# Dark photon pair production via off-shell dark Higgs at FASER

Takeshi Araki<EMAIL_ADDRESS>Faculty of Dentistry, Ohu University,
31-1 Misumidou, Tomita-machi, Koriyama, Fukushima 963-8611, Japan  Kento Asai
<EMAIL_ADDRESS>Institute for Cosmic Ray Research (ICRR), The
University of Tokyo, Kashiwa, Chiba 277–8582, Japan  Yohei Nakashima
<EMAIL_ADDRESS>Department of Physics, Kyushu
University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan  Takashi Shimomura
<EMAIL_ADDRESS>Faculty of Education, Miyazaki University,
Miyazaki, 889-2192, Japan

###### Abstract

We consider a dark photon model in which the dark U(1) gauge symmetry is
spontaneously broken by a vacuum expectation value of a new scalar boson. We
focus on the ForwArd Search ExpeRiment (FASER) and calculate its sensitivity
to the dark photon produced from the off-shell decay of the new scalar boson.
It is found that the off-shell production extends the sensitivity region
beyond the kinematical threshold of the on-shell decay of the scalar boson,
and that the sensitivity region can be spanned to unexplored region. We also
show the parameter space in which perturbative calculation is valid for the
unitarity of an S matrix.

††preprint: UME-PP-028††preprint: KYUSHU-HET-288

##  I Introduction

Feebly interacting and light dark sector has received growing interests in
recent years. It is motivated by unanswered problems in the Standard Model
(SM) such as dark matter, neutrino masses, strong CP problem, and so on (for
review see refs. Agrawal  _et al._ (2021); Feng  _et al._ (2023)). Ordinary
matters interact with the dark sector only via so-called portal particles
whose masses are postulated below GeV scale. Such light portal particles are
searched in present and future high intensity experiments. Since many
extensions beyond the minimal setup have been proposed, detailed studies on
the production of portal particles are important to determine the sensitivity
to the dark sector in experiments.

There are generally four types of portal particles; vector, scalar, fermion,
and axion-like particles. Among them, vector portal is a gauge boson of an
abelian dark symmetry U(1)D under which all SM particles are singlet. The
vector boson has a mass and kinetically mixes with the hypercharge gauge boson
Okun (1982); Galison and Manohar (1984); Holdom (1986); Foot and He (1991);
Babu  _et al._ (1998). Due to the kinetic mixing, it can interact with the SM
particles through the electromagnetic current. In this sense, it is called
dark photon Pospelov  _et al._ (2008); Huh  _et al._ (2008); Pospelov (2009).
The dark photon can be directly produced from bremsstrahlung, meson decays,
and QCD processes in terrestrial experiments. In dark photon models, the dark
Higgs boson also can exist, which gives a mass to the dark photon by
spontaneous breaking of U(1)D symmetry. As a direct consequence of the mass
generation, the dark Higgs boson can decay into a pair of dark photons. Such a
decay can be dominant in the dark Higgs boson decays due to the enhancement by
longitudinal polarization of massive dark photon. In this situation, the dark
Higgs decay becomes new production process of dark photon. In refs. Araki  _et
al._ (2021, 2023); Foguel  _et al._ (2022); Cheung  _et al._ (2024), the dark
photon production from the dark Higgs decay at the FASER and other experiments
Feng  _et al._ (2018a, b); Ariga  _et al._ (2019, 2018a) was studied. Using a
large number of the dark Higgs bosons produced by heavy meson decays through
the scalar mixing with the SM Higgs boson, it was shown that the sensitivity
to the dark photon parameters can be enlarged. The similar analyses have been
studied for inelastic dark matter Li  _et al._ (2022), lepton flavor violation
Araki  _et al._ (2023), dark Higgs Foguel  _et al._ (2022); Ferber  _et al._
(2024); Felkl  _et al._ (2023) in gauge $L_{\mu}-L_{\tau}$ Nomura and
Shimomura (2021) and $B-L$ models Dev  _et al._ (2021). In these studies, only
decay of the on-shell dark Higgs was considered.

When the dark Higgs boson is heavier than the mesons, it cannot be produced
from the meson decays. Even if the dark Higgs boson is lighter than the meson,
it cannot decay into the dark photons when it is lighter than a pair of dark
photons. Instead, the mesons can decay directly into the dark photons through
off-shell state of the dark Higgs boson. Such meson decays can be substantial
due to the enhancement by the longitudinal polarization. In fact, suppression
by the propagator of the dark Higgs boson can be compensated by the
contribution from longitudinal polarization111Similar case was studied for a
pair production of the dark Higgs boson through the off-shell SM Higgs boson
in meson decay Feng  _et al._ (2018b). . On the other hand, dark matter
contained in the dark sector can interact with SM particles through the dark
photon. When the dark matter is produced from freeze-out mechanism in the
early Universe, the gauge coupling constant of U(1)D must be of order unity to
satisfy the observed dark matter abundance Izaguirre  _et al._ (2013); Batell
_et al._ (2014); Kahn  _et al._ (2015); Izaguirre  _et al._ (2015). Such a
large gauge coupling will also result in substantial number of dark photons
via off-shell state of the dark Higgs boson. Therefore, the sensitivity to the
dark photon parameters should be studied in the case of the off-shell dark
Higgs boson. In addition, the perturbative unitarity bound Lee  _et al._
(1977a, b) suggests similar order of the mass of the dark Higgs boson to that
of the dark photon for the large gauge coupling. Then, it will be natural to
consider the light dark photon and dark Higgs boson simultaneously.

The dark photon and dark Higgs boson have been searched in collider and fixed
target/beam dump experiments as well as astrophysical observations. To date,
the parameter space of the dark photon and dark Higgs boson have been tightly
constrained. For complete lists of the experiments and constraints, we refer
the reader to review for instance Bauer  _et al._ (2020); Ilten  _et al._
(2018) and Winkler (2019); Ferber  _et al._ (2024). Further searches are
conducted in experiments such as Belle-II Altmannshofer  _et al._ (2019), NA64
Banerjee  _et al._ (2020), LHCb Aaij  _et al._ (2017, 2020). Among these, the
FASER (ForwArd Search ExpeRiment) experiment Feng  _et al._ (2018a, b); Ariga
_et al._ (2019, 2018a) exploits the proton-proton collision at LHC to search
for neutral long-lived particles. The detector is placed 480 m downstream from
the ATLAS interaction point. Utilizing a large cross-section of proton-proton
inelastic interaction in the forward direction, FASER realizes high
sensitivity to new long-lived particles. FASER started data taking since 2022
and will collect about 150 fb-1 of data with 14 TeV collision energy in LHC
Run 3. It is planned to upgrade FASER 2 with larger detector and to take 3
ab-1 at High-Luminosity LHC.

The purpose of this paper is to show that the dark photon production via off-
shell dark Higgs boson gives a substantial number of events. The sensitivity
region at FASER can be enlarged when we consider this process. It is also
shown that the SM Higgs boson invisible decay and perturbative unitarity
bounds can constrain the parameter space when the dark Higgs boson has sub-GeV
mass.

This paper is organized as follows. In section II, we introduce the dark
photon model with dark higgs, and present the relevant interaction Lagrangian
to our analyses. In section III, we give the differential branching ratio of b
quark into the dark photons via off-shell dark Higgs boson. Then, the formula
to calculate expected number of events is shown. Constraints from the
invisible decay of the SM Higgs boson and perturbative unitarity are given in
section IV. In section V, expected sensitivity plots to dark photon decay
signal at FASER2 are presented, and we summarize our analysis in section VI.

##  II Dark Photon Model

We begin our discussion by deriving the interaction Lagrangian used in our
analysis. A dark photon model considered in this study is invariant under the
gauge symmetries $G_{\mathrm{SM}}\times\mathrm{U(1)}_{D}$ where
$G_{\mathrm{SM}}$ represents the SM gauge symmetry. The new gauge symmetry
U(1)D is the symmetry of dark sector and all of the SM fields are assumed to
be uncharged under it. Although the gauge boson $X_{\mu}$ of U(1)D has no
direct interactions to the SM sector, the gauge symmetry allows $X_{\mu}$ to
form the kinetic mixing term with the hypercharge gauge boson $B_{\mu}$ Okun
(1982); Galison and Manohar (1984); Holdom (1986); Foot and He (1991); Babu
_et al._ (1998). Then, the dark photon emerges as a linear combination of
$X_{\mu}$ and $B_{\mu}$. In this study, the dark symmetry is assumed to be
spontaneously broken by a new complex scalar boson $\Phi$. For the sake of
minimality, we consider the case that $\Phi$ is charged only under U(1)D and
its gauge charge is assigned to be unity.

###  II.1 Lagrangian

The Lagrangian of the model is given by

$\displaystyle\mathcal{L}=\mathcal{L}_{\mathrm{SM}}-\frac{1}{4}X^{\mu\nu}X_{\mu\nu}+\frac{\epsilon}{2}B^{\mu\nu}X_{\mu\nu}+|D_{\mu}\Phi|^{2}-V(H,\Phi),$
(1)

where $\mathcal{L}_{\mathrm{SM}}$ denotes the SM Lagrangian without the Higgs
potential. The second and third terms are the kinetic term of $X_{\mu}$ and
kinetic mixing term between $X_{\mu}$ and $B_{\mu}$ with the constant
parameter $\epsilon$. The field strength tensors of $X_{\mu}$ and $B_{\mu}$
are given by

$\displaystyle
X_{\mu\nu}=\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu},~{}~{}~{}B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu},$
(2)

respectively. The fourth term is the kinetic term of $\Phi$. The covariant
derivative acting on $\Phi$ is given by

$\displaystyle D_{\mu}=\partial_{\mu}-ig^{\prime}X_{\mu},$ (3)

where $g^{\prime}$ is the gauge coupling constant of U(1)D. The last term
$V(H,\Phi)$ is the scalar potential in which $H$ is the electroweak SU(2)
doublet field. The concrete form of $V$ is given by

$\displaystyle
V(H,\Phi)=-\mu_{H}^{2}H^{\dagger}H-\mu_{\Phi}^{2}\Phi^{\dagger}\Phi+\frac{\lambda_{H}}{2}(H^{\dagger}H)^{2}+\frac{\lambda_{\Phi}}{2}(\Phi^{\dagger}\Phi)^{2}+\lambda_{H\Phi}(H^{\dagger}H)(\Phi^{\dagger}\Phi),$
(4)

where $\mu^{2}_{H}$ and $\mu^{2}_{\Phi}$ as well as the quartic couplings
$\lambda_{H}$, $\lambda_{\Phi}$ and $\lambda_{H\Phi}$ are taken to be
positive. The values of these parameters are assumed to be chosen
appropriately so that the elecroweak and dark symmetry are broken
spontaneously as desired.

###  II.2 Mass eigenstates of the scalar and gauge field

On the vacuum, the scalar fields $H$ and $\Phi$ are expanded around their
vacuum expectation values $v$ and $v_{\Phi}$, respectively, as

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v+H^{\prime}\end{pmatrix},~{}~{}~{}~{}\Phi=\frac{1}{\sqrt{2}}(v_{\Phi}+\Phi^{\prime}),$
(5)

where $H^{\prime}$ and $\Phi^{\prime}$ are the dynamical degrees of freedom.
Here, the Nambu-Goldstone bosons to be absorbed by the gauge bosons are
omitted. Inserting eq. (5) into the scalar potential eq. (4), we can obtain
the scalar fields $h$ and $\phi$ in the mass basis

$\displaystyle\begin{pmatrix}H^{\prime}\\\
\Phi^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\alpha h+\sin\alpha\phi\\\
-\sin\alpha h+\cos\alpha\phi\end{pmatrix},$ (6)

where the scalar mixing angle $\alpha$ is defined by

$\displaystyle\tan
2\alpha=-\frac{2\lambda_{H\Phi}vv_{\Phi}}{\lambda_{H}v^{2}-\lambda_{\Phi}v_{\Phi}^{2}}.$
(7)

The masses of $h$ and $\phi$ are given by

$\displaystyle m_{h}^{2}$
$\displaystyle=\lambda_{H}v^{2}c_{\alpha}^{2}+\lambda_{\Phi}v_{\Phi}^{2}s_{\alpha}^{2}-2\lambda_{H\Phi}vv_{\Phi}s_{\alpha}c_{\alpha},$
(8a) $\displaystyle m_{\phi}^{2}$
$\displaystyle=\lambda_{H}v^{2}s_{\alpha}^{2}+\lambda_{\Phi}v_{\Phi}^{2}c_{\alpha}^{2}+2\lambda_{H\Phi}vv_{\Phi}s_{\alpha}c_{\alpha},$
(8b)

where $s_{\alpha}$ and $c_{\alpha}$ are abbreviations for $\sin\alpha$ and
$\cos\alpha$. In our analysis, we take $\alpha$ to be small and identify $h$
as the SM Higgs boson with $m_{h}=125$ GeV.

The mass eigenstates of the neutral gauge bosons $A_{\mu},~{}Z_{\mu}$ and
$A^{\prime}_{\mu}$ are obtained by diagonalizing the mass term and kinetic
term simultaneously, which are given by

$\displaystyle\begin{pmatrix}\tilde{A}_{\mu}\\\ \tilde{Z}_{\mu}\\\
X_{\mu}\end{pmatrix}=\begin{pmatrix}1&\epsilon c_{W}rs_{\chi}&\epsilon
c_{W}rc_{\chi}\\\ 0&c_{\chi}-\epsilon s_{W}rs_{\chi}&-s_{\chi}-\epsilon
s_{W}rc_{\chi}\\\ 0&rs_{\chi}&rc_{\chi}\end{pmatrix}\begin{pmatrix}A_{\mu}\\\
Z_{\mu}\\\ A^{\prime}_{\mu}\end{pmatrix},$ (9)

where $\tilde{A}_{\mu}$ and $\tilde{Z}_{\mu}$ denote the SM photon and neutral
weak boson, respectively, in the limit of $\epsilon\to 0$. The gauge mixing
angle is abbreviated as $s_{\chi}=\sin\chi$ and $c_{\chi}=\cos\chi$, and
$r=(1-\epsilon^{2})^{-1/2}$. Similarly, $s_{W}=\sin\theta_{W}$ and
$c_{W}=\cos\theta_{W}$ for the Weinberg angle. The gauge mixing angle is
defined by

$\displaystyle\tan 2\chi=\frac{2\epsilon
rs_{W}m_{\tilde{Z}}^{2}}{r^{2}m_{X}^{2}-(1-\epsilon^{2}r^{2}s_{W}^{2})m_{\tilde{Z}}^{2}},$
(10)

where $m_{\tilde{Z}}$ is the SM $Z$ boson mass and

$\displaystyle m_{X}=g^{\prime}v_{\Phi}.$ (11)

The masses of $Z$ and $A^{\prime}$ are given by

$\displaystyle m_{Z}^{2}$
$\displaystyle=m_{\tilde{Z}}^{2}c_{\chi}^{2}+r^{2}(m_{X}^{2}+\epsilon^{2}s_{W}^{2}m_{\tilde{Z}}^{2})s_{\chi}^{2}-2\epsilon
rs_{W}m_{\tilde{Z}}^{2}s_{\chi}c_{\chi},$ (12a) $\displaystyle
m_{A^{\prime}}^{2}$
$\displaystyle=m_{\tilde{Z}}^{2}s_{\chi}^{2}+r^{2}(m_{X}^{2}+\epsilon^{2}s_{W}^{2}m_{\tilde{Z}}^{2})c_{\chi}^{2}+2\epsilon
rs_{W}m_{\tilde{Z}}^{2}s_{\chi}c_{\chi}.$ (12b)

###  II.3 Yukawa interaction of dark Higgs boson

The dark Higgs boson can interact with the SM fermions through the scalar
mixing. The interaction Lagrangian is obtained by inserting eq.(6) into the
Yukawa interaction terms as

$\displaystyle\mathcal{L}_{\mathrm{yukawa}}$
$\displaystyle=\sum_{f}\frac{m_{f}}{v}\overline{f}f(\cos\alpha
h+\sin\alpha\phi),$ (13)

where $f$ runs over all of the SM fermions and $m_{f}$ is the mass of the
fermion $f$.

###  II.4 Gauge interaction of dark photon and dark Higgs boson

In the parameter space of our interest, the conditions that $\epsilon\ll 1$
and $m_{A^{\prime}}^{2}\ll m_{Z}^{2}$ are generally satisfied. Then, the gauge
mixing angle eq. (10) is approximated as

$\displaystyle\chi\simeq-\epsilon s_{W}+\mathcal{O}(\epsilon^{3}).$ (14)

The mass eigenstates eq. (9) become

$\displaystyle\begin{pmatrix}\tilde{A}_{\mu}\\\ \tilde{Z}_{\mu}\\\
X_{\mu}\end{pmatrix}\simeq\begin{pmatrix}1&0&\epsilon c_{W}\\\ 0&1&0\\\
0&-\epsilon s_{W}&1\end{pmatrix}\begin{pmatrix}A_{\mu}\\\ Z_{\mu}\\\
A^{\prime}_{\mu}\end{pmatrix},$ (15)

and their masses eq. (12) become

$\displaystyle m_{Z}^{2}\simeq
m_{\tilde{Z}}^{2},~{}~{}~{}m_{A^{\prime}}^{2}\simeq m_{X}^{2},$ (16)

up to the order of $\epsilon$, respectively. Thus, the interaction term of
dark photon with the SM fermions is simplify obtained by replacing
$\tilde{A}_{\mu}\to A_{\mu}+\epsilon c_{W}A^{\prime}_{\mu}$ in the SM
Lagrangian as

$\displaystyle\mathcal{L}_{A^{\prime}-\mathrm{fermion}}=\varepsilon
eJ^{\mu}_{\mathrm{em}}A^{\prime}_{\mu},$ (17)

where $J^{\mu}_{\mathrm{em}}$ is the electromagnetic current for the fermions.
Here, we define $\varepsilon\equiv\epsilon c_{W}$.

The gauge interactions of the scalar bosons in mass basis are obtained from
the scalar kinetic terms by using eqs. (6) and (15). The relevant interactions
in our analysis are given by

$\displaystyle\mathcal{L}_{\mathrm{gauge-scalar}}$
$\displaystyle=\frac{1}{2}{g^{\prime}}^{2}A^{\prime}_{\mu}A^{\prime\mu}(-s_{\alpha}h+c_{\alpha}\phi)^{2}+g^{\prime}m_{A^{\prime}}A^{\prime}_{\mu}A^{\prime\mu}(-s_{\alpha}h+c_{\alpha}\phi).$
(18)

The last term represents the decay of $\phi$ into a dark photon pair, which we
analyze in this paper.

###  II.5 Scalar self-interactions

In the end of this section, we give the scalar self-interaction Lagrangian
which is derived from the scalar potential eq. (4). Inserting eqs. (5) and (6)
into eq.(4), one obtains

$\displaystyle\mathcal{L}_{\mathrm{scalar-self}}$
$\displaystyle=-\frac{\lambda_{H}}{8}\big{[}4v(c_{\alpha}h+s_{\alpha}\phi)^{3}+(c_{\alpha}h+s_{\alpha}\phi)^{4}\big{]}$
$\displaystyle\quad-\frac{\lambda_{\Phi}}{8}\big{[}4v_{\Phi}(-s_{\alpha}h+c_{\alpha}\phi)^{3}+(-s_{\alpha}h+c_{\alpha}\phi)^{4}\big{]}$
$\displaystyle\quad-\frac{\lambda_{H\Phi}}{4}\big{[}2v(c_{\alpha}h+s_{\alpha}\phi)(-s_{\alpha}h+c_{\alpha}\phi)^{2}+2v_{\Phi}(c_{\alpha}h+s_{\alpha}\phi)^{2}(-s_{\alpha}h+c_{\alpha}\phi)$
$\displaystyle\qquad\qquad+(c_{\alpha}h+s_{\alpha}\phi)^{2}(-s_{\alpha}h+c_{\alpha}\phi)^{2}\big{]}.$
(19)

The scalar self-interactions are used to derive the perturbative unitarity
constraints.

##  III Expected Number of Dark Photon Decay at FASER

The expected number of dark photon decays is given in this section. We study
the situation where the dark photons are produced from the decays of the dark
Higgs boson and consider the detection at the FASER2 experiment Feng  _et al._
(2018a, b); Kling and Trojanowski (2018); Feng  _et al._ (2018c); Ariga  _et
al._ (2018b, 2019, a); Feng  _et al._ (2023); Abreu  _et al._ (2024). If the
dark Higgs boson is light enough, it can be produced as on-shell state from
meson decays, such as $B$, $K$, $\eta$ and $\eta^{\prime}$. Among these, the
$B$ meson is dominant source for the dark Higgs boson. The $K_{L}$ meson is
sub-dominant because it is absorbed by LHC neutral target absorbers before its
decay Feng  _et al._ (2018b). The contributions from other mesons are almost
negligible due to suppressed branching fraction. For the production of the
off-shell dark Higgs boson, the $B$ meson decay also will be dominant one
because the decay vertex of the dark Higgs boson is common in all mesons as
shown in figure 1. Thus, we evaluate the $B$ meson decay in the following
analyses.

###  III.1 Differential decay width of $B$ meson into Dark Photons

Figure 1: Feynman diagram of the dark photon pair production through the off-
shell dark Higgs boson in $b$ quark decay.

The decay of $B$ meson into two dark photons through the off-shell dark Higgs
boson can occur in loop-process shown in figure 1. The elementary process of
the decay is

$\displaystyle b\to s+\phi^{\ast}\to s+A^{\prime}+A^{\prime},$ (20)

where $b$ and $s$ denote bottom and strange quark, respectively, and
$\phi^{\ast}$ off-shell dark Higgs boson. In the loop, all up-type quarks
propagate and contribute to the decay. However, the decay process is dominated
by the top quark loop due to the large Yukawa coupling. Since we consider the
situation where the dark Higgs boson is lighter than $W$ boson and top quark
$t$, the off-shell decay of $b\to s+\phi^{\ast}$ can be described by an
effective Lagrangian Grinstein  _et al._ (1988); Chivukula and Manohar (1988)
as

$\displaystyle\mathcal{L}_{\Delta B=1}^{\mathrm{eff}}=$
$\displaystyle\frac{3\alpha_{\mathrm{em}}}{32\pi
s_{W}^{2}}V_{tb}\frac{m_{t}^{2}}{m_{W}^{2}}V_{ts}^{\ast}\frac{c_{\alpha}h+s_{\alpha}\phi}{v}m_{b}\overline{s}\left(1+\gamma^{5}\right)b+\mathrm{H.c.},$
(21)

where $\alpha_{\mathrm{em}}$ is the fine structure constant and
$V_{tb},~{}V_{ts}$ are the Cabibbo-Kobayashi-Maskawa matrix element Cabibbo
(1963); Kobayashi and Maskawa (1973); Workman and Others (2022). The masses of
top, bottom quark and $W$ boson are denoted as $m_{t}$, $m_{b}$ and $m_{W}$,
respectively.

Using the effective Lagrangian eq. (21) and the gauge-scalar interaction, the
decay amplitude of the process (20) can be given by

$\displaystyle M\left(b\to sA^{\prime}A^{\prime}\right)$
$\displaystyle=\left(\frac{3\alpha_{\mathrm{em}}}{32\pi
s_{W}^{2}}V_{tb}\frac{m_{t}^{2}}{m_{W}^{2}}V_{ts}^{\ast}\frac{s_{\alpha}}{v}m_{b}\right)\left(i\frac{m^{2}_{A^{\prime}}}{v_{\Phi}}c_{\alpha}\right)$
$\displaystyle\times\overline{u}\left(p_{1}\right)\left(1+\gamma_{5}\right)u\left(k\right)\frac{i}{s_{23}-m_{\phi}^{2}+i\mathrm{Im}(\Pi(s_{23}))}\epsilon_{1}^{*\mu}\left(p_{2}\right)\epsilon_{2\mu}^{*}\left(p_{3}\right),$
(22)

where four momentum assignment to each particle is shown in figure 1 and
$s_{23}\equiv(p_{2}+p_{3})^{2}$. The Dirac spinor with four momentum $p$ is
denoted by $u(p)$, and similarly the polarization vector of dark photon is
$\epsilon_{i}^{\mu}(p),(i=1,2)$, respectively. In the propagator of the dark
Higgs boson, $\Pi(s_{23})$ represents the sum of one-particle irreducible
amplitudes for two-point function of $\phi$. When the propagator has narrow
peak around $m_{\phi}$, the imaginary part can be given by
$m_{\phi}\Gamma(m_{\phi}^{2})$. However, in our case, the imaginary part can
be comparable to $m_{\phi}$ due to the decay of $\phi\to
A^{\prime}A^{\prime}$, in which the longitudinal polarization enhances the
amplitude. Thus, the $s_{23}$ dependence of $\Pi$ should be kept. In order to
do it, we replace the imaginary part with
$\sqrt{s_{23}}~{}\Gamma_{\phi}(s_{23})$.

The differential decay width with respect to $s_{23}$ is given by

$\displaystyle\frac{d}{ds_{23}}\Gamma\left(b\to sA^{\prime}A^{\prime}\right)$
$\displaystyle=\frac{9(g^{\prime}\alpha_{\mathrm{em}}s_{\alpha}c_{\alpha})^{2}m_{A^{\prime}}^{2}m_{b}^{3}}{2^{16}\pi^{5}s_{W}^{4}v^{2}}\left|V_{tb}\right|^{2}\left|V_{ts}\right|^{2}\frac{m_{t}^{4}}{m_{W}^{4}}\left(1-\frac{s_{23}}{m_{b}^{2}}\right)^{2}$
$\displaystyle\times\sqrt{1-\frac{4m_{A}^{\prime
2}}{s_{23}}}\frac{1}{\left(s_{23}-m_{\phi}^{2}\right)^{2}+s_{23}(\Gamma_{\phi}(s_{23}))^{2}}\left[2+\left(\frac{s_{23}}{2m_{A^{\prime}}^{2}}-1\right)^{2}\right],$
(23)

where the second term in the last parenthesis represents the longitudinal
polarization. The partial decay width can be obtained by integrating over
$s_{23}$ from $4m_{A^{\prime}}^{2}$ to $m_{b}^{2}$. It is noticed that the
constribution from $K$ meson decay is negligible due to the small CKM angle or
Yukawa coupling suppression of charm quark.

###  III.2 Expected number of signal events at FASER

|  $L_{\rm min}$ (m) |  $L_{\rm max}$ (m) |  $R$ (m) |  $\mathcal{L}$ (ab-1)
---|---|---|---|---
FASER 2 | 475 | 480 | 1.0 | 3.0
Table 1:  Dimension of the FASER2 detector and integrated luminosity used for
this study. $L_{\rm min}$ and $L_{\rm max}$ are the distance to the front and
rear end of the detector from the IP, respectively. $R$ is detector radius,
respectively. $\mathcal{L}$ is the integrated luminosity.

The dark photon produced from $B$ meson decay propagates to the FASER2
detector, and then decays into the SM charged particles. The expected number
of dark photon decay is calculated by

$\displaystyle N$ $\displaystyle=\mathcal{L}\int
dp_{B}d\theta_{B}\frac{d\sigma_{pp\to
B}}{dp_{B}d\theta_{B}}~{}\int_{4m_{A^{\prime}}^{2}}^{m_{b}^{2}}ds_{23}\left[\frac{d}{ds_{23}}{\rm
Br}(B\to X_{s}A^{\prime}A^{\prime})\right]\left(\mathcal{P}_{A^{\prime}}^{\rm
det}(\bm{p}_{2})+\mathcal{P}_{A^{\prime}}^{\rm det}(\bm{p}_{3})\right)~{},$
(24)

where the integrated luminosity is written as $\mathcal{L}$, the momentum and
angle of each $B$ meson are denoted as $p_{B}$ and $\theta_{B}$, respectively.
The differential cross section of $B$ meson is taken from the data file in
FORESEE package Kling and Trojanowski (2021). The differential decay branching
ratio is given by

$\displaystyle\frac{d}{ds_{23}}{\rm Br}(B\to
X_{s}A^{\prime}A^{\prime})\equiv\frac{1}{\Gamma_{b}}\frac{d}{ds_{23}}\Gamma(b\to
sA^{\prime}A^{\prime}),$ (25)

where $\Gamma_{b}$ is the full width of $b$ quark. The probability that
$A^{\prime}$ decays inside the detector with momentum $\bm{p}$ is written by
$\mathcal{P}_{A^{\prime}}^{\rm det}(\bm{p})$ which is given by

$\displaystyle\mathcal{P}_{A^{\prime}}^{\rm
det}(\bm{p})=\left(e^{-L_{\mathrm{min}}/\beta\gamma\tau_{A^{\prime}}}-e^{-L_{\mathrm{max}}/\beta\gamma\tau_{A^{\prime}}}\right)\Theta\left(R-L_{\mathrm{min}}\tan\theta\right),$
(26)

where $\beta\gamma$ is the Lorentz factor corresponding to the momentum
$|\bm{p}|$, and $\theta$ is the angle of the momentum to the beam direction.
The last term $\Theta(x)$ is the step function. The distance to the front
(rear) of the detector from the IP is denoted as $L_{\mathrm{min(max)}}$, and
the radius of the detector is $R$. Dimension of the detector and integrated
luminosity for our study is summarized in table 1.

##  IV Constraints

The dark photon and dark Higgs boson have been searched by beam dump, collider
and neutrino experiments so far, and at present tight constraints are set
separately on the parameter space of those particles Feng  _et al._ (2023).
These constraints should be applied to the dark photon and dark Higgs boson in
our model. In addition, there are other constraints which arise from the
interactions of the dark photon with the dark Higgs boson. One of such
constraints is the decay of the SM Higgs boson into the dark Higgs bosons and
dark photons through the scalar mixing. The other one is the perturbative
unitarity constraint on the dark photon scattering, when we assume that the
perturbative expansion is valid at least at the scale of experiments. In this
section, we derive these constraints in our dark photon model.

###  IV.1 Invisible decay of SM Higgs boson

The SM Higgs boson can decay into a dark photon pair through the scalar mixing
when the dark photon is light enough. Since the dark photon is long-lived in
the parameter space of our interest, the dark photons will escape detectors.
Then, such a decay will be regarded as invisible decay of the SM Higgs boson
in experiments. The partial width of this decay can be significantly enhanced
due to the longitudinal mode of the dark photon, resulting in a large
branching ratio of the invisible decay of the SM Higgs boson.

The partial decay widths of $h\to A^{\prime}A^{\prime}$ and $h\to\phi\phi$ are
given by

$\displaystyle\Gamma(h\to A^{\prime}A^{\prime})$
$\displaystyle=\frac{g^{\prime
2}\sin^{2}\alpha}{8\pi}\frac{m_{A^{\prime}}^{2}}{m_{h}}\sqrt{1-\frac{4m_{A^{\prime}}^{2}}{m_{h}^{2}}}\left(2+\frac{m_{h}^{4}}{4m_{A^{\prime}}^{4}}\left(1-\frac{2m_{A^{\prime}}^{2}}{m_{h}^{2}}\right)^{2}\right),$
(27a) $\displaystyle\Gamma(h\to\phi\phi)$
$\displaystyle=\frac{{g^{\prime}}^{2}c_{\alpha}^{6}\tan^{2}2\alpha}{128\pi}\frac{m_{h}^{3}}{m_{A^{\prime}}^{2}}\sqrt{1-\frac{4m_{A^{\prime}}^{2}}{m_{h}^{2}}},$
(27b)

where we used the term proportional to $\lambda_{H\Phi}vc_{\alpha}^{3}$ and
neglected other terms due to the suppression by $m_{\Phi}/m_{A^{\prime}}$ and
$\alpha^{4}$. We neglected the partial decay width of $h\to\phi\phi\phi$
because of the smaller coupling constants and phase space suppression. The
branching ratio of the invisible decay of the SM Higgs boson is given by

$\displaystyle\mathrm{Br}(h\to\mathrm{invisible})=\frac{\Gamma(h\to
A^{\prime}A^{\prime})+\Gamma(h\to\phi\phi)+\Gamma(h\to
4\nu)}{\Gamma_{h}+\Gamma(h\to A^{\prime}A^{\prime})+\Gamma(h\to\phi\phi)},$
(28)

where $\Gamma_{h}=4.07$ MeV is the total decay width predicted by the SM
Workman and Others (2022), and $\Gamma(h\to 4\nu)=1.2\times 10^{-3}\Gamma_{h}$
is the partial decay width into $\nu\bar{\nu}\nu\bar{\nu}$ de Florian  _et
al._ (2016); Workman and Others (2022). The current experimental upper bounds
on the invisible decay branching ratio of the SM Higgs boson are placed by
ATLAS Aad  _et al._ (2023) and CMS Tumasyan  _et al._ (2022, 2023). ATLAS set
more stringent bound at $95$% C.L. combined Run 1 and 2 data as

$\displaystyle\mathrm{Br}^{\mathrm{comb}}_{\mathrm{inv}}<0.107.$ (29)

By future collider experiments, such as the High-Luminisity LHC (HL-LHC) and
International Linear Collider (ILC), expected 95% C.L. upper limit reach
BR${}_{\rm inv}^{\rm HL-LHC}<0.025$ ATL (2022) and BR${}_{\rm inv}^{\rm
ILC}<0.0016$ Potter  _et al._ (2022), respectively. We use these current and
expected upper limits in our analysis.

###  IV.2 Perturbative unitarity

Assuming the perturbative expansion of amplitudes is valid up to the LHC beam
energy scale, the dark Higgs boson mass can be bound by the dark photon mass
and gauge coupling. This is because the dark Higgs boson is the origin of the
dark photon mass, and known as perturbative unitarity bound Lee  _et al._
(1977a, b).

The constraint can be obtained from the amplitudes of $A^{\prime}A^{\prime}\to
A^{\prime}A^{\prime},~{}A^{\prime}A^{\prime}\leftrightarrow\phi\phi,~{}\phi\phi\to\phi\phi$
and $A^{\prime}\phi\to A^{\prime}\phi$ scatterings. In the center of mass
frame of the initial particles, the scattering amplitude can be written by
using partial wave expansion as

$\displaystyle M=\sum_{l=0}^{\infty}8\pi\frac{E_{\rm
CM}}{|\bm{p}|}(2l+1)a_{l}(|\bm{p}|)P_{l}(\cos\theta),$ (30)

where $E_{\rm CM}$ and $\bm{p}$ denote the energy and momentum of the initial
particles, respectively, and $\theta$ is the scattering angle of a final
particle with respect to $\bm{p}$. The Legendre polynomial is denoted as
$P_{l}(\cos\theta)$. The expansion coefficient $a_{l}(|\bm{p}|)$ is restricted
so that the perturbative unitarity is maintained. The constraint for the
S-wave coefficient is given by

$\displaystyle\left|\mathrm{Re}(a_{0}(\bm{p}))\right|<\frac{1}{2}.$ (31)

The coefficients for the scatterings of $A^{\prime}$ and $\phi$ are read from
the scattering amplitudes using eqs. (18) and (19),

$\displaystyle a_{0}(A^{\prime}A^{\prime}\to
A^{\prime}A^{\prime})=-\frac{3}{16\pi}\frac{g^{\prime
2}m_{\phi}^{2}}{m_{A^{\prime}}^{2}},$ (32a) $\displaystyle
a_{0}(A^{\prime}A^{\prime}\leftrightarrow\phi\phi)=\frac{1}{16\pi}\left(2g^{\prime
2}+g^{\prime 2}\frac{m_{\phi}^{2}}{m_{A^{\prime}}^{2}}-4g^{\prime
2}\log\left(\frac{s}{m_{A^{\prime}}^{2}}\right)\right),$ (32b) $\displaystyle
a_{0}(\phi\phi\to\phi\phi)=-\frac{3}{16\pi}\frac{g^{\prime
2}m_{\phi}^{2}}{m_{A^{\prime}}^{2}},$ (32c) $\displaystyle
a_{0}(A^{\prime}\phi\to A^{\prime}\phi)=\frac{1}{16\pi}\left(g^{\prime
2}-g^{\prime 2}\frac{m_{\phi}^{2}}{m_{A^{\prime}}^{2}}-2g^{\prime
2}\log\left(\frac{s}{m_{A^{\prime}}^{2}}\right)\right),$ (32d)

where $s$ is the squared center-of-mass energy. It should be noticed that the
$\log(s)$ term cannot be neglected when $m_{A^{\prime}}$ is much smaller
compared with $s$, in particular for large $g^{\prime}$. Then, diagonalizing
the matrix of the coefficients in basis of $A^{\prime}A^{\prime}$, $\phi\phi$
and $A^{\prime}\phi$, the constraint is given by the largest eigenvalue,
$|a_{0,\mathrm{max}}|$ as

$\displaystyle|a_{0,\mathrm{max}}|<\frac{1}{2}.$ (33)

As we will see in the next section, the perturbative unitarity bound can be
tighter in heavy $m_{\phi}$ spectrum than the invisible decay of the SM Higgs
boson.

##  V Sensitivity region

We now show the expected sensitivity region of dark photon decays in
$m_{A^{\prime}}\,\mathchar 45\relax\,\varepsilon$ plane, where $\varepsilon$
is defined just below eq. (17), for the FASER2 setup. The expected number of
events is calculated by using eqs. (23) and (24). As was discussed in refs.
Ariga  _et al._ (2018b); Araki  _et al._ (2021), the possible main backgrounds
are high energetic muons with radiation and neutrinos. These can be reduced to
a negligible level by assuming $99.99$% veto efficiency. In eq. (23), the on-
shell state as well as the off-shell state of the dark Higgs boson are
automatically included. As representative examples, the parameters are chosen
as $g^{\prime}=0.7$ and $0.5$ for $m_{\phi}=2,~{}0.5$ and $0.05$ GeV,222For
$g^{\prime}<0.7$, the log(s) term as well as the constant $g^{\prime 2}$ term
can be safely neglected for our choice of parameters. However, for
$g^{\prime}>0.7$, these terms becomes comparable or larger than $g^{\prime
2}m_{\phi}^{2}/m_{A^{\prime}}^{2}$. while the scalar mixing is fixed to be
$\alpha=10^{-4}$.

|
---|---

Figure 2: The $95$% C.L. sensitivity contours for the dark photon decays for
$m_{\phi}=2$ (top left), $0.5$ (top right) and $0.05$ (bottom) GeV. The gauge
coupling is taken to be $g^{\prime}=0.7$. The red curves represent our results
which include both the on-shell and off-shell dark Higgs state, while the
black solid curves are only for the on-shell dark Higgs state. Green and
purple colored regions are excluded by the perturbative unitarity and
invisible decay of the SM Higgs boson constrains, respectively. Details can be
found in the text.

Figure 2 shows the sensitivity region for $g^{\prime}=0.7$ at $95$% C.L. which
corresponds to $3$ signal events with null background. The mass of the dark
Higgs boson is taken to be $2$ (top left), $0.5$ (top right) and $0.05$
(bottom) GeV, respectively. The red solid curve in each panel represents our
result which includes the on-shell and off-shell dark Higgs state, while the
black solid one represents the same sensitivity region only for the on-shell
decay of the dark Higgs boson calculated by following ref. Araki  _et al._
(2021). The black dashed curve shows $95$% C.L. sensitivity region of the so-
called vanilla dark photon model without dark Higgs boson for comparison Feng
_et al._ (2023). The grey regions are excluded by past experiments and
observation Kling and Trojanowski (2021) (references therein). The purple
vertical lines indicate the upper limit on the dark photon mass obtained from
the invisible decay of the SM Higgs boson, eq. (28), corresponding to
$\mathrm{Br}(h\to\mathrm{inv.})=0.107$ (solid), $0.05$ (dashed) and $0.01$
(dotted). The purple colored region is excluded by the present experimental
result Aad  _et al._ (2023). The green colored region is also excluded by the
perturbative unitarity constraint in eq. (33).

From the figure, one can see that the sensitivity regions cover the wide area
of the parameter space which cannot be explored by the on-shell decay of the
dark Higgs boson. In each panel, the sensitivity regions reach to
$m_{A^{\prime}}\simeq 1.2$ GeV which is almost independent of $m_{\phi}$. This
behavior is attributed to the follows facts. In the region where
$2m_{A^{\prime}}>m_{\phi}$, $s_{23}$ is always larger than $m_{\phi}^{2}$. The
propagator in eq. (23) loses $m_{\phi}$ dependence and behaves as
$1/s_{23}^{2}$ for large $s_{23}$. That propagator is canceled by the
longitudinal contributions of the dark photon in the $b$ decay. This feature
is shown in figure 3 and explained later. For heavy $m_{\phi}$, the
perturbative unitarity constraint is more stringent than the invisible decay
of the SM Higgs boson. It has already excluded the dark photon
$m_{A^{\prime}}<0.8$ GeV.333For $g^{\prime}=1$, the upper limit on
$m_{A^{\prime}}$ from the perturbative unitarity constraint is
$\mathcal{O}(10)$ GeV. This bound is mainly determined by the $\log(s)$ term.
In such a case, higher order processes should be taken into account. On the
other hand, for light $m_{\phi}$, the constraint on the invisible decay of the
SM Higgs boson has already excluded the light dark photon from the on-shell
decay of the dark Higgs boson. Therefore, the off-shell decays of the dark
Higgs bosons should be taken into account for dark photon search. In the end,
compared with the on-shell decay of the dark Higgs boson, our sensitivity
region differs in $m_{\phi}>2m_{A^{\prime}}$ region especially for light
$m_{\phi}$. This could be because the narrow width approximation is not valid
for such spectrum. Due to the longitudinal enhancement of $\phi\to
A^{\prime}A^{\prime}$ decay, the $\phi$ decay width is comparable to its mass.

|
---|---
Figure 3: Differential branching ratio with respect to $s_{23}$. The dark
photon mass is taken to $0.1$ (left) and $1.1$ GeV. Black and red curves
correspond to the $m_{\phi}=2$ and $0.5$ GeV in both panel. The solid curve
represents the decay into both the longitudinal and transverse contribution of
the dark photon while the dashed one represents only transverse component.

Figure 3 is the comparison of the differential branching ratios of $b\to
sA^{\prime}A^{\prime}$ with that for the transverse components $b\to
sA^{\prime}_{T}A^{\prime}_{T}$ with respect to $s_{23}$. The mass of the dark
Higgs boson is chosen as $m_{\phi}=2$ (black) and $0.5$ (red) GeV, and the
dark photon mass is taken to $0.1$ (left) and $1.1$ GeV (right), respectively.
The solid and dashed curves represent the total and transverse contribution,
respectively. Thus, the difference of both curve is the longitudinal
contribution. Both panels show that the longitudinal contribution dominates
the differential branching ratio except for the lower and upper kinematical
thresholds. In the left panel, the dark photon is light so that it can be
produced by the on-shell decay of the dark Higgs boson. Such a decay can be
seen as a resonance peak in the differential branching ratio. For $m_{\phi}=2$
GeV, the resonance peak is broad due to the large $\Gamma_{\phi}(s_{23})$. On
the other hand, in the right panel, the dark photon is only produced via the
off-shell decay of the dark Higgs boson. Right panel shows that the total
differential branching ratios slowly decrease than those only with the
transverse contribution do. This is because the $s_{23}$ dependence
compensates between the propagator and the longitudinal contribution.

|
---|---

Figure 4: The same plot as Figure 2 for $g^{\prime}=0.5$.

Figure 4 is the same plot as figure 2 for $g^{\prime}=0.5$. Since the expected
number of events is proportional to $g^{\prime 2}$, the sensitivity regions
are narrower. Nonetheless, the sensitivity region from the off-shell decay of
the dark Higgs boson appears in smaller $\epsilon$ region than that from
proton bremsstrahlung and meson decay production. For $g^{\prime}<0.5$, the
sensitivity region from the off-shell decay of the dark Higgs boson coincides
with the on-shell one.

##  VI Conclusion

We have considered a dark photon model with the dark Higgs boson which gives a
mass to the dark photon through spontaneous symmetry breaking of the dark
symmetry. We studied the off-shell contribution of the dark Higgs boson to
dark photon production, and derived the expected sensitivity reach of dark
photon signal at the FASER2 experiment.

We have shown that the off-shell dark Higgs boson contribution for the dark
photon production is important at the FASER experiment. Even for the light
dark Higgs boson, the off-shell process is unsuppressed due to the
longitudinal component of the dark photon, and the sensitivity region extends
to heavier dark photon mass. We also derived the constraint from the invisible
decay of the SM Higgs boson and perturbative unitarity. The perturbative
unitarity constraint is stringent in heavy mass region of the dark Higgs boson
due to the longitudinal enhancement. For lighter dark Higgs boson, the
constraint on the invisible decay of the SM Higgs boson excluded the on-shell
production of the dark photon for $g^{\prime}>0.5$ as shown in Fig. 2 and 4.
Thus, the off-shell decay of the dark Higgs boson should be taken into account
for the dark photon search.

## Acknowledgments

This work was partially supported by JSPS KAKENHI Grant Numbers JP24K07024 (T.
A.), JP23K13097 (K. A.), JP22K03622 (T. S.), and JP23H01189 (T. A. and T. S.).

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3.0 cm3.0cm1.0 cm2.5 cm

#  Shannon entropy to quantify complexity in the financial market

Alexis Rodriguez Carranza Mathematics institute, National University of
Trujillo, Trujillo - Perú<EMAIL_ADDRESS>José Luis Ponte
Bejarano. Juan Carlos Ponte Bejarano.
Segundo Eloy Soto Abanto Department of Science, Cesar Vallejo University,
Trujillo - Perú. (jpontebe@ucvvirtual.edu.pe).Department of Science, Cesar
Vallejo University, Trujillo - Perú. (jcpontep@ucvvirtual.edu.pe).Department
of Science, Cesar Vallejo University, Trujillo - Perú.
(ssotoa@ucvvirtual.edu.pe).

Abstract

In this paper we study the complexity in the information traffic that occurs
in the peruvian financial market, using the Shannon entropy. Different series
of prices of shares traded on the Lima stock exchange are used to reconstruct
the unknown dynamics. We present numerical simulations on the reconstructed
dynamics and we calculate the Shannon entropy to measure its complexity.

Keywords. Dynamic Systems, Temporal series, Shannon entropy.

##  1 Introduction

In a organized and developed society, the economic aspect plays a fundamental
role for the well-being of the members of society, which is why understanding
and knowing the dynamics of the financial market allows sustainable social
growth over time. The crisis that has shaken the world economy should raise
questions for economists about the approach used to analyze economic
phenomena. The classical economic models that are used in the asset market,
see [8], simulate the volatile behavior of exchange rates such as the prices
of financial evaluations negotiated in efficient markets, but the current
exchange rate contains the information available instantaneously and the
observed changes reflect the effect of new events that are unpredictable by
definition.

Information theory, particularly Shannon entropy, has been used in this field
to understand financial market behavior. For example, Chen J. discusses the
similarity between information theory concepts and the economic value of
information with respect to markets [1], and uses Shannon’s entropy to explain
most of the empirical evidence about behaviors of the market, determining the
value of the information according to the number of people who know it [2].
Maasoumi and Racine applied entropy to find non-linear dependencies of stock
price returns and their predictions [7]. Garca, Cruz and Venegas use Shannon
entropy to propose a market efficiency measure to be applied to different
capital markets: DJIA, S & P500, FTSE100 and IPC [5].

These investigations show that Shannon’s entropy has been applied to different
fields of finance in an optimal way. In our case we will use it to determine
the number of bits necessary to obtain information about what happens with the
evolution of the behavior of prices in the financial market. We will do our
research on a couple of price time series: stocks of the companies TelfBC and
Credicorp to determine the complexity in the financial market. For this, it
will be necessary to determine clues about the presence of a fractal attractor
set in the dynamics of the series. Figure 1 shows the evolution of the prices
of these shares.

?figurename? 1: Series of prices of the shares of the companies TelfBC and
Credicorp.

##  2 Reconstruction theorems

######  Definition 1 (time serie).

Given a dynamic system with flow
$\\{\phi_{t}\\}_{t\in\mathop{I\\!\\!R}\nolimits}$ in a phase space $H$. A time
series $s:\mathop{I\\!\\!R}\nolimits\longrightarrow\mathop{I\\!\\!R}\nolimits$
is defined as the values that a function takes, called the observation
function, $F:H\longrightarrow\mathop{I\\!\\!R}\nolimits$ given by
$s(t)=F(\phi_{t}(x_{0}))$, $t\in\mathop{I\\!\\!R}\nolimits,x_{0}\in H$.

For the reconstruction of the hidden attractor in the time series, Takens [11]
uses the information of the dynamic system contained in the time series. For
such a reconstruction, the delay coordinates are defined, with which a single
temporal observation is needed.

######  Definition 2.

Be $\Phi$ the flow of a dynamic system over a differentiable manifold $M$ of
at least class $C^{1}$, $T$ a positive integer (called delay) and
$h:M\rightarrow\mathop{I\\!\\!R}\nolimits$ a class function $C^{1}$. The delay
coordinate application is defined
$F_{(\Phi,T,h)}:M\rightarrow\mathop{I\\!\\!R}\nolimits^{n+1}$ by:

$F_{(\Phi,T,h)}(x)=\left(h(x),h(\Phi_{T}(x)),h(\Phi_{2T}(x)),\ldots,h(\Phi_{nT}(x))\right)$

A geometric idea of the application of delay coordinates $F_{(\Phi,T,h)}$ is
shown in figure 2.

?figurename? 2: Application of delay coordinates $F_{(\Phi,T,h)}$; and
reconstruction of the attractor $A$ using the application $F_{(\Phi,T,h)}$ in
$\mathop{I\\!\\!R}\nolimits^{n+1}$ with $n=2$.

######  Teorema 1 (Takens embedding Theorem [11]).

Be $M$ a differentiable variety, at least of class $C^{2}$, compact of
dimension $m$; $\\{\Phi_{k}\\}_{k\in{\mathbb{Z}}}$ the flow of a discrete
dynamic system over $M$ with $\Phi_{k}:M\rightarrow M$ a diffeomorphism of
class $C^{2}$; and $h:M\rightarrow\mathop{I\\!\\!R}\nolimits$ a function of
class $C^{2}$. So it is a generic property that the application
$F_{(\Phi,h)}:M\rightarrow\mathop{I\\!\\!R}\nolimits^{2m+1}$ defined by

$F_{(\Phi,h)}(x)=\left(h(x),h(\Phi_{k}(x)),h(\Phi_{2k}(x)),\ldots,h(\Phi_{2mk}(x))\right)$

is a embedding of $M$.

Taken’s Theorem tells us that if we have a dynamic system that depends on $m$
variables, and if we choose an observation function $h$, for this system with
those measurements we construct the vectors:

$\left(h(x),h(\Phi_{k}(x)),h(\Phi_{2k}(x)),\ldots,h(\Phi_{2mk}(x))\right)$
$\left(h(\Phi_{k}(x)),h(\Phi_{2k}(x)),\ldots,h(\Phi_{2mk}(x)),h(\Phi_{2mk+1}(x))\right)$
$\vdots$

we can have a copy of the original dynamic system.
This result is important, since we can observe one of the variables of the
system over time and by choosing the dimension of the reconstruction vectors
properly, we can understand the evolution of the system of m variables, as
illustrated in Figure 3.

?figurename? 3: Takens embedding Theorem. Reconstruction of the phase space
$M$ and the attractor $A$ via the application of delay coordinates
$F_{(\Phi,h)}$.

The final generalization, used in this article, was given by Tim Sauer, James
A. Yorke, and Martin Casdagli [9]. They propose that it is possible to have an
injective copy of the attractor via the application of delay coordinates with
A being a fractal set. Some results and definitions are necessary.

######  Lema 1.

[9] Let $A$ a compact subset of $\mathop{I\\!\\!R}\nolimits^{k}$ and
$F_{0},F_{1},\ldots,F_{t}$ Lipschitz applications of $A$ on
$\mathop{I\\!\\!R}\nolimits^{n}$. For each integer $r\geq 0$, be $S_{r}$ the
set of pairs $x\neq y$ on $A$ for which the matrix $n\times t$

$M_{xy}=\left[F_{1}(x)-F_{1}(y),\ldots,F_{t}(x)-F_{t}(y)\right]$

has rank $r$, and $d_{r}=\dim_{B}(\overline{S_{r}})$. For each
$\alpha\in\mathop{I\\!\\!R}\nolimits^{t}$, with
$\alpha=(\alpha^{1},\dots,\alpha^{t})$, define
$F_{\alpha}=F_{0}+\displaystyle\sum_{i=1}^{t}\alpha^{i}F_{i}:A\rightarrow\mathop{I\\!\\!R}\nolimits^{n}$.
Then, for $\alpha\in\mathop{I\\!\\!R}\nolimits^{t}$ out of a subset of zero
measure of $\mathop{I\\!\\!R}\nolimits^{t}$, it is true that yes $d_{r}<r$ for
all integers $r\geq 0$, then the application $F_{\alpha}$ it is injective.

######  Definition 3.

A Borel subset A of a normed vector space V is prevalent if there exists a
subspace, E of V, of finite dimension such that for each $v\in V,v+e\in A$ for
almost everything (Lebesgue measure) $e\in E.$

We will use the term for almost every application if the set of such functions
is prevalent.

######  Definition 4.

Let $I_{n}$ be the identity matrix of $n\times n$ and $(\cdot,\cdot)$ denotes
the greatest common divisor. Let’s use the convention that $(p,0)=0$. For the
integers $p>q\geq 0$ let’s define the matrix $p\times(p-(p,q))$ by:

$\displaystyle C_{pq}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{lll}&I_{p-(p,q)}&\\\ \\\
-I_{(p,q)}&\quad\cdots&-I_{(p,q)}\\\ \end{array}\right]$

Let’s define $C^{\infty}_{pq}$ a matrix $\infty\times(p-(p,q))$ formed by
repeating the block $C_{pq}$ vertically. And for a positive integer $w$ let’s
define $C^{w}_{pq}$ the matrix formed by the $w$ top rows of
$C^{\infty}_{pq}$.

######  Remark 1.

If $p\leq n$, of the definition 4 it is observed that rank
$C^{n}_{pq}=p-(p,q)$; and then rank $C^{n}_{pq}=\min\\{n,p-(p,q)\\}$. From
this it follows that:

  * a)

rank $C^{n}_{p0}\geq\min\\{n,p\\}=p$ and

  * b)

rank $C^{n}_{pq}\geq\min\\{n,p/2\\}=p/2$.

######  Teorema 2 (Fractal Delay Embedding Prevalence Theorem [9]).

Be $\Phi$ the flow of a dynamic system over an open subset $U$ of
$\mathop{I\\!\\!R}\nolimits^{k}$, and $A$ a compact subset of $U$ of
$\dim_{B}(A)=d$. Be $n>2d$ an integer and $T>0$. Let’s assume that $A$
contains only a finite number of equilibrium points; does not contain periodic
orbits of $\Phi$ of period $T$ or $2T$ y contains a finite number of periodic
orbits of $\Phi$ of period $3T,4T,\ldots,nT$. So for almost all (in the
prevalente sense) function $h$ of class $C^{1}$ on $U$, the delay coordinate
application $F_{(\Phi,T,h)}:U\rightarrow\mathop{I\\!\\!R}\nolimits^{n}$,
defined by:

$F_{(\Phi,T,h)}(x)=\left(h(x),h(\Phi_{T}(x)),h(\Phi_{2T}(x)),\ldots,h(\Phi_{(n-1)T}(x))\right),$

it is injective on $A$.

proof: Be $\\{h_{i}\\}_{i=1}^{t}$ a basis for the polynomials of $k$ variables
of lesser degree and equal to $2n$. For $i=1,\dots,t$ let’s define

$\displaystyle F_{i}(x)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{l}h_{i}(x)\\\ h_{i}(\Phi_{T}(x))\\\
h_{i}(\Phi_{2T}(x))\\\ \quad\vdots\\\ h_{i}(\Phi_{(n-1)T}(x))\\\
\end{array}\right]$

For each $\alpha\in\mathop{I\\!\\!R}\nolimits^{t}$, with
$\alpha=(\alpha^{1},\dots,\alpha^{t})$, let’s define:

$h_{\alpha}=\sum_{i=1}^{t}\alpha^{i}h_{i}:\mathop{I\\!\\!R}\nolimits^{k}\rightarrow\mathop{I\\!\\!R}\nolimits.$

Then,

$F_{(\Phi,T,h_{\alpha})}=\sum_{i=1}^{t}F_{i}:\mathop{I\\!\\!R}\nolimits^{k}\rightarrow\mathop{I\\!\\!R}\nolimits^{n}.$

To use the lemma 1 we need to determine, for each $(x,y)\in A\times A$ with
$x\neq y$, the rank of the matrix:

$M_{xy}=[F_{1}(x)-F_{1}(y),\dots,F_{t}(x)-F_{t}(y)]_{n\times t},$

which can be expressed as:

$\displaystyle M_{xy}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{lll}h_{1}(x)-h_{1}(y)&\cdots&h_{t}(x)-h_{t}(y)\\\
h_{1}(\Phi_{T}(x))-h_{1}(\Phi_{T}(y))&\cdots&h_{t}(\Phi_{T}(x))-h_{t}(\Phi_{T}(y))\\\
\qquad\qquad\vdots&\ddots&\qquad\qquad\vdots\\\
h_{1}(\Phi_{(n-1)T}(x))-h_{1}(\Phi_{(n-1)T}(y))&\cdots&h_{t}(\Phi_{(n-1)T}(x))-h_{t}(\Phi_{(n-1)T}(y))\\\
\end{array}\right]=J\cdot H,$

where,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{llll}h_{1}(z_{1})&\cdots&h_{t}(z_{1})\\\
h_{1}(\Phi_{T}(z_{2}))&\cdots&h_{t}(\Phi_{T}(z_{2}))\\\
\qquad\vdots&\ddots&\qquad\vdots\\\
h_{1}(\Phi_{(n-1)T}(z_{q}))&\cdots&h_{t}(\Phi_{(n-1)T}(z_{q}))\\\
\end{array}\right]_{q\times t},$

$q\leq 2n$; the $z_{j}$, $j=1,\dots q$, are different and $J$ is a matrix of
$n\times q$ whose rows consist of $1,0$ y $-1$.
For each integer $p\leq n$ positive let us denote by $A_{p}$ the set of
periodic period points $p$ of $\Phi_{T}$ found in $A$, that is to say,

$A_{p}=\\{x\in A:\Phi_{T}^{p}(x)=x\\}.$

From the statement of the theorem, as $A$ contains a finite number of periodic
period points $p\leq n$ of the flow $\Phi_{T}$, then

$\dim_{B}(A_{p})<p/2.$

Now we will divide the study of the rank of $M_{xy}$ in three cases:
Case 1: The points $x$ and $y$ they are not periodicals $\leq n$.
In this case $J$ is an upper or lower triangular matrix and rank $J=n$. Later,

$\mbox{rank }M_{xy}=\mbox{rank }(J\cdot H)=n.$

Let’s define the set

$S_{n}=\\{(x,y)\in A\times A:x\neq y\mbox{ con rank }M_{xy}=n\\},$

from where you have:

$\dim_{B}(S_{n})\leq 2d<n.$

So, by the lemma 1 we conclude that for almost everything
$\alpha\in\mathop{I\\!\\!R}\nolimits^{t}$, the application
$F_{(\Phi,T,h_{\alpha})}$ it is injective on $A$.
Case 2: The points $x$ and $y$ are in different periodic orbits of period
$\leq n$.
Let’s assume that $p$ and $q$ are minimal positive integers such that
$\Phi_{T}^{p}(x)=x$, $\Phi_{T}^{q}(y)=y$ and $1\leq q\leq p\leq n$. In this
case, the matrix $J$ contains a copy of $C_{p0}^{n}$. Then,

$\mbox{rank }M_{xy}=\mbox{rank }(J\cdot H)=\mbox{rank }J\geq\mbox{rank
}C^{n}_{p0}>2\dim_{B}(A_{p}).$

Let’s define the set $S_{r}$, with $x$ and $y$ the points dealt in the case 2,
by:

$S_{r}=\\{(x,y)\in A\times A:x\neq y\mbox{ con rank }M_{xy}=r\\},$

from where:

$\dim_{B}(S_{r})\leq 2\dim_{B}(A_{p})<\mbox{rank }M_{xy}=r.$

Then, for the lemma 1, we conclude that for almost everything
$\alpha\in\mathop{I\\!\\!R}\nolimits^{t}$ the application
$F_{(\Phi,T,h_{\alpha})}$ it is injective on $A$.
Case 3: The points $x$ and $y$ are in the same periodic orbit of period $\leq
n$.
Let’s assume that $p$ and $q$ are minimal positive integers such that
$\Phi_{T}^{p}(x)=x$, $\Phi_{T}^{q}(x)=y$ and $1\leq q<p\leq n$. Given that $x$
and $y$ lie in the same periodic orbit, the column space (the set of linearly
independent columns) of $J$ contains the column space of $C^{n}_{pq}$. Thus,

$\mbox{rank }M_{xy}=\mbox{rank }(J\cdot H)=\mbox{ rank }J>\mbox{rank
}C^{n}_{pq}>\dim_{B}(A_{p}).$

Let’s define the set $S_{r}$, with $x$ and $y$ the points dealt in the case 3,
by:

$S_{r}=\\{(x,y)\in A\times A:x\neq y\mbox{ con rank }M_{xy}=r\\}.$

As $x$ and $y$ are in the same periodic orbit,

$\dim_{B}(S_{r})=\dim_{B}(A_{p})<\mbox{rank }M_{xy}=r.$

Then, for the lemma 1, we conclude that for almost everything
$\alpha\in\mathop{I\\!\\!R}\nolimits^{t}$ the application
$F_{(\Phi,T,h_{\alpha})}$ it is injective on $A$. This concludes the
demonstration. We should mention that the Theorem 2 does not give an estimate
on the smallest dimension for which almost every application of delay
coordinates is injective. However, there are numerical algorithms that allow
estimating the mergulho dimension and the delay time in the reconstructions.
These are the mutual information and the false neighbor method, which we
mention in the next section.

##  3 Mutual information and false neighbors

###  3.1 Mutual information

######  Definition 5.

Let $X$ and $Y$ two discrete random variables with probability distributions
$p(x)$ and $p(y)$, respectively. Mutual information between variables $X$ and
$Y$ is defined by:

$I(X,Y)=\sum_{x}\sum_{y}p(x,y)\log_{2}\frac{p(x,y)}{p(x)p(y)},$

where $p(x,y)$ represents the joint probability.

If we consider that the random variable $X$ take the values
$x_{1},x_{2},\ldots,x_{N}$ So what $Y$ take the values
$x_{1+T},x_{2+T},\ldots,x_{N+T}$, the mutual information for the time series
is expressed as a function of $T$:

$I(T)=\sum_{t=1}^{N}P(x_{t},x_{t+T})\log_{2}\frac{P(x_{t},x_{t+T})}{P(x_{t})P(x_{t+T})},$

where $P(x_{t})$ is the probability that $X$ take a value $x_{t}$,
$P(x_{t+T})$ is the probability that $Y$ take a value $x_{t+T}$. And
$P(x_{t},x_{t+T})$ is the probability that $X$ take a value $x_{t}$ and $Y$
take a value $x_{t+T}$.

######  Remark 2.

We comment the following:

  * a)

The mutual information acts as the nonlinear autocorrelation function that
indicates how, in a nonlinear way, the measurements at different times are
connected on average over all measurements.

  * b)

Mutual information between $x_{t}$ y $x_{t+T}$ quantifies the information you
have about the state $x_{t+T}$ assuming we have knowledge of the state
$x_{t}$.

Fraser and Swinney [4] propose to use the first minimum of the mutual
information between $x_{t}$ and $x_{t+T}$ as the optimal delay time. The idea
is that a suitable delay time has to be large enough so that the information
available in time $t+T$ is significantly different from the information over
time $t$, but, not too much so that this information is not lost.
For the calculation of the mutual information of a time series,
$\\{x_{1},x_{2},\ldots,x_{N}\\}$, the first step is to find the maximum,
$x_{\max}=\max\\{x_{1},x_{2},\ldots,x_{N}\\}$ and the minimum,
$x_{\min}=\min\\{x_{1},x_{2},\ldots,x_{N}\\}$ of the series values. Then, the
value $|x_{\max}-x_{\min}|$ is divided into $j$ intervals of the same size.
Finally the expression is calculated

$I(T)=\sum_{h=1}^{j}\sum_{k=1}^{j}P_{h,k}(T)\ln\frac{P_{h,k}(T)}{P_{h}P_{k}}$

where $P_{h}$ and $P_{k}$ denote the probabilities that the variables take the
values in the $h-$esimo and $k-$esimo interval, respectively, and $P_{h,k}(T)$
is the joint probability that $x_{t}$ is in the interval $h$ and $x_{t+T}$ is
in the interval $k$.
The first local minimum of $I(T)$ indicates the largest amount of information
we can have from the state $x_{t}$ in order to determine the status $x_{t+T}$.
To obtain the dimension of the space where the attractor is rebuilt, called
the embedding dimension, let’s see the false neighbors method.

###  3.2 False neighbors

The technique of False neighbors was introduced by Kennel [6]. It is an
efficient tool to determine the smallest required reconstruction dimension,
that is, the embedding dimension. By reconstruction theorems, we are only
guaranteed that for an adequate reconstruction dimension, say $m$, a copy of
the attractor can be obtained at $\mathop{I\\!\\!R}\nolimits^{m}$. In
addition, for $n>m$ the same thing happens. In this sense, the false neighbors
method can serve as an optimization procedure trying to have the lowest value
of $m$, in such a way as to have the reconstruction of the adequate phase
space.
The idea of false neighbors is based on the following geometric intuition:
We observe that the dimension of an attractor expresses the degrees of freedom
that are needed to extend. Likewise, if the dimension of the reconstruction
space were less than the dimension of the attractor, it would not have
sufficient degrees of freedom for it, being reduced. Then, nearby points in
that state can be mistaken as true neighbors. By increasing the reconstruction
dimension, the attractor will deform and false neighbors will no longer be
neighbors. The following algorithm estimates the value of $m$ for which a
fraction of false neighbors is null.

To calculate the fraction of false neighbors, the following algorithm is used:
Given a point
$X_{t}^{(m)}=\left(x_{t},x_{t+\tau},x_{t+2\tau},\ldots,x_{t+(m-1)\tau}\right)$
in the reconstruction space $m-$dimensional, we have to find a neighbor
$X_{i}^{(m)}$ such that $\|X_{i}^{(m)}-X_{t}^{(m)}\|<\varepsilon$, donde
$\varepsilon$ is a small constant. So, we calculate the normalized distance
$R_{i}$ between the coordinates in $\mathop{I\\!\\!R}\nolimits^{m+1}$ of the
points $X_{i}^{(m+1)}=(X_{i}^{(m)},x_{i+m\tau})$ y
$X_{t}^{(m+1)}=(X_{t}^{(m)},x_{t+m\tau})$ according to the following equation:

$R_{i}=\displaystyle\frac{|x_{i+m\tau}-x_{t+mr}|}{\|X_{i}^{(m)}-X_{t}^{(m)}\|}$
(5)

If $R_{i}$ is greater than a given referential value $R_{tol}$, then
$X_{i}^{(m)}$ is marked as a false neighbor of $X_{t}^{(m)}$. Equation (1) has
to be calculated for the whole series and for several values of
$m=1,2,\ldots,$ until the fraction of points for which $R_{i}>R_{tol}$ be
despicable. In accordance with Kennel [6], $R_{tol}=10$ It has proven to be a
good choice for a considerable number of series. A formal mathematical proof
of this fact is non-existent.

Next we will give a brief summary about how you can think about the concept of
the dimension of an object [12].
We will use the following ideas to define the information dimension, which
will be useful to detect the fractal structure.

##  4 Shannon entropy and information dimension

A way of thinking about the dimension of a set $A$, is in terms of how many
real numbers are needed to specify the position of a point in that set. For
example, the position of a point on the line is determined by one number, the
position on a plane by two. Now, the idea is to understand this concept for
more complicated sets than lines, planes, volumes, etc.
We will need the following definitions:

######  Definition 6.

Be $A$ a nonempty set of $R^{m}$. The information dimension of set A is given
by:

$D_{I}=\lim_{r\rightarrow 0}\frac{-S(r)}{\log_{2}(r)}$

where $S(r)$ is the information (in bits) needed to specify the position of a
point on the set $A$ with a precision $r$.

The calculation of $S(r)$ was given by Claude Shannon [10].

######  Definition 7.

Be $X$ a discrete random variable defined in a set A, with $n-$events whose
probabilities of occurrence are $p_{1},p_{2},\ldots,p_{n}$. The Shannon
entropy is defined from $X$, by:

$H(X)=-\sum_{i=1}^{n}p_{i}\log_{2}(p_{i})$

######  Remark 3.

For clarity of Shannon’s concept of entropy, the following observations should
be kept in mind:

  * 1)

Shannon entropy does not depend on the values that the random variable can
take $X$, but only of probabilities. Therefore, the Shannon entropy is a
function of a certain probability distribution. $p=(p_{1},p_{2},\dots,p_{n})$,
so this is usually denoted by $H(p)$.

  * 2)

Since the base of the logarithm is $2$, Shannon entropy is measured in bits.
This is an average measure of the uncertainty of the random variable, that is,
it is the average number of bits required to describe the random variable $X$.
This means that the Shannon entropy $H(p)$ represents the average amount of
information to specify the position of a point in $A$.

######  Teorema 3.

Let A be a nonempty subset of $\mathop{I\\!\\!R}\nolimits^{m}$ and $X$ a
discrete random variable defined in $A$ with $n-$events, whose probabilities
of occurrence are $p_{1},p_{2},\ldots,p_{n}$. Then, the information dimension
of the set $A$ is given by:

$D_{I}=\lim_{r\rightarrow
0}\frac{\displaystyle\sum_{i=1}^{n}p_{i}\log_{2}(p_{i})}{\log_{2}(r)},$

where $r$ represents the precision with which a point is specified in $A$.

proof: For the set $A$ let us consider a partition of it in $n$-boxes $B_{i}$
of diameter $r$. Then, the probability that a point in the set $A$ be in the
box $B_{i}$ is given by $p_{i}=\displaystyle\frac{\mu(B_{i})}{\mu(A)}$. Using
Shannon’s entropy, the information needed to specify a point in the set $A$
with precision $r$ is given by:

$S(r)=-\sum_{i=1}^{n}p_{i}\log_{2}(p_{i})$

This relationship leads directly to an expression for the information
dimension of the set $A$:

$\displaystyle D_{I}$ $\displaystyle=$ $\displaystyle\lim_{r\rightarrow
0}\frac{-S(r)}{\log_{2}(r)}$ $\displaystyle D_{I}$ $\displaystyle=$
$\displaystyle\displaystyle\lim_{r\rightarrow
0}\frac{\displaystyle\sum_{i=1}^{n}p_{i}\log_{2}(p_{i})}{\log_{2}(r)}$

This concludes the proof.
In the figure 4 a partition of the set $A$ is observed, for which

$p_{i}=\displaystyle\frac{\mbox{ $N^{0}$ of points in }B_{i}\mbox{ of
}A}{\mbox{ $N^{0}$ of points in }A}$

?figurename? 4: Partition of the set $A$ in $n$ boxes $B_{i}$ of diameter
$r<1$.

##  5 Applications to the financial market

In this section we will apply the techniques elaborated in previous sections
to reconstruct the hidden attractor in the price series of the companies
TelfBC and Credicorp. In addition, the Shannon entropy of the reconstructed
dynamics will be determined, and then the information dimension of this will
be calculated. The numerical value of the information dimension will indicate
the complexity in the financial market. Also, it will tell us that the
reconstructed attractor is a fractal set.
The first series of prices studied was from the company TelfBC, with prices
from 01/02/1992 to 11/26/2014, a total of 21 years of daily observations. And
the second series of prices was from the Credicorp company, with prices from
10/25/1995 to 11/26/2014, a total of 19 years of daily observations.

###  5.1 Reconstruction of attractors

For the reconstruction of the attractors, corresponding to the price series of
the companies TelfBC and Credicorp, some parameters must be determined. The
delay time is determined using the mutual information algorithm and the
mergulho dimension is estimated using the false neighbors algorithm.
The time delay in rebuilding the dynamics of the price series for TelfBC was
$T=$ 9 and for Credicorp it was $T=$ 6\. These results are shown in the figure
5.

?figurename? 5: Delay time $T=$ 9 and $T=$ 6, corresponding to the price
series of the companies TelfBC and Credicorp.

Embedding dimension is shown in figure 6. The percentage of false neighbors is
almost zero when $n=$ 12 for TelfBC and $n=11$ for Credicorp.

?figurename? 6: Embedding dimension $n=$ 12 and $n=$ 11, corresponding to the
price series of the companies TelfBC and Credicorp.

Knowing the delay time $T=9$ and the mergulho dimension $n=12$, we apply the
application of delay coordinates to reconstruct the hidden attractor in the
series of prices of the shares of the company TelfBC. The figure 7 shows the
projection of this attractor on the coordinate axes.

?figurename? 7: Reconstruction of attractors.

Knowing the delay time $T=6$ and the mergulho dimension $n=11$, we apply the
application of delay coordinates to reconstruct the hidden attractor in the
series of prices of the shares of the company Credicorp. The figure 8 shows
the projection of this attractor on the coordinate axes.

?figurename? 8: Reconstruction of attractors.

###  5.2 Shannon entropy

Once the attractor $A$ has been reconstructed via the delay coordinates,
corresponding to the financial market price series, the Shannon entropy is
calculated for the price dynamics of the shares of the companies TelfBC and
Credicorp. For each of them the Shannon entropy is $8.58$ bits and $8.44$ bits
of information. These values indicate the number of bits of information
necessary to specify a point in the price dynamics of the shares of financial
market companies. The figure 9 shows these values.

?figurename? 9: Shannon entropy $H=8.58$, 8.44 bits for the share price
dynamics of the companies TelfBC and Credicorp.

Using the 3 theorem it is obtained that for the reconstructed attractor, for
the price series of the company TelfBC, the information dimension is
$D_{I}=1.19$ and for Credicorp it is $D_{I}=2.38$. These values are shown in
the figure 10.

?figurename? 10: Information dimension $DI=$ 1.19, 2.38 for the attractors
corresponding to the price series of the companies TelfBC and Credicorp.

These numerical values for the information dimension, 1.19 and 2.38, show the
existence of a fractal set in the price dynamics of the shares of the
companies TelfBC and Credicorp. In addition, these values quantify the
complexity in the financial market, since it is necessary between 1 and 2
significant variables for TelfBC and between 2 and 3 significant variables for
Credicorp to understand the dynamics of the prices of the shares of the
mentioned companies.

##  6 Conclusions

Using the non-linear analysis and the information theory applied to the series
of prices of the shares of the companies TelfBC and Credicorp, we obtained the
following results:

  1. 1.

The Shannon entropy for the share price dynamics of the companies TelfBC and
Credicorp were 8.58 and 8.44 bits, respectively. These indicated the number of
bits of information necessary to specify a point in the price dynamics of the
shares of financial market companies.

  2. 2.

The information dimension for the share price dynamics of financial market
companies were 1.19 and 2.38, respectively for each company. The existence of
a fractal attractor is evidenced in the dynamics

ORCID
Jose Luis Ponte Bejarano https://orcid.org/0000-0002-4997-7950,
Alexis Rodriguez Carranza https://orcid.org/0000-0002-0290-165X,
Juan Carlos Ponte Bejarano https://orcid.org/0000-0002-8682-9682.
Segundo Eloy Soto Abanto https://orcid.org/0000-0003-1004-5520

##  ?refname?

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