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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. 

A<sc>bstract</sc> Thepp→W^±(→μ^±ν_μ)Xcross-sections are measured at a proton-proton centre-of-mass energy$$ \sqrt{s}=5.02 $$ $\sqrt{s}=5.02$TeV using a dataset corresponding to an integrated luminosity of 100 pb⁻¹recorded by the LHCb experiment. Considering muons in the pseudorapidity range 2.2< η <4.4, the cross-sections are measured differentially in twelve intervals of muon transverse momentum between 28< p_T<52 GeV. Integrated overp_T, the measured cross-sections are$$ {\displaystyle \begin{array}{c}{\sigma}_{W^{+}\to {\mu}^{+}{\nu}_{\mu }}=300.9\pm 2.4\pm 3.8\pm 6.0\ \textrm{pb},\\ {}{\sigma}_{W^{-}\to {\mu}^{-}{\overline{\nu}}_{\mu }}=236.9\pm 2.1\pm 2.7\pm 4.7\ \textrm{pb},\end{array}} $$ $\begin{array}{c}{\sigma }_{W^{+}\to {\mu }^{+}{\nu }_{\mu }}=300.9\pm 2.4\pm 3.8\pm 6.0\mathrm{pb},\\ {\sigma }_{W^{-}\to {\mu }^{-}{\overline{\nu }}_{\mu }}=236.9\pm 2.1\pm 2.7\pm 4.7\mathrm{pb},\end{array}$ where the first uncertainties are statistical, the second are systematic, and the third are associated with the luminosity calibration. These integrated results are consistent with theoretical predictions. This analysis introduces a new method to determine theW-boson mass using the measured differential cross-sections corrected for detector effects. The measurement is performed on this statistically limited dataset as a proof of principle and yields$$ {m}_W=80369\pm 130\pm 33\ \textrm{MeV}, $$ $m_W=80369\pm 130\pm 33\mathrm{MeV},$ where the first uncertainty is experimental and the second is theoretical. 

The first dedicated $Z$-boson mass measurement at the LHC with $Z\to {\mu }^{+}{\mu }^{-}$decays is reported. The dataset uses proton-proton collisions at a center-of-mass energy of 13 TeV, recorded in 2016 by the LHCb experiment, and corresponds to an integrated luminosity of $1.7{\mathrm{fb}}^{-1}$. A template fit to the ${\mu }^{+}{\mu }^{-}$mass distribution yields the following result for the $Z$-boson mass: $m_Z=91,185.7\pm 8.3\pm 3.9\mathrm{MeV}$, where the first uncertainty is statistical and the second systematic. This result is consistent with previous measurements and predictions from global electroweak fits. 

The substructure of jets in quantum chromodynamics (QCD) has garnered significant attention with the advent of infrared- and collinear-safe clustering algorithms and observables. A key question emerging from these studies is how in-jet emissions at soft and hard energy scales, across collinear and wide angles relative to the emitter, differ with the mass of the emitting parton. The Lund jet plane (LJP) is a perturbatively well-defined substructure observable that maps the radiation pattern of jets onto a plane, visually distinguishing emissions with different kinematic properties. Comparing LJP for jets containing hadrons of low versus high mass enables the testing of QCD splitting functions from first-principles calculations across both soft and hard regimes and at different radiation angles. This article presents the first measurement of the LJP for light-quark-enriched and beauty-initiated jets at a center-of-mass energy of 13 TeV at LHCb. This marks the first direct observation of the dead-cone effect in beauty-quark jets, measured in the collinear region of the LJP. 

A<sc>bstract</sc> An angular analysis ofB⁰→ K^*0e⁺e⁻decays is presented using proton-proton collision data collected by the LHCb experiment at centre-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb⁻¹. The analysis is performed in the region of the dilepton invariant mass squared of 1.1–6.0 GeV²/c⁴. In addition, a test of lepton flavour universality is performed by comparing the obtained angular observables with those measured inB⁰→ K^*0μ⁺μ⁻decays. In general, the angular observables are found to be consistent with the Standard Model expectations as well as with global analyses of otherb → sℓ⁺ℓ⁻processes, whereℓis either a muon or an electron. No sign of lepton-flavour-violating effects is observed. 

A<sc>bstract</sc> A search for the decay$$ {B}_c^{+} $$ $B_c^{+}$→ χ_c1(3872)π⁺is reported using proton-proton collision data collected with the LHCb detector between 2011 and 2018 at centre-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of 9 fb⁻¹. No significant signal is observed. Using the decay$$ {B}_c^{+} $$ $B_c^{+}$→ψ(2S)π⁺as a normalisation channel, an upper limit for the ratio of branching fractions$$ {\mathcal{R}}_{\psi (2S)}^{\chi_{c1}(3872)}=\frac{{\mathcal{B}}_{B_c^{+}\to {\chi}_{c1}(3872){\pi}^{+}}}{{\mathcal{B}}_{B_c^{+}\to \psi (2S){\pi}^{+}}}\times \frac{{\mathcal{B}}_{\chi_{c1}(3872)\to J/\psi {\pi}^{+}{\pi}^{-}}}{{\mathcal{B}}_{\psi (2S)\to J/\psi {\pi}^{+}{\pi}^{-}}}<0.05(0.06), $$ $R_{\psi \left(2S\right)}^{{\chi }_{c1}\left(3872\right)}=\frac{B_{B_c^{+}\to {\chi }_{c1}\left(3872\right){\pi }^{+}}}{B_{B_c^{+}\to \psi \left(2S\right){\pi }^{+}}}\times \frac{B_{{\chi }_{c1}\left(3872\right)\to J/\psi {\pi }^{+}{\pi }^{-}}}{B_{\psi \left(2S\right)\to J/\psi {\pi }^{+}{\pi }^{-}}}<0.05\left(0.06\right),$ is set at the 90 (95)% confidence level. 

The branching fraction of the decay $B^{+}\to \psi \left(2S\right)\varphi \left(1020\right)K^{+}$, relative to the topologically similar decay $B^{+}\to J/\psi \varphi \left(1020\right)K^{+}$, is measured using proton-proton collision data collected by the LHCb experiment at center-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of $9{\mathrm{fb}}^{-1}$. The ratio is found to be $0.061\pm 0.004\pm 0.009$, where the first uncertainty is statistical and the second systematic. Using the world-average branching fraction for $B^{+}\to J/\psi \varphi \left(1020\right)K^{+}$, the branching fraction for the decay $B^{+}\to \psi \left(2S\right)\varphi \left(1020\right)K^{+}$is found to be $(3.0\pm 0.2\pm 0.5\pm 0.2)\times {10}^{-6}$, where the first uncertainty is statistical, the second systematic, and the third is due to the branching fraction of the normalization channel. © 2025 CERN, for the LHCb Collaboration2025CERN