# Probing the unitarity of the renormalizable theory of massive conformal gravity

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## Abstract

The presence of an unstable massive spin-2 ghost state in the renormalizable theory of massive conformal gravity leads to a pair of complex poles appearing in the first sheet of the energy plane. Here we show that the positions of these poles are gauge dependent, which makes the theory unitary.

## Keywords

Gauge symmetry Unitarity Quantum Gravity## 1 Introduction

The massive conformal gravity (MCG) is a renormalizable theory of gravity [1, 2] that has, in addition to the usual positive energy massless spin-2 field, a negative energy massive spin-2 field. At the quantum level, the negative energy field translates into an unstable negative norm ghost state. The instability of the ghost state makes it necessary to use a modified perturbation expansion in terms of dressed propagators. Since the bare propagator of the ghost state has a negative residue, the original real pole split into a pair of complex-conjugate poles in the first Riemannian energy sheet of the dressed propagator. If the positions of the complex poles are gauge dependent, we can move them around by varying the corresponding gauge-fixing parameter. In this case, the *S*-matrix connects only asymptotic states with positive norm and thus it is a unitary matrix.

It is well known that fourth order derivative theories of gravity have a massive ghost pole (or rather complex poles in the dressed propagator) whose position is gauge independent [3]. The advantage of MCG over these theories is that its linearized action is invariant, independently, under coordinate and conformal gauge transformations. Thus, even if the position of the MCG massive ghost pole is independent of the coordinate gauge-fixing parameter, as happens in the fourth order derivative theories of gravity, its dependence on the conformal gauge-fixing parameter alone is sufficient to ensure the unitarity of the theory.

This paper is organized as follows. In Sect. 2, we describe the nature of the MCG massive ghost pole. In Sect. 3, we probe the gauge dependence of the positions of the MCG complex poles by using the Nielsen identities. In Sect. 4, we present our conclusions.

## 2 Massive ghost pole

^{1}[4]

*m*is a constant with dimension of mass, \(\varphi \) is a scalar field called dilaton,

*N*fermion fields to the MCG action (1), expanding in powers of 1 /

*N*, and using the Cauchy integral theorem, we can write the spin-2 part of the graviton dressed propagator in the spectral form [9]

*M*, \(M^*\), \(\mathscr {R}\), and \(\mathscr {R}^*\) are the positions and residues of a complex-conjugate pole pair, respectively, \(\rho (a)\) is a spectral function obtained by cutting all the self-energy graphs of the continuum states, and

*C*is an appropriate path in the complex plane. We can see from (16) that the pole for the unstable massive ghost has split into a pair of complex-conjugate poles in the physical Riemannian energy sheet, which supposedly breaks the unitarity of the

*S*-matrix. However, if the positions of the complex poles are gauge dependent, the unitarity of the gauge-invariant

*S*-matrix is satisfied. We will address this issue in the next section.

## 3 Gauge dependence

*c*is a scalar (anti-ghost) ghost field.

*L*are sources for the composite fields \(\delta _{1}h_{\mu \nu }\), \(\delta _{2}h_{\mu \nu }\), and \(\delta _{2}\sigma \), respectively, and \(\varGamma \) is the effective action. Differentiating (30) and (31) with respect to \(\varOmega _{1}\) and \(\varOmega _{2}\), respectively, and setting \( \varOmega _{1} = \varOmega _{2} = 0\), we obtain

*S*-matrix, such gauge-dependent poles disappear from the spectrum and unitarity is satisfied.

## 4 Final remarks

Here we presented a study on the unitarity of MCG. First, we noted that due to the presence of the unstable massive spin-2 ghost state in the linearized theory, we must use a dressed propagator perturbation expansion, which results in the emergence of a pair of complex poles in the first sheet of the energy plane. Then, by using the Nielsen identities, it was shown that the positions of the complex poles are found to be dependent on the conformal gauge-fixing parameter. This is enough to ensure that the excitations represented by the complex poles do not contribute to the gauge-invariant absorptive part of the *S*-matrix, leading to the unitarity of the theory. Therefore, we conclude that MCG is a consistent, renormalizable and unitary theory of quantum gravity.

## Footnotes

- 1.
Here we use units in which \(c=\hbar =1\).

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