Resummation of Massive Gravity (1011.1232v2)

Abstract: We construct four-dimensional covariant non-linear theories of massive gravity which are ghost-free in the decoupling limit to all orders. These theories resum explicitly all the nonlinear terms of an effective field theory of massive gravity. We show that away from the decoupling limit the Hamiltonian constraint is maintained at least up to and including quartic order in non-linearities, hence, excluding the possibility of the Boulware-Deser ghost up to this order. We also show that the same remains true to all orders in a similar toy-model.

• The paper presents a procedure to resum all nonlinear terms in the effective field theory, ensuring ghost-free behavior in the decoupling limit.
• It demonstrates the absence of the Boulware-Deser ghost up to quartic order by maintaining a crucial Hamiltonian constraint.
• The approach outperforms previous cubic-order methods and lays the groundwork for addressing the cosmological constant problem and higher-order stability challenges.

Resummation of Massive Gravity

The paper by Claudia de Rham, Gregory Gabadadze, and Andrew J. Tolley presents a significant advancement in the development of ghost-free massive gravity theories in four dimensions. It specifically addresses the issue of the Boulware-Deser (BD) ghost, which characteristically appears in nonlinear generalizations of the Fierz-Pauli massive gravity model. The authors succeed in constructing covariant non-linear theories of massive gravity that remain ghost-free in the decoupling limit, an accomplishment they achieve by resumming explicitly all the nonlinear terms of an effective field theory (EFT) of massive gravity.

Key contributions of the paper include:

1. Ghost-Free Theories in Decoupling Limit: The authors derive Lagrangians that automatically give rise to ghost-free theories up to all orders in the decoupling limit. They implement a procedure to determine the appropriate coefficients in the EFT Lagrangian at each order, ensuring that only a finite, specifically structured set of terms survive. This structure is dictated by symmetry principles.
2. Extension Beyond Decoupling Limit: Beyond the decoupling limit, the authors analyze the Hamiltonian constraint up to quartic order in non-linearities and demonstrate the absence of the BD ghost at this order. This result is crucial because retaining the Hamiltonian constraint is pivotal to excluding the sixth degree of freedom — the BD ghost. They extend their findings to a similar toy-model to prove that the constraints hold in general.
3. Comparison with Other Resummation Methods: The paper situates its findings against alternative resummation methods, such as those relying on auxiliary extra dimensions. The authors argue that while previous approaches secured a ghost-free decoupling limit only up to cubic order, their method effectively resums nonlinear terms using an algebraic nonlinear equation to achieve the continuation of the constraint structure.
4. Potential for Further Developments: By exhibiting a working example of a nonlinear theory free from the BD ghost, this paper lays the groundwork for future exploration into broader classes of ghost-free massive gravity theories beyond the quartic order in perturbations. Addressing remaining theoretical challenges, especially regarding higher-order stability and quantum consistency, remains a promising direction for future paper.
5. Implications for the Cosmological Constant Problem: The paper implicates a potential resolution to the cosmological constant problem by regulating the graviton mass. This consideration opens a new avenue for theoretical investigations within the landscape of modified gravity theories and their role in cosmological modeling.

In conclusion, this research constitutes a significant advancement in understanding and formulating ghost-free theories of massive gravity. By establishing a robust method to resum nonlinearities and suppress undesirable degrees of freedom, the findings provide a pathway to resolving long-standing theoretical challenges related to the inclusion of a small graviton mass in gravitational theories. Further exploration into extending these results beyond quartic non-linearities and ensuring stability across different scales will be instrumental for advancing our understanding of massive gravity within the broader framework of fundamental physics.