4.6 Beyond the Standard Model 223the mass comes from the gauge fields:
M∼B^2 δW^3 ∼(
g
δ^2 W) 2
δ^3 W∼mW
e^2. (4.240)
One can show that this estimate, obtained forλ>e^2 ,still applies in the more
complicated caseλ<e^2.
The existence of magnetic monopoles is an inevitable consequence of Grand Uni-
fied Theories with (semi)simple gauge groupsG. This general result follows imme-
diately from the topological interpretation of monopole solutions. Monopoles exist
if the vacuum manifoldMcontains noncontractible 2-surfaces, or equivalently, the
groupπ 2 (M)is nontrivial. In the case above,M=S^2 andπ 2
(
S^2
)
=Z.In Grand
Unified Theories, a semisimple groupGis broken toH=SU( 3 )QCD×U( 1 )em;
the vacuum manifold is the quotient groupGmoduloH,that is,M=G/H.Using
a well known result from the theory of homotopy groups,
π 2 (G/H)=π 1 (H)=π 1(
SU( 3 )QCD×U( 1 )em)
=Z, (4.241)
we conclude that monopoles are unavoidable if the unification group incorporates
electromagnetism. Furthermore, because theSU( 3 )QCDgauge fields are confined,
the monopoles carry only the magnetic charge of theU( 1 )emgroup.
Problem 4.34Why are the arguments presented above not applicable to elec-
troweak symmetry breaking?
Let us estimate the abundance of Grand Unified Theory monopoles produced at
TGUT∼ 1015 GeV.As we have already said, at least one defect per horizon volume
is created during symmetry breaking. Taking into account that the horizon scale at
this time is abouttH∼ 1 /TGUT^2 ,we immediately obtain the following estimate for
the average number of monopoles per photon:
nM
nγ1
TGUT^3 t^3 H∼TGUT^3. (4.242)
This ratio does not change significantly during the expansion of the universe and
so the present energy density of the GUT monopoles should be
ε^0 M∼MnM(t 0 )∼
mW
e^2TGUT^3 T 03
∼ 10 −^16
( mW
1015 GeV)( TGUT
1015 GeV) 3
gcm−^3. (4.243)But this is at least 10^13 times the critical density−obviously a cosmological
disaster. Either one has to abandon Grand Unified Theories or find a solution to