Interplay between Lattice and { Model and/or Dispersive Representation } for g-2 HLbL Tom Blum, Norman Christ, Masashi Hayakawa, Taku Izubuchi, Luchang Jin, Chulwoo Jung, Chrisoph Lehner (RBC&UKQCD) RIKEN BNL Research Center 2018-06-18, “Second Plenary Workshop of the Muon g-2 Theory IniBaBve” Mainz, Germany 1
Introduction [ HVP: Bernecker Meyer 2011] 2
Sweat spots of Lattice vs DR/Model n Lattice, after take continuum/infinite volume limits with all disconnected, short distance (high energy) : less noisy long distance (low energy) : very noisy n DR / Model ( or experiments ) heavy particle / multiple hadron : less control light particle, pi0 pole or pion-loop : well controlled -> Could cover sweat spots complementarily ? n For HVP , a good comparison/interplay is done in Eucliean coordinate space [ Christoph Lehner’s talk ] 3
First try [ Luchang Jin’s talk ] n LMD model in coordinate space n Fixed min {|x-y|,|x-z|,|y-z| } < R(min) n Plot as function of max {|x-y|,|x-z|,|y-z| } = R(max) n L = 9.6 fm, a=0.1fm, Nf=2+1 physical pion mass n Subtracted lepton part (to isolate the long-distant part in this exercise) n Connected only. Model is multiplied by 34/9 according to conn:disconn = 34:(-25) from charge factors 4
HLbL point source method [L. Jin et al. 1510.07100] • Anomalous magnetic moment, F 2 ( q 2 ) at q 2 ! 0 limit ( q 2 = 0) F cHLbL P ( � s 0 ,s ) i x,y,z,x op 0) F C 2 u s 0 ( ~ k ( x, y, z, x op ) u s ( ~ = ✏ i,j,k ( x op � x ref ) j · i ¯ 0) 2 2 V T m • Stochastic sampling of x and y point pairs. Sum over x and z . F C ( � ie ) 6 G ⇢ , � , ( x, y, z ) H C ⌫ ( x, y, z, x op ) = ⇢ , � , , ⌫ ( x, y, z, x op ) , x op , ν y x x, ρ y, σ z, κ z α , ρ α , ρ η , κ t src η , κ t snk x src x snk β , σ β , σ 5
cHLbL Subtraction using current conservation • From current conservation, 0 , and mass gap, ∂ ρ V ρ ( x ) = h xV ρ ( x ) O (0) i ⇠ | x | n exp( � m π | x | ) H C X X ρ , σ , κ , ν ( x, y, z, x op ) = h V ρ ( x ) V σ ( y ) V κ ( z ) V ν ( x op ) i = 0 x x X H C ρ , σ , κ , ν ( x, y, z, x op ) = 0 z at V ! 1 and a ! 0 limit (we use local currents). • We could further change QED weight G (2) G (1) ρ , σ , κ ( x, y, z ) � G (1) ρ , σ , κ ( y, y, z ) � G (1) ρ , σ , κ ( x, y, y ) + G (1) ρ , σ , κ ( x, y, z ) = ρ , σ , κ ( y, y, y ) without changing sum P ρ , σ , κ , ν ( x, y, z, x op ) . x,y,z G ρ , σ , κ ( x, y, z ) H C • Subtraction changes discretization error and finite volume error. • Similar subtraction is used for HVP case in TMR kernel, which makes FV error smaller. • Also now G (2) σ , κ , ρ ( z, z, x ) = G (2) σ , κ , ρ ( y, z, z ) = 0 , so short distance O ( a 2 ) is suppressed. • The 4 dimensional integral is calculated numerically with the CUBA library cubature ( x, y, z ) is represented by 5 parameters, compute on N 5 grid points and rules. interpolates. ( | x � y | < 11 fm). 6
Integrand : Lattice vs LVD (preliminary) R min = 1.0 fm 7 Pion TFF sub Pion TFF no-sub 6 Lattice 48D sub integrand F 2 (0) (10 − 10 ) 5 Lattice 48D no-sub 4 3 2 1 0 − 1 − 2 0 1 2 3 4 5 6 7 8 R max (fm) 7
Integrand (preliminary) R(min) = 1.0 fm 1.4 Pion TFF sub Lattice 48D sub 1.2 integrand F2(0) (1e-10) 1 0.8 0.6 0.4 0.2 0 -0.2 0 1 2 3 4 5 6 7 8 R(max) (fm) model integral is extrapolated to con2nuum/infinite volume limits extrapola2ons to be scru2nized 8
Patch-up example Preliminary 48D R(min) = 0.5 fm 50 Pion TFF sub Lattice 48D sub 40 Combine sub F2(0) (1e-10) 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 R(max) (fm) Switching point ( long: model, short: lattice ) 9
Preliminary 48D R(min) = 1.0 fm 50 Pion TFF sub Lattice 48D sub 40 Combine sub F2(0) (1e-10) 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 R(max) (fm) 10
Preliminary 48D R(min) = 2.0 fm 50 Pion TFF sub Lattice 48D sub 40 Combine sub F2(0) (1e-10) 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 R(max) (fm) 11
Preliminary 48D R(min) = 5.0 fm 50 Pion TFF sub Lattice 48D sub 40 Combine sub F2(0) (1e-10) 30 20 10 0 -10 0 1 2 3 4 5 6 7 8 R(max) (fm) 12
Is this safe ? n At given distance, there are other than pi0 contribution in DR and models [ truncation ] n Probably not large for appropriate choice n To be safer, we could try to consider subtracting pi0 contribution from Lattice GH = GH(Lat; all) - GH(Lat; pi0) + GH(DR; pi0) n How to compute GH(Lat; pi0) is non-trivial 13
Similar problem in tau HVP [ Hiroshi Ohki et al. arXiv:1803.07228 ] n In case of Vus analysis of tau -> up-strange inclusive hadronic decay n We subtract K-pole contribution from lattice by fitting HVP in the on-shell long-distance, and evaluate the rest: C(t) = A exp(- mK t) + rest( t ) [ A, mK is from fit ] ( also tau-input for g-2 : [ Mattia Brunno’s talk ]) 14
Tau decay ν τ ν τ { } hadrons Im s τ − τ − s • W − W − ¯ ¯ u u V-A current � (Hadronic) vacuum polarization function Π ( Q 2 ) • Experiment side : τ → ν + had through V-A vertex. EW correction S EW Γ ( τ − → hadrons ij ν τ ) R ij = Γ ( τ − → e − ¯ ν e ν τ ) Z m 2 ✓ ◆ ✓ ◆ � 12 π | V ij | 2 S EW s 1 + 2 s τ Im Π (1) ( s ) + Im Π (0) ( s ) = 1 − m 2 m 2 m 2 0 τ τ τ | {z } ≡ Im Π ( s ) • Lattice side : The Spin=0 and 1, vacuum polarization, Vector(V) or Axial (A) current- current two point Z d 4 xe iqx D E Π µ ν ij ; V/A ( q 2 ) = i 0 | T J µ ij ; V/A ( x ) J † µ ij ; V/A (0) | 0 = ( q µ q ν − q 2 g µ ν ) Π (1) ij ; V/A ( q 2 ) + q µ q ν Π (0) ij ; V/A 15
τ inclusive decay experiments To compare with experiments, ✓ ◆ � 1 + 2 s a conventional value of |Vus|=0.2253 is used ρ ( s ) ≡ | V us | 2 Im Π 1 ( s ) + Im Π 0 ( s ) ˜ m 2 τ 1 - π 0 , K 0 π - (Adematz) Belle K 0 π - π 0 Belle K - π + π - BaBar K - 2 π , Κ (3 − 5) π ,K η 0.1 ALEPH K pQCD, D=0 OPE (nf=3) 0.01 [K. Maltman ] 0.001 s )˜ ρ ( s ) 0.0001 1e-05 0 1 2 3 4 2 ] s [GeV γ K ω ( m 2 For K pole, we assume a delta function form K ) γ K ∼ 2 | V us | 2 f 2 obtained from either experimental value of K→μ or τ→k decay width. K γ K [ τ → K ν τ ] =0 . 0012061(167) exp (13) IB [HFAG16] 8 γ K [ K µ 2 ] =0 . 0012347( 29) exp (22) IB [PDG16] 16
Pi0 subtraction on Lattice [ N. Christ et al @ UConn ] 17
[ also Luchang Jin’s talk ] 18
Lattice implementation n lattice pi0-gamma-gamma FF could be computed separately, and if it’s accurately determined, we could replace for long-distance of the full HLbL n Or compute pi0-pole contribution simultaneously with the full HLbL on the same ensemble and subtract under the jack-knife 19
Discussion n Interplay b/w Lattice and DR/model is a useful “plan-B” for HVP . Could we apply to HLbL ? n Lattice : disconnected, continuum/infinite V limit n Another interplay for HLbL possible ? n How about the box diagram in DR ? n Sum-rule for the full HLbL from Lattice to constraint DR or model ? Int[ pole, cuts in DR ] = Int[ Euclidean Amp ] n Use of GEVP in subtracting pi0 or other specific contribution ? [ A. Meyer’s talk ] 20
Finite Energy Sum Rule (FESR) [Shifman, Vainshtein, and Zakharov ’79] The finite energy sum rule (FESR) � s 0 ω ( s ) ρ ( s ) ds = − 1 � ω ( s ) Π ( s ) ds, ( s 0 : finite energy) 2 π i 0 | s | = s 0 w(s) is an arbitrary regular function such as polynomial in s. • LHS : spectral function ρ(s) is related to the experimental τ inclusive decays ◆ 2 ✓ =12 π 2 | V us | 2 S EW dR us ; V/A ✓ ◆ � 1 − s 1 + 2 s Im Π 1 ( s ) + Im Π 0 ( s ) ds m 2 m 2 m 2 τ τ τ Im(s) pQCD ✓ ◆ � 1 + 2 s ρ ( s ) ≡ | V us | 2 Im Π 1 ( s ) + Im Π 0 ( s ) ˜ m 2 τ Re(s) • RHS … Analytic calculation ���������������������������������������������������������������������������������� s 0 s th with perturbative QCD (pQCD) and OPE 4 τ experiment 21
Our new method : Combining FESR and Lattice • If we have a reliable estimate for Π ( s ) in Euclidean (space-like) points, s = − Q 2 k < 0 , we could extend the FESR with weight function w ( s ) to have poles there, Np Z ∞ X w ( s ) Im Π ( s ) = π Res k [ w ( s ) Π ( s )] s = − Q 2 Our strategy k sth k If we have a reliable estimate for Π(s) in Euclidean (space-like) points, ✓ 1 + 2 s ◆ we could extend the FESR with weight function w(s) to have N poles there, Im Π (1) ( s ) + Im Π (0) ( s ) ∝ s ( | s | → ∞ ) Π ( s ) = m 2 τ • For N p ≥ 3 , the | s | → ∞ circle integral vanishes. (generalized dispersion relation ) Im( s ) pQCD OPE spectral data Im(s) Im(s) pQCD Lattice HVP Re(s) Re(s) Re( s ) XXX s 0 s th Lattice HVPs τ experiment τ experiment & pQCD 22
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