example for $su(2)$. The first non-trivial case, $su(3)$, will be described in more detail at the end of section 5.

After normal ordering the expressions (\[R2\]– \[R4\]), [*i.e.*]{} moving all positive modes to the right of negative modes, and with the help of the above trace formulae we find : $$\label{Rn2}
{\cal O}_{\rm reg}^{(2)}(R) = 2 \pi^2 I_R\;  J_0^a J_0^a \,,$$ $$\begin{aligned}
\label{Rn3}
{\cal O}_{\rm reg}^{(3)}(R)  &=&
{2\pi^3\over 3} \, I_R^{(3)}\,   d^{abc} \, J^a_0 J^b_0 J^c_0
\; + \;  4 \pi^2  I_R\; f^{abc} \sum_{n>0}
{1\over n}\;
J^a_{-n}J^b_0 J^c_n  \;  -
 \\
&&  \hspace*{-0.3cm}  -\;  4 \pi^2 i\; I_R\,  h^\vee\, \psi^2
\left[ \; 
\sum_{n>0} {1\over n} J^a_{-n} J^a_n \, -\, 
{1\over 2}\,  J^a_0 J^a_0  \left(
\sum_{n>0}{e^{-ns}\over n}  \right) \, + \,    
{\kappa\, \over 6}\, {\rm dim}(g) \; \left(\sum_{n>0} e^{-ns}
\right)  \right] 
\,, \nonumber\end{aligned}$$ and $$\begin{aligned}
\label{Rn4}
{\cal O}_{\rm reg}^{(4)}(R)   &=& : {\cal O}_{\rm reg}^{(4)}(R) :
-\;  2 \pi^2 \, I_R\,  h^\vee \, \psi^2\,\kappa\, \Biggl[\;
\sum_{n>0} {1\over n}\; J^a_{-n} J^a_{n}
  - \, J^a_0
J^a_0 \; \left( \sum_{n>0} {e^{-ns}\over n} \right) \, +
\nonumber \\
& & + \; { \kappa\over 4}\,   {\rm dim}(g)\,
\left(\sum_{n>0}e^{-ns} \right)
  \Biggr]\;\  + \;\ 
{\rm subleading}\,.\end{aligned}$$ For the quartic operator we have not written out explicitly the ‘subleading 