Conservative dynamics of binary systems to fourth Post-Newtonian order in the EFT approach I: Regularized Lagrangian

TL;DR

This paper derives the missing G^3 and G^4 contributions to the conservative two-body dynamics at fourth post-Newtonian order using an effective field theory approach, providing a regularized Lagrangian consistent with prior results.

Contribution

It presents the first calculation of the G^3 and G^4 sectors at 4PN order within the EFT framework, completing the conservative dynamics at this level.

Findings

Derived the G^3 and G^4 sectors at 4PN order.

Constructed a regularized Lagrangian consistent with previous methods.

Validated results against existing computations.

Abstract

Within the framework of the effective field theory approach to the post-Newtonian approximation to General Relativity, we report the so far missing G^3 and G^4 sectors of the conservative two body dynamics at fourth perturbative order for non-spinning compact objects. Following the standard procedure, the exchange of gravitational interaction is integrated out to obtain a systematic expansion in terms of Feynman graphs. Building on this result and on previously obtained ones, we are able to write the local-in-time regularized Lagrangian at fourth post-Newtonian order, which is in agreement with results previously computed with other methods.

Equations

$${\mathcal{S}}_{pp}=-\sum _{a=1,2}{m}_a\int d{\tau }_a=-\sum _{a=1,2}{m}_a\int \sqrt{-g_{\mu \nu }(x_a)d{x}_a^{\mu }d{x}_a^{\nu }},$$

$${\mathcal{S}}_{EH+GF}=2{\Lambda }^2\int d^{d+1}x\sqrt{-g}\left(R(g)-\frac{1}{2}{\Gamma }^{\mu }{\Gamma }_{\mu }\right),$$

$$\displaystyle g_{\mu\nu}=e^{2\phi/\Lambda}\left(\begin{array}[]{cc}\displaystyle-1&\displaystyle\frac{A_{j}}{\Lambda}\\ \displaystyle\frac{A_{i}}{\Lambda}&\displaystyle e^{-c_{d}\phi/\Lambda}\left(\delta_{ij}+\frac{\sigma_{ij}}{\Lambda}\right)-\frac{A_{i}A_{j}}{\Lambda^{2}}\end{array}\right)$$

$${\mathcal{S}}_{pp}=-\sum _{a=1,2}{m}_a\int \mathrm{d}{\tau }_a=-\sum _{a=1,2}{m}_a\int \mathrm{d}{t}_a{e}^{\varphi /\Lambda }\sqrt{{\left(1-\frac{A_i}{\Lambda }{v}_a^i\right)}^2-e^{-c_d\varphi /\Lambda }\left(v_a^2+\frac{{\sigma }_{ij}}{\Lambda }{v}_a^i{v}_a^j\right)},$$

$$\displaystyle\begin{array}[]{rcl}\displaystyle{\mathcal{S}}_{bulk}&\supset&\displaystyle\int{\rm d}^{d+1}x\sqrt{-\gamma}\left\{\frac{1}{4}\left[(\vec{\nabla}\sigma)^{2}-2(\vec{\nabla}\sigma_{ij})^{2}-\left(\dot{\sigma}^{2}-2(\dot{\sigma}_{ij})^{2}\right){\rm e}^{\frac{-c_{d}\phi}{\Lambda}}\right]-c_{d}\left[(\vec{\nabla}\phi)^{2}-\dot{\phi}^{2}{\rm e}^{-\frac{c_{d}\phi}{\Lambda}}\right]\right.\\ &&\displaystyle+\left[\frac{F_{ij}^{2}}{2}+\left(\vec{\nabla}\!\!\cdot\!\!\vec{A}\right)^{2}-\dot{\vec{A}}^{2}{\rm e}^{-\frac{c_{d}\phi}{\Lambda}}\right]{\rm e}^{\frac{c_{d}\phi}{\Lambda}}+\frac{2}{\Lambda}\left[\left(F_{ij}A^{i}\dot{A^{j}}+\vec{A}\!\!\cdot\!\!\dot{\vec{A}}(\vec{\nabla}\!\!\cdot\!\!\vec{A})\right){\rm e}^{\frac{c_{d}\phi}{\Lambda}}-c_{d}\dot{\phi}\vec{A}\!\!\cdot\!\!\vec{\nabla}\phi\right]\\ &&\displaystyle+2c_{d}\left(\dot{\phi}\vec{\nabla}\!\!\cdot\!\!\vec{A}-\dot{\vec{A}}\!\!\cdot\!\!\vec{\nabla}\phi\right)+\frac{\dot{\sigma}_{ij}}{\Lambda}\left(-\delta^{ij}A_{l}\hat{\Gamma}^{l}_{kk}+2A_{k}\hat{\Gamma}^{k}_{ij}-2A^{i}\hat{\Gamma}^{j}_{kk}\right)-c_{d}\frac{\dot{\phi}^{2}\vec{A}^{2}}{\Lambda^{2}}\\ &&\displaystyle-\left.\frac{1}{\Lambda}\left(\frac{\sigma}{2}\delta^{ij}-\sigma^{ij}\right)\left({\sigma_{ik}}^{,l}{\sigma_{jl}}^{,k}-{\sigma_{ik}}^{,k}{\sigma_{jl}}^{,l}+\sigma_{,i}{\sigma_{jk}}^{,k}-\sigma_{ik,j}\sigma^{,k}\right)\right\}\,,\end{array}$$

$$\exp [\text{i}{\mathcal{S}}_{eff}]=\int D\varphi D{A}_{i}D{\sigma }_{ij}\exp [\text{i}({\mathcal{S}}_{bulk}+{\mathcal{S}}_{pp})].$$

$$W_p^a(t)\equiv \int {\mathrm{d}}^{d}x{W}^a(t,x)e^{-\text{i}p\cdot x}\mathrm{with}{W}^a=\{\varphi ,A_i,{\sigma }_{ij}\}.$$

$$P[W_p^a(t_a)W_{p^{\prime }}^b(t_b)]$$

$$\mathcal{P}(p^2,t_a,t_b)=\frac{\text{i}}{p^2-{\partial }_{t_a}{\partial }_{t_b}}.$$

$$a_1^2\simeq a_1^2-{\left({\vec{a}}_1+\frac{G{m}_2\vec{r}}{r^3}\right)}^2=-2\frac{G{m}_2{a}_1^r}{r^3}-\frac{G^2{m}_2^2}{r^4},$$

$$\begin{minipage}{71.13188pt} \begin{center} \includegraphics[width=56.9055pt]{EFT_gravity} \end{center} \end{minipage}\quad=\quad\sum c_{a}M_{a}\,,$$

$$\displaystyle\begin{array}[]{rcl}M^{(2loop)}_{1}(p)&\equiv&\displaystyle\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\frac{1}{k_{1}^{2}(p-k_{2})^{2}(k_{1}-k_{2})^{2}}\,,\\ M_{2}^{(2loop)}(p)&\equiv&\displaystyle\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\frac{1}{k_{1}^{2}(p-k_{1})^{2}k_{2}^{2}(p-k_{2})^{2}}\,,\end{array}$$

$$\displaystyle\begin{array}[]{rcl}M^{(3loop)}_{1}(p)&\equiv&\displaystyle\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\frac{d^{d}k_{3}}{(2\pi)^{d}}\frac{1}{(p-k_{3})^{2}k_{2}^{2}(k_{1}-k_{2})^{2}(k_{1}-k_{3})^{2}}\,,\\ M^{(3loop)}_{2}(p)&\equiv&\displaystyle\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\frac{d^{d}k_{3}}{(2\pi)^{d}}\frac{1}{(p-k_{2})^{2}(p-k_{3})^{2}k_{1}^{2}k_{3}^{2}(k_{1}-k_{2})^{2}}\,,\\ M^{(3loop)}_{3}(p)&\equiv&\displaystyle\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\frac{d^{d}k_{3}}{(2\pi)^{d}}\frac{1}{(p-k_{2})^{2}(p-k_{3})^{2}k_{1}^{2}(k_{1}-k_{2})^{2}(k_{1}-k_{3})^{2}}\,,\end{array}$$

$${\mathcal{L}}_{G^3}^{(4PNgraphs)}=\frac{G^3{m}_1^3{m}_2}{r^3}[v_2.a_1\left(\frac{4859}{120}{v}_1^r-\frac{12187}{360}{v}_2^r\right)r+a_1^r(\frac{493}{18}{v}_1^2+\frac{5143}{720}\left(v_2^2-2v_1.v_2\right)$$

$$+\frac{6347}{240}{v_2^r}^2-\frac{11497}{120}{v}_1^r{v}_2^r)r-\frac{12187}{720}r{a}_2^r{v}_1^2+\frac{15787}{720}{v}_1^4+\frac{19927}{720}{v}_1^2\left(v_2^2-2v_1.v_2\right)+\frac{43}{2}{v}_1.v_{2}v.v_2$$

$$+\frac{91}{16}{v}_2^4+v_1^2\left(\frac{15487}{120}{v}_1^r{v}_2^r-\frac{3373}{30}{v_1^r}^2-\frac{16387}{240}{v_2^r}^2\right)+v_1.v_2\left(7v_1^r{v}_2^r+\frac{11557}{240}{v_1^r}^2-2{v_2^r}^2\right)$$

$$+v_2^2\left(\frac{7}{4}{v_2^r}^2-\frac{1}{8}{v_1^r}^2-\frac{7}{2}{v}_1^r{v}_2^r\right)+\left(\frac{1661}{18}{v_1^r}^2-\frac{12613}{144}{v}_1^r{v}_2^r-\frac{15}{4}{v_2^r}^2\right){v_1^r}^2]$$

$$+\frac{G^3{m}_1^2{m}_2^2}{r^3}\{a_1^r\left[\left(\frac{20129}{450}-\frac{21{\pi }^2}{32}\right)v_1^2+\left(\frac{23297}{600}-\frac{43}{128}{\pi }^2\right)\left(v_2^2-2v_1.v_2\right)+\left(\frac{123{\pi }^2}{128}-\frac{431}{24}\right){v_2^r}^2\right]r$$

$$+r{v}_2.a_1\left(\frac{425}{36}-\frac{41{\pi }^2}{64}\right)v_1^r+\left(\frac{3953}{90}+\frac{133{\pi }^2}{1024}\right)v_1^4+\left(\frac{31759}{720}+\frac{133{\pi }^2}{1024}\right)\left(v_1^2{v}_2^2+2(v_1.v_2{)}^2-4v_1^2{v}_1.v_2\right)$$

$$+v_1.v_2\left(\frac{90223}{300}-\frac{447{\pi }^2}{256}\right)v^r{v}_1^r-2v_1.v_2{v}_1^r{v}_2^r+v_1^2[\left(\frac{447{\pi }^2}{512}-\frac{22687}{150}\right)v_1^r(v_1^r-2v_2^r)$$

$$+\left(\frac{447{\pi }^2}{512}-\frac{90223}{600}\right){v_2^r}^2]+[\left(\frac{2155{\pi }^2}{256}-\frac{7348}{15}\right)v_1^r{v}_2^r+\left(\frac{1837}{15}-\frac{2155{\pi }^2}{1024}\right){v_1^r}^2$$

$$+\left(\frac{29617}{80}-\frac{6465{\pi }^2}{1024}\right){v_2^r}^2]{v_1^r}^2\}+{\mathcal{L}}_{G^{3}fin}^{(4PN{a}^2)}+{\mathcal{L}}_{G^{3}pole}^{(4PN)}$$

$$+\log \bar{r}\{\frac{G^3{m}_1^3{m}_2}{r^3}[v_2.a_1\left(37v_2^r-48v_1^r\right)r+r{a}_1^r\left(16v_1.v_2-32v_1^2-8v_2^2-\frac{51}{2}{v_2^r}^2+84v_1^r{v}_2^r\right)$$

$$+\frac{37}{2}{v}_1^2\left(r{a}_2^r-v^2\right)+v_1^2\left(\frac{195}{2}{v_1^r}^2-111v_1^r{v}_2^r+\frac{111}{2}{v_2^r}^2\right)-42v_1.v_2{v_1^r}^2-70{v_1^r}^3{v}^r]$$

$$+\frac{G^3{m}_1^2{m}_2^2}{r^3}\left[14r{a}_1^r{v}^2+42(v_1^2{v}^r-2v_1.v_2{v}_1^r)v^r-70{v_1^r}^2\left({v_1^r}^2-4v_1^r{v}_2^r+3{v_2^r}^2\right)\right]\},$$

$${\mathcal{L}}_{G^{3}fin}^{(4PN{a}^2)}$$

$${\mathcal{L}}_{G^{3}pole}^{(4PN)}$$

$${\mathcal{L}}_{G^{3}pole}^{(4PN{a}^2)}$$

$${\mathcal{L}}_{G^4}^{(4PNgraphs)}=\frac{G^4{m}_1^4{m}_2}{r^4}\left[\frac{50}{9}{v}_1^2-\frac{227}{36}{v}_1.v_2+\frac{15}{16}{v}_2^2-\frac{422}{9}{v_1^r}^2+\frac{817}{18}{v}_1^r{v}_2^r+\frac{9}{4}{v_2^r}^2\right]$$

$$+\frac{G^4{m}_1^3{m}_2^2}{r^4}[\left(\frac{466}{9}-\frac{157}{32}{\pi }^2\right)v_1^2+\left(\frac{69}{16}{\pi }^2-\frac{1201}{18}\right)v_1.v_2+\left(\frac{19}{32}{\pi }^2+\frac{323}{18}\right)v_2^2$$

$$\left(\frac{2363}{96}{\pi }^2-\frac{1468}{9}\right){v_1^r}^2+\left(\frac{1958}{9}-\frac{1307}{48}{\pi }^2\right)v_1^r{v}_2^r+\left(\frac{251}{96}{\pi }^2-\frac{382}{9}\right){v_2^r}^2]$$

$$+\left(\frac{1}{(d-3)}-4\log \bar{r}\right)\{\frac{G^4{m}_1^3{m}_2^2}{3r^4}\left[v_1^2-13v_1.v_2+12v_2^2+44v_1^r{v}^r\right]$$

$$+\frac{23}{3}\frac{G^4{m}_1^4{m}_2}{r^4}\left[v.v_1-4v_1^r{v}^r\right]\}.$$

$${\mathcal{L}}_{G^5}^{(4PNgraphs)}=\frac{3}{8}\frac{G^5{m}_1^5{m}_2}{r^5}+\frac{31}{3}\frac{G^5{m}_1^4{m}_2^2}{r^5}+\frac{141}{8}\frac{G^5{m}_1^3{m}_2^3}{r^5}.$$

$${\mathcal{L}}^{(3PN)}={\mathcal{L}}_{G+G^2}^{(3PN)}+{\mathcal{L}}_{G^3+G^4}^{(3PN)}+{\mathcal{L}}_{pole}^{(3PN)}$$

$${\mathcal{L}}_{G^3+G^4}^{(3PN)}=\frac{G^3{m}_1^3{m}_2}{r^3}\left[\frac{55}{9}{v}_1^2-\frac{137}{18}{v}_1.v_2+\frac{5}{4}{v}_2^2-\frac{67}{4}{v_1^r}^2+\frac{29}{2}{v}_1^r.v_2^r+\frac{3}{2}{v_2^r}^2\right]$$

$$+\frac{G^3{m}_1^2{m}_2^2}{r^3}\left[-\frac{305}{72}{v}_1^2+\frac{439}{144}{v}_1.v_2+\frac{383}{24}{v_1^r}^2-\frac{889}{48}{v}_1^r.v_2^r+\frac{41{\pi }^2}{64}\left(v.v_1-3v^r{v}_1^r\right)\right]$$

$$-\frac{3}{8}\frac{G^4{m}_1^4{m}_2}{r^4}-\frac{67}{3}\frac{G^4{m}_1^3{m}_2^2}{r^4}+\frac{11}{3}\log \bar{r}\left[-\frac{G^3{m}_1^3{m}_2}{r^3}\left(v.v_1-3v^r{v}_1^r\right)+2\frac{G^4{m}_1^3{m}_2^2}{r^4}\right],$$

$${\mathcal{L}}_{pole}^{(3PN)}=-\frac{11G^2{m}_1^2{m}_2}{3(d-3)}\left(a_1^2+2a_1.a_2\right)+\frac{11G^3{m}_1^3{m}_2}{3(d-3)r^2}{a}_1^r,$$

Full text

classical general relativity, coalescing binaries, post-Newtonian expansion

I Introduction

The present paper reports the $G^{3}$ and the local-in-time $G^{4}$ sectors of the near zone Lagrangian governing the relativistic conservative dynamics of the two non-spinning body motion at fourth post-Newtonian (PN) approximation order to General Relativity (GR), concluding the effort initiated by the authors and collaborators with the derivation of the third PN order Foffa and Sturani (2011) and then with the partial fourth PN results for the $G,G^{2}$ Foffa and Sturani (2013a) and $G^{5}$ sectors Foffa et al. (2017) by using Effective Field Theory methods (EFT) applied to gravity Goldberger and Rothstein (2006), see Goldberger (2007); Foffa and Sturani (2014); Porto (2016); Levi (2018) for reviews, being $G$ Newton’s constant (in 3+1 dimensions). At generic nPN order the dynamics is naturally split in sectors of the type $G^{j}v^{2(n-j+1)}$, with $0<j\leq n+1$, as $v^{2}\sim GM/r$ by using Kepler law, with $M$ being the total mass of the binary system. The velocity $v$ is the expansion parameter, two successive PN orders being separated by a factor $v^{2}$. We work in dimensional regularization and keep our expressions in $d$ dimensions, without distinguishing the nature of poles as $d\to 3$. Following procedure outlined in Manohar and Stewart (2007); Porto and Rothstein (2017), the companion paper Foffa et al. (2019) will complete the project by properly identifying the intermediate infrared (IR) and ultraviolet (UV) divergences, as well as providing a self-contained, ambiguity-free renormalization of the effective theory, including also conservative contributions from radiation modes.

The two body problem in GR has been studied at length in the last decades, see Blanchet (2014); Futamase and Itoh (2007); Schäfer and Jaranowski (2018); Goldberger (2007); Foffa and Sturani (2014); Porto (2016), and its phenomenological relevance relies on being an unavoidable ingredient for constructing template waveforms Taracchini et al. (2012); Schmidt et al. (2015) necessary for the detections of gravitational waves made by the laser interferometric detector LIGO Abbott et al. (2016, 2017a, 2017b, 2018) and jointly with his European homologous Virgo Abbott et al. (2017c, d). Solving this problem has further applications to numerical relativity Mroue et al. (2013), self force calculations Akcay et al. (2015), which are also related to waveform template construction, and in general the GR two-body dynamics is a problem rich of intriguing theoretical aspects, representing an highly non-trivial test-bed for classical field theory beyond its phenomenological applications.

Our work is developed within the framework of EFT methods, while other groups have obtained the complete 4PN conservative dynamics with mutually consistent results using different methods: within the Arnowitt-Deser-Misner (ADM) Hamiltonian formalism Damour et al. (2014, 2015); Jaranowski and Schäfer (2015); Damour et al. (2016) and within the Fokker action one in Bernard et al. (2016, 2017a, 2017b); Marchand et al. (2018); Bernard et al. (2018).

To make contact with standard literature, we will refer to near-zone dynamics as the one involving the interaction of massive bodies at the scale of their mutual distance $r$, mediated by off-shell, ”potential” gravitational modes, with 4-momentum $k^{\mu}$ scaling as $(k_{0},k)\sim(1/r,v/r)$, being $v$ the relative velocity. On the other hand, we will refer to far zone as the radiation region, which is at a distance from the massive bodies equal or larger than the radiation wavelength $\sim r/v$, and where the dynamics involves on-shell, radiative gravitational modes with $(k_{0},k)\sim(v/r,v/r)$.

The outline of the paper is as follows. In sec. II we give an overview of the methods employed in deriving the GR two-body effective Lagrangian in the PN approximation, the key ingredient being the perturbative functional integration of potential gravitational modes in the near zone; in the same section we also recall the invariance of gauge independent physical results under manipulation of (gauge dependent) Lagrangian terms by use of the equations of motion.

In sec. III we show the bare result of the Feynman diagram computation at order $G^{3}$ and $G^{4}$, while revisiting also the 3PN sector. In sec. IV we combine these results with those of Foffa and Sturani (2013a); Foffa et al. (2017) and we re-organize them for ease of comparison with previous works.

While in this paper we are concerned with the contribution from potential modes to the near zone Lagrangian, for directly compare with previous literature we will simply add to our computation the far-zone conservative and local-in-time contribution due to the tail, postponing to Foffa et al. (2019) a discussion for the theoretical foundations for this procedure, which relies on the proper identification of IR/UV poles. In particular we show how this procedure enables the decomposition of the final result into two structures: a first one, containing all the remaining poles, which is irrelevant for the dynamics because it vanishes on the equations of motion; and a second, finite one, which we show being equivalent to the Lagrangian reported in Bernard et al. (2018). Finally, sec. V gathers our conclusions.

II Method

We describe point particles via a world-line action $\mathcal{S}_{pp}$ which resides on the world-line of the binary constituents and a bulk part $\mathcal{S}_{EH+GF}$ which encodes the dynamics of gravity:

[TABLE]

[TABLE]

where the gauge fixing term with $\Gamma^{\mu}\equiv g^{\nu\rho}\Gamma^{\mu}_{\nu\rho}$ has been added to the standard Ricci term of the Einstein-Hilbert action. Here $\Lambda^{-2}\equiv 32\pi G^{(d)}$ is defined in terms of the $d-$dimensional Newton’s constant $G^{(d)}=G(\sqrt{4\pi}e^{\gamma_{E}/2}L_{0})^{d-3}$ being $L_{0}$ an arbitrary length scale introduced to adjust dimensions, with the presence of $\pi$ and Euler Gamma factors introduced in its definition for later convenience. The introduction of the arbitrary scale $L_{0}$ is made necessary by the use of dimensional regularization, but this length-scale does not appear in expressions involving physical observables.

As suggested in Kol and Smolkin (2008) and already done in our previous papers reporting 3PN and partial 4PN results, we find convenient to adopt the metric parametrization

[TABLE]

with $c_{d}\equiv 2\frac{d-1}{d-2}$ and Latin indices $i,j$ runs over spatial indices only. This ansatz has the virtue of diagonalizing the kinetic terms of the fields $\phi,A_{i},\sigma_{ij}$ speeding up the computations.

Substituting ansatz (5) into actions (1,2) one can rewrite $\mathcal{S}_{pp}$ as

[TABLE]

and its Taylor expansion provides the various particle-gravity vertices of the EFT.

Analogously we report ${\mathcal{S}}_{bulk}$ truncated at the order relevant for the computation in this paper

[TABLE]

where $F_{ij}\equiv A_{j,i}-A_{i,j}$, and $\hat{\Gamma}^{i}_{jk}$ is the connection of the purely spatial $d$-dimensional metric $\gamma_{ij}\equiv\delta_{ij}+\sigma_{ij}/\Lambda$, which is also used above to raise and contract spatial indices. All spatial derivatives are understood as simple (not covariant) derivatives and when ambiguities might raise gradients are always meant to act on contravariant fields, e.g. $\vec{\nabla}\!\!\cdot\!\!\vec{A}\equiv\gamma^{ij}A_{i,j}$ and $F_{ij}^{2}\equiv\gamma^{ik}\gamma^{jl}F_{ij}F_{kl}$.

The 2-body effective action is found by integrating out the gravity fields from the ${\mathcal{S}}_{bulk}+{\mathcal{S}}_{pp}$

[TABLE]

As usual in field theory, the functional integration can be perturbatively expanded in terms of Feynman diagrams involving the gravitational degrees of freedom as internal lines only. No gravitational modes will appear among external lines, with the massive bodies playing the role of non-dynamical sources and sinks of gravitational modes; only diagrams corresponding to classical contributions to the effective Lagrangian need to be considered.

To make manifest the $v$ scaling necessary to classify the results according to the PN expansion, it is convenient to work with the a mixed decomposition of the fields working in direct time-coordinate and Fourier transforming on the space ones:

[TABLE]

The amplitudes corresponding to each diagram can be built from the Feynman rules in momentum-space derived from ${\mathcal{S}}_{pp}$, ${\mathcal{S}}_{bulk}$ with propagators:

[TABLE]

where $P^{\phi\phi}=-\frac{1}{c_{d}}$, $P^{A_{i}A_{j}}=\delta_{ij}$, $P^{\sigma_{ij}\sigma_{kl}}=-\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}+(2-c_{d})\delta_{ij}\delta_{kl}\right)$ and

[TABLE]

The terms involving time derivatives in the propagator must be Taylor expanded and will end up hitting the exponential factors $\leavevmode\nobreak\ e^{ip\cdot x}$ at source points, eventually generating extra factors of $v$ and its derivatives that has to be kept at the appropriate order.

The effective potential involves kinematic variables (positions, velocities,…) of the massive bodies at the same time, retardation effects dictated by GR are reinstated in this framework by the knowledge of the trajectories via their derivatives at a point, and at any finite PN order only a finite number of trajectory derivatives are necessary to reconstruct the effective action.

Note that working in the harmonic gauge simplifies considerably the computations since linearized equations show that all polarizations satisfy D’Alembertian type of equations, but this is a gauge artifact, analogous to what happens in electromagnetism in the Feynman gauge, as gravity has only 2 propagating degrees of freedoms. Indeed the resulting effective action will not be gauge-invariant, as it is not observable, and as first noticed in Schäfer (1984) substituting into the Lagrangian the equations of motion is equivalent to a coordinate transformation. As a consequence, adding to the Lagrangian a term proportional to the equations of motion (henceforth, EOMs) does not alter the resulting dynamics.

On a technical point, we observe that the near-zone Lagrangian can contain terms involving second or higher derivatives of the particle trajectory, leading to higher than second order differential equations of motion. This problem can be solved by adding harmless terms proportional to the square of the EOMs which neither alter the resulting EOMs nor represent a change of coordinates, but have the welcome feature of getting rid of high derivative terms. Such a procedure was dubbed double zero trick Damour and Schäfer (1985) and roughly works as follows: taking for instance a $a_{1}^{2}$ term at Newtonian order one can add a harmless term proportional to the square of the equations of motion

[TABLE]

(with notation $X^{r}\equiv\vec{X}\cdot\vec{r}/r$) without altering the resulting EOMs.

In the following sections we report the results of Feynman diagrams at $G^{n}$ order with $n={3,4}$, involving integration over $n-1$ internal momenta $k_{1},\ldots,k_{n-1}$ and over the momentum $p$ exchanged between the two massive bodies. While the integration diagrams in the $G$ and $G^{2}$ sub-sector can be readily performed via the use of the formulae (43,44), at orders $G^{3}$ and $G^{4}$ diagrams with up to $7$ propagators are present, and they can be computed with the technique of integration by part identities Tkachov (1981); Chetyrkin and Tkachov (1981); Laporta (2000) enabling the reduction of all integrals to the master integrals.

The amplitudes related to any given Feynman diagram can be decomposed as

[TABLE]

where the left hand side represents a generic EFT Feynman graph (thick horizontal lines represent the classical external sources which exchange the dynamical fields $\phi$, $A$, $\sigma$ propagating in the shaded area), and the master integrals $M_{a}$ needed for our purpose are

[TABLE]

at order $G^{3}$, and the additional

[TABLE]

at order $G^{4}$.

Such master integrals can be represented in the language introduced in Foffa et al. (2017), to which we remind the reader for further details, as in figs. 1,2. The particle physics-oriented reader will appreciate the correspondence between the number of $d$-dimensional momentum integrations in the definitions of the $M_{a}$s and the number of loops in their graphical representation, as well as between the denominator factors and the internal lines of the graphs.

The integrations by parts in multi-loop computations have been performed by direct application of the integration by parts method Tkachov (1981); Chetyrkin and Tkachov (1981) and independently via its implementation in the Reduze code Studerus (2010). Master integrals have been computed by repeated application of equation (44) reported in the appendix, and the Lagrangian contribution is finally obtained by taking the Fourier transform with respect to $p$, which is the momentum carried by the external legs in figs. 1,2.

For the actual computation we use the Feyncalc software Shtabovenko (2016) in Mathematica.

III Results of diagram computation

The result for the $G$ and $G^{2}$ near zone computations are already published in eq.(13,14,17,18,19) of Foffa and Sturani (2013a), with the list of relative diagrams in fig. 1 and 2 of the same paper (respectively 3 and 23 diagrams), and will not be reported here.

III.1 The 4PN ${\bm{G}^{3}}$ sector

The $G^{3}$ sector receives contribution from 212 diagrams which are available, together with their values, at http://fiteoweb.unige.ch/$\sim$foffa/research/research.html. Their sum gives 111We denote the scalar product among two vectors $\vec{a}$, $\vec{b}$ as $a.b$, define $\vec{v}\equiv\vec{v}_{1}-\vec{v}_{2}$ and an upper $r$ stands for scalar product with unit vector separation between the two bodies, e.g. $v^{r}\equiv(v_{1}-v_{2}).(x_{1}-x_{2})/r$.:

[TABLE]

with $\bar{r}\equiv r/L_{0}$, ${\cal L}^{(4PNa^{2})}_{G^{3}fin}$ gathering the non-singular contributions quadratic in accelerations or higher derivatives

[TABLE]

and ${\cal L}^{(4PN)}_{G^{3}pole}$ gathering the singular part

[TABLE]

in which we have separated the part quadratic in acceleration or higher derivative terms

[TABLE]

III.2 The 4PN ${\bm{G}^{4}}$ and ${\bm{G}^{5}}$ sectors

The $G^{4}$ sector receives contribution from 317 diagrams, also available with their individual values at http://fiteoweb.unige.ch/$\sim$foffa/research/research.html (see also Cristofoli (8 03) for a detailed sample calculation) whose sum gives

[TABLE]

For later convenience, we recall here the also sum of the 50 $G^{5}$ diagrams in fig.1 of Foffa et al. (2017):

[TABLE]

III.3 The 3PN sector

Divergences occur in the effective Lagrangian starting from 3PN order, which have to be taken into account along with the 4PN $G^{2}$ Foffa and Sturani (2013a) , $G^{3}$ and $G^{4}$ ones.

We consider here the bare 3PN Lagrangian derived from near-zone graphs computed in Foffa and Sturani (2011), without the use of coordinate shift performed in Blanchet et al. (2004) and Foffa and Sturani (2011), (but we add double zeroes to the finite part to get rid of terms quadratic in accelerations or higher derivative) with the result that our regularized 3PN Lagrangian coincides with the one of Blanchet et al. (2004) only up to $G^{2}$ terms included. In total we have:

[TABLE]

with

[TABLE]

and

[TABLE]

while ${\cal L}^{(3PN)}_{G+G^{2}}$ is the same as reported in sec. 7.2 of Blanchet (2014).

IV The regularized 3PN and 4PN Lagrangian

We now put together the results of the previous section, transforming eq. (26) by means of the double zero trick, while leaving unaltered analogous higher derivative terms contained in the divergent pieces, given by eq.(28). We also include here the 4PN terms generated by applying the double zero trick to the 2PN, 3PN, 4PN $G$, $G^{2}$ and $G^{3}$ finite terms, according to the equations reported in Appendix D.

As we mentioned in the introduction, for the sake of comparison with previous results we add by hand the local-in-time contribution from the tail term ${\cal L}_{tail}$. The complete expression of the dimensionally regularized ${\cal L}_{tail}$, inclusive of its non-local-in-time part obtained earlier in Blanchet and Damour (1988), has first been computed in Foffa and Sturani (2013b); Galley et al. (2016), and re-derived in Bernard et al. (2017b)

[TABLE]

which takes the explicit form (45) after the quadrupole ${\cal Q}_{ij}$ is expressed in terms of orbital variables 222The part involving $\gamma_{E}$ could well be incorporated into the local-in-time part, here we decide it to keep into the non-local part..

We will retain the local-in-time part, including the pole, which we need to compare our results with those presented in Damour et al. (2014) and Marchand et al. (2018). We discuss the proper incorporation of the tail terms as well as the renormalization of the effective theory in Foffa et al. (2019), to which we refer the reader for details.

In what follows we show that the regularized local-in-time Lagrangian obtained with this procedure agrees with previous results in the literature. The sum of all the terms discussed so far can be written as

[TABLE]

The quantity $\frac{1}{d-3}{\cal L}_{poles}$ beside all divergent terms (including the tail ones) includes some ad-hoc finite pieces to ensure that it vanishes on-shell; such finite terms, which are included in eq. (53) 333These finite terms can also be read in eqs. (3.14,3.16,3.17,4.19) of Foffa et al. (2019)., have been consequently added with opposite sign into ${\cal L}^{(3PN)}_{fin}+{\cal L}^{(4PN)}_{fin}$ : the result has been organized in this way for ease of comparison with the literature, and we are now going to analyze the various contributions separately.

IV.1 ${\cal L}_{poles}$

The term containing ${\cal L}_{poles}$ has both UV and IR divergences. The former are expected to warn about failure to describe short distance features of the model, however for non-spinning bodies short-distance departures from the point particle limit enter only at 5PN order, as demonstrated by the effacement principle Damour (1982), and all UV divergences met in the present work must drop out of physical observables.

On the other hand the artificial splitting in near and far zone involving gravitational modes with hard and soft momenta introduces in the near-zone computation spurious IR divergences, some of which are contributing to ${\cal L}_{poles}$, while the remaining IR divergences are canceled by the addition of ${\cal L}_{tail}$.

However all the terms lumped together in ${\cal L}_{poles}$ do not affect physical observables (like the energy-frequency relation, or the periastron advance) because, as can be see from their explicit form reported in (53), ${\cal L}_{poles}$ is proportional to the EOMs, including ${\cal O}(d-3)$ terms and 1PN corrections to the leading order EOM when necessary.

In the Fokker action treatment of Bernard et al. (2016) this feature has been exploited to remove such poles by means of a coordinate shift. As ${\cal L}_{poles}$ contains both UV and IR divergences, removing them via a coordinate shift, although providing the correct result, is not well motivated from the point of view of field theory, where UV divergences should be removed by counterterms and all IR divergences should be treated together, irrespectively of the fact that they vanish on-shell (like the ones in ${\cal L}_{poles}$) or not (like the one canceled by ${\cal L}_{tail}$). A fully field-theory-motivated, as well as self contained and ambiguity-free treatment of such matter, is beyond the scope of this work and is the subject of Foffa et al. (2019).

IV.2 ${\cal L}^{(3PN)}_{fin}+{\cal L}^{(4PN)}_{fin}$

The two finite terms ${\cal L}^{(3PN)}_{fin}$ and ${\cal L}^{(4PN)}_{fin}$ include

• •

the finite part of the result of diagram computations

• •

the addition of double zeroes to get rid of higher derivatives

• •

the finite part of the instantaneous tail term

• •

the opposite of finite pieces appearing in ${\cal L}_{poles}/(d-3)$.

For the 3PN part one gets

[TABLE]

As to the 4PN part, it can be decomposed into

[TABLE]

where the finite terms ${\cal L}^{(4PN)}_{G}+{\cal L}^{(4PN)}_{G^{2}}$ are have already been computed and published, see eqs.(13,26) in Foffa and Sturani (2013a), and the others read:

[TABLE]

[TABLE]

[TABLE]

${\cal L}^{(3PN)}_{fin}+{\cal L}^{(4PN)}_{fin}$ can be compared directly with the result published in app. A of Bernard et al. (2018) and we find that they differ by a term proportional to the equations of motion:

[TABLE]

with

[TABLE]

and $\vec{Eq}^{0}_{1}$ is defined in (C).

We have thus proven that the regularized Lagrangian derived within the EFT framework is fully consistent with the results obtained by other groups; ${\cal L}^{3PN}_{fin}+{\cal L}^{4PN}_{fin}$ can thus de facto be used as the finite Lagrangian in that it provides the correct dynamics for physical observables, like energy of circular orbits and periastron precession.

V Conclusions

We have derived the last missing ingredients and completed the calculation of the regularized conservative, local-in-time Lagrangian at fourth post-Newtonian order with effective field theory methods.

We have also shown that our results for the $G^{3}$ and $G^{4}$ sectors are consistent with the literature since following the manipulations presented in Marchand et al. (2018) we arrive at the same answer, also consistent with Damour et al. (2014). However, we remark that the treatments presently available in literature, while providing the correct physical answer, still present delicate aspects related to the need of external inputs like the matching to self-force computations as in the ADM approach Bini and Damour (2013), or to the use of a coordinate shift to remove both UV and (some of the) IR divergences, as in the Fokker action one Bernard et al. (2016). The companion paper Foffa et al. (2019) is devoted to the concurrent resolution/clarification of both these issues.

Acknowledgments

The authors wish to thank Rafael Porto and Ira Rothstein for fruitful exchanges, which eventually led to the project reported in Foffa et al. (2019). We wish also to thank ICTP-SAIFR, supported by FAPESP grant 2016/01343-7, for the organization of the workshop “Analytic methods in General Relativity”, where many stimulating discussions took place, as well as the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work. The work of RS has been supported for part of the duration of the present work by the FAPESP grant n. 2012/14132-3 and wishes to thank the University of Geneva Physics Department for hospitality and support during his frequent visits. RS acknowledges the High Performance Computing Center at UFRN. SF is supported by the Fonds National Suisse and by the SwissMap NCCR and thanks the International Institute of Physics of UFRN in Natal for kind hospitality and support during his visit.

Appendix A Fundamental integrals

[TABLE]

[TABLE]

Appendix B Tail terms

In terms of relative variables the instantaneous part (that is, the terms not involving $k_{0}$) of eq.(IV) can be rewritten as

[TABLE]

The non-instantaneous piece becomes also instantaneous in the case of circular orbit and its contribution is necessary to recover the correct logarithmic part of the well known result of the energy of circular orbits.

Appendix C Explicit expression of ${\cal L}_{poles}$

The equations of motion up to order $\varepsilon\equiv d-3$ are

[TABLE]

with

[TABLE]

[TABLE]

and analogue expressions for $\vec{Eq}_{2}^{0,\varepsilon}$.

If we define

[TABLE]

one can rewrite $\hat{\cal L}^{poles}$ as a term proportional to the equations of motion at ${\cal O}(\varepsilon)$, which is the needed order as ${\cal L}^{poles}$ appears with a prefactor $\varepsilon^{-1}$ in eq.(35)

[TABLE]

Notice that only the last three lines contain ${\cal O}(\varepsilon)$ terms (and thus a non-vanishing finite contribution to the regularized Lagrangian), whereas in the previous lines $\vec{Eq}_{1,2}$ and $\vec{K}_{1,2}$ are combined to make the ${\cal O}(\varepsilon)$ contribution vanish.

Appendix D Results of double zero tricks

In deriving the previous results, the following double zero tricks have been used:

D.1 From 2PN

[TABLE]

D.2 From 3PN

[TABLE]

with

[TABLE]

D.3 From 4PN

[TABLE]

[TABLE]

with

[TABLE]

and finally

[TABLE]

with

[TABLE]