Worldline effective field theory of inspiralling black hole binaries in presence of dark photon and axionic dark matter

This paper employs Worldline Effective Field Theory to compute conservative and radiative corrections to the dynamics of inspiralling non-spinning black hole binaries in a dark matter environment containing axion-like particles and dark photons, determining the specific post-Newtonian orders at which axion-electromagnetic coupling and kinetic mixing influence orbital evolution and gravitational or scalar radiation.

The Gist

Imagine the universe as a giant, cosmic dance floor. Usually, when we watch two massive dancers (like black holes) spin around each other, we only care about their gravity. They pull on each other, spiral inward, and eventually crash, sending out ripples in the fabric of space-time called gravitational waves. This is what detectors like LIGO listen for.

But what if the dance floor isn't empty? What if the air around them is filled with invisible, ghostly fog?

This paper is about investigating what happens when those two black holes dance through a specific kind of "fog" made of Dark Matter. Specifically, the authors are looking at a universe where this dark matter isn't just one thing, but a mix of two mysterious ingredients:

1. Axions (The Whispering Ghosts): Ultra-light particles that act like a scalar field. They are famous for solving a puzzle in particle physics, but here they act like a subtle, invisible wind.
2. Dark Photons (The Invisible Twins): Particles that are like regular light (photons) but have a secret connection to a "dark" version of electricity. They are heavy and sluggish compared to normal light.

The Tool: The "Worldline" Lens

To study this, the authors use a technique called Worldline Effective Field Theory (WEFT).

• The Analogy: Imagine trying to understand the motion of a boat in a stormy ocean. You don't need to track every single water molecule. Instead, you treat the boat as a single point on a line (a "worldline") and simplify the ocean into two parts: the waves (radiation that flies away) and the currents (the immediate pull of the water).
• The Method: The authors "integrate out" the heavy, immediate currents to see how they change the boat's path (conservative dynamics) and how much energy the boat loses by creating waves (radiative dynamics).

The Discovery: How the Fog Changes the Dance

The paper calculates how these invisible particles change the black holes' dance in two main ways:

1. Changing the Orbit (Conservative Dynamics)

Normally, two black holes spiral in at a predictable speed. But with this dark matter fog, the rules change slightly.

• The "Mixing" Effect: The Dark Photons can "mix" with regular light. The authors found that this mixing creates a tiny extra tug on the black holes, changing their orbit. It's like if the dancers were holding a rubber band that occasionally snapped and pulled them slightly off-course.
• The "Whisper" Effect: The Axions (ghosts) interact with light in a very strange, "twisted" way (called the Chern-Simons term). The authors found that this interaction only starts to matter when the dance gets very fast and complex (at a specific "2.5 Post-Newtonian" order). It's a subtle effect, like a whisper that only becomes audible when the music gets loud.

2. Changing the Song (Radiative Dynamics)

As the black holes spiral, they lose energy by emitting waves. Usually, they only emit gravitational waves. But in this foggy universe, they start emitting other things too:

• Scalar Waves: The black holes start "screaming" in the language of the Axion ghosts.
• Dark Light Waves: They also emit waves of the Dark Photons.
• The Twist: The most surprising finding is about the Axion-Photon interaction. The authors discovered that if the black holes are dancing in a flat circle (2D), this interaction creates zero extra signal. It's like a spinning top that only wobbles if you tilt it. However, if the dance happens in 3D (like a helix or a tilted orbit), this interaction suddenly turns on and creates a unique signal.

Why Does This Matter?

Think of gravitational waves as a song played by the black holes. If the song changes slightly because of the dark matter fog, we might be able to hear it.

• The Goal: The authors are trying to figure out: "If we listen to the song of a black hole merger, can we tell if it was dancing in a fog of Axions and Dark Photons?"
• The Result: They have calculated exactly how the song changes. They found that the "Dark Photon mixing" changes the song early on, while the "Axion whisper" changes the song much later, and only if the dance is 3D.

The Big Picture

This paper is a recipe book for physicists. It doesn't say "We found Dark Matter." Instead, it says, "Here is the exact mathematical recipe for how Dark Matter would change the sound of black holes. If you listen to the next black hole collision, look for these specific notes. If you hear them, you've found the fog."

It's a bit like being a detective who has figured out exactly how a specific type of mud would change the footprints of a suspect. Now, when the police (scientists) find a muddy footprint, they can say, "Aha! This wasn't just any mud; it was this specific kind of Dark Matter!"

Technical Summary

1. Problem Statement

The paper addresses the dynamics of non-spinning inspiralling binary black hole (BBH) systems immersed in a dark matter environment composed of two specific ultra-light fields:

1. Axion-Like Particles (ALPs): Modeled as a massive scalar field ($\phi$) with an anomalous coupling to the electromagnetic field via a Chern-Simons type term ($g_{a\gamma\gamma} \phi F_{\mu\nu}\tilde{F}^{\mu\nu}$).
2. Ultra-Light Vector (ULV) / Dark Photons: Modeled as a massive Proca field ($B_\mu$) that interacts with the standard photon ($A_\mu$) via a kinetic mixing term ($\gamma F_{\mu\nu}B^{\mu\nu}$).

The primary goal is to compute the conservative dynamics (corrections to the effective potential and orbital motion) and the radiative dynamics (power radiated in gravitational, electromagnetic, scalar, and Proca modes) up to specific Post-Newtonian (PN) orders. The ultimate aim is to determine if gravitational wave (GW) observations can constrain the coupling constants $g_{a\gamma\gamma}$ and $\gamma$, in addition to the masses of the dark sector fields.

2. Methodology: Worldline Effective Field Theory (WEFT)

The authors utilize the Worldline Effective Field Theory (WEFT), also known as Non-Relativistic General Relativity (NRGR). This approach is chosen for its efficiency in separating scales and systematically organizing perturbative expansions.

• Scale Separation: The method distinguishes between three scales:
  • $R \sim GM$: Size of the compact objects.
  • $r$: Orbital separation (potential scale).
  • $\lambda$: Wavelength of radiation (radiative scale).
  • Hierarchy: $R \ll r \ll \lambda$, with $v/c \ll 1$.
• Degrees of Freedom:
  • Worldline: The black holes are treated as point particles with worldlines $x_a^\mu(\tau)$.
  • Fields: The fields are decomposed into potential modes (heavy, non-radiative, $k^0 \ll |\vec{k}|$) and radiation modes (light, radiative, $k^0 \sim |\vec{k}|$).
• Procedure:
  1. Integrate out potential modes: This generates an effective action for the worldlines and radiation modes.
  2. Conservative Dynamics: The real part of the effective action yields the effective potential, from which the equations of motion (orbit corrections) are derived.
  3. Radiative Dynamics: The imaginary part of the effective action is calculated using the Optical Theorem. This relates the imaginary part to the total power radiated ($P = \int E d\Gamma$).
• Power Counting: The authors establish specific EFT power counting rules for the radiative sector, defining orders as $N(n)LO$ (Next-to-$n$-Leading Order), where $LO \sim O(L^{1/2}v^{1/2})$.

3. Key Contributions and Results

A. Conservative Dynamics (Bound Sector)

The authors computed corrections to the effective potential up to specific PN orders for various interaction sectors:

• Gravitational, EM, and Proca Sectors: Computed up to 1PN order ($O(Lv^2)$).
• Pure Scalar Sector: Computed up to 2PN order ($O(Lv^4)$).
• Axion-Photon Coupling ($g_{a\gamma\gamma}$): The contribution from the CP-violating Theta term enters the conservative dynamics at 2.5PN order ($O(Lv^5)$). This is a novel finding, as standard scalar interactions usually appear at integer PN orders.
• Kinetic Mixing ($\gamma$): The mixing between the dark photon and standard photon contributes at 1PN order.

B. Radiative Dynamics (Dissipative Sector)

The paper calculates the power radiated in various channels, extending previous results to higher orders:

• Scalar Radiation:
  • Computed up to N(4)LO.
  • The contribution from the axion-photon coupling ($g_{a\gamma\gamma}$) to scalar radiation appears at N(5)LO.
  • Crucial Finding: The $g_{a\gamma\gamma}$ contribution to scalar radiation vanishes for orbits confined to a plane (e.g., circular orbits in 2D) due to the presence of a binormal-like term in the effective action. It only yields non-zero contributions for 3D orbits (e.g., helical or precessing orbits).
• Electromagnetic and Proca Radiation:
  • Computed up to N(3)LO for dipole radiation.
  • New results for Proca dipole radiation including mass effects.
• Gravitational Radiation:
  • The authors computed corrections to the standard quadrupole formula arising from the coupling of scalar, EM, and Proca fields to gravity.
  • Kinetic Mixing ($\gamma$): Contributes to gravitational radiation at N(2)LO and N(4)LO.
  • Axion-Photon Coupling ($g_{a\gamma\gamma}$): Contributes to gravitational radiation at N(7)LO.
  • Similar to the scalar case, the $g_{a\gamma\gamma}$ contribution to gravitational radiation vanishes for planar orbits and requires 3D orbital motion to be detectable.

C. Summary of PN Orders for Couplings

Coupling	Conservative Sector Order	Radiative Sector (Gravitational) Order	Radiative Sector (Scalar) Order
Kinetic Mixing ($\gamma$)	1PN	N(2)LO, N(4)LO	-
Axion-Photon ($g_{a\gamma\gamma}$)	2.5PN	N(7)LO	N(5)LO

4. Significance and Implications

• Constraints on Dark Matter Models: This work provides the theoretical framework necessary to use GW observations to constrain the parameter space of ultra-light dark matter models, specifically the axion-photon coupling ($g_{a\gamma\gamma}$) and dark photon kinetic mixing ($\gamma$).
• Orbital Geometry Dependence: A major theoretical insight is that the parity-violating axionic effects are suppressed for planar orbits. This implies that detecting these specific signatures via GWs might require systems with significant orbital inclination, spin-induced precession, or non-circular 3D trajectories.
• Methodological Advancement: The paper demonstrates the power of WEFT in handling complex multi-field interactions (gravity + scalar + vector + EM) and calculating high-order PN corrections (up to N(7)LO for specific terms) that would be intractable using traditional Green's function methods.
• Future Directions: The authors outline that the next step is to compute the full GW waveform phase using these results to perform parameter estimation. They also suggest generalizing the work to spinning binaries, non-Abelian gauge fields, and Post-Minkowskian (PM) regimes using Worldline Quantum Field Theory (WQFT).

In conclusion, this paper establishes a comprehensive effective field theory framework for binary black holes in a dark matter environment, deriving explicit formulas for energy loss and orbital corrections that open new avenues for testing fundamental physics with gravitational wave astronomy.