Why Extra Dimensions on the Lattice?

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) • Wilson plaquette action unchanged: Polyakov loop rotated: • Global center transformation: • Order parameter: for confinement spontaneously broken • high-T: perturbative 1-loop gluonic potential for or

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

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

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

8. Roberge-Weiss transition jumps when • Minimum of

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 • Can vary mass & nb. flavors • Centrifugal (apbc) or centripetal (pbc) force • Possibility of {deconfined, “split”, “reconfined”} minima of split reconfined

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

12. Non-perturbative issues : 2nd-order phase transitions ? • Phase diagram vs • Does the Debye mass really depend on Polyakov eigenvalues ? 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 for parity breaking (Chern-Simons term) Or pair together 2 species with mass no 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 Creutz, 1979 • Phase diagram: Coulomb vs confining (first-order) • 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:

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

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