Shear Viscosity and Electrical Conductivity of Rotating Nuclear Medium in Hadron Resonance Gas and Nambu-Jona Lasinio Models

This paper investigates how rotation influences the anisotropic shear viscosity and electrical conductivity of strongly interacting matter by employing kinetic theory within Hadron Resonance Gas and Nambu-Jona-Lasinio models, revealing that rotation suppresses chiral condensates, generates a sizable Hall-like conductivity, and reduces transport coefficients relative to isotropic cases.

The Gist

Imagine a massive, high-speed dance party where particles are the dancers. This is what happens inside a heavy-ion collision (like smashing two gold or lead atoms together at nearly the speed of light). Usually, physicists study how these particles flow, how they stick together, and how they conduct electricity.

But there's a twist in this story: The whole dance floor is spinning.

In a non-central collision (where the atoms don't hit dead-on), the debris created has a huge amount of spin or rotation, much like a figure skater pulling in their arms to spin faster. This paper asks a simple but profound question: How does this spinning affect the "stickiness" (viscosity) and the "electric flow" (conductivity) of the particle soup?

Here is the breakdown of their findings using everyday analogies.

1. The Setup: The Spinning Ice Rink

Think of the hot particle soup created in the collision as a giant, spinning ice rink.

• The Dancers: Quarks and gluons (the building blocks of matter) or protons and neutrons (hadrons).
• The Spin: The entire rink is rotating.
• The Force: Because the rink is spinning, the dancers feel a "fictitious" force called the Coriolis force. You know this if you've ever tried to walk in a straight line on a spinning merry-go-round; you get pushed sideways.

The authors wanted to know: If the whole system is spinning, does the "fluid" behave differently than if it were just sitting still?

2. The Two Main Ingredients: Viscosity and Conductivity

To understand the results, we need to define two key properties:

• Shear Viscosity (The "Sticky Honey" Factor):
  Imagine trying to slide a deck of cards. If the cards are dry, they slide easily (low viscosity). If they are covered in honey, they stick together and resist sliding (high viscosity). In physics, this measures how much the fluid resists flowing.

  • The Paper's Finding: When the rink spins, the fluid becomes less sticky in some directions and more sticky in others. It's like the honey suddenly decides to flow easier if you push it parallel to the spin, but harder if you push it across the spin.

• Electrical Conductivity (The "Traffic Flow" Factor):
  Imagine a highway. Conductivity is how easily cars (electric charges) can drive down it.

  • The Paper's Finding: The spinning rink creates a new kind of traffic jam. Not only does the spin change how easily cars move forward or sideways, but it also creates a sideways current (called the Hall effect) that wouldn't exist if the rink were still.

3. The Two Models: The "Toy Box" vs. The "Realistic Simulator"

The authors used two different ways to calculate this, like using two different video game engines:

• Model A: The HRG (Hadron Resonance Gas) & QGP (Quark-Gluon Plasma):

  • The Analogy: This is like a Toy Box. Below a certain temperature, the particles are distinct toys (protons, pions, etc.) bouncing around. Above that temperature, they melt into a "soup" of massless energy (quarks and gluons).
  • The Result: They calculated that the spinning makes the fluid slightly less viscous (less sticky) and creates that sideways electrical flow.

• Model B: The NJL (Nambu-Jona-Lasinio) Model:

  • The Analogy: This is a Realistic Simulator. It treats the particles not just as toys, but as complex entities that gain "mass" (heaviness) from their interactions, similar to how a person gains weight by eating.
  • The Result: They found that spinning actually makes the particles "lighter" (reduces their mass) and lowers the temperature at which they melt into the soup. This changes how the fluid flows.

4. The Big Surprises

A. The "Valley" Shape

Usually, if you plot how "sticky" this fluid is against temperature, you get a valley shape. It's very sticky at low temps, gets super slippery (low viscosity) right at the phase transition (when it turns from solid-like to liquid-like), and then gets sticky again.

• The Spin Effect: When they added the spin, the valley didn't disappear, but the whole curve dropped lower. The spinning fluid is even more slippery (a better fluid) than the non-spinning one.

B. The "Hall" Effect (The Sideways Push)

This is the coolest part.

• In a Magnetic Field: If you put a magnetic field on a fluid, positive charges go one way and negative charges go the other. They cancel each other out, so the "sideways" flow is often zero or very small.
• In a Spinning Field: The Coriolis force (from the spin) pushes everyone sideways, regardless of whether they are positive or negative. It's like a wind blowing everyone to the left.
• The Result: The authors found a huge sideways electrical current (Hall conductivity) purely because of the spin. This is a unique fingerprint of rotation that doesn't happen with magnetic fields.

C. The "Cooling" Reality Check

In a real collision, the system expands and cools down very fast. The spin also slows down as the system gets bigger (conservation of angular momentum).

• The authors mapped this out: As the system cools, the spin slows down, but the fluid's properties change in a specific way. They found that the "anisotropy" (the difference between flowing with the spin vs. against it) is most noticeable when the system is cooling down toward the end of the collision.

5. Why Does This Matter?

Think of the universe as a giant, spinning fluid.

• The "Perfect" Fluid: Scientists have long suspected that the Quark-Gluon Plasma is the "perfect fluid" (the least sticky substance in the universe). This paper suggests that rotation makes it even more perfect.
• Detecting the Spin: Since we can't see the spin directly, we need to look for its fingerprints. The paper suggests that if we look at the sideways flow of charged particles or the emission of light (photons) from these collisions, we might see the "Hall effect" signature. This would be the smoking gun proving that the universe was spinning during the Big Bang's aftermath.

Summary in One Sentence

Just as spinning a bucket of water changes how the water sloshes, the rotation of the subatomic soup created in particle collisions makes the fluid flow more smoothly and generates a unique sideways electrical current that acts as a signature of the universe's spin.

Technical Summary

Here is a detailed technical summary of the paper "Shear Viscosity and Electrical Conductivity of Rotating Nuclear Medium in Hadron Resonance Gas and Nambu-Jona Lasinio Models."

1. Problem Statement and Motivation

The primary objective of this research is to investigate the impact of rotation (vorticity) on the transport properties of strongly interacting matter created in off-center heavy-ion collisions (HICs).

• Context: Recent experimental observations (STAR, ALICE) have confirmed significant global spin polarization of hyperons and spin alignment of vector mesons in HICs, indicating the presence of strong fluid vorticity ($\Omega \sim 10^{21} s^{-1}$).
• Gap: While transport coefficients (shear viscosity $\eta$ and electrical conductivity $\sigma$) are well-studied in static media or under magnetic fields, their behavior in a rotating medium remains largely unexplored.
• Specific Challenge: Rotation breaks the spatial isotropy of the medium, similar to how a magnetic field does. This necessitates calculating anisotropic transport coefficients:
  • Shear Viscosity: Parallel ($\eta_{\parallel}$), Perpendicular ($\eta_{\perp}$), and Hall ($\eta_{\times}$).
  • Electrical Conductivity: Parallel ($\sigma_{\parallel}$), Perpendicular ($\sigma_{\perp}$), and Hall ($\sigma_{\times}$).
• Key Distinction: Unlike magnetic fields where the Lorentz force acts oppositely on positive and negative charges (often canceling Hall effects at zero net charge), the Coriolis force in a rotating frame acts on all particles regardless of charge, potentially generating a sizable Hall-like conductivity even at zero net baryon density.

2. Methodology

The authors employ Relativistic Kinetic Theory within the Relaxation Time Approximation (RTA) to solve the Boltzmann Transport Equation (BTE) in a rotating frame.

A. Theoretical Framework

• Rotating Metric: The BTE is formulated using the metric of a co-rotating frame ($g_{\mu\nu}$) derived from the transformation between inertial and rotating coordinates. This introduces connection coefficients ($\Gamma^{\alpha}_{\mu\lambda}$) into the BTE, which manifest as pseudo-forces (specifically the Coriolis force).
• Linearization: The distribution function is split into equilibrium ($f^0$) and a small perturbation ($\delta f$). The analysis retains terms linear in angular velocity $\Omega$, effectively ignoring centrifugal effects while focusing on the Coriolis force.
• Anisotropy: The perturbation $\delta f$ is solved to extract components proportional to velocity gradients (for viscosity) and electric fields (for conductivity), leading to distinct tensor components.

B. Model Implementations

The study utilizes two complementary frameworks to cover the full temperature range of HIC evolution:

1. Combined QGP-HRG Framework:
   • Hadronic Phase ($T < T_c$): Modeled using the Hadron Resonance Gas (HRG) model, treating hadrons and resonances as a non-interacting gas with hard-sphere scattering interactions.
   • Quark-Gluon Plasma Phase ($T > T_c$): Modeled as a massless gas of quarks and gluons with constant relaxation time.
2. Nambu-Jona-Lasinio (NJL) Model:
   • A 2-flavor effective field theory incorporating spontaneous chiral symmetry breaking.
   • Rotation is introduced via spinorial connections in the Lagrangian.
   • The constituent quark mass ($M$) is dynamically generated and calculated as a function of temperature ($T$) and angular velocity ($\Omega$) by solving the gap equation.

C. Relaxation Time Calibration

To ensure realistic estimates, the relaxation time ($\tau_c$) is calibrated against existing microscopic calculations and lattice QCD data:

• QGP/NJL High-T: Constant $\tau_c$.
• Hadronic/NJL Low-T: Calculated via hard-sphere scattering ($\tau_c \propto 1/n\sigma v$), with scattering lengths tuned to reproduce the "valley-like" temperature dependence of $\eta/s$ observed in other models.

3. Key Contributions

1. First Systematic Calculation: This work provides the first systematic characterization of the temperature dependence of anisotropic shear viscosity and electrical conductivity in a rotating nuclear medium using both HRG and NJL models.
2. Hall Conductivity in Neutral Matter: The authors demonstrate that rotation induces a non-dissipative Hall-like conductivity ($\sigma_{\times}$) even at zero net baryon density. This contrasts with magnetic fields where particle-antiparticle contributions cancel out at $\mu_B=0$.
3. Chiral Symmetry and Rotation: In the NJL model, the study shows that rotation suppresses the chiral condensate and slightly reduces the constituent quark mass, effectively lowering the pseudo-critical temperature for chiral restoration, although this effect is negligible at phenomenologically relevant HIC angular velocities.
4. Temperature-Dependent Vorticity: The study moves beyond constant $\Omega$ assumptions by mapping $\Omega$ to temperature ($T$) using hydrodynamic cooling laws (Bjorken expansion), providing a more realistic evolution of transport coefficients.

4. Key Results

• Anisotropy: Rotation induces significant anisotropy. The parallel components ($\eta_{\parallel}, \sigma_{\parallel}$) and perpendicular components ($\eta_{\perp}, \sigma_{\perp}$) differ substantially from the isotropic case.
  • Magnitude: The anisotropic components are generally suppressed compared to the isotropic values.
  • Reduction: For temperature-dependent $\Omega(T)$, the reduction in $\eta_{\parallel}$ is 20–70% (HRG) and 10–50% (NJL) in the hadronic phase. The reduction in $\eta_{\perp}$ is even more pronounced (45–90%).
• Valley-Like Structure: When using a realistic temperature-dependent angular velocity $\Omega(T)$, the ratios $\eta/s$ and $\sigma/T$ retain a valley-like temperature dependence (minimum near $T_c$), similar to the isotropic case but with reduced magnitudes.
  • Note: If a constant $\Omega$ is assumed, this valley structure is distorted.
• Hall Components:
  • Viscosity: A non-zero Hall viscosity ($\eta_{\times}$) is observed in both QGP and HRG phases.
  • Conductivity: A significant Hall conductivity ($\sigma_{\times}$) appears. Its magnitude peaks when the rotational relaxation time ($\tau_{\Omega} = 1/2\Omega$) equals the thermal relaxation time ($\tau_c$).
• Phase Transition Effects: In the NJL model, the transition from the hadronic to the partonic phase is smooth. At high $T$, constituent masses vanish (chiral restoration), and NJL results converge with the massless QGP-HRG results.

5. Significance and Future Implications

• Phenomenological Probes: The anisotropy in transport coefficients suggests that peripheral heavy-ion collisions (high vorticity) will exhibit different fluid dynamic evolution compared to central collisions. This could manifest in azimuthal anisotropy in particle spectra.
• Dilepton/Photon Spectra: Since photon and dilepton emission rates are linked to electrical conductivity, the anisotropic conductivity induced by rotation could lead to observable anisotropies in dilepton spectra.
• Differentiation from Magnetic Fields: The presence of a Hall component at zero net charge serves as a unique signature to distinguish rotational effects from magnetic field effects in experimental data.
• Future Directions: The authors suggest extending this work to include radial flow (simultaneous rotation and expansion), species-dependent hard-core radii, and more rigorous Green-Kubo calculations to refine these estimates.

In conclusion, this paper establishes that rotation is a critical factor in the transport dynamics of the quark-gluon plasma and hadronic matter, fundamentally altering the viscosity and conductivity tensors and offering new observables for understanding the vortical nature of the QCD medium.