The Hierarchy Problem and New Dimensions at a Millimeter
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Transcript of The Hierarchy Problem and New Dimensions at a Millimeter
The Hierarchy Problem and New Dimensions at a Millimeter
Ye Li
Graduate Student
UW - Madison
Hierarchy Problem
Two “Fundamental” Energy Scale Electroweak Scale: mEW ~ 103 GeV
Planck Scale: MPl = GN-1/2 ~ 1018 GeV
New Framework with Extra dimensions One Fundamental Scale: Weak Scale United Gravitational & Gauge Interaction Independent of SUSY or Technicolor
Gravitational Potential :
1 22 1
(4 )
1( )
n nPl n
mmV r
M r
( )r R
n extra spatial dimensions of radius ~ R
Effective 4-D MPl: MPl2 ~ MPl(4+n)
2+n Rn
One Scale Assumption: MPl(4+n) ~ mEW
1 22
(4 )
1( )
n nPl n
mmV r
M R r
r R
1 2/3017 1
10n
n
EW
TeVR cm
m
Experimental Consequences Gravity comparable in strength to gauge inter
action at weak scale Gravitational force law: deviation from 1/r2 o
n distance R SM particle with energy > weak scale escape t
o extra dimensionsSpecific Cases
n = 1, R ~ 1013 cm n = 2, R ~ 100 μm - 1 mm
Excluded !!! Particularly
Exciting !!!
Phenomenological & Astrophysical Constraints
Total Emission Rate of Graviton
2
1( )n
Pl
ERM
Energy available to the graviton
All Kaluza-Klein (KK) excitations of graviton recurring once every 1/R, per extra dimension n
Rate of emitting a single graviton 1/MPl
2
2
n
nEW
E
m
Branching Ratio of Emitting a graviton: 2 n
EW
E
m
2
n
nEW
E
m
High Energy Experiments: Missing Energy carried by massless particles Absence of relevant decay modes puts strong
constraints to the scale mEW and/or nAstrophysics:
Relate to Goldstone boson’s emission rate F:
Accelerate star’s cooling dynamics
22
1/n
nEW
EF
m
Sun: ΔE ~ KeV → F ~ 1012 GeV > 107 GeV (largest F probed by far)
Supernova SN1987A: F ~ 108 GeV < 1010 GeV Interesting !
Construction of a Realistic Model
Six Dimensions: g = (-1,1,1,1,1,1)The extra two dimensions
x5, x6 probed by gravitational force
→ two-sphere6-D Scalar Field: Φ
Non-zero VEV: Λ Two zeros: vortex & anti-vortex
(north & south pole) Nielsen-Olesen Solution:
1( ) , (0) 0, ( ) |ibulkr
f r e f f r
2 ,r R r
What if it’s a torus instead of a two-sphere? Equivalent to a two-torus with zero inner radius Two 4-D vortices become a single one
1. Localization of Fermions and Higgs scalar
A pair of 6-D left-handed Weyl spinors
Couple to the vortex field:
6-D Dirac Eq. in the vortex bkg
7 , , ( , ), ( , )L R L R
. .h h c
A iA bulkh e
Solutions with localized massless fermions:
( ) ( ), ( ) ( )x r x r 1,2,3,4
Written in 4-D Weyl spin
ors
Provided the spinors satisfies
have definite 4-D chirality Similar discussion for
( ), ( )x x 5 6( )
5 ( )i i ir bulke r h e
0( ) exp{ ( ') '}
rr h f r dr ,
,
The vortex supports a single 4-D massless chiral mode which can be
L R
How to generate mass ?
Higgs mechanism still works !
Higgs field potential:
2 2 2( ' ) ( )h m HH c HH
2 2( )m HH c HH 2 2 2( ' ) ( )bulkh m HH c HH
In the bulk: r >Λ-1In the vortex
core: r=0
m2,h’,c > 0
If h’φbulk-m2 > 0, positive mass
Vortex as an attractive potential
2. Localization of Gauge FieldsField confinement
Two infinite planes repelling the field lines
Coulomb’s Law: 1/r2
Coulomb’s Law: 1/r
Flux Conservation
Same sort of model in our
case
4-D Lagrangian:
Higgs mechanism applied on the string Inside the vortex:
2 out of 3 gluons: large masses ~ M the 3rd gluon: a massless photon
Outside the vortex:the photon → non-Abelian gauge theory
2 2 22
22 22 2 2 2
1( ) ( )
4
( ) '( )bulk
L TrG G Dg
h M
Confines and develops a mass gapΛ ~ mEW
3. A Realistic Theory
Standard Model embedded in Pati-Salam group:
In addition: a U(1)V factor and a singlet scalar field Φ
Gauge group spontaneously broken to
Crucial Assumption: Gauge group strongly coupled, developing a mass gap ~ Λ(cutoff)
(4) (2) (2)R LG SU SU SU
(3) (1)EMSU U
Thank you !
Reference: N. Arkani-Hamed et al., Phys. Letter B 429 (1998) 263-272