# Testing Lorentz Symmetry Using High Energy Astrophysics Observations

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## Abstract

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## 1. Introduction

## 2. The Coleman–Glashow Formalism

**M**${}^{2}$ can be transformed so that $\mathbf{Z}$ is the identity and

**M**${}^{2}$ is diagonalized to produce the standard theory of n decoupled free fields. (We adopt the standard notation of denoting four-vector indexes by Greek letters and three-vector spatial indexes by Latin letters).

**$\eta $**is a small dimensionless Hermitian matrix that commutes with

**M**${}^{2}$ so that the fields remain separable and the resulting single particle energy-momentum eigenstates go from eigenstates of

**M**${}^{2}$ at low energy to eigenstates of

**$\eta $**at high energies.

## 3. LIV Modified Kinematics for QED

## 4. Ultrahigh Energy Cosmic Rays

#### 4.1. Extragalactic Origin

#### 4.2. The Greisen–Zatsepin–Kuz’min Effect

## 5. Modification of the GZK Effect

#### 5.1. Energy Threshold

#### 5.2. Detailed GZK Kinematics with LIV

## 6. Comparison of LIV-Modified UHECR Spectra with Observations

- The CBR photon number density increases as ${(1+z)}^{3}$, and the CBR photon energies increase linearly with $(1+z)$. The corresponding energy loss for protons at any redshift z is thus given by:$$\begin{array}{c}\hfill {r}_{\gamma p}(E,z)={(1+z)}^{3}r\left[(1+z)E\right].\end{array}$$
- It is assumed that the average UHECR volume emissivity is of the form given by $q({E}_{i},z)=K\left(z\right){E}_{i}^{-\mathsf{\Gamma}}$ where ${E}_{i}$ is the initial energy of the proton at the source and $\mathsf{\Gamma}=2.55$. The source evolution is assumed to be $K\left(z\right)\propto {(1+z)}^{3.6}$ with $z\le 2.5$ so that $K\left(z\right)$ is roughly proportional to the empirically-determined z-dependence of the star formation rate. $K(z=0)$ and $\mathsf{\gamma}$ are normalized to fit the data below the GZK threshold.

## 7. Modified GZK Neutrino Spectrum from a Modified UHECR Spectrum

## 8. Generalizing the Coleman–Glashow Formalism

#### 8.1. Time of Flight from $\gamma $-ray Bursts

#### 8.2. Gamma-Ray Absorption with Planck Suppressed LIV

## 9. Vacuum Birefringence

## 10. LIV in the Neutrino Sector

#### 10.1. Fermion LIV Operators with $\left[d\right]>4$ LIV with Rotational Symmetry in SME

#### 10.2. LIV in the Neutrino Sector I: Lepton Pair Emission

#### 10.2.1. Lepton Pair Emission in the [d] = 4 Case

#### 10.2.2. Vacuum ${e}^{+}{e}^{-}$ Pair Emission in the $\left[d\right]>4$ Cases

## 11. LIV in the Neutrino Sector II: Neutrino Splitting

## 12. The Neutrinos Observed by IceCube

## 13. Extragalactic Superluminal Neutrino Propagation

## 14. The Theoretical Neutrino Energy Spectrum

#### 14.1. [d] = 4 $\mathcal{CPT}$ Conserving Operator Dominance

#### 14.2. [d] = 6 $\mathcal{CPT}$ Conserving Operator Dominance

#### 14.3. [d] = 5 CPT Violating Operator Dominance

## 15. Summary: Results for Superluminal Neutrinos

## 16. Stable Pions from LIV

## 17. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The calculated proton inelasticity modified by LIV for ${\delta}_{\pi p}=3\times {10}^{-23}$ as a function of CBR photon energy and proton energy [18].

**Figure 2.**The calculated proton attenuation lengths as a function of proton energy modified by LIV for various values of ${\delta}_{\pi p}$ (solid lines), shown with the attenuation length for pair production unmodified by LIV (dashed lines). From top to bottom, the curves are for ${\delta}_{\pi p}=1\times {10}^{-22}$, $3\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$, $2\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$, $1\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$, $3\times {10}^{-24}$, 0 (no Lorentz violation) [18].

**Figure 3.**Comparison of Auger data [20] with calculated spectra for various values of ${\delta}_{\pi p}$, taking ${\delta}_{p}=0$ (see the text). From top to bottom, the curves give the predicted spectra for ${\delta}_{\pi p}=1\times {10}^{-22}$, $6\times {10}^{-23}$, $4.5\times {10}^{-23},3\times {10}^{-23}$, $2\times {10}^{-23}$, $1\times {10}^{-23}$, $3\times {10}^{-24}$, 0 (no Lorentz violation) [18].

**Figure 5.**Diagrams for muon decay (top), charged current-mediated vacuum electron-positron pair emission (VPE) (bottom left) and neutral current-mediated neutrino splitting and VPE (bottom right). Time runs from left to right, and the flavor index i represents $e,\mu $, or $\tau $ neutrinos.

**Figure 6.**Propagated neutrino spectra including energy losses as described in the text [36]. Separately calculated n = 2 neutrino spectra with the VPE case shown in blue and the neutrino splitting case shown in green. The black spectrum takes account of all three processes (redshifting, neutrino splitting and VPE) occurring simultaneously. The rates for all cases are fixed by setting the rest frame threshold energy for VPE at 10 PeV. The neutrino spectra are normalized to the IceCube data both with (gray) and without (black) an estimated flux of prompt atmospheric neutrinos subtracted [49].

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Stecker, F.W.
Testing Lorentz Symmetry Using High Energy Astrophysics Observations. *Symmetry* **2017**, *9*, 201.
https://doi.org/10.3390/sym9100201

**AMA Style**

Stecker FW.
Testing Lorentz Symmetry Using High Energy Astrophysics Observations. *Symmetry*. 2017; 9(10):201.
https://doi.org/10.3390/sym9100201

**Chicago/Turabian Style**

Stecker, Floyd W.
2017. "Testing Lorentz Symmetry Using High Energy Astrophysics Observations" *Symmetry* 9, no. 10: 201.
https://doi.org/10.3390/sym9100201