analytical results for hadronic contributions to the muon...

Analyticalresults for themuon g − 2

Johan Bijnens

Introduction

HVP

HLbL

Conclusions

1/53

Analytical results for hadronic contributions tothe muon g − 2

Johan BijnensLund University

[email protected]

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

IFIC Valencia 17 October 2019

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


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

2/53

Introduction

Magnetic moment: ~µ = gq

2m~S

For angular momentum: g = 1

Dirac equation: g = 2

Structure and/or QFT give different values

Anomaly: a =g − 2

2QED (Schwinger):

a = 1 +α

2π+ . . .

Jacob Bourjaily/Wikipedia


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

3/53

Introduction

Experiments done for electron and muon: very precise

aexpe = 11596521.8073(0.0028) 10−10 Harvard

aexpµ = 11659209.1(5.4)(3.3) 10−10 BNL

Main difference: electron loops enhanced by log(me/mµ)

aSMµ = aQEDµ + aEWµ + ahadµ

aQEDµ = α

2π+0.765857425(17)(απ

)2+24.05050996(32)

(απ

)3

+130.8796(63)(απ

)4+ 753.3(1.0)

(απ

)5

up to 4 loops essentially analytically, 5-loops numerically

aQEDµ = 11658471.895(0.008) 10−10

aEWµ = 15.36(0.10) 10−10

aexpµ − aQEDµ − aEWµ = 7218(63) 10−11


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

4/53

Why do we do this?

The muon aµ =gµ − 2

2will be measured more precisely

J-PARC Fermilab


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

5/53

Why do we do this?

Experiment dominated by BNL, FNAL error down by four

Theory taken from PDG2018

aSMµ = aQED

µ + aEWµ + aHad

µ

aHadµ = aLO-HVP

µ + aHO-HVPµ + aHLbL

µ

Impressive agreement with gµ to 2× 10−9

Part value errors units

aEXPµ : 116 592 091.x (54)(33) ×10−11

aSMµ : 116 591 823.x (1)(34)(26) ×10−11

∆aµ: 268.x (63)(43) ×10−11

aQEDµ : 116 584 719.0 (0.1) ×10−11

aEWµ : 153.6 (1.0) ×10−11

aLO-HVPµ : 6 931.x (33)(7) ×10−11

aHO-HVPµ : −86.3 (0.9) ×10−11

aHLbLµ : 105.x (26) ×10−11


Johan Bijnens

Introduction

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HLbL

Conclusions

6/53

Numerical summary

Long standing discrepancy with SM

Theory number has had slowly shrinking error (especiallyLO HVP)

aexpµ − aSMµ = (27.0− 33.6)± (7.3− 7.8) 10−10

3.6− 4.6σ discrepancy

Many BSM papers explaining this(experiment has ≥ 1950 citations)


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

7/53

Hadronic contributions

LO-HVP HLbL

The blobs are hadronic contributions

I will present some results for LO-HVP and HLbLshort-distance as well as give an overview

There are higher order contributions of both types: knownaccurately enough


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

8/53

My recent work related to the muon g − 2

[1] J. Bijnens and J. Relefors, “Pion light-by-light contributions to the muon

g − 2,” JHEP 1609 (2016) 113 [arXiv:1608.01454 [hep-ph]]

[2] J. Bijnens, N. Hermansson-Truedsson and A. Rodriguez-Sanchez,

“Short-distance constraints for the HLbL contribution to the muon

anomalous magnetic moment,” Phys. Lett. B 798C (2019) 134994

[arxiv:1908.03331].

[3] J. Bijnens and J. Relefors, “Connected, Disconnected and Strange Quark

Contributions to HVP,” JHEP 1611 (2016) 086 [1609.01573 [hep-lat]].

[4] J. Bijnens and J. Relefors, “Vector two-point functions in finite volume

using partially quenched chiral perturbation theory at two loops,” JHEP

1712 (2017) 114 [arXiv:1710.04479 [hep-lat]].

[5] J. Bijnens, J. Harrison, N. Hermansson-Truedsson, T. Janowski, A. Juttner

and A. Portelli, “Electromagnetic finite-size effects to the hadronic vacuum

polarization,” Phys. Rev. D100 (2019) 014508 [arXiv:1903.10591 [hep-lat]].

[1,2,3]: HLbL, [4,5]: HVPI will concentrate on [2,5]


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

9/53

Muon g − 2: LO hadronic from experiment

=⇒

aLO HVPµ = 1

3

(απ

)2 ∫∞m2πds K(s)

s R(0)(s)

R(0)(s) bare cross-section ratioσ(e+e− → hadrons)

σ(e+e− → µ+µ−)

Bare, many different evaluations,. . .

e+e− versus τ -decays

Main problem: combining data from many sources anddata sometimes not compatible with errors


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

10/53

Numerical summary: HVP

LO-HVP latest evaluations (talks at workshops KEK2018/Seattle 2019 )

Paris (Davier et al.)aLO HVPµ = 693.9± 1.0± 3.4± 1.6± 0.1ψ ± 0.7QCD 10−10

Liverpool (Keshavarzi et al) aLO HVPµ = 693.8(2.4) 10−10

Jegerlehner aLO HVPµ = 689.46(3.25) 10−10

HLS (Benayoun et al) aLO HVPµ = 686.1(3.8) 10−10

Slight differences in data, how put together plus whattheoryDiscrepancy Babar/KLOE in ππ-channelCan be with or without τ -dataNumbers vary slightly whenever new experiments included

aNLO HVPµ = −9.82(0.04) 10−10

aNNLO HVPµ = 1.24(0.01) 10−10

Errors: LO-HVP and HLbL, rest negligible


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

11/53

HVP: Summary of results

Nf = 2 + 1 + 1 BMW 17

HPQCD 16

ETMC 13

Nf = 2 + 1 RBC/UKQCD 18

Nf = 2 Mainz/CLS 17

620 660 700 740ahvpµ · 1010

R ratio

HLMNT 11

DHMZ 11

DHMZ 17

Jegerlehner 17

KNT 18

RBC/UKQCD 18

Figure from1807.09370

Somediscrepancies,latticeaccuraciesimprove


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

12/53

Status lattice results

Dispersive has reached below 0.5%

Lattice accuracies a few %

Improvement continuous

Finite volume corrections known to two-loop order inChPT [4]

Next: isospin breaking

mu −md : “easy”Electromagnetism: possibly large finite volume correctionssince 1/Ln rather than exp(−mπL)

J. Bijnens, J. Harrison, N. Hermansson-Truedsson, T. Janowski,

A. Juttner and A. Portelli, “Electromagnetic finite-size effects to the

hadronic vacuum polarization,” arXiv:1903.10591 [hep-lat].

Presented at g − 2 meetings in KEK february 2018, Mainz June 2018

They are expected to be small


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

13/53

Conventions

Main object: Πµνab (q) = i

∫d4xe iq·x〈0|T (jµa (x)jν†b (0)|0〉

jµEM = (2/3)jµU − (1/3)jµD − (1/3)jµS jµQ = qγµq

Continuum/infinite volume:ΠµνEM(q) =

(qµqν − q2gµν

)ΠEM(q2)

Known positive weight functions v ,w and Q2 = −q2:

aµ =

∫ ∞threshold

dq2w(q2)1

πImΠEM(q2)

aµ =

∫ ∞0

dQ2v(Q2)(−Π(Q2) + Π(0)

)Dispersion relation:

Π(q2) = Π(0) +q2

π

∫ ∞threshold

ds1

s(s − q2)

1

πImΠ(s)


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

14/53

First: Scalar QED

Pion loops: finite volume effects suppressed by e−mπL

(if off-shell)

Photon loops have suppression only by powers of 1/L

Dynamical photons: large finite volume effects possible

Scalar QED in usual MS(µ2

ChPT = µ2MS

e, e = 2.71 . . .)

L = (∂µΦ∗ + ieAµΦ∗) (∂µΦ− ieAµΦ)−m20Φ∗Φ− 1

4FµνF

µν

(−λ (φ∗φ)2 not needed)


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

15/53

Finite volume

Finite volume for photons started in earnest withN. Hayakawa, S. Uno, Prog. Theor. Phys. 120(2008)413 [0804.2044]

Get the explicit 1/L behaviour from one-loop integrals:Z. Davoudi, M. Savage, Phys. Rev. D90(2014)054503 [1402,6471]

S. Borsyani et al, Science 347(2015)1452 [1406.4088]

Problem for photon:

∫ddk

1

k2→∫

dk0∑~k

1

(k0)2 − ~k2

Singularity no longer has the kd−1 to soften it

Solution QEDL: drop all modes with ~k = 0 Hayakawa-Uno

We extend the arguments of Davoudi-Savage to two-looporder, use QEDL and a lattice with infinite time extension

We only calculate corrections suppressed by 1/L andpowers, not exponentially suppressed contributions

Below threshold so mesons (pions) are off-shell


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

16/53

Integrals at finite volume

S = 1i2

∫dd l

(2π)dddk

(2π)d1

k2(l2−m2)((k+l−p)2−m2))

do l0, k0 integrals via contour integration~k = 2π

L ~n and expand in 1/L

Write the ~k part as

1

Ld−1

∑~n 6=~0

=

∫dd−1k

(2π)d−1+

1

Ld−1

∑~n 6=~0

−∫

dd−1k

(2π)d−1

In the first term resum the series in 1/L: infinite volumecontribution

Call the quantity in brackets(1/Ld−1

)∆′~n

Define cm = ∆~n1

|~n|mThese are known numerically


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

17/53

Example: mass

∆Vm2

m2= e2

(4c2

16π2mL+

2c1

16π2m2L2+O

(1

L4, e−mL

))Agrees with earlier results

c2 shows up since BOTH propagators can be ’on-shell’

For Πµν below threshold only photon line goes ’on-shell’=⇒ corrections start only at 1/L2

Only true if infinite volume mass is used in the expressionsat LO


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

18/53

Numerical results mass

-0.01

-0.008

-0.006

-0.004

-0.002

0

2 3 4 5 6 7 8 9 10

∆V m

2/m

2

mL

1/L

1/L+1/L2

Infinite volume isδm2

m2= 0.00852

So very large finite volume corrections to electromagnetic part


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

19/53

Twopoint function

Do the k0, l0 integral

expand for large L as explained earlier

Rest can be expressed in terms of

Zij (m2, p2) =

∫dd−1l

(2π)d−1

1

(~l2 + m2)i/2(4~l2 + 4m2 − p2)j

these correspond to one-loop integrals with masses

Ωij ≡ Zij(m2, p2)mi+2j−d+1

Calculate in center of mass frame: p = (p0,~0)

QEDL: no disconnected contribution

tµν spatial part of gµν

Π((p0)2) ≡ −13p2 tµν (Πµν(p)− Πµν(p = 0))

Infinite volume: Π(p2) = Π(p2)


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

20/53

Diagrams

Lowest order:

NLO:

contributions from counterterms and


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

21/53

Results

Π(p2) =c1

πm2L2

(16

3Ω−1,3 −

1

3Ω1,2 −

32

3Ω1,3 −

2

3Ω3,2 +

16

3Ω3,3 −

1

8Ω5,1 + Ω5,2

)+

c0

m3L3

(−

128

3Ω−2,4 +

256

3Ω0,4 −

5

3Ω2,2 +

8

3Ω2,3 −

128

3Ω2,4

−3

8Ω4,1 +

7

6Ω4,2 −

8

3Ω4,3

)+O

(1

L4, e−mL

)

Simplify using relations of the Ωij to

Π(p2) = +c0

m3L3

(−

16

3Ω0,3 −

5

3Ω2,2 +

40

9Ω2,3 −

3

8Ω4,1 +

7

6Ω4,2 +

8

9Ω4,3

)

+O(

1

L4, e−mL

)

the 1/L2 cancels: expected: far away the photon sees nocharge since it is a neutral current: only dipole effect


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

22/53

Scalar QED on a lattice

Correction for mL ∼ 5 less than 1%


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

23/53

Generality

the 1/L2 cancels: expected: far away the photon sees nocharge since it is a neutral current: only dipole effect

The blob (hadrons) is a four-point function ofelectromagnetic currents and has no singularities in theregime needed

The conclusion about 1/L3 is general


Johan Bijnens

Introduction

HVP

From experiment

lattice

sQED

Finite volume

ConclusionsHVP

HLbL

Conclusions

24/53

Conclusions HVP em finite volume

Showed you results for:

Finite volume corrections to the electromagneticcontribution as estimated in scalar QED are small

This is a universal feature: the object under study isneutral and contains no hadronic on-shell propagators

At the level of precision needed for the next level:negligible


Johan Bijnens

Introduction

HVP

HLbL

Status someyears ago

Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop

Quarkloopconstituent

SD: naive

SD: correct

SD: numerical

Conclusions

25/53

HLbL: the main object to calculate

q1 q2↑ q3

q4↑

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

Two main evaluations around 1995


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

26/53

Numerical summary: HLbL

The largest constribution is π0 (and η, η′) exchange/pole

Beware: pole/exchange not quite the sameMost evaluations are in reasonable agreement

The pion loop can be sizable but a large differencebetween the two evaluations

For the pure pion loop part, even larger numbers havebeen proposed by Engel, Ramsey-Musolf

Scalar exchange versus ππ-rescattering

There are other contributions but the sum is smaller thanthe leading pseudo-scalar exchange

BPP: (8.3± 3.2) 10−10 HKS: (8.96± 1.54) 10−10


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

27/53

Numerical summary: HLbL

π0-exchange (in units of 10−10)

BPP: 5.9(0.9)pointlike VMD 5.6Dispersive (1808.04823): 6.26+0.30

−0.25

η, η′ add about 1.5 each

Pion loop (in units of 10−10)

BPP: −1.9(1.3) large error to include other modeldispersive (1702.07347): −2.4(1)

a1: 0.7

Scalars −0.7

Quark loop: here is the main double counting problem 0–2

Short distance constraints might enhance a bit

Everything else

aHLbLµ = 10.5(2.6) 10−10 (error up to 4 10−10 defendable)


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

28/53

General properties

Πµνλσ(q1, q2, q3) =

q3

q2q1

q4

Actually we really needδΠµνλσ(q1, q2, q3)

δq4ρ

∣∣∣∣q4=0


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

29/53

General properties

Πµνλσ(q1, q2, q3):

In general 138 Lorentz structures (136 in 4 dimensions)

Using q1µΠµνλσ = q2νΠµνλσ = q3λΠµνλσ = q4σΠµνλσ = 043 (41) gauge invariant structures

41 helicity amplitudes

Bose symmetry relates some of them

Compare HVP: one function, one variable

General calculation from experiment via dispersionrelations: recent progressColangelo, Hoferichter, Kubis, Procura, Stoffer,. . .

Well defined separation between different contributions

Theory initiative: paper under preparation

One remaining problem: intermediate- and short-distances


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

30/53

General properties

Formalism of Colangelo et al., JHEP 1704 (2017) 161 [1702.07347]

Πµνλσ(q1, q2, q3) =∑

i=1,54

TµνλσΠi

Q2i = −q2

i

Q23 = Q2

1 + Q22 + 2τQ1Q3

aµ =2α3

3π2

∫ ∞0dQ1dQ2Q

31Q

32

∫ 1

−1dτ√

1− τ2∑

i=1,12

TiΠi

The 12 Πi are related to 6 Πi with q4 → 0.

Π1 = Π1 , Π2 = C23

[Π1

], Π3 = Π4 , Π4 = C23

[Π4

],

Π5 = Π7 , Π6 = C12

[C13

[Π7

]], Π7 = C23

[Π7

],

Π8 = C13

[Π17

], Π9 = Π17 , Π10 = Π39 ,

Π11 = −C23

[Π54

], Π12 = Π54 ,


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

31/53

General properties

Colangelo et al.

Dispersion relations can be written for the Πi

Can cleanly separate different contributions:π0, η, η′, ππ,. . .

Well defined separations (other ways of doing it can befine as well)

Caveat: separate contributions must satisfy certain sumrules so the dispersion relations work

White-paper in progress (actually both HVP and HLbL)


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

32/53

π0-pole

Depends on the double off-shell form-factorFπ0γ∗γ∗(Q

21 ,Q

22 )

Old days: VMD

Now: normalization from experiment, form-factors mixtureexperiment/theory/lattice

Short-distance constraints important

Fully dispersive Hoferichter et al., 1808.04823

aπ0-poleµ = 63.0+2.7

−2.1 10−11

Pade/Canterbury approximantsMasjuan, Sanchez-Puertas 1701.05829

aπ0-poleµ = 63.6(2.7) 10−11

Compatible with all other estimates (except MV)


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

33/53

η, η′-pole

Depends on the double off-shell form-factorFη(′)γ∗γ∗(Q

21 ,Q

22 )

Old days: VMD

Now: normalization from experiment, form-factors mixtureexperiment/theory/lattice

Short-distance constraints important

Pade/Canterbury approximantsMasjuan, Sanchez-Puertas 1701.05829

aη-poleµ = 16.3(1.4) 10−11

aη′-poleµ = 14.5(1.9) 10−11

Compatible with all other estimates (except MV)


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

34/53

π-loop

BPP: −1.9 10−10 (ENJL/Full-VMD)

HKS: −0.5 10−10 (HLS model)Since then:

HLS does not satisfy shortdistance JB, Prades 2007

Polarizability expected to give small enhancement−20(5) 10−11 JB, Relefors, 2016

rescattering as parametrized by scalars expected to giveenhancement beyond this −6.8(2.0) 10−11 JB, Relefors, 2016

Dispersive: Pion-box and rescattering has all contributions

Well defined separation

Pion box: needs FV (Q2): the pion form-factorColangelo, Hoferichter, Procura, Stoffer, 1702.07347

aπ-boxµ = −15.8(2) 10−11

Pion S-wave rescatteringaπ-boxµ = −8(1) 10−11

Other rescatterings: work in progress


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

35/53

Other hadrons

There are very many states above 1 GeV

Not that much is known about their photon couplings

Even less (essentially nothing) about form-factors in thecouplings to photons

Many estimatesPauk-Vanderhaeghen,Jegerlehner,BPP,Melnikov-Vainshtein,. . .

Estimates:

Scalars/tensors: ascalars+tensorsµ = −2(3) 10−11

Axials: aaxialsµ = +8(?) 10−11

Melnikov-Vainshtein, 2003 have a much large estimate (∼ 22)


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

36/53

MV short-distance

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 Q21 ≈ Q2

2 Q23 : Leading term in OPE of two vector

currents is proportional to axial current

Πρναβ ∝ PρQ2

1〈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


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

37/53

Short-distance: MV

The pointlike vertex implements shortdistance part, notonly π0-exchange

+Are these part of the quark-loop? JB, Prades, 2007

See also in Dorokhov,Broniowski, Phys.Rev. D78(2008)07301

BPP quarkloop + π0-exchange ≈ MV π0-exchange

Important short-distance constraint but numerical effectprobably overestimated


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

38/53

Quarkloop

Use (constituent) quark loop

Used for full estimates since the beginning (1970s)

Used for short-distance estimates with mass as a cut-offJB, Pallante, Prades, 1996

We recalculated:

q4q1

q2 q1

p

In agreement with quarkloop formulae fromHoferichter, Stoffer, private communication

In agreement with known numerics


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

39/53

Charm loop

Kuhn et al., hep-ph/0301151, Phys.Rev. D68 (2003) 033018

aµ =(απ

)3Nce

4q

[m2µ

M2

(33ζ3 − 19

16

)+ · · ·

]Up to m10

µ /M10 in paper

mc = 1.27 GeV

aHLbLcµ = (3.165− 0.0786− 0.00033 + · · · )× 10−11

= 3.1(1)× 10−11

mb = 4.18 GeV

aHLbLbµ = 1.8× 10−13


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

40/53

Quarkloop: u, d , s

1 1.5 2 2.5 30

5

10

15

20

Qmin [GeV]

aµ×10

11

Π1–12

Π1–2

Π3–12

figure: Hoferichter

Q1,Q2,Q3>Qmin

MQ = 0: fullMQ = 0.3 GeV

dashed

aHLbLQµ =

54× 10−11

MQ provides an infrared cut-off, MQ → 0 divergent

About 12× 10−11 from above 1 GeV for MQ = 0.3 GeV

About 17× 10−11 from above 1 GeV for MQ = 0


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

41/53

Quarkloop

Is it a first term in a systematic OPE?

OPE has been used as constraints on specificcontributions

π0γ∗γ∗ asymptotic behaviourConstraints on many other hadronic formfactorsq2

1 ≈ q22 q2

3 Melnikhov, Vainshtein 2003

These are discussed in the next two talks


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

42/53

Short-distance: first attempt

Πµνλσ = −i∫d4xd4yd4ze−i(q1·x+q2·y+q3·z)

⟨T(jµ(x)jν(y)jλ(z)jσ(0)

)⟩Usual OPE: x , y , z all small

First term in the expansion is the quark-loopno problem with ∂/∂qρ4 and q4 → 0

p in loop ⇒ no singular propagators:

q4q1

q2 q3

p

Next term problems: no loop momentum;

q4 → 0 propagator diverges:

q4q1

q2 q3


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

43/53

Short-distance: correctly

Similar problem in QCD sum rules for electromagneticradii and magnetic moments

Ioffe, Smilga, 1984

For the q4-leg use a constant background field and do theOPE in the presence of that constant background field

Use radial gauge: Aλ4 (w) = 12wµF

µλ

whole calculation is immediately with q4 = 0.

First term is exactly the usual quark loop

(even including quark masses)

⊗“q4”

q1 q2

q3

p


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

44/53

Quarkloop results: massless case

Πqli =

∑q

Nce4q

16π2

∫ 1

0dx

∫ 1−x

0dy Ii (x , y)

D = x(1− x − y)Q21 + y(1− x − y)Q2

2 + xyQ23

I1 =16

D2

(y2 − 3y3 + 2y4 − 4xy2 + 4xy3 − x2y + 2x2y2

− Q21

Q23

(y + x)(1− x − y)2(−1 + 2x + 2y))

I7 =64

D3xy(−1 + 2x)(x + y)(1− x − y)2

I39 =64

D3xy(1− x − y)(y − y2 + x − 3xy + 2xy2 − x2 + 2x2y)

I54 =16

D2

(1

Q21

y2(1− y)(1− 2y)− 1

Q22

x2(1− x)(1− 2x)

)I4 and I17 similar but slightly longer expressions


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

45/53

Short-distance: next term(s)

Do the usual QCD sum rule expansion in terms of vacuumcondensates

There are new condensates, induced by the constantmagnetic field: 〈qσαβq〉 ≡ eqFαβXq

Lattice QCD Bali et al., arXiv:1206.4205

Xu = 40.7± 1.3 MeV,Xd = 39.4± 1.4 MeV,Xs = 53.0± 7.2 MeV

Could have started at order 1/Q, only starts at 1/Q2 viamqXq corrections to the leading quark-loop result

Xq and mq are very small, only a very small correction

Next order: very many condensates contribute, work inprogress


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

46/53

Short-distance

Result derived from:

⊗“q4”

q1 q2

q3

Nc = 3 and one quark

Π1 =mqXqe4q

−4(Q21 + Q2

2 − Q23 )

Q21Q

22Q

43

Π7 = 0

Π4 =mqXqe4q

8

Q21Q

22Q

23

Π17 =mqXqe4q

8

Q21Q

22Q

43

Π54 =mqXqe4q

−4(Q21 − Q2

2 )

Q41Q

42Q

23

Π39 = 0


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

47/53

Short-distance: numerical results

preliminary

Q1,Q2,Q3 ≥ Qmin

mu = md = ms = 0 for quark-loop

mu = md = 5 MeV and ms = 100 MeV for mqXq

Qmin quarkloop muXu + mdXd msXs

1 GeV 17.3× 10−11 5.40× 10−13 8.29× 10−13

2 GeV 4.35× 10−11 3.40× 10−14 5.22× 10−14

Above 1 GeV still 15% of total value of HLbL

Quarkloop goes as 1/Q2min

mqXq goes as 1/Q4min

(dimensional analysis)

Naive suppression is mqXq/Q2min ∼ 2× 10−3

Observed is roughly that


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

48/53

Numerical results

10-14

10-13

10-12

10-11

10-10

1 1.5 2 2.5 3 3.5 4

aµ

HL

bL

Qmin

Quark loop mass correctionQuark loop

Condensate


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

49/53

Short-distance: 1/Q2min

Can we understand scaling with Qmin?

aµ =2α3

3π2

∫ ∞0dQ1dQ2Q

31Q

32

∫ 1

−1dτ√

1− τ2∑

i=1,12

TiΠi

Do Qi → λQi

overall factor goes as λ8

Quark loop has no scale thus Πi scale with their dimensionΠ1, Π4 ∼ λ−4, Π7, Π17, Π39, Π54 ∼ λ−6

=⇒ Π1,...,4 ∼ λ−4 Π5,...,12 ∼ λ−6

Expand the Ti for Qi mµ: T1 ∼ m4µ, Ti 6=1 ∼ m2

µ

T1 ∼ λ−8, T2,3,4 ∼ λ−6, T5,...,12 ∼ λ−4

Put all together: quark-loop scales as aSD qlµ ∼ λ−2

mqXq adds an overall factor =⇒ aSDXqµ ∼ λ−4


Johan Bijnens

Introduction

HVP

HLbL


Generalproperties

π0-pole

η, η′-pole

π-loop

Other hadrons

Short-distance:MV

Quarkloop

Charm loop


SD: naive

SD: correct

SD: numerical

Conclusions

50/53

Short distance HLbL Conclusions

We have shown that the quarkloop really is the first termof a proper OPE expansion for the HLbL

We have calculated the next term which is suppressed byquark masses and a small Xq: negligible

The next term contains both the usual vacuum and a largenumber of induced condensates but will not be suppressedby small quark masses

Why do this: matching of the sum over hadroniccontributions to the expected short distance domain

Finding the onset of the asymptotic domain

Still to be worked out: overlap with what is in the otherparts


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

51/53

Putting together

LO-HVP:

Paris (Davier et al.)aLO HVPµ = 6939± 10± 34± 16± 1ψ ± 7QCD 10−11

Liverpool (Keshavarzi et al) aLO HVPµ = 6938(24) 10−11

HO-HVP: aHO HVPµ = −85.8(4) 10−11

HLbL: put all numbers together

π0, η, η′: 93.8(4.0) 10−11

π-loop: −24(1) 10−11

Kaon-loop: −0.5 10−11

HO-HLbL Colangelo et al., 2014 3(2) 10−11

Charm loop: aHLbLcµ = 3.1(1) −11

Other hadronic: 6(?) 10−11

Short-distance: 10(10) 10−11

Sum: aHLbLµ = 91(17?) 10−11


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

52/53

Putting together

So what is the status?

aexpµ − aQEDµ − aEWµ = 7218(63) 10−11

LO-HVP: aLO HVPµ = 6939(40?) 10−11

Note: Babar/KLOE discrepancy

HO-HVP: aHO HVPµ = −85.8(4) 10−11

HLbL: aHLbLµ = 91(17?) 10−11

Note: error from overlap and guessing putting together

Sum: 6952(44?) 10−11

Difference: ∆aµ = 274(77?) 10−11


Johan Bijnens

Introduction

HVP

HLbL

Conclusions

53/53

Putting together

Difference: ∆aµ = 274(77?) 10−11

HVP: Babar/KLOE: Belle II, Novosibirsk

Lattice HVP: not competitive now but will get there

HLbL: improvements have happened

Lattice HLbL: numbers compatible

Theory: errors are slowly going downno large changes in numbers expected, in fact numbersquite stable since 2000

Experiment: FNAL is running;first result expected late this year or early next year

Thank you for listening

analytical results for hadronic contributions to the muon...

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