The 6th Post-Newtonian Potential Terms at $O(G_N^4)$

We calculate the potential contributions of the Hamiltonian in harmonic coordinates up 6PN for binary mass systems to $O(G_N^4)$ and perform comparisons to recent results in the literature \cite{Bern:2021dqo} and \cite{Bini:2020nsb}.

Gravitational wave signals from merging black holes and neutron stars [3,4] provide tests of precision predictions within Einstein gravity for the dynamics of binary systems of massive objects. Their dynamics is calculated using post-Newtonian [2,[5][6][7][8][9][10][11] and the post-Minkowskian [1,12] 1 methods 2 . To match present and future measurements, high order computations are necessary. In the post-Newtonian (PN) case the level of 6PN [2,10,11] has now been reached for the conservative part of the two-body motion, while in the post-Minkowskian approach the calculation of the O(G 4 N ) potential terms is the most far reaching result [1]. Here G N denotes Newton's constant. The different calculations are often performed using different gauges (harmonic, ADM, isotropic and EOB) in deriving the Lagrangian or Hamiltonian. It is important to cross check the results between the two methods, and to compare different representations in the most general way possible. One either can perform special comparisons in calculating the same observable or one performs a canonical transformation [14] between the different Hamiltonians obtained. The latter result implies that the results for all observables are the same because of the invariance of the action.
In this paper we report about the calculation of the O(G 4 N ) potential terms to 6PN, extending previous work covering the terms up to O(G 3 N ). We use the effective field theory approach [15], for details see [10]. We work in the harmonic gauge and D = 4 − 2ε dimensions. The Feynman diagrams are generated using QGRAF [16]. The Lorentz algebra is carried out using Form [17] and we perform the integration by parts (IBP) reduction to master integrals using the code Crusher [18]. Table 1 gives an overview on the present calculation.
Redundant diagrams are eliminated in a series of steps outlined in Ref. [7]. 6065 diagrams do finally contribute to the present result. The computation time amounted to a few days on an Intel(R) Xeon(R) CPU E5-2643 v4. The Lagrange function of mth order, still containing the accelerations a i and time derivatives thereof, are converted into a first order Lagrange density by applying double zero insertions [19,20] together with partial integration and the remaining linear accelerations by a shift [20,21], cf. [6]. 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.
We define H = (H real − Mc 2 )/(µc 2 ) with M = m 1 + m 2 , µ = m 1 m 2 /M, ν = µ/M, with m 1,2 the two masses of the binary system, c the velocity of light, and obtain the following result for the potential contributions 3 to the 6PN Hamiltonian to O(G 4 N ) 1 See also Ref. [12] in [10]. 2 For surveys see [13]. 3 including the kinetic terms.
working in cms coordinates and using the rescaling defined in [9], Eq. (7). In dot-products the vectors are 3-vectors. Otherwise the same symbol denotes their modulus and p.n = p.r/r. The implicit counting of the powers in η 2 ≡ 1/c 2 is defined in [6], Eq. (54).
In the limit ν → 0 we agree with the Schwarzschild solution in harmonic coordinates [22]. To compare our result with the post-Newtonian expansion of the result of Ref. [1] where {·, ·} denotes the Lie bracket and H andH are the Hamiltonians for which the transformation is performed. The corresponding expressions have to be expand to the respective post-Newtonian order. In the logarithmic terms the scale r 0 = e −γ E /2 /(2 √ πµ 1 ) appears. Here γ E denotes the Euler-Mascheroni constant and µ 1 is the rescaled mass scale appearing in G N in D dimensions.
The function g inducing the canonical transformation is in general given by with ν-dependent coefficients α ijklm . Using this ansatz and evaluating Eq. (2) the corresponding explicit transformation can be found. The generating function, g isotr harm , is given in Appendix A, Eq. (4), mapping our result to that of [1]. The potential contributions to the scattering angle have been already found to be the same up to 5PN, cf. [1], referring to the Hamiltonian derived in [9]. Here we proved that this applies to the potential contributions to all observables to 6PN.
Next we compare to the part of the local contributions in Ref. [2], Eq. (7.29) to O(G 4 N ) and the lower order terms in G N , which stem from the potential terms. These are all contributions with the exception of the purely rational terms of order ν 1 , ν 2 and ν 3 , which contain also local tail contributions [11,23]. We determine the generating functiong EOB harm , given in Appendix A, Eq. (5). Again we find full agreement in all these terms, which also will imply agreement for the corresponding contributions to the scattering angle. In this way a thorough test of all results up to 6PN stemming from the potential contributions to O(G 4 N ) has been be obtained.
Acknowledgment. We thank Z. Bern for providing a computer-readable version of the Hamiltonian obtained in [1]. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 764850, SAGEX and KMP Berlin.

A The generators of the canonical transformations
The generator of the canonical transformation from harmonic coordinates to the isotropic coordinates used in [1] is given by