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Facts of life with gamma(5)

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F. Jegerlehner

High Energy Physics - Theory · Submitted 26 May 2000 · 45 references

Pith's verdict: A clear, honest advocacy piece for NDR: the non-regularization argument in Section 3 is solid, but the Summary's scheme-equivalence language is looser than the paper's own Section 3 allows, and the explicit check is one loop in an unpublished 1979 preprint.

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arXiv IDhep-th/0005255 (version 1)
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CitationF. Jegerlehner. "Facts of life with gamma(5)." arXiv:hep-th/0005255 (2000). https://pith.science/paper/BI4HFWER
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The paper

The paper claims

The paper argues that in dimensional regularization one should keep $γ_5$ anticommuting with all Dirac matrices and handle the Adler–Bell–Jackiw triangle anomaly as an intrinsically four-dimensional object, rather than adopting the non-comm

The reading

Thirty years after dimensional regularization was invented, the treatment of $γ_5$ in chiral gauge theories was still disputed. Jegerlehner's aim is to settle the practical question: which $γ_5$ scheme should a person computing electroweak radiative corrections actually use? He argues that the 't Hooft–Veltman prescription, in which $γ_5$ fails to anticommute with the Dirac matrices in $D≠4$ dimensions, generates spurious anomalies that break the chiral symmetry and hence gauge invariance, and that these must be repaired by hand order by order through Ward–Takahashi and Slavnov–Taylor identities. His claim is that this repair work is avoidable: if one starts from chiral fields, rewrites the Feynman rules accordingly, and then takes the anticommutator $AC(μ)={γ_μ,γ_5}$ to zero, the only genuine anomaly left is the triangle anomaly, which can be computed directly in four dimensions using Adler's Bose-symmetry-plus-vector-current-conservation argument. The payoff, if correct, is that higher-order Standard Model calculations become far less tedious and far easier to automate.

What carries the argument

The load-bearing object is the anticommutator $AC(μ) = {γ_μ, γ_5}$ and the derived algebra $Γ_μ = γ_μ - ½AC(μ)γ_5$, which by construction anticommutes with $γ_5$. The paper uses it in two directions. First, it writes the chiral currents and the fermion propagator in terms of $Γ_μ$, showing that a nonzero $AC$ contaminates the free Lagrangian and destroys the dimensional continuation of fermion loops. Second, it demands that the $Γ$-algebra close, ${Γ_μ,Γ_ν} = 2G_{μν}$, and shows that the trace condition forces $G$ to be block diagonal with the extra-dimensional block $ḡ = 0$ — a singular metric, i.e. the four-dimensional Dirac algebra again. The second piece of machinery is Adler's observation that in the $VVA$ triangle the two divergent amplitudes $A_1, A_2$ have mass dimension zero and are fixed uniquely by Bose symmetry plus vector-current conservation in terms of the convergent amplitudes $A_3$ through $A_8$, so the anomaly is scheme-independent and can be inserted by hand in exactly four dimensions.

Extended reading notes

Core claim

The central claim is that the anticommuting-$γ_5$ scheme (naive dimensional regularization, NDR) is not merely convenient but is the only consistent option once one takes chiral fields seriously. The argument runs through the anticommutator $AC(μ)≡{γ_μ,γ_5}$. If $AC(μ)≠0$, the standard Feynman rules written in Dirac fields manifestly break gauge invariance, and the correct chiral currents acquire extra $AC$-dependent pieces; the fermion propagator becomes $(k!!/ - ½AC(k)γ_5 + m)/(K^2-m^2)$ with $K^2 = k^2 - ¼AC(k)AC(k)$. Imposing the trace condition $Tr(γ_μγ_νγ_ργ_σγ_5)≠0$ in the physical four-dimensional subspace forces $AC(μ)$ to be a matrix of rank $D-4$ with elements of order one, not a small correction. Then $K^2 = hat{k}^2$ and the propagator collapses to its four-dimensional form, so fermion loops cannot be regularized by continuation in $D$ at all. Jegerlehner shows that the 't Hooft–Veltman algebra, recast in terms of $Γ_μ ≡ γ_μ - ½AC(μ)γ_5$, closes only with a singular metric whose extra block vanishes, which is dimensional reduction in disguise. From this he concludes that any scheme enforcing the trace condition sacrifices either gauge invariance or regularization, and that the honest choice is to give up the trace condition, keep $γ_5$ anticommuting, and compute the hard anomaly separately in four dimensions.

Load-bearing premise

The load-bearing premise is that setting the anticommutator ${γ_μ,γ_5}$ to zero leaves no residual spurious anomalies beyond the hard triangle anomaly, so that the tedious restoration of Ward–Takahashi and Slavnov–Taylor identities can be skipped entirely; the support offered is the argument itself plus explicit one-loop checks in an unpublished 1979 preprint, with no published all-orders proof.

Editorial extensions

If this is right

  • If the argument holds, no order-by-order restoration of Ward–Takahashi or Slavnov–Taylor identities is needed in NDR calculations; only the hard anomaly diagrams require explicit attention.
  • The spurious anomalies that appear in non-anticommuting schemes are removable artefacts, so any physical amplitude computed in NDR should agree with the 't Hooft–Veltman result after the symmetry restoration has been carried out.
  • Because the extra $AC$ terms are not perturbative in $D-4$, treating them as counterterms is illegitimate; the paper's diagnosis rules out a whole family of proposed repair prescriptions.
  • The scheme is straightforward to implement in automated computer codes, which is the practical reason the author expects it to remain the working choice for multi-loop electroweak calculations.
  • The Standard Model's renormalizability still rests on anomaly cancellation between quarks and leptons, which the paper takes as a separate, already-established requirement.

Reading between the lines

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

  • The paper's argument is essentially a no-go statement about the trace condition, so it predicts that any future scheme claiming to satisfy both the trace condition and exact gauge invariance at the loop level must fail at some order — a claim that could be tested against explicit higher-order computations.
  • The one-loop checks reported in the appendix are the empirical anchor; a natural extension would be to ask whether the same cancellation of $AC$-dependence persists at three loops, where the anomalous subgraph structure becomes richer.
  • The observation that the 't Hooft–Veltman algebra is equivalent to dimensional reduction once rewritten in $Γ$ form suggests that disputes about 'consistent' versus 'inconsistent' schemes are partly terminological, and that the real dividing line is whether one keeps the trace condition.
  • If the anticommuting scheme is as clean as claimed, the remaining bottleneck for precision electroweak work shifts from symmetry restoration to the tensor reduction of the anomaly contributions from boxes and pentagons.
Original abstract

The increasing precision of many experiments in elementary particle physics leads to continuing interest in perturbative higher order calculations in the electroweak Standard Model or extensions of it. Such calculations are of increasing complexity because more loops and/or more legs are considered. Correspondingly efficient computational methods are mandatory for many calculations. One problem which affects the feasibility of higher order calculations is the problem with gamma(5) in dimensional regularization. Since the subject thirty years after its invention is still controversial I advocate here some ideas which seem not to be common knowledge but might shed some new light on the problem. I present arguments in favor of utilizing an anticommuting gamma(5) and a simple 4-dimensional treatment of the hard anomalies.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,820 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.18 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The increasing precision of many experiments in elementary particle physics leads to continuing interest in perturbative higher order calculations in the electroweak Standard Model or extensions of it. Such calculations are of increasing complexity because more loops and/or more legs are considered. Correspondingly efficient computational methods are mandatory for many calculations. One problem which affects the feasibility of higher order calculations is the problem with gamma(5) in dimensional regularization. Since the subject thirty years after its invention is still controversial I advocate here some ideas which seem not to be common knowledge but might shed some new light on the problem. I present arguments in favor of utilizing an anticommuting gamma(5) and a simple 4-dimensional treatment of the hard anomalies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The increasing precision of many experiments in elementary particle physics leads to continuing interest in perturbative higher order calculations in the electroweak Standard Model or extensions of it. background 1.00 0.04 0.02 asserted 0.99
s1 Such calculations are of increasing complexity because more loops and/or more legs are considered. background 1.00 0.05 0.02 asserted 1.00
s2 Correspondingly efficient computational methods are mandatory for many calculations. background 0.99 0.05 0.03 asserted 1.00
s3 One problem which affects the feasibility of higher order calculations is the problem with gamma(5) in dimensional regularization. background 0.98 0.05 0.03 asserted 1.00
s4 Since the subject thirty years after its invention is still controversial I advocate here some ideas which seem not to be common knowledge but might shed some new light on the problem. contribution 0.54 0.24 0.06 speculative 0.54
s5 I present arguments in favor of utilizing an anticommuting gamma(5) and a simple 4-dimensional treatment of the hard anomalies. contribution 0.89 0.74 0.85 asserted 0.74

What kind of paper the classifier reports this is: position or commentary 0.99 · evidence from argument 1.00.

What was examined, and what was not

sources read
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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • S sections
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  • 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

Figures
Figure 1. The VVA triangle diagrams. In D = 4, working as usual in momentum space, we may perform a covariant decomposition of the third rank pseudotensor which depends on the two independent momenta p1 and p2: A µνλ(p1, p2) = ε µνλα (p1α A1 + p2α A2) + ε µλαβ p1αp2β (p ν 1 A3 + p ν 2 A4) + ε νλαβ p1αp2β (p µ 1 A5 + p µ 2 A6) + ε µναβ p1αp2β  p λ 1 A7 + p λ 2 A8  (37) where the amplitudes Ai are Lorentz scalars. We now impose • Bose symmetry (i.e. consider the sum of the two diagrams of

Figures reproduced from arXiv:hep-th/0005255 by the authors.

References
How later papers used it

72 later papers cite it

Evidence: substantial. 72 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 98 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

Uses its method or tool 1 of 3 papers with this label
How the literature uses it

63 arXiv papers cite this in 84 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 3 papers
  • Here we work in the Larin-scheme [ref] and define the axial-vector current explicitly as

    Lepton-pair production at hadron colliders at N$^3$LO in QCD · Axial-vector production at N$^3$LO in QCD · 0.99

  • Furthermore we have reproduced our results using the naive-dimensional regularisation scheme as defined by Jegerlehner [ref] (the quasi self-chiral scheme), where one keeps the anti-commuting $$ [ref], while giving up on

    Two-Loop Rational Terms for Spontaneously Broken Theories · Treatment of $$ · 0.89

  • This tutorial illustrates, with the $Z^*γγ$ Green function as an example, how chiral Ward identities can be enforced using Alder's method [ref] . * Extracting Form Factors from the $μ→eγ$ amplitude – This tutorial explains the use of Projector in detail, while also explaining how can be used to verify the decoupling theorem, gauge invariance, and Ward identities. * Scattering of Light by Light – This tutorial explains how to compute the light-by-light scattering matrix element at leading order, and illustrates the capabilities and limitations of covariant methods used in .

    Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals · Expansion of the Documentation · 0.88

Compares with it 1 paper
  • The results for operators of dimensions one to six derived in Secs. [ref], [ref] and [ref] have been verified with the results provided in the Refs. [ref] [In these references, authors have employed the BMHV regularisation scheme [ref] to compute the finite contribution to the renormalisable IR Lagrangian, i.e., D1 to D4 operators.

    One-loop Effective Action up to Dimension Eight: Integrating out Heavy Fermion(s) · Dimension Four operators in FUOLEA · 0.99

Cites it as support 1 paper
Cites it as background 31 papers

Discussion

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