Chiral effects and Joule heating in hot and dense matter

This paper demonstrates that at higher temperatures, chiral plasma instability can generate strong magnetic fields even with small initial chiral imbalances, and reveals a novel mechanism where the chiral magnetic effect driven by density fluctuations causes significant Joule heating, potentially playing a critical role in the dynamics of supernovae and neutron star mergers.

The Gist

Imagine a bustling, super-hot crowd of tiny particles inside a dying star or a colliding neutron star. In this extreme environment, electrons (the tiny, fast-moving particles) have a special property called "chirality," which you can think of as a "handedness." Some electrons are "right-handed" and some are "left-handed."

Usually, the number of right-handed and left-handed electrons is perfectly balanced. But in this paper, the authors ask: What happens if there is an imbalance? What if, for a moment, there are more right-handed electrons than left-handed ones?

The paper explores two major consequences of this imbalance in the hot, dense soup of a star.

1. The "Spinning Top" Effect (Chiral Plasma Instability)

Think of the imbalance of handed electrons like a spinning top that is slightly off-balance. In a perfect vacuum, this imbalance would cause the top to wobble and grow stronger, creating a powerful magnetic field (like a giant magnet). This is called Chiral Plasma Instability (CPI).

• The Old Problem: Previous scientists thought that because real electrons have a tiny bit of "mass" (they aren't perfectly weightless), this mass acts like a friction brake. It flips the "handedness" of the electrons, turning right-handed ones into left-handed ones. They believed this friction was so strong that it would stop the magnetic field from ever growing, unless the initial imbalance was huge (as big as the total number of electrons).
• The New Discovery: The authors re-examined this using a wider range of temperatures. They found that heat changes the rules.
  • In cold, dense matter, the "friction" (mass) wins, and the magnetic field dies out.
  • But in hotter environments (like a supernova or a merging neutron star), the "friction" slows down. This allows the "spinning top" to wobble and grow even if the initial imbalance is much smaller than previously thought.
  • The Analogy: Imagine trying to spin a coin on a table. If the table is cold and sticky (cold matter), the coin stops immediately. But if the table is hot and slippery (hot matter), the coin can spin for a long time, even if you didn't push it very hard. This means strong magnetic fields can form in stars much more easily than we thought.

2. The "Electric Heater" Effect (Joule Heating)

The second part of the paper looks at what happens when this imbalance exists inside a star that already has a massive magnetic field (like a magnetar).

• The Mechanism: When you have an imbalance of "handed" electrons moving through a strong magnetic field, it creates a special electric current (called the Chiral Magnetic Effect).
• The Result: In a normal conductor, electricity flows smoothly. But in this star, the resistance of the material causes this special current to generate intense heat, similar to how a toaster wire glows red hot when electricity passes through it. This is called Joule Heating.
• The Surprise: The authors found that even a very small, modest imbalance (something that might naturally happen due to density fluctuations in the star) can generate a massive amount of heat in a very short time (milliseconds).
• The Scale: The energy released is so intense that it is comparable to the fundamental energy scale of the universe's building blocks (the QCD scale). It's like a tiny spark suddenly releasing the energy of a nuclear explosion.
• The Feedback Loop: This heat doesn't just sit there; it warms up the star, which changes how the particles move, which might create even more imbalance, creating a cycle of heating and fluctuation.

Summary

The paper tells us two main things about the physics of dying and colliding stars:

1. Hotter is better for magnets: In hot, dense stellar environments, the "brakes" on magnetic field growth are weaker than we thought. This means strong magnetic fields can form even with small initial imbalances.
2. Imbalance creates fire: A small imbalance in particle "handedness" inside a strong magnetic field acts like a powerful heater, dumping huge amounts of energy into the star in a flash. This could be a critical, previously overlooked ingredient in understanding how supernovae explode and how neutron stars merge.

The authors suggest that these effects should be included in computer simulations of these cosmic events to get a more accurate picture of what happens when stars die and collide.

Technical Summary

1. Problem Statement

The paper addresses two primary phenomena arising from the Chiral Magnetic Effect (CME) in hot and dense matter, specifically relevant to neutron stars (NS), proto-neutron stars, and neutron star mergers:

1. Chiral Plasma Instability (CPI): Can an initial electron chiral imbalance ($\mu_5$) generate strong magnetic fields via CPI? Previous literature (e.g., Refs. [8, 9, 23]) suggested that for realistic massive electrons, the chirality-flipping rate due to electron mass ($\Gamma_m$) is so high that it suppresses CPI unless the chiral chemical potential $\mu_5$ is comparable to the vector chemical potential $\mu_e$ (i.e., $\mu_5 \sim \mu_e$). This hierarchy is considered phenomenologically unlikely in many astrophysical scenarios.
2. Joule Heating: How does the CME current dissipate energy in a pre-existing strong magnetic field? While CPI focuses on field generation, the authors investigate whether the CME current, driven by density fluctuations in a magnetized medium, acts as a significant source of Ohmic (Joule) heating.

The core challenge is to re-evaluate the viability of CPI under a broader range of parameters (temperature $T$, $\mu_5$, $\mu_e$) and to quantify the energy deposition from CME-driven Joule heating in environments with density fluctuations.

2. Methodology

The authors employ a combination of kinetic theory, Boltzmann equations, and magnetohydrodynamics (MHD) principles:

• Chirality Flip Rate Calculation ($\Gamma_m$):
  • They calculate the rate of chirality flipping induced by the electron mass term ($m_e$) using the Boltzmann equation.
  • They consider two distinct regimes for protons: non-degenerate ($T \gg T_p$) and degenerate ($T \ll T_p$).
  • They analyze two limits for the chiral chemical potential relative to temperature: $T \gg \mu_5$ and $\mu_5 \gg T$.
  • Crucially, they include electron-electron scattering contributions, which were previously neglected, particularly in the regime where $\mu_5^2 \gg MT$ (where $M$ is the proton mass).
• CPI Growth Rate ($\Gamma_{CPI}$):
  • They solve the modified Maxwell's equations including the CME current ($J_{CME} = \xi B$, where $\xi = \frac{\alpha_{EM}}{\pi}\mu_5$) and finite electrical conductivity ($\sigma$).
  • They derive the growth rate for the maximally unstable mode ($k = \xi/2$).
• Joule Heating Estimation:
  • They model a scenario where density fluctuations (e.g., in turbulent merger environments) continuously pump chiral charge into the system via weak interactions (Urca processes), balancing the loss due to mass-induced flipping.
  • They calculate the power dissipation ($J \cdot E$) resulting from the CME current in a background magnetic field ($B_0$) with finite conductivity.

3. Key Contributions

A. Re-evaluation of Chiral Plasma Instability (CPI)

The authors challenge the established constraint that $\mu_5$ must be $\sim \mu_e$ for CPI to survive.

• Temperature Dependence: They find that at higher temperatures, the hierarchy of rates reverses. The CPI growth rate can exceed the chirality flip rate even when $\mu_5 \ll \mu_e$.
• Specific Regimes:
  • Non-degenerate Protons: The ratio $\Gamma_{CPI}/\Gamma_m$ scales as $\mu_5^2$. For $\mu_5 \sim 20$ MeV and $\mu_e \sim 200$ MeV, the instability can grow.
  • Degenerate Protons: In the regime $T \gg \mu_5$ and $T \ll \mu_5$ (provided $\mu_5^2 \ll MT$), the flip rate is suppressed at higher temperatures, allowing CPI to dominate for $\mu_5 \sim \mathcal{O}(10)$ MeV, which is significantly smaller than $\mu_e$.
  • Electron-Electron Scattering: They identify a new regime ($\mu_5^2 \gg MT$) where electron-electron scattering dominates the chirality flip rate. However, they find this specific regime does not sustain CPI effectively due to the resulting rate hierarchy.

B. Discovery of CME-Driven Joule Heating

The paper identifies a qualitatively new mechanism for energy dissipation in magnetized hot dense matter.

• Mechanism: In environments with density fluctuations (e.g., turbulence in NS mergers), chiral charge is continuously generated. This sustains a background $\mu_5$ (estimated at $\sim$ keV to MeV scales) which drives a CME current in the presence of strong magnetic fields ($B_0 \sim 10^{17}-10^{18}$ Gauss).
• Energy Scale: The resulting Joule heating deposits energy densities on the order of the QCD scale ($\Lambda_{QCD}^4$).
• Timescales: Significant energy deposition occurs over milliseconds to seconds.

4. Key Results

• CPI Viability:

  • For degenerate protons at temperatures $T > 1$ MeV, the instability can grow even if $\mu_5 \ll \mu_e$ (e.g., $\mu_5 \sim 10$ MeV vs $\mu_e \sim 200$ MeV).
  • The maximum magnetic field achievable via CPI is estimated as $B \sim \frac{\mu_e \mu_5}{\sqrt{2\pi}}$. For typical parameters, this yields $B \sim 10^{16}$ Gauss.
  • The constraint $\mu_5 \sim \mu_e$ is only necessary at very low temperatures; higher temperatures relax this requirement.

• Joule Heating Estimates:

  • Using parameters from merger simulations ($B_0 \sim 10^{18}$ Gauss, $\tau_B \sim 10^{-3}$ s, $T \sim 40$ MeV, $\mu_5 \sim 10^{-3}$ MeV), the deposited energy density is:
    $$E_J \sim 8 \times 10^3 (\Lambda_{QCD})^4$$
  • Even at lower temperatures ($T \sim 20$ MeV) with smaller $\mu_5$, the energy density remains significant ($\sim 0.008 \Lambda_{QCD}^4$).
  • This heating is sufficient to alter the thermodynamic state of the medium, affecting transport coefficients, Urca rates, and neutrino trapping.

5. Significance and Implications

• Astrophysical Dynamics: The findings suggest that CPI is a viable mechanism for generating strong magnetic fields in neutron stars and mergers, even without extreme initial chiral imbalances, provided the temperature is sufficiently high.
• Merger Simulations: The identification of CME-driven Joule heating as a critical energy source implies that current simulations of neutron star mergers and supernovae may be missing a key feedback loop. The heating could dampen density fluctuations, alter the equation of state, and influence the evolution of the magnetic field and neutrino emission.
• Theoretical Framework: The paper provides a more complete calculation of chirality flip rates, including electron-electron scattering and various temperature regimes, offering a refined theoretical basis for chiral magnetohydrodynamics in dense matter.
• Future Work: The authors propose that future MHD simulations for neutron star mergers should incorporate CME currents and the feedback of Joule heating on transport coefficients to accurately model the dynamics of these extreme environments.

In summary, Sen and Vaidya demonstrate that higher temperatures reverse the suppression of Chiral Plasma Instability, making it a robust mechanism for magnetic field generation, and they reveal a powerful, previously overlooked Joule heating mechanism driven by density fluctuations in magnetized dense matter.