EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-176 LHCb-PAPER-2020-020 12 October 2020

Strong constraints

on the $b\rightarrow s\gamma$ photon polarisation

from ${{B}^{0}}\rightarrow{{K}^{*0}}e^{+}e^{-}$ decays

LHCb collaboration†††Authors are listed at the end of this paper.

An angular analysis of the $B^{0}\rightarrow K^{*0}e^{+}e^{-}$ decay is
performed using a data sample corresponding to an integrated luminosity of
9${\mbox{\,fb}^{-1}}$ of $pp$ collisions collected with the LHCb experiment.
The analysis is conducted in the very low dielectron mass squared ($q^{2}$)
interval between $0.0008$ and $0.257\text{\,Ge\kern-1.00006ptV}^{2}$, where
the rate is dominated by the $B^{0}\rightarrow K^{\ast 0}\gamma$ transition
with a virtual photon. The fraction of longitudinal polarisation of the
$K^{\ast 0}$ meson, $F_{\rm L}$, is measured to be $F_{\rm L}=(4.4\pm 2.6\pm
1.4)\%$, where the first uncertainty is statistical and the second systematic.
The $A_{\rm T}^{\rm Re}$ observable, which is related to the lepton forward-
backward asymmetry, is measured to be $A_{\mathrm{T}}^{\mathrm{Re}}=-0.06\pm
0.08\pm 0.02$. The $A_{\rm T}^{(2)}$ and $A_{\rm T}^{\rm Im}$ transverse
asymmetries, which are sensitive to the virtual photon polarisation, are found
to be $A_{\rm T}^{(2)}=0.11\pm 0.10\ \pm 0.02$ and $A_{\rm T}^{\rm Im}=0.02\pm
0.10\pm 0.01$. The results are consistent with Standard Model predictions and
provide the world’s best constraint on the $b\rightarrow s\gamma$ photon
polarisation.

Published in JHEP 12 (2020) 081

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Decay processes mediated by $b\rightarrow s\gamma$ transitions are suppressed
in the Standard Model (SM) as they proceed through flavour-changing neutral
currents involving electroweak-loop Feynman diagrams. The precise study of
their properties is sensitive to small contributions from physics beyond the
Standard Model (BSM). Since in the SM the weak force only couples to left-
handed quarks, the photons emitted in $b\rightarrow s\gamma$ transitions are
predominantly left-handed. The contribution with right-handed polarisation is
suppressed by the ratio of the $s$ and $b$ quark masses. Therefore, a larger
right-handed contribution would represent a clear sign of BSM physics [1, 2,
3, 4, 5, 6, 7, 8, 9, 10]. The chirality of the $b\rightarrow s\gamma$
transition was indirectly probed at the BaBar, Belle and LHCb experiments,
using measurements of the inclusive $B\rightarrow X_{s}\gamma$ branching ratio
[11, 12, 13, 14, 15] as well as the mixing-induced $C\\!P$ asymmetries and
time-dependent decay rates of radiative ${B}^{0}$ and ${B}^{0}_{s}$ decays
[16, 17, 18]. In this paper, the $b\rightarrow s\ell\ell$ transition where the
dilepton pair originates from a virtual photon is used to measure the
$b\rightarrow s\gamma$ photon polarisation. In order to isolate
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ transitions dominated by the
$b\rightarrow s\gamma$ contribution, the analysis is restricted to a region of
very low dilepton mass squared ($q^{2}$), which can only be accessed via the
$b\rightarrow s{e^{+}e^{-}}$ transition due to the low electron mass [19, 20].
This paper presents an angular analysis of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ decay111The inclusion of
charge-conjugate processes is implied throughout this paper. Natural units
with $c=1$ are used throughout this paper. in the region of $q^{2}$ between
0.0008 and 0.257$\text{\,Ge\kern-1.00006ptV}^{2}$. The symbol ${K}^{*0}$
denotes the ${K}^{*0}(892)$ meson reconstructed via its decay
${{K}^{*0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$. An angular analysis of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ decays was performed by LHCb
in the $q^{2}$ region between 0.002 and 1.120$\text{\,Ge\kern-1.00006ptV}^{2}$
[21]. The analysis presented here uses a data sample collected between the
years 2011 and 2018. This sample comprises approximately five times as many
$B^{0}$ decays. This analysis also employs a selection technique that greatly
improves the signal purity as well as the sensitivity to the $b\rightarrow
s\gamma$ photon polarisation.

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ decay can be described
over the full $q^{2}$ range by the left (right)-handed Wilson coefficients
${C}_{7}^{(\prime)}$, ${C}_{9}^{(\prime)}$ and ${C}_{10}^{(\prime)}$ [22, 23].
These coefficients encode information about short-distance effects and are
sensitive to BSM physics. The detailed description of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ differential decay rate
involves hadronic form factors describing the ${{B}^{0}}\rightarrow{{K}^{*0}}$
transition and other long-distance effects that can be difficult to predict
[24, 25, 26, 27]. The results of the angular analysis of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ decay [28], the
measurements of the branching fractions of several
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ decays [29, 30, 31] and the ratio of
the branching fractions of the electron and muon channels of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$ and
${{B}^{+}}\\!\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$ decays [32, 33] exhibit
tensions with respect to SM predictions. Model independent fits of the
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ measurements involve all of the Wilson
coefficients mentioned above [34, 35, 36, 37, 38]. Since the very-low-$q^{2}$
region is associated with the left- and right-handed electromagnetic operators
[19], it contains unique information that can be used to determine the $C_{7}$
and $C_{7}^{\prime}$ Wilson coefficients.

For the ${{B}^{0}}\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ decay the partial decay
width can be described in terms of $q^{2}$ and three angles
$\theta_{\ell},\theta_{K}$ and $\phi$. The angle $\theta_{\ell}$ is defined as
the angle between the direction of the $e^{+}$ and the direction opposite to
that of the ${B}^{0}$ meson in the dielectron rest frame. The angle
$\theta_{K}$ is defined as the angle between the direction of the kaon and the
direction opposite to that of the ${B}^{0}$ meson in the ${K}^{*0}$ meson rest
frame. The angle $\phi$ is the angle between the plane containing the electron
and positron and the plane containing the kaon and pion in the ${B}^{0}$ meson
rest frame. The basis is chosen so that the angular definition for the ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ decay is the $C\\!P$ conjugate of
that of the ${B}^{0}$ decay. These definitions are identical to those used for
the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}}{\mu^{-}}$ analysis in Ref.
[39], including the sign flip of $\phi$ ($\phi\rightarrow-\phi$) for the
${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ decay. The angle $\phi$
is transformed such that $\tilde{\phi}=\phi+\pi$ if $\phi<0$. This
transformation cancels out terms that have a $\sin\phi$ or $\cos\phi$
dependence and simplifies the angular expression without any loss of
sensitivity to the remaining observables. In the region of $q^{2}$ considered
in this paper, where the photon is almost on-shell, the fraction of
${{K}^{+}}{{\pi}^{-}}$ pairs in an S-wave configuration is suppressed with
respect to the value measured at higher $q^{2}$ [40], because a longitudinally
polarised ${{K}^{+}}{{\pi}^{-}}$ pair cannot couple to a real photon. Using
Refs. [41, 42] it can be shown that the ratio of the S-wave fraction to the
fraction of longitudinal polarisation of the ${{K}^{*0}}$ is constant as
function of $q^{2}$ in the 0-6$\text{\,Ge\kern-1.00006ptV}^{2}$ range.
Neglecting the ${{K}^{+}}{{\pi}^{-}}$ S-wave contribution, and in the limit of
massless leptons (a very good approximation for electrons), the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ angular distribution can be
expressed as

$\begin{split}\frac{1}{\mathrm{d}(\Gamma+\bar{\Gamma})/\mathrm{d}q^{2}}\frac{\mathrm{d}^{4}(\Gamma+\bar{\Gamma})}{\mathrm{d}q^{2}\,\mathrm{d}{\cos\theta_{\ell}}\,\mathrm{d}{\cos{\theta_{K}}}\,\mathrm{d}{\tilde{\phi}}}=\frac{9}{16\pi}\Big{[}&\tfrac{3}{4}(1-F_{\mathrm{L}})\sin^{2}\theta_{K}+F_{\mathrm{L}}\cos^{2}\theta_{K}\\\
\phantom{\Big{[}}+&\tfrac{1}{4}(1-F_{\mathrm{L}})\sin^{2}\theta_{K}\cos
2\theta_{\ell}-F_{\mathrm{L}}\cos^{2}\theta_{K}\cos 2\theta_{\ell}\\\
\phantom{\Big{[}}+&(1-F_{\mathrm{L}})A_{\mathrm{T}}^{\mathrm{Re}}\sin^{2}\theta_{K}\cos{\theta_{\ell}}\\\
\phantom{\Big{[}}+&\tfrac{1}{2}(1-F_{\mathrm{L}})A_{\mathrm{T}}^{(2)}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\cos
2\tilde{\phi}\\\
\phantom{\Big{[}}+&\tfrac{1}{2}(1-F_{\mathrm{L}})A_{\mathrm{T}}^{\mathrm{Im}}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\sin
2\tilde{\phi}\Big{]}\,.\end{split}$ (1)

The four angular observables $F_{\mathrm{L}}$, $A_{\mathrm{T}}^{\mathrm{Re}}$,
$A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ are combinations of
the transversity amplitudes $A_{0}$, $A_{\perp}$ and $A_{||}$, as detailed in
Ref. [21]. The observable $F_{\mathrm{L}}$ corresponds to the longitudinal
polarisation fraction of the ${K}^{*0}$ meson and is expected to be small at
low $q^{2}$, since the virtual photon is quasi-real and therefore transversely
polarised. The observable $A_{\mathrm{T}}^{\mathrm{Re}}$ is related to the
lepton forward-backward asymmetry, $A_{\mathrm{FB}}$, by
$A_{\mathrm{T}}^{\mathrm{Re}}=\frac{4}{3}A_{\mathrm{FB}}(1-F_{\mathrm{L}})$
[43]. The observable $A_{\mathrm{T}}^{(2)}$ is averaged between ${B}^{0}$ and
${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ decays, while, given the
$\phi$ sign flip for ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$
decays, $A_{\mathrm{T}}^{\mathrm{Im}}$ corresponds to a $C\\!P$ asymmetry
[44]. The $A_{\mathrm{T}}^{\mathrm{Re}}$, $A_{\mathrm{T}}^{(2)}$ and
$A_{\mathrm{T}}^{\mathrm{Im}}$ transverse asymmetries are related to the
$P_{i}$ angular basis [45] through $A_{\mathrm{T}}^{\mathrm{Re}}=2P_{2}$,
$A_{\mathrm{T}}^{(2)}=P_{1}$ and
$A_{\mathrm{T}}^{\mathrm{Im}}=-2P_{3}^{{C\\!P}}$.

The $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ observables
depend only on the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$
transversity amplitudes, $A_{\perp}$ and $A_{||}$, and vanish if these
amplitudes are completely left-handed. In the limit ${q^{2}}\rightarrow 0$,
which is a good approximation for the $q^{2}$ region considered in this paper,
the $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ observables are
closely related to the photon polarisation in
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ transitions. In particular, the
ratio of the right- and left-handed photon amplitudes, $A_{\rm R}$ and $A_{\rm
L}$, can be related to $A_{\mathrm{T}}^{(2)}$ and
$A_{\mathrm{T}}^{\mathrm{Im}}$ through [43, 10]

$\begin{split}\tan\chi&\equiv\left|A_{\rm R}/A_{\rm L}\right|,\\\
A_{\mathrm{T}}^{(2)}&=\sin(2\chi)\cos(\phi_{\rm L}-\phi_{\rm R}),\\\
A_{\mathrm{T}}^{\mathrm{Im}}&=\sin(2\chi)\sin(\phi_{\rm L}-\phi_{\rm
R}),\end{split}$ (2)

where $\phi_{\rm L(R)}$ is the $A_{\rm L(R)}$ weak phase and the small strong
phase difference between the amplitudes is neglected. Corrections to these
approximations, due to terms proportional to ${C}_{9}$ and ${C}_{10}$, are
smaller than 0.006 even in the presence of large BSM effects in
${C}_{7}^{(\prime)}$ [46]. The mixing-induced $C\\!P$ asymmetries and time-
dependent decay rates of radiative ${B}^{0}$ and ${B}^{0}_{s}$ decays have
very similar expressions, but also involve $B$-mixing phases [10].

The $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ observables are
predicted to be very small in the SM when compared to the current experimental
sensitivity. Using the Flavio software package [46] (version 2.0.0) the
following SM predictions are calculated for the four angular observables in
the $q^{2}$ range considered

$\begin{split}F_{\mathrm{L}}({\rm SM})&=\phantom{+}0.051\pm 0.013,\\\
A_{\mathrm{T}}^{\mathrm{Re}}({\rm SM})&=-0.0001\pm 0.0004,\\\
A_{\mathrm{T}}^{(2)}({\rm SM})&=\phantom{+}0.033\pm 0.020,\\\
A_{\mathrm{T}}^{\mathrm{Im}}({\rm SM})&=-0.00012\pm 0.00034.\end{split}$ (3)

A detailed discussion of the theoretical uncertainties on the hadronic
contributions involved in these predictions can be found in Ref. [10]. The
uncertainties on the transverse asymmetries are much smaller than the
experimental sensitivity of the results presented in this paper.

##  2 The LHCb detector and data set

The study reported here is based on $pp$ collision data, corresponding to an
integrated luminosity of 9$\text{\,fb}^{-1}$, collected at the Large Hadron
Collider (LHC) with the LHCb detector [47]. The data were taken in the years
2011, 2012 and 2015–2018, at centre-of-mass energies of 7, 8 and 13 TeV,
respectively. The LHCb detector [47, 48] is a single-arm forward spectrometer
covering the pseudorapidity range $2<\eta<5$. The detector includes a tracking
system consisting of a vertex detector surrounding the $pp$ interaction region
and of tracking stations on either sides of a $4{\rm\,Tm}$ dipole magnet.
Charged particles are identified using information from two ring-imaging
Cherenkov detectors (RICH), electromagnetic (ECAL) and hadronic (HCAL)
calorimeters and muon chambers. The online event selection is performed by a
trigger, which consists of a hardware stage, based on information from the
calorimeters and muon system, followed by a software stage, which fully
reconstructs the event. The hardware electron trigger requires the presence of
an ECAL cluster with minimum transverse energy between 2.5 and 3.0 GeV,
depending on the data-taking period. Signal
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ candidates are retained if at
least one of the electrons fires the electron trigger. Alternatively,
candidates are selected if the hardware trigger requirements were passed by
objects in the rest of the event that are independent of the decay products of
the signal ${B}^{0}$ candidate. The software trigger requires a two-, three-
or four-track vertex with a significant displacement from a primary $pp$
interaction vertex (PV). At least one charged particle must have a significant
transverse momentum ($p_{\mathrm{T}}$) and be inconsistent with originating
from any PV. A multivariate algorithm [49] is used to identify displaced
vertices consistent with the decay of a $b$ hadron.

Samples of simulated events, produced with the software described in Refs.
[50, *Sjostrand:2007gs, 52, 53, 54, 55, *Agostinelli:2002hh, 57], are used to
characterise signal and background contributions. The simulated samples are
corrected for known differences between data and simulation in kinematics,
particle identification, detector occupancy, hardware trigger efficiency and
reconstruction effects, based on a general approach developed by the LHCb
collaboration for tests of lepton universality [33].

##  3 Reconstruction and selection

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ candidates are formed by
combining a ${K}^{*0}$ candidate with a pair of oppositely charged tracks
identified as electrons. For events passing the trigger, ${K}^{*0}$ candidates
are formed by combining a pair of charged tracks identified as ${K}^{+}$ and
${\pi}^{-}$ mesons. Each track is required to be of good quality and to be
inconsistent with originating from a PV. Kaons and pions are required to have
transverse momenta larger than 250 MeV and are identified using information
from the RICH detectors. Electrons with $p_{\mathrm{T}}$ exceeding 500 MeV and
with a good-quality vertex are used to form dielectron candidates. The
reconstructed invariant mass of the ${{K}^{+}}{{\pi}^{-}}$ system is required
to be within $100\text{\,Me\kern-1.00006ptV}$ of the mass of the ${{K}^{*0}}$
meson [58].

The tracks from the electrons, kaon and pion are required to form a good-
quality vertex that is significantly displaced from any PV. In events with
multiple PVs, the one with the smallest value of $\chi^{2}_{\rm IP}$ is
associated to the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ candidate.
Here $\chi^{2}_{\rm IP}$ is defined as the difference in the vertex-fit
$\chi^{2}$ of a given PV reconstructed with and without the tracks forming the
candidate under consideration. In addition, the angle between the
${B}^{0}$-candidate momentum vector and the vector between the associated PV
and the ${B}^{0}$ decay vertex is required to be small.

Electrons can lose a significant amount of their energy when interacting with
the detector material due to emission of bremsstrahlung photons. A dedicated
procedure, which searches for neutral energy deposits in the ECAL that are
compatible with being emitted by the electron upstream of the magnet, is
applied to correct for this effect [33]. The limitations of this recovery
technique degrade the resolution of the reconstructed invariant masses of both
the dielectron pair and the ${B}^{0}$ candidate.

##  4 Analysis strategy

The $q^{2}$ region under study is chosen to maximise the sensitivity to
$b\rightarrow s\gamma$ contributions (${C}_{7}^{(\prime)}$). First of all, the
reconstructed invariant-mass squared resolution of the dielectron pair is
improved by a kinematic fit that constrains the
${{K}^{+}}{{\pi}^{-}}{e^{+}}{e^{-}}$ mass to the known ${B}^{0}$ mass [58].
The reconstructed $q^{2}$ is required to be lower than
$0.25\text{\,Ge\kern-1.00006ptV}^{2}$ to minimise the sensitivity to vector
and axial-vector currents (${C}_{9}^{(\prime)}$ and ${C}_{10}^{(\prime)}$)
while retaining as many signal candidates as possible. The larger data set of
this analysis makes it possible to significantly reduce this upper bound with
respect to Ref. [21], where it was 1$\text{\,Ge\kern-1.00006ptV}^{2}$.
Low-$q^{2}$ signal candidates are most sensitive to ${C}_{7}^{(\prime)}$, but
suffer from a degradation of the resolution in $\tilde{\phi}$ due to multiple
scattering of the quasi-collinear electrons in the tracking detectors.
Furthermore, ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ decays followed by a
photon conversion in the material of the detector contaminate the lower end of
the $q^{2}$ spectrum. The reconstructed $q^{2}$ is thus required to exceed
$10^{-4}\text{\,Ge\kern-1.00006ptV}^{2}$, resulting in a $\tilde{\phi}$
resolution of $0.11\text{\,rad}$ and a
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ background fraction of about $2\%$
(see Sec. 5).

The signal selection efficiency as a function of the dielectron mass, obtained
from simulation, is presented in Fig. 1. The efficiency is approximately
uniform across the signal region. Close to the boundaries, the efficiency
drops due to the selected range of reconstructed $q^{2}$ and the effect of the
dielectron mass resolution. Therefore, following Ref. [21], effective $q^{2}$
boundaries are defined between 0.0008$\text{\,Ge\kern-1.00006ptV}^{2}$ and
0.257$\text{\,Ge\kern-1.00006ptV}^{2}$ to allow for theoretical predictions of
the angular observables without input from LHCb simulation. Using the Flavio
software package, it was checked that predictions with both SM and BSM values
for the Wilson coefficients calculated in this effective $q^{2}$ range (grey
line in Fig. 1) agree very well with those calculated taking into account a
complete description of the $q^{2}$ efficiency using LHCb simulation (points
in Fig. 1).

Figure 1: Relative efficiency as a function of the dielectron invariant mass
($q$). The points represent the efficiency obtained from simulation, while the
vertical lines represent the effective $q^{2}$ boundary defined in the text.

The region of reconstructed $q^{2}$ below
$10^{-4}\text{\,Ge\kern-1.00006ptV}^{2}$ is enriched in
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ decays and is used as a control
sample. Its kinematics and background level are very similar to the signal
$q^{2}$ region, but with much larger candidate yields.

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ branching ratio is
expected to be as small as $(2.0\pm 0.4)\times 10^{-7}$ in the $q^{2}$ range
studied in this paper [46]. Nonetheless, a pure signal sample is obtained by
greatly reducing all expected background contributions with the selection
described in Sec. 5. A fit to the reconstructed ${{K}^{*0}}{e^{+}e^{-}}$
invariant mass, $m(K^{+}\pi^{-}{e^{+}e^{-}})$, in a wide range between 4500
and 6200 MeV is used to estimate the remaining background contamination, as
explained in Sec. 6. Afterwards, a four-dimensional fit to the
${{K}^{+}}{{\pi}^{-}}{e^{+}e^{-}}$ invariant mass and the three angles
$\cos{\theta_{\ell}}$, $\cos{\theta_{K}}$ and $\tilde{\phi}$ is used to
measure the four angular observables $F_{\mathrm{L}}$,
$A_{\mathrm{T}}^{\mathrm{Re}}$, $A_{\mathrm{T}}^{(2)}$ and
$A_{\mathrm{T}}^{\mathrm{Im}}$. This fit is performed in a reduced
$m(K^{+}\pi^{-}{e^{+}e^{-}})$ window between 5000 and 5400 MeV in order to
simplify the angular modelling of the background components. The background
fractions are constrained to those obtained in the wider
$m(K^{+}\pi^{-}{e^{+}e^{-}})$ window. Signal and background angular shapes are
determined using simulation and data samples (Sec. 7) and then fitted to the
signal sample (Sec. 8).

The $m(K^{+}\pi^{-}{e^{+}e^{-}})$ mass resolution, the angular acceptance and
the background rates depend on how the event has been triggered at the
hardware level. The data sample is therefore divided into two categories:
events for which at least one of the two electron candidates fires the
electron trigger, and events triggered by activity in the event that is not
associated with any of the signal decay particles. Furthermore, the data sets
collected in 2011–12 (Run 1) and 2015–18 (Run 2) are treated separately to
account for kinematic differences due to the different $pp$ collision
energies.

##  5 Background studies

Several sources of specific background are considered, with studies performed
using samples of simulated events unless stated otherwise. All of the
identified background sources that are expected to contribute at a level of
more than $1\%$ of the signal are modelled and included in the analysis.

A large background comes from the semileptonic
${{B}^{0}}\\!\rightarrow{{D}^{-}}{e^{+}}\nu$ decay, with
${{{D}^{-}}\rightarrow{{K}^{*0}}{e^{-}}{\overline{\nu}}}$. This contribution
populates the region below the ${B}^{0}$ mass and has a combined branching
fraction four orders of magnitude larger than that of the signal. In the case
where both neutrinos have low energies, the signal selection is ineffective at
rejecting it. The positron from the ${B}^{0}$ decay tends to be more energetic
than the electron from the ${D}^{-}$ decay and hence the reconstructed
$\cos{\theta_{\ell}}$ distribution favours large values since
$\cos{\theta_{\ell}}$ is highly correlated with the $e^{+}e^{-}$ energy
asymmetry. In order to avoid any potential bias in the measurement of
$A_{\mathrm{T}}^{\mathrm{Re}}$, a symmetric requirement of
$|\cos{\theta_{\ell}}|<0.8$ is applied, resulting in a $5\%$ loss of signal
while rejecting $98\%$ of this semileptonic background.

The radiative decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ has a
branching fraction about two orders of magnitude larger than that of the
signal and has a very similar distribution in the reconstructed
${{K}^{+}}{{\pi}^{-}}{e^{+}}{e^{-}}$ mass. In the signal sample, contamination
from this background is at the level of $23\%$, but is reduced to about $2\%$
by rejecting dielectron pairs compatible with originating from detector
material [59]. A specific weighting procedure is applied to the
${{B}^{0}}\rightarrow{{K}^{*0}}{\gamma}$ simulation to match the true
$e^{+}e^{-}$ mass distribution of Ref. [60] since the Geant4 version used here
does not accurately model high-mass $e^{+}e^{-}$ pair production.

The ${{B}^{0}}\rightarrow{{K}^{*0}}\eta$ and
${{B}^{0}}\rightarrow{{K}^{*0}}{{\pi}^{0}}$ decays where the $\eta$ or
${{\pi}^{0}}$ meson decays to ${e^{+}}{e^{-}}{\gamma}$ (Dalitz decay) can pass
the selection if the photon is very soft, or if it is recovered as a
bremsstrahlung photon. In the latter case, the $m(K^{+}\pi^{-}{e^{+}e^{-}})$
mass peaks at the ${B}^{0}$ mass. The contamination from the $\eta$
(${{\pi}^{0}}$) Dalitz decay is estimated to be at the level of 4% (2%) in the
mass region used in the angular fit.

To suppress background from ${{B}^{0}_{s}}\\!\rightarrow\phi{e^{+}}{e^{-}}$
decays, where the $\phi$ meson decays to a ${{K}^{+}}{{K}^{-}}$ pair and one
of the kaons is misidentified as a pion, the two-hadron invariant mass
computed under the $K^{+}K^{-}$ hypothesis is required to be larger than 1040
MeV. Background contributions from misidentified ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow pK^{-}{e^{+}}{e^{-}}$,
${{B}^{0}}\rightarrow{{K}^{*0}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{e^{+}e^{-}}$ decays are found to
be negligible.

Partially reconstructed (PR) background contributions arising from
$B\\!\rightarrow{{K}^{*0}}\pi{e^{+}e^{-}}$ decays, where one of the pions is
not reconstructed, are suppressed by exploiting the kinematic balance of the
decay. The ratio of the ${K}^{*0}$ and dielectron momenta components
transverse to the ${B}^{0}$ direction is expected to be unity unless some
energy is lost through bremsstrahlung emission. Since at low $q^{2}$
bremsstrahlung photons do not significantly modify the dielectron direction,
this ratio can be used to recover the lost energy and recompute the corrected
reconstructed ${B}^{0}$ mass called $m_{\rm HOP}$ [33]. In the case of PR
background, however, the missing particles are not necessarily emitted in the
same direction as either electron. Therefore the requirement $m_{\rm
HOP}>4900\text{\,Me\kern-1.00006ptV}$ rejects about $70\%$ of the PR
background with a $90\%$ signal efficiency, estimated from simulation. This
background yield is found to be 5% of that of the signal in the narrow mass
window.

In order to reduce the level of combinatorial background, a multivariate
classifier based on a boosted decision tree algorithm (BDT) [61, 62] is used.
The BDT classifier is trained to separate simulated
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ events from background events
taken from the upper invariant-mass sideband
($m(K^{+}\pi^{-}{e^{+}e^{-}})>5600\text{\,Me\kern-1.00006ptV}$) in data and
uses eight kinematic and decay topology variables including the $\chi^{2}_{\rm
IP}$ of final-state particles with respect to the associated PV and the
$p_{\mathrm{T}}$ of the ${B}^{0}$ candidate and its flight distance from the
PV. The classifier achieves a background rejection of $90\%$ and a signal
efficiency of $94\%$. The semileptonic and combinatorial background sources
contribute a contamination of about 7% and are therefore modelled in the fit
to the data.

##  6 ${{K}^{+}}{{\pi}^{-}}{e^{+}e^{-}}$ invariant-mass spectra

An unbinned maximum-likelihood fit to the ${{K}^{+}}{{\pi}^{-}}{e^{+}e^{-}}$
invariant-mass distribution is performed simultaneously to the signal and
control samples in order to measure the background fractions. In both samples,
the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ components are described by a
bifurcated Crystal Ball (CB) function [63], which consists of a Gaussian core
with asymmetric power-law tails. The shapes of the ${{K}^{*0}}\eta$ and
${{K}^{*0}}{{\pi}^{0}}$ background contributions are modelled by non-
parametric probability density functions (PDFs) [64], while the shape of the
PR background is modelled by the sum of a CB function and a Gaussian function.
Finally, the shapes of the semileptonic and combinatorial background (SL/C)
are parametrised together by an exponential function. All shapes apart from
the SL/C are fixed from simulation. The widths and mean values of the signal
CB functions are corrected for differences between data and simulation using a
high-purity data sample of
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}({e^{+}e^{-}}){{K}^{*0}}$
candidates.

Since the $b\rightarrow s\gamma$ contribution dominates both the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ and $B\rightarrow{{K}^{*0}}\pi
e^{+}e^{-}$ decay rates in the $q^{2}$ region considered, the PR background is
expected to be similar for the signal and control samples. The ratio of PR
background and signal yields is therefore shared between the two samples.
Using the fit in the wider $m(K^{+}\pi^{-}{e^{+}e^{-}})$ mass window, the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ yield in the restricted
$m(K^{+}\pi^{-}{e^{+}e^{-}})$ range used for the angular fit is estimated to
be 450. In the control sample, the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$
yield is about 2950, while in both samples the signal-to-background ratio is
about 5. The invariant-mass distributions together with the PDFs resulting
from the fit are shown in Figs. 2 and 3 for the control and signal samples,
respectively.

Figure 2: Distributions of the (left) ${{K}^{+}}{{\pi}^{-}}{e^{+}e^{-}}$
invariant mass and (right) $\cos{\theta_{K}}$ of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ candidates. The black points
represent the data, while the solid blue curve shows the total PDF. The signal
component is represented by the dashed pink line and the shaded areas are the
background components, as detailed in the legend. The SL/C component is
composed of semileptonic and combinatorial background contributions. The
dashed vertical lines indicate the restricted mass range used in the angular
analysis.

Figure 3: Distributions of the (top left) ${{K}^{+}}{{\pi}^{-}}{e^{+}e^{-}}$
invariant mass, (top right) $\cos{\theta_{\ell}}$, (bottom left)
$\cos{\theta_{K}}$ and (bottom right) $\tilde{\phi}$ variables of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ candidates in the
reconstructed $q^{2}$ range between $10^{-4}\text{\,Ge\kern-0.90005ptV}^{2}$
and 0.25$\text{\,Ge\kern-0.90005ptV}^{2}$. The black points represent the
data, while the solid blue curve shows the total PDF. The signal component is
represented by the dashed red line and the shaded areas are the background
components, as detailed in the legend. The SL/C component is composed of
semileptonic and combinatorial background contributions. The dashed vertical
lines indicate the restricted mass range used in the angular analysis.

##  7 Angular modelling

The ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ angular distribution
described using Eq. 1 is multiplied by an acceptance function evaluated from
simulated ${{B}^{0}}\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ decays to take into
account the effect of the reconstruction and selection efficiency. The
acceptance function, $\varepsilon$, factorises between the angles such that

$\varepsilon(\cos{\theta_{\ell}},\cos{\theta_{K}},\tilde{\phi})\simeq\varepsilon(\cos{\theta_{\ell}})\varepsilon(\cos{\theta_{K}})\varepsilon(\tilde{\phi}).$
(4)

The $\cos{\theta_{K}}$ and $\cos{\theta_{\ell}}$ acceptance functions are
modelled with fourth-order Legendre polynomials. For the $\tilde{\phi}$ angle,
non-uniform acceptance terms proportional to $\cos(2\tilde{\phi})$ and
$\sin(2\tilde{\phi})$ are allowed for completeness, however, no significant
deviation from a uniform distribution is observed.

Since for ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ decays the presence of
the electrons is only due to the interaction of the real photon with the
detector material, the $\cos{\theta_{\ell}}$ and $\tilde{\phi}$ dependent
parts of Eq. 1 are integrated out to model purely the $\cos{\theta_{K}}$
dependent part of the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ angular
shape, which depends only on the parameter $F_{\mathrm{L}}$. The value of
$F_{\mathrm{L}}$ for the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ decay is
obtained from the fit to the control sample detailed in Sec. 8. When included
as a background in the fit to the signal sample, the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ angular shape is obtained from the
simulation sample and is assumed to factorise between the angles.

The background contributions due to ${{B}^{0}}\rightarrow{{K}^{*0}}\eta$ and
${{B}^{0}}\rightarrow{{K}^{*0}}{{\pi}^{0}}$ decays that contribute to the
signal and control samples are modelled using simulation and are also assumed
to factorise in the three angles. Since this background has $F_{\mathrm{L}}=1$
due to the ${{\pi}^{0}}$ or $\eta$ angular momentum, the $\cos{\theta_{K}}$
distribution has a very different shape compared to
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ decays. Its precise modelling
is validated by the measurement of the $F_{\mathrm{L}}$ parameter in the
control channel reported in Sec. 8.

The angular shape of the PR background is modelled using the same functional
shape as the signal, determined from ${{B}^{+}}\rightarrow
K_{1}(1270){e^{+}}{e^{-}}$ simulated events, where one of the pions from the
$K_{1}(1270)\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decay is not
reconstructed.

A sample of ${{B}^{0}}\rightarrow{{K}^{*0}}{e^{+}}{\mu^{-}}$ candidates from
LHCb data is used to determine the SL/C angular shapes. Since this decay is
forbidden in the SM due to lepton flavour conservation, this sample will
mostly comprise semileptonic and combinatorial background events. The $q^{2}$
and BDT requirements are slightly relaxed to increase the sample size. It is
checked that this sample is a good proxy for SL/C background by assigning the
muon candidate an electron mass and comparing the resulting angular and
${{K}^{+}}{{\pi}^{-}}{e^{+}e^{-}}$ invariant-mass distributions to those of
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ decays in the upper mass
sideband and low BDT output regions. The angular shapes of the three angles
are found to factorise in this sample and therefore are modelled separately.

##  8 Angular observables

To determine the four angular observables, $F_{\mathrm{L}}$,
$A_{\mathrm{T}}^{(2)}$, $A_{\mathrm{T}}^{\mathrm{Im}}$ and
$A_{\mathrm{T}}^{\mathrm{Re}}$, an unbinned maximum likelihood fit is
performed to the $m(K^{+}\pi^{-}{e^{+}e^{-}})$, $\cos{\theta_{\ell}}$,
$\cos{\theta_{K}}$ and $\tilde{\phi}$ distributions in a restricted
$m(K^{+}\pi^{-}{e^{+}e^{-}})$ window between 5000 and 5400 MeV. The inclusion
of $m(K^{+}\pi^{-}{e^{+}e^{-}})$ improves the statistical power of the fit
since, even within the restricted window, the mass shapes of the signal and of
the background contributions are very different. The fractions of the fit
components are constrained using a multivariate Gaussian function to the
results in the wide mass range extrapolated to the narrow mass range.
Pseudoexperiments are used to assess the impact of fitting the
$m(K^{+}\pi^{-}{e^{+}e^{-}})$ distribution again in the restricted range. The
resulting bias is found to be negligible.

The fitting procedure is verified using a large sample of fully simulated
events, with the obtained values of $F_{\mathrm{L}}$, $A_{\mathrm{T}}^{(2)}$,
$A_{\mathrm{T}}^{\mathrm{Im}}$ and $A_{\mathrm{T}}^{\mathrm{Re}}$ in excellent
agreement with the inputs. The
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ fit is then validated on
data by performing a similar fit to the $m(K^{+}\pi^{-}{e^{+}e^{-}})$ and
$\cos{\theta_{K}}$ distributions of the control sample. The $\cos{\theta_{K}}$
distribution for the control sample, together with the PDF projections
resulting from the fit, are shown in Fig. 2. The fitted value of
$F_{\mathrm{L}}=0.0^{+0.7}_{-0.0}\,\%$ is compatible with a completely
transverse ${K}^{*0}$ polarisation, as expected due to presence of the real
photon. Finally, the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ fit is
further validated using $10\,000$ pseudoexperiments including signal and
background components. Several input values for the angular observables,
$F_{\mathrm{L}}$, $A_{\mathrm{T}}^{(2)}$, $A_{\mathrm{T}}^{\mathrm{Im}}$ and
$A_{\mathrm{T}}^{\mathrm{Re}}$, are studied including those associated with
BSM models, and the results are in good agreement with the inputs, with the
exception of $F_{\mathrm{L}}$, where a small bias at the level of $7\%$ of its
statistical uncertainty is observed and corrected for. Furthermore, the non-
negligible size of the $\tilde{\phi}$ resolution results in an underestimation
of the magnitude of $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$
by $4\%$. Although this shift is negligibly small for the magnitudes expected
in the SM, it could be sizeable for large ${C}_{7}^{\prime}$ values and
therefore the $A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ values
are corrected for this effect. The angular distributions,
$\cos{\theta_{\ell}}$, $\cos{\theta_{K}}$ and $\tilde{\phi}$, for the signal
region, together with the PDF projections resulting from the fit, are shown in
Fig. 3. Results for the angular observables are given in Sec. 10.

##  9 Systematic uncertainties

Table 1: Summary of systematic uncertainties on the four angular observables, $A_{\mathrm{T}}^{(2)}$, $A_{\mathrm{T}}^{\mathrm{Im}}$, $A_{\mathrm{T}}^{\mathrm{Re}}$ and $F_{\mathrm{L}}$. The total systematic uncertainty is the sum in quadrature of all the contributions. For comparison, the statistical uncertainties are shown in the last row of the table. Source of systematic | $A_{\mathrm{T}}^{(2)}$ | $A_{\mathrm{T}}^{\mathrm{Im}}$ | $A_{\mathrm{T}}^{\mathrm{Re}}$ | $F_{\mathrm{L}}$
---|---|---|---|---
Simulation sample size for acceptance | $0.007$ | $0.007$ | $0.007$ | $0.003$
Acceptance function modelling | $0.004$ | $0.001$ | $0.008$ | $0.001$
${{B}^{0}}\rightarrow{{K}^{*0}}{e^{+}}{\mu^{-}}$ sample size for SL/C | $0.007$ | $0.007$ | $0.007$ | $0.003$
SL/C angular modelling | $0.012$ | $0.005$ | $0.006$ | $0.005$
PR model other than $K_{1}(1270)$ | $0.001$ | $0.003$ | $0.002$ | $0.001$
$\eta$ or ${{\pi}^{0}}$ angular modelling | $<0.001$ | $<0.001$ | $0.002$ | $0.010$
Corrections to simulation | $0.003$ | $0.001$ | $0.003$ | $0.007$
Signal mass shape | $0.002$ | $0.002$ | $0.004$ | $0.001$
Total systematic uncertainty | $0.017$ | $0.012$ | $0.015$ | $0.014$
Statistical uncertainty | $0.103$ | $0.102$ | $0.077$ | $0.026$

To evaluate systematic uncertainties resulting from the limited size of the
data and simulation samples used to determine the angular shapes and
acceptances, a bootstrapping technique is used [65]. In addition, systematic
uncertainties related to various modelling choices used in the fits are
evaluated by generating pseudoexperiments with an alternative model and
fitting with the nominal model used to fit the data. The results of the mass
and angular fits are then compared with the input values to assess the size of
the uncertainties. The alternative modelling choices considered are detailed
in the following.

The systematic uncertainties related to the corrections applied to simulated
events used to model the angular acceptance are evaluated by fitting
uncorrected simulated events. An alternative model using Legendre polynomials
of order six instead of four is used to estimate the systematic uncertainties
related to the shape of the acceptance function.

To take into account possible variations in the angular shapes of the PR
background due to states other than the $K_{1}(1270)$ meson, alternative
shapes are determined from a sample of
${{B}^{+}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{e^{+}}{e^{-}}$ simulated
events. This sample is reweighted in the ${{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$
Dalitz plane to match the distribution in
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{e^{+}}{e^{-}})K^{\rm
res}(\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}})$ data, where $K^{\rm res}$ is
any kaon resonance that decays to ${{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$.
Alternative models for the SL/C background are obtained by tightening either
the $q^{2}$ or BDT requirements used in the
${{B}^{0}}\rightarrow{{K}^{*0}}{e^{+}}{\mu^{-}}$ selection.

To estimate the systematic uncertainty due to differences between data and
simulation in the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ mass shapes,
the signal mass PDF is corrected using a fit to the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ rather than the
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}({e^{+}e^{-}}){{K}^{*0}}$
channel. The systematic uncertainties are summarised in Table 1. The total
systematic uncertainty, obtained by adding all individual sources in
quadrature, is smaller than the statistical uncertainty for all observables.

##  10 Results

The four ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ angular observables
measured in the effective $q^{2}$ range from $0.0008$ to
$0.257$$\text{\,Ge\kern-1.00006ptV}^{2}$ are found to be

$\displaystyle F_{\mathrm{L}}$ $\displaystyle=$ $\displaystyle 0.044\pm
0.026\pm 0.014,$ $\displaystyle A_{\mathrm{T}}^{\mathrm{Re}}$ $\displaystyle=$
$\displaystyle-0.06\pm 0.08\pm 0.02,$ $\displaystyle A_{\mathrm{T}}^{(2)}$
$\displaystyle=$ $\displaystyle+0.11\pm 0.10\pm 0.02,$ $\displaystyle
A_{\mathrm{T}}^{\mathrm{Im}}$ $\displaystyle=$ $\displaystyle+0.02\pm 0.10\pm
0.01,$

where the first contribution to the uncertainty is statistical and the second
systematic. Correlations between the observables are measured to be

| $F_{\rm L}$ | $A_{\mathrm{T}}^{\mathrm{Re}}$ | $A_{\mathrm{T}}^{(2)}$ | $A_{\mathrm{T}}^{\mathrm{Im}}$
---|---|---|---|---
$F_{\rm L}$ | 1.00 | $-0.02$ | $-0.01$ | 0.02
$A_{\mathrm{T}}^{\mathrm{Re}}$ |  | 1.00 | 0.05 | 0.02
$A_{\mathrm{T}}^{(2)}$ |  |  | 1.00 | 0.10
$A_{\mathrm{T}}^{\mathrm{Im}}$ |  |  |  | 1.00

These results supersede Ref. [21]. Using Eq. 2, the measured
$A_{\mathrm{T}}^{(2)}$ and $A_{\mathrm{T}}^{\mathrm{Im}}$ observables are used
to determine the photon polarisation in
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ decays

$\displaystyle{\rm Re}\left(A_{\rm R}/A_{\rm L}\right)$ $\displaystyle=$
$\displaystyle 0.05\pm 0.05$ $\displaystyle{\rm Im}\left(A_{\rm R}/A_{\rm
L}\right)$ $\displaystyle=$ $\displaystyle 0.01\pm 0.05.$

Furthermore, using the Flavio software package [46], these measurements can be
used to determine the polarisation of the $b\rightarrow s\gamma$ transition,
which can be expressed as the ratio of the right- and left-handed
$C_{7}^{(\prime)}$ Wilson coefficients. Details about the calculations of
hadronic contributions can be found in Ref. [10]. The obtained constraints are
shown in Fig. 4, where they are compared to those from previous measurements
by the Belle, BaBar and LHCb experiments [11, 12, 13, 14, 15, 16, 17, 18].
Here, the $C_{7}^{(\prime)}$ regularisation-scheme independent effective
coefficients are calculated at the scale $\mu=4.8\text{\,Ge\kern-1.00006ptV}$
[10]. The value of the left-handed $C_{7}$ coefficient is fixed to its SM
value, $C_{7}^{\rm SM}=-0.2915$. Theoretical uncertainties related to the
predictions of the experimental observables are taken into account in the
constrained areas. The results presented in this paper provide the world’s
best constraint on the $b\rightarrow s\gamma$ photon polarisation.

Figure 4: Constraints at $2\sigma$ level on the real and imaginary parts of
the ratio of right- and left-handed Wilson coefficients, $C_{7}^{\prime}$ and
$C_{7}$. The $C_{7}$ coefficient is fixed to its SM value. The measurements of
the inclusive branching fraction, ${\cal B}(B\rightarrow X_{s}\gamma)$, and
the ${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{0}}\gamma$ mixing-
induced $C\\!P$ asymmetry by the Belle and BaBar experiments [11, 12, 13, 14,
15, 16, 17] are shown in blue and yellow, respectively, the
${{B}^{0}_{s}}\rightarrow\phi{\gamma}$ measurements at LHCb [18] in purple and
the measurement presented in this paper in red. The global fit is shown in
dashed lines and the SM prediction is represented by a black star and
corresponds to the ratio of $s$\- and $b$-quark masses.

##  11 Conclusion

An angular analysis of the ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$
decay is performed using proton-proton collision data, corresponding to an
integrated luminosity of 9.0$\text{\,fb}^{-1}$, collected by the LHCb
experiment between 2011 and 2018. Angular observables are measured for the
first time in the $q^{2}$ range from $0.0008$ to
$0.257$$\text{\,Ge\kern-1.00006ptV}^{2}$.

The results are consistent with SM predictions [43, 24, 46] and are used to
measure both the real and imaginary parts of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\gamma}$ photon polarisation with a
precision of 5%. Furthermore, the results of this paper make it possible to
constrain the $b\rightarrow s\gamma$ photon polarisation with significantly
better precision than the combination of previous measurements.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, Labex P2IO
and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of
Frontier Sciences of CAS, CAS PIFI, Thousand Talents Program, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Royal Society and the Leverhulme Trust (United
Kingdom).

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala84, S. Aiola25, Z. Ajaltouni9, S. Akar64, J.
Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44, Z. Aliouche61,
G. Alkhazov37, P. Alvarez Cartelle47, S. Amato2, Y. Amhis11, L. An21, L.
Anderlini21, A. Andreianov37, M. Andreotti20, F. Archilli16, A. Artamonov43,
M. Artuso67, K. Arzymatov41, E. Aslanides10, M. Atzeni49, B. Audurier11, S.
Bachmann16, M. Bachmayer48, J.J. Back55, S. Baker60, P. Baladron Rodriguez45,
V. Balagura11, W. Baldini20, J. Baptista Leite1, R.J. Barlow61, S. Barsuk11,
W. Barter60, M. Bartolini23,i, F. Baryshnikov80, J.M. Basels13, G. Bassi28, B.
Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1, A.
Beiter67, V. Belavin41, S. Belin26, V. Bellee48, K. Belous43, I. Belov39, I.
Belyaev38, G. Bencivenni22, E. Ben-Haim12, A. Berezhnoy39, R. Bernet49, D.
Berninghoff16, H.C. Bernstein67, C. Bertella47, E. Bertholet12, A. Bertolin27,
C. Betancourt49, F. Betti19,e, M.O. Bettler54, Ia. Bezshyiko49, S. Bhasin53,
J. Bhom33, L. Bian72, M.S. Bieker14, S. Bifani52, P. Billoir12, M. Birch60,
F.C.R. Bishop54, A. Bizzeti21,s, M. Bjørn62, M.P. Blago47, T. Blake55, F.
Blanc48, S. Blusk67, D. Bobulska58, V. Bocci30, J.A. Boelhauve14, O. Boente
Garcia45, T. Boettcher63, A. Boldyrev81, A. Bondar42,v, N. Bondar37, S.
Borghi61, M. Borisyak41, M. Borsato16, J.T. Borsuk33, S.A. Bouchiba48, T.J.V.
Bowcock59, A. Boyer47, C. Bozzi20, M.J. Bradley60, J.T. Brandt16, S. Braun65,
A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A. Brossa Gonzalo55, D.
Brundu26, A. Buonaura49, C. Burr47, A. Bursche26, A. Butkevich40, J.S.
Butter31, J. Buytaert47, W. Byczynski47, S. Cadeddu26, H. Cai72, R.
Calabrese20,g, L. Calefice14, L. Calero Diaz22, S. Cali22, R. Calladine52, M.
Calvi24,j, M. Calvo Gomez83, P. Camargo Magalhaes53, A. Camboni44, P.
Campana22, D.H. Campora Perez47, A.F. Campoverde Quezada5, S. Capelli24,j, L.
Capriotti19,e, A. Carbone19,e, G. Carboni29, R. Cardinale23,i, A. Cardini26,
I. Carli6, P. Carniti24,j, K. Carvalho Akiba31, A. Casais Vidal45, G. Casse59,
M. Cattaneo47, G. Cavallero47, S. Celani48, J. Cerasoli10, A.J. Chadwick59,
M.G. Chapman53, M. Charles12, Ph. Charpentier47, G. Chatzikonstantinidis52,
C.A. Chavez Barajas59, M. Chefdeville8, C. Chen3, S. Chen26, A. Chernov33,
S.-G. Chitic47, V. Chobanova45, S. Cholak48, M. Chrzaszcz33, A. Chubykin37, V.
Chulikov37, P. Ciambrone22, M.F. Cicala55, X. Cid Vidal45, G. Ciezarek47,
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de Vries78, C.T. Dean66, W. Dean84, D. Decamp8, L. Del Buono12, B. Delaney54,
H.-P. Dembinski14, A. Dendek34, V. Denysenko49, D. Derkach81, O. Deschamps9,
F. Desse11, F. Dettori26,f, B. Dey72, P. Di Nezza22, S. Didenko80, L. Dieste
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D. Dutta61, M. Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede68,
V. Egorychev38, S. Eidelman42,v, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S.
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J. Hu16, J. Hu71, W. Hu7, W. Huang5, X. Huang72, W. Hulsbergen31, R.J.
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D. Ilin37, P. Ilten52, A. Inglessi37, A. Ishteev80, K. Ivshin37, R.
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Jones55, B. Jost47, N. Jurik47, S. Kandybei50, Y. Kang3, M. Karacson47, J.M.
Kariuki53, N. Kazeev81, M. Kecke16, F. Keizer54,47, M. Kenzie55, T. Ketel32,
B. Khanji47, A. Kharisova82, S. Kholodenko43, K.E. Kim67, T. Kirn13, V.S.
Kirsebom48, O. Kitouni63, S. Klaver31, K. Klimaszewski35, S. Koliiev51, A.
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Liang67, T. Lin60, R. Lindner47, V. Lisovskyi14, R. Litvinov26, G. Liu71, H.
Liu5, S. Liu6, X. Liu3, A. Loi26, J. Lomba Castro45, I. Longstaff58, J.H.
Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,m, S. Luchuk40, M.
Lucio Martinez31, V. Lukashenko31, Y. Luo3, A. Lupato61, E. Luppi20,g, O.
Lupton55, A. Lusiani28,r, X. Lyu5, L. Ma6, S. Maccolini19,e, F. Machefert11,
F. Maciuc36, V. Macko48, P. Mackowiak14, S. Maddrell-Mander53, O. Madejczyk34,
L.R. Madhan Mohan53, O. Maev37, A. Maevskiy81, D. Maisuzenko37, M.W.
Majewski34, S. Malde62, B. Malecki47, A. Malinin79, T. Maltsev42,v, H.
Malygina16, G. Manca26,f, G. Mancinelli10, R. Manera Escalero44, D.
Manuzzi19,e, D. Marangotto25,o, J. Maratas9,u, J.F. Marchand8, U. Marconi19,
S. Mariani21,47,h, C. Marin Benito11, M. Marinangeli48, P. Marino48, J.
Marks16, P.J. Marshall59, G. Martellotti30, L. Martinazzoli47, M.
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Materok13, R. Matev47, A. Mathad49, Z. Mathe47, V. Matiunin38, C. Matteuzzi24,
K.R. Mattioli84, A. Mauri31, E. Maurice11,b, J. Mauricio44, M. Mazurek35, M.
McCann60, L. Mcconnell17, T.H. Mcgrath61, A. McNab61, R. McNulty17, J.V.
Mead59, B. Meadows64, C. Meaux10, G. Meier14, N. Meinert75, D. Melnychuk35, S.
Meloni24,j, M. Merk31,78, A. Merli25, L. Meyer Garcia2, M. Mikhasenko47, D.A.
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Mitchell57, B. Mitreska61, D.S. Mitzel47, A. Mödden14, R.A. Mohammed62, R.D.
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D. Vom Bruch12, A. Vorobyev37, V. Vorobyev42,v, N. Voropaev37, R. Waldi75, J.
Walsh28, C. Wang16, J. Wang3, J. Wang72, J. Wang4, J. Wang6, M. Wang3, R.
Wang53, Y. Wang7, Z. Wang49, D.R. Ward54, H.M. Wark59, N.K. Watson52, S.G.
Weber12, D. Websdale60, C. Weisser63, B.D.C. Westhenry53, D.J. White61, M.
Whitehead53, D. Wiedner14, G. Wilkinson62, M. Wilkinson67, I. Williams54, M.
Williams63,68, M.R.J. Williams57, F.F. Wilson56, W. Wislicki35, M. Witek33, L.
Witola16, G. Wormser11, S.A. Wotton54, H. Wu67, K. Wyllie47, Z. Xiang5, D.
Xiao7, Y. Xie7, H. Xing71, A. Xu4, J. Xu5, L. Xu3, M. Xu7, Q. Xu5, Z. Xu5, Z.
Xu4, D. Yang3, Y. Yang5, Z. Yang3, Z. Yang65, Y. Yao67, L.E. Yeomans59, H.
Yin7, J. Yu70, X. Yuan67, O. Yushchenko43, K.A. Zarebski52, M. Zavertyaev15,c,
M. Zdybal33, O. Zenaiev47, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, Y.
Zhang47, Y. Zhang62, A. Zhelezov16, Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V.
Zhukov13,39, J.B. Zonneveld57, S. Zucchelli19,e, D. Zuliani27, G. Zunica61.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Ijclab, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
69Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
70Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 16
75Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
76INFN Sezione di Perugia, Perugia, Italy, associated to 20
77Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
78Universiteit Maastricht, Maastricht, Netherlands, associated to 31
79National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
80National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
81National Research University Higher School of Economics, Moscow, Russia,
associated to 41
82National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
83DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 44
84University of Michigan, Ann Arbor, United States, associated to 67

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Firenze, Firenze, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mUniversità di Padova, Padova, Italy
nUniversità di Pisa, Pisa, Italy
oUniversità degli Studi di Milano, Milano, Italy
pUniversità di Urbino, Urbino, Italy
qUniversità della Basilicata, Potenza, Italy
rScuola Normale Superiore, Pisa, Italy
sUniversità di Modena e Reggio Emilia, Modena, Italy
tUniversità di Siena, Siena, Italy
uMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
vNovosibirsk State University, Novosibirsk, Russia
wINFN Sezione di Trieste, Trieste, Italy