Bound on the exponential growth rate of out-of-time-ordered correlators

It has been conjectured by Maldacena, Shenker, and Stanford [J. High Energy Phys.~08 (2016) 106] that the exponential growth rate of the out-of-time-ordered correlator (OTOC) $F(t)$ has a universal upper bound $2\pi k_B T/\hbar$. Here we introduce a one-parameter family of out-of-time-ordered correlators $F_{\gamma }(t)$ ($0\le \gamma \le 1$), which has as good properties as $F(t)$ as a regularization of the out-of-time-ordered part of the squared commutator $⟨[A(t),B(0){]}^2⟩$ that diagnoses quantum many-body chaos, and coincides with $F(t)$ at $\gamma =1\mathrm{/}2$. We rigorously prove that if $F_{\gamma }(t)$ shows a transient exponential growth for all $\gamma$ in $0\le \gamma \le 1$, that is, if the OTOC shows an exponential growth regardless of the choice of the regularization, then the growth rate $\lambda$ does not depend on the regularization parameter $\gamma$, and satisfies the inequality $\lambda \le 2\pi k_{B}T\mathrm{/}\mathrm{\hslash }$.