Agravity up to infinite energy
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Abstract
The selfinteractions of the conformal mode of the graviton are controlled, in dimensionless gravity theories (agravity), by a coupling \(f_0\) that is not asymptotically free. We show that, nevertheless, agravity can be a complete theory valid up to infinite energy. When \(f_0\) grows to large values, the conformal mode of the graviton decouples from the rest of the theory and does not hit any Landau pole provided that scalars are asymptotically conformally coupled and all other couplings approach fixed points. Then agravity can flow to conformal gravity at infinite energy. We identify scenarios where the Higgs mass does not receive unnaturally large physical corrections. We also show a useful equivalence between agravity and conformal gravity plus two extra conformally coupled scalars, and we give a simpler form for the renormalization group equations of dimensionless couplings as well as of massive parameters in the presence of the most general matter sector.
1 Introduction
However, Eq. (2) means that four derivatives act on the graviton: thereby some graviton components have a negative kinetic term.^{1} Classically the theory in (2) is sick [13]: the energy is unbounded from below. A sensible quantum theory might exist, analogously to what happens with fermions: their classical energy is negative, but their quantum theory is sensible.^{2} We will not address this problem here.
We will here study whether this theory can flow up to infinite energy. The Quantum Field Theory (QFT) part can have this property. Realistic TeVscale extensions of the Standard Model (SM) can be asymptotically free [23, 24], and it is not known whether the SM itself can be asymptotically safe, in a nonperturbative regime [25]. The gravitational coupling \(f_2\) is asymptotically free. The difficulty resides in the coupling \(f_0\): a small \(f_0\) grows with energy, until it becomes large.
In this paper we will show that, despite this, the theory can flow up to infinite energy, in an unusual way. In Sect. 2 we present an alternative formulation of agravity that makes it easier to compute its renormalization group equations (RGE): \(f_0\) becomes the quartic of a special scalar, the conformal mode of the agraviton. Then a large \(f_0\) means that the conformal mode of the agraviton gets strongly selfcoupled. The rest of the theory decouples from it, if at the same time all scalars become conformally coupled, namely if all \(\xi \) parameters run to \(1/6\), and all the other couplings reach ultraviolet (UV) fixed points, where all \(\beta \)functions vanish.
In Sect. 4 we isolate the conformal mode of the graviton and show that its strong dynamics is such that \(f_0\) does not hit a Landau pole. This means that the infiniteenergy limit of agravity can be conformal gravity. The unusual phenomenon that allows one to reach infinite energy is that the conformal mode of the graviton fluctuates freely, but the rest of theory is not coupled to it: it becomes a gauge redundancy of a new local symmetry, Weyl symmetry. Since this symmetry is anomalous, conformal gravity cannot be the complete theory: going to lower energy the conformal model of the graviton starts coupling to the rest of the theory, which becomes agravity. This issue is discussed in Sect. 3. In Sect. 5 we propose scenarios where the Higgs mass does not receive unnaturally large corrections. Conclusions are given in Sect. 6. Finally, in the appendix we provide a new and simple expression for the oneloop RGE of all dimensionless parameters (Appendix A) as well as of all dimensionful parameters (Appendix B) in the presence of the most general matter sector, which was not studied before.
2 Agravity
Transformations of coordinates and fields under a Weyl transformation
Dilatation  \(\otimes \)  Diffeomorphism  =  Weyl transformation  

Coordinates  \(\mathrm{{d}}x^\mu \)  \(e^\sigma \mathrm{{d}}x^\mu \)  \(e^{\sigma } \mathrm{{d}}x^\mu \)  \(\mathrm{{d}}x^\mu \)  
Graviton  \(g_{\mu \nu }\)  \(g_{\mu \nu }\)  \(e^{2\sigma } g_{\mu \nu }\)  \(e^{2\sigma } g_{\mu \nu }\)  
Scalars  \(\phi \)  \(e^{\sigma } \phi \)  \(\phi \)  \(e^{\sigma } \phi \)  
Vectors  \(V_{\mu }\)  \(e^{\sigma } V_{\mu }\)  \(e^{\sigma } V_{\mu }\)  \(V_{\mu }\)  
Fermions  \(\psi \)  \(e^{3\sigma /2} \psi \)  \(\psi \)  \(e^{3\sigma /2} \psi \) 
These RGE show peculiar features. Only scalars (not vectors nor fermions) generate \(f_0\) at oneloop, and only if their \(\xi \)couplings have a nonconformal value, \(\xi _{ab}\ne \,\delta _{ab}/6\). The \(\xi \)couplings often appear in the RGE in the combination \(\xi _{ab}+\delta _{ab}/6\), but not always. The coupling \(f_0\) appears at the denominator in the RGE for the \(\xi \)couplings [4].
The above features can be understood noticing that a new symmetry appears in the limit \(f_0 \rightarrow \infty \) and \(\xi _{ab} \rightarrow \,\delta _{ab}/6\): the Weyl (or local conformal) symmetry. The Weyl symmetry is a local dilatation \(\mathrm{{d}}x^\mu \rightarrow e^{\sigma (x)} \mathrm{{d}}x^\mu \) compensated by the special diffeomorphism \(\mathrm{{d}}x^\mu \rightarrow e^{\sigma (x)} \mathrm{{d}}x^\mu \) such that the coordinates \(\mathrm{{d}}x^\mu \) remain unaffected. The various fields rescale under a dilatation as determined by their mass dimension, and they transform under a diffeomorphism as dictated by their Lorentz indices, as summarized in Table 1.
Agravity is invariant under global Weyl transformations: being dimensionless, it is invariant under global dilatations (for which \(\sigma \) does not depend on x); being covariant, it is invariant under local diffeomorphisms.
2.1 Equivalent formulations of agravity
The extra scalar field \(\sigma (x)\), defined in (6), will be called the ‘conformal mode of the agraviton’; for the moment it is introduced as an extra gauge redundancy. We will comment on the corresponding gauge symmetry later on.
The formulations presented in this section certainly are equivalent at the classical level. At quantum level, the equivalence needs to take into account the anomalous transformation law of the pathintegral measure, which amounts to adding an effective \(\sigma \)dependent term in the action. This amounts to \(\sigma \) starting to couple to terms that break scale invariance proportionally to their quantum \(\beta \)functions. These extra couplings only affect RGE at higher loop orders, as we will discuss in Sect. 3.
From Eq. (12) one can rederive the oneloop RGE for \(f_0\) and \(\xi _{ab}\), computed as gravitational couplings in [4]. The two results agree. Furthermore, the same RGE acquire a simpler form if rewritten in terms of the \(\lambda _{ABCD}\) coefficients. The RGE are explicitly written in Eq. (50) in Appendix A, and neither \(f_0\) nor any other coupling appear anymore at the denominator in the RGE.
2.2 The graviton propagator
3 Conformal gravity
In general, conformal gravity is not a consistent quantum theory, because its Weyl gauge symmetry is anomalous. In a simpler language, the dimensionless couplings run with energy as described by their RGE.^{8} The theory is no longer scale invariant, and the conformal mode of the graviton couples to all nonvanishing \(\beta \)functions. The Weylbreaking terms of the agravity Lagrangian are generated back by quantum corrections. The consistent quantum theory is agravity. For this reason our work differs from articles where conformal gravity is proposed as a complete theory of gravity [39, 40].
Nevertheless, conformal gravity can be the consistent infiniteenergy limit of agravity provided that all \(\beta \)functions vanish at infinite energy: the theory must be asymptotically free or asymptotically safe, in other words all couplings other than \(f_0\) have to reach a UV fixed point where all \(\beta \)functions vanish, as we will see.
3.1 Anomalous generation of \(1/f_0^2\)
However, the fact that \(f_2\) and other gauge, Yukawa and quartic couplings start having nonvanishing \(\beta \)functions means that the conformalgravity computation becomes inconsistent when going to higher orders. The conformal mode of the agraviton, \(\sigma \), is a decoupled degree of freedom in the classical Lagrangian of conformal gravity. At quantum loop level, \(\sigma \) starts coupling to all terms that break scale invariance proportionally to their \(\beta \)functions, so that \(\sigma \) can no longer be gauged away.
The \(\, \cdots \, \) in Eq. (27) denote extra terms due to Yukawa couplings (partially computed in [51]) and to gravitational terms (never computed and presumably first arising at order \(f_2^6\)). The full unknown expression might perhaps take the form of a \(\beta \)function of some combination of couplings, given that the Weyl symmetry is not broken when all \(\beta \)functions vanish. Barring this exception, which seems not to be relevant (nature is neither described by a free theory nor by a conformal theory), Eq. (27) means that conformal gravity is not a complete theory: at some loop level, quantum corrections start generating back the extra couplings \(f_0\) and \(\xi _{ab}\) present in agravity.
One important aspect of Eq. (27) is that its righthand side vanishes when all couplings sit at a fixed point, where all \(\beta \)functions vanish. This tells us that the \(f_0\rightarrow \infty \) limit is consistent when the other couplings on the righthandside approach a fixed point.
3.2 Anomalous generation of \(\xi +1/6\)
It is important to note that the righthandside of Eq. (32) vanishes when all couplings sit at a fixed point, where all \(\beta \)functions vanish. This tells us that the \(f_0\rightarrow \infty \) limit is consistent when at the same time \(\zeta _{ab}\rightarrow 0\) and the other couplings approach a fixed point. In this precise limit the conformal mode decouples from the rest of the degrees of freedom.
4 The conformal mode of the agraviton
So far we have shown that a large selfcoupling \(f_0\) of the conformal mode of the agraviton does not affect the rest of physics, provided that the nonminimal couplings \(\xi \) of scalars go to the conformal value and the remaining couplings approach a fixed point. We next address the big issue: what happens to the conformal mode of the agraviton when \(f_0\) is big?
 1.
If \(\beta (f_0)\) grows at large \(f_0\) faster than \(f_0\), then \(\int ^\infty \mathrm{{d}}f_0/\beta (f_0)\) is finite and \(f_0\) hits a Landau pole at finite energy. The theory is inconsistent.^{10}
 2.
If \(\beta (f_0)\) vanishes for some \(f_0=f_0^*\), then \(f_0\) grows to \(f_0^*\), entering into asymptotic safety.
 3.
If \(\beta (f_0)\) remains positive but grows less than or as \(f_0\), then \(f_0\) grows to \(f_0=\infty \) at infinite energy.^{11}
When \(f_0\) grows the path integral receives contributions from fluctuations of \(\sigma \) with larger and larger amplitude, probing the terms in the action of Eq. (34) with higher powers in \(\sigma \). For large \(f_0\) the action becomes dominated by the \((\partial \sigma )^4\) term that has the highest power of \(\sigma \), while the kinetic term becomes negligible. This can happen because all terms in the action have the same number of derivatives. For example, a field configuration \(\sigma (r) = \sigma _0 e^{r^2/a^2}\) contributes as \(S \sim (\sigma _0 + \sigma _0^2)^2/f_0^2\), independently of the scale a, such that for \(f_0\gtrsim 1\) the path integral is dominated by the second term.
This shows that, in the full theory, \(f_0\) can flow to large values without hitting Landau poles: \(\beta (f_0) ={{\mathcal {O}}}(1/f_0)\) at \(f_0 \gg 1\). Having distilled the nonperturbative dynamics of the conformal mode of the agraviton in a simple action, Eq. (34), it seems now feasible to fully clarify its dynamics. We have shown that it hits no Landau poles, excluding case 1. of the initial list. The theory at \(f_0 \gg 1\) should be computable by developing a perturbation theory in \(1/f_0\). We have not been able of excluding case 2: a vanishing \(\beta (f_0)\) at \(f_0 \sim 4\pi \). Nonperturbative numerical techniques seem needed to determine the behavior of the theory at the intermediate energy at which \(f_0 \sim 4\pi \), although this currently needs adding a regulator that breaks the symmetries of the theory (such as a lattice or a momentum averager [56, 57, 58]), obscuring possible general properties (such as the sign of \(\beta (f_0)\)) that could follow from the positivity of the symmetric action in Eq. (34).
5 Scenarios compatible with naturalness of the Higgs mass

\(\Lambda _0\), the energy scale at which the selfcoupling of the conformal mode equals \(f_0 \sim 4\pi \), with \(f_0 \ll 4\pi \) at \(E \ll \Lambda _0\) and \(f_0 \gg 4\pi \) at \(E\gg \Lambda _0\).

\(\Lambda _2\), the energy scale at which the graviton selfcoupling equals \(f_2 \sim 4\pi \), with \(f_2 \ll 4\pi \) at \(E \gg \Lambda _2\).

The Planck scale. As this is the largest known mass scale, in the context of dimensionless theories it can be interpreted as the largest dynamically generated vacuum expectation value or condensate.
In this section we adopt Higgs mass naturalness as a criterion to limit the possible speculations. For example, the simplest possibility in which the Planck scale is identified with \(\Lambda _2\) or \(\Lambda _0\) leads to unnaturally large physical corrections to the Higgs mass from gravity. Naturalness demands \(f_2 \ll 1\) at the Planck scale, while \(f_0\) can be either very small or very large, giving rise to two natural possibilities shown in Fig. 1: \(f_0\ll 1\) at \(M_{\mathrm{Pl}}\) (left panel) and \(f_0 \gg 1\) at \(M_{\mathrm{Pl}}\) (right).
5.1 \(f_0 \ll 1\) at the Planck scale
 (a)
Is naturalness still satisfied, or \(f_0\) becoming strongly coupled at the energy scale \(\Lambda _0\) generates a \({{\tilde{\lambda }}}_{H,S}\) of the same order?
 (b)
Can one get \(\xi _S > 0\) at the Planck scale starting from \(\xi _S = 1/6\) at infinite energy?
The answer to (a) seems to be positive: as shown in Sect. 2 perturbative corrections in \(f_0\) behave like quartic scalar couplings, and thereby renormalize the \({{\tilde{\lambda }}}_{H,S}\) couplings (mixed quartics between the scalars and the conformal mode of the graviton) only multiplicatively, like in the oneloop RGE, Eq. (50d). The same happens at \(f_0 \gg 1\): nonvanishing \({{\tilde{\lambda }}}_{H,S}\) are only generated by \(f_2\) (see Eq. (31)) and by the multiloop anomalous effects discussed in Sect. 3. Nonperturbative corrections in \(f_0 \sim 4\pi \) presumably too renormalize \({{\tilde{\lambda }}}_{H,S}\) only multiplicatively, as the scalars H, S are not involved in the strong selfcoupling of the conformal mode of the graviton.
Concerning issue (b), the answer can be positive in a theory where \(\xi _S\) is very close to \(1/6\) around and above the energy scale \(\Lambda _0\), and a positive \(\xi _S\) is only generated through anomalous running (see e.g. Eq. (32)) at a much lower energy where \(f_0 \ll 1\) by some matter coupling becoming nonperturbative.
Given that nonperturbative physics seems anyhow necessary, we propose here a simpler mechanism for the generation of the Planck mass that relies on a new strong coupling \(g_P\), rather than on a perturbative coupling \(\lambda _S\). Without introducing any extra scalar S (and thereby bypassing the issue of a small \(\lambda _{HS}\)), the Planck scale can be induced by a new gauge group G (under which the Higgs is neutral) with a gauge coupling \(g_P\) that runs to nonperturbative values around the Planck scale, such that condensates f are generated. This is shown as blue curve in Fig. 1. This scenario can be very predictive, as one coupling \(g_P\) dominates the dynamics. The sign of \(M_\mathrm{Pl}^2\) is predicted; however, it is not determined by dispersion relations and seems to depend on the detailed strong dynamics of the model (gauge group, extra matter representations) [63, 64, 65, 66, 67, 68].
5.2 \(f_0\gg 1\) at the Planck scale
A simpler alternative that avoids having a very large RGE scale at which \(f_0\) crosses \(4\pi \) is that \(f_0\) is still large at the Planck scale and never gets small.
The conformal mode of the agraviton only has small anomalous couplings, until its dynamics suddenly changes when some vacuum expectation value or condensate is first generated. We assume that the largest such effect is the Planck mass, which can be generated in the ways discussed in the previous section. Then the treelevel Lagrangian of Eq. (41) describes how \(\sigma \) splits into twoderivative modes. The SO(1,1) symmetry that prevented quantum corrections to the strongly interacting theory with \(f_0\gg 1\) gets broken by \(M_{\mathrm{Pl}}\).
6 Conclusions
In dimensionless gravity theories (agravity), the conformal mode of the agraviton consists of two fields: the usual conformal mode of the graviton and an extra scalar, jointly described by a fourderivative action for a single field \(\sigma \), defined by \(g_{\mu \nu }(x) =e^{2\sigma (x)} \eta _{\mu \nu }\). The selfinteractions of the conformal mode of the agraviton are controlled by a coupling \(f_0\) that is not asymptotically free. In Sect. 2 we recomputed its RGE, and we extended it at the twoloop level, by developing a formulation where \(f_0^2\) becomes an extra scalar quartic coupling. In the presence of scalars, their dimensionless \(\xi \)couplings to gravity become scalar quartics, and the whole agravity can be rewritten as conformal gravity plus two extra scalars with an SO(1,1) symmetry. This perturbative equivalence allowed us to recompute the oneloop RGE equations of a generic agravity theory, confirming previous results [4], writing them in an equivalent simpler form where no couplings appear at the denominator in the \(\beta \)functions, extending them at two loops.
In particular, rewriting \(f_0^2\) as a quartic scalar clarifies why a small \(f_0\) grows with energy in any agravity theory. A Landau pole would imply that agravity is only an effective theory and that the Higgs mass receives unnaturally large corrections.
In Sects. 2, 3 and 4 we have shown that, nevertheless, agravity can be a complete theory. Agravity can be extrapolated up to infinite energy, although in an unusual way: the dimensionless coupling \(f_0\) grows with energy, becomes strongly coupled above some critical RGE scale \( \Lambda _0\), and can smoothly grow to \(f_0\rightarrow \infty \) at infinite energy. Although we have excluded that \(f_0\) has a Landau pole, i.e. that it blows up at finite energy, there is another possibility which we have not studied in the present work: \(f_0\) can approach asymptotically a finite nonperturbative fixed point. Analyzing this possibility requires having control on intermediate regimes where \(f_0 \sim 4 \pi \), which is beyond our current ability.
Provided that all scalars are asymptotically conformally coupled (all \(\xi \)couplings must run approaching \(1/6\)) and all matter couplings approach a fixed point (possibly a free one, like in QCD) in the UV, the simultaneous \(f_0\rightarrow \infty \) limit turned out to be consistent. In this case and in the limit of infinite energy the conformal mode of the agraviton fluctuates freely and decouples from the rest of the theory. In the UV limit the theory can then be computed by viewing \(\sigma \) as a gauge redundancy, which can be fixed with the Faddeev–Popov procedure. One then obtains conformal gravity at infinite energy. In Sect. 3 we provided the oneloop RGE at the zero order in the expansion in \(1/f_0^2\) and \(\xi +1/6\), including the most general matter sector.
However, the conformal symmetry is anomalous and its violation is dictated by renormalization group equations that describe how the dimensionless parameters that break conformal symmetry, \(f_0\) and \(\xi +1/6\), are generated at a fewloop order. As a result, at energies much above \(\Lambda _0\) the conformal mode of the agraviton \(\sigma \) is strongly selfcoupled (\(f_0\gg 1\)) and fluctuates wildly, being negligibly coupled to other particles. In Sect. 4 we isolated its peculiar action and showed that, despite the strong coupling, it can be controlled through its symmetries. The action is sufficiently simple for its full quantum behavior to be simulated on a Euclidean lattice.
The anomalous multiloop RGE which generate \(1/f_0^2\) and \(\xi +1/6\), are not (yet) fully known, but it is already possible to discuss the physical implications of this theory. We assume that the largest mass scale dynamically generated through vacuum expectation values or condensates is the Planck scale. Two situations discussed in Sect. 5 can lead to a scenario where the Higgs mass does not receive unnaturally large corrections. If \(f_0 \ll 1\) at the Planck scale one obtains agravity at subPlanckian energies: we wrote the most general RGE for massive parameters, and we argued that a new gauge group with a fermion in the adjoint can become strongly coupled around the Planck scale and successfully generate \(\bar{M}_\mathrm{Pl}\), without generating a Planckian cosmological constant (this mechanism was never explored before in the context of agravity). Alternatively, \(f_0 \gg 1\) at the Planck scale seems to be a viable possibility: in this case the scalar component of the agraviton is above the Planck scale.
Footnotes
 1.
This can maybe be avoided introducing an infinite series of higher derivative terms [12], but the resulting gravity theories contain infinite free parameters and are not known to be renormalizable.
 2.
The ample literature of ‘ghosts’ was critically reviewed in [14]; for later work, see [15, 16, 17, 18, 19, 20, 21, 22], where it was proposed that a fourderivative variable q(t) contains two canonical degrees of freedom (d.o.f.), \(q_1 = q\) and \(q_2 = \dot{q}\), with opposite timereflection parity, such that usual Teven representation (\(q_1 x\rangle = x x\rangle \) and \(p_1x\rangle =i\frac{\mathrm{{d}}}{\mathrm{{d}}x}x\rangle \)) must be combined with the Todd representation (\(q_2 y\rangle = i y y\rangle \) and \(p_2y\rangle =\frac{\mathrm{{d}}}{\mathrm{{d}}y}y\rangle \)) obtaining consistent results (positive energy, normalizable wave functions, Euclidean continuation), although the interpretation of the resulting negative norm is unclear.
 3.
Different statements in the literature (even recent) appear either because some previous results contained wrong signs or because some authors use computational techniques that try to give a physical meaning to power divergences, obtaining gaugedependent and cutoffdependent results. Claims that a runaway potential with very small \(f_0\) can mimic Dark Energy do not take into account bounds on extra graviton components.
 4.
We omitted the topological Gauss–Bonnet term.
 5.
Similar remarks have been made in the context of Einstein gravity (rather than in agravity) in [33, 34, 35], where it was found that Einstein gravity is equivalent to conformal gravity plus a single conformally coupled scalar. Similar statements have been made in a different theory without the \(R^2/6f_0^2\) term in [36, 37].
 6.
For a recent summary see [38].
 7.
We treated the Weyl transformation as a change of variables in field space. We could equivalently have seen it as an extra gauge redundancy. In this alternative formalism, using the Fadeev–Popov procedure to fix both diffeomorphisms and the Weyl symmetry, the gauge fixing in Eq. (15) avoids mixed terms in the ghost system; the ghosts for the Weyl gauge fixing are nondynamical and integrating them out is equivalent to the modified diffeomorphism transformation law of the traceless graviton, Eq. (22).
 8.
One might hope that all couplings stay at fixed points at all energies, but this possibility is excluded because one must recover a nonconformal behavior at low energies for phenomenological reasons.
 9.The group quantities \(C_{2G}\), \(C_{2F}\) and \(T_F\) are defined as usual in terms of the generators \(t^A\) in the representation R as follows: For example, for the vector representation of SU(N) we have
 10.
 11.
For example, this behavior is realized if the \(\beta \)function has the form \(\beta (f_0) = f_0 Z(f_0) \) with \(Z(f_0) = b/(f_0^2 + f_0^{2})/(4\pi )^2\) with \(b>0\). Then at low energy \(f_0\) runs logarithmically towards \(f_0 \rightarrow 0\), and at large energy \(1/f_0\) runs logarithmically towards \(1/f_0 \rightarrow 0\). Indeed, the full solution for \(f_0^2>0\) is \( f_0^{2} =t + \sqrt{1+t^2}\) where \( t = b\ln ( \bar{\mu }/\Lambda _0)/{(4\pi )^{2}}\) and \(\Lambda _0\) is the transition scale at which \(f_0\sim 1\).
 12.
Alternatively, since conformal invariance can be seen as an inversion \(x^\mu \rightarrow y^\mu = x^\mu /x^2\) followed by a translation and by another inversion, one can more simply check that the action is invariant under the inversion: \(\mathrm{{d}}^4 x \rightarrow \mathrm{{d}}^4 y/y^8\), \( \sigma (x)\rightarrow \sigma (y) + \ln y^2\) and \([\Box _x\sigma + (\partial _x\sigma )^2]= y^4 [\Box _y\sigma + (\partial _y\sigma )^2]\). The transformation rule of \(\sigma \) under the coordinate transformation \(x^\mu \rightarrow y^\mu = x^\mu /x^2\) can be obtained by recalling its general definition in (6) and that we are assuming here a conformally flat metric, i.e. \(g_{\mu \nu }(x) = e^{2\sigma (x)} \eta _{\mu \nu }\).
 13.
Many authors refuse to view the theory with higher derivative as legitimate because of the consequent ghosts; see e.g. [59] for attempts to discard the \((\Box \sigma )^2\) term. Accepting the presence of higher derivatives allows one to describe the Weyl anomaly as ordinary RGE running of \(f_{0,2}\), rather than by modifying Einstein gravity by adding a complicated ‘quantum anomalous action’ [60, 61, 62] which encodes the anomalous behavior of generic undefined theories of gravity.
 14.
It can be rewritten in a covariant form as the mass term resulting, in the unitary gauge, from the spontaneous symmetry breaking of general coordinate invariance acting separately on ordinary fields and on composite fields, \(\hbox {GL}_h \otimes \hbox {GL}_\rho {\mathop {\rightarrow }\limits ^{f}}\hbox {GL}\) (see e.g. [69]).
Notes
Acknowledgements
We thank Andrei O. Barvinsky, Agostino Patella, Alberto Ramos, Francesco Sannino, Ilya L. Shapiro, Andreas Stergiou, Nikolaos Tetradis, Enrico Trincherini and Hardi Veermae for useful discussions. This work was supported by the ERC grant NEONAT.
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