We study out-of-time ordered four-point functions in two dimensional conformal field theories by suitably analytically continuing the Euclidean correlator. For large central charge theories with a sparse spectrum, chaotic dynamics is revealed in an exponential decay; this is seen directly in the contribution of the vacuum block to the correlation function. However, contributions from individual non-vacuum blocks with large spin and small twist dominate over the vacuum block. We argue, based on holographic intuition, that suitable summations over such intermediate states in the block decomposition of the correlator should be sub-dominant, and attempt to use this criterion to constrain the OPE data with partial success. Along the way we also discuss the relation between the spinning Virasoro blocks and the on-shell worldline action of spinning particles in an asymptotically AdS spacetime.

Article funded by SCOAP3

2$ violates the chaos bound. At first sight, this seems reasonable, for the truncation to the Virasoro vacuum block contribution is tantamount to focusing on graviton exchange in the bulk and ignoring others. A moment's reflection however reveals a problem: the cross ratio of the OTO four-point function at the scrambling time $t_*$ is exponentially close to the boundary of the radius of convergence of the operator product expansion (OPE).\footnote{This is most apparent in the pillow coordinate $q$ reviewed in section~\ref{sec:pillow}.} One expects the contributions from the non-vacuum blocks to the four-point function would become important. Furthermore, it does not follow that in the physical result only the graviton exchange dominates (the latter would require that in the $VW$ OPE can be truncated to sole vacuum contribution, which will not hold universally). Subsequent investigations have focused on subleading corrections~\cite{Fitzpatrick-ml-2016thx} while more recent work~\cite{Liu-ml-2018iki,Hampapura-ml-2018otw} explores the limitations of the vacuum block truncation ansatz for the chaos correlator.\footnote{The exponential structure~\eqref{eqn:exp_decay} of the OTO correlators can already be seen in a further truncated sector that includes only the identity operator and the stress tensor. The contribution from the stress tensor is responsible for the exponential factor in the second term of~\eqref{eqn:exp_decay}. This inspires recent attempts at constructing an effective description for chaotic dynamics~\cite{Blake-ml-2017ris}, which was specifically applied to 2d CFTs~in~\cite{Haehl-ml-2018izb}.} Our interest in the current work will be to make a careful analysis of the non-vacuum contributions in the Euclidean correlator, and implications thereof in the out-of-time-order observable of interest. We start by noting that it was already anticipated by~\cite{Roberts-ml-2014ifa} that non-vacuum blocks could in-principle have a significant contribution around the scrambling time upon analytic continuation. We will first verify this explicitly by taking into account the contributions from non-vacuum primaries. This has also been confirmed independently in recent works~\cite{Liu-ml-2018iki,Hampapura-ml-2018otw}. A technical aid we will employ is to use Zamolodchikov's recursion relation~\cite{Zamolodchikov-ml-1985ie,Zamolodchikov1987} to obtain the non-vacuum blocks numerically to a good accuracy (using results of~\cite{Fitzpatrick-ml-2016thx}) to enable make firm predictions. Specifically, we will see that the deviation from the vacuum block result depends on the spin of the light non-vacuum primaries in the spectrum. Knowing the individual blocks however is insufficient for our purposes. Had there been a finite number of non-vacuum primaries, we would have trouble in estimating the chaos correlator in Lorentz signature, especially if each individual block had large contribution to the correlator. What hopefully saves the day is the fact that the CFTs of interest have an infinite number of primaries. While each is locally more important, summing over all of them with the correct OPE coefficients should result in a sub-dominant contribution. This is for example analogous to the net phase shift induced in forward scattering amplitudes in flat space, where individual higher-spin states give more and more dominant contribution, while the resummed answer is bounded by unitarity (see eg.,~\cite{Camanho-ml-2014apa}). Assuming therefore that we have an infinite number of primaries contributing, ideally we can bound their density and their OPE coefficients by demanding that the truncation of~\cite{Roberts-ml-2014ifa} indeed gives the physically acceptable answer. While the density of primaries will have to obey the sparseness condition, we would learn of new constraints on the OPE coefficients (which should be stronger than simple factorization statements). We set-up the problem of putting bounds on OPE data, and will show that the contributions from very heavy primaries and large spin primaries in the intermediate states are innocuous. This is done by explicit evaluation of the conformal blocks to get a good estimate, and uses some recently derived bounds on the growth of OPE coefficients~\cite{Chang-ml-2015qfa,Chang-ml-2016ftb}. We are also able to estimate the contribution of the light intermediate operators, and find the need for a conspiracy between them and the moderately heavy intermediate states for the vacuum block to provide the correct answer post analytic continuation. The somewhat sticky point of our analysis is an inability to find useful bounds for intermediate primaries which are moderately heavy $h \sim c$. Here the problem we encounter is a technical obstacle --- we have not managed to obtain a useful estimate of the conformal blocks themselves. We do try to exploit semi-analytic bounds on the conformal block data (see~\cite{Kusuki-ml-2017upd, Kusuki-ml-2018wcv, Kusuki-ml-2018nms, Kusuki-ml-2018wpa}) to provide an estimate, but find that the results known thus far are too limiting to get a handle on the behaviour of the blocks on timescales of order the scrambling time (they do give a handle on the very late time behaviour of the correlator). The outline of the paper is as follows: in section~\ref{sec:review} we will review the basic features of the chaos correlator, and revisit the analysis of~\cite{Roberts-ml-2014ifa} to set the stage for the discussion. Section~\ref{sec:nonvac} is devoted to demonstrating the limitations both analytically and numerically of the truncation to the vacuum block contribution to the Euclidean OPE prior to analytic continuation. In section~\ref{sec:estimates} we attempt to put bounds on the OPE data, though as advertised, our attempts will only be partially successful. Following this in section~\ref{sec:adsgrav} we take the opportunity to revisit the geodesic computation of the conformal blocks and show how spinning particles in the bulk AdS$_3$ can be used to reproduce the results derived in section~\ref{sec:nonvac} and conclude with a brief discussion in section~\ref{sec:discuss}. The appendices~\ref{sec:BTZshockwave} and~\ref{sec:geodesics} are devoted to providing details of the spinning particle analysis, while appendix~\ref{sec:semiwline} computes the Euclidean block for completeness. \newpage ]]>

x$, the operators $W$'s and $V$'s are time-like separated, and the ordering of the operator in the four-point function is determined by the ordering of the $\epsilon_i$.
In terms of the cross ratios $z$ and $\bar z$, the light-cone singularities at $z_1=z_2$, $z_4$, $z_3$ (or $\bar z_1=\bar z_2$, $\bar z_4$, $\bar z_3$) correspond to $z=0$, $1$, $\infty$ (or $\bar z=0$, $1$, $\infty$). The function $G(z,\bar z)$ of independent complex variables $z$ and $\bar z$ has branch cuts extending from the $z=0$, $1$, and $\infty$, which can be chosen to lie on $(-\infty,0]$ and $[1,\infty)$. We are interested in the process when the $W$ operators approach the light-cone of the $V$ operators, which occurs at $t=\pm x$ and correspondingly at $z=1$ or $\bar z=1$. The cross ratio $z$ expanded at the light-cone $t=x$~as
\ie
z=\;&{\sin \pi(\epsilon_2-\epsilon_1)\sin \pi(\epsilon_4-\epsilon_3)\over \sin \pi(\epsilon_3-\epsilon_1)\sin \pi(\epsilon_4-\epsilon_2)}
\\
& - i\pi (t-x){\sin\pi(\epsilon_3+\epsilon_4-\epsilon_1-\epsilon_2)\sin\pi(\epsilon_2-\epsilon_1)\sin\pi(\epsilon_4-\epsilon_3)\over \sin^2\pi(\epsilon_4-\epsilon_2)} + {\cal O}(t-x)^2.
\fe
As the time $t$ increases from $-x

\Lambda$. This condition also implies the convergence of the Virasoro block expansion along the analytic continuation from the Euclidean four-point function~\eqref{eqn:EuclideanWWVV} to the OTO four-point function~\eqref{OtO4-pt}.
The function $H(q)$ has a nice interpretation as the Virasoro block of the four-point function on the pillow geometry $T^2/\bZ_2$~\cite{Maldacena-ml-2015iua}. The $T^2$ can be viewed as an elliptic curve inside $\widehat\bC^2$ with the coordinates $(x,y)$ is defined by
\ie
y^2=x\, (z-x) \, (1-x),
\fe
which can be viewed as a double-cover of the Riemann sphere branched at 0, $z$, 1 and $\infty$, where $x$ is the coordinate of the base Riemann sphere $\widehat\bC$. The $\bZ_2$ action is generated by $y\to -y$. The map from the Riemann sphere $\widehat\bC$ to $T^2/\bZ_2$ is explicitly given by
\ie
x\mapsto u={1\over \theta_3(q)^2}\int^x_0{dw\over \sqrt{w(1-w)(z-w)}},
\fe
where $u$ is the coordinate of $T^2/\bZ_2$, which takes complex values with identification $u\sim u+2\pi\sim u+2\pi \tau$ and the $\mathbb{Z}_2$ identification is $u\sim -u$. The positions of the operators $(0,z,1,\infty)$ are mapped to $(0,\pi,\pi+\pi\tau,\pi \tau)$. Taking the conformal anomaly factor into account, the four-point function on $T^2/\bZ_2$ is given by
\ie
&\left

1$ and an SL(2) invariant normalizable vacuum, there are infinitely many large spin primaries whose twists accumulate to ${c-1\over 12}$. By this result, for any $t$ there are always infinitely many operators with large enough ${\ell\over \tau}$ that violates the inequality~\eqref{eqn:vacuumDominateApprox}. Hence, the contribution of them to the four-point function are large compared to the contribution from the vacuum Virasoro block. However, one should notice that in deriving the inequality~\eqref{eqn:vacuumDominate}, we have used the formulas~\eqref{eqn:heavyLightBlock} for the Virasoro blocks, which is only valid when $c\gg h\ge 0$ in the large $c$ limit. In section~\ref{sec:num}, we compute the Virasoro blocks in the $q$ expansion numerically in high powers, and demonstrate that the contributions from large spin and low twist blocks individually are larger than the contribution from the vacuum block. ]]>

2$. On the other hand, for a theory with higher spin conserved currents of unbounded spin, the vacuum block of the higher spin algebra does not decay in late time~\cite{Perlmutter-ml-2016pkf}. In other words, the Lyapunov exponent is zero.\footnote{The vanishing of the Lyapunov exponent holds for theories with higher spin symmetries which nevertheless have a sparse spectrum as noted in~\cite{Perlmutter-ml-2016pkf} and reconfirmed in~\cite{Belin-ml-2017jli} for permutation orbifolds.} The vacuum block of the higher spin algebra can be decomposed as a sum over Virasoro blocks weighted by OPE coefficients. In the above two cases, we see that different sums over the Virasoro blocks can lead to very different Lyapunov exponents. ]]>

(z,\bar z) ={}& \sum_{\substack{h>h_0+\ca, \\\bar h> \bar h_0+\ca}} \, C_{VV}(h,\bar h)^2 \; {\cal F}_h(z) {\cal F}_{\bar h}(\bar z) ={} \int_{(h_0+\ca,\bar h_0+\ca)}^\infty dh\, d \bar h \; K(h,\bar h) \; {\cal F}_h(z) {\cal F}_{\bar h} (\bar z)\, . \end{align} We have simplified the sum by representing it as an integral over intermediate states, using the density of OPE coefficients \begin{align} \label{eq:Kdef} K(h,\bar h) = \sum_{\mathcal{O}_{h',\bar h'}} C_{VV}(h', \bar h')^2 \delta(h-h') \delta(\bar h - \bar h')\, . \end{align} To resum~\eqref{eq:Ggtr}, we need approximations for the density of OPE coefficients $K(h, \bar h)$ and the Virasoro blocks ${\cal F}_h(z)$, for $h, \bar h \gg c$. We use the following universal features: \begin{enumerate} \item For the conformal blocks we use the expression~\eqref{eqn:pillowF} in terms of the pillow coordinate $q(z)$. It is worth noting that that $H_h(q)\to 1$ as $h\to \infty$,~\eqref{eqn:Hinf}, so the full block in this limit is given by the universal prefactor \begin{align} {\cal F}(z) = (16q)^{h-\ca} \; z^{\ca-2h_v} \, (1-z)^{\ca-h_v-h_w} \, \theta_3(q)^{12\,\ca-8(h_v+h_w)} \, . \label{eq:Finf} \end{align} We emphasize that the only $h$ dependence in the blocks in this regime is the exponential suppression $q^h$. \item The density of OPE coefficients $K(h,\bar h)$ for asymptotically heavy intermediate states can be determined by Cardy-like arguments, as shown in~\cite{Das-ml-2017cnv}. In particular, using the $q$ coordinates, the crossing equation translates into a statement about the modular behavior of the Virasoro block decomposition under $\tau \to {-} \frac{1}{\tau}$. Defining effective temperatures by $\tau(z) = \frac{i \beta(z)}{2\pi}$ and $\bar \tau(\bar z) = {-} \frac{i \bar \beta(\bar z)}{2\pi}$, vacuum dominance in the $\beta, \bar \beta \to \infty$ limit, i.e., $z, \bar z \to 0$, and the asymptotic behavior of the blocks~\eqref{eq:Finf} leads to the following universal behavior of the OPE coefficients for heavy intermediate states $h, \bar h \gg c$: \begin{align} \label{eq:Kinf} K(h,\bar h) \equiv{}& \kappa(h) \kappa(\bar h)\, , \end{align} where \begin{align} \label{eq:kappadef} \kappa(h) ={}& 16^{{-}h} e^{2\pi \sqrt{\ca(h-\ca)}} \end{align} Here we have worked to order $\sqrt{h}$ in the exponent for simplicity; subleading corrections are readily obtained. We note that this expression is for the density of OPE coefficients, so that the average coefficients for large $h, \bar h$ can be found using the Cardy density of states: \begin{align} \overline{C^2(h, \bar h)} = \frac{K(h,\bar h)}{\rho(h, \bar h)} \sim 16^{-h-\bar h} e^{{-}2\pi \sqrt{\ca} (\sqrt{h-\ca} + \sqrt{\bar h - \ca})}\, . \end{align} Thus we are essentially including subleading corrections to the bound $C^2 \sim 16^{-h-\bar h}$ at large $h, \bar h$. \end{enumerate} Combining these two ingredients, we see that the resummation~\eqref{eq:Ggtr} holomorphically factorizes and the resummation is governed by the following integral (we recall that $|q| < 1$): \begin{align} \label{eq:Idef} I(h_0;q)\equiv{}& \int_{h_0+\ca}^\infty d h \, (16 q)^{h-\ca} \kappa(h) \sim \int_{h_0}^\infty d \tilde h \, e^{{-} \gamma \tilde h + 2\pi \sqrt{\ca \tilde h}}\, , \end{align} where we have only kept terms to the order we are working, shifted the integration variable to $\tilde h= h- \ca$, and defined $\gamma = {-} \log q$. This integral can be evaluated to yield \begin{align} I(h_0;q) \sim{}& \sqrt{\frac{\ca}{\pi \gamma}} \frac{\partial}{\partial \ca} \left[e^{\pi^2 \ca/\gamma} \erfc \left( \sqrt{\gamma h_0} - \pi \sqrt{\tfrac{\ca}{\gamma}} \right) \right] \nonumber \\ \sim{}& \frac{q^{h_0} e^{2\pi \sqrt{\ca h_0}}}{\gamma} \, . \end{align} Here $\erfc(z) = 1 - \erf(z)$ is the complementary error function, and in the second line we have kept only the leading term for large $h_0$, using the asymptotic approximation \begin{align} \erfc(z) \sim \frac{e^{{-} z^2}}{\pi z} \left(1 + {\cal O}(z^{{-}1}) \right)\, , \end{align} for $|z| \to \infty$. With this result, we can approximate~\eqref{eq:Ggtr} by \begin{align} \label{eq:Ggtrapprox} G_> \sim{}& \frac{q^{h_0} \bar q^{\bar h_0} }{\gamma \bar \gamma}\bar [z \bar z (1-z)(1-\bar z)]^{\ca - 2 h_v} [\theta_3(q) \theta_3(\bar q)]^{12 \ca - 16 h_v} \, . \end{align} We can now continue this contribution to the chaos regime using~\eqref{eq:qchaoslimit} and $\bar q \to \frac{\bar z}{16}$ as $\bar z \to 0$. The behavior of $\theta_3(q)$ as $q \to i$ can be determined by the modular properties of $\theta_3$, which tells us \begin{align} \theta_3(\tau) = [{-}i({-}2 \tau+1)]^{{-}\frac{1}{2}} \theta_3\left(\frac{\tau}{{-}2\tau+1} \right)\, . \end{align} The limit~\eqref{eq:qchaoslimit} corresponds to $\tau \sim \frac{1}{2} - \frac{i \pi}{4\log (z/16)}$ (with $z\to 0$) and therefore \begin{align} \theta_3(q\sim i) \to \left[\frac{\pi}{2\log(z/16)} \right]^{{-}\frac{1}{2}} \, , \end{align} where we have used $\theta_3(\tau \to i \infty) = 1$. Taking the limits in~\eqref{eq:Ggtrapprox}, we end up with \begin{align} G_> \sim{}& {-}\frac{2i e^{{-}2\pi i (\ca - \Delta_v)}}{\pi \log(\bar z/16)} z^{\ca - \Delta_v} \bar z^{\bar h_0 - \Delta_v} \left[ \frac{\pi}{2 \log(z/16)} \right]^{4 \Delta_v - 6 \ca} \nonumber \\ \equiv{}& P(x,t) e^{{-}\frac{2\pi}{\beta} [(\bar h_0 + \ca-2 \Delta_v )t + (\bar h_0 - \ca) x ]} \label{eq:Gheavy} \end{align} where in the last line we have grouped all of the prefactors, which are sub-exponential in $t$, into $P(x,t)$, which is explicitly given by \begin{align} P(t) ={}& {-} \frac{2i e^{{-}2\pi i (\ca - \Delta_v)}}{\pi \log(\bar z/16)} \left[ \frac{\pi}{2 \log(z/16)} \right]^{4 \Delta_v - 6 \ca} \nonumber \\ ={}& \frac{2i e^{{-}2\pi i (\ca - \Delta_v)}}{\pi \Big[ \frac{2 \pi}{\beta} (x+t) -4 \log 2 \Big]}\bg[ \frac{\pi}{\frac{4 \pi}{\beta} (x-t) - 8\log 2} \bg]^{4 \Delta_v - 6 \ca} \end{align} One can check that $P(t \to \infty)$ simplifies to a constant. The key takeaway of~\eqref{eq:Gheavy} is the manifest exponential decay in $t$, demonstrating that the contribution of the very heavy operators is suppressed in the chaos regime. ]]>

\ca$ and similarly for the antichiral part. The full operator spectrum is given by a product of the holomorphic and antiholomorphic pieces, leading to four regions in the $(h, \bar h)$ plane; this spectral data, extracted by the T-channel vacuum block, was termed Virasoro mean field theory (VMFT) in~\cite{Collier-ml-2018exn}. For unitary compact CFTs with $c>1$ and a positive lower bound on the twists of non-vacuum primaries, the spectrum and OPE coefficients of primary operators with $h\gg c$ or $\bar h\gg c$ universally approach those in the VMFT~\cite{Collier-ml-2018exn}. We have in fact already encountered a portion of this spectrum, namely in section~\ref{sec:largehhb}, where we resummed the asymptotically heavy operators, $h, \bar h \gg c$. One can check that the spectral density we used matches that extracted via the fusion kernel in~\cite{Kusuki-ml-2018wpa,Collier-ml-2018exn}. Furthermore, resumming the discrete-discrete part of the VMFT spectrum (in the large $c$ limit we are interested in) reproduces the exchange of the vacuum operator in the T-channel, i.e.~reproduces $\vert 1-z\vert^{-2\Delta_v}$, which is obviously non-singular in the chaos regime. In fact, in the large $c$ limit this portion of the spectrum reduces, to leading order, to that of the familiar double-twist operators of global MFT spectrum. In the rest of this section, we consider the asymptotically large spin portions of the VMFT spectrum, which arise from combining a discrete holomorphic operator with the continuous antiholomorphic spectrum (and vice versa). We first consider the case with fixed $h$ and varying $\bar h$. That is, we set $h = h_m \equiv 2h_v+ m +\delta h_m$ (here $\delta h_m$ is an exact anomalous dimension due to summation of all multi-traces built out of stress tensors) and integrate over $\bar h$ using the antiholomorphic half of the spectral density we used in section~\ref{sec:largehhb}. More concretely, as shown in~\cite{Collier-ml-2018exn}, the spectral density obtained from the fusion kernel factorizes into holomorphic and antiholomorphic pieces, with the antiholomorphic half given by $\kappa(\bar h)$, defined in~\eqref{eq:kappadef} and the holomorphic half given by a residue of the fusion kernel: \begin{align} C_{VV}^2(h_m, \bar h) \sim {-} 2 \pi \kappa(\bar h) \Res\limits_{\alpha_s=\alpha_m} S_{\alpha_s 1} \, . \end{align} Here $S$ is the fusion kernel and $\alpha_m$ is defined via $h_m = \alpha_m(Q-\alpha_m)$ (where $Q^2 = \frac{c-1}{6}$), which rewrites a T-channel Virasoro block in terms of S-channel Virasoro blocks; we refer the reader to~\cite{Collier-ml-2018exn} for its explicit form (which we won't need in detail). The contribution of the $\bar h \gg c$ portion of the $m$th quantum Regge trajectory is then given by \begin{align} G_{m,>} (z,\bar z) \equiv {\cal F}_{h_m}(z) \int_{\bar h_0 + \ca} d \bar h\, C_{VV}^2(h_m, \bar h) {\cal F}_{\bar h}(\bar z) \sim \left({-} 2\pi \Res_{\alpha_s=\alpha_m} S_{\alpha_s, 1} \right) {\cal F}_{h_m}(z) I(\bar h_0; \bar q)\, , \end{align} where $I(\bar h_0; \bar q)$ is given by~\eqref{eq:Idef}. We want to continue this sum to the chaos regime. The antiholomorphic continuation is straightforward, since $I(\bar h_0, \bar q) \propto \bar q^{\bar h_0}$ and $\bar q \sim \frac{\bar z}{16}$ in the chaos regime. For the holomorphic half, we can use the behavior of the semiclassical blocks in the chaos region described in section~\ref{sec:vacbl}. Combining the two pieces, we find \begin{align} G_{m,>}(z, \bar z) \propto e^{\frac{2 \pi}{\beta}(h_m t_* - \bar h_0 t)}\, , \end{align} up to ratios of polynomials in $t$. Since by construction we have taken $\bar h_0 \gg c$, we see that the tail ends of each quantum Regge trajectory are in fact exponentially suppressed in the chaos region. Here we have assumed that the holomorphic part of the OPE coefficients, i.e., $\Res S_{\alpha_s 1}$, does not have any interesting features in the large $c$ limit. This can be ensured by taking $h_v$ sufficiently small, i.e., scaling sufficiently weakly with $c$. Explicit expressions for these residues can be found in~\cite{Collier-ml-2018exn}. In this case, the leading order residues are simply the (chiral half of) global MFT double-twist OPE coefficients. We leave a more detailed analysis of perhaps more interesting semiclassical limits to future work. A similar story holds for the $\bar h$ fixed, $h > h_0 + \ca$ trajectories as well. Using the factorized form of the OPE coefficients, one finds \begin{align} G_{>,m}(z, \bar z) \sim \left({-} 2\pi \Res_{\bar \alpha_s=\bar \alpha_m} S_{\bar \alpha_s, 1} \right) {\cal F}_{\bar h_m}(\bar z) I(h_0; q)\, . \end{align} Using~\eqref{eq:qchaoslimit}, the contribution of this sector is again suppressed in the chaos limit, as the anti-holomorphic block is exponentially suppressed and one has \begin{align} G_{>,m}(z, \bar z) \propto e^{{-}\frac{2\pi}{\beta}\bar h_m t -\frac{ \pi\beta}{8\,t} \, h_0} \, . \end{align} ]]>

{\hat m}_v\, c$, where ${\hat m}_v\, c \sim \mathcal{O}(h_v)$ (cf., proposition $3$ in~\cite{Chang-ml-2015qfa}). While the precise expressions are slightly involved, in principle we have a handle on the OPE coefficients weighted by the density of states. The primary technical obstacle is determining the semiclassical Virasoro blocks for all exchanged operator dimensions, or at least the behavior in the chaos region $q \to i$. These blocks are known to leading order for $\frac{h}{c} \ll 1$ (and $\frac{h_v}{c} \ll 1$), which we have significantly exploited above. However, as far as we are aware, there are no results available thus far for intermediate-to-heavy exchanged operators for times of order the scrambling time $t_*$. In recent years there has been some very interesting progress in understanding the recursive determination of the pillow block $H_h(q)$, which allows for some progress in this front. In particular,~\cite{Kusuki-ml-2017upd, Kusuki-ml-2018wcv, Kusuki-ml-2018nms, Kusuki-ml-2018wpa} has found, at least numerically and in some cases analytically, that the series coefficients in the $q$ expansion of $H_h(q)$, \begin{align} H_h(q) = \sum_{n=0}^\infty \mathfrak{h}_n \,q^n\, , \end{align} exhibit universal behavior in the $n\to \infty$ limit. For example,\footnote{For much more detailed investigation of the Virasoro blocks utilizing this and other similar asymptotic series data, we encourage the reader to consult~\cite{Kusuki-ml-2017upd, Kusuki-ml-2018wcv, Kusuki-ml-2018nms, Kusuki-ml-2018wpa}.} if $h_w \gg \frac{3}{4} \ca \gg h_v$, one has \begin{equation} \begin{split} \mathfrak{h}_{2k} \sim{}& ({-}1)^k (2k)^{\sf a} \; e^{{\sf A} \, \sqrt{2k}} \quad \text{as $k \to \infty$, with} \\ {\sf A} ={}& \pi \sqrt{\ca-2\,h_v}\, , \\ {\sf a} ={}& 2(h_v+h_w)- \frac{c+5}{8}\, . \end{split} \end{equation} Here we recall that $\mathfrak{h}_n=0$ for odd $n$ in this OPE channel and $\ca$ is defined in~\eqref{eq:ccdef}. Using this asymptotic series data, one can approximate the behavior of the block $H_h(q)$ near the boundary of the unit disk in the $q$ plane by treating the series summation as an integral to find \begin{align} \begin{split} H_h(q) \sim{}& \int_0^\infty d k\, (2k)^{\sf a} \; \tilde q^{2k} \; e^{{\sf A} \sqrt{2k}} \\ ={}& e^{{\sf A}^2/4\gamma} \; \gamma^{-{\sf a} - \frac{3}{2}} \bigg\{ \gamma^{\frac{1}{2}} \, \Gamma(1+{\sf a} ) \; {}_1 F_1\left({-}{\sf a} - \tfrac{1}{2}, \tfrac{1}{2}, {-} \frac{{\sf A}^2}{4\gamma}\right) \\ & + {\sf A} \, \Gamma(1+{\sf a} ) {}_1 F_1\left({-}{\sf a} , \tfrac{3}{2}, {-} \frac{{\sf A}^2}{4\gamma}\right) \bigg\} \\ \sim{}& \frac{{\sf A}^{2{\sf a} +1}}{\gamma^{2{\sf a} +\frac{3}{2}}} \; e^{{\sf A}^2/4\gamma} \, , \end{split} \end{align} where we have set $q = i \,\tilde q$, $\gamma = {-} \log \tilde q$, and taken the $c\to \infty$ limit in the last line. Sending $q\to i$, one can check that $\gamma \sim \frac{{-} \pi^2}{4\log (z/16)}$ to finally arrive at \begin{align} H_h(q \to i) \sim \frac{{\sf A}^{2{\sf a} +1}}{\gamma^{2{\sf a} +\frac{3}{2}}}\, z^{{-}(\ca-2h_v)}\, . \end{align} This power law behavior in $z$ essentially cancels the power law prefactor in the full Virasoro block ${\cal F}_h(z)$ (up to logarithmic corrections in $z$) and one is left with \begin{align} \frac{{\cal F}_h(z)}{{\cal F}_0(z)} \to q^h\, . \end{align} From this, we see that the (holomorphic) contribution of the blocks to the four point function is exponentially suppressed, $z^{2h_v} {\cal F}_h \!\propto\! e^{{-}\frac{2\pi} {\beta} 2h_vt}$, for $q$ asymptotically close to $q\!=\!i$. Unfortunately, this asymptotic behavior is probing $q$ very close to the unit circle, which corresponds to very late times, while the chaotic behavior we are interested in is situated near the scrambling time. A simple way to see that we aren't quite seeing the physics we are interested in is that this limiting behavior is independent of the exchanged operator dimension $h$, whereas we expect there to be a delicate interplay among blocks near the scrambling time to be compatible with the chaos bound. It would be very interesting if one can push the asymptotic series coefficients further away from the unit circle to access times near the scrambling time $t_*$, but we leave this to future work. ]]>

0$, and the worldline action is extremized at \ie\label{eqn:minimizedValues} v_B=-{1\over 2}f(x_B),~~~e^{x_B}&=\sqrt{{2h_v-h\over 2h_v+h}\; e^{2x} \left(1+ f(x) \right) } \,. \fe We then have after plugging in the expressions for $f(x)$ from~\eqref{eqn:fxP} \ie\label{eqn:minimizedAction} e^{-S}&= \frac{\left(2\,h_v+h\right)^{h+2\,h_v} }{\left(2\,h_v-h\right)^{h-2\,h_v}\, \left( 4\, h_v^2\right)^{2\,h_v}} \frac{e^{-2\, h \, x}}{ \left( 1+ \frac{6\pi \, h_w}{c\, \sin \tau} \, e^{t-x}\right)^{h+2h_v}} \,, \fe which is consistent with CFT result, viz., \ie e^{-S}\propto G_0(x,t)G_{h, h}(x,t), \fe where $G_0(x,t)$ and $G_{h, \bar h}(x,t)$ are in~\eqref{eqn:G0Gh}. This is the desired generalization of the result in~\cite{Roberts-ml-2014ifa} for $h\neq0$. In our configuration, the worldline action~\eqref{eqn:worldline} is independent of $v_S$. To determine $v_S$, we move the point $\vec {\bf x}_S$ infinitesimally along the $u$-direction. The resulting worldline action is minimized at \ie v_S=v_B\cosh x_B. \fe ]]>

] (0,0) -- (10,0) node[right]{$h$}; \draw [thick, black, ->] (0,0) -- (0,10) node[above]{$\bar{h}$}; \draw[thick, black] (2,0) arc (0:90:2); \draw[thick,dotted] (4,0) arc (0:90:4); \draw[thick, black] (3.65,3.65) node[below,rotate=-45]{intermediate operators}; \draw[thick,black,dashed] (6,0) arc (0:35:6); \draw[thick,black,dashed] (0,6) arc (90:55:6); \draw[thick, black,dashed] (4.91,3.44) -- (10,3.44); \draw[thick, black,dashed] (3.44,4.91) -- (3.44,10); \draw[thick, black,dotted] (4.25,4.25) -- (5,10); \draw[thick, black,dotted] (4.25,4.25) -- (10,5); \draw[thick, black] (2,0) node[below]{$h_{\text{gap}}$}; \draw[thick, black] (0,2) node[left]{$\bar{h}_{\text{gap}}$}; \draw[thick, black] (0,6) node[left]{$\bar{h}_{0} + \ca$}; \draw[thick, black] (6,0) node[below]{$h_{0} +\ca$}; \draw[thick, black] (4,0) node[below]{$\epsilon\, c$}; \draw[thick, black] (0,4) node[left]{$\epsilon\,c$}; \node[label={[label distance=0.5cm,text depth=-1ex,rotate=45]right: heavy operators}] at (4.75,6) {}; \node[label={[label distance=0.5cm,text depth=-1ex,rotate=45]right: $\sim e^{-\frac{2\pi}{\beta}(\bar{h}_0 + \ca - 2\Delta_v)\,t}$}] at (5.5,5.5) {}; \node[label={[label distance=0.5cm,text depth=-1ex,rotate=0]right: large spin}] at (6,2.25) {}; \node[label={[label distance=0.5cm,text depth=-1ex,rotate=0]right: $\sim e^{{-}\frac{2\pi}{\beta}\bar h_m t -\frac{ \pi\beta}{8\,t} \, h_0}$}] at (6,1.25) {}; \node[label={[label distance=0.5cm,text depth=-1ex,rotate=90]right: large spin }] at (1.25,6) {}; \node[label={[label distance=0.5cm,text depth=-1ex,rotate=90]right: $\sim e^{\frac{2 \pi}{\beta}(h_m t_* - \bar h_0 t)}$ }] at (2.25,6) {}; \draw [thick, black] (-0.1,0.5) node[right]{$\sim e^{\frac{2\pi}{\beta} (t-t_*)}$}; \draw [thick, black] (1.4,2) node[right]{$\sim e^{\ell_\text{gap} t}$}; \end{tikzpicture} } ]]>

0$ parts of BTZ metric with a glueing condition~\cite{Shenker-ml-2013pqa,Roberts-ml-2014isa} \ie\label{eqn:vShift} v\big|_{u<0}\cong v\big|_{u>0}+f(x). \fe ]]>

0)$. The velocity vector at point ${\mathscr B}$ is \ie\label{eqn:middleVelocityAtB} \dot u\big|_{{\mathscr B}}=\dot u(s_i)=0,\quad \dot v\big|_{{\mathscr B}}=\dot v(s_i)=0,\quad \dot x\big|_{{\mathscr B}}=\dot x(s_i)=\pm 1. \fe A consistent check of our solutions is that the momentum conservation is satisfied at point ${\mathscr B}$, i.e.\ \ie 2h_v \dot{\vec x}_{\overline{{\mathscr L}{\mathscr B}}}\big|_{{\mathscr B}}+2h_v \dot{\vec x}_{\overline{{\mathscr R}{\mathscr B}}}\big|_{{\mathscr B}}=2h \dot{\vec x}_{\overline{{\mathscr B}{\mathscr S}}}\big|_{{\mathscr B}}, \fe where $\dot{\vec x}_{\overline{{\mathscr L}{\mathscr B}}}\big|_{{\mathscr B}}$, $\dot{\vec x}_{\overline{{\mathscr R}{\mathscr B}}}\big|_{{\mathscr B}}$, $\dot{\vec x}_{\overline{{\mathscr R}{\mathscr S}}}\big|_{{\mathscr B}}$ are given in~\eqref{eqn:leftVelocityAtB},~\eqref{eqn:rightVelocityAtB}, and~\eqref{eqn:middleVelocityAtB}, and we have used~\eqref{eqn:minimizedValues}. Let us consider the normal vectors $q(s)$ and $\tilde q(s)$ that satisfy the orthonormal conditions \ie &q\cdot \tilde q=q\cdot \dot{\vec x}=\tilde q\cdot \dot{\vec x}=0,\quad-q^2=\tilde q^2=1, \fe and the parallel transport equations \ie \nabla q=\nabla \tilde q=0. \fe The equation $\nabla q=0$ can be written explicitly as \ie \dot q^u&=0, \\ \dot q^v-v \dot x q^x&=0, \\ \dot q^x-2v \dot x q^u&=0. \fe The solution to the orthonormal conditions and the parallel transport equations is \ie\label{eqn:normalQ} q^x(s)&=2 q^u \left[v_S\cosh s-v_B\cosh(s-|x_B|)\right]\,{\rm csch}\,x_B, \\ q^v(s)&={1\over 4q^u} +q^u\left[v_S\cosh s-v_B\cosh(s-|x_B|)\right]^2\,{\rm csch}^2\,x_B, \fe and \ie\label{eqn:normalTq} \tilde q^x(s)&=2 \tilde q^u \left[v_S\cosh s-v_B\cosh(s-|x_B|)\right]\,{\rm csch}\,x_B, \\ \tilde q^v(s)&=-{1\over 4\tilde q^u} +q^u\left[v_S\cosh s-v_B\cosh(s-|x_B|)\right]^2\,{\rm csch}^2\,x_B. \fe ]]>