Lattice Chiral Gauge Theory

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Lattice Chiral Gauge Theory

• Richard C. Brower
• Extreme Scale Computing Workshop
• - Quantum Universe_at_ Stanford,Dec. 9-11 , 2008

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Outline for LCGT

• Motivation
• 3 Proposals for LCGT
• 1. Exact gauge chiral
  Overlap
• 2. Exact gauge decoupled Mirrors Domain Wall
• 3. Gauge fixed Counter terms Wilson
  Fermions
• Prospects at Exaflops ?

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Few references (of 143 on hep-lat)

• Method 1
• M. Luscher, Nucl. Phys. B549 (1999) 295 "Chiral
  gauge theories revisited hep-th/0102028
  regularization of chiral gauge theories to all
  orders of perturbation theory, JHEP 0006, 028
  (2000) arXivhep-lat/0006014.
• Kadoh and Kikukawa, A simple construction of
  fermion measure term in U(1) chiral lattice gauge
  theories with exact gauge invariance
  arXiv0709.3656
• Method 2
• M. Creutz, M. Tytgat, C. Rebbi and S-S
  Xue,Lattice formulation of the standard model
  Phys.Lett.B402341-345,1997
• E. Poppitz and Y. Shang, Lattice Chirality and
  the Decoupling of Mirror FermionsarXiv0706.1043
  
• J. Giedt and E. Poppitz, Chiral lattice gauge
  theories and the strong coupling dynamics of a
  Yukawa-Higgs model with Ginsparg-Wilson fermions,
  arXivhep-lat/0701004 .
• T. Bhattacharya, M. R. Martin and E.
  Poppitz,Phys. Rev. D 74, 085028 (2006)
  arXivhep-lat/0605003 .
• Method 3
• M. Golterman, Lattice chiral gauge theories,
  Nucl. Phys. Proc. Suppl. 94, 189 (2001)
  arXivhep-lat/0011027 .
• M. Golterman and Y. Shamir, SU(N) chiral gauge
  theories on the lattice Phys. Rev. D 70, 094506
  (2004) arXivhep-lat/0404011 .

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Motivation

• Standard Model is a Chiral Gauge Theory
• Electro-weak Sector Vector QCD
• DO CHIRAL GAUGE THEORIES EXIST?
• One loop quantum effects require Anomaly
  Cancellation. Gauge invariant perturbation
  theory
• But Global Anomalies can arise
  non-perturbatively.
• PROPERTIES OF STRONG CHIRAL GAUGE DYNAMICS?
• Beyond the Standard Model (BMS) may require
  strong Chiral Dynamics.
• Lattice is a possible rigorous approach to
  Chiral Gauge Theories
• But it isnt easy! (Good news really?)

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Quote from BSM whitepaper(see www.usqcd.org)

• Another major unsolved problem in lattice gauge
  theory is how to formulate a theory where the
  fermions are coupled to the gauge ?elds in a
  non-vector-like manner. We know that neutrinos
  are left handed so, a chiral formulation is
  essential. The apparent difficulty of formulating
  such theories on the lattice may be a hint at
  deep physics issues. Are mirror particles to the
  neutrinos required, perhaps at some large mass?
  Is the breaking of parity inherent in chiral
  theories of a spontaneous origin? The chiral
  coupling to weak interactions in the standard
  model exploits the Higgs mechanism is some form
  of spontaneous breaking a required feature of
  chiral theories? A non-perturbative formulation
  is crucial to even framing these questions.

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1 Exact gauge and Chiral on lattice (Luscher).

• Action Guarantee (classical) invariance with
  overlap action
• with
• Anomaly free Rep D RU
• Measure Must define a measure that is gauge
  invariant at the quantum level.

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Partition Function

• Measure
• has a gauge dependent phase ambiguity because
  of constraint
• Gauge EOM comes from local
• Locality Must choose phase in the measure so
  that ja¹(x)
• is local and gauge invariant!

From Fermion Action
From Measure
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Gauge Invariance

• Infinitesimal gauge invariance
• To get gauge invariant local current ja¹(x)
  
• In continuum limit requires zero anomaly dR
  0
• General construction only known for smooth
  U(1) lattice gauge fields with

Gauge dep of measure
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2. Chiral fermion a s1 and Mirrors at sLs
(Poppitz)

• Measure Trivial measure of a vector theory.
• Anomaly free Rep on both walls.
• Add Yukawa (or 4 Fermi ) Hopefully push Mirror
  Fermions to cut-off.

y Equivalence of DW and overlap means this can be
rewritten as before!
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3 Gauge Fixed (Golterman-Shamir)

• Continuum perturbation theory is taken as a
  guide.
• smooth gauge fixing to (hopefully) avoid
  quantum fluctuations that drive you into a
  vector-like phase.
• Add counter terms to give BRST invariance in
  continuum
• U(1) example

This is not a very easy way to proceed. Need to
tune counter terms and regaing gauge invariance
in the continuum limit.
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Prospects for Exaflops

• Method 2 to me looks the most promising.
• Teraflop-years sufficient to test LCGT proposals
• If LCGT is correct, then Exaflops would be
  useful as part of a rapid exploration of BSM
  studies.
• Remember
• 1 Teraflops year 5 hour at 0.01 Exaflop/s
  sustained!
• Exaflop/s transformational role!
• Fast turn allows rapid exploration of BSM
  proposals.
• Need flexible software tool box.