112
Baryogenesis
the SO(IO) generators M I2 , M 34 , M56 and that of SO(4) by M 78 , M 9 • 1O • The
Cartan subalgebras of SU(2)L,R are then generated by Tl.R = !(M 78 ± M9.1~.
We first note that
D=M23~7
(4.108)
where
(MGb)"j = 8~8~ - 8~8~
(4.109)
-
' J
J '
is an element of SO(IO). The action of D on the underlying 10-dimensional space
is to reflect the coordinates x G ... _x G (a = 2,3,6, 7) leaving the remainder
invariant. Thus, the effect of D on the Cartan subalgebra of SU(4) ;;:: SO(6) is
to reverse the signs of the generators
D: T 3•8•IS ~ _T 3 • 8 • 15
(4.110)
while on that of SU(2)L x SU(2)R
D: Tl ... T~.
(4.111)
Thus,
D : (4,1, 1) ... (4, 1,1)
(4.112)
as asserted.
Because of this, unlike the SU(5) GUTs, a baryon asymmetry is often not
generated at the GUT-breaking scale but rather at the lower scale MI at which
U ( I)' is broken. Whether or not this happens, of course, depends upon the Higgs
fields responsible for this symmetry breaking and their coupling to the matter
fields. The Higgs fields that couple to matter must be in one or more of the
SO(IO) representations occurring in the product
16 x 16 = 10,. + 1200 + 126s •
(4.113)
For example, with the colour-triplet Higgs particles belonging to the 10dimensional representation, the symmetry (4.112) ensures that
r(H3 ~ iQ) = r(H3 ~ lQ)
(4.114)
at one-loop level, so no baryon asymmetry results, similarly in the QQ channels
[29]. However, if the Higgs content is enlarged to include a 4S-dimensional, a
l26-dimensional and an additional 10-dimensional representations, then with a
suitable choice of (4S) we can break
S0(10) ~ G3122
(4.115)
in a way which breaks the D-symmetry (G3122 is defined in (4.103». This is
done by ensuring that the non-zero VEVs are odd under D. In addition, the
extra Higgs content splits gL and gR, the coupling constants of the SU(2)L.R
groups, and ensures that the colour-triplet Higgs mass eigenstates are complex
Baryogenesis
the SO(IO) generators M I2 , M 34 , M56 and that of SO(4) by M 78 , M 9 • 1O • The
Cartan subalgebras of SU(2)L,R are then generated by Tl.R = !(M 78 ± M9.1~.
We first note that
D=M23~7
(4.108)
where
(MGb)"j = 8~8~ - 8~8~
(4.109)
-
' J
J '
is an element of SO(IO). The action of D on the underlying 10-dimensional space
is to reflect the coordinates x G ... _x G (a = 2,3,6, 7) leaving the remainder
invariant. Thus, the effect of D on the Cartan subalgebra of SU(4) ;;:: SO(6) is
to reverse the signs of the generators
D: T 3•8•IS ~ _T 3 • 8 • 15
(4.110)
while on that of SU(2)L x SU(2)R
D: Tl ... T~.
(4.111)
Thus,
D : (4,1, 1) ... (4, 1,1)
(4.112)
as asserted.
Because of this, unlike the SU(5) GUTs, a baryon asymmetry is often not
generated at the GUT-breaking scale but rather at the lower scale MI at which
U ( I)' is broken. Whether or not this happens, of course, depends upon the Higgs
fields responsible for this symmetry breaking and their coupling to the matter
fields. The Higgs fields that couple to matter must be in one or more of the
SO(IO) representations occurring in the product
16 x 16 = 10,. + 1200 + 126s •
(4.113)
For example, with the colour-triplet Higgs particles belonging to the 10dimensional representation, the symmetry (4.112) ensures that
r(H3 ~ iQ) = r(H3 ~ lQ)
(4.114)
at one-loop level, so no baryon asymmetry results, similarly in the QQ channels
[29]. However, if the Higgs content is enlarged to include a 4S-dimensional, a
l26-dimensional and an additional 10-dimensional representations, then with a
suitable choice of (4S) we can break
S0(10) ~ G3122
(4.115)
in a way which breaks the D-symmetry (G3122 is defined in (4.103». This is
done by ensuring that the non-zero VEVs are odd under D. In addition, the
extra Higgs content splits gL and gR, the coupling constants of the SU(2)L.R
groups, and ensures that the colour-triplet Higgs mass eigenstates are complex