# Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. CPT Theorem

#### 2.1. Wightman Axioms

**Poincaré covariance of the Hilbert space where the theory is set.**This means that unitary operators $U(\Lambda ,\phantom{\rule{0.166667em}{0ex}}a)$ exist that implement Lorentz transformations and space-time translations;**Existence of a vacuum state $|0\rangle $**.There is a unique state, the vacuum, that is unaffected by Poincaré transformations up to a phase: $\left|U(\Lambda ,\phantom{\rule{0.166667em}{0ex}}a)\right|0\rangle ={e}^{i\varphi}|0\rangle $. This implies that this state can only have null four-momentum and angular momentum, since these quantities change under Lorentz transformations. The vacuum state must have even the lowest allowed energy and must be cyclic, which is acting on it via the creation operators every Hilbert space state can be constructed;**Fields constructed via operators.**All physical quantities can be constructed using polynomials of fields acting on the Hilbert space. The fields transform under Poincaré symmetry as scalars, spinors, and tensors. The fields are defined in such a way that each one corresponds to a definite physical state, i.e., a particle with defined physical quantities and quantum numbers, such as mass, spin etc.;**Energy positivity.**The Hamiltonian operator is supposed to have non negative eigenvalues. This property together with covariance under the action of Lorentz group implies that the physical four-momentum belongs to the light cone;**Microscopic causality.**Causality is imposed in the meaning of locality, that is, the field operators can commute/anticommute only if they are defined on points separated by space-like vectors: ${[\varphi \left(x\right),\phantom{\rule{0.166667em}{0ex}}\varphi \left(y\right)]}_{\pm}=0$ for all space-time points such that $({x}^{\mu}-{y}^{\mu}){\eta}_{\mu \nu}({x}^{\nu}-{y}^{\nu})<0$.

#### 2.2. Complex Lorentz Group

#### 2.3. Axiomatic CPT Theorem Demonstration

#### 2.4. Lagrangian Field Theory CPT Theorem Demonstration

- Scalar:$${\varphi}_{CPT}\left(x\right)=\theta \varphi \left(x\right){\theta}^{\u2020}={\varphi}^{\u2020}(-x);$$
- Fermionic field with spin $=\frac{1}{2}$:$${\psi}_{CPT}\left(x\right)=\theta \psi \left(x\right){\theta}^{\u2020}=-{\gamma}_{5}{\psi}^{\u2020T}(-x);$$
- Bosonic fields with spin $=1$$${A}_{\mu CPT}\left(x\right)=\theta {A}_{\mu}\left(x\right){\theta}^{\u2020}=-{A}_{\mu}^{\u2020}(-x).$$

## 3. CPT Violation Implies Lorentz Invariance Violation

## 4. Consequences of CPT Symmetry

## 5. CPT Violation Motivations

## 6. CPT Theorem in Curved Spacetime

**Locality and covariance.**The indices must transform under the action of a generic $\rho $ transformation as:$$({\rho}^{*}\times \dots {\rho}^{*}\times {\rho}_{*}^{-1}){C}_{\left({j}^{\prime}\right)}^{({i}_{1}^{\prime},\dots ,{i}_{n}^{\prime})}\left[{M}^{\prime}\right]={C}_{\left(j\right)}^{({i}_{1},\dots ,{i}_{n})}\left[M\right]$$**Identity element..**$${C}_{\left(j\right)}^{{i}_{1},\dots \mathbf{i},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)={C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{k-1},\phantom{\rule{0.166667em}{0ex}}{i}_{k+1},\dots ,{i}_{n}}({x}_{1},\dots {x}_{k-1},\phantom{\rule{0.166667em}{0ex}}{x}_{k+1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)$$**Compatibility with ∗**$${C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)\sim {C}_{\left({j}^{*}\right)}^{{i}_{1}^{*},\dots ,{i}_{n}^{*}}{\pi}_{0}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)$$**Commutativity-anticommutativity.**$${C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{k+1},\phantom{\rule{0.166667em}{0ex}}{i}_{k},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{k+1},\phantom{\rule{0.166667em}{0ex}}{x}_{k},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)=-{(-1)}^{F\left({i}_{k}\right)F\left({i}_{k+1}\right)}{C}_{\left(j\right)}^{{i}_{1},\dots ,{i}_{n}}({x}_{1},\dots ,{x}_{n},\phantom{\rule{0.166667em}{0ex}}y)$$**Scaling degree.**$$sd\{{C}_{\left(k\right)}^{\left(i\right)\left(j\right)}\}\le dim\left(i\right)+dim\left(j\right)-dim\left(k\right);$$**Asymptotic positivity.**$dim\left(i\right)\ge 0$ and $dim\left(i\right)=0$ if and only if $i=\mathbb{I}$;**Spectrum condition.**The singularities on the field product must have a positive frequency;**Associativity.**An opportunely defined notion of associativity is required;**Analytic dependence upon the metric.**The OPE coefficients must be regular functionals of the space-time metric.

#### Construction of a QFT from OPE Coefficients

**Linearity.**$\forall {a}_{i}$ and $\forall {f}_{i}$ must be true that:$${\varphi}^{i}(\sum _{j}{a}_{j}{f}_{j})=\sum _{j}{a}_{j}\varphi \left({f}_{j}\right);$$**Existence of ∗ operator.**$${\left[{\varphi}^{\left(i\right)}\left(f\right)\right]}^{*}={\varphi}^{i*}\left(\overline{f}\right);$$**Relations arising from OPE.**If $O(\xb7)$ is a smeared quantum field, then $O\left(y\right)$ is a well defined algebra element for every point y, that is if $\langle O\left(y\right)|O{\left(y\right)}^{*}\rangle =0$ then $O\left(y\right)=0$ as an algebra element;**Anticommutation Relation.**$${\varphi}^{{i}_{1}}\left({f}_{1}\right){\varphi}^{{i}_{1}}\left({f}_{2}\right)={(-1)}^{K\left({i}_{1}{i}_{2}\right)}{\varphi}^{{i}_{1}}\left({f}_{2}\right){\varphi}^{{i}_{1}}\left({f}_{1}\right)$$**Positivity.**$$\langle {A}^{*}A\rangle \ge 0\phantom{\rule{2.em}{0ex}}\forall A\in \mathcal{A}\left(M\right);$$**OPE series expansion.**$$\langle {\varphi}^{{i}_{1}}\left({x}_{1}\right),\dots ,{\varphi}^{{i}_{n}}\left({x}_{n}\right)\rangle \sim \sum _{j}{C}_{j}^{({i}_{1},\dots ,{i}_{n})}({x}_{1},\dots {x}_{n},\phantom{\rule{0.166667em}{0ex}}y)\langle {\varphi}^{j}\left(y\right)\rangle ;$$**Spectrum condition.**The previously cited spectrum condition can be written in the following form:$$WF\left(\right)open="("\; close=")">\langle {\varphi}^{{i}_{1}}({x}_{1},\dots {\varphi}^{{i}_{n}}\left({x}_{n}\right)\rangle $$

**Lemma**

**1.**

**Theorem**

**1.**

## 7. CPT and Gravity

## 8. CPT Violation and LIV Research

#### 8.1. Very Special Relativity

#### 8.2. Standard Model Extension

#### 8.3. Theories Preserving Covariance

#### 8.4. CPT Violation and LIV Geometry Framework

## 9. Search for CPT and LIV Violation in Astroparticle Physics

#### 9.1. Ultra High Energy Cosmic Rays

#### 9.2. Time Delays

#### 9.3. CPT and LIV in Neutrino Physics

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Schwinger, J. The Theory of Quantized Fields. I. Phys. Rev.
**1951**, 82, 914. [Google Scholar] [CrossRef] - Lüders, G. Proof of the TCP theorem. Ann. Phys.
**1957**, 2, 1–15. [Google Scholar] [CrossRef] - Pauli, W. Niels Bohr and the Development of Physics; MacGraw-Hill: New York, NY, USA, 1955; pp. 30–51. [Google Scholar]
- Jost, R. A remark on the C.T.P. theorem. Helv. Phys. Acta
**1957**, 30, 409–416. [Google Scholar] [CrossRef] - Bell, J.S. Time reversal in field theory. Proc. R. Soc. Lond.
**1955**, 231, 479–495. [Google Scholar] - Schwinger, J. Spin, statistic, and the TCP theorem. Proc. Natl. Acad. Sci. USA
**1958**, 44, 223–228. [Google Scholar] [CrossRef] [PubMed][Green Version] - Greenberg, O. CPT violation implies violation of Lorentz invariance. Phys. Rev. Lett.
**2002**, 89, 231602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Streater, R.; Wightman, A. PCT, Spin and Statistics, and All That; Princeton University Press: Princeton, NJ, USA, 2000. [Google Scholar]
- Bogolyubov, N.; Logunov, A.; Todorov, I. Introduction to Axiomatic Quantum Field Theory; Kluwer Academic Publishers: Boston, MA, USA, 1990. [Google Scholar]
- Haag, R. Local Quantum Physics: Fields, Particles, Algebras; Springer: Berlin, Germany, 1992. [Google Scholar]
- Lehnert, R. CPT Symmetry and Its Violation. Symmetry
**2016**, 8, 114. [Google Scholar] [CrossRef][Green Version] - Greenberg, O. Why is CPT fundamental? Found. Phys.
**2006**, 36, 1535–1553. [Google Scholar] [CrossRef][Green Version] - Hall, D.W.; Wightman, A.S. A theorem on invariant analytic functions with applications to relativistic quantum field theory. Mater. Fys. Medd. Danske Vid. Selsk.
**1957**, 31, 5. [Google Scholar] - Greaves, H.; Thomas, T. On the CPT theorem. Stud. Hist. Philos. Sci. B
**2014**, 45, 46–65. [Google Scholar] [CrossRef][Green Version] - Chaichian, M.; Dolgov, A.D.; Novikov, V.A.; Tureanu, A. CPT Violation Does Not Lead to Violation of Lorentz Invariance and Vice Versa. Phys. Lett. B
**2011**, 699, 177–180. [Google Scholar] [CrossRef][Green Version] - Tureanu, A. CPT and Lorentz Invariance: Their Relation and Violation. J. Phys. Conf. Ser.
**2013**, 474, 2031. [Google Scholar] [CrossRef][Green Version] - Chaichian, M.; Fujikawa, K.; Tureanu, A. Electromagnetic interaction in theory with Lorentz invariant CPT violation. Phys. Lett. B
**2013**, 718, 1500–1504. [Google Scholar] [CrossRef][Green Version] - Duetsch, M.; Gracia-Bondia, J.M. On the assertion that PCT violation implies Lorentz non-invariance. Phys. Lett. B
**2012**, 711, 428–433. [Google Scholar] [CrossRef][Green Version] - Greenberg, O. Remarks on a Challenge to the Relation between CPT and Lorentz Violation. arXiv
**2011**, arXiv:1105.0927. [Google Scholar] - Hawking, S. Breakdown of Predictability in Gravitational Collapse. Phys. Rev. D
**1976**, 14, 2460–2473. [Google Scholar] [CrossRef] - Hawking, S. The Unpredictability of Quantum Gravity. Commun. Math. Phys.
**1982**, 87, 395–415. [Google Scholar] [CrossRef] - Wheeler, J.; Ford, K. Geons, black holes, and quantum foam: A life in physics. Am. J. Phys.
**2000**, 68, 584. [Google Scholar] [CrossRef] - Mavromatos, N.E. CPT violation: Theory and phenomenology. arXiv
**2005**, arXiv:hep-ph/0504143. [Google Scholar] - Kostelecky, V.; Samuel, S. Spontaneous Breaking of Lorentz Symmetry in String Theory. Phys. Rev. D
**1989**, 39, 683. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kostelecky, V.; Potting, R. CPT and strings. Nucl. Phys. B
**1991**, 359, 545–570. [Google Scholar] [CrossRef] - Kostelecky, V.; Potting, R. CPT, strings, and meson factories. Phys. Rev. D
**1995**, 51, 3923–3935. [Google Scholar] [CrossRef][Green Version] - Hollands, S.; Wald, R.M. Quantum field theory in curved spacetime, the operator product expansion, and dark energy. Gen. Relat. Gavit.
**2008**, 40, 2051–2059. [Google Scholar] [CrossRef][Green Version] - Hollands, S.; Wald, R.M. Axiomatic quantum field theory in curved spacetime. Commun. Math. Phys.
**2010**, 293, 85–125. [Google Scholar] [CrossRef][Green Version] - Morrison, P. Approximate Nature of Physical Symmetries. Am. J. Phys.
**1958**, 26, 358–368. [Google Scholar] [CrossRef] - Schiff, L.I. Sign of the Gravitational Mass of a Positron. Phys. Rev. Lett.
**1958**, 1, 254–255. [Google Scholar] [CrossRef] - Myron, M.L.G. K
_{2}^{0}and the Equivalence Principle. Phys. Rev.**1961**, 121, 311–313. [Google Scholar] [CrossRef] - Nieto, M.; Goldman, J. The Arguments against ’antigravity’ and the gravitational acceleration of antimatter. Phys. Rept.
**1991**, 205, 221–281. [Google Scholar] [CrossRef][Green Version] - Chardin, G.; Rax, J. CP violation: A Matter of (anti)-gravity? Phys. Lett. B
**1992**, 282, 256–262. [Google Scholar] [CrossRef] - Chardin, G. CP violation and antigravity (revisited). Nucl. Phys. A
**1993**, 558, 477C–496C. [Google Scholar] [CrossRef] - Hajdukovic, D.S. Do we live in the universe successively dominated by matter and antimatter? Astrophys. Space Sci.
**2011**, 334, 219–223. [Google Scholar] [CrossRef] - Hajdukovic, D.S. What would be outcome of a Big Crunch? Int. J. Theor. Phys.
**2010**, 49, 1023–1028. [Google Scholar] [CrossRef][Green Version] - Noyes, H.P. On ‘Dark Energy from Antimatter’ by Walter R. Lamb; SLAC-PUB-12849; Stanford Linear Accelerator Center (SLAC): Menlo Park, CA, USA, 2007. [Google Scholar]
- Benoit-Levy, A.; Chardin, G. Observational constraints of a Milne Universe. arXiv
**2008**, arXiv:0811.2149. [Google Scholar] - Villata, M. CPT symmetry and antimatter gravity in general relativity. EPL
**2011**, 94, 20001. [Google Scholar] [CrossRef][Green Version] - Coleman, S.R.; Glashow, S.L. High-energy tests of Lorentz invariance. Phys. Rev. D
**1999**, 59, 116008. [Google Scholar] [CrossRef][Green Version] - Cohen, A.G.; Glashow, S.L. Very special relativity. Phys. Rev. Lett.
**2006**, 97, 021601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gibbons, G.W.; Gomis, J.; Pope, C.N. General very special relativity is Finsler geometry. Phys. Rev. D
**2007**, 76, 081701. [Google Scholar] [CrossRef][Green Version] - Colladay, D.; Kostelecky, V. Lorentz violating extension of the standard model. Phys. Rev. D
**1998**, 58, 116002. [Google Scholar] [CrossRef][Green Version] - Kostelecky, V.A.; Russell, N. Data Tables for Lorentz and CPT Violation. Rev. Mod. Phys.
**2011**, 83, 11. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G. Doubly special relativity. Nature
**2002**, 418, 34–35. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G. Doubly special relativity: First results and key open problems. Int. J. Mod. Phys. D
**2002**, 11, 1643. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. The principle of relative locality. Phys. Rev. D
**2011**, 84, 084010. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G.; Bianco, S.; Rosati, G. Planck-Scale-Deformed Relativistic Symmetries and Diffeomorphisms on Momentum Space. Phys. Rev. D
**2020**, 101, 026018. [Google Scholar] [CrossRef][Green Version] - Torri, M.D.C.; Antonelli, V.; Miramonti, L. Homogeneously Modified Special relativity (HMSR). Eur. Phys. J. C
**2019**, 79, 808. [Google Scholar] [CrossRef][Green Version] - Antonelli, V.; Miramonti, L.; Torri, M.D.C. Neutrino oscillations and Lorentz invariance violation in a Finslerian geometrical model. Eur. Phys. J. C
**2018**, 78, 667. [Google Scholar] [CrossRef][Green Version] - Torri, M.D.C. Lorentz Invariance Violation Effects on Ultra High Energy Cosmic Rays Propagation, a Geometrical Approach. Ph.D. Thesis, Milan University (UNIMI), Milan, Italy, 2019. [Google Scholar]
- Kostelecky, V.A. Comments on Lorentz and CPT Violation. In Proceedings of the 6th Meeting on CPT and Lorentz Symmetry, Bloomington, IN, USA, 17–21 June 2014; pp. 33–36. [Google Scholar] [CrossRef][Green Version]
- Lämmerzahl, C.; Perlick, V. Finsler geometry as a model for relativistic gravity. Int. J. Geom. Meth. Mod. Phys.
**2018**, 15, 1850166. [Google Scholar] [CrossRef][Green Version] - Hohmann, M.; Pfeifer, C.; Voicu, N. Finsler gravity action from variational completion. Phys. Rev. D
**2019**, 100, 064035. [Google Scholar] [CrossRef][Green Version] - Bubuianu, L.; Vacaru, S.I. Black holes with MDRs and BekenteinHawking and Perelman entropies for FinslerLagrangeHamilton Spaces. Ann. Phys.
**2019**, 404, 10–38. [Google Scholar] [CrossRef][Green Version] - Schreck, M. Classical Lagrangians and Finsler structures for the nonminimal fermion sector of the Standard-Model Extension. Phys. Rev. D
**2016**, 93, 105017. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G.; Barcaroli, L.; Gubitosi, G.; Liberati, S.; Loret, N. Realization of doubly special relativistic symmetries in Finsler geometries. Phys. Rev. D
**2014**, 90, 125030. [Google Scholar] [CrossRef][Green Version] - Pfeifer, C. Finsler spacetime geometry in Physics. Int. J. Geom. Meth. Mod. Phys.
**2019**, 16, 1941004. [Google Scholar] [CrossRef][Green Version] - Fuster, A.; Pabst, C.; Pfeifer, C. Berwald spacetimes and very special relativity. Phys. Rev. D
**2018**, 98, 084062. [Google Scholar] [CrossRef][Green Version] - Barcaroli, L.; Brunkhorst, L.K.; Gubitosi, G.; Loret, N.; Pfeifer, C. Hamilton geometry: Phase space geometry from modified dispersion relations. Phys. Rev. D
**2015**, 92, 084053. [Google Scholar] [CrossRef] - Zatsepin, G.; Kuzmin, V. Upper limit of the spectrum of cosmic rays. JETP Lett.
**1966**, 4, 78–80. [Google Scholar] - Greisen, K. End to the cosmic ray spectrum? Phys. Rev. Lett.
**1966**, 16, 748–750. [Google Scholar] [CrossRef] - Resconi, E.; Coenders, S.; Padovani, P.; Giommi, P.; Caccianiga, L. Connecting blazars with ultrahigh-energy cosmic rays and astrophysical neutrinos. Mon. Not. R. Astron. Soc.
**2017**, 468, 597–606. [Google Scholar] [CrossRef] - Scully, S.; Stecker, F. Lorentz Invariance Violation and the Observed Spectrum of Ultrahigh Energy Cosmic Rays. Astropart. Phys.
**2009**, 31, 220–225. [Google Scholar] [CrossRef][Green Version] - Stecker, F.W.; Scully, S.T. Searching for New Physics with Ultrahigh Energy Cosmic Rays. New J. Phys.
**2009**, 11, 085003. [Google Scholar] [CrossRef] - Torri, M.D.C.; Bertini, S.; Giammarchi, M.; Miramonti, L. Lorentz Invariance Violation effects on UHECR propagation: A geometrized approach. J. High Energy Astrophys.
**2018**, 18, 5–14. [Google Scholar] [CrossRef] - Saveliev, A.; Maccione, L.; Sigl, G. Lorentz Invariance Violation and Chemical Composition of Ultra High Energy Cosmic Rays. J. Cosmol. Astropart. Phys.
**2011**, 3, 046. [Google Scholar] [CrossRef][Green Version] - Shapiro, I. Four tests of General Relativity. Phys. Rev. Lett.
**1964**, 13, 789–791. [Google Scholar] [CrossRef] - Albert, J.; Aliu, E.; Anderhub, H.; Antonelli, L.A.; Antoranz, P.; Backes, M.; Baixeras, C.; Barrio, J.A.; Bartko, H.; Bastieri, D.; et al. Probing Quantum Gravity using Photons from a flare of the active galactic nucleus Markarian 501 Observed by the MAGIC telescope. Phys. Lett. B
**2008**, 668, 253–257. [Google Scholar] [CrossRef] - Ellis, J.; Konoplich, R.; Mavromatos, N.E.; Nguyen, L.; Sakharov, A.S.; Sarkisyan-Grinbaum, E.K. Robust Constraint on Lorentz Violation Using Fermi-LAT Gamma-Ray Burst Data. Phys. Rev. D
**2019**, 99, 083009. [Google Scholar] [CrossRef][Green Version] - Abdalla, H. The 2014 TeV γ-Ray Flare of Mrk 501 Seen with H.E.S.S.: Temporal and Spectral Constraints on Lorentz Invariance Violation. Astrophys. J.
**2019**, 870, 93. [Google Scholar] [CrossRef][Green Version] - Xu, H.; Ma, B.Q. Regularity of high energy photon events from gamma ray bursts. J. Cosmol. Astropart. Phys.
**2018**, 1, 050. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G.; Ellis, J.R.; Mavromatos, N.E.; Nanopoulos, D.V.; Sarkar, S. Tests of quantum gravity from observations of gamma-ray bursts. Nature
**1998**, 393, 763–765. [Google Scholar] [CrossRef][Green Version] - Amelino-Camelia, G.; Loret, N.; Rosati, G. Speed of particles and a relativity of locality in κ-Minkowski quantum spacetime. Phys. Lett. B
**2011**, 700, 150–156. [Google Scholar] [CrossRef][Green Version] - Loret, N. Exploring special relative locality with de Sitter momentum-space. Phys. Rev. D
**2014**, 90, 124013. [Google Scholar] [CrossRef][Green Version] - Aldrovandi, R.; Pereira, J.G. De Sitter relativity: A New road to quantum gravity. Found. Phys.
**2009**, 39, 1–19. [Google Scholar] [CrossRef] - Bolmont, J.; Perennes, C. Probing modified dispersion relations in vacuum with high-energy γ-ray sources: Review and prospects. J. Phys. Conf. Ser.
**2020**, 1586, 012033. [Google Scholar] [CrossRef] - Torri, M.D.C. Neutrino Oscillations and Lorentz Invariance Violation. Universe
**2020**, 6, 37. [Google Scholar] [CrossRef][Green Version] - Cohen, A.G.; Glashow, S.L. A Lorentz-Violating Origin of Neutrino Mass? arXiv
**2006**, arXiv:hep-ph/0605036. [Google Scholar] - Stecker, F.W.; Scully, S.T.; Liberati, S.; Mattingly, D. Searching for Traces of Planck-Scale Physics with High Energy Neutrinos. Phys. Rev. D
**2015**, 91, 045009. [Google Scholar] [CrossRef][Green Version] - Kostelecky, V.A.; Mewes, M.M. Electrodynamics with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2009**, 80, 015020. [Google Scholar] [CrossRef][Green Version] - Aartsen, M.E.; Ackermann, M.; Adams, J.; Aguilar, J.A.; Ahlers, M.; Ahrens, M.; Altmann, D.; Anderson, T.; Arguelles, C.; Arlen, T.C.; et al. Observation of High-Energy Astrophysical Neutrinos in Three Years of IceCube Data. Phys. Rev. Lett.
**2014**, 113, 101101. [Google Scholar] [CrossRef][Green Version] - Liberati, S. Tests of Lorentz invariance: A 2013 update. Class. Quantum Grav.
**2013**, 30, 133001. [Google Scholar] [CrossRef] - Stecker, F.W.; Scully, S.T. Propagation of Superluminal PeV IceCube Neutrinos: A High Energy Spectral Cutoff or New Constraints on Lorentz Invariance Violation. Phys. Rev. D
**2014**, 90, 043012. [Google Scholar] [CrossRef][Green Version] - Kostelecky, A.; Mewes, M. Fermions with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2013**, 88, 096006. [Google Scholar] [CrossRef][Green Version] - Diaz, J.S.; Kostelecky, A.; Mewes, M. Testing Relativity with High-Energy Astrophysical Neutrinos. Phys. Rev. D
**2014**, 89, 043005. [Google Scholar] [CrossRef][Green Version] - Jacobson, T.A.; Liberati, S.; Mattingly, D.; Stecker, F.W. New limits on Planck scale Lorentz violation in QED. Phys. Rev. Lett.
**2004**, 93, 021101. [Google Scholar] [CrossRef] - Montemayor, R.; Urrutia, L.F. Synchrotron radiation in Lorentz-violating electrodynamics: The Myers-Pospelov model. Phys. Rev. D
**2005**, 72, 045018. [Google Scholar] [CrossRef][Green Version] - Altschul, B. Synchrotron and inverse compton constraints on Lorentz violations for electrons. Phys. Rev. D
**2006**, 74, 083003. [Google Scholar] [CrossRef][Green Version] - Maccione, L.; Liberati, S.; Celotti, A.; Kirk, J.G. New constraints on Planck-scale Lorentz violation in QED from the Crab Nebula. J. Cosmol. Astropart. Phys.
**2007**, 10, 013. [Google Scholar] [CrossRef][Green Version] - Stecker, F.W. Limiting superluminal electron and neutrino velocities using the 2010 Crab Nebula flare and the IceCube PeV neutrino events. Astropart. Phys.
**2014**, 56, 16. [Google Scholar] [CrossRef][Green Version] - Katori, T. Test of Lorentz Violation with Astrophysical Neutrino Flavor at IceCube. In Proceedings of the 8th Meeting on CPT and Lorentz Symmetry (CPT’19), Bloomington, IN, USA, 12–16 May 2019. [Google Scholar] [CrossRef][Green Version]
- Ellis, J.; Janka, H.T.; Mavromatos, N.E.; Sakharov, A.S.; Sarkisyan, E.K.G. Probing Lorentz Violation in Neutrino Propagation from a Core-Collapse Supernova. Phys. Rev. D
**2012**, 85, 045032. [Google Scholar] [CrossRef][Green Version] - Chakraborty, S.; Mirizzi, A.; Sigl, G. Testing Lorentz invariance with neutrino bursts from supernova neutronization. Phys. Rev. D
**2013**, 87, 017302. [Google Scholar] [CrossRef][Green Version] - Datta, A.; Gandhi, R.; Mehta, P.; Sankar, S.U. Atmospheric neutrinos as a probe of CPT and Lorentz violation. Phys. Lett. B
**2004**, 597, 356. [Google Scholar] [CrossRef][Green Version] - Chatterjee, A.; Gandhi, R.; Singh, J. Probing Lorentz and CPT Violation in a Magnetized Iron Detector using Atmospheric Neutrinos. J. High Energy Phys.
**2014**, 06, 045. [Google Scholar] [CrossRef][Green Version] - Ellis, J.; Mavromatos, N.E.; Sakharov, A.S.; Sarkisyan-Grinbaum, E.K. Limits on Neutrino Lorentz Violation from Multimessenger Observations of TXS 0506+056. Phys. Lett. B
**2019**, 789, 352. [Google Scholar] [CrossRef] - Wei, J.J.; Zhang, B.B.; Shao, L.; Gao, H.; Li, Y. Multimessenger tests of Einstein’s weak equivalence principle and Lorentz invariance with a high-energy neutrino from a flaring blazar. J. High Energy Astrophys.
**2019**, 22, 1. [Google Scholar] [CrossRef][Green Version] - Kostelecky, V.A.; Mewes, M. Lorentz violation and short-baseline neutrino experiments. Phys. Rev. D
**2004**, 70, 076002. [Google Scholar] [CrossRef][Green Version] - Aguilar-Arevalo, A.A. Test of Lorentz and CPT violation with Short Baseline Neutrino Oscillation Excesses. Phys. Lett. B
**2013**, 718, 1303. [Google Scholar] [CrossRef][Green Version] - Abe, K. Search for Lorentz and CPT violation using sidereal time dependence of neutrino flavor transitions over a short baseline. Phys. Rev. D
**2017**, 95, 111101. [Google Scholar] [CrossRef][Green Version] - Diaz, J.S. Long-baseline neutrino experiments as tests for Lorentz violation. In Proceedings of the Meeting of the Division of the American Physical Society, DPF 2009, Detroit, MI, USA, 26–31 July 2009. [Google Scholar]
- Yu-Feng, L.; Zhen-Hua, Z. Tests of Lorentz and CPT Violation in the Medium Baseline Reactor Antineutrino Experiment. Phys. Rev. D
**2014**, 90, 113014. [Google Scholar] [CrossRef][Green Version] - Abe, K.; T2K Collaboration. The T2K Experiment. Nucl. Instrum. Meth. A
**2011**, 659, 106. [Google Scholar] [CrossRef] - Quilain, B. Results of Lorentz- and CPT-Invariance Violation at T2K and Future Perspectives. In Proceedings of the 7th Meeting on CPT and Lorentz Symmetry, Bloomington, IN, USA, 20–24 June 2016; pp. 125–128. [Google Scholar] [CrossRef]
- Adamson, P. Measurement of Neutrino Oscillations with the MINOS Detectors in the NuMI Beam. Phys. Rev. Lett.
**2008**, 101, 13180. [Google Scholar] [CrossRef][Green Version] - Adamson, P. A Search for Lorentz Invariance and CPT Violation with the MINOS Far Detector. Phys. Rev. Lett.
**2010**, 105, 151601. [Google Scholar] [CrossRef] - Barenboim, G.; Lykken, J.D. MINOS and CPT-violating neutrinos. Phys. Rev. D
**2009**, 80, 113008. [Google Scholar] [CrossRef][Green Version] - Acciarri, R. Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE): Conceptual Design Report, Volume 2: The Physics Program for DUNE at LBNF. arXiv
**2015**, arXiv:1512.06148. [Google Scholar] - Célio, C.A.M. Physics Beyond the Standard Model with DUNE: Prospects for Exploring Lorentz and CPT Violation. In Proceedings of the 8th Meeting on CPT and Lorentz Symmetry, Bloomington, IN, USA, 12–16 May 2019; pp. 150–153. [Google Scholar]
- Acero, M.A. First Measurement of Neutrino Oscillation Parameters using Neutrinos and Antineutrinos by NOvA. Phys. Rev. Lett.
**2019**, 123, 151803. [Google Scholar] [CrossRef][Green Version] - Abe, K. Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations. Nature
**2020**, 580, 339. [Google Scholar] - Mezzetto, M.; Terranova, F. Three-flavour oscillations with accelerator neutrino beams belongs to the Topical Collection on “Neutrino Oscillations” of the journal Universe. Universe
**2020**, 6, 32. [Google Scholar] [CrossRef][Green Version] - Majhi, R.; Chembra, S.; Mohanta, R. Exploring the effect of Lorentz invariance violation with the currently running long-baseline experiments. Eur. Phys. J. C
**2020**, 80, 364. [Google Scholar] [CrossRef] - Colladay, D.; Kostelecky, V. CPT violation and the standard model. Phys. Rev. D
**1997**, 55, 6760. [Google Scholar] [CrossRef][Green Version] - Diaz, J.S.; Kostelecky, A.; Mewes, M. Perturbative Lorentz and CPT violation for neutrino and antineutrino oscillations. Phys. Rev. D
**2009**, 80, 076007. [Google Scholar] [CrossRef][Green Version] - Kostelecky, A.; Mewes, M. Lorentz and CPT violation in neutrinos. Phys. Rev. D
**2004**, 69, 016005. [Google Scholar] [CrossRef][Green Version] - Kostelecky, V.A.; Mewes, M. Neutrinos with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D
**2012**, 85, 096005. [Google Scholar] [CrossRef][Green Version] - Abe, K.; Haga, Y.; Hayato, Y.; Ikeda, M.; Iyogi, K.; Kameda, J.; Kishimoto, Y.; Miura, M.; Moriyama, S.; Nakahata, M.; et al. Test of Lorentz invariance with atmospheric neutrinos. Phys. Rev. D
**2015**, 91, 052003. [Google Scholar] [CrossRef][Green Version] - Agarwalla, S.K.; Masud, M. Can Lorentz Invariance Violation affect the Sensitivity of Deep Underground Neutrino Experiment? arXiv
**2019**, arXiv:1912.13306. [Google Scholar] [CrossRef] - Antonelli, V.; Miramonti, L.; Torri, M.D.C. Geometrical models with Lorentz invariance violation and neutrino oscillations. Il Nuovo Cimento C
**2020**, 43, 65. [Google Scholar] [CrossRef][Green Version] - Miramonti, L.; Antonelli, V.; Torri, M.D.C. Homogeneously Modified Special Relativity applications for UHECR and Neutrino oscillations. In Proceedings of the Tenth Edition of the International Conference on High Energy and Astroparticle Physics (TIC-HEAP), Constantine, Algeria, 19–21 October 2019. [Google Scholar]
- Ageron, M. ANTARES: The first undersea neutrino telescope. Nucl. Instrum. Meth. A
**2011**, 656, 11. [Google Scholar] [CrossRef] - Capozzi, F.; Lisi, E.; Marrone, A. Probing the neutrino mass ordering with KM3NeT-ORCA: Analysis and perspectives. J. Phys. G
**2018**, 45, 024003. [Google Scholar] [CrossRef][Green Version] - Aartsen, M.G. Astrophysical neutrinos and cosmic rays observed by IceCube. Adv. Space Res.
**2018**, 62, 2902. [Google Scholar] [CrossRef][Green Version] - Schröder, F.G. High-Energy Galactic Cosmic Rays (Astro2020 Science White Paper). Bull. Am. Astron. Soc.
**2019**, 51, 131. [Google Scholar] - Abreu, P.; Aglietta, M.; Ahlers, M.; Ahn, E.; Albuquerque, I.; Allard, D.; Allekotte, I.; Allen, J.; Allison, P.; Almela, A.; et al. Search for Point-like Sources of Ultra-high Energy Neutrinos at the Pierre Auger Observatory and Improved Limit on the Diffuse Flux of Tau Neutrinos. Astrophys. J. Lett.
**2012**, 775, L4. [Google Scholar] [CrossRef][Green Version] - An, F. Neutrino Physics with JUNO. J. Phys. G
**2016**, 43, 030401. [Google Scholar] [CrossRef] - Antonelli, V.; Miramonti, L.; Ranucci, G. Present and Future Contributions of Reactor Experiments to Mass Ordering and Neutrino Oscillation Studies. Universe
**2020**, 6, 52. [Google Scholar] [CrossRef][Green Version] - Carmona, J.M.; Corts, J.L.; Relancio, J.; Javier, J.; Reyes, M.K. Lorentz Violation Footprints in the Spectrum of High-Energy Cosmic Neutrinos—Deformation of the Spectrum of Superluminal Neutrinos from Electron-Positron Pair Production in Vacuum. Symmetry
**2019**, 11, 1419. [Google Scholar] [CrossRef][Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Antonelli, V.; Miramonti, L.; Torri, M.D.C.
Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics. *Symmetry* **2020**, *12*, 1821.
https://doi.org/10.3390/sym12111821

**AMA Style**

Antonelli V, Miramonti L, Torri MDC.
Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics. *Symmetry*. 2020; 12(11):1821.
https://doi.org/10.3390/sym12111821

**Chicago/Turabian Style**

Antonelli, Vito, Lino Miramonti, and Marco Danilo Claudio Torri.
2020. "Phenomenological Effects of CPT and Lorentz Invariance Violation in Particle and Astroparticle Physics" *Symmetry* 12, no. 11: 1821.
https://doi.org/10.3390/sym12111821