Volume 1 of the FCC Feasibility Report presents an overview of the physics case, experimental programme, and detector concepts for the Future Circular Collider (FCC). This volume outlines how FCC would address some of the most profound open questions in particle physics, from precision studies of the Higgs and EW bosons and of the top quark, to the exploration of physics beyond the Standard Model. The report reviews the experimental opportunities offered by the staged implementation of FCC, beginning with an electron-positron collider (FCC-ee), operating at several centre-of-mass energies, followed by a hadron collider (FCC-hh). Benchmark examples are given of the expected physics performance, in terms of precision and sensitivity to new phenomena, of each collider stage. Detector requirements and conceptual designs for FCC-ee experiments are discussed, as are the specific demands that the physics programme imposes on the accelerator in the domains of the calibration of the collision energy, and the interface region between the accelerator and the detector. The report also highlights advances in detector, software and computing technologies, as well as the theoretical tools/reconstruction techniques that will enable the precision measurements and discovery potential of the FCC experimental programme. The content and structure of this report are guided by the scope and priorities defined in the mandate of the FCC Feasibility Study. It is therefore not intended to serve as an exhaustive review of the full physics potential of FCC. Several topics, already covered in earlier reports such as the FCC CDR, are not reiterated here or are addressed only briefly, in alignment with the study’s focus. This volume reflects the outcome of a global collaborative effort involving hundreds of scientists and institutions, aided by a dedicated community-building coordination, and provides a targeted assessment of the scientific opportunities and experimental foundations of the FCC programme.

The complementarity between chiral perturbation theory and the linear sigma model is exploited to study $\pi^0\pi^0$ production in $\phi$ radiative decays, where the effects of the f ₀ (980) scalar resonance, and those of its more controversial $\sigma (500)$ partner, should become manifest via the $\phi\rightarrow K^+ K^- (\gamma)\rightarrow\pi^0\pi^0\gamma $ decay chain. The recently reported data on $\phi\rightarrow\pi^0\pi^0\gamma$ coming from the VEPP-2M e ⁺ e ^- collider in Novosibirsk and the DA $\Phi$ NE $\phi$ -factory in Frascati can be reasonably described in our approach, which we propose as a promising first step towards more detailed analyses. The f ₀ (980) contribution, which appears as a moderately narrow peak at the high part of the dipion mass spectrum, can be interpreted as the isoscalar member of the scalar nonet with a large $f_0 K\bar{K}$ coupling and an $f_0\pi\pi$ coupling suppressed by almost ideal $\sigma$ -f₀ mixing. Indeed, the mixing angle in the flavor basis is found to be $\phi_{S}\approx - 6^\circ$ , if the f₀ propagator is approximated by a simple Breit-Wigner expression, or $\phi_{S}\approx -9^\circ$ , if an improved two-channel analysis is performed. The $\sigma (500)$ resonance, which is then strongly coupled to pion pairs, yields a tiny contribution because, in our approach, its coupling to kaon pairs is proportional to $m^2_\sigma-m^2_K$ and thus quite small.

In this work, we build a covariant basis for operators acting on the $(j,0)$\bigoplus${}(0,j)$ Lorentz group representations. The construction is based on an analysis of the covariant properties of the parity operator, which for these representations transforms as the completely temporal component of a symmetrical tensor of rank $2j$. The covariant properties of parity involve the Jordan algebra of anticommutators of the Lorentz group generators which unlike the Lie algebra is not universal. We make the construction explicit for $j=1/2$, 1 and $3/2$, reproducing well-known results for the $j=1/2$ case. We provide an algorithm for the corresponding calculations for arbitrary $j$. This covariant basis provides an inventory of all the possible interaction terms for gauge and nongauge theories of fields for these representations. In particular, it supplies a single second-rank antisymmetric structure, which in the Poincar\'e projector formalism implies a single Pauli term arising from gauge interactions and a single (free) parameter $g$, the gyromagnetic factor. This simple structure predicts that for an elementary particle in this formalism all multipole moments, ${Q}_{E}^{l}$ and ${Q}_{M}^{l}$, are dictated by the complete algebraic structure of the Lorentz generators and the value of $g$. We explicitly calculate the multipole moments, for arbitrary $j$ up to $l=8$. Comparing with results in the literature we find that only the electric charge and magnetic moment of a spin $j$ particle are independent of the Lorentz representation under which it transforms, all higher multipoles being representation dependent. Finally we show that the propagation of the corresponding spin $j$ waves in an electromagnetic background is causal.