Functional Renormalization Group Flow of Massive Gravity

Eur. Phys. J. C (2020) 80:271 https://doi.org/10.1140/epjc/s10052-020-7835-8

Functional renormalization group ﬂow of massive gravity

Maximiliano Bindera, Iván Schmidt Universidad Técnica Fedérico Santa María and Centro Cientíﬁco Tecnológico de Valparaiso (CCTVal), Valparaiso, Chile

Received: 28 October 2019 / Accepted: 13 March 2020 / Published online: 26 March 2020 © The Author(s) 2020

Abstract We apply the functional renormalization group malism. The Asymptotic Safety conditions are based on the equation to a massive Fierz–Pauli action in curved space and existence of fixed points in the flow of the couplings con- find that, even though a massive term is a modification in stants as functions of the energy scale. It has been used for the infrared sector, the mass term modifies the value of the many gravity Lagrangians, such as Einstein–Hilbert, F(R) non-gaussian fixed point in the UV sector. We obtained the type, torsion, etc. In all these cases, gravity appears to be an beta function for the scale dependent mass parameter and asymptotically safe theory [8–11]. found that the massive Fierz–Pauli case still seems to be an In quantum field theory, the particle mediating the gravi- asymptotically safe theory. tational interaction is a spin 2 massless particle [13], which is always attractive and produces a long range force. On the other hand, observations show that the universe is acceler- 1 Introduction ating in its expansion [14], and this is in contradiction with the principle of a universal attractive interaction. Possible From a phenomenological point of view, General Relativity explanations could come from modifications of general rel- [1] is one of the best physical theories that we have at the ativity in the infrared sector, such as giving a mass to the present time. It explains with great accuracy macroscopical graviton, resulting in a massive gravity theory [15]. Study- phenomena that could not be accounted for with newtonian ing massive gravity is also important in extra-dimensional mechanics, such as the bending of light by a massive object extensions of the Standard Model of fundamental interac- and the precession of Mercury’s perihelion. Problems arise tions, since in these theories the higher Kaluza–Klein modes when one tries to give a quantum mechanical description of are usually massive gravitons. gravity following well known quantization procedures. This In this paper we study how the beta functions for the New- quantum theory of gravity presents divergences in the UV ton’s and cosmological coupling constants get modified if sector [2], which render it non-renormalizable by the usual we include a mass term in the Einstein–Hilbert action. This perturbative approach, and therefore General Relativity is theory has been proven to have theoretical problems [16], usually considered an effective field theory valid up to some particularly in its ultraviolet completion, and therefore it is energy scale [3]. Several attempts have been made in order important to see whether and how these difficulties are still to obtain a quantum theory of gravity, such as string theory, present when it is considered from the point of view of the non commutative geometry, loop quantum gravity, etc., and renormalization group. Here we will not enter into details one of these is Asymptotic Safety [4–6]. about massive gravity, and for that purpose the reader can Asymptotic Safety is a set of conditions that ensures the see any of the several references [15]. Our main goal in this good behavior of a theory in the full energy scale, which work is to study how the known flow diagrams, and the fixed means that the theory under consideration can be considered points of the coupling constants in the Einstein–Hilbert trun- a fundamental theory. This approach has been used exten- cation get modified when a mass term is added to the action. sively in order to understand the UV behavior of gravity, in The m → 0 limit is not a trivial one already in non-abelian a way that avoids the non-renormalizability of the quantum gauge theories and this could be appreciated in the flow dia- theory of gravity in the perturbative quantum field theory for- grams or in the beta functions. a e-mail: [email protected] (corresponding author) 123 271 Page 2 of 6 Eur. Phys. J. C (2020) 80 :271

(2) 2 Massive Fierz–Pauli truncation forward: first obtain by expanding gμν =¯gμν + hμν k δ and then take the derivative δh(x)δh(y) . Then, consider the We follow the same procedure as for the Einstein–Hilbert (2) −1 fourier transform for the trace Tr[( + Rk) ∂t Rk]= truncation in [17], but we include a mass term in the action 2 −isD2 Tr[W(−D )]= dsW(s)Tr[e ] and use the Heat 2 √ kernel expansion to evaluate Tr e−isD . After perform- [g, g¯]=2κ2 Z dd x g(−R + 2λ¯ ) k Nk k ing a Mellin transform we can compare the coefficients of √ √ d4x g and d4x gR in the LHS and RHS of the evolu- 1 2 μν 2 − g¯ m (hμν h − h ) (1) 2 tion equation and obtain a system of two equations (a brief description is given in Appendix A). Here g, g¯ and h are related by gμν =¯gμν + hμν where One obtains the following set of equations for the cosmo- ¯ ¯ g is a fixed background metric, hμν is the fluctuating field logical and gravitational coupling constants λk and Z Nk: that enters in the path integral and Z Nk is related to the run- ≡ ¯ / ¯ ¯ 1 − d d 1 1 ning Newton constant Gk G Z Nk with G a fixed con- ∂ (Z λ ) = (4π) 2 k d d+1) 2Q −η (k)Q ¯ t Nk k κ2 d N d stant and λk is the dimensionfull cosmological constant. The 16 2 2 construction of the mass term in this cases is the only one 1 − 8dQd (0) possible that do not generates a massive scalar ghost when 2 m2 (d − 1)(d2 − 2) linearizing (1). So this action describes a single spin 2 mas- + Q2 − η (k)Q 2 2 2 d N d sive graviton. Since we are working without the presence of a k (d − 2) 2 2 source, T μν = 0, we can restore the diffeomorphism invari- 1 − d d−2 d ∂t Z Nk =− (4π) 2 k (d + 1) ance using the Stuckelberg mechanism [12] . The evolution 4κ2 12 equation for k is given by the Wetterich equation [18] 1 1 − 2Q − η (k)Q 1 d −1 N d −1 ∂ [ ¯,ξ,ξ¯;¯]=1 (2) + ∂ 2 2 t k h g Tr k Rk t Rk 2 ¯ ¯ ¯ ¯ hh hh − d ( − ) 2 − η ( ) 2 −1 d 1 2Q d N k Q d 2 2 2 − 1 (2) + Tr k Rk 2 2 2 ¯ 2 m d (d − 1) ξξ − dQ1 ( ) − Q2 ( ) − d − 0 d 0 2 −1 3 2 1 2 k 24(d − 2) ( ) − 2 + R ∂ R (2) k k t k 2 − η ( ) 2 ξξ¯ ξξ¯ 2Q d − N k Q d − (4) 2 1 2 1 where the second term is for the Fadeev–Popov ghosts (ξ¯ Here Q m and Qm are the “threshold functions” men- and ξ), which are included as in Yang–Mills theories and in n n μ tioned in [5,17], which are integrals depending on the form of ∂ μν − this case we are working with the harmonic gauge h 2 1 W[−D ] and the cutoff R . In our case we consider the sharp ∂ν h = 0. k 2 2 ˆ p2 ¯ cutoff R (p ) = lim ˆ R (1 − ), so that the integrals In order to obtain the beta functions for λk and Z Nk, k R→∞ k2 we consider the evolution equation in the case gμν =¯gμν that appear can be evaluated in analytic form. The equations (2) (4) are for the dimensionfull couplings Z and λ¯ , while for (or hμν = 0), after performing the derivatives = Nk k k λ ≡ −2λ¯ ≡ ¯ d−2 −1 δ2 ¯ the dimensionless couplings k k k, gk Gk Z Nk δ ( )δ ( ) k. The same holds for the ghost part with ξ = m h x h y and mk ≡ one obtain the beta functions ξ = 0. Thus the left hand side of the evolution equation k results in ∂ g = (d − 2 + η (k))g t k N k gk 5 ∂t λk =−(2 − ηN )λk − 5ln(1 − 2λk) − 2ζ(3) + ηN ∂ [ , ¯]= κ2 d ¯ − (∂ ) + ∂ ( λ¯ ) π 2 t k g g 2 d x g t Z Nk R 2 t Z Nk k η 2 21 N (3) − m − − 2ln(1 − 2λk) − 4ζ(3) , (5) k 8 2

As we can see, taking the limit h → 0intheLHSofthe where ηN (k) =−∂tlnZNk is the anomalous dimension of evolution equation suppresses all the information of ∂t m(k), the R operator. and therefore in this case we can consider the mass as a scale 2gk 18 dependent coupling or just a parameter with mass dimen- ηN =− + 5ln(1 − 2λk) 6π + 5g 1 − 2λ sions. k k The right hand side of (2) is much more technical to obtain. − ζ( ) + + 2 3 2 6 mk (6) For details the reader can see [17], but the recipe is straight 1 − 2λk 123 Eur. Phys. J. C (2020) 80 :271 Page 3 of 6 271

2 = , . . λ = λ = . Fig. 1 Flow diagrams for mk 0 0 1and05. Here k (gk ) is in the x (y) axis. The coordinates for the m 0 ﬁxed point are given by k 0 3296 and gk = 0.40266

Fig. 2 Coordinates of the non-gaussian ﬁxed point for different values θ 2 Fig. 3 The real part of the critical exponent is positive and grows of the dimensionless mass mk 2 with the dimensionless mass mk

asymptotically safe behavior around the non-gaussian fixed 2 → point for both the massive and for the massless case. As we can see in (5), there is no discontinuity in the mk 0 limit and one can reproduce the massless results of [17] taking the massless limit of the massive case. It is important 2 m 3 Mass beta function to notice that in (4) the mass appears as k2 , so it automatically results as a dimensionless term in the dimensionless flow equations. We can consider the mass term as a scale dependent param- As we can see in Fig. 1, the flow diagrams change contin- eter mk = m(k) and obtain its beta function by taking field 2 uously as the dimensionless mass mk grows. derivatives of the Wetterich equation following the procedure The non gaussian fixed point gets modified as the dimen- described in detail in [19] for the case of non-abelian mas- sionless mass grows (Fig. 2). In this case we have taken values sive theories (similar applications have been done for the case 2 <λ < . for mk that are consistent with the interval 0 k 0 5, of gravity [20] but not in a massive gravity context ). Since to avoid problems with the ln(1 − 2λk) term in (5). For this the mass term comes multiplied by a quadratic term in the ≤ 2 ≤ purpose we have chosen values in the range 0 mk 6. field hμν, we consider a second order functional derivative We can also obtain the behavior of the critical expo- of equation (2) to obtain nents depending on the value of the mass parameter (Fig. 3). = (2) (3) (3) The values of critical exponents for the mk 0 case are ∂t = Tr Gk Gk Gk∂t Rk θ = θ ∗ = . − . k k k 1 2 1 94091 3 31065i. Therefore, we have two complex conjugate critical exponents whose real part is pos- 1 (4) − Tr Gk Gk∂t Rk , (7) itive and grows together with the mass parameter, ensuring an 2 k 123 271 Page 4 of 6 Eur. Phys. J. C (2020) 80 :271 =[(2) + ]−1 3g 2m2 12 where Gk k Rk is the modified propagator and − k − k − (n) δn 2(5g + 6π) 1 − 2λ 1 − 2λ k is the n-order functional derivative δhδh··· k with respect k k k π 2 π 2 to the fluctuating√ field hμν√. So we need the expansion up 10 4 0 1 2 + ( − log(1 − 2λk)) − − 4 to fourth order in g → g¯(O(h ) + O(h ) + O(h ) + 3 6 9 ( 3)+ ( 4)) → O h O h and the same for the curvature R with gμν 2 2 gkm 5 π g¯μν + hμν. After taking the field derivatives and contracting + k − ( − λ ) π 2log 1 2 k the indices we obtain for the left hand side of (7) 2 3 3 2 π 2 2mk 12 4 (2) 1 2 4 2 − − − − 1 ∂t = κ d x g¯ − (∂t Z Nk)D − 2∂t (Z Nkλk) 1 − 2λ 1 − 2λ 9 k 2 k k g 2m2 ∂ ( 2) − 3 k − k − 12 ¯ t Z Nkmk + ∂t Z NkCT R + 12 , (8) (5gk + 6π) 1 − 2λk 1 − 2λk 2 10 π 2 4π 2 while for the terms in the right hand side we get + ( − log(1 − 2λk)) − − 4 (11) 3 6 9 (3) (3) Tr[Gk Gk Gk ∂t Rk ] ⎡ ⎤ In this case we obtain an absolute non-gaussian fixed 2 2 2 2 ⎢ 43626(D ) + 6504(−R + 2λk )(D ) + 272(−R + 2λk ) ∂t Rk ⎥ ⎢ ⎥ point(where βm = βλ = βg = 0) with coordinates λk = = Tr ⎢ ⎥ ⎣ 3 ⎦ . , = . = . 2 m2 0 3981 gk 0 2192 and mk 0 4221 besides the IR fixed − D2 − λ + k2 R( −D ) + C R + k 2 k k2 R 2 point for λk = gk = mk = 0 where all the beta functions Tr[G (4)G ∂ R ] vanish. k ⎡ k t k ⎤ As we can see, the original m = 0 fixed point in the ⎢ − 100(D2) − 8(−R + 2λ ) ∂ R ⎥ ⎢ k t k ⎥ = ⎢ ⎥ λk − gk plane gets slightly modified by the presence of a Tr 2 (9) ⎣ 2 ⎦ 2 2 −D2 mk mass term. Notice that in this case, in which the mass is − D − 2λk + k R( ) + CR R + k2 2 2 → scale dependent, the limit mk 0 cannot be taken since the Then we follow the same procedure described in Sect. 2, values of the critical exponents depend on the non gaussian = . expanding the denominators and performing a Mellin trans- fixed point, which is non zero for mk 0 4221. This is [ (− 2)] an indication of the problems that are present in massive form on√ the Tr W D expressions. By comparing the d4x g¯ terms in both sides of (7) we obtain gravity. It seems that both approaches correspond to different physical theories. Nevertheless, looking at Fig. 1 it can be − ∂ (Z λ ) − 3∂ (Z m2) t Nk k t Nk k seen that for m close to 0.4221, which is the value that gives 10 − d our second approach for the fixed point mass, the values of = ( π) 2 λ¯ 2 d · 3 − η 3 4 4 kk 2 272 2Qd/2 N Qd/2 κ2 λk and gk are approximately those that are obtained in the 2 second approach. This shows the consistency between both mk 4 4 − 3 · 272 2Q / − ηN Q / calculations. 2 d 2 d 2 For the critical exponents in the UV fixed point, we obtain ∗ d ¯ 2 2 the values θ1 = 342408, θ2 = 7.91 and θ = 5.09, while for + k 8 ∗ 2λk 2Q / − ηN Q / 3 d 2 d 2 the gaussian fixed point (g = λ = m = 0) we obtain the k k k 2 critical exponents θ1 = θ2 = 2 and θ3 =−2. Clearly, these mk 3 3 − 2Q / − ηN Q / (10) 2 d 2 d 2 values show the difficulties of massive gravity, which we certainly expected to encounter, since all approaches to massive This is a dimensionfull equation in the couplings. Com- gravity at a certain point produce results that have no physical bining this equation with the ones shown in (4), we can obtain significance. It could also be that there are inconsistencies in the mass beta function with the dimensionless couplings. For the truncation, which means that we need to consider non- the sharp cutoff this is given by linear terms (interactions). Another possibility could be that, 960g 544 272m2 since we are working in a theory that is not manifestly gauge ∂ m2 =− m2 + k λ2 − k t k 2 k k 2 3 invariant, which can be stated as a particular gauge choice, π (1 − 2λk) (1 − 2λk) we could restore gauge invariance through the Stueckelberg m2 − k + 16 mechanism, and analyze this more general theory. 2(1 − 2λ )2 1 − 2λ k k + 8gk ζ( ) − ( − λ ) 4 Comments and conclusions π 20 4 3 2log 1 2 k 42m2 We have analyzed massive gravity in its simplest version + k − ζ( ) 64 3 (Fierz–Pauli), within the Asymptotic Safety renormalization 1 − 2λk 123 Eur. Phys. J. C (2020) 80 :271 Page 5 of 6 271 group formalism. In a first approach, we considered the mass just from General Relativity, and also those that are present to be a fixed parameter, and found that we get different in an extra interaction potential. The first case leads to dif- fixed points for the other scale dependent coupling constants ficulties such as ghost instabilities, while the last one has a (Newton and cosmological constant) as we change this mass lot of freedom (choices of parameters). One version, named parameter. In this case there is no discontinuity in the m → 0 3, seems to solve some of the problems, and is therefore limit, so there is no vDVZ discontinuity and the massless the best present possibility of having a sound massive gravity limit appears as a continuous limit of the massive case and theory. Another possibility is that, since we are working in 2 the flow diagram changes continuously as mk grows. This a theory that is not manifestly gauge invariant, we could use may be because in order to obtain the threshold functions the Stueckelberg mechanism in order to promote this theory an expansion is usually made around the Ricci scalar and in into a gauge invariant version, which could clarify the ori- + 2 this case the expansion was made around the term R cmmk gin on the unphysical results. This is an approach that has with cm constant, so the mass term acts as a correction to the worked in similar instances, such as the clarification of the curvature. vDVZ discontinuity [15]. It is certainly important to consider The result that with this procedure there is no vDVZ dis- these extensions from the point of view of the renormaliza- continuity is important, because at present the only solution tion group and asymptotic safety, which is something that we to this problem is given by the Stueckelberg mechanism, and tend to do in future publications. also because the variation in the mass parameter affects the value of the cosmological and gravitational constants, with Acknowledgements We would like to thanks O. Castillo for his col- no discontinuity. laboration in the preparation of this manuscript. M.B. is also grate- ful to Arkady Vainshtein for hospitality and useful discussions dur- The mass term, which is an infrared modification of grav- ing his stay at university of Minnesota. This work is supported by ity, changes the coordinates of the non gaussian fixed point. MECESUP2(Chile) under grant FSM0806-D3042, DGIP-UTFSM This means that an infrared modification still produces an (Chile), UTFSM-PIIC project, Fondecyt (Chile) grant 1180232 and by effect in the UV sector of the theory. On the other hand, the ANID PIA/APOYO AFB180002. gaussian fixed point, λk = gk = 0, does not change with the Data Availability Statement This manuscript has no associated data mass term. or the data will not be deposited. [Authors’ comment: Right now there In the case where the mass term is taken as a constant, are no data that would be relevant for the purposes of this paper.] the critical exponents (Fig. 3) deviate from the “expected” Open Access This article is licensed under a Creative Commons Attri- values (∼ 2). This fact is consistent with the upper bound of bution 4.0 International License, which permits use, sharing, adaptation, the graviton mass predicted by experimental measurements, distribution and reproduction in any medium or format, as long as you which gives a maximum value of order ∼ 10−23 eV (smaller give appropriate credit to the original author(s) and the source, pro- < −18 vide a link to the Creative Commons licence, and indicate if changes than the upper limit of the photon mass 10 eV). There- were made. The images or other third party material in this article fore, if the graviton mass is not equal to zero, it must be small are included in the article’s Creative Commons licence, unless indi- so the truncation gives consistent results. cated otherwise in a credit line to the material. If material is not Nevertheless, we should stress that there is a consistency included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- check that can be made between both cases. In our first ted use, you will need to obtain permission directly from the copy- approach with a constant mass, looking at Fig. 1 it can be right holder. To view a copy of this licence, visit http://creativecomm seen that for m close to 0.4221, which is the value that gives ons.org/licenses/by/4.0/. 3 our second approach for the fixed point mass, the values of Funded by SCOAP . λk and gk are approximately those that are obtained in the second approach. This shows the consistency between both calculations. Appendix A: Denominator expansion On the other hand, for the case with the mass as a scale dependent parameter, we still obtain a non-gaussian fixed In order to obtain the right hand side for the evolution equa- point as Asymptotic Safety demands, but the values of the tion (2), after expanding the metric in its background and critical exponents indicate that there are serious problems. In fluctuating part we obtain, two terms of the form our view, these values show the expected difficulties of massive gravity, that are present in general in all approaches to ( ) − N ( 2 + R ) 1 ∂ R = (A1) massive gravity, which produce results that have no physical k ˆ ˆ t k + + 2 A Ci R Cmmk significance. This could also be related to inconsistencies in the truncation, which means that we need to consider non- 2 A =−D2 − λ N = ( − η (k))k2 R ( −D ) + with 2 k, 2 N k k2 linear terms (interactions). These have been shown to have 2 D2 R ( −D ) C C important effects in the classical theory [16], and come from 2 k k2 and i , and m two constants which depend on ˆ 2 two sources. First we have those non-linear terms that arise the dimension d and the field = (h,φ).Theterm( +Rk) 123 271 Page 6 of 6 Eur. Phys. J. C (2020) 80 :271

2 = , . . λ Fig. 4 Flow diagrams for the expansion made around Ci R for mk 0 0 1and05. Here k (gk ) is in the x (y) axis is known as the inverse modified propagator [7]. In order to 6. R. Percacci, in Approaches to Quantum Gravity: Towards a New compare the right and left hand side of (2) using the heat Understanding of Space, Time and Matter, ed. by D. Oriti (Cam- kernel expansion we need the R-term to be on the numerator bridge University Press, Cambridge, 2009). arXiv:0709.3851 7. M. Reuter, F. Saueressig, Quantum Einstein gravity. of these expressions, so an expansion is made around Ci R arXiv:1202.2274 [hep-th] −1 −1 −2 according to N(A + Cs R) = NA + NA Cs R. 8. M. Reuter, J.-E. Daum, Renormalization group flow of the Holst 710 If we perform the expansion around Ci R (as is done in action. Phys. Lett. B , 215–218 (2012) [5]), we obtain terms in the beta functions of the form log(1− 9. A. Codello, R. Percacci, Fixed points of higher derivative gravity. Phys.Rev.Lett.97, 221301 (2006). arXiv:hep-th/0607128 λ − 2) ( − λ ) 2 k Cmmk instead of log 1 2 k and the flow diagrams 10. O. Lauscher, M. Reuter, Flow equation of quantum Einstein gravity are the ones shown in Fig. 4. in a higher derivative truncation. Phys. Rev. D 66, 025026 (2002) ( − λ − 2) 11. R. Percacci, E. Sezgin, One loop beta functions in topologi- As we can see, the argument in log 1 2 k Cmmk becomes negative and we can no longer obtain informa- cally massive gravity. Class. Quantum Gravity 27, 155009 (2010). arXiv:1002.2640 [hep-th] tion from the beta functions. To avoid this problem, we per- 12. E.C.G. Stuckelberg, Helv. Phys. Act 30, 209 (1957) + 2 form the same expansion but around Ci R Cmmk to obtain 13. S. Weinberg, The Quantum Theory of Fields (Cambridge University the flow equations in (5). It is worth mentioning that both Press, Cambridge, 1995) approaches give the same flow of the couplings for the m → 0 14. Supernova Search Team Collaboration, A.G. Riess et al., Obser- vational evidence from supernovae for an accelerating universe limit. and a cosmological constant. Astron. J. 116, 1009–1038 (1998). arXiv:astro-ph/9805201 15. K. Hinterbichler, Theoretical aspects of massive gravity. References arXiv:1105.3735 [hep-th] 16. A.I. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B 39, 393–394 (1972) 1. A. Einstein, The foundation of the general theory of relativity. Ann. 17. M. Reuter, F. Saueressig, Phys. Rev. D 65, 065016 (2002). Phys. 49, 769–822 (1916) arXiv:hep-th/0110054 2. G. ’t Hooft, M.J.G. Veltman, Ann. Poincaré Phys. Theor. A 20,69 18. C. Wetterich, Exact evolution equation for the effective potential. (1974) Phys. Lett. B 301, 90 (1993) 3. J.F. Donoghue, Introduction to the effective field theory description 19. A. Codello, Phys. Rev. D 91(6), 065032 (2015). arXiv:1304.2059 of gravity. arXiv:gr-qc/9512024 [hep-th] 4. S. Weinberg, in General Relativity, an Einstein Centenary Sur- 20. N. Christiansen, B. Knorr, J.M. Pawlowski, A. Rodigast, Global vey, ed. by S.W. Hawking, W. Israel (Cambridge University Press, flows in quantum gravity. Phys. Rev. D 93(4), 044036 (2016) Cambridge, 1979) 5. M. Reuter, Phys. Rev. D 57, 971 (1998). arXiv:hep-th/9605030

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