  
\label{fig:MJTSTARcfnfit}
\end{figure*}\vspace*{-1.0pc
\subsection{The away particles from a hadron-trigger do not measure the fragmentation function~\cite{ppg029}}
It was generally assumed, as implied by Feynman, Field and Fox in 1977~\cite{FFF}, that the $p_{Ta}$ (or $x_E$, or $x_h$) distribution of away-side hadrons from a single hadron trigger with $p_{Tt}$, corrected for $\mean{z_t}$, would be the same as that from a jet-trigger and would measure the away-jet fragmentation function as it does for direct photon triggers~\cite{ppg095}. However, attempts to try this at RHIC led to the discovery~\cite{ppg029} that the $x_E$ distribution does not measure the fragmentation function. The good news was that it measured the ratio of the away jet to the trigger jet transverse momenta, $\hat{x}_h=\hat{p}_{Ta}/\hat{p}_{Tt}$, Eq.~\ref{eq:condxe2N}
    \begin{equation}
\left.{dP_{\pi} \over dx_E}\right|_{p_{T_t}}  = {N\,(n-1)}{1\over\hat{x}_h} {1\over {(1+ {x_E \over{\hat{x}_h}})^{n}}} \,  
\qquad,  
\label{eq:condxe2N}
\end{equation}   
with the value of $n=8.10$ ($\pm0.05$) fixed as determined in  Ref.~\cite{ppg080}, where $n$ is the power-law of the inclusive $\pi^0$ spectrum and is observed to be the same in p$+$p and Au+Au collisions in the $p_{T_t}$ range of interest as noted in section~\ref{sec:shiftmeasured} above. 
 

\section{How to apply this information to find $\hat{q}$ from p$+$p and Au$+$Au di-hadron measurements}
A recent STAR $\pi^0$+h di-hadron measurement in p$+$p and Au$+$Au collision at \mbox{$\sqrt{s_{_{NN}}}$}\xspace=200 GeV~\cite{STARPLB760} is used to measure  $\mean{\hat{q}L}$ by calculating $k_T$ in each case as in Eq.~\ref{eq:qhatsolution}.
For a di-jet produced in a hard scattering, the initial $\hat{p}_{Tt}$ and $\hat{p}_{Ta}$ will both be reduced by energy loss in the medium to become $\hat{p}{'}_{Tt}$ and $\hat{p}{'}_{Ta}$ that will be measured by the di-hadron correlations with $p_{Tt}$ and $p_{Ta}$ in Au$+$Au collisions. As both jets from the initial di-jet lose energy in the medium, the azimuthal angle between the di-jets  from the $\mean{k_{T}^2}$ in the original collision should not change unless the medium induces multiple scattering from $\hat{q}$. Thus, without $\hat{q}$ and assuming the same fragmentation transverse momentum $\rms{j_T}$ in the original jets and those that have lost energy, the   $p_{\rm out}$ between the away hadron with $p_{Ta}$ and the trigger hadron with  $p_{Tt}$ will not change (Fig.~\ref{fig:mjt-poutxe}), but the $\mean{k{'}_{T}^2}$ will be reduced according to Eq.~\ref{eq:kTformula2} because the ratio of the away to the trigger jets $\hat{x}{'}_h=\hat{p}{'}_{Ta}/\hat{p}{'}_{Tt}$ will be reduced. Thus the calculation of $k{'}_T$ from the di-hadron p$+$p measurement to compare with Au$+$Au measurement with the same di-hadron trigger $p_{Tt}$ and $p_{Ta}$ must use the values of $\hat{x}_h$, and $\mean{z_T}$ from the Au$+$Au measurement to compensate for the energy lost by the original dijet in p$+$p collisions.
\begin{figure*}[!th]
\includegraphics[width=0.49\linewidth]{figs/stwbfef}
\includegraphics[width=0.48\linewidth]{figs/MJTPHENIXxEfit}
\caption[]{a) (left) Fits of Eq.~\ref{eq:condxe2N} to the STAR away-side $z_T$ distributions~\cite{STARPLB760} in  Au$+$Au 0-12\% centrality and p$+$p for $12<p_{Tt}<20$ GeV/c. The Au$+$Au curve is a fit with $\hat{x}^{AA}_h=0.36\pm 0.05$ with error corrected by $\sqrt{\chi^2/\rm{dof}}$. The points with the open circles are the $y_i$ and systematic errors $\sigma_{b_i}$ of the data points while the filled points are $y_i+{\epsilon_b}\sigma_{b_i}$ with errors $\tilde{\sigma}_i$ and $\epsilon_b=-1.3\pm0.5$.  b) (right) Fits to PHENIX $x_E$ distributions~\cite{ppg106,MJTGrenoble} in p$+$p and Au$+$Au for $\pi^0$-h correlations with $9\leq p_{Tt}\leq 12$ GeV/c} 
\vspace*{-1.0pc} 
\label{fig:xEfits}  
\end{figure*}
\section{Calculation of ${\mean{\hat{q} L}}$ from the STAR measurement~\cite{STARPLB760}.} 
\subsection{Determine $\mean{p^2_{\rm out}}$ from the $\pi^0$-h correlation function} 
\label{sec:findpout}
This is accomplished by fitting the $\pi^0$-h correlation functions for $12<p_{Tt}<20$ GeV/c~\cite{STARPLB760} to a gaussian in $\sin{\Delta\phi}$ for the away-side, $\pi/2\leq \Delta\phi\leq 3\pi/2$~\cite{ppg029,ppg089,ppg195}; and a gaussian in $\Delta\phi$ for the trigger side $-\pi/2\leq \Delta\phi\leq\pi/2$ (Fig.~\ref{fig:MJTSTARcfnfit}). The results for Au$+$Au 0-12\% centrality are $\mean{p^2_{\rm out}}=0.547\pm0.163$ (GeV/c)$^2$, $\chi^2/{\rm dof}=33/21$ for $1.2<p_{Ta}<3$ GeV/c and $\mean{p^2_{\rm out}}=0.851\pm0.203$ (GeV/c)$^2$, $\chi^2/{\rm dof}=23/21$ for $3<p_{Ta}<5$ GeV/c. The same fits to the p$+$p measurements with the same $p_{Tt}$ and $p_{Ta}$ yielded $\mean{p^2_{\rm out}}=0.263\pm0.113$ (GeV/c)$^2$, $\chi^2/{\rm dof}=186/21$ for $1.2<p_{Ta}<3$ GeV/c and $\mean{p^2_{\rm out}}=0.576\pm0.167$ (GeV/c)$^2$, $\chi^2/{\rm dof}=137/21$ for $3<p_{Ta}<5$ GeV/c where the errors have been corrected up by $\sqrt{\chi^2/{\rm dof}}$.
\subsection{Determine $\hat{x}_h=\hat{p}_{Ta}/\hat{p}_{Tt}$}
This is done by a fit of Eq.~\ref{eq:condxe2N} to the STAR measurements of what they call the away-side $z_T$ distribution~\cite{STARPLB760} (called the $x_h$ or $x_E$ distribution here) for $12<p_{Tt}<20$ GeV/c in p$+$p and Au$+$Au 0-12\% centrality collisions (Fig.~\ref{fig:xEfits}). The fit~\cite{ppg079} takes account of the statistical and correlated systematic errors, $\sigma_i$ and $\sigma_{b_i}$, for each data point with $dP/dx_E=y_i$ : 
	\begin{equation}
{\chi}^2={\left[\sum_{i=1}^{n}
{{(y_i+\epsilon_b \sigma_{b_i} -y_i^{\rm fit})^2}  \over {{\tilde{\sigma}}_i^2}} \right]}+ {\epsilon_b^2} \qquad