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