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 [1]. The equations of motion of the gauge-invariant combination Zs(r) of scalar metric perturbations is shown to possess an irregular singular point at the horizon radius rh. Very interestingly, at a specific value of the imaginary frequency and momentum used to read off the analogs of the “Lyapunov exponent” λL and “butterfly velocity” vb not only does rh become a regular singular point, but truncating the incoming mode solution of Zs(r) as a power series around rh, yields a “missing pole”, i.e., Cn,n+1 = 0, det M(n) = 0, n ∈ \({\mathbb{Z}}^{+}\) is satisfied for a single n ≥ 3 depending on the values of the string coupling gs, number of (fractional) D3 branes (M)N and flavor D7-branes Nf in the parent type IIB set [2], e.g., for the QCD(EW-scale)-inspired N = 100, M = Nf = 3, gs = 0.1, one finds a missing pole at n = 3. For integral n > 3, truncating Zs(r) at \(\mathcal{O}\left({\left(r-{r}_{h}\right)}^{n}\right)\), yields Cn,n+1 = 0 at order n, ∀n ≥ 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 \(\mathcal{O}\left({R}^{4}\right)\) corrections. Hence, (the aforementioned analogs of) λL, vb are unrenormalized up to \(\mathcal{O}\left({R}^{4}\right)\) in \(\mathcal{M}\) theory.
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Acknowledgments
GY thanks the Infosys Foundation for the partial support at CMI. SSK is supported by a Junior Research Fellowship (JRF) from the Ministry of Human Resource and Development (MHRD), Govt. of India. AM is partly supported by a Core Research Grant number SER-1829-PHY from the Science and Engineering Research Board, Govt. of India. One of us (AM) would like to thank J. Maldacena, M. Mezei and K. Sil for very useful clarifications. One of us (SSK) would like to thank IIT Roorkee for high-end computational facilities. We thank the anonymous referee for the many useful comments that improved the paper’s presentation.
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Yadav, G., Kushwah, S.S. & Misra, A. Pole-skipping and chaos in hot\(\mathcal{M}{\text{QCD}}\). J. High Energ. Phys. 2024, 15 (2024). https://doi.org/10.1007/JHEP05(2024)015
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DOI: https://doi.org/10.1007/JHEP05(2024)015