Revisiting the constraints on the Supersymmetric Higgs sector at the Tevatron

We analyze the production of the neutral Higgs particles of the Minimal Supersymmetric extension of the Standard Model at the Fermilab Tevatron collider. We consider the two main production and detection channels: gluon--gluon and bottom quark fusion leading to Higgs bosons which subsequently decay into tau leptons, $gg, b\bar b\! \to\! {\rm Higgs}\! \to\! \tau^+\tau^-$. We update the production cross sections and the decay branching ratios and obtain production rates that are significantly smaller at high masses than the ones used by the CDF and D0 experiments in their search. We then evaluate the various theoretical uncertainties that affect these rates, uncertainties that have not been been considered in the experimental analyses and which turn out to be rather large. Including these two effects will significantly loosen the constraints obtained on the supersymmetric Higgs sector at the Tevatron.