Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~qfthep04/2004/Proceedings2004/Alfaro_QFTHEP04_227_243.ps Äàòà èçìåíåíèÿ: Thu Sep 17 21:20:26 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 08:26:29 2012 Êîäèðîâêà: IBM-866 Ïîèñêîâûå ñëîâà: universe

Loop quantum gravity e#ects on the high energy
cosmic ray spectrum
Jorge Alfaro
Facultad de FÒÐsica, Pontificia Universidad CatÒolica de Chile
Casilla 306, Santiago 22,
Chile
Gonzalo A. Palma
Department of Applied Mathematics and Theoretical Physics,
Center for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA,
United Kingdom
Abstract
Recent observations on ultra high energy cosmic rays (those cosmic rays with energies greater
than # 4 ½ 10 18 eV) suggest an abundant flux of incoming particles with energies above 1 ½ 10 20 eV.
These observations violate the GreiseníZatsepiníKuzmin cuto#. To explain this anomaly we argue
that quantumígravitational e#ects may be playing a decisive role in the propagation of ultra high
energy cosmic rays. We consider the loop quantum gravity approach and provide useful techniques to
establish and analyze constraints on the loop quantum gravity parameters arising from observational
data. In particular, we study the e#ects on the predicted spectrum for ultra high energy cosmic rays
and conclude that is possible to reconcile observations.
1 Introduction
In the present discussion we are concerned with the observation of ultra high energy cosmic rays (UHECR),
i.e. those cosmic rays with energies greater than # 4 ½ 10 18 eV. Although not completely clear, it has
been suggested that these high energy particles are possibly heavy nuclei [1] (we will assume here that
they are protons) and, by virtue of the isotropic distribution with which they arrive to us, that they
originate in extragalactic sources.
The propagation of UHECR in intergalactic space is subject to interaction with the cosmic microwave
background radiation (CMBR). Its presence produces friction on UHECR making them release energy in
the form of secondary particles. This a#ects their possibility to reach great distances. A first estimation of
the characteristic distance that UHECR can reach before losing most of their energy was simultaneously
made in 1966 by K. Greisen [2] and G. T. Zatsepin & V. A. Kuzmin [3]. They showed that the observation
of cosmic rays with energies greater than 4½10 19 eV should be greatly suppressed. This energy is usually
referred as to the GZK cuto# energy. A few years later, F.W. Stecker [4] computed the mean life time for
protons as a function of their energy, giving a more accurate perspective of the energy dependence of the
cuto# and showing that cosmic rays with energies above 1 ½ 10 20 eV should not travel more than # 100
Mpc. More detailed approaches to the GZKícuto# feature have been made since these first estimations.
For example V. Berezinsky & S.I. Grigorieva [5], V. Berezinsky et al. [6] and S.T. Scully & F.W. Stecker
[7] have made progress in the theoretical study of the spectrum J(E) (i.e. the flux of arriving particles
as a function of the observed energy E) that UHECR should present. As a result, the GZK cuto# exists
in the form of a suppression in the predicted flux of cosmic rays with energies above # 8 ½ 10 19 eV.
Currently there are two main di#erent sets of data for the observed flux J(E) in its most energetic
sector (E > 4 ½ 10 18 eV). On one hand we have the observations from the High Resolution Fly's Eye
(HiRes) collaboration group [8], which seem to be consistent with the predicted theoretical spectrum and,
therefore, with the presence of the GZK cuto#. Meanwhile, on the other hand, we have the observations
from the Akeno Giant Air Shower Array (AGASA) collaboration group [9], which reveal an abundant flux
of incoming cosmic rays with energies above 1 ½ 10 20 eV. The appearance of these high energy events is
opposed to the predicted GZK cuto#, and represents a great challenge that has motivated a vast amount
of new ideas and mechanisms to explain the phenomenon [10]. If the AGASA observations are correct
then, since there are no known active objects in our neighborhood (let us say within a radius R # 100
227

Mpc) able to act as sources of such energetic particles and since their arrival is mostly isotropic (without
any privileged local source), we are forced to conclude that these cosmic rays come from distances larger
than 100 Mpc. This is commonly referred as the GreiseníZatsepiníKuzmin (GZK) anomaly.
One of the interesting notions emerging from the possible existence of the GZK anomaly is that, since
ultra high energy cosmic rays involve the highest energy events registered up to now, then a possible
framework to understand and explain this phenomena could be of a quantumígravitational nature [11,
12, 13]. This possibility is indeed very exciting if we consider the present lack of empirical support for the
di#erent approaches to the problem of gravity quantization. In the context of the UHECR phenomena, all
these di#erent approaches motivated by di#erent quantum gravity formulations, have usually converged
on a common path to solve and explain the GZK anomaly: the introduction of e#ective models for the
description of high energy particle propagation. These e#ective models, pictures of the yet unknown full
quantum gravity theory, o#er the possibility to modify conventional physics through new terms in the
equations of motion (now e#ective equations of motion), leading to the eventual breakup of fundamental
symmetries such as Lorentz invariance (expected to be preserved at the fundamental level). These Lorentz
symmetry breaking mechanisms are usually referred as Lorentz invariance violations (LIV's) if the break
introduce a privileged reference frame, or Lorentz invariance deformations (LID's), if such a reference
frame is absent [14]. Its appearance on theoretical and phenomenological grounds (such as high energy
astrophysical phenomena) has been widely studied, and o#ers a large and rich amount of new signatures
that deserve attention [15].
To deepen the above ideas, we have adopted the loop quantum gravity (LQG) theory [16]. It is possible
to study LQG through e#ective theories that take into consideration matterígravity couplings. In this
line, in the works of J. Alfaro et al. [17], the e#ects of the loop structure of space at the Planck level are
treated semiclassically through a coarseígrained approximation. An interesting feature of these methods
is the explicit appearance of the Plank scale l p and the appearance of a new length scale L # l p (called the
``weave'' scale), such that for distances d # L the quantum loop structure of space is manifest, while for
distances d # L the continuous flat geometry is regained. The presence of these two scales in the e#ective
theories has the consequence of introducing LIV's to the dispersion relations E = E(p) for particles with
energy E and momentum p. It can be shown that these LIV's can significantly modify the kinematical
conditions for a reaction to take place. For instance, as shown in detail in [18], a consequence for the
UHECR phenomenology is that the kinematical conditions for a reaction between a primary cosmic ray
and a CMBR photon can be modified, leading to new e#ects and predictions such as an abundant flux
of cosmic rays well beyond the GZK cuto# energy (explaining in this way the AGASA observations).
In this work we provide some techniques to establish and analyze new constraints on the LQG paí
rameters (or any LIV parameters coming from other theories). Also, we attempt to predict a modified
UHECR spectrum consistent with the AGASA observations. The results shown here are presented in
detail in a previous work by J. Alfaro and G. Palma [13]. This paper is organized as follows: In section
2 we give a brief derivation of the conventional spectrum and analyze it with AGASA observations. In
section 3 we present a short outline of loop quantum gravity and its e#ective description of fermionic and
bosonic fields. In section 4 we analyze the e#ects of LQG corrections on the threshold conditions for the
main reactions involved in the UHECR phenomena to take place. In section 5 we show how the modified
kinematics can be relevant to the theoretical spectrum J(E) of cosmic rays (we will present the obtained
modified spectrum). In section 6 we give some final remarks.
2 Ultra High Energy Cosmic Rays
Here we review the main steps in the derivation of the UHECR spectrum. This presentation will be useful
and relevant to understand how LQG corrections a#ect the predicted flux of cosmic rays. The following
material is mainly contained in the works of F.W. Stecker [4], Berezinsky et al. [6] and S.T. Scully &
F.W. Stecker [7].
2.1 General Description
Two simple and common assumptions used in the development of the cosmic ray spectrum are: 1) sources
are uniformly distributed in the Universe, and 2) the generation flux F (E g ) of emitted cosmic rays from
228

the sources is well described by a power law behavior of the form F (E g ) # E -#g g , where E g is the energy
of the emitted particle and # g is the generation index.
One of the main quantities for the computation of the UHECR spectrum is the energy loss -E -1 dE/dt.
This quantity describes the rate at which a cosmic ray loses energy, and takes into consideration two chief
contributions: the energy loss due to the redshift attenuation and the energy loss due to collisions with
the CMBR photons. This last contribution depends, at the same time, on the cross sections # and
the inelasticities K of the interactions produced during the propagation of protons in the extragalactic
medium, as well as on the CMBR spectrum. The most important reactions taking place in the description
of proton's propagation are the pair creation and the photoípion production:
p + # # p + e - + e + , and p + # # p + #. (1)
The photoípion production happens through several channels (for example the baryonic # and N , and
mesonic # and # resonance channels, just to mention some of them) and is the main responsible in the
appearance of the GZK cuto#.
2.2 Some Kinematics
To study the interaction between protons and the CMBR, it is useful to distinguish between three
reference systems; the laboratory system K (which we identify with the Friedman Robertson Walker
(FRW) coímoving reference system), the center of mass (c.m.) system K # , and the system where the
proton is at rest K # . In terms of these systems, the photon energy will be expressed as # in K and as # in
K # . The relation between both quantities is simply # = ##(1 - # cos #), where # = E/m p is the Lorentz
factor relating K and K # , E and m p are the energy and mass of the incident proton, # = # 1 - # -2 , and
# is the angle between the momenta of the photon and the proton measured in the laboratory system
K. To determine the total energy E # tot = E # + # # in the c.m. system, it is enough to use the invariant
energy squared s # E 2
tot - p 2
tot (where E tot = E + # and p tot are the total energy and momentum in the
laboratory system). In this way, we have E # 2
tot = s = m 2
p + 2m p #. As a consequence, the Lorentz factor
# c which relates the K reference system with the K # system, is
# c = E + #
# s #
E
(m 2
p + 2m p #) 1/2 . (2)
Let us consider the relevant case in which the reaction between the proton and the CMBR photon is
of the type
p + # # a + b, (3)
where a and b are two final particles of the collision. The final energies of these particles are easily
determined by the conservation of energyímomentum. In the K # system these are
E # a, b = 1
2 # s
(s +m 2
a, b -m 2
b, a ). (4)
Transforming this quantity to the laboratory system, and averaging with respect to the angle between
the directions of the final momenta, it is possible to find that the final average energy of a (or b) in the
laboratory system is
#E a, b # = E
2
# 1 + m 2
a, b -m 2
b, a
s
# . (5)
The inelasticity K of the reaction is defined as the average fractional di#erence K = #E/E, where
#E = E -E f is the di#erence between the initial energy E and final energy E f of the proton (in a single
collision with the CMBR photons). For the particular case of the emission of an arbitrary particle a (that
is to say p + # # p + a), expression (5) allows us to write
K a (s) = 1
2
# 1 + m 2
a -m 2
p
s
# , (6)
where K a is the inelasticity of the described process. This is one of the main quantities involved in the
study of the UHECR spectrum.
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2.3 Mean Life #(E)
To derive the UHECR spectrum it is imperative to know the mean life #(E) of the cosmic ray (or proton)
with energy E propagating in space, due to the attenuation of its energy by the interactions with the
CMBR photons. The mean life #(E) is defined through the relation #(E) -1 = # -E -1 dE/dt # col
, where
the label ``col'' refers to the fact that the energy loss is due to the collisions with the CMBR photons. It
is possible to show that the mean life #(E) can be written in the form
#(E) -1 = -
kT
2# 2 # 2
# #
# th
d# #(#)K(#)# ln[1 - e -#/2#kT ], (7)
where K(#) = #E/E is the inelasticity (with #E the di#erence between the initial and final energies of
the proton before and after each collision), #(#) is the scattering cross section with the CMBR photons,
KT is the temperature of the CMBR bath, # the lorentz factor, and # th is the threshold photon energy
for reactions to take place.
2.4 Energy Loss and Spectrum
The energy loss su#ered by a very energetic proton during its journey, from a distant source to our
detectors, is not only produced by the collisions that it has with CMBR at a particular epoch. There is
also a decrease in its energy due to the redshift attenuation produced by the expansion of the Universe.
At the same time, such expansion will a#ect the collision rate through the attenuation of the photon
gas density, which can be understood as a cooling of the CMBR through the relation T = (1 + z)T 0 ,
where z is the redshift and T 0 is the temperature of the background at the present time. To calculate
the spectrum we need to consider the rate of energy loss during any epoch z of the Universe.
For the present discussion, we consider a flat Friedman Robertson Walker (FRW) universe dominated
by matter. The above assumptions give rise to the following relation between the time coordinate t and
the redshift z: dt = -dz/H 0 (1 + z) 5/2 , where H 0 is the Hubble constant at present time. Since the
momentum of a free particle in a FRW space behaves as p # (1 + z), we will have, with the additional
consideration p # m (where m is the particle mass), that the energy loss due to redshift is
# -
1
E
dE
dt
# cr
= H 0 (1 + z) 3/2 . (8)
On the other hand, the energy loss due to collisions with the CMBR evolves as the background temí
perature changes (recall T = (1 + z)T 0 ). This evolution can be parameterized through z and is given
by
# -
1
E
dE
dt
# col
= (1 + z) 3 #([1 + z]E) -1 . (9)
Thus, the total energy loss can be expressed considering both contributions (using z instead of t):
1
E
dE
dz
= (1 + z) -1 +H -1
0 (1 + z) 1/2 #([1 + z]E) -1 . (10)
Equation (10) can be integrated numerically to provide the energy E g (E, z) of a proton generated by the
source in a z epoch and that will be detected with a energy E here on Earth. Let us express this solution
formally by: E g (E, z) = #(E, z)E.
It is also possible to manipulate equation (10) to obtain an expression for the dilatation of the
energy interval dE g /dE. To accomplish this it is necessary to integrate (10) with respect to z and then
di#erentiate it with respect to E to obtain an integral equation for dE g /dE. The solution of such an
equation is found to be
dE g (z g )
dE
= (1 + z g ) exp ## zg
0
dz
H 0
(1 + z) 1/2 db(E # )
dE #
# . (11)
where E # = (1 + z)#(E, z)E.
230

The total flux dj(E) of emitted particles from a volume element dV = R 3 (z)r 2 drd# in the epoch z
and coordinate r, measured from Earth at present time with energy E is
dj(E)dE = F (E 0 , z)dE 0 n(z)dV
(1 + z)4#R 2
0 r 2 , (12)
where j(E) is the particle flux per energy, F (E 0 , z)dE 0 the emitted particle flux within the range (E 0 , E 0 +
dE 0 ), and n(z) the density of sources in z. As previously mentioned, it is convenient to study the emission
flux with a power law spectrum of the type F (E) # E -#g . It can be shown that with such assumption, the
relation between the emission flux and the total luminosity L p of the source is F (E) = (# g - 2)L p E -#g .
To describe the evolution of the sources we shall also use a power law behavior. This will be done through
the relation
L p (z)n(z) = (1 + z) (3+m)
L 0 , (13)
where L 0 = L p (0)n(0). In this way, m = 0 corresponds to the case in which sources do not evolve. If we
consider that R 0 = (1 + z)R(z) and R(z)dr = dt for flat spaces (and v # 1 for very energetic particles)
and integrating (12) from z = 0 to some z = z max for which sources are not relevant for the phenomena,
it is possible to obtain
j(E) = (# g - 2) 1
4#
L 0
H 0
E -#g # zmax
0
dz g (1 + z g ) m-5/2 # -#g (E, z g ) dE g (z g )
dE . (14)
The above expression constitutes the spectrum of UHECR. The volumetric luminosity L 0 and the # g and
m indexes are free parameters that must be fixed observationally.
2.5 Ultra High Energy Cosmic Rays Spectrum
To accomplish the computation of the theoretical spectrum we need information about the dynamical
processes taking place in the propagation of protons along the CMBR. As we already emphasized, the
most important reactions taking place in the description of a proton's propagation are, the pair creation
p + # # p + e - + e + , and the photoípion production p + # # p + #. This last reaction is mediated by
several channels. The main channels are N + #, # + #, R, N + #(770), N + #(782). The total cross
sections and inelasticities of these processes are well known and can be used in (7) to compute the main
time life of protons as a function of their energy. Then, with the help of expressions (11) and (14), we can
finally find the predicted spectrum for UHECR. Fig.1 shows the obtained spectrum J(E) of UHECR and
the AGASA observed data. Again, we have selected for the theoretical spectrum J(E) the idealized case
Emax = #. To reconcile the data of the low energy region (E < 4 ½ 10 19 eV), where the pair creation
dominates the energy loss, it is necessary to have a generation index # g = 2.7 (with the additional
supposition that sources do not evolve) and a volumetric luminosity L 0 = 4.7 ½ 10 51 ergs/Mpc 3 yr. It
can be seen that for events with energies E > 4 ½ 10 19 eV, where the energy loss is dominated by the
photoípion production, the predicted spectrum does not fit the data well. To have a statistical sense
of the discrepancy between observation and theory, we can calculate the Poisson probability P of an
excess in the five highest energy bins. This is P = 1.1 ½ 10 -8 . Another statistical measure is provided
by the Poisson # 2 [19]. Computing this quantity for the eight highest energy bins, we obtain # 2 = 29.
These quantities show how far the AGASA measurements are from the theoretical prediction given by
the curve of Fig.1. Other more sophisticated models have also been analyzed in detail [6], nevertheless, it
has turned out that the conventional standard model of physics does not have the capacity to reproduce
the observations from the AGASA collaboration group in a satisfactory way.
3 Loop Quantum Gravity
Loop quantum gravity is a canonical approach to the problem of gravity quantization. It is based on
the construction of a spin network basis, labelled by graphs embedded in a three dimensional insertion
# in spaceítime. A consequence of this approach is that the quantum structure of spaceítime will be of
a polymerílike nature, highly manifested in phenomena involving the Planck scale l p .
231

10 18 10 19 10 20 10 21
Energy E [ eV ]
10 23
10 24
10 25
J(E)
E
3
[m
í2
sec
í1
eV
2
]
Figure 1: UHECR spectrum and AGASA observations. The figure shows the UHECR spectrum J(E)
multiplied by E 3 , for uniform distributed sources, without evolution, and with a maximum generation
energy Emax = #. Also shown are the AGASA observed events. The best fit for the low energy sector
(E < 4 ½ 10 19 eV) corresponds to # g = 2.7.
The former brief description of loop quantum gravity allows us to realize how complicated a full
treatment of a physical phenomena could be when the quantum nature of gravity is considered, even for
a flat geometry. It is possible, however, to introduce a loop state which approximates a flat 3ímetric on
# at length scales greater than the length scale L # l p . For pure gravity, this state is referred to as
the weave state |W #, and the length scale L as the weave scale. A flat weave |W # will be characterized
by L in such a way that for distances d # L the quantum loop structure of space is manifest, while for
distances d # L the continuous flat geometry is regained. With this approach, for instance, the metric
operator Ó
q ab satisfies
#W |Óq ab |W # = # ab +O(l p /L). (15)
A generalization of the former idea, to include matter fields, is also possible. In this case, the loop
state represents a matter field # coupled to gravity. Such a state is denoted by |W, ## and, again, is simply
referred to as the weave. As before, it will be characterized by the weave scale L and the hamiltonian
operators Ó
H# are expected to fulfill a relation analog to (15), that is, we shall be able to define an e#ective
hamiltonian H# such that
H# = #W, #| Ó
H# |W, ##. (16)
An approach to this task has been performed by J. Alfaro et al. [17] for 1/2íspin fermions and the
electromagnetic field. In this approach the e#ects of the loop structure of space at the Planck level are
treated semiclassically through a coarseígrained approximation [21]. This method leads to the natural
appearance of LIV's in the equations of motion derived from the e#ective hamiltonian. The key feature
here is that the e#ective hamiltonian is constructed from expectation values of dynamical quantities from
both the matter fields and the gravitational field. In this way, when a flat weave is considered, the
expectation values of the gravitational part will appear in the equations of motion for the matter fields
in the form of coe#cients with dependence in both scales, L and l p . When a flat geometry is considered,
the expectation values can be interpreted as vacuum expectation values for the considered matter fields.
A significant discussion is whether the Lorentz symmetry is present in the full LQG theory or not
[22]. For the present work, we shall assume that Lorentz symmetry is indeed present in the full LQG
theory. This assumption, jointly with the consideration that the new corrective coe#cients are vacuum
expectation values, leads us to consider that the Lorentz symmetry is spontaneously broken in the e#ective
theory level.
Specially important in the development of the e#ective equations of motion for matter fields is that
they are only valid in a homogenous and isotropic system. From the point of view of a spontaneous
232

symmetry breakup such a system is unique and, therefore, a privileged reference frame). It is possible
then to put the equations of motion (and therefore the dispersion relations) in a covariant form through
the introduction of a fourívelocity vector explicitly denoting the existence of a preferred system. From the
cosmological point of view, such a privileged system does exist, and corresponds to the CMBR coímoving
reference system. For that reason, we shall assume that the preferred system denoted by the presence of
LIV's is the same CMBR coímoving reference frame, and will use it as the laboratory system.
In what follows we briefly summarize the obtained dispersion relations for both, fermions and bosons.
3.1 Modified Dispersion Relations
The dispersion relation for fermions is found to be [17]:
E 2
‘ = p 2 + 2#p 2 + #p 4
‘ 2#p +m 2 , (17)
where the ‘ signs correspond to the helicity state of the described particle (note that these signs are
produced by parity violation terms in the equations of motion). Additionally, we have defined the
parameters #, # and # in such a way that they depend on the scales L and l p in the following way:
# = ## (l p /L) 2 , # = # # l 2
p , and # = # # l p /2L 2 , (18)
where ## , # # and # # are adimensional parameters of order 1.
In the case of bosonic particles the dispersion relation consists of
E 2 = p 2 + 2#p 2 + #p 4 +m 2 , (19)
where again we have defined parameters # and # as:
# = ## (l p /L) 2 , and # = # # l 2
p (20)
where ## , # # and # # are adimensional parameters of order 1.
As shown in detail in [13], only the # correction will be relevant in the computation of the spectrum.
Therefore, in the rest of this work we shall only consider the following dispersion relation for both,
fermions and bosons:
E 2 = p 2 + 2#p 2 +m 2 . (21)
To conclude, let us mention that the dispersion relation (21) will be used for the physical description of
electrons, protons, neutrons, # and N baryonic resonances, as well as for mesons #, # and #.
4 Threshold Conditions
A useful discussion around the e#ects that LIV's can have on the propagation of UHECR can be raised
through the study of the threshold conditions for the reactions to take place [18]. To simplify our
subsequent discussions, let us use the following notation for the modified dispersion relations
E 2 = p 2 + f(p) +m 2 , (22)
where f(p) is the deformation function of the momentum p.
A decay reaction is kinematically allowed when, for a given value of the total momentum # p 0 =
# initial # p = # final # p, one can find a total energy value E 0 such that E 0 # E min . Here E min is the
minimum value attainable by the total energy of the decaying products for a given total momentum # p 0 .
To find E min , it is enough to take the individual decay product momenta to be collinear with respect to
the total momentum # p 0 and with the same direction. To see this, we can variate E 0 with the appropriate
restrictions
E 0 = # i
E i (p i ) + # j (p j
0 - # i
p j
i ), (23)
233

where # j are Lagrange multipliers, the i index specifies the ith particle and the j index the jth vectorial
component of the di#erent quantities. Doing the variation, we obtain
#E i
#p j
i
# v j
i
= # j . (24)
That is to say, the velocities of all the final produced particles must be equal to #. Since the dispersion
relations that we are treating are monotonously increasing in the range of momenta p > #, the momenta
can be taken as being collinear and with the same direction of the initial quantity # p 0 .
In this work, we will focus on those cases in which two particles (say a and b) collide to subsequently
decay in the aforementioned final states. For the present discussion, particles a and b have momenta # p a
and #p b respectively, and the total momentum of the system is # p 0 . It is easy to see from the dispersion
relations that we are considering, that the total energy of the system will depend only on p a = |#p a | and
p b = |#p b |. Therefore, to obtain the threshold condition for the mentioned kind of process, we must find
the maximum possible total energy Emax of the initial configuration, given the knowledge of p a and p b .
To accomplish this, let us fix #p a and variate the incoming direction of # p b = Ó
np b in
E 0 = E a (#p 0 - p b Ó
n) +E b (p b ) + #(Ón 2
- 1). (25)
Varying (25) with respect to Ó
n (# is a Lagrange multiplier), we find
Ó
n i = v i
a p b /2#. (26)
In this way we obtain two extremal situations # = ‘v a p b /2, or simply Ó
n i = ‘v i
a /v a . A simple inspection
shows that for the dispersion relations that we are considering, the maximum energy is given by Ó
n i =
-v i
a /v a , or in other words, when a frontal collision takes place.
Summarizing the threshold condition for a two particle (a and b) collision and subsequent decay, can
be expressed through the following requirements:
E a +E b # # final
E f , (27)
with all final particles having the same velocity (v i = v j for any to final particles i and j), and
p a - p b = # final
p f , (28)
where the sign of the momenta
# final p f is given by the direction of the highest momentum magnitude
of the initial particles.
Our interest in the next subsections is the study of the reactions involved in high energy cosmic ray
phenomena through the threshold conditions. To accomplish this goal through simple expressions that
are easyítoímanipulate, we shall further use, for the equal velocities condition, the simplification
E b m a = E a m b , (29)
valid for the study of parameters coming from the region f(p) # m 2 . This simplification will allow the
achievement of bounds over the order of magnitude of the di#erent parameters involved in the modified
dispersion relations, which are precisely our main concern.
In the following subsections we study the kinematical e#ects of LIV's through the threshold conditions
for the reactions involved in the propagation of UHECR. Since, in this phenomena, photons are present
in the form of low energy particles (the soft photons of the CMBR), the LQG corrections in the electroí
magnetic sector of the theory can be ignored. LQG corrections to the electromagnetic sector, however,
have already been studied for other high energy reactions such as the Mkn 501 #írays [12]. Finally, let
us recall that in the complete treatment of threshold conditions, it is possible to learn that only the #
coe#cients will be significant [13].
234

4.1 PhotoíPion Production # + p # p + #
Let us begin with the photoípion production # + p # p + #. Considering the corrections provided in the
dispersion relation (21) for fermions and bosons, we note that, for the photoípion production to proceed,
the following condition must be satisfied
2 ##E 2
# + 4E # # # m 2
# (2m p +m # )
m p +m #
. (30)
where E # is the produced pion energy and ## = # p - # # .
4.2 Resonant Production # + p # #
The main channel involved in the photoípion production is the resonant production of the #(1232). It
can be shown that the threshold condition for the resonant #(1232) decay reaction to occur, is
2 ##E 2 + 4#E # m 2
# -m 2
p , (31)
where E is the incident proton energy and ## = # p - ## .
4.3 Pair Creation # + p # p + e + + e -
Pair creation, # + p # p + e + + e - , is greatly abundant in the sector previous to the GZK limit. When
the dispersion relations for fermions are considered for both protons and electrons, it is possible to find
##
m e
m p
E 2 +E# # m e (m p +m e ), (32)
where E is the incident proton energy and ## = # p - # e .
4.4 Bounds
In order to study the threshold conditions (30), (31) and (32), in the context of the GZK anomaly, we
must establish some criteria. Firstly, as we have seen in section 2, the conventionally obtained theoretical
spectrum provides a very good description of the phenomena up to an energy # 4 ½ 10 19 eV. The main
reaction taking place in this well described region is the pair creation # + p # p + e + + e - and, therefore,
no modifications are present for this reaction up to # 4 ½ 10 19 eV. As a consequence, and since threshold
conditions o#er a measure of how modified kinematics is, we will require that the threshold condition
(32) for pair creation not be substantially altered by the new corrective terms.
Secondly, we have the GZK anomaly itself, which we are committed to explain. Since for energies
greater than # 8 ½ 10 19 eV the conventional theoretical spectrum does not fit the experimental data
well, we shall require that LQG corrections be able to o#er a violation of the GZKícuto#. The dominant
reaction in the violated E > 8 ½ 10 19 region is the photoípion production and, therefore, we further
require that the new corrective terms present in the kinematical calculations be able to shift the threshold
significantly to preclude the reaction.
We begin our analysis with the threshold condition for pair production. In this case we have:
## m e
m p
E 2 +E# # m e (m p +m e ), (33)
with ## = # p - # e . As is clear from the above condition, the minimum softíphoton energy # min for the
pair production to occur, is
# min = m e
E
(m p +m e ) - ##
m e
m p
E. (34)
It follows therefore that the condition for a significant increase or decrease in the threshold energy for pair
production becomes |##| # m p (m p +m e )/E 2 . In this way, if we do not want kinematics to be modified
up to a reference energy E ref = 3 ½ 10 19 , we must impose the following constraint
|# p - # e | <
(m p +m e )m p
E 2
ref
= 9.8 ½ 10 -22 . (35)
235

Similar treatments can be found for the analysis of other astrophysical signals like the Mkn 501 #írays
[23], when the absence of anomalies is considered.
Let us now consider the threshold condition for the photoípion production. Taking only the # corí
rection, we have
2 ##E 2
# + 4E # # #
m 2
# (2m p +m # )
m p +m #
. (36)
It is possible to find that for the above condition to be violated for all energies E # of the emerging pion,
and therefore no reaction to take place, the following inequality must hold
# # - # p >
2# 2 (m p +m # )
m 2
# (2m p +m # ) = 3.3 ½ 10 -24 [#/# 0 ] 2
. (37)
where # 0 = KT = 2.35 ½ 10 -4 eV is the thermal CMBR energy. If we repeat these steps for the #(1232)
resonant decay, we obtain the following condition
## - # p >
2# 2
m 2
# -m 2
p
= 1.7 ½ 10 -25 [#/# 0 ] 2 . (38)
To estimate a range for the weave scale L, let us use as a reference energy # ref = # min , where
# min is the minimum energy for the reaction to take place, in inequality (36), when the condition for
a significant increase in the threshold condition is taken into account (for a primordial proton reference
energy E ref = 2 ½ 10 20 , this is # min # 2.9 ½ # 0 ), and join the results deduced from the mentioned
requirements. Assuming that the ## parameters are of order 1, as well as the di#erence between them
for di#erent particles, we can estimate ---for the weave scale L--- the preferred range
2.6 ½ 10 -18 eV -1 # L # 1.6 ½ 10 -17 eV -1 , (39)
where the lefthand and righthand sides come from bounds (35) and (37) respectively (since the #(1232)
is just one channel of the photoípion production, we shall not consider it to set any bound).
If no GZK anomaly is confirmed in future experimental observations, then we should state a stronger
bound for the di#erence # # - # p . Using the same assumptions to set the restriction (35) when the
primordial proton reference energy is E ref = 2 ½ 10 20 eV, it is possible to find
|# # - # p | < 2.3 ½ 10 -23 . (40)
In terms of the length scale L, this last bound may be read as
L # 1.7 ½ 10 -17 eV -1 , (41)
which is a stronger bound over L than (35), o#ered by pair creation.
5 Modified Spectrum
In this section we show the way in which the LQG correction # a#ects the prediction for the theoretical
cosmic ray spectrum. Our approach will be centered on the supposition that the LQG corrections to
the main quantities for the calculation ---such as cross sections and inelasticities of processes--- are, in a
first instance, kinematical corrections, and that the Lorentz symmetry is spontaneously broken. These
assumptions will allow us to introduce the adequate corrections when a modified dispersion relation is
known.
5.1 Kinematics
Having a spontaneous Lorentz symmetry breaking we can use the still valid Lorentz transformations
to express physical quantities observed in one reference system, in another one. This is possible since,
under a spontaneous symmetry breaking, the group representations of the broken group preserves its
236

transformation properties. In particular, it will be possible to relate the observed 4ímomenta in di#erent
reference systems through the usual rule
p # ² = # #
² p # , (42)
where p ² = (-E, #p) is an arbitrary 4ímomentum expressed in a given reference system K, p # ² is the same
vector expressed in another given system K # , and # #
² is the usual Lorentz transformation connecting
both systems. Such a transformation will keep invariant the scalar product
p ²
p² = -E 2 + p 2 , (43)
as well as any other product. Let us illustrate, for transformation (42), the situation in which K # is a
reference system with the same orientation of K and which represents an observer with velocity #
# with
respect to K. In this case # #
² shall correspond to a boost in the Ó
# = # #/|#| direction, and expression (42)
will be reduced to
E # = #(E - #
# § # p), and #p # = #(#p - #
#E), (44)
where # = (1 - # 2 ) -1/2 . A particular case of this transformation will be that in which #
# has the same
direction as #p, and K # corresponds to the c.m. reference system, that is to say, the system in which
#p # = 0. In such a case we will have #
# = # p/E and # = E/(E 2
- p 2 ) -1/2 , jointly with the relation
E # = E/# = (E 2
- p 2 ) 1/2 . (45)
In other words, the c.m. energy of a particle with energy E and momentum #p in K will correspond to
the invariant (E 2
- p 2 ) 1/2 . Furthermore, such energy is the minimum measurable energy by an arbitrary
observer; this can be confirmed by solving equation #E # /## = 0 from the relation (44) and by verifying
that the solution is # = p/E. This allows us to interpret E # = (E 2
-p 2 ) 1/2 as the rest energy of the given
particle. To simplify the notation and the ensuing discussions, let us introduce the variable s = (E # ) 2 ,
where E # is given by (45).
So far in our analysis, relativistic kinematics has not been modified. Nevertheless, a di#erence with
the conventional kinematical frame is that in the present theory the product (43) will not be independent
of the particle's energy; conversely, we will have the general expression
p ²
p² = -f a (E, #
p) -m 2
a , (46)
where f a (E, #
p) is a function of the energy and the momentum, that represents the LIV provided by the
LQG e#ective theories. Let us note that expression (46) is just the modified dispersion relation
E 2 = p 2 + f a (E, #p) +m 2
a . (47)
To be consistent f a (E, # p) must be invariant under Lorentz transformations and, therefore, can be written
as a scalar function of the energy and the momentum.
We have already made mention of the fact that LIV's inevitably introduce the appearance of a
privileged system; in the present discussion we will choose as such a system the isotropic system (which
by assumption is the coímoving CMBR system), and will express f a (E, #
p) in terms of E and # p measured
in that system. As may be expected in this situation, f a will be a function only of the energy E and the
momentum norm p = |#p|, since no trace of a vectorial field could be allowed when isotropy is imposed.
For example, in the particular case of the dispersion relation for a fermion, the function f a (E, # p) depends
uniquely on the momentum, and can be written as f a (p) = 2# a p 2 . For simplicity, we shall continue using
f a (p) instead of f a (E, # p).
Through the recently introduced notation and the use of expression (45), the c.m. energy of an a
particle with mass m a and deformation f a (p) will be
s 1/2
a = # f a (p) +m 2
a . (48)
Of course, the validity of this interpretation will be subordinate to those cases in which s a = f a (p)+m 2
a >
0, or, equivalent, to those states with a timeílike 4ímomentum. In the converse, particles with energies
237

and corrections such that s a = f a (p)+m 2
a # 0 will be described by lightílike physical states if the equality
holds, or spaceílike physical states if the inequality holds.
A new e#ect provided by LIV's is that, if a reference system where p = 0 exists, then in that system
the particle will not be generally at rest. To understand this it is su#cient to verify that in general the
velocity follows v = #E/#p #= p/E, and therefore does not vanish at p = 0. Returning to the equations
in (44), we can see that when # = v = #E/#p, the following result is produced
#E #
#p #
= # v # #E
#p - v # #p
#p #
= 0, (49)
where # v = (1 - v 2 ) -1/2 . Result (49) shows that the velocity of the system where the particle is at rest is
e#ectively v. There emerges, then, an important distinction between the phase velocity # = p/E of the
c.m. system of a particle, and the group velocity v = #E/#p of the same particle.
The above results, for a single particle, can be easily generalized to a system of many particles. For
instance, the total 4ímomentum of a system of many particles p ²
tot = # i p ²
i
will transform through the
rule (42), and the scalar product (p ² p ² ) tot will be an invariant under Lorentz transformations (as well as
any other product). As in the case of individual particles, we define # s as the total rest energy measured
in the system of c.m. That is to say: s = E 2
tot -p 2
tot . In the case in which we have a system composed of a
proton with energy E and momentum p, and a photon (from the CMBR) with energy # and momentum
k (all these quantities are measured in the laboratory isotropic system K), the s quantity shall acquire
the form
s = (E + #) 2
- (#p + # k) 2
= (E # + # # ) 2
- (#p # + # k) 2 , (50)
where the E # and p # quantities are measured in an arbitrary reference system. In particular we are
interested in the system where the proton momentum # p # is null; that is to say, the system in which
E # = # s p . If # is the photon energy in such a system, then
s = ( # s p + #) 2
- # 2
= 2 # s p # + s p , (51)
where we have used the dispersion relation # = k (or # = k # ) for the CMBR photons. These results will
allow us to express the main kinematical quantities in term of # and s p = f p (p) +m 2
p . For example, the
Lorentz factor that connects the K system with the K # system where p # = 0, will be
# = E/ # s p . (52)
Meanwhile, the Lorentz factor connecting K with the c.m. system (that in which # p # + # k = 0), will be
# c = E + #
s p + 2 # s p # #
E
s p + 2 # s p #
. (53)
As a last comment, let us note that to the first order in the expansion of the dispersion relations in
terms of the scales L and l p , when we consider high energy processes such that p 2
# f(p) +m 2 we can
freely interchange the momentum p by the energy E in the deviation function f(p). That is to say, we
may consider as a valid relation the following expression
E 2 = p 2 + f(E) +m 2 , (54)
where we have made the replacement f(p) # f(E). This procedure will greatly simplify the next
discussions.
5.2 Modified Inelasticity: p + # # p + x
Following the same methods of section 2, let us obtain a modified inelasticity K for a process of the type
p + # # p + x, where x is an emitted particle that, in the present physical problem in which we are
238

interested, can be a #, # or # meson. We note that the dispersion relation for the emerging proton (after
a collision with a photon) can be written in the form:
E 2
p - p 2
p = f p (E p ) +m 2
p , (55)
where E p is the final proton energy. Since the left side of (55) is invariant under Lorentz transformations,
we can write
(E #
p ) 2
- (p # p ) 2 = f p (E p ) +m 2
p , (56)
where the * denotes the quantities measured in the c.m. system. On the other hand, in such a system,
the following conservation relations of energy and momentum are satisfied:
E #
p +E #
x = # s and (p # p ) 2 = (p # x ) 2 . (57)
Substituting both quantities in relation (55), we obtain
2 # sE #
p = s + s p (E p ) - s x (E x ). (58)
In the same way, we also have the energy conservation relation in the laboratory system: E p +E x = E tot .
Using the definition for the inelasticity K x = #E/E for a process, where #E = E i -E f # E tot -E f , it
is possible to rewrite the energy conservation relation in the laboratory system in terms of K x through
expressions E x = K x E, and E p = (1 -K x )E, where E is the initial energy of the initial proton. Having
done this, equation (58) now acquires the form:
2 # sE #
p = s + s p [(1 -K x )E] - s x [K x E]. (59)
To simplify the development of the inelasticity, let us write the former relation as E # p = F (E, K x ), where
F = F (E, K x ) is defined through
F = 1
2 # s
(s + s p [(1 -K x )E] - s x [K x E]). (60)
On the other side, the Lorentz transformation rules give us the relation between the proton energies in the
laboratory system and the c.m. system respectively. This relation is E p /# c = E #
p +# c # E #2
p - s p (E p ) cos #.
Joining this expression with (60), it is possible to find the general equation for K x :
(1 -K x ) # s = # F (E, K x ) + # F 2 (E, K x ) - s p [(1 -K x )E] cos # # . (61)
It should be noted, however, that the solution for K x from (61) will depend in the # angle. For this
reason, once this last equation is resolved, it is convenient to define the total inelasticity K as the average
of K x with respect to the # angle. That is to say
K = 1
#
# #
0
K x d#. (62)
It is relevant to mention that now, as opposed to result (6), the inelasticity K will be a function of both,
the energy E of the initial proton and the energy # of the CMBR photon.
5.3 The m 2
a # s a = m 2
a + f a (E) Prescription
Let us recall our interpretation relative to the fact that s 1/2
a = (f a (E a ) + m 2
a ) 1/2 can be understood as
the rest energy of a particle a, as a function of the energy E a that it has in the laboratory system K. As
we have already emphasized, such an interpretation will be valid for particles with time like 4ímomenta.
In the reactions given between high energy protons and the photons of the CMBR, the whole scenario
consists of the collision between two particles, p and #, with the subsequent production of a certain
number of final particles. Let us suppose that a is one of these particles in the final state. The knowledge
239

of the inelasticity K for the reaction will allow us to estimate the average energy #E a # with which such a
particle emerges (since K provides the average fraction of energy with which such a particle is produced).
That is to say, on average, the rest energy of the final particle a will be s 1/2
a = [f a (#E a #) + m 2
a ] 1/2 .
Moreover, the knowledge of the inelasticity K will allow us to express s a as a function of the energy E
of the initial proton:
s a = s a (E). (63)
Following our previous interpretation, we can view the recently described process as a reaction between a
proton with mass s 1/2
p , which loses energy emitting particles a with mass s 1/2
a calculated in the previous
form. This idealized reasoning gives us a clear prescription to kinematically modify those dynamical
quantities with which we must work and where energy conservation is involved. This prescription is:
m 2
a # s a (E) = f a (E) +m 2
a , (64)
where we have expressed correction f a as a function of the initial energy of the incident proton.
Prescription (64) establishes the notion of an e#ective mass which is dependent on the initial energetic
content of a reaction. As a consequence, given the explicit knowledge of the dependence that a cross
section has on the masses and energies of the involved states, to obtain the modified version, it will be
appropriate to use the discussed prescription.
5.4 Redshift
Another important problem related to the introduction of LIV's in the dispersion relations is whether the
redshift relation for the propagation of particles in a FRW universe is modified. This could be of great
relevance because of the large distances involved in cosmic ray propagation and, therefore, the possible
cumulative e#ects. In the case of a FRW universe with scale factor R, it is possible to find that the
propagation of particles with momentum p 2 = g ij p i p j (with g ij the spatial part of the FRW metric)
obeys the following equation
d
d#
(pR) = 0, (65)
where # is a proper parameter describing the path followed by the particle. This result is regardless of
any particular dispersion relation.
5.5 Spectrum and Results
Introducing the above modifications to the di#erent quantities involved in the propagation of protons (like
the cross section # and inelasticity K), we are able to find a modified version for the UHECR energy loss
due to collisions. Since the only relevant correction for the GZK anomaly is #, we focused our analysis
on the particular case f(p) = 2#p 2 . To simplify our model we restricted our treatment to the case # > 0
(consistent with the e#ective mass interpretation) and used only #m #= 0, where #m is assumed to have
the same value for mesons #, # and #.
Fig.2 shows the modified energy loss #(E) for UHECR obtained for di#erent values of #m . It can
be seen therefore how the corrections can a#ect the main life time of protons propagating through the
CMBR, allowing a strong improvement in the distances that protons can reach before loosing their
characteristic energy (for energies greater than 1 ½ 10 20 eV). The e#ects that the LQG corrections have
on the propagation of UHECR are manifest through a decay of the energy loss in the range E # 1 ½ 10 20
eV. To understand this, recall relation (36) for the threshold condition of photoípion production:
2 ##E 2
# + 4E # # # m 2
# (2m p +m # )
m p +m #
. (66)
As we saw in section 4, the condition for a significant increase or decrease in the energy threshold can be
calculated as |##| # (2m p +m # )(m p + m # )/2E 2 . Therefore, for a given value of ## > 0, the energy at
which the LIV e#ects start to take place is
E 2 = 1
2## (2m p +m # )(m p +m # ). (67)
240

10 18 10 19 10 20 10 21
Energy E [ eV ]
10 -18
10 -17
10 -16
10 -15
1/E
dE
[
d
sec
í1
t
]
/
1
2
3
í
Figure 2: Modified energy loss for UHECR due to collisions. The figure shows the case #m #= 0, for
three di#erent values of the weave scale L. Curve 1: #m = 9 ½ 10 -23 (L # 8.6 ½ 10 -18 eV -1 ); curve 2:
#m = 5 ½ 10 -23 (L # 1.2 ½ 10 -17 eV -1 ); curve 3: #m = 0 (without modifications).
In the case #m = 9 ½ 10 -23 (curve 1 of Fig.2), this energy is E = 1.1 ½ 10 20 eV, while in the case
#m = 5 ½ 10 -23 (curve 2) this corresponds to E = 1.5 ½ 10 20 eV. Beyond these energy scales, at about
E # 2 ½ 10 20 eV, a sharp decay is observed in the behavior of the curve. This is due to the fact that the
modified inelasticity K will strongly constraint the energyímomentum phase space accessible to the final
states depending on the initial energy E that the primary proton carries (recall that now K is a function
of the energy E of the incident proton and the energy # of the CMBR photon).
Also, we can find the modified version of the UHECR spectrum for #m #= 0. Fig.3 shows the AGASA
observations and the predicted UHECR spectrum. The Poisson probabilities of an excess in the five
highest energy bins for the three curves are P 1 = 3.6 ½ 10 -4 , P 2 = 2.6 ½ 10 -4 and P 3 = 2.3 ½ 10 -4 .
The Poisson # 2 for the eight highest energy bins are # 2
1 = 10, # 2
2 = 10.9 and # 2
3 = 11.2 respectively.
The possibility of reconciling the data with finite maximum generation energies is significant given that
conventional models require infinite maximum generation energies Emax for the best fit. For the lower
part of the spectrum (under E = 4 ½ 10 19 eV), the parameters under consideration leave the spectrum
completely una#ected. This is due to the fact that in such a region the dominant reaction is the pair
production, which has not being modified to obtain the spectrum. A more accurate study on this issue
would require the computation of a modified inelasticity for the pair creation. Meanwhile, we must
content ourself with the semiqualitative criteria given in section 4 to rule out the parameters.
6 Conclusions
We have seen how the kinematical analysis of the di#erent reaction taking place in the propagation of
ultra high energy protons can set strong bounds on the parameters to the theory. In comparison with our
previous work, we have eliminated some previously open possibilities by the particular study of the pair
creation p+# # p+e + +e - , in the energy region where this reaction dominates the proton's interactions
with the CMBR. In this way, the only possibility still open is the correction #. If this is the case, a
favored region for the scale length L estimated through the threshold analysis would be
2.6 ½ 10 -18 eV -1 # L # 1.6 ½ 10 -17 eV -1 .
Similarly, the kinematical corrections can be studied in more detail when their e#ects are considered in the
theoretical spectrum. In this regard, we have seen how to develop a modified version of the inelasticity for
the photoípion production, and its implications in the mean life time of a high energy proton as well as on
the spectrum. To accomplish this last task we have only assumed a spontaneous Lorentz symmetry break
up in the e#ective equations of motion, allowing the use of Lorentz transformations on the dispersion
241

10 18 10 19 10 20 10 21
Energy E [ eV ]
10 23
10 24
10 25
J(E)
E
3
[m
í2
sec
í1
eV
2
]
1 2 3
Figure 3: Modified UHECR spectrum and AGASA observations. The figure shows the modified spectrum
J(E) multiplied by E 3 , for uniform distributed sources and without evolution, for the case #m = 1.5 ½
10 -22 (L # 6.7 ½ 10 -18 eV -1 ). Three di#erent maximum generation energies Emax are shown. These
are, curve 1: 5 ½ 10 20 eV; curve 2: 1 ½ 10 21 eV; and curve 3: 3 ½ 10 21 eV.
relations. Therefore, result (61) can be used in a more general context than the special case o#ered by
the LQG framework.
Future experimental developments like the Auger array, the Extreme Universe Space Observatory
(EUSO) and Orbiting WideíAngle Light Collectors (OWL) satellite detectors, will increase the precision
and phenomenological description of UHECR. On the more theoretical side, progress in the direction
of a full e#ective theory, with a systematic method to compute any correction with a known value for
each coe#cient, is one of the next steps in the ``loop'' quantization programme [25]. Therefore, it is
important to trace a phenomenological understanding of the possible e#ects that could arise as well as
the constraints on LQG, in the high and low energy regimens (for other phenomenological studies of LQG
e#ects, see for example [26]).
Recently[27] the e#ective theory approach have been under criticism.
In [28], the type of terms we used to explain AGASA data are computed within the Standard model,
depending on one arbitrary parameter. The criticism of [27] does not apply to this formalism.
Acknowledgements
The work of JA is partially supported by Fondecyt 1010967. He acknowledges the hospitality of LPTENS
(Paris); and financial support from an Ecos(France)íConicyt(Chile) project. The work of GAP is partially
supported by DAMTP and MIDEPLAN (Chile). J.A. wants to thank A.A. Andrianov and the organizers
of the XVIIIth QFTHEP Workshop at SaintíPetersburg for their hospitality.
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