Nikhef-2022-012

New Physics in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ Mixing: Present Challenges,
Prospects, and Implications for $B_{q}^{0}\to\mu^{+}\mu^{-}$

Kristof De Bruyn a,b, Robert Fleischer a,c, Eleftheria Malami a
and Philine van Vliet d

aNikhef, Science Park 105, 1098 XG Amsterdam, Netherlands

bVan Swinderen Institute for Particle Physics and Gravity, University of
Groningen, 9747 Groningen, Netherlands

cFaculty of Science, Vrije Universiteit Amsterdam,
1081 HV Amsterdam, Netherlands

dDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

The phenomenon of $B^{0}_{q}$–$\bar{B}^{0}_{q}$ mixing ($q=d,s$) provides a
sensitive probe for physics beyond the Standard Model. We explore the
corresponding space for New Physics left through the current data, having a
careful look at analyses of the Unitarity Triangle that are needed for the
Standard Model predictions of the $B_{q}$ mixing parameters. In particular, we
explore the impact of tensions between inclusive and exclusive determinations
of the CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$. Moreover, we focus on
the angle $\gamma$ of the Unitarity Triangle, comparing measurements from
$B\to DK$ and $B\to\pi\pi$, $\rho\pi$, $\rho\rho$ decays, where the latter are
typically interpreted in terms of the angle $\alpha$. We discuss various
scenarios and present the corresponding state-of-the-art constraints on the
New Physics parameters of $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. We point out
that these results have an interesting application in the analysis of rare
$B_{q}^{0}\to\mu^{+}\mu^{-}$ decays, allowing us to minimise the impact of CKM
parameters in the search for New Physics. In view of the high-precision era,
we make future projections. Interestingly, we find that for the extraction of
the New Physics parameters in the $B_{d}$ system the determination of the apex
of the Unitarity Triangle results in a key limiting factor. By contrast, the
corresponding impact is negligible for the $B_{s}$ system, making it a
promising candidate to reveal sources of New Physics.

August 2022

##  1 Introduction

Mixing between neutral $B_{q}^{0}$ and $\bar{B}_{q}^{0}$ mesons, where $q$ can
be either a down or a strange quark, offers interesting opportunities to test
the Standard Model (SM) paradigm at energy scales well beyond the centre-of-
mass energy of the Large Hadron Collider (LHC). The mixing process is an
example of a flavour-changing neutral current and proceeds via box diagrams,
which are loop-suppressed. This makes it an ideal place to search for the
effects of New Physics (NP) contributions. The parameter space for NP in
$B_{q}$-meson mixing can be determined in a model-independent way [1], and has
been analysed in the past [2, 3, 4, 5, 6, 7]. But, as we will demonstrate
below, the values of the SM predictions for the mixing phases and amplitudes
strongly depend on our choices for the input measurements, and in particular
on the determination of the apex of the Unitarity Triangle (UT). The searches
for NP in $B_{q}$-meson mixing thus critically rely on resolving various
tensions between measurements of the elements of the Cabibbo–Kobayashi–Maskawa
(CKM) quark mixing matrix [8, 9]. In this paper, we will demonstrate the
impact that the different experimental measurements related to the CKM matrix
and UT have on the parameter space for NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$
mixing. In addition, we will compare the results for NP in
$B_{d}^{0}$–$\bar{B}_{d}^{0}$ and $B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing with
each other, highlighting the similarities as well as differences between the
two systems.

The allowed parameter space for NP in $B_{q}$-meson mixing is impacted by two
important subtleties regarding the determination of the UT apex, which are
both currently at risk of being overlooked. The first subtlety is due to
existing discrepancies between the measurements of individual input
parameters, while the second subtlety is related to our choice of inputs to
determine the apex of the UT. Key examples regarding the first point are the
measurements of the CKM matrix elements $|V_{cb}|$ and $|V_{ub}|$. Despite the
best efforts from experts in the field, we have not yet been able to resolve
the discrepancies between the inclusive and exclusive determinations of these
quantities (see Ref. [10] for a review). The numerical differences between
both determinations directly affect many SM predictions in flavour physics,
including those related to $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. These
differences should therefore not be ignored, as would be the case when naively
averaging the results from both measurement strategies. Instead, at each step
of our analysis, we will demonstrate what impact the difference between the
inclusive and exclusive determinations of $|V_{cb}|$ and $|V_{ub}|$ has on the
results.

Secondly, to constrain the parameter space for NP in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, it is of utmost importance to understand
how the different input measurements can be affected by NP contributions
themselves. Global analyses that assume SM expressions, like those performed
by the CKMfitter [11] or UTfit [12] groups, combine information from all
relevant input measurements to determine the UT apex. However, each of the
individual experimental inputs can, in principle, be affected by different NP
contributions, clouding the resulting picture of the UT apex. While these
global fits are a valuable test of the SM, the obtained fit solutions cannot
reliably be used to predict SM parameters for further NP searches. Global fits
allowing for NP effects in the inputs have been made [3, 6]. However, in this
work we will only focus on NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing.

Not all experimental inputs are equally sensitive or equally affected by NP
contributions. Observables which can be measured through decays dominated by
tree-level topologies, such as the UT side $R_{b}$ and the UT angle $\gamma$,
are typically considered less prone to NP effects, while observables affected
by loop-level processes, such as the UT angle $\beta$ and the mass difference
$\Delta m_{s}$ between the mass eigenstates of the $B_{s}$-meson system, are
considered to be much more sensitive. For calculations of SM parameters in the
presence of unknown NP, the use of the latter category should therefore be
avoided. Since our main goal is to study NP in $B_{q}$-meson mixing, we
especially have to make sure to use input measurements free of possible
contributions from NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. For that
reason, we will perform and discuss our own fits of the UT apex, which aim to
be completely transparent with respect to which input measurements are used
and how they could be affected by NP.

Our SM determination of the UT apex uses only two observables: the UT side
$R_{b}$ and the UT angle $\gamma$, to keep possible contamination of the input
parameters due to NP to a minimum. Both these observables are affected by the
subtleties introduced above and thus require a careful analysis. The numerical
value for $R_{b}$ strongly depends on our choice for the CKM matrix elements
$|V_{cb}|$ and $|V_{ub}|$, propagating the unresolved discrepancies between
the inclusive and exclusive determinations to the solution for the UT apex,
and to the SM predictions of the $B_{q}$-meson mixing parameters. For
$\gamma$, on the other hand, there are multiple strategies from a variety of
$B$-meson decay channels to measure its value. The most precise direct
determination of $\gamma$ is obtained from a combination of multiple CP
violation measurements in the $B\to DK$ family of decays [13]. These channels
proceed via tree topologies only. In most beyond the SM theories, the NP
contributions are strongly suppressed in these topologies, and can thus still
be neglected. Nonetheless, it remains possible for NP effects to be present in
these decays [14, 15, 16, 17, 18]. Therefore it is important to carefully
compare the results from individual measurements, now and in the future, and
in particular when including them in experimental averages. In this respect,
the decay-time-independent measurements using $B^{+}$ and $B_{d}^{0}$ decays
show an interesting difference compared to the decay-time-dependent
measurement in the $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ channel [19, 20].

In addition, a second high-precision measurement of $\gamma$ can be obtained
from the isospin analysis of the $B\to\pi\pi$, $\rho\pi$, $\rho\rho$ decays.
This analysis is commonly presented as a measurement of the UT angle $\alpha$,
but we argue that, in view of NP searches, it is natural and more advantageous
to interpret it as a measurement of $\gamma$. In contrast to the $B\to DK$
decays, the $B\to\pi\pi$, $\rho\pi$, $\rho\rho$ modes also get contributions
from loop-level penguin topologies. The isospin relations between the three
decay modes allow us to control the size of these penguin topologies, and take
into account their impact on the determination of the UT angle $\gamma$. In
the literature, this relation is usually used to obtain a measurement of
$\alpha$, assuming $\alpha+\beta+\gamma=180^{\circ}$. However, in this paper,
we will not make this assumption and instead consider the isospin relations as
an independent determination of $\gamma$, using experimental input from the
$B_{d}^{0}$–$\bar{B}_{d}^{0}$ mixing phase $\phi_{d}$, which is given by
$2\beta$ in the SM. It should be stressed that the underlying physics
processes are vastly different from the $B\to DK$ decays, and a comparison of
the results is therefore highly non-trivial. If the two obtained values for
$\gamma$ agree with each other, this would place the determination of the UT
apex via $R_{b}$ and $\gamma$ on a more robust footing, while in case they
disagree, one would have a clear indicator for possible NP. This also
illustrates that we need to be very careful when making averages of different
measurements, and this should only be done after proper justification.

After a careful analysis of the UT apex focused on providing clean SM
predictions for the $B_{q}$-meson mixing parameters, we discuss the parameter
space that is still left for NP. Here, we will consider two different
scenarios. The first uses $R_{b}$ and $\gamma$ as input for the SM predictions
and determines the parameter space for NP separately for the $B_{d}$ and
$B_{s}$ systems. This is the most general scenario and serves as a baseline
for a comparison with the other two options. The only assumptions are that
$R_{b}$ and $\gamma$ are free of possible NP. The purpose of the other
scenario is to investigate what impact additional assumptions can have on the
allowed parameter space for NP in $B_{q}$ mixing. We will name the second
scenario that we consider Flavour Universal New Physics (FUNP), where the NP
in the $B_{d}$ and the $B_{s}$ system is assumed to be equal. In this case,
also the ratio between the mixing parameters $\Delta m_{d}$ and $\Delta m_{s}$
is free from NP, and can be used to determine the side $R_{t}$ of the UT.
Combining it with the side $R_{b}$, it is then possible to obtain the SM UT
apex without relying on $\gamma$. By comparing this scenario to the baseline,
we can see how compatible the data are with the FUNP assumption.

The unresolved discrepancies between the measurements of the CKM matrix
elements not only impact the determination of the UT apex and the NP searches
in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, but also affect other high-profile NP
searches. One particularly interesting example is the branching fraction
measurement of the rare decay $B_{s}^{0}\to\mu^{+}\mu^{-}$, which originates
from $b\to s\ell\ell$ flavour changing neutral current processes. We will
illustrate how our choices for $|V_{ub}|$ and $|V_{cb}|$ propagate to the
allowed parameter space for the pseudo-scalar and scalar NP contributions in
$B_{s}^{0}\to\mu^{+}\mu^{-}$. The dependence on these CKM matrix elements can
be minimised, as was pointed out in Refs. [21, 22, 23], by taking the ratio
between the $B_{s}^{0}\to\mu^{+}\mu^{-}$ branching fraction and the mixing
parameter $\Delta m_{s}$. Using this ratio to search for NP in the decay of
$B_{s}^{0}\to\mu^{+}\mu^{-}$ does require to take into account the NP
contributions in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. We therefore apply our
results from the general, model-independent NP fit to illustrate how this
ratio offers an interesting alternative constraint on the pseudo-scalar and
scalar NP contributions. All these considerations can also be applied to the
$B_{d}^{0}\to\mu^{+}\mu^{-}$ decay once accurate branching fraction
measurements become available in the future.

The outline of this paper is as follows: We start our analysis in Section 2
with a discussion on the input measurements of $\gamma$ and $R_{b}$, fit the
UT apex, and compare our solution with the constraint following from
$|\varepsilon_{K}|$, describing indirect CP violation in the neutral kaon
system, which is highly sensitive to $|V_{cb}|$. Next, in Section 3 we
introduce the SM $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing observables, which we
need as reference to constrain the parameter space for NP. In addition, we use
the ratio between the mixing parameters $\Delta m_{s}$ and $\Delta m_{d}$ to
obtain an alternative solution for the UT apex that can be used for SM
predictions under the assumption of FUNP. This alternative solution will be
used in our NP search. The two scenarios for NP in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing are discussed in Section 4, and the
applications for the rare decay $B_{q}^{0}\to\mu^{+}\mu^{-}$ in Section 5. In
Section 6, we explore and illustrate the impact of increased precision on the
key input measurements in the future. Finally, we conclude in Section 7.

##  2 Determination of the UT Apex

The NP searches in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing require accurate SM
predictions of the mixing parameters (see Section 3). Their SM values depend
on the apex $(\bar{\rho},\bar{\eta})$ of the UT, where

$\bar{\rho}\equiv\left(1-\frac{\lambda^{2}}{2}\right)\rho\>,\qquad\bar{\eta}\equiv\left(1-\frac{\lambda^{2}}{2}\right)\eta\>,$
(1)

and $\lambda$, $\rho$ and $\eta$ are three of the Wolfenstein parameters [24,
25]. The UT apex thus plays a crucial role in the analysis, but we cannot rely
on global fits of the UT [11, 12] to calculate the SM values, as they include
input measurements that are potentially affected by NP in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, and would thus bias our results.
Instead, we will only use information on the side $R_{b}$ and the angle
$\gamma$ to determine $\bar{\rho}$ and $\bar{\eta}$. Using the Particle Data
Group [26] parametrisation of the CKM matrix, the coordinates of the UT apex
are given by

$R_{b}\ e^{i\gamma}=\bar{\rho}+i\bar{\eta}\>,$ (2)

where $R_{b}$ and $\gamma$ will be defined in more detail below. The advantage
of this approach is that $R_{b}$ and $\gamma$ can be determined from decays
that proceed via tree topologies only. In general, NP contributions are
considered to be strongly suppressed in tree topologies, which are thus less
sensitive to their effects. Measurements of the ratios of semileptonic
branching fractions $R(D)$ and $R(D^{*})$ [27] might contest this assumption
[28, 29, 30, 31]. Interestingly, NP effects can be included in these decays
and the corresponding determination of the CKM matrix elements [32, 33, 34,
35]. If NP enters only through couplings to heavy $\tau$ leptons, the
determinations of $|V_{cb}|$ and $|V_{ub}|$ from semileptonic $B$ decays into
the light leptons $\ell=e,\mu$, which are the only determinations of these CKM
matrix elements used in this paper, would be as in the SM. Since the
determination of $\gamma$ uses non-leptonic decays, and the determination of
$R_{b}$ does not rely on decays that involve $\tau$ leptons, they can also be
considered free of NP for the analysis presented here. Nonetheless, the
measurements of $R_{b}$ and $\gamma$ require a careful discussion in view of
existing tensions. Let us therefore start with a detailed discussion of the
individual inputs, before proceeding towards a fit of the UT apex.

###  2.1 The UT Angle $\gamma$

The UT angle $\gamma$ is defined as

$\gamma\equiv\arg\left(-\frac{V_{ud}^{\phantom{*}}V_{ub}^{*}}{V_{cd}^{\phantom{*}}V_{cb}^{*}}\right)\>,$
(3)

and can be measured in $B$ decays which are sensitive to interference between
CKM-favoured $b\to c$ and CKM-suppressed $b\to u$ quark-level transitions. The
angle $\gamma$ is determined with high precision from CP violation
measurements in $B\to DK$ decays. The label $B\to DK$ groups together a long
list of decay channels and final states, including $B^{+}\to DK^{+}$,
$B^{+}\to D^{*}K^{+}$, $B^{+}\to DK^{*+}$, $B_{d}^{0}\to DK^{*0}$,
$B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ and others. Here, $D$ represents an
admixture of $D^{0}$ and $\bar{D}^{0}$ mesons, and similarly for $D^{*}$. An
overview of the various experimental measurements can be found in Refs. [27,
13]. In view of the goal of this paper, to search for NP in $B_{q}$-meson
mixing, an important distinction needs to be made between the $B^{+}$ and
$B_{d}^{0}$ decays on the one hand, and the $B_{s}^{0}$ decays on the other.
The sensitivity to $\gamma$ in the $B^{+}$ and $B_{d}^{0}$ decays comes from
direct CP violation, which can be measured from decay-time-independent
analyses, while the sensitivity to $\gamma$ in $B_{s}^{0}\to
D_{s}^{\mp}K^{\pm}$ comes from mixing-induced CP violation and requires a
decay-time-dependent analysis, taking into account the effects of
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing. This illustrates why we need to be
careful when making averages of the different $\gamma$ measurements.

#### Decay-time-independent $B\to DK$

For the decay-time-independent analyses, the latest average from the LHCb
collaboration is given by [13]

$\gamma_{B\to DK}=(64.9_{-4.5}^{+3.9})^{\circ}\>.$ (4)

Since the individual $\gamma$ measurements could, in principle, be affected by
NP effects, it remains important to compare the results from the individual
measurements and different decay channels before averaging. NP effects may be
washed out in the average, which then also becomes very challenging to
interpret theoretically. Here, we will assume that the result (4) is free from
NP.

#### Decay-time-dependent $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$

The interpretation of the results from the $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$
system requires more attention. Through interference effects with the
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing process, the CP asymmetry parameters in
$B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ allow a determination of the sum
$\phi_{s}+\gamma$ that is theoretically clean. The latest result for the
$B_{s}$-meson mixing phase $\phi_{s}$ measured in the $B_{s}^{0}\to
J/\psi\phi$ channel and corrected for contributions from penguin topologies
[36, 37], is

$\phi_{s}=-0.074_{-0.024}^{+0.025}=\left(-4.2\pm 1.4\right)^{\circ}\>.$ (5)

With this experimental input, the result for $\phi_{s}+\gamma$ can be
converted into an independent measurement of $\gamma$, regardless of possible
NP in the $B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing phase $\phi_{s}$. Any NP
contributions to $\phi_{s}$ impact the CP asymmetry measurements in both
$B_{s}^{0}\to J/\psi\phi$ and $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ equally, and
thus cancel in the determination of $\gamma$. Combining the measured CP
asymmetries in $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ [38] with the above value for
$\phi_{s}$, gives [19, 20]

$\gamma_{B_{s}\rightarrow D_{s}K}=\left(131^{+17}_{-22}\right)^{\circ}\>,$ (6)

which shows a difference of 3 standard deviations in comparison with the
decay-time-independent result of $\gamma$ in Eq. (4). A dedicated discussion
of this result and the puzzling tension with Eq. (4) can be found in Refs.
[19, 20]. If this value is to be explained by NP, it requires additional
contributions to the $B_{s}^{0}\to D_{s}^{\mp}K^{\pm}$ decay amplitudes.
Remarkably, Refs. [19, 20] have also found puzzling patterns at the branching
ratio level, as one would expect with NP effects entering the decay amplitude.
What makes the intriguing situation even more exciting, is that consistent
patterns are found when complementing the analysis with decays with similar
dynamics. In Refs. [19, 20], a model-independent strategy has been presented,
showing that the data can be accommodated with NP contributions at the level
of $30\%$ of the SM amplitudes. We note that model-dependent studies have also
been discussed in Refs. [16, 17, 18].

#### Isospin $B\to\pi\pi$, $\rho\pi$, $\rho\rho$

The UT angle $\gamma$ can also be determined from an isospin analysis of the
decays $B\to\pi\pi$, $\rho\pi$, $\rho\rho$, where $B$ is either $B^{+}$ or
$B_{d}^{0}$, and we combine all possible charge combinations for the final
states. This method was originally proposed in Ref. [39] for the $B\to\pi\pi$
system and has been extensively discussed in Ref. [40]. For the remainder of
this paper, we will use a shorthand notation and refer to this method as
$\gamma$ from isospin relations, or $\gamma_{\text{iso}}$. The isospin
relations between the $B\to\pi\pi$, $\rho\pi$, $\rho\rho$ decays are exploited
to simultaneously control the contributions from penguin topologies affecting
these decays, and to determine the weak phase difference

$\phi_{d}+2\gamma=\phi_{d}^{\text{NP}}-2\alpha\>,$ (7)

where

$\phi_{d}=\phi_{d}^{\text{SM}}+\phi_{d}^{\text{NP}}=2\beta+\phi_{d}^{\text{NP}}\>.$
(8)

In the absence of NP contributions, i.e. $\phi_{d}^{\text{NP}}=0$, relation
(7) has been used in the literature to interpret the dependence on
$\phi_{d}+2\gamma$ as an independent measurement of the UT angle $\alpha$,
with the latest result given by [27]

$\alpha=(85.2_{-4.3}^{+4.8})^{\circ}\>.$ (9)

However, $\phi_{d}^{\text{NP}}$ need not be zero, as we explore in this paper.
The isospin analysis therefore does not provide a direct measurement of the
CKM angle $\alpha$, but instead, should be interpreted as an independent
measurement of the CKM angle $\gamma$, utilising external input on the
$B_{d}^{0}$–$\bar{B}_{d}^{0}$ mixing phase $\phi_{d}$. The latest result for
$\phi_{d}$, measured in the $B_{d}^{0}\to J/\psi K^{0}$ channel and corrected
for contributions from penguin topologies, is given as follows [36, 37]:

$\phi_{d}=\left(44.4_{-1.5}^{+1.6}\right)^{\circ}\>.$ (10)

It is important to stress that NP effects in $B_{d}^{0}$–$\bar{B}_{d}^{0}$
mixing lead to the same $\phi_{d}^{\text{NP}}$ contribution in the measurement
of $\phi_{d}$ from $B_{d}^{0}\to J/\psi K^{0}$ and in the determination of the
weak phase $\phi_{d}+2\gamma$ obtained through the isospin analysis. This NP
phase thus cancels when using the experimental input (10), allowing a
measurement of $\gamma$ from $B\to\pi\pi$, $\rho\pi$, $\rho\rho$. Combining
the result for $\phi_{d}+2\gamma$ in Eq. (9) with the measurement of
$\phi_{d}$ in Eq. (10), we find

$\gamma_{\text{iso}}=(72.6_{-4.9}^{+4.3})^{\circ}\>.$ (11)

In contrast to the measurement of $\gamma$ from $B\to DK$ decays, this value
can be affected by possible NP entering the decay amplitudes through penguin
topologies.

#### Average

Comparing the values of $\gamma$ in Eqs. (4) and (11), we find a difference of
$(7.5\pm 6.7)^{\circ}$. The two $\gamma$ determinations are consistent within
1.1 standard deviations. Given their different origins, this is a non-trivial
result. Since both approaches also have similar precision, there is no
preference for one or the other. Because of these two reasons, we have chosen
to average the results for our analysis in this paper, and will use

$\gamma_{\text{avg}}=(68.4\pm 3.3)^{\circ}$ (12)

for the fit of the UT apex below. However, it is important to keep in mind
that the situation may well change in the future as the measurements of
$\gamma$ become more precise. With improved precision, differences between the
approaches could become more pronounced, thereby hinting at NP effects either
at the tree-level in $B\to DK$ decays or at the loop-level in the isospin
analysis. If such a situation arises, averaging both results would no longer
be justified.

###  2.2 The UT Side $R_{b}$

The second input for the SM fit of the UT apex is the side $R_{b}$. It is
defined as

$R_{b}\equiv\left(1-\frac{\lambda^{2}}{2}\right)\frac{1}{\lambda}\left|\frac{V_{ub}}{V_{cb}}\right|=\sqrt{\bar{\rho}\,^{2}+\bar{\eta}\,^{2}}\>.$
(13)

All three CKM matrix elements $\lambda\equiv|V_{us}|,|V_{ub}|$ and $|V_{cb}|$
can be measured in semileptonic decays involving electrons or muons, which are
generally considered to be robust against NP contributions. Nonetheless, for
each of these elements, there are still unresolved discrepancies between the
various theoretical and experimental approaches. These discrepancies have a
sizable impact on the coordinates of the UT apex, and thus also on the SM
predictions and NP searches in $B_{q}$-meson mixing. Therefore, even though
the tensions have been extensively discussed in the literature (see Ref. [26]
for an overview), it is important to summarise them here again.

#### $|V_{us}|$

The element $|V_{us}|$ is most precisely measured in semileptonic kaon decays.
The experimental average from measurements in the $K\to\pi\ell\nu_{\ell}$
decay channels, collectively referred to as $K\ell 3$, is [41, 42]

$|V_{us}|=0.22309\pm 0.00056\qquad(\text{from }K\ell 3)\>,$ (14)

while the experimental average from $K\to\mu\nu_{\mu}(\gamma)$ decays,
referred to as $K\ell 2$, is [26]

$|V_{us}|=0.2252\pm 0.0005\qquad(\text{from }K\ell 2)\>.$ (15)

These two results differ by three standard deviations. The $K\ell 2$ result is
derived from the ratio of decay rates

$\frac{\Gamma(K\to\mu\nu_{\mu}(\gamma))}{\Gamma(\pi\to\mu\nu_{\mu}(\gamma))}\propto\left(\frac{|V_{us}|f_{K^{+}}}{|V_{ud}|f_{\pi^{+}}}\right)^{2}$
(16)

between the $K\to\mu\nu_{\mu}(\gamma)$ and $\pi\to\mu\nu_{\mu}(\gamma)$ decay
channels. Here, $f_{K^{+}}$ and $f_{\pi^{+}}$ are the kaon and pion decay
constants, respectively, whose ratio is well known from lattice calculations
[43]. Furthermore, the measurement of $|V_{us}|$ from Eq. (16) also depends on
the CKM matrix element $|V_{ud}|$, whose most precise measurement [26]

$|V_{ud}|=0.97373\pm 0.00031$ (17)

comes from superallowed nuclear beta decay.

The precise determination of $|V_{ud}|$ offers a third possibility to
determine $|V_{us}|$. Using the Wolfenstein parametrisation [24, 25] of the
CKM matrix elements

$|V_{ud}|=1-\frac{1}{2}\lambda^{2}-\frac{1}{8}\lambda^{4}+\mathcal{O}(\lambda^{6})\>,\qquad|V_{us}|=\lambda+\mathcal{O}(\lambda^{7})\>,$
(18)

it is possible to express $|V_{us}|$ in terms of $|V_{ud}|$, resulting in

$|V_{us}|=0.2277\pm 0.0013\qquad(\text{from }|V_{ud}|)\>.$ (19)

Even though the uncertainty on this derived result is much larger than the
direct $K\ell 3$ measurement, we again find a difference of 3.3 standard
deviations. A comparison with the $K\ell 2$ measurement is less
straightforward as they are not independent results. Alternatively, the
measurement of $|V_{us}|$ from ${K\ell 3}$ predicts a value

$|V_{ud}|=0.97481\pm 0.00013\qquad(\text{from }K\ell 3)\>,$ (20)

which differs from the result (17) by 3.2 standard deviations. A similar
tension is found when testing the CKM matrix unitarity relation

$|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}=1\>.$ (21)

Combining the measurements in Eqs. (17) and (14) with the experimental results
for $|V_{ub}|$ introduced below, a difference of 3.2 standard deviations with
respect to 1 is found. For this test, the impact of $|V_{ub}|$, which is only
proportional to $\lambda^{3}$, is negligible. Both results suggest that the
tension between the $K\ell 3$ and $K\ell 2$ measurements of $|V_{us}|$ may
find its origin in a tension between the $|V_{us}|_{K\ell 3}$ and $|V_{ud}|$
measurements.

Lastly, $|V_{us}|$ has also been determined from $\tau$ lepton decays. The
latest average is [26]

$|V_{us}|=0.2221\pm 0.0013\qquad(\tau\text{ lepton decays)}\>.$ (22)

This result is compatible with the measurement from $K\ell 3$ decays, albeit
with a much larger uncertainty, and has a similar precision as the result (19)
based on the $|V_{ud}|$ measurement.

Although the tensions between the different determinations of $|V_{us}|$ are
intriguing, they turn out to have a negligible impact in the search for NP in
$B_{q}$-meson mixing. This is illustrated by Fig. 2, which will be discussed
in more detail below. Therefore, we will only present the results for the
$K\ell 3$-type decays in the paper, avoiding the dependence on the measurement
of $|V_{ud}|$.

####  $|V_{ub}|$ and $|V_{cb}|$

The CKM matrix elements $|V_{ub}|$ and $|V_{cb}|$ are measured in semileptonic
$B$ decays. The latest experimental averages for the inclusive determinations
are [27, 44]

$|V_{ub}|_{\text{incl}}=(4.19\pm 0.17)\times
10^{-3}\>,\qquad|V_{cb}|_{\text{incl}}=(42.16\pm 0.50)\times 10^{-3}\>,$ (23)

where $|V_{ub}|$ is calculated using the Gambino–Giordano–Ossola–Uraltsev
(GGOU) [45] approach, while the determination of $|V_{cb}|$ uses the kinematic
scheme and includes the latest three-loop calculations of the total
semileptonic width [44]. These results, however, differ by respectively 3.9
and 4.3 standard deviations from the exclusive determinations from the Heavy
Flavour Averaging Group (HFLAV) [27]

$|V_{ub}|_{\text{excl}}=(3.51\pm 0.12)\times
10^{-3}\>,\qquad|V_{cb}|_{\text{excl}}=(39.10\pm 0.50)\times 10^{-3}\>,$ (24)

which include constraints on the ratio $|V_{ub}|/|V_{cb}|$ from LHCb
measurements of the decays $\Lambda_{b}^{0}\to p\mu^{-}\bar{\nu}_{\mu}$ [46]
and $B_{s}^{0}\to K^{-}\mu^{+}\nu_{\mu}$ [47]. Contrary to the situation for
$|V_{us}|$, these differences do have a significant impact on the analysis
presented in this paper, and lead to different pictures for the allowed
parameter spaces for NP in $B_{q}$-meson mixing. Therefore, we will explicitly
show all results for both the inclusive and the exclusive determinations. This
illustrates the importance of further investigating the origin of these
tensions and of eventually resolving them.

#### $R_{b}$

Combining the measurements of $|V_{us}|$, $|V_{ub}|$ and $|V_{cb}|$, we can
calculate the UT side $R_{b}$. Due to the unresolved discrepancies between the
different determinations of these CKM matrix elements, the numerical value for
$R_{b}$ will strongly depend on our choices for the individual inputs. The two
most common combinations found in the literature pair the inclusive (or
exclusive) determinations of $|V_{ub}|$ and $|V_{cb}|$ with each other,
resulting in a fully inclusive (or exclusive ) value for $R_{b}$:

$R_{b,\text{incl},K\ell 3}=0.434\pm 0.018\>,\qquad R_{b,\text{excl},K\ell
3}=0.392\pm 0.014\>.$ (25)

These two values differ from each other by 2.4 standard deviations.

Recent progress in the inclusive determination of $|V_{cb}|$ and associated
discussions in the literature [48, 10, 44, 49], aiming to understand and
resolve the tension between the inclusive and exclusive values of $|V_{ub}|$
and $|V_{cb}|$, have sparked interest in a third possibility: The hybrid
combination of the exclusive value for $|V_{ub}|$ with the inclusive value for
$|V_{cb}|$. This results in a value for $R_{b}$ of

$R_{b,\text{hybrid},K\ell 3}=0.364\pm 0.013\>,$ (26)

which differs from the inclusive (exclusive) $R_{b}$ solution by 3.7 (1.5)
standard deviations. In this paper we will consider all three options to
illustrate the impact that the numerical values of $|V_{ub}|$ and $|V_{cb}|$
have on our searches for beyond the SM physics.

###  2.3 The UT Apex

The coordinates $(\bar{\rho},\bar{\eta})$ of the UT apex are determined from a
fit to $R_{b}$ and $\gamma$, based on the numerical inputs summarised in Table
1. The fitting procedure is implemented using the GammaCombo framework [50]
and employs the frequentist method to determine the confidence interval of the
fitted parameters. It gives the following results

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\bar{\rho}$
$\displaystyle=0.160\pm 0.025\>,$ $\displaystyle\bar{\eta}$
$\displaystyle=0.404\pm 0.022\>,$ (27) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle\bar{\rho}$ $\displaystyle=0.144\pm 0.022\>,$
$\displaystyle\bar{\eta}$ $\displaystyle=0.365\pm 0.018\>,$ (28)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\bar{\rho}$
$\displaystyle=0.134\pm 0.021\>,$ $\displaystyle\bar{\eta}$
$\displaystyle=0.338\pm 0.017\>,$ (29)

with the associated two-dimensional confidence level contours shown in Fig. 1.
The three solutions for the UT apex are compared with one other in Fig. 2.

The impact of the different measurements of $|V_{us}|$ and $\gamma$ on the UT
apex is illustrated in Fig. 2, taking the exclusive scenario as an example. At
present, all results agree within uncertainties, but with improved precision,
this may no longer be the case in the future. We will explore this further in
Section 6.

| Inclusive | Exclusive | Hybrid | Reference
---|---|---|---|---
$\alpha$ | $(85.2_{-4.3}^{+4.8})^{\circ}$ | [27]
$\phi_{d}$ | $\left(44.4_{-1.5}^{+1.6}\right)^{\circ}$ | [36, 37]
$\gamma_{B\to DK}$ | $(64.9_{-4.5}^{+3.9})^{\circ}$ | [13]
$\gamma_{\text{iso}}$ | $(72.6_{-4.9}^{+4.3})^{\circ}$ | -
$\gamma_{\text{avg}}$ | $(68.4\pm 3.3)^{\circ}$ | -
$|V_{us}|$ | $0.22309\pm 0.00056$ | [26]
$|V_{ub}|\times 10^{3}$ | $4.19\pm 0.17$ | $3.51\pm 0.12$ | $3.51\pm 0.12$ | [27]
$|V_{cb}|\times 10^{3}$ | $42.16\pm 0.50$ | $39.10\pm 0.50$ | $42.16\pm 0.50$ | [44, 27]
$R_{b}$ | $0.434\pm 0.018$ | $0.392\pm 0.014$ | $0.364\pm 0.013$ | -
$\bar{\rho}$ | $0.160\pm 0.025$ | $0.144\pm 0.022$ | $0.134\pm 0.021$ | -
$\bar{\eta}$ | $0.404\pm 0.022$ | $0.365\pm 0.018$ | $0.338\pm 0.017$ | -
Table 1: Input parameters and results for the determination of the UT apex,
split between the inclusive, exclusive and hybrid determinations of $R_{b}$.
The dashes in the reference column indicate derived quantities that were
computed in this paper.

Figure 1: Determination of the coordinates $(\bar{\rho},\bar{\eta})$ of the UT
apex from measurements of the angle $\gamma$ and the side $R_{b}$, with the
two-dimensional confidence level contours from the GammaCombo fit overlaid.
For comparison, the constraint from $|\varepsilon_{K}|$, discussed in Section
2.4, is shown (but not included in the fit). The solutions based on the
inclusive (Left), exclusive (Right) and hybrid (Bottom) determination of
$R_{b}$ are shown separately.

Figure 2: Two-dimensional confidence level contours for the coordinates
$(\bar{\rho},\bar{\eta})$ of the UT apex. Left: Comparison between the
inclusive, exclusive and hybrid scenarios. Right: Comparison between the
different values for the angle $\gamma$ in the exclusive scenario. Bottom:
Comparison between different values for $|V_{us}|$ in the exclusive scenario.

###  2.4 Comparison with $|\varepsilon_{K}|$

The apex of the UT is not only constrained by information from $B$-meson
physics, but also by other flavour processes. CP violation in the neutral kaon
system constrains the UT apex through the observable $|\varepsilon_{K}|$,
which depends on both the square and fourth power of $|V_{cb}|$ and is thus
highly sensitive to its numerical value. With sufficient precision,
$|\varepsilon_{K}|$ could therefore be an excellent observable to explore the
tension between the inclusive and exclusive determinations of $|V_{cb}|$, as
it might favour one of the two. If the tension between both determinations is
resolved in the future, a discrepancy between the constraint coming from
$|\varepsilon_{K}|$ and the fit results for the UT apex could point to NP in
neutral kaon mixing.

In the SM, the dependence of $|\varepsilon_{K}|$ on the UT apex coordinates
$\bar{\rho}$ and $\bar{\eta}$ is given by [51]:

$|\varepsilon_{K}|=\frac{G_{\text{F}}^{2}m_{W}^{2}m_{K}f_{K}^{2}}{6\sqrt{2}\pi^{2}\Delta
m_{K}}\kappa_{\varepsilon}\hat{B}_{K}|V_{cb}|^{2}\lambda^{2}\bar{\eta}\left[|V_{cb}|^{2}(1-\bar{\rho})\,\eta_{tt}^{\text{EW}}\eta_{tt}\,\mathcal{S}(x_{t})-\eta_{ut}\,\mathcal{S}(x_{c},x_{t})\right]\>,$
(30)

where $G_{\text{F}}$ is the Fermi constant, $m_{W}$ and $m_{K}$ are the $W$
and kaon mass, respectively, $f_{K}$ is the kaon decay constant, $\Delta
m_{K}$ is the mass difference between the $K^{0}_{\text{S}}$ and
$K^{0}_{\text{L}}$ mass eigenstates, and $\hat{B}_{K}$ is the kaon bag
parameter. The parameter $\kappa_{\varepsilon}$ is a multiplicative correction
factor [52] due to long-distance contributions that are not included in
$\hat{B}_{K}$. The functions

$\mathcal{S}(x_{t})=S_{0}(x_{t})+S_{0}(x_{c})-2S_{0}(x_{c},x_{t})\>,\qquad\mathcal{S}(x_{c},x_{t})=S_{0}(x_{c})-S_{0}(x_{c},x_{t})$
(31)

are combinations of the following two Inami–Lim functions [53]:

$\displaystyle S_{0}(x_{i})=$ $\displaystyle
x_{i}\left[\frac{1}{4}+\frac{9}{4(1-x_{i})}-\frac{3}{2(1-x_{i})^{2}}-\frac{3x_{i}^{2}\ln{x_{i}}}{2(1-x_{i})^{3}}\right]\>,$
(32) $\displaystyle S_{0}(x_{i},x_{j})=$
$\displaystyle\frac{x_{i}x_{j}}{x_{i}-x_{j}}\left[\frac{1}{4}+\frac{3}{2(1-x_{i})}-\frac{3}{4(1-x_{i})^{2}}\right]\ln{x_{i}}$
$\displaystyle+\frac{x_{j}x_{i}}{x_{j}-x_{i}}\left[\frac{1}{4}+\frac{3}{2(1-x_{j})}-\frac{3}{4(1-x_{j})^{2}}\right]\ln{x_{j}}-\frac{3x_{i}x_{j}}{4(1-x_{i})(1-x_{j})}\>,$
(33)

where

$x_{i}\equiv\left[\frac{\bar{m}_{i}(\bar{m}_{i})}{m_{W}}\right]^{2}\qquad\text{with
}i=c,t\>,$ (34)

and $\bar{m}_{i}(\bar{m}_{i})$ is the mass of quark $i$ in the
$\overline{\text{MS}}$ scheme. For the top quark, we use the RunDec tool [54,
55] to convert the latest experimental average of the pole mass [26] into

$\bar{m}_{t}(\bar{m}_{t})=(162.19\pm 0.30)\>\text{GeV}\>,$ (35)

ignoring light-quark mass effects and using additional experimental input for
the $Z$ boson mass $m_{Z}$ and the QCD coupling constant $\alpha_{s}(m_{Z})$.
Finally, $\eta_{tt}^{\text{EW}}$, $\eta_{tt}$ and $\eta_{ut}$ are correction
factors to the modified Inami–Lim functions $\mathcal{S}(x_{t})$ and
$\mathcal{S}(x_{c},x_{t})$ due to next-to-leading order (NLO) electroweak and
QCD contributions [51, 56]. The factors $\eta_{tt}$ and $\eta_{ut}$
parametrise the QCD corrections and are known to next-to-leading-logarithmic
(NLL) and next-next-to-leading-logarithmic (NNLL) precision, respectively
[51]. The factor $\eta_{tt}^{\text{EW}}$ estimates the impact of NLO
electroweak corrections based on the first calculations of two-loop
electroweak contributions to $\varepsilon_{K}$ [56]. All numerical values for
the observables in Eq. (30) are listed in Table 2.

| Value | Unit | Reference
---|---|---|---
$G_{\text{F}}$ | $1.1663787\times 10^{-11}$ | MeV-2 | [26]
$m_{W}$ | $80\,377\pm 12$ | MeV |  [26]†
$m_{K}$ | $497.611\pm 0.013$ | MeV | [26]
$\bar{m}_{c}(\bar{m}_{c})$ | $1.278\pm 0.013$ | GeV | [43]
$m_{t}$ | $172.69\pm 0.30$ | GeV | [26]
$m_{Z}$ | $91.1876\pm 0.0021$ | GeV | [26]
$\alpha_{s}(m_{Z})$ | $0.1179\pm 0.0009$ |  | [26]
$\bar{m}_{t}(\bar{m}_{t})$ | $162.19\pm 0.30$ | GeV | -
$\Delta m_{K}$ | $0.005289\pm 0.000010$ | ps-1 | [26]
$f_{K}$ | $155.7\pm 0.3$ | MeV | [43]
$\hat{B}_{K}$ | $0.7625\pm 0.0097$ |  | [43]
$\kappa_{\varepsilon}$ | $0.94\pm 0.02$ |  | [52]
$\eta_{tt}^{\text{EW}}$ | $0.990\pm 0.004$ |  | [56]
$\eta_{tt}$ | $0.550\pm 0.023$ |  | [51]
$\eta_{ut}$ | $0.402\pm 0.005$ |  |  [51]‡
| Inclusive & Hybrid | Exclusive | Unit | Reference
$A$ | $(6.94\pm 0.40)\times 10^{-3}$ | $(5.36\pm 0.32)\times 10^{-3}$ |  | -
$B$ | $(5.11\pm 0.35)\times 10^{-3}$ | $(3.78\pm 0.27)\times 10^{-3}$ |  | -
$|\varepsilon_{K}|_{\text{SM}}$ | $(2.54\pm 0.22)\times 10^{-3}$ | $(1.74\pm 0.15)\times 10^{-3}$ |  | -
$|\varepsilon_{K}|_{\text{exp}}$ | $(2.228\pm 0.011)\times 10^{-3}$ |  | [26]
Table 2: Input parameters and results for the SM calculation of
$|\varepsilon_{K}|$, split between the inclusive and exclusive determinations
of $|V_{cb}|$. The dashes in the reference column indicate derived quantities
computed in this paper.† This average does not yet include the new measurement
from the CDF II experiment [57], but the difference with the current world
average has no significant impact on the numerical prediction for
$|\varepsilon_{K}|_{\text{SM}}$.‡ Ref. [58] has computed a correction factor
for $\eta_{ut}$ of 1.005, which is not yet included here but has a negligible
impact on $|\varepsilon_{K}|$.

In the SM, the measured value of $|\varepsilon_{K}|$ describes a hyperbola

$\bar{\eta}=\frac{|\varepsilon_{K}|}{A-B\bar{\rho}}$ (36)

in the $\bar{\rho}\\--\bar{\eta}$ plane. Using our solutions for the UT apex
in Eqs. (27)–(29), we find the SM predictions

$\displaystyle\text{Incl, }K\ell 3$
$\displaystyle|\varepsilon_{K}|_{\text{SM}}$ $\displaystyle=(2.54\pm
0.22)\times 10^{-3}\>,$ (37) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle|\varepsilon_{K}|_{\text{SM}}$ $\displaystyle=(1.74\pm
0.15)\times 10^{-3}\>,$ (38)

where the inclusive value also covers the hybrid scenario, as
$|\varepsilon_{K}|_{\text{SM}}$ only depends on $|V_{cb}|$. The values of $A$
and $B$ can be found in Table 2. The differences between the SM predictions
and the experimental result [26]

$|\varepsilon_{K}|=(2.228\pm 0.011)\times 10^{-3}$ (39)

are 1.4 and 3.2 standard deviations for the inclusive/hybrid and exclusive
scenarios, respectively. This shows a preference for the inclusive value, but
the uncertainty on the SM predictions is not yet sufficient to draw any
further conclusions.

The same conclusion also arises from the comparison between the contours
following from the measurement of $|\varepsilon_{K}|$ and the fit solution for
the UT apex fixed through $\gamma$ and $R_{b}$, shown in Fig. 1. Besides
finding the same small tensions as between Eqs. (37), (38) and (39), it is
interesting that the contour for $|\varepsilon_{K}|$ from the exclusive
determination lies completely above the fit solution from $\gamma$ and
$R_{b}$, while the contour from the inclusive determination lies mainly below
the corresponding fit solution. The contour from the hybrid determination
falls in between the inclusive and exclusive determinations, and overlaps with
the fit solution. Out of the three scenarios, it gives the most consistent
picture of the UT apex. This illustrates the strong dependence on the value of
$|V_{cb}|$, which in the future could help to understand the puzzle between
the inclusive and exclusive scenarios, assuming NP in kaon mixing can be
controlled or ignored.

##  3 $B_{q}^{0}$–$\bar{B}_{q}^{0}$ Mixing in the Standard Model

In the SM, mixing between the neutral $B_{q}^{0}$ and $\bar{B}_{q}^{0}$ mesons
originates from box diagrams. Since these are loop processes and thus strongly
suppressed, the mixing phenomenon is an excellent place for NP to enter. The
determination of the parameter space of possible NP contributions to
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing will rely on the results for the UT apex
from Section 2.3, which are needed to compute the SM predictions of the mixing
parameters characterising neutral $B_{q}$-meson mixing. Comparing their SM
predictions to the corresponding experimental values will allow us to
constrain possible NP effects in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, as we
will show in Section 4.

###  3.1 The Mixing Parameters in the Standard Model

The SM prediction of the mixing angle $\phi_{d}$ is given by:

$\phi_{d}^{\text{SM}}=2\beta=2\text{arg}\left(-\frac{V_{cd}V_{cb}^{*}}{V_{td}V^{*}_{tb}}\right)=2\tan^{-1}\left(\frac{\bar{\eta}}{1-\bar{\rho}}\right)\>.$
(40)

Using Eqs. (27)–(29), we obtain the following results:

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\phi_{d}^{\text{SM}}$
$\displaystyle=(51.4\pm 2.8)^{\circ}\>,$ (41) $\displaystyle\text{Excl, }K\ell
3$ $\displaystyle\phi_{d}^{\text{SM}}$ $\displaystyle=(46.2\pm
2.3)^{\circ}\>,$ (42) $\displaystyle\text{Hybrid, }K\ell 3$
$\displaystyle\phi_{d}^{\text{SM}}$ $\displaystyle=(42.6\pm 2.2)^{\circ}\>.$
(43)

These can be compared to the experimental measurement of $\phi_{d}$ that was
already given in Eq. (10).

The SM expression for $\phi_{s}$ is given by:

$\phi_{s}^{\text{SM}}=-2\delta\gamma=-2\lambda^{2}\bar{\eta}+\mathcal{O}(\lambda^{4})\>,$
(44)

which at this order in the Wolfenstein parametrisation [24, 25] can be
expressed in terms of either $\eta$ or $\bar{\eta}$ interchangeably. Note that
in contrast to the expression for $\phi_{d}^{\text{SM}}$, the dependence on
the UT apex is doubly Cabibbo-suppressed. The numerical predictions based on
Eqs. (27)–(29) are:

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\phi_{s}^{\text{SM}}$
$\displaystyle=-0.0402\pm 0.0022=(-2.30\pm 0.13)^{\circ}\>,$ (45)
$\displaystyle\text{Excl, }K\ell 3$ $\displaystyle\phi_{s}^{\text{SM}}$
$\displaystyle=-0.0363\pm 0.0018=(-2.08\pm 0.10)^{\circ}\>,$ (46)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\phi_{s}^{\text{SM}}$
$\displaystyle=-0.0336\pm 0.0017=(-1.93\pm 0.10)^{\circ}\>.$ (47)

These SM predictions are a factor 2.5 to 3 less precise than the value [11]

$\phi_{s}^{\text{SM}}=-0.03682_{-0.00060}^{+0.00086}=\left(-2.110_{-0.034}^{+0.049}\right)^{\circ}\>,$
(48)

which was obtained from a global fit of the UT, and is commonly referenced in
the literature. However, the global UT fits rely for their input on
information from $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, without accounting for
possible contributions from NP that may introduce biases. We will therefore
not use these results here and trade the loss in precision for a better
control of the possible NP effects that could enter this determination. The
experimental measurement of $\phi_{s}$ is given in Eq. (5).

The mass difference between the heavy and light mass eigenstates of the
neutral $B_{q}$-meson system is given by

$\Delta
m_{q}=2|M_{12}^{q}|+\mathcal{O}\left(\left|\frac{\Gamma_{12}^{q}}{M_{12}^{q}}\right|^{2}\right)\>,$
(49)

where $M_{12}^{q}-\frac{i}{2}\Gamma_{12}^{q}$ is the off-diagonal element of
the effective Hamiltonian describing $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. In
the SM, the element $M_{12}^{q}$ is given as

$\left|M_{12}^{q}\right|_{\text{SM}}=\frac{G_{\text{F}}^{2}m_{W}^{2}}{12\pi^{2}}m_{B_{q}}\left|V_{tq}V_{tb}\right|^{2}\>S_{0}(x_{t})\>\eta_{2B}\hat{B}_{B_{q}}f_{B_{q}}^{2}\>,$
(50)

where $m_{B_{q}}$ is the $B_{q}$ mass, $f_{B_{q}}$ is the $B_{q}$ decay
constant, $\hat{B}_{B_{q}}$ is the renormalisation group invariant bag
parameter, $S_{0}(x_{t})$ is the Inami–Lim function given in Eq. (32), and
$\eta_{2B}=0.551$ is a short-distance QCD correction factor [59, 60]. The
decay constants and bag parameters have been calculated using lattice QCD
methods. The latest computation using $N_{f}=2+1+1$ quark flavours gives the
following results [61, 43]:

$f_{B_{d}}\sqrt{\hat{B}_{B_{d}}}=(210.6\pm 5.5)\>\text{MeV}\>,\qquad
f_{B_{s}}\sqrt{\hat{B}_{B_{s}}}=(256.1\pm 5.7)\>\text{MeV}\>.$ (51)

The uncertainty on $\eta_{2B}$ is $\mathcal{O}(1\%)$ [60], and can therefore
still be neglected with the current precision. Eq. (50) does not take into
account next-to-leading-order electroweak effects, which have been calculated
in Ref. [62] to be at the level of 1%. Hence, they can also be omitted given
the current precision. The values of all parameters needed to compute $\Delta
m_{q}^{\text{SM}}$ are summarised in Table 3.

| Inclusive | Exclusive | Hybrid | Unit | Reference
---|---|---|---|---|---
$m_{B_{s}}$ | $5366.92\pm 0.10$ | MeV | [26]
$m_{B_{d}}$ | $5279.66\pm 0.12$ | MeV | [26]
$f_{B_{d}}\sqrt{\hat{B}_{B_{d}}}$ | $210.6\pm 5.5$ | MeV | [61]
$f_{B_{s}}\sqrt{\hat{B}_{B_{s}}}$ | $256.1\pm 5.7$ | MeV | [61]
$\xi$ | $1.212\pm 0.016$ |  | [61]
$\Delta m_{d}^{\text{SM}}$ | $0.513\pm 0.040$ | $0.439\pm 0.033$ | $0.510\pm 0.037$ | $\text{ps}^{-1}$ | -
$\Delta m_{s}^{\text{SM}}$ | $17.23\pm 0.87$ | $14.80\pm 0.76$ | $17.19\pm 0.87$ | $\text{ps}^{-1}$ | -
$\Delta m_{d}$ | $0.5065\pm 0.0019$ | $\text{ps}^{-1}$ | [27]
$\Delta m_{s}$ | $17.7656\pm 0.0057$ | $\text{ps}^{-1}$ | [27, 63]
$\phi_{d}^{\text{SM}}$ | $51.4\pm 2.8$ | $46.2\pm 2.3$ | $42.6\pm 2.2$ | Degrees | -
$\phi_{s}^{\text{SM}}$ | $-2.30\pm 0.13$ | $-2.08\pm 0.10$ | $-1.93\pm 0.10$ | Degrees | -
$\phi_{d}$ | $44.4_{-1.5}^{+1.6}$ | Degrees | [36, 37]
$\phi_{d}$ | $-4.2\pm 1.4$ | Degrees | [36, 37]
Table 3: Input values and results for the determination of the
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing parameters $\Delta m_{q}$ and $\phi_{q}$.
The dashes in the reference column indicate derived quantities that were
computed in this paper.

Using the Wolfenstein parametrisation of the CKM matrix [24, 25], we can
rewrite the CKM matrix elements appearing in the SM expression for the mass
differences $\Delta m_{q}$ in terms of the experimental inputs and the
coordinates of the UT apex that were discussed in Section 2:

$\displaystyle|V_{td}V_{tb}|$
$\displaystyle=\lambda|V_{cb}|\sqrt{(1-\bar{\rho})^{2}+\bar{\eta}\,^{2}}+\mathcal{O}\left(\lambda^{7}\right)\>,$
(52)
$\displaystyle=\lambda|V_{cb}|\sqrt{1-2R_{b}\cos\gamma+R_{b}^{2}}+\mathcal{O}\left(\lambda^{7}\right)\>,$
(53) $\displaystyle|V_{ts}V_{tb}|$
$\displaystyle=\phantom{\lambda}|V_{cb}|\left[1-\frac{\lambda^{2}}{2}\left(1-2\bar{\rho}\right)\right]+\mathcal{O}\left(\lambda^{6}\right)\>,$
(54)
$\displaystyle=\phantom{\lambda}|V_{cb}|\left[1-\frac{\lambda^{2}}{2}\left(1-2R_{b}\cos\gamma\right)\right]+\mathcal{O}\left(\lambda^{6}\right)\>.$
(55)

We see that $|V_{td}V_{tb}|$, and thus $\Delta m_{d}^{\text{SM}}$, depends at
leading order on $\bar{\rho}$ and $\bar{\eta}$, while for $|V_{ts}V_{tb}|$ and
$\Delta m_{s}^{\text{SM}}$ the dependence on the UT apex only enters at next-
to-leading order in $\lambda$. In these expressions, the difference between
$\rho$ and $\bar{\rho}$ is of the same order as the neglected terms.

We find the following predictions for the mass differences $\Delta m_{q}$:

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\Delta m_{d}^{\text{SM}}$
$\displaystyle=(0.513\pm 0.040)\>\text{ps}^{-1}\>,$ $\displaystyle\Delta
m_{s}^{\text{SM}}$ $\displaystyle=(17.23\pm 0.87)\>\text{ps}^{-1}\>,$ (56)
$\displaystyle\text{Excl, }K\ell 3$ $\displaystyle\Delta m_{d}^{\text{SM}}$
$\displaystyle=(0.439\pm 0.033)\>\text{ps}^{-1}\>,$ $\displaystyle\Delta
m_{s}^{\text{SM}}$ $\displaystyle=(14.80\pm 0.76)\>\text{ps}^{-1}\>,$ (57)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\Delta m_{d}^{\text{SM}}$
$\displaystyle=(0.510\pm 0.037)\>\text{ps}^{-1}\>,$ $\displaystyle\Delta
m_{s}^{\text{SM}}$ $\displaystyle=(17.19\pm 0.87)\>\text{ps}^{-1}\>.$ (58)

These SM predictions can be compared with the experimental measurements [27,
63]:

$\displaystyle\Delta m_{d}$ $\displaystyle=(0.5065\pm
0.0019)\>\text{ps}^{-1}\>,$ (59) $\displaystyle\Delta m_{s}$
$\displaystyle=(17.7656\pm 0.0057)\>\text{ps}^{-1}\>,$ (60)

which are already one to two orders of magnitude more precise. Similar results
have been obtained from a combined analysis of lattice and light-cone QCD sum
rules (LQSR) results in Ref. [64]. The experimental values and the SM
predictions can be visually compared with each other as in Fig. 3, or
numerically through the ratio

$\rho_{q}\equiv\frac{\Delta m_{q}}{\Delta m_{q}^{\text{SM}}}\>,$ (61)

which equals 1 in the SM. With the results in Eqs. (56)–(60), we obtain:

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\rho_{d}$
$\displaystyle=0.989\pm 0.078\>,$ $\displaystyle\rho_{s}$
$\displaystyle=1.031\pm 0.052\>,$ (62) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle\rho_{d}$ $\displaystyle=1.153\pm 0.088\>,$
$\displaystyle\rho_{s}$ $\displaystyle=1.200\pm 0.062\>,$ (63)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\rho_{d}$
$\displaystyle=0.993\pm 0.073\>,$ $\displaystyle\rho_{s}$
$\displaystyle=1.034\pm 0.052\>.$ (64)

For the inclusive and hybrid scenarios, the ratios are compatible with 1, and
thus the SM, while those from the exclusive scenario differ by (15–20)%. In
the exclusive scenario, the SM predictions for $\Delta m_{s}$ and $\Delta
m_{d}$ differ by respectively 3 and 2 standard deviations from their
experimentally measured values. The smaller central values for $\Delta
m_{s}^{\text{SM}}$ and $\Delta m_{d}^{\text{SM}}$ in the exclusive scenario
compared to the inclusive and hybrid scenarios, and hence the smaller
discrepancies with the experimental measurements, are due to the value of
$|V_{cb}|$, which shows the same pattern.

Figure 3: Comparison of the experimental averages and SM predictions for the
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing parameters $\Delta m_{d}$ (Left) and
$\Delta m_{s}$ (Right).

###  3.2 Determining the UT Apex from $R_{b}$ and Mixing

If we assume the SM expressions for the $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing
parameters $\Delta m_{d}$ and $\Delta m_{s}$, it is possible to determine the
UT apex without having to rely on information from $\gamma$. Although this
scenario is not considered as the baseline, due to the strong assumption that
needs to be made, it can be a useful alternative when discrepancies arise
between the various measurements of $\gamma$. It is therefore interesting to
explore this option as well.

In the scenario without $\gamma$, the coordinates of the UT apex are fixed by
the values of the sides $R_{b}$ and $R_{t}$. The UT side $R_{t}$ is defined as

$R_{t}\equiv\left|\frac{V_{td}V_{tb}}{V_{cd}V_{cb}}\right|=\sqrt{(1-\bar{\rho})^{2}+\bar{\eta}\,^{2}}=\frac{1}{\lambda}\left|\frac{V_{td}}{V_{ts}}\right|\left[1-\frac{\lambda^{2}}{2}\left(1-2\bar{\rho}\right)\right]+\mathcal{O}\left(\lambda^{4}\right)\>,$
(65)

where the ratio of CKM matrix elements

$\left|\frac{V_{td}}{V_{ts}}\right|=\xi\sqrt{\frac{m_{B_{s}}\Delta
m_{d}^{\text{SM}}}{m_{B_{d}}\Delta m_{s}^{\text{SM}}}}$ (66)

can be determined from the $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing parameters.
Here,

$\xi\equiv\frac{f_{B_{s}}\sqrt{\hat{B}_{B_{s}}}}{f_{B_{d}}\sqrt{\hat{B}_{B_{d}}}}$
(67)

is the ratio of bag parameters and decay constants of the $B_{d}$ and the
$B_{s}$ systems. This ratio has been calculated on the lattice, with the
latest value given by [61, 43]:

$\xi=1.212\pm 0.016\>.$ (68)

Compared to the individual results for the bag parameters and decay constants
in Table 3, it is known with much higher precision as many uncertainties
cancel in the ratio. Combining this result with the experimental values of
$\Delta m_{d}$ and $\Delta m_{s}$ in Eqs. (59) and (60) we get

$\left|\frac{V_{td}}{V_{ts}}\right|=0.2063\pm 0.0004\pm 0.0027\>,$ (69)

where the first uncertainty is due to the experimental measurements and the
second due to the lattice input.

The coordinates of the UT apex, determined from a fit to the sides $R_{b}$ and
$R_{t}$ are

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\bar{\rho}$
$\displaystyle=0.180\pm 0.014\>,$ $\displaystyle\bar{\eta}$
$\displaystyle=0.395\pm 0.020\>,$ (70) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle\bar{\rho}$ $\displaystyle=0.163\pm 0.013\>,$
$\displaystyle\bar{\eta}$ $\displaystyle=0.357\pm 0.017\>,$ (71)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\bar{\rho}$
$\displaystyle=0.153\pm 0.013\>,$ $\displaystyle\bar{\eta}$
$\displaystyle=0.330\pm 0.016\>,$ (72)

with the two-dimensional confidence level contours shown in Fig. 5. Compared
to the fit results based on $R_{b}$ and $\gamma$ in Eqs. (27)–(29), we find a
similar precision for $\bar{\eta}$, while the precision on $\bar{\rho}$ is a
factor 2 better. The solutions (70)–(72) correspond to the values

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle R_{t}$
$\displaystyle=0.910\pm 0.012\>,$ (73) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle R_{t}$ $\displaystyle=0.909\pm 0.012\>,$ (74)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle R_{t}$
$\displaystyle=0.909\pm 0.012\>,$ (75)

which can be compared to the values for $R_{t}$ calculated from Eqs.
(27)–(29):

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle R_{t}$
$\displaystyle=0.932\pm 0.024\>,$ (76) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle R_{t}$ $\displaystyle=0.930\pm 0.021\>,$ (77)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle R_{t}$
$\displaystyle=0.930\pm 0.021\>.$ (78)

Also here the scenarios with $\gamma$ are a factor 2 less precise than the
scenarios without $\gamma$.

Figure 4: Determination of the UT apex $(\bar{\rho},\bar{\eta})$ from the
measurements of the sides $R_{b}$ and $R_{t}$. For comparison, also the
constraint from $\Delta m_{d}$ alone is shown (but not included in the fit).
The solutions based on the inclusive (Left), exclusive (Right) and hybrid
(Bottom) determination of $R_{b}$ are shown separately.

Alternatively, $R_{t}$ can also be determined without relying on information
from the $B_{s}$ system, which has the advantage that its value will not be
affected by NP in $B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing. The comparison with
the results in Eqs. (73)–(75) then theoretically provides another way to
constrain NP effects in the $B_{s}$-meson system. The disadvantage of this
approach is that we cannot use the ratio $\xi$ of bag parameters and decay
constants, which will result in larger associated uncertainties. In this
alternative method, $R_{t}$ is calculated from the definition in Eq. (65),
using the inclusive or exclusive value of $|V_{cb}|$, and obtaining
$|V_{td}V_{tb}|$ from the measurement of $\Delta m_{d}$ through Eq. (50). The
approach using only the measurement of $\Delta m_{d}$, and avoiding $\Delta
m_{s}$, gives

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle R_{t}$
$\displaystyle=0.926\pm 0.027\>,$ (79) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle R_{t}$ $\displaystyle=0.999\pm 0.029\>,$ (80)

with the hybrid scenario matching the inclusive result as it does not depend
on $|V_{ub}|$. The corresponding contour is also shown in Fig. 4. Given the
current precision, the results for $R_{t}$ determined through only $\Delta
m_{d}$ are not competitive with the results for $R_{t}$ determined through
$\xi$. Hence, we will only use $R_{t}$ determined through $\xi$ in our
analysis. For the inclusive and hybrid scenarios the two determinations of
$R_{t}$ overlap completely, while a slight discrepancy is visible for the
exclusive scenario. Combined with the relatively large errors on $R_{t}$
determined through $\Delta m_{d}$, it does not yet allow us to draw any
conclusions regarding NP in $B_{s}$ mixing.

Figure 5: Comparison of the two-dimensional confidence regions for the UT apex
$(\bar{\rho},\bar{\eta})$ determined through $\gamma$ and $R_{b}$, and through
$R_{b}$ and $R_{t}$.

From the comparison in Fig. 5, we can see that the determination of the apex
from $R_{b}$ and $R_{t}$ is more precise than the one utilising $R_{b}$ and
$\gamma$. However, $R_{t}$ has been determined assuming SM expressions for
$\Delta m_{d}$ and $\Delta m_{s}$, and hence ignores contributions from
possible NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. Possible NP will
contaminate the determination of $R_{t}$, except in the special case of FUNP.
In such scenarios, possible NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing drops
out in the ratio between $\Delta m_{d}$ and $\Delta m_{s}$. To determine NP in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing in a general scenario, we cannot use the
above determination of $R_{t}$, but we need the UT apex determination of
$R_{b}$ and $\gamma$ discussed in Section 2.3. In the next Section we will
investigate in more detail what impact the different determinations of the UT
apex have on NP searches in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing.

##  4 New Physics in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ Mixing

We can use the SM predictions for the mixing parameters $\Delta m_{q}$ and
$\phi_{q}$, together with their experimental values, to constrain the
parameter space of possible NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. We
introduce the NP parameters $\kappa_{q}$ and $\sigma_{q}$ by modifying the
mixing parameters in the same way as in Ref. [1]:

$\displaystyle\Delta m_{q}$ $\displaystyle=\Delta
m_{q}^{\text{SM}}\left|1+\kappa_{q}e^{i\sigma_{q}}\right|\>,$ (81)
$\displaystyle\phi_{q}$
$\displaystyle=\phi_{q}^{\text{SM}}+\phi_{q}^{\text{NP}}=\phi_{q}^{\text{SM}}+\arg\left(1+\kappa_{q}e^{i\sigma_{q}}\right)\>.$
(82)

Here, the size of the NP effects is described by $\kappa_{q}$, while
$\sigma_{q}$ is a complex phase that accounts for additional CP-violating
effects. This parametrisation is model-independent; we are not making any
assumptions regarding the origin of the NP. We will use Eqs. (81) and (82) to
explore two different NP scenarios, which are summarised in Table 4.

| Scenario I | Scenario II
---|---|---
Assumptions | no NP in $\gamma$ | FUNP
UT apex fit |  $R_{b}$ and $\gamma$ |  $R_{b}$ and $R_{t}$
NP parameters |  $(\kappa_{d},\sigma_{d})$ and $(\kappa_{s},\sigma_{s})$ | $(\kappa_{d},\sigma_{d})=(\kappa_{s},\sigma_{s})$
Table 4: Two different scenarios for which we will determine the NP
parameters.

Scenario I represents the most general case, and has the least assumptions. We
utilise the UT apex determination in Section 2.3 for the SM predictions of the
mixing parameters $\Delta m_{q}$ and $\phi_{q}$, together with the numerical
values given in Section 3.1, to determine the NP parameters
$(\kappa_{d},\sigma_{d})$ and $(\kappa_{s},\sigma_{s})$ independently from
each other.

For the second scenario we explore other options that require additional
assumptions. In Scenario II we consider the case where NP contributions are
equal in the $B_{d}$ and the $B_{s}$ systems, i.e. FUNP. Such a scenario could
occur if NP predominantly couples to the third generation quarks and leptons,
and is explicitly realised in NP models with a $U(2)$ symmetry [65, 66, 67,
68]. Note that FUNP is different from the well-known Minimal Flavour Violation
scenario, which assumes no CP-violating NP phase at all, as was already noted
in Ref. [1, 68], where the same scenario was studied. Assuming the FUNP
framework, it is possible to employ the UT apex determination that only relies
on $R_{b}$ and the mixing parameters described in Section 3.2, without
requiring additional information on $\gamma$. In this way, possible NP in
$\gamma$ will not affect the results. However, assuming that NP affects the
$B_{d}$ and $B_{s}$ systems equally is a strong assumption. A comparison of
the FUNP scenario with the general scenario provides a test of the FUNP
assumption and gives a measure of the impact that these assumptions have on
the constraints on the parameter space of NP in $B_{q}^{0}$–$\bar{B}_{q}^{0}$
mixing.

The last scenario we will discuss is an interpolation between Scenario I and
II. We will assume FUNP to determine the UT apex, using only information from
$R_{b}$ and the mixing parameters, but will relax this assumption when
determining $(\kappa_{d},\sigma_{d})$ and $(\kappa_{s},\sigma_{s})$. This
scenario has less stringent assumptions on the form of the NP than for the
full FUNP scenario. Comparing it with Scenario I and II allows us both to
estimate the impact of the FUNP assumption on the fit to the NP parameters,
and to explore the impact of the different UT apex determinations.

###  4.1 Scenario I: General New Physics

Scenario I is a general determination of the NP parameters in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, given in Eqs. (81) and (82). We will use
our results from Section 2 including information from $R_{b}$ and $\gamma$,
together with the SM expressions and values for the mixing parameters,
summarised in Table 3, to determine the NP parameters
$(\kappa_{q},\sigma_{q})$. The only assumption that enters this scenario is
the absence of NP in $\gamma$ and the semileptonic decays needed to determine
$R_{b}$. The results are shown in Fig. 6 and given as follows:

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\kappa_{d}$
$\displaystyle=0.121_{-0.055}^{+0.056}\>,$ $\displaystyle\sigma_{d}$
$\displaystyle=\left(261_{-35}^{+37}\right)^{\circ}\>,$ (83)
$\displaystyle\kappa_{s}$ $\displaystyle=0.045_{-0.033}^{+0.048}\>,$
$\displaystyle\sigma_{s}$
$\displaystyle=\left(312_{-77}^{+37}\right)^{\circ}\>,$ (84)
$\displaystyle\text{Excl, }K\ell 3$ $\displaystyle\kappa_{d}$
$\displaystyle=0.156_{-0.084}^{+0.093}\>,$ $\displaystyle\sigma_{d}$
$\displaystyle=\left(347_{-25}^{+21}\right)^{\circ}\>,$ (85)
$\displaystyle\kappa_{s}$ $\displaystyle=0.205_{-0.059}^{+0.064}\>,$
$\displaystyle\sigma_{s}$
$\displaystyle=\left(347.6_{-9.8}^{+8.5}\right)^{\circ}\>,$ (86)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\kappa_{d}$
$\displaystyle=0.031_{-0.031}^{+0.057}\>,$ $\displaystyle\sigma_{d}$
$\displaystyle=\left(104_{-104}^{+256}\right)^{\circ}\>,$ (87)
$\displaystyle\kappa_{s}$ $\displaystyle=0.053_{-0.034}^{+0.046}\>,$
$\displaystyle\sigma_{s}$
$\displaystyle=\left(309_{-65}^{+34}\right)^{\circ}\>.$ (88)

We observe in Fig. 6 that the individual constraints from $\phi_{d}$ in the
inclusive scenario and $\Delta m_{s}$ in the exclusive scenario are in tension
with the SM, while all other constraints are compatible with $\kappa_{q}=0$.

Figure 6: Two-dimensional confidence regions of the fit for $\kappa_{d}$ and
$\sigma_{d}$ (left), and $\kappa_{s}$ and $\sigma_{s}$ (right), which
parametrise NP contributions in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. For
illustration, also the individual constraints from $\phi_{q}$ and $\Delta
m_{q}$ are shown. Top: Inclusive scenario. Middle: Exclusive scenario. Bottom:
Hybrid scenario.

Figure 7: Comparison between the inclusive, exclusive and hybrid scenario of
the two-dimensional confidence regions of the fits for $\kappa_{d}$ and
$\sigma_{d}$ (left), and $\kappa_{s}$ and $\sigma_{s}$ (right), which
parametrise NP contributions in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing.

Figure 8: Comparison between the two-dimensional confidence regions of the
Scenario I and Scenario II fits for $\kappa_{q}$ and $\sigma_{q}$, which
parametrise NP contributions in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. Left:
Inclusive scenario. Right: Exclusive scenario. Bottom: Hybrid scenario. Figure
9: Comparison between the inclusive, exclusive and hybrid scenarios of the
two-dimensional confidence regions of a combined fit for the NP parameters
$\kappa_{d}$ and $\kappa_{s}$ describing NP contributions in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. The black diagonal line represents the
FUNP scenario, where $\kappa_{d}=\kappa_{s}$.

A comparison of the two-dimensional confidence regions between the three fit
solutions based on different values of $R_{b}$ is given in Fig. 7, and a
comparison between the fit solutions for the $B_{d}$ and $B_{s}$ systems in
Fig. 8. The correlation between $\kappa_{d}$ and $\kappa_{s}$ is also shown in
Fig. 9. As illustrated in Fig. 9, all three $R_{b}$ solutions are compatible
with the assumption $\kappa_{d}=\kappa_{s}$, which is used in the FUNP
scenario. However, the conclusions regarding the presence of NP are different
for the three scenarios: The inclusive one is compatible with $\kappa_{s}=0$,
but suggests $\kappa_{d}\neq 0$; the exclusive fit result is compatible with
$\kappa_{d}=0$, but finds $\kappa_{s}$ different from zero with a significance
of 3.5 standard deviations; and the hybrid case is compatible with both
$\kappa_{d}=0$ and $\kappa_{s}=0$ and hence finds no tension with the SM.

###  4.2 Scenario II: Flavour Universal New Physics

In the second scenario, we will assume FUNP, where the size and phase of the
NP contributions, $\kappa_{q}$ and $\sigma_{q}$, are equal for the $B_{d}$ and
$B_{s}$ systems:

$\kappa_{d}=\kappa_{s}\equiv\kappa\>,\quad\sigma_{d}=\sigma_{s}\equiv\sigma\>.$
(89)

An important consequence is that the NP effects drop out in the ratio $\Delta
m_{d}/\Delta m_{s}$. This means that the side $R_{t}$ of the UT, defined in
Eq. (65), will receive no contributions from NP at order $\lambda^{2}$. We can
use the results for the UT apex coming from information from $R_{b}$ and
$R_{t}$, as given in Eqs. (70)–(72), to determine the SM values of $\Delta
m_{q}$ and $\phi_{q}$ without additional information on $\gamma$. Comparing
the SM predictions to their experimental values, we get constraints on
$\kappa$ and $\sigma$. This results in the following fit values for $\kappa$
and $\sigma$:

$\displaystyle\text{Incl, }K\ell 3$ $\displaystyle\kappa$
$\displaystyle=0.057_{-0.026}^{+0.040}\>,$ $\displaystyle\sigma$
$\displaystyle=\left(294_{-53}^{+34}\right)^{\circ}\>,$ (90)
$\displaystyle\text{Excl, }K\ell 3$ $\displaystyle\kappa$
$\displaystyle=0.203_{-0.057}^{+0.062}\>,$ $\displaystyle\sigma$
$\displaystyle=\left(347.7_{-8.3}^{+7.4}\right)^{\circ}\>,$ (91)
$\displaystyle\text{Hybrid, }K\ell 3$ $\displaystyle\kappa$
$\displaystyle=0.043_{-0.036}^{+0.049}\>,$ $\displaystyle\sigma$
$\displaystyle=\left(326_{-90}^{+32}\right)^{\circ}\>.$ (92)

These results are very similar to those arising from the fit for the $B_{s}$
system from $R_{b}$ and $\gamma$ in Eqs. (83)–(88), implying that the $B_{s}$
system dominates these fits. In Fig. 8, we can see that in the exclusive case,
the results for FUNP almost completely overlap with those of the $B_{s}$
system for Scenario I. In the inclusive and hybrid scenarios, the shape of the
contours for the $B_{d}$ and the $B_{s}$ system are very different, indicating
that FUNP might not be a correct assumption. However, the solutions still
overlap within the error margin, and one can see that the FUNP scenario
interpolates between the $B_{d}$ and $B_{s}$ systems, with the $B_{s}$ system
still dominating. In all cases, we cannot rule out FUNP, as it is still
compatible with the results from the general scenario for both $B_{d}$ and
$B_{s}$.

##  5 Applications for the Rare Decays $B_{q}^{0}\to\mu^{+}\mu^{-}$

Parameter | Value | Unit | Reference
---|---|---|---
$\hat{B}_{s}$ | $1.232\pm 0.053$ |  | [61, 43]
$\hat{B}_{s}/\hat{B}_{d}$ | $1.008\pm 0.025$ |  | [61, 43]
$2y_{s}\equiv\Delta\Gamma_{s}/\Gamma_{s}$ | $0.128\pm 0.007$ |  | [27]
$2y_{d}\equiv\Delta\Gamma_{d}/\Gamma_{d}$ | $0.001\pm 0.010$ |  | [27]
$\tau_{\mu\mu}^{s}$ | $2.07\pm 0.29$ | ps | [69]
$\tau_{B_{s}}$ | $1.520\pm 0.005$ | ps | [27]
$f_{B_{s}}$ | $230.3\pm 1.3$ | MeV | [43]
$\tau_{B_{s},H}$ | $1.624\pm 0.009$ | ps | [27]
$\eta_{Y}$ | 1.0113 |  | [70]

Table 5: Input values for the SM computation of the
$B_{q}^{0}\to\mu^{+}\mu^{-}$ branching fractions. $\tau_{B_{s},H}$ is the
lifetime of the heavy $B_{s}$ mass eigenstate and can be expressed in terms of
$\tau_{B_{s}}$ and $y_{s}$ as $\tau_{B_{s}}/(1-y_{s})$

.

The NP searches in the rare decays $B_{q}^{0}\to\mu^{+}\mu^{-}$ are also
affected by our choices for the CKM matrix elements. The SM predictions for
the branching fractions are, using unitarity of the CKM matrix, proportional
to the square of $|V_{cb}|$, and thus differ between the inclusive and
exclusive determinations. Refs. [21, 23, 22] suggested to take the ratio with
$\Delta m_{q}$ in order to eliminate this dependence on $|V_{cb}|$. However,
as we have seen above, $\Delta m_{q}$ may be affected by NP effects. Using our
analyses we are able to take these effects into account, and show the
resulting constraints on NP in $B_{s}^{0}\to\mu^{+}\mu^{-}$. In addition, we
will also explore the option of using the ratio of
$B_{d}^{0}\to\mu^{+}\mu^{-}$ and $B_{s}^{0}\to\mu^{+}\mu^{-}$ branching
fractions to determine the UT side $R_{t}$, providing us with a second
opportunity to determine the UT apex without relying on the angle $\gamma$.

###  5.1 $B_{s}^{0}\to\mu^{+}\mu^{-}$ in the Standard Model

At lowest order in the SM, the “theoretical” branching fraction of the decay
$B_{s}^{0}\to\mu^{+}\mu^{-}$ is given as [71]

$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})=\frac{\tau_{B_{s}}G_{\text{F}}^{4}m_{W}^{4}\sin^{4}\theta_{W}}{8\pi^{5}}\left|C_{10}^{\text{SM}}\right|^{2}|V_{ts}^{\phantom{*}}V_{tb}^{*}|^{2}f_{B_{s}}^{2}m_{B_{s}}m_{\mu}^{2}\sqrt{1-\frac{4m_{\mu}^{2}}{m_{B_{s}}^{2}}}\>,$
(93)

where $\theta_{W}$ is the weak mixing angle, and $\tau_{B_{s}}$ is the
$B_{s}$-meson lifetime. The theoretical branching fraction is defined at decay
time $t=0$ and thus does not include effects coming from
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing. This is not the case for the
experimentally measured time-integrated branching fraction, which is related
to the theoretical branching fraction as follows[72]:

$\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})=\frac{1+\mathcal{A}_{\Delta\Gamma}^{\mu\mu}y_{s}}{1-y_{s}^{2}}\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})\>,$
(94)

where $\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=1$ in the SM. The parameter $y_{s}$
is defined as

$y_{s}\equiv\frac{\tau_{B_{s}}}{2}\Delta\Gamma_{s}\>,$ (95)

where $\Delta\Gamma_{s}$ is the decay width difference between the $B_{s}$
mass eigenstates. It is now clear that $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing
effects will affect the time-integrated branching fraction. The Wilson
coefficient $C_{10}$ can be expressed as

$C_{10}^{\text{SM}}=\frac{\eta_{Y}Y_{0}(x_{t})}{\sin^{2}\theta_{W}}\>,$ (96)

where $\eta_{Y}$ is a QCD correction factor [70], and $Y_{0}$ is the Inami–Lim
function [53]

$Y_{0}(x)=\frac{x}{8}\left[\frac{4-x}{1-x}+\frac{3x}{(1-x)^{2}}\ln
x\right]\>.$ (97)

It is interesting to note that in this specific parametrisation the dependence
on $\sin^{2}\theta_{W}$ drops out from the theoretical prediction (93). All
numerical inputs are summarised in Table 5.

The SM branching fraction depends both on the value of $|V_{cb}|$ and on the
solution of the UT apex through the relation (54) for the CKM matrix elements
$|V_{ts}^{\phantom{*}}V_{tb}^{*}|$, although the latter dependence enters only
through higher order corrections. This results in different predictions for
the three scenarios considered in this paper:

$\displaystyle\text{Incl, }K\ell 3$
$\displaystyle\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$
$\displaystyle=(3.81\pm 0.11)\times 10^{-9}\>,$ (98) $\displaystyle\text{Excl,
}K\ell 3$ $\displaystyle\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$
$\displaystyle=(3.27\pm 0.10)\times 10^{-9}\>,$ (99)
$\displaystyle\text{Hybrid, }K\ell 3$
$\displaystyle\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$
$\displaystyle=(3.80\pm 0.10)\times 10^{-9}\>.$ (100)

In comparison with the latest prediction [73]

$\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})|_{\text{SM}}=(3.66\pm 0.14)\times
10^{-9}$ (101)

found in the literature, our uncertainties are slightly smaller. This is
primarily because the experimental measurements for the CKM matrix elements
and top quark mass have improved compared to what was used in Ref. [73]. It is
important to point out that the result (101) includes NLO electroweak and QCD
corrections, while our predictions in Eqs. (98)-(100) are based on the LO
expression only. We have not included NLO corrections since our goal is to
illustrate the dependence of the numerical prediction on the choice for the
CKM matrix elements and UT apex. This introduces a spread, as illustrated in
Fig. 10, that is still much larger and thus more important than the NLO
corrections.

Figure 10: Comparison of the experimental measurements, averages and SM
predictions for the time-integrated branching fraction
$\bar{\mathcal{B}}(B_{s}^{0}\to\mu^{+}\mu^{-}$)

On the experimental side, the $B_{s}^{0}\to\mu^{+}\mu^{-}$ branching fraction
has been measured by the LHCb [69], ATLAS [74] and CMS [75, 76]
collaborations. Their results are compared to the SM predictions in Fig. 10.
In the context of the discussion on the $b\to s\ell\ell$ anomalies, multiple
groups [77, 78, 79] have performed averages for these results, which are also
shown in Fig. 10. The recent update [76] from CMS is not yet included in these
averages. For the analysis below, we choose the result from Ref. [79]:

$\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})=(2.85_{-0.31}^{+0.34})\times
10^{-9}\>.$ (102)

This average differs from the SM prediction (99) by one standard deviation,
and from the SM predictions (98) and (100) by 2.4 standard deviations.
Individual measurements of the $B_{d}^{0}\to\mu^{+}\mu^{-}$ branching fraction
are not yet significantly different from zero. The LHCb [69], ATLAS [74] and
CMS [75] results therefore only provide an upper limit, with the most
stringent upper bound given by [69]

$\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})<0.26\times 10^{-9}\>.$ (103)

Nonetheless, based on the combined analysis with $B_{s}^{0}\to\mu^{+}\mu^{-}$,
Ref. [78] does give the average

$\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})=(0.56\pm 0.70)\times 10^{-10}\>.$
(104)

This result is still compatible with zero within one standard deviation.

###  5.2 Determining NP in $\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$

NP can modify the $B_{s}^{0}\to\mu^{+}\mu^{-}$ branching fraction through
pseudo-scalar (P) or scalar (S) contributions, or through
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing. The measured
$B_{s}^{0}\to\mu^{+}\mu^{-}$ branching fraction is given by

$\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})=\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})^{\text{SM}}\times\frac{1+\mathcal{A}^{\mu\mu}_{\Delta\Gamma_{s}}y_{s}}{1+y_{s}}\left(|P_{\mu\mu}^{s}|^{2}+|S_{\mu\mu}^{s}|^{2}\right)\>,$
(105)

where $\mathcal{A}^{\mu\mu}_{\Delta\Gamma_{s}}$ depends on
$P_{\mu\mu}^{s}\equiv|P_{\mu\mu}^{s}|e^{i\varphi_{P}}$,
$S_{\mu\mu}^{s}\equiv|S_{\mu\mu}^{s}|e^{i\varphi_{S}}$ and
$\phi_{s}^{\text{NP}}$ as

$\mathcal{A}_{\Delta\Gamma}^{\mu\mu}=\frac{|P_{\mu\mu}^{s}|^{2}\cos(2\varphi_{P}-\phi_{s}^{\text{NP}})-|S_{\mu\mu}^{s}|^{2}\cos(2\varphi_{S}-\phi_{s}^{\text{NP}})}{|P_{\mu\mu}^{s}|^{2}+|S_{\mu\mu}^{s}|^{2}}\>.$
(106)

In the SM, we have

$P^{s,\text{SM}}_{\mu\mu}=1\>,\qquad S^{s,\text{SM}}_{\mu\mu}=0\>.$ (107)

Comparing the experimental average (102) with the SM predictions (98)-(100),
we can constrain the parameters $|P_{\mu\mu}^{s}|$ and $|S_{\mu\mu}^{s}|$, as
proposed in Ref. [72]. For the scenario where the NP phases for the pseudo-
scalar and scalar contributions are zero, i.e. $\varphi_{P}=\varphi_{S}=0$,
the results are shown in Fig. 11.

Figure 11: Two-dimensional confidence regions for the constraints from
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})$ and $R_{s\mu}$ on $|P_{\mu\mu}^{s}|$ and
$|S_{\mu\mu}^{s}|$, assuming $\varphi_{P}=\varphi_{S}=0$. Left: Inclusive
scenario. Right: Exclusive scenario. Bottom: Hybrid scenario.

The dependence of the NP searches with
$\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ on the CKM matrix element
$|V_{cb}|$ and the UT apex, which is clearly visible when comparing the blue
contours in Fig. 11, can be minimised by constructing the ratio with the
$B_{s}$ mass difference $\Delta m_{s}$ [21, 23, 22]:

$\mathcal{R}_{s\mu}\equiv\left|\frac{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}{\Delta
m_{s}}\right|\>.$ (108)

At lowest order, the SM contribution is given by

$\mathcal{R}_{s\mu}^{\text{SM}}=\frac{\tau_{B_{s}}}{1-y_{s}}\frac{3G_{\text{F}}^{2}m_{W}^{2}\sin^{4}\theta_{W}}{4\pi^{3}}\frac{|C_{10}^{\text{SM}}|^{2}}{S_{0}(x_{t})\eta_{2B}\hat{B}_{B_{s}}}m_{\mu}^{2}\sqrt{1-4\frac{m_{\mu}^{2}}{m_{B_{s}}^{2}}}\>,$
(109)

resulting in the prediction

$\mathcal{R}_{s\mu}^{\text{SM}}=(2.22\pm 0.10)\times 10^{-10}\>\text{ps}\>.$
(110)

Including NLO corrections Ref. [22] finds a similar result

$\mathcal{R}_{s\mu}^{\text{SM}}=(2.042_{-0.058}^{+0.083})\times
10^{-10}\>\text{ps}\>,$ (111)

which differs from ours due to the choice for the bag parameter
$\hat{B}_{B_{s}}$. Also here, the choices made for the input parameters still
have a bigger impact than the NLO corrections.

Including NP effects in both $\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ and
$\Delta m_{s}$, the expression for the ratio $\mathcal{R}_{s\mu}$ is
generalised as follows:

$\mathcal{R}_{s\mu}=\mathcal{R}_{s\mu}^{\text{SM}}\times\frac{1+\mathcal{A}^{\mu\mu}_{\Delta\Gamma_{s}}y_{s}}{1+y_{s}}\frac{|P_{\mu\mu}^{s}|^{2}+|S_{\mu\mu}^{s}|^{2}}{\sqrt{1+2\kappa_{s}\cos\sigma_{s}+\kappa_{s}^{2}}}\>.$
(112)

It is important to stress that although the leading dependence on the CKM
matrix elements drops out in Eq. (109) because we take the ratio with respect
to $\Delta m_{s}$, the ratio in Eq. (112) introduces a dependence on the CKM
matrix elements through the NP parameters $(\kappa_{s},\sigma_{s})$. Hence,
when taking into account NP effects in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing,
one should stay careful with the choice of the input of the CKM matrix
elements.

Combining the experimental results in Eqs. (102) and (60), we get

$\mathcal{R}_{s\mu}=(1.60\pm 0.19)\times 10^{-10}\>.$ (113)

Comparing this with the SM value, we can again obtain contours in the
$|P^{s}_{\mu\mu}|$–$|S^{s}_{\mu\mu}|$ plane. These are also shown in Fig. 11
for the scenario where the NP phases for the pseudo-scalar and scalar
contributions are zero.

###  5.3 Using $\bar{\mathcal{B}}(B_{q}\to\mu^{+}\mu^{-})$ to Extract $R_{t}$

In Section 3.2, we introduced an alternative determination of the SM UT apex
based on the sides $R_{b}$ and $R_{t}$, avoiding the use of the angle
$\gamma$. This second UT fit scenario becomes interesting when searching for
potential NP contributions in the measurements of $\gamma$. However, its main
limitation is the calculation of the side $R_{t}$ based on the ratio of CKM
matrix elements $|V_{td}/V_{ts}|$, which is susceptible to NP contributions as
well. In Eq. (66), this ratio was determined using the
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing parameters. Here we would like to explore
the determination of the ratio $|V_{td}/V_{ts}|$ using the rare decays
$B_{q}^{0}\to\mu^{+}\mu^{-}$. In the SM and FUNP scenario, the result would be
consistent with the strategy based on $\Delta m_{s}$ and $\Delta m_{d}$, which
can be tested experimentally.

Based on the SM expressions in Eqs. (93) and (94), the ratio of branching
fractions between $B_{d}^{0}\to\mu^{+}\mu^{-}$ and
$B_{s}^{0}\to\mu^{+}\mu^{-}$ is given by

$\left.\frac{\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})}{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}\right|_{\text{SM}}=\lambda^{2}R_{t}^{2}\left[1+\frac{\lambda^{2}}{2}\left(1-2\bar{\rho}\right)\right]^{2}\frac{\sqrt{m_{B_{d}}^{2}-4m_{\mu}^{2}}}{\sqrt{m_{B_{s}}^{2}-4m_{\mu}^{2}}}\frac{f_{B_{d}}^{2}}{f_{B_{s}}^{2}}\frac{1-y_{s}}{1-y_{d}}\>,$
(114)

where we used Eq. (65) to rewrite the ratio $|V_{td}/V_{ts}|$ in terms of the
UT side $R_{t}$. Combining this with the SM values for $R_{t}$ given in Eqs.
(70)–(72), which are calculated using $R_{b}$ and $\gamma$, and using
$f_{B_{s}}/f_{B_{d}}=1.209\pm 0.005$ [43], we get the predictions:

$\displaystyle\text{Incl, }K\ell 3$
$\displaystyle\left.\frac{\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})}{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}\right|_{\text{SM}}$
$\displaystyle=0.0282\pm 0.0019\>,$ (115) $\displaystyle\text{Excl, }K\ell 3$
$\displaystyle\left.\frac{\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})}{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}\right|_{\text{SM}}$
$\displaystyle=0.0281\pm 0.0018\>,$ (116) $\displaystyle\text{Hybrid, }K\ell
3$
$\displaystyle\left.\frac{\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})}{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}\right|_{\text{SM}}$
$\displaystyle=0.0281\pm 0.0017\>.$ (117)

The difference between the three $R_{b}$ scenarios is almost negligible given
the current precision. This suggests a high-precision measurement of the
$B_{q}^{0}\to\mu^{+}\mu^{-}$ branching fractions will be necessary to
constrain $R_{t}$ through this method.

In contrast, the current experimental result for the ratio of branching
fractions, based on Ref. [78] that gives the average for
$\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})$ in Eq. (104), is

$\frac{\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})}{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}=0.019\pm
0.024\>,$ (118)

and has an uncertainty of 125%. This corresponds to a value for the UT side
$R_{t}$ of

$R_{t}=0.77\pm 0.48\>.$ (119)

In comparison with the determinations in Eqs. (73)–(78) the uncertainty is a
factor 20 to 40 larger. However, with the third data taking period of the LHC
just started, more precise measurements of the $B_{q}^{0}\to\mu^{+}\mu^{-}$
branching fractions can be expected in the near future, thereby making this an
interesting option to explore, and useful addition to the study of NP effects
in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing.

##  6 What Will the Future Bring?

###  6.1 Improved Precision on the NP Parameters $\kappa_{q}$ and $\sigma_{q}$

The NP searches in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing are limited by the SM
predictions for $\phi_{d}$, $\Delta m_{d}$ and $\Delta m_{s}$. Only the SM
value for $\phi_{s}$ is still more precisely known than its experimental
measurement. The value of $\phi_{d}^{\text{SM}}$ is limited by our knowledge
of the UT apex, and in particular $R_{b}$, while the largest uncertainty in
the calculations of $\Delta m_{d}^{\text{SM}}$ and $\Delta m_{s}^{\text{SM}}$
are due to the non-perturbative parameters. Improving these calculations has
turned out to be difficult, and the associated time-scale unpredictable. We
therefore refrain from making specific estimates for the future. Instead we
will only illustrate the impact that improvements on the UT apex, lattice
calculations and the CKM matrix element $|V_{cb}|$ have on the NP parameters
$\kappa_{q}$ and $\sigma_{q}$, and explore their relative importance. Table 6
numerically compares the current precision on $\kappa_{q}$ and $\sigma_{q}$
with what is achievable assuming a hypothetical reduction of the uncertainty
on each of these three inputs by 50%. For the two inputs that have the largest
impact, the two-dimensional confidence level contours are compared in Fig. 12.

For the $B_{s}$-meson system, the precision on $\kappa_{s}$ and $\sigma_{s}$
is limited by the uncertainty on the lattice calculations. Improvements to
these calculations will reduce the allowed confidence region the most, and are
therefore eagerly anticipated. In contrast, the impact from improvements in
our knowledge of the UT apex is essentially negligible. This is in particular
the case for $\phi_{s}$. In Ref. [80], discussing the prospects for LHCb
Upgrade II, an experimental precision on the measurement of
$\phi_{s}^{\text{eff}}$ from $B_{s}^{0}\to J/\psi\phi$ of 4 mrad was
forecasted. This is still a factor 2 bigger than the current precision on
$\phi_{s}^{\text{SM}}$, given in Eqs. (45)–(47). The search for NP in
$\phi_{s}$ will thus remain limited by the experimental data, irrespective of
our knowledge on the UT apex. This conclusion was already noted in Ref. [6],
and can be easily understood: Both $\phi_{s}^{\text{SM}}$ and $\Delta
m_{s}^{\text{SM}}$ only depend on $\bar{\rho}$ and $\bar{\eta}$ at next-to-
leading order, and their contributions are suppressed by a factor
$\lambda^{2}$ compared to the leading order terms, while the lattice
parameters appear quadratically in the leading order expression of $\Delta
m_{s}^{\text{SM}}$. Also $|V_{cb}|$ appears quadratically in $\Delta
m_{s}^{\text{SM}}$, but the impact of an improved measurement remains small as
long as the lattice results dominate the error budget.

The future prospects for finding NP in the $B_{s}$-meson system are highly
dependent on the assumptions made. From the considered scenarios shown in Fig.
12, the exclusive scenario assuming a 50% improvement from lattice appears
most exciting. If this were to be realised, we could claim NP in
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing with a significance of more than 5
standard deviations. In the other scenarios, the situation will be more
challenging.

For the $B_{d}$-meson system, the situation is very different. Here,
improvements in the determination of the UT apex and the reduction of the
uncertainties from lattice calculations have an equally big impact on the
allowed confidence regions for $\kappa_{d}$ and $\sigma_{d}$. In particular,
the limitation due to the UT apex is surprising, as was also pointed out in
Ref. [36] in the context of NP searches in $\phi_{d}$. The uncertainties on
the SM predictions for $\phi_{d}$ in Eqs. (41)–(43) are about 40% larger than
the uncertainty on the experimental measurement in Eq. (10). This situation
will not significantly change in the near future, as we can illustrate using
the forecasts for the LHCb Upgrade II [80]. LHCb expects to reduce the
uncertainty on $R_{b}$ to 1% and on $\gamma$ to $0.35^{\circ}$ by the end of
the HL-LHC programme. This corresponds to a precision on
$\phi_{d}^{\text{SM}}$ of $0.48^{\circ}$. By contrast, LHCb expects to be able
to measure the mixing-induced CP asymmetry of the decay $B_{d}^{0}\to J/\psi
K_{\text{S}}^{0}$ with an uncertainty of only 0.003. This corresponds to a
precision on $\phi_{d}^{\text{eff}}$ of $0.24^{\circ}$, and can be translated
to a measurement of $\phi_{d}$ after taking into account the contributions
from higher-order penguin topologies. In Ref. [36], we argued that it is
possible to control these penguin corrections with similar experimental
precision, thus resulting in an estimated precision on $\phi_{d}$ of
$0.34^{\circ}$. Thus at the end of the HL-LHC programme, the uncertainty on
the SM prediction will still be 40% larger than the experimental measurement.
Measurements from the Belle II experiment will help to reduce this difference,
but cannot completely bridge the gap in precision. This makes it all the more
important to critically analyse the UT fit and carefully select its inputs.
The determination of the SM values of $\bar{\rho}$ and $\bar{\eta}$ should
remain a high priority as it impacts the vast majority of NP searches, ranging
from $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing to rare decays like
$B_{q}^{0}\to\mu^{+}\mu^{-}$.

The future prospects for finding NP in the $B_{d}$-meson system are again
highly dependent on the assumptions made. From the considered scenarios shown
in Fig. 12, the inclusive scenario assuming a 50% improvement on the UT apex
stands out. Here, we could find hints for NP in $B_{d}^{0}$–$\bar{B}_{d}^{0}$
mixing with a significance of more than 3 standard deviations. This is less
promising than the $B_{s}$-meson system. The difference is mainly due to the
smaller value we find for $\kappa_{d}$ in the current data.

As a final remark, note that improvements on the non-perturbative parameters
can also be achieved by combining the lattice calculations with LQSR results.
However, dedicated averages like those provided by FLAG [43] for the lattice
calculations are still missing. Nonetheless, such averages, if available, can
have a significant impact on the study of $B$-meson decays, both on the NP
searches in $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing, and in the determination of
$|V_{ub}|$ and $|V_{cb}|$, and thus the UT apex.

Fit | Scenario | $\kappa_{q}$ | $\sigma(\kappa_{q})$ | $\sigma_{q}$ | $\sigma(\sigma_{q})$
---|---|---|---|---|---
$B_{d}$ I3 | Current | 0.121 | $-0.055/+0.056$ | $261^{\circ}$ | $-35^{\circ}/+37^{\circ}$
| 50% improvement $|V_{cb}|$ | 0.121 | $-0.055/+0.056$ | $261^{\circ}$ | $-35^{\circ}/+37^{\circ}$
| 50% improvement lattice | 0.121 | $-0.055/+0.056$ | $261^{\circ}$ | $-31^{\circ}/+31^{\circ}$
| 50% improvement UT apex | 0.121 | $-0.036/+0.037$ | $261^{\circ}$ | $-28^{\circ}/+31^{\circ}$
$B_{d}$ E3 | Current | 0.156 | $-0.084/+0.093$ | $347^{\circ}$ | $-25^{\circ}/+21^{\circ}$
| 50% improvement $|V_{cb}|$ | 0.156 | $-0.081/+0.089$ | $347^{\circ}$ | $-24^{\circ}/+21^{\circ}$
| 50% improvement lattice | 0.156 | $-0.069/+0.074$ | $347^{\circ}$ | $-22^{\circ}/+21^{\circ}$
| 50% improvement UT apex | 0.156 | $-0.071/+0.078$ | $347^{\circ}$ | $-15^{\circ}/+10^{\circ}$
$B_{d}$ H3 | Current | 0.031 | $-0.031/+0.057$ | $104^{\circ}$ | $-104^{\circ}/+256^{\circ}$
| 50% improvement $|V_{cb}|$ | 0.031 | $-0.031/+0.056$ | $104^{\circ}$ | $-104^{\circ}/+256^{\circ}$
| 50% improvement lattice | 0.031 | $-0.031/+0.050$ | $104^{\circ}$ | $-104^{\circ}/+256^{\circ}$
| 50% improvement UT apex | 0.031 | $-0.031/+0.046$ | $104^{\circ}$ | $-104^{\circ}/+256^{\circ}$
$B_{s}$ I3 | Current | 0.045 | $-0.033/+0.048$ | $312^{\circ}$ | $-77^{\circ}/+37^{\circ}$
| 50% improvement $|V_{cb}|$ | 0.045 | $-0.032/+0.046$ | $312^{\circ}$ | $-73^{\circ}/+37^{\circ}$
| 50% improvement lattice | 0.045 | $-0.028/+0.032$ | $312^{\circ}$ | $-47^{\circ}/+36^{\circ}$
| 50% improvement UT apex | 0.045 | $-0.033/+0.048$ | $312^{\circ}$ | $-77^{\circ}/+37^{\circ}$
$B_{s}$ E3 | Current | 0.205 | $-0.059/+0.064$ | $347.6^{\circ}$ | $-9.8^{\circ}/+8.5^{\circ}$
| 50% improvement $|V_{cb}|$ | 0.205 | $-0.053/+0.058$ | $347.6^{\circ}$ | $-9.4^{\circ}/+8.5^{\circ}$
| 50% improvement lattice | 0.205 | $-0.040/+0.042$ | $347.6^{\circ}$ | $-8.8^{\circ}/+8.5^{\circ}$
| 50% improvement UT apex | 0.205 | $-0.059/+0.064$ | $347.6^{\circ}$ | $-9.7^{\circ}/+8.5^{\circ}$
$B_{s}$ H3 | Current | 0.053 | $-0.034/+0.046$ | $309^{\circ}$ | $-65^{\circ}/+34^{\circ}$
| 50% improvement $|V_{cb}|$ | 0.053 | $-0.033/+0.044$ | $309^{\circ}$ | $-61^{\circ}/+33^{\circ}$
| 50% improvement lattice | 0.053 | $-0.029/+0.030$ | $309^{\circ}$ | $-39^{\circ}/+31^{\circ}$
| 50% improvement UT apex | 0.053 | $-0.034/+0.046$ | $309^{\circ}$ | $-64^{\circ}/+34^{\circ}$
Table 6: Numerical comparison of the NP parameters $\kappa_{q}$ and
$\sigma_{q}$ assuming a hypothetical reduction of 50% in the uncertainty on
the CKM matrix element $|V_{cb}|$, the lattice calculations, or the UT apex.
The fit results are given separately for the inclusive (I3), exclusive (E3)
and hybrid (H3) $R_{b}$ scenario with $|V_{us}|$ determined with the $K\ell 3$
approach.

Figure 12: Comparison of the two-dimensional confidence regions for the NP
parameters $\kappa_{d}$ and $\sigma_{d}$ (Left), and $\kappa_{s}$ and
$\sigma_{s}$ (Right) assuming a hypothetical reduction of 50% in the
uncertainty on the CKM matrix element $|V_{cb}|$, the lattice calculations, or
the UT apex. Top: Inclusive scenario. Middle: Exclusive scenario. Bottom:
Hybrid scenario.

###  6.2 NP in $\gamma$

As discussed in Section 2.1, the two determinations of $\gamma$, based on the
analysis of $B\to DK$ decays and based on the isospin analysis of the
$B\to\pi\pi$, $\rho\pi$, $\rho\rho$ decays, are in good agreement with each
another. Given the very different origins of these measurements, this is a
non-trivial result. In the future, improved precision on the input
measurements to these analyses could result in discrepancies between the two
$\gamma$ determinations. Averaging over both results, as we have done in
Section 2.1, would then no longer be justified, and the SM determination of
the UT apex would have to be revisited. As it is a priori unclear which of the
two measurements, if not both, is affected by NP contributions, independent
information from additional observables will then be necessary to resolve the
situation. At the same time, this will also provide exciting new opportunities
to search for NP, both in $\gamma$ itself and in $B_{q}^{0}$–$\bar{B}_{q}^{0}$
mixing, which is strongly correlated with the coordinates of the UT apex.

In Sections 3.2 and 5.3, we have explored alternative scenarios that do not
rely on the measurements of $\gamma$. However, all these scenarios have a
common bottleneck: the SM determination of $R_{t}$. Both the determination
based on the measurements of $\Delta m_{d}$ and $\Delta m_{s}$, and the
determination based on the branching fractions of the rare decays
$B_{q}^{0}\to\mu^{+}\mu^{-}$ require us to make additional assumptions about
FUNP, such that the ratios between the $B_{d}$ and $B_{s}$ observables remains
SM-like. Although this may be sufficient to find evidence of NP, it does not
allow for a general model-independent approach and exploring multiple
strategies simultaneously may be necessary to understand the origin of the NP
contributions.

###  6.3 Opportunities for $\bar{\mathcal{B}}(B_{q}\to\mu^{+}\mu^{-})$

The ratio of branching fractions between $B_{d}^{0}\to\mu^{+}\mu^{-}$ and
$B_{s}^{0}\to\mu^{+}\mu^{-}$ provides an alternative opportunity to determine
the UT side $R_{t}$. However, as the precision on $R_{t}^{2}$ scales directly
with the precision on the ratio of branching fractions, the current
experimental uncertainties are still too large to already explore this option
in detail. But we can illustrate the potential for the LHC upgrade programme.
The LHCb collaboration expects to be able to reduce the uncertainty on the
ratio of branching fractions to 34% by 2025 and to 10% by the end of the HL-
LHC era [80]. Assuming the SM value from Eqs. (76)–(78), this results in the
prospects

$R_{t}=0.93\pm 0.16\>\text{(2025)}\>,\qquad R_{t}=0.931\pm
0.047\>\text{(Upgrade II)}\>,$ (120)

where the uncertainties are still fully dominated by the branching fraction
measurements. All other uncertainties combined only contribute at the 1% level
and can thus safely be ignored. In comparison with the determinations in Eqs.
(73)–(78), the LHCb Upgrade II prospects are still a factor 2 to 4 larger, as
illustrated in Fig. 13. We thus would require a precision on the ratio of
branching fractions of 5% or better to match the current determination based
on $R_{b}$ and $\gamma$, and 2.5% or better to match the current determination
based on $R_{b}$, $\Delta m_{d}$ and $\Delta m_{s}$.

Figure 13: Comparison between the current SM determination of the UT apex and
the expected constraint for the LHCb Upgrade II on $R_{t}$ from the ratio of
branching fractions of $B_{d}^{0}\to\mu^{+}\mu^{-}$ and
$B_{s}^{0}\to\mu^{+}\mu^{-}$. Left: Inclusive scenario. Right: Exclusive
scenario. Bottom: Hybrid scenario.

A second future application for the ratio of branching fractions between
$B_{d}^{0}\to\mu^{+}\mu^{-}$ and $B_{s}^{0}\to\mu^{+}\mu^{-}$ is the quantity
[81]

$\displaystyle U_{\mu\mu}^{ds}$
$\displaystyle\equiv\sqrt{\frac{|P_{\mu\mu}^{d}|^{2}+|S_{\mu\mu}^{d}|^{2}}{|P_{\mu\mu}^{s}|^{2}+|S_{\mu\mu}^{s}|^{2}}}\>,$
(121)
$\displaystyle=\left[\frac{\tau_{B_{s}}}{\tau_{B_{d}}}\frac{1-y_{d}^{2}}{1-y_{s}^{2}}\frac{1+\mathcal{A}_{\Delta\Gamma}^{d}y_{d}}{1+\mathcal{A}_{\Delta\Gamma}^{s}y_{s}}\frac{\sqrt{m_{B_{s}}^{2}-4m_{\mu}^{2}}}{\sqrt{m_{B_{d}}^{2}-4m_{\mu}^{2}}}\left(\frac{f_{B_{s}}}{f_{B_{d}}}\right)^{2}\left|\frac{V_{ts}}{V_{td}}\right|^{2}\frac{\bar{\mathcal{B}}(B_{d}\to\mu^{+}\mu^{-})}{\bar{\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})}\right]^{1/2}\>.$
(122)

It has the advantage over the individual $B_{d}^{0}\to\mu^{+}\mu^{-}$ and
$B_{s}^{0}\to\mu^{+}\mu^{-}$ branching fraction measurements that common
parameters and uncertainties drop out in the ratio. Therefore it provides a
more powerful test of the SM, where $U_{\mu\mu}^{ds}=1$.

##  7 Conclusions

In this paper, we have presented a comprehensive study of NP in
$B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. Our main goal is to determine the
allowed parameter space for NP, having a critical look at analyses of the
determination of the UT apex, which is needed as input for the corresponding
SM predictions. In particular, we explore the impact of the discrepancies
between inclusive and exclusive measurements of the CKM matrix elements
$|V_{ub}|$ and $|V_{cb}|$, which allow us to fix the side $R_{b}$ of the UT.
Specifically, we perform separate analyses for the inclusive and exclusive
determinations, respectively, and for a hybrid scenario, which combines the
exclusive $|V_{ub}|$ with the inclusive $|V_{cb}|$ value. Generally, we find
sizeable differences between these combinations, demonstrating that it would
be very desirable and important to finally resolve the tensions between
inclusive and exclusive determinations of the CKM parameters.

Combining the $R_{b}$ side with information on the angle $\gamma$, we can
determine the apex of the UT. This angle is usually determined through $B\to
DK$ decays, which proceed only via tree topologies. In our analysis, we
utilise another avenue to obtain information on $\gamma$, which is provided by
$B\to\pi\pi$, $\rho\pi$, $\rho\rho$ modes and isospin relations allowing us to
control penguin uncertainties. Typically, these measurements are interpreted
in terms of the UT angle $\alpha$. However, using information on the mixing
phase $\phi_{d}$, they actually determine $\gamma$. Interestingly, we find
full agreement between these two determinations of $\gamma$, which is non-
trivial in view of the different dynamics governing the corresponding decays.
Within the SM, such an agreement is expected. Should a discrepancy arise in
the future, it would indicate NP contributions to the corresponding decay
amplitude, thereby requiring a more involved analysis. In view of the current
agreement, we make an average between the two values and assume that this is
the SM value of $\gamma$.

Using this information on $R_{b}$ and $\gamma$, we perform determinations of
the UT apex for the various determinations of the CKM matrix elements
$|V_{ub}|$ and $|V_{cb}|$. Furthermore, within the SM, indirect CP violation
in the neutral kaon system, described by the observable $|\varepsilon_{K}|$,
allows us to obtain another constraint on the UT apex, given by a hyperbola in
the $\bar{\rho}$–$\bar{\eta}$ plane. This curve depends strongly on
$|V_{cb}|$, thereby suggesting a probe to distinguish between the inclusive
and exclusive determinations of this CKM element. In our analysis, we find
indeed this feature: the exclusive case gives sizeable tension, the inclusive
case is more consistent, while the hybrid scenario results in the most
compatible picture with the UT apex. Due to this strong dependence, it will be
interesting to see how the interplay between the $|\varepsilon_{K}|$ hyperbola
and the UT apex will evolve in the future.

The determination of the UT from $R_{b}$ and $\gamma$ plays a key role for the
SM predictions of the $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing parameters. It is
important to stress that the corresponding UT analysis does not depend on any
information from $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing. We find that the choice
of input parameters – inclusive, exclusive or hybrid – significantly impacts
the picture emerging for the NP parameters in $B^{0}_{q}$–$\bar{B}^{0}_{q}$
mixing.

First, we constrain in a model-independent way the NP parameters separately
for $B_{s}$ and $B_{d}$ systems. The strongest evidence for NP would arise in
the $B_{s}$ system for the exclusive CKM matrix elements, corresponding to a
NP contribution of about $20\%$ with a significance of 3.5 standard
deviations.

Next, we consider a flavour universal scenario with equal NP contributions to
the $B_{d}$ and $B_{s}$ systems. In such a situation, the NP effects would
cancel in the ratio $\Delta m_{d}/\Delta m_{s}$. Consequently, we could use it
to fix the side $R_{t}$ of the UT. Combining this side with $R_{b}$ allows us
then to determine the UT apex without any information on $\gamma$. We find a
picture very similar to the situation in the fit for the $B_{s}$ system,
implying that this system dominates the analysis. While in the exclusive case
the fit regions overlap well with those for the general NP analysis for the
$B_{s}$ system, we find that the shapes of the contours for $B_{d}$ and
$B_{s}$ systems are very different for the inclusive and hybrid scenarios,
thereby indicating that the flavour universal scenario might not be realised
in nature. However, in view of the current uncertainties, we cannot rule out
such kind of NP.

Finally, we go beyond the flavour universal scenario and relax the
corresponding assumption when determining the NP parameters. While we still
use the ratio $\Delta m_{d}/\Delta m_{s}$ to determine the UT apex together
with the information on $R_{b}$, we extract $\kappa_{q}$, $\sigma_{q}$
separately using the individual measurements of $\Delta m_{q}$ and $\phi_{q}$.
We find that the solutions for the $B_{d}$ and $B_{s}$ systems are
statistically compatible with each other, but do not look the same.
Consequently, the data favour a NP scenario which is not flavour universal in
this analysis, although it cannot be excluded with the current precision. Let
us finally emphasize that this analysis does not require information on
$\gamma$. It will be interesting to see how the picture for the NP parameters
will evolve in the future.

Our results for the NP parameters in the $B_{s}$ system can be applied to the
analysis of the rare leptonic decay $B^{0}_{s}\to\mu^{+}\mu^{-}$. The
corresponding SM branching ratio, using unitarity of the CKM matrix, depends
on the CKM element $|V_{cb}|$. We have explored the impact of different
determinations of this quantity, finding that it plays the key role for the
uncertainty of the SM branching ratio. This dependence can be eliminated
through the ratio of the $B^{0}_{s}\to\mu^{+}\mu^{-}$ branching ratio with the
mass difference $\Delta m_{s}$. We generalise this approach by including
possible NP contributions to $B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing through our
analysis and constrain the space for NP in $B_{s}^{0}\to\mu^{+}\mu^{-}$
through the current measurement of the branching ratio. Following these lines,
the CKM factors and the UT apex enter only through the analysis of NP in
$B_{s}^{0}$–$\bar{B}_{s}^{0}$ mixing. Another interesting mode is the
$B^{0}_{d}\to\mu^{+}\mu^{-}$ decay which has not yet been observed. In the SM,
the ratio of the branching ratios of $B^{0}_{d}\to\mu^{+}\mu^{-}$ and
$B^{0}_{s}\to\mu^{+}\mu^{-}$ allows us to determine the $R_{t}$ side of UT. We
have illustrated this interesting option for a future scenario, comparing it
with the current picture of the UT.

We have extrapolated our analysis of the NP parameters of
$B^{0}_{q}$–$\bar{B}^{0}_{q}$ mixing to the future high-precision era by
considering various projections for the CKM matrix elements, the UT apex fit
and lattice calculations. For the $B_{d}$ system we find that the SM
determination of the UT apex represents a key limitation for the NP searches.
On the other hand, the $B_{s}$ system has no such limitation and looks
particularly interesting for revealing new sources of NP. In the future,
studies of $B_{q}^{0}$–$\bar{B}_{q}^{0}$ mixing will remain a central element
of the analysis to further constrain NP. It will be exciting to obtain the
full picture providing links to other probes such as anomalies in semileptonic
rare $B$ decays.

## Acknowledgements

This research has been supported by the Netherlands Organisation for
Scientific Research (NWO). PvV acknowledges support from the DFG through the
Emmy Noether research project 400570283, and through the German-Israeli
Project Cooperation (DIP).

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