Four-Phonon Scattering Insights

• Four-phonon scattering is defined by fourth-order anharmonic interactions in lattice dynamics that renormalize phonon frequencies and determine lifetimes.
• State-of-the-art first-principles methods, combining DFT and perturbative approaches, accurately compute four-phonon rates and their temperature-dependent scaling.
• Four-phonon processes can significantly reduce lattice thermal conductivity and alter optical linewidths, impacting energy transport in diverse materials.

Searching arXiv for recent and foundational papers on four-phonon scattering to ground the article. Four-phonon scattering denotes anharmonic phonon–phonon interactions generated by fourth-order terms in the lattice potential. Within perturbative lattice dynamics, it is the conventional higher-order correction beyond the three-phonon approximation, and it enters both the real part of the phonon self-energy, which renormalizes phonon frequencies, and the imaginary part, which determines linewidths and lifetimes. First-principles studies now evaluate these processes explicitly throughout the Brillouin zone, showing that four-phonon rates generally increase quadratically with temperature, can become comparable to three-phonon rates at medium and high temperatures, and in specific settings can dominate optical-phonon frequency shifts, Raman linewidths, or even near-field radiative heat transfer (Feng et al., 2015, Feng et al., 2018, Han et al., 2021, Feng et al., 2023).

1. Perturbative formulation and scattering channels

In anharmonic lattice dynamics, the crystal Hamiltonian is expanded as

$$\hat{H} = \hat{H}_0 + \hat{H}_3 + \hat{H}_4 + \cdots$$

with $\hat{H}_3$ and $\hat{H}_4$ representing cubic and quartic anharmonicity, respectively. Four-phonon scattering is therefore controlled by fourth-order interatomic force constants and associated quartic matrix elements (Feng et al., 2015).

For transport calculations, a standard relaxation-time form writes the total phonon lifetime as the sum of separate inverse lifetimes,

$$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$

or, in more complete treatments, with additional isotope, boundary, or electron-related channels added through Matthiessen’s rule (Gu et al., 2020, Wu et al., 2022). The corresponding lattice thermal conductivity is commonly written as

$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$

or in tensor form with mode-resolved velocities and nonequilibrium corrections in the Boltzmann transport equation (BTE) (Gu et al., 2020, Jain et al., 1 Apr 2025).

The quantum-mechanical four-phonon rate is commonly decomposed into three classes of events. In the notation used in several first-principles implementations, these are the $--$, $+-$, and $++$ channels; in physical language they correspond to splitting, redistribution, and combination processes. A representative expression is

$$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$

where the $\mathcal{L}_{\pm\pm}$ terms encode the quartic matrix elements together with momentum and energy conservation (Feng et al., 2015).

For finite-temperature frequency renormalization, the shift of a phonon mode $\hat{H}_3$0 is written as

$\hat{H}_3$1

where the four-phonon term is the first-order quartic contribution, the three-phonon term is second-order, the tadpole term is usually negligible, and the quasi-harmonic term captures thermal expansion (Feng et al., 2018).

2. First-principles realization and computational infrastructure

Modern calculations of four-phonon scattering typically combine density functional theory or density functional perturbation theory with finite-displacement extraction of second-, third-, and fourth-order interatomic force constants, followed by BTE solution for phonon transport or perturbative evaluation of phonon self-energies (Feng et al., 2018, Han et al., 2021). In silicon, for example, fourth-order force constants were computed by considering up to the fifth nearest neighbors, together with a dense $\hat{H}_3$2 $\hat{H}_3$3-mesh for frequency-shift calculations (Feng et al., 2018).

A major software development was the introduction of "FourPhonon," an extension module within the ShengBTE framework. It calculates four-phonon scattering rates, implements an adaptive energy broadening scheme, and provides "Fourthorder.py" for fourth-order IFC generation, while preserving the parallelism and workflow conventions of ShengBTE (Han et al., 2021). Subsequent performance work produced "FourPhonon_GPU," a heterogeneous CPU–GPU implementation based on OpenACC that reports over $\hat{H}_3$4 acceleration for the scattering-rate computation step and over $\hat{H}_3$5 total runtime speedup without sacrificing accuracy (Guo et al., 1 Oct 2025).

Machine-learning workflows are increasingly used when fourth-order IFC extraction becomes prohibitive. In Mg$\hat{H}_3$6GeSe$\hat{H}_3$7, a MACE message-passing neural network was trained to DFT-level accuracy, molecular-dynamics trajectories were generated, and compressed sensing lattice dynamics was used to recover third- and fourth-order IFCs before solving the Peierls–Boltzmann equation and the Wigner transport equation (You et al., 2024).

The numerical treatment is delicate. In graphene, using $\hat{H}_3$8 K IFCs severely overestimates four-phonon scattering compared with finite-temperature IFCs, by about an order of magnitude at $\hat{H}_3$9 K, and physically meaningful comparison with molecular dynamics required phonon frequency broadening through a Lorentzian treatment of the energy-conservation delta function (Gu et al., 2019). In 2D-GaN, the fourth-order IFCs were found to be highly sensitive to the atomic displacement used in finite differences; convergence was obtained only for $\hat{H}_4$0 Å, whereas the default $\hat{H}_4$1 Å severely overestimated four-phonon scattering (Sun et al., 2024). These results establish convergence of fourth-order force constants, energy broadening, and finite-temperature renormalization as central methodological issues rather than minor implementation details.

3. Frequency renormalization, linewidths, and spectral observables

Four-phonon scattering is not limited to thermal-resistance calculations. In silicon, rigorous first-principles evaluation of frequency shifts from three- and four-phonon scattering showed that the shifts are overwhelmingly controlled by the four-phonon term for both acoustic and optical modes, with agreement described as reasonable relative to inelastic neutron scattering throughout the Brillouin zone (Feng et al., 2018). A particularly important finding was that optical phonons near $\hat{H}_4$2 are highly sensitive to the cutoff radius of the fourth-order force constants: increasing the cutoff from second to third nearest neighbors substantially increases the computed shift, whereas four-phonon scattering rates nearly saturate when considering the second nearest neighbors (Feng et al., 2018). The same study found that higher-order anharmonicity beyond fourth order becomes important for the $\hat{H}_4$3-point optical phonon at high temperature when benchmarked against ab initio molecular dynamics (Feng et al., 2018).

In graphene Raman spectroscopy, explicit treatment of four-phonon anharmonicity, phonon renormalization, and electron–phonon coupling resolved a long-standing discrepancy between theory and measurement for the $\hat{H}_4$4-peak linewidth. The phonon–phonon contribution obeys

$\hat{H}_4$5

and the total intrinsic linewidth is

$\hat{H}_4$6

The reported result is that four-phonon scattering contributes a prominent linewidth that increases with temperature and exceeds the three-phonon contribution already at room temperature, while the temperature dependence from electron–phonon interaction reverses above a doping threshold $\hat{H}_4$7 (Han et al., 2021).

These studies clarify a useful distinction. Four-phonon scattering can dominate spectral renormalization or optical linewidths even when related corrections to lattice thermal conductivity from phonon renormalization or thermal expansion are reported to be small, as in high-temperature intrinsic silicon where four-phonon scattering substantially reduces $\hat{H}_4$8 above $\hat{H}_4$9 K but phonon renormalization and thermal expansion have only minor effects on $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$0 (Gu et al., 2020).

4. Role in lattice thermal transport

The strongest recurring result across the literature is that four-phonon scattering suppresses lattice thermal conductivity, often by amounts that invalidate three-phonon-only predictions. Foundational work showed directly that four-phonon rates are comparable to three-phonon rates at medium and high temperatures, scale as $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$1, and reduce $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$2 by more than $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$3 in Lennard-Jones argon at $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$4 K; even in diamond, silicon, and germanium, the reduction remains appreciable at high temperature (Feng et al., 2015). In intrinsic silicon, four-phonon scattering was identified as the dominant intrinsic mechanism reducing $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$5 above $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$6 K, while electronic conduction becomes a separate major channel above $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$7 K (Gu et al., 2020).

The quantitative impact is highly material dependent:

System	Reported four-phonon effect	arXiv
Intrinsic Si	Considerably reduces $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$8 above $$\tau_\lambda^{-1} = \tau_{3ph,\lambda}^{-1} + \tau_{4ph,\lambda}^{-1},$$9 K	(Gu et al., 2020)
Monolayer MoS$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$0	$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$1 Wm$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$2K$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$3 at $$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$4 K in RTA	(Chaudhuri et al., 2023)
AlSb	Isotopically pure $$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$5: $$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$6 W/mK at room temperature	(Yang et al., 2019)
Mg$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$7GeSe$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$8	$$\kappa = \frac{1}{V}\sum_\lambda C_\lambda v_\lambda^2 \tau_\lambda,$$9 W/mK at $--$0 K; $--$1 reduction	(You et al., 2024)
LaMoN$--$2, $--$3	$--$4 reduced by approximately $--$5	(Jain et al., 1 Apr 2025)
Ge$--$6Sb$--$7Te$--$8	$--$9 W/mK at $+-$0 K; $+-$1 reduction	(Ghosh et al., 2023)

Several studies also show that four-phonon scattering reorganizes modal contributions rather than merely scaling all lifetimes downward. In AlSb, the optical-phonon share of $+-$2 at $+-$3 K drops from $+-$4 in a three-phonon calculation to $+-$5 once four-phonon processes are included (Yang et al., 2019). In monolayer MoS$+-$6, the contribution of the ZA branch to $+-$7 is suppressed from $+-$8 to $+-$9 (Chaudhuri et al., 2023). In crystalline Ge$++$0Sb$++$1Te$++$2, quartic anharmonicity produces a faster-than-$++$3 decay of $++$4, rationalized as

$++$5

thereby challenging the conventional high-temperature $++$6 scaling and the idea of a universal lower bound to phononic thermal diffusivity (Ghosh et al., 2023).

5. Material-specific mechanisms and selection rules

The microscopic origin of strong four-phonon effects is not universal. In two-dimensional materials, a recurring motif is the out-of-plane acoustic ZA mode, especially when it retains a quadratic dispersion near $++$7. In monolayer MoS$++$8, the unusually high four-phonon rate of ZA phonons was attributed to the simultaneous effect of the acoustic–optical frequency gap, strong anharmonicity, and the reflection-symmetry selection rule that restricts three-phonon processes with an odd number of ZA phonons (Chaudhuri et al., 2023). Tensile strain makes the ZA branch more linear and dramatically reduces the importance of four-phonon scattering in the strained monolayer (Chaudhuri et al., 2023). A related low-temperature phenomenon was reported for $++$9-XN ($$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$0 B, Al, Ga): at $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$1 K, including four-phonon processes reduces $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$2 by $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$3, $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$4, and $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$5, respectively, with the effect traced to the ZA mode and tunable through strain engineering (Feng et al., 2023).

The interplay of symmetry and phase space is subtle. In 2D-GaN, reflection symmetry leads to many more allowed four-phonon processes than three-phonon processes, but the magnitude of the effect depends strongly on converged fourth-order IFCs; with converged force constants, four-phonon scattering still reduces lattice thermal conductivity by $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$6 relative to the three-phonon result (Sun et al., 2024). In monolayer $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$7-NbN, which lacks mirror symmetry, the reported strong four-phonon interactions are predominantly confined to acoustic phonons, and the reduction in $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$8 is $$\tau_{4,\lambda}^{-1}= \sum_{\lambda_1\lambda_2\lambda_3}\left\{ \frac{1}{6} \frac{n_{\lambda_1}^0 n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{--} + \frac{1}{2} \frac{(1+n_{\lambda_1}^0) n_{\lambda_2}^0 n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{+-} + \frac{1}{2}\frac{(1+n_{\lambda_1}^0) (1+n_{\lambda_2}^0) n_{\lambda_3}^0}{n_{\lambda}^0} \mathcal{L}_{++} \right\},$$9 at $\mathcal{L}_{\pm\pm}$0 K and $\mathcal{L}_{\pm\pm}$1 at $\mathcal{L}_{\pm\pm}$2 K (Murari et al., 18 May 2025).

In bulk and layered semiconductors with large acoustic–optical gaps, the mechanism is different. In AlSb and related III–V compounds, flat optical branches and large acoustic–optical gaps suppress many three-phonon processes, leaving four-phonon scattering to diminish optical thermal transport and weaken isotope effects (Yang et al., 2019). In $\mathcal{L}_{\pm\pm}$3-, $\mathcal{L}_{\pm\pm}$4-, and $\mathcal{L}_{\pm\pm}$5-PbP monolayers, the large acoustic–optical gaps of $\mathcal{L}_{\pm\pm}$6, $\mathcal{L}_{\pm\pm}$7, and $\mathcal{L}_{\pm\pm}$8 THz prohibit the dominant $\mathcal{L}_{\pm\pm}$9 three-phonon channel, while allowed four-phonon processes drive the lattice thermal conductivities down to $\hat{H}_3$00, $\hat{H}_3$01, and $\hat{H}_3$02 W/mK at $\hat{H}_3$03 K (Wu et al., 2022).

A common misconception is that pronounced four-phonon effects require strongly suppressed three-phonon scattering. High-throughput calculations on $\hat{H}_3$04 Half-Heusler materials argue otherwise: the acoustic phonon bandwidth dominates both three-phonon and four-phonon phase space, and materials with smaller acoustic bandwidths tend to have a more pronounced four-phonon effect even when three-phonon scattering is not significantly suppressed (Wu et al., 2024). This suggests that no single dispersion feature, taken in isolation, serves as a universal indicator of four-phonon dominance.

6. Extensions, exceptions, and conceptual limits

Although four-phonon scattering is usually discussed as a resistive mechanism for lattice conduction, its consequences are not always suppressive. In weakly anharmonic polar materials such as BAs and BSb, including four-phonon scattering in the phonon damping factor broadens surface-phonon-polariton resonances and activates additional radiative channels; the reported consequence is a nearly $\hat{H}_3$05-fold increase in total near-field heat flux between two BAs thin films compared with three-phonon scattering alone (Feng et al., 2023). The same work also notes that four-phonon scattering can decrease near-field radiative heat transfer in other systems (Feng et al., 2023). A plausible implication is that the usual intuition—more anharmonic scattering necessarily means less heat transfer—holds for lattice conduction but not uniformly across other bosonic transport channels.

Recent diagrammatic work has also limited the extent to which high-temperature thermal-conductivity suppression can be attributed uniquely to conventional four-phonon processes. A mixed cubic–quintic contribution, identified as a quintic-anharmonicity-assisted three-phonon channel, appears at the same perturbative order as the conventional quartic–quartic four-phonon channel, shares the temperature scaling $\hat{H}_3$06, and was found to be comparable to four-phonon scattering in Si over a broad frequency and temperature range; in strongly anharmonic AgCl it can become comparable even to ordinary three-phonon scattering at room temperature (Xia, 23 Jun 2026). That result does not negate the importance of four-phonon scattering, but it does mean that quartic processes are not always the only same-order correction relevant to high-temperature linewidths and thermal transport.

The present state of the field therefore combines mature formalism with open-ended scope. Four-phonon scattering is now a standard ingredient in accurate first-principles thermal transport for many semiconductors, oxides, phase-change materials, and two-dimensional crystals; yet the accuracy of any prediction still depends on force-constant convergence, finite-temperature renormalization, broadening schemes, and, in some regimes, inclusion of additional same-order channels beyond the conventional quartic picture (Gu et al., 2019, Sun et al., 2024, Xia, 23 Jun 2026).