Ultra High Energy Cosmic Ray in light of the Lorentz Invariance Violation Effects within the Proton Sector

Ultra-high energy cosmic rays (UHECRs), most likely of extraterrestrial Aab:2017tyv ; Abreu:2012ybu ; Tinyakov:2015qfz ; Abbasi:2016kgr ; Blasi:2014roa origin, are composed mainly by protons and other atomic nuclei with energies above 1 EeV. Nowadays, it is believed that cosmic rays (CRs) with energy larger than $\sim 5\times 10^{19}{\rm eV}$ can have only a finite free propagation path. The Universe is opaque to the propagation of such high energy particles, whose propagation is affected by their interactions with the Cosmic Microwave Background (CMB) photons. Through these processes, UHECRs will dissipate energy during their propagation. Above $10^{19}$ eV, nuclei are photo-dissociated by interactions with the CMB photons. Similarly, a $10^{20}$ eV proton loses most of its energy by photopion production over a distance of order 50 Mpc. This means that UHECRs would have a finite mean free path of order $100$ Mpc for $10^{20}$ eV protons and such values can be even shorter for higher energies or heavier nuclei. Therefore, UHECRs can only be detected on Earth below a typical energy threshold if they origin from some distant object. Such a theoretical upper bound on the energy of UHECR is known as Greisen-Zatsepin-Kuzmin (GZK)-cutoff Greisen ; Zatsepin , marking the maximum energy that cosmic ray particles can achieve as they travel through the CMB.

However, great confusion had been caused by the measurements of the UHECR spectrum in the past decades. Actually, almost at the same time the GZK cutoff was proposed, the Volcano Ranch experiment 1963-1st-uhecr-event and the SUGAR array 1968-2nd-uhecr-event observed the earlier published events with energy above $10^{20}$ eV. Subsequently, more and more events with energy scale above GZK cutoff emerged one after another 1986-3nd-uhecr-event ; uhecr-4 ; uhecr-5 ; uhecr-6 ; uhecr-7 ; uhecr-8 ; uhecr-9 . Recent experiments also reported some observed high energy events with the corresponding energies above the GZK cutoff, for example, the Akeno-AGASA experiment AGASA-1 ; AGASA-2 , HiRes CollaborationHires-1 , and High resolution measurements like HiResHires-2 , the Pierre Auger Observatory pao-1 ; pao-2 , and Telescope Array TA-1 . Therefore, we need new proposals to explain these beyond GZK cutoff signatures, such as the Z-burst mechanism, vorton or the decay of monopole-antimonoploe pairs.

Although the sources of UHECRs are still unclear, we suggest that their propagation can possibly involve some effects from new physics, such as perturbative Lorentz Invariance Violation (LIV), which acts as the effective description of some quantum gravity theory at low energies. Very tiny LIV effects can also appear in the low energy Standard Model(SM) from some UV theory, which can nonetheless account for some peculiar features observed in the physics of UHECRs, for example, extended opacity horizon for UHECRs Coleman ; Stecker ; Scully ; TorriUHECR ; TorriPhD , because such gradual effects can increase with energy. Therefore, the detection of some UHECRs events with energies exceeding the GZK cutoff can possibly be explained in such a frame.

As we know, the components of UHECRs can be classified in three categories: light components principally composed by protons and He bare nuclei, intermediate components consisting of C, N and O bare nuclei, and heavy components composed of iron type bare nuclei (e.g., Fe and Ni) 2110.09900 . Since proton is the most abundant component of UHECRs, which only deflect slightly by magnetic fields and follow more direct trajectories compared to heavier nuclei. In this paper, we would like to explore potential effects of LIV in proton sector on the propagation of UHECRs.

Numerous theoretical models of LIV in various new physics frameworks had been proposed so far 9809521-Koste1 ; Coleman ; 0601236-vsr-Cohen ; AmelinoCamelia ; AmelinoCamelia2 ; AmelinoCamelia3 ; AmelinoCamelia4 ; string ; Chengyi2021 ; Chengyi2022 ; Chengyi2023 ; Rovelli2008 ; Ashtekar2021 ; lihao2023 ; DSR1-1 ; DSR1-2 ; 0112090-DSR2-1-Smolin2 ; DSR2-2 ; 0601236-vsr-Cohen . In our analysis, we choose to adopt a straightforward and simplest extension to the Standard Model (SM) of particle physics TorriHMSR ; 2110.09900 . The introduction of LIV can slightly alter the relevant kinematic and modify the permissible phase space for the photopion production process.

In the following sections, we will concentrate on the LIV for proton sector with modified dispersion relation from several perspectives: the geometry of spacetime, the extension of the SM, and the altered kinematics, etc. Possible LIV effects for interactions between UHECRs and CMB will be explored. This work is organized as follows. In Section 2, we present a specific LIV form for proton sector and its possible origin. In Section 3, we discuss the phenomenological implications of LIV on UHECR interactions with the CMB, which can affect the propagation of LHECR and shed light on the explanation of beyond GZK cut off events. In Section 4, we outline future directions for extending the analysis to encompass a more realistic cosmic ray (CR) composition scenario.

Lorentz invariance plays a key role in Einstein’s special relativity and serves as a cornerstone for both the general relativity and quantum field theory in modern physics. However, Lorentz symmetry may not be exact near the Planck scale ($M_{Pl}=1.22\times 10^{19}$ GeV), possibly subjected to the modification of the dispersion relation. From a bottom-up point of view, such a modification can effectively describe a LIV scheme that modifies the phenomenology of particle physics and at the same time preserves space-time isotropy as well as not introduce exotic interactions. It may origin from geometrized interaction with the assumed background quantum structure, akin to the approach taken in the Homogeneously Modified Special Relativity (HMSR) theoretical framework TorriHMSR ; 2304.12767 ; 9809521-Koste1 ; Coleman ; 0601236-vsr-Cohen ; AmelinoCamelia ; AmelinoCamelia2 ; AmelinoCamelia3 ; AmelinoCamelia4 ; Smolin1 ; 0112090-DSR2-1-Smolin2 .

Conventional dispersion relations can be modified by the introduction of various LIV terms that are suppressed by a factor of ($E/M_{Pl})^{n}$, where the exponent $n$ is sensitive to the underlying UV theories. The modified dispersion relations (MDRs) for massive fermions can be expressed as follows 1406.4568 .

The four-momentum of the massive fermion, such as protons, is denoted by ($E_{p},~{}\vec{p}$), with $|\vec{p}|$ the magnitude of the three-momentum vector. The exponent $n$ in the Lorentz-violating term determines the energy dependence: $n=1$ corresponds to linear energy dependence, and $n=2$ corresponds to quadratic energy dependence. The sign of the Lorentz violation correction is given by $\eta_{n}=\pm 1$, where $\eta_{n}=+1$ indicates that the maximum attainable velocity of this massive particle is less than convential vacuum light speed (subluminal case), and $\eta_{n}=-1$ indicates the opposite (superluminal) case. The suppression scale of the $n$-th order Lorentz violation is characterized by the Lorentz violation parameter $E_{LV,n}$. A modified dispersion relation, such as eq. (1), can possibly affect significantly the propagation of cosmic particles with ultra-high energies. It is important to note that, in practice, each particle species is assumed to have its own unique dispersion relation 0812.0121 ; 2304.12767 ; 2105.07967 ; 2110.09900 .

If Lorentz symmetry is violated in the proton sector while other particles keep it, CR protons with different energies that scatter with cosmic microwave background (CMB) photons would experience different time delay ($\Delta t$) when they reach Earth. As long as the cosmological propagation distances of cosmic rays are very large and the energies of these protons are very high, different time delay ($\Delta t$) caused by the effect of Lorentz violation could become measurable. This is similar to the case of high-energy photons, which, with modified dispersion relations, exhibit vacuum dispersion effects 1406.4568 .

UHECRs undergo various attenuation processes when they interact with background photons. These processes are influenced by the specific composition and the energy levels of the UHECRs.

When protons have energies below the threshold $E_{p,th}=5\times 10^{19}\,\text{eV}$, the $\Delta$-resonant photopion production process, as described in (16), is the dominant mechanism. For protons with even high energies, the process in (17) becomes the principal mode of attenuation. The photopion production processes are the core mechanism that responsible for the GZK cut-off phenomenon for protons, setting an upper bound on the energy of UHECRs from distant sources.

By analyzing the interactions between photons and protons, we can discern the characteristics of the GZK effect and revisit the propagation of high-energy protons across the Universe. Within the framework of the SM of particle physics, the collision of a proton with a photon can proceed via an intermediate resonant $\Delta$ state, which can be either real or virtual. Such dominant processes can be depicted by $p\gamma\to\Delta\to N\pi^{+}/p\pi^{0}$. Therefore, to produce resonantly a real $\Delta$ particle, the center-of-mass energy in the collision between the proton and the CMB photon must exceed the $\Delta$ rest mass ¹¹1 Should the kinematic prerequisites for generating a real $\Delta$ particle not be satisfied, the incorporation of LIV could markedly diminish the photopion production phenomenon. This, in turn, would have a consequential impact on the GZK cutoff..

The interaction between high-energy protons and the CMB background photons impedes significantly the ability of these protons to travel vast distances across the Universe. Denoting $E_{p}$ the energy of the high-energy proton and $\epsilon_{b}$ the energy of the CMB photon, the principles of special relativity allow us to deduce the threshold condition for this process

where $m_{p}$, $m_{N}$, and $m_{\pi}$ denote the masses of protons, neutrons, and pions, respectively. Given the average energy $\epsilon_{b}\approx 6.35\times 10^{-4}$ eV for CMB photons, the threshold energy for proton can be calculated by (18) to be $E_{p,th}=5\times 10^{19}$ eV. Protons with energies exceeding this threshold could unlikely reach the Earth due to such photopion processes involving the CMB photons. It can be seen from (18) that the threshold energy for proton decreases as the energy of the background photon increases. As a rough estimation, for $\epsilon_{b}$ from $10^{-3}$ eV to $1$ eV, the resulting thresholds $E_{p,th}$ would span from $10^{19}$ eV to $10^{17}$ eV.

We anticipate that LIV effects can only result in minor deviations from ordinary GZK phenomenon. Given that the dispersion relation is altered by LIV as depicted in eq. (1), the kinematics of the reaction $p\gamma\to NX$ should be modified accordingly, resulting in a modification of the allowed proton energy spectrum in UHECR, where $N$ represents nuclei and $X$ denotes mesons. To simplify the analysis, we will focus on the process $p\gamma\to N\pi$ in the following discussion (with the relevant Feynman diagram shown in fig. 1), where $\hat{\vec{p}}_{r}$ denotes the unit vector of the photon momentum.

• •

  LIV only in the proton sector:

  • –

    Only $n=1$ order LIV term is present:

    As noted previously, each particle species can practically have its own unique dispersion relation. For simply, in the process $p\gamma\to N\pi$, we assume that there is LIV in the proton sector, while other sectors are Lorentz invariant ²²2More practical discussions would consider that both fermions and mesons are LIV, such as that in Ref.2404.15838 . However, it is also feasible to consider only the LIV from the initial proton, since in the process $p\gamma\to N\pi$, the initial particle energy is much higher, which is more accordance with the assumption that LIV occures at very high energies. More importantly, as seen later, if the proton is LIV, it will require an increase of the minimum energy of the photons in the process $p\gamma\to N\pi$, much higher than the CMB photon energy. In this case, it would be quite difficult to have this process.. The dispersion relation for protons could be modified as the form in eq. (1)

    	$\displaystyle E_{p}^{2}$	$\displaystyle=$	$\displaystyle m_{p}^{2}+|\vec{p}|^{2}\left[1-\eta_{n}\left(\frac{|\vec{p}|}{E_{LV,n}}\right)^{n}\right]$		(19)
    		$\displaystyle=$	$\displaystyle m_{p}^{2}+|\vec{p}|^{2}\left(1-\eta\frac{|\vec{p}|}{E_{LV}}\right).$		(20)

    Here we choose $n=1$, $\eta_{1}=\eta$ and $E_{LV,1}=E_{LV}$.

    The following relations are satisfied for the determination of the energy threshold of proton 0211466-thrs-theorem :

    $$\vec{p}_{N}+\vec{p}_{\pi}=0;\\ ~{}~{}~{}(p_{r}+p_{p})^{2}=(m_{\pi}+m_{N})^{2},$$

    (21)

    so we can obtain the photon energy at the threshold,

    	$\displaystyle\epsilon^{LV}_{b(th)}$	$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta_{n}|\vec{p}|^{n+2}}{4E_{p}E_{LV,n}^{n}}$		(22)
    		$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta|\vec{p}|^{3}}{4E_{p}E_{LV}},~{}~{}(n=1)$
    		$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta E_{p}^{2}}{4E_{LV}},~{}~{}(E_{p}\simeq|\vec{p}|),$
    		$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}~{},~{}~{}~{}~{}{\rm for}~{}~{}E_{LV}\to\infty$		(23)

    where $E_{p}\simeq|\vec{p}|$ to leading order. Such a result agrees with the case without Lorentz violation in eq.(18) when $E_{LV}\to\infty$.

    We can find the global minimum for $\epsilon^{LV}_{b(th)}$ from eq.(22) by requiring

    $\displaystyle\frac{\partial\epsilon^{LV}_{b(th)}}{\partial E_{p}}=0~{}.$

    (24)

    The critical proton energy, $E_{p}=E_{p(cr)}$ that corresponds to the minimum of $\epsilon^{LV}_{b(th)}$ is given by

    $\displaystyle E_{p(cr)}=\left(\frac{\left[(m_{\pi}+m_{N})^{2}-m_{p}^{2}\right]E_{LV}}{2\eta}\right)^{\frac{1}{3}}.$

    (25)

    It is obvious that the critical proton energy $E_{p(cr)}$ increases with $E_{LV}$. Therefore, a higher LIV scale leads to a higher critical proton energy. The two terms in eq.(22) contribute equally to the threshold photon energy $\epsilon^{LV}_{b(th)}$ when $E_{p}$ takes value at the critical value $E_{p(cr)}$. Since the threshold energy $\epsilon^{LV}_{b(th)}$ in eq.(22) increases with the increment of proton energy $E_{p}$ upon the critical energy $E_{p(cr)}$, the numbers of CMB photons that can interact (via phtopion process) with such energetic protons decrease rapidly. Therefore, LIV effects could lead to a ”reemergence” of ultra-energetic protons in the astrophysical spectra. Such a pattern in the UHE proton spectrum could potentially explain the observations of cosmic rays with energies exceeding the traditional GZK cut-off. Observations of ultra-high energy protons (of energy $E_{ob}$) may be a sign of LIV upon $E_{ob}^{3}/(m_{\pi}m_{N})$ .

    In eq.(25), when $\eta=-1$, the solution will not be real, forcing us to consider the case with $\eta=+1$. This choice is opposite to the case of photons that depicted in eq.(3) of 2105.07967 . The reason lies in that photons lack rest mass while protons have rest mass. In photopion process $p+\gamma\to\pi+N$, the total rest mass of the final states is larger than that of the initial states.

    In the following discussions, we will take $\eta=+1$. The linear order ($n=1$) LIV scale (for photon sector) are determined to be larger than $E_{(LV,sub)}=3.6\times 10^{17}$ GeV for subluminal photon ($\eta=-1$) from Gamma Ray Burst (GRB) constraints 1607.03203 ; 1607.08043 ; 1801.08084 ; 2108.05804 ; 2105.07967 . If we naively adopt such a characteristic LIV scale for the proton LIV case, we can obtain the critical proton energy $E_{p(cr)}$

    $$E_{p(cr)}\simeq 1.723\times 10^{5}~{}{\rm GeV},$$

    (26)

    with the corresponding lower threshold energy scale for CMB photon

    $$\epsilon^{LV}_{b(th)}|_{min}\approx 4\times 10^{2}~{}{\rm eV}.$$

    (27)

    Such a value is almost six orders of magnitude greater than the average energy of CMB photon, making it almost impossible for any CMB photon to reach this energy threshold.

    

    However, the value of $E_{LV}$ can be quite different in the proton LIV sector from that in the photon LIV sector. There are many discussions on the bounds for $E_{LV}$ 1607.03203 ; 1607.08043 ; 1801.08084 ; 2108.05804 ; 2105.07967 ; 2105.06647 , within which the strongest one is (effectively) $E_{LV}^{eff}\simeq 10^{29}$ GeV 1810.13215 (which, in fact, amounts to very small LIV effect with $\eta\simeq 10^{-10}$ for $E_{LV}=M_{Pl}$). Although an effective transplanck scale seems not natural, it can a be sign of some UV mechanism that can strongly suppress the LIV effect with very tiny value of $\eta$. So we just take $E_{LV}^{eff}\simeq 10^{29}$ as the maximum value for LIV and such a value can be consistent with all phenomenological constraints 1607.03203 ; 1607.08043 ; 1801.08084 ; 2108.05804 ; 2105.07967 .

    In Fig.2, we show the values of $\epsilon^{LV}_{b(th)}$ versus $E_{LV}$ for $E_{LV}$ taken values from $10^{17}$ GeV to $10^{29}$ GeV. It is evident from this figure that, the minimum energy threshold for CMB photon that can allow its interaction with UHECR proton via photopion processes, is significantly higher (by several orders of magnitude) than the average energy of the CMB photons, which is around $2.6\times 10^{-4}$ eV. Consequently, with LIV in the proton sector, it is difficult for these CMB photons to effectively deter the ultra-energetic protons via the photopion processes when the energy of the proton excedd the corresponding critical value (26).

    Note that the critical threshold $E_{p(cr)}$ in eq.(22) increases dramatically with the increment of the proton energy $E_{p}$. If we adopt the upper range of $E_{LV}$ at around $E^{eff}_{LV}\sim 10^{29}$ GeV, we can get $\epsilon^{LV}_{b(th)}\simeq 0.1$ eV, which is much larger than the average energy of CMB photons. The critical proton energy for $E^{eff}_{LV}\sim 10^{29}$ can be determined to be $10^{18}$ eV, which is much smaller than the event observed by Akeno-AGASA experiment AGASA-1 ; AGASA-2 that have an energy at almost $E_{p}(max)=10^{21}$ eV. So, protons in UHECRs at energies of order $10^{21}$ eV could potentially reach Earth from distant sources without experiencing significant attenuation by CMB photons during their propagation. So, such beyond GZK cut off event can potentially be detectable AGASA-1 ; AGASA-2 ; Hires-1 ; Hires-2 ; pao-1 ; pao-2 ; TA-1 .

    

    We show in fig.3 the allowed values of proton energy $E_{p}$ versus the LIV scale $E_{LV}$ when threshold energies for CMB photons needed for photopion processes are given fixed. It can be seen that, given the value of $\epsilon^{LV}_{b(th)}$, higher energy of CR protons are required for higher LIV violation scale $E_{LV}$.

    Observed UHECR data can be used to impose stringent constraints on the low energy effective LIV theory, which can emerge as an effective description of some UV extensions of the SM 9703464-sme-cpt-sm ; 9809521-Koste1 ; sme-3 . For example, with the observation of energetic CR protons around $10^{20}$ eV, one can constrain the LIV coefficient $\eta$ to be approximately $\eta\sim 10^{-10}$ if we take the LIV scale to lie near the Planck scale. Such an $\eta$ value is significantly lower than its natural choice $\eta\sim{\cal O}(1)$ that one might anticipate. Such a stringent constraint imply that the UHECR data could shed new lights on the test of LIV theories.

  • –

    Both $n=1$ and $n=2$ order LIV terms are present:

    In the case both $n=1$ and $n=2$ order LIV terms are present, the threshold energy for the photons are given as

    	$\displaystyle\epsilon^{LV}_{b(th)}$	$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta_{p;L}|\vec{p}|^{3}}{4E_{p}E_{LV}}+\frac{\eta_{p;NL}|\vec{p}|^{4}}{4E_{p}E_{LV}^{2}},$		(28)
    		$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta_{p;L}E_{p}^{2}}{4E_{LV}}+\frac{\eta_{p;NL}E_{p}^{3}}{4E_{LV}^{2}},$		(29)

    The extreme for $\epsilon^{LV}_{b(th)}$ can be determined from Eq.(29) by requiring

    $\displaystyle\frac{\partial\epsilon^{LV}_{b(th)}}{\partial E_{p}}=0~{},$

    (30)

    which leads to the quartic equation

    $\displaystyle 3\eta_{p;NL}E_{p}^{4}+2\eta_{p;L}E_{p}^{3}E_{LV}-\left[(m_{\pi}+m_{N})^{2}-m_{p}^{2}\right]E_{LV}^{2}=0~{}.$

    (31)

    Although an quartic equation can be solved analytically, the form of its roots are rather complicate. So, it is better to recourse to numerical methods for its solutions. For very large $E_{LV}$, we can always find an approximate root with $E_{p}\approx-\frac{3\eta_{p:NL}}{2\eta_{p;L}}E_{LV}$. So, from Vieta theorem, we anticipate that the remaining three solutions have magnitude of order $E_{p(cr)}\equiv(m_{\pi}m_{N}E_{LV})^{1/3}$, which is the same order as eq.(25). However, for certain fixed choices of $E_{LV}$, it is possible the values of $\epsilon^{LV}_{b(th)}$ with respect to $E_{p}$ have multiple local minimums and maximums. If the value of typical local maximum exceeds the average CMB photon energy, the GZK cut-off in this case shows a discontinous pattern. That is, protons with energies in several disconnected ranges can survive the GZK constraints. The existence of such a pattern requires that the eq.(29) should have three local minima/maxima with one saddle point, which corresponds to the requirement that eq.(31) should have four distinct real roots. Observations of such discontinuous GZK cut-off bands can be a sign of this scenario. Besides, not as naively expected, the presence of higher-order LIV terms can play an important role in determining the GZK cut-off energy bands.

  We would like to analyze the root structure of the quartic equation. The numbers of real roots of Eq.(31) determine the local minima of $\epsilon^{LV}_{b(th)}$. On the other hand, the real roots are determined by the discriminant of the quartic equation. For an quartic equation of the form

  $\displaystyle x^{4}+qx^{2}+rx+s=0~{},$

  (32)

  the discriminant is given by

  $\displaystyle\Delta=16q^{4}s-4q^{3}r^{2}-128q^{2}s^{2}+144qr^{2}s-27r^{4}+256s^{3}~{}.$

  (33)

  An necessary and sufficient condition for the existence of four distinct real roots requires that

  $\displaystyle\Delta>0,~{}q<0,~{}s>\frac{q^{2}}{4}.$

  (34)

  In our case, the equation eq.(31) can be reformulated into the form of eq.(32) after dividing the coefficient $3\eta_{p;NL}$ and replacing

  $\displaystyle E_{p}\rightarrow\tilde{E}_{p}=E_{p}+\frac{\eta_{p;L}E_{LV}}{6\eta_{p;NL}}.$

  (35)

  The quartic equation (31) reduces to

  $\displaystyle\tilde{E}_{P}^{4}+q\tilde{E}_{p}^{2}+r\tilde{E}_{P}+s=0~{},$

  (36)

  with

  	$\displaystyle q$	$\displaystyle=$	$\displaystyle-\frac{1}{6}\left(\frac{\eta_{p;L}E_{LV}}{\eta_{p;NL}}\right)^{2}~{},~{}~{}~{}~{}~{}r=\frac{1}{27}\left(\frac{\eta_{p;L}E_{LV}}{\eta_{p;NL}}\right)^{3}~{},$
  	$\displaystyle s$	$\displaystyle=$	$\displaystyle-\frac{\left[(m_{\pi}+m_{N})^{2}-m_{p}^{2}\right]E_{LV}^{2}}{3\eta_{p;NL}}-\frac{1}{432}\left(\frac{\eta_{p;L}E_{LV}}{\eta_{p;NL}}\right)^{4}~{}.$		(37)

  As physical energy $E_{p}$ should take positive values, the four real roots are required to be positive. Therefore, we should impose the following additional constraints

  $\displaystyle\eta_{p;NL}f(E_{p})>0,~{}~{}~{}{\rm for}~{}~{}~{}~{}~{}~{}E_{P}<0,$

  (38)

  in addition to eq.(34), with

  $\displaystyle f(E_{p})\equiv 3\eta_{p;NL}E_{p}^{4}+2\eta_{p;L}E_{p}^{3}E_{LV}-\left[(m_{\pi}+m_{N})^{2}-m_{p}^{2}\right]E_{LV}^{2}~{}.$

  (39)

  Existence of discontinuous GZK cut-off energy bands requires that the above equation (36) satisfies the conditions in eq.(34) and eq.(38).

• •

  LIV in both the proton and photon sectors:

  If LIV origins from the underlying spacetime structures, dispersion relations for gauge bosons, such as photon, should also be amended to incorporate the LIV effects. So, we would like to include also the LIV effects for initial state photons for completeness.

  The MDR for photons can take the following form as0510172 ; 1406.4568 ; 2105.07967

  $$E_{\gamma}^{2}=|\vec{p}|^{2}\left[1-\sum\limits_{n}\eta_{\gamma;n}\left(\frac{|\vec{p}|}{E_{LV}}\right)^{n}\right].$$

  (40)

  Therefore, with LIV effects for both proton and photon sectors, the threshold energy for the photons can be determined to be

  	$\displaystyle\epsilon^{LV}_{b(th)}$	$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta_{p;n_{p}}|\vec{p}|^{n_{p}+2}}{4E_{p}E_{p;LV}^{n_{p}}}+\frac{\eta_{\gamma;n_{\gamma}}|\vec{p}_{\gamma}|^{n_{\gamma}+2}}{4E_{p}E_{\gamma;LV}^{n_{\gamma}}},$		(41)
  		$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta_{p;n_{p}}|\vec{p}|^{n_{p}+2}}{4E_{p}E_{p;LV}^{n_{p}}}+\frac{\eta_{\gamma;n_{\gamma}}(\epsilon^{LV}_{b(th)})^{n_{\gamma}+2}}{4E_{p}E_{\gamma;LV}^{n_{\gamma}}},$

  For simplication, we focus on the case with $n_{p}=1$ and $n_{\gamma}=1$ in our following discussions.

  The LIV scales for both sector could be hierarchical with

  $$E_{LV}\equiv E_{p;LV}\gg E_{\gamma;LV}.$$

  So, we have the threshold energy for photon

  	$\displaystyle\epsilon^{LV}_{b(th)}$	$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\frac{\eta_{p}|\vec{p}|^{3}}{4E_{p}E_{p;LV}}+\frac{\eta_{\gamma}|\vec{p}_{\gamma}|^{3}}{4E_{p}E_{\gamma;LV}},$		(42)
  		$\displaystyle=$	$\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\eta_{p}\frac{E_{p}^{2}}{4E_{LV}}+\frac{\eta_{\gamma}(\epsilon^{LV}_{b(th)})^{3}}{4E_{p}E_{\gamma;LV}},$

  with $E_{p}\simeq|\vec{p}|,\epsilon^{LV}_{b(th)}\simeq|\vec{p}_{\gamma}|$ when LIV effects are present in both proton and photon sectors. The second term in eq.(43) gives the dominant contribution to $\epsilon^{LV}_{b(th)}$ when $E_{p}$ is very large with

  $$E_{p}\gg{E}_{p(cr)}\simeq\left[m_{\pi}m_{N}E_{LV}\right]^{1/3}.$$

  So we have

  $\displaystyle\epsilon^{LV}_{b(th)}$

  $\displaystyle\approx$

  $\displaystyle\frac{(m_{\pi}+m_{N})^{2}-m_{p}^{2}}{4E_{p}}+\eta_{p}\frac{E_{p}^{2}}{4E_{LV}}+\frac{\eta_{\gamma}\eta_{p}^{3}E_{p}^{5}}{32E_{LV}^{3}E_{\gamma;LV}},$

  (43)

  in the large $E_{p}$ regions. We can see that the third term can be non-negligible only if $E_{LV}\gtrsim E_{p}\gg E_{\gamma;LV}$. Current lower bound on $E^{n_{\gamma}}_{\gamma,LV}$ is rather stringent, which is constrained to be much higher than the Planck scale for $n_{\gamma}=1$. So, the presence of LIV in the photon sector with $n_{p}=1$ and $n_{\gamma}=1$ will not lead to additional non-negligible contributions to the threshold energy $\epsilon^{LV}_{b(th)}$, whose value is almost the same as the case without LIV in the photon sector.

  If the leading LIV effect for photons are of quadratic order ($n_{\gamma}=2$), by similar discussions for the case $n_{\gamma}=2$ and $n_{p}=1$, one can conclude again that the quadratic LIV term of the photon sector can lead to non-negligible contribution to the threshold energy of photon only if $E_{LV}\gtrsim E_{p}\gg E^{n_{\gamma}=2}_{\gamma;LV}$. In this case, the bound on $E^{n_{\gamma}=2}_{\gamma,LV}$ is not so stringent, which can possibly be smaller than the Planck scale. So, LIV in the photon sector can possibly increase the photon threshold energy $E^{n_{\gamma}}_{\gamma,LV}$ for given $E_{P}$ with $E_{LV}\gtrsim E_{p}\gg E^{n_{\gamma}=2}_{\gamma;LV}$, making it easier for the UHECR protons to surpass the GZK cut-off scale. Such a conclusion can hold for the cases with leading LIV effect of photons taking $n_{\gamma}\geq 2$.

Tiny LIV effects may origin from typical space-time structures in quantum gravity theories. So, it is reasonable to anticipate that tiny LIV effects can be present in the proton sector. We find that, with tiny LIV effects in the proton sector, the threshold energy of photon that can engage in the photopion interactions with protons can be pushed to much higher scales (of order 0.1 eV to $10^{3}$ eV) in comparison with the case without LIV. Therefore, the proton specie in UHECRs can possibly travel a long distance without being attenuated by the photopion processes involving the CMB photons, possibly explain the observed beyond-GZK cut-off events. We also find that, when both the leading order and next leading order LIV effects are present, the higher order LIV terms can possibly lead to discontinuous GZK cut-off energy bands. Observation of beyond-GZK cut-off UHECR events involving protons can possibly constrain the scale of LIV. Such UHECR events can act as a exquisitely probe of LIV effects and shed new lights on the UV LIV theories near the Planck scale.

The form of LIV term in the proton sector also preserves the spatial-temporal isotropy. So, correlation between the anisotropy of UHECR arrival directions and the spatial distributions of candidate sources can also be used to constrain the relevant parameters. We leave such studies for our future works.

This work was supported by the National Natural Science Foundation of China (NNSFC) under grant Nos.12075213 and 12335005, by the Natural Science Foundation for Distinguished Young Scholars of Henan Province under grant number 242300421046.