patterns of remnant discrete symmetries
TRANSCRIPT
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Patterns of Remnant Discrete Symmetries
Roland Schierenin collaboration with
Rolf Kappl, Patrick Vaudrevange, Michael Ratz and Bjorn Petersen
based on JHEP 0908:111 (2009)
Heidelberg, 28. Juni 2010
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
1 Abelian discrete symmetries
2 Anomalies
3 Applications
4 Summary
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
General considerations
A general finite, abelian group is always of the form(fundamental theorem of finite abelian groups)
Zd1× . . .× ZdN
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
General considerations
A general finite, abelian group is always of the form(fundamental theorem of finite abelian groups)
Zd1× . . .× ZdN
Main use of abelian, discrete symmetries: Forbid couplingsExample: matter parity to suppress proton decay
XXXUDD
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
General considerations
A general finite, abelian group is always of the form(fundamental theorem of finite abelian groups)
Zd1× . . .× ZdN
Main use of abelian, discrete symmetries: Forbid couplingsExample: matter parity to suppress proton decay
XXXUDD
abelian, discrete groups have only one-dimensional irreduciblerepresentationsy cannot explain mixing angles, etc.
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Origin of abelian discrete symmetries
Arguments against imposing discrete symmetries:
their breaking leads to domain walls Zeldovich et.al (1974)
y solutions:
symmetry is not exact
break before inflation
embedd in gauge symmetry
. . .
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Origin of abelian discrete symmetries
Arguments against imposing discrete symmetries:
their breaking leads to domain walls Zeldovich et.al (1974)
y solutions:
symmetry is not exact
break before inflation
embedd in gauge symmetry
. . .
quantum gravity is expected to violate global symmetries(black hole only carry gauged charges) L. Krauss and F. Wilczek (1989)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Origin of abelian discrete symmetries
Arguments against imposing discrete symmetries:
their breaking leads to domain walls Zeldovich et.al (1974)
y solutions:
symmetry is not exact
break before inflation
embedd in gauge symmetry
. . .
quantum gravity is expected to violate global symmetries(black hole only carry gauged charges) L. Krauss and F. Wilczek (1989)
in string theory all symmetries must be gauged Polchinski
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Review: U(1) → ZN
φ ψ
U(1) 3 1
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Review: U(1) → ZN
φ ψ
U(1) 3 1gets a VEV
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Review: U(1) → ZN
φ ψ
U(1) 3 1gets a VEV
Calculation
1 usual ansatz:
e i 3α(x)φ = φ
⇒ α = 2πn
3
2 transformation of ψ:
ψ → e2πi n3ψ
Visualization
1 φ defines a lattice
0bc bc bc bc bc
ψut
2 A coupling ψM is allowed ifM q(ψ) lies on the lattice.
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Review: U(1) → ZN
φ ψ
U(1) 3 1gets a VEV
Calculation
1 usual ansatz:
e i 3α(x)φ = φ
⇒ α = 2πn
3
2 transformation of ψ:
ψ → e2πi n3ψ
Visualization
1 φ defines a lattice
0bc bc bc bc bc
ψut
2 A coupling ψM is allowed ifM q(ψ) lies on the lattice.
Unbroken symmetry: Z3
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
define a charge matrix Q ij = qj(φ(i)) and similarly Qψ
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
define a charge matrix Q ij = qj(φ(i)) and similarly Qψ
ansatz: exp(i∑
j Q ij αj
)φ(i) = φ(i)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
define a charge matrix Q ij = qj(φ(i)) and similarly Qψ
ansatz: exp(i∑
j Q ij αj
)φ(i) = φ(i) ⇔ Q α = 2πn
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
define a charge matrix Q ij = qj(φ(i)) and similarly Qψ
ansatz: exp(i∑
j Q ij αj
)φ(i) = φ(i) ⇔ Q α = 2πn
Diagonalize Q with unimodular matrices (det = ±1)(Smith normal form)
AQ B = diag′(d1, . . . , dN)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
define a charge matrix Q ij = qj(φ(i)) and similarly Qψ
ansatz: exp(i∑
j Q ij αj
)φ(i) = φ(i) ⇔ Q α = 2πn
Diagonalize Q with unimodular matrices (det = ±1)(Smith normal form)
AQ B = diag′(d1, . . . , dN)
Unbroken symmetry: Zd1× . . .× ZdN
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
The general case: Breaking U(1)N
Notation: VEV fields φ(i) with U(1)j charge qj(φ(i)) and other fields
ψ(k)
define a charge matrix Q ij = qj(φ(i)) and similarly Qψ
ansatz: exp(i∑
j Q ij αj
)φ(i) = φ(i) ⇔ Q α = 2πn
Diagonalize Q with unimodular matrices (det = ±1)(Smith normal form)
AQ B = diag′(d1, . . . , dN)
Unbroken symmetry: Zd1× . . .× ZdN
charges of the ψ(i): Q ′ψ = Qψ B
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
charge matrix: Q =
8 −24 22 4
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
charge matrix: Q =
8 −24 22 4
= A−1
2 00 60 0
B−1
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
charge matrix: Q =
8 −24 22 4
= A−1
2 00 60 0
B−1
Unbroken symmetry: Z2 × Z6
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
charge matrix: Q =
8 −24 22 4
= A−1
2 00 60 0
B−1
Unbroken symmetry: Z2 × Z6
charges of the ψ(i):
(1 31 5
)B =
(1 11 3
)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example: Visualization
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
bcbc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
φ(1)
φ(2)
φ(3)
U(1)1
U(1)2
ut
ut
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
A two-dimensional example: Visualization
U(1)1 U(1)2
φ(1) 8 -2φ(2) 4 2φ(3) 2 4
U(1)1 U(1)2
ψ(1) 1 3ψ(2) 1 5
bcbc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
φ(1)
φ(2)
φ(3)
U(1)1
U(1)2
ut
ut
bc bc bc bc bc bc bc
bc bc bc bc bc bc bc
ut ψ(1)
utψ(2)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Redundancies
A trivial example
Z6 and a field ψ having charge 4:
ψ → e2πi 4 n6ψ = e2πi 2 n
3ψ
One cannot perform all symmetry transformations. n = 0, 1, 2
Conclusion: Just Z3 and ψ has charge 2
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Redundancies
A trivial example
Z6 and a field ψ having charge 4:
ψ → e2πi 4 n6ψ = e2πi 2 n
3ψ
One cannot perform all symmetry transformations. n = 0, 1, 2
Conclusion: Just Z3 and ψ has charge 2
For a general group Zd1× . . .× ZdN
there are two cases:
1 In one factor, di and all charges have a greatest common dividor > 1
2 Two factors are equal
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Redundancies
A trivial example
Z6 and a field ψ having charge 4:
ψ → e2πi 4 n6ψ = e2πi 2 n
3ψ
One cannot perform all symmetry transformations. n = 0, 1, 2
Conclusion: Just Z3 and ψ has charge 2
For a general group Zd1× . . .× ZdN
there are two cases:
1 In one factor, di and all charges have a greatest common dividor > 1
2 Two factors are equal
Complication
for a general group there a very many equivalent charge assigments(corresponds to the automorphisms of the group)
e.g. Z2 × Z4 × Z4 has 1536 equivalent charge assigments9 / 24
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Two Examples
Case 1
Z4 Z8
ψ(1) 2 4ψ(2) 3 3
Z4 Z8
ψ(1) 2 4ψ(2) 2 3
Z2 Z8
ψ(1) 1 4ψ(2) 1 3
All three tables show physically equivalent symmetry groups
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Two Examples
Case 1
Z4 Z8
ψ(1) 2 4ψ(2) 3 3
Z4 Z8
ψ(1) 2 4ψ(2) 2 3
Z2 Z8
ψ(1) 1 4ψ(2) 1 3
All three tables show physically equivalent symmetry groups
Case 2
Z2 Z6
ψ(1) 1 1ψ(2) 1 3
Z2 Z2 Z3
ψ(1) 1 1 2ψ(2) 1 1 0
Z6
ψ(1) 1ψ(2) 3
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Obtaining abelian discrete symmetriesRedundanciesGeneralization
Generalization
So far: U(1)N → Zd1× . . .× ZdN
Mixed cases: Zd × U(1) → Zd1× Zd2
Write as U(1) × U(1) and introduce dummy field φ with charge(d , 0)
R-symmetry breaking: Introduce dummy field Ω which has the samecharge as the superpotential
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Discrete Anomalies
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
ZN anomalies
The setup
gauge theory with simple gauge group Ggauge
a ZN symmetry
fields ψ(i) in irreducible representation r(i) of Ggauge; ZN charge q(i)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
ZN anomalies
The setup
gauge theory with simple gauge group Ggauge
a ZN symmetry
fields ψ(i) in irreducible representation r(i) of Ggauge; ZN charge q(i)
Under a ZN transformation
ψ(i)→ exp
(2πi
Nq(i)
)ψ(i)
DΨDΨ → exp
(2πi
Nngauge
∑
i
q(i) 2ℓ(r(i))
)DΨDΨ
T. Araki et. al. (2008)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
ZN anomalies
The setup
gauge theory with simple gauge group Ggauge
a ZN symmetry
fields ψ(i) in irreducible representation r(i) of Ggauge; ZN charge q(i)
Under a ZN transformation
ψ(i)→ exp
(2πi
Nq(i)
)ψ(i)
DΨDΨ → exp
(2πi
Nngauge
∑
i
q(i) 2ℓ(r(i))
)DΨDΨ
T. Araki et. al. (2008)
ngauge ∈ Z
Note: ngauge = 0 for Ggauge = U(1)13 / 24
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
ZN anomalies
The setup
gauge theory with simple gauge group Ggauge
a ZN symmetry
fields ψ(i) in irreducible representation r(i) of Ggauge; ZN charge q(i)
Under a ZN transformation
ψ(i)→ exp
(2πi
Nq(i)
)ψ(i)
DΨDΨ → exp
(2πi
Nngauge
∑
i
q(i) 2ℓ(r(i))
)DΨDΨ
T. Araki et. al. (2008)
Dynkin indexfactor 2 comes from normalization ℓ(M) = 1
2 for SU(M)13 / 24
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
ZN anomalies
The setup
gauge theory with simple gauge group Ggauge
a ZN symmetry
fields ψ(i) in irreducible representation r(i) of Ggauge; ZN charge q(i)
Under a ZN transformation
ψ(i)→ exp
(2πi
Nq(i)
)ψ(i)
DΨDΨ → exp
(2πi
Nngauge
∑
i
q(i) 2ℓ(r(i))
)DΨDΨ
T. Araki et. al. (2008)
Anomaly condition:∑
i q(i)ℓ(r(i)) = 0 mod N
2
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
Consequences of discrete anomalies
instantons induce an effective term in the Lagrangian
L ⊃ e−
8π2
g2 k∏
i=1
(ψ(i))2ℓ(r(i))
t’Hooft (1976)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
Consequences of discrete anomalies
instantons induce an effective term in the Lagrangian
L ⊃ e−
8π2
g2 k∏
i=1
(ψ(i))2ℓ(r(i))
t’Hooft (1976)
This term breaks any ZN symmetry unless∑
i
q(i)ℓ(r(i)) = 0 modN
2
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
Consequences of discrete anomalies
instantons induce an effective term in the Lagrangian
L ⊃ e−
8π2
g2 k∏
i=1
(ψ(i))2ℓ(r(i))
t’Hooft (1976)
This term breaks any ZN symmetry unless∑
i
q(i)ℓ(r(i)) = 0 modN
2
example
ψ(1) ψ(2)
SU(M) M MZ6 3 1
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
ZN anomalies
Consequences of discrete anomalies
instantons induce an effective term in the Lagrangian
L ⊃ e−
8π2
g2 k∏
i=1
(ψ(i))2ℓ(r(i))
t’Hooft (1976)
This term breaks any ZN symmetry unless∑
i
q(i)ℓ(r(i)) = 0 modN
2
example
ψ(1) ψ(2)
SU(M) M MZ6 3 1
anomaly
ψ(1)ψ(2)
ψ(1) ψ(2)
SU(M) M MZ2 1 1
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An SO(10) SUSY GUT example
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The model R. Mohapatra, M. Ratz (2007)
ψm H H ′ ψH ψH A S
SO(10) 16 10 10 16 16 45 54Z6 1 4 2 4 2 0 0
ψm, H and H ′ are SM matter and Higgses
ψH , ψH , A and S break SO(10) → GSM
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The model R. Mohapatra, M. Ratz (2007)
ψm H H ′ ψH ψH A S
SO(10) 16 10 10 16 16 45 54Z6 1 4 2 4 2 0 0
ψm, H and H ′ are SM matter and Higgses
ψH , ψH , A and S break SO(10) → GSM
non-anomalous Z6 suppresses proton decay
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The model R. Mohapatra, M. Ratz (2007)
ψm H H ′ ψH ψH A S
SO(10) 16 10 10 16 16 45 54Z6 1 4 2 4 2 0 0
ψm, H and H ′ are SM matter and Higgses
ψH , ψH , A and S break SO(10) → GSM
non-anomalous Z6 suppresses proton decay
There is an additional U(1)X
SO(10) → SU(5) × U(1)X
16 → 10−1 + 53 + 1−5
Singlet obtains a VEV
U(1)X × Z6 broken by fields with charge (±5,±2)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The model II
U(1)X × Z6 broken by fields with charge (±5,±2)
Naive expectation: Z5 × Z2
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The model II
U(1)X × Z6 broken by fields with charge (±5,±2)
Naive expectation: Z5 × Z2
Q U D L E HU HD
Z30 23 23 1 1 23 14 6Z5 × Z6 (3,1)(3,1)(1,5)(1,5)(3,1)(4,4)(1,0)
This field content is anomalous!
Either more light states are present or the Z6 is not exact.
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A string theory example
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The model
M. Blaszczyk, S. Groot Nibbelink, M. Ratz, F. Ruehle, M. Trapletti, P. Vaudrevange (2009)
Z2 × Z2 orbifoldcompactification of theheterotic string
massless spectrum: 3 ×
generations + vector-like
(local) SU(5) GUT structure
4D gauge group
SU(3)C × SU(2)L × U(1)Y×[SU(3) × SU(2)2 × U(1)8
]
discrete R-symmetriesZR
2 × ZR2 × ZR
2
bcbcSU(5)
bcbcSU(5)
bcbcSU(5)bcbc SU(5)
SU(6)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
The spectrum
massless spectrum:
# representation label # representation label
3 (3, 2; 1, 1, 1) 16
q 3 (3, 1; 1, 1, 1)− 23u
3 + 1 (3, 1; 1, 1, 1) 13
d 1 (3, 1; 1, 1, 1)− 13d
3 + 1(1, 2; 1, 1, 1)− 12ℓ 1 (1, 2; 1, 1, 1) 1
2ℓ
3 (1, 1; 1, 1, 1)1 e 33 (1, 1; 1, 1, 1)0 s
3 (1, 2; 1, 1, 1)− 12h 3 (1, 2; 1, 1, 1) 1
2h
4 (3, 1; 1, 1, 1) 13δ 4 (3, 1; 1, 1, 1)− 1
3δ
5 (1, 1; 3, 1, 1)0 x 5 (1, 1; 3, 1, 1)0 x
6 (1, 1; 1, 1, 2)0 y 6 (1, 1; 1, 2, 1)0 z
some SM singlets get VEV
many choices (string landscape)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
Search for a good vacuum
We look for vacua with the following properties:
all exotics decouple
non-trivial Yukawa couplings
matter parity
Our vacuum has the following properties:
21 fields φ which get a VEV
the φ break U(1)8 × ZR2 × ZR
2 × ZR2 → ZM
2 × ZR4
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZM2 × Z
R4 quantum numbers & implications
quarks and leptons Higgs and exotics
q1 1 2 u1 1 2q2 1 2 u2 1 2q3 1 0 u3 1 0
d1 1 2 ℓ1 1 2d2 1 2 ℓ2 1 2d3 1 0 ℓ3 1 0d4 1 0 ℓ4 1 0d1 1 2 ℓ1 1 2
e1 1 2e2 1 2e3 1 0
h1 0 2 h1 0 2h2 0 0 h2 0 2h3 0 0 h3 0 2
δ1 0 2 δ1 0 2δ2 0 0 δ2 0 0δ3 0 2 δ3 0 0δ4 0 2 δ4 0 0
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZM2 × Z
R4 quantum numbers & implications
quarks and leptons Higgs and exotics
q1 1 2 u1 1 2q2 1 2 u2 1 2q3 1 0 u3 1 0
d1 1 2 ℓ1 1 2d2 1 2 ℓ2 1 2d3 1 0 ℓ3 1 0d4 1 0 ℓ4 1 0d1 1 2 ℓ1 1 2
e1 1 2e2 1 2e3 1 0
h1 0 2 h1 0 2h2 0 0 h2 0 2h3 0 0 h3 0 2
δ1 0 2 δ1 0 2δ2 0 0 δ2 0 0δ3 0 2 δ3 0 0δ4 0 2 δ4 0 0
One massless Higgs pair: Mh =
0 0 0∗ ∗ ∗
∗ ∗ ∗
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZM2 × Z
R4 quantum numbers & implications
quarks and leptons Higgs and exotics
q1 1 2 u1 1 2q2 1 2 u2 1 2q3 1 0 u3 1 0
d1 1 2 ℓ1 1 2d2 1 2 ℓ2 1 2d3 1 0 ℓ3 1 0d4 1 0 ℓ4 1 0d1 1 2 ℓ1 1 2
e1 1 2e2 1 2e3 1 0
h1 0 2 h1 0 2h2 0 0 h2 0 2h3 0 0 h3 0 2
δ1 0 2 δ1 0 2δ2 0 0 δ2 0 0δ3 0 2 δ3 0 0δ4 0 2 δ4 0 0
SU(3)C exotics δ decouple
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZM2 × Z
R4 quantum numbers & implications
quarks and leptons Higgs and exotics
q1 1 2 u1 1 2q2 1 2 u2 1 2q3 1 0 u3 1 0
d1 1 2 ℓ1 1 2d2 1 2 ℓ2 1 2d3 1 0 ℓ3 1 0d4 1 0 ℓ4 1 0d1 1 2 ℓ1 1 2
e1 1 2e2 1 2e3 1 0
h1 0 2 h1 0 2h2 0 0 h2 0 2h3 0 0 h3 0 2
δ1 0 2 δ1 0 2δ2 0 0 δ2 0 0δ3 0 2 δ3 0 0δ4 0 2 δ4 0 0
Yukawas have realistic structure Yu =
∗ ∗ 0∗ ∗ 00 0 1
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZM2 × Z
R4 quantum numbers & implications
quarks and leptons Higgs and exotics
q1 1 2 u1 1 2q2 1 2 u2 1 2q3 1 0 u3 1 0
d1 1 2 ℓ1 1 2d2 1 2 ℓ2 1 2d3 1 0 ℓ3 1 0d4 1 0 ℓ4 1 0d1 1 2 ℓ1 1 2
e1 1 2e2 1 2e3 1 0
h1 0 2 h1 0 2h2 0 0 h2 0 2h3 0 0 h3 0 2
δ1 0 2 δ1 0 2δ2 0 0 δ2 0 0δ3 0 2 δ3 0 0δ4 0 2 δ4 0 0
SU(5) relations Ye = Yd : good for 3rd generation but bad for 1st & 2nd
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZM2 × Z
R4 quantum numbers & implications
quarks and leptons Higgs and exotics
q1 1 2 u1 1 2q2 1 2 u2 1 2q3 1 0 u3 1 0
d1 1 2 ℓ1 1 2d2 1 2 ℓ2 1 2d3 1 0 ℓ3 1 0d4 1 0 ℓ4 1 0d1 1 2 ℓ1 1 2
e1 1 2e2 1 2e3 1 0
h1 0 2 h1 0 2h2 0 0 h2 0 2h3 0 0 h3 0 2
δ1 0 2 δ1 0 2δ2 0 0 δ2 0 0δ3 0 2 δ3 0 0δ4 0 2 δ4 0 0
qi qj qk ℓmproton decay operators involve always 3rd generationfieldy proton stable at this level
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZR4 anomaly
ZR4 is anomalous (partly descends from the so-called ‘anomalous U(1)’)
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZR4 anomaly
ZR4 is anomalous (partly descends from the so-called ‘anomalous U(1)’)
Extra terms at the non-perturbative level:
proton decay operators
[q q q ℓ]light generations ∼ φ15e−a S
mixing between first two and third generations
(Yu)13 ∼ φ4e−a S
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZR4 anomaly
ZR4 is anomalous (partly descends from the so-called ‘anomalous U(1)’)
Extra terms at the non-perturbative level:
proton decay operators
[q q q ℓ]light generations ∼ φ15e−a S
mixing between first two and third generations
(Yu)13 ∼ φ4e−a S
‘Anomalous’ ZR4 explains suppressed µ term and relates the suppression of
proton decay operators to mixing between first two and third generations
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
An SO(10) SUSY GUT exampleA string theory example
ZR4 anomaly
ZR4 is anomalous (partly descends from the so-called ‘anomalous U(1)’)
Extra terms at the non-perturbative level:
proton decay operators
[q q q ℓ]light generations ∼ φ15e−a S
mixing between first two and third generations
(Yu)13 ∼ φ4e−a S
‘Anomalous’ ZR4 explains suppressed µ term and relates the suppression of
proton decay operators to mixing between first two and third generationsMany similar configurations. . .
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Summary
We have shown
how to obtain any discrete, abelian symmetry by spontaneousbreaking U(1)N
There are of course other ways, e.g. extra dimensions
how to eleminate redundancies for discrete, abelian symmetries
how discrete anomalies influence a model
how all this can be applied in model building
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Abelian discrete symmetriesAnomalies
ApplicationsSummary
Summary
We have shown
how to obtain any discrete, abelian symmetry by spontaneousbreaking U(1)N
There are of course other ways, e.g. extra dimensions
how to eleminate redundancies for discrete, abelian symmetries
how discrete anomalies influence a model
how all this can be applied in model building
Thank You!
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