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Hadronic resonances from the 4PI effective action of QCD

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Jonas Wessely, Markus Q. Huber

High Energy Physics - Phenomenology · Submitted 29 Oct 2026 · 117 references

Pith's verdict: 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.

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arXiv ID2610.40390 (version 1)
Pith IDAFRNKZNU, Pith's permanent record of this paper and its review
CitationJonas Wessely, Markus Q. Huber. "Hadronic resonances from the 4PI effective action of QCD." arXiv:2610.40390 (2026). https://pith.science/paper/AFRNKZNU
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The paper

The paper claims

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…

The reading

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.

What carries the argument

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 $\sqrt{P^2} = -\Gamma/2 + iM$.

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Original abstract

We discuss the 4PI effective action as a tool to study resonances in QCD. To this end, we present the 4PI effective action of QCD in linear covariant gauges up to four loops and derive from it the kernels for two-body bound states, using mesons as an example. The emergence of resonances via decay channels is built into the formalism at the level of the four-point functions, and we employ an on-shell decomposition of its physical channels to arrive at a diagrammatic representation for physical two-body decays within the Bethe-Salpeter equation framework. We apply this formalism to the $\rho$, $K^*$ and $\phi$ mesons and extract their widths.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 3,675 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We discuss the 4PI effective action as a tool to study resonances in QCD. To this end, we present the 4PI effective action of QCD in linear covariant gauges up to four loops and derive from it the kernels for two-body bound states, using mesons as an example. The emergence of resonances via decay channels is built into the formalism at the level of the four-point functions, and we employ an on-shell decomposition of its physical channels to arrive at a diagrammatic representation for physical two-body decays within the Bethe-Salpeter equation framework. We apply this formalism to the $\rho$, $K^*$ and $\phi$ mesons and extract their widths.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We discuss the 4PI effective action as a tool to study resonances in QCD. contribution 0.72 0.62 0.09 asserted 0.81
s1 To this end, we present the 4PI effective action of QCD in linear covariant gauges up to four loops and derive from it the kernels for two-body bound states, using mesons as an example. contribution 0.67 0.95 0.09 asserted 1.00
s2 The emergence of resonances via decay channels is built into the formalism at the level of the four-point functions, and we employ an on-shell decomposition of its physical channels to arrive at a diagrammatic representation for physical two-body decays within the Bethe-Salpeter equation framework. method 0.95 0.89 0.14 asserted 1.00
s3 We apply this formalism to the $\rho$, $K^*$ and $\phi$ mesons and extract their widths. result 0.79 0.96 0.42 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.68 · evidence from proof or derivation 0.94.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1: Diagrammatic representation of Γ
Figure 2. FIG. 2: Diagrammatic representation of the part of Γ
Figure 3. FIG. 3: Diagrammatic representation of the part of Γ
Figure 4. FIG. 4: Summing the four-fermion basketball diagram
Figure 5. FIG. 5: Homogeneous BSE for fermionic constituents.
Figure 6. FIG. 6: Meson kernel truncated at one-loop order. Flavour-changing interactions are implied by the summation over
Figure 7. FIG. 7: Separation of the quark four-point function
Figure 8. FIG. 8: On-shell replacements in the kernel diagrams. The equation in the second row corresponds to a switch of the
Figure 9. FIG. 9: Truncated meson BSE kernel containing only the
Figure 11. FIG. 11: Branch cuts of the pion (blue) and kaon (orange) propagators in the complex
Figure 12. FIG. 12: SPM poles of the

Showing 11 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.40390 by the authors.

References

Discussion

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