 + \frac{1}{M_1} \left[\frac{\zeta_{21}}{E}+\frac{\zeta_{21}^\prime }{E_2} \right] \nonumber \\
&  & \mbox{} - \frac{2(EE_2-p_2ly_0)}{M_1^2} \left[\frac{p_2ly_0\theta_3-\zeta_{11}}{E} - \frac{2p_2ly_0\theta_3^\prime -\zeta_{11}^\prime}{E_2} \right] - \frac{1}{2M_1^2} \left[\frac{I_n}{E}+\frac{I_n^\prime}{E_2} \right] - \frac{4mM_2}{p_2l}(J_{2n}+J_{3n}),
\end{eqnarray}
and
\begin{equation}
\frac{M_1^2}{4p_2l} \Lambda_{8n} = \frac{m^2}{2M_1^2} \left[ 2\eta_0 +\left[E(1-\beta^2)+E_2-M_1\right]\theta_3 + \left[M_1-E-\frac{M_2^2}{E_2}\right]\theta_3^\prime + \frac{\zeta_{11}^\prime}{E_2}-\frac{\zeta_{11}}{E}-2J_{1n}\right], 
\end{equation}
\begin{equation}
\Lambda_{9n} = -\frac12 \Lambda_{8n}. \label{eq:lb9a}
\end{equation}
The $\theta_m$, $\eta_0$, and $\zeta_{ij}$ functions can be found in Ref.~\cite{rfm04}. The $\theta_m^\prime$ and $\zeta_{ij}^\prime$ functions are obtained by making the replacements $p_2 \leftrightarrow l$ in the corresponding $\theta_m$ and $\zeta_{ij}$ functions. The additional functions involved in the above expressions are
\begin{equation}
J_{1n} = \frac{1}{\beta\beta_2} \left[-(1+\beta\beta_2) + (\beta+\beta_2)(\mathrm{arctanh}\, \beta+\mathrm{arctanh}\, \beta_2) - (1-\beta\beta_2y_0)(-1+\beta^\prime \mathrm{arctanh}\, \beta^\prime)\right].
\end{equation}
\begin{eqnarray}
\frac{I_n}{4p_2^2l^2} & = & \frac{E_\nu^0}{p_2^2}\eta_0 + \frac{\beta E_\nu^0+l-p_2}{2\beta p_2^2}\theta_0 - \frac{EE_\nu^0}{p_2^2}(\theta_3-\theta_4) +\frac{1}{2p_2^2\beta^2}\left[3{E_\nu^0}^2-l^2+3E(E+2E_\nu^0)\right] (\theta_3-\theta_4-\beta\theta_5) \nonumber \\
&  & \mbox{} + \left[ y_0^2 - \frac{{E_\nu^0}^2}{2p_2^2} \right] \theta_3 - \frac{3E}{2p_2^2}(E_\nu^0+E)\theta_{10} - \frac{3E}{2p_2l}(E+E_\nu^0)(\theta_{12} - \theta_{13}) + \frac{y_0}{2} \theta_{12} + \frac{3E}{2p_2} Y_1 + \frac{1}{2\beta^2}Y_3 - \frac{2y_0}{p_2l} \zeta_{11}, \nonumber \\
\end{eqnarray}
\begin{eqnarray}
\frac{8M_1^2}{mM_2} J_{2n} & = & \frac{1+{\beta^\prime}^2}{1-{\beta^\prime}^2} - 2\mathrm{arccosh} \left[\frac{1}{\sqrt{1-{\beta^\prime}^2}} \right] \left[\mathrm{arccosh} \left[\frac{1}{\sqrt{1-{\beta^\prime}^2}} \right] + \frac{2\beta^\prime}{1-{\beta^\prime}^2} \right] -\frac{2(1+\beta\beta_2)^2}{(1-\beta^2)(1-{\beta_2}^2)} + 1 \nonumber \\
&  & \mbox{} + 2\mathrm{arccosh} \left[\frac{1+\beta\beta_2}{\sqrt{1-\beta^2}\sqrt{1-\beta_2^2}}\right] \left[\mathrm{arccosh} \left[\frac{1+\beta\beta_2}{\sqrt{1-\beta^2}\sqrt{1-\beta_2 ^2}}\right] + \frac{2(\beta+\beta_2)(1+\beta\beta_2)}{(1-\beta^2)(1-\beta_2^2)}\right],
\end{eqnarray}
and
\begin{equation}
\frac{2M_1^2}{a} J_{3n} = \frac{p_2l}{mM_2}\eta_0 + \frac{a\beta^\prime}{2mM_2} \mathrm{arccosh} \left[ \frac{a}{2mM_2}\right] - \frac{EE_2}{mM_2} (\beta+\beta_2) \mathrm{arccosh} \left[\frac{1+\beta\beta_2}{\sqrt{1-\beta^2}\sqrt{1-\beta_2^2}} \right],
\end{equation}
where the factors $\beta$, $\beta^\prime$, $\beta_2$, and $a$ are defined in Eqs.~(\ref{eq:beta}), (\ref{eq:betap}), (\ref{eq:beta2}), and (\ref{eq:aa}), respectively.

\section{\label{sec:finr}Final results and discussions}

The differential decay rate of $K_{l3}^0$ decays in the variables $E$ and $E_2$ (that is, the DP) including radiative corrections to order $(\alpha/\pi)(q/M_1)$, is given by
\begin{equation}
d\Gamma(K_{l3}^0) = d\Gamma_V + d\Gamma_B. \label{eq:dkl3t}
\end{equation}
$d\Gamma_V$ is given by Eq.~(\ref{eq:dgv}). For $d\Gamma_B$ two forms are available. The first one contains the triple integration over the real photon variables standing so it can be performed numerically. It is given by 
Eq.~(\ref{eq:dgb}), which is expressed in terms of the functions $\Lambda_{kn}$ introduced in Eqs.~(\ref{eq:lb1}) and (\ref{eq:lb2})--(\ref{eq:lb3}). The infrared divergence and the finite terms that come along with it have been explicitly and analytically extracted, however; the infrared divergence is of course canceled out in the sum in (\ref{eq:dkl3t}). The second form of $d\Gamma_B$ is completely analytical; the integration over the photon variables has been explicitly computed and the analytical versions of the functions $\Lambda_{kn}$ are thus given in Eqs.~(\ref{eq:lb1a})--(\ref{eq:lb9a}).

The main result obtained here can be cast into the compact form
\begin{equation}
d\Gamma(K_{l3}^0) = \frac{G_F^2}{32\pi^3} |V_{us}|^2 M_1^3 \left[A_0^\prime + \frac{\alpha}{\pi}A_n^\prime \right] dEdE_2. \label{eq:dgfin}
\end{equation}
$A_0^\prime$ has been previously computed; it is given in Eq.~(16) of Ref.~\cite{jl2011}. On the other hand, $A_n^\prime$ can be written as
\begin{eqnarray}
A_n^\prime & = & A_0^\prime (\mathrm{Re}\, \Phi_{n} + I_{0n}) + A_{Vn}^\prime \mathrm{Re}\, \Phi_n^\prime + A_{Bn} \nonumber \\
& = & A_{10} |f_+^\prime(q^2,p_+\cdot l)|^2 + A_{20} \mathrm{Re} [f_+^\prime(q^2,p_+\cdot l) {f_-^\prime}^*(q^2,p_+\cdot l)] + A_{30} |f_-^\prime(q^2,p_+\cdot l)|^