Fundamental QCD parameters from the lattice
Transcript of Fundamental QCD parameters from the lattice
Fundamental QCD parameters from the lattice
Jochen Heitger
Westfalische Wilhelms-Universitat MunsterWilhelm-Klemm-Str. 9, D-48149 Munster
Sino-German Workshop on Frontiers in QCDDeutsches Elektronen-Synchrotron DESY, Hamburg
20. September 2006
Jochen Heitger Fundamental QCD parameters from the lattice
Does QCD describe all strong interaction phenomena ?
Complementary properties: Confinement & Asymptotic freedom
0.5
0
− 0.5
0 0.2 0.4 0.6 0.8
Lattice Gauge Theory
Model Calculation
Distance in fm
Pote
ntial in
GeV
Confinement Potential
between Quarks
0.1
0.2
0.3
0.4
0.5
!s
1 10 100
Measured
QCD
Energy in GeV
Uncovering the connectionbetween the low- and high-energy regimes of QCD:Demands non-perturbativetreatment of the theory (NP)
Jochen Heitger Fundamental QCD parameters from the lattice
Does QCD describe all strong interaction phenomena ?
Complementary properties: Confinement & Asymptotic freedom
0.5
0
− 0.5
0 0.2 0.4 0.6 0.8
Lattice Gauge Theory
Model Calculation
Distance in fm
Pote
ntial in
GeV
Confinement Potential
between Quarks
0.1
0.2
0.3
0.4
0.5
!s
1 10 100
Measured
QCD
Energy in GeV
Uncovering the connectionbetween the low- and high-energy regimes of QCD:Demands non-perturbativetreatment of the theory (NP)
Scope of lattice QCD computations:Determination of fundamental parameters
→ ΛQCD, quark massesStringent tests of QCD
→ e.g. Fπ/ΛQCD, (light) hadron spectrumPredict experimentally inaccessible physical quantities
→ e.g. hadron masses, decay constants, mixing amplitudesJochen Heitger Fundamental QCD parameters from the lattice
Lattice QCD‘Ab initio’ approach to determine phenomenologically relevant key parameters
LQCD [g0, mf] = −1
2g20
Tr FµνFµν +∑
f=u,d,s,...
ψf γµ (∂µ + g0Aµ) + mf ψf
Fπ
mπ
mK
mD
mB
︸ ︷︷ ︸
Experiment
LQCD [g0,mf]=⇒
ΛQCD12(Mu + Md)
Ms
Mc
Mb
︸ ︷︷ ︸
QCDparameters (RGIs)
+
FD
FB
BB
ξ
· · ·
︸ ︷︷ ︸
Predictions
LQCD [g0,mf]=⇒ means formulation of QCD on a Euclidean lattice with:
Gauge invariance
Locality
Unitarity
Applicable for all scalesTechnical issues/obstacles
I Continuum limit & RenormalizationI Computing resources of > 1TFlop/s
Jochen Heitger Fundamental QCD parameters from the lattice
Outline
1 Lattice QCD basics
2 Determining fundamental QCD parametersThe running of the coupling in two-flavour QCDQuenched versus unquenched: The strange quark’s mass
3 New perspectives for B-physicsNon-perturbative Heavy Quark Effective TheoryApplication: The b-quark’s mass
4 Summary & Outlook
Jochen Heitger Fundamental QCD parameters from the lattice
Lattice QCD basics
Lattice QCD basics
Jochen Heitger Fundamental QCD parameters from the lattice
Lattice QCD basics
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- 6¾?
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L
T
ψ(x)Uµ(x) = e iag0Aµ(x)
a
Lattice cutoff a−1 ∼ ΛUV
Finite volume L3 × T
Lattice action
S[U, ψ, ψ] = SG[U] + SF[U, ψ, ψ]
SG = 1g2
0
∑
p
Tr 1 − U(p)
SF = a4∑
x
ψ(x)D[U]ψ(x)
EVs: represented as path integrals
Z =
∫
D[U]D[ψ,ψ] e−S[U,ψ,ψ] =
∫
D[U]∏
f det(D/ + mf
)e−SG[U]
〈O〉 =1
Z
∫∏
x,µdUµ(x)O∏
f det(D/ + mf
)e−SG[U] = thermal average
Stochastic evaluation with Monte Carlo methods
→ Observables 〈O〉 = 1N
∑Nn=1 On ± ∆O from numerical simulations
Jochen Heitger Fundamental QCD parameters from the lattice
Lattice QCD basics
Towards realistic QCD simulations — Systematics
(1) Effects of dynamical fermionsValence quark approximation (quenching):det
(D/ + mf
)= 1
→ neglection of quark loops
Costs: ‘full’ QCDquenched QCD & 100 − 1000
(A) Quenched QCD: quark loops neglected
(B) Full QCD
Jochen Heitger Fundamental QCD parameters from the lattice
Lattice QCD basics
Towards realistic QCD simulations — Systematics
(1) Effects of dynamical fermionsValence quark approximation (quenching):det
(D/ + mf
)= 1
→ neglection of quark loops
Costs: ‘full’ QCDquenched QCD & 100 − 1000
(A) Quenched QCD: quark loops neglected
(B) Full QCD
(2) Lattice artefacts cause discretization errors
〈O〉lattice = 〈O〉continuum+ O (ap)
removable by extrapolating to the continuum limit: a → 0
accelerated convergence for p ′ > p → O(ap) improvement
Jochen Heitger Fundamental QCD parameters from the lattice
Lattice QCD basics
Towards realistic QCD simulations — Systematics
(1) Effects of dynamical fermionsValence quark approximation (quenching):det
(D/ + mf
)= 1
→ neglection of quark loops
Costs: ‘full’ QCDquenched QCD & 100 − 1000
(A) Quenched QCD: quark loops neglected
(B) Full QCD
(2) Lattice artefacts cause discretization errors
〈O〉lattice = 〈O〉continuum+ O (ap)
removable by extrapolating to the continuum limit: a → 0
accelerated convergence for p ′ > p → O(ap) improvement
(3) Quark mass restrictions
a ¿ m−1q ¿ L
→ Extrapolations to mu,d and mb , guided by ChPT and HQET
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters
Determining fundamental QCD parameters
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters
Scale dependence of QCD parameters
Renormalization group (RG) equations, g ≡ g(µ)
µ∂g
∂µ= β(g)
g→0∼ −g3
b0 + b1g
2 + b2g4 + . . .
µ∂m
∂µ= τ(g)m τ(g)
g→0∼ −g2
d0 + d1g
2 + . . .
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters
Scale dependence of QCD parameters
Renormalization group (RG) equations, g ≡ g(µ)
µ∂g
∂µ= β(g)
g→0∼ −g3
b0 + b1g
2 + b2g4 + . . .
µ∂m
∂µ= τ(g)m τ(g)
g→0∼ −g2
d0 + d1g
2 + . . .
Solution leads to exact equations in a mass-independent scheme
Λ ≡ µ(b0g
2)−b1/(2b2
0) e−1/(2b0g2) exp
−
∫ g
0
dg[ 1
β(g)+
1
b0g3−
b1
b20g
]
M ≡ m(µ)(2b0g
2)−d0/(2b0)
exp
−
∫ g
0
dg[ τ(g)
β(g)−
d0
b0g
] RG invariantquark mass
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters
Scale dependence of QCD parameters
Renormalization group (RG) equations, g ≡ g(µ)
µ∂g
∂µ= β(g)
g→0∼ −g3
b0 + b1g
2 + b2g4 + . . .
µ∂m
∂µ= τ(g)m τ(g)
g→0∼ −g2
d0 + d1g
2 + . . .
Solution leads to exact equations in a mass-independent scheme
Λ ≡ µ(b0g
2)−b1/(2b2
0) e−1/(2b0g2) exp
−
∫ g
0
dg[ 1
β(g)+
1
b0g3−
b1
b20g
]
M ≡ m(µ)(2b0g
2)−d0/(2b0)
exp
−
∫ g
0
dg[ τ(g)
β(g)−
d0
b0g
] RG invariantquark mass
Simple relations between different renormalization schemes:
Λ ′/Λ = exp c/(4πb0) , α ′ = α + cα2 + O(α3) and M ′ = M
⇒ Choose suitable physical scheme/coupling to compute Λ and M
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
The running of the coupling in two-flavour QCDLPHAA
Collaboration , NPB713(2005)378Physical couplingsI are defined for all energies µ and fix a renormalization schemeI should be independent of the regularization procedureI in a lattice computation: Problem of a hierarchy of disparate scales
L−1 ¿ 0.2 GeV ¿ µ ' 10 GeV ¿ a−1
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
The running of the coupling in two-flavour QCDLPHAA
Collaboration , NPB713(2005)378Physical couplingsI are defined for all energies µ and fix a renormalization schemeI should be independent of the regularization procedureI in a lattice computation: Problem of a hierarchy of disparate scales
L−1 ¿ 0.2 GeV ¿ µ ' 10 GeV ¿ a−1
Solution [ Luscher, Weisz & Wolff, 1991; LPHAACollaboration , 1993-2005 ]
Identify µ = 1/L =⇒ One is left with L/a À 1 =⇒ Finite-size scaling
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
The running of the coupling in two-flavour QCDLPHAA
Collaboration , NPB713(2005)378Physical couplingsI are defined for all energies µ and fix a renormalization schemeI should be independent of the regularization procedureI in a lattice computation: Problem of a hierarchy of disparate scales
L−1 ¿ 0.2 GeV ¿ µ ' 10 GeV ¿ a−1
Solution [ Luscher, Weisz & Wolff, 1991; LPHAACollaboration , 1993-2005 ]
Identify µ = 1/L =⇒ One is left with L/a À 1 =⇒ Finite-size scaling
Realization: QCD Schrodinger functional (SF) as an intermediate scheme
C
time
space
T
0
C’ T × L3 QCD with Dirichlet boundary conditions in time
Take finite-size effect as physical observable:
1/ g2SF(L) ≡ δ(free energy)/δ(boundary conditions)
NP handle on β(g) via the step scaling function:
σ(u) ≡ g2(2L)∣∣g2(L)=u ,m (L)=0
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Recursive strategy to obtain the step scaling functionFix g2(L0 ≡ Lmax), compute g2(L) for latticesalternating in L = 2−kL0 resp. a = 2−ka0
. . .
12
12
a = a
a
0
L = 0a = 0L
,L = L0
a → 0 limit of Σ(u,a/L) = σ(u) + O(a/L)
Comparison to PT and NP fit of σ(u) toσ(u) = u + s0u
2 + s1u3 + s2u
4 + . . .
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Nf = 0 (SF scheme) & Experiment + PT ( MS scheme, [ Bethke, 2004 ] )
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Start from g2(Lmax) ≡ 5.5 and evolvek steps with NP step scaling function:g2(Lmax/2k) = σ(g2(Lmax/2k+1))
I g2 6 uk small: continue with pert. β
I Result: − ln(ΛLmax) = 1.09(7)
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Start from g2(Lmax) ≡ 5.5 and evolvek steps with NP step scaling function:g2(Lmax/2k) = σ(g2(Lmax/2k+1))
I g2 6 uk small: continue with pert. β
I Result: − ln(ΛLmax) = 1.09(7)
To set the scale for Λ in MeV, useI ideally: large volume computation of FK
I here: r0 = 0.5 fm , FQQ(r0)r20 = 1.65
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Nf = 2 β–function in the SF scheme
Comparison to PT and Nf = 2 :
Non-perturbative deviations from3–loop β for α > 0.25
Perturbative series by itself doesn’tshow signs of failure at e.g. α ≈ 0.4
Non-perturbative running of αSF and the Λ–parameter for Nf = 2
Start from g2(Lmax) ≡ 5.5 and evolvek steps with NP step scaling function:g2(Lmax/2k) = σ(g2(Lmax/2k+1))
I g2 6 uk small: continue with pert. β
I Result: − ln(ΛLmax) = 1.09(7)
To set the scale for Λ in MeV, useI ideally: large volume computation of FK
I here: r0 = 0.5 fm , FQQ(r0)r20 = 1.65
ΛMSr0 = 0.62(8) ⇒ ΛMS = 245(32)MeVJochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Discussion
I Running close to PT below αSF ≈ 0.2, non-perturbative above 0.25
I ΛMSr0 : Nf dependence & Comparison to phenomenology
Nf 0 2 4 5
[ LPHAACollaboration ] 0.60(5) 0.62(4)(4)
[ Bethke, Loops&Legs04 ]‘experiment’ 0.74(10) 0.54(8)
[ Blumlein et al., Loops&Legs04 ]DIS NNLO 0.57(8)
Irregular Nf dependence ? But errors relatively large . . .
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The Nf = 2 running coupling
Discussion
I Running close to PT below αSF ≈ 0.2, non-perturbative above 0.25
I ΛMSr0 : Nf dependence & Comparison to phenomenology
Nf 0 2 4 5
[ LPHAACollaboration ] 0.60(5) 0.62(4)(4)
[ Bethke, Loops&Legs04 ]‘experiment’ 0.74(10) 0.54(8)
[ Blumlein et al., Loops&Legs04 ]DIS NNLO 0.57(8)
Irregular Nf dependence ? But errors relatively large . . .
I Improvement of lattice results: Via replacing Lmax/r0 by FK × Lmax
with small quark masses and small lattice spacingsAlso: Nf = 3, 4 (Nf = 3 on agenda of JLQCD and CP-PACS)Lattice will yield best controlled and most precise results for Λ
(minimal assumptions !)Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
A precision calculation of ms
Garden, H., Sommer & Wittig, NPB571(2000)237StrategyCombination of lattice QCD with chiral perturbation theory→ relate LQCD with experimentally known (spectroscopic) quantity, here:
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
A precision calculation of ms
Garden, H., Sommer & Wittig, NPB571(2000)237StrategyCombination of lattice QCD with chiral perturbation theory→ relate LQCD with experimentally known (spectroscopic) quantity, here:
m2PS(Mref)r
20 = (mKr0)
2∣∣exp
ChPT:
2Mref ' Ms + M` , Ms
M`= 24.4(1.5)
where M` = 12(Mu + Md) and M ≡ lim
µ→∞
[ 2b0g
2(µ) ]−d0/(2b0) m(µ)
: RGI quark mass
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
A precision calculation of ms
Garden, H., Sommer & Wittig, NPB571(2000)237StrategyCombination of lattice QCD with chiral perturbation theory→ relate LQCD with experimentally known (spectroscopic) quantity, here:
m2PS(Mref)r
20 = (mKr0)
2∣∣exp
ChPT:
2Mref ' Ms + M` , Ms
M`= 24.4(1.5)
where M` = 12(Mu + Md) and M ≡ lim
µ→∞
[ 2b0g
2(µ) ]−d0/(2b0) m(µ)
: RGI quark mass
Determination of Mref through numerical simulations
(1)PCAC relation :
FKm2K = (mu + ms) 〈 0 | uγ5s | K 〉
(uγ5s)MS = ZP(g0, aµ) (uγ5s) lattice
(2) ZP(g0, aµ) is poorly convergent in PT→ non-perturbative values for ZP & M/m(µ) needed [ LPHAA
Collaboration 1999 ]
(3) Mref fixed by K-meson matrix elements to be taken from the lattice
Ms + M`
FK
=M
m(µ)
1
ZP(g0, aµ)× m2
PS
〈 0 | `γ5s | PS 〉
∣∣∣∣mPS=mK
+ O(a2)
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Hadron physics from the Schrodinger FunctionalH., NPB557(1999)309
Guagnelli, H., Sommer & Wittig, NPB560(1999)465Volume: L & 1.5 fm , T = 2L
SF = QCD partition function with Dirichlet BCs in time(finite-V renormalization scheme, but has also some practical benefits for large-V physics)
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Hadron physics from the Schrodinger FunctionalH., NPB557(1999)309
Guagnelli, H., Sommer & Wittig, NPB560(1999)465Volume: L & 1.5 fm , T = 2L
SF = QCD partition function with Dirichlet BCs in time(finite-V renormalization scheme, but has also some practical benefits for large-V physics)
Fermionic correlation functions
fX(x0) ∝⟨ ∑
y,z
X(x) ζ(y)γ5ζ(z)⟩
X =
Aµ = ψiγµγ5ψj : axial vector currentP = ψiγ5ψj : pseudoscalar density
f1 ∝⟨ ∑
u,v,y,z
ζ ′
i(u)γ5ζ′
j(v) ζj(y)γ5ζi(z)⟩
ζ, ζ : boundary (anti-)quark fields
construct ratios with f1 to eliminate Z ζ,ζ
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Hadron physics from the Schrodinger FunctionalH., NPB557(1999)309
Guagnelli, H., Sommer & Wittig, NPB560(1999)465Volume: L & 1.5 fm , T = 2L
SF = QCD partition function with Dirichlet BCs in time(finite-V renormalization scheme, but has also some practical benefits for large-V physics)
Fermionic correlation functions
fX(x0) ∝⟨ ∑
y,z
X(x) ζ(y)γ5ζ(z)⟩
X =
Aµ = ψiγµγ5ψj : axial vector currentP = ψiγ5ψj : pseudoscalar density
f1 ∝⟨ ∑
u,v,y,z
ζ ′
i(u)γ5ζ′
j(v) ζj(y)γ5ζi(z)⟩
ζ, ζ : boundary (anti-)quark fields
construct ratios with f1 to eliminate Z ζ,ζ
⇒ for 0 ¿ x0 ¿ T :
fA√f1
≈ 〈0|A|PS〉 e−(x0−T/2)mPSfA
fP
≈ 〈0|A|PS〉〈0|P|PS〉
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Computation in similar spirit:I Nf = 0 mcharm, input: r0 = 0.5 fm, mDs
I mMSc (mc) = 1.30(3)GeV [Rolf&Sint, ’02]
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Computation in similar spirit:I Nf = 0 mcharm, input: r0 = 0.5 fm, mDs
I mMSc (mc) = 1.30(3)GeV [Rolf&Sint, ’02]
Crucial for precise predictions: Inclusion of dynamical (sea) quark effects
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Result in quenched QCD [ Garden, H., Sommer & Wittig, NPB571(2000)237 ]
mMSs (2 GeV) = 97 ± 4 MeV
NP renormalization & Continuum limitI scale set by FK = 160(2)MeV
I 3 all errors except quenching (∼ 15%?)
Computation in similar spirit:I Nf = 0 mcharm, input: r0 = 0.5 fm, mDs
I mMSc (mc) = 1.30(3)GeV [Rolf&Sint, ’02]
Crucial for precise predictions: Inclusion of dynamical (sea) quark effects
Non-perturbative running of m for Nf = 2 [ LPHAACollaboration , NPB729(2005)117 ]
mi(µ) = Zm(g0, aµ) × mbarei (g0)
whereby Zm is flavour independent
Scale evolution & RGI mass M via thesame recursive strategy as for α(µ):I exp. input: r0 = 0.5 fm, mK = 495MeV
I mMSs (2GeV) = 97 ± 22MeV
Jochen Heitger Fundamental QCD parameters from the lattice
Determining fundamental QCD parameters The strange quark’s mass
Unquenched mstrange (Nf = 2, 2 + 1) [ Status November 2005 ]
0 0.005 0.01 0.015 0.02 0.025 0.03
0
20
40
60
80
100
120
mM
Ss
(µ=
2G
eV)
[MeV
]
a2 [fm2]
ALPHA, Nf =2, W/CW, nonpert. renorm. (SF)
ALPHA, Nf =2, W/CW, nonpert. renorm. (SF), Zcon
A
CP-PACS, Nf =2, I/CW, pert. renorm.
JLQCD, Nf =2, W/CW, pert. renorm.
QCDSF/UKQCD, Nf =2, W/CW, nonpert. renorm. (RI-MOM)
SPQCDR, Nf =2, W/W, nonpert. renorm. (RI-MOM)
CP-PACS/JLQCD, Nf =3, I/CW, pert. renorm., stat. err. only
HPQCD/MILC/UKQCD, Nf =3, LW/KS, pert. renorm. 1-loop
HPQCD/MILC/UKQCD, Nf =3, LW/KS, pert. renorm. 2-loop
← Nf = 0
I Unquenched estimates have not yet stabilizedLarge cutoff effects at a ≈ 0.1 fm
Better control over continuum limit requiredI Impact of perturbative versus non-perturbative renormalization ?
Results with pert. renormalization tend to be smaller than NP onesI Presently: Hard to claim a definitive Nf dependence of mstrange
Note: QCD sum rules in agreement with range of lattice results
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics
New perspectives for B-physics
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics
B-physics & Lattice QCDI Precision CKM-physics involves hadronic matrix elements:
Determination of CKM matrix parameters, tests of its unitarityCP violation
I b-quark mass, spectrum and lifetimes of beauty-hadrons, . . .
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics
B-physics & Lattice QCDI Precision CKM-physics involves hadronic matrix elements:
Determination of CKM matrix parameters, tests of its unitarityCP violation
I b-quark mass, spectrum and lifetimes of beauty-hadrons, . . .
Illustration: Unitarity triangle analysis (‘CKM fit’) [ Okamoto @ Lattice 2005 ]
〈M | O∆M=2 |M 〉 =4
3m2
MF2MBM
〈 0 | bγµγ5q |Bq 〉 = ipµFBq, q = d, s
∆md ∝ F2Bd
BBd|VtdV
∗
tb|2
∆ms
∆md
∝F2
BsBBs
F2Bd
BBd
|Vts|2
|Vtd|2= ξ2 |Vts|
2
|Vtd|2
UT with lattice input from BK, FBd,BBd
,ξ: Today versus expected (5-3)% accuracy
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1
ρ−0.6
−0.4
−0.2
0.0
0.2
0.4
η β
~|Vtd |
~|V
ub|
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Lattice 201X
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Lattice HQET — Why ?
Consider a large lattice as possible in the quenched approximation
λπ = 1/mπ ≈ L
λB ∼ 1/mb < a
I Light quarks: too lightWidely spread objectsFinite-volume errors due tolight pions
I b-quark: too heavyExtremely localized objectB-mesons need very fineresolutions, otherwise:
I discretization errorsI ‘fall through the lattice’
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Lattice HQET — Why ?
Consider a large lattice as possible in the quenched approximation
λπ = 1/mπ ≈ L
λB ∼ 1/mb < a
I Light quarks: too lightWidely spread objectsFinite-volume errors due tolight pions
I b-quark: too heavyExtremely localized objectB-mesons need very fineresolutions, otherwise:
I discretization errorsI ‘fall through the lattice’
⇒ Propagating b on the lattice beyond today’s computing resources
⇒ Recourse to an effective theory for the b-quark:
Heavy Quark Effective Theory[ Eichten, 1988; Eichten & Hill, 1990 ]
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
HQET — An asymptotic expansion of QCD
Physical picture
q, gQ
m−1Q
a
L
Λ−1QCD
PµQ = mQvµ + kµ
v2 = 1residual momentum:
k ∼ O(ΛQCD) ¿ mQv
Momentum scales in heavy-light (Qq) mesons
Q almost at rest at bound state’s center,surrounded by the light DOFs
Motion of the heavy quark is suppressedby ΛQCD/mQ
Formally, in case of the B-system:LHQET = 1/mb–expansion of continuum QCD
ψb γµDµ + mbψb → Lstat + O(1/mb)
Lstat(x) = ψh(x) D0 + mb ψh(x)
P+ψh = ψh, P+ = 12(1+γ0) ⇒ 2 d.o.f.
Systematic & accurate for mh/ΛQCD À 1
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Non-perturbative formulation of HQETH. & Sommer, JHEP0402(2004)022Beyond the static approximation
SHQET = a4∑
x
Lstat(x) +
n∑
ν=1
L(ν)(x)
, L(ν)(x) =∑
i
ω(ν)
i L(ν)
i (x)
Lstat = ψh [∇∗0 + δm ] ψh → Eichten-Hill action
L(1)1 = ψh
(− 1
2σ · B
)ψh ≡ Ospin →
chromomagnetic interactionwith the gluon field
L(1)2 = ψh
(− 1
2D2
)ψh ≡ Okin →
kinetic energy from heavyquark’s residual motion
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Non-perturbative formulation of HQETH. & Sommer, JHEP0402(2004)022Beyond the static approximation
SHQET = a4∑
x
Lstat(x) +
n∑
ν=1
L(ν)(x)
, L(ν)(x) =∑
i
ω(ν)
i L(ν)
i (x)
Lstat = ψh [∇∗0 + δm ] ψh → Eichten-Hill action
L(1)1 = ψh
(− 1
2σ · B
)ψh ≡ Ospin →
chromomagnetic interactionwith the gluon field
L(1)2 = ψh
(− 1
2D2
)ψh ≡ Okin →
kinetic energy from heavyquark’s residual motion
Effective theory regularized on a space-time latticeδm, ω = ω(g0, m) to be determined such that HQET matches QCDAt classical level: ωspin = ωkin = 1/m + O(g2
0) , δm = 0 + O(g20)
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Non-perturbative formulation of HQETH. & Sommer, JHEP0402(2004)022Beyond the static approximation
SHQET = a4∑
x
Lstat(x) +
n∑
ν=1
L(ν)(x)
, L(ν)(x) =∑
i
ω(ν)
i L(ν)
i (x)
Lstat = ψh [∇∗0 + δm ] ψh → Eichten-Hill action
L(1)1 = ψh
(− 1
2σ · B
)ψh ≡ Ospin →
chromomagnetic interactionwith the gluon field
L(1)2 = ψh
(− 1
2D2
)ψh ≡ Okin →
kinetic energy from heavyquark’s residual motion
Effective theory regularized on a space-time latticeδm, ω = ω(g0, m) to be determined such that HQET matches QCDAt classical level: ωspin = ωkin = 1/m + O(g2
0) , δm = 0 + O(g20)
Analogously: Composite fields in the effective theory, e.g.
AHQET0 (x) = ZHQET
A︸ ︷︷ ︸1+O(g2
0)
ψl(x)γ0γ5ψh(x)︸ ︷︷ ︸
Astat0 (x)
+ cHQETA︸ ︷︷ ︸
∝ 1/m
ψl(x)γjγ5
←−Djψh(x) + . . .
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Expectation valuesPath integral representation at the quantum level
〈O〉 =1
Z
∫
D[ϕ] O[ϕ] e−(Srel+SHQET) Z =
∫
D[ϕ] e−(Srel+SHQET)
Now the integrand is expanded in a power series in 1/m
exp −SHQET =
exp−a4
∑xLstat(x)
1 − a4∑
xL(1)(x) + 12
[a4
∑xL(1)(x)
]2− a4
∑xL(2)(x) + . . .
⇒ 〈O〉 =1
Z
∫
D[ϕ] e−Srel−a4∑
x Lstat(x) O
1 − a4∑
xL(1)(x) + . . .
Important (but not automatic) implications of this definition of HQET
1/m – terms appear only as insertions of local operators⇒ Power counting: Renormalizability at any given order in 1/m
⇔ Existence of the continuum limit with universality
Effective theory = Continuum asymptotic expansion in 1/m
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
RenormalizationWe assume massless quarks except the b and no 1/m – term in O
(fA)R (Mb/Λ, x0Λ) =
ZHQETA e−mbarex0
⟨Astat
0 (x)O[1 + ωkin
∑yOkin(y) + ωspin
∑yOspin(y)
]⟩
stat
+ cHQETA
⟨δAstat
0 O⟩stat
=
ZHQETA e−mbarex0
fstatA + ωkinf
kinA + ωspinf
spinA + cHQET
A fstatδA
0
LxLxL
x
x0
0
= T
=⇒ Effective theory receives renormalizations in
mbare = mb + δm ZHQETA cHQET
A ωkin ωspin
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
RenormalizationWe assume massless quarks except the b and no 1/m – term in O
(fA)R (Mb/Λ, x0Λ) =
ZHQETA e−mbarex0
⟨Astat
0 (x)O[1 + ωkin
∑yOkin(y) + ωspin
∑yOspin(y)
]⟩
stat
+ cHQETA
⟨δAstat
0 O⟩stat
=
ZHQETA e−mbarex0
fstatA + ωkinf
kinA + ωspinf
spinA + cHQET
A fstatδA
0
LxLxL
x
x0
0
= T
=⇒ Effective theory receives renormalizations in
mbare = mb + δm ZHQETA cHQET
A ωkin ωspin
The Problem: Power-law divergences due to operator mixing
ORd=5 = ZO
Od=5 +
∑k ck Ok
d=4
, ck = a−1c
(0)
k + c(1)
k g20 + . . .
⇒ for a → 0 divergent remainder, if ck are only perturbatively calculated
Example at the static level:Linearly divergent counterterm δm ∝ 1/a originates from mixing of ψhD0ψh with ψhψh
⇒ Non-pert. renormalization of HQET needed to have a continuum limit
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Solution: Non-perturbative matching of HQET and QCDH. & Sommer, JHEP0402(2004)022Effective theory approximates QCD,
if the coefficients ck are chosen correctly such that
ΦHQET(M) = ΦQCD(M) + O(1/[ r0M ]n+1
)
M = RG-invariant (heavy) quark mass(free of renormalization scheme dependence)
Strategy to guarantee this equivalence of HQET and QCD
Fix the coefficients of regularized HQET by imposing matching conditions¨
§
¥
¦ΦHQET
k (M) = ΦQCDk (M) k = 1, . . . ,NHQET (?)
(?) determines the set ck for any lattice spacing (bare coupling)
ΦQCDk = Renormalized quantities, computable for a → 0 in QCD
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Solution: Non-perturbative matching of HQET and QCDH. & Sommer, JHEP0402(2004)022Effective theory approximates QCD,
if the coefficients ck are chosen correctly such that
ΦHQET(M) = ΦQCD(M) + O(1/[ r0M ]n+1
)
M = RG-invariant (heavy) quark mass(free of renormalization scheme dependence)
Strategy to guarantee this equivalence of HQET and QCD
Fix the coefficients of regularized HQET by imposing matching conditions¨
§
¥
¦ΦHQET
k (M) = ΦQCDk (M) k = 1, . . . ,NHQET (?)
(?) determines the set ck for any lattice spacing (bare coupling)
ΦQCDk = Renormalized quantities, computable for a → 0 in QCD
Question
How to treat the heavy quark as particle with finite mass on the lattice ?
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Idea: Employing finite volumes
Goal: Non-perturbative matching of HQET & QCD
QCD
1/mb À a
Matching conditions
ΦQCDk = ΦHQET
k
for observables Φk
(e.g. matrix elements)
HQET
1/mb ¿ L
I HQET parameters fixed by relating them to QCD observables in small V
I Legitimate: Underlying Lagrangian does not know about the finite V !
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Idea: Employing finite volumes
Goal: Non-perturbative matching of HQET & QCDObjection: Requires to simulate the b-quark as a relativistic particle
QCD
1/mb À a
Matching conditions
ΦQCDk = ΦHQET
k
for observables Φk
(e.g. matrix elements)
HQET
1/mb ¿ L
I HQET parameters fixed by relating them to QCD observables in small V
I Legitimate: Underlying Lagrangian does not know about the finite V !
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Idea: Employing finite volumes
Goal: Non-perturbative matching of HQET & QCDObjection: Requires to simulate the b-quark as a relativistic particle⇒ Trick: Start with QCD in small volume V = L4, L ≡ L0 ' 0.2 fm
QCD
1/mb À a
Matching conditions
ΦQCDk = ΦHQET
k
for observables Φk
(e.g. matrix elements)
HQET
1/mb ¿ L
I HQET parameters fixed by relating them to QCD observables in small V
I Legitimate: Underlying Lagrangian does not know about the finite V !
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Connecting small and large volumes
(?) in finite V : ΦHQETk (L, M) = ΦQCD
k (L, M) k = 1, . . . ,NHQET
Typical choice: L = L0 ' 0.2 − 0.4 fm, very small lattice spacingsLarge volumes required to extract physical observables (e.g. mB, FBs
)→ How can we bridge the gap to practicable lattice spacings ?
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics Non-perturbative HQET
Connecting small and large volumes
(?) in finite V : ΦHQETk (L, M) = ΦQCD
k (L, M) k = 1, . . . ,NHQET
Typical choice: L = L0 ' 0.2 − 0.4 fm, very small lattice spacingsLarge volumes required to extract physical observables (e.g. mB, FBs
)→ How can we bridge the gap to practicable lattice spacings ?
Recursive finite-size scaling
p p p pp p p pp p p pp p p p
-
ΦHQET(L0)
p p p pp p p pp p p p
p p p p
-
ΦHQET(8L0)
p p p p
p p p p
p p p p
p p p p
-
p p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p p
p p p p p p p
p p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p pp p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
p p p p p p p
[ Luscher, Weisz & Wolff, 1991; LPHAACollaboration , 1993-2005 ]
Non-perturbative, continuum limit exists
Use of the QCD Schrodinger Functional
Change L → 2L via step scaling functions
ΦHQETk (2L) = σk
(ΦHQET
j (L), j = 1, . . . ,N)
→ Large V , where the B fits comfortably
⇒ First fully non-perturbative formulation of HQET, including matchingJochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
Application: Computation of Mb in lowest-order HQETH. & Sommer, JHEP0402(2004)022
H. & Wennekers, JHEP0402(2004)064Introduce L– dependent energiesfrom correlators in the B-channel (and with SF boundary conditions):
Γ(L, M)
Γstat(L)
=
B-meson mass in a finite volume of extent L4
energy of a state with B-meson quantum numbers in L4
C(x0,M) :
x0 = 0
ζl2
ζl1
x0 = L
A0 → Γ(L,M) ≡ −d
dx0
ln [C(x0,M)]∣∣∣x0=
L
2
Cstat(x0) :
x0 = 0
ζh
ζl
x0 = L
Astat0 → Γstat(L) ≡ −
d
dx0
ln [Cstat(x0)]∣∣∣x0=
L
2
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
⇒ Matching condition by equating in small volume with linear extent L0 :
Γstat(L0) + mbare = Γ(L0, Mb) [ Recall: mbare = m + 1a
ln(1 + a δm) ]
As C(x0)x0→∞
∼ e−mBx0 and Cstat(x0)x0→∞
∼ e−Estatx0 in the large – L
limit, we have to connect this condition (by finite-size scaling) to
Estat + mbare = mB
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
⇒ Matching condition by equating in small volume with linear extent L0 :
Γstat(L0) + mbare = Γ(L0, Mb) [ Recall: mbare = m + 1a
ln(1 + a δm) ]
As C(x0)x0→∞
∼ e−mBx0 and Cstat(x0)x0→∞
∼ e−Estatx0 in the large – L
limit, we have to connect this condition (by finite-size scaling) to
Estat + mbare = mB
Sketch of the method
Experiment Lattice with amb ¿ 1
mB = 5.4 GeV Γ(L0, M)
Γstat(L2) Γstat(L1) Γstat(L0)
ui = g2(Li)σm(u1) σm(u0)? ?¾ ¾
A step scaling function σm(u) ≡ 2L [ Γstat(2L) − Γstat(L) ] bridgesbetween small & larger volumes by evolving L0 → L2 = 22L0 ' 1 fm
For L ' 2 fm @ same resolution:Physical situation to accommodate (& calculate) B-meson properties
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
⇒ Equation to solve for the b-quark mass:
L0Γ(L0,M)︸ ︷︷ ︸
a→0≡ ΩQCD in QCD (L40)
= L0m(exp)
B − L0
[ Γstat(L2) − Γstat(L0) ] − [Estat − Γstat(L2) ]
︸ ︷︷ ︸a→0≡ ΩHQET in HQET (contains σm)
Divergent static quark’s self-energy δm cancels in differences !Γ(L0, M) carries entire (relativistic) heavy quark mass dependence
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
⇒ Equation to solve for the b-quark mass:
L0Γ(L0,M)︸ ︷︷ ︸
a→0≡ ΩQCD in QCD (L40)
= L0m(exp)
B − L0
[ Γstat(L2) − Γstat(L0) ] − [Estat − Γstat(L2) ]
︸ ︷︷ ︸a→0≡ ΩHQET in HQET (contains σm)
Divergent static quark’s self-energy δm cancels in differences !Γ(L0, M) carries entire (relativistic) heavy quark mass dependence
QCD ( z ≡ L0M À 1 , L0 ' 0.2 fm ) Quenched static result
O(a) improved QCD & HQET
NP renormalization & Cont. limit
mMSb
(mMS
b
)= 4.12(8)GeV + O( 1
Mb)
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
Inclusion of 1/m–terms in the computation of Mb
Della Morte, Garron, Sommer & Papinutto, PoS LAT2005(2005)223&224LPHAA
Collaboration , in preparationmB at next-to-leading order of HQET
mB = Estat + mbare + ωkinEkin + ωspinEspin
Ekin, Espin associated with ψh(−12D2)ψh and ψh(−
12σ · B)ψh in L(1)
→ Three observables Φ1, Φ2, Φ3 required in the matching stepConsidering the spin-averaged B-meson instead, ωspin cancels:
m(av)
B = 14mB + 3
4m∗
B = Estat + mbare + ωkinEkin
→ Only two obervables Φ1, Φ2 necessary
The previous strategy now extends to
Experiment Lattice with amb ¿ 1
mB = 5.4 GeV Φ1(L1, M), Φ2(L1, M)
ΦHQET1 (L2), Φ
HQET2 (L2) ΦHQET
1 (L1), ΦHQET2 (L1)
L1 ' 0.4fm,L2 = 2L1
u1 = g2(L1)
σm(u1)
σkin1 (u1), σ
kin2 (u1)
? ?¾
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
Matching formula = Static part + 1/m – correction :
L2m(av)
B = L2mstatB
= L2
[Estat − Γ stat
1 (L2)]+ σm(u1) + 2Φ2(L1,M)
+ L2m(1)
B
= σkin2 (u1)Φ1(L1,M) + L2
[Ekin − Γ kin
1 (L2)]ωkin
ImplementationAs in the static case, but technical details more involved (& omitted here)
I SF boundary conditions, i.e. T × L3, Dirichlet at x0 = 0, T , fermionfields periodic in space modulo a phase: ψ(x + Lk) = e iθψ(x)
I Avoid extra term ( 1m
fstatδA ) in boundary-to-A0 correlator by boundary-
to-boundary ones with pseudoscalar or vector quantum numbers
(f1)R (T) = Zboundary e−mbareT
fstat1 + ωkinf
kin1 + ωspinf
spin1
0
LxLxL
x
x0
0
= T
=
I Φ1 : Suitable ratios of f1(θ), f1(θ′) such that Z– factors drop out
Φ2 : Spin-averaged energy Γ1 = −∂T ln f(av)1 = mbare + Γ stat
1 + ωkinΓkin1
plus corresponding step scaling functions σkin1 (u), σkin
2 (u)
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
Examples for continuum limit extrapolations (quenched)
Note: 1/m – terms approach a = 0 only with a rate ∝ a
0 0.0005 0.001 0.0015 0.002 0.0025
(a/L) 2
8
8.5
9
9.5
10
Φ2
0 0.05 0.1 0.15 0.2a/L
0
2
4
6
8
10
Σ2
kin
HYP 2HYP 10.1 % uncertainty on M
b
Left:Dimensionless spin-averaged finite-volume energy from f1 in QCD
Right:Cancellation of 1/a2 power-divergence in σkin
2 = lima/L→0 Σkin2
Jochen Heitger Fundamental QCD parameters from the lattice
New perspectives for B-physics The b-quark’s mass
10 11 12 13 14M L
1
12
13
14
15
16
17
18
19
L2 m
B
exp - L
2 (E
stat-Γ
1
stat(L
2)) - σ
m
2 Φ2(L
1, M)
Mb
stat
Left: Leading order (= static)
I mlight = 0 & L1 ' 0.4 fm
I Input: r0m(exp)
B , r0 = 0.5 fm
I Error dominated by that on ZM inLM = ZM Z (1 + bmamq) × Lmq
( NP known incl. O(a) [ LPHAACollaboration ] )
Quenched b-quark’s mass to order Λ2/mb in HQET
Result in the MS scheme (reported @ Lattice Conference 2005):
mb (mb) = mstatb + m
(1)
b mstatb = 4.35(6)GeV , m(1)
b = −0.05(3)GeV
= 4.30(7)
Though quenched approximation, well within range quoted by PDG
Difficult piece: Large-volume HQET matrix element [Ekin − Γkin1 (L2)]
Jochen Heitger Fundamental QCD parameters from the lattice
Summary & Outlook
Summary & Outlook
I Predictive power of lattice QCD demands high precision:→ Overcoming the error from the quenched approximation is under way→ Several & fine enough resolutions required to quantify cutoff effects
I New quality of the computations employing lattice HQET:X Non-perturbative formulation including the matchingX Continuum limit at large quark masses (small-volume setup !)X Renormalization of 1/m – terms can be performed non-perturbatively
[ Della Morte et al., PoS LAT2005 (2005) 223&224, LPHAACollaboration 3 H. ]
X Physics results still quenched, but extension of the methods to QCDwith dynamical quarks (Nf > 0) straightforward and already started
[ first steps towards FBs for Nf = 2: Della Morte, Fritzsch & H. ]
I Next steps, resp. related work in progress:→ Decay constant FB
→ Spin-splitting mB∗ − mB
→ NP matching for Nf = 2
Jochen Heitger Fundamental QCD parameters from the lattice