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

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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 ?


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Introduction: Why 2UED isattractive


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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...)


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Ad hoc parities






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Ad hoc parities



• Neutrinos: See-Saw

• Dark Matter: often aby-product of othermodels with additionalad hoc parity (R-parity,KK-parity...)


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Ad hoc parities






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Ad hoc parities






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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!


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• (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



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• (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



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• (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



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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


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1UEDOdd dimensions: no chiral

fermions

Classical trick on S1/Z2for a chiral zero-mode

(unique)No symmetry

2UED







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fermionsClassical trick on S1/Z2for a chiral zero-mode

(unique)

No symmetry

2UED







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(unique)No symmetry

2UED







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(unique)No symmetry

2UED





With greater dimensions comes greater freedom

No systematic survey of curved spaces


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(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.)




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(unique)No symmetry




R2/G

Exactly one flat geometry(Cacciapaglia et al.)




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(unique)No symmetry









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(unique)No symmetry








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Survey of Positively CurvedGeometries


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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))


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First question: Which of these have non-trivial isometries ?

Fundamental relation:



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First question: Which of these have non-trivial isometries ?Fundamental relation:



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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 ?


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A UED model with Bulkfermions


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.


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∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ



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∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ



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∇χ = ∂χ + (X + Ω)η ∇η = ∂η + (X − Ω)χ



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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 ?


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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


⟨X⟩ = n

2gcos θdϕ =

√2R

κcos θdϕ



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Tentative Model



L = LSM − 14

XµνXµν

+ |DMH|2 + µ2|H|2 − λ

2|H|4


⟨X⟩ = n

2gcos θdϕ =

√2R

κcos θdϕ



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Tentative Model



L = LSM − 14

XµνXµν

+ |DMH|2 + µ2|H|2 − λ

2|H|4


⟨X⟩ = n

2gcos θdϕ =

√2R

κcos θdϕ



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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


⟨X⟩ = n

2gcos θdϕ =

√2R

κcos θdϕ



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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


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|DMH|2 − µ2|H|2 + λ

2|H|4

A priori θ-dependent : Numerical Solution

.Method..


1 2 3 4 5

0.001

0.01

0.1

1

1 2 3 4 5

Mode

Αi


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|DMH|2 − µ2|H|2 + λ

2|H|4


.Method..


1 2 3 4 5

0.001

0.01

0.1

1

1 2 3 4 5

Mode

Αi


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|DMH|2 − µ2|H|2 + λ

2|H|4


.Method..


1 2 3 4 5

0.001

0.01

0.1

1

1 2 3 4 5

Mode

Αi


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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


15/25

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Effects of the symmetry breaking• Higgs: O(1/R)

• X: g/R ∼ 10 keV


1 10 100 100010-2

10-1

100

101

102

103

104

105

106

107

108

109

1010


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

.

Effects of the symmetry breaking• Higgs: O(1/R)• X: g/R ∼ 10 keV


1 10 100 100010-2

10-1

100

101

102

103

104

105

106

107

108

109

1010


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

.



1 10 100 100010-2

10-1

100

101

102

103

104

105

106

107

108

109

1010


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

.



1 10 100 100010-2

10-1

100

101

102

103

104

105

106

107

108

109

1010


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


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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


.

• 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 ?


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.


s



.Extra-Higgs..



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s



.Extra-Higgs..



.






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s



.Extra-Higgs..



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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..



.






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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


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Localizing fermions


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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.


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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)


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l=0

l=2

l=3

m=0m=-2 m=2

l=1


• 6D QCD+EW

• 4D Standard modelfermion content withgauge couplings to the4D gauge vectors




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l=0

l=2

l=3

m=0m=-2 m=2

l=1







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l=0

l=2

l=3

m=0m=-2 m=2

l=1







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l=0

l=2

l=3

m=0m=-2 m=2

l=1







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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


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Phenomenology


.

• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances:

R ≥ 1.5 TeV

.Dark matter..

.


WIMPZILLA!



• Relic density:



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Phenomenology


.

• Tier-1 excitations couple at loop level: sub-dominant a priori• Tier-2 excitations provide resonances: R ≥ 1.5 TeV

.Dark matter..

.


WIMPZILLA!



• Relic density:



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Phenomenology


.


.Dark matter..

.

Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!• Direct detection:


• Relic density:



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Phenomenology


.


.Dark matter..

.

Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!• Direct detection: hard because loop-suppressed• Relic density:



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Phenomenology


.


.Dark matter..

.

Likely DM candidate: scalar Tier-2 photon: WIMPZILLA!• Direct detection: hard because loop-suppressed• Relic density: needs loop calculations


22/25

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Conclusion: A negative future ?


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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


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• 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,



down


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curvature radius• Mass gap protected by curvature radius

• Can pull down Mpl while keeping MKK high enough• Many features attractive for cosmology (flatness, uniformity,



down


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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


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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



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radius the only thing to stabilize



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down


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Thank you for your attention


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Education

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