Chapter 3
Grand Unified Theory
3.1
SU(5) Unification
Gauge bosons:
�
SU (3)
SU (2)
�
(3.1)
U (1): (commuting with SU (3) × SU (2))
⎛
⎞
e2iλ
e2iλ
⎜
⎜
⎜
⎜
⎜
⎜
⎝
e2iλ
−3iλ
e
e−3iλ
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.2)
Or Lie algebra:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
2
2
2
−3
−3
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.3)
One gets breaking SU (5) → SU (3) × SU (2) × U (1) in this pattern with an adjoint
“Higgs” field obeying
10
3.1. SU(5) UNIFICATION
11
⎛
⎞
2
⎜
⎜
⎜
< Φ >∝ ⎜
⎜
⎜
⎝
2
2
−3
What do the functions think of this?
�
uR
dR
eR
2
3
u
d
�
,
1
6
L
�
�
v
e
(3.5)
2
3
(dC
R ) 1
3
(eC
R )1
−→
−1
(3.4)
L − 21
(uC
R )−
change conjugation
− 3
1
−3
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.6)
all L – handed .
Multiplets? Clues:
15 =
10
����
antisymmetric tensor
+ ����
5
�
Altogether?
(3.7)
vector
Y =0
(3.8)
2
1
1
1
+2×− +3×− +3× +1
=
0
6
2
3 � �3
v
(dC
make 5
R ) 31 +
e L − 1
6×
(3.9)
(3.10)
2
Actually (using
�
αβ )
⎛
⎞
dC
R
⎟
⎜
⎝ −e ⎠ = 5̄
v
Note:
⎛
⎜
⎜
⎜
g̃ ⎜
⎜
⎜
⎝
(3.11)
⎞
2
2
2
−3
−3
⎟
⎟
⎟
⎟ = g
� Y
⎟
⎟
⎠
(3.12)
12
CHAPTER 3. GRAND UNIFIED THEORY
with
g̃ =
−g �
6
(3.13)
Can the residue be identified with 10? Ψαi , α = 1, 2, 3, i = 4, 5, color: 3, SU (2) : 2.
�
(3.14)
2
1
Y = − (2 + 2) = − ⇒ uC
R
6
3
(3.15)
1
1
Y = − (2 − 3) = ⇒
6
6
�
u
d
L
Ψαβ , color: 3̄, SU (2) : singlet.
Ψij , color: singlet, SU (2) : 2
1
Y = − (−3 − 3) = 1 ⇒ eC
R
6
(3.16)
It clicks.
Normalizing g̃:
⎛
SU (3) generators
:
⎜
⎜
⎜
gun. ⎜
⎜
⎜
⎝
⎛
SU (2) generators
:
trΓa Γb =
U (1) generators
:
−
12
⎜
⎜
⎜
g̃
⎜
⎜
⎜
⎝
0
0
0
⎞
0
⎜
⎜
⎜
gun. ⎜
⎜
⎜
⎝
1
fab
2⎛
2
−
12
⎞
0
0
1
2
− 12
⎟
⎟
⎟
⎟ etc.
⎟
⎟
⎠
⎞
2
2
−3
−3
⎟
⎟
⎟
⎟ etc.
⎟
⎟
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.17)
(3.18)
(3.19)
3.1. SU(5) UNIFICATION
13
⎛
⎜
⎜
⎜
= g
� ⎜
⎜
⎜
⎝
−
3
1
−
1
3
− 3
1
⎞
1
2
1
1
1 2
�
g 2 (3 · ( )2 + 2 · ( )2 ) =
g
3
2
2
un.
5
1
2
�
g 2( ) =
g
6
2
un.
3
2
�
g2 =
g
5 un.
1
2
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.20)
(3.21)
(3.22)
(3.23)
So “naive” prediction:
5
2
gS2 = g
ω2 = gun.
3
�2
3
g
5
sin2 θω = � 2
=
g + gω2
1 +
(3.24)
3
5
3
8
=
(3.25)
Expt. ≈ 0.22.
Coupling Constant
Energy
Q0
Figure 3.1: Coupling Constant
Need substantial remaining. Q0 = observation point (e.g. M2 ).
Constraints: 3 observable → 2 quantities.
dgi
= βi (gi )
dt
(3.26)
14
CHAPTER 3. GRAND UNIFIED THEORY
d g12
i
dt
1
g32 (Q0 )2
1
−
g32 (Mun. )
1
MU
− 2βi◦ ln
2
gi (Q0 )
Q0
1
1
− gj (Q0 )2
gi (Q0 )2
βi◦ − βj◦
= −2βi◦
(3.27)
MU
Q0
(3.28)
:
independent of i
(3.29)
:
independent of i, j
(3.30)
= 2β3◦ ln
N.B.: of course
5 �
g12 = g 2
3
(3.31)
Master formula:
11
4
1
β0 = − e2 + T 1 + T0
2
3
3
3
Minimal Standard Model (MSM):
�
real × 21
weyl × 12
4
1
× 6 × = −7
3
2
11
4
1
1 1 1
19
= − × 2 + × 12 ×
× + × = −
3
3
6
2
2 3 2
����
(3.32)
β (3) = −11 +
(3.33)
β (2)
(3.34)
weyl
β (1)
3 4
1
1
2
1
1
=
{ × ����
3 ×
[6 × ( )2 + 3 × ( )2 + 3( )2 + 2 × ( )2 + 12 ]
5 3
2
6
3
3
2
����
f amilies
weyl
1
1
41
+ × 2 · ( )2 } =
3
2
10
(3.35)
MSSM:
Δβ (3) =
4
1
1
1
+ × 12 × = 4
× 3 ×
3
3
2
2
����
real
2 Higgs + Higgsions:
× 2 × 12
+ 34 × 2 × 12 ×
+ 13 × 12 × 21
+ 13 × 21
4
3
Δβ (2) =
1
2
Gaugions
Higgsions
Sf erminos
Extra Higgs
(3.36)
3.1. SU(5) UNIFICATION
15
=
25
6
(3.37)
Higgsions:
Δβ (1) =
3 4
1
1
{ × 2 × (2 × ( )2 ) ×
5 3
2
2
����
weyl
1
1
2
1
1
+ × 3[6 × ( )2 + 3 × ( )2 + 3 × ( )2 + 2 × ( )2 + 1]
3
6
3
3
2
1
1 2
+ ×( ) }
3
2
5
=
2
(3.38)
Leaving out Higgs:
MSM:
β (3) = −7
10
β (2) = −
3
(1)
= 4
β
(3.39)
(3.40)
(3.41)
MSSM:
Δβ (3) = 4
10
Δβ (2) =
3
Δβ (1) = 2
Δβ (3) − Δβ (2)
2
= −
(3)
(2)
β −β
11
(3)
(1)
Δβ − Δβ
2
= −
(3)
(1)
β −β
11
(2)
(1)
Δβ − Δβ
2
= −
(2)
(1)
β −β
11
(3.42)
(3.43)
(3.44)
(3.45)
(3.46)
(3.47)
Therefore, all correlations to unification come from one doublet ⇒ 6 effective
doublets.
Also change in MU :
16
CHAPTER 3. GRAND UNIFIED THEORY
ln
(
MU
Q
1
)SU SY
βi − βj
1
βi − βj
1
2
11
1
� (
)M SM (1 − ) � (
)M SM
βi − βj
11
9 βi − βj
∝
11
1017
→ 1012· 9 +2 � 1016.5
2
10
(3.48)
(3.49)
(3.50)
3.2. SU (5)
3.2
17
SU (5)
• Fermion Multiplets
SU (3) × SU (2) × U (1) ⊂ SU (5)
(3.51)
Needs U (1) traceless.
LH fields:
QL
CUR
CDR
LL
CER
Look for 5, 10 with
�
3
3
3¯
¯3
1
1
2 1
2 16
1 − 23
1 13
2 − 12
1 1
d
6
3
3
2
1
�
Y
1
−2
1
−1
1
(3.52)
Y = 0.
Fµ :
�
1
3
1
3
CdR
1
3
− 12 − 12
e
ν
�
(3.53)
T µν : antisymmetric
¯ 1, − 2
T αβ : 3,
3
αi
T
: 3, 2, 61
T ij : 1, 1, 1
(3.54)
• Symmetry Breaking
Adjoint (traceless)
⎛
⎞
2
⎜
⎜
⎜
⎜
⎜
⎜
⎝
2
2
−3
⎟
⎟
⎟
⎟ M
⎟
⎟
⎠
−3
SU (5) → SU (3) × SU (2) × U (1)
�
�
SU (3)
SU (2)
Still need SU (2) × U (1) ⇒ U (1).
(3.55)
(3.56)
(3.57)
18
CHAPTER 3. GRAND UNIFIED THEORY
Minimal:
⎛
⎜
⎜
⎜
5⎜
⎜
⎜
⎝
0
0
0
0
√υ
2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(3.58)
There could be others.
Mass terms:
ϕµ T νρ T στ �µνρστ
(3.59)
Note on “Majorana” mass terms:
¯ � )Ψ
Ψ� Ψ = (eΨ
1 − γ5
)Ψ
ΨL = (
2
(
(3.60)
(3.61)
1 − γ5
)ΨL = 0
2
�
�
0 1
γ5 =
1 0
ΨL =
�
η
−η
(3.62)
(3.63)
�
(3.64)
�
CΨ� = γ2 Ψ ∗
�
�
iσ2 η ∗
�
CΨ2 =
iσ2 η ∗
¯ �) =
(eΨ
�
(3.65)
(3.66)
iσ2 η � iσ2 η �
∝ ηa� �ab ηb
�
�
0 1
1 0
��
η
−η
�
(3.67)
(3.68)
where, a and b are Dirac indices.
Therefore,
�
(j)
ηa(i) �ab ηb
Note symmetric in i ⇔ j, due to Fermi statistics.
(3.69)
3.2. SU (5)
19
Link between texture and representation at unification: (gab symmetric)
ϕµ T (a)νp T (b)στ
(3.70)
allows self-mass for 2 component neutral fields.
υ
αβ γi
√ �5αβγji T
� ��T �
2
CU U
R
ϕ∗µ T µν Fν :
(3.71)
L
υ
√ T 5α Fα : DL CDR
2
υ 54
√ T F4 : CER EL
2
(3.72)
(3.73)
• B Validation
Vector bosons:
Figure 3.2: Vectro Bosons
∗ 14 2 4 ∗2
T12
T X4 X2 F F4
2
3 �
uR
uL dR
eL
(3.74)
(3.75)
(�12345 T 12 T 45 ϕ3 )(ϕ∗3 T 31 F1 )
uR̄3 e¯R uR̄2 d¯
R1
(3.76)
(3.77)
Triplet Higgs:
Single appearance of �αβγ .
Phenomenology ⇒ M large.
• Implementing SB; Hierarchy problem ϕ+ ϕ, ϕ+ Aϕ, ϕ+ A2 ϕ, trA2 ϕ+ ϕ, trA2 ,
trA3 , trA4 , tr(A2 )2 . Need big vev for A, small for ϕ. Heavy ϕα not by decou­
pling, but by conspiracy. Not inconsistent, but ugly.
20
CHAPTER 3. GRAND UNIFIED THEORY
• Normalization
⎛
⎛
⎜
⎜
⎜
�⎜
g
⎜
⎜
⎝
1
3
...
1
2
g⎜
⎝
1
3
1
3
−
1
2
1
2
⎞
⎟
⎠
(3.78)
⎞
⎟
⎟
⎟
⎟ =
g()� 2 1
⎟
di
= 2
⎟
⎠
− 2
1
1
1
1
�
g 2 (3 · ( )2 + 2 · ( )2 ) = g 2
3
2
2
1
� 2 5
g ( ) = g 2
6
2
3
2
�2
g
g =
5
�
3
2
g
g2
3
2
5
sin θw = 2
=
�2
3
2
=
g − g
8
(1 + 5
)g
(3.79)
(3.80)
(3.81)
(3.82)
(3.83)
Expct. ≈ 0.22.
Also, of course,
gSU (2)
=1
gSU (3)
(3.84)
3.3. SO(10) UNIFICATION
3.3
21
SO(10) Unification
SU (6)?
6 × 5 ab µν
,F ,T
2
SU (5) in SO(10): 5 complex components
Zj = Xj + iYj
�
< Z |Z > =
Xj� Xj + Yj� Yj
�
�
��
(3.85)
+i
�
SO(10) leaves this part invariant
�
�
Xj� Yj
−
Yj� Xj
��
(3.86)
(3.87)
�
SP (10) leaves this part invariant
SO(10) commutators (structure contents): rotation in kl plane
δ�
kl Xj = �(δjk Xl − δjl Xk )
(δ�kl δηmn − δηmn δ�kl
)Xj = ηδ�kl (δjm Xn
−δjn Xm ) − �δηmn (δjk Xl − δjl Xk )
= �η{δjm δnk Xl − δjm δnl Xk
−δjn δmk Xl + δjn δml Xk
−δjk δlm Xn − δjk δln Xm
+δjl δkm Xn − δjl δkn Xm }
= δnk T lm − δnl T km − δmk T ln + δml T kn
√
Γ matrices “=” rotation
(spinor rep.)
(3.88)
(3.89)
(3.90)
(3.91)
{Γk , Γl } = 2δkl
(3.92)
[Γk , Γl ] = 2(Γk Γl − δkl )
(3.93)
Claim: − 14 [Γk , Γl ] satisfy the SO(10) commutators.
So
1
1
[[Γk , Γl ], [Γm , Γn ]] =
[Γk Γl , Γm Γn ]
16
4
1
=
(Γk Γl Γm Γn − Γm Γn Γk Γl )
4
Now use
(3.94)
(3.95)
22
CHAPTER 3. GRAND UNIFIED THEORY
Γa Γb = −Γb Γa + 2δab
(3.96)
to pull through Tk and Tl in turn. End terms cancel.
Pick up terms:
1
1
(Γk Γl Γm Γn − Γm Γn Γk Γl ) = (−δnk Γm Γl + δmk Γn Γl − δnl Γk Γm + δml Γk Γn ) (3.97)
4
2
Now use
1
Γm Γl = ([Γm , Γl ] + 2δml ) etc.
2
(3.98)
δδ terms all cancel:
1
(−δnk Γm Γl + δmk Γn Γl − δnl Γk Tm + δml Γk Γn ) =
2
1
(−δnk [Γm , Γl ] + δmk [Γn , Γl ] − δnl [Γk , Γm ] + δml [Γk , Γn ])
4
(3.99)
Compare to
δnk Γlm − δnl Γkm − δmk Γln + δml Γkn
(3.100)
U −1 (R)T µ U (R) = Rνµ T ν
(3.101)
QED.
Construction of Γ matrices:
Γ1
Γ2
Γ3
Γ4
Γ5
=
=
=
=
=
σ1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1
σ2 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1
σ3 ⊗ σ1 ⊗ 1 ⊗ 1 ⊗ 1
σ3 ⊗ σ2 ⊗ 1 ⊗ 1 ⊗ 1
σ3 ⊗ σ2 ⊗ σ1 ⊗ 1 ⊗ 1
..
.
with Pauli σ – matrices does the job.
Note:
(3.102)
(3.103)
(3.104)
(3.105)
(3.106)
3.3. SO(10) UNIFICATION
23
i
σ3 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1
2
i
=
1 ⊗ σ3 ⊗ 1 ⊗ 1 ⊗ 1
2
..
.
R12 =
(3.107)
R34
(3.108)
So we diagonalize:
SO(2) ⊗ SO(2) ⊗ SO(2) ⊗ SO(2) ⊗ SO(2) ⊂ SO(10)
(3.109)
This gives us a 25 = 32 – dimensional representation of SO(10) by
1
R(eiθab Tab ) = eiθab (− 4 [Γa ,Γb ])
(3.110)
It is not quite irreducible.
Note
K = −iΓ1 Γ2 · · · Γ10
(3.111)
anticommutes with all the Γi .
Also K ∗ = K and K is Hermitean (exercise).
K 2 = −iΓi · · · Γ10 Γi · · · Γ10 = (−)1+
�
10×9
2
��
=1
(3.112)
�
pulling through
Therefore, the Rkl commute with K and we can project spinos.
1+K
S
(3.113)
2
These make the 16 + 16� representations, which are irreducible.
Shift-register rotation: The components of S can be labeled by their SO(2)5
eigenvalues ± 21 (supplying a −i).
S→
Rbef ore = iRstandard
(3.114)
k = +σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3 ⊗ σ3
(3.115)
Since
the 16 has even number of signs, (16) has odd number of signs.
Back to SU (5), υ � Jυ invariant, with
24
CHAPTER 3. GRAND UNIFIED THEORY
⎛
J
υ
Δυ Jυ
�
=
→
=
=
⇒spinor
Similarly we identify
0 1
⎜
⎜ −1 0
⎜
⎜
⎜
⎜
⎜
⎝
⎞
. .
.
⎟
⎟
⎟
⎟
⎟ =
T12 + · · · + T910
⎟
1
⎟
⎠
0
−1 0
υ + �Gυ
�υ � (GT J + JG)υ
�υ � (−GJ + JG)υ
�
σ2 invariant
SU (3) ⊂ SO(6)
SU (2) ⊂ SO(4)
(3.116)
(3.117)
(3.118)
(3.119)
(3.120)
(3.121)
(3.122)
and an extra U (1) for hyperchrnge. The diagonal elements (maximal torus) are
simply represented as the R2i−1,2i (−trace part). Thus,
1
R12 − (R12 + R34 + R56 ) etc. ∈ SU (3)
3
R78 − R910 ∈ SU (2)
(3.123)
(3.124)
With
1
1
(R12 + R34 + R56 ) − R78 − R910 ∝ U (1)Y
6
4
Analysis of 16 standard model Q − #s:
5 + signs
1 state
| + + + ++ >
SU (3) × SU (2) × U (1) singlet
1 + signs
5 state, 2 types
| − − + −− >
(3.125)
(3.126)
3.3. SO(10) UNIFICATION
25
| − + − −− >
| + − − −− >
�
��
�
SU (3) (3)
SU (2) singlet
1
1
1
Ỹ =
(−1) − (−2) =
6
4
3
| − − − +− >
| − − − −+ >
�
(3.127)
�
��
SU (3) singlet
SU (2) doublet
1
1
Ỹ = (−3) = −
6
2
(3.128)
With Y = Ỹ , these can be identified as
dC
R ,
�
−e
ν
�
(3.129)
L
They are our 5¯ of SU (5).
3 + signs
10 state, 3 types
| + + + −− >
SU (3) singlet
SU (2) singlet
1
1
Y = (+3) − (−2) = 1 ≡ eC
R
6
4
| + − − ++ >
| − + − ++ >
| + − − ++ >
(3.130)
SU (3) ¯3
SU (2)singlet
1
1
2
Y = (−1) − (2) = − ≡ u
C
R
6
4
3
| + + − +− >
| + − + +− >
(3.131)
�
��
�
26
CHAPTER 3. GRAND UNIFIED THEORY
| − + + +− >
| + + − −+ >
| + − + −+ >
| − + + −+ >
SU (3) 3
SU (2) 2
�
�
1
1
u
Y = (1) = ≡
d L
6
6
(3.132)
Comments:
1. The construction of Γ –matrices – anticommuting objects – used here contains
the existance of bosonization in 1 + 1d field theories. Also, the fermionization of
spin chains (from σ ⊗ σ ⊗ · · · → anticommuting quantities), known as JordanWigner trick.
2.
| + + + ++ >≡ NRC
(3.133)
plays an important role in current thinking about symmetric masses. It can get
a mass (Majorana Mass) of the type
Mij N¯Ri NRCj
(3.134)
This involves breaking SO(10), but not SU (3) × SU (2) × U (1), so M can be
large. Also, NS can connect to ordinary left-handed neutrinos through the
ordinary Higgs doublet, in the form
µij N¯Ri Lµj ϕ∗µ
(3.135)
where, i, j are formly indices and µ = SU (2) index.
By 2nd order perturbation theory we induce Majorana Masses for the νL .
<Φ>
<Φ>
M
Figure 3.3: Majorana Masses
m∼
µ2
M
(3.136)
3.3. SO(10) UNIFICATION
27
Breaking Scheme: Higgs ϕ in 16
< ϕN > =
�
0
SO(10) → SU (5)
(3.137)
(3.138)
5
5
NRC :
C
−3 −15
dR , LL :
C
uC
,
Q
,
e
:
1
10
L R
R
(3.139)
αB + βL + γY
(3.140)
−β = 5
β = −5
(3.141)
with “right” hyperchrnge.
Extra U (1):
NRC :
dC
R:
−
α γ
+
= −3
3
3
α = 5
(3.142)
LL :
β−
γ
= −3
2
γ = −4
(3.143)
uC
R:
5 2γ ?
− −
= 1
3
3
5 γ
= 1
− +
3 3
QL :
(3.144)
(3.145)
28
CHAPTER 3. GRAND UNIFIED THEORY
α γ
5 2
+
=
+
3
6
3 6
= 1
(3.146)
eC
R:
−β + γ = 1
(3.147)
αB + βL + γY = 5B − 5L − 4Y
(3.148)
Γ1 = σ1 ⊗ 1 ⊗ · · · ⊗ 1
Γ2 = σ2 ⊗ 1 ⊗ · · · ⊗ 1
..
.
(3.149)
(3.150)
There are 45adjoint and 10vectro as before.
Fermion masses: bilinears in ϕ
∗
ϕ∗ transforms as e[Γµ ,Γν ]
(Cϕ∗ )� =
=
∗
CΓµ =
C =
∗
Ce[Γµ ,Γν ] ϕ∗
∗
e[Γµ ,Γν ] Cϕ∗
±Γµ C
Γ1 Γ3 Γ5 Γ7 Γ9
(3.151)
(3.152)
(3.153)
With + sign:
C = σ1 ⊗ iσ2 ⊗ σ1 ⊗ iσ2 ⊗ σ1
(3.154)
symmetric and real.
Anticommute with K: Construct Majorana bilinears
−→
ϕϕ ∼ (ϕ∗ C)+ tensor made f rom Γ ϕ
(3.155)
to be consistent with projection add number of indices. Irreducible ⇒ Totally
antisymmetric.
16 × 16 = 10vector + 1203−tensor + 1265−tensor:self −dual
� �� �
S
�
��
A
�
�
16 × 16 = 1scale + 452−tensor + 2104−tensor
��
S
�
(3.156)
(3.157)
3.3. SO(10) UNIFICATION
29
Final comments on SO(10) vs. SU (5): 3 RH neutrinos vs. SU (5) × U (1). Charge
quantization.
Q ∝ �(B − L)
p : 1 + �, e : −1 − �, n : �, v : −�.
N.B.: mechanics of charge quantization. n → pev̄.
(3.158)