Angular analyses of rare decays at the LHC

$B^{0}\rightarrow K^{\ast 0}\ell^{+}\ell^{-}$ has been a gold-plated channel [1] since the $B$-factory era due to the relatively narrow $K^{\ast 0}(892)$ resonance. Especially in the low $q^{2}$ regime, where the recoiling $K^{\ast 0}$ has a large $\gamma$-factor in the parent $B^{0}$ frame, QCD sum rules on the light-cone (LCSR) affords control over the FF calculations. However, with the BaBar/Belle statistics, only 1-dimensional angular analyses in either the lepton or hadron helicity angles ($\cos\theta_{l}$ or $\cos\theta_{K}$) were possible [3]. Full 3-dimensional analysis in $d\theta_{l}d\theta_{K}d\chi$ in $q^{2}$ bins was possible only with the advent of LHCb. For instance, while the full BaBar dataset had $\mathcal{O}(50)$ $B^{0}\rightarrow K^{\ast 0}(\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}$ events, the existing Run1+2 LHCb dataset already includes $\mathcal{O}(10^{4})$ clean signal events for this muonic mode. On the other hand, $B$-factories, including Belle II, have complementary advantages, with better reconstruction for the $\pi^{0}$ isospin modes as well the dielectron channels.

An important result from the muonic analyses at LHCb is tension with SM predictions in the angular observable $P^{\prime}_{5}$ [4] (see Fig. 3a). Similar tensions have also been seen in $B^{+}\rightarrow K^{\ast+}\mu^{+}\mu^{-}$ [5] and $B_{s}^{0}\rightarrow\phi\mu^{+}\mu^{-}$ [6]. A related tension is observed in the overall branching fractions in several $b\rightarrow s\mu^{+}\mu^{-}$ modes which tend to consistently lie lower than the SM predictions [7, 8, 9]. Competitive results have also come from ATLAS/CMS [10, 11, 12] where the advantage is higher overall luminosity, but the disadvantage is the limited $B$-physics trigger bandwidth and lack of a RICH detector for $K^{+}/\pi^{+}/p$ separation. Numerous global fits with different data subsets, statistical methods and theory priors have been performed [2] pointing to a preferred negative $C^{\rm NP}_{9}$. The major point of contention, however, has been the effect of non-factorizable long-distance contributions due to soft+hard gluons from charm-loops that can mimic NP effects (see Fig. 3b). To constrain the non-factorizable part in a data-driven fashion, LHCb has performed an unbinned angular analysis [13, 14] using the same dataset as in Ref. [5]. The underlying transversity amplitudes are

where $\mathcal{F}_{\lambda}$ are the usual local FFs (taken from LCSR and lattice QCD) and $\mathcal{H}_{\lambda}$ are the new non-factorizable part which are extracted from a $q^{2}$-dependent parameterization. The values of $C_{i}^{\rm SM}$ are taken from theory, while allowing for $C_{9,10}^{\rm NP}$ to be floated. The results of the fit are shown in Fig. 4. Good consistency is found in the extracted binned observables compared to Ref. [5]. The overall tension with the SM is reduced to $\sim 1.8\sigma$ in $C_{9}$, and $\sim 1.4\sigma$ in global fits. The full Run 1+2 analysis and more precise theory FFs will improve upon these results.

While most of the theory and experimental investigations have focused on the ground state vector states $K^{\ast}(892)$ and $\phi(1020)$ in $B$-meson decays, LHCb has also probed the excited $K^{\ast}_{0,2}(1430)\rightarrow K^{+}\pi^{-}$ [15] and $f^{\prime}_{2}(1525)\rightarrow K^{+}K^{-}$ [7], including an angular moments analysis [16] for the former, to separate the $S$-, $P$- and $D$-wave $K\pi$ states. The results are shown in Fig. 5. The theory interpretation however will require reliable FFs for $B$ decays to these excited states.

For the 3-body final state $B\rightarrow K\mu^{+}\mu^{-}$, only the lepton helicity angle, $\theta_{l}$, can be defined and the SM predicts an almost pure $\sin^{2}\theta_{l}$ distribution save for small effects due to the muon mass. The angular distribution is be sensitive to new scalar and tensor operators via the new terms $A_{\rm FB}$ and $F_{\rm H}$:

Figure 6 shows the results of the fit to Run 1 data for both $B^{+,0}$ [17], subject to the constraints $|A_{\rm FB}|\leq F_{\rm H}/2$, $0\leq F_{\rm H}\leq 3$, such that the rate in Eq. 4 is positive. The extracted $A_{\rm FB}$ and $F_{\rm H}$ in different $q^{2}$ bins are also consistent with SM.

LHCb is unique among the $B$-factories to have access to all $b$-hadron species, including large samples of ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}$ baryons. Therefore it is possible to probe $b\rightarrow s$ penguin decays in the baryonic sector as well. For ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar 28931\relax}^{0}$ transition, the narrow ${\mathchar 28931\relax}^{0}$ state allows lattice QCD calculations [18] for the FFs. For a given value of $q^{2}$, the decay rate for polarized $\vec{{{\mathchar 28931\relax}^{0}_{\mathrm{b}}}}$ and $\vec{{\mathchar 28931\relax}^{0}}$ depends on five angles, $\vec{\mathchar 28938\relax}=\{\theta,\theta_{l},\phi_{\ell},\theta_{h},\phi_{h}\}$ (see Fig. 2) and is expanded in an orthonormal angular basis as

The $K_{i}$ moments can be related to more familiar observables such as forward-backward asymmetries in the angles: $A^{\ell}_{\rm FB}=[\frac{3}{2}K_{3}](\rightarrow\cos\theta_{l})$, $A^{h}_{\rm FB}=[K_{4}+\frac{1}{2}K_{5}](\rightarrow\cos\theta_{h})$, $A^{\ell h}_{\rm FB}=[\frac{3}{4}K_{6}](\rightarrow\cos\theta_{l}\cos\theta_{h})$. If the ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}$ is unpolarized, only the first 10 moments are non-zero [19]. Due to the long-lived nature of the ${\mathchar 28931\relax}^{0}$, its reconstruction in LHCb is somewhat non-trivial. At low $q^{2}$, the Run 1 analysis [8] found very few events. Therefore the analysis using Run 1 + partial Run 2 (collected between 2011-16) [20] focused on the high-$q^{2}$ region, $q^{2}\in[15,20]$ GeV². The results are shown in Fig. 7. The SM predictions are taken from the EOS [19] software package and show a slight tension with the data in the $K_{6}$ observable.

The first observation of ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow pK^{-}\mu^{+}\mu^{-}$ decay using LHCb Run 1 data [22] demonstrated a rich $m(pK)$ spectrum. The comparison between the resonant and non-resonant spectra is shown in Fig. 8. Employing the full Run 1+2 datasets, LHCb has examined the $m(pK^{-})$ region around the narrow $\mathchar 28931\relax(1520)$ resonance. Integrating over the angles and including the resonances ${\mathchar 28931\relax}(1405)(\frac{1}{2}^{-})$, ${\mathchar 28931\relax}(1520)(\frac{3}{2}^{-})$, ${\mathchar 28931\relax}(1600)(\frac{1}{2}^{+})$ and ${\mathchar 28931\relax}(1800)(\frac{1}{2}^{-})$, 1-dimensional fits are performed in $m(pK^{-})$. Differential cross-sections for the ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow\mathchar 28931\relax(1520)\mu^{+}\mu^{-}$ decay are also provided [23] that show large discrepancies with theory calculations in the low $q^{2}$ region (see Fig. 9).

To measure the photon polarization in $H_{b}\rightarrow H_{s}\vec{\gamma}$, the hadronic system must undergo a 3-body decay, so that the normal to the plane defines a preferred direction and the up-down asymmetry is proportional to the photon polarization, $\lambda_{\gamma}$. For example, this has been utilized in $B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}\gamma$ [25] as shown in Fig 10a. In such cases, due to poor knowledge of the resonant structures in the $H_{s}$ system and thereby the hadronic current $\mathcal{J}_{\mu}^{\rm had}$, the proportionality factor remains unknown and $\lambda_{\gamma}$ can still not be extracted out. An exception is the 2-body decay of $\vec{{\mathchar 28931\relax}^{0}}\rightarrow p\pi^{-}$, where the self-analyzing nature of the ${\mathchar 28931\relax}^{0}$ polarization provides the preferred direction. Moreover, the differential rate is $d\Gamma/d\cos\theta_{p}\propto(1-\alpha_{\mathchar 28931\relax}\lambda_{\gamma}\cos\theta)$ where the ${\mathchar 28931\relax}^{0}$ decay asymmetry parameter $\alpha_{\mathchar 28931\relax}$ is known quite precisely [28] and thereby $\lambda_{\gamma}$ can be extracted from a fit to the $\cos\theta_{p}$ slope, as shown in Fig 10b. Experimentally, ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar 28931\relax}^{0}\gamma$ is challenging due to lack of a reconstructible secondary vertex for the ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}$ decay. Employing dedicated reconstruction to reject the high background, the first observation with Run 2 2016 data was reported in Ref. [29] and the polarization measurement with full Run 2 was reported in Ref. [27]. The measured value of $\alpha_{\gamma}=0.82^{+0.17}_{-0..26}{\rm(stat.)}^{+0.04}_{-0.13}{\rm(syst.)}$ is compatible with the SM expectation of 1 and from global $C_{7}^{(^{\prime})}$ fits, reducing a 4-fold ambiguity in the $C_{7}^{\rm NP}$ phase to a 2-fold ambiguity.

As for the electroweak penguin case in Sec. 2.5, for the radiative ${{\mathchar 28931\relax}^{0}_{\mathrm{b}}}\rightarrow{\mathchar 28931\relax}^{\ast}(pK^{-})\gamma$ decay, the $[pK^{-}]$ system is dominated by a large number of overlapping resonances. Employing the full Run1+2 data, LHCb has performed a 2-dimensional angular analysis of this mode in $\{m(pK),\cos\theta_{p}\}$. All well-established ${\mathchar 28931\relax}^{\ast}$ states [28] are included in an isobar model with Breit-Wigner lineshapes incorporating mass-dependent widths. The only exception is the ${\mathchar 28931\relax}(1405)$ resonance where a two-pole Flatte-form is used. To reduce the large set of parameters, the maximum orbital angular momentum of the ${\mathchar 28931\relax}^{\ast}\rightarrow pK^{-}$ decay is taken to be $L=3$. The result from such a reduced model is shown in Fig. 11. The fit fractions and interference fit fractions are reported from the amplitude analysis.