Spectral function of a scalar boson coupled to fermions

Abstract: We present the calculation of the spectral function of an unstable scalar boson coupled to fermions as resulting from the resummation of the one loop diagrams in the scalar particle self energy. We work with a large but finite high-energy cutoff: in this way, the spectral function of the scalar field is always correctly normalized to unity, independently on the value of the cutoff. We show that this high energy cutoff affects the Breit-Wigner width of the unstable particle: the larger the cutoff, the smaller is the width at fixed coupling. Thus, the existence of a high energy cutoff (alias minimal length), and for instance the possible opening of new degrees of freedom beyond that energy scale, could then be in principle proven by measuring, at lower energy scales, the line shape of the unstable scalar state. Although the Lagrangian here considered represents only a toy-model, we discuss possible future extensions of our work which could be relevant for particle physics phenomenology.