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 Harvey Meyer HVP by Mainz-CLS
Outline ◮ Calculation in the time-momentum representation in N f = 2 + 1 QCD ◮ Technical improvements over our N f = 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¨ orz, HM, D. Mohler, K. Ottnad, H. Wittig. All numerical results in this talk are still preliminary! Harvey Meyer HVP by Mainz-CLS
HVP: definitions (Euclidean space) ◮ primary object on the lattice: G µν ( x ) = � j µ ( x ) j ν (0) � . ◮ polarization tensor: � � Q µ Q ν − δ µν Q 2 � d 4 x e iQ · x G µν ( x ) = Π( Q 2 ) . Π µν ( Q ) ≡ � ∞ ◮ dQ 2 K ( Q 2 ; m 2 a hvp = 4 α 2 µ ) [Π( Q 2 ) − Π(0)] µ 0 12 π 2 , R ( s ) ≡ σ ( e + e − → hadrons) ◮ Spectral representation: ρ ( s ) = R ( s ) , 4 πα ( s ) 2 / (3 s ) � ∞ ρ ( s ) Π( Q 2 ) − Π(0) = Q 2 ds s ( s + Q 2 ) . 4 m 2 π Lautrup, Peterman & de Rafael Phys.Rept 3 (1972) 193; Blum hep-lat/0212018 (PRL) Harvey Meyer HVP by Mainz-CLS
The time-momentum representation (TMR) ◮ mixed-representation Euclidean correlator: (natural on the lattice) � 3 � G TMR ( x 0 ) = − 1 d 3 x G kk ( x ) , 3 k =1 ◮ the spectral representation: � ∞ dω ω 2 ρ ( ω 2 ) e − ω | x 0 | , G TMR ( x 0 ) = x 0 � = 0 . 0 ◮ Finally, the quantity a hvp is given by µ � 2 � ∞ � α a hvp dx 0 G ( x 0 ) ˜ = f ( x 0 ) , µ π 0 � 2 π 2 − 2 + 8 γ E + 4 0 − 8 ˜ x 2 f ( x 0 ) = + ˆ x 0 K 1 (2ˆ x 0 ) x 2 m 2 ˆ ˆ µ 0 � � � 3 x 0 ) + G 2 , 1 x 2 +8 log(ˆ ˆ 0 | 2 1 , 3 0 , 1 , 1 2 x 0 = m µ x 0 , γ E = 0 . 577216 .. and G m,n where ˆ is Meijer’s function. p,q Bernecker & Meyer 1107.4388; Mainz-CLS 1705.01775. Harvey Meyer HVP by Mainz-CLS
Expected integrand for a hvp (using pheno. R ) µ 0.5 0.4 � � t � 0.3 10 3 � t 3 G � t � K 0.2 0.1 41% 45% 11% 3% 0.0 0 1 2 3 4 5 6 t � fm � Bernecker & Meyer 1107.4388 Harvey Meyer HVP by Mainz-CLS
Finite-size effects: discrete states on the torus 12 10 8 R 1 ( ω 2 ) 6 4 2 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω [GeV] � � 3 / 2 | F π ( s ) | 2 + other channels 1 1 − 4 m 2 Isovector final states: ρ ( s ) = π /s 48 π 2 � � � �� � � � � 3 πs 2 � � � � | F π ( s ) | 2 = qφ ′ ( q )+ k ∂δ 1 ( k ) � � d x j z ( x ) � � ππ � 0 . � L � � ∂k 2 k 5 L 3 HM 1105.1892 (PRL); figure: model for F π ( s ) . Harvey Meyer HVP by Mainz-CLS
Landscape of CLS ensembles 450 N300 N202 (H200) H101 B450 400 S400 H102 N302 N203 350 m π [MeV] 300 N200 N401 N101 (H105) J303 250 C101 D200 200 E250 150 100 0 0 . 050 2 0 . 065 2 0 . 077 2 0 . 085 2 a 2 [fm] N f = 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. Harvey Meyer HVP by Mainz-CLS
A first look at the three connected integrands on ensemble D200 G ( x 0 ) � K ( x 0 ) /m µ 0 . 016 Light Strange (x6) Charm (x6) 0 . 012 0 . 008 0 . 004 0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 x 0 [fm] Harvey Meyer HVP by Mainz-CLS
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: f K + 1 2 f π . ◮ new: non-perturbative on-shell improvement of the vector current: calculation of c V [G´ erardin, Harris et al., in prep.] . ⇒ a hvp approaches its continuum value with O( a 2 ) corrections. µ ◮ local current ¯ ψ ( x ) γ µ ψ ( x ) as well as lattice Noether current ⇒ use two discretizations of the current-current correlator (ll,lc). Perform constrained simultaneous continuum limit for a hvp . µ ◮ We benefit from a dedicated lattice calculation of the I = ℓ = 1 scattering phase and of the pion form factor at timelike q 2 : [H¨ orz et al. 1511.02351 and in prep.] . Used to control tail of isovector correlation function and for the finite-size correction. Harvey Meyer HVP by Mainz-CLS
Charm contribution Harvey Meyer HVP by Mainz-CLS
Tuning of κ c , the bare charm mass parameter Tuning of κ c : by imposing m eff s = m exp D s = 1972 MeV on each lattice ensemble. c ¯ N200 4 . 15 × 10 6 4 . 1 × 10 6 4 . 05 × 10 6 4 × 10 6 D s m 2 3 . 95 × 10 6 3 . 9 × 10 6 3 . 85 × 10 6 7 . 81 7 . 82 7 . 83 7 . 84 7 . 85 7 . 86 1 /κ c ◮ Interpolation : m 2 D s vs 1 /κ c → linear behaviour ◮ am D s = 0 . 86 at β = 3 . 40 : discretization effects are expected to be large. Harvey Meyer HVP by Mainz-CLS
y = m 2 π / (16 π 2 f 2 Chiral & continuum extrapolation of a c ( � π ) ) µ a HVP , LO × 10 − 10 µ,c 30 25 20 15 10 β = 3 . 40 β = 3 . 46 5 β = 3 . 55 β = 3 . 70 0 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 . 12 y � 10 10 · a c µ = 14 . 95(47) stat (11) χ ◮ The O ( a ) − improvement reduces lattice artefacts significantly ◮ The simultaneous continuum extrapolation (linear in a 2 ) works well. Harvey Meyer HVP by Mainz-CLS
Strange contribution Harvey Meyer HVP by Mainz-CLS
Strange contribution: data at physical quark masses (E250) G ( x 0 ) � K ( x 0 ) /m µ loc-loc 0 . 002 cons-loc 0 . 0015 0 . 001 0 . 0005 0 0 0 . 5 1 1 . 5 2 2 . 5 3 x 0 [fm] ◮ With full O ( a ) − improvement of the vector currents ◮ Two discretizations almost coincide: remaining discretization errors small. Harvey Meyer HVP by Mainz-CLS
Finite size effects on strangeness correlator at m π = 280 MeV G ( x 0 ) � K ( x 0 ) /m µ 0 . 002 L = 2 . 05 fm L = 2 . 70 fm 0 . 0015 L = 4 . 10 fm ⇒ At the level of 0 . 5 % precision, volume effects are 0 . 001 negligible for L ≥ 2 . 7 fm. 0 . 0005 0 0 0 . 5 1 1 . 5 2 2 . 5 3 x 0 [fm] a ll − imp a lc − imp CLS µ 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) Harvey Meyer HVP by Mainz-CLS
y = m 2 π / (16 π 2 f 2 Chiral & continuum extrapolation ( � π ) ) ◮ Fit : a µ ( a, � y, d ) = y exp ) + δ d a 2 + γ 1 ( � a µ (0 , � y − � y exp ) + γ 2 ( � y log � y − � y exp log � y exp ) a HVP , LO × 10 − 10 µ,s 100 90 80 70 60 50 40 β = 3 . 40 β = 3 . 46 30 β = 3 . 55 β = 3 . 70 20 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 . 12 y � a HVP , LO = 53 . 6(2 . 5) stat (0 . 8) χ µ,s ◮ The first error is the statistical error; ◮ second error : variation wrt setting the cut at m π = 360 MeV; ◮ statistical error dominated by the scale setting error. Harvey Meyer HVP by Mainz-CLS
Light-quark contributions Harvey Meyer HVP by Mainz-CLS
Auxiliary calculation: time-like pion form factor For all the ensembles with m π < 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 ( m π = 280 MeV): parametrizing the time-like pion form factor in the Gounaris-Sakurai form: m GS Γ GS = 776(4) MeV = 59(2) MeV. ρ ρ Other parametrizations are being investigated. [Bulava, H¨ orz et al., 1511.02351 (LAT2015).] Harvey Meyer HVP by Mainz-CLS
Saturation of the correlator by the low-lying states (D200) G ( x 0 ) � K ( x 0 ) /m µ 0 . 016 1 exp fit cf. slide 6: loc-cons n=4 12 n=3 0 . 012 n=2 10 n=1 8 x cut R 1 ( ω 2 ) 0 6 0 . 008 4 2 0 . 004 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω [GeV] 0 0 0 . 5 1 1 . 5 2 2 . 5 3 x 0 [fm] ∞ � A n e − E n x 0 G ( x 0 ) = n =1 ◮ Excellent cross-check that the tail is understood. [Update from H. Wittig et al. 1710.10072 (LAT2017)] Harvey Meyer HVP by Mainz-CLS
Saturation of the correlator by the low-lying states (D200) G ( x 0 ) � K ( x 0 ) /m µ 0 . 004 x cut 1 exp fit 0 cf. slide 6: loc-cons n=4 12 0 . 003 xn=3 yn=2 10 zn=1 8 R 1 ( ω 2 ) 0 . 002 6 4 2 0 . 001 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω [GeV] 0 2 2 . 2 2 . 4 2 . 6 2 . 8 3 3 . 2 x 0 [fm] ∞ � A n e − E n x 0 G ( x 0 ) = n =1 ◮ Excellent cross-check that the tail is understood. [Update from H. Wittig et al. 1710.10072 (LAT2017)] Harvey Meyer HVP by Mainz-CLS
Finite size effects: check on the lattice ( m π = 280 MeV H105 vs. N101 ) G ( x 0 ) � K ( x 0 ) /m µ 0 . 013 0 . 012 0 . 011 0 . 01 0 . 009 0 . 008 0 . 007 L/a = 32 ( L = 2 . 70 fm) 0 . 006 L/a = 48 ( L = 4 . 10 fm) L/a = 32 with FSE 0 . 005 L/a = 48 with FSE 0 . 004 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 x 0 [fm] ◮ FSE consistent with the estimate using the pion FF and L¨ uscher formalism (see [1705.01775 Mainz-CLS]) . Harvey Meyer HVP by Mainz-CLS
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