By applying a circularly polarized and slightly blue-detuned microwave field with respect to the first excited rotational state of a dipolar molecule, one can engineer a long-range, shallow potential well in the entrance channel of the two colliding partners. As the applied microwave ac-field is increased, the long-range well becomes deeper and can support a certain numbers of bound states, which in turn bring the value of the molecule-molecule scattering length from a large negative value to a large positive one. We adopt an adimensional approach where the molecules are described by a rescaled rotational constantB = B/sE 3 where sE 3 is a characteristic dipolar energy. We found that molecules withB > 10 8 are immune to any quenching losses when a sufficient ac-field is applied, the ratio elastic to quenching processes can reach values above 10 3 , and that the value and sign of the scattering length can be tuned. The ability to control the molecular scattering length opens the door for a rich, strongly correlated, many-body physics for ultracold molecules, similar than that for ultracold atoms.
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