Prospects for measurements

with strange hadrons at LHCb

A. A. Alves Junior1, M. O. Bettler2, A. Brea Rodríguez1, A. Casais Vidal1, V.
Chobanova1, X. Cid Vidal1, A. Contu3, G. D’Ambrosio4, J. Dalseno1, F.
Dettori5, V.V. Gligorov6, G. Graziani7, D. Guadagnoli8, T. Kitahara9,10, C.
Lazzeroni11, M. Lucio Martínez1, M. Moulson12, C. Marín Benito13, J. Martín
Camalich14,15, D. Martínez Santos1, J. Prisciandaro 1, A. Puig Navarro16, M.
Ramos Pernas1, V. Renaudin13, A. Sergi11, K. A. Zarebski11

1Instituto Galego de Física de Altas Enerxías (IGFAE), Santiago de Compostela,
Spain
2Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
3INFN Sezione di Cagliari, Cagliari, Italy
4INFN Sezione di Napoli, Napoli, Italy
5Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom,
now at Università degli Studi di Cagliari, Cagliari, Italy
6LPNHE, Sorbonne Université, Université Paris Diderot, CNRS/IN2P3, Paris,
France
7INFN Sezione di Firenze, Firenze, Italy
8Laboratoire d’Annecy-le-Vieux de Physique Théorique , Annecy Cedex, France
9Institute for Theoretical Particle Physics (TTP), Karlsruhe Institute of
Technology, Kalsruhe, Germany
10Institute for Nuclear Physics (IKP), Karlsruhe Institute of Technology,
Kalsruhe, Germany
11School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
12INFN Laboratori Nazionali di Frascati, Frascati, Italy
13Laboratoire de l’Accelerateur Lineaire (LAL), Orsay, France
14Instituto de Astrofísica de Canarias and Universidad de La Laguna,
Departamento de Astrofísica, La Laguna, Tenerife, Spain
15CERN, CH-1211, Geneva 23, Switzerland
16Physik-Institut, Universität Zürich, Zürich, Switzerland

This report details the capabilities of LHCb and its upgrades towards the
study of kaons and hyperons. The analyses performed so far are reviewed,
elaborating on the prospects for some key decay channels, while proposing some
new measurements in LHCb to expand its strangeness research program.

##  1 Introduction

The study of strange-hadron decays has fuelled discoveries in particle physics
for the past seventy years. For instance, experimental anomalies in the
strange sector motivated the prediction of the charm quark via the Glashow-
Iliopoulos-Maiani (GIM) mechanism, while the discovery of $C\\!P$ violation
prompted the postulation of the beauty and top quarks within the Cabibbo-
Kobayashi-Maskawa (CKM) paradigm; all now key ingredients of the Standard
Model (SM). Presently, strangeness decays are valuable probes in the search
for dynamics Beyond the Standard Model (BSM), being particularly relevant in
searches for sources of quark flavour violation beyond the CKM matrix. Since
$s\to d$ transitions have the strongest suppression factor, they can typically
probe energy scales higher than those accessible in charm or beauty-hadron
decays for couplings of comparable size [1]. Nevertheless, flavour physics
experiments have greatly enhanced such knowledge from charm and beauty decays
in recent years, while few measurements of strange-hadron decays have been
updated or performed for the first time.

Several dedicated experiments exist for specific measurements, however few
experiments possess the potential to construct a comprehensive program on the
study of strange hadrons. In this work, it is argued that the LHCb experiment
has the capacity, both in terms of detector performance and statistics, to
produce leading measurements exploiting almost all strange-hadron species,
particularly in the search for their rare decays. An overview of the current
results and prospects of strangeness decays at LHCb is given, demonstrating
LHCb’s unique reach as a strangeness factory and motivating further research
in this area. In fact, the LHCb collaboration has already published the
world’s most precise measurements in $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\mu^{+}\mu^{-}$ [2, 3] and $\Sigma^{+}\to p\mu^{+}\mu^{-}$ [4], while
projecting world-leading results for $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-}$ [5] and $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ [6]. Experiments such as BESIII [7], NA62 [8,
9], KLOE2 [10], KOTO [11, 12] and CLAS [13, 14, 15] further enrich the field
with diverse and complementary research programs of their own.

This document is organised as follows: Section 2 is dedicated to the
discussion of the production of strange-hadron decays at LHC and its detection
in LHCb. Section 3 summarises the results and prospects of LHCb for several
rare decays of strange hadrons. The capabilities for the measurement of the
$K^{+}$ mass as well as for the study of semileptonic hyperon decays are
presented in section 4, while conclusions are drawn in section 6.

##  2 Production and detection of strange hadrons

The LHCb detector [16] is a single-arm forward spectrometer, covering the
pseudorapidity range $2<\eta<5$, collecting data in proton-proton collisions
at the Large Hadron Collider at CERN. It is composed of a silicon-strip vertex
detector surrounding the $pp$ interaction region (VELO), with a length of
about 1 metre from the interaction point, a large-area silicon-strip detector
(TT) located upstream of a dipole magnet and three tracking stations of
silicon-strip detectors and straw drift tubes placed downstream of the magnet.
Particle identification is provided by two ring-imaging Cherenkov detectors,
an electromagnetic and a hadronic calorimeter, and a muon system composed of
alternating layers of iron and multiwire proportional chambers. LHCb has
collected so far an integrated luminosity of about 8 $\text{fb}^{-1}$.

The LHCb detector will be upgraded for the next run of the LHC. This upgrade,
hereafter referred to as Phase-I, includes a completely new tracking system
with a pixel-based VELO [17], the Upstream Tracker (UT) replacing the TT and
scintillating fibre detectors acting as tracking stations [17]. The Phase-I
detector will collect on the order of $50$ $\text{fb}^{-1}$ of integrated
luminosity [18]. An Expression of Interest for a second upgrade, hereafter
denoted as Phase-II, can be found in ref. [19]. It is intended that on the
order of $300$ $\text{fb}^{-1}$ of integrated luminosity will be collected
with this upgrade.

The production of strange hadrons at LHC is exceedingly abundant. Physics
projections are derived from simulated events invoking the `Pythia` software
generator [20], where proton-proton collisions are configured with a centre-
of-mass energy $\sqrt{s}=13$ TeV and an average of one interaction per
collision. The conclusions of this study are unaffected for other anticipated
LHC collision energies of $14$ TeV even up to 28 TeV. Multiplicities of
various particles are estimated from these events in a broad LHCb geometric
acceptance of pseudorapidity $\eta\in[1,6]$, prior to any simulated detector
response. This multiplicity is shown for strange hadrons in figure 1 alongside
an assortment of well-known heavy flavoured hadrons for comparison. Multiple
kaons and about one hyperon per event are expected to be produced in these
interactions, which is roughly two and three orders of magnitude greater than
for charmed and beauty hadrons, respectively. Thus, the LHCb experiment will
have at its disposal the statistics necessary both for precision measurements
of strange-hadron decays and for searches for their rare decays.

The efficiency of detecting strange-hadron decays will, however, not be the
same as for heavy flavour for several reasons. The detector layout, which is
optimised for $b$ decays, implies a relatively lower acceptance for
$K^{0}_{\mathrm{\scriptscriptstyle S}}$, with
$K^{0}_{\mathrm{\scriptscriptstyle L}}$ and $K^{+}$ efficiencies diminished
even further. This is due to the differing flight lengths of the different
mesons. The typical decay length of a $B$ meson is $\sim 1$ cm,
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ can fly a distance of nearly one
metre, while $K^{\pm}$ and $K^{0}_{\mathrm{\scriptscriptstyle L}}$ traverse
distances longer than the full LHCb detector length on average. Flight
distance distributions achieved by various strange hadrons before decaying are
also obtained from Pythia simulations, which are displayed within the context
of the LHCb detector in figure 2.

Figure 1: Multiplicity of particles produced in a single $pp$ interaction at
$\sqrt{s}=13$ TeV within LHCb acceptance.

Depending on the decay position of a given particle, its charged decay
products can be reconstructed in LHCb exploiting the relevant tracking sub-
detectors. The different track categories are defined in ref. [17] as:

  * •

long tracks: when all possible tracking information from the VELO to the T
stations is available, implying that the mother particle decayed within about
1 metre of the $pp$ interaction point;

  * •

downstream tracks: where only the TT and T stations register tracks, allowing
strange hadrons to be reconstructed with decay lengths up to about 2 metres
from the interaction point.

In order to provide an estimate of the reconstruction efficiencies for long
tracks, the published $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$
analysis from LHCb is taken as a benchmark [3]. Events with a decay time $t$
in the range of $t/\tau_{S}\in[0.10,1.45]$ were used, where $\tau_{S}$ is the
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ lifetime. From these numbers, one
could simply obtain

$\frac{\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle
L}}}}{\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle S}}}}\approx 3.5\times
10^{-3}\quad,$

for the ratio of $K^{0}_{\mathrm{\scriptscriptstyle L}}$ to
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ efficiencies,
${\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle L}}}}$ and
${\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle S}}}}$, respectively.

However, as the acceptance inside the VELO is not uniform, larger lifetimes
result in lower reconstruction efficiencies, further reducing
${\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle L}}}}$ next to
${\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle S}}}}$. This can be approximated
by an exponential acceptance or so-called ‘beta factor’ $\epsilon(t)\sim
e^{-\beta t}$ [21], with $\beta\sim 86\,$ns-1 in the case of
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ decays [22]. In this
case, the reduction factor becomes

$\frac{\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle
L}}}}{\epsilon_{K^{0}_{\mathrm{\scriptscriptstyle
S}}}}=\frac{\Gamma_{L}\bigint_{0.1\tau_{S}}^{1.45\tau_{S}}e^{-t(\Gamma_{S}+\beta)}dt}{\Gamma_{S}\bigint_{0.1\tau_{S}}^{1.45\tau_{S}}e^{-t(\Gamma_{L}+\beta)}dt}\approx
2.2\times 10^{-3},$ (1)

where $\Gamma_{S}$ and $\Gamma_{L}$ are the $K^{0}_{\mathrm{\scriptscriptstyle
S}}$ and $K^{0}_{\mathrm{\scriptscriptstyle L}}$ decay widths. Assuming that
the same acceptance parametrisation used in eq. (1) holds also for $K^{\pm}$,
the relative efficiency of $K^{\pm}$ decays with respect to
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays is then at the level of 1%. On
the other hand, the use of downstream tracks can allow for an increased
lifetime acceptance. The transverse momenta of the products of strangeness
decays, significantly softer than for $b$-hadron decays, are also detrimental
to their detection at LHCb. While $b$-hadron decay products generally have a
transverse momenta of around 1-2 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, for
$s$-hadron decays the range is more in the region of 100-200
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.

Figure 2: A side view of the LHCb detector layout [16] compared with the decay
length of strange hadrons in $pp$ collisions at $\sqrt{s}=13$ TeV. The blue
(1m) and red (2m) dashed lines indicate the approximate distance from the
interaction point at which daughter tracks can be reconstructed as long and
downstream tracks, respectively.

The acceptances for several benchmark channels, as well as invariant-mass
resolutions, are estimated in the following applying a simplified simulation
of the LHCb upgrade tracking, based on the detector descriptions found in
refs. [17, 23, 24]. The following selection criteria are applied to all decay
channels: the daughter particles are required to have a track impact parameter
to primary vertex of greater than 400 microns, a momentum greater than 3000
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ with transverse momentum greater than
80 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, while the reconstructed hadron is
required to have a decay time greater than 8.9 ps and a flight distance in the
plane transverse to the beam greater than 3 mm. These requirements are based
on the Run 2 trigger for detached soft dimuons [25] and on the lower decay
time requirement from ref. [2]. These requirements are expected to be
realistic also for future data-acquisition periods in LHCb. Acceptances are
then normalised to that of fully reconstructed
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$, which is found to be
around $1\%$. The results of this simplified simulation are given in table 1,
where the efficiency is shown using long tracks only ($\epsilon_{L}$) and
using downstream tracks only ($\epsilon_{D}$), along with the invariant-mass
resolution for each reconstruction method. The efficiency scale factors for
charged hadrons with at least 300 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
electrons with over 200 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ transverse
momenta are also normalised to fully reconstructed
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ and indicated in
parentheses. As neutral particles like the photon, neutrino and $\pi^{0}$ are
not reconstructed in this study, the invariant mass of particular strange
hadrons is calculated with additional kinematic constraints.

Absolute efficiencies depend significantly on the fidelity of the momentum
spectra provided by Pythia, hence it is preferred to quote only relative
acceptances here. As bremsstrahlung corrections are important in electron
reconstruction, such modes are shown separately in table 2, in which the
reference channel for efficiency normalisation is
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$. The
reconstruction and selection efficiency for $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ has been estimated with full LHCb simulation
to be $\sim 1\times 10^{-4}$ in ref. [6]. Lepton Flavour Violating (LFV) modes
are listed in table 3, normalised to $K^{0}_{S}\to\mu^{+}e^{-}$.

Table 1: Acceptance scale factors $\epsilon$, and mass resolutions $\sigma$, for only long ($L$) and only downstream ($D$) tracks obtained from our simplified description of the LHCb Upgrade tracking system geometry. The production ratio of the strange hadron with respect to $K_{S}^{0}$ is shown as $\mathcal{R}$. All efficiencies are normalised to that of fully reconstructed $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ and averaged over particles and anti-particles. Channels containing a photon, neutrino and $\pi^{0}$ are partially reconstructed. Channel | $\mathcal{R}$ | $\epsilon_{L}$ | $\epsilon_{D}$ |  $\sigma_{L}$(${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$) |  $\sigma_{D}$(${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)
---|---|---|---|---|---
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ | 1 | 1.0 (1.0) | 1.8 (1.8) | $\sim 3.0$ | $\sim 8.0$
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}$ | 1 | 1.1 (0.30) | 1.9 (0.91) | $\sim 2.5$ | $\sim 7.0$
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$ | 1 | 0.93 (0.93) | 1.5 (1.5) | $\sim 35$ | $\sim 45$
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\gamma\mu^{+}\mu^{-}$ | 1 | 0.85 (0.85) | 1.4 (1.4) | $\sim 60$ | $\sim 60$
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ | 1 | 0.37 (0.37) | 1.1 (1.1) | $\sim 1.0$ | $\sim 6.0$
$K^{0}_{\mathrm{\scriptscriptstyle L}}\to\mu^{+}\mu^{-}$ | $\sim 1$ |  2.7 (2.7) $\times 10^{-3}$ | 0.014 (0.014) | $\sim 3.0$ | $\sim 7.0$
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ | $\sim 2$ |  9.0 (0.75) $\times 10^{-3}$ |  41 (8.6) $\times 10^{-3}$ | $\sim 1.0$ | $\sim 4.0$
$K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ | $\sim 2$ |  6.3 (2.3) $\times 10^{-3}$ | 0.030 (0.014) | $\sim 1.5$ | $\sim 4.5$
$\Sigma^{+}\to p\mu^{+}\mu^{-}$ | $\sim 0.13$ | 0.28 (0.28) | 0.64 (0.64) | $\sim 1.0$ | $\sim 3.0$
$\Lambda\to p\pi^{-}$ | $\sim 0.45$ | 0.41 (0.075) | 1.3 (0.39) | $\sim 1.5$ | $\sim 5.0$
$\Lambda\to p\mu^{-}\bar{\nu_{\mu}}$ | $\sim 0.45$ | 0.32 (0.31) | 0.88 (0.86) | $-$ | $-$
$\Xi^{-}\to\Lambda\mu^{-}\bar{\nu_{\mu}}$ | $\sim 0.04$ |  39 (5.7) $\times 10^{-3}$ | 0.27 (0.09) | $-$ | $-$
$\Xi^{-}\to\Sigma^{0}\mu^{-}\bar{\nu_{\mu}}$ | $\sim 0.03$ |  24 (4.9) $\times 10^{-3}$ | 0.21 (0.068) | $-$ | $-$
$\Xi^{-}\to p\pi^{-}\pi^{-}$ | $\sim 0.03$ | 0.41(0.05) | 0.94 (0.20) | $\sim 3.0$ | $\sim 9.0$
$\Xi^{0}\to p\pi^{-}$ | $\sim 0.03$ | 1.0 (0.48) | 2.0 (1.3) | $\sim 5.0$ | $\sim 10$
$\Omega^{-}\to\mathchar 28931\relax\pi^{-}$ | $\sim 0.001$ |  95 (6.7) $\times 10^{-3}$ | 0.32 (0.10) | $\sim 7.0$ | $\sim 20$
Table 2: Acceptance scale factors $\epsilon$, and mass resolutions $\sigma$, for only long ($L$) and only downstream ($D$) tracks obtained from our simplified description of the LHCb Upgrade tracking system geometry. All efficiencies are normalised to that of fully reconstructed $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ and are averaged between particles and anti-particles. The invariant-mass resolutions shown in the table correspond to the ideal case of perfect bremsstrahlung recovery. Channel | $\mathcal{R}$ | $\epsilon_{L}$ | $\epsilon_{D}$ |  $\sigma_{L}$(${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$) |  $\sigma_{D}$(${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)
---|---|---|---|---|---
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ | 1 | 1.0 (0.18) | 2.83 (1.1) | $\sim 2.0$ | $\sim 10$
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}e^{+}e^{-}$ | 1 | 1.18 (0.48) | 2.93 (1.4) | $\sim 2.0$ | $\sim 11$
$K^{+}\to\pi^{+}e^{+}e^{-}$ | $\sim 2$ | 0.04 (0.01) | 0.17 (0.06) | $\sim 3.0$ | $\sim 13$
$\Sigma^{+}\to pe^{+}e^{-}$ | $\sim 0.13$ | 1.76 (0.56) | 3.2 (1.3) | $\sim 3.5$ | $\sim 11$
$\Lambda\to p\pi^{-}e^{+}e^{-}$ | $\sim 0.45$ | $<2.2\times 10^{-4}$ |  $\sim 17$ $(<2.2)$ $\times 10^{-4}$ | $-$ | $-$
Table 3: Acceptance scale factors $\epsilon$, and mass resolutions $\sigma$, for only long ($L$) and only downstream ($D$) tracks obtained from our simplified description of the LHCb Upgrade tracking system geometry. All efficiencies are normalised to that of fully reconstructed $K^{0}_{S}\to\mu^{+}e^{-}$ and averaged between particles and anti-particles. The invariant-mass resolutions shown in the table correspond to the ideal case of perfect bremsstrahlung recovery. Channel | $\mathcal{R}$ | $\epsilon_{L}$ | $\epsilon_{D}$ |  $\sigma_{L}$(${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$) |  $\sigma_{D}$(${\mathrm{\,Me\kern-0.80005ptV\\!/}c^{2}}$)
---|---|---|---|---|---
$K^{0}_{S}\to\mu^{+}e^{-}$ | 1 | 1.0 (0.84) | 1.5 (1.3) | $\sim 3.0$ | $\sim 8.0$
$K^{0}_{L}\to\mu^{+}e^{-}$ | 1 |  3.1 (2.6) $\times 10^{-3}$ |  13 (11) $\times 10^{-3}$ | $\sim 3.0$ | $\sim 7.0$
$K^{+}\to\pi^{+}\mu^{+}e^{-}$ | $\sim 2$ |  3.1 (1.1) $\times 10^{-3}$ |  16 (8.5)$\times 10^{-3}$ | $\sim 2.0$ | $\sim 8.0$

###  2.1 Trigger

The current trigger of LHCb has three stages, a hardware stage (L0) and two
software stages (HLT1 and HLT2). The L0 is practically unchangeable and
implies an efficiency loss of roughly $80\%$ of reconstructible strange-hadron
decays involving muons [25]. For non muonic final states it implies a loss of
about 90% to 99%, due to the much larger transverse energy trigger thresholds
for hadrons and electrons [26], depending on whether also events triggered by
the underlying event (and not by the considered signal) are accepted or not
[27]. During Run 1, the total trigger efficiency for strangeness decays was
$\text{1--2}\%$ or lower, depending on the final state. The main reason for
those low efficiencies is the soft transverse momentum spectrum of strange-
hadron decay products. During Run 2, dedicated software triggers for strange-
hadron decays into dimuons have been implemented with an overall improvement
of about one order of magnitude in the total trigger efficiency achieved with
respect to Run 1 [25]. In the Upgrade of the LHCb experiment, the trigger is
expected to be entirely software based with L0 removed, hence ${\cal O}(1)$
efficiencies are attainable. 111Here and in the following, trigger
efficiencies are calculated and referred to events that have passed the full
offline selection, hence perfect efficiencies are attainable when the trigger
requirements are aligned to, or looser than, the offline selection. It has
been shown in simulation that for dimuon final states, the output rate can be
kept under control for transverse momentum thresholds as low as
$80~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ without any significant signal
loss [5]. Although the dimuon final state is the cleanest signature from an
experimental perspective, trigger algorithms for other final states are
possible and are currently under investigation. As an example, a software
trigger for dielectrons from strange decays was already implemented during Run
2 [6] and will serve as a basis for the Upgrade.

###  2.2 Flavour Tagging

As pointed out in ref. [28], $K^{0}_{\mathrm{\scriptscriptstyle
S}}$-$K^{0}_{\mathrm{\scriptscriptstyle L}}$ interference has an effective
lifetime which is only twice that of the $K^{0}_{\mathrm{\scriptscriptstyle
S}}$ and thus has an enhanced acceptance in LHCb compared to pure
$K^{0}_{\mathrm{\scriptscriptstyle L}}$ decays. By tagging the initial flavour
of the $K^{0}$ , access to $K^{0}_{\mathrm{\scriptscriptstyle L}}$ physics and
$C\\!P$ phenomena in the $K^{0}_{\mathrm{\scriptscriptstyle
S}}-K^{0}_{\mathrm{\scriptscriptstyle L}}$ system is permitted through these
interference effects. 222While the present paper is focused mainly on rare and
semileptonic decays, a program of measurements of $C\\!P$ violation in the
$K^{0}_{\mathrm{\scriptscriptstyle S}}-K^{0}_{\mathrm{\scriptscriptstyle L}}$
system is in principle possible and merits further study. Though not used for
this paper, it is valuable to mention the possibility of strange-hadron
flavour tagging at LHCb through $K^{0}$ processes such as $pp\rightarrow
K^{0}K^{-}X$, $pp\rightarrow K^{*+}X\rightarrow K^{0}\pi^{+}X$ and
$pp\rightarrow K^{0}\Lambda^{0}X$.

##  3 Rare decays

Rare decays are excellent probes for BSM. On the theoretical side, the SM
background to each process is small by definition, while experimentally,
measurements are typically statistically limited, but this limitation can
constantly be improved. In this section, the status and prospects for several
benchmark rare decays of different strange-hadron species are shown.

###  3.1 Rare decays of $K^{0}_{S}$ mesons

Due to its shorter lifetime compared to $K^{0}_{\mathrm{\scriptscriptstyle
L}}$ and $K^{+}$, the $K^{0}_{\mathrm{\scriptscriptstyle S}}$ meson is the
most accessible in terms of reconstruction in LHCb. With a geometric
acceptance at the $1\%$ level and a production cross section of about 0.3
barn, the LHCb Phase-II upgrade could reach branching fraction sensitivities
down to the level of $10^{-15}$ in the ideal case of perfect selection and
trigger with no background. In the following, the channels LHCb has already
investigated are discussed in addition to new analysis suggestions.

####  3.1.1 $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$

In the SM, the $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ decay
is dominated by long-distance (LD) effects with subdominant short-distance
(SD) contributions coming from $Z$-penguin and $W$-box diagrams. Yet in
absolute terms, the long-distance contribution is still minute with the decay
rate highly suppressed [29, 30, 28]. The theoretical prediction,

${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-})_{\rm
SM}=(5.18\pm 1.50_{\rm LD}\pm 0.02_{\rm SD})\times 10^{-12}\quad,$

when compared with the current experimental upper limit [3]

${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-})<8\times
10^{-10}~{}{\rm{at}}~{}90\%~{}{\rm{CL}}\quad,$

leaves room for small BSM contributions to interfere and compete with the SM
rate. This is shown to be the case in leptoquark (LQ) models [31, 32] as well
as in the Minimal Supersymmetric Standard Model (MSSM) [33]. In the LQ case,
the enhancements can reach as high as the current experimental bound, while
within the MSSM, ${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\mu^{+}\mu^{-})$ can adopt values anywhere in the range
$[0.78,35.00]\times 10^{-12}$, even surpassing the experimental bound in
certain narrow, finely-tuned regions of the parameter space [33]. This can be
seen in figure 3, where $A^{\mu}_{L\gamma\gamma}$ indicates the long-distance
contribution to ${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle
L}}\to\mu^{+}\mu^{-})$. The $C\\!P$ asymmetry of this decay is also sensitive
to BSM contributions, but experimentally accessible only by means of a tagged
analysis.

Figure 3:  A generic scan of $\mathcal{B}(K_{S}^{0}\rightarrow\mu^{+}\mu^{-})$
vs $\mathcal{B}(K_{L}^{0}\rightarrow\mu^{+}\mu^{-})$ from Ref. [33], in an
MSSM scenario with universal gaugino masses. The cyan dots correspond to
predictions with $A^{\mu}_{L\gamma\gamma}>0$ and the orange crosses to
predictions using $A^{\mu}_{L\gamma\gamma}<0$. The vertically hatched area
corresponds to the SM prediction for $A^{\mu}_{L\gamma\gamma}>0$ while the
diagonally hatched area corresponds to the SM prediction for
$A^{\mu}_{L\gamma\gamma}<0$.

The LHCb prospects for the search for $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\mu^{+}\mu^{-}$ decays are excellent. With only 2011 data, the
experiment improved the previous world upper limit by a factor of thirty [2]
and recently gained another factor of ten [3]. In the case of an LHCb Phase-II
upgrade running during the proposed HL-LHC era, the full software trigger will
allow an exploration of branching fractions below the $10^{-11}$ regime.
Figure 4, first shown in Ref. [34], shows the expected upper limit of ${\cal
B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-})$ as a function of
the integrated luminosity scaled by the trigger efficiency, based on the
extrapolation given in Ref. [3]. This demonstrates that if the trigger
efficiency is near $\sim 1$, as can be achieved technically with the Phase-I
full software trigger, LHCb could exclude branching fractions down towards the
vicinity of the SM prediction.

Figure 4:  Expected upper limit of ${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\mu^{+}\mu^{-})$ from LHCb including upgrades, against the product of
the integrated luminosity and trigger efficiency, taken from Ref. [34]. The
LHCb upgrade is expected to collect 50$\text{fb}^{-1}$, and the Phase-II
$\approx 300\text{fb}^{-1}$.

####  3.1.2 $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$

The experimental uncertainty on ${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-})$ is the dominant uncertainty on the SM prediction
of ${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle L}}\to\pi^{0}\mu^{+}\mu^{-})$,
the latter being an important channel for BSM searches, such as extra
dimensions [35]. Currently, the only existing measurement comes from the NA48
experiment [36],

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

The upgraded LHCb experiment can quickly eclipse NA48 in terms of precision on
${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-})$ and
achieve a level of $0.25\times 10^{-9}$ with 50 $\text{fb}^{-1}$ of integrated
luminosity and assuming 100% trigger efficiency [5] (see footnote 1).

Aside from the branching fraction, the differential decay rate in the dimuon
mass possesses interesting information. As the electromagnetic structure of
this decay in the SM receives only a single contribution from the vector
current, an amplitude analysis cannot offer any advantages over a fit to the
dimuon mass spectrum alone. The decay dynamics of this channel are assumed to
be governed by a linear dependence in $q^{2}$, thus there are two free, real
parameters of the model, which can be determined from data, $a_{S}$ and
$b_{S}$, where $b_{S}$ is the coefficient of the linear term in $q^{2}$. This
complements the information available from the branching fraction, which has
the form,

${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-})\propto
0.07-4.52a_{S}-1.5b_{S}+98.7a_{S}^{2}+57.7a_{S}b_{S}+8.95b_{S}^{2}\quad,$

in the SM [37].

Importantly, $a_{S}$ is the relevant parameter for the SM determination of
${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle L}}\to\pi^{0}\mu^{+}\mu^{-})$. It
has been estimated from the NA48 measurement of ${\cal
B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-})$ that
$|a_{S}|=1.2\pm 0.2$ [35], assuming vector meson dominance (VMD), where
$b_{S}/a_{S}=m^{2}_{K}/m^{2}_{\rho}$. Without VMD, resolving $a_{S}$ with only
a single observable is not possible. Hence, as the precision in ${\cal
B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-})$ increases,
use of the $q^{2}$ dependence, which has been calculated in Ref. [37], becomes
a viable approach in avoiding this model dependence.

Two degenerate solutions are expected for both $a_{S}$ and $b_{S}$. A pseudo-
experiment study indicates that the significance of the sign-flip in $a_{S}$
is consistent with zero even up to signal yields well beyond the reach of any
proposed LHC upgrade. Although the model-dependent expectation is that the
product $a_{S}b_{S}$, is preferred to be positive, the proximity to zero of
the $b_{S}$ solution corresponding to negative $a_{S}$ renders this constraint
untenable.

A number of analysis configurations from a purely statistical point of view
are considered, neglecting systematic uncertainties. The statistical power has
been obtained from the expected sensitivity in ${\cal
B}(K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-})$, where the
signal plus background yield is translated into an effective signal-only
yield. Firstly, the scenario where both $a_{S}$ and $b_{S}$ are measured from
the $q^{2}$ distribution is considered. An additional constraint coming from
NA48 is also considered, which relates the branching fraction of
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}e^{+}e^{-}$, to $a_{S}$ and
$b_{S}$ through

${\cal B}(K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}e^{+}e^{-})=[0.01-0.76a_{S}-0.21b_{S}+46.5a_{S}^{2}+12.9a_{S}b_{S}+1.44b_{S}^{2}]\times
10^{-10}\quad.$

The uncertainty on $a_{S}$ using the value of $b_{S}$ motivated by VMD is also
investigated. In this paradigm, it becomes possible to measure $a_{S}$ from
the $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$ yield
alone, which is tested as the final case.

Table 4: Projected statistical uncertainties on $a_{S}$ under various analysis conditions. Configuration |  Phase I |  Phase II
---|---|---
BR & $q^{2}$ fit | 0.25 | 0.10
BR & $q^{2}$ fit with NA48 constraint | 0.19 | 0.10
BR & $q^{2}$ fit fixing $b_{S}$ | 0.06 | 0.024
$a_{S}$ measurement from BR alone | 0.06 | 0.024

The reach of LHCb in each of these scenarios is summarised in table 4 for
different effective yields. In the case that $b_{S}$ is measured from the
data, its uncertainties are expected to be 0.87 (0.35) for the Phase-I (Phase-
II) data samples. The results show that with the effective events from Phase-I
data, the constraint coming from NA48 on the
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}e^{+}e^{-}$ branching fraction
will play a role in reducing the uncertainty on $a_{S}$, while with Phase-II
data, the uncertainty will be entirely dominated by the LHCb
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$ measurement.
The results also indicate the vast improvement in $a_{S}$ that becomes
possible at the expense of model independence and demonstrate that the $q^{2}$
distribution has very little impact on the overall uncertainty on $a_{S}$ when
$b_{S}$ is fixed. Further improvements could, of course, come from an LHCb
measurement of $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}e^{+}e^{-}$.

####  3.1.3 $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$
and other $K^{0}_{\mathrm{\scriptscriptstyle S}}$ dielectron modes

With a relatively high branching fraction of $\sim 5\times 10^{-5}$ [38], the
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ decay
offers an excellent opportunity to study rare decays of
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ mesons to electrons at LHCb. Due to
bremsstrahlung, electrons are generally more difficult to reconstruct than
other particles, such as pions or muons. This is especially the case for low
momentum electrons, such as those expected in
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays. Given the branching fraction
of $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$, a
significant yield per $\text{fb}^{-1}$ is expected to be produced within the
LHCb acceptance, thus this decay could be used both for $C\\!P$-violation
studies [38] and to search for potential resonant structure in the
$e^{+}e^{-}$ invariant-mass spectrum. From a purely experimental standpoint,
it is interesting for the study of both the reconstruction and identification
of low momentum electrons and to harness as a normalisation channel for
various 4-body $K^{0}_{\mathrm{\scriptscriptstyle S}}$ rare decays. Examples
include decays to four leptons, which could be sensitive to the presence of
BSM contributions [39], suppressed SM decays such as
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$, or
Lepton Flavour Violating decays like $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\mu^{+}\mu^{+}e^{-}e^{-}$ and $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}\mu^{+}e^{-}$. Moreover,
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ could
present as a prominent background in these searches, ergo, a comprehensive
understanding of its expected yield and invariant-mass distribution becomes
crucial.

The $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ decay
at LHCb is studied in Ref. [6]. This analysis involves a generic study of the
decay using LHCb simulated samples and includes a search with the Run 1 data,
giving prospects for Run 2 and Run 3. The LHCb hardware trigger is found to
limit observation of this decay, with only $\sim$100 candidates per
$\text{fb}^{-1}$ expected to be reconstructed and selected in Run 1 and Run 2.
Despite this relatively low yield, it is also concluded that a purpose-built
offline selection, including the use of a Multi-Variate Analysis (MVA)
classifier, could lead to an observation of the signal. The prospects for Run
3 are much better, with an expected yield at the level of $\sim 50\times
10^{3}$ selected candidates per $\text{fb}^{-1}$. Furthermore, the presence of
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ as a
background for 4-lepton final states is also studied. Figure 5, taken from
Ref. [6], shows the invariant-mass shape of the
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ decay in
conjunction with the alternate $\mu^{+}\mu^{-}e^{+}e^{-}$ mass hypothesis, to
highlight its separation with respect to a potential
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}e^{+}e^{-}$ signal,
both obtained from simulation. While both peaks are separated, a significant
contamination from $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ is expected in the signal region due to the
long tails of the distribution and the much larger yield expected for this
mode. However this contribution can be modelled from simulation and systematic
effects controlled with data, in analogy to the contamination of
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}$ decays as a
background for $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ [3].

Figure 5:  Expected invariant-mass shape of $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ shown additionally with the
$\mu^{+}\mu^{-}e^{+}e^{-}$ mass hypothesis, taken from Ref. [6].

The presence of electron bremsstrahlung combined with the low transverse
momentum of the final state particles, makes the invariant-mass resolution of
this final state significantly worse when compared to
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$, for instance. New
reconstruction strategies could enhance the sensitivity of LHCb to
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ and other
similar final states, such as those mentioned above. Given that the position
of the $K^{0}_{\mathrm{\scriptscriptstyle S}}$ production and decay vertices
can be determined, the invariant-mass resolution of the
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ could be calculated ignoring the
absolute momentum of one of the four final state particles through
relativistic kinematic constraints. This is advantageous as the invariant-mass
resolution becomes less dependent on bremsstrahlung, given that the direction
of electrons in the VELO is barely influenced by such effects. In addition,
this technique could allow a more efficient reconstruction of these electrons,
using tracks not required to have a segment after the magnet. Taking into
account that the VELO pattern recognition efficiency is at the level of $\sim
70\%$ [40], even for tracks with $p\sim{\cal
O}(1{\mathrm{\,Me\kern-1.00006ptV\\!/}c})$, improvements in the reconstruction
efficiency up to a factor of 10 could be theoretically possible.

####  3.1.4 $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\gamma\mu^{+}\mu^{-}$,
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to X^{0}\mu^{+}\mu^{-}$ and
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to X^{0}\pi^{\pm}\mu^{\mp}$

The analysis strategy of $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-}$ can be applied to any
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to X^{0}\mu^{+}\mu^{-}$ mode, where
$X^{0}$ is an arbitrary neutral system. The performance of the search will be
strongly related to the mass of the neutral system, with heavier $X^{0}$
leading to superior invariant-mass resolution of the
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ peak. The resolution is studied here
using simulated $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\gamma\mu^{+}\mu^{-}$
decays, corresponding to the most restrictive case of a massless $X^{0}$. This
decay is predicted in the SM to occur with a branching fraction of $(1.45\pm
0.27)\times 10^{-9}$ [41]. Background from generated
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}$ is also considered
with the aforementioned simplified tracking emulation. From figure 6, the
distinction between signal and background is visibly worse for
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\gamma\mu^{+}\mu^{-}$ than it is for
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$. Nevertheless,
both peaks show clear separation and hence the search is feasible. A reduction
of the $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{+}\pi^{-}$ background is
possible by requiring the dimuon candidate to point away from the primary
vertex, in the same way as is done in $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-}$ analysis [5]. A similar strategy can be embraced
in $K^{0}_{\mathrm{\scriptscriptstyle S}}\to X^{0}\pi^{\pm}\mu^{\mp}$, where
the $X^{0}$ in this case could be some neutrino, either from the SM decay
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{\pm}\mu^{\mp}\nu$ or a heavy BSM
neutrino (see also section 4.2.1).

Figure 6:  Reconstructed invariant mass for $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\gamma\mu^{+}\mu^{-}$ (top) and $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-}$ (bottom) obtained from simulation. The
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\gamma\mu^{+}\mu^{-}$ and
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$ signal events
are shown with a solid blue line and the $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}$ background is illustrated with red filled histograms.
The left side portrays events reconstructed with long tracks, while
reconstruction with downstream tracks are depicted on the right.

###  3.2 Rare decays of $K^{+}$ mesons

From the efficiency ratios of table 1 and considering that sensitivities for
$K^{0}_{\mathrm{\scriptscriptstyle S}}$ branching fractions are at the
$10^{-10}$-$10^{-12}$ level, sensitivities from $10^{-7}$-$10^{-10}$ could be
expected for $K^{+}$ decays, depending on the background level. For $K^{+}$
mesons, which are electrically charged and long-lived, the possibility to
interact with one or more VELO stations can lead to an additional source of
discrimination against combinatorial background [42]. Single event
sensitivities could then well reach below $10^{-12}$, in the case of very
small background (muonic channels), while taking into account higher levels of
background, possible sensitivities of order $10^{-10}-10^{-11}$ are foreseen.

####  3.2.1 $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ and $K^{+}\to\pi^{+}e^{+}e^{-}$

The decays $K^{\pm}\rightarrow\pi^{\pm}\mu^{+}\mu^{-}$ are flavour-changing
processes induced at the one-loop level, which are well suited to explore SM
structure and its extensions. These decays are dominated by long-distance
contributions involving one photon exchange i.e.
$K\rightarrow\pi\gamma^{*}\rightarrow\pi\mu^{+}\mu^{-}$. The branching
fraction has been derived within the framework of Chiral Perturbation Theory
($\chi_{PT}$) in terms of a vector-interaction form factor, which describes
the single-photon exchange and characterises the dimuon invariant-mass
spectrum [37, 43, 44]. The differential decay rate can be written as a
kinematic term depending on masses and 4-momenta, multiplied by $|W(z)|^{2}$,
where $W$ is the form factor and $z=(m_{\mu\mu}/M_{K})^{2}$. The form factor
is given by $W(z)\propto W_{\rm pol}(z)W_{\pi\pi}(z)$, where the second term
represents the tiny contribution from the two-pion-loop intermediate state and
the first term is phenomenologically described by a polynomial. As the form
factor is required to vanish at lowest order in the low-energy chiral
expansion, the polynomial term takes the form $W_{\rm pol}(z)=(a_{+}+b_{+}z)$,
where $a_{+}$ and $b_{+}$ are free parameters of the model to be determined by
experiment. In a similar fashion to $b\rightarrow s$ transitions,
$s\rightarrow d$ processes can be described with an effective Lagrangian
depending on Wilson coefficients, generating only the non-zero Wilson
coefficients $C_{7A}$ and $C_{7V}$ for the semileptonic operators. Such
coefficients can be split into SM and BSM contributions. In particular,
$a_{+}$ can be written as a function of the Wilson coefficient $C_{7A}$ [45],
leading to potential constraints on BSM. A further comparison of the electron
and muon channels would provide an additional test of Lepton Flavour
Universality and further constrain BSM dynamics.

Natural extensions of the SM involve the inclusion of sterile neutrinos which
mix with ordinary neutrinos. An example is the Neutrino Minimal Standard Model
($\nu$MSM) [46], which can be further extended by adding a scalar field to
incorporate inflation and provide a common source for electroweak symmetry
breaking and right-handed neutrino masses [47]. The new particles predicted by
these models can be produced in charged kaon decays. Notably, the two-unit
Lepton Number Violating (LNV) $K^{\pm}\rightarrow\pi^{\mp}\mu^{\pm}\mu^{\pm}$
decay could proceed via an off-shell or on-shell Majorana neutrino [48], while
an inflaton could be produced in the Lepton Number Conserving (LNC)
$K^{\pm}\rightarrow\pi^{\pm}X$, decaying promptly to
$X\rightarrow\mu^{+}\mu^{-}$ [49].

The NA48/2 collaboration [50, 51] reports the most precise measurement to date
of the branching fraction and provide limits on the Majorana neutrino and
inflaton. They measured

${\cal B}(K^{\pm}\rightarrow\pi^{\pm}\mu^{+}\mu^{-})=(9.62\pm 0.21_{\rm
stat}\pm 0.13_{\rm syst})\times 10^{-8},$ ${\cal
B}(K^{\pm}\rightarrow\pi^{\mp}\mu^{\pm}\mu^{\pm})<8.6\times
10^{-11}\,\,\,(90\%\;{\rm CL}),$ ${\cal
B}(K^{\pm}\rightarrow\pi^{\pm}X)<10^{-11}{\text{-}}10^{-9}\,\,\,(90\%\;{\rm
CL}),$

where the range depends on the assumed resonance lifetime. The NA62 experiment
plans to improve on all these measurements and limits, though with positively-
charged kaons only [8]. The LHCb mass resolution is sufficient to separate
these decays from the kinematically similar $K^{+}\to\pi^{+}\pi^{+}\pi^{-}$,
as illustrated in figure 7. LHCb can acquire large
$K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ signal yields as table 1 and figure 1 clearly
indicate. Assuming ${\cal O}(1)$ trigger efficiencies, a yield of ${\cal
O}$(104) fully reconstructed and selected signal events is expected per year
of upgraded-LHCb data taking, even considering only long-track candidates.
This suggests $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$ decays would provide an early
opportunity for a measurement to demonstrate the potential of the upgraded
detector for these channels. Similar arguments apply to the
$K^{+}\to\pi^{+}e^{+}e^{-}$ mode, whose somewhat lower reconstruction
efficiency due to the presence of electrons is negated by its larger branching
fraction. Rigorous control over the systematic uncertainties will be paramount
in order to improve the current world-average precision of $3\%$ on the
electron mode. If successful, the full spectrum of both channels will afford a
highly precise test of Lepton Flavour Universality.

Figure 7:  Reconstructed invariant mass for $K^{+}\to\pi^{+}\mu^{+}\mu^{-}$,
where signal events are shown with a solid blue line and
$K^{+}\to\pi^{+}\pi^{+}\pi^{-}$ background illustrated by red filled
histograms. The left side gives events reconstructed with long tracks, while
reconstruction with downstream tracks are pictured on the right.

###  3.3 Tests of LFV

Modes with LFV, such as $K\to(n\pi)\mu^{\pm}e^{\mp}$ form null tests of the
SM. Sizeable BSM contributions to such decays have garnered increased
attention in recent times because of hints at Lepton Universality Violation
(LUV) in $B\to K^{(\ast)}\ell^{\pm}\ell^{\mp}$ processes. In fact, both
classes of processes can be generated by new contributions to the product of
two neutral currents, involving down-type quarks and leptons respectively, the
only difference being the strength of the flavour couplings involved.

From the amount of LUV alluded to in $B\to K^{(\ast)}\ell^{\pm}\ell^{\mp}$,
one may expect $B\to K^{(\ast)}$ LFV rates of the order of $10^{-8}$ using
general effective-theory (EFT) arguments [52]. More quantitative estimates
require the introduction of a flavour model [53, 54, 55, 56, 57, 45, 58, 59,
60, 61, 62, 63, 64]. As discussed in Ref. [65], such arguments can be extended
to the $K\to(\pi)\mu^{\pm}e^{\mp}$ case, with fairly general assumptions on
the different flavour couplings involved. Expected rates can be as large as
$10^{-10}$ \- $10^{-13}$ for the $K_{L}\to\mu^{\pm}e^{\mp}$ mode and a factor
of $\sim 100$ smaller for $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$. Taking into
account the suppression mechanisms at play, such ‘large’ rates are a non-
trivial finding. Their relatively wide range is due to the inherent model
dependence especially in the choice of the leptonic coupling and the overall
scale of the new interaction, typically between $5$ and $15$ TeV [65]. Since
limits on the branching fractions for the $K\to\pi e\mu$ modes were pushed
down to the level of $10^{-11}-10^{-12}$ in the 1990s, there has been no
significant further progress on the experimental side, with the current limits
at 90% CL,

(4)

being decades old.

These modes can be profitably pursued at the upgraded LHCb, benefiting from
huge strange-production yields. In fact, starting from a total $K^{\pm}$ cross
section of 0.63 barns and taking into account the fraction of kaons in the
pseudorapidity acceptance of LHCb, one can estimate a $K^{\pm}$ cross section
as large as $0.14$ barns. Ref. [65] presents a feasibility study of the modes
listed in eq. (4), taking $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$ as a benchmark. It
can be seen that LHCb may be able to update the existing limits and probe a
sizeable part of the parameter space suggested by the discrepancies in $B$
physics.

###  3.4 Rare decays of $\Sigma$ hyperons

LHCb has recently published the most precise search for $\Sigma^{+}\to
p\mu^{+}\mu^{-}$ [4], showing strong evidence for this decay with $4.1\sigma$
significance. A measurement of the branching fraction is reported along with a
dimuon invariant-mass distribution consistent with SM predictions, challenging
the so-called HyperCP-anomaly [70]. This measurement was based on Run 1 data,
where no trigger path existed specifically for this channel. As discussed in
Ref. [25], Run 2 will have a dedicated trigger both at the HLT1 and HLT2
levels, where about an order of magnitude increase in the trigger efficiency
is anticipated. With a signal yield in excess of 150 events, Run 2 data will
allow a measurement of the differential decay rate and possibly other
observables with recent predictions such as the forward-backward asymmetry
[71]. Applying similar reasoning on the trigger efficiency as with other
decays in this document, on the order of a thousand signal decays could be
measured per year of data taking with an upgraded LHCb detector, opening the
possibility for precision measurements of direct $C\\!P$ violation. Assuming
similar reconstruction and selection efficiencies, a search for the lepton and
baryon number violating $\Sigma^{+}\to\bar{p}\mu^{+}\mu^{+}$ decay could also
be performed, reaching an expected branching fraction sensitivity on the order
of $10^{-9}$.

While of great interest, it will be difficult for LHCb to improve the
precision on the branching fraction of the radiative $\Sigma^{+}\to p\gamma$
decay, whose world average is currently $\mathcal{B}(\Sigma^{+}\to
p\gamma)=(1.23\pm 0.05)\times 10^{-3}$ [72]. On the other hand, the ability to
reconstruct the $\Sigma^{+}\to p\pi^{0}$ decay, which has similar topology in
the detector, has already been demonstrated [4]. This implies that the
$\Sigma^{+}\to p\gamma$ decay could be useful as an alternative normalisation
channel, particularly in a possible search for $\Sigma^{+}\to pe^{+}e^{-}$
decays. By virtue of the electron mass, this channel receives a larger
contribution from long-distance photon contributions compared to
$\Sigma^{+}\to p\mu^{+}\mu^{-}$, for a predicted branching fraction of
$\mathcal{B}(\Sigma^{+}\to pe^{+}e^{-})\in[9.1,10.1]\times 10^{-6}$ [73]. The
only experimental information available on this channel dates back to 1969
where three events where observed leading to an upper limit of $7\times
10^{-6}$ at 90% CL [74]. Unsurprisingly, this yield is not yet distinguishable
from converted-photon $\Sigma^{+}\to p\gamma$ decays. Although electron
reconstruction is more difficult, it is expected that the LHCb experiment
could improve on this measurement and perhaps reach the SM level already with
Run 2 data. Analogously, the LFV decays $\Sigma^{+}\to pe^{\pm}\mu^{\mp}$
could also be searched for with similar sensitivity.

Owing to the extreme difficulty of reconstructing neutrons, the LHCb
experiment will most likely not contribute towards the study of the
$\Sigma^{-}$ hyperon, barring exotic channels with baryon number violation.

As far as $\Sigma^{0}$ particles are concerned, these do not have a sizeable
decay time, due to their electromagnetic decay into $\Lambda\gamma$, therefore
they would decay at the production vertex in LHCb. For this reason while our
simplified model could predict their reconstruction efficiency, the
sensitivity for $\Sigma^{0}$ decays would be dominated by primary interaction
background, which would require a full simulation to be understood. We
therefore do not provide estimates on these sensitivities. We limit ourselves
to suggest that LHCb could attempt a first search for the
$\Sigma^{0}\to\Lambda e^{+}e^{-}$ decay, for which no experimental measurement
is currently available, despite the fact that several authors proposed this
decay to study parity violation in strangeness-conserving weak currents [75,
76, 77]. In lieu of an experimental measurement the PDG reports a theoretical
calculation driven by internal photon conversions for an expected branching
fraction of about $5\times 10^{-3}$ [78], easily reachable by LHCb if
background can be controlled.

###  3.5 Rare decays of $\Lambda$ hyperons

The most compelling contribution LHCb could offer in the realm of $\Lambda$
hyperon is the improvement on the branching fraction of the radiative
$\Lambda\to p\pi^{-}\gamma$ decay, whose measured value
$\mathcal{B}(\Lambda\to p\pi^{-}\gamma)=(8.4\pm 1.4)\times 10^{-4}$, is known
only for pion centre-of-mass momenta less than
$95~{}{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ [79]. In addition, first studies
of $\Lambda\to p\pi^{-}e^{+}e^{-}$, which proceeds via flavour-changing
neutral currents could be possible, reaching branching fractions of
$10^{-6}-10^{-7}$. A major challenge for $\Lambda\to p\pi^{-}e^{+}e^{-}$ is
the extremely low transverse electron momentum as illustrated in figure 8,
translating into a meagre reconstruction efficiency in accordance with table
2. The corresponding channel with muons in this case would be phase-space
forbidden.

Figure 8: Momentum (left) and transverse momentum (right) for electrons
generated in various strangeness decays, where the dotted blue represents
$\Lambda\to p\pi^{-}e^{+}e^{-}$, solid red $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{+}\pi^{-}e^{+}e^{-}$ and filled green
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}e^{+}e^{-}$.

LHCb can also advance the study of baryon-number-violating decays, which can
be produced by virtual particles with masses at the Grand Unified Theory (GUT)
scale. For weakly decaying particles, this would imply branching fractions
suppressed proportionally to ${(m_{W}/\Lambda_{\rm GUT})^{4}}$, in principle
placing observation out of reach for LHCb and any other experiment. These
decays are also indirectly constrained by severe limits from nucleon decays.
The CLAS collaboration has recently reported searches for several baryon-
number-violating $\Lambda$ decays [80]. Most of these are in the form
$\Lambda\to h\ell$, where $h$ is a $K^{+}$ or $\pi^{+}$ meson and $\ell=e,\mu$
leptons. CLAS then provided the first direct experimental limits on such
branching fractions to be in the range $[10^{-7},10^{-6}]$. LHCb can certainly
improve on most of these limits, reaching sensitivities around the $10^{-9}$
level already with Run 2 data.

###  3.6 Rare decays of hyperons with multiple strangeness

In addition to hadrons with one strange quark or anti-quark ($|S|=1$), LHCb
will also produce a large number of baryons with more strange quarks, namely
the $\Xi$ and $\Omega$ hyperons. As can be seen from figure 1, the production
of $\Xi$ is in the region of charmed mesons, while $\Omega$ production is
further suppressed, due to the additional strange quark, to the level around
the beauty meson. Nevertheless, this provides a large dataset with which to
improve existing measurements on these hadrons.

In the context of rare decays, the main interest for $|S|>1$ hyperons is for
$\Delta S=2$ transitions, which are practically forbidden in the SM, with
branching fractions of order $10^{-17}$. Potential NP transitions mediated by
parity-odd low-energy operators may enhance the observed rates while
respecting constraints from $K^{0}-\bar{K}^{0}$ mixing [81]. In this respect,
the LHCb experiment has the capabilities to improve the branching fraction of
$\Xi^{0}\to p\pi^{-}$, which has an upper limit of $8.2\times 10^{-6}$ at 90%
CL obtained at the HyperCP experiment [82]. This decay has an experimental
signature completely reminiscent of the corresponding $\Lambda$ decay, which
is selected even without particle identification at LHCb [83], making it the
ideal calibration sample for $\Xi^{0}\to p\pi^{-}$. Therefore, there is no
doubt that the background to this channel could be rejected with high signal
retention. Branching fractions of order $10^{-9}-10^{-10}$ could be reached
with LHCb Upgrade data.

In similar vein, the $\Omega\to\Lambda\pi^{-}$ decay has an upper limit on the
branching fraction of $2.9\times 10^{-6}$ at 90% CL also placed by the HyperCP
experiment [82]. The sensitivity to this channel is again expected to be
improved over the current limit given its clean topology, down to branching
fractions of order $10^{-8}-10^{-9}$. Incidentally, the channel $\Xi^{-}\to
p\pi^{-}\pi^{-}$, which has an upper limit of only $3.7\times 10^{-4}$ at 90%
CL [84], will also be easily improved by LHCb, similarly to $\Xi^{0}\to
p\pi^{-}$, reaching sensitivities of order $10^{-9}$.

##  4 Other measurements with strange-hadron decays

###  4.1 Measurement of the $K^{+}$ meson mass

Due to its superb tracking performance, the LHCb detector is particularly
suited for a precision measurement of the charged kaon mass. The current
experimental average of the $K^{+}$ meson mass is $m_{K^{+}}=493.677\pm 0.013$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [72]. The uncertainty is dominated
by the disagreement between the two most precise measurements, both performed
using kaonic atom transitions [85, 86]. Despite the relatively low acceptance
in LHCb, the large production cross section for strange mesons in $pp$
collision allows for a large number of $K^{+}\to\pi^{+}\pi^{-}\pi^{+}$
candidates to be fully reconstructed with an excellent signal-to-background
ratio [42]. The number of fully reconstructed decays occurring within the VELO
acceptance is estimated to be of $\mathcal{O}(10^{7})/\mathrm{fb^{-1}}$ for
$pp$ collisions at $\sqrt{s}=13$ TeV with a relatively good mass resolution of
$\lesssim 4$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [4]. Therefore, the
statistical error on the mass is expected to be below $10^{-3}$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ with the entire LHCb dataset. The
main systematic uncertainty, which is expected to limit the final precision,
will most likely come from the knowledge of the momentum scale resolution,
which is proportional to the Q-value of the decay,
$m_{K^{+}}-3m_{\pi^{\pm}}\approx 75{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
For $K^{+}\to\pi^{+}\pi^{-}\pi^{+}$, this systematic should be below $0.02$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [87], making this measurement
competitive with the world average.

###  4.2 Semileptonic decays

The latest results from semileptonic $b\to c$ transitions suggest the
possibility of BSM contributions in charged-current quark decays breaking
Lepton Flavour Universality (LFU) [88]. Hence, it is natural to investigate if
similar patterns can be found in $s\to u$ transitions.

####  4.2.1 Semileptonic $K^{0}_{\mathrm{\scriptscriptstyle S}}$ decays

A search for the $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{\pm}\mu^{\mp}\nu$ process, which is as yet unobserved
experimentally, could be performed at LHCb. This would be useful as a
measurement of LFU when comparing to the well-known
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{\pm}e^{\mp}\nu$ decay [72].
Depending on the precision achieved, the measurement of this branching
fraction could also be useful in constraining the CKM matrix element
$|V_{us}|$ [89]. However, LHCb would need excellent control over the
systematics to reach the $<$ 1% level of precision that would be required for
such a measurement to be competitive. The most challenging background for this
search is expected to arise from the corresponding
$K^{0}_{\mathrm{\scriptscriptstyle L}}$ decay to the same final state. The
much larger branching fraction of the $K^{0}_{\mathrm{\scriptscriptstyle L}}$
decay, $\sim 27\%$ [72], compensates the reduction in efficiency due to the
longer $K^{0}_{\mathrm{\scriptscriptstyle L}}$ lifetime, leading to
significant yields still deposited within the LHCb acceptance: considering the
expected $K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{\pm}\mu^{\mp}\nu$
branching fraction, $(4.69\pm 0.05)\times 10^{-4}$ [72], the ratio of
$K^{0}_{\mathrm{\scriptscriptstyle L}}$ to $K^{0}_{\mathrm{\scriptscriptstyle
S}}$ events in this final state in the LHCb acceptance is expected to be about
1.5 (4.5) when using long (downstream) tracks, without further selection.
However, given the precise knowledge of the $K^{0}_{\mathrm{\scriptscriptstyle
L}}$ branching fraction, $(27.04\pm{0.07})\%$ [72], this contribution could be
statistically subtracted leaving only a small systematic uncertainty.

####  4.2.2 Semileptonic hyperon decays

Semileptonic hyperon decays have been shown to be sensitive to BSM scalar and
tensor contributions [90]. The branching fractions of such hyperon decays,
which are copiously produced at the LHC, show uncertainties at the
$20\%-100\%$ level leaving vast room for progress. For example, ${\cal
B}(\Lambda\to p\mu^{-}\bar{\nu_{\mu}})=(1.57\pm 0.35)\times 10^{-4}$, ${\cal
B}(\Xi^{-}\to\Lambda\mu^{-}\bar{\nu_{\mu}})=3.5^{+3.5}_{-2.2}\times 10^{-4}$
and ${\cal B}(\Xi^{-}\to\Sigma^{0}\mu^{-}\bar{\nu_{\mu}})<8\times 10^{-4}$ at
$90\%$ CL.

Those decays would be partially reconstructed in LHCb, as was shown in section
2, with improved measurements directly translating into tighter bounds on LFU,
since the electron modes have already been measured very precisely. Kinematic
constraints such as those applied in the $K^{0}_{\mathrm{\scriptscriptstyle
S}}\to\pi^{0}\mu^{+}\mu^{-}$ analysis can be used to reconstruct the strange-
baryon peak. Since the expected yields for strange semileptonic decays are
large, the main challenge is not the trigger efficiency, but is instead the
discrimination against peaking backgrounds like $\Lambda\to p\pi^{-}$ or
$\Xi^{-}\to\Lambda\pi^{-}$. The mass of the $p,\mu$ candidates from
$\Lambda\to p\mu^{-}\bar{\nu_{\mu}}$ and misidentified $\Lambda\to p\pi^{-}$
is shown in figure 9, which also plots the dependency of the mass against the
estimated missing momentum transverse to the $\Lambda$ flight direction.
Clearly, the signal and peaking background provide contrasting signatures. It
has to be noted, however, that neither final state radiation in the $\Lambda$
decay nor the decay in flight of the pion are included in the simulation, both
of which are effects that can partially dilute the discriminating power of the
missing transverse momentum. A similar study is performed for
$\Xi^{-}\to\Lambda\mu^{-}\bar{\nu_{\mu}}$, which also demonstrates the
separation between signal and the corresponding peaking-background
distribution from $\Xi^{-}\to\Lambda\pi^{-}$ decays, as depicted in figure 10.

Figure 9:  The left plot shows the reconstructed invariant mass for
$\Lambda\to p\mu^{-}\bar{\nu_{\mu}}$ candidates. Signal events are given by a
solid blue line, while the $\Lambda\to p\pi^{-}$ background is displayed in
filled red. The right figure shows a scatter plot of the reconstructed mass vs
missing momentum in the plane transverse to the $\Lambda$ flight direction for
signal (blue squares) and $\Lambda\to p\pi^{-}$ background (red stars). Final
state radiation in the $\Lambda$ decay vertex is not included in the
simulation.

Figure 10:  The left plot shows the reconstructed invariant mass for
$\Xi^{-}\to\Lambda\mu^{-}\bar{\nu_{\mu}}$ candidates. Signal events are given
by a solid blue line, while the $\Xi^{-}\to\Lambda\pi^{-}$ background is
displayed in filled red. The right figure shows a scatter plot of the
reconstructed mass vs missing momentum in the plane transverse to the
$\Xi^{-}$ flight direction for signal (blue squares) and $\Lambda\to p\pi^{-}$
background (red stars). Final state radiation in the $\Xi^{-}$ and $\Lambda$
decay vertices is not included in the simulation.

##  5 Competition from other experiments

Competition from other experiments on strange-hadron decays will be scarce in
the coming years. We briefly review it in the following. The NA48 experiment
has contributed significantly to the physics of strange-hadron decays, but has
already analysed their full dataset on rare $K^{0}_{S}$ and hyperon decays
(e.g. refs [38, 36, 91, 92]) and we are not aware of any plan to exploit it
further. The NA62 experiment will give fundamental results on charged kaons,
however it will not have a neutral beam at its disposal before 2026. In
particular, NA62 may reach the $10^{-12}$ ballpark in LFV kaon decays [93]
with the data collected so far. The KLOE2 experiment will most probably be
able to contribute on semileptonic measurements, in addition to its core
$C\\!P$-violation program, and possibly measure the $K^{+}$ mass, but will not
have enough statistics for rare decays. The CLAS experiment could possibly
contribute again to searches on rare hyperon decays, but will not be
competitive with LHCb below the $10^{-7}$ level in branching fraction.
Similarly it is not expected to contribute on $K^{0}_{S}$ decays. Finally,
flavour factories such as BESIII and BelleII can possibly contribute to the
physics of rare strange-hadron decays. The BESIII collaboration has for
example published a search for $\eta^{\prime}\to K\pi$ decays [94], reaching a
branching fraction limit of order $10^{-4}$. We are not aware of any published
physics result from the Belle collaboration on rare strange hadron decays and
this topic is not mentioned in BelleII physics book [95]. In addition, there
are new proposed facilities such as TauFV [96] which may be able to reach
${\cal O}(10^{19})$ kaons in the decay volume, with a detector layout
comparable to that of LHCb, for which however we are not aware of more in-
depth sensitivity studies on the decay modes discussed in this paper. However,
we would welcome an increase in the interest for strange physics and would
consider competition from these collaborations to be a very healthy
development indeed.

##  6 Conclusions

The decays of strange particles become increasingly important as the energy
scale for dynamics beyond the Standard Model increases. The LHCb experiment
has provided the world’s best measurements in
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\mu^{+}\mu^{-}$ and $\Sigma^{+}\to
p\mu^{+}\mu^{-}$ decays with excellent prospects for expanding its research
program on strangeness decays. For the first time, this paper reports
estimates of detection efficiencies for several
$K^{0}_{\mathrm{\scriptscriptstyle S}}$, $K^{\pm}$ and hyperon decay channels
and evaluates the invariant-mass resolution that could be achieved with the
full and downstream tracking systems, while demonstrating the capacity of LHCb
to resolve signal from potential peaking-background distributions. The results
show that several promising new measurements are feasible in various
$K^{0}_{\mathrm{\scriptscriptstyle S}}$, $K^{\pm}$ and hyperon decays with
diverse final states.

## Acknowledgements

The LHCb authors would like to thank their collaboration colleagues who
participated in helpful discussions on the subject of this paper. We would
like to thank Jorge Portolés for helpful discussions in
$K^{0}_{\mathrm{\scriptscriptstyle S}}\to\pi^{0}\mu^{+}\mu^{-}$. The work of
IGFAE members is supported by ERC-StG-639068 “BSMFLEET” and XuntaGal. In
addition, the work of XCV is also supported by MINECO through the Ramón y
Cajal program RYC-2016-20073. JMC acknowledges support from the Spanish MINECO
through the Ramón y Cajal program RYC-2016-20672. The work of AS is funded by
STFC grant ST/J00412X/1. The work of VVG is partially supported by ERC-
CoG-724777 “RECEPT”. The work of FD is partially supported by the Science and
Technology Facilities Council grant ST/N000331/1. AP acknowledges support by
SNF under contract 168169.

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