Background-subtracted $ m( K ^+ \pi ^- )$ distribution for $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ decays in the range $1.1< q^2 <6.0 {\mathrm{ Ge V^2 /}c^4} $. The region $1330< m( K ^+ \pi ^- ) <1530 {\mathrm{ Me V /}c^2} $ is indicated by the blue, hatched area.

Angle conventions for (a) $\overline{ B }{} {}^0 \rightarrow K ^- \pi ^+ \mu ^- \mu ^+ $ and (b) $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+ \mu ^- $, as described in Ref. [12]. The leptonic and hadronic frames are back-to-back with a common $\hat{y}$ axis. For the dihedral angle $\phi$ between the leptonic and hadronic decay planes, there is an additional sign flip $\phi\rightarrow -\phi$ compared to previous LHCb analyses [1,2,3,4].

Relative efficiency in $\cos{\theta_\ell}$ , $\cos{\theta_K}$ and $\phi'$ in the region $1.1< q^2 <6.0 {\mathrm{ Ge V^2 /}c^4} $ and $1330< m( K ^+ \pi ^- ) <1530 {\mathrm{ Me V /}c^2} $ as determined from a moment analysis of simulated $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ decays, shown as a histogram. The efficiency function is shown by the blue, dashed line.

Invariant mass $ m( K ^+ \pi ^- \mu ^+ \mu ^- )$ for (left) the control decay $ B ^0 \rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K ^{*0} $ and (right) the signal decay $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ in the bin $1.1 < q^2 < 6.0 {\mathrm{ Ge V^2 /}c^4} $. The solid black line represents the total fitted function. The individual components of the signal (blue, shaded area) and combinatorial background (red, hatched area) are also shown.

Differential branching fraction of $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ in bins of $ q^2$ for the range $1330< m( K ^+ \pi ^- ) <1530 {\mathrm{ Me V /}c^2} $. The error bars indicate the sums in quadrature of the statistical and systematic uncertainties.

Measurement of the normalised moments, $\overline{\Gamma}_{i}$, of the decay $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ in the range $1.1< q^2 <6.0 {\mathrm{ Ge V^2 /}c^4} $ and $1330< m( K ^+ \pi ^- ) <1530 {\mathrm{ Me V /}c^2} $. The error bars indicate the sums in quadrature of the statistical and systematic uncertainties.

The distributions of each of the decay angles within the signal region. The acceptance-corrected data is represented by the points with error bars. The estimated signal distribution is shown by the blue, shaded histogram. The projected background from the upper mass sideband is shown by the red, hatched histogram, which is stacked onto the signal histogram.

Invariant mass $ m( K ^+ \pi ^- \mu ^+ \mu ^- )$ distributions of the signal decay $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ in each of the $ q^2$ bins used for the differential branching fraction measurement. The solid black line represents the total fitted function. The individual components of the signal (blue, shaded area) and combinatorial background (red, hatched area) are also shown.

Animated gif made out of all figures.

Expected resonant contributions above the $ K ^{*}(892)^{0}$ mass range. For each, the spin-parity, $J^P$, and branching fraction to $ K \pi $, $\mathcal{B}(K\pi)$, are given (taken from Ref. [10]).

Differential branching fraction of $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ in bins of $ q^2$ for the range $1330< m( K ^+ \pi ^- ) <1530 {\mathrm{ Me V /}c^2} $. The first uncertainty is statistical, the second systematic and the third due to the uncertainty on the $ B ^0 \rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K ^{*}(892)^{0}$ and $ { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} \rightarrow \mu ^+\mu ^- $ branching fractions.

Measurement of the normalised moments, $\overline{\Gamma}_{i}$, of the decay $ B ^0 \rightarrow K ^+ \pi ^- \mu ^+\mu ^- $ in the range $1.1< q^2 <6.0 {\mathrm{ Ge V^2 /}c^4} $ and $1330< m( K ^+ \pi ^- ) <1530 {\mathrm{ Me V /}c^2} $. The first uncertainty is statistical and the second systematic.

Summary of the main sources of systematic uncertainty for the differential branching fraction and the angular moments analysis. Typical ranges are quoted for the different $ q^2$ bins used in the differential branching fraction measurement, and for the moments measured in the angular analysis. The systematic uncertainties are significantly smaller than the statistical ones.

The transversity-basis moments of the 41 orthonormal angular functions $f_i(\Omega)$ in Eq. 1 [12]. The amplitudes correspond to the $\overline{ B }{} {}^0 $ decay.

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