In the CRC “Nuclei: From Fundamental Interactions to Structure and Stars” we will investigate the strong and electroweak interaction physics from nuclei to stars. The strong interaction described by quantum chromodynamics (QCD) is responsible for binding neutrons and protons into nuclei and for the many facets of nuclear structure physics. Combined with the electroweak interaction, it determines the structure of all nuclei in the nuclear chart in a similar way as quantum electrodynamics shapes the periodic table of elements. While the latter is well understood, it is still unclear **how the nuclear chart emerges from the underlying forces**.

During the last decades, nuclear structure theory has evolved into a field with a systematic theoretical foundation, with nuclear forces based on QCD and advanced methods to solve the nuclear many-body problem with controlled uncertainties. Effective field theories (EFT), first proposed in a seminal article by Steven Weinberg, have played a guiding role in this process, as they reduce the complexity of the underlying QCD theory to the relevant degrees of freedom in a systematic way. While this was first demonstrated for light nuclei, considerable progress in recent years has highlighted that this approach can be extended towards heavier systems. Research performed in this direction has the ultimate goal to understand the nuclear chart from first principles. Since the properties of nuclei, their existence, excitations and decays are all encoded in the nuclear chart, it represents the boundary condition for the complete evolution of matter from the Big Bang into our days, and therefore even the basis of our own existence.

**Figure 1:** *Left panel: Nuclei in the nuclear chart. The blue squares mark all known nuclei, with stable nuclei in darker blue. More than half of the nuclear chart is unknown ^{1}. Right panel: Chiral EFT for nuclear forces, where the different contributions at successive orders are shown diagrammatically. Nucleons and pions are represented by solid and dashed lines, respectively. Many-body forces are highlighted in orange including the year they were derived.*

The new era of nuclear structure physics, where EFTs of the strong interaction provide an exciting link between experimental and theoretical frontiers, has just started. For strong interactions at low energies, chiral EFT offers a systematic basis for nuclear forces^{2}, built on the symmetries of QCD, with controlled expansions of the interactions in the inverse chiral-symmetry breaking scale. This is shown in the right panel of Figure 1, where the first column represents nucleon-nucleon interactions at different orders. The interaction between the nucleons (solid lines) is mediated by the exchange of pions (dashed lines), the Goldstone bosons of QCD, which are responsible for the long-range part of strong interactions. Consequently, chiral EFT is applicable at energy scales of the order of the pion mass. Three-nucleon (3N) forces, which emerge naturally in EFTs, enter at next-to-next-to-leading order (N^{2}LO) (see the right panel in Figure 1). Moreover, EFTs lead to a hierarchy among many-body interactions, with 4N forces at next-to-next-to-next-to-leading order (N^{3}LO). Combining EFTs with advanced many-body methods opens up a systematic path to investigate nuclear forces and their impact on nuclei and nuclear matter.

In strongly interacting systems, three-body forces are especially important and have been the target of recent theoretical and experimental work^{3}. The calculation of binding energies in light nuclei required the introduction of three-body forces and they play a key role in universal properties of halo nuclei and their connection to the Efimov effect in ultracold atoms^{4}. Three-nucleon forces are a frontier in the physics of nuclei, for shell structure and the evolution to the limits, the driplines^{5,6}. Exotic nuclei become increasingly sensitive to 3N forces and other subtle components of nuclear forces, so that experiments with rare-isotope beams can provide unique insights into strong interactions. Calculations based on chiral EFT interactions also provide systematic constraints for the properties of nuclear matter in neutron stars and core-collapse supernovae^{7}. The physics of 3N forces therefore connects nuclear structure physics with nuclear astrophysics.

The exploration of many-body forces is particularly exciting, because at N^{3}LO, 3N and 4N forces (see the right panel of Figure 1) are predicted with many new structures. These have never been applied beyond the lightest nuclei and **must still pass experimental precision tests**. These developments come in time with the establishment of major nuclear physics facilities like the Facility for Antiproton and Ion Research (FAIR) in Darmstadt and other new facilities and upgrades worldwide, including the Radioactive Ion Beam Facility (RIBF) at RIKEN in Japan, the ARIEL facility at TRIUMF in Canada, the HIE-ISOLDE upgrade at CERN, the SPES facility at the INFN laboratory in Legnaro, Italy, the future Facility for Rare Isotope Beams (FRIB) in the US, and the future SPIRAL2 facility at GANIL in France, which will give great access to the unexplored regions of the nuclear chart.

The electroweak force plays a crucial role in nuclear physics. Gauge symmetry allows using the same EFT expansion to derive electroweak operators that are consistent with the strong interaction. Therefore, the couplings in nuclear forces largely determine also electroweak processes^{8,9}. The relation between nuclear forces and electroweak processes in EFT predicts their momentum-transfer dependence due to pion physics, which enables important consistency tests. Moreover, chiral EFT provides consistent electroweak one- and two-body currents. Two-body currents, also known as meson-exchange currents, enter at higher order, just like 3N forces. For electromagnetic reactions, two-body currents have been derived recently and shown to provide significant contributions to electromagnetic processes in few-nucleon systems^{10,11}.

The **exploration of electroweak interactions in nuclei and nuclear matter** is therefore emerging as a new area of EFT research. Experimentally, this opens up exciting opportunities for an electron-beam machine. The superconducting Darmstadt electron linear accelerator S-DALINAC is unique worldwide to explore electromagnetic probes in the energy regime of chiral EFT. Its electron beam can also be used to simulate the couplings of other probes^{12}, for example for neutrino-nucleus reactions, or to test and constrain the operators and nuclear structure involved in nuclear matrix elements for fundamental symmetries. This will be applied to structure factors for the scattering of weakly interacting massive particles (WIMPs) off nuclei^{13}, which are needed in direct detection of dark matter, and to the nuclear structure aspects of nuclear matrix elements for neutrinoless double-beta decay^{14}. The relevant momentum transfers involved in WIMP scattering off nuclei and neutrinoless double-beta decay are again of the order of the pion mass, so that this is a prime regime for chiral EFT.

The strong and electroweak forces describe all reactions for the synthesis of the elements in the Universe and all the microphysics relevant for how stars shine and explode in supernovae. Electroweak neutrino processes play a pivotal role in the explosion mechanism and the nucleosynthesis in core-collapse supernovae. Combining the advances for the nuclear matter equation of state and consistent neutrino-matter interactions^{15} with simulations of core-collapse supernovae^{16} will lead to an understanding of **how nuclei, neutrinos and the equation of state impact the nucleosynthesis of elements in these explosive events**. This will connect to questions in neutrino physics and to forefront astronomical observations of the oldest stars, which shed light on the chemical history of the elements in the universe.

In summary, the physics of nucleonic matter ranges from universal properties at low densities and in ultracold atoms to the densest matter we know to exist in neutron stars. Chiral EFT provides a link between nuclear structure and matter in stars with the underlying theory of QCD, to which it is connected through lattice QCD^{17}. With the pioneering work in this CRC, accessing the medium-mass region within the first funding period, we will provide the basis for the ultimate goal of a consistent description of nuclear structure across the nuclear chart into the heavy mass region. Additionally, strong and electroweak processes will provide systematic uncertainties for nuclear matrix elements required to test fundamental symmetries and to understand the chemical contribution from core-collapse supernovae to the Universe.