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A<sc>bstract</sc> This paper presents a study of the charmless three-body decays$$ {B}_{(s)}^0\to {K}_{\mathrm{S}}^0{h}^{+}h{\prime}^{-} $$ $B_{\left(s\right)}^0\to K_S^0{h}^{+}h{\prime }^{-}$ (whereh⁽′⁾=π, K), using a sample ofppcollision data collected by the LHCb experiment during Runs 1 and 2 of the LHC, corresponding to an integrated luminosity of 9 fb⁻¹. The decay$$ {B}_s^0\to {K}_{\mathrm{S}}^0{K}^{+}{K}^{-} $$ $B_s^0\to K_S^0{K}^{+}{K}^{-}$ is observed for the first time, and the following ratios of branching fractions are measured:$$ {\displaystyle \begin{array}{l}\frac{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{K}^{+}{K}^{-}\right)}{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{\pi}^{+}{\pi}^{-}\right)}=0.578\pm 0.007\pm 0.017,\\ {}\frac{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{K}^{\pm }{K}^{\mp}\right)}{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{\pi}^{+}{\pi}^{-}\right)}=0.1363\pm 0.0035\pm 0.051,\\ {}\begin{array}{l}\frac{\mathcal{B}\left({B}_s^0\to {K}_{\mathrm{S}}^0{\pi}^{+}{\pi}^{-}\right)}{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{\pi}^{+}{\pi}^{-}\right)}=0.269\pm 0.011\pm 0.015\pm 0.008,\\ {}\frac{\mathcal{B}\left({B}_s^0\to {K}_{\mathrm{S}}^0{K}^{+}{K}^{-}\right)}{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{\pi}^{+}{\pi}^{-}\right)}=0.0303\pm 0.0041\pm 0.0025\pm 0.0009,\\ {}\frac{\mathcal{B}\left({B}_s^0\to {K}_{\mathrm{S}}^0{K}^{\pm }{\pi}^{\mp}\right)}{\mathcal{B}\left({B}^0\to {K}_{\mathrm{S}}^0{\pi}^{+}{\pi}^{-}\right)}=1.818\pm 0.021\pm 0.031\pm 0.056,\end{array}\end{array}} $$ $\begin{array}{c}\frac{B\left(B^0\to K_S^0{K}^{+}{K}^{-}\right)}{B\left(B^0\to K_S^0{\pi }^{+}{\pi }^{-}\right)}=0.578\pm 0.007\pm 0.017,\\ \frac{B\left(B^0\to K_S^0{K}^{\pm }{K}^{\mp }\right)}{B\left(B^0\to K_S^0{\pi }^{+}{\pi }^{-}\right)}=0.1363\pm 0.0035\pm 0.051,\\ \begin{array}{c}\frac{B\left(B_s^0\to K_S^0{\pi }^{+}{\pi }^{-}\right)}{B\left(B^0\to K_S^0{\pi }^{+}{\pi }^{-}\right)}=0.269\pm 0.011\pm 0.015\pm 0.008,\\ \frac{B\left(B_s^0\to K_S^0{K}^{+}{K}^{-}\right)}{B\left(B^0\to K_S^0{\pi }^{+}{\pi }^{-}\right)}=0.0303\pm 0.0041\pm 0.0025\pm 0.0009,\\ \frac{B\left(B_s^0\to K_S^0{K}^{\pm }{\pi }^{\mp }\right)}{B\left(B^0\to K_S^0{\pi }^{+}{\pi }^{-}\right)}=1.818\pm 0.021\pm 0.031\pm 0.056,\end{array}\end{array}$ where the uncertainties are statistical, systematic, and due to knowledge of the ratio of hadronisation fractions of the$$ {B}_s^0 $$ $B_s^0$ andB⁰mesons, respectively. 

Abstract An amplitude analysis of the$${{{B} ^0} \!\rightarrow {\eta _{c}} (1S) {{K} ^+} {{\pi } ^-}} $$ $B^0\to {\eta }_c\left(1S\right)K^{+}{\pi }^{-}$ decays with$${\eta _{c}} (1S) \rightarrow {p} {\overline{{p}}} $$ ${\eta }_c\left(1S\right)\to p\overline{p}$ is performed using a sample corresponding to an integrated luminosity of 9$$\,\text {fb} ^{-1}$$ ${\text{fb}}^{-1}$ of$$p $$ $p$ $$p $$ $p$ collision data collected by the LHCb detector at centre-of-mass energies of$$\sqrt{s}$$ $\sqrt{s}$ = 7, 8 and 13$$\,\text {Te\hspace{-1.00006pt}V}$$ $\text{Te}\text{V}$ . The data are described with a model including only intermediate contributions from known$${K} ^{*0}$$ $K_{}^{\ast 0}$ resonances. Evidence for an exotic resonance in the$${\eta _{c}} (1S) {{\pi } ^-} $$ ${\eta }_c\left(1S\right){\pi }^{-}$ system, reported in a previous analysis of this decay channel, is not confirmed. The inclusive branching fraction of the$${{B} ^0} \rightarrow {\eta _{c}} (1S) {{K} ^+} {{\pi } ^-} $$ $B^0\to {\eta }_c\left(1S\right)K^{+}{\pi }^{-}$ decays is measured to be$$\begin{aligned}&\mathcal {B}({{B} ^0} \rightarrow {\eta _{c}} (1S) {{K} ^+} {{\pi } ^-}) \\&\quad = (5.82 \pm 0.20 \pm 0.23 \pm 0.55) \times 10^{-4}, \end{aligned}$$ $\begin{array}{cc}& B(B^0\to {\eta }_c\left(1S\right)K^{+}{\pi }^{-})\\ & =(5.82\pm 0.20\pm 0.23\pm 0.55)\times {10}^{-4},\end{array}$ where the first uncertainty is statistical, the second systematic, and the third arises from the limited knowledge of external branching fractions. 

A<sc>bstract</sc> A measurement of the branching fraction for the decay$$ \Lambda \to p{\mu}^{-}{\overline{\nu}}_{\mu } $$ $\Lambda \to p{\mu }^{-}{\overline{\nu }}_{\mu }$ is presented usingppcollision data collected by the LHCb experiment at a centre-of-mass energy of 13 TeV. The analysis is based on data recorded between 2016 and 2018, corresponding to an integrated luminosity of 5.4 fb⁻¹. The result is obtained using Λ→ pμ⁻decays as a normalisation channel. The measured branching fraction is$$ \mathcal{B}\left(\Lambda \to p{\mu}^{-}{\overline{\nu}}_{\mu}\right)=\left(1.462\pm 0.016\pm 0.100\pm 0.011\right)\times {10}^{-4} $$ $B\left(\Lambda \to p{\mu }^{-}{\overline{\nu }}_{\mu }\right)=\left(1.462\pm 0.016\pm 0.100\pm 0.011\right)\times {10}^{-4}$ , where the uncertainties are statistical, systematic, and due to the limited knowledge of the normalisation mode branching fraction, respectively. This result improves the precision of the branching fraction measurement by a factor of two compared to the previous best measurement and sets a more stringent bound on lepton flavour universality ins → uquark transitions. It is consistent with previous measurements, and the extracted lepton flavour universality test observable,$$ {R}^{\mu e}=\frac{\Gamma \left(\Lambda \to p{\mu}^{-}{\overline{\nu}}_{\mu}\right)}{\Gamma \left(\Lambda \to p{e}^{-}{\overline{\nu}}_e\right)}=0.175\pm 0.012 $$ $R^{\mathrm{\mu e}}=\frac{\Gamma \left(\Lambda \to p{\mu }^{-}{\overline{\nu }}_{\mu }\right)}{\Gamma \left(\Lambda \to p{e}^{-}{\overline{\nu }}_e\right)}=0.175\pm 0.012$ , agrees with the Standard Model prediction. 

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. 

A<sc>bstract</sc> A model-independent determination of the CKM angleγis presented, using theB^±→ [K⁺K⁻π⁺π⁻]_Dh^±andB^±→ [π⁺π⁻π⁺π⁻]_Dh^±decays, withh=K, π. This measurement is the first phase-space binned study of these decay modes, and uses a sample of proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of 9 fb⁻¹. The phase-space bins are optimised for sensitivity toγ, and in each bin external inputs from the BESIII experiment are used to constrain the charm strong-phase parameters. The result of this binned analysis is$$ \gamma ={\left(53.{9}_{-8.9}^{+9.5}\right)}^{{}^{\circ}} $$ $\gamma ={\left(53.9_{-8.9}^{+9.5}\right)}^{°}$ , where the uncertainty includes both statistical and systematic contributions. Furthermore, when combining with existing phase-space integrated measurements of the same decay modes, a value of$$ \gamma ={\left(52.{6}_{-6.4}^{+8.5}\right)}^{{}^{\circ}} $$ $\gamma ={\left(52.6_{-6.4}^{+8.5}\right)}^{°}$ is obtained, which is one of the most precise determinations ofγto date. 

Searches are presented for $B_c^{+}\to D{h}^{+}{h}^{-}$ decays, where $D$ is a charmed meson and $h^{\pm }$ is a charged pion or kaon, using $pp$ collision data collected by the LHCb experiment corresponding to an integrated luminosity of $9{\mathrm{fb}}^{-1}$ . The decays $B_c^{+}\to D^{+}{K}^{+}{\pi }^{-}$ , $B_c^{+}\to D^{*+}{K}^{+}{\pi }^{-}$ , and $B_c^{+}\to D_s^{+}{K}^{+}{K}^{-}$ are observed for the first time. Their branching fractions, expressed as ratios relative to that of the $B_c^{+}\to B_s^0{\pi }^{+}$ decay, are determined to be $\mathcal{R}(B_c^{+}\to D^{+}{K}^{+}{\pi }^{-})=(1.96\pm 0.23\pm 0.08\pm 0.10)\times {10}^{-3}$ , $\mathcal{R}(B_c^{+}\to D^{*+}{K}^{+}{\pi }^{-})=(3.67\pm 0.55\pm 0.24\pm 0.20)\times {10}^{-3}$ , $\mathcal{R}(B_c^{+}\to D_s^{+}{K}^{+}{K}^{-})=(1.61\pm 0.35\pm 0.13\pm 0.07)\times {10}^{-3}$ , where the first uncertainty is statistical, the second is systematic, and the third is due to the limited precision on the $D$ -meson branching fractions. The decay channels proceed primarily through excited $K^0$ or $D^0$ resonances or $\varphi$ mesons, and open a new avenue for studies of charge-parity violation in beauty mesons. 

An amplitude analysis of the $B^{+}\to \left({\pi }^{+}{\pi }^{-}\right)\left(K_{\mathrm{S}}^0{\pi }^{+}\right)$ decay is performed in the mass regions $0.30<m_{{\pi }^{+}{\pi }^{-}}<1.10\mathrm{GeV}/c^2$ and $0.75<m_{K_{\mathrm{S}}^0{\pi }^{+}}<1.20\mathrm{GeV}/c^2$ , using $pp$ collision data recorded with the LHCb detector corresponding to an integrated luminosity of $9{\mathrm{fb}}^{-1}$ . The polarization fractions and $CP$ asymmetries for $B^{+}\to \rho \left(770{)}^0{K}^{*}\right(892{)}^{+}$ decays are measured. Violation of the $CP$ symmetry in the decay $B^{+}\to \rho \left(770{)}^0{K}^{*}\right(892{)}^{+}$ is observed for the first time, with a significance exceeding 9 standard deviations. The $CP$ asymmetry is measured to be ${\mathcal{A}}_{CP}=0.507\pm 0.062\left(\mathrm{stat}\right)\pm 0.024\left(\mathrm{syst}\right)$ and the $CP$ -averaged longitudinal polarization fraction of $f_L=0.720\pm 0.028\left(\mathrm{stat}\right)\pm 0.009\left(\mathrm{syst}\right)$ . The measurements help to shed light on the polarization puzzle of $B$ mesons decaying to two vector mesons. 

A<sc>bstract</sc> The ratio between the branching fractions of theB⁰→ρ(770)⁰γandB⁰→K^*(892)⁰γdecays is measured with 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 measured value is$$ \frac{\mathcal{B}\left({B}^0\to \rho {(770)}^0\gamma \right)}{\mathcal{B}\left({B}^0\to {K}^{\ast }{(892)}^0\gamma \right)}=0.0189\pm 0.0007\pm 0.0005, $$ $\frac{B\left(B^0\to \rho {\left(770\right)}^0\gamma \right)}{B\left(B^0\to K^{\ast }{\left(892\right)}^0\gamma \right)}=0.0189\pm 0.0007\pm 0.0005,$ where the first uncertainty is statistical and the second systematic. The branching fraction forB⁰→ ρ(770)⁰γdecays is hence obtained as$$ \mathcal{B}\left({B}^0\to \rho {(770)}^0\gamma \right)=\left(7.9\pm 0.3\pm 0.2\pm 0.2\right)\times {10}^{-7}, $$ $B\left(B^0\to \rho {\left(770\right)}^0\gamma \right)=\left(7.9\pm 0.3\pm 0.2\pm 0.2\right)\times {10}^{-7},$ where the last uncertainty is due to the branching fraction of the normalisation mode. This result assumes that both theρ(770)⁰andK^*(892)⁰decays saturate the dihadron mass spectra considered in the analysis. It is consistent with the current world-average value and by far the most precise measurement to date. 

A<sc>bstract</sc> A study of charm mixing andCPviolation inD⁰→K^±π^∓π^±π^∓decays is performed using data collected by the LHCb experiment in proton-proton collisions from 2015 to 2018, corresponding to an integrated luminosity of 6 fb⁻¹. The ratio of promptly producedD⁰→K⁺π⁻π⁺π⁻toD⁰→K⁻π⁺π⁻π⁺decay rates is measured as a function ofD⁰decay time, both inclusive over phase space and in bins of phase space. Taking external inputs for the$$ {D}^0-{\overline{D}}^0 $$ $D^0-{\overline{D}}^0$ mixing parametersxandyallows constraints to be obtained on the hadronic parameters of the charm decay. When combined with previous measurements from charm-threshold experiments and at LHCb, improved knowledge is obtained for these parameters, which is valuable for studies of the angleγof the Unitarity Triangle. An alternative analysis is also performed, in which external inputs are taken for the hadronic parameters, and the mixing parameters are determined, including ∆xand ∆y, which are nonzero in the presence ofCPviolation. It is found that$$ x=\left(0.{85}_{-0.24}^{+0.15}\right)\% $$ $x=\left(0.{85}_{-0.24}^{+0.15}\right)\%$ ,$$ y=\left(0.{21}_{-0.27}^{+0.29}\right)\% $$ $y=\left(0.{21}_{-0.27}^{+0.29}\right)\%$ , ∆x= (−0.02 ± 0.04) % and$$ \Delta y=\left(0.0{2}_{-0.03}^{+0.04}\right)\% $$ $\Delta y=\left(0.02_{-0.03}^{+0.04}\right)\%$ . These results are consistent with previous measurements and the hypothesis ofCPconservation.