Phase transitions and electroweak baryogenesis
131
This is called the 'sphaleron washout condition'. In non-local baryogenesis, CPviolating interactions of the particles with the bubble wall produce an asymmetry
in a quantum number other than the baryon number and the resulting particles
carry this asymmetry into the unbroken phase, away from the wall. Then baryonnumber non-conserving effects convert this asymmetry into a baryon asymmetry,
(some of) which is frozen in the broken phase after the bubble wall has passed.
If the speed of the wall is greater than the sound speed in the plasma, the former
process dominates [54]; otherwise the latter does but, in general, both may occur
and the total baryon asymmetry is the sum of that generated by the two processes.
We have already noted that the phase transition is (weakly) first order in
electroweak theory and that at the phase transition (2.61) gives
2CTe
t/le = veT!,) = -n
(4.226)
so
v(Te) = 2C < 0 17
(4.227)
Te
3>.. '" •
which does not satisfy (4.225). In fact, requiring that (4.225) is satisfied would
require
mH;S 47 GeV
(4.228)
in clear contradiction to the current lower bound (4.195).
The perturbative calculations of the corrections to the effective potential are
not a priori reliable, because of the so called 'infrared problem' that afflicts finitetemperature field theory [55,56]. It derives from the existence of an expansion
parameter of the form
g2
g2T
E" =
2
~ _2_
when m « T
(4.229)
m
where m is some bosonic mass appearing in the propagators. Then light modes,
those with m « g~T, interacting with the Higgs are a problem that should be
treated non-perturbatively. The direct method of carrying out a four-dimensional
finite temperature lattice simulation is difficult because the weak coupling entails
the existence of multiple length scales which are difficult to fit simultaneously on
a finite lattice. Also. in practice, chiral fermions cannot be handled efficiently.
However, all of these problems can be overcome by using a finite-temperature
effective field theory [57-59], obtained by integrating out (perturbatively) all nonzero Matsubara modes, which includes, in particular, all fermions. The resulting
effective theory is then three-dimensional and involves only the surviving infrared
modes, the Higgs and the spatial components of the S U (2) and U (I) gauge fields.
This theory is ideally suited for lattice simulations. It is found that, in the m HTI' plane, there is a line of first-order phase transitions that end at a critical point
after which there is only a crossover transition. The endpoint is known to high
precision and is at
mH,e = 72.3 GeV Te = 109.2 GeV.
(4.230)
131
This is called the 'sphaleron washout condition'. In non-local baryogenesis, CPviolating interactions of the particles with the bubble wall produce an asymmetry
in a quantum number other than the baryon number and the resulting particles
carry this asymmetry into the unbroken phase, away from the wall. Then baryonnumber non-conserving effects convert this asymmetry into a baryon asymmetry,
(some of) which is frozen in the broken phase after the bubble wall has passed.
If the speed of the wall is greater than the sound speed in the plasma, the former
process dominates [54]; otherwise the latter does but, in general, both may occur
and the total baryon asymmetry is the sum of that generated by the two processes.
We have already noted that the phase transition is (weakly) first order in
electroweak theory and that at the phase transition (2.61) gives
2CTe
t/le = veT!,) = -n
(4.226)
so
v(Te) = 2C < 0 17
(4.227)
Te
3>.. '" •
which does not satisfy (4.225). In fact, requiring that (4.225) is satisfied would
require
mH;S 47 GeV
(4.228)
in clear contradiction to the current lower bound (4.195).
The perturbative calculations of the corrections to the effective potential are
not a priori reliable, because of the so called 'infrared problem' that afflicts finitetemperature field theory [55,56]. It derives from the existence of an expansion
parameter of the form
g2
g2T
E" =
2
~ _2_
when m « T
(4.229)
m
where m is some bosonic mass appearing in the propagators. Then light modes,
those with m « g~T, interacting with the Higgs are a problem that should be
treated non-perturbatively. The direct method of carrying out a four-dimensional
finite temperature lattice simulation is difficult because the weak coupling entails
the existence of multiple length scales which are difficult to fit simultaneously on
a finite lattice. Also. in practice, chiral fermions cannot be handled efficiently.
However, all of these problems can be overcome by using a finite-temperature
effective field theory [57-59], obtained by integrating out (perturbatively) all nonzero Matsubara modes, which includes, in particular, all fermions. The resulting
effective theory is then three-dimensional and involves only the surviving infrared
modes, the Higgs and the spatial components of the S U (2) and U (I) gauge fields.
This theory is ideally suited for lattice simulations. It is found that, in the m HTI' plane, there is a line of first-order phase transitions that end at a critical point
after which there is only a crossover transition. The endpoint is known to high
precision and is at
mH,e = 72.3 GeV Te = 109.2 GeV.
(4.230)