  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. $N\pi$, $N\pi\pi$ 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 $\Lambda$.
