Planckian Scattering Rates

Updated 6 August 2025

Planckian scattering rates are defined by a universal dissipation scale, 1/τ ∼ kB·T/ħ, relevant across quantum materials and high-energy phenomena.
They bridge experimental findings—such as T‑linear resistivity in strange metals—with theoretical models including quantum critical and electron-phonon interactions.
The framework also extends to high-energy gravity, where scattering cross sections scale with energy, ensuring unitarity and consistent ultraviolet behavior.

Planckian scattering rates refer to quantum-limited dissipation rates characterized, in diverse contexts, by a relaxation or scattering rate of order $1/\tau \sim k_B T/\hbar$ , where $T$ is the temperature and the prefactors reflect various underlying microscopic mechanisms. This timescale, often called the “Planckian time” $\tau_{\mathrm{Pl}} = \hbar/(k_B T)$ , emerges across quantum matter systems—ranging from strongly correlated electron metals and high-temperature superconductors to gravitational scattering at and above the Planck scale—signaling a potential universal bound on inelastic dissipation processes. The concept has become central in attempts to explain anomalous transport properties such as $T$ -linear resistivity in “strange metals”, and to organize the dynamics of transplanckian gravitational collisions.

1. Foundational Formulation and Theoretical Bounds

At the core of Planckian scattering is the assertion that, in the absence of any small or large parameters other than temperature, the only available timescale for energy relaxation is $\tau_{\mathrm{Pl}} = \hbar/(k_B T)$ (Hartnoll et al., 2021). In metallic systems, this leads to a scattering rate

$\Gamma(T) = \frac{1}{\tau} \simeq \alpha \frac{k_B T}{\hbar}$

where $\alpha$ is typically of order unity, though it can deviate substantially depending on the microscopic situation (Sadovskii, 2020, Ahn et al., 2022).

In the context of quantum gravity and high-energy particle scattering, a related scaling emerges not directly from temperature, but from the fundamental constants and kinematic invariants controlling the high-energy limit. When the center-of-mass energy far exceeds the Planck mass, tree-level differential cross sections in quadratic gravity theories exhibit a scaling

$\frac{d\sigma}{d\Omega}(E, \theta) \propto \frac{1}{E^2}$

reflecting a suppression of Planckian scattering rates in the ultraviolet, necessary for the perturbative consistency (and unitarity) of the theory (Ciafaloni et al., 2014, Holdom, 2021, Cunha et al., 26 May 2025). The essential point is the absence of uncontrollable ultraviolet divergences in Planckian/ultra-Planckian scattering due to structural cancellations within the amplitude.

2. Planckian Dissipation in Electron Systems: Experimental and Theoretical Perspectives

The prototypical manifestation of Planckian dissipation in condensed matter is $T$ -linear resistivity, where experimental studies consistently extract a scattering rate of the form $\Gamma(T) \simeq \alpha k_B T/\hbar$ in strange metals, including cuprates, heavy fermion compounds, and iron-based superconductors (Grissonnanche et al., 2020, Mousatov et al., 2020, Chang et al., 2022, III et al., 1 May 2025). Detailed angle-dependent magnetoresistance and terahertz spectroscopy measurements have shown:

In materials such as Nd-LSCO (a hole-doped cuprate), the inelastic part of the scattering rate extracted from transport saturates the Planckian limit $\alpha \approx 1.2 \pm 0.4$ and is found to be isotropic around the Fermi surface, in contrast to elastic scattering that displays hot-spot anisotropy (Grissonnanche et al., 2020).
In FeTe $_{1-x}$ Se $_x$ , terahertz spectroscopy reveals two parallel conduction channels: a broad, weakly $T$ -dependent one, and a sharp Drude component with a Planckian-limited scattering rate ( $\alpha \sim 2.4 - 3$ ), from which most of the superfluid spectral weight is sourced below $T_c$ (III et al., 1 May 2025).

Theoretical models reproduce this behavior by identifying classes of interactions whose inelastic “on-shell” (energy-conserving) processes dominate the relaxation. For instance, models inspired by the Sachdev-Ye-Kitaev (SYK) paradigm, but generalized to itinerant fermions with random, momentum-conserving resonant scattering, yield spectral functions and transport lifetimes $\tau_{\mathrm{tr}}$ that are strictly $T$ -linear and essentially independent of interaction strength for a broad range of couplings. The resulting expression

$\frac{1}{\tau_{\mathrm{tr}}} = f \frac{k_B T}{\hbar}, \quad f \approx 1{-}5$

is robust even when the underlying fermion dispersion and disorder are nontrivial (Patel et al., 2019).

3. Competing and Intertwined Scattering Mechanisms

Despite the ubiquity of Planckian scaling, the microscopic origin of the quantum-limited rate varies significantly and is, in some systems, accidental rather than fundamental. Analyses have shown:

In conventional metals at high $T$ , $T$ -linear resistivity and associated Planckian rates naturally appear as the outcome of electron-phonon scattering, especially above the Debye temperature. This is not a unique feature of strongly correlated systems, as even simple metals with moderate electron-phonon coupling constants $\lambda$ can reach $\alpha\sim 1$ simply as a consequence of the analytic dependence of $\Gamma(T) = 2\pi\lambda T$ and the effective mass renormalization $m^* = m(1+\lambda)$ (Shaginyan et al., 2019, Sadovskii, 2020).
In strongly correlated systems, such as heavy-fermion metals and high- $T_c$ superconductors, the $T$ -linear scattering may be linked to critical fluctuations near a quantum critical point, such as the emergence of flat bands due to a fermion condensation quantum phase transition (FCQPT), which then produce collective zero-sound modes that act as effective phonons (Shaginyan et al., 2019, Chang et al., 2022).
In 2D semiconductor systems, Planckian bounds arise from the interplay of temperature-dependent screening of disorder and electron-electron interactions, even when the inelastic processes themselves are subdominant. The extracted $\hbar/\tau$ is consistently observed not to exceed $\sim 10 k_B T$ , enforcing a generalized (rather than strict) Planckian upper bound (Ahn et al., 2022).
In “extreme” strange metals, the breakdown of the simple Drude paradigm for optical conductivity and the presence of strong dynamical (energy-over-temperature) scaling complicate direct extraction of $\tau$ , but the scaling framework again suggests a characteristic quantum-limited relaxation rate (Li et al., 2022).

Phonon contributions, both as scatterers of electrons and vice versa, remain important. For example, thermal diffusivity and Lorenz ratio measurements in high- $T_c$ cuprates and Ru-based perovskites show robust Planckian electron-phonon scattering at high $T$ . The Lorenz ratio, $L(T)/L_0$ , which significantly exceeds unity when phonons dominate the heat current, is a direct marker that Planckian dissipation can originate from either electron-electron or electron-phonon processes, and that their contributions can be disentangled via careful thermal transport analysis (Mousatov et al., 2020, Sun et al., 2023).

Table: Scaling of Scattering Rate in Representative Systems

System/Classification	Scattering Rate $\Gamma(T)$	Dominant Mechanism
Conventional metal ( $T \gtrsim T_D$ )	$2\pi\lambda T$	Electron-phonon
Strongly-correlated “strange” metal	$\alpha k_B T/\hbar$ , $\alpha \sim 1$	Quantum critical, FCQPT
2D semiconductor (screen/disorder)	$\sim (1{-}10)k_B T/\hbar$	Screened disorder, e-e
Ultra-Planckian graviton scattering	Cross-section $\propto 1/E^{2}$ or $1/s$	Unitarity, UV-completion

4. Planckian Rates Beyond Electronic Systems: Planck-Scale and Gravity

In quantum gravity, Planckian scattering rates arise in the context of high-energy (transplanckian) collisions, graviton-mediated parton-parton scattering, and the paper of super-Planckian cross sections. Key features include:

In the ACV eikonal framework for ultra-high energy gravitational scattering, the S-matrix resums elastic ladder diagrams to all orders, resulting in an eikonal amplitude $S_{\mathrm{eik}}(b, s) = \exp\{i\delta_0(b, s)\}$ whose phase $\delta_0$ encodes deflection angles and time delays of order $R \ln(L/b)$ , with $R=2G\sqrt{s}$ the gravitational radius and $b$ the impact parameter. Subleading rescattering corrections (via H-diagrams and auxiliary fields) systematically expand in $R^2/b^2$ . Both action-based and geometric metrics yield consistent Planckian scattering corrections up to this order, with time delays and trajectory shifts calculable within the same formalism (Ciafaloni et al., 2014).
In UV-complete quantum gravity theories such as quadratic gravity and Agravity, tree-level hard scattering cross sections for elementary particles (gluons, quarks) mediated by gravitational exchange scale as $1/s$ at high energies. This scaling is crucial for unitarity and absence of ultraviolet pathologies. Intricate cancellations, including contributions from ghosts and higher-derivative terms, ensure that the cross section decreases with increasing energy—an essential feature for a consistent Planckian (and ultra-Planckian) regime (Holdom, 2021, Cunha et al., 26 May 2025).
In dynamic disordered systems, the destruction of Anderson localization by moving impurities results in a universal “Planckian diffusion” $D = \alpha \hbar/m$ even without thermal equilibrium. This quantum diffusion limit, with typical $\alpha \in [0.5, 2]$ , prescribes a minimum diffusion constant for quantum particles subjected to moving disorder, supplanting the Planckian time limit in nonthermal scenarios (Zhang et al., 27 Nov 2024).

5. Interpretation, Limitations, and Universality

Despite the recurring appearance of Planckian scaling, several caveats and subtleties must be noted:

The apparent universality of the “Planckian limit” is, in many cases, a result of how scattering rates are extracted (the analysis procedure) rather than a fundamental microscopic bound. In conventional metals, for example, empirical values of $\alpha$ align with unity largely due to the values of the electron-phonon coupling and mass renormalization (Sadovskii, 2020).
In systems with extremely renormalized electronic bandwidth (heavy fermion compounds near Kondo-destruction QCPs), measured scattering rates can be well below the naive Planckian value even when the resistivity remains strictly linear in $T$ . This discrepancy indicates that the Drude-based extraction of $\tau$ is not generically valid, and that the true dissipation mechanisms must be sought in more intricate interplay between mass renormalization, Fermi surface reconstruction, and quantum critical fluctuations (Taupin et al., 2022).
In strange metals exhibiting strong deviations from the Drude behavior, especially for optical conductivity, relaxation rates inferred from scaling collapses ( $\omega/T$ scaling) in the low-frequency regime may provide a more reliable measure of Planckian dissipation than fits to individual lifetimes (Li et al., 2022).
In dynamic disordered systems, the emergence of a Planckian diffusion coefficient $D = \alpha \hbar/m$ signals that quantum-limited transport is as universal under moving disorder as Anderson localization is for static disorder, reinforcing that Planckian-scale dissipation is not a phenomenon exclusive to close-to-equilibrium or strictly thermal regimes (Zhang et al., 27 Nov 2024).

6. Open Problems and Broader Connections

The paper of Planckian scattering rates connects a wide spectrum of research areas:

In condensed matter, establishing the extent to which Planckian-limited rates can be attributed to quantum criticality, electron-phonon coupling, or emergent collective modes remains an open issue, with ongoing studies employing ARPES, optical, and thermal transport probes across correlated electron systems (Grissonnanche et al., 2020, Fink et al., 2020, Chang et al., 2022, Zhakina et al., 2023).
The robustness of Planckian scaling in the presence of disorder, multiple scattering channels, or magnetic fields is experimentally tested in a variety of platforms, with consistent findings of isotropic rates and insensitivity to field up to very high strengths in, e.g., cuprates (Ataei et al., 2022).
In ultrahigh-energy gravitational and particle physics, Planckian regimes probe the limits of quantum field theory, unitarity without positivity, and the role of higher-derivative operators in ensuring UV-completion and finite cross-sections (Holdom, 2021, Cunha et al., 26 May 2025).
The distinction between “sub-Planckian,” “Planckian,” and “super-Planckian” regimes in different scattering contexts, and the possibility of a universal quantum-limited bound on diffusion and dissipation—either as a strict limit or an order-of-magnitude guideline—remains a topic of active discussion (Ahn et al., 2022, Sun et al., 2023, Aydin et al., 2023).

In summary, Planckian scattering rates offer a unifying language for quantum-limited processes, with empirically robust signatures in numerous condensed matter and high-energy contexts. Their physical origins, limits of applicability, and significance as a universal bound continue to be the subject of intensive exploration.