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\begin{document}

%%%%%%%%%%%%%%%%%%%%% Title %%%%%%%%%%%%%%%%%%%%%%


\title{Reggeized models for electroproduction of pion at forward angles }

%%%%%%%%%%%%%%%%%%%% Authors %%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%% Addresses %%%%%%%%%%%%%%%%%%%%%

\author{Byung Geel Yu}%
\email{bgyu@kau.ac.kr}
\affiliation{Research Institute of Basic Sciences, Korea Aerospace
University, Koyang 412-791, Korea}

\author{Tae Keun Choi}%
%\email{tkchoi@yonsei.ac.kr}
\affiliation{Department of Physics,
Yonsei University, Wonju 412-791, Korea}

\author{Kook Jin Kong}%
%\email{kong@kau.ac.kr}
\affiliation{Research Institute of Basic
Sciences, Korea Aerospace University, Koyang 412-791, Korea}

\date{\today}

%%%%%%%%%%%%%%%%%%%% Abstract %%%%%%%%%%%%%%%%%%%%%


\pacs{13.60.Le, 13.60.-r, 13.60.Rj}


\maketitle




\section{Electroproduction of pseudoscalar meson}

\subsubsection{Kinematics and differential cross sections}

We briefly introduce the kinematics in the exclusive $\pi$ electroproduction,
\begin{eqnarray}\label{ee-reac}
e(\kappa)+N(p)\to e(\kappa)+N(p')+\pi(q),
\end{eqnarray}
and denote the notations and definitions of variables.
The mandelstam variables for the reaction in the three channel are
\begin{eqnarray}
&&s=(p+k)^2 =(p' + q)^2,\nonumber\\
&&t=(q-k)^2 = (p-p')^2,\nonumber\\
&&u=(p-q)^2 =(p'-k)^2
\end{eqnarray}
with the virtual photon momentum $k^2=\kappa'^2 -\kappa^2$ in
terms of the initial and the final electron momenta $\kappa$ and
$\kappa'$. The virtuality of photon is given by $Q^2 = -k^2$.


In the electroproduction fully expressed in Eq. (\ref{ee-reac})
where the target nucleon is at rest, the $z$-axis is directed along
the three momentum of the virtual photon $\gamma^*$ which is defined as
$\vec{k}=\left(0,\,0,\,\sqrt{\nu^2+Q^2}\right)$
with the virtual photon four-momentum $k=\kappa-\kappa'=\left(\nu,\,\vec{k}\right)$,
$\nu=E_e-E_e'$ in the laboratory system.

More variables are the Bjorken scaling variable $x_B=Q^2/2M\nu$ as the fraction
of the target nucleon��s momentum carried by the struck quark, and the invariant
energy in the $\gamma^*$-$N$ system,
$W^2=M^2+2M\nu-Q^2=M^2+Q^2\left({1\over x_B}-1\right)
$.

The $S$-matrix element  for electroproduction is\footnote{For
photoproduction the $S$-matrix element is given by
\begin{eqnarray}
S=\frac{1}{(2\pi)^2}\sqrt{\frac{M M'}{4E_\gamma q_0 E E' }}
i\delta^{4} (p+k-p' -q) {\cal M}_{fi}\nonumber
\end{eqnarray}}
\begin{eqnarray}\label{smatrix1}
S=\frac{1}{(2\pi)^{7/2}}\frac{m_e \sqrt{M M' }}{\sqrt{2E_\gamma
q_0 E_eE_e' E E' }} i\delta^{4} (p+k-p' -q) {\cal
M}_{fi}\,.
\end{eqnarray}

%\subsection{kinematics}

%\begin{figure}[b]
%\begin{center}
%\includegraphics[width=0.45\textwidth]{kine.eps}
%\end{center}
%\caption{Exclusive reaction $N(e,e'\pi)N'$ in the laboratory.
%scattering $(e,e')$ plane and reaction $N(\gamma^*,\pi)N'$ plane.}
%\label{kine}
%\end{figure}

\begin{eqnarray}
\frac{d^{3} \sigma}{dE'_{e}d\Omega'_{e}d\Omega_{\pi}}= \Gamma
\frac{d\sigma}{d\Omega_{\pi}}
\end{eqnarray}
where the flux of the electron current is given by
\begin{eqnarray}
\Gamma=\frac{\alpha}{2\pi^2}
\frac{E'_{e}}{E_e}\frac{K_{H}}{Q^{2}}\frac{1}{1-\epsilon},
\end{eqnarray}
with the equivalent photon lab-energy $K_{H}=\frac{W^2 -M^{2}}{2
M}\, $.

The ratio of the longitudinal to the transverse polarization of the virtual photon is
\begin{eqnarray}\label{ell}
\epsilon=\left[1+2\frac{\nu^2+Q^2}{4E_e\left(E_e-\nu\right)-Q^2}\right]^{-1}
\end{eqnarray}
This is rewritten as
\begin{eqnarray}
\epsilon={1-y-{Q^2\over 4E_e^2}\over 1-y+{y^2\over2}+{Q^2\over 4E_e^2}}
\end{eqnarray}
in terms of $y=\nu/E_e$.

The longitudinal component $\epsilon_L=\left(Q^2/\nu^2\right)\epsilon$.
In terms of electron scattering angle, the electron polarization $
\epsilon=\left(1+\frac{2|\vec{k}|^2}{Q^2}\tan{\frac{1}{2}
\theta_e}\right)^{-1}$ measured by the angle $\theta_e$ deviated from the
reaction plane.


Separating the kinematical part of electron scattering in Eq.
(\ref{smatrix1}) the full cross section for virtual
photoproduction
\begin{eqnarray}
\gamma^*(k)+N(p)\to N(p')+\pi(q)
\end{eqnarray}
is expressed as follows\footnote{
Note that the CEA \cite{brown} and the Cornell experiments \cite{bebek,bebek1}
follow the convention for the polarization factor
$d\sigma_{LT}$ cross section in Eq. (\ref{crss1}),
$$\sqrt{\epsilon(\epsilon+1)/2}\,\frac{d\sigma_{TL}}{d\Omega_{\pi}}
\cos{\Phi_{\pi}}.$$
},
%\begin{widetext}
\begin{eqnarray}\label{crss1}
&&\frac{d \sigma_{\rm tot}}{d\Omega_{\pi}} =\frac{d
\sigma_{U}}{d\Omega_{\pi}} +\epsilon
\frac{d\sigma_{TT}}{d\Omega_{\pi}} \cos{2\Phi_{\pi}}
\nonumber\\&&\hspace{3cm}
+\sqrt{2\epsilon(\epsilon+1)}\,\frac{d\sigma_{TL}}{d\Omega_{\pi}}
\cos{\Phi_{\pi}},
%&&\frac{d \sigma_{i}}{dt}=(\frac{\pi}{qk})\frac{d \sigma_i}{d\Omega_{\pi}} ,\quad
%i=T,L,TT,TL \nonumber
\end{eqnarray}
%\end{widetext}
where the unpolarized cross section is given by
\begin{eqnarray}\label{dcs_u}
\frac{d \sigma_{U}}{d\Omega_{\pi}}=\frac{d\sigma_T}{d\Omega_{\pi}}
+\epsilon \frac{d\sigma_L}{d\Omega_{\pi}}\,,
\end{eqnarray}
and $\frac{d\sigma_T}{d\Omega_{\pi}}$,
$\frac{d\sigma_L}{d\Omega_{\pi}}$,
$\frac{d\sigma_{TT}}{d\Omega_{\pi}}$, and
$\frac{d\sigma_{LT}}{d\Omega_{\pi}}$ are the transverse,
longitudinal, transverse-transverse, and longitudinal-transverse
cross sections, respectively.

\subsubsection{Reactions}

For completeness we consider in this work the reaction proceeding via four channels as below.

\begin{eqnarray}
&&\gamma^*\,p\to \pi^+\,n\,,\label{ch1}\\
&&\gamma^*\,n\to \pi^-\,p\,,\label{ch2}\\
&&\gamma^*\,p\to \pi^0\,p\,,\label{ch3}\\
&&\gamma^*\,n\to \pi^0\,n\,,\label{ch4}
\end{eqnarray}


\section{YCK model}


We utilize the effective Lagrangians for the construction of the
Born approximation amplitude to one-photon exchange at the tree
level. It is, then, extended to reggeize the meson exchange in the
$t$-channel, following the prescription in Ref.
\cite{glv,prc2011}.

For the charged pion process, the charge conjugation $C$-even
exchange in the $t$ channel is allowed with the sign of the
photon-meson coupling constant determined from the $G$-parity
counting. Thus, the $G$-parity-even exchanges $\pi^\pm$,
$a_1^\pm$, and $a_1^\pm$ have the opposite signs when coupling to
photon between $\pi^+$ and $\pi^-$ reactions. However, in order to
avoid confusion caused by the signs of the coupling constants when
considering the phase of the Regge propagators between
exchange-degenerate pair $\{\rho-a_2\}$, for instance, we will
write the signs in absolute values in $g_{\gamma \pi a_1}$ and
$g_{\gamma \pi a_2}$, with the sign-opposite only appearing in the
pion exchanges in the production amplitudes below. The signs of
$a_1$ and $a_2$ coupling constants will then be included in the
amplitudes in Eqs. (\ref{ch12}) and (\ref{ch34}) later, which are
the final expressions containing the phases desired for the
current calculation.

Therefore, the reggeized amplitudes for charged pion
electroproduction are written as \cite{prc2011},
\begin{widetext}
\begin{eqnarray}
&&{\cal M}_{\pi^+n}=i\sqrt{2}\, eg_{\pi NN}\bar{u}'(p')
    \biggl[\widetilde{F}_{\pi}(Q^2){(2q-k)\cdot\epsilon\over t-m^2_\pi}\,\gamma_5
    %\nonumber\\&&\hspace{1cm}
    + \gamma_5 {(\rlap{/}{p}+\rlap{/}{k}+M_p)\over s-M_p^2}
    \Gamma_{\gamma^*NN}(Q^2)\biggr]u(p)(t-m_\pi^2){\cal R}^\pi(s,t)\,,
    \label{proton}\\
&&{\cal M}_{\pi^-p}=-i \sqrt{2}\,eg_{\pi NN}\bar{u}'(p')
     \biggl[\widetilde{F}_{\pi}(Q^2){(2q-k)\cdot\epsilon\over t-m^2_\pi}\,\gamma_5
     %\nonumber\\&&\hspace{1cm}
     -\Gamma_{\gamma^*NN}(Q^2){(\rlap{/}{p}-\rlap{/}{q}+M_p)\over u-M_p^2}\gamma_5
     \biggr]u(p)(t-m_\pi^2){\cal R}^\pi(s,t)\,,
     \label{neutron}\\
%
&&{\cal M}_{V}=\frac{g_{\gamma \pi
V}}{m_0}F^{\rho}(Q^2)
\varepsilon_{\mu\nu\alpha\beta}\epsilon^{\mu}k^{\nu}q'^{\alpha}
(-g^{\beta\rho}+ q'^{\beta}q'^{\rho}/m_{V}^2) \bar{u}'(p')\left[
g^v_{VNN}\,\gamma_{\rho} +
i\frac{g^t_{VNN}}{2M}\sigma_{\lambda\rho}q'^{\lambda} \right]u(p){\cal R}^V(s,t)\,,
\label{V}  \\%
%
&&{\cal M}_{T} ={2g_{\gamma\pi
T}\over
m_0^2}\,F^T(Q^2)\varepsilon_{\mu\nu\alpha\beta}\,\epsilon^\mu\,k^\nu
q'^\alpha q_\rho\Pi^{\beta\rho;\lambda\sigma}(q-k)
\bar{u}'(p')\left[{2g^{(1)}\over M}(\gamma_\lambda P_\sigma+\gamma_\sigma
P_\lambda)+{4g^{(2)}\over M^2} P_\lambda P_\sigma\right]u(p){\cal R}^T(s,t)\,
\label{T}  \\%
%
&&{\cal M}_{A}=i\frac{g_{\gamma \pi
A}}{m_0}F^A(Q^2) ( k\cdot q'\,\epsilon_{\mu} -\epsilon\cdot
q'\,k_{\mu}) (-g^{\mu\nu}+q'^{\mu}q'^{\nu}/m_A^2)
g^v_{ANN}\,\bar{u}'(p')\gamma_{\nu}\gamma_5 u(p){\cal R}^A(s,t)\,,
\label{A}  \\
%
&&{\cal M}_{B}=i\frac{g_{\gamma \pi
B}}{m_0} F^B(Q^2)( k\cdot q'\,\epsilon_{\mu} -\epsilon\cdot
q'\,k_{\mu}) (-g^{\mu\nu}+q'^{\mu}q'^{\nu}/m_B^2)\bar{u}'(p')
i\frac{g^t_{BNN}}{2M}
\sigma_{\lambda\nu}q'^{\lambda}\gamma_5 u(p){\cal R}^B(s,t)\,,
\label{B}
\end{eqnarray}
\end{widetext}
where the $\sqrt{2}$ factor is understood in the meson sector in
which vector, tensor, and axial mesons are written by the
representative symbols for charged reactions. The mass parameter
$m_0=1$ GeV and the $t$-channel momentum transfer $q'=q-k$, and
momentum sum $P={1\over2}(p+p')$.

The tensor meson spin-2 projection is given by
$\Pi^{\beta\rho;\lambda\sigma}(q')={1\over2}(\eta^{\beta\lambda}\eta^{\rho\sigma}+
\eta^{\beta\sigma}\eta^{\rho\lambda})-{1\over3}\eta^{\beta\rho}\eta^{\lambda\sigma}$
with $\eta^{\mu\nu}=-g^{\mu\nu}+{q'^\mu q'^\nu\over m_T^2}$.

The Regge propagator for an exchanged meson is written collectively for $\varphi(=V,\,T,\,A\,B)$,
\begin{eqnarray}
&&{\cal R}^\varphi(s,t)={\pi\alpha'_\varphi\times{}\rm phase\over
\Gamma(\alpha_\varphi(t)+1-J)\,\sin(\pi\alpha_\varphi(t))}\left({s\over
s_0}\right)^{\alpha_\varphi(s)-J}\,,
\nonumber\\
\end{eqnarray}
where the Regge phase ${1\over
2}((-1)^J+e^{-i\pi\alpha_\varphi(t)})$ in the canonical form is
assigned to the spin-$J$ meson for the exchange-nondegenerate
case.

\subsubsection*{The Regge phase in production channels}

%Given the same as in Ref. \cite{prc2011}
The Regge phases for the electroproduction amplitudes from Eqs.
(\ref{proton}) to (\ref{B}) for charged reactions are chosen the same as in the case
of photoproduction \cite{prc2011} which is expressed in schematic notation as
follows,
\begin{eqnarray}\label{ch12}
&&{\cal M}(\gamma^* N\to\pi^\pm N)=
\pm ie\pi\left\{\begin{array}{c}e^{-i\pi\alpha_\pi}\\1\end{array}\right\}
+b_1\left\{\begin{array}{c}e^{-i\pi\alpha_{b_1}}\\-1\end{array}\right\}
\nonumber\\&&\hspace{3cm}
+\rho\left\{\begin{array}{c}-e^{-i\pi\alpha_\rho}\\1\end{array}\right\}
+a_2\left\{\begin{array}{c}-e^{-i\pi\alpha_{a_2}}\\-1\end{array}\right\}
\nonumber\\&&\hspace{3cm}
\pm a_1{1\over2}\left(-1+e^{-i\pi\alpha_{a_1}}\right),
\end{eqnarray}
where the $\sqrt{2}$ is omitted for brevity.

%It should be noted that the Regge phases are taken as the same with those in our model for photoproduction \cite{prc2011}, %but opposite\footnote{which is equal to set $$
%-\rho\left\{\begin{array}{c}e^{-i\pi\alpha_\pi}\\-1\end{array}\right\}
%-a_2\left\{\begin{array}{c}e^{i\pi\alpha_\pi}\\1\end{array}\right\}
%.$$} to those in models of GLV, KM, and VR.

\subsubsection*{Gross-Riska prescription for nucleon and pion electromagnetic form factors}

The nucleon charge form factor in the electromagnetic vertex given by
\begin{eqnarray}
\Gamma_{\gamma^*NN}(k^2)=\widetilde{F}^N_{1}(k^{2})/\kern-6pt\epsilon-{\kappa_N\over4M}F^N_2(k^2)
\left[\rlap{/}\epsilon,\rlap{/}k\right] \label{emff}
\end{eqnarray}
and pion charge form factor are chosen by following the
Gross-Riska prescription
\begin{eqnarray}\label{gr3}
&&\widetilde{F}^N_{1}(k^{2})/\kern-6pt\epsilon
%=\widetilde{F}^{\,\prime}_{1}(k^{2})/\kern-6pt\epsilon
=F^N_1(k^2)\left(/\kern-6pt\epsilon
-/\kern-6ptk\frac{\epsilon\cdot k}{
k^2}\right)+F^N_1(0)\,/\kern-6ptk\frac{\epsilon\cdot k}{
k^2}\,,\\
%
&&\widetilde{F}^{\pi}(k^{2})(2q-k)\cdot\epsilon=F^{\pi}(k^{2})
(2q-k)\cdot\left(\epsilon-k\frac{\epsilon\cdot k}{
k^2}\right)
\nonumber\\&&\hspace{3cm}
+F^{\pi}(0)(2q-k)\cdot k \frac{\epsilon\cdot k}{ k^2}\,.
\end{eqnarray}

In the following, we suggest two versions of the YCK model, where the YCK1 employs the GPDs of the nucleon EM form factors $F_1$ and $F_2$, while the YCK2 introduces the dipole type with the cutoff mass $\Lambda_1=\Lambda_2=1.55$ GeV to these form factors.


\subsection{YCK1: The Reggeized model with GPDs for proton EMFFs}

For studying the partonic aspect of the reaction in the DIS regime, $Q^2\geq$ 1 GeV$^2$ and $W\geq2$ GeV over the resonance region, we consider the case where the nucleon EMFF form factors parameterized by the GPDs in the
$\gamma^*NN$ vertex which are written as,
\begin{eqnarray}
&&F^{p}_1(Q^2)=\int_0^1
dx\left[e_uu_v(x)+e_dd_v(x)\right]x^{\alpha'_q(1-x)Q^2},
\label{f1p}\\
&&F^{n}_1(Q^2)=\int_0^1
dx\left[e_du_v(x)+e_ud_v(x)\right]x^{\alpha'_q(1-x)Q^2},
\label{f1n}\\
&&F^{p}_2(Q^2)={1\over \kappa_p}\int_0^1
dx\biggl[{e_u\kappa_u\over N_u}(1-x)^{\eta_u}u_v(x)
\nonumber\\&&\hspace{2cm} +{e_d\kappa_d\over
N_d}(1-x)^{\eta_d}d_v(x)\biggr]x^{\alpha'_q(1-x)Q^2},
\label{f2p}\\
&&F^{n}_2(Q^2)={1\over \kappa_n}\int_0^1
dx\biggl[{e_d\kappa_u\over N_u}(1-x)^{\eta_u}u_v(x)
\nonumber\\&&\hspace{2cm} +{e_u\kappa_d\over
N_d}(1-x)^{\eta_d}d_v(x)\biggr]x^{\alpha'_q(1-x)Q^2}\,.
\label{f2n}
\end{eqnarray}
%where the normalizations for the quark charge densities, $2/N_u$ and $1/N_d$ are understood in the charge
%form factor $F_1^{p(n)}(Q^2)$.
Valence quark charge and anomalous magnetic momenta are $e_u=2/3$,
$e_d=-1/3$, $\kappa_u=1.673$, and $\kappa_d=-2.033$, respectively.
$\eta_u=1.713$ and $\eta_d=0.566$ from a fit to the nucleon form
factor data. We make the slope parameter adjusted to obtain
$\alpha'_q=0.3$ GeV$^{-2}$ which yields the same result from the
$\Lambda_1^p=1.55$ GeV.

The unpolarized valence quark densities are from Ref. Martin2002;
\begin{eqnarray}\label{ungpd}
&&u_v(x)=0.262x^{-0.69}(1-x)^{3.5}(1+3.83x^{0.5}+37.65x),\nonumber\\
\\
&&d_v(x)=0.061x^{-0.65}(1-x)^{4.03}(1+49.05x^{0.5}+8.65x)\nonumber\\
\end{eqnarray}
at the input scale $\mu^2=1$ GeV$^2$. For simplicity, the
zero-skewness GPDs are implemented in the $\gamma^* NN$ vertex
form factors throughout the work.
\\

{\bf - In epion-tch-v2.exe, the slope $\alpha'_q=0.3$ is given as an input with the $F_2$ able to be turned on with the same slope $\alpha'_q$ chosen automatically. The pion cutoff $\Lambda_\pi=775$ MeV. The positive sign of $a_1$ exchange is chosen. The output is read as the BSA.dat for both the $t$-dependence and $Q^2$-dependence }
\\

\subsection{YCK2: The conventional Reggeized model}

The proton electromagnetic form factors (EMFF) are considered in charged pion processes, which are expressed as
\begin{eqnarray}
&&F_1^p (Q^2)={1+\tau\,\mu_p\over
1+\tau}\left(1+Q^2/\Lambda_{1}^2 \right)^{-2}\,,
\label{diracff1}
\\&&\kappa_p F_2^p(Q^2)={\mu_p-1\over 1+\tau}\left(1+Q^2/\Lambda_{2}^2 \right)^{-2}
\end{eqnarray}
with $\tau =Q^2/4M_P^2$ and $\Lambda_2=\Lambda_1$.

In the description of the reaction cross sections model prediction with the magnetic term is worse than without it. Thus, it is usually neglected.
\\

{\bf - In epion-tch-v2.exe, the cutoff $\Lambda_1=1.55$ GeV is given as an input with the $F_2$ turned on with cutoff $\Lambda_2=\Lambda_1$ chosen automatically. The pion cutoff $\Lambda_\pi=775$ MeV. The positive sign of $a_1$ exchange is chosen. The output is read as the BSA.dat for both the $t$-dependence and $Q^2$-dependence}


\subsection{Meson sectors}

In both approaches the monopole type of charge form factor is shared for the meson exchange in common,
\begin{eqnarray}\label{meson-ff}
&&F_{\varphi}(Q^2)=\left(1+Q^2/\Lambda_\varphi^2 \right)^{-1}
\end{eqnarray}
for $\varphi=\pi,\,V,\,T,\,A,\,B$ and $h_1$,

For the pion exchange, the cutoff mass
\begin{eqnarray}
\Lambda_\pi=0.775\ {\rm GeV}
\end{eqnarray}
is chosen for the pion charge form factor.
In the Particle Data Group, it is reported that mesons of high spin are decaying to $a_1\to\rho\pi$, $b_1\to\omega\pi$, and $h_1\to\rho\pi$, respectively. According to the Vector Meson Dominance, $a_1\to\rho\pi\to\gamma\pi$, for instance, this leads to
\begin{eqnarray}
\Lambda_{a_1}=0.775\ {\rm GeV}\,,\
\Lambda_{b_1}=0.782\ {\rm GeV}\,,\
\Lambda_{h_1}=0.775\ {\rm GeV}\,,\nonumber\\
\end{eqnarray}
respectively.
For the tensor meson $a_2$ exchange, we assume $\rho(1450)\to a_2\pi$ rather than the $\rho(775)$ and take the cutoff mass $\Lambda_{a_2}=1.45$ GeV in order to assign its role similar to that in the case of photoproduction \cite{prc2011}.
\\




The role of the axial meson $a_1(1260)$ affects the SSA as well as the ratio of the cross sections $d(\pi^-p)/d(\pi^+n)$, so that the change of its sign leads to the different results. We once predicted $d\sigma_{LT'}/d\sigma_U$ with the sign of the $a_1$ positive. But the case of $a_1$ with the sign negative results in the better description of $d\sigma_{LT'}/d\sigma_U$ and the cross section ratio as well.
\\



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}
%\begin{figure}[t]
\begin{center}
\includegraphics[clip=true,width=2\columnwidth,angle=0.]{fig1.eps}
%\includegraphics[width=6.5cm]{fig1.eps}
\end{center}
\caption{$t$-dependence of $d\sigma_{LT'}/d\sigma_U$ for the exclusive reaction $p(e,e'\pi^+)n$ at JLab KaonLT experiment.
The solid curve is from the epion-tch-v2, where we choose the GPDs for $F_1$ and $F_2$ with $\alpha'_q$=0.3 GeV$^{-2}$ with $\Lambda_\pi$=775 MeV.
$a_1$ with the sign positive is shown. The red dashed curve comes from the case of choosing dipole FF for $F_1$ and $F_2$, $\Lambda_1=\Lambda_2=$1.55 GeV with the same cutoff $\Lambda_\pi$.
The output is BSA.dat.
{\bf The green dash-dotted curve is obtained from the VR model in the epion-km-vr-btk-bsa.f90 with BSA$\_$t.dat as an output.}
}
\label{fig1}
%\end{figure}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[clip=true,width=1\columnwidth,angle=0.]{fig2.eps}
\end{center}
\caption{$Q^2$-dependence of $d\sigma_{LT'}/d\sigma_U$
for the exclusive reaction $p(e,e'\pi^+)n$ from JLab.
{\bf In epion-tch-v2, we choose the GPDs for $F_1$ and $F_2$ with $\alpha'_q$=0.3 GeV$^{-2}$ with $\Lambda_\pi$=775 MeV.
The solid curve is from the GPDs for nucleon form factors with the $a_1$ exchange of the sign positive. The dashed results from the dipole FF for the BSA. The output is BSA.dat.}
}
\label{fig2}
\end{figure}
%%\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The $LT'$ component of the cross section relevant to the beam polarization asymmetry for the virtual photon is calculated as
\begin{eqnarray}
%A_{LU}(\phi)=\sqrt{2\epsilon_L(1-\epsilon_L)}\,
{d\sigma_{LT'}\over d\sigma_{U}},
\end{eqnarray}
and the results are compared with the data measured from the KaonLT BSA experiment.

{\bf In the next section, the KM and VR models are presented with our model BTK for comparison.
In the epion-km-vr-btk-bsa.f90, only the nucleon charge form factor $F_1$ is taken in three model calculations with the $F_2$ excluded in common.
The KM and VR models reproduce the BSA in this code to yield BSA$\_$t.dat and BSA$\_$QQ.dat as the outputs for the $t$- and $Q^2$-dependences of the BSA.
But, the btk model in this code is simply an extension of the JKPS model, and hence, results in the failure of the BSA, because of the absence of the nucleon $F_2$ FF.
For the case of reproducing the BSA in our model, only the epion-tch-v2 model is valid for both nucleon form factors $F_1$ and $F_2$ parameterized by GPDs (YCK1), or by the dipole form factors (YCK2).
}


\section{KM model}

The gauge prescription for the charged pion exchange with the nucleon pole term by the KM model is the same with ours.
The full description of the model is given in the paper, Murat M. Kaskulov and Ulrich Mosel, {\it Deep exclusive charged �� electroproduction above the resonance region}, Phys. Rev. C {\bf81}, 045202 (2010).

The model is featured by the nucleon form factor

\begin{widetext}
\begin{eqnarray}
&&F_s(Q^2,s)= \frac{
s\,\ln\left[{\xi Q^2\over M_p^2}+1\right]\frac{2\xi Q^2+s}{\xi^2 Q^4}-\frac{s(\xi Q^2+s)}{\xi Q^2(\xi Q^2+M_p^2)}+\ln\left[\frac{s-M_p^2}{M_p^2}\right]-i\pi
}
{
\left({\xi Q^2\over s}+1\right)^2 \left(\frac{s^2+2sM_p^2}{2M_p^4}+\ln\left[\frac{s-M_p^2}{M_p^2}\right]-i\pi\right)
}\,\label{kmffs}
\\
&&F_u(Q^2,u)= \frac{
u\,\ln\left[{\xi Q^2\over M_p^2}+1\right]\frac{2\xi Q^2+u}{\xi^2 Q^4}-\frac{u(\xi Q^2+u)}{\xi Q^2(\xi Q^2+M_p^2)}+\ln\left[\frac{M_p^2-u}{M_p^2}\right]
}
{
\left({\xi Q^2\over u}+1\right)^2 \left(\frac{u^2+2uM_p^2}{2M_p^4}+\ln\left[\frac{M_p^2-u}{M_p^2}\right]\right)
}\,\label{kmffu}
\end{eqnarray}
\end{widetext}
with $\xi=0.4$
for the nucleon charge form factor $F_1$ in the $s$- and $u$-channels, respectively.

For the pion exchange, the cutoff mass
\begin{eqnarray}
\Lambda_{\pi}=0.68\ {\rm GeV}
\end{eqnarray}
is chosen for the monopole form factor in Eq. (\ref{meson-ff}), averaging over the scattered values $\Lambda_{\pi}=0.768$ GeV for $Q^2<0.4$ GeV$^2$, $\Lambda_{\pi}=0.632$ for $0.6<Q^2<1.5$ GeV$^2$ and $\Lambda_{\pi}=0.678$ GeV at high $Q^2$.
To account for the anti-shrinkage effect further, due to the decrease of the slope of the parton contribution with increasing $Q^2$, the slope of the pion trajectory is modified to be
\begin{eqnarray}\label{pi-alpha}
\alpha_\pi'={0.74\over {1+2.4{Q^2\over s}}}\,{\rm GeV}^{-2}\ .
\end{eqnarray}

The beam spin azimuthal (BSA) moment is defined as
\begin{eqnarray}
A_{LU}(\phi)=\sqrt{2\epsilon(1-\epsilon)}\,{d\sigma_{LT'}\over d\sigma_{U}},
\end{eqnarray}
where the virtual photon polarization for the ratio of the longitudinal to the transverse component $\epsilon$ is
given in Eq. (\ref{ell}).

The $A_{LU}(\phi)$ is calculated with the energy of the electron beam energy for the input, $E_e=5.77$ GeV in Eq. (\ref{ell}) for the JLab experiment at 5.7 GeV.

\section{VR model}

The VR model is based on the paper, T. Vrancx and J. Ryckebusch, {\it Charged-pion electroproduction above the resonance region,} Phys. Rev.  C {\bf89}, 025203 (2014)


Simplifying the KM form factors for the nucleon charge form factor in Eqs. (\ref{kmffs}) and (\ref{kmffu}), VR consider the phenomenological $s$-dependence of the dipole form-factor
\begin{eqnarray}
F_1(Q^2,x)=\left(1+{Q^2\over \Lambda^2_{\gamma pp*}(x)}\right)^{-2}, \ \ \ (x=s,u)
\end{eqnarray}
where
\begin{eqnarray}
&&\Lambda^2_{\gamma pp*}(s)=\Lambda_{\gamma pp}+\left(\Lambda_\infty-\Lambda_{\gamma pp}\right)\left(1-{M_p^2\over s}\right),
\\
&&\Lambda^2_{\gamma pp*}(u)=\Lambda_{\gamma pp}+\left(\Lambda_\infty-\Lambda_{\gamma pp}\right)\left(1-{M_p^2\over{ 2M_p^2-u}}\right),
\end{eqnarray}
which assume $\Lambda_\infty=2.194$ and $\Lambda_{\gamma pp}=0.84$ GeV in numerical calculations.

For the pion exchange, the cutoff mass
\begin{eqnarray}
\Lambda_{\pi}=0.655\ {\rm GeV}
\end{eqnarray}
is chosen for the monopole form factor in Eq. (\ref{meson-ff}) with the slope of the pion trajectory given in Eq. (\ref{pi-alpha}).


The $LT'$ component of the cross section relevant to the beam polarization asymmetry for the virtual photon is calculated as
\begin{eqnarray}
%A_{LU}(\phi)=\sqrt{2\epsilon_L(1-\epsilon_L)}\,
{d\sigma_{LT'}\over d\sigma_{U}},
\end{eqnarray}
and the results are compared with the data measured from the $KaonLT_BSA$ experiment.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[clip=true,width=1\columnwidth,angle=0.]{fig3.eps}
\end{center}
\caption{
$t$-dependence of $A_{LU}$ for the exclusive reaction $p(e,e'\pi^+)n$ at JLab 5.7 GeV electron beam.
{\bf The program epion-km-vr-btk-bsa.exe requires the electron beam energy 5.77 as an input to yield the BSA$\_$t.dat.}
}
\label{fig3}
\end{figure}
%%\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[clip=true,width=1\columnwidth,angle=0.]{fig4.eps}
\end{center}
\caption{$t$-dependence of $d\sigma_U$ for the exclusive reaction $p(e,e'\pi^+)n$.
{\bf The program epion-km-vr-btk-bsa.exe yields the dSU.dat.} 
}
\label{fig4}
\end{figure}
%%\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\end{document}























