15. Grand Unified Theories
1
15. GRAND UNIFIED THEORIES
Revised October 2005 by S. Raby (Ohio State University).
15.1.
Grand Unification
15.1.1.
Standard Model: An Introduction
:
In spite of all the successes of the Standard Model [SM], it is unlikely to be the
final theory. It leaves many unanswered questions. Why the local gauge interactions
SU(3)
C
×
SU(2)
L
×
U(1)
Y
, and why 3 families of quarks and leptons? Moreover, why does
one family consist of the states [
Q, u
c
, d
c
;
L, e
c
] transforming as [(3
,
2
,
1
/
3)
,
(
¯
3
,
1
,
−
4
/
3)
,
(
¯
3
,
1
,
2
/
3); (1
,
2
,
−
1)
,
(1
,
1
,
2)], where
Q
= (
u, d
), and
L
= (
ν, e
) are SU(2)
L
doublets, and
u
c
, d
c
, e
c
are charge conjugate SU(2)
L
singlet fields with the U(1)
Y
quantum numbers
given? [We use the convention that electric charge
Q
EM
=
T
3
L
+
Y/
2 and all fields are
left-handed.] Note the SM gauge interactions of quarks and leptons are completely fixed
by their gauge charges. Thus, if we understood the origin of this charge quantization,
we would also understand why there are no fractionally charged hadrons. Finally, what
is the origin of quark and lepton masses, or the apparent hierarchy of family masses
and quark mixing angles? Perhaps if we understood this, we would also know the origin
of
CP
violation, the solution to the strong
CP
problem, the origin of the cosmological
matter-antimatter asymmetry, or the nature of dark matter.
The SM has 19 arbitrary parameters; their values are chosen to fit the data. Three
arbitrary gauge couplings:
g
3
, g, g
(where
g
,
g
are the SU(2)
L
, U(1)
Y
couplings,
respectively) or equivalently,
α
s
= (
g
2
3
/
4
π
)
, α
EM
= (
e
2
/
4
π
) (
e
=
g
sin
θ
W
), and
sin
2
θ
W
= (
g
)
2
/
(
g
2
+ (
g
)
2
). In addition, there are 13 parameters associated with the 9
charged fermion masses and the four mixing angles in the CKM matrix. The remaining
3 parameters are
v, λ
[the Higgs VEV (vacuum expectation value) and quartic coupling]
(or equivalently,
M
Z
, m
0
h
), and the QCD
θ
parameter. In addition, data from neutrino
oscillation experiments provide convincing evidence for neutrino masses. With 3 light
Majorana neutrinos, there are at least 9 additional parameters in the neutrino sector;
3 masses, 3 mixing angles and 3 phases. In summary, the SM has too many arbitrary
parameters, and leaves open too many unresolved questions to be considered complete.
These are the problems which grand unified theories hope to address.
15.1.2.
Charge Quantization
:
In the Standard Model, quarks and leptons are on an equal footing; both fundamental
particles without substructure. It is now clear that they may be two faces of the same
coin; unified, for example, by extending QCD (or SU(3)
C
) to include leptons as the fourth
color, SU(4)
C
[1].
The complete Pati-Salam gauge group is SU(4)
C
×
SU(2)
L
×
SU(2)
R
,
with the states of one family [(
Q, L
)
,
(
Q
c
, L
c
)] transforming as [(4
,
2
,
1)
,
(
¯
4
,
1
,
¯
2)], where
Q
c
= (
d
c
, u
c
)
, L
c
= (
e
c
, ν
c
) are doublets under SU(2)
R
. Electric charge is now given
by the relation
Q
EM
=
T
3
L
+
T
3
R
+ 1
/
2(
B
–
L
), and SU(4)
C
contains the subgroup
SU(3)
C
×
(
B
–
L
) where
B
(
L
) is baryon (lepton) number. Note
ν
c
has no SM quantum
numbers and is thus completely “sterile.” It is introduced to complete the SU(2)