On the renormalization of Poincar\'e gauge theories

Poincar\'e Gauge Theories are a class of Metric-Affine Gravity theories with a metric-compatible (i.e. Lorentz) connection and with an action quadratic in curvature and torsion. We perform an explicit one-loop calculation starting with a single term of each type and show that not only are all other terms generated, but also many others. In our particular model all terms containing torsion are redundant and can be eliminated by field redefinitions, but there remains a new term quadratic in curvature, making the model non-renormalizable. We discuss the likely behavior of more general theories of this type.


Introduction
Metric-Affine theories of Gravity (MAGs) are a vast class of theories of gravity that generalize Einstein's General Relativity (GR) by allowing the connection to be independent of the metric [1][2][3][4].In the most general case, the connection will have both torsion and nonmetricity, and the difference between the independent connection and the Levi-Civita connection is a tensor called distortion.MAGs can be presented in various equivalent ways.As in all theories of gravity, one can use either coordinate bases, or orthonormal bases (a.k.a.tetrads, or vierbeins), or even completely general frames.This amounts to a choice of GL(4) gauge [5][6][7][8][9][10][11][12][13].Furthermore, one can choose the independent variables to be either metric/frame and connection, in which case we say that the theory is in its Cartan form, or metric/frame and distortion, in which case we say that the theory is in its Einstein form [14]. 3 These are mere field redefinitions, and leave the physics unchanged.Thus a given MAG can be presented in (at least) six different equivalent ways.Which of these forms is more convenient depends on what one wants to do, and we shall see that the calculation of quantum effects can be made simpler by choosing the appropriate formulation.
In a general MAG, the number of possible terms in the Lagrangian is huge: there are 12 terms of dimension two and close to one thousand terms of dimension four [14].It is clear that in order to study their properties, one has to start by considering simpler subcases.One can either impose additional symmetries, such as projective invariance [15][16][17], or kinematical constraints.If the distortion is symmetric in the first and third index, or antisymmetric in the second and third, we call the theory a symmetric/antisymmetric MAG, respectively. 4Symmetric MAGs have zero torsion and antisymmetric MAGs have zero nonmetricity (i.e. the connection Appendices, but the final results are discussed in Section 4. The effect of field redefinitions is discussed in Section 5, and Section 6 contains a summary of possible extensions.
Preliminary results of this paper have appeared in O. M.'s Ph.D. thesis [44].

Poincaré gauge theories
In this section we collect well known facts about antisymmetric MAGs and in particular Poincaré gauge theories.This is meant mainly to establish our notation and terminology.In the end of this section we specify the action that we are going to take as a starting point for our quantum calculation.

Kinematics
We use orthonormal frames {e a } in the tangent spaces and {e a } in the cotangent spaces.They are related to the coordinate bases by e a = θ a µ ∂ µ and e a = θ a µ dx µ .The components of the metric in the orthonormal frames are η ab ≡ diag(−1, 1, 1, 1) and the components of a Lorentz connection are A λ a b .They are related to the components in the coordinate bases by 2) The components A abc = θ a λ η bd A λ d c are antisymmetric in b, c.The curvature is defined by the commutator of covariant derivatives: where ) and [v, w] is the Lie bracket.The definition is such that the derivatives of v and w cancel and (2.4) While D v X is a tensor of the same type as X, one can define also the covariant differential DX, which is a tensor with one more covariant index, and DDX, that has two more covariant indices.In components, it is customary to write (DDX) a µν = D µ D ν X a .The antisymmetric part of this tensor is which is different from (2.3).Thus, one has to be careful about the meaning of "commutator of covariant derivatives".The tetrad uniquely defines the Levi-Civita connection, whose coordinate components Γ µ ρ σ are the Christoffel symbols, and are related to the tetrad components Γ µ a b as in (2.2).For the LC connection, the covariant derivative is denoted ∇ and the curvature R defined by is the Riemann tensor.Finally, the last ingredient is the torsion tensor, which is defined as the exterior covariant derivative of the frame field: and in coordinate frames is just Given a metric g µν , a Lorentz connection can be uniquely decomposed into where Γ αβγ (in a coordinate basis) are the Christoffel symbols and K αβγ , antisymmetric in (β, γ), is called the contorsion tensor.It is related to the torsion by (2.10) The curvature tensor of the Levi-Civita connection is the Riemann tensor R µν α β and is related to the curvature tensor of the independent connection as follows: The analog of the Ricci scalar for the connection A µ α β is the unique contraction F = F µν µν , which, up to total derivatives, can be written as One can define four different notions of "total covariant derivative" of the tetrad, using either A or Γ for the latin and the greek index.We observe that the transformation (2.2) is equivalent to the statement that the total covariant derivative of the tetrad, using the same connection on both indices, is zero (this is often called the "tetrad postulate").The total covariant derivative, using the Levi-Civita connection for the greek index and the dynamical connection for the latin one, is nothing but the contorsion: The following table summarizes our notation, which is Yang-Mills-like for the independent connection, and GR-like for the Levi-Civita connection, and we stress that the same notation is used when the connection coefficients refer to a coordinate basis (greek indices) or an orthonormal basis (latin indices): coefficients cov.der.curvature Levi-Civita connection

Dynamics
The standard Lagrangian for PGT is quadratic in curvature and torsion: where T µ = T λ λ µ .The non-consecutive numbering of coefficients comes from [14] and is motivated by compatibility with the action of more general MAGs.In writing this action, we have made two choices.One can write the action in orthonormal bases, simply changing the middle index of torsion and the last two indices of curvature from greek to latin.This is a mere gauge choice and is completely inconsequential to the physical content of the theory.We have chosen to think of the action as a functional of the metric and of the independent gauge field A. This is the choice of variables that makes MAG more similar to a YM theory, and in [14] we called it the Cartan form of the theory.One can choose to present the theory in what we called the Einstein form, where the action is regarded as a functional of the metric and torsion (or equivalently metric and contorsion).This change of dynamical variables is performed by using (2.9). 6n the following we shall consider the case where the only nonzero couplings are c 1 and a 1 .The Lagrangian in orthonormal frames can then be written This writing clearly exposes the variables that have to be varied.Here the metric is to be regarded as a composite of the tetrads, as in (2.1) and F depends on A but not on the tetrad.
This action looks much more complicated when written in Einstein form: (2.18) In the basis defined in [14], we have while the Lagrangian L 3 contains several terms of the form RT T and T T ∇T , and the Lagrangian L 4 contains several terms of the form T T T T , all of which we do not write out explicitly.The terms in L E 2 affect the propagation in flat space, while those in L E 3 and L E 4 only give rise to 3-and 4-point vertices.
Even though the Lagrangian is presented in Cartan form, it is easier to work with Levi-Civita covariant derivatives ∇ rather than the A-covariant derivatives D. As for tensorial quantities such as curvatures F and R, they can coexist in formulas.To read off the divergences and beta functions, we will first bring everything to Einstein form, and finally convert everything back to Cartan form, in a particular basis of invariants.

Setup for one loop calculation
The calculations have been performed in the Euclidean regime, where the action differs from (2.17) by an overall sign.

Expansion
The basic variables in (2.17) are the tetrad and connection.Their variations will be called Then we have Varying the action and using these relations one arrives at the Hessian, which is a quadratic form in X and Z.It is convenient to rewrite all D derivatives as ∇ derivatives plus terms linear in torsion.Furthermore, since the Hessian is a combination of tensors and their covariant derivatives, we can write it in coordinate bases, by converting all latin indices to greek ones.The terms with two derivatives are where the bar over the covariant derivatives indicate that they are computed with the background metric.The occurrence of the nonminimal terms would greatly complicate the calculation, but can be avoided by choosing a suitable gauge.We observe that this is only possible because we started in the vierbein formalism and the Hessian vanishes on fields that are local Lorentz or diffeomorphism transformations.Starting in coordinate bases, one would have to fix only the diffeomorphism invariance (4 parameters) and one can fix this gauge in such a way as to remove the nonminimal terms in the X-X sector.By choosing a suitable Lorentz gauge fixing we can remove also the nonminimal terms in the Z-Z sector.We shall see this in detail in Section 3.3.

Gauge algebra
Due to the structure of the gauge group, the gauge fixing conditions for gravity in tetrad formulation (whether the connection is independent or not) is more complicated than imposing separate gauge conditions for diffeomorphisms and local Lorentz tranformations.This kind of complication already occurs in the case of Yang-Mills fields coupled to gravity [45,46].In the case of Einstein-Cartan theory it has been discussed in [47][48][49][50].We will broadly follow these references.
The fields of antisymmetric MAG are defined on OM, the bundle of orthonormal frames of the base manifold M, and its associated bundles, and the action is invariant under automorphisms of this bundle.One can parametrize locally this group by diffeomorphisms of M and local Lorentz transformations, acting in the standard way on latin and greek indices, respectively: where ω ab = −ω ba is an infinitesimal Lorentz gauge parameter and v µ is an infinitesimal diffeomorphism (a vectorfield on M).Note that the latin indices are inert under this definition of diffeomorphism.The algebra of these transformations is This shows that the local Lorentz transformations are a normal subgroup of the full gauge group, and the diffeomorphisms are the quotient of the full group by this subgroup.Now we observe that whereas the general fluctuation X a µ transforms properly under local Lorentz transformations, the gauge fluctuation δ v θ a µ does not.This would become a serious obstacle in the following, because δ v θ a µ is used in the construction of the ghost operator, and this definition would lead to a non-covariant ghost operator. 7The solution consists in defining a modified action of diffeomorphisms on the fields, which consists of the original action defined above, plus an infinitesimal Lorentz transformation with parameter The action of these modified diffeomorphisms on the fields is where we used (2.15).Their algebra is 7 One can also try to covariantize L v θ a µ by adding and subtracting v ρ Γ ρ ν µ θ a µ .However, the resulting covariant derivatives are only covariant under diffeomorphisms: the derivative acting on θ a µ is not Lorentz-covariant, with the result that L v θ a µ is not a Lorentz vector. 8Here we follow [47,49,50].Alternatively one could also use , where Γ µ a b are the components of the Levi-Civita connection in the orthonormal frame.This would lead to a simpler transformation for θ (the second term would be absent) but a more complicated one for A. where . This is just a different way of parametrizing the full gauge group of the theory, where the normal subgroup has remained untouched.
In background field calculations we have to define how to split the transformation of a field into transformations of its background and fluctuation parts.In the so called "quantum" transformations δ Q the backgrounds are invariant and the whole transformation of the field is attributed to the fluctuation: ) The "background" transformations δ B are defined in such a way that the backgrounds transform as the original field (in particular, Ā transforms as a connection).In detail, the background Lorentz transformations are and the background diffeomorphisms are given by the Lie derivative on all fields.The background diffeomorphisms can be covariantized as above, in particular

Gauge fixed Hessian
We fir the gauge by choosing the Lorentz-like gauge conditions (3.21) In the latter expression it is understood that the covariant derivative is defined in terms of the background Levi-Civita connection for both types of indices.Then, the gauge fixing action is This breaks invariance under the "quantum" transformations while preserving invariance under the "background" transformations.Since the total background covariant derivative of the background tetrad is zero, in the second term we can harmlessly transform all the latin indices to greek ones.Then we remain only with background diffeomorphism invariance, and we do not need to worry about the covariantization that was discussed in the previous section.That discussion only plays a role in the definition of the ghost action.
Setting the parameters α D = α L = 1 (Feynman gauge), integrating by parts and commuting derivatives one gets We see that the first line exactly cancels the unwanted nonminimal terms in the (3.10).At this point, rescaling and performing some integrations by parts, one can write the gauge-fixed Hessian in the form where Ψ = X Z and with where the V 's and W 's are matrices in the space of the fields, with the appropriate free indices.Note that this contains all the terms coming from the second variation of the action, plus the terms in the second line of (3.23).
The operator O must be self-adjoint, which implies the conditions In these formulae, the square brackets are only meant to highlight the grouping of indices, and do not represent antisymmetrization.
With the rescaling (3.24) the quantum fields X and Z have canonical dimension one, V has dimension 1 and W has dimension 2. We do not give the components of these tensors but just indicate the general structures that they contain: Here terms without tensors have to be understood as combinations of the metric.We note that V and W can be treated as small perturbations compared to the leading ∇2 ≈ E 2 term, provided the couplings and the background fields satisfy These conditions are necessary for the applicability of the small-time heat kernel expansion.

Ghost action
The gauge fixing has to be supplemented by the ghost action.We define the ghost operators Here we have the infinitesimal "quantum" transformations applied to the gauge fixing conditions, with the transformation parameters ω a b and v µ replaced by the ghost fields Σ a b and C µ .Then the ghost action is given by where Σa b and Cµ are the antighost fields.All indices here have been suppressed for notational clarity.When one evaluates explicitly the infinitesimal transformations in (3.37), one obtains some operators of the form ∇∇, i.e. containing both the background and the full connection.However, we are ultimately only interested in the effective action at zero fluctuation fields, so we can identify the full and background fields.This means that the ghost operators are constructed entirely with background fields.Since the total covariant derivative of the background vierbein with the background LC connection is zero, we can write all formulas using only coordinate (greek) indices, without producing new terms.The ghost operators are then: (3.39) 4 One loop divergences and beta functions

One loop divergences
The one loop effective action is given by the classical action plus a quantum contribution Γ = S + ∆Γ (1) .(4.1) The effective action could contain non-local terms, but these are related to infrared effects.We are interested here in the UV behavior of the theory and in particular in the logarithmically divergent part, which is local.Thus we can write where L i are operators constructed with the fields and their covariant derivatives, and g i are the corresponding (renormalized) dimensionless couplings.A basis for a class of dimension-four operators L i will be given in Section 4.2.There is a similar expansion for the classical action S, whose coefficients g Bi are the bare couplings.In the presence of a momentum cutoff Λ, the logarithmically divergent part of Γ can be written where µ is a reference scale that has to be introduced for dimensional reasons, and where the coefficients β i are defined by this equation.Thus Here we assume that the bare couplings depend on the UV cutoff in such a way that the renormalized couplings are finite, and as a consequence the renormalized couplings must depend on µ.Then we find that relating the coefficients of the logarithmic divergences to the beta functions.For our theory, where both O and ∆ gh are operators of the form Defining ∇ ′ = ∇ + 1 2 V and redefining W, the term with one derivative can be removed.Note that the connection ∇ ′ now has vectorial torsion and nonmetricity, but we can still apply the standard formulae for the Seeley-De Witt coefficients, which give the following logarithmically divergent part of Tr log O: This formula can also be obtained directly from the expansion of Tr log and application of the off-diagonal heat kernel techniques explained in [51].This calculation will be described in a forthcoming paper [52].
The individual terms in these expressions have been evaluated for the operators O and ∆ gh using the packages xAct and xTensor.Summing them leads to the separate X-Z and ghost contributions that are far too large to be reported here.In appendices A.1 and A.2 we give only the terms proportional to two powers of curvature and torsion, which contribute to the flat space propagator.The sum of all terms of this type generated by the fields X-Z and the ghosts are given in equations (A.5) and (A.11) respectively.These will be sufficient to illustrate some of our main points.Besides such terms, the logarithmic divergences contain most if not all terms of dimension four, as well as terms of higher dimension, as we shall see.

Bases
We will show explicitly only the logarithmically divergent terms that are proportional to two powers of curvature and torsion (and their derivatives).To exhibit them we shall make use of the bases of invariants introduced in [14].In the Einstein form of the theory, up to integrations by parts, one has the following invariants: (4.10) Here tr (12) T b = T a ab , div (3) T αβ = ∇ γ T αβγ etc.The Bianchi identities give three linear relations among the last five invariants.We choose {H RT 3 , H RT 5 } as independent invariants of type R∇T .
In the Cartan form of the theory the list of invariants is longer: (13)µν F (13)  µν , L F F 8 = F (13)µν F (13)  νµ , L F F 16 = F 2 .(4.13) (4.14) The non-consecutive numbering is due to the fact that in [14] we defined bases for more general (not necessarily antisymmetric) MAGs.Compared to the Einstein form of the theory, there are more invariants, but also more relations, so that the number of independent invariants is the same.
Since we are mainly interested in PGT, it is natural to keep in the basis all terms that contain F , and instead remove others.We can choose as a basis the six L F F invariants, that are present in (2.16), plus By listing these terms we see that restricting ourselves to the PGT Lagrangian (2.16), we ignore eight possible dimension-four term that can modify the propagation of torsion and introduce dynamic mixing between the spin-two components of torsion and metric.

Results
Summing B XZ − 2B gh from (A.5) and (A.11) and removing the linearly dependent operators, we arrive at the following expression of the divergent terms in Einstein form The ellipses stand for higher powers of R, T and their ∇-derivatives.
Then going to the basis in Cartan form we arrive at The ellipses stand for higher powers of F , T and their D-derivatives.This is our main result.It shows that starting from a single F F -type term, we not only generate all other F F terms, but also all other terms of dimension four that can affect the flat space propagator, namely terms of the form DT DT and F DT .This is a sign that PGT is non-renormalizable off shell.We shall discuss shortly what happens on shell, but let us mention here that the ellipses include most, if not all, dimension four terms, which are of the form T T F , T T DT , T T T T , as well as higherdimensonal invariants of the form F F F F .To understand the origin of the latter, we go back to equations (3.34,3.35)and note that V contains terms of the form F c 1 /a 1 and W contains terms of the form F 2 c 1 /a 1 .Then the terms in (4.9) proportional to V 4 or W 2 will contain terms of the form F 4 (c 1 /a 1 ) 2 .In the perturbative treatment of the theory, the coupling is proportional to the inverse of c 1 , so the appearance of these terms is somewhat surprising.We can understand better their origin by comparing with the situation in QG, with Lagrangian of the form (schematically) m 2 P R + αR 2 , where m P is th Planck mass and α is the inverse of the higher derivative coupling.This is just the Einstein form of our action (2.18), with T 2 replaced by the Palatini term and neglecting the torsion.In order to have a canonically normalized graviton, one has to rescale the metric fluctuation by 1/ √ α.Then, the kinetic operator is of the form 2 + V µν ∇ µ ∇ ν + W, where V contains terms proportional to R or m 2 P /α and W contains terms proportional to R 2 or Rm 2 P /α [53].In both cases (PGT and QG) the non-derivative part of the kinetic operator contains terms quadratic in curvature, that originate (for example) from the variation of the determinant of the metric.In the case of QG these terms come without any couplings, because the coupling outside the action has been absorbed by the rescaling of the metric fluctuation.In the case of our MAG, the coupling outside the action is c 1 but the graviton is rescaled by √ a 1 , and a dimensionful prefactor remains.In QG the operator is of fourth order and the logarithmic divergences contain at most trW, [54][55][56] so terms of order F 4 cannot be generated.In the case of the specific MAG considered here, the operator is of second order and the logarithmic divergences contain trW 2 , so that F 4 terms can, and do, appear.

On shell divergences
Next, we have to consider the effect of going on shell, or equivalently of performing field redefinitions to remove some of the divergences.We recall that in GR the equation of motion implies the vanishing of the Ricci tensor.This allows us to discard the divergences proportional to R µν R µν and R 2 .Similarly, in our MAG the lower order equations of motion imply that T = 0.All divergences that involve T vanish on shell, which means that all terms involving T are redundant.
To see this from another angle, consider any term of the form T abc X abc where X abc is any combination of T , ∇T , F and ∇F , antisymmetric in a, c.Varying the connection we have Simply choosing δA µ a b = 1 2 θ c µ X c a b , we have δ A T = X.Thus, applying this field redefinition of A to the term T 2 in the action, we can cancel any divergence that contains T , confirming that those terms are redundant.
Thus, in principle, the only important logarithmic divergences are those that involve curvatures, and since torsion vanishes we can consider just R-curvatures (as opposed to F -curvatures, that can form many more invariants).Thus the set of possible counterterms in this theory is the same as in a generic metric theory of gravity.Looking at (4.17), we see that all three invariants quadratic in R-curvatures are generated.The Riemann-squared term can be absorbed by renormalizing the coupling c 1 .Another term can be removed using the Gauss-Bonnet theorem.Unlike in GR, however, the equations of motion do not imply the vanishing of the Ricci tensor, and therefore one divergent term remains, implying that the theory is on shell non-renormalizable already at one loop.

More general theories
From these results one can infer the likely behavior of other MAGs.Let us start from other PGTs with action (2.16), but without the Palatini term (i.e. a 0 = 0).Since the structure of the kinetic operator (3.26), with (3.34,3.35), is generic, one would expect that starting from "almost any" PGT one will generate most dimension-four terms as well as terms up to F 4 c i /a j .It will still be the case that field redefinitions can eliminate divergences containing torsion, but now, given that generically all three RR terms will be present in the bare Lagrangian, all divergences of type RR can be absorbed in redefinitions of the respective couplings.The on-shell nonrenormalizability of the PGT will therefore be due to the higher powers of F , that do not seem likely to be removable by redefining the metric.
Introducing in MAG the Palatini term will generate new contributions proportional to the ratio a 0 /a 1 .For example, the coefficient of L F F 1 in (4.18), entering the beta function of c F F 1 , would become 71 15 + 5 8 This will not change the general structure of the divergences.The lower order equation of motion of the connection will still imply the vanishing of torsion, but the lower order equation of motion of the metric will now additionally imply that the Ricci tensor must be zero.Then the possible counterterms will be the same as in GR, except that now they will appear already at one loop.The discussions in [29] offer some additional hints.For example, it was noted that the theory with Lagrangian 1 2 is renormalizable when one sets T = 0, since then it reduces to QG, but not in general, where it amounts to GR coupled to a Yang-Mills field, a system that was studied in [40].It seems plausible that the theory will have better chances of being renormalizable when its Einstein form will contain the QG operators.In this connection it is worth noting that the known examples of ghost-and tachyon-free MAGs do not contain the QG operators in their Einstein form.Renormalizability demands that the kinetic term of Z contain two derivatives but that of X contain four derivatives [30,31,38].This can be achieved by adding to the action terms of the form (DT ) 2 , in the tetrad formalism.For example, the Lagrangian has a kinetic operator that contains in its leading X−X part an additional power of ∇ 2 compared to the theory we studied in this paper.The elimination of nonminimal terms by our gauge choice would still be possible for this kinetic operator.How generally it can be made to work is an open question.
Besides this very technical point, as already noticed in [14], the number of derivatives depends on the gauge and so could the power counting.This would be analogous to what happens in spontaneously broken gauge theories, where renormalizability is proven in one gauge and unitarity in another.Indeed we know that working with coordinate frames or orthonormal frames amounts to a choice of unitary gauge in a more general GL(4)-invariant formulation of the theory, in the sense that a field transforming under GL(4) by a shift is set to zero.Thus, it seems likely that among all MAGs there will be some that are renormalizable, but it may be necessary to choose a special gauge to see this.Then one can calculate the terms entering in (4.9).We have the following dimension-two parts trW = 10R − 3∇ µ T µ + . . .tr∇ µ V µ = 0 trV µ V µ = − 1 12 F µν µν + . . ., where once again we omit terms containing higher powers of R, T .Summing up everything we get

1
The contribution of X and ZThe commutator of the Riemannian covariant derivatives on the fields are[∇ µ , ∇ ν ] 2 d 4 x