\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}


\begin{document}


\title{Misconceptions in UHE-$\nu$ Radio Experiments}
\date{}
\maketitle

\section{Center of Mass Energy $\neq$ Neutrino Energy}

The energy of a neutrino ($E_\nu$) is it intrinsic energy ($E_\nu=\sqrt{(p_\nu c)^2+(m_\nu c^2}^2$). The center-of-mass (COM) energy, which is useful in accelerators, has to do with the amount of energy involved in the collision, including not only the neutrino, but also the target; usually a nucleon. 

Suppose a neutrino of energy $E_\nu=10^9$ TeV interacting with a proton at rest, then the center of mass energy ($\sqrt{s}$) can be approximated to be
\begin{align*}
\sqrt{s}&\approx\sqrt{2E_\nu m_p}\\
&\approx 45 \text{ TeV},
\end{align*}
where $m_p$ is the mass of the proton.

As a reference, the COM that the LHC can attain is 14 TeV, approximately 3 times smaller than what we just calculated for a neutrino. This can allow probes of the Standard Model at higher energies than what a human made accelerator can get.

\section{?}


\end{document}