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\begin{center}{\Large \bf {Enhancement and Suppression of the 
Neutrino-Nucleon Total Cross Section at Ultra-High Energies}}\\

\vspace{1in}
{\large  Jamal Jalilian-Marian}\\

\vspace{.2in}
{\it Physics Department, Brookhaven National Laboratory,
Upton NY 11973\\ }
  

\end{center}

\vspace*{25mm}


\begin{abstract}

\noindent We argue that high gluon density effects at small $x$ are  
important for calculation of ultra-high energy neutrino nucleon cross 
sections due to the phenomenon of geometric scaling. We calculate the 
cross section for $\nu \, N \rightarrow \mu \, X$, including high gluon 
density effects, using the all twist formalism of McLerran and 
Venugopalan and show that it can be related to the dipole nucleon 
cross section measured in DIS experiments. For neutrino energies of 
$E_{\nu}\sim 10^{12}$ GeV, the geometric scaling region extends all 
the way up to $Q^2 \sim M^2_{W}$. We show that geometric scaling 
can lead to an {\it enhancement} of neutrino nucleon total cross section
by $1-2$ orders of magnitude compared to the leading twist cross section 
and discuss the implications for neutrino observatories. At extremely 
high energies, gluon saturation effects suppress the neutrino nucleon 
total cross section and lead to its unitarization.

\end{abstract}
\newpage

\section{Introduction}

Ultra high energy neutrinos are a source of great mystery and excitement
and offer a possible window to beyond the standard model phenomena.
Due to their weak interactions with matter, neutrinos can travel large 
distances and therefore carry information about very distant objects.
The origins of ultra high energy neutrinos are uncertain and subject 
of intense theoretical and experimental interest and investigation \cite{fh}.
Some possible sources are active galactic nuclei, decays of super heavy
particles and gamma ray bursts. They can be detected by measuring the
air showers initiated by the muon produced in neutrino-nucleon interactions 
through charged current.
 
The total neutrino nucleon cross section can be calculated 
\cite{ptqcd} in the standard model using the various parameterizations 
of parton distribution functions \cite{parton} measured at HERA 
\cite{hera}. At very high neutrino energies, one is sensitive to the 
behavior of the parton distribution functions at small $x$. It can be 
shown that power growth of the parton distribution functions with $x$ 
would lead to power growth of the neutrino nucleon total cross section 
with neutrino energy. This would eventually lead to violation of unitarity 
at high energies. 

Unitarization of ultra high energy neutrino nucleon cross sections has
been of considerable interest lately \cite{dicus,rsssv,stasto,gkr}. 
Saturation of the gluon distribution function at very small $x$ \cite{glr} 
is expected to restore unitarity at high energies. One can make a rough 
estimate of the magnitude of unitarity corrections at a given neutrino 
energy by considering the first higher twist correction factor 
$\alpha_s xG(x,Q^2)/ \pi R^2 Q^2$. Since the neutrino nucleon cross
section is dominated by scales $Q^2 \simeq M_W^2$, the effective value
of $x$ is $\sim {M_W^2 \over 2 M_h\,E_{\nu}}$. At neutrino energy of
$E_{\nu} \sim 10^{12} $GeV, this is only a few percent effect. 
However, due to the phenomenon of geometric scaling, it is too naive to 
conclude that higher twist (high gluon density) effects can be 
disregarded. 

It is an experimental fact that the HERA data at small $x$ ($< 0,01$) and
all $Q^2$ show geometric scaling \cite{gscale}. In other words, the DIS
cross section depends only on one variable, $Q^2/Q^2_s (x)$ rather than
two independent variables $x$ and $Q^2$. Here $Q^2_s(x)$ is the saturation
scale of the nucleon, arising from high gluon density effects, which can 
be extracted from the HERA data \cite{gbw}. Geometric scaling is a property 
and prediction of the all twist formulation of QCD evolution equations for 
DIS structure functions and cross sections at small $x$ (high energy)
\cite{nonlin}. It has been shown recently that the nonlinear evolution 
equations for the the structure functions at small $x$ exhibit this 
geometric scaling property \cite{iim}. 

A simple but intuitive picture of geometric scaling at high $Q^2$ 
emerges when one considers the BFKL evolution equation and its boundary 
condition. It can be shown that matching the BFKL equation and the
full nonlinear evolution equation across the low-high gluon density
regions leads to the solutions of BFKL evolution equations having
the property of geometric scaling \cite{dino} in the transverse
momentum region much higher than one would naively expect. In other
words, geometric scaling means that high gluon density effects, 
which are dominant at scales $Q^2 \leq Q^2_s$, influence observables
at much higher scales $Q^2 \gg Q_s^2$. This was used in \cite{klm} to fit 
the RHIC data on pion spectra at $p_t^2 \gg Q_s^2$. In \cite{iim} the 
$Q^2$ region where geometric scaling holds is calculated to be 
\be
Q^2_{max} \ll \bigg[ {Q_s^2(x) \over \Lambda_{QCD}^2}\bigg]\, Q_s^2(x)
\label{eq:gsregion}
\ee
We show the geometric scaling region for different neutrino energies in 
Figure (\ref{fig:Q_max}). For easy reference, we also show $M_W^2$. We 
have used the Golec-Biernat and W\"usthoff parameterization of the 
saturation scale such that
\be
Q_s^2(x)\equiv Q_{s0}^2\,(x_0/x)^{\lambda}
\label{eq:gbw}
\ee
where $Q^2_{s0}=1.0$  $GeV^2$, $x_0=3.0 \times 10^{-4}$ and $\lambda = 0.28$,
$\Lambda_{QCD}=0.2$ GeV. Furthermore, since most of the contribution to 
the cross section comes from $Q^2 \sim M^2_W$, we have set  
$x= {M_W^2 \over 2m_p E_{\nu}}$.

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=12cm}
\centerline{\epsffile{Q_max.eps}}
\caption{The geometric scaling region.}
\label{fig:Q_max}
\end{figure}

Clearly, by $E_{\nu} \sim 10^{12} $ GeV we are in the geometric scaling 
region. Therefore, high gluon density and gluon saturation effects 
may be crucial. Since the standard expressions for the neutrino nucleon
cross sections are calculated within the leading twist perturbative QCD
formalism, they will break down due to higher twist nature of gluon
saturation. Here, we use the effective action and renormalization group
approach to high energy QCD \cite{nonlin} to calculate this cross section 
including all higher twists (high gluon density) effects.


\section{Neutrino nucleon cross section}

In leading twist perturbative QCD, the expression for the neutrino nucleon 
differential cross section is given by 

\be
{d^2 \sigma^{\nu N} \over dxdQ^2}={G_F^2 \over \pi}
\bigg({M^2_{W,Z} \over Q^2 + M^2_{W,Z}}\bigg)^2 
\bigg[q(x,Q^2) + (1- Q^2/xs)^2 {\bar q}(x,Q^2)\bigg]
\label{eq:difcs}
\ee
where $x$ and $Q^2$ are the standard DIS variables and
quark and anti-quark distributions include the appropriate 
couplings for neutral and charged currents in DIS. 
To get the total cross section, one integrates over $x$ and $Q^2$
\be
\sigma_{total}^{\nu N}(s)=\int_0^1 dx \int_0^{xs} dQ^2
{d^2 \sigma^{\nu N} \over dxdQ^2}
\label{eq:stcs}
\ee
At high neutrino energies and for vary high values of $Q^2 >> M^2_{W,Z}$, 
the integrand dies off quickly while shrinkage of phase space kills the 
contribution of low momentum ($Q^2 << M^2_{W,Z}$) region. Therefore, the 
dominant contribution to the total cross section comes from the region
of $Q^2 \sim M_{W,Z}^2$ if one uses the standard parton distribution functions 
(by standard, we mean any of the available parameterization of parton 
distributions such as MRS, CETEQ, GRV \cite{parton}). 

At small $x$, higher twist effects become important. This means that
the standard parton distribution functions, defined as the
expectation values of certain two point operators get contributions
from higher twist operators. This spoils their interpretation as a
number density. One can still define and calculate physical observables 
such as structure function $F_2$ \cite{mv} which are experimentally 
measured. However, one can not relate the all twist structure functions 
to number distributions such as the standard gluon distribution function 
$xG$. 

\subsection{Charged current exchange}

Here we use the effective action and classical field approach to
high gluon density effects to calculate the cross section for
neutrino nucleon charged current interaction
\be
\nu_{\mu}\, N \rightarrow \mu \, X
\label{process}
\ee
Since we will ignore all lepton masses, our results would also apply 
to electron and tau neutrino scattering. Also, in this
work, we will ignore the neutral current exchange but it is quite
similar to the process considered here. Our goal here is to derive
an analytic expression for the above cross section such that it
includes all higher twist effects which are expected to unitarize it. 

We start with writing the differential cross section in terms of
leptonic and hadronic tensors
\be
{d\sigma \over dx\, dQ^2} = {1 \over 4 \pi}{y \over x s}
{G_F^2 M_W^4 \over [Q^2 + M_W^2]^2} \,
L^{\mu\nu}(k_1,k_2)\,W_{\mu\nu}(q^2, P\cdot q)
\label{eq:cslw}
\ee
where $k_1$,$k_2$ are the incoming and outgoing lepton momenta, 
$P$ is the nucleon momentum while $q=k_1-k_2$ is the momentum
transfer ($Q^2=-q^2$). The leptonic tensor $L_{\mu\nu}(k_1,k_2)$ 
is standard and is not affected by high gluon density effects. It is 
\be
L^{\mu\nu}(k_1,k_2) \equiv 2\bigg[k_1^{\mu}\,k_2^{\nu} +  k_1^{\nu}\,k_2^{\mu} 
- g^{\mu\nu}\,k_1\cdot k_2 + i\,\epsilon^{\mu\nu\rho\sigma}\,k_{1\rho}\,
k_{2\sigma}\bigg]
\label{eq:lepten}
\ee 
The hadronic tensor $W_{\mu\nu}$ contains all the information about
the high gluon density effect in a hadron. It is defined as
\be
W_{\mu\nu}(q^2, P\cdot q) \equiv {1 \over 2\pi} Im \int d^4z \, e^{iqz} 
<P|T\,J_{\mu}^{\dagger}(z)\,J_{\nu}(0)|P>
\label{eq:hadten}
\ee
where $J_{\mu}\equiv \bar{u}\gamma_{\mu}(1+\gamma_5)d$ is the charged 
weak current. The all twist hadronic tensor for electron proton DIS 
with a photon exchange has been already evaluated in \cite{mv}. Our
calculation here is a straightforward generalization to $W$ exchange
relevant for the process considered here. Since we are working with
a classical background field and external sources of color charge, 
we will need to generalize (\ref{eq:hadten}). This is already done
in \cite{mv} where the hadronic tensor is defined as
\be
W_{\mu\nu}\equiv {\sigma \over 2\pi} {P^+ \over M_h} \,Im\, \int dX^-\,
\int d^4z \, e^{iqz} 
<T J_{\mu}^{\dagger}(X^- + z/2) J_{\nu}(X^- - z/2)>
\label{eq:genhadten}
\ee
where 
\be
<T J_{\mu}^{\dagger}(x) J_{\nu}(y)> = Tr\, 
\gamma_{\mu} (1+\gamma_5) S_u(x,y)\gamma_{\nu} (1+\gamma_5) S_d(y,x)
\label{eq:trjj}
\ee
and $S_{u,d}(x,y)$ is the $u$ or $d$ quark propagator in the 
background of the classical color field in coordinate space while  
$\sigma$ is the target hadron transverse area and $P^+$ is the
large component of the hadron momentum. Despite its appearance,
the hadronic tensor defined in (\ref{eq:genhadten}) is Lorenz covariant
as discussed in \cite{mv}. The propagator in the background field is given 
by\footnote{Since we are ignoring quark masses, we will
not distinguish between $u$ and $d$ quarks and drop the flavor label from
here on.} \cite{mv}  
\be
S(x,y) = S_0 (x,y) &-& i \int d^4r\,\bigg\{\bigg[ \theta (x^-)\theta (-y^-)
[V^{\dagger}(r_t) -1] -  \theta (-x^-)\theta (y^-)[V(r_t) -1]\bigg]
\nonumber \\
&&
S_0(x-r)\gamma^- \delta (r^-) S_0(r-y)\bigg\}
\label{eq:atprop}
\ee
with the free fermion propagator given by
\be
S_0(x-y)\equiv - \int {d^4\, p \over (2\pi)^4} e^{ip(x-y)} 
{{\slp} \over p^2 -i\epsilon}
\label{eq:freeprop}
\ee
and $V(r_t)$ is a matrix in fundamental representation which includes
the infinitely many gluon exchanges between the quark and the hadron.
The propagator has also other pieces which involve $\theta$ functions
on the same side in $x^-$ and $y^-$. As shown in \cite{mv}, these pieces
are pure gauges and do not contribute to the cross section. Therefore,
they are not included here. Finally, we will have to color average
our results to get the physical cross sections. We will come back to 
this point later when we discuss dipole models.

Using eqs. (\ref{eq:trjj}) and (\ref{eq:atprop}) in (\ref{eq:genhadten})
and after some lengthy algebra, we get
\be
W_{\mu\nu}= {N_c \sigma\over 2\pi} {P^+\over M_h} \,Im\, 
\int {d^4p \over (2\pi)^4}
{d^4k \over (2\pi)^4} (2\pi)\delta(k^-)\, \tilde{\gamma}(x,k_t)
{M_{\mu\nu} \over p^2\, (p-k)^2\, (p-q)^2\, (p-q-k)^2}
\label{eq:wmunuint}
\ee
where 
\be
\tilde{\gamma}(x,k_t)\equiv \int d^2 r_t \, e^{i k_t\cdot r_t} \,\gamma(x,r_t) 
\label{eq:gammaft}
\ee
with
\be
\gamma (x,r_t) \equiv {1 \over N_c} Tr \,[1 - <V(r_t)V^{\dagger}(0)>]
\label{eq:gammadef}
\ee
and 
\be
M_{\mu\nu}\equiv 2\,Tr (1-\gamma_5)\gamma_{\mu}
({\slp}-{\slq} -{\slk})
\gamma^- ({\slp}-{\slq}) \gamma_{\nu}{\slp}
\gamma^- ({\slp} -{\slk})
\label{eq:Mmunu}
\ee
The function $\gamma (x,r_t)$ is related to the (quark anti-quark) 
dipole-nucleon scattering amplitude in coordinate space. The dipole
cross section, in the double log approximation, is sometimes modeled as 
\be
\sigma_{dipole} = \sigma_0 \bigg[
1- exp[-\# \alpha_s \,r_t^2 \,xG(x,1/r_t^2)/\sigma_0]\bigg]
\label{eq:dipole}
\ee
and used in \cite{gbw} to successfully fit the HERA data below $x=0.01$. 

To get the imaginary part of the hadronic tensor, we use the Landau-Cutkosky
cutting rules. There are two distinct ways of cutting the diagram as
shown in Fig. (\ref{fig:cut}). The dotted lines are $W$ boson gauge fields
while the thin solid lines represent fermions. The thick solid line
with a filled circle represent insertion of the classical field and 
the thick dashed lines represent the possible cuts. The cuts where 
both classical field insertions are on the same side are not kinematically 
allowed. 
\begin{figure}[htp]
\centering
\setlength{\epsfxsize=10cm}
\centerline{\epsffile{cut.eps}}
\caption{Imaginary part of the hadronic tensor.}
\label{fig:cut}
\end{figure}
The cut propagator is put on shell along with a theta function to ensure 
positive (negative) energy for fermions (anti fermions). The sum of the 
two cuts is proportional to
\be
{\theta(p^+)\theta(q^+ - p^+ + k^+) \delta(p^2) 
\delta((p - k - q)^2) \over (p - k)^2\,(p - q)^2}
+
{\theta(p^+ - k^+)\theta(q^+ - p^+) \delta((p - k)^2) 
\delta((p - q)^2) \over p^2\,(p - k - q)^2} 
\nonumber
\ee
It is straightforward to show that the two contributions are actually
equal with appropriate change of variables and $\mu \leftrightarrow \nu$.
We get
\be
W_{\mu\nu}&=& {2N_c \sigma \over 2\pi}{P^+\over M_h} \,\int 
{d^4p \over (2\pi)^4}
{d^4k \over (2\pi)^4} (2\pi)\delta(k^-)\, \tilde{\gamma}(x,k_t)
\,M_{\mu\nu}
\nonumber \\
&\times&
{\theta(p^+ - k^+)\theta(q^+ - p^+) (2\pi)\delta((p - k)^2) 
(2\pi)\delta((p - q)^2) \over p^2\,(p - k - q)^2} 
\label{eq:wmunuint2}
\ee
where
\be
M_{\mu\nu} &\equiv& Tr \bigg[(1-\gamma_5)\gamma_{\mu}
({\slp}-{\slq} -{\slk})
\gamma^- ({\slp}-{\slq}) \gamma_{\nu}{\slp}
\gamma^- ({\slp} -{\slk}) \nonumber \\
&+&
(1-\gamma_5)\gamma_{\mu}
({\slp}-{\slq})
\gamma^- ({\slp}-{\slq}-{\slk}) \gamma_{\nu}({\slp} - {\slk})
\gamma^- {\slp}\bigg]
\label{eq:Mmunu1}
\ee
We can now use the delta functions to perform some of the integrals
in (\ref{eq:wmunuint2}). Defining $z\equiv p^-/q^-$, the effect of the
theta functions is to restrict the $z$ integration to the region between
$0$ and $1$. We then get
\be
M_h\,W_{\mu\nu}(q^2,P\cdot q) &=& {N_c\sigma \over 16\pi^2} 
{2P\cdot q \over q^- q^-} \int_0^1 dz \,
{d^2p_t \over (2\pi)^2}\,{d^2k_t \over (2\pi)^2}\,
\tilde{\gamma}(x,k_t)
\nonumber \\
&&
{\{[M^{sym}_{\mu\nu} + \mu \leftrightarrow \nu] + 
[M^{asym}_{\mu\nu} - \mu \leftrightarrow \nu]\} \over
[(p_t - zq_t)^2 -z(1-z)q^2][(p_t + k_t -zq_t)^2 -z(1-z)q^2]}
\label{eq:wmunuint3}
\ee
where $M^{sym}_{\mu\nu}$ and $M^{asym}_{\mu\nu}$ are now given by 
\be
M^{sym}_{\mu\nu} = Tr \, \gamma_{\mu}
({\slp} - {\slq} + {\slk})
\gamma^- ({\slp} - {\slq}) \gamma_{\nu}{\slp}
\gamma^- ({\slp} + {\slk})
\ee
and
\be
M^{asym}_{\mu\nu} = -\,Tr \, \gamma_5 \gamma_{\mu}
({\slp} - {\slq} + {\slk})
\gamma^- ({\slp} - {\slq}) \gamma_{\nu}{\slp}
\gamma^- ({\slp} + {\slk})
\ee
with $k^-=0$, $q^+=0$, $p^-=zq^-$ and 
\be
p^+ = - {(p_t - q_t)^2 \over 2(1-z)\,q^-}\;\;\;\;\;
\;\;\;\; k^+=-p^+ + {(p_t+k_t)^2 \over 2z\,q^-}
\ee
It is customary to write the hadronic tensor in terms of Lorenz invariant 
functions $W_1,W_2,W_3$ defined as \cite{rgr}
\be
M_hW_{\mu\nu}\equiv -(g_{\mu\nu} - {q_{\mu}q_{\nu} \over q^2})\,F_1 + 
{1 \over P\cdot q}(P_{\mu} - {q_{\mu}P\cdot q \over q^2})
(P_{\nu} - {q_{\nu}P\cdot q \over q^2})\,F_2 
+ i\,\epsilon_{\mu\nu\rho\sigma}{P^{\rho}q^{\sigma}\over P\cdot q}\,F_3
\label{eq:decomp}
\ee
where the structure functions are defined as $F_1=M_h\,W_1$, 
$F_2=\nu\, W_2$ and $F_3=\nu\, W_3$ with $M_h$ being the target nucleon 
mass and $P\cdot q = M_h \nu$. The differential cross section
$d\sigma/dxdQ^2$ can be written in terms of the structure functions
as
\be
{d\sigma \over dxdQ^2}= {1\over 2\pi}{G_F^2 \over [1 + {Q^2\over M_W^2}]}
\bigg\{y^2\, xF_1 + (1-y)\,F_2 + y[1-{y\over 2}]\,xF_3\bigg\}
\label{eq:dsigstruc}
\ee
which are related to the hadronic tensor via
\be
{F_2 \over 2x} = -{q^-q^- \over q^2}\, M_hW^{++}
\label{eq:f2}
\ee
\be
F_1 - {F_2 \over 2x} = {q^2 \over q^- q^-}\,M_hW^{--}
\label{eq:f1}
\ee
\be
F_3 = \Pi^{\mu\nu}\, M_hW_{\mu\nu}
\label{eq:f3}
\ee
where 
\be
\Pi^{\mu\nu} = -i\epsilon^{\mu\nu\alpha\beta}
{P_{\alpha}\, q_{\beta}\over 2 P\cdot q}
\ee
To give explicit expressions for the structure functions, we need to 
evaluate the traces\footnote{We would like to thank W. Vogelsang for his
help with evaluating these traces.}. This has been already done for the 
case of $W^{++}$ and $W^{--}$ in \cite{mv}. Using $K_0^{\prime} = -K_1$ and 
the identity
\be
\int_0^{\infty} dp \,{p\, J_0(p\,r_t) \over [p^2 + a^2]} \equiv K_0(a\,r_t)
\ee
gives 
\be
2xF_1 \!=\!
{N_c \sigma Q^2 \over 4\pi^3}\!\int^1_0 dz\!\int dr_t^2\, [1 - \gamma(x,r_t)]
\bigg\{2 z^2(1-z)^2Q^2K_0^2(ar_t) + 
a^2 [z^2 + (1-z)^2]K_1^2(ar_t) \bigg\}
\nonumber
\ee
\be
F_2= {N_c \sigma Q^2 \over 4\pi^3}\!\int^1_0 dz\!\int 
dr_t^2\, [1 - \gamma(x,r_t)]
\bigg\{4 z^2(1-z)^2Q^2K_0^2(ar_t) + 
a^2 [z^2 + (1-z)^2]K_1^2(ar_t)\bigg\}\nonumber
\ee
\be
xF_3= {N_c \sigma Q^2 \over 4\pi^3}\int^1_0 dz\int 
dr_t^2\, [1 - \gamma(x,r_t)]
\bigg\{(1-2z)\,a^2 \,K_1^2(ar_t)\bigg\}
\label{eq:f1f2f3}
\ee
with $a^2\equiv z(1-z)Q^2$ and $K_0$ and $K_1$ are the modified Bessel 
functions. This is our main result. Using these expressions
for the structure functions in (\ref{eq:dsigstruc}) and (\ref{eq:stcs}) gives 
the all twist cross section for $\nu \, N \rightarrow \mu \, X$. 

One can distinguish three distinct kinematical regions in which the
neutrino nucleon total cross section has a different behavior. In the
very high energy limit where unitarity effects are dominant ($Q_s \sim M_W$), 
the cross section is given by (\ref{eq:f1f2f3}). This is the saturation 
region and the 
total cross section grows much slower (compared to the perturbative power
growth) due to high gluon density effects. At lower energies where 
$Q_s^2$ and $Q^2_{max}$, as defined in (\ref{eq:gsregion}), are both much 
less than $M_W^2$, one can use the standard perturbative results to which 
our expressions reduce\footnote{See \cite{mv} for a discussion of high $Q^2$ 
limit of all twist calculations.}. At high energies (but not too high where
unitarity effects are dominant) where $Q^2_s \ll M_W^2$ but 
$Q^2_{max} \gg M_W^2$, we are in the geometric scaling region where
high gluon density effects modify the boundary conditions of small $x$ QCD 
(BFKL) evolution equation and must be taken into account. We discuss this
region in the next section and show that the neutrino nucleon total cross 
section is enhanced.

\section{Enhancement of neutrino-nucleon cross section}

Geometric scaling was first observed at HERA for the virtual photon
nucleon total cross section \cite{gscale}. Roughly speaking, geometric 
scaling is the phenomenon that DIS structure functions depend only
on one variable $\tau\equiv Q^2/Q^2_s(x)$ rather than two independent
variables $x$ and $Q^2$. Geometric scaling arises naturally from the
all twist formulation of small $x$ QCD \cite{nonlin,bal}. It has been 
shown that geometric scaling is a property of the non-linear 
generalizations of QCD evolution equations at small $x$ and that the 
scaling region extends way beyond the saturation region \cite{iim}, 
contrary to naive expectations.

The saturation region, shown in Figure (\ref{fig:satreg}), provides the 
boundary for the BFKL equation \cite{dino} which is the correct evolution 
equation in the geometric scaling region. High gluon density effects in 
the saturation region screen the diffusion of BFKL into the low momentum 
region \cite{gbms}.

\begin{figure}[htp]
\centering
\setlength{\epsfxsize=14cm}
\centerline{\epsffile{satreg.eps}}
\caption{The saturation region.}
\label{fig:satreg}
\end{figure}
To see how inclusion of the saturation region as the boundary of BFKL 
equation \cite{dino} affects the neutrino nucleon cross section, we 
consider the dipole nucleon cross section $\sigma (x,r_t,b_t)$ which is
the universal factor showing up in all twist cross sections. Since we
have treated the nucleon as uniform in transverse plane, the $b_t$
dependence is trivial and given by the nucleon size $\pi R^2$.
Therefore, in our notation the dipole cross section in the geometric scaling 
is
\be
\sigma_{GS} (x,r_t,b_t) = 2 \pi R^2 \gamma (x,r_t)
\label{eq:imp}
\ee
In \cite{iim} a solution to BFKL equation which includes the high gluon
density effects in the boundary condition and exhibits geometric scaling
was found. It has the form
\be
\gamma (x,r_t=2/M_W) =  \bigg[{4Q_s^2(x) \over M_W^2}\bigg]^{\lambda_s}
\exp{\bigg\{- {1\over 2\beta \bar{\alpha_s} log 1/x}
\bigg[log[M_W^2/4Q_s^2(x)]\bigg]^2\bigg\}}
\label{eq:gamma}
\ee
with $\bar{\alpha_s}={N_c\alpha_s \over \pi}$ and $\beta=34$ and we
have replaced $r_t \sim 2/Q$ with $Q\sim M_W$. A LO BFKL 
analysis leads to $\lambda_s=0.644$, while Double Log approximation
gives $\lambda_s=0.5$. A more complete analysis of the energy dependence 
of the saturation scale has been recently performed in \cite{dino} using 
the NLO BFKL equation with collinear (and anti-collinear) terms resumed.  
It is shown that the functional form of the saturation scale depends on
the energy considered and that $\lambda_s$ is energy dependent. In the 
$x$ range of relevance here, however, it is a good approximation to treat
it as a constant $\lambda_s \sim 0.28$. Furthermore, in this $x$ range, we
use the relation $Q_s^2(x)=\Lambda^2\, e^{\lambda_s log1/x}$ where 
$\Lambda$ is some hadronic scale.

One can use (\ref{eq:imp}) and (\ref{eq:gamma}) in (\ref{eq:f1f2f3})
to calculate the neutrino nucleon cross section in the geometric scaling 
region. Due to the non-trivial energy dependence of the saturation scale, this
is quite involved. Here, we will make a very rough estimate of the 
cross section and show that it is enhanced compared to the leading twist
calculations. In the leading twist approximation and using 
(\ref{eq:dipole}), the dipole cross section $\sigma (x,r_t=2/M_W)$ may be 
written as (in $fm^2$)
\be
\sigma_{LT} (x,r_t=2/M_W) \simeq {4\pi^2 \alpha_s xG(x,M_W^2) \over 75 M_W^2}
\label{eq:LT} 
\ee 
To get a feeling for the enhancement, we consider $E_{\nu}=10^{12}$ GeV.
In this case, most of the contribution to the total cross section is coming
\cite{gkr} from the region $ 10^{-9} < x < 10^{-6}$. As one goes to higher 
neutrino energies, one gets more contributions from smaller values of $x$. 
For sake of argument, we consider $x={M_W^2 \over s} = 3.2 \times 10^{-9}$.
At this value of $x$, the ratio of geometric scaling and leading twist 
cross sections is $\sim 100$. This illustrates the fact that the neutrino 
nucleon cross section is enhanced in some neutrino energy range due to the 
phenomenon of geometric scaling.

Off course, this is a very rough estimate but the enhancement should be
robust. In a quantitative analysis, one will have to include contributions 
from different regions of $x$ where one may or may not be in the geometric 
scaling region. At some neutrino energies, the dominant contribution will 
come from $x$'s where we will be fully in the geometric scaling region and 
the enhancement will be maximal\footnote{We are implicitly assuming that the
cross section is dominated by $Q \sim M_W$ even in the geometric scaling
region. It is possible that the effective $Q$ will shift to smaller values 
but this will make the enhancement even stronger.}. As one goes to yet 
higher energies, the geometric scaling region shrinks due to the fact that 
$Q_s^2\rightarrow M_W^2$ and one approaches the saturation region where 
unitarity effects will become more important and will eventually suppress 
the cross section compared to the leading twist cross section. A more quantitative analysis is in progress and will 
be reported elsewhere.

\section{Discussion}

We have calculated the total cross section for neutrino nucleon 
scattering via the charged current exchange including the high gluon 
density (higher twist) effects. We have shown that this cross section
is expressed in terms of the dipole nucleon cross section which
is the universal object appearing in all twist cross sections \cite{pA}. 
Using our expressions for the neutrino nucleon cross section and
some model of the dipole nucleon cross section (given for example in
\cite{gbw}), one can estimate at what neutrino energies protons will
look black to neutrinos (the black disk limit). This turns out to be 
at neutrino energies of $E_{\nu} \sim 10^{18}$ GeV. This may be too 
high of an energy for this effect to be observable in the near future. 

A more interesting effect happens at much smaller than energies
than the black disk limit. As shown here, the geometric scaling region
extends all the way up to and beyond the weak boson mass already at
neutrino energies of $E_{\nu} > O(10^{12})$ GeV. Therefore 
neutrino nucleon cross sections at these energies will be dominated
by scales which are within the geometric scaling region where cross 
sections are typically enhanced. This enhancement factor can be as large 
as $1-2$ orders of magnitude at  $E_{\nu} > 10^{12}$ GeV. This will have 
very interesting consequences for neutrino astronomy and cosmology 
\cite{kus}.

This enhancement will be important for the current and future neutrino 
observatories \cite{agasa}, \cite{auger}, \cite{hires}, \cite{euso}, 
\cite{owl}. It will be very interesting to see whether the observed 
cosmic ray data points beyond the GZK cutoff \cite{gzk} is due to 
neutrinos \cite{weiler}. For 
these events to be neutrinos with their cross sections enhanced due to 
geometric scaling will require a very large enhancement factor already at 
$E_\nu \sim 10^{11-12}$ where the data points are \cite{agasa}. Several
experiments will detect neutrinos through their horizontal air showers
in the Earth's atmosphere due to neutrino air interactions whose rate 
would increase if neutrino nucleon cross section is enhanced. On the
other hand, the rate of up-going air showers initiated by the leptons
produced in neutrino nucleon interactions would decrease if the neutrino
nucleon cross section is enhanced since the Earth will be less transparent
to neutrinos. A better understanding of the neutrino nucleon total cross
section is essential to these experiments which will help clarify the 
origins of ultra high energy neutrinos.



\leftline{\bf Acknowledgments} 

We would like to thank A. Dumitru, R. Harlander, K. Itakura, D. Kharzeev, 
W. Kilgore, S. Kretzer, A. Stasto, D. Teaney, D. Triantafyllopoulos, 
R. Venugopalan and W. Vogelsang for useful discussions. We would also like 
to thank L. McLerran for pointing out to us the importance of geometric 
scaling and its role in the enhancement of cross sections and many 
illuminating discussions. This work is supported by the U.S.\ Department 
of Energy under Contract No.\ DE-AC02-98CH10886 and in part by a PDF from 
BSA. 

     
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