[Paper Review] Possible Derivative Interactions in Massive bi-Gravity

Giorgi Tukhashvili|arXiv (Cornell University)|Apr 27, 2017

Cosmology and Gravitation Theories8 references3 citations

TL;DR

This paper investigates derivative interactions in massive bigravity using the metric formalism, deriving necessary (but not sufficient) conditions for ghost freedom. Despite constructing models that satisfy these criteria, the study finds that in the decoupling limit, such interactions still generate a ghost, indicating that derivative mixings in massive bi-gravity are inherently problematic and likely ghost-prone even when mass terms are ghost-free.

ABSTRACT

Using the metric formalism, we study the derivative mixings of spin-2 fields in massive bi-Gravity. Necessary (but not sufficient) criteria are given for such mixings to be ghost free. Examples satisfying those criteria are studied and it is shown that in the decoupling limit they host a ghost.

Motivation & Objective

• To identify necessary conditions for derivative mixings of spin-2 fields in massive bigravity to be ghost-free.
• To analyze whether such conditions can be realized in concrete models without introducing instabilities.
• To investigate the behavior of proposed derivative interaction terms in the decoupling limit, particularly focusing on strong coupling and ghost modes.
• To assess the viability of constructing a consistent, ghost-free massive bigravity theory with kinetic (derivative) mixings.

Proposed method

• Uses the metric formalism to derive equations of motion for a general class of derivative interaction terms between two spin-2 fields.
• Applies the decoupling limit with cutoff scale $\Lambda = (m^2 M_p)^{1/3}$ to analyze low-energy behavior.
• Introduces Stückelberg fields to restore diffeomorphism invariance and probe ghost modes in the nonlinear regime.
• Constructs specific interaction terms, such as $ \frac{M_h M_f}{2} \sqrt{-g} g^{\mu\nu} \tilde{R}_{\mu\nu} + \beta \frac{M_h M_f}{2} \sqrt{-\tilde{g}} \tilde{g}^{\mu\nu} R_{\mu\nu} $, and evaluates their ghost content.
• Performs a strong coupling and ghost analysis in the decoupling limit, computing the ghost mass explicitly.
• Imposes consistency conditions on parameters to test whether the ghost can be removed, finding them incompatible with required constraints.

Experimental results

Research questions

• RQ1 Can derivative interactions between two massive spin-2 fields in bigravity be made ghost-free while preserving diffeomorphism invariance?
• RQ2 Do the necessary conditions for ghost freedom in derivative mixings remain sufficient when nonlinear effects are included?
• RQ3 What is the behavior of such derivative interactions in the decoupling limit, particularly regarding strong coupling and ghost modes?
• RQ4 Can the ghost mass be tuned to zero by adjusting coupling parameters without violating consistency conditions?
• RQ5 Is there a consistent parameter regime where derivative mixings avoid both ghosts and strong coupling?

Key findings

• The proposed derivative interaction terms satisfy the necessary (but not sufficient) conditions for ghost freedom derived in the paper.
• In the decoupling limit, the model exhibits both strongly coupled terms and a ghost, indicating that the necessary conditions are not sufficient to ensure a healthy theory.
• The ghost mass is computed as $ M_{\text{ghost}}^{2} = -\frac{3}{2} \left[14(\alpha - \beta)a_2 - 8\frac{\beta a_1}{\kappa}\right]^{-1} \frac{\Lambda^6}{a_0^2 (a \cdot x)^2} $, which remains negative and divergent under physical parameter choices.
• The condition to eliminate the ghost, $ 14(\alpha - \beta)a_2 - 8\frac{\beta a_1}{\kappa} = 0 $, is incompatible with the requirement $ \alpha \ne \beta $, which is needed for consistency with the decoupling limit analysis.
• Even when the theory is symmetric under $ g \leftrightarrow \tilde{g} $, redefining fields does not resolve the ghost issue, as it leads to $ \beta = 0 $, which contradicts the model's assumptions.
• The presence of infinitely many such derivative terms with independent parameters makes it infeasible to prove ghost-freeness in the general case, suggesting fundamental obstructions to such interactions.