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Hadronic Vacuum Polarization with Twisted Mass Fermions Marcus Petschlies ETMC Helmholtz-Institut f ur Strahlen- und Kernphysik, Rheinische Friedrich-Willhelms-Universit at Bonn First Workshop on the g 2 Initiative QCenter, Fermilab,

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Hadronic Vacuum Polarization with Twisted Mass Fermions

Marcus Petschlies ETMC

Helmholtz-Institut f¨ ur Strahlen- und Kernphysik, Rheinische Friedrich-Willhelms-Universit¨ at Bonn

First Workshop on the g − 2 Initiative QCenter, Fermilab, June 4 2017

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 1 / 26

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Outline Motivation

◮ Twisted Mass fermions ◮ Setup for Nf = 2 and Nf = 2 + 1 + 1

HVP @ ETMC

◮ results for Nf = 2 + 1 + 1 ◮ initial results for Nf = 2 physical point ensemble ◮ current work for tm+clover at physical pion mass

Summary & Outlook

◮ g-2 @ ETMC Project status ◮ Agenda Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 2 / 26

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Motivation - Twisted Mass Lattice QCD various gauge field ensembles

◮ Nf = 2 twisted mass ◮ Nf = 2 + 1 + 1 twisted mass ◮ Nf = 2 twisted mass + clover @ mπ ◮ Nf = 2 + 1 + 1 twisted mass + clover @ mπ

comprehensive software suite for solving Dirac equation and Wick contractions

◮ tmLQCD ( solvers, exact & inexact deflation )

[Jansen and Urbach, 2009, Abdel-Rehim et al., 2013]

◮ DDalphaAMG (adaptive multi-grid solver with tm support, port to GPUs)

[Alexandrou et al., 2016]

◮ cvc code package ( various Wick contractions )

automatic O (a) improvement of physical observables in the continuum limit in particular renormalized vacuum polarization function ΠR(Q2) and ahvp

µ

parity and SU(2) isospin symmetry breaking at non-zero lattice spacing SU(2) → U(1)3

◮ m±

π = m0 π

◮ Jup

µ Jup ν = Jdn µ Jdn ν by lattice artefacts

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 3 / 26

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Motivation - Twisted Mass Lattice QCD (degenerate) light quark action (Nf = 2) Sl =

  • x

¯ χl

  • DW + mq + iµl γ5 τ 3

χl(x) (1) DW = 1 2 γµ

  • ∇f

µ + ∇b µ

  • −

a 2 ∇b

µ ∇f µ − mcr

  • O (a) improvement at maximal twist: mq = m0 − mcr → 0

Noether current for Jem

µ

Jµ(x) = 1 2

  • ¯

χl(x) (γµ − 1) Uµ(x) χl(x + aˆ µ) + ¯ χl(x + aˆ µ) (γµ + 1) U†

µ(x) χl(x + aˆ

µ)

  • ZV = 1

0 = ∂b

µ

  • Jµ(x) Jν(y) + a−3 δ(4)

x,y δµν Cν(y)

  • Πµν(x,y)

exact at non-zero lattice spacing

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 4 / 26

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Motivation - Twisted Mass Lattice QCD (non-degenerate) heavy quark action (. . . + 1 + 1) Sh =

  • x

¯ χh

  • DW + mq + iµσ γ5 τ 1 + µδ τ 3

χh(x) (2) µσ τ 1 and µδ τ 3 break isospin symmetry completely − → no conserved vector current for strange and charm Osterwalder-Seiler ( = mixed action) setup [Frezzotti and Rossi, 2004] Sval

h

=

  • f =s,c
  • x

¯ ψf

  • DW + mq + iµf γ5 τ 3

ψf (x) µs, µc tuned by physical value of 2m2

K − m2 PS and mD

(valence) Noether currents Jf

µ ∼ ¯

ψf ψf ; automatic O (a) remains valid

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 5 / 26

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HVP from tmLQCD and Nf = 2 + 1 + 1

consider hadronic leading-order ∆αQED(Q2) ∝ Πγ(Q2) ∆αhvp

QED(Q2) = −4πα0

  • 5

9 Πud

R

  • Q2 ·

H2 H2

phys

  • + 1

9 Πs

R

  • Q2

+ 4 9 Πc

R

  • Q2

. (3) inter-/extrapolation, Π(0) Πf

low(Q2) = M

  • i=1

g 2

i m2 i

m2

i + Q2 + N−1

  • j=0

aj(Q2)j Πf

high(Q2) = log(Q2) B−1

  • k=0

bk(Q2)k +

C−1

  • l=0

cl(Q2)l Πf(Q2) = (1 − Θ(Q2 − Q2

match))Πf low(Q2) + Θ(Q2 − Q2 match)Πf high(Q2)

chiral and continuum extrapolation ∆αhvp

QED(Q2)(mPS, a) = A + B1 m2 PS (+ . . .) + C a2

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 6 / 26

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HVP from tmLQCD with Nf = 2 + 1 + 1

a = 0.061 fm, L = 2.9 fm a = 0.061 fm, L = 1.9 fm a = 0.078 fm, L = 3.7 fm a = 0.078 fm, L = 1.9 fm a = 0.078 fm, L = 2.5 fm a = 0.086 fm, L = 2.8 fm Nf = 2 result, Pad´ e fit Nf = 2 result, standard fit m2

PS

  • GeV2

∆αhvp,ud(1 GeV2)

0.25 0.2 0.15 0.1 0.05 0.003 0.0025 0.002 0.0015 0.001

a[fm] mPS[MeV] L[fm] mPSL 0.061 227 2.9 3.3 0.061 318 2.9 4.7 0.061 387 1.9 3.7 0.078 274 2.5 3.5 0.078 319 2.5 4.0 0.078 314 3.7 5.9 0.078 393 2.5 5.0 0.078 456 2.5 5.8 0.078 491 1.9 4.7 0.086 283 2.8 4.0 0.086 323 2.8 4.6 0.086 361 2.8 5.1

statistics O (250/150/150) gauge configurations for up-down / strange / charm quark

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 7 / 26

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HVP from tmLQCD with Nf = 2 + 1 + 1 and ahvp

l

∆α from Jegerlehner’s alphaQED package lattice data linearly extrapolated to mπ in CL Q2 GeV2

∆αhvp

QED(Q2)

10 8 6 4 2 0.01 0.008 0.006 0.004 0.002 ahlo

l

tmLQCD

  • disp. analyses

e 1.782 (64) (85) · 10−12 1.866 (10) (05) · 10−12 [Nomura and Teubner, 2013] µ 6.78 (24) (16) · 10−8 6.91 (01) (05) · 10−8 [Jegerlehner and Szafron, 2011] τ 3.41 (8) (6) · 10−6 3.38 (4) · 10−6 [Eidelman and Passera, 2007]

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 8 / 26

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Quark-disconnected contribution

  • 0.12
  • 0.1
  • 0.08
  • 0.06
  • 0.04
  • 0.02

0.02 0.2 0.4 0.6 0.8 1 1.2 Z2

V Π(Q2)

Q2/GeV2

  • 0.004

0.004 0.2 0.4 0.6 0.8 1 connected, combined isospin components disconnected, isospin 0 component disconnected, isospin 1 component

isovector and isoscalar contribution Π3

µν(x, y) = J3 µ(x) J3 ν(y)disc

tm only, with one − end trick Π0

µν(x, y) = J0 µ(x) J0 ν(y)disc

a = 0.078 fm, mπ = 393 MeV, L = 2.5 fm, mπL = 5.0, up-down contribution 1548 × 24 + 4996 × 48 gauge configurations × stochastic volume sources

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 9 / 26

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twisted mass + clover at Nf = 2 and physical pion mass Sl =

  • x

¯ χl

  • DW [U] + mq + iµl γ5 τ 3 + i

4 csw σµν Fµν [U]

  • χl(x)

(4) clover term not added for O (a) improvement but to reduce the effects of isospin splitting pion masses range from 130 to 350 MeV 2 volumes at mPS = 130 MeV, 4.4 fm and 5.8 fm (single) lattice spacing a = 0.0914 (3) (15) fm

100 200 300 400 100 200 300 400 Mπ± [MeV] Mπ0 [MeV] L/a = 24 L/a = 32 L/a = 48

pion mass splitting compatible with zero ⇒ reduced finite size effects ⇒ reduces isospin splitting effects

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 10 / 26

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twisted mass + clover at Nf = 2 and physical pion mass, L = 4.4 fm

M2

π

  • GeV2

aud

τ [10−6]

0.25 0.2 0.15 0.1 0.05 2.7 2.3

aud

µ [10−8]

5.8 5.4 5.0

aud

e [10−12]

1.6 1.4 1.2

physical point

  • extr. Nf = 2
  • extr. Nf = 2 + 1 + 1

ahvp

e

· 1012 1.45(11) 1.51(04) 1.50(03) ahvp

µ

· 108 5.52(39) 5.72(16) 5.67(11) ahvp

τ

· 106 2.65(07) 2.65(02) 2.66(02)

[Abdel-Rehim et al., 2015]

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 11 / 26

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Restart 2017 focus on tmLQCD+clover at Nf = 2 and physical pion mass 1 lattice spacing (a = 0.091 fm), but 2 volumes (4.4 fm and 5.8 fm) twisted mass + clover Nf = 2 + 1 + 1 at physical pion mass in production more measurements and statistics Πµν(x, y) = Jµ(x) Jν(y) point-to-all, i.e. one ∼ few, fixed source locations y per gauge configuration ⇒ little of information content per gauge configuration actually used signal in disconnected contribution

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 12 / 26

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Restart 2017 - extended list of observables

HVP and π0, η, η′ → γ γ (η, η′ on upcoming Nf = 2 + 1 + 1 physical point gauge field ensemble) neutral pion decay on the lattice [Ji and Jung, 2001, Cohen et al., 2008, Shintani et al., 2009, Feng et al., 2011, Feng et al., 2012] dispersive approach to g − 2 HLbL [Colangelo et al., 2014b, Colangelo et al., 2014a, Pauk and Vanderhaeghen, 2014] ΠHLbL

µνλσ = Ππ0 µνλσ + ΠFsQED µνλσ

+ Πη

µνλσ + Πππ µνλσ + Πη′ µνλσ + . . .

π0, η, η′ µ µ µ µ

dispersive approach +. . .

γ γ γ∗ γ∗ γ∗ γ∗ γ∗ γ∗ FP→γγ∗ FP→γ∗γ∗

recent calculation by Mainz group [Nyffeler, 2016, Antoine et al., 2016]

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 13 / 26

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Restart 2017 - from point-to-all towards all-to-all use even-odd preconditioned exact low mode + stochastic high mode contributions to build correlators [Blum et al., 2016] quark propagator = inverse twisted mass Dirac matrix

D−1

tm =

Mee Meo Moe Moo −1 = 1 −M−1

ee Meo C −1

C −1

  • ×
  • M−1

ee

−γ5 Moe M−1

ee

γ5

  • exact inverse M−1

ee

available computationally intensive part: inversion of C → C −1 construct subspace from Nev eigenvectors to lowest-lying eigenvalues of CC †, CC † V = V Λ decomposition of odd sub-lattice with orthogonal projectors 1 = PV + P⊥

V ,

PV = V V † inversion on PV becomes trivial inversion on P⊥

V becomes cheap (exact deflation of lowest eigenmodes)

D−1

tm =

1 −M−1

ee Meo C −1

C −1 1 PV + P⊥

V E

  • ξ ξ†

M−1

ee

−γ5 Moe M−1

ee

γ5

  • Marcus Petschlies (HISKP)

HVP @ ETMC g − 2 Initiative 14 / 26

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Convergence test of exact low modes

0.001 0.01 0.1 1 10 5 10 15 20 25 Re (Cm−m) t/a 57 × 5 point sources lma, Nev = 200 Nev = 400 Nev = 600 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 5 10 15 20 25 Re (Cm−m) t/a 57 × 5 point sources lma, Nev = 200 lma, Nev = 400 lma, Nev = 600

connected 2-point correlation functions charged pion 2-point function (left) and connected (local) vector current 2-point function (right) Nf = 2 + 1 + 1, a = 0.086 fm, mπ = 320 MeV, L = 2 fm

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 15 / 26

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SLIDE 16

Convergence test of exact low modes

  • 0.0005

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 5 10 15 20 25 Re (Cm−m) t/a 2629 configs × NN vol. sources. one-end-trick lma, 57 configs, Nev = 200 Nev = 400 Nev = 600

  • 1e-05

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 5 10 15 20 25 Re (Cm−m) t/a lma, 57 configs, Nev = 200 Nev = 400 Nev = 600

disconnected 2-point correlators disconnected η − η 2-point function (left) and (isoscalar) vector 2-point function (right) Nf = 2 + 1 + 1, a = 0.086 fm, mπ = 320 MeV, L = 2 fm

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 16 / 26

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2- and 3-point functions HVP Jµ(x) Jν(y) P → γ γ P(x) Jµ(y) Jν(z) Jµ conserved vector current P pseudoscalar (π0, η) interpolating field

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 17 / 26

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Connected and disconnected diagrams

Γµ t ti tf t ΓP Γν Γµ t ti tf t ΓP Γν

Γµ t ti tf t ΓP Γν Γµ t ti tf t ΓP Γν

Γµ t ti tf t ΓP Γν Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 18 / 26

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Exact low modes + stochastic high modes - diagram factorization

Γµ t ti tf t ΓP Γν

exact low / stochastic high mode decomposition allows factorization of diagrams sufficient to calculate building blocks of type V † Γ W build also 4-point function → Jµ(x1) Jν(x2) Jλ(x3) Jσ(x4)

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 19 / 26

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Summary - project status Wick contractions for building blocks are implemented accumulation of measurements for Nf = 2 tm+clover physical point ensemble with L = 4.4 fm (483) first measurements for L = 5.8 fm ( III/2017) measurements for Nf = 2 + 1 + 1 tm+clover physical point ensemble with L = 5.2 fm (643) (end of 2017) generalization to inexact deflation, combination with multi-grid method

  • n-going work on leading isospin-breaking corrections by collaboration in

Rome (RM123 method): path integral expansion in small parameters mu − md and αQED

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 20 / 26

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Also on the agenda update on HVP NLO contribution with Nf = 2 + 1 + 1, Nf = 2 tm+clover

µ µ γ (a) µ µ γ e (b) µ µ γ (c) 0.1 0.2 0.3 0.4

mPS

2 [GeV 2]

  • 0.1
  • 0.09
  • 0.08
  • 0.07

aµ

(3,vp) [10

  • 8]

a=0.079 fm L=1.6 fm a=0.079 fm L=1.9 fm a=0.079 fm L=2.5 fm a=0.063 fm L=1.5 fm a=0.063 fm L=2.0 fm linear quadratic pheno, Nf=2

ETMC result for Nf = 2 twisted mass from Lattice 2011

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 21 / 26

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Also on the agenda since May 2017 pion-nucleon scattering project

∆ e e N π N γ∗ Q2

  • n Nf = 2 tm+clover physical point lattices (4.4 fm and 5.8 fm)

includes measurements for pion-pion I = 1 scattering (FV spectrum, phase shift )

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 22 / 26

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SLIDE 23

Thank you very much for your attention.

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 23 / 26

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SLIDE 24

Comparison of point-to-all and lm+point-to-all mixed current correlator

1.00e-08 1.00e-07 1.00e-06 1.00e-05 1.00e-04 1.00e-03 1.00e-02 1.00e-01 5 10 15 20 25 Jk(t) Jk(0) t/a full, point-to-all, p = 0 lm + point-to-all

full, point-to-all result from 110 × 16 measurements low-mode + point-to-all with Nev = 1200 eigenvectors

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 24 / 26

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SLIDE 25

ahvp

e

with TMR

1.00e-12 1.20e-12 1.40e-12 1.60e-12 1.80e-12 2.00e-12 2.20e-12 2.40e-12 m2

π

0.05 0.1 0.15 0.2 0.25 ahlo

e

m2

PS [GeV2]

fit, a = 0.061 fm fit, a = 0.078 fm fit, a = 0.086 fm a = 0.061 fm, L = 2.9 fm a = 0.061 fm, L = 1.9 fm a = 0.078 fm, L = 3.7 fm a = 0.078 fm, L = 2.5 fm a = 0.086 fm, L = 2.8 fm

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 25 / 26

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SLIDE 26

dΠR/dQ2(0) with TMR

0.00 0.02 0.04 0.06 0.08 0.10 m2

π

0.05 0.1 0.15 0.2 0.25 dΠR/dQ2(0) [GeV−2] m2

PS [GeV2]

alphaQED fit, a = 0.061 fm fit, a = 0.078 fm fit, a = 0.086 fm a = 0.061 fm, L = 2.9 fm a = 0.061 fm, L = 1.9 fm a = 0.078 fm, L = 3.7 fm a = 0.078 fm, L = 2.5 fm a = 0.086 fm, L = 2.8 fm

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 26 / 26

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SLIDE 27

Abdel-Rehim, A., Burger, F., Deuzeman, A., Jansen, K., Kostrzewa, B., et al. (2013). Recent developments in the tmLQCD software suite. Abdel-Rehim, A. et al. (2015). Simulating QCD at the Physical Point with Nf = 2 Wilson Twisted Mass Fermions at Maximal Twist. Alexandrou, C., Bacchio, S., Finkenrath, J., Frommer, A., Kahl, K., and Rottmann, M. (2016). Adaptive Aggregation-based Domain Decomposition Multigrid for Twisted Mass Fermions.

  • Phys. Rev., D94(11):114509.

Antoine, G., Meyer, H. B., and Nyffeler, A. (2016). Lattice calculation of the pion transition form factor π0 → γ∗γ∗. Blum, T., Boyle, P. A., Izubuchi, T., Jin, L., Jttner, A., Lehner, C., Maltman, K., Marinkovic, M., Portelli, A., and Spraggs, M. (2016). Calculation of the hadronic vacuum polarization disconnected contribution to the muon anomalous magnetic moment.

  • Phys. Rev. Lett., 116(23):232002.

Cohen, S. D., Lin, H.-W., Dudek, J., and Edwards, R. G. (2008). Light-Meson Two-Photon Decays in Full QCD. PoS, LATTICE2008:159. Colangelo, G., Hoferichter, M., Kubis, B., Procura, M., and Stoffer, P. (2014a).

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 26 / 26

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SLIDE 28

Towards a data-driven analysis of hadronic light-by-light scattering.

  • Phys. Lett., B738:6–12.

Colangelo, G., Hoferichter, M., Procura, M., and Stoffer, P. (2014b). Dispersive approach to hadronic light-by-light scattering. JHEP, 09:091. Eidelman, S. and Passera, M. (2007). Theory of the tau lepton anomalous magnetic moment.

  • Mod. Phys. Lett., A22:159–179.

Feng, X., Aoki, S., Fukaya, H., Hashimoto, S., Kaneko, T., et al. (2012). Two-photon decay of the neutral pion in lattice QCD. Phys.Rev.Lett., 109:182001. Feng, X., Aoki, S., Hashimoto, S., Kaneko, T., Noaki, J.-I., and Shintani, E. (2011). Lattice calculation of neutral pion decay form factor using two different methods. PoS, LATTICE2011:154. Frezzotti, R. and Rossi, G. (2004). Chirally improving Wilson fermions. II. Four-quark operators. JHEP, 0410:070. Jansen, K. and Urbach, C. (2009). tmLQCD: A Program suite to simulate Wilson Twisted mass Lattice QCD. Comput.Phys.Commun., 180:2717–2738.

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 26 / 26

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SLIDE 29

Jegerlehner, F. and Szafron, R. (2011). ρ0 − γ mixing in the neutral channel pion form factor F e

π and its role in comparing e+e−

with τ spectral functions.

  • Eur. Phys. J., C71:1632.

Ji, X.-d. and Jung, C.-w. (2001). Studying hadronic structure of the photon in lattice QCD.

  • Phys. Rev. Lett., 86:208.

Nomura, D. and Teubner, T. (2013). Hadronic contributions to the anomalous magnetic moment of the electron and the hyperfine splitting of muonium.

  • Nucl. Phys., B867:236–243.

Nyffeler, A. (2016). On the precision of a data-driven estimate of hadronic light-by-light scattering in the muon g-2: pseudoscalar-pole contribution. Pauk, V. and Vanderhaeghen, M. (2014). Anomalous magnetic moment of the muon in a dispersive approach.

  • Phys. Rev., D90(11):113012.

Shintani, E., Aoki, S., Hashimoto, S., Onogi, T., and Yamada, N. (2009). pi0 to two-photon decay in lattice QCD. PoS, LAT2009:246.

Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 26 / 26