%
It is well known that grand unified theories (GUTs), having only a single
gauge group,  thus possess a single gauge coupling of that group. 
The Standard Model (SM) gauge couplings are derived, after spontaneous breaking of
the GUT, from the renormalisation group evolution of the gauge couplings, from the
Grand Unified scale to the weak scale where they are measured. Thus there is
a testable GUT prediction: the 
couplings, which are measured at about the weak scale,
should all unify to a common value at
a single, very high energy scale. 
Assuming the SM as the relevant effective field theory, they
do not unify. However, assuming a supersymmetric desert above the weak scale
and using the  
minimal supersymmetric standard model (MSSM), the strong and electroweak
couplings do approximately unify at a common scale of $M_{GUT} \approx 2 \cdot
10^{16}$ GeV.
%as shown in the left-hand panel of Figure\,\ref{alfa}. 
In the specific calculation used here, the couplings do not quite
match, see Figure\,\ref{alfa} (right).
In realistic GUTs such deviations may occur and are
caused by threshold effects, for example by the prediction of heavy GUT relic
particles that lie just below $M_{GUT}$. 
An accurate inference of this deviation therefore gives important clues into
the structure of such heavy states and therefore, ultimately, the GUT itself. 
%This is particularly valuable in view of the fact that GUTs are not directly
%accessible because of the huge masses  
%involved.
It is visible that the present level of uncertainty of the strong
coupling is much larger than that of the weak coupling and the
fine structure constant, while with the LHeC a huge improvement is
expected. 
%, see the right-hand panel of Figure\,\ref{alfa}.
%
\begin{figure}[t]
\unitlength=1cm
\begin{picture}(13,7)
\put(0,0){\includegraphics[clip=,angle=0.,width=.53\textwidth]{asrun1.pdf}}
\put(9,7.){\includegraphics[clip=,angle=-90.,width=.43\textwidth]{alfbt-crop.pdf}}
\end{picture}
\caption{Left: Running of the strong coupling constant up to high scales
 $\sqrt{Q^2} = \mu$ in the absence (SM) and presence of $n$ extra 
 dimensions~\cite{wobi};
 Right: Extrapolation of the coupling constants (1/$\alpha$) within 
 the MSSM (for the parameter point CMSSM40.2.5~\cite{AbdusSalam:2011fc})
 to the Grand Unified scale as predicted by {\tt SOFTSUSY}~\cite{SOFTSUSY}. The width of the red line is the
 uncertainty of the world average of $\alpha_s$, which
 is dominated by
 the lattice QCD calculation chosen for the PDG average.
 The black band is the LHeC projected experimental 
 uncertainty~\cite{AbelleiraFernandez:2012cc}.
}
\label{alfa}
\end{figure}
