⚛️ phenomenology

Photon emission from rotating plasmas: a generalized McLerran-Toimela formula and the onset of superradiance

This paper derives a generalized McLerran-Toimela formula showing that rotating plasmas emit significantly more soft photons than non-rotating ones due to leading-order one-loop contributions and superradiant amplification of specific modes, which in turn induces an instability in the magnetic fields generated during relativistic heavy-ion collisions.

Original authors: Kirill Tuchin

Published 2026-08-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Kirill Tuchin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe where matter doesn't just sit still or flow like water, but spins like a cosmic top. This is the world of plasma, a super-hot soup of charged particles that makes up most of the visible universe, from the inside of stars to the tiny, fleeting fireballs created in particle accelerators on Earth. Usually, when scientists study how these plasmas glow or emit light, they treat them like a calm, stationary pond. But in reality, these plasmas often rotate, sometimes with incredible speed. The big question is: does this spinning change how the plasma shines? It turns out, rotation is like a secret ingredient that can drastically alter the recipe for light emission, potentially creating bursts of energy that stationary plasmas simply can't make. Understanding this isn't just about pretty pictures; it helps physicists decode the violent collisions of atomic nuclei and might even explain why we see more low-energy light than our current theories predict.

This paper dives into that spinning mystery, specifically looking at a "quark-gluon plasma"—a state of matter so hot that even protons and neutrons melt into a swirling sea of their smaller parts, quarks and gluons. The author, Kirill Tuchin, asks a simple but profound question: If we spin this plasma, how does it emit photons (particles of light), and does it absorb them differently?

The main discovery is a new mathematical rule, a "generalized formula," that describes exactly how a rotating plasma glows. The paper finds that spinning changes the rules of the game entirely. In a stationary plasma, creating a photon usually requires a complex, two-step dance involving many particles (a "two-loop" process). However, the paper shows that in a rotating plasma, the spin itself provides the extra push needed to make this happen in a single, much simpler step (a "one-loop" process). Because of this, the rotating plasma emits a massive amount of "soft" photons—low-energy light—compared to its non-spinning cousin. It's as if the spinning motion acts like a turbocharger, allowing the plasma to spit out light much more easily, especially at lower energies.

The paper also uncovers a phenomenon called "superradiance." Imagine a spinning merry-go-round. If you throw a ball at it in the same direction it's spinning, the ball might bounce back with more energy than it had when it hit. In this plasma, photons with specific properties (low energy and a specific "twist" or angular momentum) behave like that ball. Instead of being absorbed by the plasma, they are amplified and emitted at a higher rate than they are swallowed. The paper calculates that this effect creates an instability, essentially causing the magnetic field that accompanies the plasma to grow stronger in a specific, low-energy range.

Crucially, the paper rules out the idea that we can ignore the edges of the plasma. In a stationary system, you can pretend the plasma is infinitely big, but in a rotating one, the edges are essential because the outer rim can't spin faster than the speed of light. The author explicitly argues against taking the limit of an infinite radius while keeping the spin speed fixed; the plasma must have a boundary. Furthermore, while the paper suggests that this rotation might help solve a long-standing puzzle about why we see more low-energy photons in experiments than expected (the "direct photon puzzle"), it presents this as a promising lead rather than a solved mystery. The results are based on theoretical calculations and simulations, not direct measurements of this specific effect yet, but the math suggests that if you spin the plasma, the light show changes dramatically, particularly in the deep infrared, low-energy region.

Technical Summary: Photon Emission from Rotating Plasmas and the Onset of Superradiance

Problem Statement
While plasmas are known to emit electromagnetic radiation, the specific effects of constant angular velocity ( $\Omega$ ) on photon emission and absorption spectra have not been fully established in a general framework. Previous studies of rotating plasmas focused primarily on synchrotron radiation requiring strong magnetic fields, or treated rotation as a negligible perturbation. However, in the context of the quark-gluon plasma (QGP) produced in relativistic heavy-ion collisions, substantial vorticity has been observed. This paper addresses the need for a general relationship between the photon spectrum and the photon polarization tensor in a rotating medium, specifically investigating how rotation alters the perturbative order of photon production and whether it induces superradiant instabilities in concurrent magnetic fields.

Methodology
The derivation proceeds through the following steps:

Formalism: The author derives photon emission and absorption probabilities using the S-matrix formalism, expressing them in terms of Wightman functions of the current-current correlator ( $j^\mu$ ).
Basis Selection: Recognizing that a rotating system possesses axial symmetry and a physical boundary (the "light cylinder" at $R < \Omega^{-1}$ ), the author abandons the standard plane-wave basis. Instead, they employ a cylindrical wave basis characterized by longitudinal momentum $k_z$ , transverse momentum $k_\perp$ , and azimuthal quantum number $m$ .
Boundary Conditions: The finite extent of the plasma ( $R$ ) is treated as essential. The wavefunctions are constructed to satisfy orthogonality within a cylinder of radius $R$ and length $L$ . The limit $R \to \infty$ is shown to be invalid for rotating systems as it leads to divergences in the summation over $m$ .
Derivation of Rates: By evaluating the matrix elements in the rotating frame (using the rotation operator $R = \exp\{-iL_z \Omega t\}$ ), the author relates the net emission rate to the imaginary part of the retarded photon polarization tensor, $\text{Im } \Pi_R$ .
Application to QGP: The general formula is applied to a rotating QGP. The spectral function $\text{Im } \Pi_T$ is evaluated at the one-loop order, utilizing the known expressions for the current-current correlator in the chiral limit.

Key Contributions

Generalized McLerran-Toimela Formula: The primary theoretical result is Equation (27), which relates the differential emission/absorption rate of a rotating plasma to the spectral function of the current-current correlator. This serves as the rotating analog to the McLerran-Toimela formula used for non-rotating plasmas.
Shift in Perturbative Order: The paper demonstrates that for a rotating plasma, the leading contribution to photon emission arises at the one-loop order. In contrast, for a non-rotating QGP, the leading contribution appears at the two-loop order ( $\alpha \alpha_s$ ). Rotation provides the necessary energy and momentum transfer (via the term $-m\Omega$ ) to allow a photon to decay into a quark-antiquark pair at one-loop, rendering the contribution finite without requiring higher-order corrections.
Superradiance Condition: The analysis identifies a specific regime where the emission rate exceeds the absorption rate. This occurs when the photon energy $\omega$ and azimuthal quantum number $m$ satisfy the condition $\omega < m\Omega$ . In this regime, the rotating system exhibits superradiance.

Results

Photon Spectrum and Elliptic Flow: Calculations for a rotating QGP (with parameters $T=0.3$ GeV, $R=2-3$ fm, and $\Omega=0.05-0.25$ fm $^{-1}$ ) show that the rotating plasma emits significantly more soft photons (infrared region, $\omega \lesssim 1$ GeV) than a non-rotating plasma. At higher energies ( $\omega \gtrsim 1.5$ GeV), the rotating spectrum is comparable in magnitude to the leading-order non-rotating two-loop result.
Elliptic Flow ( $v_2$ ): The calculated elliptic flow is negative in the infrared region and becomes positive above $\omega \simeq 1.5$ GeV. The negative $v_2$ at low energies is attributed to the dominance of low- $m$ modes which are emitted at small angles relative to the rotation axis.
Magnetic Field Instability: The paper analyzes the interaction between the rotating plasma and the strong magnetic field generated in heavy-ion collisions. It finds that superradiant modes ( $\omega < m\Omega$ ) experience exponential growth, driving an instability in the magnetic field. For a plasma lifetime of $\sim 10$ fm, this instability is phenomenologically relevant for $m=1, 2$ modes in the deep infrared ( $\omega \lesssim 10$ MeV).
Comparison with Previous Work: The results overturn previous conclusions (e.g., Ref. [61]) that magnetic field instabilities in rotating plasmas are irrelevant due to slow growth rates. The inclusion of superradiance reveals a mechanism for rapid amplification of infrared magnetic modes.

Significance and Claims
The paper posits that the rotation of the quark-gluon plasma is a critical factor in understanding electromagnetic radiation in relativistic heavy-ion collisions.

Direct Photon Puzzle: The significant enhancement of soft photon emission by the rotating plasma suggests that accounting for rotation may provide a solution to the "direct photon puzzle" (the discrepancy between theoretical predictions and experimental measurements of photon yields).
Superradiant Instability: The study argues that superradiance is not merely a theoretical curiosity but a mechanism that can amplify magnetic fields and potentially produce observable spikes of low-energy photons ( $\omega \lesssim 10$ MeV).
Future Directions: The author suggests that the interplay between the superradiant instability discussed here and the chiral magnetic instability (which operates on longer timescales) warrants further investigation, as they may mutually reinforce each other.

The work remains modest regarding specific experimental predictions, noting that the calculated elliptic flow must be weighed against other contributions before direct comparison with experiment, and that the spectral density used is based on a one-loop approximation in the chiral limit.

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