introduction to lattice qcd ii a dedicated project

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Introduction to Lattice QCD IIA dedicated project

Karl Jansen ��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

• Leading order hadronic contributionto the muon anomalous magnetic moment

– motivation– describe the computation– new method for the lattice– results

A success of Quantum Field Theories: Electromagnetic Interaction

Quantum Electrodynamics (QED)

coupling of the electromagnetic interaction is small⇒ perturbation theory 4-loop calculation

magnetic moment of the electron

~µ = gee~

2mec~S

~S spin vector

deviation from ge = 2: ae = (ge − 2)/2

ae(theory) = 1159652201.1(2.1)(27.1) · 10−12

ae(experiment) = 1159652188.4(4.3) · 10−12

1

The question of the muon

magnetic moment of the muon

~µ = gµe~

2mµc~S

deviation from gµ = 2: aµ = (gµ − 2)/2

aµ(theory) = 1.16591790(65) · 10−3

aµ(experiment) = 1.16592080(63) · 10−3

→ there is a 3.2σ discrepancy

aµ(experiment)− aµ(theory) = 2.90(91) · 10−9

2

Motivation

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3

Why is this interesting?

→ new proposed experiments at Fermilab (USA) and JPARC (Japan) ≈ 2015

brings down error

σex = 6.3 · 10−10 → 1.6 · 10−10

challenge: bring down theoretical error to same level

⇒ possibility

aµ(experiment)− aµ(theory) > 5σ

• possible breakdown of standard model?

• aµ = aQEDµ + aweak

µ + aQCDµ + aNP

µ

aNPµ ∝ m2lepton/m

2new

→ muon ideal to discover new physics

for τ experimental results too imprecise

4

Is the lattice important?

→ size of various contributions to aµ

Contribution σth

QCD-LO (α2s) 5.3 · 10−10

QCD-NLO (α3s) 3.9 · 10−10

QED/EW 0.2 · 10−10

Total 6.6 · 10−10

⇒ non-perturbative hadronic contributions are most important

5

γ(q)µ(p′)

µ(p)

1

The contributions to gµ − 2

The vertex

– p, p′ incoming and outgoing momenta

– q = p− p′ photon momentum

• put muon in magnetic field ~B interaction Hamiltonian H = ~µ ~B

• measure interaction of muon with photon (magnetic field)

6

µ µγ⇒ a

QED(1)µ = α

2π ,

1

The simplest, QED contribution to gµ − 2

Schwinger, 1948

α QED fine structure constant

7

The leading order hadronic contribution, ahadµ

µ

γ

had

µ

• ahadµ contributes almost 60% of theoretical error

• needs to be computed non-perturbatively

• computation of vacuum polarization tensor

Πµν(q) = (qµqν − q2gµν)Π(q2)

8

Relation to experimental extraction

connection between real and imaginary part of Π(q2)

Π(q2)−Π(0) = q2

π

∫∞0ds ImΠ(s)s(s−q2)

ImΠ(s) related to experimental data of total cross section in e+e− annihilation

ImΠ(s) = α3R(s)

π−π ρ,ω φ J/ψ

0 0.5 1 1.5 2 2.5 3 3.5E [GeV]

1

2

3

4

5

R(E

)

important contributions

• ρ, ω (Nf = 2)

• Φ (Nf = 2 + 1)

• J/Ψ (Nf = 2 + 1 + 1)

9

How the data really look like

• demanding analysis of O(1000) channelsJegerlehner, Nyffeler, Phys.Rep.

10

Euclidean expression for hadronic contribution

ahadµ = α2

∫∞0

dQ2

Q2 F(Q2

m2µ

)(Π(Q2)−Π(0))

with F(Q2

m2µ

)a known function T. Blum

→ need to compute vacuum polaization function

Continuum:

Πµν(Q) = i∫d4xeiQ·(x−y)〈0|TJµ(x)Jν(y)|0〉

Jµ hadronic electromagnetic current

Jµ(x) =∑f ef ψf(x)γµψf(x) = 2

3u(x)γµu(x)− 13d(x)γµd(x) + · · ·

local vector current given is conserved

∂µJµ(x) = 0 ⇒ QµΠ(Q)µν = 0

eliminating the factor QµQν −Q2δµν

⇒ obtain Π(Q2)

11

Going to the lattice

• work with Nf = 2 twisted mass fermions

Stm =∑x χ(x) [DW +m0 + iµγ5τ3]χ(x)

• use the conserved lattice current

Exercize: derive conserved current

use vector transformation

δV χ(x) = iεV (x)τχ(x) , δV χ(x) = −iχ(x)τεV (x)

τ =

(2/3 00 −1/3

)= 1

61 + 12τ

3

Can we also use the local current?

12

Conserved lattice current

J tmµ (x) = 12

(χ(x)τ(γµ − r)Uµ(x)χ(x+ µ) + χ(x+ µ)τ(γµ + r)U†µ(x)χ(x)

)satisfying

∂∗µJtmµ (x) = 0

with ∂∗µ backward lattice derivative

13

Lattice vacuum polarization tensor

Fourier transformation of conserved vector current:

J tmµ (Q) =∑x e

iQ·(x+µ/2)J tmµ (x) , Qµ = 2 sin(Qµ2

)leading to

Πµν(Q) = 1V

∑x,y e

iQ·(x+µ/2−y−ν/2)〈J tmµ (x)J tmν (y)〉

from which we extract the vacuum polarization function

Πµν(Q) = (QµQν − Q2δµν)Π(Q2)

14

Fit to vacuum polarization function

Fit function

ΠM,N(Q2) = −59

∑Mi=1

m2i

Q2+m2i

+∑Nn=0 an(Q2)n

typically i = 1, 2, 3 , n = 0, 1, 2, 3 (systematic error)

0 10 20 30 40 50 60 70 80 90

Q2 [GeV

2]

-0.2

-0.15

-0.1

-0.05

Π(Q

2)

0 1 2 3 4

-0.2

-0.16

15

Do we control hadronic vacuum polarisation?(Xu Feng, Dru Renner, Marcus Petschlies, K.J.; Lattice 2010)

0 0.1 0.2 0.3 0.4

mπ2 [GeV

2]

0

100

200

300

400

500

600

700

had [

10

-10]

Jegerlehner 2008a=0.079 fma=0.063 fmquadratic in mπ

2• experiment: ahvp,exp

µ,Nf=2 = 5.66(05)10−8

• lattice: ahvp,oldµ,Nf=2 = 2.95(45)10−8

→ misses the experimental value→ order of magnitude larger error

• have used different volumes

• have used different values of lattice spacing

16

Dis-connected contribution

a graph representing dis-conected contributions

tsrc t

→ has been basically always be neglected

17

Can it be the dis-connected contribution?(Xu Feng, Dru Renner, Marcus Petschlies, K.J.)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

q2 [GeV

2]

-0.12

-0.08

-0.04

0

Π(q

2)

disconnectedconnected

• dedicated effort

• have included dis-connected contributions for first time

• smallness consistent with chiral perturbation theory (Della Morte, Juttner)

18

Different extrapolation to the physical point

lattice: simulations at unphysical quark masses, demand only

limmPS→mπ ahvp,lattl = ahvp,phys

l

⇒ flexibility to define ahvp,lattl

standard definitions in the continuum

ahvpl = α2

∫∞0dQ2 1

Q2ω(r)ΠR(Q2)

ΠR(Q2) = Π(Q2)−Π(0)

ω(r) = 64

r2(

1+√

1+4/r)4√

1+4/r

with r = Q2/m2l

19

Redefinition of ahvp,lattl

redefinition of r for lattice computations

rlatt = Q2 · Hphys

H

choices

• r1: H = 1; Hphys = 1/m2l

• r2: H = m2V (mPS); Hphys = m2

ρ/m2l

• r3: H = f2V (mPS); Hphys = f2

ρ/m2l

each definition of r will show a different dependence on mPS but agree byconstruction at the physical point

remark: strategy often used in continuum limit extrapolations, e.g. charm quarkmass determination

20

comparison using r1, r2, r3

0 0.1 0.2 0.3 0.4

mPS

2 [GeV

2]

1

2

3

4

5

6a

µhvp [

10

-8]

mV

fV

1

21

A new result from the lattice

• experimental value: ahvp,expµ,Nf=2 = 5.66(05)10−8

• from our old analysis: ahvp,oldµ,Nf=2 = 2.95(45)10−8

→ misses the experimental value→ order of magnitude larger error

• from our new analysis: ahvp,newµ,Nf=2 = 5.66(11)10−8

→ error (including systematics) almost matching experiment

22

Anomalous magnetic moments, a check

0 0.1 0.2 0.3 0.4

mPS

2 [GeV

2]

2.2

2.4

2.6

2.8

a τhvp [

10-6

]

4.5

5

5.5

6

a µhvp [

10-8

]

11.21.41.61.8

a ehvp [

10-1

2 ] ahloe = 1.513 (43) · 10−12 LQCDahloe = 1.527 (12)·10−12 PHENO

ahloµ = 5.72 (16) · 10−8 LQCD

ahloµ = 5.67 (05) · 10−8 PHENO

ahloτ = 2.650 (54) · 10−6 LQCDahloτ = 2.638 (88)·10−12 PHENO

23

anomalous magnetic moment of muon

0 0.1 0.2 0.3 0.4

mPS

2 [GeV

2]

3.5

4

4.5

5

5.5

aµh

vp [

10

-8]

a=0.079 fm L=1.6 fma=0.079 fm L=1.9 fma=0.079 fm L=2.5 fma=0.063 fm L=1.5 fma=0.063 fm L=2.0 fmlinearquadraticmeasured

• have used different volumes

• have used different values of lattice spacing

• have included dis-connected contributions

⇒ can control systematic effects

24

Why it works: fitting the Q2 dependence

Fit function

ΠM,N(Q2) = −59

∑Mi=1

m2i

Q2+m2i

+∑Nn=0 an(Q2)n

i = 1: ρ-meson → dominant contribution ∝ 5.010−8

i = 2: ω-meson ∝ 3.710−9

i = 3: φ-meson ∝ 3.410−9

0 10 20 30 40 50 60 70 80 90

Q2 [GeV

2]

-0.2

-0.15

-0.1

-0.05Π

(Q2)

0 1 2 3 4

-0.2

-0.16

25

Why it works

0 0.1 0.2 0.3 0.4

mπ2 [GeV

2]

800

900

1000

1100

1200

mV

[M

eV

]

a=0.079 fma=0.063 fmPDG

• mV consistent with resonance analysis(Feng, Renner, K.J.)

• strong dependence on mPS

26

Ken Wilson award

27

Next steps

Fit function

ΠM,N(Q2) = −59

∑Mi=1

m2i

Q2+m2i

+∑Nn=0 an(Q2)n

add i = 4: J/Ψ, i = 5 . . .

• ETMC is performing simulations wit dynamicalup, down, strange and charm quarks→ unique opportunity

• avoids ambiguity with experiment comparison(what counts for Nf = 2?)

• generalized boundary conditions: Ψ(L+ aµ) = eiθΨ(x)→ θ continuous momentum→ allows to realize arbitrary momenta on the lattice

28

INT Workshop onThe Hadronic Light-by-Light Contribution to the Muon Anomaly

February 28 - March 4, 2011

Some sildes from

Lee Roberts (g-2) collaboration

• new experiment E989

• move Brookhaven equipment to Fermilab

29

What is moved

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30

How and where is it moved

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31

What is expected

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32

What can be achieved

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33

When will it happen

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34

The accuracy question

We need a precision < 1%

• include explicit isospin breaking

• include electromagnetism

• need computation of light-by-light contribution

• reach small quark mass → physical point

Much ado for young people

35

Simulation setup for Nf = 2 + 1 + 1Configurations available through ILDG

β a[fm] L3T/a4 mπ[MeV] status

1.9 ≈ 0.085 24348 300 – 500 ready1.95 ≈ 0.075 32364 300 –500 ready2.0 ≈ 0.065 32364 300 ready2.1 ≈ 0.055 48396 300 – 500 running/ready

643128 230 thermalizing643128 200 queued963192 160 planned

• trajectory length always one

• 1000 trajectores for thermalization

• ≥ 5000 trajectores for measurements

36

Next, α3s, contribution

µ(p)

γ(k) kρ

had + 5 permutations of the qi

µ(p′)

q1µq2νq3λ

termed: light-by-light scattering

involves 4-point function

Πµναβ(q1, q2, q3) =∫xyz

eiq1·x+iq2·y+iq3·z 〈jµ(0)jν(x)jα(y)jβ(z)〉

jµ electromagnetic quark current

jµ = 23uγµu− 1

3dγµd− 13sγµs+ 2

3cγµc

37

Momentum sources(Alexandrou, Constantinou, Korzec, Panagopoulos, Stylianou)

← following Gockeler et.al.

for renormalization: need Green function in momentum space

G(p) = a12

V

∑x,y,z,z′ e

−ip(x−y)〈u(x)u(z)J (z, z′)d(z′)d(y)〉

e.g. J (z, z′)=δz,z′γµ corresponds to local vector current

sources:

baα(x) = eipxδαβδab

solve for

DlattG(p) = b

Advantage: very high, sub-percent precision data (only moderate statistics)

Disadvantage: need inversion for each momentum separately

→ would need 3V inversions ...

38

Illustration of precision

with O(a2) effects

O(a2) effects pert. subtracted

with O(a2) effects

O(a2) effects pert. subtracted

use the momentum source method to attack the 4-point functionas needed for light-by-light scattering (P. Rakow et.al., lattice’08)

39

Summary

• lattice calculation of muon anomalous magnetic moment

• looked hopeless first ← order of magnitude larger error than experiment

• introduced new method → start to match experimental accuracy

• outlook

– include first two quark generations– include isospin breaking and electromagnetism– attack light-by-light scattering

40

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