Pole-Skipping and Chaos in Hot $\mathcal{M}$QCD

Abstract

We address the question of whether thermal QCD at high temperature is chaotic from the $\mathcal{M}$ theory dual of QCD-like theories at intermediate coupling as constructed in arXiv: 2004.07259. The equations of motion of the gauge-invariant combination $Z_s(r)$ of scalar metric perturbations is shown to possess an irregular singular point at the horizon radius $r_h$. Very interestingly, at a specific value of the imaginary frequency and momentum used to read off the analogs of the ''Lyapunov exponent'' ${\lambda }_L$ and ''butterfly velocity'' $v_b$ not only does $r_h$ become a regular singular point, but truncating the incoming mode solution of $Z_s(r)$ as a power series around $r_h$, yields a ''missing pole'', i.e., $C_{n,n+1}=0,\det M^{(n)}=0,n\in {\mathbb{Z}}^{+}$ is satisfied for a single $n\ge 3$ depending on the values of the string coupling $g_s$, number of (fractional) $D3$ branes $(M)N$ and flavor $D7$-branes $N_f$ in the parent type IIB set (arXiv:hep-th/0902.1540), e.g., for the QCD(EW-scale)-inspired $N=100,M=N_f=3,g_s=0.1$, one finds a missing pole at $n=3$. For integral $n>3$, truncating $Z_s(r)$ at $\mathcal{O}((r-r_h{)}^n)$, yields $C_{n,n+1}=0$ at order $n,\forall n\ge 3$. Incredibly, (assuming preservation of isotropy in ${\mathbb{R}}^3$ even with the inclusion of higher derivative corrections) the aforementioned gauge-invariant combination of scalar metric perturbations receives no $\mathcal{O}(R^4)$ corrections. Hence, (the aforementioned analogs of) ${\lambda }_L,v_b$ are unrenormalized up to $\mathcal{O}(R^4)$ in $\mathcal{M}$ theory.

Comments: v2: 1+55 pages, LaTeX, Explanatory material and references added, title slightly modified, to appear in JHEP