hadronic light-by-light scattering

Hadronic light-by-light scattering

in the muon g − 2

Andreas Nyffeler

Institut fur Kernphysik

Johannes Gutenberg Universitat Mainz, Germany

[email protected]

Institut fur Kern- und TeilchenphysikTechnische Universitat Dresden, Germany

October 30, 2014

Outline

Part I: Status of the muon g − 2

• Basics of the anomalous magnetic moment

• Theory of the muon g − 2: QED, weak interactions, hadroniccontributions

Part II: Hadronic light-by-light scattering in the muon g − 2

• Starting point, quark-gluon versus hadronic picture, model-basedapproaches

• Main results

• Further partial results for single-resonance poles and exchanges,potential problems with pion-loop and quark-loop

• New development: data driven approach using dispersion relations

Conclusions

Part I: Status of the muon g − 2

Basics of the anomalous magnetic momentElectrostatic properties of charged particles:Charge Q, Magnetic moment ~µ, Electric dipole moment ~d

For a spin 1/2 particle:

~µ = ge

2m~s, g = 2| z

Dirac

(1 + a), a =1

2(g − 2) : anomalous magnetic moment

Long interplay between experiment and theory: structure of fundamental forces

In Quantum Field Theory (with C,P invariance):

γ(k)

p p’

= (−ie)u(p ′)

264γµ F1(k2)| z Dirac

+iσµνkν

2mF2(k2)| z Pauli

375 u(p)

F1(0) = 1 and F2(0) = a

ae : Test of QED. Most precise determination of α = e2/4π.aµ: Less precisely measured than ae , but all sectors of Standard Model (SM),i.e. QED, Weak and QCD (hadronic), contribute significantly.Sensitive to possible contributions from New Physics. Often (but not always !):

al ∼„

ml

mNP

«2

⇒„

mµ

me

«2

∼ 43000 more sensitive than ae [exp. precision → factor 19]

Tests of the Standard Model and search for New Physics

Search for New Physics with two complementary approaches:

1 High Energy Physics:e.g. Large Hadron Collider (LHC) at CERNDirect production of new particlese.g. heavy Z ′ ⇒ resonance peak in invariantmass distribution of µ+µ− at MZ ′ .

2 Precision physics:e.g. anomalous magnetic moments ae , aµIndirect effects of virtual particles in quantumcorrections

⇒ Deviations from precise predictions inStandard Model

For MZ ′ m` : a` ∼„

m`

MZ ′

«2

No decoupling for new light vector mesons, e.g.

“dark photon” with MV ∼ (10− 100) MeV.

ae , aµ allow to exclude some models of NewPhysics or to constrain their parameter space.

p

pq

q

µ−

µ+

Z ′

µ µZ ′

γ

Milestones in measurements of aµ

Authors Lab Muon Anomaly

Garwin et al. ’60 CERN 0.001 13(14)Charpak et al. ’61 CERN 0.001 145(22)Charpak et al. ’62 CERN 0.001 162(5)Farley et al. ’66 CERN 0.001 165(3)Bailey et al. ’68 CERN 0.001 166 16(31)Bailey et al. ’79 CERN 0.001 165 923 0(84)Brown et al. ’00 BNL 0.001 165 919 1(59) (µ+)Brown et al. ’01 BNL 0.001 165 920 2(14)(6) (µ+)Bennett et al. ’02 BNL 0.001 165 920 4(7)(5) (µ+)Bennett et al. ’04 BNL 0.001 165 921 4(8)(3) (µ−)

World average experimental value (g − 2 Collaboration at BNL, Bennett et al.’06 + CODATA 2008 value for λ = µµ/µp, ratio of muon to proton magneticmoment from muonium hyperfine splitting):

aexpµ = (116 592 089± 63)× 10−11 [0.5ppm]

Goal of new planned g − 2 experiments at Fermilab (partly recycled from BNL:moved ring magnet ! Pictures at: http://muon-g-2.fnal.gov/bigmove/ ) andJ-PARC (completely new concept, not magic γ): δaµ = 16× 10−11

Theory needs to match this precision !

Experimental value for electron ae (Hanneke et al. ’08):

aexpe = (1 159 652 180.73± 0.28)× 10−12 [0.24ppb]

Some theoretical comments

• Anomalous magnetic moments are dimensionlessTo lowest order in perturbation theory in quantum electrodynamics (QED):

= ae = aµ =α

2π[Schwinger 1947/48]

• Loops with different masses ⇒ ae 6= aµ

- Internal large masses decouple:

=

"1

45

„me

mµ

«2

+O

m4e

m4µ

lnmµ

me

!#“απ

”2

- Internal small masses give rise to large log’s of mass ratios:

=

»1

3ln

mµ

me−

25

36+O

„me

mµ

«–“απ

”2

Muon g − 2: Theory

In Standard Model (SM):

aSMµ = aQED

µ + aweakµ + ahad

µ

In contrast to ae , here now the contributions from weak and strong interactions(hadrons) are relevant, since aµ ∼ (mµ/M)2.

QED contributions

• Diagrams with internal electron loops are enhanced.

• At 2-loops: vacuum polarization from electron loops enhanced by QEDshort-distance logarithm

• At 3-loops: light-by-light scattering from electron loops enhanced by QEDinfrared logarithm [Aldins et al. ’69, ’70; Laporta + Remiddi ’93]

+ ...

e

µ

a(3)µ

˛lbl

=

»2

3π2 ln

mµ

me+ . . .

–“απ

”3= 20.947 . . .

“απ

”3

• Loops with tau’s suppressed (decoupling)

QED result up to 5 loops

Include contributions from all leptons (Aoyama et al. 2012):

aQEDµ = 0.5×

“απ

”+ 0.765 857 425 (17)|z

mµ/me,τ

×“απ

”2

+ 24.050 509 96 (32)|zmµ/me,τ

×“απ

”3

+ 130.8796 (63)|znum. int.

×“απ

”4

+ 753.29 (1.04)| z num. int.

×“απ

”5

= 116 584 718.853 (9)|zmµ/me,τ

(19)|zc4

(7)|zc5

(29)|zα(ae )

[36]× 10−11

• Earlier evaluation of 5-loop contribution yielded c5 = 662(20) (Kinoshita, Nio2006, numerical evaluation of 2958 diagrams, known or likely to be enhanced).New value is 4.5σ from this leading log estimate and 20 times more precise.

• Aoyama et al. 2012: What about the 6-loop term ? Leading contribution fromlight-by-light scattering with electron loop and insertions of vacuum-polarizationloops of electrons into each photon line ⇒ aQED

µ (6-loops) ∼ 0.1× 10−11

Contributions from weak interaction1-loop contributions [Jackiw, Weinberg, 1972; . . . ]:

W W

νµ Z Hµ

γa) b) c)

aweak, (1)µ (W ) =

√2Gµm2

µ

16π2

10

3+O(m2

µ/M2W ) = 388.70(0)× 10−11

aweak, (1)µ (Z) =

√2Gµm2

µ

16π2

(−1 + 4s2W)2 − 5

3+O(m2

µ/M2Z ) = −193.89(2)× 10−11

Contribution from Higgs negligible: aweak, (1)µ (H) ≤ 5× 10−14 for mH ≥ 114 GeV.

aweak, (1)µ = (194.82± 0.02)× 10−11

2-loop contributions (1678 diagrams):

aweak, (2)µ = (−42.08± 1.80)× 10−11, large since ∼ GFm2

µα

πln

MZ

mµ

Under control. Uncertainties from mH(100− 300 GeV), 3-loop (RG): ±1.5× 10−11,small hadronic uncertainty: ±1.0× 10−11.

Total weak contribution:

aweakµ = (153.2± 1.8)× 10−11

Recent reanalysis by Gnendiger et al ’13, using the now known Higgs mass, yields

aweakµ = (153.6± 1.0)× 10−11 (error: 3-loop, hadronic).

Hadronic contributions to the muon g − 2: largest source of error• QCD: quarks bound by strong gluonic interactions into hadronic states

• In particular for the light quarks u, d , s → cannot use perturbation theory !

Possible approaches to QCD:

• Lattice QCD: often still limited precision

• Effective quantum field theories with hadrons (ChPT): limited validity

• Simplifying hadronic models: model uncertainties not controllable

• Dispersion relations: extend validity of EFT’s, reduce model dependence, oftennot all the needed input data available

Different types of contributions to g − 2:

Zγ

(a) (b) (c)

Light quark loop not well defined→ Hadronic “blob”

(a) Hadronic vacuum polarization (HVP) O(α2),O(α3)

(b) Hadronic light-by-light scattering (HLbL) O(α3)

(c) 2-loop electroweak contributions O(αGFm2µ)

2-Loop EWSmall hadronic uncertainty from triangle diagrams.Anomaly cancellation within each generation !Cannot separate leptons and quarks !

e u,d

γ Z γ Zµ µ

Hadronic vacuum polarization

aHVPµ =

µ µ

Optical theorem (from unitarity; conservation of probability) for hadronic contribution

→ dispersion relation:

Im ∼2

∼ σ(e+e−→ γ∗→ hadrons)

aHVPµ =

1

3

“απ

”2Z ∞

0

ds

sK(s) R(s), R(s)=

σ(e+e− → γ∗ → hadrons)

σ(e+e− → γ∗ → µ+µ−)

[Bouchiat, Michel ’61; Durand ’62; Brodsky, de Rafael ’68; Gourdin, de Rafael ’69]

K(s) slowly varying, positive function ⇒ aHVPµ positive. Data for hadronic cross section

σ at low center-of-mass energies√

s important due to factor 1/s: ∼ 70% from

ππ [ρ(770)] channel, ∼ 90% from energy region below 1.8 GeV.

Other method instead of energy scan: “Radiative return”

at colliders with fixed center-of-mass energy (DAΦNE, B-

Factories, BES) [Binner, Kuhn, Melnikov ’99; Czyz et al.

’00-’03]

Hadrons

γ

γ

e−

e+

Measured hadronic cross-section

Pion form factor |Fπ(E)|2(ππ-channel)

R(s) =1

4

„1−

4m2π

s

« 32

|Fπ(s)|2

(4m2π < s < 9m2

π)

R-ratio:

Jegerlehner + Nyffeler ’09

Hadronic vacuum polarization: some recent evaluations

Authors Contribution to aHVPµ × 1011

Jegerlehner ’08; JN ’09 (e+e−) 6903.0± 52.6

Davier et al. ’09 (e+e−) [+ τ ] 6955± 41 [7053± 45]

Teubner et al. ’09 (e+e−) 6894± 40

Davier et al. ’10 (e+e−) [+ τ ] 6923± 42 [7015± 47]

Jegerlehner + Szafron ’11 (e+e−) [+ τ ] 6907.5± 47.2 [6909.6± 46.5 ]

Hagiwara et al. ’11 (e+e−) 6949.1± 42.7

Benayoun at al. ’12 (e+e− + τ : HLS improved) 6877.2± 46.3

• Precision: < 1%. Non-trivial because of radiative corrections (radiated photons).

• Even if values for aHVPµ after integration agree quite well, the systematic

differences of a few % in the shape of the spectral functions from differentexperiments (BABAR, CMD-2, KLOE, SND) indicate that we do not yet have acomplete understanding.

• Use of τ data: additional sources of isospin violation ? Ghozzi + Jegerlehner ’04;Benayoun et al. ’08, ’09; Wolfe + Maltman ’09; Jegerlehner + Szafron ’11(ρ− γ-mixing), also included in HLS-approach by Benayoun et al. ’12.

• Lattice QCD: Various groups are working on it, precision at level of 5-10%, notyet competitive with phenomenological evaluations.

Muon g − 2: current statusSummary of SM contributions to aµ (based on various recent sources):

• Leptonic QED contributions: aQEDµ = (116 584 718.853± 0.036)× 10−11

• Weak contributions: aweakµ = (153.2| z

?

± 1.8|z?

)× 10−11

• Hadronic contributions:- Vacuum Polarization:

aHVPµ (e+e−) = (6907.5| z

??

± 47.2|z??

−(100.3± 2.2))× 10−11

- Light-by-Light scattering: aHLbLµ = (116± 40| z

???

)× 10−11

• Total SM contribution:

aSMµ = (116 591 795± 47|z

v.p.

± 40|zHLbL

± 1.8|zQED + EW

[±62])× 10−11

• “New” experimental value (shifted +9.2× 10−11 due to new λ = µµ/µp):

aexpµ = (116 592 089± 63)× 10−11

⇒ aexpµ − aSM

µ = (294± 88)× 10−11 [3.3 σ]

Sign of New Physics ? Hadronic uncertainties need to be better controlled inorder to fully profit from future g − 2 experiments with δaµ = 16× 10−11.

Muon g − 2: other recent evaluations

170 180 190 200 210

aµ× 10

10 – 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)

Source: Hagiwara et al. ’11. Note units of 10−10 !

3.3 σ

2.4 σ

3.6 σ

3.3 σ

3.3 σ

Aoyama et al. ’12: aexpµ − aSM

µ = (249± 87)× 10−11 [2.9 σ]

Benayoun et al. ’13: aexpµ − aSM

µ = 394× 10−11 [4.9 σ]

Part II: Hadronic light-by-light scattering in the muon g − 2

Hadronic light-by-light scattering in the muon g − 2

O(α3) hadronic contribution to muon g − 2: four-point function〈VVVV 〉 projected onto aµ (external soft photon kµ → 0).

k

µ−(p)µ−(p’)

k = p’ − p

Consider matrix element of light-quark electromagnetic current

jρ(x) =2

3(uγρu)(x) −

1

3(dγρd)(x) −

1

3(sγρs)(x)

between muon states:

〈µ−(p ′)|(ie)jρ(0)|µ−(p)〉 = (−ie)u(p ′)Γρ(p ′, p)u(p)

=

Zd4q1

(2π)4

Zd4q2

(2π)4

(−i)3

q21 q2

2 (q1 + q2 − k)2

i

(p ′ − q1)2 −m2

i

(p ′ − q1 − q2)2 −m2

×(−ie)3 u(p ′)γµ(6p ′− 6q1 + m)γν(6p ′− 6q1− 6q2 + m)γλu(p)

×(ie)4 Πµνλρ(q1, q2, k − q1 − q2)

with k = p′ − p and the fourth-rank light-quark hadronic tensor

Πµνλρ(q1, q2, q3)=

Zd4x1

Zd4x2

Zd4x3 e i(q1·x1+q2·x2+q3·x3)〈Ω |Tjµ(x1)jν(x2)jλ(x3)jρ(0)|Ω 〉

Momentum conservation: k = q1 + q2 + q3.

Projection onto g − 2, Properties of Πµνλρ(q1, q2, q3)

Flavor diagonal current jµ(x) is conserved and one has the Ward identities:

qµ1 ; qν2 ; qλ3 ; kρΠµνλρ(q1, q2, q3) = 0

⇒ Πµνλρ(q1, q2, k − q1 − q2) = −kσ∂

∂kρΠµνλσ(q1, q2, k − q1 − q2)

Defining Γρ(p ′, p) = kσΓρσ(p ′, p) one finally obtains (Aldins et al. ’70):

aµ = F2(0) =1

48mtr ((6p + m)[γρ, γσ]( 6p + m)Γρσ(p, p))

i.e. one can formally take the limit kµ → 0 inside the loop integrals. Problem reducesto calculation of two-point function with zero-momentum insertion. One also getsbetter UV convergence properties of individual Feynman diagrams (fermion-loop).

Properties of Πµνλρ(q1, q2, q3) (Bijnens et al. ’95):

• In general 138 Lorentz structures. But only 32 contribute to g − 2.

• Using Ward identities, there are 43 gauge invariant structures.

• Bose symmetry relates some of them.

• All depend on q21 , q

22 , q

23 , qi · qj , but before taking derivative and kµ → 0, also on

k2, k · qi .

• Compare with HVP: one function, one variable.

HLbL in muon g − 2

QED: light-by-light scattering at higher orders in perturbation series via lepton-loop:

γ e

γ γ

γ

In muon g − 2:

⇒

e

µ

γ

Hadronic light-by-light scattering in muon g − 2 from strong interactions (QCD):

µ

γ

=

π0, η, η′

+ . . . + . . .

π+

ahad.LxLµ =

Coupling of photons to hadrons, e.g. π0, via form factor: π0

γ

γ

View before 2014: in contrast to HVP, no direct relation to experimental data → sizeand even sign of contribution to aµ unknown !Approach: use some hadronic model at low energies with exchanges and loops ofresonances and some form of (dressed) “quark-loop” at high energies.Problem: 〈VVVV 〉 depends on several invariant momenta ⇒ distinction between lowand high energies is not as easy as for two-point function 〈VV 〉 (HVP).Note: one can (in principle) always perform Wick rotation to Euclidean momenta,where effects of resonances are smoothed out, but there are mixed regions where Q2

1 is

small and Q22 large and vice versa.

Hadronic light-by-light scattering

Data on γγ → γγ

In any case, it is a good ideato look at actual data. For in-stance, obtained by Crystal Balldetector ’88 viae+e−→e+e−γ∗γ∗→e+e−π0:

e+

e−π0

Feynman diagram from Colan-gelo et al., arXiv:1408.2517

Invariant γγ mass spectrum:

Three spikes from light pseudoscalars in reaction:

γγ → π0, η, η′ → γγ

for (almost) real photons.

New development in 2014: use of dispersion relations for a data driven approach toHLbL in muon g − 2 from (still to be measured !) scattering of two off-shell photons:

γ∗γ∗ → π0, η, η′

γ∗γ∗ → ππ

(Colangelo et al.; Pauk, Vanderhaeghen).

Current approach to HLbL scattering in g − 2

Classification of de Rafael ’94

Chiral counting p2 (from Chiral Perturbation Theory (ChPT)) and large-NC countingas guideline to classify contributions (all higher orders in p2 and NC contribute):

k

µ−(p)µ−(p’)

k = p’ − p

=ρ

π+

+

π , η, η0 ,

+

Exchange ofother reso-nances(f0, a1, f2 . . .)

+ρ

Q

Chiral counting: p4 p6 p8 p8

NC -counting: 1 NC NC NC

pion-loop pseudoscalar exchanges quark-loop(dressed) (dressed)

Relevant scales in HLbL (〈VVVV 〉 with off-shell photons !): 0− 2 GeV, i.e. muchlarger than mµ !

Constrain models using experimental data (processes of hadrons with photons: decays,form factors, scattering) and theory (ChPT at low energies; short-distance constraintsfrom pQCD / OPE at high momenta).

Issue: on-shell versus off-shell form factors. For instance pion-pole with on-shell pionform factors versus pion-exchange with off-shell pion form factors (in both cases withone or two off-shell photons).

Pseudoscalars: numerically dominant contribution (according to most models !).Warrant special attention.

Pion-pole versus pion-exchange in aHLbL;π0

µ

• To uniquely identify contribution of exchanged neutral pion π0 in Green’sfunction 〈VVVV 〉, we need to pick out pion-pole:

q

q

q

π0

1

2

4

q3

+ crossed diagrams

lim(q1+q2)2→m2

π

((q1 + q2)2 −m2π)〈VVVV 〉

Residue of pole: on-shell vertex function 〈0|VV |π〉→ on-shell form factor Fπ0γ∗γ∗ (q2

1 , q22)

• But in contribution to muon g − 2, we evaluate Feynman diagrams, integratingover photon momenta with exchanged off-shell pions. For all the pseudoscalars:

π0 ,, η ’η ,... Shaded blobs represent off-shell formfactor FPS∗γ∗γ∗ ((q1 + q2)2, q2

1 , q22)

where PS = π0, η, η′, π0′, . . .

Off-shell form factors are either inserted “by hand” starting from constant,pointlike Wess-Zumino-Witten (WZW) form factor or using e.g. some resonanceLagrangian. Contribution with off-shell form factors is model dependent.

• Within a dispersive framework the pole contribution can be uniquely defined fromimaginary part of Feynman diagram (similarly for multi-particle intermediatestates like ππ), but this represents only part of the contribution.

HLbL in muon g − 2: main results

HLbL scattering: Summary of selected results

k

µ−(p)µ−(p’)

k = p’ − p

=ρ

π+

+

π , η, η0 ,

+

Exchange ofother reso-nances(f0, a1, f2 . . .)

+ρ

Q

Chiral counting: p4 p6 p8 p8

NC -counting: 1 NC NC NC

Contribution to aµ × 1011:

BPP: +83 (32)HKS: +90 (15)KN: +80 (40)MV: +136 (25)2007: +110 (40)PdRV:+105 (26)N,JN: +116 (40)

-19 (13)-5 (8)

0 (10)

-19 (19)-19 (13)

ud.: -45

+85 (13)+83 (6)+83 (12)

+114 (10)

+114 (13)+99 (16)

ud.: +∞

-4 (3) [f0, a1]+1.7 (1.7) [a1]

+22 (5) [a1]

+8 (12) [f0, a1]+15 (7) [f0, a1]

+21 (3)+10 (11)

0

+2.3 [c-quark]+21 (3)

ud.: +60

ud. = undressed, i.e. point vertices without form factors

BPP = Bijnens, Pallante, Prades ’96, ’02; HKS = Hayakawa, Kinoshita, Sanda ’96, ’98, ’02;KN = Knecht, Nyffeler ’02; MV = Melnikov, Vainshtein ’04; 2007 = Bijnens, Prades; Miller, deRafael, Roberts; PdRV = Prades, de Rafael, Vainshtein ’09 (compilation); N,JN = Nyffeler ’09;Jegerlehner, Nyffeler ’09 (compilation)

Recall (in units of 10−11): δaµ(HVP) ≈ 45; δaµ(exp [BNL]) = 63; δaµ(future exp) = 16

HLbL scattering: Summary of selected results (continued)

Some results for the various contributions to aHLbLµ × 1011:

Contribution BPP HKS, HK KN MV BP, MdRR PdRV N, JN

π0, η, η′ 85±13 82.7±6.4 83±12 114±10 − 114±13 99 ± 16

axial vectors 2.5±1.0 1.7±1.7 − 22±5 − 15±10 22±5

scalars −6.8±2.0 − − − − −7±7 −7±2

π, K loops −19±13 −4.5±8.1 − − − −19±19 −19±13

π,K loops+subl. NC

− − − 0±10 − − −

quark loops 21±3 9.7±11.1 − − − 2.3 (c-quark) 21±3

Total 83±32 89.6±15.4 80±40 136±25 110±40 105 ± 26 116 ± 39

BPP = Bijnens, Pallante, Prades ’95, ’96, ’02; HKS = Hayakawa, Kinoshita, Sanda ’95, ’96; HK = Hayakawa, Kinoshita ’98, ’02; KN =

Knecht, Nyffeler ’02; MV = Melnikov, Vainshtein ’04; BP = Bijnens, Prades ’07; MdRR = Miller, de Rafael, Roberts ’07; PdRV =

Prades, de Rafael, Vainshtein ’09; N = Nyffeler ’09, JN = Jegerlehner, Nyffeler ’09

• Pseudoscalar-exchanges dominate numerically. Other contributions notnegligible. Cancellation between π,K -loops and quark loops !

• PdRV: Analyzed results obtained by different groups with various models andsuggested new estimates for some contributions (shifted central values, enlargederrors). Do not consider dressed light quark loops as separate contribution !Assume it is already taken into account by using short-distance constraint of MV’04 on pseudoscalar-pole contribution. Added all errors in quadrature !

• N, JN: New evaluation of pseudoscalar exchange contribution imposing newshort-distance constraint on off-shell form factors. Took over most values fromBPP, except axial vectors from MV. Added all errors linearly.

Comments

• History: several changes in size and even sign due to errors since firstevaluation by Calmet et al. ’76. 2001: last sign change in dominantpseudoscalar contribution (KN ’02). More recently, size of dressedquark-loop and dressed pion-loop have been questioned (Fischer et al. ’11,’13; Engel et al. ’12, ’13).

• All estimates are currently based on models ! (ENJL, VMD, large-NC

QCD matched to OPE (LMD, LMD+V), HLS, (constituent, chiral) quarkmodels, AdS/QCD, Pade approximants, DSE with truncations . . . ).

• In some cases it is not obvious whether actually the same contributions arecalculated or not, e.g. pion-pole versus pion-exchange.

• Currently most used estimates:

aHLbLµ = (105± 26)× 10−11 (Prades, de Rafael, Vainshtein ’09)

(“Glasgow consensus”)

aHLbLµ = (116± 40)× 10−11 (Nyffeler ’09; Jegerlehner, Nyffeler ’09)

Compilations based on work by different groups using various models.Error estimates are mostly guesses !

• Data driven approach to HLbL with a more controlled error estimate isneeded, e.g. using dispersion relations, similarly to HVP !

• Future: maybe lattice QCD will provide one day a reliable estimate forHLbL in muon g − 2. Preliminary attempts by Blum et al. ’05, . . . , ’14.

One-particle intermediate states: resonance exchanges / poles

Note

• Within phenomenological approach with models, one particle exchanges /poles come from pseudoscalars, scalars, axial vectors and tensor mesons.

• Within dispersion relation approach only “stable” physical intermediatestates are considered: one-pion, two-pion, three-pion, photons, muons . . .

Pion-pole contribution

Pion-pole contribution determined by measurable pion transition form factorFπ0γ∗γ∗ (q2

1 , q22) (on-shell pion, one or two off-shell photons). Analogously for

η, η′-pole contributions.

Knecht, Nyffeler ’02:

aHLbL;π0

µ = −e6Z

d4q1

(2π)4

d4q2

(2π)4

1

q21q2

2 (q1 + q2)2[(p + q1)2 − m2µ][(p − q2)2 − m2

µ]

×"Fπ0γ∗γ∗ (q2

1 , (q1 + q2)2) Fπ0γ∗γ∗ (q22 , 0)

q22 − m2

π

T1(q1, q2; p)

+Fπ0γ∗γ∗ (q2

1 , q22 ) Fπ0γ∗γ∗ ((q1 + q2)2, 0)

(q1 + q2)2 − m2π

T2(q1, q2; p)

#

T1(q1, q2; p) =16

3(p · q1) (p · q2) (q1 · q2) −

16

3(p · q2)2 q2

1 −8

3(p · q1) (q1 · q2) q2

2

+ 8(p · q2) q21 q2

2 −16

3(p · q2) (q1 · q2)2 +

16

3m2µ q2

1 q22 −

16

3m2µ (q1 · q2)2

T2(q1, q2; p) =16

3(p · q1) (p · q2) (q1 · q2) −

16

3(p · q1)2 q2

2 +8

3(p · q1) (q1 · q2) q2

2

+8

3(p · q1) q2

1 q22 +

8

3m2µ q2

1 q22 −

8

3m2µ (q1 · q2)2

where p2 = m2µ and the external photon has now zero four-momentum (soft photon).

Currently, only single-virtual TFF Fπ0γ∗γ∗ (Q2, 0) has been measured by CELLO,CLEO, BABAR, Belle. Analysis ongoing at BES, measurement planned at KLOE-2.

Relevant momentum regions in aHLbL;π0

µ

• In Knecht, Nyffeler ’02, a 2-dimensional integral representation for the pion-polecontribution was derived for a certain class of form factors (VMD-like).Schematically:

aHLbL;π0

µ =

Z ∞0

dQ1

Z ∞0

dQ2

Xi

wi (Q1,Q2) fi (Q1,Q2)

with universal weight functions wi . Dependence on form factors resides in the fi .

• Plot of weight functions wi from Knecht, Nyffeler ’02:

00.5

11.5

00.5

11.5

0

2

4

6

Q1 [GeV]

wf1

(Q1,Q

2)

Q2 [GeV] 0

0.51

1.5

00.5

11.5

0

2

4

6

Q1 [GeV]

wg

1

(MV,Q

1,Q

2)

Q2 [GeV]

0 1 20

12

−0.5

0

0.5

1

Q1 [GeV]

wg

2

(Mπ,Q1,Q

2)

Q2 [GeV] 0 1 2

01

2−0.04

−0.02

0

0.02

0.04

0.06

Q1 [GeV]

wg

2

(MV,Q

1,Q

2)

Q2 [GeV]

• Relevant momentum regions around0.25− 1.25 GeV. As long as formfactors in different models lead todamping, expect comparable results

for aHLbL;π0

µ , at level of 20%.

• Jegerlehner, Nyffeler ’09 derived3-dimensional integral representationfor general form factors(hyperspherical approach).Integration over Q2

1 ,Q22 , cos θ, where

Q1 · Q2 = |Q1||Q2| cos θ.

• Idea recently taken up by Dorokhovet al. ’12 (for scalars) and Bijnens,Zahiri Abyaneh ’12, ’13 (for allcontributions, work in progress).

Impact of form factor measurements: example KLOE-2

On the possibility to measure the π0 → γγ decay width and the γ∗γ → π0 transitionform factor with the KLOE-2 experiment

D. Babusci, H. Czyz, F. Gonnella, S. Ivashyn, M. Mascolo, R. Messi, D. Moricciani,A. Nyffeler, G. Venanzoni and the KLOE-2 Collaboration ’12

2 [GeV]2Q-210 -110 1 10

)|2|F

(Q

0

0.05

0.1

0.15

0.2

0.25

0π → γ*γ

Simulation of KLOE-2 measurement of TFFF (Q2) (red triangles). MC program EKHARA2.0 (Czyz, Ivashyn ’11) and detailed detectorsimulation.Solid line: F (0) given by chiral anomaly (WZW).Dashed line: form factor according to on-shellLMD+V model (Knecht, Nyffeler ’01).CELLO (black crosses) and CLEO (blue stars)data at higher Q2.

Within 1 year of data taking, collecting5 fb−1, KLOE-2 will be able to measure:

• Γπ0→γγ to 1% statistical precision.

• γ∗γ → π0 transition form factorF (Q2) in the region of very low,space-like momenta0.01 GeV2 ≤ Q2 ≤ 0.1 GeV2 with astatistical precision of less than 6%in each bin.KLOE-2 can (almost) directlymeasure slope of form factor atorigin (note: logarithmic scale inQ2 in plot !).

Impact of form factor measurements: example KLOE-2 (continued)• Error in aHLbL;π0

µ related to model parameters determined by Γπ0→γγ(normalization of FF; not taken into account in most papers) and F (Q2) will bereduced as follows:

• δaHLbL;π0

µ ≈ 4× 10−11 (with current data for F (Q2) + ΓPDGπ0→γγ)

• δaHLbL;π0

µ ≈ 2× 10−11 (+ ΓPrimExπ0→γγ )

• δaHLbL;π0

µ ≈ (0.7− 1.1)× 10−11 (+ KLOE-2 data)

• Note that error does not account for other potential uncertainties in aHLbL;π0

µ ,e.g. related to choice of model, 2nd off-shell photon, off-shellness of pion.

• Simple models with few parameters, like VMD (two parameters: Fπ ,MV ), whichare completely determined by the data on Γπ0→γγ and F (Q2), can lead to very

small errors in aHLbL;π0

µ . For illustration:

aHLbL;π0

µ;VMD = (57.3± 1.1)× 10−11

aHLbL;π0

µ;LMD+V = (72± 12)× 10−11 (off-shell LMD+V FF, including all errors)

• But this might be misleading ! VMD and LMD+V give equally good fits totransition form factor F (Q2), but differ in doubly-off shell transition form factor

Fπ0γ∗γ∗ (q21 , q

22). Results for aHLbL;π0

µ differ by about 20% ! Reason: VMD form

factor has wrong high-energy behavior ⇒ too strong damping in aHLbL;π0

µ;VMD :

FVMDπ0γ∗γ∗

(q2, q2) ∼ 1/q4, for large q2, i.e. falls off too fast compared to OPE

prediction FOPEπ0γ∗γ∗

(q2, q2) ∼ 1/q2 which is fulfilled by LMD+V.

⇒ Dispersive approach to not rely (or less) on models.

Pseudoscalar pole / exchange in large-NC QCD

Model for FP(∗)γ∗γ∗ aµ(π0)× 1011 aµ(π0, η, η′)× 1011

VMD 57 83

LMD (on-shell) [KN] 73 −LMD+V (on-shell, h2 = 0) [KN] 58( 10 ) 83( 12 )

LMD+V (on-shell, h2 = −10 GeV2) [KN] 63( 10 ) 88( 12 )

LMD+V (on-shell, constant 2nd FF) [MV] 77( 7 ) 114( 10 )

LMD+V (off-shell) [N] 72( 12 ) 99( 16 )

LMD+P (off-shell) [KaNo] 65.8(1.2) −LMD+P (off-shell) [RGL] 66.5(1.9) 104.3(5.2)

LMD+P (on-shell) [RGL] 57.5(0.5) 82.7(2.8)

KN = Knecht, Nyffeler ’02MV = Melnikov, Vainshtein ’04N = Nyffeler ’09KaNo = Kampf, Novotny ’11 (RχT)RGL = Roig, Guevara, Lopez Castro ’14 (RχT)

Note: KN, MV, N use VMD FF for η, η′

Other recent partial evaluations (mostly pseudoscalars)

• Nonlocal chiral quark model (off-shell) [Dorokhov et al.]

2008: aLbyL;π0

µ = 65(2)× 10−11

2011: aLbyL;π0

µ = 50.1(3.7)× 10−11, aLbyL;PSµ = 58.5(8.7)× 10−11

2012: aLbyL;π0+σµ = 54.0(3.3)× 10−11, aLbyL;a0+f0

µ ∼ 0.1× 10−11

aLbyL;PS+Sµ = 62.5(8.3)× 10−11

Strong damping for off-shell form factors. Positive and small contribution fromscalar σ(600), differs from other estimates (BPP ’96, ’02; Blokland, Czarnecki,Melnikov ’02).

• Holographic (AdS/QCD) model 1 (off-shell ?) [Hong, Kim ’09]

aLbyL;π0

µ = 69× 10−11, aLbyL;PSµ = 107× 10−11

• Holographic (AdS/QCD) model 2 (off-shell) [Cappiello, Cata, D’Ambrosio ’10]

aLbyL;π0

µ = 65.4(2.5)× 10−11

Used AdS/QCD to fix parameters in ansatz by D’Ambrosio et al. ’98.

• Pade approximants (on-shell)

aLbyL;π0

µ = 54(5)× 10−11 [Masjuan ’12 (using on-shell LMD+V FF)]

aLbyL;π0

µ = 64.9(5.6)× 10−11, aLbyL;PSµ = 89(7)× 10−11

[Escribano, Masjuan, Sanchez-Puertas ’13]Fix parameters in Pade approximants from data on transition form factors.

More on single-resonance poles and exchanges• Pauk, Vanderhaeghen ’14:

aµ(f0(980), f ′0 , a0) = [(−0.9± 0.2) to (−3.1± 0.8)]× 10−11

aµ(f1, f′

1 ) = (6.4± 2.0)× 10−11

aµ(f2, f′

2 , a2, a′2) = (1.1± 0.1)× 10−11

Meson pole approximation. Using monopole form factor (VMD) for scalars anddipole form factors (strong damping !) for axial vectors and tensors:

FmonMγ∗γ∗ (q2

1 , q22 )

FMγ∗γ∗ (0, 0)=

1

(1 − q21/Λ2

mon)

1

(1 − q22/Λ2

mon)

FdipMγ∗γ∗ (q2

1 , q22 )

FMγ∗γ∗ (0, 0)=

1

(1 − q21/Λ2

dip)2

1

(1 − q22/Λ2

dip)2

First calculation of tensor meson contribution to HLbL. Forward sum rules for γγscattering (Pascalutsa, Vanderhaeghen, ’10; Pascalutsa, Pauk, Vanderhaeghen’12) used to constrain model. They link transition form factors of pseudoscalars,axial vectors and tensor mesons ⇒ Λdip = 1.5 GeV.

• Jegerlehner (Talks at g − 2 Workshops in Mainz, April 2014):

aµ(f0(980), f ′0 , a0) = (−(6.01− 6.31)± 1.20)× 10−11

aµ(f1, f′

1 , a1) = (7.55− 7.58)± 2.71)× 10−11

Meson exchange contributions. Models inspired from Melnikov, Vainshtein ’04,but correctly implement Landau-Yang theorem.

• Both calculations give contribution for axial vectors, which is substantiallysmaller than Melnikov, Vainshtein ’04: aµ(f1, f ′1 , a1) = (22± 5)× 10−11.

Two-particle intermediate states (pion-loop)

Dressed pion-loop

1. ENJL/VMD versus HLS

Model aπ−loopµ × 1011

scalar QED (no FF) -45

HLS -4.5

ENJL -19

full VMD -15

Strong damping if form factors areintroduced, very model dependent:compare ENJL (BPP ’96) versus HLS(HKS ’96). See also discussion inMelnikov, Vainshtein ’04.

Origin: different behavior of integrands in contribution to g − 2(Zahiri Abyaneh ’12; Bijnens, Zahiri Abyaneh ’12; Talks by Bijnens at MesonNet 2013, Prague; g − 2 Workshop, Mainz 2014)

p1ν

p2α qρ

p3β

p5p4p′ p

One can do 5 of the 8 integrations in the 2-loop integral for g − 2 analytically, using thehyperspherical approach / Gegenbauer polynomials (Jegerlehner, Nyffeler ’09; taken up inBijnens, Zahiri Abyaneh ’12):

aXµ =

ZdlP1

dlP2aXLLµ =

ZdlP1

dlP2dlQ aXLLQ

µ , with lP = ln(P/GeV)

Contribution of type X at given scale P1, P2, Q is directly proportional to volume under surface

when aXLLµ and a

XLLQµ are plotted versus the energies on a logarithmic scale.

0.1 1

10 0.1 1 10

-4e-11

-2e-11

0

2e-11

4e-11

6e-11

8e-11

1e-10

aµLLQ

π loop

VMDHLS a=2

P1 = P2Q

aµLLQ

Momentum distribution of full VMD andHLS pion-loop contribution for P1 = P2.

HLS: Integrand changes from positive to

negative at high momenta. Leads to

cancellation and therefore smaller absolute

value. Usual HLS model (a = 2) known to

not fullfill certain QCD short-distance

constraints.

Dressed pion-loop (continued)2. Role of pion polarizability and a1 resonance

• Engel, Patel, Ramsey-Musolf ’12: ChPT analysis of LbyL up to order p6 in limitp1, p2, q mπ . Identified potentially large contributions from pion polarizability(L9 + L10 in ChPT) which are not fully reproduced in ENJL / HLS models usedby BPP ’96 and HKS ’96. Pure ChPT approach is not predictive. Loops notfinite, would need new aµ counterterm (Knecht et al. ’02).

• Engel, Ph.D. Thesis ’13; Engel, Ramsey-Musolf ’13: tried to include a1 resonanceexplicitly in EFT. Problem: contribution to g − 2 in general not finite (loops withresonances) ⇒ Form factor approach with a1 that reproduces pion polarizabilityat low energies, has correct QCD scaling at high energies and generates a finiteresult in aµ:

LI = −e2

4Fµνπ

+

1

D2 + M2A

!Fµνπ− + h.c. + · · ·

LII = −e2

2M2A

π+π−"

M2V

∂2 + M2V

!Fµν

#2

+ · · ·

aπ−loopµ × 1011:

Model (a) (b)

I -11 -34

II -40 -71

Second and third columns in Table correspond to different values for thepolarizability LECs, (αr

9 + αr10): (a) (1.32± 1.4)× 10−3 (from radiative pion

decay π+ → e+νeγ ) and (b) (3.1± 0.9)× 10−3 (from radiative pionphotoproduction γp → γ′π+n).

Potentially large results (absolute value): aπ−loopµ ∼ −(11− 71)× 10−11.

Variation of 60× 10−11 ! Uncertainty underestimated in earlier calculations ?

Dressed pion-loop (continued)

• Issue taken up in Zahiri Abyaneh ’12; Bijnens, Zahiri Abyaneh ’12; Bijnens,Relefors (to be published); Talk by Bijnens at g − 2 Workshop, Mainz2014.

Tried various ways to include a1, but again no finite result for g − 2achieved. With a cutoff of 1 GeV:

aπ−loopµ = (−20± 5)× 10−11 (preliminary)

Very close to old result aπ−loopµ = (−19± 13)× 10−11 in BPP ’96, ’02.

• Maybe only model-independent approach with dispersion relations can givea reliable prediction for pion-loop. But to include the effects of the axialvector meson a1 and the matching with QCD short-distances, the region1− 2 GeV should be covered as well.

Dressed quark-loop

Dressed quark-loopDyson-Schwinger equation (DSE) approach [Fischer, Goecke, Williams ’11, ’13]

Claim: no double-counting between quark-loop and pseudoscalar exchanges (orexchanges of other resonances)

HLbL in Effective Field Theory (hadronic) picture:

' q + + +/− + · · ·

Quarks here may have different interpretation than below !

HLbL using functional methods (all propagators and vertices fully dressed):

' q + . . . +. . .

. . ....

... + · · ·

Expansion of quark-loop in terms ofplanar diagrams (rainbow-ladderapproximation):

q = + + · · ·

Pole representation of ladder-exchangecontribution:

. . .P2→−M2

PS−−−−−−−→

Truncate DSE using well tested model for dressed quark-gluon vertex (Maris, Tandy’99).

Large contribution from quark-loop (even after recent correction), in contrast to allother approaches, where coupling of (constituent) quarks to photons is dressed byform factors (ρ− γ-mixing, VMD).

Dressed quark-loop (continued)

Dyson-Schwinger equation approach [Fischer, Goecke, Williams ’11, ’13]

aHLbL;π0

µ = 57.5(6.9)× 10−11 (off-shell)

aHLbL;PSµ = 81(2)× 10−11

aHLbL;quark−loopµ = 107(2)× 10−11

aHLbLµ = 188(4)× 10−11

Error for PS, quark-loop and total only from numerics. Quark-loop: some partsare missing. Not yet all contributions to HLbL calculated. Systematic error ?

Note: numerical error in quark-loop in earlier paper (GFW PRD83 ’11):

aHLbL;quark−loopµ = 136(59)× 10−11

aHLbLµ = 217(91)× 10−11

Dressed quark-loop (continued)

• Constituent quark loop [Boughezal, Melnikov ’11]

aHLbLµ = (118− 148)× 10−11

Consider ratio of HVP and HLbL with pQCD corrections. Paper was reaction toearlier results using DSE yielding large values for quark-loop and total.

• Constituent Chiral Quark Model [Greynat, de Rafael ’12]

aHLbL;CQloopµ = 82(6)× 10−11

aHLbL;π0

µ = 68(3)× 10−11 (off-shell)

aHLbLµ = 150(3)× 10−11

Error only reflects variation of constituent quark mass MQ = 240± 10 MeV,fixed to reproduce HVP in g − 2. Determinations from other quantities givelarger value for MQ ∼ 300 MeV and thus smaller value for quark-loop. 20%-30%systematic error estimated. Not yet all contributions calculated.

• Pade approximants [Masjuan, Vanderhaeghen ’12]

aHLbLµ = (76(4)− 125(7))× 10−11

Quark-loop with running mass M(Q) ∼ (180− 220) MeV, where averagemomentum 〈Q〉 ∼ (300− 400) MeV is fixed from relevant momenta in 2-dim.integral representation for pion-pole in Knecht, Nyffeler ’02.

+ P

µ

q1q2q3

π

µ

Summary of recent developments• Recent evaluations for pseudoscalars:

aHLbL;π0

µ ∼ (50− 69)× 10−11

aHLbL;PSµ ∼ (59− 107)× 10−11

Most evaluations agree at level of 15%, but some are quite different.• New estimates for axial vectors (Pauk, Vanderhaeghen ’14; Jegerlehner ’14):

aHLbL;axialµ ∼ (6− 8)× 10−11

Substantially smaller than in MV ’04 !• First estimate for tensor mesons (Pauk, Vanderhaeghen ’14):

aHLbL;tensorµ = (1.1± 0.1)× 10−11

• Open problem: Dressed pion-loopPotentially important effect from pion polarizability and a1 resonance(Engel, Patel, Ramsey-Musolf ’12; Engel ’13; Engel, Ramsey-Musolf ’13):

aHLbL;π−loopµ = −(11− 71)× 10−11

Maybe large negative contribution, in contrast to BPP ’96, HKS ’96.• Open problem: Dressed quark-loop

Dyson-Schwinger equation approach (Fischer, Goecke, Williams ’11, ’13):

aHLbL;quark−loopµ = 107× 10−11 (still incomplete !)

Large contribution, no damping seen, in contrast to BPP ’96, HKS ’96.

Data driven approach to HLbL using dispersion relations (DR)

Strategy: Split contributions to HLbL into two parts

I: Data-driven evaluation using DR (hopefully numerically dominant):

(1) π0, η, η′ poles

(2) ππ intermediate state

II: Model dependent evaluation (hopefully numerically subdominant):

(1) Axial vectors (3π-intermediate state), . . .

(2) Quark-loop, matching with pQCD

Error goals: Part I: 10% precision (data driven), Part II: 30% precision.To achieve overall error of about 20% (δaHLbL

µ = 20× 10−11).

Colangelo et al., arXiv:1402.7081, arXiv:1408.2517:Classify intermediate states in four-point function. Then project onto g − 2.

Πµνλσ = Ππ0

µνλσ + ΠFsQEDµνλσ + Πππµνλσ + · · ·

Ππ0

µνλσ = pion pole (similarly for η, η′). Contribution to g − 2 discussed earlier.

ΠFsQEDµνλσ = scalar QED with vertices dressed by pion vector form factor FV

π

Πππµνλσ = remaining ππ contribution

Photon-photon processes in e+e− collisions

Feynman diagrams from Colangelo et al., arXiv:1408.2517

Space-like kinematics:

e+

e−π0

e+

e−π

π

By tagging the outgoing leptons (single-tag, double-tag), one can infer the virtual(space-like) momenta Q2

i = −q2i of the photons.

Left: process allows to measure pion-photon-photon transition form factor (TFF)Fπ0γ∗γ∗ (Q2

1 ,Q22 ).

Time-like kinematics:

e+

e− π0

e+

e−π

π

ππ intermediate stateColangelo et al., arXiv:1402.7081, arXiv:1408.2517

Extracting the pion vector form factor FVπ from data, the contribution ΠFsQED

µνλσ is

known to sufficient accuracy.

The remaining ππ contribution from Πππµνλσ is then given by

aππµ = e6Z

d4q1

(2π)4

Zd4q2

(2π)4

Pi Ii`s, q2

1 , q22

´Tππ

i

`q1, q2; p

´q2

1q22sZ1Z2

,

with known kinematical functions Tππi (q1, q2; p), while information on scattering

amplitude on the cut is given by the dispersive integrals Ii (s, q21 , q

22).

For instance, for first S-wave:

I1`s, q2

1 , q22

´=

1

π

∞Z4M2π

ds′

s′ − s

»„1

s′ − s−

s′ − q21 − q2

2

λ`s′, q2

1 , q22

´«Im h0++,++

`s′; q2

1 , q22 ; s, 0

´+

2ξ1ξ2

λ`s′, q2

1 , q22

´ Im h000,++

`s′; q2

1 , q22 ; s, 0

´–λ(x, y , z) = x2 + y2 + z2 − 2(xy + xz + yz): Kallen function

ξi : normalization of longitudinal polarization vectors of off-shell photons

hJλ1λ2,λ3λ4

(s; q21 , q

22 ; q2

3 , q24): partial-wave helicity amplitudes with angular momentum

J for process γ∗`q1, λ1

´γ∗`q2, λ2

´→ γ∗

`q3, λ3

´γ∗`q4, λ4

´.

Partial-wave unitarity relates imaginary parts in integrals Ii to helicity partial waveshJ,λ1λ2

(s; q21 , q

22) for γ∗γ∗ → ππ, which have to be determined from experiment:

Im hJλ1λ2,λ3λ4

`s; q2

1 , q22 ; q2

3 , q24

´=

p1− 4M2

π/s

16πhJ,λ1λ2

`s; q2

1 , q22

´hJ,λ3λ4

`s; q2

3 , q24

´

Another approach to HLbL using DR’sPauk, Vanderhaeghen, arXiv:1403.7503, arXiv:1409.0819

Write DR directly for form factorF2(k2):

F2(0) =1

2πi

∞Z0

dk2

k2Disck2 F2(k2)

Then study contributions fromdifferent intermediate states fromthe cuts in the unitarity diagramsrelated to measurable physicalprocesses like γ∗γ∗ → X andγ∗ → γX :

Pseudoscalar pole contribution

Considering the pseudoscalarintermediate state and VMD formfactor (ρ− γ mixing) it was shownthat from dispersion relation oneobtains precisely the result givenby direct evaluation of two-loopintegral in Knecht, Nyffeler ’02 forthe pseudoscalar pole.

Dashed: two-particle cutDotted: three-particle cutDouble-dashed: pseudoscalar pole

Double-solid: vector meson pole

HLbL @ NLOGolangelo, Hoferichter, Nyffeler, Passera, Stoffer ’14

Recently, a surprisingly large NNLO HVPcontribution was obtained by Kurz et al. ’14:

aHVP, LOµ = (6907.5± 47.2)× 10−11

aHVP, NLOµ = (−100.3± 2.2)× 10−11

aHVP, NNLOµ = (12.4± 0.1)× 10−11

Enhancement because of large ln(mµ/me),prefactors π2 in QED LbL.

(a) 3a (b) 3b (c) 3b (d) 3c

(e) 3c (f) 3c (g) 3b,lbl (h) 3d

Figure 1: Sample NNLO Feynman diagrams contributing to ahadµ .

Could there be a similarly large effect in HLbL @ NLO ?We calculated the potentially large contribution from anadditional electron loop (using simple VMD model for pion-poleto model full HLbL)

aπ0-pole, NLOµ = 1.5 · 10−11

2.6% effect of aπ0-poleµ = 57.2 · 10−11, very close to

renormalization group arguments 3× απ× 2

3log

mµme≈ 2.5%.

Estimating the not yet calculated diagrams with HLbL with additional radiativecorrections to the muon line or internal HLbL by comparing with HLbL insertions withmuon loop in QED, which are suppressed by factor 4, we obtain the estimate:

aHLbL, NLOµ = (3± 2) · 10−11,

Negligible with current precision goal for HLbL !

Conclusions• Over many decades, the (anomalous) magnetic moments of electron and muon

have played a crucial role in atomic and elementary particle physics.

• ae : Test of QED, precise determination of fine-structure constant α.aµ: Test of Standard Model, potential window to New Physics:

aexpµ − aSM

µ = (294± 88)× 10−11 [3.3 σ]

• Two new planned g − 2 experiments at Fermilab and JPARC with goal ofδaexpµ = 16× 10−11 (factor 4 improvement)

• Theory needs to match this precision !

• Hadronic vacuum polarization

Ongoing and planned experiments on σ(e+e− → hadrons) with a goal ofδaHVPµ = (20− 25)× 10−11 (factor 2 improvement)

• Hadronic light-by-light scattering

aHLbLµ = (105± 26)× 10−11 (Prades, de Rafael, Vainshtein ’09)

aHLbLµ = (116± 40)× 10−11 (Nyffeler ’09; Jegerlehner, Nyffeler ’09)

Need a much better understanding of the complicated hadronic dynamics to getreliable error estimate of ±20× 10−11.

- Better theoretical models needed; more constraints from theory (ChPT,pQCD, OPE); close collaboration of theory and experiment to measure therelevant decays, form factors and cross-sections of various hadrons withphotons.

- Promising new data driven approach using dispersion relations for π0, η, η′

and ππ. Still needed: data for scattering of off-shell photons.- Future: Lattice QCD.

And finally:

g-2 measuring the muon

In the 1950s the muon was still a

complete enigma. Physicists could

not yet say with certainty whether it

was simply a much heavier electron

or whether it belonged to another

species of particle. g-2 was set up to

test quantum electrodynamics, which

predicts, among other things, an

anomalously high value for the muon’s

magnetic moment ‘g’, hence the name of

the experiment. «The science I have experienced has been all about imagining and creating pioneering devices and observing entirely new phenomena, some of which have possibly never even been predicted by theory. That’s what invention is all about and it’s something quite extraordinary. CERN was marvellous for two reasons: it gave young people like me the opportunity to forge ahead in a new field and the chance to develop in an international environment.»

Francis Farley, Infinitely CERN, 2004

In 1959, six physicists joined forces to try to measure the muon’s magnetic moment using CERN’s first accelerator, the Synchrocyclotron. In 1961, the team published the first direct measurement of the muon’s anomalous magnetic moment to a precision of 2% with respect to the theoretical value. By 1962, this precision had been whittled down to just 0.4%. This was a great success since it validated the theory of quantum electrodynamics. As predicted, the muon turned out to be a heavy electron.

The first g-2 experiment at the SC, sitting on the experiment’s 6 m long magnet. From right to left : A. Zichichi, Th. Muller, G. Charpak, J.C. Sens. Standing : F. Farley. The sixth person was R. Garwin.

The second g-2 experiment started in 1966 under the leadership of Francis Farley and it achieved a precision 25 times higher than the previous one. This allowed phenomena predicted by the theory of quantum electrodynamics to be observed with a much greater sensitivity — vacuum polarisation for instance, which is the momentary appearance of ‘virtual’ electron and antielectron pairs with very short lifetimes. The experiment also revealed a quantitative discrepancy with the theory and thus prompted theorists to re-calculate their predictions.

“g-2 is not an experiment: it is a way of life.” John Adams

A third experiment, with a new technical approach, was launched in 1969, under the leadership of Emilio Picasso. The final results were published in 1979 and confirmed the theory to a precision of 0.0007%. They also allowed observation of a phenomenon contributing to the magnetic moment, namely the presence of ‘virtual hadrons’. After 1984, the United States took up the mantle of investigating the muon’s anomalous magnetic moment, applying the finishing touches to an edifice built at CERN.

The g-2 muon storage ring in 1974.

Source: CERN

“g − 2 is not an experiment: it is a way of life.”John Adams (Head of the Proton Synchrotron at CERN (1954-61) and Director General of CERN (1960-1961))

This statement also applies to many theorists working on the g − 2 !

Backup

Form factor Fπ0γ∗γ∗(q21 , q

22) and transition form factor F (Q2)

• Form factor Fπ0γ∗γ∗(q21 , q

22) between an on-shell pion and two off-shell

photons:

i

Zd4x e iq1·x〈0|Tjµ(x)jν(0)|π0(q1 + q2)〉 = εµναβ qα1 qβ2 Fπ0γ∗γ∗(q2

1 , q22)

jµ(x) = (ψQγµψ)(x), ψ ≡

0@ uds

1A, Q = diag(2,−1,−1)/3

(light quark part of electromagnetic current)

Bose symmetry: Fπ0γ∗γ∗(q21 , q

22) = Fπ0γ∗γ∗(q2

2 , q21)

Form factor for real photons is related to π0 → γγ decay width:

F2π0γ∗γ∗(q2

1 = 0, q22 = 0) =

4

πα2m3π

Γπ0→γγ

Often normalization with chiral anomaly is used:

Fπ0γ∗γ∗(0, 0) = − 1

4π2Fπ

• Pion-photon transition form factor:

F (Q2) ≡ Fπ0γ∗γ∗(−Q2, q22 = 0), Q2 ≡ −q2

1

Note that q22 = 0, but ~q2 6= ~0 for on-shell photon !

The LMD and LMD+V form factorsKnecht, Nyffeler, EPJC ’01; Nyffeler ’09

• Ansatz for 〈VVP〉 and thus Fπ0∗γ∗γ∗ in large-NC QCD in chiral limit with1 multiplet of lightest pseudoscalars (Goldstone bosons) and 2 multipletsof vector resonances, ρ, ρ′ (lowest meson dominance (LMD) + V)

• Fπ0∗γ∗γ∗ fulfills all leading (and some subleading) QCD short-distanceconstraint from OPE

• Reproduces Brodsky-Lepage (BL): limQ2→∞

Fπ0∗γ∗γ∗(m2π,−Q2, 0) ∼ 1/Q2

(OPE and BL cannot be fulfilled simultaneously with only one vector resonance,

LMD)

• Normalized to decay width Γπ0→γγ

LMD form factors (off-shell, on-shell, transition form factor):

FLMDπ0∗γ∗γ∗(q2

3 , q21 , q

22) =

Fπ3

q21 + q2

2 + q23 − cV

(q21 −M2

V ) (q22 −M2

V ), q2

3 = (q1 + q2)2

FLMDπ0γ∗γ∗(q2

1 , q22) =

Fπ3

q21 + q2

2 − cV

(q21 −M2

V ) (q22 −M2

V ), cV = cV −m2

π

FLMD(Q2) = − Fπ3M2

V

Q2 + cV

Q2 + M2V

Note that LMD transition form factor does not fall off like 1/Q2 for large Q2.

The LMD+V form factorOff-shell LMD+V form factor:

FLMD+Vπ0∗γ∗γ∗(q2

3 , q21 , q

22) =

Fπ3

q21 q2

2 (q21 + q2

2 + q23) + PV

H (q21 , q

22 , q

23)

(q21 −M2

V1) (q2

1 −M2V2

) (q22 −M2

V1) (q2

2 −M2V2

)

PVH (q2

1 , q22 , q

23) = h1 (q2

1 + q22)2 + h2 q2

1 q22 + h3 (q2

1 + q22) q2

3 + h4 q43

+h5 (q21 + q2

2) + h6 q23 + h7

q23 = (q1 + q2)2

Fπ = 92.4 MeV, MV1= Mρ = 775.49 MeV, MV2

= Mρ′ = 1.465 GeV, Free

parameters: hi

On-shell LMD+V form factor:

FLMD+Vπ0γ∗γ∗ (q2

1 , q22) =

Fπ3

q21 q2

2 (q21 + q2

2) + h1 (q21 + q2

2)2 + h2 q21 q2

2 + h5 (q21 + q2

2) + h7

(q21 −M2

V1) (q2

1 −M2V2

) (q22 −M2

V1) (q2

2 −M2V2

)

h2 = h2 + m2π, h5 = h5 + h3m

2π, h7 = h7 + h6m

2π + h4m

4π

Transition form factor:

FLMD+V(Q2) =Fπ3

1

M2V1

M2V2

h1Q4 − h5Q

2 + h7

(Q2 + M2V1

)(Q2 + M2V2

)

• h1 = 0 in order to reproduce Brodsky-Lepage behavior.• Can treat h1 as free parameter to fit the BABAR data, but the form factor

does then not vanish for Q2 →∞, if h1 6= 0.

Fixing the LMD+V model parameters hih1, h2, h5, h7 are quite well known:

• h1 = 0 GeV2 (Brodsky-Lepage behavior FLMD+Vπ0γ∗γ

(m2π ,−Q2, 0) ∼ 1/Q2)

• h2 = −10.63 GeV2 (Melnikov, Vainshtein ’04: Higher twist corr. in OPE)

• h5 = 6.93± 0.26 GeV4 − h3m2π (fit to CLEO data of FLMD+V

π0γ∗γ(m2

π ,−Q2, 0))

• h7 = −NCM4V1

M4V2/(4π2F 2

π)− h6m2π − h4m4

π

= −14.83 GeV6 − h6m2π − h4m4

π (or normalization to Γ(π0 → γγ))

Fit to BABAR data: h1 = (−0.17± 0.02) GeV2, h5 = (6.51± 0.20) GeV4 − h3m2π

with χ2/dof = 15.0/15 = 1.0. Result for aHLbL;π0

µ;LMD+V not affected (shifts compensate).

h3, h4, h6 are unknown / less constrained:

• New short-distance constraint ⇒ h1 + h3 + h4 = M2V1

M2V2χ (∗)

χ = quark condensate magnetic susceptibility of QCD (Ioffe, Smilga ’84).

LMD ansatz for 〈VT 〉 ⇒ χLMD = −2/M2V = −3.3 GeV−2 (Balitsky, Yung ’83)

Close to χ(µ=1 GeV) = −(3.15± 0.30) GeV−2 (Ball et al. ’03)Assume large-NC (LMD/LMD+V) framework is self-consistent⇒ χ = −(3.3± 1.1) GeV−2

⇒ vary h3 = (0± 10) GeV2 and determine h4 from relation (∗) and vice versa

Lattice QCD: χMSu (µ = 2 GeV) = −(2.08± 0.08) GeV−2 (Bali et al. ’12).

• Final result for aLbyL;π0

µ is very sensitive to h6

Assume LMD/LMD+V estimates of low-energy constants from chiral Lagrangianof odd intrinsic parity at O(p6) are self-consistent. Assume 100% error onestimate for relevant low-energy constant ⇒ h6 = (5± 5) GeV4

The VMD form factor

Vector Meson Dominance:

FVMDπ0∗γ∗γ∗((q1 + q2)2, q2

1 , q22) = − NC

12π2Fπ

M2V

q21 −M2

V

M2V

q22 −M2

V

on-shell = off-shell form factor !

Only two model parameters even for off-shell form factor: Fπ and MV

Note:

• VMD form factor factorizes FVMDπ0γ∗γ∗(q2

1 , q22) = f (q2

1)× f (q22). This might

be a too simplifying assumption / representation.

• VMD form factor has wrong short-distance behavior:FVMDπ0γ∗γ∗(q2, q2) ∼ 1/q4, for large q2, falls off too fast compared to OPE

prediction FOPEπ0γ∗γ∗(q2, q2) ∼ 1/q2.

Transition form factor:

FVMD(Q2) = − NC

12π2Fπ

M2V

Q2 + M2V

Anomalous magnetic moment in quantum field theoryQuantized spin 1/2 particle interacting with external, classical electromagnetic field

4 form factors in vertex function(conserved electromagnetic current: jµ(x) = eψ(x)γµψ(x), momentum transferk = p′ − p, not assuming parity or charge conjugation invariance)

γ(k)

p p’

≡ 〈p′, s′|jµ(0)|p, s〉

= (−ie)u(p′, s′)

"γµ F1(k2)| z

Dirac

+iσµνkν

2mF2(k2)| z Pauli

+γ5 σµνkν

2mF3(k2) + γ5

`k2γµ − k/ kµ

´F4(k2)

#u(p, s)

Real form factors for spacelike k2 ≤ 0. Working out the low-momentum(non-relativistic) limits one obtains:

F1(0) = 1 (renormalization of charge e)

µ =e

2m(F1(0) + F2(0)) (magnetic moment)

F2(0) = a (anomalous magnetic moment)

d = −e

2mF3(0) (electric dipole moment, violates CP)

F4(k2) = anapole moment (violates P)

Electron g − 2: Theory

Main contribution in Standard Model (SM) from mass-independent Feynmandiagrams in QED with electrons in internal lines (perturbative series in α):

aSMe =

5Xn=1

cn

“απ

”n

+2.7478(2)× 10−12 [Loops in QED with µ, τ ]

+0.0297(5)× 10−12 [weak interactions]

+1.682(20)× 10−12 [strong interactions / hadrons]

The numbers are based on the paper by Aoyama et al. 2012.

QED: mass-independent contributions to ae

• α: 1-loop, 1 Feynman diagram; Schwinger 1947/48:

c1 = 12

• α2: 2-loops, 7 Feynman diagrams; Petermann 1957, Sommerfield 1957:

c2 = 197144

+ π2

12− π2

2ln 2 + 3

4ζ(3) = −0.32847896557919378 . . .

• α3: 3-loops, 72 Feynman diagrams; . . . , Laporta & Remiddi 1996:

c3 =28259

5184+

17101

810π2 −

298

9π2 ln 2 +

139

18ζ(3)−

239

2160π4

+83

72π2ζ(3)−

215

24ζ(5) +

100

3

Li4

„1

2

«+

1

24ln4 2−

1

24π2 ln2 2

ff= 1.181241456587 . . .

• α4: 4-loops, 891 Feynman diagrams; Kinoshita et al. 1999, . . . , Aoyamaet al. 2008; 2012:

c4 = −1.9106(20) (numerical evaluation)

• α5: 5-loops, 12672 Feynman diagrams; Aoyama et al. 2005, . . . , 2012:

c5 = 9.16(58) (numerical evaluation)

Replaces earlier rough estimate c5 = 0.0± 4.6.New result removes biggest theoretical uncertainty in ae !

Mass-independent 2-loop Feynman diagrams in ae

1) 2) 3)

4) 5) 6)

7) 8) 9)

γ γµ e τµ

γ

7)

Mass-independent 3-loop Feynman diagrams in ae

1) 2) 3) 4) 5) 6)7) 8)

9) 10) 11) 12) 13) 14) 15) 16)

17) 18) 19) 20) 21) 22) 23) 24)

25) 26) 27) 28) 29) 30) 31) 32)

33) 34) 35) 36) 37) 38) 39) 40)

41) 42) 43) 44) 45) 46) 47) 48)

49) 50) 51) 52) 53) 54) 55) 56)

57) 58) 59) 60) 61) 62) 63) 64)

65) 66) 67) 68) 69) 70) 71) 72)

Determination of fine-structure constant α from g − 2 of electron

• Recent measurement of α via recoil-velocity of Rubidium atoms in atominterferometer (Bouchendira et al. 2011):

α−1(Rb) = 137.035 999 037(91) [0.66ppb]

This leads to (Aoyama et al. 2012):

aSMe (Rb) = 1 159 652 181.82 (6)|z

c4

(4)|zc5

(2)|zhad

(78)|zα(Rb)

[78]× 10−12 [0.67ppb]

⇒ aexpe − aSM

e (Rb) = −1.09(0.83)× 10−12 [Error from α(Rb) dominates !]

→ Test of QED !

• Use aexpe to determine α from series expansion in QED (contributions from

weak and strong interactions under control !). Assume: Standard Model“correct”, no New Physics (Aoyama et al. 2012):

α−1(ae) = 137.035 999 1657 (68)|zc4

(46)|zc5

(24)|zhad+EW

(331)| z a

expe

[342] [0.25ppb]

The uncertainty from theory has been improved by a factor 4.5 by Aoyamaet al. 2012, the experimental uncertainty in aexp

e is now the limiting factor.

• Today the most precise determination of the fine-structure constant α, afundamental parameter of the Standard Model.

The Brookhaven Muon g − 2 ExperimentThe first measurements of the anomalous magnetic moment of the muon were

performed in 1960 at CERN, aexpµ = 0.00113(14) (Garwin et al.) [12% precision] and

improved until 1979: aexpµ = 0.0011659240(85) [7 ppm] (Bailey et al.)

In 1997, a new experiment started at the Brookhaven National Laboratory (BNL):

⇒⇒

⇒⇒

⇒⇒

⇒⇒

⇒µ

⇒spin

momentum

StorageRing

ωa = aµeBmµ

actual precession × 2Source: BNL Muon g − 2 homepage

Angular frequencies for cyclotron precession ωc and spin precession ωs :

ωc =eB

mµ γ, ωs =

eB

mµ γ+ aµ

eB

mµ, ωa = aµ

eB

mµ

γ = 1/p

1− (v/c)2. With an electric field to focus the muon beam one gets:

~ωa =e

mµ

„aµ~B −

»aµ −

1

γ2 − 1

–~v × ~E

«Term with ~E drops out, if γ =

p1 + 1/aµ = 29.3: “magic γ” → pµ = 3.094 GeV/c

The Brookhaven Muon g − 2 Experiment: storage ring

Source: BNL Muon g − 2 homepage

The Brookhaven Muon g − 2 Experiment: result for aµ

Histogram with 3.6 billiondecays of µ−:

s]µs [µTime modulo 1000 20 40 60 80 100

Mil

lio

n E

ve

nts

pe

r 1

49

.2n

s

10-3

10-2

10-1

1

10

Source: Bennett et al. 2006

N(t) = N0(E) exp

„−t

γτµ

«× [1 + A(E) sin(ωat + φ(E))]

Exponential decay with mean lifetime:τµ,lab = γτµ = 64.378µs

(in lab system).

Oscillations due to angular frequencyωa = aµeB/mµ.

aµ =R

λ− Rwhere R =

ωa

ωpand λ =

µµ

µp

Brookhaven experiment measures ωa andωp (spin precession frequency for proton).

λ from hyperfine splitting of muonium

(µ+e−) (external input).

With new CODATA 2008 value for λ, one obtains a new world average(dominated by BNL g − 2 Experiment, Bennett et al. 2006):

aexpµ = 0.00116592089(63) [0.5 ppm]

Electron g − 2: Experiment

Latest experiment: Hanneke, Fogwell, Gabrielse, 2008

Cylindrical Penning trap for single electron

(1-electron quantum cyclotron)

Source: Hanneke et al.

n = 0

n = 1

n = 2

n = 0

n = 1

n = 2

ms = -1/2 ms = 1/2

top endcapelectrode

compensationelectrode

compensationelectrode

field emissionpoint

bottom endcapelectrode

nickel rings

microwave inlet

ring electrode

quartz spacer

trap cavity electron

0.5 cm

(a)

νc - 3δ/2

νc - δ/2νa = gνc / 2 - νc

fc = νc - 3δ/2

νc - 5δ/2νa

Cyclotron and spin precession levels of electron in Penning trap

Source: Hanneke et al.

ge

2=νs

νc' 1 +

νa − ν2z /(2fc )

fc + 3δ/2 + ν2z /(2fc )

+∆gcav

2

νs = spin precession frequency; νc , νc = cyclotron frequency: free electron, electron inPenning trap; δ/νc = hνc/(mec2) ≈ 10−9 = relativistic correction4 quantities are measured precisely in experiment:

fc = νc − 32δ ≈ 149 GHz; νa = g

2νc − νc ≈ 173 MHz;

νz ≈ 200 MHz = oscillation frequency in axial direction;

∆gcav = corrections due to oscillation modes in cavity

⇒ aexpe = 0.00115965218073(28) [0.24 ppb ≈ 1 part in 4 billions]

Only the 12th decimal place uncertain ! (Kusch & Foley, 1947/48: 4% precision)

Precision in ge/2 even 0.28 ppt ≈ 1 part in 4 trillions !

hadronic light-by-light scattering

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