Abstract
We find unitary and local theories of higher curvature gravity in the vielbein formalism, known as Poincar\'e gauge theory, by utilizing the equivalence to ghost-free massive bigravity. We especially focus on three and four dimensions, but extensions into a higher-dimensional spacetime are straightforward. In three dimensions, quadratic gravity $\mathcal{L}=R+{T}^{2}+{R}^{2}$, where $R$ is the curvature and $T$ is the torsion with indices omitted, is shown to be equivalent to zwei-dreibein gravity and free from the ghost at fully nonlinear orders. In a special limit, new massive gravity is recovered. When the model is applied to the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, unitarity both in the bulk theory and in the boundary theory implies that the torsion must not vanish. On the other hand, in four dimensions, the absence of a ghost at nonlinear order requires an infinite number of higher curvature terms, and these terms can be given by a schematic form $R(1+R/\ensuremath{\alpha}{m}^{2}{)}^{\ensuremath{-}1}R$, where $m$ is the mass of the massive spin-2 mode originating from the higher curvature terms and $\ensuremath{\alpha}$ is an additional parameter that determines the amplitude of the torsion. We also provide another four-dimensional ghost-free higher curvature theory that contains a massive spin-0 mode as well as a massive spin-2 mode.
Highlights
Giving a mass to the graviton is an idea from the pioneering study by Fierz and Pauli [1] and has been extensively discussed since
This ghostfree theory, often dubbed the de Rham-Gabadadze-Tolley theory, is the theory of a single massive spin-2 field extended to bigravity [7], which includes the massless spin-2 field as well, and to multigravity, which has multiple massive spin-2 fields as well as the massless one [8]
The de Rham-Gabadadze-Tolley (dRGT) theory and its extensions provide a well-defined framework for modification of general relativity (GR), and many phenomenological aspects have been investigated
Summary
Giving a mass to the graviton is an idea from the pioneering study by Fierz and Pauli [1] and has been extensively discussed since (see [2,3] for reviews). Reference [9] demonstrated that ghost-free theories can be obtained as a Kaluza-Klein compactification of the higher dimensional GR In this case, multiple massive spin-2 fields are naturally expected, corresponding to the Kaluza-Klein states of the graviton. When the Lagrangian of PGT is algebraic in field strength, the existence of first-class constraints, namely, the symmetries of PGT, concludes that there exist 18 degrees of freedom (dofs) in addition to the massless graviton in four dimensions [17,18] These 18 dofs are classified into massive spin-2Æ; 1Æ; 0Æ particle species where the number and Æ of JPðJ 1⁄4 0; 1; 2; P 1⁄4 ÆÞ denote the spin and the parity, respectively. There is a nonlinearly ghost-free PGT involving a massive spin-2 mode when an infinite number of appropriate higher curvature terms are added.
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