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\begin{document}
%
\title{Methods to Determine Neutrino Flux at Low Energies: }
\subtitle{Investigation of the Low $\nu$ Method}
%\author{A. Bodek and U. Sarica}
%\institute{Department of Physics and Astronomy, University of
%Rochester, Rochester, NY 14627-0171}
\author{A. Bodek\inst{1}, U. Sarica\inst{1}, D. Naples\inst{2} and L. Ren\inst{2} }
\institute{Department of Physics and Astronomy, University of
Rochester, Rochester, NY 14627-0171 USA
\and University of Pittsburgh, Pittsburgh, PA 15260 }
%
\date{Received: date / Revised version: date Jan 11, 2012}
% The correct dates will be entered by Springer
% $\nu_\mu, \nub_\mu$
\abstract{
We investigate the low $\nu$ method (developed by the CCFR/NUTEV collaborations) to determine the neutrino flux in a wide
band neutrino beam at very low energies, a region of interest to neutrino oscillations experiments.
Events with low hadronic final state energy $\nu<\nu_{cut}$ (of 1, 2 and 5 GeV) were used by the MINOS collaboration to determine the neutrino flux in their
measurements of neutrino ($\nu_\mu$) and antineutrino ($\nub_\mu$) total cross sections.
The lowest $\nu_\mu$ energy for which the method was used in MINOS
is 3.5 GeV, and the lowest $\nub_\mu$ energy is 6 GeV. At these energies, the cross sections are dominated
by inelastic processes.
We investigate the application of the method to determine the
neutrino flux for $\nu_\mu$,$\nub_\mu$ energies as low as 0.7 GeV where the cross sections are dominated by quasielastic scattering and $\Delta$(1232) resonance production. We find that the method can be extended to low energies by using $\nu_{cut}$ values of 0.25 and 0.50 GeV, which is feasible in fully active neutrino detectors such as MINERvA.
\PACS
{
{13.60.Hb}{Total and inclusive cross sections (including deep-inelastic processes)} \and
{13.15.+g}{ Neutrino interactions}
} % end of PACS codes
} %end of abstract
%
\maketitle
%
\section{Introduction}
\label{intro}
A detailed understanding of neutrino ($\nu_\mu$) and antineutrino ($\nub_\mu$) interaction cross sections for various final
states is required for the next generation neutrino oscillations experiments.
The relevant neutrino energy region of interest
for the large neutrino detectors such as T2K\cite{T2K}, MINOS\cite{MINOS,MINOS2},
and NOVA\cite{NOVA} is $0.50.5$ GeV). Neutrino interactions in this energy range
are currently being studied at MINERvA.
\section{The low $\nu$ method at high energies \label{lnu_high}}
If we neglect terms which are proportional to the muon mass,
the differential cross section $\frac{d^2\sigma^{\nu,\nub}}{dxdy}$ for charged current scattering of $\nu_{\mu}$ ($\nub_{\mu}$)
with an incident energy $E_{\nu}$, muon final energy
$E_{\mu}$ and scattering angle $\theta$ can be written in terms of
the structure functions ${\cal F}_1= M{\cal W}_1 (x,Q^2)$, ${\cal F}_2=\nu {\cal W}_2 (x,Q^2)$
and ${\cal F}_3=\nu {\cal W}_3 (x,Q^2)$:
\begin{eqnarray}
\label{cross1}
\frac{d^2\sigma^{\nu(\overline{\nu})}}{dxdy} &=& \frac{G_F^2 M
E_{\nu}}{\pi} {\Big(}\Big[1-y(1+\frac{Mx}{2E_{\nu}}) \nonumber\\
&+&\frac{y^2}{2}\Big(\frac{1+(\frac{2Mx}{Q})^2}{1+R}\Big)\Big] {\cal F}_2
\pm \Big[y-\frac{y^2}{2}\Big]x{\cal F}_3{\Big)},
\end{eqnarray}
where $G_F$ is the Fermi weak coupling constant, $M$ is the
proton mass, $y=\nu/E_\nu$ where $\nu=E_{had}=E_{\nu}-E_{\mu}$,
$Q^2=4E_{\nu}E_{\mu}\sin^2(\theta/2)$
is the square of four momentum transfer, and $x=Q^2/(2M\nu)$ is the Bjorken scaling variable.
Here, $R_L (x,Q^2)$ is defined as the ratio of the longitudinal
and transverse structure functions ($\sigma_L/\sigma_T$). It is related to the other structure functions by,
\begin{equation}
R(x,Q^2)
= \frac {\sigma_L }{ \sigma_T}
= \frac{{\cal F}_2 }{ 2x{\cal F}_1}(1+\frac{4M^2x^2 }{Q^2})-1
= \frac{{\cal F}_L }{ 2x{\cal F}_1}
\end{equation}
where ${\cal F}_L$ is called the longitudinal structure function.
\begin{eqnarray}
{\cal F}_L(x,Q^2) &=& {\cal F}_2 \left(1 + \frac{4 M^2 x^2 }{ Q^2}\right) - 2x{\cal F}_1
\end{eqnarray}
Other useful relations are:
\begin{equation}
2x{\cal F}_1 = {\cal F}_2 \left(1 + \frac{4 M^2 x^2 }{ Q^2}\right) - {\cal F}_L(x,Q^2).
\label{eq:fl-rel}
\end{equation}
\begin{eqnarray}
2x{\cal F}_1 (x,Q^{2}) &=& {\cal F}_2 (x,Q^{2})
%\times \\ &&
\frac{1+4M^2x^2/Q^2}{1+R(x,Q^{2})} \nonumber
\end{eqnarray}
\begin{eqnarray}
{\cal W}_1 (x,Q^{2}) &=& {\cal W}_2(x,Q^{2})
% \times \\&&
\frac{1+\nu^2/Q^2}{1+R(x,Q^{2})} \nonumber
\end{eqnarray}
The three structure functions ${\cal F}_2(x,Q^2)$, ${\cal F}_1(x,Q^2)$ and $x{\cal F}_3(x,Q^2)$
depend on $x$ and $Q^2$.
The plus sign in front of the $xF_3$ term is for neutrinos and the minus is for
antineutrinos.
Integrating over $x$, the differential dependence on $\nu$ can be written in the
simplified form
\begin{equation}
\frac{d\sigma^{\nu,\nub}}{d\nu}=A\left(1+\frac{B}{A}\frac{\nu}{E_{\nu}}-\frac{C}{A}\frac{\nu^2}{2E_{\nu}^2}\right) \label{eq:dsigmadnu}.
\end{equation}
The coefficients $A,$ $B$, and $C$ depend on integrals over
structure functions, where
%
\begin{eqnarray}
\label{Efcminos}
A &=& \frac{G^2_FM}{\pi} \displaystyle\int_{0}^{1}{\cal F}_2(x) dx,\\
\nonumber
%\end{eqnarray}
%\begin{eqnarray}
B &=& -\frac{G^2_FM}{\pi} \displaystyle\int_{0}^{1} \Big( {\cal F}_2(x) \mp x{\cal F}_3(x)\Big) dx,\\
\nonumber
%\label{eq:b}
%\end{eqnarray}
%\nonumber
C &=& B - \frac{G^2_FM}{\pi} \displaystyle\int_{0}^{1} {\cal F}_2(x) ~ \tilde{R}~ dx,
\nonumber
\end{eqnarray}
where
\begin{eqnarray*}
\tilde{R} &=& \left(\frac{1+\frac{2Mx}{\nu}}{1+R_{L}}-\frac{Mx}{\nu}-1\right).
\end{eqnarray*}
%
In the limit $\nu/E_{\nu} \rightarrow 0$, the $A$ term dominates
and the $B$ and $C$ terms are very small.
The MINOS collaboration used the number of low $\nu$ events (with $\nu<\nu_{cut}$) in the detector
to determine the relative flux of neutrinos and antineutrinos as a function of $E_{\nu}$.
In the MINOS analysis, the relative flux is determined using events with $\nu<1$ GeV for
$\nu_\mu$ energies in the range $318$ GeV.
MINOS divides the number of low $\nu$ events with $y3$ GeV and for $\nub_\mu$ interactions with $E_{\nub}>5$ GeV.
Therefore, to determine the flux for $E_{\nu}< $ 3 GeV and $E_{\nub}< 5$ GeV we need to use a $\nu_{cut}$ which is smaller than 1 GeV.
We investigate $\nu_{cut}=0.25$ GeV to be used for $ E_{\nu,\nub}>0.7$ GeV,
and $\nu_{cut}=0.5$ GeV to be used for $ E_{\nu,\nub}>1.4$ GeV. These samples can be cross calibrated against the $\nu_{cut}=1$ GeV sample
in the range $ E_{\nu}>3$ GeV for neutrinos and $ E_{\nub}>5$ GeV for antineutrinos. Similarly, they can be calibrated
against the $\nu_{cut}=2$ GeV and $\nu_{cut}=5$ GeV samples in the range $ E_{\nu,\nub}>9$ GeV and $ E_{\nu,\nub}>18$ GeV, respectively.
% BEGIN FIGURE 2
%\vspace{-0.15in}
\begin{figure}[t]
\includegraphics[width=3.5in,height=3.5in]{2nuvsq24.pdf}
\caption{The accessible kinematic region in the $Q^2$ (in GeV$^2$), $\nu=E_{had}$ (in GeV) plane for
$\nu_{\mu}$ ($\nub_{\mu}$) energies less than 4 GeV (color online).
}
\label{nuvsq24}
\vspace{-0.05in}
\end{figure}
% END FIGURE 2
\vspace{-0.1in}
%
% BEGIN FIGURE 3 Nu vs Q2 up to 1.5
\begin{figure}[t]
\includegraphics[width=3.5in,height=3.5in]{3nuvsq215.pdf}
\caption{ The accessible kinematic region in the $Q^2$ (in GeV$^2$), $\nu=E_{had}$ (in GeV) plane for
$\nu_{\mu}$ ($\nub_{\mu}$) energies less than 1.5 GeV (color online). }
\label{nuvsq22}
\vspace{-0.08in}
\end{figure}
%END FIGURE 3
%
%SECTION 3
\section{The low $\nu$ method at low energies \label{lnu}}
%
In the few GeV region, the are several types
of neutrino interaction processes as defined by
the final state invariant mass $W$. These
include quasielastic (QE) reactions ($W<1.07$ GeV), production of
the $\Delta$(1232) resonance ($1.1 2.0$ GeV ).
Fig.~\ref{nuvsq24} shows the accessible kinematic region in $Q^2$ (in GeV$^2$ and $\nu=E_{had}$ (in GeV) for $E_{\nu}<$ 4 GeV. Fig.~ \ref{nuvsq22} shows
the accessible kinematic region in $Q^2$ and $\nu$ for $E_{\nu}< $ 1.5 GeV.
For $E_{\nu}$= 3 GeV, about 1/3 of the total charged current cross section
originates from QE scattering, 1/3 from resonance production and
1/3 from inelastic scattering.
As seen in Fig.~\ref{nuvsq22} the $\nu<0.25$ GeV sample is dominated almost
entirely by QE events with $Q^2<0.45$ GeV$^2$.
The $\nu<0.5~GeV$ sample includes both QE events with $Q^2<0.95$ GeV$^2$ and
also $\Delta$(1232) resonance events with $Q^2<0.3$ GeV$^2$. Both samples includes a very small fraction of events originating from coherent pion production.
In the low $\nu$ region it is more convenient to write the expression
for the charged current differential cross sections as follows\cite{paschos,pyu}:
%\begin{widetext}
\begin{eqnarray} \di
\frac{d^2\si}{dQ^2 d\nu}= S_{cos}
\frac{1}{2E^2}{\cal W}_1 \left[Q^2+m_\mu^2\right] \nonumber \\
+ S_{cos}{\cal W}_2 \left[ (1-\frac {\nu}{E} ) - \frac {(Q^2+m_\mu^2)}{4E^2} \right] \nonumber\\
+S_{cos}{\cal W}_3 \left[ \frac{Q^2}{2ME} - \frac{\nu}{4E} \frac { Q^2+m_\mu^2}{ME} \right] \nonumber\\
+ S_{cos} {\cal W}_4 \left[ m_\mu^2 \frac{(Q^2+m_\mu^2)}{4M^2E^2}\right] \nonumber \\
- S_{cos}{\cal W}_5 \left[ \frac {m_\mu^2} {ME} \right]
%\nonumber
\label{cross-small}
\end{eqnarray}
%
where $S_{cos}=\frac{G^2}{2\pi}\cos^2\theta_C = 80\times 10^{-40}~cm^2/GeV^2$.
In the scattering process, there are additional small contributions from
strangeness and charm
non-conserving processes. In the discussion below we do not
show these terms explicitly, but
charm and strangeness changing contributions are assumed to be included in the analysis.
(The strangeness changing valence quark contributions are proportional to $\frac{G^2}{2\pi}\sin^2\theta_C$).
Each of the structure functions has a vector and axial component (except for ${\cal W}_3$ which originates from axial-vector interference).
The vector part of ${\cal W}_4$ and ${\cal W}_5$ are well known since they are related to the
vector part of ${\cal W}_2$ and ${\cal W}_1$ by the following expressions\cite{paschos}:
%
\begin{eqnarray}
{\cal W}_4^{vector}&=&{\cal W}_2^{vector}\frac{ M^2\nu^2 }{Q^4}-{\cal W}_1^{vector}\frac{ M^2 }{Q^2}
\nonumber \\
{\cal W}^{vector}_5&=&{\cal W}_2^{vector}\frac{ M\nu }{Q^2}\nonumber
\end{eqnarray}
At low $\nu$ and very high energy the charged current cross section is
only a function of ${\cal W}_2$. If we integrate the
cross section from $\nu_{min}\approx 0$ up to $\nu$= $\nu_{cut}$ (where $\nu_{cut}$ is small), we can write the expression for
the cross section in terms of ${\cal W}_2$ only, and energy dependent corrections
ratios to the ${\cal W}_2$ component:
%\begin{widetext}
\begin{eqnarray}
\displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}} \di
\frac{d^2\si}{dQ^2 d\nu}dQ^2 d\nu &=& \sigma_{{W}_2} + \sigma_{2}+\sigma_{1}+\sigma_{3}+\sigma_{4}+\sigma_{5}\nonumber
%&= & \sigma_{{W}_2}(E) \left[ 1+ f_{2} + f_{1} + f_{3}+f_{4}+f_{5} \right] \nonumber \\
%&=& \sigma_{{W}_2}({\infty}) \left[f_{C} \right]
\label{cross-w123}
\end{eqnarray}
%\begin{eqnarray}
%A &=& \frac{G^2_FM}{\pi}\displaystyle\int {\cal F}_2(x) dx,\\
%\nonumber
where we integrate the cross section up to $\nu<\nu_{cut}$ (of 0.25 or 0.5 GeV or
kinematic limit ).
Here,
$\sigma_{{W}_2} \approx \sigma_{{W}_2} (\infty)$, where
%\begin{widetext}
\begin{eqnarray}
\sigma_{{W}_2} &=& S_{cos} \displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}} {\cal W}_2 ~d\nu. \\
\sigma_{{W}_2} (\infty) &=& S_{cos} \displaystyle\int_{\nu_{min (E=\infty)}}^{\nu_{cut}} {\cal W}_2 ~d\nu.
\end{eqnarray}
and the small corrections to the QE cross section are:
\begin{eqnarray}
\sigma_{2} &=&S_{cos} \displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}}
\left[ -\frac{\nu}{E} -\frac{Q^2+m_\mu^2}{4E^2} \right]{\cal W}_2 ~d\nu
\nonumber \\
\sigma_{1} &=&S_{cos} \displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}}
- \left[ \frac{(Q^2+m_\mu^2)}{2E^2} \right]{\cal W}_1 ~d\nu \nonumber \\
\sigma_{3} &=&S_{cos} \displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}}
\left[ \frac{Q^2}{2ME} - \frac{\nu}{4E} \frac { Q^2+m_\mu^2}{ME} \right] {\cal W}_3 ~d\nu \nonumber\\
\sigma_{4} &=&S_{cos} \displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}}
\left[ m_\mu^2 \frac{(Q^2+m_\mu^2)}{4M^2E^2} \right] {\cal W}_4 ~d\nu \nonumber \\
\sigma_{5} &=&S_{cos} \displaystyle\int_{\nu_{min (E)}}^{\nu_{cut}}
\left[ \frac {-m_\mu^2} {ME} \right] {\cal W}_5 ~d\nu
\label{w123}
\end{eqnarray}
The above can be written in terms of fractional corrections:
\begin{eqnarray}
\sigma_{\nu cut}(E)&=& \sigma_{{W}_2}({\infty}) \left[f_{C} \right] \nonumber \\ \nonumber \\
f_{C}&=&\left[ f_{W2}+ f_{2} + f_{1} + f_{3}+f_{4}+f_{5} \right] \nonumber \\ \nonumber \\
f_{W2} &=&\frac {\sigma_{W2} }{\sigma_{{W}_2}(\infty)} \nonumber (\approx 1) \\
f_{2} &=&\frac {\sigma_{2} }{\sigma_{{W}_2}(\infty)} (=kinematic~correction)
\nonumber \\
f_{1} &=& \frac {\sigma_{1} }{\sigma_{{W}_2} (\infty)} (=important) \nonumber \\
f_{3}&=&\frac {\sigma_{3} }{\sigma_{{W}_2} (\infty)} (=important) \nonumber\\
f_{4}&=& \frac {\sigma_{4} }{\sigma_{{W}_2} (\infty)} (=very~small) \nonumber \\
f_{5}&=&\frac {\sigma_{4} }{\sigma_{{W}_2} (\infty))} (=very~small)
\label{fw123}
\end{eqnarray}
The energy dependent corrections $f_{W2}$,
$f_{1}$, $f_{2}$, $f_{3}$, $f_{3}$, and $f_{4}$ and $f_{5}$ can be calculated within a specific models.
The theoretical uncertainty in $f_C$ determines the systematic
uncertainty in the relative flux which can extracted from the low $\nu$ events.
\begin{enumerate}
\item $f_{W2}
=\frac {\sigma_{W2} }{\sigma_{{W}_2} (\infty)}
\approx 1$ is well known and does not contribute to the uncertainty in $f_C$.
\item The energy dependent correction $f_{2}$ is purely kinematic and
therefore does not contribute to the uncertainty in $f_C$.
\item The contributions of $f_{4}$ and $f_{5}$ are small since they are proportional to the square of the muon mass,
and therefore have a negligible contribution to the uncertainty in $f_C$. (Note that the vector parts of $f_{4}$ and $f_{5}$ are known very well since they can be expressed in terms of the vector parts of ${\cal W}_1$ and ${\cal W}_2$).
\item The only non-negligible uncertainty originates from the modeling of the contributions of $f_{1}$ and $f_{3}$ (primarily from $f_{3}$).
\end {enumerate}
%BEGIN FIG 4
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{4geniesigmanumaxcomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{4geniefractionnumaxcomparison.pdf}
\vspace{-0.3in}
\caption{Top panel: Neutrino partial charged cross sections per nucleon for low $\nu$ events (for $\nu$ cuts of 0.25, 0.5, 1, 2 and 5 GeV) as a function of energy determined from the GENIE Monte Carlo\cite{GENIE} for a carbon target. Also shown
are the measurements of MINOS on iron (for $\nu$ cuts of 1, 2 and 5 GeV). Bottom panel: The fraction of low $\nu$ neutrino events
in the GENIE\cite{GENIE} Monte Carlo as
compared with the measurements of MINOS. The MINOS cross sections for iron has been corrected for the excess number of neutrons in iron (color online). }
\label{nufrac}
\end{figure}
%END FIG 4
%BEGIN FIG 5
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{5genieantisigmanumaxcomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{5genieantifractionnumaxcomparison.pdf}
\vspace{-0.3in}
\caption{Top panel: Antineutrino partial charged cross sections per nucleon for low $\nu$ events (for $\nu$ cuts of 0.25, 0.5, 1, 2 and 5 GeV) as a function of energy determined from the GENIE\cite{GENIE} Monte Carlo for a carbon target. Also shown
are the measurements of MINOS on iron (for $\nu$ cuts of 1, 2 and 5 GeV). Bottom panel: The fraction of low $\nu$ antineutrino events
in the GENIE Monte Carlo as
compared with the measurements in MINOS. The MINOS cross sections for iron has been corrected for the excess number of neutrons in iron (color online). }
\label{nufrac2}
\end{figure}
%END FIG 5
The technique does not depend on the modeling of ${\cal W}_2$ because the $\sigma_{{W}_2}$ cross section is the same at all energies. All energy dependent corrections
are expressed in terms of ratios to $\sigma_{{W}_2}$. In quark parton language,
the uncertainty in $f_1$ is related to the uncertainty in the longitudinal structure function at low $Q^2$
and the uncertainty in $f_3$ is related to the uncertainty in level of antiquarks in the nucleon at low $Q^2$.
For QE scattering and resonance production the structure functions are expressed in terms of form factors.
\subsection{ Partial charged current cross sections}
The top panel of Fig. \ref{nufrac} shows
the partial neutrino charged current cross section per nucleon for low $\nu$ events (for $\nu$ cuts of 0.25, 0.5, 1, 2 and 5 GeV) as a function of energy as determined by the GENIE\cite{GENIE} Monte Carlo for a carbon target.
The top panel of Fig. \ref{nufrac2} shows the corresponding partial charged current cross sections for antineutrinos.
Also shown
are the measurements of the partial charged current cross sections on iron from the MINOS collaboration (for $\nu$ cuts of 1, 2 and 5 GeV).
The MINOS cross sections for iron has been corrected for the excess number of neutrons in iron. Note that the partial cross sections on carbon and on iron can be somewhat different.
At high energies (as shown in Fig. \ref{nufrac} and \ref{nufrac2}) the partial cross sections for a fixed $\nu_{cut}$ are independent of energy and are approximately equal for neutrinos and antineutrino. The fact that these partial charged current cross section are relatively independent of energy is the basis for the low $\nu$ method.
The bottom panels of figures \ref{nufrac} and \ref{nufrac2} show the fraction of low $\nu$ events predicted by the GENIE Monte Carlo as
compared with the measurements in MINOS.
In order to use the technique at low energies the fractions must be smaller than 0.6. Therefore,
at the lowest energies we must use $\nu$ cuts of 0.25 and 0.50 GeV.
MINOS is a sampling
target calorimeter which has poor resolution at low hadron energy. Therefore, low $\nu$ samples with
$\nu< 0.25~GeV$ and $\nu< 0.5~GeV$ cannot be defined reliably.
On the other hand, since the MINERvA detector is a fully active target calorimeter, low $\nu$ samples with $\nu< 0.25~GeV$ and $\nu< 0.5~GeV$ can be used.
\subsection{ Absolute normalization}
Since the neutrino energy range for MINERvA is limited to lower energies,
we propose that the MINERvA charged current cross section
measurements be normalized to the cross section in the energy range
between 10 to 20 GeV (e.g. at a mean energy of 15.1 GeV).
The absolute level of the charged current cross section
at this energy range has been measured by both the MINOS and NOMAD collaborations.
The MINOS total cross section measurement
for an isoscalar iron target at a neutrino energy of 15.1 GEV is
$$\sigma^{MINOS}_\nu /E = 0.708\pm0.020\times 10^{-38}cm^2/GeV $$
per nucleon in iron. Here the total error of 0.02 is the combined
statistical, systematic and normalization errors of $0.008\pm0.012\pm0.015$,
respectively.
The NOMAD cross section measurement
for an isoscalar carbon target at a neutrino energy of 15.1 GEV is
$$\sigma^{NOMAD}_\nu/ E= 0.698\pm0.025\times 10^{-38}cm^2/GeV $$
per nucleon in carbon.
The MINOS total cross section measurement
for an isoscalar iron target at an antineutrino energy of 15.1 GEV is
$$\sigma^{MINOS}_{\nub} /E = 0.304\pm0.012\times 10^{-38}cm^2/GeV $$
per nucleon in iron. Here, the total error of 0.012 is the combined
statistical, systematic and normalization errors of $0.007\pm0.007\pm0.006$,
respectively.
Alternatively, it may be possible for MINERvA to normalize to the partial cross sections measured by
MINOS for $\nu< 1~GeV$ and $\nu< 2~GeV$ at 15.1 GeV. These partial cross sections (which were used by MINOS to determine their relative flux)
are relatively constant between 10 and 20 GeV. However, the MINOS partial cross sections are measured on iron. The MINERvA target
is solid scintillator (i.e. carbon), and the partial cross sections for iron and carbon can be different.
For a neutrino
energy of 15.1 GeV MINOS
measured the following isoscalar partial cross sections on iron (per nucleon).
$$\sigma^{MINOS}_{\nu} (15.1)=1.729\pm 0.049 \times 10^{-38}cm^2 ( \nu<2~GeV) $$
$$\sigma^{MINOS}_{\nu}(15.1) = 0.968\pm 0.027\times 10^{-38}cm^2 ( \nu<1~GeV) $$
For an antineutrino
energy of 15.1 GeV MINOS
has measured the following isoscalar partial cross sections on iron (per nucleon).
$$\sigma^{MINOS}_{\nub} (15.1)= 1.585\pm 0.063 \times 10^{-38}cm^2 ( \nu<2~GeV) $$
$$\sigma^{MINOS}_{\nub}(15.1) = 0.939\pm 0.039 \times 10^{-38}cm^2 ( \nu<1~GeV) $$
% BEGIN FIG 6
% CONTRIBUTIONS to Sigma nu<0.25 GeV
\begin{figure}
\includegraphics[width=3.7in,height=2.6in]{6totalcarbonnumax25grandcomponents.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.15in}
\includegraphics[width=3.7in,height=2.6in]{6antitotalcarbonnumax25grandcomponents.pdf}
\vspace{-0.2in}
\caption{The $\nu<0.25~ GeV$ partial charged current cross sections (per nucleon) as a function of energy from the GENIE Monte Carlo
(for carbon target).
Shown are the QE contribution, the contribution from pion production process (e.g. $\Delta$, inelastic and coherent pion) and
the total. The $\nu<0.25~ GeV$ cross sections for $\nu_\mu$
are shown on the top panel, and the $\nu<0.25~ GeV$ cross sections for $\nub_\mu$
are shown on the bottom panel. Most of the $\nu<0.25~ GeV$ events are QE and
only a very small contribution is from pion production processes (color online).}
\label{QEvsDelta25}
\end{figure}
% End figure 6
%
\section {Using low $\nu$ events with $ \nu<0.25~GeV$ }
%
As seen in Fig.~\ref{nuvsq22} the $\nu<0.25$ region is dominated by QE events.
This is illustrated in Fig.~\ref{QEvsDelta25} which shows the relative contributions
of QE and non-QE processes to $\nu<0.25~ GeV$ cross section as a function of energy (as determined from the GENIE Monte Carlo). The $\nu<0.25~ GeV$ cross sections for $\nu_\mu$
are shown on the top panel, and the $\nu<0.25~ GeV$ cross sections for $\nub_\mu$
are shown on the bottom panel.
The QE contribution is shown in red, the contribution from pion production process (e.g. $\Delta$, inelastic and coherent pion) is shown in blue and and the total is shown in black.
Most of the events are QE and the contribution from pion production processes is
negiligible.
As mentioned earlier, the technique does not rely the modeling of ${\cal W}_2$,
or the modeling of nuclear effects (e.g. Fermi motion smearing) on ${\cal W}_2$. This is because the cross section $\sigma_{{W}_2}$ (including nuclear effects) is the same at all neutrino energies.
The uncertainty in the flux extracted from the event sample with $ \nu<0.25~GeV$ is determined by how well we can model the relative contributions of ${\cal W}_1$ and ${\cal W}_3$ for the case of QE scattering on bound nucleons, or equivalently the relative contributions of $f_{1}$ and $f_{3}$ to $f_C$. Here $f_{1}$ and $f_{3}$ are proportional to the ratios $\frac{{\cal W}_1}{{\cal W}_2}$ and $\frac{{\cal W}_3}{{\cal W}_2}$. Since the ratios $\frac{{\cal W}_1}{{\cal W}_2}$ and $\frac{{\cal W}_3}{{\cal W}_2}$ for QE scattering on free nucleons are very well known, the uncertainty in $f_C$ originates primarily
from modeling the nuclear corrections to $\frac{{\cal W}_1}{{\cal W}_2}$ and $\frac{{\cal W}_3}{{\cal W}_2}$ for nucleons bound in a nuclear target.
%
%
%
%figure 7
\begin{figure}[ht]
\includegraphics[width=3.3in,height=2.5in]{7GMPN.pdf}
\caption{ The transverse enhancement ratio\cite{transverse} (${\cal R}_{T}$) as a function of $Q^2$. Here, ${\cal R}_{T}$ is ratio of the integrated transverse response function
for QE electron scattering on nucleons bound in carbon divided by
the integrated response function for independent nucleons.
The black points are extracted from Carlson $et~al$\cite{MEC4}, and
the blue bands are extracted from a fit\cite{vahe-thesis} to QE data from the JUPITER\cite{JUPITER} experiment (Jlab experiment E04-001). The curve is a fit to the data of the form ${\cal R}_{T}=1+AQ^2e^{-Q^2/B}$. The dashed lines are the upper and lower error bands (color online).
}
\label{GMPN}
\end{figure}
%END FIG 7
% BEGIN FIGURE 8
\begin{figure}
\includegraphics[width=3.5 in,height=2.9in]{8elas_nu_log_mar_enu_meson_lagr.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.5 in,height=2.9in]{8elas_nub_log_mar_enu_meson_lagr.pdf}
\vspace{-0.3in}
\caption{Comparison of predictions for the $\nu_{\mu}$, $\bar{\nu}_\mu$ total QE cross section sections from the nominal TE model,
the "Independent Nucleon (MA=1.014)" model, the
"Larger $M_A$ ($M_A$=1.3) model", and the
"QE+np-nh RPA" MEC model of Martini et al.\cite{MEC5}
The data points are the measurements of MiniBooNE\cite{MiniBooNE} (gray stars) and NOMAD\cite{NOMAD} (purple circles) (color online).}
\label{totalMEClog}
\end{figure}
%END FIGURE 8
%
%BEGIN FIG 9
\begin{figure}
\includegraphics[width=3.7 in,height=2.9in]{9qetransversecontributionsnonumax.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7 in,height=2.9in]{9antiqetransversecontributionsnonumax.pdf}
\vspace{-0.2in}
\caption{Contribution of the various components
($\sigma_{W_2}$, $\sigma_{2}$, $\sigma_{1}$, $\sigma_{3}$, $\sigma_{4}$, $\sigma_{5}$)
to the total QE cross section (a predicted by the TE model). Top panel: Neutrinos. Bottom panel: Antineutrinos (color online).
}
\label{totalMECcomp}
\end{figure}
% END FIG 9
%
\subsection {Quasielastic $\nu_{\mu},\bar{\nu}_\mu$ scattering}
%
The relationship between the structure functions and form factors for $\nu_\mu,\bar{\nu}_\mu$
QE scattering\cite{Lle72} on free nucleons is given by\cite{transverse,steffens}:
$$W^{\nu-vector}_{1-Qelastic} =\delta(\nu-\frac{Q^2}{2M})\tau |{\cal G}_M^V (Q^2)|^2$$
$$W^{\nu-axial}_{1-Qelastic} = \delta(\nu-\frac{Q^2}{2M})(1+\tau)|{\cal F}_A (Q^2)|^2$$
$$W^{\nu-vector}_{2-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})|{
\cal F}_V (Q^2)|^2$$
$$W^{\nu-axial}_{2-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})|{\cal F}_A (Q^2)|^2$$
$$W^{\nu}_{3-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})|2 {\cal G}_M^V(Q^2) {\cal F}_A (Q^2)|$$
$$W^{\nu-vector}_{4-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})\frac{1}{4}(|{\cal F}_V (Q^2)|^2 - |{\cal G}_M^V (Q^2)|^2)$$
$$W^{\nu-axial}_{4-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})\times \frac{1}{4} \times $$
$$ \left[ {\cal F}_A^2(Q^2)+(\frac{Q^2}{M^2}+4)|{\cal F}_p(Q^2)|^2 -({\cal F}_A(Q^2) +2{\cal F}_P(Q^2))^2 \right]$$
$$W^{\nu-vector}_{5-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})\frac{1}{2}|{\cal F}_V (Q^2)|^2$$
$$W^{\nu-axial}_{5-Qelastic} =
%2\frac {M}{\nu}
\delta(\nu-\frac{Q^2}{2M})\frac{1}{2}|{\cal F}_A (Q^2)|^2$$
where
$$ {\cal G}_E^V(Q^2)=G_E^p(Q^2)-G_E^n(Q^2), $$
$$
{\cal G}_M^V(Q^2)=G_M^p(Q^2)-G_M^n(Q^2).
$$
and
$$ | {\cal F}_V(Q^2)|^{2}=
\frac{[{\cal G}_E^V(Q^2)]^2+ \tau [{\cal G}_M^V(Q^2)]^2}{1+\tau}.$$
Here, $G_E^p(Q^2)$, $G_E^n(Q^2)$, $G_M^p(Q^2)$ and $G_M^n(Q^2)$
are the electric and magnetic nucleon form factors, which are measured
in electron scattering experiments.
Note that:
$$\sigma_T^{vector}\propto \tau |{\cal G}^V_M (Q^2)|^2;~~\sigma_T^{axial}\propto (1+\tau)|{\cal F}_A (Q^2)|^2$$
$$\sigma_L^{vector}\propto {({\cal G}_E^V(Q^2))^2}; ~~~\sigma_L^{axial}= 0$$
Therefore, for QE $\nu_{\mu},\bar{\nu}_\mu$ scattering only ${\cal G}_M^V$ contributes
to the transverse virtual
boson absorption cross section.
%
%
\subsection{Transverse enhancement QE scattering from nuclei}
%
%
Studies of QE electron scattering on nuclear targets\cite{MEC4} indicate
that only the longitudinal part of the QE cross section can be described in terms of a universal response function of
independent nucleons bound in a nuclear potential (and free nucleon form factors). In contrast, a significant additional enhancement with respect to the model is observed in the transverse part of the QE cross section.
The enhancement in the transverse QE cross section has been attributed to meson exchange currents (MEC) in a nucleus. Within models of meson exchange currents the enhancement is primarily in the transverse part of the QE cross section, while the enhancement in the longitudinal QE cross section is small (in agreement with the electron scattering experimental data).
The conserved vector current hypothesis (CVC) implies that the corresponding vector structure function for the QE cross section in $\nu_{\mu},\bar{\nu}_\mu$ scattering can be expressed in terms of the structure functions measured in electron scattering on nuclear targets. Therefore, there should also be a transverse enhancement in neutrino scattering. In models of meson exchange currents the enhancement in the
axial part of $\nu_{\mu},\bar{\nu}_\mu$ QE cross section on nuclear targets is also expected small.
The transverse enhancement observed in electron scattering is a function of both $Q^2$ and
$\nu$. However, a simple way to account for the integrated transverse enhancement\cite{transverse} from nuclear
effects is to assume that $G_M^p(Q^2)$ and $G_M^n(Q^2)$ are enhanced
in a nuclear targets by factor
$\sqrt{R_{TL}}$.
Bodek, Budd and Christy\cite{transverse} have used
electron scattering data\cite{MEC4,JUPITER,vahe-thesis} to parametrize ${R_{TL}}$ as follows:
$$R_{TL}=1+AQ^2e^{-Q^2/B}$$
with $A=6.0$ and $B=0.34$ GeV$^2$. The electron scattering data indicates that the transverse enhancement is maximal near $Q^2$=0.3 GeV$^2$ and is small for $Q^2$ greater
than 1.5 GeV$^2$.
The upper error band is given by $A=6.7$ and $B=0.35$ GeV$^2$, and
the lower error band is given by $A=5.3$ and $B=0.33$ GeV$^2$.
In modeling $\nu_{\mu},\bar{\nu}_\mu$ QE scattering on nuclear targets we use $BBBA2007_{25}$ parametrization\cite{quasi}
of the free nucleon electromagnetic form factors $G_E^p(Q^2)$, $G_E^n(Q^2)$,
$G_M^p(Q^2)$ and $G_M^n(Q^2)$ (with $M_V^2=0.71$ GEV$^2$), and a dipole axial form factor with $M_A=1.014~GeV$. We apply the transverse enhancement correction to $G_M^p(Q^2)$ and $G_M^n(Q^2)$. We also apply Pauli blocking corrections to the differential QE cross section as parametrized by Paschos and Yu\cite{pyu}.
% as implemented in the $NEUGEN$ Monte Carlo\cite{GENIE}.
We refer to this model as the Transverse Enhancement (TE) model. This is the nominal model that is used
in this paper.
We also compare calculations based on the nominal TE model to two other
models. The first model is the independent nucleon model with Pauli blocking with $M_A=1.014~GeV$, without transverse enhancement. We refer to this model as the "Independent Nucleon (MA=1.014)" model. This model is very close to the model which is currently implemented in the GENIE Monte Carlo (the GENIE default value is $M_A=0.99~GeV$").
The second model is the independent nucleon model with Pauli blocking, $M_A=1.3$ GeV, without transverse enhancement. We refer to this model as the "Larger $M_A$ ($M_A$=1.3) model". We use the difference between the three models as a conservative systematic error on the flux extracted from the $\nu$ samples.
Fig. \ref{totalMEClog} shows a comparison of predictions of various model predictions for the $\nu_{\mu}$, $\bar{\nu}_\mu$ total QE cross section sections to experimental data on nuclear targets.
Shown are "Independent Nucleon (MA=1.014)" model, the
"Larger $M_A$ ($M_A$=1.3) model", and the TE model
(with upper and lower error bands).
Also shown are the predictions of the "QE+np-nh RPA" MEC model of Martini et al.\cite{MEC5} The data points are the QE cross section measurements of MiniBooNE\cite{MiniBooNE} (gray stars) and NOMAD\cite{NOMAD} (purple circles). Note that there is an overall $\approx10\%$ systematic error in the experimental QE cross sections because of uncertainties in the determination of the neutrino and antineutrino fluxes in each of the two experiments.
In this paper we use the error band in the transverse enhancement parameters as a lower limit on systematic error in the modeling. We use the "Independent Nucleon (MA=1.014)" and the "Larger $M_A$ ($M_A$=1.3) model" as conservative upper limits on the errors in the modeling.
Fig. \ref{totalMECcomp} shows the contribution of the various components
($\sigma_{W_2}$, $\sigma_{2}$, $\sigma_{1}$, $\sigma_{3}$, $\sigma_{4}$, $\sigma_{5}$) to the total QE cross section (as defined by Eq. \ref {w123}) as a function
of incident energy. These contributions are calculated using the TE model. The top panel shows the contribution of the various
components for the neutrino QE cross section, and the bottom panel shows
the contribution of the various components for the antineutrino QE cross section.
%As mentioned earlier, only the uncertainties in $\sigma_{1}$, $\sigma_{3}$ contributed to the uncertainty of % the extracted flux from
%the $nu<0.25$ GeV sample.
%
%
%BEGIN FIG 10
\begin{figure}
%[t]
\includegraphics[width=3.5in,height=3.5in]{10neutrino-cross.pdf}
%\includegraphics[width=3.75in,height=2.7in]{10neutrino-crossA-not-used.pdf}
\caption{The MINOS\cite{MINOS2}(iron), NOMAD\cite{NOMAD}(carbon) and Serp96\cite{Serp96} (aluminum) measurements of $\sigma_{total}/E$ per nucleon on isoscalar nuclear targets for
$\nu$ in units of $10^{-38}cm^2/GeV$ (with statistical, systematic and normalization errors combined in quadrature). We also show BNL82\cite{BNL82} data which was taken with a deuterium target corrected for the additional transverse enhancement in the QE contribution to the total cross section (in carbon) as discussed in the text.
The orange line shows the predictions of the unmodified GENIE Monte Carlo. The QE cross section in the GENIE MC is shown as the blue line. The QE contribution calculated with the TE model
is show as a green line. The dashed blue line shows the prediction of the modified GENIE MC (using the TE model QE cross section instead).
The thick brown line is a parametrization described in the text (color online). }
\label{cross}
\end{figure}
%END FIG 10
%
%BEGIN FIG 11
\begin{figure}
%[t]
\includegraphics[width=3.5in,height=2.7in]{11antineutrino-cross.pdf}
\includegraphics[width=3.5in,height=2.7in]{11ratio-cross.pdf}
\caption{Same as Fig. \ref{cross} but (a) for the antineutrino charged current cross section, (b) for the ratio of antineutrino and neutrino total cross sections (color online).
}
\label{cross2}
\end{figure}
%END FIG 11
%%%
\subsection{Neutrino and antineutrino total cross sections}
%
The MINOS collaboration uses the criteria that the fraction of low $\nu$ events that are used for the determination of the relative neutrino flux in an energy bin should be less than 60\% of the total number of charged current events. In order to test for this fraction, we need to use a parametrization
to estimate the energy dependence of the neutrino and antineutrino charged current total cross sections.
Fig. \ref{cross} and \ref{cross2} show the $\nu_\mu$ and $\nub_\mu$ total charged current cross sections measured on isocalar nuclear targets by the MINOS\cite{MINOS2} (iron), NOMAD\cite{NOMAD}(carbon) , and Serpukov\cite{Serp96} (Serp96, aluminum) experiments. The total cross sections
per nucleon (divided by neutrino energy) are shown in units of $10^{-38}cm^2/GeV$ (with statistical, systematic and normalization errors combined in quadrature). The ratio
of the $\nub_\mu$ and $\nu_\mu$ total charged current cross sections is shown in the bottom panel of Fig. \ref{cross2}.
The cross sections reported by the MINOS collaboration were measured using a neutrino
flux extracted from low $\nu$ samples with $\nu$ less than 1, 3, and 5 GeV.
Also shown in Fig. \ref{cross} are low energy cross sections measured by at BNL\cite {BNL82} (BNL82). Since the BNL82
cross sections were measured on a deuterium target we we apply a correction
to account for nuclear effects. The BNL82 points shown in the figure were increased
by the difference of the predictions of the TE model for the QE cross section
(which is expected to describe the cross section on a heavy nuclear target) and the
"Independent Nucleon (MA=1.014)" model (which is expected to describe the QE cross sections on deuterium).
The orange line shows the predictions of the GENIE Monte Carlo. The QE cross sections
in the GENIE MC are a calculated using the independent nucleon model
with $M_A=0.99$ GeV. The QE contribution to the cross section from GENIE is shown as a blue line.
The QE contribution calculated with the TE model
is show as a green line.
The curve labeled GENIE with QE-TE (shown as a dotted blue line)
represent the GENIE cross section increased
by the difference of the predictions of the TE model for the QE cross section
(which is expected to describe the cross section on a heavy nuclear target) and the
"Independent Nucleon (MA=0.99)" model (which is currently implemented in GENIE).
In our investigation of the low $\nu$ technique, we use a parametrization to estimate
the total $\nu_\mu$, $\nub_\mu$ charged current cross sections. The parametrization, which is shown as the thick red line in Fig. \ref{cross}, is given by
$$\frac{\sigma_\nu}{E_\nu} = [A + B~e^{-E_\nu/C1}+D~e^{-E_\nu^2/C2}](1-Ke^{-(E_\nu-0.1)/C3})$$
where for $\nu_\mu$ we use $A_\nu$=~0.675, $B_\nu$=~0.12, $C1_\nu$=~9~GeV, $D_\nu$~=0.4, $C2_\nu$=~3 GeV$^2$, $C3_\nu$=~0.22 GeV, and $K=~1.0$.
For
$\nub_\mu$ we use $A_{\nub}$=0.329, $B_{\nub}$=~-0.06 and $C1_{\nub}$=13~GeV $D_{\nub}$=~0.09, $C2_{\nub}$=~30 GeV$^2$ , $C3_\nu$=~0.8 GeV, and $K=~0.8$. Here, $\frac{\sigma_\nu}{E_\nu}$ is
total charged current cross section per nucleon
in units of $10^{-38}cm^2/GeV$.
The above form is constrained to yield the average world cross section measurements in the 30 to 50 GeV region of
of 0.675 $10^{-38}cm^2/GeV$, and 0.329 $10^{-38}cm^2/GeV$ for
$\nu_\mu$ and $\nub_\mu$, respectively.
We only use this parametrization to estimate the fractional contribution of
low $\nu$ events to the total cross section to determine the region where it is less than 60$\%$.
When improved total cross section measurements become available (e.g. from MINERvA), this parameterization can be updated to include the new data.
%%
%
%
%
%BEGIN FIG 12
\begin{figure}
\includegraphics[width=3.7 in,height=3.0in]{12qetransversecontributionsnumax25.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7 in,height=3.0in]{12antiqetransversecontributionsnumax25.pdf}
\vspace{-0.3in}
\caption{Contribution of the various components
($\sigma_{W_2}$, $\sigma_{2}$, $\sigma_{1}$, $\sigma_{3}$, $\sigma_{4}$, $\sigma_{5}$) to the $\nu< 0.25$ GeV partial charged current cross section. This sample is is dominated by QE $\nu_\mu N \rightarrow \mu^- P$ events. Top panel: Neutrinos. Bottom panel: Antineutrinos (color online).
}
\label{nu25comp}
\end{figure}
%END FIG 12
%figure 13
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{13Totalfcnumax25.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{13Totalgrandfractionnumax25.pdf}
\vspace{-0.3in}
\caption{The $\nu<0.25$ GeV sample for $\nu_\mu$ scattering on carbon. This sample is dominated by QE $\nu_\mu N \rightarrow \mu^- P$ events. Top panel: The total correction factor $f_C$ (black line), the contribution
of the kinematic correction to ${\cal W}_2$ ($f_2$) (yellow line), the contributions from ${\cal W}_1$ ($f_1$) (pink line), the contribution from ${\cal W}_3$ ($f_3$) (blue line), and the very small contributions of ${\cal W}_4$ ($f_4$), and ${\cal W}_5$ ($f_5$). Bottom panel: The fractional contribution of
$\nu<0.25$ GeV events to the total $\nu_\mu$ charged current cross section. The fraction is less than 60$\%$ for $\nu_\mu$ energies above 0.7 GeV (color online).
}
\label{neutrino25}
\end{figure}
%END FIG 13
%
% BEGIN FIG 14
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{14Antitotalfcnumax25.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{14Antitotalgrandfractionnumax25.pdf}
\vspace{-0.3in}
\caption{ The $\nu<0.25$ GeV sample for $\nub_\mu$ scattering on carbon. This sample is dominated by QE $\nub_\mu P \rightarrow \mu^+ N$ events. Top panel: The total correction factor $f_C$ (black line), the contribution of the kinematic correction to ${\cal W}_2$ ($f_2$) (yellow line), the contributions from ${\cal W}_1$ ($f_1$) (pink line), the contribution from ${\cal W}_3$ ($f_3$) (blue line), and the very small contributions of ${\cal W}_4$ ($f_4$), and ${\cal W}_5$ ($f_5$). Bottom panel: The fractional contribution of
$\nu<0.25$ GeV events to the $\nub_\mu$ total charged current cross section. The fraction is less than 60$\%$ for $\nub_\mu$ energies above 1.0 GeV (color online). }
\label{antineutrino25}
\end{figure}
% END FIG 14
%
%
% BEGIN FIG 15
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{15Totalfcerrornumax25pdf.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{15Antitotalfcerrornumax25.pdf}
\vspace{-0.3in}
\caption{ The error band in the correction factor $f_C$ for $\nu<0.25 $ GeV. Top panel: Neutrinos. Bottom panel: Antineutrinos (color online). }
\label{nudiff}
\end{figure}
% END FIG 15
%
%
%BEGIN Fig 16
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{16fcbarnumax25grandcomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{16antifcbarnumax25grandcomparison.pdf}
\vspace{-0.3in}
\caption{ Comparisons of our calculated values of the normalized
${\bar f_{C:\nu<0.25}} (15.1)(E)$ (=$\bar f_C (15.1) $ for $\nu<0.25 $ GeV)
to values from the GENIE MC.
The values calculated with the nominal TE model for QE scattering ($M_A=1.014$ GeV) are shown in black. The values calculated assuming no transverse enhancement and $M_A=1.014$ GeV are shown in red.
The GENIE prediction (which has no transverse enhancement and uses $M_A=0.99$ GeV) is close to the red curve as expected. As discussed in the text, the values for $M_A=1.014$ GeV (red line) and $M_A=1.3$ GeV (blue line) are very close to each other (color online). }
\label{fcbar25}
\end{figure}
% END FIG 16
%
%
\subsection{Results with $\nu<0.25$ GeV}
%
Fig. \ref{nu25comp} show the
contribution of the various components
($\sigma_{W_2}$, $\sigma_{2}$, $\sigma_{1}$, $\sigma_{3}$, $\sigma_{4}$, $\sigma_{5}$) to the $\nu<0.25$ GeV partial cross section. This sample is is dominated by QE $\nu_\mu N \rightarrow \mu^- P$ events. The partial cross section as a function of energy
for neutrinos is shown in the top panel and the partial cross section
for antineutrinos is shown in the bottom panel. The partial cross section (per nucleon) is calculated on a carbon target using the TE model.
The uncertainty in the relative values of the $\nu<0.25$ GeV partial cross section
as a function of energy determines the uncertainty in the determination of the relative fluxes. Here
$f_C (E) $ is the ratio of the partial cross section to the value of the partial cross section
at $E=\infty$.
Fig. \ref{neutrino25}(a) (top) shows the correction factor
$f_C$ for the $\nu<0.25$ GeV sample for neutrinos as a function energy. The error bands
in $f_C$ (originating from the uncertainty in the transverse enhancement) are shown as the dotted lines,
and represent the lower limit on errors. Also shown on the figure
is the negative contribution from the kinematic correction $f_2$ (which is well known), and the contributions
of $f_1$, $f_3$, $f_4$ and $f_5$. Here the contribution of $f_4$ and $f_5$ is
negligible. For the case of neutrino scattering, the positive contributions of $f_1$ and $f_3$ partially cancel the negative contribution of $f_2$. Fig. \ref{neutrino25}(b) (bottom) shows the fractional contribution of the
$\nu<0.25$ GeV sample to the total neutrino charged current cross section. This fraction is less than 60$\%$ for $\nu_\mu$ energies above 0.70 GeV.
Fig. \ref{antineutrino25}(a) (top) shows the correction factor
$f_C$ for the $\nu<0.25$ GeV sample for antineutrinos as a function energy. The error bands
in $f_C$ (originating from the uncertainty in the transverse enhancement) are shown as the dotted lines,
and represent the lower limit on errors. Also shown on the figure
is the negative contribution from the kinematic correction $f_2$ (which is well known), and the contributions
of $f_1$, $f_3$, $f_4$ and $f_5$. Here the contribution of $f_4$ and $f_5$ is
negligible. For the case of antineutrino scattering $f_3$ changes sign, and both
$f_2$ and $f_3$ are negative. Fig. \ref{antineutrino25}(b) (bottom) shows the fractional contribution of the
$\nu<0.25$ GeV sample to the total antineutrino charged current cross section. This fraction is less than 60$\%$ for $\nub_\mu$ energies above 1.0 GeV.
%
\subsubsection{Uncertainty in the $f_C$ correction factors}
%
It has been traditional to use the value and error in the effective $M_A$ extracted
from neutrino scattering data as an estimate of various uncertainties. Typically, the difference
between results with
$M_A=1.014$ GeV and $M_A=1.3$ GeV are used an upper limit on the error.
We find that the values of the $f_C$ correction factor
are insensitive to $M_A$. This is because
at small $Q^2$, both ratios $f_1$, and $f_3$ are insensitive to $M_A$. Specifically, both
$$\frac{{\cal W}_1^{QE}}{{\cal W}_2^{QE}} = \frac{(1+\tau)|{\cal F}_A (Q^2)|^2+\tau |{\cal G}_M^V(Q^2)|^2}{|{\cal F}_A (Q^2)|^2+[{\cal F}_V(Q^2)]^2}$$
$$\frac{{\cal W}_3^{QE}}{{\cal W}_2^{QE}} = \frac{|2 {\cal G}_M^V(Q^2) {\cal F}_A (Q^2)|}{|{\cal F}_A (Q^2)|^2+[{\cal F}_V(Q^2)]^2},$$
%where
% $$ | {\cal F}_V(Q^2)|^{2}=
%\frac{[{\cal G}_E^V(Q^2)]^2+ \tau [{\cal G}_M^V(Q^2)]^2}{1+\tau},$$
are insensitive to $M_A$ because the change in $F_A$ at small $Q^2$ is small.
Since $f_C$ is insensitive to large variations in $M_A$ one may naively surmise that the error in $f_C$ is small.
%Similarly, if we use the transverse enhancement model and use the errors in the TE
%parameters, we also get a small error in $f_C$.
However, we find that the difference between the values $f_C$ calculated
with and without transverse enhancement is larger than the error estimate
extracted from the uncertainty in $M_A$. This is because $\frac{{\cal W}_3^{QE}}{{\cal W}_2^{QE}}$ is sensitive
to ${\cal G}_M^V(Q^2)$, which depends on the magnitude of the
transverse enhancement at small $Q^2$.
Fig. \ref{nudiff} shows the errors in $f_C$ from the uncertainty in the TE parameters. The
error originating
from uncertainties in the TE parameters is also very small (less than 0.005).
We obtain a more conservative estimate of the systematic error from uncertainties in the modeling
the QE cross section by taking the difference
between $f_C$ calculated with and without transverse enhancement. At the lowest energy of
0.7 GeV, this difference is -0.05 for $\nu_\mu$. Since at 0.7 GeV $f_C^{\nu} \approx1.3$ this
corresponds to a maximum error in the determination of the $\nu_\mu$ flux of 3.8\%.
For $\nub_\mu$ the difference
between $f_C$ calculated with and without transverse enhancement at an energy of
1.0 GeV is +0.03. Since at 1.0 GeV $f_C^{\nub} \approx0.6$
this corresponds a maximum error in the determination of the $\nub_\mu$ flux of 5\%.
These differences can be used as
upper limits on the model uncertainties in $f_C$. If one takes the average of all the models, a conservative upper
limit of the model uncertainty in the relative flux extracted from the $\nu<0.25$ GeV sample is 1.9\% for $\nu_\mu$ energies above 0.7 GeV and 2.5\% for $\nub_\mu$ energies above 1.0 GeV.
A study of the $Q^2$ distributions of QE events in MINERvA can
be used to constrain the $Q^2$ dependence of the QE differential cross sections and thus reduce the model dependence in the determination of the relative flux to a negligible level (and also extend the technique to lower energies).
\vspace{-0.1in}
%
%
\subsection{Comparison to GENIE and ${\bar f_{C:\nu<0.25}}$(15.1 GeV)}
%
We have used a sample of events generated by the GENIE Monte Carlo.
Our studies are done at the generated
level and therefore do not depend on the
detector parameters or energy resolutions of any specific experiment.
We extract the energy dependence of the
$\nu<0.25$ GeV cross section from the GENIE MC sample using
the following expression:
$$ \sigma_{\nu<0.25} ^{MC}(E) = \frac {N ^{MC}_{\nu<0.25} (E) }{N^{MC}_{QE}(E)} \times \sigma_{QE} ^{MC}(E) $$
where the superscript $MC$ refers to events generated by the GENIE Monte Carlo.
Here, $N ^{MC} (E)$ is the number of events generated by the Monte Caro
with neutrino energy E, and $N ^{MC}_{\nu<0.25}(E)$ is the subset of these events
with $\nu<0.25$ GeV.
As mentioned earlier, we propose that the neutrino cross sections at low energy
be measured relative to the neutrino cross section at 15.1 GeV.
For any cross section
model we can define the normalized quantity ${\bar f_{C:\nu<0.25}}$(15.1 GeV) as:
$${\bar f_{C:\nu<0.25}} (15.1)(E)=\sigma_ {\nu<0.25} (E)/ \sigma_ {\nu<0.25} (E=15.1~ GeV)$$
which is equivalent to
$${\bar f_{C:\nu<0.25}} (15.1)(E) = f_{C:\nu<0.25} (E) /f_{C:\nu<0.25}(E=15.1~GeV)$$
We compare the values of
${\bar f_{C:\nu<0.25}} (15.1)(E) $ predicted by the GENIE MC to our calculations.
For all of the models, the value of $f_{C:\nu<0.25} ($E=15.1~GeV)=1.000 (for $\nu$) and 0.995 (for $\nub$). These values can be used to convert between ${\bar f_{C:\nu<0.25}}(E)$ and ${f_{C:\nu<0.25}} (E)$.
Comparisons of our calculated values of the normalized
${\bar f_{C:\nu<0.25}} (15.1)(E)$ to values from the GENIE MC are shown in Fig. \ref{fcbar25}.
The top panel shows the comparison for neutrinos and the bottom panel shows the comparison
for antineutrinos.
Our calculation for the TE model is shown in black. Our calculation assuming no transverse enhancement and $M_A=1.014$ GeV is shown in red.
As mentioned earlier, the values for $M_A=1.014$ GeV (red line) and $M_A=1.3$ GeV (blue line) are very close to each other. The GENIE prediction (which has no transverse enhancement and uses $M_A=0.99$ GeV) is close to the red curve as expected.
%
%
%
%
% BEGIN FIG 17
% CONTRIBUTIONS to Sigma nu<0.5 GeV
\begin{figure}
\includegraphics[width=3.7in,height=3.0in]{17totalcarbonnumax5grandcomponents.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=3.0in]{17antitotalcarbonnumax5grandcomponents.pdf}
\vspace{-0.3in}
\caption{The $\nu<0.5$ GeV partial charged current cross section as a function of energy from the GENIE Monte Carlo.
The QE contribution is shown in red, the contribution from pion production process ($\Delta$, inelastic and coherent pion production) is shown in blue, and
the total $\nu<0.5$ GeV partial cross section is shown in black. The $\nu<0.5$ GeV partial cross section for $\nu_\mu$
is shown in the top panel, and the $\nu<0.5$ GeV partial cross section for $\nub_\mu$
is shown in the bottom panel.
About 2/3 of the $\nu<0.5$ GeV events are QE and
1/3 are from pion production production processes (color online). }
\label{QEvsDelta50}
\end{figure}
\vspace{-0.1in}
% END FIGURE 17
%
%
% SECTION 5
\section {Using low $\nu$ events with $\nu<0.5$ GeV }
%
The $\nu<0.5$ GeV samples for $\nu_\mu$ and $\nub_\mu$ scattering are also dominated by QE events, but include a significant fraction (about 1/3) of events in which a single pion is produced in the final state.
As seen in Fig. \ref{nuvsq22}, the $\nu<0.5$ GeV samples are composed of
QE events with $Q^2<0.9$ GeV$^2$, and $\Delta$(1232) events with
$Q^2<0.3$ GeV$^2$.
%
%
%
% Fig 18
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{18Anupdeltanonumaxsigmacomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{18Aantinundeltanonumaxsigmacomparison.pdf}
\vspace{-0.3in}
\caption{ $\nu_\mu P \to \mu^- \Delta^{++}$ (top panel) and $\nub_\mu N \to \mu^+ \Delta^-$ (bottom panel)
cross sections (for $W<1.4$ GeV) measured
on free nucleons (H and D), compared to predictions from the GENIE MC (black points with errors).
The structure functions (form factors) for these two processes are the same.
The free nucleon cross sections are expected to be higher than the corresponding cross sections
on nuclear targets which are shown in Fig. \ref {deltanup-pauli} (because of Pauli suppression and final state interactions).
The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2), the red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) and the blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) represent three parametrization that conservatively
span the available data on H,D and nuclear targets. The cross sections predicted by GENIE on free nucleons are near the upper bound of
our three parametizations (color online). }
\label{deltanup}
\end{figure}
% Fig 18
%
%Fig 19
\begin{figure}
\includegraphics[width=3.7in,height=2.9in] {19AnupDeltagrandsigmawithpauli.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in] {19AAntinunDeltagrandsigmawithpauli.pdf}
\vspace{-0.3in}
\caption{ $\nu_\mu P \to \mu^- \Delta^{++}$ (top panel) and $\nub_\mu N \to \mu^+ \Delta^-$ (bottom panel) cross sections (for $W<1.4~GeV)$) measured on nuclear targets, compared to predictions from the GENIE MC (black points with errors).
The structure functions (form factors) for these two processes are same. The
cross sections on nuclear targets are expected to be somewhat smaller than the corresponding cross sections for free nucleons which are
shown in Fig. \ref {deltanup} (because of Pauli suppression and final state interactions).
The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2), the red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) and the blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) (all include Pauli suppression) represent three parametrization that conservatively
span the available data on
the production of $\Delta^{++}$ and $\Delta^-$ on
H,D and nuclear targets. The cross sections predicted by GENIE on nuclear targets are near the upper bound of
our three parametizations (color online).}
\label{deltanup-pauli}
\end{figure}
%\vspace{-0.1in}
% Fig 19
%
%
%
Fig. \ref{QEvsDelta50} shows
the $\nu<0.5$ GeV partial charged current cross sections as a function of energy. The partial cross sections
extracted from from the GENIE Monte Carlo are shown as black points
with MC statistical errors. The $\nu<0.5$ GeV partial cross section for $\nu_\mu$ scattering
is shown on the top panel, and the $\nu<0.5$ GeV partial cross section for $\nub_\mu$ scattering
is shown on the bottom panel.
The QE contribution to the $\nu<0.5$ GeV partial cross section is shown in red, and the contribution from pion production processes ($\Delta$, inelastic and coherent pion production) is shown in blue.
As seen in Fig. \ref{QEvsDelta50}, the pion production contribution to the $\nu<0.5$ GeV partial
cross section is relatively constant with energy, while the QE contribution has some energy
dependence. Therefore, the energy dependence of the
sum of the two contributions to the $\nu<0.5$ GeV partial cross section requires modeling of the relative magnitude of QE and pion production processes (specifically at low $Q^2$).
As shown in Fig. \ref {deltanup} and \ref {deltanup-pauli}, the consistency
among the experimental measurements of pion production cross sections in the
region of the $\Delta (1232$) resonance is of order 20\%
(depending on the neutrino energy and the nuclear target). We use this variation to get an
estimate of the model uncertainty in the determination of the neutrino flux from the $\nu<0.5$ GeV samples. This uncertainty can be greatly reduced when more precise measurements of the QE and pion production cross sections become available (e.g. from MINERvA).
%
%SECTION 6
\section {Pion production with $W<1.4$ GeV}
The antineutrino structure functions are related to the
neutrino structure functions by the following relationship.
\begin{eqnarray}
{\cal F}_{i} ^{\bar{\nu} n} & =& {\cal F}_{i} ^{\nu p} \nonumber \\
{\cal F}_{i} ^{\bar{\nu} p} & =& {\cal F}_{i} ^{\nu n}
\label{iso}
\end{eqnarray}
\subsection {$\nu_\mu P \to \mu^- \Delta^{++}$ and $\nub_\mu N \to \mu^+ \Delta^-$ (FIT-A)}
We define the cross section for $\nu_\mu P \to \mu^- \Delta^{++}$ as the integrated
cross section for $W<1.4$ GeV for the following single final state:
$$\nu_\mu P \to \mu^- P \pi ^{+}$$
We define the cross section for $\nub_\mu N \to \mu^+ \Delta^-$ as the integrated
cross section for $W<1.4$ GeV for the following single final state:
$$\nub_\mu N \to \mu^+ N \pi^-$$
Therefore, our definition includes the sum of the contributions of the resonant cross section
and the non-resonant continuum.
The structure functions (form factors) for the reactions $\nu_\mu P \to \mu^- \Delta^{++}$ and $\nub_\mu N \to \mu^+ \Delta^-$ defined above are the same (except that for antineutrinos the structure function $W_3$ changes sign).
It has been experimentally determined\cite{BEBC90} that $\nu_\mu P$ cross section for $W<1.4$ GeV is dominated by the resonant $\Delta^{++}$ production
process. Similarly, the $W<1.4$ GeV cross section for $\nub_\mu N$ is dominated by the
resonant $\Delta^-$ production process.
We parametrize the $\Delta^{++}$ and $\Delta^-$ production cross sections in terms of form factors as given by Paschos and Lalakulich\cite{paschos}, with the form factors of Paschos and
D. Schalla\cite{paschos}. In order to obtain predictions for the $W<1.4$ GeV region, we divide all theoretical $\Delta$ production cross sections by a factor of 1.2 (because 20\% of the resonant cross section is above $W=1.4$ GeV). We vary two of the parameters in the model, specifically $M_A^{\Delta}$ and $C_5^A$ to obtain a band that span the experimental data. We extract $M_A^{\Delta}$ from the measured
$Q^2$ distributions and use $C_5^A$ to set the overall normalization.
%
The top panel in Fig. \ref {deltanup} shows a summary of cross section measurements for $\nu_\mu P \to \mu^- \Delta^{++}$ on free nucleons (hydrogen or deuterium targets.) Shown are bubble chamber measurements at low energy from Argonne (ANL73\cite{ANL73}, ANL79\cite{ANL79}, ANL82\cite{ANL82}) and measurement at low energy from Brookhaven (BNL86\cite{BNL86}). Also shown are measurements at higher energies from the Fermilab bubble chamber (FNAL78\cite{FNAL78}, FNAL81\cite{FNAL81}) and high energy data from CERN (BEBC80\cite{BEBC80}, BEBC80\cite{BEBC90}).
The bottom panel in Fig. \ref {deltanup} show the BEBC90\cite{BEBC90}
cross section measurements for $\nub_\mu N \to \mu^+ \Delta^-$ on free nucleons (deuterium target). The predictions from the GENIE MC on free nucleons (shown as black points with MC statistical errors)
are near the upper bound of our three parametrizations.
The black curve labeled Paschos-2011 ($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2) uses the original
values of $M_A^{\Delta}$ and $C_5^A$ from the paper\cite{paschos} by Paschos and Lalakulich.
These values were obtained from fits to cross sections and $Q^2$ distributions
measured at low energies at Brookhaven and Argonne.
The red curve
labeled FIT-A1 ($M_A^{\Delta}$=1.93, $C_5^A$ = 0.62) is derived from
a fit to the cross sections and $Q^2$ distribution of the higher energy
BEBC90\cite{BEBC90} data for $\nub_\mu N \to \mu^+ \Delta^-$. The blue curve labeled
FIT-A2 ($M_A^{\Delta}$=1.75, $C_5^A$ = 0.49) is derived from
a fit to the cross sections and $Q^2$ distribution of the higher energy
BEBC90\cite{BEBC90} data for $\nu_\mu P \to \mu^- \Delta^{++}$.
The top panel in Fig. \ref {deltanup-pauli} shows a summary of cross section measurements for $\nu_\mu P \to \mu^- \Delta^{++}$ data on nuclear targets. Shown are the measurements of Gargamelle78\cite{GGM78} (Propane), SKAT88\cite{SKAT88} (Freon), and SKAT89\cite{SKAT89} (Freon).
The bottom panel shows measurements of
$\nub_\mu N \to \mu^+ \Delta^-$ cross sections on nuclear targets from
Gargamelle78\cite{GGM79} (Propane) and SKAT89\cite{SKAT89} (Freon).
%The predictions from the GENIE MC on nuclear targets (shown as black points with MC statistical errors)
%are near the upper bound of our three parametrizations.
Aside from Pauli suppression and final state interaction, the structure functions (form factors) for the processes in
Fig. \ref {deltanup} and Fig. \ref {deltanup-pauli} are the same.
The black (Paschos-2011), red (FIT-A1) and blue (FIT-A2) curves shown in the figures use the free nucleon form factors (but include the Pauli suppression for the case of nuclear targets). The calculations
do not include the effect of final state interaction for the nuclear targets.
The three curves (Paschos-2011, FIT-A1 and FIT-A2) conservatively
span all the available $\Delta^{++}$ and $\Delta^-$ production cross sections on hydrogen, deuterium and nuclear targets, as shown in Fig. \ref {deltanup} and \ref {deltanup-pauli}. The cross sections for
the production of $\Delta^{++}$ and $\Delta^-$ on nuclear targets
predicted by GENIE are near the upper bound of our three parametrizations.
%
%
% Fig 20
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{20Bnundeltanonumaxsigmacomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{20Bantinupdeltanonumaxsigmacomparison.pdf}
\vspace{-0.3in}
\caption{ $\nu_\mu N \to \mu^- \Delta^{+}$ (top panel) and $\nub_\mu P \to \mu^+ \Delta^0$ (bottom panel) cross sections $(W<1.4~GeV)$ measured on free nucleons (H or D). The predictions from the GENIE MC are shown as black points with errors. The structure functions (form factors) for these two processes are expected to be the same. The cross sections on free nucleons are expected to be higher than the cross sections
on nuclear targets (which are shown in Fig. \ref {deltanun-pauli}).
The green curve labeled FIT-B ($M_A^{\Delta}$=1.62, $C_5^A$ = 1.27) provides a parametrization
of structure functions which describe the for
the production of production of $\Delta^{+}$ and $\Delta^0$
free nucleon data.
The GENIE MC cross section for $\Delta^{+}$ and $\Delta^0$ production on free nucleons
are lower than the fit (color online). }
\label{deltanun}
\end{figure}
%Fig 20
%
%Fig 21
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{21BnunDeltagrandsigmawithpauli.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{21BAntinupDeltagrandsigmawithpauli.pdf}
\vspace{-0.3in}
\caption{ $\nu_\mu N \to \mu^- \Delta^{+}$ (top panel) and $\nub_\mu P \to \mu^+ \Delta^0$ (bottom panel) ($W<1.4$ GeV) cross sections on nuclear targets predicted by FIT-B ($M_A^{\Delta}$=1.62, $C_5^A$ = 1.27). The fit, which is shown as the green line, is compared to predictions from the GENIE MC, which are shown black points with MC statistical errors.
The fit uses form factors obtained from a parametrization
data on free nucleons, and includes the effect of
Pauli suppression (but not final state interaction).
The structure functions (form factors) for these two processes are expected to be the same, and
the cross sections somewhat smaller than the cross sections for free nucleons
(which are shown in Fig. \ref {deltanun}), because of Pauli suppression and final state interactions. The GENIE MC cross section for $\Delta^{+}$ and $\Delta^0$ on nuclear targets are lower than the fit (color online).
}
\label{deltanun-pauli}
\end{figure}
% Fig 21
Additional details are given in the Appendix.
\subsection {$\nu_\mu N \to \mu^- \Delta^{+}$ and $\nub_\mu P \to \mu^+ \Delta^0$ (FIT-B)}
We define the cross section for $\nu_\mu N \to \mu^- \Delta^{+}$ as the sum of the integrated
cross sections for $W<1.4$ GeV for the following two final states:
$$\nu_\mu P \to \mu^- N \pi ^{+}$$
$$\nu_\mu P \to \mu^- P \pi ^{0}$$
We define the cross section for $\nub_\mu P \to \mu^+ \Delta^0$ as the sum of the integrated
cross sections for $W<1.4$ GeV for the following two final states:
$$\nub_\mu N \to \mu^+ P \pi^-$$
$$\nub_\mu N \to \mu^+ N \pi^0$$
Therefore, our definition includes the sum of the contributions of the resonant cross section
and non-resonant continuum.
The structure functions (form factors) for the reactions $\nu_\mu N \to \mu^- \Delta^{+}$ and $\nub_\mu P \to \mu^+ \Delta^0$ defined above
are the same (except that for antineutrinos the structure function $W_3$ changes sign).
Because of Clebsch-Gordan coefficients\cite{paschos} the form factors for $\nu_\mu N$ resonant
production of the $\Delta^{+}$
are equal to the form factors for the $\nu_\mu N$ times $1/\sqrt(3)$. This implies
that the resonant cross section for $\Delta^{+}$ production in $\nu_\mu N$ collisions is a third of the resonant cross section for $\Delta^{++}$ production in $\nu_\mu P$ collisions. Similarly, the resonant cross section for $\Delta^{0}$ production in $\nub_\mu P$ collisions is a third of the cross section
for resonant $\Delta^{-}$ in $\nub_\mu N$ collisions.
However, unlike the case for $\nu_\mu P$ ($\Delta^{++}$) and $\nub_\mu N$ ($\Delta^{-}$), where
the cross sections are dominated by the resonant process, there is
a significant contribution from the non-resonant continuum to the $W<1.4$ GeV cross section
in $\nu_\mu N$ and $\nub_\mu P$ collisions.
The top panel of Fig. \ref{deltanun} shows the $\nu_\mu N \to \mu^- \Delta^{+}$ cross sections $(W<1.4~ GeV)$ measured on free nucleons (deuterium). Shown
are measurements from ANL79\cite{ANL79}, ANL82\cite{ANL82}, and BEBC90\cite{BEBC90}.
The predictions from the GENIE MC are shown as black points with MC statistical errors.
In order to describe the data (which has a large non-resonant contribution)
we changed the parameters in the
Paschos and Lalakulich\cite{paschos} resonance model to fit the
observed $Q^2$ distribution and total $W<1.4$ GeV
cross sections. The green curve labeled FIT-B ($M_A^{\Delta}$=1.62, $C_5^A$ = 1.27) is derived from
a fit to the $W<1.4$ GeV cross sections and $Q^2$ distribution of the
BEBC90\cite{BEBC90} data for $\nu_\mu N \to \mu^- \Delta^{+}$.
This curve provides a parametrization which describe the experimental data for
the production of $\Delta^{+}$ (with neutrinos) and $\Delta^0$ (for antineutrinos) on free nucleons. The GENIE MC cross sections for
the production of $\Delta^{+}$ on free nucleons are lower than the fit.
The structure functions (form factors) for the reactions $\nu_\mu N \to \mu^- \Delta^{+}$ and $\nub_\mu P \to \mu^+ \Delta^0$ $(W<1.4~GeV)$ are are same. The bottom panel of Fig. \ref{deltanun} shows a comparison
of the predictions of FIT-B ($M_A^{\Delta}$=1.62, $C_5^A$ = 1.27) (green curve) for the
$\nub_\mu P \to \mu^+ \Delta^0$ cross sections on free nucleons compared to the predictions from the GENIE MC which are shown as black points with MC statistical errors. The GENIE MC cross sections for
the production of $\Delta^{0}$ on free nucleons are lower than the fit.
Fig. \ref{deltanun} shows the prediction of FIT-B ($M_A^{\Delta}$=1.62, $C_5^A$ = 1.27) (green curve) for the $\nu_\mu N \to \mu^- \Delta^{+}$ (top panel) and $\nub_\mu P \to \mu^+ \Delta^0$ (bottom panel) $W<1.4$ GeV cross sections on nuclear targets compared to predictions from the GENIE MC (black points with MC statistical errors).
The cross sections on nuclear targets are expected to be somewhat lower than the cross sections
on free nucleons (which are shown in Fig. \ref {deltanun}).
Here, FIT-B includes the effect of
Pauli suppression (but not final state interaction). The GENIE MC cross sections for
the production of $\Delta^{+}$ and $\Delta^{0}$ on nuclear targets are lower than the fit.
%Fig 22
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{22nucarbondeltagrandsigmapernucleoncomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{22antinucarbondeltagrandsigmapernucleoncomparison.pdf}
\vspace{-0.3in}
\caption{ The total cross sections on carbon (per nucleon) predicted by GENIE for $W<1.4$ GeV (black points with MC statistical errors) for $\nu_\mu C \to (\mu^- \Delta^{++}$ or $\Delta^{+}$) are shown on the top panel, and for $\nub_\mu C \to \mu^+ (\Delta^0$ or $\Delta^{-}$) are shown on the bottom panel. The GENIE cross section predictions for the total $\Delta$(1232) production cross sections
on carbon fall near the lower bound of our three parametrizations of the experimental data (color online).}
\label{deltan-carbon}
\end{figure}
% Fig 22
%BDEGIN FIGURE 23
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{23Totalfcnumax50.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{23Totalgrandfractionnumax50.pdf}
\vspace{-0.3in}
\caption{The $\nu<0.5$ GeV sample for $\nu_\mu$. This sample includes both QE $\nu_\mu N \rightarrow \mu^- P$ events ($\approx$66\%) and $\Delta$ production events ($\approx$ 33\%). Top panel: The total corrections factor $f_C$ (with error bands) and the contributions
of the kinematic correction to ${\cal W}_2$ ($f_2$), and the contributions from ${\cal W}_1$ ($f_1$), ${\cal W}_3$ ($f_3$), ${\cal W}_4$ ($f_4$), and ${\cal W}_5$ ($f_5$). Bottom panel: The fractional contribution of
$\nu<0.5$ GeV events to the total cross section. Using our nominal model (QE with transverse enhancement and the Paschos 2011 model for $\Delta$ production, shown as the black line), we find that the fraction of $\nu<0.5$ GeV events is less than 60$\%$ for $\nu_\mu$ energies above 1.2 GeV (color online).
}
\label{neutrino50}
\end{figure}
%END FIG 23
%
% BEGIN FIG 24
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{24Antitotalfcnumax50.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{24Antitotalgrandfractionnumax50.pdf}
\vspace{-0.3in}
\caption{ The $\nu<0.5$ GeV sample for $\nub_\mu$. This sample includes both QE $\nub_\mu P \rightarrow \mu^+ N$ events ($\approx$66\%) and $\Delta$ production events ($\approx$33\%). Top panel: The total corrections factor $f_C$ and the contributions
of the kinematic correction to ${\cal W}_2$ ($f_2$),
and the contributions from ${\cal W}_1$ ($f_1$), ${\cal W}_3$ ($f_3$), ${\cal W}_4$ ($f_4$), and ${\cal W}_5$ ($f_5$). Bottom panel: The fractional contribution of
$\nu<0.5$ GeV events to the total cross section. Using our nominal model (TE model for QE scattering and the Paschos 2011 model for $\Delta$ production) we find that the fraction of $\nu<0.5$ GeV events is less than 60\% for $\nub_\mu$ energies above 2 GeV (color online).}
\label{antineutrino50}
\end{figure}
% END FIG 24
Additional details are given in the Appendix.
%subsection 6.3
\subsection{Comparisons of $W<1.4$ GeV cross sections on carbon}
A more relevant comparison is to determine how well the GENIE Monte Carlo describes
the sum of the proton and neutron cross sections on carbon, since
it is the total number of $\nu<0.5$ GeV events on carbon that are used in the determination
of the neutrino flux.
Fig. \ref {deltan-carbon} shows the predictions from the GENIE MC
for total $\Delta$ production cross section for $W<1.4$ GeV on carbon (per nucleon). The neutrino cross sections for $\nu_\mu C \to \mu^- (\Delta^{++}$ or $\Delta^{+}$) are shown in the top panel, and the antineutrino cross sections $\nub_\mu C \to (\mu^+ \Delta^0$ or $\Delta^{-}$) are shown in the bottom panel. The cross sections which are predicted by GENIE are compared to our three parametrizations.
(Paschos-2011, FIT-A1 and FIT-A2 for $\Delta^{++}$ and $\Delta^{-}$, and FIT-B for
$\Delta^{+}$ and $\Delta^{0}$).
%The predicted neutrino cross sections are shown on the top panel and the predicted antineutrino cross %sections are shown on the bottom panel.
The GENIE cross section predictions for the total $\Delta$ production cross sections
on carbon (which use the Rein and Sehgal model\cite{rs} for resonance production) fall near the lower bound of our three parametrizations of the experimental data.
%BEGIN FIG 25
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{25fcbarnumax5grandcomparison.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{25antifcbarnumax5grandcomparison.pdf}
\vspace{-0.3in}
\caption{ Comparisons of our calculated values of the normalized ${\bar f_{C:\nu<0.5}} (15.1)(E)$ (=$\bar f_C (15.1) $ for $\nu<0.5 $ GeV)
to values from the GENIE MC.
Our nominal model (shown as the solid black line) uses the TE model for QE scattering and the Paschos 2011 model for $\Delta$ production. Neutrinos are shown on the top panel and antineutrinos are shown on the bottom panel (color online). }
\label{fcbar50}
\end{figure}
% END FIG 25
% BEGIN FIG 26
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{26Totalcarbongrandfcbarerrornumax50.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{26Antitotalcarbongrandfcbarerrornumax50.pdf}
\vspace{-0.3in}
\caption{ The error band in the normalized correction factor ${\bar f_{C:\nu<0.5}} (15.1)(E)$ (=$\bar f_C (15.1) $ for $\nu<0.5 $ GeV). Our nominal model is QE with transverse enhancement and the Paschos 2011 model for $\Delta$ production. Shown are the differences between our nominal model
and other model assumptions for neutrinos (top panel) and for antineutrinos (bottom panel). For neutrinos with energies greater than 1.2 GeV, the error in
$\bar f_C (15.1) $ is less than 0.03, which corresponds to a 2.6\% upper limit on the uncertainty in
the neutrino flux. For antineutrinos with energies greater than 2 GeV the error in
$\bar f_C (15.1) $ is less than 0.01 (which corresponds to a 1.4\% upper limit on the uncertainty in the antineutrino flux (color online).
}
\label{nudiff50}
\end{figure}
% END FIG 26
%
%subsection 6.4
\subsection{Determination of neutrino and antineutrino flux using $\nu<0.5$ GeV samples on carbon}
%
The $\nu<0.5$ GeV sample includes both QE $\nu_\mu N \rightarrow \mu^- P$ events ($\approx$ 66\%) and $\Delta$ production events ($\approx$ 33\%).
The top panel of Fig. \ref {neutrino50} shows the total correction factor $f_{C}(E) $ for the $\nu<0.5$ GeV sample (defined as $f_{C:\nu<0.5}(E)$) for $\bf {neutrino}$ running.
Also shown are the various contributions to $f_{C:\nu<0.5}(E)$ including
the kinematic correction to ${\cal W}_2$ ($f_2$), and the contributions from ${\cal W}_1$ ($f_1$), ${\cal W}_3$ ($f_3$), ${\cal W}_4$ ($f_4$), and ${\cal W}_5$ ($f_5$). The bottom panel shows the fractional contribution of $\nu<0.5$ GeV events to the charged current neutrino total cross section. Using our nominal model (TE model for QE scattering and the Paschos 2011 model for $\Delta$ production) we find that the fraction of $\nu<0.5$ GeV events is less than 60\% for $\nu_\mu$ energies above 1.2 GeV.
The top panel of Fig. \ref {antineutrino50} shows the total correction factor $f_{C:\nu<0.5}(E)$) for $\bf {antineutrino}$ running. Also shown are the various contributions to $f_{C:\nu<0.5}$ including
the kinematic correction to ${\cal W}_2$ ($f_2$),
and the contributions from ${\cal W}_1$ ($f_1$), ${\cal W}_3$ ($f_3$), ${\cal W}_4$ ($f_4$), and ${\cal W}_5$ ($f_5$). The bottom panel shows the fractional contribution of $\nu<0.5$ GeV events to the charged current antineutrino total cross section. Using our nominal model (TE model for QE scattering and the Paschos 2011 model for $\Delta$ production) we find that the fraction of $\nu<0.5$ GeV events is less than 60\% for $\nub_\mu$ energies above 2 GeV.
As for the $\nu<0.25$ sample, we propose that the neutrino and antineutrino cross sections at low energy
be measured relative to the cross sections at 15.1 GeV.
Therefore, we define normalized quantity ${\bar f_{C:\nu<0.5}} (15.1)(E)$ for the $\nu<0.5$ sample as:
$${\bar f_{C:\nu<0.5}} (15.1)(E)=\sigma_ {\nu<0.5} (E)/ \sigma_ {\nu<0.5} (E=15.1~ GeV)$$
which is equivalent to
$${\bar f_{C:\nu<0.5}} (15.1)(E) = f_C (E) /f_C(E=15.1~GeV)$$
The values of $f_{C:\nu<0.5}$(E=15.1~GeV)=1.0113(for $\nu$) and 0.9507 (for $\nub$).
These values can be used to convert between ${\bar f_{C:\nu<0.5}}(E)$ and ${f_{C:\nu<0.5}} (E)$.
Fig. \ref {fcbar50} show comparisons of our calculated values of the normalized ${\bar f_{C:\nu<0.5}} (15.1)(E)$ (shown as the solid black line) to values extracted from the GENIE MC. The values are calculated
from our nominal model which uses the TE model for QE scattering and the Paschos 2011 model for $\Delta$ production. Neutrinos are shown on the top panel and antineutrinos are shown on the bottom panel (color online).
%% Here, ${\bar f_{C:\nu<0.5}} (15.1)(E)$ is defined by:
%
%$${\bar f_{C:\nu<0.5}} (15.1)(E) =\sigma_ {\nu<0.5} (E)/ \sigma_ {\nu<0.5} (E=15.1~GeV)$$
%$$= f_C (E) /f_C(E=15.1~GeV)$$
%
%
%
Figure \ref {nudiff50} shows the error band in the correction factor ${\bar f_{C:\nu<0.5}} (15.1)(E)$ for
neutrinos (top panel) and antineutrinos (bottom panel).
%Our nominal model is QE with transverse enhancement and the Paschos 2011 model for $\Delta$ production.
The error band is defined as the differences between our nominal model and other model assumptions. For neutrinos with energies greater than 1.2 GeV, the error in
$\bar f_C (15.1) $ is less than 0.03, which corresponds to a 2.6\% upper limit on the model uncertainty in the neutrino flux extracted from the $\nu<0.5$ GeV sample. For antineutrinos with energies greater than 2 GeV the error in $\bar f_C (15.1) $ is less than 0.01 (which corresponds to a 1.4\% upper limit on the model uncertainty in the antineutrino flux extracted from the $\nu<0.5$ GeV sample.
In order to go to lower neutrino
and antineutrino energies we need to use the $\nu<0.25$ GeV sample. The model uncertainty in the relative flux extracted from the $\nu<0.25$ GeV sample is 1.9\% for $\nu_\mu$ energies above 0.7 GeV and 2.5\% for $\nub_\mu$ energies above 1.0 GeV. With improved determination of $QE$ and $\Delta$ production
cross sections (e.g. in MINERvA), the model uncertainties can be further reduced, and the method
may be extended to lower energies.
% Fig 27
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{27Allasianup40_5dsigmadq2.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{27Allasiaantinun30dsigmadq2.pdf}
\vspace{-0.3in}
\caption{ $d\sigma/dQ^2$ cross sections (for $W<1.4$ GeV) measured
on deuterium at high energies by Allasia et al. (BEBC90\cite{BEBC90}). The cross sections for for $\nu_\mu P \to \mu^- \Delta^{++}$ are shown on the top panel and the cross sections for $\nub_\mu N \to \mu^+ \Delta^-$ are shown on the bottom panel. These two reactions should be described by the same form factors. The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2) is from fits to lower energy $\nu_\mu P$ data (BNL and Argonne). The red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) is a fit to the BEBC90 $\nub_\mu N$ data on the bottom panel. The blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) is a fit to the BEBC90 $\nu_\mu P$ data on the top panel. The variation among the three curves is taken as a systematic error. (color online). }
\label{dsdq2allasia}
\end{figure}
% Fig 27
% Fig 28
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{28Allennup25dsigmadq2.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{28Allasianun40_5dsigmadq2.pdf}
\vspace{-0.3in}
\caption{ Top panel: $d\sigma/dQ^2$ for $\nu_\mu P \to \mu^- \Delta^{++}$
cross sections (for $W<1.4$ GeV) measured on hydrogen at high energies by Allen et. al. (BEBEC80\cite{BEBC80}).
The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2) is from fits to low energy $\nu_\mu P$ data (BNL and Argonne). The red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) and the blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) are from fits to high energy Allasia et al. (cite{BEBC90}) data on deuterium. The variation among the three curves is taken as a systematic error.
Bottom panel: $d\sigma/dQ^2$ for $\nu_\mu N \to \mu^- \Delta^+$
cross sections (for $W<1.4$ GeV) measured by BEBEC90
on free nucleons on deuterium. The green curve labeled
FIT-B ($M_A$=1.62, $C_5^A$ = 1.27) represent a fit to the BEBEC90 $\nu_\mu N$ data (color online).
}
\label{dsda2allen}
\end{figure}
% Fig 28
\subsection{Resolution, acceptance and radiative corrections}
The ${\nu<0.25}$ events are primarily QE events with $Q^2<2M\times 0.25 \approx 0.45$ GeV$^2$. We can select either all events with ${\nu<0.25}$ GeV or only QE events with $Q^2<0.5$ GeV$^2$
The ratio of the number of reconstructed events with
${\nu<0.25}$ (or $Q^2<0.5$ GeV$^2$) in data and MC as a function
of energy is proportional to the ratio of the true flux to the simulated flux in
the MC. This ratio provides a measure of the relative neutrino flux
as a function of energy. A complete Monte Carlo should include the small contributions from
coherent pion production, strange particle production such as QE production of
hyperons\cite{E180},
and radiative corrections\cite{radcor,rujula}. The
the effects of experimental resolution and acceptance should also be simulated.
At preset the
GENIE Monte Carlo includes coherent pion production, but
does not include the QE production of hyperons, nor radiative effects.
If the GENIE Monte Carlo is used, then one may wish to weight the
rate of QE events (as a function of $Q^2$) by the ratio of events expected in the TE model to the number
of events predicted by the model
which is implemented in GENIE (i.e. the "Independent Nucleon" model with $M_A=0.99$ GeV).
In addition, QE production of hyperons and radiative effects need to be added.
\section{Conclusions}
We find the the model uncertainties in using the low $\nu$ event samples with $\nu<0.25$ and $\nu<0.5$
GeV are well under control (less than 3\%). Therefore, the low $\nu$ technique can be used at low energies (0.7 GeV for neutrinos and 1 GeV for antineutrinos).
Once data from MINErVA on QE scattering and resonance production becomes available,
the model uncertainties can be made even smaller, and the technique may be extended to
even lower energies.
Since the model uncertainties are under control, the dominant systematic error originates
from how well the detector response is understood, Specifically, the mis- reconstuction
of high $\nu$ events as low $\nu$ events must be modeled reliably.
This is because at high energies (as shown in Fig.~\ref{nuvsq24}) mis-reconstruction
of the hadron energy of high $\nu$ events can
increase the number of low $\nu$ events, while at low energies there
are fewer high $\nu$ events that can be mis-reconstructed at low $\nu$.
% Fig 29
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{29nupdsigmadq2E40.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{29antinundsigmadq2E40.pdf}
\vspace{-0.3in}
\caption{ The three $\nu_\mu P/ \nub_\mu N$ $d\sigma/dQ^2$ cross sections models (for $W<1.4$ GeV) with Pauli suppression for nuclear
targets at an energy of 40.5 GeV. The cross sections for for $\nu_\mu P \to \mu^- \Delta^{++}$ are shown on the top panel and the cross sections for $\nub_\mu N \to \mu^+ \Delta^-$ are shown on the bottom panel. These two reactions should be described by the same form factors. The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2) is from fits to lower energy $\nu_\mu P$ free nucleon data (BNL and Argonne). The red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) is a fit to the BEBC90 $\nub_\mu N$ free nucleon data. The blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) is a fit to the BEBC90 $\nu_\mu P$ free nucleon data. The variation among the three curves is taken as a systematic error. (color online). }
\label{dsdq2Amodel}
\end{figure}
% Fig 29
% Fig 30
\begin{figure}
\includegraphics[width=3.7in,height=2.9in]{30nundsigmadq2E40.pdf}
% must leave a line before and after vspace to have a space between the two figurs
\vspace{-0.2in}
\includegraphics[width=3.7in,height=2.9in]{30antinupdsigmadq2E40.pdf}
\vspace{-0.3in}
\caption{ Our $d\sigma/dQ^2$ cross sections model (for $W<1.4$ GeV) with Pauli suppression for nuclear
targets at an energy of 40.5 GeV. The cross sections for for $\nu_\mu N \to \mu^+ \Delta^{-}$ are shown on the top panel and the cross sections for $\nub_\mu P \to \mu^+ \Delta^0$ are shown on the bottom panel.
The green curve labeled
FIT-B ($M_A$=1.62, $C_5^A$ = 1.27) represent a fit to the BEBEC90 $\nu_\mu N$ free nucleon data (color online).
}
\label{dsdq2Bmodel}
\end{figure}
% Fig 30
\section{Appendix}
In appendix we illustrate the inconsistencies between the lower energy and higher energy
data for neutrino and antineutrino production of the $\Delta(1232)$ resonance. We use a range
of fits to span the systematic error in our modeling of $\Delta$ production cross sections.
The form factors for $\nu_\mu P \to \mu^- \Delta^{++}$ and $\nub_\mu N \to \mu^+ \Delta^-$ should be
the same.
The $d\sigma/dQ^2$ differential cross sections ($W<1.4$ GeV) for
$\nu_\mu P \to \mu^- \Delta^{++}$ measured at high energies are shown in the top panel
of Fig. \ref{dsdq2allasia} (Allasia et. al., BEBC90\cite{BEBC90} data on deuterium) and also
on the top panel of Fig. \ref{dsda2allen} (Allen et. al. BEBC80\cite{BEBC80} data on hydrogen)
The bottom panel of Fig. \ref{dsdq2allasia} show the $d\sigma/dQ^2$ cross sections at high energies ($W<1.4$ GeV) for $\nub_\mu N \to \mu^+ \Delta^-$ measured by Allasia et. al. (BEBC90) data on deuterium. The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2) is from fits to lower energy $\nu_\mu P \to \mu^- \Delta^{++}$ data (BNL and Argonne). The red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) is a fit to the BEBC90 $\nub_\mu N \to \mu^+ \Delta^-$ data . The blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) is a fit to the BEBC90 $\nu_\mu P \to \mu^- \Delta^{++}$ data. The variation among the three curves is taken as a systematic error.
Bottom panel of Fig. \ref {dsda2allen} shows values of $d\sigma/dQ^2$
differential cross sections for $\nu_\mu N \to \mu^- \Delta^+$ (for $W<1.4$ GeV) measured by BEBEC90
on free nucleons on deuterium. This reaction has different form factors then $\nu_\mu P \to \mu^- \Delta^{++}$.
The green curve labeled FIT-B ($M_A$=1.62, $C_5^A$ = 1.27) represent a fit to the BEBEC90 $\nu_\mu N$ data (color online).
We use the above models with the addition of Pauli suppression in order to model the differential cross sections
on nuclear targets.
Fig. \ref {dsdq2Amodel} shows our three $\nu_\mu P/ \nub_\mu N$ $d\sigma/dQ^2$ cross sections models (for $W<1.4$ GeV) with Pauli suppression for nuclear
targets at an energy of 40.5 GeV. The cross sections for for $\nu_\mu P \to \mu^- \Delta^{++}$ are shown on the top panel and the cross sections for $\nub_\mu N \to \mu^+ \Delta^-$ are shown on the bottom panel. These two reactions should be described by the same form factors. The black curve labeled Paschos-2011($M_A^{\Delta}$=1.05, $C_5^A$ = 1.2) is from fits to lower energy $\nu_\mu P$ free nucleon data (BNL and Argonne). The red curve
labeled FIT-A1 ($M_A$=1.93, $C_5^A$ = 0.62) is from a fit to the BEBC90 $\nub_\mu N$ free nucleon data. The blue curve labeled
FIT-A2 ($M_A$=1.75, $C_5^A$ = 0.49) is from a fit to the BEBC90 $\nu_\mu P$ free nucleon data. The variation among the three curves is taken as a systematic error.
Fig. \ref {dsdq2Bmodel} shows our $\nu_\mu N/ \nub_\mu P$ $d\sigma/dQ^2$ cross sections model (for $W<1.4$ GeV) with Pauli suppression for nuclear
targets at an energy of 40.5 GeV. The cross sections for for $\nu_\mu N \to \mu^+ \Delta^{-}$ are shown on the top panel and the cross sections for $\nub_\mu P \to \mu^+ \Delta^0$ are shown on the bottom panel.
The green curve labeled
FIT-B ($M_A$=1.62, $C_5^A$ = 1.27) is extracted fit to the BEBEC90 $\nu_\mu N$ free nucleon data.
% NEW INFORATION
\subsection{$\Delta$ production form factors}
For the vector contribution we use the formulae for
${\cal W}_1,{\cal W}_2,{\cal W}_3,{\cal W}_4, {\cal W}_5, and {\cal W}_6 $on free nucleons from Lalakulich and Paschos\cite{paschos}. We neglect the effect of Fermi motion. The form factors that we use
are taken from Paschos and Schalla\cite{paschos}. Specifically, the vector form factors are
\begin{eqnarray}
C_3^V (Q^2) = \frac{2.13 / D_V}{1+ \frac{Q^2}{4M_V^2}} \; & , & \; C_4^V (Q^2) = \frac{-1.51 / D_V}{1+ \frac{Q^2}{4M_V^2}} \\
C_5^V (Q^2) = \frac{0.48 / D_V}{1+\frac{Q^2}{0.776 M_V^2}} \; & \textmd{ and } & \; D_V =\left( 1 + \frac{Q^2}{M_V^2} \right)^2
\end{eqnarray}
with $M_V$ = 0.84 GeV, which have been extracted from electroproduction data.
For the vector-axial interference $W_3 (Q^2,\nu)$ Paschos and Schalla use the form factor $C_5^A (Q^2)$
$C_5^A(Q^2)= \frac{C_5^A}{(1+Q^2/M_A^2)^2}\frac{1}{1+2Q^2/M_A^2}$, where $C_5^A (0)$
is defined as $C_5^A$, and $C_4^A= -\frac{1}{4} C_5^A$
Paschos and Schalla use low energy $\pi^+ p \rightarrow \Delta^{++}$ where the non-resonant background is smallest. With $M_A$=1.05 GeV they extract value of $C_5^A (0) = 1.08$ from the data. Since this value is close to $1.20$ predicted by the Goldberger-Treiman relation, they chose to use $C_5^A$=1.2.
For $\Delta^{++}$ and $\Delta^{-}$ we define the Pachos-2011 parameterization using above
form factors with $C_5^A$=1.2 (extracted through PCAC), $M_A$=1.05 GeV, and the vector form factors described above. As mentioned earlier, FIT-A1 and FIT-A2 use the same form but with different values of
$C_5^A$ and $M_A$.
For $\Delta^{+}$ and $\Delta^{0}$ production our Fit-B uses the the same form factors multiplied by a factor of $1/\sqrt(3)$ (as expected\cite{paschos} from Clebsch-Gordan coefficients). However, in order to account for the large non-resonance background, we use different values
$C_5^A$ and $M_A$.
Paschos and Schalla mention that several recent articles also calculate $C_5^A(0)$ by fitting experimental data~\cite{Hernandez:2007qq,Lalakulich:2010ss,Leitner:2008ue,Graczyk:2009qm,Hernandez:2010bx,AlvarezRuso:1998hi,SajjadAthar:2009rc} with their values varying from 0.87 up to 1.20. Models with a resonant background~\cite{Hernandez:2007qq,Lalakulich:2010ss} prefer the power value, while the other articles~\cite{Leitner:2008ue,Graczyk:2009qm,Hernandez:2010bx,AlvarezRuso:1998hi,SajjadAthar:2009rc} prefer values closer to 1.20. The reasons for the differences is the treatment of the non-resonant background, the form of the axial form factor that is used, and the exact kinematics at small $Q^2$.
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%
\end{thebibliography}
\end{document}