1. 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

2. 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 ?

3. Introduction to Lattice Field Theory

4. 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

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

6. 2-1. Gauge Fields on a lattice Scalar siteVector link (continuum) Gauge Fields Problem:is not gauge covariant !

7. 2-1. Gauge Fields on a lattice Link variables Gauge transformation covariant

8. 2-2. Gauge Invariance Product of link variables

9. 2-3. Action for gauge fields Gauge invariance Simplest loop (plaquette)

10. 2-4. Action for gauge fields Hausdorff formula

11. 2-5. Action and Path integral Redefined Action Path integral

12. 2-6. Elitzur’s theorem “Local gauge symmetry can not be broken spontaneously”

13. 2-7. Gloval vs. Local Global symmetry: Local symmetry:

14. 2-8. Observables: Wilson Loop Wilson Loop Physical meanings area-law quark confinement perimeter-law deconfinement

15. 2-9. Strong Coupling Expansion SU(N) group integral

16. 2-9. Strong Coupling Expansion(SCE) “gauge field is random”

17. 2-10. Wilson Loop in SCE

18. Repeating this we obtain Area-law Confinement

19. 3-1. Lattice Fermions: Naive Fermion

20. 3-2. Fermion doubling propagator

21. 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

23. Remark: Chiral anomaly

24. 3-4. Solution: Wilson fermions “Wilson term” “decoupling of doublers at low energy” Caution: Wilson term violates chiral symmetry

25. Hadron mass calculation by Monte Carlo simulations

26. 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

27. 4-0. Hadron bound state of quarks:

28. 4-1. Path Integral Formula Fermion integral

29. 4-2. Example: meson 2-pt. function Here we use Wilson fermion(r=1)

30. 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

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

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

33. SU(2) lattice gauge theory

35. Algorithm “N hits pseudo heat-bath algorithm”

36. SU(3) SU(2) subgroup update

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

38. Theorem 1

39. Theorem 2 (1)

40. Theorem 3 (2)

41. 7-1. Hybrid Monte Carlo (HMC) for full QCD General case

42. Leap-frog method for MD step Initial Final Intermediate

43. Proof of detailed valance

44. Total probability Q.E.D.

45. 7-2. HMC for QCD

46. Algorithm for HMC in QCD

47. Leap-frog variation of gauge field

48. 8-1. Construction of propagator Calculation of propagator Linear equation

49. 8-2. Gradient method

50. 8-3. Conjugate Gradient (CG) method Solution

51. Q.E.D.

53. 8-3. CG algorithm

54. Example

55. 8-4. Even-odd precondition for Wilson fermion

56. 9-1. Extraction of mass from propagator

57. 9-2. Meson

58. 9-3. Baryon

59. 9-4. Effective mass

60. 10-1. Error Analysis and Fit

61. 10-2. Jack-Knife error

62. 10-3. Fit

63. 10-4. Auto-correlation

65. 10-5. Jack-Knife and auto-correlation

66. 11. Quark mass dependence of hadron masses

68. 11-2. Chiral extrapolation to physical quark mass

69. 12. Continuum extrapolation

70. Continuum limit