with
J2 sin2 (
X 1 ----- (4-156)
r + J2
and
r E22A2
62 = (4-157)
2f2(r + J2)2
Now we rewrite Eq. (4-147) by replacing the original quadratic part Pii with
/2
p2 p 22 + C3 I + 4PIV + 5V + 0(C6), (4-158)
where Piv now includes the additional quartic order terms that have resulted from
the replacement of PII by p2 through Eqs. (4-150) and (4-153). A Laurent series
expansion of 9 (1/p)|t=to is then
S1 a(2)t to -2 1 a IIlt=to 3 [aa (P2) IIItto -1
p t=to 2 2 p3 4 p5
1 a0aPiv t-t 3 [a+ (P2) IV t=t + (aPIIlI) -PII t=to
S2 p4 p5
15 [aa (p2)] I to
16 p0
1 .Pvt t=to
1 2 p
3 [a. (p2)] Pvl tto + (aaPIII) PIV t=to + (aIv) PIIIt= to
+4
4 p5
15 2 [a (p2)] PPiv lto + (aPjIII) 2II t
16 p
35 [ (p2)] 3 I tto
+I35 p1 + O(C2). (4-159)
32 p0
After the derivatives in Eq. (4-159) are taken, the dependence upon 0, 4 and
r may be removed in favor of po, X and 6 by use of Eqs. (4-155)-(4-157). Then
the three steps of (i) a Legendre polynomial expansion for the dependence,
while r and 4 are held fixed, followed by (ii) an integration over 4, while r is held
fixed, and finally (iii) taking the limit 6 -+ 0, together provide the regularization