Nikhef-2022-022

SI-HEP-2022-36

P3H-22-123

Fingerprinting CP-violating New Physics with $B\to K\mu^{+}\mu^{-}$

Robert Fleischer a,b, Eleftheria Malami a,c, Anders Rehult a, and K. Keri Vos
a,d

aNikhef, Science Park 105, NL-1098 XG Amsterdam, Netherlands

bDepartment of Physics and Astronomy, Vrije Universiteit Amsterdam,
NL-1081 HV Amsterdam, Netherlands

cCenter for Particle Physics Siegen (CPPS), Theoretische Physik 1,
Universität Siegen, D-57068 Siegen, Germany

dGravitational Waves and Fundamental Physics (GWFP),
Maastricht University, Duboisdomein 30,
NL-6229 GT Maastricht, the Netherlands

The $B\to K\ell^{+}\ell^{-}$ system is particularly interesting to explore
physics beyond the Standard Model. Such effects could provide new sources of
CP violation, thereby giving rise to Wilson coefficients which could not only
be real, as assumed in most analyses, but also be complex. We propose a new
strategy to extract the complex $C_{9\mu}$, $C_{10\mu}$ from the data,
utilising the complementarity of direct CP violation in $B^{+}\to
K^{+}\mu^{+}\mu^{-}$ and mixing-induced CP violation arising – in addition to
direct CP violation – in $B^{0}_{d}\to K_{S}\mu^{+}\mu^{-}$, as well as the
differential rate in appropriate $q^{2}$ bins. The long-distance effects,
which play a key role to generate the direct CP asymmetries, cannot be
reliably calculated due to their non-perturbative nature. In order to describe
them, we apply a model by the LHCb Collaboration employing experimental data.
Assuming specific scenarios, we demonstrate the fingerprinting of CP-violating
New Physics and the distinction between ambiguities in the model of the long-
distance contributions. Finally, New Physics could couple differently to muons
and electrons, as probed through the lepton flavour universality ratio
$R_{K}$. We discuss these effects in the presence of CP violation and present
a new way to measure the direct CP asymmetries of the $B\to Ke^{+}e^{-}$
channels.

December 2022

##  1 Introduction

The charged and neutral $B\to K\mu^{+}\mu^{-}$ decays are powerful probes to
test the Standard Model of particle physics (SM) and to search for New Physics
(NP). For several years, these decays and similar processes originating from
the same $b\to s\mu^{+}\mu^{-}$ quark-level transition have exhibited
interesting tensions between the SM predictions and experimental data [1, 2,
3, 4, 5].

A multitude of global effective field theory (EFT) analyses have been
performed to determine the short-distance physics that could cause these
anomalies (see e.g. [6, 7, 8, 9, 10, 11, 12, 13]). The EFT framework allows
for model-independent analyses in terms of operators and their short-distance
Wilson coefficients which – in most analyses – are considered to be real.
However, in general, NP could also provide new sources of CP violation, which
are encoded in complex phases of Wilson coefficients. While the usual key
players in the search for new CP violation are non-leptonic $B$ decays [14,
15], CP violation might also enter leptonic [16] and semileptonic rare $B$
decays [17, 18, 19, 20, 21, 22].

For the charged $B^{\pm}\to K^{\pm}\mu^{+}\mu^{-}$ modes, direct CP violation
could arise from the interference of two amplitudes with different CP-
conserving and CP-violating phases. In the decays at hand, the CP-conserving
phases arise through $\bar{c}c$ resonances. In the SM, the CP-violating phases
come from elements of the unitary Cabibbo–Kobayashi–Maskawa (CKM) matrix. For
the decays we study here, the matrix element $|V_{cb}|$ is particularly
relevant. Unfortunately, we are facing discrepancies between inclusive and
exclusive determinations of this quantity, which has implications for SM
calculations of the decay rate. In our analysis, we pay special attention to
these determinations and address them separately, in line with [23].
Neglecting tiny doubly Cabibbo-suppressed terms, the direct CP asymmetry is
zero. Consequently, a nonzero value would be an unambiguous sign of new
sources of CP violation.

A qualitative NP analysis of direct CP violation thus crucially relies on the
description of resonance effects [19]. These non-perturbative, i.e. long-
distance effects are challenging to compute (see e.g.[24, 25, 26, 27, 9]).
Here we use a long-distance model implemented by the LHCb collaboration based
on the Krüger–Seghal approach [28, 29]. Performing a fit to their experimental
data, four sets of fit parameters (branches) were obtained [30].

In order to obtain the full picture, we also consider the neutral
$B_{d}^{0}\to K_{S}\mu^{+}\mu^{-}$ decay. In this channel, in addition to
direct CP violation, there is also mixing-induced CP violation arising from
interference between $\bar{B}^{0}_{d}$–$B^{0}_{d}$ mixing and
$B_{d}^{0},\bar{B}_{d}^{0}$ decays into the $K_{S}\mu^{+}\mu^{-}$ final state.
In contrast to direct CP violation, mixing-induced CP violation does not
require CP-conserving phases. Therefore it is much more robust with respect to
long-distance effects [21].

In this paper, we present a new strategy that exploits the complementary
information provided by the direct and mixing-induced CP asymmetries to
determine the complex phases of the Wilson coefficients $C_{9\mu}$ and
$C_{10\mu}$. We demonstrate that different sources of new CP-violating physics
leave distinct “fingerprints” on the observable space, allowing us to
transparently determine the Wilson coefficients. This is possible even without
making specific assumptions, such as having NP only in $C_{9\mu}$ or assuming
the relation $C_{9\mu}^{\rm NP}=-C_{10\mu}^{\rm NP}$ as is frequently done in
the literature. In order for the strategy to work, we need to discriminate
between the four long-distance branches. We show how we can do this by
considering the CP asymmetries in specific $q^{2}$ bins.

A first constraint on the direct CP asymmetry has been obtained by the LHCb
Collaboration [31]. However, the mixing-induced CP asymmetry in $B_{d}^{0}\to
K_{S}\ell^{+}\ell^{-}$ has not yet been measured. Consequently, we consider
benchmark scenarios to illustrate the procedure.

CP-violating NP may also have different strengths for muons versus electrons,
which can be probed through the lepton-flavour universality ratio $R_{K}$ [32,
33, 34]. When allowing for CP-violating effects, special care has to be taken
by measuring these ratios for the $B^{-}(\bar{B}^{0}_{d})$ and
$B^{+}({B}^{0}_{d})$ modes separately. In this paper, we define several new
observables and find a new way to measure the direct CP asymmetry in $B\to
Ke^{+}e^{-}$.

This paper is organized as follows: In Section 2, we introduce the theoretical
framework and observables, as well as the model for the long-distance effects.
We calculate the SM prediction for the branching ratio of $B^{\pm}\to
K^{\pm}\mu^{+}\mu^{-}$ and, while doing so, address the uncertainty arising
from discrepancies between different determinations of the CKM element
$|V_{cb}|$. Then, in Section 3, we discuss how we can fingerprint NP with
direct and mixing-induced CP asymmetries in $B\to K\mu^{+}\mu^{-}$. We discuss
the interplay of these observables and demonstrate how we can use them to
extract the complex values of $C_{9\mu}$ and $C_{10\mu}$. In Section 4, we
discuss lepton flavour universality. Finally, we conclude in Section 5 .

##  2 Theoretical framework

###  2.1 Effective Hamiltonian

The low-energy effective Hamiltonian for $b\to s\ell^{+}\ell^{-}$ transitions
is [21, 35, 36, 37]

$\mathcal{H}_{\rm
eff}=-\frac{4G_{F}}{\sqrt{2}}\left[\lambda_{u}\Big{\\{}C_{1}(\mathcal{O}_{1}^{u}-\mathcal{O}_{1}^{c})+C_{2}(\mathcal{O}_{2}^{u}-\mathcal{O}_{2}^{c})\Big{\\}}+\lambda_{t}\sum\limits_{i\in
I}C_{i}\mathcal{O}_{i}\right]\ ,$ (1)

where $\lambda_{q}=V_{qb}V_{qs}^{*}$ and
$I=\\{1c,2c,3,4,5,6,8,7^{(\prime)},9^{(\prime)}\ell,10^{(\prime)}\ell,S^{(\prime)}\ell,P^{(\prime)}\ell,T^{(\prime)}\ell\\}$.
We neglect the terms proportional to $\lambda_{u}$ which are doubly Cabibbo
suppressed and contribute at the $\mathcal{O}(\lambda^{2})\sim 5\%$ level,
with $\lambda\equiv|V_{us}|$ in the Wolfenstein expansion of the CKM matrix
[38, 39]. We consider the following operators111In comparison with [21], we
have absorbed a factor $m_{b}$ into the definition of $\mathcal{O}_{S\ell}$
and $\mathcal{O}_{P\ell}$. :

$\displaystyle\mathcal{O}_{7^{(\prime)}}$
$\displaystyle=\frac{e}{(4\pi)^{2}}m_{b}[\bar{s}\sigma^{\mu\nu}P_{R(L)}b]F_{\mu\nu},$
$\displaystyle\mathcal{O}_{S^{(\prime)}\ell}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}m_{b}[\bar{s}P_{R(L)}b](\bar{\ell}\ell),$
(2) $\displaystyle\mathcal{O}_{9^{(\prime)}\ell}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}[\bar{s}\gamma^{\mu}P_{L(R)}b](\bar{\ell}\gamma_{\mu}\ell),$
$\displaystyle\mathcal{O}_{P^{(\prime)}\ell}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}m_{b}[\bar{s}P_{R(L)}b](\bar{\ell}\gamma_{5}\ell),$
$\displaystyle\mathcal{O}_{10^{(\prime)}\ell}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}[\bar{s}\gamma^{\mu}P_{L(R)}b](\bar{\ell}\gamma_{\mu}\gamma_{5}\ell),$
$\displaystyle\mathcal{O}_{T^{(\prime)}\ell}$
$\displaystyle=\frac{e^{2}}{(4\pi)^{2}}[\bar{s}\sigma^{\mu\nu}P_{R(L)}b](\bar{\ell}\sigma_{\mu\nu}\ell)\
,$

with $P_{R(L)}=\frac{1}{2}(1\pm\gamma_{5})$ and
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. The index $\ell$
denotes the flavour of the lepton, that we will omit for simplicity whenever
it is clear from the context if we consider a generic lepton or a specific
flavor. In this paper, we will consider only light leptons, i.e. $\ell=\mu,e$.

We use the following standard parameterization for the local form factors,

$\displaystyle\matrixelement{K^{-}(p_{K})}{\bar{s}\gamma_{\mu}b}{B^{-}(p_{B})}$
$\displaystyle=(p_{B}+p_{K})_{\mu}f_{+}(q^{2})+\frac{m_{B}^{2}-m_{K}^{2}}{q^{2}}q_{\mu}(f_{0}(q^{2})-f_{+}(q^{2})),$
$\displaystyle\matrixelement{K^{-}(p_{K})}{\bar{s}\sigma_{\mu\nu}b}{B^{-}(p_{B})}$
$\displaystyle=i\bigg{[}(p_{B}+p_{K})_{\mu}q_{\nu}-(p_{B}-p_{K})_{\nu}q_{\mu}\bigg{]}\frac{f_{T}(q^{2})}{m_{B}+m_{K}},$
$\displaystyle\matrixelement{K^{-}(p_{K})}{\bar{s}b}{B^{-}(p_{B})}$
$\displaystyle=\frac{m_{B}^{2}-m_{K}^{2}}{m_{b}-m_{s}}f_{0}(q^{2}),$ (3)

where $q=p_{B}-p_{K}$ is the momentum transfer of the final state lepton-
antilepton pair.

The angular distribution for $B^{-}\to K^{-}\ell^{+}\ell^{-}$ takes the form
(see e.g. [21, 36, 40])

$\displaystyle\frac{d\Gamma(B^{-}\to
K^{-}\ell^{+}\ell^{-})}{dq^{2}d\cos\theta_{\ell}}$ $\displaystyle=$
$\displaystyle\bar{G}_{0}(q^{2})+\bar{G}_{1}(q^{2})\cos\theta_{\ell}+\bar{G}_{2}(q^{2})\frac{1}{2}\left(3\cos\theta_{\ell}^{2}-1\right)\
,$ (4)

where $\theta_{\ell}$ is defined as the angle between the $\ell^{-}$ three-
momentum and the reversed of the $B^{-}$ three-momentum in the dilepton rest
frame. Integrating (4) over the full range $\theta_{\ell}\in[0,\pi]$, gives

$\displaystyle\frac{d\Gamma(B^{-}\to K^{-}\ell^{+}\ell^{-})}{dq^{2}}$
$\displaystyle=$ $\displaystyle 2\bar{G}_{0}(q^{2})$ (5)

with

$\displaystyle\bar{G}_{0}$ $\displaystyle=$
$\displaystyle\frac{4}{3}(1+2\hat{m}_{\ell}^{2})\absolutevalue{\bar{h}_{V}}^{2}+\frac{4}{3}\beta_{\ell}^{2}\absolutevalue{\bar{h}_{A}}^{2}+2\beta_{\ell}^{2}\absolutevalue{\bar{h}_{S}}^{2}+2\absolutevalue{\bar{h}_{P}}^{2}$
(6)
$\displaystyle+\frac{8}{3}(1+8\hat{m}_{\ell}^{2})\absolutevalue{\bar{h}_{T_{t}}}^{2}+\frac{4}{3}\beta_{\ell}^{2}\absolutevalue{\bar{h}_{T}}^{2}+16\hat{m}_{\ell}\imaginary[\bar{h}_{V}\bar{h}_{T_{t}}^{*}]\
,$

where $\hat{m}_{\ell}\equiv m_{\ell}/\sqrt{q^{2}}$ and
$\beta_{\ell}\equiv\sqrt{1-4\hat{m}_{\ell}^{2}}$ [21]. The coefficients
$\bar{G}_{1}$ and $\bar{G}_{2}$ are given in [21]. The decay rate is a
function of the amplitudes

$\displaystyle\bar{h}_{V}$
$\displaystyle=\mathcal{N}\frac{\sqrt{\lambda_{B}}}{2\sqrt{q^{2}}}\bigg{[}\frac{2m_{b}}{m_{B}+m_{K}}(C_{7}+C_{7^{\prime}})f_{T}(q^{2})+((C_{9}+C_{9^{\prime}}))f_{+}(q^{2})\bigg{]},$
(7) $\displaystyle\bar{h}_{A}$
$\displaystyle=\mathcal{N}\frac{\sqrt{\lambda_{B}}}{2\sqrt{q^{2}}}(C_{10}+C_{10^{\prime}})f_{+}(q^{2}),$
(8) $\displaystyle\bar{h}_{S}$
$\displaystyle=\mathcal{N}\frac{m_{B}^{2}-m_{K}^{2}}{2}\frac{m_{b}}{m_{b}-m_{s}}(C_{S}+C_{S^{\prime}})f_{0}(q^{2}),$
(9) $\displaystyle\bar{h}_{P}$
$\displaystyle=\mathcal{N}\frac{m_{B}^{2}-m_{K}^{2}}{2}\bigg{[}\frac{m_{b}(C_{P}+C_{P^{\prime}})}{m_{b}-m_{s}}+\frac{2m_{\ell}}{q^{2}}(C_{10}+C_{10^{\prime}})\bigg{]}f_{0}(q^{2}),$
(10) $\displaystyle\bar{h}_{T}$
$\displaystyle=-i\mathcal{N}\frac{\sqrt{\lambda_{B}}}{\sqrt{2}(m_{B}+m_{K})}(C_{T}-C_{T^{\prime}})f_{T}(q^{2}),$
(11) $\displaystyle\bar{h}_{T_{t}}$
$\displaystyle=-i\mathcal{N}\frac{\sqrt{\lambda_{B}}}{2(m_{B}+m_{K})}(C_{T}+C_{T^{\prime}})f_{T}(q^{2}),$
(12)

where $\lambda_{B}=\lambda(m_{B}^{2},m_{K}^{2},q^{2})$ is the Källén function
and the normalization factor is

$\mathcal{N}=-\frac{\alpha
G_{F}}{\pi}V_{ts}^{*}V_{tb}\sqrt{\frac{q^{2}\beta_{\ell}\sqrt{\lambda_{B}}}{2^{10}\pi^{3}m_{B}^{3}}}\
.$ (13)

The differential $B^{+}\to K^{+}\ell^{+}\ell^{-}$ decay rate follows from (5)
by replacing $\bar{h}$ with $h$. The latter are obtained from $\bar{h}$ by
taking the complex conjugate of all CP-violating phases, present in the CKM
matrix elements in $\mathcal{N}$ and (possibly) in NP contributions to the
Wilson coefficients.

In the remainder of this paper, we neglect possible NP contributions to
chirality-flipped (primed) Wilson coefficients such that
$h_{T}=\sqrt{2}h_{T_{t}}$. Primed Wilson coefficients can be easily included
by replacing $C_{i}\to C_{i}+C_{i^{\prime}}$, except in the case of $C_{T}$.

###  2.2 Hadronic long-distance effects

The decay $B^{-}\to K^{-}\ell^{+}\ell^{-}$ can also proceed through an
intermediate vector meson $V$, which subsequently decays into the lepton pair.
This process originates from the $\mathcal{O}_{1}^{c}$ and
$\mathcal{O}_{2}^{c}$ current-current operators in the low-energy effective
Hamiltonian (1) through loops and corresponding rescattering effects as
illustrated in Fig. 1. We note that the hadronic long-distance contributions
have been discussed intensively (see e.g.[24, 25, 26, 27, 41, 42, 9]). We
account for these effects by replacing $C_{9}$ by an effective coefficient

$C_{9}^{\rm eff}=C_{9}+Y(q^{2})\ ,$ (14)

where the function $Y\equiv|Y|e^{i\delta_{Y}}$ with a CP-conserving strong
phase $\delta_{Y}$.

Figure 1: Hadronic long-distance effects in $B^{-}\to K^{-}\ell^{+}\ell^{-}$.

The long-distance contributions are especially large in $q^{2}$-regions close
to a $c\bar{c}$ resonance peak, where the intermediate meson can go on shell.
The large size of non-local effects close to vector resonances is both a
blessing and a curse. On the one hand, a larger amplitude means that any
direct CP asymmetry will be enhanced near the peaks [19]. On the other, a
large amplitude means that measurements near the resonances are contaminated
by large backgrounds. For this reason, experimental studies are usually
restricted to regions far away from the peaks [3, 43]. In this work, we
consider $q^{2}$ bins near the resonances, laying the foundation for future
analyses once the challenge of the peaks can be overcome. Since the $Y$
contains non-perturbative physics, we will use experimental data to determine
it. We note that the perturbative contribution to $C_{9}^{\rm eff}$ has been
calculated and discussed in [44, 45, 46, 47, 17, 48]. Here we do not add this
contribution to our description, to avoid possible double counting because we
fit to the experimental data (see e.g. discussion in [49] for inclusive $B\to
X_{s,d}\ell^{+}\ell^{-}$ decays). We adopt a parameterization where the
$c\bar{c}$ vector resonances in $B\to K\mu^{+}\mu^{-}$ are modelled as a sum
of Breit–Wigner distributions following [28, 29]:

$Y(q^{2})=\sum\limits_{j}\eta_{j}e^{i\delta_{j}}A_{j}^{\rm res}(q^{2})\ ,$
(15)

where $\eta_{j}$ is the magnitude of a resonance amplitude, $\delta_{j}$ its
CP-conserving phase, and

$A_{j}^{\rm
res}(q^{2})=\frac{m_{0j}\Gamma_{0j}}{(m_{0j}^{2}-q^{2})-im_{0j}\Gamma_{j}(q^{2})}\
,$ (16)

where $\Gamma_{j}$ and $m_{0j}$ are the width and pole mass of the $j$th
resonance, respectively. We use the values of [30]. The sum runs over the six
$c\bar{c}$ resonances $J/\psi$, $\psi(2S)$, $\psi(3770)$, $\psi(4040)$,
$\psi(4160)$, and $\psi(4415)$. In this model, contributions from other broad
resonances and continuum contributions are ignored. There also exist
resonances involving lighter quarks, but their contribution to the branching
ratios is only around $1\%$ [30]. We neglect these contributions for
consistency as we also neglect the doubly Cabibbo-suppressed contributions to
the decay amplitude as denoted after (1).

The parameters $\eta$ and the phases $\delta$ can then be determined from
experimental data. The LHCb collaboration has measured them by fitting the
model to the full invariant mass spectrum [30]. In the fit four “branches”
were found, corresponding to different signs for the phases of the $J/\psi$
and $\psi(2S)$ resonances. We denote them by $Y_{--}$, $Y_{-+}$, $Y_{+-}$, and
$Y_{++}$, where the first (second) sign matches the sign of the CP-conserving
phase of the $J/\psi(\psi(2S))$. Fig. 2 shows the $|Y(q^{2})|$ and
$\delta_{Y}$ for each of the four branches. We can see that the absolute value
of $Y(q^{2})$ grows largest near the $c\bar{c}$ resonances, and furthermore
that it behaves similarly in all four branches. The overall phase
$\delta_{Y}(q^{2})$ differs more: up to $m_{J/\psi}^{2}\approx
10\;$\mathrm{G}\mathrm{eV}^{2}$$, it is determined almost completely by the
phase of the $J/\psi$, so the $Y_{--},Y_{-+}$ branches are similar, as are the
$Y_{+-},Y_{++}$. Near the $\psi(2S)$ peak, $\delta_{Y}(q^{2})$ is determined
by the phase of the $\psi(2S)$. At higher $q^{2}$, past the two major
$c\bar{c}$ resonances, $\delta_{Y}(q^{2})$ follows a pattern that is similar
for all branches. Because the phase $\delta_{Y}(q^{2})$ is distinct for each
branch, we can use CP-violating observables that are sensitive to
$\delta_{Y}(q^{2})$ to distinguish between the different branches. We will
return to this in Section 3.3. We note that this description of the hadronic
resonances is a model and as such could be significantly improve in the future
using either more data or a different parametrization [41, 42], see for
example the recent discussion in [50]. Nevertheless, it is instructive to
consider this model to determine the distinct patterns given by the CP
asymmetries.

(a) $\absolutevalue{Y(q^{2})}$

(b) $\delta_{Y}(q^{2})$

Figure 2: Absolute value and phase of the hadronic long-distance function
$Y(q^{2})$. The four colors show the four branches from Ref. [30] (see text).

###  2.3 Direct CP violation

In order to generate a direct CP-asymmetry, at least two interfering
amplitudes are required, as well as two different CP-violating phases and CP-
conserving phases. In the case of $B\to K\mu^{+}\mu^{-}$, a CP-violating phase
can come from tiny Cabibbo-suppressed interference terms that we neglect, or
from NP, manifest as a complex phase in a Wilson coefficient. In our approach,
the only source of a CP-conserving phase in this decay is the function
$Y(q^{2})$ that describes hadronic long-distance effects.

We denote by $\bar{\Gamma}\equiv\Gamma(B^{-}\to K^{-}\ell^{+}\ell^{-})$,
$\Gamma\equiv\Gamma(B^{+}\to K^{+}\ell^{+}\ell^{-})$, and define the
differential direct CP asymmetry as

$\displaystyle\mathcal{A}_{\rm CP}^{\rm
dir}(q^{2})\equiv\frac{d\bar{\Gamma}/dq^{2}-d\Gamma/dq^{2}}{d\bar{\Gamma}/dq^{2}+d\Gamma/dq^{2}}$
(17)

with

$\displaystyle\frac{d\bar{\Gamma}/dq^{2}-d\Gamma/dq^{2}}{2}$
$\displaystyle=\frac{4}{3}(1+2\hat{m}_{\ell}^{2})(\absolutevalue{\bar{h}_{V}}^{2}-\absolutevalue{h_{V}}^{2})+16\hat{m}_{\ell}(\imaginary[\bar{h}_{V}\bar{h}_{T_{t}}^{*}]-\imaginary[h_{V}h_{T_{t}}^{*}])$
$\displaystyle=\absolutevalue{\mathcal{N}}^{2}\frac{\lambda_{B}}{\sqrt{q^{2}}}f_{+}(q^{2})\absolutevalue{Y(q^{2})}\sin\delta_{Y}(q^{2})$
$\displaystyle\bigg{[}\frac{4(1+2\hat{m}_{\ell}^{2})}{3\sqrt{q^{2}}}f_{+}(q^{2})\absolutevalue{C_{9\ell}^{\rm
NP}}\sin\phi_{9}^{\rm NP}$ (18)
$\displaystyle+\frac{8\hat{m}_{\ell}}{m_{B}+m_{K}}f_{T}(q^{2})\absolutevalue{C_{T\ell}^{\rm
NP}}\sin\phi_{T\ell}^{\rm NP})\bigg{]}\ ,$

where the $\bar{h}_{i}$ were given in (7) and (12). In addition, we have
defined the CP-violating phase $\phi_{i}^{\rm NP}$ of a Wilson coefficient
through

$C_{i\ell}^{\rm NP}\equiv\absolutevalue{C_{i\ell}^{\rm
NP}}e^{i\phi_{i\ell}^{\rm NP}}\ ,$ (19)

where we explicitly denote by $\ell$ the lepton flavour. We note that, in
order to get direct CP violation in $B\to K\ell^{+}\ell^{-}$, an interference
between the hadronic long-distance phase $\delta_{Y}(q^{2})$ and a NP phase
$\phi_{9\ell}^{\rm NP}$ or $\phi_{T\ell}^{\rm NP}$ is required. Because the
direct CP asymmetry is proportional to $\sin\delta_{Y}(q^{2})$, it inherits
the properties of the CP-conserving phase variations as a function of $q^{2}$
given in Fig. 2a. Consequently, as has been observed in [19], the direct CP
asymmetry will vary across the $q^{2}$ spectrum, growing largest close to the
first two $c\bar{c}$ resonance peaks.

In the SM, tiny amounts of direct CP violation can be generated through
interference between the terms proportional to $\lambda_{t}$ and the doubly
Cabibbo-suppressed $\lambda_{u}$ term which we neglect. For this reason, in
the SM $\mathcal{A}_{\rm CP}^{\rm dir}$ is suppressed by a factor
$\lambda^{2}$ relative to (18). Therefore, an observed sizeable direct CP
asymmetry in $B\to K\ell^{+}\ell^{-}$ would be a clear sign of new CP-
violating physics.

The denominator of the direct CP asymmetry is

$\frac{d\bar{\Gamma}/dq^{2}+d\Gamma/dq^{2}}{2}=\bar{G}_{0}+G_{0}$ (20)

with $\bar{G}_{0}$ given in (6).

To connect with experiment, we consider the $q^{2}$-integrated direct CP
asymmetry, defined as

$\mathcal{A}_{\rm CP}^{\rm dir}[q^{2}_{\rm min},q^{2}_{\rm
max}]=\frac{\bar{\Gamma}[q^{2}_{\rm min},q^{2}_{\rm max}]-\Gamma[q^{2}_{\rm
min},q^{2}_{\rm max}]}{\bar{\Gamma}[q^{2}_{\rm min},q^{2}_{\rm
max}]+\Gamma[q^{2}_{\rm min},q^{2}_{\rm max}]}\ ,$ (21)

where

$\Gamma[q^{2}_{\rm min},q^{2}_{\rm max}]=\int_{q^{2}_{\rm min}}^{q^{2}_{\rm
max}}\frac{d\Gamma}{dq^{2}}dq^{2}\ .$ (22)

Note that in (21), all the differential decay rates have been integrated
individually – this is not the same as integrating (17) over a $q^{2}$ bin.

The numerator of the $q^{2}$-binned direct CP asymmetry can be written as

$\displaystyle\frac{\bar{\Gamma}[q^{2}_{\rm min},q^{2}_{\rm
max}]-\Gamma[q^{2}_{\rm min},q^{2}_{\rm max}]}{2}$ $\displaystyle=\rho_{9\rm
Im}^{\ell}\absolutevalue{C_{9\ell}^{\rm NP}}\sin\phi_{9\ell}^{\rm
NP}+\rho_{T\rm Im}^{\ell}\absolutevalue{C_{T\ell}^{\rm
NP}}\sin\phi_{T\ell}^{\rm NP}\ ,$ (23)

and the denominator as

$\displaystyle\frac{\bar{\Gamma}[q^{2}_{\rm min},q^{2}_{\rm
max}]+\Gamma[q^{2}_{\rm min},q^{2}_{\rm max}]}{2}=\Gamma_{\rm
SM}^{\ell}+\rho_{9}^{\ell}\absolutevalue{C_{9\ell}^{\rm
NP}}^{2}+\rho_{10}^{\ell}\absolutevalue{C_{10\ell}^{\rm
NP}}^{2}+\rho_{P}^{\ell}\absolutevalue{C_{P\ell}^{\rm
NP}}^{2}+\rho_{S}^{\ell}\absolutevalue{C_{S\ell}^{\rm
NP}}^{2}+\rho_{T}^{\ell}\absolutevalue{C_{T\ell}^{\rm NP}}^{2}$
$\displaystyle\quad\quad\quad\quad+\rho_{9\rm
Re}^{\ell}\absolutevalue{C_{9\ell}^{\rm NP}}\cos\phi_{9\ell}^{\rm
NP}+\rho_{10\rm Re}^{\ell}\absolutevalue{C_{10\ell}^{\rm
NP}}\cos\phi_{10\ell}^{\rm NP}+\rho_{P\rm
Re}^{\ell}\absolutevalue{C_{P\ell}^{\rm NP}}\cos\phi_{P\ell}^{\rm
NP}+\rho_{T\rm Re}^{\ell}\absolutevalue{C_{T\ell}^{\rm
NP}}\cos\phi_{T\ell}^{\rm NP}$ $\displaystyle\quad\quad\quad\quad+\rho_{10P\rm
Re}^{\ell}\absolutevalue{C_{10\ell}^{\rm NP}}\absolutevalue{C_{P\ell}^{\rm
NP}}\cos\phi_{10\ell}^{\rm NP}\cos\phi_{P\ell}^{\rm NP}+\rho_{10P\rm
Im}^{\ell}\absolutevalue{C_{10\ell}^{\rm NP}}\absolutevalue{C_{P\ell}^{\rm
NP}}\sin\phi_{10\ell}^{\rm NP}\sin\phi_{P\ell}^{\rm NP}$
$\displaystyle\quad\quad\quad\quad+\rho_{9T\rm
Re}^{\ell}\absolutevalue{C_{9\ell}^{\rm NP}}\absolutevalue{C_{T\ell}^{\rm
NP}}\cos\phi_{9\ell}^{\rm NP}\cos\phi_{T\ell}^{\rm NP}+\rho_{9T\rm
Im}^{\ell}\absolutevalue{C_{9\ell}^{\rm NP}}\absolutevalue{C_{T\ell}^{\rm
NP}}\sin\phi_{9\ell}^{\rm NP}\sin\phi_{T\ell}^{\rm NP}\ .$ (24)

The coefficients $\Gamma_{\rm SM}$ and $\rho$ depend on the choice of long-
distance model. Numerical values for these coefficients are given in Appendix
A, with long-distance branches separated where appropriate.

###  2.4 Mixing-induced CP violation

Charged $B$-meson decays exhibit only direct CP violation. Neutral $B^{0}_{q}$
mesons show $B^{0}_{q}-\bar{B}^{0}_{q}$ oscillations, which have profound
implications for CP violation. Due to $B^{0}_{q}-\bar{B}^{0}_{q}$ mixing,
interference effects arise between $B^{0}_{q}$ and $\bar{B}^{0}_{q}$ decaying
into the same final state $f$, thereby leading to mixing-induced CP violation.
This phenomenon has received a lot of attention in non-leptonic $B$ decays but
is also very interesting for the rare modes that we study here.

The $B^{0}_{d}\to K^{0}\ell^{+}\ell^{-}$ and
$\bar{B}^{0}_{d}\to\bar{K}^{0}\ell^{+}\ell^{-}$ modes followed by $K^{0}\to
K^{+}\pi^{-}$ and $\bar{K}^{0}\to\pi^{+}K^{-}$, respectively, are flavour
specific decays, which can be distinguished through the charges of the
subsequent kaon decays. However, if we observe the neutral kaons as $K_{S}$,
both $B^{0}$ and $\bar{B}^{0}$ can decay into the same final state. We note
that we neglect the CP violation in the neutral kaon system, which appears at
the $10^{-3}$ level, and thereby treat $K_{S}$ as a CP eigenstate.
Consequently, for these decays $B^{0}_{q}-\bar{B}^{0}_{q}$ mixing can generate
interference between these decay amplitudes, giving rise to mixing-induced CP
violation. Therefore, in the $B^{0}_{d}\to K_{S}\mu^{+}\mu^{-}$ channel these
interference effects are present. The amplitudes of $B^{0}_{d}\to
K_{S}\ell^{+}\ell^{-}$ and $\bar{B}^{0}_{d}\to K_{S}\ell^{+}\ell^{-}$ are
related through spectator quarks in the isospin limit to the charged ones of
$B^{+}\to K^{+}\ell^{+}\ell^{-}$ and $B^{-}\to K^{-}\ell^{+}\ell^{-}$,
respectively, taking the normalisation factor $1/\sqrt{2}$ at the amplitude
level coming from the $K_{S}$ state into account.

In the presence of $B^{0}_{q}-\bar{B}^{0}_{q}$ mixing, the time evolution can
be expressed as:

$\displaystyle\Gamma(B^{0}_{q}\to f)$
$\displaystyle=\Big{[}|g_{-}(t)|^{2}+|\xi_{f}|^{2}|g_{+}(t)|^{2}-2\real\\{\xi_{f}\
g_{+}(t)\ g_{-}(t)^{*}\\}\Big{]}\Gamma_{f},$
$\displaystyle\Gamma(\bar{B}^{0}_{q}\to f)$
$\displaystyle=\Big{[}|g_{+}(t)|^{2}+|\xi_{f}|^{2}|g_{-}(t)|^{2}-2\real\\{\xi_{f}\
g_{-}(t)\ g_{+}(t)^{*}\\}\Big{]}{\Gamma}_{f},$ (25)

where the normalisation $\Gamma_{f}$ denotes the “unevolved” $B_{q}\to f$ rate
and the time dependence enters through:

$\displaystyle g_{+}(t)\ g_{-}(t)^{*}$
$\displaystyle=\frac{1}{4}\left[e^{-\Gamma_{L}t}-e^{-\Gamma_{H}t}-2ie^{-\Gamma
t}\sin{\Delta Mt}\right],$ $\displaystyle|g_{\mp}(t)|^{2}$
$\displaystyle=\frac{1}{4}\left[e^{-\Gamma_{L}t}+e^{-\Gamma_{H}t}\mp
2e^{-\Gamma t}\cos{\Delta Mt}\right].$ (26)

Here $\Gamma_{H,L}$ is the width of the “heavy” and “light” mass eigenstates
and $\Delta M=M_{H}-M_{L}$ is the mass difference between those eigenstates.
The parameter $\xi$ is a process-specific physical observable that measures
the strength of the interference effects:

$\xi_{f}=e^{-i\phi_{q}}\left[e^{\phi_{CP}}\frac{A(\bar{B}^{0}_{q}\to
f)}{A({B}^{0}_{q}\to f)}\right]\ .$ (27)

Here $\phi_{CP}$ is a convention-dependent phase, which later cancels in the
ratio of the amplitudes, while $\phi_{q}$ is the $B^{0}_{q}-B^{0}_{q}$ mixing
phase. This phase is sizeable for the $B_{d}$ system and takes the value [51,
52]

$\phi_{d}=(44.4^{+1.6}_{-1.5})^{\circ}\ ,$ (28)

which is extracted from $B_{d}^{0}\to J/\psi K_{S}$ and which also takes
penguin contributions into account. We note here that the mixing phase
$\phi_{s}$ in the $B_{s}$ system, e.g. in the $B_{s}^{0}\to
f_{0}\ell^{+}\ell^{-}$ decay [21], is much smaller, for which reason
$B_{s}^{0}$ decays are more sensitive to NP effects. We can then introduce the
time-dependent decay rate asymmetry (following the notation in [53]):

$\displaystyle\frac{\Gamma(B^{0}_{q}(t)\to f)-\Gamma(\bar{B}^{0}_{q}(t)\to
f)}{\Gamma(B^{0}_{q}(t)\to f)+\Gamma(\bar{B}^{0}_{q}(t)\to f)}$
$\displaystyle=\frac{\mathcal{C}\cos(\Delta Mt)+\mathcal{S}\sin(\Delta
Mt)}{\cosh(\frac{\Delta\Gamma_{q}}{2}t)+\mathcal{A}_{\Delta\Gamma}\sinh(\frac{\Delta\Gamma_{q}}{2}t)}\
,$ (29)

where $\Delta\Gamma_{q}\equiv\Gamma_{L}-\Gamma_{H}$ denotes the decay width
difference in the $B_{q}$ system. We note that $\Delta\Gamma_{q}$ is sizeable
in the $B_{s}$ system but negligible in the $B_{d}$, therefore making the
observable $\mathcal{A}_{\Delta\Gamma}$ not accessible in $B^{0}_{d}\to
K_{S}\ell^{+}\ell^{-}$. A full angular analysis would give rise to 6 angular
observables as discussed in [21]. As the time-dependent measurement of these
observables appears to be challenging, our starting point is to consider rates
integrated over the full range $\theta$ (as in the charged case above), such
that only the $S$-wave component remains. At a decay time $t=0$, where mixing
effects are switched off, the untagged differential decay rate then yields:

$\displaystyle\frac{d\Gamma(B_{d}\to K_{S}\ell^{+}\ell^{-})\pm
d\Gamma(\bar{B}_{d}\to K_{S}\ell^{+}\ell^{-})}{ds}$
$\displaystyle=2[G_{0}\pm\bar{G}_{0}]\ ,$ (30)

with $\bar{G}_{0}$ given in (6) but with the replacements

$\bar{h}_{X}(\bar{B}_{d}\to
K_{S}\ell^{+}\ell^{-})=\frac{1}{\sqrt{2}}\bar{h}_{X}(B^{-}\to
K^{-}\ell^{+}\ell^{-})\ ,\quad{h}_{X}(\bar{B}_{d}\to
K_{S}\ell^{+}\ell^{-})=\frac{1}{\sqrt{2}}{h}_{X}(B^{-}\to
K^{-}\ell^{+}\ell^{-})\ ,$ (31)

to take the normalization of the $K_{S}$ into account.

From (2.4) and (2.4), we find

$\displaystyle G_{0}(t)-\bar{G}_{0}(t)$ $\displaystyle=e^{-\Gamma
t}\Big{[}(G_{0}-\bar{G}_{0})\cos(x\Gamma t)-s_{0}\sin(x\Gamma t)\Big{]}\ ,$
(32) $\displaystyle G_{0}(t)+\bar{G}_{0}(t)$ $\displaystyle=e^{-\Gamma
t}\Big{[}(G_{0}+\bar{G}_{0})\cosh(y\Gamma t)-h_{0}\sinh(y\Gamma t)\Big{]}\ ,$
(33)

where $y=\Delta\Gamma/(2\Gamma)$ and $x=\Delta M/\Gamma$. And [21]

$\displaystyle s_{0}=2\imaginary[e^{-i\phi_{d}}$
$\displaystyle\bigg{(}\frac{4}{3}(1+2\hat{m}_{\ell}^{2})\tilde{h}_{V}h_{V}^{*}+\frac{4}{3}\beta_{\ell}^{2}\tilde{h}_{A}h_{A}^{*}+2\beta_{\ell}^{2}\tilde{h}_{S}h_{S}^{*}+2\tilde{h}_{P}h_{P}^{*}$
(34)
$\displaystyle+\frac{8}{3}(1+8\hat{m}_{\ell}^{2})\tilde{h}_{T_{t}}h_{T_{t}}^{*}+\frac{4}{3}\beta_{\ell}^{2}\tilde{h}_{T}h_{T}^{*}\bigg{)}\bigg{]}-16\hat{m}_{\ell}\real[e^{-i\phi_{d}}\tilde{h}_{V}h_{T_{t}}^{*}-e^{i\phi_{d}}h_{V}\tilde{h}_{T_{t}}^{*}\bigg{]}\
,$

The normalization is as in (31) and $\tilde{h}_{X}=\eta_{X}\bar{h}_{X}$, with
for the $K_{S}$, $\eta_{V}=\eta_{A}=\eta_{P}=\eta_{T_{t}}=-1$ and
$\eta_{S}=\eta_{T}=1$ (see [21, 22] for a detailed discussion). The
coefficient $h_{0}$ is also given in [21].

Finally, we can write the CP asymmetries as222We note that we have
$\mathcal{A}_{\rm CP}^{\rm mix}=2\sigma_{0}$, where $\sigma_{0}$ was used in
[21].

$\displaystyle\mathcal{C}$
$\displaystyle\equiv\frac{G_{0}-\bar{G}_{0}}{G_{0}+\bar{G}_{0}}=-\mathcal{A}_{\rm
CP}^{\rm dir}(B^{-}\to K^{-}\ell^{+}\ell^{-})\ ,$ (35)
$\displaystyle\mathcal{S}$
$\displaystyle\equiv\frac{-s_{0}}{G_{0}+\bar{G}_{0}}\equiv-\mathcal{A}_{\rm
CP}^{\rm mix}\ ,$ (36)

where the charged direct CP asymmetry was already introduced in (17).

Unlike the direct CP asymmetry, which is sensitive only to a CP-violating
phase in $C_{9}$ or $C_{T}$, the mixing-induced CP asymmetry is affected by
complex phases in any of the Wilson coefficients. The mixing phase $\phi_{d}$,
which is experimentally found to be sizeable, plays a key role, leading to a
large mixing-induced CP asymmetry. Therefore, having new complex Wilson
coefficients involved, this CP asymmetry can be shifted away from its SM
value. We provide the SM prediction in the next section.

###  2.5 Standard Model predictions and input parameters

For the SM Wilson coefficients, we use [54]

$C_{7}^{\rm SM}=-0.292,\quad C_{9}^{\rm SM}=4.07,\quad C_{10}^{\rm SM}=-4.31\
,$ (37)

which are flavour universal (i.e. equal for $\ell=\mu=e$) and determined at
$\mu=m_{b}$. In addition, we use the quark masses in ${\overline{\rm MS}}$ at
$m_{b}$: $\bar{m}_{b}(m_{b})=4.18\pm 0.03$ and $\bar{m}_{s}(m_{b})=0.078\pm
0.007$ [55]. For the local form factors in (2.1), we use the recent lattice
QCD determination of [56].

To determine the CKM elements $V_{ts}V_{tb}^{*}$, we exploit CKM unitarity to
write (see [23]):

$V_{ts}V_{tb}^{*}=-V_{cb}\bigg{[}1-\frac{\lambda^{2}}{2}(1-2\bar{\rho}+2i\bar{\eta})\bigg{]}+\mathcal{O}(\lambda^{6})\
,$ (38)

which introduces the dependence of the apex of the Unitarity Triangle (UT)
through $\bar{\rho}$ and $\bar{\eta}$ at $\mathcal{O}(\lambda^{2})$ with
respect to the leading $|V_{cb}|$ contribution. For completeness, we do
include here the $\lambda^{2}$-suppressed terms such that we have an imaginary
part arising from the CKM matrix elements.

The discrepancies between the inclusive and exclusive $|V_{ub}|$ and
$|V_{cb}|$ determinations lead to different pictures for the allowed parameter
spaces for NP in $B_{q}$-meson mixing. We employ the recent results from [23]
where a CKM fit was performed using both the inclusive and exclusive case. On
top of these, a hybrid approach with the exclusive $|V_{ub}|$ and inclusive
$|V_{cb}|$ values was studied [23]. We note that for determining the CKM
factor in (38), the difference for $|V_{ub}|$ is marginal and only enters via
higher order corrections, which we neglect. In this limit, the hybrid scenario
coincides with the inclusive one. We provide our results separately, first for
the inclusive/hybrid and then for the exclusive case.

For $|V_{cb}|$, we take the result from inclusive $b\to c\ell\nu$ decays [57]:

$|V_{cb}|_{\rm incl}=(42.16\pm 0.50)\times 10^{-3}\ ,$ (39)

which agrees with the recent determination in [58]. Finally, (38) then gives

$V_{ts}V_{tb}^{*}|_{\rm incl/hybrid}=(-41.4\pm 0.5+0.7i)\times
10^{-3},\quad|V_{ts}V_{tb}^{*}|_{\rm incl/hybrid}=(41.4\pm 0.5)\times 10^{-3}\
,$ (40)

where here and in the following we take into account the tiny imaginary part
from the CKM elements. Taking the exclusive $|V_{cb}|$ given in [23] from
[59], we find

$|V_{ts}V_{tb}^{*}|_{\rm excl}=(38.4\pm 0.5)\times 10^{-3},$ (41)

which differs from (40) at the $4\sigma$ level to the different $|V_{cb}|$
values.

Finally, we find for the branching ratio in the $1.1\;{\rm
GeV}^{2}<q^{2}<6.0\;{\rm GeV}^{2}$ range:

$\mathcal{B}(B^{-}\to K^{-}\mu^{+}\mu^{-})^{\rm SM}[1.1,6.0]_{\rm
incl/hybrid}=(1.83\pm 0.14)\times 10^{-7}\ ,$ (42)

where we have added the variation of the different long distance branches as
an additional uncertainty. Even when including those, the uncertainty is still
dominated by the form factor uncertainty. We note that we do not include a
systematic uncertainty from our long-distance assumptions. Comparing this with
the experimental measurement of the branching ratio in this $q^{2}$-bin [1]
333We have converted the differential rate given in [1] by multiplying with
the appropriate bin-width.:

$\mathcal{B}(B^{\pm}\to K^{\pm}\mu^{+}\mu^{-})[1.1,6.0]=(1.19\pm 0.07)\times
10^{-7}\ ,$ (43)

we find

$\frac{\mathcal{B}(B^{-}\to K^{-}\mu^{+}\mu^{-})^{\rm
SM}[1.1,6.0]_{\rm{{incl/hybrid}}}}{\mathcal{B}(B^{\pm}\to
K^{\pm}\mu^{+}\mu^{-})[1.1,6.0]}-1=0.54\pm 0.15\ ,$ (44)

which is a $3.5\sigma$ deviation from zero. Taking the exclusive $|V_{cb}|$
determination, we find

$\mathcal{B}(B^{-}\to K^{-}\mu^{+}\mu^{-})^{\rm SM}[1.1,6.0]_{\rm
excl}=(1.57\pm 0.13)\times 10^{-7}\ ,$ (45)

and

$\frac{\mathcal{B}(B^{-}\to K^{-}\mu^{+}\mu^{-})^{\rm SM}[1.1,6.0]_{\rm
excl}}{\mathcal{B}(B^{\pm}\to K^{\pm}\mu^{+}\mu^{-})[1.1,6.0]}-1=0.33\pm 0.13\
,$ (46)

which lies $2.4\sigma$ from the SM. This study shows that the unsatisfactory
determination of the CKM factors has a profound impact on the comparison
between the data and the SM prediction for the corresponding branching ratio,
thus it has to be resolved.

Finally, we note that our determination is based on our assumptions for the
hadronic long-distance effects. As such, our prediction differs slightly from
[60] which found:

$\mathcal{B}(B^{-}\to K^{-}\mu^{+}\mu^{-})^{\rm SM}[1.1,6.0]=(2.07\pm
0.17)\times 10^{-7}\ ,$ (47)

which shows a $4.7\sigma$ deviation from the SM. In this case, the authors
used a different model for the $Y(q^{2})$ including the perturbative
contribution discussed previously. Their variation of the charm mass results
in a larger uncertainty than ours. The SM predicitions for these modes,
including long-distance effects, have been discussed in a long range of
papers, most recently in [9]. They also find a significant tension with the
SM.

In addition, we give the prediction for the branching ratio in the $7\;{\rm
GeV}^{2}<q^{2}<8\;{\rm GeV}^{2}$ range:

$\mathcal{B}(B^{+}\to K^{+}\mu^{+}\mu^{-})^{\rm SM}[7,8]=(0.367\pm
0.031)\times 10^{-7}\ .$ (48)

Finally, we note that for electrons, the branching ratios are almost
identical, differing only at the third digit through phase-space effects.

Using the value of $\phi_{d}$ in (28), we determine the values of the CP
asymmetries. Since we neglect the $\lambda^{2}$ suppressed $\lambda_{u}$
contributions, we have

$\mathcal{A}_{\rm CP}^{\rm dir}|_{\rm SM}=0,$ (49)

for all $q^{2}$ regions. The mixing-induced CP asymmetry for $1.1\;{\rm
GeV}^{2}<q^{2}<6\;{\rm GeV}^{2}$:

$\mathcal{A}_{\rm CP}^{\rm mix}|_{\rm SM}=0.72\pm 0.02,$ (50)

which is exactly equal to $\phi_{d}$ if the tiny CKM phase is neglected [21].
The tiny uncertainty stems thus mainly from $\phi_{d}$. In addition, the CKM
factors drop out and we are not affected by the difference in $|V_{cb}|$ as
for the branching ratio.

##  3 Fingerprinting New Physics with direct and mixing-induced CP asymmetries
in $B\to K\mu^{+}\mu^{-}$

###  3.1 Experimental constraints

We consider data on the branching ratio [1] and the direct CP asymmetry [31]
of the $B^{+}\to K^{+}\mu^{+}\mu^{-}$ channel in the bin of
$q^{2}\in[7,8]\>$\mathrm{G}\mathrm{eV}^{2}$$, where the LHCb collaboration
finds [1]

$\mathcal{B}(B^{+}\to K^{+}\mu^{+}\mu^{-})[7,8]=(23.1\pm 1.8)\times 10^{-9}$
(51)

and [31]

$\mathcal{A}_{\rm CP}^{\rm dir}[7,8]=0.041\pm 0.059\ .$ (52)

Here we have added the statistical and systematic uncertainties in quadrature.
In this paper, we focus on this $q^{2}$-bin because it lies close to the
$J/\psi$ resonance at
$m_{J/\psi}^{2}=$9.6\text{\,}\mathrm{G}\mathrm{eV}^{2}$$, where the direct CP
asymmetry could be enhanced [19]. In Fig. 3, we depict the experimental data
in the different $q^{2}$-bins as presented in [31]. The average of over all
the bins is [31]

$\mathcal{A}_{\rm CP}^{\rm dir}=0.012\pm 0.017\ .$ (53)

Figure 3: Illustration of the experimental measurements of the direct CP
asymmetry of $B^{-}\to K^{-}\mu^{+}\mu^{-}$ by the LHCb collaboration [31].
The blue line shows the prediction for benchmark point 1 given in (55) (see
also [19]).

We consider three scenarios:

$\displaystyle\text{Scenario 1:}\quad C_{9\mu}^{\rm NP}\neq 0\ ,$ (54)
$\displaystyle\text{Scenario 2:}\quad C_{9\mu}^{\rm NP}=-C_{10\mu}^{\rm
NP}\neq 0\ ,$ $\displaystyle\text{Scenario 3:}\quad C_{10\mu}^{\rm NP}\neq 0\
.$

All three scenarios fit the data better than the SM with pulls of
$(4$-$7)\sigma$ [7, 9, 10, 11, 12, 13].

In Fig. 4, we illustrate the $1\sigma$-experimental constraints on the complex
$C_{9\mu}$ and $C_{10\mu}$ coefficients for the three NP scenarios in (54). We
note that for Scenario 3, shown in Fig. 4c, there is no bound from the direct
CP asymmetry, as can be seen also from (18). For simplicity, we show results
using the $Y_{--}(q^{2}$) hadronic long-distance branch. In addition, we use
the hybrid scenario for the CKM factors. For the other three branches, the
constraints from the branching ratio remain similar while the allowed region
from the direct CP asymmetry changes. This dependence on the direct CP
asymmetry to the choice of the branch is an important feature that we will
exploit in the remainder of this paper. In Fig. 4, we have indicated two
benchmark points with large CP-violating phases, allowed by the current data:

Benchmark Point 1: $\displaystyle\quad\absolutevalue{C_{9\mu}^{\rm
NP}}/\absolutevalue{C_{9}^{\rm SM}}=0.75\ ,\quad\quad\phi_{9\mu}^{\rm
NP}=195^{\circ}\ ,$ (55) Benchmark Point 2:
$\displaystyle\quad\absolutevalue{C_{9\mu}^{\rm NP}}/\absolutevalue{C_{9}^{\rm
SM}}=\absolutevalue{C_{10\mu}^{\rm NP}}/\absolutevalue{C_{9}^{\rm
SM}}=0.30\quad\phi_{9\mu}^{\rm NP}=\phi_{10\mu}^{\rm NP}-\pi=220^{\circ}\ .$

We will use these parameter sets for illustrative purposes in Sec. 3.4. In
Fig. 3, we show benchmark scenario 1 to demonstrate the enhancement of the
direct CP asymmetry in the resonance region. A similar scenario was considered
in [19].

In (54), we only consider scenarios with $C_{9\mu}$ and/or $C_{10\mu}$. We do
not discuss NP entering through $C_{S\mu}$ and $C_{P\mu}$, because any
significant contribution of these coefficients to $B\to K\ell^{+}\ell^{-}$
would have a much larger, clearly visible influence on
$B_{s}^{0}\to\mu^{+}\mu^{-}$ [16] as they would lift the helicity suppression.
In addition, we do not explicitly consider tensor couplings [61]. If only
$C_{T\mu}$ were present, the $B\to K\mu^{+}\mu^{-}$ rate in (24) reduces to

$\Gamma[1.1,6.0]=\Gamma_{\rm SM}+\rho_{T}^{\ell}\absolutevalue{C_{T\ell}^{\rm
NP}}^{2}+\rho_{T\rm Re}^{\ell}\absolutevalue{C_{T\ell}^{\rm
NP}}\cos\phi_{T\ell}^{\rm NP}\ ,$ (56)

where the $\rho$ are given in Appendix A from which we find $\rho_{T\rm
Re}<\rho_{T}$. Therefore, the branching ratio will always be pushed upwards,
independent of the phase difference $\phi_{T}$. This situation changes if also
NP in other Wilson coefficients is present, in which case the lower value of
the branching ratio can be accounted for (see (24)). For the direct CP
asymmetry, we explicitly find

$\mathcal{A}_{\rm CP}^{\rm dir}=\frac{\rho_{T\rm Im}^{\ell}|C_{T\ell}^{\rm
NP}|\sin\phi_{T\ell}^{\rm NP}}{\Gamma_{\rm
SM}^{\ell}+{\rho_{T}^{\ell}}|C_{T\ell}^{\rm NP}|^{2}}\ .$ (57)

We note that in this case $\mathcal{A}_{\rm CP}^{\rm dir}<0.03$, because
$\rho_{T}^{\ell}>>\rho_{T\rm Im}^{\ell}$. Therefore, we focus on NP in $C_{9}$
and $C_{10}$.

(a) $C_{9\mu}^{\rm NP}$ only

(b) $C_{9\mu}^{\rm NP}=-C_{10\mu}^{\rm NP}$

(c) $C_{10\mu}^{\rm NP}$ only

Figure 4: Experimental $1\sigma$ constraints on the three NP scenarios
specified in (54). The star and diamond indicate the benchmark points 1 and 2
given in (55), respectively.

###  3.2 Correlations between CP-violating observables

We will now demonstrate how to distinguish between the three NP scenarios in
(54) using the correlations between the direct and mixing-induced CP
asymmetries of $B\to K\mu^{+}\mu^{-}$. To illustrate the method, we first
consider only the $Y_{--}$ branch. Considering now NP Wilson coefficients that
are within $1\sigma$ of the branching ratio measurement in (51), while adding
experimental and theoretical uncertainties in quadrature, gives the
correlations in 5. In addition, we show the experimental constraint on the
direct CP asymmetry for the $B^{+}\to K^{+}$ decay given in (52). We observe
that each scenario leaves a distinct “fingerprint” in the $\mathcal{A}_{\rm
CP}^{\rm dir}$-$\mathcal{A}_{\rm CP}^{\rm mix}$ plane. For Scenario 1, with
only $C_{9\mu}^{\rm NP}$, the direct CP asymmetry can range from $[-0.2,0.2]$,
while the mixing-induced CP asymmetry remains close to the SM prediction. On
the other hand, Scenario 2 results in mixing-induced CP asymmetries as large
as $\pm 0.4$, although parts of the allowed region are already in tension with
the measurements of the direct CP asymmetries. With a complex phase in
$C_{10\mu}$ only (Scenario 3), the direct CP asymmetry remains zero.
Nevertheless, it allows for large mixing-induced CP asymmetries.

Figure 5: Correlations between the CP asymmetries $\mathcal{A}_{\rm CP}^{\rm
dir}$ and $\mathcal{A}_{\rm CP}^{\rm mix}$ for the three NP scenarios in (54).
The SM point is marked by a square, while the current experimental bound on
the direct CP asymmetry is illustrated by a green vertical band.

These studies illustrate the power of the correlations between the CP-
violating observables to distinguish different NP scenarios.

###  3.3 Distinguishing hadronic long-distance branches

In Fig. 6a, we show the correlations between the CP asymmetries for the
$Y_{--},Y_{-+},Y_{+-}$ and $Y_{++}$ branches for Scenario 1 and 2 again in the
range of $q^{2}\in[7,8]$ GeV2. We note that Scenario 3, with only
$C_{10\mu}^{\rm NP}$, is not included as it is not influenced by the different
branches. The branch multiplicity makes it more challenging to disentangle the
different NP scenarios.

In order to distinguish the branches, we use the high sensitivity of the
direct CP asymmetry to the long-distance model and the strong dependence on
the $q^{2}$-bin. Exploiting these features, we can once again distinguish
between the branches by using the CP asymmetry correlations in different parts
of the $q^{2}$-spectrum. In Fig. 6b and 6c, we show the correlations for two
bins with $q^{2}>11$ GeV2. These correlations are obtained by varying the
Wilson coefficients as follows:

  * •

Scenario 1: $\absolutevalue{C_{9\mu}^{\rm NP}}/\absolutevalue{C_{9}^{\rm
SM}}\in[0,0.75]$, $\phi_{9\mu}^{\rm NP}\in[90,270]^{\circ}$.

  * •

Scenario 2: $\absolutevalue{C_{9\mu}^{\rm NP}}/\absolutevalue{C_{9}^{\rm
SM}}\in[0,0.50]$, $\phi_{9\mu}^{\rm NP}\in[90,270]^{\circ}$.

From 6, we observe that the regions that overlap at lower $q^{2}$ drift apart.
To understand this, we consider the sign of the direct CP asymmetry:

${\rm sign}[\mathcal{A}_{\rm CP}^{\rm dir}]=\sin\phi_{9\mu}^{\rm
NP}\times\sin\delta_{Y}(q^{2})\ .$ (58)

In the bin $q^{2}\in[1.1,6.0]\;$\mathrm{G}\mathrm{eV}^{2}$$, the $J/\psi$
resonance provides the dominant contribution to $\delta_{Y}(q^{2})$. For
$q^{2}\in[12.5,13.5]\;$\mathrm{G}\mathrm{eV}^{2}$$, the $\psi(2S)$-threshold
opens and drives the sign of $\delta_{Y}(q^{2})$ (see Fig. 2b). At even higher
$q^{2}$, the higher $c\bar{c}$ resonances dominate and we are no longer
sensitive to the phases of the first two resonances. Due to all the different
contributions to $\delta_{Y}$ at different $q^{2}$, the sign of the direct CP
asymmetry varies across the $q^{2}$ spectrum, thereby giving distinct
fingerprints. We have specifically picked the $q^{2}$-bins of Fig. 6 to
illustrate how branches that are close together in some $q^{2}$-bins separate
in other bins.

By measuring the CP asymmetries in different parts of the $q^{2}$ spectrum, we
can thus differentiate between the hadronic long-distance branches.

(a) ${q^{2}\in[7,8]\;$\mathrm{G}\mathrm{eV}^{2}$}$

(b) ${q^{2}\in[11,11.8]\;$\mathrm{G}\mathrm{eV}^{2}$}$

(c) ${q^{2}\in[14,15]\;$\mathrm{G}\mathrm{eV}^{2}$}$

Figure 6: Correlations between $\mathcal{A}_{\rm CP}^{\rm dir}$ and
$\mathcal{A}_{\rm CP}^{\rm mix}$ in different $q^{2}$ bins including all four
hadronic long-distance branches.

###  3.4 Extracting Wilson coefficients from the CP asymmetries

If the CP asymmetries in $B\to K\mu^{+}\mu^{-}$ deviate from their SM values,
the next goal is to extract the Wilson coefficients. In the general scenario
of complex and independent NP contributions to both $C_{9\mu}$ and
$C_{10\mu}$, we have four parameters entering the observables and therefore we
need at least four observables to determine them from the measured data. Here
we demonstrate a minimal scenario with four observables, but stress that also
other information from different bins could be used. To this end, we use the
direct and mixing-induced CP asymmetries and the CP-averaged integrated
branching ratio of $B\to K\mu^{+}\mu^{-}$ in two different $q^{2}$ bins.
Utilizing the strength of each observable, we consider:

  * •

the direct CP asymmetry (21) integrated over $q^{2}\in[8,9]$,

  * •

the mixing-induced CP asymmetry (36) in $q^{2}\in[1.1,6.0]$,

  * •

the branching ratio in $q^{2}\in[1.1,6.0]$ and $q^{2}\in[15,22]$.

The branching ratios and the mixing-induced CP asymmetry are considered
outside the resonance region and thus less affected by long-distance effects.
On the other hand, for the direct CP asymmetry we pick the bin close to the
$J/\psi$ peak at $9\text{\,}\mathrm{G}\mathrm{eV}^{2}$ where it may be
enhanced.

Figure 7: Constraints on $C_{9\mu}$ and $C_{10\mu}$ in the complex plane
assuming the input measurements and uncertainties given in (59). The lines
show the $68\%$ and $90\%$ C.L. contours. Figure 8: Constraints in the complex
plane on $C_{9\mu}$, assuming $C_{9\mu}$ only and $C_{9\mu}=-C_{10\mu}$ for
the uncertainties as specified by (59) and (60).

In order to illustrate the extraction of Wilson coefficients from these four
observables, we consider benchmark point 2 specified in (55). This NP scenario
corresponds to

$\displaystyle\mathcal{A}_{\rm CP}^{\rm dir}[8,9]$ $\displaystyle=0.16\pm
0.02\ ,$ $\displaystyle\mathcal{A}_{\rm CP}^{\rm mix}[1.1,6]$
$\displaystyle=0.94\pm 0.04\ ,$ (59) $\displaystyle\mathcal{B}[1.1,6.0]$
$\displaystyle=(1.15\pm 0.02)\times 10^{-7}\ ,$
$\displaystyle\mathcal{B}[15,22]$ $\displaystyle=(0.908\pm 0.018)\times
10^{-7}\ ,$

where we consider a possible future scenario for the experimental
uncertainties. The uncertainties on our inputs in (59) allows us to explore
how the precision on the CP asymmetries translates into allowed regions for
the Wilson coefficients. Due to the complexity of the system of equations, we
perform a $\chi^{2}$ fit444If NP only enters through $C_{9\mu}$ or
$C_{9\mu}=-C_{10\mu}$, we can solve the system analytically using only the
direct and mixing-induced CP asymmetries as demonstrated in Appendix B. while
setting theory uncertainties to zero. Figure!7 shows $68\%$ and $90\%$ C.L.
regions for the extracted Wilson coefficients in the complex plane. We find a
good determination of the imaginary part of $C_{9\mu}$ but a less constraining
situation for $C_{10\mu}$. This can be improved by considering additional
$q^{2}$ bins and thus over-constraining the system.

It is interesting to consider the $C_{9\mu}$-only and $C_{9\mu}=-C_{10\mu}$
scenarios to show the possible precision of such an over-constrained fit. For
the $C_{9\mu}$-only scenario, we use benchmark point 1 of (55), which gives

$\displaystyle\mathcal{A}_{\rm CP}^{\rm dir}[8,9]$ $\displaystyle=0.15\pm
0.02\ ,$ $\displaystyle\mathcal{A}_{\rm CP}^{\rm mix}[1.1,6]$
$\displaystyle=0.66\pm 0.04\ ,$ (60) $\displaystyle\mathcal{B}[1.1,6.0]$
$\displaystyle=(1.16\pm 0.02)\times 10^{-7}\ ,$
$\displaystyle\mathcal{B}[15,22]$ $\displaystyle=(0.806\pm 0.016)\times
10^{-7}\ ,$

where the uncertainties again indicate a possible future scenario. Using the
inputs in (60) for $C_{9\mu}$ only and (59) for $C_{9\mu}=-C_{10\mu}$, we find
the $68\%$ and $90\%$ C.L. regions in Fig. 8.

Finally, we illustrate our strategy in Fig. 9. By combining the direct and
mixing-induced CP asymmetries with the branching ratios in specific bins, we
can optimally exploit the complementarity of these observables. In this way,
both the hadronic long-distance effects as well as the short-distance Wilson
coefficients can be determined from the data.

Figure 9: Illustration of the strategy to determine complex Wilson
coefficients $C_{9\mu}$ and $C_{10\mu}$.

##  4 Testing lepton flavour universality

###  4.1 Setting the stage

Previously, we considered new sources of CP violation specifically for the
muonic channel. In the SM, the only difference between the muon and electron
channels is caused by tiny phase space effects, while $C_{9}$ and $C_{10}$ are
strictly lepton-flavour universal. It is possible that NP enters in a lepton-
flavour universal way as in the SM. On the other hand, NP may also affect the
different generation distinctly (see e.g. [62, 63, 64, 65, 66]). When testing
for lepton-flavour universality, special care is needed because new sources of
CP violation may play a role. To demonstrate this, we define the integrated
rates as

$\bar{\Gamma}_{\ell}[q^{2}_{\rm min},q^{2}_{\rm max}]\equiv\int_{q^{2}_{\rm
min}}^{q^{2}_{\rm max}}dq^{2}\frac{d\Gamma(B^{-}\to
K^{-}\mu^{+}\mu^{-})}{dq^{2}}$ (61)

and equivalently for $\Gamma_{\ell}$ for the $B^{+}\to K^{+}\mu^{+}\mu^{-}$.
For simplicity, we omit the explicit $q^{2}$-bin in the following. Deviations
from lepton flavour universality between muons and electrons are then probed
through the ratios

$R_{K}\equiv\frac{\Gamma_{\mu}}{\Gamma_{e}}\
,\quad\quad\quad\bar{R}_{K}\equiv\frac{\bar{\Gamma}_{\mu}}{\bar{\Gamma}_{e}}\
,$ (62)

for the $B^{+}$ and its conjugate $B^{-}$ mode, respectively.

We note that, usually, the CP-averaged ratio is quoted as $R_{K}$, although
this is often not explicitly written. To make this difference explicit, we
define

$\langle
R_{K}\rangle=\frac{\Gamma_{\mu}+\bar{\Gamma}_{\mu}}{\Gamma_{e}+\bar{\Gamma}_{e}}\
.$ (63)

We note that this averaged quantity is not the same as the sum of $R_{K}$ and
$\bar{R}_{K}$ as defined in (62) but rather

$\langle
R_{K}\rangle=\frac{1}{2}\left[R_{K}+\bar{R}_{K}+(\bar{R}_{K}-R_{K})\mathcal{A}_{\rm
CP,e}^{\rm dir}\right]\,$ (64)

where the direct CP asymmetry for the electron mode is defined analogously to
the muon mode in (17). In the SM, $R_{K}$ and $\bar{R}_{K}$ are $1$ to an
excellent precision, even when including tiny QED effects [67, 68].

The latest LHCb measurements in the
$q^{2}\in[1.1,6.0]\>$\mathrm{G}\mathrm{eV}^{2}$$-bin reads [3]

$\langle R_{K}\rangle[1.1,6.0]=0.846^{+0.044}_{-0.041}\ .$ (65)

This measurement deviates from the SM prediction with a significance of
$3.1\sigma$ [3]. Measurements of this quantity have also been given by the
BaBar [5] and Belle [4] collaborations.

Using the LHCb measurement of the muon channel in (52), combined with (65)
gives [3]:

$\mathcal{B}(B\to Ke^{+}e^{-})[1.1,6.0]=(1.40\pm 0.10)\times 10^{-7}\ .$ (66)

As said, the SM prediction only differs from the the muonic channel in (42)
through tiny phase space effects. Taking (42), we find

$\frac{\mathcal{B}(B^{-}\to K^{-}e^{+}e^{-})^{\rm
SM}[1.1,6.0]_{\rm{{incl/hybrid}}}}{\mathcal{B}(B^{\pm}\to
K^{\pm}e^{+}e^{-})[1.1,6.0]}-1=0.31\pm 0.14\ ,$ (67)

differing by $2.2\sigma$. A somewhat larger difference was found in [60].
Using the exclusive CKM factors, we find

$\frac{\mathcal{B}(B^{-}\to K^{-}e^{+}e^{-})^{\rm SM}[1.1,6.0]_{\rm
excl.}}{\mathcal{B}(B^{\pm}\to K^{\pm}e^{+}e^{-})[1.1,6.0]}-1=0.13\pm 0.12\ ,$
(68)

which differs by $1.1\sigma$. We note that even though these SM predictions
suffer from uncertainties due to the modeling of the long-distances effects,
they seem to indicate consistently that there may be NP present in the muon
and/or electron channels.

###  4.2 New CP-violating couplings

If $C_{9\mu}$ is different from $C_{9e}$, they would cause a difference
between $R_{K}$ and $\bar{R}_{K}$. For completeness, we note that if we have
NP in $C_{T}$, which enters with proportional to the mass, this would also
give an effect if $C_{T\mu}=C_{Te}$. However, precisely because they are
proportional to the mass, these effects are also suppressed (see previous
discussions). Therefore, we do not further discuss such terms.

We stress that it important to separately measure these two ratios. The amount
of CP violation in $R_{K}$ can be defined as

$\mathcal{A}_{\rm
CP}^{R_{K}}\equiv\frac{\bar{R}_{K}-R_{K}}{\bar{R}_{K}+R_{K}}\ .$ (69)

This new observable in (69) provides a measure of whether lepton flavour non-
universal NP in $B\to K\ell^{+}\ell^{-}$ are also CP violating. We can rewrite
(69) in terms of the direct CP asymmetries of the individual muonic and
electronic decay channels:

$\displaystyle\mathcal{A}_{\rm CP}^{R_{K}}$
$\displaystyle=\left[\frac{\mathcal{A}_{\rm CP}^{\rm dir,\mu}-\mathcal{A}_{\rm
CP}^{\rm dir,e}}{1-\mathcal{A}_{\rm CP}^{\rm dir,\mu}\ \mathcal{A}_{\rm
CP}^{\rm dir,e}}\right]\ ,$ (70)

where $\mathcal{A}_{\rm CP}^{\rm dir,\mu}$ and $\mathcal{A}_{\rm CP}^{\rm
dir,e}$ denote the direct CP asymmetries of $B\to K\mu^{+}\mu^{-}$ and $B\to
Ke^{+}e^{-}$, respectively defined in (17). From (70), we observe that the
$\mathcal{A}_{\rm CP}^{R_{K}}$ will be enhanced if $\mathcal{A}_{\rm CP}^{\rm
dir,\mu}$ has an opposite sign to $\mathcal{A}_{\rm CP}^{\rm dir,e}$. In
addition, if the muonic and electronic direct CP asymmetries are identical,
the observable vanishes. Therefore, any measurement of this observable is a
clear sign of CP-violating NP with different magnitude and phase for the
electron and muon channels.

Interestingly, is also possible to access the electronic direct CP asymmetry
directly through separate measurements of the ratios for the $B^{-}$ and
$B^{+}$. Specifically, we find

$\mathcal{A}_{\rm CP}^{\rm dir,e}=\frac{2\langle R_{K}\rangle-
R_{K}-\bar{R}_{K}}{\bar{R}_{K}-R_{K}}=\frac{2\langle
R_{K}\rangle}{\bar{R}_{K}-R_{K}}-\frac{1}{\mathcal{A}_{\rm CP}^{R_{K}}}\ .$
(71)

At the moment, the only limit on the $\mathcal{A}_{\rm CP}^{\rm dir,e}$ comes
from the Belle Collaboration555They use a weighted average over different
$q^{2}$ bins, both below and above the $c\bar{c}$ resonances. [69]:

$\mathcal{A}_{\rm CP}^{\rm dir,e}=0.14\pm 0.14\ ,$ (72)

therefore, it would interesting to see if the new observable could be used to
obtain a more precise determination.

##  5 Conclusions

The charged and neutral $B\to K\mu^{+}\mu^{-}$ decays offer exciting probes of
New Physics. In this paper, our main focus has been on imprints of possible
new sources of CP violation. We have performed a comprehensive analysis of
these channels, using the most recent state-of-the-art lattice QCD
calculations for the required non-perturbative form factors.

For the $q^{2}$ regions with a large impact of hadronic $c\bar{c}$ resonances,
we have implemented a model by the LHCb collaboration using experimental data.
The corresponding fit results in different branches of the hadronic parameters
describing the resonances. Interestingly, these effects generate CP-conserving
strong phases, which are necessary ingredients for direct CP violation. In the
SM, such CP asymmetries are negligibly small. However, new CP-violating phases
arising in the short-distance Wilson coefficients may lead to sizeable CP
violation, thereby signalling the presence of New Physics.

In our analysis, we have complemented these direct CP asymmetries with mixing-
induced CP violation in the $B^{0}_{d}\to K_{S}\mu^{+}\mu^{-}$ decay. We have
pointed out that the interplay of the CP asymmetries, utilising also
information from the differential decay rates, allows us to distinguish
between the different hadronic parameter sets describing the resonance
effects, as well as between different NP scenarios. In these studies, we have
considered the information for the time-dependent differential decay rates
integrated over the angle describing the kinematics of the muon pair, thereby
simplifying the analysis. Measuring angular distributions would provide even
more information.

We highlight that, in agreement with other studies, we have found that the
differential decay rates calculated in the SM are significantly smaller than
the experimental values, thereby indicating the presence of NP. We have
pointed out that the discrepancies between inclusive and exclusive
determinations of $|V_{cb}|$ – and to a smaller extent of $|V_{ub}|$ – have a
large impact on the branching ratios. While the exclusive values result in a
difference of the charged $B\to K\mu^{+}\mu^{-}$ branching ratio in the
$q^{2}$ region within [1.1,6.0] $\mathrm{G}\mathrm{eV}^{2}$ at the
$2.5\,\sigma$ level, a hybrid scenario pairing the inclusive value of
$|V_{cb}|$ with the exclusive value of $|V_{ub}|$ results in a SM branching
ratio lying $3.5\,\sigma$ below the measured result. The latter CKM
combination gives the most consistent picture of constraints on neutral $B$
mixing with the Standard Model. It will be important to finally resolve the
issues in the determination of these CKM matrix elements.

We have presented a new strategy to determine the complex Wilson coefficients
$C_{9\mu}$ and $C_{10\mu}$ from measurements of the branching ratios as well
as the direct and mixing-induced CP asymmetries in appropriate $q^{2}$
regions. The method makes use of our finding that different NP sources lead to
distinct “fingerprints”, allowing a transparent determination of these
coefficients without making specific assumptions, such as having NP only in
$C_{9\mu}$ or assuming the relation $C_{9\mu}^{\rm NP}=-C_{10\mu}^{\rm NP}$,
as is frequently done in the literature. Since not all required measurements
are yet available, we have illustrated this method through specific examples.

In the presence of NP, we may have a violation of lepton flavour universality
which is probed through the $R_{K}$ ratio. Using the different determinations
for the CKM factors, we found that the integrated SM branching ratio of the
$B^{-}\to K^{-}e^{+}e^{-}$ channel for $q^{2}$ in [1.1,6.0]
$\mathrm{G}\mathrm{eV}^{2}$ is about $2\sigma$ below the corresponding
experimental result for the hybrid/inclusive case and about $1\sigma$ for the
exclusive. We have pointed out that new sources of CP violation require
special care in studies to distinguish between $B\to K\ell^{+}\ell^{-}$ decays
and their CP conjugates for the final states with muons and electrons. We have
presented a new method to measure direct CP violation in the $B\to
Ke^{+}e^{-}$ modes using only $R_{K}$-like ratios for decays and their CP
conjugates.

It will be very interesting to monitor how the data will evolve in the future.
The methods to "fingerprint" CP-violating NP in $B\to K\mu^{+}\mu^{-}$ decays
offer an exciting playground for the future high-precision era of $B$ physics.
It will also be exciting the see whether new sources of CP violation can be
revealed in these semileptonic rare $B$ decays, and whether they will
complement current puzzles in CP violation in non-leptonic $B$ decays.

## Acknowledgements

This research has been supported by the Netherlands Organisation for
Scientific Research (NWO).

##  Appendix A Coefficients $\rho_{i}$

In Table 1 we list numerical values for the $\rho_{i}$ coefficients defined in
(23) and (24) integrated over three different $q^{2}$ bins. For the
coefficients that are sensitive to the choice of long-distance branch, we give
separate values for the four different $Y_{\pm\pm}$ branches. For the other
coefficients, we sum over the branches.

Coefficient | Value per $q^{2}$ bin ($\times 10^{2}$ except $\Gamma_{\rm SM}$)
---|---
| $[1.1-6.0]$ | $[8-9]$ | $[15-22]$
$\Gamma_{\rm SM}$ | $(7.34\pm 0.58)\times 10^{-20}$ | $(1.63\pm 0.19)\times 10^{-20}$ | $(5.61\pm 0.50)\times 10^{-20}$
$\rho_{9}/\Gamma_{\rm SM}$ | $3.20\pm 0.04$ | $2.80\pm 0.27$ | $2.67\pm 0.10$
$\rho_{10}/\Gamma_{\rm SM}$ | $3.20\pm 0.04$ | $2.80\pm 0.27$ | $2.68\pm 0.10$
$\rho_{P}/\Gamma_{\rm SM}$ | $1.03\pm 0.05$ | $2.41\pm 0.28$ | $9.81\pm 0.79$
$\rho_{S}/\Gamma_{\rm SM}$ | $1.02\pm 0.05$ | $2.40\pm 0.28$ | $9.78\pm 0.79$
$\rho_{T}/\Gamma_{\rm SM}$ | $0.702\pm 0.078$ | $1.44\pm 0.20$ | $2.79\pm 0.33$
$\rho_{9\rm Re}/\Gamma_{\rm SM}$ | $22.6\pm 0.15$ | $19.3\pm 1.7$ | $22.4\pm 0.4$
$\rho_{10\rm Re}/\Gamma_{\rm SM}$ | $-27.6\pm 0.3$ | $-24.1\pm 2.3$ | $-23.1\pm 0.8$
$\rho_{P\rm Re}/\Gamma_{\rm SM}$ | $-2.20\pm 0.11$ | $-2.16\pm 0.25$ | $-4.08\pm 0.33$
$\rho_{T\rm Re}/\Gamma_{\rm SM}$ | $1.24\pm 0.08$ | $1.06\pm 0.11$ | $1.22\pm 0.08$
$\rho_{10P\rm Re}/\Gamma_{\rm SM}$ | $0.509\pm 0.025$ | $0.502\pm 0.058$ | $0.948\pm 0.077$
$\rho_{9T\rm Re}/\Gamma_{\rm SM}$ | $0.351\pm 0.024$ | $0.306\pm 0.035$ | $0.290\pm 0.022$
$\rho_{10P\rm Im}/\Gamma_{\rm SM}$ | $-0.509\pm 0.025$ | $-0.501\pm 0.059$ | $-0.949\pm 0.077$
$\rho_{9T\rm Im}/\Gamma_{\rm SM}$ | $-0.351\pm 0.024$ | $-0.306\pm 0.036$ | $-0.289\pm 0.022$
$\rho_{9\rm Im}/\Gamma_{\rm SM}$($Y_{--}$) | $-3.59\pm 0.78$ | $-13.2\pm 3.1$ | $-2.09\pm 1.09$
$\rho_{9\rm Im}/\Gamma_{\rm SM}$($Y_{-+}$) | $-2.02\pm 0.78$ | $-10.7\pm 3.3$ | $-2.59\pm 0.83$
$\rho_{9\rm Im}/\Gamma_{\rm SM}$($Y_{+-}$) | $1.25\pm 0.77$ | $9.66\pm 3.23$ | $-1.65\pm 0.86$
$\rho_{9\rm Im}/\Gamma_{\rm SM}$($Y_{++}$) | $2.80\pm 0.80$ | $12.3\pm 3.3$ | $-5.85\pm 0.76$
$\rho_{T\rm Im}/\Gamma_{\rm SM}$($Y_{--}$) | $0.197\pm 0.045$ | $0.723\pm 0.176$ | $-0.112\pm 0.059$
$\rho_{T\rm Im}/\Gamma_{\rm SM}$($Y_{-+}$) | $0.111\pm 0.045$ | $0.586\pm 0.187$ | $0.141\pm 0.047$
$\rho_{T\rm Im}/\Gamma_{\rm SM}$($Y_{+-}$) | $-0.0673\pm 0.0436$ | $-0.524\pm 0.187$ | $0.0889\pm 0.0477$
$\rho_{T\rm Im}/\Gamma_{\rm SM}$($Y_{++}$) | $-0.155\pm 0.045$ | $-0.680\pm 0.182$ | $0.319\pm 0.047$
Table 1: Numerical values of the coefficients in Appendix A. $q^{2}$ ranges
are given in units of $\mathrm{G}\mathrm{eV}^{2}$. All results are specific to
$B^{\pm}\to K^{\pm}\mu^{+}\mu^{-}$.

##  Appendix B Analytical solution

In this section, we derive the analytic expressions for $C_{9}$ and its phase
$\phi_{9}$ in terms of the mixing-induced CP asymmetry and the direct CP
asymmetry assuming either NP only in $C_{9\mu}$ or in $C_{9\mu}=-C_{10\mu}$.
To do this, we use (18) and (34), describing the numerators of the direct and
mixing-induced CP asymmetries. To solve the system, we write:

$\displaystyle s_{0}(q^{2})$ $\displaystyle=s_{0}^{\rm
SM}(q^{2})+a(q^{2})\absolutevalue{C_{9}^{\rm
NP}}\cos\phi_{9}+b(q^{2})\absolutevalue{C_{9}^{\rm
NP}}\sin\phi_{9}+c(q^{2})\absolutevalue{C_{9}^{\rm
NP}}^{2}\bigg{(}\cos^{2}\phi_{9}-\sin^{2}\phi_{9}\bigg{)}$ (73)
$\displaystyle+d(q^{2})\absolutevalue{C_{9}^{\rm
NP}}^{2}\sin\phi_{9}\cos\phi_{9}+e(q^{2})\absolutevalue{C_{10}^{\rm
NP}}\left(\absolutevalue{C_{10}^{\rm
SM}}\cos\phi_{10}+\frac{1}{2}\absolutevalue{C_{10}^{\rm
NP}}\bigg{(}\cos^{2}\phi_{10}-\sin^{2}\phi_{10}\bigg{)}\right)$
$\displaystyle+f(q^{2})\absolutevalue{C_{10}^{\rm
NP}}\sin\phi_{10}\left(\absolutevalue{C_{10}^{\rm
SM}}+\absolutevalue{C_{10}^{\rm NP}}\cos\phi_{10}\right)\ ,$

with

$\displaystyle s_{0}^{\rm SM}=$
$\displaystyle\mathcal{C}_{n}\bigg{\\{}\frac{\lambda_{B}}{6q^{2}}(1+2\hat{m}_{\ell}^{2})\bigg{(}\bigg{(}\frac{2m_{b}}{m_{B}+m_{K}}C_{7}f_{T}(q^{2})\bigg{)}^{2}$
(74)
$\displaystyle+\frac{4m_{b}}{m_{B}+m_{K}}C_{7}f_{T}(q^{2})f_{+}(q^{2})\bigg{(}C_{9}^{\rm
SM}+\real Y(q^{2})\bigg{)}$ $\displaystyle+f_{+}^{2}(q^{2})\bigg{[}(C_{9}^{\rm
SM})^{2}+2C_{9}^{\rm SM}\real
Y(q^{2})+\absolutevalue{Y(q^{2})}^{2}\bigg{]}\bigg{)}$
$\displaystyle+\bigg{(}\frac{\lambda_{B}}{6q^{2}}\beta_{\ell}^{2}f_{+}^{2}(q^{2})+(m_{B}^{2}-m_{K}^{2})^{2}\frac{m_{\ell}^{2}}{q^{4}}f_{0}^{2}(q^{2})\bigg{)}(C_{10}^{\rm
SM})^{2}\bigg{\\}}\ ,$ $\displaystyle a(q^{2})$
$\displaystyle=\mathcal{C}_{n}\frac{\lambda_{B}}{3q^{2}}(1+2\hat{m}_{\ell}^{2})\bigg{(}\frac{2m_{b}}{m_{B}+m_{K}}C_{7}f_{T}(q^{2})f_{+}(q^{2})+f_{+}^{2}(q^{2})(C_{9}^{\rm
SM}+\real Y(q^{2}))\bigg{)}\ ,$ $\displaystyle b(q^{2})$
$\displaystyle=a(\mathcal{C}_{n}\to\hat{\mathcal{C}}_{n})\ ,$ $\displaystyle
c(q^{2})$
$\displaystyle=\mathcal{C}_{n}\frac{\lambda_{B}}{6q^{2}}(1+2\hat{m}_{\ell}^{2})f_{+}^{2}(q^{2})\
,$ $\displaystyle d(q^{2})$
$\displaystyle=c(\mathcal{C}_{n}\to\hat{\mathcal{C}}_{n})\ ,$ $\displaystyle
e(q^{2})$
$\displaystyle=\mathcal{C}_{n}\bigg{(}\frac{\lambda_{B}}{3q^{2}}\beta_{\ell}^{2}f_{+}^{2}(q^{2})+2(m_{B}^{2}-m_{K}^{2})^{2}\frac{m_{\ell}^{2}}{q^{4}}f_{0}^{2}(q^{2})\bigg{)}\
,$ $\displaystyle f(q^{2})$
$\displaystyle=e(\mathcal{C}_{n}\to\hat{\mathcal{C}}_{n})\ ,$ (75)

where we also took the small CKM phase in the normalization into account via

$\mathcal{C}_{n}\equiv\bigg{[}-4\cos\phi_{d}\real\mathcal{N}\imaginary\mathcal{N}-2\sin\phi_{d}((\real\mathcal{N})^{2}-(\imaginary\mathcal{N})^{2})\bigg{]}\
,$ (76)

and

$\hat{\mathcal{C}}_{n}\equiv\bigg{[}+4\sin\phi_{d}\real\mathcal{N}\imaginary\mathcal{N}-2\cos\phi_{d}((\real\mathcal{N})^{2}-(\imaginary\mathcal{N})^{2})\bigg{]}\
.$ (77)

Taking now the differential rate for the $B\to K_{S}$666The charged rate can
also be used by taking into account the normalization factor., we find that

$\absolutevalue{C_{9}^{\rm NP}}=\frac{(\bar{\Gamma}-\Gamma)}{\rho_{9\rm
Im}\sin\phi_{9}^{\rm NP}}=\frac{2\mathcal{B}(B_{d}^{0}\to
K_{S}\ell^{+}\ell^{-})\mathcal{A}_{\rm CP}^{\rm dir}\tau_{B}}{\rho_{9\rm
Im}\sin\phi_{9}^{\rm NP}}\ .$ (78)

Finally, we can solve the system of equations. For $C_{10}^{\rm NP}=0$, we
then find

$\displaystyle\cot\phi_{9}^{\rm NP}=-\frac{1}{2c}\bigg{(}a\frac{\rho_{9\rm
Im}}{(\bar{\Gamma}-\Gamma)}+d\bigg{)}$
$\displaystyle\pm\sqrt{\frac{1}{(2c)^{2}}\bigg{(}a\frac{\rho_{9\rm
Im}}{(\bar{\Gamma}-\Gamma)}+d\bigg{)}^{2}-\bigg{(}\frac{(s_{0}^{\rm
SM}-s_{0})}{c}\frac{\rho_{9\rm
Im}^{2}}{(\bar{\Gamma}-\Gamma)^{2}}+\frac{b}{c}\frac{\rho_{9\rm
Im}}{(\bar{\Gamma}-\Gamma)}-1\bigg{)}}\ .$ (79)

For the scenario in which $C_{9}=-C_{10}$, the solution is obtained from (B)
by replacing:

$d\to d-f\ ,\quad a\to a+C_{10}^{\rm SM}e\ ,\quad b\to b-C_{10}^{\rm SM}f\
,\quad c\to c+\frac{e}{2}\ .$ (80)

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