths are substantial: the kernels are derived algorithmically with DoFun; the non-trivial internal cross-check that the triangle couplings agree with the first-order eigenvalue perturbation (Table III) validates the implementation; the rho triangle reproduces the independent result of Mader et al.; the contour-deformation treatment, including the unequal-mass K* case, is detailed and generalizable; and the authors are unusually explicit about their limitations, including the missing phi->3pi channel (b363) and the rainbow-ladder model dependence. The central quantitative claim, however, rests on the premise that the computed box and crossed-box diagrams alone set the discontinuity of the eigenvalue on the time-like axis, which is asserted rather than demonstrated (see major comments).","major_comments":[{"comment":"The width extraction rests on the assertion that 'the sum of all missing diagrams only shifts the real part of the eigenvalue curve on the time-like axis, while the discontinuities are determined by the decay diagrams we compute.' No cutting-rule analysis is supplied for the omitted terms. The rest term of (15), the dropped vertical t-channel exchange, and the two-quark-two-gluon diagrams can acquire s-channel discontinuities after the loop integration in (28), and the authors' own admitted counterexample, phi->3pi at about 15% of the total width (b363), shows the statement is not generally true. The sub-permille suppression of rho->4pi and K*->K pi pi is asserted (b365-b366), not computed. Please either provide a quantitative estimate of the omitted-diagram discontinuities in the pi pi and K pi channels (for instance by including the non-pole part of the pi pi amplitude or the vertical meson exchange of [77]), or reframe the quoted numbers as partial widths from the computed diagrams with the missing channels explicitly quantified.","section":"V.A (before Eq. 39); Eq. (15)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The tuning ansatz (40), which absorbs the missing-diagram physics into a real rescaling of the rainbow-ladder kernel strength c, is a specific realization of the unproven premise of Major comment 1, and its effect on the width, as opposed to the mass, is not quantified. The claim that this tuning affects the magnitude of the discontinuity (39) only at second order via the iteration of the BSE should be demonstrated, since changing c modifies the vector BSA that enters the decay kernel linearly; the c-trajectories already shown in Fig. 12 contain the information needed to report Gamma(c) in the tuning window, which would indicate how sensitively the extracted width depends on the tuning prescription.","section":"V.A (Eq. 40 and Table II)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The errors quoted in Table II (e.g., +-1.2 MeV on Gamma_rho) are Schlessinger-point continuation errors only; they do not cover the dominant systematic uncertainties from the rainbow-ladder/Maris-Tandy input or from the unresolved omitted-diagram issue of Major comment 1. Since the authors themselves identify the model dependence as the largest systematic uncertainty (b220), the paper should present a combined error budget that separates the SPM (numerical) error from the model- and truncation-induced systematics, so that the reader can assess the significance of the 25-33% deviations from experiment.","section":"V.A (Table II) and F.4","authors_acknowledge":"The dependence on this model is the largest systematic uncer- tainty of the present computation.","authors_acknowledge_p":0.63}],"minor_comments":[{"comment":"The 1 sigma ellipses should be labelled as SPM-continuation errors only, in line with the error budget requested in Major comment 3, to avoid the impression that they cover the model dependence.","section":"Fig. 12, right panels","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The factor 1/2 needed to reproduce the rho result of [45] is stated but not explained; a brief comment on its origin would help the reader judge the discrepancy.","section":"V.A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The claim that the four-loop 4PI effective action of QCD is presented here for the first time should be stated with precision against the prior scalar 4PI constructions of [9,57] and the 3PI QCD results of [5].","section":"II.B","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The neglect of the u-channel diquark poles in the on-shell decomposition (15) is implicit in the discussion around Fig. 8; it should be listed explicitly in the truncation summary.","section":"III.C","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"A few references are arXiv preprints or in-preparation items (e.g., [25], [39], [69], [73], and the 'will be provided elsewhere' note on the DoFun details in Sec. II C); these should be flagged as such in the bibliography.","section":"Bibliography","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its limitations, which is to its credit, and the internal cross-checks (Table III and the Mader et al. benchmark) give confidence that the numerical implementation is correct. The editor may wish to have the novelty claim in Sec. II B (first presentation of the four-loop 4PI QCD action) verified against the prior nPI literature, since Berges [5] and Carrington [9,57] are close antecedents. The central open question, which I expect the authors to address in revision, is whether the width claim can be separated from the unproven premise that omitted diagrams do not contribute to the discontinuity of the eigenvalue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This paper is worth a serious referee. The genuinely new part is the derivation: the first explicit four-loop 4PI effective action of QCD, the algorithmic on-shell-pole decomposition of the four-quark vertex, and the resulting identification of box and crossed-box diagrams as the two-body decay channels in the BSE. The correction of the earlier rho result by a factor 1/2 and the extension to K* and phi are real additions, and the paper is refreshingly clear about what is model input and what is computed.\n\nThe internal cross-checks are the strongest evidence. The triangle couplings agree with the first-order perturbation of the eigenvalue curve (Table III), and the rho triangle reproduces the independent result of Mader et al. That consistency tells me the implementation is probably correct for the diagrams it computes.\n\nThe soft spot is exactly where the stress-test note puts it. Before Eq. (39), the authors assert that all missing diagrams only shift the real part of the eigenvalue, so the discontinuity is fixed by the computed decay diagrams. That is an assertion, not a demonstrated property. The rest term in the on-shell decomposition (15) is not guaranteed to have a purely real time-like action, and diagrams dropped because they are thought to be subleading can in principle acquire s-channel discontinuities after loop integration. The phi->3pi channel, which the authors concede carries about 15% of the width, is an admitted counterexample. For rho and K*, the multi-pion channels are argued to be sub-permille, but no quantitative check is shown. So the SPM errors in Table II are statistical errors on the extrapolation; they do not cover the dominant systematic uncertainty in the width claim.\n\nI also agree with the reader that the tuning of c per meson to reproduce the experimental mass makes the widths model-dependent extractions rather than parameter-free predictions. That is not a fatal flaw, because the authors are honest about it and the framework is the point. But the paper's conclusion should not oversell the numerical widths.\n\nBottom line: the derivation and numerical methodology deserve peer review and publication, likely with revision. The referee should push on the discontinuity assertion, either by softening the claim or by adding a cutting-rule analysis of the omitted terms. I would not desk-reject this, and I would cite it for the 4PI machinery even while treating the quoted widths with caution.","headline":"A genuinely new 4PI-based derivation of two-body decay kernels inside the BSE, with strong internal cross-checks, but the quoted widths rest on an unproven assertion that all omitted diagrams shift only the real part of the eigenvalue.","tokens_in":38526,"tokens_out":1800,"would_cite":true,"duration_ms":16158,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["12.38.Lg","11.10.St","14.40.Be"],"model":"deepseek/deepseek-v4-flash","headline":"This paper claims that the two-body decay widths of the rho, K* and phi mesons are fixed by the box and crossed-box diagrams obtained by inserting on-shell pion and kaon poles into the four-loop 4PI kernels of QCD, reporting tuned widths…","keywords":["4PI effective action","Bethe-Salpeter equation","vector meson decays","hadronic resonances","decay widths","rainbow-ladder truncation","on-shell decomposition","contour deformation"],"falsifier":"Restore a single omitted piece and check whether the discontinuity $\\operatorname{disc}\\lambda(P^2) = 2i\\operatorname{Im}\\lambda(P^2+i\\epsilon)$ on the time-like axis changes by more than the SPM error bars: for instance, include the sequential channel $\\phi\\to\\rho\\pi\\to\\pi\\pi\\pi$ with a dressed $\\rho$ propagator, or add kernels containing two-quark-two-gluon vertices. A cheaper probe is to replace the Maris-Tandy effective interaction, the paper's stated largest systematic uncertainty, with a causal interaction and recompute the tuned $\\rho$ width; a shift larger than about 10 MeV would show the reported widths are bound to the model rather than to the 4PI derivation.","tokens_in":39027,"feed_emoji":"⚛️","tokens_out":11102,"duration_ms":89373,"temperature":0.7,"pith_summary":"This paper claims that the two-body decay widths of hadronic resonances can be derived, diagram by diagram, from the 4PI (four-particle irreducible) effective action of QCD. The key step is an on-shell decomposition of the four-quark vertex: replacing the full vertex by its pion and kaon poles turns two four-loop kernel diagrams into the box and crossed-box diagrams whose branch cuts create the imaginary part of the Bethe-Salpeter eigenvalue, that is, the decay width. Solving the homogeneous Bethe-Salpeter equation at complex momenta with a tuned binding interaction, the authors obtain $\\Gamma_\\rho = 109.1\\pm1.2$ MeV, $\\Gamma_{K^*} = 31.7\\pm0.2$ MeV and $\\Gamma_\\phi = 2.7\\pm0.3$ MeV, compared with PDG values of 147.4, 47.1 and 3.556 MeV. The 24--33 percent shortfall is attributed to the model dependence of the rainbow-ladder (Maris-Tandy) input rather than to the 4PI derivation itself, and the framework is claimed to extend to glueballs and baryons.","feed_headline":"Two box diagrams fix the rho, K*, phi decay widths","feed_subtitle":"A four-loop 4PI QCD action yields rho, K* and phi widths within 24-33 percent of experiment.","key_machinery":"The load-bearing object is the on-shell decomposition of Eq. (15), which writes the four-quark scattering vertex as $\\sum_i \\bar\\Gamma_i D_i(s)\\Gamma_i$ plus crossed-channel, diquark-channel, and rest terms; inserted into the two one-loop kernel diagrams of the 4PI Bethe-Salpeter kernel, it produces the box diagram and the crossed-box diagram that carry the physical $\\pi\\pi$, $K\\pi$ and $K\\bar K$ thresholds. Two technical devices carry the computation: a contour-deformation algorithm, including the on-shell momentum-partitioning choice $\\eta_{\\rm os}(M) = (M^2 + m_K^2 - m_\\pi^2)/(2M^2)$ that aligns the unequal-mass branch cuts, which allows the BSE to be solved at complex total momenta above threshold, and a factorisation of the decay kernel into left and right quark-triangle halves that reduces the numerical cost from many six-dimensional integrals to a product of two one-dimensional matrix integrals. The resonance pole is located by the eigenvalue condition $\\lambda(P^2)=1$ on the first sheet, with the Schlessinger point method continuing $\\lambda(P^2)$ to the second sheet where the pole sits at $\\sqrt{P^2} = -\\Gamma/2 + iM$.","core_discovery":"The paper's central claim is that in the 4PI framework a resonance width is fixed, not by solving the full four-point function in the time-like region, but by its on-shell poles alone. The authors decompose the four-quark vertex as a sum of meson poles, their crossed-channel partners, and a rest term (Eq. 15), insert this decomposition into the kernel of the homogeneous Bethe-Salpeter equation, and identify two surviving diagrams, the box and the crossed-box diagram of Fig. 8, whose discontinuities generate the two-body decay thresholds. Because the Bethe-Salpeter eigenvalue reaches unity at the resonance pole, $\\sqrt{P^2} = -\\Gamma/2 + iM$, the imaginary part of the eigenvalue curve on the time-like axis directly encodes the width. With the rainbow-ladder interaction scaled by a parameter $c$ chosen so that $M(c^*)$ equals the PDG mass, the computed pole positions are $M = 774.9$ MeV with $\\Gamma = 109.1$ MeV for the $\\rho$, $895.5$ MeV with $\\Gamma = 31.7$ MeV for the $K^*$, and $1019.4$ MeV with $\\Gamma = 2.7$ MeV for the $\\phi$, against PDG widths of 147.4, 47.1 and 3.556 MeV. The paper further shows that the on-shell quark-triangle coupling $g_{VPP}$ extracted from the decay kernel agrees with the first-order shift of the rainbow-ladder eigenvalue, and that this identity is not restricted to the rainbow-ladder truncation.","pith_inferences":["If the central claim holds, the $\\phi$'s missing width from $\\phi\\to\\pi\\pi\\pi$, which the authors estimate at about 15 percent of the total, could be recovered by dressing the $\\rho$ propagator in the sequential $\\phi\\to\\rho\\pi$ channel, effectively promoting the rest term of Eq. (15) from an assumption into a computation.","The tuning parameter $c$ turns this computation into a consistency test for beyond-rainbow-ladder physics: a computation with dressed quark-gluon vertices should push $c$ back toward unity and lift the couplings $g_{\\rho\\pi\\pi}$, $g_{K^*K\\pi}$ and $g_{\\phi KK}$ by the correlated 8--13 percent needed to match experiment.","The same on-shell insertion logic applied to the two-ghost-two-gluon vertex, where the authors note BRST-quartet states might cancel unphysical cuts, would give a concrete place to test whether confinement shows up as the cancellation they describe.","The deviation pattern, largest for the $\\rho$ and smallest for the $\\phi$, correlates with available phase space, so a beyond-RL computation that raises $g_{\\rho\\pi\\pi}$ should raise the other two couplings in a predictable ordering, a correlated prediction of the framework."],"forward_implications":["The same 4PI derivation applies to arbitrary two-body hadronic decays, including glueballs through gluonic four-point channels and baryons once six-point functions are handled, because the diagrammatic step from the four-loop action is algorithmic.","Resonance widths become computable without resolving the full four-point function in the time-like region; only the on-shell poles of the decay products are needed as input.","All four pseudoscalar tensor structures must be kept: the $F$ structure reduces the $\\rho$ width by about 50 percent, while $G$ and $H$ shift the coupling by roughly $-7$ and $+5$ percent and cancel, so truncated amplitudes systematically distort widths.","The equivalence between the triangle (impulse-approximation) coupling and the first-order eigenvalue shift provides a general internal cross-check for any BSE decay computation, beyond rainbow-ladder.","The general contour-optimisation algorithm handles unequal-mass decay products, as needed for $K^*\\to K\\pi$, and so extends the method to most two-body channels of the hadron spectrum."],"supporting_citations":[{"why":"Provides the earlier rho-meson Bethe-Salpeter decay computation that this paper corrects (the factor 1/2 in the decay kernel) and extends to the K* and phi.","marker":"[45]"},{"why":"Supplies the contour-deformation arguments for scattering amplitudes that are generalised here to unequal-mass decay products.","marker":"[46]"},{"why":"Provides the four-loop 4PI topologies for a scalar theory that the QCD effective action is constructed from via the superfield formalism.","marker":"[57]"},{"why":"Supplies the Maris-Tandy effective interaction model used for all quark propagators and meson amplitudes; the paper calls it the largest systematic uncertainty.","marker":"[84]"},{"why":"Provides the experimental masses and widths of the rho, K* and phi used for the c-tuning and for the final comparison.","marker":"[98]"},{"why":"Gives the impulse-approximation triangle couplings for light vector meson decays that the present decay-kernel couplings are compared against.","marker":"[99]"},{"why":"Provides the independent rainbow-ladder triangle result g_rho-pi-pi = 5.20 used as a benchmark for the full BSE computation.","marker":"[100]"},{"why":"Supplies the Schlessinger point method used to analytically continue the eigenvalue curve to the second Riemann sheet and locate the resonance pole.","marker":"[105]"}],"fun_headline_variants":["4PI action yields meson widths from two box diagrams","On-shell poles set rho, K*, phi widths in 4PI QCD","Box and crossed-box diagrams determine resonance widths","Four-loop 4PI QCD action fixes three meson widths","Resonance widths emerge from on-shell poles in 4PI action"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that every omitted contribution, the rest term of the on-shell decomposition in Eq. (15), unquenching, two-quark-two-gluon vertices, and multi-pion final states, shifts only the real part of the Bethe-Salpeter eigenvalue, so the imaginary part, and hence the width, is entirely fixed by the computed box and crossed-box diagrams; the authors explicitly acknowledge that the $\\phi\\to\\pi\\pi\\pi$ channel carries about 15 percent of the total width and is absent from their setup.","fun_headline_variants_meta":{"raw":{"variants":["4PI action yields meson widths from two box diagrams","On-shell poles set rho, K*, phi widths in 4PI QCD","Box and crossed-box diagrams determine resonance widths","Four-loop 4PI QCD action fixes three meson widths","Resonance widths emerge from on-shell poles in 4PI action"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.00012,"raw_usage":{"cost":0.0001201018,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001201018,"upstream_inference_prompt_cost":2.91018e-05,"upstream_inference_completions_cost":9.1e-05},"total_tokens":1579,"prompt_tokens":1079,"completion_tokens":500,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":949,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":423}},"tokens_in":1079,"tokens_out":500,"cost_source":"openrouter_usage","duration_ms":5630,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":423,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-30T10:58:56.192332+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Restore a single omitted piece and check whether the discontinuity $\\operatorname{disc}\\lambda(P^2) = 2i\\operatorname{Im}\\lambda(P^2+i\\epsilon)$ on the time-like axis changes by more than the SPM error bars: for instance, include the sequential channel $\\phi\\to\\rho\\pi\\to\\pi\\pi\\pi$ with a dressed $\\rho$ propagator, or add kernels containing two-quark-two-gluon vertices. A cheaper probe is to replace the Maris-Tandy effective interaction, the paper's stated largest systematic uncertainty, with a causal interaction and recompute the tuned $\\rho$ width; a shift larger than about 10 MeV would show the reported widths are bound to the model rather than to the 4PI derivation.","supporting_citations":[{"cited_title":"Vector mesons as dynamical resonances in the Bethe-Salpeter framework","cited_arxiv_id":"1804.11161","evidence_quote":"Provides the earlier rho-meson Bethe-Salpeter decay computation that this paper corrects (the factor 1/2 in the decay kernel) and extends to the K* and phi."},{"cited_title":"Scattering amplitudes and contour deformations","cited_arxiv_id":"1907.05402","evidence_quote":"Supplies the contour-deformation arguments for scattering amplitudes that are generalised here to unequal-mass decay products."},{"cited_title":"The 4PI Effective Action for \\phi^4 Theory","cited_arxiv_id":"hep-ph/0401123","evidence_quote":"Provides the four-loop 4PI topologies for a scalar theory that the QCD effective action is constructed from via the superfield formalism."},{"cited_title":"Bethe-Salpeter Study of Vector Meson Masses and Decay Constants","cited_arxiv_id":"nucl-th/9905056","evidence_quote":"Supplies the Maris-Tandy effective interaction model used for all quark propagators and meson amplitudes; the paper calls it the largest systematic uncertainty."},{"cited_title":"Strong Decays of Light Vector Mesons","cited_arxiv_id":"nucl-th/0208019","evidence_quote":"Gives the impulse-approximation triangle couplings for light vector meson decays that the present decay-kernel couplings are compared against."},{"cited_title":"Hadronic decays of mesons and baryons in the Dyson-Schwinger approach","cited_arxiv_id":"1106.3159","evidence_quote":"Provides the independent rainbow-ladder triangle result g_rho-pi-pi = 5.20 used as a benchmark for the full BSE computation."}],"review_version":1}