Studying effective potentials is an important tool to understand, through classical concepts, the quantum behaviour of physical systems. Moreover, effective potentials are a natural way to study spontaneous symmetry breaking. Even though Supersymmetry is one of the most discussed proposals of physics beyond the standard model, to construct realistic models of particle physics, Supersymmetry should be spontaneously broken, since we have not observed the supersymmetric partners of ordinary particles. However, due to non-renormalization theorems, spontaneous supersymmetry breaking is ruled out for a wide class of models with phenomenological interest. Therefore, it is essential to understand under which conditions supersymmetryshould be spontaneously broken, and the studying of the effective potential is a natural way to deal with this problem. Furthermore, methods of effective potential has been applied to the studying of quantum gravity phenomena. In recent works, several authors support the thesis that gravitational corrections cause a running to the gauge coupling constants, even if they themselves (i.e. without gravitation) are exactly conformal. This proposal has a direct consequence on the phenomenon of asymptotic freedom, where the most notable example is the asymptotic freedom in quantum electrodynamics. Such results are obtained in the light of effective theories. From this point of view, Supergravity is an effective theory of Superstrings, so supergravitational contributions should play a key role in the studying of the running of gauge coupling constants. Using the superfield formalism, we will calculate the beta function of the Super-Yang-Mills theories coupled to Supergravity, verifying the possible existence of asymptotic freedom for the gauge couplings.
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