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Multiple Extra Dimensions andCosmic Sources of KK Particles
Erin De Pree
June 22, 2015
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Plan
1 General introduction to extra dimensions
2 Warped extra dimensions
3 Multiple extra dimensions
4 Involving students in high energy theory
5 Cosmic sources (if time)
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 2 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
S1: Wave function
Continuity requires:
Ψ (y) = Ψ (y + 2πR)
Ψ =1√
2πR
+∞∑n=−∞
ψ(n)e iny/R
E 2 = (pc)2 +(Mc2
)2De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 3 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
S1: Mass
E 2 = (pc)2 +(Mc2
)2= (p3Dc)2 + (pyc)2 +
(Mc2
)2= (p3Dc)2 +
( nR~c)2
+(Mc2
)2︸ ︷︷ ︸(mnc2)2
in particle physics, ~ = c = 1:
m2n =
n2
R2+ M2 Kaluza-Klein (KK) particles
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 4 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
S1: Mass
E 2 = (pc)2 +(Mc2
)2= (p3Dc)2 + (pyc)2 +
(Mc2
)2= (p3Dc)2 +
( nR~c)2
+(Mc2
)2︸ ︷︷ ︸(mnc2)2
in particle physics, ~ = c = 1:
m2n =
n2
R2+ M2 Kaluza-Klein (KK) particles
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 4 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Randall-Sundrum Model
ds2 = e−2krc |φ|︸ ︷︷ ︸warp factor
ηµνdxµdxν − r 2c dφ
2︸ ︷︷ ︸extra dimension
Planck
AdS5
TeV
k
ke−πkrcπrc
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 5 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Multiple Extra Dimensions
Fermions in Randall-Sundrum models with twoadditional unwarpped extra dimensions
Jeremy Perrin (2013 Apker Finalist)and Erin De PreearXiv:1310.1928 [hep-ph]
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 6 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
6D Models
ds2 = e−2krc |φ|ηµνdxµdxν − r 2c dφ
2 − R2dθ2︸ ︷︷ ︸new extra dim
ProblemNo bulk mass terms that are also 6D-chiral andhave four-components
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 7 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
6D Models
ds2 = e−2krc |φ|ηµνdxµdxν − r 2c dφ
2 − R2dθ2︸ ︷︷ ︸new extra dim
ProblemNo bulk mass terms that are also 6D-chiral andhave four-components
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 7 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
7D Warped Extra Dimensions
Instead we add two additional extra dimensions.
Choices
• Compactify on a torus (T2) or a sphere (S2)?
• Apply the warp factor to just the 4Ddimensions or all other dimensions?
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 8 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
AdS5 × T2
ds2 = e−2krc |φ|ηµνdxµdxν − r 2c dφ
2 − R2(dθ21 + dθ22
)Set of coupled differential equations for the KKwavefuntions.
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 9 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
AdS5 × S2
ds2 = e−2krc |φ|ηµνdxµdxν − r 2c dφ
2
− R2(dθ2 + sin2 θdω2
)ProblemNo zero-mode solutions without additional structure
General result for all manifolds of positive curvature.
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 10 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
AdS5 × S2
ds2 = e−2krc |φ|ηµνdxµdxν − r 2c dφ
2
− R2(dθ2 + sin2 θdω2
)ProblemNo zero-mode solutions without additional structure
General result for all manifolds of positive curvature.
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 10 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
AdS5 × S2
ds2 = e−2krc |φ|ηµνdxµdxν − r 2c dφ
2
− R2(dθ2 + sin2 θdω2
)ProblemNo zero-mode solutions without additional structure
General result for all manifolds of positive curvature.
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 10 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Research Group
1 Start early
2 Work in small groups
3 Not every project results in a publication
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 11 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Questions and Discussion
Thank you
ekdepree@smcm.eduarXiv:1505.00024 [astro-ph.HE]
arXiv:1310.1928 [hep-ph]
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 12 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Possible sources of KK particles
Colliders
• Control the source (luminosity, ECM, etc)
• Expensive
Cosmic sources• Free
• No control over the source
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 13 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Possible sources of KK particles
Colliders
• Control the source (luminosity, ECM, etc)
• Expensive
Cosmic sources• Free
• No control over the source
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 13 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Gamma Ray Bursts (GRBs)
Ian Morgan, Ted Tao, De Pree, Kevin TennysonarXiv:1505.00024 [astro-ph.HE]
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 14 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Observer’s frame
Earth
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 15 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Beaming
Earth
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 16 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Results
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 17 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
Questions and Discussion
Thank you
ekdepree@smcm.eduarXiv:1505.00024 [astro-ph.HE]
arXiv:1310.1928 [hep-ph]
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 18 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
S1: Momentum
p2 = p · p = p2x1+ p2x2
+ p2x3+ p2y = p23D + p2y
p̂ = −i~∇, p̂y = −i~ ∂∂y
p̂yΨ =−i~√2πR
ψ(n)e iny/R(in
R
)=
n~R
Ψ
Eigenvalue for p̂y : n~/R
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 19 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
S1: Momentum
p2 = p · p = p2x1+ p2x2
+ p2x3+ p2y = p23D + p2y
p̂ = −i~∇, p̂y = −i~ ∂∂y
p̂yΨ =−i~√2πR
ψ(n)e iny/R(in
R
)=
n~R
Ψ
Eigenvalue for p̂y : n~/R
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 19 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
S1: Fermion KK Spectrum
mass
0
1/R
2/R
ΨL
n ≥ 0 n < 0
ΨR
n ≥ 0 n < 0
For M = 0
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 20 / 21
Introduction Warped Extra Dimensions Multiple Extra Dimensions Student Involvement Cosmic Sources
AdS7 Model
ds2 = e−2krc |φ|[ηµνdx
µdxν − R2(dθ21 + dθ22
)]−r 2c dφ2
De Pree Jun 2015 Multiple Extra Dimensions & Cosmic Sources 21 / 21
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