Conformal anomaly transport induced by dark photon

This paper investigates how gravitational inhomogeneity affects transport phenomena in a system of massless Dirac fermions coupled to both Maxwell and dark photon fields, revealing that the resulting currents are proportional to beta functions and predicting specific corrections to scale conductivities in both the visible and dark sectors.

The Gist

Imagine the universe as a giant, invisible ocean. Usually, we think of this ocean as perfectly flat and calm (like a sheet of glass). But in reality, it's more like a trampoline that can stretch, shrink, and ripple.

This paper is about what happens when you drop a pebble into that trampoline (a gravitational wave) or when the trampoline itself starts expanding (like the Big Bang). The authors, Marek Rogatko and Karol Wysokiński, are asking a very specific question: If the fabric of space changes shape, does it make invisible "electric" currents flow in places we can't see?

Here is the breakdown of their discovery using simple analogies:

1. The Two Invisible Sectors: The "Visible" and the "Dark"

Imagine the universe has two layers of electricity:

• The Visible Layer: This is the normal electricity we know (light, magnets, your phone charger).
• The Dark Layer: This is a secret, invisible layer of electricity called "Dark Photons." We can't see it, but the authors suspect it exists.

The paper suggests these two layers are slightly "stuck" together. Think of them like two sheets of paper taped together. If you pull on one, the other moves a little bit too. This "tape" is called kinetic mixing.

2. The Problem: When Space Gets "Squished"

In physics, there's a rule called Symmetry. It's like saying, "If I stretch a rubber band, the laws of physics shouldn't change." Usually, this holds true.

However, at the quantum level (the world of tiny particles), this symmetry breaks. This is called an Anomaly.

• The Analogy: Imagine a perfect circle drawn on a balloon. If you blow up the balloon (stretch the space), the circle gets bigger. But if you look at the atoms making up the ink of the circle, they don't stretch the same way. The "ink" (quantum fields) reacts differently than the "balloon" (space). This mismatch creates a tiny, unexpected force.

3. The Discovery: The "Scale Current"

The authors calculated what happens when this "mismatch" occurs in the presence of both the Visible and Dark layers.

They found that when space stretches or shrinks (due to gravity or expansion), it doesn't just sit there. It creates a current—a flow of energy—like water flowing down a hill.

• The Twist: Because the Visible and Dark layers are "stuck" together, the stretching of space creates a current in the Dark layer that is influenced by the Visible layer, and vice versa.

They call this the Scale Anomaly Current.

• Simple Translation: If you stretch the universe, you accidentally generate a "dark electric current" that flows alongside normal electricity.

4. Two Ways This Happens (The Scenarios)

The paper looks at two specific ways space can change:

A. The Gravitational Wave (The Ripple)
Imagine a massive black hole colliding far away. It sends a ripple through space (a gravitational wave).

• The Effect: As this ripple passes through our "invisible ocean," it stretches and squeezes space back and forth.
• The Result: This squeezing creates a tiny, oscillating current in the dark sector. It's like a tiny generator being shaken by the wave. The authors calculated exactly how strong this current would be based on how "sticky" the two layers are.

B. The Expanding Universe (The Balloon)
Imagine the universe is a balloon being blown up.

• The Effect: Space is constantly stretching everywhere.
• The Result: This constant stretching creates a steady "dark current." The paper found that the strength of this current depends on a specific number (the "mixing parameter") that tells us how strongly the visible and dark worlds talk to each other.

5. Why Should We Care? (The "So What?")

You might ask, "We can't see dark photons, so why does this matter?"

The authors suggest a way to find them: Condensed Matter Physics.

• The Analogy: Think of a Dirac/Weyl Semimetal. These are special crystals (like graphene) where electrons behave as if they are massless and moving at the speed of light. They are like a "laboratory universe" on a tiny chip.
• The Experiment: If you heat up one side of this crystal (creating a temperature gradient) and put it in a magnetic field, the "scale anomaly" should create a current flowing sideways.
• The Prediction: If dark photons exist and are "stuck" to our world, they will change the size of this sideways current. By measuring this current in a lab, we might finally catch a glimpse of the dark sector without needing a giant telescope.

Summary

The paper is a theoretical recipe. It says:

1. Space isn't just empty; it's a fabric that can stretch.
2. When it stretches, it breaks a quantum rule, creating a tiny electric current.
3. Because our world is secretly connected to a "Dark World," this current flows in both.
4. We might be able to detect this invisible current by looking at special crystals in a lab, proving that the "Dark Sector" is real.

It's a beautiful example of how the biggest things in the universe (gravity, expansion) might be connected to the tiniest things (quantum particles) through a hidden handshake.

Technical Summary

1. Problem Statement

The paper investigates the influence of gravitational field inhomogeneities on quantum transport phenomena in a system comprising massless Dirac fermions coupled to both the visible electromagnetic sector (Maxwell field) and a hidden "dark" sector (Dark Photon model).

Specifically, the authors address the conformal (scale) anomaly. In classical field theory, massless systems are invariant under scale transformations (dilatations). However, at the quantum level, renormalization breaks this symmetry, leading to a non-vanishing trace of the energy-momentum tensor ($\langle T^\mu_\mu \rangle \neq 0$). The central question is how this scale anomaly, in the presence of a dark photon (coupled via kinetic mixing), induces anomalous currents when the spacetime metric undergoes a slight conformal deformation.

2. Methodology

A. Theoretical Framework: Dark Photon Model
The authors utilize the massless dark photon model where a hidden $U(1)$ gauge field ($B_\mu$) couples to the visible Maxwell field ($A_\mu$) via a kinetic mixing term $\alpha F_{\mu\nu}B^{\mu\nu}$.

• Field Redefinition: To simplify the action and remove the kinetic mixing term, they perform a linear transformation of the gauge fields ($\tilde{A}_\mu, \tilde{B}_\mu$) and the associated charges ($\tilde{e}_A, \tilde{e}_B$). This results in two decoupled gauge fields with effective charges that are mixtures of the original visible and dark charges, dependent on the mixing parameter $\alpha$.
• Matter Coupling: Massless Dirac fermions are coupled to these redefined gauge fields.

B. Spacetime Geometry and Approximation

• Conformal Transformation: Instead of solving the full Quantum Field Theory (QFT) in a curved background, the authors assume the spacetime metric is a slight Weyl-type conformal deformation of the Minkowski metric: $g_{\mu\nu} = e^{2\Omega(x)}\eta_{\mu\nu}$, where $|\Omega(x)| \ll 1$.
• Justification: This approximation allows the use of standard QFT methods in flat space, treating the conformal factor $\Omega(x)$ as a perturbation, rather than employing the complex machinery of QFT on curved manifolds.

C. Derivation of Anomalous Currents

• Scale Invariance Breaking: The authors calculate the expectation value of the trace of the energy-momentum tensor, $\langle T^\mu_\mu \rangle$, which is non-zero due to quantum fluctuations. This trace is proportional to the $\beta$-functions of the effective gauge couplings ($\tilde{e}_A, \tilde{e}_B$).
• Kubo Formalism: Using linear response theory (Kubo formula), they relate the induced current $\langle j_\mu \rangle$ to the conformal factor $\Omega(x)$ and the background gauge fields.
• Three-Point Functions: The calculation involves evaluating three-point correlation functions involving the current and the trace of the energy-momentum tensor.

3. Key Contributions

1. Generalization to Two Sectors: The work extends previous studies of conformal anomaly transport (which focused on single gauge sectors) to a system with two interacting $U(1)$ gauge sectors (visible and dark).
2. Derivation of Scale Currents: They derive explicit expressions for the anomalous currents induced by the scale anomaly in both the visible and dark sectors. These currents are shown to be proportional to the $\beta$-functions of the effective charges.
3. Identification of New Transport Effects: The authors identify two distinct phenomena arising from the spatial and temporal gradients of the conformal factor:
   • SEEDM (Scale Electric Effect with Dark Matter): A current induced by the time derivative of the conformal factor ($\partial_0 \Omega$) acting on electric fields.
   • SMEDM (Scale Magnetic Effect with Dark Matter): A current induced by the spatial gradient of the conformal factor ($\partial_j \Omega$) acting on magnetic fields.
4. Quantification of Dark Sector Influence: They demonstrate how the kinetic mixing parameter $\alpha$ modifies the conductivities in both sectors, predicting specific dependencies on $\alpha$ (linear for the dark sector, quadratic for the visible sector corrections).

4. Results

General Current Equations
The total anomalous current $\langle j_\mu \rangle_{scale}$ is a sum of contributions from the visible ($\tilde{A}$) and dark ($\tilde{B}$) sectors. The spatial components ($j_m$) and time component ($j_0$) are given by:
$$\langle j_m \rangle_{scale} = \sigma_F E^{(F)}_m + \sigma_B E^{(B)}_m + \epsilon_{mlk} K^{(F)}_l B^{(F)}_k + \epsilon_{mlk} K^{(B)}_l B^{(B)}_k$$
$$\langle j_0 \rangle_{scale} = K^{(F)}_m E^{(F)}_m + K^{(B)}_m E^{(B)}_m$$
Where:

• $E^{(F/B)}$ and $B^{(F/B)}$ are the electric and magnetic fields of the visible and dark sectors.
• $\sigma$ represents scale anomalous Ohm conductivities.
• $K$ represents vector parameters acting as magneto-conductivities.
• These coefficients depend on the derivatives of the conformal factor ($\partial_0 \Omega, \partial_j \Omega$) and the $\beta$-functions.

Specific Scenarios

• Weak Gravitational Waves:
  For a gravitational wave passing through the system, the conformal factor $\Omega$ is related to the Riemann tensor components.

  • The time derivative $\partial_0 \Omega$ is proportional to the third time derivative of the gravitational wave amplitude ($\dddot{h}_{TT}$).
  • The spatial gradient $\partial_j \Omega$ is proportional to the second time derivative ($\ddot{h}_{TT}$).
  • This implies that passing gravitational waves can induce oscillating anomalous currents in both sectors.

• Expanding Universe (Cosmological Scale):
  In a homogeneous isotropic expanding universe with scale factor $a(t)$, the conformal factor relates to the Hubble parameter $H(t) = \dot{a}/a$.

  • Assuming the dark sector is charge-neutral ($e_d = 0$), the conductivities are derived as:
    $$\sigma_F \propto -\frac{\sqrt{2}e^2}{12\pi^2} \left(1 + \frac{3}{8}\alpha^2\right) H(t)$$
    $$\sigma_B \propto -\frac{\sqrt{2}e^2}{16\pi^2} \alpha H(t)$$
  • Key Finding: The visible sector conductivity receives a correction proportional to $\alpha^2$, while the dark sector conductivity is directly proportional to $\alpha$.

5. Significance and Implications

• Experimental Prospects: The paper suggests that these scale-anomaly-induced currents could potentially be detected in condensed matter systems, specifically Dirac and Weyl semimetals. These materials host massless Dirac fermions and are known to exhibit anomalous transport effects. The authors propose looking for thermoelectric effects (like the Nernst effect) perpendicular to temperature gradients and magnetic fields, analogous to previous proposals but now including dark sector contributions.
• Dark Matter Detection: The results provide a theoretical mechanism for how dark photons could influence observable electromagnetic currents in the presence of gravitational inhomogeneities (like gravitational waves or cosmic expansion).
• Theoretical Bridge: The work bridges high-energy physics (dark sector models, anomalies) and condensed matter physics (transport in semimetals), offering a new avenue to probe physics beyond the Standard Model using tabletop experiments or astrophysical observations.

In conclusion, the paper establishes that gravitational inhomogeneities, modeled as conformal deformations, induce anomalous transport currents in a dark photon model. These currents are uniquely characterized by their dependence on the $\beta$-functions and the kinetic mixing parameter $\alpha$, offering a potential signature for dark sector physics.