Seiberg Duality

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Seiberg Duality

• James Barnard
• University of Durham

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SUSY disclaimer

• All that following assumes supersymmetry

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SQCD

• Supersymmetric generalisation of QCD
• Gauge group SU(N), chiral flavour group SU(Nf)
• Contains quarks and antiquarks
• For now
• No superpotential
• Lives in the conformal window

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SQCD RG flow

• The theory has two fixed points
• UV fixed point at g0 (i.e. asymptotic freedom)
• Non-trivial IR Seiberg fixed point at gg

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SQCDM

• Pretty similar to SQCD
• Gauge group SU(Ñ), chiral flavour group SU(Nf)
• Contains quarks, antiquarks and elementary
  mesons
• For now
• Also lives in the conformal window
• Superpotential

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SQCDM RG flow

• The theory has three fixed points
• UV fixed point at gy0 (i.e. asymptotic freedom)
• Non-trivial IR Seiberg fixed point with decoupled
  mesons at gg, y0
• Interacting meson fixed point at gg, yy

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The duality
Seibergs conjecture For the physical systems
described by these two fixed points are identical!
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Evidence for Seiberg duality

• Non-anomalous global symmetries, corresponding to
  physical Noether charges, are identical
• Gauge invariant degrees of freedom for each
  theory coincide (classical moduli space matching)
• Highly non-trivial t Hooft anomaly matching
  conditions exist between the two theories
• Duality survives under deformation of the
  theories

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Global symmetries

• Non-anomalous, global symmetry group for both
  theories is

Quark flavour groups
Baryon number
R-symmetry (specific to SUSY fermions and bosons
transform differently)
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Moduli space matching

• Equation of motion for elementary mesons in
  SQCDM removes composite mesons from moduli space
• Results from the SQCDM superpotential
• Baryon matching non-trivial

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t Hooft anomaly matching

• Standard test for dualities in gauge theories
• Imagine gauging the global symmetries
• This generally results in some of the symmetries
  becoming anomalous
• The values of these anomalies can be calculated
• If the values match in both theories it is
  generally accepted that both theories describe
  the same physics
• Highly non-trivial and fully quantum mechanical
  test

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Deformation

• Can add terms to the superpotential of SQCD
• Adding the appropriate terms to the
  superpotential of SQCDM preserves the duality
• Example Massive mesons
• Add quartic coupling to SQCD
• Corresponds to massive elementary mesons in
  SQCDM
• Breaks chiral flavour symmetry to diagonal
  subgroup in both theories
• Allows exact duality

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Deformation
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Why is it useful?

• Outside of the conformal window, Seiberg duality
  is a strong-weak duality - an asymptotically free
  gauge theory is coupled to an infrared free gauge
  theory
• Seiberg duality can be used to form a duality
  cascade - gives an infinite number of
  descriptions for a single physical system
• Duality cascades may be used to amplify the
  effect of, e.g. baryon number violation
• Seiberg duality may allow for a more natural
  unification of gauge couplings in which proton
  decay is highly suppressed
• Any result which improves our understanding of
  gauge theories is a good thing

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Building a Seiberg duality 1
Start with global symmetry group
Assign simplest representations to dual quarks
Match baryons - trivial result
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Building a Seiberg duality 2
Assign alternative representations to dual quarks
Match baryons
Need to add elementary mesons - cannot build
composite operators. Elementary mesons contribute
exactly the right amount to the anomalies for t
Hooft anomaly matching!
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Summary

• Seiberg duality provides a useful tool for
  understanding gauge theories
• Though unproven, there is a lot of highly
  non-trivial evidence supporting the idea
• The mechanisms for constructing general Seiberg
  dualities are not fully understood
• It is hoped that, by investigating these methods,
  it will be possible to construct a Seiberg
  duality for more useful models - such as the
  SU(5) GUT

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Thank you for listening

• Any questions?