Seiberg Duality

Seiberg Duality

James BarnardUniversity of Durham

SUSY disclaimer

All that following assumes supersymmetry

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

SQCD RG flow

The theory has two fixed points:1. UV fixed point at g=0 (i.e.

asymptotic freedom)2. Non-trivial IR Seiberg fixed

point at g=g*

SQCD+M

• 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

SQCD+M RG flow

The theory has three fixed points:

1. UV fixed point at g=y=0 (i.e. asymptotic freedom)

2. Non-trivial IR Seiberg fixed point with decoupled mesons at g=g*, y=0

3. Interacting meson fixed point at g=g*’, y=y*’

Seiberg’s conjecture:For

the physical systems described by these two fixed points are identical!

The duality

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

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)

Moduli space matching

• Equation of motion for elementary mesons in SQCD+M removes composite mesons from moduli space

• Results from the SQCD+M superpotential

• Baryon matching non-trivial

‘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

Deformation

• Can add terms to the superpotential of SQCD• Adding the appropriate terms to the superpotential of

SQCD+M preserves the dualityExample: Massive mesons• Add quartic coupling to SQCD

• Corresponds to massive elementary mesons in SQCD+M

• Breaks chiral flavour symmetry to diagonal subgroup in both theories

• Allows exact duality…

Deformation

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

Building a Seiberg duality 1

Start with global symmetry group

Assign simplest representations to dual quarks

Match baryons - trivial result

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!

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

Thank you for listening

Any questions?