113
group contains an element (analogous to a 2
π
rotation in the case of
SU
(2)), which reverses
the sign of spinors, but not tensors. Calling this element
r
, we have
r
:
ψ
→ −
ψ
and
H
→
H.
(14.8)
This is not the same as R parity because it lacks
θ
→ −
θ
, which amounts to reversing
the signs of spacetime spinors. This operation – which is just 2
π
rotation – is sometimes
denoted (
−
1)
F
, where
F
is fermion number. It is an exact symmetry, since fermion number
is conserved modulo two. Thus R parity is just
r
(
−
1)
F
, which means that it is equivalent to
the gauge group element
r
. This element
r
is related to baryon number and lepton number
by
r
= (
−
1)
3(
B
−
L
)
.
(14.9)
Now for the key point:
This discrete gauge symmetry is unbroken if and only if no
Spin(10) spinor fields acquire vevs as part of a Higgs mechanism. In other words, only
r

invariant fields condense and acquire vevs. In this case, 3(
B
−
L
) would be conserved mod 2,
which could be possible in a realistic model. In that case we would have related R parity to an
unbroken gauge symmetry, so that it could be exactly conserved. This would ensure that the
LSP, the lightest particle with negative R parity, is absolutely stable. Phenomenologically,
to control baryon number and lepton number violation, this seems to be quite desirable.
Nonetheless, there is a considerable literature on models with broken R parity symmetry in
which the LSP is not absolutely stable. Some of them may be viable.
14.5
Susy unification
The fact that there is a single gauge coupling constant for a simple GUT group
G
, such
as
SU
(5) or
SO
(10), means that for any properly normalized generator
T
in the standard
model subalgebra, the quantity
g
2
i
tr (
T
2
)
,
(14.10)
should be the same, when evaluated at the unification scale
m
X
.
Here, the index
i
=
1
,
2
,
3 labels the standard model groups
U
(1),
SU
(2),
SU
(3).
The trace runs over any
representation of
G
. Let’s illustrate this for a few choices of
T
:
Q
: electric charge
T
3
: third component of weak isospin
T
c
: a generator of color
SU
(3)
e
Y
=
Y/
2 : weak hypercharge (suitably normalized)
(14.11)
Unification then requires that
g
2
1
tr (
e
Y
2
) =
g
2
2
tr (
T
2
3
) =
g
2
3
tr (
T
2
c
)
.
(14.12)
114
These are also the same as
e
2
tr
Q
2
. To check this substitute
Q
=
T
3
+
e
Y
and
e
=
g
1
g
2
√
g
2
1
+
g
2
2
and use tr (
T
3
e
Y
) = 0, to get
tr
Q
2
= tr
T
2
3
+ tr
e
Y
2
=
(
g
2
1
g
2
2
+ 1
)
tr
e
Y
2
=
g
2
1
e
2
tr
e
Y
2
=
g
2
2
e
2
tr
T
2
3
,
(14.13)
Note that
e
Y
=
Y/
2 is the “properly” normalized generator.
As in the standard model, the weak mixing angle (or Weinberg angle) is sin
2
θ
W
=
e
2
/g
2
2
.
Using the formulas given above, this can be evaluated at the unification scale
m
X
.
sin
2
θ
W
=
tr
T
2
3
tr
Q
2
=
g
2
1
g
2
1
+
g
2
2
.
(14.14)
Let’s evaluate this for the
¯
5
representation of
SU
(5) which contains
¯
d
and
l
.
Any other
representation gives the same answer.
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 Summer '19
 mechanics, Quantum Field Theory, String Theory, Lie group, Lie algebra, supersymmetry