Leading electromagnetic corrections to meson massesand the HVP

Vera GulpersJames Harrison, Andreas Juttner, Antonin Portelli, Christopher Sachrajda

School of Physics and AstronomyUniversity of Southampton

July 26, 2016

LATTICE2016

RBC/UKQCD Collaboration

BNL and RBRC

Mattia BrunoTomomi IshikawaTaku IzubuchiChulwoo JungChristoph LehnerMeifeng LinTaichi KawanaiHiroshi OhkiShigemi Ohta (KEK)Amarjit SoniSergey Syritsyn

CERN

Marina Marinkovic

Columbia University

Ziyuan BaiNorman ChristLuchang JinChristopher KellyBob Mawhinney

Greg McGlynnDavid MurphyJiqun Tu

University of Connecticut

Tom Blum

Edinburgh University

Peter BoyleGuido CossuLuigi Del DebbioRichard KenwayJulia KettleAva KhamsehBrian PendletonAntonin PortelliOliver WitzelAzusa Yamaguchi

KEK

Julien Frison

Peking University

Xu Feng

Plymouth University

Nicolas Garron

University of Southampton

Jonathan FlynnVera GulpersJames HarrisonAndreas JuttnerAndrew LawsonEdwin LizarazoChris SachrajdaFrancesco SanfilippoMatthew SpraggsTobias Tsang

York University (Toronto)

Renwick Hudspith

Outline

Introduction

QED correction to meson masses

QED correction to the HVP

Introduction

Introduction

I Isospin breaking corrections− different masses of u and d quark− QED corrections

I expected to be of order of 1%

I e.g. aµ, isospin breaking effects crucial to be competitive with determinationfrom e+e− → hadrons

I QED effects

I stochastic QED using U(1) gauge configurations [J. Harrison, Tue 15:00]

I expansion of the path integral in α [RM123 Collaboration, Phys.Rev. D87, 114505 (2013)]

〈O〉 =1

Z

∫D[U]D[A]D[Ψ,Ψ] O e−SF[Ψ,Ψ,A,U] e−SA[A] e−SG[U]

→ compute the leading order QED corrections

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 1 / 12

Introduction

Diagrams at O(α)

I two insertions of the conserved vector current or one insertion of the tadpoleoperator at O(α)

I three different types of (connected) diagrams

photon exchange self energy tadpole

0

x

y

z 0

x y

z 0x

z

I e.g. photon exchange diagram for a charged Kaon

C(z0) =∑

#»z

∑

x,y

Tr[Ss(z, x) Γc

ν Ss(x, 0) γ5 Su(0, y) Γcµ Su(y, z) γ5

]∆µν(x−y)

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 2 / 12

Introduction

photon propagator

I photon propagator (Feynman gauge)

∆µν(x− y) = δµν1

V

∑

k,#»k 6=0

eik·(x−y)

4∑ρ

sin2 kρ

2

I subtract all spatial zero modes→ QEDL [Borsanyi et al., Science 347 (2015) 1452-1455]

I rewrite photon propagator

∆µν(x− y) ≈∑

u

∆µν(x− u)η(u)η†(y) = ∆µν(x)η†(y)

with a stochastic source (e.g. Z2)

1

N

N∑

i=1

ηi(u)η†i (y) ≈ δu,y

I calculate ∆µν(x) =∑u

∆µν(x− u)η(u) using Fast Fourier Transform

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 3 / 12

Introduction

construction of the correlators

I photon exchange for a charged Kaon

C(z0) =∑

#»z

∑

x,y

Tr[Ss(z, x) Γc

ν Ss(x, 0) γ5 Su(0, y) Γcµ Su(y, z) γ5

]∆µν(x)η†(y)

I sequential propagators

Γcν

∆µν(x)

Γcµ

η†(y)

I contraction

0

x

y

z

I similar for the self energy using a double sequential propagator

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 4 / 12

Introduction

setup of the run

I Nf = 2 + 1 Domain Wall Fermions

I 64× 243 lattice with a−1 = 1.78 GeV

I Ls = 16, M5 = 1.8

I 87 gauge configurations

I pion mass mπ = 340 Mev

I different masses for valence u and d quarks≈ physical mass difference [BMW Collaboration, 1604.07112]

I physical valence strange quark mass [T. Blum et al, Phys. Rev. D93, 074505 (2016)]

I one Z2 noise for the stochastic insertion of the photon propagator per gaugeconfiguration and source position

I 3 source positions

I computational cost:17 inversions per valence quark and source position

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 5 / 12

QED correction to meson masses

results - correlators

I two quarks with mu

I photon exchange

0

x

y

z

I self energy

0

x y

z

I tadpole

0x

z 1

10

100

1000

10000

0 8 16 24 32

C(t)

t

PRELIMINARY

twopt∑µ self energy∑

µ photon ex hange∑µ tadpole

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 6 / 12

QED correction to meson masses

corrections to meson masses

I extract mass correction from an O(α) diagram by[RM123 Collaboration, Phys.Rev. D87, 114505 (2013)]

C(t) = C2pt(t) + CO(α)(t) = A e−(m+δm)·t ⇒ δm = −∂tCO(α)(t)

C2pt(t)

I photon exchange self energy tadpole

0

x

y

z 0

x y

z 0x

z

I example: charged Kaon

0

0.2

0.4

0.6

0.8

1

0 8 16 24 32

C

e

x

h

(t)/C2

p

t

(t)

t

PRELIMINARY

Cex h

/C2ptlinear �t

0

0.1

0.2

0.3

0.4

0 8 16 24 32

C

s

e

l

f

,

u

(t)/C2

p

t

(t)

t

PRELIMINARY

Cself

/C2ptlinear �t

0

0.1

0.2

0 8 16 24 32C

t

a

d

,

u

(t)/C2

p

t

(t)

t

PRELIMINARY

Ctad

/C2ptlinear �t

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 7 / 12

QED correction to meson masses

results QED corrections to meson masses

I some (very preliminary) results for QED corrections to meson masses(w/o finite volume correction)

Quantity this work stochastic QED [Tue, 15:00]

Mγπ+ 2.70± 0.02 MeV 3.42± 0.02 MeV

Mγπ0 0.70± 0.02 MeV 1.52± 0.01 MeV

Mπ+ −Mπ0 2.00± 0.03 MeV 1.90± 0.02 MeVMγ

K+ 2.12± 0.02 MeV 2.70± 0.02 MeVMγ

K0 0.28± 0.01 MeV 0.55± 0.01 MeV

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 8 / 12

QED correction to meson masses

results QED corrections to meson masses

I some (very preliminary) results for QED corrections to meson masses(w/o finite volume correction)

Quantity this work stochastic QED [Tue, 15:00]

Mγπ+ 2.70± 0.02 MeV 3.42± 0.02 MeV

Mγπ0 0.70± 0.02 MeV 1.52± 0.01 MeV

Mπ+ −Mπ0 2.00± 0.03 MeV 1.90± 0.02 MeVMγ

K+ 2.12± 0.02 MeV 2.70± 0.02 MeVMγ

K0 0.28± 0.01 MeV 0.55± 0.01 MeV

I pion mass splitting is a special case [RM123 Collaboration, Phys.Rev. D87, 114505 (2013)]

→ depends only on photon exchange diagram

Mπ+ −Mπ0 =(qu − qd)2

2e2 ∂t

Cexch(t)

C2pt(t)

0

x

y

z

I problem in the self energy and/or the tadpole diagram?→ needs to be resolved

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 8 / 12

QED correction to meson masses

Comparison of statistical precision

I computational cost• perturbative method 17 inversions per quark flavor• stochastic method 3 inversions per quark flavor

I statistical error ∆ of QED contribution to effective Kaon mass

I scaled by√

# inversions

0

0.5

1

1.5

2

2.5

3

0 8 16 24 32

∆

p

e

r

t

/∆

s

t

o

h

t

PRELIMINARY

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 9 / 12

QED correction to the HVP

The hadronic vacuum polarisation

I vacuum polarisation tensor

Πµν(Q) =∑

x

ei Q·x⟨

jγµ(x) jγν(0)⟩

= (QµQν − δµνQ2) Π(Q2)

I correlatorCµν(x) = Zv q2

f

⟨Vcµ(x)V`ν(0)

⟩

I construction of the HVP tensor, see eg. [RBC/UKQCD, JHEP 1604 (2016) 063], [M. Spraggs, Tue 17:10]

Πµν(Q) =∑

x

e−iQ·xCµν(x)−∑

x

Cµν(x)

(with zero mode subtraction)

I vacuum polarisation

Π(Q2) =1

3

∑

j

Πjj(Q)

Q2

Π(Q2) ≡4

9Πu(Q2) +

1

9Πd(Q2) +

1

9Πs(Q2)

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 10 / 12

QED correction to the HVP

First look at HVP

I hadronic vacuum polarisation for the u quark

I left: HVP without QED Πu0(Q2), right: QED corrections to HVP δΠu(Q2)

Πu(Q2) = Πu0(Q2) + δΠu(Q2)

−0.2

−0.15

−0.1

−0.05

0

0 2 4 6 8

Πu(Q

2)

Q2/GeV

PRELIMINARY

Πu(Q2) w/o QED

−0.0006

−0.0004

−0.0002

0

0.0002

0.0004

0.0006

0 2 4 6 8

δΠu(Q

2)

Q2/GeV2

PRELIMINARY

self energy

ex hange

tadpole

total QED orre tion

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 11 / 12

QED correction to the HVP

Summary

I Leading order QED corrections by expansion of the path integral

I corrections to meson masses and HVP

I exploratory study

I currently, discrepancy between results from stochastic and perturbativeapproach→ needs to be resolved

Outlook

I Coulomb gauge for the photon propagator

I more gauge ensembles

I matrix elements [N. Carrasco et al , Phys. Rev. D91 (2015) 074506], [N. Tantalo, Wed 11:50], [S. Simula, Wed 12:10]

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 12 / 12

Backup

Backup

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 13 / 12

Backup

expansion of the Wilson-Dirac operator in e

I Including QED link variables the action is

SeW =

∑

x

[Ψ(x) (M + 4) Ψ(x)−

1

2Ψ(x)(1− γµ)Eµ(x)Uµ(x)Ψ(x + µ)

−1

2Ψ(x + µ)(1− γµ)U†µ(x)E†µ(x)Ψ(x)

]

with QED link variables

Eµ(x) = e−ieef Aµ(x) = 1− ieefAµ(x) +1

2(eef)

2Aµ(x)Aµ(x) + . . .

I Expanding the action in e one finds

SeW − S0

W =∑

x,µ

{−ieef Aµ(x) Vc

µ(x) +(eef)2

2Aµ(x)Aµ(x) Tµ(x)

}

with the conserved vector current Vcµ(x) and the tadpole operator Tµ(x)

Vcµ(x) =

1

2

[Ψ(x+µ)(1+γµ)U†µ(x)Ψ(x)−Ψ(x)(1−γµ)Uµ(x)Ψ(x+µ)

]

Tµ(x) =1

2

[Ψ(x)(1−γµ)Uµ(x)Ψ(x+µ)+Ψ(x+µ)(1+γµ)U†µ(x)Ψ(x)

]

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 14 / 12

Backup

photon propagator with FFT

I ∆µν(x) =∑u

∆µν(x− u)η(u) with ∆µν(x− y) = δµν1V

∑k,

#»k 6=0

eik·(x−y)

k2

I Fourier Transform of the stochastic source

η(u)FFT−→ η(k)

I divide by k2 and subtract the zero mode

η(k)

k2−→

η(k)

k2

∣∣∣∣#»k =0

= 0

I Fourier Transform

FFT−→ ∆(x) =∑

k,#»k 6=0

η(k)

k2eik·x =

∑

u

∆(x− u)η(u)

Vera Gulpers (University of Southampton) Lattice 2016 July 26, 2016 15 / 12