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

Using analytical continuation for ahvp

µ

Karl Jansen

in collaboration with Xu Feng, Shoji Hashimoto, Grit Hotzel, Marcus Petschlies, Dru Renner

• Status of standard ahvp

µ

calculation

• Analytical continuation
• Example of ahvp

µ

• Conclusion

The full four-flavour contribution for leptons

m2

PS

• GeV2

audsc

τ

0.25 0.2 0.15 0.1 0.05 3.8e-06 3.4e-06 3.0e-06 2.6e-06

audsc

µ

0.25 0.2 0.15 0.1 0.05 7.0e-08 6.5e-08 6.0e-08 5.5e-08 5.0e-08

audsc

e

0.25 0.2 0.15 0.1 0.05 2.0e-12 1.8e-12 1.6e-12 1.4e-12 1.2e-12

• fit function:

aµ(mPS, a) = A + B m2

P S + C a2

• maximal twist: only O(a2) effects
• full analysis of short distance singularities

→ O(a)-improvement not spoiled

Light contribution at the physical point

a = 0.061fm, L = 2.9fm a = 0.061fm, L = 1.9fm a = 0.078fm, L = 3.7fm a = 0.078fm, L = 2.5fm a = 0.078fm, L = 1.9fm a = 0.086fm, L = 2.8fm Nf = 2 result Preliminary m2

PS

• GeV2

aud

µ

0.25 0.2 0.15 0.1 0.05 6.0e-08 5.0e-08 4.0e-08 3.0e-08 2.0e-08 1.0e-08 0.0e+00

• modified method
• standard method
• VMD and polynomial fit

Light contribution at the physical point

a = 0.061fm, L = 2.9fm a = 0.061fm, L = 1.9fm a = 0.078fm, L = 3.7fm a = 0.078fm, L = 2.5fm a = 0.078fm, L = 1.9fm a = 0.086fm, L = 2.8fm Nf = 2 result, Pad´ e fit Nf = 2 result, standard fit Preliminary m2

PS

• GeV2

aud

µ

0.25 0.2 0.15 0.1 0.05 6.0e-08 5.0e-08 4.0e-08 3.0e-08 2.0e-08 1.0e-08 0.0e+00

• modified method
• standard method
• VMD and polynomial fit
• compare to Pad´

e fit

Light contribution all leptons

m2

PS

• GeV2

aud

τ

0.25 0.2 0.15 0.1 0.05 2.8e-06 2.4e-06 2.0e-06

aud

µ

0.25 0.2 0.15 0.1 0.05 6.2e-08 5.8e-08 5.4e-08 5.0e-08

aud

e

0.25 0.2 0.15 0.1 0.05 1.6e-12 1.4e-12 1.2e-12 1.0e-12

ahvp

e

= 1.50(03)10−12 (Nf = 2 + 1 + 1) ahvp

µ

= 5.67(11)10−8 (Nf = 2 + 1 + 1) ahvp

τ

= 2.66(02)10−6 (Nf = 2 + 1 + 1)

• fit function:

aµ(mPS, a) = A + B m2

P S + C a2

Alternative method: analytic continuation Compute HVP function via analytic continuation ¯ Π(K2)(KµKν − δµνK2) =

• dt eωt

d3 x ei

k x Ω|T{JE µ (

x, t)JE

ν (

0, 0)}|Ω

• JE

µ (X) electromagentic current

• K = (

k, −iω), k spatial momentum, ω the photon energy (input) Advantage

• vary ω → smooth values for K2 = −ω2 +

k2

• can cover space-like and time-like momentum regions
• can reach small momenta and even zero momentum
• important condition:

−K2 = ω2 − k2 < M 2

V ,

• r

ω < Evector

• 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

Cµν( k, t) =

• x e−i

k( x+aˆ µ/2−aˆ ν/2) JE µ (

x, t)JE

ν (

0, 0) ,

• discrete momenta
• k = (2π/L)

n

• transformation in time

¯ Πµν( k, ω; T) = T/2

t=−T/2 eω(t+a(δµ,t−δν,t)/2)Cµν(

k, t) ¯ Π(K2; T)

• KµKν − δµνK2

= ¯ Πµν( k, ω; T)

Correlators for different polarization

• 0.05

0.05 0.1 0.15 Re[Cµ(k,t)], n=(1,0,0), {µ,}={x,x}

• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 Im[Cµ(k,t)], n=(1,0,0), {µ,}={x,t}

• 30
• 20
• 10

10 20 30 t/a

• 0.05

0.05 0.1 0.15 Re[Cµ(k,t)], n=(1,0,0), {µ,}={y,y}

• 30
• 20
• 10

10 20 30 t/a 0.002 0.004 0.006 0.008 Re[Cµ(k,t)], n=(1,0,0), {µ,}={t,t}

• 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: tmax = ηT/2

¯ Π(K2; tmax)

• KµKν − δµνK2

= ¯ Πµν( k, ω; tmax) ¯ Πµν( k, ω; tmax) = tmax−a(δµ,t−δν,t)

t=−tmax

eω(t+a(δµ,t−δν,t)/2)Cµν( k, t) – for each fixed η method correct for T → ∞

• for η = 1 introduce a finite size effect
• for t > tmax: describe data by model
• Here:

– choice of η = 3/4

• assume ground state dominance for large t (ρ-mass)

Demonstration of n indpendence

• 0.24
• 0.22
• 0.2
• 0.18

Π(0; tmax) |n|

2=0, {µ,ν}={x,x}

|n|

2=1, {µ,ν}={x,x}

|n|

2=1, {µ,ν}={x,t}

|n|

2=1, {µ,ν}={y,y}

|n|

2=1, {µ,ν}={t,t}

|n|

2=2, {µ,ν}={x,x}

|n|

2=2, {µ,ν}={x,y}

|n|

2=2, {µ,ν}={x,t}

|n|

2=2, {µ,ν}={z,z}

|n|

2=2, {µ,ν}={t,t}

|n|

2=3, {µ,ν}={x,x}

|n|

2=3, {µ,ν}={x,y}

|n|

2=3, {µ,ν}={x,t}

|n|

2=3, {µ,ν}={t,t}

averaged result 1 2 3 |n|

2

• 0.24
• 0.22
• 0.2
• 0.18

Π(0; tmax) + Π(0; t > tmax)

• increasing error for larger

n2

HVP from analytical continuation

• 0.25
• 0.2

(K

2, tmax) + (K 2, t > tmax)

|n|2=0, {µ,}={x,x} conventional |n|2=1, {µ,}={x,x} |n|2=1, {µ,}={x,t} |n|2=1, {µ,}={y,y} |n|2=1, {µ,}={t,t}

• 0.4

0.4 0.8 K2 [GeV2]

• 0.25
• 0.2

(K

2, tmax) + (K 2, t > tmax)

|n|2=2, {µ,}={x,x} |n|2=2, {µ,}={x,y} |n|2=2, {µ,}={x,t} |n|2=2, {µ,}={z,z} |n|2=2, {µ,}={t,t}

• 0.4

0.4 0.8 K2 [GeV2]

|n|2=3, {µ,}={x,x} |n|2=3, {µ,}={x,y} |n|2=3, {µ,}={x,t} |n|2=3, {µ,}={t,t}

• 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)

• 0.01

0.01 0.02 0.03 0.04 0.05 0.2 0.4 0.6 0.8 1 ΠR Q2 / GeV2 Experiment + DR lattice ΠR, Nf=2+1+1, tmax/a=31

• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 0.025 0.2 0.4 0.6 0.8 1 D Q2 / GeV2 lattice Adler function, Nf=2+1+1, tmax/a=31

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 ahvp

µ

split in three pieces

• a(1)

¯ µ

directly calculable from lattice data a(1)

¯ µ

= α2 K2

max

dK2

1 K2f

• K2

m2

µ

m2

ρ

m2

V

• (Π(K2) − Π(0))
• a(2)

¯ µ

• nly large momentum region: model dependence

a(2)

¯ µ

= α2 ∞

K2

max dK2

1 K2f

• K2

m2

µ

m2

ρ

m2

V

• (Π(K2) − Π(K2

max))

• a(3)

¯ µ

correction term a(2)

¯ µ

= α2 ∞

K2

max dK2

1 K2f

• K2

m2

µ

m2

ρ

m2

V

• (Π(K2

max) − Π(0))

Comparison to standard calculation

3e-08 4e-08 5e-08 6e-08 7e-08 aµ

hvp(tmax)

0.1 0.2 0.3 0.4 0.5 mπ

2 [GeV 2]

3e-08 4e-08 5e-08 6e-08 7e-08 aµ

hvp(tmax) + aµ hvp(t > tmax)

a=0.079 fm, T/2=L=1.6 fm a=0.079 fm, T/2=L=1.9 fm a=0.079 fm, T/2=L=2.5 fm a=0.063 fm, T/2=L=1.5 fm a=0.063 fm, T/2=L=2.0 fm

without FSE with FSE

• 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 < Kmax with −K2 = ω2 = k2 < M 2

V (or, ω < EV )

– 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. ahvp

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 Q2 = 0 is not available?