Scalar Fields within Warped Extra Dimension Aqeel Ahmed

138 Pages·2015·2.64 MB·English

University of Warsaw
Faculty of Physics
Institute of Theoretical Physics
Scalar Fields within Warped Extra Dimension
Aqeel Ahmed
Doctoral Thesis
Thesis Advisor: Prof. Bohdan Grzadkowski
August 2015

ABSTRACT
Inthisthesis,weexploredthreedifferentimplicationsofscalarfieldsinwarpedextradimension.
First, scalar fields were employed to dynamically generate singular branes in Randall-
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Sundrum (RS)-like models by appropriate profiles — the smooth/thick-branes. In the
context of thick-branes, we constructed four different setups: (i) a smooth generalization
of RS2 where a scalar field dynamically generates a singular brane allowing symmetric
or asymmetric warped geometries on either side of the brane; (ii) a double thick-brane
scenario which mimics two positive tension branes and allows to address the hierarchy
problem; (iii)aZ symmetrictriplethick-brane; and(iv)adilatonicthick-branescenario.
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The stability of background solution is verified in all the above mentioned setups.
Second, we considered a thick-brane cosmological model with warped fifth-dimension
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where dynamics of the 4D universe is driven by time-dependent five-dimensional (5D)
background. Different scenarios were found for which the cosmic scale factor a(t,y) and
the scalar field φ(t,y) depend non-trivially on time t and fifth-dimension y.
Third, we discussed a symmetric 5D model with three D3-branes (IR–UV–IR) where the
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Higgs doublet and the other Standard Model (SM) fields are embedded in the bulk. The
Z geometric symmetry led to the warped KK-parity for all the bulk fields. Within this
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setup we investigated the low-energy effective theory for the bulk SM bosonic sector. It
turnedoutthatthezero-modescalarsectorcontainsanevenscalarwhichmimicstheSM
Higgs boson and a second, stable odd scalar particle which is a dark matter candidate.
The model that resulted from the Z -symmetric background geometry resembles the
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Inert Two Higgs Doublet Model. Implications for dark matter were discussed within this
model.
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To the memory of my grandfather
Khuda-Bakhsh
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CONTENTS
Abstract i
Acknowledgments v
Preface vii
1. Introduction 1
1.1. Structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. Conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. RS models and their generalizations 7
2.1. RS models: a brief review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1. RS1: a solution to the hierarchy problem . . . . . . . . . . . . . . . . . 7
2.1.2. RS2: an alternative to compactification . . . . . . . . . . . . . . . . . . 10
2.2. A Z symmetric generalization of RS1: the IR-UV-IR model . . . . . . . . . . . 11
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2.3. Asymmetric generalization of RS2 . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4. Localization of gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. Brane modeling in warped extra dimension 21
3.1. Thick brane generalization of RS1 in modified gravity . . . . . . . . . . . . . . 23
3.1.1. Thick branes with periodic extra dimensions . . . . . . . . . . . . . . . 24
3.1.2. Negative tension brane in modified gravity . . . . . . . . . . . . . . . . 26
3.1.3. Conclusions on thick-brane generalization of the RS1 model . . . . . . . 26
3.2. Modeling branes with scalar fields minimally coupled to gravity . . . . . . . . . 27
3.2.1. Single asymmetric thick-brane model . . . . . . . . . . . . . . . . . . . . 28
3.2.2. Double thick-brane model . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3. Triple Z -symmetric thick-brane model . . . . . . . . . . . . . . . . . . 40
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Contents
3.2.4. Dilatonic thick-brane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.5. Generalized thick-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3. Localization of a scalar field on a thick-brane . . . . . . . . . . . . . . . . . . . 45
3.4. Stability of the background solutions . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.1. Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.2. Vector perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.3. Tensor perturbations and localization of gravity . . . . . . . . . . . . . . 53
3.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4. Thick-brane cosmology 57
4.1. Brane-world cosmology: a brief review . . . . . . . . . . . . . . . . . . . . . . . 57
4.2. Thick brane cosmological solutions . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1. Static thick-brane solutions . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2. Time-dependent thick-brane solutions . . . . . . . . . . . . . . . . . . . 68
4.2.3. Generalized superpotential method . . . . . . . . . . . . . . . . . . . . . 70
4.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5. Warped Higgs dark matter 73
5.1. Warped KK-parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2. SSB in the IR-UV-IR model: the Abelian Higgs mechanism . . . . . . . . . . . 77
5.2.1. SSB by vacuum expectation values of KK modes . . . . . . . . . . . . . 79
5.2.2. SSB by a vacuum expectation value of the 5D Higgs field . . . . . . . . 83
5.3. SM EWSB by a bulk Higgs doublet . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.1. Quantum corrections to scalar masses . . . . . . . . . . . . . . . . . . . 93
5.3.2. Dark matter relic abundance . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6. Summary and conclusions 99
A. Linearized Einstein equations 103
A.1. SVT decomposition of perturbations and gauge choice . . . . . . . . . . . . . . 105
A.2. Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.3. Vector perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.4. Tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B. SSB in the IR-UV-IR model: real scalar case 111
B.1. SSB by vacuum expectation values of KK modes . . . . . . . . . . . . . . . . . 111
B.2. SSB by a vacuum expectation value of 5D scalar field . . . . . . . . . . . . . . 116
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ACKNOWLEDGMENTS
First and foremost, I thank my advisor Bohdan Grzadkowski, for his invaluable guidance,
encouragement and support. I am grateful for his advice and help not just on the technical
aspects, but also at the personal level. I am also indebted to him for supporting me to attend
many schools and conferences which broadened my horizons about particle physics.
I am greatly indebted to my mentors and collaborators, Jack Gunion (UC Davis) and Jose
Wudka (UC Riverside). I would like to thank Jack and Jose for their guidance and hospi-
tality during my visits at their universities. I greatly benefited from their insightful physics
understanding and collaboration.
I would like to thank Yun Jiang for his collaboration and introducing me to the world of
programming. ThanksarealsoduetoLukaszDulnyforhiscollaboration. Ithankallmyfriends
and colleagues at our institute with whom I exchanged many valuable thoughts on physics
and non-physics topics, especially Neda Darvishi, Aleksandra Drozd, Mateusz Duch, Mateusz
Iskrzynski, Saereh Najjari, Abdur Rehman, Bogumila Swiezewska and Pawel Szczerbiak. I
would like to thanks my officemates and colleagues: Subhaditya Bhattacharya, Blazenka Melic
(UCRiverside),MarcGillioz,EnnioSalvioni,YuhsinTsai(UCDavis)and,especiallymyfriend
Shoaib Munir, for discussions on (and off) physics. I would also like to thank the participants
of schools and conferences I have attended, especially Barry Dillon, with whom I had the
opportunity to talk and discuss physics.
I acknowledge the financial support from the Foundation for Polish Science International
PhD Projects Program co-financed by the EU European Regional Development Fund and Na-
tionalCentreforPhysics(Pakistan). IamalsogratefultoUCDavis,UCRiverside,NORDITA,
Galileo Galilei Institute for Theoretical Physics, and Mainz Institute for Theoretical Physics
for their hospitality and partial support during my visits.
Last, but not the least, I would like to thanks my family for their love as well as their in-
valuable support and encouragement throughout my academic career.
Aqeel Ahmed August 2015, Warsaw
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“Scientific thought and its creation are the common
and shared heritage of mankind.”
— Abdus Salam, Nobel Laureate 1979
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PREFACE
Theworkpresentedinthisdissertationisbasedonthefollowingpublications[1,2,3,4,5,6,7]:
1. “Brane modeling in warped extra dimension”,
Aqeel Ahmed and Bohdan Grzadkowski,
JHEP 1301 (2013) 177, [arXiv:1210.6708].
2. “Modeling branes in warped extra dimension”,
Aqeel Ahmed, Lukasz Dulny and Bohdan Grzadkowski,
Acta Phys.Polon. B44 (2013) no. 11, 2381.
3. “Generalized Randall-Sundrum model with a single thick-brane”,
Aqeel Ahmed, Lukasz Dulny and Bohdan Grzadkowski,
Eur. Phys. J. C 74 (2014) 2862 [arXiv:1312.3577].
4. “Thick-Brane Cosmology”,
Aqeel Ahmed, Bohdan Grzadkowski and Jose Wudka,
JHEP 1404 (2014) 061, [arXiv:1312.3576].
5. “Higgs dark matter from a warped extra dimension – the truncated-inert-doublet model”,
Aqeel Ahmed, Bohdan Grzadkowski, John F. Gunion and Yun Jiang,
JHEP 1510 (2015) 033, [arXiv:1504.03706].
6. “Radius stabilization and dark matter with a bulk Higgs in warped extra dimension”,
Aqeel Ahmed, Bohdan Grzadkowski, John F. Gunion and Yun Jiang,
Acta Phys.Polon. B46 (2015) no. 11, in press, [arXiv:1510.04116].
7. “Higgs dark matter from a warped extra dimension”,
Aqeel Ahmed, Bohdan Grzadkowski, John F. Gunion and Yun Jiang,
PoS PLANCK 2015 (2015) 002, [arXiv:1510.05722].
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