Lattice Gauge Theory
N.D. Hari Dass
Institute of Mathematical Sciences
Chennai

Quark Confinement, Hadronic Strings and the KABRU Supercomputer
N.D. Hari Dass
Institute of Mathematical Sciences
Chennai

The Periodic Table

The Scale of Things

Artists view of an atom

The Atom

Nuclear forces mediated pions

Neutron spallation

Are neutrons and protons elementary?

Hitting particles with particles

A particle accelerator

High energy electron proton collision

Birth of the hadronic string
Remarkably, if one plotted the angular momentum of the state vs mass(or square of the mass) they all fell on families of parallel straight lines.
Nielsen, Nambu, Goto, Susskind in 1968 made the astonishing suggestion that a relativistically rotating string had states of excitation this spectrum!

Slide 14

The quark model

Pion in the quark model
The relativity theory says that there is an antiparticle for each particle.
The antiparticle of u-quark with electric charge 2/3 is u-bar antiquark with  charge -2/3.
The pion in this model is made up of u and d-bar .

What holds the quarks together?
If there were no forces between quarks (anti – quarks) they would just fly apart without forming hadrons like protons, pions, neutrons etc…
The next important question is : what is the nature of these forces?
Are they attractive, repulsive …
How do they vary with distance and what is their dependence on the quark type?

The way electric charges interact: Quantum Electrodynamics

Quantum Chromodynamics
It turns out that fully consistency with the hadrons requires that each quark type comes in 3 species which will symbolically be denoted as red , blue and green .
Observed hadrons only correspond to combinations that are white.
For example : proton is u(red), u(blue) and d(green).
The analog of photons are gluons and they come in 8 colours.

Quarks and Gluons of SU(3) QCD

What is QCD? – More technically…
QCD is a non-Abelian Gauge Theory.
The gauge group is SU(3).
The Quarks carry the fundamental representation 3.
The Gluons, which transmit the forces between quarks, carry the adjoint representation 8.
3 has triality, 8 has no triality.
The theory has defied analytical results despite the best efforts for more than 25 years.
It is an outstanding problem of theoretical physics.

Mathematical Formulation of QCD
First let us look at electrodynamics described by Maxwell’s equations .

"Now we have 8 vector..."
Now we have 8 vector potentials and 8 scalar potentials:

Discretising QCD

More on updating

The Wilson Loop

The Great Quark Puzzle
It should be possible  to release quarks by supplying sufficient energy to protons and neutrons.
But no quarks were liberated even in very high energy collisions!

Making A Virtue Out Of An Embarassment
“Mama Mia, if you don’t see something you should be seeing, there must be a very deep reason”---- N. Testa
The theory of Quarks must be such that they can never be free!

Is Such A Thing Possible At All?
‘t Hooft, Mandelstam, Nambu: A thin flux tube connects the quarks and keeps them from being liberated.
This can happen to magnetic monopoles submerged in a superconductor.
Dual Superconductor Picture.
Can this be proven in Quantum Chromodynamics (QCD), the best candidate theory of Quarks and Gluons?

Slide 30

The dual picture
Instead of charges consider magnetic monopoles:

Numerical Simulations
Discretise the space-time manifold.
Continue from Minkowski signature to Euclidean signature.
The Quantum Field Theory is mapped exactly into Classical Statistical Mechanics in Four spatial dimensions.
Simulate that numerically.

Numerical  Evidence For Flux Tubes

The latest simulation results on flux tubes
Due to a path breaking development in algorithms it is now possible to measure extremely tiny quantities on the lattice.
The flux tube is actually a string ! It is not rigid but can actually fluctuate like a string.
Even its states of excitations can be probed numerically and they indeed look like hadronic states.

The Polyakov Loop

The string picture of protons, neutrons and pions…

Coming Back To Flux Tubes
If quarks are not too heavy, the flux tube can snap.
If quarks carry adjoint charge no flux tube should be formed.

Unresolved Issues
What causes the flux tube formation?
Colour Monopoles?
Colour Vortices(Z3)?
Instantons and topological charges?