Lattice QCD and the QCD Vacuum Structure

Lattice QCD and the QCD Vacuum Structure

Ivan Horváth

University of Kentucky

2Ivan Horváth@University of the Pacific, Apr 2006

Outline QCD = Quantum Chromodynamics

3 Why’s (What’s)

Why Quantum QCD? Why Lattice QCD? Why Vacuum?

Vacuum & Path Integral

Summation over the Paths Configurations and Vacuum

Structure Degree of Space-Time Order

Topological Vacuum

What is Topological Vacuum?

Lattice Topological Field Surprising Structure of

Topological Vacuum Fundamental Structure Global Nature Low-Dimensionality Space-Filling Feature


3 Why’s: Why Quantum Chromodynamics?

Goal of physics is to explain and predict natural phenomena

Historically this proceeded via discovering/understanding forces driving them

Gravity Electromagnetism

Weak Force

Strong Force

Long-range

Long-range

1810 meter

1510 meter


Why Quantum Chromodynamics continued…

Weak and strong force require quantum description Quest for unified description of all fundamental forces

(reductionism) At present this means gauge invariant quantum field theory


Why QCD continued… Standard Model


3 Why’s: Why Lattice QCD?

Strange behavior of QCD relative to QED

( ), ( )A x x Elementary fields of QED:

photon electron

( ), a=1,2,...,8

( ), b 1,2,3

a

b

A x

x

Elementary fields of QCD:

gluons

quarks

Elementary fields/particles of QCD are never observed!

Elementary particles of QCD are influenced by interaction strongly and approximate methods involving them do not work!


Why Lattice QCD continued…

Defining fields and interaction on space-time lattice

allows to define the theory and treat it numerically

Kenneth Wilson (1974) Michael Creutz (1979)

,( )

( )n

n

QCD LQCD

A x U

x

S S


3 Why’s: Why Vacuum?

Vacuum in Quantum Field Theory (QFT) – state in the Hilbert space with lowest energy

Pays the role of the medium where everything happens

Medium can be very important – in QFT medium is pretty much everything!

Look back at the non-observability of elementary particles in QCD: this is usually referred to as the confinement


Why Vacuum continued…




Why vacuum continued…



Understanding of QCD Vacuum is crucial for understanding of strong interactions!

Calculation of all observables in QFT involves calculating vacuum expectation values

Origin of all observables can be traced to vacuum structure!


Why Vacuum continued… (masses)

Hadron propagator


Why Vacuum continued… (masses)


Vacuum and the Path Integral (Paths)

In Quantum Theory vacuum is not a “uniform medium”. Rather it is a fluctuating medium.

This fluctuating nature is most naturally expressed in Feynman’s path integral formulation of quantum theory.

Consider a Quantum-Mechanical particle described by Hamiltonian H and corresponding classical action S.

How does one grasp the task of understanding QCD vacuum?

( )

[ ( )]

, | , | |

( ) ( )

f i

f

i

iH t t

f f i i f i

x

iS x ti i f f

x

x t x t x e x

Dx e x t x x t x


Summation over the paths continued…

Every path x(t) can be thought of as a configuration of this one-dimensional system.

Path integration is a summationover the configurations!!!



What is a generalization to Quantum field Theory?

For a QM particle the configuration/path is one possible history for the dynamical variable involved (its coordinate)

For quantum field it is the same: the history of field values in 3-d space

( ) (x,t)x t

Configuration/Path is a function of space-time variables!



( , P( ) )

But how do we sum these paths up?

There is a representation of QFT (Euclidean field theory)where this is particularly transparent!

QFT ensemble

All content is stored in the probability distribution!

= ( ) ( )P

In lattice field theory such

statistical sum is meaningfully defined


Configurations & the Vacuum Structure

VACUUM STATISTICAL ENSEMBLE OF CONFIGURATIONS

Isn’t this too much fluctuation? Can we learn anything?

BASIC ASSUMPTION of path-integral approach to vacuum structure:

The statistical sum is dominated by a specific kind of configurations with high degree of space-time order (typical

configurations)!

VACUUM STRUCTURE is associated with SPACE-TIME STRUCTURE

in typical configurations.


Degree of space-time order

( )x

( )x

How do we quantify degree of space-time order in a configuration?

01011001011010101110…

binary string S

Kolmogorov complexity of S is a measure of order in

Universal Turing

machine

P(S) S

Minimal length of P(S) in bits is the Kolmogorov complexity of ( )x


Topological Vacuum (What is…)

In QCD it is important to understand behavior of various composite fields

( ), a=1,2,...,8

( ), b 1,2,3

a

b

A x

x

( ) ( ) ( ) a a a b cabcF x A x A x g f A A

fundamental fields

compositefield

Important composite field is topological charge density

2

264( ) ( ) ( )g a aq x F x F x


What is topological vacuum? continued…

4 ( ) 0( ) ( )a a

d x q x QA z A z

Topological vacuum is the vacuum defined by the ensemble of q(x) induced by the QCD ensemble

configuration of A(x) configuration of q(x)

Understanding topological vacuum is considered an important key to understanding QCD vacuum

Topological charge density is a topological field (stable under deformations)


Lattice Topological Field

Topological properties are frequently thought to be tied to continuity of the underlying space-time. Can the lattice analog of topological field be strictly topological?

Yes it can! (Hasenfratz, Laliena, Niedermayer, 1998)

It behaves in a continuum-like manner (integer global charge, index theorem)

Related to defining lattice theory with exact chiral symmetry (Ginsparg-Wilson fermions)

F5

x,y

1( ) tr ( , ) S ( ) ( , ) ( )

2q x D x x x D x y y


Lattice topological field continued…

, ,x

( ) 0( ) ( )a b a b

q x QU z U z

Strictly topological on the space-time lattice!


Surprising Structure of Topological Vacuum

How do we examine the structure of topological vacuum?

Define gauge theory on a finite lattice Generate the ensemble via Monte-Carlo simulation

Calculate the probabilistic chain of topological density

Examine the space-time behavior in typical configurations

( , P( ) ) U U ( 1) ( ) ( 1)..., , , ,...i i iU U U

ensemble probabilistic chain

Elements of probabilistic chain are “typical configurations”

( 1) ( ) ( 1)..., , , ,...i i iq q q


Fundamental Structure

I.H. et al, 2003


Global Nature of the Structure

Characteristics of global behavior saturate faster than physical observables

Structure has to be viewed as global! I.H. et. al. 2005


Low-Dimensional Nature

Claim: It is impossible to embed 4-d manifold in sign-coherent regions of QCD topological structure (I.H. et.al. 2003)

Topological structure has low-dimensional character


Space-Filling Feature

Two seemingly contradictory facts: Coherent topological structure is low-dimensional Occupies finite fraction of space-time

In geometry there are intriguing objects defying this space-filling curves (Peano, 1890)

Finite line occupies zero fraction of a surface


Space-Filling Feature continued…





Peano curve: continuous surjection

QCD structure: continuous surjection

d is the embedding dimension of the structure

QCD topological structure is a quantum analog of space-filling object!

2[0,1] [0,1]

[0,1] [0,1]d

1 4d


Thanks to my collaborators

Andrei Alexandru University of Kentucky Jianbo Zhang University of Adelaide Ying Chen Academia Sinica Shao-Jing Dong University of Kentucky Terry Draper University of Kentucky Frank Lee George Washington

Univ. Keh-Fei Liu University of Kerntucky Nilmani Mathur Jefferson Laboratory Sonali Tamhankar Hamline University Hank Thacker University of Virginia

Documents

Lattice QCD and the QCD Vacuum Structure