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Recent discovery of superconductivity in YbRh$_2$Si$_2$ has raised particularinterest in its pairing mechanism and gap symmetry. Here we propose aphenomenological theory of its superconductivity and investigate possible gapstructures by solving the multiband Eliashberg equations combining realisticFermi surfaces from first-principles calculations and a quantum critical formof magnetic pairing interactions. The resulting gap symmetry shows sensitivedependence on the in-plane propagation wave vector of the quantum criticalfluctuations, suggesting that superconductivity in YbRh$_2$Si$_2$ is located onthe border of $(p_x+ip_y)$ and $d_{x^2-y^2}$-wave solutions. This leads to twocandidate phase diagrams: one has only a spin-triplet $(p_x+ip_y)$-wavesuperconducting phase; the other contains multiple phases with a spin-singlet$d_{x^2-y^2}$-wave state at zero field and a field-induced spin-triplet$(p_x+ip_y)$-wave state. In addition, the electron pairing is found to bedominated by the `jungle-gym' Fermi surface rather than the `doughnut'-likeone, in contrast to previous thought. This requests a more elaborate andrenewed understanding of the electronic properties of YbRh$_2$Si$_2$.

Nearly-degenerate $p_x+ip_y$ and $d_{x^2-y^2}$ pairing symmetry in the heavy fermion superconductor YbRh$_2$Si$_2$