# Rare decays at LHCb

C. Langenbruch1,2 on behalf of the LHCb collaboration  1University of Warwick,
Department of Physics, Gibbet Hill Road, Coventry CV4 7AL, UK
2RWTH Aachen, I. Physikalisches Institut B, Sommerfeldstr. 14, 52074 Aachen,
Germany

###### Abstract

Rare decays are flavour changing neutral current processes that are loop-
suppressed in the Standard Model (SM). New particles in SM extensions can
therefore give significant contributions, modifying branching fractions and
angular distributions. Consequently, rare decays are particularly sensitive
probes for New Physics (NP). These proceedings summarize the latest results
from the LHCb experiment on rare decays. While most results are in good
agreement with SM predictions, some tensions that recently appeared in rare
semileptonic $b\\!\rightarrow s\ell^{+}\ell^{-}$ decays are also discussed.

##  1 Introduction

Rare decays that proceed via $b\rightarrow s(d)$ quark level transitions
constitute flavour changing neutral currents and, in the SM, are forbidden at
tree-level and can only occur at loop-level. Contributions from NP can
therefore be comparably large and significantly affect branching fractions and
angular distributions. Precision measurements of rare decays therefore
constitute sensitive searches for NP. Furthermore, they allow to determine the
underlying operator structure of potential new contributions in global fits
[1, 2, 3, 4]. These proceedings present recent results on rare decays from
LHCb, determined using $3\mbox{\,fb}^{-1}$ of data taken during the LHC Run 1.

##  2 Observation of the very rare decay
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$

The very rare decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ is not only
loop but also helicity suppressed. Due to the fully leptonic final state, the
decay is experimentally well accessible and theory predictions are
particularly precise. The SM prediction for the branching fraction of the
decay is given in Ref. [5] as ${\cal
B}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}})_{\rm SM}=(3.66\pm 0.23)\times
10^{-9}$. The related decay ${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$ is
further suppressed by the ratio of CKM matrix elements $\left|V_{\rm
td}/V_{\rm ts}\right|^{2}$, resulting in a SM prediction of ${\cal
B}({{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}})_{\rm SM}=(1.06\pm 0.09)\times
10^{-10}$. The decays are particularly sensitive to contributions from NP in
the (pseudo)scalar sector, since these possible new contributions are enhanced
compared to the SM contribution. The ratio of the branching fractions of the
two decays is ${\cal R}={\cal B}({{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}})_{\rm
SM}/{\cal B}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}})_{\rm
SM}=0.0295^{+0.0028}_{-0.0025}$ for the SM, as well as in NP models with the
property of minimal flavour violation.

The CMS and LHCb collaborations have performed a combined analysis [6]. Figure
1 (left) shows the signal candidates overlaid with a combined fit sharing
signal and nuisance parameters. The measured branching fractions are ${\cal
B}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}})=(2.8^{+0.7}_{-0.6})\times
10^{-9}$ and ${\cal
B}({{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}})=(3.9^{+1.6}_{-1.4})\times
10^{-10}$, in agreement with the SM prediction at $1.2\,\sigma$ and
$2.2\,\sigma$, respectively. The decay
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ is observed with a significance
of $6.2\,\sigma$. First evidence is found for the decay
${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$ with a significance of
$3.0\,\sigma$. The branching fraction ratio is ${\cal R}=0.14^{+0.08}_{-0.06}$
and is shown in Fig. 1 (right). It is compatible with the SM and minimal
flavour violation at $2.3\,\sigma$.

Figure 1: (Left) $B^{0}_{(s)}\\!\rightarrow{\mu^{+}\mu^{-}}$ candidates in the
combined CMS and LHCb dataset. (Right) branching fraction ratio ${\cal
R}={\cal B}({{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}})/{\cal
B}({{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}})$. The dark (light) green
regions define the $1\,\sigma$ ($2\,\sigma$) confidence intervals.

##  3 Rare electroweak penguin decays

###  3.1 The decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$

The rare decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ gives
access to many angular observables that are sensitive to NP contributions. The
decay is fully defined by three decay angles
$\vec{\Omega}=(\theta_{l},\theta_{K},\phi)$, and $q^{2}$, the invariant mass
of the dimuon system squared. The $C\\!P$-averaged angular distribution in a
bin of $q^{2}$ is given by [7]

$\displaystyle\frac{1}{\mathrm{d}(\Gamma+\bar{\Gamma})/\mathrm{d}q^{2}}\frac{\mathrm{d}^{3}(\Gamma+\bar{\Gamma})}{\mathrm{d}\vec{\Omega}}=\frac{9}{32\pi}$
$\displaystyle\Big{[}$ $\displaystyle\frac{3}{4}(1-{F_{\rm
L}})\sin^{2}\theta_{K}+{F_{\rm L}}\cos^{2}\theta_{K}+\frac{1}{4}(1-{F_{\rm
L}})\sin^{2}\theta_{K}\cos 2\theta_{l}$ $\displaystyle-$ $\displaystyle{F_{\rm
L}}\cos^{2}\theta_{K}\cos
2\theta_{l}+{S_{3}}\sin^{2}\theta_{K}\sin^{2}\theta_{l}\cos 2\phi$
$\displaystyle+$ $\displaystyle{S_{4}}\sin 2\theta_{K}\sin
2\theta_{l}\cos\phi+{S_{5}}\sin 2\theta_{K}\sin\theta_{l}\cos\phi$
$\displaystyle+$ $\displaystyle\frac{4}{3}{A_{\rm
FB}}\sin^{2}\theta_{K}\cos\theta_{l}+{S_{7}}\sin
2\theta_{K}\sin\theta_{l}\sin\phi$ $\displaystyle+$ $\displaystyle{S_{8}}\sin
2\theta_{K}\sin
2\theta_{l}\sin\phi+{S_{9}}\sin^{2}\theta_{K}\sin^{2}\theta_{l}\sin
2\phi\Big{]},$

where $F_{\rm L}$ denotes the longitudinal polarisation fraction of the
${{K}^{*0}}$ and $A_{\rm FB}$ the forward-backward asymmetry of the dimuon
system. In Ref. [8] an alternative parametrisation using the
$P_{i}^{(\prime)}$ observables was proposed, that are designed such that
hadronic form-factor uncertainties cancel at leading order. An example is the
observable $P_{5}^{\prime}$, defined as $P_{5}^{\prime}=S_{5}/\sqrt{F_{\rm
L}(1-F_{\rm L})}$.

The LHCb collaboration performed the first full angular analysis of the decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ [9]. Figure 2 (left) shows
$A_{\rm FB}$ overlaid with SM predictions from Refs. [1, 10]. In the $q^{2}$
region $1.1<q^{2}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, the data
lies below the prediction, but overall good agreement is observed. The
observable $P_{5}^{\prime}$ is given in Fig. 2 (right), together with the SM
prediction from Ref. [11]. In the two $q^{2}$ bins
$[4,6]{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$[6,8]{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ local deviations from the
SM prediction are observed that correspond to $2.8\,\sigma$ and $3.0\,\sigma$.
A global analysis of all $C\\!P$-averaged
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ observables finds a global
deviation of $3.4\,\sigma$ from the SM prediction. The $C\\!P$ asymmetries
$A_{3,\ldots,9}$ are also measured and show good agreement with the SM
expectation.

Figure 2:  (Left) The forward-backward asymmetry $A_{\rm FB}$, overlaid with
SM predictions from Refs. [1, 10]. (Right) The angular observable
$P_{5}^{\prime}$, overlaid with SM predictions from Ref. [11].

###  3.2 The decay ${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$

The decay ${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ is the dominant
$b\\!\rightarrow s{\mu^{+}\mu^{-}}$ decay in the ${{B}^{0}_{s}}$ system. The
fact that the final state
$\phi(\rightarrow{{K}^{+}}{{K}^{-}}){\mu^{+}}{\mu^{-}}$ is not flavour
specific reduces the number of angular observables accessible in this decay
compared to the decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$.
LHCb performed a full angular analysis and measurement of the differential
branching faction [12]. The angular observables are found to be in good
agreement with SM predictions. Figure 3 (left) shows the differential
branching fraction, overlaid with the SM predictions from Refs. [1, 10, 13].
In the low $q^{2}$ region
$1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ the differential
branching fraction is found to be $3.3\,\sigma$ below the SM prediction.

Figure 3: The differential branching fraction of (left) the decay
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$, overlaid with SM
predictions from Refs. [1, 10, 13] and (right) the decay
${{B}^{+}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$, overlaid with SM
predictions from Refs. [15, 16, 17].

###  3.3 The decay ${{B}^{+}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$

The decay ${{B}^{+}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ is a
$b\rightarrow d{\mu^{+}\mu^{-}}$ transition and therefore in the SM suppressed
by $\left|V_{\rm td}/V_{\rm ts}\right|^{2}$ compared to the corresponding
$b\\!\rightarrow s{\mu^{+}\mu^{-}}$ decay
${{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$. LHCb has performed a
measurement of the differential branching fraction and the $C\\!P$ asymmetry
of this decay [14]. Figure 3 (right) shows the differential branching
fraction, overlaid with SM predictions from Refs. [15, 16, 17]. Good agreement
of the data with the SM predictions is observed, which is further improved in
Ref. [16] where contributions from light resonances are included. The $C\\!P$
asymmetry of the decay is measured to be ${\cal A}_{{C\\!P}}=-0.11\pm 0.12\pm
0.01$, in good agreement with the SM expectation.

In addition, the size of the CKM matrix elements $|V_{\rm td}|$ and $|V_{\rm
ts}|$ as well as their ratio are determined as $|V_{\rm
td}|=7.2^{+0.9}_{-0.8}\times 10^{-3}$, $|V_{\rm ts}|=3.2^{+0.4}_{-0.4}\times
10^{-2}$ and $|V_{\rm td}/V_{\rm ts}|=0.24^{+0.05}_{-0.04}$. Recent form-
factors from lattice calculations allow for a determination of the CKM matrix
elements from rare decays [14, 18] with a precision similar to the
determination from mixing measurements [19, 20].

##  4 Lepton universality tests using rare decays

The ratio ${\cal R}_{K}(q^{2}_{\rm min},q^{2}_{\rm
max})=\bigl{[}\int_{q^{2}_{\rm min}}^{q^{2}_{\rm max}}{\rm d}{\cal
B}({{B}^{+}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}})/{\rm
d}q^{2}\bigr{]}/\bigl{[}\int_{q^{2}_{\rm min}}^{q^{2}_{\rm max}}{\rm d}{\cal
B}({{B}^{+}}\\!\rightarrow{{K}^{+}}e^{+}e^{-})/{\rm d}q^{2}\bigr{]}$ is a
sensitive test of lepton universality. In the SM, the value of ${\cal R}_{K}$
in the range $1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is
precisely predicted to be $1\pm{\cal O}(10^{-3})$ [21]. Hadronic uncertainties
largely cancel in the ratio. The LHCb collaboration finds ${\cal
R}_{K}=0.745^{+0.090}_{-0.074}\pm 0.036$, in tension with the SM prediction at
$2.6\,\sigma$ [22]. Further lepton universality tests at LHCb are in
preparation, including measurements of ${\cal R}_{K^{*}}$ and ${\cal
R}_{\phi}$. Furthermore, the ${\cal R}_{K}$ measurement motivates searches for
lepton flavour violating decays [23, 24].

##  5 Conclusions

Recent results on rare decays from LHCb have been presented. While most are in
good agreement with SM predictions, some tensions exist, most notably in the
angular observables of the decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$, the branching fraction of
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ and ${\cal R}_{K}$. Global
fits of the data on rare $b\rightarrow s$ transitions indicate a significance
of this tension of around $3$–$4\,\sigma$ [1, 2]. While consistent NP
explanations for the deviations in the form of new heavy gauge bosons [24, 25,
26, 27] and leptoquarks [23, 28, 29, 30] have been discussed, underestimated
hadronic uncertainties cannot be excluded [31, 32, 33].

The presented measurements motivate further work both in theory as well as
experiment. With the upcoming Run 2 data, LHCb will perform further analyses
of rare $b\rightarrow s$ decays, including additional tests of lepton
universality and searches for lepton flavour violating decays.

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