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Full four-quark interaction basis shifts QCD critical endpoint

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2026-07-09 13:14 UTC pith:LAT7XXHA

load-bearing objection Fierz-complete four-quark channels in fRG-QCD: useful systematic study, but CEP shift is confounded by simultaneous parameter retuning the 3 major comments →

arxiv 2607.07354 v1 pith:LAT7XXHA submitted 2026-07-08 hep-ph nucl-exnucl-th

Fierz-complete four-quark interactions and the QCD phase diagram

Zi-ning Wang , Li-jun Zhou , Chuang Huang , Rui Wen , Shi Yin , Wei-jie Fu This is my paper
classification hep-ph nucl-exnucl-th PACS 12.38.Aw11.10.Wx25.75.Nq
keywords four-quarkinteractionschannelsfierz-completephasediagramdynamicsincreases
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper investigates what happens to the predicted shape of the QCD phase diagram — the map of how nuclear matter transforms under extreme temperature and density — when you include all possible ways four quarks can interact with each other simultaneously, rather than just the single dominant channel that prior calculations kept. The central object is the Fierz-complete four-quark interaction basis: a set of ten distinct tensor channels (sigma, pion, eta, a, vector, axial-vector, and adjoint scalar-pseudoscalar combinations) that together capture every independent way four light quarks can scatter. The authors embed all ten channels into a functional renormalization group computation of QCD at finite temperature and baryon chemical potential, dynamically hadronizing the two dominant channels (sigma and pion) into meson exchanges while tracking the remaining eight as raw four-quark couplings. The paper finds that in the vacuum and at low baryon density, the sigma and pion channels dominate overwhelmingly and all other channels are negligible — validating the simpler single-channel truncation used in earlier work for that regime. However, near the critical endpoint where the chiral crossover gives way to a first-order phase transition, the subleading channels grow to roughly ten percent of the dominant ones. Including them shifts the critical endpoint from (T, mu_B) = (107, 635) MeV to (102, 647) MeV — slightly colder and denser — and increases the curvature of the phase boundary from kappa_2 = 0.0142 to 0.0151. The shifts are small and remain within the acknowledged truncation errors of the functional approach, but they quantify how much the single-channel approximation was actually costing.

Core claim

The paper's core result is a quantitative accounting of how much the predicted QCD critical endpoint and phase boundary curvature change when the four-quark interaction sector is upgraded from one channel to the full Fierz-complete set of ten. The sigma and pion channels dominate everywhere except near the critical endpoint, where subleading channels (eta, a, vector, axial-vector, adjoint scalar-pseudoscalar) rise to about ten percent of the dominant couplings. The net effect moves the critical endpoint to slightly lower temperature and higher baryon chemical potential, and makes the phase boundary marginally more curved. The authors also find that distinguishing the sigma and pion Yukawacou

What carries the argument

The Fierz-complete four-quark basis of ten tensor channels for light quarks, evolved under the functional renormalization group flow with dynamical hadronization of the sigma and pion channels into meson exchanges. The flow equations couple the four-quark vertices to the quark-gluon vertex, Yukawa couplings, and meson propagators across all ten channels simultaneously. A phenomenological infrared enhancement function compensates for retaining only the classical tensor structure of the quark-gluon vertex.

Load-bearing premise

The calculation retains only the classical tensor structure of the quark-gluon vertex, compensating with a phenomenological infrared enhancement tuned to reproduce dynamical chiral symmetry breaking, and approximates the strange quark sector with a two-flavor potential rather than a full 2+1-flavor one. These approximations matter most precisely at the large baryon chemical potential where the CEP sits and where the paper claims the new four-quark channels become relevant.

What would settle it

If including non-classical quark-gluon vertex structures or a full 2+1-flavor strange quark potential shifts the CEP location by more than the ~5 MeV in T and ~12 MeV in mu_B reported here, then the Fierz-completeness correction documented in this paper is subdominant to other truncation effects and the quantitative CEP prediction remains provisional.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The result that subleading four-quark channels reach ~10% of the dominant ones near the CEP suggests that further truncation improvements — such as including non-classical quark-gluon vertex structures — could shift the CEP by comparable or larger amounts, since the CEP sits at mu_B/T ~ 6.3 where the authors acknowledge truncation errors grow substantially.
  • The phase boundary curvature kappa_2 = 0.0151 can be directly compared to lattice QCD extrapolations and to chemical freeze-out data from heavy-ion collisions, providing a consistency check on whether the functional approach tracks the true QCD phase boundary at low baryon density.
  • The growth of vector and diquark channels near the CEP points toward possible additional phase transitions or instabilities at higher baryon density (such as color superconductivity) that would require extending the truncation further.
  • The framework provides a systematic error budget for functional QCD predictions of the CEP location: the Fierz-completeness correction is quantified here, but the quark-gluon vertex truncation and the approximate strange-quark potential remain as known sources of systematic uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. This manuscript investigates the impact of Fierz-complete four-quark interactions (ten tensor channels) on the QCD phase diagram within the functional renormalization group (fRG) framework. Building on prior work (Ref. [22]) that retained only the scalar-pseudoscalar channel, the authors extend the truncation to include all ten Fierz-complete channels for light quarks, with dynamical hadronization applied to the sigma and pion channels. The flow equations for all ten four-quark couplings, the distinct sigma and pion Yukawa couplings, and the quark-gluon vertex are derived and solved. The main results are: (i) in the vacuum, the sigma and pion channels dominate overwhelmingly while other channels are negligible; (ii) near the critical endpoint (CEP), non-dominant channels grow to approximately 10% of the dominant ones; (iii) the phase boundary curvature increases slightly from kappa_2 = 0.0142 to 0.0151; and (iv) the CEP shifts from (107, 635) MeV to (102, 647) MeV. The vacuum observables (m_pi, m_sigma, m_l, m_s) are reproduced and the chiral condensate is compared favorably with lattice data at mu_B = 0.

Significance. The paper provides a systematic and technically demanding extension of the fRG approach to QCD at finite temperature and density, moving from a single four-quark channel to the full Fierz-complete basis. The explicit flow equations for all ten channels (Appendix G), the Yukawa couplings (Appendix F), and the dynamical hadronization framework represent a substantial technical contribution. The finding that non-dominant channels reach ~10% near the CEP is a useful quantitative result for assessing truncation systematics. The curvature result kappa_2 = 0.0151 is consistent with prior fRG, DSE, and lattice estimates. The work is a meaningful step toward quantifying the reliability of functional QCD predictions for the CEP location.

major comments (3)
  1. §IV, Eq. (23) and surrounding text: The CEP shift from (107, 635) MeV to (102, 647) MeV is attributed to the inclusion of Fierz-complete four-quark channels. However, as discussed in Appendix D (Eq. D2-D3), the infrared enhancement parameter 'a' is simultaneously changed from 0.034 to 0.013, and the Yukawa couplings h_sigma and h_pi are now distinguished (previously set equal in Ref. [22]). The paper states the reduction in 'a' is 'compensated by the Fierz-complete four-quark interactions' but does not quantify this compensation. Without a sensitivity analysis that isolates the channel-extension effect from the parameter retuning, the specific CEP numbers (102, 647) MeV cannot be cleanly attributed to the Fierz-complete dynamics. The authors should either provide such an analysis (e.g., showing the CEP location with the old 'a' value and the new channels, or vice versa) or explicitly re-
  2. §IV, Eq. (22): The curvature kappa_2 = 0.0151(1) is fitted in the range mu_B/T in [0,3] and [0,4]. The text states both ranges yield the same result, but the individual fit results for each range are not reported separately. Given that the CEP sits at mu_B/T ~ 6.3, the choice of fit range could affect kappa_2. Please report the fit results for each range separately, including the chi-squared or goodness of fit, to support the claim that the curvature is robust.
  3. Appendix D, Eq. (D1): The strange quark sector uses an approximate 2-flavor potential V_k(rho, rho_s) ~ V_k(rho) + (1/2)V_k(2*rho_s) rather than a full 2+1-flavor potential. This approximation is inherited from Ref. [22]. Since the CEP region is precisely where the paper claims new channels become relevant, the authors should comment on whether this approximation in the strange sector could systematically bias the CEP location, and whether the direction of such a bias can be estimated.
minor comments (7)
  1. §III.A, Fig. 8: The y-axis label in the top-left panel reads 'lambda [GeV^{-2}]' with values up to ~3000, while the top-right panel (T=130, mu_B=300) has the same label but values up to ~100. Please verify the units and scale are consistent across panels, or clarify if different normalizations are used.
  2. §IV, Fig. 19: The phase boundary line from this work (black dashed) appears to deviate from the fRG result of Ref. [22] (red) at higher mu_B. It would help to quantify the maximum deviation in T_c(mu_B) between the two calculations in the crossover region.
  3. Appendix D, Table I: The parameter 'a = 0.013' is described as providing 'only 1.3% larger quark-gluon coupling.' However, Eq. (D3) shows the enhancement factor approaches 1+a in the infrared, so the enhancement is 1.3% relative to the unenhanced coupling. This phrasing could be slightly misleading; consider rephrasing to 'an infrared enhancement of 1.3%.'
  4. §III.C, Fig. 16: The strong couplings alpha_{l l A}, alpha_{s s A}, and alpha_{A3} are shown for T = 0, 150, 300 MeV. The text mentions they 'decrease with the increasing temperature,' but the figure shows this only for the light quark coupling. Please verify the strange quark and three-gluon couplings show the same trend, or clarify.
  5. References: Several references to future work are cited as 'arXiv:2603.xxxxx' (e.g., Refs. [6], [20], [33], [34]) with 2026 dates. Please verify these are publicly available or update with published references where applicable.
  6. §V (Conclusions): The sentence 'We have studies the four-quark couplings' should read 'We have studied the four-quark couplings.'
  7. Appendix G: The fish diagram contributions (Eqs. G30-G39) involve very lengthy expressions. While their inclusion is appreciated for reproducibility, a brief comment on whether these expressions have been cross-checked (e.g., against symmetry relations or limiting cases) would strengthen confidence in their correctness.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The comments are well-taken and we address each one below. We agree that the simultaneous change of the infrared enhancement parameter and the channel extension complicates the attribution of the CEP shift, and we will add a sensitivity analysis to disentangle these effects. We will also report the individual fit results for each fit range. Regarding the strange-sector approximation, we will add an expanded discussion of its potential systematic impact.

read point-by-point responses

  1. Referee: §IV, Eq. (23): CEP shift attributed to Fierz-complete channels, but parameter 'a' simultaneously changed from 0.034 to 0.013, and h_sigma/h_pi now distinguished. No sensitivity analysis isolating channel-extension from parameter retuning. Request either such analysis or explicit qualification.

    Authors: The referee raises a valid and important point. We agree that the simultaneous change of the infrared enhancement parameter a (from 0.034 to 0.013) and the distinction of h_sigma and h_pi, alongside the inclusion of Fierz-complete channels, means that the CEP shift cannot be cleanly attributed to the channel extension alone based solely on the comparison as currently presented. We will address this by performing and reporting a sensitivity analysis in the revised manuscript. Specifically, we will compute the CEP location with the new Fierz-complete channels but using the old value a = 0.034 (and h_sigma = h_pi), and conversely, to isolate the effect of the channel extension from the parameter retuning. This will allow us to quantify the individual contributions. We will also revise the text in §IV to explicitly state that the CEP shift reflects the combined effect of channel extension and parameter retuning, rather than attributing it solely to the Fierz-complete dynamics. We note that the physical logic for the reduced 'a' is sound: the additional four-quark channels provide extra channels for chiral symmetry breaking dynamics, so less phenomenological infrared enhancement of the quark-gluon coupling is needed. However, we agree this should be demonstrated quantitatively rather than merely stated. revision: yes

  2. Referee: §IV, Eq. (22): kappa_2 = 0.0151(1) fitted in ranges mu_B/T in [0,3] and [0,4]. Individual fit results for each range not reported separately. Request separate results including chi-squared or goodness of fit.

    Authors: This is a reasonable request. We will report the individual fit results for each range (mu_B/T in [0,3] and [0,4]) separately in the revised manuscript, including the corresponding chi-squared or goodness-of-fit measures. In our current analysis, both ranges yield kappa_2 = 0.0151 and kappa_4 = 0.00023 within the quoted errors, and the chi-squared values are comparable. We will present these numbers explicitly in a table or in the text to support the robustness claim. We note that the CEP location at mu_B/T ~ 6.3 is well outside both fit ranges, so the curvature extraction is not contaminated by critical fluctuations near the CEP, which is part of the reason the result is stable across the two ranges. revision: yes

  3. Referee: Appendix D, Eq. (D1): Strange quark sector uses approximate 2-flavor potential V_k(rho, rho_s) ~ V_k(rho) + (1/2)V_k(2*rho_s) rather than full 2+1-flavor potential. Request comment on whether this could systematically bias the CEP location, and whether direction of bias can be estimated.

    Authors: We agree that this approximation warrants discussion, particularly given that the CEP region is where new channels become relevant. We will add a dedicated paragraph in Appendix D addressing the potential systematic bias. To summarize the content of this addition: The approximation V_k(rho, rho_s) ~ V_k(rho) + (1/2)V_k(2*rho_s) treats the strange quark sector as a spectator to the light-quark dynamics in the mesonic potential, while still determining the strange quark mass dynamically. Since the strange quark is considerably heavier than the light quarks and the chiral transition is primarily driven by the light-quark sector, the impact on the phase boundary and CEP location is expected to be moderate. However, near the CEP, where critical fluctuations in the sigma channel are large, the feedback from strange-quark dynamics could introduce a systematic shift. The direction of the bias is difficult to estimate without a full 2+1-flavor calculation, but we expect it to be subdominant compared to the truncation effects already quantified (e.g., the ~10% non-dominant channel contributions). We note that a comprehensive 2+1-flavor effective potential with scalar and pseudoscalar mesonic nonets is being implemented in the companion work [Ref. 33], which will allow a direct assessment of this systematic. We will reference this ongoing improvement and add a cautionary note that the strange-sector approximation is an inherited limitation whose quantitative impact on the CEP location remains to be fully assessed. revision: partial

Circularity Check

0 steps flagged

No significant circularity. The CEP location and phase boundary curvature are outputs of integrated flow equations, not fitted inputs. Self-citations provide the framework but the flow equations are explicitly derived.

full rationale

The paper's central results — the CEP location (102, 647) MeV and curvature κ₂ = 0.0151 — are obtained by integrating the explicitly written flow equations (Appendices F, G) from an ultraviolet cutoff Λ = 20 GeV down to k = 0. The input parameters (Table I: c_σ, c_σs, α_{s,Λ}) are fitted to reproduce known vacuum observables (m_π, m_σ, m_l, m_s), which is standard practice and not circular: the fitted quantities (hadron masses at T = μ = 0) are distinct from the predicted quantities (CEP location, phase boundary curvature at finite T, μ_B). The infrared enhancement parameter a = 0.013 is tuned to ensure dynamical chiral symmetry breaking — a physical requirement, not a fit to the CEP. The paper builds on prior work by overlapping authors (Refs [22], [41–43]) for the fRG framework, dynamical hadronization technique, and four-quark flow equations. However, these self-citations provide the truncation scheme and flow equation structure, which are then explicitly written out in the present paper (Appendices C, F, G). The hadronization condition (Eq. C9: λ̃_φ = 0) is a standard truncation choice from [66–68], not a self-citation. The four-quark couplings in the eight non-hadronized channels are computed from their flow equations, not set by hand. The comparison with lattice data at μ_B = 0 (Figs. 17, 18) and with independent DSE/lattice results for the CEP (Fig. 19) provides external benchmarks. The skeptic's concern about simultaneous retuning of parameter a alongside the channel extension is a valid methodology concern about confounding variables, but it is not circularity: the parameter a is not fitted to reproduce the CEP location, and the CEP shift is a genuine output of the calculation, not an input by construction. The minor self-citations raise the score to 2, but the central derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

8 free parameters · 6 axioms · 0 invented entities

The paper introduces no new particles, forces, or postulated entities. All fields (quarks, gluons, mesons) are standard QCD degrees of freedom. The Fierz-complete four-quark basis is a mathematical decomposition of standard QCD interactions, not a new physical postulate. The dynamical hadronization technique is an established method, not a new invention. The free parameters are all fitted to vacuum observables or inherited from prior work, and their values are standard within the fRG-QCD program.

free parameters (8)
  • c_σ = 3.6 GeV³
    Strength of explicit chiral symmetry breaking for light quarks, fitted to reproduce m_π = 138 MeV and m_l = 355 MeV in vacuum (Table I).
  • c_σs = 97.2 GeV³
    Strength of explicit chiral symmetry breaking for strange quark, fitted to reproduce m_s = 505 MeV (Table I).
  • α_{s,Λ} = 0.235
    Strong coupling at UV cutoff Λ = 20 GeV, fitted to reproduce vacuum observables (Table I).
  • a (IR enhancement) = 0.013
    Phenomenological infrared enhancement parameter for the quark-gluon coupling (Eq. D3), chosen to ensure dynamical chiral symmetry breaking. Smaller than a = 0.034 in prior work because Fierz-complete channels compensate.
  • b (IR enhancement scale) = 2 GeV
    Scale below which the infrared enhancement function activates (Eq. D3), taken from prior work [22].
  • δ (IR enhancement exponent) = 2
    Exponent in the infrared enhancement function (Eq. D3), taken from prior work [22].
  • α (Polyakov rescaling) = 0.57
    Linear rescaling factor relating pure-gauge and QCD Polyakov loop reduced temperatures (Eq. D8), taken from prior work.
  • T_c^glue = 225 MeV
    Pure gauge critical temperature for the Polyakov loop potential (Eq. D8), taken from prior work.
axioms (6)
  • domain assumption Landau gauge (ξ = 0) is adopted for all computations (Sec. A, Eq. A1).
    Standard gauge choice in fRG-QCD; results are gauge-dependent in principle but physical observables are gauge-invariant.
  • domain assumption Only the classical tensor structure of the quark-gluon vertex is retained; non-classical channels are neglected (Sec. III C, App. C).
    Truncation justified by prior work showing non-classical channels are subleading, but introduces systematic uncertainty.
  • domain assumption The strange quark effective potential is approximated from the 2-flavor potential via V_k(ρ,ρ_s) ≈ V_k(ρ) + ½V_k(2ρ_s) (Eq. D1).
    Approximation inherited from prior work [22]; a full 2+1-flavor potential is planned for future work [33].
  • domain assumption Dynamical hadronization condition: λ̃_φ = 0 for all k (Eq. C9), transferring four-quark dynamics in σ and π channels to meson exchanges.
    Standard technique in fRG; ensures no double-counting of bound-state dynamics.
  • domain assumption External momenta are neglected (p = 0) and only the lowest fermionic Matsubara mode is kept for external propagators (Eq. F9).
    Standard truncation in fRG-QCD at finite T; reduces computational complexity.
  • domain assumption The Polyakov loop potential is parameterized with temperature-dependent coefficients from pure-gauge lattice data (Eqs. D4-D7, Ref [69]).
    Phenomenological input; the rescaling to QCD (Eq. D8) is approximate.

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read the original abstract

The dynamics of Fierz-complete four-quark interactions and its influence on the QCD phase diagram have been investigated within the functional renormalization group approach to QCD at finite temperature and densities. It is found that in the vacuum the pion and sigma channels play the overwhelmingly dominant role, and all the other channels are negligible. However, when it is near the critical end point (CEP), the magnitude of four-quark couplings in other channels increases sizably and they become more and more important. In comparison to the single scalar-pseudoscalar channel of four-quark interactions, the dynamics of Fierz-complete four-quark interactions increases a bit the curvature of the phase boundary, and moves the CEP to location of larger baryon chemical potential and smaller temperature.

Figures

Figures reproduced from arXiv: 2607.07354 by Chuang Huang, Li-Jun Zhou, Rui Wen, Shi Yin, Wei-jie Fu, Zi-ning Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrammatic representation of the flow equation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagrammatic representation of the flow equation of the four-quark vertices, where [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Three representative points ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Four-quark flows for the pion (left panel) and sigma (right panel) channels as functions of the RG scale in the vacuum, [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Four-quark flows for the pion (left panel) and sigma (right panel) channels as functions of the RG scale with [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Four-quark flows for the pion (left panel) and sigma (right panel) channels as functions of the RG scale with [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Four-quark flows for the pion (left panel) and sigma (right panel) channels as functions of the RG scale with [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Four-quark couplings of Fierz-complete channels as functions of the RG scale in the vacuum (top-left), and at the [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Diagrammatic representation of the flow equations of quark-meson vertices, where the orange and green dashed lines [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Yukawa flows for the pion (left panel) and sigma (right panel) as functions of the RG scale in the vacuum. “Exchange [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Yukawa flows for the pion (left panel) and sigma (right panel) as functions of the RG scale in the vacuum, which are [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Left panel: Yukawa couplings for the pion and sigma as functions of the RG scale in vacuum with different initial [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Yukawa couplings for the pion and sigma as functions of the RG scale for different values of temperature at [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Diagrammatic representation of the flow equation of quark-gluon vertex. [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Running of the strong couplings for the quark-gluon [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Renormalized light quark chiral condensate ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Reduced condensate ∆ [PITH_FULL_IMAGE:figures/full_fig_p011_18.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Four-quark flows for the [PITH_FULL_IMAGE:figures/full_fig_p017_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Effective four-quark couplings of the pion and sigma channels as functions of the RG scale at different temperature [PITH_FULL_IMAGE:figures/full_fig_p017_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Four-quark couplings of the [PITH_FULL_IMAGE:figures/full_fig_p018_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Four-quark couplings of all Fierz-complete channels at [PITH_FULL_IMAGE:figures/full_fig_p018_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Yukawa flows for the pion (left panel) and sigma (right panel) as functions of the RG scale with [PITH_FULL_IMAGE:figures/full_fig_p019_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Yukawa flows for the pion (left panel) and sigma (right panel) as functions of the RG scale with [PITH_FULL_IMAGE:figures/full_fig_p019_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Yukawa flows for the pion (left panel) and sigma (right panel) as functions of the RG scale with [PITH_FULL_IMAGE:figures/full_fig_p020_26.png] view at source ↗

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