The Roper state is extracted with valence overlap fermions on a $2+1$-flavor domain-wall fermion lattice (spacing $a = 0.114$ fm and $m_{pi} = 330$ MeV) using both the Sequential Empirical Bayes (SEB) method and the variational method. The results are consistent, provided that a large smearing-size interpolation operator is included in the variational calculation to have better overlap with the lowest radial excitation. Similar calculations carried out for an anisotropic clover lattice with similar parameters find the Roper $approx 280$ MeV higher than that of the overlap fermion. The fact that the prediction of the Roper state by overlap fermions is consistently lower than those of clover fermions, chirally improved fermions, and twisted-mass fermions over a wide range of pion masses has been dubbed a Roper puzzle. To understand the origin of this difference, we study the hairpin $Z$-diagram in the isovector scalar meson ($a_0$) correlator in the quenched approximation. Comparing the $a_0$ correlators for clover and overlap fermions, at a pion mass of 290 MeV, we find that the spectral weight of the ghost state with clover fermions is smaller than that of the overlap at $a = 0.12$ fm and $0.09$ fm, whereas the whole $a_0$ correlators of clover and overlap at $a = 0.06$ fm coincide within errors. This suggests that chiral symmetry is restored for clover at $a le 0.06$ fm and that the Roper should come down at and below this $a$. We conclude that this work supports a resolution of the Roper puzzle due to $Z$-graph type chiral dynamics. This entails coupling to higher components in the Fock space (e.g. $Npi$, $Npipi$ states) to induce the effective flavor-spin interaction between quarks as prescribed in the chiral quark model, resulting in the parity-reversal pattern as observed in the experimental excited states of $N, Delta$ and $Lambda$.
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