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Abstract In this paper we review the physics opportunities at linear$$\mathrm{e}^{+}\mathrm{e}^{-} $$ $e^{+}{e}^{-}$colliders with a special focus on high centre-of-mass energies and beam polarisation, take a fresh look at the various accelerator technologies available or under development and, for the first time, discuss how a facility first equipped with a technology that is mature today could be upgraded with technologies of tomorrow to reach much higher energies and/or luminosities. In addition, we discuss detectors, alternative collider modes, as well as opportunities for beyond-collider experiments and R&D facilities as part of a linear collider facility (LCF). The material of this paper supports all plans for$$\mathrm{e}^{+}\mathrm{e}^{-} $$ $e^{+}{e}^{-}$linear colliders and the additional opportunities they offer, independently of technology choice or proposed site, as well as R&D for advanced accelerator technologies. This joint perspective on the physics goals, early technologies and upgrade strategies has been developed by the LCVision team based on an initial discussion at LCWS2024 in Tokyo and a follow-up at the LCVision Community Event at CERN in January 2025. It heavily builds on decades of achievements of the global linear collider community, in particular in the context of CLIC and ILC. 

The explicit breaking of the axial symmetry by quantum fluctuations gives rise to the so-called axial anomaly. This phenomenon is solely responsible for the decay of the neutral pion π 0 into two photons (γγ), leading to its unusually short lifetime. We precisely measured the decay width Γ of the π 0 →   γ γ process. The differential cross sections for π 0 photoproduction at forward angles were measured on two targets, carbon-12 and silicon-28, yielding Γ ( π 0 →   γ γ ) = 7.798 ± 0.056 ( stat . ) ± 0.109 ( syst . )  eV , where stat. denotes the statistical uncertainty and syst. the systematic uncertainty. We combined the results of this and an earlier experiment to generate a weighted average of Γ ( π 0 →   γ γ ) = 7.802 ± 0.052 ( stat . ) ± 0.105 ( syst . )  eV . Our final result has a total uncertainty of 1.50% and confirms the prediction based on the chiral anomaly in quantum chromodynamics. 

A bstract The NA62 experiment at CERN targets the measurement of the ultra-rare $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ K + → π + ν ν ¯ decay, and carries out a broad physics programme that includes probes for symmetry violations and searches for exotic particles. Data were collected in 2016–2018 using a multi-level trigger system, which is described highlighting performance studies based on 2018 data. 

A bstract A sample of 2 . 8 × 10 4 K + → π + μ + μ − candidates with negligible background was collected by the NA62 experiment at the CERN SPS in 2017–2018. The model-independent branching fraction is measured to be (9 . 15 ± 0 . 08) × 10 − 8 , a factor three more precise than previous measurements. The decay form factor is presented as a function of the squared dimuon mass. A measurement of the form factor parameters and their uncertainties is performed using a description based on Chiral Perturbation Theory at $$ \mathcal{O} $$ O ( p 6 ).