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
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 is therefore to relate the Hadamard-regularized quadrupole moment , for general orbits, to its pHS part:  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[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 thatwhere 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.
© Max Planck Society and the author(s)