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arXiv:0806.4351v1 [hep-ph] 26 Jun 2008

Phys. Rev. D 77, 117901 (2008) arXiv:0806.4351 [hep-ph]

Addendum: Ultrahigh-energy cosmic-ray bounds on

nonbirefringent modified-Maxwell theory

F.R. Klinkhamer∗

Institute for Theoretical Physics, University of Karlsruhe (TH),

76128 Karlsruhe, Germany

M. Risse†

University of Wuppertal, Physics Department,

Gaußstraße 20, 42097 Wuppertal, Germany

Abstract

Nonbirefringent modified-Maxwell theory, coupled to standard Dirac particles, involves nine dimen-

sionless parameters, which can be bounded by the inferred absence of vacuum Cherenkov radiation

for ultrahigh-energy cosmic rays (UHECRs). With selected UHECR events, two-sided bounds

on the eight nonisotropic parameters are obtained at the 10−18level, together with an improved

one-sided bound on the single isotropic parameter at the 10−19level.

PACS numbers: 11.30.Cp, 12.20.-m, 41.60.Bq, 98.70.Sa

Keywords: Lorentz violation, quantum electrodynamics, Cherenkov radiation, cosmic rays

∗Electronic address: frans.klinkhamer@physik.uni-karlsruhe.de

†Electronic address: risse@physik.uni-wuppertal.de

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In Ref. [1], ultrahigh-energy-cosmic-ray (UHECR) bounds have been given for the nine

Lorentz-violating “deformation parameters” of nonbirefringent modified-Maxwell theory

coupled to standard Dirac particles, where the parameters were restricted to a particular do-

main. In this addendum, we obtain corresponding results for two sets of nonisotropic param-

eters outside this domain (the two sets are, respectively, parity-odd and parity-even). These

new bounds are essentially two-sided, whereas an improved bound on the single isotropic

parameter remains one-sided. For convenience, the final bounds will be presented in terms

of the widely-used standard-model-extension (SME) parameters [2, 3].

The ‘Note Added in Proof’ of Ref. [1] used 29 UHECR events [4, 5, 6] which, for complete-

ness, are listed in Table I. From the energies and flight directions of these 29 UHECR events,

the following two–σ bound was obtained on the quadratic sum of the nine nonbirefringent

TABLE I: Selected UHECR events with energies above 57 EeV as recorded by the Pierre Auger

Observatory over the period 1 January 2004 to 31 August 2007 [4], together with an additional

320 EeV event from the Fly’s Eye detector [5] and a 210 EeV event from the AGASA array [6].

Shown are the arrival time (year and Julian day), the primary energy E, and the arrival direction

with right ascension RA and declination DEC. The estimated errors for the Auger events [4] are a

25% relative error on the energy and a 1◦error in the pointing direction (the errors for the Fly’s

Eye and AGASA events are of the same order, possibly somewhat larger [5, 6]).

YearDayE[EeV]RA[deg]DEC[deg]YearDayE[EeV]RA[deg] DEC[deg]

1991

1993

2004

2004

2004

2004

2004

2005

2005

2005

2005

2005

2005

2006

2006

288

337

125

142

282

339

343

54

63

81

295

306

306

35

55

320

210

70

84

66

83

63

84

71

58

57

59

84

85

59

85.2

18.9

267.1

199.7

208.0

268.5

224.5

17.4

331.2

199.1

332.9

315.3

114.6

53.6

267.7

48.0

21.1

−11.4

−34.9

−60.3

−61.0

−44.2

−37.9

−1.2

−48.6

−38.2

−0.3

−43.1

−7.8

−60.7

2006

2006

2006

2006

2007

2007

2007

2007

2007

2007

2007

2007

2007

2007

8179

83

69

69

201.1

350.0

52.8

200.9

192.7

331.7

200.2

143.2

47.7

219.3

325.5

212.7

185.4

105.9

−55.3

185

296

299

13

51

69

84

145

186

193

221

234

235

9.6

−4.5

−45.3

−21.0148

58

70

64

78

64

90

71

80

69

2.9

−43.4

−18.3

−12.8

−53.8

−33.5

−3.3

−27.9

−22.9

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Lorentz-violating parameters αl[1]:

? α ∈ D(open)

causal:|? α|2≡

8

?

l=0

?αl?2< A2,

?2

(1a)

A = 3 × 10−18

?

Mprim

56 GeV/c2

,(1b)

showing explicitly the dependence on the mass of the primary charged particle (taken equal

for all events). There are indications [4] that these UHECRs originate predominantly from

protons but, in order to be on the safe side, we will later take the mass Mprimto be equal to

that of iron, Mprim= 56 GeV/c2. Bound (1) as well as all other bounds in this addendum

are based on the Cherenkov threshold condition (10) in Sec. II B of Ref. [1] and the reader

is referred to this section, in particular, for further details.

The domain used in (1a) is defined by

D(open)

causal≡ {? α ∈ R9: ∀b x∈R3 (α0+ αj? xj+ ? αjk? xj? xk) > 0},

where ? x ≡ ? x/|? x| denotes a unit vector in Euclidean three-space and the traceless symmetric

parameter domain (2) allows for vacuum Cherenkov radiation in all directions and, with

boundaries added, is believed to constitute a significant part of the physical domain of the

theory, where, e.g., unitarity and microcausality hold; cf. Appendix C of Ref. [7]. It may,

nevertheless, be of interest to get bounds outside this domain, because modified-Maxwell

theory could be only part of the full Lorentz-noninvariant theory.

The crucial observation is that domain (2) shrinks to zero size in the hyperplane α0=

? αjk= 0, so that bound (1a) becomes ineffective there. Still, the data from Table I can be

this hyperplane

(2)

3 × 3 matrix ? αjkis defined in terms of the parameters αlfor l = 4,...,8 (see below). The

used to get the following two–σ bound on the three parity-odd nonisotropic parameters in

α0= α4= α5= α6= α7= α8= 0 :

3

?

j=1

(αj)2<

?

4 × 10−18?2?

Mprim

56 GeV/c2

?4

.(3)

Similarly, there is a two–σ bound on the five parity-even nonisotropic parameters in an

orthogonal hyperplane

4 × 10−18?2?

α0= α1= α2= α3= 0 :

8

?

l=4

(αl)2<

?

Mprim

56 GeV/c2

?4

.(4)

It is, in principle, possible to get other bounds for the eight nonisotropic parameters, but,

for the moment, bounds (3) and (4) suffice.

If only a single parameter αlfor l ∈ {1,...,8} is considered (all seven other nonisotropic

parameters and the isotropic parameter α0being zero), bounds (3) and (4) give a two-sided

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bound on that single isolated parameter. Setting Mprim= 56GeV/c2and showing explicitly

the approximate one–σ error, these bounds are

l ∈ {1,...,8} :|αl| < (2 ± 1) × 10−18, (5)

for α0= αm= 0 with m ∈ {1,...,8} and m ?= l. Incidentally, the possibility of getting

certain two-sided Cherenkov bounds from an isotropic set of UHECR events has already

been noted in Appendix C of Ref. [7].

For completeness, we also give the following one-sided bound on the single α0parameter

α0< (1.4 ± 0.7) × 10−19,(6)

for αm= 0 with m ∈ {1,...,8}. Bound (6) has been derived by setting Mprim= 56 GeV/c2

and using the 148 EeV Auger event from Table I, which has a reliable energy calibration

[4]. For the Fly’s Eye event with an estimated energy of 320 EeV [5], bound (6) would be

reduced by a factor of approximately 5 according to Eq. (10) in Ref. [1].

In order to facilitate the comparison with existing laboratory bounds and future ones,

we provide a dictionary between our αl(or ? αµν) parameters and the nonbirefringent SME

α8

? α23

The Cartesian coordinates employed (cf. Sec. III A of Ref. [3]) are such that the flight-

direction vector ? q of an UHECR primary at the top of the Earth atmosphere is given by

? q3

in terms of the celestial coordinates RA ≡ α and DEC ≡ δ from Table I.

Using the dictionary (7), bounds (5) and (6) give the following two–σ (95% CL) bounds

parameters defined by Eq. (11) in Ref. [3]:

? α ≡

α0

α1

α2

α3

α4

α5

α6

α7

≡

? α00

? α02

? α11

? α13

? α01

? α03

? α12

? α22

=

2? κtr

−2(? κo+)(23)

−2(? κo+)(12)

−(? κe−)(12)

−(? κe−)(22)

−2(? κo+)(31)

−(? κe−)(11)

−(? κe−)(13)

−(? κe−)(23)

.(7)

? q1

? q2

= −

sin(π/2 − δ)cosα

sin(π/2 − δ)sinα

cos(π/2 − δ)

,(8)

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on the nine isolated SME parameters of nonbirefringent modified-Maxwell theory:

??(? κo+)(ij)??

(kl)=(11),(12),(13),(22),(23)< 4 × 10−18,

(ij)=(23),(31),(12)< 2 × 10−18, (9a)

??(? κe−)(kl)??

(9b)

? κtr < 1.4 × 10−19,(9c)

for the Sun-centered Cartesian coordinate system employed in (8). The Cherenkov bounds

(9a), (9b), and (9c) are significantly stronger than the current laboratory bounds at the

10−12, 10−16, and 10−7levels, respectively; see the third paragraph of Sec. V in Ref. [1] for

further discussion and references. It is to be emphasized that these Cherenkov bounds only

depend on the measured energies and flight directions of the charged cosmic-ray primaries

at the top of the Earth atmosphere.

ACKNOWLEDGMENTS

FRK acknowledges the hospitality of The Henryk Niewodnicza´ nski Institute of Nuclear

Physics in Cracow, Poland, where part of this work was done, and the help of M. Schreck

with the signs in (7). Both authors thank V.A. Kosteleck´ y for useful suggestions regarding

the presentation of the results of this addendum.

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[4] J. Abraham et al. [Pierre Auger Collaboration], “Correlation of the highest-energy cosmic

rays with the positions of nearby active galactic nuclei,” Astropart. Phys. 29, 188 (2008),

arXiv:0712.2843v1 [astro-ph].

[5] D.J. Bird et al., “Detection of a cosmic ray with measured energy well beyond the ex-

pected spectral cutoff due to cosmic microwave radiation,” Astrophys. J. 441, 144 (1995),

arXiv:astro-ph/9410067.

[6] N. Hayashida et al., “Observation of a very energetic cosmic ray well beyond the predicted

2.7–K cutoff in the primary energy spectrum,” Phys. Rev. Lett. 73, 3491 (1994).

[7] C. Kaufhold and F.R. Klinkhamer, “Vacuum Cherenkov radiation in spacelike Maxwell–Chern–

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