Shear viscosity and electrical conductivity of rotating quark matter in Nambu--Jona-Lasinio Model

This study utilizes a two-flavor Nambu--Jona-Lasinio model within a kinetic theory framework to demonstrate that finite rotation modifies the transport properties of quark matter by reducing the chiral condensate, inducing anisotropy in shear viscosity and electrical conductivity, and generating significant non-dissipative Hall-like currents at zero net density.

The Gist

Imagine the universe right after the Big Bang. It wasn't a cold, empty void; it was a super-hot, super-dense soup of tiny particles called quarks and gluons. Physicists call this soup Quark-Gluon Plasma (QGP). It's like the "primordial smoothie" from which all matter in the universe was eventually made.

Today, scientists recreate this soup in massive particle accelerators (like the Large Hadron Collider) by smashing heavy atoms together at nearly the speed of light. When these atoms collide, they don't just get hot; they also start spinning wildly, like a figure skater pulling in their arms.

This paper is about understanding how this spinning, super-hot soup behaves. Specifically, the authors wanted to know two things:

1. How well does it conduct electricity? (Electrical Conductivity)
2. How "sticky" or fluid is it? (Shear Viscosity)

Here is a simple breakdown of their findings using everyday analogies.

1. The Spinning Soup and the "Magic Spin"

Usually, when we think of spinning, we think of a merry-go-round. But in the quantum world, spinning creates a weird force called the Coriolis force.

• The Analogy: Imagine you are walking on a spinning merry-go-round. If you try to walk in a straight line, the spinning floor pushes you sideways. You feel a "ghost force" that isn't there in a stationary room.
• In the Paper: The authors realized that this "ghost force" (Coriolis force) acts on the quarks in the plasma just like a magnetic field does. It messes with how the particles move, making the plasma behave differently depending on which direction it's spinning.

2. The "Melting Ice" Effect (Chiral Symmetry Breaking)

Inside this plasma, quarks usually have a "heavy" mass because they are stuck together in a specific way (like ice cubes in a drink). This is called the chiral condensate.

• The Discovery: The authors found that spinning makes the ice melt faster. As the rotation speed increases, the quarks become lighter and freer.
• Why it matters: When the quarks get lighter, the soup becomes a better conductor of electricity and flows more easily. It's like adding more water to your smoothie; it becomes less thick and conducts electricity better.

3. The "One-Way Street" vs. The "Roundabout"

This is the most surprising part of the paper.

• The Magnetic Field Scenario: If you put a non-spinning soup in a strong magnetic field, the positive particles go one way, and the negative particles go the other. They cancel each other out, so there is no net "Hall effect" (a sideways flow of electricity).
• The Rotation Scenario: The authors found that rotation treats all particles the same, regardless of whether they are positive or negative.
• The Analogy: Imagine a roundabout (traffic circle). Whether you are driving a red car or a blue car, the spinning road pushes you both in the same direction.
• The Result: Because the spin pushes everyone the same way, a new type of electricity flow appears that usually doesn't happen. It's like a "Hall effect" caused by spinning, which creates a unique, non-dissipative (energy-saving) current.

4. The "Valley" Shape

The authors calculated how these properties change as the soup cools down (which happens naturally after the collision).

• The Shape: They found that the ability to conduct electricity and the "stickiness" of the fluid follow a valley shape.
  • At very high temperatures (just after the crash), the soup is very fluid.
  • As it cools down to a "middle" temperature, it gets a bit "stickier" and less conductive (the bottom of the valley).
  • As it cools further, it changes again.
• The Spin's Role: The spinning makes the "valley" slightly deeper and creates a difference between flowing with the spin versus flowing across the spin. It's like running on a treadmill that is slightly tilted; it's easier to run in one direction than the other.

5. Why Should We Care?

You might ask, "Who cares about spinning quark soup?"

• Understanding the Universe: This helps us understand the very first moments of the Big Bang.
• Neutron Stars: Some neutron stars (dead stars that are incredibly dense) spin thousands of times a second. This soup might exist inside them, and understanding its "stickiness" and conductivity helps us predict how these stars behave and what signals they send out.
• New Physics: Finding that rotation creates a "Hall effect" (usually reserved for magnets) opens up new ways to think about how matter interacts with forces.

Summary

In short, this paper is a recipe for understanding what happens when you take the hottest, densest matter in the universe and spin it.

They found that spinning:

1. Melts the internal "glue" holding quarks together, making them lighter.
2. Creates a sideways flow of electricity that doesn't happen with magnets.
3. Makes the fluid anisotropic, meaning it flows differently depending on the direction relative to the spin.

It's like discovering that if you spin a bowl of soup fast enough, it doesn't just get hot; it starts to act like a completely different kind of liquid with its own unique rules!

Technical Summary

1. Problem Statement

The primary objective of this research is to investigate the transport properties of quark matter under the influence of rotation (vorticity).

• Context: In peripheral heavy-ion collisions (HIC) at facilities like RHIC and LHC, the collision of ultra-relativistic nuclei generates not only extreme temperatures and densities (creating Quark-Gluon Plasma, QGP) but also immense orbital angular momentum. This angular momentum is transferred to the medium, creating a rotating system with significant vorticity.
• Gap in Knowledge: While the effects of strong magnetic fields on the Equation of State (EoS) and transport coefficients (like electrical conductivity $\sigma$ and shear viscosity $\eta$) are well-studied, the impact of rotation on these properties remains less explored.
• Specific Challenge: Rotation introduces anisotropy into the system. Unlike magnetic fields, which distinguish between positive and negative charges via the Lorentz force, rotation affects all particles via the Coriolis force regardless of charge. The authors aim to quantify how rotation modifies the EoS and induces anisotropic transport coefficients (splitting isotropic $\sigma$ and $\eta$ into parallel, perpendicular, and Hall-like components).

2. Methodology

The authors employ a multi-step theoretical framework combining effective field theory and kinetic theory:

A. Thermodynamic Framework: NJL Model in a Rotating Frame

• Model: They utilize the two-flavor Nambu-Jona-Lasinio (NJL) model, an effective field theory that captures spontaneous chiral symmetry breaking and dynamical mass generation.
• Rotating Lagrangian: The standard NJL Lagrangian is modified to include spinorial connections ($\Gamma_\mu$) derived from the metric of a rotating frame. The metric connects fixed inertial coordinates with co-rotating coordinates, introducing the angular velocity $\vec{\Omega}$.
• Grand Potential & Gap Equation: The grand potential energy ($\tilde{\Omega}$) is derived. By minimizing this potential with respect to the constituent quark mass ($M$), they obtain a gap equation.
  • Key Finding: Rotation reduces the chiral condensate, thereby lowering the constituent quark mass $M$. This implies that rotation can lower the chiral phase transition temperature ($T_c$), although at phenomenological HIC vorticities ($\Omega \approx 0.01$ GeV), this shift is small.

B. Transport Framework: Kinetic Theory & Boltzmann Equation

• Boltzmann Transport Equation (BTE): The transport coefficients are calculated using the quasi-particle BTE within the Relaxation Time Approximation (RTA).
• Force Terms: The BTE is modified to include effective forces arising from:
  1. Rotating Geometry: The Coriolis force (and centrifugal force, though the latter is often negligible in the transport term context here).
  2. Space-time dependent Mass: Gradients in the constituent mass $M(x)$.
  3. External Fields: An electric field $\vec{E}$ is included to calculate conductivity.
• Anisotropy Handling: The Coriolis force breaks rotational symmetry. The authors propose an ansatz for the deviation from equilibrium distribution ($\delta f$) involving independent traceless tensors and vectors. This allows the derivation of distinct components for transport coefficients.

C. Numerical Calibration

• Parameters: The NJL parameters (current quark mass, cutoff $\Lambda$, coupling $G_S$) are fixed to reproduce pion mass and decay constants in the vacuum.
• Relaxation Time ($\tau_c$): Since $\tau_c$ is not directly calculable from first principles in this context, the authors calibrate it:
  • For $T < 170$ MeV (hadronic phase): $\tau_c$ is calculated using a hard-sphere scattering model with a tunable scattering length $a$.
  • For $T > 170$ MeV (QGP phase): $\tau_c$ is treated as a tunable constant.
  • The calibration ensures the calculated isotropic $\sigma$ and $\eta/s$ (at $\Omega=0$) match existing theoretical bands from literature (LQCD, DQPM, etc.).
• Phenomenological $\Omega(T)$: To simulate realistic HIC conditions, they use a time-dependent angular velocity profile $\Omega(t)$ derived from AMPT model simulations, linked to temperature via Bjorken scaling ($T \propto t^{-1/3}$).

3. Key Contributions

1. First NJL Study of Rotating Transport: This is the first work to calculate electrical conductivity and shear viscosity of rotating quark matter using the NJL model with a fixed coupling constant.
2. Anisotropic Decomposition: The authors successfully decompose the transport coefficients into five independent shear viscosity components ($\eta_0, \eta_1, \eta_2, \eta_3, \eta_4$) and three conductivity components ($\sigma_\perp, \sigma_\parallel, \sigma_\times$).
3. Hall-like Transport in Rotation: A major theoretical contribution is the identification of Hall-like transport phenomena in rotating systems at zero net quark density.
   • Contrast with Magnetic Fields: In magnetic fields, Hall currents from quarks and anti-quarks cancel out at zero chemical potential. In rotation, the Coriolis force acts identically on both, leading to a non-zero, significant Hall component ($\sigma_\times$ and $\eta_\times$).
4. Mechanism of Anisotropy: The paper clarifies that anisotropy arises from two sources:
   • Kinetic: The modification of the effective relaxation time due to the interplay between thermal relaxation ($\tau_c$) and rotational relaxation ($\tau_\Omega = 1/2\Omega$).
   • Thermodynamic: The modification of the EoS (reduced quark mass) due to rotation, which generally enhances transport coefficients.

4. Key Results

• Constituent Mass: The constituent quark mass $M$ decreases with increasing angular velocity $\Omega$ and temperature $T$, indicating chiral restoration is facilitated by rotation.
• Transport Coefficients vs. $\Omega$:
  • As $\Omega$ increases, the isotropic components split. The parallel ($\parallel$) and perpendicular ($\perp$) components diverge, and Hall components emerge.
  • The Hall components peak when the rotational relaxation time equals the thermal relaxation time ($\tau_\Omega \approx \tau_c$).
  • At very high $\Omega$, the system approaches the massless limit, and NJL results converge with massless gas estimations.
• Temperature Dependence (Realistic Scenario):
  • Using a phenomenological $\Omega(T)$, the authors observe that all transport components follow a "valley-shaped" pattern with temperature (decreasing then increasing), consistent with the behavior of $\eta/s$ in QGP.
  • Anisotropy Magnitude: The anisotropy (difference between parallel and perpendicular components) is more pronounced at lower temperatures ($T < 170$ MeV) and remains significant as the system approaches kinetic freeze-out ($T \approx 100$ MeV).
  • Hall Effect: The Hall components remain non-zero and significant across the temperature range, unlike in magnetic field scenarios where they vanish at $\mu_B=0$.
• Suppression: While rotation induces anisotropy, the magnitude of the anisotropic components is generally suppressed compared to the isotropic component found in the absence of rotation.

5. Significance and Implications

• Phenomenology of HIC: The results suggest that rotation-induced anisotropy could significantly affect anisotropic flow (elliptic and triangular flow) and particle spectra in heavy-ion collisions.
• Electromagnetic Observables: Since photon and dilepton production rates are linked to electrical conductivity, the anisotropic conductivity implies that dilepton spectra in HIC may exhibit anisotropy due to rotation, offering a potential experimental signature.
• Theoretical Insight: The discovery of robust Hall-like transport in rotating neutral matter challenges the intuition derived from magnetic field studies and highlights the unique nature of vorticity in QCD matter.
• Future Directions: The paper sets the stage for detailed studies on how this anisotropy influences the evolution of the fireball and the final state observables in experiments like STAR and ALICE.

In summary, this work provides a rigorous theoretical foundation for understanding how the rotation of the QGP modifies its fundamental transport properties, predicting observable anisotropies and a unique Hall effect that distinguishes rotational dynamics from magnetic dynamics.