One loop photon-graviton mixing in an electromagnetic field: Part 3

Technical Summary: One Loop Photon-Graviton Mixing in an Electromagnetic Field: Part 3

Problem Statement
This work addresses the calculation of the one-loop photon-graviton conversion amplitude in a constant external electromagnetic field within Einstein-Maxwell theory. While the tree-level conversion (the Gertsenshtein effect) and the leading one-loop corrections (irreducible diagrams) have been previously studied, a specific class of one-loop diagrams involving a "photon tadpole" subdiagram had been historically assumed to vanish. The prevailing argument was that a constant field emits only zero-momentum photons, leading to a factor of $\delta(k)$, which, combined with gauge invariance (linear dependence on momentum), results in a vanishing contribution ($\delta(k)k^\mu = 0$).

However, recent work by Gies and Schubert (2016) in the context of pure QED demonstrated that such tadpole diagrams can yield finite, non-zero results due to infrared divergences in the connecting photon propagator when treated via dimensional regularization. This paper extends that finding to the photon-graviton mixing problem, aiming to explicitly evaluate the previously neglected tadpole contribution and determine its impact on the total amplitude and the phenomenon of magnetic dichroism.

Methodology
The authors employ the worldline formalism, a first-quantized approach to quantum field theory that provides a unified framework for calculating loop amplitudes. The methodology involves:

1. Worldline Representation: The authors utilize the master formulas for one-loop amplitudes involving one graviton and $N$ photons. These formulas express the amplitude as a proper-time integral over a path integral of a particle moving in a background field.
2. Inclusion of Constant Fields: The formalism is generalized to include a constant external electromagnetic field $F_{\mu\nu}$. This introduces specific determinant factors (e.g., $\det^{1/2}[Z/\sin Z]$ for scalars and $\det^{1/2}[Z/\tan Z]$ for spinors, where $Z = eF T$) and modifies the worldline Green's functions ($G_B$ and $G_F$) to their constant-field versions.
3. Unified Calculation: The paper calculates three distinct contributions to the one-loop amplitude:
   • Irreducible ($\Gamma^{(irr)}$): The standard diagram where the graviton and photon attach directly to the loop.
   • Reducible/External ($\Gamma^{(ext)}$): Diagrams where the graviton attaches to a photon emitted by the external field.
   • Tadpole ($\Gamma^{(tadpole)}$): The new contribution involving a one-photon tadpole subdiagram connected to the main loop via a photon propagator.
4. Renormalization and Regularization: The calculations are performed in $D$ dimensions. Ultraviolet (UV) divergences are handled via on-shell renormalization (subtracting the zero-field and zero-momentum limit). The treatment of the tadpole specifically relies on the dimensional regularization identity $\int d^D k \, \delta^D(k) \frac{k^\mu k^\nu}{k^2} = \frac{\eta^{\mu\nu}}{D}$, which resolves the apparent vanishing of the tadpole.
5. Ward Identities: The gravitational Ward identity is used as a non-trivial consistency check for the derived amplitudes.

Key Contributions and Results

• Existence of the Tadpole Contribution: The paper explicitly demonstrates that the one-photon tadpole diagram does not vanish in a constant electromagnetic field. By applying dimensional regularization to the infrared divergence of the connecting photon propagator, the authors derive a non-zero expression for the tadpole contribution to the photon-graviton amplitude.
• Explicit Formulas: The authors provide closed-form integral representations for the renormalized tadpole contribution for both scalar and spinor loops in a general constant field. For a purely magnetic background, these are expressed in terms of the Hurwitz zeta function.
• Weak-Field Expansion: A systematic weak-field expansion ($eB \ll m^2$) is derived for the spinor tadpole contribution, showing it is parametrically suppressed compared to the irreducible contribution in this regime.
• Impact on Magnetic Dichroism: A central result is the analysis of the tadpole's effect on magnetic dichroism (the difference in conversion rates for different photon polarizations). The authors show that the tadpole contribution depends on the photon and graviton polarizations only through a global factor proportional to the tree-level amplitude tensor $C_{\mu\nu,\alpha}$. Consequently, the tadpole contributes equally to both photon polarizations.
  • Result: The tadpole diagram, while non-zero, does not contribute to magnetic dichroism. The leading-order dichroism remains determined solely by the irreducible and external reducible diagrams.
• Ward Identity Verification: The gravitational Ward identity is verified for the full set of diagrams. It is shown that the tadpole contribution is separately invariant under the gravitational gauge transformation on-shell, while the irreducible and external contributions satisfy a coupled Ward identity.

Significance and Claims
The paper claims to provide a complete and consistent one-loop description of photon-graviton conversion in external fields.

• Theoretical Completeness: The inclusion of the tadpole diagram is presented as conceptually necessary to complete the one-loop description, clarifying the connection between photon-graviton mixing and other one-particle reducible (1PR) effects in background-field QED.
• Phenomenological Modesty: The authors maintain a modest stance regarding immediate experimental impact. They reiterate that under realistic field strengths, the one-loop corrections (including the new tadpole term) are small (suppressed by $\alpha$) compared to the tree-level effect. Furthermore, since the tadpole does not alter the magnetic dichroism, the previous analysis by Ahlers et al. regarding dichroism as the leading Standard Model source of magnetic dichroism remains valid.
• Future Utility: The work is framed as providing a robust framework for future studies, particularly in strong-field regimes ($eB \gg m^2$) where the relative importance of different loop contributions may change, and in extensions of the Standard Model involving light or weakly charged particles (e.g., mini-charged particles).

In summary, the paper corrects a long-standing assumption regarding the vanishing of tadpole diagrams in constant fields, provides their explicit evaluation using the worldline formalism, and establishes that while they contribute to the total amplitude, they do not alter the polarization-dependent dichroism effects previously identified.