# Subluminal dimension-6 photon-sector bounds

Date: 2026-09-11
Thread: HAWC/Data Tables sign of c_(I)00^(6), and best published bound on the subluminal sign

## Verdict

Established: The HAWC PRL 124, 131101 (2020) bound that appears in Kostelecky-Russell Table D22 as `|c_(I)00^(6)| < 12.4e-31 GeV^-2` is physically a superluminal-photon bound. HAWC's own mechanism is photon decay/splitting; the paper says photon decay is "due to superluminal LIV," photon splitting is a "superluminal LIV decay process," and HAWC explicitly defers the n=2 subluminal Bethe-Heitler analysis to later work.

Inference: With HAWC's SME convention `c_(I)00^(d=n+4) = -sqrt(pi) alpha_n`, their superluminal `+ alpha_n` branch constrains negative `c_(I)00^(6)`, not positive/sub-luminal `c_(I)00^(6)`.

Established + Inference: The strongest published subluminal-side bound I found is from Yang, Bi, and Yin, JCAP 04 (2024) 060, arXiv:2312.09079, using LHAASO GRB 221009A time-of-flight: `E_QG,2 > 12.0 x 10^11 GeV` at 95% CL for the subluminal n=2 scenario. Using the HAWC/SME isotropic mapping gives `c_(I)00^(6) < sqrt(pi)/(12.0 x 10^11 GeV)^2 = 1.23 x 10^-24 GeV^-2` for the positive/sub-luminal sign. If only bounds published directly as `c_(I)00^(6)` are allowed, the strongest clean direct coefficient bound I found is Rubtsov, Satunin, and Sibiryakov, JCAP 05 (2017) 049, `c_(I)00^(6) < 4 x 10^-23 GeV^-2` at 95% CL, from subluminal Bethe-Heitler air-shower suppression.

Evidence classes used below: Established = directly quoted or directly tabulated in a cited paper/table. Inference = my sign/units conversion from quoted equations. Serious speculation = physically motivated but not a clean exclusion bound. Anomaly = an observational tension claim. Anecdote = none used.

## 1. Albert et al. 2020 / HAWC: what sign is constrained?

Source checked: A. Albert et al. (HAWC Collaboration), "Constraints on Lorentz Invariance Violation from HAWC Observations of Gamma Rays above 100 TeV," Phys. Rev. Lett. 124, 131101 (2020), arXiv:1911.08070. URLs checked: https://arxiv.org/abs/1911.08070, https://ar5iv.labs.arxiv.org/html/1911.08070, https://arxiv.org/e-print/1911.08070.

Established, Abstract:

> "Superluminal LIV enables the decay of photon at high energy."

Established, Lorentz Invariance Violation section:

> "The MDR for photons is ... `E_gamma^2 - p_gamma^2 = +/- |alpha_n| p_gamma^(n+2)` ... The sign usually refers to the so-called superluminal (`+`), and subluminal (`-`) dominant phenomena."

Established, Photon decays section, photon -> e+e-:

> "Considering the photon decay, `gamma -> e+ e-`, due to superluminal LIV, the resulting decay rates are fast and effective at energies where the process is allowed ... This creates a hard cutoff in the gamma-ray spectrum with no high-energy photons reaching the Earth from cosmological distances above a given threshold."

Established, Photon decays section, photon -> N gamma:

> "A second superluminal LIV decay process considered in this work is photon splitting to multiple photons, `gamma -> N gamma`."

Established, same section:

> "However, this process has no threshold, and is kinematically allowed whenever `E_gamma^2 > p_gamma^2`."

Inference: In their MDR, `E_gamma^2 > p_gamma^2` is the `+ |alpha_n|` branch, which HAWC just defined as superluminal.

Established, only subluminal-analysis statement I found in the HAWC paper:

> "Refs. [rubstov_MULTI-TEV, Astapov:2019xmt, Satunin:2019gsl] discuss a different method of setting limits on subluminal LIV with n=2 using modifications to the Bethe-Heitler interaction of photons in the atmosphere. However, unlike the photon splitting process, this does not result in a sharp effective threshold. Thus setting a limit using this effect must use different analysis techniques than the ones we have used to analyze the HAWC data, and we must defer such analysis to a later publication."

Established: I found no HAWC sentence claiming that the photon-decay or photon-splitting analysis constrains the subluminal branch. The one HAWC sentence about subluminal n=2 says that it requires a different Bethe-Heitler analysis and is deferred.

Established but sign-confusing, Supplemental Material Table VII caption:

> "HAWC Sources and 95% CL lower limits on E_c and two-sided LIV limits in the framework of the SME. (3gamma) stands for the limits derived due to photon splitting."

Inference: That caption is the source of the possible two-sided presentation, but the same supplemental section defines the SME columns as limits on `-c_(I)00^(d)` and `-sum Y c`, derived from superluminal `alpha_n`. It does not supply a subluminal likelihood or subluminal threshold calculation.

## 2. HAWC limit variables and SME sign convention

Established, HAWC Lorentz Invariance Violation section:

> "The MDR for photons is ... `E_gamma^2 - p_gamma^2 = +/- |alpha_n|p_gamma^(n+2)`, where `(E_gamma,p_gamma)` is the photon four-momentum, `alpha_n` is the LIV parameter, `n` is the leading order of the correction from the underlying theory, and `p_gamma approximately E_gamma` at first order in `alpha_n` ... The sign usually refers to the so-called superluminal (`+`), and subluminal (`-`) dominant phenomena."

Established, same section:

> "For `n>0`, limits on the LIV parameter `alpha_n` can be interpreted in terms of some LIV energy scale, `E_LIV^(n)=alpha_n^(-1/n)`."

Established, Limit Calculation section:

> "These limits are intrinsically one-sided, as we lose statistical power to identify a finite `E_c` for large values of `E_c`."

Established, Limit Calculation section:

> "The 95% CL limits are reinterpreted as limits on `E_gamma`. Then Eqs. ... directly lead to lower limits to `E_LIV^(1)` and `E_LIV^(2)`, while we derive upper limits on `alpha_0` ..."

Established, HAWC Supplemental Material, SME reinterpretation:

> "For `n=2` (or any `n` even), and considering only `zeta^0`, if there is directional independence (`jm = 0 0`), `c_(I)00^(d=n+4) = -sqrt(pi) alpha_n`, while in a directional dependent scenario, `sum_jm Y_jm(theta_k,varphi_k) c_(I)jm^(d=n+4) = -alpha_n/2` ..."

Established, same supplemental section:

> "We give the corresponding HAWC limits on SME coefficients, `tilde{kappa}_tr`, `k^(5)_(V)00`, `-c_(I)00^(d)`, and (`-sum_jm Y_jm(theta_k,varphi_k) c_(I)jm^(d)`), in the Table [tabs:SME]."

Established, HAWC Supplemental Material Table VII: the column header is `-c_(I)00^(d)`, with a `d=6 (3gamma)` value of `12.4` for eHWC J1825-134 in units `10^-49 eV^-2`, equal to `12.4 x 10^-31 GeV^-2`.

Inference: Since `c_(I)00^(6) = -sqrt(pi) alpha_2` and HAWC's superluminal branch is positive `alpha_2`, the HAWC `12.4 x 10^-31 GeV^-2` photon-splitting number is a bound on `-c_(I)00^(6)`, i.e. on negative `c_(I)00^(6)`. It is not a bound on the positive/sub-luminal sign.

## 3. Kostelecky-Russell Data Tables Table D22: one-sided vs two-sided notation

Source checked: V.A. Kostelecky and N. Russell, "Data Tables for Lorentz and CPT Violation," arXiv:0801.0287v19. URL checked: https://arxiv.org/html/0801.0287v19.

Established, general table convention:

> "Each of these data tables provides information about the results of searches for Lorentz violation for a specific sector of the SME."

Established, summary-table convention:

> "each displayed sensitivity value represents our conservative estimate of a 2sigma limit, given to the nearest order of magnitude, on the modulus of the corresponding coefficient."

Established, data-table convention:

> "The second column contains the measurements and bounds, presented in the same form as documented in the literature."

Established, data-table caution:

> "The reader is referred to the latter for details of experimental and theoretical procedures, assumptions underlying the results, definitions of unconventional notations, and other relevant information."

Established, D21-D25 photon-sector note:

> "Tables D21-D25 contain a compilation of some measurements and bounds on coefficients for Lorentz violation in the nonminimal photon sector of the SME."

Established, same note:

> "In the first columns of Tables D21-D25, the various spherical harmonics ... are evaluated at specified angles, which are the celestial coordinates of certain astrophysical sources."

Established, Table D22 entries from ref. [190] HAWC:

> `|c^(6)_(I)00| < 12.4 x 10^-31 GeV^-2`, System: `Astrophysics`, Ref. `[190]`.

Established, Table D22 direction-combination entries from ref. [190] HAWC include:

> `|sum Y_jm(103.45 deg,276.41 deg)c^(6)_(I)jm| < 3.5 x 10^-31 GeV^-2`, Ref. `[190]`.

> `|sum Y_jm(83.75 deg,286.95 deg)c^(6)_(I)jm| < 4.93 x 10^-31 GeV^-2`, Ref. `[190]`.

> `|sum Y_jm(67.96 deg,83.6 deg)c^(6)_(I)jm| < 20.1 x 10^-31 GeV^-2`, Ref. `[190]`.

> `|sum Y_jm(53.26 deg,304.94 deg)c^(6)_(I)jm| < 50.3 x 10^-31 GeV^-2`, Ref. `[190]`.

Established: I found no Table D22 convention note, footnote, superscript, or explicit sign statement attached to the HAWC-derived `c_(I)` entries saying "superluminal only" or identifying the sign of `c_(I)00^(6)`. One-sided and two-sided bounds are distinguished in the table by the mathematical form copied into the result column: intervals such as `(a to b)`, inequalities such as `<` or `>`, or absolute-value inequalities `|...| < ...`. The HAWC rows are printed as absolute-value inequalities, but the Data Tables' own caveat says the source paper controls the procedures and assumptions.

Inference: Table D22's absolute-value presentation of the HAWC number should not be read as an experimentally tested two-sided constraint on subluminal `c_(I)00^(6)`. HAWC's source mechanism and sign convention make it a superluminal/negative-c constraint.

## 4. Published bounds that do constrain the subluminal sign

Sign convention used in this section: HAWC gives `c_(I)00^(6) = -sqrt(pi) alpha_2`. Rubtsov et al. give `epsilon_gamma/M_LV,gamma^2 = -c_(I)00^(6)/sqrt(pi)`. Therefore Rubtsov's subluminal `epsilon_gamma=-1` corresponds to positive `c_(I)00^(6)` with `c_(I)00^(6) = sqrt(pi)/M_LV,gamma^2`. Time-of-flight papers often use `s_pm=+1` for high-energy photons slower than low-energy photons; that is the subluminal side and corresponds to positive `c_(I)00^(6)` under the SME group-velocity convention quoted by Du et al. and Guerrero et al.

### Rubtsov, Satunin, Sibiryakov 2017

Citation: G. Rubtsov, P. Satunin, and S. Sibiryakov, "Constraints on violation of Lorentz invariance from atmospheric showers initiated by multi-TeV photons," JCAP 05 (2017) 049, arXiv:1611.10125, https://arxiv.org/abs/1611.10125.

Established, mechanism and sign:

> "The relevant processes differ depending on whether epsilon_gamma is positive or negative. With some abuse of language, we will refer to these cases as 'superluminal' and 'subluminal' respectively."

Established, SME relation:

> "epsilon_gamma/M^2_LV,gamma = - c^(6)_(I)00/sqrt(pi)."

Established, photon decay sign:

> "In the superluminal case (epsilon_gamma = +1) a high-energy photons can decay into e+e- pairs in the vacuum."

Established, EBL pair-production threshold mechanism:

> "Subluminal LV in photons (epsilon_gamma = -1) shifts the threshold of pair production upward ... This leads to higher predictions for the VHE photon flux from extragalactic sources than in the LI case. Non-detection of large fluxes constrains LV."

Established, EBL numbers quoted by Rubtsov et al.:

> "Translating it into the bound on the quartic term one obtains, M_LV,gamma > 3 x 10^11 GeV (epsilon_gamma = -1)."

> "Recent analysis of the VHE part of the spectrum of Mrk 501 during the 2014 flare leads to a stronger limit [53]: M_LV,gamma > 7.5 x 10^11 GeV (epsilon_gamma = -1) (12) at 95% confidence level (CL)."

Established, caveat on EBL bounds:

> "It is worth noting that these bounds rely on the assumption that the observed cutoff in the Mrk 501 spectrum is not intrinsic to the source, but is fully accounted for by absorption on EBL. Besides, they require modeling of the EBL spectrum."

Established, Bethe-Heitler shower mechanism:

> "However, for subluminal photons the modification of the Bethe-Heitler cross section can be important. If m^2_gamma,eff(p_gamma)<0, |m^2_gamma,eff(p_gamma)| >> 4m_e^2 the cross section gets strongly suppressed."

Established, direct subluminal SME c bound:

> "From it one reads the constraint M_LV,gamma > 2.1 x 10^11 GeV (epsilon_gamma = -1) at 95% CL. (26a) In the effective field theory parameterization of [33] this translates into a one-sided bound on the coefficient c^(6)_(I)00, c^(6)_(I)00 < 4 x 10^-23 GeV^-2 at 95% CL. (26b)"

Established, H.E.S.S. Crab flare bound:

> "It implies the bound, M_LV,gamma > 1.3 x 10^11 GeV (epsilon_gamma = -1) at 95% CL, (27a) or c^(6)_(I)00 < 10^-22 GeV^-2 at 95% CL (27b) in the notations of [33]."

Answer: Yes, this constrains the subluminal sign, i.e. positive `c_(I)00^(6)`. Direct coefficient bound: `c_(I)00^(6) < 4 x 10^-23 GeV^-2` at 95% CL from HEGRA Crab shower data. EBL bounds quoted in the same paper are also subluminal but are model-dependent; `M_LV,gamma > 7.5 x 10^11 GeV` would correspond to `c_(I)00^(6) < 3.15 x 10^-24 GeV^-2` if converted with Rubtsov's equation. Rubtsov does not present that number as a `c_(I)00^(6)` table entry.

### Vasileiou et al. 2013

Citation: V. Vasileiou et al., "Constraints on Lorentz invariance violation from Fermi-Large Area Telescope observations of gamma-ray bursts," Phys. Rev. D 87, 122001 (2013), arXiv:1305.3463, https://arxiv.org/abs/1305.3463.

Established, sign convention:

> "s_pm is the 'sign of LIV', a theory-dependent factor equal to +1 (-1) for a decrease (increase) in photon speed with an increasing photon energy (also referred to as the 'subluminal' and 'superluminal' cases)."

Established, abstract result:

> "For the subluminal case (where high energy photons propagate more slowly than lower energy photons) and without taking into account any source-intrinsic dispersion, our most stringent limits (at 95% CL) are obtained from GRB 090510 and are E_QG,l > 7.6 times the Planck energy (E_Pl) and E_QG,q > 1.3 x 10^11 GeV for linear and quadratic leading order LIV-induced vacuum dispersion, respectively."

Established, Results Table caption:

> "Lower Limits on E_QG for linear (n=1) and quadratic (n=2) LIV for the subluminal (s_pm=+1) and superluminal (s_pm=-1) cases. The CL values are one-sided. These limits were produced using the total degree of dispersion in the data, tau_tot."

Established, Results Table n=2 subluminal rows in units `10^10 GeV` include GRB 090510: PairView `6.7`, SMM `13`, Likelihood `8.6`, so the strongest subluminal quadratic 95% entry is `E_QG,2 > 13 x 10^10 GeV = 1.3 x 10^11 GeV`.

Answer: Mechanism is time-of-flight dispersion, not threshold decay. It works for both signs; the paper reports subluminal (`s_pm=+1`) and superluminal (`s_pm=-1`) one-sided lower limits separately. Conversion for the strongest subluminal n=2 result: `c_(I)00^(6) < sqrt(pi)/(1.3 x 10^11 GeV)^2 = 1.05 x 10^-22 GeV^-2` (Inference). Kostelecky-Russell Table D22 also lists direct SME intervals from this paper; the tightest positive/isotropic side I saw is `c_(I)00^(6) < 0.57 x 10^-20 GeV^-2`, weaker than the E_QG conversion and far weaker than the LHAASO/Rubtsov bounds.

### Agrawal, Singirikonda, Desai 2021

Citation: R. Agrawal, H. Singirikonda, and S. Desai, "Search for Lorentz Invariance Violation from stacked Gamma-Ray Burst spectral lag data," JCAP 05 (2021) 029, arXiv:2102.11248, https://arxiv.org/abs/2102.11248.

Established, sign convention:

> "v(E) = c [1 - s_pm (n + 1)/2 (E/E_QG)^n], where s_pm = +/-1 denotes the sign of the Lorentz Invariance violation (LIV), corresponding to sub-luminal (s_pm = +1) or super-luminal (s_pm = -1) ..."

Established, abstract/result characterization:

> "We do not find a decisive evidence for such an energy-dependent speed of light for two different models of LIV. When we assume a constant intrinsic lag coupled with an unknown intrinsic scatter, we do not find any evidence for LIV. However, when we use GRB-dependent parameters to model the intrinsic emission, we get decisive evidence for LIV violation."

Established, Kostelecky-Russell Table D22 entry from ref. [221]: `sum_jm Y_jm(nhat)c^(6)_(I)jm = 10^(-14.2 +/- 0.1) GeV^-2`, System: `Astrophysics`.

Answer: Mechanism is time-of-flight/spectral-lag dispersion. It can constrain either sign in principle. This is not a clean one-sided subluminal upper bound; it is a model-dependent fitted direction combination at about `6.3 x 10^-15 GeV^-2`, much weaker than Rubtsov/LHAASO. Evidence class for the quoted number as a bound: Established table entry; interpretation as a subluminal-side constraint only if the fitted combination is positive is Inference.

### Du et al. 2021

Citation: S.-S. Du et al., "Lorentz Invariance Violation Limits from the Spectral Lag Transition of GRB 190114C," Astrophys. J. 906, 8 (2021), arXiv:2010.16029, https://arxiv.org/abs/2010.16029.

Established, sign convention and scope:

> "s_pm = +/-1 represents the sign of the LIV effect corresponding to the subluminal (s_pm = +1) or superluminal (s_pm = -1) scenario (i.e., s_pm=+1 or s_pm=-1 stands for a decrease or an increase in photon group velocity with an increasing photon energy). Thus, s_pm=+1 would be the case that higher-energy photons propagate more slowly relative to the lower-energy photons in a vacuum. This gives the LIV-induced negative time lags. Thus we only consider the case of s_pm = +1 in this work."

Established, SME sign statement:

> "where the coefficients c^(d)_(I)jm can be either positive or negative, leading to a decreasing or an increasing velocity of light with photon energy. Thus, a positive sum ... c ... would imply a negative spectral lag contributed by Lorentz violation."

Established, SME result:

> "Considering a negative spectral lag due to LIV in the SME framework, we obtain sum_jm _0Y_jm(theta,phi)c^(d)_(I)jm <= 6.77 x 10^-13 GeV^-2 and 1.56 x 10^-7 GeV^-4 (2sigma) for ... d=6 and 8."

Established, Table 2: for GRB 190114C, `sum_jm _0Y_jm(116.9 deg,54.5 deg)c^(6)_(I)jm = 5.05^(+1.72)_(-1.25) x 10^-13 GeV^-2`, with isotropic `c^(6)_(I)00 <= 2.40 x 10^-12 GeV^-2`.

Answer: Mechanism is time-of-flight/spectral-lag transition. It is explicitly a subluminal-sign analysis (`s_pm=+1`, positive SME combination), but it is much weaker than Rubtsov/LHAASO.

### Wei, Liu, Wei, Zhang, Wu 2022

Citation: J.-J. Wei, T. Liu, H. Wei, B.-B. Zhang, and X.-F. Wu, "Constraints on Anisotropic Lorentz Invariance Violation with Gamma-Ray Bursts," Universe 8, 519 (2022), arXiv:2210.03897, https://arxiv.org/abs/2210.03897.

Established, mechanism and coefficient target:

> "The coefficients c^(d)_(I)jm are associated with CPT-even operators causing dispersion without leading-order birefringence ... In the present work, we focus on the nonbirefringent vacuum dispersion coefficients c^(d)_(I)jm."

Established, group-velocity/sign formula:

> "Setting all other coefficients for birefringent propagation to zero, the group-velocity defect including anisotropies is given by delta v_g = - sum_djm (d - 3)E^(d-4) _0Y_jm(nhat)c^(d)_(I)jm ..."

Established, fitting assumption:

> "we require the Delta t_LIV term in Equation (5) not to dominate over Delta t_int."

Established, results caveat:

> "For the case of d = 6, our constraints are not competitive with existing bounds but can be deemed as comparatively robust."

Established, Kostelecky-Russell Table D22 isotropic entry from ref. [218]: `|c^(6)_(I)00| = 4.25^(+1.60)_(-1.63) x 10^-15 GeV^-2`, System: `Astrophysics`. Direction-combination entries for individual GRBs are typically at `10^-14` to `10^-12 GeV^-2` in Table D22.

Answer: Mechanism is time-of-flight/spectral-lag dispersion. It can include the subluminal sign because positive SME combinations correspond to slower high-energy photons under the quoted group-velocity formula. It is not competitive with Rubtsov/LHAASO.

### Guerrero, Campoy-Ordaz, Potting, Gaug 2025

Citation: M. Guerrero, A. Campoy-Ordaz, R. Potting, and M. Gaug, "Bounding anisotropic Lorentz Invariance Violation from measurements of the effective energy scale of quantum gravity," Phys. Rev. D 112, 104002 (2025), arXiv:2508.02883, https://arxiv.org/abs/2508.02883.

Established, abstract mechanism:

> "Observations of energy-dependent photon time delays from distant flaring sources provide significant constraints on Lorentz Invariance Violation (LIV). Such effects originate from modified vacuum dispersion relations, causing differences in propagation times for photons emitted simultaneously from gamma-ray bursts, active galactic nuclei, or pulsars."

Established, sign convention:

> "s_pm defines the sign of the (LIV) effect (s_pm = -1 for superluminal and s_pm = +1 for subluminal behaviour) ..."

Established, conversion target:

> "In particular, existing bounds on the quadratic case (n = 2) of E_QG,n can be systematically converted into constraints on the non-birefringent, CPT-conserving SME coefficients c^(6)_(I)jm."

Established, Table IV individual-coefficient result: `c^(6)_(I)00 = (-0.5 +/- 1.5) x 10^-15 GeV^-2`, with 95% CL lower/upper limits `-3.4` and `2.4` in units `10^-15 GeV^-2`.

Answer: Mechanism is time-of-flight dispersion; it is designed to handle both signs through two-sided Gaussianized constraints and includes the subluminal/positive side. The positive/isotropic 95% side is about `c_(I)00^(6) < 2.4 x 10^-15 GeV^-2`, much weaker than LHAASO/Rubtsov.

## 5. Subluminal pair-production transparency and GRB 221009A/LHAASO

Established: Pair-production transparency is the right sign for subluminal photon dispersion. Rubtsov et al. state:

> "Subluminal LV in photons (epsilon_gamma = -1) shifts the threshold of pair production upward ... This leads to higher predictions for the VHE photon flux from extragalactic sources than in the LI case. Non-detection of large fluxes constrains LV."

Established: Rubtsov et al. quote Mrk 501 EBL/absorption constraints on the subluminal quartic photon term: `M_LV,gamma > 3 x 10^11 GeV (epsilon_gamma=-1)` and `M_LV,gamma > 7.5 x 10^11 GeV (epsilon_gamma=-1)` at 95% CL for a 2014-flare analysis, with the caveat that these rely on interpreting the cutoff as EBL absorption rather than source-intrinsic.

Inference: Using `c_(I)00^(6)=sqrt(pi)/M_LV,gamma^2` for subluminal `epsilon_gamma=-1`, `M_LV,gamma > 7.5 x 10^11 GeV` corresponds to `c_(I)00^(6) < 3.15 x 10^-24 GeV^-2`. This is strong, but I did not find it published in that source as a direct `c_(I)00^(6)` value.

GRB 221009A transparency checks:

Established, Finke and Razzaque 2023, ApJ Lett. 942 L21, arXiv:2210.11261, https://arxiv.org/abs/2210.11261:

> "The preliminary detections of the gamma-ray burst 221009A up to 18 TeV by LHAASO and up to 251 TeV by Carpet 2 have been reported ... We show that the survival of the 18 TeV photon detected by LHAASO is not unlikely with many recent extragalactic background light models, although the detection of a 251 TeV event is still very unlikely. This can be resolved if Lorentz invariance is violated at an energy scale E_QG < 49 E_Planck in the linear (n=1) case, and E_QG < 10^-6 E_Planck in the quadratic (n=2) case (95% confidence limits) ... This could potentially be the first evidence for subluminal Lorentz invariance violation."

Classification: Serious speculation / Anomaly, not an exclusion bound on the subluminal side. It is an upper limit on `E_QG` needed to make the universe transparent enough under that interpretation, not a lower-limit constraint of the kind `c < ...`. Converted, `E_QG,2 < 10^-6 E_Pl` means LIV would need to be at least roughly `c_(I)00^(6) > 1.2 x 10^-26 GeV^-2` if `E_Pl=1.22 x 10^19 GeV`; that is an allowed/preferred region for an anomaly claim, not a bound excluding subluminal `c`.

Established, Zhao et al. 2023, Eur. Phys. J. C 83, 92, arXiv:2210.10778, https://arxiv.org/abs/2210.10778:

> "We find that the standard physics is compatible with the observations of 18 TeV photons within 3.5sigma confidence interval."

Established, Piran and Ofengeim 2024, arXiv:2308.03031, checked for GRB 221009A context:

> "When an 18 TeV photon was reported ... However, careful processing [2] of the LHAASO data resulted in lowering the highest detected photon energy to 10-13 TeV, relaxing the problem of the optical depth for this GRB."

Established, LHAASO Collaboration PRL 133, 071501 (2024), arXiv:2402.06009, https://arxiv.org/abs/2402.06009: this paper is time-of-flight, not EBL-transparency. Abstract:

> "We use this unique observation to place stringent constraints on an energy dependence of the speed of light in vacuum, a manifestation of Lorentz invariance violation (LIV) predicted by some quantum gravity (QG) theories. Our results show that the 95% confidence level lower limits on the QG energy scales are E_QG,1 > 10 times of the Planck energy E_Pl for the linear, and E_QG,2 > 6 x 10^-8 E_Pl for the quadratic LIV effects, respectively. Our limits on the quadratic LIV case improve previous best bounds by factors of 5--7."

Established, LHAASO PRL sign/results text extracted from the paper:

> "s = +/-1 is the 'sign' of the LIV effect, corresponding to the 'subluminal' or 'superluminal' scenarios ..."

> "our result on the energy scale E_QG,2 > 6.9 x 10^11 GeV (E_QG,2 > 7.0 x 10^11 GeV) for a subluminal (superluminal) LIV effect represents the best time-of-flight limit ..."

Inference: LHAASO Collaboration PRL subluminal quadratic ToF bound converts to `c_(I)00^(6) < sqrt(pi)/(6.9 x 10^11 GeV)^2 = 3.72 x 10^-24 GeV^-2` for positive/sub-luminal `c`.

Established, Yang, Bi, and Yin 2024, JCAP 04 (2024) 060, arXiv:2312.09079, https://arxiv.org/abs/2312.09079:

> "The Large High Altitude Air Shower Observatory(LHAASO) has detected the onset, rise, and decay phases of the afterglow of GRB 221009A, covering a wide energy range of photons approximately from 0.2 to 18 TeV."

> "For instance, through the maximum likelihood method, we determine the 95% confidence level lower limits to be E_QG,1 > 14.7 (6.5) x 10^19 GeV for the subluminal (superluminal) scenario of n = 1, and E_QG,2 > 12.0 (7.2) x 10^11 GeV for the subluminal (superluminal) scenario of n = 2."

> "We find that the rapid rise and slow decay behaviors of the afterglow can impose strong constraints on the subluminal scenario, while the constraints are weaker for the superluminal scenario."

Answer: GRB 221009A gives strong subluminal time-of-flight bounds, especially Yang et al. 2024 and the LHAASO Collaboration PRL. I did not find a clean published GRB 221009A EBL-transparency exclusion bound on positive `c_(I)00^(6)` stronger than the ToF bounds; the transparency papers I checked either make an anomaly/evidence-style subluminal claim requiring `E_QG` below a scale, or argue standard physics can accommodate the reported 18 TeV photon within uncertainties.

## 6. Single strongest subluminal c_(I)00^(6) bound found

| Scope | Evidence class | Mechanism | Subluminal sign constrained? | Published limit | SME c_(I)00^(6) bound for subluminal sign | Citation |
|---|---|---|---|---|---|---|
| Strongest published bound found after converting E_QG,2 to SME c | Established published E_QG limit; Inference for SME conversion | LHAASO GRB 221009A time-of-flight dispersion | Yes. Paper states subluminal n=2 separately. | `E_QG,2 > 12.0 x 10^11 GeV` at 95% CL, subluminal n=2 | `0 < c_(I)00^(6) < 1.23 x 10^-24 GeV^-2` | S. Yang, X.-J. Bi, and P.-F. Yin, "Constraints on Lorentz invariance violation from the LHAASO observation of GRB 221009A," JCAP 04 (2024) 060, arXiv:2312.09079, https://arxiv.org/abs/2312.09079 |
| Strongest direct c_(I)00^(6) coefficient bound found, if conversion-only E_QG papers are excluded | Established | Bethe-Heitler air-shower suppression for subluminal photons | Yes. `epsilon_gamma=-1`, and paper says it is a one-sided bound on `c^(6)_(I)00`. | `M_LV,gamma > 2.1 x 10^11 GeV`, `epsilon_gamma=-1`, 95% CL | `0 < c_(I)00^(6) < 4 x 10^-23 GeV^-2` | G. Rubtsov, P. Satunin, and S. Sibiryakov, "Constraints on violation of Lorentz invariance from atmospheric showers initiated by multi-TeV photons," JCAP 05 (2017) 049, arXiv:1611.10125, https://arxiv.org/abs/1611.10125 |

Final answer to the narrow HAWC question: the HAWC `12.4 x 10^-31 GeV^-2` number should be treated as a one-sided superluminal/negative-`c_(I)00^(6)` bound. It does not constrain the subluminal/positive-`c_(I)00^(6)` sign.