Lattice QCD Comes of Age y Richard C. Brower XLIst Rencontres de Moriond March QCD and Hadronic interactions at high energy

QCD Theory Space! Color Supercond (Dense quarks) Asymptotically Free (UV/Short Distances) String/Gravity Flux Tubes/Spectra (IR/Long Distances) Chiral Restored (High Temp) N color 1/g 2 g2g2 kT BB N = 1, n f = 1 N = 0 * Lattice * Strassler, Katz Orginos Schmidt,Levkova IIB IIA D=11 SGRA HO HEI N = 2 N = 1 M-theory Super String Theory Space!

Comparison of Chemistry & QCD : K. Wilson (1989 Capri): “ lattice gauge theory could also require a 10 8 increase in computer power AND spectacular algorithmic advances before useful interactions with experiment...” ab initio Chemistry = flops  10 Mflops 3.Gaussian Basis functions ab initio QCD = 2030?* 2.10 Mflops  1000 Tflops 3.Clever Multi-scale Variable? * Fast Computers +Rigorous QCD Theoretical AnalysisSmart Algorithms + = ab inition predictions “Almost 20 Years ahead of schedule!”

BNL+JLab+FNAL+BG/L= O(10 Tflop/s)

USA SciDAC Software Group * Software Coordinating Committee UK Peter Boyle Balint Joo

Optimized Dirac Operators, Inverters Level 3 QDP (QCD Data Parallel) Lattice Wide Operations, Data shifts Level 2 QMP (QCD Message Passing) QLA (QCD Linear Algebra) Level 1 QIO Binary Data Files / XML Metadata SciDAC QCD API C/C++, implemented over MPI, native QCDOC, M-via GigE mesh Optimised for P4 and QCDOC Exists in C/C++ ILDG collab

Lattice QCD

Sources of Error  Wrong “theory” --- no quark loops  solution: Keep Fermionic det & Disconnected diagrams  Finite lattice spacing a  solution: a <.1 fermi + O(a 2 ) asymptotic freedom  Light quark limit m u/d /m s  O(1/20)  solution: Chiral pert. theory + Exact Lattice Chiral Symmetry  Finite space-time volume  solution: Big memory computer  Monte Carlo 1/N 1/2 sampling error  solution: Algorithms + $’s

Staggering Results: Role of Determinant (aka Sea Quarks) This is real QCD --- No more excuses (except Staggered Fermion with Det[D] ¼ trick: 4 * ¼ taste loops. Tasteful Chiral perturbation theory to take a  0)

Strong Coupling Constant Lattice:  S (M Z ) = (12) Experiment:  S (M Z ) = (20) Lattice (data) vs Perturbation Theory (red/one sigma band)

Alpha Strong

CKM projected improvement via Lattice Gauge Before After

Properties of  and K mesons Rule out m u = 0 by 5 sigma (Strong CP problem not solved!) lattice value is |Vus| = ±0.0026,experimental results: |Vus| = (23)

Axial Charge of the Nucleon Lattice g A = (84) Experiment g A = (29)

Semi-leptonic Form Factor (prediction)

Multi-scale Algorithms  String Length 1000 Mev ( » 0.2 fm)  Quarks Masses: (197 fm Mev) 2, 8, 100, 1200, 4200, 175,000 Mev  Nuclear: scattering length/effective range a singlet = fm ( » 8 Mev) & r = 2.73 a triplet = fm ( » 36 Mev) & r = fm  Deuteron Binding = 50 Mev. (» 4 fm)  Finite T, finite  etc Log(m q ) Flavor: u,d,s,c,b,t QCD length scales:

Confinement length vs Pion Compton length ll m -1 

Quark loops: Multi-time step HMC  Hasenbush Trick:  Rational Hybrid Monte Carlo: In Hybrid Monte Carlo (HMC) simulations, the determinant acts as a potential for molecular evolutions: Equilibrium by “molecular chaos”: Speed up by separating force terms and using multiple step sizes: n times

Wilson Fermions with Multi-time step trick (moving the Berlin Wall) Wilson is Almost as efficient as Staggered BUT respects flavor sym (Urbach, Jansen, Shindler, Wegner, hep-lat/ )

Multi-grid al 1980’s failure point: Universal Autocorrelation:  = F(m l  ) Gauss-Jacobi (Diamond), CG (circle), 3 level (square & star)  = 3 (cross) 10(plus) 100( square

New fangled Algebraic-Adaptive Multigrid for Disconnected Diagrams

s = 1s = 2s = Ms = L s qLqL qRqR QLQL QRQR qLqL qRqR QRQR QLQL LEFT RIGHT Exact Lattice Chiral Fermions: ( Taking the 5th Dimension Seriously ?)

5-d Flavor Current  4-d Vector/Axial Current Vector: Axial: 4-d Ward-Takahashi Identities via decent relations:

“QCD and a Holographic Model of Hadrons” Erlich, Katz, Son, Stephanov, hep-ph/05011 (fit “  qcd, m q,  ”) Remarkably similar to AdS/CFT approach to Flavor Currents * constrained fit

Conclusions Not even the Beginning of the End..., perhaps the End of the Beginning”But II. Postdictions  Predictions I. Search for signals  Calibration of Errors Coming of Age for Lattice Field theory: III. To paraphrase W.C. “This is Not the End of Lattice Gauge Theory...,