Generalization of the Proca Action

Abstract: We consider the Lagrangian of a vector field with derivative self-interactions with a priori arbitrary coefficients. Starting with a flat space-time we show that for a special choice of the coefficients of the self-interactions the ghost-like pathologies disappear. For this we use the degeneracy condition of the Hessian. This constitutes the Galileon-type generalization of the Proca action with only three propagating physical degrees of freedom. The longitudinal mode of the vector field is associated to the usual Galileon interactions for a specific choice of the overall functions. In difference to a scalar Galileon theory, the generalized Proca field has more free parameters and purely intrinsic vector interactions. We then extend this analysis to a curved background. The resulting theory is the Horndeski Proca action with second order equations of motion on curved space-times.

• The paper extends the standard Proca action by incorporating derivative self-interactions to confine massive vector fields to three physical degrees of freedom.
• It employs degeneracy conditions on the Hessian matrix to systematically eliminate ghost-like instabilities and maintain second-order field equations.
• The analysis confirms consistency in both flat and curved spacetimes, drawing parallels with vector Galileons and Horndeski theory.

Overview of Generalization of the Proca Action

This paper presents a comprehensive study on the generalization of the Proca action for a massive vector field with derivative self-interactions. The authors investigate the Lagrangian framework necessary to retain only three physical degrees of freedom while avoiding ghost-like instabilities. By systematically analyzing derivative interactions and imposing degeneracy conditions on the Hessian matrix, they construct a theory that extends the Proca action to include vector Galileon interactions, both in flat and curved spacetime settings.

The generalized Proca action constructed in this study is inspired by the efforts to adapt the Fierz-Pauli action for a massive graviton, analogous to how de Rham and Gabadadze approached the construction of massive gravity theories. The motivation lies in finding self-interacting Lagrangians that still confine to three degrees of freedom, akin to the scalar Galileon theories introduced as infrared modifications of gravity.

Numerical Insights and Analytical Structure

For flat spacetime, the paper derives a series of Lagrangians describing the generalized Proca action with derivative self-interactions. It explores terms up to the fifth order and demonstrates that by selecting appropriate functional forms and coefficients, the theory ensures the non-propagation of an additional unphysical degree of freedom. These selections yield constraints through degeneracy conditions of the Hessian matrix, leading to a consistent set of equations.

The Lagrangians $\mathcal{L}_2$ to $\mathcal{L}_6$ are constructed with an emphasis on retaining second-order equations of motion. Notably, the functional freedom afforded to vector interactions exceeds that of scalar Galileons, as seen in the presence of additional parameters. Derived via Levi-Civita contractions and integration by parts, this framework manages to align vector Galileon interactions with the expected propagation of physical degrees of freedom.

Theoretical Implications and Future Directions

Incorporating the Stueckelberg mechanism to examine the scalar sector of the resulting Lagrangian confirms parallelism with Galileon interactions. By ensuring the Stueckelberg field equations remain second-order, the analysis supports the theoretical soundness of the generalized Proca theory as a viable description of massive vector fields in both flat and curved backgrounds.

The implications of this study are substantial for theoretical models involving vector fields interacting non-minimally with gravity, especially considering modern efforts in cosmology and particle physics that demand more intricate and stable descriptions of massive fields.

On extending this analysis to curved spacetime, the authors achieve a vector analog to the Horndeski theory. They elucidate how non-minimal couplings to gravity ensure that the equations maintain their order, thus preparing the extended theory for potential application in strong gravitational fields or cosmological models.

Conclusion

This paper contributes significantly to the repertoire of modified field theories by rigorously generalizing the Proca action and delivering a flexible framework for massive vector fields. By maintaining constraints that preserve the propagation count of physical modes, it lays the groundwork for further explorations into energetic and dynamic scenarios that might demand such theoretical sophistication. As computational methods and experimental capabilities advance, future work may investigate the phenomenological and empirical implications of vector Galileons, particularly in contexts where gravitational and quantum field interactions converge.

In summary, while the generalization of the Proca action as presented is insightful and robust, its applicability extends to theoretical paradigms where advanced mathematical treatment of vector fields could yield substantial physical insights.