Near-extremal hydrodynamics and the holographic product formula

TL;DR

The paper develops and applies the holographic product formula to near-extremal hydrodynamics, showing how holographic spectral functions factorize into IR-CFT data and pole structures in regimes where ω, k, and T are small compared to μ. It develops two main near-extremal forms: an extremal-hydrodynamic power-law factor tied to IR scaling and a near-extremal IR-CFT product with a 1+O(T/μ) correction, each expressing G_{12} as a product over poles times a soft or IR factor. Through explicit examples—probe currents, stress-tensor sound channel, and magnetized branes—the work demonstrates the power of the product formula to constrain and compute low-energy spectral functions and confirms numerically the near-extremal pole structure. It further shows how IR scaling improves neutrino-transport predictions in holographic matter, highlighting the practical impact of incorporating IR-CFT data into hydrodynamic descriptions.

Abstract

The holographic product formula is used to determine the general form taken by holographic spectral functions in the near-extremal hydrodynamic regime, with energy $ω$, momentum $k$ and temperature $T$ much smaller than a hard scale $μ$. The resulting expressions simplify in the extremal limit $T \ll ω,k\ll μ$, for which the low-temperature gapless modes and the IR conformal behavior factorize. In some cases, this factorization extends to the general near-extremal regime $ω,k,T\llμ$ at leading order in $T/μ$. Several examples are discussed with different types of gapless modes and IR CFTs, including new numerical results for low temperature quasi-normal modes. We end with a concrete application that shows how the inclusion of the IR conformal behavior improves the description of the spectral function at low energies.

Figures (13)

• Figure 1: Schematic pole structure of holographic spectral functions in the near-extremal regime $T\ll\mu$. The black dots indicate hard poles of order $\mathcal{O}(\mu)$, the blue dots soft poles of order $\mathcal{O}(T)$ and the red dot is an example of gapless pole. The near-extremal hydrodynamic region (bounded by the yellow-dashed line) contains only soft and gapless poles. This picture is close to the actual pole structure of the correlators in section \ref{['sec:sound']}, but more general behaviors are possible. In particular the soft poles of section \ref{['sec:mb']} have finite real parts.
• Figure 2: Relative difference of the numerically computed transverse (left) and longitudinal (right) spectral functions, with the leading order hydrodynamic (top) and extremal hydrodynamic (bottom) approximations. The relative differences are shown as a function of frequency $\omega$ and momentum $|\vec{k}|$, in units of the horizon radius $r_H$. The chemical potential to temeperature ratio is $\mu/T\simeq 65$.
• Figure 3: Neutrino opacity $\kappa$ and its hydrodynamic approximations as a function of the neutrino energy $E_\nu$, for $\mu/T\simeq4.7$ (left) and $\mu/T\simeq 65$ (right). $\kappa_{e,0}$ is a normalization factor that was introduced in Jarvinen:2023xrx.
• Figure 4: Same as figure \ref{['Fig:kaenu']} but for the anti-neutrino opacity $\bar{\kappa}$.
• Figure 5: Imaginary part of the first soft poles of the transverse polarization function in units of $2\pi T$ and as a function of momentum $r_e|\vec{k}|$, for $d=4$ and $r_e T = (50\pi)^{-1}$. The blue dots are the numerical results whereas the orange lines show the poles of the IR CFT$_1$ correlator \ref{['tc1']}.