# First Experimental Survey of a Whole Class of Non-Commutative Quantum Gravity Models in the VIP-2 Lead Underground Experiment

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## Abstract

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## 1. Introduction

## 2. Energy Dependence of the PEP Violation Probability in NCQG Models

- For $\kappa $-Poincaré, different quantization procedures of the particle fields lead to different predictions; we will refer in particular to the following schemes: the Arzano–Marcianò (AM) procedure [10] and the Freidel–Kowalski-Glikman–Nowak (FKN) procedure [17]. In the AM quantization procedure, PEP violations are induced with a suppression ${\delta}^{2}=E/{\Lambda}_{1}$ [10], where E is the characteristic energy of the considered transition and ${\Lambda}_{1}$ is the energy scale of space-time non-commutativity. In the FKN case, the PEP violation is actually missing. In this sense, the experimental investigation of statistics violations can also provide important down-top indications on the “right” quantization procedure to solve this ambiguity in the formulation of the theory.
- Another relevant case, corresponding to the power energy expansion with $k=3$, concerns the “triply special relativity” model, which we refer to as Kowalski-Glikman–Smolin (KS). The KS framework [13] introduces an additional infrared scale, related to the cosmological constant, and plays the role of an IR regulator. A quantum field theory endowed with the algebra of symmetries discussed in the KS framework might in principle provide IR/UV mixing, an interesting feature of some non-commutative quantum field theories. At the same time, the development of the field theoretic approach requires deepening the Hopf algebra structure of the new symmetries proposed in the KS model. Since this step is still missing at the theoretical level, our phenomenological analysis may be considered as a guidance for the theory that must be still developed for $k=3$. Indeed, a possible interplay between the UV energy scale $\kappa $ and the IR energy scale ${R}^{-1}$—related to the cosmological constant $\Lambda $ by $\Lambda ={R}^{-2}$—may induce PEP violations at orders $k=1$, $k=2$ and $k=3$. Requesting consistency for $k=1$ and $k=2$ with the current experimental bounds then provides strong limits on higher order corrections that can be allowed.
- For a generic NCQG model, deviations from the PEP in the commutation/anti-commutation relations can be parametrized [4] as$${a}_{i}{a}_{j}^{\u2020}-q\left(E\right){a}_{j}^{\u2020}{a}_{i}={\delta}_{ij}\phantom{\rule{0.166667em}{0ex}},$$The q-model requires a hyper-fine tuning of the q parameter. $q\left(E\right)$ is related to the PEP violation probability by$$q\left(E\right)=-1+2{\delta}^{2}\left(E\right).$$For a generic parametrization (${M}_{k}$), we straightforwardly obtain:$${M}_{k}:\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\delta}^{2}\left(E\right)=\frac{{E}^{k}}{{\Lambda}_{k}^{k}}+O\left({E}^{k+1}\right)\phantom{\rule{0.166667em}{0ex}},$$

## 3. The Experiment and the Strategy of the Analysis

## 4. Statistical Analysis Model

## 5. Results

## 6. Discussion

- $\kappa $-Poincaré, in the AM $\kappa $-Poincaré fields’ quantization model, is ruled out (we obtain ${\Lambda}_{1}>4.2\xb7{10}^{21}$ Planck scales).
- $\theta $-Poincaré can be excluded up to a fraction of the Planck scale (we obtain ${\Lambda}_{2}>1.6\xb7{10}^{-1}$ Planck scales).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The figure shows the measured X-ray spectrum corresponding to an acquisition time of $\Delta t\approx 6.1\xb7{10}^{6}\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ in the region of interest. For a comparison, the expected signal distribution (with arbitrary normalization) is also shown in orange for the ${A}_{3}$ analysis and the ${M}_{3}$ parametrization.

**Table 1.**This table summarizes the calculated values for the PEP-violating K${}_{\alpha}$ and K${}_{\beta}$ atomic transition energies in Pb (column labeled forbbiden). As a reference, the allowed transition energies are also quoted (allowed). Energies are in keV.

Transitions in Pb | Allowed | Forbbiden |
---|---|---|

1s - 2p${}_{3/2}$ K${}_{\alpha 1}$ | 74.969 | 73.713 |

1s - 2p${}_{1/2}$ K${}_{\alpha 2}$ | 72.805 | 71.652 |

1s - 3p${}_{3/2}$ K${}_{\beta 1}$ | 84.938 | 83.856 |

1s - 4p${}_{1/2(3/2)}$ K${}_{\beta 2}$ | 87.300 | 86.418 |

1s - 3p${}_{1/2}$ K${}_{\beta 3}$ | 84.450 | 83.385 |

**Table 2.**Resolutions ($\sigma $) in keV estimated at the energies of the PEP-violating K${}_{\alpha}$ and K${}_{\beta}$ transitions.

Transitions in Pb | $\mathit{\sigma}$ (keV) | Error (keV) |
---|---|---|

K${}_{\alpha 1}$ | 0.492 | 0.037 |

K${}_{\alpha 2}$ | 0.491 | 0.037 |

1s - 3p${}_{3/2}$ K${}_{\beta 1}$ | 0.497 | 0.036 |

1s - 4p${}_{1/2(3/2)}$ K${}_{\beta 2}$ | 0.498 | 0.036 |

1s - 3p${}_{1/2}$ K${}_{\beta 3}$ | 0.497 | 0.036 |

**Table 3.**Detection efficiencies evaluated at the energies of the K${}_{\alpha}$ and K${}_{\beta}$ PEP-violating transitions and corresponding branching ratios.

Forb. Transitions | $\mathbf{BR}$ | $\mathit{\u03f5}$ |
---|---|---|

K${}_{\alpha 1}$ | 0.462 ± 0.009 | $(5.39\pm 0.11)\xb7{10}^{-5}$ |

K${}_{\alpha 2}$ | 0.277 ± 0.006 | $\left({4.43}_{-0.09}^{+0.10}\right)\xb7{10}^{-5}$ |

K${}_{\beta 1}$ | 0.1070 ± 0.0022 | $(11.89\pm 0.24)\xb7{10}^{-5}$ |

K${}_{\beta 2}$ | 0.0390 ± 0.0008 | $\left({14.05}_{-0.28}^{+0.29}\right)\xb7{10}^{-5}$ |

K${}_{\beta 3}$ | 0.0559 ± 0.0011 | $\left({11.51}_{-0.23}^{+0.24}\right)\xb7{10}^{-5}$ |

**Table 4.**This table summarizes the upper limits $\overline{S}$ on the expected numbers of signal counts, and the corresponding lower bounds on the scales ${\Lambda}_{k}$ for each analysis ${A}_{i}$ and for the ${M}_{k}$ parametrizations corresponding to $k=1,2,3$.

${\mathit{A}}_{\mathit{i}}$, ${\mathit{M}}_{\mathit{k}}$ | $\overline{\mathit{S}}$ | Lower Limit on $\mathit{\Lambda}$ in Planck Scale Units |
---|---|---|

${A}_{1}$, $k=1$ | 11.4913 | $3.1\xb7{10}^{21}$ |

${A}_{1}$, $k=2$ | 11.3776 | $1.4\xb7{10}^{-1}$ |

${A}_{1}$, $k=3$ | 11.2610 | $4.9\xb7{10}^{-9}$ |

${A}_{2}$, $k=1$ | 15.1408 | $2.8\xb7{10}^{21}$ |

${A}_{2}$, $k=2$ | 15.1640 | $1.4\xb7{10}^{-1}$ |

${A}_{2}$, $k=3$ | 15.1859 | $5.1\xb7{10}^{-9}$ |

${A}_{3}$, $k=1$ | 18.7270 | $4.2\xb7{10}^{21}$ |

${A}_{3}$, $k=2$ | 19.1847 | $1.6\xb7{10}^{-1}$ |

${A}_{3}$, $k=3$ | 19.5993 | $5.6\xb7{10}^{-9}$ |

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**MDPI and ACS Style**

Piscicchia, K.; Marcianò, A.; Addazi, A.; Sirghi, D.L.; Bazzi, M.; Bortolotti, N.; Bragadireanu, M.; Cargnelli, M.; Clozza, A.; De Paolis, L.;
et al. First Experimental Survey of a Whole Class of Non-Commutative Quantum Gravity Models in the VIP-2 Lead Underground Experiment. *Universe* **2023**, *9*, 321.
https://doi.org/10.3390/universe9070321

**AMA Style**

Piscicchia K, Marcianò A, Addazi A, Sirghi DL, Bazzi M, Bortolotti N, Bragadireanu M, Cargnelli M, Clozza A, De Paolis L,
et al. First Experimental Survey of a Whole Class of Non-Commutative Quantum Gravity Models in the VIP-2 Lead Underground Experiment. *Universe*. 2023; 9(7):321.
https://doi.org/10.3390/universe9070321

**Chicago/Turabian Style**

Piscicchia, Kristian, Antonino Marcianò, Andrea Addazi, Diana Laura Sirghi, Massimiliano Bazzi, Nicola Bortolotti, Mario Bragadireanu, Michael Cargnelli, Alberto Clozza, Luca De Paolis,
and et al. 2023. "First Experimental Survey of a Whole Class of Non-Commutative Quantum Gravity Models in the VIP-2 Lead Underground Experiment" *Universe* 9, no. 7: 321.
https://doi.org/10.3390/universe9070321