\section{Introduction}

The Comic Rays spectrum at high energies\cite{PDG10,Gaisser} become quite elaborated
these days. In particular, the so-called "knee", slightly above $10^{15} eV$,
and the "ankle", slightly above $10^{18.5}eV$, appear rather convincingly.
To develop the field further, it is important to understand how these phenomena
occur, especially in our Universe.

In this note, we are interested in cosmic rays in the energy range greater than
$10^{14} eV$ but less than $10^{26} eV$, including those greater than $10^{20} eV$
(those unexplored regimes), the so-called "ultra high energy cosmic rays (UHECR's)".
In particular, we examine the interaction of UHECR with the $3^\circ$ cosmic microwave
background (CMB), one cosmic media in our Universe. As seen below, the center of
mass (CM) energy squared would be $1.3 \times 10^{19} eV^2$ for an UHECR of energy
$10^{22}\,eV$ intersecting or interacting a CMB photon.
So, the CM energy would be $1.15 \times 10^{11}\,eV$, or 115 GeV, if an UHECR of
energy $10^{25}\,eV$ intersects a CMB photon. It is slightly above the mass of the $Z^0$
weak boson. So, the energy range which we talk about coincides the range which the
Standard Model\cite{PDG10} is well tested - so, we shouldn't anticipate any new
physics but only a replay of the Standard Model physics in a very peculiar kinematic
setup. Isn't it?

What is an UHECR particle? The possibility may include the protons,
neutrons (time-dilated), deuterons, alphas, the heavy nuclei, neutrinos,
and others. For some reason they could be produced or accelerated to these energies.
In the bottom-up scenario, heavy nuclei from astronomical events, maybe severer than
supernovae explosions and happening (approaching) in our direction with high relative
velocities, may provide UHECR's of greater than $10^{20}\,eV$; chunks
of nuclei or protons would be the origins of those extremely high energy particles
(say, $\ge 10^{22}\, eV$).

The muon, if produced at $10^{22}\,eV$, would have a time-dilation
factor $10^{22}/10^8$, or $10^{14}$; the lifetime would be $2\times 10^{-6}
\times 10^{14} sec$, or $2\times 10^8 sec$ (about 7 years). So, 7 light years
(a muon produced and captured 7 light years away) are still too short in our
astronomical environments. Others such as pions, kaons, etc. have lifetimes
even much shorter and do not play a role here\cite{PDG10}. On the other hand,
a neutron of $10^{22}\,eV$, of which the lifetime at rest is $15\,min$, would
have a dilated lifetime $1,000\, sec \times 10^{22}/10^9$ or $10^{16}\,sec$ or
$3.17\times 10^8$ years. So, neutrons of $10^{22} eV$ would be fairly stable and
could come from 317 Mega light years away or 100 Mpc away.

In this context, we know that the electrons, positrons, photons, etc.,
could not survive beyond certain (high) energies, because of the electromagnetic
interactions. For example, $e^\pm + \gamma_{CMB} \to e^\pm + \gamma$,
$\gamma+\gamma_{CMB} \to e^- + e^+$, etc.

In our Universe, there are plenty of $3^\circ K$ cosmic microwave background (CMB) and
$1.9^\circ K$ cosmic neutrino background (C$\nu$B). Even though the energies of
these particles sound extremely low, the ultra high energy cosmic rays (UHECR), including
protons, in fact can see them if energy is high enough. In particular, the following
reactions don't have the thresholds:

\begin{equation}
e^\pm + \gamma_{CMB} \to e^\pm + \gamma,\qquad \mu^\pm + \gamma_{CMB} \to \mu^\pm
+ \gamma,
\end{equation}

\begin{equation}
p + \gamma_{CMB} \to p + \gamma,
\end{equation}

\begin{equation}
\gamma + \gamma_{CMB} \to \gamma + \gamma,
\end{equation}

\begin{equation}
\alpha + \gamma_{CMB} \to \alpha + \gamma, ....
\end{equation}

On the other hand, the reactions listed below have some thresholds and
would start to play some important roles, when UHECR's energy reaches
at the threshold:

\begin{equation}
\gamma + {\gamma}_{CMB} \to e^- + e^+,
\end{equation}

\begin{equation}
p + \gamma_{CMB} \to p + \{ e^- e^+\},
\end{equation}

\begin{equation}
d + \gamma_{CMB} \to p + n,
\end{equation}

\begin{equation}
^3He + \gamma_{CMB} \to d + p,
\end{equation}

\begin{equation}
\alpha + \gamma_{CMB} \to ^3H + p, ... etc.,
\end{equation}
plus some others. Hereafter we assume that cosmic rays, depending on the energy, would
be composition of all "stable" particles, including protons, deuterons, $e^\pm$,
$\mu^\pm$, $\gamma$, $\nu$, etc. As said earlier, $\mu^\pm$ may be the borderline of
"stable particles" when we consider the effects due to time dilation; the neutrons,
with the lifetime ($\approx$ 15 min) much longer, could be "stable" if the energy is
greater than, e.g. $10^{22}eV$. High energy $\nu$ (those higher than tens of GeV) are
presumably there but, in view of the feeble nature of weak interactions, cannot be
seen so far. These are well-known particles; thus, in this note we don't speculate
on those unknown particles, such as super-symmetric particles.

Here we assume that in the UHECR there contain $\gamma$, $p$, $d$, etc., with
certain probabilities - our definition of UHECR\cite{Gaisser}. The UHECR's
interact with the CMB photons via the above reactions and etc. This note
emphasizes the interactions of UHECR's with CMB, and emphasizes CMB's visibility
by UHECR's.

In a related context\cite{Ma}, we discussed the interplay of the cosmic neutrino background
(C$\nu$B) and UHECR's, assuming some clustering of C$\nu$B. The significant clustering of
C$\nu$B would lead to the {\it first} detection of the C$\nu$B. So far, the detection of
C$\nu$B is still in the air. In this note, we try to provide some semi-quantitative
treatments of the interplay of CMB and UHECR.

\section{Initial Thinking}

Let us quote the Compton-scattering formula\cite{Wu}:

\begin{equation}
\sigma(p+\gamma \to p +\gamma) \to {2\pi\alpha^2\over s} ln({s\over m^2}),
\quad as \quad t \to \infty.
\end{equation}
Now the initial photon is the CMB photon. Suppose that the energy of the UHECR
proton is $10^{22}$ eV and so the CM energy is $s\approx 2\times 10^{22} eV\cdot
6.5 \times 10^{-4}eV \approx 1.3 \times 10^{19} eV^2$. So, we obtain
\begin{equation}
\sigma \approx 10^{-23} eV^{-2} \approx 4 pb \to \lambda_p \approx 2\times 10^5\, Mpc,
\end{equation}
which already exceeds the size of the present Universe (about 4500 Mpc).

On the other hand, $E_p=10^{20}eV$ would imply the mean free path $\lambda\approx
2000\,Mpc$ and $E_p = 10^{18} eV$ implies $\lambda_p \approx 20 Mpc$.

Now we turn our attention to the similar formula if the UHECR is the electron
(positron). For the electron energy $E_e=10^{15}\,eV$, we have $s\approx 2\times
10^{15} eV \cdot 6.5\times 10^{-4}\,eV \approx 1.3 \times 10^{12} eV^2$ so that
$\sigma\approx 3.35 \times 10^{-4}\cdot s^{-1} ln (s/m_e^2) \approx 4.14 \times
10^{-16}\,eV^{-2}$, the cross section corresponding to $0.17\, barns$, or to
$\lambda_e \approx 50 kpc$. If nothing else happens, then the electron would be
deflected in $50\,kpc$.

For the muon of energy $10^{19}\,eV$ or $1.3 \times 10^{16} eV^2$, we have
$\sigma\approx 3.35 \times 10^{-4}\cdot s^{-1} ln (s/m_\mu^2)$ or $\sigma\approx
0.7176 \times 10^{-23}\,eV^{-2}$. For such muon, the (dilated) lifetime becomes
$2\times 10^{-6} \times 10^{19-8} sec$ or $2\times 10^5 sec$. Combining the two,
a muon of energy $10^{24}eV$ would last $2\times 10^{10} sec$ and gets negligible
effect from bremsstrahlung.

For the reaction $\gamma + \gamma_{CMB} \to \gamma + \gamma$, it comes
from the box diagrams and is of higher order, $O((\alpha/2\pi)^4)$ (and so is
small). Thus, we needn't consider it for the moment. For $\alpha + \gamma_{CMB} \to
\alpha + \gamma$ or off other nuclei, the situation is similar to that for
$p+ \gamma_{CMB} \to p+\gamma$, as described as above.

\section{UHECR physics near $10^{20}\, eV$}

Next, we consider those inelastic reactions which might leave their marks
on the UHECR physics, say, Reactions (5)-(9), etc. Reliable estimates can be
obtained by working out the peculiar kinematics and using the well-known cross
sections.

Let us consider, for example, Reaction (6), i.e. $p+ \gamma_{CMB} \to p+
(e^-e^+)$, with $(e^-e^+)$ characterized a composite mass $\bar m$. The
four momentum conservation reads
\begin{equation}
p'_\mu + k'_\mu = p_\mu + k_\mu.
\end{equation}
Or, we have
\begin{equation}
p'_\| + k'_\| = p-k,\\
p'_\bot + k'_\bot = 0,\\
p'+ m_p^2/(2p') + k' + {\bar m}^2/(2k')= p + m_p^2/(2 p) +k.
\end{equation}
We find
\begin{equation}
p'=(4k + m_p^2/p)^{-1} \{ 2p\cdot k + m_p^2- {\bar m}^2/2 \pm [(2p\cdot k)^2
- {\bar m}^2 2p\cdot k + {\bar m}^4/4 -{\bar m}^2 m_p^2]^{1\over 2}\}.
\end{equation}

\begin{equation}
k'=(4k + m_p^2/p)^{-1} \{ 2p\cdot k + m_p^2- {\bar m}^2/2 \mp [(2p\cdot k)^2
- {\bar m}^2 2p\cdot k + {\bar m}^4/4 -{\bar m}^2 m_p^2]^{1\over 2}\}.
\end{equation}

To get some ideas, we have $\sigma(p+\gamma_{CMB}\to p+\gamma) \to {2\pi\alpha^2
\over S} ln ({S\over m^2})$ as $S\to \infty$. As quoted earlier (as our benchmark), 
at $E\approx 10^{22}eV$, one has $\sigma\approx 10^{-23}eV^{-2}\approx 4 pb$ and, 
with the density of CMB photons, we find a mean free
path $\lambda\approx 2\times 10^5 Mpc$, bigger than the Universe size of 4,000 Mpc.

However, the $1\over S$ behavior indicates that at $E\approx 10^{18} eV$ we have
$\sigma \approx 0.4 nb$ or $\lambda \approx 20 Mpc$, a noticeable result. Comparing
the process $p + \gamma_{CMB} \to p + (e^-e^+)$ to $p+ \gamma_{CMB} \to p+\gamma$,
we lose a factor of $\alpha$. This means that at $10^{18} eV$ this effect is barely
visible. {\it This would be a marginal explanation of the ankle effect!!}

Fortunately, there are other reactions, such as $d+ \gamma_{CMB} \to p+ n$
(Reaction (7)) or similar, with the thresholds in the range of a couple of MeV.
The deuteron component in the UHECR flux might be small but the cross section is
much bigger - it serves as an additional reason for the "ankle".

Our explanation of the "ankle" makes some sense. In general,
the electromagnetic effects out of CMB photons, or of higher order, would
make marks in the UHECR physics. On the other hand, the weak reactions, of
cross section $10^{-42} cm^2$ ($= 10^{-6} pb$), are mostly invisible.


Now let us return to Reaction (5), i.e. $\gamma + \gamma_{CMB}\to e^-+ e^+$.
We have
\begin{equation}
S \approx 4 E \cdot E_{CMB} \ge 4 m_e^2; \quad or \quad E_\gamma \ge 4.1 \times
10^{14} eV.
\end{equation}
This means that the high energy photons, those greater than $4.1 \times 10^{14} eV$,
would be depleted from UHECR. After all, the electromagnetic reactions proceed fast
enough. The depletion of the photons from UHECR would explain the happening of the
"knee".

Channel (6) or (7) or others, as described as above in a simplified manner, would
not occur until UHECR reaches a certain threshold. This happens for UHECR at
$10^{18.5}eV$, the
place for the "ankle". In fact, the cross section for the channel $p + \gamma_{CMB}
\to p + (e^-e^+)$ would be down by a factor of $\alpha/\pi$ (as compared to,
for example, Reaction (5) or (7)), but the logarithmic
plot for the UHECR could show that - the effect of $10^{-2.5}$ if protons are
majority of UHECR.

Let come back to Reaction (7), i.e. $d + \gamma_{CMB} \to p + n$. At the threshold,
we find, UHECR identified as deuterons,

\begin{equation}
4 k(p + {m_d^2\over {2p}})+m_d^2 =2 m_n^2 + 2 m_p^2,\quad 4kp =8349.34 MeV^2,\quad
p=8.8785 \times 10^{18} eV.
\end{equation}
These numbers indicate the threshold of $(2m_e)$, or sightly above, to occur at
$10^{18.5}eV$, as explained earlier.

How about reactions (8), (9), etc.? In fact, heavy nuclei ($A\ge 3$), as seen by
the Auger Collaboration\cite{Gaisser,PDG10}, could be of some importance. As
indicated earlier, this may be so if parts of UHECR's come from the inward collapse
of Supernova explosion.

We see that a lot of nuclear reactions with effective energies less than 10 MeV
may become relevant at $10^{19}eV$, until we hit another threshold of the
famous GZK\cite{GZK}:

\begin{equation}
p+\gamma_{CMB}\to \pi + N, \quad E \approx {2m_N m_\pi+m_\pi^2 \over {2E_{CMB}} }
=1.10 \times 10^{20} eV.
\end{equation}

This is another order of magnitude - but very close in our logarithmic plot.
Clearly, interesting physics occurs for UHECR of energy $10^{18.5 - 20.5}eV$.

{\it To say it explicitly, $10^{20}eV$ is where the GZK effect occurs and $10^{18.5}eV$ is
where the "knee" appears (and where nuclear physics dominates). So, what is above
$10^{21-25}\,eV$? Particle physics is probed by CMB - that would be our answer. }

The high energy cosmic rays measured in the atmosphere are what we are interested most.
We are already in the vicinity of $10^{20}\, eV$, maybe marching toward higher and
the higher. Those UHECR's may come from the outside solar system, or from the distant
galaxies, and these would be most interesting. As we have said earlier, these UHECR's
are presumably there for a while and thus stable, composed of "stable" particles, such
as $e^\pm$, $\gamma$, $\nu$, $p$, $\bar p$, $d$, ..., $n$, $\mu^\pm$, etc. We don't
take into account $e^\pm$ because of their zigzag paths. we so far don't
take into account $\nu$'s or $\bar\nu$'s mainly due to their (weak) no-interacting
features. As indicated before, too high energy photons (greater than $4.1\times 10^{14}eV$)
could become elusive also. How to reproduce the UHECR curve\cite{Gaisser} should be one of
the most urgent questions.

\section{UHECR Physics near $10^{25}\,eV$}

An UHECR particle of energy $10^{25}\,eV$ encountering the $3^\circ$ CMB photon
would have the CM energy squared of $1.3\times 10^{22}\, eV^2$ or the CM energy
$115\, GeV$, just above the $W^\pm$ or $Z^0$ mass. This is where Weiler called
it the $Z^0$-bursts\cite{Weiler}. Clearly, both $W^\pm$ and $Z^0$ show up at
these energies.

If the UHECR particle would be a proton, an alpha particle, or one of those
familiar particles in the Standard Model, it would be a replay of the Standard
Model\cite{PDG10}, except the very odd kinematics.

What if the UHECR particle is something else, such as some supersymmetric particle?
But it interacts with the $3^\circ$ CMB photon, or with
the electromagnetic interactions; it means that it carries the electric charge.
We infer that this supersymmetric particle cannot be the lowest-mass neutral
supersymmetric particle. The open-up of new channel would be very interesting.

In other words, the initial UHECR's do not have to be the ordinary particles and
may not interact with the $3^\circ$ CMB - we should partition these UHECR's 
accordingly. It could be the supersymmetric particle to begin with - the
primary supersymmetric particle decays eventually the most stable neutral
supersymmetric particle plus a bunch of more familiar particles. Our Universe
might be full of surprises foe us.

The other aspect is well-known - the occurrence of the $W^\pm$ or $Z^0$ bursts
when we crosses the thresholds, as we know the CM energy exceeding $100\,GeV$.
This ought to be rather familiar, except that things are happening with the very
odd kinematics.

\section{Outlook}

Ultra High Energy Cosmic Rays, for the energy greater than $10^{14} eV$ but
less than $10^{26} eV$, via interactions with the Cosmic Microwave Background,
sort of map out nuclear physics in the lower end (less than $10^{20}\,eV$)
and particle physics in the higher end (near $10^{26}\,eV$). Why is this
interesting? CMB serves as as the medium, becoming the obstacle when the
comic rays are energetic enough. CMB and UHECR are both exotic and deserve
our attention.

\section*{Acknowledgments}
The Taiwan CosPA project is funded by the Ministry of Education (89-N-FA01-1-0
up to 89-N-FA01-1-5). W-Y. P. Hwang's research is also supported by National Science
Council (NSC99-2112-M-002-009-MY3). B.-Q. Ma's research is supported in part by China's
National Science Foundation.

