where p2 = X2 + Y2 + Z2 and 7 represents a length scale of the background
geometry. Following Refs. [17] and [18], the regularization parameters can be
determined from evaluating the multiple components of
Va,) = V,(q/p) + O(p2/R4), (4-146)
where a labels the Schwarzschild background coordinates x" = (t, r, 0, 0). The
remainder O(p2/R3) in the above approximation is disregarded since it gives no
contribution to V,s as we take the "coincidence limit", x x' where x"
denotes a point in the vicinity of the particle and x' the location of the particle in
the Schwarzschild background.
In evaluating the multiple components of Vabs via Eqs. (4-146) and (4-144),
singularities are expected to occur with certain terms. To help identify those
singularities, we introduce an order parameter c which is to be set to unity at the
end of the calculation: we attach e to each n th order part of p2 in Eq. (4-144)
and re-express p2 as
p2 2, II + 3,PI + c4Piv + C5PV + 0(6), (4-147)
where PII, PIII, PIV and Pv represent the quadratic, cubic, quartic and quintic
order parts of p2, respectively.
We express 0, (1/p) in a Laurent series expansion, and every denominator
of this expansion takes the form of P3/2 (n = 3, 5, 7, 9, ... ). Thus, P IIl 1 i, an
important role in the multiple decomposition, but the quadratic part PII, directly
taken from Eq. (4-144), is not yet fully ready for this task. First, 4 0o must be
decoupled from r ro so that each appears only as an independent complete square.
Coupling between t to and 4 (o does not create difficulty in the decomposition.