Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the...
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Transcript of Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the...
![Page 1: Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the Lattice, Osaka, March 2013.](https://reader035.fdocuments.net/reader035/viewer/2022062519/5697bf9c1a28abf838c935c0/html5/thumbnails/1.jpg)
Why Extra Dimensions on the
Lattice?
Philippe de ForcrandETH Zurich & CERN
Extra Dimensions on the Lattice, Osaka, March 2013
![Page 2: Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the Lattice, Osaka, March 2013.](https://reader035.fdocuments.net/reader035/viewer/2022062519/5697bf9c1a28abf838c935c0/html5/thumbnails/2.jpg)
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
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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)
![Page 4: Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the Lattice, Osaka, March 2013.](https://reader035.fdocuments.net/reader035/viewer/2022062519/5697bf9c1a28abf838c935c0/html5/thumbnails/4.jpg)
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
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Fundamental quarks: explicitly broken
• Fundamental quarks (with apbc) favor real sector
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Why does fundamental matter break ?
• Fermions (with apbc) in representation R induce term
(minus sign from apbc)
fundamental adjoint
apbc apbcpbc pbc
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Non-thermal t-boundary conditions: imaginary chem. pot.
• Now
symmetry!
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Roberge-Weiss transition
• Minimum of jumps when
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Phase diagram (non-perturbative)
• End-point of RW line can be: critical, triple or tricritical depending on
(critical, tricritical gives massless modes)
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Same with adjoint fermions
• Centrifugal (apbc) or centripetal (pbc) force
• Can vary mass & nb. flavors
•
• Possibility of {deconfined, “split”, “reconfined”} minima of
split reconfined
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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!
![Page 12: Why Extra Dimensions on the Lattice? Philippe de Forcrand ETH Zurich & CERN Extra Dimensions on the Lattice, Osaka, March 2013.](https://reader035.fdocuments.net/reader035/viewer/2022062519/5697bf9c1a28abf838c935c0/html5/thumbnails/12.jpg)
Non-perturbative issues
• Phase diagram vs
• Does the Debye mass really depend on Polyakov eigenvalues ?
: 2nd-order phase transitions ?
Arnold & Yaffe, 1995
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Lots to do in (3+1)d
• Cheaper than extra dimensions
•Can even substitute bosons for fermions (with pbc apbc)
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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
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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:
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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)
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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.