Lattice QCD and the QCD Vacuum Structure

1. Lattice QCD and the QCD Vacuum Structure Ivan Horváth University of Kentucky

2. 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 Outline QCD = Quantum Chromodynamics

3. 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

4. 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

5. Why QCD continued…Standard Model

6. 3 Why’s: Why Lattice QCD? • Strange behavior of QCD relative to QED Elementary fields of QED: photon electron gluons Elementary fields of QCD: 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!

7. 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)

8. 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

9. Why Vacuum continued…

10. Why Vacuum continued…

11. Why vacuum continued…

12. 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!

13. Why Vacuum continued… (masses) Hadron propagator

14. Why Vacuum continued… (masses)

15. Vacuum and the Path Integral (Paths) How does one grasp the task of understanding QCD vacuum? • In Quantum Theory vacuum is not a “uniform medium”. Rather it is afluctuating 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.

16. 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 summation over the configurations!!!

17. Summation over the paths continued… • 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 Configuration/Path is a function of space-time variables!

18. Summation over the paths continued… But how do we sum these paths up? There is a representation of QFT (Euclidean field theory) where this is particularly transparent! ensemble QFT All content is stored in the probability distribution! In lattice field theory such statistical sum is meaningfully defined

19. Configurations & the Vacuum Structure STATISTICAL ENSEMBLE OF CONFIGURATIONS VACUUM 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.

20. Degree of space-time order 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

21. Topological Vacuum (What is…) • In QCD it is important to understand behavior of various composite fields fundamental fields composite field • Important composite field is topological charge density

22. What is topological vacuum?continued… Topological charge density is a topological field (stable under deformations) 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

23. 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)

24. Lattice topological field continued… Strictly topological on the space-time lattice!

25. 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 ensemble probabilistic chain Elements of probabilistic chain are “typical configurations”

26. Fundamental Structure I.H. et al, 2003

27. Global Nature of the Structure Characteristics of global behavior saturate faster than physical observables I.H. et. al. 2005 Structure has to be viewed as global!

28. 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

29. 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

30. Space-Filling Feature continued…

31. Space-Filling Feature continued…

32. 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!

33. 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