4.4 “Symmetry restoration” and phase transitions 165
As we will see,CPviolation plays a very important role in baryogenesis, en-
suring the possibility of different decay rates for particles and their antiparticles in
particular decay channels. If this were not the case, generation of baryon asymmetry
would be impossible.
Note that if we accompany theCPtransformation by time reversalT(t→−t),
which reverses the direction of arrows on the diagrams and changes the handedness,
(4.90) remains invariant. The Lagrangian of the Standard Model isCPT-invariant.
This invariance guarantees that thetotaldecay rates, which include all decay chan-
nels, are the same for particles and their antiparticles.
4.4 “Symmetry restoration” and phase transitionsA classical scalar fieldχinteracts with gauge fields which influence its behavior.
In the early universe this influence can be described using an effective potential.
At very high temperatures the effective potential has only one minimum, atχ= 0 ,
and the homogeneous component ofχ disappears. As a result all fermions and
intermediate bosons become massless and one says that the symmetry is restored.
In fact, as we have pointed out, the gauge symmetry is never broken by the Higgs
mechanism. Nevertheless, in deference to the commonly used terminology we use
the term symmetry restoration to designate the disappearance of the homogeneous
scalar field.
As the universe expands the temperature decreases. Below a critical temperature
the effective potential acquires an energetically favorable local minimum, atχ(T)=
0 ,and the transition to this state becomes possible. Depending on the parameters
of the theory this can be either a phase transition or a simple cross-over.
In this section we investigate symmetry restoration and phase transitions in gauge
theories.
4.4.1 Effective potential
To introduce an idea of the effective potential we first consider a simple model
describing a self-interacting real scalar field, which satisfies the equation
χ;α;α+V′(χ)= 0 , (4.92)whereV′(χ)≡∂V/∂χ.The fieldχcan always be decomposed into homogeneous
and inhomogeneous components:
χ(t,x)=χ ̄(t)+φ(t,x), (4.93)so that the spatial average ofφ(x,t)is equal to zero.Substituting (4.93) into (4.92),
expanding the potentialVin powers ofφ,and averaging over space, we obtain the