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Employing a realistic tight-binding model describing the Fermi surface in thenormal state of $Sr_2RuO_4$ we map out magnetic field versus temperature phasediagrams 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) abicritical point at $T=0.88$K separating low field second order phasetransitions from high field Pauli limiting first order transitions. We find,however, strikingly different phase behaviour within the high field Paulilimiting region. For $d_{x^2-y^2}$ pairing symmetry an additional lowercritical line of first order transitions is found (terminating in a criticalpoint at $T=0.09-0.22$K depending on the choice of Hubbard U parameters) whilefor $d_{xz}+id_{yz}$ no such additional high field phase transitions are foundfor any choice of Hubbard U. In conjunction with our earlier finding [{\itPhysical Review B} {\bf 102} (23), 235203] for $p$-wave helical pairing of astill different high field phase structure (a lower critical field line meetingthe upper critical field line exactly at the bicritical point), we suggest highfield Pauli limiting phase structure as a possible route to distinguish pairingsymmetries in this material.

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