Lorentz Violation for Photons and Ultrahigh-Energy Cosmic Rays

Matteo Galaverni1,2,3 and Gunter Sigl4,5

1INAF-IASF Bologna, via Gobetti 101, I-40129 Bologna, Italy2Dipartimento di Fisica, Universita di Ferrara, via Saragat 1, I-44100 Ferrara, Italy

3INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna, Italy4II. Institut fur theoretische Physik, Universitat Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany5APC* (AstroParticules et Cosmologie), 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France

(Received 13 August 2007; revised manuscript received 19 November 2007; published 18 January 2008)

Lorentz symmetry breaking at very high energies may lead to photon dispersion relations of the form!2 � k2 � �nk

2�k=MPl�n with new terms suppressed by a power n of the Planck mass MPl. We show that

first and second order terms of size j�1j * 10�14 and �2 & �10�6, respectively, would lead to a photoncomponent in cosmic rays above 1019 eV that should already have been detected, if corresponding termsfor e� are significantly smaller. This suggests that LI breaking suppressed up to second order in the Planckscale is unlikely to be phenomenologically viable for photons.

DOI: 10.1103/PhysRevLett.100.021102 PACS numbers: 98.70.Sa, 11.30.Cp, 96.50.sb

Introduction.—Many quantum gravity theories suggestthe breaking of Lorentz invariance (LI) with the strength ofthe effects increasing with energy. The most promisingexperimental tests of such theories, therefore, exploit thehighest energies at our disposal which are usually achievedin violent astrophysical processes. If LI is broken in theform of nonstandard dispersion relations for various parti-cles, absorption and energy loss processes for high-energycosmic radiation can be modified [1]. Conversely, experi-mental confirmation that such processes occur at the ex-pected thresholds would allow us to put strong constraintson such LI breaking effects. This was shown in the case ofultrahigh-energy cosmic rays producing pions by theGreisen-Zatsepin-Kuzmin (GZK) effect [2] above thethreshold at �7� 1019 eV and in the case of pair produc-tion of high-energy photons with the diffuse low-energyphoton background [3].

While the thresholds of electron-positron pair produc-tion by high-energy �-rays on low-energy backgroundphotons have not yet been experimentally confirmed be-yond doubt, constraints on LI breaking for photons havebeen established based on the very existence of TeV � raysfrom astrophysical objects [4].

Here we exploit the fact that if pair production of high-energy � rays on the cosmic microwave background(CMB) would be inhibited above �1019 eV, one wouldexpect a large fraction of � rays in the cosmic ray flux atthese energies, independent on where the real pair pro-duction threshold is located. Based on the fact that nosignificant �-ray fraction is observed, we derive limits onLI violating parameters for photons that are more strin-gent than former limits. These limits do not depend onhe poorly known strength of the astrophysical radiobackground.

Hybrid detectors begin to put constraints on the compo-sition of cosmic rays at highest energies. Particularly it isalready possible to put upper limits on the fraction of

photons on the 10% level at energies above 1019 eV usingAuger hybrid observations [5], AGASA [6–8], andYakutsk [8,9] data. Above 1020 eV, the current upper limitis �40% [8]. In fact, the latest upper limits from thesurface detector data of the Pierre Auger observatory arealready at the level of�2% above 1019 eV [10]. In the nextfew years these constraints will improve with statistics:The Pierre Auger experiment can reach a sensitivity of�0:3% within a few years and �0:03% within 20 yearsaround 1019 eV, and a sensitivity at the 10% level around1020 eV within 20 years [11].

Neutral pions created by the GZK effect decay intoultrahigh energy photons. They subsequently interactwith low-energy background photons of the CMB andthe universal radio background (URB) through pair pro-duction, ��! e�e�. This leads to the development of anelectromagnetic cascade and suppresses the photon fluxabove the pair production threshold on the CMB of�1015 eV. Above �1019 eV the interaction length forphotons is smaller than a few Mpc, whereas for nucleonsabove the GZK threshold at�7� 1019 eV it is of the orderof 20 Mpc. As a result, the photon fraction theoreticallyexpected is smaller than�1% around 1019 eV, and smallerthan �10% around 1020 eV [12,13], in agreement withexperimental upper limits.

The breaking of Lorentz invariance, by modifying thedispersion relation for photons, would affect the energythreshold for pair production. If the change in the disper-sion relation is sufficiently large, pair production can be-come kinematically forbidden at ultrahigh energies, andsuch photons could reach us from cosmological distances.As a consequence, at least if ultrahigh energy cosmic raysconsist of mostly protons, one would expect a significantphoton fraction in cosmic rays above 1019 eV, in conflictwith experimental upper limits. Figures 1 and 2 which wereobtained with the CRPROPA code [17,18] show that the ratioof the integral photon to primary cosmic ray flux above

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1019 eV would be ’20%, and thus higher than the abovementioned experimental upper limits. In this scenario, wehave used a relatively steep proton injection spectrum/E�2:6. Harder injection spectra also give acceptable fitsabove�1019 eV, as well as higher photon fractions due toincreased pion production [13]. Whereas for pair produc-tion without LI breaking, the predicted photon fractionalways stays below experimental upper limits, harder in-jection spectra and larger maximal energies in the absenceof pair production would overshoot the experimental limitseven more than in Figs. 1 and 2.

Therefore, LI violating parameters for photons are con-strained by the requirement that pair production be allowedbetween low-energy background photons and photons ofenergies between 1019 eV and 1020 eV. We assume that

pion production itself is not significantly modified and thatthe modifications of the dispersion relations of electronsand positrons are significantly smaller than for photons.This is consistent since the photon content of other par-ticles is on the percent level [19].

Formalism.—We denote the 4-momenta with �!;k� forthe ultrahigh-energy photon, �!b;kb� for the backgroundphoton, and �E�;p�� for the electron and positron,respectively.

We consider the following modified dispersion relationsfor photons, electrons, and positrons:

!2 � k2 � �nk2

�kMpl

�n;

E2� � p2

� �m2e � �

�n p

2�

�p�Mpl

�n;

(1)

with n 1 and where Mpl ’ 1019 GeV and me are thePlanck mass and the electron mass, respectively.

Using the exact relation for energy-momentum conser-vation, the kinematic relation for the decay of a neutralpion of mass m� into two � rays of energy momentum�!1;k1� and �!1;k2�, respectively, and equal helicity is2!1!2 � 2k1 k2 � �n�k

n�21 � kn�2

2 �=MnPl � m2

�. Forj�nj & 1, the absolute values of the LI violating termsare always much smaller than the ones of !1!2 andk1 k2, which themselves are much larger than m2

� inmost of the phase space. Therefore, the kinematics ofpion decay is not significantly modified.

This is different for pair production by photons: Exactenergy-momentum conservation implies that �!�!b�

2 ��k� kb�

2 � �E� � E��2 � �p� � p��2. The left-hand

side is maximized for antiparallel initial photon momenta(head-on collision) and the right-hand side is minimizedfor parallel final momenta of the pair [4,20]. Writing p� �yk, p� � �1� y�k with 0 � y � 1, assuming relativisticleptons and using !� !b, after some algebra we thusobtain at the threshold

�nk2

�kMPl

�n� 4k!b �

m2e

y�1� y�� 0; (2)

where

�n �n � ��n y

n�1 � ��n �1� y�n�1 (3)

and the asymmetry y in the final momenta at threshold isdetermined by maximizing the left-hand side of Eq. (2).For example, if ��n � ��n >�2n�3�me=k�2�Mpl=k�n=n�n� 1�, then y � 1

2 .Introducing x k=k0 with k0 m2

e=�4y�1� y�!b�,Eq. (2) can be rewritten as

�nxn�2 � x� 1 � 0; (4)

where

FIG. 2. The ratio of the integral photon to primary cosmic rayflux above a given energy as a function of that energy for the twoscenarios shown in Fig. 1.

FIG. 1 (color online). Fluxes for uniform E�2:6 proton injec-tion between 1019 and 1021 eV up to redshift 3. AGASA data[14] are shown as triangles, HiRes data [15] as crosses. Solidlines: with CMB and the minimal version of the universal radiobackground, based on observations [16]: from top to bottom:protons, neutrinos per flavor, and photons. Dashed lines: withoutany pair production by photons above 1019 eV: from top tobottom: protons, photons, and neutrinos per flavor.

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�n �nk0

4!b

�k0

Mpl

�n: (5)

If �n � ��n � ��n � 0 we have �n � 0 and y � 12 and

thus the usual threshold for pair production in Lorentzinvariant theory, k � m2

e=!b. Furthermore, if the LI vio-lating terms in the electron and positron dispersion rela-tions are smaller than the photon terms, j��n j & �n, then�n ’ �n and we will obtain constraints essentially on thephoton terms �n. If not otherwise stated we will make thisassumption in the following.

If �n > 0, Eq. (4) admits one real positive solutionxln��n�< 1 for each value of �n > 0. Therefore, for pho-tons with a positive LI violating term in the modifieddispersion relation Eq. (1), pair production is kinematicallyallowed above a threshold k0xln��n�< k0.

Otherwise, if the coefficient of xn�2 in Eq. (4) is nega-tive, this equation has real solutions only if j�nj � �cr

n �n� 1�n�1=�n� 2�n�2. In particular, if j�nj � �cr

n there isonly one real solution and pair production is kinematicallyallowed only for a particular value of the momentum of theultrahigh-energy photon. If j�nj<�cr

n , there are two realsolutions, 0< xln��n�< xun��n�, and thus pair productionis allowed only in the range of energies k0xln��n� � ! �k0xun��n�. These two cases are summarized in Fig. 3.

Requiring pair production to be allowed, we obtainconstraints only from photons with a negative sign in themodified dispersion relation, because for photons with apositive LI breaking term, pair production is allowed forany value of �n above k0xln��n�< k0. We also stress thatphotons with a negative LI breaking term are stable againstphoton decay (�! e�e�) and photon splitting (�! N�).

Requiring the interaction of ultrahigh-energy photons,1019 eV & k & 1020 eV, with CMB photons of energy!b ’ 6� 10�4 eV corresponds to requiring that pair pro-duction is kinematically allowed for 2:3� 104 & x &

2:3� 105. Since photons with a negative LI breaking

term in the dispersion relation have both a lower and anupper energy threshold for pair production, denoted byxln��n� and xun��n�, respectively, we have the two condi-tions xln��n� & 2:3� 104 and 2:3� 105 & xun��n�. Thesewill lead to constraints on �n and thus �n.

Constraints on Lorentz invariance breaking to first orderin the Planck mass.—In this case n � 1 and the firstcondition xl1��1� & 2:3� 104 is always true if the lowerthreshold exists, �1 >��

cr1 � �4=27. The second condi-

tion 2:3� 105 & xu1��1� is fulfilled if �1 * �1:9� 10�11.These two necessary conditions can be translated into aconstraint for �1 using the definition for �n, Eq. (5), andk0 ’ 4:4� 1014 eV:

�1 �1k0

4!b

�k0

Mpl

�* �1:9� 10�11;

�1 * �2:4� 10�15:

(6)

For n � 1 effective field theory implies LI violating termsin the dispersion relation of equal absolute value andopposite sign for left and right polarized photons [21].Therefore, in order to avoid photon fractions in cosmicrays * 5 times higher than observed above�1019 eV, pairproduction has to be allowed for both polarizations, andthus for both signs in the dispersion relation. Thus theconstraint obtained for �1 ’ �1 < 0 is valid also for posi-tive LI violating terms: j�1j & 2:4� 10�15.

Constraints on Lorentz invariance breaking to secondorder in the Planck mass.—In this case n � 2 and the firstcondition xl2��2� & 2:3� 104 is always true if the lowerthreshold exists, �2 >��

cr2 � �27=256. The second con-

dition 2:3� 105 & xu2��2� is fulfilled if �2 * �8:2�10�17. These two necessary conditions then lead to thefollowing constraint for �2:

�2 �2k0

4!b

�k0

Mpl

�2

*�8:2�10�17; �2 *�2:4�10�7:

(7)

For interactions with the URB, k0 ’ 6� 1019 eV, we ob-tain the constraint assuming the existence of at least onesolution with xln��n� & 2. This eventually leads to theconditions j�1j & 7:2� 10�21 at first order, and ��2 &

8:5� 10�13 at second order. These are several orders ofmagnitudes more restrictive than the constraints Eqs. (6)and (7) obtained in the CMB case. Therefore, if the con-straints from interactions with the CMB are violated, therewould also be no interaction with the URB and so no pairproduction on any relevant background. Thus the con-straint from pair production with the CMB is not modifiedby the presence of the URB.

Discussion and conclusions.—To our knowledge, onlyLI breaking suppressed to first order in the Planck mass hasso far been ruled out in the electromagnetic sector [22–24].In terms of the dimensionless parameters �n, the best upperlimit was j�1j & 2� 10�7 [25], based on frequency de-

0 1 2 3 4x

-2

-1

0

1

2

FIG. 3. The left-hand side of Eq. (4) for various cases for n �1. Dashed line: photons with a positive LI breaking term �1 �2=27; continuous line: photons with unbroken LI, �1 � 0; dottedlines: photons with a negative LI breaking term, with �1 ��6=27, �4=27, �2=27, in ascending order. Pair production iskinematically allowed for values of x k=k0 for which thecurves are positive.

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pendent rotation of linear polarization (vacuum birefrin-gence) of optical or UV photons of the afterglow fromdistant �-ray bursts. A former, more stringent constraint,j�1j & 2� 10�15 [26], was based on polarization of MeV� rays which could not be confirmed [23].

Constraints based on modified reaction thresholds wereso far obtained from observations of multi-TeV � raysfrom blazars at distances *100 Mpc, over which suchphotons are expected to produce pairs on the infraredbackground. However, given that involved photon energiesare much smaller than 1019 eV, resulting constraints are ofthe order j�1j & 1 [27], much weaker than our constraintsj�1j & 2:4� 10�15 and ��2 & 2:4� 10�7. Our new con-straints suggest that LI breaking suppressed up to secondorder in the Planck scale is unlikely to be phenomenolog-ically viable for photons. Although similar constraints havebeen obtained in an independent approach based on theabsence of vacuum Cerenkov radiation of ultrahigh-energyprotons [19,28], such constraints depend on the somewhatuncertain partonic structure of these protons.

It is interesting to note that the detection of a photon of1019 eV would put strong constraints on any positive LIbreaking term in the dispersion relation �1 < 10�17 forn � 1 and �2 < 10�8 for n � 2, in order to avoid photondecay.

Our constraints, Eqs. (6) and (7), hold for the linearcombinations of LI breaking terms for photons, electrons,and positrons defined in Eq. (5). They translate directlyinto constraints on the photon terms �n if the LI breakingterms for electrons and positrons are significantly smallerthan the ones for photons. This is typically the case if theonly LI breaking terms for electrons or positrons are in-duced by their photon content [19].

Note that in supersymmetric QED, corrections to thedispersion relation of a particle of mass m are of the form�nm2�k=MPl�

n and are thus negligible in astrophysicalcontexts [29]. Therefore, our constraints apply only tothe nonsupersymmetric case.

We are grateful to Theodore A. Jacobson and DavidMattingly for valuable comments on this project. M. G.thanks APC for the hospitality during the developments ofthis work.

*UMR 7164 (CNRS, Universite Paris 7, CEA, Observatoirede Paris).

[1] S. R. Coleman and S. L. Glashow, Phys. Rev. D 59,116008 (1999).

[2] K. Greisen, Phys. Rev. Lett. 16, 748 (1966); G. T. Zatsepinand V. A. Kuzmin, Pis’ma Zh. Eksp. Teor. Fiz. 4, 114(1966) [JETP Lett. 4, 78 (1966)].

[3] R. Aloisio, P. Blasi, P. L. Ghia, and A. F. Grillo, Phys.Rev. D 62, 053010 (2000).

[4] T. Jacobson, S. Liberati, and D. Mattingly, Phys. Rev. D67, 124011 (2003).

[5] J. Abraham et al. (Pierre Auger Collaboration), Astropart.Phys. 27, 155 (2007).

[6] K. Shinozaki et al., Astrophys. J. 571, L117 (2002).[7] M. Risse et al., Phys. Rev. Lett. 95, 171102 (2005).[8] G. I. Rubtsov et al., Phys. Rev. D 73, 063009 (2006).[9] A. V. Glushkov, D. S. Gorbunov, I. T. Makarov, M. I.

Pravdin, G. I. Rubtsov, I. E. Sleptsov, and S. V. Troitsky,JETP Lett. 85, 131 (2007).

[10] M. D. Healy, for the Pierre Auger Collaboration,arXiv:0710.0025.

[11] M. Risse and P. Homola, Mod. Phys. Lett. A 22, 749(2007).

[12] G. Sigl, Phys. Rev. D 75, 103001 (2007).[13] G. B. Gelmini, O. Kalashev, and D. V. Semikoz, J. Cosmol.

Astropart. Phys. 11 (2007) 002.[14] K. Shinozaki (AGASA Collaboration), Nucl. Phys. B,

Proc. Suppl. 151, 3 (2006); see also http://www-akeno.icrr.u-tokyo.ac.jp/AGASA/.

[15] R. U. Abbasi et al. (High Resolution Fly’s EyeCollaboration), Phys. Rev. Lett. 92, 151101 (2004).

[16] T. A. Clark, L. W. Brown, and J. K. Alexander, Nature(London) 228, 847 (1970).

[17] See http://apcauger.in2p3.fr//CRPropa.[18] E. Armengaud, G. Sigl, T. Beau, and F. Miniati, Astropart.

Phys. 28, 463 (2007).[19] O. Gagnon and G. D. Moore, Phys. Rev. D 70, 065002

(2004).[20] D. Mattingly, T. Jacobson, and S. Liberati, Phys. Rev. D

67, 124012 (2003).[21] R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601

(2003).[22] T. Jacobson, S. Liberati, and D. Mattingly, Ann. Phys.

(N.Y.) 321, 150 (2006).[23] S. Liberati, Proc. Sci., P2GC (2007) 018.[24] L. Maccione, S. Liberati, A. Celotti, and J. G. Kirk,

J. Cosmol. Astropart. Phys. 10 (2007) 013.[25] Y. Z. Fan, D. M. Wei, and D. Xu, Mon. Not. R. Astron.

Soc. 376, 1857 (2007).[26] T. A. Jacobson, S. Liberati, D. Mattingly, and F. W.

Stecker, Phys. Rev. Lett. 93, 021101 (2004).[27] F. W. Stecker, Astropart. Phys. 20, 85 (2003).[28] S. Bernadotte and F. R. Klinkhamer, Phys. Rev. D 75,

024028 (2007).[29] S. Groot Nibbelink and M. Pospelov, Phys. Rev. Lett. 94,

081601 (2005).

PRL 100, 021102 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending18 JANUARY 2008

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