The Hierarchy Problem and New Dimensions at a Millimeter Ye Li Graduate Student UW - Madison

Hierarchy Problem  Two “Fundamental” Energy Scale  Electroweak Scale: m EW ~ 10 3 GeV  Planck Scale: M Pl = G N -1/2 ~ GeV  New Framework with Extra dimensions  One Fundamental Scale: Weak Scale  United Gravitational & Gauge Interaction  Independent of SUSY or Technicolor

 Gravitational Potential : n extra spatial dimensions of radius ~ R  Effective 4-D M Pl : M Pl 2 ~ M Pl(4+n) 2+n R n  One Scale Assumption: M Pl(4+n) ~ m EW

 Experimental Consequences  Gravity comparable in strength to gauge interaction at weak scale  Gravitational force law: deviation from 1/r 2 on distance R  SM particle with energy > weak scale escape to extra dimensions  Specific Cases  n = 1, R ~ cm  n = 2, R ~ 100 μm - 1 mm Excluded !!! Particularly Exciting !!!

Phenomenological & Astrophysical Constraints  Total Emission Rate of Graviton 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/M Pl 2  Branching Ratio of Emitting a graviton:

 High Energy Experiments:  Missing Energy carried by massless particles  Absence of relevant decay modes puts strong constraints to the scale m EW and/or n  Astrophysics:  Relate to Goldstone boson’s emission rate F:  Accelerate star’s cooling dynamics Sun: ΔE ~ KeV → F ~ GeV > 10 7 GeV (largest F probed by far) Supernova SN1987A: F ~ 10 8 GeV < GeV Interesting !

Construction of a Realistic Model  Six Dimensions: g = (-1,1,1,1,1,1)  The extra two dimensions x 5, x 6 probed by gravitational force → two-sphere  6-D Scalar Field: Φ  Non-zero VEV: Λ  Two zeros: vortex & anti-vortex (north & south pole)  Nielsen-Olesen Solution:

 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 Solutions with localized massless fermions: Written in 4-D Weyl spinors

 Provided the spinorssatisfies  have definite 4-D chirality   Similar discussion for The vortex supports a single 4-D massless chiral mode which can be

 How to generate mass ? Higgs mechanism still works !  Higgs field potential: In the bulk: r >Λ -1 In the vortex core: r=0 m 2,h’,c > 0 If h’φ bulk -m 2 > 0, positive mass Vortex as an attractive potential

2. Localization of Gauge Fields  Field confinement Two infinite planes repelling the field lines Coulomb’s Law: 1/r 2 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 Confines and develops a mass gapΛ ~ m EW

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)

Thank you !  Reference: N. Arkani-Hamed et al., Phys. Letter B 429 (1998)