Restrictions from Lorentz Invariance Violation on Cosmic Ray Propagation

Restrictions from Lorentz invariance violation on cosmic ray propagation

H. Mart´ınez-Huerta∗ and A. Pérez-Lorenzana† Departamento de F´ısica, Centro de Investigacióny de Estudios Avanzados del I.P.N., Apdo. Post. 14-740, 07000, Ciudad de México, México (Dated: February, 2017) Lorentz Invariance Violation introduced as a generic modification to particle dispersion relations is used to study high energy cosmic ray attenuation processes. It is shown to reproduce the same physical effects for vacuum Cherenkov radiation, as in some particular models with spontaneous breaking of Lorentz symmetry. This approximation is also implemented for the study of photon decay in vacuum, where stringent limits to the violation scale are derived from the direct observation of very high energy cosmic ray photon events on gamma telescopes. Photo production processes by cosmic ray primaries on photon background are also addressed, to show that Lorentz violation may turn off this attenuation process at energies above a well defined secondary threshold.

PACS numbers: 11.30.Cp, 12.20.-m, 96.50.sb, 98.70.Sa

I. INTRODUCTION a particular LIV model. Although, it can be compatible with the mSME, they usually express their LIV parame- Lorentz symmetry stands as one of the cornerstones ters in different ways. Here, we propose to use the derived of fundamental physics. Nonetheless, as for any other physics from the mSME as an ansatz to build a generic fundamental principle, exploring its limits of validity has approach with the same physics, in order to contrast the been an important motivation for theoretical and expe- results with experiments at the highest energetic win- rimental research on the past [1]. Currently, supported dow, that is, those observing cosmic and gamma rays. models with Lorentz Invariance Violation (LIV) have re- Astrophysical scenarios are an interesting place to test newed the interest of the community due to the oppor- the possible signatures of LIV due to the high energies tunity to observe or restrict new physics with the re- and very long distances they involve. The effects of LIV cent measurements from the Observatories at the highest are expected to increase with energy for some energetic energies. regime close to the Plank scale. Additionally, the very Lorentz Invariance Violation is motivated as a possible long distances can lead to a significant LIV effect due to consequence of beyond the Standard Model (SM) theo- accumulative processes. For the present study we have ries, such as Quantum Gravity or String Theory, just to chosen to explore vacuum Cherenkov radiation and pho- mention some (see for instance Refs. [1–10]). However, ton decay for these purpose. Both processes are forbid- the lack of observations of the LIV derived phenomena den in the SM physics, but under LIV hypothesis they imposes limits on the validity scale of these models. are permitted and can lead to extreme scenarios due to Among the approaches used to introduce LIV on par- their implications on cosmic and gamma ray propagation. ticle physics two stand out. One is generic and it is not Once our generic approach had been validated, we will (necessarily) bounded to a particular model. The other proceed to the exploration of other processes that are one is based on the spontaneous breaking of the Lorentz also relevant for cosmic ray propagation. symmetry. The former is implemented via a general mod- This article is organized as follows. On next subsec- ification of the single particle dispersion relation, whereas tion we will briefly present both, the spontaneous and the second mechanism introduces to the Standard Model the generic LIV mechanisms, that we will use along the a collection of LIV renormalizable operators that preserve paper. On Sec. II we present the derived physics for energy-momentum and microcausality, among other de- vacuum Cherenkov radiation from the mSME and de- sired properties of the Standard Model. These even have velop the generic approach for the same phenomena. In

arXiv:1610.00047v3 [astro-ph.HE] 8 Mar 2017 the advantage of being singlets of the Lorentz group at Sec. III, we focus on photon decay, using the LIV generic coordinates level. This extension is named the mini- approach, to show that this phenomenon implies a very mal Lorentz-violating extension of the Standard Model restrictive scenario for photon propagation. As a conse- (mSME) [11]. quence of it, we extract limits for the LIV scale, EQG, Both mechanisms, the mSME and the generic LIV, can to first and second order on the LIV correction, from lead to physics beyond Standard Model. Nevertheless, the direct observation of very high energy photon events the generic approach tends to converge quicker and sim- at H.E.S.S. and HEGRA cosmic gamma ray telescopes. pler to phenomenology and it is not necessarily bound to In addition, a general analysis for photo production processes on the photon cosmic background, oriented to eva- luate the effects of the generic LIV effects on the de- termination of the energy thresholds, is presented along ∗ hmartinez@fis.cinvestav.mx section IV. Here, we also discuss the appearance of a se- † aplorenz@fis.cinvestav.mx condary energy threshold, as a signature of LIV, above 2 which photo production becomes forbidden. Finally, a 2. mSME last section is dedicated to summarize our results. The mSME provides a more specific theoretical ground to study the resulting physics from the introduction of A. Lorentz-violating mechanisms the LIV hypothesis, throughout the spontaneous violation of Lorentz symmetry [1, 2]. This mechanism adds 1. Generic LIV to the Standard Model (SM) all observer-scalar terms that are products of SM and gravitational fields which are not Lorentz invariant, and generically require cou- Generic LIV mechanism is perhaps the most commonly plings that explicitly carry Lorentz indices. The mini- used on phenomenology, studies since it offers a clear mum set of such operators that maintain gauge invari- idea on the derived physics from the LIV hypothesis and ance and power-counting renormalizability conform the it is not necessarily bound to any particular LIV-model. so called mSME and was proposed by Colladay and Kost- This mechanism converges to the introduction of an extra elecky [11]. A first classification of these new terms cor- term in the dispersion relation of a single particle [10, responds to the own sectors of SM, such as the L 12–20]. Generically, this term can be motivated by the lepton or L , for lepton or gauge sectors respectively. Addi- introduction of a not explicitly Lorentz invariant term gauge tionally, such operators can be classified into those that at the free particle Lagrangian [21]. The extra term is break CPT symmetry and those that do not. They are restricted by a dimensionless coefficient that we call . named CPT-odd and CPT-even respectively. To the lowest order, the new dispersion relation for a In the present work, we have chosen the photon sector free particle takes the form: with the following additional contributions to the standard Lagrangian density, E2 − p2 ± A2 = m2, (1) 1 LCPT −even = − (k ) F ρλF µν , (3) where (E, p) stand for the four-momenta associated with photon 4 F ρλµν a given particle of mass m, and we use the short hand and notation p = |p| . A can take the form of E or p, however, for the ultra relativistic limit where m {E, p}, CPT −odd 1 ρ λ µν L = (kAF ) ρλµν A F , (4) any particular choice of A will be equivalent. We will photon 2 choose the sign of according to whether we correct en- with the gauge field Aµ and the field strength tensor ergy or momentum, leaving positive sign for energy. For Fµν = ∂µAν −∂ν Aµ. In above µνρλ is the antisymmetric simplicity, we shall take A = p in most of this article, Levi-Civita symbol. The coefficients kF and kAF in Eqs. but in section IV, we will use A = E. Asking that the (3) and (4) are the ones that break particle Lorentz In- LIV term can only be relevant at the highest energies, variance. Besides, kF is dimensionless and kAF has mass should be small, which guarantees that the physics at dimension one. On the analysis regarding this sector, it low energies remains Lorentz invariant. is usual not to assume the simultaneous presence of LIV The most general modification to the dispersion rela- corrections for fermions and we are going to do so in our tion would rather involve a general function of energy and own study along the following sections. momentum, such that one could write E2 −p2 +(A)A2 = The CPT-even term can be studied separately as in m2. Since at low energies the contribution of should the so called modified Maxwell theory (modM [31]), be negligible, one can use a Taylor expansion and keep one term of order n at once, assuming it as the leading 1 µν 1 µν ρλ LmodM = − Fµν F − κ Fµν F . (5) term, in order to study the underlying physics. On this 4 4 ρλ assumption the dispersion relation will become µν κ ρλ is a dimensionless fourth rank tensor, with 256 com- 2 2 n+2 2 ponents, but it its usually assumed to have vanishing E − p ± αnA = m , (2) µν double trace, κ µν = 0, and demanded to obey the same symmetries which hold for the Riemann curvature tensor, n (n) n where αn = 1/M = 1/(EQG) defines the energy scale −n κµνρλ = −κνµρλ = κνµλρ, κµνρλ = κρλµν , (6) associated to the LIV physics. αn units are in eV , hereafter. M is commonly associated with the energy such that only 19 independent components remain [32]. scale of Quantum Gravity, EQG, which is expected to be However, the following ansatz, used in Ref. [31], reduces 19 close to the Planck scale, EP l ≈ 10 GeV . In literature, the number of independent parameters from 19 to 1. one can find upper limits to E(n) by different techniques, QG 1 even beyond the Planck scale [22–30]. κµνρλ = (ηµρκ˜νλ − ηµλκ˜νρ + ηνλκ˜µρ − ηνρκ˜µλ), (7) 2 Notice that Eq. (2) can be found from a particular LIV model. For instance, Coleman and Glashow formalism is and compatible with the special case n = 0 [20, 21], likewise, 3 κ˜µν = κ˜ diag(1, 1/3, 1/3, 1/3), (8) for n = 1 see for instance [29] and for any n see Ref. [30]. 2 tr 3 where Eq. (7) restricts the theory to the nonbirefringent Hereafter, (ω, k) will stand for the photon four-momenta sector and Eq. (8) does it to the isotropic sector. components. Also, (E, p) will represent the initial four- On the other hand, the CPT-odd term is studied in momentum of the spin 1/2 charged particle involved in the spacelike Maxwell-Chern-Simons theory (MCS [33]), the process, while (E0, p0) will stand for the final four- with the following Lagrangian density, momentum components. Following Refs. [31, 33], the dispersion relations for a single photon from modM and 1 1 L = − F F µν + M ξµAν F ρλ, (9) MCS models are written as MCS 4 µν 4 CS µνρλ s −33 1 − κtr where MCS ∼ 1/L0 ≈ 2 × 10 eV is the Chern-Simons ω (k) = e k, k = |k|, (11) modM 1 + κ mass scale [33, 34] and ξµ = (0, 0, 0, 1) is the fixed space- etr like normalized Chern-Simons vector. It is noteworthy that in Eqs. (2), (3) and (4) the in- r M 2 M 2 volved coefficients are not necessarily the same and so, ω (k)2 = k2 ± M k2 cos2 φ + CS + CS , numerically, MCS ± CS 4 2 (12) kF 6= kAF 6= αn, (10) whereκ ˜tr is the restricted coefficient from Eq. (8). In the MCS expression, φ is the angle between the wave vector despite that all of them parametrize the LIV hypothesis. k and the space component of the Chern-Simons vector, Hence, the values and limits that can be derived for them ξ. Notice that this model is not rotationally invariant, so shall be different too. Therefore, for comparison proposes energy-momentum will be conserved but angular momen- with the LIV generic approach, we will use the limit va- tum shall not. The subscript ± denotes two polarization lues given in the literature for the involved parameters of modes. modM and MCS models. The reported emission rates for these models are

e2 r ! 4π 1 1 − κetr E II. VACUUM CHERENKOV RADIATION ΓmodM = p √ − 1 2 2 E 1 + κtr E2 − m2 3˜κtr 1 − κetr e r 1 − κ A. Emission Rate etr 2 p 2 2 × (3κetr + 5κetr + 2)E E − m 1 + κetr 2 2 2 2i When electrically charged and massive particles ruled − (3˜κtr + 3˜κtr + 2)E + (3˜κtr + 4˜κtr + 1)m , by a Lorentz invariant physics are added to the LIV (13) models presented above, the new terms allow photon and emission in vacuum to happen. This process is com- e2 ( monly called vacuum Cherenkov radiation (See for in- MCS h i Γ = 4π 4 m2 + 2(p · ξ)2 − M 2 stance [20, 21, 31, 33, 35]). In a general scenario, the MSC 16E((p · ξ)2 + m2)1/2 CS derived physics will be spin dependent, and thus, either 2k particles with spin 0 or 1/2 will be able to emit photons. × arcsinh max + 2(2|p · ξ| − k )M M max CS To further proceed with our analysis for the generic im- CS ) q plementation of LIV and the latter comparison with the 2 2 2 kmax − 4(|p · ξ| − kmax) MCS + 4kmax − 8m , outcomes of the modM and MCS models, we will restrict MCS our discussion to spin 1/2 particles. In what follows, let (14) us ﬁrst summarize the results for these models. where

p 2 2 2MCS |p · ξ|(MCS + 2 (p · ξ) + m ) kmax = , (15) 2 2 p 2 2 MCS + 4m + 4MCS (p · ξ) + m is the maximum magnitude for the photon momentum component for MCS emission rate. Next, let us elaborate the corresponding analysis for the generic case. So far, there is not a clear path to derive the emission rate from the LIV generic approach. FIG. 1. Spontaneous photon emission, or vacuum Cherenkov However, we will follow the generalities from the two pre- radiation, Feynman diagram. vious models to construct it. Starting with Eq. (2), the dispersion relation for photons can be written as

0 2 2 n The process of interest, lp → lp γk, is depicted in ω = k (1 + αnk ), (16) Fig. 1, where the vacuum Cherenkov radiation may correspond in general to leptons, protons and heavy nu- where αn is the LIV coefficient of order n. We shall clei. The subscript denotes the momentum notation. also neglect, as in previous models, any LIV effect in the 4 corresponding fermion dispersion relations. Then, we will and when develop a generic correction to the emission rate from the r ! 1 2 4 4p cos θ 2 2 2 standard LI theory, given by: k± = E ± E + (E − p cos θ) . 2p cos θ α1 3 0 3 s 1 2 d p d k (23) dΓ = |M| 3 0 0 3 2 E(p) (2π) 2E (p ) (2π) 2ω(k, n, αn) (17) In what follows we will only use the mode k+, since k0 means that there is none emission and k will be ruled × (2π)4δ(4)(p − p0 − k)Θ(k), − out by the Heaviside function in Eq. (17), besides it is nonphysical. where we have inserted a Heaviside function, Θ(k), in Finally, one can use the expressions in Eqs. (16), (18) order to preserve only physical solutions for the emitted and that for k in Eq. (23), to perform the integral over k. photon momentum. Also, s = 1 is an statistical factor, + j! Hence, the emission rate from the LIV generic approach where j counts one per each group of identical particles at first order in α (LIVgen1) will be expressed as, in the final state. |M|2 is the squared amplitude proba- n bility of the process, computed from the standard QED 2 Z θmax 2 3 e 1 |4m − α1k+| rules but making use of the correction in the photon four- Γ = 4π 4E ω(k+, α1) momenta. After the calculations we found that 0 k sin θdθ × , (1+ 3 α k ) q 2 2 2 n+2 2 1 + 2 2 |M| = e |4m − αnk |, (18) | − k+ + p cos θ − √ k+ + E − 2k+p cos θ| 1+α(1)k+ (24) where orthogonality relation is assumed for photon at In figure 2, we show the emission rate for the three first approximation, ¯ ≈ g . For a more detailed µ ν µν different models for proton mass, m = m . We have calculation see Ref. [36]. The delta function in Eq. (17) p plotted, in the (green) dashed line, the modM emission encodes the conservation of energy-momentum in the sys- rate at the first order from the expansion on E andκ ˜ tem. Using Eq. (16) and keeping only first order in LIV 00 from Eq. (13) and settingκ ˜ = 10−21, which is com- α terms, we made the integration over p0 in Eq. (17) 00 n patible with the limits reported in Ref. [24] . From a and write the remaining delta function as: phenomenological point of view, the rate has four main characteristics. First, it has a growing behavior with the δ(0)(g(p, k, m, n, α, θ)) = √ , (19) charged particle energy. Second, it has a drop ending (0) p 2 2 p 2 2 n δ ( p + m − (p − k) + m − k 1 + αnk ) threshold for the model, inversely proportional to the LIV coefficient. This threshold protects the LI physics where θ is the emission angle and g is a polynomial func- at energies below it. Third, for a given energy above the tion of order n in k. To solve the integration in k with- threshold, the emission rate decreases for smaller values out the rest frame, that is, only in the observer frame, in the LIV parameter. And fourth, it has a sensitive we used the following well known property of the delta behaviour with the mass and charge of the emitting par- function, ticle. The last two characteristics will be discussed in the next section. The MCS emission rate, in dash-dotted X δ(x − xi) δ(g(x)) = , (20) (blue) line in the same figure, shows the same behav- |g0(x )| i i ior as modM but the derived threshold is higher in energy than the previous one, since it is highly constrained where xi are the g(ki) roots, that we can found from the from the MCS. Finally, we present in a (red) continuous following expression, line the emission rate from the LIVgen1 model for very small θ and with a phenomenological threshold given by 2 2 2 2 2 2 n+1 p cos θ − (p + m ) k−(p + m )αnk 2 1/3 (21) 4m n+2 E & α . Hence, the generic approach reproduces + p cos θαnk = 0. 1 the main features of the previous two models. We have Each solution of Eq. (21) implies that the process will found that the case n = 0 has a similar behaviour [37]. satisfy energy-momentum conservation. Note that if p 2 2 αn = 0 then k = 0, since p cos θ 6= p + m , and consequently, the process is prohibited for LI scenarios. In 1. Sensitivity to UHECR mass composition other words, the photon emission process in vacuum, or vacuum Cherenkov radiation, is permitted as far as αn Considering the application of these models to the phe- is present. nomenon of cosmic rays, we have chosen m = mp and For n = 1, we found that Eq. (21) has three different m = mF e as representative masses for a light and a heavy solutions. That is, in the generic LIV scenario energy component of the cosmic ray flux, respectively. It is esti- and momentum are preserved in at least three different mated that the composition of ultra high energy cosmic situations, given by the standard case where rays (UHECR) is mixed and varies between these two limits [38]. To appreciate the differences between the k0 = 0; (22) components, in Figs. 3 and 4, the behavior of the Γ 5

22 − − 22 modM 10 LIVgen1 ( , ; ) 10 −, modM1 ( − , ) − 16 16 , 10 − 10 MCS ( , ) − , ] ] 10 − e 10 e 10 10 , [ [ − −

4 4 10 10

10-2 10-2

10-8 10-8 1017 1019 1021 1023 1025 1027 1029 1017 1019 1021 1023 1025 1027 1029

E [e] E [e]

FIG. 2. Comparison of emission rates for vacuum Cherenkov FIG. 4. Emission rates for vacuum Cherenkov radiation from radiation for a proton. Free LIV parameters were chosen as modified Maxwell theory. Corresponding rates for mF e, mp indicated and different κetr are as indicated. Both rates grow with energy, but lighter particles have a lower emission rate. The energy threshold is sensitive to particle composition. functions from LIVgen1 and modM for proton and iron are compared. From them, it can be seen that the effect 1016 of vacuum Cherenkov emission is sensitive to the compo- LIVgen1 sition of cosmic rays. In Figs. 3 and 4, we also stand −36 1 =10 , 1014 −36 1 =10 ,

22 LIVgen1 ] 10 12 − eV 10

, [

− dt 16 ,

10 0 10 − 10 , dP ]

− e 1010 , [ 8

10 − 4

10 106 1017 1018 1019 1020 1021 1022 1023 10-2 E [eV]

10-8 1017 1019 1021 1023 1025 1027 1029 FIG. 5. Radiated energy for vacuum Cherenkov radiation E [e] from LIV generic approach at first order correction, with n = −2 1 and θmax = 10 . Corresponding curves for mF e and mp are as indicated. FIG. 3. Emission rates for vacuum Cherenkov radiation from LIV generic approach at first order correction, with n = 1 and −2 θmax = 10 . Corresponding rates for mF e, mp and different α1 are as indicated. Both rates grow with energy, but lighter Moreover, in LIVgen1 and modM there are regions where particles have a lower emission rate. The energy threshold is vacuum Cherenkov radiation only affects the lighter par- sensitive to particle composition. ticles but not heavier, until the energy is high enough that vacuum Cherenkov radiation can affect all the massive out the emission rate behaviour for smaller values in the particles. Therefore, a reduction to the CR flux could be LIV parameters. The emission rates become smaller as expected due to vacuum Cherenkov radiation that starts α1 and κetr do. with the lighter components. Hence, a tendency to heavy In Fig. 5 it is shown the emitted energy from the components could be expected in the UHECR flux lo- generic approach. A strong emission rate will present a cated between both the thresholds, followed by a reduc- stronger attenuation effect. For an energy scale where tion on the contributions from the entire mass spectrum vacuum Cherenkov radiation is allowed for the entire at even higher energies. From the results in Figures 3, −36 −1 mass spectrum, between proton and iron limits, heav- 5 and 6 and for a value for α1 = 10 eV , one could ier particles will suffer vacuum Cherenkov attenuation expect that this trend would show up around an energy some orders of magnitude higher than lighter particles. range of about 1018 to 1019eV . Observatories such as Then, heavier particles will be attenuated sooner in en- Pierre Auger, Telescope Array and HiRes have measured ergy. This special behavior is clarified in Fig. 6 with and reported the spectrum of ultra-energy cosmic rays the emission probability as a function of mean free path. at EeV energies [39–41]. Thus, a dedicated analysis with 6 their data could probe such an effect or at least set limits The decay rate of the process will be thus given by to αn. 3 3 0 s 1 2 d p d p dΓ = |M| 3 3 0 0 2 ω(k, n, αn) (2π) 2E(p) (2π) 2E (p ) (25) 0 −2 4 (4) 0 LIVgen1 , =10 × (2π) δ (k − p − p )Θ(p), 0 21 =10 eV 0 −36 0 0 1 =10 where (E, p) and (E , p ) are the four-momenta compo- 0 + − nents of the final leptons, l and l , respectively. The 0 squared probability amplitude is then given by Eq. (18) 00 from the previous section. Next, we shall perform the in- 0 0 modM1 tegration over p and write the delta function as follows, 0 =1021 eV √ (0) (0) n 0 −21 δ (g(k, p, m, n, α, sin θ)) = δ 1 + αnk k ˜ =10 0 (26) p 2 2 p 2 2 2 0 − p + m − p + k + m − 2pk cos θ . 00 0 0 m As before, we need to use the g function roots, now 10 given by the expression,

n 2 2 n+1 (αnk + 2 sin θ)p − 2αnk cos θp 2 n 2 (27) FIG. 6. Vacuum Cherenkov emission probability from modi- + 2m + αnk m = 0. fied Maxwell theory and LIV generic approach at first order correction. Notice that unlike to the expressions obtained for the previous process, in here Eq. (27) is a second order polynomial function in p. To solve it, the condition k 6= 0 is assumed. It is not difficult then to see that p will satisfy energy-momentum conservation when,

n+1 III. PHOTON DECAY p± = αnk cos θ q 2 2n+2 2 2 n n 2 ± αnk cos θ − 4(sin θ + αnk )(1 + αnk )m Similarly to the process of spontaneous photon emi- n 2 −1 ssion, photon decay, as depicted in Fig. 7, will naturally × (2αnk + 2 sin θ) . arise. This special LIV process has been studied before in (28) the context of generic LIV corrections and the SME (see In order to ask for p to be real and positive, the dis- for instance [15–17, 20, 21, 31, 35]). In the present study, criminant in Eq. (28) must be real and positive too. we derive the LIV photon decay following the generic This condition will restrict the possible emission angles construction presented in section II, that preserves the for any given momentum of the photon, with regard to generalities of the particular models presented in section LIV parameters. To exemplify this restriction, we have I A. This turns out to be a very restrictive scenario that plotted the discriminant behavior for a photon energy occurs only for very high energy photons [20]. There- of 100 TeV, for m = me, n = 1 and diﬀerent values of fore, it shall be useful to derive tight upper limits for the α1. The result is depicted in Fig. 8. Due to the angu- generic LIV coeﬃcient in the photon sector. lar dependence of the function, while α1 decreases, the discriminant becomes negative for a given interval in θ and the permitted integration region decreases. At the limit when θ goes to zero, the angular integration space becomes null, and the process forbidden. Therefore, this limit establish a threshold on αn above which LIV photon decay is enable,

4m2 α ≥ . (29) n kn(k2 − 4m2)

FIG. 7. Feynman diagram for LIV photon decay to a pair of The analytic expression for the photon decay rate Γ is leptons. e2 |4m2 − α kn+2| Γ = √ n n 4π 4k 1 + αnk As we previously did for vacuum Cherenkov radiation, (30) Z θmax p2 sin θdθ we start the present analysis by considering the LIV pho- X × 0 , 2 2 n 0 |pE + (p − k cos θ)E| ton dispersion relation in Eq. (16), ω = k (1 + αnk ). p± 7

+ → k = 100 TeV θ , 1 21018 1 21021 5 0 71022 10111029

− − − −

FIG. 10. Photon decay probability as a function of the mean free path, for a 100 TeV photon and for the integration limits in Fig. 8.

FIG. 8. The discriminant behavior from Eq. (28) for k = 100 ever, in the limit, where θmax → 0, a photon of 100 TeV TeV, m = me, n = 1 and different values of α1. The positive region set the integration limits for the decay rate in Eq. (30). will be able to reach us from kpc distances, but a smaller αn is implied. This would set a critical value that fixes an energy dependent threshold for the LIV parameter. where For any αn above threshold, photon decay becomes ex- tremely efficient. Below threshold, on the other hand, the p p E = p2 + m2,E0 = k2 + p2 + m2 − 2kp cos θ, process is forbidden. That means, that a lower limit for (31) (n) 1/n EQG ≈ 1/αn in the photon sector will directly emerge Numerically solving the integral for n = 1, 2 one can from any observed high energy cosmic photon event, of found the decay rate that can be seen in Fig. 9. It can momentum kobs, expressed as be appreciated that the decay rates are steadily growing for both n = 1, 2. They show a change in the slope at an 2 2 1/n (n) kobs − 4m energy inversely proportional to the LIV coefficient α E > kobs . (32) n QG 4m2 and the decay rates become smaller as αn does. We have found that the case n=0 have a similar behaviour than for n = 1, 2 [37]. In addition, Eqs. (29), (30), for n=0, are compatible, under the assumption that LIV corrections are made only in the photon sector, with the threshold and the decay rate reported in Ref. [21].

1032 LIVgen1 − 1028 − 24 10 − ]

e − [ 20

− 10 −

1016

1012 (1) FIG. 11. EQG excluded region from LIV photon decay into 108 1010 1012 1014 1016 1018 1020 1022 1024 1026 electron positron pairs with HEGRA [24, 42] and H.E.S.S. [31, 43] photon energy measurements. Using Eq. (29) we k [e] (1) 20 (1) found that EQG ≥ 1.5 × 10 GeV from HEGRA and EQG ≥ 1.7 × 1019GeV from H.E.S.S., in the n = 1 case. FIG. 9. Photon decay rates from LIV generic approach for n = 1, 2 and θmax ≈ π/2. Following this line of thought, and using above ex- As an example, to illustrate how restrictive this process pression for photon decay into electron positron pairs, (n) is, in Fig. 10 we show the photon survival probability as m = me, we can depict the EQG limit as a function of the a function of the mean free path for photons. There we photon energy event, as it can be seen in Fig. 11, for n = 1 have used the same values we used in Fig. 8. From this and Fig. 12 for n = 2, represented by the (blue) diagonal results, it is clear that a 100 TeV photon will not be able line on the plots. The (red) continuous and horizontal −12 −18 to propagate beyond 10 m, for an α1 = 10 . How- line in both ﬁgures is one of the best limits for the LIV 8

to all involved particles, the modiﬁcation to their dispersion relation, as presented in Eq. (2). To be speciﬁc, we have chosen two representative and relevant processes for cosmic ray physics: pair photo production and pion photo production [44–47],

+ − γCR γb −→ e e , (33)

0 pCR γb −→ p π , (34) where b and CR denote background and cosmic ray primary particles, respectively. Notice that we have chosen protons for the second process, but the analysis would be equally valid for neutrons too. Unlike to the work

(2) presented in the previous sections, here we will not refer FIG. 12. EQG exclude region from LIV photon decay into to any particular model to support the main LIV correc- electron positron pairs with HEGRA [24, 42] and H.E.S.S. tions, so keeping the analysis genuinely generic. Similar [31, 43] photon energy measurements. Using Eq. (29) we approaches can be found in Refs. [16, 18, 19]. Further- found that E(2) ≥ 2.8 × 1012GeV from HEGRA and E(2) ≥ QG QG more, we shall consider a first-order correction function 11 6.5 × 10 GeV from H.E.S.S., in the n = 2 case. Ξ to energy-momentum conservation in order to explore the relevance at cosmic ray energies. This correction is motivated by the work in Ref. [48] (for an alternative generic photon sector proposed by Fermi-LAT [30]. The model see also [49]), where several LIV coefficients were dashed (red) lines are our derived limits from H.E.S.S. added in order to properly correct energy-momenta. To and HEGRA reported data [24, 31, 42, 43]. The (colored) the best of our knowledge such a general approach has bars are the uncertainty of the highest photon energy re- not been explored in this general context elsewhere be- ported. We are using the lower value as an observed pho- fore. For instance, for pair photo production, in Eq. (33), ton energy from astrophysical distances. In our analysis let (ω, ωb) and (E+,E−) be the corresponding initial and the best limits are obtained from HEGRA data, as indi- final energies of particles in the process. Therefore, in (1) 20 cated in the figures. We found that EQG ≥ 1.5×10 GeV the LIV setup, we write energy conservation relation as and E(2) ≥ 2.8 × 1012GeV , for n = 1 and n = 2, respec- QG E + E = ω + ω + Ξ. (35) tively. + − b Or equivalently,

IV. COSMIC RAY PHOTO PRODUCTION E+ = K(ω + ωb + Ξ), (36) and Another known scenario derived from the generic modification to particle dispersion relations is the one where E− = (1 − K)(ω + ωb + Ξ), (37) the thresholds for known processes are shifted to different energies than the ones expected under a LI regime. where K is the inelasticity function of the process. However, as we discuss in what follows, there is also an Next, we will use the system where the collision is head interesting feature derived from this approach that we on; since the very high and ultra high energy cosmic pho- believe deserves some further attention. It appears in tons have higher energy than the photon background, the study of particle production by inelastic collisions, ωb ω, the final momenta are collinear. Additionally, particularly those of the type AB → CD, in which, in a we will introduce the ultra relativistic regime approxi- LI regime, massive final particle states, C and D, require mation, that is m {E, p} M and we will keep first some minimal energy from the initial particles A and order LIV coefficients only, since they are expected to be B for the process to happen. As we will show next, in small. the generic LIV setup a higher energy threshold appears For pair photo production, let the invariants be where particle production becomes again forbidden. In 2 2 n+2 the scenario where the collision describes the interaction Sphoton = ω − k = −αnω , (38) of an energetic particle propagating in the cosmic photon and background, this last threshold means that at some high energy the cosmic particle shall again freely propagate, 2 2 2 n+2 S± = E − p = m − α±,nE , (39) at least free from this particular process. This is what ± ± e ± we will call a recovery scenario. where we have inserted a different LIV parameter for the For the propose of establishing the above scenario in electron sector to account for a possible particle depen- the analysis below we will apply, in the most general way dence. Previous work have studied similar corrections 9 but considering a momenta correction with the opposite Before analysing the implications of last equation, lets sign in αn [16]. Substituting and equating the initial and move towards the exploration of the general photo pro- final invariant to first approximation in LIV coefficients, duction process Aωb → CD, to determine its correspond- we found that ing threshold equation. Particular cases will be pair and pion photo production by cosmic nucleons. Following a 2 2 1 me similar path as before, we would start by writing down 4ωωb − me − 2 K(1 − K) 2K(1 − K)(ω + ωb + Ξ) the corresponding modified dispersion relations for the n+2 n n+2 ωb ωb ωb involved particles. Aside to Eq. (38) for the background = αnω 1 + n+2 − 1 + n 2 2 n+2 ω ω ω photon, we would also have Ei − pi = −αi,nEi , for n+1 n+2 i = A, C, D, respectively. A similar algebra as before + α+,nK (ω + ωb) (40) 2 would then give the following equation for the threshold, me 1 at the lower order, for high energy cosmic ray primaries, × −1 + 2 2 (1 − K)(ω + ωb) n+1 n+2 2 2 + α−,n(1 − K) (ω + ω ) Km + (1 − K)m b 4ω E + m2 − C D = Υ En+2 , (44) 2 b A A n A me 1 K(1 − K) × −1 + 2 . 2 K(ω + ωb) where, assuming that ωb EA, Note that Ξ, the LIV contributions to energy-momenta n+1 n+1 conservation, was reduce to one term. By demanding Υn = αA,n − αD,nK − αC,n(1 − K) . (45) that me, ωb, Ξ ω, the last expression becomes, at the lower correction order, Note that, if we set all LIV coefficients to zero, Eq. (44) implies the solution 2 me n+2 4ωωb − = βnω , (41) 2 2 2 K(1 − K) KmC + (1 − K)mD mA EA0 = − . (46) 4ωbK(1 − K) 4ωb where we have defined Clearly, pair photo production formulae is recovered if n+1 n+1 βn = αn − α+,nK − α−,n(1 − K) . one takes mA = 0, and mC,D = me. By taking mA,C = mπ mp for a proton, mD = mπ for pion and K = , mπ +mp Notice that the contribution of Ξ has disappeared. This we get the known LI threshold for neutral pion photo- clearly shows that, at leading order, LI momentum con- production, servation rule effectively holds for high energy cosmic particle collisions. m2 + 2m m E = π π p . (47) In the limit where α = α = 0, the RHS of Eq. (41) p0 n ±,n 4ωb vanishes, thus providing a unique solution to ω in terms 2 me of K, given by ω0 = . For K = 1/2, the stan- This particular threshold plays a relevant role in UHECR 4K(1−K)ωb dard LI pair production threshold for a cosmic photon since the end of CR flux is commonly associated to the will arrive as usual, where GZK cutoff [45–47]. Once more, by introducing dimensionless variables in 2 the general case, now given by LI me ωth = . (42) ωb n+1 EA EA0 xA = and Λn,A = Υn , (48) Otherwise, Eq. (41) is a polynomial equation of order EA0 4ωb n+2 on ω, which in general may have more than one real solution. This shall be the core of our argument below. we finally get the same form for the threshold equation as It is instructive to rewrite Eq. (41) in a more simple and in the pair photo production case. Therefore, irrespective generic form by defining the dimensionless variables of the process we can express such an equation as

n+2 n+1 Λnx − x + 1 = 0, (49) ω ω0 xγ = and Λn,γ = βn . (43) ω0 4ωb LIV IL where in general x = E /Eth and Λn contains an In such terms the polynomial threshold equation for pair energy independent linear function, {βn, Υn}, on all LIV photo production becomes parameters. Clearly, if {βn, Υn} = 0, LI regime would be recovered. n+2 Λn,γ xγ − xγ + 1 = 0 . It is worth stressing that in both the cases, Eq. (49) is, to ﬁrst approximation in LIV terms, independent from Last equation is consistent with the result presented in particle nature, the functional form of A in Eq. (2) and Ref. [16] for pair production under a generic LIV correc- Ξ in Eq. (35). Last means that the results do not depend tion and assuming LI energy-momentum conservation. on lineal corrections to the energy-momentum laws. 10

photo production process becomes forbidden and the Universe gets transparent to the propagation of the cosmic ray primary. Such an scenario is of course in plain contradiction with astrophysical observations and thus, one must impose the consis- tency condition

Λ1 < 4/27. (50)

Note that such a critical value do exist for any arbitrary n. A simple algebra shows that real roots n+1 n+2 demand that Λn < Λnc = (n + 1) /(n + 2) , which is a quickly decreasing function on n. At critical value, the only root is xnc = (n + 2)/(n + 1).

iii) Finally, for 0 < Λ1 < 4/27 the threshold equation has three real solutions. Nevertheless, only two of them are positive, and physically acceptable. A non trivial but important consequence arises here. A range of x values shall now become bounded by two thresholds. FIG. 13. Solutions to the threshold equation for n = 1, as a function of Λ . The f(x) roots separate free propagation 1 The whole effect of these scenarios can easily be un- and photo production zones as indicated. Critical point (in derstood if we plot f(x) roots as a function of Λ1, as green) indicates the upper limit on Λ1. LI case is stressed by we have done in Fig. 13, and notice that the correspon- a (red) point at Λ1 = 0. ding curve actually divides the parameter space into two regions. Clearly, by simply looking at the particular solution where LI holds (for Λ1 = 0), we can identify the A. The LIV secondary threshold free propagation zone below the curve (where E < Eth). This zone is analytically connected to the region on the Let us next consider the implications of LIV threshold RHS of the plot, where Λ1 > 4/27, in accordance to our equation (49). Clearly, to each value of Λn it shall corre- previous discussion. Furthermore, it is connected to the spond a different scenario which describes the behavior region above the curve, where 0 < Λ1 < 4/27. The con- of photo production processes. For simplicity we will first clusion is unavoidable. Whereas the lower root gives the consider in detail the lower order correction, n = 1, and expected energy threshold where photo production gets then comment on the general case. In this case we get switched on, the second root, at higher energies, rep- 3 the cubic equation f(x) = Λ1x − x + 1 = 0, which in resents a new energy threshold where photo production general has three different solutions, but for the LI case shall, once more, stop. at Λ1 = 0. As it is easy to check, the possible roots of This conclusions are consistent with the findings in f(x) provide three different physical scenarios, as follows: Ref.[16] for pair photo production, although it is worth stressing that this physics is rather common to all photo i) For Λ1 < 0 there is only one real root for f(x). production processes. Furthermore, it is not difficult to Other two roots are complex, and thus, non phy- realize that there always exist a secondary threshold for sical. Furthermore, the real root turns out to be any value of n, and for 0 < Λn < Λnc. As a matter of fact, always positive but smaller than one. This implies RHS of Eq. (49) has a linear behaviour for small x, with that the derived LIV physics produces a thresh- a negative slope, whereas, for large x, it is dominated n+2 old at lower energies than that expected in the by the single term Λnx , with a positive derivative. LIV LI LI regime (Eth < Eth ). This means an earlier Accordingly, there exist a single absolute minimum, at −1/n+1 activation of photo production in energy than ex- xmin = [(n + 2)Λn] , that lays in the energy re- pected. The gap among the two thresholds will gion where photo production is on, beyond the first root increase as Λ1 goes to more negative values. (above the first threshold). This, in turn, gives rise to a second root, and therefore, to the secondary threshold. Note that for an universal α1 we get β1 = Υ1 = That would be the signature of the generic LIV for any α1K(2−K), and thus this scenario only arises pro- photo production process. vided that α1 < 0. This phenomena would produce an opacity band for a ii) There is a critical value at Λ1 = 4/27 above which propagating primary. Out of the band ultra high energy f(x) has only one negative and two complex roots. cosmic ray particles, like protons and very high energy Therefore, for Λ1 > 4/27, there is no physical solu- photons, would be free of the photo production processes tion to the threshold equation. As a consequence, produced by the interactions with background photons. 11

Additionally, the bandwidth decreases for larger values takes into account the standard amplitudes derived from of Λn. Whereas the first threshold can only take values QED rules, but explicitly incorporating the LIV modifi- from x ∈ [1, xnc), the second threshold decreases very cation of the photon dispersion relation, to order n in the fast as Λn grows, until the photo production processes photon momentum. All LIV effects for fermions were ne- fade out, when Λn reaches its critical limit, Λnc. glected, though, in order to be able to compare the results The appearance of an opacity band should have visi- to those of models based on the spontaneous breaking of ble effects in the integrated cosmic ray flux. Although Lorentz Invariance. The analysis shows that the emis- a detailed calculation of this is out of the scope of the sion rate does present the same qualitative behaviour present work, our results suggest that potentially visible as the outcomes obtained from the modified Maxwell effects are possible. In general grounds, for individual Theory and the Maxwell-Chern-Simons Theory: emis- particles, one would expect that for cosmic ray primaries sion rate grows with the charged particle energy and be- with energies within the band, the physics should be very come smaller as αn does, but it presents a drop ending similar to the standard LI scenario. Photo production threshold which is inversely dependant on the LIV coef- processes will contribute to quickly diminish the energy ficient. This threshold sets the energy value below where of the proton primary, and to the disappearance of high LI physics is recovered. We have also analysed the mass energy primary photons. However, if energies well above and charge composition sensitivity of the emission rate, the second threshold were accessible to the primaries, since this results would have some impact on the analysis they will see a transparent Universe along its propaga- of the attenuation process along the propagation of high tion, which means no depletion on their flux would be energy cosmic ray primaries. Our results show a clear de- expected. In this sense the measured flux would show a pendency on the cosmic primary mass and charge, hav- recovery behavior at the largest energies. ing a stronger emission rate for heavier nuclei, but with −45 −1 As an example, for Υ1 ≥ 1.67 × 10 eV , primary a similar slope and a higher energy threshold. Similar cosmic protons would avoid pion photo production, and results are found when the modified Maxwell Theory is hence the GZK threshold [45–47]. From the present anal- used for the calculations. This would imply that for en- −20 −1 ysis if αp,1 = απ,1 = 0, then αγ,1 & −10 eV will be ergies below the LIV Cherennkov thresholds, cosmic ray strong enough to produce a noticeable threshold change flux should be consistent with LI physics. As we move up, below 1020 eV . However, further observations of UHE just passing the proton emission threshold, protons will cosmic and gamma rays are essential. LIV studies un- be attenuated along propagation, and the heavier mass der the Colleman and Glashow approach, through the cosmic ray component would become dominant. As en- correction of inelasticity, that contrasts with the experi- ergy passes the threshold for heavier masses, the cosmic mental data of Pierre Auger Observatory and HiRes can rays flux should move again towards the lighter compo- be found in Refs. [18, 19]. nent mass, as a stronger attenuation of heavier particles A potential smoothing of the effect is possible, once the appears. A detailed calculation of this, that take into specifics of flux injection and energy distribution of the consideration the astrophysical models for particle injec- photon background be taken into account in the calcu- tion, is needed to better understand the extension of the lation of the expected cosmic ray fluxes. But we believe effect for the ultra high energy cosmic rays. this deserves some further consideration, since it may The implementation of the generic approach to photon serve to probe the limits for the LIV generic approach. decay on vacuum proves that this process, if allowed, is very efficient and stringent for the propagation of the cosmic ray primary. Nevertheless, as we have argue, V. CONCLUSIONS phase space for outgoing fermions vanishes when photon energy is below a critical threshold energy, defined (n) 2 2 2 1/n Generic LIV, expressed as the deformation of parti- through the expression EQG > k[(k − 4m )/4m ] . cle dispersion relations, by energy/momentum dependent At such energies, the total decay rate becomes zero and functions, can be used to study the possible LIV physics, the process prohibited, so recovering the standard LI re- without relaying on specific models, and thus, in a more sults. Above this threshold, on the other hand, photon direct and simple manner. As a way to demonstrate decay rate quickly becomes very large. Thus, the process this point, we have addressed three different physical strongly restricts the possible propagation of the photon processes that are relevant in the study of the propa- to very short distances from source, to the extent that gation of the very to ultra high energy cosmic rays: vac- it becomes unlikely for a primary photon to travel as- uum Cherenkov radiation, photon decay, and photo pro- tronomical distances. The aftermath of this is a novel, duction processes. Whereas the first two are considered direct and very simple way to bound LIV energy scale, (n) as characteristic of LIV, since they are forbidden in LI EQG, through the direct observation of the very high schemes, photo production turned out to also provide in- energy photons. Using reported gamma telescopes data teresting signatures for LIV effects. and photon decay into electron positron pairs, we have First, Cherenkov radiation analysis has been used to established, for n = 1, the stringent bound for the LIV 20 validate the generic approximation. To this end, we scale at EQG > 1.5 × 10 GeV , coming from HEGRA have elaborated a calculation for the emission rate, which most energetic events. 12

Finally, generic LIV effects incorporated to all parti- pected as a consequence above the opacity band defined cles involved in the most general photo production pro- by the thresholds. cess, where a high energy cosmic particle interacts with Nevertheless, a word of caution to our above naive ob- the photon background, producing two particles at fi- servations is in order. Even though our conclusions are nal state, had been shown to modify the LI production genuine and had to be considered valid for any local cos- threshold. Interestingly enough, our analysis shows that mic ray particle photo production process, the actual cal- LIV effects are completely general, at first approxima- culation of the effects that LIV corrections, and second tion. They do not depend on the particular nature of threshold, should have on the observed cosmic ray flux the particles involved in the process, neither depend on had, yet, to be calculated with care. Actual calculation possible lineal corrections to energy momentum conser- has to take into account a number of issues. A complete vation. Regardless to the specific photo production pro- analysis of the processes including arbitrary angle emis- cess, LIV shifts the production threshold towards lower sion has to be considered, as well as the details of the or higher energies. This solely depends on the sign of injected primary flux and the photon background energy the overall LIV parametric function, Λn, that encodes distribution. Besides, second threshold happens at the LIV effects and which enters in a polynomial like term of very high energies. Few astrophysical sources are known order n + 1 that corrects the, formerly linear, thresh- to produce cosmic ray primaries at such regimes. Even if old equation. Furthermore, as we have argued, this they do so, candidates are always far enough as to make changes imply an upper bound for the LIV parameter, cosmic expansion effects to be relevant. Redshift dimin- n+1 n+2 at Λn > (n + 1) /(n + 2) . For larger values of Λn ishes by itself the energy of propagating particles, which photo production remains forbidden. Consequently, the means that even if a charged particle primary is produced shifting on the LIV production threshold is predicted to at high enough energies at source reference frame, local always be smaller than a factor of (n + 2)/(n + 1) for interactions with photon background, on our observer Λn > 0. Even more importantly, our analysis has un- reference frame, would be happening at lower energies. covered a novel signature for the LIV physics of photo On our frame, at some critical distance between source production processes, that is worth to underline. De- and observer, the cosmic primary might reach energies rived from the LIV modification to the universal thresh- where the local Universe becomes again opaque, and en- old equation, a secondary threshold at higher energies ergy attenuation would take place as usual. This may still will always appear, for Λn > 0, where particle production have, however, distinctive features. It shall effectively becomes again forbidden. Above such threshold, cosmic move the second threshold effects towards an apparently particles shall again propagate freely, giving rise to a pos- higher energy on source basis, whereas it would move sible recovery on the cosmic ray flux. the first threshold closer to the LI value. But it could Physical implications of the purely LIV secondary also modify the injected flux in a distance to source de- threshold are interesting. A cosmic particle primary shall pendant way, that, if enough data were available, could encounter an opaque Universe along its propagation, due be distinguishable on the features of the observed flux. to photon background, only for primary energies within Comparison among a set of well known sources, located the two thresholds. At such energies, photo production at different redshifts, may serve as a probe to establish will attenuate the energy of the particle, moving it to observational bounds for the LIV parameters. We believe lower values where the process is switched off, permit- this possibility deserves further study. ting the further free propagation. Of course this is in general the standard LI scenario, but for the shifting of the actual position of the threshold. On the other hand, if particle primaries were injected by astrophysical sources ACKNOWLEDGMENTS at energies well above the LIV secondary threshold, they would not suffer attenuation due to photo production. A This work was partially supported by Conacyt Mexico, recovery on the cosmic particle primary flux would be ex- under grant 237004.

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