Euler-Heisenberg actions in higher dimensions

Abstract: We extend Schwinger's proper-time formalism to provide a method for computing the one-loop effective action for both spinor and scalar quantum electrodynamics in $d=2n>4$ dimensions. The closed form expression for the six-dimensional Euler-Heisenberg action is given, along with a subsequent analysis of pair production in $d$ dimensions. In the $d=6$ case, we present a composite conformal primary field of dimension $+6$ which determines the contribution of the electromagnetic field to the Weyl anomaly in curved space.

• The paper extends Schwinger’s proper-time method to compute one-loop effective actions for spinor and scalar QED in even dimensions, focusing on explicit 6D formulations.
• It derives closed-form effective Lagrangians with detailed weak-field expansions and pair production rate formulas under constant electromagnetic fields.
• It links electromagnetic field invariants to the Weyl anomaly by constructing composite conformal primary fields in curved backgrounds.

Euler-Heisenberg Actions in Higher Dimensions

Introduction and Motivation

The paper "Euler-Heisenberg actions in higher dimensions" (2604.13636) systematically extends Schwinger's proper-time approach for the computation of one-loop effective actions in both spinor and scalar quantum electrodynamics (QED) to arbitrary even dimensions, with a focus on $d=6$. The classical Euler-Heisenberg action encapsulates non-linear effects of quantum fields in four dimensions, such as vacuum polarization and pair production, in a closed-form effective Lagrangian. Its generalization to higher dimensions addresses gaps in the literature regarding explicit component expressions, particularly in six-dimensional QED. The analysis also connects the electromagnetic sector to the Weyl anomaly in curved backgrounds by identifying composite conformal primary fields of appropriate scaling dimension.

Generalization of Schwinger's Proper-Time Formalism

The authors adapt Schwinger's operator-based proper-time formalism to compute the one-loop effective action for Dirac spinors and charged scalars in $d=2n$ dimensions. The effective action, given by the trace log of the quadratic operator in the background field, is expressed via the heat kernel machinery. The calculation employs constant electromagnetic fields and exploits the algebraic properties of gamma matrices and their representations in higher dimensions, including spinor traces decomposed into Weyl sectors.

In six dimensions ($d=6$), the key technical advance is the explicit computation of traces and determinants associated with the field strength tensor $F_{ab}$ and the spinorial matrix $\mathbb{F}$, whose eigenvalue structure is characterized via quartic and sextic equations involving three electromagnetic invariants: $\mathcal{F}$ (quadratic), $\mathcal{G}$ (cubic), and $\mathcal{H}$ (quartic). These invariants fully parametrize the weak-field expansion of the effective Lagrangians.

Closed-Form Euler-Heisenberg Actions in 6D

For spinor QED in six dimensions, the explicit one-loop effective Lagrangian is obtained:

$$\mathcal{L}_{\text{spinor}}^{(1)}(x) = -\frac{1}{2}\int^{\infty}_0 \frac{ds}{s}\frac{8}{(4\pi s)^3} e^{-m^2 s} \left[(es)^3 \lambda_1 \lambda_2 \lambda_3 \cot(es\lambda_1) \cot(es\lambda_2) \cot(es\lambda_3) - 1 - \frac{2}{3}(es)^2 \mathcal{F}\right]$$

where $\lambda_i$ are the eigenvalues of $d=2n$0, determined by the electromagnetic invariants. The weak-field expansion yields structured polynomial contributions:

• Leading order: $d=2n$1
• Next-to-leading: $d=2n$2

For scalar QED, the analogous effective Lagrangian and expansion are provided, with different numerical coefficients and structure, notably

• Leading order: $d=2n$3

The explicit results fill the gap in component-wise expressions for the six-dimensional Euler-Heisenberg action, previously unavailable, and clarify the algebraic structure of field invariants in higher dimensions.

Pair Production Rates in Arbitrary Even Dimensions

The analysis of pair production in a constant electric field uses the imaginary part of the one-loop effective action, evaluated via the residue contributions from the poles of trigonometric functions in the proper-time integrals. The generic result for spinor QED in $d=2n$4 dimensions is:

$d=2n$5

and for scalar QED, the $d=2n$6 factor is omitted. The reproduction of Schwinger's result in $d=2n$7 confirms consistency, while the dimensional generalization provides exact formulas for pair production in higher-dimensional QED, relevant for theoretical investigations of quantum field theories and string-inspired models with extra dimensions.

Weyl Anomaly and Composite Conformal Primaries

The work establishes the relation between electromagnetic contributions and the Weyl anomaly in curved backgrounds for $d=2n$8. Using heat kernel expansions, the authors compute the $d=2n$9 coefficient that controls the logarithmic divergence and thereby the relevant higher-derivative counterterm proportional to $d=6$0.

A composite conformal primary field $d=6$1 of dimension $d=6$2 is constructed:

$d=6$3

This field governs the electromagnetic contribution to the Weyl anomaly, supplementing the gravity sector's Euler density and Weyl invariants. The degeneracies and uniqueness of the primary field in $d=6$4 are discussed. The results refine the structure of conformal anomalies in higher dimensions, linking gauge field theory to mathematical aspects of conformal geometry.

Implications and Directions for Further Research

The paper provides a comprehensive and rigorous framework for effective actions in higher-dimensional QED, augmenting the toolkit of quantum field theorists working in non-standard spacetime dimensions, supersymmetric extensions, and string theory. The explicit results for the six-dimensional case open avenues for:

• Two-loop computations generalizing Ritus's work to $d=6$5 and beyond
• Applications in superconformal field theories, including component reduction from harmonic superspaces
• Systematic analysis of higher-derivative gauge theories with renormalizable and classically conformal behavior
• Refinements in anomaly computations for supergravity, conformal gravity, and their matter couplings

The methods and results provide benchmarks for checks of dualities, consistency conditions, and UV properties in extended gauge and gravity theories.

Conclusion

The extension of the Euler-Heisenberg action to arbitrary even dimensions, and its explicit formulation in six-dimensional QED, fills an important gap in the literature. The structured weak-field expansions and pair production rates contribute fundamentally to our understanding of quantum effects in gauge fields beyond four dimensions. The connection with the Weyl anomaly via composite conformal primaries advances both practical computations and the theoretical foundations of quantum field theory in diverse geometric backgrounds. The results serve as a basis for further investigations, including multi-loop computations and more general field-theoretic frameworks (2604.13636).