# CKM 2018
Summary of Working Group 4:
Mixing and mixing-related CP violation in the B system
$\Delta M$, $\Delta\Gamma$, $\phi_{s}$, $\phi_{1}/\beta$, $\phi_{2}/\alpha$,
$\phi_{3}/\gamma$

Sevda Esen
Nikhef, NL
E-mail<EMAIL_ADDRESS>

Alexander Lenz
IPPP, Durham University, UK
E-mail<EMAIL_ADDRESS>
### Abstract

We summarise the excellent talks of Working group 4 at the 10th CKM Workshop
in Heidelberg taking place from 17th till 21st September 2018.

##  1 B mixing

###  1.1 Introduction

In the Standard Model (SM) mixing of neutral $B_{q}$-mesons is governed by the
famous box-diagrams, with internal $W$-bosons and internal $up$-, $charm$\-
and $top$-quarks, see Fig. 1 for the case of $B_{s}$-mesons - for a more
detailed introduction into $B$-mixing, see e.g. [1].

Figure 1:  Standard Model diagrams for the transition between $B_{s}$ and
$\bar{B}_{s}$ mesons.

The contribution of internal on-shell particles (only the $charm$\- and the
$up$-quark can contribute) is denoted by $\Gamma_{12}^{q}$; the contribution
of internal off-shell particles (all depicted particles can contribute) is
denoted by $M_{12}^{q}$. In the $B$-system there are simple relations111This
holds not for $D$-mixing, see e.g. [2, 3, 4]. between $\Gamma_{12}^{q}$,
$M_{12}^{q}$ and the physical observables mass difference $\Delta M_{q}$, the
decay rate difference $\Delta\Gamma_{q}$ and the semi-leptonic asymmetries
$a_{sl}^{q}$:

$\Delta M_{q}\approx 2\left|M_{12}^{q}\right|\,,\hskip
28.45274pt\Delta\Gamma_{q}\approx
2\left|\Gamma_{12}^{q}\right|\cos\phi_{12}^{q}\,,\hskip
28.45274pta_{sl}^{q}\approx\left|\frac{\Gamma_{12}^{q}}{M_{12}^{q}}\right|\sin\phi_{12}^{q}\,,$
(1)

with $\phi_{12}^{q}=\arg(-M_{12}^{q}/\Gamma_{12}^{q})$.

###  1.2 The mass difference $\Delta M_{q}$

Manuel Schiller [5] presented an overview of recent measurements for the mass
difference done by LHCb. These values are known with a high precision [6]
(based on the individual measurements [7, 8, 9, 10, 11]):

$\displaystyle\Delta M_{d}^{\rm Exp.}$ $\displaystyle=$
$\displaystyle(0.5065\pm 0.0019)\,\mbox{ps}^{-1}\,.$ (2) $\displaystyle\Delta
M_{s}^{\rm Exp.}$ $\displaystyle=$ $\displaystyle(17.757\pm
0.021)\,\mbox{ps}^{-1}\,.$ (3)

The calculation of $M_{12}^{q}$ gives

$\displaystyle M_{12}^{q}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}}{12\pi^{2}}\lambda_{t}^{2}M_{W}^{2}S_{0}(x_{t})\hat{\eta}_{B}Bf_{B_{q}}^{2}M_{B_{q}}\,,$
(4)

where $\lambda_{t}$ denotes the CKM elements $V_{tq}^{*}V_{tb}$ and the Inami-
Lim function $S_{0}$ [12] contains the result of the 1-loop box diagram in the
SM and perturbative 2-loop QCD corrections [13] are compressed in the factor
$\hat{\eta}_{B}$. The bag parameter $B$ and the decay constant $f_{B_{q}}$
quantify the hadronic contribution to $B$-mixing, the uncertainties of their
numerical values make up the by far biggest uncertainty in the SM prediction
of the mass difference. In the SM only one dimension 6 operator with a V-A
Dirac structure arises. Depending on this input we get a range of predictions
for the mass difference in the $B_{s}$-system as indicated in Table 1, taken
from [14].

Source | $f_{B_{s}}\sqrt{\hat{B}}$ | $\Delta M_{s}^{\rm SM}$
---|---|---
HPQCD14 [15] | $(247\pm 12)\;{\rm MeV}$ | $(16.2\pm 1.7)\,\mbox{ps}^{-1}$
HQET-SR [16] | $(261\pm 8)\;{\rm MeV}$ | $(18.1\pm 1.1)\,\mbox{ps}^{-1}$
ETMC13 [17] | $(262\pm 10)\;{\rm MeV}$ | $(18.3\pm 1.5)\,\mbox{ps}^{-1}$
HPQCD09 [18] = FLAG13 [19] | $(266\pm 18)\;{\rm MeV}$ | $(18.9\pm 2.6)\,\mbox{ps}^{-1}$
FLAG17 [20] | $(274\pm 8)\;{\rm MeV}$ | $(20.01\pm 1.25)\,\mbox{ps}^{-1}$
FNAL/MILC 16 [21] | $(274.6\pm 4)\;{\rm MeV}$ | $(20.1\pm 0.7)\,\mbox{ps}^{-1}$
HPQCD06 [22] | $(281\pm 20)\;{\rm MeV}$ | $(21.0\pm 3.0)\,\mbox{ps}^{-1}$
RBC/UKQCD14 [23] | $(290\pm 20)\;{\rm MeV}$ | $(22.4\pm 3.4)\,\mbox{ps}^{-1}$
Fermilab11 [24] | $(291\pm 18)\;{\rm MeV}$ | $(22.6\pm 2.8)\,\mbox{ps}^{-1}$
Table 1: List of predictions for the non-perturbative parameter
$f_{B_{s}}\sqrt{\hat{B}}$ and the corresponding SM prediction for $\Delta
M_{s}$. The current FLAG average is dominated by the FERMILAB/MILC value from
2016.

Lucia di Luzio [25] pointed out the importance of the precise value of SM
prediction for the mass difference and a strict control of the corresponding
uncertainties. Lepto-quarks and $Z^{\prime}$ models are popular explanations
of the B anomalies222Due to time and space restrictions we will not attempt to
cite the numerous relevant papers in that field.; these new models would also
affect $B_{s}$-mixing - in the case of $Z^{\prime}$ models already at tree-
level. In Fig. 2 (from [14]) we show the allowed parameter range for a
$Z^{\prime}$ model: in order to explain e.g. $R_{K^{(*)}}$ the mass of the
$Z^{\prime}$ and the coupling to the $b$\- and $s$-quark should lie within the
black parabola-like shape (the 1 sigma bound is a solid line, the 2 sigma one
a dotted line). Taking the FLAG inputs from 2013 for the mass difference one
can exclude the blue region. Taking the new FLAG average, that is dominated by
the 2016 FNAL/MILC value we are left with the red exclusion region and almost
all of the possible parameter space of the $Z^{\prime}$ model is excluded.
Thus the exact numerical value of the hadronic input for $B_{s}$-mixing has
severe consequences for beyond the Standard Model (BSM) physics.

Figure 2:  Allowed parameter space of $Z^{\prime}$ models that try to explain
the B anomalies.

Thomas Mannel [26], Thomas Rauh [27] and Aida El-Khadra [28] presented in
detail three non-perturbative determinations of the hadronic input for
$B$-mixing. Thomas Mannel described [26] a series of papers of the Siegen
group where perturbative three-loop corrections to a HQET sum rule for the V-A
mixing operator for the $B_{d}$ mesons were determined [29]. Later on the
matching of the HQET result to QCD was performed at 2-loop [30, 31].
Interestingly it turns out that HQET sum rule results are competitive in
precision with the most modern lattice determinations of the Bag parameter.
Naively one might expect a precision of about $20\%$ \- $30\%$ for sum rule
determinations, but in this case it turns out that one can write down a sum
rule for $B-1$ [29, 16]. Since the value of $B$ is very close to one, a result
of e.g. $B-1\approx-0.10\pm 0.02...0.03$ transforms in a high precision for
the Bag parameter $B\approx 0.9\pm 0.02...0.03$. Unfortunately the mass
difference $\Delta M_{q}$ is proportional to $f_{B_{q}}^{2}B$ and there is no
similar trick known for the decay constant $f_{B_{q}}$ \- so one has to use
for the sum rule value of the mass difference either a precise determination
of the decay constant from lattice simulations or a much more uncertain value
from sum rules [32]. In Table 1 the precise lattice value for the decay
constant from [33] was used.
Aida El-Khadra discussed [28] in detail the lattice result and the
corresponding error budget of the FNAL/MILC collaboration [21], which
dominates the current FLAG average. Their $N_{f}=2+1$ determination of
$f_{B_{s}}\sqrt{B}$ achieves an impressive precision of about $3\%$. The
precision of elder published lattice results is limited by the use of a static
action or by smaller ensembles, larger lattice spacings, fewer
configurations,..; but there are plans from RBC/UKQCD (a first paper appeared
since the CKM conference [34]), HPQCD, ETM,.. to improve on that and thus
cross-check the value of FNAL/MILC, that has severe consequences on BSM
effects. In addition to the SM V-A operator for the mass difference, FNAL/MILC
determined also the four remaining $\Delta B=2$ dimension 6 operators of the
so-called SUSY basis. Two of these new operators contribute to the SM
prediction of $\Delta\Gamma$, that will be discussed below, all four of the
new operators can appear in BSM contributions to the mass difference. Two of
Aida’s main conclusions were:

  * •

Since the bag parameter $B$ is a derived quantity, it does not profit from
correlations. Therefore it is better for phenomenological applications to use
directly the FLAG averages for the matrix elements, i.e. $f_{B_{s}}\sqrt{B}$,
instead of reconstructing them from the averages of bag parameters and decay
constants seperatly.

  * •

There are no roadblocks to increasing the precision of the hadronic $B$-mixing
parameter on the lattice. With current technology a precision of $1-2\%$ for
the dimension 6 operators seems to be achievable.

Thomas Rauh presented [27] an extension of the sum rule calculation of the
Siegen group to all dimension 6 operators for mixing (see Fig. 3) and
lifetimes of heavy mesons [16] \- this will be discussed below - and the
determination of $m_{s}$-corrections [35] to the Siegen result, i.e. the first
sum rule determination of the Bag parameter $B$ for $B_{s}$ mixing.

Figure 3:  Comparison of different determinations of the $B_{s}$ mixing Bag
parameters. The grey areas denote the preliminary values from [35]. The red
value (GKMP’16) is the sum rule result from Siegen for $B_{d}$ mixing [26] and
the green symbols denote the FNAL/MILC results.

Compared to the FNAL/MILC result the new sum rule result for $B_{s}$ mesons
seems to prefer lower values for the Bag parameter, thus being more consistent
with the experimental result for $\Delta M_{s}$ and leading to weaker
constraints for BSM models. Here some words of caution are in order: as Arifa
pointed out FNAL/MILC so far did not perform a direct determination of the Bag
parameter - the FNAL/MILC value shown in Fig. 3 is a derived quantity with
external input for the decay constant, the sum rule determination is still
preliminary and both approaches still overlap within their uncertainties.

###  1.3 The decay rate difference $\Delta\Gamma_{s}$

The measurement of the decay rate difference was discussed in the talks of
Manuel Schiller (LHCb) [5], Pavel Reznicek (ATLAS) [36], Chandiprasad Kar
(CMS) [37] and Varvara Batozskaya (LHCb) [38]. HFLAV combines these
measurements to

$\displaystyle\Delta\Gamma_{s}$ $\displaystyle=$ $\displaystyle(0.088\pm
0.006)\,\mbox{ps}^{-1}\,,$ (5)
$\displaystyle\frac{\Delta\Gamma_{d}}{\Gamma_{d}}$ $\displaystyle=$
$\displaystyle-0.002\pm 0.010\,.$ (6)

The calculation of $\Gamma_{12}^{q}$ is more involved and is based on the
Heavy Quark Expansion (HQE) (see [39] for a review and the original
references). According to the HQE the total decay rate of a heavy hadron can
be expanded in the inverse of the heavy quark mass as

$\displaystyle\frac{1}{\tau}=\Gamma$ $\displaystyle=$
$\displaystyle\Gamma_{0}+\frac{\Lambda^{2}}{m_{b}^{2}}\Gamma_{2}+\frac{\Lambda^{3}}{m_{b}^{3}}\Gamma_{3}+\frac{\Lambda^{4}}{m_{b}^{4}}\Gamma_{4}+...\,.$
(7)

The hadronic scale $\Lambda$ is of order $\Lambda^{QCD}$, its numerical value
has to be determined by direct computation. For hadron lifetimes it turns out
that the dominant correction to $\Gamma_{0}$ is the third term $\Gamma_{3}$.
Each of the $\Gamma_{i}$’s can be split up in a perturbative part and non-
perturbative matrix elements - it can be formally written as

$\displaystyle\Gamma_{i}$ $\displaystyle=$
$\displaystyle\left[\Gamma_{i}^{(0)}+\frac{\alpha_{S}}{4\pi}\Gamma_{i}^{(1)}+\frac{\alpha_{S}^{2}}{(4\pi)^{2}}\Gamma_{i}^{(2)}+...\,,\right]\langle
O^{d=i+3}\rangle$ (8)

where $\Gamma_{i}^{(0)}$ denotes the perturbative LO-contribution,
$\Gamma_{i}^{(1)}$ the NLO one and so on; $\langle O^{d=i+3}\rangle$ is the
non-perturbative matrix element of $\Delta B=0$ operators of dimension $i+3$.
The mixing quantity $\Gamma_{12}^{q}$ obeys a very similar HQE, but now the
operators change the $b$-quantum number by two units, $\Delta B=2$:

$\displaystyle\Gamma_{12}$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{3}}{m_{b}^{3}}\Gamma_{3}+\frac{\Lambda^{4}}{m_{b}^{4}}\Gamma_{4}+...\,.$
(9)

Uli Nierste [40] gave an overview of the theoretical status of the SM
prediction for $\Gamma_{12}^{s}$. In the $\overline{MS}$ scheme he obtains

$\Delta\Gamma_{s}=\left(0.104\pm 0.008_{scale}\pm 0.007_{B}\pm
0.015_{1/m_{b}}\right)\,\mbox{ps}^{-1}\,,$ (10)

where the first uncertainty is due to an unphysical renormalisation scale
dependence at NLO-QCD ($\Gamma_{3}^{(1)}$) [41, 42, 43]. In the pole mass
scheme this uncertainty is considerably larger [40] :

$\Delta\Gamma_{s}=\left(0.091\pm 0.020_{scale}\pm 0.006_{B}\pm
0.017_{1/m_{b}}\right)\,\mbox{ps}^{-1}\,.$ (11)

The scale uncertainty can be reduced by a NNLO-QCD calculation. The second
uncertainty is due to the matrix elements of operators of dimension 6. Here
two additional operators to the one appearing in the mass difference are
arising. We have currently a HQET sum rule determination for $B_{d}$ mesons
[29, 16] and lattice determinations from 2016 [21] ($N_{f}=2+1$) and 2013 [17]
($N_{f}=2$). For Uli’s prediction the values from FNAL/MILC have been used.
The third uncertainty stems from higher orders in the HQE. The dimension 7
perturbative part has been determined already in 1996 by Buchalla and Beneke
[45] for $B_{s}$ and in [46] for $B_{d}$ \- the non-perturbative matrix
elements have so far only been estimated in vacuum insertion approximation.
Uli presented a first calculation of a sub-set of all NNLO-QCD corrections
($\Gamma_{3}^{(2)}$) [44] and promised the full $\alpha_{s}/m_{b}$ corrections
($\Gamma_{4}^{(1)}$) for CKM 2020, the full NNLO-QCD corrections
($\Gamma_{3}^{(2)}$) to $\Delta\Gamma_{s}$ for CKM 2022 and the full NNLO-QCD
corrections ($\Gamma_{3}^{(2)}$) to the semi-leptonic asymmetries (here also
subleading CKM structures have to be determined) for CKM 2024.
Matthew Wingate [47] presented the current status of the ongoing HPQCD
activities [48] to perform the first non-perturbative determination of the
matrix elements of the dimension 7 operators. These matrix elements are
currently only estimated in vacuum insertion and they therefore make up the
largest contribution to the SM uncertainty of $\Delta\Gamma_{s}$. Any profound
non-perturbative determination of the dimension 7 contribution will
considerably reduce the theory uncertainty and have many phenomenological
implications. Finally it will also be very interesting to see how well the
vacuum insertion works at dimension 7.
We finally had a talk of Gilberto Tetlalmatzi-Xolocotzi [49], who was
questioning the assumption of having no new physics effects acting in tree-
level non-leptonic decays, see e.g. [50]. According to his studies new effects
of the order of $10\%$ to the tree-level Wilson coefficients $C_{1}$ and
$C_{2}$ are clearly not ruled out yet. For some observables these small
deviations could have dramatic effects:

  * •

The decay rate difference of neutral $B_{d}$ mesons, $\Delta\Gamma_{d}$ could
be enhanced by up to $+160\%$ or $-291\%$ of its SM value [2].

$\displaystyle\Delta\Gamma_{d}^{\rm SM}$ $\displaystyle=$
$\displaystyle(2.99\pm 0.52)\cdot 10^{-3}\mbox{ps}^{-1}\,.$ (12)

  * •

The experimental extraction of the CKM angle $\gamma$ could be modified by up
to $5^{\circ}$ compared to the SM expectation \- a huge value compared to the
future planned uncertainties [51].

For completeness we also show the SM predictions for the semileptonic CP
asymmetries [2].

$\displaystyle a_{sl}^{s}=(2.27\pm 0.25)\cdot 10^{-5}\,,$ $\displaystyle
a_{sl}^{d}=-(4.90\pm 0.54)\cdot 10^{-4}\,.$ (13)

##  2 Lifetimes

Lifetime measurements were presented in the talks of Manuel Schiller (LHCb)
[5], Pavel Reznicek (ATLAS) [36], Chandiprasad Kar (CMS) [37] and Varvara
Batozskaya (LHCb) [38]. HFLAV combines these measurements to [6]

$\displaystyle\frac{\tau(B^{+})}{\tau(B_{d})}$ $\displaystyle=$ $\displaystyle
1.076\pm 0.004\,,$ (14) $\displaystyle\frac{\tau(B_{s})}{\tau(B_{d})}$
$\displaystyle=$ $\displaystyle 0.993\pm 0.004\,,$ (15)
$\displaystyle\frac{\tau(\Lambda_{b})}{\tau(B_{d})}$ $\displaystyle=$
$\displaystyle 0.967\pm 0.007\,,$ (16)

Thomas Rauh [27] presented the current status of SM predictions depicted in
Fig. 4 \- see also [52] for a detailed overview of the current theory status.
The SM values are based on the perturbative calculations [53, 54, 55, 56, 57].
The non-perturbative dimension 6 matrix elements for mesons (except for small
corrections arising in $B_{s}$ and $D_{s}$) were recently calculated via HQET
sum rules [16] \- here a complementary lattice evaluation would be very
important, either for looking for BSM effects in the very precisely predicted
ratio $\tau(B_{s})/\tau(B_{d})$ \- this could point towards new effects in
hadronic tree-level decays [50] \- , or for testing the convergence of the HQE
in the $b$\- and in particular in the $charm$-system. For baryons we do not
have a complete first principle determination of the non-perturbative matrix
elements - there are sum rule determinations of the condensate contribution
for the $\Lambda_{b}$ [58] \- we have, however, some estimates [39, 60] of the
size of the matrix elements using spectroscopy as an input (based on [59]). LO
dimension 7 contributions were determined in [57, 60, 61]. So far there exists
no non-perturbative determination of the matrix elements of dimension 7
operators. In Fig. 4, taken from [16], we compare the most solid SM
predictions for heavy lifetimes with experiment and find an excellent
agreement, as well as a strong hint for the convergence of the HQE for total
inclusive rates in the charm system.

Figure 4:  Comparison of the most solid SM predictions for heavy lifetimes
with experiment.

##  3 Mixing angles

Neglecting penguin contributions one gets a very precise SM prediction for the
mixing-induced CP-violating phase $\phi_{s}^{c\bar{c}s}$ of $-0.0370\pm
0.0006$ rad. If there is new physics acting in $B$ mixing (i.e. in $M_{12}$),
then $\phi_{s}^{c\bar{c}s}$ and $\phi_{12}^{d}$ (defined below Eq.(1)) receive
the same new contributions - historically this lead regularly to some
confusion between these two phases [62, 63]. Recent measurements of
$\phi_{s}^{c\bar{c}s}$ by LHCb and ATLAS using Run1 data were presented in
detail by Varvara Batozskaya [38] and Pavel R̆ezníc̆ek [36]. The measurements
from LHCb include $B_{s}^{0}\to J/\psi
K^{+}K^{-},J/\psi\pi^{+}\pi^{-},\Psi(2S)\phi$ and $D_{s}^{-}D_{s}^{+}$. The
current HFLAV combination including all presented results reads
$\phi_{s}^{c\bar{c}s}=-0.021\pm 0.031\,\textrm{rad}$ and as shown in Fig. 5 it
is consistent with the SM prediction.

Figure 5:  The ($\phi_{s}^{c\bar{c}s},\Delta\Gamma_{s}$) plane, the individual
68% confidence-level contours of ATLAS, CMS, CDF, D0 and LHCb, their combined
contour (white solid line and shaded area), as well as the Standard Model
predictions (very thin black rectangle) are shown.

Processes occurring purely via gluonic or electroweak penguin transitions
provide an excellent opportunity to search for new heavy particles entering in
the penguin loops and new results from LHCb for such decays were presented by
María Vieites Díaz [67]. The effective weak phase $\phi_{s}^{d\bar{d}s}$ and
$\phi_{s}^{s\bar{s}s}$ using flavour-tagged, time-dependent, amplitude
analyses of $B_{s}^{0}\to(K^{+}\pi^{-})(K^{-}\pi^{+})$ [65] and
$B_{s}^{0}\to\phi\phi$ [64] decays are measured to be

$\displaystyle\phi_{s}^{d\bar{d}s}$ $\displaystyle=$ $\displaystyle-0.10\pm
0.13\pm 0.14\,\textrm{rad}\,,$ (17) $\displaystyle\phi_{s}^{s\bar{s}s}$
$\displaystyle=$ $\displaystyle-0.06\pm 0.05\pm 0.03\,\textrm{rad}\,.$ (18)

The measurement of $\phi_{s}^{s\bar{s}s}$ is the first CP violation
measurement using run 2 data from LHCb, while $\phi_{s}^{d\bar{d}s}$ using
only run 1 is the first measurement of this phase. Both results are in
agreement with the SM prediction.
The CKM angle $\beta(\phi_{1})$ is very well measured by B-factories and in
recent years also by LHCb with a competing precision. The current world
average is $(22.2\pm 0.7)^{\circ}$ or $(67.8\pm 0.7)^{\circ}$. This two-fold
ambiguity has been recently resolved by a joint Belle and Babar analysis of
$B^{0}\to D^{(*)0}h^{0}(\pi^{0},\eta,\omega)$ decays and the larger solution
is excluded by $7.3\sigma$ [68].
The measurements of $\phi_{s,d}(-2\beta)$ from $b\to c\bar{c}s$ transitions
involves not only tree-level contributions, but also penguin diagrams. These
”penguin-pollution” contributions are very hard to be calculated directly, see
[69]. Therefore simplifying assumptions like SU(3) flavour symmetries are
commonly used to estimate the potential size of penguin pollution via decay
channels with enhanced penguin-to-tree amplitude ratios. The CP violation
measurement for the decay $B_{s}^{0}\to J/\psi K_{S}^{0}$ using a time-
dependent flavour-tagged analysis with LHCb run 1 data set was presented by
Simon Akar [66]. Combining this result with existing measurements from
$B^{+}\to J/\psi K^{+}$, $B^{+}\to J/\psi\pi^{+}$ and $B^{0}\to J/\psi\pi^{0}$
decays, the shift due to the penguin-pollution is estimated to be
$\Delta\beta=(-1.10^{+0.70}_{-0.85})^{\circ}$. This result will be further
improved using updated Belle measurements of the branching fraction and CP
asymmetries of the $B^{0}\to J/\psi\pi^{0}$ decay [68]. Additionally, a new
measurement of $\sin 2\beta^{eff}$ using $B^{0}\to K_{S}\pi^{0}\pi^{0}$ decays
was presented by Bilas Pal and found to be consistent with the measurements
from $b\to c\bar{c}s$ decays.
Benjamin Oberhof presented Belle2 prospects for the mixing and CP violation in
B decays [70]. With improved detector performance and huge amount of data to
be collected by both Belle2 and LHC experiments, the precision of the mixing
angles is expected to be better than 1% level by 2030.
Martin Jung presented the theoretical aspects of the precision determination
of the mixing angles and effects of SU(3) breaking in penguin pollution
estimates [71] \- updating earlier results obtained in [72, 73, 74]. Martin’s
conclusion was that $b\to c\bar{c}s$ transitions remain golden modes and that
$SU(3)_{F}$ methods will improve with better data.

##  4 Conclusion

During CKM 2018 we witnessed an impressive improvement both in experiment and
theory for mixing and mixing related CP observables. A continuation of this
work will hopefully lead to a deeper understanding of the fundamental
principles of nature.

## Acknowledgement

We would like to thank all speakers of our session for their excellent talks
and the organisers of CKM 2018 for their amazing work.

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