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\begin{center}
{\Large{Comparison between Wiser $\pi^-$ rates calculation and data}}\\
{\Large{from transversity and PVDIS experiments}}
\bigskip
Seamus Riordan, Nguyen Ton, Zhiwen Zhao, Xiaochao Zheng\\
\today
%\author{Seamus Riordan}
%\affiliation{xx}
%\author{Nguyen Ton}
%\affiliation{University of Virginia, Chaottesville, Virginia 22904, USA}
%\author{Zhiwen Zhao}
%\affiliation{Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA}
%\author{Xiaochao Zheng}
%\affiliation{University of Virginia, Charlottesville, Virginia 22904, USA}
\end{center}
\begin{abstract}
The Wiser code has been used widely to estimate the pion background in electron scattering
experiments. However, there has been little effort to understand how it was adapted to electron scattering.
We hope to fill this gap in this writeup. We also provide comparison between Wiser calculation and
data from the 6 GeV transversity and PVDIS experiments.
\end{abstract}
\section{Introduction -- The Wiser Code}
Parametrization of hadron production in electron scattering $(e,N)$ is a convenient tool for the design
of nuclear physics experiments and the analysis of the resulting data. Parametrized cross sections can
be used in radiative correction, calculating experimental background, and experimental feasibility
design such as for $(e,e'N)$ coincidence measurements.
For experiments at (earlier) SLAC and at JLab, a widely used code is called Wiser, based on a parameterization of
inclusive photoproduction of protons, kaons, and pions (for kaons and pions both charges were included)~\cite{Wiser:1977yw}.
The Wiser code is based on the early SLAC data using: (1) a hydrogen target, (2) a bremsstrahlung photon
beam made from the electron beam, with endpoints of 5, 7, 9, 11, 15, and 19 GeV, and
(3) the SLAC 8 GeV/$c$ spectrometer. The lowest momentum coverage for $\pi^-$ is either 1 or 2~GeV/$c$,
for $K^-$ between 1 and 3 GeV/$c$, and for $p$ or $\bar p$ between 1 and 4 GeV/$c$, depending on the
beam energy.
Because bremsstralung photon beam was used, a method called bremsstralung subtraction was used to extract the
cross section for a monochromatic photon beam. The final results were given for
the pion, kaon, proton and anti-proton in the form of [Eqs.~(IV-A-2) and (IV-A-2) of \cite{Wiser:1977yw}]:
\begin{eqnarray}
E\frac{d^3\sigma}{dp'^3}&& \label{eq:wiser_fit}
\end{eqnarray}
where $E$ is the electron beam energy and $p'$ the hadron momentum.
The Fortran routine WISER\_ALL\_FIT returns the value of $E\frac{d^3\sigma}{dp'^3}/K$ with $K$ a specific photon energy. Data from a deuterium target were available, but were not bremsstrahlung-subtracted and only integrated
cross sections were included in Ref.~\cite{Wiser:1977yw}.
Because of the specific kinematic regime of the data, it is expected that the Wiser code should work reasonably well for the
multi-GeV beam energy at JLab.
To obtain the hadron cross section for an electron beam, we essentially do an integral which is the inverse
of Wiser's analysis, see Eq.~(III-A-4) of \cite{Wiser:1977yw}:
\begin{eqnarray}
\frac{d\sigma}{dp'd\Omega} &=& \frac{p'^2}{E} (\frac{K_\gamma^\mathrm{total}}{E}) \int_{K_\mathrm{min}}^{E} E\frac{d^3\sigma}{dp'^3}\frac{\alpha(K,K_0)}{K}dK \label{eq:IIIA4}
\end{eqnarray}
where $K_\mathrm{min}$ is the minimum photon energy required for
producing the hadron at the given $p'$ and $\alpha(K,K_0)$ is a bremsstrahlung factor.
The upper limit of the integral is the maximum bremsstralung photon energy $K_0$ and can be taken as the electron beam energy
$E$. The fraction ${K_\gamma^\mathrm{total}}/{E}$ describes the total energy contained in the bremsstrahlung beam relative
to the electron beam energy, which if multiplied by the running times gives $EQ$, the number of equivalent quanta in
Ref.~\cite{Wiser:1977yw}.
The number of photons as a function of the photon energy $K$ for a given $E$ falls roughly as $1/K$, and is generally written as
\begin{eqnarray}
\frac{dn}{dK}(K,E) &=& b\frac{\alpha(K,K_0)}{K}dK
\end{eqnarray}
where $\alpha$ describes the deviation of the spectrum from a $1/K$ shape, and $b$ is proven to be equal to $EQ$. The
bremsstrahlung photon beam current can thus be written as
\begin{eqnarray}
\frac{dn}{dtdK}(K,E) &=& \frac{K_\gamma^\mathrm{total}}{E} \frac{\alpha(K,K_0)}{K}dK~.
\end{eqnarray}
The normalization is such that $\int_0^{K_0}\alpha(K,K_0)dK=K_0$ and to a good approximation we can take $\alpha(K,K_0)=1$.
\bigskip
The integral is done in the code
\smallskip
\centerline{WISER\_ALL\_SIG(E1,PTP,THP,RAD\_LEN,ITYPE,TOTAL)}
\smallskip
\noindent
where E1 is the electron beam energy or max of bremsstralung spectra in MeV;
PTP and THP are the momentum and the scattering angle of the
outgoing hadron, in MeV and degrees respectively; RAD\_LEN is the input radiation length
in percent including both internal and external, ITYPE specifies the hadron type, and TOTAL is
the output cross section $d\sigma/dE'd\Omega$ in nb/GeV-str. The calculationof
$d\sigma/dE'd\Omega$ is done as
%
\begin{eqnarray}
\frac{d\sigma}{dp'd\Omega} &=& \frac{p'^2}{E} \frac{\mathrm{RAD\_LEN}}{100.}
\int_{K_{min}}^{E} E\frac{d^3\sigma}{dp'^3}\frac{1}{K}dK
\end{eqnarray}
where we can see that the approximation $\alpha(K,K_0)=1$ was already taken and the input
radiation length should describe the fractional energy that has been converted to
bremsstrahlung photons, ${K_\gamma^\mathrm{total}}/{E}$.
\bigskip
We refer to WISER\_ALL\_SIG as the {\it Wiser} code hereafter.
\subsection{The Equivalent Photon Radiator}
We denote $t$ as the total radiation length input to the {\it Wiser} code, RAD\_LEN.
It must include both internal and external radiators.
The bremsstrahlung conversion of electron beam passing through material
is given by Eq.~(32.30) and (32.31) of Ref.~\cite{Agashe:2014kda}, which for
small thickness (fraction of the radiation length $X_0$) is given by
\begin{eqnarray}
\frac{d\sigma}{dK} &=& \frac{A}{X_0N_AK}\left(\frac{4}{3}-\frac{4}{3}y+y^2\right)\label{eq:brem_pdg1}
\end{eqnarray}
with $A$ the atomic number of the target and $N_A$ the Avogadro number. This can be integrated to give the total number of photons
\begin{eqnarray}
N_\gamma &=& \frac{t_\mathrm{material}}{X_0}\left[\frac{4}{3}\ln\left(\frac{k_\mathrm{max}}{k_\mathrm{min}}\right)
-\frac{4(k_\mathrm{max}-k_\mathrm{min})}{3E}+\frac{(k_\mathrm{max}^2-k_\mathrm{min}^2)}{2E^2}\right)\label{eq:brem_pdg2}
\end{eqnarray}
where $y=K/E$. To simplifies the calculation, we note that $0