Matthew Black

#  $B$ meson Decay Constants Using Relativistic Heavy Quarks

Oliver Witzel

###### Abstract

We present an update on ongoing work to extract pseudoscalar and vector decay
constants for $B^{(*)}$, $B^{(*)}_{s}$ and $B^{(*)}_{c}$ mesons and determine
phenomenologically-interesting ratios such as $f_{B_{s}}/f_{B}$ or
$f_{B^{*}}/f_{B}$. Our calculation is based on ${\rm N_{f}}=2+1$ dynamical
flavour gauge field ensembles generated by the RBC/UKQCD collaborations using
domain-wall fermions and the Iwasaki gauge action. Using domain-wall light,
strange, and charm quarks and relativistic $b$ quarks, we obtain results at
multiple lattice spacings and valence quark masses.

##  1 Introduction

Leptonic decays of $B$ mesons are some of the simplest flavour-changing
processes in which a $b$ quark can participate. In the Standard Model (SM),
the decay rate of a $B^{+}$ meson decaying into a charged-lepton–neutrino pair
is given by

$\Gamma(B^{+}\to\ell^{+}\nu_{\ell})=\frac{G_{F}^{2}m_{B}m_{\ell}^{2}}{8\pi}\left(1-\frac{m_{\ell}^{2}}{m_{B}^{2}}\right)^{2}\,|V_{ub}|^{2}\,f_{B}^{2}.$
(1)

From the perspective of QCD, this is a very clean process where all of the
nonperturbative, hadronic physics is contained within the decay constant
$f_{B}$. The simplicity of Eq.(1) would motivate using these decay rates to
determine the CKM matrix element $|V_{ub}|$, however this requires high
precision knowledge of both the measured decay rate and the decay constant
$f_{B}$. The limiting factor in this respect is still experiment where thus
far $B^{+}\to\tau^{+}\nu_{\tau}$ and $B^{+}\to\mu^{+}\nu_{\mu}$ have been
observed with precision $\sim 30\%$ and $40\%$ respectively [1, 2, 3, 4, 5,
6]; this precision is anticipated to be improved upon in the near-future using
the new data being collected by Belle II. The pseudoscalar decay constants of
$B$ mesons with a light/strange valence quark are well-studied using
nonperturbative methods. From QCD sum rules, the best determinations have a
precision $\sim 10\%$ [7, 8, 9]. With lattice QCD, these are currently known
to a precision of less than $1\%$ [10, 11, 12, 13, 14]; from the pseudoscalar
decay constants also comes the important SU(3)-breaking ratio
$f_{B_{s}}/f_{B}$. Furthermore, these decay constants are important in other
processes such as in extracting bag parameters of neutral $B$-meson mixing
from hadronic matrix elements or in predicting the rare leptonic decays
$B_{q}\to\ell^{+}\ell^{-}$.

As well studied as the pseudoscalar decay constants are, one can also consider
the decay constants of more ‘exotic’ $B$ states. As the next generation of
experiments begin, such as HL-LHC and Belle II, and further theoretical
techniques allow the community to predict more ‘exotic’ decay channels, there
is more motivation than ever to develop high-precision lattice calculations of
new decay constants, with lower-hanging ‘exotic’ fruits being for example the
heavier $B_{c}$ meson and the vector states $B_{q}^{*},\,q=l,s,c$. To date,
only one calculation of the pseudoscalar $B_{c}$ decay constant has been
performed by the HPQCD collaboration [15]. Similarly, there exists only two
lattice calculations of the decay constant for the vector meson state
$B_{q}^{*}$ from the HPQCD [15] and ETM [16] collaborations. Of additional
interest in the case of the vector decay constant is that these two
calculations (with $q=l,s$) are in tension with one another and in fact their
ratios $f_{B^{*}_{(s)}}/f_{B_{(s)}}$ lie to opposite sides of unity. The
single calculation of $f_{B_{c}}$ and the tension in $f_{B^{*}_{(s)}}$
motivates further lattice calculations of these quantities to first reach
consensus on their values and then push to higher precision. We will
contribute to this by analysing the pseudoscalar and vector decay constants
for $B,\,B_{s},$ and $B_{c}$ mesons.

##  2 Lattice calculation

We use seven RBC/UKQCD $2{+}1$-flavour domain-wall fermion (DWF) and Iwasaki
gauge field ensembles with four lattice spacings $a\sim 0.11$, $0.08$, $0.07$,
$0.06\,\text{fm}$ (determined from RBC/UKQCD analyses [17, 18, 19]) and pion
masses $\in[267,433)\,\text{MeV}$. Light and strange quarks are simulated with
the Shamir DWF action with $M_{5}=1.8$. These ensembles are listed in Table 1.

$\begin{array}[]{cccccccccccc}&L&T&L_{s}&a^{-1}\\!/\\!\,\text{GeV}&am_{l}^{\text{sea}}&am_{s}^{\text{sea}}&M_{\pi}/\\!\,\text{MeV}&\text{\\#
cfgs}&\text{\\# sources}\\\
\hline\cr\text{C1}&24&64&16&1.785&0.005&0.040&340&1636&1\\\
\text{C2}&24&64&16&1.785&0.010&0.040&433&1419&1\\\\[5.16663pt]
\text{M1}&32&64&16&2.383&0.004&0.030&302&628&2\\\
\text{M2}&32&64&16&2.383&0.006&0.030&362&889&2\\\
\text{M3}&32&64&16&2.383&0.008&0.030&411&544&2\\\\[5.16663pt]
\text{F1S}&48&96&12&2.785&0.002144&0.02144&267&98&24\\\\[5.16663pt]
\text{X1}&32&64&12&3.148&0.0047&0.0186&371&40&16\end{array}$
Table 1: Ensembles used for the simulations reported here [20, 21, 17, 18].
$am_{l}^{\text{sea}}$ and $am_{s}^{\text{sea}}$ are the light and strange sea
quark masses and $M_{\pi}$ is the unitary pion mass. Valence strange quarks
are near their physical mass.

As bottom quarks are too heavy to be simulated directly at their physical
value on these lattices, we choose the effective action approach in order to
simulate physical $b$ quarks. Specifically, we choose the relativistic heavy
quark (RHQ) action, which is the Columbia variant [22, 23] of the Fermilab
heavy-quark action [24], with three nonperturbatively-tuned parameters
$m_{0}a,\,c_{P},$ and $\zeta$ [25]. The tuning of these parameters on each
ensemble was initially carried out in Ref. [25]; we report here first results
for tuning the RHQ parameters on the X1 ensemble. Charm quarks are simulated
with the Möbius DWF action and $M_{5}=1.6$ [26, 18, 27, 19]. We use three
masses below $m_{c}^{\text{phys}}$ on C and two masses bracketing
$m_{c}^{\text{phys}}$ on M, F, and X. Light and strange quarks have point
sources, while the $b$ and $c$ quarks use Gaussian-smeared sources and point
or smeared sinks; to improve statistics on X, instead of point sources we use
Z2-noise wall sources which are also Gaussian-smeared for the $b$ quarks.

Here we consider the vacuum-to-meson matrix elements of $B_{q}$ mesons at rest
(where $q=l,s,c$ denotes the valence quark) $\langle
0|J_{\mu}|B_{q}(0)\rangle$ for the axial-vector $A_{\mu}$ and vector $V_{\mu}$
currents to yield the decay constants of the pseudoscalar and vector meson
states respectively, where for the pseudoscalar we require only the temporal
component $A_{0}$ and for the vector the spatial components $V_{i},\,i=1,2,3$.
For these matrix elements, we can relate a lattice current $J_{\mu}$ to the
renormalised current $\mathcal{J}_{\mu}$ via the multiplicative
renormalisation factor

$Z_{J_{\mu}}^{bq}=\rho_{J_{\mu}}^{bq}\sqrt{Z^{qq}_{V}\,Z^{bb}_{V}},$ (2)

which follows the partially-nonperturbative approach [28, 29] where
$Z_{V}^{bb}$ and $Z_{V}^{qq}$ are nonperturbatively calculated and
$\rho_{J_{\mu}}^{bq}$ is the remaining correction calculated perturbatively at
one-loop and is expected to only deviate from unity by ${\cal O}(1\%)$ [30,
31]. For DWFs, $Z_{V}^{qq}$ can be computed via the relation

$Z_{V}^{qq}=Z_{A}^{qq}+{\cal O}(am_{\rm res}),$ (3)

and $Z_{V}^{bb}$ is calculated from the matrix element

$Z_{V}^{bb}\langle B_{q}|V_{0}|B_{q}\rangle=2M_{B_{q}}.$ (4)

Using the RHQ action, in order to reduce cut-off effects we require for
lattice (axial-)vector currents to include ${\cal O}(a)$-improvement terms
before this renormalisation. For the spatial components of the vector current
$V_{i}=\bar{b}\gamma_{i}q$, the relevant subset of the ${\cal
O}(a)$-improvement operators for calculating the decay constant are

$\displaystyle V_{i}^{0}$ $\displaystyle=\bar{b}\gamma_{i}q,$ $\displaystyle
V_{i}^{2}$ $\displaystyle=\bar{b}(2\overleftarrow{D}_{i})q,$ $\displaystyle
V_{i}^{4}$
$\displaystyle=\bar{b}\gamma_{i}\sum_{j}\gamma_{j}(2\overleftarrow{D}_{j})\,q,$
(5)

and similarly for the axial-vector current
$A_{0}=\bar{b}\gamma_{0}\gamma_{5}q$ replacing
$\gamma_{i}\to\gamma_{0}\gamma_{5}$. By considering only the temporal
component, $A_{0}^{2}$ is eliminated using the equations of motion. Then, we
can write the required ${\cal O}(a)$-improved currents as

$\displaystyle A_{0}$ $\displaystyle=A_{0}^{0}+c_{A}^{4}\,A_{0}^{4},$ (6)
$\displaystyle V_{i}$
$\displaystyle=V_{i}^{0}+c_{V}^{2}V_{i}^{2}+c_{V}^{4}V_{i}^{4}.$ (7)

The coefficients $c_{A,V}^{k}$ of the ${\cal O}(a)$-improvement operators are
computed at one-loop perturbation theory [31, 30].

Figure 1: Two-point correlator used to determine a pseudoscalar decay
amplitude. $t_{0}$ indicates the position of the source with pseudoscalar
current $\bar{b}\gamma_{5}q$, and $t_{A_{\mu}}$ is the time of insertion of
the axial-vector current $A_{\mu}$.

The renormalised vacuum-to-meson matrix elements are related to the decay
constants as

$\displaystyle\langle 0|{\cal A}_{0}|B_{q}(0)\rangle$
$\displaystyle=f_{B_{q}}M_{B_{q}},$ (8) $\displaystyle\langle 0|{\cal
V}_{i}|B_{q,j}^{*}(0)\rangle$
$\displaystyle=f_{B_{q}^{*}}M_{B_{q}^{*}}\delta_{ij}.$ (9)

It can be simpler first to consider the decay amplitude
$\Phi_{B_{q}}=f_{B_{q}}\sqrt{M_{B_{q}}}$ which can be defined as a ratio of
two-point correlators in the limit of $t_{0}\ll t\ll T+t_{0}$ such that
excited states have decayed and the ground state of the system is clearly
resolved. By ‘folding’ correlators such that they only have forwards-
propagating states up to extent $T/2$ and $t_{0}=0$, we need only consider the
limit $t\gg 0$ to find the ground state. For pseudoscalar and vector decay
amplitudes respectively, these ratios are defined as

$\displaystyle\Phi_{B_{q}}$
$\displaystyle=\sqrt{2}\,Z_{A_{0}}^{bq}\;\lim_{t\gg
0}\frac{|\tilde{C}_{A_{0}P}(t,0)|}{\sqrt{\tilde{\tilde{C}}_{{P}{P}}(t,0)\,e^{-M_{B_{q}}t}}},$
(10) $\displaystyle\Phi_{B_{q}^{*}}$
$\displaystyle=\sqrt{2}\,Z_{V_{i}}^{bq}\;\lim_{t\gg
0}\frac{|\tilde{C}_{V_{i}V_{i}}(t,0)|}{\sqrt{\tilde{\tilde{C}}_{V_{i}^{0}V_{i}^{0}}(t,0)\,e^{-M_{B_{q}^{*}}t}}},$
(11)

where we define $P$ the pseudoscalar current $\bar{b}\gamma_{5}q$ and a two-
point correlation function $C_{XY}(t,t_{0})$ with currents $X,Y$ inserted at
times $t,t_{0}$ respectively. $\tilde{C}$ indicates the correlator is smeared
at the source, and $\tilde{\tilde{C}}$ indicates both source and sink are
smeared. A schematic of the two-point correlator $C_{A_{0}P}(t,t_{0})$ is
shown in Figure 1.

Finally, we present the decay amplitudes as dimensionless ratios over the
$B_{s}$ meson mass $M_{B_{s}}^{3/2}$, which is used to tune the RHQ parameters
to the physical value of the $b$ quark. To gain our final results in physical
units after performing continuum extrapolations, we can simply multiply by the
experimental value of $M_{B_{s}}$ [32]. In Figure 2, we show examples of the
effective mass $M_{B_{s}^{(*)}}$ and dimensionless bare decay amplitude
$\Phi_{B_{q}^{(*)}}/M_{B_{s}}^{3/2}$.

Figure 2: Determination of the effective mass (top) and dimensionless decay
amplitude (bottom) for the pseudoscalar on F1S (left) and for the vector on C2
(right). The data are fitted to a constant value in the region where excited
state contamination is minimised and we have a clear ground state signal.

##  3 Chiral and Continuum Extrapolations

Here we show the current status of our analysis to extrapolate to the physical
values for the decay amplitudes of pseudoscalar $B_{(s)}$ systems. Analysis of
the $B_{c}$ follows analogously with an extra-/interpolation to physical charm
mass using the $\eta_{c}$ meson mass. Measurements are still in progress for
the vector states and thus extra-/interpolations to physical values cannot yet
be performed.

For the determination of $\Phi_{B_{s}}$ we currently consider a two step
procedure. First we correct for slight mistunings of the used strange quark
mass by inter-/extrapolating to the physical value. That way we obtain values
for $\Phi_{B_{s}}$ for each of our six different ensembles and use these
values to also form SU(3) symmetry breaking ratios $\Phi_{B_{s}}/\phi_{B}$. In
order to determine $f_{B_{s}}$ we need to renormalize our $\Phi_{B_{s}}$
values and then perform in a second step a continuum limit extrapolation in
$a^{2}$.

Skipping the renormalisation of the decay amplitudes for now, we continue
looking at the SU(3) symmetry breaking ratio $\Phi_{B_{s}}/\Phi_{B}$ right
away. To perform an extrapolation of our data shown in Figure 3, we use NLO
SU(2) Heavy Meson Chiral Perturbation Theory (HM$\chi$PT). For sufficiently
light valence quark masses, this will provide an effective description of the
decay amplitude as a function of the pion mass. This allows us to take the
joint continuum and physical pion mass limit from our data. Similarly we use
HM$\chi$PT to also calculate the SU(3)-breaking ratio $f_{B_{s}}/f_{B}$ In the
limit of unitary pions, our fit Ansätze for these extrapolations are

$\displaystyle\frac{\Phi_{B_{x}}}{M_{B_{s}}^{3/2}}$
$\displaystyle=\Phi_{0}\left\\{1-\frac{1+3g_{b}^{2}}{(4\pi
f_{\pi})^{2}}\frac{3}{4}\cdot
M_{xx}^{2}\ln(M_{xx}^{2}/\Lambda_{\chi}^{2})+c_{1}\cdot\frac{2M_{xx}^{2}}{(4\pi
f_{\pi})^{2}}+c_{2}\cdot\frac{a^{2}}{(4\pi f_{\pi})^{2}}\right\\},$ (12)
$\displaystyle\frac{\Phi_{B_{s}}}{\Phi_{B_{x}}}$
$\displaystyle=R_{\Phi}\left\\{1+\frac{1+3g_{b}^{2}}{(4\pi
f_{\pi})^{2}}\frac{3}{4}\cdot
M_{xx}^{2}\ln(M_{xx}^{2}/\Lambda_{\chi}^{2})\right\\}+d_{1}\cdot\frac{2M_{xx}^{2}}{(4\pi
f_{\pi})^{2}}+d_{2}\cdot\frac{a^{2}}{(4\pi f_{\pi})^{2}},$ (13)

where $\Phi_{0},c_{1},c_{2},R_{\Phi},d_{1},d_{2}$ are fit coefficients,
$M_{xx}$ is the mass of a pion with two valence quarks of mass $m_{x}$,
$g_{b}=0.56(3)$ the $B^{*}B\pi$ coupling constant [33], and
$f_{\pi}=130.2(1.2)\,$MeV the pion decay constant [34]. The chiral-continuum
extrapolation for $\Phi_{B_{s}}/\Phi_{B_{x}}$ is shown in Figure 3.

Figure 3: Chiral-continuum extrapolation of $\Phi_{B_{s}}/\Phi_{B_{q}}$ using
NLO SU(2) HM$\chi$PT. The coloured data are associated with different gauge
configurations as indicated in the legend. Only the filled points (unitary)
enter the fits. The coloured lines indicate the fit function at each
corresponding lattice spacing, the black line indicates the function at zero
lattice spacing. The black dashed line indicates the isospin-averaged pion
mass, $m_{\bar{\pi}}\sim 138\,$MeV. Only statistical errors are shown.

It is also possible for the above fits to choose different Ansätze to perform
our extrapolations. The variation of results from different Ansätze will be
included in a full systematic error analysis later on. Further systematic
effects to be considered include, for example, sea quark mass mistuning and
RHQ parameter uncertainties.

##  4 Tuning RHQ on X1

In early analysis of the $B_{c}$ decay constant using C, M, and F ensembles,
large cut-off effects are found, particularly on the coarse C ensembles.
Therefore, to improve extrapolation to a physical value, we choose to include
the extra-fine ensemble in the RHQ analysis: X1. To use this additional
ensemble, we perform the nonperturbative tuning procedure for the RHQ
parameters $m_{0},\,c_{P},$ and $\zeta$ as previously done for the other
ensembles [25]. Measurements performed on the X1 ensemble make use of Grid
[35, 36] and Hadrons [37].

This RHQ tuning process is not yet completed for this analysis, however at the
current stage of this procedure, we find promising signals of the potential of
X1 to aid in the analysis of the $B_{c}$ decay constants, and potentially the
light $B_{(s)}$ states as well. Examples of fits entering the tuning analysis
are show in Figure 4.

Figure 4: Examples of fits included in the tuning analysis on X1. Left plot:
extraction of the ground state meson mass with $\vec{p}^{2}=0$ for
pseudoscalar $B_{s}$ and vector $B_{s}^{*}$. The valence quark is set to the
unitary strange mass, $am_{q}=0.0186$. Two correlators with different
Gaussian-smearing radii are used in a combined fully-correlated fit and a
first excited state is also included for the smaller smearing radius. Right
plot: a fit to the slope of the continuum dispersion relation where ground
state energies are first extracted for each integer square momentum $n^{2}$,
and we fit to $n^{2}\leq 3$. Tuned RHQ parameters enforce a slope equal to 1.

##  5 $B_{c}$ Decay Constant

In analysing the $B_{c}$ system, one finds a difference from $B$ mesons with
light/strange valence quarks in that the heavier $c$ valence quark leads to
higher statistical precision in the correlators, given the same number of
measurements. However, we are instead more sensitive to systematic effects, in
particular charm discretisation errors. In the left plot of Figure 5, one can
see that the statistical precision is a factor $\sim 2$ higher than
previously.

On each ensemble, we extract the decay amplitude at multiple charm quark
masses and then on C we extrapolate to the physical charm mass, while on M, F,
and X we bracket the physical value and interpolate. We define the physical
charm point through the $\eta_{c}(1S)$ meson mass (neglecting disconnected
diagrams), matching to the measured value $M_{\eta_{c}}=2983.9(4)\,$MeV [32].
Until the RHQ tuning is complete and we have finalised data on X1, we do not
settle on a fit Ansatz for the combined continuum and physical charm
extrapolation. In the right plot of Figure 5, we present the pseudoscalar
decay amplitudes of the $B_{c}$ meson on each ensemble. The different slopes
one can see between ensembles indicate cut-off effects which need to be
controlled in the final extrapolation. We use open symbols for the X1 data to
indicate that these are not finalised until the RHQ tuning is finished, and
therefore their vertical positioning will change in the final analysis.
However the slope between the two data points indicates this ensemble will
indeed be important in extrapolating to physical values.

Figure 5: Analysis of the $B_{c}$ pseudoscalar dimensionless decay amplitudes.
Left plot: determination of the effective mass on F1S for a single charm mass.
Right plot: collection of decay amplitudes on each ensemble for multiple charm
quark masses, represented through the $\eta_{c}(1S)$ mass (excluding
disconnected contributions). The open-symbol data are only representative
while the X1 RHQ tuning is still in process.

##  6 Summary

$B$ meson decay constants are important quantities for understanding the
hadronic physics in weak decays of the SM. Here we have presented ongoing work
to extract decay constants for both pseudoscalar and vector $B$ mesons with
light, strange, and charm valence quarks using the effective RHQ action to
simulate physical $b$ quarks. Analysis of pseudoscalar $f_{B_{q}}$ for $q=l,s$
including isospin-breaking effects in the light quark is ongoing. Systematic
effects for the $B_{c}$ meson motivated the inclusion of the X1 ensemble where
the nonperturbative tuning of the RHQ action parameters is in progress. In
addition, we are working on extracting $B^{*}$ vector meson states and
determining their decay constants. Once we have collected all measurements, we
will obtain renormalised continuum results for all decay constants, as well as
interesting ratios such as the SU(3)-breaking estimate $f_{B_{s}}/f_{B^{0,+}}$
and the vector-to-pseudoscalar ratio $f_{B_{l,s,c}^{*}}/f_{B_{l,s,c}}$ where
an important question remains as to which side of unity this quantity should
lie.

## Acknowledgments

We thank our RBC/UKQCD colleagues for helpful discussions and suggestions.
Measurements performed on the X1 ensemble make use of Grid [35, 38] and
Hadrons [37]. Computations used resources provided by the USQCD Collaboration,
funded by the Office of Science of the US Department of Energy and by the
ARCHER UK National Supercomputing Service, as well as computers at Columbia
University, Brookhaven National Laboratory, and the OMNI cluster at the
University of Siegen. We used gauge field configurations generated on the
DiRAC Blue Gene Q system at the University of Edinburgh, part of the DiRAC
Facility, funded by BIS National E-infrastructure grant ST/K000411/1 and STFC
grants ST/H008845/1, ST/K005804/1 and ST/K005790/1. M.B. is supported by
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through
grant 396021762 - TRR 257 “Particle Physics Phenomenology after the Higgs
Discovery”.

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Nikhef-2024-008

SI-HEP-2024-10

P3H-24-027

Targeting (Pseudo)-Scalar CP Violation with $B_{s}\to\mu^{+}\mu^{-}$

Robert Fleischer a,b, Eleftheria Malami a,c, Anders Rehult a,b, 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 leptonic decay $B_{s}\to\mu^{+}\mu^{-}$ is both rare and theoretically
clean, making it an excellent probe for New Physics searches. Due to its
helicity suppression in the Standard Model, this decay is particularly
sensitive to new (pseudo)-scalar contributions. We present a new strategy for
detecting CP-violating New Physics contributions of this kind, exploiting two
observables: $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$, which is accessible
due to the sizeable decay width difference of the $B_{s}$ system, and the
mixing-induced CP asymmetry $\mathcal{S}_{\mu\mu}$. We also utilize
information from semileptonic $b\to s\mu^{+}\mu^{-}$ decays. Using the
currently available experimental information, we find remarkably constrained
regions in the
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$ plane that
serve as promising targets for future measurements.

May 2024

##  1 Introduction

The leptonic decay $B_{s}\to\mu^{+}\mu^{-}$ is exceptionally rare – out of
every billion $B_{s}$ mesons produced, only about three decay into the
$\mu^{+}\mu^{-}$ final state [1]. This rarity has made the decay an elusive
target. First constraints on its branching ratio were obtained at the Tevatron
by the CDF [2] and DØ [3] collaborations, followed by an observation at the
LHC, in a joint effort by LHCb and CMS [4]. Recent measurements by LHCb [5],
CMS [6], and ATLAS [7] have pushed the experimental precision on the branching
ratio to the ten-percent level, a precision set to improve even further in the
coming decade [8]. Pioneering analyses have also been performed by LHCb [5],
CMS [9], and ATLAS [10] on the $B^{0}_{s}\to\mu^{+}\mu^{-}$ effective lifetime
$\tau_{s}^{\mu\mu}$, measurable due to the sizeable decay width difference
$\Delta\Gamma_{s}$ of the $B_{s}$ system [11]. The effective lifetime can be
converted into the observable $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$.

In the Standard Model (SM), the decay is a flavor-changing neutral current
process and is helicity suppressed: its branching ratio is proportional to the
muon mass squared. Possible new axial-vector, scalar and pseudo-scalar
leptonic currents would affect this decay, where the (pseudo)-scalar
interactions would lift the helicity suppression. The rarity of the
$B_{s}\to\mu^{+}\mu^{-}$ decay thus makes it an excellent choice for indirect
New Physics (NP) searches.

In these explorations, both the branching ratio and
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ are important observables, and
further data on them are highly valuable. However, $B_{s}\to\mu^{+}\mu^{-}$
has even more information to offer. Theoretical work has identified the
mixing-induced CP asymmetry $\mathcal{S}_{\mu\mu}$ as a promising target of
study [12, 13]. This observable, accessible through a time-dependent, flavour-
tagged analysis, is generated through interference between
$\bar{B}^{0}_{s}\to\mu^{+}\mu^{-}$ and $B^{0}_{s}\to\mu^{+}\mu^{-}$ decay
processes that originates from $B^{0}_{s}$–$\bar{B}^{0}_{s}$ mixing.

In this paper, we explore new ways to probe possible (pseudo)-scalar
contributions by fully exploiting the rare $B_{s}\to\mu^{+}\mu^{-}$ decay. To
accomplish this, we need input on possible NP axial-vector interactions. If
present, such contributions as well as new (pseudo)-scalar terms would also
enter semileptonic rare $b\to s$ transitions. We demonstrate that
(pseudo)-scalar interactions are very suppressed in the semileptonic rare
$B$-meson decays, thereby allowing us to use these decays to determine the
axial-vector NP contributions. This information can then be used to focus the
analysis of $B_{s}\to\mu^{+}\mu^{-}$ on (pseudo)-scalar effects.

To achieve this, we present a new strategy to determine whether such possible
(pseudo)-scalar NP terms violate CP symmetry. This method exploits the
complementary dependence of $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and
$\mathcal{S}_{\mu\mu}$ on such NP effects. In the case of CP-conserving
(pseudo)-scalar NP, these observables are strongly correlated: we identify
remarkably constrained regions in the
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$ plane that
serve as interesting target regions for future measurements. Any future
measurement of these observables outside this target region would indicate CP-
violating phases in the (pseudo)-scalar interactions.

This paper is structured as follows: In Sec. 2, we introduce our theoretical
framework for $B_{s}\to\mu^{+}\mu^{-}$, construct the relevant observables,
and provide SM predictions. We focus in particular on CKM uncertainties. In
Sec. 3, we demonstrate how rare semileptonic $B_{(s)}$ decays can be used to
determine the axial-vector NP contribution. In Sec. 4, we consider
(pseudo)-scalar NP coefficients and show how we obtain our target regions for
the $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and $\mathcal{S}_{\mu\mu}$
observables. We conclude and give a brief outlook in Sec. 5.

##  2 Theoretical Framework for the $\bar{B}^{0}_{s}\to\mu^{+}\mu^{-}$ Decay

###  2.1 Effective Hamiltonian

In order to describe the $\bar{B}^{0}_{s}\to\mu^{+}\mu^{-}$ decay, we apply
effective quantum field theory, in where heavy SM and NP degrees of freedom
are “integrated out”, leading to a low-energy effective Hamiltonian of the
following form [14, 11, 15, 12]:

$\mathcal{H}_{\rm eff}=-\frac{G_{\rm
F}}{\sqrt{2}\pi}V_{ts}^{*}V_{tb}\alpha_{\rm
em}\left[C_{10}O_{10}+C_{S}O_{S}+C_{P}O_{P}+C_{10}^{\prime}O_{10}^{\prime}+C_{S}^{\prime}O_{S}^{\prime}+C_{P}^{\prime}O_{P}^{\prime}\right]\
.$ (1)

Here, $G_{\rm F}$ is the Fermi constant, $V_{tb}$ and $V_{ts}^{*}$ are
elements of the Cabibbo–Kobayashi–Maskawa (CKM) matrix, and $\alpha_{\rm em}$
is the QED fine structure constant. The four-fermion operators111We note that
our convention for $O_{S}^{(\prime)}$ and $O_{S}^{(\prime)}$ differs by a
factor $m_{b}$ from [16] but agrees with [11, 15, 12]. are given as follows:

$\displaystyle O_{10}$
$\displaystyle=(\bar{s}\gamma_{\mu}P_{L}b)(\bar{\mu}\gamma^{\mu}\gamma_{5}\mu)\
,$ $\displaystyle O_{10}^{\prime}$
$\displaystyle=(\bar{s}\gamma_{\mu}P_{R}b)(\bar{\mu}\gamma^{\mu}\gamma_{5}\mu)\
,$ (2) $\displaystyle O_{S}$
$\displaystyle=m_{b}(\bar{s}P_{R}b)(\bar{\mu}\mu)\ ,$ $\displaystyle
O_{S}^{\prime}$ $\displaystyle=m_{b}(\bar{s}P_{L}b)(\bar{\mu}\mu)\ ,$
$\displaystyle O_{P}$
$\displaystyle=m_{b}(\bar{s}P_{R}b)(\bar{\mu}\gamma_{5}\mu)\ ,$ $\displaystyle
O_{P}^{\prime}$ $\displaystyle=m_{b}(\bar{s}P_{L}b)(\bar{\mu}\gamma_{5}\mu)\
,$

where $m_{b}$ is the bottom-quark mass, and
$P_{R(L)}=\frac{1}{2}(1\pm\gamma_{5})$. In general, the Wilson coefficients
depend on the flavour of the final-state leptons [13]. In this paper, we
consider only decays into two muons and the corresponding coefficients. In the
effective Hamiltonian describing the CP-conjugate processes, all CP-violating
phases associated with the Wilson coefficients change their signs.

For convenience, we define the combinations

$\displaystyle\mathcal{C}_{i}^{\pm}$ $\displaystyle\equiv C_{i}\pm
C_{i}^{\prime}\ ,\quad i\in\\{10,S,P\\}\ ,$ (3)

for the axial-vector, scalar and psuedo-scalar $\mu\bar{\mu}$ currents. In the
SM, only the axial-vector operator $O_{10}$ arises, and we introduce

$C_{10}\equiv C_{10}^{\rm SM}+C_{10}^{\rm NP}\ .$ (4)

###  2.2 Observables and Inputs

Let us discuss the observables of $\bar{B}^{0}_{s}\to\mu^{+}\mu^{-}$ in this
subsection. We use (1) to calculate the decay amplitude through evaluating the
hadronic matrix element between an initial $B^{0}_{s}$ meson and the final
$\mu^{+}\mu^{-}$ state. The decay constant $f_{B_{s}}$ encodes all the
hadronic physics and is known with impressive precision from Lattice QCD
(LQCD) computations. The HFLAG average [17] utilising the results of [18, 19,
20, 21] yields

$f_{B_{s}}=(230.3\pm 1.3)\;$\mathrm{MeV}$\ .$ (5)

The branching ratio of this decay channel in the SM can be written as follows
[22]:

$\displaystyle{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$
$\displaystyle=\frac{G_{F}^{2}\absolutevalue{V_{ts}V_{tb}}^{2}\alpha_{\rm
em}^{2}m_{B_{s}}f_{B_{s}}^{2}\tau_{B_{s}}}{64\pi^{3}}\sqrt{1-\frac{4m_{\mu}^{2}}{m_{B_{s}}^{2}}}4m_{\mu}^{2}|C_{10}^{\rm
SM}|^{2}\bigg{[}\absolutevalue{P}^{2}+\absolutevalue{S}^{2}\bigg{]}\ ,$ (6)

which is a CP-averaged quantity. Alternatively, we may replace $\alpha_{\rm
em}$ by

$\alpha_{\rm em}=\frac{2G_{\rm F}^{2}m_{W}^{4}\sin^{4}\theta_{\rm
W}}{\pi^{2}}\ ,$ (7)

where $\theta_{\rm W}$ is the Weinberg angle. Numerically, we use $\alpha_{\rm
em}=1/132$, and for the Wilson coefficient we use

$C_{10}^{\rm SM}=-4.194\ ,$ (8)

obtained in flavio [23] at the scale $\mu=m_{b}$.

The expression in (6), which holds in the presence of interactions from beyond
the SM, introduces the quantities [11]

$P\equiv\frac{C_{10}-C_{10}^{\prime}}{C_{10}^{\rm
SM}}+\frac{M_{B_{s}}^{2}}{2m_{\mu}}\left(\frac{m_{b}}{m_{b}+m_{s}}\right)\left(\frac{C_{P}-C_{P}^{\prime}}{C_{10}^{\rm
SM}}\right)\equiv\absolutevalue{P}e^{i\varphi_{P}}\ ,$ (9)
$S\equiv\sqrt{1-\frac{4m_{\mu}^{2}}{M_{B_{s}}^{2}}}\frac{M_{B_{s}}^{2}}{2m_{\mu}}\left(\frac{m_{b}}{m_{b}+m_{s}}\right)\left(\frac{C_{S}-C_{S}^{\prime}}{C_{10}^{\rm
SM}}\right)\equiv\absolutevalue{S}e^{i\varphi_{S}}\ ,$ (10)

where $\varphi_{P}$ and $\varphi_{S}$ are CP-violating phases, and
$m_{s}=0.093$ MeV denotes the mass of the strange quark. In the SM, we have by
definition the following values:

$P_{\rm SM}=1\ ,\quad\quad S_{\rm SM}=0\ .$ (11)

The branching ratio ${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ in (6) exhibits
helicity suppression, which is reflected by the $m_{\mu}^{2}$ factor. Looking
at (9) and (10), we observe that the helicity suppression can be lifted by the
$\mathcal{C}_{P}^{(^{\prime})}$ and
$\mathcal{C}_{S}^{(^{\prime})}$coefficients, entering with a factor
$1/m_{\mu}$.

The expression of (6) defines the “theoretical” branching ratio at $t=0$.
However, after this point the meson starts to oscillate into its antiparticle.
What is then measured is the time-integrated branching ratio
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$, which is related to the
theoretical one through [11]

$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})_{\rm
theo}=\left[\frac{1-y_{s}^{2}}{1+\mathcal{A}_{\Delta\Gamma_{s}}y_{s}}\right]\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})\
,$ (12)

where

$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})_{\rm
theo}=\frac{1}{2}\bigg{[}\mathcal{B}(\bar{B}^{0}_{s}\to\mu^{+}\mu^{-})+\mathcal{B}(B^{0}_{s}\to\mu^{+}\mu^{-})\bigg{]}$
(13)

is the CP-averaged (“untagged”) theoretical branching ratio. The parameter
[24]

$y_{s}\equiv\frac{\Gamma_{\rm L}^{(s)}-\Gamma_{\rm H}^{(s)}}{\Gamma_{\rm
L}^{(s)}+\Gamma_{\rm H}^{(s)}}=\frac{\Delta\Gamma_{s}}{2\Gamma_{s}}=0.062\pm
0.004$ (14)

depends on the decay width difference $\Delta\Gamma_{s}$ between the $B_{s}$
mass eigenstates and on the $B_{s}$ lifetime ${\tau_{B_{s}}}=1/\Gamma_{s}$.
Due to $\Delta\Gamma_{s}$, we gain access to

$\mathcal{A}_{\Delta\Gamma_{s}}=\frac{\absolutevalue{P}^{2}\cos(2\varphi_{P}-\phi_{s}^{\rm
NP})-\absolutevalue{S}^{2}\cos(2\varphi_{S}-\phi_{s}^{\rm
NP})}{\absolutevalue{P}^{2}+\absolutevalue{S}^{2}}\ .$ (15)

This observable is given by $+1$ in the SM but can generally take any value
between $-1$ and $+1$. The phase $\phi_{s}^{\rm NP}$ is related to CP-
violating contributions from beyond the SM to $B^{0}_{s}$–$\bar{B}^{0}_{s}$
mixing:

$\phi_{s}^{\rm NP}=\phi_{s}-\phi_{s}^{\rm SM}\ .$ (16)

Following [25, 26], we combine the value of $\phi_{s}$ with the SM prediction
$\phi_{s}^{\rm SM}$. The $\phi_{s}$ comes from decays of the kind
$B^{0}_{s}\to J/\psi\phi$, where contributions from penguin topologies have
been taken into account [27]:

$\phi_{s}=(-4.2\pm 1.4)^{\circ}\ .$ (17)

The SM value arises from an analysis of the Unitarity Triangle (UT) utilising
only information from the angle $\gamma$ and the side $R_{b}$ [27]:

$\phi_{s}^{\rm SM}=(-2.01\pm 0.12)^{\circ}\ .$ (18)

Finally, using (19), the NP phase is determined as [27]

$\phi_{s}^{\rm NP}=(2.2\pm 1.4)^{\circ}\ ,$ (19)

which is the value we use in order to include NP contributions to
$B^{0}_{s}$–$\bar{B}^{0}_{s}$ mixing.

We note that $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ can be obtained from
measurements of the $B_{s}\to\mu^{+}\mu^{-}$ effective lifetime, defined as
follows (see also [11, 12]):

$\tau_{\mu\mu}^{s}\equiv\frac{\int_{0}^{\infty}t\langle\Gamma(B_{s}(t)\to\mu^{+}\mu^{-})\rangle
dt}{\int\int_{0}^{\infty}\langle\Gamma(B_{s}(t)\to\mu^{+}\mu^{-})\rangle dt}\
.$ (20)

Measuring this quantity requires time information for untagged data samples.
ATLAS [10], CMS [6] and the LHCb Collaboration [5] have all recently reported
measurements of the effective lifetime $\tau_{\mu\mu}^{s}$:

$\displaystyle\tau_{\mu\mu}^{s}|_{\rm LHCb\;2021}$ $\displaystyle=(2.07\pm
0.29)\;$\mathrm{p}\mathrm{s}$\ ,$ (21) $\displaystyle\tau_{\mu\mu}^{s}|_{\rm
CMS\;2022}$ $\displaystyle=(1.83\pm 0.23)\;$\mathrm{p}\mathrm{s}$\ ,$
$\displaystyle\tau_{\mu\mu}^{s}|_{\rm ATLAS\;2023}$ $\displaystyle=(0.99\pm
0.45)\;$\mathrm{p}\mathrm{s}$\ .$

Using the following relation [11]:

$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}=\frac{1}{y_{s}}\left[\frac{(1-y_{s}^{2})\tau_{\mu\mu}^{s}-(1+y_{s}^{2})\tau_{B_{s}}}{2\tau_{B_{s}}-(1-y_{s}^{2})\tau_{\mu\mu}^{s}}\right]\
,$ (22)

the above results can be translated into bounds on
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$, yielding:

$\displaystyle\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}|_{\rm LHCb\;2021}$
$\displaystyle=8.44\pm 7.11\ ,$ (23)
$\displaystyle\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}|_{\rm CMS\;2022}$
$\displaystyle=3.75\pm 3.68\ ,$
$\displaystyle\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}|_{\rm ATLAS\;2023}$
$\displaystyle=-4.08\pm 2.52\ .$

Unfortunately, the uncertainties are still too large to constrain the
observable within its model-independent range of
$-1\leq\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}\leq 1$.

Finally, we can also consider the full time-dependent rate of the $B_{s}$
mesons decaying into two muons with helicity $\lambda=L,R$. Even though the
expression in (6) corresponds to a summation over helicities, it is now useful
to work on the case for specific helicities. A flavour-tagged analysis would
then give access to the CP-violating decay rate asymmetry [11, 22]:

$\frac{\Gamma(B^{0}_{s}(t)\to\mu_{\lambda}^{+}\mu_{\lambda}^{-})-\Gamma(\bar{B}^{0}_{s}(t)\to\mu_{\lambda}^{+}\mu_{\lambda}^{-})}{\Gamma(B^{0}_{s}(t)\to\mu_{\lambda}^{+}\mu_{\lambda}^{-})+\Gamma(\bar{B}^{0}_{s}(t)\to\mu_{\lambda}^{+}\mu_{\lambda}^{-})}=\frac{\mathcal{C}_{\mu\mu}^{\lambda}\cos(\Delta
M_{s}t)+\mathcal{S}_{\mu\mu}\sin(\Delta
M_{s}t)}{\cosh(y_{s}t/\tau_{B_{s}})+\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}\sinh(y_{s}t/\tau_{B_{s}})}\
,$ (24)

with the CP asymmetries

$\mathcal{C}_{\mu\mu}^{\lambda}=-\eta_{\lambda}\left[\frac{2\absolutevalue{PS}\cos(\varphi_{P}-\varphi_{S})}{\absolutevalue{P}^{2}+\absolutevalue{S}^{2}}\right]\equiv-\eta_{\lambda}\mathcal{C}_{\mu\mu}\
,$ (25)
$\mathcal{S}_{\mu\mu}=\frac{\absolutevalue{P}^{2}\sin(2\varphi_{P}-\phi_{s}^{\rm
NP})-\absolutevalue{S}^{2}\sin(2\varphi_{S}-\phi_{s}^{\rm
NP})}{\absolutevalue{P}^{2}+\absolutevalue{S}^{2}}\ ,$ (26)

both of which are zero in the SM. The observable
$\mathcal{C}_{\mu\mu}^{\lambda}$ depends on the helicity of the lepton pair
($\eta_{L}=+1$ and $\eta_{R}=-1$), making it difficult to measure. However, we
stress that even knowing its sign would already be useful to resolve discrete
ambiguities that arise when determining (pseudo)-scalar Wilson coefficients
[12]. We note that no data are yet available for $\mathcal{S}_{\mu\mu}$ and
$\mathcal{C}_{\mu\mu}$.

Keeping the time dependence but summing over the muon helicities, (24) becomes

$\frac{\Gamma(B^{0}_{s}(t)\to\mu^{+}\mu^{-})-\Gamma(\bar{B}^{0}_{s}(t)\to\mu^{+}\mu^{-})}{\Gamma(B^{0}_{s}(t)\to\mu^{+}\mu^{-})+\Gamma(\bar{B}^{0}_{s}(t)\to\mu^{+}\mu^{-})}=\frac{\mathcal{S}_{\mu\mu}\sin(\Delta
M_{s}t)}{\cosh(y_{s}t/\tau_{B_{s}})+\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}\sinh(y_{s}t/\tau_{B_{s}})}\
.$ (27)

The three observables $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$,
$\mathcal{S}_{\mu\mu}$, and $\mathcal{C}_{\mu\mu}$ are theoretically clean
since the decay constant $f_{B_{s}}$ cancels in all of them. They satisfy the
relation

$(\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu})^{2}+(\mathcal{S}_{\mu\mu})^{2}+(\mathcal{C}_{\mu\mu})^{2}=1\
,$ (28)

and are therefore not independent of one another.

###  2.3 Information from Branching Ratios

The SM prediction for the branching ratio in (6) depends on the CKM matrix
elements entering $|V_{ts}V_{tb}|$. Using the Wolfenstein parametrization, we
obtain

$\displaystyle|V_{ts}V_{tb}|=\phantom{\lambda}|V_{cb}|\left[1-\frac{\lambda^{2}}{2}\left(1-2R_{b}\cos\gamma\right)\right]+\mathcal{O}\left(\lambda^{6}\right)\>,$
(29)

where $\lambda\equiv|V_{us}|$, the parameter

$R_{b}\equiv\left|\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}\right|=\left(1-\frac{\lambda^{2}}{2}\right)\frac{1}{\lambda}\left|\frac{V_{ub}}{V_{cb}}\right|\
,$ (30)

is the usual side of the UT from the origin to the apex, and $\gamma$ is the
corresponding angle. Expression (29) is governed by $|V_{cb}|$ up to
$\mathcal{O}({\lambda^{2}})$ corrections, which involve $R_{b}$ and $\gamma$.

To determine $|V_{ts}V_{tb}|$, we thus require values for $|V_{ub}|$ and
$|V_{cb}|$. This poses a problem, as these CKM factors carry a substantial,
hidden theoretical uncertainty. There are two methods to determine $|V_{ub}|$
and $|V_{cb}|$. One uses exclusive decay processes and the other inclusive
measurements. Ideally, both determinations should agree with each other, but
in practice, tensions arise. These long-standing tensions constitute the key
uncertainty for the branching ratio, and special care is needed. Therefore, we
study separately the inclusive and exclusive case, as discussed in [26]. In
addition to these two determinations, a hybrid approach combining the
exclusive $|V_{ub}|$ with the inclusive $|V_{cb}|$ values was also studied
[26].

In the inclusive and hybrid scenarios, the results for $|V_{ts}V_{tb}|$ are
the same. Using input from inclusive $b\to c\ell\nu_{\ell}$ decays [28], we
obtain:

$|V_{ts}V_{tb}|_{\rm incl/hybrid}=(41.4\pm 0.5)\times 10^{-3}\ .$ (31)

Taking the exclusive $|V_{cb}|$ from the HFLAV collaboration [29], we find

$|V_{ts}V_{tb}|_{\rm excl}=(38.4\pm 0.5)\times 10^{-3},$ (32)

which differs from (31) at the $4\sigma$ level due to the different $|V_{cb}|$
values. Finally, we find the following predictions for the SM values of the
branching ratios:

$\displaystyle\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})_{\rm SM}^{\rm
incl/hybrid}$ $\displaystyle=(3.71\pm 0.04|_{f_{B_{s}}}\pm 0.09|_{\rm CKM}\pm
0.02|_{\rm source})\cdot 10^{-9}$ $\displaystyle=(3.71\pm 0.10)\cdot 10^{-9}\
,$ (33) $\displaystyle\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})_{\rm
SM}^{\rm excl}$ $\displaystyle=(3.19\pm 0.04|_{f_{B_{s}}}\pm 0.08|_{\rm
CKM}\pm 0.02|_{\rm source})\cdot 10^{-9}$ $\displaystyle=(3.19\pm 0.09)\cdot
10^{-9}\ .$ (34)

It becomes clear that there is a significant variation depending on the CKM
parametrization. Thus, special attention is needed for the CKM uncertainties.
The results in (33) and (34) are in the ballpark of other theoretical
predictions [30], and we note that the spread between inclusive and exclusive
results is much wider than the quoted uncertainties.

The current experimental world average stands at [31]

$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})=(3.45\pm 0.29)\times 10^{-9}\
,$ (35)

relying mainly on recent measurements by LHCb [5], CMS [6], and ATLAS [7].
This experimental average is smaller than the inclusive SM prediction in (33)
but larger than the exclusive SM prediction in (34).

Moving now towards studies of NP, we introduce the ratio between the
experimental and SM branching ratios:

$\overline{R}=\frac{\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}{\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})_{\rm
SM}}\ .$ (36)

In the SM, by definition $\overline{R}=1$. Combining the current experimental
results in (35) with the inclusive/hybrid and exclusive values for the SM
predictions in (33) and (34), we obtain

$\displaystyle\overline{R}_{\rm incl/hybrid}$ $\displaystyle=0.93\pm 0.082\ ,$
(37) $\displaystyle\overline{R}_{\rm excl}$ $\displaystyle=1.08\pm 0.096\ ,$
(38)

where we see again that the experimental result can be either larger or
smaller than the SM prediction, depending on the choice of the CKM
parameterization.

Taking into account the difference between the time-integrated and theoretical
branching ratios, $\overline{R}$ can be written as follows [22]:

$\displaystyle\overline{R}$
$\displaystyle=\bigg{[}\frac{1+\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}y_{s}}{1+y_{s}}\bigg{]}(|P|^{2}+|S|^{2})$
$\displaystyle=\left[\frac{1+y_{s}\cos(2\varphi_{P}-\phi_{s}^{\rm
NP})}{1+y_{s}}\right]|P|^{2}+\left[\frac{1-y_{s}\cos(2\varphi_{S}-\phi_{s}^{\rm
NP})}{1+y_{s}}\right]|S|^{2}\ ,$ (39)

where $S$ and $P$ are given in (10) and (9), respectively. Using the value of
$\phi_{s}^{\rm NP}$ in (19) and assuming real (pseudo)-scalar couplings, i.e.
$\varphi_{P}=\varphi_{S}=0$, this ratio constrains a circle in the $|P|$–$|S|$
plane. In Fig. 1, we show the circles that result from using the
inclusive/hybrid and exclusive values of $\overline{R}$. We note that within
their respective uncertainties these circles overlap. The SM point, with
$P_{\rm SM}=1$ and $S_{\rm SM}=0$, is also indicated and lies between both
predictions. Allowing for physics beyond the SM, entering through
$\mathcal{C}^{-}_{10}$ and/or through new (pseudo)-scalar couplings in
$\mathcal{C}^{-}_{S,P}$, the branching ratio thus constrains a combination of
these possible new couplings. In the following, we discuss how to disentangle
these possible NP effects. For our analysis, we will use the incl/hybrid value
for the CKM elements.

Figure 1: Constraints in the $|P|$–$|S|$ plane from the current world average
experimental measurement of $\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})$ for the
inclusive/hybrid and exclusive CKM parameterizations.

##  3 Extracting $\mathcal{C}^{-}_{10}$ from Semileptonic Rare $B_{(s)}$
Decays

The leptonic $B_{s}\to\mu^{+}\mu^{-}$ decay is governed by combinations of the
scalar and pseudo-scalar Wilson coefficients. As seen in (9) and (10),
$\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$ enter in $P$ and $S$,
respectively. In addition, $P$ contains the coefficient
$\mathcal{C}^{-}_{10}$. The quantities $P$ and $S$ can be determined through
observables of the $B_{s}\to\mu^{+}\mu^{-}$ channel, as discussed in [13].
However, in order to constrain the corresponding Wilson coefficients, we have
to disentangle the $\mathcal{C}^{-}_{10}$ and $\mathcal{C}^{-}_{P}$
contributions which both enter $P$. Fortunately, we can do this by extracting
$\mathcal{C}^{-}_{10}$ from semileptonic decays of the kind $B\to
K^{(*)}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$. Let us now have a
closer look at how we can accomplish this task.

###  3.1 Theoretical Framework

Decay | Wilson Coefficients
---|---
$B_{s}\to\phi\mu^{+}\mu^{-}$ | $\mathcal{C}^{+}_{9},\mathcal{C}^{-}_{9},\mathcal{C}^{+}_{10},\mathcal{C}^{-}_{10},(\mathcal{C}^{-}_{P},\mathcal{C}^{-}_{S})$
$B\to K^{*}\mu^{+}\mu^{-}$ | $\mathcal{C}^{+}_{9},\mathcal{C}^{-}_{9},\mathcal{C}^{+}_{10},\mathcal{C}^{-}_{10},(\mathcal{C}^{-}_{P},\mathcal{C}^{-}_{S})$
$B\to K\mu^{+}\mu^{-}$ | $\mathcal{C}^{+}_{9},\mathcal{C}^{+}_{10},(\mathcal{C}^{+}_{P},\mathcal{C}^{+}_{S})$
$B_{s}\to\mu^{+}\mu^{-}$ | $\mathcal{C}^{-}_{10},\mathcal{C}^{-}_{P},\mathcal{C}^{-}_{S}$
Table 1: Rare $B$ decays and their dependence on the Wilson coefficients.
Parentheses indicate negligible dependencies.

The semileptonic $B_{(s)}\to K^{\ast}(\phi)\mu^{+}\mu^{-}$ and $B\to
K\mu^{+}\mu^{-}$ modes are described by the weak Hamiltonian

$\mathcal{H}_{\rm eff}=-\frac{G_{F}}{\sqrt{2}\pi}V_{ts}^{*}V_{tb}\alpha_{\rm
em}\sum\limits_{i\in I}C_{i}\mathcal{O}_{i}\ ,$ (40)

where we have neglected doubly Cabibbo-suppressed terms. The sum runs over the
index
$I=\\{1c,2c,3,4,5,6,8,7^{(\prime)},9^{(\prime)}\mu,10^{(\prime)}\mu,S^{(\prime)}\mu,P^{(\prime)}\mu,T^{(\prime)}\mu\\}$
and includes all operators that can contribute to the modes. In particular, in
contrast with $B_{s}\to\mu^{+}\mu^{-}$, now also the following operators
enter:

$\mathcal{O}_{9^{(\prime)}\mu}=[\bar{s}\gamma^{\mu}P_{L(R)}b](\bar{\mu}\gamma_{\mu}\mu)\
,\quad\mathcal{O}_{7^{(\prime)}}=\frac{1}{e}m_{b}[\bar{s}\sigma^{\mu\nu}P_{R(L)}b]F_{\mu\nu}\
,$ (41)

with $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. In the SM, we
have [23]

$C_{9}^{\rm SM}=4.114\ ,\quad\quad C_{7}^{\rm SM}=-0.296\ .$ (42)

Expressions for the differential rate of the $B_{(s)}\to
K^{*}(\phi)\mu^{+}\mu^{-}$ modes are given in e.g., [16], while those for
$B\to K\mu^{+}\mu^{-}$ can be found in e.g., [32]. In the following, we
neglect NP contributions to $C_{7}$ and the tensor coefficient $C_{T}$. The
branching ratios then depend on specific combinations of $C_{9,10,S,P}$, given
in Table 1, where we have defined

$\mathcal{C}_{9}^{\pm}\equiv C_{9}\pm C_{9}^{\prime}$ (43)

in analogy with (3).

In the $B_{(s)}\to K^{*}(\phi)$ modes, there are several additional
observables besides the branching ratio (see, e.g., [16] for an overview). It
is important to note that, unlike the $B_{s}\to\mu^{+}\mu^{-}$ channel, the
semileptonic modes are not helicity suppressed. As a result, they are much
less sensitive than the $B_{s}\to\mu^{+}\mu^{-}$ decay to $C_{P}$ and $C_{S}$.

Recent predictions for the branching ratio of $B\to K\mu^{+}\mu^{-}$ can be
found in [32]. For the $B_{s}\to\phi$ and $B\to K^{*}$ modes, we present the
SM predictions using the inputs above, the LQCD and Light-cone sum rule (LCSR)
form factors from [33] and perturbative long-distance effects from [16]. We
find the following SM predictions in the
$1.1<~{}q^{2}<~{}6.0\;$\mathrm{G}\mathrm{eV}^{2}$$ range:

$\displaystyle\overline{\mathcal{B}}(B_{s}\to\phi\mu^{+}\mu^{-})|_{\rm
incl/hybrid}{}$ $\displaystyle=(5.63\pm 0.58)\cdot 10^{-8}\ ,$ (44)
$\displaystyle\overline{\mathcal{B}}(B_{s}\to\phi\mu^{+}\mu^{-})|_{\rm
excl}{}$ $\displaystyle=(4.85\pm 0.49)\cdot 10^{-8}\ .$ (45)

Analogous to the discussion in Sec. 2.2, we converted the theoretical rate to
the experimental branching ratio corresponding to time-integrated untagged
data samples following [16]. Again we note the significant difference in the
SM predictions depending on the CKM matrix element determinations, which was
also noted, e.g., in [32]. For the $K^{*}$ modes, we find

$\displaystyle\mathcal{B}(B\to K^{*\pm}\mu^{+}\mu^{-})|_{\rm incl/hybrid}{}$
$\displaystyle=(5.35\pm 0.72)\cdot 10^{-8}\ ,$ $\displaystyle\mathcal{B}(B\to
K^{(*\pm)}\mu^{+}\mu^{-})|_{\rm excl}{}$ $\displaystyle=(4.61\pm 0.61)\cdot
10^{-8}\ ,$ (46)

and

$\displaystyle\overline{\mathcal{B}}(B\to K^{*0}\mu^{+}\mu^{-})|_{\rm
incl/hybrid}{}$ $\displaystyle=(5.00\pm 0.64)\cdot 10^{-8}\ ,$
$\displaystyle\overline{\mathcal{B}}(B\to K^{*0}\mu^{+}\mu^{-})|_{\rm excl}{}$
$\displaystyle=(4.31\pm 0.55)\cdot 10^{-8}\ .$ (47)

These results agree within uncertainties with [34, 33, 23]. The experimental
values measured by the LHCb Collaboration [35, 36] are given as follows:

$\overline{\mathcal{B}}(B_{s}\to\phi\mu^{+}\mu^{-})|_{\rm exp}=(2.88\pm
0.21)\cdot 10^{-8}\ ,$ (48)

and

$\overline{\mathcal{B}}(B^{\pm}\to K^{(*\pm)}\mu^{+}\mu^{-})|_{\rm
exp}=(3.66\pm 0.87)\cdot 10^{-8}\ .$ (49)

We find that the experimental branching ratios are smaller than the SM
predictions, a trend that is also reflected in the semileptonic $B\to
K\mu^{+}\mu^{-}$ channel (see e.g., [32]). These anomalies are well known and
have been the focus of many global fits, see e.g., [37, 38, 39, 40, 41, 34,
42, 43, 44], where usually the coefficient $C_{9}^{(\prime)}$ shows the
largest statistical pull away from its SM value. However, NP may well also
enter through $C_{10}^{(\prime)}$.

###  3.2 Extracting $\mathcal{C}^{-}_{10}$

The first step of our analysis is to determine $\mathcal{C}^{-}_{10}$ from the
rare semileptonic decays and then $\mathcal{C}^{-}_{P,S}$ from
$B_{s}\to\mu^{+}\mu^{-}$. However, $\mathcal{C}^{-}_{P}$ and
$\mathcal{C}^{-}_{S}$ also enter the semileptonic modes. We will show that the
impact of $\mathcal{C}^{-}_{P,S}$ on the semileptonic decays is actually
small, and consequently we can determine $\mathcal{C}^{-}_{10}$ through these
decays even though they in principle also depend on $\mathcal{C}^{-}_{P,S}$.

We simplify the analysis by only allowing for real NP coefficients and
assuming that no NP enters in $C_{9}^{(\prime)}$. We illustrate in Fig. 2 the
current constraint from the branching ratio of $B_{s}\to\mu^{+}\mu^{-}$ in the
$\mathcal{C}^{-}_{P}$–$\mathcal{C}^{-}_{S}$ plane. In addition, we show the
branching ratio of $B_{s}\to\phi\mu^{+}\mu^{-}$ as a function of
$\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$. The three ellipses represent
the SM value in (44) and allowing for a range the SM value $\pm$ 0.5 %,
specifically $[5.60,5.66]\times 10^{-8}$. For Fig. 2, we assume
$\mathcal{C}_{10}^{-\rm NP}=0$, but we find similar results when
$\mathcal{C}_{10}^{-\rm NP}\neq 0$. We observe that the current
$B_{s}\to\mu^{+}\mu^{-}$ measurement already constrains
$\absolutevalue{\mathcal{C}^{-}_{P,S}}\leq
0.06\;$\mathrm{G}\mathrm{eV}^{-1}$$. On the other hand, we find that a NP
contribution of that size affects the semileptonic branching ratio only at the
percent level. This demonstrates that the leptonic decay is indeed
significantly more sensitive than the semileptonic one to (pseudo)-scalar NP
and that $\mathcal{C}^{-}_{P,S}$ can only affect the branching ratio of
$B_{s}\to\phi\mu^{+}\mu^{-}$ at the level of less than one percent. We obtain
similar results for the $B^{\pm}\to K^{\pm\ast}\mu^{+}\mu^{-}$ and $B^{\pm}\to
K^{\pm}\mu^{+}\mu^{-}$ decays.

The small sensitivity of the semileptonic decays with respect to new
(pseudo)-scalar contributions can also be seen by looking at the observables
$R_{S}$ and $R_{W}$, defined in (66) and (68) of [45]. These are constructed
to be sensitive to scalar NP contributions, and $R_{W}$ vanishes for
$m_{\ell}=0$ and $\mathcal{C}^{-}_{S}=\mathcal{C}^{-}_{T}=0$. However, as was
also noted in [45], $R_{S}$ only changes minimally when allowing for new
scalar contributions. In addition, $R_{W}$ does get affected by new scalar
couplings, but its value remains of $\mathcal{O}(10^{-2})$. Therefore, only
very precise experimental information on these observables, which are
currently not measured, might help in constraining $\mathcal{C}^{-}_{P,S}$.

(a)

(b)

Figure 2: Correlation between $\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$
for $\mathcal{C}_{10}^{-\rm NP}=0$. The blue circular band is the $1\sigma$
constraint from $\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-}$) in (35), and
the ellipses indicate values of
$\overline{\mathcal{B}}(B_{s}\to\phi\mu^{+}\mu^{-})$ in the
$q^{2}\in[1.1,6.0]\;$\mathrm{G}\mathrm{eV}^{2}$$ bin.

We have demonstrated that semileptonic rare $B$ decays are much less sensitive
to (pseudo)-scalar NP than the leptonic $B_{s}\to\mu^{+}\mu^{-}$ channel. For
this reason, we may neglect $\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$
and extract $\mathcal{C}^{-}_{10}$ from the rare semileptonic $b\to s$ decays.
In Fig. 3 we illustrate the results of such an extraction as contours in the
$C_{10}^{\rm NP}$–$C_{10}^{\prime\rm NP}$ plane, assuming (a)
$C_{9}^{(\prime)\rm NP}=0$ and (b) $C_{9}^{(\prime)\rm
NP}=-C_{10}^{(\prime)\rm NP}$. Here we show constraints from the branching
ratios of $B_{s}\to\phi\mu^{+}\mu^{-}$ (blue), $B\to K^{*}\mu^{+}\mu^{-}$
(orange), and $B\to K\mu^{+}\mu^{-}$ (green). The contour following from $B\to
K\mu^{+}\mu^{-}$ is a pair of straight diagonal lines because this decay
depends only on $\mathcal{C}^{+}_{10}$. The contours from the vector modes, on
the other hand, are ellipses, since those decays depend on both
$\mathcal{C}^{+}_{10}$ and $\mathcal{C}^{-}_{10}$. By using data on the three
branching ratios together, we can determine $C_{10}$ and $C_{10}^{\prime}$ up
to a fourfold ambiguity. We mark the resulting four points by card suits and
give the corresponding values of $\mathcal{C}^{-}_{10}$ and
$\mathcal{C}^{+}_{10}$ in Tab. 2. We note that especially the interplay
between $B^{\pm}\to K^{\pm}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$
strongly constrains the allowed region for $C_{10}^{(^{\prime})}$, i.e. at the
$1\sigma$ level, the size of the symbols basically corresponds to the allowed
regions. In the case of CP-violating NP in $\mathcal{C}^{-}_{10}$, the rare
semileptonic $B$ decays can also be used to constraint its complex
coefficients using CP-asymmetries [32] and possible angular observables for
the vector modes.

(a)

(b)

Figure 3: Bounds on $C_{10}^{\rm NP}$ and $C_{10}^{\prime\rm NP}$ from
semileptonic $b\to s\mu^{+}\mu^{-}$ decays, assuming real Wilson coefficients.
The four points mark the overlapping region and are given in Table 2.

Symbol | $C_{10}^{\rm NP}$ | $C_{10}^{\prime\rm NP}$
---|---|---
$\heartsuit$ | $2.3$ | $-0.7$
$\spadesuit$ | $3.7$ | $-2.1$
$\diamondsuit$ | $5.0$ | $2.1$
$\clubsuit$ | $6.4$ | $0.7$

Symbol | $C_{10}^{\rm NP}$ | $C_{10}^{\prime\rm NP}$
---|---|---
$\heartsuit$ | $1.1$ | $-0.3$
$\spadesuit$ | $3.7$ | $-3.0$
$\diamondsuit$ | $4.6$ | $2.7$
$\clubsuit$ | $6.7$ | $0.4$

Table 2: Values of $C_{10}^{\rm NP}$ and $C_{10}^{\prime\rm NP}$ at the four
marked points in Fig. 3. Left: $C_{9}^{(\prime)\rm NP}=0$. Right:
$C_{9}^{(\prime)\rm NP}=-C_{10}^{(\prime)\rm NP}$.

We can compare these constraints with those obtained $B_{s}\to\mu^{+}\mu^{-}$.
Assuming no (pseudo)-scalar NP contributions, the experimental data on
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ imposes two diagonal lines in
the $C_{10}^{\rm NP}$–$C_{10}^{\prime\rm NP}$ plane as shown in Fig. 4, where
we have used as our experimental input (35). Due to the precision of the
experimental results, there is no overlap at the $1\sigma$-level with the
points determined from the semileptonic decays. This suggests that NP
contributions only to $C_{10}$ and $C_{10}^{\prime}$, or to
$C_{10}^{(\prime)}=-C_{9}^{(\prime)}$, cannot simultaneously accommodate the
current data on both semileptonic and leptonic decays. In the following, we
therefore discuss the possibility of at least one additional NP contribution.
Assuming that NP enters the semileptonic decays only through
$C_{10}^{(\prime)}$ or through $C_{10}^{(\prime)\rm NP}=-C_{9}^{(\prime)\rm
NP}$ additional NP contribution must then enter the leptonic decay
$B_{s}\to\mu^{+}\mu^{-}$ through $\mathcal{C}^{-}_{P}$ and/or
$\mathcal{C}^{-}_{S}$.

(a)

(b)

Figure 4: Constraints from the $B_{s}\to\mu^{+}\mu^{-}$ branching ratio on
$C^{(\prime)\rm NP}_{10}$, assuming real coefficients and no (pseudo)-scalar
NP. The inputs are specified in Sec. 2.2, and we use the hybrid/incl. CKM
parameters in (31). The four points are the same as in Fig. 3.

##  4 Target Regions for New CP Violation in the (Pseudo)-Scalar Sector

###  4.1 Utilizing the Branching Ratio and
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$

Having obtained $\mathcal{C}^{-}_{10}$ through the semileptonic rare $B_{(s)}$
decays, we can now use the branching ratio and
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ of the
$B_{s}~{}\to~{}\mu^{+}\mu^{-}$ mode to determine $\mathcal{C}^{-}_{P}$ and
$\mathcal{C}^{-}_{S}$, assuming both are real. As shown in Fig. 1, the
branching ratio fixes a circular relation between $S$ and $P$. By using
$\mathcal{C}^{-}_{10}$ as input, we obtain a relation between
$\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$, illustrated in Fig. 5 for the
four benchmark points in Table 2. Here we used the hybrid/incl. scenario for
the CKM factors in (31) and the input parameters in Sec. 2.2. Since
$B_{s}\to\mu^{+}\mu^{-}$ depends only on $\mathcal{C}^{-}_{10}$, points in the
$C_{10}$–$C_{10}^{\prime}$ plane that lie on the same ascending diagonal will
give the same circle. Thus, the four points only give two circles.

Figure 5: Allowed values of $\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$
following from the $B_{s}\to\mu^{+}\mu^{-}$ branching ratio using the
hybrid/incl. CKM values in (31) The two circles correspond to different values
of $\mathcal{C}_{10}^{-\rm NP}$, taken from Fig. 3(a), and represent the
different intersections between contours coming from semileptonic rare $B$
decays.

Utilizing this relationship between $\mathcal{C}^{-}_{P}$ and
$\mathcal{C}^{-}_{S}$, we can employ $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$
to determine their values. When $\mathcal{C}^{-}_{10}$ is real,
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ fixes straight lines in the
$\mathcal{C}^{-}_{P}$–$\mathcal{C}^{-}_{S}$ plane, as shown in [13] and
illustrated in Fig. 66(a). Here we only show one value for
$\mathcal{C}^{-}_{10}$, corresponding to the point marked by a heart in Fig.
3. We conclude that a measurement of $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$
can determine $\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$ up to a fourfold
ambiguity. At the moment, we cannot constrain these contributions any further,
although pioneering measurements of $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$
are already available through the $B_{s}\to\mu^{+}\mu^{-}$ effective lifetime
(23). It will be interesting to see how future measurements of this observable
will develop.

(a)

(b)

Figure 6: Values of $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ in the
$\mathcal{C}^{-}_{P}$–$\mathcal{C}^{-}_{S}$ plane for a real (a) and complex
(b) $\mathcal{C}^{-}_{10}$. The circular bands show the $1\sigma$ regions
consistent with the measurement of
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ using the hybrid/incl. CKM
values in (31).

If $\mathcal{C}^{-}_{10}$ is complex, the picture changes as we have
illustrated in Fig. 66(b). Here, we assume the following benchmark scenario
for $\mathcal{C}^{-}_{10}$:

$\displaystyle\mathcal{C}_{10}^{-\rm NP}$
$\displaystyle=0.30\absolutevalue{C_{10}^{\rm SM}}e^{i\>60^{\circ}}\ .$ (50)

The circle indicates again the contour from the branching ratio, where in this
case the point $\mathcal{C}^{-}_{S,P}=0$ is allowed because our benchmark
point for $\mathcal{C}^{-}_{10}$ accommodates the current experimental
branching ratio of $B_{s}\to\mu^{+}\mu^{-}$. Adding
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ gives curved contours in the
$\mathcal{C}^{-}_{P}$–$\mathcal{C}^{-}_{S}$ plane. We note that the value
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}=1$ is excluded due to the complex
value fixed for $\mathcal{C}^{-}_{10}$. The fact that now certain values of
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ are excluded is an interesting new
feature. Turning things around, with only real (pseudo)-scalar couplings, we
can thus obtain a bound on $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ from
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ using (15). In this case,
assuming $\varphi_{S}=0$ and using (9) we see that $\mathcal{C}^{-}_{10}$ will
generate a non-trivial phase $\varphi_{P}\neq 0$. For the example in Fig.
66(b), we find $|\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}|<0.87$ within the
branching ratio bound. We note that also in this case a measurement of
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ can still be used to determine the
Wilson coefficients $\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$.

###  4.2 Utilizing Mixing-Induced CP Violation

Additional information on possible new sources of CP violation comes from the
mixing-induced CP asymmetry $\mathcal{S}_{\mu\mu}$ in (26). No measurements of
$\mathcal{S}_{\mu\mu}$ are currently available; extracting it would require a
time-dependent, flavour-tagged analysis as can be seen in (27).

We first consider real $\mathcal{C}^{-}_{10},\;\mathcal{C}^{-}_{P}$ and
$\mathcal{C}^{-}_{S}$. In that case, we can only have a non-zero
$\mathcal{S}_{\mu\mu}$ due to $\phi_{s}^{\rm NP}$ entering in (10). We then
have

$\mathcal{S}_{\mu\mu}=-\sin\phi_{s}^{\rm
NP}\left(\frac{\absolutevalue{P}^{2}-\absolutevalue{S}^{2}}{\absolutevalue{P}^{2}+\absolutevalue{S}^{2}}\right)\
=[-0.062,0.062]\ ,$ (51)

where we take $\phi_{s}^{\rm NP}$ from (19). The range indicates the extreme
values reached for either $|S|=0$ or $|P|=0$. A measurement outside this range
would clearly indicate additional sources of CP violation which could enter as
complex phases in $\mathcal{C}^{-}_{10},\;\mathcal{C}^{-}_{P}$ and
$\mathcal{C}^{-}_{S}$. By the time such a measurement is available, we will
have a much sharper picture of $\phi_{s}^{\rm NP}$ from $B_{s}\to J/\psi\phi$
[26]. In addition, the semileptonic rare $B$ decays will allow a determination
of $\mathcal{C}^{-}_{10}$. If this coupling is then found to be real, a
measurement of $\mathcal{S}_{\mu\mu}$ would be crucial to find possible CP-
violating (pseudo)-scalar interactions. In this case, a measurement outside
the (updated) range in (51) would unambiguously indicate additional sources of
new (pseudo)-scalar interactions.

However, measurements in the rare semileptonic $B$ decays could also result in
a complex $\mathcal{C}^{-}_{10}$ (see e.g., [32]). This would be a more
exciting situation, as then a new source of CP violation would already have
been established. In this case, it would be important to explore whether also
new (pseudo)-scalar interactions are present. In order to address this
question, an analysis of $\mathcal{S}_{\mu\mu}$ and
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ in $B_{s}\to\mu^{+}\mu^{-}$ will be
crucial. As discussed in the previous subsection, in this case
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ constrains a circle in the
$\mathcal{C}^{-}_{P}$–$\mathcal{C}^{-}_{S}$ plane. Figure 77(a) shows this
circle for our complex benchmark value of $\mathcal{C}_{10}^{-\rm NP}$ in
(50). This constraint can be translated into the
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$ plane,
yielding the red and blue C-shaped region in Fig. 77(b). The region shows
which values of $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and
$\mathcal{S}_{\mu\mu}$ are consistent with data on
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ when $\mathcal{C}^{-}_{P}$
and $\mathcal{C}^{-}_{S}$ are real but $\mathcal{C}^{-}_{10}$ is complex. The
resulting strong correlation between $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$
and $\mathcal{S}_{\mu\mu}$ provides a narrow target region for the search for
new CP-violating sources that enter through the (pseudo)-scalar Wilson
coefficients.

Moreover, if $\mathcal{C}^{-}_{P}$ and/or $\mathcal{C}^{-}_{S}$ have complex
phases, $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and $\mathcal{S}_{\mu\mu}$
are no longer restricted to this region but can take any values within the
unit disk. If future measurements were to fall outside the target region, we
would have evidence for a complex phase in $\mathcal{C}^{-}_{P}$ and/or
$\mathcal{C}^{-}_{S}$. In order to illustrate this exciting feature, we have
marked four coloured points in Fig. 77(b) for several complex coefficients
$\mathcal{C}^{-}_{P}$ and/or $\mathcal{C}^{-}_{S}$.

We conclude that through the surprisingly strong correlation between
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and $\mathcal{S}_{\mu\mu}$, we may
reveal possible CP-violating effects in the (pseudo)-scalar sector.

(a)

(b)

Figure 7: Correlations for the complex $\mathcal{C}^{-}_{10}$ benchmark
scenario (50). (a) Constraints on real $\mathcal{C}^{-}_{P}$ and
$\mathcal{C}^{-}_{S}$ from $\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$,
and (b) the corresponding correlation between
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and $\mathcal{S}_{\mu\mu}$ indicated
with the same colour coding. The coloured points in (b) show the correlation
for complex $\mathcal{C}^{-}_{P}$ and/or $\mathcal{C}^{-}_{S}$ which spans the
whole unit disk as discussed in the text.

The method is robust with respect to the value of $\mathcal{C}^{-}_{10}$
extracted from the semileptonic $B$ decays. In Fig. 8, we show the target
regions that result from three different values of $\mathcal{C}_{10}^{-\rm
NP}$. This shows the power of this approach: after fixing
$\mathcal{C}^{-}_{10}$, the branching ratio
$\overline{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ constrains a very specific
region in the $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$
plane. Without fixing $\mathcal{C}^{-}_{10}$, the whole unit disk is allowed.

For completeness, we also consider the correlations with
$\mathcal{C}_{\mu\mu}$, defined in (25). This observable, which is very
challenging to measure, is only non-zero if scalar couplings are present. In
Fig. 9, we show the unit spheres in the
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$–$\mathcal{C}_{\mu\mu}$
space for three different values of $\mathcal{C}^{-}_{10}$ given by (28).
Again, the coloured contours are those allowed for real (pseudo)-scalar
couplings coefficients. The projection along the $\mathcal{C}_{\mu\mu}$ axes
corresponds to the 2D plots presented above.

(a)

(b)

(c)

Figure 8: Correlations between $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and
$\mathcal{S}_{\mu\mu}$ for different values of $\mathcal{C}^{-}_{10}$.

(a)

(b)

(c)

Figure 9: Illustration of the 3D correlations between
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$, $\mathcal{S}_{\mu\mu}$, and
$\mathcal{C}_{\mu\mu}$ for different values of $\mathcal{C}^{-}_{10}$.

In summary, the strong correlation between
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and $\mathcal{S}_{\mu\mu}$, in
combination with the branching ratio $\mathcal{B}(B_{s}\to\mu^{+}\mu^{-}$),
allows us to detect the presence of possible CP-violating phases in
$\mathcal{C}^{-}_{P}$ and/or $\mathcal{C}^{-}_{S}$. We illustrate the
procedure in the flowchart of Fig. 10. Starting from the rare semileptonic
decays, we can extract $\mathcal{C}^{-}_{10}$. This allows us to constrain a
region in the $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$
plane. Once future data is available, as a final step, we can compare our
target region with data to determine if there are new CP-violating
contributions in the $\mathcal{C}^{-}_{S}$ and/or $\mathcal{C}^{-}_{P}$
coefficients.

Figure 10: Illustration of our strategy to detect the presence of CP-violating
phases in $\mathcal{C}^{-}_{P}$ and/or $\mathcal{C}^{-}_{S}$.

##  5 Conclusions and Outlook

We have presented a new strategy to determine whether there are new CP-
violating (pseudo)-scalar interactions by fully exploiting the rare
$B_{s}\to\mu^{+}\mu^{-}$ decay. We find that especially future measurements of
the $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$ and $\mathcal{S}_{\mu\mu}$
observables have great potential in revealing New Physics.

In our method, we use $\mathcal{C}^{-}_{10}$ as an external input. We have
demonstrated that the rare semileptonic $B_{(s)}\to K\mu^{+}\mu^{-}$ and $B\to
K^{*}(\phi)\mu^{+}\mu^{-}$ decays are insensitive to new (pseudo)-scalar
coefficients, which allows the extraction of $\mathcal{C}^{-}_{10}$. We then
find surprisingly constrained target regions in the
$\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$–$\mathcal{S}_{\mu\mu}$ plane
assuming NP entering in $\mathcal{C}^{-}_{P}$ and $\mathcal{C}^{-}_{S}$ to be
CP-conserving. Moreover, any measurement outside these regions would indicate
new CP-violating phases in the (pseudo)-scalar coefficients. This method
provides an efficient way to identify possible new sources of CP violation in
the (pseudo)-scalar sector.

Given the potential of CP-violating observables in $B_{s}\to\mu^{+}\mu^{-}$ to
reveal NP, we strongly encourage the experimental community to perform the
corresponding measurements. Pioneering, untagged time-dependent analyses of
the effective lifetime by LHCb, ATLAS, and CMS are already available, leading
to first determinations of $\mathcal{A}_{\Delta\Gamma_{s}}^{\mu\mu}$. As a
next step, it would be crucial to perform also tagged time-dependent analyses,
allowing the extraction of the observable $\mathcal{S}_{\mu\mu}$.

By the time these challenging measurements are available, we will have a
clearer picture of possible NP effects. In particular, $\phi_{s}^{\rm NP}$
will have been constrained with much higher precision, which may even
establish a non-zero value of that phase. Moreover, measurements in the
semileptonic rare $B$ decays may already have brought us into the exciting
scenario where NP effects are revealed in the coefficient
$\mathcal{C}^{-}_{10}$. Either way, the observables in
$B_{s}\to\mu^{+}\mu^{-}$ will play a key role in probing new CP-violating
effects in (pseudo)-scalar interactions and determining the full dynamics of
this rare decay.

## Acknowledgements

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

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