8.3 Dimensional regularization of the equations of motion

As reviewed in Section 8.2, work at 3PN order using Hadamard’s self-field regularization showed the appearance of ambiguity parameters, due to an incompleteness of the Hadamard regularization employed for curing the infinite self field of point particles. We give here more details on the determination using dimensional regularization of the ambiguity parameter which appeared in the 3PN equations of motion (recall that is equivalent to the static ambiguity parameter , see Equation (133)).

Dimensional regularization was invented as a means to preserve the gauge symmetry of perturbative quantum field theories [202515773]. Our basic problem here is to respect the gauge symmetry associated with the diffeomorphism invariance of the classical general relativistic description of interacting point masses. Hence, we use dimensional regularization not merely as a trick to compute some particular integrals which would otherwise be divergent, but as a powerful tool for solving in a consistent way the Einstein field equations with singular point-mass sources, while preserving its crucial symmetries. In particular, we shall prove that dimensional regularization determines the kinetic ambiguity parameter (and its radiation-field analogue ), and is therefore able to correctly keep track of the global Lorentz-Poincaré invariance of the gravitational field of isolated systems.

The Einstein field equations in space-time dimensions, relaxed by the condition of harmonic coordinates , take exactly the same form as given in Equations (9, 14). In particular denotes the flat space-time d’Alembertian operator in dimensions. The gravitational constant is related to the usual three-dimensional Newton’s constant by

where denotes an arbitrary length scale. The explicit expression of the gravitational source term involves some -dependent coefficients, and is given by
When we recover Equation (15). In the following we assume, as usual in dimensional regularization, that the dimension of space is a complex number, , and prove many results by invoking complex analytic continuation in . We shall pose .

We parametrize the 3PN metric in dimensions by means of straightforward -dimensional generalizations of the retarded potentials , , , , and of Section 7. Those are obtained by post-Newtonian iteration of the -dimensional field equations, starting from the following definitions of matter source densities

which generalize Equations (116). As a result all the expressions of Section 7 acquire some explicit -dependent coefficients. For instance we find [30]
Here means the retarded integral in space-time dimensions, which admits, though, no simple expression in physical space.

As reviewed in Section 8.1, the generic functions we have to deal with in 3 dimensions, say , are smooth on except at and , around which they admit singular Laurent-type expansions in powers and inverse powers of and , given by Equation (121). In spatial dimensions, there is an analogue of the function , which results from the post-Newtonian iteration process performed in dimensions as we just outlined. Let us call this function , where . When the function admits a singular expansion which is a little bit more complicated than in 3 dimensions, as it reads

The coefficients depend on , and the powers of involve the relative integers and whose values are limited by some , , and as indicated. Here we will be interested in functions which have no poles as (this will always be the case at 3PN order). Therefore, we can deduce from the fact that is continuous at the constraint

For the problem at hand, we essentially have to deal with the regularization of Poisson integrals, or iterated Poisson integrals (and their gradients needed in the equations of motion), of the generic function . The Poisson integral of , in dimensions, is given by the Green’s function for the Laplace operator,

where is a constant related to the usual Eulerian -function by
We need to evaluate the Poisson integral at the point where it is singular; this is quite easy in dimensional regularization, because the nice properties of analytic continuation allow simply to get by replacing by in the explicit integral form (145). So we simply have
It is not possible at present to compute the equations of motion in the general -dimensional case, but only in the limit where  [9630]. The main technical step of our strategy consists of computing, in the limit , the difference between the -dimensional Poisson potential (147), and its Hadamard 3-dimensional counterpart given by , where the Hadamard partie finie is defined by Equation (122). Actually, we must be very precise when defining the Hadamard partie finie of a Poisson integral. Indeed, the definition (122) stricto sensu is applicable when the expansion of the function , when , does not involve logarithms of ; see Equation (121). However, the Poisson integral of will typically involve such logarithms at the 3PN order, namely some where formally tends to zero (hence is formally infinite). The proper way to define the Hadamard partie finie in this case is to include the into its definition, so we arrive at [36]
The first term follows from Hadamard’s partie finie integral (124); the second one is given by Equation (122). Notice that in this result the constant entering the partie finie integral (124) has been “replaced” by , which plays the role of a new regularization constant (together with for the other particle), and which ultimately parametrizes the final Hadamard regularized 3PN equations of motion. It was shown that and are unphysical, in the sense that they can be removed by a coordinate transformation [3738]. On the other hand, the constant remaining in the result (148) is the source for the appearance of the physical ambiguity parameter , as it will be related to it by Equation (150). Denoting the difference between the dimensional and Hadamard regularizations by means of the script letter , we pose (for the result concerning the point 1)
That is, is what we shall have to add to the Hadamard-regularization result in order to get the -dimensional result. However, we shall only compute the first two terms of the Laurent expansion of when , say . This is the information we need to clear up the ambiguity parameter. We insist that the difference comes exclusively from the contribution of terms developing some poles in the -dimensional calculation.

Next we outline the way we obtain, starting from the computation of the “difference”, the 3PN equations of motion in dimensional regularization, and show how the ambiguity parameter is determined. By contrast to and which are pure gauge, is a genuine physical ambiguity, introduced in Refs. [3638] as the single unknown numerical constant parametrizing the ratio between and (where is the constant left in Equation (148)) as

where and are the two masses. The terms corresponding to the -ambiguity in the acceleration of particle 1 read simply
where the relative distance between particles is denoted (with being the unit vector pointing from particle 2 to particle 1). We start from the end result of Ref. [38] for the 3PN harmonic coordinates acceleration in Hadamard’s regularization, abbreviated as HR. Since the result was obtained by means of the specific extended variant of Hadamard’s regularization (in short EHR, see Section 8.1) we write it as
where is a fully determined functional of the masses and , the relative distance , the coordinate velocities and , and also the gauge constants and . The only ambiguous term is the second one and is given by Equation (151).

Our strategy is to express both the dimensional and Hadamard regularizations in terms of their common “core” part, obtained by applying the so-called “pure-Hadamard-Schwartz” (pHS) regularization. Following the definition of Ref. [30], the pHS regularization is a specific, minimal Hadamard-type regularization of integrals, based on the partie finie integral (124), together with a minimal treatment of “contact” terms, in which the definition (124) is applied separately to each of the elementary potentials (and gradients) that enter the post-Newtonian metric in the form given in Section 7. Furthermore, the regularization of a product of these potentials is assumed to be distributive, i.e.  in the case where and are given by such elementary potentials (this is in contrast with Equation (123)). The pHS regularization also assumes the use of standard Schwartz distributional derivatives [199]. The interest of the pHS regularization is that the dimensional regularization is equal to it plus the “difference”; see Equation (155).

To obtain the pHS-regularized acceleration we need to substract from the EHR result a series of contributions, which are specific consequences of the use of EHR [3639]. For instance, one of these contributions corresponds to the fact that in the EHR the distributional derivative is given by Equations (129, 130) which differs from the Schwartz distributional derivative in the pHS regularization. Hence we define

where the ’s denote the extra terms following from the EHR prescriptions. The pHS-regularized acceleration (153) constitutes essentially the result of the first stage of the calculation of , as reported in Ref. [109].

The next step consists of evaluating the Laurent expansion, in powers of , of the difference between the dimensional regularization and the pHS (3-dimensional) computation. As we reviewed above, this difference makes a contribution only when a term generates a pole , in which case the dimensional regularization adds an extra contribution, made of the pole and the finite part associated with the pole (we consistently neglect all terms ). One must then be especially wary of combinations of terms whose pole parts finally cancel (“cancelled poles”) but whose dimensionally regularized finite parts generally do not, and must be evaluated with care. We denote the above defined difference by

It is made of the sum of all the individual differences of Poisson or Poisson-like integrals as computed in Equation (149). The total difference (154) depends on the Hadamard regularization scales and (or equivalently on and , ), and on the parameters associated with dimensional regularization, namely and the characteristic length scale introduced in Equation (139). Finally, our main result is the explicit computation of the -expansion of the dimensional regularization (DR) acceleration as
With this result we can prove two theorems [30]:

Theorem 8 The pole part of the DR acceleration (155) can be re-absorbed (i.e. renormalized) into some shifts of the two “bare” world-lines: and , with, say, , so that the result, expressed in terms of the “dressed” quantities, is finite when .

The situation in harmonic coordinates is to be contrasted with the calculation in ADM-type coordinates within the Hamiltonian formalism, where it was shown that all pole parts directly cancel out in the total 3PN Hamiltonian: No renormalization of the world-lines is needed [96]. A central result is then as follows:

Theorem 9 The renormalized (finite) DR acceleration is physically equivalent to the Hadamard-regularized (HR) acceleration (end result of Ref. [38]), in the sense that

where denotes the effect of the shifts on the acceleration, if and only if the HR ambiguity parameter entering the harmonic-coordinates equations of motion takes the unique value (135).

The precise shifts and needed in Theorem 9 involve not only a pole contribution (which would define a renormalization by minimal subtraction (MS)), but also a finite contribution when . Their explicit expressions read:

where is Newton’s constant, is the characteristic length scale of dimensional regularization (cf. Equation (139)), is the Newtonian acceleration of the particle 1 in dimensions, and depends on Euler’s constant .