"Curved extra-dimensions" by Nicolas Deutschmann (Institut de Physique Nucleaire de Lyon, France)

..
CURVEDEXTRA-DIMENSIONS
Nicolas
Deutschmann
Work in progress with Giacomo Cacciapaglia and Aldo Deandrea
University of the Witwatersrand, Johannesburg
NITheP, July 22th
2014
Nicolas Deutschmann Curved Extra-Dimensions 1/25
...
1/25

.
Outline
Introduction: Why 2UED is attractive
Survey of Positively Curved Geometries
A UED model with Bulk fermions
Localizing fermions
Conclusion: A negative future ?
...
2/25

.
Introduction: Why 2UED is
attractive
...
3/25

.
Ad hoc parities
Many theoretically satisfying solutions to the short-comings of
the Standard Model
...
5/25

.
Ad hoc parities
the Standard Model
• Hierarchy: SuSy, RS,
Little Higgs
...
5/25

.
Ad hoc parities
the Standard Model
Little Higgs
• Neutrinos: See-Saw
...
5/25

.
Ad hoc parities
the Standard Model
Little Higgs
• Neutrinos: See-Saw
• Dark Matter: often a
by-product of other
models with additional
ad hoc parity (R-parity,
KK-parity...)
...
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
...
6/25

.
• The eigenmodes of all ﬁelds in Xn create a KK-tower.
...
6/25

.
• Isometries of Xn: Noether theorem imposes selection rules for
decays
...
6/25

.
• Isometries of Xn: Noether theorem imposes selection rules for
decays
A stable excitation of a neutral SM ﬁeld could be Dark Matter!
...
6/25

.
Why go to curved (n ≥ 2) UED ?
Two limiting factors: Isometries and fermions
1UED
2UED
...
7/25

.
1UED
Odd dimensions: no chiral
fermions
2UED
...
7/25

.
1UED
fermions
Classical trick on S1
/Z2
for a chiral zero-mode
(unique)
2UED
...
7/25

.
1UED
fermions
/Z2
(unique)
No symmetry
2UED
...
7/25

.
1UED
fermions
/Z2
(unique)
No symmetry
2UED
With greater dimensions comes greater freedom
...
7/25

.
1UED
fermions
/Z2
(unique)
No symmetry
2UED
Chiral fermions, but
constructed from both
left- and right-handed
Weyls
...
7/25

.
1UED
fermions
/Z2
(unique)
No symmetry
2UED
Weyls
Similar tricks for some
R2
/G
...
7/25

.
1UED
fermions
/Z2
(unique)
No symmetry
2UED
Weyls
R2
/G
Exactly one ﬂat geometry
(Cacciapaglia et al.)
...
7/25

.
1UED
fermions
/Z2
(unique)
No symmetry
2UED
Weyls
R2
/G
Exactly one ﬂat geometry
(Cacciapaglia et al.)
No systematic survey of curved spaces
...
7/25

.
Survey of Positively Curved
Geometries
...
8/25

.
Positively curved geometries
.
Uniformization theorem..
.
All positively curved 2D surfaces can be described as S2
/G with
G a discrete subgroup of O(3)
...
9/25

.
.
.
/G with
First question: Which of these have non-trivial isometries ?
...
9/25

.
.
.
/G with
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))
...
9/25

.
Orbifolds with symmetries
(a) S2
/Cn (b) S2
/Cnh (c) S2
/Sn (d) S2
/Dn
...
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 ?
...
10/25

.
A UED model with Bulk
fermions
...
11/25

.
A gauge ﬁeld to kill the connection
Method from Randjbar-Daemi, Salam and Strathdee
...
12/25

.
Add an extra U(1) gauge ﬁeld X
...
12/25

.
The connection term in the two Weyl spinors of a chiral 6D
spinors become diﬀerent:
χ = ∂χ + (X + Ω)η η = ∂η + (X − Ω)χ
...
12/25

.
The connection term in the two Weyl spinors of a chiral 6D
spinors become diﬀerent:
χ = ∂χ + (X + Ω)η η = ∂η + (X − Ω)χ
If X cancels ±Ω one of the chiralities has a zero-mode.
...
12/25

.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D
L = LSM
...
13/25

.
Tentative Model
Start by writing the Standard Model Lagrangian in 6D with the
new gauge ﬁeld
L = LSM −
1
4
XµνXµν
...
13/25

.
Tentative Model
new gauge ﬁeld
L = LSM −
1
4
XµνXµν
With ⟨X⟩ a magnetic monopole, ﬁxed by GR:
⟨X⟩ =
n
2g
cos θdϕ =
√
2R
κ
cos θdϕ
...
13/25

.
Tentative Model
new gauge ﬁeld
L = LSM −
1
4
XµνXµν
⟨X⟩ =
n
2g
cos θdϕ =
√
2R
κ
cos θdϕ
How to hide this new gauge boson ?
...
13/25

.
Tentative Model
new gauge ﬁeld and an additional Higgs ﬁeld :
L = LSM −
1
4
XµνXµν
+ |DM H|2
+ µ2
|H|2
−
λ
2
|H|4
⟨X⟩ =
n
2g
cos θdϕ =
√
2R
κ
cos θdϕ
How to hide this new gauge boson ?
...
13/25

.
Higgs Mechanism in a Monopole Background
Need to ﬁnd the minimum
|DM H|2
− µ2
|H|2
+
λ
2
|H|4
A priori θ-dependent :
...
14/25

.
|DM H|2
− µ2
|H|2
+
λ
2
|H|4
A priori θ-dependent : Numerical Solution
...
14/25

.
|DM H|2
− µ2
|H|2
+
λ
2
|H|4
.
Method..
.
Minimization using
Fourier coeﬃcients
...
14/25

.
|DM H|2
− µ2
|H|2
+
λ
2
|H|4
.
Method..
.
Minimization using
Fourier coeﬃcients
1 2 3 4 5
0.001
0.01
0.1
1
1 2 3 4 5
Mode
Αi
...
14/25

.
Eﬀects of the symmetry breaking
• Higgs:
...
15/25

.
• Higgs: O(1/R)
...
15/25

.
• Higgs: O(1/R)
• X: g/R ∼ 10 keV
...
15/25

.
• Higgs: O(1/R)
• X: g/R ∼ 10 keV
Too weak coupling for collider physics.
Compatible with short-range gravity tests
...
15/25

.
• Higgs: O(1/R)
• X: g/R ∼ 10 keV
Too weak coupling for collider physics.
Compatible with short-range gravity tests
1 10 100 1000
10-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
Excludedby
experiment
Lamoreaux
U.Colorado
Stanford2
Stanford1
U.Washington2
gauged
B#
Yukawamessengers
dilaton
KKgravitons
strange
modulus
gluon
modulus
heavyq
moduli
Stanford3
α
λ (µm)
U.Washington1
...
15/25

.
Spectrum of the model
SM Gauge scalar X vector Extra Higgs
0
1
2
3
4
5
6
...
16/25

.
Some Experimental Remarks
.
X boson properties
..
.
• Interacts too weakly for colliders
• Can decay into neutrinos
(Γ ≃ 10−11 eV)
• Not a good DM candidate
...
17/25

.
.
X boson properties
..
.
(Γ ≃ 10−11 eV)
.
Extra-Higgs
..
.
No direct SM interaction so not expected at
colliders
Decay mostly into X pairs
...
17/25

.
.
X boson properties
..
.
(Γ ≃ 10−11 eV)
.
Extra-Higgs
..
.
colliders
.
Tier-1 Excitations..
.
• Need to be pair-produced
• Gluon most likely (massless zero-mode &
QCD)
• Signature: jj + ̸ET
• Loop calculation needed to raise
degeneracy with γ
• Open question: prompt decay to LKK ?
...
17/25

.
.
X boson properties
..
.
(Γ ≃ 10−11 eV)
.
.
• 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
.
Extra-Higgs
..
.
colliders
.
.
QCD)
degeneracy with γ
...
17/25

.
.
X boson properties
..
.
(Γ ≃ 10−11 eV)
.
.
• 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
..
.
colliders
.
.
QCD)
degeneracy with γ
...
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
...
18/25

.
Localizing fermions
...
19/25

.
Branes, fat branes and all that
Bottom-up
All but one symmetric
orbifolds have singular
points
• QFT argument:
localized counter-terms
• GR argument: branes
Top-Bottom ∃ explicit
examples of localization:
• RS (not relevant)
• Georgi mechanism on
S1
/Z2
Nice for later if the model
is phenomenologically
relevant.
...
20/25

.
What the model looks like
• Speciﬁc orbifold
choice: S2
/S4
...
21/25

.
l=0
l=2
l=3
m=0m=-2 m=2
l=1
choice: S2
/S4
• 6D QCD+EW
...
21/25

.
l=0
l=2
l=3
m=0m=-2 m=2
l=1
choice: S2
/S4
• 6D QCD+EW
• 4D Standard model
fermion content with
gauge couplings to the
4D gauge vectors
...
21/25

.
l=0
l=2
l=3
m=0m=-2 m=2
l=1
choice: S2
/S4
• 6D QCD+EW
4D gauge vectors
• ﬁrst tier: unstable
scalars (loop-level)
...
21/25

.
l=0
l=2
l=3
m=0m=-2 m=2
l=1
choice: S2
/S4
• 6D QCD+EW
4D gauge vectors
• ﬁrst tier: unstable
scalars (loop-level)
• second tier: unstable
vectors, stable scalars
(DM)
...
21/25

.
Phenomenology
Two competing constraints: LHC v.s. Dark Matter
.
LHC constraints..
.
• Tier-1 excitations couple at loop level:
• Tier-2 excitations provide resonances:
.
Dark matter..
.
Likely DM candidate: scalar Tier-2 photon:
• Direct detection:
• Relic density:
...
22/25

.
Phenomenology
.
LHC constraints..
.
• Tier-1 excitations couple at loop level: sub-dominant a priori
• Tier-2 excitations provide resonances:
.
Dark matter..
.
• Relic density:
...
22/25

.
Phenomenology
.
LHC constraints..
.
• Tier-2 excitations provide resonances: R ≥ 1.5 TeV
.
Dark matter..
.
• Relic density:
...
22/25

.
Phenomenology
.
LHC constraints..
.
.
Dark matter..
.
Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!
• Relic density:
...
22/25

.
Phenomenology
.
LHC constraints..
.
.
Dark matter..
.
• Direct detection: hard because loop-suppressed
• Relic density:
...
22/25

.
Phenomenology
.
LHC constraints..
.
.
Dark matter..
.
• Direct detection: hard because loop-suppressed
• Relic density: needs loop calculations
...
22/25

.
Conclusion: A negative future ?
...
23/25

.
A Brighter Horizon: Hyperbolic Extra-Dimensions
• Fermions behave much more nicely: there is a massless mode
...
24/25

.
• Much more freedom: arbitrary high volumes for a given
curvature radius
...
24/25

.
curvature radius
• Mass gap protected by curvature radius
...
24/25

.
curvature radius
• Can pull down Mpl while keeping MKK high enough
...
24/25

.
curvature radius
• Many features attractive for cosmology (ﬂatness, uniformity,
inﬂation, ...)
...
24/25

.
curvature radius
inﬂation, ...)
• Topological constraints (genus↔ V/Rn
) make the curvature
radius the only thing to stabilize
...
24/25

.
curvature radius
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
...
24/25

.
Thank you for your attention
...
25/25

"Curved extra-dimensions" by Nicolas Deutschmann (Institut de Physique Nucleaire de Lyon, France)

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (20)

Similar to "Curved extra-dimensions" by Nicolas Deutschmann (Institut de Physique Nucleaire de Lyon, France)

Similar to "Curved extra-dimensions" by Nicolas Deutschmann (Institut de Physique Nucleaire de Lyon, France) (12)

More from Rene Kotze

More from Rene Kotze (20)

Recently uploaded

Recently uploaded (20)

"Curved extra-dimensions" by Nicolas Deutschmann (Institut de Physique Nucleaire de Lyon, France)