Structural identities in the first order formulation of quantum gravity

We study the self-consistency of the first order formulation of quantum gravity, which may be attained by introducing, apart from the graviton field, another auxiliary quantum field. By comparing the forms of the generating functional $Z$ before and after integrating out the additional field, we derive a set of structural identities which must be satisfied by the Green's functions at all orders. These are distinct from the usual Ward identities, being necessary for the self-consistency of the first order formalism. They relate the Green's functions involving the additional quantum field to those containing a certain composite graviton field, which corresponds to its classical value. Thereby, the structural identities lead to a simple interpretation of the auxiliary field.


I. INTRODUCTION
The first order formulation of gauge theories has a simple form involving only cubic interactions, which are momentum-independent. This simplifies the computations of the quantum corrections in the usual second-order gauge theories, that involve momentum dependent three-point as well as higher-point vertices [1][2][3][4][5][6][7][8][9][10][11][12]. In quantum gravity, for example, the first order formulation allows to replace an infinite number of complicated multiple graviton couplings present in the second-order Einstein-Hilbert (EH) action, by a small number of simple cubic vertices [7,8]. The EH action has the form (1.1) where G N is Newton's constant and the affine connection Γ λ µν may be written in terms of the metric g µν as Γ λ µν = 1 2 g λσ (g µσ,ν + g νσ,µ − g µν,σ ) . (1. 2) The Ricci tensor R µν (Γ) is given by As noted by Einstein and Palatini [13] at the classical level, it is possible to treat both g µν and Γ λ µν as being independent quantities. In this first order action, the equation of motion for Γ λ µν yields Eq. (1.2). At the quantum level, it has been shown [7,8] that the radiative corrections computed using the first-order and second-order EH actions are the same.
In a previous paper [14], we examined a set of structural identities which are necessary for the consistency of the first order formulation of the Yang-Mills theory. The purpose of the present work is to extend this analysis to quantum gravity, where the corresponding structural identities ensure the self-consistency of the first order formulation.
To this end, we introduce a source j µν for the graviton field and also a source J λ µν for the other auxiliary field which is treated as an independent field, and consider the generating functional Z[J, j] of Green's functions. We then compare the functional dependence of Z[J, j] on the sources in the original first order formalism with that obtained after making a suitable shift which enables to integrate out the auxiliary field. The equality of these functional forms leads to a set of structural identities among the Green's functions which must be satisfied to all orders. Such identities are complementary but distinct from usual Ward identities, being necessary for the internal consistency of the first order formulation of quantum gravity.
These identities show that in the first order formalism the Green's functions containing only external graviton fields are the same as the corresponding ones which occur in the second-order formulation. Furthermore, these identities relate the Green's functions involving external auxiliary fields to those involving a certain composite graviton field. This combination, which corresponds to the classical value of the auxiliary field, contains graviton fields which are pinched at the same spacetime point. It is well known [15][16][17] that composite fields can lead to short-distance singularities. In the present case, such singularities are important for the cancellations of ultraviolet (UV) divergences arising from loop diagrams, which are necessary for the implementation of the structural identities.
Since calculations in quantum gravity have a great algebraic complexity, in section 2 we recast the analysis done in [14] into an alternative form, which is based on a simpler diagonal representation of the first order formulation of the Yang-Mills theory [18]. Such a representation exhibits similar features to those in quantum gravity, yet it is easier to handle algebraically. With this insight, we consider in section 3 the Lagrangian and the generating functional of Green's functions in a corresponding diagonal representation of the first order formulation of quantum gravity [8]. In section 4 we derive a transparent structural identity, which has been explicitly verified to one-loop order, that clarifies the meaning of the auxiliary field in this formulation. In section 5, we study another structural identity satisfied by the Green's functions and examine the cancellations between the loop UV divergences and the short distance singularities arising from the tree diagrams involving composite fields. A brief discussion of the results is given in section 6. Several details of the one-loop calculations are outlined in the Appendix.

II. STRUCTURAL IDENTITIES IN YANG-MILLS THEORY
The first order formulation of the Yang-Mills theory involves the gluon A a µ and the auxiliary fields F a µν whose dynamics is described by the Lagrangiañ This form has a single vertex F AA but leads to a rather involved non-diagonal matrix-propagator, containing the (AA), (F F ) and the mixed (F A), (AF ) propagators. On the other hand, if we make in Eq. (2.1) the change of variable one obtains the Lagrangiañ which involves two cubic vertices AAA , H AA as well as two simple propagators (AA), (HH). The BRST renormalization of this diagonal formulation of the Yang-Mills theory has been implemented to all orders in reference [18] (see also [19][20][21]). The complete Lagrangian density for this formulation in covariant gauges is where ξ is a gauge-fixing parameter andη a , η b are ghost fields. In addition, we will also introduce the external sources J a µν andj a µ as follows: L source =J a µνH a µν +j a µ A a µ . (2.5) The generating functional for Green's functions is given by the path integral where N is a normalization factor and S = d d x L I YM . This equation has a form which is suitable for functional differentiation with respect toJ andj, and therefore for obtaining the Green's functions.
If we were to setJ a µν = 0 at the outset (so that we would consider Green's functions with only external fields A a µ ) and make the change of variable in the functional integral then one can integrate out theH a µν field and find that is the generating functional for the second-order theory, characterized by the Lagrangian density together with the source termj a µ A a µ . This establishes the important property that the Green's functions with only external gluon fields are the same in both approaches.
We now consider using Z[J,j] withJ = 0 and examine what changes occur in the first order formalism when there are external fieldsH a µν . To this end, we consider in place of Eq. (2.7), the shift This leads, after integrating out theH a µν field, to the alternative form of the generating functional Comparing the forms Eq. (2.6) and Eq. (2.11) of the generating functionals and differentiating these with respect toJ andj, leads to a set of structural identities among the Green's functions, which must be satisfied to all orders. Such structural identities lead to relations between the Green's functions involvingH fields and the Green's functions that contain the composite fields gf abc A b µ (x)A c ν (x). These identities hold both for the finite as well as for the UV divergent parts of the Green's functions. We have verified them explicitly for the divergent contributions to one-loop order, using dimensional regularization in 4 − 2 dimensions.
Taking the functional derivatives of Eq. (2.6) and Eq. (2.11) with respect toJ a µν (x) andj b α (y) atJ =j = 0 and equating the results, we obtain the relation Eq. (2.12) represents a quantum mechanical extension of the relation:H a µν = gf abc A b µ A c ν which holds at the classical level. This structural identity is clearly satisfied in the tree approximation, since the mixed (HA) propagator vanishes in our theory. The right hand side of Eq. (2.12) also vanishes at the tree level. To order g 2 , we find that the divergent parts on both sides of Eq. (2.12) are, in momentum space, equal to Applying δ 2 /δJ a µν (x)δJ b αβ (y) to Eqs. (2.6) and (2.11) and equating the results, leads to where This identity is also manifestly satisfied at three level, where the first term on the right hand side of Eq. (2.14) is just equal to the tree propagator (HH). To order g 2 , one can verify, in momentum space, that the divergent part on both sides of Eq. (2.14) are equal to It is worth to point out that in the identities Eq. (2.12) and Eq. (2.14), the origin of the divergent contributions is different. On the left hand side of these equations, UV divergences come from one-loop graphs, whereas on their right hand side short-distance singularities arise from the pinched tree graphs.
Further differentiations of Eqs. (2.6) and (2.11) with respect toJ andj yield a set of structural identities which are complementary to the usual Ward identities. One can compare the above identities with the ones found in reference [14] in the usual first order formulation of the Yang-Mills theory. (see, for example, equations (3.1); (3.2) and (4.1); (4.2) in [14]). One can see that the structural identities obtained in the diagonal representation have a much simpler form. This feature will be especially useful for the derivation of the corresponding identities in quantum gravity.

III. DIAGONAL FORMULATION OF FIRST ORDER PALATINI ACTION
Instead of using g µν and Γ λ µν as independent fields in the action Eq. (1.1), it turns out to be more useful to employ the independent combinations [7] h µν = √ −gg µν (3.1) and Thus, we arrive at the following Lagrangian density in d spacetime dimensions In order to proceed, h µν is expanded about a flat metric η µν (κ = √ 16πG N ) h µν (x) = η µν + κφ µν (x). . However, it leads to an involved nondiagonal matrix propagator containing (φφ), (GG) and the mixed propagator (φG). As in the Yang-Mills theory, it proves convenient to use a diagonal formulation of the first order EH action [8]. This may be achieved by making the change of variable (compare with Eq. (2.2)) (3.6) In this way, the Lagrangian density Eq. (3.3) may be written in the form where M µν λ πτ σ (φ) is given by Thus, we see that the Lagrangian Eq. (3.7) involves three cubic vertices HφH , φHφ and φφφ . On the other hand, it leads only to two uncoupled propagators (φφ) and (HH). Using the Lagrangian in Eq. (3.7) in the Euler-Lagrange equation for the field H λ µν we obtain the classical solution Since M (η) + κM (φ) = M (η + κφ), this can be written as which is just the classical second-order Einstein-Hilbert Lagrangian. This demonstrate the classical equivalence of the two formalisms [8].
In order to obtain the propagator of the φ µν field, we use the gauge fixing Lagrangian With this gauge fixing, the contributions coming from the vector ghost fields d ν ,d µ are [22] Thus, the complete diagonal first order Lagrangian density becomes (3.14) Next, we will also introduce the external sources J µν λ and j µν as follows Using the above results, the generating functional for Green's functions will be given by the Feynman path integral where N is a normalization factor and S = d d xL I EH . This equation has a form which is appropriate for generating the Green's functions, through the application of functional differentiations with respect to J µν λ and j µν . Performing the following shift in the functional integral (3.16) we obtain This enables to integrate out the auxiliary field H λ µν and leads to the alternative form of the generating functional 1 The Green's functions which involves the field H λ µν necessarily will have H λ µν appearing in a closed loop. But the propagator for the field H λ µν is momentum independent and hence the associated loop momentum integrals vanish if we use dimensional regularization [8]. where L II EH is the second-order EH Lagrangian, with ghosts and gauge fixing, which may be written as We remark that the alternative generating functional (3.19) has a certain similarity to the corresponding functional in the Yang-Mills theory given by (2.11). The analogy is even more pronounced if we note that the coefficient of the source J is just the result found at the classical level, given in Eq. (3.10) for the auxiliary field. We also note though that unlike Eq. (2.11), the term quadratic in the source J for the auxiliary field contains field dependency. Using a similar procedure to that employed in the Yang-Mills theory [see Eqs. (2.7)-(2.9)], one can show that the Green's function with only external gravitons are the same in the first and second-order formulations.

IV. CONSISTENCY CONDITION FOR THE AUXILIARY FIELD
Taking the functional derivatives of Eqs. (3.16) and (3.19) with respect to J µν λ and j πτ at J µν λ = j πτ = 0 and equating the results, we obtain the structural identity where (M −1 ) λ µν ρ αβ is defined in Eq. (3.6). Equation (4.1) is manifestly satisfied at tree level because its left hand side vanishes since there is no mixed Hφ propagator in the theory. Similarly, the right hand side of (4.1) vanishes in the tree approximation (order zero in κ).
To one-loop order, the contribution to the left hand side of Eq. (4.1) arises from the Feynman diagrams shown in Fig. 1. Using dimensional regularization, the contribution from the graph in Fig. (1a) actually vanishes, while the divergent contribution coming from graph in Fig. (1b) is given in momentum space, in the gauge ξ = 1 by where . . . stands for terms with other tensor structures which are given in a general gauge in the Appendix. In order to obtain the contribution coming from the right hand side of Eq. (4.1), one must expand the expression in the square bracket in a power series of κφ. Using for simplicity a schematic notation, we obtain where M (φ) is a linear function of φ which is given by Eq. (3.8). Substituting this result in the right hand side of Eq. (4.1) one gets, up to order κ 2 , two terms that involve, respectively, a product of three and four φ fields. Using Wick's theorem we can verify that the contribution from the cubic term comes from the Feynman graph shown in Fig. 2a.
This diagram corresponds to a three-point tree Green's function which has however two coordinates pinched at the same space time point x.
As we have mentioned earlier, such a composite field leads to an ultra-violet (short distance) contribution. Using the appropriate expression for the three-point graviton vertex [8], one can evaluate in momentum space the contribution from Fig. (2a). The result turns out to be in agreement with the one given in Eq. (4.2). One must also consider the contribution involving four φ fields in (4.1), which arises due to the last term in Eq. (4.3). This is represented by the Feynman diagram shown in figure 2b. However, such a pinched contribution vanishes upon using dimensional regularization. Thus, we see that the features which appear in the structural identity (4.1) are similar to those which occur in the Yang-Mills theory via the identity (2.12). It is straightforward to generalize Eq. (4.1) to an arbitrary number of graviton fields, namely This relation may be interpreted as being, in quantum gravity, a quantum-mechanical extension of the relation (3.10) which holds at the classical level.

V. A SECOND STRUCTURAL IDENTITY
Applying δ 2 /δJ µν λ (x)δJ πτ ρ (y) to Eqs. (3.16) and (3.19) and equating the results, yields where we have introduced the shorthand notation The structural identity (5.1) is clearly satisfied at tree level, where the (HH) propagator is precisely equal to We will now examine the perturbative expansion of each side of Eq. (5.1). To one-loop order, the contributions to the left hand side of this equation arise from the Feynman diagrams shown in Fig. 3.
Using dimensional regularization, the contribution from graph in Fig. (3a) vanishes, while the divergent part of the contribution from graph in Fig. (3b) is given in momentum space, in the gauge ξ = 1, by where . . . denote terms with other tensorial structures which are explicitly given in the Appendix. Next, let us examine the contributions of order κ 2 which come from the terms on the right hand side of the Eq. (5.1). Such a contribution could arise from the first term, but this vanishes upon using dimensional regularization. We note here that these composite fields contributions are pinched at the spacetime points x and y. The Feynman diagrams associated with such Green's functions are shown in Fig. 4. The divergent contributions coming from Fig.  (4a) (there is an additional graph with x ↔ y on the left side), turn out to add up to a result which agrees with that given in Fig. (3b). We have also verified this identity at one loop order for any dimension d in a general gauge (see the Appendix). On the other hand, the contributions coming from Fig. (4b) vanish upon using dimensional regularization in momentum space. Verifying this result beyond order κ 2 becomes exceedingly difficult, as it would involve going beyond one loop order.
We remark that the structural identity (5.1) resembles the identity (2.14) which holds in the Yang-Mills theory. Therefore, as we have seen in the previous examples, the structural identities in the diagonal representation of the first order Yang-Mills and gravity theories exhibit many similar features, though they are not identical.

VI. DISCUSSION
We have examined the structural identities which ensure the self-consistency of the first order formulation of quantum gravity. Since calculations in this theory are quite involved even at one loop order, we have studied first the structural identities in the diagonal representation of Yang-Mills theory, which are simpler. It turns out that these identities in Yang-Mills theory have many features similar to the ones which occur in the diagonal representation of the first order quantum gravity. With this insight, we have compared the forms of the generating functionals Z[J, j] of Green's functions in quantum gravity, before and after integrating out the auxiliary field H λ µν . Differentiations of these two forms with respect to J λ µν and j µν yield a set structural identities given in Eqs. (4.1) and (5.1) which are complementary but distinct from the usual Ward identities. These identities show that the Green's functions containing only external external graviton (gluon) fields are the same in the first and second-order formulations.
These identities also lead to connections between the Green's functions involving the field H λ µν and the Green's functions in second-order formulation containing a composite graviton field that corresponds to the classical value of the auxiliary field. Eq. (4.4) provides a simple interpretation of the auxiliary field H λ µν . An interesting feature is that the implementation of the structural identities requires cancellations between UV divergences which appear in one-loop diagrams, and the short-distance singularities that occur in the tree graphs which are "pinched" at the same spacetime points. This shows that the singularities arising at the tree level from the composite graviton field are necessary for the first order formulation of quantum gravity to be consistent. These identities have also a practical utility as they allow us to compute more efficiently, in the second-order formulation, some involved composite field expectation values in terms of those containing the local auxiliary field.
Recently [23,24], we have introduced a Lagrange multiplier field which restricts the path integral in quantum gravity to the field configurations that satisfy the classical equations of motion. It was shown that such a method has the effect of eliminating all multi-loop corrections beyond the one-loop order and doubling of the usual one-loop contributions. This makes it possible to renormalize the EH action while retaining unitarity. Such a treatment was employed both in the second-order as well as in the first order formulations of quantum gravity. In the later case, one may also expect to have a corresponding set of structural identities which are necessary for the consistency of the theory. This is an interesting issue which requires further study.

Appendix: One loop results
We employ the same Feynman rules, procedures and conventions as in section 3 of [8], with the replacements φ µν → κφ µν , G λ µν → κH λ µν and S → S/κ 2 (S is the action), so that the coupling constant κ is shown explicitly in the vertices and in the resulting Green's functions.

The general approach for the calculation of massless one-loop self-energies
Let us consider some generic field theory for fields φ a , where a represents a collection of Lorentz indices, or indices for internal degrees of freedom such as in the case of Yang-Mills theories. The most general form of the momentum space massless self-energy is where n is the number of independent tensors which can be obtained from the general symmetry properties of Π ab (k) (for instance, in the case of the photon self-energy there are the two independent tensors η µν and k µ k ν ). Upon contracting Eq. (A.1) with each of the n tensors, we obtain n linear equations for the coefficients C i , containing several scalar integrals of the following type (using Einstein summation convention for the labels a and b) Next, we simplify the n scalars I ab (p, q)T i ab (k), using the relations so that all the scalar integrals acquire the form In the simplest cases r = s = 1. When considering gauge theories with a general gauge fixing parameter, we can have r = 1, 2 and s = 1, 2. Since we are using a dimensional regularization procedure, the only non-vanishing integrals are the following we have a factor of i relative to Eq. (3.31a) of [8] which takes into account that we are Wick rotating back to Minkowski space).
Once we have all the relevant scalar integrals in Eq. (A.5), we may solve the linear system of algebraic equations for constants C i in (A.1). In general this procedure would be of no practical use unless we make use of computer algebra algorithms, as we have done in the present work (for example the tensor basis for the self-energy of the H field has 22 rank 6 tensors). Using this procedure, we have previously obtained the expression for the graviton self-energy in the diagonalized first order formalism [8].
For d = 4 − 2 Eq. (A.5) yields the following UV pole part It is worth mentioning that the present approach is as an example of the Passarino-Veltman reduction method [25].
2. The Hφ self-energy Figures (1a) and (1b), without the external free propagators, are the two contributions for the mixed Hφ-fields self-energy. Since the internal H field propagator in Fig (1a) has no momentum dependence, the loop momentum integration vanishes when using dimensional regularization. The self-energy contribution from Fig. (1b) can be expressed as where the tensors (T Hφ i ) λ µν πτ , i = 1 . . . 12, are given by Using the Feynman rules given in Ref. [8], we obtain the equivalent of I ab (p, q) in Eq. (A.1). Next, using the general approach described in subsection ( 1) we obtain the coefficients for the Hφ self-energy shown in Table (I). These expressions have an UV part, which arises when d = 4 − 2 and → 0, given by the numbers in Table (II  3. The H-field self-energy Figures (3a) and (3b), without the external free propagators, are the two contributions for the H-field self-energy. Since the internal H field propagator in Fig (3a) has no momentum dependence, the loop momentum integration vanishes when using dimensional regularization. The self-energy contribution from Fig. (3b) can be expressed as δ ρ π δ λ ν η µτ + δ ρ π δ λ µ η ντ + η πν δ λ µ δ ρ τ + η πµ δ λ ν δ ρ τ , (A.10a) Using the Feynman rules given in Ref. [8], we obtain the equivalent of I ab (p, q) in Eq. (A.1). Next, using the general approach described in subsection ( 1) we obtain the coefficients for the HH self-energy shown in Table (III). These expressions have an UV part, which arises when d = 4 − 2 and → 0, given by the numbers in Table (IV    between P Hφ (i) and C Hφ (i) . Then, using the results for C Hφ (i) given in Table (I) we obtain the entries of Table (V) for the mixed Hφ propagator. Table ( and C HH (i) . Then, using the results for C HH (i) given in Table (III) we obtain the entries of Table (VII)  for the H-field propagator.

Explicit verification of the structural identities
The right side of Eq. (4.1), at order κ 2 , can be written as We are using the same notation employed for the self-energies (p is the integration momentum, k is an external momentum and q = p + k); D µνρσ (p) is the graviton propagator and V µναβγδ (p, q, r) is the cubic graviton vertex given respectively by the Eqs. (3.25a) and (3.25e) of [8] 2 . Since M is just a combination of products of ηs and δs, each of the several terms in Eq. (A.15) can be cast in the same form as (A.11), in terms of the tensor basis given by Eqs. (A.8).
After a straightforward calculation, we have obtained a result which coincides with the one-loop contribution to the mixed Hφ propagator (the same structure constants shown in Table (V)), which confirms the identity (4.1) for any dimension and gauge parameter. Similarly, the second term on the right side of Eq. (5.1), at order κ 2 , can be written as Eq. (A.17) can also be cast in the same form as (A.12), in terms of the tensor basis given by Eqs. (A.10). After a straightforward calculation, we have obtained a result which coincides with the one-loop contribution to the H-field propagator (the same structure constants shown in Table (VII)), which confirms the identity (5.1) for any dimension and gauge parameter. We point out that these structural identities relate elements of the basic Feynman rules, in each formalism, in a non-trivial way. There is also a practical implication since these identities allow one to compute some rather involved composite field expectation values in a much more efficient way by using the auxiliary field instead.