Dimensional regularization of the IR divergences in the Fokker action of point-particle binaries at the fourth post-Newtonian order
Laura Bernard, Luc Blanchet, Alejandro Bohé, Guillaume Faye, Sylvain Marsat
aa r X i v : . [ g r- q c ] J a n Dimensional regularization of the IR divergencesin the Fokker action of point-particle binariesat the fourth post-Newtonian order
Laura Bernard, ∗ Luc Blanchet, † AlejandroBoh´e, ‡ Guillaume Faye, § and Sylvain Marsat ¶ CENTRA, Departamento de F´ısica,Instituto Superior T´ecnico – IST, Universidade de Lisboa – UL,Avenida Rovisco Pais 1, 1049 Lisboa, Portugal G R ε C O , Institut d’Astrophysique de Paris — UMR 7095 du CNRS,Universit´e Pierre & Marie Curie, 98 bis boulevard Arago, 75014 Paris, France Max Planck Institute for Gravitational Physics (Albert Einstein Institute),Am Muehlenberg 1, 14476 Potsdam-Golm, Germany (Dated: July 31, 2018)
Abstract
The Fokker action of point-particle binaries at the fourth post-Newtonian (4PN) approximationof general relativity has been determined previously. However two ambiguity parameters associatedwith infra-red (IR) divergencies of spatial integrals had to be introduced. These two parameterswere fixed by comparison with gravitational self-force (GSF) calculations of the conserved energyand periastron advance for circular orbits in the test-mass limit. In the present paper togetherwith a companion paper, we determine both these ambiguities from first principle, by meansof dimensional regularization. Our computation is thus entirely defined within the dimensionalregularization scheme, for treating at once the IR and ultra-violet (UV) divergencies. In particular,we obtain crucial contributions coming from the Einstein-Hilbert part of the action and from thenon-local tail term in arbitrary dimensions, which resolve the ambiguities.
PACS numbers: 04.25.Nx, 04.30.-w, 97.60.Jd, 97.60.Lf ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] . INTRODUCTION In previous works [1, 2] (respectively referred to as Papers I and II), we determinedthe Fokker Lagrangian of the motion of compact binary systems (without spins) in har-monic coordinates at the fourth post-Newtonian (4PN) approximation of general relativity. Equivalent results had been previously achieved using the ADM Hamiltonian formalism, inADM-like coordinates, developed at 4PN order [3–7]. Partial results have been obtained atthe 4PN order using the effective field theory (EFT) approach [8–11]. A prominent feature ofthis order is the non-locality (in time) due to the imprint of gravitational wave tails startingat that approximation (see also [12]).We start with the gravitation-plus-matter action, made of the gauged-fixed Einstein-Hilbert action of general relativity plus the matter terms describing point particles. TheFokker action governing the motion of compact binaries is then obtained by replacing thegeneric metric in the complete action by an explicit post-Newtonian solution of the cor-responding Einstein field equations. The PN metric is parametrized by appropriate PNpotentials, which are obtained as explicit functionals of the particles’ parameters and tra-jectories. The maximal PN order to which each of the components of the metric is to becontrolled and inserted into the action is determined by the method called “ n + 2” in Paper I(see Sec. IV A there). The spatial integrals coming from the Einstein-Hilbert part of theaction are computed in the physical domain, using elementary solutions of the Poisson equa-tion (the Fock kernel [13] as well as its generalizations) and, in a first stage, the Hadamard“partie finie” integral in 3 dimensions, which is equivalent to a Riesz integration [14].In a second stage, as will be reported in the present paper, we correct the calculation so asto take into account dimensional regularization and the presence of poles in the dimension, inthe limit where d − →
0, for both UV and IR type divergences. Finally, as the non-local tailterm is not included into the “ n + 2” method, we have to compute it in d dimensions and addit separately to the action. The resulting Fokker Lagrangian is a generalized one, dependingon accelerations and derivatives of accelerations. We reduce it to a simpler Lagrangianlinear in accelerations (in harmonic coordinates) by adding suitable multi-zero terms andtotal time derivatives.Carefully choosing and implementing regularizations play a crucial role in this field. InPapers I and II, we adopted a dimensional regularization scheme for treating the ultra-violet(UV) divergences associated with point-particles, as well as a Hadamard regularization forcuring the infra-red (IR) divergences occurring at the bound at infinity of integrals in thegravitational part of the Fokker action (as we know, IR divergences start occurring preciselyat the 4PN order). Unfortunately, we had to introduce in Paper I an “ ambiguity parameter ”reflecting some incompleteness in the Hadamard treatment of the IR divergences. Thisambiguity was then fixed by matching the conserved energy in the case of circular orbitsto known results obtained from gravitational self-force (GSF) calculations in the test-masslimit [15–18]. Note that an equivalent ambiguity parameter had also to be included in theADM Hamiltonian formalism [6]. Furthermore, we were forced to add in Paper II a second ambiguity parameter in order to match the periastron advance for circular orbits with theresults coming from GSF calculations. The latter results are known from numerical [19–21]and analytical [7, 12, 22, 23] studies. As we conjectured in Paper II, this second ambiguity As usual the n PN order means the terms of order ( v/c ) n in the equations of motion relatively to theNewtonian acceleration. i.e. , to compute their values from first principles . Todo so, we employ the powerful dimensional regularization [25–27] (instead of Hadamard’s)for resolving the IR divergences of the Fokker action occurring at the bound at infinity ofspatial integrals. Therefore, our Fokker action will now be entirely based on dimensionalregularization, for both the IR and UV divergences. We have two main tasks:1. Computing the difference between the dimensional regularized and the Hadamardregularized gravitational ( i.e. , Einstein-Hilbert) parts of the Fokker action. For thiscalculation we shall use known formulas for the “difference” between these two reg-ularizations coming from Refs. [28, 29]. The needed accuracy of the post-Newtoniancalculation will follow the rules of the method n + 2 in Paper I;2. Evaluating the non-local tail term in d dimensions or, rather, an associated homoge-neous solution that is to be added to the “difference” computed from the n +2 method.The precise way in which the 4PN tail effect enters our calculation is through the“matching” equation, whose solution gives a connection between the near zone andthe far zone where tails propagate. This equation is the key to the final completionof the problem and the computation of the ambiguity parameters. We find that thecalculation reduces to that of a series of elementary non-local integrals, multipliedby some non trivial numerical coefficient, which is computed in closed analytic formwith Euler gamma functions in App. D. As we shall see (and in agreement with EFTworks [9, 10, 30, 31]), such a tail-induced homogeneous solution contains a UV-likepole in d dimensions. We shall prove that this pole precisely cancels the IR-like poleremaining from the n + 2 method after applying suitable shifts, while the finite partgives a suplementary contribution of the form of the ambiguity parameters of Paper II.Adding up the contributions from the latter two steps (and also, subtracting off a particularsurface term in our previous Hadamard IR regularization scheme), we finally find that themodification of the Lagrangian takes exactly the form postulated in Paper II. Moreover, wefind that the two ambiguity parameters δ and δ (following exactly the definition in Sec. IIof Paper II) are in complete agreement with the result of Paper II [see Eq. (2.6) there],so that the corresponding conserved energy and periastron advance for circular orbits at4PN order are correct. We conclude that our 4PN dynamics based on the Fokker action inharmonic coordinates is now complete.We see that the calculation crucially relies on dimensional regularization and one maywonder why this regularization finally gives the correct answer. We are in fact borrowingthis technique to quantum field theory and EFT [32], since dimensional regularization was In a first version of the present paper, due to an incomplete implementation of the regularization procedurefor the matching between near and far zones in the computation of the tail term, we could only solve forthe second ambiguity parameter. In the companion paper [24], we carefully implement this regularization,and show that it yields the correct value of the first ambiguity parameter. We refer to the companion paper [24] for more details about the matching equation and the overallcalculation. d dimensions. These formulas are thenapplied in Sec. IV to the derivation of the tail term in the near zone metric and then inthe Fokker action. We obtain a UV pole that exactly cancels the IR pole coming from thegravitational part of the Fokker action. This determines the second ambiguity (Sec. V).Moreover, thanks to a careful matching between the near zone and the far zone in ourformalism, we are able to determine the first ambiguity parameter as well, as shown in [24].Technical appendices provide important material on: the homogeneous solutions of the waveequation and their PN expansion in App. A; the multipole expansion of elementary functionsand potentials in d dimensions in App. B; some distributional limits of Green’s functions inApp. C; the computation of some particular intricate coefficient in App. D. II. DIMENSIONAL REGULARIZATION OF INFRA-RED DIVERGENCES
In Paper I [1] it was shown that IR divergences, due to the behaviour of spatial integralsat infinity, start to appear at the 4PN order in the Fokker action of general matter systems.These IR divergences are associated with non-local tail effects in the dynamics occuring at4PN order [34, 35]. In Paper I it was found that two arbitrary scales respectively associatedwith tails (denoted s in Paper I) and the IR cut-off (denoted r ) combine to give an“ambiguity” parameter α = ln( r /s ) which could not be determined within the method.Equivalent results had been obtained with the Hamiltonian formalism in Ref. [6]. However,in contrast to the Hamiltonian formalism, we had to introduce in Paper II a second ambiguityparameter and argued that it was due to our particular treatment of the IR divergences basedon the Hadamard “partie finie” integral. On the other hand, the UV divergences associatedwith point particles were cured by dimensional regularization.In the present paper we shall employ dimensional regularization for both the IR and UVdivergences. As we shall see, using dimensional regularization does modify the end result forthe Fokker Lagrangian (and associated Hamiltonian), but in a way that is fully consistentwith the conjecture put forward in Paper II. Therefore this justifies the final 4PN dynamicsobtained in Paper II and in particular, we confirm that the 4PN dynamics is compatiblewith existing GSF computations of the energy and periastron advance for circular orbits.We want to regularize the three-dimensional divergent integral I = Z d x F ( x ) , (2.1)where the function F is obtained by following the PN iteration procedure of the field equa-4ions using the method n +2 (see Sec. IV A of Paper I). The integral (2.1) represents a genericterm in the gravitational (Einstein-Hilbert) part of the Fokker Lagrangian L g . Specifically,since we are dealing with the IR bound at infinity, we consider I R = Z r> R d x F ( x ) , (2.2)where the integration domain is restricted to be r = | x | > R , with R being a sufficientlylarge constant radius. The divergences occur from the expansion of F when r → + ∞ , whichis of the type (for any N ∈ N ) F ( x ) = N X p = − p r p ϕ p ( n ) + o (cid:18) r N (cid:19) . (2.3)The coefficients ϕ p depend on the unit direction n = x /r and on p ∈ Z ; the minimal value of p corresponds to some highly divergent behaviour with growing power ∼ r p of the distance.In what follows we shall write for simplicity some formal expansion series without explicitingthe remainder term, that is F ( x ) = X p > − p r p ϕ p ( n ) . (2.4)In Paper I a regularization factor ( r/r ) B was introduced into the integrand and theintegral was considered in the sense of analytic continuation in B ∈ C . Then the regularizedvalue of the integral was defined as the finite part (FP), i.e. , the coefficient of the zero-th power of B , in the Laurent expansion of the regularized integral when B →
0. Thisprescription, which is equivalent to a Hadamard regularization (HR), reads I HR R = FP B =0 Z r> R d x (cid:16) rr (cid:17) B F ( x ) . (2.5)A straightforward calculation, plugging (2.4) into (2.2) (where R is a large radius), yieldsthe HR-regularized version of the integral as I HR R = − X p =3 R − p − p Z dΩ ϕ p ( n ) − ln (cid:18) R r (cid:19) Z dΩ ϕ ( n ) , (2.6)where dΩ denotes the standard surface element in the direction n . As we see the crucialcoefficient in the expansion (2.4) is that for p = 3; it corresponds to a logarithmic divergenceof the original integral (2.2).In the present paper, motivated by the success of dimensional regularization when treat-ing the UV divergencies, we treat the IR divergences of the integral (2.1) with the sameregularization procedure. In d spatial dimensions the equivalent of F ( x ), i.e. , arising fromthe same PN iteration of the field equations but performed in d dimensions, will be a function F ( d ) ( x ) with a more general expansion when r → + ∞ of the type F ( d ) ( x ) = X p > − p q X q = − q r p (cid:18) ℓ r (cid:19) qε ϕ ( ε ) p,q ( n ) . (2.7) In Appendix B we shall refer to the far zone expansion when r → + ∞ as a “multipole” expansion andconveniently denote it as M ( F ( d ) ). /r now depend linearly on ε = d −
3, with p ∈ Z as before and with also q ∈ Z , bounded from below and from above by − q and q . Here ℓ denotes the usual constant scale associated with dimensional regularization.Assuming that the coefficients ϕ ( ε ) p,q have a well-defined limit when ε → i.e. , that they donot contain any pole ∝ /ε (such an assumption is always verified at 4PN order), we obtainthe following relation with the coefficients ϕ p in the limit ε → ϕ p ( n ) = q X q = − q ϕ ( ε =0) p,q ( n ) . (2.8)The dimensional regularization (DR) prescription, to be considered as usual in the senseof complex analytic continuation in d ∈ C , reads now I DR R = Z r> R d d x ℓ d − F ( d ) ( x ) . (2.9)Working in the limit where ε → i.e. , keeping only the pole ∝ /ε followed by the finitepart ∝ ε , and using also the relation (2.8), we readily obtain I DR R = − X p =3 R − p − p Z dΩ ϕ p ( n ) + X q (cid:20) q − ε − ln (cid:18) R ℓ (cid:19)(cid:21) Z dΩ ε ϕ ( ε )3 ,q ( n ) + O ( ε ) . (2.10)Very important in this formula, is that the angular integration in the second term, becauseof the presence of the pole, is to be performed over the ( d − ε ( n ), up to order ε .We shall thus add to the computations of Papers I and II the difference between thetwo prescriptions, say D I = I DR R − I HR R . Note that the first term in (2.10) is identical tothe corresponding term in (2.6), and thus cancels out in the difference. We thus obtain, todominant order when ε → D I = X q (cid:20) q − ε − ln (cid:18) r ℓ (cid:19)(cid:21) Z dΩ ε ϕ ( ε )3 ,q ( n ) + O ( ε ) , (2.11)where, as expected, the scale R has disappeared from the difference.We have applied the formula (2.11) to each of the terms composing the gravitational part L g of the Fokker Lagrangian. Thus, we have computed the expansion when r → + ∞ of thevarious potentials parametrizing the metric in d dimensions as given by Eqs. (4.14)–(4.15) inPaper I. These potentials are those needed at the 4PN order following the method “ n + 2”described in Sec. IVA of Paper I. For this calculation we use the far-zone expansion of someelementary functions in d dimensions (notably the elementary Fock kernel g [13]); this willbe described in Appendix B. Once we have computed the expansions of all the potentials weplug them into L g and obtain the coefficients ϕ ( ε )3 ,q ( n ) corresponding to all the terms. Then A priori the result also contains terms that diverge at infinity. These terms correspond to the coefficients ϕ ( ε ) p,q with q = 1 and p
3, but do not appear in our computation. Extensive use is made of the software Mathematica together with the tensor package xAct [36].
6e simply evaluate Eq. (2.11) for each of the terms and obtain the Fokker action with IRdivergences correctly regularized by means of DR.The total difference will actually be called D L inst g = P D I . Indeed it is composed of allthe terms obtained following the method n + 2, which keeps track of the “instantaneous”terms, but neglects the “tail” term which will be investigated in Sec. IV. Thus, D L inst g iscomposed of a pole part ∝ /ε followed by a finite part ∝ ε which depends on the arbitraryIR scale r as well as on ℓ . We next look for a (physically irrelevant) shift that will removemost of the poles 1 /ε and eliminate most of the dependence on the constant r . We find,after applying a suitable shift, that the difference becomes (irreducibly) D L inst g = G m c (cid:20) ε − (cid:18) √ ¯ q r ℓ (cid:19)(cid:21) (cid:16) I (3) ij (cid:17) + G m m m c r (cid:18) − n v ) + 123475 v (cid:19) + O ( ε ) , (2.12)where we pose ¯ q = 4 π e γ E with γ E being the Euler constant. The other notations are exactlythe same as in Papers I and II, e.g. , m = m + m is the total mass and ( n v ) is theEuclidean scalar between the relative direction between the two bodies and their relativevelocity.As we see there is a remaining pole in Eq. (2.12), and we shall prove in Sec. IV thatit will be cancelled by a corresponding pole coming from the 4PN tail term evaluated in d dimensions. The pole is proportional to the square of the third time-derivative of thequadrupole moment I ij . In a small 4PN term, the quadrupole can be taken to be theNewtonian one; however, consistently with the pole 1 /ε in front, it is to be evaluated in d dimensions, up to order ε included. For completeness we show here the complete expressionup to that order, (cid:16) I (3) ij (cid:17) = G m m r (cid:18) −
883 ( n v ) + 32 v (cid:19) (cid:20) − ε (cid:18) √ ¯ q r ℓ (cid:19)(cid:21) + ε G m m r (cid:18) − n v ) + 96 v (cid:19) + O (cid:0) ε (cid:1) . (2.13)Gladly, we discover that the two terms in the second line of Eq. (2.12) have exactly thestructure of the two “ambiguity” parameters δ and δ that were introduced in Paper II. Aswe shall see, this will permit to confirm the conjecture advocated in Paper II, namely thatdifferent IR regularizations have merely the effect of changing the values of two and onlytwo ambiguity parameters δ and δ (modulo, of course, irrelevant world-line shifts).Next, in addition to Eq. (2.12), we must also consider another “instantaneous” contribu-tion when working in full DR. This is due to the fact that in HR it matters if we start froma gravitational Lagrangian at quadratic order of the type ∼ ∂h∂h or of the type ∼ h (cid:3) h ( i.e. , the propagator form). Indeed, the two Lagrangians differ by a surface term ∼ ∂ ( h∂h )coming from the integration by part, and this surface term does contribute in HR. On thecontrary, in DR it does not matter whether one starts with the Lagrangian in the form ∼ ∂h∂h or with the Lagrangian in propagator form ∼ h (cid:3) h because the surface term is In practical calculations we always verify that the coefficient ϕ ( ε )3 , ( n ) averages to zero, so that there is noproblem with the value q = 1 in Eq. (2.11). d . The fact that the two La-grangians are equivalent in DR constitutes a very nice feature of DR as opposed to HR. InPaper I we initially performed our HR calculation with the ∼ ∂h∂h Lagrangian and thencorrected it by adding the appropriate surface term so that our HR prescription starts witha Lagrangian having the propagator form ∼ h (cid:3) h . On the other hand our calculation ofthe difference yielding (2.12) has been done with the prescription ∼ ∂h∂h , so we now haveto subtract off the latter surface term. After applying an appropriate shift, this gives thefollowing contribution to be subtracted from the HR result in order to control the full DR: D L surf g = G m m m c r (cid:20) − n v ) + 6415 v (cid:21) . (2.14)Again we find it to have the form of the ambiguity parameters modulo shifts.In the language of EFT (see for instance Ref. [30]) our “instantaneous” calculation whichhas been done in the present section and yields Eq. (2.12), corresponds to the so-called“potential mode” contribution, say V pot . As emphasized in [30, 31], the pole it containsis an IR pole, thus ε ≡ ε IR . However, there is now to take into account the contributioncoming from the conservative part of the 4PN tail effect in d dimensions, which correspondsin the EFT language to the “radiation” contribution, say V rad . As we shall prove in Sec. IVthe IR pole in Eq. (2.12) will be cancelled by a corresponding UV pole ε ≡ ε UV coming fromthe radiation term in d dimensions. III. FORMULAS FOR THE NEAR-ZONE EXPANSION IN d DIMENSIONS
In this section and the following one we shall prove that there is another contributionin the difference between DR and HR, coming from the tail effect in d dimensions. Indeedthe computation in the previous section was based on the method “ n + 2” (see Sec. IV A ofPaper I) which is valid for symmetric terms defined from the usual symmetric propagator.However the tail effect at 4PN order is to be added separately since it is in the form of anhereditary type homogeneous solution of the wave equation, which is of the anti-symmetric type ( i.e. , advanced minus retarded), thus regular when r →
0, and which has not beentaken into account in the method n + 2.We start by general considerations on the near-zone expansion of the solution of the flatscalar wave equation in d + 1 space-time dimensions (thus, with x ∈ R d ), (cid:3) h ( x , t ) = N ( x , t ) . (3.1)The source of such an equation will represent a generic term in the source of the equa-tion (4.2) that we shall solve in the next section. The retarded Green’s function G ret ( x , t )of that scalar wave equation, thus satisfying (cid:3) G ret ( x , t ) = δ ( t ) δ ( d ) ( x ), explicitly reads G ret ( x , t ) = − ˜ k π θ ( t − r ) r d − γ − d (cid:18) tr (cid:19) , (3.2)where ˜ k = π − d Γ( d −
1) (with Γ being the usual Eulerian function) denotes a pure constantso defined that lim d → ˜ k = 1, and θ ( t − r ) denotes the usual Heaviside step function. The General conventions from earlier works [28, 29] are adopted. We pose G = c = 1 in this section. G adv ( x , t ) is given by the same expression but with θ ( − t − r ) instead of θ ( t − r ). We have introduced for convenience the function γ s ( z ) definedfor any s ∈ C and | z | > γ s ( z ) = 2 √ π Γ( s + 1)Γ( − s − ) (cid:0) z − (cid:1) s = Γ( − s )2 s +1 Γ( s + 1)Γ( − s − (cid:0) z − (cid:1) s , (3.3)where the normalisation has been chosen so that Z + ∞ d z γ s ( z ) = 1 . (3.4)The latter integral converges when − < ℜ ( s ) < − and can be extended to any s ∈ C bycomplex analytic continuation. For strictly negative integer values (say s ∈ − − N ) thefunction (3.3) is zero in an ordinary sense, but is actually a distribution; for instance wecan check that γ − ( z ) = δ ( z −
1) (see Appendix C). Notice that the Green’s function (3.2)depends only on t and the d -dimensional Euclidean norm r = | x | . Its Fourier transform isalso known [see e.g. Eq. (2.4) in Ref. [29]]. The retarded solution of the wave equation (3.1)is given by h ( x , t ) = Z + ∞−∞ d t ′ Z d d x ′ G ret ( x − x ′ , t − t ′ ) N ( x ′ , t ′ )= − ˜ k π Z + ∞ d z γ − d ( z ) Z d d x ′ N ( x ′ , t − z | x − x ′ | ) | x − x ′ | d − . (3.5)Now, we want to identify a piece in this solution, that will be a homogeneous anti-symmetric solution of the wave equation which is regular when x →
0. It may be obtainedby performing the formal near-zone expansion of h ( x , t ). Later we shall use this homogeneoussolution to control the tail effect in the near zone. Thus, for this application we considerthat N ( x , t ) represents a particular term in the quadratic part of the Einstein field equationsoutside the matter source, i.e. , a generic term of N [ h ] in Eq. (4.2) below. In particular N ( x , t ) is to be thought as already “multipole-expanded” outside the matter source. The function γ s ( z ) is the natural generalization of the function γ ℓ ( z ) (for ℓ ∈ N ) introduced in [37,38] to parametrize “radiation-reaction” STF multipole moments. In a similar way one can introduce afunction δ s ( z ) which would be a generalization of the function parametrizing the “source-type” multipolemoments [39], δ s ( z ) = Γ( s + ) √ π Γ( s + 1) (cid:0) − z (cid:1) s , and satisfying R − d z δ s ( z ) = 1. One can show that γ s ( z ) = − (1 + e − πs ) δ s ( z ), thus γ ℓ ( z ) = − δ ℓ ( z )when ℓ ∈ N . Note also that the Riesz [14] kernels Z α ( t, r ) in Minkowski d + 1 space-time (satisfying theconvolution algebra Z α ∗ Z β = Z α + β ) are given in terms of the function γ s ( z ) by Z α ( t, r ) = Γ( d − α )Γ( α ) r α − d − α π d γ α − d − (cid:18) tr (cid:19) .
9e start from Eq. (3.5) in which we swap the time and space integrals, defining˜ N ret ( x ′ , | x − x ′ | , t ) = − ˜ k π Z + ∞ d z γ − d ( z ) N ( x ′ , t − z | x − x ′ | ) | x − x ′ | d − , (3.6)which is a homogeneous solution of the wave equation with respect to the field point x : (cid:3) ˜ N ret ( x ′ , | x − x ′ | , t ) = 0. Homogeneous solutions of the wave equation are investigatedin general terms in Appendix A. The d -dimensional integral (3.5) is defined by complexanalytic continuation in d = 3 + ε , and we are looking to the neighbourhood of ε = 0, thelatter point being excluded. However, we shall find that for some particular terms in ourcalculation, the analytic continuation cannot be performed as the ε ’s cancel out. In order tobe protected when such a cancellation happens, we introduce a regulator r ′ η in factor of thesource (where r ′ = | x ′ | ), and carry out all calculations with some finite parameter η ∈ C ,invoking the analytic continuation in η when necessary. At the end of our calculation weshall compute the limit when η →
0, and find that this limit is finite for any ε . Finally weapply the DR prescription on the result, looking for the neighbourhood of ε = 0 and thepresence of poles 1 /ε . For more details see the companion paper [24], where this method isreferred to as the “ εη ” regularization. From now on we thus consider (with implicit limitwhen η → h ( x , t ) = Z d d x ′ r ′ η ˜ N ret ( x ′ , | x − x ′ | , t ) . (3.7)The d -dimensional integral is then split into two pieces, each of which corresponds to theregions of integration | x ′ | < R and | x ′ | > R , respectively, for some positive R . If we choose R equal to the near-zone radius, we are allowed to replace the source ˜ N ret ( x ′ , | x − x ′ | , t ) ofthe inner integral by its own PN expansion, as given by Eqs. (A7)–(A8) in Appendix A. Theresult may be written as an integral over the whole space, minus the same integral over theregion | x ′ | < R . This yields h ( x , t ) = Z d d x ′ r ′ η ˜ N ret ( x ′ , | x − x ′ | , t )+ Z | x ′ | >R d d x ′ r ′ η h ˜ N ret ( x ′ , | x − x ′ | , t ) − ˜ N ret ( x ′ , | x − x ′ | , t ) i , (3.8)where the overbar refers to the PN expansion. Next, in the second integral, extending overthe exterior zone ( | x ′ | > R ), we can perform a formal Taylor expansion when | x ′ | → + ∞ .After expressing the result in terms of symmetric-trace-free (STF) tensors, we find˜ N ret ( x ′ , | x − x ′ | , t ) = + ∞ X q =0 ( − ) q q ! + ∞ X j =0 ∆ − j ˆ x Q (cid:16) ˆ ∂ ′ Q ˜ N (2 j )ret ( y , r ′ , t ) (cid:17) y = x ′ , (3.9)where ˆ ∂ ′ Q denotes the STF projection of a product of q partial derivatives ∂ ′ Q = ∂ ′ i · · · ∂ ′ i q with respect to x ′ i ( i.e. , ∂ ′ i = ∂/∂x ′ i ), where Q = i · · · i q is a multi-index with q indices, andwhere the time multi-derivatives are indicated with the superscript index (2 j ). Furthermorewe employ the useful short-hand notation (with r = | x | ) [37, 38]∆ − j ˆ x Q = Γ( q + d )Γ( q + j + d ) r j ˆ x Q j j ! , (3.10)10or the iterated inverse Poisson operator acting on the STF product ˆ x Q of q source points x i , such a notation being motivated by the fact that ∆(∆ − j ˆ x Q ) = ∆ − j +1 ˆ x Q . Notice that inEq. (3.9) the point y is held constant when applying the partial derivatives, and is to bereplaced by x ′ only afterwards. The same treatment applies also for the overbared quantityin the last term of (3.8). At this stage we obtain the near-zone or PN expansion h ( x , t ) = Z d d x ′ r ′ η ˜ N ret ( x ′ , | x − x ′ | , t )+ + ∞ X q =0 ( − ) q q ! + ∞ X j =0 ∆ − j ˆ x Q Z | x ′ | >R d d x ′ r ′ η (cid:16) ˆ ∂ ′ Q ˜ N (2 j )ret − ˆ ∂ ′ Q ˜ N (2 j )ret (cid:17) y = x ′ . (3.11)Applying the same idea as before, i.e. , decomposing the second term as an integral over thewhole space minus the same integral restricted to the inner region | x ′ | < R , we can furtherrewrite the above expression as h = Z d d x ′ r ′ η ˜ N ret ( x ′ , | x − x ′ | , t ) + + ∞ X q =0 ( − ) q q ! + ∞ X j =0 ∆ − j ˆ x Q Z d d x ′ r ′ η (cid:16) ˆ ∂ ′ Q ˜ N (2 j )ret (cid:17) y = x ′ + ∆ . (3.12)This takes almost the requested form, but there is still the last term with a peculiar unwantedform, given by∆ = − + ∞ X q =0 ( − ) q q ! + ∞ X j =0 ∆ − j ˆ x Q (cid:20)Z | x ′ |
0, the second term in (3.15) is a homogeneous solution; let us call it h asym for a reasonto soon become clear. In more details it reads h asym = − ˜ k π + ∞ X q =0 ( − ) q q ! + ∞ X j =0 ∆ − j ˆ x Q Z + ∞ d z γ − d ( z ) Z d d x ′ r ′ η ˆ ∂ ′ Q (cid:20) N (2 j ) ( y , t − zr ′ ) r ′ d − (cid:21) y = x ′ . (3.16)This is our looked-for homogeneous solution; it is clearly of the form h asym = P + ∞ q =0 P + ∞ j =0 ∆ − j ˆ x Q F (2 j ) Q ( t ), on which form we can directly check that (cid:3) h asym = 0. Fur-thermore, that solution is manifestly regular when r →
0, and so it must be identified witha homogeneous anti-symmetric solution of the wave equation in d dimensions, of the typehalf-retarded minus advanced. In particular, Eq. (3.16) must be identified with an anti-symmetric solution H asym whose general form is given by Eq. (A15). Bearing unimportantfactors, this means that we should always be able to find a function f Q ( t ) such that F Q ( t ) = Z + ∞ d τ τ − ε h f (2 ℓ +2) Q ( t − τ ) − f (2 ℓ +2) Q ( t + τ ) i . (3.17)We prove this statement by going to the Fourier domain. Given the Fourier transform ˆ F Q ( ω )of F Q ( t ), Eq. (3.17) will be verified provided that the Fourier transform ˆ f Q ( ω ) of f Q ( t ) takesthe expression ˆ f Q ( ω ) = 2i( − ) ℓ cos( πε )Γ(1 − ε ) sign( ω ) | ω | ℓ +1+ ε ˆ F Q ( ω ) . (3.18)Next, we consider the case of a source term which has a definite multipolarity ℓ , namely N ( x , t ) = ˆ n L N ( r, t ), where ˆ n L is the STF projection of the product of ℓ unit vectors n i , andlike before L = i · · · i ℓ . We shall denote the corresponding solution by h asym L ( x , t ). Usingˆ ∂ ′ Q f ( r ′ ) = ˆ n ′ Q r ′ q ( r ′− d / d r ′ ) q f ( r ′ ) in (3.16), we can explicitly perform the angular integrationin d dimensions [see e.g. Eqs. (B23) in [28]], and get h asym L = ( − ) ℓ +1 Γ( d )2 ℓ ( d − d + ℓ ) + ∞ X j =0 ∆ − j ˆ x L Z + ∞ d z γ − d ( z ) × Z + ∞ d r ′ r ′ d + ℓ − η (cid:18) r ′ dd r ′ (cid:19) ℓ (cid:20) N (2 j ) ( | y | , t − zr ′ ) r ′ d − (cid:21) | y | = r ′ . (3.19)Still this formula can be substantially simplified by means of a series of integrations by partsover the z -variable, and we nicely obtain h asym L = − d + 2 ℓ − + ∞ X j =0 ∆ − j ˆ x L Z + ∞ d z γ − d − ℓ ( z ) Z + ∞ d r ′ r ′− ℓ +1+ η N (2 j ) ( r ′ , t − zr ′ ) . (3.20)We now specialize Eq. (3.20) to the case of a source term made of a quadratic interactionbetween a monopolar static solution ∝ r d − and some homogeneous multipolar retarded solution, namely, a spatial multi-derivative of a monopolar retarded solution [see Eq. (4.5)].12ndeed, such source term will be the one we meet when computing the tail effect as seen inthe near zone ( r → ε = d − N ( r, t ) = r − k − ε Z + ∞ d y y p γ − − ε ( y ) F ( t − yr ) , (3.21)where k, p ∈ N and the function F ( t ) stands for some time derivative of a component ofa multipole moment, namely the source quadrupole moment I ij ( t ) that we shall considerin Sec. IV. Plugging (3.21) into (3.20), and performing the change of integration variable r ′ → τ = ( y + z ) r ′ , we readily obtain h asym L = − C p,kℓ ℓ + 1 + ε + ∞ X j =0 ∆ − j ˆ x L Z + ∞ d τ τ − ℓ − k +1 − ε + η F (2 j ) ( t − τ ) , (3.22)with the following purely numerical coefficient (also depending on the dimension) C p,kℓ = Z + ∞ d y y p γ − − ε ( y ) Z + ∞ d z ( y + z ) ℓ + k − ε − η γ − ℓ − − ε ( z ) . (3.23)We are ultimately interested in the limit ε →
0, but it is clear that the integral over τ in (3.22) becomes ill-defined in this limit because of the bound τ = 0 of the integral. Onthe other hand since F ( t ) is a time derivative of a multipole moment, we can assume thatit is zero in a neighbourhood of t = −∞ so there is no problem with the bound τ = + ∞ of the integral. We thus make explicit the generic presence of a pole ∝ /ε when ε → ℓ + k − ε ≡ ε UV . All surface terms vanish by analytic continuationin ε and because F ( t − τ ) is zero when τ → ∞ , so we arrive at h asym L = ( − ) ℓ + k C p,kℓ ℓ + 1 + ε Γ(2 ε − η )Γ( ℓ + k − ε − η ) + ∞ X j =0 ∆ − j ˆ x L Z + ∞ d τ τ − ε + η F (2 j + ℓ + k − ( t − τ ) . (3.24)Note the retarded character of this solution, which comes directly from the retardedcharacter of the source term postulated in Eq. (3.21). In our approach, we are iteratingthe Einstein field equations by means of retarded potentials. Thus, at some given non-linear order, for instance quadratic, we obtain a retarded source term which represents the physical solution, containing both conservative and radiation-reaction dissipative effects.Only at this stage do we identify an “anti-symmetric” piece which is a part of the physicalretarded solution generated by that source term, and which will be associated with the taileffect in the near zone.The equation (3.24) is our final formula for this section, with which we can directly controlthe looked-for limit when ε →
0. In generic cases a pole ∼ /ε will show up, while the finitepart beyond the pole will contain an ordinary tail integral with the usual logarithmic kernel.The numerical coefficient C p,kℓ defined by Eq. (3.23) is a priori not trivial to control, butfortunately we have found a way to compute it analytically as described in Appendix D. IV. DERIVATION OF THE TAIL TERM IN d DIMENSIONS
We shall compute the tail term in d dimensions directly in the near zone metric of generalmatter sources, then obtain its contribution in the equations of motion of compact binaries13nd finally in the Fokker action. The Einstein field equations in harmonic gauge in thevaccum region outside an isolated source read (cid:3) h µν = Λ µν [ h ] , (4.1a) ∂ ν h µν = 0 , (4.1b)where (cid:3) is the flat d’Alembertian operator, h µν = √− gg µν − η µν is the “gothic” metricdeviation from flat space-time, and Λ µν denotes the non-linear gravitational source term,which is at least quadratic in h and its derivatives. As we shall see, to control the 4PN taileffect we can limit ourselves to the quadratic non-linear order, say h µν = Gh µν + G h µν + O ( G ). Denoting by N µν [ h ] the quadratic piece in the non-linear source term Λ µν theequations to be solved are thus (cid:3) h µν = N µν [ h ] , (4.2)together with ∂ ν h µν = 0. At this stage we know that the tail effect is an interaction betweenthe constant mass of the system M and its time-varying mass-type STF quadrupole moment I kl ( t ). Accordingly the linearized metric is composed of two pieces, say h µν = h µνM + h µνI kl .The static one corresponding to the mass reads h M = − M , h iM = 0 , h ijM = 0 , (4.3)while the dynamical one for the quadrupole moment in harmonic gauge is given by h I kl = − ∂ ij ˜ I ij , (4.4a) h iI kl = 2 ∂ j ˜ I (1) ij , (4.4b) h ijI kl = − I (2) ij . (4.4c)We are essentially following the notation of Eqs. (3.44) in [29]. In particular we denote ahomogeneous retarded solution of the d’Alembertian equation as˜ I ij ( t, r ) = − π Z + ∞−∞ d t ′ G ret ( x , t − t ′ ) I ij ( t ′ )= ˜ kr d − Z + ∞ d z γ − d ( z ) I ij ( t − zr ) . (4.5)See the retarded Green’s function of the d’Alembertian equation in Eq. (3.2) above. For thestatic mass this reduces to a homogeneous solution of the Laplace equation,˜ M ( r ) = − π M Z + ∞−∞ d t ′ G ret ( x , t − t ′ ) = ˜ kMr d − . (4.6)The quadratic source term N µν [ h ] built out of the linearized metrics (4.3)–(4.4) reads N M × I kl = − h M ∂ h I kl − h ijI kl ∂ ij h M − d − d − ∂ i h M ∂ i h I kl + ∂ i h M ∂ h iI kl , (4.7a) N iM × I kl = − h M ∂ h iI kl + d d − ∂ i h M ∂ h I kl + ∂ j h M ∂ h ijI kl + ∂ j h M (cid:0) ∂ i h jI kl − ∂ j h iI kl (cid:1) , (4.7b)14 ijM × I kl = − h M ∂ h ijI kl + d − d − ∂ ( i h M ∂ j ) h I kl − d − d − δ ij ∂ k h M ∂ k h I kl − δ ij ∂ k h M ∂ h kI kl + 2 ∂ ( i h M ∂ h j )0 I kl . (4.7c)As we have investigated in Sec. III, the tail effect we are looking for comes from a suitablehomogeneous anti-symmetric solution of the wave equations (4.2). We have therefore appliedour end result given by Eq. (3.24), together with the explicit method for the computationof the coefficients C p,kℓ as explained in Appendix D, to each of the terms of Eqs. (4.7). Weconsider only the pole part ∝ /ε followed by the finite part when ε →
0, and re-expandwhen c → + ∞ in order to keep only the terms contributing at the 4PN order. We thenobtain the homogeneous solution responsible for the tails as h ii asym = 8 G M c x ij Z + ∞ d τ (cid:20) ln (cid:18) c √ ¯ q τ ℓ (cid:19) − ε + 6160 (cid:21) I (7) ij ( t − τ ) + O (cid:18) c (cid:19) , (4.8a) h i asym = − G M c x j Z + ∞ d τ (cid:20) ln (cid:18) c √ ¯ q τ ℓ (cid:19) − ε + 107120 (cid:21) I (6) ij ( t − τ ) + O (cid:18) c (cid:19) , (4.8b) h ij asym = 8 G Mc Z + ∞ d τ (cid:20) ln (cid:18) c √ ¯ q τ ℓ (cid:19) − ε + 45 (cid:21) I (5) ij ( t − τ ) + O (cid:18) c (cid:19) , (4.8c)where we have introduced the usual variable h ii = d − [( d − h + h ii ] (see Paper I). Inthis standard harmonic gauge the tail integrals and their associated (UV) poles are spreadout in all components of the metric. In Eqs. (4.8) we have inserted the correct numericalcoefficients computed in the companion paper [24], which are crucial in the end in order toobtain the “first” ambiguity.Alternatively, we can do the calculation starting from the linear quadrupole metric in atransverse-tracefree (TT) harmonic gauge. Thus, instead of Eqs (4.4), we may consider thelinear quadrupole TT metric h ′ I kl = 0 , (4.9a) h ′ iI kl = 0 , (4.9b) h ′ ijI kl = − I (2) ij + 4 ∂ k ( i ˜ I j ) k − d − δ ij ∂ kl ˜ I kl − d − d − ∂ ijkl ˜ I ( − kl . (4.9c)In the TT gauge the quadratic source term is especially simple, N ′ M × I kl = − ∂ ij h M h ′ ijI kl , (4.10a) N ′ iM × I kl = ∂ j h M ∂ h ′ ijI kl , (4.10b) N ′ ijM × I kl = − h M ∂ h ′ ijI kl , (4.10c)and, relaunching our calculation (with inputs from [24]), we readily obtain h ′ ii asym = − G Mc x ij I (6) ij ( t ) + O (cid:18) c (cid:19) , (4.11a) We suppress the mention “ M × I kl ”, and restore the factors of c and G . Here G denotes the usualNewtonian constant, such that G ( d ) = G ℓ d − in d dimensions. We recall also that ¯ q = 4 π e γ E . ′ i asym = 45 G Mc x j I (5) ij ( t ) + O (cid:18) c (cid:19) , (4.11b) h ′ ij asym = 165 G Mc Z + ∞ d τ (cid:20) ln (cid:18) c √ ¯ q τ ℓ (cid:19) − ε + 940 (cid:21) I (5) ij ( t − τ ) + O (cid:18) c (cid:19) . (4.11c)In the TT gauge the tail integral and the associated pole appear only in the spatial compo-nents of the metric (notice also that h ′ ii asym = 0 in this case).Finally the tails in the harmonic metric (4.8) or its TT counterpart (4.11) will yield amodification of the equations of motion. To compute it in the simplest way we perform agauge transformation (this time, at quadratic order), so designed as to transfer all relevantterms in the “00 ii ” component of the metric. In the new gauge the 4PN tail effect is thusentirely described by the single scalar potential h ′′ ii asym , or equivalently by the 00 componentof the usual covariant metric, given by g ′′ asym00 = − h ′′ ii asym . We finally obtain g ′′ asym00 = − G M c x ij Z + ∞ d τ (cid:20) ln (cid:18) c √ ¯ q τ ℓ (cid:19) − ε + 4160 (cid:21) I (7) ij ( t − τ ) + O (cid:18) c (cid:19) . (4.12)This result properly recovers the known tail integral in 3 dimensions [see Eqs. (5.24) in [34]].In addition, there appear the pole and a certain numerical coefficient, say κ = . The(UV-type) pole is in agreement with the result of Ref. [10]. On the other hand the constant κ = has the form of the ambiguity α introduced in Paper I, which is itself equivalent tothe ambiguity C of the Hamiltonian formalism [6], and is now determined. Note that thevalue of this constant is the same in both our calculations, in harmonic and TT gauge.As shown here and in the companion paper [24], the value of the numerical coefficient, i.e. , κ = , comes out directly from the matching equation between the near zone and theradiation zone, and a consistent application of the εη regularization together with the closedform expressions of the coefficients C p,kℓ obtained in Appendix D. Of course the value of κ was already known from comparison with GSF calculations, but we now directly obtain thecorrect value (see Sec. V), which also agrees with the published coefficient obtained in thecomputation of the d -dimensional tail effect by Galley et al [10] using EFT methods [seetheir Eq. (3.3)].Once we have the single scalar effect (4.12) at the level of the metric, it is straightforwardto obtain the equivalent effect at the level of the Lagrangian or Fokker action. Recall thatthe corresponding piece in the Fokker action will describe only the conservative part ofthe dynamics associated with the tail effect (see Paper I for discussion). We thus find themanifestly time-symmetric contribution to the gravitational part of the action, S tail g = G M c Z + ∞−∞ d t I (3) ij ( t ) Z + ∞ d τ (cid:20) ln (cid:18) c √ ¯ q τ ℓ (cid:19) − ε + 4160 (cid:21) (cid:16) I (4) ij ( t − τ ) − I (4) ij ( t + τ ) (cid:17) , (4.13)which can elegantly be rewritten by means of the Hadamard partie finie (Pf) integral as S tail g = G M c Pf τ DR0
Z Z d t d t ′ | t − t ′ | I (3) ij ( t ) I (3) ij ( t ′ ) , (4.14)where τ DR0 = ℓ c √ ¯ q e ε − plays the role of the Hadamard cut-off scale. Finally, when con-sidering the difference between the DR and HR results, we have to correct for the differenttreatments of the tail term in the two procedures. In Sec. III of Paper I we obtained the tail16erm in the same form as Eq. (4.14) but with a different Hadamard scale τ HR0 = 2 s . Thedifference of Lagrangians to be added to the result of Paper I concerning the tail is thus D L tail g = − G M c ln (cid:18) τ DR0 τ HR0 (cid:19) (cid:16) I (3) ij (cid:17) = G m c (cid:20) − ε + 4130 + 2 ln (cid:18) √ ¯ q s ℓ (cid:19)(cid:21) (cid:16) I (3) ij (cid:17) , (4.15)where we approximated M = m + O (1 /c ) in the second equality. Thus, the pole in (4.15)will indeed cancel out the pole in the instantaneous part of the Fokker action [see Eq. (2.12)]. V. DETERMINATION OF THE AMBIGUITY PARAMETERS
We gather and recapitulate our results from the previous sections. Recall that in Paper I,the 4PN Fokker Lagrangian constructed in harmonic coordinates initially depended on thearbitrary constant parameter α = ln (cid:18) r s (cid:19) , (5.1)which was then adjusted to the value α = by comparison to the circular orbit limit ofthe binary’s conserved energy in the small mass ratio limit. We therefore have to:1. Restore the arbitrariness of the parameter α by adding to the end result of Paper Ithe contribution D L αg = 2 G m c (cid:18) α − (cid:19) (cid:16) I (3) ij (cid:17) ; (5.2)2. Add the difference between the DR and HR evaluations of the IR divergences in theinstantaneous part of the gravitational action, as computed using the method n + 2,and whose result has been obtained in Eq. (2.12);3. Subtract off the particular surface term given by Eq. (2.14) and which was necessaryin the HR scheme for having a Lagrangian starting at the quadratic order with thepropagator form ∝ h (cid:3) h ;4. Finally, add the difference between the radiation non-local tails in DR and HR asobtained in (4.15).Concerning the matter part L m of the Fokker Lagrangian, nothing is to be changed withrespect to the result of Paper I since there are no IR divergences therein and L m standscorrect in DR. Finally our full DR Lagrangian reads L = L Paper I + D L αg + D L inst g − D L surf g + D L tail g . (5.3)Inserting our explicit results we find that the poles properly cancel out as announced;furthermore the constants r , s and ℓ also correctly disappear, and so does the irrationalnumber ¯ q = 4 π e γ E . The modification of the Lagrangian then takes exactly the form postu-lated in Eq. (2.4) of Paper II, namely L = L Paper I + G m m m c r (cid:16) δ ( n v ) + δ v (cid:17) , (5.4)17ut where the two ambiguity parameters δ and δ are now unambiguously determined, as δ = − , δ = 19235 . (5.5)This is exactly the values we obtained in Paper II by demanding that the conserved energyand periastron advance for circular orbits recover the GSF calculations in the small mass-ratio limit. This result confirms the soundness of the postulated form of the ambiguities inPaper II and shows the power of dimensional regularization for handling both UV and IRdivergences in the problem of motion in classical GR.Remarkably, the value κ = we have obtained in our result for the tail [see Eq. (4.13)],agrees with the result found by Galley et al [10] in their computation of the tail term in d dimensions (including both conservative and dissipative effects) by means of EFT methods.This indicates that when the EFT calculation will be fully completed at the 4PN order [8–11],their result will be free of any ambiguity like ours. Acknowledgments
It is a pleasure to thank Tanguy Marchand for having checked the calculation of the taileffect in Sec. IV. L.Bl. and G.F. acknowledge a very useful and productive “Workshop onanalytical methods in General Relativity” organized by Rafael Porto and Riccardo Sturaniat ICTP/SAIFR in S˜ao Paulo, Brazil. L.Be. acknowledges financial support provided underthe European Union’s H2020 ERC Consolidator Grant “Matter and strong-field gravity:New frontiers in Einstein’s theory” grant agreement no. MaGRaTh646597.
Appendix A: Homogeneous solutions of the wave equation in d + 1 dimensions The general “monopolar” homogeneous retarded solution of the wave equation in d + 1dimensions (where d = 3 + ε ), such that (cid:3) ˜ f ret ( t, r ) = 0, reads, following the notation (4.5),˜ f ret ( r, t ) = − π Z + ∞−∞ d t ′ G ret ( x , t − t ′ ) f ( t ′ )= ˜ kr d − Z + ∞ d y γ − d ( y ) f (cid:16) t − ryc (cid:17) , (A1)or, in more details, recalling ˜ k = Γ( d − /π d − and the function γ s ( y ) displayed in Eq. (3.3),˜ f ret ( r, t ) = 2 π ε r − − ε Γ( − ε ) Z + ∞ d y ( y − − − ε f (cid:16) t − ryc (cid:17) . (A2)In this Appendix we shall mostly investigate the post-Newtonian expansion of that solu-tion. We notice that by posing τ = ry/c we are fixing the argument of the function f in (A2),and then the formal PN expansion c → + ∞ becomes equivalent to a formal expansion when y → + ∞ , which can simply be evaluated by inserting into (A2) the series( y − − − ε = + ∞ X k =0 ( − ) k k ! Γ( − ε )Γ( − k − ε ) y − − k − ε . (A3)18n this way we readily obtain˜ f ret = 2 π ε c ε + ∞ X k =0 ( − ) k k ! ( r/c ) k Γ( − k − ε ) Z + ∞ r/c d τ τ − − k − ε f ( t − τ ) . (A4)At this stage we split the integral according to R + ∞ r/c = − R r/c + R + ∞ . The two pieces willrespectively yield the decomposition of Eq. (A4) into “even” and “odd” pieces in the limit ε →
0, where we are following the standard PN terminology, i.e. , meaning the parity of thepower of 1 /c in front. Thus, ˜ f ret = ˜ f even + ˜ f oddret . (A5)In the even piece, corresponding to (minus) the integral from 0 to r/c , we are allowedto formally expand the integrand when τ →
0, since by definition r/c → + ∞ X k =0 ( − ) k k ! 1 (cid:0) k + − p + ε (cid:1) Γ( − k − ε ) = Γ( − p + ε )Γ( − p ) . (A6)Although it is valid for any p ∈ N , this formula gives zero whenever p is an odd integer.Thus only will contribute the even values p = 2 j , reflecting the even character, in the PNsense, of that term. Furthermore we get a “local” expansion in any dimensions, given by˜ f even = r − − ε π ε + ∞ X j =0 ( − ) j j j ! Γ (cid:0) ε − j (cid:1) (cid:16) rc (cid:17) j f (2 j ) ( t ) . (A7)As for the odd piece, corresponding to the integral from 0 to + ∞ , it will irreducibly be givenby a non-local integral (“violation of Huygens’ principle”), except when ε = 0. We performa series of integrations by parts to arrive at an expression which is manifestly finite in thelimit ε → f oddret = − π ε c ε Γ( ε )Γ(1 − ε ) + ∞ X j =0 j j ! ( r/c ) j Γ( j + ε ) Z + ∞ d τ τ − ε f (2 j +2) ( t − τ ) . (A8)Notice that this expression, unlike (A7), is regular when r → i.e. , ˜ f oddret ∈ C ∞ ( R ). Westraightforwardly check that Eqs. (A7) and (A8) recover in the limit ε → f oddret becomes localin this limit): ˜ f ret ( r, t ) (cid:12)(cid:12)(cid:12) ε =0 = f ( t − r/c ) r , (A9a)˜ f even ( r, t ) (cid:12)(cid:12)(cid:12) ε =0 = + ∞ X j =0 r j − (2 j )! c j f (2 j ) ( t ) (A9b)˜ f oddret ( r, t ) (cid:12)(cid:12)(cid:12) ε =0 = − + ∞ X j =0 r j (2 j + 1)! c j +1 f (2 j +1) ( t ) . (A9c)19he same analysis but done for the advanced monopolar homogeneous solution, i.e. , usingthe advanced Green’s function [given by Eq. (3.2) with θ ( − t − r ) in place of θ ( t − r )], gives˜ f adv = ˜ f even + ˜ f oddadv , (A10)where the even part is the same as before, and with the advanced odd part˜ f oddadv = − π ε c ε Γ( ε )Γ(1 − ε ) + ∞ X j =0 j j ! ( r/c ) j Γ( j + ε ) Z + ∞ d τ τ − ε f (2 j +2) ( t + τ ) . (A11)In the limit ε → f oddadv (cid:12)(cid:12) ε =0 = − ˜ f oddret (cid:12)(cid:12) ε =0 . Further, we define the associatedsymmetric and anti-symmetric solutions,˜ f sym = 12 (cid:0) ˜ f ret + ˜ f adv (cid:1) = ˜ f even + 12 (cid:0) ˜ f oddret + ˜ f oddadv (cid:1) , (A12a)˜ f asym = 12 (cid:0) ˜ f ret − ˜ f adv (cid:1) = 12 (cid:0) ˜ f oddret − ˜ f oddadv (cid:1) . (A12b)In particular, the anti-symmetric solution is non-local (except when ε = 0), regular when r →
0, and becomes purely odd in the PN sense when ε = 0,˜ f asym = − π ε c ε Γ( ε )Γ(1 − ε ) + ∞ X j =0 j j ! ( r/c ) j Γ( j + ε ) Z + ∞ d τ τ − ε h f (2 j +2) ( t − τ ) − f (2 j +2) ( t + τ ) i . (A13)The most general “multipolar” homogeneous retarded solution will be obtained by re-peatedly applying spatial differentiations on the latter monopolar solution, hence˜ H ret ( x , t ) = + ∞ X ℓ =0 ˆ ∂ L ˜ f L ret ( r, t ) , (A14)where ˆ ∂ L denotes the STF product of ℓ spatial derivatives (and L = i · · · i ℓ ). Similarly onecan define the advanced, symmetric and anti-symmetric multipolar solutions. For instance,the anti-symmetric solution can be re-written in the manifestly regular form˜ H asym = − π ε Γ( ε )Γ(1 − ε ) + ∞ X ℓ =0 ℓ Γ( ℓ + ε ) + ∞ X j =0 ∆ − j ˆ x L c j +2 ℓ +1+ ε × Z + ∞ d τ τ − ε h f (2 j +2 ℓ +2) L ( t − τ ) − f (2 j +2 ℓ +2) L ( t + τ ) i , (A15)where we recall the short-hand notation∆ − j ˆ x L = Γ( ℓ + ε )Γ( ℓ + j + ε ) r j ˆ x L j j ! . (A16)The homogeneous solution investigated in Sec. III, and that we computed directly from anear-zone expansion, is precisely of the previous anti-symmetric type (A15). We showed thisby going to the Fourier domain [see Eqs. (3.17) and (3.18)].20 ppendix B: Multipole expansion of elementary functions in d dimensions For our computation of the difference between the DR and HR prescriptions for the IRregularization of integrals at infinity in Sec. II, we need to control the expansion at infinity( r → + ∞ ) of non-linear potentials in d dimensions. These potentials are defined by meansof elementary solutions of the Poisson or d’Alembert equation in d dimensions, the simplestone being the famous Fock kernel obeying in d dimensions∆ g = r − d r − d . (B1)The exact expression in 3 dimensions is g ( ε =0) = ln( r + r + r ) [13]. The explicit formof the solution in d dimensions has been obtained in the Appendix C of Ref. [28]. In theAppendix B of Paper I we have given the local expansion of that function in d dimensionsnear the singularities (when r or r → r → + ∞ , that we shall refer to as a multipole expansion denoted M ( g ).Suppose we want to compute the multipole expansion M ( P ) of some elementary potential P , solution of the wave equation (cid:3) P = σ , where σ is some source term with non compactsupport like in (B1). In the usual post-Newtonian (or near zone) iteration scheme, neglectingtime-odd contributions, the potential is given by P = I − σ where the usual symmetricpropagator reads I − = + ∞ X p =0 (cid:18) c ∂∂t (cid:19) p ∆ − p − . (B2)Now the far-zone expansion M ( P ) will be obtained from the far-zone expansion M ( σ ) ofthe corresponding source term by application of (B2), but for a non-compact support sourceit is known that there is also a homogeneous solution of the symmetric type to be added,and which is specified by Eq. (3.23) of Ref. [37]. Generalizing the formula to d dimensions,this means that the solution is the sum of a particular solution obtained by application ofEq. (B2), plus a specific homogeneous symmetric one, M ( P ) = I − [ M ( σ )] − π + ∞ X ℓ =0 ( − ) ℓ ℓ ! ∂ L ˜ σ L sym , (B3)where the overbar on the homogeneous solution means the PN or near-zone expansion, and,following the Appendix A, the homogeneous symmetric solution reads˜ σ L sym ( r, t ) = ˜ kr d − Z + ∞ d y γ − d ( y ) h σ L (cid:0) t − ry/c (cid:1) + σ L (cid:0) t + ry/c (cid:1)i . (B4)Here σ L denotes the ℓ -th multipole moment of the source σ given (in non-STF guise) by σ L ( t ) = Z d d x ′ x ′ L σ ( x ′ , t ) . (B5)Note that we are performing a full DR calculation, so the multipole moment σ L is definedwithout invoking a finite part regularization (based on some regulator ( r/r ) B with B ∈ C );instead, DR is taking care of the IR divergences, appearing here due to the fact that the21ource σ has a non-compact spatial support. Similarly, the particular solution or first termin Eq. (B3), is defined in a pure DR way, with the iterated Poisson operator ∆ − p − in (B2)acting on each term of the multipole expansion of the source M ( σ ), whose general structurein d dimensions is provided by Eq. (2.7). It is clear that the Poisson operator and its iteratedversion make sense when applied to such terms [see, e.g. , (A16)].Finally, because of the overbar prescription in Eq. (B3), we need the post-Newtonian ornear-zone expansion of the object ˜ σ L sym . The PN expansion of the homogeneous symmetricsolution has been investigated in the previous App. A. It consists essentially of even con-tributions but also, in d dimensions, or some residual non-local odd terms [see Eqs. (A12)].The odd terms will disappear in 3 dimensions; we neglect these since they are dissipativecontributions. Thus we simply assimilate the symmetric part with the even part, and weget, from Eq. (A7), ˜ σ L sym = r − − ε π ε + ∞ X j =0 ( − ) j j j ! Γ (cid:0) ε − j (cid:1) (cid:16) rc (cid:17) j σ (2 j ) L ( t ) . (B6)We have applied the previous formulas to the source term σ = r − d r − d in Eq. (B1).Defining g and f such that, up to the 1PN order, P = g + 12 c ∂ t f + O (cid:18) c (cid:19) , (B7)we have ∆ g = σ and ∆ f = 2 g in this convention. We obtain M ( g ) = r − ε − ε + ∞ X ℓ =0 ℓ − ( ℓ + 1)! Γ( ℓ + ε +12 )Γ( ε +12 ) ˆ n L r ℓ +1+ ε ℓ X s =0 y h L − S y S i + 1 (cid:2) Γ( ε ) (cid:3) ∞ X m =0 m − r m +2 ε [ m ] X s =0 Γ( ε + m − s )Γ( ε + m − s ) ˆ n M − S ( m − s + ε )( s + ε − )(2 s )!! × m X ℓ =0 Γ( ℓ + ε +12 )( ℓ − s )! Γ( m − ℓ + ε +12 )( m − ℓ − s )! ˆ y L − S,S ′ ˆ y M − LS,S ′ , (B8a) M ( f ) = r − ε (1 − ε ) ∞ X ℓ =0 ℓ − ( ℓ + 1)! Γ( ℓ + ε − )Γ( ε − ) (cid:20) ℓ X s =0 y h L − S y S i (cid:18) r − (2 ℓ + ε − ℓ + ε + 3)( ℓ + 2) × (cid:16) y ( ℓ − s + 1) + y ( s + 1) − r − ε ( ℓ − s + 1)( s + 1) (cid:17)(cid:19)(cid:21) ˆ n L r ℓ +1+ ε + 1 (cid:2) Γ( ε ) (cid:3) ∞ X m =0 m − r m − ε [ m ] X s =0 Γ( ε + m − s )Γ( m − s − ε )Γ( s + ε − )Γ( ε + m − s )Γ( m − s + 1 + ε )Γ( s + ε +12 ) ˆ n M − S (2 s )!! With our notation for multi-indices meaning, for instance,ˆ n M − S = STF[ n i · · · n i m − s ] , ˆ n M − S ˆ y L − S,S ′ ˆ y M − LS,S ′ = ˆ n i ··· i m − s ˆ y i ··· i ℓ − s j ··· j s ˆ y i ℓ − s +1 ··· i m − s j ··· j s . m X ℓ =0 Γ( ℓ + ε +12 )( ℓ − s )! Γ( m − ℓ + ε +12 )( m − ℓ − s )! ˆ y L − S,S ′ ˆ y M − LS,S ′ . (B8b)Similarly, in our calculations we have also to consider the potentials f and f obeying∆ f = r − d r − d , ∆ f = r − d r − d , (B9)and we obtain, for instance, M ( f ) = r − ε − ε + ∞ X ℓ =0 ℓ − ( ℓ + 2)! Γ( ℓ + ε +12 )Γ( ε +12 ) ˆ n L r ℓ +1+ ε ℓ X s =0 ( s + 1) y h L − S y S i − − ε ) (cid:2) Γ( ε − ) (cid:3) ∞ X m =0 m − r m +2 ε [ m ] X s =0 Γ( ε + m − s )Γ( ε + m − s ) ˆ n M − S (2 s )!! × m X ℓ =0 Γ( ℓ + ε − )( ℓ − s )! Γ( m − ℓ + ε +12 )( m − ℓ − s )! ˆ y L − S,S ′ ˆ y M − LS,S ′ × (cid:20) r ( m − s − ε )( s + ε − ) − (2 ℓ + ε − y (2 ℓ + ε + 3)( m − s + ε )( s + ε − ) (cid:21) . (B10)The above formulas have been extensively used to control the IR divergences in the gravita-tional part of the Fokker action in Sec. (II). However we have found that in fact, the resultof our computation of the difference DR − HR does not depend on the detailed prescriptionwe followed to control the homogeneous anti-symmetric solution in Eq. (B3). The indepen-dence with respect to the added homogeneous solution in Eq. (B3) is certainly a good signof the solidness of our result.
Appendix C: Distributional limits of the function γ s ( z ) The function γ s ( z ) defined by Eq. (3.3) is zero in an ordinary sense for strictly negativeinteger values s = − − ℓ (where ℓ ∈ N ). In this Appendix we compute γ − − ℓ ( z ) in the senseof distributions. From Eq. (3.3) we have γ − − ℓ − ε ( z ) = 2 √ π Γ( − ℓ − ε )Γ( ℓ + ε ) (cid:0) z − (cid:1) − − ℓ − ε θ ( z − . (C1)We added the Heaviside step function θ ( z −
1) to recall that this expression is defined onlyfor z >
1. Considered as a distribution (indexed by a parameter ε ∈ C ), Eq. (C1) is to beapplied on test functions ϕ ( z ) that are at once smooth, i.e. , ϕ ∈ C ∞ ( R ), and with compactsupport. Hence, h γ − − ℓ − ε , ϕ i = 2 √ π Γ( − ℓ − ε )Γ( ℓ + ε ) Z + ∞ d z (cid:0) z − (cid:1) − − ℓ − ε ϕ ( z ) . (C2)Under this form we see that the limit ε → z = 1, but can madefinite by performing some integrations by parts. The surface terms will always be zero by23nalytic continuation in ε at the bound z = 1, and because the test function has a compactsupport. After ℓ + 1 integrations by parts we obtain h γ − − ℓ − ε , ϕ i = ( − ) ℓ +1 √ π Γ(1 − ε )Γ( ℓ + ε ) Z + ∞ d z (cid:0) z − (cid:1) − ε (cid:18) dd z (cid:19) ℓ +1 h(cid:0) z + 1 (cid:1) − − ℓ − ε ϕ ( z ) i , (C3)and, under that form, we can directly take the limit ε → h γ − − ℓ , ϕ i = ( − ) ℓ ℓ +1 (2 ℓ − (cid:18) dd z (cid:19) ℓ (cid:20) ϕ ( z ) (cid:0) z + 1 (cid:1) ℓ +1 (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) z =1 . (C4)More explicitly this gives h γ − − ℓ , ϕ i = ℓ X i =0 ( − ) i α ℓi ϕ ( i ) (1) , (C5a)where α ℓi = 2 i − ℓ (2 ℓ − ℓ − i )! i !( ℓ − i )! . (C5b)So, finally the result for γ − − ℓ when viewed as a distribution reads γ − − ℓ ( z ) = ℓ X i =0 α ℓi δ ( i ) ( z − , (C6)with δ ( i ) being the i -th derivative of the Dirac function. In particular γ − ( z ) = δ ( z − G ( ε =0)ret ( x , t ) = − δ ( t − r )4 π r . (C7) Appendix D: Computation of the coefficients C p,kℓ These coefficients, defined in d = 3 + ε dimensions by Eq. (3.23), are written in the form C p,kℓ = 4 π Γ( − ε )Γ( − ℓ − ε )Γ( ε )Γ( ℓ + ε ) L pa,b,c , (D1)together with the following definition of the double integral, L pa,b,c = Z + ∞ d y y p ( y − a Z + ∞ d z ( z − b ( y + z ) c , (D2)and the particular set of coefficients a = − − ε , b = − ℓ − − ε , and c = ℓ + k − ε − η ,where the parameter η was introduced in Eq. (3.7).The integral (D2) is computed by first relating it to the simpler integral correspondingto p = 0, namely K a,b,c = L a,b,c or K a,b,c = Z + ∞ d y ( y − a Z + ∞ d z ( z − b ( y + z ) c . (D3)24he latter integral in turn converges for ℜ ( a ) > − ℜ ( b ) > − ℜ (2 a + c ) < − ℜ (2 b + c ) < − ℜ (2 a + 2 b + c ) < −
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