Automated Calculation Scheme for αAutomated Calculation Scheme for α ｎ ｎ Contributions of QED to Lepton g-2: Contributions of QED to Lepton g-2:

Diagrams without Lepton LoopsDiagrams without Lepton Loops

M. Nio ( RIKEN)M. Nio ( RIKEN)

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

w/ T. Kinoshita@Cornell Universityw/ T. Kinoshita@Cornell UniversityT. Aoyama and M. Hayakawa@RIKENT. Aoyama and M. Hayakawa@RIKEN

hep-ph/0512288 hep-ph/0512288 hep-ph/0512330, 0507249, 0402206,0210322hep-ph/0512330, 0507249, 0402206,0210322

What is electron g-2 ?What is electron g-2 ? experiment and theoryexperiment and theory importance in physicsimportance in physics

　　　　　　　　　　　　　　　　　　　　 fine structure constantfine structure constant α α

Automation of g-2 calculationAutomation of g-2 calculation why the 10why the 10thth-order term is needed-order term is needed our automation scheme our automation scheme

§1. Electron anomalous magnetic moment§1. Electron anomalous magnetic moment

The g factor of the electron is modified by radiative corrections:The g factor of the electron is modified by radiative corrections:

The forward scattering amplitude of the electron:The forward scattering amplitude of the electron:

The Pauli form factor is a source of the electron anomaly:The Pauli form factor is a source of the electron anomaly:

is a dimensionless constant.is a dimensionless constant.　　　　　　

Experiments: UW87 and HV05Experiments: UW87 and HV05Penning trap measurement:Penning trap measurement: “ “geonium”=confinement of a single electrongeonium”=confinement of a single electron by means of the electro-magnetic fieldsby means of the electro-magnetic fields in a metallic cavity.in a metallic cavity.

B. Odom ’0B. Odom ’0 ４４ Harvard UHarvard U Ph. D thesis Ph. D thesis

ωa anomaly frequency anomaly frequency

ω ｓ　　 spin frequencyspin frequency

ωc cyclotron frequency cyclotron frequency

★★U. of Washington measurement: 1987 H. Dehmelt et al.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 cavitydue to a hyperbola cavity ★★Harvard University measurement: 2005 G. Gabrielse et al. on going 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. 2004B. Odom Ph.D thesis, Harvard U. 2004 　　 Cylindrical cavity, whose resonance structure is analytically known,Cylindrical cavity, whose resonance structure is analytically known, 　　　　 is used. is used.

Electron g-2Electron g-2 Muon g-2Muon g-2

QED mass independentQED mass independent 999999996ppb999999996ppb 994623ppm994623ppm

QED mass dependent QED mass dependent 2.3ppb2.3ppb 5313ppm5313ppm

HadronicHadronic 1.4ppb1.4ppb about about 60ppm60ppm

WeakWeak 0.03ppb0.03ppb 1ppm1ppm

Electron g-2 v.s. Muon g-2Electron 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§2. Theoretical formula for Electron g-2Perturbation series of the fine structure constant α:Perturbation series of the fine structure constant α:

Up to 8Up to 8thth-order contributions have been analytically-order contributions have been analytically 　　and/orand/or

numerically known:numerically known: 　　 　　 TK & MN hep-ph/0507249 PRD73,013003(2005)TK & MN hep-ph/0507249 PRD73,013003(2005)

★★88thth-order contribution:-order contribution:

uncertainty of UW87 measurementuncertainty of UW87 measurement

So, we need the accurate value of ASo, we need the accurate value of A11(8)(8)..

★★1010thth-order contribution:-order contribution: Educated guess |AEducated guess |A11

(10)(10)| < 4.0 | < 4.0 P. Mohr and B. Taylor P. Mohr and B. Taylor CODATA 2002 CODATA 2002

RMP77,1(’05)RMP77,1(’05)

uncertainty of HV05 measurementuncertainty 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 AWe want the value A11(10)(10) ! !

not necessary to be very accurate.not necessary to be very accurate.

Theoretical prediction of electron g-2:Theoretical prediction of electron g-2: need the fine structure constant valueneed the fine structure constant value Cs atomic recoil expt.Cs atomic recoil expt. S. Chu et al. 2001S. Chu et al. 2001

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 　　　　　　　　　　　　　　　　　　　　　　　　　　　　 88thth- - 　　 1010thth- -

　　　　 αα

Difference between experiment and theory: Difference between experiment and theory:

expt theoryexpt theory

Need more precise value of the fine structure constant α.Need more precise value of the fine structure constant α.

The world-best value of the fine structure The world-best value of the fine structure constant from the electron g-2constant from the electron g-2

obtain αobtain α

Preliminary! Please do not quote it.Preliminary! Please do not quote it.

Various determination of the fine structure constant.Various determination of the fine structure constant.

They must coincide if our understanding of physics is correct.They must coincide if our understanding of physics is correct.

§3. 10§3. 10thth-order term-order term1267212672 diagrams are divided into 5 groups. They are further divided diagrams are divided into 5 groups. They are further divided

into 32 gauge invariant sets:into 32 gauge invariant sets: # of sets # of FD # of sets # of FD II. 2. 2ndnd-order photon correction+vp’s 10 208-order photon correction+vp’s 10 208IIII. 4. 4thth-order photon correction+vp’s 6 600 -order photon correction+vp’s 6 600

　　　　　　　　　　　　　　　　　　　　　　　　 and/or light-by-lightand/or light-by-lightIIIIII. 6. 6thth-order photon correction+vp’s 3 1140-order photon correction+vp’s 3 1140 or light-by-light or light-by-light IV.IV. 8 8thth-order photon correction+vp’s 1 2072-order photon correction+vp’s 1 2072V.V. 10 10thth-order without fermion loop 1 6354-order without fermion loop 1 6354VI.VI. (external) light-by-light 11 2298(external) light-by-light 11 2298 The leading contribution to muon g-2 is reported by T. Kinoshita and MNThe leading contribution to muon g-2 is reported by T. Kinoshita and MN hep-ph/0512330 to appear PRDhep-ph/0512330 to appear PRD

set I set II set IIIset I set II set III 208 diagrams 600 diagrams 1140 diagrams208 diagrams 600 diagrams 1140 diagrams

set IV set V set VIset IV set V set VI 2072 diagrams 6354 diagrams 2298 diagrams2072 diagrams 6354 diagrams 2298 diagrams

None of them dominates. None of them dominates. Need to evaluate Need to evaluate ALLALL 12672 diagrams. 12672 diagrams.

Set V: Set V: 6354 6354 diagrams w/o fermion loopdiagrams w/o fermion loopThe most difficult set among 6 sets.The most difficult set among 6 sets.

★★ 　　 # of diagrams are many..!# of diagrams are many..! Amalgamate the Ward-Takahashi related diagrams:Amalgamate the Ward-Takahashi related diagrams: 6354 6354 6354 / 9 = 706 6354 / 9 = 706

Time reversal symmetry:Time reversal symmetry: 706 706 389 389 independent independent self-energy like diagramsself-energy like diagrams　 　 6354 diagrams form one gauge invariant set.6354 diagrams form one gauge invariant set. need to calculate all need to calculate all 389389 to get a physical number. to get a physical number.

389 self-energy like diagrams

★★ 　　 Renormalization structure is very complicated.Renormalization structure is very complicated.

Calculation by hand with no mistake seems impossible. Calculation by hand with no mistake seems impossible.

An automation scheme is desired !An automation scheme is desired !

X-ProjectX-Project: automatic code generation: automatic code generation 　　　　　　　　　　　　 T. Aoyama , M. Hayakawa, T. Kinoshita, and MNT. Aoyama , M. Hayakawa, T. Kinoshita, and MN hep-ph/0512288 to appear Nucl. Phys. Bhep-ph/0512288 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 output: FORTRAN code ready to numerical integration including UV renormalization terms. integration including UV renormalization terms.

IR div. is handled by a finite photon mass.IR div. is handled by a finite photon mass.

Ｄｉａｇｒａｍ　Ｄｉａｇｒａｍ　 w/o fermion loopw/o fermion loop

Its specific properties enable us to automate the Its specific properties enable us to automate the code generation:code generation:

１．１． AA ｌｌ　ｌｅｐｔｏｎ　ｌｌ　ｌｅｐｔｏｎ　 propagators form a single path.propagators form a single path.2. All vertices lie on the lepton path.2. All vertices lie on the lepton path.

3. Photon propagators contract pair of vertices3. Photon propagators contract pair of vertices. . not 1PInot 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 The contraction pattern is onlyonly the input information. the input information.Everything about a diagram is contained in this pattern.Everything about a diagram is contained in this pattern.

Evaluating a diagram:Evaluating a diagram:★★Amplitude is expressed in terms of the function of Amplitude is expressed in terms of the function of

Feynman parameters U, BFeynman parameters U, B ijij, A, Aii, and V., and V. zzaa zzbb

zz11 　　 zz ２２ 　 　 　 　 zz ３３

Feynman parameters: a parameter zFeynman parameters: a parameter zii 　　（（ 0<z0<zii<1) assigned to each <1) assigned to each 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 fermion/photon line “i”.fermion/photon line “i”. 　　 BBijij ( z ( zii) : “correlation” between loop momenta “I” and “j” .) : “correlation” between loop momenta “I” and “j” . determined solely by determined solely by the topologythe topology of a diagram. of a diagram. U(zU(zii): Jacob determinant from the momentum space to): Jacob determinant from the momentum space to the Feynman parameter space.the Feynman parameter space. AAijij(z(zii): Related to flow of external momenta.): Related to flow of external momenta. Once BOnce Bijij is obtained, one can construct U and A is obtained, one can construct U and Aii, then V., then V.

★★Construct UV subtraction terms:Construct UV subtraction terms: 1.1. 　　 List up all UV divergent sub diagrams.List up all UV divergent sub diagrams. self-energy sub-diagramself-energy sub-diagram vertex sub-diagramvertex 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 eg. M4a abaabab b sub-diagram: 2 g1= aba, g2=babsub-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, B3. Perform K-operation for the amplitude, B ijij, U, V, and A, U, V, and Aii.. Power counting limit of the Feynman parameters.Power counting limit of the Feynman parameters. Forest 1 (g1): K12 operation z1Forest 1 (g1): K12 operation z10, z20, z20, za0, za00

zaza

z1 zbz1 zb

FORMFORM

Maple

FORM

Perl

Perl Perl

Shell Script

Code generation is on a HP α machine:Code generation is on a HP α machine: ~5min. for one code generation.~5min. for one code generation. A few day for all A few day for all 389389 diagrams diagrams

Fortran codes consist of more than 80,000 lines.Fortran codes consist of more than 80,000 lines.13dim. integration by 13dim. integration by VEGASVEGAS adaptive iterative Monte Carlo integrationadaptive iterative Monte Carlo integration

One diagram evaluation:One diagram evaluation: 10107 7 sampling points with 20 iteration sampling points with 20 iteration 5-7 hours on the Xeon 32 CPU PC cluster5-7 hours on the Xeon 32 CPU PC cluster Need 10Need 108 8 pts ×100 it to reach the desired precision.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 We wish to evaluate 389 389 diagrams…diagrams…

The numerical calculation has been carried on The numerical calculation has been carried on RRiken iken SSuper uper CCombined ombined CCluster Systemluster System.. Linux PC cluster system.Linux PC cluster system. 2048 cpu 12.4 TFlops.2048 cpu 12.4 TFlops. operation started April 2005.operation started April 2005.

RICH experimental data analysisRICH experimental data analysis BIO information serverBIO information server 　　　　　　 We use 500~700 CPU everyday. We use 500~700 CPU everyday. Ａ　Ａ　 Peta-flops computer will be introduced as a national projecPeta-flops computer will be introduced as a national projec

tt in 2010 in 2010 京速コンピュータ開発プロジェクト（準備室＠Ｒ京速コンピュータ開発プロジェクト（準備室＠Ｒ

ＩＫＥＮ）　　ＩＫＥＮ）　　

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

Diagrams including a self-energy sub diagram isDiagrams including a self-energy sub diagram iscurrently being evaluated.currently being evaluated. * * IR divergence is handled by IR divergence is handled by a finite photona finite photon mass. mass.

Can we really get a correct answer with a finite Can we really get a correct answer with a finite photon mass calculation ? photon mass calculation ?

* 6* 6thth-order test has been done.-order test has been done. Yes, we can.Yes, we can. * 8* 8thth-order test is now going on.-order test is now going on. Need to understand the IR structure more.Need to understand the IR structure more.

What to do next:What to do next:Need to Need to automateautomate constructionconstruction ofof IR IR subtractionsubtraction t terms to realize the zero photon mass limit. erms to realize the zero photon mass limit.

in progressin progressNeed to Need to automate calculationautomate calculation of the of the residual renorresidual renormalizationmalization. .

K-operation does not generate the K-operation does not generate the On-Shell On-Shell renormalization renormalization constants.constants.

in progressin progressExtend our code generation to diagrams w/ fermioExtend our code generation to diagrams w/ fermion loop n loop

not yet donenot yet done

Remarks:Remarks:

We will get the first number of the 10We will get the first number of the 10thth- - order term from order term from 12672 12672 diagrams in a few diagrams in a few years. years. With a few % uncertainty.With a few % uncertainty.

The precise number of the 10The precise number of the 10thth-order term -order term will be evaluated on a super-computer inwill be evaluated on a super-computer in

the next generation, the next generation, 京速計算機京速計算機 ..