PoS(DSU 2012)021 b http://pos.sissa.it/ . The values of Λ and and the cosmological G G , Ilya L. Shapiro a Júlio C. Fabris a inside and additional Λ and , Oliver F. Piattella, a ∗ G in the Einstein-Hilbert action, as induced by the Renormalization Group (RG). The Λ [email protected] Speaker. We explore the phenomenology of nontrivialfects quantum come effects from on low-energy the gravity. running These of ef- the gravitational coupling parameter constant Renormalization Group corrected General Relativity (RGGR model)quantum is effects, used and to parametrize it these isdue assumed to that these nontrivial the RG dominant effects. darkcation, Here in matter-like we particular present effects on additional inside the details Poisson galaxies on equationre-analyse is the extension the RGGR that model ordinary defines appli- the elliptical effective potential, NGC alsoonic 4494 we contribution, using and a explicit solutions slightly are differentthe presented model NGC for for 4494 the its parameters running bary- as of shown herefor have galaxies, a and better suggest agreement a with larger the radial general anisotropy RGGR than picture the previously published result. ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. Departamento de Física, Universidade Federal do Espírito Santo, 29075-910, Vitória, ES, Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, c

VIII International Workshop on the DarkJune Side 10-15, of 2012 the Universe, Rio de Janeiro, Brazil Davi C. Rodrigues details on the elliptical NGC 4494 Renormalization Group approach to Gravity: the running of a Brazil b Brazil E-mail: PoS(DSU 2012)021 7 − (2.1) ]. Also, 16 Davi C. Rodrigues ] the existence of a 1 Λ . 1 − 0 and ], a certain logarithmic running of G , namely a variation of about 10 4 ν G G , 2 7 = running inside galaxies, which was used in ¯ h 2 Planck c Λ M is a running parameter in the infrared. However, ν 2 G ]. 2 and by evaluating the consequences considering the may behave as a true constant in the far IR limit, 10 = ] galaxies. We proposed in Ref. [ , µ 2 1 G ]). We extended previous considerations by identifying 9 − 8 , µ , 8 d 7 dG µ ≡ ], and are consistent with the phenomenological consequences 1 7 − G β ] and elliptical [ ] we presented new results on the application of renormalization group 1 6 , ]. The previous attempts to apply this picture to galaxies have considered 5 , 14 4 , , and the local value of the Newtonian potential (this relation was reinforced af- 3 13 , µ , 2 , 12 1 , ]). With this choice, the renormalization group-based approach (RGGR) was capable 11 15 ]. Another approach to find the effective potential, which was used previously, is to use a 2 The gravitational coupling parameter Currently, in the context of quantum field theory in curved space time, it is impossible to con- In Refs. [ This section is, in part, a review on the RGGR dynamics, but the route to deduce the effec- is a direct consequence of covariance and must hold in all loop orders. Hence the situation is as leading to standard General Relativity inspace such time, limit. there Nevertheless, in is the no context proof of on QFT that. in curved According to Refs. [ this possibility can not befar ruled infrared out. due to The theinstance, possibility renormalization [ of group General (RG) Relativity has being been modified considered in in the different contexts, for Ref. [ conformal transformation. 2.1 Effective potential and the modified Poisson equation struct a formal proof that the coupling parameter a proper renormalization group energy scale observational data of disk [ relation between to mimic effects withvery great small precision. variation Also, on it the is gravitational remarkable coupling that parameter this picture induces a from this route comes the equation that governs the G follows: either there is no newinfrared, gravitational or effect there induced are by such the deviations and renormalization the group gravitational in coupling the runs far as for simplicity point-like galaxies (e.g., [ terwards [ of diverse approaches toquantum the gravity, see subject, in including particular Refs. the [ related to the asymptotic safety scenario of (RG) corrections to Generalrelation Relativity between in RG the large astrophysicalphenomenological scale model domain, effects was in named and RGGR. particular darkapplication These on matter-like developments to were a effects gravity directly possible in of based galaxies. on Ref. the RG The [ resulting tive potential is new, andsymmetry) an is explicit here expression presented. for Additional the details modified on Poisson this equation deduction (for will appear spherical in Ref. [ Renormalization Group approach to Gravity 1. Introduction of its value across areference galaxy to (depending Renormalization on Group the corrected matter General Relativity. distribution). We call this model RGGR,2. in The RGGR effective potential and the running of PoS(DSU 2012)021 is G ν 1. does (2.4) (2.5) (2.7) (2.3) (2.6) (2.2) κ appears |  ). For the 1 G − 2.2 0 G can be stated as / Davi C. Rodrigues ) N is the gravitational µ 0 Φ ( G . G satisfies ( | . 2 µν G φ T d ) 2. G 4 / θ , leads to the following field π c ( 2 8 c . µν ; moreover, within the weak field x . κ sin T ), satisfies = 4 G ρ 2 1 ≡ r 0 . The constant 2.2 , , gd − 0 2 G ) + Φ c G −  2 0 π G 2 ν 1 8 µ √ θ function, µ µ ∇ / to the Newtonian one d ) are much smaller than one. Also it will be 2 µ Λ = ) = 2 ) and  µ 0 r 2 µ Φ 0 0 ∇ η ( ( µ G µ ln G + − (  G G 2 ln 2 0 G R c µ and − ν µ 3 ν dr , as given by ( ) Z 2 κ )  + r µν ( µ g π 1 − 3 η ( ln 1 c e N − 16 G ν acts as the effective potential, in the sense that its gradient 2 + Φ G ) = 2 κ µ  + = ] = ( dt g κ G [ 2 G Φ  c ) + 2 r ( RGGR ∇ κ µν S e g − Λ 1 = + ]) 2 2 are external scalar fields, that is, their dynamics do not come from pure , due to the smallness of the variation of space-time signature. , µν 0 ds Λ G is dynamically irrelevant, as it will be shown. The dimensionless constant G 17 0. 0 ]). µ 2 = and + ++) ν − G ( ) leads to the logarithmically varying 2.1 is an arbitrary constant (it needs not to be is a reference scale introduced such that 1 0 µ µ We use the From the geodesics, one finds that The form of the action in question is simply the Einstein-Hilbert one, in which The above, together with some energy-momentum tensor To find solutions for static spherically symmetric space-times (a natural symmetry for some , ). Since we have no means to compute the latter from first principles, its value should be fixed 1 ) r ( 2.2 assumed “small RG corrections”, i.e. that is proportional to the acceleration felt by a test particle. But contrary to General Relativity, limit, the value of equations (see, e.g. [ To be clear, by weak field it is mean that both not satisfy a Poisson equation.ρ By solving the field equations in the presence of “dust” of density where classical arguments, but are given from the RG equations; in particular ellipticals, including NGC 4494), the following line element will be considered, constant as measured in the Solarassumes System. the Actually, value there of is no need to be very precise on where a phenomenological parameter which depends( on the details of the quantum theory leading to eq. inside the integral, namely, In the above, Therefore the relation between the effective potential from observations. The first possibility,corresponds namely to of no new gravitational effects in the far infrared, Renormalization Group approach to Gravity Equation ( where problem of internal galaxy kinematics, the cosmologicaland constant effects as are shown negligible in (as expected [ PoS(DSU 2012)021 r ], / 2 for 1 . , the (2.8) (2.9) ) µ N ∝ (2.11) (2.10) 0) [ r ( Φ µ = Λ ν ; µν alone. From G R ]. Another one ) also appeared 7 Davi C. Rodrigues , leads to and N 2.10 G . ∝ Φ µ  ]. and consequently 2 2 N 0 N N 18 as argued above. The relevant Φ Φ , Φ 0 2 µ − , 6 N N Φ 2 , Φ 1 ∇ − . is irrelevant for the dynamics in the weak  , G 0 ν α  0 ∇ Φ να  N 3 0 R Φ N Φ 1 2 Φ + Φ  4 f Λ  = + 0), together with the Bianchi identities ( = =  ρ 0 0 Λ G = ), µ µ 2 µ µ G ν  c ; π . The above result is exact. Equation ( ν µν , but should be found from the Newtonian potential 1 4 2.10 T r ∇ −  , / G 0 N N κ 2.9 Φ Φ ∇ νκ να R 2 identification that seems better justified both from the theoretical and = = µ 0 1 is a phenomenological parameter that needs to depend on the mass of the can be stated as a differential equation that depends on Λ − Λ G α Λ the dynamics in this limit does not depend on ) 0  are constants. The precise value of µ ν ]. And from eqs.( . α limit; which is unsatisfactory on observational grounds (bad Newtonian limit and ∇ 20 α r − , ] ν and 1 19 ∇ 0  Φ ( ] we introduced a The running of In order to derive a test particle acceleration, we have to specify the proper energy scale The parameter would be a complicated function with dependence on diverse constants, that would lead to a 1 f Hence, from the knowledge of the matter distribution, one can find 2.3 The running of in Refs. [ If theory with small (or null) prediction power. The simplest assumption, is to use [ the energy-momentum conservation ( since observational points of view.comes from The the characteristic geometric scaling weak-field 1 gravitational energy scale does not Renormalization Group approach to Gravity 2.2 The energy scale setting latter is the field that characterises gravity in such limit. Therefore, where system, and it must go toa zero good when Newtonian the limit. mass of Fromthe the the increase system Tully-Fisher of goes law, the to it mass zero. is of This expected disk is to galaxies. necessary increase For to monotonically more have with details, see [ the problem setting in question, whichfield is limit. a time-independent gravitational This phenomena isthe in a the usual weak recent procedures area for ofIn high exploration energy [ of scattering the of renormalization particles group cannot application, be where applied straightforwardly. field limit, since parameter is correspondence to the Tully-Fisher law).suitable One cut-off, but way this to rough recover procedure the does Newtonian not limit solves is the to Tully-Fisher issues impose [ a in the large PoS(DSU 2012)021 . ) r ( ∗ (2.13) (2.14) (2.15) (2.12) stands M ]. Here r 22 , is the luminos-  is the anisotropy ) 1 2 Davi C. Rodrigues , R β ( 2 1 I ] are robust to small ), 2 error bars, the results − ) R β σ ( , I ], and which has received 2 1 2 u is the total (effective) mass is the baryonic mass )  . r r 0 B ( , is the Gamma function, dr − M 0 dr r Γ  ) ) r 0 1 2 ], namely we do not directly use the ( r , , ( 2 ) 1 2 r M ∗ r ) ( ρ r + r ( ∞ r β ` R , RGGR )  2 r r 1 M u ( r π R ∗ 4  M  ) + B would stand, within a good approximation, as the 5 K r + β ( ) ∞ 1 ∗ r R + ( 2 Z M ) c 0 M 2 ) 0 . / G R ) ) ] and references therein for additional details. In the case G 1 ( αν ) = for the projected (line of sight) radius, r 2 β 2 I r ( ( − ( ∗ R Γ 44) (see further comments in the Conclusions). β . M ) = M ( ) = r Γ ( R π ( is the incomplete beta function, 2 p √ , i.e., RGGR σ dt r  ) M . See Ref. [ 1 β r − − b t 3 2 . − ] 1 2 1  ( 1 1 − is the luminosity density (found from the deprojection of − β a 2 ) t r u x ( 0 2 1 ` R ]), and Fig. ≡ 1 21 ) ) = u b ( , K a For RGGR without dark matter, the total mass inside the radius In part this is relevant to evaluate to what extent our results presented in [ NGC 4494 is an ordinary elliptical recently analysed within RGGR [ Since elliptical galaxies are mainly supported by velocity dispersions (VD), the main equation , x ( B of the system at the radius parameter (it is zero if the galaxy has an isotropic VD profile) and Nevertheless, the non-Newtonian gravitational effects of RGGRcan for be spherically understood symmetric systems from the Newtoniantotal perspective effective as mass if [ the total mass was given by the following considerable attention previously, in particular due to its apparent lackphotometric data of with dark the matter Sérsic [ extension tothat model best the fit stellar the mass, instead surfaceSee only brightness the Table of Sérsic this profile galaxy is used (an approach here labeled “Full Sérsic”). stellar mass inside the radius with The above is the theapproximation) essential within equation RGGR. to model elliptical galaxies (compatible with the spherical 3. NGC 4494 3.1 Mass models we present a variation of the analysis presented in that Ref. [ changes on the baryonic mass content. The main results are i) within the 1 for galaxy kinematics in this(see case also is [ the following expression for the projected (line-of-sight) VD Renormalization Group approach to Gravity 2.4 Application to spherically symmetric Jeans equation where for the deprojected (spherical) radius, ity intensity, of Newtonian gravity without dark matter, are compatible; ii) the best fit valueanisotropy for (from the about anisotropy parameter isotropy considerably to moves 0 towards radial PoS(DSU 2012)021 ] 1 , 0), 0 [ β = β ), K.IMF 2 ] red 1 χ , inside this 1 . The found − ] Λ [ 2

∈ χ ]. 5.10 0.22 M Davi C. Rodrigues β and 0 CDM case, where 24 . definition (it was used , ) Λ G 2

12 χ 22 , 3 9.09 0.36 5 9.02 0.38 M 5 7.06 0.29 . . . 0 9

. . indicates isotropic VD, 1 1 10 1 3 7 ] ËË 2.52.5 M 0 ± ± + − [ ± 0 0 0 1 2020 β . . 7 . . . 8 8 ËË [ ∈ Mass (10 ËË ∗ , all the others display a “dearth of ËË 10101010 2 M 99 was used in the ) 21 10 32 9 31 10 ËË 88 B . . . ∗ ,

98 1 . . 2.02.0 0 0 0

77 ϒ M 0 3 ËË L 66 + − ± ± ± ( 4 / ËË . 55 B ∗ 23 09 27 3 . . . ¯¯ ϒ ]. 44 ¯¯ 2 ], namely ¯¯ ËË ¯¯ ¯¯ [arcsec] 33 23 ¯¯ 19 4 [kpc] ¯¯ 41 52 . . . 6 ¯¯ R 0 0 0 ¯¯ 1.51.5 ¯¯ R − + ¯¯ β 22 ± ¯¯ 10 ¯¯ lead to total stellar masses that are inside the range 44 ¯¯ NGC 4494 . 19 . ¯¯ 0 1 log ]. On the other hand, only the second model has a value ¯¯ ¯¯ 23 1111 RGGR without dark matter 1.1. (except for the error bars, the above also holds for 6 0 4 7 70 0 43 0 4 ¯¯ . . . ] ], see this reference for further details. (1) 0 99 0.90.9 0 . . 0 0 10 1 2 2 0 0.80.8 , ¯¯ 1.01.0 ± − + ± ± × 0 0.70.7 [ 6 . 77 0.60.6 17 84 . ∈ 2 ¯¯ . . α ν inside NGC 4494 0 1 0 0.50.5 β . The vertical dashed line signs the radius above which the observational data ¯¯ Λ ] 0.40.4 1 00 ]

, 1 ¯¯ 5050 , ]

1 0 0 100100 200200 150150 [ [km/s] VD sight of Line [ and β β − [ G ∈ ] ) 1 ]). For details on this procedure see Ref. [ ] , β 1 0 0 ( [ [ 23 β β [ B ,

L / Two NGC 4494 mass models within RGGR and without dark matter. The curves refer to mass NGC 4494 results for RGGR, assuming that the stellar mass profile is only given by a Sérsic profile. This

M 0 Full Sérsic+K.IMF+ Full Sérsic+K.IMF+ Full Sérsic+ Stellar model Full Sérsic+ . compatible with the baryonic matter amount of this galaxy 1 In accordance with expectations driven by the analysis of other galaxies, see the conclusions. From the NGC 4494 results above, here the corresponding running of The four models presented in Table 2 ± α 2 . indicates constant anisotropy with is a reference to Kroupa IMF, and it means that the expected value of 4 RG effects” for the given baryonic content.some This radial has anisotropy some is similarities needed with to the find reasonable amounts of3.2 dark matter The [ running of expected from the Kroupa (and Chabrier) IMF’s [ Figure 1: is considered for the fitting procedure (10 arcsec). models composed by the stellarlines component are (given the by resulting VD a for SérsicNewtonian each profile) and model, and the non-Newtonian yellow contributions RGGR dashed to gravity. and thethe blue The total second dotted black assumes VD. lines solid are One respectively of the the stellar models assumes isotropy ( anisotropy values are also reasonableof [ Table 1: table extends a table on this galaxy in Ref.[ Renormalization Group approach to Gravity PoS(DSU 2012)021 ; 0 (or , at G Λ 0 50.0 50.0 ” (cyan Φ ) = ] ) which 1 0 , ( α 0 [ G β [kpc] r 10.0 10.0 Davi C. Rodrigues ]). The solid curve 5.0 2 5.0 ) is a natural, if not 2.1 relation to other physical 1.0 1.0 α 0.5 0.5 depends on the value of ], and fits much better within the Galaxy deprojected radius ) r 23 value for its mass (considering the ( G . In both cases, the absolute value of αν ), 0 0.1 0.1 Λ 9 7 5 7 5 9 / (right) inside NGC 4494, both plots considers 10 10 2.9 0.1 0.1 ) 10

10 10 10 10 10 ]. In particular, we are unveiling a correlation

r , plots in the left are normalised with oteeeg density energy 1000 the to 1000 (for further details, see also [ Λ ( for each model. The right plot compares the ratio

18 )

Λ G omlzdcontribution Normalized 0 e 2.2 R Φ 7 25 70 ( Λ (left) and to ]. 60 G 2 may not be zero in the far infrared. Currently, there is no c [kpc] 18 / G r ∗ ρ 50 G π , the dashed curve to 40 ) 0 ]). Here it is commented that with a small change on the modelling of 2 and 4 Λ ” (black on the left plot, blue on the right plot) and “Full Sérsic+ ] 2 ) 0 30 c [ r was found (model with free constant anisotropy). The corresponding mass ( β ( / α Λ ) r ( 20 ∗ ρ G Galaxy deprojected radius π 10 and the galaxy baryonic mass. The galaxy that was in worst agreement with such 0 The plots show the variation of 7 6 6 α 0 - - - 10 10 10

¥ ¥ ¥ RGGR without dark matter is a model with one phenomenological free parameter ( In summary, RGGR is a model based on the theoretical possibility that the beta function of

1. 5.

), nevertheless dynamically such constant plays no role (it can be anything and it will not have 1.5 0 1 /G ) r ( 0 G µ solutions presented in Ref.[ general picture that is being unveiled [ the baryonic mass (i.e., modelling itsthe mass errors on exclusively the from conversion its from light Sérsic tolower profile, mass), mass which a and greater is higher tendency well towards higher inside radialis anisotropy, still inside the mass range associated with the Kroupa IMF [ way to directly deduce this behaviourunique, from first possibility principles, that nevertheless has eq. appeared many ( curved times space-time. before in the context of Quantum Field Theory in is capable of dealingparameters with is being the disclosed in kinematics a work of inbetween progress diverse [ galaxies. The correlation was NGC 4494, since it displayed a too low the gravitational coupling parameter 4. Conclusions corresponds to 4 any radius, is too small towith significantly the change approximations the used average internal to galaxy derive dynamics, the which effective potential is in of accordance RGGR. Figure 2: dynamical impact whenever the weak field approximation holds). Renormalization Group approach to Gravity galaxy are shown. A priori, according to eqs. ( two models, “Full Sérsic+ on the left plot,the black latter on fixes the the right (dynamicallyof plot). irrelevant) the constant absolute Both value of of the PoS(DSU 2012)021 ]. ]. Int. J. , ]. A27 Phys. ]. Quantum , ]. ]. (2004) 001, Davi C. Rodrigues hep-ph/0410095 0412 Int.J.Mod.Phys. , arXiv:1102.2188 gr-qc/9402003 arXiv:1101.5611 JCAP , ]. (2005) 012, [ Disk and elliptical galaxies within astro-ph/9502010 The Dark matter problem and (2011) 98–102, arXiv:0911.4967 0501 . ]. 1471 (2011) 151–159, [ ]. (2011) 124037, [ Dynamics of the Laplace-Runge-Lenz vector in JCAP , Quantum gravity and the large scale structure of (1994) 1001–1009, [ D83 Galaxy rotation curves from General Relativity with Galaxy Rotation Curves from General Relativity hep-th/0410117 (2010) 020, [ 8 D50 ]. Proceedings of the International Conference on Two (1996) 363–374, [ , 1004 Work in progress D5 AIP Conf.Proc. ]. , Phys.Rev. Running G and Lambda at low energies from physics at M(X): , hep-th/0702051 Running coupling constants, Newtonian potential and JCAP Astrophysical and cosmological constraints on a scale dependent Phys. Rev. , (1992) 219–224. , arXiv:1203.2286 (2004) 124028, [ (1993) 27–33. Renormalization group scale-setting in astrophysical systems B281 D70 Quantum gravity at astrophysical distances? Running Newton constant, improved gravitational actions, and galaxy On the Possibility of Quantum Gravity Effects at Astrophysical Scales B311 arXiv:1010.3585 Int. J. Mod. Phys. ]. , ]. (2012) 031, [ arXiv:1203.2695 Phys. Lett. Elliptical galaxies kinematics within general relativity with renormalization group (2006) 2011–2028, [ , Phys. Rev. , 1209 Phys. Lett. , D15 (2011) 1–6, [ JCAP , B703 arXiv:1209.0504 hep-th/0410119 Possible cosmological and astrophysical implications [ renormalization group improved gravity [ gravitational coupling rotation curves Mod. Phys. quantum gravity Lett. Cosmological Models, ed.: Plaza y Valdés, S.A. de C.V. the quantum-corrected Newton gravity the universe Renormalization Group corrections effects with Infrared Renormalization Group Effects corrections to gravity and their implications(2012) for 1260006, cosmology [ and astrophysics nonlocalities in the effective action [8] M. Reuter and H. Weyer, [9] M. Reuter and H. Weyer, [7] I. L. Shapiro, J. Sola, and H. Stefancic, [6] D. C. Rodrigues, P. L. de Oliveira, J. C. Fabris, and I. L. Shapiro, [2] D. C. Rodrigues, [3] D. C. Rodrigues, P. S. Letelier, and I. L. Shapiro, [4] C. Farina, W. Kort-Kamp, S. Mauro, and I. L. Shapiro, [5] J. C. Fabris, P. L. de Oliveira, D. C. Rodrigues, A. M. Velasquez-Toribio, and I. L. Shapiro, [1] D. C. Rodrigues, P. S. Letelier, and I. L. Shapiro, [16] D. C. Rodrigues, B. Koch, and O. Piattella [10] M. Reuter and H. Weyer, [14] O. Bertolami and J. Garcia-Bellido, [11] J. T. Goldman, J. Perez-Mercader, F. Cooper, and M. M. Nieto, [12] O. Bertolami, J. M. Mourao, and J. Perez-Mercader, [15] S. Domazet and H. Stefancic, [13] D. A. R. Dalvit and F. D. Mazzitelli, Renormalization Group approach to Gravity Acknowledgements DCR thanks the DSU organisersthank for CNPq the and invitation FAPES and forported for partial by the financial CNPq, very FAPEMIG support. and nice ICTP. meeting. The work DCR of and I.Sh. JCF hasReferences been partially sup- PoS(DSU 2012)021 ]. ]. (2012). A dearth of , Davi C. Rodrigues et. al. (2005) 705–722, Work in progress astro-ph/0308518 363 arXiv:0810.1291 ]. ]. ]. (Mar., 2012) 2, Masses Out to Five 748 (2003) 1696–1698, [ (Feb., 2009) 329–353, [ 301 arXiv:1010.2799 Mon.Not.Roy.Astron.Soc. , 393 9 arXiv:1107.5815 Astronphys. J. Science , hep-th/0311196 , (2011) 055008, [ Confronting lambda-CDM with the optical observations of elliptical 28 (2011) 103502, [ Asymptotically safe gravity as a scalar-tensor theory and its cosmological The Planetary Spectrograph elliptical galaxy survey: the dark (2004) 104022, [ Renormalization group improved gravitational actions: A Brans-Dicke Exact renormalization group with optimal scale and its application to ]. D84 ]. Mon. Not. R. Astron. Soc. D69 , Phys.Rev. , Class.Quant.Grav. Phys.Rev. , , arXiv:1110.0833 astro-ph/0405491 [ cosmology [ dark matter in ordinary elliptical galaxies O. Gerhard, M. Arnaboldi, F. deP. Lorenzi, Das, K. and Kuijken, K. M. C. R. Freeman, matter Merrifield, in E. NGC O’Sullivan, A. 4494 Cortesi, galaxies. 2. Weighing the dark matter component Effective Radii: The Realm of Dark Matter approach implications [19] B. Koch and I. Ramirez, [24] N. R. Napolitano, A. J. Romanowsky, L. Coccato, M. Capaccioli, N. G. Douglas, E. Noordermeer, [18] P. L. de Oliveira, D. C. Rodrigues, J. C. Fabris, G. Gentile, and I. L. Shapiro [21] G. A. Mamon and E. L. Lokas, [23] A. J. Deason, V. Belokurov, N. W. Evans, and I. G. McCarthy, [22] A. J. Romanowsky, N. D. Douglas, M. Arnaboldi, K. Kuijken, M. R. Merrifield, Renormalization Group approach to Gravity [17] M. Reuter and H. Weyer, [20] Y.-F. Cai and D. A. Easson,