$q{\bar q}$ Potential at Finite T and Weak Coupling in ${\cal N}=4$

We compute the potential between a $q{\bar q}$ singlet for ${\cal N}=4$ $SUSY$ with gauge group $SU(N)$ at finite temperature $T$, large distances $rT \gg 1$ and weak coupling $g$. As a first step, we only consider at the electric modes and we compute the Debye mass $m_D$ where we find that each of the $8N$ bosonic degrees of freedom contributes to $m_D^2$ with $\frac{1}{6}g^2 T^2$ while each of the $8N$ fermionic degrees of freedom contributes with $\frac{1}{12}g^2 T^2$ yielding to $m_D^2=2Ng^2T^2$. Then, motivated by results obtained in the literature from both, a weak coupling approach for $QCD$ and a large coupling investigation of ${\cal N}=4$ $SUSY$ through $AdS/CFT$, we attempt to include magnetic mode corrections.

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