72
Topological defects
which is broken by the VEV of~. Using cylindrical polar coordinates, we look
for solutions for ~ of the form (3.15), as before, but we must now also determine
AI" For large p, with the boundary condition (3.19) so that ~ approaches one of
its minima,
~ ..... Tle intJ
as p ~ 00.
(3.28)
If we also arrange that AI' has the boundary condition
A" ..... -ie- I 01' In(~/TI)
asp ~ 00
(3.29)
then D,,~ and F",IJ both approach zero for large values of p and the energy density
vanishes for large p. Indeed, when the complete solution with these boundary
conditions is constructed numerically, in a way that we shall discuss shortly, the
energy density approaches zero fast enough as p ~ 00 that this cosmic string has
finite energy density per unit length.
The local cosmic string or gauge string carries magnetic flux. The amount
of flux may be determined by integrating over the area of a circle of large radius
R in the (p, 0) plane with the asymptotic form (3.29). Then (exercise 5)
f B . dS = fA. dl = 211' ne- l •
(3.30)
Thus, the local cosmic string characterized by winding number n carries n units
of magnetic flux ae- I . Local cosmic strings are, therefore, quantized tubes of
magnetic flux analogous to flux lines in a superconductor.
To construct the required (static) solution for ~ and AI" we take
~ = Tle ill9 f(p)
(3.31)
as for the global cosmic string. Then, since
O~
I
-p+--6+-1c
o~ ~
O~A
V~=
(3.32)
(Jp
p iJe
iJz
the boundary condition (3.29) suggests that we should take A" to have non-zero
components only in the p and iJ directions. Working in a gauge in which the
component Ap = 0, we take
=
n
~
A
-a(p)6.
(3.33)
ep
The functions a (p) and f (p) are then determined numerically by solving the field
equations
DI'D"~ + l.(~.~ - Tl2)~ = 0
(3.34)
oIJF"IJ + ie(~·D"~ - ~(D"~)·) = 0
(3.35)
subject to the boundary conditions (3.28) and (3.29).