Virtual Photons in Chiral Perturbation Theory

R. Urech

REVIEW

Virtual Photons in Chiral Perturbation Theory

R. Urech

High Energy Physics - Phenomenology · Submitted 20 May 1994 · 18 references

Pith's verdict: The standard reference for electromagnetic counterterms in three-flavor CHPT; the K_i basis and its beta functions are the durable product, and the Dashen application is honestly presented as a partial result.

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arXiv IDhep-ph/9405341 (version 2)

Pith IDT673PCQ2, Pith's permanent record of this paper and its review

CitationR. Urech. "Virtual Photons in Chiral Perturbation Theory." arXiv:hep-ph/9405341 (1994). https://pith.science/paper/T673PCQ2

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Referee report

Significance

This is the standard reference for the electromagnetic counterterm Lagrangian of three-flavour CHPT, and the K_i basis together with the Sigma_i of Eq. (35) is what later calculations of e^2 m_q corrections in the meson sector build on. The derivation is self-contained given standard heat-kernel technology, and the Q = 0 limit correctly reproduces the Gasser-Leutwyler L_i, H_i sector, which is a meaningful check on the reduction. The explicit display of the divergent coefficients, rather than only the counterterm list, is a genuine strength: it makes the renormalization of electromagnetic corrections checkable at this order. The Dashen-theorem application is honest about what is and is not determined, and the comparison with the existing literature in Sec. 3.4 is useful. The paper does not overclaim a numerical prediction where the finite parts of the K_i^r are unknown.

Major objections (3)

[Eqs. (8), (36), (45), (53), (74), (89)-(97)] The combination written C F_o^4 (and C F_o^2 in Eq. 8) is dimensionally inconsistent with C having dimension (mass)^4, and the inconsistency is numerically checkable. With C = 61.1 x 10^-6 GeV^4 from Eq. (48) and F_o = 93.3 MeV, Eq. (8) as printed gives 2 e^2 C F_o^2 = 9.8 x 10^-8 GeV^2, i.e. an electromagnetic pion mass of about 0.3 MeV, whereas Eq. (49) and the surrounding discussion require the 4.8 MeV splitting. Presumably C/F_o^2 and C/F_o^4 are intended throughout. This matters beyond typesetting because Z enters the central result: Eq. (36) defines Z = C F_o^4, which is not dimensionless, while the Sigma_i of Eq. (35) must be dimensionless coefficients of lambda. Please confirm the intended definitions and correct Eqs. (8), (36), (45), (53), (74) and (89)-(97) consistently, since Z = C/F_o^4 is what fixes Sigma_15, Sigma_16 and Sigma_17.
[Sec. 2, Eq. (31) and Appendix A] The claim that L(Q)_4 'can be simplified to' Eq. (31) is the load-bearing step for the central result, but the reduction is not displayed. The checks listed in Appendix A (the constant e^4 term, the masses, the e^2 m_q and e^4 scattering amplitudes, and the matrix elements in Eq. 85) establish that the 17 K_i are independent, not that no eighteenth independent O(p^4) structure exists. The acknowledgements record that the original set was overcomplete and that H. Neufeld identified the redundancy, which shows the counting is delicate. Please either show the reduction explicitly, using Eqs. (79)-(82), or tabulate which structure of Tr(1/12 Y Y + 1/2 Lambda^2) in Eq. (27) is absorbed by which K_i. Such a table would convert the completeness assertion into a verifiable statement at modest length.
[Sec. 3.2, Eq. (56)] The bound |K_i| <~ 1/(16 pi^2) = 6.3 x 10^-3 is stated as an assumption, and the paper is explicit that it is not derived; that acknowledgement is appropriate. What remains unresolved is whether the bound is even self-consistent with the divergent coefficients of Eq. (35). With Z = C/F_o^4 = 0.81 from Eqs. (36) and (48), Sigma_15 = 3/2 + 3Z + 20Z^2 is about 17, and the one-loop running K_i^r(mu) = K_i^r(mu') + (Sigma_i/16 pi^2) ln(mu'/mu) changes K_i^r by roughly 0.075 between mu = 0.5 and 1 GeV, an order of magnitude above the assumed bound. The bound therefore cannot hold at both scale points, and the quoted uncertainty in Eqs. (51), (57) and (58) may be underestimated. Please either justify the bound against the size of the Sigma_i or show how the conclusions of Sec. 3.2 change under a relaxed bound.

Minor objections (6)

[Eq. (51) versus Eq. (76)] The central value at mu = M_rho is quoted as 0.52 x 10^-3 GeV^2 in Eq. (51) and as 0.49 x 10^-3 GeV^2 in the Summary, Eq. (76). The other two scale points agree, and Eq. (52) is consistent with 0.52, so the Summary value appears to be the outlier. Please reconcile.
[Eq. (35)] The vanishing of Sigma_7, Sigma_13 and Sigma_14 is a nontrivial statement, namely that K_7, K_13 and K_14 are finite at one loop. A sentence noting this and, if possible, a brief explanation of why these particular structures are not renormalized would be informative.
[Eqs. (42)-(43)] The charged-meson two-point function is stated to have a cut rather than a pole, with the exponent 1 + e^2 f_a(p^2). Since the photon is massless, the infrared regularization that defines M_a^2 in this expression should be stated explicitly, even if only by reference.
[Sec. 3.1 and Sec. 3.3] The symbol Delta M_P^2 denotes the electromagnetic splitting in Eq. (41) but the full experimental splitting in Eqs. (59)-(61). The distinction is recoverable from context but the notation is overloaded; a subscript or a sentence at the start of Sec. 3.3 would remove the ambiguity.
[Sec. 2, Eq. (31)] The text states that terms proportional to polynomials in the background photon field A_mu are omitted. A sentence explaining why no divergence in Eq. (27) requires such a term would preempt the natural question, since the omission is otherwise only asserted.
[Sec. 3.2] The numerical inputs F_pi = 93.3 MeV, M_pi = 135 MeV and M_K = 495 MeV are stated, but the value of B_0 or of the quark mass combination used in Eq. (40) is not. Since Eq. (40) is the starting point for the mass decomposition, the numerical input there should be given.

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The paper

The paper claims

Urech extends chiral perturbation theory to include virtual photons, computes the one-loop divergences of the generating functional, and writes down the order-$p^4$ local action that absorbs them, then uses it to estimate the order-$e^2 m_q

The reading

Chiral perturbation theory describes pions and kaons as Goldstone bosons of broken chiral symmetry, order by order in momenta and quark masses. This paper asks what happens when photons are treated on the same footing: the charge matrix is promoted to a spurion so that the electromagnetic coupling respects the chiral symmetry, and the photon field is counted so that the usual power counting survives. Urech then computes the one-loop divergences of the generating functional and shows they are cancelled by a local action of order $p^4$ containing 29 couplings, ten of them the familiar strong ones and seventeen new ones carrying the electromagnetic effects. The payoff is a definite statement about renormalization: any calculation of electromagnetic corrections in three-flavour chiral perturbation theory at this order must run through these counterterms. As an application, the paper asks whether Dashen's theorem — the equality of the pion and kaon electromagnetic squared-mass splittings in the chiral limit — survives once quark masses are turned on, and finds that the answer is not fixed until the finite parts of the new couplings are known.

What carries the argument

The load-bearing device is the spurion trick applied to the charge matrix: $Q$ is split into separate left and right spurions $Q_L, Q_R$ that transform under $SU(3)_L × SU(3)_R$, with the photon field assigned dimension $O(1)$ and the spurions dimension $O(p)$, so that the electromagnetic interaction does not upset the derivative counting. The one-loop functional is then obtained by expanding $U$ and $A_μ$ around the classical solutions, collecting the meson and photon fluctuations into a single twelve-component object, and evaluating the resulting Gaussian determinant in dimensional regularization. The divergent piece is the standard Seeley–DeWitt form $-(1/16π^2)(1/(d-4)) ∫ d^4x Tr( Y_{μν}Y^{μν}/12 + Λ^2/2 )$, with $Y_μ$ the combined connection for flavour and photon indices. Matching that trace against a basis of chirally invariant operators — simplified using the $3×3$ Cayley–Hamilton trace identity and the relation $Q^2 = (e/3)Q + (2e^2/9)𝟙$ — produces the 29-coupling counterterm lagrangian and its $\beta$ functions.

Extended reading notes

Core claim

The paper establishes that the one-loop divergences generated by the chiral lagrangian with virtual photons can be absorbed by a local action at order $p^4$ built from the strong couplings $L_1$–$L_{10}$ and $H_1,H_2$ together with seventeen new couplings $K_1$–$K_{17}$ that appear only when the charge matrix is present. The divergent coefficients $K_i = 16π^2 Σ_i λ/(d-4) + K_i^r(μ)$ are given explicitly, with $Σ_i$ expressed in terms of a single combination $Z = C F_o^4$ of the electromagnetic low-energy constant $C$ and the pion decay constant in the chiral limit. Setting the charge to zero recovers the Gasser–Leutwyler strong-sector result, so the new terms are a clean extension rather than a modification. Applied to the meson masses at order $e^2 m_q$, the result is a formula for $ΔM_K^2 - ΔM_π^2$ containing calculable logarithms, a term proportional to $L_5^r(μ)$, and two combinations $R_π M_π^2 + R_K M_K^2$ of the unknown finite couplings. Numerically the calculable part ranges from $-0.26$ to $0.99 × 10^{-3}$ GeV$^2$ as the scale moves from 0.5 to 1 GeV, and the paper's bound on the unknown part is larger than that, so no firm correction to Dashen's theorem is claimed.

Load-bearing premise

The numerical statement about Dashen's theorem rests on the assumption that the unknown finite parts of the new couplings can be bounded by naive power counting, $|K_i| ≲ 1/(16π^2) = 6.3 × 10^{-3}$; that bound is an order-of-magnitude expectation rather than a derived result, and the paper itself shows it permits a correction larger than the computed loop terms.

Editorial extensions

If this is right

Any future one-loop computation of electromagnetic corrections to meson masses, decay constants, or scattering amplitudes in three-flavour chiral perturbation theory can be renormalized using the tabulated $Σ_i$, rather than rederiving the counterterm basis.
The scale dependence of the electromagnetic corrections is fixed by the coefficients $Σ_i$, so the residual scale dependence of a physical quantity becomes a check on whether all relevant $K_i^r$ have been included.
The correction to Dashen's theorem at order $e^2 m_q$ is reduced to the numerical evaluation of seven finite couplings $K_3^r, K_4^r, K_5^r, K_6^r, K_9^r, K_{10}^r, K_{11}^r$ together with $L_5^r$, which the paper leaves as an explicit open task.
The alternative route through the QCD kaon mass splitting, using $R = 43.7 ± 2.7$ and an $η$–$η'$ mixing angle of $22° ± 4°$, gives $ΔM_K^2 - ΔM_π^2 = -(0.06 ± 1.01) × 10^{-3}$ GeV$^2$, a small central value with an uncertainty large enough to accommodate most published estimates.

Reading between the lines

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

Because the $K_i$ beta functions depend on $C$ only through the single combination $Z = C F_o^4$, a lattice determination of the electromagnetic contribution to the pion mass in the chiral limit would pin $Z$ and thereby fix the entire divergent structure numerically, turning the counterterm basis into a predictive scheme rather than a classification.
The same spurion construction should extend to the $ΔS = 1$ weak sector already treated at one loop without photons, which would let electromagnetic and weak counterterms be separated cleanly in $K → ππ$ amplitudes; the paper does not attempt this.
The large spread between the paper's own bound-based estimate and the vector-meson-dominance and large-$N_c$ estimates in the literature suggests that a lattice computation of $M_{π^±}^2 - M_{π^0}^2$ at unphysical quark masses, extrapolated through the formula in Appendix B, would discriminate among them without needing the $K_i^r$ individually.

Original abstract

In the framework of chiral perturbation theory virtual photons are included. We calculate the divergences of the generating functional to one loop and determine the structure of the local action that incorporates the counterterms which cancel the divergences. As an application we discuss the corrections to Dashen's theorem at order [e^2*m(quark)].

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.16 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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In the framework of chiral perturbation theory virtual photons are included. We calculate the divergences of the generating functional to one loop and determine the structure of the local action that incorporates the counterterms which cancel the divergences. As an application we discuss the corrections to Dashen's theorem at order [e^2*m(quark)].

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	In the framework of chiral perturbation theory virtual photons are included.	contribution 0.36	0.57	0.06	asserted 0.99
s1	We calculate the divergences of the generating functional to one loop and determine the structure of the local action that incorporates the counterterms which cancel the divergences.	contribution 0.55	0.96	0.25	asserted 1.00
s2	As an application we discuss the corrections to Dashen's theorem at order [e^2*m(quark)].	contribution 0.76	0.86	0.30	asserted 0.84

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

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References

Works cited (18)

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S.Weinberg, Physica 96A (1979) 327

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J.Gasser, H.Leutwyler, Nucl. Phys. B250 (1985) 465

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

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H.Leutwyler, Nucl. Phys. B337 (1990) 108

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R.Dashen, Phys. Rev. 183 (1969) 1245

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T.Das, G.S.Guralnik, V.S.Mathur, F.E.Low, J.E.Young, Phys. Rev. Lett. 18 (1967) 759

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S.Weinberg, Phys. Rev. Lett. 18 (1967) 507

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J.Gasser, H.Leutwyler, Phys. Rep. 87 (1982) 77

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Particle Data Group, K.Hikasa et al., Phys. Rev. D45 (1992) S1

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J.F.Donoghue, B.R.Holstein, Y.-C.R.Lin, Phys. Rev. L ett. 55 (1985) 2766; F.J.Gilman, R.Kauﬀman, Phys. Rev. D36 (1987) 2761; Riazuddin, Fayyazuddin, Phys. Rev. D37 (1988) 149

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J.Bijnens, Phys. Lett. B306 (1993) 343 20 This figure "fig1-1.png" is available in "png" format from: http://arxiv.org/ps/hep-ph/9405341v2

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56 later papers cite it

Evidence: substantial. 56 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 19 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

6 later papers by other authors are listed below: 3 used its method or tool, 1 used its data or values and 2 compared their own results with it.

Uses its method or tool 3 of 12 papers with this label

Followed this paper to include electromagnetism in chiral perturbation theory by adding virtual photons as dynamical degrees of freedom.

Electromagnetic corrections in eta --> 3 pi decays (2008) · 0.91 · paraphrased: the licence does not allow quoting
Introduced quark charges as spurion fields to take care of the effects of hard photons, much as this paper did.

$C$ and $CP$ violation in effective field theories and applications to $\eta$-meson decays (2025) · 0.98 · paraphrased: the licence does not allow quoting
Used the leading order chiral Lagrangian including isospin-breaking and electromagnetic effects as given by this paper.

Investigation of a0-f0 mixing (2007) · 0.89 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 4 papers with this label

Used the chiral logarithm contribution calculated in this paper to make a rough estimate of the size of the electromagnetic contribution to the pion mass.

Lattice computation of the electromagnetic contributions to kaon and pion masses (2018) · 0.90 · paraphrased: the licence does not allow quoting

Compares with it 2 papers

Compared its own dimensionless constants K with this paper's, noting that this paper's K's are mu_ChPT^(d-4) times its own.

Precise Estimate of Charged Wino Decay Rate (2022) · 0.89 · paraphrased: the licence does not allow quoting
Compared its own results for the anomalous dimensions of the couplings with this paper's and found them in agreement.

Neutrinoless double beta decay in effective field theory: the light Majorana neutrino exchange mechanism (2017) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

55 arXiv papers cite this in 98 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 · 65 sentences without a confident label

Uses its method or tool 12 papers

Following Ref. [ref] to include electromagnetism in the framework of ChPT by adding virtual photons as additional dynamical degrees of freedom, the full NLO effective Lagrangian can be written as [equation] The new local electromagnetic interactions in the Lagrangians $ℒ_em^(2)$, $ℒ_em^(4)$ contain the quark charge matrix $=e diag(2,-1,-1)/3$.

Electromagnetic corrections in eta --> 3 pi decays · Lagrangians · 0.91
A very elegant and convenient tool to perform the calculation is the effective field theory framework set up in Refs. [ref] .

On the pion decay constant · The effective Lagrangian · 0.99
Then, following Refs. [ref] , it is possible to build a complete basis of effective operators involving leptons, photons, light mesons, and spurion sources up to order $p^4$ in the chiral expansion and respecting chiral symmetry.

Pion $\beta$ decay and $\tau\to\pi\pi\nu_\tau$ beyond leading logarithms · Matching with the spurion method · 0.96

Uses its data or values 3 papers

In fact, experiments have reached a level of precision which makes it mandatory to include radiative corrections, and to consider the rate for π→ℓν(γ)According to PDG [ref] , f_π=(130.4±0.04±0.2) This value is based on data [branching fraction for $π^-→μ^-ν̅_μ(γ)$ and lifetime], and on theoretical work performed in Refs. [ref] .

On the pion decay constant · Introduction · 1.00
The divergent parts of all these couplings ($c_i=Γ_i, n_i, d_i, κ_i, z_i$) are known and can be found in Ref. [ref] , respectively.

Isospin-Violating Contributions to $\epsilon'/\epsilon$ · $χ$PT formulation · 0.84
The values of ${Σ_i,Ξ_i}$ are given in Refs. [ref] , respectively.

New method for calculating electromagnetic effects in semileptonic beta-decays of mesons · The EFT representation · 0.98

Compares with it 2 papers

Our results for the anomalous dimensions of these couplings are in agreement with Ref. [ref] , d g_ν^ππ/dlnμ = - 36/5 , d g_ν^πN/dlnμ = -2(1-g_A^2) .

Neutrinoless double beta decay in effective field theory: the light Majorana neutrino exchange mechanism · Estimating the Low Energy Constants · 0.98
Thus, $K$'s in Ref. [ref] are $μ_ChPT^d-4$ times $K$'s in this paper.

Precise Estimate of Charged Wino Decay Rate · Chiral Lagrangian · 0.89

Cites it as background 13 papers

Let us recall here that there exist explicit one-loop calculations of the various $ππ$ amplitudes [ref] and form factors [ref] considered here, obtained within the full QCD+QED effective theory [ref] .

Two-loop representations of low-energy pion form factors and pi-pi scattering phases in the presence of isospin breaking · Determination of the subtraction parameters · 0.87
At order $p^4$, there will be additional terms, classified in Refs. [ref] .

Isospin breaking in Kl4 decays · Isospin breaking: the framework · 0.87
The electromagnetic Lagrangian is given in Ref. [ref] and the weak Lagrangian is given in Ref. [ref] .

Baryon chiral perturbation theory with virtual photons and leptons · Building blocks · 0.85

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.

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