1. Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the Lattice, Osaka, March 2013

2. Motivation BSM phenomenology (while we can...) Grand Unification Make sense of a non-renormalizable theory Learn about confinement Non-perturbative questions: Lattice is only known gauge-invariant non-perturbative regulator of QFT

3. Dimensional reduction (3+1)d Fourier decomposition: Thermal boundary conditions:for bosons, fermions Kaluza-Klein tower: static modes for bosons; fermions decouple Additional d.o.f.:or (with extra dim, other b.c. possible, esp. orbifold)

4. Center Symmetry SU(3) Global center transformation: Wilson plaquette action unchanged: Polyakov loop rotated: Order parameter:for confinement high-T: perturbative 1-loop gluonic potential foror spontaneously broken

5. Fundamental quarks: explicitly broken Fundamental quarks (with apbc) favor real sector

6. Why does fundamental matter break ? Fermions (with apbc) in representation R induce term (minus sign from apbc) fundamentaladjoint apbc pbc

7. Non-thermal t-boundary conditions: imaginary chem. pot. Now symmetry!

8. Roberge-Weiss transition Minimum of jumps when

9. Phase diagram (non-perturbative) End-point of RW line can be: critical, triple or tricritical depending on (critical, tricritical gives massless modes)

10. Same with adjoint fermions Centrifugal (apbc) or centripetal (pbc) force Can vary mass & nb. flavors Possibility of {deconfined, “split”, “reconfined”} minima of splitreconfined

11. Observable (gauge-invariant) consequences? At 1-loop, depends on phases of eigenvalues different masses Polyakov loop eigenvalues are gauge- invariant: deconfined splitreconfined invariant under Gauge-symmetry breaking!

12. Non-perturbative issues Phase diagram vs Does the Debye mass really depend on Polyakov eigenvalues ? : 2nd-order phase transitions ? Arnold & Yaffe, 1995

13. Lots to do in (3+1)d Cheaper than extra dimensions Can even substitute bosons for fermions (with pbc apbc)

14. Additional complications in (4+1)d Fermions in odd dimensions: Two inequivalent choices forparity breaking (Chern-Simons term) Or pair together 2 species with massno sign pb (no interesting physics?) Non-renormalizability: Non-perturbative fixed point (Peskin) ? 4d localization (“layered phase”, Fu & Nielsen, etc..) ? Or take lattice as effective description: ~ independent of UV-completion if

15. Lattice SU(2) Yang-Mills in (4+1)d Phase diagram: Coulomb vs confining (first-order) Creutz, 1979 Coulomb phase: dim.red. to 4d for any Tree-level: Lattice spacing shrinks exponentially fast with continuum limit at fixed, non-zero : increase (Wiese et al) anisotropic couplings:

17. Possible continuum limits (w/ Kurkela & Panero) Continuum limit is always 4d All “northeast” directions in plane give 4d continuum Yang- Mills By fine-tuning, can keep adjoint Higgs with “light” mass in 4d theory (Del Debbio et al)

18. The Hosotani mechanism made simple Gluons Fund. quarks apbc Fund. quarks pbc

19. Outlook: 6 dimensions One massless adjoint fermion in 6d after dim. red. In the background of k units of flux: k chiral fermions SM mass hierarchy? No pb. with fermions and parity Possibility of stable flux: Hosotani’s “other mechanism” Flux >0 or <0left- or right-handed fermions in 4d ? Libanov et al.