  We investigate the ground state properties of the spin-$1/2$ pyrochlore
Heisenberg antiferromagnet using pseudofermion functional renormalization group
techniques. The first part of our analysis is based on an enhanced parton
mean-field approach which takes into account fluctuation effects from
renormalized vertex functions. Our implementation of this technique extends
earlier approaches and resolves technical difficulties associated with a
diagrammatic overcounting. Using various parton ans\"atze for quantum spin
liquids, dimerized and nematic states our results indicate a tendency for
lattice symmetry breaking in the ground state. While overall quantum spin
liquids seem unfavorable in this system, the recently proposed monopole state
still shows the strongest support among all spin liquid ans\"atze that we have
tested, which is further confirmed by our complementary variational Monte Carlo
calculations. In the second part of our investigation, we probe lattice
symmetry breaking more directly by applying the pseudofermion functional
renormalization group to perturbed systems. Our results from this technique
confirm that the system's ground state either exhibits broken $C_3$ rotation
symmetry, or a combination of inversion and $C_3$ symmetry breaking.
