Ultra-High-Energy Astroparticles as Probes for Lorentz Invariance Violation

: Compelling evidence for Lorentz invariance violation (LIV) would demand a complete revision of modern physics. Therefore, searching for a signal or extending the validity of the invariance is fundamental for building our understanding of the extreme phenomena in the Universe. In this paper, we review the potential of ultra-high-energy astroparticles in setting limits on LIV. The standard framework of LIV studies in astroparticle physics is reviewed and its use on the electromagnetic and hadronic sectors are discussed. In particular, the current status of LIV tests using experimental data on ultra-high-energy photons and cosmic rays is addressed. A detailed discussion with improved argumentation about the LIV kinematics of the relevant interactions is shown. The main previous results are presented together with new calculations based on recently published astrophysical models.


Introduction
Lorentz invariance is one of the fundamental symmetries in the standard model (SM) of elementary particles. There is renewed interest in the community in testing the limits of its validity given that grand unified theories, such as quantum gravity and string theory, speculate about a certain degree of Lorentz invariance violation (LIV) [1][2][3][4][5][6][7][8][9][10].
Most such LIV effects are expected to be small. However, the incredibly high energies of ultra-high-energy (UHE, E > 10 18 eV) astroparticles, the extreme distances they travel to Earth, and the current level of experimental precision and of the high exposure of observatories in operation makes them promising probes for LIV.
In this review, we summarize these efforts, discussing the current state-of-the-art analyses and their limitations. First, in Section 2, we present the most common phenomenological framework used in astrophysical tests of LIV. Next, in Sections 3 and 4, we discuss the most up-to-date results and limits obtained with UHE photons and cosmic rays, respectively. The dependence of these analyses on astrophysical models is also explored. Finally, in Section 5, we address our final remarks and prospects for future LIV studies using UHE astroparticles.

LIV Framework
The astrophysical corrections of LIV can be addressed via a family of modified dispersion relations (MDR) for each type of particle (see, for instance, [1] and Refs. therein), where E a and p a are the energy and momenta, and δ is the LIV parameter for the a-particle.
In the special case of photons (m γ = 0), the MDR is, where E γ and p γ is the energy and momenta and δ n,γ stands for the photon LIV parameter. The positive sign corresponds to the so-called superluminal phenomena and the minus to the subluminal. Using one leading order n at a time, when n = 0, the LIV correction corresponds the so-called maximal velocity of photons. If n = 1 or n = 2, the LIV scale of the correction can be expressed by E (n) LIV = (|1/δ n a,n |). Astrophysical tests had settled strong constraints on the different LIV parameters [1]. For instance, for the subluminal side, the LIV modified pair production interaction on EBL (γγ EBL → e + e − ) bounded δ γ,1 ≥ −8.3 × 10 −30 eV −1 E LIV ≥ −2.4 × 10 21 eV , using TeV gamma rays [13], and ref. [27] constrained δ γ,2 > 3. 46 LIV ≥ 1.7 × 10 22 eV through LIV air shower suppression. On the other hand, astrophysical superluminal tests usually derive stronger limits, see, for instance, [28], which by the absence of LIV photon decay (γ → e + e − ) effects on gamma rays reports δ γ,0 ≤ 7.1 × 10 −19 , as well as δ γ, 1 LIV ≥ 2.5 × 10 25 eV by the lack of photon splitting (γ → 3γ). Furthermore, for a collection of Lorentz violating limits in the context of the standard model extension [5] including other SM sectors and dimensions, see the tables in [29].

Testing LIV with UHE Photons
No UHE photon has been detected so far [30][31][32]. The identification of a UHE photon source would lead to very precise and strong limits on LIV. UHE cosmic rays (UHECR) interact with the photon background producing pions. The neutral ones rapidly decay, producing UHE photons which could, in principle, be detected on Earth [33,34] as a secondary flux of UHE photons.
As first proposed in [17,35] and further developed by in [14], if subluminal LIV (δ γ,n < 0) is considered, the interaction rate of UHECRs with background photons decreases, leading to an increase in the expected number of UHE photons arriving on Earth.

Propagation of UHE Photons Considering LIV
Photons may be absorbed on their way to Earth, producing an electron-positron pair when interacting with the photon background (γγ bg → e − e + ) [36]. The mean free path of the interaction is determined by the energy distribution of the background photons and the kinematics of the pair production. In an LI scenario, the mean free path is such that most extragalactic UHE photons are expected to be absorbed. If LIV in the electromagnetic sector is considered, on the other hand, the kinematics of the interaction is expected to be modified [11]. The energy threshold for the pair production considering LIV effects is given by [12,[14][15][16][17]20] th where E γ and are, respectively, the energies of the UHE and background photons, m e is the electron mass, and δ n,γ is the LIV coefficient. The LIV energy threshold ( th) can be larger or smaller than the Lorentz-invariant (LI) case, depending on the sign of δ n,γ in the last term of the equation. Figure 1 shows the threshold energy for the background photon as a function of the energy of the UHE photon for different subluminal LIV coefficients of order n = 0. Similar results are found for higher orders. In a LI scenario, the dominant fields are the extragalactic background light (EBL) for E 10 14 eV, the cosmic microwave background (CMB) for 10 14 E/eV 10 17 , and the radio background for E 10 17 eV. When subluminal (superluminal) LIV is considered, the threshold increases (decreases) with respect to the LI scenario and, thus, fewer (more) interactions are expected. The energy above which this effect becomes significant decreases with increasing absolute values of the LIV coefficient. The survival probability, P surv , of a photon emitted at redshift z, with energy E γ is obtained by the following expression, where H 0 and h(z) are the cosmological terms, σ(E γ , , z) is the Breit-Wheeler crosssection [37], and n( , z) is the distribution of background photons. For the EBL, the model in [38] was used as an example. For the RB, the parameterization from [39] was used.

Searching for LIV in UHE Photon Data
When superluminal LIV in considered in photon propagation, the survival probability of the UHE photon is reduced. Since no UHE photon event has yet been detected, it is not straightforward to use current UHE data to search for a decrease in the expected flux.
When subluminal LIV is considered, the survival probability of the UHE photon increases allowing the calculation of upper limits on the LIV coefficients based on the measured upper limits of UHE photons. Several works have used this technique [14,17,35,40]. Since no UHE photon source was detected, only secondary photons produced in the propagation of UHECR are considered in the calculation. As such, the flux of UHE photons arriving on Earth is highly dependent on the UHECR astrophysical model: spacial distribution of the sources, emission energy spectra and chemical composition [41]. Thereby, as first discussed in [14], LIV limits imposed by this technique are highly dependent on the astrophysical assumptions about the UHECR sources.
In phenomenological studies [42], homogeneously distributed sources of UHECRs are usually considered to emit cosmic rays isotropically with energy spectra described by where A and Z denote, respectively, the atomic mass and charge for each species of emitted nuclei. The fraction of a given species, f A , the spectral index, Γ, and the maximum rigidity attainable by the sources, R max , are free parameters and vary between different astrophysical models. Chemical composition is usually divided into five major groups almost equally separated in ln A, with five representative nuclei, 1 H, 4 He, 14 N, 28 Si and 56 Fe. Such a simplistic description is justified by the lack of knowledge about the sources of UHECR.
Following the procedure proposed in [14], we calculated LIV limits for several astrophysical models covering the phase space of Equation (5) to illustrate their influence on the LIV limits. UHECR propagation is simulated with the state-of-the-art Monte Carlo package CRPropa 3 [43]. The propagation of the secondary UHE photons is then performed using the package EleCa [44], with an implementation of the LIV corrections described in Section 3.1. The simulated energy spectrum of UHECR is normalized to the energy spectrum measured by the Pierre Auger Observatory [45] at E = 10 18.75 eV. Finally, the LIV limit is obtained as the smallest absolute value of the LIV coefficient for which the predicted flux is more intense than the upper limits imposed by the Pierre Auger Observatory [30][31][32]. Figure 3 shows the logarithmic order of the LIV limit imposed for each combination of maximum attainable rigidity at the sources and spectral index for the two extreme compositions at the sources, pure proton and pure iron. The results cover all the possibilities, ranging from not being sensitive to LIV up to the case where the model would be inconsistent with data even under a LI assumption. The largest dependence comes from R max . For small maximum rigidities, no cosmic ray will be accelerated up to enough energies to produce UHE photons and, thus, modifying the propagation of UHE photons will have no effect. For large maximum rigidities, on the other hand, too many photons are produced, surpassing the upper limits on the UHE photon flux measured by the Pierre Auger Observatory even without increasing the threshold of the pair production. The composition also plays an important role. For heavy compositions a larger fraction of the phase space of models is allowed under an LI assumption. On the other hand, a larger fraction of the phase space is also not sensitive to LIV.
LIV Limits from Different UHECR Astrophysical Models UHECR astrophysical models are expected to describe current energy spectrum and composition measurements [45][46][47][48][49], reducing the allowed phase space of Figure 3. In Table 1, we present a collection of phenomenological models for the UHECR sources which can either be described by Equation (5) or be approximated by it.  LIV > 10 30 eV , which, for n = 1, would be 13 orders of magnitude more restrictive than current subluminal LIV limits obtained in the photon sector using TeV gamma rays [11,13]. This emphasizes once more the strong model dependence of these results. A study comparing the capability of these models to describe UHECR data is outside the scope of this work.
A further reduction of the allowed phase space of UHECR models is expected in the near future with the ongoing upgrade of current facilities, in particular, the Auger-Prime [57,58] at the Pierre Auger Observatory and the TAx4 [59] at the Telescope Array experiment. Future data obtained with the upgraded experiments will improve current statistics, allowing a better description of the spectral features, specially the suppression, which has a direct impact on the possible values for the spectral index and for the maximum power of acceleration of the sources. AugerPrime will also present the capability of investigating the presence of a proton component at the highest energies. If this is confirmed, models which predict a larger flux of UHE photons will be preferred and, consequently, the possibility of imposing restrictive LIV limits is favored. Future experiments such as the Probe of Extreme Multi-Messenger Astrophysics (POEMMA) [60] and Global Cosmic Ray Observatory (GCOS) [61] are expected to shrink the phase space even further in the years to come, softening the model dependence of current LIV studies using UHECR.      [62,63], the propagation of UHECR at energies of a few tens of EeV is limited by interactions with the photon background. A sharp decrease in the attenuation length of protons is expected above E ∼ 10 19.5 eV due to photopion production (p + γ bg → ∆ + → p/n + π 0/+ ). Furthermore, at these energies, the propagation of heavier nuclei is restricted to a few tens of Mpc by the photodisintegration, through which the nucleus emits a nucleon and becomes a lighter chemical species ( A N + γ bg → A−1 N + p/n). The emission of two or more nucleons is also possible, but rates are considerably smaller. At lower energies, 10 18 eV E 10 19.5 eV, the propagation is dominated by pair production (p + γ bg → p + e − + e + ).

As first proposed independently by Greisen and by Zatsepin & Kuzmin
These interactions create a propagation horizon, which, combined with the maximum power of acceleration of the sources, leads to suppression at the highest energy end of the spectrum [42] and increases the relative contribution of local sources located up to a few tens of Mpc [64,65]. The relative contribution to the suppression coming from the propagation horizon and from the acceleration limitations is yet to be clarified and differs for each UHECR phenomenological models (see, e.g., Table 1).
If LIV is considered in the hadronic sector, the kinematics of these interactions is modified, leaving imprints that could be detected in UHECR data. Below, we develop further the previous calculations [15,66] of photopion production and photodisintegration under LIV assumption.

Kinematics of Interactions of UHECR with the Photon Background Considering LIV
The kinematics of any interaction in the form of (a + b → c + d) can be solved by imposing energy-momentum conservation such as where s is the square of the total rest energy in the center-of-mass reference frame (CMS). Following the MDR in Equation (1), we also define the square of the energy of the particle i in the reference frame where its momentum is zero as The last term incorporates the LIV effects and can be seen as a shift in the effective mass of the particle. For the cases treated here, b represents the background photon, with energy , for which no LIV is considered and, thus, m b = 0 and δ b = 0. The energies of the outgoing particles, E c and E d , are unknown and can be expressed as a function of the inelasticity, K, of the interaction, In order to calculate s = s init = s final , we use the reference frame of the propagating UHECR (a), denoted with , Joining Equations (6)-(9) leads to where θ is the angle between the outgoing particles, c and d.
For the photodisintegration, the bound energy of the nucleus also needs to be taken into account. This can be achieved by shifting Equation (9) [66], where LI thr is the energy threshold in the NRF in a LI scenario. Finally, the phase space of the interaction can be obtained as the combinations of E a , , δ a,n , δ c,n and δ d,n for which Equation (10) has at least one real solution with 0 < K < 1.
While LIV could be introduced independently for each particle species motivated by the underlying theory, kinematically and in a phenomenological approach for astroparticles [15], one can consider a MDR to hadronic particles (δ had,n ), such as protons (p), nuclei (A: the atomic mass), and pions (π), then, Equation (1) reads, Considering heavier nuclei as a superposition of A-protons, Then, using Equation (12) in Equation (15), Following that LIV effects on pion production process relies on δ π,n − δ p,n [15], and using the latest derivation in Equation (16), we consider, δ p,n = A n δ A,n = δ π,n /2 := δ had,n . (17) Figures 7 and 8 show the energy threshold in the nucleus reference frame (NRF) for the pion production (a ↔ p, c ↔ p, d ↔ π) and for the photodisintegration (a ↔ A N, c ↔ A−1 N, d ↔ p) for different positive LIV hadronic coefficients (δ had,n ). Similar to the case of UHE photons, the main effect of LIV is a pronounced increase in the energy threshold (and consequent decrease in the phase space), which becomes significant above given energy that depends on δ had,n . Nevertheless, the effect is saturated for the highest energies differently from the pair production. This is because considered particles affected by LIV are present on both sides of the interaction. The effect of negative values of δ had,n is the opposite, decreasing the energy threshold. It is, however, suppressed by a decrease in the cross-section for lower background photon energies.
For the case of the photodisintegration, [66] presents a parameterization for the energy threshold in the NRF in the form of   Top and bottom panels correspond to an initial nuclues of helium and iron, respectively.

Searching for LIV in UHECR Data
The pioneering work [70] on testing LIV with UHECR was motivated by the lack of the flux suppression at the highest energies as published by the Akeno Giant Air Shower Array (AGASA) [71]. The absence of a cut-off in the energy spectrum was interpreted as a hint of LIV in the highest energies which would cause the decrease in the interaction rate of UHECRs. However, the data measured by the High-Resolution Fly's Eye Cosmic Ray Detector (HiRes) [72] and by the Pierre Auger Observatory [73] have shown a suppression at the expected energy range. More recent results confirmed the suppression of the energy spectrum at 46 EeV with more than 5σ confidence level [45,46]. Accordingly, LIV studies [15,74,75] have changed their approach and focused on estimating the LIV scale for which the lack of suppression would be in contradiction with measurements and, thus, LIV limits could be imposed. These works considered a pure proton composition and found that the data at that time could be well described up to some small LIV coefficients, and limits ranging from δ p,0 ∼10 −22 were imposed.
The continuous operation of the Pierre Auger Observatory imposed another challenge to the LIV studies. The distributions of the depth in which the shower has its maximum energy deposit, X max , cannot be described by a pure proton composition and point towards an intermediate composition [47,48,76].
Over the last years, the Pierre Auger Collaboration has worked on searching for LIV imprints in their data [40,77,78] taking into account the measured energy spectrum and X max distributions. A combined fit of both energy spectrum and X max distributions was performed under different LIV assumptions. The effects of LIV on the photodisintegration were also considered to treat the chemical composition properly. The combined fit follows the procedure in [42], where a homogeneous distribution of UHECR sources emitting cosmic rays isotropically is considered, and the propagation is simulated with Monte Carlo methods.
The parameters of the energy spectrum used in the fit were shown to be sensitive to the LIV assumption. In particular, for intermediate to strong LIV coefficients, the spectral index shifts towards −2 as predicted in mechanisms based on Fermi acceleration [79][80][81][82][83]. The deviance of the fit decreases for LIV coefficients of the order δ had,0 = 10 −20 to −21 . Nevertheless, the absolute value of the deviance of the fit is quite large, with a reduced value of 300/101, indicating overall poor description of the data by the considered model. Therefore, even though the results point to a preference for models with fewer interactions during propagation, it was not possible to claim whether that could be caused by LIV or by other astrophysical features not covered in the simplistic model considered, such as a larger contribution of local sources, effects of structured extragalactic magnetic fields or differences in emission from different sources.

Conclusions and Future Prospects
In this review, we explore the potential of UHE astroparticles as probes for LIV, presenting the current status of phenomenological research in this area. In particular, we discuss LIV in two sectors, photonic and hadronic, using the latest data of UHE photons and UHECR, respectively.
For the case of UHE photons, we review a commonly used analytical approach for the shift in the energy threshold of the pair production due to LIV. This interaction drives the photon propagation, leading to an absorption of the initial flux that modulates the expected measurements on Earth. When LIV is considered, an increase in the energy threshold is predicted, resulting in fewer interactions and, thus, a higher expected flux on Earth. For some astrophysical and subluminal LIV scenarios, the predicted flux is stronger than current upper limits on the photon flux, and, consequently, limits on LIV can be imposed. We discuss the model dependence of these results. We show that, within a range of reasonable astrophysical assumptions for the UHECR sources, the results can vary from complete insensitivity to LIV to the most restrictive LIV limits in the literature, δ γ,0 > −10 −22 , LIV > 10 30 eV . We demonstrate that such a dependence is dominated by the maximum attainable energy at the sources, followed by the chemical composition. Low maximum power of acceleration of the sources contributes to insensitivity to LIV since most UHECR do not achieve enough energy to produce pions and, therefore, fewer UHE photons are emitted. Heavier compositions tend to lead to weaker LIV limits in the same direction, as the photodisintegration dominates over the pion production. In Figure 3, we present the LIV limits imposed by a collection of astrophysical models (Table 1). A comparison between the capabilities of each model to describe UHECR data is beyond the scope of this work, and, thus, neither of the limits (or absence of limits) are favored.
For the case of UHECR, we review the calculation of the effects of LIV in the kinematics of interactions which can be expressed in the form a + b → c + d. In particular, we discuss the cases of the pion production and the photodisintegration, the leading energy losses for the highest energies. The inelasticity of the interaction can be reduced to a transcendental equation. For the photodisintegration, the energy threshold can be described by the parameterization proposed in [66], significantly speeding up the simulation of the UHECR propagation with LIV. Once more, the main LIV effect is an increase in the energy threshold, which leads to fewer interactions.
We briefly summarize the efforts of the community to describe the UHECR data considering LIV. Pioneer works have considered LIV in the proton sector and proposed to describe the energy spectrum measured by the AGASA, HiRes, and Pierre Auger Observatory experiments with a pure proton composition. Restrictive limits on the proton LIV coefficient were imposed. Nevertheless, a pure proton composition conflicts with current X max measurements performed by the Pierre Auger Observatory. Recently, the Pierre Auger Collaboration has published a fit of the energy spectrum and X max distributions under LIV assumptions. The results indicate a smaller deviance for a LIV scenario, however, a bad fit performance is seen overall and a more realistic model for the UHECR sources is needed for conclusive claims.
In summary, UHE astroparticles have proven to be a crucial tool in the search for a possible violation of Lorentz invariance. Current experiments provide precise and high statistics measurements, and a robust and widely spread framework was developed by many authors for the calculation of expected signatures of LIV in UHE data. Nevertheless, modeling UHECR is intricate due to their charged nature, the lack of knowledge about galactic and extragalactic magnetic fields, and the intrinsic fluctuations of extensive airshowers, which prevents an event-by-event composition measurement. Results on LIV strongly differ for different classes of models that can describe current UHE data. Planned upgrades of current experiments, such as AugerPrime [57,58] for the Pierre Auger Observatory and TAx4 [59] for the Telescope Array experiment, as well as future experiments such as the Probe of Extreme Multi-Messenger Astrophysics (POEMMA) [60] and Global Cosmic Ray Observatory (GCOS) [61] will be able to shrink the phase space of UHECR models, by constraining a possible proton component and better measuring the energy spectrum beyond the suppression. We are, thus, yet to see the full potential of UHE astroparticles as probes on LIV in the upcoming years.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations
The following abbreviations are used in this manuscript: