Superconducting double transition and substantial Knight shift in $Sr_2RuO_4$

Recent nuclear magnetic resonance experiments measuring the Knight shift in $Sr_2RuO_4$ have challenged the widely accepted picture of chiral pairing in this superconductor. Here we study the implications of helical pairing on the superconducting state while comparing our results with the available experimental data on the upper critical field and Knight shift. We solve the Bogoliubov-de-Gennes equation employing a realistic three-dimensional tight-binding model that captures the experimental Fermi surface very well. In agreement with experiments we find a Pauli limiting to the upper critical field and, for a finite in-plane magnetic field, a double superconducting transition below a temperature $T^*$. These transitions are first-order in nature and merge into a single second-order transition at a bicritical point $(T^\ast,H^\ast$), for which we find (0.8~K, 2.35~T) with experiment reporting (0.8~K, 1.2~T) [\textit{J. Phys. Soc. Jpn.} \textbf{71},2839 (2002)]. Furthermore, we find a substantial drop in the Knight shift in quantitative agreement with recent experiments.