The Muon g − 2: present and futureFred Jegerlehner, DESY Zeuthen/Humboldt University Berlin$'“Strong Coupling Gauge Theories Beyond the Standard Model”(SCGT14mini)
March 5 - March 7, 2014
Nagoya University, Nagoya, Japan&

F. Jegerlehner%

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

Outline of Talk:vIntroduction
vStandard Model Prediction for aµvThe hadronic effects and precision limitationsvEffective field theory: the Resonance Lagrangian Approach
vThe hadronic LbL: setup and problems
vTheory vs experiment: do we see New Physics?
vFutureF. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 20141

Introduction
Particle with spin ~s ⇒ magnetic moment ~µ (internal current circulating)e~~µ = gµ~s ;2mµc

gµ = 2 (1 + aµ)αDirac: gµ = 2 , aµ = 2π
+ · · · muon anomalyγ(q)µ(p′ )

Electromagnetic Lepton Vertex= (−ie) ū(p0)σµν qν
µ2
γ F1(q ) + i 2mµ F2(q2)


u(p)

µ(p)

F1(0) = 1 ;F2(0) = aµaµ responsible for the Larmor precession

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

2

µ

⇒

StorageRing⇒

momentum⇒⇒eBωa = aµ mc

⇒

⇒
spin

⇒

⇒

⇒
⇒

Million Events per 149.2ns~~ of beam of spin particles in a homogeneous magnetic field B
Larmor precession ω
10

1

10

10

10

-1

-2

-3

0

20

40

actual precession × 2 ∼ 120 /circle

60
80
100
Time modulo 100 µ s [µ s]~ at magic energy ∼ 3.1 GeV~ is directly proportional to B
Magic Energy: ωE∼3.1GeV
h
i
1
e
e
~
~
~
~
~ a = m aµ B − aµ − γ2−1 β × Eω' m aµ B
at ”magic γ”

CERN, BNL g-2 experiments
Stern, Gerlach 22: ge = 2; Kusch, Foley 48: ge = 2 (1.00119 ± 0.00005)
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 20143

In a uniform magnetic field, as in muon g − 2 experimental setup:ω /ωRaµ = γωωac = µµ/µ pa−ωap/ω p = λ−Rr ω p = (e/m pc)hBi free proton NMR frequency
r R = ωa/ω p = 0.003 707 2063(20) from E-821
r λ = ωL/ω p = µµ/µ p = from hyperfine splitting of muonium
value used by E-821
new value

3.18334539(10)3.183345107(84)CODATA 2011: [raXiv:1203.5425v1]⇒change in aµ: +1.10 × 10−10




exp

aµ = (11 659 209.1 ± 5.4 ± 3.3[6.3]) × 10−10 updated

F. Jegerlehner



SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

4

Standard Model Prediction for aµ
What is new?
l new CODATA values for lepton mass ratios mµ/me, mµ/mτ
l spectacular progress by Aoyama, Hayakawa, Kinoshita and Nio on 5–loop QED
calculation (as well as improved 4–loop results)r O(α5) electron g − 2, substantially more precise α(ae)r Complete O(α5) muon g − 2, settles better the QED partr QED Contribution
The QED contribution to aµ has been computed through 5 loopsmGrowing coefficients in the α/π expansion reflect the presence of large ln mµe ' 5.3
terms coming from electron loops. Input:
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

5

exp

ae = 0.001 159 652 180 73(28)Gabrielse et al. 2008

α−1(ae) = 137.0359991657(331)(68)(46)(24)[0.25 ppb]

Aoyama et al 2012

−11
aQED
= 116 584 718.851 |(0.029)(0.009)(0.018)(0.007)[0.36]×10µ{z } | {z } | {z } | {z }αinp

me /mµα4α5The current uncertainty is well below the ±60 × 10−11 experimental error from E821
# n of loops
1
2
3
4
5
tot
F. JegerlehneraQED× 1011µCi [(α/π)n]+0.5+0.765 857 426(16)+24.050 509 88(32)+130.8796(63)+753.290(1.04)

116140973.289 (43)413217.628 (9)30141.9023 (4)381.008 (18)
5.094 (7)116584718.851 (0.036)

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 20146

❶

1 diagram

Schwinger 1948❷

7 diagramsPeterman 1957, Sommerfield 1957

❸ 72 diagrams

Lautrup, Peterman, de Rafael 1974,
Laporta, Remiddi 1996

❹ 871 diagrams

Kinoshita 1999, Kinoshita, Nio 2004,
Ayoama et al. 2009/2012❺ estimates of leading terms

Karshenboim 93,
Czarnecki, Marciano 00, Kinoshita, Nio 05❏ all 12672 diagrams (fully automated numerical)
Ayoama et al. 2012

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 20147

r Weak contributions

W

Brodsky, Sullivan 67, ...,
Bardeen, Gastmans, Lautrup 72
Higgs contribution tiny!

W
+

νµ

+

Z

H

= (194.82 ± 0.02) × 10−11
aweak(1)µKukhto et al 92
M
potentially large terms ∼ G F m2µ απ ln mµZ

µ
Z

•γ

µ
+

W
νµ

W
νµ
W

+

Z

e, u,d,· · ·
γ
+ ···
µ

Peris, Perrottet, de Rafael 95
quark-lepton (triangle anomaly) cancellation
Czarnecki, Krause, Marciano 96

Heinemeyer, Stöckinger, Weiglein 04, Gribouk, Czarnecki 05 full 2–loop result

Most recent evaluations: improved hadronic part (beyond QPM)

aweak
= (154.0 ± 1.0[had] ± 0.3[mH, mt, 3 − loop]) × 10−11µnew: mH known!

(Knecht, Peris, Perrottet, de Rafael 02, Czarnecki, Marciano, Vainshtein 02, FJ 12,
Gnendiger, Stöckinger, Stöckinger-Kim 13)
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 20148

r Hadronic stuff: the limitation to theory
General problem in electroweak precision physics:
contributions from hadrons (quark loops) at low energy scales
Leptons
Quarksge, µ, τ <α : weak coupling
pQED✓

γ
γ

u, d, s, · · <
·

γ

γ

>

γ
>
αs : strong coupling
✗
pQCD

(a)

(b)

(c)
u,d,· · ·u,d,· · ·
µ
µ
γ γ
γ
γ
Z
+
+ ···
γ γ(Z) + µµ•(a) Hadronic vacuum polarization O(α2), O(α3)(b) Hadronic light-by-light scattering O(α3)(c) Hadronic effects in 2-loop EWRC O(αG F m2µ)F. JegerlehnerLight quark loops↓
Hadronic “blobs”

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 20149

r Evaluation of ahadµLeading non-perturbative hadronic contributions ahad
µ can be obtained in terms of2

Rγ (s) ≡ σ(0)(e+e− → γ∗ → hadrons)/ 4πα3s data via dispersion integral:
2

ahad
=
µ

 αm 2 
µ

3π

Z∞
ZEcut
pQCD
data
Rγ (s) K̂(s)Rγ (s) K̂(s) 
+
ds
ds
2
s
s2
2
Ecut

4m2π

 





l Experimental error implies theoretical uncertainty!
l Low energy contributions enhanced: ∼ 75% come from region 4m2π < m2ππ < MΦ2
Data: CMD-2, SND, KLOE, BaBarahad(1)= (690.7 ± 4.7)[695.5 ± 4.1] 10−10µe+e−–data based [incl. BaBar MD09]
1.0 GeV
ρ, ω

ρ, ω
Υ
ψ

φ, . . .

0.0 GeV, ∞9.5 GeV
3.1 GeV
2.0 GeV

0.0 GeV, ∞3.1 GeV
2.0 GeVφ, . . .

1.0 GeVF. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201410

γ hard

φ

e+

+ −

π π , ρ0γe−s = Mφ2 ; s′ = s (1 − k), k = Eγ /Ebeama)hadronsb)a) Radiative return, b) Standard energy scan.

vGood old idea: use isospin symmetry to include existing high quality τ–data(including isospin corrections)
e+

γ

γ

ū, d¯π +π −, · · · [I = 1]

e−

u, d
⇑
isospin rotation
⇓

τ−

W

W

ū
π 0π −, · · ·ν̄µdCorrected data: large discrepancy [∼ 10%] persists! τ vs. e+e− problem! [manifest
since 2002]
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201411

Recent: τ (charged channel) vs. e+e− (neutral channel) puzzle resolved
F.J.& R. Szafron, ρ − γ interference(absent in charged channel):(π)−i Πµν
(q) =
γρ

+

.





30(s) = rργ (s) RIB(s) 3−(s)r τ require to be corrected for missing ρ − γ mixing!
r results obtained from e+e− data is what goes into aµr off-resonance tiny for ω, φ in ππ channel (scaled up ΓV /Γ(V → ππ)
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201412τ decayse+ e− +CVC

ALEPH 1997

390.75 ± 2.65 ± 1.94

ALEPH 2005

388.74 ± 4.00 ± 2.07

OPAL 1999

380.25 ± 7.27 ± 5.06

CLEO 2000

391.59 ± 4.11 ± 6.27

Belle 2008

394.67 ± 0.53 ± 3.66

τ combined

391.06 ± 1.42 ± 2.06

CMD-2 2006

386.58 ± 2.76 ± 2.59

SND 2006

383.99 ± 1.40 ± 4.99

KLOE 2008

380.21 ± 0.34 ± 3.27

KLOE 2010377.35 ± 0.71 ± 3.50

BABAR 2009389.35 ± 0.37 ± 2.00

e+ e− combined

385.12 ± 0.87 ± 2.18aµ [ππ], I = 1, (0.592 − 0.975) GeV

380 390 400

×10−10I=1 part of ahad
µ [ππ]
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201413τ decayse+ e− +CVC

ALEPH 1997

385.63 ± 2.65 ± 1.94

ALEPH 2005

383.54 ± 4.00 ± 2.07

OPAL 1999

375.39 ± 7.27 ± 5.06

CLEO 2000

386.61 ± 4.11 ± 6.27

Belle 2008

389.62 ± 0.53 ± 3.66

τ combined

385.96 ± 1.40 ± 2.10

CMD-2 2006

386.58 ± 2.76 ± 2.59

SND 2006

383.99 ± 1.40 ± 4.99

KLOE 2008

380.21 ± 0.34 ± 3.27

KLOE 2010377.35 ± 0.71 ± 3.50

BABAR 2009389.35 ± 0.37 ± 2.00

e+ e− combined

385.12 ± 0.87 ± 2.18aµ [ππ], I = 1, (0.592 − 0.975) GeV

380 390 400

×10−10I=1 part of ahad
µ [ππ]
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201414

|Fπ(E)|2 in units of e+e− I=1 (CMD-2 GS fit)
Best “proof”:

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201415

|Fπ(E)|2 in units of e+e− I=1 (CMD-2 GS fit)
Best “proof”:

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201416

Is our model viable?

How photons couple to pions? Use γγ → π+π−, π0π0 as a probe: what wesee: 1) below about 1 GeV photons couple to pions as point-like objects(i.e. to the charged ones overwhelmingly), 2) at higher energies thephotons see the quarks exclusively and form the prominent tensor
resonance f2(1270). Plotted 2 σ(π0π0) vs. σ(π+π−)Strong tensor meson resonance in ππ channel f2(1270) with photons directly
LbL
probe the quarks! Contribution to ahad?
µF. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201417

Effective field theory: the Resonance Lagrangian Approach
HVP dominated by spin 1 resonance physics! need theory of ρ, ω, φ, · · ·r Principles to be included: Chiral Structure of QCD, VMD & electromagnetic
gauge invariance.

vGeneral framework: resonance Lagrangian extension of chiral perturbation
theory (CHPT), i.e. implement VMD model with Chiral structure of QCD. Specificversion Hidden Local Symmetry (HLS) effective Lagrangian Bando, Kugo,
Yamawaki. First applied to HLbL of muon g − 2 Hayakawa, Kinoshita, Sanda.
Global Fit strategy:
Data below E0 = 1.05 GeV (just above the φ) constrain effective Lagrangian
couplings, using 45 different data sets (6 annihilation channels and 10 partial
width decays).

r Effective theory predicts cross sections:
π+π−, π0γ, ηγ, η0γ, π0π+π−, K + K −, K 0 K̄ 0 (83.4%),
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201418

l Missing part:
4π, 5π, 6π, ηππ, ωπ and regime E > E0
evaluated using data directly and pQCD for perturbative region and taill Including self-energy effects is mandatory (γρ-mixing, ρω-mixing ..., decays
with proper phase space, energy dependent width etc)l Method works in reducing uncertainties by using indirect constraints
l Able to reveal inconsistencies in data, e.g. KLOE vs BaBar
Main goal:r Single out representative effective resonance Lagrangian by global fit
is expected to help in improving EFT calculations of hadronic
light-by-light scattering (such concept so far missing)r could help improving uncertainty on hadronic VP (besides e+e− and τ decay
data other experimental information
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201419

o Fit of τ +PDG vs π+π−–data

Benayoun et al 2012/13
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201420

The ππ scattering phase of our HLS prediction
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201421

incl. ISR
DHMZ10 (e+ e− )180.2 ± 4.9
DHMZ10 (e+ e− +τ )189.4 ± 5.4
JS11 (e+ e− +τ )179.7 ± 6.0
HLMNT11 (e+ e− )182.8 ± 4.9
DHMZ10/JS11 (e+ e− +τ )181.1 ± 4.6
BDDJ13∗ (e+ e− +τ )177.7 ± 5.8
excl. ISR
+ −DHea09 (e e )178.8 ± 5.8
BDDJ12∗ (e+ e− +τ )175.4 ± 5.3
experiment
BNL-E821 (world average)
208.9 ± 6.3
150[3.6 σ][2.4 σ][3.4 σ][3.3 σ][3.6 σ][3.7 σ][3.5 σ][4.1 σ]∗HLS fits

200

250
10

aµ ×10 -11659000

Comparison with other Results. Note: results depend on which value is taken for
HLbL. JS11 and BDDJ13 includes 116(39) × 10−11 [JN], DHea09, DHMZ10,
HLMNT11 and BDDJ12 use 105(26) × 10−11 [PdRV].
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201422

Main issues
r region 1.2 to 2 GeV bad data; test-ground exclusive vs inclusive R
measurements (more than 30 channels!) Who will do it? BES III
radiative return!

r discrepancy BaBar vs KLOE ππ data. Who can clarify it?

Davier&Malaescu 2013
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201423

Davier&Malaescu 2013
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201424

The hadronic LbL: setup and problems
γ(k)Hadrons in h0|T {Aµ(x1)Aν(x2)Aρ(x3)Aσ(x4)}|0i

kρ

had
q3λ

Key object full rank-four hadronic
vacuum polarization tensor

Πµνλρ(q1, q2, q3) =v non-perturbative physics

Z

q2ν

q1µ

µ(p)

µ(p′ )

d4 x1 d4 x2 d4 x3 ei (q1 x1+q2 x2+q3 x3)
×h 0 | T { jµ(x1) jν(x2) jλ(x3) jρ(0)} | 0 i .

v general covariant decomposition involves 138 Lorentz structures of which
v 32 can contribute to g − 2v fortunately, dominated by the pseudoscalar exchanges π0, η, η0, ... described by
the effective Wess-Zumino Lagrangian
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201425

v generally, pQCD useful to evaluate the short distance (S.D.) tailv the dominant long distance (L.D.) part must be evaluated using some low energy
effective model which includes the pseudoscalar Goldstone bosons as well as
the vector mesons which play a dominant role (vector meson dominance mechanism); HLS, ENJL, general RLA, large Nc inspired ansätze, and others
Need appropriate low energy effective theory ⇒amount to calculate the following
type diagrams
83(12) × 10−11−19(13) × 10−11+62(3) × 10−11
q = (u, d, s, ...)

π± , K ±π 0 , η, η ′

L.D.

L.D.

S.D.

LD contribution requires low energy effective hadronic models: simplest case π0γγ
vertexF. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201426

Crystal Ball 1988
u, d
+

+

g

+···

+

u, d
π

→
|

0

+

π±
{z

L.D.

+···+
}

+···
|

{z

S.D.

}

Data show almost background free spikes of the PS mesons! Substantial
background form quark loop is absent. Clear message from data: fully
non-perturbative, evidence for PS dominance. However, no information about axial
mesons (Landau-Yang theorem). Illustrates how data can tell us where we are.
Low energy expansion in terms of hadronic components: theoretical models vsexperimental data à KLOE, KEDR, BES, BaBar, Belle, ?
Basic problem: (s, s1, s2)–domain of Fπ0∗γ∗γ∗ (s, s1, s2); here (0, s1, s2)–plane
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201427

Two scale problem: “open regions”

???

pQCD

RLA

???

???

– Data + Dispersion Relation, OPE,
– QCD factorization,– Brodsky-Lepage approach– Models constrained by data

One scale problem: “no problem”

RLA

pQCD

Novel approach: refer to quark–hadron duality of large-Nc QCD, hadron spectrum
known, infinite series of narrow spin 1 resonances ’t Hooft 79 ⇒no matchingproblem (resonance representation has to match quark level representation)
De Rafael 94, Knecht, Nyffeler 02
Constraints for on-shell pions (pion pole approximation)v General form–factor Fπ0∗γ∗γ∗ (s, s1, s2) is largely unknown
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

28

2

e2 Nc
2
Fπ0γγ (mπ, 0, 0) = 12π2 f = παfπ ≈ 0.025 GeV−1 well determined byπv The constant e
π0 → γγ decay rate (from Wess-Zumino Lagrangian); experimental improvement
needed!

v Information on Fπ0γ∗γ (m2π, −Q2, 0) from e+e− → e+e−π0 experiments′e − (pt )0970597-008

0.30
CELLO
CLEO

γ

Q2 > 0

∗

π

0

2
2
Q IIF(Q ) II (GeV)

e− (pb )
0.20

2f

0.10

γ
e+

2

q ∼0
′

e+

0

2.5

5.0
2

7.5

10.0

2

Q (GeV )CELLO and CLEO measurement of the π0 form factor Fπ0γ∗γ (m2π, −Q2, 0) at highspace–like Q2. Outdated by BaBar? Belle conforms with theory expectations!

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201429

Brodsky–Lepage interpolating formula gives an acceptable fit.

Fπ0γ∗γ (m2π, −Q2, 0) '

1
2 fπ
1
∼
4π2 fπ 1 + (Q2/8π2 fπ2) Q2

Inspired by pion pole dominance idea this FF has been used mostly(HKS,BPP,KN) in the past, but has been criticized recently (MV and FJ07).

r Melnikov, Vainshtein: in chiral limit vertex with external photon must be
non-dressed! i.e. use Fπ0γ∗γ (0, 0, 0), which avoids eventual kinematicinconsistency, thus no VMD damping ⇒result increases by 30% !

r In g − 2 external photon at zero momentum ⇒ only Fπ0∗γ∗γ (−Q2, −Q2, 0) not
Fπ0γ∗γ (m2π, −Q2, 0) is consistent with kinematics. Unfortunately, this off–shell form
factor is not known and in fact not measurable and CELLO/CLEO constraint does
not apply!. Obsolete far off-shell pion (in space-like region).
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201430′e − (pt )

a)

b)
′

µ−

e− (pb )hard γ∗hard γe+

Q2 > 0“soft”πq2 ∼ 00µ−hard γ∗soft γQ2
hard

π0q2 ∼ 0′e+

Measured is Fπ0γ∗γ (m2π, −Q2, 0) at high space–like Q2, needed at externalvertex is Fπ0∗γ∗γ (−Q2, −Q2, 0) or Fπ0∗γ∗γ (q2, q2, 0) if integral to be evaluated in
Minkowsky space.

r I still claim using Fπ0∗γ∗γ (0, 0, 0) in this case is not a reliable approximation!
Need realistic “model” for off–shell form–factor Fπ0∗γ∗γ (q2, q2, 0) via DR from data!
Note: Fπ0∗γ∗γ (−Q2, −Q2, 0) is a one-scale problem. Self-energy type of problem ⇒can get it via dispersion relation from appropriate data
Is it really to be identified with Fπ0∗γ∗γ (0, 0, 0)?
Can we check such questions experimentally or in lattice QCD?
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201431

Evaluation of aLbL
in the large-Nc frameworkµv Knecht & Nyffeler and Melnikov & Vainshtein were using pion-pole approximation together with large-Nc π0γγ–form-factor
v FJ & A. Nyffeler: relax from pole approximation, using KN off-shell LDM+V formfactor

Fπ0∗γ∗γ∗ (p2π, q21, q22) =Fπ P(q21, q22, p2π)
3 Q(q21, q22)

P(q21, q22, p2π) = h7 + h6 p2π + h5 (q22 + q21) + h4 p4π + h3 (q22 + q21) p2π+h2 q21 q22 + h1 (q22 + q21)2 + q21 q22 (p2π + q22 + q21))
Q(q21, q22) = (q21 − M12) (q21 − M22) (q22 − M12) (q22 − M22)
all constants are constraint by SD expansion (OPE). Again, need data to fix
parameters!
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201432

My estimations
Leading LbL contribution from PS mesons:aµ[π0, η, η0] ∼ (93.91= [63.14 + 14.87 + 15.90] ± 12.40) × 10−11
Expected contribution from axial mesons:aµ[a1, f10, f1] ∼ (28.13= [7.02 + 19.38 + 1.74] ± 5.63) × 10−11
Expected contribution from qq̄ scalars:aµ[a0, f00, f0] ∼ (−5.98= [−0.17 − 2.96 − 2.85] ± 1.20) × 10−11
depending slightly on assuming nonet symmetry, ideal mixing
V. Pauk, M. Vanderhaeghen: meson pole conributionsaµ[a0, f00, f0] ∼ (−3.1= [−0.63 − 1.84 − 0.61] ± 0.8) × 10−11aµ[ f10, f1] ∼ (6.4= [5.0 + 1.4] ± 2.0) × 10−11aµ[ f20, f2, a02, a2] ∼ (1.1= [0.79 + 0.07 + 0.22 + 0.02] ± 0.1) × 10−11
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201433

LbL: Present
JN09 based on Nyffeler 09: the only result relaxing from pole approximation

aLbL;had= (116 ± 39) × 10−11µSummary of results
BPP
KN

Contribution

HKS

π0, η, η0π, K loops
axial vectors
scalars
quark loops

82.7±6.4−4.5±8.1
1.7±1.7−
9.7±11.1

85±13−19±132.5±1.0−6.8±2.0
21± 3

total

89.6±15.4

83±32

MV

PdRV

N/JN

83±12
−
−
−
−

114±10
0±10
22± 5
−
−

114±13−19±19
15±10
−7± 7
2.3

99±16
−19±13
22± 5
−7± 2
21± 3

80±40

136±25105±26116±39Is this the final answer? How to improve? A limitation to more precise g − 2 tests?
Looking for new ideas to get ride of model dependence
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201434

r Need better constrained effective resonance Lagrangian (e.g. HSL and ENJL
models vs. RLA of Ecker et al). “Global effort” needed!
recent: HLS global fit available Benayoun et al 2010/12

r Lattice QCD will provide an answer [take time (“yellow” region only?)]!
r Amplitudes in terms of Dispersion Relations (Cutkosky-rules technique)exploiting data! Which data needed?

r Using DESER-GILBERT-SUDARSHAN representation for vertex functions(analog to Kallen-Lehmann representation for two-point function)
may be of help.

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201435

r Try exploiting possible new experimental constraints from γγ → hadronse+γP, V, Sqq̄ππ̄, K K̄, T, Sqqq̄ q̄γe−mostly single-tag events: KLOE, KEDR (taggers), BaBar, Belle, BES III (highluminosity)
e+
γ

e+

P

γ

V

P
V

γ
e−

e−

γ

e+ e−

Dalitz-decays: ρ, ω, φ → π0(η)e+e− Novosibirsk, NA60,JLab, Mainz, Bonn, Jülich,
BES
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

36

e+
γ

γ
P

γ

γ

e−

would be interesting, but is buried in the backgroundr all in conjunction with DR Vanderhaeghen et al 2012/14

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

37

Theory vs experiment: do we see New Physics?
Contribution

Value

Error

Reference

QED incl. 4-loops+5-loops
Leading hadronic vac. pol.
Subleading hadronic vac. pol.
Hadronic light–by–light
Weak incl. 2-loops

11 658 471.8851
688.60-9.832
11.6
15.40

0.036
4.24
0.082
3.9
0.10

Remiddi, Kinoshita ...
HLS driven
2012 update
evaluation (J&N 09)
CMV06/FJ12/BSS13

TheoryExperimentExp.- The. 3.7 standard deviations

11 659 177.65
11 659 209.1
31.25

5.76
6.3
8.54

–
BNL Updated–

Standard model theory and experiment comparison [in units 10−10]. What
represents the 3 σ deviation: r new physics? r a statistical fluctuation? runderestimating uncertainties (experimental, theoretical)?
vdo experiments measure what theoreticians calculate?
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

38

Most natural New Physics contributions: (examples)
a)

γ

M
M
f
f
mµ M0 [S,P] mµ

b)

c)

d)
H−

M0 [V,A]

H+

X−

X0

X+
X0

neutral boson exchange: a) scalar or pseudoscalar and c) vector or axialvector, flavor changing or
not, new charged bosons: b) scalars or pseudoscalars, d) vector or axialvector

Left: mµ = M  M0F. Jegerlehner

Right: mµ  M0 = M

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201439

In general:
2
NP
NP mµ
∆aµ = α M2

NP

exp−theNP searches (LEP, Tevatron, LHC): typically MNP >> MW , then ∆aµ= ∆aNPµNP
requires α ∼ 1 spoiling perturbative arguments. Exception: 2HDM, SUSY tan βenhanced coupling! Note: NP sensitivity enhanced for muon by ∼ 40 000 relative
to electron, while ae is only 2250 times more precise than aµ.
Problem: LEP, Tevatron and LHC direct bounds on masses of possible new states[typically MX > 800 GeV ]F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201440

FutureThe big challenge: two complementary experiments: Fermilab with ultra hot
muons and J-PARC with ultra cold muons (very different radiation) to come
Provided deviation is real 3σ → 9σ possible? Provided theory and needed crosssection data improves the same as the muon g − 2 experiments!
Key: need substantial progress in non-perturbative QCD
For muon g − 2:vmain obstacle: hadronic light-by-light [data, lattice QCD, RLA]
vprogress in evaluating HVP: more data (BaBar, Belle, VEPP 2000, BESIII,...),
lattice QCD in reach (recent progress Jansen et al, Wittig et al, Blum et al)in both cases lattice QCD will be the answer one day ,valso low energy effective RL and DR approach need be further developed.
F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201441

And here we are:
FNAL
2017

BNL
2004

CERN III1976

CERN II1968

CERN I
1961

4th
QED 6th
8th
10th
hadronic VP
hadronic LBL
weak
???

New Physics
SM precision
0.03

0.1

1

10
102
aµ uncertainty [ppm]

103

104

Sensitivity of g − 2 experiments to various contributions. The increase in precision
with the BNL g − 2 experiment is shown as a cyan vertical band. New Physics is
exp
exp
illustrated by the deviation (aµ − athe)/aµµF. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 2014

42

The
challenge:'$

ahad,VP[LO]µ(6923 ± 42) × 10−11

ahad,VP
[NLO]
µ
aEW
µ

(−98 ± 1) × 10−11(154 ± 1) × 10−11

ahad,LbLµexp
δaµ presentexp
δaµ future[(105 ÷ 115) ± (26 ÷ 40)] × 10−11
63 × 10−11
16 × 10−11+58.82 ±0.36 ppm+0.90 ±0.22 ppm±0.54 ppm
±0.14 ppm

&

%

$

'

Next generation experiments require a factor 4 reduction of the uncertainty
optimistically feasible is factor 2 we hope&

F. Jegerlehner%

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201443

What do we see in the muon g − 2 ??? You may find what it is!

l Still a question: do we calculate what experiments measure?
Resent: Arbuzov and Kopylowa 2013: effect of real radiation on aµ:α(1,κ)∆a f =2π



1 + δd(κ)
f



!

δa(κ)
f =

1 1
+ ln |κ| κ + O(κ2)
4 2

as it should smooth as κ → 0 (“offshellness” of the muon)
Assumeexp−SM∆aµ⇒∼ 3 × 10−9 'κ ' −3.5 × 10−7 ; κmµ ∼ 35 eV ,F. Jegerlehner

α (κ)
δaµ
2π

remember pµ ' 9.1 GeV

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201444

l Do we need new ideas to pin down more precisely the hadronic effects?
The muon g − 2 story continues!

Thank you for your attention!
Thanks to KIM for the invitation and the kind hospitality!

F. Jegerlehner

SCGT14Mini, Nagoya, Japan , March 5 - March 7, 201445