Invariant mass of the $ K ^+ \pi ^- \mu ^+\mu ^- $ system versus $q^2$. The decay $ B ^0 \rightarrow K ^{*0} \mu ^+\mu ^- $ is clearly visible inside the dashed vertical lines. The horizontal lines denote the charmonium regions, where the tree-level decays $ B ^0 \rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K ^{*0} $ and $ B ^0 \rightarrow \psi {(2S)} K ^{*0} $ dominate. These candidates are excluded from the analysis.

Relative efficiency in $\cos\theta_{l} $, $\cos\theta_{K} $, $\phi$ and $ q^2$ , as determined from a principal moment analysis of simulated three-body $ B ^0 \rightarrow K ^{*0} \mu ^+\mu ^- $ phase-space decays. The efficiency as a function of $\cos\theta_{l} $, $\cos\theta_{K} $ and $\phi$ is shown for the regions $0.1 < q^2 < 0.98\mathrm{ Ge V} ^{2}/c^{4}$ (black solid line) and $18.0 < q^2 <19.0 \mathrm{ Ge V} ^{2}/c^{4}$ (red dashed line). The efficiency as a function of $ q^2$ is shown after integrating over the decay angles. The histograms indicate the distribution of the simulated three-body $ B ^0 \rightarrow K ^{*0} \mu ^+\mu ^- $ phase-space decays used to determine the acceptance.

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 ^{*0} \mu ^+\mu ^- $, integrated over the full $q^2$ range (see text). Overlaid are the projections of the total fitted distribution (black line) and the signal and background components. The signal is shown by the blue shaded area and the background by the red hatched area.

Angular and mass distributions for $1.1<q^2<6.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. The candidates have been weighted to account for the acceptance. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area. \vspace{2cm}

Angular and $ q^2$ distribution of candidates overlaid by the result of the amplitude fit. The distribution of candidates in $ q^2$ and the three decay angles is given in a $\pm50 {\mathrm{ Me V /}c^2} $ window around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area.

The $ C P$ -averaged observables in bins of $q^2$, determined from a maximum likelihood fit to the data. The shaded boxes show the SM predictions based on the prescription of Ref. [19].

The $ C P$ -asymmetric observables in bins of $q^2$, determined from a maximum likelihood fit to the data.

The optimised angular observables in bins of $q^2$, determined from a maximum likelihood fit to the data. The shaded boxes show the SM prediction taken from Ref. [14].

The $ C P$ -averaged observables in bins of $q^2$, determined from a moment analysis of the data. The shaded boxes show the SM predictions based on the prescription of Ref. [19].

The $ C P$ -asymmetric observables in bins of $q^2$, determined from a moment analysis of the data.

The optimised angular observables in bins of $q^2$, determined from a moment analysis of the data. The shaded boxes show the SM predictions taken from Ref. [14].

The observable $S_{6c}$ in bins of $q^2$, as determined from a moment analysis of the data.

The observables $S_4$, $S_5$ and $A_{\rm FB}$ determined by fitting for the $ q^2$ dependent decay amplitudes. The line indicates the best-fit to the dataset. The band indicates the 68% interval on the bootstraps at each point in $ q^2$ . Note that, the correlation between points in the bands means it is not possible to extract the uncertainty on the zero-crossing points from these figures.

The $\Delta\chi^2$ distribution for the real part of the generalised vector-coupling strength, ${\cal C}_9$. This is determined from a fit to the results of the maximum likelihood fit of the $ C P$ -averaged observables. The SM central value is ${\rm Re}({\cal C}_9^{\rm SM}) = 4.27$ [11]. The best fit point is found to be at $\Delta{\rm Re}({\cal C}_9)= -1.04 \pm 0.25$.

Angular and mass distribution of $ B ^0 \rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K ^{*0} $ candidates in data. A small signal component is also included in the fit to account for $\overline{ B }{} {}^0_ s \rightarrow { J \mskip -3mu/\mskip -2mu\psi \mskip 2mu} K ^{*0} $ decays. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the solid blue component and the background by the red hatched component.

Angular and mass distributions for $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the solid blue component and the background by the red hatched component.

Angular and mass distributions for $1.1<q^2<2.5 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the solid blue component and the background by the red hatched component.

Angular and mass distributions for $2.5<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the solid blue component and the background by the red hatched component.

Angular and mass distributions for $4.0<q^2<6.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the solid blue component and the background by the red hatched component.

Angular and mass distributions for $6.0<q^2<8.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area.

Angular and mass distributions for $11.0<q^2<12.5 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area.

Angular and mass distributions for $15.0<q^2<17.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area.

Angular and mass distributions for $17.0<q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area.

Angular and mass distributions for $15.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $. The distributions of $m( K ^+ \pi ^- )$ and the three decay angles are given for candidates in the signal mass window $\pm50 {\mathrm{ Me V /}c^2} $ around the known $ B ^0 $ mass. Overlaid are the projections of the total fitted distribution (black line) and its different components. The signal is shown by the blue shaded area and the background by the red hatched area.

Animated gif made out of all figures.

Angular observables $I_j$ and their corresponding angular terms for dimuon masses that are much larger than twice the muon mass. The terms in the lower part of the table arise from the $ K ^+ \pi ^- $ S-wave contribution to the $ K ^+ \pi ^- \mu ^+\mu ^- $ final state. The $\bar{I}_i$ coefficients are obtained by making the substitution ${\cal A} \rightarrow \bar{{\cal A}}$, i.e. by complex conjugation of the weak phases in the amplitudes.

Summary of the different sources of systematic uncertainty on the angular observables. Upper limits or typical ranges are quoted for the different groups of observables. The column labelled $q^2_0$ corresponds to the zero-crossing points of $S_4$, $S_5$ and $A_{\rm FB}$.

$ C P$ -averaged angular observables evaluated by the unbinned maximum likelihood fit, in the range $1.1 < q^2 < 6.0 \mathrm{ Ge V} ^{2}/c^{4}$ and $15.0 < q^2 < 19.0 \mathrm{ Ge V} ^{2}/c^{4}$. The first uncertainties are statistical and the second systematic.

$ C P$ -averaged angular observables evaluated by the unbinned maximum likelihood fit. The first uncertainties are statistical and the second systematic.

$ C P$ -asymmetric angular observables evaluated by the unbinned maximum likelihood fit. The first uncertainties are statistical and the second systematic.

Optimised angular observables evaluated by the unbinned maximum likelihood fit. The first uncertainties are statistical and the second systematic.

$ C P$ -averaged angular observables evaluated using the method of moments. The first uncertainties are statistical and the second systematic.

$ C P$ -asymmetries evaluated using the method of moments. The first uncertainties are statistical and the second systematic.

Optimised observables evaluated using the method of moments. The first uncertainties are statistical and the second systematic.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $ .

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $1.1<q^2<2.5 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $2.5<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $4.0 <q^2< 6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $6.0<q^2<8.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $11.0 <q^2< 12.5 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $15.0 <q^2< 17.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $17.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $1.1 <q^2< 6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables from the maximum likelihood fit in the bin $15.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $1.1<q^2<2.5 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $2.5<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $4.0 <q^2< 6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $6.0 <q^2< 8.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $11.0 <q^2< 12.5 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $15.0 <q^2< 17.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $17.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $1.1 <q^2< 6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables from the maximum likelihood fit in the bin $15.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $1.1<q^2<2.5 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $2.5<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $4.0 <q^2< 6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $6.0 <q^2< 8.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $11.0 <q^2< 12.5 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $15.0 <q^2< 17.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $17.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $1.1 <q^2< 6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables from the maximum likelihood fit in the bin $15.0 <q^2< 19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $1.1<q^2<2.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $2.0<q^2<3.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $3.0<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $4.0<q^2<5.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $5.0<q^2<6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $6.0<q^2<7.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $7.0<q^2<8.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $11.00 <q^2<11.75 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $11.75 <q^2<12.50 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $15.0 <q^2<16.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $16.0 <q^2<17.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $17.0 <q^2<18.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $18.0 <q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -averaged observables obtained for the method of moments in the bin $15.0 <q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $1.1<q^2<2.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $2.0<q^2<3.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $3.0<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $4.0<q^2<5.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $5.0<q^2<6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $6.0<q^2<7.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $7.0<q^2<8.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $11.00 <q^2<11.75 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $11.75 <q^2<12.50 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $15.0 <q^2<16.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $16.0 <q^2<17.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $17.0 <q^2<18.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $18.0 <q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the $ C P$ -asymmetric observables obtained for the method of moments in the bin $15.0 <q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $0.10<q^2<0.98 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $1.1<q^2<2.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $2.0<q^2<3.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $3.0<q^2<4.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $4.0<q^2<5.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $5.0<q^2<6.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $6.0<q^2<7.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $7.0<q^2<8.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $11.00 <q^2<11.75 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $11.75 <q^2<12.50 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $15.0 <q^2<16.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $16.0 <q^2<17.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $17.0 <q^2<18.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $18.0 <q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $.

Correlation matrix for the optimised angular observables obtained for the method of moments in the bin $15.0 <q^2<19.0 {\mathrm{ Ge V^2 /}c^4} $.

Supplementary material full pdf

This ZIP file contains supplementary material for the publication LHCb-PAPER-2015-051. The files are: supplementary.pdf: An overview of the extra figures. *.pdf, *.png, *.eps, *.C: The figures in various formats.