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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

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

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 ’0 ４
Harvard U
Ph. D thesis

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

★ 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.

Electron g-2 v.s. Muon g-2
Electron g-2 Muon g-2
QED mass independent 999999996ppb 994623ppm
QED mass dependent 2.3ppb
5313ppm
Hadronic 1.4ppb
about
60ppm
Weak 0.03ppb
1ppm
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/0507249 PRD73,013003(2005)

★ 8 th -order contribution: uncertainty of UW87 measurement
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
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 A
1
(10) !
not necessary to be very accurate.

Theoretical prediction of electron g-2: need the fine structure constant value
Cs atomic recoil expt.
S. Chu et al. 2001
8 th 10 th α
Difference between experiment and 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
Preliminary! Please do not quote it.

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

th
12672 diagrams are divided into 5 groups. They are further divided into 32 gauge invariant sets:
# of sets # of FD
I . 2 nd order photon correction+vp’s 10 208
II . 4 th order photon correction+vp’s 6 600 and/or light-by-light
III . 6 th order photon correction+vp’s 3 1140 or light-by-light
IV.
8 th order photon correction+vp’s 1 2072
V.
10 th -order without fermion loop 1 6354
VI.
(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

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.

6354
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.

389 self-energy like diagrams

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

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.

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
{(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
1 z
２ z
３
Feynman parameters: a parameter z i
（ 0<z i
<1) assigned to each fermion/photon line “i”.
B ij
( z i
) : “correlation” between loop momenta “I” and “j” .
determined solely by the topology of a diagram.
U(z i
): Jacob determinant from the momentum space to the Feynman parameter space.
A ij
(z i
): Related to flow of external momenta.
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.
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, B ij
, U, V, and A i
.
Power counting limit of the Feynman parameters.
Forest 1 (g1): K12 operation z1  0, z2  0, za  0 za z1 zb

Perl
Perl
Maple
FORM
Shell Script
Perl

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:
10 7 sampling points with 20 iteration
5-7 hours on the Xeon 32 CPU PC cluster
Need 10 8 pts × 100 it to reach the desired precision.
A few month to complete one diagram.
We wish to evaluate 389 diagrams…

The numerical calculation has been carried on
R iken S uper C ombined
C luster 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）

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.
Can we really get a correct answer with a finite photon mass calculation ?
* 6 th -order test has been done.
Yes, we can.
* 8 th -order test is now going on.
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
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

We will get the first number of the 10 th order term from 12672 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 .