Pion mass and decay constant

C. D. Roberts; P. C. Tandy; P. Maris

REVIEW

Pion mass and decay constant

P. Maris, C. D. Roberts, P. C. Tandy

Nuclear Theory · Submitted 2 Jul 1997 · 17 references

Pith's verdict: A model-independent derivation of the chiral-limit pion constraints from the axial-vector WTI; the central chain holds, with two small overreaches in the peripheral claims.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 1 major objection · 6 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDnucl-th/9707003 (version 1)

Pith IDRJ5TATE3, Pith's permanent record of this paper and its review

CitationC. D. Roberts, P. C. Tandy, P. Maris. "Pion mass and decay constant." arXiv:nucl-th/9707003 (1997). https://pith.science/paper/RJ5TATE3

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Short version: this is the paper that pins down the chiral-limit consequences of the axial-vector Ward-Takahashi identity for the pion, and it does so without assuming a form for the quark-antiquark scattering kernel. The central results are correct and worth having on the shelf. What is new: Eqs. (10)-(13) tie the pion Bethe-Salpeter amplitude to the quark self-energy — E_pi fixed by B(k^2), and the subleading F_pi, G_pi, H_pi tied to A and A'. The identification r_A = f_pi (25) via the axial-vector vacuum polarisation is clean, and the residue r_P = condensate/f_pi (29) plus the exact form of GMOR (31) follow from it. The observation that H_pi is nonzero even in Feynman gauge, contra Delbourgo-Scadron, is a fair correction, and the authors are honest that (31) is qualitatively equivalent to Miransky/Munczek [16]. Soft spots, in proportion: the pole decompositions (7) and (20) are argued rather than proved, but the unit residue is the standard LSZ statement following from the canonical normalisation [3], so I do not lose sleep over it. The genuinely under-argued step is Eq. (26), N_pi = f_pi: the text says only 'using (10)-(13) it is clear', and the worked demonstration is given for rainbow-ladder alone. It is true, but the general case is asserted. Second, the remark that A identically 1 forces F_pi = G_pi = H_pi = 0 does not follow from (11)-(13) alone — those are satisfied by nonzero G_pi, H_pi with compensating G_R, H_R, so an extra dynamical argument is needed. Both points are peripheral to the main chain, but they sit inside the abstract's claims, so a referee should ask for a line or two. The 40 percent eigenvalue reduction and 20 percent m_pi underestimate are explicitly flagged as work in progress, so no complaint there. The citation pattern leans on the authors' own framework papers, but those are the right references for the DSE machinery and the AV-WTI conventions. Who it is for: DSE practitioners and anyone building truncations for pion and kaon observables — it functions as a correctness checklist. Not for a general audience; no figures, dense, and you need the framework papers to follow it. Recommendation: send it out. The derivation is checkable line by line and the results are load-bearing for the field. Ask for the two clarifications above; neither threatens the verdict.

Referee report

Significance

The results are model-independent constraints on the pion vertex, obtained without truncating the scattering kernel, and they tie the pion's internal structure directly to the quark self-energy. Eqs. (10)-(13), (25), (29) and (31) are parameter-free and falsifiable: any truncation that fails to preserve Eq. (14) is predicted to miss them, and the paper states this as a concrete diagnostic. The identification of the axial-vector pole residue with fπ and of the pseudoscalar residue with the condensate gives a clean, exact form of the GMOR relation valid beyond leading order in the current-quark mass. The paper is a compact letter, but the derivation is explicit at each step of the central chain, and the results have been widely used as consistency conditions in subsequent Dyson-Schwinger studies.

Major objections (1)

[Discussion following Eq. (21), leading to Eq. (26)] The equivalence Nπ = fπ in the chiral limit is advertised in the abstract and is one of the paper's stated results, but the general-case derivation is compressed to the single sentence 'Using (10)-(13) it is clear that in the chiral limit Nπ = fπ.' The difficulty is specific: the normalisation integral (21) contains a third term involving ∂K/∂Pμ, and Eqs. (10)-(13) say nothing about K. The text demonstrates the cancellation only for rainbow-ladder, where ∂Pμ K ≡ 0 makes that term vanish identically, and the integration-by-parts method is attributed to Ref. [2] in that same restricted setting. For a general kernel the third term is nonzero and its contribution to (21) must be shown to combine with the first two so that the result matches Eq. (22) with rA = fπ. As written, a reader cannot verify the general claim without reconstructing the argument. Please add the few lines of algebra that treat the ∂K/∂Pμ term, or state explicitly that the general proof follows Ref. [2] and indicate how the kernel-derivative term is handled there. This is a local addition and does not affect the central chain (10)-(13), (25), (29), (31).

Minor objections (6)

[Paragraph following Eq. (26)] The claim that any kernel yielding A(k²) ≡ 1 must also yield Fπ ≡ Gπ ≡ Hπ ≡ 0 does not follow from Eqs. (11)-(13) alone: Eq. (11) reads FR + 2 rA Fπ = A(k²) and Eq. (12) reads GR + 2 rA Gπ = 2A′(k²), both of which are satisfied by nonzero Fπ, Gπ, Hπ with compensating regular parts FR, GR, HR. An additional dynamical input is required, for example that the regular part of the axial-vector vertex is saturated by its tree-level value when A ≡ 1. Please supply that argument or soften the claim to a statement about the models considered.
[Abstract, first sentence] The abstract states that Γπ 'necessarily involves a non-negligible γ5γ·P term'. By the paper's own Eq. (11) and the A ≡ 1 remark, Fπ vanishes identically when A ≡ 1, so 'necessarily' holds only when A ≢ 1; and 'non-negligible' is a quantitative statement that rests on the numerical study described as underway rather than on anything derived here. Consider qualifying the abstract to 'when A ≢ 1' and reserving the quantitative adjective for the numerical work.
[Penultimate paragraph, the 40% and 20% figures] The statement that omitting Fπ, Gπ and Hπ reduces the Bethe-Salpeter eigenvalue by about 40%, corresponding to a 20% underestimate of mπ, is presented as a result but no calculation, model specification, or reference is given; the accompanying sentence says only that a detailed numerical study is underway. Either provide the model details and the numbers behind these figures, or label them explicitly as preliminary expectations.
[Eqs. (6), (7), (17) and (28)] There are small notational inconsistencies. In Eq. (6) the regular remainder is written with a superscript j while in Eq. (7) the same object appears with a superscript i; the flavour index should be uniform. In Eq. (17) the left-hand side is written iΓj5(k;P) although Γj5 was defined without the factor of i in Eq. (15), and Eq. (28) similarly carries a stray i on the left-hand side. Please make the factors of i consistent with the definitions.
[Eq. (29) and the definition of the condensate] The chiral-limit condensate is defined as ⟨q̄q⟩⁰μ ≡ Z4(μ,Λ) Nc ∫ trD[S(q)]. Since the conventional definition carries an overall minus sign and a specific trace convention, please state the convention explicitly so that the sign of Eq. (31) can be checked against standard normalisations.
[Eqs. (7), (17) and (20)] The derivation assumes that the remainders in the pole decompositions are analytic in P² at P² = -mπ². This is the standard LSZ structure and is not in doubt, but a single sentence stating the assumption would make the term-by-term matching that produces Eqs. (10)-(13) and (25) self-contained for a reader outside the Dyson-Schwinger literature.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Chiral symmetry alone fixes the pion's internal structure, tying its Bethe-Salpeter amplitude to the quark propagator, and makes the pion decay constant equal to the amplitude's normalisation.

The reading

The paper asks how a bound state of two very heavy constituent quarks can come out nearly massless, and answers that the pion's internal structure is not an independent input but a consequence of how the quark dresses itself. Working with the Dyson-Schwinger equations of QCD and making no assumption about the form of the quark-antiquark scattering kernel, the authors show that the axial-vector Ward-Takahashi identity forces the chiral-limit pion Bethe-Salpeter amplitude to be tied term by term to the quark propagator's two scalar functions, $A(k^2)$ and $B(k^2)$. From the same identity they obtain exact expressions for the pion decay constant and mass, prove that the decay constant equals the Bethe-Salpeter normalisation constant in the chiral limit, and recover the Gell-Mann-Oakes-Renner relation as a corollary rather than an assumption. The practical stake is a rule for model builders: any truncation that breaks the axial-vector Ward-Takahashi identity will misdescribe the pion, and the paper points to the rainbow-ladder truncation as the simplest scheme that keeps it.

What carries the argument

The load-bearing object is the axial-vector Ward-Takahashi identity, the statement that the divergence of the isovector axial-vector vertex is fixed by the inverse quark propagator (plus a current-quark mass term away from the chiral limit). Because the identity is exact and involves the same scattering kernel that appears in the quark Dyson-Schwinger equation and in the Bethe-Salpeter equation, it acts as a bridge: whatever the kernel is, the pion Bethe-Salpeter amplitude and the quark propagator must satisfy the same relation. The authors exploit this by writing the axial-vector vertex as a pion pole of residue $r_A$ plus a part regular at $P^2=-m_��^2$, substituting into the identity, and equating coefficients of each Dirac structure. The unit residue of the pole in the scattering amplitude follows from the canonical Bethe-Salpeter normalisation, which is what lets the pole-matching go through without further assumptions.

Extended reading notes

Core claim

The central claim is that the axial-vector Ward-Takahashi identity, combined with a single-pion-pole decomposition of the axial-vector and pseudoscalar vertices, yields the chiral-limit relations $r_A E_��(k;0)=B(k^2)$, $F_R(k;0)+2r_A F_��(k;0)=A(k^2)$, $G_R(k;0)+2r_A G_��(k;0)=2A'(k^2)$ and $H_R(k;0)+2r_A H_��(k;0)=0$, where $r_A$ is the residue of the pion pole. Matching the same pole in the axial-vector vacuum polarisation identifies that residue with the pion decay constant, $r_A=f_��$, and matching it in the pseudoscalar vertex identifies the pseudoscalar residue as $r_P^0 = -��0_��/f_��$ in the chiral limit. Combining the two gives the exact, kernel-independent statement $f_��^2 m_��^2 = 2m(��)��^{��}_{��}$, of which the familiar Gell-Mann-Oakes-Renner relation is the leading term in the current-quark mass. A corollary the authors stress is that the Bethe-Salpeter amplitude must contain a non-negligible $��_5��!·!P$ piece; dropping $F_��$, $G_��$ and $H_��$ in a model calculation reduces the BSE eigenvalue by about 40 percent, which means the pion mass comes out roughly 20 percent too low.

Load-bearing premise

The derivation assumes that both the axial-vector vertex and the quark-antiquark scattering amplitude consist of exactly one pion pole of unit residue plus a remainder that is genuinely regular at the pion mass, so that the two sides of the Ward identity can be matched term by term.

Editorial extensions

If this is right

Any truncation of the quark-antiquark scattering kernel that preserves the axial-vector Ward-Takahashi identity automatically reproduces the exact relation $f_��^2 m_��^2 = 2m(��)��^{��}_{��}$, with the Gell-Mann-Oakes-Renner relation following as its leading term.
In the chiral limit the pion decay constant and the Bethe-Salpeter normalisation constant coincide, so the two ways of fixing the amplitude's normalisation are not independent.
The pion Bethe-Salpeter amplitude must retain the $��_5��!·!P$ Dirac structure; omitting it in a model calculation lowers the bound-state eigenvalue by about 40 percent and the pion mass by about 20 percent.
The rainbow-ladder truncation, in which the kernel is a single dressed-gluon exchange and the quark-gluon vertex is bare, is the simplest scheme consistent with the identity and is therefore a defensible starting point for quantitative pion calculations.
A kernel that produces $A(k^2) \equiv 1$ must also produce $F_�� \equiv G_�� \equiv H_�� \equiv 0$ if it preserves the identity, so the common practice of setting $A \equiv 1$ and keeping only the $��_5$ term is internally inconsistent.

Reading between the lines

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

The paper's logic suggests a diagnostic for existing model calculations: any published pion calculation that reports a fine-tuned or mismatched pion mass can be checked by asking whether its kernel satisfies the axial-vector Ward-Takahashi identity, which is a cheaper test than re-solving the model.
Because the relations are derived without assuming a kernel, they could be used as constraints in lattice-informed reconstructions of the quark propagator and the pion vertex, where the identity must hold as a matter of QCD and can serve as a consistency check on extrapolations to the chiral limit.
The same pole-matching technique should extend to the flavour-singlet channel and to the $��'$, where an additional gluon-related singularity complicates the pole structure; the paper does not attempt this, but the method as stated does not obviously forbid it.
The claim that $N_�� = f_��$ fails in earlier model studies unless $A \equiv 1$ implies that some published values of $f_��$ from those models carry a systematic error whose size could be estimated by reinstating $F_��$, $G_��$ and $H_��$.

Original abstract

Independent of assumptions about the form of the quark-antiquark scattering kernel we derive the explicit relation between the pion Bethe-Salpeter amplitude, Gamma_pi, and the quark propagator in the chiral limit; Gamma_pi necessarily involves a non-negligible gamma_5 gamma.P term (P is the pion four-momentum). We also obtain exact expressions for the pion decay constant, f_pi, and mass, both of which depend on Gamma_pi; and demonstrate the equivalence between f_pi and the pion Bethe-Salpeter normalisation constant in the chiral limit. We stress the importance of preserving the axial-vector Ward-Takahashi identity in any study of the pion itself, and in any study whose goal is a unified understanding of the properties of the pion and other hadronic bound states.

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, 2026-09-23: 1 request, 3,148 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Independent of assumptions about the form of the quark-antiquark scattering kernel we derive the explicit relation between the pion Bethe-Salpeter amplitude, Gamma_pi, and the quark propagator in the chiral limit; Gamma_pi necessarily involves a non-negligible gamma_5 gamma.P term (P is the pion four-momentum). We also obtain exact expressions for the pion decay constant, f_pi, and mass, both of which depend on Gamma_pi; and demonstrate the equivalence between f_pi and the pion Bethe-Salpeter normalisation constant in the chiral limit. We stress the importance of preserving the axial-vector Ward-Takahashi identity in any study of the pion itself, and in any study whose goal is a unified understanding of the properties of the pion and other hadronic bound states.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Independent of assumptions about the form of the quark-antiquark scattering kernel we derive the explicit relation between the pion Bethe-Salpeter amplitude, Gamma_pi, and the quark propagator in the chiral limit; Gamma_pi necessarily involves a non-negligible gamma_5 gamma.P term (P is the pion four-momentum).	result 0.91	0.94	0.51	asserted 1.00
s1	We also obtain exact expressions for the pion decay constant, f_pi, and mass, both of which depend on Gamma_pi; and demonstrate the equivalence between f_pi and the pion Bethe-Salpeter normalisation constant in the chiral limit.	result 0.89	0.95	0.26	asserted 1.00
s2	We stress the importance of preserving the axial-vector Ward-Takahashi identity in any study of the pion itself, and in any study whose goal is a unified understanding of the properties of the pion and other hadronic bound states.	interpretation 0.84	0.21	0.18	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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

References

Works cited (17)

[16]

V. A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories (World Scientiﬁc, Singapore, 1993), pp. 202-207; H. J. Munc zek, Phys. Rev. D 52 (1995) 4736

View on Pith 1995
[3]

C. H. Llewellyn-Smith, Ann. Phys. (NY) 53 (1969) 521

View on Pith 1969
[2]

R. Jackiw and K. Johnson, Phys. Rev. D 8 (1973) 2386

View on Pith 1973
[1]

C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477

View on Pith 1994
[4]

R. Delbourgo and M. D. Scadron, J. Phys. G 5 (1979) 1621

View on Pith 1979
[5]

C. J. Burden, et al ., Phys. Rev. C 55 (1997) 2649

View on Pith 1997
[6]

P. Pascual and R. Tarrach, “QCD: Renormalisation for the Practitioner” (Springer-Verlag, Berlin, 1984)

View on Pith 1984
[7]

M. R. Frank and C. D. Roberts, Phys. Rev. C 53 (1996) 390

View on Pith 1996
[8]

F. T. Hawes, T. Sizer and A. G. Williams, Phys. Rev. D 55 (1997) 3866

View on Pith 1997
[9]

M. R. Pennington, “Calculating hadronic properties in s trong QCD”, archive: hep-ph/9611242

View on Pith Pith review arXiv
[10]

M. Baker, J. S. Ball and F. Zachariasen, Nucl. Phys. B 186 (1981) 531; ibid, 560; D. Atkinson, et al ., Nuovo Cimento A 77 (1983) 197; N. Brown and M. R. Pennington, Phys. Rev. D 39 (1989) 2723

View on Pith 1981
[11]

Z. Dong, H. J. Munczek and C. D. Roberts, Phys. Lett. B 333 (1994) 536; A. Bashir and M. R. Pennington, Phys. Rev. D 50 (1994) 7679

View on Pith 1994
[12]

A. G. Williams, G. Krein and C. D. Roberts, Ann. Phys. 210 (1991) 464; H. J. Munczek and P. Jain, Phys. Rev. D 46 (1992) 438

View on Pith 1991
[13]

G. Preparata and W. I. Weisberger, Phys. Rev. D 175 (1968) 1965

View on Pith 1968
[14]

D. W. McKay, H. J. Munczek and B.-L. Young, Phys. Rev. D 37 (1988) 195

View on Pith 1988
[15]

C. D. Roberts, Nucl. Phys. A 605 (1996) 475. 11

View on Pith 1996
[17]

A. Bender, C. D. Roberts and L. v. Smekal, Phys. Lett. B 380 (1996) 7. 12

View on Pith 1996

Forward citations

Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Hadronic resonances from the 4PI effective action of QCD
hep-ph 2026-10 conditional novelty 7.0 of 10

Meson decay widths are extracted from Bethe-Salpeter kernels built from the 4PI effective action of QCD, after the rainbow-ladder kernel is tuned to the experimental masses.
Spectra of light and heavy mesons with $J \le 5$ in a relativistic Bethe-Salpeter approach
hep-ph 2025-10 conditional novelty 6.0 of 10

First J=4,5 meson Bethe-Salpeter tensor bases plus exploratory rainbow-ladder spectra and Regge trajectories from light through bottom quarkonia.
Distribution amplitudes of heavy-light pseudo-scalar and vector mesons from Dyson-Schwinger equations framework
hep-ph 2025-01 conditional novelty 6.0 of 10

First DSE predictions for B*, B*_s, and B*_c distribution amplitudes show peaks near the Euclidean constituent quark mass ratio and a universal spin ordering.
Chiral symmetry breaking for large "Nf" and the properties of the "rho" meson near the conformal window
hep-ph 2026-09 accept novelty 5.0 of 10

The rho meson's mass is nearly constant as flavor number increases, but its size grows and it ultimately dissociates due to loss of confinement, not by crossing a threshold.
$D^* D \pi$ and $B^* B \pi$ couplings from Dyson-Schwinger equations framework
hep-ph 2024-11 conditional novelty 5.0 of 10

A Dyson-Schwinger/Bethe-Salpeter calculation predicts g_D*Dpi = 16.22 and g_B*Bpi = 40.09, with static couplings consistent with lattice results.

Show 3 more citing papers

The quark-gluon vertex in Landau gauge bound-state studies
hep-ph 2014-04 conditional novelty 5.0 of 10

A fitted, flavour-dependent model of the quark-gluon vertex indicates that anomalous chromomagnetic moment components dominate over the conventional scalar component in the quark self-energy.
pi- and K-meson Bethe-Salpeter Amplitudes
nucl-th 1997-08 conditional novelty 5.0 of 10

The axial-vector Ward-Takahashi identity forces pseudoscalar meson Bethe-Salpeter amplitudes to contain gamma5 gamma.P and gamma5 gamma.k k.P terms, and ties the axial-vector and pseudoscalar pole residues so that the...
Infrared Properties of QCD from Dyson-Schwinger equations
hep-ph 2006-05 accept novelty 3.0 of 10

Landau-gauge Dyson-Schwinger studies indicate an infrared-finite running coupling, a vanishing gluon propagator, and dynamically generated quark mass, though lattice and continuum results still differ at low momentum.

How later papers used it

97 later papers cite it

Evidence: substantial. 94 of the citing papers share no author with it, and 18 of those carry a confident label from Pith's citation classifier.

Pith read 18 of the 163 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 6 papers with this label

Took explicit parameterisations for vertex and propagator from this paper, including the pion pole in the axial vector vertex, to deduce the form of the dominant amplitude for the pi0.

Hadronic light-by-light scattering in the muon g-2: a Dyson-Schwinger equation approach (2010) · 0.88 · paraphrased: the licence does not allow quoting
Approximated the full pion Bethe-Salpeter wave function in the quark-SDE and the kernel of the BSE by the leading amplitude in the chiral limit given by this paper.

Beyond the rainbow: effects from pion back-coupling (2008) · 0.90 · paraphrased: the licence does not allow quoting

How the literature uses it

76 arXiv papers cite this in 124 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 98 sentences without a confident label

Contests it 1 paper

Nonetheless, Ref. [ref] utilizes a one-covariant model for the corresponding BSAs, which, while illustrative, compromises Poincaré invariance by truncating the structure of such [ref] .

$D^* D \pi$ and $B^* B \pi$ couplings from Dyson-Schwinger equations framework · INTRODUCTION · contest 0.72

Builds on it 2 papers

Equation [ref] can be extended to the above mentioned generalized GMOR relation [ref] [equation] where the residue of the pion mass pole in the pseudoscalar vertex of the AWW model is given by [ref] [equation] Equation [ref] is valid for all current-quark masses and pseudoscalar states.

Poincare covariant pseudoscalar and scalar meson spectroscopy in Wigner-Weyl phase · Results and discussion · 0.92
Together with the isovector ground state pseudoscalars [ref] , first excited pseudoscalar states [ref] and ground state vectors [ref] , this is a new case among the few where the Euclidian Bethe Salpeter equation has been solved so far.

Microscopic quark study of the eta and eta' masses · Conclusion · 0.89

Uses its method or tool 5 papers

In order to get hold of, at least, one of the desired solutions, we exploit the relationship between the full quark propagator $S(p)$ — obtainable as solution to the quark Dyson–Schwinger equation — and the Bethe–Salpeter amplitude $Φ(p,0)$ of (flavour-nonsinglet) pseudoscalar mesons arising from the (renormalized) axial-vector Ward–Takahashi identity of QCD in the chiral limit [ref] : the sought relationship (in its Euclidean-space formulation indicated by underlined quantities) reads [ref] [equation]

Bethe-Salpeter-Motivated Modelling of Pseudo-Goldstone Pseudoscalar Mesons · Momentum Dependence of our Bethe–Salpeter Interaction Kernel by Utilizing Inversion · 0.92
Taking explicit parameterisations for vertex and propagator (see [ref] ) that include the pion pole in the axial vector vertex, the following form of the dominant amplitude for the $π^0$ can be deduced:

Hadronic light-by-light scattering in the muon g-2: a Dyson-Schwinger equation approach · Off-shell prescription · 0.88
We study the pion back reaction on the quarks in the Green's function approach to Landau gauge QCD using Schwinger-Dyson and Bethe-Salpeter equations (SDE/BSE) [ref] .

Hadronic unquenching effects in the quark propagator · Introduction · 0.94

Cites it as support 5 papers

According to [ref] , the condensates have local support restricted to the hadrons' interior.

Post-Newtonian Feasibility of the Closed String Massless Sector · Introduction · 0.97
And although these structures are, in principle, subdominant, its presence is required to satisfy crucial symmetries [ref] , and its effects in different quantities can manifest themselves in a noticeable way.

QCD anomalies in electromagnetic processes: A solution to the $\gamma\to3\pi$ puzzle · introduction · 0.96
The strong support of the LFWFs at infrared $k_T^2$ originates from the strength of the covariant Bethe-Salpeter wave functions at low $|k_T|$, which is closely connected to DCSB, as illustrated model-independently in Ref. [ref] .

Spatial and Momentum Imaging of the Pion and Kaon · PION AND KAON LIGHT FRONT WAVE FUNCTIONS · 0.99

Cites it as background 11 papers

A prominent example is the axial-vector Ward-Takahashi identity (AVWTI) [ref] which relates the two-quark kernel $K^(2)$ to the kernel of the quark DSE.

Hadron properties from QCD bound-state equations · Bound-state equations · 0.89
In the context of BSE analyses [ref] , the dressed axial-vector vertex is represented by the structure $Γ^fg_5μ(K;P)$, which is understood to connect an incoming quark of flavor $g$ and momentum $K_- = K - (1-η)P$ to an outgoing quark of flavor $f$ and momentum $K_+ = K + ηP$; here $P$ and $K$ represent the total and relative momentum of the quark pair, and $η$ is a dimensionless parameter upon which calculations cannot depend.

A Euclidean bridge to the relativistic constituent quark model · Gluon Dressing Effect · 0.93
Recent progress in lattice QCD [ref] , Dyson–Schwinger method [ref] , and effective field theories [ref] , together with high-energy scattering experiments [ref] , has enabled increasingly precise investigations of pion and kaon structure.

Feasibility Study of Pion and Kaon Structure via the Sullivan Process at EicC · Introduction · 0.83

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [16]

V. A. Miransky, Dynamical Symmetry Breaking in Quantum Field Theories (World Scientiﬁc, Singapore, 1993), pp. 202-207; H. J. Munc zek, Phys. Rev. D 52 (1995) 4736

View on Pith 1995

[2] [3]

C. H. Llewellyn-Smith, Ann. Phys. (NY) 53 (1969) 521

View on Pith 1969

[3] [2]

R. Jackiw and K. Johnson, Phys. Rev. D 8 (1973) 2386

View on Pith 1973

[4] [1]

C. D. Roberts and A. G. Williams, Prog. Part. Nucl. Phys. 33 (1994) 477

View on Pith 1994

[5] [4]

R. Delbourgo and M. D. Scadron, J. Phys. G 5 (1979) 1621

View on Pith 1979

[6] [5]

C. J. Burden, et al ., Phys. Rev. C 55 (1997) 2649

View on Pith 1997

[7] [6]

P. Pascual and R. Tarrach, “QCD: Renormalisation for the Practitioner” (Springer-Verlag, Berlin, 1984)

View on Pith 1984

[8] [7]

M. R. Frank and C. D. Roberts, Phys. Rev. C 53 (1996) 390

View on Pith 1996

[9] [8]

F. T. Hawes, T. Sizer and A. G. Williams, Phys. Rev. D 55 (1997) 3866

View on Pith 1997

[10] [9]

M. R. Pennington, “Calculating hadronic properties in s trong QCD”, archive: hep-ph/9611242

View on Pith Pith review arXiv

[11] [10]

M. Baker, J. S. Ball and F. Zachariasen, Nucl. Phys. B 186 (1981) 531; ibid, 560; D. Atkinson, et al ., Nuovo Cimento A 77 (1983) 197; N. Brown and M. R. Pennington, Phys. Rev. D 39 (1989) 2723

View on Pith 1981

[12] [11]

Z. Dong, H. J. Munczek and C. D. Roberts, Phys. Lett. B 333 (1994) 536; A. Bashir and M. R. Pennington, Phys. Rev. D 50 (1994) 7679

View on Pith 1994

[13] [12]

A. G. Williams, G. Krein and C. D. Roberts, Ann. Phys. 210 (1991) 464; H. J. Munczek and P. Jain, Phys. Rev. D 46 (1992) 438

View on Pith 1991

[14] [13]

G. Preparata and W. I. Weisberger, Phys. Rev. D 175 (1968) 1965

View on Pith 1968

[15] [14]

D. W. McKay, H. J. Munczek and B.-L. Young, Phys. Rev. D 37 (1988) 195

View on Pith 1988

[16] [15]

C. D. Roberts, Nucl. Phys. A 605 (1996) 475. 11

View on Pith 1996

[17] [17]

A. Bender, C. D. Roberts and L. v. Smekal, Phys. Lett. B 380 (1996) 7. 12

View on Pith 1996