ModMax Parameter: Conformal Nonlinear Electrodynamics

• ModMax parameter is the real, dimensionless deformation defining a unique nonlinear, conformally invariant extension of Maxwell electrodynamics with duality symmetry.
• It controls exponential screening of electromagnetic charge, modifying black hole metrics, photon spheres, and quasinormal frequencies.
• Beyond classical dynamics, ModMax impacts quantum, supersymmetric, and holographic models, offering measurable astrophysical signatures and novel soliton solutions.

The ModMax parameter is the single real, dimensionless deformation parameter (commonly denoted γ, η, or a) in the unique conformal and SO(2) duality-invariant extension of Maxwell electrodynamics known as ModMax theory. ModMax electrodynamics generalizes the classical Maxwell Lagrangian to a non-linear theory characterized by highly symmetric field equations and a tunable nonlinearity that leaves the light cone and duality structure of the vacuum intact. When coupled to gravity, the ModMax parameter controls the strength of nonlinearity, the effective screening of electromagnetic charges, and the physical features of black hole solutions and their perturbations.

1. Formal Definition and Range

The ModMax Lagrangian is given by

$$\mathcal{L}_{\rm ModMax}(F,G;\gamma) = -\tfrac12\,F\,\cosh\gamma + \tfrac12\,\sqrt{F^2 + G^2}\,\sinh\gamma$$

where

• $F = \tfrac12 F_{\mu\nu} F^{\mu\nu}$ is the usual Maxwell invariant,
• $G = \tfrac12 F_{\mu\nu}\widetilde{F}^{\mu\nu}$, with $\widetilde{F}^{\mu\nu}$ the Hodge dual,
• $\gamma \geq 0$ is the ModMax deformation parameter.

The theory reduces smoothly to Maxwell electrodynamics in the limit $\gamma \to 0$, with all higher-order (nonlinear) corrections controlled by $\gamma$. For all physically acceptable field configurations, causality (subluminal propagation), positivity of energy, and unitarity require $\gamma \geq 0$; negative values are excluded due to the emergence of superluminal birefringent modes and loss of convexity (Bandos et al., 2021). There is no upper bound on $\gamma$ from argument of local field theory or gravitational consistency.

2. Symmetry and Physical Interpretation

ModMax theory is engineered to simultaneously preserve:

• Poincaré and conformal invariance: The Lagrangian is homogeneous in the field invariants, ensuring a traceless stress tensor: $$F \partial_F\mathcal{L} + G \partial_G\mathcal{L} -\mathcal{L} = 0$$.
• SO(2) electromagnetic duality: Under electric-magnetic duality rotations, ModMax field equations remain invariant due to a specific differential relation involving the derivatives of $\mathcal{L}$ with respect to $F$ and $G$.

The ModMax parameter $\gamma$ therefore parametrizes a unique family of nonlinear, conformally invariant, duality-preserving theories. Physically, increasing $\gamma$ continuously interpolates from pure Maxwell electrodynamics ($\gamma = 0$) to a regime dominated by nonlinear, self-dual interactions. The primary effect is a uniform exponential "screening" of all terms arising from the electromagnetic charge in the classical and semiclassical regime.

3. Gravitational Coupling and Black Hole Solutions

In the context of Einstein–ModMax gravity, the parameter $\gamma$ modifies the Reissner–Nordström-like black hole metric by introducing a simple exponential screening factor on the charge term: $$ds^2 = f(r)dt^2 - f(r)^{-1}dr^2 - r^2 d\Omega^2, \quad f(r) = 1 - \frac{2M}{r} + \frac{Q^2 e^{-\gamma}}{r^2}$$ The effective squared charge is thus $Q_{\rm eff}^2 = Q^2 e^{-\gamma}$.

Extremality and horizon locations are controlled by this screened charge: $r_{\pm} = M \pm \sqrt{M^2 - Q^2 e^{-\gamma}}$ The bound for the charge-to-mass ratio at extremality is

$(Q/M)_{\rm max} = e^{\gamma/2}$

thus, for fixed $M$ and $Q$, increasing $\gamma$ brings the solution closer to Schwarzschild geometry and allows larger total physical charge before the onset of extremality.

4. Physical Manifestations and Observable Effects

The exponential damping of electromagnetic charge by $e^{-\gamma}$ has several critical observational implications:

• Photon-sphere and light deflection: The photon sphere radius, lensing deflection angles, and absorption cross section are all functions of $Q_{\rm eff}$. For instance, in the BTZ-ModMax black hole, the light deflection angle in the weak field regime is uniformly rescaled by $e^{-\gamma}$ (Kala, 27 Jan 2025).
• Quasinormal modes and ringdown: The effective potential for wave propagation acquires $Q^2 e^{-\gamma}$ dependence, shifting the quasinormal frequencies, Lyapunov, and instability exponents, typically reducing both real and imaginary parts (lower frequencies, slower decay) as $\gamma$ is increased (Panah et al., 5 Nov 2024, Siahaan, 5 Sep 2024).
• Magnetization and test-particle dynamics: In magnetized black holes, the induced moment and ISCO locations are shifted as $\gamma$ increases, with orbits and frequency-resonant radii moving outward, and Larmor/anti-Larmor symmetry breaking observable through $Q e^{-\gamma/2}$ (Siahaan, 21 Sep 2024, Awal et al., 10 Nov 2025).
• Thermodynamics and phase structure: All charge-dependent thermodynamic quantities (Hawking temperature, heat capacity, free energies) couple to $\gamma$ via $Q^2 e^{-\gamma}$. Critical points and the occurrence of Hawking–Page and van der Waals–like transitions are strongly $\gamma$-dependent; at large $\gamma$, small black holes become thermodynamically stable and phase transitions shift or disappear (Panah, 19 Feb 2024, Panah, 8 Jul 2025, Panah et al., 30 May 2024, Sucu et al., 8 Aug 2025).

ModMax parameter	Metric charge term	Physical effect
$\gamma=0$	$Q^2$	Reissner–Nordström limit
$\gamma>0$	$Q^2 e^{-\gamma}$	Charge screening, suppresses electromagnetic repulsion
$\gamma \to \infty$	0	Schwarzschild limit

5. Nonlinear Dynamics Beyond Black Holes

The ModMax parameter not only controls black hole geometry, but also strongly influences:

• Electromagnetic solitons and knots: In flat spacetime, ModMax admits deformed versions of the null Hopfion–Rañada knotted solutions. These cease to be strictly null, becoming “almost null” with $E^2 - B^2\propto \sinh\gamma$, thus resolving the null singularity of the original Lagrangian (Dassy et al., 2021).
• Generalized field equations and corrections: In the weak-field regime, an expansion in γ shows that ModMax supplies the leading (dual-invariant, conformal) nonlinear correction to the Maxwell action: $$\mathcal{L} \simeq -\tfrac12 F + \gamma \sqrt{F^2+G^2}$$.
• Transport in holographic models: In AdS/CFT setups, γ controls the deviation of magnetotransport coefficients from ordinary Maxwellian scaling. Notably, γ can tune the position and height of the Nernst dome, the Hall angle profile, and even drive the system to an exotic, quasiparticle-dominated regime (Barrientos et al., 3 Jun 2025).
• Higher-order deformations: In frameworks generalizing ModMax (e.g. (Kruglov, 2021)), γ remains the "mixing" or "modulating" parameter in the four-parameter family that unifies ModMax, Born–Infeld, and exponential nonlinear electrodynamics.

6. Supersymmetric and Quantum Extensions

ModMax admits a unique N=1 supersymmetric extension (“superModMax”) in which γ weighs the relative strength of the chiral kinetic versus higher-derivative superfield interactions. The resulting action remains superconformal and duality-invariant for all γ≥0 (Bandos et al., 2021). In T$\bar{T}$-deformed mechanical analogs, γ plays an identical role as a hyperbolic deformation parameter, preserving duality and conformal symmetry (García et al., 2022).

7. Astrophysical Constraints and Observational Status

Recent MCMC parameter estimation analyses of QPO data from multiple black hole sources consistently prefer moderate, nonzero values of the ModMax parameter (η ~ 4–6), excluding the Maxwell limit at several sigma confidence in most objects (Awal et al., 10 Nov 2025). This points to measurable signatures of ModMax-type nonlinear electrodynamics in strong-field astrophysics, pending further observational confirmation. Increasing γ tends to shift ISCO radii outward and lowers predicted frequencies for QPOs, partially mimicking the effect of black hole spin in Kerr models.

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In summary, the ModMax parameter provides a single, mathematically robust control over the deviation from Maxwell electrodynamics under the strictures of conformal and duality invariance. It enters gravitational dynamics, classical and quantum wave propagation, electromagnetic soliton structure, and holographic transport exclusively through hyperbolic ($\cosh\gamma, \sinh\gamma$) and exponential ($e^{-\gamma}$) combinations, acting as a uniform “screening” or deformation dial in all physical sectors derived from the theory. Its effects are manifest in the transition from Reissner–Nordström to Schwarzschild limits, the stabilization of black hole thermodynamics, the suppression of non-linear electromagnetic self-interactions, the modification of astrophysical observables, and the mathematical extension to generalized nonlinear electrodynamic and supersymmetric theories.