 the top of the band is located at a higher energy than the maximum energy of the measurement. Note that $t'<0$ lifts (lowers) the band around the nodes (antinodes). Enhancing $-t'$ shifts the Fermi surface to $(0,0)$ as well, as we will discuss in Sec. \[ssec:tp\]. Then a better agreement with the ARPES data may be reached.

In Ref.  the back-bending behavior around the antinodes was interpreted as an evidence of preformed Cooper pairs in the pseudogap state because it resembles a band dispersion in the BCS superconductors.[@bc57] However, our result implies another simple interpretation: The band is pushed down by the neighboring zero surface, which cuts the Fermi level around $(\pi,\pi)$, due to the large self-energy around it. This picture is confirmed in Fig. \[fig:fig2\](d), where we plot the energy dependence of the self-energy at the Fermi momentum $\Vec{k}_{\text{F2}}$ closest to the zero surface. We see that the real part of the self-energy is negatively large below the zero at $\w\simeq0.06t$, which should push down the band around the momentum. The result indicates that the pair formation is not necessary and the zeros of $G$ resulting directly from strong correlation effects give an alternative interpretation of the back-bending dispersions.

Recently, an evidence against the scenario that the back bending is a consequence of preformed pair was further reported.[@hh10] In this ARPES observation, the back-bending momentum, i.e., the location of the pole at the lowest binding energy along momentum cuts, is clearly deviated from the Fermi momentum at above the pseudogap opening temperature. In contrast, we note that the two momenta should agree in the BCS theory and in the preformed pair scenario as well. Furthermore, the ARPES reported that the back-bending momentum along $(\pi,0)$-$(\pi,\pi)$ shifts closer to $(\pi,\pi)$ than the Fermi momentum. This indicates a zero surface around $(\pi,\pi)$ at the Fermi level and a center of the gap residing above $E_\text{F}$, in full consistency with our pole-