zero structure Fig.2(a).

We note that a back-bending dispersion was already observed in an early QMC study,[@ph97] where antiferromagnetic fluctuations were proposed as the origin of the pseudogap. Although our numerical data do not exclude the antiferromagnetic fluctuations from the possible mechanisms, the less-$\Vec{k}$-dependent pseudogap as well as the asymmetric location of the hole pocket (see Sec. \[ssec:arc\]) seems to oppose the mechanism; instead it rather supports that the pseudogap is a direct consequence of the proximity to the Mott insulator.

Energy-distribution curve {#ssec:edc}
-------------------------

Zeros of $G$ are not directly seen in spectra. Their footprints may, however, be detected through a sudden suppression of spectra due to a large Im$\S$ around the zeros. Figure \[fig:edc\](a) shows energy-distribution curves (EDC) of the spectral function along momentum cuts $(0,0)\text{-}(\pi,0)\text{-}(\pi,\pi)$, calculated by the CDMFT+ED with $\eta=0.1t$ for $t'=-0.2t$, $U=12t$, and $n=0.93$. The results exhibit a coherent peak around $(\pi,0)$ just below the Fermi level, its shift to lower energy from $(\pi,0)$ to $(\frac{\pi}{2},0)$, and the incoherent feature around $(0,0)$ and $(\pi,\pi)$. All these features are consistent with ARPES data \[Fig. 5 in Ref. , reproduced in Fig. \[fig:edc\](b)\]. This agreement supports our zero mechanism and shows that the incoherent feature can be interpreted as the effect of zeros of $G$: Since the zero surface exists just above the band around $(\pi,\pi)$ \[Fig. \[fig:fig2\](a)\] and since another one is located just below the band around $(0,0)$ \[not shown in Fig. \[fig:fig2\](a), but can refer to Fig. \[fig:rkw\_tp\](b) below\], the spectrum associated with the band is smeared around these momenta due to the large Im$\S$ around the zeros. While the suppression around $(\pi,\pi)$ is relevant to the emergence of Fermi arc,[@sk06; @sm09; @sm09-2] that around $(0,0)$ is related to the waterfall behavior discussed in the next subsection.

Waterfal