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\begin{document}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\mbox{ } \\[-1cm]
\mbox{ }\hfill TUM--HEP--488/02\\%[-1mm] 
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\mbox{ }\hfill \today\\%[-1mm]

\begin{center}
  {\Large\bf Ultra-high energy cosmic rays from super-heavy X
    particle decay} \\[8mm]
           
Cyrille Barbot \\[4mm]

{\it Physik Dept., TU M\"unchen, James Franck Str., D--85748
 Garching, Germany} \\[1mm]
\end{center}

\bigskip
\bigskip 
\bigskip 

%
\begin{abstract}

\vskip 0.5cm

\noindent
%PACS number(s): 11.30.Pb, 11.30.Er

In this talk, I present the last and more precise results obtained in
the computation of the final spectra of stable particles issued from
the decay of super-heavy X particles ($M_X \sim 10^{21}$ to $10^{25}$
eV). Such very energetic decay products, carrying a fraction of the
mass of the X particle, are believed to be a plausible explanation for
the observed ultra-high energy cosmic rays (UHECR). Combining these
results with X-particle models and with a code describing the
propagation effects for UHECRs through the interstellar medium, it
becomes possible to make some predictions on the fluxes expected on
Earth, hopefully detectable in the next generation of experiments.

\end{abstract}
%

\setcounter{footnote}{0}

In the second part of the 20th century, the spectrum of cosmic rays
(CRs) has been mesured over more than 12 decades of energy. Even if
our understanding of it has grown a lot in the last few decades, many
enigma are remaining. One of them concerns the extremity of this
spectrum, at the highest energies, where theorists were expecting a
strong cut-off to occur at energies of the order of $5.10^{19}$ eV :
indeed, at these energies, CRs should be of extragalactic origin, and
probably coming from distances further than the local cluster of
galaxies, because we know no astrophysical object able to accelerate
particles enough to give them this energy in our vicinity. But the
point is that particles carrying energies above $10^{20}$ eV traveling
over cosmological distances should loose their energy through propagation
effects ; for example, a proton will interact with the cosmological
microwave background (CMB) and photoproduce pions, with an interaction
length of a few tens of Mpc, loosing around 20 \% of its energy at
each interaction. Similar processes occur with nuclei, photons or
electrons. Thus particles with initial energy $\sim 10^{20}$ eV should reach
the Earth with a maximal energy $\sim 5.10^{19}$ eV, the so-called GZK
cut-off \cite{GZK:1,GZK:2}.\footnote{A notable exception are, of
  course, the neutrinos, which can travel over cosmological distances
  without loosing their energy. But the events observed on Earth
  cannot be attributed to primary neutrinos.} The fact is that events
have been registered above this cut-off in very different experiments
over the last few decades \cite{HaverahPark,FlysEye,HIRES,AGASA}. Such
an observation is almost impossible to reconcile with any model of
acceleration of charged particles in any astrophysical object.
Moreover, there is another strong indication against these models :
UHECRs are expected to travel rather straight away
in the universe, without being deviated by the (inter)galactic
magnetic fields. Thus they should point to there sources within a few
degrees. Yet, excepted the existence of a few doublets and triplets in
the experimental data, the observations are compatible with an almost
perfect isotropy \cite{Anisotropy}\footnote{Nevertheless, it should be
  noted that there are still attempts to explain the UHECRs with these
  classical ``bottom-up'' theories, see for example
  \cite{WaxmanGRB,Biermann}.}.

These remarks lead to the development of another class of models for
explaining the existence of UHECRs, namely the ``top-down'' theories,
which are considering that the observed events could be generated
through the decay of some mysterious super-heavy ``X'' particles. The
existence of such X particles is predicted in number of GUT theories
or in relation with topological defects collapsing or annihilating,
and they can be created rather naturally at the end of the inflation
\cite{Allahverdi:1,Allahverdi:2}.  Among other more ``model
dependant'' properties, top-down models require that these particles
should have a mass bigger than the highest energies observed in UHECR
events, $M_X > 10^{21}$ eV, and a lifetime of the order of (or greater
than) the age of the universe.  They could be trapped homogeneously in
the galaxies\footnote{or possibly in some topological defects
  protecting them against decay, with distributions very different
  from matter distributions in the galaxies.}, explaining the isotropy
of the data, and would constitute semi-local sources for UHECRs,
avoiding the GZK problem.  Moreover, if they are abundant enough and
trapped in the galaxies, they could be a very good candidate for the
dark matter problem \cite{SHDM:1}.


Excellent reviews on the UHECRs can be found in the litterature
\cite{reviewSigl,reviewSarkar,reviewStecker}. In this talk I will
focus on the top-down theories, and give general results for the decay
of ultra-heavy particles (first presented in \cite{BarbotDrees:1}),
independantly of any particular model.\newline


\setcounter{footnote}{0}

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\Line(-370,1)(-340,26)
\Line(-370,0)(-340,25) \Text(-360,25)[]{$\tilde{g}$} % gluino
\Line(-340,26)(-310,46)
\Line(-340,25)(-310,45) \Text(-330,45)[]{$\tilde{g}$}% gluino
\Gluon(-340,25)(-310,10){-3}{4}\Text(-320,25)[]{$g$} % gluon
\DashLine(-310,45)(-280,55){3}  \Text(-295,60)[]{$\tilde{q}_L$} %squark
\Line(-310,45)(-280,35) \Text(-295,30)[]{$q_L$} % quark
\DashLine(-280,-120)(-280,120){4} \Text(-280,-130)[c]{1 TeV}% 1 TeV
\Text(-280,-145)[c]{(SUSY}
\Text(-280,-160)[c]{+ $SU(2)\otimes U(1)$}
\Text(-280,-175)[c] {breaking)}

\Line(-280,55)(-250,80) \Text(-265,80)[]{$q$} % quark
\Line(-250,80)(-190,70) \Text(-220,68)[]{$q$} % quark
\Gluon(-250,80)(-220,90){3}{4} \Text(-235,97)[]{$g$} % gluon
\Line(-220,90)(-190,100) \Text(-205,103)[]{$q$} % quark
\Line(-220,90)(-190,80) \Text(-205,80)[]{$q$} % quark

\Line(-280,56)(-250,46)
\Line(-280,55)(-250,45) \Text(-265,42)[]{$\tilde{\chi}_2^0$}%neutralino 2 
\Line(-250,45)(-220,55) \Text(-235,57)[]{$q_L$}% quark
\Line(-250,46)(-170,36)
\Line(-250,45)(-170,35) \Text(-166,35)[l]{$\tilde{\chi}_1^0$}% neutralino 1 
\Line(-250,45)(-220,25) \Text(-235,27)[]{$q$} % quark
\Line(-220,55)(-190,65) % \Text(-205,65)[]{$q_L$}% quark
\Gluon(-220,55)(-190,45){-3}{4} %gluon
\DashLine(-190,-120)(-190,120){4} \Text(-190,-130)[c]{1 GeV}% 1 GeV
\Text(-190,-145)[c]{(hadronization)}

\Line(-370,0)(-340,-25) \Text(-360,-18)[]{$q_L$} % quark
\Photon(-340,-25)(-310,-5){3}{5} \Text(-330,-5)[]{$B$} % B
\DashLine(-310,-5)(-280,5){3} \Text(-295,8)[]{$\tilde{q}_R$}% squark
\DashLine(-310,-5)(-280,-15){3} \Text(-295,-18)[]{$\tilde{q}_R$}% squark
\Line(-340,-25)(-310,-45) \Text(-330,-40)[]{$q$} % quark
\Photon(-310,-45)(-280,-25){3}{5} \Text(-308,-31)[]{$W$} % W
\Line(-280,-25)(-190,-5) \Text(-240,-8)[]{$\tau$} %tau
\Line(-190,-5)(-160,5) \Text(-175,9)[]{$a_1^{-}$}% a1-
\Line(-160,5)(-130,15) \Text(-145,17)[]{$\rho^{-}$}% rho-
\Line(-130,15)(-100,15) \Text(-115,11)[]{$\pi^{-}$}% pi-
\Line(-100,15)(-70,15) \Text(-66,15)[l]{$\nu_\mu$} %nu_mu
\Line(-100,15)(-70,5) \Text(-78,2)[]{$\mu^{-}$} %mu
\Line(-70,5)(-50,5) \Text(-46,5)[l]{$\nu_\mu$} %nu_mu
\Line(-70,5)(-50,-5) \Text(-46,-7)[l]{$\nu_e$} %nu_e
\Line(-70,5)(-50,-15) \Text(-46,-16)[l]{$e^{-}$} %e


\Line(-130,15)(-100,35) \Text(-115,35)[]{$\pi^0$} % pi0
\Photon(-100,35)(-70,35){3}{5} \Text(-66,35)[]{$\gamma$} % gamma
\Photon(-100,35)(-70,55){3}{5} \Text(-66,57)[]{$\gamma$} % gamma

\Line(-160,5)(-130,-5) \Text(-145,-8)[]{$\pi^0$}% pi0
\Photon(-130,-5)(-100,-25){3}{5} \Text(-96,-27)[]{$\gamma$} % gamma
\Photon(-130,-5)(-100,-5){3}{5} \Text(-96,-3)[]{$\gamma$} % gamma
\Line(-190,-5)(-160,-15) \Text(-175,-16)[]{$\nu_\tau$} % nu_tau
\Line(-280,-25)(-100,-45) \Text(-96,-45)[l]{$\nu_\tau$} %nu_tau

\Line(-310,-45)(-280,-65) \Text(-300,-62)[]{$q$} % quark
\Line(-280,-65)(-250,-55) \Text(-265,-53)[]{$q$} % quark
\Gluon(-280,-65)(-250,-75){-3}{4} \Text(-265,-80)[]{$g$}% gluon
\Gluon(-250,-75)(-220,-95){-3}{4} \Text(-240,-95)[]{$g$} % gluon
\Line(-220,-95)(-190,-105) \Text(-205,-105)[]{$q$} % quark
\Line(-220,-95)(-190,-85) \Text(-205,-85)[]{$q$}% quark

\Gluon(-250,-75)(-220,-65){3}{4} \Text(-240,-62)[]{$g$} % gluon
\Line(-220,-65)(-190,-55)  \Text(-205,-55)[]{$q$}% quark
\Line(-220,-65)(-190,-75)  \Text(-205,-75)[]{$q$}% quark

\GOval(-190,65)(20,7)(0){0} % neutron
\Text(-175,65)[r]{$n$}
\Line(-170,65)(-130,80) \Text(-126,82)[l]{$p$}% proton
\Line(-170,65)(-130,65) \Text(-126,65)[l]{$e^{-}$}% electron
\Line(-170,65)(-130,50) \Text(-126,48)[l]{$\nu_e$} % nu_e

\GOval(-190,-80)(15,7)(0){0} % pi_0
\Text(-170,-80)[r]{$\pi^0$}
\Photon(-168,-83)(-130,-65){3}{5} \Text(-126,-63)[]{$\gamma$} % gamma
\Photon(-168,-83)(-130,-95){3}{5} \Text(-126,-98)[]{$\gamma$} % gamma
\end{picture} 
\caption{Schematic MSSM cascade for an initial squark with
  a virtuality $Q = M_X$. The full circles indicate decays of massive
  particles, in distinction to fragmentation vertices. See the text
  for further details.}
\end{center}
\label{Cascade}
\end{figure}


We first briefly describe the physical steps involved in the decay
cascade of an ultra-heavy X particle in the framework of the MSSM, as
they are illustrated on fig~\ref{cascade}.  Our basic assumption is
that the X particle decays in N very virtual particles of the
MSSM\footnote{The existence of an energy scale as high as $M_X$
  strongly suggests the existence of superparticles with masses not
  much above 1 TeV, in order to guarantee the perturbative stability
  of the hierarchy between $M_X$ and the weak scale. We therefore
  usually allow superparticles as well as ordinary particles to be
  produced in $X$ decays, as described by the minimal supersymmetric
  extension of the Standard Model (MSSM).}, each of them initiating a
decay cascade, following the known physics at lower energy. At high
virtuality, in the region of asymptotic freedom, each of the primaries
will initiate a perturbative shower, splitting into two allowed
particles of smaller virtuality, according to the Feynman laws
\footnote{In contrast to previous works
  \cite{Berezinsky:2000,Coriano:2001,Sarkar:2001,ToldraLSP},
  we considered in our treatment all gauge interactions as well as
  third generation Yukawa interactions, rather than only SUSY--QCD;
  note that at energies above $10^{20}$ eV all gauge interactions are
  of comparable strength. The inclusion of electroweak gauge
  interactions in the shower gives rise to a significant flux of very
  energetic photons and leptons, which had not been identified in
  earlier studies.}. These products will split at their turn too, and
the process will continue until the virtuality has decreased enough,
at a scale where both SUSY and $SU(2) \otimes U(1)$ will break (for
simplicity we are considering a unique SUSY mass scale $M_{SUSY} \sim
1$ TeV for all sparticles). $M_{SUSY}$ is symbolized by the first
vertical dash-line in fig 1. All the on-shell massive
sparticles produced at this stage will then decay into Standard Model
(SM) particles and the only (eventually) stable sparticle, the
so-called Lightest Supersymmetric Particle (LSP). The heavy SM
particles, like the top quarks and the massive bosons, will decay too,
but the lighter quarks and gluons will continue a perturbative
partonic shower until they have reached either their on-shell mass
scale or the typical scale of hadronization (say 1 GeV), the second
vertical dash-line of fig 1.  At this stage, the color
effects become too strong and the partons cannot propagate freely
anymore, being forced to combine into colorless hadronic states.
Finally, the unstable hadrons will also decay, and only the stable
particles will remain and propagate in the intergalactic space, namely
the protons, photons, electrons, the three species of neutrinos and
the LSP (and their antiparticles).

\begin{figure}[h!]
\setlength{\unitlength}{0.3 cm}
\begin{minipage}[t]{4.5cm}
\input{Low_G_uL.tex}
\end{minipage}

\noindent
\begin{minipage}[b]{4.5cm}
\input{Low_G_eL_.tex}
\end{minipage} \hfill
\caption{$x^3 \times$ Fragmentation functions of a first generation SU(2) doublet quark (top) and a slepton (bottom) into stable particles. }
\label{FFs}
\end{figure}


The perturbative part of the shower is treated through the numerical
resolution of DGLAP evolution equations \cite{AP} extended to the
complete spectrum of the MSSM. These equations describe the evolution
of the so-called ``fragmentation functions'' (FFs), which are
describing the fragmentation of any fondamental particle into any
other ; the DGLAP evolution equations describe more specifically the
impact of all 3-legs MSSM Feynman diagrams on these FFs, and
their evolution with energy through the running of the associated
coupling constants. We worked out all the FFs of the MSSM by solving
these equations for all the unbroken fields between $M_{SUSY}$ and
$M_X$. At the breaking scale $M_{SUSY}$ we applied the canonical
unitary transformation to the FFs of the unbroken fields in order to
obtain those of the broken ones, and computed the decay cascade of the
supersymmetric part of the spectrum. We used here the results of the
public code Isasusy \cite{Isasusy} to describe the allowed decays and
their branching ratios, for a given set of SUSY parameters. If
R-parity is conserved, we obtain the final spectrum of the stable LSP
at this step, and the rest of the available energy is distributed
between the SM particles. After a longer perturbative cascade down to
$Q \sim \max(m_{quark},1 GeV)$, as stated before, the quarks and
gluons will hadronize. We used the results of \cite{Poetter} as input
functions for describing the hadronization and convoluted them with
our previous results for the FFs of quarks and gluons (according to
the factorization theorem of QCD, see for example \cite{QCDrev}).
During the complete cascade, we paid a special attention to the
conservation of energy (what was not doable in previous studies,
because of the incomplete treatment of the cascade). We are able to
follow the energy conservation on the complete evolution up to a few
per thousand.\newline

\begin{figure}[t!]
\setlength{\unitlength}{.3 cm}
\begin{minipage}[t]{4.5cm}
\input{Low_G_ratio_uL.tex}
\end{minipage}

\noindent
\begin{minipage}[b]{4.5cm}
\input{Low_G_ratio_eL_.tex}
\end{minipage} \hfill
\caption{Ratios of FFs $D_I^h/D_I^p$ for different stable particles $h$,
for an initial first or second generation $SU(2)$ doublet quark, $I =
q_L$, (top) or slepton, $I = \tilde e_L$, (bottom).} 
\label{ratio}
\end{figure}

As a result of the code we have written, we can obtain the spectrum of
any stable particle (protons, photons, electrons, neutrinos,
LSPs)\footnote{We summed over particles and antiparticles.} at the end
of the cascade, for {\it any} given N-body decay mode of the X
particle into particles of the MSSM\footnote{To describe any N body
  decay from the FFs of the decay products of X, we just need one more
  convolution between these FFs and the phase space of the decay.}. We
choose to show here the results obtained for two initial SU(2)
doublets of the first generation, a quark and a slepton.
Fig~\ref{FFs} shows the FFs themselves and fig~\ref{ratio} the ratios
of the different FFs on the proton flux.  Among the new general
features appearing on these spectra, I would like to describe here a
few points of particular interest :
\begin{itemize}
\item[1.] The spectra at small and large x are quite different ;
  especially, as can be seen on fig~\ref{ratio}, the small x physics
  is to a strong extent independant on the nature of the primaries,
  and the ratios between the different stable particles always remain
  the same. It allows us to order the fluxes from the strongest to the
  lowest one in the small x region : $\nu_\mu$, $\gamma$, $\nu_e$,
  electrons, and protons, within one order of magnitude; the two
  smallest fluxes, LSP and finally $\nu_\tau$, can be one order of
  magnitude lower than the proton flux at small x. We see that at
  large x, the fluxes are much more model dependent and change with
  the nature of the X decay products, but that LSP and $\gamma$ are
  generally the strongest ones.
\item[2.]  Due to the strength of the electroweak couplings at very
  high energy, the photon and neutrino spectra are even stronger than
  what was expected before ; these primaries have to be added to the
  flux of secondaries expected from propagation effects (especially
  through decay of pions after pion photo-production over the CMB). It
  is a puzzling result, because there are already strong indications
  that no photon event has been observed in the extremity of the CR
  spectrum (see for example \cite{noGamma:1}).\newline
\end{itemize}


In this talk, I have shown the type of results that can be obtained
with the code ``SHdecay'', which allows to compute the final spectra
of stable particles for any N-body decay mode of an initial
super-heavy X particle, in the framework of the MSSM. This code will
be soon made available. Combined with a model describing the nature,
the cosmic distribution, and the decaying properties of an X particle,
on one hand, and with a propagation code describing the losses of
energy of the stable particles traveling through the interstellar medium,
on the other hand, it allows to study many particular scenarios and to
make quantitative predictions for the fluxes to be observed on Earth
for each of these scenarios. We already developped a partial approach
of these issues, in collaboration with F. Halzen and D. Hooper, for
computing the expected fluxes of neutrinos \cite{bdhh:1} and
neutralinos \cite{bdhh:2} on Earth for different models, in the next
generation of experiments.


%--------------------------- References -------------------------------

\bibliographystyle{utphys}
\bibliography{references}

\clearpage
\noindent

\end{document}

