A comprehensive study of the local and nonlocal amplitudes contributing to
the decay $B^0\rightarrow K^{*0}(\to K^+\pi^-) \mu^+\mu^-$ is performed by
analysing the phase-space distribution of the decay products. The analysis is
based on $pp$ collision data corresponding to an integrated luminosity of
8.4fb$^{-1}$ collected by the LHCb experiment. This measurement employs for the
first time a model of both one-particle and two-particle nonlocal amplitudes,
and utilises the complete dimuon mass spectrum without any veto regions around
the narrow charmonium resonances. In this way it is possible to explicitly
isolate the local and nonlocal contributions and capture the interference
between them. The results show that interference with nonlocal contributions,
although larger than predicted, only has a minor impact on the Wilson
Coefficients determined from the fit to the data. For the local contributions,
the Wilson Coefficient $C_9$, responsible for vector dimuon currents, exhibits
a $2.1\sigma$ deviation from the Standard Model expectation. The Wilson
Coefficients $C_{10}$, $C_{9}'$ and $C_{10}'$ are all in better agreement than
$C_{9}$ with the Standard Model and the global significance is at the level of
$1.5\sigma$. The model used also accounts for nonlocal contributions from
$B^{0}\to K^{*0}\left[\tau^+\tau^-\to \mu^+\mu^-\right]$ rescattering,
resulting in the first direct measurement of the $b s\tau\tau$ vector
effective-coupling $C_{9\tau}$.
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One-dimensional projections of the acceptance function determined from simulation.
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The mass distribution $m( K ^+ \pi ^- \mu ^+\mu ^- )$ of candidates in the data in five separate $ q^2$ regions. The data is overlaid with the results of a simultaneous fit to determine the signal fractions.
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The $ q^2$ distribution in the data, overlaid with the PDF projection from the baseline data fit. The total PDF is decomposed into signal and background components, with the signal contributions further decomposed into local and nonlocal contributions as described in Sec. 2.5.1. Note the hybrid linear/log scale to incorporate the very tall peaks from the charmonium states.
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One-dimensional confidence intervals for the Wilson Coefficients, obtained using a likelihood profile method. The shaded regions consider only statistical uncertainties, while the dashed vertical lines indicate the same regions with systematic uncertainties included. The vertical black dashed lines show the Standard Model values.
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Two-dimensional confidence regions for selected combinations of the Wilson Coefficients, obtained using a likelihood profile method. The shaded regions indicate the $1\sigma$ and $3\sigma$ confidence regions considering only statistical uncertainties, while the dashed contours indicate the same regions with systematic uncertainties included. The horizontal and vertical dashed lines show the Standard Model values.
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Comparison of form factors (orange) prefit and (maroon) postfit. The bands denote the 68% intervals from varying the form factors according to the postfit and prefit covariance matrices, respectively. Only the statistical uncertainty is accounted for in the postfit intervals.
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The nonlocal contributions from (maroon) this analysis that includes one- and two-particle hadronic amplitudes expressed as shifts to \C9. The contributions from the $\Delta \mathcal{C}_7^{\lambda}$ terms are also included, but the tau-loop contribution is excluded. The shaded bands indicate 68% confidence regions from varying the fit parameters according to the covariance matrix accounting for both statistical and systematic uncertainties. The results of $z$-expansion fits [39] from the $4.7\text{ fb} ^{-1} $ LHCb analysis [43] are also shown (pink) with and (yellow) without theory input from $ q^2 <0$. See text for more detail.
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Plots of the angular observables in the optimised basis showing both the total and the contributions from local amplitudes only.
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Plots of the angular observables in the standard basis showing both the total and the contributions from local amplitudes only.
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Plots of the unbinned angular observables in the standard basis shown for the both the baseline fit to data, and with the Wilson Coefficients (WCs) set to their Standard Model (SM) values. These are compared against SM predictions from Ref. [36].
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Result of the fit to candidates in the signal mass region. The four rows correspond to the distributions of $\cos{\theta_{K}}$, $\cos{\theta_{\ell}}$, $\phi$ and $ q^2$ . The three columns correspond to the low-, mid- and high- $ q^2$ regions. The total PDF is shown in blue, the signal PDF in red and the background PDF in dotted black. The impact of the neglected exotic states is visible in the $\cos{\theta_{K}}$ distributions.
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The $ q^2$ distribution in the data, overlaid with the PDF projection from the baseline data fit. The total PDF is decomposed into signal and background components, with the signal contributions further decomposed into contributions from the different transversity amplitudes.
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The $ q^2$ distribution in the data, overlaid with the PDF projection from the baseline data fit. The total PDF is decomposed into signal and background components, with the signal contributions further decomposed into contributions from different Lorentz structures.
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Comparison of unbinned observables constructed out of the signal parameters with the measurements from the dedicated LHCb binned analyses [5,81] that used 4.7 $\text{ fb} ^{-1}$ of data for the angular analysis (black). The shaded bands indicate 68% confidence regions from varying the fit parameters according to the covariance matrix accounting for both statistical and systematic uncertainties.
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Comparison of unbinned observables constructed out of the signal parameters with the measurements from the dedicated LHCb binned analyses [5,81] that used 4.7 $\text{ fb} ^{-1}$ of data for the angular analysis and 3 $\text{ fb} ^{-1}$ for the branching fraction (black). The shaded bands indicate 68% confidence regions from varying the fit parameters according to the covariance matrix accounting for both statistical and systematic uncertainties.
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Three $q^2$ regions defining the simultaneous fit categories when determining the Wilson Coefficients.
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The signal fraction in the full mass range $5220 \leq m( K ^+ \pi ^- \mu ^+\mu ^- ) \leq 5840 \text{ Me V /}c^2 $ determined in five $q^2$ regions chosen to isolate different combinatorial background contributions.
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The means and widths of the pull distributions in pseudoexperiments for the Wilson Coefficients. The bias is quoted as a fraction of the statistical uncertainty on the parameter.
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Results for the Wilson Coefficients. The first uncertainty is statistical, while the second is systematic.
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[Error creating the table]
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Results for the (left column) magnitudes and (right column) phases of the dominant one-particle nonlocal contributions. The first uncertainty is statistical, while the second is systematic. The magnitudes, $|A_{j}^{\lambda}|$, and phases, $\delta_{j}^{\lambda}$, are defined in Eq. 24. The values of amplitude parameters that are fixed in the fit to the data appear with a dash.
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Results for the (left column) real and (right column) imaginary parts of the higher charmonium resonance nonlocal amplitudes as defined in Eq. 24. The first uncertainty is statistical, while the second is systematic.
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Results for the parameters of the two-particle and nonresonant nonlocal contributions for the (left) real and (right) imaginary components as defined in Eqs. 29 and Sec. 2.5.3. The first uncertainty is statistical, while the second is systematic.
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Results for the local form factors. The first uncertainty is statistical, while the second is systematic. The dashed entries represent the parameters being fixed in the fit due to their degeneracy with the nonlocal $\Delta \mathcal{C}_{7}^{\perp,0}$ parameters.
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