1. Lattice QCD: Past, Present and Future Norman H. Christ RHIC Physics in the Context of the Standard Model June 20, 2006

2. RBRC Symposium June 20, 2006 (2) Outline Lattice gauge theory –Formulation –Errors Computers Lattice fermions Monte Carlo methods Techniques & Results –Heavy quarks –K =>  –B K Conclusion

3. RBRC Symposium June 20, 2006 (3) RBC Collaboration RBRC –Yasumichi Aoki (Wuppertal) –Tom Blum –Chris Dawson –Takumi Doi –Koichi Hashimoto (Kanazawa) –Taku Izubuchi (Kanazawa) –Yukio Nemoto –Jun-Ichi Noaki (Southamption) –Shigemi Ohta (KEK) –Kostas Orginos (WM) –Shoichi Sasaki (Tokyo) –Matthew Wingate (U. of Wash.) –Norikazu Yamada (KEK) –Takeshi Yamazaki BNL –Frederico Berruto –Michael Creutz –Saumen Datta –Frithjof Karsch –Jack Laiho (Fermilab) –Peter Petreczki –Konstantin Petrov –Sasa Prelovsek (Ljubljana) –Christian Schmidt –Enno Scholz –Amajit Soni –Takashi Umeda Columbia –Christopher Aubin –Michael Cheng –Norman Christ –Calin Cristian –Saul Cohen –George Fleming (Yale) –Chulwoo Jung –Changhoan Kim (Southampton) –Ludmila Levkova (Indiana) –Xiaodong Liao –Meifeng Lin –HueyWen Lin –Oleg Loktik –Robert Mawhinney –Samuel Shu –Lingling Wu –Azusa Yamaguchi (Glasgow) –Yurie Zhestkov

4. RBRC Symposium June 20, 2006 (4) UKQCD Collaboration Edinburgh –David Antonio –Peter Boyle –Mike Clark (BU) –Alistair Hart –Tony Kennedy –Richard Kenway –Balint Joo (JLab) –Chris Maynard –Brian Pendleton –Robert Tweedie –James Zanoti Glasgow –Azusa Yamaguchi Southampton –Jonathan Flynn –Andreas Juettner –Changhoan Kim –Jun Noaki –Chris Sachrajda Swansea –Chris Alton –Aurora Trivini

5. RBRC Symposium June 20, 2006 (5) Lattice QCD Introduce a space-time lattice. Perform the Euclidean Feynman path integral. –Precise non-perturbative formulation. –Capable of numerical evaluation. Evaluate using Monte Carlo, importance sampling, with hybrid molecular dynamics/Langevin evolution. “First principles” ability to determine the consequences of QCD with controlled errors.

6. RBRC Symposium June 20, 2006 (6) Well understood sources of error: Finite Volume: Kinematic: p = 2  n/L  r  < L: errors ~ e -r/L Compare different volumes, Effort ~ ln(Error) Quenching:Include det(D+m) ! Effort ~ 100X Finite MC statistics: Search for long-time Autocorrelations (A. Hart, UKQCD) Effort ~ 1/(Error) 2

7. RBRC Symposium June 20, 2006 (7) Sources of error: Perturbation theory: Step scaling: a ~ (0.5) N, g 4 ~ 1/N 2 Effort ~ 1/Error 1/2 Non-zero lattice spacing: Symanzik improvement: L cont +    c i L i dim 5 +…   q…qq…q  lattice Tune action to make c i to vanish. Effort ~ 1/(Error) 8/n Quark masses too large: Effort ~ 1/(Error) 2 + 1/(Error) 8/n

8. RBRC Symposium June 20, 2006 (8) Computers Unlike conventional theory, LGT requires equipment and funding. Partially driven by technology: what was impossible 10 year ago is easy today: –1983 VAX 11/780 10 6 flops  – 2005 QCDOC 10 13 flops The RBRC lattice program is enabled by RIKEN’s investment in QCDSP and QCDOC.

9. RBRC Symposium June 20, 2006 (9) RBRC Computers QCDSP –0.6 Tflops –Completed 1998 –Gordon Bell Prize QCDOC –10 Tflops –Completed 2005 –Parent of Bluegene Overview of the QCDSP and QCDOC computers, IBM Research Journal, Vol 49, No. 2/3, p 351 (2005)

10. RBRC Symposium June 20, 2006 (10) QCDOC Collaboration Columbia (DOE) –Norman Christ –Saul Cohen –Calin Cristian –Zhihua Dong –Changhoan Kim –Ludmila Levkova –Xiaodong Liao –Meifeng Lin –Guofeng Liu –Robert Mawhinney –Shu Li –Azusa Yamaguchi BNL (SciDAC) –Chulwoo Jung –Konstantin Petrov –Stratos Efstathiadis UKQCD (PPARC) –Peter Boyle –Mike Clark –Balint Joo RBRC (RIKEN) –Shigemi Ohta –Tilo Wettig IBM –Dong Chen –Alan Gara –Design groups: Yorktown Heights, NY Rochester, MN Raleigh, NC

11. RBRC Symposium June 20, 2006 (11) Critical contributions from many people BNL staff: –Ed Mcfadden –Eric Blum –Ed Brosnan –Christopher Channing –Andrew Como –Joe Depace –Don Gates –Paul Poleski

12. RBRC Symposium June 20, 2006 (12) UKQCD Machine (12,288 nodes/10 Tflops)

13. RBRC Symposium June 20, 2006 (13) QCDOC2 ! A possible for a follow-on machine. There are competitive alternatives: –Workstation clusters –BlueGene/L Target: $0.01/Mflops or $10M/Petaflops Future technologies: –“Mother board” on-a-chip: highly integrated, 128 processors on a chip. –Cheap Intel mobile processor, densely packaged. –10 6 single-chip processor + memory (5mm) 2

14. RBRC Symposium June 20, 2006 (14) Domain Wall Fermions 5-D theory with 4-D surface states. Typical 5-D extent of 16. “Revolution” in the lattice treatment of fermions.

15. RBRC Symposium June 20, 2006 (15) Domain Wall Fermions Chiral symmetry breaking ~ exp(-  L s ).  is the smallest eigenvalue of a 5-D transfer matrix: Where C and B are the helicity conserving and violating parts of the 4-D Wilson lattice operator.

16. RBRC Symposium June 20, 2006 (16) Domain Wall Fermions Zero modes of the 4-D Dirac operator => unit eigenvectors of T Rough, strong-coupling lattices have many such modes. Peter Boyle, UKQCD

17. RBRC Symposium June 20, 2006 (17) RHMC Algorithm (Mike Clark & Tony Kennedy) Express: For our DWF calculation use: Different poles can be evolved with different step sizes and different precisions. Factor 3-6 speed-up! (Boyle/Mawhinney)

18. RBRC Symposium June 20, 2006 (18)

19. RBRC Symposium June 20, 2006 (19) K =>  Decays (Chanhoan Kim and Takeshi Yamazaki) Miani-Testa theorem: Large times yield  at rest:  (t)  H wk (0) K(-t)  (p=0)|H wk |K  –Use finite box with anti-periodic boundary conditions to force p   0. –Give the K a finite momentum. Must correct for the finite-box  state being a superposition of different J: Only J=0 couples to decaying K but C(0,0)  1. Fixed by Lellouch and Luscher.

20. RBRC Symposium June 20, 2006 (20) K =>  Decays Center of mass system Lab system Lab:  I=2 = 11.9(3.2) CM:  I=2 = 12.3(1.6)  phase shifts

21. RBRC Symposium June 20, 2006 (21) K 0 – K 0 Mixing (B K ) (Chris Dawson, Saul Cohen, Jun Noaki) New Nf=2+1 data: 1/a = 1.6 GeV 24 3 x 64, L s = 16 16 configurations (Saul Cohen) CKM matrix

22. RBRC Symposium June 20, 2006 (22) Charm quark physics At present m charm a  ½, too large to be neglected. Exploit approach of Fermilab and Tsukuba groups –Rest system effective action (up to O(a) errors): –Tune lattice parameters to remove these and other unphysical terms of order (ma) n and (ma) n |p|a. –Four are not redundant: m 0, , c B and c E 

23. RBRC Symposium June 20, 2006 (23) Charm quark physics (HueyWen Lin) Use step scaling to reach a << 1/m charm. Match on-shell quantities between: 16 3 24 3 1/a = 3.6GeV 1/a = 5.4GeV Find: One parameter is redundant!

24. RBRC Symposium June 20, 2006 (24) Charmed Physics Using 3 lattice parameters: Finally scale down to 1/a=2.4 GeV (16 3 => 24 3 again)

25. RBRC Symposium June 20, 2006 (25) Outlook QCDOC is barely a year old: much exciting work lies ahead. –24 3 x 64, 1/a=1.6 GeV, 2+1 flavor DWF calculations underway. –With RHMC: 32 3 x 64, 2+1, 1/a=2.1 GeV next. New LGT finite temperature effort at BNL and the RBRC Many important topics can be tackled in full QCD: –  ’  nucleon structure / Kl3 decays / neutron EDM / g  -2 /  ’ mass –Hyperons beta decay / electromagnetic splittings / heavy quarks We are now beginning to plan for petaflops computers. Low energy QCD is now a quantitative science! Relativistic field theory has been tamed!