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Hadronic light-by-light scattering in the muon anomalous magnetic - - PowerPoint PPT Presentation

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Hadronic light-by-light scattering in the muon anomalous magnetic moment on the lattice Nils Asmussen in Collaboration with Antoine G erardin, Harvey Meyer, Andreas Nyffeler University of Southampton 30 October 2018 Nils Asmussen (SOTON)

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SLIDE 1

Hadronic light-by-light scattering in the muon anomalous magnetic moment on the lattice

Nils Asmussen in Collaboration with Antoine G´ erardin, Harvey Meyer, Andreas Nyffeler

University of Southampton

30 October 2018

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 1 / 29

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SLIDE 2

Gyromagnetic Moment (History)

gyromagnetic moment: µ = g e 2mS

1924

Stern-Gerlach experiment

  • bserved µ

1928

Dirac theory: g = 2

1947

g ≈ 2 × (1 + 0.00118(3)) Foley and Kush

1948

g = 2 × (1 + α

2π)

≈ 2 × (1 + 0.001161) Schwinger

Where are we today?

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 2 / 29

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SLIDE 3

Hadronic Light-by-Light Contribution

anomalous magnetic moment aµ = gµ−2

2

contribution aµ[10−10] reference QED 11 658 471.895 ± 0.008 Aoyama et al ’12 HVP LO 693.1 ± 3.4 Davier et al ’17 HVP NLO −9.84 ± 0.07 Hagiwara et al ’11 HVP NNLO 1.24 ± 0.01 Kurz et al ’14 HLBL LO 10.5 ± 2.6 Prades et al ’09 HLBL NLO 0.3 ± 0.2 Colangelo et al ’14 EW 15.36 ± 0.10 Gnendiger et al ’13 total 11 659 182.3 ± 4.3 Davier et al ’17 experimental 11 659 208.9 ± 6.3 Bennett et al ’06

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 3 / 29

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SLIDE 4

Anomalous Magnetic Moment of the Muon

≈ 3 to 4 standard deviations discrepancy between aexp

µ

and atheo

µ

→ new physics?

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 4 / 29

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SLIDE 5

Anomalous Magnetic Moment of the Muon

≈ 3 to 4 standard deviations discrepancy between aexp

µ

and atheo

µ

→ new physics?

reduce uncertainties

experiment theory for HLbL J-PARC phenomenology lattice QCD Fermilab reduce model uncertainties model independent estimates for dominant contribution Blum et al. (’05,. . . )’15,. . . ,’17 (π0 , η , η′ ; ππ) Mainz lattice group using experimental input Colangelo et al. ’14,. . . ,’17 Pauk and Vanderhaeghen ’14

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 4 / 29

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SLIDE 6

Euclidean position-space approach to aHLbL

µ

x y 0 z

master formula

aHLbL

µ

= me6 3

  • d4y

d4x ¯ L[ρ,σ];µνλ(x, y)

  • QED

i Πρ;µνλσ(x, y)

  • QCD
  • .

i Πρ;µνλσ(x, y) = −

  • d4z zρ
  • jµ(x) jν(y) jσ(z) jλ(0)
  • .

¯ L[ρ,σ];µνλ(x, y) computed in the continuum & infinite-volume no power-law finite-volume effects from the photons manifest Lorentz covariance

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 5 / 29

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SLIDE 7

Outline

1

Tests of the QED Kernel

2

Tests of the Lattice Gauge Theory Code

3

Lattice QCD

4

Conclusion

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 6 / 29

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SLIDE 8

Stages of the Computation

tests of the QED kernel

continuum and infinite volume π0 pole and lepton loop test different choices for the QED kernel

tests of the lattice gauge theory code

Lattice QED compare to lepton loop results

Lattice QCD

first results for the fully connected contribution study pion mass dependence and discretisation/finite volume effects

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 7 / 29

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SLIDE 9

1

Tests of the QED Kernel

2

Tests of the Lattice Gauge Theory Code

3

Lattice QCD

4

Conclusion

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 8 / 29

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SLIDE 10

Tests in Continuum, Infinite Volume

master formula

aHLbL

µ

= me6 3 8π3 ∞ d|y||y|3 ∞ d|x||x|3 π dβ sin2β ¯ L[ρ,σ];µνλ(x, y) i Πρ;µνλσ(x, y)

  • .

i Πρ;µνλσ(x, y) = −

  • d4z zρ
  • jµ(x) jν(y) jσ(z) jλ(0)
  • .
  • d|x|,
  • d|y| and
  • dβ evaluated numerically
  • d4z evaluated (semi-)analytically

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 9 / 29

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SLIDE 11

Contribution of the π0 to aHLbL

µ

(Model!)

1 2 3 4 2 4 |y|/fm f(|y|) × 1010fm mπ = 300 MeV mπ = 600 MeV mπ = 900 MeV 1 2 3 4 1 2 3 |y|max/fm aHlbl

µ

(|y|max) × 1010 mπ = 300 MeV mπ = 600 MeV mπ = 900 MeV

dashed line = result from momentum-space integration we reproduce the known result contribution is perhaps surprisingly long-range integrand peaked at short distances

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 10 / 29

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SLIDE 12

Lepton loop integrand contribution to aHLbL

µ

2 4 6 8 10 2 4 6 8 |y|/fm f(|y|) × 109fm ml = mµ/2 ml = mµ ml = 2mµ

we reproduce the known result contribution is long-range integrand sharply peaked at short distances

The QED kernel is correct

we reproduce the π0-pole in VMD model we reproduce the lepton loop

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 11 / 29

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SLIDE 13

What next?

achievements

method for aHLbL

µ

  • n the lattice

verified the QED kernel learned about the integrand

challenges in the view of lattice computations

contributions are quite long range integrand peaked at small distances

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 12 / 29

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SLIDE 14

What next?

achievements

method for aHLbL

µ

  • n the lattice

verified the QED kernel learned about the integrand

challenges in the view of lattice computations

contributions are quite long range integrand peaked at small distances

a way to improve

do subtractions on the kernel (first proposed by Blum et al. ’17) exploit

  • x i ˆ

Π(x, y) =

  • y i ˆ

Π(x, y) = 0 example:

L(0) = ¯ L[ρ,σ];µνλ(x, y) L(1) = ¯ L[ρ,σ];µνλ(x, y) − 1

2 ¯

L[ρ,σ];µνλ(x, x) − 1

2 ¯

L[ρ,σ];µνλ(y, y) aHLbL

µ

= me6

3

  • x,y L(0)(x, y)i ˆ

Π(x, y) = me6

3

  • x,y L(1)(x, y)i ˆ

Π(x, y)

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 12 / 29

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SLIDE 15

Continuum, Infinite Volume

master formula

aHLbL

µ

= me6 3 8π3 ∞ d|y||y|3 ∞ d|x||x|3 π dβ sin2β ¯ L[ρ,σ];µνλ(x, y) i Πρ;µνλσ(x, y)

  • .

subtractions on the kernel

we try (short notation): L(0) = ¯ L(x, y) (standard kernel) L(1) = ¯ L(x, y) − 1

2 ¯

L(x, x) − 1

2 ¯

L(y, y) L(2) = ¯ L(x, y) − ¯ L(0, y) − ¯ L(x, 0) L(3) = ¯ L(x, y)−¯ L(0, y)−¯ L(x, x)+¯ L(0, x) L(0)(0, 0) = 0 L(1)(x, x) = 0 L(2)(0, y) = L(2)(x, 0) = 0 L(3)(x, x) = L(3)(0, y) = 0

y integrand lepton loop ml = mµ

−0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 f(|y|) × 108 fm |y| fm L(0) L(1) L(2) L(3)

with all kernels L(0,1,2,3) we can reproduce the known result we expect L(2,3) to be advantageous on the Lattice

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 13 / 29

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SLIDE 16

1

Tests of the QED Kernel

2

Tests of the Lattice Gauge Theory Code

3

Lattice QCD

4

Conclusion

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 14 / 29

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SLIDE 17

Lattice QED Computation

master formula

aHLbL

µ

= me6 3 2π2

|y|

a|y||y|3 a4

x∈Λ

¯ L[ρ,σ];µνλ(x, y) i Πρ;µνλσ(x, y)

  • .

i Πρ;µνλσ(x, y) = −a4

z∈Λ

zρ

  • jµ(x) jν(y) jσ(z) jλ(0)
  • .

i ˆ Π in Lattice QED

goal

reproduce known lepton loop result validate Lattice QCD code focus on standard kernel L(0) and subtracted kernel L(2)

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 15 / 29

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SLIDE 18

Lattice QED Computation with Wilson Fermions

lattice gauge theory

O = 1 Z

  • D[U]e−SG [U]D[ψ, ¯

ψ]e−SF [ψ, ¯

ψ,U]O[ψ, ¯

ψ, U] SG[U] =β

  • n∈Λ
  • µ<ν

Re Tr[1 − Uµν(n)] SF[ψ, ¯ ψ, U] = Wilson fermions local vector currents: jl

λ(x) = ¯

qxγλqx conserved vector currents: jc

λ(x) = 1 2

  • ¯

qx+ˆ

λ(γλ + 1)U† λ,xqx + ¯

qx(γλ − 1)Uλ,xqx+ˆ

λ

  • Nils Asmussen (SOTON)

HLbL g-2 on the lattice 30 October 2018 16 / 29

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SLIDE 19

Lattice QED Computation with Wilson Fermions

− →

lepton loop

lattice gauge theory

O = 1 Z

  • ✘✘✘✘✘

✘ ❳❳❳❳❳ ❳ D[U]e−SG [U]D[ψ, ¯ ψ]e−SF [ψ, ¯

ψ,U]O[ψ, ¯

ψ, U] SG[U] =β

  • n∈Λ
  • µ<ν

Re Tr[1 −✟✟✟ ✟ ✯1 Uµν(n)] = 0 SF[ψ, ¯ ψ, U] = Wilson fermions QED leading order local vector currents: jl

λ(x) = ¯

qxγλqx conserved vector currents: jc

λ(x) = 1 2

  • ¯

qx+ˆ

λ(γλ + 1)U† λ,xqx + ¯

qx(γλ − 1)Uλ,xqx+ˆ

λ

  • Nils Asmussen (SOTON)

HLbL g-2 on the lattice 30 October 2018 16 / 29

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SLIDE 20

Lattice QED Computation with Wilson Fermions

continuum extrapolation lepton loop (ml = 2mµ) L(0)

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.02 0.04 0.06 0.08 0.1 0.12 L(0)(x, y) = L(x, y)

aLbL

µ

× 108 amµ

aLbL,ll

µ

= 0.1659 aLbL,cc

µ

= 0.1596

dashed line: continuum extrapolation for mµ = 7.2 using a quadratic fit solid line: volume extrapolation: curve shifted by the difference between the results for lattice extents mµL = 7.2 and 14.4 at fixed a

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 17 / 29

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SLIDE 21

Lattice QED Computation with Wilson Fermions

continuum extrapolation lepton loop (ml = 2mµ) L(2)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.02 0.04 0.06 0.08 0.1 0.12 L(2)(x, y) = L(x, y) − L(x, 0) − L(0, y)

aLbL

µ

× 108 amµ

aLbL,ll

µ

= 0.1503 aLbL,cc

µ

= 0.1498

less discretisation effects it is advantageous to use the subtracted kernel L(2)

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 18 / 29

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SLIDE 22

1

Tests of the QED Kernel

2

Tests of the Lattice Gauge Theory Code

3

Lattice QCD

4

Conclusion

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 19 / 29

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SLIDE 23

Lattice Setup

CLS Nf = 2 + 1 ensembles

CLS L3 × T a [fm] mπ [MeV] mπL L [fm] #confs H105 323 × 96 0.086 285 3.9 2.7 1000 N101 483 × 128 285 5.9 4.1 400 N203 483 × 128 0.064 340 5.4 3.1 750 N200 483 × 128 285 4.4 3.1 800 D200 643 × 128 200 4.2 4.2 1100 O(a) improved Wilson fermions

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 20 / 29

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SLIDE 24

Nf = 2 + 1 CLS Ensembles

100 150 200 250 300 350 400 450 0.0502 0.0642 0.0762 0.0862

mπ [MeV] a2 [fm2]

H101 H102 N101 (H105) C101 B450 S400 N401 N202 (H200) N203 N200 D200 E250 N300 N302 J303

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 21 / 29

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SLIDE 25

Integrand of acHLbL

µ

with L(2), mπ =340 MeV, a=0.064 fm

−20 20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

acHLbL

µ

= 82(9) × 10−11 f(|y|) × 109 fm

|y| [fm] N203

lattice data

fully connected contribution only we already observe a good signal integrand non-zero up to 2 fm

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 22 / 29

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SLIDE 26

Integrand of acHLbL

µ

with L(2), mπ =340 MeV, a=0.064 fm

−20 20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

acHLbL

µ

= 82(9) × 10−11 f(|y|) × 109 fm

|y| [fm] N203

lattice data VMD - 340 MeV

for long distances the simple VMD Model seems to provide a good approximation to the full QCD computation the size of the box L = 3.1 fm is large enough to capture the HLbL contribution for this pion mass

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 23 / 29

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SLIDE 27

Pion Mass Dependence of acHLbL

µ

20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

acHLbL

µ

× 1011

|y| [fm] a = 0.064 [fm]

mπ = 340 MeV mπ = 285 MeV mπ = 200 MeV

the results show an upward trend for decreasing pion mass currently collecting more statistics in long distance regime

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 24 / 29

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SLIDE 28

Pion Mass Dependence of acHLbL

µ

20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

acHLbL

µ

× 1011

|y| [fm] a = 0.064 [fm]

mπ = 340 MeV mπ = 285 MeV mπ = 200 MeV

the results show an upward trend for decreasing pion mass currently collecting more statistics in long distance regime

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 24 / 29

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SLIDE 29

Discretisation Effects, mπ = 285 MeV

20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

acHLbL

µ

× 1011

|y| [fm]

a = 0.086 fm a = 0.064 fm

discretisation effects seem to be small (we are increasing statistics)

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 25 / 29

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SLIDE 30

Finite Size Effects, a = 0.086 fm

20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

acHLbL

µ

× 1011

|y| [fm]

mπL = 4.0 mπL = 6.0

finite size effects seem to be small (we are increasing statistics)

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 26 / 29

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SLIDE 31

Integrand: Effect of Different Subtractions

−20 20 40 60 80 100 120 140 0.5 1 1.5 2 2.5 3

f(|y|) × 109 fm

|y| [fm] N203

L(2) L(3)

mπ = 340 MeV a = 0.064 fm L = 3.1 fm The subtracted kernels L(2) and L(3) both work well

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 27 / 29

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SLIDE 32

1

Tests of the QED Kernel

2

Tests of the Lattice Gauge Theory Code

3

Lattice QCD

4

Conclusion

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 28 / 29

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SLIDE 33

Conclusions

Explicit formula for aHLbL

µ

QED kernel function multiplying the position-space QCD correlation function

Tests

QED kernel: reproduce known results for π0 pole and lepton loop in the continuum for the standard kernel L(0) and subtracted kernels L(1,2,3) Lattice implementation: Reproduce lepton loop result in Lattice QED

Lattice QCD

First Mainz results for the fully connected contribution (in QED∞) Subtractions are needed to obtain a signal at long distances The discretisation and finite-size effects seem to be small

Future

We are collecting more statistics Perform chiral and continuum extrapolations Implement disconnected contribution

Nils Asmussen (SOTON) HLbL g-2 on the lattice 30 October 2018 29 / 29