..
CURVEDEXTRA-DIMENSIONS
Nicolas Deutschmann
Work in progress with Giacomo Cacciapaglia and Aldo Deandrea
University of the Witwatersrand, JohannesburgNITheP, July 22th 2014
Nicolas Deutschmann Curved Extra-Dimensions 1/25...
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Outline
Introduction: Why 2UED is attractive
Survey of Positively Curved Geometries
A UED model with Bulk fermions
Localizing fermions
Conclusion: A negative future ?
Nicolas Deutschmann Curved Extra-Dimensions 2/25...
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Introduction: Why 2UED isattractive
Nicolas Deutschmann Curved Extra-Dimensions 3/25...
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Dark Matter!
.
Ad hoc parities
Many theoretically satisfying solutions to the short-comings ofthe Standard Model
• Hierarchy: SuSy, RS,Little Higgs
• Neutrinos: See-Saw• Dark Matter: often a
by-product of othermodels with additionalad hoc parity (R-parity,KK-parity...)
Nicolas Deutschmann Curved Extra-Dimensions 5/25...
5/25
.
Ad hoc parities
Many theoretically satisfying solutions to the short-comings ofthe Standard Model
• Hierarchy: SuSy, RS,Little Higgs
• Neutrinos: See-Saw• Dark Matter: often a
by-product of othermodels with additionalad hoc parity (R-parity,KK-parity...)
Nicolas Deutschmann Curved Extra-Dimensions 5/25...
5/25
.
Ad hoc parities
Many theoretically satisfying solutions to the short-comings ofthe Standard Model
• Hierarchy: SuSy, RS,Little Higgs
• Neutrinos: See-Saw
• Dark Matter: often aby-product of othermodels with additionalad hoc parity (R-parity,KK-parity...)
Nicolas Deutschmann Curved Extra-Dimensions 5/25...
5/25
.
Ad hoc parities
Many theoretically satisfying solutions to the short-comings ofthe Standard Model
• Hierarchy: SuSy, RS,Little Higgs
• Neutrinos: See-Saw• Dark Matter: often a
by-product of othermodels with additionalad hoc parity (R-parity,KK-parity...)
Nicolas Deutschmann Curved Extra-Dimensions 5/25...
5/25
.
Ad hoc parities
Many theoretically satisfying solutions to the short-comings ofthe Standard Model
• Hierarchy: SuSy, RS,Little Higgs
• Neutrinos: See-Saw• Dark Matter: often a
by-product of othermodels with additionalad hoc parity (R-parity,KK-parity...)
Nicolas Deutschmann Curved Extra-Dimensions 5/25...
5/25
.
A potential solution in the UED frame work
Universal Extra-Dimensions: all fields propagate in the bulk
• (4 + n)D space: M4 × Xn a compact space
• The eigenmodes of all fields in Xn create a KK-tower.• Isometries of Xn: Noether theorem imposes selection rules for
decays
A stable excitation of a neutral SM field could be Dark Matter!
Nicolas Deutschmann Curved Extra-Dimensions 6/25...
6/25
.
A potential solution in the UED frame work
Universal Extra-Dimensions: all fields propagate in the bulk
• (4 + n)D space: M4 × Xn a compact space• The eigenmodes of all fields in Xn create a KK-tower.
• Isometries of Xn: Noether theorem imposes selection rules fordecays
A stable excitation of a neutral SM field could be Dark Matter!
Nicolas Deutschmann Curved Extra-Dimensions 6/25...
6/25
.
A potential solution in the UED frame work
Universal Extra-Dimensions: all fields propagate in the bulk
• (4 + n)D space: M4 × Xn a compact space• The eigenmodes of all fields in Xn create a KK-tower.• Isometries of Xn: Noether theorem imposes selection rules for
decays
A stable excitation of a neutral SM field could be Dark Matter!
Nicolas Deutschmann Curved Extra-Dimensions 6/25...
6/25
.
A potential solution in the UED frame work
Universal Extra-Dimensions: all fields propagate in the bulk
• (4 + n)D space: M4 × Xn a compact space• The eigenmodes of all fields in Xn create a KK-tower.• Isometries of Xn: Noether theorem imposes selection rules for
decays
A stable excitation of a neutral SM field could be Dark Matter!
Nicolas Deutschmann Curved Extra-Dimensions 6/25...
6/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UED
Odd dimensions: no chiralfermions
Classical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UED
Chiral fermions, butconstructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedomNo systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermions
Classical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UED
Chiral fermions, butconstructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedomNo systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)
No symmetry
2UED
Chiral fermions, butconstructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedomNo systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UED
Chiral fermions, butconstructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedomNo systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UED
Chiral fermions, butconstructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedom
No systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UEDChiral fermions, but
constructed from bothleft- and right-handed
Weyls
Similar tricks for someR2/G
Exactly one flat geometry(Cacciapaglia et al.)
With greater dimensions comes greater freedom
No systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UEDChiral fermions, but
constructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/G
Exactly one flat geometry(Cacciapaglia et al.)
With greater dimensions comes greater freedom
No systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UEDChiral fermions, but
constructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedom
No systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Why go to curved (n ≥ 2) UED ?Two limiting factors: Isometries and fermions
1UEDOdd dimensions: no chiral
fermionsClassical trick on S1/Z2for a chiral zero-mode
(unique)No symmetry
2UEDChiral fermions, but
constructed from bothleft- and right-handed
WeylsSimilar tricks for some
R2/GExactly one flat geometry
(Cacciapaglia et al.)
With greater dimensions comes greater freedomNo systematic survey of curved spaces
Nicolas Deutschmann Curved Extra-Dimensions 7/25...
7/25
.
Survey of Positively CurvedGeometries
Nicolas Deutschmann Curved Extra-Dimensions 8/25...
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Positively curved geometries
.Uniformization theorem..
.All positively curved 2D surfaces can be described as S2/G withG a discrete subgroup of O(3)
First question: Which of these have non-trivial isometries ?Fundamental relation:
S ∈ Isom(S2/G) ⇐⇒ ∀g ∈ G, ∃h ∈ G|S(g(x)) = h(S(x))
Nicolas Deutschmann Curved Extra-Dimensions 9/25...
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Positively curved geometries
.Uniformization theorem..
.All positively curved 2D surfaces can be described as S2/G withG a discrete subgroup of O(3)
First question: Which of these have non-trivial isometries ?
Fundamental relation:
S ∈ Isom(S2/G) ⇐⇒ ∀g ∈ G, ∃h ∈ G|S(g(x)) = h(S(x))
Nicolas Deutschmann Curved Extra-Dimensions 9/25...
9/25
.
Positively curved geometries
.Uniformization theorem..
.All positively curved 2D surfaces can be described as S2/G withG a discrete subgroup of O(3)
First question: Which of these have non-trivial isometries ?Fundamental relation:
S ∈ Isom(S2/G) ⇐⇒ ∀g ∈ G, ∃h ∈ G|S(g(x)) = h(S(x))
Nicolas Deutschmann Curved Extra-Dimensions 9/25...
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Orbifolds with symmetries
(a) S2/Cn (b) S2/Cnh (c) S2/Sn (d) S2/Dn
Next question: Which of this can embed 4D chiral fermions ?
Nicolas Deutschmann Curved Extra-Dimensions 10/25...
10/25
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Orbifolds with symmetries
(a) S2/Cn (b) S2/Cnh (c) S2/Sn (d) S2/Dn
Next question: Which of this can embed 4D chiral fermions ?
Nicolas Deutschmann Curved Extra-Dimensions 10/25...
10/25
.
Orbifolds with symmetries
(a) S2/Cn (b) S2/Cnh (c) S2/Sn (d) S2/Dn
Next question: Which of this can embed 4D chiral fermions ?
Nicolas Deutschmann Curved Extra-Dimensions 10/25...
10/25
.
A UED model with Bulkfermions
Nicolas Deutschmann Curved Extra-Dimensions 11/25...
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A gauge field to kill the connection
Method from Randjbar-Daemi, Salam and Strathdee
Add an extra U(1) gauge field X
The connection term in the two Weyl spinors of a chiral 6Dspinors become different:
∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ
If X cancels ±Ω one of the chiralities has a zero-mode.
Nicolas Deutschmann Curved Extra-Dimensions 12/25...
12/25
.
A gauge field to kill the connection
Method from Randjbar-Daemi, Salam and Strathdee
Add an extra U(1) gauge field X
The connection term in the two Weyl spinors of a chiral 6Dspinors become different:
∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ
If X cancels ±Ω one of the chiralities has a zero-mode.
Nicolas Deutschmann Curved Extra-Dimensions 12/25...
12/25
.
A gauge field to kill the connection
Method from Randjbar-Daemi, Salam and Strathdee
Add an extra U(1) gauge field X
The connection term in the two Weyl spinors of a chiral 6Dspinors become different:
∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ
If X cancels ±Ω one of the chiralities has a zero-mode.
Nicolas Deutschmann Curved Extra-Dimensions 12/25...
12/25
.
A gauge field to kill the connection
Method from Randjbar-Daemi, Salam and Strathdee
Add an extra U(1) gauge field X
The connection term in the two Weyl spinors of a chiral 6Dspinors become different:
∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ
If X cancels ±Ω one of the chiralities has a zero-mode.
Nicolas Deutschmann Curved Extra-Dimensions 12/25...
12/25
.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D
with thenew gauge field and an additional Higgs field :
L = LSM
− 14
XµνXµν + |DMH|2 + µ2|H|2 − λ
2|H|4
With ⟨X⟩ a magnetic monopole, fixed by GR:
⟨X⟩ = n
2gcos θdϕ =
√2R
κcos θdϕ
How to hide this new gauge boson ?
Nicolas Deutschmann Curved Extra-Dimensions 13/25...
13/25
.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D with thenew gauge field
and an additional Higgs field :
L = LSM − 14
XµνXµν
+ |DMH|2 + µ2|H|2 − λ
2|H|4
With ⟨X⟩ a magnetic monopole, fixed by GR:
⟨X⟩ = n
2gcos θdϕ =
√2R
κcos θdϕ
How to hide this new gauge boson ?
Nicolas Deutschmann Curved Extra-Dimensions 13/25...
13/25
.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D with thenew gauge field
and an additional Higgs field :
L = LSM − 14
XµνXµν
+ |DMH|2 + µ2|H|2 − λ
2|H|4
With ⟨X⟩ a magnetic monopole, fixed by GR:
⟨X⟩ = n
2gcos θdϕ =
√2R
κcos θdϕ
How to hide this new gauge boson ?
Nicolas Deutschmann Curved Extra-Dimensions 13/25...
13/25
.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D with thenew gauge field
and an additional Higgs field :
L = LSM − 14
XµνXµν
+ |DMH|2 + µ2|H|2 − λ
2|H|4
With ⟨X⟩ a magnetic monopole, fixed by GR:
⟨X⟩ = n
2gcos θdϕ =
√2R
κcos θdϕ
How to hide this new gauge boson ?
Nicolas Deutschmann Curved Extra-Dimensions 13/25...
13/25
.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D with thenew gauge field and an additional Higgs field :
L = LSM − 14
XµνXµν + |DMH|2 + µ2|H|2 − λ
2|H|4
With ⟨X⟩ a magnetic monopole, fixed by GR:
⟨X⟩ = n
2gcos θdϕ =
√2R
κcos θdϕ
How to hide this new gauge boson ?
Nicolas Deutschmann Curved Extra-Dimensions 13/25...
13/25
.
Higgs Mechanism in a Monopole Background
Need to find the minimum
|DMH|2 − µ2|H|2 + λ
2|H|4
A priori θ-dependent :
Numerical Solution
.Method..
.Minimization usingFourier coefficients
1 2 3 4 5
0.001
0.01
0.1
1
1 2 3 4 5
Mode
Αi
Nicolas Deutschmann Curved Extra-Dimensions 14/25...
14/25
.
Higgs Mechanism in a Monopole Background
Need to find the minimum
|DMH|2 − µ2|H|2 + λ
2|H|4
A priori θ-dependent : Numerical Solution
.Method..
.Minimization usingFourier coefficients
1 2 3 4 5
0.001
0.01
0.1
1
1 2 3 4 5
Mode
Αi
Nicolas Deutschmann Curved Extra-Dimensions 14/25...
14/25
.
Higgs Mechanism in a Monopole Background
Need to find the minimum
|DMH|2 − µ2|H|2 + λ
2|H|4
A priori θ-dependent : Numerical Solution
.Method..
.Minimization usingFourier coefficients
1 2 3 4 5
0.001
0.01
0.1
1
1 2 3 4 5
Mode
Αi
Nicolas Deutschmann Curved Extra-Dimensions 14/25...
14/25
.
Higgs Mechanism in a Monopole Background
Need to find the minimum
|DMH|2 − µ2|H|2 + λ
2|H|4
A priori θ-dependent : Numerical Solution
.Method..
.Minimization usingFourier coefficients
1 2 3 4 5
0.001
0.01
0.1
1
1 2 3 4 5
Mode
Αi
Nicolas Deutschmann Curved Extra-Dimensions 14/25...
14/25
.
Effects of the symmetry breaking• Higgs:
O(1/R)• X: g/R ∼ 10 keV
Too weak coupling for collider physics.Compatible with short-range gravity tests
1 10 100 100010-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
Excludedbyexperiment
Lamoreaux
U.Colorado
Stanford2
Stanford1
U.Washington2
gauged
B#
Yukawamessengers
dilaton
KKgravitons
strange
modulus
gluon
modulus
heavyq
moduli
Stanford3
α
λ (µm)
U.Washington1
Nicolas Deutschmann Curved Extra-Dimensions 15/25...
15/25
.
Effects of the symmetry breaking• Higgs: O(1/R)
• X: g/R ∼ 10 keV
Too weak coupling for collider physics.Compatible with short-range gravity tests
1 10 100 100010-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
Excludedbyexperiment
Lamoreaux
U.Colorado
Stanford2
Stanford1
U.Washington2
gauged
B#
Yukawamessengers
dilaton
KKgravitons
strange
modulus
gluon
modulus
heavyq
moduli
Stanford3
α
λ (µm)
U.Washington1
Nicolas Deutschmann Curved Extra-Dimensions 15/25...
15/25
.
Effects of the symmetry breaking• Higgs: O(1/R)• X: g/R ∼ 10 keV
Too weak coupling for collider physics.Compatible with short-range gravity tests
1 10 100 100010-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
Excludedbyexperiment
Lamoreaux
U.Colorado
Stanford2
Stanford1
U.Washington2
gauged
B#
Yukawamessengers
dilaton
KKgravitons
strange
modulus
gluon
modulus
heavyq
moduli
Stanford3
α
λ (µm)
U.Washington1
Nicolas Deutschmann Curved Extra-Dimensions 15/25...
15/25
.
Effects of the symmetry breaking• Higgs: O(1/R)• X: g/R ∼ 10 keV
Too weak coupling for collider physics.Compatible with short-range gravity tests
1 10 100 100010-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
Excludedbyexperiment
Lamoreaux
U.Colorado
Stanford2
Stanford1
U.Washington2
gauged
B#
Yukawamessengers
dilaton
KKgravitons
strange
modulus
gluon
modulus
heavyq
moduli
Stanford3
α
λ (µm)
U.Washington1
Nicolas Deutschmann Curved Extra-Dimensions 15/25...
15/25
.
Effects of the symmetry breaking• Higgs: O(1/R)• X: g/R ∼ 10 keV
Too weak coupling for collider physics.Compatible with short-range gravity tests
1 10 100 100010-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
Excludedbyexperiment
Lamoreaux
U.Colorado
Stanford2
Stanford1
U.Washington2
gauged
B#
Yukawamessengers
dilaton
KKgravitons
strange
modulus
gluon
modulus
heavyq
moduli
Stanford3
α
λ (µm)
U.Washington1
Nicolas Deutschmann Curved Extra-Dimensions 15/25...
15/25
.
Spectrum of the model
SM Gauge scalar X vector Extra-Higgs
0
1
2
3
4
5
6
Nicolas Deutschmann Curved Extra-Dimensions 16/25...
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Some Experimental Remarks.X boson properties..
.
• Interacts too weakly for colliders• Can decay into neutrinos
(Γ ≃ 10−11 eV)• Not a good DM candidate
.Tier-2 Excitations..
.
• As for Tier-1: Gluon• If Isom(S2/G) = Z2: Single production• Need less √
s
• Striking jj resonance easier to look for• Sets a limit around 1.5 − 2 TeV
• Constraints can be escaped if S2/G hasa continuous symmetry
.Extra-Higgs..
.No direct SM interaction so not expected atcollidersDecay mostly into X pairs
.Tier-1 Excitations..
.
• Need to be pair-produced• Gluon most likely (massless zero-mode &
QCD)• Signature: jj + ET
• Loop calculation needed to raisedegeneracy with γ
• Open question: prompt decay to LKK ?
Nicolas Deutschmann Curved Extra-Dimensions 17/25...
17/25
.
Some Experimental Remarks.X boson properties..
.
• Interacts too weakly for colliders• Can decay into neutrinos
(Γ ≃ 10−11 eV)• Not a good DM candidate
.Tier-2 Excitations..
.
• As for Tier-1: Gluon• If Isom(S2/G) = Z2: Single production• Need less √
s
• Striking jj resonance easier to look for• Sets a limit around 1.5 − 2 TeV
• Constraints can be escaped if S2/G hasa continuous symmetry
.Extra-Higgs..
.No direct SM interaction so not expected atcollidersDecay mostly into X pairs
.Tier-1 Excitations..
.
• Need to be pair-produced• Gluon most likely (massless zero-mode &
QCD)• Signature: jj + ET
• Loop calculation needed to raisedegeneracy with γ
• Open question: prompt decay to LKK ?
Nicolas Deutschmann Curved Extra-Dimensions 17/25...
17/25
.
Some Experimental Remarks.X boson properties..
.
• Interacts too weakly for colliders• Can decay into neutrinos
(Γ ≃ 10−11 eV)• Not a good DM candidate
.Tier-2 Excitations..
.
• As for Tier-1: Gluon• If Isom(S2/G) = Z2: Single production• Need less √
s
• Striking jj resonance easier to look for• Sets a limit around 1.5 − 2 TeV
• Constraints can be escaped if S2/G hasa continuous symmetry
.Extra-Higgs..
.No direct SM interaction so not expected atcollidersDecay mostly into X pairs
.Tier-1 Excitations..
.
• Need to be pair-produced• Gluon most likely (massless zero-mode &
QCD)• Signature: jj + ET
• Loop calculation needed to raisedegeneracy with γ
• Open question: prompt decay to LKK ?
Nicolas Deutschmann Curved Extra-Dimensions 17/25...
17/25
.
Some Experimental Remarks.X boson properties..
.
• Interacts too weakly for colliders• Can decay into neutrinos
(Γ ≃ 10−11 eV)• Not a good DM candidate
.Tier-2 Excitations..
.
• As for Tier-1: Gluon• If Isom(S2/G) = Z2: Single production• Need less √
s
• Striking jj resonance easier to look for• Sets a limit around 1.5 − 2 TeV
• Constraints can be escaped if S2/G hasa continuous symmetry
.Extra-Higgs..
.No direct SM interaction so not expected atcollidersDecay mostly into X pairs
.Tier-1 Excitations..
.
• Need to be pair-produced• Gluon most likely (massless zero-mode &
QCD)• Signature: jj + ET
• Loop calculation needed to raisedegeneracy with γ
• Open question: prompt decay to LKK ?
Nicolas Deutschmann Curved Extra-Dimensions 17/25...
17/25
.
Some Experimental Remarks.X boson properties..
.
• Interacts too weakly for colliders• Can decay into neutrinos
(Γ ≃ 10−11 eV)• Not a good DM candidate
.Tier-2 Excitations..
.
• As for Tier-1: Gluon• If Isom(S2/G) = Z2: Single production• Need less √
s
• Striking jj resonance easier to look for• Sets a limit around 1.5 − 2 TeV• Constraints can be escaped if S2/G has
a continuous symmetry
.Extra-Higgs..
.No direct SM interaction so not expected atcollidersDecay mostly into X pairs
.Tier-1 Excitations..
.
• Need to be pair-produced• Gluon most likely (massless zero-mode &
QCD)• Signature: jj + ET
• Loop calculation needed to raisedegeneracy with γ
• Open question: prompt decay to LKK ?
Nicolas Deutschmann Curved Extra-Dimensions 17/25...
17/25
.
Summing up the model
Spherical Extra-Dimensions are hard to construct.• Chiral fermions do not come easily• Need to add two extra-fields: X and H ′
• Unsatisfying because X and H ′ are rather untestable They are ugly
Nicolas Deutschmann Curved Extra-Dimensions 18/25...
18/25
.
Localizing fermions
Nicolas Deutschmann Curved Extra-Dimensions 19/25...
19/25
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Branes, fat branes and all that
Bottom-upAll but one symmetricorbifolds have singularpoints• QFT argument:
localized counter-terms• GR argument: branes
Top-Bottom ∃ explicitexamples of localization:• RS (not relevant)• Georgi mechanism on
S1/Z2
Nice for later if the modelis phenomenologicallyrelevant.
Nicolas Deutschmann Curved Extra-Dimensions 20/25...
20/25
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What the model looks like
l=0
l=2
l=3
m=0m=-2 m=2
l=1
• Specific orbifoldchoice: S2/S4
• 6D QCD+EW• 4D Standard model
fermion content withgauge couplings to the4D gauge vectors
• first tier: unstablescalars (loop-level)
• second tier: unstablevectors, stable scalars(DM)
Nicolas Deutschmann Curved Extra-Dimensions 21/25...
21/25
.
What the model looks like
l=0
l=2
l=3
m=0m=-2 m=2
l=1
• Specific orbifoldchoice: S2/S4
• 6D QCD+EW
• 4D Standard modelfermion content withgauge couplings to the4D gauge vectors
• first tier: unstablescalars (loop-level)
• second tier: unstablevectors, stable scalars(DM)
Nicolas Deutschmann Curved Extra-Dimensions 21/25...
21/25
.
What the model looks like
l=0
l=2
l=3
m=0m=-2 m=2
l=1
• Specific orbifoldchoice: S2/S4
• 6D QCD+EW• 4D Standard model
fermion content withgauge couplings to the4D gauge vectors
• first tier: unstablescalars (loop-level)
• second tier: unstablevectors, stable scalars(DM)
Nicolas Deutschmann Curved Extra-Dimensions 21/25...
21/25
.
What the model looks like
l=0
l=2
l=3
m=0m=-2 m=2
l=1
• Specific orbifoldchoice: S2/S4
• 6D QCD+EW• 4D Standard model
fermion content withgauge couplings to the4D gauge vectors
• first tier: unstablescalars (loop-level)
• second tier: unstablevectors, stable scalars(DM)
Nicolas Deutschmann Curved Extra-Dimensions 21/25...
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.
What the model looks like
l=0
l=2
l=3
m=0m=-2 m=2
l=1
• Specific orbifoldchoice: S2/S4
• 6D QCD+EW• 4D Standard model
fermion content withgauge couplings to the4D gauge vectors
• first tier: unstablescalars (loop-level)
• second tier: unstablevectors, stable scalars(DM)
Nicolas Deutschmann Curved Extra-Dimensions 21/25...
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.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter.LHC constraints..
.
• Tier-1 excitations couple at loop level:
sub-dominant a priori
• Tier-2 excitations provide resonances:
R ≥ 1.5 TeV
.Dark matter..
.
Likely DM candidate: scalar Tier-2 photon:
WIMPZILLA!
• Direct detection:
hard because loop-suppressed
• Relic density:
needs loop calculations
Nicolas Deutschmann Curved Extra-Dimensions 22/25...
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.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter.LHC constraints..
.
• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances:
R ≥ 1.5 TeV
.Dark matter..
.
Likely DM candidate: scalar Tier-2 photon:
WIMPZILLA!
• Direct detection:
hard because loop-suppressed
• Relic density:
needs loop calculations
Nicolas Deutschmann Curved Extra-Dimensions 22/25...
22/25
.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter.LHC constraints..
.
• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances: R ≥ 1.5 TeV
.Dark matter..
.
Likely DM candidate: scalar Tier-2 photon:
WIMPZILLA!
• Direct detection:
hard because loop-suppressed
• Relic density:
needs loop calculations
Nicolas Deutschmann Curved Extra-Dimensions 22/25...
22/25
.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter.LHC constraints..
.
• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances: R ≥ 1.5 TeV
.Dark matter..
.
Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!• Direct detection:
hard because loop-suppressed
• Relic density:
needs loop calculations
Nicolas Deutschmann Curved Extra-Dimensions 22/25...
22/25
.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter.LHC constraints..
.
• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances: R ≥ 1.5 TeV
.Dark matter..
.
Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!• Direct detection: hard because loop-suppressed• Relic density:
needs loop calculations
Nicolas Deutschmann Curved Extra-Dimensions 22/25...
22/25
.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter.LHC constraints..
.
• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances: R ≥ 1.5 TeV
.Dark matter..
.
Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!• Direct detection: hard because loop-suppressed• Relic density: needs loop calculations
Nicolas Deutschmann Curved Extra-Dimensions 22/25...
22/25
.
Conclusion: A negative future ?
Nicolas Deutschmann Curved Extra-Dimensions 23/25...
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.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode
• Much more freedom: arbitrary high volumes for a givencurvature radius
• Mass gap protected by curvature radius• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,
inflation, ...)• Topological constraints (genus↔ V/Rn) make the curvature
radius the only thing to stabilize• Quantum corrections could play a significant role as Mpl goes
down
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode• Much more freedom: arbitrary high volumes for a given
curvature radius
• Mass gap protected by curvature radius• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,
inflation, ...)• Topological constraints (genus↔ V/Rn) make the curvature
radius the only thing to stabilize• Quantum corrections could play a significant role as Mpl goes
down
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode• Much more freedom: arbitrary high volumes for a given
curvature radius• Mass gap protected by curvature radius
• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,
inflation, ...)• Topological constraints (genus↔ V/Rn) make the curvature
radius the only thing to stabilize• Quantum corrections could play a significant role as Mpl goes
down
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode• Much more freedom: arbitrary high volumes for a given
curvature radius• Mass gap protected by curvature radius• Can pull down Mpl while keeping MKK high enough
• Many features attractive for cosmology (flatness, uniformity,inflation, ...)
• Topological constraints (genus↔ V/Rn) make the curvatureradius the only thing to stabilize
• Quantum corrections could play a significant role as Mpl goesdown
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode• Much more freedom: arbitrary high volumes for a given
curvature radius• Mass gap protected by curvature radius• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,
inflation, ...)
• Topological constraints (genus↔ V/Rn) make the curvatureradius the only thing to stabilize
• Quantum corrections could play a significant role as Mpl goesdown
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode• Much more freedom: arbitrary high volumes for a given
curvature radius• Mass gap protected by curvature radius• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,
inflation, ...)• Topological constraints (genus↔ V/Rn) make the curvature
radius the only thing to stabilize
• Quantum corrections could play a significant role as Mpl goesdown
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode• Much more freedom: arbitrary high volumes for a given
curvature radius• Mass gap protected by curvature radius• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,
inflation, ...)• Topological constraints (genus↔ V/Rn) make the curvature
radius the only thing to stabilize• Quantum corrections could play a significant role as Mpl goes
down
Nicolas Deutschmann Curved Extra-Dimensions 24/25...
24/25
.
Thank you for your attention
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