3h$, decays allow the study of the distribution of the $C\\!P$ asymmetry
across a two-dimensional phase space (usually represented by the Dalitz plot).

The $D\rightarrow 3h$ decays proceed mostly through intermediate resonant
states. With a relatively small phase space, the resonances are spread over
the entire Dalitz plot. The different resonant amplitudes overlap and
interfere, providing the strong phase difference needed for direct $C\\!P$
violation to occur. This is a unique feature of multi-body decays. In some
regions of the phase space the difference in strong phase between the various
resonant amplitudes is very large, which may cause relatively large and
localised $C\\!P$ asymmetries. The difference in strong phases often change
sign across the Dalitz plot, causing eventual $C\\!P$ asymmetries to change
sign as well, reducing the sensitivity of phase-space-integrated measurements.

The most appealing aspect of this study is that one does not need to rely on
models for the resonant structure of $D\rightarrow 3h$ decays. The $C\\!P$
violation may be observed in a direct, model-independent comparison between
$D^{+}$ and $D^{-}$ Dalitz plots. In both $K^{-}K^{+}\pi^{+}$ and
$\pi^{-}\pi^{+}\pi^{+}$ spectra the $D^{+}_{s}$ signals can be used as control
channels to investigate possible phase-space-dependent asymmetries in
detection efficiency. This is the most sensitive technique for a first
observation of $C\\!P$ violation in charged $D$ mesons.

In the SM, small $C\\!P$ asymmetries in charged $D$ mesons are expected in CS
decays, but are highly suppressed in doubly-Cabibbo-suppressed (DCS) and
Cabibbo-favoured decays. New mechanisms of $C\\!P$ violation would require
subleading amplitudes involving new particles — a charged Higgs, for instance.
This mechanism would generate $C\\!P$ violation in Cabibbo-favoured decays as
well, but the observation of the $C\\!P$ asymmetries would be obscured by the
vastly dominant SM amplitudes. In DCS decays, however, the SM amplitudes are
suppressed by a factor $\sim\tan^{4}\theta$, increasing the possibility of an
observation. Therefore, DCS decays, such as $D^{+}\rightarrow K^{-}K^{+}K^{+}$
and $D^{+}\rightarrow\pi^{-}K^{+}\pi^{+}$, offer an unique opportunity to
search for new sources of $C\\!P$ violation in an almost “background free”
environment.

The estimated signal yields in future upgrades are summarised in Table 6.6.
The yields are based on an extrapolation of the Run 2 yields per unit
luminosity, made under the same assumptions of Section 6.1. The estimated
sensitivities to observation of $C\\!P$ violation, using the decay
$D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ as example, are presented in Table
6.7.

Table 6.6: Extrapolated signal yields, in units of 106, of the Cabibbo-suppressed decays $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$, $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$, and of the doubly Cabibbo-suppressed decays $D^{+}\rightarrow K^{-}K^{+}K^{+}$, $D^{+}\rightarrow\pi^{-}K^{+}\pi^{+}$.  Sample ($\mathcal{L}$) | $D^{+}\rightarrow K^{-}K^{+}\pi^{+}$ | $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ | $D^{+}\rightarrow K^{-}K^{+}K^{+}$ | $D^{+}\rightarrow\pi^{-}K^{+}\pi^{+}$
---|---|---|---|---
Run 1–2 (9$\mbox{\,fb}^{-1}$) | 200 | 100 | 14 | 8
Run 1–4 (23$\mbox{\,fb}^{-1}$) | 1,000 | 500 | 70 | 40
Run 1–4 (50$\mbox{\,fb}^{-1}$) | 2,600 | 1,300 | 182 | 104
Run 1–6 (300$\mbox{\,fb}^{-1}$) | 17,420 | 8,710 | 1,219 | 697
Table 6.7: Sensitivities to $C\\!P$-violation scenarios for $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decays. Simulated $D^{+}$ and $D^{-}$ Dalitz plots are generated with relative changes in the phase of the $R\pi^{\pm}$ amplitude, $R=\rho^{0}(770)$, $f_{0}(500)$ or $f_{2}(1270)$. The values of the phase differences are given in degrees and correspond to a 5$\sigma$ $C\\!P$-violation effect. Simulations are performed with $3\mbox{\,fb}^{-1}$ and extrapolated to the expected integrated luminosities. resonant channel | $9\mbox{\,fb}^{-1}$ | $23\mbox{\,fb}^{-1}$ | $50\mbox{\,fb}^{-1}$ | $300\mbox{\,fb}^{-1}$
---|---|---|---|---
$f_{0}(500)\pi$ | 0.30 | 0.13 | 0.083 | 0.032
$\rho^{0}(770\pi$ | 0.50 | 0.22 | 0.14 | 0.054
$f_{2}(1270)\pi$ | 1.0 | 0.45 | 0.28 | 0.11

#### 6.2.3 Measurements with ${{D}^{0}}\\!\rightarrow h^{+}h^{-}h^{+}h^{-}$
decays

Standard-Model $C\\!P$ violation could be observed in the Cabibbo-suppressed
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays,
while New Physics is needed to justify any observation in the doubly Cabibbo-
suppressed ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays.

Many techniques can be adopted to search for $C\\!P$ violation, all of them
exploiting the rich resonant structure of the decays. Those that have been so
far used at LHCb are based on ${\widehat{T}}$-odd asymmetries and the energy
test, while studies are ongoing to measure model-dependent $C\\!P$ asymmetries
in the decay amplitudes.

The study of ${\widehat{T}}$-odd asymmetries exploits potential $P$-odd
$C\\!P$ violation from the interference of the different amplitude structures
in the decay, as described in Ref. [255]. This uses a triple product
$C_{T}=\vec{p}_{A}\cdot(\vec{p}_{B}\times\vec{p}_{C})$ constructed from the
momenta of three of the final state particles
$\vec{p}_{A},\,\vec{p}_{B},\,\vec{p}_{C}$. LHCb has studied
${\widehat{T}}$-odd asymmetries using data corresponding to
3$\mbox{\,fb}^{-1}$ from the Run 1 dataset, obtaining a sensitivity of
$2.9\times 10^{-3}$ with very small systematic uncertainties [256]. The
peculiarity of this measurement is the absence of instrumental asymmetries,
since it is given by the difference of two asymmetries measured separately on
${D}^{0}$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays,

$\displaystyle
A_{T}=\frac{(\Gamma({{D}^{0}},C_{T}>0)-\Gamma({{D}^{0}},C_{T}<0))}{(\Gamma({{D}^{0}},C_{T}>0)+\Gamma({{D}^{0}},C_{T}<0))}\quad\bar{A}_{T}=\frac{(\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}},\bar{C}_{T}>0)-\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}},\bar{C}_{T}<0))}{(\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}},\bar{C}_{T}>0)+\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}},\bar{C}_{T}<0))}$ (6.6)

and $a_{{C\\!P}}=(A_{T}-\bar{A}_{T})/2$. One can therefore expect the
measurement to scale with luminosity to reach a sensitivity down to $2.9\times
10^{-5}$ ($9.4\times 10^{-5}$) for
${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
(${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$) decays, as
detailed in Table 6.8.

Table 6.8: Extrapolated signal yields, and statistical precision on ${\widehat{T}}$-odd $C\\!P$-violation observables. | ${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ | ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
---|---|---
Sample ($\mathcal{L}$) | Yield ($\times 10^{6}$) | $\sigma(a_{{C\\!P}}^{\widehat{T}\text{-odd}})$ | Yield ($\times 10^{6}$) | $\sigma(a_{{C\\!P}}^{\widehat{T}\text{-odd}})$
Run 1–2 (9$\mbox{\,fb}^{-1}$) | 13.5 | $2.4\times 10^{-4}$ | 4.7 | $5.4\times 10^{-4}$
Run 1–3 (23$\mbox{\,fb}^{-1}$) | 69 | $1.1\times 10^{-4}$ | 12 | $3.4\times 10^{-4}$
Run 1–4 (50$\mbox{\,fb}^{-1}$) | 150 | $7.5\times 10^{-5}$ | 26 | $2.3\times 10^{-4}$
Run 1–5 (300$\mbox{\,fb}^{-1}$) | 900 | $2.9\times 10^{-5}$ | 156 | $9.4\times 10^{-5}$

The energy test method is an unbinned model-independent two-sample test that
assesses the p-value for the hypothesis that two distributions are sampled
from the same underlying distribution. Applied to the phase-space distribution
of $D$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}$ decays, this gives
a probability for the hypothesis of $C\\!P$ symmetry. The energy test is
described in detail in Ref. [257, 258] and has been applied at LHCb to
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ [201] and
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays
[226]. In the latter, two variants were applied, one testing for $P$-even and
the other for $P$-odd $C\\!P$ violation, with the latter giving an intriguing
$2.7\sigma$ deviation from the no-$C\\!P$ violation hypothesis.

The method is insensitive to global asymmetries; however, it is expected that
it will become sensitive to variations in phase space of production and
detection asymmetries. These can be controlled in data by application of the
method to Cabibbo-favoured decays such as
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$. Assuming
scaling with the square-root of the ratio of sample sizes, the same p-values
can be expected for the $C\\!P$ asymmetries given in Table 6.9.

The increased sensitivity of the Upgrade II dataset is demonstrated in Fig.
6.5, where the binned $S_{{C\\!P}}$ test [257] is applied to data from a
simplified simulation. Inserting $C\\!P$ violation in the same resonances as
indicated by the current $2.7\sigma$ anomaly, it is illustrated that in the
simulated scenario the Upgrade II dataset is required to observe and study the
$C\\!P$ violation contribution.

Table 6.9: Overview of sensitivities to various $C\\!P$-violation scenarios for ${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays as extrapolated from Ref. [226]. The relative changes in magnitude and phase of the amplitude of the resonance $R$ to which sensitivity is expected are given in $\%$ and ∘, respectively. The $P$-wave $\rho^{0}(770)$ is a $P$-odd component. The phase change in this resonance is tested with the $P$-odd $C\\!P$-violation test. Results for all other scenarios are given with the standard $P$-even test. $R$ (partial wave) | $9\mbox{\,fb}^{-1}$ | $23\mbox{\,fb}^{-1}$ | $50\mbox{\,fb}^{-1}$ | $300\mbox{\,fb}^{-1}$
---|---|---|---|---
$a_{1}\\!\rightarrow{{\rho}^{0}}\pi$ (S) | $1.4\%$ | $0.6\%$ | $0.4\%$ | $0.17\%$
$a_{1}\\!\rightarrow{{\rho}^{0}}\pi$ (S) | $0.8^{\circ}$ | $0.35^{\circ}$ | $0.24^{\circ}$ | $0.10^{\circ}$
${\rho}^{0}$ ${\rho}^{0}$ (D) | $1.4\%$ | $0.6\%$ | $0.4\%$ | $0.17\%$
${\rho}^{0}$ ${\rho}^{0}$ (P) | $0.8^{\circ}$ | $0.35^{\circ}$ | $0.24^{\circ}$ | $0.10^{\circ}$

Figure 6.5: The binned $S_{C\\!P}$ test [257] applied to simplified simulation
for the ${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decay. The left hand plot shows the significance of the (normalised) asymmetry
in each bin for the event yield expected following Runs 1 and 2, with the
right hand plot using the expected event yield corresponding to
300$\mbox{\,fb}^{-1}$. The contributions from the four main intermediate
resonances associated with the ${{D}^{0}}$ meson decay (as determined using
CLEO-c data [259]) are simulated; $C\\!P$-violation is introduced to the
matter and anti-matter samples in the test by varying the relative yield of
the ${{D}^{0}}\rightarrow\rho^{0}\rho^{0}$ contribution in the samples being
compared by $\pm 0.4\%$. Large bin significances and a significant $\chi^{2}$
value are only observed in the dataset corresponding to 300$\mbox{\,fb}^{-1}$;
in order to observe this $C\\!P$-violation the Upgrade II dataset is required.

#### 6.2.4 Prospects with neutral final states

Radiative charm decays are discussed in Sect. 7.4.4, with decays containing
neutral hadrons discussed here. Charm decays with neutrals in the final state
can help to shed light on the SM or beyond-SM origin of possible
$C\\!P$-violation signals by testing correlations between $C\\!P$ asymmetries
measured in various flavour-$\mathrm{SU}(3)$ related decays [244, 245, 243].
They are, however, particularly challenging in hadronic collisions, where the
calorimeter background for low energy clusters is high, while the trigger
retention rate needs to be kept low to allow for affordable rates.

Nevertheless, good performances are achieved when considering decays, such as
${{D}^{0}}\\!\rightarrow\pi^{+}\pi^{-}\pi^{0}$, with at least two charged
particles in the final states that can help to identify the displaced decay
vertex of the charm meson. In only 2$\mbox{\,fb}^{-1}$ of data collected
during 2012, LHCb has reconstructed about 660,000
${{D}^{0}}\\!\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decays [201], i.e. about five
times larger than that from the full BaBar data set [260], with comparable
purity. Preliminary estimates for Run 2 data, give about 500 000 signal decays
per $\mbox{\,fb}^{-1}$, making future $C\\!P$-violation searches in this
channel very promising. Similarly, large samples of
${D}_{({s})}^{+}\\!\rightarrow\eta^{(\prime)}\pi^{+}$ decays, with
$\eta^{(\prime)}\\!\rightarrow\pi^{+}\pi^{-}\gamma$, or
${{D}^{+}}\\!\rightarrow\pi^{+}\pi^{0}$ decays, with $\pi^{0}\\!\rightarrow
e^{+}e^{-}\gamma$, are already possible with the current detector. The
${D}_{({s})}^{+}\\!\rightarrow\eta^{\prime}\pi^{+}$ mode, as an example, has
been used by LHCb during Run 1 to perform the most precise measurement of
$C\\!P$ asymmetries in these channels to date, with uncertainties below the 1%
level [261].

More challenging final states consisting only of neutral particles, such as
$\pi^{0}\pi^{0}$ or $\eta\eta$, can still be reconstructed with
$\pi^{0}\\!\rightarrow\gamma\gamma$ or $\eta\\!\rightarrow\gamma\gamma$
candidates made of photons which, after interacting with the detector
material, have converted into an $e^{+}e^{-}$ pair. Such conversions must
occur before the tracking system to have electron tracks reconstructed.
Although the reconstruction efficiency of these “early” converted photons in
the current detector reaches only a few percent of the calorimetric photon
efficiency, their purity is much higher. This approach may become interesting
only with the large data sets that are expected to be collected by the end of
Upgrade II. The $\eta$ decays can also be reconstructed through the
$\pi^{+}\pi^{-}\gamma$ final state.

Unlike the ${D}^{0}$ decays which are usually tagged with a soft pion from
${D}^{*+}$ decays, there is no easy tagging of the ${D}^{+}$ modes, which thus
often suffer from a high combinatorial background. Employing a $\pi^{0}$ tag
using ${{D}^{*+}}\\!\rightarrow{{D}^{+}}\pi^{0}$ decays could facilitate
future studies of ${D}^{+}$ decays, in particular those with challenging
and/or high multiplicity final states.

The study of these modes may be challenging in Upgrade I due to the cluster
pile-up at higher luminosities and radiation damage of the current
calorimeter. A central element of the Upgrade II detector is the proposed new
calorimeter system (see Sect. 2.3.2). The proposed increase in granularity
would improve the efficiency for $\pi^{0}\\!\rightarrow\gamma\gamma$ decays
and, in particular, could make the $\pi^{0}$ tag feasible.

#### 6.2.5 Prospects with charmed baryons

The first charmed baryon, ${\mathchar 28931\relax}^{+}_{c}$, was discovered
over 40 years ago, but its properties are still poorly understood. Even less
is known about the heavier states such as the $\mathchar 28932\relax_{{c}}$
and $\mathchar 28938\relax_{{c}}$ baryons. This is due to a historical lack of
large experimental datasets, which has in turn disfavoured theoretical study
[262]. Given the large ${\mathchar 28931\relax}^{+}_{c}$ fiducial cross-
section measured by LHCb at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ [263],
and the recent discovery of the doubly charmed $\mathchar
28932\relax_{{c}{c}}^{++}$ baryon [264] there is now the experimental
possibility to execute a broad charmed baryon programme. Along with the
evidence for $C\\!P$ violation in beauty baryon decays [193], there is also
theoretical interest in understanding if similar behaviour exists in the charm
sector.

In general, charmed baryon decays offer a rich laboratory within which to
study matter-antimatter asymmetries. The most easily accessible modes for LHCb
are multibody decays containing one proton and several kaons or pions, such as
the Cabibbo-favoured ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decay and the
singly Cabibbo-suppressed ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{K}^{+}}$ and
${p}{{\pi}^{-}}{{\pi}^{+}}$ decays. Unlike the analogous final states of the
spinless neutral and charged $D$ mesons, such charmed baryon final states have
at least five degrees of freedom, allowing for a complex variation of the
strength of $C\\!P$ violation across the phase space. A complete description
of such a space is experimentally challenging. This is compounded by the
relatively short lifetime of the ${\mathchar 28931\relax}^{+}_{c}$ baryon with
respect to those of the $D$ mesons, which decreases the power of selections
based on the displacement of the charm vertex, increasing the prompt
combinatorial background, which mainly comprises low-momentum pions and kaons.
Selections of charm baryons then heavily rely on good discrimination between
proton, kaon, and pion hypotheses in the reconstruction of charged tracks, and
on a precise secondary vertex reconstruction. The latter is also necessary
when reconstructing charm baryons originating from semileptonic $b$-hadron
decays, which has the advantage of both a simple trigger path (a
high-$p_{\mathrm{T}}$, displaced muon) and a cleaner experimental signature
due to the large $b$-hadron lifetime. The presence of the proton in the final
state, whilst a useful handle for selections, poses experimental challenges
for $C\\!P$-violation measurements in addition to those present for
measurements with $D$ mesons, as the proton-antiproton detection asymmetry
must be accounted for. This has not yet been measured at LHCb due to the lack
of a suitable control mode.

The most precise measurement of $C\\!P$ violation in charm baryons was made
recently by the LHCb collaboration using Run 1 data, corresponding to
$3$$\mbox{\,fb}^{-1}$ of integrated luminosity [200]. The difference between
the phase-space-averaged $C\\!P$ asymmetries in ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{K}^{+}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}$ decays was found
to be consistent with $C\\!P$ symmetry to a statistical precision of
$0.9\,\%$. This difference is largely insensitive to the proton detection
asymmetry, but masks like-sign $C\\!P$ asymmetries between the two modes.
Further studies must then gain a precise understanding of the proton detection
asymmetry, in addition to measuring the variation of $C\\!P$ violation across
the decay phase space.

Although there is little literature on the subject, the magnitude of direct
$C\\!P$ violation in charm baryon decays is expected to be similar to that for
charmed mesons [262], and so the first step in furthering our understanding is
to reach a precision of $0.5\times 10^{-4}$, which can be met given the
$300$$\mbox{\,fb}^{-1}$ of integrated luminosity collected by the end of Run
5. The acquisition of more data is vital in enabling studies of states heavier
than the ${\mathchar 28931\relax}^{+}_{c}$ baryon, as their alternate
compositions may permit considerably different dynamics.

The $\mathchar 28932\relax_{{c}{c}}^{++}$ baryon was discovered using the
${{\mathchar 28931\relax}^{+}_{c}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ final state
with data taken in 2016 corresponding to an integrated luminosity of
$1.7$$\mbox{\,fb}^{-1}$. A signal yield of $313\pm 33$ was determined, which
can be extrapolated to around $100,000$ such decays obtainable with data
corresponding to $300$$\mbox{\,fb}^{-1}$. This will allow asymmetry
measurements with a precision of $0.4$ %.

Consequently the high statistics of Upgrade II offers the possibility to open
up the detailed study of this new research area.

## 7 Rare decays

The precision on measurements with rare decays of beauty and charm hadrons, or
$\tau$ leptons, is dominated by the limited available data samples. This will
still be the case after the Upgrade I data taking period. For example, to
reach a precision $\sim 10\%$ on the ratio $R={\cal
B}(\mbox{${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$})/{\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$})$ will provide
strong constraints (or evidence) for new flavour structures beyond the SM.
Moreover, the Upgrade II yields will allow meaningful measurements of new
observables, namely the effective lifetime ($\tau_{\mu\mu}^{\rm eff}$) and the
time-dependent $C\\!P$ asymmetry of
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays, and will allow for
improved searches for other (semi-)leptonic $B$ and $D$ decays. If the current
indications of lepton non-universality NP contributions to
${b}\rightarrow{s}\ell^{+}\ell^{-}$ processes are confirmed with more data,
Upgrade II will be needed to be able to discriminate between potential NP
models, by increasing the precision, but more importantly, by expanding the
range of complementary decay modes studied (including the less abundant
${b}\rightarrow{d}\ell^{+}\ell^{-}$ processes). In this case, a more
performant calorimeter system with improved granularity, as discussed in this
document, should be able to improve the efficiency for bremsstrahlung recovery
in ${b}\rightarrow{s}e^{+}e^{-}$ and therefore provide a significant
improvement in sensitivity. A better calorimeter performance should also help
to improve the precision of the Upgrade II in the measurements of radiative
beauty and charm decays. In the following sections a more detailed description
of the potential of the Upgrade II in rare and radiative decays is given.

### 7.1 Leptonic $B$ decays

#### 7.1.1 Measurements with $B\rightarrow{\mu^{+}\mu^{-}}$ decays

##### 7.1.1.1 Branching fractions

The decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ represents one of
the most theoretically clean and sensitive probes to reveal effects of New
Physics beyond the SM unreachable by direct searches. This decay is both loop-
and helicity-suppressed and is therefore very rare, with an expected branching
fraction of ${\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})=(3.66\pm 0.23)\times
10^{-9}$ in the SM [265]. The related mode
${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ is further suppressed by the ratio
of CKM matrix elements $|{V_{{t}{d}}}|^{2}/|{V_{{t}{s}}}|^{2}$, resulting in a
SM prediction ${\cal
B}(\mbox{${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$})=(1.06\pm 0.09)\times
10^{-10}$ [265]. These decays are particularly sensitive probes for new scalar
or pseudoscalar contributions.

In 2017 the LHCb collaboration reported the first observation by a single
experiment of the decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ [266]
and measured ${\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})=\left(3.0\pm
0.6^{+0.3}_{-0.2}\right)\times 10^{-9}$ and ${\cal
B}(\mbox{${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$})=\left(1.5^{+1.2+0.2}_{-1.0-0.1}\right)\times
10^{-10}$ using data collected in $pp$ collisions corresponding to a total
integrated luminosity of 4.4$\mbox{\,fb}^{-1}$ (Fig. 7.1, left). Both
measurements are compatible with the SM predictions but affected by large
statistical uncertainties. The analysis exploits the excellent performance of
the LHCb detector in terms of muon identification and vertex and mass
resolution. Compared to the previous analysis [8], the contamination of
combinatorial and ${B}\\!\rightarrow h^{+}h^{\prime-}$ backgrounds were
reduced by more than a factor two thanks to the development of improved
isolation variables and the optimisation of the muon identification
requirements. This has led to a reduction of the statistical uncertainty of
about 30% for ${\cal B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$
for a given integrated luminosity.

Figure 7.1: (Left) A two-dimensional representation of the LHCb branching
fraction measurements for ${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. The central values are
indicated with the black plus sign. The profile likelihood contours for
1,2,3…$\sigma$ are shown as blue contours. The Standard Model value is shown
as the red cross labelled SM. (Right) Background-subtracted
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decay-time distribution with
the fit result superimposed.

While the uncertainty of ${\cal
B}(\mbox{${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$ will be dominated by the
statistical uncertainty, the projected uncertainty of ${\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$ with a
300$\mbox{\,fb}^{-1}$ dataset depends on the assumptions made for the
systematic uncertainties. The current systematic uncertainty is dominated by
the 5.8% relative uncertainty associated with the $b$-quark fragmentation
probability ratio ($f_{s}/f_{d}$) [267] followed by approximately 3% from the
branching fractions of the normalisation modes, 2% from particle
identification and 2% from track reconstruction. The projected relative
statistical uncertainty of the current analysis on a dataset of
300$\mbox{\,fb}^{-1}$ is 1.8%. At the end of the Upgrade II data taking
period, a conservative assumption for the systematic uncertainty on ${\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$ is about 4%, which
would imply an uncertainty of ${\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$ to be approximately
$0.30\times 10^{-9}$ with 23$\mbox{\,fb}^{-1}$ and $0.16\times 10^{-9}$ with
$300\mbox{\,fb}^{-1}$. The increased precision will be able to cover larger
part of the unconstrained parameter space of MSSM models. The ratio of
branching fractions, ${\cal
B}(\mbox{${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$})/{\cal
B}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$, is a powerful
observable to test minimal flavour violation. The relative uncertainty of this
ratio is expected to remain limited only by statistics and decrease from
$90\%$ for the current measurement to about $34\%$ with $23\mbox{\,fb}^{-1}$
and $10\%$ with $300\mbox{\,fb}^{-1}$. All estimates use the quoted SM
predictions as central values for the branching ratios and similar detector
and analysis performance as in Ref. [266].

The estimated experimental uncertainties at 300$\mbox{\,fb}^{-1}$ are close to
the uncertainty of the current SM prediction from theory, which is dominated
by the uncertainty of the ${{B}^{0}_{s}}$ decay constant, determined from
lattice QCD calculations, and the CKM matrix elements. Both are expected to
improve in precision in the future.

##### 7.1.1.2 Effective lifetime and $C\\!P$ violation

With a 300$\mbox{\,fb}^{-1}$ dataset, precise measurements of additional
observables are possible, namely the effective lifetime ($\tau_{\mu\mu}^{\rm
eff}$) and the time-dependent $C\\!P$ asymmetry of
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays. Both quantities are
sensitive to possible new contributions from the scalar and pseudo-scalar
sector in a way complementary to the branching ratio measurement[268].

The effective lifetime is related to the mean ${B}^{0}_{s}$ lifetime
$\tau_{B_{s}}$ through the relation

$\tau_{\mu\mu}^{\rm
eff}=\frac{\tau_{B_{s}}}{1-y^{2}_{s}}\frac{1+2A^{\mu\mu}_{\Delta\Gamma}y_{s}+y^{2}_{s}}{1+A^{\mu\mu}_{\Delta\Gamma}y_{s}}\,,$
(7.1)

where $y_{s}=\tau_{B_{s}}\Delta\Gamma_{s}/2$ and
$\Delta\Gamma_{s}=\Gamma_{B^{0}_{sL}}-\Gamma_{B^{0}_{sH}}$. The parameter
$A^{\mu\mu}_{\Delta\Gamma}$ is equal to 1 in the SM, where only the heavy mass
eigenstate decays to $\mu^{+}\mu^{-}$, but can take any value between $-1$ and
$+1$ in scenarios beyond the SM. LHCb has performed the first measurement of
the ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ effective lifetime by
fitting the signal decay-time distribution using a dataset of
4.4$\mbox{\,fb}^{-1}$, resulting in $\tau_{\mu\mu}^{\rm eff}=2.04\pm 0.44\pm
0.05{\mathrm{\,ps}}$ [266] (Fig. 7.1, right). The relative uncertainty on
$\tau_{\mu\mu}^{\rm eff}$ is expected to decrease to approximately 8% with
23$\mbox{\,fb}^{-1}$ and 2% with 300$\mbox{\,fb}^{-1}$ of data. While the
current experimental uncertainty is larger than
$\tau_{B^{0}_{sH}}-\tau_{B^{0}_{sL}}$, a 2% uncertainty on $\tau_{\mu\mu}^{\rm
eff}$ allow stringent constraints to be set on $A^{\mu\mu}_{\Delta\Gamma}$ and
in particular would allow the degeneracy between any possible contribution
from new scalar and pseudoscalar mediators to be broken.

Assuming a tagging power of about 3.7%[46], a dataset of 300$\mbox{\,fb}^{-1}$
allows a pure sample of more than 100 flavour-tagged
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decays (effective yield) to be
reconstructed and their time-dependent $C\\!P$ asymmetry to be measured. From
the relation

$\frac{\Gamma(B^{0}_{s}(t)\rightarrow\mu^{+}\mu^{-})-\Gamma(\bar{B^{0}_{s}}\rightarrow\mu^{+}\mu^{-})}{\Gamma(B^{0}_{s}(t)\rightarrow\mu^{+}\mu^{-})+\Gamma(\bar{B^{0}_{s}}\rightarrow\mu^{+}\mu^{-})}=\frac{S_{\mu\mu}\sin(\Delta
M_{s}t)}{\cosh(y_{s}t/\tau_{B_{s}})+A^{\mu\mu}_{\Delta\Gamma}\sinh(y_{s}t/\tau_{B_{s}})}\,,$
(7.2)

where $t$ is the signal proper time and $\Delta M_{s}$ is the mass difference
of the heavy and light ${B}^{0}_{s}$ mass eigenstates, $S_{\mu\mu}$ can be
measured with an uncertainty of about 0.2. On the other hand, the signal yield
expected in a 23$\mbox{\,fb}^{-1}$ dataset is too low to allow a meaningful
constraint to be set on $S_{\mu\mu}$. A nonzero value for $S_{\mu\mu}$ would
automatically indicate evidence of $C\\!P$-violating phases beyond the SM.

#### 7.1.2 Search for $B\rightarrow{\tau^{+}\tau^{-}}$ decays

LHCb has performed a search for the $B\rightarrow{\tau^{+}\tau^{-}}$ decays
using a data sample corresponding to an integrated luminosity of
$3\,\mbox{\,fb}^{-1}$ [269]. Assuming no contribution from
${{B}^{0}}\rightarrow{\tau^{+}\tau^{-}}$ decays, an upper limit is set on the
${{B}^{0}_{s}}\rightarrow{\tau^{+}\tau^{-}}$ branching fraction of $6.8\times
10^{-3}$ at 95% CL. This is the first direct limit on
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\tau^{+}\tau^{-}})$. If instead no
contribution from ${{B}^{0}_{s}}\rightarrow{\tau^{+}\tau^{-}}$ decays is
assumed, the limit is
${\mathcal{B}}({{B}^{0}}\rightarrow{\tau^{+}\tau^{-}})<2.1\times 10^{-3}$ at
95% CL. This is the world’s best limit on
${\mathcal{B}}({{B}^{0}}\rightarrow{\tau^{+}\tau^{-}})$ and constitutes a
factor 2.6 improvement with respect to the BaBar result [270].

New Physics models proposed to explain the tensions currently observed in
$b\rightarrow s\ell^{+}\ell^{-}$ and $b\rightarrow c\ell^{+}\nu_{\ell}$
transitions allow for a potential enhancement of the
$B\rightarrow{\tau^{+}\tau^{-}}$ branching fractions by several orders of
magnitude [271, 272, 273, 274, 275] with respect to their SM predictions,
${\mathcal{B}}({{B}^{0}}\rightarrow{\tau^{+}\tau^{-}})=(2.22\pm 0.19)\times
10^{-8}$ and
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow{\tau^{+}\tau^{-}})=(7.73\pm
0.49)\times 10^{-7}$ [265]. With potential enhancements of the branching
fractions close to the current best upper limits it therefore remains
interesting to pursue a search for the $B\rightarrow{\tau^{+}\tau^{-}}$
decays, even though, based on the current experimental precision, a
measurement of the SM value will remain out of reach for LHCb. Naively scaling
our obtained limits with the collected luminosity, and taking into account the
factor two increase in the $b\bar{b}$ cross-section when going from
7–8$\mathrm{\,Te\kern-1.00006ptV}$ to 14$\mathrm{\,Te\kern-1.00006ptV}$ we
expect to improve the limit on the
${{B}^{0}_{s}}\rightarrow{\tau^{+}\tau^{-}}$ branching fraction by a factor 5
to $1.3\times 10^{-3}$ by the end of Upgrade I and by another factor 2.6 to
$5\times 10^{-4}$ by the end of Upgrade II.

The understanding of the intermediate resonance structure of the
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ decay, which is exploited
in the current analysis to define a region with enhanced signal sensitivity,
forms a potential systematic limitation in the
$B\rightarrow{\tau^{+}\tau^{-}}$ searches. Here, we most likely will have to
rely on Belle II to provide improved input measurements.

### 7.2 Lepton-flavour, lepton-number and baryon-number violating decays

#### 7.2.1 Search for ${B}\rightarrow e^{\pm}\mu^{\mp}$ decays

The lepton-flavour-violating (LFV) process
${B}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ is forbidden within the SM, however
recent hints of lepton-flavour non-universality in $B$-meson decays [276, 277]
suggest the presence of a mechanism at higher energies able to differentiate
the flavour of charged leptons. In relation to the anomalies observed, several
models [278, 274, 279] have been developed where the branching fractions of
the ${B}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ can be as large as
$\mathcal{O}(10^{-11})$.

The LHCb collaboration has recently published [280] the world’s best limits on
the branching fractions of the
${{B}^{0}_{s}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ decays using the first
3$\mbox{\,fb}^{-1}$ collected in 2011 and 2012 at 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$ respectively. The decay-time acceptance of
the ${{B}^{0}_{s}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ decays can be affected
by the relative contribution of the two ${B}^{0}_{s}$ mass eigenstates to the
total decay amplitude, due to their large lifetime difference. Therefore, the
upper limit on the branching fraction of
${{B}^{0}_{s}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ decays is evaluated in two
extreme hypotheses: where the amplitude is completely dominated by the heavy
eigenstate or by the light eigenstate. The results are
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$})<6.3\,(5.4)\times
10^{-9}$ and
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$})<7.2\,(6.0)\times
10^{-9}$ at $95\%\,(90\%)$ CL, respectively. The limit for the branching
fraction of the ${B}^{0}$ mode is
${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$})<1.3\,(1.0)\times
10^{-9}$ at $95\%\,(90\%)$ CL.

Assuming similar performances in background rejection and signal retention as
in the current analysis, at the end of the Upgrade I data taking period the
LHCb experiment will be able to probe branching fractions of
${{B}^{0}_{s}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{e^{\pm}}{\mu^{\mp}}$ decays down to $8\times
10^{-10}$ and $2\times 10^{-10}$, respectively. The additional samples
accumulated during the Upgrade II data taking period will push down these
limits to $3\times 10^{-10}$ and $9\times 10^{-11}$ respectively, close to the
interesting region where NP effects may appear. The improvement at Upgrade II
over Upgrade I in electron reconstruction will be very important in attaining,
or exceeding, this goal.

#### 7.2.2 Search for ${B}\rightarrow\tau^{\pm}\mu^{\mp}$ decays

A wide variety of Beyond Standard Model (BSM) scenarios predict the occurrence
of lepton-flavour violating $B$ decays, with the highest rates where third
generation leptons are involved, such as
${B}^{0}_{\left(s\right)}\\!\rightarrow{\tau^{\pm}}{\mu^{\mp}}$. For different
models, not excluded by the current experimental constraints, the expected
branching fraction for the LFV process
${{B}^{0}_{s}}\\!\rightarrow{\tau^{\pm}}{\mu^{\mp}}$ is in the range
$10^{-9}-10^{-4}$ [281, 282, 283, 284, 285, 286, 275, 287, 288, 289, 290, 271,
272].

On the experimental side, an upper limit on the
${{B}^{0}}\\!\rightarrow{\tau^{\pm}}{\mu^{\mp}}$ channel has been already set
by BaBar:
${\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{\tau^{\pm}}{\mu^{\mp}}$}\right)<2.2\times
10^{-5}$ at $90\%$ CL [291]. The first search on the
${{B}^{0}_{s}}\\!\rightarrow{\tau^{\pm}}{\mu^{\mp}}$ channel is in progress in
LHCb and the results are expected soon on data recorded in 2011 and 2012 using
the ${\tau^{\pm}}\\!\rightarrow\pi^{\pm}\pi^{\mp}\pi^{\pm}\nu$ and
${\tau^{\pm}}\\!\rightarrow\pi^{\pm}\pi^{\mp}\pi^{\pm}{{\pi}^{0}}\nu$ decay
modes. Given the presence of a neutrino that escapes detection this kind of
analysis is much more complicated than those investigating electron or muon
final states. A specific reconstruction technique is used in order to infer
the energy of the $\nu$, taking advantage of the known $\tau$ decay position
given by the $3\pi$ reconstructed vertex. This way, the complete kinematics of
the process can be solved up to a two-fold ambiguity. LHCb expects to reach
sensitivities of a few times $10^{-5}$ with the Run 1 and 2 data sets.
Extrapolating the current measurements to the Upgrade II LHCb could reach
${\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{\tau^{\pm}}{\mu^{\mp}}$}\right)<3\times
10^{-6}$ at $90\%$ CL. The mass reconstruction technique depends heavily on
the resolution of the primary and the $\tau$ decay vertices, hence improvement
in the tracking system in Upgrade II, including a removal or reduction in
material of the VELO RF foil, will be very valuable.

#### 7.2.3 Search for $B\rightarrow Ke\mu$ and $B\rightarrow K\tau\mu$ decays

If New Physics allows for charged lepton-flavour violation then the branching
fractions of $B\rightarrow K\ell\ell^{{}^{\prime}}$ or ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}\ell\ell^{{}^{\prime}}$
will be enhanced with respect to their purely leptonic counterparts, since the
helicity suppression is lower. Furthermore, if observed, they would allow the
measurement of more observables with respect to the lepton-flavour violating
decays discussed in the previous sections, thanks to their multi-body final
states and, in the case of ${{\mathchar 28931\relax}^{0}_{b}}$, to the non-
zero initial spin.

In many generic new physics models with lepton flavour universality violation,
charged lepton-flavour violating decays of $b$-hadrons can be linked with the
anomalies recently measured in $b\rightarrow s\ell\ell$ decays [276, 277,
292]. The current limits set by the $B$ Factories on the branching fractions
of $B\rightarrow Ke\mu$ and $B\rightarrow K\tau\mu$ decays are $<13\times
10^{-8}$ [293] and $<4.8\times 10^{-5}$ [294] at 90% confidence level,
respectively.

At LHCb, searches for ${{B}^{+}}\\!\rightarrow{{K}^{+}}e^{\pm}\mu^{\mp}$,
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$,
${{B}^{+}}\\!\rightarrow{{K}^{+}}\tau^{\pm}\mu^{\mp}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}e^{\pm}\mu^{\mp}$
are ongoing. These searches are complementary, as charged lepton-flavour
violation couplings among different families are expected to be different. The
analyses involving $\tau$ leptons reconstruct candidates via the
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}\nu_{\tau}$ channel,
which allows the reconstruction of the $\tau$ decay vertex.111It should be
noted that searches for ${{B}^{+}}\\!\rightarrow{{K}^{+}}\tau^{\pm}\mu^{\mp}$
from $B_{s2}^{*}$ without $\tau$ reconstruction can give complementary
information. All these decays contain at least one muon, which is used to
efficiently trigger on the event. Usually, since these decays involve
combinations of leptons that are not allowed in the Standard Model, the
backgrounds can be kept well under control, leaving very clean samples only
polluted by candidates formed by the random combinations of tracks. This
combinatorial effect is higher for the channel with a $\tau$ lepton in the
final state decaying into three charged pions. The other relevant background
comes from chains of semileptonic decays, where two or more neutrinos are
emitted and therefore combinations of leptons of different flavours are
possible. These decays have typically a low reconstructed invariant mass, due
to the energy carried away by the neutrinos, and so they do not significantly
pollute the signal region.

The expected upper limits at LHCb using the first $9\mbox{\,fb}^{-1}$ of data
taken are $\mathcal{O}(10^{-9})$ and $\mathcal{O}(10^{-6})$ for the
${{B}^{+}}\\!\rightarrow{{K}^{+}}e^{\pm}\mu^{\mp}$ and
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$ decays respectively, at
the 90% confidence level. The limit for
${{B}^{+}}\\!\rightarrow{{K}^{+}}\tau^{\pm}\mu^{\mp}$ is expected to be
similar to ${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$. The
sensitivity of these analyses scales almost linearly with integrated
luminosity for ${{B}^{+}}\\!\rightarrow{{K}^{+}}e^{\pm}\mu^{\mp}$, and with
the square root of the integrated luminosity for
${{B}^{0}}\\!\rightarrow{{K}^{*0}}\tau^{\pm}\mu^{\mp}$. In both cases, the
expected limits using the Upgrade II data are in the region of interest of the
models currently developed for explaining the $B$ anomalies, so they will
provide strong constraints on the New Physics scenarios with charged lepton-
flavour violation.

#### 7.2.4 Search for ${\tau^{+}}\rightarrow{\mu^{+}\mu^{-}}{\mu^{+}}$ decays

An important test of the SM is the search for the lepton-flavour-violating
process ${\tau^{\pm}}\\!\rightarrow{\mu^{\pm}}\mu^{+}\mu^{-}$. Within the SM
with zero neutrino masses this process is strictly forbidden. Depending on the
mechanism of neutrino mass generation, many theories [295, 296, 297, 298]
beyond the SM predict this branching ratio to be in the region
$(10^{-9}-10^{-8})$. The current experimental limit [299, 300, 301],
$\cal{B}(\mbox{${\tau^{\pm}}\\!\rightarrow{\mu^{\pm}}\mu^{+}\mu^{-}$})<$
$1.2\times 10^{-8}$, is a combination of the results from LHCb and the $B$
Factories and reaches the starting point of this range. Belle II will probe
this interesting region of sensitivity when accumulating up to
$50~{}\rm{ab}^{-1}$. The LHC proton collisions at 13 TeV produces $\tau$
leptons, primarily in the decay of heavy flavour hadrons, with a cross-section
five orders of magnitude larger than at Belle II. This compensates for the
higher background levels and lower integrated luminosity, and means that
during the Upgrade II LHCb would also be able to probe down to ${\cal
O}(10^{-9})$, and independently confirm any Belle II discovery or
significantly improve the combined limit. In addition, the proposed
improvements to the LHCb calorimeter during the Upgrade II will be helpful in
suppressing backgrounds such as
${{D}^{+}_{s}}\\!\rightarrow\eta(\\!\rightarrow\mu^{+}\mu^{-}\gamma)\mu^{+}\nu_{\mu}$,
enabling LHCb to make best use of its statistical power.

#### 7.2.5 Search for lepton-number and baryon-number violating decays

In the Standard Model, baryon and lepton number conservation are accidental,
low-temperature symmetries that can be violated through the Bell-Jackiw
(chiral) anomaly [302]. Despite this, neither baryon number violation (BNV)
nor lepton number violation (LNV) has been observed experimentally, with BNV
in particular being constrained by stringent lower limits on the mean
lifetimes of protons and of bound neutrons [303]. Physics beyond the SM may
introduce additional sources of LNV and BNV, so that searches for these
effects are sensitive to new physics and can constrain—or potentially
reveal—its presence.

Experimentally, since these are null searches, sensitivity is assumed to scale
linearly with integrated luminosity ${\cal L}$ when the background is
negligible and as $\sqrt{{\cal L}}$ if the background is significant. LHCb has
already published searches in certain channels, and others are in progress:

  * •

Searches for LNV in various $B$-meson decays of the form $B\rightarrow
X\mu^{+}\mu^{+}$, where $X$ is a system of one or more hadrons. The principal
motivation is the sensitivity to contributions from Majorana neutrinos [304],
which may be on-shell or off-shell, depending on the decay mode. The published
results consist of searches for
${{{B}^{+}}}\\!\rightarrow{{K}^{-}}\mu^{+}\mu^{+}$,
${{{B}^{+}}}\\!\rightarrow{{\pi}^{-}}\mu^{+}\mu^{+}$ and
${{B}^{+}}\\!\rightarrow D^{+}_{(s)}{\mu^{-}}{\mu^{-}}$ [305, 306, 307]. A
limit of $\mathcal{B}(B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+})<4\times 10^{-9}$
is set at the 95% confidence level, along with more detailed limits as a
function of the Majorana neutrino mass. Since the combinatorial background was
found to be low but not negligible with the Run 1 data, we estimate that the
limit can be improved by a factor of ten with the full Upgrade II dataset.

  * •

Search for BNV in ${\mathchar 28932\relax}^{0}_{b}$ oscillations [308]. Six-
fermion, flavour-diagonal operators, involving two fermions from each
generation, could give rise to BNV/LNV without violating the nucleon stability
limit [309, 310]. Since the ${\mathchar 28932\relax}^{0}_{b}$ ($bsd$) baryon
has one valence quark from each generation, it could couple directly to such
an operator and oscillate to a ${\overline{\mathchar 28932\relax}}^{0}_{b}$
baryon. The published search used the Run1 data and set a lower limit on the
oscillation period of 80${\mathrm{\,ps}}$. Since events are tagged by decays
of the ${\mathchar 28932\relax}^{\prime-}_{b}$ and ${\mathchar
28932\relax}^{*-}_{b}$ resonances, with the former being particularly clean,
and since the analysis also uses the decay-time distribution of events, the
sensitivity is expected to scale linearly. Although the decay mode used in the
published analysis is hadronic (${\mathchar
28932\relax}^{0}_{b}\rightarrow{{\mathchar 28932\relax}^{+}_{c}}{{\pi}^{-}}$),
future work could also benefit from the lower-purity but higher-yield
semileptonic mode ${\mathchar 28932\relax}^{0}_{b}\rightarrow{{\mathchar
28932\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$.

  * •

${{\mathchar 28931\relax}^{+}_{c}}\rightarrow{\overline{p}}\mu^{+}\mu^{+}$.
This channel has previously been investigated at the $e^{+}e^{-}$ $B$
Factories. The current upper limit, obtained by BaBar [311], is
$\mathcal{B}({{\mathchar
28931\relax}^{+}_{c}}\rightarrow{\overline{p}}{\mu^{+}}{\mu^{+}})<9.4\times
10^{-6}$ at the 90% confidence level. With Run 1 and 2 data alone, it should
be possible to reduce this to $1\times 10^{-6}$. Further progress depends on
the background level, but an additional factor of 5–10 with the full Upgrade
II statistics is likely.

  * •

${{\mathchar 28931\relax}^{+}_{c}}\rightarrow\mu^{+}\mu^{-}\mu^{+}$.
Experimentally, this is a particularly promising decay mode: the final state
with three muons is very clean, and there are no known sources of peaking
background. This search could be added for little extra effort to the
${\tau^{-}}\rightarrow{\mu^{+}}{\mu^{-}}{\mu^{-}}$ search described in the
preceding section.

The discussion in this section is certainly not exhaustive; the list of
possible few-body decay modes that violate BNV or LNV is long. In order to
make best use of the LHCb data, it will be important to have input from theory
to identify those modes that could best evade the bounds from nucleon
stability.

### 7.3 Flavour-changing $b\rightarrow s\ell^{+}\ell^{-}$ and $b\rightarrow
d\ell^{+}\ell^{-}$ transitions

#### 7.3.1 Introduction

Recent LHCb measurements of rare semileptonic decays show discrepancies with
respect to SM predictions. None of these deviations is by itself significant
enough to be considered as evidence for NP but global fits [312, 313, 314]
show that they can be interpreted in a consistent picture, with an ${\cal
O}(1)$ NP contribution to the vector (and potentially axial-vector) coupling
strength of the decays. Regardless of whether these discrepancies are
confirmed with additional data, the fact that ${\cal O}(1)$ NP contributions
are still allowed demonstrates the importance of making precise measurements
of ${b}\rightarrow{s}\ell^{+}\ell^{-}$ and ${b}\rightarrow{d}\ell^{+}\ell^{-}$
processes. The NP contribution can be associated with new particles at mass
scales well above the LHC energy reach, e.g. by a
multi-$\mathrm{\,Te\kern-1.00006ptV}$-scale $Z^{\prime}$ boson or a
leptoquark. A precise determination of the effective couplings, through
measurements of different ${b}\rightarrow{s}\ell^{+}\ell^{-}$ and
${b}\rightarrow{d}\ell^{+}\ell^{-}$ processes, is therefore critical to
understand or constrain the structure of any NP model.

In the rest of this section, a number of benchmark NP scenarios are considered
(see Table 7.1). Scenarios I and II are inspired by the current discrepancies.
The first scenario is that which best explains the present rare semileptonic
decay data. The second scenario best explains the rare semileptonic
measurements if a purely left-handed coupling to quarks and leptons is
required for NP. This requirement is theoretically well motivated and arises
in models designed to simultaneously explain the discrepancies seen in both
tree-level semitauonic and loop-level semileptonic decays. The third and
fourth scenarios assume that the current discrepancies are not confirmed but
there is instead a small contribution from right-handed currents that would
not be visible with the current level of experimental precision. These
scenarios will serve to illustrate the power of the large Upgrade II data set
to distinguish between different NP models. This power relies critically on
the ability to exploit multiple related decay channels.

Table 7.1:  Benchmark NP scenarios. The first scenario is inspired by the present discrepancies in the rare decays, including the angular distributions of the decay ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ and the measurements of the branching fraction ratios $R_{K}$ and $R_{K^{*}}$. The second scenario is inspired by the possibility of explaining the rare decays discrepancies and those measured in the observables $R(D^{(*)})$. The third and fourth scenarios assume a small right-handed chirality coupling. The Wilson coefficients ($C_{i}$) are discussed in Sec. 7.3.2 scenario | $C_{9}^{\rm NP}$ | $C_{10}^{\rm NP}$ | $C_{9}^{\prime}$ | $C_{10}^{\prime}$
---|---|---|---|---
I | $-1.4$ | 0 | 0 | 0
II | $-0.7$ | 0.7 | 0 | 0
III | 0 | 0 | 0.3 | 0.3
IV | 0 | 0 | 0.3 | $-0.3$

#### 7.3.2 Theoretical framework

Flavour-changing neutral-current decays involving $b\rightarrow
s\ell^{+}\ell^{-}$ and $b\rightarrow d\ell^{+}\ell^{-}$ transitions are
suppressed by the GIM mechanism in the SM and are therefore promising places
to search for effects of NP. New particles that arise in extensions of the SM
can contribute to the amplitude of these decays with a similar strength to the
SM processes. Feynman diagrams for SM and possible NP extensions are shown in
Fig. 7.2.

The ${b}\rightarrow{s}$ and ${b}\rightarrow{d}$ FCNC processes are described
by the effective Hamiltonian,

${\cal H}_{\rm
eff}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{tq}^{*}\left[\frac{\alpha_{e}}{4\pi}\sum_{i}{\cal
C}_{i}(\mu){\cal O}_{i}(\mu)\right],$ (7.3)

where $G_{F}$ is the Fermi constant, $V_{ij}$ are elements of the CKM matrix
and $\alpha_{e}$ is the fine structure constant [315]. The $C_{i}$
coefficients are the so-called Wilson coefficients that encode short-distance
contributions to the decay amplitude at scales above $\mu$. The Wilson
coefficients are universal and apply to both the ${b}\rightarrow{s}$ and
${b}\rightarrow{d}$ processes. Finally, the ${\cal O}_{i}$ are local operators
with different Lorentz structures. The most important operators for
${b}\rightarrow{s}\ell^{+}\ell^{-}$ and ${b}\rightarrow{d}\ell^{+}\ell^{-}$
decays are

$\displaystyle\begin{split}{\cal
O}_{7}^{(\prime)}&=\frac{m_{b}}{e}\left(\overline{s}\sigma^{\mu\nu}P_{\rm
R(L)}b\right)F_{\mu\nu}~{},\\\ {\cal
O}_{9}^{(\prime)}&=\left(\overline{s}P_{\rm
L(R)}b\right)\left(\overline{\ell}\gamma^{\mu}\ell\right)~{},\\\ {\cal
O}_{10}^{(\prime)}&=\left(\overline{s}P_{\rm
L(R)}b\right)\left(\overline{\ell}\gamma^{\mu}\gamma^{5}\ell\right)~{},\end{split}$
(7.4)

where $m_{b}$ is the $b$-quark mass and $P_{\rm L,R}$ are the left- and right-
handed projection operators. The coefficients $C_{9}$ and $C_{10}$ correspond
to the coupling to a vector and an axial-vector leptonic current,
respectively. The V–A structure of the weak interaction leads to the
expectation that $C_{9}\approx-C_{10}$ and $C^{\prime}_{9,10}\approx 0$.
Hadronic matrix elements are parametrised in terms of form factors that can be
determined with non-perturbative methods, such as QCD sum rules or lattice
QCD. Contributions from new particles can modify the Wilson coefficients,
leading to observable effects in the rate and angular distribution of the
decays or introducing new sources of $C\\!P$ violation.

Figure 7.2:  Feynman diagrams representing (top) SM contributions to
$b\rightarrow s{\mu^{+}\mu^{-}}$ transitions and (bottom) possible
contributions in NP models. The bottom left diagram shows the contribution of
a $Z^{\prime}$ boson, while the bottom right shows that of a leptoquark.

Different regions of dilepton mass squared ($q^{2}$) and decays with different
final-state hadrons provide sensitivity to different combinations of the
Wilson coefficients. New physics contributions at energy scales above $m_{b}$
can modify the Wilson coefficients from their SM-values, or introduce entirely
new Lorentz structures. Decomposing possible NP contributions in terms of
effective operators with a well defined Lorentz structure allows the model-
independent correlations between different observables and decays to be
exploited. The low-$q^{2}$ region of the decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ (and other vector meson
final-states) is particularly sensitive to NP effects, due to the large
interference effects between $C_{7}^{(\prime)}$ and $C_{9}^{(\prime)}$. The
large number of different ${b}\rightarrow{s}\ell^{+}\ell^{-}$ and
${b}\rightarrow{d}\ell^{+}\ell^{-}$ decay modes that will be accessible in
Upgrade II will improve the sensitivity to the Wilson coefficients and will
allow the consistency between different measurements to be tested
comprehensively.

The interpretation of observables in terms of Wilson coefficients requires
knowledge of QCD matrix elements in the non-perturbative regime and these are
difficult to calculate. For example, long-distance contributions involving
intermediate charm-loops have generated much debate about the interpretation
of global fits for the Wilson coefficients [316, 317, 318]. Fortunately, the
correlation with different observables allows theoretical uncertainties to be
reduced to a significant extent, either by constraining the matrix elements
from the data, or by forming ratios where the matrix elements cancel such as
in lepton universality tests.

#### 7.3.3 Branching fractions and angular observables in $b\rightarrow
s\ell^{+}\ell^{-}$ transitions

The branching fractions of a number of exclusive
${b}\rightarrow{s}\ell^{+}\ell^{-}$ processes have been studied using the LHCb
Run 1 data set. The experimental measurements are systematically lower than
their corresponding SM predictions. The largest discrepancy appears for
${\mathcal{B}}(\mbox{${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$})$
which, in the $1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ region,
is more than $3\,\sigma$ from the SM predictions [319]. For both the Upgrade I
and Upgrade II datasets, the precision of the measurement of the branching
fractions will be limited by the knowledge of the
${B}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X$ decay modes
that are used to normalise the observed signals. The knowledge of these
branching fractions will be improved by the Belle II collaboration but will
inevitably limit the precision of the absolute branching fractions of rare
${b}\rightarrow{s}\ell^{+}\ell^{-}$ processes. The comparison between the
predicted and measured branching fractions will in any case be limited by the
theoretical knowledge of the form factors, even if in the future these are
determined parameterically using the data. A better comparison between theory
and experiment can be achieved by studying isospin and $C\\!P$ asymmetries,
which with the Upgrade II data set will be experimentally probed with percent
level precision. The Upgrade II data set will also enable new decay modes to
be studied, for example higher-spin ${K}^{*}$ states and modes with larger
numbers of decay products.

It is also possible to reduce theoretical and experimental uncertainties by
comparing regions of angular phase-space of
${b}\rightarrow{s}\ell^{+}\ell^{-}$ decays. The angular distribution of
${B}\\!\rightarrow V\ell^{+}\ell^{-}$ decays, where $V$ is a vector meson, can
be expressed in terms of eight $q^{2}$-dependent angular coefficients that
depend on the Wilson coefficients and the form-factors. Measurements of
angular observables in ${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$
decays show a discrepancy with respect to SM predictions [320, 292, 321, 322,
293, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336,
337]. This discrepancy is largest in the form-factor-independent observable
$P^{\prime}_{5}$ [292]. The decay
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ can also be described by the
same angular formalism as the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decay. However, in this
case the ${B}^{0}_{s}$ and the ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ mesons decay to a common
final state and it is not possible to determine the full set of observables
without tagging the initial flavour of the ${B}^{0}_{s}$. This is discussed
further in Sec. 7.3.6.

With the large data set that will be collected with Upgrade II, corresponding
to around 440 000 fully reconstructed
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decays, it will be
possible to make a precise determination of the angular observables in narrow
bins of $q^{2}$ or using a $q^{2}$-unbinned approach [338, 339]. The expected
precision of an unbinned determination of $P_{5}^{\prime}$ in the SM and in
Scenarios I and II is illustrated in Fig. 7.3. Upgrade II will enable these
scenarios to be clearly separated from the SM and from each other. By
combining information from all of the angular observables in the decay, it
will also be possible to distinguish models with much smaller NP
contributions. Figure 7.4 shows the expected $3\,\sigma$ sensitivity for the
Wilson coefficients for $C_{9,10}^{\prime}$ for the SM, Scenario III and
Scenario IV. These scenarios are also clearly distinguishable with the
precision that will be available with the Upgrade II data set.

Figure 7.3: Experimental sensitivity to the $P_{5}^{\prime}$ angular
observable in the SM, Scenarios I and II for (left) the Runs 1–3 and (right)
the Upgrade II data sets. The sensitivity is computed assuming that the charm-
loop contribution is determined from the data.

Figure 7.4: Expected sensitivity for the Wilson coefficients $C_{9}^{\prime}$
and $C_{10}^{\prime}$ from the analysis of the decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$. The ellipses correspond
to 3 $\sigma$ contours for the SM, Scenario III and Scenario IV for (left) the
Runs 1–3 and (right) the Upgrade II data sets.

The major challenge for ${B}\\!\rightarrow V\ell^{+}\ell^{-}$ decays is to
disentangle NP effects from SM contributions. With a large data set it will be
possible to probe the SM contributions, under the premise that a genuine NP
contribution is expected to have no $q^{2}$ dependence, while e.g. a charm
loop contribution is expected to grow approaching the pole of the charmonia
resonances. A measurement using Breit-Wigner functions to parametrise the
resonances, and their interference with the short-distance contributions to
the decay, is proposed in Ref. [340]. A similar technique has already been
applied to the Run 1 data for the
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}}$ decay [341]. An
alternative approach using additional phenomenological inputs has also been
proposed [342]. A precise knowledge of the charm loop contribution and a
parametric determination of the form factors, will come from a combination of
phenomenological and experimental methods and will allow $C_{9}$ and $C_{10}$
to be determined with great precision in ${b}\rightarrow{s}{\mu^{+}\mu^{-}}$
transitions.

#### 7.3.4 Branching fractions and angular observables in $b\rightarrow
d\ell^{+}\ell^{-}$ transitions

The Upgrade II data set will provide a unique opportunity to make precise
measurements of ${b}\rightarrow{d}\ell^{+}\ell^{-}$ processes. Using the Run 1
and 2 data sets, LHCb data have been used to observe the decays
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ [343, 344] and
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow p{{\pi}^{-}}{\mu^{+}\mu^{-}}$
[345] and find evidence for the decays
${{B}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{\mu^{+}\mu^{-}}$ (in a
${{\pi}^{+}}{{\pi}^{-}}$ mass region that is expected to be dominated by
${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{\mu^{+}\mu^{-}}$) and
${{B}^{0}_{s}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}{\mu^{+}\mu^{-}}$ [346] with
branching fractions at the $\mathcal{O}(10^{-8})$ level. The existing data
samples comprise $\mathcal{O}(10)$ decays in these decay modes. The Upgrade II
will provide samples of thousands, or tens of thousands of such decays. The
ability to measure the properties of these processes depends heavily on the
PID performance of the LHCb subdetectors. In the case of the
${{B}^{0}_{s}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}{\mu^{+}\mu^{-}}$ decay,
excellent mass resolution is also critical to separate ${B}^{0}_{s}$ and
${B}^{0}$ decays.

The ratio of branching fractions between the CKM-suppressed $b\rightarrow
d\ell^{+}\ell^{-}$ transitions and their CKM-favoured
${b}\rightarrow{s}\ell^{+}\ell^{-}$ counterparts, together with theoretical
input on the ratio of the relevant form factors, enables the ratio of CKM
elements $|V_{td}|/|V_{ts}|$ to be determined. The precision on
$|V_{td}|/|V_{ts}|$ from such decays is presently dominated by the statistical
uncertainty on the experimental measurements of
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$, and is significantly
worse than the determination from mixing measurements. The theoretical
uncertainty is at the level of 4% and is expected to improve with further
progress on the form-factors from lattice QCD. Around 17 000
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ decays are expected in
the full 300$\mbox{\,fb}^{-1}$ dataset, allowing an experimental precision
better than 2%.

The current set of measurements of ${b}\rightarrow{s}\ell^{+}\ell^{-}$
processes have demonstrated the importance of angular measurements in the
precision determination of Wilson coefficients. With the Upgrade II dataset,
where a sample of 4300 ${{B}^{0}_{s}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}{\mu^{+}\mu^{-}}$ decays is
expected, it will be possible to make a full angular analysis of a
${b}\rightarrow{d}\ell^{+}\ell^{-}$ transition. The
${{B}^{0}_{s}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}{\mu^{+}\mu^{-}}$ decay is both
self-tagging and has a final state involving only charged particles. The
Upgrade II data set will allow the angular observables in this decay to be
measured with better precision than the existing measurements of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ angular distribution.

The Upgrade II dataset will also give substantial numbers of
${B}^{0,+}\\!\rightarrow\rho^{0,+}{\mu^{+}\mu^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow N{\mu^{+}\mu^{-}}$ decays. Although the
${{B}^{0}}\\!\rightarrow{{\rho}^{0}}{\mu^{+}\mu^{-}}$ decay does not give the
flavour of the initial $B$ meson, untagged measurements will give sensitivity
to a subset of the interesting angular observables. Analysis of the
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow N{\mu^{+}\mu^{-}}$ decay will
require statistical separation of overlapping $N\\!\rightarrow p{{\pi}^{-}}$
resonances with different $J^{P}$ by performing an amplitude analysis of the
final-state particles.

The combination of information from
${\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$})$, the
differential branching fraction of the
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$ decay, and angular
measurements, notably of ${{B}^{0}_{s}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}{\mu^{+}\mu^{-}}$, will
indicate whether NP effects are present in ${b}\rightarrow{d}$ transitions at
the level of 20% of the SM amplitude with more than $5\sigma$ significance.

#### 7.3.5 Lepton-flavour universality tests

In the SM, the electroweak couplings of leptons are flavour independent, or
lepton “universal”. The ratios of branching fractions measured with different
lepton families are therefore free from hadronic corrections and such
quantities can be precisely predicted. For example, the ratios

$\displaystyle R_{X}=\int\frac{\mathrm{d}\Gamma(\mbox{${B}\\!\rightarrow
X{\mu^{+}\mu^{-}}$})}{\mathrm{d}{q^{2}}}\,\mathrm{d}{q^{2}}\Bigg{/}\int\frac{\mathrm{d}\Gamma(\mbox{${B}\\!\rightarrow
X{e^{+}e^{-}}$})}{\mathrm{d}{q^{2}}}\,\mathrm{d}{q^{2}}\,,$ (7.5)

between $B$ decays to final states with muons and electrons, where $X$ is a
hadron containing a $s$ or a $d$ quark, are predicted to be very close to
unity in the SM [347, 348, 349]. The uncertainties from QED corrections are
found to be at the percent level [350].

The Run 1 LHCb data have been used to perform the most precise measurements of
$R_{K}$ and $R_{{{K}^{*}}}$ to-date [277, 276] (see Fig. 7.5). The
measurements show some tension with the SM, all deviating from predictions at
the level of 2.1–2.6 standard deviations. Assuming the current detector
performance, approximately 46 000
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{e^{+}e^{-}}$ and 20 000
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ candidates are expected in the
range $1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ in the
Upgrade II data set. The ultimate precision on $R_{K}$ and $R_{{{K}^{*}}}$
will be better than 1%. The importance of the Upgrade II data set in
distinguishing between different NP scenarios is highlighted in Fig. 7.5. With
this data set all four NP scenarios could be distinguished at more than 5
$\sigma$ significance.

The Upgrade II data set will also enable the measurement of other $R_{X}$
ratios e.g. $R_{\phi},R_{p{K}}$ and the ratios in CKM suppressed decays. For
example, with 300$\mbox{\,fb}^{-1}$, it will be possible to determine
$R_{\pi}={\mathcal{B}}(\mbox{${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{\mu^{+}\mu^{-}}$})/{\mathcal{B}}(\mbox{${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{e^{+}e^{-}}$})$
with a few percent statistical precision. A summary of the expected
performance for a number of different $R_{X}$ ratios is indicated in Table
7.2.

Figure 7.5:  Projected sensitivity for the $R_{K}$, $R_{K^{*}}$ and $R_{\phi}$ measurements in different NP scenarios with the Upgrade II data set. The existing Run 1 measurements of $R_{K}$ and $R_{K^{*}}$ are shown for comparison.  Table 7.2:  Estimated yields of ${b}\rightarrow{s}{e^{+}}{e^{-}}$ and ${b}\rightarrow{d}{e^{+}}{e^{-}}$ processes and the statistical uncertainty on $R_{X}$ in the range $1.1<{q^{2}}<6.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ extrapolated from the Run 1 data. A linear dependence of the ${b}{\overline{b}}$ production cross section on the $pp$ centre-of-mass energy and unchanged Run 1 detector performance are assumed. Where modes have yet to be observed, a scaled estimate from the corresponding muon mode is used.  Yield | Run 1 result | 9$\mbox{\,fb}^{-1}$ | 23$\mbox{\,fb}^{-1}$ | 50$\mbox{\,fb}^{-1}$ | 300$\mbox{\,fb}^{-1}$
---|---|---|---|---|---
${{B}^{+}}\\!\rightarrow{{K}^{+}}{e^{+}e^{-}}$ |  $254\pm 29$[276] | 1 120 | 3 300 | 7 500 | 46 000
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ |  $111\pm 14$[277] | 490 | 1 400 | 3 300 | 20 000
${{B}^{0}_{s}}\\!\rightarrow\phi{e^{+}e^{-}}$ | – | 80 | 230 | 530 | 3 300
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{K}{e^{+}e^{-}}$ | – | 120 | 360 | 820 | 5 000
${{{B}^{+}}}\\!\rightarrow{{\pi}^{+}}{e^{+}e^{-}}$ | – | 20 | 70 | 150 | 900
$R_{X}$ precision | Run 1 result | 9$\mbox{\,fb}^{-1}$ | 23$\mbox{\,fb}^{-1}$ | 50$\mbox{\,fb}^{-1}$ | 300$\mbox{\,fb}^{-1}$
$R_{{K}}$ |  $0.745\pm 0.090\pm 0.036$[276] | 0.043 | 0.025 | 0.017 | 0.007
$R_{{{K}^{*0}}}$ |  $0.69\pm 0.11\pm 0.05$[277] | 0.052 | 0.031 | 0.020 | 0.008
$R_{\phi}$ | – | 0.130 | 0.076 | 0.050 | 0.020
$R_{{p}{K}}$ | – | 0.105 | 0.061 | 0.041 | 0.016
$R_{\pi}$ | – | 0.302 | 0.176 | 0.117 | 0.047

In addition to improvements in the $R_{X}$ measurements, the enlarged Upgrade
II data set will give access to new observables. For example, the data will
allow precise comparisons of the angular distribution of dielectron and dimuon
final-states. Differences between angular observables in ${B}\\!\rightarrow
X{\mu^{+}\mu^{-}}$ and ${B}\\!\rightarrow X{e^{+}}{e^{-}}$ decays are
theoretically pristine [351, 352] and are sensitive to different combinations
of Wilson coefficients compared to the $R_{X}$ measurements. Figure 7.6 shows
that an upgraded LHCb detector will enable such decays to be used to
discriminate between different NP models, for example separating between
Scenarios I and II [353]. Excellent NP sensitivity can be achieved
irrespective of the assumptions made about the hadronic contributions to the
decays.

Figure 7.6:  Constraints on the difference in the $C_{9}$ and $C_{10}$ Wilson
coefficients from electron and muon modes with the Run 3 and Upgrade II data
sets. The $3\sigma$ regions for the Run 3 data sample are shown for the SM
(solid blue), a vector-axial-vector new physics contribution (red dotted) and
for a purely vector new physics contribution (green dashed). The shaded
regions denote the corresponding constraints for the Upgrade II data set.

In the present LHCb detector, electron modes have an approximately factor five
lower efficiency than the corresponding muon modes, owing to the tendency for
the electrons to lose a significant fraction of their energy through
bremsstrahlung in the detector. This loss impacts on the ability to
reconstruct, trigger and select the electron modes. The precision with which
observables can be extracted therefore depends primarily on the electron modes
and not the muon modes. In order for $R_{X}$ measurements to benefit from the
large Upgrade II data samples, it will be necessary to reduce systematic
uncertainties to the percent level. These uncertainties can be controlled by
taking a double ratio between $R_{X}$ and the decays
${B}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}X$, where the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays to $\mu^{+}\mu^{-}$ and
${e^{+}}{e^{-}}$. This approach is expected to work well, even with very large
data sets.

Other sources of systematic uncertainty can be mitigated through design
choices for the upgraded detector. The recovery of bremsstrahlung photons is
inhibited by the ability to find the relevant photons in the ECAL (over
significant backgrounds) and by the energy resolution. A reduced amount of
material before the magnet would reduce the amount of bremsstrahlung and hence
would increase the electron reconstruction efficiency and improve the electron
momentum resolution. Higher transverse granularity would aid signal selection
and help reduce the backgrounds. With a large number of primary $pp$
collisions, the combinatorial background will increase and will need to be
controlled with the use of timing information. However, the Run 1 data set
indicates that it may be possible to tolerate a significant (i.e. larger than
a factor two) increase in combinatorial backgrounds without destroying the
signal selection ability.

#### 7.3.6 Time-dependent analysis of
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}}\rightarrow\rho^{0}{\mu^{+}}{\mu^{-}}$

Time-dependent analyses of rare decays into $C\\!P$ eigenstates can deliver
orthogonal experimental information to time-integrated observables. So far, no
time-dependent measurement of the
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ decay has been performed due
to the limited signal yield of $432\pm 24$ in the Run 1 data sample [319].
However, the larger data samples available in Upgrade II will enable time-
dependent studies. The framework describing ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}$ and ${B}\rightarrow
V\ell^{+}\ell^{-}$ transitions to a common final-state is discussed in Ref.
[354], where several observables are discussed that can be accessed with and
without flavour tagging. Two observables called $s_{8}$ and $s_{9}$, which are
only accessible through a time-dependent flavour-tagged analysis, are of
particular interest. These observables are proportional to the mixing term
$\sin\left(\Delta m_{s}t\right)$ and provide information that is not available
through flavour specific decays. Assuming a time resolution of around
$45\mathrm{\,fs}$ and an effective tagging power of $5\%$ results in an
effective signal yield of $2000$ decays for the Upgrade II data set.

As a first step towards a full time-dependent analysis, the effective lifetime
of the decay ${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ can be studied.
The untagged time-dependent decay rate is given by

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}t}$ $\displaystyle\propto
e^{-\Gamma_{s}}\left[\cosh\left(\frac{\Delta\Gamma_{s}t}{2}\right)+A^{\Delta\Gamma}\sinh\left(\frac{\Delta\Gamma_{s}t}{2}\right)\right].$
(7.6)

The observable $A^{\Delta\Gamma}$ can be related to the angular observables
$F_{\rm L}$ and $S_{3}$ via $A^{\Delta\Gamma}=2S_{3}-F_{\rm L}$. Due to the
significant lifetime difference $\Delta\Gamma_{s}$ in the ${B}^{0}_{s}$
system, even an untagged analysis can probe right-handed currents. For the
combined low- and high-$q^{2}$ regions, preliminary studies suggest a
statistical sensitivity to $A^{\Delta\Gamma}$ of $0.05$ can be achieved with a
$300\mbox{\,fb}^{-1}$ data set.

With the Upgrade II data set it will also be possible to perform a time-
dependent angular analysis of the ${b}\rightarrow{d}$ process
${{B}^{0}}\\!\rightarrow\rho^{0}{\mu^{+}\mu^{-}}$. This process differs from
${{B}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}\mu^{-}}$ in two important regards: it
is CKM suppressed and therefore has a smaller SM branching fraction; and
$\Delta\Gamma_{d}\approx 0$, removing sensitivity to $A^{\Delta\Gamma}$. The
uncertainties on the angular observables are expected to be on the order of
$0.1$ for this case.

The time-dependent angular analyses will still be statistically limited even
with $300\mbox{\,fb}^{-1}$. It will be important to maintain good decay-time
resolution and the performance of the particle identification will be crucial
to control backgrounds, as well as to improve flavour tagging performance.

### 7.4 Radiative beauty and charm decays

In the SM, $W^{\pm}$ bosons couple only left-handedly to quarks. As a
consequence, photons emitted in FCNC $b\\!\rightarrow s\gamma$ transitions are
predominantly left handed, but several extensions of the SM, see e.g. Ref.
[355], predict that the photon can acquire a significant right-handed
component through a new heavy particle that couples right-handedly to quarks.
Indirect constraints on right-handed currents leave a large fraction of the
space for non-SM right-handed currents unexplored. The photon polarisation—the
asymmetry between the right- and left-handed components of the photon—as well
as measurements of $C\\!P$ violation or the ratio between $b\\!\rightarrow
s\gamma$ and $b\\!\rightarrow d\gamma$ decays can be used to constrain these
right-handed currents.

Experimentally, the study of these transitions at LHCb is dominated by the
limitations of the electromagnetic calorimeter, which result in (a) a limited
mass resolution that prevents good separation of partially reconstructed
backgrounds, (b) a limited ability to distinguish signal from charmless
backgrounds in which the photon is replaced by a ${\pi}^{0}$ meson decaying
into two photons, which are reconstructed as a single cluster in the ECAL, and
(c) a degraded decay-time resolution coming from the photon momentum
resolution. The Upgrade II detector is expected to improve the mass resolution
for ${\pi}^{0}$ mesons thanks to a better spatial segmentation, which would be
crucial to match with the expected statistical precision of several of the
measurements discussed in this section.

A precise measurement of the photon polarisation has not been performed yet,
but several methods to achieve the result have been proposed:

  * •

The time-dependent $C\\!P$ asymmetry of ${{B}^{0}_{({s})}}\\!\rightarrow
f_{{C\\!P}}\gamma$, where $f_{{C\\!P}}$ is a $C\\!P$ eigenstate, discussed in
Sec. 7.4.1.

  * •

The angular correlations among the three-body decay products of a kaonic
resonance in ${{{B}^{+}}}\\!\rightarrow
K_{\mathrm{res}}^{+}(\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}})\gamma$,
presented in Sec. 7.4.1.

  * •

The transverse asymmetries of ${{B}^{0}}\\!\rightarrow{{K}^{*0}}e^{+}e^{-}$,
highly sensitive to the $b\\!\rightarrow s\gamma$ process at low dielectron
invariant mass squared ($q^{2}$), and which will be discussed in Sec. 7.4.2.

  * •

The angular distributions of radiative $b$-baryon decays, detailed in Sec.
7.4.3.

Each of these measurements constrains right-handed currents differently [356]
and they thus give complementary constraints in global fits of Wilson
coefficients.

#### 7.4.1 Radiative $B$ decays

The time dependent $C\\!P$ asymmetry of ${{B}^{0}_{({s})}}\\!\rightarrow
f_{{C\\!P}}\gamma$ arises from the interference between decay amplitudes with
and without ${{B}_{({s})}^{0}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}_{({s})}^{0}}$ mixing and is
predicted to be small in the SM [357, 358, 359]. As a consequence, a large
asymmetry due to interference between the $B$ mixing and decay diagrams can
only be present if the two photon helicities contribute to both $B$ and
${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}$ decays. From the time
dependent decay rate

$\displaystyle\Gamma(\mbox{${{B}_{({s})}^{0}}({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}_{({s})}^{0}})\\!\rightarrow
f_{{C\\!P}}\gamma$})(t)\sim e^{-\Gamma_{s}t}$
$\displaystyle\big{[}\cosh\left(\frac{\Delta\Gamma_{(s)}}{2}\right)-\mathcal{A}^{\Delta}\sinh\left(\frac{\Delta\Gamma_{(s)}}{2}\right)\pm$
$\displaystyle\pm\mathcal{C}_{C\\!P}\cos\left(\Delta
m_{(s)}t\right)\mp\mathcal{S}_{C\\!P}\sin\left(\Delta m_{(s)}t\right)\big{]},$
(7.7)

where $A^{\Delta}$, $\mathcal{C}_{C\\!P}$ and $\mathcal{S}_{C\\!P}$ depend on
the photon polarisation [360]. Two strategies can be devised: one studying the
decay rate independently of the flavour of the $B$ meson, which allows
$A^{\Delta}$ to be accessed, and one tagging the flavour of the $B$ meson,
which accesses $\mathcal{S}_{C\\!P}$ and $\mathcal{C}_{C\\!P}$. The first
strategy has been exploited at LHCb to study the $4000$
${{B}^{0}_{s}}\\!\rightarrow\phi\gamma$ candidates collected in Run 1 to
obtain
$A^{\Delta}=-0.98^{+0.46}_{-0.52}\mathrm{\,(stat)}^{+0.23}_{-0.20}\mathrm{\,(syst)}$
[361], compatible at two standard deviations with the prediction of
$A^{\Delta}_{\text{SM}}=0.047^{+0.029}_{-0.025}$.

With $\sim 60$k signal candidates expected with $50\,\mbox{\,fb}^{-1}$, the
full analysis, including flavour tagging information, will improve the
statistical uncertainty on $A^{\Delta}$ to $\sim 0.07$, and will need a
careful control of the systematic uncertainties. An analysis performed with
$\sim 800$k signal decays expected with $300\,\mbox{\,fb}^{-1}$, with a
statistical uncertainty to $\sim 0.02$, requires some of the possible
improvements in ${\pi}^{0}$ reconstruction of the Upgrade II detector to be
able to use the full statistical power of the data.

In addition to studying the ${B}^{0}_{s}$ system, LHCb can study the time-
dependent decay rate of
${{B}^{0}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}\gamma$ decays, which permits access of the photon
polarisation through the $\mathcal{S}_{C\\!P}$ term. With $\mathcal{O}(1000)$
signal events in Run 1, around $35$k and $200$k are expected at the end of
Upgrade I and Upgrade II, respectively ($1.75$k and $10$k when considering the
flavour tagging efficiency), opening the doors to a very competitive
measurement of $\mathcal{S}_{C\\!P}$ in the ${B}^{0}$ system.

Another way to study the photon polarisation is through the angular
correlations among the three-body decay products of a kaonic resonance in
${{{B}^{+}}}\\!\rightarrow
K_{\mathrm{res}}^{+}(\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}})\gamma$, which
allows the direct measurement of the photon polarisation parameter in the
effective radiative weak Hamiltonian [362]. As a first step towards the photon
polarisation measurement, LHCb observed nonzero photon polarisation for the
first time by studying the photon angular distribution in bins of
${{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ invariant mass [363], but the determination
of the value of this polarisation could not be performed due to the lack of
knowledge of the hadronic system. To overcome this problem, a method to
measure the photon polarisation using a full amplitude analysis of
${{{B}^{+}}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}\gamma$ decays is
currently under development, with an expected statistical sensitivity on the
photon polarisation parameter of $\sim 5\%$ in the charged mode with the Run 1
dataset. The extrapolation of the precision to $300\,\mbox{\,fb}^{-1}$ results
in a statistical precision better than $1\%$, and hence control of the
systematic uncertainties will be crucial.

#### 7.4.2 Angular analysis with $B^{0}\rightarrow K^{\ast 0}e^{+}e^{-}$
decays in the low-$q^{2}$ region

The polarisation of the photon emitted in $b\rightarrow s\gamma$ transitions
can also be accessed via semileptonic $b\rightarrow s\ell\ell$ transitions,
for example in the decay $B^{0}\rightarrow{{K}^{*0}}\ell^{+}\ell^{-}$. Indeed,
as mentioned in previous sections, at very low $q^{2}$ these decays are
dominated by the electromagnetic dipole operator ${\cal O}_{7}^{(\prime)}$.
Namely, the longitudinal polarisation fraction ($F_{\mathrm{L}}$) is expected
to be below $20\%$ for
$q^{2}<0.2{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. In this $q^{2}$
region, the angle $\phi$ between the planes defined by the dilepton system and
the ${{K}^{*0}}\rightarrow K^{+}\pi^{-}$ decay is sensitive to the
$b\rightarrow s\gamma$ photon helicity.

While the ${{K}^{*0}}{\mu^{+}\mu^{-}}$ final state is experimentally easier to
select and measure at LHCb, the ${{K}^{*0}}{e^{+}e^{-}}$ final state allows
$q^{2}$ values below $4m_{\mu}^{2}$ to be probed, where the sensitivity to the
photon helicity is maximal. Compared to the radiative channels used for
polarisation measurements, the $B^{0}\rightarrow{{K}^{*0}}{e^{+}e^{-}}$ final
state is fully charged and gives better mass resolution and therefore better
separation from partially reconstructed backgrounds.

The sensitivity of this decay channel at LHCb was demonstrated by an angular
analysis performed with Run 1 data [364]. The angular observables most
sensitive to the photon polarisation at low $q^{2}$ are $A_{\rm T}^{(2)}$ and
$A_{\rm T}^{\rm Im}$, as defined in Ref. [364]. Indeed, in the limit
$q^{2}\rightarrow 0$, these observables can be expressed by the following
functions of $C_{7,7^{\prime}}$ (assuming NP contributions to be much smaller
than $|C_{7}^{\rm SM}|$):

$A_{\rm T}^{(2)}({q^{2}}\rightarrow 0)\simeq
2\frac{\mathcal{R}e(C_{7}^{{}^{\prime}\ast})}{|C_{7}|}\quad{\rm and}\quad
A_{\rm T}^{\rm Im}({q^{2}}\rightarrow 0)\simeq
2\frac{\mathcal{I}m(C_{7}^{{}^{\prime}\ast})}{|C_{7}|}.$ (7.8)

In order to maximise the sensitivity to the photon polarisation, the angular
analysis should be performed as close as possible to the low $q^{2}$ endpoint.
However, the events at extremely low $q^{2}$ have worse $\phi$ resolution
(because the two electrons are almost collinear) and are polluted by
$B^{0}\rightarrow{{K}^{*0}}\gamma$ decays with the $\gamma$ converting in the
VELO material. In the Run 1 analysis [364] the minimum required
$m({e^{+}e^{-}})$ was set at 20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, but
this should be reduced as the Upgrade II VELO detector will have a
significantly lower material budget (multiple scattering is the main effect
worsening the $\phi$ resolution). Similarly, the background from $\gamma$
conversions will be reduced with a lighter RF-foil or with the complete
removal of it in Upgrade II [1].

Assuming the signal yields given in Table 7.2 leads to the following
statistical sensitivities to $A_{\rm T}^{(2)}$ and $A_{\rm T}^{\rm Im}$: 11%
with 9$\mbox{\,fb}^{-1}$, 7% with 23$\mbox{\,fb}^{-1}$ and 2% with
300$\mbox{\,fb}^{-1}$. The theoretical uncertainty induced when this
observable is translated into a photon polarisation measurement is currently
at the level of $2\%$ but should improve by the time of the Upgrade II
analyses. The current measurements performed with Run 1 data have a systematic
uncertainty of order $5\%$ coming mainly from the modelling of the angular
acceptance and from the uncertainty on the angular shape of the combinatorial
background. The acceptance is independent of $\phi$ at low $q^{2}$ and its
modelling can be improved with larger simulation samples and using the proxy
channel ${{B}^{0}}\rightarrow{{K}^{*0}}{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow{e^{+}e^{-}})$.

#### 7.4.3 Radiative $b$-baryon decays

Weak radiative decays of $b$ baryons are largely unexplored, with the best
limits coming from CDF: $\mathcal{B}(\mbox{${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}\gamma$})<1.3\times
10^{-3}$ at $90\%\,$CL [365]. They offer a unique sensitivity to the photon
polarisation through the study of their angular distributions, and will
constitute one of the main topics in the radiative decays programme in the
LHCb Upgrade II.

With predicted branching fractions of $O(10^{-5}-10^{-6})$, the first
challenge for LHCb will be their observation, as the production of long-lived
particles in their decay, in addition to the photon, means in most cases that
the $b$-baryon secondary vertex cannot be reconstructed. This makes their
separation from background considerably more difficult than in the case of
regular radiative $b$ decays.

The most abundant of these decays is ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar 28931\relax}(\rightarrow
p{{\pi}^{-}})\gamma$, which is sensitive to the photon polarisation
mainly222In the following, we assume that the ${\mathchar
28931\relax}^{0}_{b}$ (and any other beauty baryon) polarisation is zero
[366], removing part of the photon-polarisation dependence. through the
distribution of the angle between the proton and the $\mathchar 28931\relax$
momentum in the rest frame of the $\mathchar 28931\relax$ ($\theta_{p}$),

$\frac{\operatorname{d}\Gamma}{\operatorname{d}\cos\theta_{p}}\propto
1-\alpha_{\gamma}\alpha_{p,1/2}\cos\theta_{p},$ (7.9)

where $\alpha_{\gamma}$ is the asymmetry between left- and right-handed
amplitudes and $\alpha_{p,1/2}=0.642\pm 0.013$ [303] is the ${\mathchar
28931\relax}\\!\rightarrow p{{\pi}^{-}}$ decay parameter. Using specialised
trigger lines for this mode $15-150$ signal events are expected using the Run
2 dataset. Preliminary studies show that a statistical sensitivity to
$\alpha_{\gamma}$ of $(20-25)\%$ is expected with these data, which would be
reduced to $\sim 15\%$ with $23\,\mbox{\,fb}^{-1}$ and below $4\%$ with
$300\,\mbox{\,fb}^{-1}$. In the LHCb Upgrade II, the addition of timing
information in the calorimeter will be important to be able to study this
combinatorial-background dominated decay; additionally, improved downstream
reconstruction would allow the use of downstream $\mathchar 28931\relax$
decays, which make up more than $2/3$ of the total signal.

The ${{\mathchar 28932\relax}_{b}}\\!\rightarrow\Xi^{-}(\rightarrow{\mathchar
28931\relax}(\rightarrow p{{\pi}^{-}}){{\pi}^{-}})\gamma$ decay presents a
richer angular distribution, with dependence to the photon polarisation in
both the $\mathchar 28931\relax$ angle ($\theta_{\mathchar 28931\relax}$) and
proton angle ($\theta_{p}$),

$\frac{\operatorname{d}\Gamma}{\operatorname{d}\cos\theta_{\mathchar
28931\relax}\cos\theta_{p}}\propto
1-\alpha_{\gamma}\alpha_{\Xi}\cos\theta_{\mathchar
28931\relax}+\alpha_{p,1/2}\cos\theta_{p}\left(\alpha_{\Xi}-\alpha_{\gamma}\cos\theta_{\mathchar
28931\relax}\right),$ (7.10)

but the lower production cross-section for these baryons, combined with a
lower reconstruction efficiency due to the presence of one extra track,
results in an order of magnitude fewer events than in the ${\mathchar
28931\relax}^{0}_{b}$ case, making the increase of statistics from the Upgrade
II even more relevant. With a similar sensitivity to the photon polarisation
to that of ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{\mathchar
28931\relax}(\rightarrow p{{\pi}^{-}})\gamma$, ${{\mathchar
28932\relax}_{b}}\\!\rightarrow\Xi^{-}\gamma$ decays will allow this parameter
to be probed with a precision of $40\%$ and $10\%$ with $23$ and
$300\,\mbox{\,fb}^{-1}$, respectively.

#### 7.4.4 Radiative $D$ decays

Charm decays with a photon in the final state are complementary to radiative
beauty and strange decays, due to the different quark content of the
underlying loop amplitudes. Radiative charm decays, although rare
($\mathcal{B}$ $\sim 10^{-5}$) and overwhelmed by long-distance QCD effects,
still offer theoretically clean probes of the SM, like for instance the
$C\\!P$ asymmetry or photon polarisation. The most promising channels are
$D^{0}\rightarrow V\gamma$, where $V$ stands for a light vector meson,
$\rho^{0}$ or $\phi$. The SM predictions for $C\\!P$ violation in these decays
is small at the order of a few $10^{-3}$, however $A_{{C\\!P}}$ could be
enhanced by NP contributions to up to $10\%$ [367]. The best sensitivities so
far achieved have been obtained with the full Belle data set and are limited
by the statistical uncertainty, reaching a precision of $(7-15)\%$ [368]. No
observation of photon polarisation has yet been performed in charm decays.

Charm decays involve photons of significantly lower energies than their beauty
counterparts and thus suffer from higher combinatorial background and lower
energy resolution. Moreover, $D^{0}\rightarrow V\gamma$ decays are also
subject to a significant, peaking and irreducible background from the
$D^{0}\rightarrow V\pi^{0}$ decays, whose branching fractions exceed the
radiative ones by two orders of magnitude. In particular, merged $\pi^{0}$
mesons, where both photons form a single energy cluster, are difficult to
suppress. Nevertheless, a feasibility study performed for the
$D^{0}\rightarrow\phi\gamma$ decay has shown that LHCb will improve Belle
sensitivities by factor $2-3$ using Run 2 data.

Having an improved ${\pi}^{0}$ reconstruction in the Upgrade II would
certainly allow for background reduction. This could allow studies of decays
with more complicated topologies e.g. $D^{+}\rightarrow K\pi\pi\gamma$. Such
four-body processes give a direct access to $P$-parity and the photon
polarisation, which can be measured through an up-down asymmetry of the photon
direction relative to the $K\pi\pi$ decay plane. In the ${{D}^{0}}\rightarrow
V\gamma$ decays the photon polarisation can be accessed indirectly, by probing
a pattern of the ${{D}^{0}}-{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ oscillations and, thus,
requires a time-dependent analysis. Such measurements should become feasible
with the Upgrade II data sample.

### 7.5 Rare (semi-)leptonic charm decays

The study of very rare FCNC charm decays is a unique probe for NP in the up-
quark sector and a relatively unexplored area of research, both theoretically
and experimentally. Moreover, the anomalies seen in the $b$-sector make
progress on charm rare decays involving $c\rightarrow u$ transitions even more
pressing.

#### 7.5.1 Search for ${{D}^{0}}\rightarrow{\mu^{+}\mu^{-}}$ decays

The decay of a $D$ meson to a dimuon pair is a key very rare charm decay.
Within the Standard Model two kinds of contributions to the
$D^{0}\rightarrow\mu^{+}\mu^{-}$ decay are present: the _long-distance_ (LD)
contribution due to long-range propagation of intermediate states, of non-
perturbative nature, and the _short-distance_ (SD) contribution due to
perturbatively calculable amplitudes. In the down-type quark sector the GIM
suppression is less effective than in the up-quark sector, due to the presence
of $top$-quarks running in the loops which leads to an effective decoupling
with the other diagrams, owing to the large top quark mass. In the _charm_
sector FCNCs are instead more effectively suppressed due to the absence of a
large mass down-type quark and therefore, within the SM, branching fractions
of rare $D$ decays have very small values. However, these processes can be
enhanced in New Physics scenarios by up to several orders of magnitude when
compared to the SM. Predictions for the SD to
$\mathcal{B}(D^{0}\rightarrow\mu^{+}\mu^{-})$ are of $\mathcal{O}(10^{-18})$
while the LD contribution from $\gamma\gamma$ recombination, based on the
current limit for $D^{0}\rightarrow\gamma\gamma$ [369], brings the expected
branching fraction up to $10^{-11}$.

The world’s best limit was obtained by LHCb with $0.9\mbox{\,fb}^{-1}$ of 2011
data [370], resulting in

$\mathcal{B}(D^{0}\rightarrow\mu^{+}\mu^{-})<6.2\times 10^{-9}\text{ at 90\%
CL. }$ (7.11)

Extrapolating the current detector performance, the expected limit is about
$5.9\times 10^{-10}$ with $23~{}\mbox{\,fb}^{-1}$ and $1.8\times 10^{-10}$
with $300~{}\mbox{\,fb}^{-1}$ of integrated luminosity, covering a large part
of the unambiguous space to search for NP without being affected by LD
uncertainties in the SM predictions.

#### 7.5.2 Search for $D\rightarrow h{\mu^{+}\mu^{-}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{\mu^{+}\mu^{-}}$ decays

As already stated ${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ electroweak penguin
transitions have proven to be powerful tools to search for New Physics effects
[312, 313]. One can study the same New Physics effects in the up-quark sector
with the ${c}\\!\rightarrow{u}\ell^{+}\ell^{-}$ transitions. Until recently,
limits on the branching fractions of these types of decays were only set at
the ${\cal O}(10^{-6})$ level [311]. Significant progress has also been made
in the SM predictions for $c\rightarrow u$ FCNCs providing more robust
calculations [371, 372, 373].

Figure 7.7: Differential branching fraction for the decay ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{\mu^{+}\mu^{-}}$ as a function of $q^{2}$
(left) in the SM and (right) in a New Physics scenario with ${\cal C}_{9}^{\rm
NP}=-0.6$ and ${\cal C}_{10}^{\rm NP}=-0.6$ [373]. The current LHCb limit is
marked with green. The extrapolated upper limit with the $23$ and
$300~{}\mbox{\,fb}^{-1}$ data sets is marked in red.

Figure 7.8:  (Left) Differential branching fraction of
${D}^{+}\rightarrow\pi^{+}{\mu^{+}\mu^{-}}$ as a function of $q^{2}$ in the
SM. The current LHCb limit is marked with a dashed black line. (Middle and
right) Predictions for $A_{{C\\!P}}$ in a NP scenario with different strong
phases $\delta_{\rho,\phi}$ [372].

Particularly suited for experiments at hadron colliders is the study of decays
with a pair of leptons in the final state, such as $D^{+}_{(s)}/{{\mathchar
28931\relax}^{+}_{c}}\rightarrow h^{+}\ell^{+}\ell^{-}$ and $D^{0}\rightarrow
h^{+}h^{-}\ell^{+}\ell^{-}$. The SD contributions to branching fractions of
these decays involves FCNC processes which are heavily suppressed in the SM
through the GIM mechanism and are at the level of $10^{-9}$ or below,
depending on the decay [373, 372]. On the other hand, LD contributions
dominate the branching fraction via vector resonances
$\left(\rho/\omega/\phi\right)$ decaying into a dilepton pair. In some New
Physics scenarios the SD contributions can be enhanced by several orders of
magnitude allowing NP to manifest as an enhancement of the branching fraction.
An example of such a model is shown in Fig. 7.7. In the benchmark scenario,
the modification of Wilson coefficients are: $C_{9}^{\rm NP}=-0.6$ and
$C_{10}^{\rm NP}=0.6$. Outside of the resonance regions the SD contributions
are enhanced to similar levels as the LD effects. This benchmark point is
currently just below the current LHCb limit [374]

${\mathcal{B}}({{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{\mu^{+}\mu^{-}})<5.9\times 10^{-8}~{}{\rm at}~{}90\%~{}{\rm CL}$ (7.12)

In the Upgrade II era LHCb is expected to improve the limit to:

${\mathcal{B}}({{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{\mu^{+}\mu^{-}})<4.4\times 10^{-9}~{}{\rm at}~{}90\%~{}{\rm CL}.$ (7.13)

An order of magnitude improvement is expected for the limit on
${\mathcal{B}}(D^{+}_{(s)}\rightarrow\pi^{+}\mu^{+}\mu^{-})$ decays, which
currently is set at $7.3\times 10^{-8}$ at 90% CL [375]. The expected upper
limits are about $1.3\times 10^{-8}$ with $23\mbox{\,fb}^{-1}$ and $0.37\times
10^{-8}$ with $300\mbox{\,fb}^{-1}$. In addition, LHCb will have the ability
to measure angular observables such as the forward-background asymmetry
$A_{\rm FB}$ or time integrated $A_{{C\\!P}}$, which will provide additional
handles to separate the LD from the SD and for which some theoretical
predictions in NP scenarios already exist [372], as shown in Fig. 7.8.

Last but not least with a sample corresponding to an integrated luminosity of
$300~{}\mbox{\,fb}^{-1}$ LHCb will be able to perform searches for the LFV
decays $D^{+}_{(s)}/{{\mathchar 28931\relax}^{+}_{c}}\rightarrow
h^{+}\ell^{+}\ell^{\prime-}$ and perform tests of lepton universality via the
ratios ${\cal B}(\mbox{$D^{+}_{(s)}/{{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow h^{+}\mu^{+}\mu^{-}$})/{\cal
B}(\mbox{$D^{+}_{(s)}/{{\mathchar 28931\relax}^{+}_{c}}\\!\rightarrow
h^{+}e^{+}e^{-}$})$.

#### 7.5.3 Measurements with $D\rightarrow h^{+}h^{-}\ell^{+}\ell^{-}$ decays

The decays $D^{0}\rightarrow h^{+}h^{-}\ell^{+}\ell^{-}$ have a richer
dynamics compared to two- and three-body decays allowing for a variety of
differential distributions to be investigated. The SD contributions to the
branching fraction proceeds through FCNC processes, which are very suppressed
in the SM through the GIM mechanism at the level of $10^{-9}$ or below [376].
On the other hand, LD contributions dominate with branching fractions up to
$\mathcal{O}(10^{-6})$ via intermediate vector resonances ($\rho/\omega/\phi$)
decaying into a dilepton pair [377, 376, 378]. In some BSM scenarios the SD
contribution can be enhanced by several orders of magnitude allowing NP to
manifest as an enhancement of the branching fraction away from resonances or
in asymmetries such as $C\\!P$ and angular asymmetries [379, 380, 371, 378,
381, 372, 382].

Due to the huge charm production cross-section at the LHC, and LHCb’s ability
to trigger on low $p_{\mathrm{T}}$ dimuons, LHCb has unique physics reach in
studying these decays. In fact, significant progress has already been made
with the observation of the Cabibbo-favoured decay
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{\mu^{-}}{\mu^{+}}$ (with the
dimuon mass in the $\rho/\omega$ region) with a branching fraction $(4.17\pm
0.42)\times 10^{-6}$ [383] and the singly Cabibbo-suppressed decays
${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{\mu^{+}}{\mu^{-}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{\mu^{+}}{\mu^{-}}$ with branching
fractions of $(9.64\pm 1.20)\times 10^{-7}$ and $(1.54\pm 0.33)\times
10^{-7}$, respectively [384]. For the latter, also the differential branching
fraction as a function of the dimuon mass squared, $q^{2}$, was measured.
Furthermore, LHCb has performed a first measurement of $C\\!P$\- and angular
asymmetries in ${{D}^{0}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{\mu^{+}\mu^{-}}$
and ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{\mu^{+}\mu^{-}}$ decays using
Run 2 data. This resulted in the first determination of the forward-backward
asymmetry $A_{\rm FB}$, the triple-product asymmetry $A_{2\phi}$ and the
$C\\!P$-asymmetry $A_{\rm CP}$ with uncertainties at the percent-level [385].
Moreover, the implementation of triggers for dielectron modes opens the
possibility of measuring branching-fraction ratios between dimuon and
dielectron modes. Since the main limit for these studies comes from the
available statistics, excellent prospects are foreseen for the LHCb Upgrade I.
Projected signal yields for the muonic modes of $\mathcal{(}10^{4})$ will
allow more sensitive studies of angular asymmetry and first amplitude analyses
to attempt to disentangle SD and LD components. However, it is with the
$300$$\mbox{\,fb}^{-1}$ upgrade that the full potential for these decays will
be exploited. The enormous event yields and improved calorimetry of Upgrade II
will allow to perform studies of both the dimuon and dielectron modes with
high precision.

### 7.6 Rare strange decays

Rare strange decays can probe BSM physics that could escape all other
experimental tests. Indeed, if BSM physics is beyond the LHC direct searches
reach and thus only accessible through indirect measurements, searches for
non-Minimal Flavour Violation BSM physics become more relevant. In this
context, $s\rightarrow d$ transitions are of paramount importance, since they
have the strongest CKM suppression. Examples of these transitions are
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\mu^{+}\mu^{-}$ and
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\pi^{0}\mu^{+}\mu^{-}$.

#### 7.6.1 Rare kaon decays

In the SM, the ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\rightarrow\mu^{+}\mu^{-}$ decay is long-distance dominated, with
subdominant short-distance contributions. However, the long-distance
contribution is still very small in absolute terms, and the decay rate is very
suppressed. For example, the SM prediction [386, 387, 388] ${\cal
B}({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\mu^{+}\mu^{-})_{\rm
SM}=(5.18\pm 1.50_{\rm LD}\pm 0.02_{\rm SD})\times 10^{-12}$ can be compared
with the current experimental upper limit [389] ${\cal
B}({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\mu^{+}\mu^{-})_{\rm
Exp}<8\times 10^{-10}~{}{\rm{at}}~{}90\%~{}{\rm{CL}}$. Therefore, even small
BSM contributions and BSM–SM interferences can compete with the SM rate. This
has been proven to be the case in leptoquark models [390, 391] as well as
supersymmetric models [392]. In the latter, ${\cal
B}({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\mu^{+}\mu^{-})$ can
have values anywhere in the range $[0.78-35]\times 10^{-12}$ (see Fig. 7.9,
left) or even saturate the current experimental bound in certain narrow
regions of the parameter space [392]. The $C\\!P$ asymmetry of the
$K^{0}\rightarrow\mu^{+}\mu^{-}$ decay is also sensitive to BSM contributions
and experimentally accessible by means of a tagged analysis.

Figure 7.9:  (Left) $\mathcal{B}(K_{S}^{0}\rightarrow\mu^{+}\mu^{-})$ vs
$\mathcal{B}(K_{L}^{0}\rightarrow\mu^{+}\mu^{-})$ in an MSSM scenario, from
Fig. 2 of Ref. [392]. The cyan dots correspond to predictions using positive
sign for the long distance contribution to
$\mathcal{B}(K_{L}^{0}\rightarrow\mu^{+}\mu^{-})$ (i.e.
$A^{\mu}_{L\gamma\gamma}>0$) and the orange crosses to predictions using
$A^{\mu}_{L\gamma\gamma}<0$. The vertically hatched area corresponds to the SM
prediction for $A^{\mu}_{L\gamma\gamma}>0$ and the inclined hatched area
corresponds to the SM prediction for $A^{\mu}_{L\gamma\gamma}<0$. (Right)
Expected limit in ${\cal B}({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\rightarrow\mu^{+}\mu^{-})$ from LHCb and upgrades, as a function of
integrated luminosity times trigger efficiency. LHCb Upgrade II will collect
at least $300\mbox{\,fb}^{-1}$. Thus, it can be seen that if the trigger
efficiency is high, as expected from a full software trigger, LHCb can exclude
branching fractions down to near the SM prediction

The LHCb prospects for the search for ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\rightarrow\mu^{+}\mu^{-}$ decays are excellent. With 2011 data the
experiment overtook the previous world best upper limit by a factor of thirty
[393], and has recently gained another order of magnitude [389]. The right
hand side of Fig. 7.9 shows the expected upper limit for ${\cal
B}({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\mu^{+}\mu^{-})$ as a
function of the integrated luminosity multiplied by the trigger efficiency. It
can be seen that if the trigger efficiency is high, as expected from a full
software trigger, LHCb can explore branching fractions down to near the SM
prediction.

Currently, the only existing measurement of ${\cal
B}({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow\pi^{0}\mu^{+}\mu^{-})$
comes from the NA48 experiment [394]

${\cal B}({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\rightarrow\pi^{0}\mu^{+}\mu^{-})=(2.9_{-1.2}^{+1.5}\pm 0.2)\times
10^{-9}.$

In the Upgrade II LHCb can achieve a statistical precision of $0.11\times
10^{-9}$ with $300\mbox{\,fb}^{-1}$ of integrated luminosity, assuming a
trigger efficiency of 100% [395]. Assuming a trigger efficiency of $50\%$ LHCb
will still be able to significantly improve on the NA48 measurement. Apart
from the branching fraction, the differential decay rate in the dimuon mass
contains interesting information about the form factor parameters $a_{S}$ and
$b_{S}$ [396]. The LHCb Upgrade II can reach a $10\%$ statistical precision on
the form factor term $|a_{S}|$ with free $b_{S}$ [397].

Other kaon decays that can be studied at LHCb include
$K^{+}\rightarrow\pi\mu\mu$ (both with opposite-sign and same-sign muon
pairs), ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\rightarrow 4\mu$, or decays
involving electrons in order to test Lepton Universality [397]. The LHCb
acceptance is such that it favours the sensitivity to
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ modes over
${K}^{0}_{\mathrm{\scriptscriptstyle L}}$ modes by about a factor 1000 due to
the longer ${K}^{0}_{\mathrm{\scriptscriptstyle L}}$ lifetime.333However, it
must be noted that the ${K}^{0}_{\mathrm{\scriptscriptstyle
S}}$–${K}^{0}_{\mathrm{\scriptscriptstyle L}}$ interference component has an
effective lifetime of $\sim 2\tau_{{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}}$, with higher chances of decay inside LHCb tracking system. The
interference component can be accessed in principle by means of a tagged
analysis.

#### 7.6.2 Rare hyperon decays

The large production of strange hadrons at LHCb, with at least one strange
meson or baryon per minimum bias event, gives rise to unprecedented datasets
for the study of rare hyperon decays. The full spectrum of hyperons will be
available [397]. Studies of rare hyperon decays in the Upgrade II would
significantly profit from low momentum proton identification provided by the
TORCH detector.

##### 7.6.2.1 Rare decays of ${\mathchar 28934\relax}$ hyperons

LHCb has recently published a search for the ${\mathchar
28934\relax}^{+}\rightarrow p\mu^{+}\mu^{-}$ decay [398], with a strong
evidence for this decay with a significance of $4.0\,\sigma$; a measurement of
the branching fraction was reported and a dimuon invariant mass distribution
found to be consistent with SM predictions, challenging the HyperCP anomaly
[399].

This measurement will be improved in Run 2 with dedicated triggers [400, 401],
leading most probably to a first observation of this channel and detailed
differential branching fraction measurement. With the additional dataset of
Upgrade II a full angular analysis will be possible, probing complementary
couplings to those probed by $K^{+}\rightarrow\pi^{+}\mu^{-}\mu^{+}$ decays.

Assuming a similar sensitivity, a search for the lepton and baryon number
violating decay ${\mathchar 28934\relax}^{+}\rightarrow\bar{p}\mu^{+}\mu^{+}$
could be performed and it would be expected to reach a sensitivity of order
$10^{-10}$ ($10^{-9}$) on the upper limit of the correspondent branching
fraction with 300 (50) $\mbox{\,fb}^{-1}$ of integrated luminosity. Similarly,
a search for ${\mathchar 28934\relax}^{+}\rightarrow pe^{+}e^{-}$ decays will
be possible with dedicated triggers. The latter mode is similar to the
${\mathchar 28934\relax}^{+}\rightarrow p\mu^{+}\mu^{-}$ case, but owing to
the lower mass of the electrons receives a larger contribution from long
distance photon contributions, for a predicted branching fraction [402] of
$\mathcal{B}({\mathchar 28934\relax}^{+}\rightarrow
pe^{+}e^{-})\in[9.1,10.1]\times 10^{-6}$. Experimentally only an upper limit
of $7\times 10^{-6}$ at 90% CL is reported [403]. It is expected to reach the
SM level already with the Upgrade I, but only with the statistics provided by
the Upgrade II it would be possible to perform a detailed study of the
dielectron invariant mass distribution. Analogously, the LFV decays
${\mathchar 28934\relax}^{+}\rightarrow pe^{\pm}\mu^{\mp}$ could be searched
for with similar sensitivity, where the SM rate is negligible.

##### 7.6.2.2 Rare decays of ${\mathchar 28931\relax}$ hyperons

An interesting study in the domain of ${\mathchar 28931\relax}$ hyperons would
be to measure the radiative ${\mathchar 28931\relax}\rightarrow
p\pi^{-}\gamma$ decay. The branching fraction of this decay is known only with
an uncertainty of 16% ($\mathcal{B}=(8.4\pm 1.4)\times 10^{-4}$ [404]) and
only for pion centre-of-mass momenta less than
$95{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. LHCb can improve this measurement
significantly. In addition a first measurement of the FCNC decay ${\mathchar
28931\relax}\rightarrow p\pi e^{+}e^{-}$ will be possible.

Several baryon-number-violating ${\mathchar 28931\relax}$ decays have been
recently searched for by the CLAS collaboration, as reported in Ref. [405].
Most of these are in the form ${\mathchar 28931\relax}\rightarrow m\ell$ where
$m$ is a $K^{+}$ or $\pi^{+}$ meson, and $\ell=e,\mu$ leptons. Typically,
these decays have strong indirect constraints from the limits on nucleon
decays, but direct experimental limits in an interesting range of sensitivity
were absent so far. The reported limits on the respective branching fractions
are in the range $[10^{-7},10^{-6}]$. The LHCb experiment can certainly
improve most of these limits, reaching a single event sensitivities of ${\cal
O}(10^{-11})$ for an integrated luminosity of $300\mbox{\,fb}^{-1}$.

## 8 Forward and high-$p_{\mathrm{T}}$ physics

The unique forward spectrometer configuration of LHCb permits a rich programme
of particle production studies at high momentum, allowing precision
measurements of Standard Model physics, and direct and indirect searches for
new phenomena. LHCb has excellent charged particle reconstruction and muon
identification, which can be exploited in muonic $W$ and $Z/\gamma^{\ast}$
(denoted $Z$ for brevity) decays. However, the current ability of LHCb to
reconstruct electrons with large transverse momenta is restricted, owing to
the saturation of cells in the ECAL when the transverse energy deposited is
above $\sim 10\mathrm{\,Ge\kern-1.00006ptV}$. The yields for
high-$p_{\mathrm{T}}$ electron signatures, for example $Z\rightarrow
e^{+}e^{-}$ decays, are up to a factor of 3 smaller than can be achieved for
the equivalent muon channels. With an upgraded ECAL, we expect similar
precision for muon and electron reconstruction at high transverse momentum. In
addition, the integrated luminosity offered by Upgrade II also offers
important new opportunities. A selection of the measurements made possible
with Upgrade II are discussed in this chapter, with the focus placed on the
different distinct areas within the forward high-$p_{\mathrm{T}}$ programme of
research at LHCb in turn.

Top quark production in the LHCb acceptance accesses larger Bjorken-$x$
values, and enhanced contributions from quark-initiated subprocesses, than at
the central rapidities covered by ATLAS and CMS. This results in a greater
sensitivity to observables depending on the presence of sizeable $q\bar{q}$
amplitudes [406]. One such observable is the $t\bar{t}$ charge asymmetry
which, in the Standard Model, rises from approximately 1% in the central
region to as high as 8% within the LHCb acceptance [407]. A number of new
physics models [408] predicting extra contributions to quark-initiated top
production result in enhancements in both the $t\bar{t}$ production cross-
section and charge asymmetry in the forward region. While measurements of the
asymmetry at ATLAS and CMS already provide limits on these models [409], high
yield measurements in the forward region have the potential to provide even
stronger constraints. In addition, as the dominant contribution to top pair
production in the forward region still arises from gluon-gluon fusion, precise
measurements of the $t\bar{t}$ cross-section at LHCb will place important
constraints on the poorly known gluon parton distribution function (PDF) at
high-$x$ values.

Measurements of gauge boson production at LHCb uniquely probe the PDFs at
smaller and larger values of Bjorken-$x$ than are accessible by ATLAS and CMS
[410]. The LHCb vector-boson production measurements have been included within
fits to data to determine PDFs [411, 412, 413, 414]. Detailed studies of the
LHCb data have shown that the measurements of gauge-boson production have
placed stringent constraints on the nonperturbative charm content of the
proton and have halved the uncertainties associated with the distribution of
valence quarks at large values of Bjorken-$x$ (above 0.1) [415]. The physics
of PDFs at large $x$ is an important study within QCD, but is also crucial to
understand the production of any potential new high-mass states detected at
ATLAS and CMS.

Precision electroweak tests provide a powerful probe of new physics. Two
parameters stand out as demanding higher precision direct measurements, to
better exploit the high precision of their corresponding indirect
determinations through global fits. Those are the effective leptonic weak
mixing angle, ${\sin}^{2}\theta_{\mathrm{W}}^{\mathrm{eff}}$, and the $W$
boson mass, $m_{W}$. The forward rapidity design of LHCb presents several
advantages with respect to the central detector experiments in measurements of
these fundamental parameters of the SM.

Studies performed thus far on the Higgs boson are consistent with the particle
predicted by the Standard Model. The vector couplings of the Higgs are already
reasonably well measured. However, the fermion couplings are more challenging
to access. New physics could easily be hiding in the, currently unmeasurable,
second-generation couplings. While no currently approved experiment is capable
of reaching sensitivity to the Higgs-charm coupling at the Standard Model
level, the unique charm-tagging capabilities of the LHCb detector mean that
LHCb Upgrade II has the potential to make the most stringent constraint at the
HL-LHC.

LHCb’s excellent lepton and jet reconstruction, as well as the flexible
trigger scheme, allow direct searches for new physics phenomena associated
with long lived particles, considering, for example, dark photons or particles
in hidden valley models. Existing searches at LHCb provide unique sensitivity
to many exotic signatures, with LHCb having published some of the most
sensitive searches in such models for new particles with masses below the
electroweak scale. Sensitivities in such searches following Upgrade II are
therefore also explored in this chapter.

### 8.1 Top physics in the forward region

Three measurements of top production have been performed by LHCb during Runs 1
and 2 of the LHC with a precision of (20–40)%, limited by the available data
samples. The first observation in the forward region was made in the $\mu b$
final state, where the top quark is identified by the presence of a muon and a
$b$-jet, using 3$\mbox{\,fb}^{-1}$ of data collected in Run 1 [416]. This
final state has the highest signal yield, but suffers from the largest
backgrounds, in particular from $W$ boson production in association with a
$b$-jet. It is also impossible to separate single top and top pair production,
which both contribute to the final state. A measurement was also made in the
$\ell bb$ final state[417] using 2$\mbox{\,fb}^{-1}$ of data, where a muon or
electron (indicated by $\ell$), in addition to two $b$-jets, is used to
identify $t\bar{t}$ events. The first measurement in Run 2 was made in the
$\mu eb$ final state, where a muon and electron, together with a $b$-jet, were
reconstructed [418]. This final state has the highest purity, but the lowest
yield and was only made possible in Run 2 due to the increase of approximately
a factor of ten, between $\sqrt{s}=13$ $\mathrm{\,Te\kern-1.00006ptV}$ and
$\sqrt{s}=8$ $\mathrm{\,Te\kern-1.00006ptV}$, in the top pair cross-section
within the LHCb acceptance.

Figure 8.1:  The predicted SM asymmetry at LHCb as a function of lepton pseudorapidity in the $\ell b$ final state at 14 TeV. The bands show the uncertainty on the theoretical predictions due to scale variations (green) and due to combined scale, PDF and $\alpha_{s}$ variations (yellow). The expected statistical precision on measurements performed by LHCb using 23 and 300$\mbox{\,fb}^{-1}$ of data is indicated by the uncertainty bars on the points. Table 8.1:  The number of $t\bar{t}$ events reconstructed at LHCb per final state in current measurements and obtained using extrapolations to datasets corresponding to integrated luminosities of 23, 50 and 300$\mbox{\,fb}^{-1}$. The mean value of Bjorken-$x$ of the most energetic initiating parton is also shown for each final state.  final state | current | 23$\mbox{\,fb}^{-1}$ | 50$\mbox{\,fb}^{-1}$ | 300$\mbox{\,fb}^{-1}$ | $<x>$
---|---|---|---|---|---
$\ell b$ | 220 [416] | 54k | 117k | 830k | 0.295
$\ell b\bar{b}$ |  024 [417] | 8k | 17k | 130k | 0.368
$\mu eb$ |  038 [418] | 1k | 2k | 12k | 0.348
$\mu eb\bar{b}$ | - | 120 | 260 | 1.5k | 0.415

While uncertainties on current measurements in the top sector at LHCb are
dominated by statistically uncertainties, the Upgrade II data will permit
precision measurements of both the top quark production cross-section and
charge asymmetry. The expected number of top-pair events reconstructed at LHCb
are given in Table 8.1, where the yields are obtained by extrapolating from
current measurements, with increases of between 10 and 50% in total efficiency
assumed due to improvements in the $b$-tagging algorithm and selection
criteria. In addition, both muons and electrons are assumed to be employed for
Upgrade II due to anticipated improvements in electron performance. The cross-
sections are calculated at next-to-leading order in perturbative QCD using the
aMC@NLO generator with the parton shower provided by Pythia 8, and electroweak
corrections are approximated as described in Ref. [407]. They indicate that
fiducial cross-section and asymmetry measurements will be made at subpercent
statistical precision in the $\ell b$ final state, and at the percent level in
the $\mu eb$ final state. The large yields will also allow differential
measurements of the asymmetry to be made as a function of lepton
pseudorapidity in the $\ell b$ and $\ell b\bar{b}$ final states to achieve the
greatest sensitivity, with the former shown in Fig. 8.1. A good knowledge of
the background contributions and their asymmetry will also be required to
reach the ultimate precision.

While the statistical precision of the $\mu eb$ final state will be lower, the
higher purity of the sample, and the unambiguous identification of top pair
events, presents the opportunity to make the most precise cross-section
measurement at LHCb with an overall precision at the level of a few percent.
An inclusive cross-section measurement at a precision of 4% will provide
reductions of over 20% on the gluon PDF at large-$x$ [419], and more stringent
constraints can be obtained by performing total and normalised differential
$t\bar{t}$ production cross-section measurements, which will be possible with
the Upgrade II data.

### 8.2 Gauge-boson production and implications for PDFs

LHCb has made an important series of measurements of $W$ and $Z$ boson
production in proton-proton collisions at a range of energies [420, 421, 422,
423, 424, 425, 426, 427, 428, 429, 430, 431, 432]. These include measurements
using leptonic gauge boson decays (to final states including electrons, muons,
or taus), and hadronic decays of the $Z$ boson. Both inclusive and associated
production measurements have been performed. These measurements rely on
excellent luminosity determination (with the precision achieved below 1.5%
[433]), powerful particle identification and tracking, jet reconstruction, and
efficient jet-flavour identification [434]. In the most precise LHCb cross-
section measurements to date the knowledge of the luminosity determination is
the dominant uncertainty, though this and other uncertainties cancel in
measurements of cross-section ratios. In the following we explore the
impressive reach of Upgrade II in this important area of physics.

Figure 8.2: The regions probed in the $x-Q^{2}$ plane with $Z$ boson
measurements at LHCb in 14 TeV proton-proton collisions. Integrated
luminosities considered are (left) a dataset of $23\mbox{\,fb}^{-1}$,
considering only dimuon final states and (right) $300\mbox{\,fb}^{-1}$,
considering both dimuon and dielectron final states. Bins are only shown if
the expected yield, determined using Pythia 8 [435], is greater than 100
events. The solid red line shows the kinematic limit set by the collision
energy; the dashed lines show the values of $x$ and $Q^{2}$ associated with
$Z$ bosons with zero transverse momentum, and rapidities of 0, 2, and 4.5.

The expected event yields and related information, are set out in Tables 8.2
and 8.3 for $W$ and $Z$ boson production, including high mass Drell-Yan, where
one region ($m(\ell\ell)>200{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) is used
to highlight typical quantities. Figure 8.2 also shows the regions probed in
the $x-Q^{2}$ plane for different integrated luminosities by measurements of
(neutral) Drell-Yan production. The Upgrade II dataset is needed in order to
make double-differential measurements as a function of the dilepton invariant
mass and rapidity at high statistical precision if one wishes to maximise
sensitivity in the regions where the PDFs have Bjorken-$x$ values close to 1,
and to reach higher values of $Q^{2}$ than are currently accessible at high
and low Bjorken-$x$. The same is also true for triple-differential
measurements that probe the boson transverse momentum or measurements that
probe high momentum jets. Differential measurements will also be made
involving both $W^{+}$ and $W^{-}$ boson production and the ratio of their
cross-sections, providing further constraints on PDFs. While such measurements
of $W$ boson production are already systematically limited, the new large
dataset will allow the study and reduction of these uncertainties. In
addition, following Upgrade II a significantly improved ECAL (without low
energy saturation) will allow better understanding of sample compositions in
electron channels, further reducing systematic uncertainties on the ratio of
the $W$ boson cross-section in electron and muon channels, allowing a more
precise test of lepton universality in $W$ boson decays.

Measurements of gauge boson production in association with high momentum jets
can also be used to place stringent constraints on the down-quark PDF at
large-$x$, with a measurement at the 1% level allowing a factor 3 reduction on
the uncertainty [436]. Differential measurements of such processes will give
yet further sensitivity to higher values of $x$, possible with the high yields
expected following Upgrade Ib and Upgrade II.

Table 8.2: Expected yields in measurements at LHCb, and the higher of the two Bjorken-$x$ values probed. The yields for single boson processes are from NNLO calculations in pQCD, calculated for Ref. [429]. The values of Bjorken-$x$ probed are determined using Pythia8. Ref. [430] and Pythia8 are also used to extrapolate these cross-sections to those in the jet final states. The yields from electron and muon final states are summed for the 300$\mbox{\,fb}^{-1}$ yields (which assume the data collected is at 14 TeV). For the yields up to 50$\mbox{\,fb}^{-1}$ only muons are considered, as the current LHCb performance at high momentum is far superior for muons than it is for electrons. Other reconstruction effects are neglected, since the efficiencies are $\mathcal{O}(1)$. Process |  | Additional | Signal yields in | $\langle x\rangle$
---|---|---|---|---
|  | Criteria | LHCb with 300$\mbox{\,fb}^{-1}$ | for $\sqrt{s}=14\mathrm{\,Te\kern-1.00006ptV}$
|  |  | (50$\mbox{\,fb}^{-1}$,23$\mbox{\,fb}^{-1}$) |
$Z/\gamma^{*}$ |  | $60<m(\ell\ell)<120\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 130\,(9,4)\times 10^{6}$ | 0.17
+jet |  | $p_{\text{T}}(\text{jet})>20\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 16\,(1,0.5)\times 10^{6}$ | 0.24
|  | $p_{\text{T}}(\text{jet})>100\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 500\,(30,14)\times 10^{3}$ | 0.38
High-mass DY |  | $m(\ell\ell)>200\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 120\,(9,4)\times 10^{3}$ | 0.32
$W^{+}$ |  |  | $\sim 1400\,(109,50)\times 10^{6}$ | 0.16
+jet |  | $p_{\text{T}}(\text{jet})>20\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 170\,(13,6)\times 10^{6}$ | 0.24
|  | $p_{\text{T}}(\text{jet})>100\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 4\,(0.4,0.2)\times 10^{6}$ | 0.35
$W^{-}$ |  |  | $\sim 1000\,(76,35)\times 10^{6}$ | 0.11
+jet |  | $p_{\text{T}}(\text{jet})>20\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 100\,(6.5,3)\times 10^{6}$ | 0.18
|  | $p_{\text{T}}(\text{jet})>100\mathrm{\,Ge\kern-1.00006ptV}$ | $\sim 2\,(0.13,0.06)\times 10^{6}$ | 0.31
Table 8.3: This table contains similar information to Table 8.2, with the same dilepton invariant mass cuts, but considers inclusive measurements made differentially as a function of the $Z/\gamma^{*}$ boson rapidity. Bjorken-$x$ values probed in $W$ boson events are similar to those shown here for Drell-Yan events at the Z peak. Process | Rapidity Bin | Signal yields in | $\langle x\rangle$
---|---|---|---
|  | LHCb with 300$\mbox{\,fb}^{-1}$ |
$Z/\gamma^{*}$ | [2.0,2.5] | $\sim 20\times 10^{6}$ | 0.08
| [2.5,3.0] | $\sim 40\times 10^{6}$ | 0.12
| [3.0,3.5] | $\sim 50\times 10^{6}$ | 0.18
| [3.5,4.0] | $\sim 20\times 10^{6}$ | 0.28
| [4.0,4.5] | $\sim 4\times 10^{6}$ | 0.42
High-mass DY | [2.0,2.5] | $\sim 30\times 10^{3}$ | 0.21
| [2.5,3.0] | $\sim 60\times 10^{3}$ | 0.30
| [3.0,3.5] | $\sim 30\times 10^{3}$ | 0.43
| [3.5,4.0] | $\sim 3\times 10^{3}$ | 0.61
| [4.0,4.5] | $10^{1}$ | 0.84

Measurements of electroweak boson production in association with heavy-flavour
jets (and mesons) can similarly be made. Existing LHCb measurements of these
cross-sections have been performed, integrated over the full LHCb acceptance.
With the larger sample sizes expected following the LHCb upgrades,
differential measurements can be made in these final states that probe the
variation of strange PDFs with Bjorken-$x$. Measurements of $Z$ +charm
production at LHCb will also be made differentially. The ratio of the cross-
sections of $Z$ +charm to $Z$ +jet events at LHCb varies between about 8% and
3% as a function of the $Z$ boson rapidity; measurements made using the Run 3
dataset will typically be sensitive to intrinsic charm with a mean value of
$x$ greater than 0.3% (sea-like intrinsic charm) and 1% (valence-like
intrinsic charm) [437]. Improvements in sensitivity are expected following
Upgrades Ib and II, since the larger sample yields will enable double-
differential measurements that probe higher $p_{\text{T}}$ jets (placing
constraints on higher $x$ values) while binning in the $Z$ boson rapidity.
Such measurements will allow the study of the $x$ and $Q^{2}$ dependence of