the qed contribution to g 2 of the electron and muon ...g2pc1.bu.edu/lept14/talks/kinoshita.pdfthe...

The QED Contribution to g − 2

of the Electron and Muon:

Current Statuspresented at

the LEPTON MOMENT International Symposium

Cape Cod, July 21 - 24, 2014

Toichiro Kinoshita

Laboratory for Elementary-Particle Physics, Cornell University

supported by U. S. National Science Foundation under Grant NSF-PHY-0757868

July 19, 2014

T. Kinoshita July 19, 2014 1 / 42

I. Introduction

In 1947 Kusch and Foley, studying Zeeman splitting of Ga atom, found

that electron’s g-factor is slightly larger than 2 predicted by Dirac eq.:

P. Kusch, H.M. Foley, PR 72, 1256 (1947)

ae(47) = 1.19 (5)× 10−3.

where ae ≡ (g − 2)/2 is the anmalous magnetic moment of the electron.

Schwinger showed that it is caused by the radiative correction:J. Schwinger, PR 73, 416L (1948); PR 75, 898 (1949)

a(2)e =

α

2π= 1.161 . . .× 10−3,

applying explicitly covariant and renormalized QED, obtained by

Tomonaga and Schwinger independently.Z. Koba, S. Tomonaga, Prog.Theor.Phys. 2, 218 (1947)

S. Tomonaga, PR74, 224 (1948)J. Schwinger, PR 74, 1439 (1948)


Introduction - cont.

ae, being the simplest quantity calculable in QED, became the subject of

intense experimental and theoretical study.

Progress in theory and experiment (spin precession, Penning trap) over

60 years improved their precision by six orders of magnitude.

By means of cylindrical Penning trap and numerous innovations Harvard

group achieved the precision of 0.24 ppb:

ae−(HV08) = 1 159 652 180.73 (28) × 10−12 [0.24ppb].

D. Hanneke, S. Fogwell, G. Gabrielse, PRL 100, 120801 (2008)D. Hanneke, S Fogwell Hoogerheide, G. Gabrielse, PRA 83, 052122 (2011)

Next generation of experiment of ae+ is being prepared at Harvard.

Reports by Fogwell-Hoogerheide and Novitzki, this conference.


Introduction - cont.

Muon spin-precession experiment at BNL has the precision of 0.5 ppm:

aµ(BNL) = 116 592 089 (63) × 10−11 [0.5ppm].

G. W. Bennett et al., PRL 92, 161802 (2004).

L. Roberts, Chinese Phys. C 34, 741 (2010).

Next generation of experiments to measure aµ are being prepared at

Fermilab and J-PARC.

Talks by Lancaster (Fermilab) and Mibe (J-PARC), this conference.


2. Theory of Electron g − 2: Current Status

In SM, ae consists of pure QED, electroweak, and hadronic parts:

ae = ae(QED) + ae(EW) + ae(had).

Assuming that muon and tau behave exactly like electron in their

interaction with photon, we can write ae(QED) as

ae(QED) = A1 + A2(me/mµ) + A2(me/mτ ) + A3(me/mµ,me/mτ ).

Feynman-Dyson rules enables us to write Ai as a power series

Ai = A(2)i

(α

π

)

+ A(4)i

(α

π

)2

+ A(6)i

(α

π

)3

+ . . . , i = 1,2,3,

where expansion coefficients are finite after renormalization.


Calculation of ae(QED)

A(n)1 is known up to n = 10.

A(2)1 = 0.5 1 diagram (analytic)

A(4)1 = −0.328 478 965 . . . 7 diagrams (analytic)

A(6)1 = 1.181 241 456 . . . 72 diagrams (analytic)

A(8)1 = −1.912 89 (90) 891 diagrams (numerical, July 2014)

A(10)1 = 7.651 (353) 12672 diagrams (numerical, July 2014)


Comment on A(8)1 .

The code of A(8)1 was written in 3 independent ways.

I: primarily by hand.

II: by automatic code generator GENCODEN in scalar mode.

III: by automatic code generator GENCODEN in vector mode.

11 Feynman parameters are mapped onto 10-dim unit cube of integration

variables of VEGAS, an iterative-adaptive integration routine based on

random sampling of integrand.

G. P. Lepage, J. Comput. Phys. 27, 192 (1978)

In most cases remapping is required to improve convergence of iteration.

The value −1.9106 (20) given in

T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRL 109, 111807 (2012).

was combination of the first two.

New result is combination of three, which are consistent with each other.


Comment on A(10)1 .

Contribution to A(10)1 comes mainly from diagrams of Set 5, which

consists of 6354 proper vertices with no closed lepton loop.

They are compressed with the help of Ward-Takahashi identity and

time-reversal symmetry into 389 self-energy-like diagrams.

Each integral occupies more than 105 lines of FORTRAN code.

Figure : One of 389 W-T-summed diagrams of Superset V.


Comment on A(10)1 .

Set 5 is divided into two classes:

XL: 153 integrals having vertex subdiagrams only:

XB: 236 integrals containing at least one self-energy subdiagrams.

14 Feynman parameters are mapped onto 13-dim. unit cube of VEGAS.

Same default mapping for all integrals initially. Then remapped to improve

convergence of iteration process.

XL-integrals were evaluated in two ways:

1) preliminary runs (default mapping, real*8)

2) 2nd run (adjusted mapping, real*8)

XB-integrals were evaluated in 3 ways:

1) preliminary runs (default mapping, real*8)

2a) 2nd run (adjusted mapping, double-double precision, 162 integrals)

2b) remaining 74 were evaluated after prelim. result was published.

3) 3rd run (real*8 for 179 of 236 integrals, d-d precision for 57 integrals).


Comment on A(10)1 - cont.

The PRL result was obtained by combining XL1, XB1, and XB2a.

Update consists of XL2, XB2a, XB2b, and XB3, excluding XL1 and XB1.

The PRL value of setV was 9.173 (570), while update is 7.656 (353).

The difference is 1.517, and combined uncertainty is√0.5702 + 0.3532 = 0.670.

Thus the shift is 1.517/0.670 = 2.3 σ.

This shift is not small, but judging from the consistency of recent

intermediate results, it is most likely that uncertainty of new result is

reliable, whereas the PRL result was not yet stable, causing

underestimate of the size of errors.


Possible contribution of positronium poles to ae(QED).

Recently it was reported that positronium pole contribution to the

vacuum-polarization loop has a nonvanishing contribution to the α5 term.

G. Mishima, arXiv:1311.7109[hep-ph].

However, this effect is cancelled by half of the e−e+ scattering

contribution near threshold, and its remaining half is included in the

perturbative calculation of the tenth-order diagram set I(i).K. Melnikov, A. Vainshtein, M. Voloshin, PRD 90, 017301 (2014) [arXiv:1402.5690]

M. I. Eides, arXiv:1402.5860[hep-ph].

M. Fael, M. Passera, arXiv:1402.1575v2 [hep-ph].

Thus no additional contribution beyond the perturbative calculation of

QED is present at the α5 level of ae.

I(d)

I(i)


4th- and 6th-order contributions of muon and tau to ae

A2 and A3 of 4th- and 6th-orders are known analytically or as power

series in me/mµ or me/mτ :

H.H. Elend PL 20, 682 (1966)M.A. Samuel, G. Li, PRD 44, 3935 (1991)

G. Li, R. Mendell, M.A. Samuel, PRD 47, 1723 (1993)S. Laporta, E. Remiddi, PLB 30l, 440 (1993)

S. Laporta, Nuovo Cim. A 106, 675 (1993)

M. Passera, PRD 75, 013002 (2007)

A(4)2 (me/mµ) = 5.197 386 67 (26)× 10−7

A(4)2 (me/mτ ) = 1.837 98 (34)× 10−9

A(6)2 (me/mµ) = −7.373 941 55 (27)× 10−6

A(6)2 (me/mτ ) = −6.583 0 (11)× 10−8

A(6)3 (me/mµ,me/mτ ) = 1.909 (1)× 10−13

Uncertainties are only those of measured ratios me/mµ or me/mτ .


8th- and 10th-order contributions of muon and tau to

ae

Eighth- and tenth-order mass-dependent terms are mostly obtained by

numerical integration.

A(8)2 (me/mµ) = 9.222 (66)× 10−4

A(8)2 (me/mτ ) = 7.38 (12)× 10−6

A(8)3 (me/mµ,me/mτ ) = 7.465 (17)× 10−7

A(10)2 (me/mµ) = −3.82 (39)× 10−3

Uncertainties arise only from numerical integration by VEGAS.

A(10)2 (me/mτ ) and A

(10)3 (me/mµ,me/mτ ) are negligibly small.


Hadronic and electroweak terms of ae

At present hadronic term is derived mostly from experimental data related

to hadronic vacuum polarization. Recent evaluations areae(had.vp) = 1.866 (10)exp(5)rad × 10−12

ae(had.vp.NLO) = −0.223 4 (12)exp(7)rad × 10−12

ae(had.vp.NNLO) = 0.028 (1) × 10−12

ae(had.lbyl) = 0.035 (10)× 10−12

D. Nomura, T. Teubner, NPB 867, 236 (2013)A. Kurz, T. Liu, P. Marquard, M. Steinhauser,PLB 734, 144 (2014)

J. Prades et al., in Lepton Dipole Moments, edited by B. L. Roberts and W. J. Marciano(World Scientific, Singapore,2009), p.303

Electroweak contribution is small but not negligible:

ae(EW) = 0.029 7 (5) × 10−12

K. Fujikawa, B.W. Lee, A.I. Sanda, PRD 6, 2923 (1972).A. Czarnecki, B. Krause, W.J. Marciano, PRL 76, 3267 (1996).

M. Knecht, S. Peris, A. Perrottet, E. de Rafael, J. High Energy Phys. 11 (2002) 003.A. Czarnecki, W.J. Marciano, A. Vainshtein, PRD 67, 073006 (2003).


ae(Standard Model)

Putting these results together we obtain

ae(SM) = ae(QED) + ae(hadron) + ae(weak),

where

ae(QED) = 0.5(α

π

)

−0.328 478 444(α

π

)2

+1.181 234 017(α

π

)3

−1.911 96 (90)(α

π

)4

+7.647 (353)(α

π

)5

ae(hadron) = 0.170 56 (151)× 10−11,

ae(weak) = 0.297 (5)× 10−13.


Most precise non-QED α

To evaluate ae(SM) precisely the most accurate α obtained by a method

independent of QED is needed.

Such α at present is one obtained by optical lattice method:

α−1(Rb11) = 137.035 999 049 (90). [0.66 ppb]

R. Bouchendira, P. Clade, S. Guellati-Khelifa, F. Nez, F. Biraben, PRL 106, 080801 (2011)

P.J. Mohr, B.N. Taylor, D.B. Newell, RMP 84, 1527 (2012)


Theoretical value of ae evaluated with α(Rb11)

Using α(Rb11) we obtain

ae(Rb11) = 1 159 652 181.636 (27)(24)(16)(763)× 10−12 [0.66 ppb],

where uncertainties are from 8th-order term, 10th-order term, hadronic

and electroweak terms, and α(Rb11), in that order.

It is well within the uncertainty of PRL value of ae(Rb11).

Intrinsic theoretical uncertainty is ∼ 0.05 × 10−12, which is 5.6 times

smaller than the uncertainty 0.28 × 10−12 of ae(HV08).

Agreement with the measured ae is somewhat improved over the

previous case −1.05 (82)× 10−12:

ae(HV08)− ae(Rb11) = −0.91 (82)× 10−12.


ae as the best source of α

Largest uncertainty 0.763 × 10−12 in ae(Rb11) comes from α(Rb11).

Non-QED α, even the best one, is too crude to test QED to the precision

achieved by theory and measurement of ae.

An alternative and more effective way to test QED (and SM) is

to assume the validity of SM and solve the equation for α:

ae(experiment) = ae(theory).

This leads to

α−1(ae) = 137.035 999 156 2 (31)(29)(18)(331) [0.25 ppb],

where uncertainties are from 8th-order, 10th-order, hadronic and

electroweak terms, and from the measurement of ae(HV08).

Its uncertainty is dominated by that of ae(HV08). Thus further

improvement of theory affects α−1 only slightly until measurement

improves substantially.


High prcision test of Quantum Mechanics

α−1(ae) is 2.6 times more precise, and in agreement with α−1(Rb11):

α−1(ae)− α−1(Rb11) ∼ 107(96)× 10−9.

This may be regarded as confirmation of validity of QED (SM) but also of

internal consistency of quantum mechanics itself.

This is because nonrelativistic QM, by which these (non-QED)

measurements of α are interpreted, must include radiative corrections as

well as effects of new physics.

Otherwise, mass and charge parameters of ordinary QM cannot be

correctly identified as physical mass and physical charge.

Effect of possible new physics may not be detectable as far as it affects

only charge form factor since it is hidden by the charge renormalization.

From this point of view α(ae) must be identical with α(Rb) or any other αmeasured in ordinary QM to any decimal point.


Non-QED measurements of α

Non-QED α obtained thus far confirm the above observation, although

majority are much less accurate than α(ae).P.J. Mohr, B.N. Taylor, D.B. Newell, RMP 80, 633 (2008)

B.J. Mount, M. Redshaw, E.G. Myers, PRA 82, 042513 (2010)R. Bouchendira, P. Clade, S. Guellati-Khelifa, F. Nez, F. Biraben, PRL 106, 080801 (2011)

P.J. Mohr, N.N. Taylor, D.B. Newell, RMP 84, 1527 (2012)

α−1(ac Josephson) = 137.035 987 5 (43) [31 ppb]

α−1(quantum Hall) = 137.036 003 0 (25) [18 ppb]

α−1(neutron wavelength) = 137.036 007 7 (28) [21 ppb]

α−1(atom interferometry) = 137.036 000 0 (11) [7.7 ppb]

α−1(Rb on optical lattice) = 137.035 999 049 (90) [0.66 ppb]


Measurement of α(Rb11)

At present most promising for further improvement is α(Rb11).

It is derived from the formula

α(Rb11) =

[

2R∞

c

mRb

me

h

mRb

]1/2

,

c = 299 792 458 ms−1 (exact by definition)

m(87Rb) = 86.909 180 535 (10) (in atomic mass units) [1.2 × 10−10]me = 0.000 548 579 909 46 (22) [4.0 × 10−10]

R∞ ≡ α2mec2h

= 10 973 731.568 539 (55) m−1 [5.0 × 10−12]h

m(87Rb)= 4.591 359 272 9 (57)× 10−9m2s−1 [1.2 × 10−9]

B.J. Mount, M. Redshaw, E.G. Myers, PRA 82, 042513 (2010)R. Bouchendira, P. Clade, S. Guellati-Khelifa, F. Nez, F. Biraben, PRL 106, 080801 (2011)

P.J. Mohr, N.N. Taylor, D.B. Newell, RMP 84, 1527 (2012)


Measurement of α(Rb11), cont.

Precision of α(Rb11) is limited by the last factor.

French group is trying to improve precision of h/mRb by means of

Bose-Einstein condensation which may increase the number of

coherent Rb atoms by 3 orders of magnitude.

Note also that the precision of me has been improved recently.

me = 0.000 548 579 909 067 (14)(9)(2) [3 × 10−11]

14 and 9 are statistical and systematic uncertainties. 2 comes from

uncertainty in theory.

Sturm, S., Koehler, F., Zatorski, J., Wagner, A., Harman, Z., Werth, G., Quint, W., Keitel, C. H., Blaum, K.,

Nature 506, 467 - 470 (2014)


Comparison of theory and experiment of ae began historically as a test of

the validity of QED.

But it has evolved into precision test of internal consistency of quantum

mechanics in general, not just test of QED or Standard Model.

Further improvement of theory and measurements will improve test of

quantum mechanics (Standard Model).

This will impose more strict constraint on possible theories beyond SM,

such as theory of dark photon.R. Essig, et al., Proceedings of "Community Summer Study 2013: Snowmass on the Mississippi

[ArXiv:1311.0029] .H. Merkel, et al., PRL 112, 221802 (2014) [ArXiv:1404.5502] .

M. Endo, K. Hamaguchi, G. Mishima, PRD 86, 095029 (2012).


3. Theory of Muon g − 2

aµ may be expressed as

aµ = aµ(QED) + aµ(EW) + aµ(had),

where

aµ(QED) = A1 + A2(mµ/me) + A2(mµ/mτ ) + A3(mµ/me,mµ/mτ ).

Feynman-Dyson rules enables us to write Ai as a power series

Ai = A(2)i

(α

π

)

+ A(4)i

(α

π

)2

+ A(6)i

(α

π

)3

+ . . . , i = 1,2,3,

with finite expansion coefficients.


Calculation of aµ(QED)

aµ(QED) including mass-dependent terms, is known up to n = 10.

a(2)µ (QED) = 0.5

a(4)µ (QED) = 0.765 857 425 (17)

a(6)µ (QED) = 24.050 509 96 (32)

a(8)µ (QED) = 130.877 4 (61) (A

(8)1 is updated)

a(10)µ (QED) = 751.77 (93) (A

(10)1 is updated)

τ -loop contribution to a(8)µ (QED) is calculated by

A. Kurz, T. Liu, P. Marquard, M. Steinhauser, arXiv:1407.0267 (2014)

Leading contribution to a(12)µ (QED) will come from diagrams which

contain one light-by-light subdiagram and three vacuum-polriztion loops.

A crude estimate gives

a(12)µ (QED)(α/π)6 ∼ 0.08 × 10−11.

This is about the same order of magnitude as the uncertainty in a(10)µ . It

would be desirable to obtain at least a crude evaluation of this term.


Hadronic and electroweak terms of aµ

Hadronic term is the source of largest uncertainty in aµ.

At present it is derived mostly from experimental data related to hadronic

vacuum polarization. Recent evaluations areaµ(had.vp) = 6949.1 (37.2)exp(21.0)rad × 10−11

aµ(had.vp.NLO) = −98.4 (0.6)exp(0.4)rad × 10−11

aµ(had.vp.NNLO) = 12.4 (10) × 10−11

aµ(had.lbyl) = 116 (40)× 10−11

aµ(had.lbyl.NLO) = 3 (2)× 10−11

K. Hagiwara, R. Liao, A. D. Martin, D. Nomura, T. Teubner, J. Phys. G 38, 085003 (2011)[arXiv:1105.3149]

A. Kurz, T. Liu, P. Marquard, M. Steinhauser,PLB 734, 144 (2014)J. Prades et al., in Lepton Dipole Moments, eds. B. L. Roberts and W. J. Marciano

(World Scientific, Singapore,2009), p.303.

G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, P. Stoffer, PLB 735, 90 ( 2014)

Electroweak contribution has been calculated up to 2-loop order:

aµ(EW) = 154 (2) × 10−11.

K. Fujikawa, B.W. Lee, A.I. Sanda, PRD 6, 2923 (1972).A. Czarnecki, B. Krause, W.J. Marciano, PRL 76, 3267 (1996).

M. Knecht, S. Peris, A. Perrottet, E. de Rafael, J. High Energy Phys. 11 (2002) 003.A. Czarnecki, W.J. Marciano, A. Vainshtein, PRD 67, 073006 (2003).


Theoretical value of aµ evaluated with α(Rb11)

From these results and α(Rb11) we obtain

aµ(Rb11) = 116 591 855.03 (0.08)(58.56)(0.02)(0.08) × 10−11,

where the uncertainties are from QED, hadronic contribution, weak

contribution, and α(Rb11), respectively.

Clearly, the uncertainty in aµ(Rb11) is dominated by the hadronic

contribution.


Comparison with experiment

Comparing aµ(Rb11) with aµ(BNL04) we obtain

aµ(BNL04)− aµ(Rb11) = 234 (87)× 10−11.

To understand the cause of this 2.8 σ discrepancy between theory and

experiment is the urgent problem right now.

If this discrepancy persists in spite of further improvement of theory and

measurement, it may be regarded as indication of new physics beyond

SM.

Of particular interest is possible contribution of "dark photon".R. Essig, et al., ArXiv:1311.0029v1 [Hep-ph] 31 Oct 2013.H. Merkel, et al., ArXiv:1404.5502v1 [Hep-ex] 22 Ap 2014.

M. Endo, K. Hamaguchi, G. Mishima, PRD 86, 095029 (2012).


4. Evaluation of tenth-order term

Let me now sketch evaluation of A(10)1 .

We have to deal with so many Feynman diagrams, each very large and

complicated, that systematic and highly automated approach is needed.

Fortunately numerical renormalization method developed in 1974 for the

α3 case turned out to be readily adaptable to automation.

P. Cvitanovic, T. Kinoshita, PRD 10, 3978, 3991, 4007 (1974)

First step: Classify them into gauge-invariant sets.

We found 32 gauge-invariant sets which may be regarded as subsets of 6

supersets.


Diagrams of Superset I.

I(a) I(b) I(c)

I(d) I(e) I(f)

I(g) I(h) I(i)

I(j)

Figure : Diagrams of Superset I.

Set I consists of 10 subsets, all built from a second-order vertex. Solid lines

represent electron propagating in magnetic field. Wavy lines represent

photons. 208 diagrams contribute to A(10)1 . 498 contribute to A

(10)2 (me/mµ).


Diagrams of Supersets II and III.

II(a) II(b) II(c)

II(d) II(e) II(f)

Figure : Diagrams of Superset II.

Set II is built from fourth-order proper vertices. 600 diagrams contribute to

A(10)1 . 1176 diagrams contribute to A

(10)2 (me/mµ).

III(a) III(b) III(c)

Figure : Diagrams of Superset III.

Set III is built from sixth-order proper vertices. 1140 diagrams contribute to

A(10)1 . 1740 diagrams contribute to A

(10)2 (me/mµ).


Diagrams of Supersets IV and V.

Figure : Diagrams of Superset IV.

Set IV is built from eighth-order proper vertices. 2072 diagrams contribute to

both A(10)1 and A

(10)2 (me/mµ).

Figure : Diagrams of Superset V.

Set V consists of 10th-order proper vertices with no closed lepton loop. 6354

diagrams contribute to A(10)1 . No contribution to A

(10)2 (me/mµ).


Diagrams of Superset VI.

VI(a) VI(b) VI(c)

VI(d) VI(e) VI(f)

VI(g) VI(h) VI(i)

VI(j) VI(k)

Figure : Diagrams of Superset VI.

This set has 11 subsets, all containing light-by-light-scattering subdiagrams.

2298 diagrams contribute to A(10)1 . 3594 contribute to A

(10)2 (me/mµ).


Numerical renormalization of tenth-order integrands

Only a small fraction of diagrams have been evaluated analytically.

At present numerical approach is the only viable option for others.

But numerical method does not work if integral is divergent, which may

arise from large momentum region (UV) and/or vanishing of virtual

photon momentum (IR).

Our integrals are full of these divergences which must be removed from

the integrand before they are integrated.

Namely, renormalization must be applied to the integrands.

Once divergences are removed numerical integration is carried out by

VEGAS.


References

Numerical results of all gauge-invariant sets have been published over

years:T. Kinoshita, M. Nio, PRD 73, 053007 (2006)

T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, N. Watanabe, PRD 78, 053005 (2008)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 78, 113006 (2008)

T. Aoyama, K. Asano, M. Hayakawa, T. Kinoshita, M. Nio, N. Watanabe, PRD 81, 053009 (2010)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 82, 113004 (2011)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 83, 053002 (2011)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 83, 053003 (2011)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 84, 053003 (2011)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 85, 033007 (2012)T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRD 85, 093013 (2012)

T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRL 109, 111808 (2012)

T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, to be published.

Next Table summarizes our published results except for the latest value

of Set V, which is not yet published.


10th-order terms of ae and aµ from 32 subsets of Set V

set nF A(10)1 A

(10)2 (me/mµ) A

(10)2 (mµ/me) A

(10)2 (mµ/mτ ) A

(10)3 (mµ/me,mµ/mτ )

I(a) 1 0.000 470 9 (1) 0.000 000 28 (0) 22.566 973 (3) 0.000 038 (0) 0.017 312 (1)

I(b) 9 0.007 010 8 (7) 0.000 001 88 (0) 30.667 091 (3) 0.000 269 (0) 0.020 179 (1)

I(c) 9 0.023 468 0 (20) 0.000 002 67 (0) 5.141 395 (1) 0.000 397 (0) 0.002 330 (0)

I(d) 6 0.003 801 7 (6) 0.000 005 46 (0) 8.8921 (11) 0.000 388 (0) 0.024 487 (2)

I(e) 30 0.010 296 (4) 0.000 001 60 (0) -0.9312 (24) 0.000 232 (0) 0.002 370 (0)

I(f) 3 0.007 568 4 (20) 0.000 047 59 (1) 3.685 049 (90) 0.002 162 (0) 0.023 390 (2)

I(g) 9 0.028 569 0 (60) 0.000 024 45 (1) 2.607 87 (72) 0.001 698 (0) 0.002 729 (1)

I(h) 30 0.001 696 0 (130) -0.000 010 14 (3) -0.5686 (11) 0.000 163 (1) 0.001 976 (3)

I(i) 105 0.017 471 0 (1040) 0.000 001 67 (2) 0.0871 (59) 0.000 024 (0) 0

I(j) 6 0.000 397 5 (18) 0.000 002 41 (6) -1.263 72 (14) 0.000 168 (1) 0.000 110 (5)

II(a) 24 -0.109 495 0 (230) -0.000 737 69 (95) -70.4717 (38) -0.018 882 (8) -0.290 853 (85)

II(b) 108 -0.473 559 0 (840) -0.000 645 62 (53) -34.7715 (26) -0.035 615 (20) -0.127 369 (60)

II(c) 36 -0.116 489 0 (320) -0.000 380 25 (46) -5.385 75 (99) -0.016 348 (14) -0.040 800 (51)

II(d) 180 -0.242 998 0 (2890) -0.000 098 17 (41) 0.4972 (65) -0.007 673 (14) 0

II(e) 180 -1.344 860 0 (9900) -0.000 465 00 (400) 3.265 (12) -0.038 06 (13) 0

II(f) 72 -2.433 550 8 (14073) -0.005 867 50 (3870) -77.465 (12) -0.267 23 (73) -0.502 95 (68)

III(a) 300 2.127 330 0 (1700) 0.007 511 15 (1081) 109.116 (33) 0.283 000 (32) 0.891 40 (44)

III(b) 450 3.327 120 0 (4500) 0.002 794 00 (97) 11.9367 (45) 0.143 600 (10) 0

III(c) 390 4.923 440 0 (50600) 0.003 703 00 (36090) 7.43 (14) 0.1999 (28) 0

IV 2072 -7.729 600 0 (48000) -0.011 360 00 (6400) -38.79 (17) -0.4357 (24) 0

V 6354 8.582 (353) 0 0 0 0

VI(a) 36 1.041 319 0 (1870) 0.006 151 50 (1110) 629.141 (12) 0.246 10 (18) 2.3590 (18)

VI(b) 54 1.346 992 0 (2710) 0.001 778 90 (350) 181.1285 (51) 0.096 522 (93) 0.194 76 (26)

VI(c) 144 -2.528 900 0 (28000) -0.005 953 00 (5900) -36.58 (12) -0.2601 (28) -0.5018 (89)

VI(d) 492 1.846 720 0 (69200) 0.001 276 40 (7580) -7.92 (60) 0.0818 (17) 0

VI(e) 48 -0.431 200 0 (6000) -0.000 750 00 (800) -4.32 (14) -0.035 94 (32) -0.1122 (24)

VI(f) 180 0.770 300 0 (22000) 0.000 033 00 (700) -38.16 (15) 0.043 47 (85) 0.0659 (31)

VI(g) 480 -1.590 420 0 (62400) -0.000 497 20 (2860) 6.96 (48) -0.044 51 (96) 0

VI(h) 630 0.179 220 0 (38700) 0.000 044 50 (900) -8.55 (23) 0.004 85 (46) 0

VI(i) 60 -0.043 800 0 (11000) -0.000 326 00 (100) -27.34 (12) -0.003 45 (33) -0.0027 (11)

VI(j) 54 -0.228 800 0 (17000) -0.000 127 10 (1250) -25.505 (20) -0.011 49 (33) -0.016 03 (58)

VI(k) 120 0.680 200 0 (38000) 0.000 015 60 (400) 97.123 (62) 0.002 17 (16) 0

sum 12672 7.651 (353) -0.003 822 (383) 742.24 (87) -0.0681 (52) 2.011 (10)


Comment on the Table

Third and fourth columns are for ae. Fifth to seventh columns are for

mass-dependent terms of aµ.

All ae and aµ of Set I have now been confirmed by analytic or Padé

methods.

P. A. Baikov, A. Maier, P. Marquard, NPB 877, 647 (2013) [arXiv:1307.6105v2]

Our calculation of complete α5 term of aµ reduces uncertainty due to

QED significantly, making it easier to identify possible new physics in the

forthcoming measurements of aµ.

T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, PRL 109, 111808 (2012)


Comment on the Table-cont.

Largest uncertainty of aµ comes from hadronic contribution. This will be

the subject of several speakers:

Vainshtein, Vanderhaeghen, Blum, Denig, this conference.

Strange and charm quark contributions to aµ have been calculated in

LATTICE QCD:

B. Chakraborty, C. T. H. Davies, G. C. Donald, R. J. Dowdall, J. Koponen, G. P. Lepage, T. Teubner

(HPQCD Collaboration), PRD 89, 114501 (2014) [arXiv: 1403.1778].

LATTICE calculation of up and down quark contributions is making good

progress, too. (Private com. from Lepage.)

LATTICE calculation of hadronic light-by-light-scattering is in progress:

T. Blum, S. Chowdhury, M. Hayakawa, T. Izubuchi, arXiv: 1407.2923.


What’s next ?

At present A(8)1 is largest source of theoretical uncertainty for ae. For

current status of analytic work, seeS. Laporta, E. Remiddi, in "Lepton Dipole Moments",

eds. B. L. Roberts, W. J. Marciano, (World Scientific, Singapore 2010), p. 119.

Meanwhile A(8)1 and A

(10)1 are being improved slowly but steadily by

numerical means.

When measurements of ae and α are improved further, SM may be

tested beyond the level of 0.1 ppb.

Together with new measurements of aµ they will impose stricter

constraint on possible new physics.


Discussion

Discoverers of QED regarded the renormalization procedure as a

jerry-built temporary fix to be replaced later by something better.S. Tomonaga: Private communication.

Letter of Dyson to Gabrielse quoted in Physics Today (August 2006), p.15.

What bothered them was mathematically dubious treatment of

divergences: renormalization just hides infinity without solving it.

K. Wilson treated this problem by means of renormalization group.

But I am not sure whether it satisfied Tomonaga and Dyson.

Anyway, the terms Wilson ignored as irrelevant are now relevant for

evaluating ae.

As I have shown, jerry-built structure still works up to tenth order,

surviving as the basic framework of QED (and Standard Model).


Discussion - cont.

Comparison of theory and experiment of ae began historically as a test of

the validity of QED.

But it has evolved into precision test of internal consistency of quantum

mechanics in general, not just test of QED or Standard Model.

With further improvements of theory and measurement, ae and aµ will

provide sharpest tools in the study of physics beyond SM.

I thank M. Nio for her very careful reading and useful suggestions.


Thank you.


the qed contribution to g 2 of the electron and muon ...g2pc1.bu.edu/lept14/talks/kinoshita.pdfthe...

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