Dressing the post-Newtonian two-body problem and classical effective field theory

We apply a dressed perturbation theory to better organize and economize the computation of high orders of the 2-body effective action of an inspiralling post-Newtonian (PN) gravitating binary. We use the effective field theory approach with the nonrelativistic field decomposition (NRG fields). For that purpose we develop quite generally the dressing theory of a nonlinear classical field theory coupled to pointlike sources. We introduce dressed charges and propagators, but unlike the quantum theory there are no dressed bulk vertices. The dressed quantities are found to obey recursive integral equations which succinctly encode parts of the diagrammatic expansion, and are the classical version of the Schwinger-Dyson equations. Actually, the classical equations are somewhat stronger since they involve only finitely many quantities, unlike the quantum theory. Classical diagrams are shown to factorize exactly when they contain nonlinear worldline vertices, and we classify all the possible topologies of irreducible diagrams for low loop numbers. We apply the dressing program to our post-Newtonian case of interest. The dressed charges consist of the dressed energy-momentum tensor after a nonrelativistic decomposition, and we compute all dressed charges (in the harmonic gauge) appearing up to 2PN in the 2-body effective action (and more). We determine the irreducible skeleton diagrams up to 3PN and we employ the dressed charges to compute several terms beyond 2PN.

Figure 1

The diagram which represents the Newtonian-potential interaction mediated through the Newtonian scalar field $\varphi$. The notations will be fully defined later in Sec. 3.

Figure 2

A class of diagrams that we interpret as the Newtonian-potential interaction between dressed energy distributions of each body through a dressed propagator. The dark blobs represent any subdiagram with an arbitrary number of vertices on the worldline and a single external leg for the Newtonian potential. There are no bulk loops, as always in classical physics. The light blob represents any subdiagram with two external legs of the Newtonian potential, which amounts to a propagator with an arbitrary number of retardation insertions.

Figure 3

The diagrammatic definition of the dressed energy distribution (top) and the dressed propagator (bottom). The dressed energy distribution is defined through the one point function for $\varphi$ in the presence of a single source (after stripping external propagators), while the dressed propagator is defined through the full two point function for $\varphi$.

Figure 4

A schematic diagrammatic representation of the Schwinger-Dyson recursive integral equation satisfied by the dressed couplings. More details are given in the body of the paper.

Figure 5

The Feynman rules involving $\varphi$ for the static scalar field theory whose action is given by (2.3). The rules describe: the propagator and the quadratic perturbation vertex—retardation (top), the cubic bulk vertex (middle), and the worldline vertices (bottom) where the ellipsis stands for additional nonlinear worldline vertices. Hereafter $k$ denotes a spatial wave number.

Figure 6

An example of a factorizable diagram (the simplest). The two factors are circled (in red) and they intersect in a NL-WL vertex. A diagram of this type appears at 1PN—see Fig. 23.

Figure 7

The diagrammatic definition of the dressed charge distribution (top) as the one point function for $\varphi$ in the presence of a single source and the dressed propagator (bottom) as the full two point function for $\varphi$ in vacuum. Retardation vertices are forbidden on the external leg of the dressed charge.

Figure 8

The diagrammatic representation for the value of the field in the presence of the point-particle source. The only difference with the definition of the dressed charge in Fig. 7 (top) is an added propagator on the external leg.

Figure 9

An example for the correspondence between diagrams of the bare theory, such as (a) and the corresponding dressed diagram (b). Note a standard subtlety: retardation insertions are allowed inside a dressed charge vertex, but not on its external leg. See text for further discussion.

Figure 10

The dressed Feynman rules contain changes relative to the bare Feynman rules in Fig. 5. The bare worldline vertex (charge) Feynman rule changes into the dressed one, the bare propagator changes into the dressed one, and the retardation rule is omitted. The rest of the rules remain unchanged.

Figure 11

An overcounting problem with an alternative definition of the (nonstatic) dressed perturbation theory, as discussed in the text.

Figure 12

The diagrammatic representation of the recursive integral equation satisfied by the dressed charge in the static limit.

Figure 13

A diagrammatic representation of the full set of (the two) recursive equations satisfied by the dressed charge (top) and propagator (bottom) in the general, nonstatic case. Note a change in the recursive relation for the dressed charge relative to the static equation in Fig. 12.

Figure 14

The top line shows a possible definition for a 2-field dressed worldline vertex. The second line demonstrates the overcounting problem which occurs. The bare diagram on the left can be cut into subdiagrams in two different ways denoted (a) and (b).

Figure 15

Classification of topologies for irreducible diagrams of the 2-body effective action, both nonfactorizable and dressing-irreducible. The top line shows the possibilities at 2-loop, and the bottom line shows the possibilities at 3-loop.

Figure 16

Feynman rules obtained from the expansion of (3.4) up to linear order in $\varphi$, $A_i$ and ${\sigma }_{ij}$. The undetermined wave numbers flow into the vertex.

Figure 17

Static bulk vertices obtained from the expansion of the Hilbert-Einstein action (3.3) and gauge fixing term (3.6) but restricting to time-independent terms. The undetermined wave numbers flow into the vertex.

Figure 18

Time-dependent bulk vertices obtained from the expansion of the Hilbert-Einstein action (3.3) and gauge fixing term (3.6). Time derivative above the propagator indicates the direction on which it acts. The undetermined wave numbers flow into the vertex.

Figure 19

The diagrammatic definition of the dressed charge distributions in PN $\rho$, $j^i$ and $s^{ij}$ as the one point function for $\varphi$, $A_i$ and ${\sigma }_{ij}$ respectively in the presence of a single source. Retardation vertices are forbidden from the external leg.

Figure 20

Feynman diagrams which contribute to the dressed energy distribution up to 2PN. Diagram (a) is leading (order 0PN) while the rest are 2PN.

Figure 21

Feynman diagrams which contribute to the dressed momentum distribution up to 1.5PN. Diagram (a) is leading (order 0.5PN) while the rest are $+1\mathrm{PN}$ (1.5PN).

Figure 22

Feynman diagrams which contribute to the dressed stress tensor distribution up to 2PN. Diagrams (a),(b) are leading (order 1PN) while the rest are $+1\mathrm{PN}$ (2PN).

Figure 23

The four diagrams which contribute to the two-body effective action at 1PN—the Einstein-Infeld-Hoffmann Lagrangian.

Figure 24

2PN Diagrams contributing at order $\mathcal{O}(G{m}^2{v}^4)$ (following 9).

Figure 25

2PN Diagrams contributing at order $\mathcal{O}(G^2{m}^3{v}^2)$ (following 9).

Figure 26

2PN Diagrams contributing at order $\mathcal{O}(G^3{m}^4{v}^0)$ (following 9).

Figure 27

A listing of all nonfactorizable skeletons appearing in the dressed post-Newtonian perturbation theory up to 3PN, listed by the PN order at which they first appear.

Figure 28

The diagrammatic representation of the recursive integral equation satisfied by the dressed energy.

Figure 29

The diagrammatic representation of terms beyond 2PN in the 2-body effective action: (a) a 3PN $\rho \mathrm{\text{−}}\rho$ term, (b) a 3PN $j\mathrm{\text{−}}j$ term, (c) a 3PN $s\mathrm{\text{−}}s$ term and finally (d) a 4PN $\rho \mathrm{\text{−}}\rho$ term.