Nikhef-2021-017

Revealing New Physics in $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ Decays

Robert Fleischer a,b and Eleftheria Malami a

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

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

The $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ system offers a determination of the
Unitarity Triangle angle $\gamma$. Intrigued by an LHCb analysis showing a
surprisingly large result in tension with information on the Unitarity
Triangle and other $\gamma$ measurements, we make a transparent study of the
measured observables, confirming the LHCb picture. The corresponding $\gamma$
puzzle at the $3\sigma$ level would require CP-violating contributions of New
Physics, which should also manifest themselves in the corresponding decay
branching ratios. Indeed, we find that the rates of the individual
$B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ channels show puzzling patterns, in
accordance with similar decays, with tensions up to $4.8\sigma$, thereby
making the situation much more exciting. We present a formalism to include
New-Physics effects in a model-independent way and apply it to the data to
constrain the corresponding parameters. Interestingly, new contributions of
moderate size could accommodate the data. Utilising this formalism in the
future high-precision $B$ physics era may allow us to finally establish new
sources of CP violation.

September 2022

##  1 Introduction

Decays of $B$ mesons provide a wide spectrum of probes for testing the quark-
flavour sector of the Standard Model (SM). A particularly interesting aspect
is given by the subtle difference between weak interactions of particles and
their antiparticles, which is described by the phenomenon of CP violation.
Here C stands for charge conjugation and P refers to parity, i.e. space
inversion. In the SM, CP violation is described through a complex phase in the
Cabibbo–Kobayashi–Maskawa (CKM) matrix. Since the SM cannot explain the
matter–antimatter asymmetry of the Universe, failing by many order of
magnitude, new sources of CP violation are suggested.

The decays $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ and $B^{0}_{s}\to
D_{s}^{+}K^{-}$ (with their CP conjugates) offer a powerful probe for testing
the SM description of CP violation [1, 2, 3]. Due to quantum-mechanical
$B^{0}_{s}$–$\bar{B^{0}_{s}}$ oscillations, interference effects are generated
between these decay channels, leading to the following time-dependent rate
asymmetry:

$\frac{\Gamma(B^{0}_{s}(t)\to D_{s}^{+}K^{-})-\Gamma(\bar{B}^{0}_{s}(t)\to
D_{s}^{+}K^{-})}{\Gamma(B^{0}_{s}(t)\to
D_{s}^{+}K^{-})+\Gamma(\bar{B}^{0}_{s}(t)\to
D_{s}^{+}K^{-})}=\frac{{C}\,\cos(\Delta M_{s}\,t)+{S}\,\sin(\Delta
M_{s}\,t)}{\cosh(y_{s}\,t/\tau_{B_{s}})+{\cal
A}_{\Delta\Gamma}\,\sinh(y_{s}\,t/\tau_{B_{s}})};$ (1)

an analogous expression holds for the CP-conjugate $D_{s}^{-}K^{+}$ final
state, where $C$, $S$ and ${\cal A}_{\Delta\Gamma}$ are replaced by
$\overline{C}$, $\overline{S}$ and $\overline{{\cal A}}_{\Delta\Gamma}$,
respectively.

The $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ and $B^{0}_{s}\to D_{s}^{+}K^{-}$
channels arise from $b\to c\bar{u}s$ and $\bar{b}\to\bar{u}c\bar{s}$ quark-
level processes, respectively. In the SM, the observables of Eq. (1) and its
CP conjugate allow a theoretically clean determination of the angle $\gamma$
of the Unitarity Triangle (UT) of the CKM matrix [1, 2, 3]. Performing these
measurements and assuming certain SM relations, LHCb [4] has reported the
result $\gamma=\left(128^{+17}_{-22}\right)^{\circ}$ modulo $180^{\circ}$.
This finding is puzzling as global SM analyses of the UT give values around
$70^{\circ}$ [5, 6, 7, 8]. This regime is also consistent with a recent
simultaneous analysis of various $B$-meson decays by the LHCb collaboration
[9]. However, these modes have dynamics different from the $B^{0}_{s}\to
D_{s}^{\mp}K^{\pm}$ channels and show sensitivity on $\gamma$ through very
different interference effects.

Intrigued by the challenge of interpreting this result, we have a transparent
look at the determination of $\gamma$ and the associated parameters,
confirming the LHCb picture and excluding the modulo $180^{\circ}$ ambiguity
[10]. If confirmed through future more precise measurements, this result could
only be explained through CP-violating contributions of New Physics (NP) at
the $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ decay amplitude level, and should then
also leave imprints on the corresponding branching ratios. Consequently, we
move on to extract the individual $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ and
$B^{0}_{s}\to D_{s}^{+}K^{-}$ branching ratios and compare them with the SM
picture, developing a method to minimise dependences on non-perturbative
hadronic and CKM parameters. We in fact arrive at yet another puzzling
situation, complementing the $\gamma$ measurement. The situation is even more
exciting, as we observe similar patterns also in branching ratios for other
$B$ decays with similar dynamics. Surprisingly small branching ratios in the
latter modes have already received attention [11, 12], also within NP
interpretations [13, 14, 15].

In view of these findings, we present a generalized description of the
$B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ decays and corresponding analysis of CP
violation to include NP contributions in a model-independent way. Applying
this formalism to the current data, we calculate correlations between the NP
parameters, involving in particular new sources of CP violation. This compact
paper complements a much more detailed discussion given in Ref. [10].

##  2 Standard Model Framework

The interference effects between $B^{0}_{s}$ or $\bar{B}^{0}_{s}$ mesons
decaying into the final states $D_{s}^{+}K^{-}$ and $D_{s}^{-}K^{+}$ caused by
the $B^{0}_{s}$–$\bar{B}^{0}_{s}$ oscillations are described by physical
observables $\xi$ and $\bar{\xi}$, respectively. In the SM, their product
provides a theoretically clean expression [2]:

${\xi}\times\bar{\xi}=e^{-i2(\phi_{s}+\gamma)},$ (2)

where $\phi_{s}$ is the CP-violating $B^{0}_{s}$–$\bar{B}^{0}_{s}$ mixing
phase. It should be noted in particular that non-perturbative hadronic matrix
elements cancel in (2). Since $\xi$ and $\bar{\xi}$ can be determined from
$C$, $S$, ${\cal A}_{\Delta\Gamma}$ and $\overline{C}$, $\overline{S}$,
$\overline{{\cal A}}_{\Delta\Gamma}$, respectively [3], and $\phi_{s}$ through
measurements of CP violation in $B^{0}_{s}\rightarrow J/\psi\phi$ [16], this
relation allows a clean extraction of $\gamma$.

Furthermore, the following relations hold in the SM framework [2]:

$|\bar{\xi}|={1}/{|\xi|}=\sqrt{(1+C)/{(1-C)}},\ \ \ \ \ C+\overline{C}=0,$ (3)

which were assumed by the LHCb collaboration [4]. Performing a sophisticated
fit to their data, LHCb found

$|\bar{\xi}|=0.37^{+0.10}_{-0.09}$ (4)

with the following results [4]:

$\phi_{s}+\gamma=\left(126^{+17}_{-22}\right)^{\circ},\ \
\delta_{s}=(-2^{+13}_{-14})^{\circ}\ \ {\text{[modulo $180^{\circ}$],}}$ (5)

were $\delta_{s}$ describes the CP-conserving strong phase difference between
the $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ and $B^{0}_{s}\to D_{s}^{+}K^{-}$
amplitudes. Here we have used $\phi_{s}=(-1.7\pm 1.9)^{\circ}$, which was
employed in the LHCb analysis, to convert $\gamma$ into $\phi_{s}+\gamma$.
Using the updated result $\phi_{s}=\left(-5^{+1.6}_{-1.5}\right)^{\circ}$,
which includes also penguin corrections in $B^{0}_{s}\to J/\psi\phi$ modes
[16], we obtain

$\gamma=\left(131^{+17}_{-22}\right)^{\circ}.$ (6)

In view of the tension of this value with the SM and the complex LHCb
analysis, it is crucial to transparently understand the situation. How can we
achieve that? Using (3) with the measured value of $C$, we find
$|\bar{\xi}|=0.40\pm 0.13$, which is in excellent agreement with (4).
Introducing the combinations

$\langle S\rangle_{\pm}\equiv(\overline{S}\pm
S)/2,\quad\langle\mathcal{A}_{\Delta\Gamma}\rangle_{+}\equiv(\overline{{\cal
A}}_{\Delta\Gamma}+{\cal A}_{\Delta\Gamma})/2,$ (7)

we obtain the following relations [2, 3, 10]:

$\tan(\phi_{s}+\gamma)=-\langle
S\rangle_{+}/\langle\mathcal{A}_{\Delta\Gamma}\rangle_{+}=-1.45^{+0.73}_{-2.76}$
(8) $\tan\delta_{s}=\langle
S\rangle_{-}/\langle\mathcal{A}_{\Delta\Gamma}\rangle_{+}=0.04^{+0.70}_{-0.40},$
(9)

where the numerical values correspond to the LHCb measurements of the
corresponding observables, yielding
$\phi_{s}+\gamma=(125^{+18}_{-22})^{\circ}$ with
$\delta_{s}=(2^{+34}_{-22})^{\circ}$. Here we have excluded the solutions
modulo $180^{\circ}$ as they would be in huge conflict with factorization
predicting $\delta_{s}\sim 0^{\circ}$ [10]. Consequently, we find excellent
agreement between this simple – but transparent – analysis and the complex
LHCb fit.

The result for $\gamma$ in (6) is much larger than the $70^{\circ}$ regime,
and shows a discrepancy at the $3\sigma$ level. It is important to stress that
this intriguing tension could not come from any long distance effects, as the
$\gamma$ determination is theoretically clean in SM. Should this puzzle remain
once the experimental picture sharpens further, it would require new sources
of CP violation. Since the experimental value of $\phi_{s}$ used in the
analysis includes possible CP-violating NP effects in
$B^{0}_{s}$–$\bar{B}^{0}_{s}$ mixing, new contributions entering directly at
the decay amplitudes of the $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ system would be
required.

Such NP effects should manifest themselves also in the corresponding branching
ratios. Let us therefore focus on these observables next. Due to the time-
dependent $B^{0}_{s}$–$\bar{B}^{0}_{s}$ oscillations, we have to distinguish
between time-integrated “experimental” branching ratios [17] and their
“theoretical” counterparts where such mixing effects are “switched off” [3,
18]. Moreover, we have to disentangle the interference effects between the two
decay paths that arise from the $B^{0}_{s}$–$\bar{B}^{0}_{s}$ oscillations. We
obtain the following expressions for for the theoretical branching ratios
[10]:

$\displaystyle\mathcal{B}(B^{0}_{s}\to D_{s}^{+}K^{-})_{\text{th}}$
$\displaystyle=2\left(\frac{1}{1+|\xi|^{2}}\right)\mathcal{B}_{\text{th}}$
(10) $\displaystyle\mathcal{B}(\bar{B}^{0}_{s}\to D_{s}^{+}K^{-})_{\text{th}}$
$\displaystyle=2\left(\frac{|\xi|^{2}}{1+|\xi|^{2}}\right)\mathcal{B}_{\text{th}}=|\xi|^{2}\mathcal{B}(B^{0}_{s}\to
D_{s}^{+}K^{-})_{\text{th}}.$ (11)

Analogous expressions hold for the branching ratios of the $B^{0}_{s}$ and
$\bar{B}^{0}_{s}$ decays into the final state $D_{s}^{-}K^{+}$. Unfortunately,
separate measurements of the experimental branching ratios for these final
states are not available, just the average

$\langle\mathcal{B}_{\text{exp}}\rangle\equiv\frac{1}{2}\left(\mathcal{B}_{\text{exp}}+\bar{\mathcal{B}}_{\text{exp}}\right)\equiv\frac{1}{2}\,\mathcal{B}^{\text{exp}}_{\Sigma}$
(12)

with $\mathcal{B}^{\text{exp}}_{\Sigma}=(2.27\pm 0.19)\times 10^{-4}$ [6].
Assuming SM expressions for the decay amplitudes, we have [2, 3]

$\mathcal{B}_{\text{th}}=\bar{\mathcal{B}}_{\text{th}}=\left[\frac{1-y_{s}^{2}}{1+y_{s}\langle{\cal
A}_{\Delta\Gamma}\rangle_{+}}\right]\langle\mathcal{B}_{\text{exp}}\rangle$
(13)

with $y_{s}=0.062\pm 0.004$ [6] and
$\langle\mathcal{A}_{\Delta\Gamma}\rangle_{+}=0.35\pm 0.23$ [10]. Finally, we
obtain [10]:

$\displaystyle\mathcal{B}(B^{0}_{s}\rightarrow D_{s}^{+}K^{-})_{\rm th}$
$\displaystyle=(0.26\pm 0.12)\times 10^{-4}$ (14)
$\displaystyle\mathcal{B}(\bar{B}^{0}_{s}\rightarrow D_{s}^{+}K^{-})_{\rm th}$
$\displaystyle=(1.94\pm 0.21)\times 10^{-4}.$ (15)

For the calculation of these branching ratios and the underlying decay
amplitudes, factorization provides the theoretical framework. Here the
corresponding hadronic matrix elements of four-quark operators entering the
low-energy effective Hamiltonian are factorized into the product of the matrix
elements of quark currents.

The decay ${\bar{B}^{0}_{s}\rightarrow D_{s}^{+}K^{-}}$ originating from $b\to
c\bar{u}s$ processes is a prime example where “QCD factorization” is expected
to work excellently for the colour-allowed tree topologies [19, 20, 21, 22,
23]. We obtain the following amplitude:

$A_{D_{s}^{+}K^{-}}^{\text{SM}}=\frac{G_{\rm
F}}{\sqrt{2}}V_{us}^{\ast}V_{cb}f_{K}F_{0}^{B_{s}\rightarrow
D_{s}}(m_{K}^{2})(m_{B_{s}}^{2}-m_{D_{s}}^{2})a_{\rm 1\,eff}^{D_{s}K},$ (16)

where $G_{\rm F}$ is the Fermi constant, $V_{us}^{\ast}V_{cb}$ a factor of CKM
matrix elements, $f_{K}$ the kaon decay constant and $F_{0}^{B_{s}\rightarrow
D_{s}}(m_{K}^{2})$ a form factor parametrising the hadronic $b\rightarrow c$
quark-current matrix element. The parameter

$a_{\rm
1\,eff}^{D_{s}K}=a_{1}^{D_{s}K}\left(1+\frac{E_{D_{s}K}}{T_{D_{s}K}}\right)$
(17)

describes the deviation from naive factorization: $a_{1}^{D_{s}K}$
characterises non-factorisable effects entering the colour-allowed tree
amplitude $T_{D_{s}K}$, whereas $E_{D_{s}K}$ denotes non-factorisable exchange
topologies. The current state-of-the-art results within QCD factorization are
found as $|a_{1}|\approx 1.07$ with a quasi-universal behaviour [12, 24], with
uncertainties at the percent level. Recently, even QED effects have been
studied [25], which are small and fully included within the uncertainties.
Specifically, for the $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ mode, we have

$|a_{1}^{D_{s}K}|=1.07\pm 0.02.$ (18)

Utilizing experimental data, as discussed in detail in Ref. [10], the exchange
topology is constrained through

$r_{E}^{D_{s}K}\equiv\left|1+\frac{E_{D_{s}K}}{T_{D_{s}K}}\right|=1.00\pm
0.08.$ (19)

It is important to stress that no anomalous behaviour of this and similar
exchange topologies that could be caused by large rescattering or other non-
factorizable effects is indicated by the data [10], as was also found and
emphasized in Ref. [11].

Our next step is to extract the $|a_{1}|$ parameters from the data in the
cleanest possible way and to compare them with the theoretical SM predictions.
In this respect, semileptonic decays provide a very useful tool [19, 11, 22].
In the case of the $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ channel, we have the
partner decay $\bar{B}^{0}_{s}\rightarrow D_{s}^{+}\ell^{-}\bar{\nu}_{\ell}$,
and introduce the ratio

$R_{D_{s}^{+}K^{-}}\equiv\frac{\mathcal{B}(\bar{B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-})_{\rm
th}}{{\mathrm{d}\mathcal{B}\left(\bar{B}^{0}_{s}\rightarrow
D_{s}^{+}\ell^{-}\bar{\nu}_{\ell}\right)/{\mathrm{d}q^{2}}}|_{q^{2}=m_{K}^{2}}},$
(20)

which takes the form

$R_{D_{s}^{+}K^{-}}=6\pi^{2}f_{K}^{2}|V_{us}|^{2}|a_{\rm
1\,eff}^{D_{s}K}|^{2}X_{D_{s}K}$ (21)

with

$X_{D_{s}K}\equiv{\Phi_{\text{ph}}}\left[{F_{0}^{B_{s}\rightarrow
D_{s}}(m_{K}^{2})}/{F_{1}^{B_{s}\rightarrow D_{s}}(m_{K}^{2})}\right]^{2},$
(22)

where $\Phi_{\text{ph}}$ is a phase-space factor which is equal to 1 with
excellent precision [10]. The CKM matrix element $|V_{cb}|$ cancels in
$R_{D_{s}^{+}K^{-}}$. Moreover, due to the normalization condition
$F_{0}^{B_{s}\rightarrow D_{s}}(0)=F_{1}^{B_{s}\rightarrow D_{s}}(0)$, we have
an essentially negligible impact of the non-perturbative hadronic form factors
[26, 27, 28]. Using a recent LHCb measurement [29] of the differential rate of
the $\bar{B}^{0}_{s}\rightarrow D_{s}^{+}\ell^{-}\bar{\nu}_{\ell}$ channel and
(15) with (19) yields

$|a_{\rm 1}^{D_{s}K}|=0.82\pm 0.11.$ (23)

We observe that this result is in tension with the theoretical prediction in
(18) at the $2.2\,\sigma$ level. Consequently, we indeed find another tension
at the decay amplitude level with respect to the SM, as we would expect in
view of the puzzling result for $\gamma$. This exciting situation is further
strengthened by the fact that a similar pattern of the $|a_{1}|$ parameters –
with surprisingly small values – arises also for other $B_{(s)}$ decays with
similar dynamics. We have extracted these quantities from the data in an
analogous way with semileptonic decay information [10], and show the results
in the left panel of Fig. 1. Here $\bar{B}^{0}_{d}\to D_{d}^{+}K^{-}$ stands
out, showing even a discrepancy of $4.8\,\sigma$. Puzzlingly small branching
ratios for this channel and the $\bar{B}^{0}_{d}\to D_{d}^{+}\pi^{-}$,
$\bar{B}^{0}_{s}\to D_{s}^{+}\pi^{-}$ modes were also pointed out in Refs.
[11, 12].

Let us next have a look at the $\bar{B}^{0}_{s}\rightarrow K^{+}D_{s}^{-}$
decay. The amplitude of this channel, which is caused by $b\to u\bar{c}s$
processes, can be expressed in a way similar to the $b\to c\bar{u}s$ case, as
we discuss in detail in [10]. In analogy to the ${\bar{B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-}}$ mode, this transition is also a colour-allowed tree decay.
However, as the roles of the heavy $c$ and light $u$ quarks are interchanged,
the heavy-quark arguments which can be used to prove factorization up to tiny
corrections for the $b\to c$ modes do not apply, and there may be larger non-
factorizable effects. In view of this less favourable theoretical situation,
we will use the following range as a reference for our analysis:

$|a_{1}^{KD_{s}}|=1.1\pm 0.1.$ (24)

Here we have followed the QCD renormalisation group analysis in Ref. [30] as
guidance, where $|a_{1}|=1.01\pm 0.02$ was found for the variation of the
global $|a_{1}|$ parameter for colour-allowed tree decays with respect to
variations of the renormalization scale and schemes. In Eq. (24), we allow for
a five times larger uncertainty [10]. Interestingly, the experimental value of
$\delta_{s}$ in (5) is found in excellent agreement with factorization [2].
Since this strong phase difference characterizes the interference between the
$b\to c\bar{u}s$ and $b\to u\bar{c}s$ decay paths, it supports factorisation –
where such phases vanish – also in the $b\to u\bar{c}s$ channel.

In analogy Eq. (20), the partner decay for the clean extraction of the
$|a_{1}^{KD_{s}}|$ parameter is $\bar{B}^{0}_{s}\rightarrow
K^{+}\ell\bar{\nu}_{\ell}$. Although this channel has been observed by LHCb
[31], a measurement of the differential rate has not yet been reported.
Consequently, we have applied the $SU(3)$ flavor symmetry and have utilised
the $\bar{B}^{0}_{d}\rightarrow\pi^{+}\ell^{-}{\bar{\nu}_{\ell}}$ mode, for
which we do have information from the BaBar and Belle collaborations [5, 6].
We find

$|a_{\rm 1}^{KD_{s}}|=0.77\pm 0.21,$ (25)

as discussed in more detail in [10]. As in the $b\to c\bar{u}s$ case, this
result favours again a value smaller than our theoretical reference in Eq.
(24). The result is illustrated in the panel on the right-hand side of Fig. 1,
where we include also the $\bar{B}^{0}_{d}\to\pi^{+}D_{s}^{-}$ mode, which
differs only through the spectator quark from the $\bar{B}^{0}_{s}\rightarrow
K^{+}D_{s}^{-}$ channel. The current uncertainties are too large to draw
further conclusions on these modes.

Figure 1: Experimental and theoretical SM values of the $|a_{1}|$ parameters
for various decay processes. The left panel illustrates decays which are
caused by $b\to c\bar{u}s$ and $b\to c\bar{u}d$ processes while the right
panel shows decays originating from $b\to u\bar{c}s$ transitions.

Within the SM, universal power-suppressed corrections of order $\Lambda_{\rm
QCD}/m_{b}$ could in principle lead to a suppression of the $|a_{1}|$
parameters [24]. However, such effects would not allow us to accommodate the
puzzling result for $\gamma$ arising from the CP-violating observables of the
$B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ system. This phenomenon would require new
sources for CP violation. The exciting possibility of NP effects in non-
leptonic tree-level decays of $B$ mesons was discussed in Refs. [32, 33], and
models for physics beyond the SM addressing the puzzles of the small branching
ratios were studied in Refs. [13, 14]. We consider these first models, which
face challenges from direct NP searches at ATLAS and CMS [15], as interesting
illustrations of specific scenarios. In the remainder of this paper, we will
work with a model-independent parametrization of NP contributions.

##  3 New Physics Analysis

Let us now extend the analysis of the $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ system
to include NP effects. As we have already noted, the NP contributions to
accommodate the puzzling measurement of $\gamma$ and the $|a_{1}|$ values
would have to enter at the decay amplitude level. For the $b\to c\bar{u}s$
quark-level transition, we generalise the decay amplitude as

$A(\bar{B}^{0}_{s}\rightarrow D_{s}^{+}K^{-})=A(\bar{B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-})_{{\text{SM}}}\left[1+\bar{\rho}\,e^{i\bar{\delta}}e^{+i\bar{\varphi}}\right]$
(26)

with the NP parameter

$\bar{\rho}\,e^{i\bar{\delta}}e^{i\bar{\varphi}}\equiv\frac{A(\bar{B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-})_{{\text{NP}}}}{A(\bar{B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-})_{{\text{SM}}}},$ (27)

where $\bar{\varphi}$ and $\bar{\delta}$ denote CP-violating and CP-conserving
phases, respectively. A similar expression can be written for the
$\bar{b}\to\bar{u}c\bar{s}$ transition, with parameters $\rho$, $\varphi$ and
$\delta$.

It is useful to introduce the following quantities [10]:

$\bar{b}\equiv\frac{\langle
R_{D_{s}K}\rangle}{6\pi^{2}f_{K}^{2}|V_{us}|^{2}|a_{\rm
1\,eff}^{D_{s}K}|^{2}X_{D_{s}K}}=1+2\,\bar{\rho}\cos\bar{\delta}\cos\bar{\varphi}+\bar{\rho}^{2}$
(28) $b\equiv\frac{\langle
R_{KD_{s}}\rangle}{6\pi^{2}f_{D_{s}}^{2}|V_{cs}|^{2}|a_{\rm
1\,eff}^{KD_{s}}|^{2}X_{KD_{s}}}=1+2\,\rho\cos\delta\cos\varphi+\rho^{2}.$
(29)

Using the experimental results for the decay ratios and the theoretical
predictions for the $|a_{\rm 1}|$ with (19) yields

$\bar{b}=0.58\pm 0.16,\ \ \ \ \ {b}=0.50\pm 0.26.$ (30)

In the SM, these observables would be equal to 1. The numerical values reflect
the puzzling patterns in Fig. 1.

For the observable $\bar{\xi}$, we obtain the generalisation

$\bar{\xi}=\bar{\xi}_{\text{SM}}\left[\frac{1+{\rho}\,e^{i{\delta}}e^{+i{\varphi}}}{1+\bar{\rho}\,e^{i\bar{\delta}}e^{-i\bar{\varphi}}}\right]=-|\bar{\xi}|e^{+i\delta_{s}}e^{-i(\phi_{s}+\gamma)}e^{i\Delta\bar{\varphi}}.$
(31)

Similarly, we rewrite $\xi$ with $\Delta{\varphi}$, where we interchange the
NP parameters $\bar{\rho}$, $\bar{\delta}$, $\bar{\varphi}$ and ${\rho}$,
${\delta}$, ${\varphi}$.

Figure 2: Correlations for the central values of the current data in the
$\bar{\varphi}$–$\varphi$ plane (left) and the $\bar{\rho}$–$\rho$ plane
(right) of of NP parameters.

The product $\xi\times\bar{\xi}$ introduced in Eq. (2), which plays the key
role for CP violation in the SM, is generalized as follows:

$\xi\times\bar{\xi}=e^{-i2(\phi_{s}+\gamma)}\Biggl{[}\frac{1+{\rho}\,e^{i{\delta}}e^{+i{\varphi}}}{1+{\rho}\,e^{i{\delta}}e^{-i{\varphi}}}\Biggr{]}\Biggl{[}\frac{1+\bar{\rho}\,e^{i\bar{\delta}}e^{+i\bar{\varphi}}}{1+\bar{\rho}\,e^{i\bar{\delta}}e^{-i\bar{\varphi}}}\Biggr{]}.$
(32)

In contrast to the SM, NP may generate non-vanishing direct CP asymmetries:

${\cal A}^{\rm dir}_{\rm CP}\equiv\frac{|A({B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-})|^{2}-|A(\bar{B}^{0}_{s}\rightarrow
D_{s}^{-}K^{+})|^{2}}{|A({B}^{0}_{s}\rightarrow
D_{s}^{+}K^{-})|^{2}+|A(\bar{B}^{0}_{s}\rightarrow
D_{s}^{-}K^{+})|^{2}}=\frac{2\,\rho\sin\delta\sin\varphi}{1+2\,\rho\cos\delta\cos\varphi+\rho^{2}}.$
(33)

An analogous expression holds for the CP-conjugate asymmetry $\bar{{\cal
A}}^{\rm dir}_{\rm CP}$, involving $\bar{\rho}$ with $\bar{\delta}$ and
${\bar{\varphi}}$. We may then write the first ratio entering (32) as

$\frac{1+{\rho}\,e^{i{\delta}}e^{+i{\varphi}}}{1+{\rho}\,e^{i{\delta}}e^{-i{\varphi}}}=e^{-i\Delta\Phi}\sqrt{\frac{1-{{\cal
A}}^{\rm dir}_{\rm CP}}{1+{{\cal A}}^{\rm dir}_{\rm CP}}},$ (34)

where

$\tan\Delta\Phi=-\left[\frac{2\rho\cos\delta\sin\varphi+\rho^{2}\sin
2\varphi}{1+2\rho\cos\delta\cos\varphi+\rho^{2}\cos 2\varphi}\right];$ (35)

the second ratio takes a similar form, involving $\bar{\rho}$, $\bar{\varphi}$
and $\bar{{\cal A}}^{\rm dir}_{\rm CP}$ with a phase $\Delta\bar{\Phi}$. We
then obtain

$\left|\xi\times\bar{\xi}\right|^{2}=\left[\frac{1-{{\cal A}}^{\rm dir}_{\rm
CP}}{1+{{\cal A}}^{\rm dir}_{\rm CP}}\right]\left[\frac{1-{\bar{\cal A}}^{\rm
dir}_{\rm CP}}{1+{\bar{\cal A}}^{\rm dir}_{\rm CP}}\right]=1+\epsilon,$ (36)

where

$-\frac{1}{2}\,\epsilon=\frac{C+\bar{C}}{\left(1+C\right)\left(1+\bar{C}\right)}={\cal
A}^{\rm dir}_{\rm CP}+{\bar{\cal A}}^{\rm dir}_{\rm CP}+{\cal O}(({\cal
A}^{\rm dir}_{\rm CP})^{2}),$ (37)

generalising the SM relations in (3). Finally, we arrive at

$\xi\times\bar{\xi}=\sqrt{1-2\left[\frac{C+\bar{C}}{\left(1+C\right)\left(1+\bar{C}\right)}\right]}e^{-i\left[2(\phi_{s}+\gamma_{\rm
eff})\right]},$ (38)

where the UT angle $\gamma$ enters as the “effective” angle

$\gamma_{\rm
eff}\equiv\gamma+\frac{1}{2}\left(\Delta\Phi+\Delta\bar{\Phi}\right)=\gamma-\frac{1}{2}\left(\Delta\varphi+\Delta\bar{\varphi}\right).$
(39)

Consequently, (6) actually corresponds to $\gamma_{\rm eff}$.

Note that in combined fits to the data for various $B$ decays to extract
$\gamma$, such as in Ref. [9], NP effects may average out to some extend,
thereby yielding an effective angle with NP contributions which – in contrast
to (39) – cannot transparently be quantified. It will rather be crucial to
search for patterns in the individual $\gamma$ determinations, aiming at the
highest precision.

Figure 3: Correlations in the $\bar{\rho}$–$\rho$ plane of NP parameters
including uncertainties.

Let us now apply our formalism to the current data. In order to be consistent
with the LHCb assumption, we set the strong phases $\delta$ and $\bar{\delta}$
to $0^{\circ}$. This implies vanishing direct CP asymmetries, in agreement
with $B\rightarrow DK$ data within the uncertainties [6], and the following
relation [10]:

$\Delta\varphi=\Delta\bar{\varphi}=\gamma-\gamma_{\rm eff}=-(61\pm
20)^{\circ},$ (40)

where

$\tan\Delta\varphi=\frac{\rho\sin\varphi+\bar{\rho}\sin\bar{\varphi}+\bar{\rho}\rho\sin(\bar{\varphi}+\varphi)}{1+\rho\cos\varphi+\bar{\rho}\cos\bar{\varphi}+\bar{\rho}\rho\cos(\bar{\varphi}+\varphi)}.$
(41)

The numerical value refers to $\gamma=(70\pm 7)^{\circ}$, which is consistent
with UT analyses, and the result in (6).

In order to convert the measured observables into constraints on the NP
parameters, we first employ $\bar{b}$ and $b$ to determine $\bar{\rho}$ and
${\rho}$ as functions of $\bar{\varphi}$ and $\varphi$, respectively. Using
then (6), we may calculate ${\varphi}$ as a function of $\bar{\varphi}$,
fixing a contour in the ${\varphi}$–$\bar{\varphi}$ plane. Finally, using
again $\bar{\rho}(\bar{\varphi})$ and ${\rho}(\varphi)$ allows us to calculate
a correlation in the $\bar{\rho}$–$\rho$ plane, where each point is linked
with $\bar{\varphi}$ and $\varphi$.

In Fig. 2, we show the corresponding correlations for the central values of
the current data. In Fig. 3, we show the impact of the uncertainties of the
input quantities $\Delta\varphi$, $b$ and $\bar{b}$, varying them separately.
We can nicely see that the SM point corresponding to the origin in the
$\bar{\rho}$–$\rho$ plane is excluded, and notice that NP contributions with
CP-violating phases are simultaneously required in the $b\to c\bar{u}s$ and
$\bar{b}\to\bar{u}c\bar{s}$ decay paths. Interestingly, we can describe the
measurements with NP contributions as small as about 30% of the SM amplitudes.

##  4 Concluding Remarks

Figure 4: Illustration of the two puzzles and the strategy we presented in
this paper.

The $\bar{B}^{0}_{s}\to D_{s}^{+}K^{-}$ and $\bar{B}^{0}_{s}\to
D_{s}^{-}K^{+}$ decays with their CP conjugates are key players in the testing
of the SM. We have demonstrated that the intriguing picture arising from the
measured CP-violating observables, which results in a tension with the SM at
the $3\sigma$ level, is complemented by a puzzling pattern of the individual
branching ratios. The latter finding is actually in accordance with
measurements of rates of $B_{(s)}$ modes with similar dynamics, where we find
tensions with up to $4.8\sigma$ significance. While the $\gamma$ measurement
cannot be explained through non-factorizable effects, the branching ratio
puzzles could in principle be accommodated through such contributions. We
would like to stress that the experimental result of the strong phase
difference $\delta_{s}$ between the $b\to c\bar{u}s$ and $b\to u\bar{c}s$
decay paths is in excellent agreement with factorization. It is exciting to
reveal and link these new puzzles, complementing indications of NP in other
corners of the flavour sector, where currently rare decays arising from $b\to
s\ell^{+}\ell^{-}$ quark-level processes involving also leptons are in the
spotlight [34].

We have presented a model-independent description of the $B^{0}_{s}\to
D_{s}^{\mp}K^{\pm}$ system to include NP effects and give a generalized
expression for an effective angle $\gamma$ in terms of NP parameters that can
be determined in a theoretically clean way from the measured CP-violating
observables. Utilizing furthermore the information from the decay branching
ratios, the NP parameters can be constrained, as we have illustrated for the
current data. In Fig. 4, we give a schematic overview. In the future high-
precision $B$ physics era, this formalism can be fully exploited through more
sophisticated experimental analyses, searching for direct CP violation and
measuring the experimental $B_{s}$ branching ratios for the separate
$D_{s}^{+}K^{-}$ and $D_{s}^{-}K^{+}$ final states. The measurement of the
differential $\bar{B}^{0}_{s}\rightarrow K^{+}\ell\bar{\nu}_{\ell}$ rate would
be another important ingredient to complement the analysis. The NP parameters
resulting from this analysis will serve as benchmarks for the model building
community.

It will be exciting to monitor the evolution of the data and the sharper
picture emerging from the application of our strategy. The central question is
whether these studies will finally allow us to establish the presence of new
CP-violating contributions to $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ decays,
thereby raising the question of whether such new sources of CP violation could
open a window to understand the matter–antimatter asymmetry of the Universe.

## Acknowledgements

We would like to thank Ruben Jaarsma, Philine van Vliet and Kristof De Bruyn
for useful discussions. This research has been supported by the Netherlands
Organisation for Scientific Research (NWO).

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