( 1 A 2 2Da
 lim as + A + L + B, + C) + 0(-4), (4-33)
 p,-p' 2 (2f- 1)(2f + 3)

where the first two terms look just identical to those in Eq. (4-28), but the third

term for Ca looks different from its counterpart. Our regularization parameters are

classified in terms of singularity and differentiability of FS, in the limit p p',

namely into e-2-order, e-l-order, co-order, el-order terms, etc. (see Section 4.4),

and all the f-dependences associated with them, as seen in Eq. (4-33), naturally re-

sult from the multiple decomposition of JS, via Legendre polynomial expansions.

We will see later in Section 4.4 that our co-order term has no clear dependence

on But in Barack and Ori [9] L + 1 is introduced as a perturbation factor

and limp_,p/ J~~F is expanded as a power series in L, in which the third term gains

L in its denominator as shown in Eq. (4-28). This discrepancy between the two

approaches, however, is resolved by the fact that Ca vanishes alv--iv. We will prove

this in Section 4.4. Also, one should note that our last parameter Da is defined dif-

ferently from D' in Eq. (4-29) (note the difference in notation). Our Da originates

from the non-singular but non-differentiable behaviors of the field in the neighbor-

hood of the world-line of the particle. Again, its coefficient -2-/2/(2L 1)(2L + 3)

results from the Legendre polynomial multiple expansions. The use of this param-

eter in the mode sum calculation will result in more rapid convergence and more

accurate final outcome.

 In Section 4.4 we present in detail the derivations of all these regularization

parameters. The results are summarized as below:


 At sgn(A) 2 A (434)

 q2 E
 A, -sgn(A) fA (4-35)