# Overview of the Phenomenology of Lorentz and CPT Violation in Atomic Systems

## Abstract

**:**

## 1. Introduction

## 2. Classification of the Lorentz- and CPT-Violating Dirac in the Quadratic Lagrange Density for a Dirac Fermion

## 3. Hierarchy and the Lorentz-Violating Perturbation

## 4. Hyperfine Transitions and Anisotropic Terms

## 5. Isotropic Terms and Optical Transitions

## 6. The Problem of Testing CPT Symmetry Using Different Frames

## 7. Difference in the Signals for Minimal and Nonminimal Lorentz-Violating Terms

## 8. Best Bounds on and Prospects for Coefficients for Lorentz Violation from Spectroscopy Experiments

## 9. Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Symbol | Description |
---|---|

d | Mass dimension of the Lorentz-violating operator contracted with the coefficient in the Lagrangian density. Used in effective Cartesian and spherical coefficients. |

w | Specifies the flavor of the Lorentz-violating operator contracted with the coefficient in the Lagrangian density. Used in all coefficients. |

j | Specifies the rank of the spherical tensor contracted with the coefficient in the one-particle Hamiltonian; $j>0$. Used in nonrelativistic and spherical coefficients. |

m | Specifies the component of the spherical tensor contracted with the coefficient in the one-particle Hamiltonian; $m\in \{-j,-j+1,\dots ,j-1,j\}$. Used in nonrelativistic and spherical coefficients. |

n | Specifies the power of the three-momentum when the the one-particle Hamiltonian is expressed in terms of ${E}_{0}$ and $\left|\mathit{p}\right|$. Used in spherical coefficients; see Equation (8). |

k | Specifies the power of the three-momentum when the the one-particle Hamiltonian is expressed in terms of ${m}_{w}$ and $\left|\mathit{p}\right|$. Used in nonrelativistic coefficients; see Equation (10). |

Terminology | Description | Types of Coefficients |
---|---|---|

Effective Cartesian | Coefficients for Lorentz-violating operators | ${{\mathcal{V}}_{w}}_{\mathrm{eff}}^{\left(d\right)\mu {\alpha}_{1}\dots {\alpha}_{d-3}}$ |

expressed as Lorentz tensors. | ${\tilde{\mathcal{T}}}_{w}\phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{-0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(d\right)\mu \nu {\alpha}_{1}\dots {\alpha}_{d-3}}$ | |

Spherical | Coefficients for Lorentz-violating operators | ${{\mathcal{V}}_{w}}_{njm}^{\left(d\right)}$ |

expressed as spherical tensors | ${{\mathcal{T}}_{w}}_{njm}^{\left(d\right)\left(0B\right)}$, ${{\mathcal{T}}_{w}}_{njm}^{\left(d\right)\left(1B\right)}$ | |

Nonrelativistic | Linear combinations of spherical coefficients | ${{\mathcal{V}}_{w}}_{kjm}^{\mathrm{NR}}$ |

of arbitrary mass dimension d. | ${{\mathcal{T}}_{w}}_{kjm}^{\mathrm{NR}\left(0B\right)}$, ${{\mathcal{T}}_{w}}_{kjm}^{\mathrm{NR}\left(1B\right)}$ | |

Minimal | Coefficients for minimal operators | Coefficients with $d\le 4$ |

Nonminimal | Coefficients for nonminimal operators | Coefficients with $d>4$ |

CPT-even | Coefficients for CPT-invariant operators | $\mathcal{V}$-type with even d or c-type |

$\mathcal{T}$-type with odd d or H-type | ||

CPT-odd | Coefficients for CPT-violating operators | $\mathcal{V}$-type with odd d or a-type |

$\mathcal{T}$-type with even d or g-type | ||

Spin-dependent | Coefficients proportional to the Pauli matrices | $\mathcal{T}$-type; or equivalently |

in the one-particle Hamiltonian | g-type and H-type | |

Spin-independent | Coefficients not proportional to the Pauli | $\mathcal{V}$-type; or equivalently |

matrices in the one-particle Hamiltonian | a-type and c-type | |

Isotropic | Coefficients for rotational scalar | Spherical or nonrelativistic |

Lorentz-violating operators | coefficients with $j=0$ | |

Anisotropic | Coefficients for Lorentz-violating operators | Spherical or nonrelativistic |

that are not rotational scalars | coefficients with $j>0$ |

**Table 3.**Best bounds on the imaginary and real part of the spin-dependent anisotropic nonrelativistic coefficients in the Sun-centered frame for electron, proton, neutron, and muon operators.

Coefficients | Neutron [36] from Xe-He Comagnetometer | Proton and Electron [35] from Hydrogen 1S Splitting | Muon [34] from Muonium 1S Splitting |
---|---|---|---|

${{H}_{w}}_{011}^{\mathrm{NR}\left(0B\right)},\phantom{\rule{0.166667em}{0ex}}{{g}_{w}}_{011}^{\mathrm{NR}\left(0B\right)}$ | $4\times {10}^{-33}$ GeV | $9\times {10}^{-27}$ GeV | $2\times {10}^{-22}$ GeV |

${{H}_{w}}_{011}^{\mathrm{NR}\left(1B\right)},\phantom{\rule{0.166667em}{0ex}}{{g}_{w}}_{011}^{\mathrm{NR}\left(1B\right)}$ | $2\times {10}^{-33}$ GeV | $5\times {10}^{-27}$ GeV | $7\times {10}^{-23}$ GeV |

${{H}_{w}}_{211}^{\mathrm{NR}\left(0B\right)},\phantom{\rule{0.166667em}{0ex}}{{g}_{w}}_{211}^{\mathrm{NR}\left(0B\right)}$ | $4\times {10}^{-31}$ GeV${}^{-1}$ | $7\times {10}^{-16}$ GeV${}^{-1}$ | $1\times {10}^{-11}$ GeV${}^{-1}$ |

${{H}_{w}}_{211}^{\mathrm{NR}\left(1B\right)},\phantom{\rule{0.166667em}{0ex}}{{g}_{w}}_{211}^{\mathrm{NR}\left(1B\right)}$ | $2\times {10}^{-31}$ GeV${}^{-1}$ | $4\times {10}^{-16}$ GeV${}^{-1}$ | $6\times {10}^{-12}$ GeV${}^{-1}$ |

${{H}_{w}}_{411}^{\mathrm{NR}\left(0B\right)},\phantom{\rule{0.166667em}{0ex}}{{g}_{w}}_{411}^{\mathrm{NR}\left(0B\right)}$ | $4\times {10}^{-29}$ GeV${}^{-3}$ | $9\times {10}^{-6}$ GeV${}^{-3}$ | $2\times {10}^{-1}$ GeV${}^{-3}$ |

${{H}_{w}}_{411}^{\mathrm{NR}\left(1B\right)},\phantom{\rule{0.166667em}{0ex}}{{g}_{w}}_{411}^{\mathrm{NR}\left(1B\right)}$ | $2\times {10}^{-29}$ GeV${}^{-3}$ | $5\times {10}^{-6}$ GeV${}^{-3}$ | $8\times {10}^{-2}$ GeV${}^{-3}$ |

Coefficient | Electron [35] | Proton [35] | Coefficient | Neutron [36] | Coefficient | Neutron [36] |
---|---|---|---|---|---|---|

GeV${}^{4-\mathbf{d}}$ | GeV${}^{4-\mathbf{d}}$ | GeV${}^{4-\mathbf{d}}$ | GeV${}^{4-\mathbf{d}}$ | |||

${{a}_{w}}_{\mathrm{eff}}^{\left(5\right)TTX}$ | <$3.4\times {10}^{-8}$ | <$3.4\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)X\left(TXT\right)}$ | <$1\times {10}^{-27}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)X\left(TXTT\right)}$ | <$9\times {10}^{-28}$ |

${{a}_{w}}_{\mathrm{eff}}^{\left(5\right)TTY}$ | <$5.6\times {10}^{-8}$ | <$5.6\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)X\left(TYT\right)}$ | <$8\times {10}^{-28}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)X\left(TYTT\right)}$ | <$7\times {10}^{-28}$ |

${{a}_{w}}_{\mathrm{eff}}^{\left(5\right)TTY}$ | <$1.3\times {10}^{-7}$ | <$1.3\times {10}^{-7}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)X\left(TZT\right)}$ | <$2\times {10}^{-27}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)X\left(TZTT\right)}$ | <$2\times {10}^{-27}$ |

${{a}_{w}}_{\mathrm{eff}}^{\left(5\right)KKX}$ | <$6.7\times {10}^{-8}$ | <$6.7\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)Y\left(TXT\right)}$ | <$8\times {10}^{-28}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)Y\left(TXTT\right)}$ | <$6\times {10}^{-28}$ |

${{a}_{w}}_{\mathrm{eff}}^{\left(5\right)KKY}$ | <$1.1\times {10}^{-7}$ | <$1.1\times {10}^{-7}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)Y\left(TYT\right)}$ | <$8\times {10}^{-28}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)Y\left(TYTT\right)}$ | <$7\times {10}^{-28}$ |

${{a}_{w}}_{\mathrm{eff}}^{\left(5\right)KKZ}$ | <$2.5\times {10}^{-7}$ | <$2.5\times {10}^{-7}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)Y\left(TZT\right)}$ | <$2\times {10}^{-27}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)Y\left(TZTT\right)}$ | <$2\times {10}^{-27}$ |

${{c}_{w}}_{\mathrm{eff}}^{\left(6\right)TTTX}$ | <$3.3\times {10}^{-5}$ | <$1.8\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)X\left(JXJ\right)}$ | <$4\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)X\left(JXJT\right)}$ | <$9\times {10}^{-26}$ |

${{c}_{w}}_{\mathrm{eff}}^{\left(6\right)TTTY}$ | <$5.5\times {10}^{-5}$ | <$3.0\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)X\left(JYJ\right)}$ | <$3\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)X\left(JYJT\right)}$ | <$7\times {10}^{-26}$ |

${{c}_{w}}_{\mathrm{eff}}^{\left(6\right)TTTZ}$ | <$1.3\times {10}^{-4}$ | <$6.9\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)X\left(JZJ\right)}$ | <$6\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)X\left(JZJT\right)}$ | <$2\times {10}^{-25}$ |

${{c}_{w}}_{\mathrm{eff}}^{\left(6\right)TKKX}$ | <$3.3\times {10}^{-5}$ | <$1.8\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)Y\left(JXJ\right)}$ | <$2\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)Y\left(JXJT\right)}$ | <$2\times {10}^{-25}$ |

${{c}_{w}}_{\mathrm{eff}}^{\left(6\right)TKKY}$ | <$5.5\times {10}^{-5}$ | <$3.0\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)Y\left(JYJ\right)}$ | <$3\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)Y\left(JYJT\right)}$ | <$7\times {10}^{-26}$ |

${{c}_{w}}_{\mathrm{eff}}^{\left(6\right)TKKZ}$ | <$1.3\times {10}^{-4}$ | <$6.9\times {10}^{-8}$ | ${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)Y\left(JZJ\right)}$ | <$6\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)Y\left(JZJT\right)}$ | <$2\times {10}^{-25}$ |

${\tilde{H}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(5\right)TJTJ}$ | <$6\times {10}^{-25}$ | ${\tilde{g}}_{w}\phantom{\rule{-2.0pt}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{\mathrm{eff}}^{\left(6\right)TJTJT}$ | <$5\times {10}^{-25}$ |

**Table 5.**Best bounds on the spin-independent isotropic nonrelativistic coefficients in the Sun-centered frame for electron, proton, and muon operators.

Constraint; Electron | Constraint; Proton | Constraint; Muon | |||
---|---|---|---|---|---|

$|{{a}_{e}}_{200}^{\mathrm{NR}}|$ | ∼$4\times {10}^{-9}$ GeV${}^{-1}$ [36] | $|{{a}_{p}}_{200}^{\mathrm{NR}}|$ | ∼$4\times {10}^{-9}$ GeV${}^{-1}$ [36] | $|{{a}_{\mu}}_{200}^{\mathrm{NR}}|$ | ∼$3\times {10}^{-5}$ GeV${}^{-1}$ [34] |

$|{{c}_{e}}_{200}^{\mathrm{NR}}|$ | ∼$2\times {10}^{-5}$ GeV${}^{-1}$ [35] | $|{{c}_{\mu}}_{200}^{\mathrm{NR}}|$ | ∼$3\times {10}^{-5}$ GeV${}^{-1}$ [34] | ||

$|{{a}_{e}}_{400}^{\mathrm{NR}}|$ | ∼50 GeV${}^{-3}$ GeV${}^{-3}$ [36] | $|{{a}_{p}}_{400}^{\mathrm{NR}}|$ | ∼50 GeV${}^{-3}$ [36] | $|{{a}_{\mu}}_{400}^{\mathrm{NR}}|$ | ∼$4\times {10}^{5}$ GeV${}^{-3}$ [34] |

$|{{c}_{e}}_{400}^{\mathrm{NR}}|$ | ∼$3\times {10}^{5}$ GeV${}^{-3}$ [35] | $|{{c}_{\mu}}_{400}^{\mathrm{NR}}|$ | ∼$4\times {10}^{5}$ GeV${}^{-3}$ [34] |

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Vargas, A.J.
Overview of the Phenomenology of Lorentz and CPT Violation in Atomic Systems. *Symmetry* **2019**, *11*, 1433.
https://doi.org/10.3390/sym11121433

**AMA Style**

Vargas AJ.
Overview of the Phenomenology of Lorentz and CPT Violation in Atomic Systems. *Symmetry*. 2019; 11(12):1433.
https://doi.org/10.3390/sym11121433

**Chicago/Turabian Style**

Vargas, Arnaldo J.
2019. "Overview of the Phenomenology of Lorentz and CPT Violation in Atomic Systems" *Symmetry* 11, no. 12: 1433.
https://doi.org/10.3390/sym11121433