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FIG. 1. Branch cuts of the FH m-spoke model. m =4. Solid lines, spokes (u =0, 6=2ka/m); dashed lines, ghost spokes [u =2u9, 6=(2k +1)7/m]. du /d¢ is discontinuous across any branch cut. it fulfills undesired boundary conditions on m addi- tional “ghost” spokes {(7,¢)|r Ry, 6=(2k +1)r/m; k =0,1,...,m—1}, on which u(r,6)=2uy. Subtract- ing ug to normalize the potential to zero at infinity, we realize that the 2m spokes carry positive and negative charges by turns. Figure 1 displays the eight spokes of the case m =4.

Figure 1 Branch cuts of the FH m-spoke model. m =4. Solid lines, spokes (u =0, 6=2ka/m); dashed lines, ghost spokes [u =2u9, 6=(2k +1)7/m]. du /d¢ is discontinuous across any branch cut. it fulfills undesired boundary conditions on m addi- tional “ghost” spokes {(7,¢)|r Ry, 6=(2k +1)r/m; k =0,1,...,m—1}, on which u(r,6)=2uy. Subtract- ing ug to normalize the potential to zero at infinity, we realize that the 2m spokes carry positive and negative charges by turns. Figure 1 displays the eight spokes of the case m =4.

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FIG. 2. Branch cuts of the present m-spoke model. m =5. Solid lines, essential branch cuts produced by the square root (Z,, =0, 6=2k7/m); dz,,/d¢ is discontinuous. Dashed lines: nonessential cuts produced by the logarithm [¢=(2k + 1)7/m]; di, /O¢ is continuous across these and AZ,, =0 on them. on them and exhibits the correct logarithmic behavior as rs>e®_=8
FIG. 2. Branch cuts of the present m-spoke model. m =5. Solid lines, essential branch cuts produced by the square root (Z,, =0, 6=2k7/m); dz,,/d¢ is discontinuous. Dashed lines: nonessential cuts produced by the logarithm [¢=(2k + 1)7/m]; di, /O¢ is continuous across these and AZ,, =0 on them. on them and exhibits the correct logarithmic behavior as rs>e®_=8
Now the lattice cutoff has been reintroduced at the upper bound to keep the integral convergent for all (non- negative) values of 7. For 7 <2, it is immaterial and the upper bound can be set equal to 1. In this case, the in- tegral can be evaluated exactly, whereas for larger 7 its behavior is determined by the cutoff because of the strong divergence of the integrand. The result of the calculation
Now the lattice cutoff has been reintroduced at the upper bound to keep the integral convergent for all (non- negative) values of 7. For 7 <2, it is immaterial and the upper bound can be set equal to 1. In this case, the in- tegral can be evaluated exactly, whereas for larger 7 its behavior is determined by the cutoff because of the strong divergence of the integrand. The result of the calculation
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