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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]).
Year DayE[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
81 79
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
−21.0 148
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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