# Amplitude analysis of the $B^0_{(s)} \to K^{*0} \overline{K}^{*0}$ decays and measurement of the branching fraction of the $B^0 \to K^{*0} \overline{K}^{*0}$ decay

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## Abstract

The $B^0 \to K^{*0} \overline{K}^{*0}$ and $B^0_s \to K^{*0} \overline{K}^{*0}$ decays are studied using proton-proton collision data corresponding to an integrated luminosity of 3fb$^{-1}$. An untagged and time-integrated amplitude analysis of $B^0_{(s)} \to (K^+\pi^-)(K^-\pi^+)$ decays in two-body invariant mass regions of 150 MeV$/c^2$ around the $K^{*0}$ mass is performed. A stronger longitudinal polarisation fraction in the ${B^0 \to K^{*0} \overline{K}^{*0}}$ decay, ${f_L = 0.724 \pm 0.051 ({\rm stat}) \pm 0.016 ({\rm syst})}$, is observed as compared to ${f_L = 0.240 \pm 0.031 ({\rm stat}) \pm 0.025 ({\rm syst})}$ in the ${B^0_s\to K^{*0} \overline{K}^{*0}}$ decay. The ratio of branching fractions of the two decays is measured and used to determine $\mathcal{B}(B^0 \to K^{*0} \overline{K}^{*0}) = (8.0 \pm 0.9 ({\rm stat}) \pm 0.4 ({\rm syst})) \times 10^{-7}$.

## Figures and captions

 Leading order Feynman diagrams for the $B ^0 \rightarrow K ^{*0} \overline{ K } {}^{*0}$ and $B ^0_ s \rightarrow K ^{*0} \overline{ K } {}^{*0}$ decays. Both modes are dominated by a gluonic-penguin diagram. Fig1a.pdf [9 KiB] HiDef png [292 KiB] Thumbnail [175 KiB] *.C file Fig1b.pdf [9 KiB] HiDef png [293 KiB] Thumbnail [176 KiB] *.C file Definition of the helicity angles, employed in the angular analysis of the $B ^0_{( s )} \rightarrow K ^{*0} \overline{ K } {}^{*0}$ decays. Each angle is defined in the rest frame of the decaying particle. Fig2.pdf [28 KiB] HiDef png [146 KiB] Thumbnail [92 KiB] *.C file Aggregated four-body invariant-mass fit result of the 2011 and 2012 data. The solid red distribution corresponds to the $B ^0_ s \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )$ decay, the solid cyan distribution to $B ^0 \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )$ , the dotted dark blue line to $\Lambda ^0_ b \rightarrow ( p \pi ^- )( K ^- \pi ^+ )$ , the dotted yellow line to $B ^0 \rightarrow ( K ^+ \pi ^- )( K ^- K ^+ )$ and the dotted cyan line represents the partially reconstructed background. The tiny combinatorial background contribution is not represented. The black points with error bars correspond to data to which the $B ^0 \rightarrow \rho ^0 K ^{*0}$ contribution has been subtracted with negatively weighted simulation, and the overall fit is represented by the thick blue line. Fig3.pdf [24 KiB] HiDef png [329 KiB] Thumbnail [245 KiB] *.C file Projections of the amplitude fit results for the $B ^0 \rightarrow K ^{*0} \overline{ K } {}^{*0}$ decay mode on the helicity angles (top row: $\cos \theta_1$ left, $\cos \theta_2$ centre and $\phi$ right) and on the two-body invariant masses (bottom row: $M( K ^+ \pi ^- )$ left and $M( K ^- \pi ^+ )$ centre). The contributing partial waves: $VV$ (dashed red), $VS$ (dashed green) and $SS$ (dotted grey) are shown with lines. The black points correspond to data and the overall fit is represented by the blue line. Fig4.pdf [24 KiB] HiDef png [333 KiB] Thumbnail [285 KiB] *.C file Projections of the amplitude fit results for the $B ^0_ s \rightarrow K ^{*0} \overline{ K } {}^{*0}$ decay mode on the helicity angles (top row: $\cos \theta_1$ left, $\cos \theta_2$ centre and $\phi$ right) and on the two-body invariant masses (bottom row: $M( K ^+ \pi ^- )$ left and $M( K ^- \pi ^+ )$ centre). The contributing partial waves: $VV$ (dashed red), $VS$ (dashed green) and $SS$ (dotted grey) are shown with lines. The black points correspond to data and the overall fit is represented by the blue line. Fig5.pdf [23 KiB] HiDef png [348 KiB] Thumbnail [294 KiB] *.C file Animated gif made out of all figures. PAPER-2019-004.gif Thumbnail

## Tables and captions

 Parameters of the mass propagators employed in the amplitude analysis. Table_1.pdf [50 KiB] HiDef png [88 KiB] Thumbnail [43 KiB] tex code Amplitudes, $A_i$, and angle-mass functions, $g_i(m_1,m_2,\theta_1,\theta_2,\phi)$, of the differential decay rate of 5. In particular, $A_0$, $A_\parallel$ and $A_\perp$ are the longitudinal, parallel and transverse helicity amplitudes of the {P--wave} whereas $A_S^+$ and $A_S^-$ are the combinations of $C P$ eigenstate amplitudes of the $SV$ and $VS$ states and $A_{SS}$ is the double {S--wave} amplitude. The table indicates the corresponding $C P$ eigenvalue, $\eta_i$. The mass propagators, $\mathcal{M}_{0,1}(m)$, are discussed in the text. Table_2.pdf [52 KiB] HiDef png [82 KiB] Thumbnail [40 KiB] tex code Signal and background yields for the 2011 and 2012 data samples, obtained from the fit to the four-body mass spectrum of the selected candidates. Statistical and systematic uncertainties are reported, the latter are estimated as explained in Sect. ???. Table_3.pdf [53 KiB] HiDef png [69 KiB] Thumbnail [36 KiB] tex code Results of the amplitude analysis of $B ^0 \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )$ and $B ^0_ s \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )$ decays. The observables above the line are directly obtained from the maximum-likelihood fit whereas those below are obtained from the former, as explained in the text, with correlations accounted for in their estimated uncertainties. For each result, the first quoted uncertainty is statistical and the second systematic. The estimation of the latter is described in Sect. ???. Table_4.pdf [76 KiB] HiDef png [189 KiB] Thumbnail [101 KiB] tex code Alternative parameters of the LASS mass propagator used in the {S--wave} systematic uncertainty estimation. Table_5.pdf [43 KiB] HiDef png [126 KiB] Thumbnail [51 KiB] tex code Systematic uncertainties for the parameters of the amplitude-analysis fit of the $B ^0_{( s )} \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )$ decay. The bias related to differences between data and simulation is included in the results shown in Table ???. Table_6.pdf [81 KiB] HiDef png [96 KiB] Thumbnail [38 KiB] tex code Systematic uncertainties for the derived observables of the amplitude-analysis fit of the $B ^0_{( s )} \rightarrow ( K ^+ \pi ^- )( K ^- \pi ^+ )$ decay. The bias related to differences between data and simulation is included in the results shown in Table ???. Table_7.pdf [58 KiB] HiDef png [99 KiB] Thumbnail [39 KiB] tex code Systematic uncertainties in the factor $\kappa$ defined in 12 split in categories. The bias originated in differences between data and simulation is corrected for in the $\kappa$ results shown in Table ???. Table_8.pdf [56 KiB] HiDef png [135 KiB] Thumbnail [49 KiB] tex code Parameters used to determine $\mathcal{B} ( B ^0 \rightarrow K ^{*0} \overline{ K } {}^{*0} )/\mathcal{B} ( B ^0_ s \rightarrow K ^{*0} \overline{ K } {}^{*0} )$. When two uncertainties are quoted, the first is statistical and the second systematic. The value of $f_s/f_d$ is taken from Ref. [44]. Table_9.pdf [61 KiB] HiDef png [45 KiB] Thumbnail [19 KiB] tex code

Created on 19 October 2019.