[a,b]Andrey Saveliev

# On Numerical Simulations of Intergalactic Electromagnetic Cascades with
Lorentz Invariance Violation

Rafael Alves Batista

###### Abstract

Lorentz invariance violation (LIV) is a proposed phenomenon where Lorentz
symmetry is violated at high energies, potentially affecting particle dynamics
and interactions. We use numerical simulations with the CRPropa framework to
investigate LIV in gamma-ray-induced electromagnetic cascades, specifically
studying how it impacts cascading electrons and photons undergoing pair
production and inverse Compton scattering. Our detailed analysis of the
simulation results, compared with existing theoretical models, reveals that
LIV can significantly alter the behavior of both components of the cascade,
photons and electrons, resulting in specific signatures in measured fluxes
that could be observed in high-energy gamma-ray observations. These insights
are crucial for ongoing searches for LIV and for the development of
theoretical models incorporating LIV effects.

##  1 Introduction

The Standard Model of particle physics (SM) is a highly successful theory that
elegantly describes the electromagnetic, weak, and strong interactions [1, 2].
It has been incredibly effective in explaining a wide range of physical
phenomena. However, there remain several unresolved issues, both from
experimental observations and theoretical considerations, that the SM fails to
account for. These include such phenomena as Dark Matter, Dark Energy,
neutrino masses, and the hierarchy problem [3].

One of the most fundamental challenges in physics is the unification of the SM
and gravity, which is one of the remaining fundamental interactions in nature.
Numerous theories have been proposed to address this issue, including Quantum
Gravity (QG) [4], which aims to bridge the gap between the quantum field
theory approach of the SM and the differential geometry framework used in
General Relativity (GR). GR itself faces several issues, such as singular
solutions and the Black Hole information loss problem. Therefore, it is
crucial to establish a connection between these theories in order to
potentially resolve these problems and gain a deeper understanding of the
fundamental forces in the universe.

However, QG is expected to be measurable directly only at the Planck Scale,
i.e. either at distances of the order of the Planck Length, given by
$\lambda_{\rm Pl}\simeq 1.62\times 10^{-35}$ m, or at energies of the order of
the Planck Mass, $M_{\rm Pl}\simeq 1.22\times 10^{28}$ eV. This is 15 orders
of magnitude larger than what the Large Hadron Collider can probe with its
design center of mass energy of 14 TeV. But if QG is tested specifically, then
also other quantities can become important, such as the energy of the particle
in some frame or a cosmological propagation distance. These quantities can be
large enough to effectively offset the Planck suppression to a physical
observable, such that very small corrections are magnified. The most promising
candidates for tests of QG [4] are Ultra High Energy Cosmic Rays (UHECR) [5,
6, 7] and very high energy (VHE) gamma rays [5, 8, 9], since they represent
the most energetic particles which have been ever observed, their energies
reaching values above $10^{20}$ eV.

Specifically, there has been a proposal suggesting that QG could potentially
lead to a phenomenon called Lorentz invariance violation (LIV) [10]. This idea
suggests that LIV may arise from the quantum fluctuations of spacetime itself.
However, the existence or absence of LIV cannot be definitively proven at this
point, as there is no compelling evidence supporting either case.
Additionally, it is not possible to completely exclude the possibility of LIV
because the Lorentz group, which describes the symmetries of spacetime, is
non-compact. Therefore, it is conceivable that LIV could manifest at energies
beyond the current limits of observation. In such scenarios, various
theoretical approaches, such as effective field theories and SM extensions
[11], propose that fundamental aspects of physics, such as dispersion
relations and hence threshold energies for different reactions, could undergo
modifications. By carefully interpreting relevant observations, these changes
may be detected and studied.

In this work we are investigating the consequences of introducing a modified
dispersion relation. In particular, we consider so-called electromagnetic
cascades which propagate in intergalactic space [12]. This describes the
process when a VHE gamma-ray photon is emitted by an energetic astrophysical
object (e.g. a blazar or a gamma-ray burst) and then interacts with a low-
energy photon of a background photon field, producing an electron/positron
pair. Such a charged lepton then upscatters a low-energy photon via Inverse
Compton (IC) scattering, resulting in a high energy photon which then can
restart the whole process which continues until the particle energy drops
below the threshold of the corresponding reaction.

LIV may result in a significant modification of these processes and hence
impact the corresponding observations. In particular, [13] showed that LIV
might dramatically increase the interaction length of pair production for
energies above $100$ TeV and therefore suppress the cascade development. On
the other hand, LIV might also imply that the photons’ speed is energy-
dependent, thus resulting in energy-dependent time delays [9].

In this work we carry out simulations using the CRPropa code of the
propagation of electromagnetic cascades including the modified dispersion
relations for VHE photons and electrons for their propagation. We then analyze
the influence of the corresponding LIV parameters on the possible observables,
such as the measured gamma-ray spectra. To do so, we first describe the
formalism of the propagation of photons and electrons/positrons and its
modifications due to the modified dispersion relations in Sec. 2. Then, in
Sec. 3, we describe the setup for the simulations of electromagnetic cascades
with LIV and present the corresponding results. Finally, in Sec. 4, we discuss
our results, while drawing the conclusions from these results and giving an
outlook for future investigations we plan to carry out for this topic in Sec.
5.

##  2 Particle Propagation with Modified Dispersion Relations

The aforementioned dispersion relations relevant for electromagnetic cascades
are in general given by [14]

$\displaystyle
E_{e}^{2}=p_{e}^{2}\left[1+\frac{m_{e}^{2}}{p_{e}^{2}}+\sum_{j=0}^{N}\chi_{j}^{e}\left(\frac{p_{e}}{M_{\rm
Pl}}\right)^{j}\right]\,,$ (1) $\displaystyle
E_{\gamma}^{2}=k_{\gamma}^{2}\left[1+\sum_{j=0}^{N}\chi_{j}^{\gamma}\left(\frac{k_{\gamma}}{M_{\rm
Pl}}\right)^{j}\right]$ (2)

for electrons and photons, respectively. Here, $m_{e}$ is the electron mass,
while $p_{e}$ and $k_{\gamma}$ is the electron and photon momentum,
respectively. On the other hand, $\chi_{j}^{\gamma}$ and $\chi_{j}^{e}$ are
the corresponding LIV parameters of the $j$th order.

Usually one assumes that one of the terms of the sums (of order $n=0,1,2,...$)
present in Eqs. (1) and (2) is dominant, such that these equations may be
reduced to

$\displaystyle
E_{e}^{2}=p_{e}^{2}\left[1+\frac{m_{e}^{2}}{p_{e}^{2}}+\chi_{n}^{e}\left(\frac{p_{e}}{M_{\rm
Pl}}\right)^{n}\right]\,,$ (3) $\displaystyle
E_{\gamma}^{2}=k_{\gamma}^{2}\left[1+\chi_{n}^{\gamma}\left(\frac{k_{\gamma}}{M_{\rm
Pl}}\right)^{n}\right]\,,$ (4)

respectively. In case of $\xi_{n}^{i}<0$ the corresponding LIV scenario is
also called subluminal, while for $\xi_{n}^{i}>0$ it is denoted as
superluminal.

The consequences of these modifications are twofold [15]: Firstly, the
threshold of a given reaction is altered, leading to a modification in the
associated propagation length. Secondly, the presence of LIV can enable new
reactions that would not be possible within the confines of Lorentz
invariance. Examples of such reactions include the spontaneous decay of
photons into pairs or photons, as well as the occurrence of the vacuum
Cherenkov effect for electrons.

Here, we primarily consider the threshold modification. In order to calculate
the threshold of a given reaction, one has to solve the equation

$\max_{0\leq\theta\leq\pi}s^{\rm in}=\min_{0\leq y\leq 1}s^{\rm out}\,,$ (5)

where $s^{\rm in}$ and $s^{\rm out}$ is the invariant mass of the incoming and
outgoing particles, respectively, given by the symbolic reaction equations

$\gamma_{\rm VHE}+\gamma_{\rm BP}\rightarrow e^{+}_{\rm VHE}+e^{-}_{\rm VHE}$
(6)

for pair production, and

$e^{\pm}_{\rm VHE}+\gamma_{\rm BP}\rightarrow e^{\pm}+\gamma_{\rm VHE}$ (7)

for Inverse Compton scattering, where in both cases $\gamma_{\rm BP}$ denotes
a background photon. The maximum on the left side of the equation should be
searched regarding the angle $\theta$ between the propagation directions of
the two incoming particles. On the other hand, since the directions outgoing
particles for very high energies may be regarded as nearly parallel, the
minimum on the right hand side should be searched regarding the fraction $y$
of the total energy carried by one of the outgoing particles.

This results in the background photon threshold energy, $\epsilon_{\rm thr}$,
given by

$\epsilon_{\rm
thr}=\begin{cases}k_{\gamma}\left[\left(\frac{m_{e}}{k_{\gamma}}\right)^{2}+\frac{1}{4}\left(\frac{\chi_{n}^{e}}{2^{n}}-\chi_{n}^{\gamma}\right)\left(\frac{k_{\gamma}}{M_{\rm
Pl}}\right)^{n}\right]&\text{for pair production,}\\\ 0&\text{for inverse
Compton scattering,}\end{cases}$ (8)

or, equivalently, in the threshold invariant mass, $s_{\rm thr}$, which may be
written as

$s_{\rm
thr}=\begin{cases}k_{\gamma}^{2}\left[4\left(\frac{m_{e}}{k_{\gamma}}\right)^{2}+\frac{\chi_{n}^{e}}{2^{n}}\left(\frac{k_{\gamma}}{M_{\rm
Pl}}\right)^{n}\right]&\text{for pair production,}\\\
p_{e}^{2}\left[\left(\frac{m_{e}}{p_{e}}\right)^{2}+\chi_{n}^{e}\left(\frac{p_{e}}{M_{\rm
Pl}}\right)^{n}\right]&\text{for inverse Compton scattering.}\end{cases}$ (9)

Using these results, one then can calculate the mean free path $\lambda_{\rm
MFP}$ by [4]

$\begin{split}\lambda_{\rm MFP}=&\frac{1}{2p_{\rm in}}\int_{s_{\rm
thr}}^{\infty}\frac{n_{\rm BP}\left(\frac{s^{*}-p_{\rm
in}^{2}\left[\left(\frac{m_{\rm in}}{p_{\rm in}}\right)^{2}+\chi_{n}^{\rm
in}\left(\frac{p_{\rm in}}{M_{\rm Pl}}\right)^{n}\right]}{4p_{\rm
in}}\right)}{\left\\{s^{*}-p_{\rm in}^{2}\left[\left(\frac{m_{\rm in}}{p_{\rm
in}}\right)^{2}+\chi_{n}^{\rm in}\left(\frac{p_{\rm in}}{M_{\rm
Pl}}\right)^{n}\right]\right\\}^{2}}\\\ &\times\int_{p_{\rm
in}^{2}\left[\left(\frac{m_{\rm in}}{p_{\rm in}}\right)^{2}+\chi_{n}^{\rm
in}\left(\frac{p_{\rm in}}{M_{\rm
Pl}}\right)^{n}\right]}^{s^{*}}\sigma(s)\left\\{s-p_{\rm
in}^{2}\left[\left(\frac{m_{\rm in}}{p_{\rm in}}\right)^{2}+\chi_{n}^{\rm
in}\left(\frac{p_{\rm in}}{M_{\rm Pl}}\right)^{n}\right]\right\\}\,{\rm
d}s\,{\rm d}s^{*}\,,\end{split}$ (10)

where $p_{\rm in}$, $m_{\rm in}$ and $\chi_{n}^{\rm in}$ is the momentum, mass
and LIV parameter of the incoming VHE particle, respectively, $n_{\rm
BP}(\epsilon)$ is the ambient photon density of the background photon filed,
and $\sigma(s)$ is the cross-section of the considered reaction, while $s_{\rm
thr}$ is given by Eq. (9).

Figure 1: Examples of mean free paths for pair production including LIV
calculated according to Eq. (10). The left panel corresponds to the CMB,
whereas the right one corresponds to the EBL model from [16]. LIV parameters
are indicated in the legend.

In Fig. 1 we show some examples of mean free paths for pair production by
gamma rays according to Eq. (10).

##  3 Simulations of Electromagnetic Cascades with Lorentz Invariance
Violation

We have created an advanced simulation tool called LIVPropa that seamlessly
integrates with the CRPropa code [17, 18]. This plugin enables detailed
simulations of gamma-ray propagation while considering LIV. Developed using
C++, LIVPropa offers a range of essential features.

One of the key functionalities is the provision of tabulated interaction rate
tables tailored for different LIV parameters. To ensure versatility and
adaptability, we have also included Python scripts as part of LIVPropa. These
scripts facilitate the generation of data for scenarios that may not be
readily available within the existing library. This flexibility allows studies
of a wide range of LIV scenarios and customization of the simulation
parameters to suit the user’s specific needs.

Fig. 2 illustrates the application of the code, considering pair production
interactions with the CMB and the EBL. We considered two sources: the extreme
blazar 1ES 0229+200 (right panel) and the Galactic center (left panel). Their
injection spectrum of both sources follows a power law with index $\alpha$ and
an exponential cut-off at an energy $E_{\text{max}}$.

Figure 2: Simulations of gamma-ray propagation consider pair production,
including LIV effects (see Eq. (10)). The left panel corresponds to a
cosmologically-distant blazar, whereas the right one is for a gamma-ray source
at the Galactic center. The injected spectral parameters assumed are indicated
in the panels. The black dot-dashed lines represent the Lorentz invariant
scenario.

In the left panel of Fig. 2, if the Galactic center is a PeVatron, emitting
gamma rays with energies up to $\sim 1\text{PeV}$, characteristic breaks in
the spectrum would be visible. These could be observable with with experiments
such the Large High Altitude Air Shower Observatory (LHAASO) [19] or the
upcoming Southern Wide-field Gamma-ray Observatory (SWGO) [20]. Note, however,
that photons from the Galactic interstellar radiation field also act as
targets, potentially modifying the observed fluxes. Nevertheless, the
qualitative arguments presented above would still hold.

The right panel of Fig. 2 is very instructive to understand the effect of LIV
on the spectra of gamma-ray sources. A suppression due to the EBL is visible
at energies around $\sim 1\;\text{TeV}$. While the exact threshold for this
interaction to occur would imply an earlier onset of suppression in the
superluminal case, the flux would quickly recover at higher energies,
following the behavior of the mean free paths shown in Fig. 1. In the
subluminal case, on the other hand, the behavior closely resembles that of the
Lorentz invariant scenario up to an energy related to the energy scale of LIV.
Beyond this energy, the flux increases again as a consequence of the large
mean free path.

##  4 Discussion

For now, LIVPropa takes into account effects related to the modified photon
dispersion relation of pair production. However, we are actively working on
implementing LIV for inverse Compton scattering as well. This enhancement will
enable us to conduct detailed three-dimensional studies of electromagnetic
cascades [12] that incorporate intergalactic magnetic fields and expand the
scope of our simulations.

It is important to note that the phenomenology of LIV encompasses more than
just simple modifications of interaction thresholds, which have been the
primary focus thus far. Within the framework of a single effective field
theory, new processes that are typically forbidden in the Lorentz invariant
case begin to emerge. Notable examples include photon decay
($\gamma\rightarrow e^{+}+e^{-}$), photon splitting
($\gamma\rightarrow\gamma+\gamma+\gamma$) and vacuum Cherenkov radiation
($e^{\pm}\rightarrow e^{\pm}+\gamma$) [15, 21]. Previous investigations
utilizing observations of objects like the Crab nebula at TeV energies have
placed constraints on LIV in the QED sector through searches for these effects
[22, 23].

In its current implementation, LIVPropa does not account for time delays
resulting from LIV. However, this consideration is mandatory when constraining
LIV through time-of-flight measurements [24]. Such strategies have been
commonly employed in the search for LIV using gamma-ray observations of
flaring blazars with imaging air Cherenkov telescopes (IACTs) [25, 26, 27].
Future facilities like the Cherenkov Telescope Array (CTA) [28, 29] and the
ASTRI Mini-Array [30] are expected to provide more precise measurements,
leading to improved constraints. Consequently, accurate simulation tools like
LIVPropa will play a crucial role in predicting the signals expected by these
advanced instruments.

##  5 Conclusions and Outlook

In this work we presented the results of Monte Carlo simulations of gamma-ray
propagation considering Lorentz invariance violation. To this end, we
developed a new software, LIVPropa, which interfaces with the CRPropa code.
For now, we implemented LIV for pair production interactions, but this will
soon be extended to include inverse Compton scattering.

Ultimately, our goal is to go beyond the usual approach of computing LIV
effects by simply folding in the optical depth of the photon field. Instead,
we aim to offer a versatile and robust tool for performing Monte Carlo
simulations that enable the inclusion of various other relevant factors. These
include, for instance, the contributions from Compton-produced photons, the
influence of magnetic fields, among others.

## Acknowledgements

The work of AS is supported by the Russian Science Foundation under grant no.
22-11-00063. RAB is funded by the “la Caixa” Foundation (ID 100010434) and the
European Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie grant agreement No 847648, fellowship code
LCF/BQ/PI21/11830030.

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