A Review of $C\\!P$ Violation Measurements in Charm at LHCb

Federico<EMAIL_ADDRESS>
CERN (European Organization for Nuclear Research)
Espl. des Particules 1, 1211 Meyrin, Switzerland

The LHCb experiment has been able to collect the largest sample ever produced
of charm-hadron decays, performing a number of measurements of observables
related to $C\\!P$ violation in the charm sector. In this document, the most
recent results from LHCb on the search of direct $C\\!P$ violation in
$D^{0}\to K_{s}^{0}K_{s}^{0}$, $D_{(s)}^{+}\to h^{+}\pi^{0}$ and
$D_{(s)}^{+}\to h^{+}\eta$ decays are summarised, in addition to the most
precise measurement of time-dependent $C\\!P$ asymmetry in $D^{0}\to
h^{+}h^{-}$ decays and the first observation of mass difference between
neutral charm-meson eigenstates.

Published in Symmetry 13 (2021) 8, 1482

##  1 Introduction

For a generic hadron $M$ and its $C\\!P$ conjugate $\overline{M}$, the
following decay amplitudes in the final state $f$ and its $C\\!P$ conjugate
$\overline{f}$ are defined

$A_{f}=\langle
f|\mathcal{H}|M\rangle,\qquad\overline{A}_{\overline{f}}=\langle\overline{f}|\mathcal{H}|\overline{M}\rangle,$
(1)

where $\mathcal{H}$ is the effective decay Hamiltonian. The $C\\!P$ asymmetry
of the decay $M\to f$ is therefore defined as

$\mathcal{A}_{C\\!P}(M\to f)\equiv\frac{\Gamma(M\to
f)-\Gamma(\overline{M}\to\overline{f})}{\Gamma(M\to
f)+\Gamma(\overline{M}\to\overline{f})}=\frac{1-|\overline{A}_{\overline{f}}/A_{f}|^{2}}{1+|\overline{A}_{\overline{f}}/A_{f}|^{2}}.$
(2)

Direct $C\\!P$ violation, or $C\\!P$ violation in the decay, takes place when
$|\overline{A}_{\overline{f}}|^{2}\neq|A_{f}|^{2}$. For neutral mesons such as
$D^{0}$, mixing must be taken into account. The eigenstates $|D_{H}\rangle$
and $|D_{L}\rangle$ of the effective Hamiltonian, which have defined masses
($m_{H}$ and $m_{L}$) and decay widths ($\Gamma_{H}$ and $\Gamma_{L}$), can be
written as a superposition of flavour eigenstates

$\displaystyle|D_{H}\rangle$
$\displaystyle=p|D^{0}\rangle-q|\overline{D}^{0}\rangle,$ (3)
$\displaystyle|D_{L}\rangle$
$\displaystyle=p|D^{0}\rangle+q|\overline{D}^{0}\rangle,$ (4)

where $p$ and $q$ are complex with $|p|^{2}+|q|^{2}=1$. The mixing is
described by the parameters

$\displaystyle x$ $\displaystyle\equiv\frac{m_{H}-m_{L}}{\Gamma},$ (5)
$\displaystyle y$ $\displaystyle\equiv\frac{\Gamma_{H}-\Gamma_{L}}{2\Gamma},$
(6)

with $\Gamma=(\Gamma_{H}+\Gamma_{L})/2$. If $|q/p|\neq 1$, $C\\!P$ violation
in the mixing occurs, and the probability of the $D^{0}$ meson to oscillate
after a time $t$ into $\overline{D}^{0}$ is different from that of the
$C\\!P$-conjugate process. If $f=\overline{f}$ and the phase

$\phi_{f}\equiv\operatorname{arg}\left(\frac{q\overline{A}_{f}}{pA_{f}}\right)$
(7)

is different from 0, $C\\!P$ violation manifests in the interference between
mixing and decay. The time-dependent $C\\!P$ asymmetry between the probability
of an initially pure $D^{0}$ meson and an initially pure $\overline{D}^{0}$
meson to decay into the final state $f$ after a time $t$

$\mathcal{A}_{C\\!P}(D^{0}(t)\to f)\equiv\frac{\Gamma(D^{0}(t)\to
f)-\Gamma(\overline{D}^{0}(t)\to f)}{\Gamma(D^{0}(t)\to
f)+\Gamma(\overline{D}^{0}(t)\to f)}$ (8)

is different from 0 if $C\\!P$ violation occurs either in the mixing or in the
interference between mixing and decay. In the $D^{0}$ meson system, the time-
dependent $C\\!P$ asymmetry can be written as

$\mathcal{A}_{C\\!P}(D^{0}(t)\to f)\approx
a^{d}_{f}+\frac{4|A_{f}|^{2}|\overline{A}_{f}|^{2}}{(|A_{f}|^{2}+|\overline{A}_{f}|^{2})^{2}}\Delta
Y_{f}\frac{t}{\tau_{D^{0}}},$ (9)

where $a^{d}_{f}$ is equal to the direct $C\\!P$ asymmetry, $\tau_{D^{0}}$ is
the $D^{0}$ lifetime and $\Delta Y_{f}$, the observable describing the amount
of time-dependent $C\\!P$ violation, is defined according to the “theoretical”
parametrisation introduced in Refs. [1, 2]. The coefficient in front of
$\Delta Y_{f}$ differs from unity by approximately $\mathcal{O}(10^{-6})$,
resulting in $\Delta Y_{f}$ to be equal to the slope of
$\mathcal{A}_{C\\!P}(D^{0}(t)\to f)$.

The combination of Cabibbo–Kobayashi–Maskawa matrix elements responsible for
$C\\!P$ violation in charm decays is
$\operatorname{Im}(V_{cb}V_{ub}^{*}/V_{cs}V_{us}^{*})\simeq-6\times 10^{-4}$,
resulting in asymmetries typically of the order of $10^{-4}$–10-3 in the
Standard Model (SM). Due to the presence of low-energy strong-interaction
effects, theoretical predictions are difficult to compute reliably [3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Some calculations are
performed using dynamical methods of QCD, such as Light-Cone Sum Rules [8, 10,
19]. Other studies rely on perturbative parameterisations of the branching
fractions and the $C\\!P$ asymmetries as a function of the topological
amplitudes contributing to the dominant and subleading amplitudes in
$SU(3)_{F}$ or $U$-spin breaking, fitting them to the measured values [3, 4,
11, 12, 13, 14, 15, 16, 18, 20].

Almost all the measurements of $C\\!P$ violation in charm hadrons have been
performed at the $B$-factories Belle and BaBar and at the hadron-collider
experiments LHCb and CDF. The production cross-section of charm in
proton–proton ($pp$) (or proton–antiproton) collisions is much larger than
that at the $B$-factories, for example $\sigma(pp\to c\overline{c}X)=(2369\pm
12)~{}\upmu\mathrm{b}$ at $\sqrt{s}=13~{}\mathrm{TeV}$ in the LHCb acceptance
[21], whereas $\sigma(e^{+}e^{-}\to\gamma^{*}\to
c\overline{c})=1.3~{}\mathrm{nb}$ at $\sqrt{s}=10.58~{}\mathrm{GeV}$. Despite
the large hadronic background, not present in the low-multiplicity environment
of the $e^{+}e^{-}$ machines, LHCb was therefore able to collect the largest
sample ever produced of charm-hadron decays, performing a number of
measurements of observables related to $C\\!P$ violation in the charm sector,
such as: searches for direct $C\\!P$ asymmetry in $D_{(s)}^{+}$ decays [22,
23, 24]; time-dependent $C\\!P$ violation in singly Cabibbo suppressed (SCS)
$D^{0}$ decays [25, 26, 27, 28]; direct $C\\!P$ asymmetries in SCS $D^{0}$
decays [29, 30, 31]; $C\\!P$ violation in mixing using $D^{0}\to K^{+}\pi^{-}$
decays [32, 33]; $C\\!P$ violation in multi-body $D^{0}$ decays [34, 35, 36,
37, 38, 39]; $C\\!P$ violation in $D^{0}$ decays into final states with
$K_{s}^{0}$ [40, 41]; and $C\\!P$ violation in baryon decays [42, 43]. The
measurement of $\Delta\mathcal{A}_{C\\!P}\equiv\mathcal{A}_{C\\!P}(D^{0}\to
K^{+}K^{-})-\mathcal{A}_{C\\!P}(D^{0}\to\pi^{+}\pi^{-})$ with the full data
sample collected by the LHCb detector led to the first observation of $C\\!P$
violation in the decay of charm hadrons [44]. The result, that challenges
perturbative and sum-rule estimates of $\Delta\mathcal{A}_{C\\!P}$ [45, 19],
prompted a renewed interest of the theory community in the field, sparking a
discussion whether the signal is consistent with the Standard Model or if it
is a sign of new physics [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]. While
according to some studies the discrepancy is due to an enhancement of
rescattering beyond expectation [12, 13, 14, 16, 18, 47, 49], for other
authors the reason of the discrepancy must be found in new interactions beyond
the SM [46]. Possible contributions of new physics to
$\Delta\mathcal{A}_{C\\!P}$, such as models involving $Z^{\prime}$, have been
studied [46, 51, 53].

Additional investigations are therefore crucial to clarify the picture and
solve open theoretical puzzles. More precise measurements of branching
fractions and $C\\!P$ asymmetries will allow the sum rules relating the
$C\\!P$ asymmetries of different decay channels to be tested, and a better
precision on the predictions obtained with the topological amplitudes to be
reached [16, 18, 48, 50].

In the following Section, the most recent results obtained by the LHCb
collaboration in the field of $C\\!P$ violation in charm will be summarised,
namely the measurements of: $C\\!P$ asymmetry in $D^{0}\to K_{s}^{0}K_{s}^{0}$
decays; $C\\!P$ asymmetry in $D_{(s)}^{+}\to h^{+}\pi^{0}$ and $D_{(s)}^{+}\to
h^{+}\eta$ decays, where $h^{+}$ denotes a $\pi^{+}$ or $K^{+}$ meson; $\Delta
Y$ in $D^{0}\to K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ decays; and mixing
parameters by using $D^{0}\to K_{s}^{0}\pi^{+}\pi^{-}$ decays. In Section 3,
the conclusions and the prospects for future measurements of $C\\!P$ violation
in charm at LHCb are given.

##  2 Recent results from LHCb

The LHCb detector [57, 58] is a single-arm forward spectrometer that covers
the pseudorapidity range between 2 and 5, whose main goal is the study of
particles containing $b$ or $c$ quarks. The tracking system consists of a
silicon-strip vertex detector (VELO), a large-area silicon-strip detector, a
dipole magnet and three stations of silicon-strip detectors and straw drift
tubes. The momentum of charged particles is measured with a relative
uncertainty varying from 0.5% to 1.0%. The impact parameter (IP) of a track,
defined as its minimum distance to a primary $pp$ collision vertex (PV), is
measured with a resolution of $(15\pm 29/p_{\mathrm{T}})~{}\upmu\mathrm{m}$,
where $p_{\mathrm{T}}$ is the momentum transverse to the beam, expressed in
$\mathrm{GeV}/c$. Particle identification of charged hadrons is performed
using information from two ring-imaging Cherenkov detectors. Electrons,
photons and hadrons are identified by a calorimeter system made of
scintillating-pad and preshower detectors, an electromagnetic and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers. The online event selection is
performed by a trigger, which consists of a hardware stage followed by a
software stage. The hardware stage is based on information from the
calorimeter and muon systems, whereas the software stage applies a full event
reconstruction.

In the measurements of $C\\!P$ asymmetries in the decays of neutral $D^{0}$
mesons to $C\\!P$ eigenstates, the $D^{0}$ flavour, when produced, is
determined by the charge of the accompanying “tagging” pion in the strong
$D^{*+}\to D^{0}\pi^{+}$ (Inclusion of the charge-conjugate process is implied
throughout this document unless explicitly specified.) decay promptly produced
in the $pp$ collisions, or by the charge of the muon in the semileptonic
$b$-hadron secondary decays. Due to the higher production cross section of
charm with respect to that of beauty, the data samples selected with prompt
tag usually convey a signal yield which is a factor $\sim$5 larger than that
with semileptonic tag.

The “raw” asymmetry obtained by counting the positively- and negatively-tagged
signal candidates is not equal to the physical asymmetry
$\mathcal{A}_{C\\!P}$, but it includes also experimental effects due to
production asymmetry $\mathcal{A}_{\mathrm{P}}$ and detection asymmetry
$\mathcal{A}_{\mathrm{D}}$. The production asymmetry is due to different
production rates of the charm meson and its anti-meson, whereas different
interaction cross-sections of particle and antiparticles with the detector,
which is not perfectly symmetric, result in the detection asymmetry. These
nuisance asymmetries, averaged over phase space for selected events, are
usually $\mathcal{O}(10^{-2})$ or less [59, 60, 61, 62], allowing the measured
raw asymmetry $\mathcal{A}_{\mathrm{raw}}$ to be written as a sum of the
physical and the experimental asymmetries. For example, the raw asymmetry of
promptly-tagged $D^{0}$ mesons that decay in a $C\\!P$ eigenstate $f$ can be
approximated as

$\mathcal{A}_{\mathrm{raw}}(D^{0}\to f)\approx\mathcal{A}_{C\\!P}(D^{0}\to
f)+\mathcal{A}_{\mathrm{P}}(D^{*+})+\mathcal{A}_{\mathrm{D}}(\pi^{+}),$ (10)

where $\mathcal{A}_{\mathrm{P}}(D^{*+})$ is the production asymmetry of the
$D^{*+}$ meson and $\mathcal{A}_{\mathrm{D}}(\pi^{+})$ is the detection
asymmetry of the tagging pion. The approximation is valid up to corrections of
$\mathcal{O}(10^{-6})$. Calibration samples which share the same production
and detection asymmetries of the signal sample are typically used to cancel
the nuisance asymmetries and obtain the $C\\!P$ asymmetry.

###  2.1 Measurement of $C\\!P$ asymmetry in $D^{0}\to K_{s}^{0}K_{s}^{0}$
decays

In the $D^{0}\to K_{s}^{0}K_{s}^{0}$ decay, the only contributing amplitudes
proceed via tree-level exchange and loop-suppressed diagrams which are of
similar size and vanish in the flavour-SU(3) limit. Their interference could
result in a $C\\!P$ asymmetry up to the percent level in the SM [63]. The
CLEO, Belle and LHCb (with data collected during the LHC Run 1) collaborations
have measured in the past the $C\\!P$ asymmetry $\mathcal{A}_{C\\!P}(D^{0}\to
K_{s}^{0}K_{s}^{0})$ [64, 65, 66], with the best precision reached by Belle,
at the level of 1.5%. The update performed by LHCb with data collected during
the LHC Run 2, corresponding to an integrated luminosity of
$6~{}\mathrm{fb}^{-1}$, is reported here.

From an experimental point of view, the relatively large lifetime of
$K_{s}^{0}$ meson is challenging, because the majority of $K_{s}^{0}$ mesons
decay after the LHCb Vertex Locator and their decay products can be
reconstructed only in the tracking stations further downstream along the beam
axis. The first software trigger stage is designed to accept the $K_{s}^{0}$
mesons whose pions are reconstructed also in the Vertex Locator (referred to
as “long” and indicated by “L” in the following), resulting in a worse trigger
efficiency for $K_{s}^{0}$ whose pions are reconstructed only in the
downstream part of the tracking system (referred to as “D”). Depending on how
the decay products of the two $K_{s}^{0}$ mesons are reconstructed, the data
sample is split in three categories that are analysed separately: two long
(LL), one long and one downstream (LD) and two downstream (DD) $K_{s}^{0}$
mesons. Once the two $K_{s}^{0}$ mesons are combined in a $D^{0}$ candidate,
its flavour is determined by looking at the charge of the accompanying pion in
the reconstructed strong decay $D^{*+}\to D^{0}\pi^{+}$. The data sample is
further split in two categories, according to whether the $D^{*+}$ candidate
is compatible or incompatible with having originated from the PV. The PV
compatibility classification is not performed for the DD sample, because of
its low statistics and signal purity. A multivariate classifier sensitive to
the signal-to-background ratio is used to select high signal purity $D^{*+}$
candidates. Finally, data collected in 2015 and 2016 are analysed separately
from those collected in 2017–2018, because of differences in trigger
requirements between these two periods.

In order to measure the raw asymmetry of the signal, an unbinned maximum-
likelihood fit is performed to the joint distribution of the invariant masses
of the $K_{s}^{0}$ mesons and the difference $\Delta m\equiv
m(K_{s}^{0}K_{s}^{0}\pi^{+})-m(K_{s}^{0}K_{s}^{0})$, simultaneously to
candidates of both flavours. The total probability density function is
parameterised by the sum of the signal component, peaking in the three
observables, and seven additional components, which describe different kinds
of background. Fit projections on $\Delta m$ distribution are reported in
Figure 1, which shows how the mass resolution and the signal purity change
between the different LL, LD and DD categories and between PV-compatible and
PV-incompatible categories.

A data sample of reconstructed $D^{*+}\to D^{0}(\to K^{+}K^{-})\pi^{+}$
decays, whose $C\\!P$ asymmetry is known to a precision much higher than that
of the signal decay [31], is used as a calibration sample to correct for
$D^{*+}$ production asymmetry and $\pi^{+}$ detection asymmetry. As
experimental asymmetries are a function of pion and $D^{*+}$ kinematics, the
kinematics of the calibration samples are weighted to match those of the
signal by using a multivariate classifier. Various contributions to the
systematic uncertainty are evaluated, namely related to weighting procedure,
difference between signal and calibration modes and fit model.

All the results obtained in each subsample are statistically compatible with
each other and their combination gives:

$\mathcal{A}_{C\\!P}(D^{0}\to K_{s}^{0}K_{s}^{0})=(-3.1\pm 1.2\pm 0.4\pm
0.2)\%,$

where the first uncertainty is statistical, the second systematic and the
third is related to the knowledge of $\mathcal{A}_{C\\!P}(D^{0}\to
K^{+}K^{-})$ [67]. This measurement, the most precise of this quantity to
date, is in agreement with the previous determinations [64, 65, 66] and is
compatible with no $C\\!P$ violation at the level of 2.4 standard deviations.

Figure 1: Distributions and fit projections of the $\Delta m$ distribution for
representative candidate categories (2017–2018 data sample).

###  2.2 Search for $C\\!P$ violation in $D_{(s)}^{+}\to h^{+}\pi^{0}$ and
$D_{(s)}^{+}\to h^{+}\eta$ decays

The study of $C\\!P$ asymmetry in the two-body $D_{(s)}^{+}\to h^{+}\pi^{0}$
and $D_{(s)}^{+}\to h^{+}\eta$ decays provides interesting tests of the SM. In
particular, SCS $D_{s}^{+}\to K^{+}\pi^{0}$, $D^{+}\to\pi^{+}\eta$ and
$D_{s}^{+}\to K^{+}\eta$ decays receive contributions from two different weak
phases allowing $C\\!P$ violation at tree-level, expected to be of the order
of $10^{-3}$–$10^{-4}$ according to the SM [8]. In the SM, the $C\\!P$
asymmetry of the SCS $D^{+}\to\pi^{+}\pi^{0}$ mode is expected to be zero as a
result of isospin constraints [12, 20, 17, 53]. It is possible to define the
isospin sum rule

$R=\frac{\mathcal{A}_{C\\!P}(D^{0}\to\pi^{+}\pi^{-})}{1+\frac{\tau_{D^{0}}}{\mathcal{B}_{+-}}\left(\frac{\mathcal{B}_{00}}{\tau_{D^{0}}}+\frac{2}{3}\frac{\mathcal{B}_{+0}}{\tau_{D^{+}}}\right)}+\frac{\mathcal{A}_{C\\!P}(D^{0}\to\pi^{0}\pi^{0})}{1+\frac{\tau_{D^{0}}}{\mathcal{B}_{00}}\left(\frac{\mathcal{B}_{+-}}{\tau_{D^{0}}}+\frac{2}{3}\frac{\mathcal{B}_{+0}}{\tau_{D^{+}}}\right)}-\frac{\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\pi^{0})}{1+\frac{3}{2}\frac{\tau_{D^{+}}}{\mathcal{B}_{+0}}\left(\frac{\mathcal{B}_{00}}{\tau_{D^{0}}}+\frac{\mathcal{B}_{+-}}{\tau_{D^{0}}}\right)}\;,$
(11)

where $\tau_{D}^{+}$ and $\tau_{D}^{0}$ represent the $D^{+}$ and $D^{0}$
lifetimes and $\mathcal{B}_{+-}$, $\mathcal{B}_{00}$ and $\mathcal{B}_{+0}$
are the branching fractions of the $D^{0}\to\pi^{+}\pi^{-}$,
$D^{0}\to\pi^{0}\pi^{0}$ and $D^{+}\to\pi^{+}\pi^{0}$ decays, respectively. A
non-zero value of $\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\pi^{0})$, associated
with a verification that $R$ is consistent with 0, would represent a signal of
new physics.

The Belle collaboration has reported the measurement of
$\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\pi^{0})=(2.31\pm 1.24\pm 0.23)\%$,
corresponding to $R=(-2.2+2.7)\times 10^{-3}$ [68], and, more recently,
precise measurements of $\mathcal{A}_{C\\!P}(D_{s}^{+}\to K^{+}\pi^{0})$,
$\mathcal{A}_{C\\!P}(D_{s}^{+}\to\pi^{+}\eta)$ and
$\mathcal{A}_{C\\!P}(D_{s}^{+}\to K^{+}\eta)$ with uncertainties ranging from
0.4% to 4.5% [69]. The measurements of the $C\\!P$ asymmetries in the
$D_{(s)}^{+}\to h^{+}\pi^{0}$ and $D_{(s)}^{+}\to h^{+}\eta$ decays (except
for $D_{s}^{+}\to\pi^{+}\pi^{0}$, that proceeds via an annihilation topology
decay and is therefore highly suppressed) performed by LHCb with an integrated
luminosity of $9~{}\mathrm{fb}^{-1}$ and $6~{}\mathrm{fb}^{-1}$, respectively,
are summarised here.

The $\pi^{0}$ and $\eta$ mesons are reconstructed in the $e^{+}e^{-}\gamma$
final state, which allows the secondary decay vertex to be reconstructed,
suppressing background originating from the $pp$ collisions. This final state
receives contributions from the $\pi^{0}\to\gamma\gamma$ and
$\eta\to\gamma\gamma$ decays, where one photon is converted to an $e^{+}e^{-}$
pair after interacting with the detector material, and the suppressed Dalitz
$\pi^{0}\to e^{+}e^{-}\gamma$ and $\eta\to e^{+}e^{-}\gamma$ decays. During
the signal reconstruction process, a bremsstrahlung-recovery algorithm
associates additional deposits from soft photons to those produced by the
electrons in the electromagnetic calorimeter. Decays with a total of either
zero or one bremsstrahlung photon per $e^{+}e^{-}$ pair are selected. After
the full selection is applied, about 86% of signal decays are found to be due
to photon conversions, and the remaining part to the Dalitz decays.

The raw asymmetries are measured by means of two-dimensional extended
simultaneous unbinned maximum-likelihood fits to the invariant mass
$m(e^{+}e^{-}\gamma)$ and the invariant mass difference $m(h^{+}h^{0})\equiv
m(h^{+}e^{+}e^{-}\gamma)-m(e^{+}e^{-}\gamma)+M(h^{0})$, where $M(h^{0})$ is
the known $\pi^{0}$ and $\eta$ mass [70]. Two-dimensional probability density
functions are used to model different contributions, namely signal decays,
pure combinatorial background, combinatorial background due to real $\pi^{0}$
meson combined with an unrelated track, misidentification background and
partially reconstructed low-mass background. The fits are performed
simultaneously in categories depending on the running period, the number of
recovered bremsstrahlung photons, the $h^{+}$ type (pion or kaon) and the
candidate charge. The results of the fits are shown in Figures 2 and 3.

Figure 2: Distribution of (left) $m(h^{+}\eta)$ and (right)
$m(e^{+}e^{-}\gamma)$ for (top) $D_{(s)}^{+}\to\pi^{+}\eta$ and (bottom)
$D_{(s)}^{+}\to K^{+}\eta$ candidates, summed over all categories of the
simultaneous fit, with projections of the fit result overlaid. $D^{+}\to
h^{+}\eta$ contribution is shown in dashed red, $D^{+}_{s}\to h^{+}\eta$ in
solid grey, pure combinatorial decays in dashed black and partially
reconstructed background in dotted magenta. The misidentification background
is too small to be seen.

Figure 3: Distribution of (left) $m(h^{+}\pi^{0})$ and (right)
$m(e^{+}e^{-}\gamma)$ for (top) $D_{(s)}^{+}\to\pi^{+}\pi^{0}$ and (bottom)
$D_{(s)}^{+}\to K^{+}\pi^{0}$ candidates, summed over all categories of the
simultaneous fit, with projections of the fit result overlaid. $D^{+}\to
h^{+}\pi^{0}$ contribution is shown in dashed red, $D^{+}_{s}\to h^{+}\pi^{0}$
in solid grey, pure combinatorial decays in dashed black and real-$\pi^{0}$
combinatorial background in dotted green. The misidentification background is
too small to be seen.

In order to cancel the production and detection asymmetries and obtain the
$C\\!P$ asymmetries, a calibration sample of $D_{(s)}^{+}\to K_{s}^{0}h^{+}$
decays is used, after weighting its kinematic distributions to match those of
the signal candidates. The values of $\mathcal{A}_{C\\!P}(D_{(s)}^{+}\to
K_{s}^{0}h^{+})$ are known with sub-percent precision [24], significantly
higher than the precision of this measurement. Systematic uncertainties
related to the choice of the fit model, presence of $D_{(s)}^{+}$ candidates
produced in secondary decays, different selection between signal and
calibration sample, kinematic weighting and uncertainties of $C\\!P$
asymmetries of the calibration decays are evaluated.

The results are

$\displaystyle\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\pi^{0})$ $\displaystyle=(-$
$\displaystyle 1.3$ $\displaystyle\pm 0.9$ $\displaystyle\pm 0.6$
$\displaystyle)\%,$ $\displaystyle\mathcal{A}_{C\\!P}(D^{+}\to K^{+}\pi^{0})$
$\displaystyle=(-$ $\displaystyle 3.2$ $\displaystyle\pm 4.7$
$\displaystyle\pm 2.1$ $\displaystyle)\%,$
$\displaystyle\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\eta)$ $\displaystyle=(-$
$\displaystyle 0.2$ $\displaystyle\pm 0.8$ $\displaystyle\pm 0.4$
$\displaystyle)\%,$ $\displaystyle\mathcal{A}_{C\\!P}(D^{+}\to K^{+}\eta)$
$\displaystyle=(-$ $\displaystyle 6$ $\displaystyle\pm 10$ $\displaystyle\pm
4$ $\displaystyle)\%,$ $\displaystyle\mathcal{A}_{C\\!P}(D^{+}_{s}\to
K^{+}\pi^{0})$ $\displaystyle=(-$ $\displaystyle 0.8$ $\displaystyle\pm 3.9$
$\displaystyle\pm 1.2$ $\displaystyle)\%,$
$\displaystyle\mathcal{A}_{C\\!P}(D^{+}_{s}\to\pi^{+}\eta)$ $\displaystyle=($
$\displaystyle 0.8$ $\displaystyle\pm 0.7$ $\displaystyle\pm 0.5$
$\displaystyle)\%,$ $\displaystyle\mathcal{A}_{C\\!P}(D^{+}_{s}\to K^{+}\eta)$
$\displaystyle=($ $\displaystyle 0.9$ $\displaystyle\pm 3.7$ $\displaystyle\pm
1.1$ $\displaystyle)\%,$

where the first uncertainty is statistical and the second systematic [71]. All
of the results are consistent with previous determinations [68, 69] and with
no $C\\!P$ violation, and the first five constitute the most precise
measurements to date of those observables. An average of
$\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\pi^{0})$ between this result, the
measurement by Belle [68] and a measurement by CLEO [72] gives
$\mathcal{A}_{C\\!P}(D^{+}\to\pi^{+}\pi^{0})=(0.43\pm 0.79)\%$, corresponding
to a value of ${R=(0.1\pm 2.4)\times 10^{-3}}$, consistent with 0.

###  2.3 Search for time-dependent $C\\!P$ violation in $D^{0}\to K^{+}K^{-}$
and $D^{0}\to\pi^{+}\pi^{-}$ decays

The magnitude of the $\Delta Y_{f}$ parameter is expected to be about $2\times
10^{-5}$ [56, 73]. The $\Delta Y_{f}$ parameter is approximately equal, up to
1% relative corrections (For a detailed description of the theoretical
parametrisation of mixing in $D^{0}$ decays, see Refs. [56, 74, 75]), to the
negative of the parameter $A_{\Gamma}^{f}$ that has been measured by the BaBar
[76], CDF [77], Belle [78] and LHCb [25, 26, 28] collaborations, and the world
average, assuming no difference is present between the $K^{+}K^{-}$ and
$\pi^{+}\pi^{-}$ final states, is $\Delta Y=(3.1\pm 2.0)\times 10^{-4}$ [79].
The measurement of $\Delta Y_{f}$ performed by LHCb with data collected during
the LHC Run 2, corresponding to an integrated luminosity of
$6~{}\mathrm{fb}^{-1}$, using $D^{*+}\to D^{0}\pi^{+}$ decays originated from
primary $pp$ interactions, is reported here.

Three decay modes are reconstructed: $D^{0}\to K^{+}K^{-}$,
$D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to K^{-}\pi^{+}$, where the $D^{0}$ meson
is associated with the accompanying pion in the $D^{*+}\to D^{0}\pi^{+}$ decay
to determine its flavour when produced. The $D^{0}\to K^{-}\pi^{+}$ mode has
the same topology and kinematic distributions very similar to those of the
signal channels and its $C\\!P$ asymmetry is known to be smaller than the
current experimental uncertainty. For these reasons, the $D^{0}\to
K^{-}\pi^{+}$ mode is used as a control sample to develop and validate the
analysis. The data sample is divided in 21 intervals of $D^{0}$ decay times in
the range of (0.45–8) $\tau_{D^{0}}$, where $\tau_{D^{0}}$ is the nominal
$D^{0}$ lifetime [70]. In each decay time interval the signal yield is
obtained by means of background subtraction in the $m(D^{0}\pi^{+})$
distribution, where the signal window is defined as [2009.2, 2011.3]
MeV/$c^{2}$ and the background candidates are taken from the lateral window
[2015, 2018] MeV/$c^{2}$. The weight assigned to the background candidates is
determined with a binned maximum-likelihood fit to the $m(D^{0}\pi^{+})$
distribution. The fit relies on an empirical model to describe the signal and
the combinatorial background contributions. The $m(D^{0}\pi^{+})$
distributions and the fit results are shown in Figure 4. After the full
selection and background subtraction, the number of candidates in the signal
region are 519, 58 and 18 millions for the $D^{0}\to K^{-}\pi^{+}$, $D^{0}\to
K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ decay channels, respectively. The
candidates are split in subsamples according to year of data taking and magnet
polarities. The analysis is performed on each subsample and the results are
combined at the end.

Figure 4: Distribution of $m(D^{0}\pi^{+})$ for (left) $D^{0}\to
K^{-}\pi^{+}$, (center) $D^{0}\to K^{+}K^{-}$ and (right)
$D^{0}\to\pi^{+}\pi^{-}$ candidates. The signal window and the lateral window
used to remove the combinatorial background (grey filled area) are delimited
by the vertical dashed lines. Fit projections are overlaid.

In principle, the measurement of $\Delta Y_{f}$ is insensitive to time-
independent asymmetries such as the production and detection asymmetries, that
depend only on the kinematics of the $D^{*+}$ meson and the tagging pion.
However, an indirect time dependence of the nuisance asymmetries exists
because of the correlation between the kinematic variables and the $D^{0}$
decay time introduced by the selection requirements. The nuisance asymmetries
are therefore removed by equalising the kinematics of tagging $\pi^{+}$ and
$\pi^{-}$ and of $D^{0}$ and $\overline{D}^{0}$ candidates, by weighting their
kinematic distributions to their average. Figure 5 shows how the $D^{0}$
transverse momentum is correlated with the decay time and the effect of the
weighting procedure to the time-dependent raw asymmetry. A side effect of the
kinematic weighting is that, because of the correlation between the decay time
and momentum of the $D^{0}$ meson, a possible time-dependent asymmetry due to
a non-zero value of $\Delta Y_{f}$ would be partially cancelled by the
kinematic weighting, inducing a dilution in the measurement. By introducing
different artificial values of $\Delta Y_{K^{-}\pi^{+}}$ in the $D^{0}\to
K^{-}\pi^{+}$ sample, up to values as large as 100 times the statistical
uncertainty of the final measurement, the dilution is found to have a linear
effect on the measured value of $\Delta Y_{K^{-}\pi^{+}}$ equal to $(96.9\pm
0.1)\%$ of the introduced value. The results of all decay channels are
therefore corrected to account for this dilution factor.

Figure 5: (Left) Normalised distributions of the $D^{0}$ transverse momentum,
in different colours for each decay-time interval. Decay time increases from
blue to yellow colour. (Right) Linear fit to the time-dependent asymmetry
(red) before and (black) after the kinematic weighting. The plots correspond
to the $D^{0}\to K^{-}\pi^{+}$ candidates recorded in 2016 with the magnet
polarity pointing upwards.

The presence of $D^{*+}$ mesons originating from $B$ mesons in the selected
sample can potentially induce a bias in the measured time-dependent asymmetry,
due to a difference in production asymmetry between $D^{*+}$ mesons
originating from secondary decays and $D^{*+}$ mesons produced in $pp$
collisions. In addition, the measurement of the decay time is performed with
respect to the PV, resulting in a biased decay time towards larger values for
secondary $D^{*+}$ mesons. Although the secondary decays are suppressed by a
requirement on the IP of the $D^{0}$ meson, some contamination is still
present in the sample. The size and asymmetry of this background is therefore
assessed by means of a binned maximum-likelihood fit to the bidimensional
distribution of IP and decay time of the $D^{0}\to K^{-}\pi^{+}$ candidates,
and the effect of secondary decays on the final measurement is corrected for.
Systematic uncertainties are assessed related to background removal, mass
difference between $D^{*+}$ and $D^{*-}$, correction due to secondary decays,
presence of misidentified background peaking in $m(D^{0}\pi^{+})$ distribution
and kinematic weighting.

The slope of the time-dependent asymmetry of the control sample is measured to
be $(-0.4\pm 0.5\pm 0.2)\times 10^{-4}$, compatible with 0 as expected. The
time-dependent asymmetries of the $D^{0}\to K^{+}K^{-}$ and
$D^{0}\to\pi^{+}\pi^{-}$ channels, after the full selection, weighting and
corrections, are displayed in Figure 6, with linear fits superimposed. The
resulting slopes are

$\displaystyle\Delta Y_{K^{+}K^{-}}$ $\displaystyle=(-2.3\pm 1.5\pm 0.3)\times
10^{-4},$ $\displaystyle\Delta Y_{\pi^{+}\pi^{-}}$ $\displaystyle=(-4.0\pm
2.8\pm 0.4)\times 10^{-4},$

where the first uncertainties are statistical and the second are systematic,
compatible with each other and with previous determinations. Assuming no
final-state dependency, and taking account of the correlation between the
systematic uncertainties, the combination of the two values gives

$\Delta Y=(-2.7\pm 1.3\pm 0.3)\times 10^{-4},$

consistent with zero within two standard deviations [80]. The combination with
previous LHCb measurements [25, 26, 28] leads to

$\displaystyle\Delta Y_{K^{+}K^{-}}$ $\displaystyle=(-0.3\pm 1.3\pm 0.3)\times
10^{-4},$ $\displaystyle\Delta Y_{\pi^{+}\pi^{-}}$ $\displaystyle=(-3.6\pm
2.4\pm 0.4)\times 10^{-4},$ $\displaystyle\Delta Y$ $\displaystyle=(-1.0\pm
1.1\pm 0.3)\times 10^{-4}.$

This result, consistent with no time-dependent $C\\!P$ violation, is the most
precise measurement of $C\\!P$ violation in charm decays and improves by
nearly a factor of two the precision of the previous world average of $\Delta
Y$ [79].

Figure 6: Asymmetry as a function of $D^{0}$ decay time, for (left) $D^{0}\to
K^{+}K^{-}$ and (right) ${D^{0}\to\pi^{+}\pi^{-}}$ samples, with linear fit
superimposed.

###  2.4 Observation of the mass difference between neutral charm-meson
eigenstates

There are currently no precise SM estimates of the mixing parameters of the
$D^{0}$ meson. The inclusive approach, that relies on the heavy-quark
expansion, predicts a value of $y$ in large disagreement with experimental
results [81, 82, 83, 84, 85]. The disagreement can be due to a violation of
the quark-hadron duality or to a lifting of the GIM suppression for higher-
dimension operators, even if these effects seem too small to explain the
discrepancy [86, 87, 88, 89, 90, 91]. The exclusive approach, which calculates
the mixing amplitudes at the hadron level by summing the contributions of all
possible intermediate virtual states, predicts the size of $x$ and $y$ to be
$\mathcal{O}(10^{-3})$ [92, 93, 94, 95, 96, 97].

According to the current world averages, the values of the mixing and
$C\\!P$-violating parameters of the $D^{0}$ meson are $x=(3.7\pm 1.2)\times
10^{-3}$, $y=(6.8^{+0.6}_{-0.7})\times 10^{-3}$,
$|q/p|=(0.951^{+0.053}_{-0.042})$ and $\phi=(-0.092^{+0.085}_{-0.079})$ [79].
These parameters have been measured by studying mainly the $D^{0}\to
K^{+}\pi^{-}$ decay, that allowed precise determinations of $y$ and the
observation of mixing in the $D^{0}$ meson, and the $D^{0}\to
K_{s}^{0}\pi^{+}\pi^{-}$ decay, which is particularly sensitive to $x$ [98,
99, 100, 101, 41]. Here, the recent measurement of the mixing and $C\\!P$
violation parameters in $D^{0}\to K_{s}^{0}\pi^{+}\pi^{-}$ decays performed by
the LHCb collaboration with the data collected between 2016 and 2018,
corresponding to $5.4~{}\mathrm{fb}^{-1}$ of integrated luminosity, is
summarised.

The analysis uses the “bin-flip” method [102], that consists in measuring, as
a function of the $D^{0}$ decay time, the ratio of the number of decays
between symmetric bins in the Dalitz plot defined by the two squared invariant
masses $m^{2}(K_{s}^{0}\pi^{\pm})$. The “bin-flip” method is a model-
independent approach and does not require an accurate modelling of the
efficiency. The Dalitz bins, illustrated in Figure 7, are defined in such a
way that the strong-phase difference between the $D^{0}$ and
$\overline{D}^{0}$ amplitudes within each bin is nearly constant [103]. The
set of bins for which $m^{2}_{+}>m^{2}_{-}$, where $m^{2}_{\pm}$ denotes
$m^{2}(K_{s}^{0}\pi^{\pm})$, is given a positive index $+b$, whereas the other
set, symmetric about the $m^{2}_{+}=m^{2}_{-}$ bisector, is given a negative
index $-b$. For initially produced $D^{0}$ ($\overline{D}^{0}$) mesons, if the
$C\\!P$ conserving amplitudes and mixing parameters are small, the expected
ratios $R^{+}_{bj}$ ($R^{-}_{bj}$) of the number of decays in each negative
Dalitz-plot bin ($-b$) to its positive counterpart ($+b$), for each decay-time
interval $j$, is

$R^{\pm}_{bj}\approx\frac{r_{b}+r_{b}\frac{\left\langle
t^{2}\right\rangle_{j}}{4}\operatorname{Re}(z^{2}_{C\\!P}-\Delta
z^{2})+\frac{\left\langle t^{2}\right\rangle_{j}}{4}\left|z_{C\\!P}\pm\Delta
z\right|^{2}+\sqrt{r_{b}}\left\langle
t\right\rangle_{j}\operatorname{Re}[X_{b}^{*}(z_{C\\!P}\pm\Delta
z)]}{1+\frac{\left\langle
t^{2}\right\rangle_{j}}{4}\operatorname{Re}(z^{2}_{C\\!P}-\Delta
z^{2})+r_{b}\frac{\left\langle
t^{2}\right\rangle_{j}}{4}\left|z_{C\\!P}\pm\Delta
z\right|^{2}+\sqrt{r_{b}}\left\langle
t\right\rangle_{j}\operatorname{Re}[X_{b}(z_{C\\!P}\pm\Delta z)]},$ (12)

where $r_{b}$ is the value of $R^{\pm}_{bj}$ at decay time $t=0$, $X_{b}$ is
the amplitude-weighted strong-phase difference between opposite bins and
$\left\langle t\right\rangle_{j}$ ($\left\langle t^{2}\right\rangle_{j}$) is
the average (squared) decay time in each positive Dalitz-plot region in units
of the $D^{0}$ lifetime. The other parameters are defined according to

$z_{C\\!P}\pm\Delta z\equiv-\left(\frac{q}{p}\right)^{\pm 1}(y+ix),$ (13)

and are obtained from a simultaneous fit of the observed $R^{\pm}_{bj}$
ratios, where external information on $c_{b}\equiv\operatorname{Re}(X_{b})$
and $s_{b}\equiv-\operatorname{Im}(X_{b})$ is used as a constraint [103, 104].
The $C\\!P$-even mixing parameters $x_{C\\!P}$ and $y_{C\\!P}$ and the
$C\\!P$-violating parameters $\Delta x$ and $\Delta y$ are therefore obtained

$\displaystyle x_{C\\!P}$ $\displaystyle\equiv-\operatorname{Im}(z_{C\\!P}),$
(14) $\displaystyle y_{C\\!P}$
$\displaystyle\equiv-\operatorname{Re}(z_{C\\!P}),$ (15) $\displaystyle\Delta
x$ $\displaystyle\equiv-\operatorname{Im}(\Delta z),$ (16)
$\displaystyle\Delta y$ $\displaystyle\equiv-\operatorname{Re}(\Delta z).$
(17)

If $C\\!P$ is conserved, $x_{C\\!P}=x$, $y_{C\\!P}=y$ and $\Delta x=\Delta
y=0$.

Figure 7: Binning scheme of the $D^{0}\to K_{s}^{0}\pi^{+}\pi^{-}$ Dalitz
plot.

The data sample is split in two categories depending on whether the
$K_{s}^{0}$ meson is reconstructed using long or downstream pions, as already
explained in Section 2.1. The flavour of the $D^{0}$ candidate is determined
by the charge of the accompanying pion in the reconstructed $D^{*+}\to
D^{0}\pi^{+}$ decay. To determine the signal yields, a fit is performed on the
invariant mass difference $\Delta m\equiv m(D^{*+})-m(D^{0})$ in each Dalitz-
plot bin and decay-time interval, separately for $D^{0}$ flavour and
$K_{s}^{0}$ category. The fit uses an empirical model that takes into account
contributions from signal and combinatorial background. A total of about 31
million signal decays is found in the whole data sample.

The signal selection includes requirements on the displacement and momenta of
the $D^{0}$ decay products that introduce efficiency variations correlated
between the phase-space coordinates and the $D^{0}$ decay time, resulting in a
possible bias on the measurement. A data-driven correction is therefore
applied to make the decay-time acceptance uniform in the phase space, removing
the correlation. A further correction is performed to cancel detection
asymmetries of the pions produced in the $D^{0}$ decays, since their
kinematics depend on the Dalitz-plot coordinate and the $D^{0}$ flavour: the
two-track $\pi^{+}\pi^{-}$ detection asymmetry is evaluated by measuring the
raw asymmetries in the control samples $D^{+}_{s}\to\pi^{+}\pi^{+}\pi^{-}$ and
$D^{+}_{s}\to\phi\pi^{+}$ in each Dalitz-plot bin, after weighting their
kinematics to that of the signal sample.

A fit is performed on all the corrected $R^{\pm}_{bj}$ ratios to determine the
mixing and $C\\!P$ violating parameters. The result is illustrated in Figure
8. Various sources of systematic uncertainties are considered and assessed
from ensembles of pseudoexperiments. They take into account contributions
related to reconstruction and selection effects, decay-time and $m_{\pm}$
resolution, presence of secondary $D^{*+}$ decays, $\pi^{+}\pi^{-}$ detection
asymmetry, fit model and approximation of strong phase input to be constant in
each bin. The mixing and $C\\!P$ violation parameters are measured to be

$\displaystyle x_{C\\!P}$ $\displaystyle=(3.97\pm 0.46\pm 0.29)\times
10^{-3},$ $\displaystyle y_{C\\!P}$ $\displaystyle=(4.59\pm 1.20\pm
0.85)\times 10^{-3},$ $\displaystyle\Delta x$ $\displaystyle=(-0.27\pm 0.18\pm
0.01)\times 10^{-3},$ $\displaystyle\Delta y$ $\displaystyle=(0.20\pm 0.36\pm
0.13)\times 10^{-3},$

where the first uncertainties are statistical and the second systematic [105].

Figure 8: (Left) $C\\!P$-averaged yield ratios and (right) differences of
$D^{0}$ and $\overline{D}^{0}$ yield ratios as a function of $D^{0}$ decay
time, shown for each Dalitz-plot bin. Fit projections are overlaid.

A likelihood function of $x$, $y$, $|q/p|$ and $\phi$ is built from the
results using a likelihood-ratio ordering that assumes the observed
correlations to be independent of the true parameter values [106], whose best
fit point is

$\displaystyle x$ $\displaystyle=(3.98^{+0.56}_{-0.54})\times 10^{-3},$
$\displaystyle y$ $\displaystyle=(4.6^{+1.5}_{-1.4})\times 10^{-3},$
$\displaystyle|q/p|$ $\displaystyle=0.996\pm 0.052,$ $\displaystyle\phi$
$\displaystyle=0.056^{+0.047}_{-0.051}.$

This result corresponds to the first observation of a non-zero value of the
mass difference $x$ of neutral charm meson mass eigenstates with a
significance of more than seven standard deviations. This measurement
significantly improves limits on $C\\!P$ violation in mixing in the charm
sector.

##  3 Conclusions and future prospects

The most recent results from LHCb concerning $C\\!P$ violation in charm sector
have been summarised here. The analysis of data collected during the LHC Run 2
allowed the most precise measurement of time-dependent $C\\!P$ asymmetry in
$D^{0}$ decays to date to be obtained and a non-zero mass difference between
neutral charm-meson eigenstates to be observed. Furthermore, the most precise
measurement of $C\\!P$ asymmetry in $D^{0}\to K_{s}^{0}K_{s}^{0}$ decays has
been performed. $C\\!P$ asymmetries in two-body decays of charged charm mesons
in final states with light neutral mesons have been determined with precisions
of the order of $10^{-2}$. Many key measurements are still to be performed
with the full data sets collected by LHCb, such as the determination of
$\mathcal{A}_{C\\!P}(D^{0}\to K^{+}K^{-})$, which is expected to be measured
with an uncertainty of the order of $7\times 10^{-4}$, and of
$y_{C\\!P}^{h^{+}h^{-}}$, defined as the deviation from unity of the average
of the effective decay widths of $D^{0}\to h^{+}h^{-}$ and
$\overline{D}^{0}\to h^{+}h^{-}$ decays.

Starting from 2022, the Upgrade I of the LHCb experiment will increase the
total integrated luminosity to $23~{}\mathrm{fb}^{-1}$
($50~{}\mathrm{fb}^{-1}$) by the end of its third (fourth) data-taking period
[107, 108]. An Upgrade II of the LHCb experiment was recently proposed and, if
approved, it will allow the total integrated luminosity to be increased to
$300~{}\mathrm{fb}^{-1}$ [109]. With such a large data set, the statistical
uncertainty of $\Delta Y$ will be reduced to a value below $2\times 10^{-5}$,
and that of $\mathcal{A}_{C\\!P}(D^{0}\to K^{+}K^{-})$ will become smaller
than $10^{-4}$ [110]. The statistical uncertainty of the mixing parameters
measured with the $D^{0}\to K_{s}^{0}\pi^{+}\pi^{-}$ decay mode is expected to
decrease by an order of magnitude. For what concerns the $C\\!P$ asymmetry of
the $D^{0}\to K_{s}^{0}K_{s}^{0}$ decay, its projected statistical uncertainty
is expected to decrease by a factor 4, reaching $\mathcal{O}(10^{-3})$. For
all the mentioned decay modes, there are no systematic uncertainties that are
known to have irreducible contributions which exceed the ultimate statistical
precision. The LHCb Upgrade I and II will therefore have a strong potential of
probing the Standard Model contribution and an impressive power to
characterise new physics contributions to $C\\!P$ violation in the charm
sector.

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# Marketing resource allocation in duopolies over social networks††thanks:
This work has been funded by the CNRS PEPS project YPSOC

Vineeth S. Varma, Irinel-Constantin Morărescu, Samson Lasaulce, and Samuel
Martin V. S. Varma, I-C. Morărescu and S. Martin are with the Université de
Lorraine, CNRS, CRAN, F-54000 Nancy, France, {vineeth.satheeskumar-
<EMAIL_ADDRESS>Lasaulce is with the Laboratoire des Signaux et
Systemes (L2S, CNRS-CentraleSupelec-Univ. Paris Sud), Gif-sur-Yvette, France.

###### Abstract

One of the key features of this paper is that the agents’ opinion of a social
network is assumed to be not only influenced by the other agents but also by
two marketers in competition. One of our contributions is to propose a
pragmatic game-theoretical formulation of the problem and to conduct the
complete corresponding equilibrium analysis (existence, uniqueness, dynamic
characterization, and determination). Our analysis provides practical insights
to know how a marketer should exploit its knowledge about the social network
to allocate its marketing or advertising budget among the agents (who are the
consumers). By providing relevant definitions for the agent influence power
(AIP) and the gain of targeting (GoT), the benefit of using a smart budget
allocation policy instead of a uniform one is assessed and operating
conditions under which it is potentially high are identified.

###### Index Terms:

Game theory; Network analysis and control; Agents-based systems

##  I Introduction

Duopoly is a standard scenario in economics, politics, and marketing that
considers the competition between two (dominant) players over a market, for
example, see [1]. Illustrative examples of real-life duopolies are
Airbus/Boeing in the market of large commercial airplanes,
Republican/Democratic parties in the American politics.

Traditional research on competitive games between marketers assumes a
homogeneous population of consumers [2, 3]. Unlike these works, we propose a
marketing resource allocation based on the influence power that each
individual has over the (physical or digital) social network. Basically, we
consider that the advertising is done in two steps: the first is done by the
marketer that allocate her resources to sway some individuals/agents on her
opinion and the second is done by the agents of the social network who
influence each other. Consequently, each marketer has to target appropriate
influential agents in the network in order to optimize her revenue. Since the
focus of the paper is on the resource allocation of the marketer, the second
step is modeled by a simple opinion dynamics model introduced in [4].

There is an obvious gap between the literature of economics and the literature
on formal opinion dynamics seen from the control community perspective. On one
hand, the literature of economics on the considered problems contains many
ideas, concepts, approaches but is not very formal, see [5, 6] which
illustrate this point. On the other hand, the control literature is formal but
does not address aspects such as the problem of competition over social
networks as we do in the manuscript. Therefore, our approach can be seen as a
contribution to bridging this gap and the model we propose can be seen as the
first step in this direction. Although some recent studies propose the control
of one or few agents (see [7, 8]) in order to enforce consensus, there are
very few that deal with the control of opinion dynamics. Besides these methods
of controlling opinion dynamics towards consensus, we also find recent
attempts to control the discrete-time dynamics of opinions such that as many
agents as possible reach a certain set after a finite number of influences
[9]. In the literature on _viral marketing_ , the idea that members of a
social network influence each other’s purchasing decisions have been studied,
with the goal being to select the best set of people, such that marketing to
this set would maximize the overall profit by propagation of influence through
the network [10]. This problem has since received much attention, including
both empirical and theoretical results [11], but these results often consider
a single entity influencing the network.

In this paper, we consider two competing marketers who want to use their
marketing budget in order to sway on their side as many individuals of the
network as possible. Thus, the natural framework to exploit is that of game
theory and a reasonable solution concept (for arguments see _e.g._ , [12]) for
analyzing such a competition situation is the Nash equilibrium (NE). In [13],
the authors consider multiple influential entities competing to control the
opinion of consumers under a game theoretical setting. However, this work
assumes an undirected graph and a voter model for opinion dynamics resulting
in strategies that are independent of the node centrality (_i.e.,_ agent
influence power). On the other hand, the recently published work [14]
considers a similar competition with opinion dynamics over a directed graph
but with no budget constraints and by considering the average agents’ opinion
instead of the final one; these two differences change the problem
significantly.

The main contributions of this paper are the followings. We introduce and
analyze a new generic model for marketing over social networks. We conduct a
complete equilibrium analysis (existence, uniqueness, determination) for the
corresponding model. We conduct a numerical performance analysis that allows
one, in particular, to obtain very useful insights in terms of investment for
the marketers.

Notation. Let $\mathbb{R}_{\geq 0}:=[0,\infty)$ denote the set of non-negative
real numbers. If $f(t)$ is a lower semi-continuous function at $t_{0}$, we use
the notation $f(t^{+}_{0})$ to imply $f(t^{+}_{0}):=\lim_{t\to
t_{0},t>t_{0}}f(t)$. Since we are concerned with a duopoly in this work, for
ease of exposition, we will denote by $-i$ when $i\in\\{1,2\\}$ is a player
index, to refer to the index of the other player, i.e. $-i:=1+i(mod\ 2)$.

##  II Problem statement

We consider a market with Firms $1$ and $2$ that are interested in attracting
consumers (referred to as agents) to their product. We consider a set of $N$
agents that continuously interact over a fixed social network. In the sequel
we denote by $\mathcal{V}=\\{1,2,\dots,N\\}$ the set of agents and the social
network is represented by a fixed weighted directed graph
$(\mathcal{V},\mathcal{E},\Omega)$, with $\mathcal{E}$ and $\Omega$
respectively representing the set of edges of the graph and the matrix of
corresponding weights. To agent $n\in\mathcal{V}$ we assign a normalized
scalar opinion $x_{n}(t)\in(0,1)$ which denotes the opinion in favor of the
product or ideas of Firm $1$. The revenue obtained by a firm is assumed to be
proportional to its average market share. Thus, for agent $n$ at time
$t\in\mathbb{R}_{\geq 0}$, the revenue of Firm $1$ is proportional to
$x_{n}(t)$ and for Firm $2$ the revenue is proportional to $1-x_{n}(t)$. We
use $x(t)=(x_{1}(t),x_{2}(t),\dots,x_{N}(t))^{\top}$ to denote the state of
the network at any time $t$, where $x(t)\in\mathcal{X}_{0}$ and
$\mathcal{X}_{0}=(0,1)^{N}$. We can define $x_{n;i}(t)$ as the opinion of
agent $n$ in favor of Firm $i$, with $x_{n;1}(t)=x_{n}(t)$ and
$x_{n;2}(t)=1-x_{n}(t)$, i.e., the two firms have competing products.

###  II-A External influence model (through marketing campaigns)

In order to obtain a larger market share, Firm $i$ invests according to the
investment or action vector which corresponds to the marketing campaign.
Without loss of generality we can consider that the campaign is reduced to a
time instant and it has an impulsive instantaneous effect on the opinion of
the agents. At the campaign instant, Firm $i$ invests according to the vector
$a_{i}\in\mathcal{A}_{i}$ where the action space for Firm $i$ is

$\mathcal{A}_{i}:=\left\\{a_{i}\in[0,b_{i}]^{N}|\sum_{n=1}^{N}a_{i,n}\leq
B_{i}\right\\}$ (1)

where $b_{i}\leq B_{i}\in\mathbb{R}_{\geq 0}$ for $i\in\\{1,2\\}$ represent
the maximum influence/discount for one specific agent and the total budget,
respectively. The vector $a_{i},\ i\in\\{1,2\\}$ is called the _action_ of
Firm $i$, with $a_{i,n}$ being the marketing expenditure targeted at agent
$n$. The campaign modify the opinions of agents according to a function
$\Phi(x_{0},a_{i},a_{-i}):\mathcal{X}_{0}\times\mathcal{A}_{i}\times\mathcal{A}_{-i}\to\mathcal{X}_{0}$
where $x_{0}=(x_{0,1},...,x_{0,N})^{\mathrm{T}}\in\mathcal{X}_{0}$ is the
vector collecting the initial opinions of the agents before the campaign in
favour of Firm $i$, i.e. $x_{0}=x(0)$. Without any loss of generality, we
consider that the campaign occurs at $t=0$. In the sequel we consider the
function
$\Phi(x_{0;i},a_{i},a_{-i})=(\phi(x_{0,1;i},a_{i,1},a_{-i,1}),...,\phi(x_{0,N;i},a_{i,N},a_{-i,N}))^{\mathrm{T}}$
with

$\phi(x_{0,n;i},a_{i,n},a_{-i,n})=\frac{x_{0,n;i}+a_{i,n}}{1+a_{i,n}+a_{-i,n}},\
\forall n\in\\{1,...,N\\}.$ (2)

If $x_{n}(t)$ is seen as the probability of agent $n$ picking the product of
Firm $1$, $\phi(\cdot)$ corresponds to a Bayessian update rule on the opinion
as used in [15]. This probability is updated at the campaign instant $t=0$,
with $a_{1,n}$ increasing the odds of choosing Firm $1$, and $a_{2,n}$
increasing the odds of choosing Firm $2$. Note that

$\phi(x_{n;i}(0),a_{i,n},a_{-i,n})=1-\phi(x_{n;-i}(0),a_{-i,n},a_{i,n}),$

and therefore, the opinion’s change is symmetric w.r.t the two firms. This
results in the campaign opinion’s change

$x_{n;i}(0^{+})=\phi(x_{n;i}(0),a_{i,n},a_{-i,n})$ (3)

for all $n\in\mathcal{V}$. The proposed function also satisfies the following
properties. If both firms spend very little on agent $n$, its opinion at $0$
is preserved after the campaign as we have

$\lim_{a_{i,n}\to 0,a_{-i,n}\to 0}\phi(x_{0,n;i},a_{i,n},a_{-i,n})=x_{0,n;i}.$
(4)

Additionally, when the resources allocated to agent $n$ are large by both
firms (provided that $b_{i}$ is large), the final opinion depends only on the
ratio of investments and not on the initial opinion.

$\lim_{a_{i,n}\to\infty,a_{-i,n}\to\infty,\frac{a_{-i,n}}{a_{i,n}}\to
c}\phi(x_{0,n;i},a_{i,n},a_{-i,n})=\frac{1}{1+c}$ (5)

###  II-B Internal influence model (within the social network)

After the campaign, the consumer’s opinion is only influenced by the other
consumers of the networks. In this work we consider that opinion dynamics in
the social network is characterized by a linear model described by the
following differential equation:

$\dot{x}(t)=-\mathbf{L}x(t)$ (6)

where $\mathbf{L}\in\mathbb{R}^{N\times N}$ is the Laplacian matrix associated
with the graph $(\mathcal{V},\mathcal{E},\Omega)$, whose components are
defined as:

$L_{m,n}=\left\\{\begin{array}[]{c}\displaystyle\sum_{n=1}^{N}\Omega_{m,n}\mbox{
if }m=n\\\\[5.69054pt] -\Omega_{m,n}\mbox{ if }m\neq n\end{array}\right..$ (7)

As a result of the marketing campaigns, we have the following hybrid opinion
dynamics model

$\left\\{\begin{array}[]{llll}\dot{x}(t)&=&-\mathbf{L}x(t)&\forall
t\in\mathbb{R}_{\geq 0}\setminus\\{0\\}\\\
x_{n}(0^{+})&=&\phi(x_{0,n},a_{1,n},a_{2,n})&\forall
n\in\mathcal{V}\end{array}\right..$ (8)

###  II-C Revenue model

In this work, we assume that the profit is based only on the opinion the
agents have after some time $T>0$. Other profit models such as an integral
over time for the opinion have been studied in [14]. In either case, we
observe that the profit can be expressed as a linear combination of the
opinions after the campaign. We define the _agent influence power_ (AIP) of
Agent $n$ as follows.

###### Definition 1

The AIP of Agent $n$ is given by $\rho_{n}>0$ where

$\rho={1}_{N}^{\top}\exp(-\mathbf{L}T)$ (9)

where ${1}_{N}$ is a column vector of ones.

The net revenue earned by Firm $i$ can therefore be written as the difference
between the profit and the marketing expenses resulting in

$u_{1}(x_{0},a_{1},a_{2}):=\gamma_{1}\rho
x(0^{+})-\lambda_{1}{1}_{N}^{\top}a_{1},$ (10)
$u_{2}(x_{0},a_{1},a_{2}):=\gamma_{2}\rho[{1}_{N}-x(0^{+})]-\lambda_{2}{1}_{N}^{\top}a_{2}.$
(11)

where $x_{n}(0^{+})=\phi(x_{0,n},a_{1,n},a_{2,n})$, $\gamma_{i}\geq 0$ is the
revenue generated per consumer for Firm $i$, and $\lambda_{i}\geq 0$ is the
advertising efficiency or pricing factor for Firm $i$.

###  II-D Game-theoretic formulation of the problem

The ingredients introduced in this section allow us to formulate the problem
as a game under strategic form that is, a triplet which is defined as follows:

$\mathcal{G}=\left(\\{1,2\\},\\{\mathcal{A}_{1},\mathcal{A}_{2}\\},\\{u_{1},u_{2}\\}\right),$
(12)

where:

  * •

$\\{1,2\\}$ is the set of players (i.e., Firms $1$ and $2$);

  * •

$\mathcal{A}_{i}$ defined in (1) is the set of pure actions for Player $i$;

  * •

$u_{i}$ as defined per (10) (11) is the utility function for Firm $i$.

Throughout the paper we assume non-aligned utility functions and partial
control for the players. In other words, in the game theoretical framework,
the meaning of optimality is not clear and there is a need for defining a
solution concept for the considered problem (see _e.g_., [12] for further
discussion). The solution retained here is the Nash equilibrium (NE) [16], as
commonly assumed for duopolies. The definition of a pure NE is as follows.

######  Definition 2 (Pure NE)

A strategy profile
$(a_{1}^{\star},a_{2}^{\star})\in\mathcal{A}_{1}\times\mathcal{A}_{2}$ is a
pure NE for $\mathcal{G}$ for a given $x_{0}$ if $\forall i\in\\{1,2\\},$

$\ \forall a_{i}\in\mathcal{A}_{i},\
u_{i}(x_{0},a_{i}^{\star},a_{-i}^{\star})\geq
u_{i}(x_{0},a_{i},a_{-i}^{\star}).$ (13)

##  III Game-theoretic analysis

This section analyse whether the solution of the problem formulated above
exists, is unique and how it can be attained.

###  III-A Nash equilibrium analysis

We start by showing that the game $\mathcal{G}$ does have a pure and unique
NE. First, we provide some auxiliary lemmas that will help us to characterize
the utility function.

###### Lemma 1

The utility function $u_{i}(x_{0;i},a_{i},a_{-i})$ is concave w.r.t $a_{i}$.

###### Proof:

Recall that $x_{0,n;i}$ stands for the initial opinion of agent $n$ w.r.t.
Firm $i$ and that $x_{0,n;i}=1-x_{0,n;-i}$. We evaluate

$\begin{array}[]{llr}\frac{\partial u_{i}}{\partial
a_{i,n}}&=&\frac{\gamma_{i}\rho_{n}}{1+a_{i,n}+a_{-i,n}}-\frac{\gamma_{i}\rho_{n}(x_{0,n;i}+a_{i,n})}{(1+a_{i,n}+a_{-i,n})^{2}}-\lambda_{i}\end{array}$
(14)

Next, we evaluate

$\begin{array}[]{lll}\frac{\partial^{2}u_{i}}{\partial
a_{i,n}^{2}}&=&-\frac{2\gamma_{i}\rho_{n}}{(1+a_{i,n}+a_{-i,n})^{2}}+\frac{2\gamma_{i}\rho_{n}(x_{0,n;i}+a_{i,n})}{(1+a_{i,n}+a_{-i,n})^{3}}\\\
&=&-\frac{2\gamma_{i}\rho_{n}(1-x_{0,n;i}+a_{-i,n})}{(1+a_{i,n}+a_{-i,n})^{3}}\end{array}$
(15)

Note that $a_{i,n},a_{-i,n}\geq 0$ and $0<x_{0,n;i}<1$ for all
$i\in\\{1,2\\}$, $n\in\\{1,2,\dots,N\\}$. Therefore, we have
$\frac{\partial^{2}u_{i}}{\partial a_{i,n}^{2}}<0$ for all $n$. We can also
easily see that $\frac{\partial^{2}u_{i}}{\partial a_{i,n}\partial a_{i,m}}=0$
for all $m\neq n$. The Hessian of the utility function is therefore a diagonal
matrix with all entries negative. Therefore, $u_{i}(\cdot)$ is concave w.r.t.
$a_{i}\in[0,b_{i}]^{N}$. ∎

Next, consider the weighted sum of utilities for the game $\mathcal{G}$ for a
given initial opinion $x_{0}$ defined as

$\sigma(a,r):=r_{1}u_{1}(x_{0},a_{1},a_{2})+r_{2}u_{2}(x_{0},a_{1},a_{2}).$
(16)

for some $r_{1},r_{2}>0$. We look at the pseudo-gradient of $\sigma(a,r)$,
used by Rosen in [17], and defined as

$g(a,r)=\left(\begin{array}[]{l}r_{1}\nabla_{a_{1}}u_{1}(a),r_{2}\nabla_{a_{2}}u_{2}(a)\end{array}\right)^{\top}.$
(17)

This allows us to look at the generalized diagonally strict concavity (DSC)
condition [17], which can be exploited to prove the uniqueness of the NE.

###### Definition 3

A function $\sigma(a,r)$ is said to satisfy the DSC for a given $r>0$ if for
every distinct pairs of action profiles $a$, $a^{\prime}$, i.e., with
$(a_{1},a_{2})\neq(a_{1}^{\prime},a_{2}^{\prime}),x_{0}=x_{0}^{\prime}$, we
have

$(a-a^{\prime})^{\mathrm{T}}(g(a,r)-g(a^{\prime},r))<0.$ (18)

###### Lemma 2

The function $\sigma(a,r)$ satisfies the DSC property w.r.t $(a_{1},a_{2})$.

###### Proof:

When the utility functions are twice differentiable, a sufficient condition
for DSC (see [17, Th.6]) is that

$G(a,r)+G(a,r)^{\mathrm{T}}<0$ (19)

where $G(a,r)$ is the Jacobian of $g(a,r)$ w.r.t. $(a_{1},a_{2})$ and (19)
means that $G(a,r)+G(a,r)^{\mathrm{T}}$ is negative definite.

We choose $r=\left(\frac{1}{\gamma_{1}},\frac{1}{\gamma_{2}}\right)$ and we
look at the elements of $G(a,r)$. The diagonal elements are
$r_{i}\frac{\partial^{2}u_{i}}{\partial a_{i,n}^{2}}$ which are negative as we
have already shown in Lemma 1. The off-diagonal elements are in general given
by $r_{i}\frac{\partial^{2}u_{i}}{\partial a_{i,n}\partial a_{j,m}}$. If
$m\neq n$, we notice from (14) that $\frac{\partial^{2}u_{i}}{\partial
a_{i,n}\partial a_{j,m}}=0$ for any $j\neq i$.
However, if $m=n$, then the off-diagonal terms of $G$ at positions $(n,N+n)$
and $(N+n,n)$ for $1\leq i\leq N$ are non zero and can be evaluated as

$\frac{1}{\gamma_{i}}\frac{\partial^{2}u_{i}}{\partial a_{i,n}\partial
a_{-i,n}}=\rho_{n}\frac{-1-a_{-i,n}+2x_{0,n;i}+a_{i,n}}{(1+a_{i,n}+a_{-i,n})^{3}}$
(20)

Note that

$-1+2x_{0,n;i}=2-2x_{0,n;-i}-1=-(-1+2x_{0,n;-i}).$

Therefore, the term at $(n,N+n)$ of $G(a,r)$ is opposite to the term at
$(N+n,n)$. As a result, $G(a,r)+G(a,r)^{\mathrm{T}}$ becomes a diagonal matrix
with all diagonal entries negative, which is therefore a negative definite
matrix. ∎

###### Theorem 1

The game $\mathcal{G}$ has a pure and unique NE.

###### Proof:

Notice that the action space $\mathcal{A}_{i}$ defined as
$a_{i,n}\in[0,b_{i}]$ with $\sum_{n=1}^{N}a_{i,n}\leq B_{i}$ is a convex and
compact subset of $\mathbb{R}_{\geq 0}^{N}$. The utility function is therefore
concave w.r.t $a_{i}\in\mathcal{A}_{i}$ from Lemma 1, with $\mathcal{A}_{i}$
being a compact and convex set and is also (jointly) continuous in $a$. This
allows us to use the result in [17, Th.1] and prove that the game
$\mathcal{G}$ has at least one pure Nash equilibrium. We can use the result in
[17, Th.2] to prove that the NE is unique since the DSC condition shown in
Lemma 2 is met if the NE exists.∎

###  III-B Dynamic characterization of the NE

We have shown that $\mathcal{G}$ has a unique pure NE. To evaluate the
performance of the two firms at equilibrium it is necessary to determine the
NE. While it is not always possible to express the equilibrium actions, it is
possible to fully characterize the equilibrium through a set of dynamic
equations, which is given next. The corresponding system of equations can
either be seen as a way of numerically determining the NE or as a way of
modeling the firms economical behavior. Indeed, we propose to exploit the
smooth or continuous time best-response dynamics introduced in [18] and used
more recently _e.g._ , in [19, 20]. A continuous-time best-response dynamics
is a set of differential equations which allows one to generate a trajectory
converging to the NE. It is given by

$\dot{a}_{i,n}=\beta_{i,n}(x_{0;i},a_{-i})-a_{i,n}$ (21)

for all $i\in\\{1,2\\}$ and $n\in\mathcal{V}$. The quantity
$\beta_{i,n}(x_{0;i},a_{-i})$ corresponds to the best-response, which is
defined by:
$\beta_{i}(x_{0;i},a_{i}):=\arg\max_{a_{i}}\\{u_{i}(x_{0;i},a_{i},a_{-i})\\}.$
We note that (21) should be a differential inclusion since the best-response
function is in general a set-valued function. However, the argmax operation
returns singleton sets (due to strict concavity). By definition, if
$(a_{1}^{\star},a_{2}^{\star})$ is a NE of $\mathcal{G}$ at $x_{0;i}$ then

$a_{1}^{\star}=\beta_{1}(x_{0;1},a_{2}^{\star})\,,\,a_{2}^{\star}=\beta_{2}(x_{0;1},a_{1}^{\star})$
(22)

and can be found by computing the unique point in
$\mathcal{A}_{1}\times\mathcal{A}_{2}$ where
$\beta_{i,n}(x_{0;i},a_{-i})-a_{i,n}=0$ holds for all $i,n$.
Given an action profile $a_{-i}$, the best-response by Firm $i$ can be
evaluated by optimizing $u_{i}$ w.r.t. $a_{i}$ while respecting the budget
constraint. Our next proposition gives a method of evaluating $\beta_{i}$.
Denote $\beta_{i,n}(x_{0;i},a_{-i})$ as the $n$-th component of $\beta_{i}$.
Then, we have

###### Proposition 1

The best-response functions are given by

$\beta_{i,n}(x_{0;i},a_{-i})=\min\\{b_{i},\max\\{0,\alpha_{i,n}(x_{0;i},a_{-i})\\}\\}$
(23)

where

$\alpha_{i,n}(x_{0;i},a_{-i})=\sqrt{\frac{\gamma_{i}\rho_{n}(x_{0,n;-i}+a_{-i,n})}{\mu_{0;i}+\lambda_{i}}}-1-a_{-i,n}$
(24)

for all $n\in\mathcal{V}$, and $\mu_{0}\in\mathbb{R}_{\geq 0}$ is such that

$\sum_{n=1}^{N}\beta_{i,n}(x_{0;i},a_{-i})\leq
B_{i},\mu_{0;i}(\sum_{n}\beta_{i,n}(x_{0;i},a_{-i})-B_{i})=0$

###### Proof:

The optimization problem to be solved in order to evaluate $\beta_{i}$ can be
written as

$\begin{array}[]{lr}\text{Maximize}_{a_{i}}u_{i}(a_{i},a_{-i})&\\\
\text{subject to }\sum_{n=1}^{N}a_{i,n}\leq B_{i}\>\>\>\&&\\\ 0\leq
a_{i,n}\leq b_{i}&\forall n\end{array}$ (25)

Problem (25) necessarily has an optimal solution as $u_{i}$ is continuous and
the optimization space is a compact set. Since all the constraint functions
are linear, Karush-Kuhn-Tucker (KKT) conditions [21] can be applied.
Additionally, the problem being convex, KKT conditions are not only necessary
but also sufficient conditions for optimality. For all $n\in\\{-N,\dots,N\\}$
let $\mu_{n}\geq 0$ be the KKT multipliers. We use $\mu_{n}$ for the
constraint $a_{i,n}\geq 0$, $\mu_{-n}$ for the constraint $a_{i,-n}\leq b_{i}$
for all $n\in\mathcal{V}$ and $\mu_{0}\geq 0$ as the multiplier for the total
budget constraint in (25). Then, the KKT conditions can be written as

$\left\\{\begin{array}[]{l}\begin{array}[]{ll}\nabla_{a_{i}}u_{i}(a_{i}^{\star})=&\sum_{n=1}^{N}-\mu_{n}\nabla_{a_{i}}a_{i,n}^{\star}+\mu_{0}\nabla_{a_{i}}\sum_{n=1}^{N}a_{i,n}^{\star}\\\
&+\mu_{-n}\nabla_{a_{i}}a_{i,n}^{\star}\geq 0,\\\ a_{i,n}^{\star}\leq
b_{i},&\sum_{n=1}^{N}a_{i,n}^{\star}-B_{i}\leq 0,\\\ \mu_{n},\mu_{-n}\geq
0,&\forall n\in\\{-N,\dots,N\\},\end{array}\\\
\begin{array}[]{ll}\mu_{n}(a_{i,n}^{\star}-b_{i})=0,\
\mu_{-n}a_{i,n}^{\star}=0&\forall n\in\\{1,\dots,N\\},\\\
\mu_{0}(\sum_{n=1}^{N}a_{i,n}^{\star}-B_{i})=0.\\\
\end{array}\end{array}\right.$

In order to evaluate $\nabla_{a_{i}}u_{i}(a_{i},a_{-i})$, we have

$\begin{array}[]{llr}\frac{\partial u_{i}(a_{i},a_{-i})}{\partial
a_{i,n}}&=&\gamma_{i}\rho_{n}\left(\frac{x_{0,n;i}+a_{i,n}}{1+a_{i,n}+a_{-i,n}}\right)^{\prime}-\lambda_{i}\\\
&=&\gamma_{i}\rho_{n}\frac{x_{0,n;-i}+a_{-i,n}}{(1+a_{i,n}+a_{-i,n})^{2}}-\lambda_{i}.\end{array}$
(26)

As $x_{0,n;-i}$ and $a_{-i,n}$ are constants, we define
$d_{i,n}:=\rho_{n}(x_{0,n;-i}+a_{-i,n})$. Thus, the first KKT condition can be
now written as

$\frac{d_{i,n}}{(1+a_{i,n}^{\star}+a_{-i,n})^{2}}=\lambda_{1}+\mu_{0}+\mu_{n}-\mu_{-n}$
(27)

which must be satisfied for all $n\in\mathcal{V}$. Let
$\mathcal{W}_{0}\subseteq\mathcal{V}$ be the set of agents for which
$a_{i,n}=0$ which leaves $\mu_{-n}\geq 0$ free in order to satisfy the KKT
condition for $n\in\mathcal{W}_{0}$. Similarly, let
$\mathcal{W}_{1}\subset\mathcal{V}$ such that $a_{i,n}=b_{i}$ for all
$n\in\mathcal{W}_{1}$. Finally, let
$\mathcal{W}_{3}:=\mathcal{V}\setminus\mathcal{W}_{1}\setminus\mathcal{W}_{0}$.
Then, we have

$\beta_{i,n}(x_{0;i},a_{-i})=\left\\{\begin{array}[]{llr}\sqrt{\frac{d_{i,n}}{\mu_{0}+\lambda_{i}}}-1-a_{-i,n}&\forall
n\in\mathcal{W}_{2}\\\ 0&\forall n\in\mathcal{W}_{0}\\\ b_{i}&\forall
n\in\mathcal{W}_{1}\end{array}\right.$ (28)

which must be positive. We can use the final complementary slackness condition
to solve for $\mu_{0}$ and get

$\mu_{0}=\left(\frac{\sum_{n\in\mathcal{W}_{2}}\sqrt{d_{i,n}}}{B_{i}-b_{i}|\mathcal{W}_{1}|+|\mathcal{W}_{2}|+\sum_{n\in\mathcal{W}_{2}}a_{-i,n}}\right)^{2}-\lambda_{i}.$
(29)

Therefore $\mathcal{W}_{2}$ must be chosen such that
$0\leq\beta_{i,n}(a_{-i})\leq b_{i}$ for all $n\in\mathcal{V}$. This gives us
(23). We write $\mu_{0;i}$ in the BR function as this parameter is player
dependent. Finally, since the problem is concave as shown in Lemma 1, we know
that any point in which the KKT conditions are satisfied must also be the
global maximum. ∎

Practically, the water level $\mu_{0;i}$ can be found with a lower complexity
by first sorting all the agents based on $\frac{1+a_{-i,n}}{\sqrt{d_{i,n}}}$.
This is because for a given $a_{-i}$, the agent with a lower value of
$\frac{1+a_{-i,n}}{\sqrt{d_{i,n}}}$ will necessarily have a larger
$\beta_{i,n}$. This lets us enforce that any agent with $0$ resource
allocation to have a lower sorting index than the ones based on (24), which in
turn will have a lower sorting index than the ones with allocation $b_{i}$.

###  III-C Expressing the NE

Although we have provided a dynamical characterization of the NE from which a
discrete-time iterative algorithm that converges to the NE, we have no
insights on how this NE depends on the various key parameters of the problem.
In particular, we would like to determine the relationship between the
structure of the resource allocation policy at NE, the AIP parameter
$\rho_{n}$, and the initial opinion $x_{0,n;i}$. The following proposition
shows that the amount of budget allocated to Agent $n$ at NE increases
proportionally with $\rho_{n}$, and will decrease if the initial opinion of
Agent $n$ is already in favor of Firm $i$. The allocation also depends on
budget constraints, the profit multiplier $\gamma_{i}$, and the advertising
efficiency $\lambda_{i}$.

###### Proposition 2

For each $n\in\mathcal{V}$, the NE $(a_{1,n}^{\star},a_{2,n}^{\star})$ is
given by

  * •

$(y,0)$ (or $(0,y)$) if $\exists y\in[0,b_{1}]$ (or $[0,b_{2}]$ respectively)
such that (22) is satisfied by one of these pairs,

  * •

or $(y,b_{2})$ (or $(b_{1},y)$) if $\exists y\in[0,b_{1}]$ (or $[0,b_{2}]$
respectively) such that (22) is satisfied by one of these pairs,

  * •

or $(a_{1,n}^{\star},a_{2,n}^{\star})\in(0,b_{1})\times(0,b_{2})$ and is given
by

$a_{i,n}^{\star}=\left(\frac{k_{i}}{k_{i}+k_{-i}}\right)^{2}k_{-i}\rho_{n}-x_{0,n;i},$
(30)

where $k_{i}=\frac{\gamma_{i}}{\lambda_{i}+\mu_{0;i}}$ and $\mu_{0;i}$ is a
common constant for all $n\in\mathcal{V}$ given by (1).

###### Proof:

We observe from (22) and (23) that if $a_{i,n}^{\star}$ for some
$i\in\\{1,2\\},n\in\mathcal{V}$ is in the interval $(0,b_{i})$, then, it must
be equal to (24). Looking at (24), we observe that at NE, we have

$\begin{array}[]{l}a_{1,n}^{\star}=\sqrt{k_{1}\rho_{n}(1-x_{0,n}+a_{2,n}^{\star})}-1-a_{2,n}^{\star}\\\
a_{2,n}^{\star}=\sqrt{k_{2}\rho_{n}(x_{0,n}+a_{1,n}^{\star})}-1-a_{1,n}^{\star}\\\
\Rightarrow k_{1}(1-x_{0,n}+a_{2,n}^{\star})=k_{2}(x_{0,n}+a_{1,n}^{\star})\\\
\Rightarrow
a_{2,n}^{\star}=\frac{k_{2}}{k_{1}}(x_{0,n}+a_{1,n}^{\star})-1+x_{0,n}\end{array}$
(31)

for all $n\in\mathcal{V}$ such that $a_{i,n}^{\star}\in[0,b_{i}]$, where
$k_{i}=\frac{\gamma_{i}}{\lambda_{i}+\mu_{0;i}}$. Substituting back, we get

$\begin{array}[]{l}\left[\frac{k_{2}}{k_{1}}(x_{0,n}+a_{1,n}^{\star})+x_{0,n}+a_{1,n}^{\star}\right]^{2}=k_{1}\rho_{n}(x_{0,n}+a_{1,n}^{\star})\\\
\Rightarrow(x_{0,n}+a_{1,n}^{\star})\left(k_{2}+k_{1}\right)^{2}=k_{1}^{2}k_{2}\rho_{n}\\\
\Rightarrow
a_{1,n}^{\star}=\left(\frac{k_{1}}{k_{1}+k_{2}}\right)^{2}k_{2}\rho_{n}-x_{0,n}\\\
a_{2,n}^{\star}=\left(\frac{k_{2}}{k_{1}+k_{2}}\right)^{2}k_{1}\rho_{n}-(1-x_{0,n}).\end{array}$
(32)

If the resulting $x_{0,i}^{\star}\notin(0,b_{i})$ from (32) for some $i$, then
we have the other two cases of the proposition. ∎

##  IV Numerical performance analysis

For all simulations we fix $\gamma_{1}=\gamma_{2}=1$, $B_{1}=B_{2}=10$ and
$\lambda_{1}=\lambda_{2}=0.1$ implying symmetry between the two firms in
advertising efficiency and profit ratios.

###  IV-A Network aware marketing versus broadcasting

For a comparison of the proposed graph aware marketing policy with classical
policies, we define the uniform budget allocation (UBA) policy, _i.e.,_ a
broadcast strategy as

$a^{\mathrm{UBA}}_{i}:=\frac{B_{i}}{N}.$ (33)

This strategy is of particular interest as a broadcast strategy is
traditionally used by firms to advertise over media such as television or
radio, while social media marketing is targeted to particular users but
requires some investment in order to learn the initial opinion of users as
well as the network structure.

In Fig. 1, we study the gain a player can make by implementing the best-
response strategy (23) instead of a UBA strategy with uniform resource
allocation. For this simulation, we consider $N=100$ with
$\rho_{n}\in\left\\{1,C\right\\}$. This can be interpreted as the AIP of a
collection of disconnected star graphs with the internal nodes being the
leaders and $C$ representing the influence the leaders have on the rest. If
$n$ is a “leader”, he will have $\rho_{n}=C$, and $1$ otherwise. We directly
consider this configuration for $\rho$ instead of looking at the exact graph
structure in order to highlight the message of this numerical example. The
resulting difference in utility between the two strategies is referred to as
the _gain of targeting_ (GoT), and is measured as

$\text{GoT}:=\frac{u_{1}(x_{0},\beta_{1}(a_{2}^{\mathrm{UBA}}),a_{2}^{\mathrm{UBA}})-u_{1}(x_{0},a_{1}^{\mathrm{UBA}},a_{2}^{\mathrm{UBA}})}{u_{1}(x_{0},a_{1}^{\mathrm{UBA}},a_{2}^{\mathrm{UBA}})}.$
(34)

We take $x_{n}(0)=0.5$ for all $n\in\mathcal{V}$ to remove any bias due to
initial opinions. As expected, we observe that a larger disparity in the AIP
($C$) leads to a larger gain by allocating more resources to the leaders. If
there are too few leaders, and $C$ is not large enough (as in
$C\in\\{5,10\\}$), this profit saturates resulting in a lower gain.

Figure 1: Gain by implementing the best response strategy over the uniform
budget allocation strategy (34) based on the fraction of leaders.

###  IV-B Influence of the initial opinions and graph structure

For our next numerical analysis, we use the graph structure shown in Fig. 2
with $15$ agents which results in

$\begin{array}[]{r}\rho=(0.27,0.37,0.98,0.48,0.59,1.58,0.81,2.30,1.09,1.40\\\
,0.59,0.81,0.81,1.46,1.46)\end{array}$

by taking the duration $T=10$ and calculating the AIP as
$\rho={1}_{N}^{\top}\exp(-10\mathbf{L})$. We initialize the starting opinions
to a random opinion given by

$\begin{array}[]{r}x(0)=(0.26,0.76,0.82,0.10,0.18,0.26,0.6,0.52,0.34\\\
,0.18,0.21,0.61,0.68,0.47,0.31)^{T}.\end{array}$

Figure 2: The size of the nodes are scaled based on the AIP $\rho_{n}$.

Figure 3: The sub-figure on top shows the AIP $\rho_{n}$ and initial opinion
$x_{n}(0)$ for Agent $n$, $n\in\\{1,...,N\\}$. Corresponding to this initial
configuration, the sub-figure on the bottom shows the resource allocation
strategies at the NE.

In Fig. 3, we compare the allocation of budget by the two players at the NE.
We observe that if $x_{n}(0)$ is closer to $1$, _i.e.,_ initially biased
towards Firm $1$, Firm $2$ will invest more to bring it closer to $0$ while a
starting opinion close to $0$ makes Firm $1$ invest more. Both firms invest
more on agents with a larger AIP $\rho_{n}$. This allocation corresponds to
the inference from Proposition 2.

##  V Conclusion

In this paper, we introduce a novel static game model which studies the
competition between firms trying to capture a market share by advertising over
social media. The consumers of the social network are therefore not only under
influence of the other consumers of the network but also of the firms. We
conduct a complete equilibrium analysis for the proposed game. Our analysis
provides concrete insight into how firms should allocate their budget. In
particular, the amount of budget a firm should invest is shown under typical
conditions to be proportional to a quantity which we define as the agent
influence power (AIP). Interestingly, our analysis allows one to quantify the
gain of targeting (GoT) _i.e.,_ the benefit of implementing a smart allocation
policy (based on best-response) instead of allocating the available budget
uniformly over the consumers. The corresponding gain is clearly shown to be
related to the fraction of dominant consumers and the value of their agent
influence power. All these encouraging results show the strong interest in
developing the proposed framework. One very relevant extension might be given
by a stochastic formulation of the problem.

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