  We promote the Immirzi parameter to be a minimally coupled scalar field and
we analyzed the Hamiltonian constraints in the framework of Loop Quantum
Gravity without the time gauge. Proper SU(2) connections can be defined and a
term containing derivatives of the field $\beta$ enters into their definition.
Furthermore, boost degrees of freedom are non-dynamical, while the
super-momentum constraints coincide with the scalar field case. Hence, the
kinematical Hilbert space can be defined as for gravity in presence of a
minimally coupled scalar field. Then, we analyzed the dynamical implications of
this scenario and we outlined how a the dynamical relaxation to a non-vanishing
vacuum expectation value is predicted.
