introduction to lattice qcd ii a dedicated project

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Introduction to Lattice QCD II A dedicated project Karl Jansen Leading order hadronic contribution to the muon anomalous magnetic moment motivation describe the computation new method for the lattice results

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Page 1: Introduction to Lattice QCD II A dedicated project

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

Page 2: Introduction to Lattice QCD II A dedicated project

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

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Page 3: Introduction to Lattice QCD II A dedicated project

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

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Page 4: Introduction to Lattice QCD II A dedicated project

Motivation

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3

Page 5: Introduction to Lattice QCD II A dedicated project

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

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Page 6: Introduction to Lattice QCD II A dedicated project

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

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Page 7: Introduction to Lattice QCD II A dedicated project

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)

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Page 8: Introduction to Lattice QCD II A dedicated project

µ µγ⇒ a

QED(1)µ = α

2π ,

1

The simplest, QED contribution to gµ − 2

Schwinger, 1948

α QED fine structure constant

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Page 9: Introduction to Lattice QCD II A dedicated project

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)

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Page 10: Introduction to Lattice QCD II A dedicated project

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

Page 11: Introduction to Lattice QCD II A dedicated project

How the data really look like

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

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Page 12: Introduction to Lattice QCD II A dedicated project

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)

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Page 13: Introduction to Lattice QCD II A dedicated project

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?

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Page 14: Introduction to Lattice QCD II A dedicated project

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

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Page 15: Introduction to Lattice QCD II A dedicated project

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)

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Page 16: Introduction to Lattice QCD II A dedicated project

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

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Page 17: Introduction to Lattice QCD II A dedicated project

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

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Page 18: Introduction to Lattice QCD II A dedicated project

Dis-connected contribution

a graph representing dis-conected contributions

tsrc t

→ has been basically always be neglected

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Page 19: Introduction to Lattice QCD II A dedicated project

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)

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Page 20: Introduction to Lattice QCD II A dedicated project

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

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Page 21: Introduction to Lattice QCD II A dedicated project

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

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Page 22: Introduction to Lattice QCD II A dedicated project

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

Page 23: Introduction to Lattice QCD II A dedicated project

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

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Page 24: Introduction to Lattice QCD II A dedicated project

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

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Page 25: Introduction to Lattice QCD II A dedicated project

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

Page 26: Introduction to Lattice QCD II A dedicated project

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

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Page 27: Introduction to Lattice QCD II A dedicated project

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

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Page 28: Introduction to Lattice QCD II A dedicated project

Ken Wilson award

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Page 29: Introduction to Lattice QCD II A dedicated project

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

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Page 30: Introduction to Lattice QCD II A dedicated project

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

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Page 31: Introduction to Lattice QCD II A dedicated project

What is moved

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30

Page 32: Introduction to Lattice QCD II A dedicated project

How and where is it moved

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Page 33: Introduction to Lattice QCD II A dedicated project

What is expected

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Page 34: Introduction to Lattice QCD II A dedicated project

What can be achieved

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Page 35: Introduction to Lattice QCD II A dedicated project

When will it happen

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Page 36: Introduction to Lattice QCD II A dedicated project

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

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Page 37: Introduction to Lattice QCD II A dedicated project

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

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Page 38: Introduction to Lattice QCD II A dedicated project

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

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Page 39: Introduction to Lattice QCD II A dedicated project

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 ...

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Page 40: Introduction to Lattice QCD II A dedicated project

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)

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Page 41: Introduction to Lattice QCD II A dedicated project

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

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