Ghost-free higher-order theories of gravity with torsion

In this manuscript we will present the theoretical framework of the recently proposed infinite derivative theory of gravity with a non-symmetric connection. We will explicitly derive the field equations at the linear level and obtain new solutions with a non-trivial form of the torsion tensor in the presence of a fermionic source, and show that these solutions are both ghost and singularity-free.

case [12]. In this work, we will show that having a charged fermion as a source, and assuming also the trace of the torsion to be different from zero, we can find new solutions with a non-trivial torsion tensor. The rest of the components of the torsion tensor have been considered to be negligible motivated by the fact that in the standard cosmological scenario, i.e. having a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, this component is identically zero [40].
We have divided the manuscript in the following sections. In Section II we will provide the action of the theory and its linearised form. In Section III we will derive the field equations of the linearized theory, performing variations with respect to both metric and contorsion. In Section IV, we will decompose the contorsion tensor into its three irreducible Lorentz invariants, and rewrite the field equations in terms of them. In Section V, we will provide new solutions of the linearized field equations in presence of a fermionic source. Finally, in Section VI, we will conclude our analysis with future outlook. For the sake of simplicity, we have written the calculations to obtain the linearized Lagrangian of the theory in the Appendices A and B. Moreover, in Appendix C we calculate the local limit of the theory and show that we can recover a local Poincaré gauge gravity.

II. THE LINEARISED ACTION
In the standard IDG theories the connection is metric and symmetric, i.e., the Levi-Civita one. Therefore, the linear action of IDG is built with the gravitational invariants and derivatives, considering only up to order O(h 2 ), where h is the linear pertubation around the Minkowski metric 1 g µν = η µν + h µν . (1) After substituting the linear expressions of the curvature tensors (Riemann, Ricci and curvature scalar) we can obtain the linearised action as first was shown in Ref. [12]. With this in mind, we wish to generalize the expressions of the curvature tensor when we consider non-symmetric connection. First, we must take into account that the torsion tensor is not geometrically related to the metric, therefore the conditions that are imposed in h µν are not enough to construct the linear action in connection. In order to so lve this problem, we will have to impose that the total connection must be of order O(h), i.e., the same as the Levi-Civita one 2 . Then, by using the relation between the Levi-Civita Γ, and the total connection Γ, we can write where the so-called contortion tensor K must be of the same order as the metric perturbation. This may seem as a strong assumption, nevertheless, as it has been known in the literature, the current constraints on torsion suggest that its influence is very small compared to the purely metric gravitational effects [41,42]. Therefore, considering a higher order perturbation in the torsion sector would make no sense physically. The way to generalize the IDG action will be to consider all the quadratic Lorentz invariant terms that can be constructed with the curvature tensors, the contorsion, and infinite derivatives operators, namely [11] where O denote differential operators containing covariant derivatives and the Minkowski metric η µν , so also the contractions of the Riemann and contorsion tensors are considered in the action. Moreover, the tilde represents the quantities calculated with respect to the total connection Γ. We will expand the previous expression to obtain the general form for the gravitational Lagrangian 3 where the F i ( )'s are functions of the D'Alambertian = η µν ∂ µ ∂ ν , of the form where M S holds for the mass defining the scale at which non-localities starts to play a role. Also, in the previous expression n can be a finite (finite higher-order derivatives theories), or infinite (IDG) number, as we will consider from now onwards unless stated otherwise. However, finite derivatives will incur ghosts and other instabilities. In the last section we shall show how only considering an infinite number of derivatives in Eq.(5) one can avoid the ghosts appearance for the torsion sector, which extends the current results on the metric one [12].
Since one needs to recover the purely metric IDG action when the torsion is zero, there are some constraints in the form of the F functions. In order to obtain these relations, let us write the action of the metric theory around a Minkowski background as presented in [12] and compare it with the Lagrangian in Eq. (4) in the limit when torsion goes to zero Then a straightforward comparison between Eqs.(6) and (7) make it clear that the following relations need to hold In order to check which are the terms that are of order O(h 2 ) in the Lagrangian Eq.(4), we still need to substitute the linearized expressions of the curvature tensors where (µν) and [µν] represent the symmetrisation and anti-symmetrisation of indices respectively. We have computed each term appearing in the Lagrangian Eq.(4) separately. These explicit calculations can be found in Appendix A. Finally, using the above expressions obtained and further simplification would yield the linearized action for metric, torsion and the mixed terms: where In order to get a deeper insight about how the functions involved in Eqs. (13), (14) and (15) are related with thẽ F i ( )'s in Eq.(4), we refer the readers to Appendix B. At this stage, it is interesting to note that L M in Eq. (13) possesses metric terms only and coincides with the Lagrangian of the non-torsion case [12], as expected. On the other hand, L MT in Eq. (14) contains the mixed terms between metric and torsion, whereas L T contains only torsion expressions. It is also worth calculating the local limit by taking M S → ∞. For the detailed calculations we refer the reader to Appendix C. Here we will just summarise that the local limit of the theory is given that the conditions in (C7) meet. The fact that the terms of the form ∇ µ K µ νρ ∇ σ K σνρ are part of the Lagrangian has been proven recently to be a sufficient condition to make the vector modes present in the theory ghost-free in the IR limit [43].
In our previous work [11] the authors showed that one of these modes can be ghost-free also in the UV limit. In this work we shall prove that the two modes can be made ghost-free in this limit.

III. FIELD EQUATIONS
Since the connection under consideration is different from Levi-Civita one, and consequently the metric and the connections are a priori independent, we need to apply Palatini formalism to obtain the field equations. We will have two set of equations; • Einstein Equations: Variation of the action, Eq.(12), with respect to the metric: • Cartan Equations: Variation of the action, Eq. (12), with respect to the contorsion It is interesting to note that δg SM δg µν has already been calculated in Ref. [12], although, calculations involving such a term were performed again as a consistency check. Let us sketch the calculations leading towards the field equations.

A. Einstein Equations
Performing variations with respect to the metric in S M , we find which is compatible with the results in Ref. [12]. For S MT , we have Therefore, the resulting Einstein's equations are: where τ µν = δS matter /δg µν is the usual energy-momentum tensor for matter fields. At this stage, we can resort to the conservation of the energy-momentum tensor, ∂ µ τ µν = 0, to find the following constraints on the functions involved in Eq. (21) We can also prove these constraints by looking at the explicit expression of the functions in Eq. (22) in the Appendix B.

B. Cartan Equations
On the other hand, performing variations with respect to the contorsion, we find and This leads us to the Cartan Equations; where Σ νρ µ = δS matter /δK µ νρ . From these field equations Eqs. (21) and (25) exact solutions cannot be obtained so easily. In order to solve them, we will decompose the contorsion field K µνρ into its three irreducible components.

IV. TORSION DECOMPOSITION
In four dimensions, the torsion field T µνρ , as well as the contorsion field K µνρ , can be decomposed into three irreducible Lorentz invariant terms [6], yielding Tensor q µ νρ , such that q ν µν = 0 and ε ρσνµ q ρσν = 0, such that the contorsion field becomes This decomposition turns out to be very useful, thanks to the fact that the three terms in Eq. (26) propagate different dynamical off-shell degrees of freedom. Hence, it is better to study them separately, compared to all the torsion contribution at the same time. Also interaction with matter, more specifically with fermions, is only made via the axial vector [6]. That is why the two remaining components are usually known as inert torsion. Under this decomposition we will study how the torsion related terms in the linearised Lagrangian in Eq.(12) change, and how to derive the corresponding field equations. Introducing (27), and the constrains of the functions in (22), in (14) we find that the mixed term of the Lagrangian becomes Now, integrating by parts and using the linearised expression for the Ricci scalar we find The first term accounts for a non-minimal coupling of the trace vector with the curvature, that is known for producing ghostly degrees of freedom [44]. Therefore, for stability reasons we impose v 2 ( ) = −3u ( ), finally obtaining In order to obtain the pure torsion part of the Lagrangian we substitute (27) into (15) Now we can proceed to calculate the field equations under the torsion decomposition. Varying the Lagrangian with respect to the metric we find the Einstein Equations: where we can see that the vectorial parts of the torsion tensor do not contribute. On the other hand, performing variations with respect to the three invariants we find three Cartan Equations • Variations with respect to the axial vector S µ • Variations with respect to the trace vector T µ • Variations with respect to the tensor part q µνρ These decomposed equations will help us to find exact solutions of the theory, as we will see in the following section.

V. SOLUTIONS
In Ref. [11], the authors found a particular solution of the IDG with torsion with a fermionic source, where only the axial torsion was considered to be dynamic. Therefore, the Einstein and Cartan equations decoupled and the solutions of the metric were the same as that in the case of IDG. In the following, provided there exists a fermion as a source, and assuming that both axial and trace torsion are different from zero 4 , we will show that we can find additional solutions that were not present in the metric IDG theory. For the IDG theory, solutions were presented in [30]. In order to make our case more clear, we have divided the calculations in the following two subsections. In the first one, we will solve Cartan equations to obtain the torsion tensor, while in the second one we will solve Einstein equations for the metric tensor.

A. Cartan Equations
Let us write down the linearised Lagrangian decomposed into the two vector invariants, where the tensor component of the torsion has been set to zero. Thus, where we have taken into account the constraints on the functions in (22) and the stability condition for the trace vector found in the previous section, namely v 2 ( ) = −3u ( ). Due to these conditions, there are no mixed terms between metric and torsion, so the Cartan and Einstein Equations would be decoupled. Despite these constraints, the torsion part of the Lagrangian (36) is far from being stable, so before finding some solutions we need to explore under which form of the functions the theory does not have any pathologies. By taking a closer look at (36) we realise that, as it is usual in metric IDG, we can make the combinations of the non-local functions to be described by an entire function, which does not introduce any new poles in the propagators, so that we can use the same stability arguments as in the local theory. This means that where the C i are constants we have used the exponential as a paradigmatic example of an entire function. This gives us the following Lagrangian S T µ . From the standard theory of vector fields we know that the last term introduces ghostly degrees of freedom, therefore we need to impose that C 5 = 0. Moreover, the kinetic terms of both vectors need to be positive, hence we also have the conditions C 2 > 0 and C 4 < 0.
At this time we know that our theory is absent of ghosts, and we are ready to find some possible solutions, that we will show that can be singularity-free. We will study the solutions of the trace and axial vector separately in the following Subsections. This is indeed possible since parity breaking terms in the action are not considered, so there are no mixed trace-axial terms.

Axial vector
First, we will consider the Cartan Equation for the axial vector (33) where A µ = δL f ermion δS µ represents the internal spin of the fermion, that minimally couples to the axial vector [6].
We realise that Eq.(39) for the axial vector, S µ , is very similar to the one in Ref. [11], and can be solved analogously. Concretly, the Equations are the same if we impose C 1 = 0 and choose the gauge ∂ µ S µ = 0. Hence, it provides a non-singular solution for the axial torsion, see Ref. [11] for details of the derivation. More specifically, if we assume that the radial component of the axial vector is zero, we will then find that a spherically symmetric solution for a rotating ring singularity is indeed regularised, as found in [11] where A µ is constant, J 0 is the Bessel function and Erfc(z) = 1 − Erf(z) is the complementary error function.

Trace vector
Let us now explore the Cartan Equation for the trace vector (34) We observe that this is just the local Proca Equation for a vector field. Therefore, it will have the same plane wave solutions propagating three stable degrees of freedom. Now, with all the components for the torsion tensor calculated, we will solve Einstein's equations to obtain the corresponding metric h µν .

B. Einstein Equations
Let us recall that Einstein's equations for a fermionic source, where the tensor component of the torsion has been set to zero are given by (32): where τ µν = η σν F µρ F σρ − 1 4 η µν F σρ F σρ , F µν being the electromagnetic tensor. It is clear that this equation is the same as in the pure metric case, since the torsion terms do not contribute. Now, if we apply the constraints that we obtained from the energy-momentum conservation, and ghost-free conditions in the metric sector, see Eq. (22), we are left with the following expression It is interesting to note that this equation has already been studied in Ref. [30], where a non-singular Reissner-Nordström solution were obtained for the same choice of the entire function in ghost free IDG, namely where Φ (r) and Ψ (r) take the following form [30] in which Erf(x) is the error function and F(x) the Dawson function. This solution is non-singular when r → 0 and recasts a Reissner-Nordström when r ≫ M −1 s .

,
We have provided the foundations of the theory of gravitation that can be constructed out infinite covariant derivatives and non-symmetric connection.The main advantages of this theory are the fact that one can introduce effective quantum effects, such as non-locality and internal spin of the particles, which ammeliorate the ring singularities present in the local theory, while preserving the stability of the spacetime. The disadvantage of course is that the calculations are quite tedious compared with GR ones. This issue can be solved if one sticks to the linear level, in which we have shown that solutions that are ghost and singularity free can be found, even with a non-vanishing torsion tensor. The method that we have used to obtain solutions is based on the decomposition of the torsion tensor into its Lorentz invariants, in particular the trace and axial vectors and the tensor part. We have assumed the latter to be zero due to symmetry arguments. Then, we have obtained the field equations of these two vectors and solve them for a fermionic source, see Eqs. (40) and (42), finding the ghost and singularity-free conditions.
We have shown that in Einstein equations the torsion vectors decouple from the metric under the stability conditions, hence obtaining the same metric solutions as in the case of a torsion-free IDG, see Eqs. (46) and (47), that are nonsingular and free of ghosts. Nevertheless, since the axial part of the torsion is different from zero, the phenomenology of the solution would be different to the one in the null torsion case [30], despite sharing the same metric solution. This is because totally antisymmetric part of the torsion, i.e. the axial vector, couples with the internal spin of the fermionic source, which produces a non-geodesical behavior in the fermions, that it is not observed when torsion is set zero (see [45] for details).
For future work in this theory, it will be interesting to calculate the next to leading order of the field equations, so that one can find torsion effects in the effective energy-momentum tensor of Einstein's equations, allowing us to make the UV extension of Poincaré gauge solutions. On the other hand, the influence of torsion and non-locality in quantum experiments remains of interest, so experimental constraints on the viability of the theory could be provided, see ref. [46].
It can be observe that these limits do not impose new relations between the functions. Hence, we have proved that if the previous limits apply, the local limit of our theory is a local PG theory, concretly the one described by the Lagrangian (C5).