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. University T. Aoyama and M. hep-ph/ hep-ph/ , , , hep-ph/ , , ,

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

§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. is a dimensionless constant.

Experiments: UW87 and HV05 Penning trap measurement: “geonium”=confinement of a single electron “geonium”=confinement of a single electron by means of the electro-magnetic fields by means of the electro-magnetic fields in a metallic cavity. in a metallic cavity. B. Odom ’0 ４ B. Odom ’0 ４ Harvard U Harvard U Ph. D thesis Ph. D thesis

anomaly frequency ω a anomaly frequency spin frequency ω ｓ spin frequency cyclotron frequency ω c cyclotron frequency

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

Electron g-2 Muon g-2 QED mass independent ppb994623ppm QED mass dependent 2.3ppb 2.3ppb 5313ppm 5313ppm Hadronic 1.4ppb 1.4ppb about 60ppm Weak 0.03ppb 0.03ppb 1ppm 1ppm 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

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

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

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

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

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

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

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

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: Amalgamate the Ward-Takahashi related diagrams: 6354  6354 / 9 =  6354 / 9 = 706 Time reversal symmetry: Time reversal symmetry: 706  389 independent 706  389 independent self-energy like diagrams self-energy like diagrams 6354 diagrams form one gauge invariant set diagrams form one gauge invariant set. need to calculate all 389 to get a physical number. need to calculate all 389 to get a physical number.

389 self-energy like diagrams

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

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

Ｄｉａｇｒａｍ w/o fermion loop Its specific properties enable us to automate the code generation: Its specific properties enable us to automate the code generation: １． A ｌｌ ｌｅｐｔｏｎ propagators form a single path. 2. All vertices lie on the lepton path. 3. Photon propagators contract pair of vertices. not 1PI not 1PI {(1,3) (2,4)} {(1,4)(2,3)} {(1,2)(3,4)} {(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.

Evaluating a diagram: ★ Amplitude is expressed in terms of the function of Feynman parameters U, B ij, A i, and V. z a z b z a z b z 1 z ２ z ３ z 1 z ２ z ３ Feynman parameters: a parameter z i （ 0<z i <1) assigned to each Feynman parameters: a parameter z i （ 0<z i <1) assigned to each fermion/photon line “i”. fermion/photon line “i”. B ij ( z i ) : “correlation” between loop momenta “I” and “j”. B ij ( z i ) : “correlation” between loop momenta “I” and “j”. determined solely by the topology of a diagram. determined solely by the topology of a diagram. U(z i ): Jacob determinant from the momentum space to U(z i ): Jacob determinant from the momentum space to the Feynman parameter space. the Feynman parameter space. A ij (z i ): Related to flow of external momenta. A ij (z i ): Related to flow of external momenta. Once B ij is obtained, one can construct U and A i, then V. Once B ij is obtained, one can construct U and A i, then V.

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

FORM Maple FORM Perl Shell Script

Code generation is on a HP α machine: ~5min. for one code generation. ~5min. for one code generation. A few day for all 389 diagrams 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 adaptive iterative Monte Carlo integration One diagram evaluation: One diagram evaluation: 10 7 sampling points with 20 iteration 10 7 sampling points with 20 iteration 5-7 hours on the Xeon 32 CPU PC cluster 5-7 hours on the Xeon 32 CPU PC cluster Need 10 8 pts ×100 it to reach the desired precision. Need 10 8 pts ×100 it to reach the desired precision. A few month to complete one diagram. A few month to complete one diagram. We wish to evaluate 389 diagrams… We wish to evaluate 389 diagrams…

The numerical calculation has been carried on Riken Super Combined Riken Super Combined Cluster System. Cluster System. Linux PC cluster system. Linux PC cluster system cpu 12.4 TFlops cpu 12.4 TFlops. operation started April operation started April RICH experimental data analysis RICH experimental data analysis BIO information server BIO information server We use 500~700 CPU everyday. 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＠Ｒ ＩＫＥＮ） in 2010  京速コンピュータ開発プロジェクト（準備室＠Ｒ ＩＫＥＮ）

Diagrams with vertex corrections only. No IR divergence.

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

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

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