# Testing the Pauli Exclusion Principle with the VIP-2 Experiment

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## Abstract

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_{s}technique.

## 1. Introduction

## 2. The VIP-2 Experiment

^{2}, thickness of 450 μm, and efficiency of 99% at 8 keV, the SDD cell satisfy these requirements, allowing a sensible improvement with respect to charge coupled devices.

#### Data

## 3. Data Analysis

#### 3.1. Statistical Model

#### 3.2. Bayesian Analysis

#### 3.3. Frequentist Analysis

_{s}analysis [19] of the VIP-2 experimental data also employs the same statistical description as in Section 3.1. A one-sided test statistic in the asymptotic approximation [20] based on the profile likelihood ${\Lambda}\left(\mathcal{S}\right)$ is constructed as:

_{s}is a more robust way to represent exclusion limits. It is expressed in terms of the p-values for the signal ${p}_{\mathcal{S}}$ and background ${p}_{0}$ hypothesis:

_{s}limit is stronger than the expected limit because of an under fluctuation of events in the region of the violating line; however, the two exclusion points are well withing one sigma. The 90% CL

_{s}limit then results to be:

#### 3.4. Discussion

_{s}. The upper limit on the number of signal events can be translated to an upper limit on the PEP violation probability, ${\beta}^{2}/2$, considering the data taking time with current, the current intensity, and the electron diffusion model [11,12,22]. Traditionally, since the experiment performed by Ramberg and Snow, the number of the electron-atom interactions in copper is expressed as ${N}_{int}=D/\mu $, with D the length of the copper target, and $\mu =3.9\times {10}^{-6}$ cm the scattering length of electrons in copper. With this simple model, the upper limit on PEP violation is:

## 4. Conclusions

_{s}technique. The VIP-2 experiment continues the data-taking, aiming to further improve the limit set in this article. At the same time, the VIP-2 collaboration is investigating with theoreticians for the interpretation of the results and for future possible measurements.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Pauli, W. Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren. Z. Für Phys.
**1925**, 31, 765. [Google Scholar] [CrossRef] - Pauli, W. The Connection between Spin and Statistics. Phys. Rev.
**1940**, 58, 716–722. [Google Scholar] [CrossRef] - Pauli, W. Nobel Lecture: Exclusion Principle and Quantum Mechanics; Physics 1942–1962; Elsevier Publishing Company: Amsterdam, The Netherlands, 1964. [Google Scholar]
- Lüders, G.; Zumino, B. Connection between Spin and Statistics. Phys. Rev.
**1958**, 110, 1450–1453. [Google Scholar] [CrossRef] [Green Version] - Abgrall, N.; Arnquist, I.J.; Avignone, F.T.; Barabash, A.S.; Bertrand, F.E.; Bradley, A.W.; Brudanin, V.; Busch, M.; Buuck, M.; Caldwell, A.S.; et al. Search for Pauli exclusion principle violating atomic transitions and electron decay with a p-type point contact germanium detector. Eur. Phys. J. C
**2016**, 76, 619. [Google Scholar] [CrossRef] [Green Version] - Bernabei, R.; Belli, P.; Cappella, F.; Cerulli, R.; Dai, C.J.; d’Angelo, A.; He, H.L.; Incicchitti, A.; Kuang, H.H.; Ma, X.H.; et al. New search for processes violating the Pauli exclusion principle in sodium and in iodine. Eur. Phys. J. C
**2009**, 62, 327–332. [Google Scholar] [CrossRef] [Green Version] - Ejiri, H.; Kinoshita, H.; Sano, H.; Ohsumi, H. Search for exotic K X-rays from neutral iodine atoms and limits on charge non-conservation. Phys. Lett. B
**1992**, 282, 281–287. [Google Scholar] [CrossRef] - Messiah, A.M.L.; Greenberg, O.W. Symmetrization Postulate and Its Experimental Foundation. Phys. Rev.
**1964**, 136, B248–B267. [Google Scholar] [CrossRef] - Ignatiev, A.Y.; Kuzmin, V. Search for slight violation of the Pauli principle. JETP Lett.
**1987**, 47, 6–8. [Google Scholar] - Greenberg, O.W.; Mohapatra, R.N. Local Quantum Field Theory of Possible Violation of the Pauli Principle. Phys. Rev. Lett.
**1987**, 59, 2507–2510. [Google Scholar] [CrossRef] - Ramberg, E.; Snow, G.A. Experimental limit on a small violation of the Pauli principle. Phys. Lett. B
**1990**, 238, 438–441. [Google Scholar] [CrossRef] - Shi, H.; Milotti, E.; Bartalucci, S.; Bazzi, M.; Bertolucci, S.; Bragadireanu, A.M.; Cargnelli, M.; Clozza, A.; Paolis, L.D.; Matteo, S.D.; et al. Experimental search for the violation of Pauli exclusion principle. Eur. Phys. J. C
**2018**, 78, 319. [Google Scholar] [CrossRef] [PubMed] - Curceanu, C.; De Paolis, L.; Di Matteo, S.; Di Matteo, H.; Sperandio, S. Evaluation of the X-ray Transitions Energies for the Pauli-Principle-Violating Atomic Transitions in Several Elements by Using the DIrac-Fock Method; Technical Report, LNF Preprint; INFN-13-21/LNF, 21 November 2013; SIDS-Pubblicazioni Laboratori Nazionali di Frascati: Frascati, Italy, 2013. [Google Scholar]
- Curceanu (Petrascu), C.; Bartalucci, S.; Bertolucci, S.; Bragadireanu, M.; Cargnelli, M.; Di Matteo, S.; Egger, J.P.; Guaraldo, C.; Iliescu, M.; Ishiwatari, T.; et al. Experimental tests of quantum mechanics-Pauli exclusion principle violation (the VIP experiment) and future perspective. J. Phys. Conf. Ser.
**2011**, 306, 012036. [Google Scholar] [CrossRef] - Quaglia, R.; Bombelli, L.; Busca, P.; Fiorini, C.; Occhipinti, M.; Giacomini, G.; Ficorella, F.; Picciotto, A.; Piemonte, C. Silicon Drift Detectors and CUBE Preamplifiers for High-Resolution X-ray Spectroscopy. IEEE Trans. Nucl. Sci.
**2015**, 62, 221–227. [Google Scholar] [CrossRef] [Green Version] - Schulz, O.; Beaujean, F.; Caldwell, A.; Grunwald, C.; Hafych, V.; Kröninger, K.; Cagnina, S.L.; Röhrig, L.; Shtembari, L. BAT.jl: A Julia-Based Tool for Bayesian Inference. SN Comput. Sci.
**2021**, 2, 210. [Google Scholar] [CrossRef] - Caldwell, A.; Eller, P.; Hafych, V.; Schick, R.; Schulz, O.; Szalay, M. Integration with an adaptive harmonic mean algorithm. Int. J. Mod. Phys. A
**2020**, 35, 2050142. [Google Scholar] [CrossRef] - Hafych, V.; Eller, P.; Caldwell, A.; Schulz, O. Parallelizing MCMC Sampling via Space Partitioning. arXiv
**2018**, arXiv:2008.03098. [Google Scholar] - Read, A.L. Presentation of search results: The CLs technique. J. Phys. G Nucl. Part. Phys.
**2002**, 28, 2693–2704. [Google Scholar] [CrossRef] - Cowan, G.; Cranmer, K.; Gross, E.; Vitells, O. Asymptotic formulae for likelihood-based tests of new physics. Eur. Phys. J. C
**2011**, 71, 1554. [Google Scholar] [CrossRef] [Green Version] - Verkerke, W.; Kirkby, D.P. The RooFit toolkit for data modeling. eConf
**2003**, C0303241, MOLT007. [Google Scholar] - Milotti, E.; Bartalucci, S.; Bertolucci, S.; Bazzi, M.; Bragadireanu, M.; Cargnelli, M.; Clozza, A.; Curceanu, C.; Paolis, L.D.; Egger, J.P.; et al. On the Importance of Electron Diffusion in a Bulk-Matter Test of the Pauli Exclusion Principle. Entropy
**2018**, 20, 515. [Google Scholar] [CrossRef] [Green Version] - Piscicchia, K.; Marton, J.; Bartalucci, S.; Bazzi, M.; Bertolucci, S.; Bragadireanu, M.; Cargnelli, M.; Clozza, A.; Grande, R.D.; Paolis, L.D.; et al. VIP-2—High-Sensitivity Tests on the Pauli Exclusion Principle for Electrons. Entropy
**2020**, 22, 1195. [Google Scholar] [CrossRef] [PubMed] - Piscicchia, K.; Milotti, E.; Amirkhani, A.; Bartalucci, S.; Bertolucci, S.; Bazzi, M.; Bragadireanu, M.; Cargnelli, M.; Clozza, A.; Grande, R.D.; et al. Search for a remnant violation of the Pauli exclusion principle in a Roman lead target. Eur. Phys. J. C
**2020**, 80, 508. [Google Scholar] [CrossRef]

**Figure 1.**Overview of a standard ${K}_{\alpha}$ transition,

**left**, and a PEP violating one,

**right**. This process has a different X-ray emission energy, and can be searched for experimentally.

**Figure 2.**Lateral and frontal sections (on the left and on the right of the figure, respectively) of the VIP-2 apparatus. The copper strip target where the electrons are circulated is indicated on the left lateral section. The SDDs X-ray detectors are visible in the frontal section around the target. The apparatus is enclosed by the vacuum chamber.

**Figure 3.**VIP-2 calibrated data in the region-of-interest 7000–8500 eV, corresponding to approximately six months of data taking between December 2019 and May 2020. The spectrum of the data acquired with a current of 180 A circulating in the target is shown in blue. The PEP violation signal is expected at around 7700 eV. Data acquired without current circulating in the target is used for reference and as control region in the data analysis and is shown in red. Both spectra show the copper and nickel ${K}_{\alpha}$ lines.

**Figure 4.**The posterior distribution for the signal yield $\mathcal{S}$ obtained by marginalization on all the parameters. Red, yellow, and green show the 95%, 90%, and 66% intervals, respectively.

**Figure 5.**Observed (solid) and expected (dashed) CL

_{s}limits as a function of the signal yield $\mathcal{S}$. The $\pm 1\sigma $ and $\pm 2\sigma $ around the expected limits are shown in green and yellow, respectively. The solid red line marks the 0.10 p-value.

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**MDPI and ACS Style**

Napolitano, F.; Bartalucci, S.; Bertolucci, S.; Bazzi, M.; Bragadireanu, M.; Capoccia, C.; Cargnelli, M.; Clozza, A.; De Paolis, L.; Del Grande, R.;
et al. Testing the Pauli Exclusion Principle with the VIP-2 Experiment. *Symmetry* **2022**, *14*, 893.
https://doi.org/10.3390/sym14050893

**AMA Style**

Napolitano F, Bartalucci S, Bertolucci S, Bazzi M, Bragadireanu M, Capoccia C, Cargnelli M, Clozza A, De Paolis L, Del Grande R,
et al. Testing the Pauli Exclusion Principle with the VIP-2 Experiment. *Symmetry*. 2022; 14(5):893.
https://doi.org/10.3390/sym14050893

**Chicago/Turabian Style**

Napolitano, Fabrizio, Sergio Bartalucci, Sergio Bertolucci, Massimiliano Bazzi, Mario Bragadireanu, Cesidio Capoccia, Michael Cargnelli, Alberto Clozza, Luca De Paolis, Raffaele Del Grande,
and et al. 2022. "Testing the Pauli Exclusion Principle with the VIP-2 Experiment" *Symmetry* 14, no. 5: 893.
https://doi.org/10.3390/sym14050893