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M. Nio ( RIKEN) Feb. 7, 2006 KEK 大型シミュレーション研究ワークショップ 「超高速計算機が切り開く計算物理学の展望」

Automated Calculation Scheme for α ｎ Contributions of QED to Lepton g-2: Diagrams without Lepton Loops. M. Nio ( RIKEN) Feb. 7, 2006 KEK 大型シミュレーション研究ワークショップ 「超高速計算機が切り開く計算物理学の展望」 w/ T. Kinoshita@Cornell University T. Aoyama and M. Hayakawa@RIKEN hep-ph/0512288

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M. Nio ( RIKEN) Feb. 7, 2006 KEK 大型シミュレーション研究ワークショップ 「超高速計算機が切り開く計算物理学の展望」

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  1. Automated Calculation Scheme for αｎ Contributions of QED to Lepton g-2: Diagrams without Lepton Loops M. Nio ( RIKEN) Feb. 7, 2006 KEK大型シミュレーション研究ワークショップ 「超高速計算機が切り開く計算物理学の展望」 w/ T. Kinoshita@Cornell University T. Aoyama and M. Hayakawa@RIKEN hep-ph/0512288 hep-ph/0512330, 0507249, 0402206,0210322

  2. What is electron g-2 ? experiment and theory importance in physics fine structure constant α • Automation of g-2 calculation why the 10th-order term is needed our automation scheme

  3. §1. Electron anomalous magnetic moment The g factor of the electron is modified by radiative corrections: The forward scattering amplitude of the electron: The Pauli form factor is a source of the electron anomaly: is a dimensionless constant.

  4. Experiments: UW87 and HV05 Penning trap measurement: “geonium”=confinement of a single electron by means of the electro-magnetic fields in a metallic cavity. B. Odom ’0４ Harvard U Ph. D thesis

  5. ωa anomaly frequency ωｓ　　spin frequency ωc cyclotron frequency

  6. ★U. of Washington measurement: 1987 H. Dehmelt et al. Source of the uncertainty <= unknown resonance shift due to a hyperbola cavity ★Harvard University measurement: 2005 G. Gabrielse et al. on going Preliminary! Please don’t quote it. B. Odom Ph.D thesis, Harvard U. 2004 Cylindrical cavity, whose resonance structure is analytically known, is used.

  7. Electron g-2 v.s. Muon g-2 Muon g-2 is more sensitive to a heavy particle than Electron g-2. Electron g-2 is an almost pure QED system. photon + electron

  8. §2. Theoretical formula for Electron g-2 Perturbation series of the fine structure constant α: Up to 8th-order contributions have been analyticallyand/or numerically known:TK & MN hep-ph/0507249 PRD73,013003(2005)

  9. ★8th-order contribution: uncertainty of UW87 measurement So, we need the accurate value of A1(8). ★10th-order contribution: Educated guess |A1(10)| < 4.0 P. Mohr and B. Taylor CODATA 2002 RMP77,1(’05) uncertainty of HV05 measurement The error will be cut down by a factor 3 in a few years. We want the value A1(10) ! not necessary to be very accurate.

  10. Theoretical prediction of electron g-2: need the fine structure constant value Cs atomic recoil expt.S. Chu et al. 2001 8th- 10th- α Difference between experiment and theory: expt theory Need more precise value of the fine structure constant α.

  11. The world-best value of the fine structure constant from the electron g-2 obtain α Preliminary! Please do not quote it.

  12. Various determination of the fine structure constant. They must coincide if our understanding of physics is correct.

  13. §3. 10th-order term 12672 diagrams are divided into 5 groups. They are further divided into 32 gauge invariant sets: # of sets # of FD I. 2nd-order photon correction+vp’s 10 208 II. 4th-order photon correction+vp’s 6 600 and/or light-by-light III. 6th-order photon correction+vp’s 3 1140 or light-by-light IV. 8th-order photon correction+vp’s 1 2072 V. 10th-order without fermion loop 1 6354 • (external) light-by-light 11 2298 The leading contribution to muon g-2 is reported by T. Kinoshita and MN hep-ph/0512330 to appear PRD

  14. set I set II set III 208 diagrams 600 diagrams 1140 diagrams set IV set V set VI 2072 diagrams 6354 diagrams 2298 diagrams None of them dominates. Need to evaluate ALL 12672 diagrams.

  15. Set V: 6354 diagrams w/o fermion loop The most difficult set among 6 sets. ★# of diagrams are many..! Amalgamate the Ward-Takahashi related diagrams: 6354  6354 / 9 = 706 Time reversal symmetry: 706  389 independent self-energy like diagrams 6354 diagrams form one gauge invariant set. need to calculate all 389 to get a physical number.

  16. 389 self-energy like diagrams

  17. ★Renormalization structure is very complicated. Calculation by hand with no mistake seems impossible. An automation scheme is desired !

  18. X-Project: automatic code generation T. Aoyama , M. Hayakawa, T. Kinoshita, and MN hep-ph/0512288 to appear Nucl. Phys. B ★input: A diagram name which specifies the sequence of vertices. eg. X001 abacbdcede {(1,3)(2,5)(4,7)(6,9)(8,10)} ★output: FORTRAN code ready to numerical integration including UV renormalization terms. IR div. is handled by a finite photon mass.

  19. Ｄｉａｇｒａｍ　w/o fermion loop Its specific properties enable us to automate the code generation: １．Aｌｌ　ｌｅｐｔｏｎ　propagators form a single path. 2. All vertices lie on the lepton path. 3. Photon propagators contract pair of vertices. not 1PI {(1,3) (2,4)} {(1,4)(2,3)} {(1,2)(3,4)} The contraction pattern is only the input information. Everything about a diagram is contained in this pattern.

  20. Evaluating a diagram: ★Amplitude is expressed in terms of the function of Feynman parameters U, Bij, Ai, and V. zazb z1z２z３ Feynman parameters: a parameter zi（0<zi<1) assigned to each fermion/photon line “i”. Bij ( zi) : “correlation” between loop momenta “I” and “j” . determined solely by the topology of a diagram. U(zi): Jacob determinant from the momentum space to the Feynman parameter space. Aij(zi): Related to flow of external momenta. Once Bij is obtained, one can construct U and Ai, then V.

  21. ★Construct UV subtraction terms: 1.List up all UV divergent sub diagrams. self-energy sub-diagram vertex sub-diagram Identification is easy for a setV diagarm. 2. Construct Zimmerman’s Forests for renormalization. eg. M4a abab sub-diagram: 2 g1= aba, g2=bab Forests: 2 Forest1(g1), Forest2(g2) 3. Perform K-operation for the amplitude, Bij, U, V, and Ai. Power counting limit of the Feynman parameters. Forest 1 (g1): K12 operation z10, z20, za0 za z1 zb

  22. Perl FORM FORM Maple Perl Perl FORM Shell Script

  23. Code generation is on a HP α machine: ~5min. for one code generation. A few day for all 389 diagrams Fortran codes consist of more than 80,000 lines. 13dim. integration by VEGAS adaptive iterative Monte Carlo integration One diagram evaluation: 107 sampling points with 20 iteration 5-7 hours on the Xeon 32 CPU PC cluster Need 108 pts ×100 it to reach the desired precision. A few month to complete one diagram. We wish to evaluate 389 diagrams…

  24. The numerical calculation has been carried on Riken Super Combined Cluster System. Linux PC cluster system. 2048 cpu 12.4 TFlops. operation started April 2005. RICH experimental data analysis BIO information server We use 500~700 CPU everyday. Ａ　Peta-flops computer will be introduced as a national project in 2010  京速コンピュータ開発プロジェクト（準備室＠ＲＩＫＥＮ）　　

  25. Diagrams with vertex corrections only. No IR divergence.

  26. Diagrams including a self-energy sub diagram is currently being evaluated. * IR divergence is handled by a finite photon mass. Can we really get a correct answer with a finite photon mass calculation ? * 6th-order test has been done. Yes, we can. * 8th-order test is now going on. Need to understand the IR structure more.

  27. What to do next: • Need to automateconstructionof IRsubtraction terms to realize the zero photon mass limit.  in progress • Need to automate calculation of the residual renormalization. K-operation does not generate the On-Shell renormalization constants.  in progress • Extend our code generation to diagrams w/ fermion loop  not yet done

  28. Remarks: • We will get the first number of the 10th- order term from 12672 diagrams in a few years. With a few % uncertainty. • The precise number of the 10th-order term will be evaluated on a super-computer in the next generation, 京速計算機.

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