Local cosmic strings
73
There are now two length scales in the problem instead of one as in the case
of the global string. There is the mass m41 of the scalar field after spontaneous
spontaneous symmetry breaking which is obtained by substituting (t/J) in V (t/J).
This gives
m~ = ,,2)..
(3.36)
It is m41 that controls the rate of variation of t/J for large P and. therefore. the
variation of !(p). There is also the mass mA of the gauge field after spontaneous
symmetry breaking which is obtained by substituting (t/J) in the (D",t/J)*(D"'t/J)
term. This gives
m~ = e 2 ,,2.
(3.37)
It is m A that controls the rate of variation of A for large p and. therefore. the
variation of a(p). The approximate solution is found to be of the form
!....., I -l1r 1 / 2 exp (- ':::)
as ~ -+ 00
(3.38)
a'" I - al~1/2exp(_~)
as ~ -+ 00
(3.39)
where 11 and al are constants and
(3.40)
~ == mAP·
It can be seen from (3.38) and (3.39) that t/J is localized on a scale m; 1 and A
is localized on a scale m AI. As a consequence. the energy density is localized
without introducing a cut-off.
The energy per unit length of a local cosmic string may be estimated as
follows. The cosmic string has an inner core where t/J is approximately zero (i.e.
a core of false vacuum) with radius
-I
.-1/2-1
R
(3.41)
41 ~ m41 = A
"
and a tube of magnetic flux of radius
R ....., m-I - e-1n-1
(3.42)
A -
A -
. , .
The energy density obtained by putting t/J = 0 in V (t/J) is !).,,4. Thus. there is an
energy density per unit length from the inner core of the cosmic string of order
! ).,,47r R~ '" ,,2. There is also an energy per unit length from the tube of magnetic
flux of order 8 2 R~ where 8 is the magnitude of the magnetic field strength. From
(3.30). with one unit of flux. we estimate
8 ....., R-2 -I
A e .
(3.43)
Thus. using (3.42). we find that 8 2 R~ ....., ,,2 and both the magnetic and inner-core
contributions to the energy per unit length JL are of the same order. The total has
order of magnitude
JL ....., ,,2.
(3.44)
73
There are now two length scales in the problem instead of one as in the case
of the global string. There is the mass m41 of the scalar field after spontaneous
spontaneous symmetry breaking which is obtained by substituting (t/J) in V (t/J).
This gives
m~ = ,,2)..
(3.36)
It is m41 that controls the rate of variation of t/J for large P and. therefore. the
variation of !(p). There is also the mass mA of the gauge field after spontaneous
symmetry breaking which is obtained by substituting (t/J) in the (D",t/J)*(D"'t/J)
term. This gives
m~ = e 2 ,,2.
(3.37)
It is m A that controls the rate of variation of A for large p and. therefore. the
variation of a(p). The approximate solution is found to be of the form
!....., I -l1r 1 / 2 exp (- ':::)
as ~ -+ 00
(3.38)
a'" I - al~1/2exp(_~)
as ~ -+ 00
(3.39)
where 11 and al are constants and
(3.40)
~ == mAP·
It can be seen from (3.38) and (3.39) that t/J is localized on a scale m; 1 and A
is localized on a scale m AI. As a consequence. the energy density is localized
without introducing a cut-off.
The energy per unit length of a local cosmic string may be estimated as
follows. The cosmic string has an inner core where t/J is approximately zero (i.e.
a core of false vacuum) with radius
-I
.-1/2-1
R
(3.41)
41 ~ m41 = A
"
and a tube of magnetic flux of radius
R ....., m-I - e-1n-1
(3.42)
A -
A -
. , .
The energy density obtained by putting t/J = 0 in V (t/J) is !).,,4. Thus. there is an
energy density per unit length from the inner core of the cosmic string of order
! ).,,47r R~ '" ,,2. There is also an energy per unit length from the tube of magnetic
flux of order 8 2 R~ where 8 is the magnitude of the magnetic field strength. From
(3.30). with one unit of flux. we estimate
8 ....., R-2 -I
A e .
(3.43)
Thus. using (3.42). we find that 8 2 R~ ....., ,,2 and both the magnetic and inner-core
contributions to the energy per unit length JL are of the same order. The total has
order of magnitude
JL ....., ,,2.
(3.44)