Consider the class of functions which are smooth () on except for the
two points and , around which they admit a power-like singular expansion of the
type^{22}

The second notion of Hadamard partie finie () concerns that of the integral , which is generically divergent at the location of the two singular points and (we assume that the integral converges at infinity). It is defined by

The first term integrates over a domain defined as from which the two spherical balls and of radius and centered on the two singularities, denoted and , are excised: . The other terms, where the value of a function at point 1 takes the meaning (122) are such that they cancel out the divergent part of the first term in the limit where (the symbol means the same terms but corresponding to the other point 2). The Hadamard partie-finie integral depends on two strictly positive constants and , associated with the logarithms present in Equation (124). These constants will ultimately yield some gauge-type constants, denoted by and , in the 3PN equations of motion and radiation field. See Ref. [36] for alternative expressions of the partie-finie integral.We now come to a specific variant of Hadamard’s regularization called the extended Hadamard regularization and defined in Refs. [36, 39]. The basic idea is to associate to any a pseudo-function, called the partie finie pseudo-function , namely a linear form acting on functions of , and which is defined by the duality bracket

When restricted to the set of smooth functions (i.e. ) with compact support (obviously we have ), the pseudo-function is a distribution in the sense of Schwartz [199]. The product of pseudo-functions coincides, by definition, with the ordinary pointwise product, namely . In practical computations, we use an interesting pseudo-function, constructed on the basis of the Riesz delta function [190], which plays a role analogous to the Dirac measure in distribution theory, . This is the so-called delta-pseudo-function defined by where is the partie finie of as given by Equation (122). From the product of with any we obtain the new pseudo-function , that is such that As a general rule, we are not allowed, in consequence of the “non-distributivity” of the Hadamard partie finie, Equation (123), to replace within the pseudo-function by its regularized value: in general. It should be noticed that the object has no equivalent in distribution theory.Next, we treat the spatial derivative of a pseudo-function of the type , namely . Essentially, we require (in Ref. [36]) that the so-called rule of integration by parts holds. By this we mean that we are allowed to freely operate by parts any duality bracket, with the all-integrated (“surface”) terms always zero, as in the case of non-singular functions. This requirement is motivated by our will that a computation involving singular functions be as much as possible the same as if we were dealing with regular functions. Thus, by definition,

Furthermore, we assume that when all the singular coefficients of vanish, the derivative of reduces to the ordinary derivative, i.e. . Then it is trivial to check that the rule (128) contains as a particular case the standard definition of the distributional derivative [199]. Notably, we see that the integral of a gradient is always zero: . This should certainly be the case if we want to compute a quantity (e.g., a Hamiltonian density) which is defined only modulo a total divergence. We pose where represents the “ordinary” derivative and the distributional term. The following solution of the basic relation (128) was obtained in Ref. [36]: where for simplicity we assume that the powers in the expansion (121) of are relative integers. The distributional term (130) is of the form (plus ). It is generated solely by the singular coefficients of (the sum over in Equation (130) is always finite since there is a maximal order of divergency in Equation (121)). The formula for the distributional term associated with the th distributional derivative, i.e. , where , reads We refer to Theorem 4 in Ref. [36] for the definition of another derivative operator, representing the most general derivative satisfying the same properties as the one defined by Equation (130), and, in addition, the commutation of successive derivatives (or Schwarz lemma)The distributional derivative (129, 130, 131) does not satisfy the Leibniz rule for the derivation of a product, in accordance with a general result of Schwartz [198]. Rather, the investigation [36] suggests that, in order to construct a consistent theory (using the “ordinary” product for pseudo-functions), the Leibniz rule should be weakened, and replaced by the rule of integration by part, Equation (128), which is in fact nothing but an “integrated” version of the Leibniz rule. However, the loss of the Leibniz rule stricto sensu constitutes one of the reasons for the appearance of the ambiguity parameters at 3PN order.

The Hadamard regularization is defined by Equation (122) in a preferred spatial hypersurface of a coordinate system, and consequently is not a priori compatible with the Lorentz invariance. Thus we expect that the equations of motion in harmonic coordinates (which manifestly preserve the global Lorentz invariance) should exhibit at some stage a violation of the Lorentz invariance due to the latter regularization. In fact this occurs exactly at the 3PN order. Up to the 2.5PN level, the use of the regularization is sufficient to get some unambiguous equations of motion which are Lorentz invariant [42]. To deal with the problem at 3PN order, a Lorentz-invariant variant of the regularization, denoted , was introduced in Ref. [39]. It consists of performing the Hadamard regularization within the spatial hypersurface that is geometrically orthogonal (in a Minkowskian sense) to the four-velocity of the particle. The regularization differs from the simpler regularization by relativistic corrections of order at least. See Ref. [39] for the formulas defining this regularization in the form of some infinite power series in . The regularization plays a crucial role in obtaining the equations of motion at the 3PN order in Refs. [37, 38]. In particular, the use of the Lorentz-invariant regularization permits to obtain the value of the ambiguity parameter in Equation (132) below.

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