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8.4 Dimensional regularization of the radiation field

We now address the similar problem concerning the binary’s radiation field (3PN beyond the Einstein quadrupole formalism), for which three ambiguity parameters, q, k, z, have been shown to appear [45Jump To The Next Citation Point44Jump To The Next Citation Point] (see Section 8.2).

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

{ (d) d - 1 integral d 4(d + 2l- 2) (1) IL (t) = --------F P d x ^xL S (x, t)- -2------------------ ^xaL S a (x,t) 2(d - 2) [l] c (d + l- 2)(d + 2l) [l+1] } -----------2(d-+-2l--2)----------- (2) + c4(d + l- 1)(d + l- 2)(d + 2l + 2)^xabL [Sl+2]ab(x, t) , (158)
where we denote (generalizing Equations (86View Equation))
2 (d - 2)t00 + tii S = --------------2------, d-- 1 c t-0i (159) Si = c , -ij Sij = t ,
and where for any source densities the underscript [l] means the infinite series
+ oo ( ) ( )2k sum --1----G--d2-+-l--- |x|-@- S[l](x,t) = 22kk!G(d- + l + k) c @t S(x,t). (160) k=0 2
The latter definition represents the d-dimensional version of the post-Newtonian expansion series (91View Equation). At Newtonian order, Equation (158View Equation) reduces to the standard result (d) integral d - 2 IL = d x rx^L + O(c ) with 00 2 r = T /c.

The ambiguity parameters q, k, and z come from the Hadamard regularization of the mass quadrupole moment Iij at the 3PN order. The terms corresponding to these ambiguities were found to be

2 3 [( ) ] DIij[q, k,z] = 44-GN-m-1- q + k m1-+-m2- y<iaj>+ z v <ivj> + 1 <--> 2, (161) 3 c6 m1 1 1 1 1
where y1, v1, and a1 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 [44Jump To The Next Citation Point] is therefore to relate the Hadamard-regularized quadrupole moment I(iHjR), for general orbits, to its pHS part:
[ ] (HR) (pHS) 1-- -9-- Iij = Iij + DIij q + 22,k, z + 110 . (162)
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. [45Jump To The Next Citation Point] and the pHS one. The pHS part is free of ambiguities but depends on the gauge constants ' r1 and ' r2 introduced in the harmonic-coordinates equations of motion [37Jump To The Next Citation Point38Jump To The Next Citation Point].

We next use the d-dimensional moment (158View Equation) to compute the difference between the dimensional regularization (DR) result and the pHS one [31Jump To The Next Citation Point32Jump To The Next Citation Point]. 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. r1 = |x - y1|--> 0 or r2 = |x - y2|--> 0) that give rise to poles oc 1/e, corresponding to logarithmic ultra-violet (UV) divergences in 3 dimensions. The infra-red (IR) region at infinity (i.e. |x|--> +o o) does not contribute to the difference DR - pHS. The compact-support terms in the integrand of Equation (158View Equation), proportional to the matter source densities s, sa, and sab, 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 (158View Equation) near the singularities that produce poles in d dimensions.

Let F(d)(x) be the non-compact part of the integrand of the quadrupole moment (158View Equation) (with indices L =_ ij), where F (d) includes the appropriate multipolar factors such as ^x ij, so that

integral (d) d (d) Iij = d xF (x). (163)
We do not indicate that we are considering here only the non-compact part of the moments. Near the singularities the function F (d)(x) admits a singular expansion of the type (143View Equation). In practice, the various coefficients f (e) 1 p,q are computed by specializing the general expressions of the non-linear retarded potentials ^ V,Va,Wab, ... (valid for general extended sources) to the point particles case in d dimensions. On the other hand, the analogue of Equation (163View Equation) in 3 dimensions is
integral I = Pf d3x F (x), (164) ij
where Pf refers to the Hadamard partie finie defined in Equation (124View Equation). The difference DI between the DR evaluation of the d-dimensional integral (163View Equation), and its corresponding three-dimensional evaluation, i.e. the partie finie (164View Equation), reads then
DIij = I(idj)- Iij. (165)
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 r1 --> 0 and r2 --> 0. We find that Equation (165View Equation) depends on two constant scales s1 and s2 coming from Hadamard’s partie finie (124View Equation), and on the constants belonging to dimensional regularization, which are e = d - 3 and the length scale l0 defined by Equation (139View Equation). 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 (165View Equation), namely
(DR) (pHS) Iij = Iij + DIij. (166)
An important fact, hidden in our too-compact notation (166View Equation), is that the sum of the two terms in the right-hand side of Equation (166View Equation) does not depend on the Hadamard regularization scales s1 and s2. Therefore it is possible without changing the sum to re-express these two terms (separately) by means of the constants r'1 and r'2 instead of s1 and s2, where r'1, r'2 are the two fiducial scales entering the Hadamard-regularization result (162View Equation). This replacement being made the pHS term in Equation (166View Equation) is exactly the same as the one in Equation (162View Equation). At this stage all elements are in place to prove the following theorem [31Jump To The Next Citation Point32Jump To The Next Citation Point]:

Theorem 10 The DR quadrupole moment (166View Equation) is physically equivalent to the Hadamard-regularized one (end result of Refs. [45Jump To The Next Citation Point, 44Jump To The Next Citation Point]), in the sense that

[ ] (HR) (DR) Iij = lie-->m0 Iij + dqIij , (167)
where dqIij denotes the effect of the same shifts as determined in Theorems 8 and 9, if and only if the HR ambiguity parameters q, k, and z take the unique values (136View Equation). Moreover, the poles 1/e separately present in the two terms in the brackets of Equation (167View Equation) cancel out, so that the physical (“dressed”) DR quadrupole moment is finite and given by the limit when e --> 0 as shown in Equation (167View Equation).

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 [96Jump To The Next Citation Point30Jump To The Next Citation Point133Jump To The Next Citation Point132Jump To The Next Citation Point], independent confirmations for the four ambiguity parameters c, q, k, and z, and confirm the consistency of dimensional regularization and its validity for describing the general-relativistic dynamics of compact bodies.


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