Hadronic light-by-light scatering from latice QCD Jeremy Green in - - PowerPoint PPT Presentation

Hadronic light-by-light scatering from latice QCD

Jeremy Green in collaboration with Nils Asmussen, Oleksii Gryniuk, Georg von Hippel, Harvey Meyer, Andreas Nyffeler, Vladimir Pascalutsa, Hartmut Witig

Institut für Kernphysik, Johannes Gutenberg-Universität Mainz

Humboldt Universität – NIC, DESY Zeuthen Join Latice Seminar June 27, 2016

Outline

• 1. Introduction
• 2. Latice four-point function
• 3. Light-by-light scatering amplitude
• 4. Strategy for g − 2
• 5. Summary and outlook

Some of these results were published in

• Phys. Rev. Let. 115, 222003 (2015) [1507.01577]

PoS(Latice 2015)109 [1510.08384]

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 2 / 30

Precision low-energy physics

◮ Direct way to search for new physics: try to create new particles at

high energy in a collider. E.g., Higgs boson at LHC.

◮ Indirect way: measure a low-energy observable very precisely, and look

for small deviations from a similarly precise Standard Model

• prediction. This can impose strong constraints on new physics that are

complementary to direct searches.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 3 / 30

Magnetic moment of the electron

In an external magnetic field, a particle will have potential energy U = − µ · B, where µ is its magnetic moment. An electron’s magnetic moment is given by

• µ = ge

−e 2me

• S.

A classical rotating charged body has gyromagnetic ratio g = 1, whereas the Dirac equation predicts g = 2. QED and the Standard Model produce deviations from g = 2: the anomalous magnetic moment, ae ≡ ge − 2 2 = + . . . = α 2π + . . .

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 4 / 30

Anomalous magnetic moment of the electron

• T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, Phys. Rev. Let. 109, 111807 (2012)

aSM

e

=

• n

α π n a(2n)

e

+ a(had)

e

+ a(EW)

e

= α 2π − (0.328 . . . ) α π 2 + (1.181 . . . ) α π 3 − (1.910 . . . ) α π 4 + (9.16 . . . ) α π 5 + 1.68(2) × 10−12 + 0.0297(5) × 10−12 Using α from atomic physics experiments yields aSM

e

= 1 159 652 181.78(6)(4)(2)(77) × 10−12 [0.67 ppb], where the dominant uncertainty comes from α. This agrees well with the experimental value, aexpt

e

= 1 159 652 180.73(28) × 10−12 [0.24 ppb].

• D. Hanneke, S. Fogwell, G. Gabrielse, Phys. Rev. Let. 100, 120801 (2008)

This can be reversed: use the experiment as a determination of α, which results in the current most precise value of the fine structure constant.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 5 / 30

Sensitivity to heavy particles

◮ The Standard Model prediction for ae is dominated by QED diagrams

containing only photons and electrons.

◮ Loops with heavier particles (µ, τ, hadrons) contribute only ∼ 5 × 10−12. ◮ Generically the contribution from a heavy particle with mass M is

suppressed by ( me

M )2. ◮ In the muon g − 2, contributions from heavy particles will therefore be

enhanced by ( mµ

me )2 ≈ 40 000, relative to their contributions to ae.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 6 / 30

Muon g − 2: experiment

BNL E821: G. W. Bennet et al. (Muon g − 2 Collaboration), Phys. Rev. D 73, 072003 (2006)

◮ Polarized muons injected into a 14 m

diameter storage ring.

◮ The muon’s spin precesses relative to its

velocity:

• ωa = e

mµ

• aµ

B −

• aµ −

1 γ 2 − 1

• v ×

E

• .

◮ “Magic” muon energy E ≈ 3.098 GeV

used to eliminate the dependence on E.

◮ The electron produced in muon decay

has direction correlated with the muon spin. aµ = 116 592 089(63) × 10−11 [0.54 ppm] BNL storage ring moved to Fermilab for new muon g − 2 experiment: E989. Goal: reduce uncertainty by a factor of four. New experiment using ultra-cold muons also planned by J-PARC E34.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 7 / 30

Muon g − 2: theory

• T. Blum et al. (US Physics “Snowmass” Self Study), 1311.2198
• F. Jegerlehner, A. Nyffeler, Phys. Rept. 477 (2009) 1–110

QED O(α) 116 140 973.32(8) = 0.5(α/π) QED O(α2) 413 217.63(1) = 0.765 857 425(17)(α/π)2 QED O(α3) 30 141.90(0) = 24.050 509 96(32)(α/π)3 QED O(α4) 381.01(2) = 130.8796(63)(α/π)4 QED O(α5) 5.09(1) = 753.29(1.04)(α/π)5 QED combined 116 584 718.95(8)

• T. Aoyama et al., Phys. Rev. Let. 109, 111808 (2012)

Electroweak 154(1)

• J. P. Miller et al., Ann. Rev. Nucl. Part. Sci. 62 (2012) 237–264

HVP (LO) 6949(43)

• K. Hagiwara et al., J. Phys. G 38 (2011) 085003

HVP (HO) −98.4(7) HLbL 105(26)

“Glasgow consensus” or 116(39)

Standard Model 116 591 828(50) ×10−11

Jegerlehner + Nyffeler

∆aµ (expt − SM) 261(78) ×10−11

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 8 / 30

Muon g − 2: theory uncertainty

The two dominant sources of uncertainty are hadronic effects: Hadronic vacuum polarization: aHVP,LO

µ

= 6949(43) × 10−11.

◮ Determined using experimental data on cross section for

e+e− → hadrons.

◮ Very active field for latice QCD calculations working

toward an ab initio prediction with competitive uncertainty. Hadronic light-by-light scatering: aHLbL

µ

= 105(26) or 116(39) × 10−11.

◮ Determined using models that include meson

exchange terms, charged meson loops, etc.

◮ Could benefit significantly with reliable input from the

latice.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 9 / 30

π 0 contribution to HLbL scatering

About 2/3 of theory prediction for aHLbL

µ

comes from π 0 exchange diagrams, which dominate at long distances. Large contributions also come from η, η′.

+ +

Their contribution to the four-point function: ΠE,π 0

µ1µ2µ3µ4(p4; p1,p2)

= −p1αp2βp3σ p4τ

• F12ϵµ1µ2α β F34ϵµ3µ4στ

(p1 + p2)2 + m2

π

+ F13ϵµ1µ3ασ F24ϵµ2µ4βτ (p1 + p3)2 + m2

π

+ F14ϵµ1µ4ατ F23ϵµ2µ3βσ (p2 + p3)2 + m2

π

• ,

where p3 = −(p1 + p2 + p4) and Fij = F (p2

i ,p2 j ) is the π 0γ ∗γ ∗ form factor.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 10 / 30

Light-by-light scatering

Before computing aHLbL

µ

, start by studying light-by-light scatering by itself. This has much more information than just aHLbL

µ

. We can:

◮ Compare against phenomenology. ◮ Test models used to compute aHLbL µ

.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 11 / 30

Latice four-point function

Directly compute four-point function of vector currents

◮ Use one local current ZVJl µ at the source point. ◮ Use three conserved currents Jc µ.

In position space: Πpos

µ1µ2µ3µ4(x1,x2,0,x4) =

• ZVJl

µ3(0)[Jc µ1(x1)Jc µ2(x2)Jc µ4(x4)

+ δµ1µ2δx1x2Tµ1(x1)Jc

µ4(x4)

+ δµ1µ4δx1x4Tµ4(x4)Jc

µ2(x2)

+ δµ2µ4δx2x4Tµ4(x4)Jc

µ1(x1)

+ δµ1µ4δµ2µ4δx1x4δx2x4Jc

µ4(x4)]

• ,

where Tµ (x) is a “tadpole” contact operator. This satisfies the conserved-current relations, ∆x1

µ1Πpos µ1µ2µ3µ4 = ∆x2 µ2Πpos µ1µ2µ3µ4 = ∆x4 µ4Πpos µ1µ2µ3µ4 = 0.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 12 / 30

Qark contractions

Compute only the fully-connected contractions, with fixed kernels summed

• ver x1 and x2:

Πpos′

µ1µ2µ3µ4(x4; f1,f2) =

• x1,x2

f1(x1)f2(x2)Πpos

µ1µ2µ3µ4(x1,x2,0,x4)

1

X X2 X4

1

X X2 X4

1

X X2 X4

Generically, need the following propagators:

◮ 1 point-source propagator from x3 = 0 ◮ 8 sequential propagators through x1, for each µ1 and f1 or f ∗ 1 ◮ 8 sequential propagators through x2 ◮ 32 double-sequential propagators through x1 and x2, for each (µ1,µ2)

and (f1,f2) or (f ∗

1 ,f ∗ 2 )

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 13 / 30

Kinematical setup

Obtain momentum-space Euclidean four-point function using plane waves: ΠE

µ1µ2µ3µ4(p4; p1,p2) =

• x4

e−ip4·x4Πpos′

µ1µ2µ3µ4(x4; f1,f2)

• fa(x)=e−ipa·x .

Thus, we can efficiently fix p1,2 and choose arbitrary p4.

◮ Full 4-point tensor is very complicated: it can be decomposed into 41

scalar functions of 6 kinematic invariants.

◮ Forward case is simpler:

Q1 ≡ p2 = −p1, Q2 ≡ p4. Then there are 8 scalar functions that depend on 3 kinematic invariants.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 14 / 30

Latice ensembles

Use CLS ensembles: Nf = 2 O(a)-improved Wilson, with a = 0.063 fm.

• 1. mπ = 451 MeV, 64 × 323
• 2. mπ = 324 MeV, 96 × 483
• 3. mπ = 277 MeV, 96 × 483
• 4. mπ = 200 MeV, 128 × 643

Keep only u and d quarks in the electromagnetic current, i.e., Jl

µ = 2

3¯ uγµu − 1 3 ¯ dγµd. Focus on the forward scatering case, but also try to examine some generic kinematics.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 15 / 30

Forward LbL amplitude

Take the amplitude for forward scatering of transversely polarized virtual photons, MTT (−Q2

1,−Q2 2,ν) = e4

4 Rµ1µ2Rµ3µ4ΠE

µ1µ2µ3µ4(−Q2; −Q1,Q1),

where ν = −Q1 · Q2 and Rµν projects onto the plane orthogonal to Q1,Q2. A subtracted dispersion relation at fixed spacelike Q2

1,Q2 2 relates this to the γ ∗γ ∗ → hadrons cross

sections σ0,2:

ν

MTT (q2

1,q2 2,ν)−MTT (q2 1,q2 2,0) = 2ν2

π ∞

ν0

dν ′

• ν ′2 − q2

1q2 2

ν ′(ν ′2 − ν2 − iϵ) [σ0(ν ′) + σ2(ν ′)] This is model-independent and will allow for systematically improvable comparisons between latice and experiment.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 16 / 30

Model for σ (γ ∗γ ∗ → hadrons)

• V. Pascalutsa, V. Pauk, M. Vanderhaeghen, Phys. Rev. D 85 (2012) 116001

Include single mesons and π +π − final states: σ0 + σ2 =

• M

σ (γ ∗γ ∗ → M) + σ (γ ∗γ ∗ → π +π −) Mesons:

◮ pseudoscalar (π 0, η′) ◮ scalar (a0, f0) ◮ axial vector (f1) ◮ tensor (a2, f2)

σ (γ ∗γ ∗ → M) depends on the meson’s:

◮ mass m and width Γ ◮ two-photon decay width Γ γγ ◮ two-photon transition form factor

F (q2

1,q2 2)

assume F (q2

1,q2 2) = F (q2 1,0)F (0,q2 2)/F (0,0)

Use scalar QED dressed with form factors for σ (γ ∗γ ∗ → π +π −).

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 17 / 30

MTT: dependence on ν and Q2

2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Q2

2 (GeV2)

2 4 6 8 10 MTT(−Q2

1, −Q2 2, ν) − MTT(−Q2 1, −Q2 2, 0)

×10−5 mπ = 324 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2)

For scalar, tensor mesons there is no data from expt; we use F (q2,0) = F (0,q2) = 1 1 − q2/Λ2 with Λ set by hand to 1.6 GeV Changing Λ by ±0.4 GeV adjusts curves by up to ±50%. Points: latice data. Curves: dispersion relation + model for cross section.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 18 / 30

MTT: dependence on ν and mπ

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ν (GeV2) 5 10 15 MTT(−Q2

1, −Q2 2, ν) − MTT(−Q2 1, −Q2 2, 0)

×10−5

Q2

1 = Q2 2 = 0.377 GeV2

mπ (MeV) 277 324 451 200

Points: latice data. Curves: dispersion relation + model for cross section. In increasing order:

◮ π 0 ◮ π 0 + η′ ◮ full model ◮ full model + high-energy

σ (γγ → hadrons) at physical mπ

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 19 / 30

Eight forward-scatering amplitudes

• V. M. Budnev, V. L. Chernyak and I. F. Ginzburg, Nucl. Phys. B 34, 470 (1971)
• V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, Phys. Rept. 15, 181 (1975)

Mµ′ν ′,µν (q1,q2) = Rµµ′Rνν ′MTT + 1

2

• RµνRµ′ν ′ + Rµν ′Rµ′ν − Rµµ′Rνν ′

Mτ

TT

+

• RµνRµ′ν ′ − Rµν ′Rµ′ν

Ma

TT

+ Rµµ′kν

2 kν ′ 2 MTL + kµ 1 kµ′ 1 Rνν ′MLT + kµ 1 kµ′ 1 kν 2 kν ′ 2 MLL

−

• Rµνkµ′

1 kν ′ 2 + Rµν ′kµ′ 1 kν 2 + (µν ↔ µ′ν ′)

• Ma

TL

−

• Rµνkµ′

1 kν ′ 2 − Rµν ′kµ′ 1 kν 2 + (µν ↔ µ′ν ′)

• Mτ

TL,

where R is a projector onto transverse polarizations and ka are longitudinal polarization vectors.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 20 / 30

Eight forward-scatering amplitudes: data

1 2 3 4 Q2

2 (GeV2)

−5 −4 −3 −2 −1 1 2 3 4 5 6 7 MTT(−Q2

1, −Q2 2, ν)

×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) −1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Mτ TT(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) −6 −5 −4 −3 −2 −1 1 Ma TT(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) −1 1 2 3 4 5 6 7 8 9 10 11 12 13 MLL(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) −6 −5 −4 −3 −2 −1 1 MTL(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) −4 −3 −2 −1 1 MLT(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) −6 −5 −4 −3 −2 −1 1 Mτ TL(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 1 2 3 4 Q2 2 (GeV2) 1 2 3 4 5 6 Ma TL(−Q2 1, −Q2 2, ν) ×10−5

mπ = 324 MeV, Q2

1 = 0.377 GeV2 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 21 / 30

Eight forward-scatering amplitudes: model analysis

• V. Pascalutsa, V. Pauk, M. Vanderhaeghen, Phys. Rev. D 85 (2012) 116001

Each amplitude is related by a dispersion relation to a (interference) cross section for γ ∗γ ∗ → hadrons. E.g.: Mτ

TT (q2 1,q2 2,ν) − Mτ TT (q2 1,q2 2,0) = 4ν2

π ∞

ν0

dν ′

• ν ′2 − q2

1q2 2

ν ′(ν ′2 − ν2 − iϵ) [σ(ν ′) − σ⊥(ν ′)] Ma

TT (q2 1,q2 2,ν) = 2ν

π ∞

ν0

dν ′

• ν ′2 − q2

1q2 2

ν ′2 − ν2 − iϵ [σ0(ν ′) − σ2(ν ′)] Work is underway to extend the model to all eight amplitudes; these will impose much stronger constraints on the model.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 22 / 30

General kinematics case

To study off-forward kinematics, we fix p2

1 = p2 2 = (p1 + p2)2 = 0.33 GeV2

and consider contractions of ΠE

µ1µ2µ3µ4(p4; p1,p2) with two different tensors:

• 1. δµ1µ2δµ3µ4 yields π 0 contribution

− 2 (p1 · p2)(p3 · p4) − (p1 · p4)(p2 · p3) (p1 + p3)2 + m2

π

F (p2

1,p2 3)F (p2 2,p2 4)

+ (p1 · p2)(p3 · p4) − (p1 · p3)(p2 · p4) (p2 + p3)2 + m2

π

F (p2

1,p2 4)F (p2 2,p2 3)

• ,

where F (0,0) = −1/(4π 2Fπ ) (Wess-Zumino-Witen) and we use vector meson dominance for dependence on p2.

• 2. δµ1µ2δµ3µ4 + δµ1µ3δµ2µ4 + δµ1µ4δµ2µ3, which is totally symmetric and thus

has no π 0 contribution. We also fix p2

3 = p2 4 to two different values and plot versus the one

remaining kinematic variable.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 23 / 30

Off-forward kinematics

0.0 0.5 1.0 1.5 2.0 |(P2 + P4)2 − (P1 + P4)2| (GeV2) 0.00 0.02 0.04 0.06 0.08 0.10 (δµ1µ2δµ3µ4 + λδµ1µ3δµ2µ4 + λδµ1µ4δµ2µ3)ΠE

µ1µ2µ3µ4

mπ = 324 MeV, P 2

1 = P 2 2 = (P1 + P2)2 = 0.33 GeV2

(λ, P 2

3 = P 2 4 [GeV2])

(0, 0.82) (1, 0.82) (0, 0.49) (1, 0.49)

Squares: contraction without π 0 contribution. Circles: contraction containing π 0 contribution. Curves: π 0 contribution assuming model for F (p2

1,p2 2).

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 24 / 30

Qark contractions: relative importance

Consider the charge factors, with qu = 2/3, qd = qs = −1/3: diagram factor Nf = 2 Nf = 3 (4)

• f q4

f

17/81 18/81 (2,2) (

f q2 f )2

25/81 36/81 (3,1) (

f q3 f )( f qf )

7/81 (2,1,1) (

f q2 f )( f qf )2

5/81 (1,1,1,1) (

f qf )4

1/81 It is also argued that with only the fully-connected diagrams, the η′ falsely appears with the mass of the pion, so that effectively the π 0 contribution is enhanced by a factor of 34/9. (J Bijnens, EPJ Web Conf. 118 (2016) 01002 [1510.05796])

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 25 / 30

(2,2) quark-disconnected contractions

1

X X2 X4

1

X X2 X4

1

X X2 X4

Evaluate one of the quark loops using stochastic estimation. Need the following propagators:

◮ 1 point-source propagator from x3 = 0 ◮ 1 noise-source propagator ◮ 1 noise-momentum-source propagator

Preliminary results for unsubtracted MTT: Large finite-volume effect!

1 2 3 4 Q2

2 (GeV2)

−4 −3 −2 −1 1 2 3 4 5 6 7 MTT(−Q2

1, −Q2 2, ν)

×10−5

mπ = 451 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

fully-connected

1 2 3 4 Q2

2 (GeV2)

−1 1 2 3 4 5 MTT(−Q2

1, −Q2 2, ν)

×10−5

mπ = 451 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

(2,2)-disconnected

1 2 3 4 Q2

2 (GeV2)

−2 −1 1 2 3 4 5 6 7 8 9 10 MTT(−Q2

1, −Q2 2, ν)

×10−5

mπ = 451 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

sum

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 26 / 30

(2,2) quark-disconnected contractions

1

X X2 X4

1

X X2 X4

1

X X2 X4

Evaluate one of the quark loops using stochastic estimation. Need the following propagators:

◮ 1 point-source propagator from x3 = 0 ◮ 1 noise-source propagator ◮ 1 noise-momentum-source propagator

Preliminary results for subtracted MTT:

1 2 3 4 Q2

2 (GeV2)

1 2 3 4 5 6 7 8 9 10 MTT(−Q2

1, −Q2 2, ν) − MTT(−Q2 1, −Q2 2, 0)

×10−5

mπ = 451 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

fully-connected

1 2 3 4 Q2

2 (GeV2)

−1 1 MTT(−Q2

1, −Q2 2, ν) − MTT(−Q2 1, −Q2 2, 0)

×10−5

mπ = 451 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

(2,2)-disconnected

1 2 3 4 Q2

2 (GeV2)

1 2 3 4 5 6 7 8 9 10 MTT(−Q2

1, −Q2 2, ν) − MTT(−Q2 1, −Q2 2, 0)

×10−5

mπ = 451 MeV, Q2

1 = 0.377 GeV2

0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2) 0.0 0.5 1.0 ν (GeV2)

sum

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 26 / 30

Hadronic contributions to the muon g − 2

O(α2): Leading order hadronic vacuum polarization. O(α3): Higher-order contributions from leading order HVP. Leading-order contribution from O(α) correction to HVP. The hadronic ingredient is forward light-by-light scatering. Hadronic light-by-light.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 27 / 30

Strategy for muon g − 2: kernel

In Euclidean space, give muon momentum p = imˆ ϵ, ˆ ϵ2 = 1. Apply QED Feynman rules and isolate F2(0); obtain aHLbL

µ

=

• d4x d4y L[ρ,σ];µνλ(ˆ

ϵ,x,y)i ˆ Πρ;µνλσ (x,y), where ˆ Πρ;µνλσ (x1,x2) =

• d4x4 (ix4)ρ
• Jµ (x1)Jν (x2)Jλ(0)Jσ (x4)
• .

The integrand for aµ is a scalar function of 5 invariants: x2, y2, x · y, x · ϵ, and y · ϵ, so 3 of the 8 dimensions in the integral are trivial. Five dimensions is still too many. Result is independent of ˆ ϵ, so we can eliminate it by averaging in the integrand: L(ˆ ϵ,x,y) → ¯ L(x,y) ≡ L(ˆ ϵ,x,y)

ˆ ϵ

Then the integrand depends only on x2, y2, and x · y.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 28 / 30

Strategy for muon g − 2: latice

aHLbL

µ

=

• d4x
• d4y d4z ¯

L[ρ,σ];µνλ(x,y)(−z)ρ

• Jµ (x)Jν (y)Jλ(0)Jσ (z)
• = 2π 2

∞ x3dx

• d4y d4z ¯

L[ρ,σ];µνλ(x,y)(−z)ρ

• Jµ (x)Jν (y)Jλ(0)Jσ (z)
• .

Evaluate the y and z integrals in the following way:

• 1. Fix local currents at the origin and x, and compute point-source

propagators.

• 2. Evaluate the integral over z using sequential propagators.
• 3. Contract with ¯

L[ρ,σ];µνλ(x,y) and sum over y. The above has similar cost to evaluating scatering amplitudes at fixed p1,p2. Do this several times to perform the one-dimensional integral over |x|.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 29 / 30

Summary and outlook

◮ The contribution from fully-connected four-point function to the

light-by-light scatering amplitude can be efficiently evaluated if two of the three momenta are fixed.

◮ Forward-scatering case is related to σ (γ ∗γ ∗ → hadrons); latice is

consistent with phenomenology, within the later’s large uncertainty.

◮ For typical Euclidean kinematics the π 0 contribution is not dominant. ◮ A strategy is in place for computing the leading-order HLbL

contribution to the muon g − 2. Work is ongoing to evaluate the kernel ¯ L[ρ,σ];µνλ(x,y) and integrate it into the latice calculation.

◮ Phenomenology indicates the π 0 contribution is dominant for g − 2;

reaching this regime (physical mπ , large volumes) may be challenging

• n the latice.

Jeremy Green (Mainz) Hadronic light-by-light scatering from latice QCD DESY Latice Seminar 30 / 30