  We study the nonlinear energy diffusion through the SYK chain in the
framework of Schwinger-Keldysh effective field theory. We analytically
construct the interacting effective Lagrangian up to $40^{th}$ order in the
derivative expansion. According to this effective Lagrangian, we calculate the
first order loop correction of the energy density response function, the pole
of which is the dispersion relation of energy diffusion. As expected, we see
that the standard derivative expansion of that dispersion relation, $\omega=-i
D_{(1)} k^2- i D_{(2)} k^4+\mathcal{O}(k^6)$, breaks down due to the long-time
tails. However, we find that the nonlinear contribution of order $n$ to the
self-energy is proportional to $\left(k^{2}\right)^{n+1/2}$. This suggests to
modify the dispersion relation by splitting it into two dispersion relations
and double the number of transport coefficients at any order as $\omega=-i
k^2\big( D_{(1,1)}\pm i D_{(1,2)} \left(k^2\right)^{1/2}\big)-i k^4\big(
D_{(2,1)}\pm i D_{(2,2)} \left(k^2\right)^{1/2}\big)+\mathcal{O}(k^6)$. We find
that the modified series, which include the effect of long-time tails, are
convergent. The radius of convergence is proportional to the ratio of thermal
conductivity to diffusion constant.
