Perspectives in Lattice QCD: Proceedings of the Workshop, Nara International Seminar House, Nara, Japan, 31 October-11 November 2005

CONTENTS......Page 8
Preface......Page 6
1. Introduction......Page 10
2. Cut-off effects: a numerical experiment on the running coupling......Page 11
4. Saddle point equation for the fixed point action i n QCD......Page 15
5. Perfect classical lattice theories......Page 17
6. The FP Dirac operator satisfies the Ginsparg-Wilson relation......Page 18
7. Lattice regularization and symmetry transformations......Page 19
References......Page 22
1. Introduction......Page 24
2.1. Monte Carlo Methods......Page 26
2.2. Central Limit Theorem......Page 27
2.4. Convergence of Markov Chains......Page 30
2.5. Detailed Balance and the Metropolis Algorithm......Page 33
2.6. Composition of Markov Steps......Page 34
2.7. Coupling from the Past......Page 35
2.8.2. Integrated Autocorrelation......Page 37
2.9. Hybrid Monte Carlo......Page 38
2.10. MDMC......Page 39
2.11. Partial Momentum Refreshment......Page 40
2.12. Baker-Campbell-Hausdorff (B CH) formula......Page 41
2.13. Symplectic Integrators......Page 42
2.14. Multiple Timescales......Page 44
2.15. Dynamical Fewnions......Page 45
2.16. Reversibility......Page 46
3.1. Polynomial Approximation......Page 48
3.2. Chebyshev’s theorem......Page 49
3.3. Chebyshev Polynomials......Page 51
3.4. Chebyshev Optimal Rational Approximation......Page 52
3.5. Non-Linearity of CG Solver......Page 54
3.7. \'No Free Lunch\' Theorem......Page 55
3.8. Multiple Pseudofermions......Page 56
3.9. Violation of NFL theorem......Page 57
3.10. Rational Hybrid Monte Carlo......Page 58
3.11.1. Finite Temperature QCD......Page 60
3.11.2. 2+1 Domain Wall Fermions......Page 62
3.12. Multiple Pseudofermions with Multiple Timescales......Page 63
3.13. Lz versus I;, Force Norms......Page 65
3.15. Conclusions (RHMC)......Page 66
4. 5D Algorithms for Chiral Lattice Fermions......Page 67
4.2. Neuberger’s Operator......Page 68
4.3. 5D Chiral Fermions......Page 69
4.5. Schur Complement......Page 71
4.6. Constraint......Page 72
4.7.1. Hyperbolic Tangent......Page 73
4.7.2. Zolotarev’s Formula......Page 75
4.7.3. Errors......Page 76
4.8.1. Continued Fraction......Page 77
4.8.2. Partial Fraction......Page 80
4.8.3. Euclidean Cayley Transform......Page 81
4.8.4. Relation to Domain Wall Fermions......Page 83
4.9. Chiral Symmetry Breaking......Page 86
4.11. Future Work......Page 87
Acknowledgments......Page 89
References......Page 90
1. Overview and Aims......Page 92
2.1. Eflective Field Theories in general......Page 95
2 . 2 . Chiral symmetry in QCD and its breaking......Page 98
2.3.1. Building blocks for C e......Page 102
2.3.2. Brief aside on vacuum structure......Page 104
2.3.3. Properties of pseudo-Nambu-Goldstone bosons at leading order......Page 105
2.3.4. Lessons for lattice simulations......Page 107
2.3.5. Power counting an XPT (M = 0)......Page 108
2.3.6. Lessons for lattice simulations (continued)......Page 110
2.3.7. Technical aside: adding sources......Page 111
2.3.8. Final form of chiral Lagrangian......Page 113
2.4. Examples of NLO results......Page 114
2.4.1. Lessons for lattice simulations (continued)......Page 116
2.4.2. Volume dependence from XPT......Page 117
2.4.3. Convergence of XPT......Page 118
2.4.4. Extension to “heavy” particles......Page 119
3.1. Why incorporate discretization errors?......Page 120
3.2. General strategy......Page 121
3.3. Application to Wilson & twisted mass fermions......Page 122
3.4. Determining the local effective Lagrangian......Page 123
3.5. Symanzik effective action for tmLQCD......Page 127
3.6.1. Power counting and terminology......Page 130
3.6.2. Mapping L$io and Lgio into XPT......Page 131
3.7.1. TmXPT at LO......Page 135
3.7.2. tmXPT at NLO......Page 137
3.8. Defining m, and the twist angle......Page 142
3.9. Results for m, N a2A& (Aoki regime)......Page 146
3.9.1. Applications to lattice simulations......Page 148
3.9.2. Bending near maximal twist......Page 153
3.9.3. Does tmLQCD work in practice?......Page 155
4.1. What is PQQCD and why is it useful?......Page 156
4.2. A field theoretic formulation of PQQCD......Page 157
4.3. Developing PQXPT......Page 160
4.3.2. Brief primer on graded Lie groups......Page 161
4.3.3. Chiral symmetry breaking......Page 162
4.3.4. Constructing the EFT......Page 163
4.3.5. What about C pQ ?......Page 166
4.4. PQXPT at LO......Page 167
4.5. NLO calculations in PQXPT and outlook......Page 170
4.5.1. Status of PQXPT calculations......Page 171
4.5.2. A final example: L7......Page 172
References......Page 173
1. Introduction......Page 178
2. Continuum QCD and chiral transformations......Page 179
3. Standard Wilson quarks......Page 181
3.1. Wilson quarks and unphysical fernionic zero modes......Page 182
3.1.1. Quenched and partially quenched approximations......Page 183
3.1.2. Potential problems in the Hybrid Monte Carlo algorithm......Page 184
4. Twisted mass lattice QCD......Page 185
4.1. Equivalence between tmQCD and QCD......Page 186
4.2. Beyond the formal continuum theory......Page 188
4.3. Lattice tmQCD with Wilson quarks......Page 189
5.1. Computation of F,......Page 191
5.2. Direct determination of the chiral condensate......Page 192
5.3. The computation of BK......Page 193
5.3.1. Renorrnalisation of OVA+A......Page 195
5.4. Further applications......Page 196
6. O(a) improvement and tmQCD......Page 197
6.2. Automatic O(a) improvement of tmQCD in a finite volume......Page 198
6.2.1. Uncertainty of the chiral limit......Page 201
6.3. Automatic O(a) improvement in infinite volume......Page 202
7. Consequences of Parity and Flavour breaking......Page 203
7.1. Non-degenerate quarks and additional flavours......Page 204
8. A chiral twist to the QCD Schrodinger functional......Page 205
8.1. The QCD Schrodinger functional......Page 206
8.2. Decoupling of heavy quarks in SF schemes......Page 208
8.2.1. Free quarks with SF boundary conditions......Page 209
8.3. SF boundary conditions and chiral rotations......Page 210
8.4. SF schemes with Wilson quarks and O(a) improvement......Page 211
8.5. The Schrodinger functional and O(a) improvement......Page 212
8.6.1. Symmetries and Counterterms......Page 213
8.7. An example from perturbation theory......Page 214
Acknowledgments......Page 215
References......Page 216
1. Introduction......Page 218
1.1. Basic renormalization: hadron spectrum......Page 219
1.3. Irrelevant operators......Page 221
1.4. Heavy Quark Effective Theory......Page 222
1.1.1. Definition......Page 223
1.1.2. Quantum mechanical interpretation......Page 224
1.1.3. Background field......Page 225
1.1.4. Perturbative expansion......Page 226
1.1.5. General renormalization properties......Page 227
1.1.6. Renormalized coupling......Page 228
1.1.7. Quarks......Page 229
1.1.7.1. Correlation functions......Page 230
1.1.7.2. Renormalized mass......Page 231
1.1.8. Lattice formulation......Page 233
1.1.8.1. Boundary conditions and the background field......Page 234
1.1.8.3. Explicit expression for......Page 235
1.1.9. More literature......Page 236
1.2. Chiral symmetry and O(a)-improvement......Page 237
1.2.1. Chiml Ward identities......Page 238
1.2.2.1. Motivation......Page 240
1.2.2.2. A warning from two dimensions......Page 242
1.2.2.3. Symanzik’s local eflective theory (SET)......Page 243
1.2.2.4. Improved lattice action and fields......Page 245
1.2.3. The PCAC relation......Page 246
1.2.4. Non-perturbative improvement......Page 247
1.2.4.1. The constant physics condition......Page 250
1.2.4.2. Summary of results......Page 253
1.2.4.3. O(a2) effects after improvement for Nf = 2......Page 255
1.2.4.4. Do we need more?......Page 256
11.1.1. The extraction of (Y f r o m experiments......Page 258
11.1.2. Reaching large scales in lattice QCD......Page 261
11.2.1. The step scaling function......Page 264
11.2.2. Lattice spacing eflects in perturbation theory......Page 266
11.2.3. The continuum limit - universality......Page 269
11.2.4. The running of the coupling......Page 271
11.2.5. The A parameter......Page 272
11.2.6. Discussion......Page 273
11.2.7. Improvements are necessary......Page 275
11.3. Renormalization group invariant quark masses......Page 276
11.4. Renormalization scale dependence of other composite operators......Page 279
111.1. Introduction......Page 281
111.1.1. Derivation of the classical theory......Page 283
111.2.1. The static approximation and its symmetries......Page 285
111.2.1.1. Lattice formulation......Page 286
111.2.1.2. Renormalization......Page 287
111.2.2. Including l/mb corrections......Page 288
111.2.2.1. limb-expansion of correlation functions and matrix elements......Page 290
111.2.2.2. Renormalization and continuum limit......Page 291
111.2.2.3. The flavor currents in the effective theory......Page 292
111.2.3. Schrodinger functional correlation functions......Page 293
111.3.1.1. Renormalization and matching in perturbation theory......Page 295
111.3.1.2. Beyond the leading order: the need for non-perturbative conversion functions......Page 300
111.3.2. A second example: mass formulae......Page 301
111.4. Non-perturbative tests of HQET......Page 303
111.5.1. Matching in small volume......Page 305
111.5.2. Step scaling functions......Page 306
111.5.3. Example: The mass of the b quark......Page 307
111.5.3.1. Static approximation......Page 308
111.5.3.2. Including l/mb corrections.......Page 310
111.5.3.3. An alternative strategy.......Page 312
111.7. Summary and perspectives......Page 313
References......Page 315