Quark and gluon production in the presence of the time-varying chiral magnetic current

This paper investigates how the time-dependent chiral magnetic conductivity in quark-gluon plasma influences particle spectra and energy loss through chiral Cherenkov processes, revealing that these effects lead to strong jet polarization in relativistic heavy-ion collisions.

The Gist

Imagine you are watching a high-speed race, but instead of cars, the racers are tiny particles called quarks and gluons. These particles are zooming through a super-hot, super-dense soup of energy known as quark-gluon plasma (the kind of stuff that existed just microseconds after the Big Bang).

Usually, when these particles race through this soup, they lose energy by bumping into things or shooting out smaller particles, kind of like a car losing speed by driving through mud or throwing sand out the back window.

This paper, written by physicist Kirill Tuchin, explores a very strange and new way these particles lose energy. It's not about bumping into mud; it's about the "magnetic weather" of the universe itself changing as they fly through it.

Here is the breakdown of the paper's big ideas using simple analogies:

1. The "Chiral Magnetic Effect": A One-Way Street for Electricity

Imagine a crowd of people (the particles) in a room. Usually, they move randomly. But in this specific "chiral" soup, there is a rule: if you turn on a magnetic field, it acts like a giant, invisible wind that forces the electric charge to flow in one specific direction, like a river flowing downstream.

This is called the Chiral Magnetic Effect (CME). The strength of this "wind" is measured by a number the author calls $b_0$.

2. The Twist: The Wind is Changing Speed

In most physics papers, scientists assume this "wind" ($b_0$) is steady. But in a real heavy-ion collision (like smashing gold atoms together), this wind doesn't stay steady. It starts strong, then fades away, or maybe even flips direction. It's time-varying.

The author asks: What happens to our racing particles if the magnetic wind they are riding is constantly changing speed?

3. The "Chiral Cherenkov" Effect: The Sonic Boom of the Quantum World

You know how a jet plane creates a sonic boom when it flies faster than the speed of sound? Or how a boat creates a wake when it moves through water?

In this paper, the author shows that when a particle moves through this changing magnetic wind, it can trigger a similar effect called Chiral Cherenkov Radiation.

• The Analogy: Imagine a surfer riding a wave. If the wave suddenly changes shape or speed, the surfer might be thrown off balance or shoot a spray of water in a specific direction.
• The Result: The racing particles (quarks and gluons) start "shooting" out new particles (photons, gluons, or quark-antiquark pairs) in a very specific, organized way. They aren't just losing energy randomly; they are losing it in a way that depends on their "spin" (a quantum property like a spinning top).

4. The "Spin" Matters (Polarization)

This is the most exciting part. Because the magnetic wind is changing, the particles that get shot out are polarized.

• The Analogy: Imagine a factory making arrows. Usually, they make arrows pointing in all directions. But in this scenario, the factory only makes arrows pointing North or South, depending on how the wind is blowing.
• The Discovery: The paper calculates exactly how many particles are made and shows that they are highly "polarized." This means if you could look at the debris from a heavy-ion collision, you would see a strong preference for particles spinning one way versus the other. This is a smoking gun for the Chiral Magnetic Effect.

5. The "Plasma" Factor

The author also considers that the "soup" isn't empty space; it's a plasma (like the inside of a neon sign or the sun). This plasma has its own natural "vibration" (plasma frequency).

• The Analogy: Imagine the surfer is in a pool. If the pool water is still, the wake is one shape. If the pool water is already rippling (plasma oscillations), the wake changes.
• The Finding: These ripples in the soup change how many particles are produced. Interestingly, it suppresses the number of gluons (the "glue" particles) but increases the number of quark pairs.

Why Does This Matter?

The author concludes that this energy loss is huge. It's comparable to the energy loss from normal collisions.

• The Big Picture: If we smash atoms together in a lab (like at the Large Hadron Collider), and we see these specific patterns of "polarized" jets (sprays of particles), it proves that the Chiral Magnetic Effect is real and that the "magnetic wind" in the early universe was changing rapidly.

In a nutshell:
This paper is a mathematical map showing how particles lose energy when they race through a magnetic field that is speeding up and slowing down. It predicts that this process creates a very specific, organized spray of new particles that spin in a particular direction. Finding this pattern in real experiments would be like finding a fossil that proves the Earth's magnetic field flipped in the distant past—it would confirm some of the most exotic laws of physics governing our universe.

Technical Summary

1. Problem Statement

The paper addresses the energy loss mechanisms of ultra-relativistic particles (quarks and gluons) propagating through a chiral medium, specifically the Quark-Gluon Plasma (QGP) produced in relativistic heavy-ion collisions.

• Context: The Chiral Magnetic Effect (CME) induces an electric (or color) current parallel to an external magnetic field ($j = b_0 B$), where the transport coefficient $b_0$ (chiral magnetic conductivity) is proportional to the chiral imbalance ($\mu_5$).
• The Gap: Previous studies on CME-induced energy loss (such as Chiral Cherenkov radiation) largely assumed a constant chiral magnetic conductivity ($b_0$). However, in realistic heavy-ion collisions, the chiral imbalance is time-dependent. It is generated by parallel chromoelectric and chromomagnetic fields in the initial stage or by sphaleron transitions during evolution, and subsequently relaxes.
• Objective: To derive the particle production spectra and energy loss rates for processes involving time-varying $b_0(t)$ in both Abelian (QED) and non-Abelian (QCD) systems, and to apply these results to a specific model of chirality relaxation in heavy-ion collisions.

2. Methodology

The author employs an effective field theory approach combined with perturbative calculations in the ultra-relativistic limit.

• Effective Lagrangian:
  • QED: The Maxwell Lagrangian is augmented with a Chern-Simons term coupled to a pseudo-scalar field $\theta$: $L = -\frac{1}{4}F^2 + c_1 \theta F \tilde{F}$. The time derivative of $\theta$ relates to the chiral chemical potential ($\dot{\theta} = \mu_5$), yielding a time-dependent $b_0(t)$.
  • QCD: A similar non-Abelian Chern-Simons term is added to the Yang-Mills Lagrangian. This introduces an anomalous contribution to the triple-gluon vertex.
• Wave Function Derivation:
  • The equations of motion for gauge bosons (photons/gluons) are solved in the presence of the time-dependent $b_0(t)$.
  • In the ultra-relativistic limit ($k \gg b_0$), the wave functions acquire a time- and helicity-dependent phase factor: $e^{i\lambda \int^t b_0(t') dt'}$.
  • Collective plasma effects are incorporated via a plasma frequency $\omega_p$, which modifies the dispersion relation.
• Scattering Matrix Calculations:
  • The S-matrix elements are calculated for three primary splitting processes:
    1. $q \to q + \gamma$ (and $q \to q + g$)
    2. $\gamma \to q + \bar{q}$ (and $g \to q + \bar{q}$)
    3. $g \to g + g$
  • The calculation involves the Fourier transform of the time-dependent amplitude, leading to integrals involving the specific time profile of $b_0(t)$.
• Specific Model:
  • The time dependence is modeled as a relaxation function: $b_0(t) = A + B \tanh(t/\tau)$.
  • This models the decay of an initial P-odd domain (where $A = -B$) or the generation of chirality in a neutral medium.
  • Analytical solutions are derived using integral tables (involving Gamma functions) for the specific $\tanh$ profile.

3. Key Contributions

• General Formulas for Time-Dependent $b_0$: The paper derives general expressions for the differential probabilities of parton splitting for any time-dependent $b_0(t)$, moving beyond the constant approximation.
• Analytical Solutions for Relaxation Models: Explicit analytical formulas (Eqs. 39, 51, 69) are provided for the spectra when $b_0(t)$ follows a hyperbolic tangent relaxation profile.
• Inclusion of Plasma Oscillations: The study systematically incorporates the plasma frequency $\omega_p$ into the wave functions and phase space calculations, analyzing how collective medium effects modify the anomalous radiation.
• Non-Abelian Extension: The work extends previous Abelian results to QCD, deriving the specific contributions of the anomalous triple-gluon vertex to the $g \to gg$ splitting process.

4. Results

The paper presents numerical results for the spectra and energy loss based on parameters relevant to heavy-ion collisions ($E \approx 20$ GeV, $\tau \approx 5$ fm/c, initial $b_0 \approx 15$ MeV).

• Particle Spectra (Figs. 1-3):
  • Strong Polarization: The spectra exhibit a strong dependence on the helicity ($\lambda$) of the emitted particles. For example, in $q \to qg$, the emission rates for $\lambda=1$ and $\lambda=-1$ differ significantly.
  • Resonance Structure: The spectra show characteristic resonance structures sensitive to the time-variation of $b_0$.
  • Effect of $\omega_p$:
    • Gluon Splitting ($g \to gg$): The inclusion of a finite plasma frequency ($\omega_p$) suppresses the total gluon multiplicity. This is because $\omega_p$ acts as a mass cutoff, removing the infrared divergences (at $x \to 0, 1$) that dominate the perturbative splitting.
    • Pair Production ($g \to q\bar{q}$): Conversely, $\omega_p$ enhances the quark multiplicity. Here, $\omega_p$ acts effectively as a photon mass, expanding the available phase space for pair production.
• Energy Loss (Fig. 4):
  • The energy loss per unit length ($-dE/dz$) is computed for various channels.
  • The magnitude of energy loss due to the chiral magnetic effect is comparable to conventional radiative processes, making it phenomenologically significant.
  • The gluon production channels ($g \to gg$ and $g \to q\bar{q}$) dominate the energy loss, exhibiting logarithmic enhancements similar to standard QCD splitting but with polarization dependence.

5. Significance and Conclusions

• Jet Polarization: The most significant physical implication is the prediction of strong polarization of jets in the Quark-Gluon Plasma. Because the energy loss and particle production rates depend heavily on the helicity of the emitted gluons/quarks, the resulting jet will be highly polarized.
• Probe of Parity Violation: This strong polarization serves as a potential observable tool for studying parity-violating processes and topological charge fluctuations in hot nuclear matter.
• Phenomenological Relevance: By modeling the relaxation of the chiral imbalance, the paper bridges the gap between theoretical anomaly-induced transport and experimental observables in heavy-ion collisions (e.g., at RHIC or LHC).
• Theoretical Advancement: The work provides a comprehensive framework for calculating anomalous transport phenomena in dynamic, time-varying chiral media, correcting previous static approximations.

In summary, Tuchin demonstrates that the time-variation of the chiral magnetic conductivity significantly alters the particle spectra and energy loss mechanisms in QGP, leading to observable effects like strong jet polarization that could be used to probe the chiral anomaly in experimental settings.