A momentum-dependent term of the potential can be added to the Skyrme parametrisation. In total the potential has the following form:

\[ U(\mathbf{r}, \mathbf{p}) = A\frac{\rho(\mathbf{r})}{\rho_0} + B\left(\frac{\rho(\mathbf{r})}{\rho_0}\right)^\tau + \frac{2C}{\rho_0}g\int\frac{d^3p'}{(2\pi)^3}\frac{f(\mathbf{r}, \mathbf{p}')}{1+\left(\frac{\mathbf{p}-\mathbf{p}'}{\Lambda}\right)^2} \]

This shape of the potential is taken from Welke:1988zz [61] and includes an integral over momentum. This integral is quite costly to evaluate during runtime and to reduce numerical cost, following the GiBUU implementation Buss:2011mx [13], we make the assumption that the distribution function takes the form of cold nuclear matter \( f(\mathbf{r}, \mathbf{p}) = \Theta(p-p_F)\), where \( p_F \) is the Fermi momentum. Note that the Fermi momentum depends on the density and therefore on the position in general. With this assumption the integral has an analytic solution and can be evaluated relatively quickly. When choosing the parameters \( C \) and \( \Lambda\) it is important to make sure that nuclear ground state properties are realistic. In other words the momentum dependence parameters have to be constrained together with the Skyrme potential parameters.

C — double, required

Parameter \( C \) of the momentum-dependent term of the potential in MeV.

Lambda — double, required

Parameter \( \Lambda \) of the momentum-dependent term in the potential in 1/fm.