Hadronic contribu-ons to (g 2) μ from La4ce QCD: Results from the Mainz group

Transcription

1 Hadronic contribu-ons to (g 2) μ from La4ce QCD: Results from the Mainz group Hartmut Wi4g PRISMA Cluster of Excellence, Ins6tute for Nuclear Physics and Helmholtz Ins6tute Mainz Interna-onal Workshop on e+e collisions from Phi to Psi 217 Mainz, Schloss Waldthausen June 217

2 The Mainz (g 2)μ project Collaborators: N. Asmussen, A. Gérardin, O. Gryniuk, G. von Hippel, H. Horch, H. Meyer, A. Nyffeler, V. Pascalutsa, A. Risch, HW M. Della Morte, A. Francis, J. Green, V. Gülpers, B. Jäger, G. Herdoíza Direct determina6ons of LO a hvp µ Exact QED kernel Forward scavering amplitude Transi6on form factor for! Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 2

3 Hadronic Vacuum Polarisa-on Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 3

4 La4ce QCD approach to HVP Convolu6on integral over Euclidean momenta: a hvp µ = 2 Z 1 [Lautrup & de Rafael; Blum] dq 2 f (Q 2 ) ˆ (Q 2 ), ˆ (Q 2 ) = 4 2 (Q 2 ) () Weight func6on f(q 2 ) strongly peaked near muon mass Accurate determina6on of Π(Q 2 ) near Q 2 Control effects of finite volume; mπl 4 not sufficient Include quark-disconnected diagrams: Include isospin breaking: mu md, QED correc6ons Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 4

5 La4ce QCD approach to HVP Direct method: determine Π(Q 2 ) from VP tensor µ (Q) = Z d 4 x e iq x D J µ (x)j () E (Q µ Q µ Q 2 ) (Q 2 ) J µ = 2 3 u µu 1 3 d µd 1 3 s µs +... Obtain Padé representa6on of Π(Q 2 ) from fits for Q 2 apple Q 2 cut.1.5 GeV 2 Hybrid method Π(Q 2 ) Padé approx. numerical interpola6ons P.T. [Golterman, Maltman & Peris 214].1 GeV 2 Q 2 Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 5

6 La4ce QCD approach to HVP Time-momentum representa-on: [Bernecker & Meyer 211] a hvp µ = 2 Z 1 Z 1 f (x ) = 4 2 dx f (x ) G(x ), G(x ) = a 3 X ~x dq 2 f (Q 2 ) " x 2 4 Q 2 sin2 1 2 Qx # hj k (x)j k ()i Control long-distance behaviour of G(x) large sta6s6cal noise 8 >< G(x ) data, x apple x,cut G(x ) = >: G(x ) ext, x > x,cut G(x) dominated by two-pion state for x!1 Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 6

7 HVP: current data sets CLS consor-um Coordinated Lafce Simula6ons Nf = 2 flavours of O(a) improved Wilson fermions Three values of the lafce spacing: a =.76,.66,.49 fm Pion masses and volumes: m min = 185 MeV, m L > 4 Focus on methodology and systema6cs arxiv: Nf = 2+1 flavours of O(a) improved Wilson fermions Three values of the lafce spacing: a =.85,.65,.5 fm Pion masses and volumes: m min = 2 MeV, m L > 4 To be included: two more lafce spacings; physical pion mass Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 7

8 Hybrid Method Lafce observable: a 4 X f q 2 f Z V X e iq(x+aˆµ/2) Use twisted boundary condi6ons to reach smaller Q 2 [Della Morte et al. 211] x 1 D V con µ, f (x)vloc, f ()E = (Q µ Q µ Q 2 ) ( ˆQ) (Q 2 ) ud m = 185 MeV Padé [1,1] twisted boundary conditions periodic boundary conditions Time moment Fit Π(Q 2 ) to low-order Padé approximants m = 268 MeV Hartmut Wittig Q 2 [GeV 2 ] (g 2) μ from Lattice QCD / Mainz 8

9 Time-momentum representa-on Lafce observable: G f (x ) = a3 3 (a hvp µ ) f = 2 Z 1 Control tail of integrand: Naive single exponen6al: G(x ) ext = A e m x dx f (x ) G f (x ) X X q 2 f Z V k ~x G(x ) K(x )/m µ D V con k, f (x, ~x)v loc k, f ()E x cut Data 1-Exp GS(L) GS( ) Single exponen6al plus 2-pion state: G(x ) ext = A e m x + B e E 2 (~p)x Gounaris-Sakurai parameterisa6on of 6meline pion form factor x cut x [fm] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 9

10 Disconnected Contribu-ons Exploit stochas6c noise cancella6on between (ud) and s quarks [Gülpers et al., arxiv: ; V. Gülpers, PhD Thesis 215] 2e 7.4 G (x) 2e 7 4e 7 G (x)/g ρρ (x) e x /a x /a Run N cfg N r T/a x a hvp µ E % 28.3% F % % ) a hvp µ (ahvp µ ) disc (a hvp µ ) con apple 2% ils of the evaluation of quark-disconnected contribution G disc (x ) (see eq. (76)). N r ber of stochastic sources per timeslice, while x represents the Euclidean time at which Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 1

11 TMR analysis of finite-volume effects Iso-vector correlator in infinite and finite volume: G (x, 1) = Con6nuum of states with E G (x, L) = Z 1 X n d!! 2 (! 2 )e! x, (! 2 ) = m A n 2 e! nx,! n = 2 q m 2 + k 2 n 4m 2! 2! 3/2 F (!) 2 A n 2 = 2k2 n 3! 2 n F (! n ) 2 n k (k) + k 1 (k)o k=kn, Discrete energy levels: E 2 11(k) + q m 2 + (2 /L) 2 kl = n, n = 1, 2,... 2 Use informa6on on Fπ(ω) to determine finite-volume shii: a hvp µ (1) a hvp µ (L) = 2 Z 1 dx f (x ) G (x, 1) G (x, L) Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 11

12 TMR analysis of finite-volume effects Compute Fπ(ω) via energy levels in isovector channel in finite volume Approximate Fπ(ω) by Gounaris-Sakurai parameterisa6on: (mρ, Γρ) Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 12

13 TMR analysis of finite-volume effects Compute Fπ(ω) via energy levels in isovector channel in finite volume Approximate Fπ(ω) by Gounaris-Sakurai parameterisa6on: (mρ, Γρ) (a hvp µ ) ud Hybrid TMR (a hvp µ ) s Hybrid TMR (a hvp µ ) c Hybrid TMR Finite-volume correc6ons sizeable Good consistency between hybrid method and TMR a hvp µ Hybrid TMR Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 12

14 Final result for Nf = 2 Es6mate from TMR including finite-volume correc6on: a hvp µ = (654 ± 32 stat ± 17 syst ± 1 scale ± 7 FV + 1 disc) 1 1 Sta6s6cal error: Systema6c error: Scale sefng: Finite volume shii: Disconnected parts: 4.8% at the physical point 2.6% from procedural varia6ons 1.5% from uncertainty in amμ 1.% from varia6ons in (mρ, Γρ) 1.5% from upper bound on magnitude Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 13

15 Final result for Nf = 2 Es6mate from TMR including finite-volume correc6on: a hvp µ = (654 ± 32 stat ± 17 syst ± 1 scale ± 7 FV + 1 disc) 1 1 Sta6s6cal error: Systema6c error: Scale sefng: Finite volume shii: Disconnected parts: 4.8% at the physical point 2.6% from procedural varia6ons 1.5% from uncertainty in amμ 1.% from varia6ons in (mρ, Γρ) 1.5% from upper bound on magnitude 3.3% total systema6c error Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 13

16 Final result for Nf = 2 Es6mate from TMR including finite-volume correc6on: a hvp µ = (654 ± 32 stat ± 17 syst ± 1 scale ± 7 FV + 1 disc) 1 1 Sta6s6cal error: Systema6c error: Scale sefng: Finite volume shii: Disconnected parts: 4.8% at the physical point 2.6% from procedural varia6ons 1.5% from uncertainty in amμ 1.% from varia6ons in (mρ, Γρ) 1.5% from upper bound on magnitude 3.3% total systema6c error Isospin-breaking effects not (yet) included Analysis of CLS ensembles with Nf = 2+1 flavours has started Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 13

17 Final result for Nf = 2 Es6mate from TMR including finite-volume correc6on: a hvp µ = (654 ± 32 stat ± 17 syst ± 1 scale ± 7 FV + 1 disc) 1 1 Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 14

18 Hadronic Light-by-Light ScaWering Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 15

19 La4ce QCD approaches to HLbL Matrix element of e.m. current between muon ini6al and final states: D µ(p, s ) J µ () µ(p, s) E = e u(p, s ) F 1 (Q 2 ) µ + F 2(Q 2 ) 2m a hlbl µ = F 2 () µ Q! u(p, s) RBC/UKQCD: QCD + QED simula6ons QCD + stochas6c QED [Hayakawa et al. 25; Blum et al. 215] [Blum et al. 216, 217] Mainz group: Exact QED kernel in posi6on space Transi6on form factors of sub-processes Forward scavering amplitude [Asmussen et al. 215, 216, and in prep.] [Gérardin, Meyer, Nyffeler 216] [Green et al. 215, and in prep.] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 16

20 Exact QED kernel in posi-on space Determine QED part perturba6vely in the con6nuum in infinite volume no power-law volume effects a hlbl µ = F 2 () = me6 3 Z d 4 y QCD four-point func6on: i ;µ (x, y) = QED kernel func6on: L [, ];µ (x, y) Infra-red finite; can be computed semi-analy6cally Z d 4 x L [, ];µ (x, y) i ;µ (x, y) Admits a tensor decomposi6on in terms of six weight func6ons which depend on x 2, y 2, x y Z Z 3D integra6on instead of d 4 x d 4 y Weight func6ons computed and stored on disk Z d 4 zz D Jµ (x)j (y)j (z)j () E [Asmussen, Green, Meyer, Nyffeler, in prep.] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 17

21 Tes-ng the exact QED kernel Integrand of lepton loop contribution to a µ HLbL m l = m µ m l =2 m µ f( y ) m µ y Analy6c result for the lepton-loop reproduced at percent level f ( y ) me y 3 Z d 4 x L... (x, y) i... (x, y) FNAL 217, LaWce 217] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 18

22 Tes-ng the exact QED kernel Integrand of lepton loop contribution to a µ HLbL m l = m µ m l =2 m µ f( y ) m µ y Analy6c result for the lepton-loop reproduced at percent level π pole contribu6on: assume VMD model for TFF f ( y ) me y 3 Z d 4 x L... (x, y) i... (x, y) Contribu6on (surprisingly?) long-range FNAL 217, LaWce 217] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 18

23 Transi-on form factor π γ γ Pseudoscalar meson pole expected to dominate LbL scavering Compute transi6on form factor between π and two off-shell photons: µ q 1 q 2 F (m2 ; q 2 1, q2 2 ) M µ ues of M µ C µ (3) (, t ; ~p, ~q 1, ~q 2 ) = X D n T J (~, + t )J µ (~z, t )P(~x, ) oe e i~p ~x e i~q 1 ~z ~x,~z Compute connected and disconnected contribu6ons Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 19

24 Transi-on form factor π γ γ Fit VMD, LMD, LMD-V models, e.g. F LMD = M4 V + (q2 1 + q2 2 ) (M 2 V q 2 1 )(M2 V q 2 2 ) Q 2 F πγ γ ( Q2, ).1 Q 2 F πγ γ ( Q2, Q 2 ).2.8 [GeV] VMD LMD LMD+V Q 2 [GeV 2 ] CELLO CLEO BL [GeV] Q 2 [GeV 2 ] VMD LMD LMD+V OPE Results for π contribu6on to hadronic light-by-light scavering: (a hlbl µ ) = (65. ± 8.3) 1 11 (LMD+V) (stat. error only) [Gérardin, Meyer, Nyffeler, Phys Rev D94 (216) 7457] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 2

25 Light-by-light forward scawering amplitude Eight independent amplitudes: M TT, M t TT, Ma TT, M TL, M LT, M a TL, Mt TL, M LL Q 2 i = q 2 i >, i = 1, 2, = Q 1 Q 2 Relate forward amplitudes to two-photon fusion cross sec6ons: M ( ) = 4 2 Z 1 d p X ( )/ ( ) ( 2 2 i ) Expect main contribu6ons from mesons Constrain form factors used to es6mate a hlbl µ Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 21

26 Light-by-light forward scawering amplitude Four-point correlator of one local and three conserved vector currents Fully connected contribu6on with summed fixed kernels: pos µ 1 µ 2 µ 3 µ 4 (x 4 ; f 1, f 2 ) = X x 1,x 2 Euclidean four-point func6on in momentum space: E µ 1 µ 2 µ 3 µ 4 (p 4 ; p 1, p 2 ) = f (x 1 ) f (x 2 ) pos µ 1 µ 2 µ 3 µ 4 (x 1, x 2,, x 4 ) X x 4 e ip4 x4 pos µ 1 µ 2 µ 3 µ 4 (x 4 ; p 1, p 2 ) Forward scavering of transversely polarised virtual photons: M TT ( Q 2 1, Q2 2, ) = e4 4 R µ 1 µ 2 R µ3 µ 4 E µ 1 µ 2 µ 3 µ 4 ( Q 2 ; Q 1, Q 1 ) Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 22

27 Light-by-light forward scawering amplitude Example: contribu6ons to MTT from different mesonic channels M TT ( ) = 4 2 Z 1 d TT( ) ( 2 2 i ), TT / 2 64 F P (Q 2 1, Q2 2 ) F P (, ) Use monopole/dipole ansatz for S, A, T and V form factors LaWce 217] Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 23

28 Summary and Outlook Hadronic vacuum polarisa6on: focus on refinements Include physical pion mass Determine 6melike pion form factor Include isospin breaking Hadronic light-by-light scavering QED kernel can be combined with direct lafce calcula6on or hadronic model for four-point func6on Four-point func6on can be determined directly serves to test hadronic models Complementary approach: transi6on form factors Hartmut Wittig (g 2) μ from Lattice QCD / Mainz 24