On the Foundations of Chiral Perturbation Theory

H. Leutwyler

REVIEW

On the Foundations of Chiral Perturbation Theory

H. Leutwyler

High Energy Physics - Phenomenology · Submitted 11 Nov 1993 · 16 references

Pith's verdict: A real foundational paper: it converts the symmetry of the chiral Lagrangian from an assumption into a theorem, conditional on pion pole dominance, and the proof holds up.

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arXiv IDhep-ph/9311274 (version 1)

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CitationH. Leutwyler. "On the Foundations of Chiral Perturbation Theory." arXiv:hep-ph/9311274 (1993). https://pith.science/paper/CZMMN23W

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Editor's note

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This is the paper that closes the gap everyone in the field had been stepping over. Weinberg, Coleman–Wess–Zumino, Gasser–Leutwyler all built the chiral Lagrangian assuming it inherits the symmetry of the underlying theory; Leutwyler shows that this follows from the Ward identities for Lorentz-invariant, anomaly-free theories, to all orders in the derivative expansion. That is the whole content, and it is worth having in print. The abstract claim [abstract:s1] is exactly what gets proved, no more. The author is also upfront that nonrelativistic theories are excluded [abstract:s2]. The leading-order step is the cleanest part: reducing the two differential equations (6.5)–(6.6) to the Killing-vector and metric conditions (6.8) is standard coset geometry done carefully, and the elimination argument is checkable line by line. The induction in §8 is structurally sound — loops from lower orders are gauge invariant by hypothesis, so only tree graphs of S^{(n+1)} matter at order p^{n+1} — and appendices A, B, D supply what they claim. The anomaly and spurion sections are short but correctly scoped: the Wess–Zumino term absorbs the noninvariance, the rest behaves as in the anomaly-free case. Soft spots, in order of size. First, pion pole dominance in §3 is a physical input about the spectrum, not a consequence of the Ward identities; the theorem is conditional on it. The paper says so plainly, so this is a scope statement rather than a hidden gap, but it is the load-bearing assumption and a reader should know it. Second, appendix C is compressed: the general solution (C.14) of the representation property is asserted to be a coboundary removable by (C.17), and the gauge-fixing plus Haar-average argument is gestured at rather than worked through. I believe it, but a referee should ask for two more pages there. Third, the claim that the theorem fails nonrelativistically rests on reference [8], listed as in preparation; that is a promissory note, not a proof, and should be flagged as such. None of this touches the main result. This is for anyone who uses chiral perturbation theory and has wondered whether the symmetry assumption is earned — which is most of low-energy QCD. Send it to a serious referee; the appendix C question is exactly the kind of thing peer review should pin down.

Referee report

Significance

If correct, the paper closes a genuine gap in the logic of an extensively used framework: the symmetry of the effective Lagrangian becomes a consequence of the Ward identities rather than an independent postulate, and the geometric determination of the leading-order Lagrangian from the decay constants alone follows. The result is parameter-free in the sense that no new couplings are introduced by the theorem itself; the only quantities entering L^(2) are the decay constants, whose number is fixed by the representation theory of H. The treatment of anomalies in §9 and of spurions in §10 is a useful structural statement, and the identification of the d=3 Chern-Simons term as the sole obstruction to gauge invariance of the Lagrangian (Eq. (5.7)) is a sharp, checkable exception. The paper is explicit that it does not cover nonrelativistic theories (abstract, [abstract:s2]) and that the companion treatment is in preparation [8]; that exclusion is acknowledged and does not weaken the relativistic claim.

Major objections (1)

[Abstract and §5, assertion D] The abstract states the result for 'anomaly free, Lorentz invariant theories' without qualification, whereas the theorem as formulated in §5 is explicitly four-dimensional: assertion D restricts to d=4, and the accompanying text and Eq. (5.7) record that in three dimensions the Lagrangian is gauge invariant only up to a Chern-Simons term. The abstract's phrasing is therefore broader than what is proved. A second, related point is that the abstract presents the result as a characterisation of the theories, while §3 makes clear that the bridge from the Ward identities to the path integral rests on the pion-pole-dominance hypotheses (Goldstone bosons as the only massless asymptotic states, pole dominance at low energy, Taylor expandability of the vertices). §11 is candid about this ('does not involve assumptions beyond the pion pole dominance hypothesis'), but the abstract is not. Since the abstract is the claim most readers will carry away, I would ask that it name both the four-dimensional restriction and the pole-dominance precondition. The derivation itself is not in question here; this is a matter of the claim's stated domain.

Minor objections (9)

[§4, power-counting formula preceding Eq. (4.5)] The displayed counting rule O_γ = Σ_{v∈γ}(O_v − 2) + (d − 2)L appears to omit the overall +2. As printed it assigns order p^0 to a tree graph built entirely from L^(2) vertices, which contradicts the sentence immediately following it ('the leading contribution stems from the tree graphs of L^(2)'). The standard Weinberg count is 2 + Σ(O_v − 2) + (d − 2)L. This may be a typesetting slip, but it should be corrected since the induction in §8 relies on the rule.
[§2, Eq. (2.2) versus §4, Eq. (4.5)] Eq. (2.2) defines e^{iΓ{f}} as a sum over n including n = 0, so Γ as written contains the disconnected vacuum-to-vacuum amplitude, whereas Eq. (4.5) normalises the path integral by Z^{-1}. It would help to state explicitly at Eq. (2.2) that the disconnected piece is to be divided out, so that the two definitions are manifestly the same object.
[Appendix C, Eqs. (C.14)–(C.17)] The step from the general solution (C.14) of the representation property to the statement that it is removed by the change of variables (C.17) is the least transparent point in the proof of assertion C. The argument is correct, but a short paragraph spelling out why (C.14) is a coboundary in the relevant sense, and why the change of variables (C.17) is a legitimate point transformation within the class allowed in §11, would make the induction self-contained for a reader who does not already know the cohomological triviality argument.
[Appendix D] The author describes the derivation as 'rather clumsy', and the reader is asked to follow a long sequence of decompositions (D.5)–(D.14) in d=3 and (D.15)–(D.22) in d=4 before reaching the conclusion that the Chern-Simons term is the only obstruction. A brief roadmap at the start of D.1, stating which steps establish existence of the potential, which establish its gauge behaviour, and which isolate the Chern-Simons piece, would substantially reduce the burden on the reader without changing the content.
[Appendix E, final paragraph] The parenthetical remark that one must 'enlarge the ball slightly, cutting out a small shell from V+ and V−' is the one place where the support argument is asserted rather than shown. Since the locality of the potential is used in appendix D, a sentence confirming that the enlarged region can be taken arbitrarily close to V, and that this suffices for the application, would close the gap.
[Notation, §§2 and 6] The symbol f carries three distinct meanings: the external field matrix f_μ = Σ_i t_i f^i_μ, its components f^i_μ, and the structure constants f^k_{ij}. The collision is manageable in context but recurs in Eqs. (2.4)–(2.6) and (6.6), and a notational remark at first use would help.
[§9] The statement that the Wess-Zumino term contributes a non-invariant piece to the generating functional only through tree graphs of O(p^4), loops being gauge invariant, is attributed to [16]. A one-line indication of the mechanism (why the loop contributions close gauge-invariantly) would let the reader follow the argument without consulting four separate references.
[§10] The bookkeeping that treats m_α as O(p^2) is introduced without comment on its limits. It would be worth stating explicitly that this counting fails if some spurion is of order p, in which case the derivative expansion must be reorganised and the pole-dominance hypothesis itself may fail.
[References and typography] Reference [11] is printed as 'Phys. Lev. Lett.' and should read 'Phys. Rev. Lett.'. There are also residual typographical errors in the running text, e.g. 'metic' for 'metric' in §7 and 'repectively' in appendix A.

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The paper

The paper claims

Leutwyler shows that, for anomaly-free Lorentz-invariant theories, the Ward identities force the low-energy effective Lagrangian for Goldstone bosons to be gauge invariant to all orders, up to field redefinitions and total derivatives.

The reading

The paper asks whether the symmetry of the effective Lagrangian used in chiral perturbation theory is an independent postulate or a consequence of the underlying theory. Chiral perturbation theory describes the low-energy interactions of the Goldstone bosons, the pions, that appear when a continuous symmetry is spontaneously broken. Leutwyler argues that, for anomaly-free Lorentz-invariant theories in four dimensions, the Ward identities obeyed by the currents are strong enough to force the effective Lagrangian to be gauge invariant, order by order in the derivative expansion, once one allows field redefinitions and total derivatives. If correct, this closes a conceptual gap: the standard symmetric chiral Lagrangian is not merely a heuristic guess but a controlled consequence of the symmetries of the underlying theory.

What carries the argument

The load-bearing device is the generating functional $Γ{f}$ of current Green functions, whose Ward identities are recast as the statement that $Γ$ is invariant under local gauge transformations of the external fields, $Γ{T(g)f}=Γ{f}$. The bridge from the underlying theory to the effective theory is pion pole dominance: the Goldstone bosons are the only massless asymptotic states, and their poles and cuts dominate the low-energy Green functions. This yields the path-integral representation $e^{iΓ{f}} = Z^{-1}∫[dπ] e^{iS_{eff}}$, with the effective action expanded in derivatives. At leading order, the most general Lorentz-invariant Lagrangian is written with a metric $g_{ab}(π)$, couplings $h_{ai}(π)$ to the external fields, and $k_{ik}(π)$; consistency of the pion equation of motion with the Ward identity forces $k_{ik}=g^{ab}h_{ai}h_{bk}$ and forces $h_{ai}$ to be Killing vectors of $g_{ab}$. At higher orders, the proof shows that the transformation law can be brought to canonical form by a field redefinition and that the remaining gauge-field action is gauge invariant except for a possible Chern-Simons term in three dimensions.

Extended reading notes

Core claim

The central claim is an invariance theorem for the effective field theory of Goldstone bosons. In a Lorentz-invariant theory in four dimensions whose Ward identities are anomaly-free, the generating functional of current Green functions is invariant under local gauge transformations of the external fields that couple to the currents. Leutwyler proves that this invariance can be transferred to the effective Lagrangian: after a suitable change of pion field variables and after dropping total derivatives, the Lagrangian can be written in a form that is invariant under the simultaneous gauge transformation of the external fields and the pion fields, to all orders of the derivative expansion. The proof treats the leading order geometrically, identifies the coupling functions as Killing vectors of the coset $G/H$, and extends the result by induction; in three dimensions a Chern-Simons term is the one allowed exception. Anomalies are handled by a Wess-Zumino term, and approximate symmetries by promoting the symmetry-breaking parameters to spurions.

Load-bearing premise

The argument's physical bridge is the pion pole dominance hypothesis: the Goldstone bosons are the only massless asymptotic states and, at low energies, the current Green functions are dominated by their poles, with vertices that admit a Taylor expansion in momenta.

Editorial extensions

If this is right

The symmetric form of the chiral Lagrangian at each order is justified by the Ward identities, so the standard practice of writing all terms allowed by chiral symmetry is not an extra assumption.
Anomalies are isolated in a Wess-Zumino term; the rest of the effective Lagrangian can be taken gauge invariant, which organizes anomaly calculations order by order.
Explicit symmetry breaking can be treated by spurions that transform contragrediently to the operators that break the symmetry, so the same invariance argument applies to approximate symmetries.
In four dimensions there is no Chern-Simons-like obstruction; in three dimensions the only non-invariant local term is the Chern-Simons term, which involves only external fields.
The theorem explains why nonrelativistic effective theories can fail the conclusion: without Lorentz invariance, the effective Lagrangian may be invariant only up to a total derivative, as in a ferromagnet.

Reading between the lines

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

The result makes the spectral assumption, not the symmetry, the real bottleneck: any Lorentz-invariant theory with extra light states or non-Taylor low-energy vertices would evade the theorem, so the symmetric Lagrangian is a low-energy statement conditional on pion pole dominance.
The same Ward-identity-plus-geometry strategy could classify non-invariant terms in effective theories with spontaneously broken spacetime symmetries, where the Lorentz-invariance step is unavailable.
A finite algebraic check at order $p^6$, enumerating local terms and quotienting by total derivatives and point transformations, would test whether any non-invariant term survives in a given coset; the theorem predicts none in the anomaly-free four-dimensional case.
The three-dimensional Chern-Simons exception suggests that topological terms are the natural loopholes to the theorem, and in four dimensions the Wess-Zumino term plays the analogous controlled role.

Original abstract

The properties of the effective field theory relevant for the low energy structure generated by the Goldstone bosons of a spontaneously broken symmetry are reexamined. It is shown that anomaly free, Lorentz invariant theories are characterized by a gauge invariant effective Lagrangian, to all orders of the low energy expansion. The paper includes a discussion of anomalies and approximate symmetries, but does not cover nonrelativistic effective theories.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.95 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The properties of the effective field theory relevant for the low energy structure generated by the Goldstone bosons of a spontaneously broken symmetry are reexamined. It is shown that anomaly free, Lorentz invariant theories are characterized by a gauge invariant effective Lagrangian, to all orders of the low energy expansion. The paper includes a discussion of anomalies and approximate symmetries, but does not cover nonrelativistic effective theories.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The properties of the effective field theory relevant for the low energy structure generated by the Goldstone bosons of a spontaneously broken symmetry are reexamined.	contribution 0.51	0.64	0.06	unclear 0.65
s1	It is shown that anomaly free, Lorentz invariant theories are characterized by a gauge invariant effective Lagrangian, to all orders of the low energy expansion.	result 0.99	0.88	0.77	asserted 1.00
s2	The paper includes a discussion of anomalies and approximate symmetries, but does not cover nonrelativistic effective theories.	limitation 1.00	0.72	0.05	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

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References

Works cited (16)

[8]

H. Leutwyler, in preparation

View on Pith
[1]

S. Weinberg, Phys. Rev. Lett. 18 (1967) 188, 507; Phys. Rev. 166 (1968) 1568

View on Pith 1967
[2]

S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247

View on Pith 1969
[3]

R. Dashen, Phys. Rev. 183 (1969) 1245; R. Dashen and M. Weinstein, Phys. Rev. 183 (1969) 1291

View on Pith 1969
[4]

L.-F. Li and H. Pagels, Phys. Rev. Lett. 26 (1971) 1204; H. Pagels, Phys. Rep. C 16 (1975) 219

View on Pith 1971
[5]

S. Weinberg, Physica A96 (1979) 327

View on Pith 1979
[6]

J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158 (1984) 142

View on Pith 1984
[7]

For a discussion of the current status, see Proc. Workshop on Eﬀective Field Theories of the Standard Mo del, Dobogok¨ o, Hungary, ed. U. Meissner (World Scientiﬁc, Singapore, 1992 ); H. Leutwyler, Proc. XXVI Int. Conf. on High Energy Physics, D allas, Aug. 1992, ed. J. R. Sanford (AIP Conf. Proc. No. 272, 1993)

View on Pith 1992
[9]

J. Goldstone, Nuovo Cim. 19 (1961) 154

View on Pith 1961
[10]

D.G. Boulware and L. S. Brown, Ann. Phys. (N.Y.) 138 (1982) 392. M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164 (1988) 217

View on Pith 1982
[11]

S. Weinberg, Phys. Lev. Lett. 17 (1966) 616

View on Pith 1966
[12]

J. Honerkamp and K. Meetz, Phys. Rev. D 3 (1971) 1996; I. S. Gerstein, R. Jackiw, B. W. Lee and S. Weinberg, Phys. Rev . D 3 (1971) 2486; G. Ecker and J. Honerkamp, Nucl. Phys. B 35 (1971) 481; B 52 (1973) 211; B 51 62 (1973) 509; T. Appelquist and C. Bernard, Phys. Rev. D 23 (1981) 425. For a recent discussion and references to the literature, se e D. Es...

View on Pith 1971
[13]

W. A. Bardeen, Phys. Rev. 184 (1969) 1848

View on Pith 1969
[14]

J. Wess and B. Zumino, Phys. Lett. 37 B (1971) 95

View on Pith 1971
[15]

J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985) 465

View on Pith 1985
[16]

J. F. Donoghue and D. Wyler, Nucl. Phys. B 316 (1989) 289; J. Bijnens, Nucl. Phys. 367 (1991) 709; R. Akhoury and A. Alfakih, Ann. Phys. 210 (1991) 81; D. Issler, Nonrenormalization of the chiral anomaly in chir al perturbation theory, preprint SLAC-PUB-4943-REV (1990). 52

View on Pith 1989

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How later papers used it

61 later papers cite it

Evidence: substantial. 61 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 99 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

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 3 papers with this label

Gave a systematic construction of EFTs for NG bosons in the gradient expansion based on the strategy outlined in this paper.

Topological interactions of Nambu-Goldstone bosons in quantum many-body systems (2014) · 0.84 · paraphrased: the licence does not allow quoting

How the literature uses it

49 arXiv papers cite this in 77 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 · 46 sentences without a confident label

Uses its method or tool 2 papers

This can be done by the source method [ref] , starting from the LO mass Lagrangian [equation] where $B_0 = -q̅q/F_π^2$ and the quark mass matrix is $M = 1_$ in our case.

QCD with an Infrared Fixed Point -- Pion Sector · Leading order chiral perturbation theory · 0.98
Here and in the companion paper [ref] , we give a systematic construction of EFTs for NG bosons in the gradient expansion, based on the strategy outlined in Ref. [ref] .

Topological interactions of Nambu-Goldstone bosons in quantum many-body systems · Introduction · 0.84

Cites it as support 1 paper

However, we stress that it has been shown that the Greens functions of QCD are indeed exactly reproduced by the ones in chiral perturbation theory [ref] , which leaves us with the LECs, as different theories can in principle lead to the same operator-structure but are characterized by different LECs.

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Cites it as background 24 papers

Its foundations are discussed in Ref. [ref] .

Chiral perturbation theory · Chiral perturbation theory for mesons · 0.94
Chiral perturbation theory (ChPT) [ref] , as an effective field theory of QCD at energies below the chiral symmetry breaking scale $Λ_χ$ of approximately 1 GeV, provides a suitable framework for calculating the model-independent pion-nucleon ($πN$) scattering amplitude.

Pion-nucleon scattering with decuplet contribution in heavy baryon SU(3) chiral perturbation theory · Introduction · 0.98
Moreover, these functions obey symmetry relations, which are known as the chiral Ward identities and which are equivalent to the invariance property of the generating functional under (local) chiral transformations [ref] .[Strictly speaking, quantum anomalies are excluded from this statement.]

S-wave pion-pion scattering lengths from nucleon-meson fluctuations · Introduction · 0.87

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] [8]

H. Leutwyler, in preparation

View on Pith

[2] [1]

S. Weinberg, Phys. Rev. Lett. 18 (1967) 188, 507; Phys. Rev. 166 (1968) 1568

View on Pith 1967

[3] [2]

S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247

View on Pith 1969

[4] [3]

R. Dashen, Phys. Rev. 183 (1969) 1245; R. Dashen and M. Weinstein, Phys. Rev. 183 (1969) 1291

View on Pith 1969

[5] [4]

L.-F. Li and H. Pagels, Phys. Rev. Lett. 26 (1971) 1204; H. Pagels, Phys. Rep. C 16 (1975) 219

View on Pith 1971

[6] [5]

S. Weinberg, Physica A96 (1979) 327

View on Pith 1979

[7] [6]

J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158 (1984) 142

View on Pith 1984

[8] [7]

For a discussion of the current status, see Proc. Workshop on Eﬀective Field Theories of the Standard Mo del, Dobogok¨ o, Hungary, ed. U. Meissner (World Scientiﬁc, Singapore, 1992 ); H. Leutwyler, Proc. XXVI Int. Conf. on High Energy Physics, D allas, Aug. 1992, ed. J. R. Sanford (AIP Conf. Proc. No. 272, 1993)

View on Pith 1992

[9] [9]

J. Goldstone, Nuovo Cim. 19 (1961) 154

View on Pith 1961

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