The Behavior of the Yukawa Potential in the Presence of a Natural Momentum Cutoff: An Analytical Study

In this paper the Proca field equations for a massive gauge particle are obtained in the presence of a natural momentum cutoff “\(p_{\max }\)” based on a covariant generalization of a one-parameter extension of the Heisenberg algebra. The Yukawa potential for a static point source in the presence of \(p_{\max }\) (generalized Yukawa potential) is obtained analytically and it is shown that in contrast with the Yukawa potential for a static point source in Proca electrodynamics, the generalized Yukawa potential has a finite value at the location of the static point source. Our calculations demonstrate that the Coulomb potential, the Yukawa potential, and the Coulomb potential in the presence of \(p_{\max }\) can be derived from the generalized Yukawa poitential. We show that the free space solutions of Proca electrodynamics in the presence of \(p_{\max }\) describe a massive gauge particle with the effective mass \(m_{eff} = \frac{m}{{\sqrt {1 - \left( {\frac{mc}{{p_{\max } }}} \right)^{2} } }}\), where \(m\) is the rest mass of the ordinary Proca particle. Numerical estimations in Sect. 5, show that the lower bound for \(p_{\max }\) must take the value \(\left( {p_{\max } } \right)_{\min } = (91.187 \pm 0.007)\,\,\frac{GeV}{c}\) in order to avoid complex values for the effective mass \(m_{eff}\). This lower bound for \(p_{\max }\) is near to the momentum scale of the electroweak interactions. It should be mentioned that for the very large values of \(p_{\max }\) the results of this work reduce to the well-known results of standard Proca electrodynamics.