Distinguishing $d_{xz}+i d_{yz}$ and $d_{x^2-y^2}$ pairing in $Sr_2RuO_4$ by high magnetic field H-T phase diagrams.

Employing a realistic tight-binding model describing the Fermi surface in the normal state of $Sr_2RuO_4$ we map out magnetic field versus temperature phase diagrams for $d_{x^2-y^2} (B_{1g})$ and $d_{xz}+id_{yz} (E_g)$ pairing types. Both produce (i) a similar Knight shift suppression of $\sim\!80\%$ and (ii) a bicritical point at $T=0.88$K separating low field second order phase transitions from high field Pauli limiting first order transitions. We find, however, strikingly different phase behaviour within the high field Pauli limiting region. For $d_{x^2-y^2}$ pairing symmetry an additional lower critical line of first order transitions is found (terminating in a critical point at $T=0.09-0.22$K depending on the choice of Hubbard U parameters) while for $d_{xz}+id_{yz}$ no such additional high field phase transitions are found for any choice of Hubbard U. In conjunction with our earlier finding [{\it Physical Review B} {\bf 102} (23), 235203] for $p$-wave helical pairing of a still different high field phase structure (a lower critical field line meeting the upper critical field line exactly at the bicritical point), we suggest high field Pauli limiting phase structure as a possible route to distinguish pairing symmetries in this material.