Gap Symmetry in KFe$_2$As$_2$

We revisit the issue of the gap symmetry in K$Fe_2$As$_2$, which is an Fe-pnictide superconductor with only hole pockets. Previous theoretical studies mostly argued for a $d-$wave gap in K$Fe_2$As$_2$ since transport and thermodynamic measurements point to the presence of the gap nodes. However, a $d-$wave gap is inconsistent with recent laser-based angle-resolved photoemission measurements. We propose the scenario for a nodal $s-$wave superconductivity induced by a non-magnetic intra-band and inter-band interactions between fermions near the two hole pockets at $\Gamma$ point. The superconducting gap that we find changes sign between the two hole pockets at $\Gamma$ point and has $\cos {4\theta}$ angular dependence and can have accidental nodes on one or several hole pockets. We argue that strong angle dependence is the consequence of near-degeneracy between inter-pocket and intra-pocket interaction on the hole pockets. We also provide a connection between the the relative phase of 4$\theta$ oscillations and the shapes of the Fermi surface and discuss the implications in the light of photoemission and tunneling experiments.