4.5 Instantons, sphalerons and the early universe 195− 1 0T= 0instantonMsphsphaleron1
T= 0Fig. 4.16.The existence of instantons of arbitrary size is not surprising since pure Yang–
Mills theory is scale-invariant. Naively integrating over all instantons leads to a
divergent transition probability. Moreover, the height of the barrier between two
minima, which can be estimated on dimensional grounds as
Vm∼SI
ρ, (4.188)
tends to zero as the instanton sizeρgrows. Hence, one might expect the transi-
tion rate to be already very large at very small temperatures. In reality this does
not happen because our consideration fails for instantons of large size. In fact, in
pure non-Abelian Yang–Mills theory the gauge fields are confined and, hence, the
instanton size cannot exceed the confinement scale.
Let us now turn to theories with the Higgs mechanism, where scale invariance is
broken. For instance, in electroweak theory one has a natural infrared cut-off scale
determined by the typical mass of the gauge bosonsMW, and a maximal instanton
size equal to
ρm∼M−W^1. (4.189)Vacuum transitions are strongly suppressed at zero temperature because the weak
coupling constant is small. At high temperatures the transition rate is determined
by sphalerons corresponding to the maximum of the potential (Figure 4.16). We
can make a rough estimate of the sphaleron mass by considering the height of the
potential in the “direction” where tunneling occurs via an instanton of the largest
possible sizeρm∼MW−^1. Then
MsphVm∼SI/ρm 2 πMW
αw∼15 TeV. (4.190)