Ghosts in metric-affine higher order curvature gravity

We disprove the widespread belief that higher order curvature theories of gravity in the metric-affine formalism are generally ghost-free. This is clarified by considering a sub-class of theories constructed only with the Ricci tensor and showing that the non-projectively invariant sector propagates ghost-like degrees of freedom. We also explain how these pathologies can be avoided either by imposing a projective symmetry or additional constraints in the gravity sector. Our results put forward that higher order curvature gravity theories generally remain pathological in the metric-affine (and hybrid) formalisms and highlight the key importance of the projective symmetry and/or additional constraints for their physical viability and, by extension, of general metric-affine theories.


I. INTRODUCTION
Higher order curvature theories of gravity in the metric formalism exhibit pathologies caused by the higher order nature of their field equations, thus introducing Ostrogradski ghosts 1 [1,2]. It is broadly believed that the metric-affine formalism (also referred to as first-order or Palatini) avoids these pathologies. The invoked reason is that the independence between the metric and the affine connection leads to second order field equations, so that one could naively expect to avoid Ostrogradski instabilities. The aim of this Letter is to demonstrate the persistence of ghosts even in the metric-affine formalism and that care must be taken for higher order curvature theories regardless the employed formalism.
As a simplified proxy for higher order curvature gravity, we will consider theories depending only on the Ricci tensor, thus called Ricci-Based Gravity (RBG) theories. These theories have received considerable attention, with some prevailing examples like the Eddington-inspired-Born-Infeld theory [3], with its numerous extensions [4] (see [5] for a recent review) and the Ricci-square theories [6]. Although sometimes not explicitly stated, most of the literature on RBG further assumes projective symmetry in the gravity sector by imposing that only the symmetric part of the Ricci tensor contributes to the action. It is well-understood that these symmetric RBG theories do not propagate additional degrees of freedom (dof's) associated to the connection, and this fact can be traced back to the action having a projective symmetry. As a matter of fact, these theories are arguably nothing but General Relativity (GR) in disguise, since they admit an Einstein frame where gravity is described by the usual Einstein-Hilbert action. This frame is achieved after integrating out the non-dynamical connection, whose * jose.beltran@usal.es † adria.delhom@uv.es 1 Throughout this work we refer to arbitrary higher order curvature gravity. Of course, theories constructed in terms of Lovelock invariants are special cases evading this general statement.
effect is then to generate new interactions in the matter sector (see [5,7] for a more detailed discussion). These matter interactions have in turn been used to place stringent constraints to symmetric RBG theories up to date [8,9]. In this Letter we will clarify what happens when the full Ricci tensor contributes to the action so that the projective symmetry is explicitly broken. We will show how these theories relate to the so-called Nonsymmetric Gravity Theories (NGT) introduced by Moffat 2 [10] where the metric carries an antisymmetric part. These NGTs have been shown to exhibit certain pathologies [12][13][14] that will then be inherited by the general RBG. Moreover, we will analyse in a more illuminating and manifest manner the presence of ghosts and Ostrogradski instabilities in RBG, what possess a serious drawback and signals the importance of the projective symmetry as a guide in the search for physically acceptable gravity theories within a metric-affine approach. After properly identifying the ghosts in RBG, we will show that constraining the connection to be torsion-free restores stability, with one extra massive vector field. Finally, we briefly discuss how the pathologies also transcend to the hybrid framework, thus showing the generic pathological nature of higher order curvature theories in any formalism, unless additional symmetries and/or restrictions are incorporated.

II. RICCI-BASED METRIC-AFFINE THEORIES
The RBG theories are described by the action where F is any analytic scalar function depending on the inverse metric g µν and the Ricci tensor R µν of the connection Γ α µβ . The matter sector is assumed to be a collection of minimally coupled fields represented by Ψ. Notice that here, unlike in previous studies where only the symmetric part of the Ricci tensor was considered [3,4,6], we allow for its antisymmetric piece as well, which explicitly breaks the projective symmetry 3 . An important result of this Letter is that there are good reasons to respect projective symmetry and only include the symmetric part of the Ricci because those theories do not exhibit additional dof's, while an explicit breaking of projective symmetry by including the antisymmetric part leads to new pathologies associated to the connection.
The field equations obtained by varying (1) with respect to the metric and the connection are respectively where T µν = − 2 √ −g δSm δgµν is the usual stress energy tensor of the matter sector, T α µν = 2Γ α [µν] is the torsion tensor and we have defined Since we are assuming that matter fields do not couple to the connection, the corresponding hypermomentum that would appear on the RHS of the connection equation vanishes. Including a non-vanishing hypermomentum would not change our conclusions so we will not consider it here for simplicity. Although we could work directly with Eqs.
(2) and (3) in order to understand the number and properties of the dof's, we will do it in a much more transparent manner by going to an Einstein-like frame.

III. NON-SYMMETRIC GRAVITY FRAME
In this section we will show the relation of (1) with the NGT [10]. We start by performing a Legendre transformation of the action (1) as follows where Σ µν is an auxiliary field whose equations of motion are R µν = Σ µν . Plugging them back into (4), one recovers (1), which shows that both actions (1) and (4) are on-shell equivalent. We can now perform the field redefinition 3 A projective transformation shifts the connection by an arbitrary 1-form field Γ α µβ → Γ α µβ + ζµδ α β . Under such a transformation, the Riemann tensor changes as δ ζ R α βµν = 2δ α β ∂ [µ ζ ν] so that R (µν) remains invariant but Rµν does not. The projective symmetry of RBG actions is therefore ensured by not including the antisymmetric part of the Ricci tensor. that will give Σ µν = Σ µν (q, g), and allows us to express the action as where we have defined the potential We notice now that the metric g µν only enters as an auxiliary field for minimally coupled matter fields. The field equations ∂U ∂g µν = √ −gT µν can then be algebraically solved to obtain g µν in terms of q µν and the matter energy-momentum tensor T µν . This solution can be used to integrate g µν out in (6) to obtain (8) It is worth making some comments here before proceeding in order to appreciate the crucial differences between the projective and non-projective invariant theories. In theories with projective symmetry, the metric q µν is symmetric and the q -sector exactly reproduces the first order formulation of GR. Hence, the connection is given by the Levi-Civita connection of q µν , while the matter sector receives new interactions as a consequence of integrating out the space-time metric g µν . The importance of enjoying the projective symmetry lies in that it ensures no new propagating dof's associated to the gravitational sector, and forces the connection to be an auxiliary field that acts as a classical source, generating new matter interactions after being integrated out. We have thus the Einstein frame of these projectively invariant theories (see e.g. [5,7] for a more detailed explanation and also [15] where it was already recognised the appearance of new matter interactions within metric-affine gravities).
The explicit breaking of projective symmetry crucially changes the situation since it translates into the propagation of new dof's that, generally, render the theories unstable. Let us illustrate this by considering vacuum configurations, so that the action is given by where we have restored the Planck mass M Pl for convenience. It is then apparent that the vacuum version of these theories reproduce the NGT of Moffat [10] with a potential U. Former analysis of NSG theories showed that the antisymmetric part of the metric carries a pathological 2-form field that jeopardises their physical viability [12,13]. The instabilities can be seen by considering the antisymmetric sector perturbatively up to quadratic order so that q µν =q µν + √ 2 M Pl (B µν +αB µα B α ν +βB 2q µν ), withq µν an arbitrary symmetric metric, B µν a 2-form field that encodes q [µν] , and where the parameters α and β reflect the possibility of field redefinitions at quadratic order. When expanding around such a background at second order in B µν we have [13]: where H 2 is the usual 2-form field kinetic term, m 2 the mass generated by U, and Γ µ is the projective mode of the connection. It has been argued that the mass can cure some pathologies associated to the curvature couplings in NGT [13], but some instabilities persist [14].
In order to make apparent the presence and nature of the instabilities, we will follow a different approach from those used in analysis of NGT that will allow us to clearly pinpoint the problems, namely we will resort to the Stueckelberg trick (see e.g. [16]). Let us first consider a flat background so the couplings to curvature in (10) disappear. Then, we can restore the gauge symmetry of the 2-form by introducing Stueckelberg fields b µ via the replacement B µν →B µν + 2 m ∂ [µ b ν] , and take the decoupling limit m → 0. There will still be the scalar mode of the gauge invariant 2-form sector described bŷ B µν that we do not need to consider to show the presence of a ghost. The relevant sector in the decoupling limit of the action in a flat background is then where B µν = 2∂ [µ b ν] and Γ µν = 2∂ [µ Γ ν] and we have rescaled Γ µ → 3m √ 2M Pl Γ µ that has been kept finite in the decoupling limit. That the theory is plagued by ghostlike instabilities is then apparent from the mixing B µν Γ µν without the pure Γ-sector kinetic term Γ µν Γ µν . This signals the presence of a ghost owed to the fact that the two eigenvalues of the kinetic matrix have opposite signs. More explicitly, if we diagonalise by means of b µ = A µ + ξ µ , Γ µ = λA µ − (2 + λ)ξ µ , the action (11) reads showing that the presence of a ghost is unavoidable.
On the other hand, the non-minimal couplings to the curvature in (10) will present additional pathologies that have also been discussed for NGT in [13]. Within our approach we can readily see and interpret the nature of these pathologies as Ostrogradski instabilities [2] associated to having higher order equations of motion for the Stueckelberg field 4 . The appropriate decoupling 4 The Ostrogradski instabilities have not been properly identified within NGT and represent yet another problem for NGT besides the pathological asymptotic fall off behaviour identified in [13].
limit now needs to take into account that the curvature scales as R ∼ M −2 Pl and the appropriate limit to be taken is m → 0 and M Pl → ∞ with Λ ≡ mM Pl fixed. In this limit, the Stueckelberg field b µ will feature non-minimal couplings with the schematic form ∼ 1 Λ 2 RBB. Unless these couplings precisely correspond to the Horndeski vector-tensor interactions [17], they will lead to higher order field equations and, consequently, prone to Ostrogradski instabilities. It is interesting to note that one can in fact reach the healthy Horndeski interaction by means of a field redefinition at quadratic order (with an appropriate choice of α and β). However, even in this case, pathologies around important cosmological and astrophysical backgrounds will be expected [18]. Furthermore, the non-existence of healthy higher dimension operators involving curvatures and B µν in 4 space-time dimensions [17,19] signals that higher order terms in B µν will reintroduce the Ostrogradski instabilities. Let us note that, while the ghost around Minkowski can be easily cured by adding a term Γ µν Γ µν (permitted if we allow for a more general framework beyond RBG), the ghosts associated to the non-minimal couplings will be more difficult to evade, if possible at all.
Our discussion clearly shows the presence of five new propagating fields contained in the connection (the two gauge vector fields with two polarisations each plus the scalar in the decoupling limit), in sharp contrast to the projectively invariant case where there are no new propagating modes. It is precisely these modes, arising from the explicit breaking of projective symmetry, that root the pathologies present in general RBG theories. Finally, it is important to realise that these instabilities arise already in the gravitational sector without including matter. However, as explained above, new interactions in the matter sector will be generated after integrating out g µν , and in particular, matter couplings to B µν . It is difficult to envision how these couplings could alleviate the stability problems, and in fact, we would rather expect additional pathologies.

IV. EXORCISING THE GHOSTS: TORSION-FREE THEORIES
So far we have seen how the projective symmetry is of paramount importance to avoid ghost-like instabilities in RBG.Wee will show now how to avoid such instabilities without imposing projective symmetry, but rather constraining the theory to be torsion-free. This can be easily implemented by adding suitable Lagrange multiplier fields enforcing T α µν = 0 so the connection field equations are where we have introduced f µν ≡ ∂f /∂R µν . Let us decompose it as √ −gf µν = √ −hh µν + √ −hB µν with h µν ≡ f (µν) and B µν ≡ f [µν] the symmetric and antisymmetric parts respectively. Since the torsion is constrained to vanish, we can conveniently decompose the connection in terms of the Levi-Civita of h µν and a disformation part L α µν as The above splitting allows to obtain the following relations that we will use below whereL ν ≡ L ν αβ h αβ is one of the two independent traces of the disformation tensor. The trace of the connection equation (13) together with (15) and (13) yields which implies the dynamical constraint On the other hand, contracting the connection equation (13) with h µν , defined as the inverse of h µν , leads to where L µ ≡ L α µα . Thus, we see that there is only one independent trace of the disformation tensor. When inserting the above relations into the connection equation, we arrive at We need to recall now the definition of the non-metricity tensor Q λ µν ≡ −∇ λ h µν = −2h α(µ L ν) λα which also gives the relation L µ = − 1 2 h αβ Q µ αβ ≡ − 1 2Q µ . These relations allow us to express the connection equation (20) as We have then solved for the full connection as the Levi-Civita of h µν plus a disformation part determined by the above non-metricity tensor. We see that the nonmetricity is fully determined by its trace so that there is only one additional vector field associated to the connection. Furthermore, from the constraint (18) we conclude that this vector field propagates 3 degrees of freedom, corresponding to a Proca field. The resolution of the problem will then be completed by considering the nonsymmetric Einstein equations. The symmetric part will allow to algebraically solve for h µν in terms of the matter fields (possibly including the vectorQ µ ). A particular case was considered in [20,21] where it was shown that this action exactly reproduces the Proca Lagrangian for the connection sector. In the more general case under consideration here, there will be more involved interactions for the Proca field, as it was also found in [22]. We can gain some clearer intuition by reformulating these theories in the Einstein frame. For that, let us rewrite our original action as follows: where Σ µν and A µν are two symmetric and antisymmetric auxiliary fields respectively and λ α µν a Lagrange multiplier field enforcing T α µν = 0. We can again perform field redefinitions analogous to those in sec. III, and integrate out the space-time metric g µν by solving its algebraic equation of motion in terms of h µν , B µν , and the matter energy-momentum tensor. After doing that, (22) becomes The connection equations for this action are exactly the same as the ones we solved above (13), so we can simply take the solution for the connection, essentially the splitting (14) with the solution (21), and insert it into the action. Since the solution for the connection satisfies our final action can be expressed as where we have dropped the boundary term∇ µQ µ . Notice that this form of the action reproduces (17) as which recuperates the constraint∇ αQ α = 0. On the other hand, the equation for B µν yields which gives the (non-linear) relation between the field strength ofQ µ and the 2-form B µν , also involving the matter fields. This is a reflection of the fact that our final action (26) is the first order form of a massive vector field with self-interactions and couplings to the matter fields. We can easily reproduce the result in [20,21] for f ∝ R + c 1 R [µν] R [µν] . In that case, the metric h µν is directly g µν , while the effective potential reduces to U ∝ B 2 so that (26) exactly reproduces the first order form of a free Proca fieldQ µ . The same result was found in [22] for theories built with the Ricci-squared scalar, and we have reached here the same conclusion for a general RBG with vanishing torsion in a more explicit form.

V. HYBRID THEORIES
In order to give a more complete discussion of Riccibased theories, we will finally consider them within the hybrid metric-affine framework [23,24] whose action is constructed with the Ricci tensor in both the metric and affine sectors. More explicitly, we will consider actions as where R µν = R µν (Γ) and R µν = R µν (g) are the Ricci tensors of the affine connection and the Levi-Civita connection of g µν respectively. The general pathologies exhibited by these theories can be straightforwardly identified by going to a bimetric version of the above action (see [25] for a discussion on pathologies of hybrid theories). Having a non-linear dependence with R µν (g) already introduces additional ghostly degrees of freedom by itself, so we will restrict the metric sector to the Einstein-Hilbert term and will focus on actions of the form that will suffice in order to illustrate the problems with these theories. We can then follow the same procedure as above for the affine sector by writing the hybrid action in the bimetric form which resembles (6), but it presents some crucial differences that make it even more pathological. If we take the decoupling limit of the g-sector (technically by sending the corresponding Planck mass to infinity), we would still have the NSG sector with the same problems. However, the hybrid theories are generally pathological even if the projective symmetry is imposed on the affine sector so that q µν is a symmetric metric. In that case, the action (31) describes a bimetric theory with an interaction potential given by U(g, q) (see also [5]). As it is well-known, only a very specific tuning of the potential allows to remove the Boulware-Deser ghost [26] of these theories [27] and, consequently, Ricci-based hybrid theories are even more prone to instabilities than their metric-affine formulation. The bi-metric construction fails for theories of the type f (R, R) so our conclusion does not apply to them (see however [25] for pathologies of those theories as well).

VI. DISCUSSION
We have shown that general RBG theories suffer from ghost-like instabilities in the additional dof's associated to the connection and which arise from the explicit breaking of projective symmetry. Having the projective symmetry then proves to be crucial for the viability of RBG, in which case the theories reduce to GR with some new matter interactions. Additionally, we have shown that the projective symmetry is not required if the connection is constrained to be torsion-free and the theory then contains one additional massive vector field. We have extended our discussion to the hybrid framework, where we have found that even with the projective symmetry, the theories typically propagate a Boulware-Deser ghost.
It is worth emphasizing that, although we have only considered RBGs, our results extend to general metricaffine theories, since including more geometrical objects in the action will typically introduce even more potentially unstable propagating modes. Let us stress however that there will be non-pathological higher order curvature theories, like e.g. theories for which the metric and metric-affine formalisms are equivalent [28,29], but the results presented in this Letter clarify that resorting to the metric-affine formalism for higher order curvature theories does not, in general, guarantee the absence of ghosts, thus sharing analogous pathologies with the metric approach. In this respect, one needs to be cautious when considering higher order curvature theories in the metric-affine formalism in order to avoid ghosts (by imposing symmetries and/or constraints), similarly to the metric framework where only judicious combinations of curvatures like the Lovelock terms lead to physically sensible theories. and the Severo Ochoa grant SEV-2014-0398 (Spain). This article is based upon work from COST Action CA15117, supported by COST (European Cooperation in Science and Technology). AD also wants to thank hospitality to the Departamento de Física Teórica de la Universidad de Salamanca.