introduction to lattice qcd ii a dedicated project
TRANSCRIPT
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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
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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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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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Motivation
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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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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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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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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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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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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)
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How the data really look like
• demanding analysis of O(1000) channelsJegerlehner, Nyffeler, Phys.Rep.
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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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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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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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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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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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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
aµ
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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Dis-connected contribution
a graph representing dis-conected contributions
tsrc t
→ has been basically always be neglected
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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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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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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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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
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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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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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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
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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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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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Ken Wilson award
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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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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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What is moved
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How and where is it moved
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What is expected
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What can be achieved
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When will it happen
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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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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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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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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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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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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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