\documentclass{article}

\def\dmatm{$\Delta m^2_{atm}$} 
\def\sinatm{$\sin^22\theta_{23}$}
\def\numunumu{$\nu_\mu\to\nu_\mu$}
\newcommand{\fixme}[1]{FIXME: {\it #1}}


\begin{document}


\title{Dependence of the Measurement of \numunumu{} Oscillation
  Parameters on Energy Resolution and its Uncertainty.}

\author{B. Viren, {\it BNL}}
\maketitle
\begin{abstract}
  The effects of energy resolution and its uncertainty on the
  measurement of \numunumu{} parameters are studied.
\end{abstract}



\section{Introduction}

One result that MINOS can provide is the measurement of any deviation
from non-maximal mixing in \numunumu{} disappearance.  This manifests
itself in a non-zero number of events at the disappearance node.
However, energy resolution will smear the measured energy spectrum and
fill in the node region with events with true neutrino energies below
and above the node.

Knowning the amount and form of the energy resolution, at some level
it is possible to unfold the measured spectrum to get the true
neutrino spectrum.  However, uncertainties in the form and magnitude
of the detector response function will translated to uncertainties in
this unfolding and thus in the resulting measurement of \sinatm.

This note uses a simple toy Monte Carlo to simulate the effects of
energy resolution and uncertainties due to incorrect unfolding.  The
note is arranged in the following sections:

\begin{itemize}
\item Oscillation framework.
\item A simple model of energy resolution.
\item Expected energy spectra with different energy resolutions.
\item Deconvolving the spectra.
\item Conclusions
\end{itemize}

\fixme{What about observing multiple nodes?  How does this help?}



\section{Oscillation Framework}

The generated spectra shown below use a full 3-$\nu$, 2-$\Delta
m^2$-scale calculation which takes into account non-uniform earth
matter density~\cite{prem} when calculating matter effects.  However,
for \numunumu{} calculations at moderate distances and the first few
oscillation nodes, it is sufficient to consider the vacuum case with a
single $\Delta m^2$-scale.  For fitting the oscillation parameters,
this simplified $\nu_\mu$ survival probability is used,

\begin{equation}
  \label{eq:survival-probability}
  P(\nu_\mu\to\nu_\mu) \approx 1 - \sin^2(2\theta_{23})sin^2\left(\frac{\Delta m^2_{atm} L}{4E}\right).
\end{equation}


At nodes, the oscillating term is unity, thus any deviation from
maximal mixing will manifest as a non-zero survival probability.



\section{Energy Resolution and Uncertainties}

The model for energy resolution considered is a Gaussian.  That is,
the probability to measure $E_m$ given incident neutrino energy $E_i$
is,

\begin{equation}
  \label{eq:energy-gaussian}
  P(E_m|E_i) \approx \frac{1}{\sqrt{2\pi}\sigma}e^{-(E_i - E_m)^2/2\sigma^2}
\end{equation}

Note that this hides a lot of the details of what actually goes in to
the resolution.  The resolution $\sigma \equiv aE_i + b\sqrt{E_i} + c$
is broken up into 3 parts~\cite{briefbook}:

\begin{description}
\item[a] Calibration errors, non-uniformities, non-linearities in
  light collection.
\item[b] Poisson statistical fluctuations.  This comes primarily from
  the statistics of counting photons.
\item[c] Constant resolutions from noise, pedestals, etc.
\end{description}

%For MINOS, the baseline resolution~\cite{minos-design-book} is taken
%to be,

% From parameter booklet:
% Det energy scale calib: 5% abs, 2% near-far
% Det EM en. res. 23%/sqrt(E), <5% constant term
% Det had en. res. 55%/sqrt(E), <7% const term
% Det mu en. res <12% from curvature or range

%\begin{equation}
%  \label{eq:minos-resolution}
%  \sigma = 5\%E_i + 
%\end{equation}

For a Super-Kamiokande like detector, the energy resolution is $\sigma = 


Energy resolution refers to the statistical and physical broadening in
the distribution of possible measured energies relative to the
distribution of true energies of the incoming particles.  This
broadening can come from counting statistics when measuring energy
directly via scintilator or cherenkov photons.  It can also come from
fitting resolution when using muon range or curvature.  Uncertainties
in detector characteristics, such as light attenuation lengths,
magnetic field strengths, non-uniformities in light detector
efficiencies can also lead to a broadening of the response of the
detector.  

Since the energy deposited is that of the interaction products and not
the primary particle, missed or poorly reconstructed products will
also lead to a broadening of the energy measurement.  This ties back
in to knowledge of detector characteristics.

The uncertainty in the energy resolution as well as the energy scale
also plays a part in the measurement of the oscillation parameters and
associated uncertainties.  If the energy response of the detector was
perfectly known then in principle one can unfold this smearing and
obtain the true energy (see below).  However, in the real world, the
best we can do is estimate the magnitude and form of the uncertainties
and see how this effects our result.

So, in general, measuring energy can be a thorny issue.  For this
study the easy road is taken and it is assumed that the detector is an
ideal calorimeter such that only statistical broadening and simple
systematics are considered and all energy from the incoming particle
is visible ({\it ie}, no missed interaction products, perfect fitting
resolution).  In this case, an incident particle of energy $E_i$ will
produce a mean number of photons $\mu_\gamma$.  In a particular event,
the actuall number of photons produced $n_\gamma$ follows the Poisson
distribution:

\begin{equation}
  \label{eq:poisson-photon-production}
  P(n_\gamma|\mu_\gamma) = \frac{\mu_\gamma^{n_\gamma}e^{-\mu_\gamma}}{n_\gamma!}
\end{equation}

Ignoring light attenuation, these photons are then collected in a
device with some efficiency $\epsilon$.  This collection is also a
Poisson process with probability to count $n_c$ photons given by:

\begin{equation}
  \label{eq:poisson-photo-collection}
  P(n_c|\epsilon,n_\gamma) = \frac{(\epsilon n_\gamma)^{n_c}e^{-\epsilon n_\gamma}}{n_c!}
\end{equation}

The measured energy is $E_m \propto n_c$.  It is distributed by the
product of these two probabilities integrated over all possible number
of intermidiate photons, $n_\gamma$.  

Taking the case of a perfect light detector, all produced photons are
collected and in the case of a large number of photons, the Poisson
can be approximated by a Gaussian,

\begin{equation}
  \label{eq:energy-gaussian}
  P(E_m|E_i) \approx \frac{1}{\sqrt{2\pi}\sigma}e^{-(aE_i - aE_m)^2/2\sigma^2}
\end{equation}

The constant $a$ is the number of photons per unit of incident energy
$E_i$ or measured energy $E_m$.  The width of the distribution is
$\sigma = \sqrt{aE_i}$ which increases with the squre root of the
energy.  An energy independent measure is $\sigma/\sqrt{E_i}$,
typically expressed as \%/$\sqrt{\mbox{GeV}}$.

In the usual case of a less than perfect light detector and other
effects the second as well as other probability distributions
complicate the matter.  However, the Gaussian approximation will
continue to be used for simplicity and the parameter $b = \epsilon a$
will give the scaling from counted photons to measured energy so that:

\begin{equation}
  \label{eq:energy-gaussian-with-eff}
  P(E_m|E_i) \approx \frac{1}{\sqrt{2\pi}\sigma}e^{-(aE_i - bE_m)^2/2\sigma^2}
\end{equation}




\fixme{Define energy res.  Justify gaussian form.  Explain
  XX\%/$\sqrt{e}$ form.  Give estimates for MINOS, water cernkov}


\section{Refresher on Convolution}

\fixme{Def of convolution, Fourier thms, deconvolution}


\section{Analytic example}

\fixme{Assume constant gaussian resolution, reconstruct with estimated
  gaussian, dependence on estimated sigma}



\section{MC results}

\fixme{Run some NuMI+MINOS and P889+H2O MCs with smearing, deconvolve
  and fit for osc. params.}


\section{Conclusion}


\end{document}

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