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We present a coil system designed to generate a highly uniform magnetic field for the n2EDM experiment at the Paul Scherrer Institute. It consists of a main $B_0$ coil and a set of auxiliary coils mounted on a cubic structure with a side length of 273 cm, inside a large magnetically shielded room (MSR). We have assembled this system and characerized its performances with a mapping robot. The apparatus is able to generate a 1 $\mu$ T vertical field with a relative root mean square deviation $\sigma$ ($B_z$)/$B_z$ = 3 $\times$ $10^{-5}$ over the volume of interest, a cylinder of radius 40 cm and height 30 cm. This level of uniformity overcomes the n2EDM requirements, allowing a measurement of the neutron Electric Dipole Moment with a sensitivity better than 1 $\times$ $10^{-27}$ ecm., Comment: 20 pages, 13 figures

The Young-Koppel model (YK) describes comprehensively the interaction of slow neutrons with diatomic gases such as H$_2$ and D$_2$. This paper reports on the first experimental results of ultracold neutron (UCN) scattering over a wide temperature range vindicating the YK model for gaseous D$_2$ and showing an important difference in the temperature dependence to a low-energy low-temperature approximation (LETA). LETA is confirmed, however, to be valid for monoatomic gases such as Ne. Calculated cross sections for other noble gases were also confirmed for ultracold neutrons. Finally, the total cross section of UCNs in H$_2$ gas was measured and analyzed applying the Young-Koppel model, however, in a more limited temperature range, confirming the theoretical prediction.

The aim of this paper is to prove a general Lebesgue decomposition theorem for positive operators on so-called anti-dual pairs, following the iterative approach introduced by Arlinskii. This procedure and the resulting theorem encompass several special cases, including positive operators on Hilbert spaces, non-negative forms on vector spaces, and representable functionals over *-algebras., Comment: 13 pages

Understanding causal relationships within a system is crucial for uncovering its underlying mechanisms. Causal discovery methods, which facilitate the construction of such models from time-series data, hold the potential to significantly advance scientific and engineering fields. This study introduces the Cross-Mapping Coherence (CMC) method, designed to reveal causal connections in the frequency domain between time series. CMC builds upon nonlinear state-space reconstruction and extends the Convergent Cross-Mapping algorithm to the frequency domain by utilizing coherence metrics for evaluation. We tested the Cross-Mapping Coherence method using simulations of logistic maps, Lorenz systems, Kuramoto oscillators, and the Wilson-Cowan model of the visual cortex. CMC accurately identified the direction of causal connections in all simulated scenarios. When applied to the Wilson-Cowan model, CMC yielded consistent results similar to spectral Granger causality. Furthermore, CMC exhibits high sensitivity in detecting weak connections, demonstrates sample efficiency, and maintains robustness in the presence of noise. In conclusion, the capability to determine directed causal influences across different frequency bands allows CMC to provide valuable insights into the dynamics of complex, nonlinear systems.

Let $F_{k,d}(n)$ be the maximal size of a set ${A}\subseteq [n]$ such that the equation \[a_1a_2\dots a_k=x^d, \; a_1

The GERmanium Detector Array (GERDA) experiment operated enriched high-purity germanium detectors in a liquid argon cryostat, which contains 0.33% of $^{36}$Ar, a candidate isotope for the two-neutrino double-electron capture (2$\nu$ECEC) and therefore for the neutrinoless double-electron capture (0$\nu$ECEC). If detected, this process would give evidence of lepton number violation and the Majorana nature of neutrinos. In the radiative 0$\nu$ECEC of $^{36}$Ar, a monochromatic photon is emitted with an energy of 429.88 keV, which may be detected by the GERDA germanium detectors. We searched for the $^{36}$Ar 0$\nu$ECEC with GERDA data, with a total live time of 4.34 yr (3.08 yr accumulated during GERDA Phase II and 1.26 yr during GERDA Phase I). No signal was found and a 90% C.L. lower limit on the half-life of this process was established T$_{1/2}$ > 1.5x10$^{22}$ yr, Comment: 10 pages, 5 figures, 1 table

High-precision searches for an electric dipole moment of the neutron (nEDM) require stable and uniform magnetic field environments. We present the recent achievements of degaussing and equilibrating the magnetically shielded room (MSR) for the n2EDM experiment at the Paul Scherrer Institute. We present the final degaussing configuration that will be used for n2EDM after numerous studies. The optimized procedure results in a residual magnetic field that has been reduced by a factor of two. The ultra-low field is achieved with the full magnetic-field-coil system, and a large vacuum vessel installed, both in the MSR. In the inner volume of ~1.4 m^3, the field is now more uniform and below 300 pT. In addition, the procedure is faster and dissipates less heat into the magnetic environment, which in turn, reduces its thermal relaxation time from 12 h down to ~1.5 h.

We present the measurement of the two-neutrino double-$\beta$ decay rate of $^{76}$Ge performed with the GERDA Phase II experiment. With a subset of the entire GERDA exposure, 11.8 kg$\cdot$yr, the half-life of the process has been determined: $T^{2\nu}_{1/2} = (2.022 \pm 0.018_{stat} \pm 0.038_{sys})\times10^{21}$ yr. This is the most precise determination of the $^{76}$Ge two-neutrino double-$\beta$ decay half-life and one of the most precise measurements of a double-$\beta$ decay process. The relevant nuclear matrix element can be extracted: $M^{2\nu}_{\text{eff}} = (0.101\pm0.001).$, Comment: 7 pages, 4 figures, 2 tables

We search for tri-nucleon decays of $^{76}$Ge in the dataset from the GERmanium Detector Array (GERDA) experiment. Decays that populate excited levels of the daughter nucleus above the threshold for particle emission lead to disintegration and are not considered. The ppp-, ppn-, and pnn-decays lead to $^{73}$Cu, $^{73}$Zn, and $^{73}$Ga nuclei, respectively. These nuclei are unstable and eventually proceed by the beta decay of $^{73}$Ga to $^{73}$Ge (stable). We search for the $^{73}$Ga decay exploiting the fact that it dominantly populates the 66.7 keV $^{73m}$Ga state with half-life of 0.5 s. The nnn-decays of $^{76}$Ge that proceed via $^{73m}$Ge are also included in our analysis. We find no signal candidate and place a limit on the sum of the decay widths of the inclusive tri-nucleon decays that corresponds to a lower lifetime limit of 1.2x10$^{26}$ yr (90% credible interval). This result improves previous limits for tri-nucleon decays by one to three orders of magnitude., Comment: 8 pages, 9 figures, 3 tables

We present a novel Active Magnetic Shield (AMS), designed and implemented for the n2EDM experiment at the Paul Scherrer Institute. The experiment will perform a high-sensitivity search for the electric dipole moment of the neutron. Magnetic-field stability and control is of key importance for n2EDM. A large, cubic, 5m side length, magnetically shielded room (MSR) provides a passive, quasi-static shielding-factor of about 10^5 for its inner sensitive volume. The AMS consists of a system of eight complex, feedback-controlled compensation coils constructed on an irregular grid spanned on a volume of less than 1000m^3 around the MSR. The AMS is designed to provide a stable and uniform magnetic-field environment around the MSR, while being reasonably compact. The system can compensate static and variable magnetic fields up to +-50muT (homogeneous components) and +-5muT (first-order gradients), suppressing them to a few muT in the sub-Hertz frequency range. The presented design concept and implementation of the AMS fulfills the requirements of the n2EDM experiment and can be useful for other applications, where magnetically silent environments are important and spatial constraints inhibit simpler geometrical solutions.

Our purpose in this note is to investigate the order properties of positive operators from a locally convex space into its conjugate dual. We introduce a natural generalization of the Busch-Gudder strength function and we prove Kadison's anti-lattice theorem and Ando's result on the infimum of positive operators in that context., Comment: 9 pages

We report on a search for a new, short-range, spin-dependent interaction using a modified version of the experimental apparatus used to measure the permanent neutron electric dipole moment at the Paul Scherrer Institute. This interaction, which could be mediated by axion-like particles, concerned the unpolarized nucleons (protons and neutrons) near the material surfaces of the apparatus and polarized ultracold neutrons stored in vacuum. The dominant systematic uncertainty resulting from magnetic-field gradients was controlled to an unprecedented level of approximately 4 pT/cm using an array of optically-pumped cesium vapor magnetometers and magnetic-field maps independently recorded using a dedicated measurement device. No signature of a theoretically predicted new interaction was found, and we set a new limit on the product of the scalar and the pseudoscalar couplings $g_sg_p\lambda^2 < 8.3 \times 10^{-28}\,\text{m}^2$ (95% C.L.) in a range of $5\,\mu\text{m} < \lambda < 25\,\text{mm}$ for the monopole-dipole interaction. This new result confirms and improves our previous limit by a factor of 2.7 and provides the current tightest limit obtained with free neutrons.

Unsupervised Relation Extraction (RE) aims to identify relations between entities in text, without having access to labeled data during training. This setting is particularly relevant for domain specific RE where no annotated dataset is available and for open-domain RE where the types of relations are a priori unknown. Although recent approaches achieve promising results, they heavily depend on hyperparameters whose tuning would most often require labeled data. To mitigate the reliance on hyperparameters, we propose PromptORE, a ''Prompt-based Open Relation Extraction'' model. We adapt the novel prompt-tuning paradigm to work in an unsupervised setting, and use it to embed sentences expressing a relation. We then cluster these embeddings to discover candidate relations, and we experiment different strategies to automatically estimate an adequate number of clusters. To the best of our knowledge, PromptORE is the first unsupervised RE model that does not need hyperparameter tuning. Results on three general and specific domain datasets show that PromptORE consistently outperforms state-of-the-art models with a relative gain of more than 40% in B 3 , V-measure and ARI. Qualitative analysis also indicates PromptORE's ability to identify semantically coherent clusters that are very close to true relations.

In nonperturbative formulation of quantum field theory (QFT), the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation (RGE). Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations., Comment: v4: final published version, 15pages + 4 page appendix

The ultracold neutron (UCN) source at the Paul Scherrer Institute (PSI) provides high intensities of storable neutrons for fundamental physics experiments. The neutron velocity spectrum parallel to the beamline axis was determined by time-of-flight spectroscopy using a neutron chopper. In particular, the temporal evolution of the spectrum during neutron production and UCN storage in the source storage volume was investigated and compared to Monte Carlo simulation results. A softening of the measured spectrum from a mean velocity of 7.7(1) m s$^{-1}$ to 5.1(1) m s$^{-1}$ occurred within the first 30 s after the proton beam pulse had impinged on the spallation target. A spectral hardening was observed over longer time scales of one measurement day, consistent with the effect of surface degradation of the solid deuterium moderator.

The ability to detect liquid argon scintillation light from within a densely packed high-purity germanium detector array allowed the GERDA experiment to reach an exceptionally low background rate in the search for neutrinoless double beta decay of $^{76}$Ge. Proper modeling of the light propagation throughout the experimental setup, from any origin in the liquid argon volume to its eventual detection by the novel light read-out system, provides insight into the rejection capability and is a necessary ingredient to obtain robust background predictions. In this paper, we present a model of the GERDA liquid argon veto, as obtained by Monte Carlo simulations and constrained by calibration data, and highlight its application for background decomposition.

Ultracold neutrons (UCNs) are used in experiments investigating fundamental interactions, testing the Standard Model of particle physics and searching for phenomena beyond it. Knowledge of the energy spectrum of UCNs is very often a key ingredient to determine the systematic effects in precision measurements utilizing UCNs. The proposed novel method allows for the in-situ measurements of the UCN velocity distribution and its time evolution. In addition, the proposed UCN spectrometer can be a handy diagnostic tool for monitoring the spectrum in critical places in the UCN transport system connecting an UCN source with experiments. In this paper, we show the results from the measurements performed at the Paul Scherrer Institute's UCN source., Comment: 8 pages, 7 figures

Ultra-low-mass axions are a viable dark matter candidate and may form a coherently oscillating classical field. Nuclear spins in experiments on Earth might couple to this oscillating axion dark-matter field, when propagating on Earth's trajectory through our Galaxy. This spin coupling resembles an oscillating pseudo-magnetic field which modulates the spin precession of nuclear spins. Here we report on the null result of a demonstration experiment searching for a frequency modulation of the free spin-precession signal of \magHg in a \SI{1}{\micro\tesla} magnetic field. Our search covers the axion mass range $10^{-16}~\textrm{eV} \lesssim m_a \lesssim 10^{-13}~\textrm{eV}$ and achieves a peak sensitivity to the axion-nucleon coupling of $g_{aNN} \approx 3.5 \times 10^{-6}~\textrm{GeV}^{-1}$., Comment: 18 pages, 4 images, submitted to SciPost Physics

We determined the ultracold neutron (UCN) production rate by superthermal conversion in the solid deuterium (sD$_2$) moderator of the UCN source at the Paul Scherrer Institute (PSI). In particular, we considered low amounts of less than $20\,$mol of D$_2$, deposited on the cooled moderator vessel surfaces in thin films of a few mm thickness. We measured the isotopic ($ c_\text{HD} < 0.2 \, \% $) and isomeric ($ c_\text{para} \le 2.7 \, \% $) purity of the deuterium to conclude that absorption and up-scattering at $5\,$K have a negligible effect on the UCN yield from the thin films. We compared the calculated UCN yield based on the previously measured thermal neutron flux from the heavy water thermal moderator with measurements of the UCN count rates at the beamports. We confirmed our results and thus demonstrate an absolute characterization of the UCN production and transport in the source by simulations.

Sharpness-aware minimization (SAM) aims to improve the generalisation of gradient-based learning by seeking out flat minima. In this work, we establish connections between SAM and Mean-Field Variational Inference (MFVI) of neural network parameters. We show that both these methods have interpretations as optimizing notions of flatness, and when using the reparametrisation trick, they both boil down to calculating the gradient at a perturbed version of the current mean parameter. This thinking motivates our study of algorithms that combine or interpolate between SAM and MFVI. We evaluate the proposed variational algorithms on several benchmark datasets, and compare their performance to variants of SAM. Taking a broader perspective, our work suggests that SAM-like updates can be used as a drop-in replacement for the reparametrisation trick.

The electron drift in germanium detectors is modeled making many assumptions. Confronted with data, these assumptions have to be revisited. The temperature dependence of the drift of electrons was studied in detail for an n-type segmented point-contact germanium detector. The detector was mounted in a temperature controlled, electrically cooled cryostat. Surface events were induced with collimated 81 keV photons from a $^{133}$Ba source. A detailed analysis of the rise time of pulses collected in surface scans, performed at different temperatures, is presented. The longitudinal anisotropy of the electron drift decreases with rising temperature. A new approach, making use of designated rise-time windows determined by simulations using SolidStateDetectors.jl, was used to isolate the longitudinal drift of electrons along different axes to quantify this observation. The measured temperature dependence of the longitudinal drift velocities combined with the standard electron drift model as widely used in relevant simulation packages results in unphysical predictions. A first suggestion to modify the electron-drift model is motivated and described. The results of a first implementation of the modified model in SolidStateDetectors.jl are shown. They describe the data reasonably well. A general review of the model and the standard input values for mobilities is suggested.

A search for Beyond the Standard Model double-$\beta$ decay modes of $^{76}$Ge has been performed with data collected during the Phase II of the GERmanium Detector Array (GERDA) experiment, located at Laboratori Nazionali del Gran Sasso of INFN (Italy). Improved limits on the decays involving Majorons have been obtained, compared to previous experiments with $^{76}$Ge, with half-life values on the order of 10$^{23}$ yr. For the first time with $^{76}$Ge, limits on Lorentz invariance violation effects in double-$\beta$ decay have been obtained. The isotropic coefficient $\mathring{a}_\text{of}^{(3)}$, which embeds Lorentz violation in double-$\beta$ decay, has been constrained at the order of $10^{-6}$ GeV. We also set the first experimental limits on the search for light exotic fermions in double-$\beta$ decay, including sterile neutrinos.

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $T^*T$ of a densely defined linear transformation $T$ and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint extensions are also revised., Comment: 22 pages

Events from alpha interactions on the surfaces of germanium detectors are a major contribution to the background in germanium-based searches for neutrinoless double-beta decay. Surface events are subject to charge trapping, affecting their pulse shape and reconstructed energy. A study of alpha events on the passivated end-plate of a segmented true-coaxial n-type high-purity germanium detector is presented. Charge trapping is analysed in detail and an existing pulse-shape analysis technique to identify alpha events is verified with mirror pulses observed in the non-collecting channels of the segmented test detector. The observed radial dependence of charge trapping confirms previous results. A dependence of the probability of charge trapping on the crystal axes is observed for the first time. A first model to describe charge trapping effects within the framework of the simulation software SolidStateDetectors.jl is introduced. The influence of metalisation on events from low-energy gamma interactions close to the passivated surface is also presented.

We present the magnetically shielded room (MSR) for the n2EDM experiment at the Paul Scherrer Institute which features an interior cubic volume with each side of length 2.92m, thus providing an accessible space of 25m3. The MSR has 87 openings up to 220mm diameter to operate the experimental apparatus inside, and an intermediate space between the layers for sensitive signal processing electronics. The characterization measurements show a remanent magnetic field in the central 1m3 below 100pT, and a field below 600pT in the entire inner volume, up to 4\,cm to the walls. The quasi-static shielding factor at 0.01\,Hz measured with a sinusoidal 2muT peak-to-peak signal is about 100,000 in all three spatial directions and rises fast with frequency to reach 10^8 above 1Hz., Comment: 10 pages, 15 Figures, submitted to Review of Scientific Instruments