Static two-body potential at fifth post-Newtonian order

We determine the gravitational interaction between two compact bodies up to the sixth power in Newton's constant GN, in the static limit. This result is achieved within the effective field theory approach to General Relativity, and exploits a manifest factorization property of static diagrams which allows to derive static post Newtonian (PN) contributions of (2n+1)-order in terms of lower order ones. We recompute in this fashion the 1PN and 3PN static potential, and present the novel 5PN contribution.


INTRODUCTION
The study of the conservative dynamics of the twobody problem in General Relativity (GR) is one of the pillars which allows to determine the gravitational waveform templates for the LIGO/Virgo [1,2] data analysis pipeline [3,4].This will be the case also for the future generations of detectors, although it is now difficult to assess precisely which level of accuracy will be needed for future more sensitive observatories like the Einstein Telescope [5] or LISA [6].
Deviations from the Newton potential due to GR effects can be studied in the so called post-Newtonian (PN) framework, that is by expanding in powers of the two virial-related quantities, such as the compactness R S /r ∼ G N m/r and the relative (squared) velocity v 2 ∼ G N m/r, where R S , m, r and G N are the Schwarzschild radius of the system, its mass and size, and Newton's constant, respectively.The first complete 1PN computation was preformed by Einstein, Infeld and Hoffmann in [7]; since then, the evaluation of the higherorder terms has been a formidable effort, whose current state-of-the-art, after the calculation of the 2PN [8,9] and 3PN [10][11][12][13] contributions, is represented by the determination of the energy at the fourth-PN order, which was achieved for the first time in refs.[14][15][16][17] and later confirmed in [18][19][20][21][22] and in [23][24][25][26].
The next complexity level, namely the fifth post-Newtonian approximation (5PN), is a very challenging goal, nevertheless several partial results have become available in the so-called post-Minkowskian expansion, i.e. the expansion in G N only, for any given order in v, up to the third order in Newton's constant G N [27][28][29][30][31].
In the present work, we provide a novel contribution to the 5PN dynamics by tackling the determination of the highest possible power in G N , namely G 6  N at 5PN, which amounts to determine the potential in the static limit.This goal is achieved by building on the ideas and the method of ref. [24], where we computed the static potential at 4PN by adopting the effective field theory (EFT) approach to GR [32][33][34][35][36], in combination with techniques for the evaluation of multiloop scattering amplitudes in momentum space (see also [37] for a related computation in direct space).The computation of 5PN static corrections turns out to be actually less demanding than the corresponding 4PN ones, owing to a factorization property of the static contributions, yielding a drastic simplification at odd-PN orders, which is explicit and intuitive in the EFT approach, and is formalized in the current work.

EFFECTIVE FIELD THEORY APPROACH
The evaluation of post-Newtonian corrections to the dynamics of binary systems can be addressed within the by-now established EFT framework [32], reviewed in [34,36].We summarize here the basic features of this arXiv:1902.10571v1[gr-qc] 27 Feb 2019 approach, along the lines and notations of [13,23], while referring the reader to the literature for a more complete treatment.The action of the system is given by in terms of the world-line point particle action, representing the binary components (for spinless point masses and neglecting tidal effects) and of the canonical Einstein-Hilbert action plus a gaugefixing term which corresponds to the same harmonic gauge condition adopted in refs.[18,38], where Γ µ ≡ g ρσ Γ µ ρσ .In the above formula, Λ −2 ≡ 32πG N L d−3 , where G N is the 3-dimensional Newton constant, and L is an arbitrary length scale which keeps the correct dimensions of Λ in dimensional regularization, and always cancels out in the expression of physical observables.In this framework, a Kaluza-Klein (KK) parametrization of the metric [39][40][41] is usually adopted: with, and i, j running over the d spatial dimensions.Accordingly, the degrees of freedom of the graviton field are reparametrized in terms of a scalar field φ, a vector field A i , and a symmetric tensor field σ ij .The field A i is not actually needed in the static limit because it always comes in association with the velocity of one of the compact bodies, so it will henceforth be set to zero; we refer to [13] for the general treatment and formulae including A i .
In terms of the metric parametrization (4), with A i = 0, each world-line coupling to the gravitational degrees of freedom φ, σ ij reads and its Taylor expansion provides the various particlegravity vertices involving φ, like the coupling of φ to matter fields, i.e. the mφ n -vertex, where black lines stands for matter and dashed blue lines indicate φ modes.The complete set of Feynman rules, also involving the fields σ ij and A i (respectively indicated by green and red lines), can be found in [24].Also the pure gravity sector S bulk can be explicitly written in terms of the KK variables; for the purpose of this paper, it is sufficient to report here only the structure of the static terms not containing the field A [42]: where f is a function depending on the field σ ij only.
The two-body effective action can be found by integrating out the gravity fields from the above-derived actions Within the field-theoretical approach, the functional integration can be perturbatively expanded in terms of Feynman diagrams involving the gravitational degrees of freedom as internal lines only, viewed as dynamical fields emitted and absorbed by the point particles which are taken as non-dynamical sources.Each diagram shows a manifest power counting both in the bodies' relative velocities and in G N (any bulk vertex involving k fields carries a factor (G N ) k 2 −1 and any mφ n -vertex carries a factor (G N ) n 2 ), thus allowing for a systematic PN classification.To be more precise, the G N power-counting is determined by the topology of a diagram, that is by its shape, irrespectively of which polarization (φ, A i or σ ij ) is running on each propagator: the polarizations are the elements that determine the power-counting of the time derivatives.
The most elementary diagram in the EFT approach is represented by the Newton-potential graph naively dubbed as 0PN-diagram.

FACTORIZATION THEOREM
Definition: Static EFT-gravity diagrams can be classified according to the type of couplings between matter and φ fields.We can distinguish between: factorizable graphs, that contain at least one mφ n -vertex with n > 1, and prime graphs, that contain only matter-φ vertices of the type mφ, namely where each φ, coming from the bulk (and not propagating between bulk vertices), couples individually to matter (see fig. 1

left).
Factorizable graphs can be obtained by sewing together two, or more, sub-graphs which, upon merging, share a mφ n -vertex (n > 1) (see fig. 1

right).
Proposition: Inspection of eq.( 7) shows that the only bulk gravity vertices allowed in a static graph are those containing (a) zero or two φ's and (b) any number of σ ij 's; the latter cannot however be attached to any particle, see eq.( 5), so they can just propagate between bulk vertices.This observation is crucial to prove an important property of prime graphs, which constitute the first novel result of this communication: Theorem: Static prime graphs exist only at even 2n-PN orders.Equivalently, static graphs at odd (2n+1)-PN orders are factorizable.
Proof: This statement can be proven by showing that any prime static graph must have an even number of φ fields attached to the particles.
For the Newtonian graph, it is trivially true by construction.Graphs generated by PN-corrections, O(G 2 N ), necessarily contain bulk vertices φφσ k (with k ≥ 1), coming from the expansions of the graviton self-interaction terms.For these diagrams, two cases may occur: i) each internal φ propagator is contracted on the one side with a matter-φ vertex, and, on the other side, with a φφσ k vertex, therefore it contributes with one power of m i to the mass-dimensions of the graph; ii) a φ propagator, not coupled with matter, must necessarily connect two φφσ k vertices, therefore it does not contribute to the massdimensions of the graph.Since the bulk vertices between φ and σ fields (φφσ, φφσσ, . . . ) are quadratic in φ, and because prime graphs are by either (i) or (ii) , we can conclude that the total number of φ fields that depart from the bulk vertices and couple to matter (either m 1 or m 2 ) is an even number.
This implies that, being n i the number of φ fields coupled to the matter m i (i = 1, 2), the total mass-like power of static prime graphs is m n1 1 m n2 2 , with n 1 + n 2 = 2n and n ∈ N + .On the other side, they correspond to static classical contributions, therefore, they must consequently scale as 1 m n2 2 /r (2n−1) (classical diagrams do not contain loops in the dynamical fields), finally implying that they belong to an even-PN order.
Due to the factorization theorem, the general structure of the contribution to the potential of a given n-PN factorizable diagram, in terms of the product of lower PN-order graphs, reads where: i) the PN-orders, n 1 of the left graph V L and n 2 of the right graph V R , are such that n 1 + n 2 + 1 = n; ii) K accounts for the new matter-φ k vertex of V n (emerging from the sewing) out of the ones included in the lower order contributions, V L,n1 and V R,n2 ; and iii) ) where the C's are the combinatoric factors associated with each graph.

GRAVITY AND FIELD THEORY DIAGRAMS
In a quantum field theory approach, any EFT-gravity graph can be interpreted as four-particle scattering amplitude [24].The contribution of each amplitude to the two-body potential V can be obtained by taking its Fourier transform, ).Since the sources, represented by black lines, are static and do not propagate, any EFT-gravity amplitude at order G N can be mapped into an ( − 1)-loop 2-point function with massless internal lines and external momentum p (p 2 = 0) [24].This observation was crucial to perform the 4PN static calculation, by employing computational techniques developed for the evaluation of multi-loop Feynman integrals in high-energy particle physics.Moreover, in the current work, we observe that the integration on p can be seen as an additional loop integration, hence it can be represented by an -loop vacuum diagram, obtained by joining the external legs into a propagator-like line (indicated by an inner black line), as In the last step, we introduce a suggestive diagrammatic representation of the Fourier integral as an -loop vacuum graph by pinching the internal black line.The presence of the dot "•" indicates the residual r-dependence of the contribution (not to be confused by fully massless, hence scaleless vacuum diagrams that vanish in dimensional regularization).For instance, Newton's potential can be represented as a one-loop vacuum graph: In the case of factorizable EFT-diagrams, the pinching generates the product of factorized vacuum diagrams.
For example, the contribution to the 5PN-potential of the diagram in fig. 1 directly representing the product of the Newton potential and of a (5-loop) 4PN-term.

RESULTS
In this section, we apply the factorization theorem to the known static potential at 1PN and 3PN, as well as to determine the novel contribution to the 5PN-order.

1PN Static Potential
Let us verify that the 1PN potential, due to a single static diagram, can be decomposed in terms of the Newton potential.According to the factorization theorem, = 2 × 2 (15) which indeed amounts to the identity, In this example, the factor K is given by the fraction on the r.h.s. of eq.( 15), while C = 1; we have adopted the convention that m 1(2) refers to the bottom (top) line in the diagrams.

3PN Static Potential
At the 3PN order, there are 8 static graphs, shown in fig. 2.
But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves But let's complete first the case j=1.I report here the results, which you should now be able to reproduce by yourselves Their contributions to the Lagrangian were computed in [13].In light of the factorization theorem, we identify two classes of diagrams: 1.In this set, we consider 3 diagrams composed of 4 Newtonian factors, corresponding to the first three diagrams of fig. 2. We represent them as 4 . ( Their contribution to the 3PN potential is: 2. The other 5 diagrams of fig.2, are built as product of 1 Newtonian term and the 3 static 2PN prime graphs, combined in all possible ways, schematically represented as For illustration purposes, the factorization theorem can be verified for one them: (C = 2 here) amounting to in agreement with [13].The contribution to the potential from the five diagrams in class 2 is: Total contribution at 3PN.The total static 3PN contribution of the diagrams belonging to classes 1 and 2 is, in agreement with the literature:

5PN Static Potential
We now apply the factorization theorem to the 5PN case, computed for the first time in the present work.There are 154 diagrams to evaluate and it is convenient to divide them in four classes, according to their factorization patterns.For ease of notation, we arrange the 5PN static graphs in four subsets, displaying the lower-PN corrections they stem out from, and give the corresponding factors K and C as understood.
1.There are 11 diagrams composed of 6 Newtonian factors, combined in different ways, and schematically represented as 6 . ( The contribution to the 5PN potential coming from this set of diagrams is: 2. One can build static factorizable diagrams as products of 3 Newtonian graphs, and either of the 2PN prime graphs, schematically represented as: This set contains 49 diagrams, 9 of which are vanishing, because one of the 2PN-factors is indeed zero.The combined contribution of the remaining diagrams is: 3. In this class, we consider 5PN diagrams schematically represented by the product of one Newtonian graph with each of the 25 static prime 4PN diagrams studied in [24], (the cardinal number attached to each graph is the same as in [24], for ease of comparison) This set contains 79 diagrams, 16 of which are vanishing (due to vanishing 4PN-factors).The remaining 63 diagrams give: Interestingly, let us observe that although this set contains contributions which are individually divergent in the d → 3 limit, as well as factors of π 2 , within their sum all poles and irrational factors cancel, and the result is indeed finite and rational.4. Finally, we consider static 5PN diagram formed by the product of two 2PN-graphs, schematically represented as This term contains 15 5PN graphs, 5 of which are manifestly vanishing, while the contribution of the remaining 10 diagrams reads: Total 5PN static potential.By combining all the previous results, the expression for the static sector of the 5PN potential finally reads, This expression contains the genuine G 6 N contribution coming from graphs, without contributions generated from lower-G N terms when using the equations of motion to eliminate terms quadratic at least in the accelerations.Together with the factorization theorem, the above result constitutes the second important result of this manuscript.Check: test-particle limit.It is possible to verify that the coefficient of the term m 6  1 m 2 agrees with what can be expected from the extreme mass ratio limit m 2 m 1 .In this limit, where only the graphs displayed in fig. 3 contribute, it is possible to consider the body with mass m 2 as a test particle in the Schwarzschild metric generated by the body with mass m 1 .The action describing the dynamics of the test body has still the form S pp described in eq.( 2), but with g µν given by the Schwarzschild metric in harmonic coordinates (which is obtained from the traditional form by the simple radial coordinate shift r → r + G N m 1 ) instead of the Minkowski one.
In the static limit, v 2 = 0, only the term g 00 survives, and the effective Lagrangian reads By expanding this expression in (where the potential is correctly reported with opposite sign w.r.t. to the lagrangian term).

CONCLUSION
We studied the two-body conservative dynamics at fifth post-Newtonian order (5PN) in the static limit within the effective field theory (EFT) approach to General Relativity.We determined an essential contribution of the complete 5PN potential at O(G 6 N ), coming from 154 Feynman diagrams.We proved a factorization property of the static diagrams at odd-PN order, and exploited it to show that their contribution can be determined recursively, from lower PN-orders.The result of the static potential at order G 6 N is found to be finite and rational -a property clearly inherited from the static G 5 N sector -and exhibits the expected Schwarzschild-like behaviour in the extreme mass ratio limit.

Figure 1 .
Figure 1.Examples of a prime 4PN-graph (left) and of a factorizable 5PN-graph (right): the latter, can be obtained by sewing the former and the Newton potential diagram.
where, p ≡ d d p/(2π) d , the box diagram stands for a generic EFT-gravity diagram, and p is the momentum transfer of the source (assuming momentum conservation p 1 + p 2 = p 3 + p 4 , then p = p 3 − p 2 = p 1 − p 4

Figure 3 .
Figure 3. 5PN-graphs contributing to the test-particle limit.The last graph (bottom-right) does not contribute to the 5PN potential, because its 4PN subdiagram vanishes.