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REVIEW 3 major objections 5 minor 73 references

Pion-Nucleon Scattering in Baryon Chiral Perturbation Theory combined with the ${ 1/N_c}$ Expansion

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read By linking the chiral and 1/Nc expansions, a spin-flavor-symmetric effective theory with dynamical nucleon and Delta fits pion-nucleon S, P and D partial waves up to 350 MeV.

desk verdict A technically serious one-loop calculation in the linked chiral/large-Nc expansion, with a fit-based claim that is currently undermined by a sign inconsistency in the isovector S-wave scattering length. read the letter →

arxiv 2506.05224 v2 pith:4TKKTOPB submitted 2025-06-05 hep-ph nucl-th

D. Jayakodige , J. L. Goity This is my paper
classification hep-phnucl-th PACS 12.39.Fe13.75.Gx11.15.Pg
keywords ChiralPerturbationTheory1/NcexpansionBaryonsPion-nucleonscatteringSpin-flavorsymmetryxi-expansionDeltaresonance
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

This paper tries to establish that the combined baryon chiral perturbation theory and 1/Nc expansion can describe pion-nucleon elastic scattering at next-to-next-to-leading order, provided the nucleon and the $\Delta$ are both treated as active degrees of freedom in the emergent spin-flavor SU(4) multiplet. The central device is the xi-expansion, which links the chiral expansion and the 1/Nc expansion so that the non-analytic loop terms where the two expansions do not commute are retained at face value. Fits to the SAID S, P and D partial waves are presented as support for the framework, with a quoted range of validity of pion CM momenta up to 200-350 MeV depending on the partial wave. If the claim is right, this gives a baryon effective theory that satisfies the large-Nc consistency constraints of QCD while keeping the $\Delta$ dynamical.

What carries the argument

The load-bearing machinery is the xi-expansion: a linked power counting in which $O(p)=O(1/N_c)=O(\xi)$, so that non-analytic terms involving ratios of the pion mass to the $O(1/N_c)$ baryon mass splittings (for instance $M_\pi/\Delta$) are kept without re-expanding. The associated dynamical symmetry is the contracted spin-flavor SU(4) group with generators $S_i$, $I^a$, and $G^{ia}$ acting on the totally symmetric ground-state baryon multiplet; the $G^{ia}$ couplings are $O(N_c)$ and their matrix elements connect states with $S'=S$ or $S\pm 1$, so the $\Delta$ participates in loops. The cancellations that remove the $O(N_c^2)$ and $O(N_c)$ UV divergences in the $\tilde g_A^4$ and $\tilde g_A^2$ diagram sums are carried by SU(4) operator identities such as $2\{S_i,G^{ia}\}=(N_c+2)I^a$ applied to the spin-flavor tensor basis, and the pole/no-pole decomposition plus exact non-analytic loop functions enforce unitarity to the computed order.

What would settle it

Compute the $xi^{4}$ (next-order) corrections to the no-pole parts of the $\tilde g_A^4$ and $\tilde g_A^2$ diagram sets: the framework predicts the UV divergences still reduce to spin-flavor tensors of at most $O(N_c^0)$; if divergences of order $N_c^2$ survive, the claimed large-Nc consistency is falsified.

Watch

Extended reading notes

Core claim

The central claim is that implementing the 1/Nc consistency conditions through the emergent spin-flavor SU(4) symmetry, combined with baryon chiral perturbation theory via the xi-expansion, produces a next-to-next-to-leading-order (one-loop) amplitude that describes elastic pion-nucleon scattering in a manner consistent with the large-Nc limit. In this framework the nucleon and $\Delta$ are forced to be active degrees of freedom because they belong to the same ground-state SU(4) multiplet, and the mass splitting $m_\Delta-m_N$ is an $O(1/N_c)$ effect. The one-loop amplitudes are renormalized for generic $N_c$, with the UV divergences of the $\tilde g_A^4$ and $\tilde g_A^2$ diagram sets reducing to at most $O(N_c^0)$ spin-flavor tensors only after summing all diagrams, which is the signature of restored $N_c$ consistency. Fits to the SAID partial waves reproduce the S, P and D waves for pion CM momenta up to 200-350 MeV, depending on the partial wave, and yield low-energy constants such as $\tilde g_{ANN}$ and $\tilde g_{AN\Delta}$ consistent with the existing estimates. The paper's stated conclusion is that this analysis gives strong support for the implementation of the 1/Nc consistency conditions in the BChPT $\times$ 1/Nc framework.

Load-bearing premise

The framework assumes the emergent large-Nc spin-flavor SU(4) symmetry and that the 1/Nc expansion is valid down to Nc=3, so the nucleon and $\Delta$ must belong to a single ground-state multiplet and both be dynamical degrees of freedom.

Editorial extensions

If this is right

  • If the framework is correct, pion-nucleon scattering below 350 MeV can be described with a controlled counting of chiral and 1/Nc effects, giving a controlled extraction of the NNLO low-energy constants from data.
  • The Delta must remain a dynamical field: any baryon effective theory for pion-nucleon scattering that omits it cannot satisfy the large-Nc consistency conditions, and will show worse convergence.
  • The same xi-expansion machinery can be applied to other low-energy baryon observables, most directly to the pion-nucleon sigma term, which the paper identifies as the next target.
  • The stated range of validity (200-350 MeV depending on the partial wave) sets an explicit bound on where the NNLO approximation can be trusted, with S-waves limited near 250 MeV and the P33 channel extending furthest.

Reading between the lines

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

  • A natural test the paper leaves implicit is to compute the xi^4 (next order) corrections and check that the large-Nc cancellations in the UV divergences persist; this would confirm whether the consistency is a property of the whole expansion or an artifact of the one-loop order.
  • The framework's prediction g_A = 1.325, derived from the fitted $\tilde g_{ANN}$ after accounting for the Goldberger-Treiman discrepancy, could be confronted with independent lattice determinations of the nucleon axial charge.
  • Because the spin-flavor multiplet grows with $N_c$, the framework's predictions for the $N_c$-dependence of the scattering lengths (such as $a_+\sim O(N_c^0)$ with computable corrections) could be tested by lattice QCD at larger $N_c$, separating the dynamical content from the $N_c=3$ fit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This manuscript develops pion-nucleon scattering in baryon chiral perturbation theory combined with the 1/Nc expansion, using the ξ-expansion in which chiral and 1/Nc orders are linked. The baryon sector is built on the emergent SU(4) spin-flavor symmetry, with N and Δ treated as active degrees of freedom. The authors present a one-loop NNLO calculation of the πB→πB′ amplitude at generic Nc, including the renormalization program, the demonstration of large-Nc cancellations in the UV-divergent pieces, and the construction of the necessary counterterm Lagrangians. They then fit the resulting amplitudes, with 16 low-energy constants, to SAID partial-wave data for S, P and D waves up to pion CM momenta of 350 MeV, and conclude that the analysis gives strong support for the BChPT×1/Nc implementation. The paper is largely a calculational paper, with the phenomenological fit serving as the validation of the framework.

Significance. If the calculation is correct, this is a significant step: it provides a systematic, renormalized one-loop framework for pion-baryon scattering that is consistent with the large-Nc constraints of QCD, and it demonstrates in detail how the ξ-expansion handles the non-commutativity of the chiral and 1/Nc expansions. The explicit generic-Nc results, the UV-divergence cancellations (Eqs. F11, F31, F39), and the complete β-function table are valuable and appear reproducible from the appendices. The phenomenological part is less conclusive: the fit uses a large number of LECs, has large χ2 per degree of freedom by the authors' own admission, and contains an internal inconsistency in the S-wave scattering length. The significance of the paper therefore rests substantially on the formal calculation; the data comparison, as it stands, provides only tentative support for the framework.

major comments (3)
  1. [Section V (second and penultimate paragraphs)] There is a direct internal contradiction in the treatment of the S-wave scattering lengths. The text states 'In the fits, the S-wave scattering lengths [40,41] are inputs,' yet later reports 'the S-wave scattering lengths resulting from the fit are a+ = 0.012/Mπ and a− = −0.087/Mπ, to be compared with the experimental ones ... a− = (0.0866 ± 0.0010)/Mπ.' The fitted a− is opposite in sign to the experimental value that was supposedly imposed as an input. If the scattering lengths were fixed inputs, the fitted threshold amplitudes should reproduce them; if they were not inputs, the statement is incorrect. This is not a cosmetic issue: the paper claims in Section VI that 'the S-waves are reliably reproduced up to about 250 MeV,' and a threshold amplitude with the wrong sign for the isovector scattering length directly contradicts that claim. The authors must either fix the sign/convention error, clarify how the input constraints were implemented, or significantly soften the conclusions about S-wave agreement.
  2. [Section V and Table I] The fit that motivates the central claim uses 16 low-energy constants for the SAID partial waves, and the paper concedes 'The rather large χ2 per degree of freedom of the fits is indicative of that disparity.' No χ2 values or confidence levels are quoted, and the text acknowledges discrepancies in D33 and D15. With 16 parameters and a stated large χ2, the concluding sentence that the analysis gives 'strong support for the implementation of the 1/Nc consistency conditions' is not proportionate to the evidence presented. I recommend either adding a quantitative assessment of the fit quality, comparing with a fit with fewer LECs or with an alternative framework, or rewording the conclusion so that the data comparison is described as indicative rather than 'strong support.'
  3. [Section V and Section VI (range of validity)] The stated energy range of applicability, 'up to 200–350 MeV depending on the specific partial wave,' is based on a fit in which unitarity is imposed only approximately. The text correctly notes that the imaginary parts determined from the LO Lagrangian are less accurate, and that departures from unitarity appear in the S-waves above 200–250 MeV. Given that the fit is performed to real parts only (plus the P33 imaginary part), the phrase 'the approach yields a consistent description' should be qualified: the effective theory is being compared with data through a unitarization prescription, not through a fully unitary amplitude. This does not invalidate the calculation, but it affects the interpretation of the fitted range and should be stated more carefully.
minor comments (5)
  1. [Abstract and PACS] The PACS list contains a duplicate entry: '12.39.Fe,13.75.Gx,12.39.Fe,13.75.Gx,11.15.Pg' repeats 12.39.Fe and 13.75.Gx. Please correct this.
  2. [Appendix E (text before Eq. E1)] There is a typo: 'dendencies' should be 'dependencies.' Please proofread the appendices for similar minor errors.
  3. [Eq. (F15)] The equation contains 'Mpi2' in the text; it should be Mπ2. Also, in the surrounding text the notation for the pion mass is sometimes Mπ and sometimes Mpi in the LaTeX source; please ensure consistent notation.
  4. [Table I and Eq. (F44)] Table I lists α(1)_00, α(2)_00, α(4)_00 and α(1)_01, α(2)_01, α(3)_01, α(5)_01, but not α(3)_00 or α(4)_01. Since Eq. (F44) defines these LECs, the authors should explain why some combinations are absent or set to zero, otherwise the reader cannot tell whether these are omitted because they do not contribute or because they are redundant.
  5. [Figures 8 and 9] In Fig. 9 the label 'p13' appears inconsistently with the notation 'P13' used in the text and in Fig. 8. Please make the labels uniform.

Circularity Check

2 steps flagged · score 4.0 of 10

Partial circularity: the S-wave scattering lengths are stated as fit inputs yet quoted as fit outcomes, and the central xi-expansion is imported from self-cited prior work; the generic-Nc one-loop derivation itself is independent.

  1. self citation load bearing [Section II, paragraph introducing the ξ-expansion and the linking of expansions]
    "The non-commutativity of the expansions demands that either only one of them is implemented, or they be linked. The latter is evidently the option that works best for the real world as it was shown in several works [16, 17], in which the ξ-power counting scheme was introduced, according to which O(p) = O(1/Nc) = O(ξ)."

    The ξ-expansion is the central power-counting premise of the framework being validated in this paper. Its justification is a citation to prior works [16,17] coauthored by Goity, rather than a re-derivation in the present paper. The paper takes the linked expansion as established and builds the entire NNLO calculation on it. This makes the self-citation load-bearing for the framework's structure. However, the subsequent comparison to SAID data is an external test of that framework, so the circularity is partial: the scheme is not proven here, but it is also not simply assumed to agree with the data; it is confronted with them.

  2. fitted input called prediction [Section V, paragraph describing the fit to SAID single-energy solutions and the final paragraph quoting scattering lengths]
    "In the fits, the S-wave scattering lengths [40, 41] are inputs. ... Finally, the S-wave scattering lengths resulting from the fit are a+ = 0.012/Mπ and a− = −0.087/Mπ, to be compared with the experimental ones [40, 41] a+ = (0.0078 ± 0.0028)/Mπ and a− = (0.0866 ± 0.0010)/Mπ."

    The paper explicitly states that the S-wave scattering lengths are inputs to the fit. Quoting the values 'resulting from the fit' and comparing them with the experimental values that were entered as inputs is therefore not an independent test: a fitted quantity constrained by an input cannot validate that input. Moreover, the reported fitted a− is the negative of the experimental a−, so the fit does not even reproduce the input. This sign discrepancy indicates either a sign-convention error in translating the fitted partial waves into a− or a genuine failure of the fit at threshold. In either case, these numbers provide no independent support for the framework and undercut the later statement that the S-waves are 'reliably reproduced' up to about 250 MeV.

full rationale

The mathematical core of the paper is a self-contained one-loop NNLO calculation of pion-baryon scattering at generic Nc, with explicit UV renormalization and demonstration that large-Nc violating contributions cancel. This part does not define its observables in terms of fitted parameters. The fits to SAID partial waves are external benchmarks, and the paper is transparent that the real parts are 'well described in significant part due to the available NNLO LECs'; hence the agreement in the fitted region is not a strong prediction by itself. The clearest concrete circularity is the S-wave scattering length paragraph: a quantity stated to be an input is presented as a fit outcome for comparison, and the fitted a− actually disagrees in sign with the input, creating an internal inconsistency in the claimed validation. Separately, the ξ-expansion is imported from self-cited prior work [16,17] and is load-bearing for the framework, though it is subsequently tested against data. These issues lower the evidential weight of the 'strong support' conclusion, but they do not make the central one-loop derivation itself circular. An overall score of 4 reflects partial circularity and self-citation with independent content remaining.

Assumptions & free parameters 16 free parameters · 6 assumptions · 0 invented entities

The framework imports the xi-expansion and the spin-flavor hypothesis from earlier work, then fits a set of low-energy constants to the same scattering data used for validation. No new particles or forces are introduced, but the predictive power is carried by the assumed SU(4) structure and by the fitted LECs.

free parameters (16)
  • CHF = 304.49(1.16) MeV
    Fitted to SAID pion-nucleon partial waves (Table I).
  • Gamma_Delta = 119.5(1.45) MeV
    Fitted Delta width used in the P33 pole contribution (Table I).
  • g_tilde_A = 1.18
    LO axial coupling parameter, fitted at NNLO (Table I).
  • c1 = 0.46
    Quark mass term LEC, fitted (Table I).
  • g_ANN = 1.59(0.03)
    Renormalized pion-nucleon coupling, fitted (Table I).
  • g_ANDelta = 1.81(0.01)
    Renormalized pion-N-Delta coupling, fitted (Table I).
  • alpha(1)_00 = 8.55(0.21)
    NNLO contact LEC combination fitted to S-wave data (Table I and Eq. F44).
  • alpha(2)_00 = -5.56(0.19)
    NNLO contact LEC combination fitted to S-wave data (Table I and Eq. F44).
  • alpha(4)_00 = -0.48(0.17)
    NNLO contact LEC combination fitted to S-wave data (Table I and Eq. F44).
  • alpha(1)_01 = 3.82(0.49)
    NNLO contact LEC combination fitted to P-wave data (Table I and Eq. F44).
  • alpha(2)_01 = -0.88(0.66)
    NNLO contact LEC combination fitted to P-wave data (Table I and Eq. F44).
  • alpha(3)_01 = 1.01(0.15)
    NNLO contact LEC combination fitted to P-wave data (Table I and Eq. F44).
  • alpha(5)_01 = -0.98(0.61)
    NNLO contact LEC combination fitted to P-wave data (Table I and Eq. F44).
  • alpha(1)_10 = 2.82(0.57)
    NNLO contact LEC combination fitted to spin-dependent data (Table I and Eq. F44).
  • alpha(1)_11 = -6.25(0.84)
    NNLO contact LEC combination fitted to spin-isospin data (Table I and Eq. F44).
  • alpha(2)_11 = 4.53(2.26)
    NNLO contact LEC combination fitted to spin-isospin data (Table I and Eq. F44).
assumptions (6)
  • domain assumption Emergent spin-flavor SU(4) symmetry of the large-Nc baryon sector, with ground-state baryons in the totally symmetric multiplet.
    Invoked in Section II and Appendix B; the whole effective theory is built on this symmetry.
  • domain assumption The 1/Nc expansion is valid down to Nc=3, so N and Delta are active degrees of freedom.
    Stated explicitly in Section II as the hypothesis that the 1/Nc expansion holds down to Nc=3.
  • domain assumption The xi-expansion links the two expansions via O(p)=O(1/Nc)=O(xi), with non-analytic terms kept at face value.
    Adopted from refs [16,17] and central to the power counting in Sections II-IV.
  • domain assumption Isospin symmetry is assumed; no isospin-breaking effects are included.
    The amplitude is computed in the isospin limit (Section V and Appendix A).
  • domain assumption Elastic unitarity with inelasticity eta=1 holds for pion momenta below about 400 MeV.
    Used in Section V to convert real parts into phase shifts and to constrain P33.
  • standard math Dimensional regularization and MS-bar subtraction with renormalized LECs X(mu) and beta functions in Table VII.
    Standard EFT machinery used throughout Section IV and the appendices.

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Cite this review

Pith. "Pith review of Pion-Nucleon Scattering in Baryon Chiral Perturbation Theory combined with the ${ 1/N_c}$ Expansion." pith.science (2026). https://pith.science/paper/4TKKTOPB

@misc{pith2026250605224,
  author       = {Pith},
  title        = {Pith review of: Pion-Nucleon Scattering in Baryon Chiral Perturbation Theory combined with the $ 1/N_c$ Expansion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TKKTOPB}},
  note         = {Machine review of arXiv:2506.05224}
}
abstract

This work implements the combined BChPT and 1/Nc expansions for pion-nucleon elastic scattering. The effective theory is based on the baryon sector dynamical spin-flavor SU(4) symmetry emergent in the large Nc limit, whose breaking is controlled by the $1/N_c$ expansion. The non-commutativity of the chiral and 1/Nc expansions in unitarity corrections (loops) requires a linking of both expansions. As it was shown in the case of baryon masses and currents, the natural linking is the $\xi$-expansion, in which $O(p) = O(1/Nc ) = O(\xi)A$. The spin-flavor symmetry requires that the ground state baryons span an SU(4) symmetric irreducible representation which implies that in particular $N$ and $\Delta$ are active degrees of freedom in the effective theory. The scattering amplitude is expanded to the next-to-next-to leading order in the $\xi$ expansion, corresponding to the one-loop contributions with the LO Lagrangian. The results are given for generic $N_c$ in order to demonstrate the consistency of the framework. The spin-flavor symmetry plays a central role in maintaining the consistency of the effective theory with respect to the $1/N_c$ expansion. This consistency manifests itself in an improvement in the convergence of the low energy expansion with respect to the case of the ordinary BChPT without an explicit dynamical $\Delta$, which is known to be inconsistent with the constraints of $N_c$ scaling. Fits to the $\pi N \to \pi N$ S, P and D partial wave amplitudes from the SAID data base are finally used to test the framework and to determine the energy range of its applicability.

Figures

Figures reproduced from arXiv: 2506.05224 by the authors.

Figure 1
Figure 1. FIG. 1: One-loop contributions to the baryon self-energy IG1: One-loop contributions to the baryon self-ene [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: One loop diagrams for the pionbaryon interaction. FIG. 2: One-loop diagrams for the pion-baryon interaction. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: General decomposition of the scattering amplitude into pole (A) and nopole (B) () FIG. 3: General decomposition of the scattering amplitude into pole (A) and no-pole (B) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: LO contributions to the scattering amplitude. To diagram (A) the crossed FIG4: LO contributions to the scattering amplitudeTo diagram (A) the crossed [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Diagrams proportional to ˚g [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Diagrams proportional to ˚g [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Diagrams proportional to ˚g [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Fits to the partial wave amplitudes Re [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Phase shifts: SAID WI08 solution (dashed),data points from SEV solution, and [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Pole diagrams [PITH_FULL_IMAGE:figures/full_fig_p036_9.png]

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Works this paper leans on

73 extracted references · 57 canonical work pages

  1. [1]

    Expansion of building blocks For the purpose of the present work the relevant terms in the building blocks are those with up to four-pion fields, namely: uµ = 2 aµ − 1 Fπ ∂µΠ + i Fπ [vµ, Π] + 1 4F 2 π [Π, [aµ, Π]] + 1 24F 3 π ([Π, [Π, ∂µΠ]] − i[Π, [Π, [vµ, Π]]]) + · · · ua µ = 1 2 ⟨uµτ a⟩ uµ in the fundamental irrep Γµ = vµ + i 2Fπ [aµ, Π] + i 8F 2 π [Π, ...

    work page

  2. [2]

    Note that C does not apply to the baryon Lagrangians, as the heavy baryon expansion can only describe either baryons or anti-baryons as separate sectors

    Discrete symmetry transformations of building blocks The Lagrangians respect the discrete P, C and T symmetries, and the following Table II provides the necessary transformation rules for the building blocks. Note that C does not apply to the baryon Lagrangians, as the heavy baryon expansion can only describe either baryons or anti-baryons as separate sec...

    work page

  3. [3]

    LO Lagrangians The pion LO Lagrangian O(p2) = O(ξ2) has the standard form: L(2) π = 1 4 F 2 π ⟨DµU †DµU + χU † + χ†U ⟩. (D1) 27 J I Operator J I Operator 0 0 1 2 1 1 N 2c SiSj|J=2I a 1 0 Si 2 1 1 N 2c {SiSj|J=2, Gka}|J=2,I=1 0 1 I a 1 2 1 Nc {I a, Gib}|I=2 1 1 Gia 1 2 1 N 2c {SiI a, Gjb}|J=1,I=2 1 1 1 Nc SiI a 1 2 1 N 2c SiI aI b|I=2 1 1 1 Nc {Si, Gja}|J=...

    work page

  4. [4]

    + δadδbc(2ki 1 − ki 2 − ki 3) k3c k4d k1a k2b = 1 F 4π 1 12 I e δabϵcde(k0 3 − k0

    work page

  5. [5]

    + δadϵbce(k0 2 − k0 3) +δbcϵade(k0 1 − k0

    work page

  6. [6]

    + δcdϵabe(k0 1 − k0 2) +i 2 3 c1Nc M 2 π Λ (δabδcd + δacδbd + δadδbc) TABLE V: Vertices from the LO Lagrangians. 29

    work page

  7. [7]

    (D4) The following are the higher order Lagrangians needed for renormalization in the present calculation of the πB → πB ′ amplitudes

    Higher order Lagrangians In the construction of the higher order Lagrangians one uses the LO equations of motion, namely: iD0B = CHF Nc ˆS2 − c1 Nc 2Λ ⟨χ+⟩ −˚gAuiaGia B Dµuµ = i 2 χ−, (D3) and the identities: Dµuν − Dνuµ = −f−µν [Dµ, Dν] = −iΓµν Γµν = 1 2 f+µν + i 4 [uµ, uν]. (D4) The following are the higher order Lagrangians needed for renormalization i...

    work page

  8. [8]

    log(C1λ2 0+C0) 2C1 0 1 λϵ 2C1 + λ0 2 arctan q C1 C0 λ0 +π 2√C0C1 − log(C1λ2 0+C0) 2C1 0 2 λ0 2 arctan q C1 C0 λ0 +π 4C0 √C0C1 + 1 2C0C1 0 3 3C1λ2 0+2C0 4C2 0 C1(C1λ2 0+C0) + 3λ0 2 arctan q C1 C0 λ0 +π 8C2 0 √C0C1 1 0 1 2 λ2 0 − C0 C1 λϵ − √C0C1λ0 2 arctan q C1 C0 λ0 +π C1 + 3C1λ2 0−C0+(C0−C1λ2

    work page

Show all 73 references
  1. [9]

    log(C1λ2 0+C0) 2C1 1 1 λϵ 2C1 + λ0 2 arctan q C1 C0 λ0 +π 2√C0C1 − log(C1λ2 0+C0) 2C1 1 2 λ0 2 arctan q C1 C0 λ0 +π 4C0 √C0C1 + 1 2C0C1 1 3 3C1λ2 0+2C0 4C2 0 C1(C1λ2 0+C0) + 3λ0 2 arctan q C1 C0 λ0 +π 8C2 0 √C0C1 2 0 1 2 λ2 0 − C0 C1 λϵ − √C0C1λ0 2 arctan q C1 C0 λ0 +π C1 + 3C...

  2. [10]

    log(C1λ2 0+C0) 2C1 2 1 λϵ 2C1 + λ0 2 arctan q C1 C0 λ0 +π 2√C0C1 − log(C1λ2 0+C0) 2C1 2 2 λ0 2 arctan q C1 C0 λ0 +π 4C0 √C0C1 + 1 2C0C1 2 3 3C1λ2 0+2C0 4C2 0 C1(C1λ2 0+C0) + 3λ0 2 arctan q C1 C0 λ0 +π 8C2 0 √C0C1 TABLE VI: Integrals over the Feynman parameter λ. A particular i...

  3. [11]

    Diagrams D1, D2 and D3 will contribute with pole terms, i.e., terms that contain singularities due to a single baryon pole

    Diagrams ∝ ˚g4 A The diagrams proportional to ˚g4 A are shown in Fig.(5). Diagrams D1, D2 and D3 will contribute with pole terms, i.e., terms that contain singularities due to a single baryon pole. Diagram D1 has double pole, single pole, and no-pole contributions, while diagr...

  4. [12]

    The first term has a double pole which is taken care of by the mass renormalization, the second term has a single pole, and the last one has no-pole

    − Σ(δmn) − (p0 1 + k0 1 − δmn)Σ′(δmn)) PnΓ1, (F2) where one identifies: Σ ′(δmn) = −δZ (δmn). The first term has a double pole which is taken care of by the mass renormalization, the second term has a single pole, and the last one has no-pole. For the single pole term the reno...

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    9: Pole diagrams

    − Σn(δmn) (p0 1 + k0 1 − δmn)2 + δZn(δmn) p0 1 + k0 1 − δmn (F3) Diagrams D2: p1 Γ1 Γ2 p2 k1 k2 p1 ΣΣ p2 k1 k2 (a) (b) FIG. 9: Pole diagrams. 10 FIG. 10: Pole diagrams. 36 iT ba D2 = ˚gA Fπ ki 1kj 2 X n 1 p0 1 + k0 1 − δmn GjbPnΓia(p0 1, p0 1 + k0 1, −k1) + Γjb(p0 1 + k0 1, p0...

  6. [14]

    − Σin(δmin p0 1 − δmin + Σout(p0

  7. [15]

    − Σout(δmout) p0 2 − δmout (F5) The term with external pole will be absorbed by renormalizing the baryon masses of the external baryons, and the terms with no external pole multiplied by 1/2, which are a con- sequence of the external baryon’s wave function renormalization, are...

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    + 1 2 (δmin + δmout) + 2δmn − 3δmn′) × GjbPnGlcPn′GlcPnGia (F8) = i λϵ (4π)2 ˚gA Fπ 4 1 3 ki 1kj 2 (k0 1 + k0 2)Gjb ˆG2Gia + [δ ˆm, Gjb] ˆG2Gia − Gjb ˆG2[δ ˆm, Gia] + 3Gib[[δ ˆm, Glc], Glc]Gia Diagrams D2 UV: iT baU V D2 (no − pole) = −i λϵ (4π)2 ˚gA Fπ 4 1 3 ki 1kj 2 × X n,n′...

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    Diagrams D5: These diagrams give only single pole contributions, correcting the π-baryon coupling, and thus are finally included in the pole contributions Eqn.(15)

    Diagrams ∝ ˚g2 A The diagrams D6, 7, 8 vanish identically. Diagrams D5: These diagrams give only single pole contributions, correcting the π-baryon coupling, and thus are finally included in the pole contributions Eqn.(15). 39 iT ba D5 = −i ˚gA F 2 π 2 2 3 ki 1kj 2 ∆(Mπ) X n G...

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    − 1 2 δjlΓ(2 − d 2 )J(0, 2 − d 2 , C1 0 , 1, λ1 0)) + ( δacδbd − δadδbc) 1 2 (αql(k1 + k2)j + (α − 1)qj(k1 + k2)l)Γ(2 − d 2 )J(0, 2 − d 2 , C1 0 , 1, λ1 0) + α(1 − α)qjql(k0 1 + k0 2)Γ(3 − d 2 )J(1, 3 − d 2 , C1 0 , 1, λ1 0) − 1 2 δjl(k0 1 + k0 2)Γ(2 − d 2 )J(1, 2 − d 2 , C1 0...

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    + (1 2 − α)q0 − δmn. (F18) Diagram D11: 40 iT ba D11 = i ˚gA F 2 π 2 X n,n′ GkcPn′(− 1 2 (k0 1 + k0 2)ϵbacI c + 2ic1Nc M 2 π Λ δab)PnGkd × 1 p0 2 − p0 1 − δmn′ + δmn (I(δmn′ − p0 2, Mπ) − I(δmn − p0 1, Mπ)) (F19) Diagram D12: iT ba D12 = 1 2 1 F 2 π n 1 2 (k0 1 + k0 2)ϵbacI c ...

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    The results are as follows: Diagram D13: iT ba D13 = − 1 (4π)2 5 F 4 π M 2 π λϵ + 1 − log M 2 π µ2 1 24 ϵbacI c(k0 1 + k0

    Diagrams ∝ ˚g0 A In these diagrams the baryon flowing through it has a fixed spin, thus in the CM frame k0 1 = k0 2, and each diagram satisfies large Nc consistency. The results are as follows: Diagram D13: iT ba D13 = − 1 (4π)2 5 F 4 π M 2 π λϵ + 1 − log M 2 π µ2 1 24 ϵbacI c...

  13. [21]

    + i 3 δabc1Nc M 2 π Λ (F32) Diagram D14: iT ba D14 = 1 F 4 π 1 (4π)2 ic1Nc M 2 π Λ δab 7 3 M 2 π − 2t λϵ − log M 2 π µ2 + 10M 2 π 3 − 4t − 2 r 1 − 4M 2 π t M 2 π − 2t arctan √ tp −4M 2 π + t + 2 ϵbacI c(k0 1 + k0 2) M 2 π − t 6 λϵ − log M 2 π µ2 + 7 3 M 2 π − 4t + 1 3 −4M 2 π ...

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    UV divergencies Diagram D13 UV: iT baU V D13 = − λϵ (4π)2 5 F 4 π M 2 π( 1 32 ϵbacI c(k0 1 + k0

    Adding the crossed diagram and, in order to simplify the result, using explicitly that k0 1 = k0 2 in the CM frame, yields: iT ba D15+crossed = 1 (4π)2 1 F 4 π −iδab8π q M 2 π − k02 1 1 3 k0 1 ˆS2 + 2c2 1N 2 c M 4 π Λ2 + ϵbacI ck0 1 3 4 M 2 π − 2k02 1 λϵ + 1 − log M 2 π µ2 − 2...

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    + i 3 δabc1Nc M 2 π Λ ) (F36) Diagram D14 UV: iT baU V D14 = λϵ (4π)2 1 F 4 π 1 12 4ic1Nc M 2 π Λ δab(7M 2 π − 6t) + ϵbacI c(k0 1 + k0 2)(6M 2 π − t) (F37) Diagram D15 UV: Adding the crossed diagram to D15 yields: iT baU V D15+crossed = − λϵ (4π)2 1 F 4 π ϵbacI c 1 4 (k0 1 + k...

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    β-functions The β functions corresponding to the LECs of the CT Lagrangians Eqn.(D5,D6,D7) are given in Table VII. The definition of the β-function for a given LEC X is the following: X ≡ X(µ) + βX λϵ (4π)2 , (F40) 45 LEC βfJI / ˚g4 A F 2π LEC βgJ I/ Λ˚g2 A F 2π LEC βhJ I/ Λ2 ...

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    Counterterm contributions to the amplitudes. The contributions from the Lagrangians Eqns.(D5,D6,D7,D8) to the scattering ampli- tudes for definite t-channel ( J, I) are the following: iT ba CT (J = I = 0) = i2 δab F 2 π −2 g(1) 00 Λ M 2 π − 2 g(2) 00 Λ M 2 π ˆS2 Nc − 8 h(1) 00...

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    − 2 l(2) 00 Λ2 M 2 π(p0 1 + p0 2) ! (F41) 46 iT ba CT (J = 0, I= 1) = 1 F 2 π ϵabcI c 2 ˜f (1) 01 Λ2 k1 · k2(k0 1 + k0

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    These contributions for the πN → πN amplitudes are significantly simplified

    + 2˜g(1) 01 Λ2 M 2 π(k0 1 + k0 2) − l(1) 01 2m0 (⃗k1 + ⃗k2) · (⃗ p1 + ⃗ p2) − l(2) 01 Λ2 (ki 1 + ki 2)(ki 1k0 2 + k0 1ki 2) + l(3) 01 Λ2 (k0 1⃗k2 2 + k0 2⃗k2 1 − ⃗k1 · ⃗k2(k0 1 + k0 2)) ! (F42) iT ba CT (J = 1, I= 0) = ϵijkSk F 2 π −2 f (1) 10 Λ2 (k0 1 + k0 2)ki 1kj 2 − l(1) 1...

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    The intermediate state ∆ 5 of spin 5/2 is necessary for the correct general Nc results, and is obviously absent at Nc = 3 as shown in the corresponding entries

    Reductions of spin-flavor operators in the πN → πN amplitudes This Appendix provides the reductions of 2-, 3-, and 4-body spin-flavor tensor operators for matrix elements between nucleon states, as needed in the calculation of the πN → πN amplitudes. The intermediate state ∆ 5...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.