Relics of Supersymmetry in Ordinary 1-flavor QCD: Hairpin Diagrams and Scalar-Pseudoscalar Degeneracy

The large-$N_c$ orientifold planar equivalence between $\mathcal{N}=1$ SUSY Yang-Mills theory and ordinary 1-flavor QCD suggests that low-energy quark-gluon dynamics in QCD should be constrained by the supersymmetry of the parent theory. One SUSY relic expected from orientifold equivalence is the approximate degeneracy of the scalar and pseudoscalar mesons in 1-flavor QCD. Here we study the role of the $q\bar{q}$ annihilation (hairpin) contributions to the meson correlators. These annihilation terms induce mass shifts of opposite sign in the scalar and pseudoscalar channels, making degeneracy plausible. Calculations of valence and hairpin correlators in quenched lattice QCD are consistent with approximate degeneracy, although the errors on the scalar hairpin are large. We also study the role of $q\bar{q}$ annihilation in the 1- and 2-flavor Nambu-Jona Lasinio model, where annihilation terms arise from the chiral field determinant representing the axial U(1) anomaly. Scalar-pseudoscalar degeneracy for the 1-flavor case reduces to a constraint on the relative size of the anomalous and non-anomalous 4-fermion couplings.