$C_{{{K}^{+}}{{K}^{-}}}$, $S_{{{K}^{+}}{{K}^{-}}}$ and
$A^{{\Delta\Gamma}}_{{{K}^{+}}{{K}^{-}}}$) using the full sample of $p$ $p$
collisions collected during Run 1 corresponding to $3.0\mbox{\,fb}^{-1}$ of
integrated luminosity [114]. The results are reported in Table 3.2, together
with the projections of the statistical uncertainties to larger samples. The
scaling of statistical uncertainties assumes the same detector performances as
in Run 1, in particular regarding the flavour tagging, the decay-time
resolution and the particle identification performance, which are particularly
important for the determination of these observables. The main sources of
systematic uncertainties are due to limited knowledge of: the variation of the
selection efficiency as a function of the $b$ meson decay time, the parameters
$\Gamma_{{s}}$ and $\Delta\Gamma_{{s}}$, and the calibration of the decay-time
resolution. The evaluation of these uncertainties is based on the study of
control modes, and hence they are expected to decrease in a statistical manner
as the available sample size grows.

Table 3.2: Current LHCb measurements of $C_{{{\pi}^{+}}{{\pi}^{-}}}$,
$S_{{{\pi}^{+}}{{\pi}^{-}}}$, $C_{{{K}^{+}}{{K}^{-}}}$,
$S_{{{K}^{+}}{{K}^{-}}}$ and $A^{{\Delta\Gamma}}_{{{K}^{+}}{{K}^{-}}}$ using
the full sample of $p$ $p$ collisions collected during Run 1, where the first
uncertainty is statistical and the second systematic. The projection of
statistical precisions for each variables are also reported.

Data sample | $C_{{{\pi}^{+}}{{\pi}^{-}}}$ | $S_{{{\pi}^{+}}{{\pi}^{-}}}$ | $C_{{{K}^{+}}{{K}^{-}}}$ | $S_{{{K}^{+}}{{K}^{-}}}$ | $A^{{\Delta\Gamma}}_{{{K}^{+}}{{K}^{-}}}$
---|---|---|---|---|---
Run 1 ($3\mbox{\,fb}^{-1}$ [114]) | $-0.34\pm 0.06\pm 0.01$ | $-0.63\pm 0.05\pm 0.01$ | $\phantom{-}0.20\pm 0.06\pm 0.02$ | $\phantom{-}0.18\pm 0.06\pm 0.02$ | $-0.79\pm 0.07\pm 0.10$
|  $\sigma$ (stat.)
Run 1-3 ($23\mbox{\,fb}^{-1}$) | $0.015$ | $0.013$ | $0.015$ | $0.015$ | $0.018$
Run 1-6 ($300\mbox{\,fb}^{-1}$) | $0.004$ | $0.004$ | $0.004$ | $0.004$ | $0.005$

#### 3.4.4 $\alpha$ from $B$ decays to $\pi\pi$, $\rho\rho$ and
$\pi^{+}\pi^{-}\pi^{0}$

For over a decade, it has been common for conference speakers to introduce the
angle $\gamma$ as the least precisely measured of the three unitarity triangle
angles. However, the uncertainty on the world average for $\gamma$ is now
around $4^{\circ}$, slightly below that for $\alpha$, which is $5^{\circ}$
[27]. As discussed in this document, LHCb has a strong programme to continue
to improve the determination of both $\beta$ and $\gamma$, and hence it is
important to continue to reduce the uncertainty also on
$\alpha\equiv\arg\left[-(V^{\phantom{*}}_{td}V_{tb}^{*})/(V^{\phantom{*}}_{ud}V_{ub}^{*})\right]$
in order to make precision tests of the SM in the CKM sector.

The method discussed in the previous subsection to determine $\gamma$ and
$-2\beta_{s}$ from $B\rightarrow\pi\pi$ and
${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decays is an extension of the
well-known isospin analysis to determine $\alpha$ from $B\rightarrow\pi\pi$
decays alone [115, 116, 117] (the unitarity relation $\alpha+\beta+\gamma=\pi$
is used implicitly). The main input from LHCb will be world-leading
measurements of the $C\\!P$-violating parameters in
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decay, but important input can
also be obtained on the ${{B}^{+}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{0}}$ decay,
using the method pioneered in Ref. [118] for
${{B}^{+}}\\!\rightarrow{{K}^{+}}{{\pi}^{0}}$ decays. Clearly good performance
of the electromagnetic calorimeter will be critical for such a measurement.
Progress on the ${{B}^{0}}\\!\rightarrow{{\pi}^{0}}{{\pi}^{0}}$ mode, which is
currently limiting the precision of the $\alpha$ determination from
$B\rightarrow\pi\pi$ decays, will mainly come from the Belle II experiment.

The situation is quite different for the $B\rightarrow\rho\rho$ system, which
currently provides the strongest constraints on $\alpha$. Although the
presence of two vector particles in the final state makes the analysis more
complicated in principle [115, 119, 120, 121], the observed dominance of
longitudinal polarisation and the smaller penguin contribution compared to
$B\rightarrow\pi\pi$ lead to good sensitivity being achieved. The rarest of
the three isospin-partner modes is the
${{B}^{0}}\\!\rightarrow\rho^{0}\rho^{0}$ decay, which has a final state of
four charged tracks following $\rho^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decays, making it well suited for study at LHCb. Indeed, this decay was first
observed by LHCb, and a time-integrated angular analysis on Run 1 data was
performed [122]. With larger data samples it will be possible not only to
improve the measurements of the branching fraction and longitudinal
polarisation fraction, but to make precise determinations of the
$C\\!P$-violating parameters. Consequently, the determination of $\alpha$ from
$B\rightarrow\rho\rho$ decays will benefit from the additional information
inherent in $S_{{{\rho}^{0}}{{\rho}^{0}}}$, compared to the case for the
$B\rightarrow\pi\pi$ system for which $S_{{{\pi}^{0}}{{\pi}^{0}}}$ is barely
measurable (Belle II plans to measure $S_{{{\pi}^{0}}{{\pi}^{0}}}$ using
candidates involving a ${{\pi}^{0}}$ Dalitz decay, but only rather limited
sensitivity is possible).

For ${{B}^{0}}\rightarrow\rho\pi$ decays, it is possible to determine $\alpha$
from a decay-time-dependent amplitude analysis of
${{B}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ decays alone, without
the need to include other channels in the analysis [123]. This is possible
since amplitudes for the decays ${{B}^{0}}\rightarrow{{\rho}^{+}}{{\pi}^{-}}$,
${{\rho}^{-}}{{\pi}^{+}}$ and ${{\rho}^{0}}{{\pi}^{0}}$ interfere in different
regions of the Dalitz plot, so that their relative phases can be measured.
First measurements have been published by BaBar and Belle [124, 125, 126], but
do not yet provide strong constraints on $\alpha$. LHCb has not yet published
any result on this channel, but it is expected that large yields will be
available, and that it should be possible to control backgrounds with good
understanding of ${{\pi}^{0}}$ reconstruction. It is worth noting that a
significant proportion of the photons from neutral pions produced at the $B$
decay vertex convert into ${e^{+}}{e^{-}}$ pairs in the LHCb detector
material, with approximately half of these conversions occurring before the
magnet. Tracks from these converted photons provide additional information
with which to constrain the $B$ decay vertex and the neutral pion momentum,
resulting in improved resolution and background rejection.

An important source of systematic uncertainty for amplitude analyses, such as
that for ${{B}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ decays, is
likely to be due to lack of knowledge of the resonant structure. Studies of
the ${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ decay will
provide useful information that can help to minimise this uncertainty [127].
Moreover, for all $\alpha$ measurements at subdegree precision, isospin-
breaking effects such as electroweak penguin contributions must also be
considered [117].

#### 3.4.5 $C\\!P$ violation in $B^{0}_{s}\rightarrow
K^{0}_{\mathrm{S}}h^{+}h^{-}$ and $B^{0}_{s}\rightarrow h^{+}h^{-}\pi^{0}$

Charmless three-body decays of ${{B}_{({s})}^{0}}$ mesons provide further
opportunities for studies of $C\\!P$ violation in penguin-dominated
transitions, and are sensitive to contributions from physics beyond the SM
(BSM) scenarios [77, 78, 79, 80]. Thus, measurements of the CKM phases
$\beta$, $\beta_{{s}}$ and $\gamma$ in these modes may differ from their SM
benchmark values. A good example is the determination of $\beta$ using the
${{B}^{0}}\rightarrow\phi{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ channel,
obtained from decay-time-dependent amplitude analysis of
${{B}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ decays. Results from BaBar and Belle [128, 129] show that
$S_{\phi{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}$ is consistent with
expectation, but with a large uncertainty. Improved measurements are
anticipated from Belle II, but LHCb will also have a relevant reach since
large yields of ${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}$ decays are available [130]. For the
${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ mode, a decay-time-integrated analysis has been
performed resulting in the first observation of $C\\!P$ violation in the
${{B}^{0}}\rightarrow{{K}^{*+}}{{\pi}^{-}}$ channel [131]; with more data this
analysis can be updated to include decay-time-dependence and determine also
$C\\!P$-violating parameters for the
${{B}^{0}}\rightarrow{{\rho}^{0}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$
and $f_{0}(980){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ channels.

While both LHCb and Belle II will study
${{B}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$
decays, only LHCb can make measurements of $C\\!P$-violating parameters of the
counterpart ${B}^{0}_{s}$ decays. Preliminary sensitivity studies of a decay-
time-dependent flavour-tagged Dalitz-plot analysis of
${{B}^{0}_{s}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays [132] indicate that the precision
achievable on $\phi_{{s}}^{{d}{\overline{u}}{u}}$ with the full Run 1 + Run 2
dataset is approximately $0.4\mathrm{\,rad}$. Extrapolation to
$300\mbox{\,fb}^{-1}$ indicates potential for a precision of around
$0.07\mathrm{\,rad}$. Similar studies for the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ decay mode can be performed, which requires a
more complicated analysis since both ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{-}}{{\pi}^{+}}$ and ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}}{{\pi}^{-}}$ final states are accessible to both ${B}^{0}_{s}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ decays with
comparable magnitude. Although currently the precision of such measurements
are dominated by the statistical uncertainty [133], the expected yields of
more than $10^{6}$ signal decays for $300\mbox{\,fb}^{-1}$ will allow a
complete understanding of the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{\pm}}{{\pi}^{\mp}}$ phase space.

Studies of ${{B}_{({s})}^{0}}\rightarrow h^{+}h^{-}{{\pi}^{0}}$ decays will
provide further sensitivity. For example, resonant contributions of the type
${{K}^{*\pm}}h^{\mp}$ will decay to the final state
${{K}^{\pm}}h^{\mp}{{\pi}^{0}}$ in addition to
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}h^{\mp}$, and therefore
a combined analysis of both can provide additional information that helps to
test the SM prediction [134, 135, 136, 137]. Large yields will be available
and the improved capabilities of the ECAL will allow background to be
controlled. Two particularly important features of these decays are that
background from $B\rightarrow V\gamma$ decays, where $V\rightarrow
h^{+}h^{-}$, must be suppressed and that it must be possible to resolve
${B}^{0}$ and ${B}^{0}_{s}$ decays to the same final state. Thus, it will be
important to have both good $\gamma$–${{\pi}^{0}}$ separation and good mass
resolution.

### 3.5 Measurements of $\gamma$ from $B^{0}_{s}\rightarrow
D_{s}^{\mp}K^{\pm}$ and $B^{0}\rightarrow D^{\mp}\pi^{\pm}$

Measurement of decay-time-dependent $C\\!P$ violation in
${B}^{0}_{s}\rightarrow{D}_{s}^{\mp}K^{\pm}$ and
${B}^{0}\rightarrow{D}^{\mp}\pi^{\pm}$ decays provide information on the
combination of the CKM angles $\gamma-2\beta_{s}$ and $\gamma+2\beta$,
respectively. Information on the angle $\gamma$ can be therefore determined by
using independent determinations of the value of $\beta_{(s)}$ as input.

Sensitivity to $C\\!P$ violation, and hence to the combination of the CKM
angles, depends on the ratio between the suppressed
$\bar{b}\rightarrow\bar{u}c\bar{q}$ ($q=d,s$) and the favoured
$\bar{b}\rightarrow\bar{c}u\bar{q}$ decay amplitudes:
$r_{D_{(s)}h}\equiv|A(B^{0}_{(s)}\rightarrow
D^{+}_{(s)}h^{-})/A(B^{0}_{(s)}\rightarrow D^{-}_{(s)}h^{+})|$, where
$h=\pi,K$. This ratio is sizeable ($\approx 0.4$) for
${B}_{s}^{0}\rightarrow{D}_{s}^{\mp}K^{\pm}$ decays, and small ($\approx
0.02$) for ${B}^{0}\rightarrow{D}^{\mp}\pi^{\pm}$ decays. Thus, for
${B}_{s}^{0}\rightarrow{D}_{s}^{\mp}K^{\pm}$ decays it is possible to
determine all five of the observables $C_{f}$, $S_{f}$, $S_{\bar{f}}$,
$A^{\Delta\Gamma}_{f}$ and $A^{\Delta\Gamma}_{\bar{f}}$ in a flavour-tagged
decay-time-dependent analysis ($C_{f}=-C_{\bar{f}}$ due to absence of $C\\!P$
violation in decay in these tree-dominated modes). In the
${B}^{0}\rightarrow{D}^{\mp}\pi^{\pm}$ case, however, $C_{f}$ is
indistinguishable from unity, and so only $S_{f}$ and $S_{\bar{f}}$ can be
measured bearing in mind that $\Delta\Gamma_{d}$ is negligible. In addition to
$r_{D_{(s)}h}$ and the weak-phase combination $\gamma-2\beta_{s}$ or
$\gamma+2\beta$, these observables depend also on the strong-phase difference
between the suppressed and favoured amplitudes, denoted $\delta_{D_{(s)}h}$.

Using Run 1 data, LHCb has determined the $B_{s}^{0}\rightarrow
D_{s}^{\mp}K^{\pm}$ coefficients from a sample of about 6000 signal decays
[138]. From the measured parameters, a value of $\gamma$ of
$(128\,^{+17}_{-22})^{\circ}$ (modulo $180^{\circ}$) is determined, using as
input $-2\beta_{s}=-0.030\pm 0.033\mathrm{\,rad}$ [27]. The result is
dominated by statistical uncertainties thanks to the wide use of data-driven
methods to determine the decay-time acceptance and resolution, and to
calibrate the flavour tagging. The systematic uncertainties are, in decreasing
order of importance, related to background from $b$-hadron decays, to
uncertainty on the value of $\Delta m_{s}$, to the calibration of the decay-
time resolution and to the flavour tagging. The first contribution can be
significantly reduced by a tighter signal selection or by using a different
fitting approach; the remaining contributions are expected to scale with the
statistics accumulated due to their data-driven nature.

In the case of ${{B}^{0}}\rightarrow{D}^{\mp}\pi^{\pm}$ decays, the smallness
of the ratio of amplitudes $r_{D\pi}$, which limits the sensitivity to $S_{f}$
and $S_{\bar{f}}$, is compensated for by a large signal yield. About 480 000
flavour-tagged signal decays are available in the Run 1 LHCb data sample
[139]. Analysis of this sample gives measurements of $S_{f}$ and $S_{\bar{f}}$
that are more precise than those from BaBar and Belle [140, 141]. Also in this
case the precision is limited by the statistical uncertainty. The dominant
sources of systematic uncertainty, such as due to knowledge of $\Delta m_{d}$
and of background subtraction, are expected to be reducible with larger
samples.

Since ${{B}^{0}}\rightarrow{D}^{\mp}\pi^{\pm}$ decays there are only two
observables, $S_{f}$ and $S_{\bar{f}}$, that depend on three unknown
quantities, $r_{D\pi}$, $\delta_{D\pi}$ and $2\beta+\gamma$, external input
must be used to obtain a constraint on $2\beta+\gamma$. A common approach is
to determine $r_{D\pi}$ from the branching fraction of
${{B}^{0}}\rightarrow{{D}^{+}_{s}}{{\pi}^{-}}$ decays, assuming
$\mathrm{SU}(3)$ symmetry,

$r_{D\pi}=\tan\theta_{c}\frac{f_{D^{+}}}{f_{D_{s}}}\sqrt{\frac{\mathcal{B}({{B}^{0}}\rightarrow{{D}^{+}_{s}}{{\pi}^{-}})}{\mathcal{B}({{B}^{0}}\rightarrow{{D}^{-}}{{\pi}^{+}})}}\,,$
(3.8)

where $\tan\theta_{c}$ is the tangent of the Cabibbo angle and
$\frac{f_{D_{s}}}{f_{D^{+}}}$ is the ratio of decay constants. Using the
obtained value of $r_{D\pi}=0.0182\pm 0.0012\pm 0.0036$, where the second
uncertainty accounts for possible nonfactorisable $\mathrm{SU}(3)$-breaking
effects [142], the intervals $|\sin(2\beta+\gamma)|\in[0.77,1.00]$ and
$\gamma\in[5,86]^{\circ}\cup[185,266]^{\circ}$ are obtained, at the 68%
confidence level. The interval of $\gamma$ is calculated using $\beta=(22.2\pm
0.7)^{\circ}$ [27]. The uncertainties on $r_{D\pi}$ and $\beta$ have
negligible impact on these intervals, as the dominant uncertainties are from
the $S_{f}$ and $S_{\bar{f}}$ measurement.

The expected statistical sensitivities for the $C\\!P$ violation parameters in
$B^{0}_{s}\rightarrow D_{s}^{\mp}K^{\pm}$ and $B^{0}\rightarrow
D^{\mp}\pi^{\pm}$ decays are shown in Table 3.3. These are based on scaling of
yields, and as such assume that the same detector performance as achieved in
Run 1 can be maintained. In particular, the sensitivity depends strongly on
the performance of the particle identification, decay time resolution and
flavour tagging. The results can be complemented by studies of the related
$B^{0}_{s}\rightarrow D_{s}^{*\mp}K^{\pm}$ and $B^{0}\rightarrow
D^{*\mp}\pi^{\pm}$ channels. In particular, the $D^{*\mp}\pi^{\pm}$ mode has
an all charged final state, and with a possible gain in the acceptance of slow
pions from $D^{*}$ decays from the addition of magnet side stations,
comparable precision to that for $B^{0}\rightarrow D^{\mp}\pi^{\pm}$ may be
possible.

The corresponding expected sensitivities of $\gamma$ from
${{B}^{0}_{s}}\rightarrow{D}_{s}^{\mp}K^{\pm}$ decays are about $4^{\circ}$,
$2.5^{\circ}$ and $1^{\circ}$ after collecting 23, 50 and
300$\mbox{\,fb}^{-1}$, respectively. It is more challenging to estimate the
constraints on $\sin(2\beta+\gamma)$ and $\gamma$ from ${{B}^{0}}\rightarrow
D^{\mp}\pi^{\pm}$ decays, since the precision of the external value of
$r_{D\pi}$ will become the dominant source of systematic uncertainty.
Theoretical advancements on understanding the nonfactorisable
$\mathrm{SU}(3)$-breaking effects are thus required.

Table 3.3: Expected statistical uncertainties on parameters of ${{B}^{0}_{s}}\rightarrow{D}_{s}^{\mp}K^{\pm}$ and ${{B}^{0}}\rightarrow{D}^{\mp}\pi^{\pm}$ decays. | ${{B}^{0}_{s}}\rightarrow{D}_{s}^{\mp}K^{\pm}$ | ${{B}^{0}}\rightarrow{D}^{\mp}\pi^{\pm}$
---|---|---
Parameters | Run 1 | 23$\mbox{\,fb}^{-1}$ | 50$\mbox{\,fb}^{-1}$ | 300$\mbox{\,fb}^{-1}$ | Run 1 | 23$\mbox{\,fb}^{-1}$ | 50$\mbox{\,fb}^{-1}$ | 300$\mbox{\,fb}^{-1}$
$S_{f}$, $S_{\bar{f}}$ | 0.20 | 0.043 | 0.027 | 0.011 | 0.02 | 0.0041 | 0.0026 | 0.0010
$A^{\Delta\Gamma}_{f}$, $A^{\Delta\Gamma}_{\bar{f}}$ | 0.28 | 0.065 | 0.039 | 0.016 | – | – | – |
$C_{f}$ | 0.14 | 0.030 | 0.017 | 0.007 | – | – | – |

## 4 Time-integrated ${C\\!P}$-violation measurements

Measurements of decay-time-integrated $C\\!P$ violation in the $B$ system
complement decay-time-dependent measurements in testing the SM picture of
quark mixing and the unitarity of the CKM matrix. They are particularly
important in giving the most precise possible tree-level measurement of the
angle
$\gamma\equiv\arg\left[-(V_{ub}^{*}V_{ud}^{\phantom{*}})/(V_{cb}^{*}V_{cd}^{\phantom{*}})\right]$
of the CKM unitarity triangle, which is unique in not depending on a top-quark
coupling. Within the SM, the irreducible theoretical uncertainty on tree-level
measurements of $\gamma$ is $|\delta_{\gamma}/{\gamma}|\leq 10^{-7}$ [143],
This means that with Upgrade II statistics, the measurement of $\gamma$ from
decay-time-integrated $C\\!P$ violation in $B\rightarrow DK$ decays will be
the most precise standard candle of the CKM picture and the unitarity triangle
apex, against which all other CKM observables can be compared. Crucially,
Upgrade II will allow for precision measurements of $\gamma$ in many
individual final states, particularly those with high multiplicity or neutral
particles, which will have complementary experimental systematic uncertainties
with the currently dominant modes. Such an overconstrained measurement of
$\gamma$ will give us particular confidence when confronting the experimental
picture of $C\\!P$ violation with the CKM expectation.

The theoretical cleanliness of the determination of $\gamma$ in the SM is
based on the fact that only tree-level processes contribute, and that it is
expected that there is no new physics contribution to such amplitudes.
However, if this assumption is dropped, Upgrade II offers the opportunity,
arising from the different decay topologies in $B^{0}$ and $B^{\pm}$ decays,
to probe for new physics effects within tree-level decays themselves. It is
important to underline that there is still considerable scope for new-physics
effects in tree-level decays [144], which would be signified by modifications
to the tree-level Wilson coefficients $C_{1}$ and $C_{2}$. These can be probed
at the few percent level with the Upgrade II dataset. Furthermore, with the
inclusion of decay-time-dependent methods (discussed in Sec. 3.5) such as
${{B}^{0}_{s}}\rightarrow{{D}^{\mp}_{s}}{{K}^{\pm}}$ and
${{B}^{0}}\rightarrow{{D}^{\mp}}{{\pi}^{\pm}}$ and very accurate knowledge of
$\gamma$, one can obtain penguin-free measurements of $\phi_{s}$ and $\beta$.
Upgrade II will also enable high-precision measurements of decay-time-
integrated $C\\!P$ violation in three-body decays of ${B}$ mesons and in the
decays of $b$ baryons and ${B}_{c}$ mesons. As with other areas of the LHCb
physics programme involving baryons, Upgrade II may be a truly unique
opportunity to systematically and precisely establish the picture of $C\\!P$
violation. Furthermore, many of these three-body and baryonic measurements
proceed through loop-level transitions and are therefore sensitive to NP, but
a proper interpretation of any observed $C\\!P$ violation requires a precise
understanding of the hadronic phases entering these transitions, which will
only be possible by a global analysis of the possible transitions enabled by
the Upgrade II statistical reach.

From an experimental point of view, the main challenges and opportunities
arising from Upgrade II are similar and can be illustrated using $B\rightarrow
DK$ decays as an example. These proceed via both $b\rightarrow c\bar{u}s$ and
$b\rightarrow u\bar{c}s$ transitions, whose phase difference has a
$C\\!P$-violating ($\gamma$) and $C\\!P$-conserving ($\delta_{B}$) part. The
ratio of these amplitudes $=r_{B}e^{i(\delta_{B}\pm\gamma)}$ and the general
rate equation (for ${B}^{\pm}$ mesons) is,

$\Gamma\left({{B}^{\pm}}\rightarrow D[\rightarrow
f]K\right)=\left|r_{D}e^{-i\delta_{D}}+r_{B}e^{i(\delta_{B}\pm\gamma)}\right|^{2}=r_{D}^{2}+r_{B}^{2}+2\kappa
r_{D}r_{B}\cos(\delta_{B}-\delta_{D}\pm\gamma)\,.$ (4.1)

Here, $r_{D}$ and $\delta_{D}$ are the amplitude ratio and $C\\!P$-conserving
phase difference between the ${C\\!P}$-conjugate final states of the $D$
decay, $D\rightarrow f$ and ${D}\rightarrow\bar{f}$. For two-body $B$ and $D$
decays, the coherence factor, $\kappa=1$, while for multibody decays, $\kappa$
reduces as the interference is diluted by incoherence amongst contributing
intermediate hadronic resonances. In both cases, the efficiency-corrected
time-integrated signal yields in each final state are interpreted in terms of
the above physical observables, notably the ${C\\!P}$-violating phase
$\gamma$. Since the bulk of the sensitivity to $\gamma$ comes from the
difference in rates of the ${B}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}$ processes, a precise control of
asymmetries in charged-particle identification and detection is crucial. Based
on our knowledge of the present detector, these systematic uncertainties will
scale with integrated luminosity through the Upgrade II period. An excellent
vertex resolution is equally important to separate the hadronic final states
from combinatorial backgrounds, and will benefit from the expected reduction
in the vertex detector material in Upgrade II , while the upgrade of the
calorimeter will greatly expand LHCb’s capabilities for modes with neutrals in
the final state. For baryonic states, the addition of the TORCH system may be
particularly helpful in allowing for low-momentum separation of protons and
kaons; this will also reduce the physics backgrounds from baryonic decays
which are often hard to model in other analyses due to their poorly known
absolute branching fractions. Finally, the addition of magnet-side stations
(see Sect. 2.3.1) may lead to important signal-yield improvements,
particularly for high-multiplicity final states where there is a non-
negligible probability of creating at least one low-momentum particle.

### 4.1 Determination of the $\gamma$ angle from tree-level decays

#### 4.1.1 $B\rightarrow DK$ GLW/ADS

The simplest $B\rightarrow DK$ topology is ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$, where the $D$ is reconstructed in a two charged-track final
state. In this case, the ratio of $b\rightarrow u\bar{c}s$ to $b\rightarrow
c\bar{u}s$ amplitudes, $r_{B}^{DK}\approx 0.10$ [27]. This is enough to
generate $\mathcal{O}(10\%)$ direct $C\\!P$ violation in $C\\!P$-eigenstates
decays $D\rightarrow{{K}^{+}}{{K}^{-}},{{\pi}^{+}}{{\pi}^{-}}$ that are
equally accessible from the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$. This is an example of the GLW
method [145] for which Eq. 4.1 simplifies with $r_{D}=1$ and $\delta_{D}=0$.
The ADS method [146] makes full use of Eq. 4.1 and is best exemplified by the
case $f={{\pi}^{\pm}}{{K}^{\mp}}$, where $r_{D}\approx 0.06$. The similar
magnitude of $r_{B}$ and $r_{D}$ gives large interference effects and high
sensitivity to the phase information.

Such modes have been observed and studied during Run 1 and Run 2 [147, 148];
all ADS/GLW asymmetries are statistically limited. The systematic
uncertainties are small and arise predominantly from sources that will
naturally decrease with increasing data, notably knowledge of instrumentation
asymmetries. Methods for measuring and correcting for the $B$-meson production
asymmetry and the ${K}^{\pm}$/${\pi}^{\pm}$ reconstruction asymmetries are
established using calibration samples; such samples will continue to be
collected. The dominant systematic uncertainties for GLW decays are due to
background contributions from ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}K^{-}$ decays
and charmless decays, while the dominant uncertainty for ADS decays arises
from the $B_{s}^{0}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$
background. All will be better determined with dedicated studies as the sample
size increases.

The ADS/GLW technique is being expanded to the other ${B}\rightarrow DK$
decays which share the same quark-level transition. An analysis of GLW
observables in ${{B}^{\pm}}\rightarrow D^{*0}{{K}^{\pm}}$ decays has been
developed for the case where the $D^{*0}$ vector meson is not fully
reconstructed [148]. This partial reconstruction has larger background
uncertainties but these will improve with more data as dedicated studies of
the background are performed. Furthermore, ADS/GLW analyses have been
developed and published in quasi-two-body modes, ${{B}^{\pm}}\rightarrow
D{{K}^{*\pm}}$ [149] and ${{B}^{0}}\rightarrow DK^{*0}$ [150]. As in the case
of ${{B}^{\pm}}\rightarrow D^{(*)0}{{K}^{\pm}}$ decays, these modes have no
limiting systematic and they will make competitive contributions with the LHCb
Upgrade datasets.

Under the assumption that systematic uncertainties decrease in parallel with
the statistical uncertainties, that is $\propto 1/\sqrt{\mathcal{L}}$, the
future precision on $\gamma$ is predicted in Fig. 4.1. In isolation, one
ADS/GLW analysis suffers from the trigonometric ambiguities in Eq. 4.1, but
the multiple solutions are generally separated and resolved in combination
with other $B\rightarrow DK$ results. Fig. 4.1 uses central values and
uncertainties in the published analyses of $B^{\pm}\rightarrow DK^{\pm}$,
$B^{\pm}\rightarrow DK^{*\pm}$ and $B^{0}\rightarrow DK^{*0}$ decays. For
$B^{\pm}\rightarrow D^{*0}K^{\pm}$ both the partial and full reconstruction
techniques are used in this study albeit with unpublished central values and
uncertainties.

Figure 4.1: Extrapolation of $\gamma$ sensitivity from the ADS/GLW analyses
where disfavoured ambiguities are ignored. The expected Belle II $\gamma$
precision at an integrated luminosity of 50 ab-1 is shown by the horizontal
grey lines.

#### 4.1.2 $B\rightarrow DK$ GGSZ

Measurements of $\gamma$ in which the $D$ meson is reconstructed using the
three-body, self-conjugate ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and
${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$
final states provide powerful input to the overall determination of $\gamma$,
as they select a single, narrow solution (see Fig. 4.5). Referred to as the
GGSZ method [151, 152], sensitivity to $\gamma$ is obtained by comparing the
distributions of ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{h}^{+}{h}^{-}$ decays across the Dalitz plane for opposite-flavour
initial-state ${{B}^{+}}$ and ${{B}^{-}}$ mesons. The partial decay rates as a
function of Dalitz position depend mainly on the amplitudes of the ${{D}^{0}}$
decay, with only small deviations introduced from interference and
$C\\!P$-violation. These deviations are most conveniently probed through the
measurement of the $C\\!P$-violating observables
$x_{\pm}=r_{B}\cos(\delta_{B}\pm\gamma)$ and
$y_{\pm}=r_{B}\sin(\delta_{B}\pm\gamma)$. For this it is necessary to have a
good understanding of the magnitudes of the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decay amplitudes, as well as
the strong-phase differences between them, $\delta_{D}$.

The description of $\delta_{D}$ has historically been treated in two ways. One
approach is to use an amplitude model determined from flavour-tagged
${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{h}^{+}{h}^{-}$
decays. This relies on assumptions about the nature of the intermediate
resonances that contribute to the
${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{h}^{+}{h}^{-}$ final
state, and choices made about these contributions to the amplitude model lead
to systematic uncertainties which are not certain to scale with luminosity in
the Upgrade II period. An alternative method divides the data into bins
according to the Dalitz plane coordinates, and can then make use of direct
measurements of the strong-phase differences averaged over each bin [151,
152]. One such set of binning definitions for the
${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ decay mode, obtained by optimising the sensitivity
to $\gamma$, is shown in Fig. 4.2.

Figure 4.2:  (Left) Bin definitions for
${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ as a function of $m_{-}^{2}$ and $m_{+}^{2}$, the
invariant masses squared of the ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{-}$ and ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}$. Figure from
Ref [153]. (Right) Asymmetry between yields for ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ decays, with ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ in bin $i$ (for ${{B}^{+}}$) and $-i$ (for
${{B}^{-}}$). The data points are obtained from simulation with the expected
sample size at $300\mbox{\,fb}^{-1}$, assuming the current performance of the
LHCb experiment. The black histogram shows the predicted asymmetry based on
the current world average values of $\gamma$ and relevant hadronic parameters,
the red dots show the result of a single pseudoexperiment, while the red bands
show the expected uncertainties from an ensemble. The green bands show the
corresponding uncertainties with the current LHCb data set.  Figure 4.3:
Expected evolution of $\gamma$ sensitivity with the GGSZ method with (red
crosses) current CLEO inputs and with (blue $\times$-marks) $\sqrt{N}$
improvements on their uncertainty.

Specifically, the values of $c_{i}$ and $s_{i}$, which correspond to
amplitude-weighted average values of $\cos(\delta_{D})$ and $\sin(\delta_{D})$
in each Dalitz bin $i$, are required. These can be measured in a model-
independent manner by using quantum-correlated ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ pairs from $\psi(3770)$
decays, as was done previously with data taken at the CLEO-c experiment [154].
This allows the determination of $\gamma$ to also be carried out in a
completely model-independent fashion, the price for which is a small loss in
sensitivity from binning the data, and a systematic uncertainty which depends
on the precision with which the strong-phase measurements can be determined.

The model-independent method is therefore expected to be the baseline in
Upgrade II , and its uncertainty is currently statistically dominated.
Although systematic uncertainties will already become significant compared to
the statistical uncertainty in Run 3, studies performed so far give confidence
that the systematic uncertainties will generally scale with the statistical
reach well into the Upgrade II period.

The largest systematic uncertainty is due to the precision of the external
strong-phase inputs coming from the CLEO-c data which currently contribute
approximately $2^{\circ}$ to the overall uncertainty on $\gamma$ [155]. The
impact of this uncertainty on GGSZ measurements is estimated in Fig. 4.3 which
compares a $\sqrt{N}$ improvement with the expected yield increase and the
projected uncertainty if the current external information on $c_{i}$ and
$s_{i}$ is not improved. It can be seen that this starts to approach a limit
originating from the fixed size of the quantum-correlated charm input from
CLEO-c. This contribution will naturally decrease with larger
${{B}^{\pm}}\rightarrow{{D}^{0}}{{K}^{\pm}}$ samples, however, as the $B$
decays themselves also have sensitivity to $c_{i}$ and $s_{i}$. Studies
performed using pseudoexperiments with different size quantum correlated $D$
samples and LHCb $B$ data suggest that the optimal sensitivity is only reached
when the size of the input $D$ sample is at least as big as the overall $B$
sample. More precise measurements with data already recorded by the BESIII
experiment will be able to reduce the external contribution to the uncertainty
by around 50% but analysis of future larger datasets with BESIII and at LHCb
will be vital in order to avoid the external input compromising the ultimate
sensitivity to $\gamma$.

The current second largest source of systematic uncertainty comes from the
knowledge of the distribution of $D$ decays over the Dalitz plane in the
flavour-specific ${B}$ final state, with reconstruction and efficiency effects
incorporated. These are determined with a flavour-specific control decay mode
${{B}^{0}}\rightarrow{{D}^{*\pm}}\mu^{\mp}\nu_{\mu}X$, where the ${D}^{*-}$
decays to ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ ${\pi}^{-}$
and $X$ represents any unreconstructed particles. The ultimate systematic
uncertainty is particularly sensitive to data-simulation agreement and size of
simulated samples because of a need to model unavoidable differences in the
signal and control modes. Fast simulation techniques which are being deployed
and further developed at LHCb will therefore be crucial for keeping up with
the large data samples, while a fully software-based trigger will allow for a
better alignment of the signal and control channel selections compared to
today. Uncertainties from sources such as low mass backgrounds can be expected
to remain subdominant with higher statistics, as further studies will give
better understanding to their rates and shapes. Some additional complications
will present themselves as the yields will eventually be high enough that it
will become necessary to take into account effects induced from asymmetries in
the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ system, and eventually through
mixing in the ${{D}^{0}}$ system. However these are tractable problems, and
studies have already been done to understand when these effects will become
important.

#### 4.1.3 Prospects with modes involving neutrals

There are many prospects for adding orthogonal information on $\gamma$ by
applying the ADS/GLW and GGSZ techniques to modes that have an additional
${\pi}^{0}$ meson. The first use of ${\pi}^{0}$ mesons in a $\gamma$ analysis
occurred with the Run 1 ADS/GLW analysis of ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ decays with $D\rightarrow K\pi{{\pi}^{0}},\
KK{{\pi}^{0}},\pi\pi{{\pi}^{0}}$ final states [156]. The $K\pi{{\pi}^{0}}$ and
$\pi\pi{{\pi}^{0}}$ modes have branching fractions 3 and 10 times larger than
their two-body equivalents. However, the net ${\pi}^{0}$ reconstruction and
selection efficiency in these decays is low, around 3% with the current
calorimeter. Also the analysis is complicated by a combinatorial background
arising from random ${{\pi}^{0}}$ association, which will certainly worsen in
Upgrade I . Improvements to the Upgrade II calorimeter granularity and energy
resolution will therefore be crucial in achieving the ultimate sensitivity
with these modes especially by improving the ${\pi}^{0}$ mass resolution.
Enhancements in the reconstruction of ${\pi}^{0}$ mesons reconstructed from a
single cluster due to the small spatial separation of their decay photons in
the calorimeter, and the ability to separate such candidates from single
photons, will also be important.

An important mode under development for the upgrade era is
${{B}^{\pm}}\rightarrow D^{*0}{{K}^{\pm}}$ decays, with
$D^{*0}\rightarrow{{D}^{0}}{{\pi}^{0}}$ and $D^{*0}\rightarrow{{D}^{0}}\gamma$
decays. These twin modes provide an excellent sensitivity to $\gamma$ as an
exact phase difference between the two $D^{*0}$ modes can be exploited [157].
For this case, the efficient distinction of ${\pi}^{0}$ and $\gamma$
calorimeter objects is critical as the two $D^{*0}$ modes exhibit opposite
$C\\!P$ asymmetries. The initial studies show small, but clean signals with
potential sensitivity to $\gamma$ comparable to the partial reconstruction
technique described in Sec. 4.1.1. As long as the fully and partially
reconstructed datasets are kept statistically independent, the final
sensitivity from ${{B}^{\pm}}\rightarrow D^{*0}{{K}^{\pm}}$ decays will be
around $0.5^{\circ}$ as seen in Fig. 4.1.

A GGSZ-like analysis of ${{B}^{\pm}}\rightarrow{D}[\rightarrow
K_{s}^{0}\pi\pi{{\pi}^{0}}]{{K}^{\pm}}$ decays has recently been proposed for
Belle II, where a sensitivity approaching that of the $D\rightarrow
K_{s}^{0}\pi\pi$ GGSZ analysis is expected [158]. With improved ${\pi}^{0}$
efficiency, LHCb Upgrade II can exploit this mode competitively. Lastly,
higher ${\pi}^{0}$ efficiency will merit the analysis of $B^{\pm}\rightarrow
DK^{*\pm}[\rightarrow{{K}^{\pm}}{{\pi}^{0}}]$ decays. The ${\pi}^{0}$
reconstruction efficiency is typically a factor 3-4 lower than that of the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ so the
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{\pm}}$ mode is preferred.
However the $B^{\pm}\rightarrow D{{K}^{\pm}}{{\pi}^{0}}$ Dalitz analysis for
$\gamma$ should share many advantages of the isospin-conjugate decays
${{B}^{0}}\rightarrow D{{K}^{+}}{{\pi}^{-}}$ analysis (discussed next) but
with reduced ${B}^{0}_{s}$ feed down and large asymmetries in the ADS-like
region of the Dalitz space.

#### 4.1.4 Prospects with high-multiplicity modes

A variety of high-multiplicity $B$ and $D$ modes are already being established
and will play an important role in a future determination of $\gamma$. The
most developed multibody $B$ decay channel is ${{B}^{0}}\rightarrow
D{{K}^{+}}{{\pi}^{-}}$ decays, where the $D$ meson is found in an ADS/GLW-like
($K\pi,\ KK,\ \pi\pi,\ K3\pi,\ ...$) or GGSZ-like
(${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi\pi$,
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}KK,\ ...$) final states. These modes
are less abundant than the equivalent ${B}^{\pm}$ modes but the fact that both
the $b\rightarrow c\bar{u}s$ and $b\rightarrow u\bar{c}s$ transitions proceed
by colour-suppressed amplitudes means the GLW asymmetries can be very large.
Furthermore, understanding the pattern of asymmetry across the ${B}^{0}$
Dalitz plane quashes the trigonometric ambiguities. This multibody technique
will eventually surpass the quasi-two-body analysis introduced in Sec. 4.1.1.
This analysis has been established with the $KK$ and $\pi\pi$ modes using Run
1 data in Ref. [159] after the development of the ${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{+}}{{\pi}^{-}}$ amplitude
model [160]. Although the statistical sensitivity to $C\\!P$ violation using
Run 1 data alone was not significant, the method remains promising for future
analysis given the high value of $r_{B}$ in this mode ($\sim 0.25$). The
extension to ${{B}^{0}}\rightarrow
D[\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}]{{K}^{+}}{{\pi}^{-}}$ decays is of particular
importance as it allows for a so-called ‘double Dalitz’ model-independent
analysis to be performed [28, 161]. Extrapolating yields from a Run 1
${{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}}$ analysis to a dataset corresponding
to an integrated luminosity of 23 $\mbox{\,fb}^{-1}$, sensitivity studies
indicate that a precision on $\gamma$ of $3^{\circ}$ can be expected [162],
thus by extension, a sub-degree precision is expected for Upgrade II. Another
high-multiplicity mode that holds promise is $B^{\pm}\rightarrow
D{{K}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$, which has been studied with Run 1 data
with two-body $D$ decays [163], though due to the unknown Dalitz structure of
the $B$ decay a much larger dataset is needed to fully exploit this mode.

Another exciting extension to the standard ADS technique uses
${{B}^{\pm}}\rightarrow D[\rightarrow\pi K\pi\pi]{{K}^{\pm}}$ decays where the
five-dimensional (5D) Dalitz volume of the $D$ decay is split into bins. In
each bin, ADS/GLW relations derived from in Eq. 4.1 can be measured. Excellent
sensitivity to $\gamma$ is achieved as long as the $D$-system parameters
($r_{D},\,\delta_{D},\,\kappa$) in each bin are known. An optimal binning
scheme is soon to be defined from the amplitude analysis of
$D^{0}\rightarrow\pi K\pi\pi$ and $D^{0}\rightarrow K\pi\pi\pi$ decays [164].
The expected sensitivity is shown in Fig. 4.4. An additional irreducible
uncertainty from the $D$ system measurements of $<1^{\circ}$ is expected.

Other multibody $D$ decays are under development with reciprocal charm-system
measurements underway: a ${{B}^{\pm}}\rightarrow D[\rightarrow
4\pi]{{K}^{\pm}}$ analysis can build on the $D$-system knowledge gathered in
Ref. [165]; an ADS analysis of ${{B}^{\pm}}\rightarrow
D[\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}K\pi]{{K}^{\pm}}$ decays
has been demonstrated with Run 1 [166]; and work on ${{B}^{\pm}}\rightarrow
D[\rightarrow KK\pi\pi]{{K}^{\pm}}$ decays [167] is envisaged.

Figure 4.4: Projected sensitivity for a binned analysis of
${{B}^{\pm}}\rightarrow D[\rightarrow\pi K\pi\pi]{{K}^{\pm}}$ decays.

### 4.2 Ultimate $\gamma$ sensitivity and considerations on external inputs to
the analyses

The best sensitivity to $\gamma$ is achieved through a combination of
measurements, as highlighted in Fig. 4.5 (left), where the contributions from
three different methods are shown. The GLW method provides four narrow
solutions for $\gamma$ whilst the ADS and GGSZ methods have poorer sensitivity
but, importantly, give a single solution. Combining this information makes it
possible to pick out the most consistent narrow solution. Furthermore, the
various methods are complementary because their dominant sources of systematic
uncertainty have different origins. A detailed description of the LHCb
combination strategy is given in Ref. [168]. The latest LHCb $\gamma$
combination, which dominates the world average, gives
$\gamma=(74.0^{+5.0}_{-5.8})^{\circ}$ [169].

Projections for the expected precision of the LHCb $\gamma$ combination are
shown in Fig 4.6, estimating the uncertainty on $\gamma$ to be $1.5^{\circ}$
and $0.35^{\circ}$ with 23 $\mbox{\,fb}^{-1}$ and 300 $\mbox{\,fb}^{-1}$ data
samples, respectively. The LHCb projections assume that the statistical
uncertainty scales with the data sample size and include the effect of the
increased centre-of-mass energy, increased trigger performance and increased
integrated luminosity. Most of the systematic uncertainties are driven by the
size of the data samples and corresponding simulation samples. The overall
sensitivity is predominantly driven by the GLW modes which provide the
narrowest solutions for $\gamma$. The dominant systematic uncertainty for
these modes will depend on knowledge of both the shape and rate of the
background from ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{-}}$ as well as uncertainties arising from particle
identification calibration and instrumental charge asymmetries. In order to
obtain the best possible precision on $\gamma$ it will be necessary to
ascertain the correlation of these uncertainties between the different GLW
modes. The ultimate sensitivity to these modes from Belle II will be
considerably less than that at LHCb, however detailed analysis from both
experiments will provide an important cross check. For decays with neutrals in
the final state, particularly the $C\\!P$-odd GLW mode with ${D}^{0}$
$\rightarrow$ ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ ${\pi}^{0}$, the LHCb
detector has considerable disadvantages over Belle II. However these would be
mitigated by an improved electromagnetic calorimeter for the LHCb Upgrade Ib
and Upgrade II. The GGSZ modes are a powerful way to unambiguously resolve the
multiple solutions of the GLW method and furthermore offer considerable
standalone sensitivity to $\gamma$. Accurate knowledge of the selection
efficiency across the Dalitz plane is vital for these modes and contributes a
considerable systematic uncertainty. This will naturally reduce with larger
datasets as it is obtained via semi-leptonic
${{B}^{0}}\rightarrow{{D}^{*+}}{\mu^{-}}{{\nu}_{\mu}}X$ control modes but
requires large simulation samples. More precise measurement of important
external parameters, particularly $c_{i}$ and $s_{i}$ from BESIII, will be
required to reduce the uncertainty associated with the model independent GGSZ
method. The uncertainties of inputs from charm threshold data collected by
CLEO-c will begin to limit the sensitivity by the end of Run 2, so it is
essential to work together with BESIII to provide updated measurements for the
suite of charm decays and ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{h}^{+}{h}^{-}$ in particular. Provided that the charm inputs are improved
sub-degree level precision on $\gamma$ is attainable. Understanding the
correlations between different $B$ decay modes that all use these external
parameters will be vitally important as they are likely to contribute one of
the largest overall systematic uncertainties in the combination. A comparison
between the current LHCb GGSZ and GLW/ADS measurements [153, 148] with their
future projections at 300$\mbox{\,fb}^{-1}$ is shown in Fig. 4.5 (right). The
order of magnitude increase in precision is very apparent and the importance
of the combination clear, given the multiple ambiguous solutions for GLW/ADS
measurements is not resolved with increased luminosity.

Figure 4.5: Left: Comparison of different methods used to determine the angle
$\gamma$, reproduced from Ref. [27]. Right: Comparison between the current
LHCb 3-body GGSZ and 2-body GLW/ADS measurements alongside their future
projections with 300$\mbox{\,fb}^{-1}$ in the plane of $\gamma$ vs.
$r_{B}^{DK}$ (note the curtailed $y$-axis for $r_{B}^{DK}$). The scan is
produced using a pseudo-experiment, centred at $\gamma=70^{\circ}$,
$r_{B}^{DK}=0.1$, with ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays only.

The GGSZ modes are considered the _golden modes_ at Belle II and drive the
overall uncertainty on $\gamma$ which is expected to reach $1.5^{\circ}$ with
a data sample of $50~{}{\rm ab}^{-1}$. This is comparable to the sensitivity
that the LHCb $\gamma$ combination will achieve with a data sample
corresponding to approximately $23\mbox{\,fb}^{-1}$. Subsequently input from
Belle II will still contribute towards the world average by the end of LHCb’s
Upgrade I but LHCb will dominate $\gamma$ measurements with Upgrade II
(300$\mbox{\,fb}^{-1}$) contributing entirely towards a world average
precision of $\sim 0.35^{\circ}$. The impact of this measurement on the
unitarity triangle fit is shown in Fig. 10.2. It should be emphasised that
this projection includes only the currently used strategies, and does not
include improvements from other approaches. A comparison between the projected
uncertainties for LHCb and the world average as a function of integrated
luminosity is shown in Fig. 4.6.

Figure 4.6: Projected sensitivity for the LHCb $\gamma$ combination with the
currently used strategies and the world average using projections from Belle
II in addition.

### 4.3 Amplitude analysis of $B^{+}\rightarrow h^{+}h^{+}h^{-}$ decays

\begin{overpic}[width=208.13574pt]{LHCb-PAPER-2014-044-Fig3a.pdf}
\put(73.0,58.0){\bf{(a)}}
\end{overpic}\begin{overpic}[width=208.13574pt]{LHCb-PAPER-2014-044-Fig3b.pdf}
\put(73.0,58.0){\bf{(b)}} \end{overpic}

\begin{overpic}[width=208.13574pt]{LHCb-PAPER-2014-044-Fig3c.pdf}
\put(73.0,58.0){\bf{(c)}}
\end{overpic}\begin{overpic}[width=208.13574pt]{LHCb-PAPER-2014-044-Fig3d.pdf}
\put(73.0,58.0){\bf{(d)}} \end{overpic}

Figure 4.7:  Asymmetry in the number of background-subtracted and efficiency-
corrected candidates as a function of position in phase space for (a)
${{{B}^{+}}}\rightarrow{{K}^{+}}{{K}^{+}}{{K}^{-}}$, (b)
${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, (c)
${{{B}^{+}}}\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, and (d)
${{{B}^{+}}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{K}^{-}}$ decays [170].

The decays of charged $B$ mesons to three-body final states containing charged
pions and kaons have elicited much recent interest due to the observation
[170] of extremely large $C\\!P$-violating asymmetries. While the phase-space
integrated asymmetries are of order a few percent, the asymmetries as a
function of the position in phase space are considerably larger, even
approaching $\pm 1$ in some areas, as illustrated in Fig. 4.7. The pattern in
which the $C\\!P$ asymmetries vary is also intriguing. While some of the
asymmetries appear to be associated with well established resonances, such as
the $\rho(770)^{0}$ meson, there are others in regions of the phase space
where there are no such structures. It may be that additional strong phase
differences are being generated by rescattering processes, the effects of
which should be particularly apparent at the opening of thresholds.
Furthermore, it is possible that new physics processes could be contributing
and perhaps enhancing the size of the asymmetries. It is vital therefore to
perform amplitude analyses to try to disentangle the different contributions
and hence to better understand the mechanisms behind these asymmetries. Such
amplitude analyses are currently ongoing at LHCb and have previously been
performed for a subset of modes, albeit with much smaller event samples, at
the $B$ factories [171, 172, 173, 174, 128].

Extrapolating the observed yields in Ref. [170] to the data sample expected to
be collected at the end of Upgrade II gives signal yields of between 1.2 and
36 million depending on the final state. These can be compared to an
extrapolation of the yields obtained by the $B$ factories [173, 175] to the
expected 50$\mbox{\,ab}^{-1}$ sample to be collected by Belle II, which gives
approximately 47,000 to 660,000 events for the same range of modes. The far
larger yields, combined with the much better signal to background ratios, mean
that LHCb will continue to dominate the precision in these modes. Even with
such large data samples, model-independent analyses of the $C\\!P$ asymmetries
as a function of phase space can, with appropriate reduction of the bin sizes,
avoid becoming systematically limited. The results of these analyses can be
compared with predictions from theoretical models, for example as developed in
Ref. [176]. Such theoretical models for charmless hadronic $B$ decays are in
their relative infancy at present and it is hard to predict how far they can
develop on the timescale of the Upgrade II. However, it is clear that
significant advances will be needed in order for these asymmetry measurements
to be related back to CKM parameters.

There are numerous methods that allow the extraction of CKM information by
combining amplitude measurements made in different decay channels, in
particular the ${B^{0}_{(s)}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}}$, ${B^{0}_{(s)}\rightarrow h^{+}h^{-}{{\pi}^{0}}}$, and
${B^{+}\rightarrow h^{+}h^{+}h^{-}}$ families. Some such methods exploit the
interference between various ${{K}^{*}}{\pi}$ or $\rho{K}$ contributions,
which can be related using isospin symmetry [134, 135, 136, 137, 177, 178].
Constructing such relations enables the determination of the angle $\gamma$,
up to corrections for contributions from electroweak penguins. These methods
are particularly promising when using decays such as
${{{B}^{0}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}}$ and
${{{B}^{0}_{s}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{0}}}$, since here the
electroweak penguin contributions are expected to be negligible. Other methods
use the whole Dalitz plot amplitude and relate numerous decays via flavour
symmetries [179, 180, 181] or exploit interference between the charmless
decays and those that proceed via intermediate charmonium states such as
${{{B}^{+}}}\rightarrow{\chi_{{c}0}}{{\pi}^{+}}$ [182, 183].

Amplitude analyses cannot benefit from cancellations of systematic effects to
the same degree as the binned measurements of asymmetries, and are more likely
to become systematically limited. On the other hand, the extremely large
${B^{+}\rightarrow h^{+}h^{+}h^{-}}$ signal samples will allow new information
to be extracted. In particular, by performing coupled-channel analyses of
these decay modes, contributions from
${{\pi}^{+}}{{\pi}^{-}}\leftrightarrow{{K}^{+}}{{K}^{-}}$ rescattering
processes can be better understood and constrained. The development of such
new, improved models will also be of enormous benefit to analyses of many
other decay modes, such as the closely related
${B}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ and
${B}\rightarrow h^{+}h^{-}{{\pi}^{0}}$ families mentioned above and discussed
in more detail in Secs. 3.4.4 and 3.4.5. This will help to reduce the
corresponding uncertainty on the CKM phases that can be determined from those
channels.

The class of very rare hadronic decays such as
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{K}^{+}}{{\pi}^{-}}$ and
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{+}}{{K}^{-}}$ [184] proceed via
diagrams that are extremely suppressed in the SM and hence have predicted
branching fractions of order $10^{-14}$–$10^{-11}$ [185]. However, many BSM
scenarios predict significant enhancements to these decays [186, 187] to
levels of between $10^{-9}$–$10^{-6}$. Preliminary studies indicate that
sensitivities better than $10^{-9}$ can be achieved in an Upgrade II scenario
assuming similar detector performance to the present one can be maintained. Of
particular importance is the use of particle identification criteria to
suppress the backgrounds from the favoured decays such as
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ that have branching
fractions of order $10^{-5}$, and the addition of TORCH may play a
particularly important role in this.

### 4.4 $C\\!P$ violation in $b$-baryon decays

In contrast with the study of $C\\!P$ violation in beauty-meson decays, the
sector of beauty baryons remains almost unexplored. Previous to the LHC era,
only a measurement of direct $C\\!P$ asymmetries in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays was available with
${\mathcal{O}}(0.1)$ precision [188]. Thanks to the large production cross-
section of beauty baryons in $p$ $p$ collisions at the LHC, the LHCb
experiment is the only experiment capable of expanding our knowledge in this
sector, as these decays are not accessible at the $e^{+}$ $e^{-}$ KEK
collider. Hence, even though a handful of $C\\!P$ asymmetries of ${\mathchar
28931\relax}^{0}_{b}$ decays have been measured so far by LHCb [189, 190, 191,
192], the landscape of $C\\!P$ violation in the decays of beauty baryons is
expected to change rapidly in the next few years. The first observation of
$C\\!P$ violation in a baryon decay is already within the reach of LHCb with
the data collected during the Run 2 of the LHC, considering that a first
evidence for $C\\!P$ violation in baryon decays has been reported in
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays [193].

The unprecedented number of beauty baryons available with the data sample
expected to be collected in the LHCb Upgrade II phase, will allow a precision
measurement programme of $C\\!P$ violation observables in $b$-baryon decays to
be pursued, analogously to $b$-meson decays. A very interesting example is the
study of decays governed by ${b}\\!\rightarrow{u}$ and ${b}\\!\rightarrow{c}$
tree-level transitions, like the decays ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{D}^{0}}{\mathchar 28931\relax}$ and
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{D}^{0}}{p}{{K}^{-}}$. These
decays can be used to measure the angle $\gamma$ of the unitarity triangle
[194, 195, 196] in a similar way to what can be done with
${{B}^{0}}\\!\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ decays. The LHCb experiment
reported the first observation of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){p}{{K}^{-}}$,
based on a signal yield of $163\pm 18$ using a sample of $p$ $p$ collisions
corresponding to 1$\mbox{\,fb}^{-1}$ of integrated luminosity at a centre-of-
mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$ [197]. Extrapolating to
Upgrade II approximately 95000 signal decays are expected. However,
extrapolating the sensitivity to $\gamma$ is not easy, since it strongly
depends on the values of the hadronic parameters involved in the process. In
addition, even though the determination of $\gamma$ from the analysis of these
decays is expected to be theoretically very clean, the possible polarisation
of ${\mathchar 28931\relax}^{0}_{b}$ baryons produced in $p$ $p$ collisions
has to be taken into account and might represent a limiting factor for high-
precision measurements.

Another very interesting sector is that of beauty baryons decaying to final
states without a charm quark. These decays receive relevant contributions from
${b}\\!\rightarrow{d},\,{s}$ loop-level transitions, where new physics beyond
the SM may appear. Also in this case, similar quantities to those measured
with $B$-meson decays are available. For example, statistical precisions of
${\mathcal{O}}(10^{-3})$ and ${\mathcal{O}}(10^{-2})$ are expected for the
$C\\!P$ asymmetries of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}h^{-}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}h^{+}h^{-}$ decays
(with $h={K},\,{\pi}$), respectively. Very large signal yields are also
expected in several multibody final states of ${\mathchar
28931\relax}^{0}_{b}$ and ${\mathchar 28932\relax}_{b}$ decays: about $10^{6}$
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}{{K}^{+}}{{K}^{-}}$ decays, and
about $10^{5}$ ${\mathchar
28932\relax}^{0}_{b}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}{{K}^{-}}$ decays
[193, 198]. Such a signal yield will allow very precise measurements of
$C\\!P$-violating quantities to be made over the phase space of these decays,
characterised by a rich set of resonances. Unfortunately, as for the charmless
decays of $B$ mesons, the interpretation of these quantities in terms of CKM
parameters is still unclear from the theoretical point of view. Hence, more
theoretical work is crucial to exploit the full potential of beauty baryons.

Experimentally, the main issues are the determination of particle-antiparticle
production asymmetries and detection asymmetries that could mimic
$C\\!P$-violation effects. This task is generally more difficult for heavy
baryons, with respect to $B$ mesons, since methods used for measuring meson
production asymmetries [199] cannot be applied. In addition, different
interactions of baryons and antibaryons with the detector material are
difficult to calibrate. Nonetheless, several quantities can be measured in
$b$-baryon decays that are sensitive to different manifestations of $C\\!P$
violation and are largely unaffected by experimental effects. A few examples
are the difference of $C\\!P$-violating asymmetries of particles decaying to a
similar final state, $\Delta A_{{C\\!P}}$ [200], triple-product asymmetries
(TPA) [193] and energy-test (ET) [201]. It is important to note that TPA and
ET are important tools for discovery of $C\\!P$ violation in multibody decays,
while an amplitude analysis is required to study the source of $C\\!P$
violation.

## 5 Measurements of unitarity triangle sides and semileptonic decays

Tree-level charged-current semileptonic decays of the type $b\rightarrow
c\ell\nu$ and $b\rightarrow u\ell\nu$ play a critical role in the search for
new physics through quark flavour mixing. That is because these decays can
provide a theoretically clean determination of the CKM matrix elements
$|V_{{u}{b}}|$ and $|V_{{c}{b}}|$, which are essential ingredients in global
fits that assume the unitarity of the CKM matrix. These measurements have had
rather a long and puzzling history, in which those made with inclusive and
exclusive final states have differed by over three standard deviations. Given
these controversies it is clear that precision measurements with the full
suite of $b$ hadron species, only accessible to LHCb, will be crucial to
provide the maximum confidence if a discrepancy appears in the unitarity
triangle fits.

Semileptonic decays, being flavour specific, provide a unique probe of
${B}^{0}_{q}$, where $q=s,d$, meson mixing phenomena. In particular, $C\\!P$
violation in ${B}^{0}_{q}$ meson mixing can be expressed through the
semileptonic asymmetries

$a_{\rm sl}^{q}=\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{q}}\rightarrow
f)-\Gamma({{B}^{0}_{q}}\rightarrow\bar{f})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{q}}\rightarrow
f)+\Gamma({{B}^{0}_{q}}\rightarrow\bar{f})}\approx\frac{\Delta\Gamma_{q}}{\Delta
M_{q}}\tan\phi_{12}^{q},$ (5.1)

where $f$ is a flavour-specific final state that is only accessible through
the decay of the $B_{q}^{0}$ state. Mixing is required to mediate the
transition ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{q}}\rightarrow{{B}^{0}_{q}}\rightarrow
f$, and its conjugate. Semileptonic decays of the type
${{B}^{0}_{q}}\rightarrow D_{q}^{-}\mu^{+}\nu_{\mu}X$ are well suited because
(i) they are immune to any unknown $C\\!P$ violation in decay and (ii) they
have large branching ratios. These asymmetries are precisely predicted to be
tiny in the SM [202, 36], while being highly sensitive to new mixing
amplitudes in NP scenarios.

In the SM, the ${B}\\!\rightarrow D^{(*)}\mu\nu$ and ${B}\\!\rightarrow
D^{(*)}\tau\nu$ decays differ only through the effect of the lepton mass on
the available phase space. The ratio of their decay rates,

$\mathcal{R}(D^{(*)})\equiv\frac{\mbox{${B}\\!\rightarrow
D^{(*)}\tau\nu$}}{\mbox{${B}\\!\rightarrow D^{(*)}\mu\nu$}},$ (5.2)

is theoretically extremely clean, since most form factor uncertainties cancel,
while being sensitive to a range of NP models with preferential couplings to
the third generation. At the time of writing, the world averages for these
quantities are in $\sim 4\sigma$ tension with the SM expectation [27]. This
situation will be clarified with more precise measurements in the relatively
near future. If the discrepancy with the SM expectation persists, new
observables with a range of different decay modes are needed to distinguish
between different NP explanations.

### 5.1 Determination of $|V_{{u}{b}}|$ and $|V_{{c}{b}}|$

LHCb is well suited to measuring ratios of $b\rightarrow u\ell\nu$ to
$b\rightarrow c\ell\nu$ decays rates, in which the unknown $b$ production
cross sections, and to some extent also efficiency corrections, cancel. LHCb
reported a first study of the decays ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow p{\mu^{-}}{{\overline{\nu}}_{\mu}}$ and
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ with Run 1 data, which
resulted in a determination of $|V_{{u}{b}}|$/$|V_{{c}{b}}|$ [203], exploiting
precise lattice QCD calculations of the decay form factors [204, 205].

LHCb Upgrade II presents an exciting opportunity for new measurements of this
type. An excellent example is the analogous analysis with
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decays. The
relatively large spectator $s$ quark mass allows the form factors of these
decays to be computed with lattice QCD to higher precision than decays of
other $b$ species. There are also good prospects to extend the approach of
[203], with a single $q^{2}$ bin, to perform a differential measurement in
many fine bins of $q^{2}$ [206], which clearly demands substantially larger
sample sizes. Furthermore, there are several reasons to expect that, compared
to the study of ${\mathchar 28931\relax}^{0}_{b}$ decays [203], far larger
integrated luminosities are required for the ultimate precision with
${B}^{0}_{s}$ decays. Firstly the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ signal rate is
roughly one order of magnitude smaller compared to ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow p{\mu^{-}}{{\overline{\nu}}_{\mu}}$.
Secondly the ${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$
decay is subject to backgrounds from all $b$ hadron species whereas
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow
p{\mu^{-}}{{\overline{\nu}}_{\mu}}$ is primarily contaminated by other
${\mathchar 28931\relax}^{0}_{b}$ decays. Thirdly the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ decay rate is
further suppressed with respect to ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow p{\mu^{-}}{{\overline{\nu}}_{\mu}}$ at the
higher $q^{2}$ values at which the lattice QCD uncertainties are smallest.

Figure 5.1:  The factor by which the effective integrated luminosity would be
increased in an analysis of
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ decays, as a
function of the thickness of the RF foil. This factor is defined with respect
to the vertex resolution of the current detector. It is determined using
pseudoexperiments in which the
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ signal yield is
determined through a likelihood fit to the corrected mass distribution. The
other variables (primarily the first hit radius and resolution) affecting the
vertex resolution are fixed to the Upgrade I values, which should be possible
to maintain for Upgrade II as discussed in Sect. 2.3.1. The improvement in
these variables, with respect to the current detector, is the reason that the
blue trend line would cross the lower horizontal line at a larger foil
thickness than that of the real (current) detector (while crossing the upper
horizontal line at the Upgrade I thickness of 250 $\mu$m).

Upgrade II will also include several potential gains in detector performance
which are highly relevant to the reconstruction of decays like
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$. The key variable
which distinguishes the signal from background processes is the corrected
mass, which depends on the reconstructed line-of-flight between the PV and the
${B}^{0}_{s}$ decay vertex. It is the resolution on this direction which
dominates the corrected mass resolution. The removal, or further thinning, of
the RF foil is therefore a really appealing prospect, since this would reduce
the multiple scattering contribution to the corrected mass resolution. Fig.
5.1 shows the potential gain in effective integrated luminosity that can be
achieved through a reduction of the RF foil thickness. Hence, in the limit the
RF foil could be removed entirely, then there would be almost a 1.6 increase
in effective integrated luminosity. This analysis only considers the effect of
the improved corrected mass resolution, while further improvements are
expected in the selection efficiency, purity and $q^{2}$ resolution.

The dominant systematic uncertainty in the ${\mathchar 28931\relax}^{0}_{b}$
analysis [203] (material budget and its effect on the charged hadron
reconstruction efficiency) can be tightly constrained with new methods, the
performance of which will be greatly enhanced by any reduction in the RF foil,
as discussed in Sect. 5.2. The lattice QCD form factors are most precise at
large $q^{2}$ values, which correspond to low momentum kaons that are not
efficiently identified with the RICH approach of the current detector. The low
momentum PID performance of the proposed TORCH detector would greatly enhance
our capabilities with
${{B}^{0}_{s}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ decays at high
$q^{2}$.

The combination of these detector improvements with the Upgrade II dataset is
expected to reduce the experimental systematic uncertainties to the 0.5% level
on $|V_{{u}{b}}|$/$|V_{{c}{b}}|$. External branching ratio uncertainties will
also be greatly reduced at the BESIII experiment. For example
$\mathcal{B}(D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+})$ will be determined to
the 1% level, translating to a $\sim 0.5\%$ uncertainty on
$|V_{{u}{b}}|$/$|V_{{c}{b}}|$. This combined with the differential shape
information of the signals and continued improvements to the lattice
calculations will lead to a $|V_{{u}{b}}|$/$|V_{{c}{b}}|$ measurement
uncertainty of less than 1% with the Upgrade II dataset.

Upgrade II will allow several currently inaccessible decays, not least those
of the rarely produced ${B}_{c}^{+}$ mesons, to be studied in detailed. A
prime example is the decay
${{B}_{c}^{+}}\\!\rightarrow{{D}^{0}}{\mu^{+}}{{\nu}_{\mu}}$, which is
potentially very clean from a theory point of view, due to the heavy charm
spectator. Approximately 30,000 reconstructed decays can be expected with the
300$\mbox{\,fb}^{-1}$ Upgrade II dataset, which could lead to a competitive
measurement of $|V_{{u}{b}}|$ in this currently unexplored system. Purely
leptonic decays such as $B^{+}\rightarrow\mu^{+}\mu^{-}\mu^{+}\nu_{\mu}$ will
also become competitive, with similar signal yields, and can provide
information on the $B$-meson light cone distribution amplitude which is a
crucial input to the widely used theoretical tool of QCD factorisation.

The standalone determination of $|V_{cb}|$ will also be increasingly important
as other measurements get more precise, as it will become the limiting factor
in many SM predictions such as the branching fraction of
${{B}^{0}_{s}}\rightarrow{\mu^{+}\mu^{-}}$ [207]. The current uncertainty is
inflated due to the disagreement between measurements from inclusive and
exclusive final states and currently appears to critically depend on the
parameterisation used to fit the form factors [208, 209]. LHCb has already
performed a measurement of the differential rate of the decay ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}\mu\nu$,
allowing a determination of the form factors of that decay [210]. A first
determination of the absolute value of $|V_{cb}|$, exploiting theoretical
predictions for the ratios of semileptonic decay widths between different $b$
hadron species [211], is in progress. The LHCb Upgrade II dataset would
provide large samples of exclusive $b\rightarrow c\ell\nu$ decays, with the
full range of $b$ hadron species, with which incredibly precise shape
measurements could be performed, as a crucial ingredient to reach the ultimate
precision of $|V_{{c}{b}}|$.

### 5.2 Semileptonic asymmetries $a^{d}_{\rm sl}$ and $a^{s}_{\rm sl}$

The observable defined in Eq. 5.1 would require flavour tagging to be
measurable. Including the effect of an unknown production asymmetry, $a_{p}$,
the time-dependent untagged asymmetry is defined as:

$A_{\rm sl}^{q}(t)\equiv\frac{N-\bar{N}}{N+\bar{N}}=\frac{a_{\rm
sl}^{q}}{2}-\left[a_{p}+\frac{a_{\rm
sl}^{q}}{2}\right]\cdot\left[\frac{\cos\Delta
M_{q}t}{\cosh\Delta\Gamma_{q}t/2}\right],$ (5.3)

where $N$ and $\bar{N}$ denote the number of observed decays to the $f$ and
$\bar{f}$ final states, respectively. A decay-time-dependent fit can
disentangle the $B_{d}^{0}-\bar{B}_{d}^{0}$ production asymmetry from $a_{\rm
sl}^{d}$ [212]. In the $B_{s}^{0}$ case the time-integrated asymmetry is
employed [213]. Owing to the large value of $\Delta M_{s}$ the term containing
$a_{p}$ is suppressed to a calculable correction of a few $\times 10^{-4}$,
after integrating over decay time. These approaches have been applied in the
measurements with the Run 2 dataset that are listed in Table 5.1. These are
already the world’s best single measurements of these parameters. However
these measurements are far from any uncertainty floor in the SM predictions,
while being sensitive to a vast number of anomalous NP contributions to
$\Gamma_{12}^{q}$ and $M_{12}^{q}$.

The following briefly reviews the dominant sources of uncertainty on the
current LHCb measurements, and considers strategies to fully exploit the
potential of the Upgrade II. All uncertainties are as evaluated on $a_{\rm
sl}^{q}$ (i.e. all sources of raw asymmetry, and their uncertainties, are
scaled by a factor of two as expected from Eq. 5.3).

Statistical precision:

The statistical uncertainty on $a^{s}_{\rm sl}$ straightforwardly extrapolates
to $2.1\times 10^{-4}$. In the case of $a^{d}_{\rm sl}$ it should be noted
that stringent fiducial cuts and weights were imposed on the signal sample to
control certain tracking asymmetries that were not well known at the time. By
the time of the subsequent $a^{s}_{\rm sl}$ measurement, a new method with
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decays had been developed, and others are in
the pipeline. A factor of two further increase in yields is therefore assumed,
which extrapolates to an uncertainty of $1.1\times 10^{-4}$.

Detection asymmetries:

The single largest contributor is the $K^{-}\pi^{+}$ asymmetry in $a^{d}_{\rm
sl}$. This asymmetry was determined with a single method using $D^{+}$ decays
to the $K^{-}\pi^{+}\pi^{+}$ and ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\pi^{+}$ final states, with a precision of around $2.0\times 10^{-3}$
[212]. Thanks to trigger improvements, a factor of two increase in the
effective yield of the most limiting ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\pi^{+}$ final state [214] can be anticipated. This will extrapolate to an
uncertainty of $1.1\times 10^{-4}$. With improvements in the reconstruction of
downstream tracks in the first HLT stage, we may also be able to exploit
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{+}$ final states with
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decays downstream of the VELO.
Further methods have since been proposed using partially reconstructed
$D^{*+}$ decays and $D^{0}\rightarrow K^{-}K^{+}$ decays. The partial
reconstruction method will be greatly improved by the reduction of material
before the first VELO measurement point. The systematic uncertainties in these
approaches will be controlled by using ultra high statistics fast simulations
to track the kinematic dependencies in the asymmetries. We target an
uncertainty of $1.0\times 10^{-4}$ including systematic uncertainties. The
equivalent $K^{+}K^{-}$ asymmetry in the $a^{s}_{\rm sl}$ measurement will be
smaller and more precisely controlled. The $\mu^{+}\pi^{-}$ asymmetry will be
controlled by a combination of $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays,
partially reconstructed $D^{*+}$ decays, $D^{0}\rightarrow h^{-}h^{+}$ decays,
and high statistics fast simulations.

Background asymmetries:

These measurements are challenging because the ${{B}^{0}_{q}}\rightarrow
D_{q}^{-}\mu^{+}\nu_{\mu}X$ final states can be fed by the decays of other $b$
hadron species. This dilutes the relation between the raw asymmetry and
$a_{\rm sl}^{q}$ and leads to a cocktail of production asymmetry corrections.
We assume that these backgrounds can be statistically subtracted by extending
the signal fits to include the $D_{q}^{-}\mu^{+}$ corrected mass dimension. It
is assumed that the background asymmetry uncertainties can be controlled to
the $1.0\times 10^{-4}$ level.

Other considerations:

We must assess the impact of having unequal sample sizes in the two
polarities. This can be partially compensated for by assigning weights to one
polarity [215]. We note that the choice of crossing angles should be carefully
considered [216]. While we do not account for them in the current estimation,
we utilise other $D^{+}_{q}$ decay modes to better align the detection
asymmetries between $a_{\rm sl}^{s}$ and $a_{\rm sl}^{d}$. For example,
$D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ and
$D^{+}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{+}$ decays can
be used, taking advantage of possible improvements in the trigger efficiency
for ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decays in Upgrade II. While the
former decay is singly Cabibbo suppressed, its $C\\!P$ asymmetry could be
measured using promptly produced $D^{+}$ mesons.

In summary the Upgrade II dataset should allow both asymmetries to be measured
to the level of a few parts in $10^{-4}$, as listed in Table 5.1, which will
give unprecedented new physics sensitivity, and is still far from saturating
the current theory uncertainties in the SM predictions. Figure 5.2 shows the
prospective Upgrade II measurement, arbitrarily centred at a value that
differs from the SM prediction at the $10^{-3}$ level.

Table 5.1: Current theoretical and experimental determinations of the semileptonic asymmetries $a^{d}_{\rm sl}$ and $a^{s}_{\rm sl}$. Sample ($\mathcal{L}$) | $\delta a^{s}_{\rm sl}[10^{-4}]$ | $\delta a^{d}_{\rm sl}[10^{-4}]$
---|---|---
Run 1 (3 $\mbox{\,fb}^{-1}$) [213, 212] | $33$ | $36$
Run 1-3 (23 $\mbox{\,fb}^{-1}$) | $10$ | $8$
Run 1-3 (50 $\mbox{\,fb}^{-1}$) | $7$ | $5$
Run 1-5 (300 $\mbox{\,fb}^{-1}$) | $3$ | $2$
Current theory [202, 36] | $0.03$ | $0.6$
Figure 5.2:  Current and future landscape for the semileptonic asymmetries.
The grey vertical band indicates the current $B$-Factory average for $a_{\rm
sl}^{d}$ [27]. The blue ellipse represents the current LHCb Run 1 measurements
[213, 212]. The red ellipse, which is arbitrarily centred, delineates the
Upgrade II projected precision. The black ellipse shows the SM prediction, the
uncertainty of which is barely visible with these axes.

### 5.3 Lepton-flavour universality tests with $b\rightarrow cl\nu$
transitions

#### 5.3.1 $R(D)$ and $R(D^{\ast})$ with muonic and hadronic $\tau$ decays

Figure 5.3:  The projected absolute uncertainties on $\mathcal{R}(D^{\ast})$
and $\mathcal{R}(J/\psi)$ (see Sect. 5.3.2) from the current sensitivities (at
3$\mbox{\,fb}^{-1}$) to 23$\mbox{\,fb}^{-1}$, 50$\mbox{\,fb}^{-1}$, and
300$\mbox{\,fb}^{-1}$.

LHCb has made measurements of $\mathcal{R}(D^{(*)})$ using both muonic
($\tau^{+}\rightarrow{\mu^{+}}\nu\nu$) and hadronic
($\tau^{+}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}\nu$) decays of the tau
lepton [217, 218, 219]. Due to the presence of multiple neutrinos these decays
are extremely challenging to measure. The measurements rely on isolation
techniques to suppress partially reconstructed backgrounds, $B$ meson flight
information to constrain the kinematics of the unreconstructed neutrinos, and
a multidimensional template fit to determine the signal yield. Figure 5.3
shows how the absolute uncertainties on the LHCb muonic and hadronic
$\mathcal{R}(D^{\ast})$ measurements are projected to evolve with respect to
the current status. The major uncertainties are the statistical uncertainty
from the fit, the uncertainties on the background modelling and the limited
size of simulated samples. A major effort is already underway to commission
fast simulation tools. The background modelling is driven by a strategy of
dedicated control samples in the data, and so this uncertainty will continue
to improve with larger data samples. From Run 3 onward it is assumed that,
taking advantage of the full software HLT, the hadronic analysis can normalise
directly to the $B^{0}\rightarrow D^{\ast-}\mu^{+}\nu_{\mu}$ decay, thus
eliminating the uncertainty from external measurements of
$\mathcal{B}(B^{0}\rightarrow D^{\ast-}\pi^{+}\pi^{-}\pi^{+})$. It is assumed
that all other sources of systematic uncertainty will scale as
$\sqrt{\mathcal{L}}$. With these assumptions, an absolute uncertainty on
$\mathcal{R}(D^{\ast})$ of $0.003$ will be achievable for the muonic and
hadronic modes with the 300$\mbox{\,fb}^{-1}$ Upgrade II dataset.

On the timescale of Upgrade II, interest will shift toward new observables
beyond the branching fraction ratio [220]. The kinematics of the
${B}\\!\rightarrow D^{*}\tau\nu$ decays is fully described by the dilepton
mass, and three angles which are denoted $\chi$, $\theta_{L}$ and
$\theta_{D}$. LHCb is capable of resolving these three angles, as can be seen
in Fig. 5.4. However, the broad resolutions demand very large samples to
extract the underlying physics. The decay distributions within this kinematic
space are governed by the underlying spin structure, and precise measurements
of these distributions will allow the different helicity amplitudes to be
disentangled. This can be used both to constrain the spin structure of any
potential new physics contribution, and to measure the hadronic parameters
governing the ${B}\\!\rightarrow D^{*}\tau\nu$ decay, serving as an essential
baseline for SM and non-SM studies. The helicity-suppressed amplitude which
presently dominates the theoretical uncertainty on $\mathcal{R}(D^{(*)})$ is
too strongly suppressed in the ${B}\\!\rightarrow D^{(*)}\mu\nu$ decays to be
measurable, however this can be accessed in the ${B}\\!\rightarrow
D^{(*)}\tau\nu$ decay directly. If any potential new physics contributions are
assumed not to contribute via the helicity-suppressed amplitude then the
combined measurements of ${B}\\!\rightarrow D^{(*)}\mu\nu$ and
${B}\\!\rightarrow D^{(*)}\tau\nu$ decays will allow for a fully data-driven
prediction for $\mathcal{R}(D^{(*)})$ under the assumption of lepton
universality, eliminating the need for any theory input relating to hadronic
form factors. However, these measurements have yet to be demonstrated with
existing data. This exciting programme of differential measurements needs to
be developed on Run 1 and 2 data before any statement is made about the
precise sensitivity, but it offers unparalleled potential to fully
characterise both the SM and non-SM contributions to the $b\rightarrow
c\tau\nu$ transition.

Figure 5.4:  Angular resolution for simulated ${B}\\!\rightarrow D^{*}\mu\nu$
(black) and ${B}\\!\rightarrow D^{*}\tau\nu$ (red) decays, with
$\tau^{+}\rightarrow{\mu^{+}}\nu\nu$. This demonstrates our ability to resolve
the full angular distribution, with some level of statistical dilution.

#### 5.3.2 Prospects with other $b$ hadrons

As measurements in $\mathcal{R}({{D}^{*}})$ become more statistically precise,
it will become increasingly more important to provide supplementary
measurements in other $b$-hadron species with different background structure
and different sources of systematic uncertainties. For example, the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{D}^{+}_{s}}{\tau^{-}}{\overline{\nu}}$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{D}^{*+}_{s}}{\tau^{-}}{\overline{\nu}}$
decays will allow supplementary measurements at high yields, and do not suffer
as badly from cross-feed backgrounds from other mesons, unlike, for example,
${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{D}^{*+}}{\tau^{-}}{\overline{\nu}}$,
where the ${{B}^{+}}$ and ${B}^{0}_{s}$ both contribute to the ${{D}^{*+}}\mu
X$ or ${{D}^{*+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}X$ final states.
Furthermore, the comparison of decays with different spins of the $b$ and $c$
hadrons can enhance the sensitivity to different NP scenarios [221, 210]. No
published measurements exist for the ${{B}^{0}_{s}}$ case yet, but based on
known relative efficiencies and assuming the statistical power of this mode
tracks $\mathcal{R}(D^{(*)})$, we expect less than $6\%$ relative uncertainty
after Run 3, and $2.5\%$ with the Upgrade II data, where limiting systematic
uncertainties are currently expected to arise from corrections to simulated
pointing and vertex resolutions, from knowledge of particle identification
efficiencies, and from knowledge of the backgrounds from random combinations
of charm and muons. It is conceivable that new techniques and control samples
could further increase the precision of these measurements.

Methods are currently under development for separating the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{D}^{*+}_{s}}\ell^{-}{\overline{\nu}}$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{D}^{+}_{s}}\ell^{-}{\overline{\nu}}$
modes, and given the relative slow pion
(${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$) and soft photon
(${{D}^{*+}_{s}}\\!\rightarrow{{D}^{+}_{s}}\gamma$) efficiencies, the
precision in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{D}^{+}_{s}}\tau\nu$
decays can be expected to exceed that in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{{D}^{*+}_{s}}\tau\nu$,
the reverse of the situation for $\mathcal{R}(D^{(*)})$. An upgraded ECAL
would extend the breadth and sensitivity of $\mathcal{R}(D^{*(*)+}_{s})$
measurements possible in the Upgrade II scenario above and beyond the possible
benefits of improved neutral isolation in $\mathcal{R}(D)$ or
$\mathcal{R}({{D}^{+}_{s}})$ measurements.

Of particular interest are the semitauonic decays of $b$ baryons and of
${B}_{c}^{+}$ mesons. The former provides probes of entirely new Lorentz
structures of NP operators which pseudoscalar to pseudoscalar or vector
transitions simply do not access. The value of probing this supplementary
space of couplings has already been demonstrated by LHCb with its Run 1
measurement of $|{V_{{u}{b}}}|$ via the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{\mu^{-}}{\overline{\nu}}$, which places
strong constraints on right-handed currents sometimes invoked to explain the
inclusive-exclusive tensions in that quantity. By the end of Run 3, it is
expected that the relative uncertainty for $\mathcal{R}({{\mathchar
28931\relax}^{+}_{c}})$ will reach below $4\%$, and $2.5\%$ by the end of
Upgrade II. A further exciting prospect is lepton universality tests with
$b\rightarrow u\ell\nu$ decays, with $\ell=\mu,\tau$, which have been beyond
experimental reach thus far. For example the decay $B^{+}\rightarrow
p\bar{p}\ell\nu$ offers a clean experimental signature. Our capabilities with
this decay could benefit from the enhanced low momentum proton identification
with the TORCH subdetector.

Meanwhile, the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\tau^{-}}{\overline{\nu}}$ decay is an entirely unique state among the
flavoured mesons as the bound state of two distinct flavors of heavy quark,
and, through its abundant decays to charmonium final states, provides a highly
efficient signature for triggering and reconstruction at high instantaneous
luminosities. Measurements of
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\ell{\overline{\nu}}$ decays involve a trade-off between the
approximately 100 times smaller production cross-section for ${B}_{c}^{+}$
verses the extremely efficient ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ signature in the LHCb trigger. For
illustration, in Run 1, LHCb reconstructed and selected 19 000
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mu^{-}}{\overline{\nu}}$ decays, compared with 360 000 ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{D}^{*+}}{\mu^{-}}{\overline{\nu}}$
decays. This resulted in a measurement of
$\mathcal{R}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})=0.71\pm 0.17\pm
0.18$ [222]. As a result of the smaller production cross-section, the muonic
measurements have large backgrounds from $h\\!\rightarrow\mu$
misidentification from the relatively abundant
${B}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X_{h}$ decays,
where $X_{h}$ is any collection of hadrons, and so they are very sensitive to
the performance of the muon system and PID algorithms in the future. Here it
is assumed that it will be possible to achieve similar performance to Run 1 in
the upgraded system.

To project the sensitivity for
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\tau^{-}}{\overline{\nu}}$ based on Ref. [222], it is assumed that all
the systematic uncertainties can be reduced with the size of the input data
except for those that were assumed not to scale with data for the previous
predictions. For these, we assume that they can be reduced up until they reach
the same absolute size as the corresponding systematic uncertainties in the
Run 1 muonic $\mathcal{R}({{D}^{*}})$ analysis. In addition, it is assumed
that sometime in the 2020s lattice QCD calculations of the form factors for
this process will allow the systematic uncertainty due to signal form factors
to be reduced by an additional factor of two. This results in a projected
absolute uncertainty for the muonic mode of $0.07$ at the end of Run 3 and
$0.02$ by the end of Upgrade II, as can be seen in Fig. 5.3. Measurements in
the hadronic mode can be expected to reach similar sensitivities.

## 6 Mixing and ${C\\!P}$ violation in charm

The mixing of neutral $D$ mesons probes flavour changing neutral currents
between up-type quarks, which could be affected by beyond-the-SM physics in
fundamentally different ways to the down-type quarks that make up the
${K}^{0}$, ${B}^{0}$ and ${B}^{0}_{s}$ systems.

A fully software trigger will allow LHCb to take full advantage of the
enormous production rate of charm hadrons at the HL-LHC, and collect the
largest sample, by far, of charm decays ever recorded: two orders or magnitude
more than Belle II in the key channels discussed here. The flavour-tagged
sample will be further boosted for Upgrade II by the addition of magnet
stations (see Section 6.1.1).

$C\\!P$ violation in mixing-related phenomena are predicted to be very small,
${\mathcal{O}}(10^{-4})$ or less in the SM [223, 224]: there are thus
excellent prospects for observing new physics contributions beyond the SM
level. In the absence of new physics contributions to charm $C\\!P$ violation,
the LHCb Upgrade II may well be the only facility with a realistic probability
of observing these phenomena as it will be able to reach a sensitivity of
${\mathcal{O}}(10^{-5})$. The LHCb programme is also robust, as strong
sensitivity is obtained in a range of modes with complementarity in their
dominant systematic uncertainties. The use of the doubly Cabibbo-suppressed
${{D}^{0}}\\!\rightarrow K^{+}\pi^{-}$ (Sect. 6.1.1) and
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ decays
(Sect. 6.1.2), $D^{0}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\pi^{+}\pi^{-}$ (Sect. 6.1.3) and $A_{\Gamma}$ in Cabibbo suppressed
${{D}^{0}}\\!\rightarrow{h}^{+}{h}^{-}$ $C\\!P$ eigenstates (Sect. 6.1.4) are
all discussed here. Sect. 6.1.5 and Fig. 6.3 summarise the remarkable impact
that Upgrade II will have on charm indirect $C\\!P$ violation.

Effects of $C\\!P$ violation in decay are less cleanly predicted, and could be
as large as ${\mathcal{O}}(10^{-3})$ [225] in charm, close to the current
levels of sensitivity. If current hints [226] are confirmed, or if other
signals emerge with future data, it will be possible to launch a comprehensive
survey of such effects in charm decays. The higher yields of the two-body
${{D}^{0}}\\!\rightarrow{h}^{+}{h}^{-}$ (Sect. 6.2.1) modes, and the ability
to probe the interfering structures in the phase space of multi-body decays,
such as $D_{(s)}^{+}\\!\rightarrow h^{+}h^{+}h^{-}$ (Sect. 6.2.1) and
${{D}^{0}}\\!\rightarrow h^{+}h^{-}h^{+}h^{-}$ (Sect. 6.2.3), provide
interestingly complementary programmes. The potential for LHCb to study
neutral modes (Sect. 6.2.4), and perform a thorough exploration of the largely
unstudied charm-baryon sector (Sect. 6.2.5) are also discussed.

### 6.1 Neutral $D$-meson mixing and indirect $C\\!P$ violation

The precision measurement of the mixing parameters and above all the discovery
of $C\\!P$ violation in the ${D}^{0}$ system, and the precise determination of
its parameters, will be the key goals of Upgrade II.

The neutral $D$ mesons mix to form mass eigenstates
$|{D}_{1}\rangle=p|{{D}^{0}}\rangle+q|{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rangle$ and
$|{D}_{2}\rangle=p|{{D}^{0}}\rangle-q|{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rangle$ with masses $M_{1}$,
$M_{2}$ and widths $\Gamma_{1}$, $\Gamma_{2}$, respectively. The complex-
valued parameters $q$ and $p$ are normalised such that $|p|^{2}+|q|^{2}=1$. We
use the phase convention ${C\\!P}|{{D}^{0}}\rangle=+|{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rangle$ implying that, in the
limit of $C\\!P$ symmetry with $p=q$, ${D}_{1}$ is the $C\\!P$-even eigenstate
and ${D}_{2}$ is the $C\\!P$-odd state. We define the dimensionless mixing
parameters, $x=(M_{1}-M_{2})/\Gamma$ and
$y=(\Gamma_{1}-\Gamma_{2})/(2\Gamma)$, where
$\Gamma=(\Gamma_{1}+\Gamma_{2})/2$ is the average width. The deviation of
$|q/p|$ from unity parameterises $C\\!P$ violation in mixing. The relative
phase $\phi=\arg(q\overline{A}_{f}/(pA_{f}))$ between $q/p$ and the ratio of
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ and ${D}^{0}$ decay
amplitudes to a common final state $f$, $\overline{A}_{f}/A_{f}$, is sensitive
to $C\\!P$ violation in the interference between mixing and decay. Within the
SM $\phi$ is approximately independent of the decay-mode.

#### 6.1.1 Measurements with ${{D}^{0}}\\!\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}$

The mixing and $C\\!P$-violation parameters in ${D}^{0}$–${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ oscillations can be accessed
through the comparison of the decay-time-dependent ratio of
${{D}^{0}}\\!\rightarrow K^{+}\pi^{-}$ to ${{D}^{0}}\\!\rightarrow
K^{-}\pi^{+}$ rates with the corresponding ratio for the charge-conjugate
processes.

The neutral $D$-meson flavour at production can be determined from the charge
of the low-momentum pion (slow pion) produced in the flavour-conserving
strong-interaction decay ${{D}^{*+}}\\!\rightarrow{{D}^{0}}\pi^{+}$. This
flavour-tagging technique is used throughout many measurements in this
chapter. These low-momentum pions are strongly deflected by the magnetic field
in the experiment. The addition of the Magnet Stations (see Sect. 2.3.1) will
allow the momentum and charge of previously lost pions to be determined. As
illustrated in Fig. 6.1 the flavour-tagged charm sample can be increased by
40% in size by the inclusion of Magnet Station information.

Figure 6.1: The momentum spectra in LHCb simulation of slow pions from the
decay ${{D}^{*+}}\\!\rightarrow{{D}^{0}}\pi^{+}$ that leave hits in the UT.
The slow pions observed in the SciFi in Upgrade I and those that will be added
by the Magnet Stations (MS) for Upgrade II are indicated.

The ${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow K^{-}\pi^{+})\pi^{+}$
process, which is dominated by a Cabibbo-favoured amplitude, is denoted as
right-sign (RS). Wrong-sign (WS) decays,
${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow K^{+}\pi^{-})\pi^{+}$, arise
from the doubly Cabibbo-suppressed ${{D}^{0}}\\!\rightarrow K^{+}\pi^{-}$
decay and the Cabibbo-favoured ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow K^{+}\pi^{-}$
decay that follows ${D}^{0}$–${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ oscillation. Since the mixing
parameters are small, $|x|,|y|\ll 1$, the $C\\!P$-averaged decay-time-
dependent ratio of WS-to-RS rates is approximated as [227, 228, 229]

$R_{K\pi}(t)\approx R^{K\pi}_{D}+\sqrt{R^{K\pi}_{D}}\ y_{K\pi}^{\prime}\
(\Gamma t)+\frac{x_{K\pi}^{\prime 2}+y_{K\pi}^{\prime 2}}{4}\left(\Gamma
t\right)^{2},$ (6.1)

where $t$ is the $D$ proper decay time. The parameter $R_{D}^{K\pi}$ is the
ratio of suppressed-to-favoured decay rates at $t=0$. The parameters
$x_{K\pi}^{\prime}$ and $y_{K\pi}^{\prime}$ depend linearly on the mixing
parameters, $x_{K\pi}^{\prime}\equiv x\cos\delta_{K\pi}+y\sin\delta_{K\pi}$
and $y_{K\pi}^{\prime}\equiv y\cos\delta_{K\pi}-x\sin\delta_{K\pi}$, through
the strong-phase difference $\delta_{K\pi}$ between the suppressed and
favoured amplitudes. If $C\\!P$ violation occurs, the decay-rate ratios
$R^{+}(t)$ and $R^{-}(t)$ of mesons produced as ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$, respectively, are functions of
independent sets of parameters, $R_{D}^{K\pi\,\pm}$,
$(x_{K\pi}^{\prime\pm})^{2}$ and $y_{K\pi}^{\prime\pm}$. The parameters
$R^{K\pi\,+}_{D}$ and $R^{K\pi\,-}_{D}$ differ if the ratio between the
suppressed and favoured decay amplitudes is not $C\\!P$ symmetric, indicating
direct $C\\!P$ violation, which is commonly described by the asymmetry
$A_{D}^{K\pi}=(R_{D}^{K\pi\,+}-R_{D}^{K\pi\,-})/(R_{D}^{K\pi\,+}+R_{D}^{K\pi\,-})$.
Violation of $C\\!P$ symmetry either in mixing or in the interference between
mixing and decay amplitudes generates differences between
$(x_{K\pi}^{\prime\pm})^{2}=|q/p|^{\pm 2}(x_{K\pi}^{\prime}\cos\phi\pm
y_{K\pi}^{\prime}\sin\phi)^{2}$ and $y_{K\pi}^{\prime\pm}=|q/p|^{\pm
1}(y_{K\pi}^{\prime}\cos\phi\mp x_{K\pi}^{\prime}\sin\phi)$. The
$C\\!P$-violation parameters ($|q/p|$ and $\phi$) can be determined in the
analysis without the need of external inputs.

The latest measurement from LHCb [230] uses Run 1 and early Run 2 (2015–2016)
data, corresponding to a total sample of about $\mathcal{L}=5\mbox{\,fb}^{-1}$
of integrated luminosity. Assuming $C\\!P$ conservation, the mixing parameters
are measured to be $x_{K\pi}^{\prime 2}=(3.9\pm 2.7)\times 10^{-5}$,
$y_{K\pi}^{\prime}=(5.28\pm 0.52)\times 10^{-3}$, and $R_{D}^{K\pi}=(3.454\pm
0.031)\times 10^{-3}$. Studying ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays separately shows no
evidence for $C\\!P$ violation and provides the current most stringent bounds
on the parameters $A_{D}^{K\pi}$ and $|q/p|$ from a single measurement,
$A_{D}^{K\pi}=(-0.1\pm 9.1)\times 10^{-3}$ and $1.00<|q/p|<1.35$ at the
$68.3\%$ confidence level.

Table 6.1: Extrapolated signal yields, and statistical precision on the mixing and $C\\!P$-violation parameters, from the analysis of promptly produced WS ${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow K^{+}\pi^{-})\pi^{+}$ decays. Signal yields of promptly produced RS ${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays are typically $250$ times larger. Sample ($\mathcal{L}$) | Yield ($\times 10^{6}$) | $\sigma(x_{K\pi}^{\prime 2})$ | $\sigma(y_{K\pi}^{\prime})$ | $\sigma(A_{D})$ | $\sigma(|q/p|)$ | $\sigma(\phi)$
---|---|---|---|---|---|---
Run 1–2 (9$\mbox{\,fb}^{-1}$) | 1.8 | $1.5\times 10^{-5}$ | $2.9\times 10^{-4}$ | 0.51% | 0.12 | 10∘
Run 1–3 (23$\mbox{\,fb}^{-1}$) | 10 | $6.4\times 10^{-6}$ | $1.2\times 10^{-4}$ | 0.22% | 0.05 | 4∘
Run 1–4 (50$\mbox{\,fb}^{-1}$) | 25 | $3.9\times 10^{-6}$ | $7.6\times 10^{-5}$ | 0.14% | 0.03 | 3∘
Run 1–5 (300$\mbox{\,fb}^{-1}$) | 170 | $1.5\times 10^{-6}$ | $2.9\times 10^{-5}$ | 0.05% | 0.01 | 1∘

In Table 6.1 the signal yields and the statistical precision from Ref. [230]
are extrapolated to the end of Run 2 and to the end of Upgrade II, assuming
that the central values of the measurements stay the same. This assumption is
particularly important for the $C\\!P$-violation parameters, as their
precision may depend on the measured values.

Systematic uncertainties are estimated using control samples of data and none
of them are foreseen to have irreducible contributions that exceed the
ultimate statistical precision, if the detector performance (particularly in
terms of vertexing/tracking and particle identification capabilities) is kept
at least in line with what is currently achieved at LHCb.

#### 6.1.2 Measurements with
${{D}^{0}}\\!\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$

Like ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$, the decays
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ are a pair
of RS and WS decays with high sensitivity to charm mixing. However, the rich
amplitude structure across the five dimensional phase space that describes
these decays offers unique opportunities (and challenges) in these four-body
modes.

In a phase-space integrated analysis using $3\mathrm{fb}^{-1}$ of data, LHCb
made the first observation of mixing in this decay mode, and measured the
quantities $R_{D}^{K3\pi}=(3.21\pm 0.014)\cdot 10^{-3}$, as well as
$R^{K3\pi}_{\mathrm{coher}}y^{\prime}_{K3\pi}=(0.3\pm 1.8)\cdot 10^{-3}$,
$\frac{1}{4}(x^{2}+y^{2})=(4.8\pm 1.8)\cdot 10^{-5}$ [231]; the parameter
$R^{K3\pi}_{\mathrm{coher}}$ is the coherence factor, which takes into account
the effect of integrating over the entire four-body phase space [232, 233].

The unique power of multibody decays lies to a large extent in the fact that
the strong-phase difference between the interfering ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ amplitudes varies across phase
space. This can only be fully exploited through analyses of phase space
distributions (either in bins or unbinned), rather than a phase-space-
integrated approach. Such a “phase space resolved” approach allows a direct
measurement of $x_{K\pi\pi\pi}^{\prime}$ and $y_{K\pi\pi\pi}^{\prime}$ (rather
than only to $x^{\prime 2}$ and $y^{\prime}$ as in the 2-body case), and
crucially provides high sensitivity to the $C\\!P$-violating variables $\phi$
and $|q/p|$.

However, the same phase variations that make multibody decays so powerful, are
also a major challenge, as they need to be known in order to cleanly determine
the mixing and $C\\!P$-violation parameters of interest. In principle, the
relevant phases can be inferred from an amplitude model such as that obtained
from $3\mathrm{fb}^{-1}$ of LHCb data [164]. Such models may introduce
unacceptable theoretical uncertainties for the precision era of LHCb Upgrade
II unless there are significant innovations in the theoretical description of
four-body amplitudes. Model-independent approaches will also be applied. These
use quantum-correlated events at the charm threshold to infer the required
phase information in a model-unbiased way. BESIII is working closely with LHCb
[234] to provide the necessary model-independent input for
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ across
different regions of phase space for measurements of the $\gamma$ angle (see
Chapter 4) as well as charm mixing and $C\\!P$-violation measurements.

Simulation sensitivity studies with model-dependent approaches give a useful
indication of the precision that can be achieved. A recent such study [235]
uses LHCb’s latest
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ amplitude
model [164]. Table 6.2 gives the yields and sensitivities scaled from this
study, illustrating the impressive sensitivity of this decay mode. The study
is based on promptly produced ${D}^{*+}$ mesons decaying in the flavour-
conserving decay mode ${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$. Several
systematic uncertainties require improvements in the analysis method in order
to scale with increasing sample sizes. However, given the huge potential of
this channel, sufficient effort is expected to be dedicated to this challenge,
such that adequate methods can be developed, and the necessary input from
threshold measurements is both generated at BESIII and exploited optimally at
LHCb. Indeed, once these are in place, this channel has the potential for
probing ${\mathcal{O}}(10^{-5})$ $C\\!P$ violation, assuming the current world
average value of $x$.

Table 6.2: Extrapolated signal yields, and sensitivity to the mixing and $C\\!P$-violation parameters, from the analysis of ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ decays (statistical uncertainties, only). Sample ($\mathcal{L}$) | Yield ($\times 10^{6}$) | $\sigma(x_{K\pi\pi\pi}^{\prime})$ | $\sigma(y_{K\pi\pi\pi}^{\prime})$ | $\sigma(|q/p|)$ | $\sigma(\phi)$
---|---|---|---|---|---
Run 1–2 (9$\mbox{\,fb}^{-1}$) | 0.22 | $2.3\times 10^{-4}$ | $2.3\times 10^{-4}$ | 0.020 | 1.2∘
Run 1–3 (23$\mbox{\,fb}^{-1}$) | 1.29 | $0.9\times 10^{-4}$ | $0.9\times 10^{-4}$ | 0.008 | 0.5∘
Run 1–4 (50$\mbox{\,fb}^{-1}$) | 3.36 | $0.6\times 10^{-4}$ | $0.6\times 10^{-4}$ | 0.005 | 0.3∘
Run 1–5 (300$\mbox{\,fb}^{-1}$) | 22.500 | $0.2\times 10^{-4}$ | $0.2\times 10^{-4}$ | 0.002 | 0.1∘

#### 6.1.3 Amplitude analysis of
${{D}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$

The self-conjugate decay
$D^{0}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{+}\pi^{-}$
includes both the Cabibbo-favoured and doubly Cabibbo-suppressed as well as
${C\\!P}$-eigenstate processes reconstructed in the same final state. This
allows the relative strong phase between these contributions to be determined
from the data, and in turn enables both the mixing parameters $x$ and $y$ and
the $C\\!P$-violation parameters $|q/p|$ and $\phi$ to be directly measured
without need for external input. As a result, this channel provides the
dominant constraint on the parameter $x$ in the global fits.

The mixing and CPV parameters modulate the time-dependence of the complex
amplitudes, which also depend on the two-dimensional phase-space of the decay.
As such, the measurement relies on both a precise understanding of the
detector acceptance effects as a function of phase-space and decay time, and
on an accurate description of the evolution of the underlying decay amplitudes
over the Dalitz plane. As discussed in the previous section both model-
dependent and model-independent approaches using quantum-correlated $D\bar{D}$
pairs from $\psi(3770)$ decays can be applied.

Previous measurements from the CLEO [236], BaBar [237], and Belle [238]
collaborations have used the model-dependent approach, with the Belle
measurement having the best precision to date, $x=(0.56^{+0.20}_{-0.23})\%$,
$y=(0.30^{+0.16}_{-0.17})\%$ (assuming $C\\!P$ symmetry), and
$|q/p|=0.90^{+0.18}_{-0.16}$, $\phi=(-6\pm 12)^{\circ}$. The one published
LHCb result was based on 1 fb-1 of Run 1 data [239], and used a model-
independent approach with strong phases taken from a CLEO measurement [154] to
determine $x=(-0.86\pm 0.56)\%$, $y=(0.03\pm 0.48)\%$. This analysis used
around $2\times 10^{5}$ $D^{*+}\\!\rightarrow D^{0}\pi^{+}$,
$D^{0}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{+}\pi^{-}$
decays from 2011, which suffered from low ${K}^{0}_{\mathrm{\scriptscriptstyle
S}}$ trigger efficiencies that were significantly increased for 2012 and
beyond, and will benefit further from software trigger innovations in the
Upgrade era.

At LHCb these decays can be reconstructed either through semileptonic decays
such as $B^{-}\\!\rightarrow D^{0}\mu^{-}\bar{\nu_{\mu}}$ (where the muon
charge is used to tag the initial $D^{0}$ flavour), or through prompt charm
production (where the charge of the slow pion in the decay
$D^{*+}\\!\rightarrow D^{0}\pi^{+}$ tags the initial flavour). The two
channels have complementary properties and both will be important components
of future mixing and $C\\!P$ violation analyses at LHCb.

The prompt charm yields are significantly larger than for the semileptonic
channel, due to the increased production cross-section; however, triggering
signal candidates is much more efficient, and introduces fewer non-
uniformities in acceptance, for the semileptonic channel. The estimated future
yields are presented in Table 6.3. Also shown are projected statistical
precisions on the four mixing and $C\\!P$-violating parameters, which have
been extrapolated from complete analyses of the Run 1 data for both SL and
prompt cases.

Table 6.3: Extrapolated signal yields, and statistical precision on the mixing and $C\\!P$ violation parameters, for the analysis of the decay $D^{0}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{+}\pi^{-}$. Candidates tagged by semileptonic $B$ decay (SL) and those from prompt charm meson production are shown separately. Sample (lumi $\mathcal{L}$) | Tag | Yield | $\sigma(x)$ | $\sigma(y)$ | $\sigma(|q/p|)$ | $\sigma(\phi)$
---|---|---|---|---|---|---
Run 1–2 (9 fb${}^{-1})$ | SL | 10M | 0.07% | 0.05% | 0.07 | 4.6∘
Prompt | 36M | 0.05% | 0.05% | 0.04 | 1.8∘
Run 1–3 (23 fb${}^{-1})$ | SL | 33M | 0.036% | 0.030% | 0.036 | 2.5∘
Prompt | 200M | 0.020% | 0.020% | 0.017 | 0.77∘
Run 1–4 (50 fb${}^{-1})$ | SL | 78M | 0.024% | 0.019% | 0.024 | 1.7∘
Prompt | 520M | 0.012% | 0.013% | 0.011 | 0.48∘
Run 1–5 (300 fb${}^{-1})$ | SL | 490M | 0.009% | 0.008% | 0.009 | 0.69∘
Prompt | 3500M | 0.005% | 0.005% | 0.004 | 0.18∘

The dominant systematic uncertainties on mixing parameters in this channel
come from two main sources. Firstly, the precision with which the non-
uniformities in detector-acceptance versus decay time and as a function of
phase space can be determined; secondly, the knowledge of the strong-phase
evolution across the Dalitz plane. For the Run 2 analysis, both contributions
are significantly lower than the statistical precision, while in the longer
term new approaches will be necessary to further reduce these systematic
uncertainties. Trigger and event selection techniques should be adapted to
emphasise uniform acceptance, even if this comes at the cost of significant
efficiency loss. New techniques, such as the bin-flip method [240], can
further reduce dependence on the non-uniform acceptance, although at the cost
of degraded statistical precision on the mixing and $C\\!P$-violation
parameters. In the model-dependent approach many of the model systematic
uncertainties may reduce or vanish with increased integrated luminosity, as
currently fixed parameters are incorporated into the data fit, and the data
become increasingly capable of rejecting unsuitable models provided that there
is suitable evolution in the model descriptions. For the model-independent
approach, the uncertainty from external inputs (currently from CLEO-c, later
from BESIII) will also reduce with luminosity as the LHCb data starts to
provide constraining power. There are no systematic uncertainties which are
known to have irreducible contributions that exceed the ultimate statistical
precision.

For the $C\\!P$ violation parameters additional sources of systematic
uncertainty come from the knowledge of detector-induced asymmetries. In
particular, there is a known asymmetry between ${K}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ in their interactions with
material. The limitation here will be the precision with which the material
traversed by each ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ meson can be
determined. The Upgrade II detector will be constructed to minimise material,
and to allow precise evaluation of the remaining contributions. This channel
thus has comparable power on $C\\!P$ violating parameters, but with a simpler
two-dimensional phase space and complementary detector systematic
uncertainties, as the four-body decay discussed in the previous section.

#### 6.1.4 Measurement of $A_{\Gamma}$

A further sensitive probe of indirect $C\\!P$ violation in the charm sector is
given by the time evolution of decays of ${D}^{0}$ mesons into $C\\!P$
eigenstates. Due to the smallness of the charm mixing parameters, the $C\\!P$
asymmetry as a function of decay time can be written as

$\displaystyle A_{C\\!P}(t)$
$\displaystyle=\frac{\Gamma(\mbox{${{D}^{0}}(t)\\!\rightarrow
f$})-\Gamma(\mbox{${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\\!\rightarrow
f$})}{\Gamma(\mbox{${{D}^{0}}(t)\\!\rightarrow f$})+\Gamma(\mbox{${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\\!\rightarrow
f$})}=A_{C\\!P}^{\mathrm{dir}}+\frac{t}{\tau_{D}}A_{C\\!P}^{\mathrm{indir}}+{\mathcal{O}}\left(\left(\frac{t}{\tau_{D}}\right)^{2}\right)\simeq
A_{C\\!P}^{\mathrm{dir}}-\frac{t}{\tau_{D}}A_{\Gamma},$ (6.2)

where $\Gamma({{D}^{0}}(t)\rightarrow f)$ and $\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\rightarrow f)$ indicate the
time-dependent decay rates of an initial ${D}^{0}$ or ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decaying to a final state $f$
at decay time $t$, $\tau_{D}=1/\Gamma$ is the average lifetime of the
${D}^{0}$ meson, and $A_{C\\!P}^{\mathrm{dir}}$ is the asymmetry related to
direct $C\\!P$ violation. The parameter $A_{\Gamma}$ is related to indirect
$C\\!P$ violation ($\simeq-A_{C\\!P}^{\mathrm{indir}}$) and is defined as

$\displaystyle A_{\Gamma}$
$\displaystyle\equiv\frac{\hat{\Gamma}(\mbox{${{D}^{0}}\\!\rightarrow{h}^{+}{h}^{-})$}-\hat{\Gamma}(\mbox{${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow{h}^{+}{h}^{-})$}}{\hat{\Gamma}(\mbox{${{D}^{0}}\\!\rightarrow{h}^{+}{h}^{-})$}+\hat{\Gamma}(\mbox{${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow{h}^{+}{h}^{-})$}}$
(6.3)

where $\hat{\Gamma}$, the effective lifetime, is defined by

$1/\hat{\Gamma}=\frac{\int t\,\Gamma(t)\,{\rm d}t}{\int\Gamma(t)\,{\rm d}t}.$
(6.4)

Neglecting contributions from subleading amplitudes, $A_{\Gamma}$ is
independent of the final state $f$. It is therefore possible to measure
indirect $C\\!P$ violation by either measuring the two lifetimes separately,
or by the time evolution of the asymmetry.

Neglecting $C\\!P$ violation in mixing, it can be shown that
$A_{\Gamma}\approx-x\sin\phi$, implying that $|A_{\Gamma}|<|x|{~{}\raise
1.49994pt\hbox{$<$}\kern-8.50006pt\lower 3.50006pt\hbox{$\sim$}~{}}5\times
10^{-3}$ [27],thus setting the scale of the size of the effect that needs to
be measured. While this means that the quantity to be measured is necessarily
small, the large yields available in the Cabibbo-suppressed modes
$f=\pi^{+}\pi^{-}$ or $f=K^{+}K^{-}$, and tagging from the ${D}^{*\pm}$ decay,
allow precise measurements provided the systematic uncertainties can be
controlled with a high degree of precision. Tagging based on semileptonic
decays of a parent beauty hadron is also possible and has been used in a
published LHCb measurement [241], but contributes significantly lower yields.

Most potential systematic effects are essentially constant in $t$ and
therefore cause little uncertainties in the observed decay time evolution of
the asymmetry. However, second-order effects and detector–induced correlation
between momentum and proper decay time are sufficient to produce spurious
asymmetries, that must be appropriately corrected. In addition, contamination
from secondary decays is a first-order effect in time that must be suppressed,
and its residual bias accounted for. Both corrections are dependent on the
availability of a large number of favoured
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays as calibration, and can be
expected to scale with statistics; collection of this sample with the same
trigger as for the signal modes is therefore a crucial tool for performing
this measurement with high precision in the future.

The Run 1 LHCb measurement of this quantity gave consistent results in the two
$h^{+}h^{-}$ modes, averaging $A_{\Gamma}=(-0.13\pm 0.28\pm 0.10)\times
10^{-3}$ [242], which is still statistically dominated. For the reasons
mentioned above, this precision is at the threshold of becoming physically
interesting, making it a high-priority target to pursue with more data. It
seems highly likely that LHCb Upgrade II will be the only experiment built in
the foreseeable future that will be able to do this.

Table 6.4 shows expected yields and precisions attainable in LHCb Upgrade II,
under the same assumptions on efficiencies adopted in the previous sections;
this must include provisions for acquiring and storing $5\times 10^{10}$
Cabibbo-favoured decays. The ultimate combined precision is $1\times 10^{-5}$.

Table 6.4: Extrapolated signal yields, and statistical precision on indirect $C\\!P$ violation from $A_{\Gamma}$.  Sample ($\mathcal{L}$) | Tag | Yield $K^{+}K^{-}$ | $\sigma(A_{\Gamma})$ | Yield $\pi^{+}\pi^{-}$ | $\sigma(A_{\Gamma})$
---|---|---|---|---|---
Run 1–2 (9 fb${}^{-1})$ | Prompt | 60M | 0.013% | 18M | 0.024%
Run 1–3 (23 fb${}^{-1})$ | Prompt | 310M | 0.0056% | 92M | 0.0104 %
Run 1–4 (50 fb${}^{-1})$ | Prompt | 793M | 0.0035% | 236M | 0.0065 %
Run 1–5 (300 fb${}^{-1})$ | Prompt | 5.3G | 0.0014% | 1.6G | 0.0025 %

#### 6.1.5 Summary of indirect $C\\!P$ violation

A comparison of the potential precision of the analyses presented in the
previous sections is shown in Fig. 6.2, and compared with the expected
precision from Belle II. The expected LHCb constraints on $\phi$ are
translated into asymmetry constraints ($A_{{C\\!P}}^{\rm ind.}\approx
x\sin(\phi)$) by multiplying by the current HFLAV average of $x$ and
neglecting the uncertainty on this. This comparison assumes that $x$ is non-
zero, $\phi$ not overly large, and $x$ will be determined comparatively well
in the future. This comparison neglects the additional constraining power from
$|q/p|$.

The analyses presented in the previous sections are also combined to establish
the sensitivity to the $C\\!P$-violating parameters $|q/p|$ and $\phi$. The
combination is performed using the method described in Ref. [168]. At an
integrated luminosity of $300\mbox{\,fb}^{-1}$ the sensitivity to $|q/p|$ is
expected to be $0.001$ and that to $\phi$ to be $0.1^{\circ}$. This remarkable
sensitivity is contrasted with the HFLAV world average as of 2017 in Fig. 6.3.
We thus conclude that the LHCb Upgrade II will have impressive power to
characterise new physics contributions to $C\\!P$ violation and is the only
foreseen facility with a strong potential of probing the Standard Model
contribution.

Figure 6.2: The predicted constraints on the indirect $C\\!P$ violation
asymmetry in charm from the decay channels indicated in the labels at the
bottom of the columns. Predictions are shown in LS2 (2020) from LHCb, LS3
(2025) from LHCb, at the end of Belle II (2025), and at the end of the HL-LHC
LHCb Upgrade II programme.  Figure 6.3: The estimated constraints for LHCb
Upgrade II on $\phi$, $|q/p|$ from the combination of the analyses in the
previous section (red) compared to the current world-average precision [27]
(light blue).

### 6.2 Direct $C\\!P$ violation

LHCb has a comprehensive programme of searches for direct $C\\!P$ violation in
charm. Neutral $D$ mesons, where indirect ${C\\!P}$ violation can also
contribute, and charged ${D}^{\pm}_{s}$ meson decays, where only direct
effects can occur, are both studied. The higher statistics of two-body decays
and the interfering amplitudes of multi-body decays are both probed. Four-body
decays and the new field of baryon decays, allow the complementary studies of
$P$-even and $P$-odd $C\\!P$ violation contributions. Singly and doubly
Cabibbo-suppressed decays are both studied and channels with new physics
sensitive loop processes or those with exchange diagrams where larger SM
contributions are expected. The precision study of modes containing neutral
particles will be opened up by the proposed calorimeter of Upgrade II.

Direct $C\\!P$ violation effects in the charm system could be larger than
those in indirect $C\\!P$ violation, and Upgrade II will be able to
characterise the direct $C\\!P$ sources. Alternatively $C\\!P$ violation
effects may be very small and Upgrade II will be needed to probe them. In
either scenario the experiment will have a strong programme in this field.

#### 6.2.1 Measurement of $A_{C\\!P}$ in
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and $C\\!P$ violation in other
two-body modes

The singly Cabibbo-suppressed $D^{0}\rightarrow{{K}^{-}}{{K}^{+}}$ and
$D^{0}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays discussed in Sect. 6.1.4 for
indirect $C\\!P$ violation studies, also play a critical role in the
measurement of time-integrated direct $C\\!P$ violation. The amount of $C\\!P$
violation in these decays is expected to be below the percent level [243, 244,
245, 246, 247, 248, 249, 250], but large theoretical uncertainties due to
long-distance interactions prevent precise SM predictions. In the presence of
physics beyond the SM, the expected $C\\!P$ asymmetries could be enhanced
[251], although an observation near the current experimental limits would be
consistent with the SM expectation. The direct $C\\!P$ violation is associated
with the breaking of $C\\!P$ symmetry in the decay amplitude. It is measured
through the time-integrated $C\\!P$ asymmetry in the $h^{-}h^{+}$ decay rates

$A_{{C\\!P}}({{D}^{0}}\rightarrow
h^{-}h^{+})\equiv\frac{\Gamma(D^{0}\rightarrow h^{-}h^{+})-\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow
h^{-}h^{+})}{\Gamma(D^{0}\rightarrow h^{-}h^{+})+\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow h^{-}h^{+})}.$ (6.5)

The sensitivity to direct $C\\!P$ violation is enhanced through a measurement
of the difference in $C\\!P$ asymmetries between ${D}^{0}$ $\rightarrow$
${K}^{-}$ ${K}^{+}$ and ${D}^{0}$ $\rightarrow$ ${\pi}^{-}$ ${\pi}^{+}$
decays, $\Delta
A_{{C\\!P}}=A_{{C\\!P}}({{K}^{-}}{{K}^{+}})-A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})$,
in which detector asymmetries largely cancel.

The individual asymmetries $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ and
$A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})$ can also be measured. A measurement of
the time-integrated $C\\!P$ asymmetry in $D^{0}\rightarrow{{K}^{-}}{{K}^{+}}$
has been performed at LHCb with 3$\mbox{\,fb}^{-1}$ collected at centre-of-
mass energies of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$. The flavour of the
charm meson at production is determined from the charge of the pion in
${{D}^{*+}}\rightarrow{{D}^{0}}\pi^{+}$ decays, or via the charge of the muon
in semileptonic $b$-hadron decays (${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rightarrow{{D}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$).
The analysis strategy so far relies on the $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{-}$, $D^{+}\rightarrow K_{s}^{0}\pi^{+}$ and
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays as
control samples [252]. In this case, due to the weighting procedures aiming to
fully cancel the production and reconstruction asymmetries, the effective
prompt signal yield for $A_{{C\\!P}}({{K}^{-}}{{K}^{+}})$ is reduced. The
expected signal yields and the corresponding statistical precision in Upgrade
II are summarised in Table 6.5.

Table 6.5: Extrapolated signal yields and statistical precision on direct $C\\!P$ violation observables for the promptly produced samples. Sample ($\mathcal{L}$) | Tag | Yield | Yield | $\sigma(\Delta A_{{C\\!P}})$ | $\sigma(A_{{C\\!P}}(hh))$
---|---|---|---|---|---
|  |  ${D}^{0}$ $\rightarrow$ ${K}^{-}$ ${K}^{+}$ |  ${D}^{0}$ $\rightarrow$ ${\pi}^{-}$ ${\pi}^{+}$ | [$\%$] | [%]
Run 1–2 (9 $\mbox{\,fb}^{-1}$) | Prompt | 52M | 17M | $0.03$ | $0.07$
Run 1–3 (23 $\mbox{\,fb}^{-1}$) | Prompt | 280M | 94M | $0.013$ | $0.03$
Run 1–4 (50 $\mbox{\,fb}^{-1}$) | Prompt | 1G | 305M | $0.007$ | $0.015$
Run 1–5 (300 $\mbox{\,fb}^{-1}$) | Prompt | 4.9G | 1.6G | $0.003$ | $0.007$

The $\Delta A_{{C\\!P}}$ observable is robust against systematic
uncertainties. The main sources of systematic uncertainties are inaccuracies
in the fit model, the weighting procedure, the contamination of the prompt
sample with secondary ${{D}^{0}}$ mesons and the presence of peaking
backgrounds. There are no systematic uncertainties which are expected to have
irreducible contributions which exceed the ultimate statistical precision.
This channel is already entering the upper range of the physically interesting
sensitivities, and will likely continue to provide the world’s best
sensitivity to direct $C\\!P$ violation in charm in Upgrade II. The power of
these two-body $C\\!P$ eigenstates at LHCb Upgrade II is illustrated in Fig.
6.4, which shows the indirect (see Sect. 6.1.4) and direct $C\\!P$ constraints
that will come from these modes.

Figure 6.4: The estimated constraints for LHCb Upgrade II on indirect and
direct charm $C\\!P$ violation from the analysis of two-body $C\\!P$
eigenstates. The current world-average precision [27] is $\pm 2.6\times
10^{-4}$ for indirect and $\pm 18\times 10^{-4}$ for direct $C\\!P$ violation
and thus larger than the full scale of this plot.

There are a significant number of other two-body modes of strong physics
interest where Upgrade II will also make important contributions. These
include the decay modes
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ (0.28%),
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ $(0.6\times 10^{-4})$,
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{*0}}$
$(0.8\times 10^{-4})$,
${{D}^{+}_{s}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}$
$(3.2\times 10^{-4})$,
${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}$
$(1.2\times 10^{-4})$, ${{D}^{+}}\rightarrow\phi{{\pi}^{+}}$ $(6\times
10^{-5})$, ${{D}^{+}}\rightarrow\eta^{\prime}{{\pi}^{+}}$ $(3.2\times
10^{-5})$, ${{D}^{+}_{s}}\rightarrow\eta^{\prime}{{\pi}^{+}}$ $(3.2\times
10^{-4})$, where the projected statistical only $C\\!P$ asymmetry
sensitivities are given in parentheses. The first three modes mentioned are
notable as they receive sizeable contributions from exchange amplitudes at
tree-level and could have a relatively enhanced contribution from penguin
annihilation diagrams which are sensitive to New Physics. Consequently some
authors have highlighted them as potential $C\\!P$ violation discovery
channels [253, 254].

#### 6.2.2 Measurements with $D^{+}\rightarrow h^{+}h^{-}h^{+}$ and
$D_{s}^{+}\rightarrow h^{+}h^{-}h^{+}$ decays

Searches for direct $C\\!P$ violation in the phase space of Cabibbo-suppressed
(CS) $D^{+}\rightarrow h_{1}^{+}h_{2}^{-}h_{3}^{+}$ decays, hereafter referred
to as $D\rightarrow 3h$, are complementary to that of $D^{(0,+)}\rightarrow
h_{1}h_{2}$ ($h_{i}=\pi,K$). In charged $D$ systems, only $C\\!P$ violation in
the decay is possible. The main observable is the $C\\!P$ asymmetry, which, in
the case of two-body decays, is a single number. In contrast, $D\rightarrow