The hadronic vacuum polarization contribution to ( g 2) : status - - PowerPoint PPT Presentation

The hadronic vacuum polarization contribution to (g − 2)µ: status of the Mainz-CLS calculation

Harvey Meyer g-2 workshop, Mainz, 20 June 2018

Cluster of Excellence

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Outline

◮ Calculation in the time-momentum representation in Nf = 2 + 1 QCD ◮ Technical improvements over our Nf = 2 calculation [1705.01775 (JHEP)]. ◮ Results for the strange and charm connected contributions. ◮ Status of the light-quark contribution.

CLS-Mainz HVP collaboration: A. G´ erardin, T. Harris, G. von Hippel, B. H¨

• rz,

HM, D. Mohler, K. Ottnad, H. Wittig. All numerical results in this talk are still preliminary!

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HVP: definitions (Euclidean space)

◮ primary object on the lattice: Gµν(x) = jµ(x)jν(0). ◮ polarization tensor:

Πµν(Q) ≡

• d4x eiQ·xGµν(x) =
• QµQν − δµνQ2

Π(Q2).

◮

ahvp

µ

= 4α2 ∞ dQ2 K(Q2; m2

µ) [Π(Q2) − Π(0)] ◮ Spectral representation: ρ(s) = R(s) 12π2 , R(s) ≡ σ(e+e−→hadrons) 4πα(s)2/(3s)

, Π(Q2) − Π(0) = Q2 ∞

4m2

π

ds ρ(s) s(s + Q2).

Lautrup, Peterman & de Rafael Phys.Rept 3 (1972) 193; Blum hep-lat/0212018 (PRL)

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The time-momentum representation (TMR)

◮ mixed-representation Euclidean correlator: (natural on the lattice)

GTMR(x0) = −1 3

3

• k=1
• d3x Gkk(x),

◮ the spectral representation:

GTMR(x0) = ∞ dω ω2ρ(ω2) e−ω|x0|, x0 = 0.

◮ Finally, the quantity ahvp µ

is given by ahvp

µ

= α π 2 ∞ dx0 G(x0) ˜ f(x0), ˜ f(x0) = 2π2 m2

µ

• − 2 + 8γE + 4

ˆ x2 + ˆ x2

0 − 8

ˆ x0 K1(2ˆ x0) +8 log(ˆ x0) + G2,1

1,3

• ˆ

x2

0| 3 2

0, 1, 1

2

where ˆ x0 = mµx0, γE = 0.577216.. and Gm,n

p,q

is Meijer’s function.

Bernecker & Meyer 1107.4388; Mainz-CLS 1705.01775.

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Expected integrand for ahvp

µ

(using pheno. R)

1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 t fm 103t3 Gt K t

41% 45% 11% 3%

Bernecker & Meyer 1107.4388

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Finite-size effects: discrete states on the torus

2 4 6 8 10 12 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1(ω2) ω [GeV]

Isovector final states: ρ(s) = 1 48π2

• 1 − 4m2

π/s

3/2 |Fπ(s)|2 + other channels |Fπ(s)|2 =

• qφ′(q)+k ∂δ1(k)

∂k

3πs 2k5L3

• L
• ππ
• dx jz(x)
• 2

.

HM 1105.1892 (PRL); figure: model for Fπ(s).

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Landscape of CLS ensembles

100 150 200 250 300 350 400 450 0.0502 0.0652 0.0772 0.0852

mπ [MeV] a2 [fm]

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

Nf = 2 + 1 ensembles, O(a) improved Wilson quarks, treelevel-improved L¨ uscher-Weisz gauge action. Algorithm uses twisted-mass reweighting. Exact isospin symmetry on the reweighted configurations. I will often illustrate our calculations using ensemble D200.

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A first look at the three connected integrands on ensemble D200

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3 3.5

x0 [fm] G(x0) K(x0)/mµ

Light Strange (x6) Charm (x6)

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Technical aspects of the calculation

◮ scale setting: we use the lattice spacing values from

[1608.08900 (PRD) Bruno, Korzec, Schaefer].

For instance, a[fm] = 0.06440(65)(15) for D200: 1% precision. Dimensionful quantity used for scale-setting: fK + 1

2fπ. ◮ new: non-perturbative on-shell improvement of the vector current:

calculation of cV [G´

erardin, Harris et al., in prep.].

⇒ ahvp

µ

approaches its continuum value with O(a2) corrections.

◮ local current ¯

ψ(x)γµψ(x) as well as lattice Noether current ⇒ use two discretizations of the current-current correlator (ll,lc). Perform constrained simultaneous continuum limit for ahvp

µ

.

◮ We benefit from a dedicated lattice calculation of the I = ℓ = 1 scattering

phase and of the pion form factor at timelike q2: [H¨

• rz et al. 1511.02351 and in

prep.]. Used to control tail of isovector correlation function and for the

finite-size correction.

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Charm contribution

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Tuning of κc, the bare charm mass parameter

Tuning of κc : by imposing meff

c¯ s = mexp Ds = 1972 MeV on each lattice ensemble.

3.85 × 106 3.9 × 106 3.95 × 106 4 × 106 4.05 × 106 4.1 × 106 4.15 × 106 7.81 7.82 7.83 7.84 7.85 7.86

m2

Ds

1/κc N200

◮ Interpolation : m2 Ds vs 1/κc → linear behaviour ◮ amDs = 0.86 at β = 3.40 : discretization effects are expected to be large.

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Chiral & continuum extrapolation of ac

µ

( y = m2

π/(16π2f2 π))

5 10 15 20 25 30 0.02 0.04 0.06 0.08 0.1 0.12

• y

aHVP,LO

µ,c

× 10−10

β = 3.40 β = 3.46 β = 3.55 β = 3.70

1010 · ac

µ = 14.95(47)stat(11)χ ◮ The O(a)−improvement reduces lattice artefacts significantly ◮ The simultaneous continuum extrapolation (linear in a2) works well.

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Strange contribution

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Strange contribution: data at physical quark masses (E250)

0.0005 0.001 0.0015 0.002 0.5 1 1.5 2 2.5 3

x0 [fm] G(x0) K(x0)/mµ

loc-loc cons-loc

◮ With full O(a)−improvement of the vector currents ◮ Two discretizations almost coincide: remaining discretization errors small.

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Finite size effects on strangeness correlator at mπ = 280 MeV

0.0005 0.001 0.0015 0.002 0.5 1 1.5 2 2.5 3

x0 [fm] G(x0) K(x0)/mµ

L = 2.05 fm L = 2.70 fm L = 4.10 fm

⇒ At the level of 0.5 % precision, volume effects are negligible for L ≥ 2.7 fm.

CLS all−imp

µ

alc−imp

µ

U101 (L = 2.05 fm) 69.5(0.6) 65.8(0.6) H105 (L = 2.70 fm) 71.8(0.4) 68.0(0.4) N101 (L = 4.10 fm) 71.9(0.3) 68.0(0.3)

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Chiral & continuum extrapolation ( y = m2

π/(16π2f2 π))

◮ Fit : aµ(a,

y, d) = aµ(0, yexp) + δd a2 + γ1 ( y − yexp) + γ2 ( y log y − yexp log yexp)

20 30 40 50 60 70 80 90 100 0.02 0.04 0.06 0.08 0.1 0.12

• y

aHVP,LO

µ,s

× 10−10

β = 3.40 β = 3.46 β = 3.55 β = 3.70

aHVP,LO

µ,s

= 53.6(2.5)stat(0.8)χ

◮ The first error is the statistical error; ◮ second error : variation wrt setting the cut at mπ = 360 MeV; ◮ statistical error dominated by the scale setting error.

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Light-quark contributions

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Auxiliary calculation: time-like pion form factor

For all the ensembles with mπ < 300 MeV: dedicated study (except for E250)

◮ Overlap and energy levels to constrain the tail of the correlation function ◮ Time-like pion form factor to estimate finite-size effects.

On N200 (mπ = 280 MeV): parametrizing the time-like pion form factor in the Gounaris-Sakurai form: mGS

ρ

= 776(4) MeV ΓGS

ρ

= 59(2) MeV. Other parametrizations are being investigated.

[Bulava, H¨

• rz et al., 1511.02351 (LAT2015).]

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Saturation of the correlator by the low-lying states (D200)

• cf. slide 6:

2 4 6 8 10 12 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1(ω2) ω [GeV]

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3

xcut x0 [fm] G(x0) K(x0)/mµ

1 exp fit loc-cons n=4 n=3 n=2 n=1

G(x0) =

∞

• n=1

An e−Enx0

◮ Excellent cross-check that the tail is understood.

[Update from H. Wittig et al. 1710.10072 (LAT2017)]

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Saturation of the correlator by the low-lying states (D200)

• cf. slide 6:

2 4 6 8 10 12 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1(ω2) ω [GeV]

0.001 0.002 0.003 0.004 2 2.2 2.4 2.6 2.8 3 3.2

xcut x0 [fm] G(x0) K(x0)/mµ

1 exp fit loc-cons n=4 xn=3 yn=2 zn=1

G(x0) =

∞

• n=1

An e−Enx0

◮ Excellent cross-check that the tail is understood.

[Update from H. Wittig et al. 1710.10072 (LAT2017)]

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Finite size effects: check on the lattice (mπ = 280 MeV H105 vs. N101)

0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x0 [fm] G(x0) K(x0)/mµ

L/a = 32 (L = 2.70 fm) L/a = 48 (L = 4.10 fm) L/a = 32 with FSE L/a = 48 with FSE

◮ FSE consistent with the estimate using the pion FF and L¨

uscher formalism

(see [1705.01775 Mainz-CLS]).

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Integrand at the physical pion mass (E250)

0.004 0.008 0.012 0.016 0.5 1 1.5 2 2.5 3

xcut x0 [fm] G(x0) K(x0)/mµ

1 exp fit loc-cons

◮ statistics being increased ◮ FSE: for mπ = 140 MeV and mπL = 4, our estimate:

1010 · [aµ(∞) − aµ(L)] = 20.4 ± 4.1

[1705.01775 Mainz-CLS];

◮ For mπL = 6, the estimate goes down by a factor ≈ 10.

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Chiral extrapolations (light contribution): y = m2

π/(16π2f2 π)

300 400 500 600 700 800 0.02 0.04 0.06 0.08 0.1 0.12

• y

aHVP,LO

µ,ud

× 10−10

β = 3.40 β = 3.46 β = 3.55 β = 3.70

Fit ansatz: aµ(a, y, d) = aµ(0, yexp) + δd a2 + γ1 ( y − yexp) + γ2 ( y log y − yexp log yexp)

• 1010 · aHVP,LO

µ,s

= 643(21)stat(××)syst.

◮ chiral extrapolation in good agreement with direct calculation at the

physical point (E250);

◮ other ensembles dictate the lattice-spacing dependence.

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Disconnected contributions

−10 −8 −6 −4 −2 0.5 1 1.5 2 2.5

x0 [fm] adisc

µ (x0) × 1010 mπ = 340 MeV mπ = 280 MeV mπ = 200 MeV

Integrated contribution Since 2mℓ + ms = constant in these ensembles, we envisage an extrapolation in (mℓ − ms)2.

ratio of disc and conn −1/9 t [fm] C2pt,disc

V l−s

L

V l−s

L

(t)/C2pt,conn V l

LV l L (t)

1.5 1 0.5 0.2 0.15 0.1 0.05

• 0.05
• 0.1
• 0.15
• 0.2

N200 (mπ = 280 MeV) At what distance do we reach the asymptotic behavior [1306.2532 Mainz-CLS] Gdisc(t) Gisovector(t)

t→∞

− → −1 9 ? Caveat: the relative finite-size effect on adisc

µ

could be very large.

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Disconnected contribution using a new Lorentz-covariant method

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 mµ|x| −0.00010 −0.00008 −0.00006 −0.00004 −0.00002 0.00000 0.00002 0.00004 CCS TMR −5 −4 −3 −2 −1 1 2 1010ahvp

µ

0.0 0.5 1.0 1.5 2.0 2.5 |x| [fm] ◮ at |x|CCS = |x0|TMR, much smaller uncertainties in the covariant version. ◮ ensemble D200: 128 × 643, mπ = 200 MeV, a = 0.064 fm.

HM 1706.01139; M. C` e, K. Ottnad et al.

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Conclusion

◮ Compared to our previous calculation [1705.01775]

◮ O(a) improvement implemented + four lattice spacings

0.050 < a/fm < 0.087 ⇒ exquisite control over cutoff effects

◮ much higher statistics ◮ dedicated calculation of the timelike pion form factor available.

◮ much better control of the tail of the correlators and of their

finite-volume effects.

◮ Main sources of uncertainty in the calculation :

→ Statistics & scale setting → Disconnected diagrams → Chiral fits → Finite-size effects.

◮ gearing up for QED+isospin breaking terms.

[A. Risch, H. Wittig 1710.06801].

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Backup Slides

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Finite-size effects on M(ν) = ∞

0 dt tν G(t), (ν > 2)

Non-interacting pions: (ω ≡ 2√k2 + m2

π)

M(ν, L) − M(ν, ∞) = 4 3 Γ(ν + 1)

• 1

L3

• k

−

• d3k

(2π)3

• k2

ων+3 . Interacting case: M(ν, L) − M(ν, ∞) = Γ(ν + 1) ∞ 4k dk ων [ρ(ω, L) − ρ(ω)], ρ(ω) = 1 12π2 R(ω2) = 1 48π2

• 1 − 4m2

π/ω23/2 |Fπ(ω2)|2 + . . . ,

ρ(ω, L) =

∞

• n=1

An E2

n

δ(ω − En) where (kn, An) are related to the pion form factor F(ω2) = |F(ω2)|eiδ11 by the L¨ uscher formalism.

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Diagnostics of the estimated finite-size correction

Key question: for given L, what states dominate in the calculation of the finite-size effects? Compute f∆(L; kπ) = Γ(ν + 1) ∞ 4k dk ων [ρ(ω, L) − ρ(ω)] · ψ(k, kπ), ψ(k, kπ) = N · exp

• − (k − kπ)2

2∆2

• ,

∞ dk ψ(k, kπ) = 1.

• 0.0002

0.0002 0.0004 0.0006 0.0008 0.001 0.5 1 1.5 2 2.5 3 kπ / mπ Γ(ν) mπ

ν−1Integralk>0[(4k dk/ων) [ρ(ω)-ρL(ω)].N exp(-(k-kπ)2/(2∆2))]

ν=4, ∆=mπ/2 ω=2[k2+mπ

2]

N-1= Integralk>0[dk exp(-(k-kπ)2/(2∆2))] kρ/mπ mπL=4 free mπL=4 GS mπL=6 free mπL=6 GS

GS= Gounaris-Sakurai form of Fπ(ω2): yields a larger FSE than free pions Larger volume ⇒ 1) reduced finite-size effect 2) dominated by softer pions, hence better predicted by ChPT. mπL = 4: the 1-loop ChPT prediction is not quantitatively reliable yet.

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A Lorentz-covariant coordinate-space approach

◮ primary object: Gµν(x) = jµ(x)jν(0). ◮ ahvp µ

=

• d4x Gµν(x) Hµν(x) = 2π2 ∞

d|x| |x|3 [Gµν(x) Hµν(x)], Hµν(x) = −δµνH1(|x|) + xµxν x2 H2(|x|) a transverse tensor with Hi(|x|) =

8α2 3m2

µ fi(mµ|x|) and

f2(z) = G2,2

2,4

• z2|

7 2 , 4

4, 5, 1, 1

• − G2,2

2,4

• z2|

7 2 , 4

4, 5, 0, 2

• 8√πz4

, f1(z) = f2(z) − 3 16√π ·

• G2,3

3,5

• z2|

1, 3

2 , 2

2, 3, −2, 0, 0

• − G2,3

3,5

• z2|

1, 3

2 , 2

2, 3, −1, −1, 0 .v

fi(z)/z4

f1(z)/z4 f2(z)/z4 2 4 6 8 10 12 0.0000 0.0002 0.0004 0.0006 0.0008 z

HM, 1706.01139.

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