Binary Black Holes in Einstein-Maxwell-Dilaton Theory: Second Post-Newtonian Dynamics from Effective Field Theory

Technical Summary: Binary Black Holes in Einstein-Maxwell-Dilaton Theory: Second Post-Newtonian Dynamics from Effective Field Theory

Problem Statement
The detection of gravitational waves (GWs) from compact binary coalescences offers a unique opportunity to test General Relativity (GR) in the strong-field regime and search for signatures of alternative gravity theories. Einstein-Maxwell-Dilaton (EMd) theory, which arises naturally in the low-energy limit of string theory and Kaluza-Klein compactifications, posits that black holes can carry both electric and scalar (dilatonic) charges. While the conservative dynamics of EMd binaries were previously known up to the first post-Newtonian (1PN) order, higher-order corrections are necessary to develop accurate waveform models for GW observations. Specifically, the conservative sector is a critical ingredient for constructing Post-Newtonian (PN) inspiral waveforms and effective-one-body (EOB) models. This work addresses the gap in knowledge by deriving the conservative two-body Lagrangian for charged black-hole binaries in EMd theory through the second post-Newtonian (2PN) order.

Methodology
The author employs the Effective Field Theory (EFT) approach to the relativistic two-body problem, originally developed by Goldberger and Rothstein for non-relativistic GR (NRGR). This framework is extended to include the electromagnetic and scalar degrees of freedom inherent to EMd theory.

Key methodological components include:

• Kaluza-Klein Decomposition: The metric is parametrized using a temporal Kaluza-Klein ansatz, decomposing it into non-relativistic gravitational (NRG) fields: the Newtonian potential ($\phi$), the gravito-magnetic potential ($A_i$), and spatial metric perturbations ($\sigma_{ij}$). This decomposition exploits the hierarchy of scales in the PN expansion ($r_c \ll r \ll \lambda$) and significantly reduces the complexity of Feynman diagrams.
• Worldline and Bulk Interactions: Compact objects are modeled as point particles with scalar-field-dependent masses. The matter action is expanded to include couplings to the metric, electromagnetic field, and dilaton field, characterized by scalar response parameters ($\alpha, \beta, \beta'$).
• Diagrammatic Expansion: The conservative dynamics are determined by near-zone potential modes. The author systematically constructs Feynman diagrams up to 2PN order, categorizing contributions by their scaling in Newton's constant ($G$), charges ($q$), and velocity ($v$).
• Loop Calculations: The calculation involves evaluating one-loop and two-loop integrals. Divergent integrals are handled using dimensional regularization. The two-loop topologies are classified as factorizable, nested, or irreducible, with the latter reduced using integration-by-parts identities.
• Power Counting: A systematic power-counting scheme tracks powers of $G$, $q$, and $v$ to organize the diagrammatic expansion, ensuring all relevant terms up to 2PN order are included.

Key Contributions

1. Derivation of the 2PN Lagrangian: The primary contribution is the complete derivation of the conservative two-body Lagrangian for EMd binaries through 2PN order. This extends previous results limited to 1PN.
2. Inclusion of Mixed Interactions: The result incorporates gravitational, electromagnetic, and dilaton interactions, as well as their cross-couplings, up to the required order. This includes terms scaling as $O(Gv^4)$, $O(G^2v^2)$, $O(Gq^2v^2)$, $O(G^3)$, $O(G^2q^2)$, and $O(Gq^4)$.
3. Feynman Rules and Vertices: The paper explicitly derives the worldline and bulk interaction vertices and propagators required for the PN calculation within the EMd framework, utilizing the NRG formalism to reveal a clear coupling hierarchy between gravitational, gauge, and scalar interactions.

Results
The author presents the full conservative Lagrangian (Eq. 7.1) expressed in terms of the relative separation vector, velocities, accelerations, masses, charges, and scalar response parameters. The result is validated through several consistency checks:

• Limiting Cases: The derived Lagrangian correctly reduces to the known 2PN Einstein-Maxwell Lagrangian (when scalar couplings are zero), the 2PN scalar-tensor Lagrangian (when electric charges are zero), and the 2PN GR Lagrangian (when both charges and scalar couplings vanish).
• Independent Verification: The result agrees with calculations performed using the EFTofPNG package.
• Static Test-Body Limit: An independent check was performed by treating one body as an exact EMd black hole and the other as a test particle. The weak-field expansion of the exact solution was shown to exactly reproduce the static sector of the derived two-body Lagrangian through 2PN order.

Significance
The paper claims that these results provide the necessary conservative dynamics to develop higher-accuracy waveform models for testing EMd theory with gravitational-wave observations. By extending the conservative dynamics beyond 1PN, the work enables more precise constraints on the underlying parameters of modified gravity theories, such as the dilaton coupling constant $a$ and the scalar charges of compact objects. The author notes that while this work focuses on the conservative near-zone dynamics, a complete waveform model will eventually require the corresponding radiative sector (including scalar and electromagnetic radiation), which lies beyond the scope of the current calculation. The derived equations of motion and gauge-invariant quantities serve as a foundation for future work, including the development of effective-one-body descriptions and the investigation of observational signatures of charged black-hole binaries.