An Introduction to Lattice QCD and Monte Carlo Simulations Sinya Aoki Institute of Physics, University of Tsukuba 2005 Taipei Summer Institute on Particles and Fields (TSI2005) May 30 - September 16, 2005, NTU, Taipei, Taiwan

Lattice Field Theories 1. Continuum (quantum) Field theories Perturbative expansion needed to define the theory Divergences Regularization/Renormalization Gauge volume Gauge fixing Path-integral quantization, canonical quantization 2. Lattice (quantum) field theories does not rely on perturbation theory lattice spacing regularization continuum limit( ) has to be taken (renormalization) Path-integral in Euclidean space Strong or weak coupling expansions, Monte Carlo method Definition of Quantum Field Theories ?

Introduction to Lattice Field Theory

1. Scalar Field 2. Gauge Field Link variables and action Elitzur’s theorem and Wilson Loop Strong coupling expansion 3. Fermion Field Naive Fermion and doubling problem Nielesen-Ninomiya’s theorem Wilson fermion Contents

1. Scalar Fields on a lattice Scalar field lives on a lattice site Derivative Dimensionless field

2-1. Gauge Fields on a lattice Scalar siteVector link (continuum) Gauge Fields Problem:is not gauge covariant !

2-1. Gauge Fields on a lattice Link variables Gauge transformation covariant

2-2. Gauge Invariance Product of link variables

2-3. Action for gauge fields Gauge invariance Simplest loop (plaquette)

2-4. Action for gauge fields Hausdorff formula

2-5. Action and Path integral Redefined Action Path integral

2-6. Elitzur’s theorem “Local gauge symmetry can not be broken spontaneously”

2-7. Gloval vs. Local Global symmetry: Local symmetry:

2-8. Observables: Wilson Loop Wilson Loop Physical meanings area-law quark confinement perimeter-law deconfinement

2-9. Strong Coupling Expansion SU(N) group integral

2-9. Strong Coupling Expansion(SCE) “gauge field is random”

2-10. Wilson Loop in SCE

Repeating this we obtain Area-law Confinement

3-1. Lattice Fermions: Naive Fermion

3-2. Fermion doubling propagator

3-3. Nielsen-Ninomiya’s theorem (a)Translational invariance (b)Chiral symmetry (c)Hermiticity (d)Fermion bilinear (e)Locality # of poles with + chirality = # of poles with - chirality (Doubling problem exists !) Sketch of the proof Poincare-Hopf’s theorem

Remark: Chiral anomaly

3-4. Solution: Wilson fermions “Wilson term” “decoupling of doublers at low energy” Caution: Wilson term violates chiral symmetry

Hadron mass calculation by Monte Carlo simulations

Path-integral formula Generation of gauge fields U (Heat-bath method) Foundation of Monte Carlo methods Hybrid Monte Carlo (HMC) for full QCD Construction of propagator Extraction of hadron masses from propagator Error analysis and Fit Quark mass dependence of hadron masses Continuum extrapolation Contents

4-0. Hadron bound state of quarks:

4-1. Path Integral Formula Fermion integral

4-2. Example: meson 2-pt. function Here we use Wilson fermion(r=1)

4-3. Procedure of the calculation 1. Generate U with probability 2. Calculate 3. use 4. Construct operator 5. Go to 1 and repeat 6. Finally

Remark “quenched approximation” neglect creation of a quark-antiquark pair from vacuum

5. Generation of U (Monte-Carlo simulations) Pseudo Heat-bath for quenched QCD 1. uniformly generate 2. by analytically solving 3. average Principle

SU(2) lattice gauge theory

Algorithm “N hits pseudo heat-bath algorithm”

SU(3) SU(2) subgroup update

6. Foundation for Monte-Carlo method Markov process Markov chain Definition

Theorem 1

Theorem 2 (1)

Theorem 3 (2)

7-1. Hybrid Monte Carlo (HMC) for full QCD General case

Leap-frog method for MD step Initial Final Intermediate

Proof of detailed valance

Total probability Q.E.D.

7-2. HMC for QCD

Algorithm for HMC in QCD

Leap-frog variation of gauge field

8-1. Construction of propagator Calculation of propagator Linear equation

8-2. Gradient method

8-3. Conjugate Gradient (CG) method Solution

Q.E.D.

8-3. CG algorithm

Example

8-4. Even-odd precondition for Wilson fermion

9-1. Extraction of mass from propagator

9-2. Meson

9-3. Baryon

9-4. Effective mass

10-1. Error Analysis and Fit

10-2. Jack-Knife error

10-3. Fit

10-4. Auto-correlation

10-5. Jack-Knife and auto-correlation

11. Quark mass dependence of hadron masses

11-2. Chiral extrapolation to physical quark mass

12. Continuum extrapolation

Continuum limit