The full four-flavour contribution for leptons 2.0e-12 1.8e-12 a - PowerPoint PPT Presentation

Using analytical continuation for a hvp µ Karl Jansen in collaboration with Xu Feng, Shoji Hashimoto, Grit Hotzel, Marcus Petschlies, Dru Renner • Status of standard a hvp calculation µ • Analytical continuation • Example of a hvp µ • Conclusion

The full four-flavour contribution for leptons 2.0e-12 1.8e-12 a udsc • fit function: 1.6e-12 e 1.4e-12 a µ ( m PS , a ) = A + B m 2 P S + C a 2 1.2e-12 0 0.05 0.1 0.15 0.2 0.25 • maximal twist: only O ( a 2 ) effects 7.0e-08 6.5e-08 • full analysis of short distance singularities a udsc 6.0e-08 µ → O ( a ) -improvement not spoiled 5.5e-08 5.0e-08 0 0.05 0.1 0.15 0.2 0.25 3.8e-06 a udsc 3.4e-06 τ 3.0e-06 2.6e-06 0 0.05 0.1 0.15 0.2 0.25 GeV 2 � m 2 � PS

Light contribution at the physical point 6.0e-08 • modified method 5.0e-08 4.0e-08 a ud µ 3.0e-08 • standard method N f = 2 result 2.0e-08 a = 0 . 086fm, L = 2 . 8fm a = 0 . 078fm, L = 1 . 9fm a = 0 . 078fm, L = 2 . 5fm 1.0e-08 a = 0 . 078fm, L = 3 . 7fm a = 0 . 061fm, L = 1 . 9fm Preliminary a = 0 . 061fm, L = 2 . 9fm 0.0e+00 0 0.05 0.1 0.15 0.2 0.25 m 2 � GeV 2 � PS • VMD and polynomial fit

Light contribution at the physical point 6.0e-08 • modified method 5.0e-08 4.0e-08 a ud µ 3.0e-08 • standard method N f = 2 result, standard fit N f = 2 result, Pad´ e fit 2.0e-08 a = 0 . 086fm, L = 2 . 8fm a = 0 . 078fm, L = 1 . 9fm a = 0 . 078fm, L = 2 . 5fm 1.0e-08 a = 0 . 078fm, L = 3 . 7fm a = 0 . 061fm, L = 1 . 9fm Preliminary a = 0 . 061fm, L = 2 . 9fm 0.0e+00 0 0.05 0.1 0.15 0.2 0.25 m 2 � GeV 2 � PS • VMD and polynomial fit • compare to Pad´ e fit

Light contribution all leptons 1.6e-12 1.4e-12 a ud a hvp = 1 . 50(03)10 − 12 ( N f = 2 + 1 + 1) e e 1.2e-12 1.0e-12 0 0.05 0.1 0.15 0.2 0.25 6.2e-08 = 5 . 67(11)10 − 8 ( N f = 2 + 1 + 1) a hvp a ud 5.8e-08 µ µ 5.4e-08 5.0e-08 0 0.05 0.1 0.15 0.2 0.25 2.8e-06 a hvp = 2 . 66(02)10 − 6 ( N f = 2 + 1 + 1) a ud τ τ 2.4e-06 2.0e-06 0 0.05 0.1 0.15 0.2 0.25 GeV 2 � m 2 � PS • fit function: a µ ( m PS , a ) = A + B m 2 P S + C a 2

Alternative method: analytic continuation Compute HVP function via analytic continuation x � Ω | T { J E x e i� ¯ ν ( � Π( K 2 )( K µ K ν − δ µν K 2 ) = dt e ωt � d 3 � k� x, t ) J E � µ ( � 0 , 0) }| Ω � • J E µ ( X ) electromagentic current • K = ( � k, − iω ) , � k spatial momentum, ω the photon energy (input) Advantage • vary ω → smooth values for K 2 = − ω 2 + � k 2 • can cover space-like and time-like momentum regions • can reach small momenta and even zero momentum • important condition: − K 2 = ω 2 − � k 2 < M 2 V , or ω < E vector • make use of ideas: (Ji; Meyer; X. Feng, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko, J. Noaki, E. Shintani; G. de Divitiis, R. Petronzio, N. Tantalo)

Fourier Transformation • spatial transformation ν/ 2) � J E x e − i� C µν ( � ν ( � k ( � x + a ˆ µ/ 2 − a ˆ x, t ) J E k, t ) = � µ ( � 0 , 0) � , � • discrete momenta � k = (2 π/L ) � n • transformation in time k, ω ; T ) = � T/ 2 Π µν ( � ¯ t = − T/ 2 e ω ( t + a ( δ µ,t − δ ν,t ) / 2) C µν ( � k, t ) ¯ = ¯ Π µν ( � Π( K 2 ; T ) K µ K ν − δ µν K 2 � � k, ω ; T )

Correlators for different polarization Re[C µ � (k,t)], n=(1,0,0), { µ, � }={x,x} Im[C µ � (k,t)], n=(1,0,0), { µ, � }={x,t} 0.03 0.02 0.15 0.01 0.1 0 0.05 -0.01 0 -0.02 -0.05 -0.03 Re[C µ � (k,t)], n=(1,0,0), { µ, � }={y,y} Re[C µ � (k,t)], n=(1,0,0), { µ, � }={t,t} 0.008 0.15 0.006 0.1 0.004 0.05 0.002 0 -0.05 0 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 t/a t/a • very different behaviour for different µ, ν • all lead to the same result eventually

Truncating of timeline transformation: introducing a finite size effect • problem for large t : correlator very noisy • truncate time summation: t max = ηT/ 2 ¯ = ¯ Π µν ( � Π( K 2 ; t max ) K µ K ν − δ µν K 2 � � k, ω ; t max ) k, ω ; t max ) = � t max − a ( δ µ,t − δ ν,t ) Π µν ( � e ω ( t + a ( δ µ,t − δ ν,t ) / 2) C µν ( � ¯ k, t ) t = − t max – for each fixed η method correct for T → ∞ - for η � = 1 introduce a finite size effect • for t > t max : describe data by model • Here: – choice of η = 3 / 4 - assume ground state dominance for large t ( ρ -mass)

Demonstration of � n indpendence 2 =0, { µ,ν }={x,x} |n| -0.18 2 =1, { µ,ν }={x,x} |n| 2 =1, { µ,ν }={x,t} Π (0; t max ) |n| -0.2 2 =1, { µ,ν }={y,y} |n| 2 =1, { µ,ν }={t,t} |n| -0.22 2 =2, { µ,ν }={x,x} |n| 2 =2, { µ,ν }={x,y} |n| -0.24 2 =2, { µ,ν }={x,t} |n| 2 =2, { µ,ν }={z,z} Π (0; t max ) + Π (0; t > t max ) |n| -0.18 2 =2, { µ,ν }={t,t} |n| 2 =3, { µ,ν }={x,x} |n| -0.2 2 =3, { µ,ν }={x,y} |n| 2 =3, { µ,ν }={x,t} |n| -0.22 2 =3, { µ,ν }={t,t} |n| averaged result -0.24 0 1 2 3 2 |n| n 2 • increasing error for larger �

HVP from analytical continuation 2 , t > t max ) -0.2 2 , t max ) + � (K |n| 2 =1, { µ, � }={x,x} |n| 2 =1, { µ, � }={x,t} -0.25 |n| 2 =1, { µ, � }={y,y} |n| 2 =0, { µ, � }={x,x} � (K |n| 2 =1, { µ, � }={t,t} conventional 2 , t > t max ) -0.2 2 , t max ) + � (K |n| 2 =2, { µ, � }={x,x} |n| 2 =2, { µ, � }={x,y} |n| 2 =3, { µ, � }={x,x} |n| 2 =2, { µ, � }={x,t} |n| 2 =3, { µ, � }={x,y} -0.25 |n| 2 =2, { µ, � }={z,z} |n| 2 =3, { µ, � }={x,t} � (K |n| 2 =2, { µ, � }={t,t} |n| 2 =3, { µ, � }={t,t} -0.4 0 0.4 0.8 -0.4 0 0.4 0.8 K 2 [GeV 2 ] K 2 [GeV 2 ] • different � n lead to consistent results • agreement with standard calculation • however, larger errors for | � n | > 0

Direct application to vacuum polarization function parameters: ( a ≈ 0 . 078 fm , V = (2 . 5fm) 3 ) 0.05 0.025 0.02 0.04 0.015 0.03 0.01 0.02 Π R D 0.005 0.01 0 -0.005 0 -0.01 Experiment + DR lattice Π R , N f =2+1+1, t max /a=31 -0.01 lattice Adler function, N f =2+1+1, t max /a=31 -0.015 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Q 2 / GeV 2 Q 2 / GeV 2 renormalized HVP Adlerfunction dispersion relation (Jegerlehner, 2011)

Mixed time-momentum representation (A. Francis, B. J¨ ager, H. Meyer, H. Wittig) renormalized HVP Adlerfunction

Application to a hvp µ split in three pieces • a (1) directly calculable from lattice data µ ¯ � � = α 2 � K 2 m 2 K 2 a (1) 1 dK 2 ρ (Π( K 2 ) − Π(0)) max K 2 f µ ¯ m 2 m 2 0 µ V • a (2) only large momentum region: model dependence µ ¯ � � m 2 = α 2 � ∞ K 2 a (2) 1 max dK 2 ρ (Π( K 2 ) − Π( K 2 K 2 f max )) µ ¯ K 2 m 2 m 2 µ V • a (3) correction term µ ¯ � � m 2 = α 2 � ∞ K 2 a (2) 1 max dK 2 (Π( K 2 ρ K 2 f max ) − Π(0)) µ ¯ K 2 m 2 m 2 µ V

Comparison to standard calculation without FSE 7e-08 6e-08 hvp (t max ) 5e-08 a µ 4e-08 a=0.079 fm, T/2=L=1.6 fm a=0.079 fm, T/2=L=1.9 fm 3e-08 a=0.079 fm, T/2=L=2.5 fm a=0.063 fm, T/2=L=1.5 fm 7e-08 hvp (t > t max ) a=0.063 fm, T/2=L=2.0 fm with FSE 6e-08 hvp (t max ) + a µ 5e-08 4e-08 a µ 3e-08 0 0.1 0.2 0.3 0.4 0.5 2 [GeV 2 ] m π • open symbols: analytic continuation • filled symbols: standard calculation of a ¯ µ • averaged over different polarizations

Summary • Tested idea of analytical continuation method for computing vacuum polarisation function – validity of method demonstrated in 1305.5878 – method works in practise • difficulties – had to truncate time summation → induce finite size effect – method only applicable for momenta K < K max with − K 2 = ω 2 = k 2 < M 2 V (or, ω < E V ) – larger errors than standard method for | � n | > 0 • my present view on analytical continuation method: it is clearly an alternative for cross-checking, e.g. a hvp – µ – it allows a direct comparison to the hvp function from phenomenological analysis of data – maybe method of choice at physical pion mass? • can it be applied to describe momentum dependence where value at Q 2 = 0 is not available?