AAgravity
Alberto Salvio a and Alessandro Strumia b a Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madridand Instituto de F´ısica Te´orica IFT-UAM/CSIC, Madrid, Spain b Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italyand National Institute of Chemical Physics and Biophysics, Tallinn, Estonia
Abstract
We explore the possibility that the fundamental theory of nature does not con-tain any scale. This implies a renormalizable quantum gravity theory where thegraviton kinetic term has 4 derivatives, and can be reinterpreted as gravity minusan anti-graviton. We compute the super-Planckian RGE of adimensional grav-ity coupled to a generic matter sector. The Planck scale and a flat space canarise dynamically at quantum level provided that a quartic scalar coupling andits β function vanish at the Planck scale. This is how the Higgs boson behavesfor M h ≈
125 GeV and M t ≈
171 GeV . Within agravity, inflation is a generic phe-nomenon: the slow-roll parameters are given by the β -functions of the theory, andare small if couplings are perturbative. The predictions n s ≈ . and r ≈ . arise if the inflaton is identified with the Higgs of gravity. Furthermore, quadrati-cally divergent corrections to the Higgs mass vanish: a small weak scale is naturaland can be generated by agravity quantum corrections. Contents a r X i v : . [ h e p - ph ] M a y Introduction
We propose a general principle that leads to a renormalizable and predictive theory of quan-tum gravity where all scales are generated dynamically, where a small weak scale coexistswith the Planck scale, where inflation is a natural phenomenon. The price to pay is a ghost-like anti-graviton state.The general principle is: nature does not possess any scale . We start presenting how thisprinciple is suggested by two recent experimental results, and next discuss its implementationand consequences.
1) Naturalness
In the past decades theorists assumed that Lagrangian terms with positive mass dimension(the Higgs mass M h and the vacuum energy) receive big power-divergent quantum correc-tions, as suggested by Wilsonian computation techniques that attribute physical meaning tomomentum shells of loop integrals [1]. According to this point of view, a modification of theSM at the weak scale is needed to make quadratically divergent corrections to M h naturallysmall. Supersymmetry seems the most successful possibility, but naturalness got increasinglychallenged by the non-observation of any new physics that keeps the weak scale naturallysmall [2].The naturalness problem can be more generically formulated as a problem of the effectivetheory ideology, according to which nature is described by a non-renormalizable Lagrangianof the form L ∼ Λ + Λ | H | + λ | H | + | H | Λ + · · · (1)where, for simplicity, we wrote only the Higgs potential terms. The assumption that Λ (cid:29) M h explains why at low energy E ∼ M h we only observe those terms not suppressed by Λ : therenormalizable interactions. Conservation of baryon number, lepton number, and other suc-cessful features of the Standard Model indicate a large Λ > ∼ GeV . In this context, gravitycan be seen as a non-renormalizable interaction suppressed by Λ ∼ M Pl = 1 .
22 10 GeV .However, this scenario also leads to the expectation that particles cannot be light unlessprotected by a symmetry. The Higgs mass should be M h ∼ Λ and the vacuum energyshould be V ∼ Λ . In nature, they are many orders of magnitude smaller, and no protectionmechanism is observed so far.We assume that this will remain the final experimental verdict and try to derive the theo-retical implications.Nature is maybe telling us that both super-renormalizable terms and non-renormalizableterms vanish and that only adimensional interactions exists.2 ) Inflation Cosmological observations suggest inflation with a small amount of anisotropies. How-ever, this is a quite unusual outcome of quantum field theory: it requires special mod-els with flat potentials, and often field values above the Planck scale. Let us discuss thisissue in the context of Starobinsky-like inflation models [3]: a class of inflation modelsfavoured by Planck data [4]. Such models can be described in terms of one scalar S (possi-bly identified with the Higgs H ) with a potential V ( S ) and a coupling to gravity − f ( S ) R .Going to the Einstein frame (i.e. making field redefinitions such that the graviton kineticterm R gets its canonical coefficient) the potential gets rescaled into V E = ¯ M V /f , where ¯ M Pl = M Pl / √ π = 2 . GeV is the reduced Planck mass. Special assumptions such as V ( S ) ∝ f ( S ) make the Einstein-frame potential V E flat at S (cid:29) M Pl , with predictions com-patible with present observations [3]. However this flattening is the result of a fine-tuning:in presence of generic Planck-suppressed operators V and f and thereby V E are generic func-tions of S/M Pl .Nature is maybe telling us that V E becomes flat at S (cid:29) M Pl because only adimensionalterms exists. The principle
These observations vaguely indicate that nature prefers adimensional terms, so that ideasalong these lines are being discussed in the literature [5].We propose a simple concrete principle: the fundamental theory of nature does not pos-sess any mass or length scale and thereby only contains ‘renormalizable’ interactions — i.e.interactions with dimensionless couplings.This simple assumption solves the two issues above and has strong consequences.First, a quasi-flat inflationary potential is obtained because the only adimensional poten-tial is a quartic term V ( S ) = λ S | S | and the only adimensional scalar/gravity coupling is − ξ S | S | R , so that V E = ¯ M ( λ S | S | ) / ( ξ S | S | ) = ¯ M λ S /ξ S is flat at tree level. At quantumlevel the parameters λ S and ξ S run, such that the slow-roll parameters are the beta-functionsof the theory, as discussed in section 4.Second, power divergences vanish just because of dimensional reasons: they would havemass dimension, but there are no masses. Vanishing of quadratic divergences leads to a mod-ified version of naturalness, where the weak scale can be naturally small even in absence ofnew physics at the weak scale [6] designed to protect the Higgs mass, such as supersymmetryor technicolor.In this context scale invariance is just an accidental symmetry, present at tree level be-cause there are no masses. Just like baryon number (a well known accidental symmetry of3he Standard Model), scale invariance is broken by quantum corrections. Then, the log-arithmic running of adimensional couplings can generate exponentially different scales viadimensional transmutation. This is how the QCD scale arises.The goal of this paper is exploring if the Planck scale and the electro-weak scale can arisein this context.
The theory
The adimensional principle leads us to consider renormalizable theories of quantum gravitydescribed by actions of the form: S = (cid:90) d x (cid:113) | det g | (cid:20) R f + R − R µν f + L adimSM + L adimBSM (cid:21) . (2)The first two terms, suppressed by the adimensional gravitational couplings f and f , are thegraviton kinetic terms. The third term, L adimSM , is the adimensional part of the usual StandardModel (SM) Lagrangian: L adimSM = − F µν g + ¯ ψi /Dψ + | D µ H | − ( yHψψ + h.c. ) − λ H | H | − ξ H | H | R (3)where H is the Higgs doublet. The last term, L adimBSM , describes possible new physics Beyondthe SM (BSM). For example adding a scalar singlet S one would have L adimBSM = | D µ S | − λ S | S | + λ HS | S | | H | − ξ S | S | R. (4)We ignore topological terms. Non renormalizable terms, the Higgs mass term M h | H | andthe Einstein-Hilbert term − M R/ π are not present in the agravity Lagrangian, becausethey need dimensionful parameters. The Planck mass can be generated dynamically if, atquantum level, S gets a vacuum expectation value such that ξ S (cid:104) S (cid:105) = M / π [7]. Theadimensional parameters of a generic agravity in 3+1 dimensions theory are:1. the two gravitational couplings f and f ;2. quartic scalar couplings λ ;3. scalar/scalar/graviton couplings ξ ;4. gauge couplings g ; Other attempts along similar lines assume that scale or conformal invariance are exact symmetries at quan-tum level. However, computable theories do not behave in this way. The second term is also known as ‘conformal gravity’.
4. Yukawa couplings y . The graviton g µν has dimension zero, and eq. (2) is the most generic adimensional actioncompatible with general relativistic invariance. The purely gravitational action just containstwo terms: the squared curvature R and the Weyl term R − R µν . They are suppressed bytwo constants, f and f , that are the true adimensional gravitational couplings, in analogyto the gauge couplings g that suppress the kinetic terms for vectors, − F µν /g . Thereby, thegravitational kinetic terms contain 4 derivatives, and the graviton propagator is proportionalto /p . Technically, this is how gravity becomes renormalizable. In presence of an inducedPlanck mass, the graviton propagator becomes M p − p = 1 M (cid:20) p − p − M (cid:21) (5)giving rise to a massless graviton with couplings suppressed by the Planck scale, and to a spin-2 state with mass M = f ¯ M and negative norm. Effectively, it behaves as an anti-gravityPauli-Villars regulator for gravity [8]. The Lagrangian can be rewritten in a convoluted formwhere this is explicit [9] (any field with quartic derivatives can be rewritten in terms of twofields with two derivatives). The f coupling gives rise to a spin-0 graviton with positive normand mass M = f ¯ M + · · · . Experimental bounds are satisfied as long as M , > ∼ eV .At classical level, theories with higher derivative suffer the Ostrogradski instability: theHamiltonian is not bounded from below [10]. At quantum level, creation of negative energycan be reinterpreted as destruction of positive energy: the Hamiltonian becomes positive,but some states have negative norm and are called ‘ghosts’ [11]. This quantization choiceamounts to adopt the same i(cid:15) prescription for the graviton and for the anti-graviton, suchthat the cancellation that leads to renormalizability takes place.We do not address the potential problem of a negative contribution to the cross-section forproducing an odd number of anti-gravitons with mass M above their kinematical threshold.Claims in the literature are controversial [12]. Sometimes in physics we have the right equa-tions before having their right interpretation. In such cases the strategy that pays is: proceedwith faith, explore where the computations lead, if the direction is right the problems willdisappear.We here compute the one loop quantum corrections of agravity, to explore its quantum be-haviour. Can the Planck scale be dynamically generated? Can the weak scale be dynamicallygenerated? The list would be much shorter for d (cid:54) = 4 . Gauge couplings are adimensional only at d = 4 . Adimensionalscalar self-interactions exist at d = { , , } . Adimensional interactions between fermions and scalars exist at d = { , } . Adimensional fermion interactions exist at d = 2 . Quantum agravity
The quantum corrections to a renormalizable theory are mostly encoded in the renormal-ization group equations (RGE) for its parameters. Ignoring gravity, a generic adimensionaltheory of real scalars φ a , Weyl fermions ψ j and vectors V A can be written as L = −
14 ( F Aµν ) + ( D µ φ a ) ψ j i /Dψ j −
12 ( Y aij ψ i ψ j φ a + h.c. ) − λ abcd φ a φ b φ c φ d (6)and its RGE have been computed up to 2 loops [13]. We here compute the one-loop β functions β p ≡ dp/d ln ¯ µ of all parameters p of a generic agravity theory, obtained addingto (6) the generic scalar/graviton coupling − ξ ab φ a φ b R (7)as well as the graviton kinetic terms of eq. (2) and the minimal gravitational interactionsdemanded by general relativity.Previous partial computations found contradictory results and have been performed withad hoc techniques in a generic background. We instead follow the usual Feynman diagram-matic approach, expanding around a flat background (the background is just an infra-redproperty which does not affect ultra-violet divergences). Eq. (2) is the most general action containing adimensional powers of the fundamental fields.Concerning the purely gravitational sector, apparently there are extra terms such as D R or R µναβ . However the first one is a pure derivative; and the second one can be eliminated usingthe topological identity R αβµν − R µν + R = 14 (cid:15) µνρσ (cid:15) αβγδ R αβµν R γδρσ (cid:39) . (8)The combination suppressed by f in eq. (2) is the square of the Weyl or conformal tensor,defined by subtracting all traces to the Riemann tensor: W µναβ ≡ R µναβ + 12 ( g µβ R να − g µα R νβ + g να R µβ − g νβ R µα ) + 16 ( g µα g νβ − g να g µβ ) R. (9)Indeed W αβµν = 12 R αβµν − R µν + 16 R (cid:39) R µν − R . (10) One of the authors (A. Salvio) adapted the public tools [14]; the other author (A. Strumia) employed hisown equivalent codes.
6e expand around the flat-space metric η µν = diag(1 , − , − , − as g µν = η µν + h µν suchthat R (cid:113) | det g | = ( ∂ µ ∂ ν − η µν ∂ ) h µν + 14 ( h µν ∂ h µν − h αα ∂ h αα + 2 h µν ∂ µ ∂ ν h αα − h µα ∂ α ∂ β h βµ ) + · · · . (11)To quantise the theory we follow the Fadeev-Popov procedure adding the gauge fixing term S gf = − ξ g (cid:90) d x f µ ∂ f µ , f µ = ∂ ν h µν . (12)We choose a non-covariant term quadratic in h µν , such that gauge fixing does not affect thegraviton couplings. At quadratic level the purely gravitational action is S = 12 (cid:90) d k k h µν (cid:20) − f P (2) µνρσ + 1 f P (0) µνρσ + 12 ξ g ( P (1) µνρσ + 2 P (0 w ) µνρσ ) (cid:21) h ρσ (13)where P (2) µνρσ = 12 T µρ T νσ + 12 T µσ T νρ − T µν T ρσ d − (14a) P (1) µνρσ = 12 ( T µρ L νσ + T µσ L νρ + T νρ L µσ + T νσ L µρ ) (14b) P (0) µνρσ = T µν T ρσ d − (14c) P (0 w ) µνρσ = L µν L ρσ (14d)are projectors over spin-2, spin-1 and spin-0 components of h µν , written in terms of d = 4 − (cid:15) , T µν = η µν − k µ k ν /k and L µν = k µ k ν /k . Their sum equals unity: ( P (2) + P (1) + P (0) + P (0 w ) ) µνρσ = ( η µν η ρσ + η µσ η ρν ) . Inverting the kinetic term of eq. (13) we find the gravitonpropagator D µν ρσ = ik (cid:20) − f P (2) µνρσ + f P (0) µνρσ + 2 ξ g ( P (1) µνρσ + 12 P (0 w ) µνρσ ) (cid:21) . (15) Gravitational ghost couplings
One needs to path-integrate over Fadeev-Popov ghosts η α and ¯ η µ with action S ghost = (cid:90) d x d y ¯ η µ ( x ) δf µ ( x ) δξ α ( y ) η α ( y ) . (16)By performing an infinitesimal transformation x µ → x (cid:48) µ = x µ + ξ µ ( x ) one finds the transfor-mation of h µν at first order in ξ µ and at a fixed point x µ : δh µν = − ( ∂ µ ξ ν + ∂ ν ξ µ ) − ( h αµ ∂ ν + h αν ∂ µ + ( ∂ α h µν )) ξ α . (17)The ghost action then is S ghost = (cid:90) d x { ∂ α ¯ η µ ( ∂ α η µ + ∂ µ η α ) + ∂ ν ¯ η µ [ h αµ ∂ ν η α + h αν ∂ µ η α + ( ∂ α h µν ) η α ] } . (18)7n order to verify gravitational gauge-independence we will perform all computations usinga more general gauge fixing, given by eq. (12) with f µ = ∂ ν ( h µν − c g η µν h αα ) . We fix gauge invariance of vectors V µ adding to the Lagrangian the standard ξ -gauge term − f V / − ξ V ) with f V = ∂ µ V µ , such that the vector propagator is i ( − g µν + ξ V k µ k ν /k ) /k .Such term does not depend on gravitons, so that the gauge-invariances of spin-2 and spin-1particles are fixed independently. We write the gauge-covariant derivatives as D µ φ a = ∂ µ φ a + iθ Aab V Aµ φ b when acting on scalarsand as D µ ψ j = ∂ µ ψ j + it Ajk V Aµ ψ k when acting on fermions (the gauge couplings are containedin the matrices θ A and t A ). In this paper we adopt dimensional regularization and the mod-ified minimal subtraction renormalization scheme MS , with energy scale ¯ µ . The anomalousdimensions γ of the fields are defined as γ = d ln Z/d ln ¯ µ in terms of the wave-functionrenormalization constants Z = 1 + δZ . Gravitational couplings of fermions are derived fol-lowing the formalism of [15]. By computing the one-loop corrections to the matter kineticterms we find the one-loop anomalous dimension of scalars (4 π ) γ Sab = Tr Y a Y † b − (2 + ξ V ) θ Aac θ Acb + (19) + f (cid:32) c g − c g − δ ab + 3 c g ξ ab c g − (cid:33) + 3 c g − c g + 134( c g − ξ g δ ab and of fermions (4 π ) γ F = 12 Y a Y † a + (1 − ξ V ) t A t A + (20) − f + f − c g + 7 c g c g − + 3 ξ g − c g + 5 c g c g − The graviton propagator becomes D µν ρσ = ik (cid:20) − f P (2) + f (cid:18) P (0) + √ c g T (0) − c g + 3 c g P (0 w ) (2 − c g ) (cid:19) + 2 ξ g (cid:18) P (1) + 2 P (0 w ) (2 − c g ) (cid:19)(cid:21) µνρσ where T (0) µνρσ = ( T µν L ρσ + L µν T ρσ ) / √ d − . The gauge of eq. (12) corresponds to c g = 0 ; the gauge used in [8]corresponds to c g = 1 , which is a convenient choice in Einstein gravity. In the generic gauge c g (cid:54) = 0 the ghostLagrangian is ¯ η µ (cid:20) ∂ η αµ + (1 − c g ) ∂ α ∂ µ + ∂ ν h αµ ∂ ν + ∂ ν h αν ∂ µ + ∂ ν ( ∂ α h µν ) − c g ∂ µ h αν ∂ ν − c g ∂ µ ( ∂ α h νν ) (cid:21) η α so that the ghost propagator is − ik (cid:20) η µν + c g − − c g k µ k ν k (cid:21) . VgV V Vg
Figure 1:
Gravitational corrections to the running of the gauge couplings. where the first lines show the well known matter terms [13], and the second lines showthe new terms due to agravity, as computed for generic gauge-fixing parameters ξ g and c g .Vectors are discussed in the next section. The one-loop correction to the kinetic term of vectors describes the RGE for the gauge cou-plings. The two new gravitational Feynman diagrams are shown in figure 1. They haveopposite logarithmic divergences, so that their sum is finite. This cancellation was noticedin [16, 17] in more general contexts, and seems due to the fact that gravitons have no gaugecharge. In conclusion, the one-loop RGE for the gauge couplings do not receive any gravitationalcorrection and the usual one-loop RGE for the gauge couplings remain valid also above thePlanck scale. Within the SM, the hypercharge gauge couplings hits a Landau pole at ¯ µ ∼ GeV . Summing the diagrams of fig. 2 gives a divergent correction that depends on the gauge fixingparameters and on the scalar/graviton couplings ξ . Adding the fermion and scalar wavefunction renormalizations of section 2.2 such dependencies cancel. We find the one-loopRGE: (4 π ) dY a d ln ¯ µ = 12 ( Y † b Y b Y a + Y a Y † b Y b ) + 2 Y b Y † a Y b ++ Y b Tr( Y † b Y a ) − { C F , Y a } + 158 f Y a . (21) Other authors try to interpret ambiguous power-divergent corrections to gauge couplings from Einsteingravity as gravitational power-running RGE, with possible physical consequences such as an asymptotically freehypercharge [18]. We instead compute the usual unambiguous logarithmic running, in the context of theorieswhere power divergences vanish. ΨΨΨ g S ΨΨ g S ΨΨΨΨ g S ΨΨ gS Ψ S Ψ Ψ gS Figure 2:
Gravitational corrections to the running of the Yukawa couplings. where C F = t A t A , and the latter term is the contribution due to agravity and has the oppositesign with respect to the analogous multiplicative term due to gauge interactions. Specializingeq. (21) to the SM, we find the one-loop RGE for the top quark Yukawa coupling: (4 π ) dy t d ln ¯ µ = 92 y t + y t ( 158 f − g − g − g ) . (22)We know of no previous computation in the literature. Tens of Feynman diagrams contribute to the scalar 4-point function at one loop. After sum-ming them and taking into account the scalar wave-function renormalization of eq. (19) thegauge dependence disappears and we find the one-loop RGE: (4 π ) dλ abcd d ln ¯ µ = (cid:88) perms (cid:20) λ abef λ efcd + 38 { θ A , θ B } ab { θ A , θ B } cd − Tr Y a Y † b Y c Y † d ++ 58 f ξ ab ξ cd + f ξ ae ξ cf ( δ eb + 6 ξ eb )( δ fd + 6 ξ fd ) + (23) + f
4! ( δ ae + 6 ξ ae )( δ bf + 6 ξ bf ) λ efcd (cid:21) + λ abcd (cid:20) (cid:88) k ( Y k − C k S ) + 5 f (cid:21) , where the first sum runs over the permutations of abcd and the second sum over k = { a, b, c, d } , with Y k and C k S defined by Tr( Y † a Y b ) = Y a δ ab , θ Aac θ Acb = C a S δ ab . (24)RGE for quartics have been computed in the literature in some models [19]; we find a simplerresult where the f term does not depend on ξ and where the f term vanishes if ξ ab = − δ ab / .Specializing our general eq. (23) to the case of the SM Higgs doublet plus a complexscalar singlet S with action given by eq. (2), the RGE become: (4 π ) dλ S d ln ¯ µ = 20 λ S + 2 λ HS + ξ S (cid:16) f + f (1 + 6 ξ S ) (cid:17) + λ S (cid:16) f + f (1 + 6 ξ S ) (cid:17) , (25) (4 π ) dλ HS d ln ¯ µ = − ξ H ξ S (cid:16) f + f (6 ξ S + 1)(6 ξ H + 1) (cid:17) − λ HS + λ HS (cid:18) λ S + 12 λ H + 6 y t + f + f (cid:104) (6 ξ S + 1) + (6 ξ H + 1) + 4(6 ξ S + 1)(6 ξ H + 1) (cid:105) (cid:19) , (26) (4 π ) dλ H d ln ¯ µ = 98 g + 920 g g + 27200 g − y t + 24 λ H + λ HS + ξ H (cid:16) f + f (1 + 6 ξ H ) (cid:17) ++ λ H (cid:18) f + f (1 + 6 ξ H ) + 12 y t − g − g (cid:19) . (27) We extract the one-loop RGE for the ξ parameters from the one-loop correction to thegraviton µν /scalar/scalar vertex. At tree level, two different Lagrangian terms contribute tosuch vertex:a) one contribution comes from the scalar kinetic term (when the graviton momentum isvanishing and the two scalars have momenta ± p );b) one contribution comes from ξ terms (when the graviton has a non-vanishing momen-tum k and one scalar has zero momentum).We compute both contributions. We find that the correction a) reproduces the scalar wave-function renormalization already computed in section 2.2, including the correct tensorialstructure p η µν − p µ p ν , provided that the graviton field is renormalized as follows: h µν → √ Z T L (cid:18) h µν − η µν h αα (cid:19) + 1 √ Z T η µν h αα (28)The wave-function renormalization Z T and Z T L differ because we used a simple gravitationalgauge-fixing term that breaks general relativity but respects special relativity: thereby distinctrepresentations of the Lorentz group (the trace and the traceless part of h µν ), get differentrenormalizations. We find the one-loop results Z T = 1 + 1(4 π ) (cid:15) c g − − c g ( c g − ξ g , (29) Z T L = 1 + 1(4 π ) (cid:15) (cid:20) c g − c g − f −
29 4 − c g + 2 c g ( c g − f −
23 9 − c g + 2 c g ( c g − ξ g (cid:21) . (30)We verified that we find the same Z T L by computing the graviton renormalization constantfrom the one-loop graviton/vector/vector vertex.Next, we compute the correction b). After adding to it the scalar and the graviton wave-function renormalization, we find that the total correction has the correct tensorial structure k η µν − k µ k ν and corresponds to the following one-loop RGE for the ξ parameters: (4 π ) dξ ab d ln ¯ µ = 16 λ abcd (6 ξ cd + δ cd ) + (6 ξ ab + δ ab ) (cid:88) k (cid:34) Y k − C k S (cid:35) + − f f ξ ab + f ξ ac (cid:18) ξ cd + 23 δ cd (cid:19) (6 ξ db + δ db ) (31)11here the sum runs over k = { a, b } . RGE for the ξ term have been computed in the literaturein some models [20, 21, 19]; we find different and simpler gravitational terms.Specialising our general eq. (31) to the case of the SM Higgs doublet plus a complex scalarsinglet S with action given by eq. (2), the RGE become: (4 π ) dξ S d ln ¯ µ = (1 + 6 ξ S ) 43 λ S − λ HS ξ H ) + f ξ S (1 + 6 ξ S )(2 + 3 ξ S ) − f f ξ S , (32) (4 π ) dξ H d ln ¯ µ = (1 + 6 ξ H )( y t − g − g + 2 λ H ) − λ HS ξ S ) ++ f ξ H (1 + 6 ξ H )(2 + 3 ξ H ) − f f ξ H . (33) The RGE for the couplings f , f are computed summing the one loop corrections to thegraviton kinetic term at 4th order in the external momentum k from: a) the graviton rainbowand seagull diagrams; b) the gravitational ghost; c) the graviton wave function renormal-ization of eq. (28). After combining all these ingredients we find a tensorial structure equalto the structure of the graviton kinetic term of eq. (13) that thereby can be interpreted as arenormalization of f and f . Adding also the matter contributions (that separately have thecorrect tensorial structure) we obtain: (4 π ) df d ln ¯ µ = − f (cid:18) N V N f
20 + N s (cid:19) , (34a) (4 π ) df d ln ¯ µ = 53 f + 5 f f + 56 f + f
12 ( δ ab + 6 ξ ab )( δ ab + 6 ξ ab ) . (34b)Here N V , N f , N s are the number of vectors, Weyl fermions and real scalars. In the SM N V = 12 , N f = 45 , N s = 4 . Unlike in the gauge case, all the contributions to the RGEof f have the same sign, such that f is always asymptotically free. This result agreeswith [16, 22]. The RGE for f agrees with [22]. See [20, 17] for results with different signs.The matter contributions had been computed in [23] leading to the concept of ‘induced grav-ity’ [24], which in our language corresponds to the RGE running of f . Concerning the puregravitational effect, agravity differs from gravity. The coupling f is asymptotically free onlyfor f < which leads to a tachionic instability M < . Having determined the quantum behaviour of agravity, we can now study if the Planck scalecan be generated dynamically. The possible ways are:12)
Non-perturbative : the couplings f or ξ become non-perturbative when running downto low energy: the Planck scale can be generated in a way similar to how the QCD scaleis generated.b) Perturbative : a quartic λ S runs in such a way that S gets a vacuum expectation value ξ (cid:104) S (cid:105) = ¯ M / .We focus on the second, perturbative, mechanism. In the usual Coleman-Weinberg case, S acquires a vev if its quartic λ S becomes negative when running down to low energy, and (cid:104) S (cid:105) is roughly given by the RGE scale at which λ (¯ µ ) becomes negative. This can be understood bynoticing that the quantum effective potential is roughly given by V eff = λ S (¯ µ ≈ S ) | S | . Thismeans that the vacuum energy is always negative, in contrast to the observed near-vanishingvacuum energy.In the gravitational case the situation is different, precisely because the effective La-grangian contains the − f ( S ) R term, that should generate the Planck scale. The field equa-tion for the scalar S in the homogeneous limit is V (cid:48) + f (cid:48) ( S )2 R = 0 (35)and the (trace of the) gravitational equation is f R + 4 V = O ( R /f , ) (36)where, around the phenomenologically desired flat-space solution, we can neglect the R term with respect to the induced Einstein term. By eliminating R we obtain the minimumequation for S : V (cid:48) − f (cid:48) f V = 0 (37)or, equivalently, V (cid:48) E = 0 where V E = ¯ M Vf (38)is called Einstein-frame potential because eq. (38) can be obtained by performing a fieldredefinition g Eµν = g µν × f / ¯ M such that the coefficient of R E in the Lagrangian has thecanonical Einstein value. Under this transformation the Lagrangian for the modulus of thescalar | S | = s/ √ becomes (cid:113) det g (cid:20) ( ∂ µ s ) − f R − V (cid:21) = (cid:113) det g E (cid:20) K ( ∂ µ s ) − ¯ M R E − V E (cid:21) . (39) Quantum corrections change when changing frame. In a generic context this leads to ambiguities. In theagravity context the fundamental action is given by eq. (2), thereby quantum corrections must be computed inthe ‘Jordan frame’. K = ¯ M (cid:18) f + 3 f (cid:48) f (cid:19) . (40)The non-canonical factor K in the kinetic term for s can be reabsorbed by defining a canon-ically normalised Einstein-frame scalar s E ( s ) as ds E /ds = √ K such that the Lagrangian be-comes (cid:113) det g E (cid:20) ( ∂ µ s E ) − ¯ M R E − V E ( s E ) (cid:21) . (41)In the agravity scenario f ( S ) is approximatively given by f ( S ) = ξ S (¯ µ ≈ s ) s and V S by λ S (¯ µ ≈ s ) s / . Thereby the Einstein-frame potential is given by V E ( S ) = ¯ M λ S ( s ) ξ S ( s ) (42)and the vacuum equation is β λ S ( s ) λ S ( s ) − β ξ S ( s ) ξ S ( s ) = 0 (43)where β p = dp/d ln µ are the β functions of the couplings p . This equation is significantly dif-ferent from the analogous equation of the usual non-gravitational Coleman-Weinberg mech-anism, which is λ S ( s ) = 0 .Furthermore, we want a nearly-vanishing cosmological constant.Unlike in the non-gravitational Coleman-Weinberg case, where V is always negative atthe minimum, in the agravity context V ( s ) = λ S ( s ) s / can be vanishing at the minimum,provided that λ S ( s ) = 0 at the minimum. This equation has the same form as the Coleman-Weinberg minimum condition, but its origin is different: it corresponds to demanding a neg-ligible cosmological constant.In summary, agravity can generate the Planck scale while keeping the vacuum energyvanishing provided that λ S ( s ) = 0 (vanishing cosmological constant), β λ S ( s ) = 0 (minimum condition), ξ S ( s ) s = ¯ M (observed Planck mass). (44)The minimum equation of eq. (43) has been simplified taking into account that λ S nearlyvanishes at the minimum.In the present scenario the cosmological constant can be naturally suppressed down toabout M : even making it as light as possible, M ∼ eV , the cosmological constant is atleast orders of magnitude larger than the observed value. Thereby we just invoke a hugefine-tuning without trying to explain the smallness of the cosmological constant.14 (cid:45) (cid:45) (cid:45) Μ in GeV Λ H Σ bands in M t (cid:61) (cid:177) (cid:72) gray (cid:76) Α s (cid:72) M Z (cid:76) (cid:61) (cid:177) (cid:72) red (cid:76) M h (cid:61) (cid:177) (cid:72) blue (cid:76) (cid:45) (cid:45) (cid:45) Μ in GeV Β Λ H RGE running of the MS quartic Higgs coupling in the SM
Figure 3:
Running of the quartic Higgs coupling in the SM [25]. Agravity corrections canincrease β λ H = dλ H /d ln ¯ µ and thereby λ H at scales above M , . Parameterizing the effectivepotential as V eff ( h ) ≡ λ eff ( h ) h / , the β function of λ eff vanishes at a scale a factor of few higherthan the β function of λ H . Models
In words, the quartic λ S must run in such a way that it vanishes together with its β functionaround the Planck mass. Is such a behaviour possible? The answer is yes; for example thisis how the Higgs quartic λ H can run in the Standard Model (see fig. 3a, upper curve). Its β function vanishes at the scale where the gauge coupling contribution to β λ H in eq. (27)compensates the top Yukawa contribution. Fig. 3b shows that this scale happens to be closeto the Planck mass. Although we cannot identify the Higgs field with the S field — the Higgsvev is at the weak scale so that the possible second minimum of the Higgs potential at thePlanck mass is not realised in nature — the fact that the conditions of eq. (44) are realisedin the SM is encouraging in showing that they can be realised and maybe points to a deeperconnection.By considering the generic RGE of agravity, one can see that in the pure gravitationallimit the conditions of eq. (44) cannot be satisfied, so the scalar S must have extra gauge andYukawa interactions, just like the Higgs. Clearly, many models are possible.A predictive model with no extra parameters is obtained by introducing a second copyof the SM and by imposing a Z symmetry, spontaneously broken by the fact that the mirrorHiggs field, identified with S , lies in the Planck-scale minimum while the Higgs field lies in theweak scale minimum. The mirror SM photon would be massless. Depending on the thermal15istory of the universe, a heavy mirror SM particle, such as a mirror neutrino or electron,could be a Dark Matter candidate. The interactions between such dark matter candidatewith the visible sector are suppressed by λ HS ; as we will discuss in section 5, the smallnessof λ HS is implied by a mechanism to understand the hierarchy between the Planck and theelectroweak scales. Inflation with a small amplitude of perturbations is not a typical outcome of quantum fieldtheory: it needs potentials with special flatness properties and often super-Planckian vacuumexpectation values. Agravity allows to compute the effective action at super-Planckian vac-uum expectation values, and potentials are flat at tree level, when expressed in the Einsteinframe. At loop level quantum corrections lead to deviations from flatness, proportional tothe β functions of the theory computed in section 2: thereby perturbative couplings lead toquasi-flat potentials, as suggested by inflation.All scalar fields in agravity are inflaton candidates: the Higgs boson h , the Higgs of grav-ity s , the scalar component of the graviton χ . In order to make χ explicit, we eliminatethe R / f term in the Lagrangian by adding an auxiliary field χ and its vanishing action − (cid:113) | det g | ( R + 3 f χ/ / f . Next, by performing a Weyl rescaling g Eµν = g µν × f / ¯ M with f = ξ S s + ξ H h + χ one obtains a canonical Einstein-Hilbert term, mixed kinetic terms forthe scalars, L = (cid:113) det g E (cid:20) − ¯ M R E + ¯ M (cid:18) ( ∂ µ s ) + ( ∂ µ h ) f + 3( ∂ µ f ) f (cid:19) − V E (cid:21) + · · · (45)as well as their effective potential: V E = ¯ M f (cid:32) V ( h, s ) + 3 f χ (cid:33) . (46)Disentangling the kinetic mixing between s and χ around the minimum of V E , we find masseigenstates M ± = m s + m χ ± (cid:118)(cid:117)(cid:117)(cid:116) ( m s + m χ ) − m s m χ ξ S (47)where m s ≡ (cid:104) s (cid:105) b/ , m χ ≡ (cid:104) s (cid:105) f (1 + 6 ξ S ) ξ S / and b = β ( β λ S ) is the β function of the β function of λ S evaluated at (cid:104) s (cid:105) .Various regimes for inflation are possible. In the limit where h or χ feel the vacuumexpectation value of s as a constant mass term, one obtains Starobinsky inflation and Higgs The agravity Lagrangian of eq. (2) also contains the square of the Weyl tensor, ( R − R µν ) /f , that remainsunaffected and does not contribute to classical cosmological evolution equations. A similar procedure allows tomake also the massive spin-2 state explicit [9]. -inflation [3]: agravity dictates how they can hold above the Planck scale. Leaving a fullanalysis to a future work, we here want to explore the possibility that the inflaton is the field s that dynamically generates the Planck scale, as discussed in section 3.In the limit where the spin 0 graviton χ is heavy enough that we can ignore its kineticmixing with s , we can easily convert s into a scalar s E with canonical kinetic term, as dis-cussed in eq. (41). Then, the usual formalism of slow-roll parameters allows to obtain theinflationary predictions. The slow-roll parameters (cid:15) and η are given by the β functions of thetheory. At leading order we find: (cid:15) ≡ ¯ M (cid:32) V E ∂V E ∂s E (cid:33) = 12 ξ S ξ S (cid:20) β λ S λ S − β ξ S ξ S (cid:21) , (48a) η ≡ ¯ M V E ∂ V E ∂s E = ξ S ξ S (cid:20) β ( β λ S ) λ S − β ( β ξ S ) ξ S + 5 + 36 ξ S ξ S β ξ S ξ S − ξ S ξ S β λ S β ξ S λ S ξ S (cid:21) . (48b)The scalar amplitude A s , its spectral index n s and the tensor-to-scalar ratio r = A t /A s arepredicted as n s = 1 − (cid:15) + 2 η, A s = V E /(cid:15) π ¯ M , r = 16 (cid:15) (49)where all quantities are evaluated at about N ≈ e -folds before the end of inflation, whenthe inflation field s E ( N ) was given by N = 1¯ M (cid:90) s E ( N )0 V E ( s E ) V (cid:48) E ( s E ) ds E . (50)In any given agravity model the running of λ S and of ξ S and consequently the inflationarypredictions can be computed numerically. We consider a simple analytic approximation that encodes the main features of this scenario.As discussed in section 3 and summarised in eq. (44), dynamical generation of the Planckscale with vanishing cosmological constant demands that the quartic λ S as well as its β func-tion vanish at a scale (cid:104) s (cid:105) = ¯ M Pl / √ ξ S . Thereby, around such minimum, we can approximatethe running parameters as λ S (¯ µ ≈ s ) ≈ b s (cid:104) s (cid:105) , ξ S (¯ µ ) ≈ ξ S . (51)We neglect the running of ξ S , given that it does not need to exhibit special features. Thecoefficient b = β ( β λ S ) can be rewritten as b ≡ g / (4 π ) , where g is the sum of quartic powersof the adimensional couplings of the theory. It can be computed in any given model. Withthis approximated running the slow-roll parameters of eq.s (48) simplify to (cid:15) ≈ η ≈ ξ S ξ S s/ (cid:104) s (cid:105) = 2 ¯ M s E . (52)17he latter equality holds because, within the assumed approximation, the explicit expressionfor the Einstein-frame scalar s E is s E = ¯ M Pl (cid:115) ξ S ξ S ln s (cid:104) s (cid:105) . (53)The Einstein-frame potential gets approximated, around its minimum, as a quadratic poten-tial: V E = ¯ M λ S ξ S ≈ M s s E with M s = g ¯ M Pl π ) (cid:113) ξ S (1 + 6 ξ S ) . (54)Notice that the eigenvalue M − of eq. (47) indeed reduces to M s , in the limit where it is muchlighter than the other eigenvalue M (cid:39) f ¯ M (1 + 6 ξ S ) .Inserting the value of s E at N ≈ e -folds before the end of inflation, s E ( N ) ≈ √ N ¯ M Pl ,we obtain the predictions: n s ≈ − N ≈ . , r ≈ N ≈ . , A s ≈ bN π ξ S (1 + 6 ξ S ) . (55)Such predictions are typical of quadratic potentials, and this is a non-trivial fact.Indeed, vacuum expectation values above the Planck scale, s E ≈ √ N ¯ M Pl , are needed forinflation from a quadratic potential and, more generically, if the tensor/scalar ratio is abovethe Lyth bound [26]. This means that, in a generic context, higher order potential termssuppressed by the Planck scale become important, so that the quadratic approximation doesnot hold.Agravity predicts physics above the Planck scale, and a quadratic potential is a good ap-proximation, even at super-Planckian vev, because coefficients of higher order terms are dy-namically suppressed by extra powers of the loop expansion parameters, roughly given by g / (4 π ) . Higher order terms are expected to give corrections of relative order g √ N / (4 π ) ,which are small if the theory is weakly coupled. We here consider the specific model presented in section 3, where the scalar S is identifiedwith the Higgs doublet of a mirror sector which is an exact copy of the SM, with the onlydifference that S sits in the Planck-scale minimum of the SM effective potential.This model predicts that the β function coefficient in eq. (51) equals g ≈ . providedthat we can neglect the gravitational couplings f , f with respect to the known order-one SMcouplings y t , g , g , g .Thereby the observed scalar amplitude A s = 2 . − [4] is reproduced for ξ S ≈ . Alarge ξ S is perturbative as long as it is smaller than /f , .We notice that ξ S is not a free parameter, within the context of the SM mirror model: thevev of the Higgs mirror s is given by the RGE scale at which β λ S vanishes (see fig. 3b), and18n order to reproduce the correct Planck scale with ξ S ≈ one needs (cid:104) s (cid:105) = ¯ M Pl / √ ξ S =1 . GeV . The fact that this condition can be satisfied (within the uncertainties) is a testof the model.The inflaton mass M s ≈ . GeV is below the Planck scale because suppressed by the β -functions of the theory, see eq. (54).The model allows to compute the full inflationary potential from the full running of λ S (shown in fig. 3) and of ξ S . The computation is conveniently performed in the Landau gauge ξ V = 0 , given that the gauge-dependence of the effective potential gets canceled by thegauge-dependence of the scalar kinetic term [27]. By performing a numerical computationwe find a more precise prediction r ≈ . for N ≈ . This is compatible with the expectedaccuracy of the quadratic approximation, estimated as g √ N / (4 π ) ≈ in section 4.1.In conclusion, we identified the inflaton with the field that dynamically generates thePlanck scale. In the agravity context, such field must have a dimensionless logarithmic po-tential: this is why our predictions for r ≈ /N ≈ . differ from the tentative prediction r ≈ /N ≈ . of a generic ξ -inflation model with mass parameters in the potential [3]. In section 3 we discussed how the Planck scale can be dynamically generated. We nowdiscuss how it is possible to generate also the electro-weak scale, such that it naturally ismuch below the Planck scale. ‘Naturally’ here refers to the modified version of naturalnessadopted in [6], where quadratically divergent corrections are assumed to vanish, such thatno new physics is needed at the weak scale to keep it stable. The present work proposed atheoretical motivation for the vanishing of power divergences: they have mass dimension,and thereby must vanish if the fundamental theory contains no dimensionful parameters.This is the principle that motivated our study of adimensional gravity.In this scenario, the weak scale can be naturally small, and the next step is exploring whatcan be the physical dynamical origin of the small ratio M h /M ∼ − . The dynamics thatgenerates the weak scale can be:a) around the weak scale , with physics at much high energy only giving negligible finitecorrections to the Higgs mass. Models of this type have been proposed in the litera-ture [5], although the issue of gravitational corrections has not been addressed. Suchmodels lead to observable signals in weak-scale experiments.b) much above the weak scale . For example, Einstein gravity naively suggests thatany particle with mass M gives a finite gravitational correction to the Higgs mass at19hree [28] and two loops: δM h ∼ y t M (4 π ) M + ξ H M (4 π ) M (56)which is of the right order of magnitude for M ∼ GeV .In the context of agravity we can address the issue of gravitational corrections, and proposea scenario where the weak scale is generated from the Planck scale. It is convenient to dividethe computation into 3 energy ranges1)
Low energies : at RGE scales below the mass M , of the heavy gravitons, agravity canbe neglected and the usual RGE of the SM apply. The Higgs mass parameter receives amultiplicative renormalization: (4 π ) dM h d ln ¯ µ = M h β SM M h , β SM M h = 12 λ H + 6 y t − g − g . (57)2) Intermediate energies between M , and M Pl : agravity interactions cannot be neglectedbut M h and M Pl appear in the effective Lagrangian as apparent dimensionful param-eters. We find that their RGE are gauge-dependent because the unit of mass is gaugedependent. The RGE for adimensional mass-ratios are gauge-independent and we find (4 π ) dd ln ¯ µ M h ¯ M = − ξ H [5 f + f (1 + 6 ξ H )] − (cid:18) M h ¯ M (cid:19) (1 + 6 ξ H ) ++ M h ¯ M (cid:20) β SM M h + 5 f + 53 f f + f ( 13 + 6 ξ H + 6 ξ H ) (cid:21) . (58)The first term is crucial: it describes corrections to M h proportional to M Pl . A naturallysmall [6] weak scale arises provided that the agravity couplings are small: f , f ≈ (cid:115) πM h M Pl ∼ − . (59)The mass of the spin-2 graviton ghost is M = f ¯ M Pl / √ ≈ GeV . The spin-0massive component of the graviton mixes with the other scalars giving rise to the masseigenvalues of eq. (47). Experimental bounds are safely satisfied.3)
Large energies above the Planck mass: the theory is adimensional and the RGE of sec-tion 2 apply. According to the Lagrangian of eq. (4), the quartic coupling λ HS | H | | S | leads to a Higgs mass term M h | H | given by M h = λ HS (cid:104) s (cid:105) . Ignoring gravity, λ HS We also verified that the RGE for the ratio of scalar to fermion masses is gauge invariant. We cannotcomparare our eq. (58) with gauge-depend RGE for M Pl computed in the literature [17, 20, 22, 29] withdiscrepant results, given that we use a different gauge. S sector. Within agravity, a non vanishing λ HS is unavoidably generatedby RGE running at one-loop order, as shown by its RGE in eq. (26), which contains thenon-multiplicative contribution: (4 π ) dλ HS d ln ¯ µ = − ξ H ξ S [5 f + f (6 ξ S + 1)(6 ξ H + 1)] + · · · . (60)For ξ S = 0 this equation is equivalent to (58). We need to assume that the mixedquartic acquires its minimal natural value, λ HS ∼ f , (for simplicity we do not considerthe possibility of values of ξ H,S = { , − / } that lead to special cancellations).In conclusion, agravity unavoidably generates a contribution to the Higgs mass given by M h ≈ ¯ M ξ H (4 π ) [5 f + f (6 ξ S + 1)(6 ξ H + 1)] (cid:96) (61)where (cid:96) is a positive logarithmic factor.This alternative understanding of the Higgs mass hierarchy problem relies on the small-ness of some parameters. All parameters assumed to be small are naturally small, just likethe Yukawa coupling of the electron, y e ∼ − , is naturally small. These small parameters donot receive unnaturally large quantum corrections. No fine-tuned cancellations are necessary.At perturbative level, this is clear from the explicit form of the one-loop RGE equationsderived in section 2: quantum corrections to f , f are proportional to cubic powers of f , f ,and higher order loop corrections are even more suppressed.At non perturbative level, a black hole of mass M might give a quantum correction of order δM h ∼ M e − S BH where S BH = M / M is the black hole entropy. Black holes with Planck-scale mass might give an unnaturally large correction, δM h (cid:29) M h , ruining naturalness.Planck-scale black holes do not exist in agravity, where the minimal mass of a black hole is M BH > ∼ ¯ M Pl /f , , as clear from the fact that the massive anti-gravitons damp the /r Newtonbehaviour of the gravitational potential at r < ∼ /M , : V = − Gmr (cid:20) − e − M r + 13 e − M r (cid:21) . (62)Thereby, non-perturbative quantum corrections are expected to be negligible in agravity, be-cause exponentially suppressed as e − /f , .The Higgs of gravity s has a mass M s which can be anywhere between the weak scaleand the Planck scale, depending on how large are the gauge and Yukawa couplings withinits sector. Its couplings to SM particles are always negligibly small. In the model where s isthe Higgs of a mirror copy of the SM, its mass is a few orders of magnitude below the Planckscale. 21s a final comment, we notice that accidental global symmetries (a key ingredient of axionmodels) are a natural consequence of the dimensionless principle. In the usual scenario, adhoc model building is needed in order to suppress explicit breaking due to mass terms ornon-renormalizable operators [30]. An axion can be added to agravity compatibly with finitenaturalness along the lines of [6]. In conclusion, we proposed that the fundamental theory contains no dimensionful parameter.Adimensional gravity (agravity for short) is renormalizable because gravitons have a kineticterm with 4 derivatives and two adimensional coupling constants f and f .The theory predicts physics above the Planck scale. We computed the RGE of a genericagravity theory, see eq.s (21), (23), (31) and (34). We found that quantum corrections candynamically generate the Planck scale as the vacuum expectation value of a scalar s , thatacts as the Higgs of gravity. The cosmological constant can be tuned to zero. This happenswhen a running quartic coupling and its β function both vanish around the Planck scale, assummarised in eq. (44). The quartic coupling of the Higgs in the SM can run in such a way,see fig. 3.The graviton splits into the usual massless graviton, into a massive spin 2 anti-graviton,and into a scalar. The spin 2 state is a ghost, to be quantised as a state with positive kineticenergy but negative norm.The lack of dimensional parameters implies successful quasi-flat inflationary potentials atsuper-Planckian vacuum expectation values: the slow-roll parameters are the β functions ofthe theory. Identifying the inflaton with the Higgs of gravity leads to predictions n s ≈ . for the spectral index and r ≈ . for the tensor/scalar amplitude ratio.The Higgs of gravity can also be identified with the Higgs of the Higgs: if f , f ∼ − are small enough, gravitational loops generate the observed weak scale. In this context, aweak scale much smaller than the Planck scale is natural: all small parameters receive smallquantum corrections. In particular, quadratic divergences must vanish in view of the lack ofany fundamental dimensionful parameter, circumventing the usual hierarchy problem. Acknowledgments
We thank P. Benincasa, S. Dimopoulos, J. Garcia-Bellido, G. Giudice, J. March-Russell, P. Menotti, M.Redi, M. Shaposhnikov for useful discussions. This work was supported by the ESF grant MTT8 andby the SF0690030s09 project and by the European Programme PITN-GA-2009-237920 (UNILHC). Weare grateful to CERN for hospitality. The work of Alberto Salvio has been also supported by the SpanishMinistry of Economy and Competitiveness under grant FPA2012-32828, Consolider-CPAN (CSD2007-00042), the grant SEV-2012-0249 of the “Centro de Excelencia Severo Ochoa” Programme and thegrant HEPHACOS-S2009/ESP1473 from the C.A. de Madrid. eferences [1] K.G. Wilson, Phys. Rev. D3 (1971) 1818. Wilson itself later changed idea writing “This claim makes nosense” in K.G. Wilson, Nucl. Phys. Proc. Suppl. 140 (2005) 3 [arXiv:hep-lat/0412043].[2] L. Giusti, A. Romanino and A. Strumia, Nucl. Phys. B 550 (1999) 3 [arXiv:hep-ph/9811386]. R. Barbieri,A. Strumia, arXiv:hep-ph/0007265. A. Strumia, JHEP 1104, 073 (2011) [arXiv:1101.2195]. A. Arvanitaki,M. Baryakhtar, X. Huang, K. van Tilburg and G. Villadoro, JHEP 1403 (2014) 022 [arXiv:1309.3568].[3] A.A. Starobinsky, Phys. Lett. B 91 (1980) 99. R. Fakir, W. Unruh, Phys. Rev. D41 (1990) 1783. D.I.Kaiser, Phys. Rev. D 52 (1995) 4295 [arXiv:astro-ph/9408044]. S. Tsujikawa, B. Gumjudpai, Phys.Rev. D 69 (2004) 123523 [arXiv:astro-ph/0402185]. F. L. Bezrukov, M. Shaposhnikov, Phys. Lett. B659 (2008) 703 [arXiv:0710.3755]. F.L. Bezrukov, A. Magnin, M. Shaposhnikov, S. Sibiryakov, JHEP1101 (2011) 016 [arXiv:1008.5157]. C. P. Burgess, H. M. Lee and M. Trott, JHEP 0909 (2009) 103[arXiv:0902.4465]. J.L.F. Barbon, J.R. Espinosa, Phys. Rev. D 79 (2009) 081302 [arXiv:0903.0355]. A.Linde, arXiv:1402.0526. C. P. Burgess, S. P. Patil, M. Trott, arXiv:1402.1476. G.F. Giudice, H.M. Lee,arXiv:1402.2129. T. Prokopec, J. Weenink, arXiv:1403.3219. J. Joergesen, F. Sannino, O. Svendesen,arXiv:1403.3289.[4] Planck Collaboration, arXiv:1303.5082.[5] F. Englert, C. Truffin and R. Gastmans, Nucl. Phys. B 117 (1976) 407. W Bardeen, FERMILAB-CONF-95-391-T. C. T. Hill, arXiv:hep-th/0510177. R. Hempfling, Phys. Lett. B 379 (1996) 153 [arXiv:hep-ph/9604278]. K. A. Meissner and H. Nicolai, Phys. Lett. B 648 (2007) 312 [arXiv:hep-th/0612165].J. P. Fatelo, J. M. Gerard, T. Hambye and J. Weyers, Phys. Rev. Lett. 74 (1995) 492. T. Hambye, Phys.Lett. B 371 (1996) 87 [arXiv:hep-ph/9510266]. W. -F. Chang, J. N. Ng and J. M. S. Wu, Phys. Rev.D 75 (2007) 115016 [arXiv:hep-ph/0701254]. R. Foot, A. Kobakhidze and R. R. Volkas, Phys. Lett. B655 (2007) 156 [arXiv:0704.1165]. R. Foot, A. Kobakhidze, K. .L. McDonald and R. .R. Volkas, Phys.Rev. D 76 (2007) 075014 [arXiv:0706.1829]. R. Foot, A. Kobakhidze, K. L. McDonald, R. R. Volkas,Phys. Rev. D 77 (2008) 035006 [arXiv:0709.2750]. S. Iso, N. Okada, Y. Orikasa, Phys. Rev. D 80 (2009)115007 [arXiv:0909.0128]. T. Hur and P. Ko, Phys. Rev. Lett. 106 (2011) 141802 [arXiv:1103.2571].L. Alexander-Nunneley and A. Pilaftsis, JHEP 1009 (2010) 021 [arXiv:1006.5916]. S. Iso and Y. Orikasa,PTEP 2013 (2013) 023B08 [arXiv:1210.2848]. C. Englert, J. Jaeckel, V. V. Khoze and M. Spannowsky,arXiv:1301.4224. E.J. Chun, S. Jung, H.M. Lee, arXiv:1304.5815. M. Heikinheimo, A. Racioppi, M. Raidal,C. Spethmann, K. Tuominen, arXiv:1304.7006. T. Henz, J. M. Pawlowski, A. Rodigast and C. Wetterich,Phys. Lett. B 727 (2013) 298 arXiv:1304.7743. T. Hambye, A. Strumia, Phys. Rev. D 88 (2013) 055022[arXiv:1306.2329]. C. D. Carone and R. Ramos, Phys. Rev. D 88, 055020 (2013) [arXiv:1307.8428]. V.V.Khoze, arXiv:1308.6338. I. Quiros, arXiv:1312.1018, arXiv:1401.2643. C.T. Hill, arXiv:1401.4185.[6] M. Farina, D. Pappadopulo and A. Strumia, JHEP 1308 (2013) 022 [arXiv:1303.7244].[7] Models with induced and/or varying Planck mass have been discussed in the literature, see e.g.P. Minkowski, Phys. Lett. B 71 (1977) 419. A. Zee, Phys. Rev. Lett. 42 (1979) 417. S.L. Adler, Phys.Rev. Lett. 24 (1980) 1567. C. Wetterich, Nucl. Phys. B 302 (1988) 668. C. Wetterich, Phys. Rev. Lett. 90(2003) 231302 [arXiv:hep-th/0210156].[8] K. S. Stelle, Phys. Rev. D 16 (1977) 953.[9] G. Magnano, M. Ferraris, M. Francaviglia, Gen. Rel. Grav. 19 (1987), 465. G. Magnano, M. Ferraris, M.Francaviglia, Class. Quantum Grav. 7 (1990), 557. A. Jakubiec, J. Kijowski, Phys. Rev. D 37 (1988) 1406. J.C. Alonso, F. Barbero, J. Julve, A. Tiemblo, Class. Quantum Grav. 11 (1994), 865. A. Hindawi, B. A. Ovrut,D. Waldram, Phys. Rev. D 53 (1996) 5583 [arXiv:hep-th/9509142].
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