Scalar gravity: Post-Newtonian corrections via an effective field theory approach Rafael A. Porto1 and Riccardo Sturani2 1Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 2Physics Department, Univ. Genva, 24 Quai E. Ansermet, 1211 Geneve, Switzerland and INFN, Presidenza dell’INFN, Roma, Italy The problem of motion in General Relativity has lost its academic status and become an active research area since the next generation of gravity wave detectors will rely upon its solution. Here we will show within scalar gravity, how ideas borrowed from Quantum Field Theory can be used to solve the problem of motion in a systematic fashion. We will concentrate in Post-Newtonian corrections. We will calculate the Einstein-Infeld-Hoffmann action and show how a systematic perturbative expansion puts strong constraints on the couplings of non-derivative interactions in thetheory. 7 0 0 I. INTRODUCTION 2 n Consider the apparently simple problem of the earth motion around the sun. The Newtonian solution is an a J excellent approximationbut suppose we wish to be more accurate. A closer look reveals that there are many sources 8 of complication. Einstein theory teaches us how to correct for relativistic effects. However, the earth is clearly not a 1 point particle, and will thus deform under the influence of tidal forces. In addition the whole sun-earth system will radiate energy away in the form of gravitational waves. The inclusion of all these effects can make the problem of 1 solving for the trajectory intractable. In the past, solving this problem was only of academic interest, but the next v generationofgravitywavedetectorswillrelyuponits solution[1]. The constructionofaccuratetemplates forgravity 5 wave interferometersis a daunting task. After more than ten years of work the templates have been completed up to 0 third Post-Newtonian (PN) order for non-spinning compact binaries [2]. However, it was not clear how to: proceed 1 1 tohigherordersina systematicfashion,include finite sizeeffects due tospinorspin-spincorrections. During the last 0 years a new framework has emerged, coined NRGR (Non-Relativistic General Relativity) due to its similarities with 7 EffectiveFieldTheory(EFT)ideasinparticlephysics,whereallofthe apparentobstaclesofthe traditionalapproach 0 canbeensuccessfullyovercome[3,4]. NRGRnaturallyallowsforasystematicaccountoftheinternalstructureofthe / c binary constitutes and permits us to calculate back reaction as well as dissipative effects [5]. Moreover, new results q for spinning compact binaries have been recently reported [6]. In this short contribution we will show within scalar - gravity, how an EFT approach can be used to solve the problem of motion in a systematic fashion [7]. In particular r g we will calculate the Einstein-Infeld-Hoffmann (EIH) actionfor the case of two scalar-gravitating bodies, accurate up : to 1PN.The purposeof this contributionis pedagogical,allowingus to concentrateon the conceptualaspects. As we v i shall see a systematic perturbative expansion puts strong constraints on the couplings of non-derivative interactions X in the theory. r a II. SCALAR GRAVITY The starting point of the EFT approach consists of a theory of point particles coupled to a real scalar field φ we shall call the “s-graviton”. For simplicity we will consider here a massive φ3 theory in a Minkowski background, though we will discuss other type of models later on. The action will be given by S =S +S , with g pp φ S = d4x ∂ φ∂µφ µ2φ2 λφ3 , S = m dτ 1+ (1) g µ pp a a − − − M Z a Z r (cid:0) (cid:1) X describing the s-gravitondynamics and motion of the binary system (a=1,2). In this equation M sets the coupling to matter, and λ,µ, the self-interaction and s-graviton mass respectively. Also dτ = √ηµνdxνdxµ represents the proper time along the a-th particle and ηµν diag(+, , , ), we work in h¯ = c = 1 units. The choice of matter ≡ − − − coupling is meant to resemble Einstein case, at least for the h mode, with M playing the role of the Planck Mass. 00 The normalization is also chosen to mimic the graviton propagator. We could in principle add a set of higher order operatorsintheworldlineactiontoaccountforfinite sizeeffects. However,φ3 theoryissuper-renormalizableanditis 2 possible to show that the n-point function is UV finite and no higher order operatorsareneeded 1. One other aspect of the super-renormalizability is the fact that a φ3 self-interaction in four spacetime dimensions has a dimensionful coupling and the perturbative approach breaks down at distances of order 1/λ. This is connected to IR divergences (in the massless limit) which appear in the perturbative expansion due to factors of λ/E, with E the energy of the s-graviton. These IR divergences must cancel in any physical observable, such as the binding energy of the binary system. However, a resummation procedure is in general needed in order to achieve a finite result [8]. There are a few ways to overcome this. We could work in six dimensions where λ is dimensionless, or with a IR cutoff. Instead we adopted a small s-graviton mass. Notice that a mass term can be produced by a tadpole mechanism, therefore a s-gravitonmass,µ, wouldbe naturally generatedby quantumfluctuations since no symmetry preventsit. One would then expect µ λ. We will see in what follows how a well defined perturbation theory puts strong constraints in ∼ the self-interaction coupling of the theory. We will discuss later on under which circumstances this is a more generic phenomena. III. NRGR The powerofthe EFT formalismresidesin a manifest powercounting inthe expansionparameterofthe theory,in this case the relative velocity v. Here we will pinpoint the necessary steps and refer to Goldberger’s contribution for further details [7]. The expansion of the worldline Lagrangianleads to m v2 φ 1 1 φ2 L = a v2 1 a + v4+ +..., (2) pp 2 a− − 2 M 4 a 4M2 a (cid:20) (cid:18) (cid:19) (cid:21) X where we have chosen x0 as the worldline parameter. The propagator for the field φ appearing in L is still fully pp relativistic,andthereforeasmallvelocityexpansionhasyettobeperformed. Todealwiththisproblemitisconvenient to decompose the s-graviton field into potential modes (φ¯) with momentum scaling kµ (v/r,1/r) (notice they can never go on shell), and radiation modes (Φ) whose momentum scale as kµ (v/r,v/r).∼In the EFT spirit potential ∼ modes do not propagate and can be thus integrated out at each order in perturbation theory. Radiation s-gravitons on the other hand can appear on shell and must be kept as propagating degrees of freedom in order to reproduce the correct long distance physics. A. Power counting In the EFT approachone computes the effective action perturbatively,in our case in v, basedon systematic power counting rules. Inorderto obtainthe latter one starts with the scalinglaws for the (φ¯,Φ)fields. For convenience one first introduces Φ , where the large momentum piece of the potential s-graviton is factored out [3, 7]. By expanding k (1) we get i d4k i hΦk(x0)Φq(0)i=(2π)3δ3(k+q)δ(x0)2(k2−+µ2), hφ¯(x)φ¯(0)i= (2π)42(k2+µ2)eikx (3) Z for the propagators. Notice that we have decided to keep the mass “non-perturbatively” to cure IR divergences, although we will assume µr < v in what follows, and treat it as a perturbation when allowed, in order to resemble the massless power counting rules and a 1/r leading order potential. If we assign the scaling x0 v/r we obtain the ∼ following leading order power counting rules v2 φ¯ v/r, Φ √vr2 Φ M , (4) k ∼ ∼ → ∼ √L where L = mvr. The last arrow follows from the assumption that the leading order potential is given by 1/r and hence the virial theorem, v2 m , applies 2. This assumption is true in the case of λ = 0,µr < v, however λ = 0 ∼ M2r 6 can easily jeopardize our power counting due to the introduction of a dimensionful coupling. In order to tackle this 1 Notice also that using field re-definitions ( equations of motion) we can always trade ∂2φ by a polynomial, so higher dimensional operatorarealwaysoftheformφα andcou∼ldbeabsorbedintotheworldlinecouplings. 2 Noticeitalsoimpliesm/M √Lv ∼ 3 problem let us exercise our scaling rules and power count the first correction due to λ. The diagram is shown in fig. 2a and it scales as m 3 r m v2 3 r λ v2 3 fig. 2a dx0 Φ dx0δ3(k)λΦ3 M M Lv2(λMr2), (5) ∼ M ∼ vM √L vr3 √L ∼ h i (cid:2) (cid:3) (cid:20) (cid:21) (cid:20) (cid:21) v2(λMr2)times the leading ordertermwhichscalesas L [3]. It is easyto see higher orderterms inλ followthe same pattern. For λ =g2M, with g a dimensionless coupling, we end up with r < 1 for the validity of the perturbative gM approximation and power counting. In order to make sense of the perturbative approach we had to cure this IR singularities before expecting any power counting to work, and that is what the s-graviton mass is doing. If we demandourleadingorderpotentialtomatchtheNewtoniancasewewillsetM m andthereforethe perturbative Pl expansion is valid for r < l /g, with l = 10−33 cm, the Planck length. To av≡oid entering the quantum realm3 we Pl Pl willhavetofinetuneg toanextremelysmallnumberoftheorderof10−40fortypicalbinarysystemsinthesolarmass range in the inspiral regime. This obviously defies naturalness arguments and puts a flag on the phenomenological viabilityofsuchtheorysinceitimpliesaridiculouslysmallself-coupling,λ 10−80m 10−60GeV! Noticethatthe Pl ∼ ∼ problemdoesnotlieintheself-couplingitselfbutinthestrengthoftheworldlinecouplingwhichdeterminestheleading order scaling laws. In Einstein theory this is taken care of by the three graviton coupling, g k2/M g Mr2 1. 3 3 The condition µ v/r also implies a stunningly tiny s-graviton mass of the order of 10−30ev∼. These →are consist∼ent, ≤ and somehow equivalent, to solar system constraints [9], whereas by naturalness arguments µ λ would produce an ∼ even smaller value. We will hereon assume µr v,λr 1, and proceed with this theory as a playground. ∼ ≪ IV. EINSTEIN-INFELD-HOFFMANN Let us concentrate now in the calculation of the 1PN correction to the gravitational potential. The leading order one s-graviton exchange can be easily seen to reproduce Newtonian gravity [3]. We also need to take into account diagrams with one single s-graviton exchange which are down by v2 shown in figures 1a and 1b plus the non linear terms depicted infigures2a and2b. We willtreatthe s-gravitonmass asperturbationin the one s-gravitonexchange (µr v) and that is shown in diagram 1c. The computation proceeds systematically by using the Feynman rules of ∼ the EFT orderby order. We will concentrate in detail in fig. 2a,we will display the full resultlater on. For the three s-gravitondiagram we will have 2 1 im im 2 1 fig. 2a= − − dt1dt2dt2′ T(Φ(x1)Φ(x2)Φ(x2′)) . (6) 2 2M 2M h i (cid:18) (cid:19) Z Our task now is to compute the three-point function. For a φ3 theory one obtains 3 d3k 3 3 i hT(Φ(x1)Φ(x2)Φ(x2′))i=3!(−iλ)δ(t1−t2)δ(t1−t2′)Z r (2π)r3e−iPiki·xi(2π)3δ3 i ki! j 2(k2j−+µ2). (7) Y X Y Thenextstepwouldbetoplugthisexpressionbackinto(6),getafiniteresultwhichwewillhavetofurtherexpand in powers of µr v and keep the leading order piece, already at 1PN for λMr2 1. In the EFT spirit a better way ∼ ∼ to proceedis to treatµas aperturbationinthe same waytime derivativesaretreated,by expandingthe propagators in powers of µ/k. For the one s-graviton exchange this represents no harm. In general one faces the problem that | | IRdivergenceswillonlycanceloutafterallthetermsareincluded. Ifwearewillingtoacceptthatisthe caseonecan calculate the 1PN correction by taking the massless limit of (7) and keep the (non-constant) finite piece. Therefore, introducing d=3+ǫ and taking the limit ǫ 0 one gets → 3m m2 d3k d3k 1 3πG m m2 ddk 1 fig. 2a = iλ 1 2 dt 2 1 e−ik1·(x1−x2) =iλ N 1 2 dt 1 e−ik1·(x1−x2) 64M3 (2π)3(2π)3k2k2(k +k )2 16M (2π)3(k2)3/2 Z 1 2 1 2 Z 1 G m m2 x x 2 −ǫ/2 = i3λ N 1 2Γ(ǫ/2) dt | 1− 2| i3λMG2 m2m dtlog(µx x )+constant. (8) 64πM 4 →− N 2 1 | 1− 2| Z (cid:18) (cid:19) Z 3 Recallthatloopeffects inNRGRforgravitonsaresuppressedby1/Landcanbethusignoredintheclassicalscenario[3]. 4 v2 1 µ2 2 a) b) c) FIG.1: Ones-gravitonexchangecontributionat1PN.TheNrepresentsacorrectiontothepropagator,and×amassinsertion. 1 2 (a) (b) FIG. 2: Non linear contributions at 1PN. with G = 1 , and the “constant” piece also contains the 1 IR pole4. N 32πM2 ǫ Our final task consists in collecting the other few pieces. We refer to [3, 7] for details since the calculations are almost identical. Let us compute the result for the new term in fig. 1c due to the s-gravitonmass insertion, m m d3k µ2 i fig. 1c= i 1 2 dt dt δ(t t ) e−ik·(x1−x2) = dt G m m µ2 x x , (9) − 8M2 1 2 1− 2 (2π)3k4 2 N 1 2 | 1− 2| Z Z Z which is nothing but the (v2) piece in the expansion of the Yukawa potential, e−µr µ 1(1+ µ2r2 +...). O − r ∼ − r 2 Putting everything together, including mirror images, we finally obtain 1 G m m (v x )(v x ) G2 m m (m +m ) L = m v4+ N 1 2 v2+v2+(v v ) 1· 12 2· 12 + N 1 2 1 2 EIH 8 a a 2x x 1 2 1· 2 − x x 2 2x x 2 a | 1− 2|(cid:20) | 1− 2| (cid:21) | 1− 2| X 1 + G m m µ2 x x 3λ G2 M m m (m +m ) log(µx x ) (10) 2 N 1 2 | 1− 2|− N 2 1 1 2 | 1− 2| where we have also included the relativistic corrections to the kinetic energy of the point particles. The logarithmic potential introduces a very interesting feature, namely a 1 force into the equations of motion and therefore v2 r ∼ a/r+b+..., which implies a dark matter type of effect for the galaxy rotation curves. This is however by no means a serious candidate and we mention this only as a curiosity. V. DISCUSSION - CONCLUSIONS The EFT approach is a powerful tool within the PN framework. Using no more than dimensional analysis many conclusions can be already drawn before dwelling into the details of the calculations. We applied the techniques in the case of a massive φ3 theory as a playground but the ideas can be easily extended to more complicated scenarios. From the NRGR power counting rules we learned that in order to produce a well defined perturbative expansion, λ had to be fine tuned to a ridiculously small (compared with m ) scale. One might however ask whether this is a Pl feature of a φ3 theory or will this be faced in other scenarios. Let us consider a more general case, S = d4x (γ(φ)∂ φ∂µφ+B(φ)) (11) φ µ Z In what follows we will consider two distinct case. 4 In the massive case this is translated into a logµ factor. The full result is a Bessel function, K0(µr), whose leading order piece in µr reproducesthelogarithmicpotential. 5 A. B(φ)=0 n If we expand γ(φ) 1+φ/M +..., we will get a kinetic piece plus a potential V(φ) with terms like φ φ∂2φ ∼ M n 1. This theory is not renormalizable, and it is easy to show it resembles Einstein case. We can also(cid:16)sh(cid:17)ow that ≥ the perturbative approach is under control by power counting the contribution from a generic term in V(φ). For an (n+2) s-gravitondiagram we will get, n+2 v2 n+2 √L M2r2/v Lv2n. (12) √L ∼ (cid:18) (cid:19) For instance the first term in the expansion, φ2∂2φ, resembles the three graviton coupling in Einstein theory (up to tensor structure). Had we chosen this interaction we would have ended up with a similar 1PN correctionas in the original Einstein-Infeld-Hoffmann action [3, 7]. B. B(φ)6=0 This case is substantially different. Let us study a generic term, gM4(φ/M)n, with g a dimensionless coupling and n 4. The n s-gravitondiagram will scale as ≥ gM4r4v2n−1 gL2v2n−7 L g(m/M)2v2n−8. (13) ∼ ∼ To have a controlled perturbative expansion we would have to impose gv2n−8(m/M)2 < 1. For the marginal case n = 4, setting M = m one needs g < mPl 2 10−70! for solar mass binary constitutes. We can improve this Pl m ∼ number byconsideringhigherdimensionalterms,namely largern,but the enhancementis reallyminute. Notice that (cid:0) (cid:1) this problem arises at the classical level since the coupling to elementary particles is already too small to represent anytrouble. Is only inthe superpositionof terms,whichbuild upthe massivelumpof the star,thatthe perturbative expansionbreaksdown. Fromthisanalysisweconcludethatinpurescalargravitynon-derivativeself-interactionsare extremely constrained. One couldthen wonderabout moregeneralmodels including scalarfields, like tensor-scalargravity[10]. In the latter in addition to the graviton field a scalar interaction is added with an action similar to (11) in a curved spacetime background. Within this type of scenarios the problems we encountered here can be cured by modifying the power counting. For instance, a large mass can be added to the scalar field (larger than the inverse of the solar system distance), which will render the field a negligible short range interaction. Another possibility would be to keep it nearly massless but weaken the coupling to matter to a much feeble strength M m . In this case the 3-scalar Pl diagram (fig 2a) will now scale as L λ m 3. For λ gM one needs g(m/M)≫2 < v2 in order to be treat as a mv2 M ∼ perturbation. By naturalness argument one would expect g 1, and we will then have a very tiny coupling to elementary particles. For instance, the co(cid:0)upl(cid:1)ing to a proton w∼ill be of the order of m /M 10−60! For a φ4 proton ∼ theory the 4-scalar diagram would now scale as λ˜(m/M)4. Compared to the leading Newtonian potential we get a v suppression of order λ˜(m/M)3(Mrv2)−1, which can be seen to be effectively small for λ˜ 1, M m. Again the ∼ ∼ couplingtoelementaryparticlesishighlysuppressed. Bothsolutionswillkeepthetheoryconsistentwithexperimental data, both rely however in the introduction of a high mass scale into the theory, much higher than the Planck scale or the scale of particle physics. Perhaps this is an indication that non-derivative self-interactions are not present in nature 5 Acknowledgments We wouldlike to thank Walter GoldbergerandIraRothsteinfor helpful comments anddiscussions. Special thanks to Christophe Grojean and Francis Bernardeau for organizing such a great summer school and give us the chance to 5 The possibility that the dilaton, the scalar Goldstone mode of the SSB of conformal invariance, could provide an analog of Einstein theorywasrecentlyraisedin[11]. Contrarytothetraditionalloreshiftinvariancedoesnotforbidanon-derivativeinteractionasinthe caseofinternalsymmetries. Asimilarfinetunning,whichwasarguedin[11]tobeofthesamenatureasthecosmologicalconstant,was invokedtoavoidunstableconfigurations. 6 contribute to this volume. The work of R.A.P was supported by DOE contracts DOE-ER-40682-143and DEAC02- 6CH03000. [1] http://www.ligo.caltech.edu [2] L. Blanchet, T. Damour, G. Esposito-Farese and B. R.Iyer, Phys.Rev.Lett.93, 091101 (2004). [3] W. Goldberger and I.Rothstein, Phys. Rev.D 73, 104029 (2006) [4] R.A. Porto, Phys.Rev.D 73, 104031 (2006) [5] W. Goldberger and I.Rothstein, Phys. Rev.D 73, 104030 (2006) [6] R.A. Porto and I. Rothstein, Phys.Rev.Lett.97, 021101 (2006) [7] For a thorough development we encourage the reader to look at Walter Goldberger’s contribution to this volume. arXiv:hep-ph/0701129. [8] R.Jackiw and S. Templeton, Phys. Rev.D 23, 2291 (1981) [9] B. Bertotti, L. Iess and P. Tortora, Nature425, 374 (2003) [10] T. Damour and G. Esposito-Farese, Class. Quant.Grav. 9, 2093 (1992). [11] R.Sundrum,arXiv:hep-th/0312212.