Status of the STATUSOFTHEMUONg-2LIGHTBY …home.thep.lu.se/~bijnens/talks/prague13.pdfOverview HLbL General properties π0-exchange π-loop Quark-loop Summary Future Conclusions 6/48

Status of themuon g-2

light by lightcontribution

Johan Bijnens

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STATUS OF THE MUON g-2 LIGHT BY

LIGHT CONTRIBUTION

Johan Bijnens

Lund University

[email protected]

http://www.thep.lu.se/∼bijnens

http://www.thep.lu.se/∼bijnens/chpt.html

MesonNet meeting Prague – 17 June 2013

http://www.thep.lu.se/~bijnens/

http://www.thep.lu.se/~bijnens/chpt.html



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Overview

1 Overview

2 HLbLGeneral propertiesπ0-exchangeπ-loop: new stuff is hereQuark-loopSummary

3 Future

4 Conclusions



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Muon g − 2: overview

in terms of the anomaly aµ = (g − 2)/2

Data dominated by BNL E821 (statistics)(systematic)aexp

µ+ = 11659204(6)(5) × 10−10

aexp

µ−= 11659215(8)(3) × 10−10

aexpµ = 11659208.9(5.4)(3.3) × 10−10

Theory is off somewhat (electroweak)(LO had)(HO had)aSMµ = 11659180.2(0.2)(4.2)(2.6) × 10−10

∆aµ = aexpµ − aSMµ = 28.7(6.3)(4.9) × 10−10 (PDG)

E821 goes to Fermilab, expect factor of four in precision

Note: g agrees to 3 10−9 with theory

Many BSM models CAN predict a value in this range(often a lot more or a lot less)

Numbers taken from PDG2012, see references there



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Summary of Muon g − 2 contributions

1010aµexp 11 659 208.9 6.3

theory 11 659 180.2 5.0

QED 11 658 471.8 0.0EW 15.4 0.2LO Had 692.3 4.2HO HVP −9.8 0.1HLbL 10.5 2.6

difference 28.7 8.1

Error on LO hadall e+e− based OKτ based 2 σ

Error on HLbL

Errors addedquadratically

3.5 σ

Difference:4% of LO Had270% of HLbL1% of leptonic LbL

Generic SUSY: 12.3× 10−10(

100 GeVMSUSY

)2tanβ

MSUSY ≈ 66 GeV√tanβ



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HLbL: the main object to calculate

p1ν

p2α qρ

p3β

p5p4p′ p

Muon line and photons: well known

The blob: fill in with hadrons/QCD

Trouble: low and high energy very mixed

Double counting needs to be avoided: hadron exchangesversus quarks



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A separation proposal: a start

E. de Rafael, “Hadronic contributions to the muon g-2 and low-energy QCD,”

Phys. Lett. B322 (1994) 239-246. [hep-ph/9311316].

Use ChPT p counting and large Nc

p4, order 1: pion-loop

p8, order Nc : quark-loop and heavier meson exchanges

p6, order Nc : pion exchange

Does not fully solve the problemonly short-distance part of quark-loop is really p8

but it’s a start



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A separation proposal: a start

E. de Rafael, “Hadronic contributions to the muon g-2 and low-energy QCD,”

Phys. Lett. B322 (1994) 239-246. [hep-ph/9311316].

Use ChPT p counting and large Nc

p4, order 1: pion-loop

p8, order Nc : quark-loop and heavier meson exchanges

p6, order Nc : pion exchange

Implemented by two groups in the 1990s:

Hayakawa, Kinoshita, Sanda: meson models, pion loop usinghidden local symmetry, quark-loop with VMD, calculationin Minkowski space

JB, Pallante, Prades: Try using as much as possible aconsistent model-approach, ENJL, calculation in Euclideanspace



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General properties

Πρναβ(p1, p2, p3) =

p3

p2p1

q

Actually we really needδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0



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General properties

Πρναβ(p1, p2, p3):

In general 138 Lorentz structures (but only 32 contributeto g − 2)

Using qρΠρναβ = p1νΠ

ρναβ = p2αΠρναβ = p3βΠ

ρναβ = 043 gauge invariant structures

Bose symmetry relates some of them

All depend on p21, p22 and q2, but before derivative and

p3 → 0 also p23, p1 · p2, p1 · p3, p2 · p3Compare HVP: one function, one variable

General calculation from experiment difficult to see how

In four photon measurement: lepton contribution



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General properties

∫

d4p1(2π)4

∫

d4p2(2π)4

plus loops inside the hadronic part

8 dimensional integral, three trivial,

5 remain: p21 , p22 , p1 · p2, p1 · pµ, p2 · pµ

Rotate to Euclidean space:

Easier separation of long and short-distanceArtefacts (confinement) in models smeared out.

More recent: can do two more using Gegenbauertechniques Knecht-Nyffeler,Jegerlehner-Nyffeler,JB–Zahiri-Abyaneh–Relefors

P21 , P

22 and Q2 remain

study aXµ =∫

dlP1dlP2

aXLLµ =

∫

dlP1dlP2

dlQaXLLQµ

lP = ln (P/GeV ), to see where the contributions are

Study the dependence on the cut-off for the photons



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General properties

∫

d4p1(2π)4

∫

d4p2(2π)4

plus loops inside the hadronic part

8 dimensional integral, three trivial,

5 remain: p21 , p22 , p1 · p2, p1 · pµ, p2 · pµ

Rotate to Euclidean space:

Easier separation of long and short-distanceArtefacts (confinement) in models smeared out.

More recent: can do two more using Gegenbauertechniques Knecht-Nyffeler,Jegerlehner-Nyffeler,JB–Zahiri-Abyaneh–Relefors

P21 , P

22 and Q2 remain

study aXµ =∫

dlP1dlP2

aXLLµ =

∫

dlP1dlP2

dlQaXLLQµ

lP = ln (P/GeV ), to see where the contributions are

Study the dependence on the cut-off for the photons



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π0 exchange

“π0”

“π0”= 1/(p2 −m2π)

The blobs need to be modelled, and in e.g. ENJL containcorrections also to the 1/(p2 −m2

π)

Pointlike has a logarithmic divergence

Numbers π0, but also η, η′



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π0 exchange

BPP: aπ0

µ = 5.9(0.9) × 10−10

Nonlocal quark model: aπ0

µ = 6.27× 10−10

A. E. Dorokhov, W. Broniowski, Phys.Rev.D78 (2008)073011. [0805.0760]

DSE model: aπ0

µ = 5.75× 10−10

Goecke, Fischer and Williams, Phys.Rev.D83(2011)094006[1012.3886]

LMD+V: aπ0

µ = (5.8 − 6.3)× 10−10

M. Knecht, A. Nyffeler, Phys. Rev. D65(2002)073034, [hep-ph/0111058]

Formfactor inspired by AdS/QCD: aπ0

µ = 6.54× 10−10

Cappiello, Cata and D’Ambrosio, Phys.Rev.D83(2011)093006 [1009.1161]

Chiral Quark Model: aπ0

µ = 6.8× 10−10

D. Greynat and E. de Rafael, JHEP 1207 (2012) 020 [1204.3029].

Constraint via magnetic susceptibility: aπ0

µ = 7.2× 10−10

A. Nyffeler, Phys. Rev. D 79 (2009) 073012 [0901.1172].

All in reasonable agreement



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MV short-distance: π0 exchange

K. Melnikov, A. Vainshtein, Hadronic light-by-light scattering contribution

to the muon anomalous magnetic moment revisited, Phys. Rev. D70

(2004) 113006. [hep-ph/0312226]

take P21 ≈ P2

2 ≫ Q2: Leading term in OPE of two vectorcurrents is proportional to axial current

Πρναβ ∝ Pρ

P21〈0|T (JAνJVαJVβ) |0〉

JA comes from

+

AVV triangle anomaly: extra info

Implemented via setting one blob = 1

“π0”=⇒ “π0”

aπ0

µ = 7.7 × 10−10



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π0 exchange

The pointlike vertex implements shortdistance part, notonly π0-exchange

+Are these part of the quark-loop? See also inDorokhov,Broniowski, Phys.Rev. D78(2008)07301

BPP quarkloop + π0-exchange ≈ MV π0-exchange



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π0 exchange

Which momentum regimes important studied: JB and

J. Prades, Mod. Phys. Lett. A 22 (2007) 767 [hep-ph/0702170]

aµ =

∫

dl1dl2aLLµ with li = log(Pi/GeV )

0.1

1

10 0.1

1 10

0

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

7e-10

aµLL(VMD)

P1

P2

aµLL(VMD)

0.1

1

10 0.1

1 10

0

1e-10

2e-10

3e-10

4e-10

5e-10

6e-10

7e-10

aµLL(KN)

P1

P2

aµLL(KN)

Which momentum regions do what:volume under the plot ∝ aµ



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Pseudoscalar exchange

Point-like VMD: π0 η and η′ give 5.58, 1.38, 1.04.

Models that include U(1)A breaking give similar ratios

Pure large Nc models use this ratio

The MV argument should give some enhancement overthe full VMD like models

Total pseudo-scalar exchange is aboutaPSµ = 8− 10× 10−10

AdS/QCD estimate (includes excited pseudo-scalars)aPSµ = 10.7 × 10−10

D. K. Hong and D. Kim, Phys. Lett. B 680 (2009) 480 [0904.4042]



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π-loop

The ππγ∗ vertex is always done using VMD

ππγ∗γ∗ vertex two choices:

Hidden local symmetry model: only one γ has VMDFull VMDBoth are chirally symmetricThe HLS model used has problems with π+-π0 massdifference (due to not having an a1)

Final numbers quite different: −0.45 and −1.9 (×10−10)

For BPP stopped at 1 GeV but within 10% of higher Λ



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π and K -loop

Cut-off 1010aµGeV π bare π VMD π ENJL π HLS K ENJL0.5 −1.71(7) −1.16(3) −1.20(0.03) −1.05(0.01) −0.020(0.001)0.6 −2.03(8) −1.41(4) −1.42(0.03) −1.15(0.01) −0.026(0.001)0.7 −2.41(9) −1.46(4) −1.56(0.03) −1.17(0.01) −0.034(0.001)0.8 −2.64(9) −1.57(6) −1.67(0.04) −1.16(0.01) −0.042(0.001)1.0 −2.97(12) −1.59(15) −1.81(0.05) −1.07(0.01) −0.048(0.002)2.0 −3.82(18) −1.70(7) −2.16(0.06) −0.68(0.01) −0.087(0.005)4.0 −4.12(18) −1.66(6) −2.18(0.07) −0.50(0.01) −0.099(0.005)

HLS JB-Zahiri-Abyaneh

note the suppression by the propagators

K -loop neglected in the remainder



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π loop: Bare vs VMD

0.1

1

10 0.1

1 10

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loop

VMDbare

P1 = P2

Q

-aµLLQ

plotted aLLQµ for P1 = P2

aµ =∫

dlP1dlP2

dlQ aLLQµ

lQ = log(Q/1 GeV)



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π loop: VMD vs HLS

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

Usual HLS, a = 2



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π loop: VMD vs HLS

0.1 1

10 0.1 1 10

0

2e-11

4e-11

6e-11

8e-11

1e-10

1.2e-10

-aµLLQ

π loop

VMDHLS a=1

P1 = P2Q

-aµLLQ

HLS with a = 1, satisfies more short-distance constraints



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π loop

ππγ∗γ∗ for q21 = q22 has a short-distance constraint fromthe OPE as well.

HLS does not satisfy it

full VMD does: so probably better estimate

Ramsey-Musolf suggested to do pure ChPT for the π loopK. T. Engel, H. H. Patel and M. J. Ramsey-Musolf, “Hadronic

light-by-light scattering and the pion polarizability,” Phys. Rev. D 86

(2012) 037502 [arXiv:1201.0809 [hep-ph]].

So far ChPT at p4 done for four-point function in limitp1, p2, q ≪ mπ (Euler-Heisenberg plus next order)

Polarizability (L9 + L10) up to 10%, charge radius 30%

Both HLS and VMD have charge radius effect but notpolarizability



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π loop






(2012) 037502 [arXiv:1201.0809 [hep-ph]].






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π loop






(2012) 037502 [arXiv:1201.0809 [hep-ph]].






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π loop: L9, L10

ChPT for muon g − 2 at order p6 is not powercountingfinite so no prediction for aµ exists.

But can be used to study the low momentum end of theintegral over P1,P2,Q

The four-photon amplitude is finite still at two-loop order(counterterms start at order p8)

Add L9 and L10 vertices to the bare pion loopJB-Zahiri-Abyaneh

Program the Euler-Heisenberg plus NLO result ofRamsey-Musolf et al. into our programs for aµ

Bare pion-loop and L9, L10 part in limit p1, p2, q ≪ mπ

agree with Euler-Heisenberg plus next order analytically



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π loop: L9, L10

ChPT for muon g − 2 at order p6 is not powercountingfinite so no prediction for aµ exists.

But can be used to study the low momentum end of theintegral over P1,P2,Q

The four-photon amplitude is finite still at two-loop order(counterterms start at order p8)

Add L9 and L10 vertices to the bare pion loopJB-Zahiri-Abyaneh

Program the Euler-Heisenberg plus NLO result ofRamsey-Musolf et al. into our programs for aµ

Bare pion-loop and L9, L10 part in limit p1, p2, q ≪ mπ

agree with Euler-Heisenberg plus next order analytically



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π loop: VMD vs charge radius

0.1

0.2

0.4 0.1

0.2 0.4

-4e-11

-2e-11

0

2e-11

4e-11

6e-11

8e-11

1e-10

-aµLLQ

π loop

VMDL9=-L10

P1 = P2Q

-aµLLQ

low scale, charge radius effect well reproduced



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π loop: VMD vs L9 and L10

0.1

0.2

0.4 0.1

0.2 0.4

0

2e-11

4e-11

6e-11

8e-11

1e-10

1.2e-10

-aµLLQ

π loop

VMDL10,L9

P1 = P2Q

-aµLLQ

L9 + L10 6= 0 gives an enhancement of 10-15%

To do it fully need to get a model: include a1



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Include a1

L9 + L10 effect is froma1

But to get gauge invariance correctly need

a1 a1



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Include a1

Consistency problem: full a1-loop?

Treat a1 and ρ classical and π quantum: there must be aπ that closes the loopArgument: integrate out ρ and a1 classically, then do pionloops with the resulting Lagrangian

To avoid problems: representation without a1-π mixing

Check for curiosity what happens if we add a1-loop



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Include a1

Use antisymmetric vector representation for a1 and ρ

Fields Aµν , Vµν (nonets)

Kinetic terms: −12

⟨

∇λVλµ∇νVνµ − 1

2VµνVµν⟩

−12

⟨

∇λAλµ∇νAνµ − 1

2AµνAµν⟩

Terms that give contributions to the Lri :

FV

2√2〈f+µνV

µν〉+ iGV√2〈V µνuµuν〉+ FA

2√2〈f−µνA

µν〉

L9 =FVGV

2M2V

, L10 = − F 2V

4M2V

+F 2A

4M2A

Weinberg sum rules: (Chiral limit)

F 2V = F 2

A + F 2π F 2

VM2V = F 2

AM2A

VMD for ππγ: FVGV = F 2π



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Vµν only

Πρναβ(p1, p2, p3) is not finite(but was also not finite for HLS)

ButδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0

also not finite

(but was finite for HLS)

Derivative one finite for GV = FV /2

Surprise: g − 2 identical to HLS with a =F 2V

F 2π

Yes I know, different representations are identical BUTthey do differ in higher order terms and even in what ishigher order

Same comments as for HLS numerics



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Vµν only

Πρναβ(p1, p2, p3) is not finite(but was also not finite for HLS)

ButδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0

also not finite

(but was finite for HLS)

Derivative one finite for GV = FV /2

Surprise: g − 2 identical to HLS with a =F 2V

F 2π

Yes I know, different representations are identical BUTthey do differ in higher order terms and even in what ishigher order

Same comments as for HLS numerics



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Vµν and Aµν

Add a1

Calculate a lotδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0

finite for:

GV = FV = 0 and F 2A = −2F 2

π

If adding full a1-loop GV = FV = 0 and F 2A = −F 2

π

Clearly unphysical (but will show some numerics anyway)



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Vµν and Aµν

Add a1

Calculate a lotδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0

finite for:

GV = FV = 0 and F 2A = −2F 2

π

If adding full a1-loop GV = FV = 0 and F 2A = −F 2

π

Clearly unphysical (but will show some numerics anyway)



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Vµν and Aµν

Start by adding ρa1π vertices

λ1 〈[V µν ,Aµν ]χ−〉 +λ2 〈[V µν ,Aνα] hµν〉

+λ3 〈i [∇µVµν ,Aνα] uα〉 +λ4 〈i [∇αVµν ,Aαν ] uµ〉

+λ5 〈i [∇αVµν ,Aµν ] uα〉 +λ6 〈i [V µν ,Aµν ] f−αν〉

+λ7 〈iVµνAµρAν

ρ〉All lowest dimensional vertices of their respective type

Not all independent, there are three relations

Follow from the constraints on Vµν and Aµν (thanksStefan)



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Vµν and Aµν : big disappointment

Work a whole lotδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0

not obviously finite

Work a lot more

Prove thatδΠρναβ(p1, p2, p3)

δp3λ

∣

∣

∣

∣

p3=0

finite, only same

solutions as before

Try the combination that show up in g − 2 only

Work a lot

Again, only same solutions as before

Small loophole left: after the integration for g − 2 couldbe finite but many funny functions of mπ,mµ,MV and MA

show up.



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π loop: add a1 and F 2A = −2F 2

π

0.1

1

10

0.1

1

10

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loop

bare

FA2 = -2 F2

P1 = P2

Q

-aµLLQ

Lowers at low energies, L9 + L10 < 0 here

funny peak at a1 mass



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π loop: add a1 and F 2A = −F 2

π plus a1-loop

0.1

1

10

0.1

1

10

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loop

bare

FA2 = -F2

P1 = P2

Q

-aµLLQ

Lowers at low energies, L9 + L10 < 0 here

funny peak at a1 mass canceled

Still unphysical case



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a1-loop: cases with good L9 and L10

0.1

1

10

0.1

1

10

-2e-10

-1.5e-10

-1e-10

-5e-11

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loopbare

Weinberg no a1-loop

P1 = P2

Q

-aµLLQ

Add FV , GV and FA

Fix values by Weinberg sum rules and VMD in γ∗ππ

no a1-loop



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0.1

1

10

0.1

1

10

-2e-10

-1.5e-10

-1e-10

-5e-11

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loopbare

Weinberg with a1-loop

P1 = P2

Q

-aµLLQ

Add FV , GV and FA

Fix values by Weinberg sum rules and VMD in γ∗ππ

With a1-loop (is different plot!!)



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0.1

1

10

0.1

1

10

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loop

bare

a1 no a1-loop,VMD

P1 = P2

Q

-aµLLQ

Add a1 with F 2A = +F 2

π

Add the full VMD as done earlier for the bare pion loop



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0.1

1

10

0.1

1

10

0

5e-11

1e-10

1.5e-10

2e-10

-aµLLQ

π loop

bare

a1 with a1-loop,VMD

P1 = P2

Q

-aµLLQ

Add a1 with F 2A = +F 2

π and a1-loop

Add the full VMD as done earlier for the bare pion loop



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Integration results

0

5e-11

1e-10

1.5e-10

2e-10

2.5e-10

3e-10

3.5e-10

4e-10

4.5e-10

5e-10

0.1 1 10

-aµΛ

Λ

sQED

sQED π0

VMD

HLS

HLS a=1

ENJL

P1,P2,Q ≤ Λ



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Integration results

0

5e-11

1e-10

1.5e-10

2e-10

2.5e-10

3e-10

3.5e-10

4e-10

0.1 1 10

-aµΛ

Λ

a1 FA2= -2F2

a1 FA2 = -1 a1-loop

HLSHLS a=1

VMDa1 VMD

a1 Weinberg

P1,P2,Q ≤ Λ (computer still running)



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Integration results with a1

Problem: get high energy behaviour good enough

But all models with reasonable L9 and L10 fall way insidethe error quoted earlier (−1.9± 1.3) 10−10

Tentative conclusion: Use hadrons only below about 1GeV: aπ−loop

µ = (−2.0 ± 0.5) 1010



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Pure quark loop

Cut-off aµ × 107 aµ × 109 aµ × 109

Λ Electron Muon Constituent Quark(GeV) Loop Loop Loop0.5 2.41(8) 2.41(3) 0.395(4)0.7 2.60(10) 3.09(7) 0.705(9)1.0 2.59(7) 3.76(9) 1.10(2)2.0 2.60(6) 4.54(9) 1.81(5)4.0 2.75(9) 4.60(11) 2.27(7)8.0 2.57(6) 4.84(13) 2.58(7)

Known Results 2.6252(4) 4.65 2.37(16)

MQ : 300 MeV

now known fully analytically

Us: 5+(3-1) integrals extra are Feynman parameters

Slow convergence:

electron: all at 500 MeVMuon: only half at 500 MeV, at 1 GeV still 20% missing300 MeV quark: at 2 GeV still 25% missing



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ENJL quark-loop

Cut-off aµ × 1010 aµ × 1010 aµ × 1010 aµ × 1010

Λ sumGeV VMD ENJL masscut ENJL+masscut

0.5 0.48 0.78 2.46 3.20.7 0.72 1.14 1.13 2.31.0 0.87 1.44 0.59 2.02.0 0.98 1.78 0.13 1.94.0 0.98 1.98 0.03 2.08.0 0.98 2.00 .005 2.0

Very stable

ENJL cuts off slower than pure VMD

masscut: MQ = Λ to have short-distance and no problemwith momentum regions

Quite stable in region 1-4 GeV



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Quark loop DSE

DSE model: aqlµ = 13.6(5.9) × 10−10 T. Goecke, C. S. Fischer

and R. Williams, Phys. Rev. D 83 (2011) 094006 [arXiv:1012.3886 [hep-ph]]

Not a full calculation (yet) but includes an estimate ofsome of the missing parts

a lot larger than bare quark loop with constituent mass

I am puzzled: this DSE model (Maris-Roberts) doesreproduces a lot of low-energy phenomenology. I wouldhave guessed that it would give numbers very similar toENJL.

Can one find something in between full DSE and ENJLthat is easier to handle?

Error found in calculation, still not finalized: preliminaryaqlµ = 10.7(0.2) × 10−10 T. Goecke, C. S. Fischer and R. Williams,

arXiv:1210.1759



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Other quark loop

de Rafael-Greynat 1210.3029 (7.6 − 8.9) 10−10

Boughezal-Melnikov 1104.4510 (11.8 − 14.8) 10−10

Masjuan-Vanderhaeghen 1212.0357 (7.6 − 12.5) 10−10

Various interpretations: the full calculation or not

All (even DSE) have in common that a low quark mass isused for a large part of the integration range



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Summary: ENJL vc PdRV

BPP PdRV arXiv:0901.0306

quark-loop (2.1 ± 0.3) · 10−10 —pseudo-scalar (8.5 ± 1.3) · 10−10 (11.4 ± 1.3) · 10−10

axial-vector (0.25 ± 0.1) · 10−10 (1.5 ± 1.0) · 10−10

scalar (−0.68 ± 0.2) · 10−10 (−0.7± 0.7) · 10−10

πK -loop (−1.9 ± 1.3) · 10−10 (−1.9± 1.9) · 10−10

errors linearly quadraticallysum (8.3 ± 3.2) · 10−10 (10.5 ± 2.6) · 10−10



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What can we do more?

The ENJL model can certainly be improved:

Chiral nonlocal quark-model (like nonlocal ENJL): so faronly π0-exchange doneDSE: π0-exchange similar to everyone else, quark-loop verydifferent, looking forward to final results

More resonances models should be tried, AdS/QCD is oneapproach, RχT (Valencia et al.) possible,. . .

Note short-distance matching must be done in manychannels, there are theorems JB,Gamiz,Lipartia,Pradesthat with only a few resonances this requires compromises

π-loop: HLS smaller than double VMD (understood)models with ρ and a1: difficulties with infinities



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Constraints from experiment:J. Bijnens and F. Persson, hep-ph/hep-ph/0106130

Studying three formfactors Pγ∗γ∗ in P → ℓ+ℓ−ℓ′+ℓ′−,e+e− → e+e−P exact tree level and for g − 2 (butbeware sign):

Conclusion: possible but VERY difficultTwo γ∗ off-shell not so important for our choice ofform-factor

All information on hadrons and 1-2-3-4 off-shell photons iswelcome: constrain the models

More short-distance constraints: MV, Nyffeler integratewith all contributions, not just π0-exchange

Need a new overall evaluation with consistent approach.

Lattice has done first steps



Johan Bijnens

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Conclusions

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Constraints from experiment:J. Bijnens and F. Persson, hep-ph/hep-ph/0106130

Studying three formfactors Pγ∗γ∗ in P → ℓ+ℓ−ℓ′+ℓ′−,e+e− → e+e−P exact tree level and for g − 2 (butbeware sign):

Conclusion: possible but VERY difficultTwo γ∗ off-shell not so important for our choice ofform-factor

All information on hadrons and 1-2-3-4 off-shell photons iswelcome: constrain the models

More short-distance constraints: MV, Nyffeler integratewith all contributions, not just π0-exchange

Need a new overall evaluation with consistent approach.

Lattice has done first steps



Johan Bijnens

Overview

HLbL

Future

Conclusions

48/48

Summary of Muon g − 2 contributions

1010aµexp 11 659 208.9 6.3

theory 11 659 180.2 5.0

QED 11 658 471.8 0.0EW 15.4 0.2LO Had 692.3 4.2HO HVP −9.8 0.1HLbL 10.5 2.6

difference 28.7 8.1

Error on LO hadall e+e− based OKτ based 2 σ

Error on HLbL

Errors addedquadratically

3.5 σ

Difference:4% of LO Had270% of HLbL1% of leptonic LbL

Generic SUSY: 12.3× 10−10(

100 GeVMSUSY

)2tanβ

MSUSY ≈ 66 GeV√tanβ

Documents

Status of the STATUSOFTHEMUONg-2LIGHTBY …home.thep.lu.se/~bijnens/talks/prague13.pdfOverview HLbL General properties π0-exchange π-loop Quark-loop Summary Future Conclusions 6/48