Can the gamma-ray bursts travelling through the interstellar space be explained without invoking the drastic assumption of Lorentz invariance violation?
aa r X i v : . [ a s t r o - ph . H E ] J a n Can the gamma-ray bursts travelling through theinterstellar space be explained without invoking thedrastic assumption of Lorentz invariance violation?
M. Chaichian a, , I. Brevik b, and M. Oksanen a, a Department of Physics, University of Helsinki, P.O.Box 64,FI-00014 Helsinki, Finland b Department of Energy and Process Engineering, Norwegian University of Scienceand Technology, N-7491 Trondheim, Norway
Abstract
Experimental observations indicate that gamma-ray bursts (GRB) and high-energy neutrino bursts may travel at different speeds with a typical delay mea-sured at the order of hours or days. We discuss two potential interpretationsfor the GRB delay: dispersion of light in interstellar medium and violation ofLorentz invariance due to quantum gravitational fluctuations. Among a fewother media, we consider dispersion of light in an axion plasma, obtaining theaxion plasma frequency and the dispersion relation from quantum field theoryfor the first time. We find that the density of axions inferred from observationsis far too low to produce the observed GRB delay. However, a more precise es-timation of the spatial distribution of axions is required for a conclusive result.Other known media are also unable to account for the GRB delay, althoughthere remains uncertainties in the observations of the delays. The interpretationin terms of Lorentz invariance violation and modified dispersion relation suffersfrom its own problems: since the modification of the dispersion relation shouldnot be dependent on particle type, delays between photons and neutrinos arehard to explain. Thus neither interpretation is sufficient to explain the observa-tions. We conclude that a crucial difference between the two interpretations isthe frequency dependence of the propagation speed of radiation: in dispersiveplasma the group speed increases with higher frequency, while Lorentz invari-ance violation implies lower speed at higher frequency. Future experiments shallresolve which one of the two frequency dependencies of GRB is actually the case.
Talk given in 40th International Conference on High Energy Physics - ICHEP2020July 28 – August 6, 2020Prague, Czech Republic (virtual meeting) masud.chaichian@helsinki.fi [email protected] markku.oksanen@helsinki.fi Introduction
Gamma-ray bursts (GRB) are highly energetic and diverse events, which are thoughtto be produced by violent stellar processes, in particular supernovas and mergers ofbinary neutron stars. Those events may also produce high-energy cosmic rays andconsequently bursts of high-energy neutrinos [1]. Neutrino bursts have been observedto be shifted in time with respect to the GRB (see [2, 3] and references therein). Thetime window τ = t GRB − t ν between the arrival times of a GRB t GRB and a neutrinoburst t ν can vary between an hour or several days. Assuming that a GRB and thecorresponding neutrino burst are produced at the same time or within a short period,a significant delay τ would indicate that the electromagnetic and neutrino signals havetravelled at different speeds. Note, however, that the recent experimental studies showonly faint neutrino signals associated with GRB [2, 3], and hence the observed delaysmay be inaccurate.It is equally challenging to interpret the GRB delay within standard physics whereGRB are delayed due to the interaction of photons with interstellar media, a phe-nomenon which always occurs. In this way, one can also shed additional light on the“microstructure” of the Universe or a part of it and its constituents. The interactionof neutrinos with any interstellar medium is extremely weak and hence the dispersionof neutrinos is negligible. Secondly, while the neutrinos are massive and oscillating,the effect on the speed of high-energy neutrinos is very small. Consider a GRB withphoton energy 1 TeV and neutrinos with the same energy, E = 1 TeV. The dispersionrelation E = p c + m c gives the speed of the neutrinos as v ν = dEdp ≈ c (1 − d ν ),where d ν = m c E . Averaging over 3 neutrinos, h m c i = (1 / . , where themasses are estimated with the heaviest neutrino mass. The speed of neutrinos isgiven by d ν = 0 . × − . Thus the delay compared to a signal travelling at thespeed c would be measured in nanoseconds even for signals from furthest galaxies: τ = D × d ν /c . − s, using a maximal travelling distance D = 10 m (across thewhole universe). That is negligible compared to the observed GRB delays. Theoriesof neutrino production in GRB actually predict neutrinos with even higher energy oforder 10 –10 TeV [1], which means v ν is even closer to c . This justifies v ν = c in ourestimates.Violation of Lorentz invariance and the associated modification of the dispersionrelation has been considered as a potential interpretation of the delay of high-energyGRB [4, 5]. This approach is motivated by various approaches to quantum gravity,since quantum-gravitational fluctuations may lead to a non-trivial refractive index [6].We shall comment the Lorentz invariance violation interpretation in Sec. 4.Before seeking to modify fundamental principles such as Lorentz invariance we pre-fer to consider possible explanations for the observed phenomena by means of standardphysics. We consider the dispersion of light in several media and assess the producedGRB delay when photons and neutrinos are assumed to be emitted from the samesource at the same time. Neither electron plasma nor photon plasma can account forthe observed GRB delay. Then we consider axions. Axions are pseudoscalar particlesthat may both provide a solution to the strong CP problem and constitute cold darkmatter. Axions are not electrically charged, since a charged axion would be luminuous,but can still interact with photons. Axion electrodynamics has been studied activelyand it is connected to topological insulators [7, 8]. Therefore an axion plasma is a2lausible cosmic medium that would have an effect on the propagation of light fromdistant galaxies. We derive the dispersion relation in an axion plasma and assess itseffect on the GRB delay. A plasma can support both longitudinal and transverse waves. We are interested intransverse waves. Dispersion relation for light in a plasma is ω = c k + ω p , (1)where ω p is the plasma frequency. The angular frequency is also given as ω = v (ˆ k ) · k = c | k | n , where n is the refraction index and v = cn ˆ k is the phase velocity, where ˆ k = k | k | .Thus the refraction index is related to the plasma frequency as n = 1 − ω p ω . (2)In an isotropic medium, ω = ω ( | k | ) and ω p = ω p ( | k | ), group velocity is parallelto phase velocity. When the photon momentum is large compared to the plasmafrequency, c k ≫ ω p , we obtain that group velocity is only slightly lower than c , v g = ∂ω ( | k | ) ∂ | k | ≃ c (cid:18) − ω p ω (cid:19) ≡ c (1 − d ) . (3)Now we explain how the plasma frequency and refraction index can be derived fromquantum field theory. From here on we assume units ~ = c = ǫ = 1. The refractionindex is related to the forward scattering amplitude f (0) as [9] n = 1 + 2 π N f (0) ω , (4)where N is the number density of scatterers. The relation (4) is valid when n is closeto one, | n − | ≪
1, and follows from the inteference between incident and scatteredwaves. Inserting (2) into (4), we obtain a relation between the plasma frequency andthe scattering amplitude. When the photon frequency is large compared to the plasmafrequency, ω ≫ ω p , the relation is given as ω p = − πN f (0) . (5)The scattering amplitude f ( θ ) is defined as a part of the wavefunction at large distance r from the scatterer, ψ ( r ) = C (cid:0) e i k · r + f ( θ ) e ikr /r (cid:1) , where C is a normalization factor.The differential cross section is given in terms of the scattering amplitude as dσ ( θ ) = | f ( θ ) | d Ω. The differential cross section can as well be obtained from quantum fieldtheory. Hence we obtain the differential cross section dσ at angle θ = 0 in quantumfield theory and identify the forward scattering amplitude as | f (0) | = (cid:18) dσ (0) d Ω (cid:19) . (6)3or an electron plasma, we obtain the differential cross section for scattering of aphoton on an electron in quantum field theory. In the rest frame of the initial electron,we obtain dσ (0) = ( e / π m e ) d Ω, and according to (6) we get | f (0) | = α/m e ,where α is the fine-structure constant, α = e / π , and m e is the electron mass. Thus theplasma frequency (5) is given as ω p = N e m e , (7)which is the same result that is obtained from classical electrodynamics [10, 11]. With m e = 0 .
511 MeV, we obtain the group velocity (3) for the photon energy 1 TeV, v GRB = c (1 − . × − × N × metre ) . (8)Therefore, dispersive properties of an electron gas are not significant enough to accountfor a time delay of the order of several hours as observed.Dispersion of light in light plasma also produces a too small delay. We obtainfrom light on light scattering ω p = const . × N γ e /ω , where N γ is the number densityof photons. For a delay of the order of few hours, we would need photon density N γ = 10 m − , while according to the Planck data on the CMB (Cosmic MicrowaveBackground) radiation: N γ = (4–5) × m − . Interaction Lagrangian of axion electrodynamics is [12] L aγγ = − gaF µν ˜ F µν = − gaǫ µνρσ ∂ µ A ν ∂ ρ A σ , (9)where g is a coupling constant, a is the axion pseudoscalar field, the electromagneticfield strength tensor is F µν = ∂ µ A ν − ∂ ν A µ , and its dual ˜ F µν = ǫ µνρσ F ρσ .Consider scattering of photon on axion γ + a → γ + a at tree level. The scatteringamplitude M is a sum of two terms represented by the diagrams in Fig. 1. We are a ( p ) γ ( k, λ ) γ a ( p ′ ) γ ( k ′ , λ ′ ) a ( p ) γ ( k, λ ) γ ( k ′ , λ ′ ) γ a ( p ′ )Figure 1: Feynman diagrams for scattering of photon on axion (drawn from left toright)interested in scattering with parallel momenta for initial and final photons, i.e. withangle θ = 0. The differential cross section for unpolarized photons with angle θ = 0 isobtained in the rest frame of the initial axion as dσ (0) = 164 π m a X λ,λ ′ |M (0) | ! d Ω = (cid:18) g π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ω (2 ω + m a ) + ω ( − ω + m a ) (cid:12)(cid:12)(cid:12)(cid:12) d Ω . (10)4here ω is the energy of the initial photon. The scattering amplitude | f (0) | is obtainedaccording to (6). Since the axions are very light, m a ∼ − eV, and we are interestedin very high energy photons, we consider the limit ω ≫ m a . For photons with energieswell above the axion mass, the plasma frequency is nearly constant, i.e. independentof the frequency of the incoming light: ω p = 38 N g m a (cid:18) m a ω + O (cid:18) m a ω (cid:19)(cid:19) ≃ N g m a . (11)Estimating the effective coupling constant to be g = 10 − GeV − , and the axionmass m a = 10 − eV, we obtain the group velocity for the photon of energy 1 TeV inan axion plasma, v GRB = c (1 − d ) , d = 316 N g m a ω = 1 . × − × N × metre . (12)Thus the delay of the GRB in the galactic plasma is τ = D × dc = 316 N g m a ω Dc = 4 . × − × D × N × metre × second , (13)where D is the distance traveled by the photons. Typical value of the delay τ takenfrom ANTARES data is τ = 3 .
25 hours. The effective distance travelled by photonsin expanding Universe depends on the redshift z [2], D ( z ) = cH R z z ) dz √ Ω m (1+ z ) +Ω Λ . If D is taken as the diameter of observable Universe, D = 8 . × m, we need an axionnumber density N ≃ m − . This is a very large number density that apparentlycontradicts experimental data.A more realistic scenario is to consider that axions are concentrated in galactichalos (constituting cold dark matter). The mass density of axions in a galactic halois estimated D m = 0 .
45 GeV / cm , and the radius of the halo is 5 × m [14].Number density of axions is N GH = 0 . × m − . In order to produce the delay τ = 3 .
25 hours, the axion number density in galactic halo should be N ≃ m − ,which is much higher than from data N GH multiplied by any number of farther galaxieswithin the diameter of the Universe.Details of the derivation of (11) will be presented elsewhere [13]. For related workson the bending of light in axion backgrounds but not considering the issues concerningGRB, see [15] and references therein. In the quantum gravity motivated interpretation that violates Lorentz invariance [4, 5],the dispersion relation is modified to contain higher-power energy terms (or higher-power momentum terms), E [1 + P n =1 ξ n ( E/E QG ) n ] = p c + m c . Then the groupvelocity of light is v g = c − ξ EE QG + O E E QG !! , (14)5here E QG is an effective quantum gravity energy scale, usually of order E QG =10 GeV. Hence the modification of the dispersion relation implies that the slowdownof radiation is increased with higher energy. Thus this approach has mainly beenused to consider the delay between higher energy photons and lower energy photonsproduced in GRB. A delay between neutrinos and photons produced in GRB mightbe possible in this interpretation only if the energy of neutrinos is several orders ofmagnitude higher than the energy of photons [5].The key feature that differentiates the dispersive plasma interpretation from theLorentz invariance violation (LIV) interpretation is the energy dependence of the signaldelay. In plasma the delay decreases with higher photon frequency, τ ∝ ω − , whilein the LIV case it increases with frequency, τ ∝ ω . It would be crucial to test thefrequency/energy dependence of the delay experimentally. That requires the exactmeasurement time of observation of GRB and spectral resolutions and therefore, theplanned broad energy range measurements are utmost crucial [16].Since neither LIV nor dispersion of light in a plasma can explain such a large delaybetween GRB and neutrinos, one could even suspect the existence of the delay withsuch an amount. References [1] E. Waxman and J. N. Bahcall,
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