To apply dimensional regularization, we must use as in Section 8.3 the -dimensional post-Newtonian iteration [leading to equations such as (142)]; and, crucially, we have to generalize to dimensions some key results of the wave generation formalism of Part A. Essentially we need the -dimensional analogues of the multipole moments of an isolated source and , Equations (85). The result we find in the case of the mass-type moments is

where we denote (generalizing Equations (86)) and where for any source densities the underscript means the infinite series The latter definition represents the -dimensional version of the post-Newtonian expansion series (91). At Newtonian order, Equation (158) reduces to the standard result with .The ambiguity parameters , , and come from the Hadamard regularization of the mass quadrupole moment at the 3PN order. The terms corresponding to these ambiguities were found to be

where , , and denote the first particle’s position, velocity, and acceleration. We recall that the brackets surrounding indices refer to the symmetric-trace-free (STF) projection. Like in Section 8.3, we express both the Hadamard and dimensional results in terms of the more basic pHS regularization. The first step of the calculation [44] is therefore to relate the Hadamard-regularized quadrupole moment , for general orbits, to its pHS part: In the right-hand side we find both the pHS part, and the effect of adding the ambiguities, with some numerical shifts of the ambiguity parameters coming from the difference between the specific Hadamard-type regularization scheme used in Ref. [45] and the pHS one. The pHS part is free of ambiguities but depends on the gauge constants and introduced in the harmonic-coordinates equations of motion [37, 38].We next use the -dimensional moment (158) to compute the difference between the dimensional regularization (DR) result and the pHS one [31, 32]. As in the work on equations of motion, we find that the ambiguities arise solely from the terms in the integration regions near the particles (i.e. or ) that give rise to poles , corresponding to logarithmic ultra-violet (UV) divergences in 3 dimensions. The infra-red (IR) region at infinity (i.e. ) does not contribute to the difference . The compact-support terms in the integrand of Equation (158), proportional to the matter source densities , , and , are also found not to contribute to the difference. We are therefore left with evaluating the difference linked with the computation of the non-compact terms in the expansion of the integrand in (158) near the singularities that produce poles in dimensions.

Let be the non-compact part of the integrand of the quadrupole moment (158) (with indices ), where includes the appropriate multipolar factors such as , so that

We do not indicate that we are considering here only the non-compact part of the moments. Near the singularities the function admits a singular expansion of the type (143). In practice, the various coefficients are computed by specializing the general expressions of the non-linear retarded potentials (valid for general extended sources) to the point particles case in dimensions. On the other hand, the analogue of Equation (163) in 3 dimensions is where refers to the Hadamard partie finie defined in Equation (124). The difference between the DR evaluation of the -dimensional integral (163), and its corresponding three-dimensional evaluation, i.e. the partie finie (164), reads then Such difference depends only on the UV behaviour of the integrands, and can therefore be computed “locally”, i.e. in the vicinity of the particles, when and . We find that Equation (165) depends on two constant scales and coming from Hadamard’s partie finie (124), and on the constants belonging to dimensional regularization, which are and the length scale defined by Equation (139). The dimensional regularization of the 3PN quadrupole moment is then obtained as the sum of the pHS part, and of the difference computed according to Equation (165), namely An important fact, hidden in our too-compact notation (166), is that the sum of the two terms in the right-hand side of Equation (166) does not depend on the Hadamard regularization scales and . Therefore it is possible without changing the sum to re-express these two terms (separately) by means of the constants and instead of and , where , are the two fiducial scales entering the Hadamard-regularization result (162). This replacement being made the pHS term in Equation (166) is exactly the same as the one in Equation (162). At this stage all elements are in place to prove the following theorem [31, 32]:Theorem 10 The DR quadrupole moment (166) is physically equivalent to the Hadamard-regularized one (end result of Refs. [45, 44]), in the sense that

where denotes the effect of the same shifts as determined in Theorems 8 and 9, if and only if the HR ambiguity parameters , , and take the unique values (136). Moreover, the poles separately present in the two terms in the brackets of Equation (167) cancel out, so that the physical (“dressed”) DR quadrupole moment is finite and given by the limit when as shown in Equation (167).This theorem finally provides an unambiguous determination of the 3PN radiation field by dimensional regularization. Furthermore, as reviewed in Section 8.2, several checks of this calculation could be done, which provide, together with comparisons with alternative methods [96, 30, 133, 132], independent confirmations for the four ambiguity parameters , , , and , and confirm the consistency of dimensional regularization and its validity for describing the general-relativistic dynamics of compact bodies.

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