# IR finite graviton propagators in de Sitter spacetime

## Abstract

The graviton propagator diverges in certain gauges in de Sitter spacetime. We address this problem in this work by generalizing the infinitesimal BRST transformations in de Sitter spacetime to finite field-dependent BRST (FFBRST) transformations. These FFBRST transformations are a symmetry of the classical action, but they do not leave the path integral measure invariant for the graviton theory in de Sitter spacetime. Due to the non-trivial Jacobian of such a finite transformation the path integral measure changes and hence the FFBRST transformation is capable of relating theories in two different gauges. We explicitly construct the FFBRST transformation which relates the theory with a diverging graviton two-point function to a theory with an infrared finite graviton. The FFBRST transformation thus establishes that the divergence in a graviton two-point function may be only a gauge artifact.

## Keywords

Gauge Parameter BRST Transformation Ghost Propagator Inflaton Field Initial Gauge## 1 Introduction

The observations from type I supernovae indicate that our universe has a positive cosmological constant and may approach de Sitter spacetime asymptotically [1, 3, 4, 5, 6]. The de Sitter spacetime is also relevant in inflationary cosmology [7, 8, 9, 10]. Inflaton fields corresponding to open strings have been studied in brane–antibrane models [11, 12] and *D*3 / *D*7 systems [13, 14], and the inflaton fields corresponding to closed strings have been studied in Kähler moduli [15, 16] and fiber inflation [17]. However, in all these models the realization of inflation depends crucially on the uplifting mechanism for de Sitter moduli stabilization [18]. This uplifting mechanism occurs in the presence of *D*3-branes. It may be noted that even the Wilson line approach crucially depends on the uplifting mechanism for de Sitter moduli stabilization [19]. Due to the relevance of de Sitter spacetime to inflation, it is important to study perturbative quantum gravity in de Sitter spacetime. However, the graviton propagator in de Sitter spacetime found by Antoniadis et al. suffered from IR divergences [60, 65]. In fact, these IR divergences occur in the covariant gauge for certain choices of the gauge parameters, \(\beta = - n(n+3)/3\) with \(n = 1, 2, 3 \ldots \) [22]. However, it is also possible to construct an IR finite graviton propagator [25, 63, 64]. So, there are strong indications to assume that the IR divergence that occurs in the propagator by Antoniadis et al. is a gauge artifact. This is supported by the fact that the free graviton propagator in covariant gauge is equivalent to the IR finite graviton propagator [26]. However, in that analysis the role of interactions was not considered. What really needs to be demonstrated is that the generating functionals for different values of the parameter \(\beta \) are related to one another. We argue that the IR divergent graviton propagators with \(\beta = - n(n+3)/3\) are related to the IR finite graviton propagators with other values of \(\beta \). However, to show that explicitly, we will need a formalism to connect the generating functionals for the graviton propagators in the covariant gauge with different values of the parameter \( \beta \). As the Euclidean approach has been used for a calculation of different propagators in de Sitter spacetime [27], including the graviton propagator [28], we will also use the Euclidean approach for calculating the graviton propagator. So, we will obtain the function on a four-dimensional sphere, and these Green functions are related to the Feynman propagator in the de Sitter spacetime through analytic continuation. It may be noted that we could also have used the planar patch of Lorentzian de Sitter spacetime for performing these calculations, however, the advantage of using the Euclidean approach is that it is easier to perform the FFBRST transformations in this approach. We will use the Euclidean vacuum as the vacuum state for performing these calculations [29].

The FFBRST transformation [30] was constructed systematically by integrating the usual BRST transformation [31]. Such generalized BRST transformations have the same form and properties as the usual BRST transformations, except these do not leave the path integral measure invariant. The non-trivial Jacobian enables such a formulation to connect theories with different effective actions, hence FFBRST transformations have found an enormous number of applications in various branches of high energy physics [30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. A similar generalization with the same motivation and goal has also been done recently in a slightly different manner [48, 49, 50]. In this work we extend the FFBRST formulation [30] in de Sitter spacetime and construct an appropriate finite field-dependent parameter to relate the generating functionals corresponding to the effective theories with graviton propagator for various values of \(\beta \). It may be noted that even though we build this formalism motivated by the IR divergences in de Sitter spacetime, this formalism is very general and can be used to relate a generating functional for the graviton propagator with any arbitrary value of \(\beta \). It may also be noted that there are real IR divergences that occur in the ghost propagator in de Sitter spacetime. However, modes responsible for these divergences do not contribute to loop diagrams in computations of the scattering amplitudes in perturbative quantum gravity and can thus be neglected [51]. It is possible to construct an effective IR finite ghost propagator for de Sitter spacetime utilizing the FFBRST transformation. In this connection we would like to comment that the gaugeon formulation [52, 53, 54, 55, 56, 57], which also connects different effective actions in perturbative quantum gravity [58, 59], could be another possibility to construct theories with an IR finite graviton propagator. However, the gaugeon formalism has certain drawbacks, in that one needs to introduce unphysical gaugeon fields in the theory and later extra conditions are required to extract the physical states.

It may be noted that there are various issues that are related the IR divergences in the graviton propagator. Furthermore, there are also several problems with the average gauges in de Sitter space and any space with linearizion instabilities [60]. It has also been argued that the main problem with certain values of the gauge parameter is that for these values of the gauge parameter logarithmic divergences rather than power law divergences occur [61, 62]. The power law divergences get automatically subtracted for the allowed values of the gauge parameter. In fact, it has been demonstrate using this line of argument that certain IR divergences also occur for the allowed values of the gauge parameter [63, 64]. Furthermore, IR divergences which appear in certain gauges have the local form of a gauge transformation, but they need not be a symmetry of the theory because the needed gauge transformation diverges at infinity and therefore invalidates the usual integration by parts and discarding of surface terms is needed to prove invariance even of the classical action [65, 66, 67, 68]. Even though we have neglected such terms in our paper by dropping a total divergence, however, we would like to point out that there are many non-trivial issues relating to the occurrence of such divergences. It may be noted that even though there are various different sources of IR divergences, in this paper, we will not address many of these issues. We will rather demonstrate that a graviton propagator in a certain gauge, in which a certain kind of IR divergences occurs, can be related to the graviton propagator in a different gauge where such IR divergences do not occur. This can be done using the FFBRST transformations, as the FFBRST transformations are a symmetry of the generating functional and not of the effective action, which is obtained by adding the gauge fixing and ghosts terms to the original action. In fact, it is this property of the FFBRST transformation that has made it possible to use the FFBRST transformation for analyzing various interesting physical systems [30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Thus, motivated by such uses of the FFBRST transformations, we will analyze the occurrence of a certain kind of IR divergences in this paper.

In this paper, we first study the perturbative quantum gravity on curved space time where we particularly emphasize the de Sitter spacetime. The effective action of perturbative quantum gravity on de Sitter spacetime respects a fermionic rigid BRST invariance. The BRST symmetry further generalizes by making the parameter finite and field-dependent following the techniques of Ref. [30]. The FFBRST transformation generalized in such a way leads to a non-trivial Jacobian for a functional measure. We show that for a particular choice of the finite field-dependent parameter the Jacobian relates the gauge parameters, stimulating IR divergent and IR finite graviton propagators. So, in Sect. 2, we analyze the perturbative quantum gravity in de Sitter spacetime, and in Sect. 3 we study the FFBRST transformation in de Sitter spacetime. Then in Sect. 4 we relate the IR divergent graviton two-point function to the IR finite graviton propagators using the FFBRST transformations. In the final section we summarize the results.

## 2 Perturbative quantum gravity

*H*is the Hubble constant. In terms of the variable \(\tau \equiv \pi /2-iHt\), the line element gets the following form:

*k*is written as \(1 + \beta ^{-1}\) for a finite value of \(\beta \). The gauge fixing condition can be incorporated at a quantum level by the addition of a gauge fixing term to the original Lagrangian,

## 3 FFBRST transformation

## 4 Recovering IR finite the graviton propagators

## 5 Conclusion

In this paper, we have analyzed perturbative quantum gravity on de Sitter spacetime. The BRST and FFBRST transformations for the perturbative quantum gravity were explicitly constructed in de Sitter spacetime. The FFBRST transformations were used to relate the generating functionals with different values of the parameter \(\beta \). We construct an appropriate finite field-dependent parameter such that the Jacobian contribution of the path integral measure relates the graviton propagator with an IR divergence to the IR finite graviton propagator. Thus, it was argued that it might be possible that a certain kind of IR divergence in the graviton propagator is only a gauge artifact. However, we would like to point out that there are arguments to try to argue that the removal of such divergences is only an artifact of the regularization procedure [61, 62]. Since the spacetime noncommutativity changes the IR behavior of quantum field theories [69, 70], and perturbative quantum gravity has been studied on noncommutative spacetime [71, 72, 73, 74, 75], it would also be interesting to analyze the IR divergences in de Sitter spacetime in noncommutative spacetime.

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