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Strontium ruthenate Sr$_2$RuO$_4$ is an unconventional superconductor whosepairing symmetry has not been fully clarified, despite more than two decades ofintensive research. Recent NMR Knight shift experiments have rekindled theSr$_2$RuO$_4$ pairing debate by giving strong evidence against all odd-paritypairing states, including chiral $p$-wave pairing that was for a long time theleading pairing candidate. Here, we exclude additional pairing states byanalyzing recent elastocaloric measurements [YS. Li et al., Nature 607,276--280 (2022)]. To be able to explain the elastocaloric experiment, we findthat unconventional even-parity pairings must include either large $d_{x^2 -y^2}$-wave or large $\{d_{xz} \mid d_{yz}\}$-wave admixtures, where the latterpossibility arises because of the body-centered point group symmetry. These$\{d_{xz} \mid d_{yz}\}$-wave admixtures take the form of distinctivelybody-centered-periodic harmonics that have horizontal line nodes. Hence$g_{xy(x^2-y^2)}$-wave and $d_{xy}$-wave pairings are excluded as possibledominant even pairing states.

Constraints on the superconducting state of Sr$_2$RuO$_4$ from elastocaloric measurements