Lorentz Invariance Violation (LIV)

• Lorentz Invariance Violation (LIV) is a deviation from exact Lorentz symmetry, characterized by modified dispersion relations and anisotropic effects observable in multiple sectors.
• LIV is explored through techniques such as photon time-of-flight, vacuum birefringence measurements, and threshold anomaly analyses in high-energy astrophysics and laboratory experiments.
• Current research balances stringent null results with intriguing positive claims from GRB observations, cosmic rays, and neutrino oscillation studies, driving joint multi-observable analyses.

Lorentz invariance violation (LIV) denotes a departure from exact Lorentz symmetry, a cornerstone of special relativity, and is commonly treated as a possible low-energy signature of quantum gravity or of effective theories beyond the Standard Model. In current research practice, LIV is parameterized through modified dispersion relations, anisotropic kinetic operators, polarization-dependent propagation, or shifted reaction thresholds, and is tested with photon time-of-flight, vacuum birefringence, gamma-ray opacity, neutrino oscillations, ultrahigh-energy cosmic rays, collider resonances, and atomic spectroscopy (Desai, 2023, Scully et al., 2010, Shafeei et al., 2022). Across these domains, the dominant empirical pattern is the accumulation of stringent lower bounds rather than a settled positive detection, although recent interpretations of GRB 221009A and the Carpet $300\ \mathrm{TeV}$ event have reopened the question in a concrete way (Galanti et al., 2 Apr 2025).

1. Formal frameworks and parameterizations

Three parameterizations recur throughout the literature. In phenomenological modified-dispersion approaches, one writes

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$

with $n=1,2$ the most commonly tested orders, $s=\pm1$ tracking subluminal or superluminal behavior, and $E_{\mathrm{QG}}$ the effective LIV scale (Desai, 2023). Closely related photon-sector studies use

$$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$

for subluminal photon propagation at high energy (Li et al., 2023). In the Coleman–Glashow framework, species-dependent maximum attainable velocities are encoded by

$$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$

which is particularly useful for hadronic and neutrino-threshold problems (Scully et al., 2010).

The Standard-Model Extension (SME) organizes LIV by operator dimension, CPT parity, and spacetime index structure. In the neutrino sector, for example, the effective Hamiltonian may be written

$$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$

with $(a_L)$ CPT-odd and $(c_L)$ CPT-even (Desai, 2023). In photon studies, CPT-odd dimension-5 operators are closely tied to birefringence, whereas quadratic $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$0 terms are often treated as CPT-even and nonbirefringent (Terzić et al., 18 Sep 2025). This suggests that sign conventions and sector assignments are model-dependent across the literature rather than universal.

A more geometric, isotropy-preserving alternative appears in Homogeneously Modified Special Relativity, where the neutrino-sector modified dispersion relation is written

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$1

leading, in the ultra-relativistic limit, to oscillation-phase shifts

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$2

In this construction, LIV effects arise from differences among the effective dispersion parameters of the mass eigenstates (Torri, 2021).

2. Principal observables: time delays, birefringence, and threshold anomalies

For transient sources, the standard cosmological time-of-flight observable is

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$3

with $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$4 for subluminal and $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$5 for superluminal conventions in that formulation (Abdalla et al., 2023). The cosmological dependence of this integral has been tested against $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$6CDM, CPL, quadratic-$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$7, and Padé dark-energy parameterizations, with the relative difference in predicted time lags found to be $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$8, smaller than current or near-future systematics (Abdalla et al., 2023). Cosmology is therefore not presently the limiting uncertainty in most GRB time-of-flight LIV analyses.

Vacuum birefringence probes a different sector of the theory. In a representative EFT form,

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$9

so polarization rotation accumulates over cosmological baselines (Desai, 2023). The review literature highlights $n=1,2$0 from GRB 061122 as the most stringent quoted prompt-polarization constraint in that dataset (Desai, 2023).

Threshold anomalies are central to LIV tests with high-energy photons. In Lorentz-invariant propagation, attenuation by $n=1,2$1 is written

$n=1,2$2

with $n=1,2$3 obtained by integrating the Breit–Wheeler cross section over redshift, angles, and background photons (Li et al., 2023, Galanti et al., 2 Apr 2025). In a leading-order LIV treatment, the pair-production threshold shifts according to

$n=1,2$4

where $n=1,2$5 encodes the net photon and lepton-sector LIV contribution (Martínez-Huerta et al., 2019). Subluminal photon-sector LIV shifts the threshold upward, reduces opacity, and can produce a recovery or hardening in attenuated spectra at the highest energies.

A systematic optical-depth search using $n=1,2$6 measured TeV spectra from $n=1,2$7 sources, of which $n=1,2$8 spectra from $n=1,2$9 sources had sufficient highest-energy leverage, reported no significant evidence for LIV and derived lower limits

$s=\pm1$0

$s=\pm1$1

at $s=\pm1$2, $s=\pm1$3, and $s=\pm1$4, respectively, for subluminal photon-sector scenarios (Martínez-Huerta et al., 2019).

3. Gamma-ray astronomy, GRB 221009A, and the current transparency debate

GRB 221009A has become the central modern case study because it combines a well-measured redshift, $s=\pm1$5 or $s=\pm1$6, with very-high-energy photons observed by multiple instruments (Li et al., 2023, Galanti et al., 2 Apr 2025). LHAASO recorded more than $s=\pm1$7 photons in $s=\pm1$8–$s=\pm1$9 with WCDA, while a preliminary KM2A report indicated possibly more than $E_{\mathrm{QG}}$0 photons up to $E_{\mathrm{QG}}$1 (Li et al., 2023). Under standard EBL attenuation with the Domínguez (2011) model and a WCDA-normalized intrinsic power law, the critical energy for at least one expected event above $E_{\mathrm{QG}}$2 is $E_{\mathrm{QG}}$3 for $E_{\mathrm{QG}}$4, whereas requiring “no more than $E_{\mathrm{QG}}$5 photon” yields $E_{\mathrm{QG}}$6 (Li et al., 2023). In the same analysis, a photon LIV scale $E_{\mathrm{QG}}$7 was found sufficient to raise $E_{\mathrm{QG}}$8 to $E_{\mathrm{QG}}$9–$$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$0, making $$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$1 transparency quantitatively viable within that model, though explicitly as an explanatory requirement rather than a statistical upper limit (Li et al., 2023).

A separate time-of-flight analysis of the publicly available $$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$2–$$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$3 LHAASO data found no evidence for an energy-dependent delay and derived $$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$4 CL lower limits

$$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$5

for $$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$6, and

$$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$7

for $$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$8 (Piran et al., 2023). These bounds are independent of the transparency anomaly and instead use the absence of detectable spectral-temporal distortions across the rising, peak, and decay phases of the TeV afterglow.

The most controversial development is the Carpet result. Using the full-detector, one-day analysis, Carpet reported a single photon-like event at

$$\omega(k)^2 \approx k^2 - \xi k^3, \qquad \xi^{-1}\equiv LIV,$$9

temporally and directionally coincident with GRB 221009A, with chance probability $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$0 and probability of a misidentified hadron $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$1 (Galanti et al., 2 Apr 2025). Taking this event at face value, the inferred expected counts in the $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$2–$$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$3 Carpet band are

$$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$4

for a photon-sector linear LIV scale

$$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$5

(Galanti et al., 2 Apr 2025). In that interpretation, conventional propagation is effectively excluded, the ALP parameter region that can account for LHAASO photons up to $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$6 fails at $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$7, and LIV becomes the only mechanism among those tested that reproduces the Carpet count. The same paper outlines a coexistence scenario in which ALPs explain transparency up to $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$8 while LIV controls the $$E^2 = p^2 + m^2 + 2\delta p^2, \qquad c_{MAV}\simeq 1+\delta, \qquad \delta_{ij}\equiv c_i-c_j,$$9 regime (Galanti et al., 2 Apr 2025).

At present, the GRB 221009A literature therefore splits into three logically distinct claims: robust lower limits from time-of-flight analyses, model-dependent transparency requirements at $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$0–$$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$1, and a conditional claim of first evidence for LIV tied to a single Carpet event at $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$2 (Piran et al., 2023, Li et al., 2023, Galanti et al., 2 Apr 2025).

4. UHECRs, neutrinos, cascades, and source physics

In the hadronic sector, a classic observable is photomeson production by ultrahigh-energy protons on the CMB. In the Coleman–Glashow framework, the relevant parameter is

$$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$3

and the modified threshold condition is

$$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$4

so positive $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$5 inhibits $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$6 interactions at the highest energies (Scully et al., 2010). In the corresponding neutrino flux, increasing $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$7 lowers the peak of $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$8 and suppresses the highest-energy tail; for $$(H_{\mathrm{eff}})_{ab} = \frac{(m^2)_{ab}}{2E} + (a_L)^\mu_{ab}p_\mu - (c_L)^{\mu\nu}_{ab}\frac{p_\mu p_\nu}{E},$$9, the peak lies near $(a_L)$0 (Scully et al., 2010). The same analysis argued that ARIANNA, with $(a_L)$1 years of exposure, can distinguish LIV effects if $(a_L)$2 (Scully et al., 2010).

Neutrino oscillation phenomenology supplies a separate precision channel. In SME-based long-baseline analyses the flavor-basis Hamiltonian is written

$(a_L)$3

with CPT-odd coefficients $(a_L)$4 and CPT-even coefficients $(a_L)$5 in $(a_L)$6 (Sarker et al., 2023). DUNE studies in the isotropic limit found that the appearance channel is most sensitive to $(a_L)$7 and $(a_L)$8, with $(a_L)$9 tending to enhance CP-violation sensitivity and $(c_L)$0 tending to degrade it (Sarker et al., 2023). In the isotropy-preserving HMSR framework, the modified phase

$(c_L)$1

produces visible effects at $(c_L)$2 for $(c_L)$3, while effects are not visible at the plotted scale for $(c_L)$4 (Torri, 2021).

Electromagnetic cascades extend LIV tests beyond single interactions. CRPropa-based simulations with LIV-modified photon and electron dispersion relations show that altered $(c_L)$5 thresholds and mean free paths can produce anomalous transparency, shifted cutoffs, and suppression-recovery patterns in intergalactic cascades (Saveliev et al., 2023, Saveliev et al., 2023). In the $(c_L)$6 phenomenology of photon decay and vacuum Cherenkov emission, the mere observation of very-high-energy photons implies

$(c_L)$7

from HEGRA and HESS events in the framework where photons are LIV-modified and charged fermions remain Lorentz invariant (Martínez-Huerta et al., 2017). UHECR photopion inelasticity calculations further show that LIV in the pion sector raises the photopion threshold and lengthens the attenuation length, whereas LIV in the proton sector lowers the threshold and shortens it (Lang et al., 2017).

Source physics itself is also LIV-sensitive. In synchrotron, inverse Compton, and first-order Fermi acceleration models with rotationally invariant modified dispersion relations,

$(c_L)$8

which can generate high-energy excesses and perturbative divergences near the LIV barrier in synchrotron-self-Compton spectra (Duarte et al., 9 Jul 2025). A separate treatment of Fermi acceleration found that first-order Fermi spectra are strongly suppressed above the LIV break, while second-order Fermi spectra harden at high energy (Duarte et al., 2024). These source-frame effects complicate any attempt to attribute a hard observed spectrum uniquely to propagation-induced LIV.

5. Laboratory, collider, and atomic searches

Atomic tests access the static photon sector of the minimal SME. Starting from the modified Green’s function, one obtains the anisotropic Coulomb potential

$(c_L)$9

which induces corrections to hydrogen and helium spectra, as well as Stark-, Zeeman-, and spin-orbit-type observables (Shafeei et al., 2022). Using spectroscopic accuracies adopted in that analysis, the quoted bounds on the common effective parameter $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$00 are

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$01

from hydrogen,

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$02

from helium,

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$03

from the permanent Stark effect, and

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$04

from the spin–orbit effect (Shafeei et al., 2022). These are much weaker than astrophysical birefringence bounds but probe different combinations of SME coefficients and very different systematics.

Loop calculations in QED show that LIV introduced in one sector is not radiatively confined there. A photon-sector or interaction-sector LIV insertion induces SME-like kinetic LIV in other sectors through self-energy and vacuum-polarization graphs, so constraints on $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$05 propagate into otherwise weakly constrained parameters (Kepuladze, 22 Apr 2025). In that study, interaction-based LIV effects would require unrealistically large parameters, $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$06, for direct detection through cross-section distortions, whereas dispersion modifications can be probed through resonance observables down to

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$07

at the LHC (Kepuladze, 22 Apr 2025).

A concrete collider implementation modifies the $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$08-boson mass shell as

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$09

with $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$10 timelike, spacelike, or lightlike (Jejelava et al., 15 Apr 2025). The resulting momentum-direction-dependent effective mass and width distort the Drell–Yan resonance most strongly near the forward region $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$11, and spacelike or lightlike $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$12 generate sidereal-time modulations through Earth’s rotation (Jejelava et al., 15 Apr 2025). The proposed ATLAS/CMS strategy targets sensitivity to

$$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$13

or $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$14 optimistically, making collider resonance studies a direct electroweak-sector complement to astrophysical propagation tests (Jejelava et al., 15 Apr 2025).

6. Status, tensions, and emerging directions

The contemporary status of LIV is defined by tension between very strong null results and a small number of provocative, mutually inconsistent positive claims. Review-level assessments note that some GRB spectral-lag “detections” imply $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$15 far below the Planck scale yet conflict with tighter bounds from GRB 090510, AGN flares, polarization, and threshold tests; stacked analyses yield poor goodness-of-fit and generally favor intrinsic spectral evolution over universal LIV (Desai, 2023). This is the main reason claims of positive detection in time-of-flight data are not regarded as established.

The GRB 221009A situation is more nuanced. The burst supports stringent lower limits from time-of-flight analyses, can be modeled with sub-Planckian photon LIV to explain $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$16 transparency, and has been interpreted as possible first evidence for LIV if the Carpet $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$17 event is confirmed (Piran et al., 2023, Li et al., 2023, Galanti et al., 2 Apr 2025). A plausible implication is that the field has shifted from generic “Planck-scale phenomenology” to highly instrument-specific, cross-validated tests in which source modeling, background rejection, EBL assumptions, and inter-experiment consistency dominate the interpretation.

Methodologically, the direction of travel is toward joint inference across observables rather than one-effect-at-a-time analyses. A recent ANN-based study trained on simulated Mrk 501 data combined energy-dependent time-of-flight and modified $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$18 absorption in a single quadratic, subluminal LIV model. A dense ANN reconstructed the common LIV parameter poorly, with $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$19, whereas a sequence-to-regression transformer showed strong correlation between injected and reconstructed $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$20 up to $$E^2 \simeq p^2 c^2 + m^2 c^4 + s\,\eta_n\,\frac{p^{2+n}c^{2+n}}{E_{\mathrm{QG}}^{\,n}}, \qquad v(E) \approx c\left[1 - s\,\frac{n+1}{2}\left(\frac{E}{E_{\mathrm{QG}}}\right)^n\right],$$21, then correctly entered an LI-like plateau beyond the dataset’s reach (Terzić et al., 18 Sep 2025). This suggests that future analyses will increasingly rely on joint timing, spectral, and possibly polarization information.

The most concrete future tests already identified in the literature are additional GRBs at known redshift with multi-instrument coverage, cross-checks of energy reconstruction and background rejection, polarization searches for ALP–LIV coexistence scenarios, and time-of-flight studies at higher energies than were publicly available for GRB 221009A (Galanti et al., 2 Apr 2025, Piran et al., 2023). More broadly, CTA, LHAASO, SWGO, IceCube-Gen2, next-generation CMB polarization experiments, and future GW interferometers extend LIV tests across photons, neutrinos, and gravity (Desai, 2023). At present, the balance of evidence still favors exact Lorentz invariance within experimental reach, but the subject remains active because the surviving parameter space is now probed simultaneously by threshold physics, propagation effects, oscillation phases, and resonance spectroscopy rather than by any single class of observable.