#  $CP$ violation in $b\to s\ell\ell$: a model independent analysis

Neetu Raj Singh Chundawat<EMAIL_ADDRESS>Indian Institute of
Technology Jodhpur, Jodhpur 342037, India

###### Abstract

We perform a model-independent global fit to all germane and updated $b\to
s\ell\ell$ data assuming new physics couplings to be complex. Under the
approximation that new physics only affects the muon sector and that either
one or two related operators contribute at a time, we identify scenarios which
provide a good fit to the data. It turns out that the favored scenarios remain
the same as obtained for the real fit but the order of solutions based on the
statistical significance has changed. Further, the magnitude of complex
couplings can be as large as that of their real counterparts and these are
reflected in direct $CP$ asymmetry, $A_{\rm CP}$, in $B\to K\mu^{+}\mu^{-}$
along with a number of angular $CP$ asymmetries, $A_{i}$, in $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ decay. The sensitivities of these observables to various
solutions are different in the low and high-$q^{2}$ bins. We also identify
important correlations between $A_{\rm CP}$ and several $A_{i}$ observables. A
precise measurement of $A_{\rm CP}$ and $A_{i}$ observables can not only
confirm the existence of additional weak phases but can also enable an unique
determination of Lorentz structure of possible new physics in $b\to
s\mu^{+}\mu^{-}$ transition.

##  I Introduction

One of the key open problems in particle physics is the observed recalcitrant
disparity between the amount of matter and antimatter in the Universe. It is
expected that the Big Bang explosion would have created matter and antimatter
in equal amounts. However, it is still not understood how one type of matter
triumphed over another in the early Universe. Sakharov’s conditions provide
three necessary ingredients required to create the observed baryon asymmetry
[1]. One of these conditions requires a $CP$ violation which can be generated
through a complex phase in the Lagrangian that cannot be reabsorbed through
the rephasing of the apposite fields.

The SM of electroweak interactions allows for $CP$ violations owing to a
complex phase in the quark mixing matrix. The 3 $\times$ 3 CKM matrix can be
parametrized by three angles and a single complex phase. This single phase of
the CKM matrix is the only source of $CP$ violation in the SM. This phase
evinces itself in several observables in the decays of $K$ and $B$ mesons. In
fact, the BaBar and Belle experiments established the CKM paradigm of $CP$
violation through several measurements of observables in the decays of $B$
mesons. However, unlike parity violation, which is maximal, the observed $CP$
violation is small and cannot account for the observed baryon asymmetry. The
amount of predicted baryons in the Universe using the CKM formalism falls
several orders of magnitude short of the observed value. Therefore one needs
to explore beyond the CKM paradigm of the SM.

The $CP$ violating observables in the decays induced by the quark level
transition $b\to s\mu^{+}\mu^{-}$ are particularly important in probing new
physics. This is because these observables are highly suppressed in the SM.
i.e., they are predicted to be less than a percent level in the SM [2, 3].
Even after including the next-to-leading order QCD corrections and hadronic
uncertainties, the $CP$ asymmetries are not expected to transcend 1% [4, 5,
6]. The $CP$-violating observables can be measured at LHC or at Belle-II
provided new physics enhances them to a level of a few percent. Therefore
measurement of any $CP$ violating observable in $b\to s\mu^{+}\mu^{-}$ sector
will provide a luculent signature of new physics.

The decay mode $b\to s\mu^{+}\mu^{-}$ is already in spotlight for a decade due
to the fact that it has provide a number of observables whose measurements are
in contention with the predictions of the SM. These include measurement of the
ratio $R_{K}\equiv\Gamma(B^{+}\to K^{+}\mu^{+}\mu^{-})/\Gamma(B^{+}\to
K^{+}e^{+}e^{-})$ which shows a scantiness of 3.1$\sigma$ as compared to the
SM value in the (1.1 $\mathrm{GeV}^{2}\leq q^{2}\leq$ 6.0 $\mathrm{GeV}^{2}$)
bin. Here $q^{2}$ is the dilepton invariant mass-squared [7]. The measurements
of analogous ratio, $R_{K^{*}}$, in the (0.045 $\rm{GeV}^{2}\leq q^{2}\leq$
1.1 $\rm{GeV}^{2}$) and (1.1 $\rm{GeV}^{2}\leq q^{2}\leq$ 6.0 $\rm{GeV}^{2}$)
bins also nonconcur with the SM at level of $\sim$ 2.5$\sigma$ [8]. These
contestations, known as lepton flavour universality violation (LFUV), can be
accredited to new physics in $b\to s\,\mu^{+}\mu^{-}$ or/and $b\to
s\,e^{+}e^{-}$ decays. Apart from these LFU ratios, the LHCb collaboration has
also provided measurements of new LFU ratios in $B^{0}\to
K_{S}^{0}\mu^{+}\mu^{-}$ and $B^{+}\to K^{*+}\mu^{+}\mu^{-}$ the channels [9].
These measurements concur with the SM at $\sim 1.5\sigma$ level.

There are number of other observables, related only to the muon sector, whose
predictions swerve from the SM. Many such observables are in
$B_{s}\to\phi\,\mu^{+}\,\mu^{-}$ and $B\to K^{*}\,\mu^{+}\,\mu^{-}$ decays.
For e.g., the experimental value of the branching ratio of
$B_{s}\to\phi\,\mu^{+}\,\mu^{-}$ decay ostentates tension with the SM at
3.5$\sigma$ level [10, 11]. The measurement of the optimized angular
observable $P^{\prime}_{5}$ in $B\to K^{*}\,\mu^{+}\,\mu^{-}$ decay in the
(4.0 $\mathrm{GeV}^{2}\leq q^{2}\leq$ 6.0 $\mathrm{GeV}^{2}$) bin deviates
from the SM prediction at the level of 3$\sigma$ [12, 13, 14, 15]. The
measured value of the branching ratio of the decay $B_{s}\to\mu^{+}\mu^{-}$
also regales tension with the SM at 2$\sigma$ level [16, 17, 18, 19, 20, 21].

In order to determine the Lorentz structure of possible new physics that can
accommodate the anomalous measurements in $b\to s\ell\ell$ decays, a model
independent analysis can be performed using the language of effective field
theory [22, 23, 24, 25, 21, 26, 27, 28, 29, 30, 31]. Barring a few [24, 26],
most of these analyses assume new physics Wilson Coefficients (WCs) to be
real. In this work, we allow the new physics WCs to be complex and perform a
global analysis of all $CP$-conserving $b\to s\ell\ell$ data under the
assumption that the beyond SM contributions affect only the muon sector. Apart
from LFU ratios mentioned above, the branching ratios of $B\to
X_{s}\mu^{+}\mu^{-}$, $B^{0}\to K^{0}\mu^{+}\mu^{-}$, $B^{+}\to
K^{+}\mu^{+}\mu^{-}$, $B^{0}\to K^{*0}\mu^{+}\mu^{-}$, $B^{+}\to
K^{*+}\mu^{+}\mu^{-}$ and $B_{s}^{0}\to\phi\mu^{+}\mu^{-}$ in several $q^{2}$
bins along with $B(B_{s}^{0}\to\mu^{+}\mu^{-})$ are included in the fits.
Further, we include a number of $CP$-conserving angular observables in
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$, $B^{+}\to K^{*+}\mu^{+}\mu^{-}$ and
$B_{s}^{0}\to\phi\mu^{+}\mu^{-}$ decays. These observables are obtained by
averaging over the angular distributions of $B$ and ${\bar{B}}$ decays.

We take the most frugal approach by considering only one operator or two
related operators at a time. For statistically favoured scenarios, we then
obtain predictions for several $CP$-violating observables. For $B^{+}\to
K^{+}\mu^{+}\mu^{-}$ decay, we calculate the direct $CP$ asymmetry, $A_{\rm
CP}$, whereas for $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay, a number of
$CP$-asymmetries are analyzed. These are obtained by comparing the angular
distributions of the corresponding $B$ and ${\bar{B}}$ decays.

For these solutions, we also obtain predictions of a few LFU ratios
$R_{\phi}\equiv\Gamma(B_{s}\to\phi\mu^{+}\mu^{-})/\Gamma(B_{s}\to\phi
e^{+}e^{-})$ in both low and high-$q^{2}$ bins. We also predict $R_{K^{(*)}}$
in the high $q^{2}$ bin. Further, we also study the LFU violating observables
$Q_{4,5}$ which can be constructed as
$Q_{4,5}=P^{{}^{\prime}\mu}_{4,5}-P^{{}^{\prime}e}_{4,5}$ [32]. These LFU
differences are inconsequentially small in the SM and hence measurement of any
of these observables will provide an univocal signals of physics beyond SM.
These predictions may allow us to discriminate between the various favored
solutions. Finally we study correlations of a number of observables, both
$CP$-violating as well as conserving with $A_{\rm CP}$ which is expected to be
measured with the highest statistical significance amongst all $CP$
asymmetries. These correlations would not only reveal the impact of new
physics phase on various quantities but would also help in sequestering
between the allowed scenarios.

Plan of the work is as follows. In Sec. II, we discuss the methodology adopted
in this work. We then provide the fit results. Moreover, we obtain predictions
of additional LFU ratios which are yet to me measured along with $Q_{4,\,5}$
observables. Using the fit results, we calculate the direct $CP$ asymmetry in
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ in Sec. 4. Here we also study correlations
between $A_{CP}$ and LFU ratio in this decay mode. In the following section,
we obtain predictions of a number of $CP$-violating observables in $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ decay. We also study correlations between $A_{CP}$ and
several $CP$-conserving and $CP$-violating observables related to $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ decay. Finally, we conclude in Sec. V.

##  II A fit to all $b\to s\ell\ell$ data

We start by performing a global fit to all $CP$ conserving data in $b\to
s\ell\ell$ by assuming new physics WCs to be complex. The data includes the
new measurements of $R_{K^{0}_{S}}$, $R_{K^{*+}}$ [9] and the updated
measurements for several $B_{s}\to\phi\mu^{+}\mu^{-}$ observables [11, 33]. We
closely follow the methodology adopted in [31] where the new physics couplings
were assumed to be real. In this section, we intend to espy the following:

  * •

The impact of the assumption of the complex coupling on the fit, i.e to spell
out the differences between the real and complex fits by making use of the
most updated data.

  * •

The upper limit on the allowed parameter space of the new weak phases
accredited by the current data. This will enable us to identify various $CP$
violating observables where large enhancement over the SM value is possible.

  * •

To discriminate between the allowed solutions by analysing additional LFU
violating observables.

###  II.1 Methodology

We include following $CP$ conserving observables in out fit:

  1. 1.

LFU ratios:  Within the SM, $R_{K}$ is predicted to be close to unity owing to
LFU which is deeply instilled in the symmetry structure of the SM. To be more
specific, $R_{K}^{\rm SM}=1\pm 0.01$ [34]. This ratio was first measured in
2014 by the LHCb collaboration [35]. The current measured value of $R_{K}$ in
$1.0\,{\rm GeV}^{2}\leq q^{2}\leq 6.0\,{\rm GeV}^{2}$ bin is $R_{K}^{\rm
exp}=0.846^{+0.044}_{-0.041}$ [7] which retrogresses from the SM prediction at
the level of 3.1$\sigma$. This can be considered as an inkling of LFUV.

In 2017, the notion of LFUV in $b\to s\ell\ell$ was substantiated by the
observation of the ratio $R_{K^{*}}$ by the LHCb collaboration [8]. This
measurement was performed in two $q^{2}$ bins. The measured values

$R_{K^{*}}^{\rm exp}=\left\\{\begin{array}[]{cc}0.660^{+0.110}_{-0.070}~{}\pm
0.024~{},{}{}&0.045\leq q^{2}\leq 1.1~{}{\rm GeV}^{2}~{},\\\
0.685^{+0.113}_{-0.069}~{}\pm 0.047~{},{}{}&1.1\leq q^{2}\leq 6.0~{}{\rm
GeV}^{2}~{},\end{array}\right.$ (1)

detour from the SM predictions [36, 37] at the level of $\sim$ 2.5$\sigma$. We
also include measurements of $R_{K^{*}}$ by the Belle Collaboration in
$0.045\,{\rm GeV}^{2}<q^{2}<1.1\,{\rm GeV}^{2},\,1.1\,{\rm
GeV}^{2}<q^{2}<6.0\,{\rm GeV}^{2},$ and $15.0\,{\rm GeV}^{2}<q^{2}<19.0\,{\rm
GeV}^{2}$ bins [38].

The recent new measurements of the ratios $R_{K_{S}^{0}}\equiv\Gamma(B^{0}\to
K_{S}^{0}\mu^{+}\mu^{-})/\Gamma(B^{0}\to K_{S}^{0}e^{+}e^{-})$ and
$R_{K^{*+}}\equiv\Gamma(B^{+}\to K^{*+}\mu^{+}\mu^{-})/\Gamma(B^{+}\to
K^{*+}e^{+}e^{-})$ are also included. The measured value of
$R_{K_{S}^{0}}^{\rm exp}=0.66^{+0.20+0.02}_{-0.14-0.04}$ and $R_{K^{*+}}^{\rm
exp}=0.70^{+0.18+0.03}_{-0.13-0.04}$ in $1.1\,{\rm GeV}^{2}\leq q^{2}\leq
6.0\,{\rm GeV}^{2}$ bin acquiesces with the SM at 1.5$\sigma$ level [9].

  2. 2.

Branching ratios: We include the updated world average of the branching ratio
of the purely leptonic decay $B_{s}\to\mu^{+}\mu^{-}$ which is $(2.93\pm
0.35)\times 10^{-9}$ [21]. This entails measurements performed by the LHCb
[16, 20], ATLAS [18] and CMS [19] collaborations. This average value is
2.3$\sigma$ away from the SM prediction. We also consider the branching
fraction of inclusive decay mode $B\to X_{s}\mu^{+}\mu^{-}$ [39] in the fit in
the low and high-$q^{2}$ bins.

We also ensheathe measurements of the differential branching fraction of
several semileptonic decays. The recently updated measurements of the
differential branching fraction of $B_{s}\to\phi\mu^{+}\mu^{-}$ by LHCb in
various $q^{2}$ intervals are included in the fit [11]. Further, the
differential branching ratios of $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ [40, 42, 41],
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$, $B^{0}\to K^{0}\mu^{+}\mu^{-}$and
$B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ [43, 42] in different $q^{2}$ bins are
encapsulated in the analysis.

  3. 3.

Angular Observables: We consider a plentitude of $CP$ conserving angular
observables in the fit. This entails longitudinal polarisation fraction
$F_{L}$, forward-backward asymmetry $A_{FB}$ and observables $S_{3}$, $S_{4}$,
$S_{5}$, $S_{7}$, $S_{8}$, $S_{9}$ in various $q^{2}$ bins, as measured by the
LHCb collaboration [14]. We also include their experimental correlations. We
also encompass the angular observables $F_{L}$, $P_{1}$, $P_{4}^{\prime}$,
$P_{5}^{\prime}$, $P_{6}^{\prime}$ and $P_{8}^{\prime}$ measured by ATLAS [44]
along with $P_{1}$, $P_{5}^{\prime}$ measured by CMS [45]. Further, the
measurements of $F_{L}$ and $A_{FB}$ by CDF and CMS collaborations are also
included [42, 41] in our analysis.

We then consider full set of angular observables in the decay $B^{+}\to
K^{*+}\mu^{+}\mu^{-}$ which was measured for the first time by the LHCb
collaboration in 2020 [46]. The optimized angular observables
$P_{1}-P_{8}^{\prime}$ and longitudinal polarisation fraction $F_{L}$, along
with their experimental correlations are included in the fit [46]. Finally, we
include $CP$ conserving angular observables in $B_{s}\to\phi\mu^{+}\mu^{-}$
decay mode. There are $F_{L}$, $S_{3}$, $S_{4}$ and $S_{7}$ as measured by the
LHCb in 2021. The available experimental correlations are also subsumed in the
fit [33].

In order to identify the Lorentz structure of possible new physics that can
account for the discrepancies in $b\to s\mu^{+}\mu^{-}$ data, we perform a
model independent analysis within the framework of effective field theory. For
this we consider new physics in the form of vector and axial-vector operators.
The effective Hamiltonian for $b\to s\mu^{+}\mu^{-}$ transition is then given
by

$\displaystyle\mathcal{H}_{\mathrm{eff}}(b\rightarrow
s\mu\mu)=\mathcal{H}^{\rm SM}+\mathcal{H}^{\rm VA}\;.$ (2)

Here the SM effective Hamiltonian can be written as

$\displaystyle\mathcal{H}^{\rm SM}$
$\displaystyle=-\frac{4G_{F}}{\sqrt{2}\pi}V_{ts}^{*}V_{tb}\bigg{[}\sum_{i=1}^{6}C_{i}\mathcal{O}_{i}+C_{8}{\mathcal{O}}_{8}+C_{7}\frac{e}{16\pi^{2}}[\overline{s}\sigma_{\alpha\beta}(m_{s}P_{L}+m_{b}P_{R})b]F^{\alpha\beta}$
$\displaystyle+C_{9}^{\rm SM}\frac{\alpha_{\rm
em}}{4\pi}(\overline{s}\gamma^{\alpha}P_{L}b)(\overline{\mu}\gamma_{\alpha}\mu)+C_{10}^{\rm
SM}\frac{\alpha_{\rm
em}}{4\pi}(\overline{s}\gamma^{\alpha}P_{L}b)(\overline{\mu}\gamma_{\alpha}\gamma_{5}\mu)\bigg{]}\;,$
(3)

where $V_{ij}$ are the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
The short-distance contributions are enciphered in the WCs $C_{i}$ of the
four-fermi operators ${\cal O}_{i}$ where the scale-dependence is implicit,
i.e. $C_{i}\equiv C_{i}(\mu)$ and ${\cal O}_{i}\equiv{\cal O}_{i}(\mu)$. The
operators ${\cal O}_{i}$ ($i=1,...,6,8$) contribute through the modifications
$C_{7}(\mu)\rightarrow C_{7}^{\mathrm{eff}}(\mu,q^{2})$ and
$C_{9}(\mu)\rightarrow C_{9}^{\mathrm{eff}}(\mu,q^{2})$. The new physics
effective Hamiltonian can be written as

$\displaystyle\mathcal{H}^{\rm VA}$ $\displaystyle=-\frac{\alpha_{\rm
em}G_{F}}{\sqrt{2}\pi}V_{ts}^{*}V_{tb}\bigg{[}C_{9}^{\rm
NP}(\overline{s}\gamma^{\alpha}P_{L}b)(\overline{\mu}\gamma_{\alpha}\mu)+C_{10}^{\rm
NP}(\overline{s}\gamma^{\alpha}P_{L}b)(\overline{\mu}\gamma_{\alpha}\gamma_{5}\mu)$
$\displaystyle+C_{9}^{\prime}(\overline{s}\gamma^{\alpha}P_{R}b)(\overline{\mu}\gamma_{\alpha}\mu)+C_{10}^{\prime}(\overline{s}\gamma^{\alpha}P_{R}b)(\overline{\mu}\gamma_{\alpha}\gamma_{5}\mu)\bigg{]}\,.$
(4)

Here $C_{9}^{\rm NP},C_{10}^{\rm NP},C_{9}^{\prime}$ and $C_{10}^{\prime}$ are
the new physics WCs which are assumed to be complex in the current analysis.
Following a penurious approach, we ruminate only those scenarios where either
only one new physics operator or two operators whose WCs are linearly related,
contributes. We call them as “1D” scenarios. Under this assumption, we perform
a $\chi^{2}$ fit to identify solutions which can accommodate the current $b\to
s\ell\ell$ measurements. The fit is performed using the CERN minimization code
MINUIT [47]. The $\chi^{2}$ which is a function of new physics WCs is defined
as

$\chi^{2}(C_{i},C_{j})=\big{[}\mathcal{O}_{\rm
th}(C_{i},C_{j})-\mathcal{O}_{\rm
exp}\big{]}^{T}\,\mathcal{C}^{-1}\,\big{[}\mathcal{O}_{\rm
th}(C_{i},C_{j})-\mathcal{O}_{\rm exp}\big{]}\,,$ (5)

where $\mathcal{O}_{\rm th}(C_{i},C_{j})$ are the theoretical predictions of
the N=156 observables used in the fit and $\mathcal{O}_{\rm exp}$ are the
corresponding central values of the experimental measurements. The total
$N\times N$ covariance matrix is obtained by adding the individual theoretical
and experimental covariance matrices. The theoretical predictions of N=156
observables along with the theoretical covariance matrix are evaluated using
flavio [36]. The experimental correlations, $\mathcal{O}_{\rm exp}$, are
admitted for the angular observables in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ [14],
$B^{+}\to K^{*+}\mu^{+}\mu^{-}$ [46] and $B_{s}\to\phi\mu^{+}\mu^{-}$ [33].
Further, for asymmetric errors, we use the larger error on both sides of the
central value.

The $\chi^{2}$ value in the SM is represented by $\chi^{2}_{\rm SM}$ whereas
$\chi^{2}_{\rm bf}$ represents the value at the the best-fit point in the
presence of new physics. We then quantitate the goodness of fit by
$\Delta\chi^{2}\equiv\chi^{2}_{\rm SM}-\chi^{2}_{\rm bf}$ for each new physics
scenario. This means that, under the given assumptions, the largest value of
this quantity would represent the best possible new physics scenario to
accommodate the entire $b\to s\mu^{+}\mu^{-}$ data.

Wilson Coefficient(s) | Real | Complex
---|---|---
| Best fit value(s) | $\Delta\chi^{2}_{\rm real}$ | Best fit value(s) | $\Delta\chi^{2}_{\rm complex}$
$C_{i}=0\,\,\rm(SM)$ | - | 0 | - | 0
1D Scenarios: |  |  |  |
$C_{9}^{\rm NP}$ | $-0.98\pm 0.15$ | 44.35 | $(-0.98\pm 0.15,\,0.01\pm 0.72)$ | 44.34
$C_{10}^{\rm NP}$ | $0.65\pm 0.12$ | 33.63 | $(0.92\pm 0.23,\,1.36\pm 0.50)$ | 35.44
$C_{9}^{\rm NP}=-C_{10}^{\rm NP}$ | $-0.46\pm 0.07$ | 44.73 | $(-0.51\pm 0.20,\,0.56\pm 1.07)$ | 41.15
$C_{9}^{\rm NP}=-C_{9}^{{}^{\prime}}$ | $-0.85\pm 0.15$ | 29.48 | $(-0.87\pm 0.15,\,-0.39\pm 0.38)$ | 29.73
Table 1: The best fit values of new WCs in various 1D scenarios. Here
$\Delta\chi^{2}=\chi^{2}_{\rm SM}-\chi^{2}_{\rm bf}$ where $\chi^{2}_{\rm bf}$
is the $\chi^{2}$ at the best fit point and $\chi^{2}_{\rm SM}$ corresponds to
the SM which is $\chi^{2}_{\rm SM}\approx$ 201\. Here we only provide those
scenarios which provide a good fit to the data.

Figure 1: Allowed parameter space for new physics Scenarios $C_{9}^{\rm NP}$
(upper left panel), $C_{10}^{\rm NP}$ (upper right panel), $C_{9}^{\rm
NP}=-C_{10}$ (lower left panel) and $C_{9}^{\rm NP}=-C_{9}^{{}^{\prime}}$
(lower right panel).

###  II.2 Fit results

It is perspicuous from Table1 that the scenarios that were favoured by
assuming new physics WCs to be real, still remains the preferred ones.
However, the order of solutions based on the value of $\Delta\chi^{2}$
changes. For real WCs, $C_{9}^{\rm NP}$ and $C_{9}^{\rm NP}=-C_{10}^{\rm NP}$
solutions had similar values of $\Delta\chi^{2}$ whereas now, $\Delta\chi^{2}$
reduces by $\sim$ 3 for the later scenario. Thus for the complex WCs,
$C_{9}^{\rm NP}$ scenario turns out to be the most viable scenario to
accommodate all $b\to s\ell\ell$ data. The $\Delta\chi^{2}$ for the other two
scenarios remains unchanged.

It is also tangible from the Table. 1 that apart from the $C_{10}^{\rm NP}$
solution, the best fit values of Re(WCs) remain almost the same. For
$C_{10}^{\rm NP}$ solution, the best fit value increases by 50% as compared to
the real-fit. Further, apart from ${\rm Im}(C_{10}^{\rm NP})$, the imaginary
part of all WCs are consistent with zero. The ${\rm Im}(C_{10}^{\rm NP})$
becomes zero consistent at 3$\sigma$. However, these WCs are allowed to have
values similar to that of their real counterparts. This is evident from the
1$\sigma$ range of complex WCs shown in Fig. 1. Therefore, it will be
intriguing to see whether some of the $CP$ violating observables can be
enhanced up to the current or planned sensitivity of LHCb or Belle-II.

In order to discriminate between the possible new physics solutions obtained
above, we now discuss some of the additional LFU violating observables. There
are either in forms of ratios or differences. In particular, we consider the
LFU ratio $R_{\phi}$ in both low as well as high $q^{2}$ bin of
$B_{s}\to\phi\mu^{-}\mu^{-}$ decay, $R_{K^{(*)}}$ in the high $q^{2}$ bin and
$Q_{4,\,5}$ observables in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay mode. The LFU
rations $R_{\phi}$ and $R_{K^{(*)}}$ in the high $q^{2}$ bin are yet to be
measured whereas $Q_{4,\,5}$ observables have been measured by Belle
collaboration but with large errors [48]. The predictions for these
observables are given in Tables 2 and 3.

Wilson Coefficient(s) | ${R_{K}}_{[15-19]}$ | ${R_{K^{*}}}_{[15-19]}$ | ${R_{\phi}}_{[1-6]}$ | ${R_{\phi}}_{[15-19]}$
---|---|---|---|---
$C_{i}=0\,\,\rm(SM)$ | $1.00\pm 0.0004$ | $0.99\pm 0.00009$ | $0.99\pm 0.0002$ | $0.99\pm 0.00008$
1D Scenarios: |  |  |  |
$C_{9}^{NP}$ | (0.76, 0.84) | (0.76, 0.84) | (0.81, 0.88) | (0.76, 0.84)
$C_{10}^{NP}$ | (0.81, 0.89) | (0.79, 0.88) | (0.78, 0.88) | (0.80, 0.89)
$C_{9}^{NP}=-C_{10}^{NP}$ | (0.73, 0.84) | (0.72, 0.83) | (0.74, 0.85) | (0.73, 0.83)
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ | (1.00, 1.00) | (0.69, 0.81) | (0.72, 0.84) | (0.68, 0.81)
Table 2: Predictions of additional LFU ratios (1$\sigma$ range).

It can be descried from Table 2 that the measurement of $R_{K}$ in the
high-$q^{2}$ region can secern scenario $C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ from
others. All other scenarios predict $R_{K}<1$ whereas $R_{K}\approx 1$ for
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ solution. Further, only this solution
predicts $R_{{\phi,\,K}[15-19]}<0.70$. The ratios $R_{K}$, $R_{K^{*}}$ and
$R_{\phi}$ in the high-$q^{2}$ bin do not have any additional discriminating
endowment. A measurement of $R_{\phi}<0.75$ can distinguish $C_{9}^{NP}$
scenario from others.

Wilson Coefficient(s) | ${Q^{\prime}_{4}}_{[4-6]}$ | ${Q^{\prime}_{4}}_{[15-19]}$ | ${Q^{\prime}_{5}}_{[4-6]}$ | ${Q^{\prime}_{5}}_{[15-19]}$
---|---|---|---|---
$C_{i}=0\,\,\rm(SM)$ | $(-2.5\pm 0.96)\times 10^{-5}$ | $(-5.8\pm 0.5)\times 10^{-6}$ | $(-3.4\pm 0.29)\times 10^{-3}$ | $(-7.7\pm 0.56)\times 10^{-4}$
1D Scenarios: |  |  |  |
$C_{9}^{NP}$ | (0.015, 0.024) | (0.000, 0.000) | (0.160, 0.295) | (0.034, 0.082)
$C_{10}^{NP}$ | (0.003, 0.004) | (0.000, 0.000) | (-0.061, 0.084) | (-0.005, 0.097)
$C_{9}^{NP}=-C_{10}^{NP}$ | (0.010, 0.022) | (0.000, 0.000) | (0.031, 0.061) | (0.002, 0.005)
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ | (0.018, 0.040) | (0.016, 0.030) | (0.081, 0.242) | (-0.039, 0.026)
Table 3: Predictions of LFU observables $Q_{4,5}$ (1$\sigma$ range).

The LFU observables $Q_{4,\,5}$ seem to have better discriminational
adroitness. A non-zero measurement of ${Q^{\prime}_{4}}$ in (15-19)
$q^{2}$-bin can confirm new physics in the form of $O_{9}$ and
$O^{\prime}_{9}$ operators related by $C_{9}^{NP}=-C_{9}^{{}^{\prime}}$. This
is because for all other viable scenarios, ${Q^{\prime}_{4}}_{[15-19]}$ is
predicted to be zero. Further, this is the only scenario which can enhance
${Q^{\prime}_{4}}_{[4-6]}$ up to 0.40. Moreover, $C_{10}^{NP}$ can be
differentiated from others through ${Q^{\prime}_{5}}$ in (4-6) bin as this is
the exclusive solution which predicts a large negative value of this
observable. The $C_{9}^{NP}$ solution can be cloistered from others if
${Q^{\prime}_{5}}_{[4-6]}$ is measured with a value $\sim$ 0.3. Comparing
these results with that obtained in ref. [31] for the real-fit, we find that
the conclusions are almost the same. This thus implies that the discriminatory
capability of these additional LFU observables, in particular $Q_{4,5}$,
remains unabridged even if the new physics WCs are allowed to be complex.

##  III Direct $CP$ asymmetry in $B^{+}\to K^{+}\mu^{+}\mu^{-}$

The $CP$ violation can be classified into two types: the direct $CP$ violating
asymmetries and triple product $CP$ asymmetries. Assume that there are two
contributions, $A_{1}\propto e^{i\alpha_{1}}e^{i\beta_{1}}$ and $A_{2}\propto
e^{i\alpha_{2}}e^{i\beta_{2}}$, in $b\to s\mu^{+}\mu^{-}$ decay. Here
$\alpha_{1,2}$ and $\beta_{1,2}$ are weak and strong phases, respectively. It
can be easily shown that the direct $CP$ asymmetries are proportional to
$\sin(\alpha_{1}-\alpha_{2})\,\sin(\beta_{1}-\beta_{2})$. This implies that
these asymmetries can have non-zero values only if the two interfering
amplitudes have a relative weak as well as strong phase. On the other hand, as
triple product asymmetries are proportional to
$\sin(\alpha_{1}-\alpha_{2})\,\cos(\beta_{1}-\beta_{2})$, a relative weak
phase between the amplitudes is sufficient to provide a non-zero value.

Wilson Coefficients | $A^{K}_{{\rm CP}{[1-6]}}\,(\%)$ | $A^{K}_{{\rm CP}{[15-19]}}\,(\%)$ | $A^{K^{*}}_{{\rm CP}{[1-6]}}\,(\%)$ | $A^{K^{*}}_{{\rm CP}{[15-19]}}\,(\%)$
---|---|---|---|---
$C_{i}=0\,\,\rm(SM)$ |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0
1D Scenarios: |  |  |  |
$C_{9}^{NP}$ | (- 0.35, 0.58) | (-3.69, 3.44) | (0.00, 0.11) | (-1.87, 1.72)
$C_{10}^{NP}$ | (0.14, 0.15) | (-0.19, -0.18) | (0.05, 0.06) | (-0.12, -0.11)
$C_{9}^{NP}=-C_{10}^{NP}$ | (- 0.66, 0.91) | (-6.24, 5.93) | (-0.05, 0.16) | (-3.12, 3.00)
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ | ( 0.12, 0.12) | (-0.16, -0.16) | (-0.05, 0.13) | (-2.92 ,2.79)
Table 4: Predictions of $A_{\rm CP}$ in $B^{+}\to K^{+}\mu^{+}\mu^{-}$ and
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay (1$\sigma$ range). Here $A^{K}_{{\rm
CP}}\equiv A_{\rm CP}(B^{+}\to K^{+}\mu^{+}\mu^{-})$ and $A^{K^{*}}\equiv
A_{\rm CP}(B^{0}\to K^{*0}\mu^{+}\mu^{-})$.

For $B\to K\mu^{+}\mu^{-}$ decays, we only have direct $CP$ asymmetry,
$A_{CP}$, which is defined as

$A_{CP}=\frac{\Gamma-\bar{\Gamma}}{\Gamma+\bar{\Gamma}}\,$ (6)

where $\Gamma$ and $\bar{\Gamma}$ are the decay rates of $B^{+}\to
K^{+}\mu^{+}\mu^{-}$ and $\bar{B}\to\bar{K}\mu^{+}\mu^{-}$ decays.
$\bar{\Gamma}$ is obtained from $\Gamma$ by changing the sign of the weak
phases. The sign of strong phases remain unchanged. The theoretical expression
of $\Gamma(B\to K\mu^{+}\mu^{-})$ is provided in Appendix A.

Effectively, various contributions to $b\to s\mu^{+}\mu^{-}$ decay are
proportional to CKM factors $V^{*}_{tb}V_{ts}$ and $V^{*}_{ub}V_{us}$. The
term proportional to $V^{*}_{cb}V_{cs}$ is eliminated using the unitarity of
the CKM mixing matrix. Although the phase of $V^{*}_{ub}V_{us}$ is large but
its magnitude is suppressed in comparison to $V^{*}_{tb}V_{ts}$. Therefore,
within the SM, various contributions to $b\to s\mu^{+}\mu^{-}$ have almost
similar weak phase. Further, the $c\bar{c}$ and $u\bar{u}$ quark loop generate
strong phase in the WC $C_{9}^{\rm eff}$. However, this is not substantial and
hence the $CP$ asymmetries in $b\to s\mu^{+}\mu^{-}$ are highly suppressed
within the SM. The fact that the new physics strong phase is negligibly small,
one requires a large value of new physics weak phase to provide an enhancement
at the level of few percent [49].

Including the dielectron mode, the Belle and BaBar collaborations measured
direct $CP$ asymmetries in $B^{+}\to K^{+}\ell\ell$ and $B^{0}\to
K^{*0}\ell\ell$ decay modes. Based on a data sample of 657 million $B\bar{B}$
pairs, Belle reported $A_{\rm CP}$ in $B^{+}\to K^{+}\ell\ell$ and $B^{0}\to
K^{*0}\ell\ell$ to be $0.04\pm 0.10$ and $-0.10\pm 0.10$, respectively [50].
These values for BaBar collaboration are $-0.03\pm 0.14$ and $0.03\pm 0.13$,
respectively [51]. This corresponds to a sample of 471 million $B\bar{B}$
events. These asymmetries, in the muonic channel, was measured by the LHCb
collaboration by making use of a data set corresponding to an integrated
luminosity of 3.0 $\rm fb^{-1}$ collected in 2011 and 2012 at centre of mass
energies of 7 and 8 TeV , respectively. The measured values are $A_{\rm
CP}(B^{+}\to K^{+}\ell\ell)=0.012\pm 0.017$ and $A_{\rm CP}(B^{0}\to
K^{*0}\ell\ell)=-0.035\pm 0.024$ [52, 53, 54]. Owing to large errors, these
measurements are consistent with the SM. But on the other hand, a possibility
of $A_{\rm CP}$ at a level of a few percent is not ruled out. It would be
interesting to see whether such an enhancement is allowed by the current data.

Figure 2: The left panel delineates correlations between $A_{\rm CP}$ in
$B^{+}\to K^{+}\mu^{+}\mu^{-}$ and $R_{K}$ for all favored “1D” solutions in
the high-$q^{2}$ region. A similar correlations between $A_{\rm CP}$ in
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ and $R_{K^{*}}$ is displayed in the right
panel.

The predictions of $A_{\rm CP}$ in $B^{+}\to K^{+}\mu^{+}\mu^{-}$ for all 1D
solutions are given in Table 4. It is apparent that none of the new physics
solutions can enhance $A_{\rm CP}$ in the low-$q^{2}$ bin at the level of a
few percent. However, such an enhancement is feasible in the high-$q^{2}$
region for $C_{9}^{NP}$ and $C_{9}^{NP}=-C_{10}^{NP}$ solutions, the
enhancement being more prominent for the later solution. Therefore any
observation of $A_{\rm CP}$ in $B^{+}\to K^{+}\mu^{+}\mu^{-}$ can be
attributed to either of these solutions. The other two favoured scenarios
predict $A_{\rm CP}<1\%$.

Here one should emphasize that although the enhancement in the high-$q^{2}$
bin is more prominent, the measurement of $A_{\rm CP}$ in the low-$q^{2}$
region appears to be more attractive as the branching ratio in the low-$q^{2}$
region is larger as compared to the high-$q^{2}$ bin. Belle-II experiment is
expected to collect a sample of a few thousand events of
$B\to(K,\,K^{*})\mu^{+}\mu^{-}$ [55]. With such an event sample, it would be
possible to have a 3$\sigma$ determination of the $CP$ asymmetries which are
of a few percent level.

Moreover, we also inquest correlations between $A_{\rm CP}$ and $R_{K}$ in the
high-$q^{2}$ region. This is elucidated in Fig. 2. For scenario
$C_{9}^{NP}=-C_{10}^{NP}$, $A_{\rm CP}$ can be as high as $\sim$ 5-6% for
which $R_{K}$ can lie between 0.73 and 0.82. The correlations are almost the
same for scenario $C_{9}^{NP}$ barring the fact that the maximum allowed value
of $A_{\rm CP}$ is $\sim$3%. The value of $A_{\rm CP}$ is $\sim$ 0.2% for the
new physics scenario $C_{10}^{NP}$. For this value of $A_{\rm CP}$, $R_{K}$
can lie anywhere between 0.74 and 0.85. Thus it is evident that any additional
discrimination, as compared to the individual predictions, between the new
physics scenarios would not be possible by analyzing the correlations between
$A_{\rm CP}$ and $R_{K}$ in the high-$q^{2}$ region.

##  IV $CP$ violating observables in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$

The differential distribution of $B\to K^{*}(\to K\pi)\mu^{+}\mu^{-}$ decay
can be parametrized in terms of one kinematic and three angular variables. The
kinematic variable is $q^{2}=(p_{B}-p_{K^{*}})^{2}$, where $p_{B}$ and
$p_{K^{*}}$ are the four-momenta of $B$ and $K^{*}$ mesons, respectively. The
angular variables are usually defined in the rest frame of the vector meson
$K^{*}$. These angles are

  * •

$\theta_{K}$ the angle between $B$ and $K$ mesons where $K$ meson emerges from
the decay of a $K^{*}$,

  * •

$\theta_{\mu}$ the angle between $\mu^{-}$ and $B$ momenta,

  * •

$\phi$ the angle between $K^{*}$ decay plane and the plane defined by the
$\mu^{+}-\mu^{-}$ momenta.

The four-fold decay distribution can be expounded as [4, 5]

$\frac{d^{4}\Gamma}{dq^{2}d\cos\theta_{\mu}d\cos\theta_{K}d\phi}=\frac{9}{32\pi}I(q^{2},\theta_{\mu},\theta_{K},\phi),$
(7)

where

$\displaystyle I(q^{2},\theta_{\mu},\theta_{K},\phi)$ $\displaystyle=$
$\displaystyle
I^{s}_{1}\sin^{2}\theta_{K}+I^{c}_{1}\cos^{2}\theta_{K}+(I^{s}_{2}\sin^{2}\theta_{K}+I^{c}_{2}\cos^{2}\theta_{K})\cos
2\theta_{\mu}$ (8)
$\displaystyle+I_{3}\sin^{2}\theta_{K}\sin^{2}\theta_{\mu}\cos 2\phi+I_{4}\sin
2\theta_{K}\sin 2\theta_{\mu}\cos\phi$ $\displaystyle+I_{5}\sin
2\theta_{K}\sin\theta_{\mu}\cos\phi$
$\displaystyle+(I^{s}_{6}\sin^{2}\theta_{K}+I^{c}_{6}\cos^{2}\theta_{K})\cos\theta_{\mu}+I_{7}\sin
2\theta_{K}\sin\theta_{\mu}\sin\phi$ $\displaystyle+I_{8}\sin 2\theta_{K}\sin
2\theta_{\mu}\sin\phi+I_{9}\sin^{2}\theta_{K}\sin^{2}\theta_{\mu}\sin 2\phi.$

The expressions of these twelve angular coefficients $I^{(a)}_{i}$ are
provided in Appendix B. These coefficients depend on the $q^{2}$ variable and
on various hadronic form factors. The corresponding expression for the four-
fold decay distribution of the $CP$ conjugate decay mode can be obtained by
substituting $\theta_{\mu}$ by $(\pi-\theta_{\mu})$ and $\phi$ by $-\phi$.
This results in the following transformations of angular coefficients

$I^{(a)}_{1,2,3,4,7}\Longrightarrow\bar{I}^{(a)}_{1,2,3,4,7},\quad
I^{(a)}_{5,6,8,9}\Longrightarrow-\bar{I}^{(a)}_{5,6,8,9}\,.$ (9)

Here $\bar{I}^{(a)}_{i}$ are the complex conjugate of $I^{(a)}_{i}$.
Therefore, one can define twelve $CP$ averaged angular observables as [4, 5]

$S^{(a)}_{i}(q^{2})=\frac{I^{(a)}_{i}(q^{2})+\bar{I}^{(a)}_{i}(q^{2})}{d(\Gamma+\bar{\Gamma})/dq^{2}},$
(10)

along with twelve $CP$ asymmetries

$A^{(a)}_{i}(q^{2})=\frac{I^{(a)}_{i}(q^{2})-\bar{I}^{(a)}_{i}(q^{2})}{d(\Gamma+\bar{\Gamma})/dq^{2}}\,.$
(11)

The $CP$ asymmetry in the dimuon mass spectrum is defined as

$A_{\rm
CP}(q^{2})=\frac{d\Gamma/dq^{2}-d\bar{\Gamma}/dq^{2}}{d\Gamma/dq^{2}+d\bar{\Gamma}/dq^{2}},$
(12)

where $d\Gamma/dq^{2}$ can be expressed in terms of angular coefficients as

$\frac{d\Gamma}{dq^{2}}=\frac{3}{4}\left(2I_{1}^{s}+I_{1}^{c}\right)-\frac{1}{4}\left(2I_{2}^{s}+I_{2}^{c}\right)\,.$
(13)

Wilson Coefficient(s) | ${A_{3}}_{[1-6]}\,(\%)$ | ${A_{4}}_{[1-6]}\,(\%)$ | ${A_{5}}_{[1-6]}\,(\%)$ | ${A_{6}^{s}}_{[1-6]}\,(\%)$ | ${A_{7}}_{[1-6]}\,(\%)$ | ${A_{8}}_{[1-6]}\,(\%)$ | ${A_{9}}_{[1-6]}\,(\%)$
---|---|---|---|---|---|---|---
$C_{i}=0\,\,\rm(SM)$ |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0
1D Scenarios: |  |  |  |  |  |  |
$C_{9}^{NP}$ | (0.00, 0.01) | (-0.11, 0.15) | (0.03, 0.04) | (-0.08, -0.07) | (0.26, 0.28) | (-3.17, 3.36) | (-0.32, 0.35)
$C_{10}^{NP}$ | (0.00, 0.00) | (0.02, 0.03) | (-0.63, 0.68) | (-1.09, 0.98) | (-14.31, 14.58) | (0.12, 0.14) | (0.01, 0.02)
$C_{9}^{NP}=-C_{10}^{NP}$ | (-0.01, 0.02) | (-0.20, 0.25) | (-0.48, 0.54) | (-0.86, 0.73) | (-10.92, 11.36) | (-5.64, 5.85) | (-0.57, 0.60)
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ | (-0.22, 0.21) | (-0.22, 0.25 ) | (0.04, 0,04) | (-0.09, -0.07) | (0.28, 0.32) | (-0.83, 0.94 ) | (-0.20, 0.13)

Table 5: Prediction of various $CP$ violating angular observables (1$\sigma$
range) in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ in the low-$q^{2}$ region.

Wilson Coefficient(s) | ${A_{3}}_{[15-19]}\,(\%)$ | ${A_{4}}_{[15-19]}\,(\%)$ | ${A_{5}}_{[15-19]}\,(\%)$ | ${A_{6}^{s}}_{[15-19]}\,(\%)$ | ${A_{7}}_{[15-19]}\,(\%)$ | ${A_{8}}_{[15-19]}\,(\%)$ | ${A_{9}}_{[15-19]}\,(\%)$
---|---|---|---|---|---|---|---
$C_{i}=0\,\,\rm(SM)$ |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0 |  $\approx$ 0
1D Scenarios: |  |  |  |  |  |  |
$C_{9}^{NP}$ | (-0.70, 0.76) | (-1.03, 1.12) | (0.10, 0.11) | (-0. 20, -0.18) | (0.01, 0.01) | (-0.14, 0.15) | (-0.11, 0.13)
$C_{10}^{NP}$ | (0.04, 0.05) | (0.06, 0.07) | (-2.07, 2.23) | (-3.93, 3.63) | (-0.59, 0.60) | (0.00, 0.00) | (0.00, 0.00)
$C_{9}^{NP}=-C_{10}^{NP}$ | (-1.22, 1.30) | (-1.80, 1.91) | (-1.68, 1.86) | (-3.28, 2.95) | (-0.48, 0.49) | (-0.24, 0.26) | (-0.21, 0.22)
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ | (-1.73, 1.76) | (-1.92, 1.99) | (0.10, 0.12) | (-0.22, -0.19) | (0.01, 0.01) | (-3.25, 3.26) | (-5.80, 5. 80)

Table 6: Prediction of various $CP$ violating angular observables (1$\sigma$
range) in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ in the high-$q^{2}$ region.

Apart from $A_{\rm CP}$, in this work we consider consider $A_{3,\,4,\,5}$,
$A_{6}^{s}$ and $A_{7,\,8,\,9}$ observables. These observables are measured by
the LHCb collaboration, however, with large errors [13]. The angular
observables $A_{3,\,4,\,5}$ and $A_{6}^{s}$ are direct $CP$ like asymmetries
whereas $A_{7,\,8,\,9}$ are triple product CP asymmetries [4]. Therefore
$A_{7,\,8,\,9}$ observables seem to be more sensitive to new weak phases as
compared to the other observables.

First of all, we examine $A_{\rm CP}$. Based on predictions obtained in Table
4 for various favored scenarios, it is pellucid that none of the solutions can
enhance $A_{\rm CP}$ to a level of a percent in the low-$q^{2}$ bin.
Therefore, the current $b\to s\ell\ell$ data suggests that the measurement of
$A_{\rm CP}$ in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay in the low-$q^{2}$
region would be a hellacious task. In the high-$q^{2}$ bin, except scenario
$C_{10}^{NP}$, all other scenarios can ameliorate $A_{\rm CP}$ up to 2-3%.
Therefore as far as $A_{\rm CP}$ is concerned, $C_{10}^{NP}$ doesn’t seem to
be a propitious candidate.

The predictions of several $CP$ violating angular observables in $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ in the low and high-$q^{2}$ regions are exhibited in
Table 5 and 6, respectively. From Table 5, it is unambivalent that all four
allowed solutions predict $A_{3,\,4,\,5}$, $A_{6}^{s}$ and $A_{9}$ asymmetries
to be less than a percent in the low-$q^{2}$ bin and hence making their
observation an arduous endeavor. However, the predictions of observable
$A_{7}$ provides encouraging signs as for $C_{10}^{NP}$ and
$C_{9}^{NP}=-C_{10}^{NP}$ solutions, ${A_{7}}_{[1-6]}$ can be enhanced up to
15% and 12%. respectively. This is due to the fact that this observable is
extremely sensitive to the $C_{10}^{NP}$ WC. Therefore measurement of
${A_{7}}_{[1-6]}$ with a value $\sim$ 0.15 (or 15%) can confirm the presence
of new physics in the form of $O_{10}$ operator. The $C_{9}^{NP}$ and
$C_{9}^{NP}=-C_{10}^{NP}$ solutions can bolster ${A_{8}}_{[1-6]}$ at the level
of a 4-5%.

Figure 3: The left (right) panel portrays correlations between $A_{\rm CP}$ in
$B^{0}\to K^{*0}\mu^{+}\mu^{-}$ and $A_{7}$ ($A_{8}$) for all favored “1D”
solutions in the low-$q^{2}$ region.

For a finer scrutinization of possible physics contributions to $b\to
s\ell\ell$, we study the correlations between $A_{\rm CP}$ and
${A_{7,8}}_{[1-6]}$ as for all other angular observables, enhancements are too
small to be observed in near future. The correlations between observables in
the low-$q^{2}$ bin are vignetted in Fig. 3. For $C_{9}^{NP}=-C_{10}^{NP}$
solution, even if $A_{\rm CP}$ is extremely small, ${A_{7}}_{[1-6]}$ can still
have a large value. For e.g., for $A_{\rm CP}\approx 0$, ${A_{7}}_{[1-6]}\sim
5\%$. Further, there is an anti-correlation between these two observables,
i.e., an increase in $A_{\rm CP}$ would result in decrease in the value of
$A_{7}$. For $C_{10}^{NP}$ solution, $A_{\rm CP}$ varies in a narrow range
around 0.05%, for which $A_{7}$ can lie anywhere between -15% to +15%.
Therefore a precise measurement of $A_{\rm CP}$ and $A_{7}$ has the potential
to differentiate between $C_{9}^{NP}=-C_{10}^{NP}$ and $C_{10}^{NP}$
solutions.

The correlations between $A_{\rm CP}$ and ${A_{8}}_{[1-6]}$ are explicated in
the right panel of Fig. 3. For all scenarios, $A_{\rm CP}$ can be non-zero
even if $A_{\rm CP}\approx 0$. For $C_{9}^{NP}=-C_{10}^{NP}$ and $C_{9}^{NP}$
solutions, ${A_{8}}_{[1-6]}\approx 2\%$ for $A_{\rm CP}\approx 0$. Other two
solutions also have non-zero values of $A_{8}$ but less than a percent for
$A_{\rm CP}\approx 0$. For $A_{\rm CP}\approx+0.15\%$, $A_{8}$ can be as large
as -6% for $C_{9}^{NP}=-C_{10}^{NP}$ solution. This anti-correlation exists in
the entire low-$q^{2}$ region for $C_{9}^{NP}=-C_{10}^{NP}$ as well as
$C_{9}^{NP}$ scenario.

Figure 4: These plots reveal correlations between $A_{\rm CP}$ in $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ and several $A_{i}$ observables for all favored “1D”
solutions in the high-$q^{2}$ bin.

We now consider predictions of $A_{i}$ observables in the high-$q^{2}$ bin as
given in Table 6. The most conspicuous feature of predictions in the
high-$q^{2}$ region is related to the observable $A_{7}$ which insinuated to
be a potential observable to verbalize the signatures of weak phase related to
the new physics solutions $C_{10}^{NP}$ and $C_{9}^{NP}=-C_{10}^{NP}$. Unlike
in the low-$q^{2}$ bin, all allowed 1D solutions fail to provide an
enhancement in $A_{7}$ above a percent level. The
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ scenario failed to make any noticeable
indentations in the low-$q^{2}$ bin as it was unable to provide any detectable
enhancements in any of the considered $A_{i}$ observables. However, in the
high-$q^{2}$ bin, this solution appears to make a riveting impact as it can
enhance $A_{8}$ and $A_{9}$ observables up to a level of a few percent. All
other favored scenarios fail to provide any meaningful enhancement. This thus
implies that the observation of either $A_{8}$ or $A_{9}$ asymmetries at the
level of a few percent in the high-$q^{2}$ bin may provide confirmatory
evidence in support of the $C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ scenario.

The observable $A_{6}^{s}$ failed to make any imprint in the low-$q^{2}$ bin.
However in [15-19] bin, this can be enhanced up to $\sim$ 3-4% by
$C_{10}^{NP}$ and $C_{9}^{NP}=-C_{10}^{NP}$ scenarios. The same is true for
$A_{3,\,4,\,5}$ angular observables for which none of the 1D solutions were
able to provide any significant enhancements in the [1-6] bin. On the
contrary, all of these observables can be enhanced up to a level of a few
percent in the high-$q^{2}$ bin. The $A_{5}$ observable can be brought to a
level of $\sim$ 2% by $C_{10}^{NP}$ and $C_{9}^{NP}=-C_{10}^{NP}$ solutions.
All solutions except $C_{10}^{NP}$ have the potential to bring $A_{4}$
observable up to a level of a percent or more. A similar job for
${A_{3}}_{[15-19]}$ observable can be done by $C_{9}^{NP}=-C_{10}^{NP}$ and
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ solutions.

We now delve correlations in the high-$q^{2}$ region. The correlations between
$A_{\rm CP}$ and $R_{K^{*}}$ in the high-$q^{2}$ bin is portrayed in the right
panel of Fig. 2. It is seeable that the plot predominantly reflects the
individual predictions rather than any noticeable pattern on the basis of
which one cannot contradistinguish between the allowed solutions.The
interrelations between $A_{\rm CP}$ and $A_{i}$ are demonstrated in Fig. 4.
Here we do not consider $A_{7}$ observable as none of the allowed 1D scenarios
can enhance it up to a level of a percent.

The $(A_{\rm CP}-A_{3})$ and $(A_{\rm CP}-A_{4})$ correlations are almost
similar. These may allow us to discriminate between the
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ and other allowed solutions, in particular,
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ and $C_{9}^{NP}=-C_{10}^{NP}$. This is due
to the fact that there is an anti-correlation between these observables for
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ whereas for other solutions, $(A_{\rm
CP}-A_{3,4})$ exhibits positive correlations. Also, for zero $A_{\rm CP}$,
$A_{3,4}\sim$ 1% for $C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ and
$C_{9}^{NP}=-C_{10}^{NP}$ . From $(A_{\rm CP}-A_{5,6}^{(s)})$ plots, it is
obvious that, for $C_{9}^{NP}=-C_{10}^{NP}$, $A_{5,6}^{(s)}$ can be $\sim 1\%$
even if $A_{\rm CP}$ is zero. Further, $A_{5}$ ($A_{6}^{s}$) has positive
(negative) correlations with $A_{\rm CP}$ for this solution. The
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ scenario can be differentiated from other
solutions through the simultaneous measurements of $A_{\rm CP}$ and $A_{9}$.
For this scenario, a zero value of $A_{\rm CP}$ predicts $A_{9}$ to be $\sim
5\%$ . A similar relation exists between $A_{\rm CP}$ and $A_{8}$ but in this
case $A_{8}$ can be as high as 2%. For all other solutions, $A_{8,9}\sim 0$ in
the allowed range of $A_{\rm CP}$.

##  V Conclusions

Assuming new physics Wilson coefficients to be complex, we perform a model-
independent global fit to all apropos $b\to s\ell\ell$ data. This include new
measurements of $R_{K^{0}_{S}}$, $R_{K^{*+}}$ together with the updated
measurements for several $B_{s}\to\phi\mu^{+}\mu^{-}$ observables. We work
under the assumption that the new physics affects only the muon sector.
Considering only one operator or two related operators at a time, we obtain
the following:

  * •

The allowed solutions remain the same as obtained for the real fits, i.e.
$C_{9}^{\rm NP}$, $C_{9}^{\rm NP}=-C_{10}^{\rm NP}$ , $C_{10}^{\rm NP}$ and
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ scenarios provide a good fit to the data.

  * •

The $C_{9}^{\rm NP}$ scenario now becomes the most preferred one as the
$\Delta\chi^{2}$ for $C_{9}^{\rm NP}=-C_{10}^{\rm NP}$ solution reduces by
$\sim$ 3\. It should be noted that for real-fit, $\Delta\chi^{2}$ for these
scenarios were almost the same.

  * •

The values of $\Delta\chi^{2}$ for the other two scenarios remain unchanged.

In order to discriminate between the above new physics solutions, we then
obtain predictions for some of the additional LFU observables. These include
$R_{\phi}$ in both low as well as high $q^{2}$ bins, $R_{K^{(*)}}$ in the high
$q^{2}$ bin and $Q_{4,\,5}$ observables in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$
decay mode. We find that accurate measurements of these observables, in
particular $Q_{4,\,5}$ , can distinguish between some of the allowed
solutions.

We find that the current data allows complex couplings to exist with an upper
bound similar to that of their real counterparts. The effect of such a weak
phase can show up in some of the $CP$ asymmetries. For the favoured solutions,
we obtained predictions of several $CP$-violating observables in $B^{0}\to
K^{*0}\mu^{+}\mu^{-}$ and direct $CP$ asymmetry in $B^{+}\to
K^{+}\mu^{+}\mu^{-}$. These asymmetries can be observed at the current or
planned experimental facilities provided new physics enhances them to a level
of a few percent. Following are our main observations:

  * •

None of the new physics solutions can enhance $A_{\rm CP}$ in the low-$q^{2}$
bin at the level of a few percent. Such an enhancement is feasible only in the
high-$q^{2}$ region. This is true for $B^{+}\to K^{+}\mu^{+}\mu^{-}$ as well
as $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay. For $B^{+}\to K^{+}\mu^{+}\mu^{-}$,
such an enhancement can be provided by $C_{9}^{NP}$ or
$C_{9}^{NP}=-C_{10}^{NP}$ solutions, the enhancement being more pronounced for
the later solution. For $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay, all solutions
except $C_{10}^{\rm NP}$ can serve this purpose.

  * •

All allowed solutions predict $A_{3,\,4,\,5}$, $A_{6}^{s}$ and $A_{9}$
asymmetries to be less than a percent level in the low-$q^{2}$ bin. However,
the predictions of observable $A_{7}$ provides eupeptic signs as for
$C_{10}^{NP}$ and $C_{9}^{NP}=-C_{10}^{NP}$ solutions, ${A_{7}}_{[1-6]}$ can
be enhanced up to 15% and 12%. respectively. The $C_{9}^{NP}$ and
$C_{9}^{NP}=-C_{10}^{NP}$ solutions can bolster ${A_{8}}_{[1-6]}$ at the level
of 4-5%.

  * •

The observation of either $A_{8}$ or $A_{9}$ observable at a level of a few
percent in the high-$q^{2}$ bin may provide confirmatory evidence in support
of the $C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ scenario.

  * •

The observable $A_{3,\,4,\,5}$ and $A_{6}^{s}$ failed to make any impact in
the low-$q^{2}$ bin. However in [15-19] bin, all of these observables can be
enhanced up to a level of a few percent.

Finally, we study correlations between $A_{\rm CP}$ and other $CP$
asymmetries. Our findings are as follows:

  * •

Even if $A_{\rm CP}$ is meagerly small, ${A_{7}}$ can still have large values
for $C_{9}^{NP}=-C_{10}^{NP}$ solution in the low-$q^{2}$ bin. Further, there
is an anti-correlation between $A_{\rm CP}$ and ${A_{7}}$ for this scenario,
i.e., larger the value of $A_{\rm CP}$, smaller is $A_{7}$.

  * •

For all scenarios,${A_{8}}_{[1-6]}$ can be non-zero even if $A_{\rm CP}\approx
0$.

  * •

The $(A_{\rm CP}-A_{3})$, $(A_{\rm CP}-A_{4})$ and $(A_{\rm CP}-A_{9})$
correlations in the high-$q^{2}$ bin can allow us to differentiate between the
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ and other allowed solutions. In particular,
$(A_{\rm CP}-A_{3,4})$ correlations may enable discrimination between
$C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ and $C_{9}^{NP}=-C_{10}^{NP}$ solutions.

  * •

For $C_{9}^{NP}=-C_{9}^{{}^{\prime}}$ solution, the angular $CP$ asymmetries
$A_{3,4,8,9}$ can have values above a percent level even if $A_{\rm CP}$ is
close to zero. The same applies to $C_{9}^{NP}=-C_{10}^{NP}$ for $A_{3,4,5}$
and $A_{6}^{s}$ observables.

Therefore the observation of $A_{\rm CP}$ as well as $CP$ violating angular
observables will not only provide an evidence of new physics with complex
phase but their accurate measurements would also facilitate the unique
identification of possible new physics in the decays induced by the $b\to
s\ell\ell$ transition. The direct asymmetry can be measured at the LHCb or
Belle-II, however the measurements of $CP$ violating angular observables
require higher statistics which can be attained at the HL-LHC [56].

##  Appendix A Decay rate of $B\to K\mu^{+}\mu^{-}$ decay

The decay rate of $B\to K\mu^{+}\mu^{-}$ is given by [57, 58]

$\Gamma(B\to K\mu^{+}\mu^{-})=\int_{q^{2}_{\rm
min}}^{{q^{2}_{max}}}dq^{2}\left(2a_{\mu}(q^{2})+\frac{2}{3}c_{\mu}(q^{2})\right)\,,$
(14)

where

$\displaystyle a_{\mu}(q^{2})$ $\displaystyle=$ $\displaystyle
E(q^{2})\Bigg{[}q^{2}|F_{P}|^{2}+\frac{\lambda}{4}\left(|F_{V}|^{2}+|F_{A}|^{2}\right)$
(15) $\displaystyle+2m_{\mu}\left(m_{B}^{2}-m_{K}^{2}+q^{2}\right){\rm
Re}(F_{P}F_{A}^{*})+4m_{\mu}^{2}m_{B}^{2}|F_{A}|^{2}\Bigg{]}\,,$
$\displaystyle c_{\mu}(q^{2})$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{4}\beta_{\mu}^{2}E(q^{2})\left(|F_{V}|^{2}+|F_{A}|^{2}\right)\,,$
(16)

with

$E(q^{2})=\frac{G_{F}^{2}\alpha|V_{tb}V_{ts}^{*}|^{2}}{512\pi^{5}m_{B}^{3}}\beta_{\mu}\lambda_{K}.$
(17)

Here
$\lambda=m^{4}_{B}+m^{4}_{K}+q^{4}-2(m^{2}_{B}m^{2}_{K}+m^{2}_{B}q^{2}+m^{2}_{K}q^{2})$
and $\beta_{\mu}=\sqrt{1-4m^{2}_{\mu}/q^{2}}$. In the low-$q^{2}$ region, all
form-factors reduce to one soft form-factor [59, 60]. In the high-$q^{2}$
region too, symmetry relations among the form factors can be delved with the
improved Isgur-Wise relation [61].

##  Appendix B Angular coefficients in $B^{0}\to K^{*0}\mu^{+}\mu^{-}$ decay

The angular coefficients appearing in the four-fold distribution of $B\to
K^{*}(\to K\pi)\mu^{+}\mu^{-}$ decay can be expressed in terms of transversity
amplitudes as [5]

$\displaystyle I_{1}^{s}$ $\displaystyle=$
$\displaystyle\frac{(2+\beta^{2}_{\mu})}{4}\left[|A^{L}_{\perp}|^{2}+|A^{L}_{\parallel}|^{2}+(L\to
R)\right]+\frac{4m^{2}_{\mu}}{q^{2}}{\rm
Re}\left(A^{L}_{\perp}A^{R*}_{\perp}+A^{L}_{\parallel}A^{R*}_{\parallel}\right),$
$\displaystyle I^{c}_{1}$ $\displaystyle=$
$\displaystyle|A^{L}_{0}|^{2}+|A^{R}_{0}|^{2}+\frac{4m^{2}_{\mu}}{q^{2}}\left[|A_{t}|^{2}+2{\rm
Re}\left(A^{L}_{0}A^{R*}_{0}\right)\right]+\beta^{2}_{\mu}|A_{S}|^{2},$
$\displaystyle I_{2}^{s}$ $\displaystyle=$
$\displaystyle\frac{\beta^{2}_{\mu}}{4}\left[|A^{L}_{\perp}|^{2}+|A^{L}_{\parallel}|^{2}+(L\to
R)\right],$ $\displaystyle I^{c}_{2}$ $\displaystyle=$
$\displaystyle-\beta^{2}_{\mu}\left[|A^{L}_{0}|^{2}+|A^{R}_{0}|^{2}\right],$
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\frac{\beta^{2}_{\mu}}{2}\left[|A^{L}_{\perp}|^{2}-|A^{L}_{\parallel}|^{2}+(L\to
R)\right],$ $\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\frac{\beta^{2}_{\mu}}{\sqrt{2}}\left[{\rm
Re}(A^{L}_{0}A^{L*}_{\parallel})+(L\to R)\right],$ $\displaystyle I_{5}$
$\displaystyle=$ $\displaystyle\sqrt{2}\beta_{\mu}\left[{\rm
Re}(A^{L}_{0}A^{L*}_{\perp})-(L\to R)-\frac{m_{\mu}}{\sqrt{q^{2}}}{\rm
Re}(A^{L}_{\parallel}A^{*}_{S}+A^{R}_{\parallel}A^{*}_{S})\right],$
$\displaystyle I^{s}_{6}$ $\displaystyle=$ $\displaystyle
2\beta_{\mu}\left[{\rm Re}(A^{L}_{\parallel}A^{L*}_{\perp})-(L\to R)\right],$
$\displaystyle I^{c}_{6}$ $\displaystyle=$ $\displaystyle
4\beta_{\mu}\frac{m_{\mu}}{\sqrt{q^{2}}}{\rm Re}\left[A^{L}_{0}A^{*}_{S}+(L\to
R)\right],$ $\displaystyle I_{7}$ $\displaystyle=$
$\displaystyle\sqrt{2}\beta_{\mu}\left[{\rm
Im}(A^{L}_{0}A^{L*}_{\parallel})-(L\to R)+\frac{m_{\mu}}{\sqrt{q^{2}}}{\rm
Im}(A^{L}_{\perp}A^{*}_{S}+A^{R}_{\perp}A^{*}_{S})\right],$ $\displaystyle
I_{8}$ $\displaystyle=$
$\displaystyle\frac{\beta^{2}_{\mu}}{\sqrt{2}}\left[{\rm
Im}(A^{L}_{0}A^{L*}_{\perp})+(L\to R)\right],$ $\displaystyle I_{9}$
$\displaystyle=$ $\displaystyle\beta^{2}_{\mu}\left[{\rm
Im}(A^{L*}_{\parallel}A^{L}_{\perp})+(L\to R)\right].$ (18)

The transversity amplitudes can be written as

$\displaystyle A_{\perp L,R}$ $\displaystyle=$ $\displaystyle
N\sqrt{2\lambda}\left[\left[(C^{\rm eff}_{9}+C^{\rm
eff\prime}_{9})\mp(C_{10}+C^{\prime}_{10})\right]\frac{V(q^{2})}{m_{B}+m_{K^{*}}}+\frac{2m_{b}}{q^{2}}C^{\rm
eff}_{7}T_{1}(q^{2})\right],$ $\displaystyle A_{\parallel L,R}$
$\displaystyle=$
$\displaystyle-N\sqrt{2}(m^{2}_{B}-m^{2}_{K^{*}})\left[\left[(C^{\rm
eff}_{9}-C^{\rm
eff\prime}_{9})\mp(C_{10}-C^{\prime}_{10})\right]\frac{A_{1}(q^{2})}{m_{B}-m_{K^{*}}}+\frac{2m_{b}}{q^{2}}C^{\rm
eff}_{7}T_{2}(q^{2})\right],$ $\displaystyle A_{0L,R}$ $\displaystyle=$
$\displaystyle-\frac{N}{2m_{K^{*}}\sqrt{q^{2}}}\left[[(C^{\rm eff}_{9}-C^{\rm
eff\prime}_{9})\mp(C_{10}-C^{\prime}_{10})]\right.$
$\displaystyle\left\\{(m^{2}_{B}-m^{2}_{K^{*}}-q^{2})(m_{B}+m_{K^{*}})A_{1}(q^{2})-\lambda\frac{A_{2}(q^{2})}{m_{B}+m_{K^{*}}}\right\\}$
$\displaystyle\left.+2m_{b}C^{\rm
eff}_{7}\left\\{(m^{2}_{B}+3m^{2}_{K^{*}}-q^{2})T_{2}(q^{2})-\frac{\lambda}{m^{2}_{B}-m^{2}_{K^{*}}}T_{3}(q^{2})\right\\}\right],$
$\displaystyle A_{t}$ $\displaystyle=$
$\displaystyle\frac{N\,\sqrt{\lambda}}{\sqrt{q^{2}}}\left[2(C_{10}-C^{\prime}_{10})\right]A_{0}(q^{2}),$
(19)

where

$N=V_{tb}V^{*}_{ts}\left[\frac{G^{2}_{F}\alpha^{2}}{3.2^{10}\pi^{5}m^{3}_{B}}q^{2}\sqrt{\lambda}\beta_{\mu}\right]^{1/2},$
(20)

with
$\lambda=m^{4}_{B}+m^{4}_{K^{*}}+q^{4}-2(m^{2}_{B}m^{2}_{K^{*}}+m^{2}_{B}q^{2}+m^{2}_{K^{*}}q^{2})$
and $\beta_{\mu}=\sqrt{1-4m^{2}_{\mu}/q^{2}}$. The expressions of form-factors
$V(q^{2})$, $A_{0,1,2}(q^{2})$ and $T_{1,2,3}(q^{2})$ are provided in ref.
[62]. These form factors are calculated within the framework of light cone sum
rule along with lattice QCD [62]. The non-factorizable corrections are
incorporated using the parametrizations of [62, 36] which are amicable with
ref. [63].

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