# The Relation between Fundamental Constants and Particle Physics Parameters

## Abstract

**:**

## 1. Introduction

## 2. Observational Constraints

#### 2.1. $\mu $ Constraints

#### 2.2. α Constraints

## 3. The Dependence of Fundamental Constants on the Physics Parameters

#### 3.1. The Proton to Electron Mass Ratio

#### 3.2. The Fine Structure Constant

#### The Physics of R

## 4. Observational Constraints on $\frac{\mathit{d}{\mathbf{\Lambda}}_{\mathit{QCD}}}{{\mathbf{\Lambda}}_{\mathit{QCD}}}$

#### Individual Constraints on $\frac{\mathrm{\Delta}\nu}{\nu}$ and $\frac{\mathrm{\Delta}h}{h}$

## 5. Constraints from a Given Model

## 6. A Model Dependent Limit on $\frac{\mathbf{\Delta}\mathbf{\alpha}}{\mathbf{\alpha}}$

## 7. Conclusions

## Conflicts of Interest

## References

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**Figure 1.**All of the observational constraints on $\mathrm{\Delta}\mu /\mu $ from radio ($z<1$) and optical ($z>1$) observations plotted versus the scale factor $a=1/(1+z)$. All constraints are at the $1\sigma $ level. The low redshift radio constraints are difficult to see at the scale of this plot. The age of the universe in gigayears is plotted on the top axis and in Figure 2.

**Figure 2.**The low redshift radio $\mathrm{\Delta}\mu /\mu $ constraints at $z=0.6874$ and $z=0.88582$ plotted versus the scale factor $a=1/(1+z)$. The error bar at $z=0.6874$ is $1\sigma $, however, the error bar at $z=0.88582$ ($a=0.53$) includes systematic effects that increase the error to $\pm {10}^{-7}$. That is the primary constraint utilized in this work.

**Figure 3.**The figure indicates the 1σ variation of the limit on $d{\mathrm{\Lambda}}_{QCD}/{\mathrm{\Lambda}}_{QCD}$ as a function of the model parameter R. The dashed line indicates the limit on $d{\mathrm{\Lambda}}_{QCD}/{\mathrm{\Lambda}}_{QCD}$ if the measured values of $d\alpha /\alpha $ and $d\mu /\mu $ are used rather than the limits. The dot is at $R=36$ which is the example value. Note that although it is not apparent at the scale of the figure the limit on $d{\mathrm{\Lambda}}_{QCD}/{\mathrm{\Lambda}}_{QCD}$ at $R=0$ is not zero but rather the small $\frac{2}{7a}\frac{d\mu}{\mu}$ term in (11) that does not depend on R.

Object | Redshift | $\mathit{\Delta}\mathit{\mu}/\mathit{\mu}$ | $1\mathit{\sigma}$ error | Ref. |
---|---|---|---|---|

J1443+2724 | 4.224 | $-9.5\times {10}^{-6}$ | $\pm 7.6\times {10}^{-6}$ | [13] |

Q0347-383 | $3.0249$ | $2.1\times {10}^{-6}$ | $\pm 6.\times {10}^{-6}$ | [14] |

Q0528-250 | $2.811$ | $3.0\times {10}^{-7}$ | $\pm 3.7\times {10}^{-6}$ | [15] |

Q J0643-5041 | $2.659$ | $7.4\times {10}^{-6}$ | $\pm 6.7\times {10}^{-6}$ | [16] |

Q0405-443 | $2.5974$ | $10.1\times {10}^{-6}$ | $\pm 6.2\times {10}^{-6}$ | [17] |

Q2348-011 | $2.426$ | $-6.8\times {10}^{-6}$ | $\pm 27.8\times {10}^{-6}$ | [18] |

He0027-1836 | $2.402$ | $-7.6\times {10}^{-6}$ | $\pm 1.0\times {10}^{-5}$ | [19] |

Q01232+082 | $2.34$ | $1.9\times {10}^{-5}$ | $\pm 1.0\times {10}^{-5}$ | [20] |

J2123-005 | $2.059$ | $5.6\times {10}^{-6}$ | $\pm 6.2\times {10}^{-6}$ | [21] |

PKS1830-211 | $0.88582$ | $-2.9\times {10}^{-8}$ | $\pm 5.7\times {10}^{-8}$ | [12] |

B0218+357 | $0.6847$ | $-3.5\times {10}^{-7}$ | $\pm 1.2\times {10}^{-7}$ | [22] |

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**MDPI and ACS Style**

Thompson, R.I.
The Relation between Fundamental Constants and Particle Physics Parameters. *Universe* **2017**, *3*, 6.
https://doi.org/10.3390/universe3010006

**AMA Style**

Thompson RI.
The Relation between Fundamental Constants and Particle Physics Parameters. *Universe*. 2017; 3(1):6.
https://doi.org/10.3390/universe3010006

**Chicago/Turabian Style**

Thompson, Rodger I.
2017. "The Relation between Fundamental Constants and Particle Physics Parameters" *Universe* 3, no. 1: 6.
https://doi.org/10.3390/universe3010006