  The past years have seen a revived interest in the diagrammatic Monte Carlo
(DiagMC) methods for interacting fermions on a lattice. A promising recent
development allows one to now circumvent the analytical continuation of dynamic
observables in DiagMC calculations within the Matsubara formalism. This is made
possible by symbolic algebra algorithms, which can be used to analytically
solve the internal Matsubara frequency summations of Feynman diagrams. In this
paper, we take a different approach and show that it yields improved results.
We present a closed-form analytical solution of imaginary-time integrals that
appear in the time-domain formulation of Feynman diagrams. We implement and
test a DiagMC algorithm based on this analytical solution and show that it has
numerous significant advantages. Most importantly, the algorithm is general
enough for any kind of single-time correlation function series, involving any
single-particle vertex insertions. Therefore, it readily allows for the use of
action-shifted schemes, aimed at improving the convergence properties of the
series. By performing a frequency-resolved action-shift tuning, we are able to
further improve the method and converge the self-energy in a non-trivial
regime, with only 3-4 perturbation orders. Finally, we identify time integrals
of the same general form in many commonly used Monte Carlo algorithms and
therefore expect a broader usage of our analytical solution.
