Spontaneous generation of geometry in four dimensions

We present the extension to 4 dimensions of an euclidean 2-dimensional model that exhibits spontaneous generation of a metric. In this model gravitons emerge as Goldstone bosons of a global SO(D) X GL(D) symmetry broken down to SO(D). The microscopic theory can be formulated without having to appeal to any particular space-time metric and only assumes the pre-existence of a manifold endowed with an affine connection. We emphasize that not even a flat metric needs to be assumed; in this sense the microscopic theory is quasi-topological. The vierbein appears as a condensate of the fundamental fermions. In spite of having non-standard characteristics, the microscopic theory appears to be renormalizable. The effective long-distance theory is obtained perturbatively around a vacuum that, if the background affine connection is set to zero, is (euclidean) de Sitter space-time. If perturbatively small connections are introduced on this background, fluctuations of the metric (i.e. gravitons) appear; they are described by an effective theory at long distances whose more relevant operators correspond to the Einstein-Hilbert action with a cosmological constant. This effective action is derived in the large N limit, N being the number of fermion species in the fundamental theory. The counterterms required by the microscopic theory are directly related to the cosmological constant and Newton constant and their couplings could eventually be adjusted to the physical values of Mp and \Lambda.


Introduction
It has been pointed out several times in the literature (see e.g. [1,2,3,4]) that gravitons could be considered as Goldstone bosons of some broken symmetry. The non-renormalizability of gravity would therefore be analogous to the bad ultraviolet properties of effective models of hadrons, for instance. This is exactly the point of view that we adopt in this paper. This work is the extension to four dimensions of a previous analysis in two dimensions [5].
In our view, concrete implementations of this idea have been lacking so far (see however [7,8]). By concrete proposal we mean a field theory that does not contain the graviton field as an elementary degree of freedom. It should not even contain the tensor η µν either, as this already implies the use of some background metric and thus the notion of geometry; gravitons being fluctuations around this flat background. Instead, one would like to see all the metric degrees of freedom emerging dynamically, like pions appear dynamically after chiral symmetry breaking in QCD. We would also like the underlying theory to be in some sense 'simpler' than gravity, in particular it should be renormalizable. It was shown in [5] that a theory with all these characteristics can be found in two dimensions.
The purpose of the present work is to extend the 2D model to the far more interesting case of four dimensions. In spite of the complications brought up by the higher dimensionality, the nice features of the 2D case persist. The amount of divergent terms remains under control, at least on shell, and the use of the equations of motion ensures that the final effective theory is precisely that of Einstein-Hilbert plus a cosmological term, complemented by higher dimensional terms normalized by a computable dimensionful constant.
The topic of 'emergent gravity' has recently been a popular one [6] but with the exceptions of the earlier works of [7] and [8] we have been unable to find any proposal meeting our rather restricted criteria of not assuming any pre-existing metric structure whatsoever. On the other hand the proposals in [7,8] appear to be untractable and quantitative results are hard or impossible to get.
Traditionally a major stumbling block in the program that will be developed below is the so-called Weinberg-Witten theorem [9] (see also [10]). The apparent pathology of theories intending to generate dynamically gauge bosons (including gravitons in this category) lies in the fact that the energy-momentum tensor has to be identically zero if massless particles with spin ≥ 1 appear and one insists in the energy momentum tensor being Lorentz covariant. However, our results, while not constituting a mathematical proof, strongly indicate that one can indeed get in 4D an effective low-energy theory with massless composite gravitons, so it is legitimate to ask why Weinberg and Witten theorem would not apply. We note something peculiar in our model, namely the energy-momentum tensor (derived in [5] for 2D and which is exactly the same in 4D) does not have tangent (Lorentz) indices. In fact Lorentz indices are of an internal nature in the present approach as we will see below. The connection between Lorentz and world indices appears only after a vierbein is dynamically generated. But then one is exactly in the same situation as General Relativity where the applicability of [9] is excluded.
The dynamical generation of geometry, combined with the usual renormalization group arguments have rather interesting consequences. Geometry and distance are induced rather than fundamental concepts. At sufficiently short scales, when the effective action does not make sense anymore, the physical degrees of freedom are fermionic. Below that scale there is not even the notion of distance: in a sense that is the shortest scale that can exist. This precludes the existence of an ultraviolet fixed point advocated by some [11] but also indicates that at short distances gravity is non-Wilsonian, suggested by others [12] in an holographic context. This paper is organized as follows: in Section 2 we review the similarities and differences between the 2D and 4D theories and discuss possible counterterms. In Section 3 we study the equations of motion and revise the calculational set-up. Section 4 is devoted to the explicit 1-loop calculation of the one, two and three point functions of the model. In Section 5 we summarize the divergent content of the effective theory and make use of the equations of motion to prove the on-shell renormalizability of the model at leading order. In Section 6 the final effective action is written down and the connection between the free constants of the theory and the physical universal constants of gravity is discussed. Finally in Section 7 we summarize our results and discuss possible extensions.

2-dimensional model review and 4 dimensional extension
We showed in [5] that a consistent and renormalizable 2D model reproducing gravity at long distances could be built. The same model can be considered in 4D. The free Lagrangian density is where ψ a and χ µ are two species of fermions transforming, respectively, under Lorentz (a, b... are tangent space indices) and Diffeomorphisms (µ, ν... are world indices). A spin connection is added to the derivative to preserve the Lorentz×Diff symmetry 1 under local coordinate transformations. It is important to notice that there is no metric to start with; there is no need of one as long as χ µ transforms as a spinorial density. The interaction term in the model, in euclidean conventions, is provided by which obviously does not require any metric to be formulated either. We will assume that we have N species of the previous fermions but we will not add an additional index to avoid complicating the notation. Note that for all practical purposes, Lorentz symmetry is an internal symmetry at this point. The object of the interaction (2) is to trigger the spontaneous breaking of the global symmetry via fermion condensation. Upon use of the equations of motion for the auxiliary field B a µψ a χ µ +χ µ ψ a = −ic and thus ψ a χ µ +χ µ ψ a = 0 ←→ B a µ = 0.
If a non-zero value for the fermion condensate appears the field B a µ acquires an expectation value. Such condensation was seen to happen in 2D (in the large N limit) and it will be present in 4D even for finite values of N as we will see in a moment. Small perturbations above this vacuum expectation value will yield the effective theory of quantum excitations of the theory. Our approach will be a perturbative one and it will correspond to a weak field expansion around the solution for w ab µ = 0.
On translational invariance grounds, for w ab µ = 0 the vacuum of the theory is obtained from the gap equation for the potential Note that the 2N preceding the integral comes from the 2N species of fermions present.
This equation has a general non-trivial solution corresponding to B a µ = M δ a µ (or any SO(D)× GL(D) global transformations of this). This is analogous to the more familiar phenomenon of chiral symmetry breaking in strong interactions and any value of B a µ in the SO(D) × GL(D) orbit is equivalent. For simplicity we will take B a µ = M δ a µ and then the gap equation reduces to an equation for M . In 4D whose formal solution is where µ dc dµ = − N 4π 2 , making M a renormalization-group invariant. In the previous, we introduced the usual mass scale µ to preserve the correct dimensionality of the D-dimensional integral as dimensional regularization is used. For the solution to actually exist we have to require c > 0 if M > µ. If µ > M the solution exists only if c < 0. Therefore c > 0 will be the case we are interested in on physical grounds.
Note that B a µ has the right structure to be identified as the vierbein, and as it was shown in [5], it consistently reemerges in the 2D effective theory to form the determinant of the spontaneously generated metric.
The free fermion propagator of the theory in the broken phase can then be easily found after replacing B a µ by its vacuum expectation value. With a 4D matrix notation A particularity of 2D was that the most general form for B a µ (in euclidean conventions) is a conformal factor times a scale M times a δ a µ . This means that perturbations around the minimum of the potential can only have one physical degree of freedom, the conformal parameter. The other degrees of freedom in B a µ can be removed by suitable coordinate transformations and are thus unphysical (recall that the microscopic theory is fully generally covariant -even without a metric).
The main difference of the 4D case with respect to the 2D case is that the maximum number of possible physical degrees of freedom for a perturbation around the value B a µ = M δ a µ grows up to six instead of one, making the calculation much more complex. Clearly, considering a uni-parametric family of perturbations is far too simple in 4D and does not yield enough information to find the long-distance effective action unambiguously. To by-pass this difficulty, but still keeping the calculation manageable, we have chosen to restrict our considerations to diagonal perturbations, where This form contains four degrees of freedom (rather than six) but is rich enough for our purposes. The validity of our conclusions rely on the assumption that the effective action should be covariant (exactly as the microscopic theory is). This was actually checked in the 2D case using heat kernel techniques. Here we have performed partial checks but we have to assume that covariance holds to draw our conclusions. Another difference with respect to the 2D case is that the integrals involved in the perturbative calculation have potentially a much worse ultraviolet behavior in 4D. We postpone to Sections 3 and 4 the explicit calculations that indicate that the nice characteristics found in the 2D model, in particular renormalizability, persist in the 4D case. However the ultimate reason for renormalizability lies in the very limited number of counterterms that can be written without a metric (and the usual assumption that the ultraviolet behavior is unaltered by the phenomenon of spontaneous symmetry breaking).

Possible counterterms
Before tackling the perturbative derivation of the effective action it is important to list the possible invariants one can write in this theory without making use of a metric. In 2D we had, as already explained, two invariants that could be constructed without having to appeal to a metric, namely L 0 and L I . The latter, upon use of the parametrization B a µ = M e − σ 2 , reduces to i.e. is the cosmological term. In addition, there is the curvature term which in 2D, which in terms of the connection is simply d 2 x dw, purely topological; therefore we do not expect it to appear in the perturbative calculation. Then, apart from the free kinetic term for the fermions, there is only one invariant term that can be written down without a metric. Or what it is tantamount, only one possible counterterm remains to absorb any divergence appearing in the perturbative calculation after integrating out the fermions. This fact enforces the renormalizability of the 2D model in the large N limit in spite of the bad ultraviolet behavior of the integrals. This argument was supported by the explicit calculations presented in [5]. Now in 4D we can write, in addition to L 0 and L I , three more counterterms Note that S D corresponds to the integral over the manifold of the determinant of B a µ . After integrating the fermion fields only S D and S R can appear if general covariance is preserved. We will denote by L D and L R the respective Lagrangian densities.
Finally, there is yet another counterterm one could write without making use of a metric, namely the Gauss-Bonnet topological invariant in 4D, which is of O(p 4 ) in the usual momentum counting.
We did not include the term S R in our action to start with because it does not contain the fermionic fields. It modifies neither the equation of motion (3) for the auxiliary field B a µ nor the gap equation (7) if the connection w µ is set to zero as we initially do (recall that we use a weak field expansion and w µ = 0 is used to determine the vacuum). However, we see that S R is an allowed counterterm in 4D and therefore it needs to be included in the initial action. In fact, any divergence in the theory must be reabsorbable in the two terms S D and S R , as they are the only local counterterms one can write before the symmetry breaking, i.e. before the generation of the metric.
When the auxiliary field B a µ is identified with the vierbein, the parametrization (10) and the equations of motion are used the two counterterms reduce to respectively; i.e. the familiar cosmological and Einstein terms. This will be explained in more detail in the next section.

Equations of motion
Let us write explicitly what the 4D counterterms look like once we replace B a µ by its vacuum expectation value plus perturbations around it. To keep the notation simple, let us consider the case in (10) and Note that because e a µ = δ a µ we can use indistinctively greek and latin indices; they are lowered and raised with a trivially flat metric. This is somewhat similar to what happens in linearized gravity where indices are lowered and raised with η µν .
Let us now work out the equations of motion for the full Lagrangian L 0 + L I + L R . In Section 2 we already discussed the equations of motion for the field B a µ when w ab µ = 0. In addition we have To solve (17) we will only consider the lowest order term in the 1/M 2 expansion, following the usual counting rules in effective Lagrangians based on a momentum expansion. The general solution for the connection is then This is precisely the usual condition between the spin connection and the vierbein in General Relativity (19), characteristic of the Palatini formalism [13] particularized to a conformally flat metric given by e a µ = δ a µ (E ρb is the inverse vierbein). Making use of (18) in (16) we are now allowed to identify the curvature in terms of the scalar field σ Note that in the particular case of a vierbein corresponding to a conformally flat metric one can integrate by parts either of the terms in (20) to obtain the other This term plus a constant times (15) are the only divergences that should appear in the final effective theory upon integration of the fermionic fields for this particular type of perturbations above the vacuum (i.e. those interpretable as a conformally flat metric).
As previously mentioned we shall consider a more general type of perturbations; namely, we will use the diagonal parametrization of the perturbations around the vacuum solution given by (10). This is not the most general one in 4D, but it is enough for our purposes. After the identification of B a µ with the vierbein, this corresponds to a metric This parametrization provides enough generality to the calculation. We can now derive the equivalent expression to (18) for the general diagonal perturbation using (19) to obtain 2 (23) Making use of the equations of motion one can compute the corresponding L R for the general 2 Again we emphasize that although it may seem strange to see latin indices in the derivatives this should not confuse the reader. After the symmetry breaking a vierbein is generated relating world indices with tangent space ones through δ a µ . In the expression (23) we have compiled the entries for w ab µ in a bi-matrix form, but they should not be multiplied; only the index ρ is summed up.
case and expand it in the σ fields. The result up to two sigma fields reads (24) More details on the calculation of (24) can be found in the Appendix. Ignoring for a moment the Gauss-Bonnet invariant, the divergent terms from the perturbative calculation for the general perturbation should match on shell either with (24) or with (25) An extension to the most general perturbation with the full six degrees of freedom should be possible but would require much more effort, which we consider unnecessary at this point as the above parametrization provides enough redundancy. Since the coefficients for the terms in the effective action are universal there should be no loss of generality in the present approach. This of course assumes that general covariance is kept all along the derivation of the effective action and by the regulator, as it should be the case in dimensional regularization.
So far we have explained how the 2D model can be consistently extended to 4D preserving the key features. We study small perturbations around a constant vacuum expectation value for the field B a µ (which does not need to be small itself) corresponding to the solution of the gap equation for w ab µ = 0. In such a theory one can write a limited number of counterterms without making use of a metric. These counterterms are consistent with the usual terms of GR once used the equations of motion. With all these ingredients we are ready to move to the actual perturbative derivation of the effective action.

1-loop structure for a general diagonal perturbation
The effective action that describes perturbations above the trivial vacuum will be given by a polynomial expansion in powers of w µ (x), σ i (x) and their derivatives obtained after integration of the fundamental degrees of freedom. In this section we will derive this effective action diagrammatically. We shall use the diagonal perturbation (10) with 4 degrees of freedom for the vierbein perturbations. For simplicity, we will calculate only the one-point and two-point functions for this rather general case and then particularize to the conformal case (σ i (x) = σ(x)) to compute some three-point functions.
Since perturbation theory in this model has some peculiar features (note in particular the behavior of the fermion propagator) in what follows we shall provide enough details so that the diagrammatic calculation can be reproduced.
Starting from the Lagrangian density L 0 +L I described in Section 2 (note that L R plays no role whatsoever in the integration of the N species of fermions), and using a parametrization of B a µ given by (10), the interaction vertices are

One and two point functions for the fields σ i
With the rules described above and using the propagator (9) we can calculate the first 1-loop diagrams for D = 4 − ǫ. We will not include the factor N in the diagrammatic results presented below. The vacuum bubble diagram is We also compute the one-point function for the different vertices (30) Let us, for this particular diagram, clarify what the origin of the numerical factor is. In the numerator, the 2! comes from the combinatorial possible connections of the external fields σ i . The other 2 is due to the two species of fermions and it is present in all diagrams. In the denominator, 2!2 · 2 comes from the vertex. Since it is a one point function there are no additional factors, however for n-point functions the corresponding n! will be present in the denominator.
(31) Next diagram is the two-point function The numerical factor in this case is composed by 2! 2 in the numerator from the possible contractions of external fields times the two species of fermions. And 2! in the denominator from the diagram being a two-point function and finally the 1/2 from each vertex. We will not elaborate on the combinatorial factors anymore but we write all factors explicitly, even if the notation may be a bit cumbersome, in order to facilitate the check of our results. By O(p 4 ) we mean finite higher order in p 2 contributions.

Diagrams with w ab µ
Now we turn to the diagrams that contain a field w ab µ . The corresponding vertex is The one and two point functions yield Suggesting that diagrams containing only one field w ab µ are zero. For two w ab µ fields we have Where F µν bcef is a complicated structure composed of external momenta and Kronecker deltas of order O(p 2 ). This divergence is of higher order, in the 1/M 2 expansion, than the one of D µν bcef . Now, taking into account that w bc µ = −w cb µ , we can show that More details can be found in the appendix.

Three-point functions
In order to keep the calculations simple, we particularize to the case B a µ = M e − σ(x) 2 δ a µ . The previous results, (28-37) are all valid taking σ i = σ, i = 1, 2, 3, 4. With this simplification we can easily further compute more diagrams. For the field σ we have And also On the other hand, for the field w ab µ we can compute

Summary of divergences
In the previous section we obtained the results of the one-, two-and three-point functions for a general diagonal perturbation, sometimes particularizing to a conformally flat metric to ease the notation. Let us now summarize the results. The divergent part of diagrams (28-31) together with the M 4 piece of diagram (32) add up in the effective action to 3 Note that this term's dimensionality matches (15). Furthermore, it can be proved that the divergent terms in (32) proportional to M 2 p 2 are precisely those corresponding to (24) in momentum space, thus allowing us to recover the first orders of L R for the general diagonal perturbation, which on shell correspond to √ gR.
Diagram (36) has two divergent terms and neither seem to immediately correspond to any of the possible counterterms discussed so far. The first one, that corresponding to D µν bcef , is of order M 2 √ gR. The other, corresponding to F µν bcef is of higher order, i.e. O( √ gR 2 ). Before addressing these apparent new divergences let us particularize to the case of a conformally flat perturbation above the vacuum. Taking σ i = σ, (28-31) plus (32) add up in the effective action to An important remark is in order at this point. Note the peculiar form of e −2σ+ σ 2 ǫ : this factor corresponds to the determinant of a conformally flat metric in D = 4 − ǫ dimensions and it is a remanent of the fact that we used dimensional regularization to calculate the momentum integrals. Of course lim ǫ→0 √ g D = √ g (where g D is the determinant of the D-dimensional metric), but this is telling us that in order to regularize our integrals it is not enough to add a mass scale to match the dimensionality; an ǫ power of the determinant of the metric is also needed to ensure diffeomorphism invariance. That is µ 2 → µ 2 e −σ . Then (43) would read Continuing with the divergences, diagrams (38) and (39) contain terms of order M 4 that are the subsequent orders of the expansion of (43) in terms of σ. As for the terms of order p 2 , one has to express them in position space. The result of diagram (32) for instance is that in position space reads The next diagrams we consider are (38) plus (39) or in position space Now it is clear that the full calculation at order M 2 p 2 resums to the following term in the effective action Note that this term has the same structure that √ gR for a conformally flat metric. This divergence can be absorbed by redefining L R and using the equations of motion. This is already telling us that the theory is renormalizable only on shell 4 . Namely when the spin connection w ab µ corresponds to the Levi-Civita one. In our approach this identification is forced by the use of the equations of motion.
For a general diagonal perturbation one has to consider the momentum dependent O(p 2 ) divergent pieces in (32) and similar diagrams with more external scalar legs. As a check we can see that the momentum dependent terms with two σ i fields faithfully reproduce the O(σ 2 ) piece in the curvature term (24) thus confirming the general covariance of the effective action. Details are relegated to the Appendix.
Let us now retake the issue of the apparent new divergences emerging from (36). To see if they really contribute to the final effective action we have to express them in terms of the σ fields using the available equations of motion. Then in principle, they must either vanish or correspond to a valid counterterm. We argued in Section 2 that there is a third possible counterterm in 4D, the Gauss-Bonnet term, which is a total derivative and should not contribute to the dynamics. But it is reasonable that the higher order divergence in (32) constitutes a piece of that term that will ultimately cancel. Full details of this calculation can be found in the appendix showing how the lower order divergence vanishes and how the higher order term indeed corresponds to a piece of the Gauss-Bonnet term, and in particular vanishes for the conformally flat parametrization of the metric.

Effective action and physical constants
We are now ready to write the effective action we obtain on shell; that is once the spin connection is set to the value obtained after use of the equations of motion and the gap equation is used. We shall present details only for a vierbein corresponding to a conformally flat metric but as previously discussed we have a good check of its validity for the divergent parts of a general diagonal perturbation above the vacuum.
We recall our conventions. We have used euclidean conventions so that the (emerging) metric has signature (+,+,+,+). The effective action at long distances is defined by the functional integral where according to our discussion in Section 2.
The effective action obtained after the diagrammatic calculation of the previous sections is where ǫ + log 4π − γ are renormalized coupling constants that have absorbed the divergences. The M S subtraction scheme is assumed. Note that the finite part of the term proportional to M 2 has received a contribution from the diagrams containing only w ab µ fields, see the appendix. Making use of the gap equation, we can write the previous expression as The resulting effective theory thus describes a geometry with a cosmological term. Sometimes it is stated in the literature [15] that if gravity is an emergent phenomenon and gravitons are Goldstone bosons all interactions should be of a derivative nature and the cosmological constant problem would be in a sense solved. This is not so, as we see a cosmological terms is generated necessarily (both in 2D and 4D), at least in the present approach.
The previous result is not exact of course. The effective action is in fact an infinite series containing higher order derivatives, starting with terms of O( √ gR 2 ) and so on, which are represented by the dots in the previous expression. In fact, as we have seen, a counterterm proportional to Gauss-Bonnet (of order O( √ gR 2 )) is required; finite terms will appear too.
The effective action should also contain a non-local finite piece corresponding to the conformal anomaly (of dimension four in 4D [14]). The conformal anomaly was indeed reproduced in the previous work in 2D [5]. Note that any dimension four term that is generated will be accompanied by a factor of N . The dimension six terms will be of O(N/M 2 ) and so forth. It would be natural to redefine the constant A ′ to include this factor of N in order to keep the counting of powers of N homogeneous. Appealing to covariance arguments we can now express (52) in terms of invariants Next we recall that the classical Einstein action corresponding to the euclidean conventions is [16] Now identifying N 16π 2 M 4 = 2Λ we indeed obtain As we see from the previous discussion, the integration of the fermions (assumed to be the fundamental degrees of freedom in the theory) yields a positive cosmological constant. As for the value of M 2 P , the Planck mass squared, the sign is not really automatically defined. More on this latter.

Fine-tuning and running of the constants
To ensure that the action is renormalization group invariant, thus observable, the following beta function for each free constant in the theory must be obeyed This running has nothing to do with the one generated by graviton exchange and it is thus unrelated to the presence or absence of asymptotic safety that some authors advocate for gravity. At scales µ ≫ M the relevant degrees of freedom are not gravitons, but the 2N fermions appearing in the microscopic Lagrangian. On the other hand, at the moment that fermions become the relevant degrees of freedom, geometry loses its meaning. There is then no 'shorter' distance than M −1 , or at the very least this regime cannot be probed. Note that to realize our physical assumption of having the fermions as fundamental degrees of freedom we should have c > 0 as discussed in Section 2.
These equations do not reflect the complete running of the dimensionless couplings associated to L D and L R , i.e. the constants associated to the cosmological and Einstein-Hilbert terms, but only the one obtained at leading order in N . That is, the 'graviton' loops are not included here; they are suppressed by one power of N if N is large. To see this last statement we recall that the usual power counting rules show that the exchange of the vierbein degrees of freedom would be accompanied by a factor of M −2 P , suppressed by 1/N . Leaving these corrections aside, we note that the two free couplings of the theory have a running that is opposed in sign to the one found in 2D.
To understand the issue it is probably useful to appeal to the QCD analogy. At long distances strong interactions are well described by the pion chiral Lagrangian, parametrized by f π or the O(p 4 ) coefficients, generically named low energy constants (LEC). The LEC are a complicated function of α s , the coupling constant of QCD. The microscopic theory proposed in this paper is the analogous of QCD, while the resulting effective theory (57) is the counterpart of the chiral effective Lagrangian. Then M P and Λ are the LEC of the present theory. The running of α s does not have an immediate translation on the LEC while in the present model, because of its simplicity, the consequences of the running in the microscopic particle reflects directly in Λ and M P . But in addition these constants have an additional running (analogous to doing pion loops in the chiral Lagrangian). The counting of powers of N disentangles both types of running.
At some scale, q ∼ M the effective theory stops making sense. At that moment the relevant degrees of freedom change and, as a result, the metric disappears. Exactly in the same way as for large momentum transfers we do not see pions but quarks. Of course, if there is no metric there is no geometry and, in particular, the notion of distance disappears altogether at length scales below M −1 . From this point of view, gravity is non-Wilsonian.
Let us now try to make contact with the value that the LEC take in gravity. Clearly, there is enough freedom in the theory (by adjusting A ′ and M ) to reproduce any values of Λ and M P . But we also want higher order terms to be small for the effective theory to make sense in a reasonable range of momenta. We may even get rid of all of the high order (O(p 4 ) and beyond) if we take M → ∞ and at the same time we take N → 0 in a prescribed form. Then, in the actual limit, which corresponds to a 'quenched' approximation, we exactly reproduce Einstein-Hilbert Lagrangian, with a cosmological constant, and nothing else. Of course in this limit, the presumed fundamental degrees of freedom disappear completely and we have all the way up to µ = ∞ Einstein's theory -with all its ultraviolet problems; there are no fundamental degrees of freedom providing form factors to cut off the offending divergent integrals.
Of course the N → 0 limit is just the opposite one to the one we have used. All our diagrammatic results are exact in the N → ∞ limit and presumably get large corrections as N approaches zero, but the general features of the model should survive.
Note that M is a fixed quantity in the model and if M 2 P increases, Λ decreases. Taking the actual observed or estimated values of these two parameters we get the value N M 4 ∼ 10 18 m −4 , which is a very low scale. One may think that this may already represent unacceptably large corrections from higher order operators. However, this is not necessarily so because the bounds on R 2 terms are very weak. For instance, the bound k < 10 74 has been quoted for a generic coefficient [17] k of the O(p 4 ) terms. Thus, a relatively low scale for M cannot be really excluded observationally by studying gravitational effects alone and one should be aware of this. However, our own perception tells us that M should be much larger than the value quoted above as the notion of metric certainly makes sense at much shorter distances. We can increase the value of M as much as we want by decreasing the value of N , as previously indicated. We shall not elaborate further on this as it seems too premature an speculation.
Finally we note that the sign of Newton's constant is not determined a priori in this theory due to the subtraction required from the counterterm in L R . This ties nicely with some of the early discussions on induced gravity [18].

Summary
Let us summarize our findings and comment on possible implications of our work. We have proposed a model where 4D gravity emerges from a theory without any predefined metric. The minimal input is provided by assuming a differential manifold structure endowed with an affine connection. Nothing more. The Lagrangian can be defined without having to appeal to a particular metric or vierbein.
Gravity and distance are induced rather than fundamental concepts in this proposal. At sufficiently short scales, when the effective action does not make sense anymore, the physical degrees of freedom are fermionic. At such short scales there is not even the notion of a smaller scale.
The relative technical simplicity of this proposal constitutes its main virtue when compared with previous proposals [7,8], where even semiquantitative discussions appear impossible. Here one is able to derive in full detail the effective action.
A very important aspect of the model is the improvement of the ultraviolet behavior. After integration of the fundamental degrees of freedom all the divergences can be absorbed in the redefinition of the cosmological constant and the Planck mass (as seen from the effective theory point of view, even though the respective counterterms do not have this meaning in the underlying theory. With the running, dictated by the corresponding beta functions, both quantities are renormalization group invariant. In addition the Gauss-Bonnet invariant is renormalized too. This happens in spite of the bad ultraviolet behavior of the propagator and the ultimate reason, we think, is that these are the only counterterms that can be written without having to assume an underlying metric that does not exist before spontaneous symmetry breaking takes place.
At long distances the fluctuations around the broken vacuum are the relevant degrees of freedom and are described by an effective theory whose lowest dimensional operators are just those of ordinary 4D gravity. They of course exhibit the usual divergences of quantum gravity but this now poses in principle no problem as we know that at very high energies this is not the right theory. For q ∼ M one starts seeing the fundamental degrees of freedom. Gravitons are the Goldstone bosons of a broken global symmetry. We have seen how the barrier of the Weinberg-Witten no-go theorem could be overcome.
In a sense the fundamental fermions resolve the point-like 3-graviton, 4-graviton, etc. interactions into extended form factors and this is the reason for the mitigation of the terrible ultraviolet behavior of quantum gravity. However this is only part of the story, because this could be equally achieved by using Dirac fermions coupled to gravity (or any other field for that matter). This would in fact be just a reproduction of the old program of induced gravity [18] and therefore not that interesting. The really novel point in this proposal is that the microscopic fermion action does not contain any metric tensor at all. Then not only is the metric and its fluctuations -the gravitons-spontaneously generated, but the possible counterterms are severely limited in number.
We stop short of making any strong claims about the renormalizability of the model. We can just say that, from our calculations and our experience with the model, renormalizability is a plausible hypothesis (our results actually amount to an heuristic proof in the large N limit). Likewise we do not insist in that the one presented is the sole possibility to carry out the present program, although it looks fairly unique. Clearly a number of issues need further study before the present proposal can be taken seriously but we think that the results presented here are of sufficient interest to make them known to researchers in the field.
A number of extensions and applications come to our mind. Perhaps the most intriguing one from a physical point of view would be to investigate in this framework singular solutions in GR such as black holes. A more in-depth study of the renormalizability issue is certainly required too as there are issues related to the renormalization group to be addressed in the present setting.
The last term we have to explore is the piece 1

(A.12)
This last expression, when expressed in position space, corresponds exactly to (A.11) except for a numerical factor and minus the second derivatives of the fields which are total derivatives and do not appear in the perturbative calculation.