High-Energy Tests of Lorentz Invariance

Sheldon L. Glashow; Sidney Coleman

REVIEW

High-Energy Tests of Lorentz Invariance

Sidney Coleman, Sheldon L. Glashow

High Energy Physics - Phenomenology · Submitted 17 Dec 1998 · 38 references

Pith's verdict: The structural machinery is sound and the §2.4 decay-kinematics theorem is a real result; the cosmic-ray payoff in §3.3 is the part that is asserted rather than shown, and one headline bound rests on an unpublished air-shower analysis.

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 →

· 2 major objections · 11 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-ph/9812418 (version 3)

Pith ID6ANKB5GU, Pith's permanent record of this paper and its review

CitationSheldon L. Glashow, Sidney Coleman. "High-Energy Tests of Lorentz Invariance." arXiv:hep-ph/9812418 (1998). https://pith.science/paper/6ANKB5GU

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

Two things worth knowing before you spend an afternoon on this. First, the framework half is solid. The representation-theoretic growth law (Eq. 2.6), the anomaly argument via the coordinate rescaling of Eq. 2.15, and the sign analysis of Eq. 2.29 all check out, and the §2.4 result is genuine: every stationary point of Y is a local maximum, so a decay can be allowed low, forbidden in a band, allowed high, and no more complicated pattern is possible. That theorem is the best thing in the paper and it does not depend on any of the phenomenology. Second, the parameter counting and the anomaly argument overlap with Colladay and Kostelecky, and the authors say so in §2.1 rather than hiding it. Credit where due: they also flag in the §3 preamble that the hadronic c_a are not really energy independent, and in the appendix that the induction proof is sketched. Those are the right disclosures to make and they are made in the right places. The appendix proof is a sketch, but it is a sketch of a true statement and the authors are honest that it is one. Now the soft spot, and it is the load-bearing one. Everything in §3.3 — GZK evasion at Eq. 3.26, neutron stability at 3.22, proton instability at 3.23 — assumes constant c_a. At 10^20 eV the expansion parameter is order 10^-2 to order unity depending on the coefficient, so the linear approximation is marginal exactly where the interesting numbers live. The authors' claim that the error shifts the parameter values but not the phenomena is plausible — the effect is a threshold shift and what matters is the sign of the correction — but it is asserted, not demonstrated. If c_Δ - c_p runs with energy, 'forbidden for all E' becomes 'forbidden in a band,' which is a weaker claim than the one advertised. That is a fixable gap, not a fatal one, but a referee should make them close it. Separately, the strongest entry in the conclusions table, 68ε_L² + ε_R² < 10^-39, comes from an air-shower analysis listed as 'to appear' plus a Kolar Gold Fields number delivered by private communication. That is a very strong bound resting on sources a reader cannot inspect. I would not weight it heavily until Cowsik and Sreekantan is in print. Who should read it: anyone working on UHE cosmic rays or neutrino oscillations who wants the kinematic vocabulary, and anyone who wants a clean worked example of how to add renormalizable Lorentz violation to the Standard Model without breaking gauge invariance. The §2.4 theorem alone is worth the read. Send it out. The structural claims will survive refereeing; the hadronic section will come back with comments, and it should.

Referee report

Significance

The paper is a substantial and, in its structural parts, convincing contribution. The anomaly argument of §2.2 is elegant and, as far as I can check, correct: the rescaling (2.15) is a field redefinition with a constant Jacobian, and the anomaly functional is invariant under general coordinate transformations, so the cancellation is indeed ε-independent. The §2.4 theorem is a genuine structural result with a clean proof; I verified the algebra of Eqs. (2.27)–(2.29) and the monotonicity of u(p0), and the extension to massless products is sound apart from a harmless factor noted below. The classification and the E^{2j} growth rule give the field a quantitative organizing principle, and the applications are falsifiable: a bump just below the GZK cutoff versus no cutoff at all, velocity oscillations of TeV neutrinos at baselines of order 1000 km sensitive to δc ~ 10^{-25}, and a lifetime anomaly of ultrahigh-energy muons. The appendix's correction of the authors' own earlier claim [4] is exemplary practice and is presented with a real, if compressed, proof. The numerical coefficients in §3.1 check out: αMτ0/4 ≃ 6.4×10^{14}, and the bounds (3.7), (3.10), (3.11), (3.12) follow from it as

Major objections (2)

[§3.3, Eq. (3.26)] The abstract lists 'undoing the GZK cutoff' among the paper's striking results, and Eq. (3.26) is the quantitative basis for it: if c_Δ − c_p exceeds 1.7×10^{−25}(ω/ω0)², Δ(1232) formation is said to be kinematically forbidden at all proton energies. The §3 preamble already acknowledges the difficulty — at E ~ 10^{20} eV the expansion parameter of §2.3, εE²/Λ_QCD², is of order unity for ε near 4×10^{−24} — and asserts that the error 'affects the values of the Lorentz-violating parameters needed to produce novel phenomena, but not the phenomena themselves.' That assertion is the load-bearing step and is not demonstrated. The paper's own §2.3 model supplies the tool: the spatial rescaling of Eq. (2.15) gives an explicit energy-dependent effective velocity, p² → p² − εp² with the expansion running in εp²/Λ², so one can compute c_Δ(E) − c_p(E) and re-examine the discriminant of the quadratic in Eq. (3.25). If the effective difference becomes energy-dependent, the 'forbidden for all E' statement of (3.26) may weaken to 'forbidden in a band', which is a suppressed rather than an absent cutoff. Either carry this estimate through for the Δ channel, or state the hedge quantitatively in §3.3 and in the abstract.
[§2.2, parameter counting] The count of 46 independent parameters is a headline result (abstract, sentence 2) and rests on two reductions that receive one sentence each. First, the spatial rescaling of Eq. (2.15) acts on every spatial derivative in the theory, so it shifts all of the ε_a by a common additive amount rather than eliminating one gauge-field B² term in isolation; the reduction is legitimate only because, as §2.4 and §3.3 repeatedly emphasize, only differences of the c_a are physical. The text should say this explicitly, since as written a reader may think a single bosonic parameter has been gauged away while the fermionic ones are untouched. Second, the lepton-phase reduction ('make two of the off-diagonal elements in one of the lepton velocity-mixing matrices real') presumes the minimal standard model with conserved lepton flavor, whereas §3.2 subsequently endows neutrinos with Majorana masses; those masses would consume the same phases. Please state the scope of the count (massless neutrinos, no leptonic mixing) and note how the number changes when neutrino masses are turned on. It would also help to say in one clause why the analogous quark phases are unavailable, namely that they are already spent putting the CKM matrix in standard form.

Minor objections (11)

[Eq. (3.5)] Both coefficients read ǫ_R + ǫ_R and ǫ_R − ǫ_R; the second would vanish identically. Presumably these should be ǫ_R ± ǫ_L, which is what the subsequent discussion of helicity-dependent rates requires.
[Eq. (3.20)] The chain 'c_γ − c_π0 = c_γ − c_e = c_γ − c_π0 = 10^{−22}' repeats the pion entry; the third term is presumably c_γ − c_p.
[Eq. (2.34)] The right-hand side carries a factor c_1 that does not follow from the preceding line: in regime (2), p′_0 is fixed by u′(p′_0) = c_1 independently of p_0, so E_min is linear in p_0 and (1/2)d²Y/dp_0² = c_1² − c_0². The sign, which is all that is used, is unaffected, but the factor should be removed or the normalization of p_1 explained.
[Notation, Eqs. (1.1), (2.7), (3.4)] The symbol ε denotes the Lorentz-violating coefficient in Eqs. (1.1) and (2.7) and then, in Eq. (3.4), the quite different combination ε_a² ≡ |δc_a sin 2θ_a|². Since the latter drives the muon and cosmic-ray bounds, a distinct symbol would prevent confusion.
[Abstract and §2] The abstract uses 'TCP-even' and 'TCP-odd' while the body consistently uses 'CPT'; the two should be standardized.
[§3.1, after Eq. (3.11)] The sentence 'a bound on b_R yields a 69-fold weaker bound on b_R + b_L' is unclear: b_R is defined in (3.9) as proportional to 68ε_L² + ε_R², so a bound on b_R is a bound on that combination, not on b_R + b_L. Please restate the intended comparison.
[Conclusions table and Refs. [14], [15]] The table lists |c′ − c|_{μe} < 4×10^{−21} from Ref. [15], which is cited as 'to appear in Physics Letters B', and the Kolar Gold Fields analysis of Ref. [14] is a private communication. Since these are among the five headline bounds, the table should mark which entries rest on published versus preliminary results.
[§3.2, Eqs. (3.17) and (3.19)] These bounds are quoted from Refs. [22] and [24] without showing the step from the experimental parameters (energy, baseline, assumed sin²2θ) to the quoted numbers. One line each would let the reader check the arithmetic.
[Eq. (3.16c)] The text says 'we set all three mixing angles equal' but does not say equal to what; presumably θ_m = θ_b = θ_c = θ, but the value is not given and the reader cannot check that (3.16c) follows from (3.15).
[Appendix] The parenthetical 'We're being a bit sloppy here: subgraphs of a 1PI graph need not be 1PI... it's easy to check that these have no effect on the power-counting' is the one step of the induction that is asserted rather than shown. Since the theorem is the paper's correction of its own earlier claim [4], a sentence spelling out the tree-graph case would make the correction self-contained.
[Typographical] 'mode mode' and 'the d muon lifetime' in §3.1, 'ous elves' for 'ourselves' in §2, 'partcular' in §3.3, 'occuring' and 'pertubation' in the Appendix, and 'Kostalecký' for Kostelecký in the text and Ref. [5] should all be corrected.

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

The paper

The paper claims

The paper builds a 46-parameter, anomaly-safe framework for tiny Lorentz violation and argues that high energy amplifies it enough to undo the GZK cutoff and generate neutrino oscillations.

The reading

Coleman and Glashow ask what happens if special relativity is not exact but is broken by tiny, renormalizable terms that single out a preferred frame while preserving rotation and translation invariance and the Standard Model gauge group. They classify all such CPT-even perturbations, count 46 independent parameters, and show that the existing anomaly cancellation is not spoiled. The key physical consequence is that each particle species acquires its own maximum attainable velocity, so that at high energy the world behaves like a medium with species-dependent light speeds. The authors then argue that cosmic-ray and neutrino observations are sensitive enough to detect or constrain velocity differences many orders of magnitude below ordinary laboratory bounds.

What carries the argument

The machinery is a preferred-frame effective Lagrangian built from operators with two spatial derivatives: for scalars, $��_i ��_a ��_{ab} ��_i ��_b$; for fermions, $i ��_{ab} u_a^�� u_b$, or in Dirac form the velocity-mixing matrices $C_L$ and $C_R$; and for gauge bosons, one $B^2$ term per gauge factor. These are the $j=1$, CPT-even, dimension-four operators; they preserve gauge invariance and anomaly cancellation because a rescaling of spatial coordinates removes $��$ from each irreducible multiplet and leaves the anomaly integral unchanged. The resulting single-particle kinematics is summarized by per-species maximal velocities $c_a$, and the decay analysis reduces to whether $E_0 �� E_{��min}(p_0)$, with the theorem that every stationary point of $Y(p_0)=E_{��min}^2-c_0^2 p_0^2$ is a local maximum.

Extended reading notes

Core claim

The central claim is that the most general renormalizable, gauge-invariant, rotationally and translationally invariant CPT-even Lorentz violation in the Standard Model is a finite, controllable deformation: 46 real parameters, all anomaly-safe, whose leading high-energy effects grow like $E^2$ in the preferred frame rather than like $E$ as CPT-odd effects do. This deformation defines energy-momentum eigenstates and maximal attainable velocities $c_a$, with dispersion relation $E_a^2 = c_a^2 p_a^2 + m_a^2 c_a^4$. Once different species have different $c_a$, decay thresholds become energy-dependent: a decay can be allowed at low energy, forbidden in a band, and allowed again, but the paper proves no more complicated alternation is possible. Applied to hadrons, a difference $c_�� - c_p$ of order $10^{-25}$ is enough to forbid $��(1232)$ formation off cosmic microwave photons at all energies, which would remove the GZK cutoff; applied to neutrinos, the same velocity matrices generate oscillation patterns that depend on energy in ways ordinary mass oscillations do not.

Load-bearing premise

The hadronic predictions assume each species' maximal attainable velocity is an energy-independent constant, which the authors acknowledge is not justified when $E��(��c^2)$ becomes comparable to the QCD scale.

Editorial extensions

If this is right

If the photon's maximum speed exceeds the electron's, photons above $2m_e/\sqrt{c_��^2-c_e^2}$ decay into electron-positron pairs, and the observed 20 TeV cosmic-ray photons bound $c_��-c_e<10^{-15}$.
Flavor-changing velocity mixing makes $��\to e+��$ grow like the cube of the muon energy, with a rest-frame branching ratio $B\simeq 6.4\times10^{14}(��_L^2+��_R^2)$, so the current limit $B<4.9\times10^{-11}$ gives $��_L^2+��_R^2<8\times10^{-26}$.
Neutrino oscillations acquire a term controlled by the velocity matrix $c$ and the CPT-odd matrix $b$, so high-energy long-baseline experiments can constrain $|��c|$ between electron and muon neutrinos to about $6\times10^{-22}$.
If $c_��-c_p>1.7\times10^{-25}$, $��(1232)$ formation off cosmic microwave photons is forbidden at all energies, removing the GZK cutoff; if $c_p>c_n$, ultra-high-energy neutrons become stable and can travel from cosmological sources.
If $c_��>c_p$, non-resonant photo-pion production is also suppressed, so the highest-energy cosmic rays could arrive from distances far beyond 50 Mpc.

Reading between the lines

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

The framework recasts Lorentz violation as a spectroscopy of maximum speeds: any species-dependent threshold anomaly in cosmic rays or neutrinos would measure differences among the $c_a$ directly, rather than merely setting a null bound.
If the GZK cutoff is absent, the same $c_��-c_p$ term should shift thresholds for multi-pion production and nuclear photodisintegration in a correlated way; looking for that correlation would separate this mechanism from exotic-source explanations.
Because electroweak gauge invariance ties $C_L$ in charged leptons to neutrinos, a TeV long-baseline oscillation anomaly should come with a correlated high-energy muon lifetime anomaly; a joint search would test the framework more sharply than either channel alone.
The 46-parameter description is a low-energy effective theory, so if Lorentz violation originates at a high scale, renormalization-group running could make the hadronic $c_a$ energy-dependent precisely where the GZK predictions are applied; a matching calculation would be needed to turn the order-of-magnitude estimates into sharp predictions.

Original abstract

We develop a perturbative framework with which to discuss departures from exact Lorentz invariance and explore their potentially observable ramifications. Tiny non-invariant terms introduced into the standard model Lagrangian are assumed to be renormalizable (dimension $\le 4$), invariant under $SU(3)\otimes SU(2)\otimes U(1)$ gauge transformations, and rotationally and translationally invariant in a preferred frame. There are a total of 46 independent TCP-even perturbations of this kind, all of which preserve anomaly cancellation. They define the energy-momentum eigenstates and their maximal attainable velocities in the high-energy limit. The effects of these perturbations increase rapidly with energy in the preferred frame, more rapidly than those of TCP-odd perturbations. Our analysis of Lorentz-violating kinematics reveals several striking new phenomena that are relevant both to cosmic-ray physics ({\it e.g.,} by undoing the GZK cutoff) and neutrino physics ({\it e.g.,} by generating novel types of neutrino oscillations). These may lead to new and sensitive high-energy tests of special relativity.

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, 5,704 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.89 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We develop a perturbative framework with which to discuss departures from exact Lorentz invariance and explore their potentially observable ramifications. Tiny non-invariant terms introduced into the standard model Lagrangian are assumed to be renormalizable (dimension $\le 4$), invariant under $SU(3)\otimes SU(2)\otimes U(1)$ gauge transformations, and rotationally and translationally invariant in a preferred frame. There are a total of 46 independent TCP-even perturbations of this kind, all of which preserve anomaly cancellation. They define the energy-momentum eigenstates and their maximal attainable velocities in the high-energy limit. The effects of these perturbations increase rapidly with energy in the preferred frame, more rapidly than those of TCP-odd perturbations. Our analysis of Lorentz-violating kinematics reveals several striking new phenomena that are relevant both to cosmic-ray physics ({\it e.g.,} by undoing the GZK cutoff) and neutrino physics ({\it e.g.,} by generating novel types of neutrino oscillations). These may lead to new and sensitive high-energy tests of special relativity.

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 develop a perturbative framework with which to discuss departures from exact Lorentz invariance and explore their potentially observable ramifications.	contribution 0.89	0.92	0.07	asserted 0.84
s1	Tiny non-invariant terms introduced into the standard model Lagrangian are assumed to be renormalizable (dimension $\le 4$), invariant under $SU(3)\otimes SU(2)\otimes U(1)$ gauge transformations, and rotationally and translationally invariant in a preferred frame.	method 0.93	0.23	0.03	asserted 0.74
s2	There are a total of 46 independent TCP-even perturbations of this kind, all of which preserve anomaly cancellation.	result 0.64	0.76	0.06	asserted 1.00
s3	They define the energy-momentum eigenstates and their maximal attainable velocities in the high-energy limit.	contribution 0.40	0.73	0.05	asserted 0.98
s4	The effects of these perturbations increase rapidly with energy in the preferred frame, more rapidly than those of TCP-odd perturbations.	result 0.98	0.69	0.13	asserted 1.00
s5	Our analysis of Lorentz-violating kinematics reveals several striking new phenomena that are relevant both to cosmic-ray physics ({\it e.g.,} by undoing the GZK cutoff) and neutrino physics ({\it e.g.,} by generating novel types of neutrino oscillations).	interpretation 0.55	0.86	0.53	asserted 0.95
s6	These may lead to new and sensitive high-energy tests of special relativity.	interpretation 0.84	0.28	0.43	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.88 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v3

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 (38)

[4]

S. Coleman and S. L. Glashow, Phys. Lett. B405 (1997) 249

View on Pith 1997
[1]

E. g., M. P. Hagen and C. M. Will, Phys. Today 40 (May 1987), 69

View on Pith 1987
[2]

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and E. N. Fortson, Phys. Rev. Lett. 57 (1986) 3125

View on Pith 1986
[3]

C.J. Berglund, L.R. Hunter, D. Krause, Jr., E.O. Prigge, and M.S. Romfeldt, Phys. Rev. Lett. 75 (1995) 1879

View on Pith 1995
[5]

D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D55 (1997) 6 760, D58 (1998) 116002, and earlier references therein

View on Pith 1997
[6]

E.g., S. Weinberg, The Quantum Theory of Fields I , §4.6 and §4.8, Cambridge U. Press, 1995

View on Pith 1995
[7]

W. Bardeen, Phys. Rev. 184 (1969) 1848; S. Adler and W. Bar deen, Phys. Rev. 182 (1969) 1517; W. Bardeen and B. Zumino, Nucl. Phys. B244 (1984 ) 421

View on Pith 1969
[8]

Versions of this section appear in: S.L. Glashow, in Proc . TAUP-97 Nucl.Phys. B70 Proc. Suppl. (1999) 70; S. L. Glashow, in Proc. First Tropica l Workshop, San Juan, Puerto Rico, ed. Jos´ e F. Nieves, AIP Conf. Proc. 444 (1998) 1 19

View on Pith 1999
[9]

A.I. Derbin et al., Phys. At. Nucl. 57 (1994) 222

View on Pith 1994
[10]

S. Coleman and S.L. Glashow, HUTP-98/A076

View on Pith
[11]

S. Coleman and S.L. Glashow, HUTP-98/A080, to appear in Proc. Neutrino ’98, Takayama, Japan

View on Pith
[12]

R.D. Bolton, Phys. Rev D38 (1988) 2077

View on Pith 1988
[13]

E.g., F. Combley, F. J. M. Farley and E. Picasso, Phys. Rep. 68 (1981 ) 93

View on Pith 1981
[14]

As reported to us by B.V. Sreekantan, private communica tion

View on Pith
[15]

R. Cowsik and B.V. Sreekantan, hep-ph/9811241 (to appear in Physics Letters B), and private communication

View on Pith Pith review arXiv
[16]

L. Stodolsky, Phys. Lett. B201 (1988) 353

View on Pith 1988
[17]

E. Waxman and J. Bahcall, Phys. Rev. Lett. 78 (1997) 2292

View on Pith 1997
[18]

M. Gasperini, Phys. Rev. D38 (1988) 2635; A. Halprin and C.N. Leung, Phys. Rev. Lett. 67 (1991) 1833

View on Pith 1988
[19]

R. Foot, C.N. Leung, and O. Yasuda, hep-ph/9809458

View on Pith Pith review arXiv
[20]

T. Kajita (for the SuperKamiokande Collaboration), to appear in Proc. Neutrino ’98, Takayama, Japan

View on Pith
[21]

C. Angelini et al., Phys. Lett. B179 (1986) 307

View on Pith 1986
[22]

E.B. Brucker et al., Phys. Rev. D34 (1986) 2183

View on Pith 1986
[23]

L.H. Hunter, private communication

View on Pith
[24]

T. Hambye, R.B. Mann and U. Sarkar, Phys. Lett. B421 (199 8) 105

View on Pith
[25]

V. A. Kosteleck´ y, Phys. Rev. Lett. 80 (1998) 1818 and In diana University preprint IUHET 396 hep-ph/9810352. 31

View on Pith Pith review arXiv 1998
[26]

K. Greisen, Phys. Rev. Lett. 16 (1966) 748

View on Pith 1966
[27]

G.T. Zatsepin and V.A. Kuz’min, JETP Lett. 41 (1966) 78

View on Pith 1966
[28]

AGASA Collab. e.g., M. Takeda et.al., Phys. Rev. Lett. 81 (1998) 1163; Fly’s Eye Collab. e.g., D.J. Bird et al., Astrophys. J. 424 (1995 144; Haverah Park Collab. See: M.A. Lawrence et al., J. Phys. G17 (1991) 733; Yakutsk Collab. e.g., N.N. Eﬁmov et al., in 22nd Intl. Cosmic Ray Conf. (1991) Dublin

View on Pith 1998
[29]

See M. Takeda [28] and V. Berezinsky, Nucl. Phys. Proc. S uppl. 70 (1999) 419 for further references in these connections

View on Pith 1999
[30]

T.W. Kephart and T.J. Weiler, Astro. Part. Phys. 4 (1996 ) 271

View on Pith 1996
[31]

T.J. Weiler, hep-ph/9710431

View on Pith Pith review arXiv
[32]

V. Berezinsky, M. Kachelrieß, and V. Vilenkin, Phys. Re v. Lett. 79 (1997) 4302

View on Pith 1997
[33]

A preliminary version of this section appears as S. Cole man and S.L. Glashow, HUTP- 98/A075, hep-ph/9808446

View on Pith Pith review arXiv
[34]

G.R. Farrar and P.L. Biermann, Phys. Rev. Lett. 81 (1998 ) 3579

View on Pith 1998
[35]

C.T. Hill and D.N. Schramm, Phys. Rev. D31 (1985) 564

View on Pith 1985
[36]

E.N. Fortson, private communication

View on Pith
[37]

S. Carroll, G. Field and R. Jackiw, Phys. Rev. D41 (1990) 1231; M. Goldhaber and V. Trimble, J. Astrophys. Astr. 17 (1996) 17 . See also: J.F.C. Wardle, R.A. Perley and M.H. Cohen, Phys. Rev. Lett. 79 (1997 ) 1801; J.P. Leahy, astro-ph/9704285; S. Carroll and G. Field, Phys. Rev. Lett. 79 (1997) 2394

View on Pith Pith review arXiv 1990
[38]

R. Eden, P. Landshoﬀ, D. Olive, and J. Polkinghorne, The Analytic S-Matrix , Cam- bridge University Press, 1966. 32

View on Pith 1966

Forward citations

Cited by 29 Pith papers

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

Very Special Relativity
hep-ph 2006-01 accept novelty 8.0 of 10

Certain proper subgroups of the Lorentz group, dubbed Very Special Relativity, can serve as nature's exact spacetime symmetry while mimicking almost every prediction of special relativity.
Ghost Condensation and a Consistent Infrared Modification of Gravity
hep-th 2003-12 conditional novelty 8.0 of 10

A scalar field rolling at constant speed forms a stable fluid with the equation of state of a cosmological constant, and its excitations modify gravity in the infrared with an oscillatory Newtonian potential.
A Material Frame: Hard Recoils from Slow Force Carriers
hep-ph 2026-07 conditional novelty 7.0 of 10

Making the photon's longitudinal mode a physical, ultra-slow particle turns gauge invariance into a smooth limit and predicts rare, hard recoils of charges against a preferred frame, bounded to c_L ≲ 10^-60 by xenon d...
The Neutrino Kaleidoscope: Searches for Non-Standard Neutrino Oscillations at Neutrino Telescopes with a TeV Muon Accelerator Source
hep-ph 2025-08 conditional novelty 7.0 of 10

A TeV muon accelerator aimed at a cubic-kilometer neutrino telescope could probe sterile neutrinos and Lorentz violation orders of magnitude beyond current limits.
Dark matter as integration constant in Horava-Lifshitz gravity
hep-th 2009-05 conditional novelty 7.0 of 10

Because Horava-Lifshitz gravity imposes its Hamiltonian constraint only globally, its field equations leave an integration function that behaves as cold dark matter, reproducing GR plus dark matter without a dark matt...

Show 24 more citing papers

Spontaneous Lorentz Breaking at High Energies
hep-th 2006-03 conditional novelty 7.0 of 10

Gauging the ghost condensate yields a vector-field theory of spontaneous Lorentz violation that evades the 100 GeV bound and allows a breaking scale up to about 10^12 GeV, with distinctive spin-dependent forces.
On perturbative aspects of a nonminimal Lorentz-violating QED with CPT-odd dimension-6 terms
hep-th 2026-09 conditional novelty 6.0 of 10

One-loop corrections from dimension-6 CPT-odd Lorentz-violating operators generate a Carroll-Field-Jackiw term that diverges, but is rendered finite and five-derivative when the coefficients obey special relations.
Constraints on Lorentz Invariance Violation from Fermi-Large Area Telescope Observations of Gamma-Ray Bursts
astro-ph.HE 2013-05 conditional novelty 6.0 of 10

No energy-dependent speed of light is found in four bright Fermi gamma-ray bursts; the 95% limits place the linear Lorentz-violation scale above 7.6 Planck energies for GRB 090510, and above about 2 Planck energies un...
New Constraints on Neutrino Velocities
hep-ph 2011-09 accept novelty 6.0 of 10

Superluminal neutrinos would rapidly lose energy by electron-positron pair emission, which rules out the OPERA anomaly and yields new limits on neutrino speed from Super-Kamiokande and IceCube.
Models of non-relativistic quantum gravity: the good, the bad and the healthy
hep-th 2010-07 accept novelty 6.0 of 10

Of three non-relativistic gravity models, only the non-projectable Horava extension has a stable, weakly coupled scalar mode, and its predicted preferred-frame PPN parameters satisfy existing solar-system limits.
Renormalization group in Lifshitz-type theories
hep-th 2009-06 accept novelty 6.0 of 10

Lorentz symmetry is not automatically recovered at low energies in Lifshitz-type theories: the speed-of-light couplings run logarithmically and require extreme fine-tuning to satisfy experimental bounds.
Neutrino-antineutrino pair production by a photon in a dense matter
hep-ph 2006-02 conditional novelty 6.0 of 10

A photon in dense matter can produce a neutrino-antineutrino pair through the neutrino magnetic moment even when the medium refractive index equals one.
Lorentz violation at high energy: concepts, phenomena and astrophysical constraints
astro-ph 2005-05 conditional novelty 6.0 of 10

Helicity-resolved effective-field-theory bookkeeping tightens some and weakens other astrophysical bounds on Planck-suppressed Lorentz violation, with photon decay in 50 TeV Crab gamma rays giving |eta| below about 0.2.
Lorentz-Violating Vector Fields Slow the Universe Down
hep-th 2004-07 conditional novelty 6.0 of 10

A Lorentz-violating timelike vector field tracks matter with negative energy density and slows cosmic expansion, and primordial nucleosynthesis bounds its norm to below about 10^18 GeV.
The phantom menaced: constraints on low-energy effective ghosts
hep-ph 2003-11 conditional novelty 6.0 of 10

Gravitational vacuum decay of ghost fields, compared with the EGRET diffuse gamma-ray flux, requires a Lorentz-violating cutoff below about 3 MeV, and a Lorentz-invariant cutoff is excluded outright.
Can Historic Measurements cast light on the Spectral Differences found between the Pierre Auger Observatory and the Telescope Array?
astro-ph.HE 2026-10 conditional novelty 5.0 of 10

Historic AGASA, Volcano Ranch and Haverah Park measurements, after empirical energy adjustments, agree with the Auger spectrum above 30 EeV, keeping the Auger versus Telescope Array tension intact.
Spherical photon orbits around the Kerr-like black hole in Einstein-Bumblebee gravity
gr-qc 2025-07 conditional novelty 5.0 of 10

In Einstein-Bumblebee Kerr-like spacetimes, spherical photon orbits are governed by a sixth-order polynomial whose roots and critical inclination angle depend on the Lorentz-violation parameter.
New Physics in Astrophysical Neutrino Flavor
hep-ph 2015-06 conditional novelty 5.0 of 10

Adding effective new-physics operators to standard neutrino oscillations, the authors map where astrophysical neutrino flavor ratios can land and argue that a precise flavor measurement would improve new-physics bound...
Physics Potential of the ICAL detector at the India-based Neutrino Observatory (INO)
physics.ins-det 2015-05 conditional novelty 5.0 of 10

Simulation-based projections indicate that a 50 kt magnetised iron calorimeter could reject the wrong neutrino mass hierarchy at roughly 3 sigma after ten years of atmospheric neutrino data, and measure the atmospheri...
Constraints and tests of the OPERA superluminal neutrinos
hep-ph 2011-09 conditional novelty 5.0 of 10

If neutrinos really travel faster than light by the OPERA amount, pion decay becomes kinematically forbidden above about 5 GeV, so OPERA's 40 GeV events bound the effect below 3e-7 and IceCube atmospheric data bound i...
OPERA-reassessing data on the energy dependence of the speed of neutrinos
hep-ph 2011-09 conditional novelty 5.0 of 10

Combining OPERA, MINOS and 1979 Fermilab muon-neutrino speed data, a momentum-independent superluminal shift fits best, while tachyon and quadratic energy-dependence models are disfavored.
Physics Beyond Colliders at CERN: Beyond the Standard Model Working Group Report
hep-ex 2019-01 accept novelty 4.0 of 10

The report compiles CERN's beyond-the-Standard-Model proposals and shows their projected sensitivity to hidden-sector particles and rare processes across sub-eV to multi-TeV mass scales.
Tests of Lorentz invariance: a 2013 update
gr-qc 2013-04 accept novelty 3.0 of 10

Lorentz-violation constraints from particle physics and astrophysics are collected and tabulated, and applying them to Horava-Lifshitz gravity is argued to force that theory's Lorentz-breaking scale well below the Pla...
Large and infinite extra dimensions
hep-ph 2001-04 accept novelty 3.0 of 10

A pedagogical survey of brane-world models in which extra dimensions are large or infinite, covering how matter is trapped on a brane and what observable signatures large, warped or infinite extra dimensions would produce.
Revisiting Varying Speed of Light in Cosmology: Insights from the Friedmann-Lema\^itre-Robertson-Walker Metric
physics.gen-ph 2025-05 reject novelty 2.0 of 10

A varying speed of light in FLRW is presented as gauge freedom via the lapse function, but the key variational derivation omits the √-g measure and the claimed Hubble-tension resolution contradicts Eq. (30).
Light Sterile Neutrinos: A White Paper
hep-ph 2012-04 accept novelty 2.0 of 10

Three neutrino anomalies are each consistent with an eV-scale sterile neutrino but sit in 2-3 sigma tension with disappearance data, so the paper argues for a dedicated new short-baseline program rather than a discove...
Quantum Spacetime Phenomenology
gr-qc 2008-06 accept novelty 2.0 of 10

A comprehensive review argues that gamma-ray, cosmic-ray, and interferometric data analyses have reached genuine Planck-scale sensitivity to candidate quantum-spacetime effects, while acknowledging that most resulting...
Various constraints on BSM physics from extensive air showers and from ultra-high energy gamma-ray and neutrino searches
astro-ph.HE 2025-01 conditional novelty 1.0 of 10

A faithful review of the strongest current bounds on Lorentz violation, superheavy dark matter, cosmic strings, and sterile neutrinos from ultra-high-energy cosmic-ray, gamma-ray, and neutrino searches.
Testing the universality of quantum gravity theories with cosmic messengers in the context of DSR theories
gr-qc 2025-01

How later papers used it

146 later papers cite it

Evidence: substantial. 145 of the citing papers share no author with it, and 18 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 242 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 4 papers with this label

Derived the threshold condition by getting the effective mass of superluminal neutrino following this paper, taking the LIV parameters of pion and muon to be zero.

Constraints and tests of the OPERA superluminal neutrinos (2011) · 0.81 · paraphrased: the licence does not allow quoting
Chose to work in the preferred frame in which the CMBR is isotropic following this paper, where rotational invariance implies no directional dependence for b.

CPT violation in long baseline neutrino experiments: a three flavor analysis (2008) · 0.80 · paraphrased: the licence does not allow quoting

How the literature uses it

121 arXiv papers cite this in 197 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 · 112 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
This can be considered as a generalization of the discussion in [ref] , where the rotational invariant case has been studied.

CPT, Lorentz invariance and anomalous clash of symmetries · A semiclassical argument · 0.93

Uses its method or tool 4 papers

To derive the threshold condition we get the effective mass of superluminal neutrino following CG [ref] as [equation] where we take the LIV parameters of pion and muon to be zero as they can be much smaller than the neutrino LIV parameter.

Constraints and tests of the OPERA superluminal neutrinos · Introduction · 0.81
The method described in this section was originally proposed in [ref] and further developed in [ref] .

Non-minimal Lorentz invariance violation in light of muon anomalous magnetic moment and long-baseline neutrino oscillation data · Isotropic violation · 0.98
Following [ref] , we choose to work in the preferred frame in which the CMBR is isotropic, where the rotational invariance implies no directional dependence for $b$.

CPT violation in long baseline neutrino experiments: a three flavor analysis · Introduction · 0.80

Uses its data or values 5 papers

However, we know [ref] that $v_e=1$ to a precision of at least $10^-15$.

New Constraints on Neutrino Velocities · Introduction and Conclusions · 0.85
The experimental bounds on $δc^2$ for ordinary particles are of the order of $10^-(21÷23)$ [ref] , which give an idea of the order of magnitude of the fine–tuning that is needed.

Renormalization group in Lifshitz-type theories · Introduction · 0.94
We know that for the electron, $v_e/c_0≤1$ to a precision of at least $10^-15$ [ref] .

Bimetric Relativity and the Opera Neutrino Experiment · Background Gauge Field and the OPERA Neutrino Speeds · 0.83

Compares with it 4 papers

This gives, upon identifying the pion energy $E∼2 ω$, the following cutoff energy for neutral-pion decay: E_π^0^ (cutoff) ∼m_π^0/√(-2 ) , which agrees with the kinematic result of Ref. [ref] .

Changes in extensive air showers from isotropic Lorentz violation in the photon sector · Heuristics of Lorentz-violating decays · 0.96
It should be noticed that, here the power dependence of $k_th$ on $ξ_1$, $-1/3$, differs from that of Ref. [ref] , $-1/2$, because there the energy dependence of LV in terms of the maximal attainable velocity [ref] is different from that of the space-time foam model.

Lorentz violation effects on astrophysical propagation of very high energy photons · Photon decay: a cutoff in the $γ$-ray spectra · 0.97
This distinguishing feature sets meVSL apart from traditional VSL models, which often assume or necessitate an explicit mechanism driving the evolution of the speed of light [ref] .

Revisiting Varying Speed of Light in Cosmology: Insights from the Friedmann-Lema\^itre-Robertson-Walker Metric · Introduction · 0.83

Cites it as support 3 papers

However, even if we assume CPT invariance we need a strong fine tuning on dimension 2 and dimension 3 operators at low energies to be in agreement with the strict constraints on Lorentz violation [ref] .

On Lorentz-Violating Supersymmetric Quantum Field Theories · Introduction · 0.97
The cutoff is predominant, instead of a suppression, because the time scale of the photon decay is quite short and the process takes place in no time compared to their time-of-flight [ref] .

Lorentz violation effects on astrophysical propagation of very high energy photons · Photon decay: a cutoff in the $γ$-ray spectra · 0.98
Indeed, for renormalizable operators, where the phase space does not grow quickly enough, reactions of the type we consider here never achieve the necessary rate given current bounds on their coefficients, even for UHE neutrinos over cosmological distances [ref] .

Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation · Introduction · 0.96

Cites it as background 57 papers

The paper [ref] develops a model with small violation of the Lorentz invariance which does not respect flavour conservation allowing the Maximum Attainable Velocity (MAV) for each massive particle to be different from the speed of light and different between different particles.

Constraints on Lorentz invariance from the event KM3-230213A · Introduction · 0.85
Recently it has been suggested that Lorentz-invariance and $CPT$ violating interactions originating at the Planck scale can also lead to neutrino oscillations, with or without neutrino mass [ref] .

Challenging Lorentz noninvariant neutrino oscillations without neutrino masses · Introduction · 0.88
In this paper, we take into account the impacts of the Lorentz violation (LV) [ref] on the flux of $ν̅_e$ around the resonance energy and show an interesting observation that the GR events may be damped by the LV.

Connect the Lorentz Violation to the Glashow Resonance Event · Introduction · 0.82

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] [4]

S. Coleman and S. L. Glashow, Phys. Lett. B405 (1997) 249

View on Pith 1997

[2] [1]

E. g., M. P. Hagen and C. M. Will, Phys. Today 40 (May 1987), 69

View on Pith 1987

[3] [2]

S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and E. N. Fortson, Phys. Rev. Lett. 57 (1986) 3125

View on Pith 1986

[4] [3]

C.J. Berglund, L.R. Hunter, D. Krause, Jr., E.O. Prigge, and M.S. Romfeldt, Phys. Rev. Lett. 75 (1995) 1879

View on Pith 1995

[5] [5]

D. Colladay and V.A. Kosteleck´ y, Phys. Rev. D55 (1997) 6 760, D58 (1998) 116002, and earlier references therein

View on Pith 1997

[6] [6]

E.g., S. Weinberg, The Quantum Theory of Fields I , §4.6 and §4.8, Cambridge U. Press, 1995

View on Pith 1995

[7] [7]

W. Bardeen, Phys. Rev. 184 (1969) 1848; S. Adler and W. Bar deen, Phys. Rev. 182 (1969) 1517; W. Bardeen and B. Zumino, Nucl. Phys. B244 (1984 ) 421

View on Pith 1969

[8] [8]

Versions of this section appear in: S.L. Glashow, in Proc . TAUP-97 Nucl.Phys. B70 Proc. Suppl. (1999) 70; S. L. Glashow, in Proc. First Tropica l Workshop, San Juan, Puerto Rico, ed. Jos´ e F. Nieves, AIP Conf. Proc. 444 (1998) 1 19

View on Pith 1999

[9] [9]

A.I. Derbin et al., Phys. At. Nucl. 57 (1994) 222

View on Pith 1994

[10] [10]

S. Coleman and S.L. Glashow, HUTP-98/A076

View on Pith

[11] [11]

S. Coleman and S.L. Glashow, HUTP-98/A080, to appear in Proc. Neutrino ’98, Takayama, Japan

View on Pith

[12] [12]

R.D. Bolton, Phys. Rev D38 (1988) 2077

View on Pith 1988

[13] [13]

E.g., F. Combley, F. J. M. Farley and E. Picasso, Phys. Rep. 68 (1981 ) 93

View on Pith 1981

[14] [14]

As reported to us by B.V. Sreekantan, private communica tion

View on Pith

[15] [15]

R. Cowsik and B.V. Sreekantan, hep-ph/9811241 (to appear in Physics Letters B), and private communication

View on Pith Pith review arXiv

[16] [16]

L. Stodolsky, Phys. Lett. B201 (1988) 353

View on Pith 1988

[17] [17]

E. Waxman and J. Bahcall, Phys. Rev. Lett. 78 (1997) 2292

View on Pith 1997

[18] [18]

M. Gasperini, Phys. Rev. D38 (1988) 2635; A. Halprin and C.N. Leung, Phys. Rev. Lett. 67 (1991) 1833

View on Pith 1988

[19] [19]

R. Foot, C.N. Leung, and O. Yasuda, hep-ph/9809458

View on Pith Pith review arXiv

[20] [20]

T. Kajita (for the SuperKamiokande Collaboration), to appear in Proc. Neutrino ’98, Takayama, Japan

View on Pith

[21] [21]

C. Angelini et al., Phys. Lett. B179 (1986) 307

View on Pith 1986

[22] [22]

E.B. Brucker et al., Phys. Rev. D34 (1986) 2183

View on Pith 1986

[23] [23]

L.H. Hunter, private communication

View on Pith

[24] [24]

T. Hambye, R.B. Mann and U. Sarkar, Phys. Lett. B421 (199 8) 105

View on Pith

[25] [25]

V. A. Kosteleck´ y, Phys. Rev. Lett. 80 (1998) 1818 and In diana University preprint IUHET 396 hep-ph/9810352. 31

View on Pith Pith review arXiv 1998

[26] [26]

K. Greisen, Phys. Rev. Lett. 16 (1966) 748

View on Pith 1966

[27] [27]

G.T. Zatsepin and V.A. Kuz’min, JETP Lett. 41 (1966) 78

View on Pith 1966

[28] [28]

AGASA Collab. e.g., M. Takeda et.al., Phys. Rev. Lett. 81 (1998) 1163; Fly’s Eye Collab. e.g., D.J. Bird et al., Astrophys. J. 424 (1995 144; Haverah Park Collab. See: M.A. Lawrence et al., J. Phys. G17 (1991) 733; Yakutsk Collab. e.g., N.N. Eﬁmov et al., in 22nd Intl. Cosmic Ray Conf. (1991) Dublin

View on Pith 1998

[29] [29]

See M. Takeda [28] and V. Berezinsky, Nucl. Phys. Proc. S uppl. 70 (1999) 419 for further references in these connections

View on Pith 1999

[30] [30]

T.W. Kephart and T.J. Weiler, Astro. Part. Phys. 4 (1996 ) 271

View on Pith 1996

[31] [31]

T.J. Weiler, hep-ph/9710431

View on Pith Pith review arXiv

[32] [32]

V. Berezinsky, M. Kachelrieß, and V. Vilenkin, Phys. Re v. Lett. 79 (1997) 4302

View on Pith 1997

[33] [33]

A preliminary version of this section appears as S. Cole man and S.L. Glashow, HUTP- 98/A075, hep-ph/9808446

View on Pith Pith review arXiv

[34] [34]

G.R. Farrar and P.L. Biermann, Phys. Rev. Lett. 81 (1998 ) 3579

View on Pith 1998

[35] [35]

C.T. Hill and D.N. Schramm, Phys. Rev. D31 (1985) 564

View on Pith 1985

[36] [36]

E.N. Fortson, private communication

View on Pith

[37] [37]

S. Carroll, G. Field and R. Jackiw, Phys. Rev. D41 (1990) 1231; M. Goldhaber and V. Trimble, J. Astrophys. Astr. 17 (1996) 17 . See also: J.F.C. Wardle, R.A. Perley and M.H. Cohen, Phys. Rev. Lett. 79 (1997 ) 1801; J.P. Leahy, astro-ph/9704285; S. Carroll and G. Field, Phys. Rev. Lett. 79 (1997) 2394

View on Pith Pith review arXiv 1990

[38] [38]

R. Eden, P. Landshoﬀ, D. Olive, and J. Polkinghorne, The Analytic S-Matrix , Cam- bridge University Press, 1966. 32

View on Pith 1966