# Lorentz Invariance Violation Tests in Astroparticle Physics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modified Dispersion Relation for Astroparticle Tests

## 3. Energy-Dependent Time Delay

## 4. Pair-Production Threshold Shift

## 5. Photon Decay

**Table 1.**Strong and recent astrophysical LIV limits. $|{\delta}_{0}|$ are upper limits meanwhile ${E}_{\mathrm{LIV}}^{\left(n\right)}$ are lower limits. PP stands for pair production, $\Delta t$ for energy-dependent time delay, AS for suppression of air shower formation, ($3\gamma $) for photons splitting, and PD for photon decay. In addition, (+) indicates ${\delta}_{\gamma ,n}>0$, while (−) is for ${\delta}_{\gamma ,n}<0$.

Type | $|{\mathit{\delta}}_{0}|$ ${10}^{-17}$ | ${\mathit{E}}_{\mathbf{LIV}}^{\left(1\right)}$ ${10}^{28}$eV | ${\mathit{E}}_{\mathbf{LIV}}^{\left(2\right)}$ ${10}^{21}$eV | Bound | Source | Reference |
---|---|---|---|---|---|---|

Limit * | - | 12.08 | 2.38 | PP (−) | MultiSrc | Lang, Martínez, and de Souza (2019) [41] |

Limit * | - | 3.3 | 0.87 | PP (−) | Mrk501 | H.E.S.S. and FACT (2017) [73] |

Limit * | - | 2.6 | 0.78 | PP (−) | Mrk 501 | H.E.S.S. (2019) [28] |

Limit * | - | 1.9 | 0.31 | PP (−) | MultiSrc | Biteau and Williams (2015) [7] |

Limit ${}^{\u2020}$ | ∼0.001 | ∼${10}^{10}$ | ∼${10}^{7}$ | PP (−) | UHECR | Lang, Martínez, and de Souza (2018) [40] |

Sens.lim. | - | ∼1.22 | ∼0.97 | PP (−) | - | CTA Consortium (2019) [53,54] |

Limit * | - | 9.3 | 0.13 | $\Delta t$ (−) | GRB090510 | Vasileiou et al. (2013) [24] |

Limit * | - | 0.055 | 0.059 | $\Delta t$ (−) | Crab | MAGIC Collaboration (2017) [26] |

Limit * | - | 0.036 | 0.085 | $\Delta t$ (−) | Mrk 501 | H.E.S.S. (2019) [28] |

Limit * | - | 0.021 | 0.026 | $\Delta t$ (−) | Mrk 501 | MAGIC Collaboration (2008) [74] |

Sens.lim. | - | 25 | 0.54 | $\Delta t$ (−) | GRBs | LHAASO Collaboration (2019) [34] |

Limit * | - | - | 1.4 | AS (−) | Crab (Tibet) | Satunin (2019) [10] |

Limit * | - | - | 0.97 | AS (−) | Crab (HAWC) | Satunin (2019) [10] |

Limit * | - | - | 0.21 | AS (−) | Crab (HEGRA) | Rubtsov, Satunin, and Sibiryakov (2017) [9] |

Limit * | - | - | 1200 | $3\gamma $ (+) | eHWC J1825-134 | HAWC Collaboration (2020) [68] |

Limit * | - | - | 1010 | $3\gamma $ (+) | eHWC J1907+063 | HAWC Collaboration (2020) [68] |

Limit * | - | - | 499 | $3\gamma $ (+) | Crab(HAWC) | HAWC Collaboration (2020) [68] |

Limit * | - | - | 410 | $3\gamma $ (+) | Crab (Tibet) | Satunin (2019) [10] |

Limit * | - | - | 315 | $3\gamma $ (+) | eHWC J2019+368 | HAWC Collaboration (2020) [68] |

Limit ** | - | - | 300 | $3\gamma $ (+) | Crab (HAWC) | Satunin (2019) [10] |

Limit * | - | - | 130 | $3\gamma $ (+) | Crab (HEGRA) | Astapov, Kirpichnikov, and Satunin (2019) [75] |

Limit * | 1.29 | 2220 | 80 | PD (+) | MultiSrc | HAWC Collaboration (2020) [68] |

Limit * | 1.75 | 1390 | 58 | PD (+) | eHWC J1825-134 | HAWC Collaboration (2020) [68] |

Limit * | 2.2 | 990 | 47 | PD (+) | eHWC J1907+063 | HAWC Collaboration (2020) [68] |

Limit * | 4.52 | 340 | 23 | PD (+) | Crab(HAWC) | HAWC Collaboration (2020) [68] |

Limit ** | - | - | 19 | PD (+) | Crab (Tibet) | Satunin (2019) [10] |

Limit * | 7.25 | 170 | 14 | PD (+) | eHWC J2019+368 | HAWC Collaboration (2020) [68] |

Limit ** | - | - | 14 | PD (+) | Crab (HAWC) | Satunin (2019) [10] |

Limit | - | 15 | 2.8 | PD (+) | Crab (HEGRA) | Martínez and Lorenzana (2017) [8] |

Limit | - | 1.7 | 0.65 | PD (+) | RX J1713.7-3946 (H.E.S.S.) | Martínez and Lorenzana (2017) [8] |

Limit | $6\times {10}^{5}$ | - | - | PD (+) | Tevatron | A. Hohensee et al. (2016) [61] |

Limit * | 40 | - | - | PD (+) | Crab (HEGRA) | Schreck (2013) [65] |

Limit | 50 | - | - | PD (+) | Crab (CANGAROO) | Stecker and Glashow (2001) [49] |

Limit * | 180 | - | - | PD (+) | RX J1713.7-3946 (H.E.S.S.) | Klinkhamer and Schreck (2008) [6] |

Limit | 300 | - | - | PD (+) | Crab (Themistocle) | Coleman and Glashow (1997) [13] |

Sens.lim. | - | ∼${10}^{2}$ | ∼10 | PD (+) | - | SGSO Alliance [71,72] |

Limit * | - | 13.4 | 0.09 | $\Delta t$ (+) | GRB090510 | Vasileiou et al. (2013) [24] |

Limit * | - | 0.026 | 0.073 | $\Delta t$ (+) | Mrk 501 | H.E.S.S. (2019) [28] |

Limit * | - | 0.045 | 0.053 | $\Delta t$ (+) | Crab | MAGIC Collaboration (2017) [26] |

Sens.lim. | - | 25 | 0.54 | $\Delta t$ (+) | GRBs | LHAASO Collaboration (2019) [34] |

## 6. Photon Splitting

## 7. Suppression of Air Shower Formation

## 8. Final Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LI | Lorentz invariance |

LIV | Lorentz invariance violation |

EBL | Extragalactic background bight |

CMB | Cosmic microwave background |

GRB | Gamma-Ray Bursts |

AGN | Active Galactic Nucleus |

CTA | Cherenkov Telescope Array |

H.E.S.S. | High Energy Stereoscopic System |

MAGIC | Major Atmospheric Gamma ray Imaging Cherenkov |

HAWC | Hight Altitud Water Cherenkov |

GZK | Kenneth Greisen, Vadim Kuzmin and Georgiy Zatsepin |

UHECR | Ultra-high-energy Cosmic Ray |

UHE | Ultra-high-energy |

IACT | Imaging Air Cherenkov Telescopes |

SGSO | Southern Gamma-Ray Survey Observatory |

SWGO | The Southern Wide-field Gamma ray Observatory |

LHAASO | Large High Altitude Air Shower Observatory |

PP | Photon pair production |

PD | Photon decay |

$\left(3\gamma \right)$ | Photon splitting into three photons |

$\Delta t$ | Energy-dependent time delay |

AS | Air shower |

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**Figure 1.**Comparison of strong and recent exclusion bounds on LIV coming from the lack of signatures of energy-dependent time delay, photon decay, photon splitting, suppression of air shower formations, and pair-production threshold shifts in astrophysical data. At the top (blue, ${v}_{\gamma}<c$), there are subluminal limits and at the bottom (green, ${v}_{\gamma}>c$) superluminal ones. The left and right panels are for the approximation orders $n=1$ and 2, respectively. See Table 1 for further information about these limits.

**Figure 2.**Gamma ray absorption for LI and LIV scenarios. When LIV is subluminal, there is a recovery in the flux while the superluminal scenario has the effect to reduce the gamma ray flux. The effect depends on the LIV energy scale, the gamma ray energy and the redshift to the source, as it can be compared between panels (

**a**,

**b**) for different red shifts but the same LIV value.

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Martínez-Huerta, H.; Lang, R.G.; de Souza, V.
Lorentz Invariance Violation Tests in Astroparticle Physics. *Symmetry* **2020**, *12*, 1232.
https://doi.org/10.3390/sym12081232

**AMA Style**

Martínez-Huerta H, Lang RG, de Souza V.
Lorentz Invariance Violation Tests in Astroparticle Physics. *Symmetry*. 2020; 12(8):1232.
https://doi.org/10.3390/sym12081232

**Chicago/Turabian Style**

Martínez-Huerta, Humberto, Rodrigo Guedes Lang, and Vitor de Souza.
2020. "Lorentz Invariance Violation Tests in Astroparticle Physics" *Symmetry* 12, no. 8: 1232.
https://doi.org/10.3390/sym12081232