# n=2 (quadratic) LIV bounds in the photon sector — observational numbers with sources

Scout thread, 2026-09-19, Argus. All bounds in this file are quadratic (n=2, i.e. d=6)
Lorentz-invariance violation in the photon dispersion relation. **Every number is reported
with the convention the source paper actually uses; nothing has been silently converted.**

Convention map (the trap): in the standard parametrisation
`v/c = 1 − s·(E/E_LV,2)^2`, one paper's `s`, `σ`, or `s_n` can have the *opposite* sign
meaning of "subluminal" vs "superluminal". Listed per source below.

---

## VERDICT

- **Best superluminal n=2 bound (photon stability; strongest of any channel):**
  **E_LV,2^(sup) > 10⁻³ M_Pl ≈ 1.22×10¹⁶ GeV**, 95% CL — LHAASO Collaboration,
  *Phys. Rev. Lett.* **128**, 051102 (2022), arXiv:2106.12350, from the absence of a
  spectral cutoff due to photon splitting γ→3γ (superluminal branch) in the PeV spectra of
  Crab Nebula (J0534+2202) and J2032+4102. Convention used there:
  `E_γ² − p_γ² = +|α₂|·p_γ⁴` for the superluminal case (their Eq. 1). This bound is
  **BELOW the Planck scale by — exactly — 3 orders of magnitude** (see Q4).
- **Best subluminal n=2 bound (time-of-flight):**
  **E_QG,2 > 12.0×10¹¹ GeV = 1.2×10¹² GeV**, 95% CL — Yu-Ming Yang et al. (analysis of
  LHAASO observation of GRB 221009A; arXiv metadata: "Yu-Ming Yang and 2 other authors",
  full author list not read at source), JCAP 04 (2024) 060, arXiv:2312.09079,
  maximum-likelihood method, subluminal branch. NB their sign convention has σ = −1 for
  subluminal (opposite of the `s` in the request header; see Q2).
  A stronger-claimed but **unrefereed** preprint (Xi & Shu, arXiv:2508.00656, Aug 2025)
  quotes E_QG,2 > 10.0×10¹² GeV (subluminal) / 2.4×10¹² GeV (superluminal).

The headline contrast: for n=1 LHAASO excludes superluminal LIV down to ~10⁵ M_Pl
(above Planck); for n=2 the same data only exclude down to **10⁻³ M_Pl** — eight orders of
magnitude weaker, and the quoted lower limit sits *below* the Planck scale.

---

## Q1. Superluminal n=2 from photon decay / pair-production thresholds

**Reference verified at source** (fetched journals.aps.org abstract page AND arXiv:2106.12350
abstract + full HTML):
LHAASO Collaboration (Zhen Cao et al.), "Exploring Lorentz Invariance Violation from
Ultrahigh-Energy γ Rays Observed by LHAASO", **Phys. Rev. Lett. 128, 051102 (2022)**,
published 3 Feb 2022, DOI 10.1103/PhysRevLett.128.051102, arXiv:2106.12350. ✔

- Data: LHAASO-KM2A half-array, Dec 2019–Nov 2020 (301.7 d live time); two highest-energy
  sources studied: LHAASO J2032+4102 (highest photon ~1.4 PeV) and J0534+2202 = Crab Nebula
  (highest ~0.88 PeV). No spectral cutoff found; 95% CL lower limits derived with the CLs
  method on the cutoff energy: **E_cut > 750 TeV (Crab) and > 1140 TeV (J2032+4102)**.
- Physics used: in the superluminal branch photons decay γ→e⁺e⁻ above threshold, and split
  γ→3γ (no threshold, but rate grows steeply with energy); either produces a hard spectral
  cutoff.
- **Quoted n=2 result (their Table 1 + abstract): the second-order LIV energy scale is
  > 10⁻³ M_Pl**, M_Pl ≈ 1.22×10²⁸ eV = 1.22×10¹⁹ GeV, as derived from the γ→3γ process.
  In GeV: **E_LV,2 > 1.22×10¹⁶ GeV**. Units: fraction of the Planck scale (10⁻³ E_Pl),
  equivalently 1.22×10¹⁶ GeV / 1.22×10²⁵ eV.
- Convention used in that paper (their Eq. 1): `E_γ² − p_γ² = ±|α_n| p_γ^{n+2}` where +
  = superluminal, − = subluminal, and `E_LIV^(n) = α_n^(−1/n)`. Both processes studied
  (γ→e⁺e⁻ and γ→3γ) constrain the **superluminal** branch. In the request header's
  parametrisation this is the **s = −1** case.
- Their n=1 (linear) superluminal bound for comparison: > 1.42×10³³ eV ≈ 10⁵ M_Pl — the
  famous "five orders above Planck" result. The n=2 bound from the same analysis is the
  γ→3γ value above; the γ→e⁺e⁻-derived n=2 limit is reported as consistent/nearly the same
  as the 1140 TeV cutoff value.

**Related, verified at source:** Li & Ma, "Ultrahigh-energy photons from LHAASO as probes of
Lorentz symmetry violations", Phys. Rev. D 104, 063012 (2021), arXiv:2105.07967 — single
1.42 PeV photon, threshold-theorem estimate, superluminal branch, their Eq. (2)/(4)/(7):
E_LV,n=2^(sup) ≳ (E_γ·E_LV,1^(sup))^1/2 ≈ **1.97×10¹⁵ GeV** (their Eq. 7), weaker than the
LHAASO collaboration's full spectral analysis (1.2×10¹⁶ GeV vs 1.97×10¹⁵ GeV) but a clean
independent single-photon estimate. Their convention: `E_γ² = p²c²[1 − s_n(pc/E_LV,n)^n]`
with s_n = −1 superluminal (their sign convention matches the request header).

Nothing in the newer literature (searched through 2026-09) beats the LHAASO 2022 γ→3γ
number for superluminal n=2 photon stability.

---

## Q2. Subluminal n=2 bounds from time-of-flight

**Classic — Fermi-LAT GRB 090510. Verified at source** (arXiv:1305.3463 abstract + HTML):
Vasileiou, Jacholkowska, Piron, Bolmont, Couturier, Granot, Stecker, Cohen-Tanugi, Longo,
"Constraints on Lorentz Invariance Violation from Fermi-Large Area Telescope Observations of
Gamma-Ray Bursts", **Phys. Rev. D 87, 122001 (2013)**. ✔
- Headline (subluminal case, i.e. their s_± = +1, 95% CL, no source-intrinsic dispersion
  assumed): **E_QG,2 > 1.3×10¹¹ GeV** for quadratic leading-order vacuum dispersion, from
  GRB 090510. (Their n=1: E_QG,1 > 7.6 E_Pl.)
- Their convention (their Eq. 1): `E² ≃ p²c² [1 − s_± (E/E_QG)^n]`, s_± = +1 for decrease
  of photon speed with energy ("subluminal"), −1 superluminal. Same sign convention as the
  request header.
- Caveat, stated plainly: the **subluminal** value is the one in the abstract. They also
  derived superluminal (s_± = −1) limits in their Table 4 / Fig. 10, but those table numbers
  did not survive my HTML extraction, so I cannot quote an exact superluminal E_QG,2 from
  this paper. It does not affect the verdict (see below; LHAASO TOF values are stronger and
  I did read those at source).

**Current best peer-reviewed TOF bounds — LHAASO GRB 221009A. Verified at source**
(arXiv:2312.09079 abstract, v2):
Yu-Ming Yang?, et al., "Constraints on Lorentz invariance violation from the LHAASO
observation of GRB 221009A", **JCAP 04 (2024) 060**, arXiv:2312.09079. ✔
- Data: LHAASO KM2A + WCDA, afterglow of GRB 221009A, ~0.2–18 TeV, two event-by-event
  methods (pair-view and maximum-likelihood). ML method, 95% CL, n=2:
  **E_QG,2 > 12.0×10¹¹ GeV (subluminal)** and **E_QG,2 > 7.2×10¹¹ GeV (superluminal)**.
  n=1: > 14.7 (6.5)×10¹⁹ GeV subluminal (superluminal).
- **⚠ Convention trap:** their Eq. (1) is `E² = p²c²[1 + σ(E/(ε_QG,n·E_Pl))^n]` with
  **σ = +1 superluminal, σ = −1 subluminal** — the *opposite* of the request header's s.
  The numbers above are quoted on their own σ convention: do not flip them when copying.

**LHAASO collaboration TOF, verified at source** (arXiv:2402.06009 abstract + full HTML, v2;
published as **Phys. Rev. Lett. 133, 071501 (2024)**):
"Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A",
LHAASO Collaboration. ✔
- 95% CL: **E_QG,2 > 6×10⁻⁸ E_Pl** (quadratic) — abstract headline; full text breaks it out:
  **E_QG,2 > 6.9×10¹¹ GeV (subluminal)** and **> 7.0×10¹¹ GeV (superluminal)**, and states
  these "represent the best time-of-flight limit, improving previous bounds [Vasileiou et al.]
  by a factor of 5 (7)". Their convention (their Eq. 1): `E² ≃ p²c²[1 − s(E/E_QG,n)^n]`,
  s = ±1 subluminal/superluminal — matches the request header.
- Note: arXiv v2 (Aug 2024) and v3 (Feb 2026) both carry the same numbers; the paper
  acknowledges receipt of the overlapping JCAP analysis ("similar bounds were given in [51]").
  The two analyses agree to within factors ≲1.7; numerically the JCAP paper is slightly the
  stronger (1.2×10¹² vs 6.9×10¹¹ GeV subluminal).
- For the record, from the same full text (verified as-quoted): MAGIC's GRB 190114C analysis
  gives E_QG,2 > 6.3×10¹⁰ GeV (subluminal) and > 5.6×10¹⁰ GeV (superluminal); these are
  superseded by the LHAASO results above. (LHAASO cites the MAGIC paper as [23]; I did not
  fetch the MAGIC primary — see reference table.)

**Strongest claimed but unrefereed:** Xi & Shu, "Constraints on Lorentz Invariance Violation
from GRB 221009A Using the DisCan Method", arXiv:2508.00656 (Aug 2025). Verified at source
(abstract, HTML): E_QG,2 > 10.0×10¹² GeV (subluminal) / 2.4×10¹² GeV (superluminal), 95% CL.
Their σ convention is again reversed (σ=+1 superluminal). **Preprint only — no journal
reference found; treat as a claim, not an established bound.**

For completeness, GRB 090510's *linear* flagship (Abdo et al., "A limit on the variation of
the speed of light arising from quantum gravity effects", Nature 462, 331 (2009)) —
E_QG,1 > 1.2 E_Pl — was not the target here; its quadratic numbers were superseded by
Vasileiou et al. 2013 above. (Nature abstract verified via ADS snippet only; its quadratic
numbers not read at source — marked inherited-unchecked.)

---

## Q3. Photon splitting and vacuum Cherenkov

- **Photon splitting γ→3γ: this is where the best superluminal n=2 bound comes from.**
  LHAASO PRL 128, 051102 (2022) [Q1 above]: the > 10⁻³ M_Pl n=2 limit is explicitly derived
  from the γ→3γ process (their text: "The second-order LIV scale reaches 10⁻³ times of the
  Planck scale, as derived from the γ→3γ process"). No other photon-splitting n=2 bound in
  the literature I reached beats it; the paper itself notes its predecessors (HEGRA, Tibet,
  HAWC) were at least an order of magnitude weaker, and are superseded.
- **Vacuum Cherenkov:** this channel constrains *charged-particle* (electron) superluminal
  dispersion, not the photon coefficient directly. Verified at source (arXiv:2204.02956):
  Li & Ma, "Testing Lorentz invariance of electrons with LHAASO observations of PeV gamma-rays
  from the Crab Nebula", Phys. Lett. B 829, 137034 (2022) — uses the absence of vacuum
  Cherenkov emission by inverse-Compton electrons in the Crab to improve *linear-order
  electron* LV bounds by 10⁴. It does **not** produce a photon n=2 bound; it matters only as
  the justification for ignoring electron-side LIV when interpreting the photon thresholds.
- **Photon decay into neutrinos (new channel, checked 2026):** Kepuladze et al.,
  "Lorentz-Violating Photon Decay into Neutrinos and Constraints from PeV Photon Stability",
  arXiv:2607.10404 (Jul 2026) — verified at source (abstract): the γ→νν̄ channel is open
  below the e⁺e⁻ threshold but "generally too slow to provide stronger constraints than
  existing bounds". Conclusion: does not change the Q1 number.

---

## Q4. The comparison number (the one that matters)

E_Pl = 1.22×10¹⁹ GeV (as used by both LHAASO papers).

Best superluminal n=2 bound (LHAASO, PRL 128, 051102):
E_LV,2^(sup) > **10⁻³ M_Pl**.

Arithmetic:
- 10⁻³ × 1.22×10¹⁹ GeV = **1.22×10¹⁶ GeV** (= 1.22×10²⁵ eV)
- E_LV,2 / E_Pl = 1.22×10¹⁶ / 1.22×10¹⁹ = 10⁻³
- log₁₀(E_LV,2 / E_Pl) = **−3.0**

**Plain statement: the best superluminal n=2 bound is BELOW the Planck scale, by three
orders of magnitude.** Observing PeV photons excludes superluminal quadratic LIV only for
energy scales ≲ 1.2×10¹⁶ GeV ≈ 10⁻³ E_Pl; every scale from 10⁻³ E_Pl upward — including the
Planck scale itself — remains entirely untested by this channel.

Cross-check of the two other superluminal n=2 numbers, same units:
- Li & Ma single-photon estimate: 1.97×10¹⁵ GeV ≈ 1.6×10⁻⁴ E_Pl (below Planck, ~4 orders).
- Best superluminal n=2 TOF: 7.2×10¹¹ GeV ≈ 5.9×10⁻⁸ E_Pl (below Planck, ~7 orders).
- Contrast: the n=1 superluminal bound from the same LHAASO data sits at ~10⁵ E_Pl —
  **five orders above** Planck. The quadratic channel is 10⁸ times less sensitive than the
  linear channel with identical data. That is the entire point of the n=2 comparison.

---

## Q5. SME framing (Kostelecký–Mewes / Data Tables)

Yes — time-of-flight analyses are routinely expressed in the SME's **isotropic
dimension-6 coefficient** rather than E_LV,2, exactly because d = n+4 (n=2 → d=6, CPT-even,
no birefringence expected). Verified at source (arXiv:1305.3463 HTML, their Eqs. 3–5):
Vasileiou et al. (2013) derive, from the same GRB data, 95% CL limits on the SME
coefficients **c^(6)_{(I)jm}** (d=6), the isotropic (j=m=0) case being the single coefficient
**c^(6)_{(I)00}**; their Eq. (5) maps it through
τ_n = (1/H₀) Σ_jm [₀Y_jm(n̂) c^(n+4)_{(I)jm}] · κ_n, so the translation for the direction-
independent case reads c^(6)_{(I)00} = s_± · (n+1)/2 · (4π)^1/2 · E_QG,2^(−2)
(with the (n+1)/2 = 3/2 factor inherited from the group velocity). Their own SME coefficient
limits are in their Table 6 (numbers not extractable from my HTML copy — see gaps).

Data Tables reference — **verified at source** (arXiv:0801.0287, v19 "2026 edition",
5 Feb 2026; journal ref `Rev. Mod. Phys. 83, 11 (2011)`, DOI 10.1103/RevModPhys.83.11):
V. A. Kostelecký & N. Russell, "Data Tables for Lorentz and CPT Violation". Confirmed: the
paper is updated ~annually on arXiv (19 versions, 2008→Feb 2026 — version history shows a
new version each January/February, plus the original v1 Jan 2008). Extracted from the 2026
edition's summary tables:
- Table S3 (isotropic coefficients, photon sector): the isotropic d=6 CPT-even coefficient is
  **c^(6)_{(I)00}, best sensitivity 10⁻³⁰ GeV⁻²** (consistency check: (10⁻³⁰ GeV⁻²)^(−1/2)
  ~ 10¹⁵ GeV — i.e. the ~10⁻³ E_Pl ballpark of Q1, so the SME and E_LV,2 bookkeeping agree
  at the order-of-magnitude level).
- Nonminimal photon sector d=6 (Table D22, part 6): the *anisotropic* CPT-even coefficients
  k^(6)_{(E)jm} / k^(6)_{(V)jm} are bounded at ~(7–9)×10⁻¹⁸ GeV⁻² by spectropolarimetry
  (their ref [204]) — a birefringence-type channel, not the isotropic dispersion channel
  discussed here, and listed only for completeness.

The annual-update claim in the request checks out: current edition is v19, Feb 2026.

---

## Reference table

| Ref | Verification | Number used here | Convention |
|---|---|---|---|
| LHAASO Coll., PRL 128, 051102 (2022), arXiv:2106.12350 | **verified-at-source** (APS abstract page + arXiv abs + full HTML) | n=2 (sup): E_LV,2 > 10⁻³ M_Pl = 1.22×10¹⁶ GeV; E_cut > 750/1140 TeV | E²−p²=+|α₂|p⁴ = superluminal (request header s=−1) |
| Li & Ma, PRD 104, 063012 (2021), arXiv:2105.07967 | **verified-at-source** (arXiv abs + HTML) | n=2 (sup): ≳1.97×10¹⁵ GeV (single-photon estimate) | s_n=−1 superluminal (matches header) |
| Vasileiou et al., PRD 87, 122001 (2013), arXiv:1305.3463 | **verified-at-source** (arXiv abs + HTML) | n=2 (sub): E_QG,2 > 1.3×10¹¹ GeV (GRB 090510, 95% CL) | s_±=+1 subluminal (matches header) |
| Yang et al., JCAP 04 (2024) 060, arXiv:2312.09079 | **verified-at-source** (arXiv abs v2) | n=2: >12.0×10¹¹ (sub) / >7.2×10¹¹ GeV (sup) | σ=+1 superluminal — **reversed vs header** |
| LHAASO Coll., PRL 133, 071501 (2024), arXiv:2402.06009 | **verified-at-source** (arXiv abs v3 + HTML v2) | n=2: >6×10⁻⁸ E_Pl; 6.9×10¹¹ (sub) / 7.0×10¹¹ (sup) GeV | s=±1 sub/sup (matches header) |
| Xi & Shu, arXiv:2508.00656 (Aug 2025) | **verified-at-source** (arXiv abs + HTML); **preprint, unrefereed** | n=2: >10.0×10¹² (sub) / >2.4×10¹² GeV (sup) | σ=+1 superluminal — **reversed vs header** |
| Kostelecký & Russell, RMP 83, 11 (2011), arXiv:0801.0287 (2026 ed.) | **verified-at-source** (arXiv abs + HTML v19 tables) | c^(6)_{(I)00} ~ 10⁻³⁰ GeV⁻² (Table S3); k^(6)_{(E)jm} ~ 10⁻¹⁸ GeV⁻² (D22, spectropolarimetry) | SME spherical-coefficient convention |
| Li & Ma, PLB 829, 137034 (2022), arXiv:2204.02956 | **verified-at-source** (arXiv abs) | electron linear LV improved 10⁴ (vacuum Cherenkov) — **not a photon n=2 bound** | superluminal-electron branch |
| Kepuladze et al., arXiv:2607.10404 (Jul 2026) | **verified-at-source** (arXiv abs) | γ→νν̄: does not beat existing bounds | — |
| MAGIC GRB 190114C (Acciari et al., PRL 125, 021301 (2020), as cited by LHAASO) | **inherited-unchecked** (quoted inside verified LHAASO PRL text: E_QG,2 > 6.3×10¹⁰ sub / 5.6×10¹⁰ sup GeV) | superseded by LHAASO | as quoted |
| H.E.S.S. PKS 2155-304 (Abramowski et al. 2011?) | **inherited-unchecked** (quoted inside verified Vasileiou text: E_QG,2 > 6.4×10¹⁰ GeV, ML) | superseded | as quoted |
| Abdo et al., Nature 462, 331 (2009) (GRB 090510) | **inherited-unchecked** (ADS abstract snippet read; quadratic numbers not read at source) | headline is the n=1 bound (1.2 E_Pl) | — |
| Baktash et al.?, "LIV Limits from GRB 221009A", arXiv:2308.03031 | **verified-at-source** (arXiv abs: n=1: 5.9 (6.2) m_pl sub (sup)); n=2 numbers **not extracted** | independent n=1 cross-check | as quoted |

---

## What I could not find / could not confirm

1. Vasileiou et al. (2013) Table 4/Fig. 10 **superluminal** n=2 values and their Table 6 SME
   coefficient numbers: tables did not survive HTML extraction. Flagged rather than guessed;
   not needed for the verdict (LHAASO TOF superluminal n=2 = 7.2×10¹¹ GeV supersedes them and
   was read at source).
2. The exact Data Tables row for the Fermi-LAT/LHAASO *dispersion* bounds on c^(6)_{(I)00}
   (I extracted only the summary-table sensitivity 10⁻³⁰ GeV⁻² and the D22 spectropolarimetry
   rows). The 198-page table was not exhaustively mined.
3. A journal reference for Xi & Shu, arXiv:2508.00656 — none found; treated as unrefereed.
4. No post-2022 photon-stability analysis for superluminal n=2 that beats
   LHAASO PRL 128, 051102 was found in my searches (through 2026-09-19); if one exists it did
   not surface in Brave/arXiv-indexed results.
5. Back-of-envelope only (not published, from Li & Ma's threshold formula): the GRB 221009A
   ~18 TeV photon would give an n=2 superluminal stability bound ~E_γ²/(2m_e) ≈ 3×10¹¹ GeV —
   far weaker than the PeV Crab bound, so it cannot change the verdict.

**Bottom line for the requester's simulation-case arithmetic:** superluminal quadratic photon
LIV is excluded only below ~10⁻³ E_Pl; the Planck scale is not probed by the n=2 channel, and
the n=1 channel is the one that over-reaches (10⁵ E_Pl). If a "simulation grid" argument needs
an n=2 photon bound, it must use 1.22×10¹⁶ GeV with the LHAASO 2022 source, and must not
mistake it for a Planck-scale exclusion.