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We present virtual contributions up to two loop level in perturbative QuantumChromodynamic (QCD) to the decay of pseudo-scalar Higgs boson ($A$) to threegluons ($g$) and also to quark ($q$), anti-quark ($\overline q$) and a gluon.With appropriate crossing, they are well suited for predicting the differentialdistribution of $A$ in association with a jet in hadron colliders up tonext-to-next-to-leading order (NNLO) in strong coupling constant and also forthe subsequent decay of $A$ to hadrons. We use effective field theory approachto integrate out the top quarks in the heavy top limit. The resulting theoryinvolves two pseudo-scalar composite operators describing the interaction of$A$ with gluons as well as with quark and anti-quark. We perform ourcomputation in dimensional regularisation and use minimal subtraction($\overline{MS}$) scheme to renormalise strong coupling constant as well as thecomposite operators. The ultraviolet (UV) finite amplitudes contain infrared(IR) divergences that are found to be in agreement with the predictions byCatani. For both the amplitudes namely $A \rightarrow g g g$ and $A \rightarrowq \overline q g$, the leading transcendental terms at one and two loops arefound to be identical to those in a three point form factor (FF) of thehalf-BPS operator in ${\cal N}=4$ Supersymmetric Yang Mills (SYM) theory whenthe QCD color factors are adjusted in a specific way. We present our results interms of harmonic polylogs well suited for further numerical study.

Two loop QCD corrections for the process Pseudo-scalar Higgs $\rightarrow 3$ partons