standard model prediction for muon g 2 - riken

Standard Model predictionfor muon g − 2

Daisuke Nomura (YITP, Kyoto U.)

talk at the workshop

‘Storage Rings and Tests of Fundamental Symmetries’

at RIKEN, Wako

March 2, 2016

D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 1 / 35

Muon g − 2: introduction


Introduction: Standard Model prediction for muon g − 2

QED contribution 11 658 471.808 (0.015) Kinoshita & Nio, Aoyama et al

EW contribution 15.4 (0.2) Czarnecki et al

Hadronic contributions

LO hadronic 694.9 (4.3) HLMNT11

NLO hadronic −9.8 (0.1) HLMNT11

light-by-light 10.5 (2.6) Prades, de Rafael & Vainshtein

Theory TOTAL 11 659 182.8 (4.9)

Experiment 11 659 208.9 (6.3) world avg

Exp − Theory 26.1 (8.0) 3.3 σ discrepancy

(in units of 10−10. Numbers taken from HLMNT11, arXiv:1105.3149)

n.b.: hadronic contributions:. .

. .

had.

LO

µ

had.

NLO

µγ

had.

l-by-l

µ


Improvements in the past few years (1)

QED contribution5-loop calculation completed:

now aµ(QED) = 11 658 471.895(8) × 10−10

(number from Aoyama et al (2012))

See Nio-san’s talk

EW contributionHiggs-boson mass just fixed:

now aµ(EW) = 15.4(1) × 10−10 Gnendiger et al ’13

was aµ(EW) = 15.4(2) × 10−10 Czarnecki et al ’02

cf: aexpµ − aSM

µ =

{(26.1 ± 8.0) × 10−10 HLMNT11

(28.7 ± 8.0) × 10−10 Davier et al ’10



NNLO hadronic contributionaµ(had, NNLO) = (1.24 ± 0.01) × 10−10

Numbers and Figs. fromA. Kurz et al, arXiv:1403.6400

3a: +0.80 × 10−10

3b: −0.41 × 10−10

3b,lbl: +0.91 × 10−10

3c: −0.06 × 10−10

3d: +0.0005 × 10−10

cf: aexpµ − aSM

µ =

{(26.1 ± 8.0) × 10−10 HLMNT11

(28.7 ± 8.0) × 10−10 Davier et al ’10

aµ(had, NLO) = (−9.8 ± 0.1) × 10−10 HLMNT11



lbyl-NLO hadronic contribution

aµ(had, lbyl-NLO) = (0.3 ± 0.2) × 10−10

had

Number and Fig. from Colangelo et al,arXiv:1403.7512

=⇒ Negligible. But it is always goodto confirm that higher order termsare really negligible.

cf: aexpµ − aSM

µ =

{(26.1 ± 8.0) × 10−10 HLMNT11

(28.7 ± 8.0) × 10−10 Davier et al ’10

ahad,lbylµ =

{(10.5 ± 2.6) × 10−10 Prades et al, ’09

(11.6 ± 3.9) × 10−10 Jegerlehner+Nyffeler, ’09D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 6 / 35

Updated Standard Model prediction for muon g − 2

Aoyama et al ’12Gnendiger et al ’13

HLMNT11

Davier et al ’10

HLMNT11Kurz et al ’14Prades et al ’09Colangelo et al ’14

BNL

Table from TDR for g − 2/EDM experiment at J-PARC (J-PARC E34)


Introduction for ahad,LOµ

The diagram to be evaluated:

.

.

.

.

had.

µ

pQCD not useful. Use the dispersionrelation and the optical theorem.

.

.

.

.

had.

=∫

ds

π(s−q2)Im

had.

2 Im

had.

=∑

dΦ

∫ 2

had.

ahad,LOµ =m2

µ

12π3

∫ ∞

sth

ds1

sK̂(s)σhad(s)

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

1 10

J/ψ ψ(2S) ϒ

√s (GeV)

(2 α

2/

3 π

2)

K(s

) R

(s)

× 1

010

0.2 –0.2 1 10√s (GeV)

• Weight function K̂(s)/s = O(1)/s=⇒ Lower energies more important=⇒ π+π− channel: 73% of total ahad,LOµ


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Y�[


Important Channels

Contributions from various channels to aµ(LO,had) for√s < 1.8GeV:

channel HLMNT11 Davier et al ’10 diffπ+π− 505.65 ± 3.09 507.80 ± 2.84 −2.15π+π−π0 47.38 ± 0.99 46.00 ± 1.48 1.38K+K− 22.09 ± 0.46 21.63 ± 0.73 0.46π+π−2π0 18.62 ± 1.15 18.01 ± 1.24 0.612π+2π− 13.50 ± 0.44 13.35 ± 0.53 0.15K0

SK0L 13.32 ± 0.16 12.96 ± 0.39 0.36

π0γ 4.54 ± 0.14 4.42 ± 0.19 0.12...

......

...Sum 634.28 ± 3.53 633.93 ± 3.61 0.35

table taken from HLMNT11D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 10 / 35

π+π− channel: Low Energy Tail

0

50

100

150

200

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

σ0 (e+e- →

π+π- )

[nb]

√s [GeV]

New FitBaBar (09)KLOE (10)

CMD-2 (06)SND (06)

OLYA-VEPP2TOF-VEPP 2M

NA7CMD-VEPP 2M

ChPT


π+π− channel: New Radiative Return Data

0

200

400

600

800

1000

1200

1400

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

σ0 (e+e- →

π+π- )

[nb]

√s [GeV]

BaBar (09)New Fit

KLOE (10)KLOE (08)


π+π− channel: Zoom-In at ρ-ω Region

600

700

800

900

1000

1100

1200

1300

1400

0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82

σ0 (e+e- →

π+π- )

[nb]

√s [GeV]

BaBar (09)New Fit

KLOE (10)KLOE (08)

CMD-2 (07)SND (06)

CMD-2 (04)


Rad. Rtn. Data (for π+π−) and Our Combined Result

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

(σ0 R

adR

et S

ets

- σ0 F

it)/σ

0 Fit

√s [GeV]

New FitBaBar (09)

New Fit (local χ2 inf)KLOE (08)KLOE (10)


Results: ahad,LOµ , ahad,NLO

µ and ∆α(5)had(M

2Z)

aµhad,LO VP

∆α(5)had (M 2

Z)

value (error)2

mπ

0.6

0.9

1.42

∞rad.

mπ0.6

0.91.4

2∞

mπ 0.60.9

1.42

4

11

∞rad.

mπ 0.60.91.4

2

4

11

∞

ahad,LOµ =(694.91 ± 3.72exp ± 2.10rad) × 10−10

ahad,NLOµ =(−9.84 ± 0.06exp ± 0.04rad) × 10−10

∆α(5)had(M

2Z) =(276.26 ± 1.16exp ± 0.74rad) × 10−4


Full SM Result and Comparison with Other Groups

170 180 190 200 210

aµ × 1010 – 11659000

HMNT (06)

JN (09)

Davier et al, τ (10)

Davier et al, e+e– (10)

JS (11)

HLMNT (10)

HLMNT (11)

experiment

BNL

BNL (new from shift in λ)


SUSY Contributions to Muon g − 2

Suppose that the 3.3 σ deviation is due to SUSY...Leading SUSY contributions in the mZ/mSUSY

expansion:. .

W̃ − H̃−

µL µRν̃µ

(a)

B̃

µL µ̃L

m2LR

µ̃R µR

(b)

B̃ H̃0

µL µRµ̃L

(c)

W̃ 0 H̃0

µL µRµ̃L

(d)

H̃0 B̃

µL µRµ̃R

(e)

In most cases, the χ̃±-ν̃ diagram (a) and/or the

B̃-µ̃L/R diagram (b) dominate. (Lopez-Nanopoulos-Wang,

Chattopadhyay-Nath, Moroi, · · · )D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 17 / 35

MSSM Contributions to Muon g − 2

x-axis: M2

(gaugino mass)

y-axis: ml̃

(slepton mass)

Figs from Cho,Hagiwara, Matsumoto,

and DN


Very recent updates


BESIII new pion form factor data (1)

[GeV]s'0.6 0.65 0.7 0.75 0.8 0.85 0.9

2 | π|F

10

15

20

25

30

35

40

45 BESIII fit

BESIII

Fig. from M. Ablikim et al (BESIII Collaboration), arXiv:1507.08188v3

Data taken by using the

ISR technique



[GeV]s'0.6 0.65 0.7 0.75 0.8 0.85 0.9

/ B

ES

III fi

t - 1

2 | π|F

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

BESIII fit

BaBar

BESIII

Fig. from M. Ablikim et al (BESIII Collaboration), arXiv:1507.08188v3

[GeV]s'0.6 0.65 0.7 0.75 0.8 0.85 0.9

/ B

ES

III fi

t - 1

2 | π|F

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

BESIII fit

KLOE 08

KLOE 10

KLOE 12



]-10(600 - 900 MeV) [10,LOππµa

360 365 370 375 380 385 390 395

BaBar 09

KLOE 12

KLOE 10

KLOE 08

BESIII

1.9± 2.0 ±376.7

0.8± 2.4 ± 1.2 ±366.7

2.2± 2.3 ± 0.9 ±365.3

2.2± 2.3 ± 0.4 ±368.1

3.3± 2.5 ±368.2

Prediction for aππµ from BESIII data is more consistent

with those from KLOE data sets than that from BaBar data

Fig. from M. Ablikim et al (BESIII Collaboration), arXiv:1507.08188v3D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 22 / 35

Slide by T. Teubner (Liverpool), talk at ‘High-precision QCD at low energy,’ Aug. ’15


Near future (5-7yrs) prospects


Slide by V. B. Golubev (BINP), talk at “Tau 2014”, Sept. 2014D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 25 / 35

Near Future Prospects (Blum et al, arXiv:1311.2198)

Current status and near-future(∼ 5-7 yrs) improvementsin δaµ (in units of 10−11)

[20]: DHMZ[21]: HLMNT

Near-future improvements in δaHLOµ

mainly from VEPP-2000 and BES-III.

Table and Fig. from T. Blum et al, arXiv:1311.2198


Near Future Prospects, Lattice (Blum et al)

As for the hadronic vacuum polarization (HVP),

“...the lattice-QCD uncertainty on aµ(HVP), currently at the5%-level, can be reduced to 1 or 2% within the next few years.”

“With increasing experience and computer power, it should bepossible to compete with the e+e− determination of aµ(HVP) bythe end of the decade, perhaps sooner with additional technicaladvances.”

As for l-by-l,

“... a lattice calculation with even a solid 30% error would alreadybe very interesting. Such a result, ..., is not out of the questionduring the next 3-5 years.”

From T. Blum et al, arXiv:1311.2198D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 27 / 35

Byproducts


Byproducts: QED coupling at the Z-boson mass

⋆ α(M2Z): the least well known among {Gµ,MZ, α(M2

Z)},which are used as input to precision electroweak fits.⋆ Running of α

α(M2Z) =

α

1 − ∆αlep(M2Z) − ∆α

(5)had(M

2Z) − ∆αtop(M2

Z)

where ∆αlep(M2Z) = 0.03149769 (Steinhauser),

∆αtop(M2Z) = − 0.0000728(14) and α = 1/137.035999679(94)

(PDG10).

⋆ Similar dispersion relation: (=⇒ byproduct of ahad,LOµ )

∆α(5)had(s) = −

αs

3πP

∫R(s′)ds′

s′(s′ − s)

⋆ Our results: ∆α(5)had(M

2Z) = (276.3 ± 1.4) × 10−4,

α(M2Z)

−1 = 128.944 ± 0.019.D. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 29 / 35

Byproducts: running QED coupling α(q2)

The hadronic contribution ∆α(5)had(q

2) to the runningQED coupling for q2 > 0 (left) and q2 < 0 (right)

-2

-1

0

1

2

3

4

10-1

1 10√s (GeV)

∆α(5

) had (

s)/α

0

0.5

1

1.5

2

2.5

3

3.5

4

10-1

1 10√s (GeV)

∆α(5

) had (

– s)

/α

Fortran subroutine to compute the above is available from us upon

requestD. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 30 / 35

Byproduct: Standard Model prediction for electron g − 2

QED contribution 115 965 218 00.7 (0.6)4loop(0.4)5loop(7.6)α (using α(87Rb)) Aoyama et al

EW contribution 0.2973 (0.0052)Czarnecki et al

Hadronic contributions

LO hadronic 18.66 (0.11) DN & TT (was 18.75(0.18) Davier et al ’98)

NLO hadronic −2.23 (0.01) DN & TT (was −2.25(0.05) Krause ’97)

light-by-light 0.39 (0.13) Jegerlehner+Nyffeler

Theory total 115 965 218 17.8 (0.6)4loop(0.4)5loop(0.2)had(7.6)α

Experiment 115 965 218 07.3 (2.8) Gabrielse et al

Theory − Exp 10.5 (8.1) 1.3σ discrepancy

(in units of 10−13. Numbers taken from Giudice et al, JHEP11(2012)113)

n.b.: hadronic contributions:. .

. .

had.

LO

e

had.

NLO

e

γhad.

l-by-l

e


Hyperfine Splitting?

Fig. from WikipediaD. Nomura (YITP, Kyoto U.) Standard Model prediction for muon g − 2 March 2, 2016 32 / 35

Mu HFS: Leading Hadronic Contributions

Diagrams:

Can be written as an integral over σ(e+e− → hadrons)with weight function KMu(s):

∆ν(had-vp) =1

2π3

me

mµ

νF

∫dsσ(s)KMu(s)

= 232.7(1.4)Hz DN & TT

(cf: 231.2(2.9) Hz, Eidelman et al ’02)

Fig. from Karshenboim+Shelyuto

PLB517(2001)32


Mu HFS: Summary Table

contribution (Hz)νHFS(QED) 4 463 302 720(253)mµ/me

(98)QED(3)ανHFS(weak) − 65νHFS(had-vp) 231(3) → 232.7(1.4) DN & TTνHFS(had-h.o.) 5(2)νHFS(theory) 4 463 302 891(272)νHFS(exp) 4 463 302 776(51) → (8)expected at MuSEUM

Black numbers are from CODATA10, arXiv:1203.5425


Summary

Hadronic contrib. to the muon g − 2: key to improve theStandard Model prediction

≳ 3 σ discrepancy in (g− 2)µ between experiment and theory=⇒ New physics? “Light” new particles like µ̃, µKK, . . . ???=⇒ Worth studying µ-EDM, µ → eγ, µ-e conv., . . .!(⇔ No new physics seen at the LHC so far. What does thismean?)

Two new experiments to measure the muon g − 2 planned atJ-PARC and Fermilab.

To establish this discrepancy more firmly, important to resolvethe tension between the KLOE and BaBar data=⇒ new data from BES-III just appeared!=⇒ new precise data from VEPP-2000 and SuperKEKBstrongly awaited.


standard model prediction for muon g 2 - riken

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