  This thesis studies the pseudoscalar Higgs boson production via gluon fusion
in the EFT framework in a CP-conserving model. First, it presents the
di-pseudoscalar Higgs boson production cross-section via gluon fusion till NNLO
with the results valid for the pseudoscalar Higgs boson of MSSM and 2HDM with
small tan $\beta$ by adjusting the top Yukawa coupling. We have used
dimensional regularization to regulate the UV and IR divergences while
carefully treating the Levi-Civita tensor and $\gamma_5$. Unlike the amplitudes
involving a pair of scalar Higgs bosons, we do not need UV contact counter
terms. We used Catani's predictions to factorize the IR singularities, and our
IR poles agreed with Catani's predictions. Our results are essential for
studies on producing a pair of pseudoscalar Higgs bosons at the LHC up to NNLO
accuracy. This thesis also includes the next-to-soft-virtual (NSV) resummed
corrections to $\overline{\text{NNLL}}$ accuracy, which are also matched to the
NNLO cross-sections for a pseudoscalar Higgs boson production via gluon fusion.
These NSV corrections are potentially significant compared to the conventional
soft-virtual (SV) logarithms. We have estimated the uncertainties due to the
choice of various PDFs and those due to the renormalization and factorization
scales. The scale uncertainties show improvement for renormalization scale
variation only, which suggests that NSV contributions from other parton
channels and beyond NSV contributions in the gluon fusion channel are necessary
for improved stability. We also studied the production cross-sections for mixed
scalar-pseudoscalar states and their impact on QCD cross-sections for different
values of the mixing angle $\alpha$. The study indicates the necessity of
improving the precision results for the pseudoscalar Higgs boson up to an order
comparable to that of the scalar Higgs boson.
