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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. 

The time-dependent $CP$ asymmetry in $B^0\to J/\psi \rho (770{)}^0$ decays is measured using proton-proton collision data corresponding to an integrated luminosity of $6{\mathrm{fb}}^{-1}$ , collected with the LHCb detector at a center-of-mass energy of 13 TeV during the years 2015–2018. The $CP$ -violation parameters for this process are determined to be $2{\beta }_{c\overline{c}d}^{\mathrm{eff}}=0.710\pm 0.084\pm 0.051\mathrm{rad}$ and $\left|\lambda \right|=1.019\pm 0.034\pm 0.024$ , where the first uncertainty is statistical and the second systematic. This constitutes the first observation of time-dependent $CP$ violation in $B^0\to J/\psi \rho (770{)}^0$ decays. Assuming approximate SU(3) flavor symmetry, these results are combined with the previous consistent LHCb measurement to set the most stringent constraint on the penguin contribution, $\mathrm{\Delta }{\varphi }_s$ , to the $CP$ -violating phase ${\varphi }_s$ in $B_s^0\to J/\psi \varphi \left(1020\right)$ decays, yielding $\mathrm{\Delta }{\varphi }_s=5.0\pm 4.6\mathrm{mrad}$ . 

Muon-electron universality is tested in $B^0\to K^{*0}{\ell }^{+}{\ell }^{-}$ decays, in the dilepton-invariant-mass region above the $\psi \left(2S\right)$ resonance. The analysis uses beauty mesons produced in proton-proton collisions recorded by the LHCb detector at center-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of $9{\mathrm{fb}}^{-1}$ . The ratio of branching fractions between the muon and electron channels $R_{K^{*0}}$ is measured to be ${1.08}_{-0.12}^{+0.14}\left(\mathrm{stat}\right)\pm 0.07\left(\mathrm{syst}\right)$ for a dilepton-invariant-mass squared above $14.0{\mathrm{GeV}}^2/c^4$ , consistent with the standard model prediction. This result represents the most precise measurement of $R_{K^{*0}}$ in this region and the first such measurement performed at a hadron collider. 

The first observation of the doubly charmed baryon ${\mathrm{\Xi }}_{cc}^{+}$ is reported through its decay to the ${\mathrm{\Lambda }}_c^{+}{K}^{-}{\pi }^{+}$ final state, with a statistical significance exceeding seven standard deviations. The observation is made using proton-proton collision data collected in 2024 with the LHCb Run 3 detector at a center-of-mass energy of 13.6 TeV, corresponding to a total integrated luminosity of $6.9{\mathrm{fb}}^{-1}$ . The ${\mathrm{\Xi }}_{cc}^{+}$ mass is measured to be $3619.97\pm 0.83\pm {0.26}_{-1.30}^{+1.90}\mathrm{MeV}/c^2$ , where the first uncertainty is statistical, the second is systematic, and the third is due to the unknown ${\mathrm{\Xi }}_{cc}^{+}$ lifetime, which is assumed to lie in the range 15–160 fs with a baseline value of 45 fs. The difference between the masses of the ${\mathrm{\Xi }}_{cc}^{+}$ and ${\mathrm{\Xi }}_{cc}^{++}$ baryons is determined to be $-1.77\pm 0.84\pm {0.15}_{-1.30}^{+1.90}\mathrm{MeV}/c^2$ . This is the first observation of a new particle made with the LHCb Run 3 detector. 

A search for the rare decay $B^{+}\to \overline{\mathrm{\Lambda }}p{\mu }^{+}{\mu }^{-}$ is performed using proton-proton collision data recorded by the LHCb experiment at a center-of-mass energy of $\sqrt{s}=13\mathrm{TeV}$ , corresponding to an integrated luminosity of $5.4{\mathrm{fb}}^{-1}$ . An excess of events is found with respect to the background-only expectation, with a signal significance of 3.5 standard deviations, in the low invariant-mass region of $m\left(\overline{\mathrm{\Lambda }}p\right)<2.8\mathrm{GeV}/c^2$ . The branching fraction is measured to be ${\mathcal{B}}_{\mathrm{low}}(B^{+}\to \overline{\mathrm{\Lambda }}p{\mu }^{+}{\mu }^{-})=\left(1.70_{-0.56}^{+0.65}\right(\mathrm{stat})\pm 0.17(\mathrm{syst})\pm 0.14(\mathrm{ext}\left)\right)\times {10}^{-8}$ , where the last uncertainty is due to external inputs on $\mathcal{B}(B^{+}\to J/\psi \overline{\mathrm{\Lambda }}p)\times \mathcal{B}(J/\psi \to {\mu }^{+}{\mu }^{-})$ . With no significant signal observed in the high $m\left(\overline{\mathrm{\Lambda }}p\right)$ region above $2.8\mathrm{GeV}/c^2$ , an upper limit is set to be ${\mathcal{B}}_{\mathrm{high}}(B^{+}\to \overline{\mathrm{\Lambda }}p{\mu }^{+}{\mu }^{-})<2.8\left(3.7\right)\times {10}^{-9}$ at the 90% (95%) confidence level. 

An analysis of the $B^0\to K^{*0}(\to K^{+}{\pi }^{-}){\mu }^{+}{\mu }^{-}$ decay is presented using proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of $8.4{\mathrm{fb}}^{-1}$ . The full set of $CP$ -averaged and $CP$ -asymmetric angular observables is determined in bins of the invariant mass squared of the dimuon system, as well as the branching fraction relative to the $B^0\to J/\psi (\to {\mu }^{+}{\mu }^{-})K^{+}{\pi }^{-}$ decay. For the first time, the full set of observables pertaining to the $K^{+}{\pi }^{-}$ S-wave contribution to the final state are presented and consideration is given to effects arising from the mass of the muons. The extracted $CP$ -asymmetry observables show no significant deviations from zero. The measurements of the $CP$ -averaged observables and the branching fractions continue to exhibit the pattern of tensions with the standard model predictions that have been seen in previous analyses that use part of the dataset considered in this Letter. 

The decay $B^0\to {\mathrm{\Lambda }}_c^{+}{\overline{\mathrm{\Lambda }}}_c^{-}{K}_{\mathrm{S}}^0$ is studied at LHCb for the first time using proton-proton collision data recorded by the LHCb experiment at a center-of-mass energy of $\sqrt{s}=13\mathrm{TeV}$ , corresponding to an integrated luminosity of $5.4{\mathrm{fb}}^{-1}$ . The branching ratio relative to the decay $B^{+}\to {\mathrm{\Lambda }}_c^{+}{\overline{\mathrm{\Lambda }}}_c^{-}{K}^{+}$ is measured to be [ $\mathcal{B}(B^0\to {\mathrm{\Lambda }}_c^{+}{\overline{\mathrm{\Lambda }}}_c^{-}{K}_{\mathrm{S}}^0)]/[\mathcal{B}(B^{+}\to {\mathrm{\Lambda }}_c^{+}{\overline{\mathrm{\Lambda }}}_c^{-}{K}^{+})]=0.53\pm 0.05\pm 0.05$ where the first uncertainty is statistical and the second is systematic. Evidence is found for contributions from two resonant states, ${\mathrm{\Xi }}_c(2923{)}^{+}$ and ${\mathrm{\Xi }}_c(2939{)}^{+}$ , in the ${\mathrm{\Lambda }}_c^{+}{K}_{\mathrm{S}}^0$ system. The two states show a significance of $3.9\sigma$ relative to the nonresonant hypothesis. These two ${\mathrm{\Xi }}_c^{+}$ states are consistent with being the isospin partners of the states observed in the ${\mathrm{\Lambda }}_c^{+}{K}^{-}$ system. 

We present SPARC: Simulation Package for Ab-initio Real-space Calculations. SPARC can perform Kohn-Sham density functional theory calculations for isolated systems such as molecules as well as extended systems such as crystals and surfaces, in both static and dynamic settings. It is straightforward to install/use and highly competitive with state- of-the-art planewave codes, demonstrating comparable performance on a small number of processors and increasing advantages as the number of processors grows. Notably, SPARC brings solution times down to a few seconds for systems with O(100 − 500) atoms on large- scale parallel computers, outperforming planewave counterparts by an order of magnitude and more. 

The collinear and transverse-momentum-dependent jet fragmentation function and the radial profile for $B^{\pm }$ mesons in jets are measured. The $B^{\pm }$ mesons are reconstructed through the $J/\psi (\to {\mu }^{+}{\mu }^{-})K^{\pm }$ decay channel using proton-proton collision data collected during 2016–2018 with the LHCb detector at a center-of-mass (c.m.) energy of $\sqrt{s}=13\mathrm{TeV}$ , corresponding to an integrated luminosity of $5.4{\mathrm{fb}}^{-1}$ . The results complement recent measurements of jet fragmentation functions for heavy-flavor hadrons and suggest a growing contribution of gluon fragmentation to $B^{\pm }$ mesons as the jet transverse momentum increases.