Lorentz violation at high energy: concepts, phenomena and astrophysical constraints

David Mattingly; Stefano Liberati; Ted Jacobson

REVIEW

Lorentz violation at high energy: concepts, phenomena and astrophysical constraints

Ted Jacobson, Stefano Liberati, David Mattingly

Astrophysics · Submitted 12 May 2005 · 110 references

Pith's verdict: A solid review with real new results bolted on; the one two-sided bound in section 5.8 leans on a 'presumably' about Crab lepton populations, and the authors know it.

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CitationDavid Mattingly, Stefano Liberati, Ted Jacobson. "Lorentz violation at high energy: concepts, phenomena and astrophysical constraints." arXiv:astro-ph/0505267 (2005). https://pith.science/paper/3J56AZOU

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Editor's note

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Worth your time, but read it for the appendices, not the headline numbers. The genuinely new content is in Appendix C: explicit helicity-resolved matrix elements and phase-space integrals for photon decay, vacuum Cerenkov, and helicity decay, plus the corrected near-threshold scaling of the photon decay rate as E²/M. That correction is stated in the text as fixing their own earlier claim (refs 64, 53), which is the kind of thing you want to see. The helicity bookkeeping — that EFT forces η_positron = −η_electron for the opposite helicity, so a single η enters each process — is done properly and changes the bounds relative to the earlier literature. The birefringence bound |ξ| ≲ 10⁻⁴ and the photon decay bound |η±| ≲ 0.2 stand on their own and are checkable line by line from C.11–C.13. They also throw out the GRB021206 polarization claim as unreliable and fall back to Gleiser–Kozameh, which is the right call and not one every author in this space makes. Citation pattern is fine; the self-citations are to published work that contains what is claimed. The stress-test note lands. Section 5.8 is the only place two one-sided constraints get combined into a two-sided |η| < 10⁻², and the load-bearing step is the sentence about the Cerenkov-satisfying charges being 'presumably at least equally numerous' and overproducing the IC flux. That assumes comparable number densities across the four lepton species and enough 50 TeV charges among them to overshoot the IC luminosity. No pulsar-wind or cascade model backs it. The paper concedes as much in section 7, where it says a full spectral model could tighten the bound by ~10⁻⁴ — which tells you the present number is not from a model. If the wind is polarized or the pair multiplicities differ, that bound collapses to the one-sided ones. The naturalness caveat in 3.4 is acknowledged by the authors and is not a flaw to hold against them, but it does mean the numbers constrain the parametrization, not the fundamental theory. For a reader who wants the threshold and rate machinery, this is the reference. For a reader who wants the Crab bound, treat 5.8 as a plausibility argument. Send it out; the appendices deserve referee attention even if 5.8 gets pushed back.

Referee report

Significance

The paper is a useful and timely synthesis, and several of its original pieces are likely to be reused. The explicit rate computations in Appendix C, in particular the threshold-adjacent rate for photon decay (Eq. C.20) and the two-regime helicity-decay rate (Eqs. C.30 and C.31), go beyond what was previously available and are stated in a form other workers can apply directly. The systematic treatment of the helicity and particle/antiparticle dependence forced by the effective field theory is a genuine improvement on earlier bounds, and the paper is candid where earlier claims were too strong: it corrects the prior E-scaling of the photon-decay rate, and it explicitly withdraws reliance on the GRB021206 polarization report in light of the reanalyses. The bounds are falsifiable in a concrete sense: a 50 TeV photon-decay threshold and a 10^-4 birefringence coefficient are both within reach of named instruments. The naturalness discussion in §3.4 is a fair statement of the main conceptual vulnerability of the whole programme, and the authors do not hide it.

Major objections (2)

[§5.8 and Fig. 1] The two-sided bound |η±| < 10^-2, which is one of the three headline numbers of §5 and the only two-sided bound in Fig. 1, rests on an argument that is not quantified. The text states that if the synchrotron-producing charges violated the Čerenkov constraint their energy would be under 50 TeV, and then that 'the charges that do satisfy the Čerenkov constraint would presumably be at least equally numerous ... and so would produce too many IC gamma-rays.' That step assumes comparable number densities across the four lepton species and assumes that the Čerenkov-satisfying species can reach 50 TeV and upscatter enough seed photons to overproduce the observed IC flux. Neither assumption is derived from a model of the pulsar wind or the pair cascade, and the word 'presumably' appears twice in the paragraph. This sits in some tension with §7, where the authors state that they have deliberately 'tolerated the possibility of four different populations of charges behaving differently'; if that tolerance is genuine, the two-sided bound does not follow. Either supply an order-of-magnitude estimate of the IC overproduction (even a crude one would settle the question), or present the bell-shaped region in Fig. 1 and the |η±| < 10^-2 sentence in §5.8 explicitly as conditional on the single-population assumption, so that it is not read as having the same status as the birefringence and photon-decay bounds.
[§4.3.4 versus §5 and Fig. 1] Section 4.3.4 discusses the photon-splitting constraint and reports that the analysis of Ref. [67], which includes the large (Eγ/me)^8 enhancement to the rate, would give ξ ≲ 10^-3 for 50 TeV Crab photons. That number is numerically competitive with the birefringence bound of 2×10^-4 that dominates Fig. 1, yet it appears neither in Fig. 1 nor in the §5 summary of constraints, and no reason for its exclusion is given there. The likely reason is that the [67] analysis assumes no Lorentz violation in the electron-positron sector and neglects the helicity dependence of the photon dispersion, so it is not consistent with the framework used elsewhere in the paper; but the reader is left to infer this. Please state the exclusion criterion explicitly in §5, and either add the constraint to Fig. 1 with a clear label or say in one sentence why it is not directly comparable to the others.

Minor objections (7)

[§5.9] The inference Δη < 0.4 from the photon-decay bound requires that both |η+| and |η-| be bounded by 0.2. Section 5.4, however, describes the allowed region as the union over the two photon helicities, i.e. the bound applies to at least one of the two pairs (η±, ξ). If the 50 TeV Crab photons are unpolarized, both bounds apply and Δη < 0.4 follows; if not, it does not. Please add the one sentence that makes this explicit.
[§4.3.1 and Eq. (C.20)] Recomputing Eq. (C.20) at α = 0.01 with kth = 10 TeV and m = 0.511 MeV, I obtain a lifetime closer to 10^-9 s than the 10^-8 s quoted in §4.3.1 and in the paragraph following Eq. (C.20). The qualitative conclusion, that the lifetime is many orders of magnitude below the 10^11 s required for a Crab photon to reach us, is unaffected, but the quoted number should be checked.
[§3.4 and §7 versus the abstract and §5] The authors state clearly in §3.4 that if no symmetry or other mechanism protects the lower-dimension operators, the parametrization (3) is misleading and the observational limits 'do not significantly constrain the fundamental theory', and §7 repeats the caveat. This is an important qualification that a reader of the abstract and of the §5 opening summary would not encounter. Consider carrying one clause of it into the abstract and into the first paragraph of §5.
[References] There are duplicate entries: Refs. [4] and [27] are the same Kostelecký-Samuel paper, and Refs. [59] and [61] are the same Bolokhov-Nibbelink-Pospelov paper. Reference [94] (Mattingly and McElrath) is listed as unpublished but is the sole support for the claim in §6.3 that the neutrino Čerenkov rate is too low at 100 TeV; either cite a published source or mark the claim as preliminary in the text.
[Fig. 1] The figure is difficult to read as printed: the sign folding, the excision of a 10^-10 strip around each axis, and the mixture of solid, dashed and bell-shaped boundaries require the long caption plus several paragraphs of §5 to decode. A compact table listing each constraint, the observable it uses, the parameter combination it bounds, and whether it applies to one or both helicities would make the summary much easier to use, and would also make the conditional status of the bell-shaped region visible at a glance.
[§5.3 and Eq. (23)] The text explains that the largest field estimate, 0.6 mG, is adopted because it yields the weakest constraint. It would help to say explicitly that Eq. (23) therefore gives the most conservative synchrotron bound, and to note how the bound would scale if the field were at the lower end of the quoted 0.15-0.6 mG range.
[Typographical] A number of small errors should be corrected: 'If If η+ = 0' in Appendix C.3; 'theshold' in the first line of Appendix C.6; 'supers eded' in §5.5; 'the bound shown in is Fig. 1' in §5.1; and 'Generaliation' in the title of Ref. [37]. The spelling of Kluźniak is inconsistent between the text and the reference list, as is 'Mkn501' versus 'Mkn 501'.

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The paper

The paper claims

Planck-suppressed Lorentz violation in quantum electrodynamics is now bounded to roughly one part in ten thousand in the photon sector, and to a few percent in the electron sector, by high-energy astrophysical observations.

The reading

The paper asks a sharp question: if quantum gravity breaks Lorentz symmetry at a level suppressed by one power of the Planck mass, could we already see it? Its answer is yes — the suppression is not so strong that existing data are blind, and the authors spend most of the article showing how gamma rays from the Crab nebula and polarized light from distant galaxies convert that possibility into numbers. The framework is an effective field theory in which a small set of dimension-five operators add terms of order $E/M$ to the photon and electron dispersion relations, and the authors argue this is the only setting in which threshold reactions and decay rates can be computed rather than guessed. What a sympathetic reader should take away is a two-sided result: the same EFT that makes the bounds derivable also forces a helicity and particle/antiparticle structure on the coefficients, so earlier bounds that ignored that structure were either too strong or too weak. The paper is a review plus new calculations — explicit rates for photon decay, vacuum Cherenkov radiation, helicity decay and pair emission, a new photon-decay constraint, and a careful re-derivation of the synchrotron and Cherenkov limits with four distinct lepton populations. The headline numbers are $|ξ| ≲ 10^{-4}$ from vacuum birefringence, $|η_±| ≲ 0.2$ from the absence of decay of 50 TeV Crab photons, and $|η| ≲ 10^{-2}$ for at least one of the two electron coefficients from the combined synchrotron and Cherenkov argument.

What carries the argument

The load-bearing object is the Myers-Pospelov extension of QED: three dimension-five, rotation-invariant, gauge-invariant operators that shift the free dispersion relations to $ω_±^2 = k^2 ± ξ k^3/M$ for right- and left-circular photons and $E_±^2 = p^2 + m^2 + η_± p^3/M$ for the two electron helicities. The sign flip between photon helicities is what produces vacuum birefringence, and the relation $η^{positron}_± = -η^{electron}_∓$, forced by the hole interpretation of the positron, is what makes every constraint apply to a pair of coefficients rather than a single number. On top of this sit the threshold theorems: for any two-to-two reaction with fixed target momentum, thresholds occur at antiparallel initial momenta and parallel final momenta, but unlike the Lorentz-invariant case an upper threshold can exist and pair creation can partition momentum asymmetrically. The rate calculations in the appendix then show the rates rise steeply enough above threshold that a threshold-only analysis loses almost nothing.

Extended reading notes

Core claim

The central claim is that Lorentz violation suppressed by the inverse Planck mass is already testable, and that when the helicity dependence implied by effective field theory is taken seriously the observational bounds become both defensible and, in places, stronger than earlier estimates. Concretely, the authors derive $|ξ| ≲ 10^{-4}$ on the dimension-five photon coefficient from the survival of linear polarization in ultraviolet light from distant galaxies; $|η_±| ≲ 0.2$ on the electron coefficients from the requirement that 50 TeV photons from the Crab nebula not decay into electron-positron pairs; a one-sided synchrotron bound that at least one of $±η_±$ exceed $-7 × 10^{-8}$; and, by combining the synchrotron and Cherenkov arguments under a single-population model of the Crab, a two-sided bound $|η_±| < 10^{-2}$ on one of the two electron parameters. The paper also establishes that the helicity-flip decay $e^- → e^-γ$ has no kinematic threshold but an effective one near 10 TeV, so that above that energy a 10 TeV electron would flip helicity in about $10^{-9}$ s — which is what makes threshold-only reasoning trustworthy for the Crab constraints.

Load-bearing premise

The two-sided $|η| < 10^{-2}$ bound rests not on a measurement but on a model: that one population of leptons in a 0.15–0.6 mG field produces both the 100 MeV synchrotron emission and the 50 TeV inverse-Compton gamma rays of the Crab nebula, with no additional charge population able to supply the synchrotron flux.

Editorial extensions

If this is right

If the bounds hold, any quantum-gravity scenario that generates dimension-five Lorentz violation in QED with coefficients of order unity is excluded, so such scenarios must either suppress these operators further or protect low-energy Lorentz symmetry by a symmetry.
Vacuum birefringence at order $E^2/M$ becomes the sharpest single handle on the photon sector, scaling with the square of photon energy and with propagation distance, so a polarized gamma-ray observation from a distant source would improve the bound by many orders of magnitude.
The absence of vacuum Cherenkov radiation from 50 TeV leptons in the Crab constrains the same coefficients that govern the maximum synchrotron frequency, tying two otherwise independent observations to one parameter.
If ultra-high-energy cosmic rays are protons, the observed absence of the GZK cutoff in one experiment and its presence in another translate into bounds of order $10^{-14}$ on proton coefficients from the absence of Cherenkov emission, and of order $10^{-11}$ on proton and pion coefficients if the cutoff is eventually located in its predicted place.
Because the dimension-five operators violate CPT as well as Lorentz invariance, a confirmed CPT symmetry in the ultraviolet would remove this entire class of signals and push attention to dimension-six operators at order $E^2/M^2$.

Reading between the lines

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

The paper's own logic suggests a cleaner discriminant than it exploits: because the birefringence signal is quadratic in photon energy while the time-of-flight signal is linear, a source with a measured polarization angle as a function of energy could separate the two without any assumption about emission simultaneity — the authors note the quadratic scaling but do not propose the fit.
The naturalness discussion cuts against the paper's own framing in a way the text only partly acknowledges: if no symmetry protects the lower-dimension operators, the coefficients bounded here are not the fundamental ones, and the observational limits constrain a parametrization rather than a theory — which makes the search for a protecting symmetry the real bottleneck, not the next order of magni
The helicity-decay channel is treated as a constraint tool, but the same rate calculation implies a propagation effect: a population of polarized TeV electrons crossing astrophysical distances would depolarize on $10^{-9}$ s timescales above the effective threshold, so any future claim of polarized TeV electron emission from a nearby source is itself a Lorentz-violation test.
The Crab-based bounds depend on a single-population emission model; a targeted multi-zone spectral fit, rather than more observing time, is what would convert the $|η| < 10^{-2}$ bound into the $10^{-6}$ bound the authors project as the reachable target.

Original abstract

We consider here the possibility of quantum gravity induced violation of Lorentz symmetry (LV). Even if suppressed by the inverse Planck mass such LV can be tested by current experiments and astrophysical observations. We review the effective field theory approach to describing LV, the issue of naturalness, and many phenomena characteristic of LV. We discuss some of the current observational bounds on LV, focusing mostly on those from high energy astrophysics in the QED sector at order E/M_Planck. In this context we present a number of new results which include the explicit computation of rates of the most relevant LV processes, the derivation of a new photon decay constraint, and modification of previous constraints taking proper account of the helicity dependence of the LV parameters implied by effective field theory.

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, 4,274 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.75 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We consider here the possibility of quantum gravity induced violation of Lorentz symmetry (LV). Even if suppressed by the inverse Planck mass such LV can be tested by current experiments and astrophysical observations. We review the effective field theory approach to describing LV, the issue of naturalness, and many phenomena characteristic of LV. We discuss some of the current observational bounds on LV, focusing mostly on those from high energy astrophysics in the QED sector at order E/M_Planck. In this context we present a number of new results which include the explicit computation of rates of the most relevant LV processes, the derivation of a new photon decay constraint, and modification of previous constraints taking proper account of the helicity dependence of the LV parameters implied by effective field theory.

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	We consider here the possibility of quantum gravity induced violation of Lorentz symmetry (LV).	background 0.69	0.27	0.07	hedged possibility 0.92
s1	Even if suppressed by the inverse Planck mass such LV can be tested by current experiments and astrophysical observations.	background 0.98	0.06	0.17	asserted 0.63
s2	We review the effective field theory approach to describing LV, the issue of naturalness, and many phenomena characteristic of LV.	contribution 0.61	0.36	0.06	asserted 0.79
s3	We discuss some of the current observational bounds on LV, focusing mostly on those from high energy astrophysics in the QED sector at order E/M_Planck.	contribution 0.64	0.18	0.10	asserted 0.91
s4	In this context we present a number of new results which include the explicit computation of rates of the most relevant LV processes, the derivation of a new photon decay constraint, and modification of previous constraints taking proper account of the helicity dependence of the LV parameters implied by effective field theory.	contribution 0.91	0.96	0.27	asserted 1.00

What kind of paper the classifier reports this is: review 0.94 · evidence from proof or derivation 0.43.

What was examined, and what was not

sources read

arXiv listing abstract v2

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

Figures

**Figure 1.** Constraints on LV in QED at O(E/M) on a log-log plot. For negative parameters minus the logarithm of the absolute value is plotted, and region of width 10−10 is excised around each axis. The constraints in solid lines apply to ξ and both η±, and are symmetric about both the ξ and the η axis. At least one of the two pairs (η±, ξ) must lie within the union of the dashed bell-shaped region and its reflection about the ξ axis. The IC and synchrotron Cerenkov lines are truncated ˇ where they cross. parameter space corresponds to the dark region of the Figure. These constraints strongly…

Figures reproduced from arXiv:astro-ph/0505267 by the authors.

References

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Forward citations

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How later papers used it

101 later papers cite it

Evidence: substantial. 91 of the citing papers share no author with it, and 5 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 195 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 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 paper · 1 by the same authors

By the same authors

Used the same procedure as this paper to obtain the critical frequency of the synchrotron radiation in the Lorentz invariance violation context.

Constraints on the deformation scale of a geometry in the cotangent bundle (2020) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

96 arXiv papers cite this in 190 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 · 128 sentences without a confident label

Builds on it 2 papers

This has been done for $α= β∈ℤ^+$ [ref] but a generalization to $α≠β$ solving the threshold condition [ref] for the momentum of the particle $a$ could be performed for small values of the exponents $α$ and $β$.

Gravitational and electromagnetic Cherenkov radiation constraints in modified dispersion relations · Electromagnetic Cherenkov radiation · 0.85
We follow the same analysis as in [ref] , for pion as well as photon emission, but considering our operators.

Planck-scale Lorentz violation constrained by Ultra-High-Energy Cosmic Rays · Vacuum Čerenkov emission · 0.85

Uses its method or tool 1 paper

In order to obtain the critical frequency of the synchrotron radiation, we use the same procedure followed in Ref. [ref] in the LIV context.

Constraints on the deformation scale of a geometry in the cotangent bundle · Astrophysical constraints: synchrotron radiation · 1.00

Uses its data or values 2 papers

In [ref] $|η_±| 0.2$ was derived using the fact that 50 TeV $γ$-rays were measured from the CN.

New constraints on Planck-scale Lorentz Violation in QED from the Crab Nebula · Photon decay · 0.87
The lifetimes of the negative helicity electron decrease around this effective energy; lifetimes for a 1 TeV electron is about 1 s while for a 50 TeV electron it is reduced to about $10^-9$ s [ref] .

On Modified Dispersion Relations and the Chandrasekhar Mass Limit · Modified Dispersion Relations · 0.80

Compares with it 3 papers

This result completely differs with the one obtained in [ref] for the LIV case, in which the high-energy scale should be $Λ≳10^7 M_P$ for a deviation of $𝒪(E/M_P)$.

Constraints on the deformation scale of a geometry in the cotangent bundle · Astrophysical constraints: synchrotron radiation · 1.00
Similar MDRs have been used to study vacuum electromagnetic CR [ref] and it was shown that for $α= β≠0$ a significant fraction of the energy of the particle $a$ is emitted almost immediately when $a$ has an energy above the threshold of the process.

Gravitational and electromagnetic Cherenkov radiation constraints in modified dispersion relations · Electromagnetic Cherenkov radiation · 0.98
However, the type of violation is different from the one discussed in [ref] in the sense that decoherence is a process that cannot be described in the context of local effective lagrangians by means of non-renormalisable higher-dimension local field-theory operators in an effectively flat space time.

Decoherence and CPT Violation in a Stringy Model of Space-Time Foam · Introduction · 0.95

Cites it as support 2 papers

The linear energy dependence in EFT is largely unfavored [ref] .

Lorentz violation effects on astrophysical propagation of very high energy photons · Some Selected Theoretical Models · 0.99
On the theoretical side, a reasonably good motivation for focusing on $O(E^2/M^2)$ LV corrections is related to the so called “naturalness problem" [ref] .

GZK photon constraints on Planck scale Lorentz violation in QED · Introduction · 0.97

Cites it as background 45 papers

This stresses the importance of a clear choice of framework when discussing this sort of phenomenological constraints (see e.g. [ref] ).] Then, if synchrotron emission up to some maximal frequency $ω_obs$ is observed, one can deduce that the LV coefficient for the corresponding leptons cannot be more negative than the value for which $ω_c^max=ω_obs$.

New constraints on Planck-scale Lorentz Violation in QED from the Crab Nebula · Synchrotron radiation · 0.90
The determination of some of the best current bounds on Lorentz violation in the various sectors of the SM has taken advantage of the high energies of cosmic rays and gamma rays (see, e.g., Refs. [ref] for three research papers and Refs. [ref] for two reviews).

Improved bound on isotropic Lorentz violation in the photon sector from extensive air showers · Introduction · 0.96
Aside from these theoretical restrictions, there are phenomenological constraints stemming from experimental bounds on Lorentz violation (LV), see e.g. [ref] .

Towards a unitary, renormalizable and ultraviolet-complete quantum theory of gravity · Particle spectrum, dispersion relations and phenomenological constraints · 0.92

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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