Hadronic Vacuum Polarization with Twisted Mass Fermions Marcus - PowerPoint PPT Presentation

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

Outline Motivation ◮ Twisted Mass fermions ◮ Setup for N f = 2 and N f = 2 + 1 + 1 HVP @ ETMC ◮ results for N f = 2 + 1 + 1 ◮ initial results for N f = 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

Motivation - Twisted Mass Lattice QCD various gauge field ensembles ◮ N f = 2 twisted mass ◮ N f = 2 + 1 + 1 twisted mass ◮ N f = 2 twisted mass + clover @ m π ◮ N f = 2 + 1 + 1 twisted mass + clover @ m π 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 Π R ( Q 2 ) and a hvp µ parity and SU (2) isospin symmetry breaking at non-zero lattice spacing SU (2) → U (1) 3 ◮ m ± π � = m 0 π ◮ � J up µ J up ν � � = � J dn µ J dn ν � by lattice artefacts Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 3 / 26

Motivation - Twisted Mass Lattice QCD (degenerate) light quark action ( N f = 2) � D W + m q + i µ l γ 5 τ 3 � � S l = χ l ¯ χ l ( x ) (1) x D W = 1 � a � ∇ f µ + ∇ b 2 ∇ b µ ∇ f � � 2 γ µ − µ − m cr µ O ( a ) improvement at maximal twist : m q = m 0 − m cr → 0 Noether current for J em µ � J µ ( x ) = 1 χ l ( x ) ( γ µ − 1) U µ ( x ) χ l ( x + a ˆ ¯ µ ) 2 � µ ) ( γ µ + 1) U † + ¯ χ l ( x + a ˆ µ ( x ) χ l ( x + a ˆ µ ) Z V = 1 � � � J µ ( x ) J ν ( y ) � + a − 3 δ (4) 0 = ∂ b x , y δ µν � C ν ( y ) � µ � �� � Π µν ( x , y ) exact at non-zero lattice spacing Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 4 / 26

Motivation - Twisted Mass Lattice QCD (non-degenerate) heavy quark action ( . . . + 1 + 1) D W + m q + i µ σ γ 5 τ 1 + µ δ τ 3 � � � S h = χ h ¯ χ h ( x ) (2) x µ σ τ 1 and µ δ τ 3 break isospin symmetry completely − → no conserved vector current for strange and charm Osterwalder-Seiler ( = mixed action) setup [Frezzotti and Rossi, 2004] � � ¯ D W + m q + i µ f γ 5 τ 3 � S val � = ψ f ψ f ( x ) h f = s , c x µ s , µ c tuned by physical value of 2 m 2 K − m 2 PS and m D µ ∼ ¯ (valence) Noether currents J f ψ f ψ f ; automatic O ( a ) remains valid Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 5 / 26

HVP from tmLQCD and N f = 2 + 1 + 1 consider hadronic leading-order ∆ α QED ( Q 2 ) ∝ Π γ ( Q 2 ) � � � Q 2 �� H 2 � Q 2 � � 5 + 1 + 4 Q 2 · ∆ α hvp QED ( Q 2 ) = − 4 πα 0 9 Π ud 9 Π s 9 Π c . R R R H 2 phys (3) inter-/extrapolation, Π(0) M N − 1 g 2 i m 2 � � Π f low ( Q 2 ) = a j ( Q 2 ) j i i + Q 2 + m 2 i =1 j =0 B − 1 C − 1 � � b k ( Q 2 ) k + Π f high ( Q 2 ) = log( Q 2 ) c l ( Q 2 ) l k =0 l =0 Π f ( Q 2 ) = (1 − Θ( Q 2 − Q 2 low ( Q 2 ) + Θ( Q 2 − Q 2 match ))Π f match )Π f high ( Q 2 ) chiral and continuum extrapolation ∆ α hvp QED ( Q 2 )( m PS , a ) = A + B 1 m 2 PS (+ . . . ) + C a 2 Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 6 / 26

HVP from tmLQCD with N f = 2 + 1 + 1 a [ fm ] m PS [MeV] L [fm] m PS L 0.003 0 . 061 227 2.9 3.3 0 . 061 318 2.9 4.7 0 . 061 387 1.9 3.7 ∆ α hvp , ud (1 GeV 2 ) 0.0025 0 . 078 274 2.5 3.5 0 . 078 319 2.5 4.0 0.002 0 . 078 314 3.7 5.9 N f = 2 result, standard fit 0 . 078 393 2.5 5.0 N f = 2 result, Pad´ e fit 0.0015 a = 0 . 086 fm, L = 2 . 8 fm 0 . 078 456 2.5 5.8 a = 0 . 078 fm, L = 2 . 5 fm a = 0 . 078 fm, L = 1 . 9 fm 0 . 078 491 1.9 4.7 a = 0 . 078 fm, L = 3 . 7 fm 0 . 086 283 2.8 4.0 a = 0 . 061 fm, L = 1 . 9 fm 0.001 a = 0 . 061 fm, L = 2 . 9 fm 0 . 086 323 2.8 4.6 0 0.05 0.1 0.15 0.2 0.25 0 . 086 361 2.8 5.1 m 2 � GeV 2 � PS statistics O (250 / 150 / 150) gauge configurations for up-down / strange / charm quark Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 7 / 26

HVP from tmLQCD with N f = 2 + 1 + 1 and a hvp l 0.01 0.008 QED ( Q 2 ) 0.006 ∆ α hvp 0.004 0.002 lattice data linearly extrapolated to m π in CL ∆ α from Jegerlehner’s alphaQED package 0 0 2 4 6 8 10 Q 2 � GeV 2 � a hlo tmLQCD disp. analyses l 1 . 866 (10) (05) · 10 − 12 [Nomura and Teubner, 2013] 1 . 782 (64) (85) · 10 − 12 e 6 . 91 (01) (05) · 10 − 8 [Jegerlehner and Szafron, 2011] 6 . 78 (24) (16) · 10 − 8 µ 3 . 38 (4) · 10 − 6 [Eidelman and Passera, 2007] 3 . 41 (8) (6) · 10 − 6 τ Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 8 / 26

Quark-disconnected contribution connected, combined isospin components 0.02 disconnected, isospin 0 component disconnected, isospin 1 component 0 0.004 -0.02 V Π( Q 2 ) 0 -0.04 -0.004 Z 2 -0.06 0 0.2 0.4 0.6 0.8 1 -0.08 -0.1 -0.12 0 0.2 0.4 0.6 0.8 1 1.2 Q 2 / GeV 2 isovector and isoscalar contribution Π 3 µν ( x , y ) = � J 3 µ ( x ) J 3 ν ( y ) � disc tm only , with one − end trick Π 0 µν ( x , y ) = � J 0 µ ( x ) J 0 ν ( y ) � disc a = 0 . 078 fm , m π = 393 MeV , L = 2 . 5 fm , m π L = 5 . 0, up-down contribution 1548 × 24 + 4996 × 48 gauge configurations × stochastic volume sources Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 9 / 26

twisted mass + clover at N f = 2 and physical pion mass � D W [ U ] + m q + i µ l γ 5 τ 3 + i � � S l = χ l ¯ 4 c sw σ µν F µν [ U ] χ l ( x ) (4) x 400 L/a = 24 clover term not added for O ( a ) L/a = 32 L/a = 48 improvement 300 but to reduce the effects of isospin splitting M π 0 [MeV] 200 pion masses range from 130 to 350 MeV 100 2 volumes at m PS = 130 MeV , 4 . 4 fm and 5 . 8 fm 0 (single) lattice spacing 0 100 200 300 400 a = 0 . 0914 (3) (15) fm M π ± [MeV] pion mass splitting compatible with zero ⇒ reduced finite size effects ⇒ reduces isospin splitting effects Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 10 / 26

twisted mass + clover at N f = 2 and physical pion mass, L = 4 . 4 fm e [10 − 12 ] 1.6 1.4 1.2 a ud µ [10 − 8 ] 5.8 5.4 a ud 5.0 τ [10 − 6 ] 2.7 a ud 2.3 0 0.05 0.1 0.15 0.2 0.25 M 2 GeV 2 � � π physical point extr. N f = 2 extr. N f = 2 + 1 + 1 · 10 12 a hvp 1 . 45(11) 1 . 51(04) 1 . 50(03) e a hvp · 10 8 5 . 52(39) 5 . 72(16) 5 . 67(11) µ a hvp · 10 6 2 . 65(07) 2 . 65(02) 2 . 66(02) τ [Abdel-Rehim et al., 2015] Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 11 / 26

Restart 2017 focus on tmLQCD+clover at N f = 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 N f = 2 + 1 + 1 at physical pion mass in production more measurements and statistics Π µν ( x , y ) = � J µ ( x ) J ν ( 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

Restart 2017 - extended list of observables HVP and π 0 , η, η ′ → γ γ ( η, η ′ on upcoming N f = 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] µνλσ = Π π 0 µνλσ + Π η ′ Π HLbL µνλσ + Π FsQED + Π η µνλσ + Π ππ µνλσ + . . . µνλσ γ γ dispersive approach F P → γγ ∗ + . . . π 0 , η, η ′ F P → γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ µ µ µ µ recent calculation by Mainz group [Nyffeler, 2016, Antoine et al., 2016] Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 13 / 26

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 � M ee � − 1 � 1 � � � − M − 1 ee M eo C − 1 M − 1 M eo 0 D − 1 ee tm = = × C − 1 − γ 5 M oe M − 1 M oe M oo 0 γ 5 ee exact inverse M − 1 available ee computationally intensive part: inversion of C → C − 1 construct subspace from N ev eigenvectors to lowest-lying eigenvalues of CC † , CC † V = V Λ decomposition of odd sub-lattice with orthogonal projectors 1 = P V + P ⊥ V , P V = V V † inversion on P V becomes trivial inversion on P ⊥ V becomes cheap (exact deflation of lowest eigenmodes) � 1 � � 1 � � � − M − 1 ee M eo C − 1 M − 1 0 0 D − 1 ee tm = � ξ ξ † � C − 1 P V + P ⊥ − γ 5 M oe M − 1 0 0 γ 5 V E ee Marcus Petschlies (HISKP) HVP @ ETMC g − 2 Initiative 14 / 26