Pith. sign in

REVIEW 4 major objections 4 minor 31 references

Dynamical Up-quark Mass Generation in QCD-like theories

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A QCD-like supersymmetric theory can dynamically generate the up-quark mass at order one, potentially solving the strong-CP problem.

desk verdict The F=N=3 result is genuinely new, but the paper itself concedes the central approximation is uncontrolled; the promised F=N+1 consistency check is the load-bearing missing piece. read the letter →

arxiv 2505.07953 v1 pith:WMTHETSG submitted 2025-05-12 hep-ph nucl-th

Csaba Csáki , Tuhin S. Roy , Maximilian Ruhdorfer , Taewook Youn This is my paper · ORCID
classification hep-phnucl-th
keywords strongCPproblemupquarkmassdynamicalgenerationchiralperturbationtheoryanomalymediatedsupersymmetrybreakingsupersymmetricQCDKaplan-Manoharidentitylow-energyconstants
topics The Strong CP Problem
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

The paper aims to compute, from a calculable supersymmetric analogue of QCD, the dynamical contribution to the up-quark mass—the piece that could make the QCD strong-CP parameter unphysical even if the up-quark Yukawa coupling vanishes. Working with three light flavors and anomaly-mediated supersymmetry breaking, the authors match the low-energy theory to the standard chiral Lagrangian at next-to-leading order and extract the low-energy constants $L_4$ and $L_6$. They find that for equal numbers of colors and flavors, $F=N=3$, the dynamically generated ratio is $\mu_u/\mu_d \simeq (7\alpha/18)(\mu_s/m)$, an order-one quantity for $\alpha\sim 1$, whereas for $F

What carries the argument

The machinery is a tree-level matching calculation: starting from the AMSB scalar potential for supersymmetric QCD, built from the Affleck-Dine-Seiberg and Seiberg superpotentials and the Kähler potentials for $F<N$ and $F=N$, the radial modes $H,T$ and the $\eta'$ are integrated out to produce a nonlinear $\sigma$ model whose coefficients are identified with the Gasser-Leutwyler low-energy constants $L_1,\dots,L_8$, $F_0$, and $B_0$ at next-to-leading order. The load-bearing identity is the Kaplan-Manohar relation connecting the physical mass-ratio shift to the combination $2L_6-L_4$: $\mu_u/\mu_d \simeq 16B_0\mu_s(2L_6-L_4)/F_0^2$; inserting Table I gives the advertised result.

What would settle it

Compute the full $F=N+1$ theory with a heavy flavor, integrate out the heavy flavor explicitly, and compare the resulting $L_4$ and $L_6$ with the quadratic-truncation values in Table I; an order-one difference would invalidate the $F=N$ central result. Alternatively, a high-precision lattice determination of $\mu_u/\mu_d$ (or equivalently of $2L_6-L_4$) in real QCD that firmly pins the ratio below roughly $0.1$ would rule out the extrapolation that makes the dynamical up mass the whole up mass.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the coefficient combination $2L_6-L_4$, which controls the strange-mass-dependent shift of the up-quark mass, is unsuppressed in the $F=N=3$ AMSB supersymmetric QCD theory. Through Eq. (11), $\mu_u/\mu_d \simeq (16B_0 \mu_s/F_0^2)(2L_6-L_4)$, the computed low-energy constants give Eq. (27): the ratio scales as $(7\alpha/18)(\mu_s/m)$, an order-one quantity for $\alpha\sim 1$ and $\mu_s/m$ not too small. This contrasts with the $F<N$ case, where the ratio is parametrically suppressed in $1/N$ and vanishes at large $N$. The authors emphasize that numerical values obtained by extrapolating to $m\sim\Lambda$ lie outside the rigorous validity of their calculation, but with $\alpha\sim 2$ and $\mu_s/m\sim 0.5$ the extrapolated ratio is $\simeq 0.4$, large enough to account for the entire physical up-quark mass and render the QCD $\theta$ angle unphysical.

Load-bearing premise

The load-bearing premise is that the $F=N$ Kähler potential truncated to quadratic order in $M$, $X$, $B$, and $\bar B$ still captures the physics, even though the paper itself notes those fields are not small compared to $\Lambda$, higher-order terms are as important as the leading term, and the claimed consistency check via $F=N+1$ is not shown.

Editorial extensions

If this is right

  • In the $F=N=3$ case, the dynamical up-mass contribution is of order one, so a QCD-like theory with a vanishing up-quark Yukawa coupling need not contradict the observed pion spectrum.
  • At large $N$, the dynamically generated ratio vanishes at leading order, consistent with existing large-$N$ estimates that the dynamical up-mass is small.
  • For $F=3<N$, the contribution is parametrically suppressed, for example $\mu_u/\mu_d\simeq 0.01$ for $N=4$ with $\Lambda_c=m$ and $\mu_s/m=0.5$.
  • The combination $2L_6-L_4$ becomes a concrete target for lattice and phenomenological chiral perturbation theory fits; measuring it directly would test the scenario.
  • If the $m\sim\Lambda$ extrapolation is accepted, the dynamical up mass alone can account for the full observed up-quark mass, making the strong-CP parameter unphysical.
  • The result isolates which low-energy constants carry the dynamical mass shift, showing that a single linear combination $2L_6-L_4$ controls the up-mass ratio.
  • The $F=N+1$ consistency check mentioned in the text, if completed, would place the quadratic Kähler truncation on firmer ground; the paper states it has been verified but does not display it.

Reading between the lines

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

  • My extension: if the $F=N$ result survives beyond the quadratic Kähler-truncation assumption, the AMSB supersymmetric QCD framework provides a concrete, calculable realization of the Kaplan-Manohar mass-shift mechanism, suggesting the strong-CP puzzle may admit a solution with a massless up-quark Yukawa but no axion.
  • My extension: a direct check would be to compute the same low-energy constants from the $F=N+1$ theory with a heavy flavor for a sequence of heavy masses and extrapolate; an order-one disagreement with Table I would signal truncation breakdown.
  • My extension: the same matching method could be applied to $F=N$ with other numbers of flavors; the paper fixes $F=3$, so whether the order-one enhancement is special to three flavors or generic remains open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper computes the dynamically generated up-quark mass in a QCD-like theory constructed from supersymmetric QCD with F=3 light flavors, perturbed by anomaly-mediated supersymmetry breaking. The authors match the low-energy scalar potential to the chiral Lagrangian at next-to-leading order, derive the low-energy constants L1-L8 (Table I), and use the combination 2L6-L4 to obtain a dynamical contribution to mu_u/mu_d. For F<N they find a result suppressed in the large-N limit, Eq. (27). For F=N=3 they obtain mu_u/mu_d ≃ (7 alpha/18)(mu_s/m), an O(1) contribution, and observe that a naive extrapolation to m ~ Lambda could produce a dynamical up-quark mass equal to the physical one, rendering the QCD theta angle unphysical. The manuscript explicitly acknowledges that the F=N calculation relies on a Kähler-potential truncation that is not parametrically controlled, and that the extrapolation to large SUSY breaking is outside the regime of validity of the calculation.

Significance. If the F=N result were under control, the paper would present a concrete QCD-like mechanism for solving the strong CP problem without an axion, making it of considerable interest. The F<N results provide a controlled, large-N-expandable computation of chiral low-energy constants in a QCD-like theory, and the observation that dynamical up-quark mass effects are suppressed at large N matches existing expectations. However, the headline F=N=3 claim is not a controlled prediction: it rests on an explicitly uncontrolled Kähler truncation and on unknown O(1) coefficients, and the strong-CP conclusion requires an extrapolation that the authors themselves label as beyond validity. The paper is therefore more valuable as a quantitative model study than as a definitive statement about QCD, and its central claim requires substantial additional support before the strong-CP implication can be stated as more than speculation.

major comments (4)
  1. [Sec. 'The scalar potential in AMSB QCD', Eq. (14)] The F=N Kähler potential is truncated to quadratic order in M, X, B, and B-bar, and the manuscript explicitly concedes that M/Lambda^2, X/Lambda^3, B/Lambda^N are not small, so higher-order Kähler terms are as important as the leading term. Since Table I and Eq. (27) rely entirely on this truncated potential, the O(1) value of mu_u/mu_d is a truncation-dependent quantity. The claimed consistency check via F=N+1 with a heavy flavor is not shown anywhere in the manuscript. Please provide the full details of that check, including the intermediate steps and the resulting L4, L6, and mu_u/mu_d, or otherwise quantify the uncertainty from the omitted Kähler terms. As written, the central F=N claim is unsupported.
  2. [Eq. (27) and Sec. 'Dynamical up quark mass in AMSB QCD'] The F=N result depends linearly on the incalculable coefficient alpha, and the numerical example in Eq. (29) chooses alpha=2 to obtain mu_u/mu_d ~ 0.4. Because alpha is an unconstrained O(1) parameter, the paper's statement that the result 'could potentially account for the entire mass of the up quark' is not a prediction of the theory unless a determination of alpha is provided. The abstract and conclusions should explicitly state that the size of the effect is contingent on this unknown coefficient and on the uncontrolled truncation, not merely on the extrapolation in m/Lambda.
  3. [Sec. 'Dynamical up quark mass from chiPT', Eqs. (9)-(11)] The identification mu_u/mu_d ≃ (beta2/beta1) mu_s assumes mu_d ≃ m_d and mu_s ≃ m_s after a specific choice of the chiral reparameterization (7). Since Eq. (7) shifts the quark masses and the LECs, the numerical values of L4 and L6 in Table I are gauge-dependent, and the ratio mu_u/mu_d computed from them inherits that dependence. The paper should justify that the chosen gauge is the physical one, or quantify the resulting uncertainty in Eq. (27), particularly for the F=N case where the effect is claimed to be O(1).
  4. [Abstract and Sec. 'Dynamical up quark mass in AMSB QCD', Eqs. (28)-(29)] The abstract states that 'extrapolating the F=N result to large supersymmetry breaking would lead to a dynamical up quark mass that is large enough to account for its entire physical mass.' The manuscript itself labels this extrapolation a 'naive guess' and 'outside the regime of validity of our theory.' This is a qualitative leap from a controlled small-m calculation to the strong-CP conclusion. The abstract and conclusions should clearly flag this as speculation, separate from the reliable small-SUSY-breaking results, so that the headline claim is not misread as a rigorous consequence of the computation.
minor comments (4)
  1. [Table I] The column heading 'F=3<N' is confusing; it should read 'F=3 < N' or 'F<N with F=3'. The heading 'F=N=3' is clear.
  2. [Throughout] There are several typographical artifacts in the text, such as 'K¨ ahler' and 'Kolumn'. The manuscript would benefit from a careful proofreading pass.
  3. [Sec. 'The chiral Lagrangian in AMSB QCD', Eqs. (15)-(19)] The notation for S, H, and T is introduced but the relation to the later integration out of radial modes could be clearer; specifically, the statement that H and T are of order O(M_Q) and O(partial^2) appears before the mass expansion is defined. A short sentence explaining the counting would improve readability.
  4. [Sec. 'Dynamical up quark mass in AMSB QCD', Eq. (28)] The numerical values in Eq. (28) are presented with a parameter choice 'Lambda_c=m, mu_s/m=0.5' but no justification for mu_s/m=0.5 is given; please note that this is an illustrative input, not a derived value.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the up-quark-mass ratio is derived from L4, L6, and B0 computed by matching AMSB SQCD to chiral perturbation theory, not from a fit or from the target result.

full rationale

The central claim, Eq. (27), is an output of Eq. (11) applied to the low-energy constants in Table I. Those constants are obtained by tree-level matching of the AMSB SQCD scalar potential to the chiral Lagrangian (Section "The chiral Lagrangian in AMSB QCD"), not by assuming the up-quark mass. The F<N result is expanded in powers of 1/N and the F=N result in powers of 1/β; both are parametric predictions with the incalculable Kähler coefficients α,β,γ,δ entering as inputs. No parameter is fitted to the up-quark mass or to µu/µd, so the "fitted input called prediction" pattern does not apply. The Kaplan-Manohar reparameterization is used to define the dynamical mass in the standard way, but the paper computes the combination 2L6−L4 from the matched scalar potential rather than postulating it; the identity is a translation, not a construction of the result. The self-citations ([13], [14], [28], [29]) supply the AMSB framework and θ/η′ context; they do not contain Eq. (27), and no uniqueness theorem from the authors is invoked to force the choice. The main caveats are the admitted truncation of the F=N Kähler potential to quadratic order ("Note that truncating the Kähler potential ... which is not the case") and the unshown F=N+1 consistency check ("It has been verified that this approach yields results consistent with our simplified method"); these are uncontrolled-approximation and missing-proof concerns, not circularity. The extrapolation to m∼Λ is explicitly labeled as beyond the regime of validity, so it is not presented as a derived prediction. Overall, the derivation chain is self-contained given the SQCD/AMSB assumptions, and no step reduces by construction to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The incalculable coefficients alpha and beta are parameters of the existing F=N SQCD Kähler potential, and mu_s/m is an input mass ratio. The main conceptual input is the uncontrolled quadratic truncation of the Kähler potential and the unknown origin of the quantum modified constraint.

free parameters (3)
  • alpha (F=N Kähler normalization) = unknown; assumed O(1); illustrative alpha=2
    Incalculable coefficient in the F=N Kähler potential, Eq. (14). Eq. (27) for F=N=3 is proportional to alpha, so the central O(1) claim depends on this choice.
  • beta (F=N Kähler normalization) = not fixed; large-beta limit used
    Incalculable coefficient in Eq. (14); it affects individual LECs in Table I, though the central ratio in Eq. (27) is quoted in the beta to infinity limit.
  • mu_s/m (input mass ratio) = illustrative 0.5
    The numerical benchmarks in Eqs. (28) and (29) set mu_s/m = 0.5. The O(1) magnitude of mu_u/mu_d depends on this ratio being not too small.
assumptions (5)
  • domain assumption The AMSB scalar potential takes the conformal compensator form of Eq. (12).
    Taken from prior work on AMSB (Refs. [14,25]) and used to derive the tree-level potential for both F<N and F=N.
  • ad hoc to paper The F=N Kähler potential can be truncated to quadratic order even though M/Lambda^2, X/Lambda^3, B/Lambda^N, and B-bar/Lambda^N are not small.
    Explicitly acknowledged in the scalar potential section as an assumption that is not valid, yet it underpins Table I and the central F=N result.
  • domain assumption The F=N quantum modified constraint det M - B B-bar = Lambda^(2N), implemented by the Lagrange multiplier X, is the correct dynamical input.
    The final paragraph admits the exact origin of this constraint remains unknown; the F=N result relies on this structure.
  • domain assumption Integrating out the heavy radial modes at tree level, neglecting fermion superpartner loops, yields the NLO chiral Lagrangian.
    Invoked in the chiral Lagrangian section with the perturbative ordering Lambda >> m >> M_Q. No quantitative check of loop suppression is given.
  • standard math The Kaplan-Manohar identity and the standard eight-operator NLO ChPT basis are valid.
    Used to define the dynamical up quark mass through the combination 2L6-L4 in Eq. (11).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamical Up-quark Mass Generation in QCD-like theories." pith.science (2026). https://pith.science/paper/WMTHETSG

@misc{pith2026250507953,
  author       = {Pith},
  title        = {Pith review of: Dynamical Up-quark Mass Generation in QCD-like theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMTHETSG}},
  note         = {Machine review of arXiv:2505.07953}
}
abstract

We calculate the dynamically generated up quark mass in some QCD-like theories with $F=3$ light flavors, obtained from supersymmetric QCD perturbed via anomaly mediated supersymmetry breaking. We match the low-energy effective theory to the traditional chiral Lagrangian of QCD and determine the coefficients to next-to-leading order in chiral perturbation theory, while also varying the number of colors $N$. We find that the dynamically generated up quark mass vanishes in the large $N$ limit, and is small for $F<N$, however for $F=N$ there is a sizeable $O(1)$ contribution. While our results are reliable only for small supersymmetry breaking, we observe that extrapolating the $F=N$ result to large supersymmetry breaking would lead to a dynamical up quark mass that is large enough to account for its entire physical mass.

Figures

Figures reproduced from arXiv: 2505.07953 by the authors.

Figure 1
Figure 1. FIG. 1. Feynman diagram for the ’t Hooft operator corre [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 9 canonical work pages

  1. [1]

    D. B. Kaplan and A. V. Manohar, Current Mass Ratios of the Light Quarks, Phys. Rev. Lett. 56, 2004 (1986)

    work page 1986

  2. [2]

    Cs´ aki, R

    C. Cs´ aki, R. T. D’Agnolo, E. Kuflik, and M. Ruhdorfer, Instanton NDA and applications to axion models, JHEP 04, 074, arXiv:2311.09285 [hep-ph]

    arXiv

  3. [3]

    Fodor, C

    Z. Fodor, C. Hoelbling, S. Krieg, L. Lellouch, T. Lippert, A. Portelli, A. Sastre, K. K. Szabo, and L. Varnhorst, Up and down quark masses and corrections to Dashen’s theorem from lattice QCD and quenched QED, Phys. Rev. Lett. 117, 082001 (2016), arXiv:1604.07112 [hep- lat]

    arXiv 2016

  4. [4]

    Aoki et al

    Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG)), FLAG Review 2024 (2024), arXiv:2411.04268 [hep-lat]

    arXiv 2024

  5. [5]

    $N_f=1+2$ mass dependence of the topological susceptibility

    J. Frison, R. Kitano, and N. Yamada, Nf = 1 + 2 mass dependence of the topological susceptibility, PoS LA T- TICE2016, 323 (2016), arXiv:1611.07150 [hep-lat]

    work page Pith review arXiv 2016

  6. [6]

    Gasser and H

    J. Gasser and H. Leutwyler, Chiral Perturbation The- ory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250, 465 (1985)

    1985

  7. [7]

    Bijnens and I

    J. Bijnens and I. Jemos, A new global fit of the Lr i at next-to-next-to-leading order in Chiral Perturbation Theory, Nucl. Phys. B 854, 631 (2012), arXiv:1103.5945 [hep-ph]

    arXiv 2012

  8. [8]

    Bijnens and G

    J. Bijnens and G. Ecker, Mesonic low-energy con- stants, Ann. Rev. Nucl. Part. Sci. 64, 149 (2014), arXiv:1405.6488 [hep-ph]

    arXiv 2014

Show all 31 references
  1. [9]

    M. Dine, P. Draper, and G. Festuccia, Instanton Effects in Three Flavor QCD, Phys. Rev. D 92, 054004 (2015), arXiv:1410.8505 [hep-ph]

    2015 arXiv

  2. [10]

    Leutwyler, Bounds on the light quark masses, Phys

    H. Leutwyler, Bounds on the light quark masses, Phys. Lett. B 374, 163 (1996), arXiv:hep-ph/9601234. 7

    1996 arXiv

  3. [11]

    Kaiser and H

    R. Kaiser and H. Leutwyler, Large N(c) in chiral pertur- bation theory, Eur. Phys. J. C 17, 623 (2000), arXiv:hep- ph/0007101

    2000

  4. [12]

    Davies, M

    D. Davies, M. Dine, and B. V. Lehmann, Light Quarks at Large N (2022), arXiv:2201.05719 [hep-ph]

    2022 arXiv

  5. [13]

    Murayama, Some Exact Results in QCD-like Theories, Phys

    H. Murayama, Some Exact Results in QCD-like Theories, Phys. Rev. Lett. 126, 251601 (2021), arXiv:2104.01179 [hep-th]

    2021 arXiv

  6. [14]

    Cs´ aki, A

    C. Cs´ aki, A. Gomes, H. Murayama, B. Noether, D. R. Varier, and O. Telem, Guide to anomaly-mediated supersymmetry-breaking QCD, Phys. Rev. D 107, 054015 (2023), arXiv:2212.03260 [hep-th]

    2023 arXiv

  7. [15]

    Gell-Mann, R

    M. Gell-Mann, R. J. Oakes, and B. Renner, Behavior of current divergences under SU(3) x SU(3), Phys. Rev. 175, 2195 (1968)

    1968

  8. [16]

    Banks, Y

    T. Banks, Y. Nir, and N. Seiberg, Missing (up) mass, accidental anomalous symmetries, and the strong CP problem, in 2nd IFT Workshop on Yukawa Couplings and the Origins of Mass (1994) pp. 26–41, arXiv:hep- ph/9403203

    1994

  9. [17]

    Aoki et al

    S. Aoki et al. (Flavour Lattice Averaging Group), FLAG Review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80, 113 (2020), arXiv:1902.08191 [hep- lat]

    2020 arXiv

  10. [18]

    Alexandrou, J

    C. Alexandrou, J. Finkenrath, L. Funcke, K. Jansen, B. Kostrzewa, F. Pittler, and C. Urbach, Ruling Out the Massless Up-Quark Solution to the Strong CPCPCP Prob- lem by Computing the Topological Mass Contribution with Lattice QCD, Phys. Rev. Lett. 125, 232001 (2020), arXiv:200...

    2020 arXiv

  11. [19]

    Affleck, M

    I. Affleck, M. Dine, and N. Seiberg, Dynamical Su- persymmetry Breaking in Supersymmetric QCD, Nucl. Phys. B 241, 493 (1984)

    1984

  12. [20]

    Seiberg, Exact results on the space of vacua of four- dimensional SUSY gauge theories, Phys

    N. Seiberg, Exact results on the space of vacua of four- dimensional SUSY gauge theories, Phys. Rev. D 49, 6857 (1994), arXiv:hep-th/9402044

    1994 arXiv

  13. [21]

    Seiberg, Electric - magnetic duality in supersymmet- ric nonAbelian gauge theories, Nucl

    N. Seiberg, Electric - magnetic duality in supersymmet- ric nonAbelian gauge theories, Nucl. Phys. B 435, 129 (1995), arXiv:hep-th/9411149

    1995 arXiv

  14. [22]

    K. A. Intriligator and N. Seiberg, Lectures on super- symmetric gauge theories and electric-magnetic duality, Nucl. Phys. B Proc. Suppl. 45BC, 1 (1996), arXiv:hep- th/9509066

    1996

  15. [23]

    Randall and R

    L. Randall and R. Sundrum, Out of this world supersym- metry breaking, Nucl. Phys. B557, 79 (1999), arXiv:hep- th/9810155

    1999

  16. [24]

    G. F. Giudice, M. A. Luty, H. Murayama, and R. Rat- tazzi, Gaugino mass without singlets, JHEP 12, 027, arXiv:hep-ph/9810442

    arXiv

  17. [25]

    Pomarol and R

    A. Pomarol and R. Rattazzi, Sparticle masses from the superconformal anomaly, JHEP 05, 013, arXiv:hep- ph/9903448

  18. [26]

    A. C. Davis, M. Dine, and N. Seiberg, The Massless Limit of Supersymmetric QCD, Phys. Lett. B 125, 487 (1983)

    1983

  19. [27]

    M. Dine, P. Draper, L. Stephenson-Haskins, and D. Xu, θ and the η′ in Large N Supersymmetric QCD, JHEP 05, 122, arXiv:1612.05770 [hep-th]

    arXiv

  20. [28]

    Cs´ aki, R

    C. Cs´ aki, R. Tito D’Agnolo, R. S. Gupta, E. Kuflik, T. S. Roy, and M. Ruhdorfer, On the dynamical origin of the η′ potential and the axion mass, JHEP 10, 139, arXiv:2307.04809 [hep-ph]

    arXiv

  21. [29]

    Cs´ aki, M

    C. Cs´ aki, M. Ruhdorfer, and T. Youn, Spontaneous CP breaking in a QCD-like theory, JHEP 12, 066, arXiv:2407.06252 [hep-ph]

    arXiv

  22. [30]

    D. U. Jungnickel and C. Wetterich, The Linear meson model and chiral perturbation theory, Eur. Phys. J. C 2, 557 (1998), arXiv:hep-ph/9704345

    1998 arXiv

  23. [31]

    Ecker, J

    G. Ecker, J. Gasser, A. Pich, and E. de Rafael, The Role of Resonances in Chiral Perturbation Theory, Nucl. Phys. B 321, 311 (1989)

    1989

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.