5G-NR

    5G New Radio
3GPP

    3rd Generation Partnership Project
ABS

    aerial base station
AC

    address coding
ACF

    autocorrelation function
ACR

    autocorrelation receiver
ADC

    analog-to-digital converter
AIC

    Analog-to-Information Converter
AIC

    Akaike information criterion
ARIC

    asymmetric restricted isometry constant
ARIP

    asymmetric restricted isometry property
ARQ

    Automatic Repeat Request
AUB

    asymptotic union bound
AWGN

    Additive White Gaussian Noise
AWGN

    additive white Gaussian noise
PSK

    asymmetric PSK
AWRICs

    asymmetric weak restricted isometry constants
AWRIP

    asymmetric weak restricted isometry property
BCH

    Bose, Chaudhuri, and Hocquenghem
BCHSC

    BCH based source coding
BEP

    bit error probability
BFC

    block fading channel
BG

    Bernoulli-Gaussian
BGG

    Bernoulli-Generalized Gaussian
BPAM

    binary pulse amplitude modulation
BPDN

    Basis Pursuit Denoising
BPPM

    binary pulse position modulation
BPSK

    Binary Phase Shift Keying
BPZF

    bandpass zonal filter
BSC

    binary symmetric channels
BU

    Bernoulli-uniform
BER

    bit error rate
BS

    base station
BW

    BandWidth
BLLL

     binary log-linear learning
CP

    Cyclic Prefix
CDF

    cumulative distribution function
CDF

    Cumulative Distribution Function
CDF

    cumulative distribution function
CCDF

    complementary cumulative distribution function
CCDF

    complementary CDF
CCDF

    complementary cumulative distribution function
CD

    cooperative diversity
CDMA

    Code Division Multiple Access
ch.f.

    characteristic function
CIR

    channel impulse response
CoSaMP

    compressive sampling matching pursuit
CR

    cognitive radio
CS

    compressed sensing
CS

    Compressed sensing
CS

    compressed sensing
CSI

    channel state information
CCSDS

    consultative committee for space data systems
CC

    convolutional coding
COVID-19

    Coronavirus disease
DAA

    detect and avoid
DAB

    digital audio broadcasting
DCT

    discrete cosine transform
DFT

    discrete Fourier transform
DR

    distortion-rate
DS

    direct sequence
DS-SS

    direct-sequence spread-spectrum
DTR

    differential transmitted-reference
DVB-H

    digital video broadcasting – handheld
DVB-T

    digital video broadcasting – terrestrial
DL

    DownLink
DSSS

    Direct Sequence Spread Spectrum
DFT-s-OFDM

    Discrete Fourier Transform-spread-Orthogonal Frequency Division Multiplexing
DAS

    Distributed Antenna System
DNA

    DeoxyriboNucleic Acid
EC

    European Commission
EED

    exact eigenvalues distribution
EIRP

    Equivalent Isotropically Radiated Power
ELP

    equivalent low-pass
eMBB

    Enhanced Mobile Broadband
EMF

    ElectroMagnetic Field
EU

    European union
EI

    Exposure Index
eICIC

    enhanced Inter-Cell Interference Coordination
FC

    fusion center
FCC

    Federal Communications Commission
FEC

    forward error correction
FFT

    fast Fourier transform
FH

    frequency-hopping
FH-SS

    frequency-hopping spread-spectrum
FS

    Frame synchronization
FS

    frame synchronization
FDMA

    Frequency Division Multiple Access
GA

    Gaussian approximation
GF

    Galois field
GG

    Generalized-Gaussian
GIC

    generalized information criterion
GLRT

    generalized likelihood ratio test
GPS

    Global Positioning System
GMSK

    Gaussian Minimum Shift Keying
GSMA

    Global System for Mobile communications Association
GS

    ground station
GMG

     Grid-connected MicroGeneration
HAP

    high altitude platform
HetNet

    Heterogeneous network
IDR

    information distortion-rate
IFFT

    inverse fast Fourier transform
IHT

    iterative hard thresholding
i.i.d.

    independent, identically distributed
IoT

    Internet of Things
IR

    impulse radio
LRIC

    lower restricted isometry constant
LRICt

    lower restricted isometry constant threshold
ISI

    intersymbol interference
ITU

    International Telecommunication Union
ICNIRP

    International Commission on Non-Ionizing Radiation Protection
IEEE

    Institute of Electrical and Electronics Engineers
ICES

    IEEE international committee on electromagnetic safety
IEC

    International Electrotechnical Commission
IARC

    International Agency on Research on Cancer
IS-95

    Interim Standard 95
KPI

    Key Performance Indicator
LEO

    low earth orbit
LF

    likelihood function
LLF

    log-likelihood function
LLR

    log-likelihood ratio
LLRT

    log-likelihood ratio test
LoS

    Line-of-Sight
LRT

    likelihood ratio test
LWRIC

    lower weak restricted isometry constant
LWRICt

    LWRIC threshold
LPWAN

    Low Power Wide Area Network
LoRaWAN

    Low power long Range Wide Area Network
NLoS

    Non-Line-of-Sight
Li-Fi

    light-fidelity
LED

    light emitting diode
LABS

    LoS transmission with each ABS
NLABS

    NLoS transmission with each ABS
MB

    multiband
MC

    macro cell
MDS

    mixed distributed source
MF

    matched filter
m.g.f.

    moment generating function
MI

    mutual information
MIMO

    Multiple-Input Multiple-Output
MISO

    multiple-input single-output
MJSO

    maximum joint support cardinality
ML

    maximum likelihood
MMSE

    minimum mean-square error
MMV

    multiple measurement vectors
MOS

    model order selection
${M}$-PSK

     $M$-ary phase shift keying
${M}$-PSK

     $M$-ary asymmetric PSK
MP

     multi-period
MINLP

    mixed integer non-linear programming
$M$-QAM

     $M$-ary quadrature amplitude modulation
MRC

    maximal ratio combiner
MSO

    maximum sparsity order
M2M

    Machine-to-Machine
MUI

    multi-user interference
mMTC

    massive Machine Type Communications
mm-Wave

    millimeter-wave
MP

    mobile phone
MPE

    maximum permissible exposure
MAC

    media access control
NB

    narrowband
NBI

    narrowband interference
NLA

    nonlinear sparse approximation
NLOS

    Non-Line of Sight
NTIA

    National Telecommunications and Information Administration
NTP

    National Toxicology Program
NHS

    National Health Service
LOS

    Line of Sight
OC

    optimum combining
OC

    optimum combining
ODE

    operational distortion-energy
ODR

    operational distortion-rate
OFDM

    Orthogonal Frequency-Division Multiplexing
OMP

    orthogonal matching pursuit
OSMP

    orthogonal subspace matching pursuit
OQAM

    offset quadrature amplitude modulation
OQPSK

    offset QPSK
OFDMA

    Orthogonal Frequency-division Multiple Access
OPEX

    Operating Expenditures
OQPSK/PM

    OQPSK with phase modulation
PAM

    pulse amplitude modulation
PAR

    peak-to-average ratio
PDF

    probability density function
PDF

    probability density function
PDF

    probability distribution function
PDP

    power dispersion profile
PMF

    probability mass function
PMF

    probability mass function
PN

    pseudo-noise
PPM

    pulse position modulation
PRake

    Partial Rake
PSD

    power spectral density
PSEP

    pairwise synchronization error probability
PSK

    phase shift keying
PD

    power density
$8$-PSK

     $8$-phase shift keying
PPP

    Poisson point process
PCP

    Poisson cluster process
FSK

    Frequency Shift Keying
QAM

    Quadrature Amplitude Modulation
QPSK

    Quadrature Phase Shift Keying
OQPSK/PM

    OQPSK with phase modulator
RD

    raw data
RDL

    ”random data limit”
RIC

    restricted isometry constant
RICt

    restricted isometry constant threshold
RIP

    restricted isometry property
ROC

    receiver operating characteristic
RQ

    Raleigh quotient
RS

    Reed-Solomon
RSSC

    RS based source coding
r.v.

    random variable
R.V.

    random vector
RMS

    root mean square
RFR

    radiofrequency radiation
RIS

    Reconfigurable Intelligent Surface
RNA

    RiboNucleic Acid
RRM

    Radio Resource Management
RUE

    reference user equipments
RAT

    radio access technology
RB

    resource block
SA-Music

    subspace-augmented MUSIC with OSMP
SC

    small cell
SCBSES

    Source Compression Based Syndrome Encoding Scheme
SCM

    sample covariance matrix
SEP

    symbol error probability
SG

    sparse-land Gaussian model
SIMO

    single-input multiple-output
SINR

    signal-to-interference plus noise ratio
SIR

    signal-to-interference ratio
SISO

    Single-Input Single-Output
SMV

    single measurement vector
SNR

    signal-to-noise ratio
SP

    subspace pursuit
SS

    spread spectrum
SW

    sync word
SAR

    specific absorption rate
SSB

    synchronization signal block
SR

    shrink and realign
tUAV

    tethered Unmanned Aerial Vehicle
TBS

    terrestrial base station
uUAV

    untethered Unmanned Aerial Vehicle
PDF

    probability density functions
PL

    path-loss
TH

    time-hopping
ToA

    time-of-arrival
TR

    transmitted-reference
TW

    Tracy-Widom
TWDT

    TW Distribution Tail
TCM

    trellis coded modulation
TDD

    Time-Division Duplexing
TDMA

    Time Division Multiple Access
Tx

    average transmit
UAV

    Unmanned Aerial Vehicle
URIC

    upper restricted isometry constant
URICt

    upper restricted isometry constant threshold
UWB

    ultrawide band
UWB

    Ultrawide band
URLLC

    Ultra Reliable Low Latency Communications
UWRIC

    upper weak restricted isometry constant
UWRICt

    UWRIC threshold
UE

    User Equipment
UL

    UpLink
WiM

    weigh-in-motion
WLAN

    wireless local area network
WM

    Wishart matrix
WMAN

    wireless metropolitan area network
WPAN

    wireless personal area network
WRIC

    weak restricted isometry constant
WRICt

    weak restricted isometry constant thresholds
WRIP

    weak restricted isometry property
WSN

    wireless sensor network
WSS

    Wide-Sense Stationary
WHO

    World Health Organization
Wi-Fi

    Wireless Fidelity
SpaSoSEnc

    sparse source syndrome encoding
VLC

    Visible Light Communication
VPN

    Virtual Private Network
RF

    Radio Frequency
FSO

    Free Space Optics
IoST

    Internet of Space Things
GSM

    Global System for Mobile Communications
2G

    Second-generation cellular network
3G

    Third-generation cellular network
4G

    Fourth-generation cellular network
5G

    Fifth-generation cellular network
gNB

    next-generation Node-B Base Station
NR

    New Radio
UMTS

    Universal Mobile Telecommunications Service
LTE

    Long Term Evolution
QoS

    Quality of Service

#  Conditional Contact Angle Distribution in LEO Satellite-Relayed
Transmission

Ruibo Wang, Anna Talgat, Student Member, IEEE Mustafa A. Kishk, Member, IEEE
and Mohamed-Slim Alouini, Fellow, IEEE Ruibo Wang, Anna Talgat and Mohamed-
Slim Alouini are with King Abdullah University of Science and Technology
(KAUST), CEMSE division, Thuwal 23955-6900, Saudi Arabia. Mustafa A. Kishk is
with the Department of Electronic Engineering, National University of Ireland,
Maynooth, W23 F2H6, Ireland. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>slim.alouini@kaust.edu.sa).

###### Abstract

This letter characterizes the contact angle distribution based on the
condition that the relay low earth orbit (LEO) satellite is in the
communication range of both the ground transmitter and the ground receiver. As
one of the core distributions in stochastic geometry-based routing analysis,
the analytical expression of the Cumulative Distribution Function (CDF) of the
conditional contact angle is derived. Furthermore, the conditional contact
angle is applied to analyze the inaccessibility of common satellites between
the ground transmitter and receiver. Finally, with the help of the conditional
contact angle, coverage probability and achievable data rate in LEO satellite-
relayed transmission are studied.

###### Index Terms:

Conditional contact angle distribution, stochastic geometry, satellite-relayed
transmission, LEO satellite networks.

##  I Introduction

In recent years, the explosive growth of the number of LEO satellites has
opened the door for many opportunities and challenges for the development of
non-terrestrial networks [1], [2]. Because of the significant increase in the
information-carrying capacity of the satellite network, more ground
communications services can be transferred to space [3]. In the face of low
latency and long-distance communication requirements, LEO satellites play an
important role [4]. In a ground-satellite-ground transmission, the ground
transmitter needs to associate with a reliable satellite as a relay. Compared
with a high orbit satellite, one of the challenges a LEO satellite has to face
is its relatively small coverage [5]. In addition, the interference from
massive satellite constellations further decreases the communication region.
Therefore, in addition to providing stronger power [6], the associated relay
satellite should also locate within the reliable communication range of both
the ground transmitter and receiver. The distribution of the distance between
a strictly selected relay satellite and the ground transmitter is challenging
but also meaningful. In relay transmission and routing problems [7], this
distribution is the basis of many system parameter analyses [8].

Several related studies have appeared in recent years. They provide effective
mathematical methods, accurate models, and practical methods for deriving the
distance distribution [9, 4, 10]. Traditional deterministic network models
that extend from cellular networks (such as the spherical Voronoi model) are
not be suitable for large-scale dynamic network modeling. In addition to
limiting the user’s distance under single-hop communication, the spherical
Voronoi model is not accurate to study the distance distribution [11].
Compared to the above modeling methods, the stochastic geometry-based method
is undoubtedly more practical for irregular network modeling [12]. Among the
existing stochastic geometry models, binomial point process (BPP) is
relatively accurate for closed area networks with a fixed number of satellites
[13, 14]. Some studies have given different forms of distance distribution
between the ground receiver and the nearest satellite [6, 14, 10]. Since the
satellites are distributed on the sphere, using angle to express the contact
distance distribution is more concise. Therefore, contact angle distribution
is introduced to analyze the coverage probability and latency of satellite
networks [15, 16]. Based on the existing research, the contributions of this
letter are summarized as follows.

  * •

We derive an analytical expression of the contact angle distribution under the
condition that the satellite is within the transmitter and receiver’s
communication range and its accuracy is verified.

  * •

The influences of the number of satellites and the distance between the
transmitter and receiver on the conditional contact angle are studied.

  * •

Based on the conditional contact angle distribution, the satellite
inaccessibility in single relay routing is analyzed and extended to multiple
relays routing.

  * •

We explain how to obtain coverage probability and achievable data rate of
uplink from transmitter to relay satellite in routing by applying conditional
contact angle.

Figure 1: Satellite-relayed communication model.

##  II System Model

In this section, we build a ground-satellite-ground relay communication model.
$N_{\rm{Sat}}$ satellites are distributed on a spherical surface with radius
$R_{\rm{Sat}}$ and form a homogeneous BPP [14]. The transmitter and receiver
are located on the Earth with distance $d$. The radius of the Earth is denoted
as $R_{\oplus}=6371\rm{km}$. We start with a simple scenario where a single
satellite is selected as a relay, because the results obtained from a single
satellite relay routing can be easily extended to a multiple one. A concrete
example of the extension of satellite inaccessibility is provided in
subsection IV-B.

We consider a satellite that maximizes the minimum quality of service of
transmitter-satellite, and satellite-receiver links is suggested to be
selected. Whether the data rate, coverage probability, or latency are
negatively correlated with the communication distance, the quality of service
can be measured by the reciprocal of the communication distance. However, when
the relay satellites form a BPP, analysis of this strategy is intractable.
Therefore, we consider a slightly suboptimal but tractable selection strategy:
the transmitter is assumed to choose the closest satellite as a relay among
the satellites that can provide reliable communication for both the
transmitter and receiver [17].

######  Definition 1 (Dome Angle).

Connect the two points with the center of the Earth, respectively, and the
angle between the two connected lines is called the dome angle of two points.

Due to Earth blockage and maximum reliable communication distance, the
transmitter and receiver can only communicate with satellites within a certain
area. The certain area is a spherical cap which is the shaded area
($\mathcal{A}_{1}$ for transmitter, and $\mathcal{A}_{2}$ for receiver) in the
top half of Fig. 1. The maximum dome angles of any two points in the spherical
cap are denoted as $\theta_{m}^{(1)}$ and $\theta_{m}^{(2)}$, respectively,
which are also called the maximum dome angles of two spherical caps
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$. To ensure that $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ have intersecting region, the following equation should be
satisfied

$\theta_{m}^{(1)}+\theta_{m}^{(2)}>4\arcsin\left(\frac{d}{2R_{\oplus}}\right).$
(1)

##  III Conditional Contact Angle Distribution

In this section, we derive the analytical expression of the CDF of the
conditional contact angle distribution. The definition of the conditional
contact angle is given below.

######  Definition 2 (Conditional Contact Angle).

Among the satellites that can provide reliable communication for the receiver,
the dome angle between the transmitter and its nearest satellite is called the
conditional contact angle.

To derive the distribution of the conditional contact angle $\theta_{c}$, the
following steps are taken: (i) fixed the region $\mathcal{A}_{2}$ and dome
angle $\theta_{d}^{(2)}$, continuously increase the dome angle
$\theta_{d}^{(1)}$ of the spherical cap corresponding to the transmitter, (ii)
calculate the intersecting area of the two spherical caps, and (iii) calculate
the probability that there are no satellites in the intersection region.

To get the the intersecting area in step (ii), the irregular intersecting area
can be divided into two parts by the red line, shown at the bottom of Fig. 1.
These two parts correspond to the right part of the left spherical cap and the
left part of the right spherical cap. The following lemma can obtain the area
of both regions. The two dome angles involved in the lemma are marked at the
bottom of Fig. 1.

Figure 2: Decomposition diagrams of the geometry combined by a cone and
spherical cap.

######  Lemma 1.

For a spherical cap with dome angle $\theta_{d}$, intercept the portion of one
side of the cap, with dome angle $\theta_{o}$. The area of the cut part
$S\left(\theta_{d},\theta_{o}\right)$ is given by,

$\small\begin{split}S\left(\theta_{d},\theta_{o}\right)&=\int_{{R_{\rm{Sat}}\cos\frac{\theta_{d}}{2}}\tan\left(\frac{\theta_{d}}{2}-\theta_{o}\right)}^{R_{\rm{Sat}}\sin\left(\frac{\theta_{d}}{2}\right)}2{R_{\rm{Sat}}}\arcsin\left(\frac{\sqrt{{R_{\rm{Sat}}^{2}\sin\left(\frac{\theta_{d}}{2}\right)}^{2}-l^{2}}}{R_{\rm{Sat}}}\right)\mathrm{d}l.\end{split}$
(2)

###### Proof.

See appendix A. ∎

Based on lemma 1, the CDF of the conditional contact distance is given in the
following theorem.

######  Theorem 1.

Given that the maximum dome angle of the receiver’s spherical cap is
$\theta_{m}^{(2)}$, the approximate CDF of the conditional contact angle
$F_{\theta_{c}}\left(\theta\right)$ is given by,

$\begin{split}F_{\theta_{c}}\left(\theta\right)=1-\left(1-\frac{S\left(2\theta,\theta_{o}^{(1)}\left(\theta\right)\right)+S\left(\theta_{m}^{(2)},\theta_{o}^{(2)}\left(\theta\right)\right)}{4\pi
R_{\rm{Sat}}^{2}}\right)^{N_{\rm{Sat}}},\end{split}$ (3)

where $\theta_{o}^{(1)}\left(\theta\right)$ and
$\theta_{o}^{(2)}\left(\theta\right)$ are defined as,

$\begin{split}\theta_{o}^{(1)}\left(\theta\right)&=\theta-\frac{a(\theta)c-\sqrt{2a(\theta)^{2}-4a(\theta)b+2b^{2}+a(\theta)bc^{2}}}{a(\theta)-b},\\\
\theta_{o}^{(2)}\left(\theta\right)&=\frac{1}{2}\theta_{m}^{(2)}-\frac{-bc+\sqrt{2a(\theta)^{2}-4a(\theta)b+2b^{2}+a(\theta)bc^{2}}}{a(\theta)-b},\end{split}$
(4)

where

$\begin{split}a(\theta)=\cos\theta,\ b&=\cos\frac{\theta_{m}^{(2)}}{2},\
c=2\arcsin\left(\frac{d}{2R_{\oplus}}\right),\end{split}$ (5)

and the domain of the contact angle is,

$\begin{split}\max\bigg{\\{}0,2\arcsin&\left(\frac{d}{2R_{\oplus}}\right)-\frac{1}{2}\theta_{m}^{(2)}\bigg{\\}}\leq\theta_{c}\\\
&\leq\min\bigg{\\{}\frac{1}{2}\theta_{m}^{(1)},2\arcsin\left(\frac{d}{2R_{\oplus}}\right)+\frac{1}{2}\theta_{m}^{(2)}\bigg{\\}}.\end{split}$
(6)

###### Proof.

See appendix B ∎

##  IV Potential Applications

###  IV-A Satellite Inaccessibility in Single Relay Routing

One of the key issues in real-time and ultra long-distance routing is how far
the distance between the transmitter and receiver can be or how many
satellites are required to ensure that the routing will not be interrupted due
to no available satellites. From theorem 1, an intuitive corollary about LEO
relay outage probability can be obtained. The definition of LEO relay outage
probability is given as follows.

######  Definition 3 (LEO Relay Outage Probability).

The LEO relay outage probability is defined as the probability that there are
no available satellites located in the communication range of both ground
transmitter and receiver.

######  Corollary 1.

Given that the Euclidean distance between the transmitter and receiver is $d$,
the LEO relay outage probability $P_{e}^{S}\left(d\right)$ is given by,

$\begin{split}P_{e}^{S}\left(d\right)=1-F_{\theta_{c}}\bigg{(}\min\bigg{\\{}\frac{1}{2}\theta_{m}^{(1)},2\arcsin\left(\frac{d}{2R_{\oplus}}\right)+\frac{1}{2}\theta_{m}^{(2)}\bigg{\\}}\bigg{)},\end{split}$
(7)

where the CDF of the conditional contact angle
$F_{\theta_{c}}\left(\theta\right)$ is defined in (3).

###  IV-B Satellite Inaccessibility in Multiple Relays Routing

With the above corollary, designers can keep the outage probability below a
threshold by adjusting $d$ and $N_{\rm{Sat}}$. However, only studying the
satellite inaccessibility in single satellite relay routing is quite limited.
The transmitter requires multiple relay satellites to send the message to the
receiver in most cases. Therefore, we consider the satellite inaccessibility
in a routing consisting of multiple satellite relays with a bent pipe
architecture [18].

In such an architecture, $N_{h}-1$ terrestrial relays are required when
$N_{h}$ relay satellites are selected. In order to reduce latency and power
consumption, we assume that every two adjacent terrestrial relays have the
same dome angle. Therefore, the multiple relays outage probability
$P_{e}^{M}\left(N_{h},d\right)$, which is the probability that there are no
satellites available in any hop, is given in the following corollary. Since
$P_{e}^{M}\left(N_{h},d\right)$ can be derived from $P_{e}^{S}\left(d\right)$
by simple geometric relations, the proof is omit here.

######  Corollary 2.

Given that the Euclidean distance between the transmitter and receiver is $d$,
the multiple relays outage probability $P_{e}^{M}\left(d\right)$ is expressed
as,

$\small\begin{split}P_{e}^{M}\left(N_{h},d\right)=1-\left(1-P_{e}^{S}\left(2R_{\oplus}\sin\left(\frac{1}{N_{h}}\arcsin\left(\frac{d}{2R_{\oplus}}\right)\right)\right)\right)^{N_{h}},\end{split}$
(8)

where $P_{e}^{S}\left(d\right)$ is defined in (7) and $N_{h}$ is the number of
selected relay satellites.

The $N_{h}$ needs to be carefully designed according to the multiple relays
outage probability. Generally speaking, a long-hop strategy (with a small
$N_{h}$) leads to a larger outage probability, while a short-hop strategy
(with a large $N_{h}$) leads to a larger latency [16].

###  IV-C Uplink Coverage and Rate Analysis with Suboptimal Relay Selection

Coverage probability and achievable maximum data rate are highlighted metrics
of satellite network analysis. In the case of satellites providing coverage to
ground users, the authors in [13] and [14] provide analytical expressions of
downlink coverage probability and achievable data rate, respectively. The
above expressions can be modified into uplink coverage probability and
achievable data rate from transmitter to satellite in the routing scenario.
The modification process is straightforward, and the only task is replacing
the contact distance with the conditional contact distance. Notice that the
domain of contact distance also needs to be replaced. Conditional contact
distance is defined as the distance from the transmitter to the closest
satellite that can provide reliable communication for both the transmitter and
receiver. The relationship between the conditional contact distance $d_{c}$
and conditional contact angle $\theta_{c}$ is,

$2R_{\oplus}R_{\rm{Sat}}\cos\theta_{c}=R_{\oplus}^{2}+R_{\rm{Sat}}^{2}-d_{c}^{2}.$
(9)

##  V Numerical Results

In this section, numerical results of the CDF of the conditional contact angle
are provided. As shown in Fig. 3 and Fig. 4, the analytical results perfectly
match the simulation results, which proves the accuracy of theorem 3. The
height of satellites is fixed at 550km and $R_{\rm{Sat}}=6921\rm{km}$. The
maximum dome angle of the receiver is $\theta_{m}^{(2)}=\frac{\pi}{4}$.

Figure 3: CDF of the conditional contact angle under different number of
satellites and the maximum dome angle of the transmitter.

In Fig. 3, the distributions of the conditional contact angle under the
different number of satellites $N_{\rm{Sat}}$ and the maximum dome angle of
the transmitter $\theta_{m}^{(1)}$ are studied. The distance between the
ground transmitter and receiver is fixed as $d=3000\rm{km}$. Reducing
$N_{\rm{Sat}}$ causes the CDF curve to move down. Changing $\theta_{m}^{(1)}$
does not have significant effects on the CDF.

Figure 4: CDF of the conditional contact angle under different distance
between the transmitter and receiver.

Fig. 4 describes the influence of the distance between the transmitter and
receiver on the distribution of the conditional contact angle. The number of
satellites is fixed as $N_{\rm{Sat}}=3000$, and the maximum dome angle of the
transmitter is $\theta_{m}^{(1)}=\frac{\pi}{4}$. As the distance between the
transmitter and receiver decreases, the CDF curve is shifted from right to
left, with little change in shape. Finally, the conditional contact angle
distribution converges to the unconditional contact angle distribution.

Figure 5: LEO relay outage probability under different distance between the
transmitter and receiver and constellation altitudes.

In Fig. 5, the LEO relay outage probability, $P_{e}^{S}\left(d\right)$, is
studied under different values for the distance between the transmitter and
receiver for companies OneWeb and SpaceX with altitudes $h=1200$km and
$h=550$km, respectively. The outage probability gets larger as we increase the
distance between the transmitter and receiver. We observe that for the larger
altitude of OneWeb, we have less LEO relay outage probability compared with
SpaceX for each distance.

##  VI Conclusion

In this letter, we derived the approximate conditional contact angle
distribution based on the stochastic geometry framework. Three Potential
applications of conditional contact angles are further given. Finally, we
provide the numerical results about the influence of the number of satellites,
the distance between the transmitter and receiver on the conditional contact
angle. For the lower altitude of the satellite constellation, it is better to
keep the distance less than 3000km to ensure that there are available
satellites between the transmitter and receiver for a single-hop transmission.

##  Appendix A Proof of Lemma 1

As shown in Fig. 2, the middle part is a geometry composed of a spherical cap
and a cone. The right (dark blue) part of the spherical cap in the geometry
corresponds to the dome angle $\theta_{o}$. According to the description in
the lemma, the area of the dark blue part is the desired
$S\left(\theta_{d},\theta_{o}\right)$. Divide the dome angle $\theta_{o}$ into
infinitesimal $\Delta\theta_{o}$. The dark blue area can be divided into
numerous arcs. $S\left(\theta_{d},\theta_{o}\right)$ is calculated by
multiplying the sum of these arcs by $\Delta\theta_{o}$. Project the spherical
cap on the circle in the upper right corner of Fig. 2. The relationship
between the arc length $l_{\rm{Arc}}$ and the chord length obtained by
projection is given by,

$l_{\rm{Arc}}=2{R_{\rm{Sat}}}\arcsin\frac{l_{\perp}}{R_{\rm{Sat}}},$ (10)

where $l_{\perp}$ is half of the chord length. Since there is a one-to-one
mapping between chord length and arc length,
$S\left(\theta_{d},\theta_{o}\right)$ can be obtained by integrating the
region on the right side of the circle. As shown in the upper right part of
Fig. 2, we choose to integrate in the direction perpendicular to the chord.
Easy to know that

$R_{\rm{Cone}}^{2}=l_{\varparallel}^{2}+l_{\perp}^{2}=R_{\rm{Sat}}\sin\left(\frac{\theta_{d}}{2}\right).$
(11)

where $R_{\rm{Cone}}$ is the radius of the projected circle. Set the center of
the circle as the origin. The upper bound of the integral is $R_{\rm{Cone}}$,
and the lower bound is given by,

$\tan\left(\frac{\theta_{d}}{2}-\theta_{o}\right)=\frac{l_{\varparallel}^{\rm{low}}}{R_{\rm{Sat}}\cos\frac{\theta_{d}}{2}}.$
(12)

The integral of $S\left(\theta_{d},\theta_{o}\right)$ is calculated as
follows,

$\begin{split}S\left(\theta_{d},\theta_{o}\right)&=\int_{l_{\varparallel}^{\rm{low}}}^{R_{\rm{Cone}}}l_{\rm{Arc}}\left(l\right)\mathrm{d}l\\\
&=\int_{l_{\varparallel}^{\rm{low}}}^{R_{\rm{Cone}}}2{R_{\rm{Sat}}}\arcsin\left(\frac{l_{\perp}}{R_{\rm{Sat}}}\right)\mathrm{d}l\\\
&=\int_{l_{\varparallel}^{\rm{low}}}^{R_{\rm{Cone}}}2{R_{\rm{Sat}}}\arcsin\left(\frac{\sqrt{{R_{\rm{Cone}}}^{2}-l^{2}}}{R_{\rm{Sat}}}\right)\mathrm{d}l.\end{split}$
(13)

Substitute (11) and (12) into (13), the final result of the lemma is obtained.
In addition, $l_{\varparallel}^{\rm{low}}$ might be less than 0. It’s
guaranteed that the lower bound of the integral is always less than the upper
bound because,

${\cos\frac{\theta_{d}}{2}}\tan\left(\frac{\theta_{d}}{2}-\theta_{o}\right)<{\cos\frac{\theta_{d}}{2}}\tan\left(\frac{\theta_{d}}{2}\right)=\sin\left(\frac{\theta_{d}}{2}\right).$
(14)

##  Appendix B Proof of Theorem 1

By definition, the CDF of the conditional contact angle can be expressed as,

$\begin{split}F_{\theta_{c}}\left(\theta\right)&=1-{\mathbb{P}}\left[\theta_{c}>\theta\right]=1-\mathbb{P}\left[{\mathcal{N}\left(\mathcal{A}_{o}\right)=0}\right]\\\
&\overset{(a)}{=}1-\left(1-\frac{S\left(\frac{1}{2}\theta_{d}^{(1)},\theta_{o}^{(1)}\right)+S\left(\frac{1}{2}\theta_{d}^{(2)},\theta_{o}^{(2)}\right)}{4\pi
r^{2}}\right)^{N_{\rm{Sat}}},\end{split}$ (15)

where $\mathcal{N}\left(\mathcal{A}_{o}\right)$ counts the number of the
satellites in the overlap region $\mathcal{A}_{o}$. For a homogeneous BPP, the
probability of the satellite locates in $\mathcal{A}_{o}$ is equal to the
ratio of the area of $\mathcal{A}_{o}$ to the total surface area of the
sphere. As shown in the bottom of Fig. 1, $\mathcal{A}_{o}$ is divided into
two parts, with dome angles $\theta_{o}^{(1)}$ and $\theta_{o}^{(2)}$. In step
(a), the area of $\mathcal{A}_{o}$ is equal to the sum of
$S\left(\frac{1}{2}\theta_{d}^{(1)},\theta_{o}^{(1)}\right)$ and
$S\left(\frac{1}{2}\theta_{d}^{(2)},\theta_{o}^{(2)}\right)$, which are
defined in (2).

When dome angles $\theta_{o}^{(1)}$, $\theta_{o}^{(2)}$ and the distance
between the transmitter and receiver $d$ are given, $\theta_{o}^{(1)}$ and
$\theta_{o}^{(2)}$ can be represented by them. From the relationship between
these dome angles,

$\begin{split}\frac{\theta_{d}^{(1)}}{2}+\frac{\theta_{d}^{(2)}}{2}-\theta_{o}^{(1)}-\theta_{o}^{(2)}=2\arcsin\left(\frac{d}{2R_{\oplus}}\right).\end{split}$
(16)

Since two right triangles share the red cutting line, the following equation
can be obtained,

$\begin{split}\frac{R_{\rm{Sat}}\cos\left(\frac{1}{2}\theta_{d}^{(1)}\right)}{\cos\left(\frac{1}{2}\theta_{d}^{(1)}-\theta_{o}^{(1)}\right)}=\frac{R_{\rm{Sat}}\cos\left(\frac{1}{2}\theta_{d}^{(2)}\right)}{\cos\left(\frac{1}{2}\theta_{d}^{(2)}-\theta_{o}^{(2)}\right)}\end{split}$
(17)

Combine (16) and (17), $\theta_{o}^{(1)}$ and $\theta_{o}^{(2)}$ can be
derived theoretically. However, it isn’t easy to obtain an analytical solution
for the two dome angles for this system of trigonometric equations. So we
approximate the cosine function by a second-order Taylor expansion,

$\cos\left(\frac{\theta_{d}}{2}-\theta_{o}\right)\approx
1-\left(\frac{1}{2}\theta_{d}-\theta_{o}\right)^{2},$ (18)

Substitute (18) into (17), we get

$\frac{\cos\left(\frac{1}{2}\theta_{d}^{(1)}\right)}{1-\left(\frac{1}{2}\theta_{d}^{(1)}-\theta_{o}^{(1)}\right)^{2}}=\frac{\cos\left(\frac{1}{2}\theta_{d}^{(2)}\right)}{1-\left(\frac{1}{2}\theta_{d}^{(2)}-\theta_{o}^{(2)}\right)^{2}}.$
(19)

Combine (16) and (19), the approximate solution is expressed as,

$\begin{split}\theta_{o}^{(1)}\left(\theta_{d}^{(1)}\right)&\approx\frac{1}{2}\theta_{d}^{(1)}-\frac{ac-\sqrt{2a^{2}-4ab+2b^{2}+abc^{2}}}{a-b},\\\
\theta_{o}^{(2)}\left(\theta_{d}^{(2)}\right)&\approx\frac{1}{2}\theta_{d}^{(2)}-\frac{-bc+\sqrt{2a^{2}-4ab+2b^{2}+abc^{2}}}{a-b},\end{split}$
(20)

where $a$, $b$ and $c$ are defined as follows,

$\begin{split}a=\cos\left(\frac{1}{2}\theta_{d}^{(1)}\right),\
b=\cos\left(\frac{1}{2}\theta_{d}^{(2)}\right),\
c=2\arcsin\left(\frac{d}{2R_{\oplus}}\right).\end{split}$ (21)

In this case, $\theta_{d}^{(2)}$ is fixed as a constant $\theta_{m}^{(2)}$,
while $\theta_{d}^{(1)}$ is twice the conditional contact angle $\theta_{c}$.
In formula (15), substitute $\theta_{m}^{(2)}$ into $\theta_{d}^{(2)}$ and
$2\theta_{c}$ into $\theta_{d}^{(1)}$ to get the final result.

The remaining problem is to determine the range of $\theta_{c}$. The relay
satellite must be within the reliable communication range of the transmitter,
so $0\leq\theta_{c}\leq\theta_{m}^{(1)}$ is required. To ensure that the two
caps intersect,

$2\theta_{c}+\theta_{m}^{(2)}\geq 4\arcsin\left(\frac{d}{2R_{\oplus}}\right).$
(22)

However, when $\theta_{c}$ is too large, the transmitter’s cap may contain the
receiver’s cap, and increasing $\theta_{c}$ further is meaningless. Therefore,
we have,

$2\theta_{c}\leq 4\arcsin\left(\frac{d}{2R_{\oplus}}\right)+\theta_{m}^{(2)}.$
(23)

In addition, to ensure that the relay satellite is always within the
receiver’s line of sight, the following inequalities always need to be
satisfied,

$\theta_{m}^{(i)}<2\arcsin{\frac{d}{2R_{\oplus}}},\ \ i=1,2.$ (24)

## References

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  *[CDF]: Cumulative Distribution Function


SNSN-323-63

CKM studies from $b$ physics at hadron machines

Patrick Koppenburg

Nikhef, Science Park 105
1098 XG, Amsterdam, The Netherlands

> In absence of direct signs of new physics at the LHC, flavour physics
> provides an ideal laboratory to look for deviations from the Standard Model
> and explore an energy regime beyond the LHC reach. Here, new results in
> $C\\!P$ violation and rare decays are presented.

> PRESENTED AT
>
>
>
>
> CKM workshop 2014, Vienna

##  1 Introduction

The first run of the Large Hadron Collider with 7 and 8
$\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions has allowed to discover the
Higgs boson [1, *Chatrchyan:2012ufa], but not to find any hint of the
existence of new particles. Neither supersymmetry nor any other sign of new
physics has popped out of the LHC. This situation may change during Run II
with higher centre-of-mass energies of 13 $\mathrm{\,Te\kern-1.00006ptV}$. In
the meantime it is worth investigating what the data at the lower energies
involved in the decays of $b$ hadrons can tell us about new physics.

Studies of $C\\!P$ violation in heavy flavour decays are both sensitive to the
above-mentioned high mass scales and to potential new phases beyond the phase
of the CKM matrix. Also of particular interest are rare decays that are
strongly suppressed in the Standard Model (SM), where new physics amplitudes
could be sizable.

##  2 $C\\!P$ violation measurements

Owing to the legacy of the $B$ factories [3], we have entered the era of
precision tests of the Cabibbo-Kobayashi-Maskawa (CKM) paradigm [4]. Precise
measurements of the angles of the unitarity triangle(s) are needed to search
for new sources of $C\\!P$ violation beyond the single phase of the CKM
matrix.

The LHC is often considered as a ${B}^{0}_{s}$ meson factory, owing to its
large cross-section and the unprecedented capabilities of the LHC experiments
to precisely resolve its oscillations. This opens the door to precision
measurements of the $C\\!P$-violating phase
$\varphi_{s}^{{c}{\overline{c}}{s}}$, which is equal to
$-2\beta_{s}\equiv-2\arg\left(-{V_{{t}{s}}}{V_{{t}{b}}^{\ast}}/{V_{{c}{s}}}{V_{{c}{b}}^{\ast}}\right)=-0.0363\pm
0.0013$ in the SM, neglecting sub-leading penguin contributions. It was
measured at the LHC using the flavour eigenstate decays
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{\mu^{+}\mu^{-}}$ and
$\phi\\!\rightarrow{{K}^{+}}{{K}^{-}}$ [5, *CMS-PAS-BPH-13-012, *LHCb-
PAPER-2013-002] and
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [8]. Recently LHCb used the decay
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ for the first time in a polarisation-dependent way
[9]. Combined with
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$, LHCb obtains
$\varphi_{s}^{{c}{\overline{c}}{s}}=-0.010\pm 0.039$ rad. The constraints on
$\varphi_{s}^{{c}{\overline{c}}{s}}$ and the decay width difference
$\Delta\Gamma_{s}$ are shown in Fig. 1 (left). The same quantity was also
measured with a fully hadronic final state using the decay
${{B}^{0}_{s}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{-}_{s}}$ with
${{D}^{\pm}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{\pm}}$, yielding
$0.02\pm 0.17\pm 0.02$ [10]. The effective tagging power $\epsilon{\cal
D}^{2}$ in excess of 5% for this channel is unprecedented at a hadron
collider.

With the precision on some CKM phases reaching the degree level, the effects
of suppressed penguin topologies cannot be neglected any more [11,
*Fleischer:1999zi, *Fleischer:1999sj, *Faller:2008zc, *DeBruyn:2010hh, 16].
Cabibbo-suppressed decay modes, where these topologies are relatively more
prominent can be used to constrain such effects. This programme has started
with studies of the decays
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\rm\scriptscriptstyle S}}$ [17],
${{B}^{0}_{s}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ [18] and more
recently with ${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ [19]. The measurement of $\sin
2\beta^{\text{eff}}$ in the latter mode allows to constrain the shift to
$\varphi_{s}^{{c}{\overline{c}}{s}}$ due to penguin topologies to the range
$[-0.018,0.021]$ radians at 68% CL. Considering the present uncertainty of
$\pm 0.039$ radians, such a shift needs to be to constrained further.

An interesting test of the Standard Model is provided by the measurement of
the mixing phase $\varphi_{s}^{{s}{\overline{s}}{s}}$ with a purely penguin-
induced mode as ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$. In this case the
measured value is $-0.17\pm 0.15\pm 0.03$ [20], which is compatible with the
SM expectation.

Similarly, the decays ${B}\\!\rightarrow hh$ with $h={\pi},{K}$ are also
sensitive to penguin topologies (as well as trees) and are sensitive to the
CKM phases $\gamma$ and $\beta_{s}$. LHCb for the first time measured time-
dependent $C\\!P$-violating observables in ${B}^{0}_{s}$ decays using the
decay ${{B}^{0}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ [21]. Using methods
outlined in Refs. [22, *Fleischer:2007hj, 16], a combination of this and other
results from ${B}\\!\rightarrow hh$ modes allows to determine
$-2\beta_{s}=-0.12{\>}^{+{\>}0.14}_{-{\>}0.16}$ $\rm\,rad$ using as input the
angle $\gamma$ from tree decays (see below), or
$\gamma=(63.5{\>}^{+{\>}7.2}_{-{\>}6.7})^{\circ}$ constraining $-2\beta_{s}$
to the SM value [24]. These values are in principle sensitive to the amount of
U-spin breaking that is allowed in this decay and are given here for a maximum
allowed breaking of 50%.

This value of $\gamma$ can be compared to that obtained from tree-dominated
${B}\\!\rightarrow{D}{K}$ decays, where the $C\\!P$-violating phase appears in
the interference of the ${b}\\!\rightarrow{c}$ and ${b}\\!\rightarrow{u}$
topologies. It is the least precisely known angle of the unitarity matrix, and
its determination from tree decays is considered free from contributions
beyond the Standard Model and unaffected by hadronic uncertainties. Yet its
precise determination is important to test the consistency of the CKM
paradigm, and to allow comparisons with determinations from modes dominated by
penguin topologies.

Figure 1: (left) Constraints on $\Delta\Gamma_{s}$ and
$\varphi_{s}^{{c}{\overline{c}}{s}}$ from various experiments. (right) Fitted
values of $x$ and $y$ for ${{B}^{-}}$ and ${{B}^{+}}$ in
${{B}^{\pm}}\\!\rightarrow{D}{{K}^{\pm}}$ with
${D}\\!\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ decays [25].

The most precise determination of $\gamma$ from a single decay mode is
achieved with ${{{B}^{+}}}\\!\rightarrow D{{K}^{+}}$ followed by
$D\\!\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ with
$h={\pi},{K}$ [25]. Here the interference of the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decay to
${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ is exploited to measure
$C\\!P$ asymmetries [26]. The method needs external input in the form of a
measurement of the strong phase over the Dalitz plane of the $D$ decay, coming
from CLEO-c data [27]. The determined $C\\!P$-violating parameters are shown
in Fig. 1 (right), and the value of $\gamma$ is
$(62{\>}^{+{\>}15}_{-{\>}14})^{\circ}$. The same decay mode is also used in a
model-dependent measurement [28] using an amplitude model.

An experimentally very different way of determining $\gamma$ is provided by
the decay ${{B}^{0}_{s}}\\!\rightarrow{{D}^{+}_{s}}^{\pm}{K}^{\mp}$ [29,
*Aleksan:1991nh, *Fleischer:2003yb, 32]. In this case the phase is measured in
a time-dependent tagged $C\\!P$-violation analysis. Using a dataset
corresponding to 1 $\mbox{\,fb}^{-1}$, LHCb determines
$\gamma=(115{\>}^{+{\>}28}_{-{\>}43})^{\circ}$, which is not competitive with
other methods but will provide important cross-checks with more data.

Figure 2: (left) LHCb combination of ${B}\\!\rightarrow D{K}$ decays measuring
the CKM phase $\gamma$ [33]. (right) Experimental constraints on
$A_{\text{sl}}^{d}$ and $A_{\text{sl}}^{s}$.

The $\gamma$ measurements of Refs. [25, 32, 34, *LHCb-PAPER-2012-055, *LHCb-
PAPER-2013-068, *LHCb-PAPER-2014-028] are then combined in an LHCb combination
prepared for the CKM conference [33]. Using only ${B}\\!\rightarrow D{K}$
decay modes one finds $\gamma=(73{\>}^{+{\>}9}_{-{\>}10})^{\circ}$, which is
more precise than the corresponding combination of measurements from the $B$
factories [3]. The likelihood profile is shown in Fig. 2 (left).

The same-sign dimuon asymmetry measured by the D0 collaboration [38] and
interpreted as a combination of the semileptonic asymmetries
$A_{\text{sl}}^{d}$ and $A_{\text{sl}}^{s}$ in ${B}^{0}$ and ${B}^{0}_{s}$
decays, respectively, remains puzzling. The measured values differ from the SM
expectation by $3\sigma$. So far LHCb has not been able to confirm or disprove
this. The measurement from LHCb follows a different approach, looking at the
$C\\!P$ asymmetry between partially reconstructed
${B}\\!\rightarrow{{D}^{+}_{s}}\mu\nu$ decays, where the flavour of the $D$
identifies that of the $B$. The measured value of $A_{\text{sl}}^{s}$ [39] and
the newly reported $A_{\text{sl}}^{d}$ [40] are both consistent with the SM
and the D0 value. The world average including measurements from the $B$
factories and D0 is not more conclusive. See Fig. 2 (right).

Large $C\\!P$ violation has also been found in charmless $b$-hadron decays
like ${{{B}^{+}}}\\!\rightarrow h^{+}h^{-}h^{+}$ [41] ($h={\pi},{K}$) and
${{{B}^{+}}}\\!\rightarrow{p}{\overline{p}}h^{+}$ [42]. In the former case
$C\\!P$ asymmetries, integrated over the phase-space, ranging from $-12\%$
(${{B}^{\pm}}\\!\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}$) to $+6\%$
(${{B}^{\pm}}\\!\rightarrow{{\pi}^{\pm}}{{K}^{+}}{{K}^{-}}$) are measured.
Particularly striking features of these decays are very the large asymmetries
in small regions of the phase-space not related to any known resonance, which
are of opposite sign for ${{B}^{\pm}}\\!\rightarrow h^{\pm}{{K}^{+}}{{K}^{-}}$
and ${{B}^{\pm}}\\!\rightarrow h^{\pm}{{\pi}^{+}}{{\pi}^{-}}$ decays. These
could be a sign of long-distance
${{\pi}^{+}}{{\pi}^{-}}\leftrightarrow{{K}^{+}}{{K}^{-}}$ rescattering.

Finally, another important field is the study of $C\\!P$ violation in beauty
baryons. The ${\mathchar 28931\relax}^{0}_{b}$ hadronisation fraction was
measured to be surprisingly large at the LHC in the forward region [43],
almost half of that of ${B}^{0}$ mesons. These baryons can be used for
measurements of $C\\!P$ violation with better precision than ${B}^{0}_{s}$
mesons. Searches have been performed by LHCb with the decays ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}{{\pi}^{-}}$ [44], ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{K}^{0}}{p}{{\pi}^{-}}$ [45], and by CDF
with ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}h^{-}$ [46]. It is to
be noted that to date no evidence of $C\\!P$ violation in any decay of a
baryon has ever been reported.

##  3 ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ and
${{B}^{0}}\\!\rightarrow{\mu^{+}\mu^{-}}$

The rare decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ proceeds in the
SM by a box-type annihilation diagram involving the $W$ boson and the $t$
quark. It is furthermore helicity-suppressed. The most recent standard model
prediction of its branching fraction is $(3.66\pm 0.23)\times 10^{-9}$, [47],
where the uncertainty is dominated about equally by CKM matrix elements and
the ${B}^{0}_{s}$ decay constant. In this calculation the branching fraction
is evaluated as an average over all decay times, see Refs. [48,
*DeBruyn:2012wk] for more details.

The decay ${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}\mu^{-}}$ has been searched for
over three decades, with most recent results from the Tevatron [50,
*Abazov:2013wjb] and the LHC [52, *LHCb-PAPER-2011-025, *Chatrchyan:2011kr,
*LHCb-PAPER-2012-007, *Chatrchyan:2012rga, *ATLAS-CONF-2013-076]. The first
evidence was reported in Summer 2012 by LHCb [58] using 2 $\mbox{\,fb}^{-1}$
from the 2011 and half of the 2012 data. A year later LHCb and CMS updated
their results to the full Run I dataset [59, *Chatrchyan:2013bka].

The data-sets in these two publications were then combined in a joint fit to
the data of both experiments [61].

Figure 3: (left) Example mass distribution from the combined LHCb and CMS
dataset. (right) Confidence level contours of the
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ measured branching fractions. The
SM expectation is shown for comparison. Figures from Ref. [61].

This is the first time such a joint fit is done at the LHC. The selections
(and thus the data-sets) are left unchanged, but the fit models are aligned to
be based on the same assumptions. Notably the treatment of the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow p{\mu^{+}}\nu$ background was different
between the two experiments, and the effect of the lifetime acceptance on the
admixture of heavy and light ${B}^{0}_{s}$ states had been neglected in the
CMS publications. As for the individual publications, the fits are performed
in each bin of boosted-decision-tree output. An example fit is shown in Fig. 3
(left). The result of the combination is a clear first observation of the
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decay and a $3\sigma$ excess
over the background for the ${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ decay.
In the latter case, the significance obtained from Wilk’s theorem is
$3.2\sigma$, while a likelihood scan following the Feldman-Cousins method [62]
yields $3.0\sigma$. This is a deviation from the SM expectation by
$2.2\sigma$. The measured ${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ branching fractions are
compared to the SM expectation in Fig. 3 (right). More data from Run II and
III will tell if the excess of ${{B}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ is
a statistical fluctuation or the indication of new physics.

##  4 Rare electroweak decays

The family of decays ${b}\\!\rightarrow{s}{\ell^{+}}{\ell^{-}}$ is a
laboratory of new physics on its own. In particular the exclusive decay
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\ell^{+}}{\ell^{-}}$ ($\ell=e,\mu$)
provides a very rich set of observables with different sensitivities to new
physics and for which the theoretical predictions are available and affected
by varying levels of hadronic uncertainties. In the case of some ratios of
observables most of these uncertainties cancel, thus providing a clean test of
the SM [63, *Kruger:2005ep, *Altmannshofer:2008dz, *Egede:2008uy,
*Bobeth:2008ij, 68].

The differential decay width with respect to the dilepton mass squared
$q^{2}$, the well-known forward-backward asymmetry $A_{\text{FB}}$, and the
longitudinal polarisation fraction $F_{\text{L}}$ of the ${K}^{*}$ resonance
have been measured by many experiments [69, *Wei:2009zv, *Aaltonen:2011ja,
*Chatrchyan:2013cda, *ATLAS-CONF-2013-038, 74] with no significant sign of
deviations from the SM expectation.

In a second analysis of the already published [74] 2011 data, LHCb published
another set of angular observables [75] suggested by Ref. [68]. In particular
a $3.7\sigma$ local deviation of the $P_{5}^{\prime}$ observable from the
Standard Model expectation was observed in one bin of $q^{2}$, shown in Fig. 4
(right).

Figure 4: (left) LHCb measurement of $P_{5}^{\prime}$ [75], (right) mass peak
of ${{{B}^{+}}}\\!\rightarrow{e^{+}e^{-}}{{K}^{+}}$ for event triggered by one
of the signal electrons at hardware level [76].

This measurement triggered a lot of interest in the theory community, with
interpretation articles being very quickly submitted to journals. See Refs.
[77, *Descotes-Genon:2013wba, *Altmannshofer:2013foa, *Datta:2013kja,
*Mahmoudi:2014mja] for a small subset. It is not clear if this discrepancy is
an experimental fluctuation, is due to under-estimated form factor
uncertainties (See Ref. [82]), or the manifestation of a heavy $Z^{\prime}$
boson, among many other suggested explanations. The contribution of
$c$$\overline{c}$ resonances is also being questioned [83] after the LHCb
observation of ${{{B}^{+}}}\\!\rightarrow\psi(4160){{K}^{+}}$ with
$\psi(4160)\\!\rightarrow{\mu^{+}\mu^{-}}$ [84], where the $\psi(4160)$ and
its interference with the non-resonant component accounts for 20% of the rate
for dimuon masses above 3770 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Such
a large contribution was not expected.

Given the hint of abnormal angular distributions, LHCb tried to look for other
deviations in several asymmetry measurements. The $C\\!P$ asymmetry in
${{B}^{0}}\\!\rightarrow{{K}^{*}}{\mu^{+}\mu^{-}}$ and
${{B}^{\pm}}\\!\rightarrow{{K}^{\pm}}{\mu^{+}\mu^{-}}$ turns out to be
compatible with zero as expected [85], as does the isospin asymmetry between
${{B}^{0}}\\!\rightarrow K^{(*)0}{\mu^{+}\mu^{-}}$ and
${{B}^{+}}\\!\rightarrow K^{(*)+}{\mu^{+}\mu^{-}}$ [86]. The lepton
universality $R_{K}=\frac{{\cal
B}({{{B}^{+}}}\\!\rightarrow{{K}^{+}}{\mu^{+}\mu^{-}})}{{\cal
B}({{{B}^{+}}}\\!\rightarrow{{K}^{+}}{e^{+}e^{-}})}$ is measured to be
$0.745{\>}^{+{\>}0.090}_{-{\>}0.074}\pm 0.036$ [76] in the
$1<q^{2}<6\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ range, which indicates a
$2.6\sigma$ tension with unity. Unlike at the $B$ factories, where electrons
and muons contribute equally to $b\\!\rightarrow s\ell^{+}\ell^{-}$ modes, at
hadron colliders only muonic modes are normally used. Yet, LHCb has
demonstrated its ability to use rare decays to electrons [87]. The use of
electrons is difficult due to the lower trigger efficiency and the poorer mass
resolution due to bremsstrahlung. Figure 4 shows the ${{K}^{+}}{e^{+}e^{-}}$
mass distribution for candidates with
$1<q^{2}<6\>{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which is safely far
from the radiative tail of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\\!\rightarrow{e^{+}e^{-}}$ decay. This result is being interpreted as
an indication of a new vector current ($Z^{\prime}$?) that would couple more
strongly to muons and interfere destructively with the SM vector current [88,
*Ghosh:2014awa].

##  5 Conclusions

The LHC is the new $b$-hadron factory and will be dominating flavour physics
until the start of Belle II, and beyond in many decay modes. Atlas, CMS and
LHCb have presented interesting new results in rare decays, that set strong
constraints on models beyond that SM and exhibit some discrepancies with the
SM predictions. It is not yet clear whether these are fluctuations, poorly
understood form factors, or new physics. The upcoming Run II, with its higher
centre-of-mass energy translating into higher $b$$\overline{b}$ cross-
sections, may tells us.

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