Why Extra Dimensions on the

Lattice?

Philippe de ForcrandETH Zurich & CERN

Extra Dimensions on the Lattice, Osaka, March 2013

Motivation

•BSM phenomenology (while we can...)

•Grand Unification

•Make sense of a non-renormalizable theory

•Learn about confinementNon-perturbative questions: Lattice is only known gauge-invariant non-perturbative

regulator of QFT

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)

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 for or spontaneously broken

Fundamental quarks: explicitly broken

• Fundamental quarks (with apbc) favor real sector

Why does fundamental matter break ?

• Fermions (with apbc) in representation R induce term

(minus sign from apbc)

fundamental adjoint

apbc apbcpbc pbc

Non-thermal t-boundary conditions: imaginary chem. pot.

• Now

symmetry!

Roberge-Weiss transition

• Minimum of jumps when

Phase diagram (non-perturbative)

• End-point of RW line can be: critical, triple or tricritical depending on

(critical, tricritical gives massless modes)

Same with adjoint fermions

• Centrifugal (apbc) or centripetal (pbc) force

• Can vary mass & nb. flavors

•

• Possibility of {deconfined, “split”, “reconfined”} minima of

split reconfined

Observable (gauge-invariant) consequences?

• At 1-loop, depends on phases of eigenvalues different masses

•Polyakov loop eigenvalues are gauge-invariant:

deconfined split reconfined

invariant under

Gauge-symmetry breaking!

Non-perturbative issues

• Phase diagram vs

• Does the Debye mass really depend on Polyakov eigenvalues ?

: 2nd-order phase transitions ?

Arnold & Yaffe, 1995

Lots to do in (3+1)d

• Cheaper than extra dimensions

•Can even substitute bosons for fermions (with pbc apbc)

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

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:

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)

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 <0 left- or right-handed fermions in 4d ?

Libanov et al.