142
Baryogenesis
which gives an acceptable ratio for a reasonable range of the parameters.
As long as the t/J condensate decays via (B - L)-conserving decay processes
after the inflaton, the previous estimate of the ratio is insensitive to the details of
the decay. However, even though most of the inftaton energy is not converted to
radiation until H = HR < m3/2, the condensate might also decay via thermal
scattering as soon as t/J is small enough, which is when gt/J < T, where g is the
gauge or Yukawa coupling constant. This must be unimportant when H '" m3/2.
However [72], the value of t/J/ T is rather sensitive to the integer n. For n = 6, the
required condition is comfortably satisfied but the n = 4 case is borderline.
The only flat direction having B - L non-zero and having n = 4 corresponds
to
A.
2
W = M (LHu)
(4.268)
which might arise directly at the string scale or be generated from a GUT by
integrating out a heavy standard-model singlet field N with coupling g LHIIN. At
low energies, this operator generates neutrino masses
A. 2
'" - v
(4.269)
m\l 2M
where v is the Higgs VEV. Then
nB-L '" 10-10 (
TR
) (IOB
eV)
(4.270)
s
1()6 GeV
m\l
where m\l is the lightest neutrino mass. Taking TR < 10 9 GeV then requires at
least one neutrino to be lighter than about 10- 5 eV, with an even stronger bound
of 10- 8 eV if the constraint TR < 10 6 GeV is enforced to ensure that thermal
scattering is unimportant when H '" m3/2. Interestingly, in this case, nonminimal Kiihler terms are not needed to ensure a negative mass-squared for t/J.
This is because such terms can arise for the Higgs field via radiative corrections.
4.13 Exercises
I. Show that the nucleon-antinucleon annihilation rate falls below the
expansion rate when the temperature T :::::: 20 MeV.
2. Show that in a time-reversal invariant theory baryon number is conserved.
3. Verify that for the SUeS) GUT N. has the value given in equation (4.41),
and that for the supersymetric theory, it has the value given in (4.42).
4. Show that the superbeavy gauge bosons in the SUeS) GUT have the decay
modes given in equation (4.67) and that the colour-triplet Higgs has the
decay modes given in (4.71).
5. Verify that in the SUeS) GUT there are no two-loop contributions to
r(H3) - r(ii3) and that the three-loop contribution given in equation (4.81)
does arise in the theory.
Baryogenesis
which gives an acceptable ratio for a reasonable range of the parameters.
As long as the t/J condensate decays via (B - L)-conserving decay processes
after the inflaton, the previous estimate of the ratio is insensitive to the details of
the decay. However, even though most of the inftaton energy is not converted to
radiation until H = HR < m3/2, the condensate might also decay via thermal
scattering as soon as t/J is small enough, which is when gt/J < T, where g is the
gauge or Yukawa coupling constant. This must be unimportant when H '" m3/2.
However [72], the value of t/J/ T is rather sensitive to the integer n. For n = 6, the
required condition is comfortably satisfied but the n = 4 case is borderline.
The only flat direction having B - L non-zero and having n = 4 corresponds
to
A.
2
W = M (LHu)
(4.268)
which might arise directly at the string scale or be generated from a GUT by
integrating out a heavy standard-model singlet field N with coupling g LHIIN. At
low energies, this operator generates neutrino masses
A. 2
'" - v
(4.269)
m\l 2M
where v is the Higgs VEV. Then
nB-L '" 10-10 (
TR
) (IOB
eV)
(4.270)
s
1()6 GeV
m\l
where m\l is the lightest neutrino mass. Taking TR < 10 9 GeV then requires at
least one neutrino to be lighter than about 10- 5 eV, with an even stronger bound
of 10- 8 eV if the constraint TR < 10 6 GeV is enforced to ensure that thermal
scattering is unimportant when H '" m3/2. Interestingly, in this case, nonminimal Kiihler terms are not needed to ensure a negative mass-squared for t/J.
This is because such terms can arise for the Higgs field via radiative corrections.
4.13 Exercises
I. Show that the nucleon-antinucleon annihilation rate falls below the
expansion rate when the temperature T :::::: 20 MeV.
2. Show that in a time-reversal invariant theory baryon number is conserved.
3. Verify that for the SUeS) GUT N. has the value given in equation (4.41),
and that for the supersymetric theory, it has the value given in (4.42).
4. Show that the superbeavy gauge bosons in the SUeS) GUT have the decay
modes given in equation (4.67) and that the colour-triplet Higgs has the
decay modes given in (4.71).
5. Verify that in the SUeS) GUT there are no two-loop contributions to
r(H3) - r(ii3) and that the three-loop contribution given in equation (4.81)
does arise in the theory.