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)