HVP lattice long-distance contributions and sQED vs. GEVP · 2018-06-21 · and sQED vs. GEVP Georg...

HVP lattice long-distance contributionsand sQED vs. GEVP

Georg von Hippel

Institute of Nuclear Physics,University of Mainz, Germany

Second Workshop of the Muon g − 2 Initiative

Mainz, 18–22 June 2018

Motivation

Time-momentum representation [Meyer and Bernecker, 2011]

aHVPµ =

(απ

)2 ∫ ∞

0dt K (mµ, t)G (t)

with a known kernel K , and G (t) =∑

x〈j3(t, x)j3(0)〉 the vectorcorrelator.

Integration runs to infinity, lattice data do not

Noise rapidly increases at large times

particularly at small pion mass and large volume

Some modelling of long-distance behaviour is required

G.M. von Hippel Long-Distance Contributions

Large-time modelling

Large-time modelling of the vector correlator:

Extract naive ground-state mass from fit to a (smeared)correlator, model G (t) as single exponential at large t;

Perform multi-exponential fits to (smeared and/or local)correlator(s), model G (t) by fitted correlator at large t;

Use the large-time approximation

G ρρ∞ (t) =

1

48π2

∫ ∞

0dω ω2

(1− 4m2

π

ω2

) 32

|Fπ(ω)|2e−ωt

by eithermodelling Fπ using the Gounaris-Sakurai parameterization witha mass and width extracted semi-naively from smeared andlocal correlators [Mainz Nf = 2, 2017] , orextracting Fπ using the Luscher method [B. Horz et al.]

This also allows for correcting finite-volume effects→ talk/discussion by D. Giusti


Modelling the integrand (mπ = 280) MeV

0

0.004

0.008

0.012

0.016

0 0.5 1 1.5 2 2.5 3

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fit2 exp fit

loc-loc



0

0.001

0.002

0.003

0.004

2 2.2 2.4 2.6 2.8 3 3.2

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fit2 exp fit

loc-loc



0

0.004

0.008

0.012

0.016

0 0.5 1 1.5 2 2.5 3

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fit2 exp fit

loc-loc



0

0.001

0.002

0.003

0.004

2 2.2 2.4 2.6 2.8 3 3.2

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fit2 exp fit

loc-loc


Reconstructing the integrand (mπ = 280) MeV

0

0.004

0.008

0.012

0.016

0 0.5 1 1.5 2 2.5 3

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fitloc-loc

n=3n=2n=1



0

0.001

0.002

0.003

0.004

2 2.2 2.4 2.6 2.8 3 3.2

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fitloc-loc

n=3n=2n=1



0

0.004

0.008

0.012

0.016

0 0.5 1 1.5 2 2.5 3

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fitloc-loc

n=4n=3n=2n=1



0

0.001

0.002

0.003

0.004

2 2.2 2.4 2.6 2.8 3 3.2

xcut0

x0 [fm]

G(x0)K(x0)/mµ

1 exp fitloc-loc

n=4n=3n=2n=1


RBC/UKQCD

Correlation Function Reconstruction

0 5 10 15 20 25 30t

0

10

20

30

40

50

60

70

80

wtC(

t)1-state reconstruction2-state reconstruction3-state reconstruction4-state reconstructionlocal vector currentscalar QED

aµ =∑

t wtC(t)

Correlated fit to reconstruct long-distance behavior oflocal vector current correlation function C(t)

More states =⇒ better reconstructionScalar QED underestimates lowest state (∼ ππ)

4 / 5

→ talk by A. Meyer in the discussion session


FNAL/MILC/HPQCD

R. Van de Water HVP contribution to muon g-2 with (2+1+1) HISQ quarks

ππ levels and noisy data

10

Equivalent Fits (χ2/dof ≤ 1)

• Fit unable to resolve difference between one or multiple rho mesons; spreads contribution over multiple terms.

• Negligible difference for t<4fm, and irrelevant for ɑμ.

3 fits

E0 \ a0 E1 \ a1 E2 \ a2 E3 \ a3 ɑμ x 1010

Fit 1 0.770 \ 0.132 1.72 \ 0.35 2.5 \ 0.1 602(7)

Fit 2 0.75 \ 0.09 0.80 \ 0.09 1.75 \ 0.36 602(7)

Fit 3 0.71 \ 0.08 0.78 \ 0.04 0.84 \ 0.10 1.76 \ 0.36 608(8)

0.15 fm

Take-away: can parameterize data over a finite t range (<3-4fm) by a fit with a single ρ

[R. Van de Water]


FNAL/MILC/HPQCD

R. Van de Water HVP contribution to muon g-2 with (2+1+1) HISQ quarks

0 1 2 3 4 5 6

t* (fm)

400

500

600

700

800

900

a µ

1.5 fm

a~0.15 fma~0.15 fma~0.12 fma~0.09 fma~0.06 fm

Selection of t*

For t* = 1.5 fm, aμHVP comes primarily from data region, and errors still controlled

14

1 2 3 4

t* (fm)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a µdat

a /aµto

tal 1.5 fm

a~0.15 fma~0.15 fma~0.12 fma~0.09 fma~0.06 fm

aµ

u,d: m

l=m

phys. ensembles

Data contribution Stability

2 fm

80% data

1.5 fm

1.5 fm

[R. Van de Water]


Some observations

Multi-exponential fits may not be able to properly resolvelow-lying states

Scalar QED seems not to be a good model for the two-pioncontribution

At small pion mass, the ground state (∼ ππ) is weakly coupledto the vector correlator and only dominates at t & 3 fm

A dedicated spectroscopic study (with multi-particle states)appears to be a requirement in order to fully understand thelarge-distance behaviour


The End = The Beginning

The floor is open for discussion . . .


HVP lattice long-distance contributions and sQED vs. GEVP · 2018-06-21 · and sQED vs. GEVP Georg...

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