The top quark vacuum polarization from two electromagnetic currents is an important input to the top quark pair production cross section at lepton colliders.
High precision of it is necessary for comparisons with experimental data at future lepton colliders.
At the top quark pair production threshold, the Coulomb singularity makes a resummation using the Schrödinger equation necessary.
At first order, it can be solved analytically. At second and third order it can only be solved numerically.
In the first part of the talk, I will present an updated version of TOPPIK [1], a Fortran code solving the Schrödinger equation numerically.
In our updated version Toppik++ we improved speed, precision, and made it numerically more stable.
Above and below the top quark production threshold, the vacuum polarization is known from QCD loop calculations at one, two, and three loops.
At four loops, only the expansions in the low energy, high energy, and threshold limit are known. From these expansions an approximation using a Padé function can be constructed.
In the second part of the talk, I present our results for an improved approximation function at four loops, where we include additional terms from the threshold expansion, which have not been considered before.
[1] A. H. Hoang and T. Teubner, "Top quark pair production close to threshold: Top mass, width and momentum distribution," Phys. Rev. D, vol. 60, p. 114027, 1999.
