The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: potential contributions

We calculate the potential contributions of the motion of binary mass systems in gravity to the fifth post--Newtonian order ab initio using coupling and velocity expansions within an effective field theory approach based on Feynman amplitudes starting with harmonic coordinates and using dimensional regularization. Furthermore, the singular and logarithmic tail contributions are calculated. We also consider the non--local tail contributions. Further steps towards the complete calculation are discussed and first comparisons are given to results in the literature.


Introduction
The measurement of gravitational wave signals from merging black holes and neutron stars [1] has been a recent milestone in astrophysics. The different gravitational wave detectors are reaching higher and higher sensitivity [2], which requests to provide more detailed predictions at the theoretical side. Currently in binary Hamiltonian dynamics the level of the 4th post-Newtonian (PN) order has been fully understood and agreeing results have been obtained using a variety of different computation techniques in quite a series of gauges which lead to identical predictions in all key observables [3][4][5][6][7][8][9]. Moreover, it has been shown by applying canonical transformations [9], that all descriptions are dynamically equivalent. The different approaches can only be compared either by using canonical transformations, which requires local representations, or by calculating observables.
At the level of the 5th post-Newtonian order, first two agreeing results on the static potential in the harmonic gauge were calculated [10,11]. Later partial results were derived using different matching techniques for the Hamiltonian in the effective one body (EOB) approach in [12,13]. 1 Here two parameters,d 5 and a 6 , which are of O(ν 2 ), with ν = m 1 m 2 /(m 1 + m 2 ) 2 , remained yet undetermined.
The conserved Hamiltonian of the motion of binary mass systems in gravity has the following expansion where k labels the post-Newtonian order, 2 with H 0PN ≡ H N . From k = 4 onward H kPN consists out of the term due to potential interactions, H pot , and the tail terms, H tail , In effective field theory approaches 3 based on Feynman diagrams this is the most natural decomposition. In [12,13] another decomposition has been chosen into the so-called non-local terms H nl kPN and the local terms H loc kPN , H kPN = H loc kPN + H nl kPN .
The non-local terms are fully contained in the tail terms and the local contributions are given by the local parts of the tail terms and the potential contributions.
In the present paper we calculate the 5PN potential corrections and some first parts of the 5PN tail terms using an effective field theory (EFT) approach; for related reviews see [24][25][26][27][28].
Here we follow Ref. [29]. 4 A series of technical details for the calculation of the potential terms have already been given in Refs. [9,11] before. In the case of the tail terms one first applies the multi-pole expansion valid for the far zone [3,13,15,25,27,[31][32][33][34][35][36][37][38][39] to the respective post-Newtonian order and then applies EFT methods to calculate their contribution, cf. [40]. Expansions of this type generally belong to the operator product expansions [41]. In the calculation one also applies the method of expansion by regions [42,43].
In the present paper observables at 5PN such as the energy and periastron advance at circular orbits could not yet be calculated in complete form, since a series of differences with the literature have still to be fully clarified. This concerns rational terms contributing to the tail 1 First results at 6PN order have been given in [14][15][16] recently. There is also a lot of activity in calculating post-Minkowskian corrections, cf. [14] Ref. [12], and [17][18][19][20][21][22][23]. 2 Here we do not deal with conserved half PN contributions occurring from 5.5 PN onward. 3 For the 4PN calculations see [3,4,11]. 4 Following the ideas in [30].
term. However, we obtain all other contributions, including the π 2 contributions to the yet undetermined constantsd 5 and a 6 , in [13]. Furthermore, quite a series of comparisons could be performed with the literature. The paper is organized as follows. In Section 2 we describe the calculation of the 5PN potential terms and present the associated Hamiltonian H pot 5PN in the harmonic gauge. We use dimensional regularization in D = 4 − 2ε dimensions. It is this method which allows a particular elegant merging of the potential and tail contributions in the conservative Hamiltonian, as we will show below. Already at 3PN the contributions to H pot have poles in 1/ε, cf. [44]. From 4PN corresponding poles also appear in the tail terms. We will discuss the main aspects of the 5PN tail term in Section 3 and construct a pole-free Hamiltonian in Section 4. Here we show that the poles in the combined Hamiltonian can be transformed away by a canonical transformation. In Section 5 we compare to results given in the literature and discuss open questions. A canonical transformation from harmonic to EOB coordinates is performed. We derive the non-local tail contributions within our approach and calculate their contribution to the binding energy and to periastron advance in the circular case. We then turn to the contributions to periastron advance from the potential terms and derive the π 2 contributions to the previously unknown constantsd 5 and a 6 and summarize our present results for the circular binding energy and periastron advance. Furthermore we briefly discuss the remaining contributions to the tail term. Section 6 contains the conclusions. In the appendices some technical aspects are presented on the merging of the potential and tail terms using the method of expansion by regions. As well we present longer formulae, which are used in the present calculation.

The potential contributions to the Hamiltonian
The calculation of the 5PN corrections is performed in the same way as has been described in Refs. [9,11]. Starting from the Einstein-Hilbert Lagrangian, we parameterize the metric g µν according to Ref. [29] in terms of scalar, vector and tensor fields, and work in the harmonic gauge. 5 The Feynman diagrams are generated using QGRAF [45]. The Lorentz algebra is carried out using Form [46] and we perform the integration by parts (IBP) reduction to master integrals using the code Crusher [47]. Table 1 gives an overview on the present calculation.   #loops QGRAF source irred. no source loops no tadpoles masters   0  3  3  3  3  0  1  72  72  72  72  1  2  3286  3286  3286  2702  1  3  81526  62246  60998  41676  1  4  545812  264354  234934  116498  7  5  332020  128080  101570  27582  4   Table 1: Numbers of contributing diagrams at the different loop levels and master integrals.
From the graphs generated by QGRAF one has to remove the source reducible graphs, graphs with source loops and tadpoles. In this way the 962719 initial diagrams reduce to 188533 diagrams.
The computation time amounts to about one week, including the time for the IBP reduction, on an Intel(R) Xeon(R) CPU E5-2643 v4 and it grows exponentially with the loop order. Most of the CPU time is needed to perform the time derivatives. Only one non-trivial master integral contributes, see [8,11]. One first obtains a Lagrange function of mth order still containing the accelerations a i and time derivatives thereof. They are removed by using first double zero insertions [48,49] together with partial integration and the remaining linear accelerations by a shift [25,[49][50][51], cf. [9]. By this operation we leave harmonic coordinates. A Legendre transformation leads then to the potential contributions of the Hamiltonian, which still contains pole terms in the dimensional parameter ε. 6 The reduced Hamiltonian in the cms is given bŷ with c the velocity of light, M = m 1 + m 2 the rest mass of the binary system and µ = m 1 m 2 /M, with where γ E is the Euler-Mascheroni constant and µ 1 the mass scale accounting for Newton's constant in D dimensions. The corresponding contributions up to 4PN have been presented in [9] before. We rescale where p and r are now the rescaled (dimensionless) cms momentum and the distance of the two masses, with n = r/r. In the following we will as widely as possible work with dimensionless quantities.
Pole and logarithmic contributions appear at O(ν), O(ν 2 ) and O(ν 3 ), in accordance with the lower PN orders, where also always one more order in ν contributes from 3PN onward. We will see in Section 4 that the tail term is only singular for O(ν) and O(ν 2 ) at 5PN.

Remarks on the tail term
We will derive a pole-free Hamiltonian at 5PN in Section 4. For this we will add the singular and logarithmic terms of the tail term,Ĥ tail,sing,log

5PN
, to the potential termĤ pot 5PN . Since these contributions are calculated, by different methods, either in the far zone (FZ) or the near zone (NZ), the question arises whether potential overlap contributions have to be considered. We remind that the calculation is performed in D dimensions, not using any other regularization.
One may apply the method of expansion by regions, which has been introduced for the asymptotic expansion of Feynman integrals for bound states in the non-relativistic limit in [42,43].
Here each loop integral is split into four distinct momentum regions, which are denoted as hard, soft, potential, and ultrasoft. Integrals over the hard and soft region correspond to quantum corrections and are not considered in the context of classical gravity.
The potential region, characterized by the momentum scaling with v ∈ [v 1 , v 2 ] and v = v phys /c the typical velocities, is also referred to as orbital region.
Here k i and R are not rescaled. However, we set the associated action variable to 1. 7 It can be identified with the near zone of the literature (i.e. the potential terms). In the ultrasoft (or radiation region), corresponding to the far zone (i.e. the tail terms), momenta exhibit the uniform four-momentum scaling The kinematic region of the potential term is with R of the order of separation the binary system. Likewise, the one of the tail term is In the former region the exchanged fields are potential gravitons and in the latter region ultrasoft gravitons. One performs a Taylor expansion of the integrands according to the respective momentum scaling in v up to the respective post-Newtonian order by observing that For the tail terms this expansion includes the multi-pole expansion, which we will discuss below. 8 Let us introduce the operators T pot and T us , which describe the Taylor expansions (with a few Laurent-terms in some cases) in the potential region and the ultrasoft region. In the post-Newtonian expansion they are given, more precisely, by with the quantifier θ(N) truncating the series at a maximal term v N , N ∈ N, which is idempotent θ l (N) ≡ θ(N). Here the coefficients T i,k denote the expansion coefficients of the function I 0 (v). The corresponding integrals have the following form for each of the D components k i , where I denotes the original integrand. One further obtains In the respective domains D pot and D us one may further apply the operators T pot and T us given the post-Newtonian accuracy one is working in. One then obtains Here the 2nd and 4th term are the overlap integrals. Eq. (17) can be further arranged to provided that holds, which we prove in Appendix A. Furthermore, the operation T us T pot leads to scaleless integrands, implying that the last term in Eq. (18) vanishes in D dimensions, see Appendix A.
We finally would would like to make some remarks on the relation on the multi-pole expansion [30] in the far zone to the ultrasoft region. One is starting from the full theory of general relativity in harmonic coordinates, i.e. the bulk action Here (20) is coupled to compact objects via the action with proper times τ 1 , τ 2 , one decomposes the metric into where η µν is the Minkowski metric. The momenta associated with H µν are of the potential type and the momenta of the h µν fields are ultrasoft, cf. Eqs. (11) and (12). The resulting loop integrals therefore have the same form as Eq. (15), as obtained from the asymptotic expansion. The action of the full theory is matched to the non-relativistic general relativity (NRGR) action by with where there are no potential modes anymore. Here T denotes the kinetic term and V NZ the nearzone potential. S NRGR,bulk is the same as the general relativity bulk action S GR,bulk from Eq. (20), but without the potential contributions to the metric. Both V NZ and the effective stress-energy tensor T µν are fixed by requiring that the NRGR action produces the same predictions as the asymptotic expansion of the full theory. In other words, the integrals over the potential region are absorbed into V NZ and T µν . We now elaborate on the relation to the multi-pole expansion. Consider where the momentum is ultrasoft by definition, i.e.
We can Taylor expand the exponential in Eq. (26) to obtain Rewriting now yields Inserting this expression back into the linear ultrasoft action Eq. (25) we retrieve the familiar starting point of the multi-pole expansion with an explicit velocity power counting. The remaining steps in the multi-pole expansion are standard. In short, one defines moments and decomposes them into irreducible SO(3) spherical tensors, choosing a symmetric trace-free (STF) basis, e.g. [3,36]. The tail terms are represented by the multi-pole expansion valid in the far zone. In our treatment we will follow Refs. [38,40].

The pole-free Hamiltonian at 5PN
It is convenient to work with pole-free Hamiltonians and we add the singular and logarithmic pieces of the Hamiltonian of the tail term in 5PN,Ĥ tail,sing,log 5PN , H tail,sing,log toĤ pot 5PN . For all contributions resulting into Eq. (33) we agree with the integrals in the multi-pole expansion in [40].
The sum of the potential term and this contribution is not pole-free yet, as is the case from 3PN onward, cf. [9]. However, after performing the following canonical transformation a polefree Hamiltonian is obtained, which is not the case for H pot 5PN and H tail,sing

5PN
individually. By this transformation one further moves away from the harmonic coordinates, which were used at the starting point of the calculation. Still a prediction of all observables is possible. Moreover, the comparison with EOB results becomes simpler, since they are given in pole-free form [13].
Following the formalism described in Ref. [9], Eqs. (38)(39)(40)(41), one obtains the corresponding generating function G(p 2 , p.n, r; ε) = p.n 1 ε −t 3 17ν 6r 2 + t 4 1 r 2 Furthermore, we transform the logarithmic part to explicitly match the structure of the non-local contribution from the tail term in harmonic coordinates, see also Section 5.2. Here the multi-pole moments I ab , O abc and J ab are those of Eq. (2.4) in [13], with indices contracted, and E is the total energy. The corresponding transformation reads G(p 2 , p.n, r; ln(r/r 0 )) = p.n ln ν 7557 − 65496ν Here t i , i = 1...5 labels the ith post-Newtonian order. The pole-free Hamiltonian based on the above contributions is then given by 9 By this we have shown in explicit form the cancellation of the singularities originally occurring in harmonic coordinates, for reasons of regularization only. In the case of the binary point-mass problem up to 5PN order no singularities survive requiring another method to be removed. At 4PN this has also been shown in Ref. [9], see also [56]. Furthermore, logarithmic terms do now only occur at O(ν) and O(ν 2 ).

Comparison to the literature
In the following we perform a series of comparisons with results in the literature.

Canonical transformation to EOB
Let us first compare to the EOB results of Ref. [13], Eq. (11.8), for the contributions at O(ν 0 ) and O(ν 3 ) and higher given in EOB coordinates in complete form. 10 These terms do not receive contributions due to tail terms and one can therefore just refer to the pole-free Hamiltonian of Section 4 to construct the canonical transformation. It is given by . (38) In this way we confirm all the contributions of O(ν 0 ) and O(ν 3 ) or higher given in [13] by an explicit Feynman diagram calculation ab initio.

The non-local terms
Next we turn to the non-local terms defined in [13], cf. Eq. (3). We perform the eccentricity expansion of the non-local contributions for δH nl 4+5PN with starting from harmonic coordinates. Here a r is the semimajor axis of the orbit, which we rescaled by a r = a r,phys c 2 /(G N M). It appears in the parameterization of the radial coordinate distance r in the form r = a r [1 − e r cos(u)], where e r denotes the "radial eccentricity" of the orbit and u the "eccentric anomaly". The Kepler equation reads n · t = 1 − e t sin(u), with n = 2π/P . Here P is the orbital period and t the coordinate time defines the eccentricity e t and one uses standard relations otherwise, cf. [57].
In the limit of vanishing eccentricity e t we obtain the following contribution for δH nl 5PN , Eq. (2.12), [ which agrees with [13]. The terms up to O(e 20 t ) are given in Appendix B. They agree with the expansion coefficients of Table I with with j = J phys c/(G N M). Here we have also introduced the dimensionless quantity η 2 , accounting for 1/c 2 . The contributions up to O(e 2 t ) are needed below to derive periastron advance for circular motion. One may now further express the variables a r and e t in terms of the normalized Delaunay variables [58] i r , i φ and i rφ , cf. [16], Eq. (A11), with i rφ = i r + i φ , i φ = j. By this one obtainŝ The variablesĤ, i r and i φ are related by Euler's chain rule sinceĤ depends only on i r and i φ and therefore a function f exists with f (Ĥ, i r , i φ ) = 0. By applying the chain rule one obtains the periastron advance, K, defined in (46) 13 One obtains HereĤ denotes the complete Hamiltonian. One may express where K nl starts at 4PN and Ω R receives non-local (nl) contributions from 4PN on, cf. (47). The contributions to K loc are calculated in Section 5.3. For K nl 4+5PN the 4PN non-local contributions to Ω R are necessary beyond the 1PN (local) correction 12 Note a difference to Eq. (8.27), [16] in the ln(2)ν 2 term at 5PN. 13 Note that also a related quantity, k = K − 1, is sometimes denoted by periastron advance. [16] with Ω R = (G N M/c 3 )Ω R,phys .
The Newtonian term of Ω R for circular orbits is 1/j 3 . Since we are only considering the 4 and 5PN contributions, the post-Newtonian expansion of 1/Ω R can be done separately for (50) and (51) which is calculated in a different way than the local contributions. The representations are, however, equivalent. Here one has cf. [16]. Eq. (54) turns into (43) for circular orbits (i r → 0). Eq. (52) agrees with Eq. (5.7) in [6], see also the expression of the related function ρ(x) in [5] and Eqs. (52,53) agree with Eq. (8.29) of [16].

Periastron advance: local terms
The local contribution to periastron advance is obtained by HereÊ results from (4) by H → E and R(r,Ê, j) It is convenient to refer to the local terms rather than to a separation of potential and tail terms. The former ones have no logarithmic terms and the corresponding integrals are therefore somewhat simpler. The logarithmic terms have already been dealt with in Section 5.2.
The integrand of (56) has the form The relationÊ is solved iteratively for R(r,Ê, j) = (p.n) 2 by applying through which the functions A, B, C and D k become polynomials inÊ and j. The integral (56) is usually solved by a mapping to a contour integral [59] applying the residue theorem, expanding in η 2 up to 5PN. Except of the integral for the Newtonian term, involving only A, B and C, all other integrals have only one residue at r = 0, see Appendix C.
We calculate the local contribution to periastron advance starting from harmonic coordinates and compare to Eq. (F5) of [16] resulting from the local EOB Hamiltonian Eq. (11.8) [13]. This is necessary to fix the notion of the parametersd 5 and a 6 in K(E, j) loc,f to 5PN. We rather use K(E, j) loc,f than K circ loc,5PN to test three relations between the parametersd 5 and a 6 , which is advantageous. To We also rederived the periastron advance starting with the ADM Hamiltonian [3] and confirm the result given in [6,16]. For the 5PN terms we obtain the partial result leaving out the few rational terms at O(ν) and O(ν 2 ) still to be calculated in complete form. The terms given in (62) agree with those of Ref. [16] considering our results in (63) and (64). Eq. (62) allows to derive the π 2 contributions tod 5 and a 6 to which we turn now.

Comparison to the other contributions to the tail terms
Let us mention that we have recalculated the contributions to the tail term given in [40] in D dimensions, but do not agree with all terms in the rational (local) contributions. A further detailed comparison has to be performed, which will be given elsewhere.
There are multi-pole moment contributions containing products of Levi-Civita tensors ε ijk , which have to be dealt with in D −1 dimensions in the present approach, see also [40,62]. Despite the fact that products of two (or more) Levi-Civita symbols, here in Euclidean space, are turned into determinants of D − 1 Kronecker symbols [63] it is known from almost all applications in Quantum Field Theory, that a so-called finite renormaliztion has to be performed to re-establish the Ward-identities. 14 The Larin method [64] is one consistent way (i.e. a non-degenerative way) to perform this analytic continuation to D dimensions. 15 Also this aspect still needs further study.
We finally mention that for the circular binding energy and periastron advance the yet differing results in the tail terms yield a numerical effect on the difference of O(1%) or less. This remaining difference still has to be settled analytically.

Conclusions
We have presented the 5PN potential contributions to the Hamiltonian of binary motion in gravity starting form the harmonic gauge and a part of the 5PN tail term. The calculation has thoroughly been performed in D dimensions, based on 188533 Feynman diagrams using effective field theory methods, as a calculation ab initio. The singular and logarithmic contributions to the 5PN tail terms have been calculated. We have shown the explicit cancellation of the singularities between both contributions, performing an additional canonical transformation to a pole-free Hamiltonian. We have shown in an explicit calculation how to match the potential and the tail terms, using dimensional regularization. Here the overlap-terms are canceling.
Comparisons to the literature have been performed. Firstly, we have shown that all terms of O(ν 0 ) and O(ν 3 ) and higher agree with the results presented in the literature. At O(ν 2 ) we determined the π 2 contributions tod 5 and a 6 . Furthermore, we also agree with the logarithmic tail and potential terms and the π 2 -terms at O(ν) and the effect of the non-local terms on E circ,5PN and K circ,5PN . We still observe a few differences in the purely rational (local) contributions to the 14 In the present application to gravity one should not call this 'finite renormalization', since no renormalization is performed. However, the contribution to the observables will be different comparing the 3-dimensional properly regulated result with that obtained in D − 1 dimensions. 15 Other prescriptions were given in [65].
tail term comparing to the present literature, which have to be clarified to obtain the complete 5PN result.
A Joining the potential and the tail term and the method of expansion by regions We now prove that the Taylor expansions in the potential and ultrasoft region commute and that the occurring overlap integrals are indeed scaleless and vanish in D dimensions. Here we use arguments given in [43]. The general form of the contributing integrands I is Apart from the loop momentum k, I depends on ultrasoft momenta p i , 1 ≤ i ≤ P us and potential momenta q i , 1 ≤ i ≤ P pot . The loop is associated with one of the two worldlines, whose fourposition at the time t is given by x(t). J is a function with the same structure as I itself, but independent of k. Finally, P denotes a polynomial in the momenta and worldline velocities v 1 , v 2 .
Our aim is to show that according to the power counting in the respective region. The P i are polynomials in the components of the four-vectors appearing in Eq. (69) and the J i are independent of k. This structure implies We first note that and similar for T N us . This allows us to expand each factor in Eq. (69) separately. J is independent of k, and P is a polynomial, so trivially For the propagators without additional momenta, we obtain and similar With respect to the remaining propagators, we first observe the absence of poles in v, i.e.
and note the following algebraic properties of the Taylor expansion operators.
We now show that by induction over N + M. The case N + M = 0 is straightforward. For N + M ≥ 1 we observe and similar where we have used the induction hypothesis in the last step. Furthermore, we find where the numerators are simply polynomials in the components of k and q i .
Finally, the exponential (69) can be expanded by observing i.e. k x(t) ∼ 1 (89) in the potential region and in the ultrasoft region. This yields We have now shown that the Taylor

B The eccentricity expansion of the non-local terms
We have recalculated the eccentricity expansion of the non-local terms using standard representations given in [13,16,66] (13) +O(e 22 t ). (94) C The contour integral for the Delaunay variable i r The integral J 1 , describing effect of the Newton dynamics, is given by All the remaining integrals are directly obtained from the residue at x = 0. The integral in (56) reads to 5PN The coefficients A to D 9 are determined iteratively expanding (57) in powers of η 2 . They depend on the respective Hamiltonian for which one may choose a pole-and log-free form.